LMFDB - Embedded newform 1512.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1512,2,Mod(1,1512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1512, 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("1512.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1512 = 2^{3} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1512.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:	$12.0733807856$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	1512.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64575 q^{5} -1.00000 q^{7} -4.64575 q^{11} -5.29150 q^{13} -1.00000 q^{19} -6.64575 q^{23} +2.00000 q^{25} -6.00000 q^{29} -8.29150 q^{31} -2.64575 q^{35} +8.29150 q^{37} -3.35425 q^{41} +3.29150 q^{43} -0.708497 q^{47} +1.00000 q^{49} -12.2915 q^{55} -7.29150 q^{59} +14.5830 q^{61} -14.0000 q^{65} +9.29150 q^{67} +13.2288 q^{71} +11.2915 q^{73} +4.64575 q^{77} -0.708497 q^{79} -10.0000 q^{83} -1.93725 q^{89} +5.29150 q^{91} -2.64575 q^{95} -2.58301 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{7} - 4 q^{11} - 2 q^{19} - 8 q^{23} + 4 q^{25} - 12 q^{29} - 6 q^{31} + 6 q^{37} - 12 q^{41} - 4 q^{43} - 12 q^{47} + 2 q^{49} - 14 q^{55} - 4 q^{59} + 8 q^{61} - 28 q^{65} + 8 q^{67} + 12 q^{73}+ \cdots + 16 q^{97}+O(q^{100}) $ 2 * q - 2 * q^7 - 4 * q^11 - 2 * q^19 - 8 * q^23 + 4 * q^25 - 12 * q^29 - 6 * q^31 + 6 * q^37 - 12 * q^41 - 4 * q^43 - 12 * q^47 + 2 * q^49 - 14 * q^55 - 4 * q^59 + 8 * q^61 - 28 * q^65 + 8 * q^67 + 12 * q^73 + 4 * q^77 - 12 * q^79 - 20 * q^83 + 12 * q^89 + 16 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.64575	1.18322	0.591608	−	0.806226i	$-0.298493\pi$
$5$	2.64575	1.18322	0.591608	+	0.806226i	$0.298493\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.64575	−1.40075	−0.700373	−	0.713777i	$-0.746983\pi$
$11$	−4.64575	−1.40075	−0.700373	+	0.713777i	$0.746983\pi$
$12$	0	0
$13$	−5.29150	−1.46760	−0.733799	−	0.679366i	$-0.762255\pi$
$13$	−5.29150	−1.46760	−0.733799	+	0.679366i	$0.762255\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.64575	−1.38573	−0.692867	−	0.721065i	$-0.743653\pi$
$23$	−6.64575	−1.38573	−0.692867	+	0.721065i	$0.743653\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−8.29150	−1.48920	−0.744599	−	0.667512i	$-0.767359\pi$
$31$	−8.29150	−1.48920	−0.744599	+	0.667512i	$0.767359\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.64575	−0.447214
$36$	0	0
$37$	8.29150	1.36311	0.681557	−	0.731765i	$-0.261303\pi$
$37$	8.29150	1.36311	0.681557	+	0.731765i	$0.261303\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.35425	−0.523846	−0.261923	−	0.965089i	$-0.584357\pi$
$41$	−3.35425	−0.523846	−0.261923	+	0.965089i	$0.584357\pi$
$42$	0	0
$43$	3.29150	0.501949	0.250975	−	0.967994i	$-0.419249\pi$
$43$	3.29150	0.501949	0.250975	+	0.967994i	$0.419249\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.708497	−0.103345	−0.0516725	−	0.998664i	$-0.516455\pi$
$47$	−0.708497	−0.103345	−0.0516725	+	0.998664i	$0.516455\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−12.2915	−1.65739
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.29150	−0.949273	−0.474636	−	0.880182i	$-0.657420\pi$
$59$	−7.29150	−0.949273	−0.474636	+	0.880182i	$0.657420\pi$
$60$	0	0
$61$	14.5830	1.86716	0.933581	−	0.358366i	$-0.116666\pi$
$61$	14.5830	1.86716	0.933581	+	0.358366i	$0.116666\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−14.0000	−1.73649
$66$	0	0
$67$	9.29150	1.13514	0.567569	−	0.823326i	$-0.307884\pi$
$67$	9.29150	1.13514	0.567569	+	0.823326i	$0.307884\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.2288	1.56996	0.784982	−	0.619518i	$-0.212672\pi$
$71$	13.2288	1.56996	0.784982	+	0.619518i	$0.212672\pi$
$72$	0	0
$73$	11.2915	1.32157	0.660785	−	0.750575i	$-0.270223\pi$
$73$	11.2915	1.32157	0.660785	+	0.750575i	$0.270223\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.64575	0.529432
$78$	0	0
$79$	−0.708497	−0.0797122	−0.0398561	−	0.999205i	$-0.512690\pi$
$79$	−0.708497	−0.0797122	−0.0398561	+	0.999205i	$0.512690\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.0000	−1.09764	−0.548821	−	0.835940i	$-0.684923\pi$
$83$	−10.0000	−1.09764	−0.548821	+	0.835940i	$0.684923\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.93725	−0.205349	−0.102674	−	0.994715i	$-0.532740\pi$
$89$	−1.93725	−0.205349	−0.102674	+	0.994715i	$0.532740\pi$
$90$	0	0
$91$	5.29150	0.554700
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.64575	−0.271448
$96$	0	0
$97$	−2.58301	−0.262264	−0.131132	−	0.991365i	$-0.541861\pi$
$97$	−2.58301	−0.262264	−0.131132	+	0.991365i	$0.541861\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	0	0
$103$	6.29150	0.619920	0.309960	−	0.950750i	$-0.399684\pi$
$103$	6.29150	0.619920	0.309960	+	0.950750i	$0.399684\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.2915	−1.28494	−0.642469	−	0.766312i	$-0.722090\pi$
$107$	−13.2915	−1.28494	−0.642469	+	0.766312i	$0.722090\pi$
$108$	0	0
$109$	−16.8745	−1.61628	−0.808142	−	0.588987i	$-0.799527\pi$
$109$	−16.8745	−1.61628	−0.808142	+	0.588987i	$0.799527\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−19.8745	−1.86964	−0.934818	−	0.355127i	$-0.884438\pi$
$113$	−19.8745	−1.86964	−0.934818	+	0.355127i	$0.884438\pi$
$114$	0	0
$115$	−17.5830	−1.63962
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	10.5830	0.962091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−7.93725	−0.709930
$126$	0	0
$127$	−6.70850	−0.595283	−0.297641	−	0.954678i	$-0.596200\pi$
$127$	−6.70850	−0.595283	−0.297641	+	0.954678i	$0.596200\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.2915	1.33602	0.668012	−	0.744150i	$-0.267145\pi$
$131$	15.2915	1.33602	0.668012	+	0.744150i	$0.267145\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.58301	0.733296	0.366648	−	0.930360i	$-0.380505\pi$
$137$	8.58301	0.733296	0.366648	+	0.930360i	$0.380505\pi$
$138$	0	0
$139$	−10.5830	−0.897639	−0.448819	−	0.893622i	$-0.648155\pi$
$139$	−10.5830	−0.897639	−0.448819	+	0.893622i	$0.648155\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	24.5830	2.05573
$144$	0	0
$145$	−15.8745	−1.31831
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.8745	1.30049	0.650245	−	0.759724i	$-0.274666\pi$
$149$	15.8745	1.30049	0.650245	+	0.759724i	$0.274666\pi$
$150$	0	0
$151$	−20.5830	−1.67502	−0.837511	−	0.546421i	$-0.815990\pi$
$151$	−20.5830	−1.67502	−0.837511	+	0.546421i	$0.815990\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−21.9373	−1.76204
$156$	0	0
$157$	−1.29150	−0.103073	−0.0515366	−	0.998671i	$-0.516412\pi$
$157$	−1.29150	−0.103073	−0.0515366	+	0.998671i	$0.516412\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.64575	0.523759
$162$	0	0
$163$	−1.29150	−0.101158	−0.0505791	−	0.998720i	$-0.516107\pi$
$163$	−1.29150	−0.101158	−0.0505791	+	0.998720i	$0.516107\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.70850	−0.519119	−0.259560	−	0.965727i	$-0.583577\pi$
$167$	−6.70850	−0.519119	−0.259560	+	0.965727i	$0.583577\pi$
$168$	0	0
$169$	15.0000	1.15385
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−22.5203	−1.71218	−0.856092	−	0.516824i	$-0.827114\pi$
$173$	−22.5203	−1.71218	−0.856092	+	0.516824i	$0.827114\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	0	0
$178$	0	0
$179$	21.2915	1.59140	0.795701	−	0.605690i	$-0.207103\pi$
$179$	21.2915	1.59140	0.795701	+	0.605690i	$0.207103\pi$
$180$	0	0
$181$	24.5830	1.82724	0.913620	−	0.406569i	$-0.133275\pi$
$181$	24.5830	1.82724	0.913620	+	0.406569i	$0.133275\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	21.9373	1.61286
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−21.3542	−1.54514	−0.772570	−	0.634930i	$-0.781029\pi$
$191$	−21.3542	−1.54514	−0.772570	+	0.634930i	$0.781029\pi$
$192$	0	0
$193$	0.583005	0.0419656	0.0209828	−	0.999780i	$-0.493320\pi$
$193$	0.583005	0.0419656	0.0209828	+	0.999780i	$0.493320\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.29150	−0.0920158	−0.0460079	−	0.998941i	$-0.514650\pi$
$197$	−1.29150	−0.0920158	−0.0460079	+	0.998941i	$0.514650\pi$
$198$	0	0
$199$	6.87451	0.487321	0.243660	−	0.969861i	$-0.421652\pi$
$199$	6.87451	0.487321	0.243660	+	0.969861i	$0.421652\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.00000	0.421117
$204$	0	0
$205$	−8.87451	−0.619823
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.64575	0.321353
$210$	0	0
$211$	6.00000	0.413057	0.206529	−	0.978441i	$-0.433783\pi$
$211$	6.00000	0.413057	0.206529	+	0.978441i	$0.433783\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.70850	0.593915
$216$	0	0
$217$	8.29150	0.562864
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−13.7085	−0.917989	−0.458994	−	0.888439i	$-0.651790\pi$
$223$	−13.7085	−0.917989	−0.458994	+	0.888439i	$0.651790\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.70850	0.179769	0.0898846	−	0.995952i	$-0.471350\pi$
$227$	2.70850	0.179769	0.0898846	+	0.995952i	$0.471350\pi$
$228$	0	0
$229$	−5.41699	−0.357965	−0.178983	−	0.983852i	$-0.557281\pi$
$229$	−5.41699	−0.357965	−0.178983	+	0.983852i	$0.557281\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.70850	−0.177440	−0.0887198	−	0.996057i	$-0.528278\pi$
$233$	−2.70850	−0.177440	−0.0887198	+	0.996057i	$0.528278\pi$
$234$	0	0
$235$	−1.87451	−0.122279
$236$	0	0
$237$	0	0
$238$	0	0
$239$	21.2915	1.37723	0.688616	−	0.725126i	$-0.258218\pi$
$239$	21.2915	1.37723	0.688616	+	0.725126i	$0.258218\pi$
$240$	0	0
$241$	23.2915	1.50034	0.750169	−	0.661246i	$-0.229972\pi$
$241$	23.2915	1.50034	0.750169	+	0.661246i	$0.229972\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.64575	0.169031
$246$	0	0
$247$	5.29150	0.336690
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.58301	−0.163038	−0.0815189	−	0.996672i	$-0.525977\pi$
$251$	−2.58301	−0.163038	−0.0815189	+	0.996672i	$0.525977\pi$
$252$	0	0
$253$	30.8745	1.94106
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.77124	0.297622	0.148811	−	0.988866i	$-0.452455\pi$
$257$	4.77124	0.297622	0.148811	+	0.988866i	$0.452455\pi$
$258$	0	0
$259$	−8.29150	−0.515209
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.22876	−0.569070	−0.284535	−	0.958666i	$-0.591839\pi$
$263$	−9.22876	−0.569070	−0.284535	+	0.958666i	$0.591839\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	21.3542	1.30199	0.650996	−	0.759081i	$-0.274352\pi$
$269$	21.3542	1.30199	0.650996	+	0.759081i	$0.274352\pi$
$270$	0	0
$271$	−26.5830	−1.61480	−0.807401	−	0.590003i	$-0.799127\pi$
$271$	−26.5830	−1.61480	−0.807401	+	0.590003i	$0.799127\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−9.29150	−0.560299
$276$	0	0
$277$	−3.70850	−0.222822	−0.111411	−	0.993774i	$-0.535537\pi$
$277$	−3.70850	−0.222822	−0.111411	+	0.993774i	$0.535537\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	9.87451	0.589064	0.294532	−	0.955642i	$-0.404836\pi$
$281$	9.87451	0.589064	0.294532	+	0.955642i	$0.404836\pi$
$282$	0	0
$283$	18.5830	1.10465	0.552323	−	0.833631i	$-0.313742\pi$
$283$	18.5830	1.10465	0.552323	+	0.833631i	$0.313742\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.35425	0.197995
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−26.5830	−1.55300	−0.776498	−	0.630120i	$-0.783006\pi$
$293$	−26.5830	−1.55300	−0.776498	+	0.630120i	$0.783006\pi$
$294$	0	0
$295$	−19.2915	−1.12319
$296$	0	0
$297$	0	0
$298$	0	0
$299$	35.1660	2.03370
$300$	0	0
$301$	−3.29150	−0.189719
$302$	0	0
$303$	0	0
$304$	0	0
$305$	38.5830	2.20926
$306$	0	0
$307$	24.1660	1.37923	0.689614	−	0.724178i	$-0.257780\pi$
$307$	24.1660	1.37923	0.689614	+	0.724178i	$0.257780\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−0.583005	−0.0330592	−0.0165296	−	0.999863i	$-0.505262\pi$
$311$	−0.583005	−0.0330592	−0.0165296	+	0.999863i	$0.505262\pi$
$312$	0	0
$313$	2.70850	0.153093	0.0765467	−	0.997066i	$-0.475611\pi$
$313$	2.70850	0.153093	0.0765467	+	0.997066i	$0.475611\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−31.2915	−1.75751	−0.878753	−	0.477277i	$-0.841624\pi$
$317$	−31.2915	−1.75751	−0.878753	+	0.477277i	$0.841624\pi$
$318$	0	0
$319$	27.8745	1.56067
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−10.5830	−0.587040
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.708497	0.0390607
$330$	0	0
$331$	10.5830	0.581695	0.290847	−	0.956769i	$-0.406063\pi$
$331$	10.5830	0.581695	0.290847	+	0.956769i	$0.406063\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	24.5830	1.34311
$336$	0	0
$337$	−27.5830	−1.50254	−0.751271	−	0.659994i	$-0.770559\pi$
$337$	−27.5830	−1.50254	−0.751271	+	0.659994i	$0.770559\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	38.5203	2.08599
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.6458	0.678859	0.339430	−	0.940631i	$-0.389766\pi$
$347$	12.6458	0.678859	0.339430	+	0.940631i	$0.389766\pi$
$348$	0	0
$349$	20.5830	1.10178	0.550892	−	0.834577i	$-0.314288\pi$
$349$	20.5830	1.10178	0.550892	+	0.834577i	$0.314288\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	17.9373	0.954704	0.477352	−	0.878712i	$-0.341597\pi$
$353$	17.9373	0.954704	0.477352	+	0.878712i	$0.341597\pi$
$354$	0	0
$355$	35.0000	1.85761
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−15.8745	−0.837824	−0.418912	−	0.908027i	$-0.637589\pi$
$359$	−15.8745	−0.837824	−0.418912	+	0.908027i	$0.637589\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	0	0
$364$	0	0
$365$	29.8745	1.56370
$366$	0	0
$367$	−15.7085	−0.819977	−0.409988	−	0.912091i	$-0.634467\pi$
$367$	−15.7085	−0.819977	−0.409988	+	0.912091i	$0.634467\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−20.8745	−1.08084	−0.540421	−	0.841395i	$-0.681735\pi$
$373$	−20.8745	−1.08084	−0.540421	+	0.841395i	$0.681735\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	31.7490	1.63516
$378$	0	0
$379$	−13.8745	−0.712686	−0.356343	−	0.934355i	$-0.615976\pi$
$379$	−13.8745	−0.712686	−0.356343	+	0.934355i	$0.615976\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	13.2915	0.679164	0.339582	−	0.940576i	$-0.389714\pi$
$383$	13.2915	0.679164	0.339582	+	0.940576i	$0.389714\pi$
$384$	0	0
$385$	12.2915	0.626433
$386$	0	0
$387$	0	0
$388$	0	0
$389$	28.0000	1.41966	0.709828	−	0.704375i	$-0.248773\pi$
$389$	28.0000	1.41966	0.709828	+	0.704375i	$0.248773\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.87451	−0.0943167
$396$	0	0
$397$	10.0000	0.501886	0.250943	−	0.968002i	$-0.419259\pi$
$397$	10.0000	0.501886	0.250943	+	0.968002i	$0.419259\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−23.1660	−1.15686	−0.578428	−	0.815734i	$-0.696333\pi$
$401$	−23.1660	−1.15686	−0.578428	+	0.815734i	$0.696333\pi$
$402$	0	0
$403$	43.8745	2.18554
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−38.5203	−1.90938
$408$	0	0
$409$	−17.8745	−0.883838	−0.441919	−	0.897055i	$-0.645702\pi$
$409$	−17.8745	−0.883838	−0.441919	+	0.897055i	$0.645702\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.29150	0.358791
$414$	0	0
$415$	−26.4575	−1.29875
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−14.7085	−0.718557	−0.359279	−	0.933230i	$-0.616977\pi$
$419$	−14.7085	−0.718557	−0.359279	+	0.933230i	$0.616977\pi$
$420$	0	0
$421$	−6.29150	−0.306629	−0.153315	−	0.988177i	$-0.548995\pi$
$421$	−6.29150	−0.306629	−0.153315	+	0.988177i	$0.548995\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−14.5830	−0.705721
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−33.1033	−1.59453	−0.797264	−	0.603631i	$-0.793720\pi$
$431$	−33.1033	−1.59453	−0.797264	+	0.603631i	$0.793720\pi$
$432$	0	0
$433$	7.87451	0.378425	0.189212	−	0.981936i	$-0.439407\pi$
$433$	7.87451	0.378425	0.189212	+	0.981936i	$0.439407\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.64575	0.317909
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8.77124	−0.416734	−0.208367	−	0.978051i	$-0.566815\pi$
$443$	−8.77124	−0.416734	−0.208367	+	0.978051i	$0.566815\pi$
$444$	0	0
$445$	−5.12549	−0.242972
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−26.4575	−1.24861	−0.624304	−	0.781182i	$-0.714617\pi$
$449$	−26.4575	−1.24861	−0.624304	+	0.781182i	$0.714617\pi$
$450$	0	0
$451$	15.5830	0.733775
$452$	0	0
$453$	0	0
$454$	0	0
$455$	14.0000	0.656330
$456$	0	0
$457$	1.00000	0.0467780	0.0233890	−	0.999726i	$-0.492554\pi$
$457$	1.00000	0.0467780	0.0233890	+	0.999726i	$0.492554\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−39.6863	−1.84837	−0.924187	−	0.381940i	$-0.875256\pi$
$461$	−39.6863	−1.84837	−0.924187	+	0.381940i	$0.875256\pi$
$462$	0	0
$463$	−29.7490	−1.38255	−0.691277	−	0.722590i	$-0.742952\pi$
$463$	−29.7490	−1.38255	−0.691277	+	0.722590i	$0.742952\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.8745	1.19733	0.598665	−	0.801000i	$-0.295698\pi$
$467$	25.8745	1.19733	0.598665	+	0.801000i	$0.295698\pi$
$468$	0	0
$469$	−9.29150	−0.429042
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−15.2915	−0.703104
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	0	0
$477$	0	0
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−43.8745	−2.00051
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.83399	−0.310315
$486$	0	0
$487$	12.7085	0.575877	0.287938	−	0.957649i	$-0.407030\pi$
$487$	12.7085	0.575877	0.287938	+	0.957649i	$0.407030\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17.8118	0.803834	0.401917	−	0.915676i	$-0.368344\pi$
$491$	17.8118	0.803834	0.401917	+	0.915676i	$0.368344\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13.2288	−0.593391
$498$	0	0
$499$	24.7085	1.10610	0.553052	−	0.833147i	$-0.313463\pi$
$499$	24.7085	1.10610	0.553052	+	0.833147i	$0.313463\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0.583005	0.0259949	0.0129975	−	0.999916i	$-0.495863\pi$
$503$	0.583005	0.0259949	0.0129975	+	0.999916i	$0.495863\pi$
$504$	0	0
$505$	−10.5830	−0.470938
$506$	0	0
$507$	0	0
$508$	0	0
$509$	42.5830	1.88746	0.943729	−	0.330719i	$-0.107291\pi$
$509$	42.5830	1.88746	0.943729	+	0.330719i	$0.107291\pi$
$510$	0	0
$511$	−11.2915	−0.499507
$512$	0	0
$513$	0	0
$514$	0	0
$515$	16.6458	0.733499
$516$	0	0
$517$	3.29150	0.144760
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.93725	−0.0848726	−0.0424363	−	0.999099i	$-0.513512\pi$
$521$	−1.93725	−0.0848726	−0.0424363	+	0.999099i	$0.513512\pi$
$522$	0	0
$523$	27.5830	1.20612	0.603060	−	0.797696i	$-0.293948\pi$
$523$	27.5830	1.20612	0.603060	+	0.797696i	$0.293948\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	21.1660	0.920261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	17.7490	0.768795
$534$	0	0
$535$	−35.1660	−1.52036
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.64575	−0.200107
$540$	0	0
$541$	9.70850	0.417401	0.208700	−	0.977980i	$-0.433077\pi$
$541$	9.70850	0.417401	0.208700	+	0.977980i	$0.433077\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−44.6458	−1.91241
$546$	0	0
$547$	−14.5830	−0.623524	−0.311762	−	0.950160i	$-0.600919\pi$
$547$	−14.5830	−0.623524	−0.311762	+	0.950160i	$0.600919\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	0.708497	0.0301284
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−34.4575	−1.46001	−0.730006	−	0.683441i	$-0.760483\pi$
$557$	−34.4575	−1.46001	−0.730006	+	0.683441i	$0.760483\pi$
$558$	0	0
$559$	−17.4170	−0.736660
$560$	0	0
$561$	0	0
$562$	0	0
$563$	19.1660	0.807751	0.403876	−	0.914814i	$-0.367663\pi$
$563$	19.1660	0.807751	0.403876	+	0.914814i	$0.367663\pi$
$564$	0	0
$565$	−52.5830	−2.21218
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−46.4575	−1.94760	−0.973800	−	0.227406i	$-0.926976\pi$
$569$	−46.4575	−1.94760	−0.973800	+	0.227406i	$0.926976\pi$
$570$	0	0
$571$	22.0000	0.920671	0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000	0.920671	0.460336	+	0.887745i	$0.347729\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−13.2915	−0.554294
$576$	0	0
$577$	0.583005	0.0242708	0.0121354	−	0.999926i	$-0.496137\pi$
$577$	0.583005	0.0242708	0.0121354	+	0.999926i	$0.496137\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	10.0000	0.414870
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27.8745	1.15050	0.575252	−	0.817976i	$-0.304904\pi$
$587$	27.8745	1.15050	0.575252	+	0.817976i	$0.304904\pi$
$588$	0	0
$589$	8.29150	0.341645
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−19.3542	−0.794784	−0.397392	−	0.917649i	$-0.630085\pi$
$593$	−19.3542	−0.794784	−0.397392	+	0.917649i	$0.630085\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	11.8118	0.482616	0.241308	−	0.970449i	$-0.422424\pi$
$599$	11.8118	0.482616	0.241308	+	0.970449i	$0.422424\pi$
$600$	0	0
$601$	35.8745	1.46335	0.731676	−	0.681653i	$-0.238738\pi$
$601$	35.8745	1.46335	0.731676	+	0.681653i	$0.238738\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	28.0000	1.13836
$606$	0	0
$607$	34.5830	1.40368	0.701840	−	0.712334i	$-0.252362\pi$
$607$	34.5830	1.40368	0.701840	+	0.712334i	$0.252362\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.74902	0.151669
$612$	0	0
$613$	−40.2915	−1.62736	−0.813679	−	0.581314i	$-0.802539\pi$
$613$	−40.2915	−1.62736	−0.813679	+	0.581314i	$0.802539\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.70850	0.350591	0.175295	−	0.984516i	$-0.443912\pi$
$617$	8.70850	0.350591	0.175295	+	0.984516i	$0.443912\pi$
$618$	0	0
$619$	21.5830	0.867494	0.433747	−	0.901035i	$-0.357191\pi$
$619$	21.5830	0.867494	0.433747	+	0.901035i	$0.357191\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.93725	0.0776144
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	27.0405	1.07647	0.538233	−	0.842796i	$-0.319092\pi$
$631$	27.0405	1.07647	0.538233	+	0.842796i	$0.319092\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−17.7490	−0.704348
$636$	0	0
$637$	−5.29150	−0.209657
$638$	0	0
$639$	0	0
$640$	0	0
$641$	13.2915	0.524983	0.262491	−	0.964934i	$-0.415456\pi$
$641$	13.2915	0.524983	0.262491	+	0.964934i	$0.415456\pi$
$642$	0	0
$643$	7.00000	0.276053	0.138027	−	0.990429i	$-0.455924\pi$
$643$	7.00000	0.276053	0.138027	+	0.990429i	$0.455924\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.7085	−0.499623	−0.249811	−	0.968295i	$-0.580369\pi$
$647$	−12.7085	−0.499623	−0.249811	+	0.968295i	$0.580369\pi$
$648$	0	0
$649$	33.8745	1.32969
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−27.2915	−1.06800	−0.534000	−	0.845485i	$-0.679312\pi$
$653$	−27.2915	−1.06800	−0.534000	+	0.845485i	$0.679312\pi$
$654$	0	0
$655$	40.4575	1.58081
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−17.8118	−0.693848	−0.346924	−	0.937893i	$-0.612774\pi$
$659$	−17.8118	−0.693848	−0.346924	+	0.937893i	$0.612774\pi$
$660$	0	0
$661$	11.8745	0.461865	0.230932	−	0.972970i	$-0.425822\pi$
$661$	11.8745	0.461865	0.230932	+	0.972970i	$0.425822\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.64575	0.102598
$666$	0	0
$667$	39.8745	1.54395
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−67.7490	−2.61542
$672$	0	0
$673$	−16.5830	−0.639228	−0.319614	−	0.947548i	$-0.603553\pi$
$673$	−16.5830	−0.639228	−0.319614	+	0.947548i	$0.603553\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−21.2288	−0.815887	−0.407944	−	0.913007i	$-0.633754\pi$
$677$	−21.2288	−0.815887	−0.407944	+	0.913007i	$0.633754\pi$
$678$	0	0
$679$	2.58301	0.0991266
$680$	0	0
$681$	0	0
$682$	0	0
$683$	46.9778	1.79755	0.898777	−	0.438406i	$-0.144457\pi$
$683$	46.9778	1.79755	0.898777	+	0.438406i	$0.144457\pi$
$684$	0	0
$685$	22.7085	0.867647
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−13.1660	−0.500859	−0.250429	−	0.968135i	$-0.580572\pi$
$691$	−13.1660	−0.500859	−0.250429	+	0.968135i	$0.580572\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−28.0000	−1.06210
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−44.5830	−1.68388	−0.841938	−	0.539574i	$-0.818585\pi$
$701$	−44.5830	−1.68388	−0.841938	+	0.539574i	$0.818585\pi$
$702$	0	0
$703$	−8.29150	−0.312720
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.00000	0.150435
$708$	0	0
$709$	6.87451	0.258178	0.129089	−	0.991633i	$-0.458795\pi$
$709$	6.87451	0.258178	0.129089	+	0.991633i	$0.458795\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	55.1033	2.06363
$714$	0	0
$715$	65.0405	2.43238
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−43.0405	−1.60514	−0.802570	−	0.596558i	$-0.796535\pi$
$719$	−43.0405	−1.60514	−0.802570	+	0.596558i	$0.796535\pi$
$720$	0	0
$721$	−6.29150	−0.234308
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−12.0000	−0.445669
$726$	0	0
$727$	−22.5830	−0.837557	−0.418779	−	0.908088i	$-0.637542\pi$
$727$	−22.5830	−0.837557	−0.418779	+	0.908088i	$0.637542\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−18.1255	−0.669480	−0.334740	−	0.942310i	$-0.608649\pi$
$733$	−18.1255	−0.669480	−0.334740	+	0.942310i	$0.608649\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−43.1660	−1.59004
$738$	0	0
$739$	34.0000	1.25071	0.625355	−	0.780340i	$-0.284954\pi$
$739$	34.0000	1.25071	0.625355	+	0.780340i	$0.284954\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−11.9373	−0.437935	−0.218968	−	0.975732i	$-0.570269\pi$
$743$	−11.9373	−0.437935	−0.218968	+	0.975732i	$0.570269\pi$
$744$	0	0
$745$	42.0000	1.53876
$746$	0	0
$747$	0	0
$748$	0	0
$749$	13.2915	0.485661
$750$	0	0
$751$	−27.2915	−0.995881	−0.497941	−	0.867211i	$-0.665910\pi$
$751$	−27.2915	−0.995881	−0.497941	+	0.867211i	$0.665910\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−54.4575	−1.98191
$756$	0	0
$757$	7.41699	0.269575	0.134788	−	0.990875i	$-0.456965\pi$
$757$	7.41699	0.269575	0.134788	+	0.990875i	$0.456965\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	16.8745	0.610898
$764$	0	0
$765$	0	0
$766$	0	0
$767$	38.5830	1.39315
$768$	0	0
$769$	−29.2915	−1.05628	−0.528139	−	0.849158i	$-0.677110\pi$
$769$	−29.2915	−1.05628	−0.528139	+	0.849158i	$0.677110\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−39.8118	−1.43193	−0.715965	−	0.698136i	$-0.754013\pi$
$773$	−39.8118	−1.43193	−0.715965	+	0.698136i	$0.754013\pi$
$774$	0	0
$775$	−16.5830	−0.595679
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.35425	0.120178
$780$	0	0
$781$	−61.4575	−2.19912
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3.41699	−0.121958
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	19.8745	0.706656
$792$	0	0
$793$	−77.1660	−2.74025
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−28.0627	−0.994033	−0.497017	−	0.867741i	$-0.665571\pi$
$797$	−28.0627	−0.994033	−0.497017	+	0.867741i	$0.665571\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−52.4575	−1.85119
$804$	0	0
$805$	17.5830	0.619720
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−7.41699	−0.260768	−0.130384	−	0.991464i	$-0.541621\pi$
$809$	−7.41699	−0.260768	−0.130384	+	0.991464i	$0.541621\pi$
$810$	0	0
$811$	−19.5830	−0.687652	−0.343826	−	0.939033i	$-0.611723\pi$
$811$	−19.5830	−0.687652	−0.343826	+	0.939033i	$0.611723\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.41699	−0.119692
$816$	0	0
$817$	−3.29150	−0.115155
$818$	0	0
$819$	0	0
$820$	0	0
$821$	35.0405	1.22292	0.611461	−	0.791274i	$-0.290582\pi$
$821$	35.0405	1.22292	0.611461	+	0.791274i	$0.290582\pi$
$822$	0	0
$823$	−11.1660	−0.389223	−0.194611	−	0.980880i	$-0.562345\pi$
$823$	−11.1660	−0.389223	−0.194611	+	0.980880i	$0.562345\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−14.0627	−0.489010	−0.244505	−	0.969648i	$-0.578625\pi$
$827$	−14.0627	−0.489010	−0.244505	+	0.969648i	$0.578625\pi$
$828$	0	0
$829$	29.8745	1.03758	0.518792	−	0.854900i	$-0.326382\pi$
$829$	29.8745	1.03758	0.518792	+	0.854900i	$0.326382\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−17.7490	−0.614230
$836$	0	0
$837$	0	0
$838$	0	0
$839$	42.0000	1.45000	0.725001	−	0.688748i	$-0.241839\pi$
$839$	42.0000	1.45000	0.725001	+	0.688748i	$0.241839\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	39.6863	1.36525
$846$	0	0
$847$	−10.5830	−0.363636
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−55.1033	−1.88892
$852$	0	0
$853$	−47.7490	−1.63490	−0.817448	−	0.576003i	$-0.804612\pi$
$853$	−47.7490	−1.63490	−0.817448	+	0.576003i	$0.804612\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	15.2288	0.520204	0.260102	−	0.965581i	$-0.416244\pi$
$857$	15.2288	0.520204	0.260102	+	0.965581i	$0.416244\pi$
$858$	0	0
$859$	5.83399	0.199053	0.0995266	−	0.995035i	$-0.468267\pi$
$859$	5.83399	0.199053	0.0995266	+	0.995035i	$0.468267\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−42.7085	−1.45381	−0.726907	−	0.686736i	$-0.759043\pi$
$863$	−42.7085	−1.45381	−0.726907	+	0.686736i	$0.759043\pi$
$864$	0	0
$865$	−59.5830	−2.02588
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.29150	0.111657
$870$	0	0
$871$	−49.1660	−1.66593
$872$	0	0
$873$	0	0
$874$	0	0
$875$	7.93725	0.268328
$876$	0	0
$877$	47.1660	1.59268	0.796342	−	0.604847i	$-0.206766\pi$
$877$	47.1660	1.59268	0.796342	+	0.604847i	$0.206766\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.93725	0.0652677	0.0326339	−	0.999467i	$-0.489610\pi$
$881$	1.93725	0.0652677	0.0326339	+	0.999467i	$0.489610\pi$
$882$	0	0
$883$	−51.7490	−1.74149	−0.870747	−	0.491732i	$-0.836364\pi$
$883$	−51.7490	−1.74149	−0.870747	+	0.491732i	$0.836364\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.4170	0.383345	0.191673	−	0.981459i	$-0.438609\pi$
$887$	11.4170	0.383345	0.191673	+	0.981459i	$0.438609\pi$
$888$	0	0
$889$	6.70850	0.224996
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0.708497	0.0237090
$894$	0	0
$895$	56.3320	1.88297
$896$	0	0
$897$	0	0
$898$	0	0
$899$	49.7490	1.65922
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	65.0405	2.16202
$906$	0	0
$907$	−17.7490	−0.589346	−0.294673	−	0.955598i	$-0.595211\pi$
$907$	−17.7490	−0.589346	−0.294673	+	0.955598i	$0.595211\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−39.8745	−1.32110	−0.660551	−	0.750781i	$-0.729677\pi$
$911$	−39.8745	−1.32110	−0.660551	+	0.750781i	$0.729677\pi$
$912$	0	0
$913$	46.4575	1.53752
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−15.2915	−0.504970
$918$	0	0
$919$	51.7490	1.70704	0.853521	−	0.521058i	$-0.174462\pi$
$919$	51.7490	1.70704	0.853521	+	0.521058i	$0.174462\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−70.0000	−2.30408
$924$	0	0
$925$	16.5830	0.545246
$926$	0	0
$927$	0	0
$928$	0	0
$929$	54.5830	1.79081	0.895405	−	0.445252i	$-0.146886\pi$
$929$	54.5830	1.79081	0.895405	+	0.445252i	$0.146886\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−15.7490	−0.514498	−0.257249	−	0.966345i	$-0.582816\pi$
$937$	−15.7490	−0.514498	−0.257249	+	0.966345i	$0.582816\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−46.5203	−1.51652	−0.758259	−	0.651954i	$-0.773950\pi$
$941$	−46.5203	−1.51652	−0.758259	+	0.651954i	$0.773950\pi$
$942$	0	0
$943$	22.2915	0.725911
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−35.2288	−1.14478	−0.572390	−	0.819981i	$-0.693984\pi$
$947$	−35.2288	−1.14478	−0.572390	+	0.819981i	$0.693984\pi$
$948$	0	0
$949$	−59.7490	−1.93954
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−35.1660	−1.13914	−0.569569	−	0.821943i	$-0.692890\pi$
$953$	−35.1660	−1.13914	−0.569569	+	0.821943i	$0.692890\pi$
$954$	0	0
$955$	−56.4980	−1.82823
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.58301	−0.277160
$960$	0	0
$961$	37.7490	1.21771
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.54249	0.0496544
$966$	0	0
$967$	−4.83399	−0.155451	−0.0777253	−	0.996975i	$-0.524766\pi$
$967$	−4.83399	−0.155451	−0.0777253	+	0.996975i	$0.524766\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−35.1660	−1.12853	−0.564265	−	0.825593i	$-0.690841\pi$
$971$	−35.1660	−1.12853	−0.564265	+	0.825593i	$0.690841\pi$
$972$	0	0
$973$	10.5830	0.339276
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.41699	−0.0453337	−0.0226668	−	0.999743i	$-0.507216\pi$
$977$	−1.41699	−0.0453337	−0.0226668	+	0.999743i	$0.507216\pi$
$978$	0	0
$979$	9.00000	0.287641
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−0.457513	−0.0145924	−0.00729620	−	0.999973i	$-0.502322\pi$
$983$	−0.457513	−0.0145924	−0.00729620	+	0.999973i	$0.502322\pi$
$984$	0	0
$985$	−3.41699	−0.108875
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−21.8745	−0.695569
$990$	0	0
$991$	21.2915	0.676347	0.338173	−	0.941084i	$-0.390191\pi$
$991$	21.2915	0.676347	0.338173	+	0.941084i	$0.390191\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	18.1882	0.576606
$996$	0	0
$997$	−38.0000	−1.20347	−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000	−1.20347	−0.601736	+	0.798695i	$0.705524\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1512.2.a.o.1.2	✓	2
3.2	odd	2		1512.2.a.p.1.1	yes	2
4.3	odd	2		3024.2.a.bk.1.2		2
12.11	even	2		3024.2.a.bh.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.a.o.1.2	✓	2	1.1	even	1	trivial
1512.2.a.p.1.1	yes	2	3.2	odd	2
3024.2.a.bh.1.1		2	12.11	even	2
3024.2.a.bk.1.2		2	4.3	odd	2

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:	\( N \)	\(=\)	\( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1512.a (trivial)

\(f(q)\)	\(=\)	\(q+2.64575 q^{5} -1.00000 q^{7} -4.64575 q^{11} -5.29150 q^{13} -1.00000 q^{19} -6.64575 q^{23} +2.00000 q^{25} -6.00000 q^{29} -8.29150 q^{31} -2.64575 q^{35} +8.29150 q^{37} -3.35425 q^{41} +3.29150 q^{43} -0.708497 q^{47} +1.00000 q^{49} -12.2915 q^{55} -7.29150 q^{59} +14.5830 q^{61} -14.0000 q^{65} +9.29150 q^{67} +13.2288 q^{71} +11.2915 q^{73} +4.64575 q^{77} -0.708497 q^{79} -10.0000 q^{83} -1.93725 q^{89} +5.29150 q^{91} -2.64575 q^{95} -2.58301 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7} - 4 q^{11} - 2 q^{19} - 8 q^{23} + 4 q^{25} - 12 q^{29} - 6 q^{31} + 6 q^{37} - 12 q^{41} - 4 q^{43} - 12 q^{47} + 2 q^{49} - 14 q^{55} - 4 q^{59} + 8 q^{61} - 28 q^{65} + 8 q^{67} + 12 q^{73}+ \cdots + 16 q^{97}+O(q^{100}) \) 2 * q - 2 * q^7 - 4 * q^11 - 2 * q^19 - 8 * q^23 + 4 * q^25 - 12 * q^29 - 6 * q^31 + 6 * q^37 - 12 * q^41 - 4 * q^43 - 12 * q^47 + 2 * q^49 - 14 * q^55 - 4 * q^59 + 8 * q^61 - 28 * q^65 + 8 * q^67 + 12 * q^73 + 4 * q^77 - 12 * q^79 - 20 * q^83 + 12 * q^89 + 16 * q^97

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