LMFDB - Embedded newform 6036.2.a.i.1.9

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:	$0$
Dimension:	$26$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.78585 q^{5} -2.07926 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.78585 q^{5} -2.07926 q^{7} +1.00000 q^{9} +2.20363 q^{11} -6.61519 q^{13} +1.78585 q^{15} -6.93993 q^{17} -3.40790 q^{19} +2.07926 q^{21} +2.31931 q^{23} -1.81076 q^{25} -1.00000 q^{27} -1.77508 q^{29} +5.22504 q^{31} -2.20363 q^{33} +3.71323 q^{35} -10.8349 q^{37} +6.61519 q^{39} +3.10677 q^{41} -3.95587 q^{43} -1.78585 q^{45} +6.71070 q^{47} -2.67669 q^{49} +6.93993 q^{51} -9.33131 q^{53} -3.93534 q^{55} +3.40790 q^{57} -8.55552 q^{59} +8.78763 q^{61} -2.07926 q^{63} +11.8137 q^{65} -8.71231 q^{67} -2.31931 q^{69} -10.1955 q^{71} -9.60865 q^{73} +1.81076 q^{75} -4.58191 q^{77} +12.9198 q^{79} +1.00000 q^{81} +3.56790 q^{83} +12.3936 q^{85} +1.77508 q^{87} +11.0077 q^{89} +13.7547 q^{91} -5.22504 q^{93} +6.08599 q^{95} +13.1768 q^{97} +2.20363 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * 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
$5$	−1.78585	−0.798654	−0.399327	−	0.916809i	$-0.630756\pi$
$5$	−1.78585	−0.798654	−0.399327	+	0.916809i	$0.630756\pi$
$6$	0	0
$7$	−2.07926	−0.785885	−0.392942	−	0.919563i	$-0.628543\pi$
$7$	−2.07926	−0.785885	−0.392942	+	0.919563i	$0.628543\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.20363	0.664419	0.332209	−	0.943206i	$-0.392206\pi$
$11$	2.20363	0.664419	0.332209	+	0.943206i	$0.392206\pi$
$12$	0	0
$13$	−6.61519	−1.83472	−0.917362	−	0.398055i	$-0.869685\pi$
$13$	−6.61519	−1.83472	−0.917362	+	0.398055i	$0.869685\pi$
$14$	0	0
$15$	1.78585	0.461103
$16$	0	0
$17$	−6.93993	−1.68318	−0.841590	−	0.540117i	$-0.818380\pi$
$17$	−6.93993	−1.68318	−0.841590	+	0.540117i	$0.818380\pi$
$18$	0	0
$19$	−3.40790	−0.781827	−0.390913	−	0.920427i	$-0.627841\pi$
$19$	−3.40790	−0.781827	−0.390913	+	0.920427i	$0.627841\pi$
$20$	0	0
$21$	2.07926	0.453731
$22$	0	0
$23$	2.31931	0.483609	0.241804	−	0.970325i	$-0.422261\pi$
$23$	2.31931	0.483609	0.241804	+	0.970325i	$0.422261\pi$
$24$	0	0
$25$	−1.81076	−0.362151
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.77508	−0.329624	−0.164812	−	0.986325i	$-0.552702\pi$
$29$	−1.77508	−0.329624	−0.164812	+	0.986325i	$0.552702\pi$
$30$	0	0
$31$	5.22504	0.938444	0.469222	−	0.883080i	$-0.344534\pi$
$31$	5.22504	0.938444	0.469222	+	0.883080i	$0.344534\pi$
$32$	0	0
$33$	−2.20363	−0.383602
$34$	0	0
$35$	3.71323	0.627650
$36$	0	0
$37$	−10.8349	−1.78125	−0.890625	−	0.454739i	$-0.849732\pi$
$37$	−10.8349	−1.78125	−0.890625	+	0.454739i	$0.849732\pi$
$38$	0	0
$39$	6.61519	1.05928
$40$	0	0
$41$	3.10677	0.485197	0.242598	−	0.970127i	$-0.422000\pi$
$41$	3.10677	0.485197	0.242598	+	0.970127i	$0.422000\pi$
$42$	0	0
$43$	−3.95587	−0.603264	−0.301632	−	0.953424i	$-0.597531\pi$
$43$	−3.95587	−0.603264	−0.301632	+	0.953424i	$0.597531\pi$
$44$	0	0
$45$	−1.78585	−0.266218
$46$	0	0
$47$	6.71070	0.978857	0.489428	−	0.872044i	$-0.337205\pi$
$47$	6.71070	0.978857	0.489428	+	0.872044i	$0.337205\pi$
$48$	0	0
$49$	−2.67669	−0.382385
$50$	0	0
$51$	6.93993	0.971784
$52$	0	0
$53$	−9.33131	−1.28175	−0.640877	−	0.767644i	$-0.721429\pi$
$53$	−9.33131	−1.28175	−0.640877	+	0.767644i	$0.721429\pi$
$54$	0	0
$55$	−3.93534	−0.530641
$56$	0	0
$57$	3.40790	0.451388
$58$	0	0
$59$	−8.55552	−1.11383	−0.556917	−	0.830568i	$-0.688016\pi$
$59$	−8.55552	−1.11383	−0.556917	+	0.830568i	$0.688016\pi$
$60$	0	0
$61$	8.78763	1.12514	0.562571	−	0.826749i	$-0.309812\pi$
$61$	8.78763	1.12514	0.562571	+	0.826749i	$0.309812\pi$
$62$	0	0
$63$	−2.07926	−0.261962
$64$	0	0
$65$	11.8137	1.46531
$66$	0	0
$67$	−8.71231	−1.06438	−0.532189	−	0.846626i	$-0.678630\pi$
$67$	−8.71231	−1.06438	−0.532189	+	0.846626i	$0.678630\pi$
$68$	0	0
$69$	−2.31931	−0.279212
$70$	0	0
$71$	−10.1955	−1.20998	−0.604989	−	0.796234i	$-0.706823\pi$
$71$	−10.1955	−1.20998	−0.604989	+	0.796234i	$0.706823\pi$
$72$	0	0
$73$	−9.60865	−1.12461	−0.562304	−	0.826931i	$-0.690085\pi$
$73$	−9.60865	−1.12461	−0.562304	+	0.826931i	$0.690085\pi$
$74$	0	0
$75$	1.81076	0.209088
$76$	0	0
$77$	−4.58191	−0.522157
$78$	0	0
$79$	12.9198	1.45359	0.726794	−	0.686855i	$-0.241009\pi$
$79$	12.9198	1.45359	0.726794	+	0.686855i	$0.241009\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.56790	0.391628	0.195814	−	0.980641i	$-0.437265\pi$
$83$	3.56790	0.391628	0.195814	+	0.980641i	$0.437265\pi$
$84$	0	0
$85$	12.3936	1.34428
$86$	0	0
$87$	1.77508	0.190309
$88$	0	0
$89$	11.0077	1.16682	0.583408	−	0.812179i	$-0.301719\pi$
$89$	11.0077	1.16682	0.583408	+	0.812179i	$0.301719\pi$
$90$	0	0
$91$	13.7547	1.44188
$92$	0	0
$93$	−5.22504	−0.541811
$94$	0	0
$95$	6.08599	0.624409
$96$	0	0
$97$	13.1768	1.33790	0.668950	−	0.743308i	$-0.266744\pi$
$97$	13.1768	1.33790	0.668950	+	0.743308i	$0.266744\pi$
$98$	0	0
$99$	2.20363	0.221473
$100$	0	0
$101$	1.21326	0.120723	0.0603617	−	0.998177i	$-0.480775\pi$
$101$	1.21326	0.120723	0.0603617	+	0.998177i	$0.480775\pi$
$102$	0	0
$103$	15.8674	1.56346	0.781728	−	0.623619i	$-0.214338\pi$
$103$	15.8674	1.56346	0.781728	+	0.623619i	$0.214338\pi$
$104$	0	0
$105$	−3.71323	−0.362374
$106$	0	0
$107$	−17.6627	−1.70751	−0.853757	−	0.520671i	$-0.825682\pi$
$107$	−17.6627	−1.70751	−0.853757	+	0.520671i	$0.825682\pi$
$108$	0	0
$109$	−15.2491	−1.46060	−0.730301	−	0.683126i	$-0.760620\pi$
$109$	−15.2491	−1.46060	−0.730301	+	0.683126i	$0.760620\pi$
$110$	0	0
$111$	10.8349	1.02840
$112$	0	0
$113$	5.21830	0.490896	0.245448	−	0.969410i	$-0.421065\pi$
$113$	5.21830	0.490896	0.245448	+	0.969410i	$0.421065\pi$
$114$	0	0
$115$	−4.14192	−0.386236
$116$	0	0
$117$	−6.61519	−0.611574
$118$	0	0
$119$	14.4299	1.32279
$120$	0	0
$121$	−6.14402	−0.558547
$122$	0	0
$123$	−3.10677	−0.280128
$124$	0	0
$125$	12.1630	1.08789
$126$	0	0
$127$	−14.3938	−1.27724	−0.638620	−	0.769522i	$-0.720495\pi$
$127$	−14.3938	−1.27724	−0.638620	+	0.769522i	$0.720495\pi$
$128$	0	0
$129$	3.95587	0.348295
$130$	0	0
$131$	−10.5082	−0.918110	−0.459055	−	0.888408i	$-0.651812\pi$
$131$	−10.5082	−0.918110	−0.459055	+	0.888408i	$0.651812\pi$
$132$	0	0
$133$	7.08591	0.614426
$134$	0	0
$135$	1.78585	0.153701
$136$	0	0
$137$	−11.5382	−0.985771	−0.492886	−	0.870094i	$-0.664058\pi$
$137$	−11.5382	−0.985771	−0.492886	+	0.870094i	$0.664058\pi$
$138$	0	0
$139$	15.2200	1.29094	0.645472	−	0.763784i	$-0.276661\pi$
$139$	15.2200	1.29094	0.645472	+	0.763784i	$0.276661\pi$
$140$	0	0
$141$	−6.71070	−0.565143
$142$	0	0
$143$	−14.5774	−1.21902
$144$	0	0
$145$	3.17002	0.263256
$146$	0	0
$147$	2.67669	0.220770
$148$	0	0
$149$	12.0350	0.985945	0.492973	−	0.870045i	$-0.335910\pi$
$149$	12.0350	0.985945	0.492973	+	0.870045i	$0.335910\pi$
$150$	0	0
$151$	−15.9197	−1.29553	−0.647763	−	0.761842i	$-0.724295\pi$
$151$	−15.9197	−1.29553	−0.647763	+	0.761842i	$0.724295\pi$
$152$	0	0
$153$	−6.93993	−0.561060
$154$	0	0
$155$	−9.33110	−0.749492
$156$	0	0
$157$	−1.08167	−0.0863263	−0.0431632	−	0.999068i	$-0.513744\pi$
$157$	−1.08167	−0.0863263	−0.0431632	+	0.999068i	$0.513744\pi$
$158$	0	0
$159$	9.33131	0.740021
$160$	0	0
$161$	−4.82243	−0.380061
$162$	0	0
$163$	5.26307	0.412235	0.206118	−	0.978527i	$-0.433917\pi$
$163$	5.26307	0.412235	0.206118	+	0.978527i	$0.433917\pi$
$164$	0	0
$165$	3.93534	0.306366
$166$	0	0
$167$	−20.0357	−1.55041	−0.775206	−	0.631708i	$-0.782354\pi$
$167$	−20.0357	−1.55041	−0.775206	+	0.631708i	$0.782354\pi$
$168$	0	0
$169$	30.7607	2.36621
$170$	0	0
$171$	−3.40790	−0.260609
$172$	0	0
$173$	4.63384	0.352305	0.176152	−	0.984363i	$-0.443635\pi$
$173$	4.63384	0.352305	0.176152	+	0.984363i	$0.443635\pi$
$174$	0	0
$175$	3.76503	0.284609
$176$	0	0
$177$	8.55552	0.643072
$178$	0	0
$179$	19.0524	1.42404	0.712021	−	0.702158i	$-0.247780\pi$
$179$	19.0524	1.42404	0.712021	+	0.702158i	$0.247780\pi$
$180$	0	0
$181$	9.45601	0.702860	0.351430	−	0.936214i	$-0.385696\pi$
$181$	9.45601	0.702860	0.351430	+	0.936214i	$0.385696\pi$
$182$	0	0
$183$	−8.78763	−0.649600
$184$	0	0
$185$	19.3495	1.42260
$186$	0	0
$187$	−15.2930	−1.11834
$188$	0	0
$189$	2.07926	0.151244
$190$	0	0
$191$	−7.54585	−0.545999	−0.272999	−	0.962014i	$-0.588016\pi$
$191$	−7.54585	−0.545999	−0.272999	+	0.962014i	$0.588016\pi$
$192$	0	0
$193$	11.5660	0.832541	0.416271	−	0.909241i	$-0.363337\pi$
$193$	11.5660	0.832541	0.416271	+	0.909241i	$0.363337\pi$
$194$	0	0
$195$	−11.8137	−0.845997
$196$	0	0
$197$	−8.50436	−0.605910	−0.302955	−	0.953005i	$-0.597973\pi$
$197$	−8.50436	−0.605910	−0.302955	+	0.953005i	$0.597973\pi$
$198$	0	0
$199$	5.92204	0.419802	0.209901	−	0.977723i	$-0.432686\pi$
$199$	5.92204	0.419802	0.209901	+	0.977723i	$0.432686\pi$
$200$	0	0
$201$	8.71231	0.614519
$202$	0	0
$203$	3.69085	0.259047
$204$	0	0
$205$	−5.54822	−0.387504
$206$	0	0
$207$	2.31931	0.161203
$208$	0	0
$209$	−7.50975	−0.519461
$210$	0	0
$211$	−2.66403	−0.183399	−0.0916996	−	0.995787i	$-0.529230\pi$
$211$	−2.66403	−0.183399	−0.0916996	+	0.995787i	$0.529230\pi$
$212$	0	0
$213$	10.1955	0.698581
$214$	0	0
$215$	7.06457	0.481800
$216$	0	0
$217$	−10.8642	−0.737509
$218$	0	0
$219$	9.60865	0.649292
$220$	0	0
$221$	45.9089	3.08817
$222$	0	0
$223$	11.6207	0.778178	0.389089	−	0.921200i	$-0.372790\pi$
$223$	11.6207	0.778178	0.389089	+	0.921200i	$0.372790\pi$
$224$	0	0
$225$	−1.81076	−0.120717
$226$	0	0
$227$	−5.73829	−0.380864	−0.190432	−	0.981700i	$-0.560989\pi$
$227$	−5.73829	−0.380864	−0.190432	+	0.981700i	$0.560989\pi$
$228$	0	0
$229$	−2.94839	−0.194835	−0.0974177	−	0.995244i	$-0.531058\pi$
$229$	−2.94839	−0.194835	−0.0974177	+	0.995244i	$0.531058\pi$
$230$	0	0
$231$	4.58191	0.301467
$232$	0	0
$233$	16.0655	1.05249	0.526243	−	0.850334i	$-0.323600\pi$
$233$	16.0655	1.05249	0.526243	+	0.850334i	$0.323600\pi$
$234$	0	0
$235$	−11.9843	−0.781768
$236$	0	0
$237$	−12.9198	−0.839229
$238$	0	0
$239$	−17.3794	−1.12418	−0.562091	−	0.827076i	$-0.690003\pi$
$239$	−17.3794	−1.12418	−0.562091	+	0.827076i	$0.690003\pi$
$240$	0	0
$241$	−20.3416	−1.31031	−0.655157	−	0.755493i	$-0.727398\pi$
$241$	−20.3416	−1.31031	−0.655157	+	0.755493i	$0.727398\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	4.78016	0.305393
$246$	0	0
$247$	22.5439	1.43444
$248$	0	0
$249$	−3.56790	−0.226107
$250$	0	0
$251$	−23.3382	−1.47310	−0.736548	−	0.676386i	$-0.763545\pi$
$251$	−23.3382	−1.47310	−0.736548	+	0.676386i	$0.763545\pi$
$252$	0	0
$253$	5.11089	0.321319
$254$	0	0
$255$	−12.3936	−0.776119
$256$	0	0
$257$	2.27427	0.141865	0.0709326	−	0.997481i	$-0.477402\pi$
$257$	2.27427	0.141865	0.0709326	+	0.997481i	$0.477402\pi$
$258$	0	0
$259$	22.5286	1.39986
$260$	0	0
$261$	−1.77508	−0.109875
$262$	0	0
$263$	14.3925	0.887477	0.443738	−	0.896156i	$-0.353652\pi$
$263$	14.3925	0.887477	0.443738	+	0.896156i	$0.353652\pi$
$264$	0	0
$265$	16.6643	1.02368
$266$	0	0
$267$	−11.0077	−0.673661
$268$	0	0
$269$	12.1315	0.739673	0.369837	−	0.929097i	$-0.379414\pi$
$269$	12.1315	0.739673	0.369837	+	0.929097i	$0.379414\pi$
$270$	0	0
$271$	−26.5739	−1.61425	−0.807126	−	0.590380i	$-0.798978\pi$
$271$	−26.5739	−1.61425	−0.807126	+	0.590380i	$0.798978\pi$
$272$	0	0
$273$	−13.7547	−0.832471
$274$	0	0
$275$	−3.99023	−0.240620
$276$	0	0
$277$	16.6649	1.00130	0.500649	−	0.865650i	$-0.333095\pi$
$277$	16.6649	1.00130	0.500649	+	0.865650i	$0.333095\pi$
$278$	0	0
$279$	5.22504	0.312815
$280$	0	0
$281$	18.0299	1.07558	0.537788	−	0.843080i	$-0.319260\pi$
$281$	18.0299	1.07558	0.537788	+	0.843080i	$0.319260\pi$
$282$	0	0
$283$	22.0846	1.31280	0.656398	−	0.754415i	$-0.272079\pi$
$283$	22.0846	1.31280	0.656398	+	0.754415i	$0.272079\pi$
$284$	0	0
$285$	−6.08599	−0.360503
$286$	0	0
$287$	−6.45978	−0.381309
$288$	0	0
$289$	31.1626	1.83309
$290$	0	0
$291$	−13.1768	−0.772436
$292$	0	0
$293$	2.24011	0.130869	0.0654344	−	0.997857i	$-0.479157\pi$
$293$	2.24011	0.130869	0.0654344	+	0.997857i	$0.479157\pi$
$294$	0	0
$295$	15.2788	0.889568
$296$	0	0
$297$	−2.20363	−0.127867
$298$	0	0
$299$	−15.3427	−0.887288
$300$	0	0
$301$	8.22526	0.474096
$302$	0	0
$303$	−1.21326	−0.0696997
$304$	0	0
$305$	−15.6934	−0.898599
$306$	0	0
$307$	21.0330	1.20041	0.600207	−	0.799845i	$-0.295085\pi$
$307$	21.0330	1.20041	0.600207	+	0.799845i	$0.295085\pi$
$308$	0	0
$309$	−15.8674	−0.902662
$310$	0	0
$311$	23.7129	1.34464	0.672319	−	0.740262i	$-0.265298\pi$
$311$	23.7129	1.34464	0.672319	+	0.740262i	$0.265298\pi$
$312$	0	0
$313$	16.7181	0.944963	0.472482	−	0.881341i	$-0.343358\pi$
$313$	16.7181	0.944963	0.472482	+	0.881341i	$0.343358\pi$
$314$	0	0
$315$	3.71323	0.209217
$316$	0	0
$317$	−8.17881	−0.459368	−0.229684	−	0.973265i	$-0.573769\pi$
$317$	−8.17881	−0.459368	−0.229684	+	0.973265i	$0.573769\pi$
$318$	0	0
$319$	−3.91162	−0.219008
$320$	0	0
$321$	17.6627	0.985834
$322$	0	0
$323$	23.6506	1.31595
$324$	0	0
$325$	11.9785	0.664447
$326$	0	0
$327$	15.2491	0.843279
$328$	0	0
$329$	−13.9533	−0.769269
$330$	0	0
$331$	24.5580	1.34983	0.674915	−	0.737895i	$-0.264180\pi$
$331$	24.5580	1.34983	0.674915	+	0.737895i	$0.264180\pi$
$332$	0	0
$333$	−10.8349	−0.593750
$334$	0	0
$335$	15.5588	0.850070
$336$	0	0
$337$	−8.14991	−0.443954	−0.221977	−	0.975052i	$-0.571251\pi$
$337$	−8.14991	−0.443954	−0.221977	+	0.975052i	$0.571251\pi$
$338$	0	0
$339$	−5.21830	−0.283419
$340$	0	0
$341$	11.5140	0.623520
$342$	0	0
$343$	20.1203	1.08640
$344$	0	0
$345$	4.14192	0.222994
$346$	0	0
$347$	−17.4526	−0.936903	−0.468451	−	0.883489i	$-0.655188\pi$
$347$	−17.4526	−0.936903	−0.468451	+	0.883489i	$0.655188\pi$
$348$	0	0
$349$	28.2331	1.51129	0.755643	−	0.654984i	$-0.227325\pi$
$349$	28.2331	1.51129	0.755643	+	0.654984i	$0.227325\pi$
$350$	0	0
$351$	6.61519	0.353093
$352$	0	0
$353$	−32.5433	−1.73211	−0.866053	−	0.499953i	$-0.833351\pi$
$353$	−32.5433	−1.73211	−0.866053	+	0.499953i	$0.833351\pi$
$354$	0	0
$355$	18.2075	0.966354
$356$	0	0
$357$	−14.4299	−0.763710
$358$	0	0
$359$	10.1755	0.537042	0.268521	−	0.963274i	$-0.413465\pi$
$359$	10.1755	0.537042	0.268521	+	0.963274i	$0.413465\pi$
$360$	0	0
$361$	−7.38619	−0.388747
$362$	0	0
$363$	6.14402	0.322478
$364$	0	0
$365$	17.1596	0.898173
$366$	0	0
$367$	31.6754	1.65344	0.826721	−	0.562612i	$-0.190204\pi$
$367$	31.6754	1.65344	0.826721	+	0.562612i	$0.190204\pi$
$368$	0	0
$369$	3.10677	0.161732
$370$	0	0
$371$	19.4022	1.00731
$372$	0	0
$373$	12.9400	0.670009	0.335004	−	0.942217i	$-0.391262\pi$
$373$	12.9400	0.670009	0.335004	+	0.942217i	$0.391262\pi$
$374$	0	0
$375$	−12.1630	−0.628092
$376$	0	0
$377$	11.7425	0.604769
$378$	0	0
$379$	24.8771	1.27785	0.638926	−	0.769269i	$-0.279379\pi$
$379$	24.8771	1.27785	0.638926	+	0.769269i	$0.279379\pi$
$380$	0	0
$381$	14.3938	0.737415
$382$	0	0
$383$	−31.0011	−1.58408	−0.792042	−	0.610467i	$-0.790982\pi$
$383$	−31.0011	−1.58408	−0.792042	+	0.610467i	$0.790982\pi$
$384$	0	0
$385$	8.18258	0.417023
$386$	0	0
$387$	−3.95587	−0.201088
$388$	0	0
$389$	−4.84548	−0.245676	−0.122838	−	0.992427i	$-0.539199\pi$
$389$	−4.84548	−0.245676	−0.122838	+	0.992427i	$0.539199\pi$
$390$	0	0
$391$	−16.0958	−0.814000
$392$	0	0
$393$	10.5082	0.530071
$394$	0	0
$395$	−23.0727	−1.16091
$396$	0	0
$397$	23.9252	1.20077	0.600387	−	0.799710i	$-0.295013\pi$
$397$	23.9252	1.20077	0.600387	+	0.799710i	$0.295013\pi$
$398$	0	0
$399$	−7.08591	−0.354739
$400$	0	0
$401$	−1.03354	−0.0516125	−0.0258062	−	0.999667i	$-0.508215\pi$
$401$	−1.03354	−0.0516125	−0.0258062	+	0.999667i	$0.508215\pi$
$402$	0	0
$403$	−34.5646	−1.72179
$404$	0	0
$405$	−1.78585	−0.0887394
$406$	0	0
$407$	−23.8761	−1.18350
$408$	0	0
$409$	−18.6327	−0.921328	−0.460664	−	0.887575i	$-0.652389\pi$
$409$	−18.6327	−0.921328	−0.460664	+	0.887575i	$0.652389\pi$
$410$	0	0
$411$	11.5382	0.569135
$412$	0	0
$413$	17.7891	0.875345
$414$	0	0
$415$	−6.37172	−0.312775
$416$	0	0
$417$	−15.2200	−0.745327
$418$	0	0
$419$	−8.19130	−0.400171	−0.200086	−	0.979778i	$-0.564122\pi$
$419$	−8.19130	−0.400171	−0.200086	+	0.979778i	$0.564122\pi$
$420$	0	0
$421$	−4.27517	−0.208359	−0.104180	−	0.994559i	$-0.533222\pi$
$421$	−4.27517	−0.208359	−0.104180	+	0.994559i	$0.533222\pi$
$422$	0	0
$423$	6.71070	0.326286
$424$	0	0
$425$	12.5665	0.609566
$426$	0	0
$427$	−18.2717	−0.884231
$428$	0	0
$429$	14.5774	0.703804
$430$	0	0
$431$	29.0027	1.39701	0.698506	−	0.715604i	$-0.253848\pi$
$431$	29.0027	1.39701	0.698506	+	0.715604i	$0.253848\pi$
$432$	0	0
$433$	−0.842181	−0.0404727	−0.0202363	−	0.999795i	$-0.506442\pi$
$433$	−0.842181	−0.0404727	−0.0202363	+	0.999795i	$0.506442\pi$
$434$	0	0
$435$	−3.17002	−0.151991
$436$	0	0
$437$	−7.90398	−0.378098
$438$	0	0
$439$	−7.92802	−0.378384	−0.189192	−	0.981940i	$-0.560587\pi$
$439$	−7.92802	−0.378384	−0.189192	+	0.981940i	$0.560587\pi$
$440$	0	0
$441$	−2.67669	−0.127462
$442$	0	0
$443$	−16.3090	−0.774862	−0.387431	−	0.921899i	$-0.626637\pi$
$443$	−16.3090	−0.774862	−0.387431	+	0.921899i	$0.626637\pi$
$444$	0	0
$445$	−19.6581	−0.931882
$446$	0	0
$447$	−12.0350	−0.569236
$448$	0	0
$449$	12.7992	0.604031	0.302015	−	0.953303i	$-0.402341\pi$
$449$	12.7992	0.604031	0.302015	+	0.953303i	$0.402341\pi$
$450$	0	0
$451$	6.84617	0.322374
$452$	0	0
$453$	15.9197	0.747972
$454$	0	0
$455$	−24.5637	−1.15156
$456$	0	0
$457$	11.1398	0.521097	0.260548	−	0.965461i	$-0.416097\pi$
$457$	11.1398	0.521097	0.260548	+	0.965461i	$0.416097\pi$
$458$	0	0
$459$	6.93993	0.323928
$460$	0	0
$461$	−16.5693	−0.771710	−0.385855	−	0.922559i	$-0.626094\pi$
$461$	−16.5693	−0.771710	−0.385855	+	0.922559i	$0.626094\pi$
$462$	0	0
$463$	−30.4349	−1.41443	−0.707215	−	0.706999i	$-0.750049\pi$
$463$	−30.4349	−1.41443	−0.707215	+	0.706999i	$0.750049\pi$
$464$	0	0
$465$	9.33110	0.432720
$466$	0	0
$467$	16.7129	0.773379	0.386689	−	0.922210i	$-0.373619\pi$
$467$	16.7129	0.773379	0.386689	+	0.922210i	$0.373619\pi$
$468$	0	0
$469$	18.1151	0.836478
$470$	0	0
$471$	1.08167	0.0498405
$472$	0	0
$473$	−8.71726	−0.400820
$474$	0	0
$475$	6.17089	0.283140
$476$	0	0
$477$	−9.33131	−0.427251
$478$	0	0
$479$	10.3321	0.472084	0.236042	−	0.971743i	$-0.424150\pi$
$479$	10.3321	0.472084	0.236042	+	0.971743i	$0.424150\pi$
$480$	0	0
$481$	71.6750	3.26810
$482$	0	0
$483$	4.82243	0.219428
$484$	0	0
$485$	−23.5317	−1.06852
$486$	0	0
$487$	27.3555	1.23959	0.619797	−	0.784762i	$-0.287215\pi$
$487$	27.3555	1.23959	0.619797	+	0.784762i	$0.287215\pi$
$488$	0	0
$489$	−5.26307	−0.238004
$490$	0	0
$491$	1.46231	0.0659933	0.0329967	−	0.999455i	$-0.489495\pi$
$491$	1.46231	0.0659933	0.0329967	+	0.999455i	$0.489495\pi$
$492$	0	0
$493$	12.3189	0.554816
$494$	0	0
$495$	−3.93534	−0.176880
$496$	0	0
$497$	21.1990	0.950904
$498$	0	0
$499$	−10.0351	−0.449231	−0.224616	−	0.974447i	$-0.572113\pi$
$499$	−10.0351	−0.449231	−0.224616	+	0.974447i	$0.572113\pi$
$500$	0	0
$501$	20.0357	0.895131
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−2.16669	−0.0964163
$506$	0	0
$507$	−30.7607	−1.36613
$508$	0	0
$509$	14.9267	0.661616	0.330808	−	0.943698i	$-0.392679\pi$
$509$	14.9267	0.661616	0.330808	+	0.943698i	$0.392679\pi$
$510$	0	0
$511$	19.9788	0.883812
$512$	0	0
$513$	3.40790	0.150463
$514$	0	0
$515$	−28.3366	−1.24866
$516$	0	0
$517$	14.7879	0.650371
$518$	0	0
$519$	−4.63384	−0.203403
$520$	0	0
$521$	11.4560	0.501896	0.250948	−	0.968001i	$-0.419258\pi$
$521$	11.4560	0.501896	0.250948	+	0.968001i	$0.419258\pi$
$522$	0	0
$523$	−31.2698	−1.36733	−0.683667	−	0.729794i	$-0.739616\pi$
$523$	−31.2698	−1.36733	−0.683667	+	0.729794i	$0.739616\pi$
$524$	0	0
$525$	−3.76503	−0.164319
$526$	0	0
$527$	−36.2614	−1.57957
$528$	0	0
$529$	−17.6208	−0.766122
$530$	0	0
$531$	−8.55552	−0.371278
$532$	0	0
$533$	−20.5519	−0.890201
$534$	0	0
$535$	31.5428	1.36371
$536$	0	0
$537$	−19.0524	−0.822171
$538$	0	0
$539$	−5.89844	−0.254064
$540$	0	0
$541$	−26.0403	−1.11956	−0.559780	−	0.828641i	$-0.689114\pi$
$541$	−26.0403	−1.11956	−0.559780	+	0.828641i	$0.689114\pi$
$542$	0	0
$543$	−9.45601	−0.405796
$544$	0	0
$545$	27.2326	1.16652
$546$	0	0
$547$	3.18501	0.136181	0.0680907	−	0.997679i	$-0.478309\pi$
$547$	3.18501	0.136181	0.0680907	+	0.997679i	$0.478309\pi$
$548$	0	0
$549$	8.78763	0.375047
$550$	0	0
$551$	6.04930	0.257709
$552$	0	0
$553$	−26.8635	−1.14235
$554$	0	0
$555$	−19.3495	−0.821340
$556$	0	0
$557$	−43.3455	−1.83661	−0.918304	−	0.395876i	$-0.870441\pi$
$557$	−43.3455	−1.83661	−0.918304	+	0.395876i	$0.870441\pi$
$558$	0	0
$559$	26.1688	1.10682
$560$	0	0
$561$	15.2930	0.645672
$562$	0	0
$563$	−5.97360	−0.251757	−0.125879	−	0.992046i	$-0.540175\pi$
$563$	−5.97360	−0.251757	−0.125879	+	0.992046i	$0.540175\pi$
$564$	0	0
$565$	−9.31907	−0.392056
$566$	0	0
$567$	−2.07926	−0.0873205
$568$	0	0
$569$	−33.9243	−1.42218	−0.711090	−	0.703101i	$-0.751798\pi$
$569$	−33.9243	−1.42218	−0.711090	+	0.703101i	$0.751798\pi$
$570$	0	0
$571$	7.50760	0.314183	0.157092	−	0.987584i	$-0.449788\pi$
$571$	7.50760	0.314183	0.157092	+	0.987584i	$0.449788\pi$
$572$	0	0
$573$	7.54585	0.315232
$574$	0	0
$575$	−4.19970	−0.175140
$576$	0	0
$577$	−0.765181	−0.0318549	−0.0159274	−	0.999873i	$-0.505070\pi$
$577$	−0.765181	−0.0318549	−0.0159274	+	0.999873i	$0.505070\pi$
$578$	0	0
$579$	−11.5660	−0.480668
$580$	0	0
$581$	−7.41858	−0.307775
$582$	0	0
$583$	−20.5627	−0.851622
$584$	0	0
$585$	11.8137	0.488437
$586$	0	0
$587$	−30.1637	−1.24499	−0.622494	−	0.782624i	$-0.713881\pi$
$587$	−30.1637	−1.24499	−0.622494	+	0.782624i	$0.713881\pi$
$588$	0	0
$589$	−17.8064	−0.733701
$590$	0	0
$591$	8.50436	0.349823
$592$	0	0
$593$	29.9808	1.23116	0.615581	−	0.788074i	$-0.288921\pi$
$593$	29.9808	1.23116	0.615581	+	0.788074i	$0.288921\pi$
$594$	0	0
$595$	−25.7695	−1.05645
$596$	0	0
$597$	−5.92204	−0.242373
$598$	0	0
$599$	−13.0064	−0.531428	−0.265714	−	0.964052i	$-0.585608\pi$
$599$	−13.0064	−0.531428	−0.265714	+	0.964052i	$0.585608\pi$
$600$	0	0
$601$	−18.3763	−0.749587	−0.374793	−	0.927108i	$-0.622286\pi$
$601$	−18.3763	−0.749587	−0.374793	+	0.927108i	$0.622286\pi$
$602$	0	0
$603$	−8.71231	−0.354793
$604$	0	0
$605$	10.9723	0.446086
$606$	0	0
$607$	−9.30961	−0.377866	−0.188933	−	0.981990i	$-0.560503\pi$
$607$	−9.30961	−0.377866	−0.188933	+	0.981990i	$0.560503\pi$
$608$	0	0
$609$	−3.69085	−0.149561
$610$	0	0
$611$	−44.3926	−1.79593
$612$	0	0
$613$	41.3076	1.66840	0.834200	−	0.551462i	$-0.185930\pi$
$613$	41.3076	1.66840	0.834200	+	0.551462i	$0.185930\pi$
$614$	0	0
$615$	5.54822	0.223726
$616$	0	0
$617$	37.1629	1.49612	0.748062	−	0.663629i	$-0.230985\pi$
$617$	37.1629	1.49612	0.748062	+	0.663629i	$0.230985\pi$
$618$	0	0
$619$	1.50754	0.0605932	0.0302966	−	0.999541i	$-0.490355\pi$
$619$	1.50754	0.0605932	0.0302966	+	0.999541i	$0.490355\pi$
$620$	0	0
$621$	−2.31931	−0.0930706
$622$	0	0
$623$	−22.8879	−0.916983
$624$	0	0
$625$	−12.6674	−0.506695
$626$	0	0
$627$	7.50975	0.299911
$628$	0	0
$629$	75.1935	2.99816
$630$	0	0
$631$	−31.0906	−1.23770	−0.618849	−	0.785510i	$-0.712401\pi$
$631$	−31.0906	−1.23770	−0.618849	+	0.785510i	$0.712401\pi$
$632$	0	0
$633$	2.66403	0.105886
$634$	0	0
$635$	25.7050	1.02007
$636$	0	0
$637$	17.7068	0.701570
$638$	0	0
$639$	−10.1955	−0.403326
$640$	0	0
$641$	−6.14338	−0.242649	−0.121324	−	0.992613i	$-0.538714\pi$
$641$	−6.14338	−0.242649	−0.121324	+	0.992613i	$0.538714\pi$
$642$	0	0
$643$	24.2097	0.954739	0.477369	−	0.878703i	$-0.341590\pi$
$643$	24.2097	0.954739	0.477369	+	0.878703i	$0.341590\pi$
$644$	0	0
$645$	−7.06457	−0.278167
$646$	0	0
$647$	35.2774	1.38690	0.693449	−	0.720506i	$-0.256090\pi$
$647$	35.2774	1.38690	0.693449	+	0.720506i	$0.256090\pi$
$648$	0	0
$649$	−18.8532	−0.740052
$650$	0	0
$651$	10.8642	0.425801
$652$	0	0
$653$	−37.6040	−1.47156	−0.735780	−	0.677221i	$-0.763184\pi$
$653$	−37.6040	−1.47156	−0.735780	+	0.677221i	$0.763184\pi$
$654$	0	0
$655$	18.7661	0.733253
$656$	0	0
$657$	−9.60865	−0.374869
$658$	0	0
$659$	−45.2872	−1.76414	−0.882070	−	0.471119i	$-0.843850\pi$
$659$	−45.2872	−1.76414	−0.882070	+	0.471119i	$0.843850\pi$
$660$	0	0
$661$	43.3602	1.68652	0.843258	−	0.537510i	$-0.180635\pi$
$661$	43.3602	1.68652	0.843258	+	0.537510i	$0.180635\pi$
$662$	0	0
$663$	−45.9089	−1.78295
$664$	0	0
$665$	−12.6543	−0.490714
$666$	0	0
$667$	−4.11695	−0.159409
$668$	0	0
$669$	−11.6207	−0.449282
$670$	0	0
$671$	19.3647	0.747565
$672$	0	0
$673$	−34.4056	−1.32624	−0.663119	−	0.748514i	$-0.730768\pi$
$673$	−34.4056	−1.32624	−0.663119	+	0.748514i	$0.730768\pi$
$674$	0	0
$675$	1.81076	0.0696961
$676$	0	0
$677$	−24.1246	−0.927184	−0.463592	−	0.886049i	$-0.653440\pi$
$677$	−24.1246	−0.927184	−0.463592	+	0.886049i	$0.653440\pi$
$678$	0	0
$679$	−27.3979	−1.05143
$680$	0	0
$681$	5.73829	0.219892
$682$	0	0
$683$	−29.6435	−1.13428	−0.567139	−	0.823622i	$-0.691950\pi$
$683$	−29.6435	−1.13428	−0.567139	+	0.823622i	$0.691950\pi$
$684$	0	0
$685$	20.6054	0.787290
$686$	0	0
$687$	2.94839	0.112488
$688$	0	0
$689$	61.7284	2.35166
$690$	0	0
$691$	−34.9279	−1.32872	−0.664359	−	0.747414i	$-0.731295\pi$
$691$	−34.9279	−1.32872	−0.664359	+	0.747414i	$0.731295\pi$
$692$	0	0
$693$	−4.58191	−0.174052
$694$	0	0
$695$	−27.1806	−1.03102
$696$	0	0
$697$	−21.5608	−0.816673
$698$	0	0
$699$	−16.0655	−0.607653
$700$	0	0
$701$	34.2776	1.29465	0.647323	−	0.762216i	$-0.275889\pi$
$701$	34.2776	1.29465	0.647323	+	0.762216i	$0.275889\pi$
$702$	0	0
$703$	36.9244	1.39263
$704$	0	0
$705$	11.9843	0.451354
$706$	0	0
$707$	−2.52267	−0.0948747
$708$	0	0
$709$	−28.3856	−1.06605	−0.533023	−	0.846101i	$-0.678944\pi$
$709$	−28.3856	−1.06605	−0.533023	+	0.846101i	$0.678944\pi$
$710$	0	0
$711$	12.9198	0.484529
$712$	0	0
$713$	12.1185	0.453840
$714$	0	0
$715$	26.0330	0.973579
$716$	0	0
$717$	17.3794	0.649047
$718$	0	0
$719$	26.1303	0.974495	0.487247	−	0.873264i	$-0.338001\pi$
$719$	26.1303	0.974495	0.487247	+	0.873264i	$0.338001\pi$
$720$	0	0
$721$	−32.9923	−1.22870
$722$	0	0
$723$	20.3416	0.756510
$724$	0	0
$725$	3.21424	0.119374
$726$	0	0
$727$	24.3997	0.904935	0.452467	−	0.891781i	$-0.350544\pi$
$727$	24.3997	0.904935	0.452467	+	0.891781i	$0.350544\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	27.4534	1.01540
$732$	0	0
$733$	−3.49438	−0.129068	−0.0645340	−	0.997916i	$-0.520556\pi$
$733$	−3.49438	−0.129068	−0.0645340	+	0.997916i	$0.520556\pi$
$734$	0	0
$735$	−4.78016	−0.176319
$736$	0	0
$737$	−19.1987	−0.707193
$738$	0	0
$739$	19.0377	0.700312	0.350156	−	0.936691i	$-0.386129\pi$
$739$	19.0377	0.700312	0.350156	+	0.936691i	$0.386129\pi$
$740$	0	0
$741$	−22.5439	−0.828172
$742$	0	0
$743$	23.2950	0.854611	0.427306	−	0.904107i	$-0.359463\pi$
$743$	23.2950	0.854611	0.427306	+	0.904107i	$0.359463\pi$
$744$	0	0
$745$	−21.4926	−0.787429
$746$	0	0
$747$	3.56790	0.130543
$748$	0	0
$749$	36.7252	1.34191
$750$	0	0
$751$	6.79988	0.248131	0.124066	−	0.992274i	$-0.460407\pi$
$751$	6.79988	0.248131	0.124066	+	0.992274i	$0.460407\pi$
$752$	0	0
$753$	23.3382	0.850492
$754$	0	0
$755$	28.4301	1.03468
$756$	0	0
$757$	−13.7885	−0.501151	−0.250576	−	0.968097i	$-0.580620\pi$
$757$	−13.7885	−0.501151	−0.250576	+	0.968097i	$0.580620\pi$
$758$	0	0
$759$	−5.11089	−0.185514
$760$	0	0
$761$	−34.3678	−1.24583	−0.622916	−	0.782288i	$-0.714052\pi$
$761$	−34.3678	−1.24583	−0.622916	+	0.782288i	$0.714052\pi$
$762$	0	0
$763$	31.7068	1.14786
$764$	0	0
$765$	12.3936	0.448093
$766$	0	0
$767$	56.5964	2.04358
$768$	0	0
$769$	−5.34192	−0.192635	−0.0963173	−	0.995351i	$-0.530706\pi$
$769$	−5.34192	−0.192635	−0.0963173	+	0.995351i	$0.530706\pi$
$770$	0	0
$771$	−2.27427	−0.0819059
$772$	0	0
$773$	−11.7999	−0.424413	−0.212206	−	0.977225i	$-0.568065\pi$
$773$	−11.7999	−0.424413	−0.212206	+	0.977225i	$0.568065\pi$
$774$	0	0
$775$	−9.46127	−0.339859
$776$	0	0
$777$	−22.5286	−0.808208
$778$	0	0
$779$	−10.5876	−0.379340
$780$	0	0
$781$	−22.4670	−0.803932
$782$	0	0
$783$	1.77508	0.0634362
$784$	0	0
$785$	1.93169	0.0689449
$786$	0	0
$787$	−13.7088	−0.488667	−0.244333	−	0.969691i	$-0.578569\pi$
$787$	−13.7088	−0.488667	−0.244333	+	0.969691i	$0.578569\pi$
$788$	0	0
$789$	−14.3925	−0.512385
$790$	0	0
$791$	−10.8502	−0.385788
$792$	0	0
$793$	−58.1318	−2.06432
$794$	0	0
$795$	−16.6643	−0.591021
$796$	0	0
$797$	−26.0602	−0.923099	−0.461549	−	0.887115i	$-0.652706\pi$
$797$	−26.0602	−0.923099	−0.461549	+	0.887115i	$0.652706\pi$
$798$	0	0
$799$	−46.5718	−1.64759
$800$	0	0
$801$	11.0077	0.388939
$802$	0	0
$803$	−21.1739	−0.747211
$804$	0	0
$805$	8.61212	0.303537
$806$	0	0
$807$	−12.1315	−0.427051
$808$	0	0
$809$	5.10703	0.179554	0.0897769	−	0.995962i	$-0.471385\pi$
$809$	5.10703	0.179554	0.0897769	+	0.995962i	$0.471385\pi$
$810$	0	0
$811$	2.93796	0.103166	0.0515829	−	0.998669i	$-0.483573\pi$
$811$	2.93796	0.103166	0.0515829	+	0.998669i	$0.483573\pi$
$812$	0	0
$813$	26.5739	0.931988
$814$	0	0
$815$	−9.39902	−0.329233
$816$	0	0
$817$	13.4812	0.471648
$818$	0	0
$819$	13.7547	0.480627
$820$	0	0
$821$	30.2615	1.05613	0.528066	−	0.849203i	$-0.322917\pi$
$821$	30.2615	1.05613	0.528066	+	0.849203i	$0.322917\pi$
$822$	0	0
$823$	41.9510	1.46232	0.731160	−	0.682206i	$-0.238979\pi$
$823$	41.9510	1.46232	0.731160	+	0.682206i	$0.238979\pi$
$824$	0	0
$825$	3.99023	0.138922
$826$	0	0
$827$	14.4416	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$827$	14.4416	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$828$	0	0
$829$	−20.8232	−0.723220	−0.361610	−	0.932330i	$-0.617773\pi$
$829$	−20.8232	−0.723220	−0.361610	+	0.932330i	$0.617773\pi$
$830$	0	0
$831$	−16.6649	−0.578100
$832$	0	0
$833$	18.5761	0.643622
$834$	0	0
$835$	35.7807	1.23824
$836$	0	0
$837$	−5.22504	−0.180604
$838$	0	0
$839$	25.3513	0.875225	0.437613	−	0.899164i	$-0.355824\pi$
$839$	25.3513	0.875225	0.437613	+	0.899164i	$0.355824\pi$
$840$	0	0
$841$	−25.8491	−0.891348
$842$	0	0
$843$	−18.0299	−0.620984
$844$	0	0
$845$	−54.9339	−1.88978
$846$	0	0
$847$	12.7750	0.438954
$848$	0	0
$849$	−22.0846	−0.757943
$850$	0	0
$851$	−25.1295	−0.861428
$852$	0	0
$853$	35.1216	1.20254	0.601271	−	0.799045i	$-0.294661\pi$
$853$	35.1216	1.20254	0.601271	+	0.799045i	$0.294661\pi$
$854$	0	0
$855$	6.08599	0.208136
$856$	0	0
$857$	−50.0517	−1.70973	−0.854867	−	0.518848i	$-0.826361\pi$
$857$	−50.0517	−1.70973	−0.854867	+	0.518848i	$0.826361\pi$
$858$	0	0
$859$	−16.8713	−0.575641	−0.287820	−	0.957684i	$-0.592931\pi$
$859$	−16.8713	−0.575641	−0.287820	+	0.957684i	$0.592931\pi$
$860$	0	0
$861$	6.45978	0.220149
$862$	0	0
$863$	0.616318	0.0209797	0.0104899	−	0.999945i	$-0.496661\pi$
$863$	0.616318	0.0209797	0.0104899	+	0.999945i	$0.496661\pi$
$864$	0	0
$865$	−8.27533	−0.281370
$866$	0	0
$867$	−31.1626	−1.05834
$868$	0	0
$869$	28.4704	0.965791
$870$	0	0
$871$	57.6336	1.95284
$872$	0	0
$873$	13.1768	0.445966
$874$	0	0
$875$	−25.2899	−0.854955
$876$	0	0
$877$	5.27092	0.177986	0.0889931	−	0.996032i	$-0.471635\pi$
$877$	5.27092	0.177986	0.0889931	+	0.996032i	$0.471635\pi$
$878$	0	0
$879$	−2.24011	−0.0755571
$880$	0	0
$881$	−19.7534	−0.665508	−0.332754	−	0.943014i	$-0.607978\pi$
$881$	−19.7534	−0.665508	−0.332754	+	0.943014i	$0.607978\pi$
$882$	0	0
$883$	−27.3820	−0.921478	−0.460739	−	0.887536i	$-0.652416\pi$
$883$	−27.3820	−0.921478	−0.460739	+	0.887536i	$0.652416\pi$
$884$	0	0
$885$	−15.2788	−0.513592
$886$	0	0
$887$	22.3374	0.750017	0.375008	−	0.927021i	$-0.377640\pi$
$887$	22.3374	0.750017	0.375008	+	0.927021i	$0.377640\pi$
$888$	0	0
$889$	29.9283	1.00376
$890$	0	0
$891$	2.20363	0.0738243
$892$	0	0
$893$	−22.8694	−0.765296
$894$	0	0
$895$	−34.0246	−1.13732
$896$	0	0
$897$	15.3427	0.512276
$898$	0	0
$899$	−9.27485	−0.309334
$900$	0	0
$901$	64.7586	2.15742
$902$	0	0
$903$	−8.22526	−0.273720
$904$	0	0
$905$	−16.8870	−0.561342
$906$	0	0
$907$	−44.2716	−1.47002	−0.735008	−	0.678059i	$-0.762821\pi$
$907$	−44.2716	−1.47002	−0.735008	+	0.678059i	$0.762821\pi$
$908$	0	0
$909$	1.21326	0.0402411
$910$	0	0
$911$	−0.801055	−0.0265401	−0.0132701	−	0.999912i	$-0.504224\pi$
$911$	−0.801055	−0.0265401	−0.0132701	+	0.999912i	$0.504224\pi$
$912$	0	0
$913$	7.86233	0.260205
$914$	0	0
$915$	15.6934	0.518806
$916$	0	0
$917$	21.8493	0.721529
$918$	0	0
$919$	20.5873	0.679114	0.339557	−	0.940586i	$-0.389723\pi$
$919$	20.5873	0.679114	0.339557	+	0.940586i	$0.389723\pi$
$920$	0	0
$921$	−21.0330	−0.693059
$922$	0	0
$923$	67.4449	2.21997
$924$	0	0
$925$	19.6194	0.645082
$926$	0	0
$927$	15.8674	0.521152
$928$	0	0
$929$	−3.26242	−0.107036	−0.0535182	−	0.998567i	$-0.517044\pi$
$929$	−3.26242	−0.107036	−0.0535182	+	0.998567i	$0.517044\pi$
$930$	0	0
$931$	9.12192	0.298959
$932$	0	0
$933$	−23.7129	−0.776327
$934$	0	0
$935$	27.3110	0.893164
$936$	0	0
$937$	−14.1655	−0.462766	−0.231383	−	0.972863i	$-0.574325\pi$
$937$	−14.1655	−0.462766	−0.231383	+	0.972863i	$0.574325\pi$
$938$	0	0
$939$	−16.7181	−0.545575
$940$	0	0
$941$	−39.8614	−1.29945	−0.649723	−	0.760171i	$-0.725115\pi$
$941$	−39.8614	−1.29945	−0.649723	+	0.760171i	$0.725115\pi$
$942$	0	0
$943$	7.20556	0.234645
$944$	0	0
$945$	−3.71323	−0.120791
$946$	0	0
$947$	−6.00969	−0.195289	−0.0976444	−	0.995221i	$-0.531131\pi$
$947$	−6.00969	−0.195289	−0.0976444	+	0.995221i	$0.531131\pi$
$948$	0	0
$949$	63.5630	2.06334
$950$	0	0
$951$	8.17881	0.265216
$952$	0	0
$953$	7.59841	0.246137	0.123068	−	0.992398i	$-0.460727\pi$
$953$	7.59841	0.246137	0.123068	+	0.992398i	$0.460727\pi$
$954$	0	0
$955$	13.4757	0.436064
$956$	0	0
$957$	3.91162	0.126445
$958$	0	0
$959$	23.9908	0.774703
$960$	0	0
$961$	−3.69901	−0.119323
$962$	0	0
$963$	−17.6627	−0.569172
$964$	0	0
$965$	−20.6551	−0.664913
$966$	0	0
$967$	−7.25446	−0.233288	−0.116644	−	0.993174i	$-0.537214\pi$
$967$	−7.25446	−0.233288	−0.116644	+	0.993174i	$0.537214\pi$
$968$	0	0
$969$	−23.6506	−0.759767
$970$	0	0
$971$	−48.6297	−1.56060	−0.780301	−	0.625404i	$-0.784934\pi$
$971$	−48.6297	−1.56060	−0.780301	+	0.625404i	$0.784934\pi$
$972$	0	0
$973$	−31.6463	−1.01453
$974$	0	0
$975$	−11.9785	−0.383619
$976$	0	0
$977$	−4.78591	−0.153115	−0.0765574	−	0.997065i	$-0.524393\pi$
$977$	−4.78591	−0.153115	−0.0765574	+	0.997065i	$0.524393\pi$
$978$	0	0
$979$	24.2569	0.775254
$980$	0	0
$981$	−15.2491	−0.486867
$982$	0	0
$983$	51.9979	1.65848	0.829238	−	0.558896i	$-0.188775\pi$
$983$	51.9979	1.65848	0.829238	+	0.558896i	$0.188775\pi$
$984$	0	0
$985$	15.1875	0.483913
$986$	0	0
$987$	13.9533	0.444138
$988$	0	0
$989$	−9.17487	−0.291744
$990$	0	0
$991$	−8.61960	−0.273811	−0.136905	−	0.990584i	$-0.543716\pi$
$991$	−8.61960	−0.273811	−0.136905	+	0.990584i	$0.543716\pi$
$992$	0	0
$993$	−24.5580	−0.779325
$994$	0	0
$995$	−10.5759	−0.335277
$996$	0	0
$997$	38.7783	1.22812	0.614060	−	0.789259i	$-0.289535\pi$
$997$	38.7783	1.22812	0.614060	+	0.789259i	$0.289535\pi$
$998$	0	0
$999$	10.8349	0.342802

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.i.1.9	✓	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.i.1.9	✓	26	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.78585 q^{5} -2.07926 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.78585 q^{5} -2.07926 q^{7} +1.00000 q^{9} +2.20363 q^{11} -6.61519 q^{13} +1.78585 q^{15} -6.93993 q^{17} -3.40790 q^{19} +2.07926 q^{21} +2.31931 q^{23} -1.81076 q^{25} -1.00000 q^{27} -1.77508 q^{29} +5.22504 q^{31} -2.20363 q^{33} +3.71323 q^{35} -10.8349 q^{37} +6.61519 q^{39} +3.10677 q^{41} -3.95587 q^{43} -1.78585 q^{45} +6.71070 q^{47} -2.67669 q^{49} +6.93993 q^{51} -9.33131 q^{53} -3.93534 q^{55} +3.40790 q^{57} -8.55552 q^{59} +8.78763 q^{61} -2.07926 q^{63} +11.8137 q^{65} -8.71231 q^{67} -2.31931 q^{69} -10.1955 q^{71} -9.60865 q^{73} +1.81076 q^{75} -4.58191 q^{77} +12.9198 q^{79} +1.00000 q^{81} +3.56790 q^{83} +12.3936 q^{85} +1.77508 q^{87} +11.0077 q^{89} +13.7547 q^{91} -5.22504 q^{93} +6.08599 q^{95} +13.1768 q^{97} +2.20363 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more