LMFDB - Embedded newform 6096.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6096,2,Mod(1,6096)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6096, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6096.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6096 = 2^{4} \cdot 3 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6096.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:	$48.6768050722$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.293061.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 14x^{2} + 9x + 36 $ x^4 - 2x^3 - 14x^2 + 9*x + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1524)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.94856$ of defining polynomial
Character	$\chi$	$=$	6096.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -4.26127 q^{5} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -4.26127 q^{5} +1.00000 q^{9} +5.20983 q^{11} -0.948562 q^{13} +4.26127 q^{15} +4.83896 q^{17} +2.83896 q^{19} -6.15839 q^{23} +13.1584 q^{25} -1.00000 q^{27} +8.26127 q^{29} -7.31942 q^{31} -5.20983 q^{33} +9.10023 q^{37} +0.948562 q^{39} -7.09351 q^{41} -12.3033 q^{43} -4.26127 q^{45} +1.20983 q^{47} -7.00000 q^{49} -4.83896 q^{51} -8.05551 q^{53} -22.2005 q^{55} -2.83896 q^{57} +13.9392 q^{59} -1.10023 q^{61} +4.04207 q^{65} +1.89712 q^{67} +6.15839 q^{69} +0.838965 q^{71} +4.94856 q^{73} -13.1584 q^{75} -3.41558 q^{79} +1.00000 q^{81} -14.4617 q^{83} -20.6201 q^{85} -8.26127 q^{87} +4.48046 q^{89} +7.31942 q^{93} -12.0976 q^{95} -4.62541 q^{97} +5.20983 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - q^{5} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - q^5 + 4 * q^9	$ 4 q - 4 q^{3} - q^{5} + 4 q^{9} - q^{11} + 2 q^{13} + q^{15} + 7 q^{17} - q^{19} + 3 q^{23} + 25 q^{25} - 4 q^{27} + 17 q^{29} - 14 q^{31} + q^{33} + 8 q^{37} - 2 q^{39} - 5 q^{41} - 4 q^{43} - q^{45} - 17 q^{47} - 28 q^{49} - 7 q^{51} + 7 q^{53} - 32 q^{55} + q^{57} + 15 q^{59} + 24 q^{61} - 13 q^{65} - 4 q^{67} - 3 q^{69} - 9 q^{71} + 14 q^{73} - 25 q^{75} - 15 q^{79} + 4 q^{81} + 15 q^{83} + 18 q^{85} - 17 q^{87} + 15 q^{89} + 14 q^{93} + 20 q^{95} + 2 q^{97} - q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - q^5 + 4 * q^9 - q^11 + 2 * q^13 + q^15 + 7 * q^17 - q^19 + 3 * q^23 + 25 * q^25 - 4 * q^27 + 17 * q^29 - 14 * q^31 + q^33 + 8 * q^37 - 2 * q^39 - 5 * q^41 - 4 * q^43 - q^45 - 17 * q^47 - 28 * q^49 - 7 * q^51 + 7 * q^53 - 32 * q^55 + q^57 + 15 * q^59 + 24 * q^61 - 13 * q^65 - 4 * q^67 - 3 * q^69 - 9 * q^71 + 14 * q^73 - 25 * q^75 - 15 * q^79 + 4 * q^81 + 15 * q^83 + 18 * q^85 - 17 * q^87 + 15 * q^89 + 14 * q^93 + 20 * q^95 + 2 * 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
$5$	−4.26127	−1.90570	−0.952848	−	0.303448i	$-0.901862\pi$
$5$	−4.26127	−1.90570	−0.952848	+	0.303448i	$0.901862\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.20983	1.57082	0.785411	−	0.618975i	$-0.212452\pi$
$11$	5.20983	1.57082	0.785411	+	0.618975i	$0.212452\pi$
$12$	0	0
$13$	−0.948562	−0.263084	−0.131542	−	0.991311i	$-0.541993\pi$
$13$	−0.948562	−0.263084	−0.131542	+	0.991311i	$0.541993\pi$
$14$	0	0
$15$	4.26127	1.10025
$16$	0	0
$17$	4.83896	1.17362	0.586811	−	0.809724i	$-0.300383\pi$
$17$	4.83896	1.17362	0.586811	+	0.809724i	$0.300383\pi$
$18$	0	0
$19$	2.83896	0.651303	0.325652	−	0.945490i	$-0.394416\pi$
$19$	2.83896	0.651303	0.325652	+	0.945490i	$0.394416\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.15839	−1.28411	−0.642056	−	0.766657i	$-0.721919\pi$
$23$	−6.15839	−1.28411	−0.642056	+	0.766657i	$0.721919\pi$
$24$	0	0
$25$	13.1584	2.63168
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.26127	1.53408	0.767039	−	0.641600i	$-0.221729\pi$
$29$	8.26127	1.53408	0.767039	+	0.641600i	$0.221729\pi$
$30$	0	0
$31$	−7.31942	−1.31461	−0.657304	−	0.753626i	$-0.728303\pi$
$31$	−7.31942	−1.31461	−0.657304	+	0.753626i	$0.728303\pi$
$32$	0	0
$33$	−5.20983	−0.906915
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.10023	1.49607	0.748034	−	0.663660i	$-0.230998\pi$
$37$	9.10023	1.49607	0.748034	+	0.663660i	$0.230998\pi$
$38$	0	0
$39$	0.948562	0.151891
$40$	0	0
$41$	−7.09351	−1.10782	−0.553910	−	0.832576i	$-0.686865\pi$
$41$	−7.09351	−1.10782	−0.553910	+	0.832576i	$0.686865\pi$
$42$	0	0
$43$	−12.3033	−1.87624	−0.938121	−	0.346308i	$-0.887435\pi$
$43$	−12.3033	−1.87624	−0.938121	+	0.346308i	$0.887435\pi$
$44$	0	0
$45$	−4.26127	−0.635232
$46$	0	0
$47$	1.20983	0.176471	0.0882357	−	0.996100i	$-0.471877\pi$
$47$	1.20983	0.176471	0.0882357	+	0.996100i	$0.471877\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	−4.83896	−0.677591
$52$	0	0
$53$	−8.05551	−1.10651	−0.553255	−	0.833012i	$-0.686614\pi$
$53$	−8.05551	−1.10651	−0.553255	+	0.833012i	$0.686614\pi$
$54$	0	0
$55$	−22.2005	−2.99351
$56$	0	0
$57$	−2.83896	−0.376030
$58$	0	0
$59$	13.9392	1.81473	0.907364	−	0.420345i	$-0.138091\pi$
$59$	13.9392	1.81473	0.907364	+	0.420345i	$0.138091\pi$
$60$	0	0
$61$	−1.10023	−0.140870	−0.0704351	−	0.997516i	$-0.522439\pi$
$61$	−1.10023	−0.140870	−0.0704351	+	0.997516i	$0.522439\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.04207	0.501357
$66$	0	0
$67$	1.89712	0.231770	0.115885	−	0.993263i	$-0.463030\pi$
$67$	1.89712	0.231770	0.115885	+	0.993263i	$0.463030\pi$
$68$	0	0
$69$	6.15839	0.741383
$70$	0	0
$71$	0.838965	0.0995668	0.0497834	−	0.998760i	$-0.484147\pi$
$71$	0.838965	0.0995668	0.0497834	+	0.998760i	$0.484147\pi$
$72$	0	0
$73$	4.94856	0.579185	0.289593	−	0.957150i	$-0.406480\pi$
$73$	4.94856	0.579185	0.289593	+	0.957150i	$0.406480\pi$
$74$	0	0
$75$	−13.1584	−1.51940
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−3.41558	−0.384283	−0.192141	−	0.981367i	$-0.561543\pi$
$79$	−3.41558	−0.384283	−0.192141	+	0.981367i	$0.561543\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.4617	−1.58738	−0.793690	−	0.608322i	$-0.791843\pi$
$83$	−14.4617	−1.58738	−0.793690	+	0.608322i	$0.791843\pi$
$84$	0	0
$85$	−20.6201	−2.23657
$86$	0	0
$87$	−8.26127	−0.885701
$88$	0	0
$89$	4.48046	0.474928	0.237464	−	0.971396i	$-0.423684\pi$
$89$	4.48046	0.474928	0.237464	+	0.971396i	$0.423684\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	7.31942	0.758989
$94$	0	0
$95$	−12.0976	−1.24119
$96$	0	0
$97$	−4.62541	−0.469639	−0.234820	−	0.972039i	$-0.575450\pi$
$97$	−4.62541	−0.469639	−0.234820	+	0.972039i	$0.575450\pi$
$98$	0	0
$99$	5.20983	0.523607
$100$	0	0
$101$	4.04207	0.402201	0.201101	−	0.979571i	$-0.435548\pi$
$101$	4.04207	0.402201	0.201101	+	0.979571i	$0.435548\pi$
$102$	0	0
$103$	−17.6295	−1.73708	−0.868542	−	0.495615i	$-0.834943\pi$
$103$	−17.6295	−1.73708	−0.868542	+	0.495615i	$0.834943\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.8390	1.62788	0.813942	−	0.580946i	$-0.197317\pi$
$107$	16.8390	1.62788	0.813942	+	0.580946i	$0.197317\pi$
$108$	0	0
$109$	7.89712	0.756407	0.378204	−	0.925722i	$-0.376542\pi$
$109$	7.89712	0.756407	0.378204	+	0.925722i	$0.376542\pi$
$110$	0	0
$111$	−9.10023	−0.863756
$112$	0	0
$113$	0.468102	0.0440354	0.0220177	−	0.999758i	$-0.492991\pi$
$113$	0.468102	0.0440354	0.0220177	+	0.999758i	$0.492991\pi$
$114$	0	0
$115$	26.2425	2.44713
$116$	0	0
$117$	−0.948562	−0.0876945
$118$	0	0
$119$	0	0
$120$	0	0
$121$	16.1423	1.46748
$122$	0	0
$123$	7.09351	0.639600
$124$	0	0
$125$	−34.7651	−3.10948
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	0	0
$129$	12.3033	1.08325
$130$	0	0
$131$	6.21919	0.543373	0.271687	−	0.962386i	$-0.412419\pi$
$131$	6.21919	0.543373	0.271687	+	0.962386i	$0.412419\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	4.26127	0.366751
$136$	0	0
$137$	22.6809	1.93776	0.968881	−	0.247527i	$-0.0796180\pi$
$137$	22.6809	1.93776	0.968881	+	0.247527i	$0.0796180\pi$
$138$	0	0
$139$	16.6201	1.40970	0.704850	−	0.709356i	$-0.251014\pi$
$139$	16.6201	1.40970	0.704850	+	0.709356i	$0.251014\pi$
$140$	0	0
$141$	−1.20983	−0.101886
$142$	0	0
$143$	−4.94184	−0.413258
$144$	0	0
$145$	−35.2035	−2.92349
$146$	0	0
$147$	7.00000	0.577350
$148$	0	0
$149$	21.3615	1.75000	0.875001	−	0.484121i	$-0.160861\pi$
$149$	21.3615	1.75000	0.875001	+	0.484121i	$0.160861\pi$
$150$	0	0
$151$	5.89712	0.479901	0.239951	−	0.970785i	$-0.422869\pi$
$151$	5.89712	0.479901	0.239951	+	0.970785i	$0.422869\pi$
$152$	0	0
$153$	4.83896	0.391207
$154$	0	0
$155$	31.1900	2.50524
$156$	0	0
$157$	16.7714	1.33851	0.669253	−	0.743034i	$-0.266614\pi$
$157$	16.7714	1.33851	0.669253	+	0.743034i	$0.266614\pi$
$158$	0	0
$159$	8.05551	0.638844
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−9.95264	−0.779551	−0.389775	−	0.920910i	$-0.627447\pi$
$163$	−9.95264	−0.779551	−0.389775	+	0.920910i	$0.627447\pi$
$164$	0	0
$165$	22.2005	1.72830
$166$	0	0
$167$	−14.2139	−1.09990	−0.549952	−	0.835196i	$-0.685354\pi$
$167$	−14.2139	−1.09990	−0.549952	+	0.835196i	$0.685354\pi$
$168$	0	0
$169$	−12.1002	−0.930787
$170$	0	0
$171$	2.83896	0.217101
$172$	0	0
$173$	−11.7808	−0.895678	−0.447839	−	0.894114i	$-0.647806\pi$
$173$	−11.7808	−0.895678	−0.447839	+	0.894114i	$0.647806\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−13.9392	−1.04773
$178$	0	0
$179$	−14.5169	−1.08504	−0.542522	−	0.840042i	$-0.682530\pi$
$179$	−14.5169	−1.08504	−0.542522	+	0.840042i	$0.682530\pi$
$180$	0	0
$181$	−10.3168	−0.766840	−0.383420	−	0.923574i	$-0.625254\pi$
$181$	−10.3168	−0.766840	−0.383420	+	0.923574i	$0.625254\pi$
$182$	0	0
$183$	1.10023	0.0813314
$184$	0	0
$185$	−38.7785	−2.85105
$186$	0	0
$187$	25.2102	1.84355
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.78081	0.707714	0.353857	−	0.935299i	$-0.384870\pi$
$191$	9.78081	0.707714	0.353857	+	0.935299i	$0.384870\pi$
$192$	0	0
$193$	10.7417	0.773206	0.386603	−	0.922246i	$-0.373648\pi$
$193$	10.7417	0.773206	0.386603	+	0.922246i	$0.373648\pi$
$194$	0	0
$195$	−4.04207	−0.289459
$196$	0	0
$197$	9.51317	0.677785	0.338893	−	0.940825i	$-0.389948\pi$
$197$	9.51317	0.677785	0.338893	+	0.940825i	$0.389948\pi$
$198$	0	0
$199$	−15.9936	−1.13376	−0.566879	−	0.823801i	$-0.691849\pi$
$199$	−15.9936	−1.13376	−0.566879	+	0.823801i	$0.691849\pi$
$200$	0	0
$201$	−1.89712	−0.133813
$202$	0	0
$203$	0	0
$204$	0	0
$205$	30.2273	2.11117
$206$	0	0
$207$	−6.15839	−0.428038
$208$	0	0
$209$	14.7905	1.02308
$210$	0	0
$211$	3.90948	0.269140	0.134570	−	0.990904i	$-0.457035\pi$
$211$	3.90948	0.269140	0.134570	+	0.990904i	$0.457035\pi$
$212$	0	0
$213$	−0.838965	−0.0574849
$214$	0	0
$215$	52.4278	3.57555
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−4.94856	−0.334393
$220$	0	0
$221$	−4.59006	−0.308761
$222$	0	0
$223$	2.10288	0.140819	0.0704095	−	0.997518i	$-0.477569\pi$
$223$	2.10288	0.140819	0.0704095	+	0.997518i	$0.477569\pi$
$224$	0	0
$225$	13.1584	0.877226
$226$	0	0
$227$	9.57505	0.635519	0.317759	−	0.948171i	$-0.397070\pi$
$227$	9.57505	0.635519	0.317759	+	0.948171i	$0.397070\pi$
$228$	0	0
$229$	−24.9287	−1.64734	−0.823669	−	0.567071i	$-0.808077\pi$
$229$	−24.9287	−1.64734	−0.823669	+	0.567071i	$0.808077\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−15.3138	−1.00324	−0.501620	−	0.865088i	$-0.667262\pi$
$233$	−15.3138	−1.00324	−0.501620	+	0.865088i	$0.667262\pi$
$234$	0	0
$235$	−5.15540	−0.336301
$236$	0	0
$237$	3.41558	0.221866
$238$	0	0
$239$	8.26127	0.534377	0.267188	−	0.963644i	$-0.413905\pi$
$239$	8.26127	0.534377	0.267188	+	0.963644i	$0.413905\pi$
$240$	0	0
$241$	8.10288	0.521952	0.260976	−	0.965345i	$-0.415956\pi$
$241$	8.10288	0.521952	0.260976	+	0.965345i	$0.415956\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	29.8289	1.90570
$246$	0	0
$247$	−2.69293	−0.171347
$248$	0	0
$249$	14.4617	0.916475
$250$	0	0
$251$	−13.5266	−0.853792	−0.426896	−	0.904301i	$-0.640393\pi$
$251$	−13.5266	−0.853792	−0.426896	+	0.904301i	$0.640393\pi$
$252$	0	0
$253$	−32.0841	−2.01711
$254$	0	0
$255$	20.6201	1.29128
$256$	0	0
$257$	7.51954	0.469056	0.234528	−	0.972109i	$-0.424646\pi$
$257$	7.51954	0.469056	0.234528	+	0.972109i	$0.424646\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	8.26127	0.511360
$262$	0	0
$263$	2.63850	0.162697	0.0813485	−	0.996686i	$-0.474077\pi$
$263$	2.63850	0.162697	0.0813485	+	0.996686i	$0.474077\pi$
$264$	0	0
$265$	34.3267	2.10867
$266$	0	0
$267$	−4.48046	−0.274200
$268$	0	0
$269$	31.5620	1.92437	0.962183	−	0.272403i	$-0.0878183\pi$
$269$	31.5620	1.92437	0.962183	+	0.272403i	$0.0878183\pi$
$270$	0	0
$271$	6.36414	0.386594	0.193297	−	0.981140i	$-0.438082\pi$
$271$	6.36414	0.386594	0.193297	+	0.981140i	$0.438082\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	68.5529	4.13390
$276$	0	0
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	0	0
$279$	−7.31942	−0.438202
$280$	0	0
$281$	23.3806	1.39477	0.697384	−	0.716698i	$-0.254347\pi$
$281$	23.3806	1.39477	0.697384	+	0.716698i	$0.254347\pi$
$282$	0	0
$283$	−8.31678	−0.494381	−0.247191	−	0.968967i	$-0.579507\pi$
$283$	−8.31678	−0.494381	−0.247191	+	0.968967i	$0.579507\pi$
$284$	0	0
$285$	12.0976	0.716599
$286$	0	0
$287$	0	0
$288$	0	0
$289$	6.41558	0.377387
$290$	0	0
$291$	4.62541	0.271146
$292$	0	0
$293$	26.9422	1.57398	0.786990	−	0.616966i	$-0.211638\pi$
$293$	26.9422	1.57398	0.786990	+	0.616966i	$0.211638\pi$
$294$	0	0
$295$	−59.3986	−3.45832
$296$	0	0
$297$	−5.20983	−0.302305
$298$	0	0
$299$	5.84161	0.337829
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−4.04207	−0.232211
$304$	0	0
$305$	4.68838	0.268456
$306$	0	0
$307$	18.6388	1.06378	0.531888	−	0.846815i	$-0.321483\pi$
$307$	18.6388	1.06378	0.531888	+	0.846815i	$0.321483\pi$
$308$	0	0
$309$	17.6295	1.00291
$310$	0	0
$311$	21.9392	1.24406	0.622029	−	0.782994i	$-0.286309\pi$
$311$	21.9392	1.24406	0.622029	+	0.782994i	$0.286309\pi$
$312$	0	0
$313$	13.3671	0.755555	0.377778	−	0.925896i	$-0.376688\pi$
$313$	13.3671	0.755555	0.377778	+	0.925896i	$0.376688\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.2899	0.914932	0.457466	−	0.889227i	$-0.348757\pi$
$317$	16.2899	0.914932	0.457466	+	0.889227i	$0.348757\pi$
$318$	0	0
$319$	43.0398	2.40976
$320$	0	0
$321$	−16.8390	−0.939859
$322$	0	0
$323$	13.7377	0.764383
$324$	0	0
$325$	−12.4815	−0.692351
$326$	0	0
$327$	−7.89712	−0.436712
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.45874	−0.0801794	−0.0400897	−	0.999196i	$-0.512764\pi$
$331$	−1.45874	−0.0801794	−0.0400897	+	0.999196i	$0.512764\pi$
$332$	0	0
$333$	9.10023	0.498690
$334$	0	0
$335$	−8.08415	−0.441684
$336$	0	0
$337$	−21.1367	−1.15139	−0.575694	−	0.817665i	$-0.695268\pi$
$337$	−21.1367	−1.15139	−0.575694	+	0.817665i	$0.695268\pi$
$338$	0	0
$339$	−0.468102	−0.0254238
$340$	0	0
$341$	−38.1329	−2.06501
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−26.2425	−1.41285
$346$	0	0
$347$	−25.3198	−1.35924	−0.679618	−	0.733566i	$-0.737855\pi$
$347$	−25.3198	−1.35924	−0.679618	+	0.733566i	$0.737855\pi$
$348$	0	0
$349$	−14.8393	−0.794330	−0.397165	−	0.917747i	$-0.630006\pi$
$349$	−14.8393	−0.794330	−0.397165	+	0.917747i	$0.630006\pi$
$350$	0	0
$351$	0.948562	0.0506305
$352$	0	0
$353$	−2.63850	−0.140433	−0.0702167	−	0.997532i	$-0.522369\pi$
$353$	−2.63850	−0.140433	−0.0702167	+	0.997532i	$0.522369\pi$
$354$	0	0
$355$	−3.57505	−0.189744
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7.25827	0.383077	0.191539	−	0.981485i	$-0.438652\pi$
$359$	7.25827	0.383077	0.191539	+	0.981485i	$0.438652\pi$
$360$	0	0
$361$	−10.9403	−0.575804
$362$	0	0
$363$	−16.1423	−0.847251
$364$	0	0
$365$	−21.0871	−1.10375
$366$	0	0
$367$	−19.3194	−1.00847	−0.504233	−	0.863568i	$-0.668225\pi$
$367$	−19.3194	−1.00847	−0.504233	+	0.863568i	$0.668225\pi$
$368$	0	0
$369$	−7.09351	−0.369273
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−12.2139	−0.632412	−0.316206	−	0.948691i	$-0.602409\pi$
$373$	−12.2139	−0.632412	−0.316206	+	0.948691i	$0.602409\pi$
$374$	0	0
$375$	34.7651	1.79526
$376$	0	0
$377$	−7.83632	−0.403591
$378$	0	0
$379$	23.7942	1.22223	0.611114	−	0.791542i	$-0.290722\pi$
$379$	23.7942	1.22223	0.611114	+	0.791542i	$0.290722\pi$
$380$	0	0
$381$	1.00000	0.0512316
$382$	0	0
$383$	30.4144	1.55410	0.777051	−	0.629438i	$-0.216715\pi$
$383$	30.4144	1.55410	0.777051	+	0.629438i	$0.216715\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−12.3033	−0.625414
$388$	0	0
$389$	29.3074	1.48594	0.742972	−	0.669322i	$-0.233415\pi$
$389$	29.3074	1.48594	0.742972	+	0.669322i	$0.233415\pi$
$390$	0	0
$391$	−29.8002	−1.50706
$392$	0	0
$393$	−6.21919	−0.313717
$394$	0	0
$395$	14.5547	0.732327
$396$	0	0
$397$	8.89448	0.446401	0.223201	−	0.974773i	$-0.428350\pi$
$397$	8.89448	0.446401	0.223201	+	0.974773i	$0.428350\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−0.467020	−0.0233219	−0.0116609	−	0.999932i	$-0.503712\pi$
$401$	−0.467020	−0.0233219	−0.0116609	+	0.999932i	$0.503712\pi$
$402$	0	0
$403$	6.94292	0.345852
$404$	0	0
$405$	−4.26127	−0.211744
$406$	0	0
$407$	47.4106	2.35006
$408$	0	0
$409$	−10.9609	−0.541983	−0.270991	−	0.962582i	$-0.587351\pi$
$409$	−10.9609	−0.541983	−0.270991	+	0.962582i	$0.587351\pi$
$410$	0	0
$411$	−22.6809	−1.11877
$412$	0	0
$413$	0	0
$414$	0	0
$415$	61.6253	3.02507
$416$	0	0
$417$	−16.6201	−0.813891
$418$	0	0
$419$	23.4778	1.14697	0.573483	−	0.819217i	$-0.305592\pi$
$419$	23.4778	1.14697	0.573483	+	0.819217i	$0.305592\pi$
$420$	0	0
$421$	4.31080	0.210095	0.105048	−	0.994467i	$-0.466500\pi$
$421$	4.31080	0.210095	0.105048	+	0.994467i	$0.466500\pi$
$422$	0	0
$423$	1.20983	0.0588238
$424$	0	0
$425$	63.6730	3.08859
$426$	0	0
$427$	0	0
$428$	0	0
$429$	4.94184	0.238594
$430$	0	0
$431$	−14.1460	−0.681390	−0.340695	−	0.940174i	$-0.610662\pi$
$431$	−14.1460	−0.681390	−0.340695	+	0.940174i	$0.610662\pi$
$432$	0	0
$433$	−9.78644	−0.470307	−0.235153	−	0.971958i	$-0.575559\pi$
$433$	−9.78644	−0.470307	−0.235153	+	0.971958i	$0.575559\pi$
$434$	0	0
$435$	35.2035	1.68788
$436$	0	0
$437$	−17.4835	−0.836347
$438$	0	0
$439$	−2.52999	−0.120750	−0.0603749	−	0.998176i	$-0.519230\pi$
$439$	−2.52999	−0.120750	−0.0603749	+	0.998176i	$0.519230\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	0	0
$443$	37.4725	1.78037	0.890187	−	0.455596i	$-0.150574\pi$
$443$	37.4725	1.78037	0.890187	+	0.455596i	$0.150574\pi$
$444$	0	0
$445$	−19.0924	−0.905068
$446$	0	0
$447$	−21.3615	−1.01036
$448$	0	0
$449$	1.32016	0.0623022	0.0311511	−	0.999515i	$-0.490083\pi$
$449$	1.32016	0.0623022	0.0311511	+	0.999515i	$0.490083\pi$
$450$	0	0
$451$	−36.9560	−1.74019
$452$	0	0
$453$	−5.89712	−0.277071
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.4320	0.487989	0.243994	−	0.969777i	$-0.421542\pi$
$457$	10.4320	0.487989	0.243994	+	0.969777i	$0.421542\pi$
$458$	0	0
$459$	−4.83896	−0.225864
$460$	0	0
$461$	16.7552	0.780366	0.390183	−	0.920737i	$-0.372412\pi$
$461$	16.7552	0.780366	0.390183	+	0.920737i	$0.372412\pi$
$462$	0	0
$463$	11.2925	0.524809	0.262405	−	0.964958i	$-0.415485\pi$
$463$	11.2925	0.524809	0.262405	+	0.964958i	$0.415485\pi$
$464$	0	0
$465$	−31.1900	−1.44640
$466$	0	0
$467$	−20.5915	−0.952860	−0.476430	−	0.879212i	$-0.658069\pi$
$467$	−20.5915	−0.952860	−0.476430	+	0.879212i	$0.658069\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−16.7714	−0.772787
$472$	0	0
$473$	−64.0983	−2.94724
$474$	0	0
$475$	37.3562	1.71402
$476$	0	0
$477$	−8.05551	−0.368837
$478$	0	0
$479$	−7.39685	−0.337971	−0.168985	−	0.985619i	$-0.554049\pi$
$479$	−7.39685	−0.337971	−0.168985	+	0.985619i	$0.554049\pi$
$480$	0	0
$481$	−8.63213	−0.393591
$482$	0	0
$483$	0	0
$484$	0	0
$485$	19.7101	0.894990
$486$	0	0
$487$	−3.98656	−0.180648	−0.0903242	−	0.995912i	$-0.528790\pi$
$487$	−3.98656	−0.180648	−0.0903242	+	0.995912i	$0.528790\pi$
$488$	0	0
$489$	9.95264	0.450074
$490$	0	0
$491$	4.94748	0.223277	0.111638	−	0.993749i	$-0.464390\pi$
$491$	4.94748	0.223277	0.111638	+	0.993749i	$0.464390\pi$
$492$	0	0
$493$	39.9760	1.80043
$494$	0	0
$495$	−22.2005	−0.997837
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−33.1426	−1.48367	−0.741834	−	0.670583i	$-0.766044\pi$
$499$	−33.1426	−1.48367	−0.741834	+	0.670583i	$0.766044\pi$
$500$	0	0
$501$	14.2139	0.635030
$502$	0	0
$503$	−1.85397	−0.0826643	−0.0413322	−	0.999145i	$-0.513160\pi$
$503$	−1.85397	−0.0826643	−0.0413322	+	0.999145i	$0.513160\pi$
$504$	0	0
$505$	−17.2243	−0.766473
$506$	0	0
$507$	12.1002	0.537390
$508$	0	0
$509$	−19.5676	−0.867318	−0.433659	−	0.901077i	$-0.642778\pi$
$509$	−19.5676	−0.867318	−0.433659	+	0.901077i	$0.642778\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2.83896	−0.125343
$514$	0	0
$515$	75.1239	3.31036
$516$	0	0
$517$	6.30299	0.277205
$518$	0	0
$519$	11.7808	0.517120
$520$	0	0
$521$	17.6514	0.773322	0.386661	−	0.922222i	$-0.373628\pi$
$521$	17.6514	0.773322	0.386661	+	0.922222i	$0.373628\pi$
$522$	0	0
$523$	7.88260	0.344682	0.172341	−	0.985037i	$-0.444867\pi$
$523$	7.88260	0.344682	0.172341	+	0.985037i	$0.444867\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−35.4184	−1.54285
$528$	0	0
$529$	14.9258	0.648946
$530$	0	0
$531$	13.9392	0.604910
$532$	0	0
$533$	6.72863	0.291449
$534$	0	0
$535$	−71.7553	−3.10225
$536$	0	0
$537$	14.5169	0.626450
$538$	0	0
$539$	−36.4688	−1.57082
$540$	0	0
$541$	−25.7942	−1.10898	−0.554491	−	0.832190i	$-0.687087\pi$
$541$	−25.7942	−1.10898	−0.554491	+	0.832190i	$0.687087\pi$
$542$	0	0
$543$	10.3168	0.442735
$544$	0	0
$545$	−33.6517	−1.44148
$546$	0	0
$547$	5.47218	0.233973	0.116987	−	0.993133i	$-0.462677\pi$
$547$	5.47218	0.233973	0.116987	+	0.993133i	$0.462677\pi$
$548$	0	0
$549$	−1.10023	−0.0469567
$550$	0	0
$551$	23.4534	0.999150
$552$	0	0
$553$	0	0
$554$	0	0
$555$	38.7785	1.64606
$556$	0	0
$557$	−26.1952	−1.10992	−0.554962	−	0.831875i	$-0.687267\pi$
$557$	−26.1952	−1.10992	−0.554962	+	0.831875i	$0.687267\pi$
$558$	0	0
$559$	11.6705	0.493608
$560$	0	0
$561$	−25.2102	−1.06437
$562$	0	0
$563$	−6.85804	−0.289032	−0.144516	−	0.989502i	$-0.546163\pi$
$563$	−6.85804	−0.289032	−0.144516	+	0.989502i	$0.546163\pi$
$564$	0	0
$565$	−1.99471	−0.0839180
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12.3712	0.518628	0.259314	−	0.965793i	$-0.416504\pi$
$569$	12.3712	0.518628	0.259314	+	0.965793i	$0.416504\pi$
$570$	0	0
$571$	17.0204	0.712279	0.356140	−	0.934433i	$-0.384093\pi$
$571$	17.0204	0.712279	0.356140	+	0.934433i	$0.384093\pi$
$572$	0	0
$573$	−9.78081	−0.408599
$574$	0	0
$575$	−81.0345	−3.37937
$576$	0	0
$577$	29.9395	1.24640	0.623200	−	0.782063i	$-0.285832\pi$
$577$	29.9395	1.24640	0.623200	+	0.782063i	$0.285832\pi$
$578$	0	0
$579$	−10.7417	−0.446411
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−41.9678	−1.73813
$584$	0	0
$585$	4.04207	0.167119
$586$	0	0
$587$	−0.206100	−0.00850664	−0.00425332	−	0.999991i	$-0.501354\pi$
$587$	−0.206100	−0.00850664	−0.00425332	+	0.999991i	$0.501354\pi$
$588$	0	0
$589$	−20.7796	−0.856208
$590$	0	0
$591$	−9.51317	−0.391319
$592$	0	0
$593$	−38.0615	−1.56300	−0.781499	−	0.623906i	$-0.785545\pi$
$593$	−38.0615	−1.56300	−0.781499	+	0.623906i	$0.785545\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	15.9936	0.654576
$598$	0	0
$599$	−22.6540	−0.925619	−0.462810	−	0.886458i	$-0.653159\pi$
$599$	−22.6540	−0.925619	−0.462810	+	0.886458i	$0.653159\pi$
$600$	0	0
$601$	34.9100	1.42401	0.712005	−	0.702175i	$-0.247787\pi$
$601$	34.9100	1.42401	0.712005	+	0.702175i	$0.247787\pi$
$602$	0	0
$603$	1.89712	0.0772568
$604$	0	0
$605$	−68.7867	−2.79658
$606$	0	0
$607$	4.04771	0.164292	0.0821458	−	0.996620i	$-0.473823\pi$
$607$	4.04771	0.164292	0.0821458	+	0.996620i	$0.473823\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.14760	−0.0464268
$612$	0	0
$613$	19.3618	0.782017	0.391009	−	0.920387i	$-0.372126\pi$
$613$	19.3618	0.782017	0.391009	+	0.920387i	$0.372126\pi$
$614$	0	0
$615$	−30.2273	−1.21888
$616$	0	0
$617$	−33.0871	−1.33204	−0.666019	−	0.745935i	$-0.732003\pi$
$617$	−33.0871	−1.33204	−0.666019	+	0.745935i	$0.732003\pi$
$618$	0	0
$619$	−28.3033	−1.13761	−0.568804	−	0.822473i	$-0.692594\pi$
$619$	−28.3033	−1.13761	−0.568804	+	0.822473i	$0.692594\pi$
$620$	0	0
$621$	6.15839	0.247128
$622$	0	0
$623$	0	0
$624$	0	0
$625$	82.3513	3.29405
$626$	0	0
$627$	−14.7905	−0.590676
$628$	0	0
$629$	44.0357	1.75582
$630$	0	0
$631$	13.5810	0.540653	0.270326	−	0.962769i	$-0.412868\pi$
$631$	13.5810	0.540653	0.270326	+	0.962769i	$0.412868\pi$
$632$	0	0
$633$	−3.90948	−0.155388
$634$	0	0
$635$	4.26127	0.169103
$636$	0	0
$637$	6.63993	0.263084
$638$	0	0
$639$	0.838965	0.0331889
$640$	0	0
$641$	49.0585	1.93769	0.968847	−	0.247659i	$-0.0796613\pi$
$641$	49.0585	1.93769	0.968847	+	0.247659i	$0.0796613\pi$
$642$	0	0
$643$	32.4875	1.28118	0.640591	−	0.767882i	$-0.278689\pi$
$643$	32.4875	1.28118	0.640591	+	0.767882i	$0.278689\pi$
$644$	0	0
$645$	−52.4278	−2.06434
$646$	0	0
$647$	−6.80253	−0.267435	−0.133718	−	0.991019i	$-0.542691\pi$
$647$	−6.80253	−0.267435	−0.133718	+	0.991019i	$0.542691\pi$
$648$	0	0
$649$	72.6208	2.85062
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.95155	−0.0763702	−0.0381851	−	0.999271i	$-0.512158\pi$
$653$	−1.95155	−0.0763702	−0.0381851	+	0.999271i	$0.512158\pi$
$654$	0	0
$655$	−26.5016	−1.03550
$656$	0	0
$657$	4.94856	0.193062
$658$	0	0
$659$	−13.6359	−0.531178	−0.265589	−	0.964086i	$-0.585566\pi$
$659$	−13.6359	−0.531178	−0.265589	+	0.964086i	$0.585566\pi$
$660$	0	0
$661$	12.2450	0.476275	0.238137	−	0.971231i	$-0.423463\pi$
$661$	12.2450	0.476275	0.238137	+	0.971231i	$0.423463\pi$
$662$	0	0
$663$	4.59006	0.178263
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−50.8761	−1.96993
$668$	0	0
$669$	−2.10288	−0.0813019
$670$	0	0
$671$	−5.73201	−0.221282
$672$	0	0
$673$	−13.1002	−0.504977	−0.252488	−	0.967600i	$-0.581249\pi$
$673$	−13.1002	−0.504977	−0.252488	+	0.967600i	$0.581249\pi$
$674$	0	0
$675$	−13.1584	−0.506467
$676$	0	0
$677$	42.3168	1.62637	0.813183	−	0.582008i	$-0.197733\pi$
$677$	42.3168	1.62637	0.813183	+	0.582008i	$0.197733\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−9.57505	−0.366917
$682$	0	0
$683$	−25.7868	−0.986704	−0.493352	−	0.869830i	$-0.664229\pi$
$683$	−25.7868	−0.986704	−0.493352	+	0.869830i	$0.664229\pi$
$684$	0	0
$685$	−96.6494	−3.69279
$686$	0	0
$687$	24.9287	0.951091
$688$	0	0
$689$	7.64115	0.291105
$690$	0	0
$691$	37.1292	1.41246	0.706231	−	0.707982i	$-0.250394\pi$
$691$	37.1292	1.41246	0.706231	+	0.707982i	$0.250394\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−70.8227	−2.68646
$696$	0	0
$697$	−34.3253	−1.30016
$698$	0	0
$699$	15.3138	0.579221
$700$	0	0
$701$	37.8154	1.42827	0.714134	−	0.700009i	$-0.246821\pi$
$701$	37.8154	1.42827	0.714134	+	0.700009i	$0.246821\pi$
$702$	0	0
$703$	25.8352	0.974394
$704$	0	0
$705$	5.15540	0.194164
$706$	0	0
$707$	0	0
$708$	0	0
$709$	4.06787	0.152772	0.0763860	−	0.997078i	$-0.475662\pi$
$709$	4.06787	0.152772	0.0763860	+	0.997078i	$0.475662\pi$
$710$	0	0
$711$	−3.41558	−0.128094
$712$	0	0
$713$	45.0759	1.68810
$714$	0	0
$715$	21.0585	0.787543
$716$	0	0
$717$	−8.26127	−0.308523
$718$	0	0
$719$	47.1348	1.75783	0.878917	−	0.476975i	$-0.158267\pi$
$719$	47.1348	1.75783	0.878917	+	0.476975i	$0.158267\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−8.10288	−0.301349
$724$	0	0
$725$	108.705	4.03720
$726$	0	0
$727$	−4.85206	−0.179953	−0.0899764	−	0.995944i	$-0.528679\pi$
$727$	−4.85206	−0.179953	−0.0899764	+	0.995944i	$0.528679\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−59.5354	−2.20200
$732$	0	0
$733$	−11.7200	−0.432888	−0.216444	−	0.976295i	$-0.569446\pi$
$733$	−11.7200	−0.432888	−0.216444	+	0.976295i	$0.569446\pi$
$734$	0	0
$735$	−29.8289	−1.10025
$736$	0	0
$737$	9.88368	0.364070
$738$	0	0
$739$	22.3313	0.821470	0.410735	−	0.911755i	$-0.365272\pi$
$739$	22.3313	0.821470	0.410735	+	0.911755i	$0.365272\pi$
$740$	0	0
$741$	2.69293	0.0989274
$742$	0	0
$743$	17.2222	0.631821	0.315910	−	0.948789i	$-0.397690\pi$
$743$	17.2222	0.631821	0.315910	+	0.948789i	$0.397690\pi$
$744$	0	0
$745$	−91.0270	−3.33497
$746$	0	0
$747$	−14.4617	−0.529127
$748$	0	0
$749$	0	0
$750$	0	0
$751$	27.0398	0.986695	0.493348	−	0.869832i	$-0.335773\pi$
$751$	27.0398	0.986695	0.493348	+	0.869832i	$0.335773\pi$
$752$	0	0
$753$	13.5266	0.492937
$754$	0	0
$755$	−25.1292	−0.914546
$756$	0	0
$757$	18.9839	0.689982	0.344991	−	0.938606i	$-0.387882\pi$
$757$	18.9839	0.689982	0.344991	+	0.938606i	$0.387882\pi$
$758$	0	0
$759$	32.0841	1.16458
$760$	0	0
$761$	17.4032	0.630866	0.315433	−	0.948948i	$-0.397850\pi$
$761$	17.4032	0.630866	0.315433	+	0.948948i	$0.397850\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−20.6201	−0.745522
$766$	0	0
$767$	−13.2222	−0.477425
$768$	0	0
$769$	33.8918	1.22217	0.611085	−	0.791565i	$-0.290733\pi$
$769$	33.8918	1.22217	0.611085	+	0.791565i	$0.290733\pi$
$770$	0	0
$771$	−7.51954	−0.270810
$772$	0	0
$773$	3.18791	0.114661	0.0573306	−	0.998355i	$-0.481741\pi$
$773$	3.18791	0.114661	0.0573306	+	0.998355i	$0.481741\pi$
$774$	0	0
$775$	−96.3118	−3.45962
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−20.1382	−0.721527
$780$	0	0
$781$	4.37086	0.156402
$782$	0	0
$783$	−8.26127	−0.295234
$784$	0	0
$785$	−71.4676	−2.55079
$786$	0	0
$787$	−13.8639	−0.494194	−0.247097	−	0.968991i	$-0.579477\pi$
$787$	−13.8639	−0.494194	−0.247097	+	0.968991i	$0.579477\pi$
$788$	0	0
$789$	−2.63850	−0.0939332
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1.04364	0.0370606
$794$	0	0
$795$	−34.3267	−1.21744
$796$	0	0
$797$	−28.6127	−1.01351	−0.506756	−	0.862089i	$-0.669156\pi$
$797$	−28.6127	−1.01351	−0.506756	+	0.862089i	$0.669156\pi$
$798$	0	0
$799$	5.85431	0.207111
$800$	0	0
$801$	4.48046	0.158309
$802$	0	0
$803$	25.7812	0.909797
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−31.5620	−1.11103
$808$	0	0
$809$	−16.0845	−0.565501	−0.282750	−	0.959194i	$-0.591247\pi$
$809$	−16.0845	−0.565501	−0.282750	+	0.959194i	$0.591247\pi$
$810$	0	0
$811$	11.9936	0.421153	0.210577	−	0.977577i	$-0.432466\pi$
$811$	11.9936	0.421153	0.210577	+	0.977577i	$0.432466\pi$
$812$	0	0
$813$	−6.36414	−0.223200
$814$	0	0
$815$	42.4108	1.48559
$816$	0	0
$817$	−34.9287	−1.22200
$818$	0	0
$819$	0	0
$820$	0	0
$821$	27.9452	0.975293	0.487647	−	0.873041i	$-0.337855\pi$
$821$	27.9452	0.975293	0.487647	+	0.873041i	$0.337855\pi$
$822$	0	0
$823$	49.7684	1.73482	0.867409	−	0.497596i	$-0.165784\pi$
$823$	49.7684	1.73482	0.867409	+	0.497596i	$0.165784\pi$
$824$	0	0
$825$	−68.5529	−2.38671
$826$	0	0
$827$	−11.1080	−0.386264	−0.193132	−	0.981173i	$-0.561865\pi$
$827$	−11.1080	−0.386264	−0.193132	+	0.981173i	$0.561865\pi$
$828$	0	0
$829$	−29.4647	−1.02335	−0.511676	−	0.859178i	$-0.670975\pi$
$829$	−29.4647	−1.02335	−0.511676	+	0.859178i	$0.670975\pi$
$830$	0	0
$831$	−18.0000	−0.624413
$832$	0	0
$833$	−33.8728	−1.17362
$834$	0	0
$835$	60.5692	2.09608
$836$	0	0
$837$	7.31942	0.252996
$838$	0	0
$839$	10.0099	0.345580	0.172790	−	0.984959i	$-0.444722\pi$
$839$	10.0099	0.345580	0.172790	+	0.984959i	$0.444722\pi$
$840$	0	0
$841$	39.2485	1.35340
$842$	0	0
$843$	−23.3806	−0.805270
$844$	0	0
$845$	51.5623	1.77380
$846$	0	0
$847$	0	0
$848$	0	0
$849$	8.31678	0.285431
$850$	0	0
$851$	−56.0428	−1.92112
$852$	0	0
$853$	43.0076	1.47255	0.736276	−	0.676681i	$-0.236582\pi$
$853$	43.0076	1.47255	0.736276	+	0.676681i	$0.236582\pi$
$854$	0	0
$855$	−12.0976	−0.413729
$856$	0	0
$857$	−40.5406	−1.38484	−0.692420	−	0.721495i	$-0.743455\pi$
$857$	−40.5406	−1.38484	−0.692420	+	0.721495i	$0.743455\pi$
$858$	0	0
$859$	14.1983	0.484440	0.242220	−	0.970221i	$-0.422124\pi$
$859$	14.1983	0.484440	0.242220	+	0.970221i	$0.422124\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−23.4226	−0.797316	−0.398658	−	0.917100i	$-0.630524\pi$
$863$	−23.4226	−0.797316	−0.398658	+	0.917100i	$0.630524\pi$
$864$	0	0
$865$	50.2012	1.70689
$866$	0	0
$867$	−6.41558	−0.217885
$868$	0	0
$869$	−17.7946	−0.603640
$870$	0	0
$871$	−1.79954	−0.0609750
$872$	0	0
$873$	−4.62541	−0.156546
$874$	0	0
$875$	0	0
$876$	0	0
$877$	17.3368	0.585422	0.292711	−	0.956201i	$-0.405443\pi$
$877$	17.3368	0.585422	0.292711	+	0.956201i	$0.405443\pi$
$878$	0	0
$879$	−26.9422	−0.908737
$880$	0	0
$881$	−14.1853	−0.477914	−0.238957	−	0.971030i	$-0.576805\pi$
$881$	−14.1853	−0.477914	−0.238957	+	0.971030i	$0.576805\pi$
$882$	0	0
$883$	28.7647	0.968010	0.484005	−	0.875065i	$-0.339182\pi$
$883$	28.7647	0.968010	0.484005	+	0.875065i	$0.339182\pi$
$884$	0	0
$885$	59.3986	1.99666
$886$	0	0
$887$	−55.0684	−1.84902	−0.924508	−	0.381162i	$-0.875524\pi$
$887$	−55.0684	−1.84902	−0.924508	+	0.381162i	$0.875524\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	5.20983	0.174536
$892$	0	0
$893$	3.43466	0.114936
$894$	0	0
$895$	61.8604	2.06776
$896$	0	0
$897$	−5.84161	−0.195046
$898$	0	0
$899$	−60.4677	−2.01671
$900$	0	0
$901$	−38.9803	−1.29862
$902$	0	0
$903$	0	0
$904$	0	0
$905$	43.9625	1.46136
$906$	0	0
$907$	−28.7347	−0.954119	−0.477059	−	0.878871i	$-0.658297\pi$
$907$	−28.7347	−0.954119	−0.477059	+	0.878871i	$0.658297\pi$
$908$	0	0
$909$	4.04207	0.134067
$910$	0	0
$911$	2.74173	0.0908374	0.0454187	−	0.998968i	$-0.485538\pi$
$911$	2.74173	0.0908374	0.0454187	+	0.998968i	$0.485538\pi$
$912$	0	0
$913$	−75.3431	−2.49349
$914$	0	0
$915$	−4.68838	−0.154993
$916$	0	0
$917$	0	0
$918$	0	0
$919$	50.8549	1.67755	0.838773	−	0.544481i	$-0.183273\pi$
$919$	50.8549	1.67755	0.838773	+	0.544481i	$0.183273\pi$
$920$	0	0
$921$	−18.6388	−0.614171
$922$	0	0
$923$	−0.795810	−0.0261944
$924$	0	0
$925$	119.744	3.93717
$926$	0	0
$927$	−17.6295	−0.579028
$928$	0	0
$929$	14.5300	0.476713	0.238357	−	0.971178i	$-0.423391\pi$
$929$	14.5300	0.476713	0.238357	+	0.971178i	$0.423391\pi$
$930$	0	0
$931$	−19.8728	−0.651303
$932$	0	0
$933$	−21.9392	−0.718257
$934$	0	0
$935$	−107.427	−3.51325
$936$	0	0
$937$	−0.817724	−0.0267139	−0.0133569	−	0.999911i	$-0.504252\pi$
$937$	−0.817724	−0.0267139	−0.0133569	+	0.999911i	$0.504252\pi$
$938$	0	0
$939$	−13.3671	−0.436220
$940$	0	0
$941$	−34.4472	−1.12295	−0.561473	−	0.827495i	$-0.689765\pi$
$941$	−34.4472	−1.12295	−0.561473	+	0.827495i	$0.689765\pi$
$942$	0	0
$943$	43.6846	1.42257
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.90311	0.126834	0.0634170	−	0.997987i	$-0.479800\pi$
$947$	3.90311	0.126834	0.0634170	+	0.997987i	$0.479800\pi$
$948$	0	0
$949$	−4.69401	−0.152374
$950$	0	0
$951$	−16.2899	−0.528236
$952$	0	0
$953$	−20.2602	−0.656292	−0.328146	−	0.944627i	$-0.606424\pi$
$953$	−20.2602	−0.656292	−0.328146	+	0.944627i	$0.606424\pi$
$954$	0	0
$955$	−41.6786	−1.34869
$956$	0	0
$957$	−43.0398	−1.39128
$958$	0	0
$959$	0	0
$960$	0	0
$961$	22.5740	0.728193
$962$	0	0
$963$	16.8390	0.542628
$964$	0	0
$965$	−45.7734	−1.47350
$966$	0	0
$967$	15.3806	0.494606	0.247303	−	0.968938i	$-0.420456\pi$
$967$	15.3806	0.494606	0.247303	+	0.968938i	$0.420456\pi$
$968$	0	0
$969$	−13.7377	−0.441317
$970$	0	0
$971$	37.5266	1.20429	0.602143	−	0.798388i	$-0.294314\pi$
$971$	37.5266	1.20429	0.602143	+	0.798388i	$0.294314\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	12.4815	0.399729
$976$	0	0
$977$	36.8669	1.17948	0.589738	−	0.807595i	$-0.299231\pi$
$977$	36.8669	1.17948	0.589738	+	0.807595i	$0.299231\pi$
$978$	0	0
$979$	23.3424	0.746027
$980$	0	0
$981$	7.89712	0.252136
$982$	0	0
$983$	25.0861	0.800121	0.400060	−	0.916489i	$-0.368989\pi$
$983$	25.0861	0.800121	0.400060	+	0.916489i	$0.368989\pi$
$984$	0	0
$985$	−40.5381	−1.29165
$986$	0	0
$987$	0	0
$988$	0	0
$989$	75.7688	2.40931
$990$	0	0
$991$	20.6314	0.655378	0.327689	−	0.944786i	$-0.393730\pi$
$991$	20.6314	0.655378	0.327689	+	0.944786i	$0.393730\pi$
$992$	0	0
$993$	1.45874	0.0462916
$994$	0	0
$995$	68.1531	2.16060
$996$	0	0
$997$	−26.4144	−0.836551	−0.418276	−	0.908320i	$-0.637365\pi$
$997$	−26.4144	−0.836551	−0.418276	+	0.908320i	$0.637365\pi$
$998$	0	0
$999$	−9.10023	−0.287919

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6096.2.a.y.1.1		4
4.3	odd	2		1524.2.a.f.1.1	✓	4
12.11	even	2		4572.2.a.r.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1524.2.a.f.1.1	✓	4	4.3	odd	2
4572.2.a.r.1.4		4	12.11	even	2
6096.2.a.y.1.1		4	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:	\( N \)	\(=\)	\( 6096 = 2^{4} \cdot 3 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6096.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -4.26127 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -4.26127 q^{5} +1.00000 q^{9} +5.20983 q^{11} -0.948562 q^{13} +4.26127 q^{15} +4.83896 q^{17} +2.83896 q^{19} -6.15839 q^{23} +13.1584 q^{25} -1.00000 q^{27} +8.26127 q^{29} -7.31942 q^{31} -5.20983 q^{33} +9.10023 q^{37} +0.948562 q^{39} -7.09351 q^{41} -12.3033 q^{43} -4.26127 q^{45} +1.20983 q^{47} -7.00000 q^{49} -4.83896 q^{51} -8.05551 q^{53} -22.2005 q^{55} -2.83896 q^{57} +13.9392 q^{59} -1.10023 q^{61} +4.04207 q^{65} +1.89712 q^{67} +6.15839 q^{69} +0.838965 q^{71} +4.94856 q^{73} -13.1584 q^{75} -3.41558 q^{79} +1.00000 q^{81} -14.4617 q^{83} -20.6201 q^{85} -8.26127 q^{87} +4.48046 q^{89} +7.31942 q^{93} -12.0976 q^{95} -4.62541 q^{97} +5.20983 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - q^{5} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - q^5 + 4 * q^9	\( 4 q - 4 q^{3} - q^{5} + 4 q^{9} - q^{11} + 2 q^{13} + q^{15} + 7 q^{17} - q^{19} + 3 q^{23} + 25 q^{25} - 4 q^{27} + 17 q^{29} - 14 q^{31} + q^{33} + 8 q^{37} - 2 q^{39} - 5 q^{41} - 4 q^{43} - q^{45} - 17 q^{47} - 28 q^{49} - 7 q^{51} + 7 q^{53} - 32 q^{55} + q^{57} + 15 q^{59} + 24 q^{61} - 13 q^{65} - 4 q^{67} - 3 q^{69} - 9 q^{71} + 14 q^{73} - 25 q^{75} - 15 q^{79} + 4 q^{81} + 15 q^{83} + 18 q^{85} - 17 q^{87} + 15 q^{89} + 14 q^{93} + 20 q^{95} + 2 q^{97} - q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - q^5 + 4 * q^9 - q^11 + 2 * q^13 + q^15 + 7 * q^17 - q^19 + 3 * q^23 + 25 * q^25 - 4 * q^27 + 17 * q^29 - 14 * q^31 + q^33 + 8 * q^37 - 2 * q^39 - 5 * q^41 - 4 * q^43 - q^45 - 17 * q^47 - 28 * q^49 - 7 * q^51 + 7 * q^53 - 32 * q^55 + q^57 + 15 * q^59 + 24 * q^61 - 13 * q^65 - 4 * q^67 - 3 * q^69 - 9 * q^71 + 14 * q^73 - 25 * q^75 - 15 * q^79 + 4 * q^81 + 15 * q^83 + 18 * q^85 - 17 * q^87 + 15 * q^89 + 14 * q^93 + 20 * q^95 + 2 * q^97 - q^99

Introduction

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