LMFDB - Embedded newform 9408.2.a.em.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9408,2,Mod(1,9408)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9408, 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("9408.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9408 = 2^{6} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9408.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:	$75.1232582216$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.2624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 3x^{2} + 2x + 1 $ x^4 - 2x^3 - 3x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 4704)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.360409$ of defining polynomial
Character	$\chi$	$=$	9408.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -4.13503 q^{5} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -4.13503 q^{5} +1.00000 q^{9} +2.29857 q^{11} +2.43361 q^{13} -4.13503 q^{15} +3.71279 q^{17} +7.84782 q^{19} -1.70143 q^{23} +12.0985 q^{25} +1.00000 q^{27} -3.25067 q^{29} +1.80904 q^{31} +2.29857 q^{33} +5.65685 q^{37} +2.43361 q^{39} -6.81128 q^{41} -5.44164 q^{43} -4.13503 q^{45} +13.2895 q^{47} +3.71279 q^{51} -9.09849 q^{53} -9.50467 q^{55} +7.84782 q^{57} -14.9463 q^{59} -2.16354 q^{61} -10.0630 q^{65} -10.2540 q^{67} -1.70143 q^{69} +3.95543 q^{71} +9.68428 q^{73} +12.0985 q^{75} -10.0388 q^{79} +1.00000 q^{81} -11.6956 q^{83} -15.3525 q^{85} -3.25067 q^{87} +12.7725 q^{89} +1.80904 q^{93} -32.4510 q^{95} +12.0274 q^{97} +2.29857 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{5} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 4 * q^5 + 4 * q^9	$ 4 q + 4 q^{3} - 4 q^{5} + 4 q^{9} + 4 q^{11} - 8 q^{13} - 4 q^{15} + 4 q^{17} + 8 q^{19} - 12 q^{23} + 12 q^{25} + 4 q^{27} + 8 q^{31} + 4 q^{33} - 8 q^{39} + 20 q^{41} - 8 q^{43} - 4 q^{45} + 16 q^{47} + 4 q^{51} + 8 q^{55} + 8 q^{57} - 16 q^{61} - 8 q^{65} - 8 q^{67} - 12 q^{69} - 12 q^{71} + 8 q^{73} + 12 q^{75} - 16 q^{79} + 4 q^{81} + 8 q^{85} + 28 q^{89} + 8 q^{93} - 24 q^{95} + 40 q^{97} + 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^5 + 4 * q^9 + 4 * q^11 - 8 * q^13 - 4 * q^15 + 4 * q^17 + 8 * q^19 - 12 * q^23 + 12 * q^25 + 4 * q^27 + 8 * q^31 + 4 * q^33 - 8 * q^39 + 20 * q^41 - 8 * q^43 - 4 * q^45 + 16 * q^47 + 4 * q^51 + 8 * q^55 + 8 * q^57 - 16 * q^61 - 8 * q^65 - 8 * q^67 - 12 * q^69 - 12 * q^71 + 8 * q^73 + 12 * q^75 - 16 * q^79 + 4 * q^81 + 8 * q^85 + 28 * q^89 + 8 * q^93 - 24 * q^95 + 40 * q^97 + 4 * 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
$5$	−4.13503	−1.84924	−0.924621	−	0.380888i	$-0.875618\pi$
$5$	−4.13503	−1.84924	−0.924621	+	0.380888i	$0.875618\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.29857	0.693046	0.346523	−	0.938042i	$-0.387362\pi$
$11$	2.29857	0.693046	0.346523	+	0.938042i	$0.387362\pi$
$12$	0	0
$13$	2.43361	0.674961	0.337480	−	0.941333i	$-0.390425\pi$
$13$	2.43361	0.674961	0.337480	+	0.941333i	$0.390425\pi$
$14$	0	0
$15$	−4.13503	−1.06766
$16$	0	0
$17$	3.71279	0.900483	0.450241	−	0.892907i	$-0.351338\pi$
$17$	3.71279	0.900483	0.450241	+	0.892907i	$0.351338\pi$
$18$	0	0
$19$	7.84782	1.80041	0.900207	−	0.435463i	$-0.143415\pi$
$19$	7.84782	1.80041	0.900207	+	0.435463i	$0.143415\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.70143	−0.354772	−0.177386	−	0.984141i	$-0.556764\pi$
$23$	−1.70143	−0.354772	−0.177386	+	0.984141i	$0.556764\pi$
$24$	0	0
$25$	12.0985	2.41970
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.25067	−0.603635	−0.301817	−	0.953366i	$-0.597593\pi$
$29$	−3.25067	−0.603635	−0.301817	+	0.953366i	$0.597593\pi$
$30$	0	0
$31$	1.80904	0.324912	0.162456	−	0.986716i	$-0.448058\pi$
$31$	1.80904	0.324912	0.162456	+	0.986716i	$0.448058\pi$
$32$	0	0
$33$	2.29857	0.400130
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.65685	0.929981	0.464991	−	0.885316i	$-0.346058\pi$
$37$	5.65685	0.929981	0.464991	+	0.885316i	$0.346058\pi$
$38$	0	0
$39$	2.43361	0.389689
$40$	0	0
$41$	−6.81128	−1.06374	−0.531871	−	0.846825i	$-0.678511\pi$
$41$	−6.81128	−1.06374	−0.531871	+	0.846825i	$0.678511\pi$
$42$	0	0
$43$	−5.44164	−0.829842	−0.414921	−	0.909857i	$-0.636191\pi$
$43$	−5.44164	−0.829842	−0.414921	+	0.909857i	$0.636191\pi$
$44$	0	0
$45$	−4.13503	−0.616414
$46$	0	0
$47$	13.2895	1.93847	0.969233	−	0.246144i	$-0.0791637\pi$
$47$	13.2895	1.93847	0.969233	+	0.246144i	$0.0791637\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	3.71279	0.519894
$52$	0	0
$53$	−9.09849	−1.24977	−0.624887	−	0.780715i	$-0.714855\pi$
$53$	−9.09849	−1.24977	−0.624887	+	0.780715i	$0.714855\pi$
$54$	0	0
$55$	−9.50467	−1.28161
$56$	0	0
$57$	7.84782	1.03947
$58$	0	0
$59$	−14.9463	−1.94584	−0.972922	−	0.231134i	$-0.925756\pi$
$59$	−14.9463	−1.94584	−0.972922	+	0.231134i	$0.925756\pi$
$60$	0	0
$61$	−2.16354	−0.277013	−0.138506	−	0.990362i	$-0.544230\pi$
$61$	−2.16354	−0.277013	−0.138506	+	0.990362i	$0.544230\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−10.0630	−1.24817
$66$	0	0
$67$	−10.2540	−1.25273	−0.626363	−	0.779532i	$-0.715457\pi$
$67$	−10.2540	−1.25273	−0.626363	+	0.779532i	$0.715457\pi$
$68$	0	0
$69$	−1.70143	−0.204828
$70$	0	0
$71$	3.95543	0.469423	0.234711	−	0.972065i	$-0.424585\pi$
$71$	3.95543	0.469423	0.234711	+	0.972065i	$0.424585\pi$
$72$	0	0
$73$	9.68428	1.13346	0.566730	−	0.823904i	$-0.308208\pi$
$73$	9.68428	1.13346	0.566730	+	0.823904i	$0.308208\pi$
$74$	0	0
$75$	12.0985	1.39701
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−10.0388	−1.12945	−0.564726	−	0.825279i	$-0.691018\pi$
$79$	−10.0388	−1.12945	−0.564726	+	0.825279i	$0.691018\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.6956	−1.28376	−0.641881	−	0.766804i	$-0.721846\pi$
$83$	−11.6956	−1.28376	−0.641881	+	0.766804i	$0.721846\pi$
$84$	0	0
$85$	−15.3525	−1.66521
$86$	0	0
$87$	−3.25067	−0.348509
$88$	0	0
$89$	12.7725	1.35388	0.676941	−	0.736037i	$-0.263305\pi$
$89$	12.7725	1.35388	0.676941	+	0.736037i	$0.263305\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.80904	0.187588
$94$	0	0
$95$	−32.4510	−3.32940
$96$	0	0
$97$	12.0274	1.22120	0.610600	−	0.791939i	$-0.290928\pi$
$97$	12.0274	1.22120	0.610600	+	0.791939i	$0.290928\pi$
$98$	0	0
$99$	2.29857	0.231015
$100$	0	0
$101$	−1.30661	−0.130012	−0.0650060	−	0.997885i	$-0.520707\pi$
$101$	−1.30661	−0.130012	−0.0650060	+	0.997885i	$0.520707\pi$
$102$	0	0
$103$	11.0033	1.08419	0.542095	−	0.840317i	$-0.317631\pi$
$103$	11.0033	1.08419	0.542095	+	0.840317i	$0.317631\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.8387	−1.04782	−0.523908	−	0.851775i	$-0.675527\pi$
$107$	−10.8387	−1.04782	−0.523908	+	0.851775i	$0.675527\pi$
$108$	0	0
$109$	9.44164	0.904345	0.452172	−	0.891931i	$-0.350649\pi$
$109$	9.44164	0.904345	0.452172	+	0.891931i	$0.350649\pi$
$110$	0	0
$111$	5.65685	0.536925
$112$	0	0
$113$	9.31371	0.876160	0.438080	−	0.898936i	$-0.355659\pi$
$113$	9.31371	0.876160	0.438080	+	0.898936i	$0.355659\pi$
$114$	0	0
$115$	7.03546	0.656060
$116$	0	0
$117$	2.43361	0.224987
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.71656	−0.519688
$122$	0	0
$123$	−6.81128	−0.614152
$124$	0	0
$125$	−29.3525	−2.62537
$126$	0	0
$127$	−13.1373	−1.16574	−0.582872	−	0.812564i	$-0.698071\pi$
$127$	−13.1373	−1.16574	−0.582872	+	0.812564i	$0.698071\pi$
$128$	0	0
$129$	−5.44164	−0.479109
$130$	0	0
$131$	8.81236	0.769940	0.384970	−	0.922929i	$-0.374212\pi$
$131$	8.81236	0.769940	0.384970	+	0.922929i	$0.374212\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−4.13503	−0.355887
$136$	0	0
$137$	0.367398	0.0313889	0.0156945	−	0.999877i	$-0.495004\pi$
$137$	0.367398	0.0313889	0.0156945	+	0.999877i	$0.495004\pi$
$138$	0	0
$139$	15.3137	1.29889	0.649446	−	0.760408i	$-0.275001\pi$
$139$	15.3137	1.29889	0.649446	+	0.760408i	$0.275001\pi$
$140$	0	0
$141$	13.2895	1.11917
$142$	0	0
$143$	5.59382	0.467779
$144$	0	0
$145$	13.4416	1.11627
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.4122	1.34454	0.672270	−	0.740306i	$-0.265319\pi$
$149$	16.4122	1.34454	0.672270	+	0.740306i	$0.265319\pi$
$150$	0	0
$151$	7.78478	0.633517	0.316758	−	0.948506i	$-0.397406\pi$
$151$	7.78478	0.633517	0.316758	+	0.948506i	$0.397406\pi$
$152$	0	0
$153$	3.71279	0.300161
$154$	0	0
$155$	−7.48042	−0.600842
$156$	0	0
$157$	17.2050	1.37311	0.686555	−	0.727078i	$-0.259122\pi$
$157$	17.2050	1.37311	0.686555	+	0.727078i	$0.259122\pi$
$158$	0	0
$159$	−9.09849	−0.721557
$160$	0	0
$161$	0	0
$162$	0	0
$163$	6.71656	0.526082	0.263041	−	0.964785i	$-0.415275\pi$
$163$	6.71656	0.526082	0.263041	+	0.964785i	$0.415275\pi$
$164$	0	0
$165$	−9.50467	−0.739938
$166$	0	0
$167$	1.21189	0.0937789	0.0468894	−	0.998900i	$-0.485069\pi$
$167$	1.21189	0.0937789	0.0468894	+	0.998900i	$0.485069\pi$
$168$	0	0
$169$	−7.07757	−0.544428
$170$	0	0
$171$	7.84782	0.600138
$172$	0	0
$173$	10.2772	0.781359	0.390679	−	0.920527i	$-0.372240\pi$
$173$	10.2772	0.781359	0.390679	+	0.920527i	$0.372240\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−14.9463	−1.12343
$178$	0	0
$179$	12.1221	0.906051	0.453026	−	0.891498i	$-0.350345\pi$
$179$	12.1221	0.906051	0.453026	+	0.891498i	$0.350345\pi$
$180$	0	0
$181$	−23.2460	−1.72786	−0.863930	−	0.503613i	$-0.832004\pi$
$181$	−23.2460	−1.72786	−0.863930	+	0.503613i	$0.832004\pi$
$182$	0	0
$183$	−2.16354	−0.159934
$184$	0	0
$185$	−23.3913	−1.71976
$186$	0	0
$187$	8.53411	0.624076
$188$	0	0
$189$	0	0
$190$	0	0
$191$	23.2691	1.68370	0.841848	−	0.539715i	$-0.181468\pi$
$191$	23.2691	1.68370	0.841848	+	0.539715i	$0.181468\pi$
$192$	0	0
$193$	−4.28613	−0.308522	−0.154261	−	0.988030i	$-0.549300\pi$
$193$	−4.28613	−0.308522	−0.154261	+	0.988030i	$0.549300\pi$
$194$	0	0
$195$	−10.0630	−0.720629
$196$	0	0
$197$	11.9818	0.853666	0.426833	−	0.904331i	$-0.359629\pi$
$197$	11.9818	0.853666	0.426833	+	0.904331i	$0.359629\pi$
$198$	0	0
$199$	22.1970	1.57350	0.786751	−	0.617270i	$-0.211761\pi$
$199$	22.1970	1.57350	0.786751	+	0.617270i	$0.211761\pi$
$200$	0	0
$201$	−10.2540	−0.723261
$202$	0	0
$203$	0	0
$204$	0	0
$205$	28.1649	1.96712
$206$	0	0
$207$	−1.70143	−0.118257
$208$	0	0
$209$	18.0388	1.24777
$210$	0	0
$211$	0.540129	0.0371840	0.0185920	−	0.999827i	$-0.494082\pi$
$211$	0.540129	0.0371840	0.0185920	+	0.999827i	$0.494082\pi$
$212$	0	0
$213$	3.95543	0.271021
$214$	0	0
$215$	22.5013	1.53458
$216$	0	0
$217$	0	0
$218$	0	0
$219$	9.68428	0.654403
$220$	0	0
$221$	9.03546	0.607791
$222$	0	0
$223$	−10.1194	−0.677646	−0.338823	−	0.940850i	$-0.610029\pi$
$223$	−10.1194	−0.677646	−0.338823	+	0.940850i	$0.610029\pi$
$224$	0	0
$225$	12.0985	0.806566
$226$	0	0
$227$	27.7587	1.84241	0.921204	−	0.389080i	$-0.127207\pi$
$227$	27.7587	1.84241	0.921204	+	0.389080i	$0.127207\pi$
$228$	0	0
$229$	−14.8962	−0.984366	−0.492183	−	0.870492i	$-0.663801\pi$
$229$	−14.8962	−0.984366	−0.492183	+	0.870492i	$0.663801\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.5644	−0.954144	−0.477072	−	0.878864i	$-0.658302\pi$
$233$	−14.5644	−0.954144	−0.477072	+	0.878864i	$0.658302\pi$
$234$	0	0
$235$	−54.9523	−3.58469
$236$	0	0
$237$	−10.0388	−0.652089
$238$	0	0
$239$	−1.70143	−0.110056	−0.0550281	−	0.998485i	$-0.517525\pi$
$239$	−1.70143	−0.110056	−0.0550281	+	0.998485i	$0.517525\pi$
$240$	0	0
$241$	27.0141	1.74013	0.870064	−	0.492939i	$-0.164077\pi$
$241$	27.0141	1.74013	0.870064	+	0.492939i	$0.164077\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	19.0985	1.21521
$248$	0	0
$249$	−11.6956	−0.741181
$250$	0	0
$251$	12.8269	0.809626	0.404813	−	0.914399i	$-0.367337\pi$
$251$	12.8269	0.809626	0.404813	+	0.914399i	$0.367337\pi$
$252$	0	0
$253$	−3.91085	−0.245873
$254$	0	0
$255$	−15.3525	−0.961410
$256$	0	0
$257$	−1.99892	−0.124689	−0.0623445	−	0.998055i	$-0.519858\pi$
$257$	−1.99892	−0.124689	−0.0623445	+	0.998055i	$0.519858\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−3.25067	−0.201212
$262$	0	0
$263$	−1.07215	−0.0661118	−0.0330559	−	0.999454i	$-0.510524\pi$
$263$	−1.07215	−0.0661118	−0.0330559	+	0.999454i	$0.510524\pi$
$264$	0	0
$265$	37.6226	2.31114
$266$	0	0
$267$	12.7725	0.781664
$268$	0	0
$269$	−31.7188	−1.93393	−0.966965	−	0.254910i	$-0.917954\pi$
$269$	−31.7188	−1.93393	−0.966965	+	0.254910i	$0.917954\pi$
$270$	0	0
$271$	0.614744	0.0373430	0.0186715	−	0.999826i	$-0.494056\pi$
$271$	0.614744	0.0373430	0.0186715	+	0.999826i	$0.494056\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	27.8093	1.67696
$276$	0	0
$277$	14.6747	0.881718	0.440859	−	0.897576i	$-0.354674\pi$
$277$	14.6747	0.881718	0.440859	+	0.897576i	$0.354674\pi$
$278$	0	0
$279$	1.80904	0.108304
$280$	0	0
$281$	12.1406	0.724248	0.362124	−	0.932130i	$-0.382052\pi$
$281$	12.1406	0.724248	0.362124	+	0.932130i	$0.382052\pi$
$282$	0	0
$283$	−7.54346	−0.448412	−0.224206	−	0.974542i	$-0.571979\pi$
$283$	−7.54346	−0.448412	−0.224206	+	0.974542i	$0.571979\pi$
$284$	0	0
$285$	−32.4510	−1.92223
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−3.21522	−0.189130
$290$	0	0
$291$	12.0274	0.705060
$292$	0	0
$293$	−21.4648	−1.25399	−0.626994	−	0.779024i	$-0.715715\pi$
$293$	−21.4648	−1.25399	−0.626994	+	0.779024i	$0.715715\pi$
$294$	0	0
$295$	61.8035	3.59834
$296$	0	0
$297$	2.29857	0.133377
$298$	0	0
$299$	−4.14060	−0.239457
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1.30661	−0.0750625
$304$	0	0
$305$	8.94631	0.512264
$306$	0	0
$307$	−7.41738	−0.423333	−0.211666	−	0.977342i	$-0.567889\pi$
$307$	−7.41738	−0.423333	−0.211666	+	0.977342i	$0.567889\pi$
$308$	0	0
$309$	11.0033	0.625957
$310$	0	0
$311$	14.7881	0.838557	0.419278	−	0.907858i	$-0.362283\pi$
$311$	14.7881	0.838557	0.419278	+	0.907858i	$0.362283\pi$
$312$	0	0
$313$	−9.95434	−0.562653	−0.281326	−	0.959612i	$-0.590774\pi$
$313$	−9.95434	−0.562653	−0.281326	+	0.959612i	$0.590774\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.97907	−0.167321	−0.0836607	−	0.996494i	$-0.526661\pi$
$317$	−2.97907	−0.167321	−0.0836607	+	0.996494i	$0.526661\pi$
$318$	0	0
$319$	−7.47191	−0.418347
$320$	0	0
$321$	−10.8387	−0.604957
$322$	0	0
$323$	29.1373	1.62124
$324$	0	0
$325$	29.4430	1.63320
$326$	0	0
$327$	9.44164	0.522124
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−6.80571	−0.374076	−0.187038	−	0.982353i	$-0.559889\pi$
$331$	−6.80571	−0.374076	−0.187038	+	0.982353i	$0.559889\pi$
$332$	0	0
$333$	5.65685	0.309994
$334$	0	0
$335$	42.4006	2.31659
$336$	0	0
$337$	−7.69564	−0.419208	−0.209604	−	0.977786i	$-0.567217\pi$
$337$	−7.69564	−0.419208	−0.209604	+	0.977786i	$0.567217\pi$
$338$	0	0
$339$	9.31371	0.505851
$340$	0	0
$341$	4.15820	0.225179
$342$	0	0
$343$	0	0
$344$	0	0
$345$	7.03546	0.378776
$346$	0	0
$347$	−10.2094	−0.548071	−0.274035	−	0.961720i	$-0.588359\pi$
$347$	−10.2094	−0.548071	−0.274035	+	0.961720i	$0.588359\pi$
$348$	0	0
$349$	27.6742	1.48137	0.740684	−	0.671854i	$-0.234502\pi$
$349$	27.6742	1.48137	0.740684	+	0.671854i	$0.234502\pi$
$350$	0	0
$351$	2.43361	0.129896
$352$	0	0
$353$	−2.51849	−0.134046	−0.0670230	−	0.997751i	$-0.521350\pi$
$353$	−2.51849	−0.134046	−0.0670230	+	0.997751i	$0.521350\pi$
$354$	0	0
$355$	−16.3558	−0.868077
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.58470	0.241971	0.120986	−	0.992654i	$-0.461394\pi$
$359$	4.58470	0.241971	0.120986	+	0.992654i	$0.461394\pi$
$360$	0	0
$361$	42.5883	2.24149
$362$	0	0
$363$	−5.71656	−0.300042
$364$	0	0
$365$	−40.0448	−2.09604
$366$	0	0
$367$	−9.38462	−0.489873	−0.244937	−	0.969539i	$-0.578767\pi$
$367$	−9.38462	−0.489873	−0.244937	+	0.969539i	$0.578767\pi$
$368$	0	0
$369$	−6.81128	−0.354581
$370$	0	0
$371$	0	0
$372$	0	0
$373$	11.0027	0.569698	0.284849	−	0.958572i	$-0.408057\pi$
$373$	11.0027	0.569698	0.284849	+	0.958572i	$0.408057\pi$
$374$	0	0
$375$	−29.3525	−1.51576
$376$	0	0
$377$	−7.91085	−0.407430
$378$	0	0
$379$	26.7050	1.37174	0.685871	−	0.727723i	$-0.259421\pi$
$379$	26.7050	1.37174	0.685871	+	0.727723i	$0.259421\pi$
$380$	0	0
$381$	−13.1373	−0.673043
$382$	0	0
$383$	−28.3940	−1.45086	−0.725432	−	0.688294i	$-0.758360\pi$
$383$	−28.3940	−1.45086	−0.725432	+	0.688294i	$0.758360\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.44164	−0.276614
$388$	0	0
$389$	−20.4450	−1.03660	−0.518300	−	0.855199i	$-0.673435\pi$
$389$	−20.4450	−1.03660	−0.518300	+	0.855199i	$0.673435\pi$
$390$	0	0
$391$	−6.31704	−0.319466
$392$	0	0
$393$	8.81236	0.444525
$394$	0	0
$395$	41.5107	2.08863
$396$	0	0
$397$	15.8752	0.796756	0.398378	−	0.917221i	$-0.369573\pi$
$397$	15.8752	0.796756	0.398378	+	0.917221i	$0.369573\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−19.2507	−0.961333	−0.480666	−	0.876904i	$-0.659605\pi$
$401$	−19.2507	−0.961333	−0.480666	+	0.876904i	$0.659605\pi$
$402$	0	0
$403$	4.40248	0.219303
$404$	0	0
$405$	−4.13503	−0.205471
$406$	0	0
$407$	13.0027	0.644519
$408$	0	0
$409$	−1.27963	−0.0632737	−0.0316368	−	0.999499i	$-0.510072\pi$
$409$	−1.27963	−0.0632737	−0.0316368	+	0.999499i	$0.510072\pi$
$410$	0	0
$411$	0.367398	0.0181224
$412$	0	0
$413$	0	0
$414$	0	0
$415$	48.3618	2.37399
$416$	0	0
$417$	15.3137	0.749916
$418$	0	0
$419$	−16.8754	−0.824417	−0.412209	−	0.911090i	$-0.635242\pi$
$419$	−16.8754	−0.824417	−0.412209	+	0.911090i	$0.635242\pi$
$420$	0	0
$421$	−23.6774	−1.15397	−0.576983	−	0.816756i	$-0.695770\pi$
$421$	−23.6774	−1.15397	−0.576983	+	0.816756i	$0.695770\pi$
$422$	0	0
$423$	13.2895	0.646155
$424$	0	0
$425$	44.9191	2.17890
$426$	0	0
$427$	0	0
$428$	0	0
$429$	5.59382	0.270072
$430$	0	0
$431$	−11.8208	−0.569390	−0.284695	−	0.958618i	$-0.591892\pi$
$431$	−11.8208	−0.569390	−0.284695	+	0.958618i	$0.591892\pi$
$432$	0	0
$433$	−21.9771	−1.05615	−0.528075	−	0.849198i	$-0.677086\pi$
$433$	−21.9771	−1.05615	−0.528075	+	0.849198i	$0.677086\pi$
$434$	0	0
$435$	13.4416	0.644477
$436$	0	0
$437$	−13.3525	−0.638736
$438$	0	0
$439$	35.3137	1.68543	0.842716	−	0.538359i	$-0.180956\pi$
$439$	35.3137	1.68543	0.842716	+	0.538359i	$0.180956\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	17.1248	0.813625	0.406813	−	0.913512i	$-0.366640\pi$
$443$	17.1248	0.813625	0.406813	+	0.913512i	$0.366640\pi$
$444$	0	0
$445$	−52.8147	−2.50366
$446$	0	0
$447$	16.4122	0.776270
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	−15.6562	−0.737223
$452$	0	0
$453$	7.78478	0.365761
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4.80571	0.224802	0.112401	−	0.993663i	$-0.464146\pi$
$457$	4.80571	0.224802	0.112401	+	0.993663i	$0.464146\pi$
$458$	0	0
$459$	3.71279	0.173298
$460$	0	0
$461$	21.0798	0.981785	0.490892	−	0.871220i	$-0.336671\pi$
$461$	21.0798	0.981785	0.490892	+	0.871220i	$0.336671\pi$
$462$	0	0
$463$	14.2861	0.663933	0.331966	−	0.943291i	$-0.392288\pi$
$463$	14.2861	0.663933	0.331966	+	0.943291i	$0.392288\pi$
$464$	0	0
$465$	−7.48042	−0.346896
$466$	0	0
$467$	22.5159	1.04191	0.520955	−	0.853584i	$-0.325576\pi$
$467$	22.5159	1.04191	0.520955	+	0.853584i	$0.325576\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	17.2050	0.792765
$472$	0	0
$473$	−12.5080	−0.575118
$474$	0	0
$475$	94.9468	4.35646
$476$	0	0
$477$	−9.09849	−0.416591
$478$	0	0
$479$	−8.90753	−0.406995	−0.203498	−	0.979075i	$-0.565231\pi$
$479$	−8.90753	−0.406995	−0.203498	+	0.979075i	$0.565231\pi$
$480$	0	0
$481$	13.7665	0.627701
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−49.7338	−2.25829
$486$	0	0
$487$	23.7848	1.07779	0.538896	−	0.842372i	$-0.318842\pi$
$487$	23.7848	1.07779	0.538896	+	0.842372i	$0.318842\pi$
$488$	0	0
$489$	6.71656	0.303733
$490$	0	0
$491$	14.1524	0.638689	0.319345	−	0.947639i	$-0.396537\pi$
$491$	14.1524	0.638689	0.319345	+	0.947639i	$0.396537\pi$
$492$	0	0
$493$	−12.0691	−0.543563
$494$	0	0
$495$	−9.50467	−0.427203
$496$	0	0
$497$	0	0
$498$	0	0
$499$	22.4122	1.00331	0.501654	−	0.865068i	$-0.332725\pi$
$499$	22.4122	1.00331	0.501654	+	0.865068i	$0.332725\pi$
$500$	0	0
$501$	1.21189	0.0541432
$502$	0	0
$503$	6.61538	0.294965	0.147483	−	0.989065i	$-0.452883\pi$
$503$	6.61538	0.294965	0.147483	+	0.989065i	$0.452883\pi$
$504$	0	0
$505$	5.40285	0.240424
$506$	0	0
$507$	−7.07757	−0.314326
$508$	0	0
$509$	−1.93588	−0.0858064	−0.0429032	−	0.999079i	$-0.513661\pi$
$509$	−1.93588	−0.0858064	−0.0429032	+	0.999079i	$0.513661\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	7.84782	0.346490
$514$	0	0
$515$	−45.4991	−2.00493
$516$	0	0
$517$	30.5468	1.34345
$518$	0	0
$519$	10.2772	0.451118
$520$	0	0
$521$	10.7079	0.469123	0.234561	−	0.972101i	$-0.424635\pi$
$521$	10.7079	0.469123	0.234561	+	0.972101i	$0.424635\pi$
$522$	0	0
$523$	−32.0896	−1.40318	−0.701590	−	0.712581i	$-0.747526\pi$
$523$	−32.0896	−1.40318	−0.701590	+	0.712581i	$0.747526\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.71656	0.292578
$528$	0	0
$529$	−20.1051	−0.874137
$530$	0	0
$531$	−14.9463	−0.648615
$532$	0	0
$533$	−16.5760	−0.717985
$534$	0	0
$535$	44.8184	1.93767
$536$	0	0
$537$	12.1221	0.523109
$538$	0	0
$539$	0	0
$540$	0	0
$541$	17.4804	0.751542	0.375771	−	0.926713i	$-0.377378\pi$
$541$	17.4804	0.751542	0.375771	+	0.926713i	$0.377378\pi$
$542$	0	0
$543$	−23.2460	−0.997580
$544$	0	0
$545$	−39.0415	−1.67235
$546$	0	0
$547$	−34.6243	−1.48043	−0.740215	−	0.672370i	$-0.765276\pi$
$547$	−34.6243	−1.48043	−0.740215	+	0.672370i	$0.765276\pi$
$548$	0	0
$549$	−2.16354	−0.0923377
$550$	0	0
$551$	−25.5107	−1.08679
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−23.3913	−0.992904
$556$	0	0
$557$	−11.9042	−0.504397	−0.252199	−	0.967676i	$-0.581154\pi$
$557$	−11.9042	−0.504397	−0.252199	+	0.967676i	$0.581154\pi$
$558$	0	0
$559$	−13.2428	−0.560111
$560$	0	0
$561$	8.53411	0.360310
$562$	0	0
$563$	18.1339	0.764255	0.382127	−	0.924110i	$-0.375192\pi$
$563$	18.1339	0.764255	0.382127	+	0.924110i	$0.375192\pi$
$564$	0	0
$565$	−38.5125	−1.62023
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−19.7102	−0.826293	−0.413147	−	0.910665i	$-0.635570\pi$
$569$	−19.7102	−0.826293	−0.413147	+	0.910665i	$0.635570\pi$
$570$	0	0
$571$	−5.96122	−0.249469	−0.124735	−	0.992190i	$-0.539808\pi$
$571$	−5.96122	−0.249469	−0.124735	+	0.992190i	$0.539808\pi$
$572$	0	0
$573$	23.2691	0.972082
$574$	0	0
$575$	−20.5847	−0.858441
$576$	0	0
$577$	34.9249	1.45394	0.726971	−	0.686668i	$-0.240927\pi$
$577$	34.9249	1.45394	0.726971	+	0.686668i	$0.240927\pi$
$578$	0	0
$579$	−4.28613	−0.178125
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−20.9135	−0.866151
$584$	0	0
$585$	−10.0630	−0.416055
$586$	0	0
$587$	9.05369	0.373686	0.186843	−	0.982390i	$-0.440174\pi$
$587$	9.05369	0.373686	0.186843	+	0.982390i	$0.440174\pi$
$588$	0	0
$589$	14.1970	0.584977
$590$	0	0
$591$	11.9818	0.492864
$592$	0	0
$593$	−0.129013	−0.00529795	−0.00264897	−	0.999996i	$-0.500843\pi$
$593$	−0.129013	−0.00529795	−0.00264897	+	0.999996i	$0.500843\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	22.1970	0.908462
$598$	0	0
$599$	21.0357	0.859495	0.429747	−	0.902949i	$-0.358603\pi$
$599$	21.0357	0.859495	0.429747	+	0.902949i	$0.358603\pi$
$600$	0	0
$601$	4.67307	0.190619	0.0953093	−	0.995448i	$-0.469616\pi$
$601$	4.67307	0.190619	0.0953093	+	0.995448i	$0.469616\pi$
$602$	0	0
$603$	−10.2540	−0.417575
$604$	0	0
$605$	23.6382	0.961028
$606$	0	0
$607$	32.3164	1.31168	0.655841	−	0.754899i	$-0.272314\pi$
$607$	32.3164	1.31168	0.655841	+	0.754899i	$0.272314\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	32.3413	1.30839
$612$	0	0
$613$	42.6480	1.72254	0.861268	−	0.508152i	$-0.169671\pi$
$613$	42.6480	1.72254	0.861268	+	0.508152i	$0.169671\pi$
$614$	0	0
$615$	28.1649	1.13572
$616$	0	0
$617$	−22.3741	−0.900745	−0.450373	−	0.892841i	$-0.648709\pi$
$617$	−22.3741	−0.900745	−0.450373	+	0.892841i	$0.648709\pi$
$618$	0	0
$619$	31.3428	1.25977	0.629886	−	0.776687i	$-0.283102\pi$
$619$	31.3428	1.25977	0.629886	+	0.776687i	$0.283102\pi$
$620$	0	0
$621$	−1.70143	−0.0682759
$622$	0	0
$623$	0	0
$624$	0	0
$625$	60.8810	2.43524
$626$	0	0
$627$	18.0388	0.720400
$628$	0	0
$629$	21.0027	0.837432
$630$	0	0
$631$	−2.79413	−0.111233	−0.0556163	−	0.998452i	$-0.517712\pi$
$631$	−2.79413	−0.111233	−0.0556163	+	0.998452i	$0.517712\pi$
$632$	0	0
$633$	0.540129	0.0214682
$634$	0	0
$635$	54.3231	2.15574
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3.95543	0.156474
$640$	0	0
$641$	−2.17042	−0.0857262	−0.0428631	−	0.999081i	$-0.513648\pi$
$641$	−2.17042	−0.0857262	−0.0428631	+	0.999081i	$0.513648\pi$
$642$	0	0
$643$	−8.91604	−0.351614	−0.175807	−	0.984425i	$-0.556254\pi$
$643$	−8.91604	−0.351614	−0.175807	+	0.984425i	$0.556254\pi$
$644$	0	0
$645$	22.5013	0.885990
$646$	0	0
$647$	−18.1018	−0.711656	−0.355828	−	0.934551i	$-0.615801\pi$
$647$	−18.1018	−0.711656	−0.355828	+	0.934551i	$0.615801\pi$
$648$	0	0
$649$	−34.3552	−1.34856
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0.826893	0.0323588	0.0161794	−	0.999869i	$-0.494850\pi$
$653$	0.826893	0.0323588	0.0161794	+	0.999869i	$0.494850\pi$
$654$	0	0
$655$	−36.4394	−1.42381
$656$	0	0
$657$	9.68428	0.377820
$658$	0	0
$659$	32.4956	1.26585	0.632924	−	0.774214i	$-0.281855\pi$
$659$	32.4956	1.26585	0.632924	+	0.774214i	$0.281855\pi$
$660$	0	0
$661$	−27.4127	−1.06623	−0.533115	−	0.846043i	$-0.678979\pi$
$661$	−27.4127	−1.06623	−0.533115	+	0.846043i	$0.678979\pi$
$662$	0	0
$663$	9.03546	0.350908
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.53078	0.214153
$668$	0	0
$669$	−10.1194	−0.391239
$670$	0	0
$671$	−4.97306	−0.191983
$672$	0	0
$673$	31.9884	1.23306	0.616531	−	0.787330i	$-0.288537\pi$
$673$	31.9884	1.23306	0.616531	+	0.787330i	$0.288537\pi$
$674$	0	0
$675$	12.0985	0.465671
$676$	0	0
$677$	40.6979	1.56415	0.782073	−	0.623186i	$-0.214162\pi$
$677$	40.6979	1.56415	0.782073	+	0.623186i	$0.214162\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	27.7587	1.06371
$682$	0	0
$683$	−17.0357	−0.651852	−0.325926	−	0.945395i	$-0.605676\pi$
$683$	−17.0357	−0.651852	−0.325926	+	0.945395i	$0.605676\pi$
$684$	0	0
$685$	−1.51920	−0.0580458
$686$	0	0
$687$	−14.8962	−0.568324
$688$	0	0
$689$	−22.1421	−0.843548
$690$	0	0
$691$	25.8926	0.985002	0.492501	−	0.870312i	$-0.336083\pi$
$691$	25.8926	0.985002	0.492501	+	0.870312i	$0.336083\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−63.3227	−2.40197
$696$	0	0
$697$	−25.2888	−0.957882
$698$	0	0
$699$	−14.5644	−0.550875
$700$	0	0
$701$	25.8781	0.977402	0.488701	−	0.872451i	$-0.337471\pi$
$701$	25.8781	0.977402	0.488701	+	0.872451i	$0.337471\pi$
$702$	0	0
$703$	44.3940	1.67435
$704$	0	0
$705$	−54.9523	−2.06962
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−30.2660	−1.13666	−0.568332	−	0.822799i	$-0.692411\pi$
$709$	−30.2660	−1.13666	−0.568332	+	0.822799i	$0.692411\pi$
$710$	0	0
$711$	−10.0388	−0.376484
$712$	0	0
$713$	−3.07794	−0.115270
$714$	0	0
$715$	−23.1306	−0.865036
$716$	0	0
$717$	−1.70143	−0.0635410
$718$	0	0
$719$	−15.2652	−0.569296	−0.284648	−	0.958632i	$-0.591877\pi$
$719$	−15.2652	−0.569296	−0.284648	+	0.958632i	$0.591877\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	27.0141	1.00466
$724$	0	0
$725$	−39.3282	−1.46061
$726$	0	0
$727$	−19.0154	−0.705241	−0.352620	−	0.935766i	$-0.614709\pi$
$727$	−19.0154	−0.705241	−0.352620	+	0.935766i	$0.614709\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−20.2036	−0.747259
$732$	0	0
$733$	9.21353	0.340309	0.170155	−	0.985417i	$-0.445573\pi$
$733$	9.21353	0.340309	0.170155	+	0.985417i	$0.445573\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−23.5696	−0.868196
$738$	0	0
$739$	−38.1078	−1.40182	−0.700910	−	0.713250i	$-0.747222\pi$
$739$	−38.1078	−1.40182	−0.700910	+	0.713250i	$0.747222\pi$
$740$	0	0
$741$	19.0985	0.701601
$742$	0	0
$743$	−2.46528	−0.0904425	−0.0452213	−	0.998977i	$-0.514399\pi$
$743$	−2.46528	−0.0904425	−0.0452213	+	0.998977i	$0.514399\pi$
$744$	0	0
$745$	−67.8650	−2.48638
$746$	0	0
$747$	−11.6956	−0.427921
$748$	0	0
$749$	0	0
$750$	0	0
$751$	28.2843	1.03211	0.516054	−	0.856556i	$-0.327400\pi$
$751$	28.2843	1.03211	0.516054	+	0.856556i	$0.327400\pi$
$752$	0	0
$753$	12.8269	0.467438
$754$	0	0
$755$	−32.1903	−1.17153
$756$	0	0
$757$	32.0691	1.16557	0.582785	−	0.812627i	$-0.301963\pi$
$757$	32.0691	1.16557	0.582785	+	0.812627i	$0.301963\pi$
$758$	0	0
$759$	−3.91085	−0.141955
$760$	0	0
$761$	−3.99136	−0.144687	−0.0723434	−	0.997380i	$-0.523048\pi$
$761$	−3.99136	−0.144687	−0.0723434	+	0.997380i	$0.523048\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−15.3525	−0.555071
$766$	0	0
$767$	−36.3734	−1.31337
$768$	0	0
$769$	21.8082	0.786423	0.393212	−	0.919448i	$-0.371364\pi$
$769$	21.8082	0.786423	0.393212	+	0.919448i	$0.371364\pi$
$770$	0	0
$771$	−1.99892	−0.0719892
$772$	0	0
$773$	18.9956	0.683224	0.341612	−	0.939841i	$-0.389027\pi$
$773$	18.9956	0.683224	0.341612	+	0.939841i	$0.389027\pi$
$774$	0	0
$775$	21.8866	0.786190
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−53.4537	−1.91518
$780$	0	0
$781$	9.09184	0.325332
$782$	0	0
$783$	−3.25067	−0.116170
$784$	0	0
$785$	−71.1433	−2.53921
$786$	0	0
$787$	−45.2063	−1.61143	−0.805716	−	0.592302i	$-0.798219\pi$
$787$	−45.2063	−1.61143	−0.805716	+	0.592302i	$0.798219\pi$
$788$	0	0
$789$	−1.07215	−0.0381696
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−5.26520	−0.186973
$794$	0	0
$795$	37.6226	1.33433
$796$	0	0
$797$	−36.5517	−1.29473	−0.647364	−	0.762181i	$-0.724129\pi$
$797$	−36.5517	−1.29473	−0.647364	+	0.762181i	$0.724129\pi$
$798$	0	0
$799$	49.3409	1.74556
$800$	0	0
$801$	12.7725	0.451294
$802$	0	0
$803$	22.2600	0.785539
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−31.7188	−1.11655
$808$	0	0
$809$	−36.0829	−1.26861	−0.634304	−	0.773083i	$-0.718713\pi$
$809$	−36.0829	−1.26861	−0.634304	+	0.773083i	$0.718713\pi$
$810$	0	0
$811$	−26.6565	−0.936036	−0.468018	−	0.883719i	$-0.655032\pi$
$811$	−26.6565	−0.936036	−0.468018	+	0.883719i	$0.655032\pi$
$812$	0	0
$813$	0.614744	0.0215600
$814$	0	0
$815$	−27.7732	−0.972853
$816$	0	0
$817$	−42.7050	−1.49406
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−42.0369	−1.46710	−0.733549	−	0.679636i	$-0.762138\pi$
$821$	−42.0369	−1.46710	−0.733549	+	0.679636i	$0.762138\pi$
$822$	0	0
$823$	26.4692	0.922659	0.461329	−	0.887229i	$-0.347373\pi$
$823$	26.4692	0.922659	0.461329	+	0.887229i	$0.347373\pi$
$824$	0	0
$825$	27.8093	0.968194
$826$	0	0
$827$	55.8663	1.94266	0.971330	−	0.237733i	$-0.0764044\pi$
$827$	55.8663	1.94266	0.971330	+	0.237733i	$0.0764044\pi$
$828$	0	0
$829$	0.394822	0.0137127	0.00685637	−	0.999976i	$-0.497818\pi$
$829$	0.394822	0.0137127	0.00685637	+	0.999976i	$0.497818\pi$
$830$	0	0
$831$	14.6747	0.509060
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−5.01120	−0.173420
$836$	0	0
$837$	1.80904	0.0625294
$838$	0	0
$839$	−20.9851	−0.724486	−0.362243	−	0.932084i	$-0.617989\pi$
$839$	−20.9851	−0.724486	−0.362243	+	0.932084i	$0.617989\pi$
$840$	0	0
$841$	−18.4331	−0.635625
$842$	0	0
$843$	12.1406	0.418145
$844$	0	0
$845$	29.2660	1.00678
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−7.54346	−0.258891
$850$	0	0
$851$	−9.62473	−0.329931
$852$	0	0
$853$	−8.76517	−0.300114	−0.150057	−	0.988677i	$-0.547946\pi$
$853$	−8.76517	−0.300114	−0.150057	+	0.988677i	$0.547946\pi$
$854$	0	0
$855$	−32.4510	−1.10980
$856$	0	0
$857$	9.45879	0.323106	0.161553	−	0.986864i	$-0.448350\pi$
$857$	9.45879	0.323106	0.161553	+	0.986864i	$0.448350\pi$
$858$	0	0
$859$	32.7377	1.11700	0.558499	−	0.829505i	$-0.311378\pi$
$859$	32.7377	1.11700	0.558499	+	0.829505i	$0.311378\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−8.73086	−0.297202	−0.148601	−	0.988897i	$-0.547477\pi$
$863$	−8.73086	−0.297202	−0.148601	+	0.988897i	$0.547477\pi$
$864$	0	0
$865$	−42.4964	−1.44492
$866$	0	0
$867$	−3.21522	−0.109194
$868$	0	0
$869$	−23.0749	−0.782762
$870$	0	0
$871$	−24.9542	−0.845540
$872$	0	0
$873$	12.0274	0.407067
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−36.8814	−1.24540	−0.622698	−	0.782462i	$-0.713964\pi$
$877$	−36.8814	−1.24540	−0.622698	+	0.782462i	$0.713964\pi$
$878$	0	0
$879$	−21.4648	−0.723990
$880$	0	0
$881$	10.7543	0.362320	0.181160	−	0.983454i	$-0.442015\pi$
$881$	10.7543	0.362320	0.181160	+	0.983454i	$0.442015\pi$
$882$	0	0
$883$	−13.8266	−0.465303	−0.232652	−	0.972560i	$-0.574740\pi$
$883$	−13.8266	−0.465303	−0.232652	+	0.972560i	$0.574740\pi$
$884$	0	0
$885$	61.8035	2.07750
$886$	0	0
$887$	34.8657	1.17067	0.585337	−	0.810790i	$-0.300962\pi$
$887$	34.8657	1.17067	0.585337	+	0.810790i	$0.300962\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2.29857	0.0770051
$892$	0	0
$893$	104.293	3.49004
$894$	0	0
$895$	−50.1254	−1.67551
$896$	0	0
$897$	−4.14060	−0.138251
$898$	0	0
$899$	−5.88058	−0.196128
$900$	0	0
$901$	−33.7808	−1.12540
$902$	0	0
$903$	0	0
$904$	0	0
$905$	96.1228	3.19523
$906$	0	0
$907$	26.9202	0.893871	0.446935	−	0.894566i	$-0.352515\pi$
$907$	26.9202	0.893871	0.446935	+	0.894566i	$0.352515\pi$
$908$	0	0
$909$	−1.30661	−0.0433374
$910$	0	0
$911$	−16.3288	−0.540999	−0.270499	−	0.962720i	$-0.587189\pi$
$911$	−16.3288	−0.540999	−0.270499	+	0.962720i	$0.587189\pi$
$912$	0	0
$913$	−26.8833	−0.889707
$914$	0	0
$915$	8.94631	0.295756
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1.51920	0.0501139	0.0250570	−	0.999686i	$-0.492023\pi$
$919$	1.51920	0.0501139	0.0250570	+	0.999686i	$0.492023\pi$
$920$	0	0
$921$	−7.41738	−0.244411
$922$	0	0
$923$	9.62595	0.316842
$924$	0	0
$925$	68.4394	2.25027
$926$	0	0
$927$	11.0033	0.361397
$928$	0	0
$929$	8.39057	0.275286	0.137643	−	0.990482i	$-0.456047\pi$
$929$	8.39057	0.275286	0.137643	+	0.990482i	$0.456047\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	14.7881	0.484141
$934$	0	0
$935$	−35.2888	−1.15407
$936$	0	0
$937$	−28.6019	−0.934382	−0.467191	−	0.884156i	$-0.654734\pi$
$937$	−28.6019	−0.934382	−0.467191	+	0.884156i	$0.654734\pi$
$938$	0	0
$939$	−9.95434	−0.324848
$940$	0	0
$941$	−30.8743	−1.00647	−0.503237	−	0.864148i	$-0.667858\pi$
$941$	−30.8743	−1.00647	−0.503237	+	0.864148i	$0.667858\pi$
$942$	0	0
$943$	11.5889	0.377386
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25.0151	0.812883	0.406441	−	0.913677i	$-0.366770\pi$
$947$	25.0151	0.812883	0.406441	+	0.913677i	$0.366770\pi$
$948$	0	0
$949$	23.5677	0.765040
$950$	0	0
$951$	−2.97907	−0.0966031
$952$	0	0
$953$	4.11942	0.133441	0.0667205	−	0.997772i	$-0.478746\pi$
$953$	4.11942	0.133441	0.0667205	+	0.997772i	$0.478746\pi$
$954$	0	0
$955$	−96.2186	−3.11356
$956$	0	0
$957$	−7.47191	−0.241533
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.7274	−0.894432
$962$	0	0
$963$	−10.8387	−0.349272
$964$	0	0
$965$	17.7233	0.570533
$966$	0	0
$967$	−23.5901	−0.758607	−0.379303	−	0.925272i	$-0.623836\pi$
$967$	−23.5901	−0.758607	−0.379303	+	0.925272i	$0.623836\pi$
$968$	0	0
$969$	29.1373	0.936024
$970$	0	0
$971$	14.8833	0.477627	0.238814	−	0.971065i	$-0.423242\pi$
$971$	14.8833	0.477627	0.238814	+	0.971065i	$0.423242\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	29.4430	0.942929
$976$	0	0
$977$	−44.6353	−1.42801	−0.714005	−	0.700141i	$-0.753121\pi$
$977$	−44.6353	−1.42801	−0.714005	+	0.700141i	$0.753121\pi$
$978$	0	0
$979$	29.3585	0.938302
$980$	0	0
$981$	9.44164	0.301448
$982$	0	0
$983$	13.1167	0.418359	0.209179	−	0.977877i	$-0.432921\pi$
$983$	13.1167	0.418359	0.209179	+	0.977877i	$0.432921\pi$
$984$	0	0
$985$	−49.5450	−1.57863
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9.25855	0.294405
$990$	0	0
$991$	37.5968	1.19430	0.597150	−	0.802129i	$-0.296300\pi$
$991$	37.5968	1.19430	0.597150	+	0.802129i	$0.296300\pi$
$992$	0	0
$993$	−6.80571	−0.215973
$994$	0	0
$995$	−91.7852	−2.90979
$996$	0	0
$997$	−26.1044	−0.826733	−0.413367	−	0.910565i	$-0.635647\pi$
$997$	−26.1044	−0.826733	−0.413367	+	0.910565i	$0.635647\pi$
$998$	0	0
$999$	5.65685	0.178975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9408.2.a.em.1.1		4
4.3	odd	2		9408.2.a.ek.1.1		4
7.6	odd	2		9408.2.a.el.1.4		4
8.3	odd	2		4704.2.a.bz.1.4	yes	4
8.5	even	2		4704.2.a.bx.1.4	yes	4
28.27	even	2		9408.2.a.en.1.4		4
56.13	odd	2		4704.2.a.by.1.1	yes	4
56.27	even	2		4704.2.a.bw.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4704.2.a.bw.1.1	✓	4	56.27	even	2
4704.2.a.bx.1.4	yes	4	8.5	even	2
4704.2.a.by.1.1	yes	4	56.13	odd	2
4704.2.a.bz.1.4	yes	4	8.3	odd	2
9408.2.a.ek.1.1		4	4.3	odd	2
9408.2.a.el.1.4		4	7.6	odd	2
9408.2.a.em.1.1		4	1.1	even	1	trivial
9408.2.a.en.1.4		4	28.27	even	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:	\( N \)	\(=\)	\( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9408.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -4.13503 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -4.13503 q^{5} +1.00000 q^{9} +2.29857 q^{11} +2.43361 q^{13} -4.13503 q^{15} +3.71279 q^{17} +7.84782 q^{19} -1.70143 q^{23} +12.0985 q^{25} +1.00000 q^{27} -3.25067 q^{29} +1.80904 q^{31} +2.29857 q^{33} +5.65685 q^{37} +2.43361 q^{39} -6.81128 q^{41} -5.44164 q^{43} -4.13503 q^{45} +13.2895 q^{47} +3.71279 q^{51} -9.09849 q^{53} -9.50467 q^{55} +7.84782 q^{57} -14.9463 q^{59} -2.16354 q^{61} -10.0630 q^{65} -10.2540 q^{67} -1.70143 q^{69} +3.95543 q^{71} +9.68428 q^{73} +12.0985 q^{75} -10.0388 q^{79} +1.00000 q^{81} -11.6956 q^{83} -15.3525 q^{85} -3.25067 q^{87} +12.7725 q^{89} +1.80904 q^{93} -32.4510 q^{95} +12.0274 q^{97} +2.29857 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{5} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 4 * q^5 + 4 * q^9	\( 4 q + 4 q^{3} - 4 q^{5} + 4 q^{9} + 4 q^{11} - 8 q^{13} - 4 q^{15} + 4 q^{17} + 8 q^{19} - 12 q^{23} + 12 q^{25} + 4 q^{27} + 8 q^{31} + 4 q^{33} - 8 q^{39} + 20 q^{41} - 8 q^{43} - 4 q^{45} + 16 q^{47} + 4 q^{51} + 8 q^{55} + 8 q^{57} - 16 q^{61} - 8 q^{65} - 8 q^{67} - 12 q^{69} - 12 q^{71} + 8 q^{73} + 12 q^{75} - 16 q^{79} + 4 q^{81} + 8 q^{85} + 28 q^{89} + 8 q^{93} - 24 q^{95} + 40 q^{97} + 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^5 + 4 * q^9 + 4 * q^11 - 8 * q^13 - 4 * q^15 + 4 * q^17 + 8 * q^19 - 12 * q^23 + 12 * q^25 + 4 * q^27 + 8 * q^31 + 4 * q^33 - 8 * q^39 + 20 * q^41 - 8 * q^43 - 4 * q^45 + 16 * q^47 + 4 * q^51 + 8 * q^55 + 8 * q^57 - 16 * q^61 - 8 * q^65 - 8 * q^67 - 12 * q^69 - 12 * q^71 + 8 * q^73 + 12 * q^75 - 16 * q^79 + 4 * q^81 + 8 * q^85 + 28 * q^89 + 8 * q^93 - 24 * q^95 + 40 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more