LMFDB - Embedded newform 4012.2.a.i.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, 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("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 8 x^{17} - 4 x^{16} + 178 x^{15} - 265 x^{14} - 1405 x^{13} + 3503 x^{12} + 4295 x^{11} + \cdots - 16 $ x^18 - 8x^17 - 4x^16 + 178x^15 - 265x^14 - 1405x^13 + 3503x^12 + 4295x^11 - 17374x^10 - 893x^9 + 38112x^8 - 18700x^7 - 32137x^6 + 26381x^5 + 3964x^4 - 5788x^3 - 108x^2 + 232*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.74374$ of defining polynomial
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.74374 q^{3} +0.394783 q^{5} -1.78199 q^{7} +0.0406123 q^{9} +O(q^{10})$	$q-1.74374 q^{3} +0.394783 q^{5} -1.78199 q^{7} +0.0406123 q^{9} -3.20764 q^{11} -5.98811 q^{13} -0.688398 q^{15} -1.00000 q^{17} -3.98074 q^{19} +3.10732 q^{21} -1.34108 q^{23} -4.84415 q^{25} +5.16039 q^{27} +2.92606 q^{29} -2.76536 q^{31} +5.59327 q^{33} -0.703501 q^{35} -7.06597 q^{37} +10.4417 q^{39} -3.64767 q^{41} +10.2528 q^{43} +0.0160331 q^{45} -2.68646 q^{47} -3.82450 q^{49} +1.74374 q^{51} -12.7059 q^{53} -1.26632 q^{55} +6.94136 q^{57} -1.00000 q^{59} +7.03474 q^{61} -0.0723708 q^{63} -2.36401 q^{65} +15.3838 q^{67} +2.33848 q^{69} -2.56238 q^{71} +10.4910 q^{73} +8.44691 q^{75} +5.71599 q^{77} +12.4908 q^{79} -9.12019 q^{81} +9.29383 q^{83} -0.394783 q^{85} -5.10227 q^{87} +10.6641 q^{89} +10.6708 q^{91} +4.82205 q^{93} -1.57153 q^{95} -7.04349 q^{97} -0.130270 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9	$ 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9} + 12 q^{11} + 2 q^{13} - 18 q^{17} + 5 q^{19} - 3 q^{21} + 21 q^{23} + 16 q^{25} + 26 q^{27} + 14 q^{29} + 15 q^{31} + 19 q^{33} + 20 q^{35} + 2 q^{37} - 14 q^{39} + 34 q^{41} + 21 q^{43} + 49 q^{45} + 69 q^{47} + 28 q^{49} - 8 q^{51} - 4 q^{53} + 18 q^{55} + 5 q^{57} - 18 q^{59} + 11 q^{61} + 35 q^{63} + 27 q^{65} + 34 q^{67} - 4 q^{69} + 37 q^{71} + 18 q^{73} + 72 q^{75} + 11 q^{77} + 11 q^{79} + 30 q^{81} + 28 q^{83} - 4 q^{85} + 7 q^{87} + 44 q^{89} - 23 q^{91} - 3 q^{93} - 11 q^{95} + 11 q^{97} + 56 q^{99}+O(q^{100}) $ 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9 + 12 * q^11 + 2 * q^13 - 18 * q^17 + 5 * q^19 - 3 * q^21 + 21 * q^23 + 16 * q^25 + 26 * q^27 + 14 * q^29 + 15 * q^31 + 19 * q^33 + 20 * q^35 + 2 * q^37 - 14 * q^39 + 34 * q^41 + 21 * q^43 + 49 * q^45 + 69 * q^47 + 28 * q^49 - 8 * q^51 - 4 * q^53 + 18 * q^55 + 5 * q^57 - 18 * q^59 + 11 * q^61 + 35 * q^63 + 27 * q^65 + 34 * q^67 - 4 * q^69 + 37 * q^71 + 18 * q^73 + 72 * q^75 + 11 * q^77 + 11 * q^79 + 30 * q^81 + 28 * q^83 - 4 * q^85 + 7 * q^87 + 44 * q^89 - 23 * q^91 - 3 * q^93 - 11 * q^95 + 11 * q^97 + 56 * 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
$3$	−1.74374	−1.00675	−0.503373	−	0.864069i	$-0.667908\pi$
$3$	−1.74374	−1.00675	−0.503373	+	0.864069i	$0.667908\pi$
$4$	0	0
$5$	0.394783	0.176553	0.0882763	−	0.996096i	$-0.471864\pi$
$5$	0.394783	0.176553	0.0882763	+	0.996096i	$0.471864\pi$
$6$	0	0
$7$	−1.78199	−0.673530	−0.336765	−	0.941589i	$-0.609333\pi$
$7$	−1.78199	−0.673530	−0.336765	+	0.941589i	$0.609333\pi$
$8$	0	0
$9$	0.0406123	0.0135374
$10$	0	0
$11$	−3.20764	−0.967139	−0.483570	−	0.875306i	$-0.660660\pi$
$11$	−3.20764	−0.967139	−0.483570	+	0.875306i	$0.660660\pi$
$12$	0	0
$13$	−5.98811	−1.66080	−0.830401	−	0.557166i	$-0.811889\pi$
$13$	−5.98811	−1.66080	−0.830401	+	0.557166i	$0.811889\pi$
$14$	0	0
$15$	−0.688398	−0.177744
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−3.98074	−0.913245	−0.456622	−	0.889661i	$-0.650941\pi$
$19$	−3.98074	−0.913245	−0.456622	+	0.889661i	$0.650941\pi$
$20$	0	0
$21$	3.10732	0.678074
$22$	0	0
$23$	−1.34108	−0.279634	−0.139817	−	0.990177i	$-0.544651\pi$
$23$	−1.34108	−0.279634	−0.139817	+	0.990177i	$0.544651\pi$
$24$	0	0
$25$	−4.84415	−0.968829
$26$	0	0
$27$	5.16039	0.993117
$28$	0	0
$29$	2.92606	0.543355	0.271677	−	0.962388i	$-0.412422\pi$
$29$	2.92606	0.543355	0.271677	+	0.962388i	$0.412422\pi$
$30$	0	0
$31$	−2.76536	−0.496673	−0.248337	−	0.968674i	$-0.579884\pi$
$31$	−2.76536	−0.496673	−0.248337	+	0.968674i	$0.579884\pi$
$32$	0	0
$33$	5.59327	0.973663
$34$	0	0
$35$	−0.703501	−0.118913
$36$	0	0
$37$	−7.06597	−1.16164	−0.580819	−	0.814032i	$-0.697268\pi$
$37$	−7.06597	−1.16164	−0.580819	+	0.814032i	$0.697268\pi$
$38$	0	0
$39$	10.4417	1.67201
$40$	0	0
$41$	−3.64767	−0.569670	−0.284835	−	0.958577i	$-0.591939\pi$
$41$	−3.64767	−0.569670	−0.284835	+	0.958577i	$0.591939\pi$
$42$	0	0
$43$	10.2528	1.56353	0.781765	−	0.623573i	$-0.214320\pi$
$43$	10.2528	1.56353	0.781765	+	0.623573i	$0.214320\pi$
$44$	0	0
$45$	0.0160331	0.00239007
$46$	0	0
$47$	−2.68646	−0.391861	−0.195930	−	0.980618i	$-0.562773\pi$
$47$	−2.68646	−0.391861	−0.195930	+	0.980618i	$0.562773\pi$
$48$	0	0
$49$	−3.82450	−0.546357
$50$	0	0
$51$	1.74374	0.244172
$52$	0	0
$53$	−12.7059	−1.74529	−0.872647	−	0.488352i	$-0.837598\pi$
$53$	−12.7059	−1.74529	−0.872647	+	0.488352i	$0.837598\pi$
$54$	0	0
$55$	−1.26632	−0.170751
$56$	0	0
$57$	6.94136	0.919406
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	7.03474	0.900706	0.450353	−	0.892851i	$-0.351298\pi$
$61$	7.03474	0.900706	0.450353	+	0.892851i	$0.351298\pi$
$62$	0	0
$63$	−0.0723708	−0.00911787
$64$	0	0
$65$	−2.36401	−0.293219
$66$	0	0
$67$	15.3838	1.87943	0.939714	−	0.341961i	$-0.111091\pi$
$67$	15.3838	1.87943	0.939714	+	0.341961i	$0.111091\pi$
$68$	0	0
$69$	2.33848	0.281520
$70$	0	0
$71$	−2.56238	−0.304098	−0.152049	−	0.988373i	$-0.548587\pi$
$71$	−2.56238	−0.304098	−0.152049	+	0.988373i	$0.548587\pi$
$72$	0	0
$73$	10.4910	1.22788	0.613941	−	0.789352i	$-0.289583\pi$
$73$	10.4910	1.22788	0.613941	+	0.789352i	$0.289583\pi$
$74$	0	0
$75$	8.44691	0.975365
$76$	0	0
$77$	5.71599	0.651397
$78$	0	0
$79$	12.4908	1.40532	0.702661	−	0.711524i	$-0.251995\pi$
$79$	12.4908	1.40532	0.702661	+	0.711524i	$0.251995\pi$
$80$	0	0
$81$	−9.12019	−1.01335
$82$	0	0
$83$	9.29383	1.02013	0.510065	−	0.860136i	$-0.329621\pi$
$83$	9.29383	1.02013	0.510065	+	0.860136i	$0.329621\pi$
$84$	0	0
$85$	−0.394783	−0.0428203
$86$	0	0
$87$	−5.10227	−0.547020
$88$	0	0
$89$	10.6641	1.13039	0.565197	−	0.824956i	$-0.308800\pi$
$89$	10.6641	1.13039	0.565197	+	0.824956i	$0.308800\pi$
$90$	0	0
$91$	10.6708	1.11860
$92$	0	0
$93$	4.82205	0.500024
$94$	0	0
$95$	−1.57153	−0.161236
$96$	0	0
$97$	−7.04349	−0.715158	−0.357579	−	0.933883i	$-0.616398\pi$
$97$	−7.04349	−0.715158	−0.357579	+	0.933883i	$0.616398\pi$
$98$	0	0
$99$	−0.130270	−0.0130926
$100$	0	0
$101$	−11.6497	−1.15919	−0.579593	−	0.814906i	$-0.696789\pi$
$101$	−11.6497	−1.15919	−0.579593	+	0.814906i	$0.696789\pi$
$102$	0	0
$103$	−11.8432	−1.16695	−0.583474	−	0.812131i	$-0.698307\pi$
$103$	−11.8432	−1.16695	−0.583474	+	0.812131i	$0.698307\pi$
$104$	0	0
$105$	1.22672	0.119716
$106$	0	0
$107$	−13.5810	−1.31293	−0.656463	−	0.754358i	$-0.727948\pi$
$107$	−13.5810	−1.31293	−0.656463	+	0.754358i	$0.727948\pi$
$108$	0	0
$109$	−15.1029	−1.44660	−0.723299	−	0.690535i	$-0.757375\pi$
$109$	−15.1029	−1.44660	−0.723299	+	0.690535i	$0.757375\pi$
$110$	0	0
$111$	12.3212	1.16948
$112$	0	0
$113$	−3.65176	−0.343528	−0.171764	−	0.985138i	$-0.554947\pi$
$113$	−3.65176	−0.343528	−0.171764	+	0.985138i	$0.554947\pi$
$114$	0	0
$115$	−0.529435	−0.0493701
$116$	0	0
$117$	−0.243191	−0.0224830
$118$	0	0
$119$	1.78199	0.163355
$120$	0	0
$121$	−0.711064	−0.0646421
$122$	0	0
$123$	6.36057	0.573513
$124$	0	0
$125$	−3.88631	−0.347602
$126$	0	0
$127$	12.5986	1.11795	0.558974	−	0.829185i	$-0.311195\pi$
$127$	12.5986	1.11795	0.558974	+	0.829185i	$0.311195\pi$
$128$	0	0
$129$	−17.8781	−1.57408
$130$	0	0
$131$	−3.78638	−0.330817	−0.165409	−	0.986225i	$-0.552894\pi$
$131$	−3.78638	−0.330817	−0.165409	+	0.986225i	$0.552894\pi$
$132$	0	0
$133$	7.09365	0.615098
$134$	0	0
$135$	2.03724	0.175337
$136$	0	0
$137$	1.40009	0.119618	0.0598089	−	0.998210i	$-0.480951\pi$
$137$	1.40009	0.119618	0.0598089	+	0.998210i	$0.480951\pi$
$138$	0	0
$139$	−6.06131	−0.514114	−0.257057	−	0.966396i	$-0.582753\pi$
$139$	−6.06131	−0.514114	−0.257057	+	0.966396i	$0.582753\pi$
$140$	0	0
$141$	4.68448	0.394504
$142$	0	0
$143$	19.2077	1.60623
$144$	0	0
$145$	1.15516	0.0959307
$146$	0	0
$147$	6.66892	0.550043
$148$	0	0
$149$	−10.0507	−0.823388	−0.411694	−	0.911322i	$-0.635063\pi$
$149$	−10.0507	−0.823388	−0.411694	+	0.911322i	$0.635063\pi$
$150$	0	0
$151$	1.78561	0.145311	0.0726554	−	0.997357i	$-0.476853\pi$
$151$	1.78561	0.145311	0.0726554	+	0.997357i	$0.476853\pi$
$152$	0	0
$153$	−0.0406123	−0.00328331
$154$	0	0
$155$	−1.09172	−0.0876889
$156$	0	0
$157$	6.31503	0.503994	0.251997	−	0.967728i	$-0.418913\pi$
$157$	6.31503	0.503994	0.251997	+	0.967728i	$0.418913\pi$
$158$	0	0
$159$	22.1558	1.75707
$160$	0	0
$161$	2.38979	0.188342
$162$	0	0
$163$	−10.3505	−0.810713	−0.405357	−	0.914159i	$-0.632853\pi$
$163$	−10.3505	−0.810713	−0.405357	+	0.914159i	$0.632853\pi$
$164$	0	0
$165$	2.20813	0.171903
$166$	0	0
$167$	−5.12785	−0.396805	−0.198403	−	0.980121i	$-0.563575\pi$
$167$	−5.12785	−0.396805	−0.198403	+	0.980121i	$0.563575\pi$
$168$	0	0
$169$	22.8574	1.75826
$170$	0	0
$171$	−0.161667	−0.0123630
$172$	0	0
$173$	−0.0717100	−0.00545201	−0.00272600	−	0.999996i	$-0.500868\pi$
$173$	−0.0717100	−0.00545201	−0.00272600	+	0.999996i	$0.500868\pi$
$174$	0	0
$175$	8.63223	0.652535
$176$	0	0
$177$	1.74374	0.131067
$178$	0	0
$179$	−14.4220	−1.07795	−0.538977	−	0.842321i	$-0.681189\pi$
$179$	−14.4220	−1.07795	−0.538977	+	0.842321i	$0.681189\pi$
$180$	0	0
$181$	1.33276	0.0990630	0.0495315	−	0.998773i	$-0.484227\pi$
$181$	1.33276	0.0990630	0.0495315	+	0.998773i	$0.484227\pi$
$182$	0	0
$183$	−12.2667	−0.906782
$184$	0	0
$185$	−2.78953	−0.205090
$186$	0	0
$187$	3.20764	0.234566
$188$	0	0
$189$	−9.19577	−0.668894
$190$	0	0
$191$	−13.0657	−0.945402	−0.472701	−	0.881223i	$-0.656721\pi$
$191$	−13.0657	−0.945402	−0.472701	+	0.881223i	$0.656721\pi$
$192$	0	0
$193$	11.3262	0.815281	0.407640	−	0.913143i	$-0.366352\pi$
$193$	11.3262	0.815281	0.407640	+	0.913143i	$0.366352\pi$
$194$	0	0
$195$	4.12220	0.295197
$196$	0	0
$197$	19.9946	1.42456	0.712279	−	0.701896i	$-0.247663\pi$
$197$	19.9946	1.42456	0.712279	+	0.701896i	$0.247663\pi$
$198$	0	0
$199$	20.5469	1.45653	0.728265	−	0.685295i	$-0.240327\pi$
$199$	20.5469	1.45653	0.728265	+	0.685295i	$0.240327\pi$
$200$	0	0
$201$	−26.8252	−1.89211
$202$	0	0
$203$	−5.21421	−0.365966
$204$	0	0
$205$	−1.44004	−0.100577
$206$	0	0
$207$	−0.0544642	−0.00378553
$208$	0	0
$209$	12.7688	0.883235
$210$	0	0
$211$	−6.91465	−0.476024	−0.238012	−	0.971262i	$-0.576496\pi$
$211$	−6.91465	−0.476024	−0.238012	+	0.971262i	$0.576496\pi$
$212$	0	0
$213$	4.46811	0.306150
$214$	0	0
$215$	4.04762	0.276045
$216$	0	0
$217$	4.92785	0.334524
$218$	0	0
$219$	−18.2936	−1.23616
$220$	0	0
$221$	5.98811	0.402804
$222$	0	0
$223$	18.2122	1.21958	0.609788	−	0.792565i	$-0.291255\pi$
$223$	18.2122	1.21958	0.609788	+	0.792565i	$0.291255\pi$
$224$	0	0
$225$	−0.196732	−0.0131155
$226$	0	0
$227$	11.7013	0.776643	0.388322	−	0.921524i	$-0.373055\pi$
$227$	11.7013	0.776643	0.388322	+	0.921524i	$0.373055\pi$
$228$	0	0
$229$	−7.89736	−0.521872	−0.260936	−	0.965356i	$-0.584031\pi$
$229$	−7.89736	−0.521872	−0.260936	+	0.965356i	$0.584031\pi$
$230$	0	0
$231$	−9.96717	−0.655791
$232$	0	0
$233$	10.8896	0.713403	0.356702	−	0.934218i	$-0.383901\pi$
$233$	10.8896	0.713403	0.356702	+	0.934218i	$0.383901\pi$
$234$	0	0
$235$	−1.06057	−0.0691840
$236$	0	0
$237$	−21.7806	−1.41480
$238$	0	0
$239$	17.6826	1.14379	0.571896	−	0.820326i	$-0.306208\pi$
$239$	17.6826	1.14379	0.571896	+	0.820326i	$0.306208\pi$
$240$	0	0
$241$	−9.35875	−0.602850	−0.301425	−	0.953490i	$-0.597462\pi$
$241$	−9.35875	−0.602850	−0.301425	+	0.953490i	$0.597462\pi$
$242$	0	0
$243$	0.422027	0.0270730
$244$	0	0
$245$	−1.50985	−0.0964608
$246$	0	0
$247$	23.8371	1.51672
$248$	0	0
$249$	−16.2060	−1.02701
$250$	0	0
$251$	25.4647	1.60732	0.803660	−	0.595089i	$-0.202883\pi$
$251$	25.4647	1.60732	0.803660	+	0.595089i	$0.202883\pi$
$252$	0	0
$253$	4.30169	0.270445
$254$	0	0
$255$	0.688398	0.0431091
$256$	0	0
$257$	22.7095	1.41658	0.708289	−	0.705923i	$-0.249467\pi$
$257$	22.7095	1.41658	0.708289	+	0.705923i	$0.249467\pi$
$258$	0	0
$259$	12.5915	0.782399
$260$	0	0
$261$	0.118834	0.00735563
$262$	0	0
$263$	17.3355	1.06895	0.534476	−	0.845184i	$-0.320509\pi$
$263$	17.3355	1.06895	0.534476	+	0.845184i	$0.320509\pi$
$264$	0	0
$265$	−5.01609	−0.308136
$266$	0	0
$267$	−18.5954	−1.13802
$268$	0	0
$269$	15.1246	0.922163	0.461081	−	0.887358i	$-0.347462\pi$
$269$	15.1246	0.922163	0.461081	+	0.887358i	$0.347462\pi$
$270$	0	0
$271$	3.22793	0.196083	0.0980414	−	0.995182i	$-0.468742\pi$
$271$	3.22793	0.196083	0.0980414	+	0.995182i	$0.468742\pi$
$272$	0	0
$273$	−18.6070	−1.12615
$274$	0	0
$275$	15.5383	0.936993
$276$	0	0
$277$	−18.7906	−1.12902	−0.564510	−	0.825426i	$-0.690935\pi$
$277$	−18.7906	−1.12902	−0.564510	+	0.825426i	$0.690935\pi$
$278$	0	0
$279$	−0.112308	−0.00672368
$280$	0	0
$281$	25.9628	1.54881	0.774406	−	0.632689i	$-0.218049\pi$
$281$	25.9628	1.54881	0.774406	+	0.632689i	$0.218049\pi$
$282$	0	0
$283$	−0.780316	−0.0463849	−0.0231925	−	0.999731i	$-0.507383\pi$
$283$	−0.780316	−0.0463849	−0.0231925	+	0.999731i	$0.507383\pi$
$284$	0	0
$285$	2.74033	0.162323
$286$	0	0
$287$	6.50012	0.383690
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	12.2820	0.719982
$292$	0	0
$293$	18.0978	1.05728	0.528642	−	0.848845i	$-0.322701\pi$
$293$	18.0978	1.05728	0.528642	+	0.848845i	$0.322701\pi$
$294$	0	0
$295$	−0.394783	−0.0229852
$296$	0	0
$297$	−16.5527	−0.960482
$298$	0	0
$299$	8.03051	0.464417
$300$	0	0
$301$	−18.2703	−1.05308
$302$	0	0
$303$	20.3140	1.16701
$304$	0	0
$305$	2.77720	0.159022
$306$	0	0
$307$	−21.2656	−1.21369	−0.606847	−	0.794819i	$-0.707566\pi$
$307$	−21.2656	−1.21369	−0.606847	+	0.794819i	$0.707566\pi$
$308$	0	0
$309$	20.6515	1.17482
$310$	0	0
$311$	−0.981141	−0.0556354	−0.0278177	−	0.999613i	$-0.508856\pi$
$311$	−0.981141	−0.0556354	−0.0278177	+	0.999613i	$0.508856\pi$
$312$	0	0
$313$	−3.08277	−0.174248	−0.0871242	−	0.996197i	$-0.527768\pi$
$313$	−3.08277	−0.174248	−0.0871242	+	0.996197i	$0.527768\pi$
$314$	0	0
$315$	−0.0285708	−0.00160978
$316$	0	0
$317$	−34.8700	−1.95849	−0.979246	−	0.202673i	$-0.935037\pi$
$317$	−34.8700	−1.95849	−0.979246	+	0.202673i	$0.935037\pi$
$318$	0	0
$319$	−9.38573	−0.525500
$320$	0	0
$321$	23.6817	1.32178
$322$	0	0
$323$	3.98074	0.221494
$324$	0	0
$325$	29.0073	1.60903
$326$	0	0
$327$	26.3355	1.45636
$328$	0	0
$329$	4.78726	0.263930
$330$	0	0
$331$	3.72231	0.204597	0.102298	−	0.994754i	$-0.467380\pi$
$331$	3.72231	0.204597	0.102298	+	0.994754i	$0.467380\pi$
$332$	0	0
$333$	−0.286965	−0.0157256
$334$	0	0
$335$	6.07326	0.331818
$336$	0	0
$337$	−20.7061	−1.12793	−0.563966	−	0.825798i	$-0.690725\pi$
$337$	−20.7061	−1.12793	−0.563966	+	0.825798i	$0.690725\pi$
$338$	0	0
$339$	6.36770	0.345846
$340$	0	0
$341$	8.87027	0.480352
$342$	0	0
$343$	19.2892	1.04152
$344$	0	0
$345$	0.923195	0.0497031
$346$	0	0
$347$	24.1959	1.29890	0.649451	−	0.760404i	$-0.274999\pi$
$347$	24.1959	1.29890	0.649451	+	0.760404i	$0.274999\pi$
$348$	0	0
$349$	−14.6012	−0.781586	−0.390793	−	0.920479i	$-0.627799\pi$
$349$	−14.6012	−0.781586	−0.390793	+	0.920479i	$0.627799\pi$
$350$	0	0
$351$	−30.9010	−1.64937
$352$	0	0
$353$	9.93424	0.528746	0.264373	−	0.964420i	$-0.414835\pi$
$353$	9.93424	0.528746	0.264373	+	0.964420i	$0.414835\pi$
$354$	0	0
$355$	−1.01158	−0.0536894
$356$	0	0
$357$	−3.10732	−0.164457
$358$	0	0
$359$	−2.45697	−0.129674	−0.0648370	−	0.997896i	$-0.520653\pi$
$359$	−2.45697	−0.129674	−0.0648370	+	0.997896i	$0.520653\pi$
$360$	0	0
$361$	−3.15369	−0.165984
$362$	0	0
$363$	1.23991	0.0650782
$364$	0	0
$365$	4.14168	0.216786
$366$	0	0
$367$	−3.23528	−0.168880	−0.0844401	−	0.996429i	$-0.526910\pi$
$367$	−3.23528	−0.168880	−0.0844401	+	0.996429i	$0.526910\pi$
$368$	0	0
$369$	−0.148140	−0.00771187
$370$	0	0
$371$	22.6419	1.17551
$372$	0	0
$373$	−1.40712	−0.0728580	−0.0364290	−	0.999336i	$-0.511598\pi$
$373$	−1.40712	−0.0728580	−0.0364290	+	0.999336i	$0.511598\pi$
$374$	0	0
$375$	6.77669	0.349947
$376$	0	0
$377$	−17.5215	−0.902405
$378$	0	0
$379$	−16.9009	−0.868139	−0.434069	−	0.900879i	$-0.642923\pi$
$379$	−16.9009	−0.868139	−0.434069	+	0.900879i	$0.642923\pi$
$380$	0	0
$381$	−21.9687	−1.12549
$382$	0	0
$383$	−17.2195	−0.879874	−0.439937	−	0.898029i	$-0.644999\pi$
$383$	−17.2195	−0.879874	−0.439937	+	0.898029i	$0.644999\pi$
$384$	0	0
$385$	2.25658	0.115006
$386$	0	0
$387$	0.416388	0.0211662
$388$	0	0
$389$	−4.58738	−0.232589	−0.116295	−	0.993215i	$-0.537102\pi$
$389$	−4.58738	−0.232589	−0.116295	+	0.993215i	$0.537102\pi$
$390$	0	0
$391$	1.34108	0.0678212
$392$	0	0
$393$	6.60244	0.333049
$394$	0	0
$395$	4.93115	0.248113
$396$	0	0
$397$	−6.05065	−0.303673	−0.151837	−	0.988406i	$-0.548519\pi$
$397$	−6.05065	−0.303673	−0.151837	+	0.988406i	$0.548519\pi$
$398$	0	0
$399$	−12.3695	−0.619247
$400$	0	0
$401$	−16.2260	−0.810289	−0.405145	−	0.914253i	$-0.632779\pi$
$401$	−16.2260	−0.810289	−0.405145	+	0.914253i	$0.632779\pi$
$402$	0	0
$403$	16.5593	0.824875
$404$	0	0
$405$	−3.60050	−0.178910
$406$	0	0
$407$	22.6651	1.12347
$408$	0	0
$409$	−19.8521	−0.981624	−0.490812	−	0.871266i	$-0.663300\pi$
$409$	−19.8521	−0.981624	−0.490812	+	0.871266i	$0.663300\pi$
$410$	0	0
$411$	−2.44139	−0.120425
$412$	0	0
$413$	1.78199	0.0876861
$414$	0	0
$415$	3.66905	0.180107
$416$	0	0
$417$	10.5693	0.517582
$418$	0	0
$419$	−8.48179	−0.414363	−0.207181	−	0.978303i	$-0.566429\pi$
$419$	−8.48179	−0.414363	−0.207181	+	0.978303i	$0.566429\pi$
$420$	0	0
$421$	2.65075	0.129189	0.0645947	−	0.997912i	$-0.479425\pi$
$421$	2.65075	0.129189	0.0645947	+	0.997912i	$0.479425\pi$
$422$	0	0
$423$	−0.109103	−0.00530479
$424$	0	0
$425$	4.84415	0.234976
$426$	0	0
$427$	−12.5359	−0.606653
$428$	0	0
$429$	−33.4931	−1.61706
$430$	0	0
$431$	−6.82731	−0.328860	−0.164430	−	0.986389i	$-0.552578\pi$
$431$	−6.82731	−0.328860	−0.164430	+	0.986389i	$0.552578\pi$
$432$	0	0
$433$	5.18145	0.249004	0.124502	−	0.992219i	$-0.460267\pi$
$433$	5.18145	0.249004	0.124502	+	0.992219i	$0.460267\pi$
$434$	0	0
$435$	−2.01429	−0.0965778
$436$	0	0
$437$	5.33848	0.255374
$438$	0	0
$439$	17.9339	0.855938	0.427969	−	0.903793i	$-0.359229\pi$
$439$	17.9339	0.855938	0.427969	+	0.903793i	$0.359229\pi$
$440$	0	0
$441$	−0.155322	−0.00739628
$442$	0	0
$443$	23.2090	1.10269	0.551346	−	0.834277i	$-0.314115\pi$
$443$	23.2090	1.10269	0.551346	+	0.834277i	$0.314115\pi$
$444$	0	0
$445$	4.21001	0.199574
$446$	0	0
$447$	17.5258	0.828942
$448$	0	0
$449$	16.2738	0.768007	0.384003	−	0.923332i	$-0.374545\pi$
$449$	16.2738	0.768007	0.384003	+	0.923332i	$0.374545\pi$
$450$	0	0
$451$	11.7004	0.550950
$452$	0	0
$453$	−3.11363	−0.146291
$454$	0	0
$455$	4.21264	0.197492
$456$	0	0
$457$	10.1284	0.473787	0.236894	−	0.971536i	$-0.423871\pi$
$457$	10.1284	0.473787	0.236894	+	0.971536i	$0.423871\pi$
$458$	0	0
$459$	−5.16039	−0.240866
$460$	0	0
$461$	−8.62691	−0.401795	−0.200898	−	0.979612i	$-0.564386\pi$
$461$	−8.62691	−0.401795	−0.200898	+	0.979612i	$0.564386\pi$
$462$	0	0
$463$	−1.16385	−0.0540885	−0.0270443	−	0.999634i	$-0.508610\pi$
$463$	−1.16385	−0.0540885	−0.0270443	+	0.999634i	$0.508610\pi$
$464$	0	0
$465$	1.90367	0.0882804
$466$	0	0
$467$	12.4028	0.573934	0.286967	−	0.957940i	$-0.407353\pi$
$467$	12.4028	0.573934	0.286967	+	0.957940i	$0.407353\pi$
$468$	0	0
$469$	−27.4138	−1.26585
$470$	0	0
$471$	−11.0117	−0.507394
$472$	0	0
$473$	−32.8871	−1.51215
$474$	0	0
$475$	19.2833	0.884778
$476$	0	0
$477$	−0.516017	−0.0236268
$478$	0	0
$479$	−31.2425	−1.42750	−0.713752	−	0.700398i	$-0.753006\pi$
$479$	−31.2425	−1.42750	−0.713752	+	0.700398i	$0.753006\pi$
$480$	0	0
$481$	42.3118	1.92925
$482$	0	0
$483$	−4.16716	−0.189612
$484$	0	0
$485$	−2.78065	−0.126263
$486$	0	0
$487$	8.77272	0.397530	0.198765	−	0.980047i	$-0.436307\pi$
$487$	8.77272	0.397530	0.198765	+	0.980047i	$0.436307\pi$
$488$	0	0
$489$	18.0485	0.816182
$490$	0	0
$491$	−18.9348	−0.854515	−0.427257	−	0.904130i	$-0.640520\pi$
$491$	−18.9348	−0.854515	−0.427257	+	0.904130i	$0.640520\pi$
$492$	0	0
$493$	−2.92606	−0.131783
$494$	0	0
$495$	−0.0514283	−0.00231153
$496$	0	0
$497$	4.56614	0.204819
$498$	0	0
$499$	3.27548	0.146630	0.0733152	−	0.997309i	$-0.476642\pi$
$499$	3.27548	0.146630	0.0733152	+	0.997309i	$0.476642\pi$
$500$	0	0
$501$	8.94162	0.399482
$502$	0	0
$503$	−10.8499	−0.483772	−0.241886	−	0.970305i	$-0.577766\pi$
$503$	−10.8499	−0.483772	−0.241886	+	0.970305i	$0.577766\pi$
$504$	0	0
$505$	−4.59910	−0.204657
$506$	0	0
$507$	−39.8573	−1.77012
$508$	0	0
$509$	−9.69903	−0.429902	−0.214951	−	0.976625i	$-0.568959\pi$
$509$	−9.69903	−0.429902	−0.214951	+	0.976625i	$0.568959\pi$
$510$	0	0
$511$	−18.6949	−0.827015
$512$	0	0
$513$	−20.5422	−0.906959
$514$	0	0
$515$	−4.67551	−0.206028
$516$	0	0
$517$	8.61720	0.378984
$518$	0	0
$519$	0.125043	0.00548879
$520$	0	0
$521$	10.0082	0.438467	0.219234	−	0.975672i	$-0.429644\pi$
$521$	10.0082	0.438467	0.219234	+	0.975672i	$0.429644\pi$
$522$	0	0
$523$	2.39246	0.104615	0.0523075	−	0.998631i	$-0.483342\pi$
$523$	2.39246	0.104615	0.0523075	+	0.998631i	$0.483342\pi$
$524$	0	0
$525$	−15.0523	−0.656937
$526$	0	0
$527$	2.76536	0.120461
$528$	0	0
$529$	−21.2015	−0.921805
$530$	0	0
$531$	−0.0406123	−0.00176242
$532$	0	0
$533$	21.8426	0.946109
$534$	0	0
$535$	−5.36156	−0.231801
$536$	0	0
$537$	25.1482	1.08522
$538$	0	0
$539$	12.2676	0.528404
$540$	0	0
$541$	−19.3395	−0.831472	−0.415736	−	0.909485i	$-0.636476\pi$
$541$	−19.3395	−0.831472	−0.415736	+	0.909485i	$0.636476\pi$
$542$	0	0
$543$	−2.32397	−0.0997312
$544$	0	0
$545$	−5.96239	−0.255401
$546$	0	0
$547$	−30.4726	−1.30291	−0.651457	−	0.758686i	$-0.725842\pi$
$547$	−30.4726	−1.30291	−0.651457	+	0.758686i	$0.725842\pi$
$548$	0	0
$549$	0.285697	0.0121933
$550$	0	0
$551$	−11.6479	−0.496216
$552$	0	0
$553$	−22.2585	−0.946527
$554$	0	0
$555$	4.86420	0.206474
$556$	0	0
$557$	0.407080	0.0172485	0.00862427	−	0.999963i	$-0.497255\pi$
$557$	0.407080	0.0172485	0.00862427	+	0.999963i	$0.497255\pi$
$558$	0	0
$559$	−61.3946	−2.59671
$560$	0	0
$561$	−5.59327	−0.236148
$562$	0	0
$563$	14.8631	0.626407	0.313203	−	0.949686i	$-0.398598\pi$
$563$	14.8631	0.626407	0.313203	+	0.949686i	$0.398598\pi$
$564$	0	0
$565$	−1.44165	−0.0606508
$566$	0	0
$567$	16.2521	0.682524
$568$	0	0
$569$	−19.8428	−0.831851	−0.415926	−	0.909399i	$-0.636542\pi$
$569$	−19.8428	−0.831851	−0.415926	+	0.909399i	$0.636542\pi$
$570$	0	0
$571$	2.41348	0.101001	0.0505005	−	0.998724i	$-0.483918\pi$
$571$	2.41348	0.101001	0.0505005	+	0.998724i	$0.483918\pi$
$572$	0	0
$573$	22.7831	0.951780
$574$	0	0
$575$	6.49637	0.270918
$576$	0	0
$577$	−42.8191	−1.78258	−0.891292	−	0.453430i	$-0.850200\pi$
$577$	−42.8191	−1.78258	−0.891292	+	0.453430i	$0.850200\pi$
$578$	0	0
$579$	−19.7500	−0.820781
$580$	0	0
$581$	−16.5615	−0.687088
$582$	0	0
$583$	40.7560	1.68794
$584$	0	0
$585$	−0.0960077	−0.00396943
$586$	0	0
$587$	−27.7658	−1.14602	−0.573009	−	0.819549i	$-0.694224\pi$
$587$	−27.7658	−1.14602	−0.573009	+	0.819549i	$0.694224\pi$
$588$	0	0
$589$	11.0082	0.453584
$590$	0	0
$591$	−34.8654	−1.43417
$592$	0	0
$593$	21.4782	0.882005	0.441002	−	0.897506i	$-0.354623\pi$
$593$	21.4782	0.882005	0.441002	+	0.897506i	$0.354623\pi$
$594$	0	0
$595$	0.703501	0.0288407
$596$	0	0
$597$	−35.8283	−1.46636
$598$	0	0
$599$	−3.97199	−0.162291	−0.0811455	−	0.996702i	$-0.525858\pi$
$599$	−3.97199	−0.162291	−0.0811455	+	0.996702i	$0.525858\pi$
$600$	0	0
$601$	40.6715	1.65902	0.829512	−	0.558489i	$-0.188619\pi$
$601$	40.6715	1.65902	0.829512	+	0.558489i	$0.188619\pi$
$602$	0	0
$603$	0.624771	0.0254426
$604$	0	0
$605$	−0.280716	−0.0114127
$606$	0	0
$607$	8.46351	0.343523	0.171762	−	0.985139i	$-0.445054\pi$
$607$	8.46351	0.343523	0.171762	+	0.985139i	$0.445054\pi$
$608$	0	0
$609$	9.09220	0.368435
$610$	0	0
$611$	16.0868	0.650803
$612$	0	0
$613$	20.0416	0.809471	0.404735	−	0.914434i	$-0.367364\pi$
$613$	20.0416	0.809471	0.404735	+	0.914434i	$0.367364\pi$
$614$	0	0
$615$	2.51105	0.101255
$616$	0	0
$617$	8.83947	0.355863	0.177932	−	0.984043i	$-0.443059\pi$
$617$	8.83947	0.355863	0.177932	+	0.984043i	$0.443059\pi$
$618$	0	0
$619$	−29.0041	−1.16577	−0.582886	−	0.812554i	$-0.698077\pi$
$619$	−29.0041	−1.16577	−0.582886	+	0.812554i	$0.698077\pi$
$620$	0	0
$621$	−6.92048	−0.277709
$622$	0	0
$623$	−19.0034	−0.761354
$624$	0	0
$625$	22.6865	0.907459
$626$	0	0
$627$	−22.2654	−0.889193
$628$	0	0
$629$	7.06597	0.281739
$630$	0	0
$631$	−0.884709	−0.0352197	−0.0176099	−	0.999845i	$-0.505606\pi$
$631$	−0.884709	−0.0352197	−0.0176099	+	0.999845i	$0.505606\pi$
$632$	0	0
$633$	12.0573	0.479235
$634$	0	0
$635$	4.97373	0.197377
$636$	0	0
$637$	22.9015	0.907391
$638$	0	0
$639$	−0.104064	−0.00411671
$640$	0	0
$641$	−43.9125	−1.73444	−0.867219	−	0.497926i	$-0.834095\pi$
$641$	−43.9125	−1.73444	−0.867219	+	0.497926i	$0.834095\pi$
$642$	0	0
$643$	−9.93522	−0.391807	−0.195903	−	0.980623i	$-0.562764\pi$
$643$	−9.93522	−0.391807	−0.195903	+	0.980623i	$0.562764\pi$
$644$	0	0
$645$	−7.05797	−0.277907
$646$	0	0
$647$	47.9063	1.88339	0.941696	−	0.336465i	$-0.109231\pi$
$647$	47.9063	1.88339	0.941696	+	0.336465i	$0.109231\pi$
$648$	0	0
$649$	3.20764	0.125911
$650$	0	0
$651$	−8.59286	−0.336781
$652$	0	0
$653$	−15.2641	−0.597329	−0.298665	−	0.954358i	$-0.596541\pi$
$653$	−15.2641	−0.597329	−0.298665	+	0.954358i	$0.596541\pi$
$654$	0	0
$655$	−1.49480	−0.0584066
$656$	0	0
$657$	0.426065	0.0166224
$658$	0	0
$659$	24.5104	0.954789	0.477394	−	0.878689i	$-0.341581\pi$
$659$	24.5104	0.954789	0.477394	+	0.878689i	$0.341581\pi$
$660$	0	0
$661$	35.8559	1.39463	0.697316	−	0.716764i	$-0.254378\pi$
$661$	35.8559	1.39463	0.697316	+	0.716764i	$0.254378\pi$
$662$	0	0
$663$	−10.4417	−0.405521
$664$	0	0
$665$	2.80046	0.108597
$666$	0	0
$667$	−3.92407	−0.151941
$668$	0	0
$669$	−31.7572	−1.22780
$670$	0	0
$671$	−22.5649	−0.871108
$672$	0	0
$673$	−16.9600	−0.653760	−0.326880	−	0.945066i	$-0.605997\pi$
$673$	−16.9600	−0.653760	−0.326880	+	0.945066i	$0.605997\pi$
$674$	0	0
$675$	−24.9977	−0.962161
$676$	0	0
$677$	−30.2673	−1.16327	−0.581633	−	0.813451i	$-0.697586\pi$
$677$	−30.2673	−1.16327	−0.581633	+	0.813451i	$0.697586\pi$
$678$	0	0
$679$	12.5514	0.481680
$680$	0	0
$681$	−20.4040	−0.781883
$682$	0	0
$683$	−11.9695	−0.458001	−0.229000	−	0.973426i	$-0.573546\pi$
$683$	−11.9695	−0.458001	−0.229000	+	0.973426i	$0.573546\pi$
$684$	0	0
$685$	0.552733	0.0211188
$686$	0	0
$687$	13.7709	0.525393
$688$	0	0
$689$	76.0844	2.89859
$690$	0	0
$691$	−34.7662	−1.32257	−0.661284	−	0.750135i	$-0.729988\pi$
$691$	−34.7662	−1.32257	−0.661284	+	0.750135i	$0.729988\pi$
$692$	0	0
$693$	0.232139	0.00881825
$694$	0	0
$695$	−2.39291	−0.0907681
$696$	0	0
$697$	3.64767	0.138165
$698$	0	0
$699$	−18.9886	−0.718216
$700$	0	0
$701$	−41.9649	−1.58499	−0.792497	−	0.609876i	$-0.791219\pi$
$701$	−41.9649	−1.58499	−0.792497	+	0.609876i	$0.791219\pi$
$702$	0	0
$703$	28.1278	1.06086
$704$	0	0
$705$	1.84936	0.0696508
$706$	0	0
$707$	20.7597	0.780747
$708$	0	0
$709$	12.5431	0.471065	0.235532	−	0.971867i	$-0.424317\pi$
$709$	12.5431	0.471065	0.235532	+	0.971867i	$0.424317\pi$
$710$	0	0
$711$	0.507279	0.0190245
$712$	0	0
$713$	3.70856	0.138887
$714$	0	0
$715$	7.58287	0.283583
$716$	0	0
$717$	−30.8337	−1.15151
$718$	0	0
$719$	−0.929782	−0.0346750	−0.0173375	−	0.999850i	$-0.505519\pi$
$719$	−0.929782	−0.0346750	−0.0173375	+	0.999850i	$0.505519\pi$
$720$	0	0
$721$	21.1046	0.785975
$722$	0	0
$723$	16.3192	0.606917
$724$	0	0
$725$	−14.1742	−0.526418
$726$	0	0
$727$	5.15476	0.191180	0.0955898	−	0.995421i	$-0.469526\pi$
$727$	5.15476	0.191180	0.0955898	+	0.995421i	$0.469526\pi$
$728$	0	0
$729$	26.6247	0.986099
$730$	0	0
$731$	−10.2528	−0.379212
$732$	0	0
$733$	−21.8352	−0.806503	−0.403252	−	0.915089i	$-0.632120\pi$
$733$	−21.8352	−0.806503	−0.403252	+	0.915089i	$0.632120\pi$
$734$	0	0
$735$	2.63278	0.0971115
$736$	0	0
$737$	−49.3456	−1.81767
$738$	0	0
$739$	34.6523	1.27471	0.637353	−	0.770572i	$-0.280029\pi$
$739$	34.6523	1.27471	0.637353	+	0.770572i	$0.280029\pi$
$740$	0	0
$741$	−41.5656	−1.52695
$742$	0	0
$743$	26.8900	0.986499	0.493250	−	0.869888i	$-0.335809\pi$
$743$	26.8900	0.986499	0.493250	+	0.869888i	$0.335809\pi$
$744$	0	0
$745$	−3.96786	−0.145371
$746$	0	0
$747$	0.377444	0.0138100
$748$	0	0
$749$	24.2013	0.884296
$750$	0	0
$751$	−50.2530	−1.83376	−0.916878	−	0.399167i	$-0.869299\pi$
$751$	−50.2530	−1.83376	−0.916878	+	0.399167i	$0.869299\pi$
$752$	0	0
$753$	−44.4038	−1.61816
$754$	0	0
$755$	0.704929	0.0256550
$756$	0	0
$757$	−18.6437	−0.677616	−0.338808	−	0.940856i	$-0.610024\pi$
$757$	−18.6437	−0.677616	−0.338808	+	0.940856i	$0.610024\pi$
$758$	0	0
$759$	−7.50101	−0.272269
$760$	0	0
$761$	53.4334	1.93696	0.968480	−	0.249091i	$-0.0801317\pi$
$761$	53.4334	1.93696	0.968480	+	0.249091i	$0.0801317\pi$
$762$	0	0
$763$	26.9133	0.974327
$764$	0	0
$765$	−0.0160331	−0.000579677 0
$766$	0	0
$767$	5.98811	0.216218
$768$	0	0
$769$	4.23995	0.152896	0.0764482	−	0.997074i	$-0.475642\pi$
$769$	4.23995	0.152896	0.0764482	+	0.997074i	$0.475642\pi$
$770$	0	0
$771$	−39.5993	−1.42613
$772$	0	0
$773$	31.2769	1.12495	0.562476	−	0.826813i	$-0.309849\pi$
$773$	31.2769	1.12495	0.562476	+	0.826813i	$0.309849\pi$
$774$	0	0
$775$	13.3958	0.481191
$776$	0	0
$777$	−21.9563	−0.787677
$778$	0	0
$779$	14.5204	0.520248
$780$	0	0
$781$	8.21918	0.294105
$782$	0	0
$783$	15.0996	0.539615
$784$	0	0
$785$	2.49307	0.0889815
$786$	0	0
$787$	−25.5280	−0.909976	−0.454988	−	0.890498i	$-0.650356\pi$
$787$	−25.5280	−0.909976	−0.454988	+	0.890498i	$0.650356\pi$
$788$	0	0
$789$	−30.2285	−1.07616
$790$	0	0
$791$	6.50740	0.231377
$792$	0	0
$793$	−42.1248	−1.49589
$794$	0	0
$795$	8.74673	0.310215
$796$	0	0
$797$	8.41127	0.297943	0.148971	−	0.988842i	$-0.452404\pi$
$797$	8.41127	0.297943	0.148971	+	0.988842i	$0.452404\pi$
$798$	0	0
$799$	2.68646	0.0950402
$800$	0	0
$801$	0.433094	0.0153026
$802$	0	0
$803$	−33.6514	−1.18753
$804$	0	0
$805$	0.943450	0.0332522
$806$	0	0
$807$	−26.3733	−0.928384
$808$	0	0
$809$	−37.5685	−1.32084	−0.660420	−	0.750897i	$-0.729622\pi$
$809$	−37.5685	−1.32084	−0.660420	+	0.750897i	$0.729622\pi$
$810$	0	0
$811$	−5.12852	−0.180087	−0.0900433	−	0.995938i	$-0.528701\pi$
$811$	−5.12852	−0.180087	−0.0900433	+	0.995938i	$0.528701\pi$
$812$	0	0
$813$	−5.62865	−0.197406
$814$	0	0
$815$	−4.08620	−0.143133
$816$	0	0
$817$	−40.8136	−1.42789
$818$	0	0
$819$	0.433364	0.0151430
$820$	0	0
$821$	38.3255	1.33757	0.668785	−	0.743456i	$-0.266815\pi$
$821$	38.3255	1.33757	0.668785	+	0.743456i	$0.266815\pi$
$822$	0	0
$823$	42.3651	1.47676	0.738378	−	0.674387i	$-0.235592\pi$
$823$	42.3651	1.47676	0.738378	+	0.674387i	$0.235592\pi$
$824$	0	0
$825$	−27.0946	−0.943313
$826$	0	0
$827$	−39.5530	−1.37539	−0.687696	−	0.725999i	$-0.741378\pi$
$827$	−39.5530	−1.37539	−0.687696	+	0.725999i	$0.741378\pi$
$828$	0	0
$829$	−26.6980	−0.927261	−0.463631	−	0.886029i	$-0.653454\pi$
$829$	−26.6980	−0.927261	−0.463631	+	0.886029i	$0.653454\pi$
$830$	0	0
$831$	32.7659	1.13664
$832$	0	0
$833$	3.82450	0.132511
$834$	0	0
$835$	−2.02439	−0.0700570
$836$	0	0
$837$	−14.2703	−0.493255
$838$	0	0
$839$	18.7878	0.648626	0.324313	−	0.945950i	$-0.394867\pi$
$839$	18.7878	0.648626	0.324313	+	0.945950i	$0.394867\pi$
$840$	0	0
$841$	−20.4382	−0.704765
$842$	0	0
$843$	−45.2723	−1.55926
$844$	0	0
$845$	9.02373	0.310426
$846$	0	0
$847$	1.26711	0.0435384
$848$	0	0
$849$	1.36066	0.0466979
$850$	0	0
$851$	9.47602	0.324834
$852$	0	0
$853$	−35.1508	−1.20354	−0.601770	−	0.798669i	$-0.705538\pi$
$853$	−35.1508	−1.20354	−0.601770	+	0.798669i	$0.705538\pi$
$854$	0	0
$855$	−0.0638235	−0.00218272
$856$	0	0
$857$	31.2357	1.06699	0.533496	−	0.845803i	$-0.320878\pi$
$857$	31.2357	1.06699	0.533496	+	0.845803i	$0.320878\pi$
$858$	0	0
$859$	−22.0929	−0.753800	−0.376900	−	0.926254i	$-0.623010\pi$
$859$	−22.0929	−0.753800	−0.376900	+	0.926254i	$0.623010\pi$
$860$	0	0
$861$	−11.3345	−0.386278
$862$	0	0
$863$	−22.6019	−0.769377	−0.384689	−	0.923046i	$-0.625691\pi$
$863$	−22.6019	−0.769377	−0.384689	+	0.923046i	$0.625691\pi$
$864$	0	0
$865$	−0.0283099	−0.000962566 0
$866$	0	0
$867$	−1.74374	−0.0592204
$868$	0	0
$869$	−40.0659	−1.35914
$870$	0	0
$871$	−92.1197	−3.12136
$872$	0	0
$873$	−0.286052	−0.00968140
$874$	0	0
$875$	6.92537	0.234120
$876$	0	0
$877$	−30.9491	−1.04508	−0.522539	−	0.852615i	$-0.675015\pi$
$877$	−30.9491	−1.04508	−0.522539	+	0.852615i	$0.675015\pi$
$878$	0	0
$879$	−31.5577	−1.06442
$880$	0	0
$881$	−44.2415	−1.49053	−0.745266	−	0.666767i	$-0.767678\pi$
$881$	−44.2415	−1.49053	−0.745266	+	0.666767i	$0.767678\pi$
$882$	0	0
$883$	9.09068	0.305926	0.152963	−	0.988232i	$-0.451118\pi$
$883$	9.09068	0.305926	0.152963	+	0.988232i	$0.451118\pi$
$884$	0	0
$885$	0.688398	0.0231402
$886$	0	0
$887$	1.88635	0.0633375	0.0316688	−	0.999498i	$-0.489918\pi$
$887$	1.88635	0.0633375	0.0316688	+	0.999498i	$0.489918\pi$
$888$	0	0
$889$	−22.4507	−0.752971
$890$	0	0
$891$	29.2543	0.980054
$892$	0	0
$893$	10.6941	0.357865
$894$	0	0
$895$	−5.69358	−0.190315
$896$	0	0
$897$	−14.0031	−0.467550
$898$	0	0
$899$	−8.09159	−0.269870
$900$	0	0
$901$	12.7059	0.423296
$902$	0	0
$903$	31.8586	1.06019
$904$	0	0
$905$	0.526150	0.0174898
$906$	0	0
$907$	51.6528	1.71510	0.857551	−	0.514399i	$-0.171985\pi$
$907$	51.6528	1.71510	0.857551	+	0.514399i	$0.171985\pi$
$908$	0	0
$909$	−0.473121	−0.0156924
$910$	0	0
$911$	−59.5888	−1.97426	−0.987132	−	0.159907i	$-0.948880\pi$
$911$	−59.5888	−1.97426	−0.987132	+	0.159907i	$0.948880\pi$
$912$	0	0
$913$	−29.8112	−0.986608
$914$	0	0
$915$	−4.84270	−0.160095
$916$	0	0
$917$	6.74729	0.222815
$918$	0	0
$919$	6.39893	0.211081	0.105541	−	0.994415i	$-0.466343\pi$
$919$	6.39893	0.211081	0.105541	+	0.994415i	$0.466343\pi$
$920$	0	0
$921$	37.0816	1.22188
$922$	0	0
$923$	15.3438	0.505047
$924$	0	0
$925$	34.2286	1.12543
$926$	0	0
$927$	−0.480981	−0.0157975
$928$	0	0
$929$	46.8009	1.53549	0.767743	−	0.640758i	$-0.221380\pi$
$929$	46.8009	1.53549	0.767743	+	0.640758i	$0.221380\pi$
$930$	0	0
$931$	15.2244	0.498958
$932$	0	0
$933$	1.71085	0.0560107
$934$	0	0
$935$	1.26632	0.0414132
$936$	0	0
$937$	54.7768	1.78948	0.894740	−	0.446587i	$-0.147361\pi$
$937$	54.7768	1.78948	0.894740	+	0.446587i	$0.147361\pi$
$938$	0	0
$939$	5.37553	0.175424
$940$	0	0
$941$	−4.09829	−0.133601	−0.0668003	−	0.997766i	$-0.521279\pi$
$941$	−4.09829	−0.133601	−0.0668003	+	0.997766i	$0.521279\pi$
$942$	0	0
$943$	4.89181	0.159299
$944$	0	0
$945$	−3.63034	−0.118095
$946$	0	0
$947$	8.92792	0.290119	0.145059	−	0.989423i	$-0.453663\pi$
$947$	8.92792	0.290119	0.145059	+	0.989423i	$0.453663\pi$
$948$	0	0
$949$	−62.8213	−2.03927
$950$	0	0
$951$	60.8040	1.97170
$952$	0	0
$953$	49.9243	1.61721	0.808603	−	0.588354i	$-0.200224\pi$
$953$	49.9243	1.61721	0.808603	+	0.588354i	$0.200224\pi$
$954$	0	0
$955$	−5.15813	−0.166913
$956$	0	0
$957$	16.3662	0.529045
$958$	0	0
$959$	−2.49495	−0.0805662
$960$	0	0
$961$	−23.3528	−0.753316
$962$	0	0
$963$	−0.551557	−0.0177737
$964$	0	0
$965$	4.47141	0.143940
$966$	0	0
$967$	35.0449	1.12697	0.563485	−	0.826127i	$-0.309460\pi$
$967$	35.0449	1.12697	0.563485	+	0.826127i	$0.309460\pi$
$968$	0	0
$969$	−6.94136	−0.222989
$970$	0	0
$971$	−44.7537	−1.43621	−0.718107	−	0.695933i	$-0.754991\pi$
$971$	−44.7537	−1.43621	−0.718107	+	0.695933i	$0.754991\pi$
$972$	0	0
$973$	10.8012	0.346271
$974$	0	0
$975$	−50.5810	−1.61989
$976$	0	0
$977$	10.1953	0.326175	0.163088	−	0.986612i	$-0.447855\pi$
$977$	10.1953	0.326175	0.163088	+	0.986612i	$0.447855\pi$
$978$	0	0
$979$	−34.2066	−1.09325
$980$	0	0
$981$	−0.613365	−0.0195832
$982$	0	0
$983$	15.3900	0.490865	0.245433	−	0.969414i	$-0.421070\pi$
$983$	15.3900	0.490865	0.245433	+	0.969414i	$0.421070\pi$
$984$	0	0
$985$	7.89355	0.251509
$986$	0	0
$987$	−8.34771	−0.265711
$988$	0	0
$989$	−13.7497	−0.437216
$990$	0	0
$991$	−48.4840	−1.54014	−0.770072	−	0.637956i	$-0.779780\pi$
$991$	−48.4840	−1.54014	−0.770072	+	0.637956i	$0.779780\pi$
$992$	0	0
$993$	−6.49073	−0.205977
$994$	0	0
$995$	8.11157	0.257154
$996$	0	0
$997$	1.30265	0.0412554	0.0206277	−	0.999787i	$-0.493434\pi$
$997$	1.30265	0.0412554	0.0206277	+	0.999787i	$0.493434\pi$
$998$	0	0
$999$	−36.4632	−1.15364

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.i.1.4	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.i.1.4	✓	18	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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q-1.74374 q^{3} +0.394783 q^{5} -1.78199 q^{7} +0.0406123 q^{9} +O(q^{10})\)	\(q-1.74374 q^{3} +0.394783 q^{5} -1.78199 q^{7} +0.0406123 q^{9} -3.20764 q^{11} -5.98811 q^{13} -0.688398 q^{15} -1.00000 q^{17} -3.98074 q^{19} +3.10732 q^{21} -1.34108 q^{23} -4.84415 q^{25} +5.16039 q^{27} +2.92606 q^{29} -2.76536 q^{31} +5.59327 q^{33} -0.703501 q^{35} -7.06597 q^{37} +10.4417 q^{39} -3.64767 q^{41} +10.2528 q^{43} +0.0160331 q^{45} -2.68646 q^{47} -3.82450 q^{49} +1.74374 q^{51} -12.7059 q^{53} -1.26632 q^{55} +6.94136 q^{57} -1.00000 q^{59} +7.03474 q^{61} -0.0723708 q^{63} -2.36401 q^{65} +15.3838 q^{67} +2.33848 q^{69} -2.56238 q^{71} +10.4910 q^{73} +8.44691 q^{75} +5.71599 q^{77} +12.4908 q^{79} -9.12019 q^{81} +9.29383 q^{83} -0.394783 q^{85} -5.10227 q^{87} +10.6641 q^{89} +10.6708 q^{91} +4.82205 q^{93} -1.57153 q^{95} -7.04349 q^{97} -0.130270 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9	\( 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9} + 12 q^{11} + 2 q^{13} - 18 q^{17} + 5 q^{19} - 3 q^{21} + 21 q^{23} + 16 q^{25} + 26 q^{27} + 14 q^{29} + 15 q^{31} + 19 q^{33} + 20 q^{35} + 2 q^{37} - 14 q^{39} + 34 q^{41} + 21 q^{43} + 49 q^{45} + 69 q^{47} + 28 q^{49} - 8 q^{51} - 4 q^{53} + 18 q^{55} + 5 q^{57} - 18 q^{59} + 11 q^{61} + 35 q^{63} + 27 q^{65} + 34 q^{67} - 4 q^{69} + 37 q^{71} + 18 q^{73} + 72 q^{75} + 11 q^{77} + 11 q^{79} + 30 q^{81} + 28 q^{83} - 4 q^{85} + 7 q^{87} + 44 q^{89} - 23 q^{91} - 3 q^{93} - 11 q^{95} + 11 q^{97} + 56 q^{99}+O(q^{100}) \) 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9 + 12 * q^11 + 2 * q^13 - 18 * q^17 + 5 * q^19 - 3 * q^21 + 21 * q^23 + 16 * q^25 + 26 * q^27 + 14 * q^29 + 15 * q^31 + 19 * q^33 + 20 * q^35 + 2 * q^37 - 14 * q^39 + 34 * q^41 + 21 * q^43 + 49 * q^45 + 69 * q^47 + 28 * q^49 - 8 * q^51 - 4 * q^53 + 18 * q^55 + 5 * q^57 - 18 * q^59 + 11 * q^61 + 35 * q^63 + 27 * q^65 + 34 * q^67 - 4 * q^69 + 37 * q^71 + 18 * q^73 + 72 * q^75 + 11 * q^77 + 11 * q^79 + 30 * q^81 + 28 * q^83 - 4 * q^85 + 7 * q^87 + 44 * q^89 - 23 * q^91 - 3 * q^93 - 11 * q^95 + 11 * q^97 + 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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