LMFDB - Embedded newform 6036.2.a.f.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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} - 12 x^{12} + 112 x^{11} + 7 x^{10} - 710 x^{9} + 281 x^{8} + 1850 x^{7} - 830 x^{6} + \cdots + 1 $ x^14 - 6x^13 - 12x^12 + 112x^11 + 7x^10 - 710x^9 + 281x^8 + 1850x^7 - 830x^6 - 1918x^5 + 623x^4 + 480x^3 - 29x^2 - 17*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.05416$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -3.05416 q^{5} -1.32476 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.05416 q^{5} -1.32476 q^{7} +1.00000 q^{9} -0.565626 q^{11} +4.42367 q^{13} +3.05416 q^{15} -2.73910 q^{17} -4.40430 q^{19} +1.32476 q^{21} +3.57521 q^{23} +4.32791 q^{25} -1.00000 q^{27} +3.76223 q^{29} -1.80179 q^{31} +0.565626 q^{33} +4.04604 q^{35} -4.00778 q^{37} -4.42367 q^{39} +0.606575 q^{41} +0.758566 q^{43} -3.05416 q^{45} +4.40556 q^{47} -5.24500 q^{49} +2.73910 q^{51} +2.43817 q^{53} +1.72751 q^{55} +4.40430 q^{57} +15.0477 q^{59} +4.05674 q^{61} -1.32476 q^{63} -13.5106 q^{65} +11.9840 q^{67} -3.57521 q^{69} +15.7394 q^{71} -11.3135 q^{73} -4.32791 q^{75} +0.749320 q^{77} -4.54886 q^{79} +1.00000 q^{81} -16.9930 q^{83} +8.36566 q^{85} -3.76223 q^{87} +15.6720 q^{89} -5.86031 q^{91} +1.80179 q^{93} +13.4515 q^{95} +0.579199 q^{97} -0.565626 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9	$ 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9} + q^{11} - q^{13} + 6 q^{15} - 6 q^{17} + q^{19} + 7 q^{21} + 10 q^{23} - 10 q^{25} - 14 q^{27} - 6 q^{29} - 5 q^{31} - q^{33} + 17 q^{35} - 12 q^{37} + q^{39} - 21 q^{41} + 8 q^{43} - 6 q^{45} + 18 q^{47} - 19 q^{49} + 6 q^{51} - q^{53} - q^{57} + 14 q^{59} - 19 q^{61} - 7 q^{63} + 17 q^{65} + 17 q^{67} - 10 q^{69} - 13 q^{71} - 12 q^{73} + 10 q^{75} - 9 q^{77} - 8 q^{79} + 14 q^{81} + 11 q^{83} - 17 q^{85} + 6 q^{87} - 9 q^{89} - 5 q^{91} + 5 q^{93} + 8 q^{95} - 25 q^{97} + q^{99}+O(q^{100}) $ 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9 + q^11 - q^13 + 6 * q^15 - 6 * q^17 + q^19 + 7 * q^21 + 10 * q^23 - 10 * q^25 - 14 * q^27 - 6 * q^29 - 5 * q^31 - q^33 + 17 * q^35 - 12 * q^37 + q^39 - 21 * q^41 + 8 * q^43 - 6 * q^45 + 18 * q^47 - 19 * q^49 + 6 * q^51 - q^53 - q^57 + 14 * q^59 - 19 * q^61 - 7 * q^63 + 17 * q^65 + 17 * q^67 - 10 * q^69 - 13 * q^71 - 12 * q^73 + 10 * q^75 - 9 * q^77 - 8 * q^79 + 14 * q^81 + 11 * q^83 - 17 * q^85 + 6 * q^87 - 9 * q^89 - 5 * q^91 + 5 * q^93 + 8 * q^95 - 25 * q^97 + 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
$5$	−3.05416	−1.36586	−0.682932	−	0.730482i	$-0.739295\pi$
$5$	−3.05416	−1.36586	−0.682932	+	0.730482i	$0.739295\pi$
$6$	0	0
$7$	−1.32476	−0.500713	−0.250357	−	0.968154i	$-0.580548\pi$
$7$	−1.32476	−0.500713	−0.250357	+	0.968154i	$0.580548\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.565626	−0.170543	−0.0852713	−	0.996358i	$-0.527176\pi$
$11$	−0.565626	−0.170543	−0.0852713	+	0.996358i	$0.527176\pi$
$12$	0	0
$13$	4.42367	1.22690	0.613452	−	0.789732i	$-0.289780\pi$
$13$	4.42367	1.22690	0.613452	+	0.789732i	$0.289780\pi$
$14$	0	0
$15$	3.05416	0.788582
$16$	0	0
$17$	−2.73910	−0.664330	−0.332165	−	0.943221i	$-0.607779\pi$
$17$	−2.73910	−0.664330	−0.332165	+	0.943221i	$0.607779\pi$
$18$	0	0
$19$	−4.40430	−1.01042	−0.505208	−	0.862998i	$-0.668584\pi$
$19$	−4.40430	−1.01042	−0.505208	+	0.862998i	$0.668584\pi$
$20$	0	0
$21$	1.32476	0.289087
$22$	0	0
$23$	3.57521	0.745483	0.372741	−	0.927935i	$-0.378418\pi$
$23$	3.57521	0.745483	0.372741	+	0.927935i	$0.378418\pi$
$24$	0	0
$25$	4.32791	0.865583
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.76223	0.698628	0.349314	−	0.937006i	$-0.386415\pi$
$29$	3.76223	0.698628	0.349314	+	0.937006i	$0.386415\pi$
$30$	0	0
$31$	−1.80179	−0.323610	−0.161805	−	0.986823i	$-0.551732\pi$
$31$	−1.80179	−0.323610	−0.161805	+	0.986823i	$0.551732\pi$
$32$	0	0
$33$	0.565626	0.0984628
$34$	0	0
$35$	4.04604	0.683906
$36$	0	0
$37$	−4.00778	−0.658874	−0.329437	−	0.944177i	$-0.606859\pi$
$37$	−4.00778	−0.658874	−0.329437	+	0.944177i	$0.606859\pi$
$38$	0	0
$39$	−4.42367	−0.708353
$40$	0	0
$41$	0.606575	0.0947311	0.0473656	−	0.998878i	$-0.484917\pi$
$41$	0.606575	0.0947311	0.0473656	+	0.998878i	$0.484917\pi$
$42$	0	0
$43$	0.758566	0.115680	0.0578401	−	0.998326i	$-0.481579\pi$
$43$	0.758566	0.115680	0.0578401	+	0.998326i	$0.481579\pi$
$44$	0	0
$45$	−3.05416	−0.455288
$46$	0	0
$47$	4.40556	0.642618	0.321309	−	0.946974i	$-0.395877\pi$
$47$	4.40556	0.642618	0.321309	+	0.946974i	$0.395877\pi$
$48$	0	0
$49$	−5.24500	−0.749286
$50$	0	0
$51$	2.73910	0.383551
$52$	0	0
$53$	2.43817	0.334908	0.167454	−	0.985880i	$-0.446445\pi$
$53$	2.43817	0.334908	0.167454	+	0.985880i	$0.446445\pi$
$54$	0	0
$55$	1.72751	0.232938
$56$	0	0
$57$	4.40430	0.583364
$58$	0	0
$59$	15.0477	1.95904	0.979520	−	0.201346i	$-0.0645317\pi$
$59$	15.0477	1.95904	0.979520	+	0.201346i	$0.0645317\pi$
$60$	0	0
$61$	4.05674	0.519412	0.259706	−	0.965688i	$-0.416374\pi$
$61$	4.05674	0.519412	0.259706	+	0.965688i	$0.416374\pi$
$62$	0	0
$63$	−1.32476	−0.166904
$64$	0	0
$65$	−13.5106	−1.67578
$66$	0	0
$67$	11.9840	1.46408	0.732039	−	0.681262i	$-0.238569\pi$
$67$	11.9840	1.46408	0.732039	+	0.681262i	$0.238569\pi$
$68$	0	0
$69$	−3.57521	−0.430405
$70$	0	0
$71$	15.7394	1.86792	0.933960	−	0.357377i	$-0.116329\pi$
$71$	15.7394	1.86792	0.933960	+	0.357377i	$0.116329\pi$
$72$	0	0
$73$	−11.3135	−1.32415	−0.662075	−	0.749437i	$-0.730324\pi$
$73$	−11.3135	−1.32415	−0.662075	+	0.749437i	$0.730324\pi$
$74$	0	0
$75$	−4.32791	−0.499745
$76$	0	0
$77$	0.749320	0.0853929
$78$	0	0
$79$	−4.54886	−0.511787	−0.255894	−	0.966705i	$-0.582370\pi$
$79$	−4.54886	−0.511787	−0.255894	+	0.966705i	$0.582370\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−16.9930	−1.86523	−0.932614	−	0.360877i	$-0.882478\pi$
$83$	−16.9930	−1.86523	−0.932614	+	0.360877i	$0.882478\pi$
$84$	0	0
$85$	8.36566	0.907384
$86$	0	0
$87$	−3.76223	−0.403353
$88$	0	0
$89$	15.6720	1.66123	0.830616	−	0.556845i	$-0.187988\pi$
$89$	15.6720	1.66123	0.830616	+	0.556845i	$0.187988\pi$
$90$	0	0
$91$	−5.86031	−0.614327
$92$	0	0
$93$	1.80179	0.186837
$94$	0	0
$95$	13.4515	1.38009
$96$	0	0
$97$	0.579199	0.0588088	0.0294044	−	0.999568i	$-0.490639\pi$
$97$	0.579199	0.0588088	0.0294044	+	0.999568i	$0.490639\pi$
$98$	0	0
$99$	−0.565626	−0.0568475
$100$	0	0
$101$	−9.41763	−0.937089	−0.468545	−	0.883440i	$-0.655221\pi$
$101$	−9.41763	−0.937089	−0.468545	+	0.883440i	$0.655221\pi$
$102$	0	0
$103$	12.1791	1.20004	0.600019	−	0.799985i	$-0.295159\pi$
$103$	12.1791	1.20004	0.600019	+	0.799985i	$0.295159\pi$
$104$	0	0
$105$	−4.04604	−0.394853
$106$	0	0
$107$	11.7351	1.13448	0.567239	−	0.823553i	$-0.308012\pi$
$107$	11.7351	1.13448	0.567239	+	0.823553i	$0.308012\pi$
$108$	0	0
$109$	−8.24290	−0.789527	−0.394763	−	0.918783i	$-0.629173\pi$
$109$	−8.24290	−0.789527	−0.394763	+	0.918783i	$0.629173\pi$
$110$	0	0
$111$	4.00778	0.380401
$112$	0	0
$113$	−3.74019	−0.351847	−0.175924	−	0.984404i	$-0.556291\pi$
$113$	−3.74019	−0.351847	−0.175924	+	0.984404i	$0.556291\pi$
$114$	0	0
$115$	−10.9193	−1.01823
$116$	0	0
$117$	4.42367	0.408968
$118$	0	0
$119$	3.62866	0.332639
$120$	0	0
$121$	−10.6801	−0.970915
$122$	0	0
$123$	−0.606575	−0.0546930
$124$	0	0
$125$	2.05266	0.183595
$126$	0	0
$127$	−10.4218	−0.924786	−0.462393	−	0.886675i	$-0.653009\pi$
$127$	−10.4218	−0.924786	−0.462393	+	0.886675i	$0.653009\pi$
$128$	0	0
$129$	−0.758566	−0.0667880
$130$	0	0
$131$	17.8238	1.55727	0.778637	−	0.627474i	$-0.215911\pi$
$131$	17.8238	1.55727	0.778637	+	0.627474i	$0.215911\pi$
$132$	0	0
$133$	5.83466	0.505929
$134$	0	0
$135$	3.05416	0.262861
$136$	0	0
$137$	2.83591	0.242288	0.121144	−	0.992635i	$-0.461344\pi$
$137$	2.83591	0.242288	0.121144	+	0.992635i	$0.461344\pi$
$138$	0	0
$139$	−20.0156	−1.69771	−0.848853	−	0.528630i	$-0.822706\pi$
$139$	−20.0156	−1.69771	−0.848853	+	0.528630i	$0.822706\pi$
$140$	0	0
$141$	−4.40556	−0.371015
$142$	0	0
$143$	−2.50214	−0.209239
$144$	0	0
$145$	−11.4905	−0.954230
$146$	0	0
$147$	5.24500	0.432601
$148$	0	0
$149$	−11.2957	−0.925379	−0.462690	−	0.886520i	$-0.653116\pi$
$149$	−11.2957	−0.925379	−0.462690	+	0.886520i	$0.653116\pi$
$150$	0	0
$151$	−20.8857	−1.69966	−0.849829	−	0.527059i	$-0.823295\pi$
$151$	−20.8857	−1.69966	−0.849829	+	0.527059i	$0.823295\pi$
$152$	0	0
$153$	−2.73910	−0.221443
$154$	0	0
$155$	5.50295	0.442008
$156$	0	0
$157$	4.05870	0.323920	0.161960	−	0.986797i	$-0.448218\pi$
$157$	4.05870	0.323920	0.161960	+	0.986797i	$0.448218\pi$
$158$	0	0
$159$	−2.43817	−0.193359
$160$	0	0
$161$	−4.73630	−0.373273
$162$	0	0
$163$	−10.3356	−0.809546	−0.404773	−	0.914417i	$-0.632650\pi$
$163$	−10.3356	−0.809546	−0.404773	+	0.914417i	$0.632650\pi$
$164$	0	0
$165$	−1.72751	−0.134487
$166$	0	0
$167$	−8.44886	−0.653792	−0.326896	−	0.945060i	$-0.606003\pi$
$167$	−8.44886	−0.653792	−0.326896	+	0.945060i	$0.606003\pi$
$168$	0	0
$169$	6.56882	0.505294
$170$	0	0
$171$	−4.40430	−0.336805
$172$	0	0
$173$	2.22758	0.169359	0.0846797	−	0.996408i	$-0.473013\pi$
$173$	2.22758	0.169359	0.0846797	+	0.996408i	$0.473013\pi$
$174$	0	0
$175$	−5.73346	−0.433409
$176$	0	0
$177$	−15.0477	−1.13105
$178$	0	0
$179$	−16.4241	−1.22759	−0.613797	−	0.789464i	$-0.710359\pi$
$179$	−16.4241	−1.22759	−0.613797	+	0.789464i	$0.710359\pi$
$180$	0	0
$181$	5.98288	0.444704	0.222352	−	0.974966i	$-0.428627\pi$
$181$	5.98288	0.444704	0.222352	+	0.974966i	$0.428627\pi$
$182$	0	0
$183$	−4.05674	−0.299883
$184$	0	0
$185$	12.2404	0.899933
$186$	0	0
$187$	1.54931	0.113297
$188$	0	0
$189$	1.32476	0.0963623
$190$	0	0
$191$	−14.6461	−1.05976	−0.529878	−	0.848074i	$-0.677762\pi$
$191$	−14.6461	−1.05976	−0.529878	+	0.848074i	$0.677762\pi$
$192$	0	0
$193$	12.5397	0.902626	0.451313	−	0.892366i	$-0.350956\pi$
$193$	12.5397	0.902626	0.451313	+	0.892366i	$0.350956\pi$
$194$	0	0
$195$	13.5106	0.967514
$196$	0	0
$197$	−1.61794	−0.115273	−0.0576367	−	0.998338i	$-0.518356\pi$
$197$	−1.61794	−0.115273	−0.0576367	+	0.998338i	$0.518356\pi$
$198$	0	0
$199$	−0.746509	−0.0529186	−0.0264593	−	0.999650i	$-0.508423\pi$
$199$	−0.746509	−0.0529186	−0.0264593	+	0.999650i	$0.508423\pi$
$200$	0	0
$201$	−11.9840	−0.845286
$202$	0	0
$203$	−4.98406	−0.349812
$204$	0	0
$205$	−1.85258	−0.129390
$206$	0	0
$207$	3.57521	0.248494
$208$	0	0
$209$	2.49119	0.172319
$210$	0	0
$211$	19.2200	1.32316	0.661580	−	0.749875i	$-0.269886\pi$
$211$	19.2200	1.32316	0.661580	+	0.749875i	$0.269886\pi$
$212$	0	0
$213$	−15.7394	−1.07844
$214$	0	0
$215$	−2.31678	−0.158003
$216$	0	0
$217$	2.38694	0.162036
$218$	0	0
$219$	11.3135	0.764499
$220$	0	0
$221$	−12.1169	−0.815069
$222$	0	0
$223$	−5.21035	−0.348911	−0.174455	−	0.984665i	$-0.555817\pi$
$223$	−5.21035	−0.348911	−0.174455	+	0.984665i	$0.555817\pi$
$224$	0	0
$225$	4.32791	0.288528
$226$	0	0
$227$	−12.8112	−0.850309	−0.425155	−	0.905121i	$-0.639780\pi$
$227$	−12.8112	−0.850309	−0.425155	+	0.905121i	$0.639780\pi$
$228$	0	0
$229$	−23.9078	−1.57987	−0.789937	−	0.613189i	$-0.789887\pi$
$229$	−23.9078	−1.57987	−0.789937	+	0.613189i	$0.789887\pi$
$230$	0	0
$231$	−0.749320	−0.0493016
$232$	0	0
$233$	−16.3227	−1.06933	−0.534667	−	0.845063i	$-0.679563\pi$
$233$	−16.3227	−1.06933	−0.534667	+	0.845063i	$0.679563\pi$
$234$	0	0
$235$	−13.4553	−0.877728
$236$	0	0
$237$	4.54886	0.295480
$238$	0	0
$239$	17.8549	1.15494	0.577470	−	0.816412i	$-0.304040\pi$
$239$	17.8549	1.15494	0.577470	+	0.816412i	$0.304040\pi$
$240$	0	0
$241$	14.5829	0.939368	0.469684	−	0.882834i	$-0.344368\pi$
$241$	14.5829	0.939368	0.469684	+	0.882834i	$0.344368\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	16.0191	1.02342
$246$	0	0
$247$	−19.4832	−1.23968
$248$	0	0
$249$	16.9930	1.07689
$250$	0	0
$251$	−11.7860	−0.743924	−0.371962	−	0.928248i	$-0.621315\pi$
$251$	−11.7860	−0.743924	−0.371962	+	0.928248i	$0.621315\pi$
$252$	0	0
$253$	−2.02223	−0.127137
$254$	0	0
$255$	−8.36566	−0.523878
$256$	0	0
$257$	−27.4180	−1.71029	−0.855144	−	0.518390i	$-0.826532\pi$
$257$	−27.4180	−1.71029	−0.855144	+	0.518390i	$0.826532\pi$
$258$	0	0
$259$	5.30935	0.329907
$260$	0	0
$261$	3.76223	0.232876
$262$	0	0
$263$	18.2900	1.12781	0.563906	−	0.825839i	$-0.309298\pi$
$263$	18.2900	1.12781	0.563906	+	0.825839i	$0.309298\pi$
$264$	0	0
$265$	−7.44656	−0.457438
$266$	0	0
$267$	−15.6720	−0.959113
$268$	0	0
$269$	−5.00674	−0.305266	−0.152633	−	0.988283i	$-0.548775\pi$
$269$	−5.00674	−0.305266	−0.152633	+	0.988283i	$0.548775\pi$
$270$	0	0
$271$	−29.0367	−1.76385	−0.881926	−	0.471387i	$-0.843753\pi$
$271$	−29.0367	−1.76385	−0.881926	+	0.471387i	$0.843753\pi$
$272$	0	0
$273$	5.86031	0.354682
$274$	0	0
$275$	−2.44798	−0.147619
$276$	0	0
$277$	22.3510	1.34294	0.671470	−	0.741032i	$-0.265663\pi$
$277$	22.3510	1.34294	0.671470	+	0.741032i	$0.265663\pi$
$278$	0	0
$279$	−1.80179	−0.107870
$280$	0	0
$281$	3.09947	0.184899	0.0924493	−	0.995717i	$-0.470530\pi$
$281$	3.09947	0.184899	0.0924493	+	0.995717i	$0.470530\pi$
$282$	0	0
$283$	−30.5508	−1.81606	−0.908028	−	0.418910i	$-0.862412\pi$
$283$	−30.5508	−1.81606	−0.908028	+	0.418910i	$0.862412\pi$
$284$	0	0
$285$	−13.4515	−0.796796
$286$	0	0
$287$	−0.803568	−0.0474331
$288$	0	0
$289$	−9.49732	−0.558666
$290$	0	0
$291$	−0.579199	−0.0339533
$292$	0	0
$293$	−21.3986	−1.25012	−0.625060	−	0.780577i	$-0.714925\pi$
$293$	−21.3986	−1.25012	−0.625060	+	0.780577i	$0.714925\pi$
$294$	0	0
$295$	−45.9581	−2.67578
$296$	0	0
$297$	0.565626	0.0328209
$298$	0	0
$299$	15.8155	0.914636
$300$	0	0
$301$	−1.00492	−0.0579226
$302$	0	0
$303$	9.41763	0.541029
$304$	0	0
$305$	−12.3899	−0.709446
$306$	0	0
$307$	27.5431	1.57197	0.785985	−	0.618245i	$-0.212156\pi$
$307$	27.5431	1.57197	0.785985	+	0.618245i	$0.212156\pi$
$308$	0	0
$309$	−12.1791	−0.692843
$310$	0	0
$311$	−31.8449	−1.80576	−0.902881	−	0.429891i	$-0.858552\pi$
$311$	−31.8449	−1.80576	−0.902881	+	0.429891i	$0.858552\pi$
$312$	0	0
$313$	−23.1536	−1.30872	−0.654359	−	0.756184i	$-0.727061\pi$
$313$	−23.1536	−1.30872	−0.654359	+	0.756184i	$0.727061\pi$
$314$	0	0
$315$	4.04604	0.227969
$316$	0	0
$317$	−9.82803	−0.551997	−0.275999	−	0.961158i	$-0.589009\pi$
$317$	−9.82803	−0.551997	−0.275999	+	0.961158i	$0.589009\pi$
$318$	0	0
$319$	−2.12801	−0.119146
$320$	0	0
$321$	−11.7351	−0.654991
$322$	0	0
$323$	12.0638	0.671250
$324$	0	0
$325$	19.1452	1.06199
$326$	0	0
$327$	8.24290	0.455833
$328$	0	0
$329$	−5.83633	−0.321767
$330$	0	0
$331$	−31.7588	−1.74562	−0.872811	−	0.488059i	$-0.837705\pi$
$331$	−31.7588	−1.74562	−0.872811	+	0.488059i	$0.837705\pi$
$332$	0	0
$333$	−4.00778	−0.219625
$334$	0	0
$335$	−36.6011	−1.99973
$336$	0	0
$337$	−15.0813	−0.821531	−0.410765	−	0.911741i	$-0.634738\pi$
$337$	−15.0813	−0.821531	−0.410765	+	0.911741i	$0.634738\pi$
$338$	0	0
$339$	3.74019	0.203139
$340$	0	0
$341$	1.01914	0.0551894
$342$	0	0
$343$	16.2217	0.875891
$344$	0	0
$345$	10.9193	0.587874
$346$	0	0
$347$	3.94561	0.211811	0.105906	−	0.994376i	$-0.466226\pi$
$347$	3.94561	0.211811	0.105906	+	0.994376i	$0.466226\pi$
$348$	0	0
$349$	−26.0682	−1.39540	−0.697698	−	0.716392i	$-0.745792\pi$
$349$	−26.0682	−1.39540	−0.697698	+	0.716392i	$0.745792\pi$
$350$	0	0
$351$	−4.42367	−0.236118
$352$	0	0
$353$	−35.5313	−1.89114	−0.945570	−	0.325419i	$-0.894495\pi$
$353$	−35.5313	−1.89114	−0.945570	+	0.325419i	$0.894495\pi$
$354$	0	0
$355$	−48.0706	−2.55132
$356$	0	0
$357$	−3.62866	−0.192049
$358$	0	0
$359$	27.9076	1.47291	0.736453	−	0.676489i	$-0.236499\pi$
$359$	27.9076	1.47291	0.736453	+	0.676489i	$0.236499\pi$
$360$	0	0
$361$	0.397881	0.0209411
$362$	0	0
$363$	10.6801	0.560558
$364$	0	0
$365$	34.5534	1.80861
$366$	0	0
$367$	19.6979	1.02822	0.514112	−	0.857723i	$-0.328122\pi$
$367$	19.6979	1.02822	0.514112	+	0.857723i	$0.328122\pi$
$368$	0	0
$369$	0.606575	0.0315770
$370$	0	0
$371$	−3.22999	−0.167693
$372$	0	0
$373$	8.76633	0.453904	0.226952	−	0.973906i	$-0.427124\pi$
$373$	8.76633	0.453904	0.226952	+	0.973906i	$0.427124\pi$
$374$	0	0
$375$	−2.05266	−0.105999
$376$	0	0
$377$	16.6428	0.857149
$378$	0	0
$379$	35.9454	1.84639	0.923196	−	0.384330i	$-0.125567\pi$
$379$	35.9454	1.84639	0.923196	+	0.384330i	$0.125567\pi$
$380$	0	0
$381$	10.4218	0.533925
$382$	0	0
$383$	7.47481	0.381945	0.190973	−	0.981595i	$-0.438836\pi$
$383$	7.47481	0.381945	0.190973	+	0.981595i	$0.438836\pi$
$384$	0	0
$385$	−2.28855	−0.116635
$386$	0	0
$387$	0.758566	0.0385601
$388$	0	0
$389$	−7.79495	−0.395220	−0.197610	−	0.980281i	$-0.563318\pi$
$389$	−7.79495	−0.395220	−0.197610	+	0.980281i	$0.563318\pi$
$390$	0	0
$391$	−9.79286	−0.495246
$392$	0	0
$393$	−17.8238	−0.899093
$394$	0	0
$395$	13.8930	0.699031
$396$	0	0
$397$	32.1264	1.61238	0.806190	−	0.591657i	$-0.201526\pi$
$397$	32.1264	1.61238	0.806190	+	0.591657i	$0.201526\pi$
$398$	0	0
$399$	−5.83466	−0.292098
$400$	0	0
$401$	−29.7828	−1.48728	−0.743640	−	0.668580i	$-0.766902\pi$
$401$	−29.7828	−1.48728	−0.743640	+	0.668580i	$0.766902\pi$
$402$	0	0
$403$	−7.97050	−0.397039
$404$	0	0
$405$	−3.05416	−0.151763
$406$	0	0
$407$	2.26690	0.112366
$408$	0	0
$409$	6.96374	0.344335	0.172167	−	0.985068i	$-0.444923\pi$
$409$	6.96374	0.344335	0.172167	+	0.985068i	$0.444923\pi$
$410$	0	0
$411$	−2.83591	−0.139885
$412$	0	0
$413$	−19.9346	−0.980917
$414$	0	0
$415$	51.8995	2.54765
$416$	0	0
$417$	20.0156	0.980171
$418$	0	0
$419$	33.3736	1.63041	0.815204	−	0.579174i	$-0.196625\pi$
$419$	33.3736	1.63041	0.815204	+	0.579174i	$0.196625\pi$
$420$	0	0
$421$	−5.24399	−0.255576	−0.127788	−	0.991801i	$-0.540788\pi$
$421$	−5.24399	−0.255576	−0.127788	+	0.991801i	$0.540788\pi$
$422$	0	0
$423$	4.40556	0.214206
$424$	0	0
$425$	−11.8546	−0.575033
$426$	0	0
$427$	−5.37421	−0.260076
$428$	0	0
$429$	2.50214	0.120804
$430$	0	0
$431$	−9.77713	−0.470948	−0.235474	−	0.971881i	$-0.575664\pi$
$431$	−9.77713	−0.470948	−0.235474	+	0.971881i	$0.575664\pi$
$432$	0	0
$433$	−24.3712	−1.17121	−0.585603	−	0.810598i	$-0.699142\pi$
$433$	−24.3712	−1.17121	−0.585603	+	0.810598i	$0.699142\pi$
$434$	0	0
$435$	11.4905	0.550925
$436$	0	0
$437$	−15.7463	−0.753248
$438$	0	0
$439$	−6.53108	−0.311711	−0.155856	−	0.987780i	$-0.549813\pi$
$439$	−6.53108	−0.311711	−0.155856	+	0.987780i	$0.549813\pi$
$440$	0	0
$441$	−5.24500	−0.249762
$442$	0	0
$443$	20.1682	0.958222	0.479111	−	0.877754i	$-0.340959\pi$
$443$	20.1682	0.958222	0.479111	+	0.877754i	$0.340959\pi$
$444$	0	0
$445$	−47.8650	−2.26902
$446$	0	0
$447$	11.2957	0.534268
$448$	0	0
$449$	−10.3010	−0.486134	−0.243067	−	0.970009i	$-0.578154\pi$
$449$	−10.3010	−0.486134	−0.243067	+	0.970009i	$0.578154\pi$
$450$	0	0
$451$	−0.343095	−0.0161557
$452$	0	0
$453$	20.8857	0.981297
$454$	0	0
$455$	17.8983	0.839087
$456$	0	0
$457$	8.33489	0.389889	0.194945	−	0.980814i	$-0.437547\pi$
$457$	8.33489	0.389889	0.194945	+	0.980814i	$0.437547\pi$
$458$	0	0
$459$	2.73910	0.127850
$460$	0	0
$461$	40.7387	1.89739	0.948695	−	0.316192i	$-0.102404\pi$
$461$	40.7387	1.89739	0.948695	+	0.316192i	$0.102404\pi$
$462$	0	0
$463$	−5.06803	−0.235531	−0.117766	−	0.993041i	$-0.537573\pi$
$463$	−5.06803	−0.235531	−0.117766	+	0.993041i	$0.537573\pi$
$464$	0	0
$465$	−5.50295	−0.255193
$466$	0	0
$467$	10.3827	0.480456	0.240228	−	0.970717i	$-0.422778\pi$
$467$	10.3827	0.480456	0.240228	+	0.970717i	$0.422778\pi$
$468$	0	0
$469$	−15.8760	−0.733084
$470$	0	0
$471$	−4.05870	−0.187015
$472$	0	0
$473$	−0.429065	−0.0197284
$474$	0	0
$475$	−19.0614	−0.874599
$476$	0	0
$477$	2.43817	0.111636
$478$	0	0
$479$	30.9416	1.41376	0.706879	−	0.707335i	$-0.250103\pi$
$479$	30.9416	1.41376	0.706879	+	0.707335i	$0.250103\pi$
$480$	0	0
$481$	−17.7291	−0.808376
$482$	0	0
$483$	4.73630	0.215509
$484$	0	0
$485$	−1.76897	−0.0803247
$486$	0	0
$487$	−13.2054	−0.598395	−0.299198	−	0.954191i	$-0.596719\pi$
$487$	−13.2054	−0.598395	−0.299198	+	0.954191i	$0.596719\pi$
$488$	0	0
$489$	10.3356	0.467392
$490$	0	0
$491$	11.7250	0.529140	0.264570	−	0.964366i	$-0.414770\pi$
$491$	11.7250	0.529140	0.264570	+	0.964366i	$0.414770\pi$
$492$	0	0
$493$	−10.3051	−0.464119
$494$	0	0
$495$	1.72751	0.0776460
$496$	0	0
$497$	−20.8509	−0.935292
$498$	0	0
$499$	39.3071	1.75963	0.879815	−	0.475317i	$-0.157667\pi$
$499$	39.3071	1.75963	0.879815	+	0.475317i	$0.157667\pi$
$500$	0	0
$501$	8.44886	0.377467
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	28.7630	1.27994
$506$	0	0
$507$	−6.56882	−0.291731
$508$	0	0
$509$	−7.10182	−0.314783	−0.157391	−	0.987536i	$-0.550308\pi$
$509$	−7.10182	−0.314783	−0.157391	+	0.987536i	$0.550308\pi$
$510$	0	0
$511$	14.9878	0.663020
$512$	0	0
$513$	4.40430	0.194455
$514$	0	0
$515$	−37.1969	−1.63909
$516$	0	0
$517$	−2.49190	−0.109594
$518$	0	0
$519$	−2.22758	−0.0977797
$520$	0	0
$521$	−15.6224	−0.684428	−0.342214	−	0.939622i	$-0.611177\pi$
$521$	−15.6224	−0.684428	−0.342214	+	0.939622i	$0.611177\pi$
$522$	0	0
$523$	−18.8273	−0.823260	−0.411630	−	0.911351i	$-0.635040\pi$
$523$	−18.8273	−0.823260	−0.411630	+	0.911351i	$0.635040\pi$
$524$	0	0
$525$	5.73346	0.250229
$526$	0	0
$527$	4.93528	0.214984
$528$	0	0
$529$	−10.2179	−0.444255
$530$	0	0
$531$	15.0477	0.653013
$532$	0	0
$533$	2.68329	0.116226
$534$	0	0
$535$	−35.8410	−1.54954
$536$	0	0
$537$	16.4241	0.708751
$538$	0	0
$539$	2.96671	0.127785
$540$	0	0
$541$	25.2437	1.08531	0.542655	−	0.839956i	$-0.317419\pi$
$541$	25.2437	1.08531	0.542655	+	0.839956i	$0.317419\pi$
$542$	0	0
$543$	−5.98288	−0.256750
$544$	0	0
$545$	25.1752	1.07839
$546$	0	0
$547$	9.46793	0.404820	0.202410	−	0.979301i	$-0.435123\pi$
$547$	9.46793	0.404820	0.202410	+	0.979301i	$0.435123\pi$
$548$	0	0
$549$	4.05674	0.173137
$550$	0	0
$551$	−16.5700	−0.705905
$552$	0	0
$553$	6.02616	0.256259
$554$	0	0
$555$	−12.2404	−0.519576
$556$	0	0
$557$	35.0377	1.48459	0.742297	−	0.670071i	$-0.233736\pi$
$557$	35.0377	1.48459	0.742297	+	0.670071i	$0.233736\pi$
$558$	0	0
$559$	3.35564	0.141929
$560$	0	0
$561$	−1.54931	−0.0654118
$562$	0	0
$563$	−37.6899	−1.58844	−0.794219	−	0.607632i	$-0.792120\pi$
$563$	−37.6899	−1.58844	−0.794219	+	0.607632i	$0.792120\pi$
$564$	0	0
$565$	11.4231	0.480576
$566$	0	0
$567$	−1.32476	−0.0556348
$568$	0	0
$569$	33.0847	1.38698	0.693491	−	0.720465i	$-0.256072\pi$
$569$	33.0847	1.38698	0.693491	+	0.720465i	$0.256072\pi$
$570$	0	0
$571$	45.1767	1.89059	0.945294	−	0.326221i	$-0.105775\pi$
$571$	45.1767	1.89059	0.945294	+	0.326221i	$0.105775\pi$
$572$	0	0
$573$	14.6461	0.611850
$574$	0	0
$575$	15.4732	0.645277
$576$	0	0
$577$	−18.6736	−0.777393	−0.388697	−	0.921366i	$-0.627075\pi$
$577$	−18.6736	−0.777393	−0.388697	+	0.921366i	$0.627075\pi$
$578$	0	0
$579$	−12.5397	−0.521131
$580$	0	0
$581$	22.5117	0.933944
$582$	0	0
$583$	−1.37909	−0.0571161
$584$	0	0
$585$	−13.5106	−0.558594
$586$	0	0
$587$	8.21966	0.339261	0.169631	−	0.985508i	$-0.445743\pi$
$587$	8.21966	0.339261	0.169631	+	0.985508i	$0.445743\pi$
$588$	0	0
$589$	7.93561	0.326981
$590$	0	0
$591$	1.61794	0.0665531
$592$	0	0
$593$	−21.1344	−0.867885	−0.433943	−	0.900940i	$-0.642878\pi$
$593$	−21.1344	−0.867885	−0.433943	+	0.900940i	$0.642878\pi$
$594$	0	0
$595$	−11.0825	−0.454339
$596$	0	0
$597$	0.746509	0.0305526
$598$	0	0
$599$	−1.33446	−0.0545245	−0.0272622	−	0.999628i	$-0.508679\pi$
$599$	−1.33446	−0.0545245	−0.0272622	+	0.999628i	$0.508679\pi$
$600$	0	0
$601$	−15.1244	−0.616936	−0.308468	−	0.951235i	$-0.599816\pi$
$601$	−15.1244	−0.616936	−0.308468	+	0.951235i	$0.599816\pi$
$602$	0	0
$603$	11.9840	0.488026
$604$	0	0
$605$	32.6187	1.32614
$606$	0	0
$607$	−45.9119	−1.86351	−0.931753	−	0.363094i	$-0.881720\pi$
$607$	−45.9119	−1.86351	−0.931753	+	0.363094i	$0.881720\pi$
$608$	0	0
$609$	4.98406	0.201964
$610$	0	0
$611$	19.4887	0.788430
$612$	0	0
$613$	20.9892	0.847744	0.423872	−	0.905722i	$-0.360671\pi$
$613$	20.9892	0.847744	0.423872	+	0.905722i	$0.360671\pi$
$614$	0	0
$615$	1.85258	0.0747032
$616$	0	0
$617$	9.72108	0.391356	0.195678	−	0.980668i	$-0.437309\pi$
$617$	9.72108	0.391356	0.195678	+	0.980668i	$0.437309\pi$
$618$	0	0
$619$	14.5532	0.584941	0.292470	−	0.956275i	$-0.405523\pi$
$619$	14.5532	0.584941	0.292470	+	0.956275i	$0.405523\pi$
$620$	0	0
$621$	−3.57521	−0.143468
$622$	0	0
$623$	−20.7617	−0.831801
$624$	0	0
$625$	−27.9087	−1.11635
$626$	0	0
$627$	−2.49119	−0.0994885
$628$	0	0
$629$	10.9777	0.437710
$630$	0	0
$631$	−8.89502	−0.354105	−0.177053	−	0.984201i	$-0.556656\pi$
$631$	−8.89502	−0.354105	−0.177053	+	0.984201i	$0.556656\pi$
$632$	0	0
$633$	−19.2200	−0.763926
$634$	0	0
$635$	31.8299	1.26313
$636$	0	0
$637$	−23.2021	−0.919302
$638$	0	0
$639$	15.7394	0.622640
$640$	0	0
$641$	−17.1953	−0.679174	−0.339587	−	0.940575i	$-0.610287\pi$
$641$	−17.1953	−0.679174	−0.339587	+	0.940575i	$0.610287\pi$
$642$	0	0
$643$	29.4212	1.16026	0.580130	−	0.814524i	$-0.303002\pi$
$643$	29.4212	1.16026	0.580130	+	0.814524i	$0.303002\pi$
$644$	0	0
$645$	2.31678	0.0912233
$646$	0	0
$647$	−40.9292	−1.60909	−0.804546	−	0.593890i	$-0.797591\pi$
$647$	−40.9292	−1.60909	−0.804546	+	0.593890i	$0.797591\pi$
$648$	0	0
$649$	−8.51135	−0.334100
$650$	0	0
$651$	−2.38694	−0.0935515
$652$	0	0
$653$	−27.4902	−1.07577	−0.537887	−	0.843017i	$-0.680777\pi$
$653$	−27.4902	−1.07577	−0.537887	+	0.843017i	$0.680777\pi$
$654$	0	0
$655$	−54.4369	−2.12702
$656$	0	0
$657$	−11.3135	−0.441383
$658$	0	0
$659$	11.0575	0.430738	0.215369	−	0.976533i	$-0.430905\pi$
$659$	11.0575	0.430738	0.215369	+	0.976533i	$0.430905\pi$
$660$	0	0
$661$	−17.8396	−0.693881	−0.346940	−	0.937887i	$-0.612779\pi$
$661$	−17.8396	−0.693881	−0.346940	+	0.937887i	$0.612779\pi$
$662$	0	0
$663$	12.1169	0.470580
$664$	0	0
$665$	−17.8200	−0.691030
$666$	0	0
$667$	13.4507	0.520815
$668$	0	0
$669$	5.21035	0.201444
$670$	0	0
$671$	−2.29460	−0.0885819
$672$	0	0
$673$	39.5441	1.52431	0.762157	−	0.647392i	$-0.224140\pi$
$673$	39.5441	1.52431	0.762157	+	0.647392i	$0.224140\pi$
$674$	0	0
$675$	−4.32791	−0.166582
$676$	0	0
$677$	−18.9130	−0.726887	−0.363443	−	0.931616i	$-0.618399\pi$
$677$	−18.9130	−0.726887	−0.363443	+	0.931616i	$0.618399\pi$
$678$	0	0
$679$	−0.767301	−0.0294463
$680$	0	0
$681$	12.8112	0.490926
$682$	0	0
$683$	−42.2020	−1.61481	−0.807407	−	0.589994i	$-0.799130\pi$
$683$	−42.2020	−1.61481	−0.807407	+	0.589994i	$0.799130\pi$
$684$	0	0
$685$	−8.66133	−0.330932
$686$	0	0
$687$	23.9078	0.912140
$688$	0	0
$689$	10.7856	0.410900
$690$	0	0
$691$	4.63635	0.176375	0.0881876	−	0.996104i	$-0.471893\pi$
$691$	4.63635	0.176375	0.0881876	+	0.996104i	$0.471893\pi$
$692$	0	0
$693$	0.749320	0.0284643
$694$	0	0
$695$	61.1311	2.31883
$696$	0	0
$697$	−1.66147	−0.0629327
$698$	0	0
$699$	16.3227	0.617380
$700$	0	0
$701$	−39.3068	−1.48460	−0.742299	−	0.670068i	$-0.766265\pi$
$701$	−39.3068	−1.48460	−0.742299	+	0.670068i	$0.766265\pi$
$702$	0	0
$703$	17.6515	0.665738
$704$	0	0
$705$	13.4553	0.506756
$706$	0	0
$707$	12.4761	0.469213
$708$	0	0
$709$	25.9339	0.973970	0.486985	−	0.873410i	$-0.338097\pi$
$709$	25.9339	0.973970	0.486985	+	0.873410i	$0.338097\pi$
$710$	0	0
$711$	−4.54886	−0.170596
$712$	0	0
$713$	−6.44176	−0.241246
$714$	0	0
$715$	7.64194	0.285793
$716$	0	0
$717$	−17.8549	−0.666805
$718$	0	0
$719$	−51.4383	−1.91832	−0.959162	−	0.282858i	$-0.908718\pi$
$719$	−51.4383	−1.91832	−0.959162	+	0.282858i	$0.908718\pi$
$720$	0	0
$721$	−16.1344	−0.600875
$722$	0	0
$723$	−14.5829	−0.542345
$724$	0	0
$725$	16.2826	0.604720
$726$	0	0
$727$	10.6253	0.394072	0.197036	−	0.980396i	$-0.436868\pi$
$727$	10.6253	0.394072	0.197036	+	0.980396i	$0.436868\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.07779	−0.0768498
$732$	0	0
$733$	−19.6008	−0.723971	−0.361985	−	0.932184i	$-0.617901\pi$
$733$	−19.6008	−0.723971	−0.361985	+	0.932184i	$0.617901\pi$
$734$	0	0
$735$	−16.0191	−0.590873
$736$	0	0
$737$	−6.77846	−0.249688
$738$	0	0
$739$	−52.3020	−1.92396	−0.961980	−	0.273120i	$-0.911944\pi$
$739$	−52.3020	−1.92396	−0.961980	+	0.273120i	$0.911944\pi$
$740$	0	0
$741$	19.4832	0.715732
$742$	0	0
$743$	53.9043	1.97756	0.988778	−	0.149391i	$-0.0477313\pi$
$743$	53.9043	1.97756	0.988778	+	0.149391i	$0.0477313\pi$
$744$	0	0
$745$	34.4989	1.26394
$746$	0	0
$747$	−16.9930	−0.621742
$748$	0	0
$749$	−15.5463	−0.568048
$750$	0	0
$751$	−22.1378	−0.807820	−0.403910	−	0.914799i	$-0.632349\pi$
$751$	−22.1378	−0.807820	−0.403910	+	0.914799i	$0.632349\pi$
$752$	0	0
$753$	11.7860	0.429505
$754$	0	0
$755$	63.7885	2.32150
$756$	0	0
$757$	−31.2120	−1.13442	−0.567211	−	0.823573i	$-0.691977\pi$
$757$	−31.2120	−1.13442	−0.567211	+	0.823573i	$0.691977\pi$
$758$	0	0
$759$	2.02223	0.0734024
$760$	0	0
$761$	−32.1001	−1.16363	−0.581814	−	0.813322i	$-0.697657\pi$
$761$	−32.1001	−1.16363	−0.581814	+	0.813322i	$0.697657\pi$
$762$	0	0
$763$	10.9199	0.395326
$764$	0	0
$765$	8.36566	0.302461
$766$	0	0
$767$	66.5659	2.40355
$768$	0	0
$769$	−1.10353	−0.0397942	−0.0198971	−	0.999802i	$-0.506334\pi$
$769$	−1.10353	−0.0397942	−0.0198971	+	0.999802i	$0.506334\pi$
$770$	0	0
$771$	27.4180	0.987436
$772$	0	0
$773$	−17.3651	−0.624581	−0.312290	−	0.949987i	$-0.601096\pi$
$773$	−17.3651	−0.624581	−0.312290	+	0.949987i	$0.601096\pi$
$774$	0	0
$775$	−7.79798	−0.280112
$776$	0	0
$777$	−5.30935	−0.190472
$778$	0	0
$779$	−2.67154	−0.0957179
$780$	0	0
$781$	−8.90260	−0.318560
$782$	0	0
$783$	−3.76223	−0.134451
$784$	0	0
$785$	−12.3959	−0.442430
$786$	0	0
$787$	−2.61767	−0.0933099	−0.0466549	−	0.998911i	$-0.514856\pi$
$787$	−2.61767	−0.0933099	−0.0466549	+	0.998911i	$0.514856\pi$
$788$	0	0
$789$	−18.2900	−0.651143
$790$	0	0
$791$	4.95486	0.176175
$792$	0	0
$793$	17.9456	0.637269
$794$	0	0
$795$	7.44656	0.264102
$796$	0	0
$797$	9.01415	0.319298	0.159649	−	0.987174i	$-0.448964\pi$
$797$	9.01415	0.319298	0.159649	+	0.987174i	$0.448964\pi$
$798$	0	0
$799$	−12.0673	−0.426910
$800$	0	0
$801$	15.6720	0.553744
$802$	0	0
$803$	6.39923	0.225824
$804$	0	0
$805$	14.4654	0.509840
$806$	0	0
$807$	5.00674	0.176245
$808$	0	0
$809$	−6.16930	−0.216901	−0.108451	−	0.994102i	$-0.534589\pi$
$809$	−6.16930	−0.216901	−0.108451	+	0.994102i	$0.534589\pi$
$810$	0	0
$811$	−14.0168	−0.492197	−0.246099	−	0.969245i	$-0.579149\pi$
$811$	−14.0168	−0.492197	−0.246099	+	0.969245i	$0.579149\pi$
$812$	0	0
$813$	29.0367	1.01836
$814$	0	0
$815$	31.5666	1.10573
$816$	0	0
$817$	−3.34095	−0.116885
$818$	0	0
$819$	−5.86031	−0.204776
$820$	0	0
$821$	−31.1740	−1.08798	−0.543990	−	0.839092i	$-0.683087\pi$
$821$	−31.1740	−1.08798	−0.543990	+	0.839092i	$0.683087\pi$
$822$	0	0
$823$	−35.4949	−1.23727	−0.618637	−	0.785677i	$-0.712315\pi$
$823$	−35.4949	−1.23727	−0.618637	+	0.785677i	$0.712315\pi$
$824$	0	0
$825$	2.44798	0.0852278
$826$	0	0
$827$	26.0090	0.904422	0.452211	−	0.891911i	$-0.350635\pi$
$827$	26.0090	0.904422	0.452211	+	0.891911i	$0.350635\pi$
$828$	0	0
$829$	−55.7964	−1.93789	−0.968945	−	0.247276i	$-0.920465\pi$
$829$	−55.7964	−1.93789	−0.968945	+	0.247276i	$0.920465\pi$
$830$	0	0
$831$	−22.3510	−0.775347
$832$	0	0
$833$	14.3666	0.497773
$834$	0	0
$835$	25.8042	0.892991
$836$	0	0
$837$	1.80179	0.0622788
$838$	0	0
$839$	−54.6107	−1.88537	−0.942685	−	0.333685i	$-0.891708\pi$
$839$	−54.6107	−1.88537	−0.942685	+	0.333685i	$0.891708\pi$
$840$	0	0
$841$	−14.8457	−0.511919
$842$	0	0
$843$	−3.09947	−0.106751
$844$	0	0
$845$	−20.0622	−0.690162
$846$	0	0
$847$	14.1486	0.486150
$848$	0	0
$849$	30.5508	1.04850
$850$	0	0
$851$	−14.3286	−0.491180
$852$	0	0
$853$	3.21001	0.109909	0.0549543	−	0.998489i	$-0.482499\pi$
$853$	3.21001	0.109909	0.0549543	+	0.998489i	$0.482499\pi$
$854$	0	0
$855$	13.4515	0.460030
$856$	0	0
$857$	16.8638	0.576056	0.288028	−	0.957622i	$-0.407000\pi$
$857$	16.8638	0.576056	0.288028	+	0.957622i	$0.407000\pi$
$858$	0	0
$859$	7.80808	0.266408	0.133204	−	0.991089i	$-0.457473\pi$
$859$	7.80808	0.266408	0.133204	+	0.991089i	$0.457473\pi$
$860$	0	0
$861$	0.803568	0.0273855
$862$	0	0
$863$	−4.04215	−0.137596	−0.0687981	−	0.997631i	$-0.521916\pi$
$863$	−4.04215	−0.137596	−0.0687981	+	0.997631i	$0.521916\pi$
$864$	0	0
$865$	−6.80338	−0.231322
$866$	0	0
$867$	9.49732	0.322546
$868$	0	0
$869$	2.57296	0.0872815
$870$	0	0
$871$	53.0132	1.79628
$872$	0	0
$873$	0.579199	0.0196029
$874$	0	0
$875$	−2.71928	−0.0919286
$876$	0	0
$877$	−28.4974	−0.962290	−0.481145	−	0.876641i	$-0.659779\pi$
$877$	−28.4974	−0.962290	−0.481145	+	0.876641i	$0.659779\pi$
$878$	0	0
$879$	21.3986	0.721757
$880$	0	0
$881$	49.6719	1.67349	0.836745	−	0.547593i	$-0.184456\pi$
$881$	49.6719	1.67349	0.836745	+	0.547593i	$0.184456\pi$
$882$	0	0
$883$	−41.4835	−1.39603	−0.698015	−	0.716083i	$-0.745933\pi$
$883$	−41.4835	−1.39603	−0.698015	+	0.716083i	$0.745933\pi$
$884$	0	0
$885$	45.9581	1.54486
$886$	0	0
$887$	11.6823	0.392253	0.196126	−	0.980579i	$-0.437164\pi$
$887$	11.6823	0.392253	0.196126	+	0.980579i	$0.437164\pi$
$888$	0	0
$889$	13.8064	0.463052
$890$	0	0
$891$	−0.565626	−0.0189492
$892$	0	0
$893$	−19.4034	−0.649311
$894$	0	0
$895$	50.1618	1.67672
$896$	0	0
$897$	−15.8155	−0.528065
$898$	0	0
$899$	−6.77873	−0.226083
$900$	0	0
$901$	−6.67839	−0.222489
$902$	0	0
$903$	1.00492	0.0334416
$904$	0	0
$905$	−18.2727	−0.607405
$906$	0	0
$907$	28.2682	0.938629	0.469315	−	0.883031i	$-0.344501\pi$
$907$	28.2682	0.938629	0.469315	+	0.883031i	$0.344501\pi$
$908$	0	0
$909$	−9.41763	−0.312363
$910$	0	0
$911$	26.0884	0.864346	0.432173	−	0.901791i	$-0.357747\pi$
$911$	26.0884	0.864346	0.432173	+	0.901791i	$0.357747\pi$
$912$	0	0
$913$	9.61170	0.318101
$914$	0	0
$915$	12.3899	0.409599
$916$	0	0
$917$	−23.6123	−0.779748
$918$	0	0
$919$	8.61840	0.284295	0.142147	−	0.989845i	$-0.454599\pi$
$919$	8.61840	0.284295	0.142147	+	0.989845i	$0.454599\pi$
$920$	0	0
$921$	−27.5431	−0.907577
$922$	0	0
$923$	69.6257	2.29176
$924$	0	0
$925$	−17.3453	−0.570311
$926$	0	0
$927$	12.1791	0.400013
$928$	0	0
$929$	−2.45246	−0.0804627	−0.0402314	−	0.999190i	$-0.512809\pi$
$929$	−2.45246	−0.0804627	−0.0402314	+	0.999190i	$0.512809\pi$
$930$	0	0
$931$	23.1006	0.757091
$932$	0	0
$933$	31.8449	1.04256
$934$	0	0
$935$	−4.73184	−0.154748
$936$	0	0
$937$	50.7268	1.65717	0.828586	−	0.559861i	$-0.189146\pi$
$937$	50.7268	1.65717	0.828586	+	0.559861i	$0.189146\pi$
$938$	0	0
$939$	23.1536	0.755588
$940$	0	0
$941$	−22.5781	−0.736025	−0.368012	−	0.929821i	$-0.619962\pi$
$941$	−22.5781	−0.736025	−0.368012	+	0.929821i	$0.619962\pi$
$942$	0	0
$943$	2.16863	0.0706204
$944$	0	0
$945$	−4.04604	−0.131618
$946$	0	0
$947$	8.38116	0.272351	0.136176	−	0.990685i	$-0.456519\pi$
$947$	8.38116	0.272351	0.136176	+	0.990685i	$0.456519\pi$
$948$	0	0
$949$	−50.0473	−1.62461
$950$	0	0
$951$	9.82803	0.318696
$952$	0	0
$953$	21.4518	0.694891	0.347445	−	0.937700i	$-0.387049\pi$
$953$	21.4518	0.694891	0.347445	+	0.937700i	$0.387049\pi$
$954$	0	0
$955$	44.7316	1.44748
$956$	0	0
$957$	2.12801	0.0687889
$958$	0	0
$959$	−3.75691	−0.121317
$960$	0	0
$961$	−27.7536	−0.895276
$962$	0	0
$963$	11.7351	0.378159
$964$	0	0
$965$	−38.2982	−1.23286
$966$	0	0
$967$	−0.245444	−0.00789295	−0.00394648	−	0.999992i	$-0.501256\pi$
$967$	−0.245444	−0.00789295	−0.00394648	+	0.999992i	$0.501256\pi$
$968$	0	0
$969$	−12.0638	−0.387546
$970$	0	0
$971$	13.9424	0.447433	0.223716	−	0.974654i	$-0.428181\pi$
$971$	13.9424	0.447433	0.223716	+	0.974654i	$0.428181\pi$
$972$	0	0
$973$	26.5160	0.850063
$974$	0	0
$975$	−19.1452	−0.613139
$976$	0	0
$977$	53.9349	1.72553	0.862766	−	0.505604i	$-0.168730\pi$
$977$	53.9349	1.72553	0.862766	+	0.505604i	$0.168730\pi$
$978$	0	0
$979$	−8.86451	−0.283311
$980$	0	0
$981$	−8.24290	−0.263176
$982$	0	0
$983$	−14.1301	−0.450682	−0.225341	−	0.974280i	$-0.572350\pi$
$983$	−14.1301	−0.450682	−0.225341	+	0.974280i	$0.572350\pi$
$984$	0	0
$985$	4.94145	0.157448
$986$	0	0
$987$	5.83633	0.185772
$988$	0	0
$989$	2.71203	0.0862376
$990$	0	0
$991$	−20.9575	−0.665738	−0.332869	−	0.942973i	$-0.608017\pi$
$991$	−20.9575	−0.665738	−0.332869	+	0.942973i	$0.608017\pi$
$992$	0	0
$993$	31.7588	1.00783
$994$	0	0
$995$	2.27996	0.0722796
$996$	0	0
$997$	−20.3219	−0.643600	−0.321800	−	0.946808i	$-0.604288\pi$
$997$	−20.3219	−0.643600	−0.321800	+	0.946808i	$0.604288\pi$
$998$	0	0
$999$	4.00778	0.126800

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.f.1.2	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.f.1.2	✓	14	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.05416 q^{5} -1.32476 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.05416 q^{5} -1.32476 q^{7} +1.00000 q^{9} -0.565626 q^{11} +4.42367 q^{13} +3.05416 q^{15} -2.73910 q^{17} -4.40430 q^{19} +1.32476 q^{21} +3.57521 q^{23} +4.32791 q^{25} -1.00000 q^{27} +3.76223 q^{29} -1.80179 q^{31} +0.565626 q^{33} +4.04604 q^{35} -4.00778 q^{37} -4.42367 q^{39} +0.606575 q^{41} +0.758566 q^{43} -3.05416 q^{45} +4.40556 q^{47} -5.24500 q^{49} +2.73910 q^{51} +2.43817 q^{53} +1.72751 q^{55} +4.40430 q^{57} +15.0477 q^{59} +4.05674 q^{61} -1.32476 q^{63} -13.5106 q^{65} +11.9840 q^{67} -3.57521 q^{69} +15.7394 q^{71} -11.3135 q^{73} -4.32791 q^{75} +0.749320 q^{77} -4.54886 q^{79} +1.00000 q^{81} -16.9930 q^{83} +8.36566 q^{85} -3.76223 q^{87} +15.6720 q^{89} -5.86031 q^{91} +1.80179 q^{93} +13.4515 q^{95} +0.579199 q^{97} -0.565626 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9	\( 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9} + q^{11} - q^{13} + 6 q^{15} - 6 q^{17} + q^{19} + 7 q^{21} + 10 q^{23} - 10 q^{25} - 14 q^{27} - 6 q^{29} - 5 q^{31} - q^{33} + 17 q^{35} - 12 q^{37} + q^{39} - 21 q^{41} + 8 q^{43} - 6 q^{45} + 18 q^{47} - 19 q^{49} + 6 q^{51} - q^{53} - q^{57} + 14 q^{59} - 19 q^{61} - 7 q^{63} + 17 q^{65} + 17 q^{67} - 10 q^{69} - 13 q^{71} - 12 q^{73} + 10 q^{75} - 9 q^{77} - 8 q^{79} + 14 q^{81} + 11 q^{83} - 17 q^{85} + 6 q^{87} - 9 q^{89} - 5 q^{91} + 5 q^{93} + 8 q^{95} - 25 q^{97} + q^{99}+O(q^{100}) \) 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9 + q^11 - q^13 + 6 * q^15 - 6 * q^17 + q^19 + 7 * q^21 + 10 * q^23 - 10 * q^25 - 14 * q^27 - 6 * q^29 - 5 * q^31 - q^33 + 17 * q^35 - 12 * q^37 + q^39 - 21 * q^41 + 8 * q^43 - 6 * q^45 + 18 * q^47 - 19 * q^49 + 6 * q^51 - q^53 - q^57 + 14 * q^59 - 19 * q^61 - 7 * q^63 + 17 * q^65 + 17 * q^67 - 10 * q^69 - 13 * q^71 - 12 * q^73 + 10 * q^75 - 9 * q^77 - 8 * q^79 + 14 * q^81 + 11 * q^83 - 17 * q^85 + 6 * q^87 - 9 * q^89 - 5 * q^91 + 5 * q^93 + 8 * q^95 - 25 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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