LMFDB - Embedded newform 6016.2.a.p.1.3

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			1.3
Root			$-2.08298$ 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)

$f(q)$	$=$	$q-2.08298 q^{3} +4.13384 q^{5} -1.67766 q^{7} +1.33879 q^{9} +O(q^{10})$	$q-2.08298 q^{3} +4.13384 q^{5} -1.67766 q^{7} +1.33879 q^{9} +2.59125 q^{11} -6.34190 q^{13} -8.61069 q^{15} +6.39805 q^{17} -0.134195 q^{19} +3.49453 q^{21} -8.53296 q^{23} +12.0886 q^{25} +3.46026 q^{27} -0.201617 q^{29} +0.349544 q^{31} -5.39751 q^{33} -6.93519 q^{35} -5.26709 q^{37} +13.2100 q^{39} +5.79268 q^{41} -8.54038 q^{43} +5.53434 q^{45} +1.00000 q^{47} -4.18545 q^{49} -13.3270 q^{51} +8.38624 q^{53} +10.7118 q^{55} +0.279526 q^{57} +8.24219 q^{59} +12.6612 q^{61} -2.24603 q^{63} -26.2164 q^{65} +2.18646 q^{67} +17.7740 q^{69} -4.61603 q^{71} +7.54452 q^{73} -25.1804 q^{75} -4.34724 q^{77} -0.107860 q^{79} -11.2240 q^{81} +16.8942 q^{83} +26.4485 q^{85} +0.419963 q^{87} -2.22240 q^{89} +10.6396 q^{91} -0.728091 q^{93} -0.554742 q^{95} +6.89369 q^{97} +3.46914 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.08298	−1.20261	−0.601303	−	0.799021i	$-0.705352\pi$
$3$	−2.08298	−1.20261	−0.601303	+	0.799021i	$0.705352\pi$
$4$	0	0
$5$	4.13384	1.84871	0.924355	−	0.381533i	$-0.124604\pi$
$5$	4.13384	1.84871	0.924355	+	0.381533i	$0.124604\pi$
$6$	0	0
$7$	−1.67766	−0.634096	−0.317048	−	0.948409i	$-0.602692\pi$
$7$	−1.67766	−0.634096	−0.317048	+	0.948409i	$0.602692\pi$
$8$	0	0
$9$	1.33879	0.446263
$10$	0	0
$11$	2.59125	0.781291	0.390646	−	0.920541i	$-0.372252\pi$
$11$	2.59125	0.781291	0.390646	+	0.920541i	$0.372252\pi$
$12$	0	0
$13$	−6.34190	−1.75893	−0.879464	−	0.475966i	$-0.842099\pi$
$13$	−6.34190	−1.75893	−0.879464	+	0.475966i	$0.842099\pi$
$14$	0	0
$15$	−8.61069	−2.22327
$16$	0	0
$17$	6.39805	1.55175	0.775877	−	0.630884i	$-0.217307\pi$
$17$	6.39805	1.55175	0.775877	+	0.630884i	$0.217307\pi$
$18$	0	0
$19$	−0.134195	−0.0307865	−0.0153933	−	0.999882i	$-0.504900\pi$
$19$	−0.134195	−0.0307865	−0.0153933	+	0.999882i	$0.504900\pi$
$20$	0	0
$21$	3.49453	0.762569
$22$	0	0
$23$	−8.53296	−1.77925	−0.889623	−	0.456696i	$-0.849033\pi$
$23$	−8.53296	−1.77925	−0.889623	+	0.456696i	$0.849033\pi$
$24$	0	0
$25$	12.0886	2.41773
$26$	0	0
$27$	3.46026	0.665928
$28$	0	0
$29$	−0.201617	−0.0374393	−0.0187196	−	0.999825i	$-0.505959\pi$
$29$	−0.201617	−0.0374393	−0.0187196	+	0.999825i	$0.505959\pi$
$30$	0	0
$31$	0.349544	0.0627799	0.0313899	−	0.999507i	$-0.490007\pi$
$31$	0.349544	0.0627799	0.0313899	+	0.999507i	$0.490007\pi$
$32$	0	0
$33$	−5.39751	−0.939586
$34$	0	0
$35$	−6.93519	−1.17226
$36$	0	0
$37$	−5.26709	−0.865905	−0.432952	−	0.901417i	$-0.642528\pi$
$37$	−5.26709	−0.865905	−0.432952	+	0.901417i	$0.642528\pi$
$38$	0	0
$39$	13.2100	2.11530
$40$	0	0
$41$	5.79268	0.904665	0.452332	−	0.891849i	$-0.350592\pi$
$41$	5.79268	0.904665	0.452332	+	0.891849i	$0.350592\pi$
$42$	0	0
$43$	−8.54038	−1.30240	−0.651198	−	0.758908i	$-0.725733\pi$
$43$	−8.54038	−1.30240	−0.651198	+	0.758908i	$0.725733\pi$
$44$	0	0
$45$	5.53434	0.825011
$46$	0	0
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	−4.18545	−0.597922
$50$	0	0
$51$	−13.3270	−1.86615
$52$	0	0
$53$	8.38624	1.15194	0.575970	−	0.817471i	$-0.304625\pi$
$53$	8.38624	1.15194	0.575970	+	0.817471i	$0.304625\pi$
$54$	0	0
$55$	10.7118	1.44438
$56$	0	0
$57$	0.279526	0.0370241
$58$	0	0
$59$	8.24219	1.07304	0.536521	−	0.843887i	$-0.319738\pi$
$59$	8.24219	1.07304	0.536521	+	0.843887i	$0.319738\pi$
$60$	0	0
$61$	12.6612	1.62110	0.810549	−	0.585670i	$-0.199169\pi$
$61$	12.6612	1.62110	0.810549	+	0.585670i	$0.199169\pi$
$62$	0	0
$63$	−2.24603	−0.282974
$64$	0	0
$65$	−26.2164	−3.25175
$66$	0	0
$67$	2.18646	0.267119	0.133559	−	0.991041i	$-0.457359\pi$
$67$	2.18646	0.267119	0.133559	+	0.991041i	$0.457359\pi$
$68$	0	0
$69$	17.7740	2.13973
$70$	0	0
$71$	−4.61603	−0.547822	−0.273911	−	0.961755i	$-0.588317\pi$
$71$	−4.61603	−0.547822	−0.273911	+	0.961755i	$0.588317\pi$
$72$	0	0
$73$	7.54452	0.883020	0.441510	−	0.897256i	$-0.354443\pi$
$73$	7.54452	0.883020	0.441510	+	0.897256i	$0.354443\pi$
$74$	0	0
$75$	−25.1804	−2.90758
$76$	0	0
$77$	−4.34724	−0.495414
$78$	0	0
$79$	−0.107860	−0.0121351	−0.00606757	−	0.999982i	$-0.501931\pi$
$79$	−0.107860	−0.0121351	−0.00606757	+	0.999982i	$0.501931\pi$
$80$	0	0
$81$	−11.2240	−1.24711
$82$	0	0
$83$	16.8942	1.85438	0.927192	−	0.374587i	$-0.122216\pi$
$83$	16.8942	1.85438	0.927192	+	0.374587i	$0.122216\pi$
$84$	0	0
$85$	26.4485	2.86875
$86$	0	0
$87$	0.419963	0.0450247
$88$	0	0
$89$	−2.22240	−0.235574	−0.117787	−	0.993039i	$-0.537580\pi$
$89$	−2.22240	−0.235574	−0.117787	+	0.993039i	$0.537580\pi$
$90$	0	0
$91$	10.6396	1.11533
$92$	0	0
$93$	−0.728091	−0.0754995
$94$	0	0
$95$	−0.554742	−0.0569154
$96$	0	0
$97$	6.89369	0.699948	0.349974	−	0.936759i	$-0.386190\pi$
$97$	6.89369	0.699948	0.349974	+	0.936759i	$0.386190\pi$
$98$	0	0
$99$	3.46914	0.348661
$100$	0	0
$101$	−3.45969	−0.344252	−0.172126	−	0.985075i	$-0.555064\pi$
$101$	−3.45969	−0.344252	−0.172126	+	0.985075i	$0.555064\pi$
$102$	0	0
$103$	−4.77433	−0.470429	−0.235214	−	0.971943i	$-0.575579\pi$
$103$	−4.77433	−0.470429	−0.235214	+	0.971943i	$0.575579\pi$
$104$	0	0
$105$	14.4458	1.40977
$106$	0	0
$107$	19.0224	1.83897	0.919484	−	0.393128i	$-0.128607\pi$
$107$	19.0224	1.83897	0.919484	+	0.393128i	$0.128607\pi$
$108$	0	0
$109$	−0.566953	−0.0543042	−0.0271521	−	0.999631i	$-0.508644\pi$
$109$	−0.566953	−0.0543042	−0.0271521	+	0.999631i	$0.508644\pi$
$110$	0	0
$111$	10.9712	1.04134
$112$	0	0
$113$	5.72670	0.538722	0.269361	−	0.963039i	$-0.413187\pi$
$113$	5.72670	0.538722	0.269361	+	0.963039i	$0.413187\pi$
$114$	0	0
$115$	−35.2739	−3.28931
$116$	0	0
$117$	−8.49047	−0.784944
$118$	0	0
$119$	−10.7338	−0.983962
$120$	0	0
$121$	−4.28543	−0.389584
$122$	0	0
$123$	−12.0660	−1.08796
$124$	0	0
$125$	29.3034	2.62097
$126$	0	0
$127$	−16.8669	−1.49669	−0.748346	−	0.663309i	$-0.769151\pi$
$127$	−16.8669	−1.49669	−0.748346	+	0.663309i	$0.769151\pi$
$128$	0	0
$129$	17.7894	1.56627
$130$	0	0
$131$	−9.51694	−0.831499	−0.415749	−	0.909479i	$-0.636481\pi$
$131$	−9.51694	−0.831499	−0.415749	+	0.909479i	$0.636481\pi$
$132$	0	0
$133$	0.225134	0.0195216
$134$	0	0
$135$	14.3042	1.23111
$136$	0	0
$137$	−5.64310	−0.482122	−0.241061	−	0.970510i	$-0.577495\pi$
$137$	−5.64310	−0.482122	−0.241061	+	0.970510i	$0.577495\pi$
$138$	0	0
$139$	13.3355	1.13110	0.565550	−	0.824714i	$-0.308664\pi$
$139$	13.3355	1.13110	0.565550	+	0.824714i	$0.308664\pi$
$140$	0	0
$141$	−2.08298	−0.175418
$142$	0	0
$143$	−16.4335	−1.37423
$144$	0	0
$145$	−0.833452	−0.0692144
$146$	0	0
$147$	8.71820	0.719065
$148$	0	0
$149$	1.39289	0.114110	0.0570551	−	0.998371i	$-0.481829\pi$
$149$	1.39289	0.114110	0.0570551	+	0.998371i	$0.481829\pi$
$150$	0	0
$151$	5.04041	0.410182	0.205091	−	0.978743i	$-0.434251\pi$
$151$	5.04041	0.410182	0.205091	+	0.978743i	$0.434251\pi$
$152$	0	0
$153$	8.56564	0.692491
$154$	0	0
$155$	1.44496	0.116062
$156$	0	0
$157$	23.3319	1.86209	0.931043	−	0.364910i	$-0.118900\pi$
$157$	23.3319	1.86209	0.931043	+	0.364910i	$0.118900\pi$
$158$	0	0
$159$	−17.4683	−1.38533
$160$	0	0
$161$	14.3154	1.12821
$162$	0	0
$163$	−9.22250	−0.722362	−0.361181	−	0.932496i	$-0.617626\pi$
$163$	−9.22250	−0.722362	−0.361181	+	0.932496i	$0.617626\pi$
$164$	0	0
$165$	−22.3125	−1.73702
$166$	0	0
$167$	3.13257	0.242406	0.121203	−	0.992628i	$-0.461325\pi$
$167$	3.13257	0.242406	0.121203	+	0.992628i	$0.461325\pi$
$168$	0	0
$169$	27.2197	2.09383
$170$	0	0
$171$	−0.179659	−0.0137389
$172$	0	0
$173$	−14.4434	−1.09811	−0.549054	−	0.835787i	$-0.685012\pi$
$173$	−14.4434	−1.09811	−0.549054	+	0.835787i	$0.685012\pi$
$174$	0	0
$175$	−20.2807	−1.53307
$176$	0	0
$177$	−17.1683	−1.29045
$178$	0	0
$179$	6.63476	0.495905	0.247952	−	0.968772i	$-0.420242\pi$
$179$	6.63476	0.495905	0.247952	+	0.968772i	$0.420242\pi$
$180$	0	0
$181$	17.0814	1.26965	0.634825	−	0.772656i	$-0.281072\pi$
$181$	17.0814	1.26965	0.634825	+	0.772656i	$0.281072\pi$
$182$	0	0
$183$	−26.3729	−1.94954
$184$	0	0
$185$	−21.7733	−1.60081
$186$	0	0
$187$	16.5789	1.21237
$188$	0	0
$189$	−5.80515	−0.422262
$190$	0	0
$191$	−6.04916	−0.437702	−0.218851	−	0.975758i	$-0.570231\pi$
$191$	−6.04916	−0.437702	−0.218851	+	0.975758i	$0.570231\pi$
$192$	0	0
$193$	−1.36235	−0.0980641	−0.0490320	−	0.998797i	$-0.515614\pi$
$193$	−1.36235	−0.0980641	−0.0490320	+	0.998797i	$0.515614\pi$
$194$	0	0
$195$	54.6082	3.91057
$196$	0	0
$197$	−4.24341	−0.302330	−0.151165	−	0.988509i	$-0.548303\pi$
$197$	−4.24341	−0.302330	−0.151165	+	0.988509i	$0.548303\pi$
$198$	0	0
$199$	21.0970	1.49553	0.747765	−	0.663964i	$-0.231127\pi$
$199$	21.0970	1.49553	0.747765	+	0.663964i	$0.231127\pi$
$200$	0	0
$201$	−4.55435	−0.321239
$202$	0	0
$203$	0.338245	0.0237401
$204$	0	0
$205$	23.9460	1.67246
$206$	0	0
$207$	−11.4238	−0.794011
$208$	0	0
$209$	−0.347734	−0.0240532
$210$	0	0
$211$	3.34604	0.230351	0.115176	−	0.993345i	$-0.463257\pi$
$211$	3.34604	0.230351	0.115176	+	0.993345i	$0.463257\pi$
$212$	0	0
$213$	9.61508	0.658814
$214$	0	0
$215$	−35.3046	−2.40775
$216$	0	0
$217$	−0.586416	−0.0398085
$218$	0	0
$219$	−15.7151	−1.06193
$220$	0	0
$221$	−40.5758	−2.72942
$222$	0	0
$223$	8.07805	0.540946	0.270473	−	0.962728i	$-0.412820\pi$
$223$	8.07805	0.540946	0.270473	+	0.962728i	$0.412820\pi$
$224$	0	0
$225$	16.1841	1.07894
$226$	0	0
$227$	−8.41230	−0.558344	−0.279172	−	0.960241i	$-0.590060\pi$
$227$	−8.41230	−0.558344	−0.279172	+	0.960241i	$0.590060\pi$
$228$	0	0
$229$	6.94864	0.459179	0.229589	−	0.973288i	$-0.426262\pi$
$229$	6.94864	0.459179	0.229589	+	0.973288i	$0.426262\pi$
$230$	0	0
$231$	9.05519	0.595788
$232$	0	0
$233$	−9.78032	−0.640730	−0.320365	−	0.947294i	$-0.603806\pi$
$233$	−9.78032	−0.640730	−0.320365	+	0.947294i	$0.603806\pi$
$234$	0	0
$235$	4.13384	0.269662
$236$	0	0
$237$	0.224669	0.0145938
$238$	0	0
$239$	9.29432	0.601199	0.300600	−	0.953750i	$-0.402813\pi$
$239$	9.29432	0.601199	0.300600	+	0.953750i	$0.402813\pi$
$240$	0	0
$241$	−26.1601	−1.68512	−0.842560	−	0.538603i	$-0.818952\pi$
$241$	−26.1601	−1.68512	−0.842560	+	0.538603i	$0.818952\pi$
$242$	0	0
$243$	12.9986	0.833858
$244$	0	0
$245$	−17.3020	−1.10538
$246$	0	0
$247$	0.851054	0.0541513
$248$	0	0
$249$	−35.1903	−2.23009
$250$	0	0
$251$	9.39104	0.592757	0.296379	−	0.955071i	$-0.404221\pi$
$251$	9.39104	0.592757	0.296379	+	0.955071i	$0.404221\pi$
$252$	0	0
$253$	−22.1110	−1.39011
$254$	0	0
$255$	−55.0916	−3.44997
$256$	0	0
$257$	−31.1541	−1.94334	−0.971671	−	0.236338i	$-0.924053\pi$
$257$	−31.1541	−1.94334	−0.971671	+	0.236338i	$0.924053\pi$
$258$	0	0
$259$	8.83640	0.549067
$260$	0	0
$261$	−0.269922	−0.0167078
$262$	0	0
$263$	−29.1211	−1.79568	−0.897842	−	0.440318i	$-0.854866\pi$
$263$	−29.1211	−1.79568	−0.897842	+	0.440318i	$0.854866\pi$
$264$	0	0
$265$	34.6674	2.12960
$266$	0	0
$267$	4.62922	0.283303
$268$	0	0
$269$	18.3423	1.11835	0.559174	−	0.829051i	$-0.311119\pi$
$269$	18.3423	1.11835	0.559174	+	0.829051i	$0.311119\pi$
$270$	0	0
$271$	24.5734	1.49273	0.746365	−	0.665537i	$-0.231798\pi$
$271$	24.5734	1.49273	0.746365	+	0.665537i	$0.231798\pi$
$272$	0	0
$273$	−22.1620	−1.34130
$274$	0	0
$275$	31.3247	1.88895
$276$	0	0
$277$	−32.4660	−1.95069	−0.975345	−	0.220686i	$-0.929171\pi$
$277$	−32.4660	−1.95069	−0.975345	+	0.220686i	$0.929171\pi$
$278$	0	0
$279$	0.467965	0.0280163
$280$	0	0
$281$	−4.87583	−0.290867	−0.145434	−	0.989368i	$-0.546458\pi$
$281$	−4.87583	−0.290867	−0.145434	+	0.989368i	$0.546458\pi$
$282$	0	0
$283$	10.3709	0.616486	0.308243	−	0.951308i	$-0.400259\pi$
$283$	10.3709	0.616486	0.308243	+	0.951308i	$0.400259\pi$
$284$	0	0
$285$	1.15551	0.0684468
$286$	0	0
$287$	−9.71816	−0.573645
$288$	0	0
$289$	23.9350	1.40794
$290$	0	0
$291$	−14.3594	−0.841762
$292$	0	0
$293$	−14.8749	−0.868998	−0.434499	−	0.900672i	$-0.643075\pi$
$293$	−14.8749	−0.868998	−0.434499	+	0.900672i	$0.643075\pi$
$294$	0	0
$295$	34.0719	1.98374
$296$	0	0
$297$	8.96641	0.520284
$298$	0	0
$299$	54.1152	3.12956
$300$	0	0
$301$	14.3279	0.825845
$302$	0	0
$303$	7.20645	0.414000
$304$	0	0
$305$	52.3393	2.99694
$306$	0	0
$307$	29.7722	1.69919	0.849594	−	0.527437i	$-0.176847\pi$
$307$	29.7722	1.69919	0.849594	+	0.527437i	$0.176847\pi$
$308$	0	0
$309$	9.94482	0.565741
$310$	0	0
$311$	24.9918	1.41715	0.708577	−	0.705634i	$-0.249338\pi$
$311$	24.9918	1.41715	0.708577	+	0.705634i	$0.249338\pi$
$312$	0	0
$313$	7.81246	0.441586	0.220793	−	0.975321i	$-0.429135\pi$
$313$	7.81246	0.441586	0.220793	+	0.975321i	$0.429135\pi$
$314$	0	0
$315$	−9.28475	−0.523136
$316$	0	0
$317$	−7.81302	−0.438823	−0.219411	−	0.975632i	$-0.570414\pi$
$317$	−7.81302	−0.438823	−0.219411	+	0.975632i	$0.570414\pi$
$318$	0	0
$319$	−0.522439	−0.0292510
$320$	0	0
$321$	−39.6233	−2.21155
$322$	0	0
$323$	−0.858588	−0.0477731
$324$	0	0
$325$	−76.6651	−4.25261
$326$	0	0
$327$	1.18095	0.0653066
$328$	0	0
$329$	−1.67766	−0.0924925
$330$	0	0
$331$	27.7435	1.52492	0.762460	−	0.647035i	$-0.223991\pi$
$331$	27.7435	1.52492	0.762460	+	0.647035i	$0.223991\pi$
$332$	0	0
$333$	−7.05152	−0.386421
$334$	0	0
$335$	9.03849	0.493825
$336$	0	0
$337$	8.15891	0.444444	0.222222	−	0.974996i	$-0.428669\pi$
$337$	8.15891	0.444444	0.222222	+	0.974996i	$0.428669\pi$
$338$	0	0
$339$	−11.9286	−0.647871
$340$	0	0
$341$	0.905755	0.0490494
$342$	0	0
$343$	18.7654	1.01324
$344$	0	0
$345$	73.4747	3.95575
$346$	0	0
$347$	−5.41869	−0.290890	−0.145445	−	0.989366i	$-0.546461\pi$
$347$	−5.41869	−0.290890	−0.145445	+	0.989366i	$0.546461\pi$
$348$	0	0
$349$	15.3105	0.819553	0.409776	−	0.912186i	$-0.365607\pi$
$349$	15.3105	0.819553	0.409776	+	0.912186i	$0.365607\pi$
$350$	0	0
$351$	−21.9447	−1.17132
$352$	0	0
$353$	5.74323	0.305681	0.152841	−	0.988251i	$-0.451158\pi$
$353$	5.74323	0.305681	0.152841	+	0.988251i	$0.451158\pi$
$354$	0	0
$355$	−19.0819	−1.01276
$356$	0	0
$357$	22.3582	1.18332
$358$	0	0
$359$	14.6531	0.773362	0.386681	−	0.922214i	$-0.373621\pi$
$359$	14.6531	0.773362	0.386681	+	0.922214i	$0.373621\pi$
$360$	0	0
$361$	−18.9820	−0.999052
$362$	0	0
$363$	8.92644	0.468516
$364$	0	0
$365$	31.1879	1.63245
$366$	0	0
$367$	14.9926	0.782605	0.391303	−	0.920262i	$-0.372025\pi$
$367$	14.9926	0.782605	0.391303	+	0.920262i	$0.372025\pi$
$368$	0	0
$369$	7.75518	0.403718
$370$	0	0
$371$	−14.0693	−0.730440
$372$	0	0
$373$	38.1193	1.97374	0.986872	−	0.161505i	$-0.0516349\pi$
$373$	38.1193	1.97374	0.986872	+	0.161505i	$0.0516349\pi$
$374$	0	0
$375$	−61.0382	−3.15200
$376$	0	0
$377$	1.27863	0.0658530
$378$	0	0
$379$	27.4039	1.40765	0.703823	−	0.710376i	$-0.251475\pi$
$379$	27.4039	1.40765	0.703823	+	0.710376i	$0.251475\pi$
$380$	0	0
$381$	35.1332	1.79993
$382$	0	0
$383$	−18.5152	−0.946080	−0.473040	−	0.881041i	$-0.656843\pi$
$383$	−18.5152	−0.946080	−0.473040	+	0.881041i	$0.656843\pi$
$384$	0	0
$385$	−17.9708	−0.915877
$386$	0	0
$387$	−11.4338	−0.581211
$388$	0	0
$389$	12.1326	0.615147	0.307574	−	0.951524i	$-0.400483\pi$
$389$	12.1326	0.615147	0.307574	+	0.951524i	$0.400483\pi$
$390$	0	0
$391$	−54.5943	−2.76095
$392$	0	0
$393$	19.8236	0.999966
$394$	0	0
$395$	−0.445874	−0.0224344
$396$	0	0
$397$	36.0640	1.81000	0.904999	−	0.425413i	$-0.139871\pi$
$397$	36.0640	1.81000	0.904999	+	0.425413i	$0.139871\pi$
$398$	0	0
$399$	−0.468949	−0.0234768
$400$	0	0
$401$	13.0015	0.649263	0.324632	−	0.945841i	$-0.394760\pi$
$401$	13.0015	0.649263	0.324632	+	0.945841i	$0.394760\pi$
$402$	0	0
$403$	−2.21677	−0.110425
$404$	0	0
$405$	−46.3983	−2.30555
$406$	0	0
$407$	−13.6484	−0.676524
$408$	0	0
$409$	18.9297	0.936014	0.468007	−	0.883725i	$-0.344972\pi$
$409$	18.9297	0.936014	0.468007	+	0.883725i	$0.344972\pi$
$410$	0	0
$411$	11.7544	0.579803
$412$	0	0
$413$	−13.8276	−0.680412
$414$	0	0
$415$	69.8381	3.42822
$416$	0	0
$417$	−27.7775	−1.36027
$418$	0	0
$419$	15.5816	0.761212	0.380606	−	0.924737i	$-0.375715\pi$
$419$	15.5816	0.761212	0.380606	+	0.924737i	$0.375715\pi$
$420$	0	0
$421$	−28.4233	−1.38527	−0.692634	−	0.721289i	$-0.743550\pi$
$421$	−28.4233	−1.38527	−0.692634	+	0.721289i	$0.743550\pi$
$422$	0	0
$423$	1.33879	0.0650941
$424$	0	0
$425$	77.3438	3.75172
$426$	0	0
$427$	−21.2412	−1.02793
$428$	0	0
$429$	34.2305	1.65266
$430$	0	0
$431$	24.0131	1.15667	0.578335	−	0.815799i	$-0.303703\pi$
$431$	24.0131	1.15667	0.578335	+	0.815799i	$0.303703\pi$
$432$	0	0
$433$	−14.6643	−0.704721	−0.352360	−	0.935864i	$-0.614621\pi$
$433$	−14.6643	−0.704721	−0.352360	+	0.935864i	$0.614621\pi$
$434$	0	0
$435$	1.73606	0.0832377
$436$	0	0
$437$	1.14508	0.0547768
$438$	0	0
$439$	1.59089	0.0759290	0.0379645	−	0.999279i	$-0.487913\pi$
$439$	1.59089	0.0759290	0.0379645	+	0.999279i	$0.487913\pi$
$440$	0	0
$441$	−5.60344	−0.266830
$442$	0	0
$443$	−16.2751	−0.773254	−0.386627	−	0.922236i	$-0.626360\pi$
$443$	−16.2751	−0.773254	−0.386627	+	0.922236i	$0.626360\pi$
$444$	0	0
$445$	−9.18707	−0.435509
$446$	0	0
$447$	−2.90136	−0.137230
$448$	0	0
$449$	20.1754	0.952138	0.476069	−	0.879408i	$-0.342061\pi$
$449$	20.1754	0.952138	0.476069	+	0.879408i	$0.342061\pi$
$450$	0	0
$451$	15.0103	0.706807
$452$	0	0
$453$	−10.4990	−0.493288
$454$	0	0
$455$	43.9823	2.06192
$456$	0	0
$457$	1.96021	0.0916948	0.0458474	−	0.998948i	$-0.485401\pi$
$457$	1.96021	0.0916948	0.0458474	+	0.998948i	$0.485401\pi$
$458$	0	0
$459$	22.1389	1.03336
$460$	0	0
$461$	13.0447	0.607553	0.303777	−	0.952743i	$-0.401752\pi$
$461$	13.0447	0.607553	0.303777	+	0.952743i	$0.401752\pi$
$462$	0	0
$463$	34.9680	1.62510	0.812551	−	0.582890i	$-0.198078\pi$
$463$	34.9680	1.62510	0.812551	+	0.582890i	$0.198078\pi$
$464$	0	0
$465$	−3.00981	−0.139577
$466$	0	0
$467$	3.92704	0.181722	0.0908608	−	0.995864i	$-0.471038\pi$
$467$	3.92704	0.181722	0.0908608	+	0.995864i	$0.471038\pi$
$468$	0	0
$469$	−3.66814	−0.169379
$470$	0	0
$471$	−48.5997	−2.23936
$472$	0	0
$473$	−22.1303	−1.01755
$474$	0	0
$475$	−1.62224	−0.0744335
$476$	0	0
$477$	11.2274	0.514068
$478$	0	0
$479$	34.1517	1.56043	0.780217	−	0.625509i	$-0.215109\pi$
$479$	34.1517	1.56043	0.780217	+	0.625509i	$0.215109\pi$
$480$	0	0
$481$	33.4034	1.52306
$482$	0	0
$483$	−29.8187	−1.35680
$484$	0	0
$485$	28.4974	1.29400
$486$	0	0
$487$	−27.7330	−1.25670	−0.628352	−	0.777929i	$-0.716270\pi$
$487$	−27.7330	−1.25670	−0.628352	+	0.777929i	$0.716270\pi$
$488$	0	0
$489$	19.2102	0.868717
$490$	0	0
$491$	33.7861	1.52474	0.762372	−	0.647139i	$-0.224035\pi$
$491$	33.7861	1.52474	0.762372	+	0.647139i	$0.224035\pi$
$492$	0	0
$493$	−1.28995	−0.0580966
$494$	0	0
$495$	14.3409	0.644574
$496$	0	0
$497$	7.74413	0.347372
$498$	0	0
$499$	−14.2433	−0.637616	−0.318808	−	0.947819i	$-0.603282\pi$
$499$	−14.2433	−0.637616	−0.318808	+	0.947819i	$0.603282\pi$
$500$	0	0
$501$	−6.52508	−0.291519
$502$	0	0
$503$	−20.0670	−0.894743	−0.447372	−	0.894348i	$-0.647640\pi$
$503$	−20.0670	−0.894743	−0.447372	+	0.894348i	$0.647640\pi$
$504$	0	0
$505$	−14.3018	−0.636422
$506$	0	0
$507$	−56.6981	−2.51805
$508$	0	0
$509$	−30.8129	−1.36576	−0.682878	−	0.730533i	$-0.739272\pi$
$509$	−30.8129	−1.36576	−0.682878	+	0.730533i	$0.739272\pi$
$510$	0	0
$511$	−12.6572	−0.559920
$512$	0	0
$513$	−0.464351	−0.0205016
$514$	0	0
$515$	−19.7363	−0.869687
$516$	0	0
$517$	2.59125	0.113963
$518$	0	0
$519$	30.0852	1.32059
$520$	0	0
$521$	−15.8385	−0.693896	−0.346948	−	0.937884i	$-0.612782\pi$
$521$	−15.8385	−0.693896	−0.346948	+	0.937884i	$0.612782\pi$
$522$	0	0
$523$	−16.5674	−0.724442	−0.362221	−	0.932092i	$-0.617981\pi$
$523$	−16.5674	−0.724442	−0.362221	+	0.932092i	$0.617981\pi$
$524$	0	0
$525$	42.2441	1.84368
$526$	0	0
$527$	2.23640	0.0974190
$528$	0	0
$529$	49.8114	2.16571
$530$	0	0
$531$	11.0346	0.478859
$532$	0	0
$533$	−36.7366	−1.59124
$534$	0	0
$535$	78.6357	3.39972
$536$	0	0
$537$	−13.8200	−0.596379
$538$	0	0
$539$	−10.8456	−0.467151
$540$	0	0
$541$	11.6764	0.502006	0.251003	−	0.967986i	$-0.419240\pi$
$541$	11.6764	0.502006	0.251003	+	0.967986i	$0.419240\pi$
$542$	0	0
$543$	−35.5801	−1.52689
$544$	0	0
$545$	−2.34369	−0.100393
$546$	0	0
$547$	−24.6170	−1.05255	−0.526274	−	0.850315i	$-0.676411\pi$
$547$	−24.6170	−1.05255	−0.526274	+	0.850315i	$0.676411\pi$
$548$	0	0
$549$	16.9507	0.723436
$550$	0	0
$551$	0.0270560	0.00115263
$552$	0	0
$553$	0.180952	0.00769485
$554$	0	0
$555$	45.3533	1.92514
$556$	0	0
$557$	−8.37022	−0.354657	−0.177329	−	0.984152i	$-0.556746\pi$
$557$	−8.37022	−0.354657	−0.177329	+	0.984152i	$0.556746\pi$
$558$	0	0
$559$	54.1623	2.29082
$560$	0	0
$561$	−34.5335	−1.45801
$562$	0	0
$563$	6.45054	0.271858	0.135929	−	0.990719i	$-0.456598\pi$
$563$	6.45054	0.271858	0.135929	+	0.990719i	$0.456598\pi$
$564$	0	0
$565$	23.6733	0.995942
$566$	0	0
$567$	18.8301	0.790789
$568$	0	0
$569$	26.3464	1.10450	0.552250	−	0.833679i	$-0.313770\pi$
$569$	26.3464	1.10450	0.552250	+	0.833679i	$0.313770\pi$
$570$	0	0
$571$	−2.32625	−0.0973503	−0.0486752	−	0.998815i	$-0.515500\pi$
$571$	−2.32625	−0.0973503	−0.0486752	+	0.998815i	$0.515500\pi$
$572$	0	0
$573$	12.6003	0.526384
$574$	0	0
$575$	−103.152	−4.30173
$576$	0	0
$577$	28.1789	1.17310	0.586552	−	0.809912i	$-0.300485\pi$
$577$	28.1789	1.17310	0.586552	+	0.809912i	$0.300485\pi$
$578$	0	0
$579$	2.83774	0.117932
$580$	0	0
$581$	−28.3428	−1.17586
$582$	0	0
$583$	21.7308	0.900000
$584$	0	0
$585$	−35.0983	−1.45113
$586$	0	0
$587$	31.4946	1.29992	0.649960	−	0.759969i	$-0.274786\pi$
$587$	31.4946	1.29992	0.649960	+	0.759969i	$0.274786\pi$
$588$	0	0
$589$	−0.0469071	−0.00193277
$590$	0	0
$591$	8.83891	0.363584
$592$	0	0
$593$	−24.6655	−1.01289	−0.506445	−	0.862272i	$-0.669041\pi$
$593$	−24.6655	−1.01289	−0.506445	+	0.862272i	$0.669041\pi$
$594$	0	0
$595$	−44.3717	−1.81906
$596$	0	0
$597$	−43.9446	−1.79853
$598$	0	0
$599$	−28.0702	−1.14692	−0.573459	−	0.819234i	$-0.694399\pi$
$599$	−28.0702	−1.14692	−0.573459	+	0.819234i	$0.694399\pi$
$600$	0	0
$601$	−26.4050	−1.07708	−0.538541	−	0.842600i	$-0.681024\pi$
$601$	−26.4050	−1.07708	−0.538541	+	0.842600i	$0.681024\pi$
$602$	0	0
$603$	2.92721	0.119205
$604$	0	0
$605$	−17.7153	−0.720228
$606$	0	0
$607$	1.25152	0.0507978	0.0253989	−	0.999677i	$-0.491914\pi$
$607$	1.25152	0.0507978	0.0253989	+	0.999677i	$0.491914\pi$
$608$	0	0
$609$	−0.704555	−0.0285500
$610$	0	0
$611$	−6.34190	−0.256566
$612$	0	0
$613$	25.2946	1.02164	0.510820	−	0.859687i	$-0.329342\pi$
$613$	25.2946	1.02164	0.510820	+	0.859687i	$0.329342\pi$
$614$	0	0
$615$	−49.8790	−2.01132
$616$	0	0
$617$	17.4433	0.702240	0.351120	−	0.936331i	$-0.385801\pi$
$617$	17.4433	0.702240	0.351120	+	0.936331i	$0.385801\pi$
$618$	0	0
$619$	−25.3702	−1.01971	−0.509857	−	0.860259i	$-0.670302\pi$
$619$	−25.3702	−1.01971	−0.509857	+	0.860259i	$0.670302\pi$
$620$	0	0
$621$	−29.5263	−1.18485
$622$	0	0
$623$	3.72844	0.149377
$624$	0	0
$625$	60.6922	2.42769
$626$	0	0
$627$	0.724321	0.0289266
$628$	0	0
$629$	−33.6991	−1.34367
$630$	0	0
$631$	−29.2676	−1.16513	−0.582563	−	0.812785i	$-0.697950\pi$
$631$	−29.2676	−1.16513	−0.582563	+	0.812785i	$0.697950\pi$
$632$	0	0
$633$	−6.96973	−0.277022
$634$	0	0
$635$	−69.7249	−2.76695
$636$	0	0
$637$	26.5437	1.05170
$638$	0	0
$639$	−6.17989	−0.244473
$640$	0	0
$641$	26.3538	1.04091	0.520457	−	0.853888i	$-0.325762\pi$
$641$	26.3538	1.04091	0.520457	+	0.853888i	$0.325762\pi$
$642$	0	0
$643$	38.5636	1.52080	0.760400	−	0.649456i	$-0.225003\pi$
$643$	38.5636	1.52080	0.760400	+	0.649456i	$0.225003\pi$
$644$	0	0
$645$	73.5386	2.89558
$646$	0	0
$647$	−9.55391	−0.375603	−0.187802	−	0.982207i	$-0.560136\pi$
$647$	−9.55391	−0.375603	−0.187802	+	0.982207i	$0.560136\pi$
$648$	0	0
$649$	21.3576	0.838358
$650$	0	0
$651$	1.22149	0.0478740
$652$	0	0
$653$	25.1668	0.984852	0.492426	−	0.870354i	$-0.336110\pi$
$653$	25.1668	0.984852	0.492426	+	0.870354i	$0.336110\pi$
$654$	0	0
$655$	−39.3415	−1.53720
$656$	0	0
$657$	10.1005	0.394059
$658$	0	0
$659$	−17.9044	−0.697457	−0.348728	−	0.937224i	$-0.613386\pi$
$659$	−17.9044	−0.697457	−0.348728	+	0.937224i	$0.613386\pi$
$660$	0	0
$661$	−21.1629	−0.823143	−0.411571	−	0.911378i	$-0.635020\pi$
$661$	−21.1629	−0.823143	−0.411571	+	0.911378i	$0.635020\pi$
$662$	0	0
$663$	84.5184	3.28242
$664$	0	0
$665$	0.930670	0.0360898
$666$	0	0
$667$	1.72039	0.0666137
$668$	0	0
$669$	−16.8264	−0.650546
$670$	0	0
$671$	32.8083	1.26655
$672$	0	0
$673$	2.38253	0.0918399	0.0459200	−	0.998945i	$-0.485378\pi$
$673$	2.38253	0.0918399	0.0459200	+	0.998945i	$0.485378\pi$
$674$	0	0
$675$	41.8299	1.61003
$676$	0	0
$677$	−16.9521	−0.651523	−0.325761	−	0.945452i	$-0.605621\pi$
$677$	−16.9521	−0.651523	−0.325761	+	0.945452i	$0.605621\pi$
$678$	0	0
$679$	−11.5653	−0.443834
$680$	0	0
$681$	17.5226	0.671468
$682$	0	0
$683$	22.7960	0.872266	0.436133	−	0.899882i	$-0.356348\pi$
$683$	22.7960	0.872266	0.436133	+	0.899882i	$0.356348\pi$
$684$	0	0
$685$	−23.3277	−0.891304
$686$	0	0
$687$	−14.4738	−0.552212
$688$	0	0
$689$	−53.1848	−2.02618
$690$	0	0
$691$	−49.1410	−1.86941	−0.934706	−	0.355421i	$-0.884338\pi$
$691$	−49.1410	−1.86941	−0.934706	+	0.355421i	$0.884338\pi$
$692$	0	0
$693$	−5.82004	−0.221085
$694$	0	0
$695$	55.1268	2.09108
$696$	0	0
$697$	37.0619	1.40382
$698$	0	0
$699$	20.3722	0.770546
$700$	0	0
$701$	26.2844	0.992748	0.496374	−	0.868109i	$-0.334664\pi$
$701$	26.2844	0.992748	0.496374	+	0.868109i	$0.334664\pi$
$702$	0	0
$703$	0.706819	0.0266582
$704$	0	0
$705$	−8.61069	−0.324297
$706$	0	0
$707$	5.80418	0.218289
$708$	0	0
$709$	−18.5740	−0.697560	−0.348780	−	0.937205i	$-0.613404\pi$
$709$	−18.5740	−0.697560	−0.348780	+	0.937205i	$0.613404\pi$
$710$	0	0
$711$	−0.144401	−0.00541547
$712$	0	0
$713$	−2.98264	−0.111701
$714$	0	0
$715$	−67.9333	−2.54056
$716$	0	0
$717$	−19.3598	−0.723006
$718$	0	0
$719$	23.4694	0.875261	0.437631	−	0.899155i	$-0.355818\pi$
$719$	23.4694	0.875261	0.437631	+	0.899155i	$0.355818\pi$
$720$	0	0
$721$	8.00971	0.298297
$722$	0	0
$723$	54.4908	2.02654
$724$	0	0
$725$	−2.43727	−0.0905181
$726$	0	0
$727$	0.891501	0.0330639	0.0165320	−	0.999863i	$-0.494737\pi$
$727$	0.891501	0.0330639	0.0165320	+	0.999863i	$0.494737\pi$
$728$	0	0
$729$	6.59635	0.244309
$730$	0	0
$731$	−54.6418	−2.02100
$732$	0	0
$733$	1.31418	0.0485402	0.0242701	−	0.999705i	$-0.492274\pi$
$733$	1.31418	0.0485402	0.0242701	+	0.999705i	$0.492274\pi$
$734$	0	0
$735$	36.0396	1.32934
$736$	0	0
$737$	5.66567	0.208698
$738$	0	0
$739$	21.4947	0.790696	0.395348	−	0.918531i	$-0.370624\pi$
$739$	21.4947	0.790696	0.395348	+	0.918531i	$0.370624\pi$
$740$	0	0
$741$	−1.77272	−0.0651227
$742$	0	0
$743$	−5.96687	−0.218903	−0.109452	−	0.993992i	$-0.534909\pi$
$743$	−5.96687	−0.218903	−0.109452	+	0.993992i	$0.534909\pi$
$744$	0	0
$745$	5.75799	0.210957
$746$	0	0
$747$	22.6178	0.827543
$748$	0	0
$749$	−31.9132	−1.16608
$750$	0	0
$751$	3.48439	0.127147	0.0635736	−	0.997977i	$-0.479750\pi$
$751$	3.48439	0.127147	0.0635736	+	0.997977i	$0.479750\pi$
$752$	0	0
$753$	−19.5613	−0.712854
$754$	0	0
$755$	20.8362	0.758309
$756$	0	0
$757$	−4.28526	−0.155751	−0.0778753	−	0.996963i	$-0.524814\pi$
$757$	−4.28526	−0.155751	−0.0778753	+	0.996963i	$0.524814\pi$
$758$	0	0
$759$	46.0567	1.67175
$760$	0	0
$761$	−29.7276	−1.07762	−0.538812	−	0.842426i	$-0.681127\pi$
$761$	−29.7276	−1.07762	−0.538812	+	0.842426i	$0.681127\pi$
$762$	0	0
$763$	0.951155	0.0344341
$764$	0	0
$765$	35.4090	1.28021
$766$	0	0
$767$	−52.2712	−1.88740
$768$	0	0
$769$	−38.7569	−1.39761	−0.698805	−	0.715312i	$-0.746284\pi$
$769$	−38.7569	−1.39761	−0.698805	+	0.715312i	$0.746284\pi$
$770$	0	0
$771$	64.8933	2.33708
$772$	0	0
$773$	−30.0163	−1.07961	−0.539806	−	0.841789i	$-0.681502\pi$
$773$	−30.0163	−1.07961	−0.539806	+	0.841789i	$0.681502\pi$
$774$	0	0
$775$	4.22551	0.151785
$776$	0	0
$777$	−18.4060	−0.660312
$778$	0	0
$779$	−0.777351	−0.0278515
$780$	0	0
$781$	−11.9613	−0.428008
$782$	0	0
$783$	−0.697647	−0.0249319
$784$	0	0
$785$	96.4502	3.44246
$786$	0	0
$787$	45.9817	1.63907	0.819535	−	0.573029i	$-0.194232\pi$
$787$	45.9817	1.63907	0.819535	+	0.573029i	$0.194232\pi$
$788$	0	0
$789$	60.6586	2.15950
$790$	0	0
$791$	−9.60746	−0.341602
$792$	0	0
$793$	−80.2960	−2.85140
$794$	0	0
$795$	−72.2114	−2.56107
$796$	0	0
$797$	37.9227	1.34329	0.671646	−	0.740872i	$-0.265588\pi$
$797$	37.9227	1.34329	0.671646	+	0.740872i	$0.265588\pi$
$798$	0	0
$799$	6.39805	0.226347
$800$	0	0
$801$	−2.97533	−0.105128
$802$	0	0
$803$	19.5497	0.689895
$804$	0	0
$805$	59.1777	2.08574
$806$	0	0
$807$	−38.2065	−1.34493
$808$	0	0
$809$	17.5644	0.617530	0.308765	−	0.951138i	$-0.400084\pi$
$809$	17.5644	0.617530	0.308765	+	0.951138i	$0.400084\pi$
$810$	0	0
$811$	12.8602	0.451582	0.225791	−	0.974176i	$-0.427503\pi$
$811$	12.8602	0.451582	0.225791	+	0.974176i	$0.427503\pi$
$812$	0	0
$813$	−51.1859	−1.79517
$814$	0	0
$815$	−38.1244	−1.33544
$816$	0	0
$817$	1.14608	0.0400962
$818$	0	0
$819$	14.2441	0.497730
$820$	0	0
$821$	−44.7868	−1.56307	−0.781535	−	0.623862i	$-0.785563\pi$
$821$	−44.7868	−1.56307	−0.781535	+	0.623862i	$0.785563\pi$
$822$	0	0
$823$	15.8572	0.552747	0.276374	−	0.961050i	$-0.410867\pi$
$823$	15.8572	0.552747	0.276374	+	0.961050i	$0.410867\pi$
$824$	0	0
$825$	−65.2486	−2.27167
$826$	0	0
$827$	−31.3254	−1.08929	−0.544645	−	0.838666i	$-0.683336\pi$
$827$	−31.3254	−1.08929	−0.544645	+	0.838666i	$0.683336\pi$
$828$	0	0
$829$	−32.8565	−1.14115	−0.570577	−	0.821244i	$-0.693281\pi$
$829$	−32.8565	−1.14115	−0.570577	+	0.821244i	$0.693281\pi$
$830$	0	0
$831$	67.6258	2.34591
$832$	0	0
$833$	−26.7787	−0.927828
$834$	0	0
$835$	12.9496	0.448138
$836$	0	0
$837$	1.20951	0.0418069
$838$	0	0
$839$	5.62580	0.194224	0.0971120	−	0.995273i	$-0.469039\pi$
$839$	5.62580	0.194224	0.0971120	+	0.995273i	$0.469039\pi$
$840$	0	0
$841$	−28.9594	−0.998598
$842$	0	0
$843$	10.1562	0.349799
$844$	0	0
$845$	112.522	3.87088
$846$	0	0
$847$	7.18949	0.247034
$848$	0	0
$849$	−21.6023	−0.741390
$850$	0	0
$851$	44.9439	1.54066
$852$	0	0
$853$	31.4234	1.07592	0.537959	−	0.842971i	$-0.319196\pi$
$853$	31.4234	1.07592	0.537959	+	0.842971i	$0.319196\pi$
$854$	0	0
$855$	−0.742683	−0.0253992
$856$	0	0
$857$	−7.86305	−0.268597	−0.134298	−	0.990941i	$-0.542878\pi$
$857$	−7.86305	−0.268597	−0.134298	+	0.990941i	$0.542878\pi$
$858$	0	0
$859$	−32.4392	−1.10681	−0.553405	−	0.832912i	$-0.686672\pi$
$859$	−32.4392	−1.10681	−0.553405	+	0.832912i	$0.686672\pi$
$860$	0	0
$861$	20.2427	0.689869
$862$	0	0
$863$	−22.3985	−0.762453	−0.381227	−	0.924482i	$-0.624498\pi$
$863$	−22.3985	−0.762453	−0.381227	+	0.924482i	$0.624498\pi$
$864$	0	0
$865$	−59.7066	−2.03008
$866$	0	0
$867$	−49.8561	−1.69320
$868$	0	0
$869$	−0.279491	−0.00948108
$870$	0	0
$871$	−13.8663	−0.469843
$872$	0	0
$873$	9.22919	0.312361
$874$	0	0
$875$	−49.1611	−1.66195
$876$	0	0
$877$	9.20323	0.310771	0.155385	−	0.987854i	$-0.450338\pi$
$877$	9.20323	0.310771	0.155385	+	0.987854i	$0.450338\pi$
$878$	0	0
$879$	30.9840	1.04506
$880$	0	0
$881$	27.4426	0.924566	0.462283	−	0.886733i	$-0.347031\pi$
$881$	27.4426	0.924566	0.462283	+	0.886733i	$0.347031\pi$
$882$	0	0
$883$	−26.5552	−0.893653	−0.446827	−	0.894621i	$-0.647446\pi$
$883$	−26.5552	−0.893653	−0.446827	+	0.894621i	$0.647446\pi$
$884$	0	0
$885$	−70.9710	−2.38566
$886$	0	0
$887$	−34.5178	−1.15899	−0.579496	−	0.814975i	$-0.696751\pi$
$887$	−34.5178	−1.15899	−0.579496	+	0.814975i	$0.696751\pi$
$888$	0	0
$889$	28.2969	0.949046
$890$	0	0
$891$	−29.0842	−0.974358
$892$	0	0
$893$	−0.134195	−0.00449068
$894$	0	0
$895$	27.4270	0.916785
$896$	0	0
$897$	−112.721	−3.76363
$898$	0	0
$899$	−0.0704739	−0.00235043
$900$	0	0
$901$	53.6556	1.78753
$902$	0	0
$903$	−29.8446	−0.993166
$904$	0	0
$905$	70.6118	2.34722
$906$	0	0
$907$	−32.5404	−1.08049	−0.540243	−	0.841509i	$-0.681668\pi$
$907$	−32.5404	−1.08049	−0.540243	+	0.841509i	$0.681668\pi$
$908$	0	0
$909$	−4.63179	−0.153627
$910$	0	0
$911$	30.1928	1.00033	0.500165	−	0.865930i	$-0.333273\pi$
$911$	30.1928	1.00033	0.500165	+	0.865930i	$0.333273\pi$
$912$	0	0
$913$	43.7772	1.44881
$914$	0	0
$915$	−109.022	−3.60414
$916$	0	0
$917$	15.9662	0.527250
$918$	0	0
$919$	−26.9216	−0.888063	−0.444032	−	0.896011i	$-0.646452\pi$
$919$	−26.9216	−0.888063	−0.444032	+	0.896011i	$0.646452\pi$
$920$	0	0
$921$	−62.0147	−2.04345
$922$	0	0
$923$	29.2744	0.963579
$924$	0	0
$925$	−63.6720	−2.09352
$926$	0	0
$927$	−6.39182	−0.209935
$928$	0	0
$929$	−20.2025	−0.662822	−0.331411	−	0.943486i	$-0.607525\pi$
$929$	−20.2025	−0.662822	−0.331411	+	0.943486i	$0.607525\pi$
$930$	0	0
$931$	0.561668	0.0184079
$932$	0	0
$933$	−52.0573	−1.70428
$934$	0	0
$935$	68.5347	2.24133
$936$	0	0
$937$	−17.8328	−0.582573	−0.291286	−	0.956636i	$-0.594083\pi$
$937$	−17.8328	−0.582573	−0.291286	+	0.956636i	$0.594083\pi$
$938$	0	0
$939$	−16.2732	−0.531054
$940$	0	0
$941$	26.6418	0.868498	0.434249	−	0.900793i	$-0.357014\pi$
$941$	26.6418	0.868498	0.434249	+	0.900793i	$0.357014\pi$
$942$	0	0
$943$	−49.4287	−1.60962
$944$	0	0
$945$	−23.9976	−0.780641
$946$	0	0
$947$	−38.2122	−1.24173	−0.620865	−	0.783918i	$-0.713218\pi$
$947$	−38.2122	−1.24173	−0.620865	+	0.783918i	$0.713218\pi$
$948$	0	0
$949$	−47.8466	−1.55317
$950$	0	0
$951$	16.2743	0.527731
$952$	0	0
$953$	−31.8311	−1.03111	−0.515555	−	0.856856i	$-0.672414\pi$
$953$	−31.8311	−1.03111	−0.515555	+	0.856856i	$0.672414\pi$
$954$	0	0
$955$	−25.0063	−0.809184
$956$	0	0
$957$	1.08823	0.0351774
$958$	0	0
$959$	9.46720	0.305712
$960$	0	0
$961$	−30.8778	−0.996059
$962$	0	0
$963$	25.4670	0.820663
$964$	0	0
$965$	−5.63174	−0.181292
$966$	0	0
$967$	37.9221	1.21949	0.609746	−	0.792597i	$-0.291272\pi$
$967$	37.9221	1.21949	0.609746	+	0.792597i	$0.291272\pi$
$968$	0	0
$969$	1.78842	0.0574523
$970$	0	0
$971$	4.62710	0.148491	0.0742454	−	0.997240i	$-0.476345\pi$
$971$	4.62710	0.148491	0.0742454	+	0.997240i	$0.476345\pi$
$972$	0	0
$973$	−22.3724	−0.717227
$974$	0	0
$975$	159.691	5.11422
$976$	0	0
$977$	30.0173	0.960339	0.480170	−	0.877176i	$-0.340575\pi$
$977$	30.0173	0.960339	0.480170	+	0.877176i	$0.340575\pi$
$978$	0	0
$979$	−5.75881	−0.184052
$980$	0	0
$981$	−0.759030	−0.0242340
$982$	0	0
$983$	26.3433	0.840221	0.420110	−	0.907473i	$-0.361991\pi$
$983$	26.3433	0.840221	0.420110	+	0.907473i	$0.361991\pi$
$984$	0	0
$985$	−17.5416	−0.558921
$986$	0	0
$987$	3.49453	0.111232
$988$	0	0
$989$	72.8747	2.31728
$990$	0	0
$991$	−23.5391	−0.747743	−0.373872	−	0.927480i	$-0.621970\pi$
$991$	−23.5391	−0.747743	−0.373872	+	0.927480i	$0.621970\pi$
$992$	0	0
$993$	−57.7890	−1.83388
$994$	0	0
$995$	87.2118	2.76480
$996$	0	0
$997$	−12.4705	−0.394945	−0.197472	−	0.980308i	$-0.563273\pi$
$997$	−12.4705	−0.394945	−0.197472	+	0.980308i	$0.563273\pi$
$998$	0	0
$999$	−18.2255	−0.576630

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6016.2.a.p.1.3	yes	13
4.3	odd	2		6016.2.a.n.1.11	yes	13
8.3	odd	2		6016.2.a.o.1.3	yes	13
8.5	even	2		6016.2.a.m.1.11	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6016.2.a.m.1.11	✓	13	8.5	even	2
6016.2.a.n.1.11	yes	13	4.3	odd	2
6016.2.a.o.1.3	yes	13	8.3	odd	2
6016.2.a.p.1.3	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)

\(f(q)\)	\(=\)	\(q-2.08298 q^{3} +4.13384 q^{5} -1.67766 q^{7} +1.33879 q^{9} +O(q^{10})\)	\(q-2.08298 q^{3} +4.13384 q^{5} -1.67766 q^{7} +1.33879 q^{9} +2.59125 q^{11} -6.34190 q^{13} -8.61069 q^{15} +6.39805 q^{17} -0.134195 q^{19} +3.49453 q^{21} -8.53296 q^{23} +12.0886 q^{25} +3.46026 q^{27} -0.201617 q^{29} +0.349544 q^{31} -5.39751 q^{33} -6.93519 q^{35} -5.26709 q^{37} +13.2100 q^{39} +5.79268 q^{41} -8.54038 q^{43} +5.53434 q^{45} +1.00000 q^{47} -4.18545 q^{49} -13.3270 q^{51} +8.38624 q^{53} +10.7118 q^{55} +0.279526 q^{57} +8.24219 q^{59} +12.6612 q^{61} -2.24603 q^{63} -26.2164 q^{65} +2.18646 q^{67} +17.7740 q^{69} -4.61603 q^{71} +7.54452 q^{73} -25.1804 q^{75} -4.34724 q^{77} -0.107860 q^{79} -11.2240 q^{81} +16.8942 q^{83} +26.4485 q^{85} +0.419963 q^{87} -2.22240 q^{89} +10.6396 q^{91} -0.728091 q^{93} -0.554742 q^{95} +6.89369 q^{97} +3.46914 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 4 q^{3} + 6 q^{5} + 2 q^{7} + 21 q^{9}+O(q^{10}) \) 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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