LMFDB - Embedded newform 6036.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, 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("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 4 x^{14} - 27 x^{13} + 106 x^{12} + 266 x^{11} - 1004 x^{10} - 1105 x^{9} + 4076 x^{8} + \cdots - 44 $ x^15 - 4x^14 - 27x^13 + 106x^12 + 266x^11 - 1004x^10 - 1105x^9 + 4076x^8 + 1501x^7 - 7100x^6 - 134x^5 + 5356x^4 - 1041x^3 - 1381x^2 + 543x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.82901$ of defining polynomial
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.82901 q^{5} +3.22079 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.82901 q^{5} +3.22079 q^{7} +1.00000 q^{9} +3.21987 q^{11} -3.97711 q^{13} -3.82901 q^{15} -2.27833 q^{17} +0.0375216 q^{19} +3.22079 q^{21} +1.29407 q^{23} +9.66131 q^{25} +1.00000 q^{27} -4.68621 q^{29} -4.65052 q^{31} +3.21987 q^{33} -12.3324 q^{35} -2.10968 q^{37} -3.97711 q^{39} -9.37784 q^{41} +10.1960 q^{43} -3.82901 q^{45} -9.03702 q^{47} +3.37347 q^{49} -2.27833 q^{51} -0.602797 q^{53} -12.3289 q^{55} +0.0375216 q^{57} +3.58965 q^{59} -3.24029 q^{61} +3.22079 q^{63} +15.2284 q^{65} +12.9269 q^{67} +1.29407 q^{69} -5.99438 q^{71} +0.896490 q^{73} +9.66131 q^{75} +10.3705 q^{77} +2.02321 q^{79} +1.00000 q^{81} +0.335149 q^{83} +8.72373 q^{85} -4.68621 q^{87} +6.96419 q^{89} -12.8094 q^{91} -4.65052 q^{93} -0.143670 q^{95} -5.63494 q^{97} +3.21987 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) $ 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9	$ 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9} - 5 q^{11} - 14 q^{13} - 11 q^{15} - 5 q^{17} + 3 q^{19} - 4 q^{21} - 32 q^{23} + 2 q^{25} + 15 q^{27} - 23 q^{29} - 13 q^{31} - 5 q^{33} - 16 q^{35} - 10 q^{37} - 14 q^{39} - 14 q^{41} + 4 q^{43} - 11 q^{45} - 20 q^{47} - 9 q^{49} - 5 q^{51} - 30 q^{53} - 10 q^{55} + 3 q^{57} - 14 q^{59} - 38 q^{61} - 4 q^{63} - 24 q^{65} - 8 q^{67} - 32 q^{69} - 41 q^{71} - 19 q^{73} + 2 q^{75} - 39 q^{77} - 27 q^{79} + 15 q^{81} - 17 q^{83} - 6 q^{85} - 23 q^{87} - 23 q^{89} + 4 q^{91} - 13 q^{93} - 30 q^{95} - 18 q^{97} - 5 q^{99}+O(q^{100}) $ 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9 - 5 * q^11 - 14 * q^13 - 11 * q^15 - 5 * q^17 + 3 * q^19 - 4 * q^21 - 32 * q^23 + 2 * q^25 + 15 * q^27 - 23 * q^29 - 13 * q^31 - 5 * q^33 - 16 * q^35 - 10 * q^37 - 14 * q^39 - 14 * q^41 + 4 * q^43 - 11 * q^45 - 20 * q^47 - 9 * q^49 - 5 * q^51 - 30 * q^53 - 10 * q^55 + 3 * q^57 - 14 * q^59 - 38 * q^61 - 4 * q^63 - 24 * q^65 - 8 * q^67 - 32 * q^69 - 41 * q^71 - 19 * q^73 + 2 * q^75 - 39 * q^77 - 27 * q^79 + 15 * q^81 - 17 * q^83 - 6 * q^85 - 23 * q^87 - 23 * q^89 + 4 * q^91 - 13 * q^93 - 30 * q^95 - 18 * q^97 - 5 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−3.82901	−1.71239	−0.856193	−	0.516657i	$-0.827176\pi$
$5$	−3.82901	−1.71239	−0.856193	+	0.516657i	$0.827176\pi$
$6$	0	0
$7$	3.22079	1.21734	0.608672	−	0.793422i	$-0.291703\pi$
$7$	3.22079	1.21734	0.608672	+	0.793422i	$0.291703\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.21987	0.970826	0.485413	−	0.874285i	$-0.338669\pi$
$11$	3.21987	0.970826	0.485413	+	0.874285i	$0.338669\pi$
$12$	0	0
$13$	−3.97711	−1.10305	−0.551526	−	0.834157i	$-0.685954\pi$
$13$	−3.97711	−1.10305	−0.551526	+	0.834157i	$0.685954\pi$
$14$	0	0
$15$	−3.82901	−0.988646
$16$	0	0
$17$	−2.27833	−0.552575	−0.276288	−	0.961075i	$-0.589104\pi$
$17$	−2.27833	−0.552575	−0.276288	+	0.961075i	$0.589104\pi$
$18$	0	0
$19$	0.0375216	0.00860804	0.00430402	−	0.999991i	$-0.498630\pi$
$19$	0.0375216	0.00860804	0.00430402	+	0.999991i	$0.498630\pi$
$20$	0	0
$21$	3.22079	0.702834
$22$	0	0
$23$	1.29407	0.269833	0.134917	−	0.990857i	$-0.456923\pi$
$23$	1.29407	0.269833	0.134917	+	0.990857i	$0.456923\pi$
$24$	0	0
$25$	9.66131	1.93226
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.68621	−0.870207	−0.435103	−	0.900380i	$-0.643288\pi$
$29$	−4.68621	−0.870207	−0.435103	+	0.900380i	$0.643288\pi$
$30$	0	0
$31$	−4.65052	−0.835258	−0.417629	−	0.908618i	$-0.637139\pi$
$31$	−4.65052	−0.835258	−0.417629	+	0.908618i	$0.637139\pi$
$32$	0	0
$33$	3.21987	0.560507
$34$	0	0
$35$	−12.3324	−2.08456
$36$	0	0
$37$	−2.10968	−0.346830	−0.173415	−	0.984849i	$-0.555480\pi$
$37$	−2.10968	−0.346830	−0.173415	+	0.984849i	$0.555480\pi$
$38$	0	0
$39$	−3.97711	−0.636848
$40$	0	0
$41$	−9.37784	−1.46457	−0.732287	−	0.680997i	$-0.761547\pi$
$41$	−9.37784	−1.46457	−0.732287	+	0.680997i	$0.761547\pi$
$42$	0	0
$43$	10.1960	1.55488	0.777439	−	0.628958i	$-0.216518\pi$
$43$	10.1960	1.55488	0.777439	+	0.628958i	$0.216518\pi$
$44$	0	0
$45$	−3.82901	−0.570795
$46$	0	0
$47$	−9.03702	−1.31819	−0.659093	−	0.752062i	$-0.729060\pi$
$47$	−9.03702	−1.31819	−0.659093	+	0.752062i	$0.729060\pi$
$48$	0	0
$49$	3.37347	0.481925
$50$	0	0
$51$	−2.27833	−0.319029
$52$	0	0
$53$	−0.602797	−0.0828005	−0.0414002	−	0.999143i	$-0.513182\pi$
$53$	−0.602797	−0.0828005	−0.0414002	+	0.999143i	$0.513182\pi$
$54$	0	0
$55$	−12.3289	−1.66243
$56$	0	0
$57$	0.0375216	0.00496985
$58$	0	0
$59$	3.58965	0.467333	0.233667	−	0.972317i	$-0.424928\pi$
$59$	3.58965	0.467333	0.233667	+	0.972317i	$0.424928\pi$
$60$	0	0
$61$	−3.24029	−0.414876	−0.207438	−	0.978248i	$-0.566513\pi$
$61$	−3.24029	−0.414876	−0.207438	+	0.978248i	$0.566513\pi$
$62$	0	0
$63$	3.22079	0.405781
$64$	0	0
$65$	15.2284	1.88885
$66$	0	0
$67$	12.9269	1.57927	0.789635	−	0.613577i	$-0.210270\pi$
$67$	12.9269	1.57927	0.789635	+	0.613577i	$0.210270\pi$
$68$	0	0
$69$	1.29407	0.155788
$70$	0	0
$71$	−5.99438	−0.711402	−0.355701	−	0.934600i	$-0.615758\pi$
$71$	−5.99438	−0.711402	−0.355701	+	0.934600i	$0.615758\pi$
$72$	0	0
$73$	0.896490	0.104926	0.0524631	−	0.998623i	$-0.483293\pi$
$73$	0.896490	0.104926	0.0524631	+	0.998623i	$0.483293\pi$
$74$	0	0
$75$	9.66131	1.11559
$76$	0	0
$77$	10.3705	1.18183
$78$	0	0
$79$	2.02321	0.227629	0.113815	−	0.993502i	$-0.463693\pi$
$79$	2.02321	0.227629	0.113815	+	0.993502i	$0.463693\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.335149	0.0367874	0.0183937	−	0.999831i	$-0.494145\pi$
$83$	0.335149	0.0367874	0.0183937	+	0.999831i	$0.494145\pi$
$84$	0	0
$85$	8.72373	0.946222
$86$	0	0
$87$	−4.68621	−0.502414
$88$	0	0
$89$	6.96419	0.738202	0.369101	−	0.929389i	$-0.379666\pi$
$89$	6.96419	0.738202	0.369101	+	0.929389i	$0.379666\pi$
$90$	0	0
$91$	−12.8094	−1.34279
$92$	0	0
$93$	−4.65052	−0.482236
$94$	0	0
$95$	−0.143670	−0.0147403
$96$	0	0
$97$	−5.63494	−0.572141	−0.286071	−	0.958209i	$-0.592349\pi$
$97$	−5.63494	−0.572141	−0.286071	+	0.958209i	$0.592349\pi$
$98$	0	0
$99$	3.21987	0.323609
$100$	0	0
$101$	0.592996	0.0590053	0.0295027	−	0.999565i	$-0.490608\pi$
$101$	0.592996	0.0590053	0.0295027	+	0.999565i	$0.490608\pi$
$102$	0	0
$103$	−8.58197	−0.845606	−0.422803	−	0.906222i	$-0.638954\pi$
$103$	−8.58197	−0.845606	−0.422803	+	0.906222i	$0.638954\pi$
$104$	0	0
$105$	−12.3324	−1.20352
$106$	0	0
$107$	−16.6553	−1.61013	−0.805064	−	0.593188i	$-0.797869\pi$
$107$	−16.6553	−1.61013	−0.805064	+	0.593188i	$0.797869\pi$
$108$	0	0
$109$	1.60814	0.154032	0.0770159	−	0.997030i	$-0.475461\pi$
$109$	1.60814	0.154032	0.0770159	+	0.997030i	$0.475461\pi$
$110$	0	0
$111$	−2.10968	−0.200242
$112$	0	0
$113$	8.30402	0.781176	0.390588	−	0.920566i	$-0.372272\pi$
$113$	8.30402	0.781176	0.390588	+	0.920566i	$0.372272\pi$
$114$	0	0
$115$	−4.95502	−0.462058
$116$	0	0
$117$	−3.97711	−0.367684
$118$	0	0
$119$	−7.33801	−0.672674
$120$	0	0
$121$	−0.632465	−0.0574968
$122$	0	0
$123$	−9.37784	−0.845572
$124$	0	0
$125$	−17.8482	−1.59639
$126$	0	0
$127$	−8.87450	−0.787484	−0.393742	−	0.919221i	$-0.628820\pi$
$127$	−8.87450	−0.787484	−0.393742	+	0.919221i	$0.628820\pi$
$128$	0	0
$129$	10.1960	0.897709
$130$	0	0
$131$	−7.30235	−0.638009	−0.319005	−	0.947753i	$-0.603349\pi$
$131$	−7.30235	−0.638009	−0.319005	+	0.947753i	$0.603349\pi$
$132$	0	0
$133$	0.120849	0.0104789
$134$	0	0
$135$	−3.82901	−0.329549
$136$	0	0
$137$	−6.64349	−0.567591	−0.283796	−	0.958885i	$-0.591594\pi$
$137$	−6.64349	−0.567591	−0.283796	+	0.958885i	$0.591594\pi$
$138$	0	0
$139$	−1.58350	−0.134311	−0.0671554	−	0.997743i	$-0.521392\pi$
$139$	−1.58350	−0.134311	−0.0671554	+	0.997743i	$0.521392\pi$
$140$	0	0
$141$	−9.03702	−0.761055
$142$	0	0
$143$	−12.8058	−1.07087
$144$	0	0
$145$	17.9435	1.49013
$146$	0	0
$147$	3.37347	0.278239
$148$	0	0
$149$	10.1028	0.827657	0.413829	−	0.910355i	$-0.364191\pi$
$149$	10.1028	0.827657	0.413829	+	0.910355i	$0.364191\pi$
$150$	0	0
$151$	−1.88225	−0.153175	−0.0765875	−	0.997063i	$-0.524402\pi$
$151$	−1.88225	−0.153175	−0.0765875	+	0.997063i	$0.524402\pi$
$152$	0	0
$153$	−2.27833	−0.184192
$154$	0	0
$155$	17.8069	1.43028
$156$	0	0
$157$	−18.9549	−1.51276	−0.756381	−	0.654131i	$-0.773035\pi$
$157$	−18.9549	−1.51276	−0.756381	+	0.654131i	$0.773035\pi$
$158$	0	0
$159$	−0.602797	−0.0478049
$160$	0	0
$161$	4.16794	0.328479
$162$	0	0
$163$	−14.5057	−1.13617	−0.568086	−	0.822969i	$-0.692316\pi$
$163$	−14.5057	−1.13617	−0.568086	+	0.822969i	$0.692316\pi$
$164$	0	0
$165$	−12.3289	−0.959803
$166$	0	0
$167$	−20.4075	−1.57918	−0.789589	−	0.613636i	$-0.789706\pi$
$167$	−20.4075	−1.57918	−0.789589	+	0.613636i	$0.789706\pi$
$168$	0	0
$169$	2.81743	0.216725
$170$	0	0
$171$	0.0375216	0.00286935
$172$	0	0
$173$	5.65867	0.430221	0.215110	−	0.976590i	$-0.430989\pi$
$173$	5.65867	0.430221	0.215110	+	0.976590i	$0.430989\pi$
$174$	0	0
$175$	31.1170	2.35223
$176$	0	0
$177$	3.58965	0.269815
$178$	0	0
$179$	1.95966	0.146472	0.0732359	−	0.997315i	$-0.476667\pi$
$179$	1.95966	0.146472	0.0732359	+	0.997315i	$0.476667\pi$
$180$	0	0
$181$	14.2761	1.06113	0.530566	−	0.847644i	$-0.321980\pi$
$181$	14.2761	1.06113	0.530566	+	0.847644i	$0.321980\pi$
$182$	0	0
$183$	−3.24029	−0.239529
$184$	0	0
$185$	8.07799	0.593906
$186$	0	0
$187$	−7.33590	−0.536454
$188$	0	0
$189$	3.22079	0.234278
$190$	0	0
$191$	−8.14830	−0.589590	−0.294795	−	0.955560i	$-0.595251\pi$
$191$	−8.14830	−0.589590	−0.294795	+	0.955560i	$0.595251\pi$
$192$	0	0
$193$	3.89931	0.280679	0.140339	−	0.990103i	$-0.455181\pi$
$193$	3.89931	0.280679	0.140339	+	0.990103i	$0.455181\pi$
$194$	0	0
$195$	15.2284	1.09053
$196$	0	0
$197$	−2.27571	−0.162138	−0.0810690	−	0.996708i	$-0.525833\pi$
$197$	−2.27571	−0.162138	−0.0810690	+	0.996708i	$0.525833\pi$
$198$	0	0
$199$	−1.79134	−0.126984	−0.0634922	−	0.997982i	$-0.520224\pi$
$199$	−1.79134	−0.126984	−0.0634922	+	0.997982i	$0.520224\pi$
$200$	0	0
$201$	12.9269	0.911792
$202$	0	0
$203$	−15.0933	−1.05934
$204$	0	0
$205$	35.9078	2.50791
$206$	0	0
$207$	1.29407	0.0899443
$208$	0	0
$209$	0.120814	0.00835691
$210$	0	0
$211$	7.37064	0.507416	0.253708	−	0.967281i	$-0.418350\pi$
$211$	7.37064	0.507416	0.253708	+	0.967281i	$0.418350\pi$
$212$	0	0
$213$	−5.99438	−0.410728
$214$	0	0
$215$	−39.0406	−2.66255
$216$	0	0
$217$	−14.9783	−1.01680
$218$	0	0
$219$	0.896490	0.0605792
$220$	0	0
$221$	9.06116	0.609520
$222$	0	0
$223$	−5.17257	−0.346381	−0.173191	−	0.984888i	$-0.555408\pi$
$223$	−5.17257	−0.346381	−0.173191	+	0.984888i	$0.555408\pi$
$224$	0	0
$225$	9.66131	0.644088
$226$	0	0
$227$	−19.4401	−1.29028	−0.645142	−	0.764063i	$-0.723202\pi$
$227$	−19.4401	−1.29028	−0.645142	+	0.764063i	$0.723202\pi$
$228$	0	0
$229$	1.94619	0.128608	0.0643040	−	0.997930i	$-0.479517\pi$
$229$	1.94619	0.128608	0.0643040	+	0.997930i	$0.479517\pi$
$230$	0	0
$231$	10.3705	0.682329
$232$	0	0
$233$	−17.0762	−1.11870	−0.559348	−	0.828933i	$-0.688949\pi$
$233$	−17.0762	−1.11870	−0.559348	+	0.828933i	$0.688949\pi$
$234$	0	0
$235$	34.6028	2.25724
$236$	0	0
$237$	2.02321	0.131422
$238$	0	0
$239$	−23.2634	−1.50478	−0.752391	−	0.658717i	$-0.771099\pi$
$239$	−23.2634	−1.50478	−0.752391	+	0.658717i	$0.771099\pi$
$240$	0	0
$241$	−0.733843	−0.0472710	−0.0236355	−	0.999721i	$-0.507524\pi$
$241$	−0.733843	−0.0472710	−0.0236355	+	0.999721i	$0.507524\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−12.9171	−0.825241
$246$	0	0
$247$	−0.149228	−0.00949512
$248$	0	0
$249$	0.335149	0.0212392
$250$	0	0
$251$	−20.4878	−1.29318	−0.646590	−	0.762837i	$-0.723806\pi$
$251$	−20.4878	−1.29318	−0.646590	+	0.762837i	$0.723806\pi$
$252$	0	0
$253$	4.16674	0.261961
$254$	0	0
$255$	8.72373	0.546301
$256$	0	0
$257$	22.6875	1.41521	0.707604	−	0.706610i	$-0.249776\pi$
$257$	22.6875	1.41521	0.707604	+	0.706610i	$0.249776\pi$
$258$	0	0
$259$	−6.79484	−0.422211
$260$	0	0
$261$	−4.68621	−0.290069
$262$	0	0
$263$	−17.5337	−1.08117	−0.540586	−	0.841289i	$-0.681797\pi$
$263$	−17.5337	−1.08117	−0.540586	+	0.841289i	$0.681797\pi$
$264$	0	0
$265$	2.30811	0.141786
$266$	0	0
$267$	6.96419	0.426201
$268$	0	0
$269$	−22.2240	−1.35502	−0.677511	−	0.735513i	$-0.736941\pi$
$269$	−22.2240	−1.35502	−0.677511	+	0.735513i	$0.736941\pi$
$270$	0	0
$271$	26.9483	1.63699	0.818495	−	0.574513i	$-0.194809\pi$
$271$	26.9483	1.63699	0.818495	+	0.574513i	$0.194809\pi$
$272$	0	0
$273$	−12.8094	−0.775262
$274$	0	0
$275$	31.1081	1.87589
$276$	0	0
$277$	19.2086	1.15413	0.577067	−	0.816697i	$-0.304197\pi$
$277$	19.2086	1.15413	0.577067	+	0.816697i	$0.304197\pi$
$278$	0	0
$279$	−4.65052	−0.278419
$280$	0	0
$281$	−12.1438	−0.724437	−0.362218	−	0.932093i	$-0.617980\pi$
$281$	−12.1438	−0.724437	−0.362218	+	0.932093i	$0.617980\pi$
$282$	0	0
$283$	−29.3348	−1.74377	−0.871887	−	0.489708i	$-0.837103\pi$
$283$	−29.3348	−1.74377	−0.871887	+	0.489708i	$0.837103\pi$
$284$	0	0
$285$	−0.143670	−0.00851030
$286$	0	0
$287$	−30.2040	−1.78289
$288$	0	0
$289$	−11.8092	−0.694661
$290$	0	0
$291$	−5.63494	−0.330326
$292$	0	0
$293$	−7.09482	−0.414484	−0.207242	−	0.978290i	$-0.566449\pi$
$293$	−7.09482	−0.414484	−0.207242	+	0.978290i	$0.566449\pi$
$294$	0	0
$295$	−13.7448	−0.800254
$296$	0	0
$297$	3.21987	0.186836
$298$	0	0
$299$	−5.14668	−0.297640
$300$	0	0
$301$	32.8392	1.89282
$302$	0	0
$303$	0.592996	0.0340668
$304$	0	0
$305$	12.4071	0.710428
$306$	0	0
$307$	13.6465	0.778847	0.389424	−	0.921059i	$-0.372674\pi$
$307$	13.6465	0.778847	0.389424	+	0.921059i	$0.372674\pi$
$308$	0	0
$309$	−8.58197	−0.488211
$310$	0	0
$311$	16.9544	0.961396	0.480698	−	0.876886i	$-0.340383\pi$
$311$	16.9544	0.961396	0.480698	+	0.876886i	$0.340383\pi$
$312$	0	0
$313$	−24.9576	−1.41068	−0.705342	−	0.708867i	$-0.749207\pi$
$313$	−24.9576	−1.41068	−0.705342	+	0.708867i	$0.749207\pi$
$314$	0	0
$315$	−12.3324	−0.694854
$316$	0	0
$317$	27.4120	1.53961	0.769807	−	0.638277i	$-0.220352\pi$
$317$	27.4120	1.53961	0.769807	+	0.638277i	$0.220352\pi$
$318$	0	0
$319$	−15.0890	−0.844819
$320$	0	0
$321$	−16.6553	−0.929608
$322$	0	0
$323$	−0.0854864	−0.00475659
$324$	0	0
$325$	−38.4241	−2.13139
$326$	0	0
$327$	1.60814	0.0889303
$328$	0	0
$329$	−29.1063	−1.60468
$330$	0	0
$331$	24.5608	1.34999	0.674993	−	0.737824i	$-0.264147\pi$
$331$	24.5608	1.34999	0.674993	+	0.737824i	$0.264147\pi$
$332$	0	0
$333$	−2.10968	−0.115610
$334$	0	0
$335$	−49.4972	−2.70432
$336$	0	0
$337$	5.00301	0.272531	0.136266	−	0.990672i	$-0.456490\pi$
$337$	5.00301	0.272531	0.136266	+	0.990672i	$0.456490\pi$
$338$	0	0
$339$	8.30402	0.451012
$340$	0	0
$341$	−14.9741	−0.810890
$342$	0	0
$343$	−11.6803	−0.630675
$344$	0	0
$345$	−4.95502	−0.266769
$346$	0	0
$347$	6.16120	0.330751	0.165375	−	0.986231i	$-0.447116\pi$
$347$	6.16120	0.330751	0.165375	+	0.986231i	$0.447116\pi$
$348$	0	0
$349$	−14.7991	−0.792179	−0.396089	−	0.918212i	$-0.629633\pi$
$349$	−14.7991	−0.792179	−0.396089	+	0.918212i	$0.629633\pi$
$350$	0	0
$351$	−3.97711	−0.212283
$352$	0	0
$353$	−10.9374	−0.582139	−0.291069	−	0.956702i	$-0.594011\pi$
$353$	−10.9374	−0.582139	−0.291069	+	0.956702i	$0.594011\pi$
$354$	0	0
$355$	22.9525	1.21819
$356$	0	0
$357$	−7.33801	−0.388368
$358$	0	0
$359$	−11.1133	−0.586537	−0.293269	−	0.956030i	$-0.594743\pi$
$359$	−11.1133	−0.586537	−0.293269	+	0.956030i	$0.594743\pi$
$360$	0	0
$361$	−18.9986	−0.999926
$362$	0	0
$363$	−0.632465	−0.0331958
$364$	0	0
$365$	−3.43267	−0.179674
$366$	0	0
$367$	7.37225	0.384828	0.192414	−	0.981314i	$-0.438368\pi$
$367$	7.37225	0.384828	0.192414	+	0.981314i	$0.438368\pi$
$368$	0	0
$369$	−9.37784	−0.488191
$370$	0	0
$371$	−1.94148	−0.100797
$372$	0	0
$373$	31.2641	1.61879	0.809396	−	0.587263i	$-0.199795\pi$
$373$	31.2641	1.61879	0.809396	+	0.587263i	$0.199795\pi$
$374$	0	0
$375$	−17.8482	−0.921678
$376$	0	0
$377$	18.6376	0.959884
$378$	0	0
$379$	−25.1947	−1.29417	−0.647083	−	0.762420i	$-0.724011\pi$
$379$	−25.1947	−1.29417	−0.647083	+	0.762420i	$0.724011\pi$
$380$	0	0
$381$	−8.87450	−0.454654
$382$	0	0
$383$	7.95689	0.406578	0.203289	−	0.979119i	$-0.434837\pi$
$383$	7.95689	0.406578	0.203289	+	0.979119i	$0.434837\pi$
$384$	0	0
$385$	−39.7088	−2.02375
$386$	0	0
$387$	10.1960	0.518293
$388$	0	0
$389$	−30.9458	−1.56902	−0.784508	−	0.620119i	$-0.787084\pi$
$389$	−30.9458	−1.56902	−0.784508	+	0.620119i	$0.787084\pi$
$390$	0	0
$391$	−2.94832	−0.149103
$392$	0	0
$393$	−7.30235	−0.368355
$394$	0	0
$395$	−7.74690	−0.389789
$396$	0	0
$397$	2.32846	0.116862	0.0584310	−	0.998291i	$-0.481390\pi$
$397$	2.32846	0.116862	0.0584310	+	0.998291i	$0.481390\pi$
$398$	0	0
$399$	0.120849	0.00605002
$400$	0	0
$401$	−17.3526	−0.866548	−0.433274	−	0.901262i	$-0.642642\pi$
$401$	−17.3526	−0.866548	−0.433274	+	0.901262i	$0.642642\pi$
$402$	0	0
$403$	18.4956	0.921334
$404$	0	0
$405$	−3.82901	−0.190265
$406$	0	0
$407$	−6.79289	−0.336711
$408$	0	0
$409$	4.41439	0.218277	0.109139	−	0.994027i	$-0.465191\pi$
$409$	4.41439	0.218277	0.109139	+	0.994027i	$0.465191\pi$
$410$	0	0
$411$	−6.64349	−0.327699
$412$	0	0
$413$	11.5615	0.568905
$414$	0	0
$415$	−1.28329	−0.0629943
$416$	0	0
$417$	−1.58350	−0.0775444
$418$	0	0
$419$	11.6874	0.570966	0.285483	−	0.958384i	$-0.407846\pi$
$419$	11.6874	0.570966	0.285483	+	0.958384i	$0.407846\pi$
$420$	0	0
$421$	−14.5203	−0.707677	−0.353839	−	0.935306i	$-0.615124\pi$
$421$	−14.5203	−0.707677	−0.353839	+	0.935306i	$0.615124\pi$
$422$	0	0
$423$	−9.03702	−0.439395
$424$	0	0
$425$	−22.0116	−1.06772
$426$	0	0
$427$	−10.4363	−0.505047
$428$	0	0
$429$	−12.8058	−0.618268
$430$	0	0
$431$	−30.8625	−1.48659	−0.743297	−	0.668961i	$-0.766739\pi$
$431$	−30.8625	−1.48659	−0.743297	+	0.668961i	$0.766739\pi$
$432$	0	0
$433$	−12.6824	−0.609478	−0.304739	−	0.952436i	$-0.598569\pi$
$433$	−12.6824	−0.609478	−0.304739	+	0.952436i	$0.598569\pi$
$434$	0	0
$435$	17.9435	0.860327
$436$	0	0
$437$	0.0485557	0.00232273
$438$	0	0
$439$	−8.87354	−0.423511	−0.211756	−	0.977323i	$-0.567918\pi$
$439$	−8.87354	−0.423511	−0.211756	+	0.977323i	$0.567918\pi$
$440$	0	0
$441$	3.37347	0.160642
$442$	0	0
$443$	−13.0293	−0.619041	−0.309521	−	0.950893i	$-0.600169\pi$
$443$	−13.0293	−0.619041	−0.309521	+	0.950893i	$0.600169\pi$
$444$	0	0
$445$	−26.6659	−1.26409
$446$	0	0
$447$	10.1028	0.477848
$448$	0	0
$449$	1.23414	0.0582426	0.0291213	−	0.999576i	$-0.490729\pi$
$449$	1.23414	0.0582426	0.0291213	+	0.999576i	$0.490729\pi$
$450$	0	0
$451$	−30.1954	−1.42185
$452$	0	0
$453$	−1.88225	−0.0884356
$454$	0	0
$455$	49.0475	2.29938
$456$	0	0
$457$	24.2156	1.13276	0.566378	−	0.824145i	$-0.308344\pi$
$457$	24.2156	1.13276	0.566378	+	0.824145i	$0.308344\pi$
$458$	0	0
$459$	−2.27833	−0.106343
$460$	0	0
$461$	−36.8704	−1.71723	−0.858613	−	0.512624i	$-0.828674\pi$
$461$	−36.8704	−1.71723	−0.858613	+	0.512624i	$0.828674\pi$
$462$	0	0
$463$	0.800018	0.0371800	0.0185900	−	0.999827i	$-0.494082\pi$
$463$	0.800018	0.0371800	0.0185900	+	0.999827i	$0.494082\pi$
$464$	0	0
$465$	17.8069	0.825775
$466$	0	0
$467$	−7.63578	−0.353342	−0.176671	−	0.984270i	$-0.556533\pi$
$467$	−7.63578	−0.353342	−0.176671	+	0.984270i	$0.556533\pi$
$468$	0	0
$469$	41.6348	1.92251
$470$	0	0
$471$	−18.9549	−0.873394
$472$	0	0
$473$	32.8298	1.50952
$474$	0	0
$475$	0.362508	0.0166330
$476$	0	0
$477$	−0.602797	−0.0276002
$478$	0	0
$479$	40.3555	1.84389	0.921945	−	0.387321i	$-0.126600\pi$
$479$	40.3555	1.84389	0.921945	+	0.387321i	$0.126600\pi$
$480$	0	0
$481$	8.39045	0.382571
$482$	0	0
$483$	4.16794	0.189648
$484$	0	0
$485$	21.5762	0.979726
$486$	0	0
$487$	24.7958	1.12361	0.561803	−	0.827271i	$-0.310108\pi$
$487$	24.7958	1.12361	0.561803	+	0.827271i	$0.310108\pi$
$488$	0	0
$489$	−14.5057	−0.655970
$490$	0	0
$491$	37.3416	1.68520	0.842602	−	0.538537i	$-0.181023\pi$
$491$	37.3416	1.68520	0.842602	+	0.538537i	$0.181023\pi$
$492$	0	0
$493$	10.6767	0.480855
$494$	0	0
$495$	−12.3289	−0.554143
$496$	0	0
$497$	−19.3066	−0.866021
$498$	0	0
$499$	26.9940	1.20842	0.604208	−	0.796827i	$-0.293490\pi$
$499$	26.9940	1.20842	0.604208	+	0.796827i	$0.293490\pi$
$500$	0	0
$501$	−20.4075	−0.911739
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−2.27059	−0.101040
$506$	0	0
$507$	2.81743	0.125127
$508$	0	0
$509$	16.8694	0.747725	0.373863	−	0.927484i	$-0.378033\pi$
$509$	16.8694	0.747725	0.373863	+	0.927484i	$0.378033\pi$
$510$	0	0
$511$	2.88741	0.127731
$512$	0	0
$513$	0.0375216	0.00165662
$514$	0	0
$515$	32.8604	1.44800
$516$	0	0
$517$	−29.0980	−1.27973
$518$	0	0
$519$	5.65867	0.248388
$520$	0	0
$521$	13.9325	0.610393	0.305196	−	0.952289i	$-0.401278\pi$
$521$	13.9325	0.610393	0.305196	+	0.952289i	$0.401278\pi$
$522$	0	0
$523$	−34.5959	−1.51277	−0.756387	−	0.654124i	$-0.773037\pi$
$523$	−34.5959	−1.51277	−0.756387	+	0.654124i	$0.773037\pi$
$524$	0	0
$525$	31.1170	1.35806
$526$	0	0
$527$	10.5954	0.461543
$528$	0	0
$529$	−21.3254	−0.927190
$530$	0	0
$531$	3.58965	0.155778
$532$	0	0
$533$	37.2967	1.61550
$534$	0	0
$535$	63.7733	2.75716
$536$	0	0
$537$	1.95966	0.0845656
$538$	0	0
$539$	10.8621	0.467865
$540$	0	0
$541$	36.4723	1.56807	0.784034	−	0.620717i	$-0.213159\pi$
$541$	36.4723	1.56807	0.784034	+	0.620717i	$0.213159\pi$
$542$	0	0
$543$	14.2761	0.612644
$544$	0	0
$545$	−6.15758	−0.263762
$546$	0	0
$547$	21.4727	0.918107	0.459053	−	0.888409i	$-0.348189\pi$
$547$	21.4727	0.918107	0.459053	+	0.888409i	$0.348189\pi$
$548$	0	0
$549$	−3.24029	−0.138292
$550$	0	0
$551$	−0.175834	−0.00749077
$552$	0	0
$553$	6.51634	0.277103
$554$	0	0
$555$	8.07799	0.342892
$556$	0	0
$557$	9.72148	0.411913	0.205956	−	0.978561i	$-0.433970\pi$
$557$	9.72148	0.411913	0.205956	+	0.978561i	$0.433970\pi$
$558$	0	0
$559$	−40.5507	−1.71511
$560$	0	0
$561$	−7.33590	−0.309722
$562$	0	0
$563$	21.5040	0.906285	0.453142	−	0.891438i	$-0.350303\pi$
$563$	21.5040	0.906285	0.453142	+	0.891438i	$0.350303\pi$
$564$	0	0
$565$	−31.7962	−1.33767
$566$	0	0
$567$	3.22079	0.135260
$568$	0	0
$569$	−31.1471	−1.30576	−0.652878	−	0.757463i	$-0.726439\pi$
$569$	−31.1471	−1.30576	−0.652878	+	0.757463i	$0.726439\pi$
$570$	0	0
$571$	30.5954	1.28038	0.640189	−	0.768218i	$-0.278856\pi$
$571$	30.5954	1.28038	0.640189	+	0.768218i	$0.278856\pi$
$572$	0	0
$573$	−8.14830	−0.340400
$574$	0	0
$575$	12.5025	0.521388
$576$	0	0
$577$	19.3581	0.805887	0.402943	−	0.915225i	$-0.367987\pi$
$577$	19.3581	0.805887	0.402943	+	0.915225i	$0.367987\pi$
$578$	0	0
$579$	3.89931	0.162050
$580$	0	0
$581$	1.07945	0.0447829
$582$	0	0
$583$	−1.94092	−0.0803849
$584$	0	0
$585$	15.2284	0.629617
$586$	0	0
$587$	5.16337	0.213115	0.106557	−	0.994307i	$-0.466017\pi$
$587$	5.16337	0.213115	0.106557	+	0.994307i	$0.466017\pi$
$588$	0	0
$589$	−0.174495	−0.00718993
$590$	0	0
$591$	−2.27571	−0.0936104
$592$	0	0
$593$	−15.7080	−0.645052	−0.322526	−	0.946561i	$-0.604532\pi$
$593$	−15.7080	−0.645052	−0.322526	+	0.946561i	$0.604532\pi$
$594$	0	0
$595$	28.0973	1.15188
$596$	0	0
$597$	−1.79134	−0.0733145
$598$	0	0
$599$	15.2564	0.623360	0.311680	−	0.950187i	$-0.399108\pi$
$599$	15.2564	0.623360	0.311680	+	0.950187i	$0.399108\pi$
$600$	0	0
$601$	7.48881	0.305475	0.152738	−	0.988267i	$-0.451191\pi$
$601$	7.48881	0.305475	0.152738	+	0.988267i	$0.451191\pi$
$602$	0	0
$603$	12.9269	0.526424
$604$	0	0
$605$	2.42171	0.0984566
$606$	0	0
$607$	6.86764	0.278749	0.139374	−	0.990240i	$-0.455491\pi$
$607$	6.86764	0.278749	0.139374	+	0.990240i	$0.455491\pi$
$608$	0	0
$609$	−15.0933	−0.611611
$610$	0	0
$611$	35.9413	1.45403
$612$	0	0
$613$	5.68794	0.229734	0.114867	−	0.993381i	$-0.463356\pi$
$613$	5.68794	0.229734	0.114867	+	0.993381i	$0.463356\pi$
$614$	0	0
$615$	35.9078	1.44794
$616$	0	0
$617$	16.5248	0.665263	0.332632	−	0.943057i	$-0.392063\pi$
$617$	16.5248	0.665263	0.332632	+	0.943057i	$0.392063\pi$
$618$	0	0
$619$	−32.3812	−1.30151	−0.650756	−	0.759287i	$-0.725548\pi$
$619$	−32.3812	−1.30151	−0.650756	+	0.759287i	$0.725548\pi$
$620$	0	0
$621$	1.29407	0.0519294
$622$	0	0
$623$	22.4302	0.898646
$624$	0	0
$625$	20.0344	0.801377
$626$	0	0
$627$	0.120814	0.00482486
$628$	0	0
$629$	4.80654	0.191649
$630$	0	0
$631$	−5.33380	−0.212335	−0.106168	−	0.994348i	$-0.533858\pi$
$631$	−5.33380	−0.212335	−0.106168	+	0.994348i	$0.533858\pi$
$632$	0	0
$633$	7.37064	0.292957
$634$	0	0
$635$	33.9805	1.34848
$636$	0	0
$637$	−13.4167	−0.531589
$638$	0	0
$639$	−5.99438	−0.237134
$640$	0	0
$641$	−0.285022	−0.0112577	−0.00562885	−	0.999984i	$-0.501792\pi$
$641$	−0.285022	−0.0112577	−0.00562885	+	0.999984i	$0.501792\pi$
$642$	0	0
$643$	12.1657	0.479767	0.239883	−	0.970802i	$-0.422891\pi$
$643$	12.1657	0.479767	0.239883	+	0.970802i	$0.422891\pi$
$644$	0	0
$645$	−39.0406	−1.53722
$646$	0	0
$647$	1.95350	0.0767999	0.0383999	−	0.999262i	$-0.487774\pi$
$647$	1.95350	0.0767999	0.0383999	+	0.999262i	$0.487774\pi$
$648$	0	0
$649$	11.5582	0.453699
$650$	0	0
$651$	−14.9783	−0.587047
$652$	0	0
$653$	−4.87297	−0.190694	−0.0953471	−	0.995444i	$-0.530396\pi$
$653$	−4.87297	−0.190694	−0.0953471	+	0.995444i	$0.530396\pi$
$654$	0	0
$655$	27.9608	1.09252
$656$	0	0
$657$	0.896490	0.0349754
$658$	0	0
$659$	−18.8365	−0.733764	−0.366882	−	0.930267i	$-0.619575\pi$
$659$	−18.8365	−0.733764	−0.366882	+	0.930267i	$0.619575\pi$
$660$	0	0
$661$	0.228625	0.00889248	0.00444624	−	0.999990i	$-0.498585\pi$
$661$	0.228625	0.00889248	0.00444624	+	0.999990i	$0.498585\pi$
$662$	0	0
$663$	9.06116	0.351906
$664$	0	0
$665$	−0.462732	−0.0179440
$666$	0	0
$667$	−6.06430	−0.234811
$668$	0	0
$669$	−5.17257	−0.199983
$670$	0	0
$671$	−10.4333	−0.402773
$672$	0	0
$673$	−27.4630	−1.05862	−0.529310	−	0.848428i	$-0.677549\pi$
$673$	−27.4630	−1.05862	−0.529310	+	0.848428i	$0.677549\pi$
$674$	0	0
$675$	9.66131	0.371864
$676$	0	0
$677$	9.07694	0.348855	0.174428	−	0.984670i	$-0.444192\pi$
$677$	9.07694	0.348855	0.174428	+	0.984670i	$0.444192\pi$
$678$	0	0
$679$	−18.1489	−0.696492
$680$	0	0
$681$	−19.4401	−0.744946
$682$	0	0
$683$	−32.5554	−1.24570	−0.622849	−	0.782342i	$-0.714025\pi$
$683$	−32.5554	−1.24570	−0.622849	+	0.782342i	$0.714025\pi$
$684$	0	0
$685$	25.4380	0.971935
$686$	0	0
$687$	1.94619	0.0742518
$688$	0	0
$689$	2.39739	0.0913333
$690$	0	0
$691$	34.6163	1.31687	0.658434	−	0.752639i	$-0.271219\pi$
$691$	34.6163	1.31687	0.658434	+	0.752639i	$0.271219\pi$
$692$	0	0
$693$	10.3705	0.393943
$694$	0	0
$695$	6.06324	0.229992
$696$	0	0
$697$	21.3658	0.809287
$698$	0	0
$699$	−17.0762	−0.645880
$700$	0	0
$701$	10.3569	0.391175	0.195588	−	0.980686i	$-0.437339\pi$
$701$	10.3569	0.391175	0.195588	+	0.980686i	$0.437339\pi$
$702$	0	0
$703$	−0.0791586	−0.00298552
$704$	0	0
$705$	34.6028	1.30322
$706$	0	0
$707$	1.90992	0.0718298
$708$	0	0
$709$	42.4567	1.59450	0.797248	−	0.603652i	$-0.206288\pi$
$709$	42.4567	1.59450	0.797248	+	0.603652i	$0.206288\pi$
$710$	0	0
$711$	2.02321	0.0758764
$712$	0	0
$713$	−6.01812	−0.225380
$714$	0	0
$715$	49.0334	1.83375
$716$	0	0
$717$	−23.2634	−0.868786
$718$	0	0
$719$	−7.08423	−0.264197	−0.132099	−	0.991237i	$-0.542172\pi$
$719$	−7.08423	−0.264197	−0.132099	+	0.991237i	$0.542172\pi$
$720$	0	0
$721$	−27.6407	−1.02939
$722$	0	0
$723$	−0.733843	−0.0272919
$724$	0	0
$725$	−45.2749	−1.68147
$726$	0	0
$727$	−35.8391	−1.32920	−0.664599	−	0.747200i	$-0.731398\pi$
$727$	−35.8391	−1.32920	−0.664599	+	0.747200i	$0.731398\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−23.2299	−0.859187
$732$	0	0
$733$	−8.06411	−0.297855	−0.148927	−	0.988848i	$-0.547582\pi$
$733$	−8.06411	−0.297855	−0.148927	+	0.988848i	$0.547582\pi$
$734$	0	0
$735$	−12.9171	−0.476453
$736$	0	0
$737$	41.6228	1.53320
$738$	0	0
$739$	−34.5458	−1.27079	−0.635394	−	0.772188i	$-0.719162\pi$
$739$	−34.5458	−1.27079	−0.635394	+	0.772188i	$0.719162\pi$
$740$	0	0
$741$	−0.149228	−0.00548201
$742$	0	0
$743$	−1.99681	−0.0732560	−0.0366280	−	0.999329i	$-0.511662\pi$
$743$	−1.99681	−0.0732560	−0.0366280	+	0.999329i	$0.511662\pi$
$744$	0	0
$745$	−38.6839	−1.41727
$746$	0	0
$747$	0.335149	0.0122625
$748$	0	0
$749$	−53.6432	−1.96008
$750$	0	0
$751$	16.4344	0.599701	0.299851	−	0.953986i	$-0.403063\pi$
$751$	16.4344	0.599701	0.299851	+	0.953986i	$0.403063\pi$
$752$	0	0
$753$	−20.4878	−0.746618
$754$	0	0
$755$	7.20714	0.262295
$756$	0	0
$757$	8.36051	0.303868	0.151934	−	0.988391i	$-0.451450\pi$
$757$	8.36051	0.303868	0.151934	+	0.988391i	$0.451450\pi$
$758$	0	0
$759$	4.16674	0.151243
$760$	0	0
$761$	36.3653	1.31824	0.659121	−	0.752037i	$-0.270929\pi$
$761$	36.3653	1.31824	0.659121	+	0.752037i	$0.270929\pi$
$762$	0	0
$763$	5.17947	0.187510
$764$	0	0
$765$	8.72373	0.315407
$766$	0	0
$767$	−14.2765	−0.515493
$768$	0	0
$769$	−29.9736	−1.08088	−0.540438	−	0.841384i	$-0.681742\pi$
$769$	−29.9736	−1.08088	−0.540438	+	0.841384i	$0.681742\pi$
$770$	0	0
$771$	22.6875	0.817070
$772$	0	0
$773$	−3.71052	−0.133458	−0.0667291	−	0.997771i	$-0.521256\pi$
$773$	−3.71052	−0.133458	−0.0667291	+	0.997771i	$0.521256\pi$
$774$	0	0
$775$	−44.9301	−1.61394
$776$	0	0
$777$	−6.79484	−0.243764
$778$	0	0
$779$	−0.351871	−0.0126071
$780$	0	0
$781$	−19.3011	−0.690648
$782$	0	0
$783$	−4.68621	−0.167471
$784$	0	0
$785$	72.5784	2.59043
$786$	0	0
$787$	25.1808	0.897599	0.448800	−	0.893632i	$-0.351852\pi$
$787$	25.1808	0.897599	0.448800	+	0.893632i	$0.351852\pi$
$788$	0	0
$789$	−17.5337	−0.624215
$790$	0	0
$791$	26.7455	0.950960
$792$	0	0
$793$	12.8870	0.457630
$794$	0	0
$795$	2.30811	0.0818604
$796$	0	0
$797$	−28.7617	−1.01879	−0.509396	−	0.860532i	$-0.670131\pi$
$797$	−28.7617	−1.01879	−0.509396	+	0.860532i	$0.670131\pi$
$798$	0	0
$799$	20.5893	0.728397
$800$	0	0
$801$	6.96419	0.246067
$802$	0	0
$803$	2.88658	0.101865
$804$	0	0
$805$	−15.9591	−0.562483
$806$	0	0
$807$	−22.2240	−0.782322
$808$	0	0
$809$	19.2338	0.676224	0.338112	−	0.941106i	$-0.390212\pi$
$809$	19.2338	0.676224	0.338112	+	0.941106i	$0.390212\pi$
$810$	0	0
$811$	50.6302	1.77787	0.888933	−	0.458037i	$-0.151447\pi$
$811$	50.6302	1.77787	0.888933	+	0.458037i	$0.151447\pi$
$812$	0	0
$813$	26.9483	0.945117
$814$	0	0
$815$	55.5424	1.94557
$816$	0	0
$817$	0.382571	0.0133844
$818$	0	0
$819$	−12.8094	−0.447598
$820$	0	0
$821$	50.7685	1.77183	0.885916	−	0.463846i	$-0.153531\pi$
$821$	50.7685	1.77183	0.885916	+	0.463846i	$0.153531\pi$
$822$	0	0
$823$	−14.6603	−0.511026	−0.255513	−	0.966806i	$-0.582244\pi$
$823$	−14.6603	−0.511026	−0.255513	+	0.966806i	$0.582244\pi$
$824$	0	0
$825$	31.1081	1.08305
$826$	0	0
$827$	−11.0548	−0.384413	−0.192206	−	0.981355i	$-0.561564\pi$
$827$	−11.0548	−0.384413	−0.192206	+	0.981355i	$0.561564\pi$
$828$	0	0
$829$	−35.6993	−1.23989	−0.619943	−	0.784647i	$-0.712844\pi$
$829$	−35.6993	−1.23989	−0.619943	+	0.784647i	$0.712844\pi$
$830$	0	0
$831$	19.2086	0.666339
$832$	0	0
$833$	−7.68588	−0.266300
$834$	0	0
$835$	78.1404	2.70416
$836$	0	0
$837$	−4.65052	−0.160745
$838$	0	0
$839$	4.10974	0.141884	0.0709420	−	0.997480i	$-0.477399\pi$
$839$	4.10974	0.141884	0.0709420	+	0.997480i	$0.477399\pi$
$840$	0	0
$841$	−7.03946	−0.242740
$842$	0	0
$843$	−12.1438	−0.418254
$844$	0	0
$845$	−10.7880	−0.371117
$846$	0	0
$847$	−2.03703	−0.0699933
$848$	0	0
$849$	−29.3348	−1.00677
$850$	0	0
$851$	−2.73008	−0.0935861
$852$	0	0
$853$	−28.7630	−0.984827	−0.492413	−	0.870361i	$-0.663885\pi$
$853$	−28.7630	−0.984827	−0.492413	+	0.870361i	$0.663885\pi$
$854$	0	0
$855$	−0.143670	−0.00491343
$856$	0	0
$857$	41.5457	1.41917	0.709587	−	0.704618i	$-0.248882\pi$
$857$	41.5457	1.41917	0.709587	+	0.704618i	$0.248882\pi$
$858$	0	0
$859$	−31.6333	−1.07931	−0.539657	−	0.841885i	$-0.681446\pi$
$859$	−31.6333	−1.07931	−0.539657	+	0.841885i	$0.681446\pi$
$860$	0	0
$861$	−30.2040	−1.02935
$862$	0	0
$863$	38.0934	1.29672	0.648358	−	0.761336i	$-0.275456\pi$
$863$	38.0934	1.29672	0.648358	+	0.761336i	$0.275456\pi$
$864$	0	0
$865$	−21.6671	−0.736704
$866$	0	0
$867$	−11.8092	−0.401062
$868$	0	0
$869$	6.51448	0.220988
$870$	0	0
$871$	−51.4117	−1.74202
$872$	0	0
$873$	−5.63494	−0.190714
$874$	0	0
$875$	−57.4853	−1.94336
$876$	0	0
$877$	22.1667	0.748518	0.374259	−	0.927324i	$-0.377897\pi$
$877$	22.1667	0.748518	0.374259	+	0.927324i	$0.377897\pi$
$878$	0	0
$879$	−7.09482	−0.239302
$880$	0	0
$881$	29.1275	0.981329	0.490665	−	0.871349i	$-0.336754\pi$
$881$	29.1275	0.981329	0.490665	+	0.871349i	$0.336754\pi$
$882$	0	0
$883$	14.6396	0.492662	0.246331	−	0.969186i	$-0.420775\pi$
$883$	14.6396	0.492662	0.246331	+	0.969186i	$0.420775\pi$
$884$	0	0
$885$	−13.7448	−0.462027
$886$	0	0
$887$	54.3378	1.82449	0.912243	−	0.409650i	$-0.134349\pi$
$887$	54.3378	1.82449	0.912243	+	0.409650i	$0.134349\pi$
$888$	0	0
$889$	−28.5829	−0.958639
$890$	0	0
$891$	3.21987	0.107870
$892$	0	0
$893$	−0.339083	−0.0113470
$894$	0	0
$895$	−7.50356	−0.250816
$896$	0	0
$897$	−5.14668	−0.171843
$898$	0	0
$899$	21.7933	0.726847
$900$	0	0
$901$	1.37337	0.0457535
$902$	0	0
$903$	32.8392	1.09282
$904$	0	0
$905$	−54.6632	−1.81706
$906$	0	0
$907$	26.5682	0.882183	0.441091	−	0.897462i	$-0.354591\pi$
$907$	26.5682	0.882183	0.441091	+	0.897462i	$0.354591\pi$
$908$	0	0
$909$	0.592996	0.0196684
$910$	0	0
$911$	−45.5306	−1.50850	−0.754249	−	0.656589i	$-0.771999\pi$
$911$	−45.5306	−1.50850	−0.754249	+	0.656589i	$0.771999\pi$
$912$	0	0
$913$	1.07914	0.0357142
$914$	0	0
$915$	12.4071	0.410166
$916$	0	0
$917$	−23.5193	−0.776677
$918$	0	0
$919$	−26.3710	−0.869898	−0.434949	−	0.900455i	$-0.643234\pi$
$919$	−26.3710	−0.869898	−0.434949	+	0.900455i	$0.643234\pi$
$920$	0	0
$921$	13.6465	0.449668
$922$	0	0
$923$	23.8403	0.784714
$924$	0	0
$925$	−20.3823	−0.670166
$926$	0	0
$927$	−8.58197	−0.281869
$928$	0	0
$929$	−6.15781	−0.202031	−0.101016	−	0.994885i	$-0.532209\pi$
$929$	−6.15781	−0.202031	−0.101016	+	0.994885i	$0.532209\pi$
$930$	0	0
$931$	0.126578	0.00414843
$932$	0	0
$933$	16.9544	0.555062
$934$	0	0
$935$	28.0892	0.918617
$936$	0	0
$937$	−7.22212	−0.235936	−0.117968	−	0.993017i	$-0.537638\pi$
$937$	−7.22212	−0.235936	−0.117968	+	0.993017i	$0.537638\pi$
$938$	0	0
$939$	−24.9576	−0.814459
$940$	0	0
$941$	−10.5661	−0.344446	−0.172223	−	0.985058i	$-0.555095\pi$
$941$	−10.5661	−0.344446	−0.172223	+	0.985058i	$0.555095\pi$
$942$	0	0
$943$	−12.1356	−0.395190
$944$	0	0
$945$	−12.3324	−0.401174
$946$	0	0
$947$	20.4058	0.663100	0.331550	−	0.943438i	$-0.392428\pi$
$947$	20.4058	0.663100	0.331550	+	0.943438i	$0.392428\pi$
$948$	0	0
$949$	−3.56544	−0.115739
$950$	0	0
$951$	27.4120	0.888897
$952$	0	0
$953$	30.2413	0.979613	0.489806	−	0.871831i	$-0.337067\pi$
$953$	30.2413	0.979613	0.489806	+	0.871831i	$0.337067\pi$
$954$	0	0
$955$	31.1999	1.00961
$956$	0	0
$957$	−15.0890	−0.487757
$958$	0	0
$959$	−21.3973	−0.690954
$960$	0	0
$961$	−9.37266	−0.302344
$962$	0	0
$963$	−16.6553	−0.536709
$964$	0	0
$965$	−14.9305	−0.480630
$966$	0	0
$967$	−38.2159	−1.22894	−0.614470	−	0.788940i	$-0.710630\pi$
$967$	−38.2159	−1.22894	−0.614470	+	0.788940i	$0.710630\pi$
$968$	0	0
$969$	−0.0854864	−0.00274622
$970$	0	0
$971$	59.7778	1.91836	0.959180	−	0.282797i	$-0.0912622\pi$
$971$	59.7778	1.91836	0.959180	+	0.282797i	$0.0912622\pi$
$972$	0	0
$973$	−5.10012	−0.163502
$974$	0	0
$975$	−38.4241	−1.23056
$976$	0	0
$977$	6.74871	0.215910	0.107955	−	0.994156i	$-0.465570\pi$
$977$	6.74871	0.215910	0.107955	+	0.994156i	$0.465570\pi$
$978$	0	0
$979$	22.4237	0.716666
$980$	0	0
$981$	1.60814	0.0513439
$982$	0	0
$983$	−36.4644	−1.16303	−0.581517	−	0.813534i	$-0.697541\pi$
$983$	−36.4644	−1.16303	−0.581517	+	0.813534i	$0.697541\pi$
$984$	0	0
$985$	8.71373	0.277643
$986$	0	0
$987$	−29.1063	−0.926465
$988$	0	0
$989$	13.1944	0.419557
$990$	0	0
$991$	−51.8622	−1.64746	−0.823729	−	0.566984i	$-0.808110\pi$
$991$	−51.8622	−1.64746	−0.823729	+	0.566984i	$0.808110\pi$
$992$	0	0
$993$	24.5608	0.779414
$994$	0	0
$995$	6.85905	0.217446
$996$	0	0
$997$	−15.6529	−0.495732	−0.247866	−	0.968794i	$-0.579729\pi$
$997$	−15.6529	−0.495732	−0.247866	+	0.968794i	$0.579729\pi$
$998$	0	0
$999$	−2.10968	−0.0667474

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.g.1.2	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.g.1.2	✓	15	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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.82901 q^{5} +3.22079 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.82901 q^{5} +3.22079 q^{7} +1.00000 q^{9} +3.21987 q^{11} -3.97711 q^{13} -3.82901 q^{15} -2.27833 q^{17} +0.0375216 q^{19} +3.22079 q^{21} +1.29407 q^{23} +9.66131 q^{25} +1.00000 q^{27} -4.68621 q^{29} -4.65052 q^{31} +3.21987 q^{33} -12.3324 q^{35} -2.10968 q^{37} -3.97711 q^{39} -9.37784 q^{41} +10.1960 q^{43} -3.82901 q^{45} -9.03702 q^{47} +3.37347 q^{49} -2.27833 q^{51} -0.602797 q^{53} -12.3289 q^{55} +0.0375216 q^{57} +3.58965 q^{59} -3.24029 q^{61} +3.22079 q^{63} +15.2284 q^{65} +12.9269 q^{67} +1.29407 q^{69} -5.99438 q^{71} +0.896490 q^{73} +9.66131 q^{75} +10.3705 q^{77} +2.02321 q^{79} +1.00000 q^{81} +0.335149 q^{83} +8.72373 q^{85} -4.68621 q^{87} +6.96419 q^{89} -12.8094 q^{91} -4.65052 q^{93} -0.143670 q^{95} -5.63494 q^{97} +3.21987 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) \) 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9	\( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9} - 5 q^{11} - 14 q^{13} - 11 q^{15} - 5 q^{17} + 3 q^{19} - 4 q^{21} - 32 q^{23} + 2 q^{25} + 15 q^{27} - 23 q^{29} - 13 q^{31} - 5 q^{33} - 16 q^{35} - 10 q^{37} - 14 q^{39} - 14 q^{41} + 4 q^{43} - 11 q^{45} - 20 q^{47} - 9 q^{49} - 5 q^{51} - 30 q^{53} - 10 q^{55} + 3 q^{57} - 14 q^{59} - 38 q^{61} - 4 q^{63} - 24 q^{65} - 8 q^{67} - 32 q^{69} - 41 q^{71} - 19 q^{73} + 2 q^{75} - 39 q^{77} - 27 q^{79} + 15 q^{81} - 17 q^{83} - 6 q^{85} - 23 q^{87} - 23 q^{89} + 4 q^{91} - 13 q^{93} - 30 q^{95} - 18 q^{97} - 5 q^{99}+O(q^{100}) \) 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9 - 5 * q^11 - 14 * q^13 - 11 * q^15 - 5 * q^17 + 3 * q^19 - 4 * q^21 - 32 * q^23 + 2 * q^25 + 15 * q^27 - 23 * q^29 - 13 * q^31 - 5 * q^33 - 16 * q^35 - 10 * q^37 - 14 * q^39 - 14 * q^41 + 4 * q^43 - 11 * q^45 - 20 * q^47 - 9 * q^49 - 5 * q^51 - 30 * q^53 - 10 * q^55 + 3 * q^57 - 14 * q^59 - 38 * q^61 - 4 * q^63 - 24 * q^65 - 8 * q^67 - 32 * q^69 - 41 * q^71 - 19 * q^73 + 2 * q^75 - 39 * q^77 - 27 * q^79 + 15 * q^81 - 17 * q^83 - 6 * q^85 - 23 * q^87 - 23 * q^89 + 4 * q^91 - 13 * q^93 - 30 * q^95 - 18 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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