LMFDB - Embedded newform 4896.2.a.bj.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4896,2,Mod(1,4896)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4896, 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("4896.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4896 = 2^{5} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4896.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:	$39.0947568296$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13768.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 2 $ x^4 - x^3 - 5x^2 + 2x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.89761$ of defining polynomial
Character	$\chi$	$=$	4896.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.49854 q^{5} +4.74127 q^{7} +2.44459 q^{11} +1.24272 q^{13} -1.00000 q^{17} -0.444588 q^{19} -0.188768 q^{23} +7.23981 q^{25} -4.84918 q^{29} -2.84918 q^{31} +16.5875 q^{35} +4.84918 q^{37} -0.865188 q^{41} -0.552500 q^{43} +6.99709 q^{47} +15.4796 q^{49} +4.10791 q^{53} +8.55250 q^{55} -0.107912 q^{59} -13.9542 q^{61} +4.34772 q^{65} +13.2129 q^{67} -12.3317 q^{71} -9.59045 q^{73} +11.5904 q^{77} +6.25582 q^{79} +11.4825 q^{83} -3.49854 q^{85} -6.88918 q^{89} +5.89209 q^{91} -1.55541 q^{95} +14.1021 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{5} + 4 q^{7} + 2 q^{13} - 4 q^{17} + 8 q^{19} + 2 q^{25} + 8 q^{31} + 8 q^{35} - 2 q^{41} + 12 q^{43} + 4 q^{47} + 8 q^{49} + 12 q^{53} + 20 q^{55} + 4 q^{59} - 8 q^{61} - 14 q^{65} + 20 q^{67}+ \cdots + 4 q^{97}+O(q^{100}) $ 4 * q + 2 * q^5 + 4 * q^7 + 2 * q^13 - 4 * q^17 + 8 * q^19 + 2 * q^25 + 8 * q^31 + 8 * q^35 - 2 * q^41 + 12 * q^43 + 4 * q^47 + 8 * q^49 + 12 * q^53 + 20 * q^55 + 4 * q^59 - 8 * q^61 - 14 * q^65 + 20 * q^67 - 4 * q^73 + 12 * q^77 + 16 * q^79 + 16 * q^83 - 2 * q^85 - 8 * q^89 + 28 * q^91 - 16 * q^95 + 4 * 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
$5$	3.49854	1.56460	0.782298	−	0.622904i	$-0.214047\pi$
$5$	3.49854	1.56460	0.782298	+	0.622904i	$0.214047\pi$
$6$	0	0
$7$	4.74127	1.79203	0.896015	−	0.444023i	$-0.146449\pi$
$7$	4.74127	1.79203	0.896015	+	0.444023i	$0.146449\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.44459	0.737071	0.368536	−	0.929614i	$-0.379859\pi$
$11$	2.44459	0.737071	0.368536	+	0.929614i	$0.379859\pi$
$12$	0	0
$13$	1.24272	0.344670	0.172335	−	0.985038i	$-0.444869\pi$
$13$	1.24272	0.344670	0.172335	+	0.985038i	$0.444869\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−0.444588	−0.101996	−0.0509978	−	0.998699i	$-0.516240\pi$
$19$	−0.444588	−0.101996	−0.0509978	+	0.998699i	$0.516240\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.188768	−0.0393608	−0.0196804	−	0.999806i	$-0.506265\pi$
$23$	−0.188768	−0.0393608	−0.0196804	+	0.999806i	$0.506265\pi$
$24$	0	0
$25$	7.23981	1.44796
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.84918	−0.900470	−0.450235	−	0.892910i	$-0.648660\pi$
$29$	−4.84918	−0.900470	−0.450235	+	0.892910i	$0.648660\pi$
$30$	0	0
$31$	−2.84918	−0.511728	−0.255864	−	0.966713i	$-0.582360\pi$
$31$	−2.84918	−0.511728	−0.255864	+	0.966713i	$0.582360\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	16.5875	2.80381
$36$	0	0
$37$	4.84918	0.797200	0.398600	−	0.917125i	$-0.369496\pi$
$37$	4.84918	0.797200	0.398600	+	0.917125i	$0.369496\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.865188	−0.135120	−0.0675599	−	0.997715i	$-0.521521\pi$
$41$	−0.865188	−0.135120	−0.0675599	+	0.997715i	$0.521521\pi$
$42$	0	0
$43$	−0.552500	−0.0842555	−0.0421278	−	0.999112i	$-0.513414\pi$
$43$	−0.552500	−0.0842555	−0.0421278	+	0.999112i	$0.513414\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.99709	1.02063	0.510315	−	0.859987i	$-0.329529\pi$
$47$	6.99709	1.02063	0.510315	+	0.859987i	$0.329529\pi$
$48$	0	0
$49$	15.4796	2.21138
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.10791	0.564265	0.282133	−	0.959375i	$-0.408958\pi$
$53$	4.10791	0.564265	0.282133	+	0.959375i	$0.408958\pi$
$54$	0	0
$55$	8.55250	1.15322
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−0.107912	−0.0140490	−0.00702449	−	0.999975i	$-0.502236\pi$
$59$	−0.107912	−0.0140490	−0.00702449	+	0.999975i	$0.502236\pi$
$60$	0	0
$61$	−13.9542	−1.78665	−0.893325	−	0.449411i	$-0.851634\pi$
$61$	−13.9542	−1.78665	−0.893325	+	0.449411i	$0.851634\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.34772	0.539269
$66$	0	0
$67$	13.2129	1.61421	0.807107	−	0.590405i	$-0.201032\pi$
$67$	13.2129	1.61421	0.807107	+	0.590405i	$0.201032\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.3317	−1.46351	−0.731753	−	0.681570i	$-0.761297\pi$
$71$	−12.3317	−1.46351	−0.731753	+	0.681570i	$0.761297\pi$
$72$	0	0
$73$	−9.59045	−1.12248	−0.561239	−	0.827654i	$-0.689675\pi$
$73$	−9.59045	−1.12248	−0.561239	+	0.827654i	$0.689675\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	11.5904	1.32085
$78$	0	0
$79$	6.25582	0.703835	0.351917	−	0.936031i	$-0.385530\pi$
$79$	6.25582	0.703835	0.351917	+	0.936031i	$0.385530\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.4825	1.26037	0.630186	−	0.776444i	$-0.282979\pi$
$83$	11.4825	1.26037	0.630186	+	0.776444i	$0.282979\pi$
$84$	0	0
$85$	−3.49854	−0.379470
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.88918	−0.730251	−0.365126	−	0.930958i	$-0.618974\pi$
$89$	−6.88918	−0.730251	−0.365126	+	0.930958i	$0.618974\pi$
$90$	0	0
$91$	5.89209	0.617659
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.55541	−0.159582
$96$	0	0
$97$	14.1021	1.43185	0.715925	−	0.698177i	$-0.246005\pi$
$97$	14.1021	1.43185	0.715925	+	0.698177i	$0.246005\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−15.9622	−1.58829	−0.794147	−	0.607725i	$-0.792082\pi$
$101$	−15.9622	−1.58829	−0.794147	+	0.607725i	$0.792082\pi$
$102$	0	0
$103$	−6.03504	−0.594650	−0.297325	−	0.954776i	$-0.596094\pi$
$103$	−6.03504	−0.594650	−0.297325	+	0.954776i	$0.596094\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.66041	0.257192	0.128596	−	0.991697i	$-0.458953\pi$
$107$	2.66041	0.257192	0.128596	+	0.991697i	$0.458953\pi$
$108$	0	0
$109$	−17.1130	−1.63913	−0.819563	−	0.572989i	$-0.805784\pi$
$109$	−17.1130	−1.63913	−0.819563	+	0.572989i	$0.805784\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−16.8332	−1.58353	−0.791766	−	0.610825i	$-0.790838\pi$
$113$	−16.8332	−1.58353	−0.791766	+	0.610825i	$0.790838\pi$
$114$	0	0
$115$	−0.660413	−0.0615838
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.74127	−0.434631
$120$	0	0
$121$	−5.02399	−0.456726
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.83608	0.700881
$126$	0	0
$127$	12.5525	1.11385	0.556927	−	0.830561i	$-0.311980\pi$
$127$	12.5525	1.11385	0.556927	+	0.830561i	$0.311980\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12.4388	−1.08678	−0.543390	−	0.839480i	$-0.682860\pi$
$131$	−12.4388	−1.08678	−0.543390	+	0.839480i	$0.682860\pi$
$132$	0	0
$133$	−2.10791	−0.182779
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.3746	−0.971800	−0.485900	−	0.874014i	$-0.661508\pi$
$137$	−11.3746	−0.971800	−0.485900	+	0.874014i	$0.661508\pi$
$138$	0	0
$139$	19.8863	1.68673	0.843366	−	0.537340i	$-0.180571\pi$
$139$	19.8863	1.68673	0.843366	+	0.537340i	$0.180571\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.03795	0.254046
$144$	0	0
$145$	−16.9651	−1.40887
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.67335	0.546702	0.273351	−	0.961914i	$-0.411868\pi$
$149$	6.67335	0.546702	0.273351	+	0.961914i	$0.411868\pi$
$150$	0	0
$151$	13.1867	1.07312	0.536560	−	0.843862i	$-0.319723\pi$
$151$	13.1867	1.07312	0.536560	+	0.843862i	$0.319723\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.96798	−0.800648
$156$	0	0
$157$	0.946908	0.0755715	0.0377857	−	0.999286i	$-0.487970\pi$
$157$	0.946908	0.0755715	0.0377857	+	0.999286i	$0.487970\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.894999	−0.0705358
$162$	0	0
$163$	−1.99418	−0.156196	−0.0780980	−	0.996946i	$-0.524885\pi$
$163$	−1.99418	−0.156196	−0.0780980	+	0.996946i	$0.524885\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.6946	−1.44663	−0.723315	−	0.690518i	$-0.757383\pi$
$167$	−18.6946	−1.44663	−0.723315	+	0.690518i	$0.757383\pi$
$168$	0	0
$169$	−11.4556	−0.881203
$170$	0	0
$171$	0	0
$172$	0	0
$173$	16.9811	1.29105	0.645524	−	0.763740i	$-0.276639\pi$
$173$	16.9811	1.29105	0.645524	+	0.763740i	$0.276639\pi$
$174$	0	0
$175$	34.3259	2.59479
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3.24493	−0.242537	−0.121269	−	0.992620i	$-0.538696\pi$
$179$	−3.24493	−0.242537	−0.121269	+	0.992620i	$0.538696\pi$
$180$	0	0
$181$	−14.7093	−1.09333	−0.546665	−	0.837351i	$-0.684103\pi$
$181$	−14.7093	−1.09333	−0.546665	+	0.837351i	$0.684103\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	16.9651	1.24730
$186$	0	0
$187$	−2.44459	−0.178766
$188$	0	0
$189$	0	0
$190$	0	0
$191$	21.0730	1.52479	0.762394	−	0.647113i	$-0.224024\pi$
$191$	21.0730	1.52479	0.762394	+	0.647113i	$0.224024\pi$
$192$	0	0
$193$	14.7754	1.06356	0.531780	−	0.846883i	$-0.321523\pi$
$193$	14.7754	1.06356	0.531780	+	0.846883i	$0.321523\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.50437	0.107182	0.0535909	−	0.998563i	$-0.482933\pi$
$197$	1.50437	0.107182	0.0535909	+	0.998563i	$0.482933\pi$
$198$	0	0
$199$	−21.3230	−1.51155	−0.755773	−	0.654834i	$-0.772739\pi$
$199$	−21.3230	−1.51155	−0.755773	+	0.654834i	$0.772739\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−22.9913	−1.61367
$204$	0	0
$205$	−3.02690	−0.211408
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.08684	−0.0751780
$210$	0	0
$211$	2.51746	0.173309	0.0866547	−	0.996238i	$-0.472382\pi$
$211$	2.51746	0.173309	0.0866547	+	0.996238i	$0.472382\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.93295	−0.131826
$216$	0	0
$217$	−13.5087	−0.917032
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.24272	−0.0835947
$222$	0	0
$223$	−9.95332	−0.666523	−0.333262	−	0.942834i	$-0.608149\pi$
$223$	−9.95332	−0.666523	−0.333262	+	0.942834i	$0.608149\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.1138	−1.13589	−0.567943	−	0.823068i	$-0.692261\pi$
$227$	−17.1138	−1.13589	−0.567943	+	0.823068i	$0.692261\pi$
$228$	0	0
$229$	11.7784	0.778335	0.389168	−	0.921167i	$-0.372763\pi$
$229$	11.7784	0.778335	0.389168	+	0.921167i	$0.372763\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−19.3186	−1.26560	−0.632802	−	0.774313i	$-0.718095\pi$
$233$	−19.3186	−1.26560	−0.632802	+	0.774313i	$0.718095\pi$
$234$	0	0
$235$	24.4796	1.59687
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−21.2071	−1.37177	−0.685886	−	0.727709i	$-0.740585\pi$
$239$	−21.2071	−1.37177	−0.685886	+	0.727709i	$0.740585\pi$
$240$	0	0
$241$	−8.37171	−0.539269	−0.269635	−	0.962963i	$-0.586903\pi$
$241$	−8.37171	−0.539269	−0.269635	+	0.962963i	$0.586903\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	54.1562	3.45991
$246$	0	0
$247$	−0.552500	−0.0351548
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.0701	−1.51929	−0.759645	−	0.650338i	$-0.774627\pi$
$251$	−24.0701	−1.51929	−0.759645	+	0.650338i	$0.774627\pi$
$252$	0	0
$253$	−0.461460	−0.0290117
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−29.5846	−1.84544	−0.922719	−	0.385473i	$-0.874038\pi$
$257$	−29.5846	−1.84544	−0.922719	+	0.385473i	$0.874038\pi$
$258$	0	0
$259$	22.9913	1.42861
$260$	0	0
$261$	0	0
$262$	0	0
$263$	12.8092	0.789848	0.394924	−	0.918714i	$-0.370771\pi$
$263$	12.8092	0.789848	0.394924	+	0.918714i	$0.370771\pi$
$264$	0	0
$265$	14.3717	0.882847
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.76817	0.351691	0.175846	−	0.984418i	$-0.443734\pi$
$269$	5.76817	0.351691	0.175846	+	0.984418i	$0.443734\pi$
$270$	0	0
$271$	22.1150	1.34339	0.671696	−	0.740827i	$-0.265566\pi$
$271$	22.1150	1.34339	0.671696	+	0.740827i	$0.265566\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	17.6984	1.06725
$276$	0	0
$277$	31.7005	1.90470	0.952350	−	0.305008i	$-0.0986591\pi$
$277$	31.7005	1.90470	0.952350	+	0.305008i	$0.0986591\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.86007	0.230273	0.115136	−	0.993350i	$-0.463270\pi$
$281$	3.86007	0.230273	0.115136	+	0.993350i	$0.463270\pi$
$282$	0	0
$283$	26.5875	1.58047	0.790233	−	0.612807i	$-0.209960\pi$
$283$	26.5875	1.58047	0.790233	+	0.612807i	$0.209960\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.10209	−0.242139
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9.02910	0.527486	0.263743	−	0.964593i	$-0.415043\pi$
$293$	9.02910	0.527486	0.263743	+	0.964593i	$0.415043\pi$
$294$	0	0
$295$	−0.377536	−0.0219810
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.234586	−0.0135665
$300$	0	0
$301$	−2.61955	−0.150989
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−48.8193	−2.79539
$306$	0	0
$307$	8.97090	0.511996	0.255998	−	0.966677i	$-0.417596\pi$
$307$	8.97090	0.511996	0.255998	+	0.966677i	$0.417596\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5.58247	−0.316553	−0.158276	−	0.987395i	$-0.550594\pi$
$311$	−5.58247	−0.316553	−0.158276	+	0.987395i	$0.550594\pi$
$312$	0	0
$313$	5.29463	0.299270	0.149635	−	0.988741i	$-0.452190\pi$
$313$	5.29463	0.299270	0.149635	+	0.988741i	$0.452190\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.5984	−0.876095	−0.438048	−	0.898952i	$-0.644330\pi$
$317$	−15.5984	−0.876095	−0.438048	+	0.898952i	$0.644330\pi$
$318$	0	0
$319$	−11.8542	−0.663711
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.444588	0.0247376
$324$	0	0
$325$	8.99709	0.499069
$326$	0	0
$327$	0	0
$328$	0	0
$329$	33.1751	1.82900
$330$	0	0
$331$	14.5467	0.799558	0.399779	−	0.916612i	$-0.369087\pi$
$331$	14.5467	0.799558	0.399779	+	0.916612i	$0.369087\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	46.2260	2.52559
$336$	0	0
$337$	11.8601	0.646059	0.323030	−	0.946389i	$-0.395299\pi$
$337$	11.8601	0.646059	0.323030	+	0.946389i	$0.395299\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−6.96507	−0.377180
$342$	0	0
$343$	40.2042	2.17082
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−36.9593	−1.98408	−0.992038	−	0.125937i	$-0.959806\pi$
$347$	−36.9593	−1.98408	−0.992038	+	0.125937i	$0.959806\pi$
$348$	0	0
$349$	7.94400	0.425232	0.212616	−	0.977136i	$-0.431802\pi$
$349$	7.94400	0.425232	0.212616	+	0.977136i	$0.431802\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.18090	0.0628528	0.0314264	−	0.999506i	$-0.489995\pi$
$353$	1.18090	0.0628528	0.0314264	+	0.999506i	$0.489995\pi$
$354$	0	0
$355$	−43.1431	−2.28980
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.86007	−0.309283	−0.154641	−	0.987971i	$-0.549422\pi$
$359$	−5.86007	−0.309283	−0.154641	+	0.987971i	$0.549422\pi$
$360$	0	0
$361$	−18.8023	−0.989597
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−33.5526	−1.75622
$366$	0	0
$367$	−19.9542	−1.04160	−0.520800	−	0.853679i	$-0.674366\pi$
$367$	−19.9542	−1.04160	−0.520800	+	0.853679i	$0.674366\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	19.4767	1.01118
$372$	0	0
$373$	25.4767	1.31913	0.659567	−	0.751646i	$-0.270740\pi$
$373$	25.4767	1.31913	0.659567	+	0.751646i	$0.270740\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.02619	−0.310365
$378$	0	0
$379$	3.65448	0.187718	0.0938590	−	0.995585i	$-0.470080\pi$
$379$	3.65448	0.187718	0.0938590	+	0.995585i	$0.470080\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4.39963	0.224811	0.112405	−	0.993662i	$-0.464144\pi$
$383$	4.39963	0.224811	0.112405	+	0.993662i	$0.464144\pi$
$384$	0	0
$385$	40.5497	2.06660
$386$	0	0
$387$	0	0
$388$	0	0
$389$	22.9709	1.16467	0.582335	−	0.812949i	$-0.302139\pi$
$389$	22.9709	1.16467	0.582335	+	0.812949i	$0.302139\pi$
$390$	0	0
$391$	0.188768	0.00954640
$392$	0	0
$393$	0	0
$394$	0	0
$395$	21.8863	1.10122
$396$	0	0
$397$	20.5196	1.02985	0.514925	−	0.857235i	$-0.327820\pi$
$397$	20.5196	1.02985	0.514925	+	0.857235i	$0.327820\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	24.5257	1.22476	0.612378	−	0.790565i	$-0.290213\pi$
$401$	24.5257	1.22476	0.612378	+	0.790565i	$0.290213\pi$
$402$	0	0
$403$	−3.54074	−0.176377
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.8542	0.587593
$408$	0	0
$409$	−7.26309	−0.359137	−0.179568	−	0.983745i	$-0.557470\pi$
$409$	−7.26309	−0.359137	−0.179568	+	0.983745i	$0.557470\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−0.511641	−0.0251762
$414$	0	0
$415$	40.1722	1.97197
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3.78418	−0.184869	−0.0924345	−	0.995719i	$-0.529465\pi$
$419$	−3.78418	−0.184869	−0.0924345	+	0.995719i	$0.529465\pi$
$420$	0	0
$421$	8.48765	0.413663	0.206831	−	0.978377i	$-0.433685\pi$
$421$	8.48765	0.413663	0.206831	+	0.978377i	$0.433685\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7.23981	−0.351182
$426$	0	0
$427$	−66.1605	−3.20173
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15.3608	0.739905	0.369952	−	0.929051i	$-0.379374\pi$
$431$	15.3608	0.739905	0.369952	+	0.929051i	$0.379374\pi$
$432$	0	0
$433$	−30.9353	−1.48665	−0.743327	−	0.668928i	$-0.766753\pi$
$433$	−30.9353	−1.48665	−0.743327	+	0.668928i	$0.766753\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.0839240	0.00401463
$438$	0	0
$439$	−7.38843	−0.352630	−0.176315	−	0.984334i	$-0.556418\pi$
$439$	−7.38843	−0.352630	−0.176315	+	0.984334i	$0.556418\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−17.6663	−0.839353	−0.419677	−	0.907674i	$-0.637857\pi$
$443$	−17.6663	−0.839353	−0.419677	+	0.907674i	$0.637857\pi$
$444$	0	0
$445$	−24.1021	−1.14255
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9.93597	0.468907	0.234454	−	0.972127i	$-0.424670\pi$
$449$	9.93597	0.468907	0.234454	+	0.972127i	$0.424670\pi$
$450$	0	0
$451$	−2.11503	−0.0995928
$452$	0	0
$453$	0	0
$454$	0	0
$455$	20.6137	0.966387
$456$	0	0
$457$	−20.6656	−0.966698	−0.483349	−	0.875428i	$-0.660580\pi$
$457$	−20.6656	−0.966698	−0.483349	+	0.875428i	$0.660580\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10.5116	0.489576	0.244788	−	0.969577i	$-0.421282\pi$
$461$	10.5116	0.489576	0.244788	+	0.969577i	$0.421282\pi$
$462$	0	0
$463$	27.2609	1.26692	0.633460	−	0.773775i	$-0.281634\pi$
$463$	27.2609	1.26692	0.633460	+	0.773775i	$0.281634\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.1530	1.16394	0.581970	−	0.813210i	$-0.302282\pi$
$467$	25.1530	1.16394	0.581970	+	0.813210i	$0.302282\pi$
$468$	0	0
$469$	62.6460	2.89272
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.35064	−0.0621023
$474$	0	0
$475$	−3.21874	−0.147686
$476$	0	0
$477$	0	0
$478$	0	0
$479$	26.3171	1.20246	0.601228	−	0.799077i	$-0.294678\pi$
$479$	26.3171	1.20246	0.601228	+	0.799077i	$0.294678\pi$
$480$	0	0
$481$	6.02619	0.274771
$482$	0	0
$483$	0	0
$484$	0	0
$485$	49.3368	2.24027
$486$	0	0
$487$	11.8183	0.535541	0.267770	−	0.963483i	$-0.413713\pi$
$487$	11.8183	0.535541	0.267770	+	0.963483i	$0.413713\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	8.70709	0.392946	0.196473	−	0.980509i	$-0.437051\pi$
$491$	8.70709	0.392946	0.196473	+	0.980509i	$0.437051\pi$
$492$	0	0
$493$	4.84918	0.218396
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−58.4680	−2.62265
$498$	0	0
$499$	35.9563	1.60963	0.804813	−	0.593528i	$-0.202265\pi$
$499$	35.9563	1.60963	0.804813	+	0.593528i	$0.202265\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−24.9980	−1.11460	−0.557302	−	0.830310i	$-0.688163\pi$
$503$	−24.9980	−1.11460	−0.557302	+	0.830310i	$0.688163\pi$
$504$	0	0
$505$	−55.8443	−2.48504
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−27.2347	−1.20716	−0.603578	−	0.797304i	$-0.706259\pi$
$509$	−27.2347	−1.20716	−0.603578	+	0.797304i	$0.706259\pi$
$510$	0	0
$511$	−45.4709	−2.01151
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−21.1138	−0.930387
$516$	0	0
$517$	17.1050	0.752277
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−24.8012	−1.08656	−0.543279	−	0.839552i	$-0.682817\pi$
$521$	−24.8012	−1.08656	−0.543279	+	0.839552i	$0.682817\pi$
$522$	0	0
$523$	33.3647	1.45894	0.729468	−	0.684015i	$-0.239768\pi$
$523$	33.3647	1.45894	0.729468	+	0.684015i	$0.239768\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.84918	0.124112
$528$	0	0
$529$	−22.9644	−0.998451
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.07519	−0.0465717
$534$	0	0
$535$	9.30757	0.402401
$536$	0	0
$537$	0	0
$538$	0	0
$539$	37.8413	1.62994
$540$	0	0
$541$	−13.6043	−0.584892	−0.292446	−	0.956282i	$-0.594469\pi$
$541$	−13.6043	−0.584892	−0.292446	+	0.956282i	$0.594469\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−59.8705	−2.56457
$546$	0	0
$547$	23.8746	1.02081	0.510403	−	0.859936i	$-0.329496\pi$
$547$	23.8746	1.02081	0.510403	+	0.859936i	$0.329496\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.15589	0.0918439
$552$	0	0
$553$	29.6605	1.26129
$554$	0	0
$555$	0	0
$556$	0	0
$557$	29.9084	1.26726	0.633629	−	0.773637i	$-0.281565\pi$
$557$	29.9084	1.26726	0.633629	+	0.773637i	$0.281565\pi$
$558$	0	0
$559$	−0.686606	−0.0290403
$560$	0	0
$561$	0	0
$562$	0	0
$563$	37.7405	1.59057	0.795287	−	0.606234i	$-0.207320\pi$
$563$	37.7405	1.59057	0.795287	+	0.606234i	$0.207320\pi$
$564$	0	0
$565$	−58.8916	−2.47759
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−19.4784	−0.816579	−0.408289	−	0.912853i	$-0.633875\pi$
$569$	−19.4784	−0.816579	−0.408289	+	0.912853i	$0.633875\pi$
$570$	0	0
$571$	−21.0713	−0.881805	−0.440902	−	0.897555i	$-0.645341\pi$
$571$	−21.0713	−0.881805	−0.440902	+	0.897555i	$0.645341\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.36664	−0.0569930
$576$	0	0
$577$	23.6365	0.984002	0.492001	−	0.870595i	$-0.336266\pi$
$577$	23.6365	0.984002	0.492001	+	0.870595i	$0.336266\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	54.4418	2.25863
$582$	0	0
$583$	10.0422	0.415904
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.9593	0.534886	0.267443	−	0.963574i	$-0.413821\pi$
$587$	12.9593	0.534886	0.267443	+	0.963574i	$0.413821\pi$
$588$	0	0
$589$	1.26671	0.0521940
$590$	0	0
$591$	0	0
$592$	0	0
$593$	36.0584	1.48074	0.740371	−	0.672198i	$-0.234650\pi$
$593$	36.0584	1.48074	0.740371	+	0.672198i	$0.234650\pi$
$594$	0	0
$595$	−16.5875	−0.680023
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−42.7597	−1.74711	−0.873557	−	0.486722i	$-0.838193\pi$
$599$	−42.7597	−1.74711	−0.873557	+	0.486722i	$0.838193\pi$
$600$	0	0
$601$	−31.6663	−1.29170	−0.645849	−	0.763466i	$-0.723496\pi$
$601$	−31.6663	−1.29170	−0.645849	+	0.763466i	$0.723496\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−17.5766	−0.714592
$606$	0	0
$607$	−40.2980	−1.63564	−0.817822	−	0.575471i	$-0.804819\pi$
$607$	−40.2980	−1.63564	−0.817822	+	0.575471i	$0.804819\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.69545	0.351780
$612$	0	0
$613$	−7.43236	−0.300190	−0.150095	−	0.988672i	$-0.547958\pi$
$613$	−7.43236	−0.300190	−0.150095	+	0.988672i	$0.547958\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	38.1242	1.53482	0.767411	−	0.641155i	$-0.221545\pi$
$617$	38.1242	1.53482	0.767411	+	0.641155i	$0.221545\pi$
$618$	0	0
$619$	−37.0410	−1.48880	−0.744401	−	0.667733i	$-0.767265\pi$
$619$	−37.0410	−1.48880	−0.744401	+	0.667733i	$0.767265\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−32.6634	−1.30863
$624$	0	0
$625$	−8.78418	−0.351367
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.84918	−0.193349
$630$	0	0
$631$	−28.1633	−1.12116	−0.560582	−	0.828099i	$-0.689423\pi$
$631$	−28.1633	−1.12116	−0.560582	+	0.828099i	$0.689423\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	43.9155	1.74273
$636$	0	0
$637$	19.2369	0.762194
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−29.1290	−1.15053	−0.575263	−	0.817969i	$-0.695100\pi$
$641$	−29.1290	−1.15053	−0.575263	+	0.817969i	$0.695100\pi$
$642$	0	0
$643$	36.4156	1.43609	0.718045	−	0.695996i	$-0.245037\pi$
$643$	36.4156	1.43609	0.718045	+	0.695996i	$0.245037\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	43.4988	1.71011	0.855057	−	0.518533i	$-0.173522\pi$
$647$	43.4988	1.71011	0.855057	+	0.518533i	$0.173522\pi$
$648$	0	0
$649$	−0.263801	−0.0103551
$650$	0	0
$651$	0	0
$652$	0	0
$653$	27.5145	1.07673	0.538363	−	0.842713i	$-0.319043\pi$
$653$	27.5145	1.07673	0.538363	+	0.842713i	$0.319043\pi$
$654$	0	0
$655$	−43.5176	−1.70037
$656$	0	0
$657$	0	0
$658$	0	0
$659$	19.9959	0.778930	0.389465	−	0.921041i	$-0.372660\pi$
$659$	19.9959	0.778930	0.389465	+	0.921041i	$0.372660\pi$
$660$	0	0
$661$	−38.9570	−1.51525	−0.757627	−	0.652688i	$-0.773641\pi$
$661$	−38.9570	−1.51525	−0.757627	+	0.652688i	$0.773641\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7.37462	−0.285976
$666$	0	0
$667$	0.915369	0.0354432
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−34.1122	−1.31689
$672$	0	0
$673$	−11.8281	−0.455938	−0.227969	−	0.973668i	$-0.573209\pi$
$673$	−11.8281	−0.455938	−0.227969	+	0.973668i	$0.573209\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	15.2347	0.585519	0.292759	−	0.956186i	$-0.405427\pi$
$677$	15.2347	0.585519	0.292759	+	0.956186i	$0.405427\pi$
$678$	0	0
$679$	66.8618	2.56592
$680$	0	0
$681$	0	0
$682$	0	0
$683$	4.82795	0.184736	0.0923681	−	0.995725i	$-0.470556\pi$
$683$	4.82795	0.184736	0.0923681	+	0.995725i	$0.470556\pi$
$684$	0	0
$685$	−39.7946	−1.52047
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5.10500	0.194485
$690$	0	0
$691$	21.6867	0.825002	0.412501	−	0.910957i	$-0.364655\pi$
$691$	21.6867	0.825002	0.412501	+	0.910957i	$0.364655\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	69.5730	2.63905
$696$	0	0
$697$	0.865188	0.0327713
$698$	0	0
$699$	0	0
$700$	0	0
$701$	38.7638	1.46409	0.732044	−	0.681257i	$-0.238566\pi$
$701$	38.7638	1.46409	0.732044	+	0.681257i	$0.238566\pi$
$702$	0	0
$703$	−2.15589	−0.0813109
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−75.6809	−2.84627
$708$	0	0
$709$	21.4105	0.804089	0.402045	−	0.915620i	$-0.368300\pi$
$709$	21.4105	0.804089	0.402045	+	0.915620i	$0.368300\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0.537834	0.0201420
$714$	0	0
$715$	10.6284	0.397480
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−16.8621	−0.628851	−0.314425	−	0.949282i	$-0.601812\pi$
$719$	−16.8621	−0.628851	−0.314425	+	0.949282i	$0.601812\pi$
$720$	0	0
$721$	−28.6137	−1.06563
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−35.1072	−1.30385
$726$	0	0
$727$	−45.9942	−1.70583	−0.852915	−	0.522050i	$-0.825167\pi$
$727$	−45.9942	−1.70583	−0.852915	+	0.522050i	$0.825167\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0.552500	0.0204350
$732$	0	0
$733$	−17.0451	−0.629574	−0.314787	−	0.949162i	$-0.601933\pi$
$733$	−17.0451	−0.629574	−0.314787	+	0.949162i	$0.601933\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	32.3001	1.18979
$738$	0	0
$739$	−20.8122	−0.765589	−0.382795	−	0.923833i	$-0.625038\pi$
$739$	−20.8122	−0.765589	−0.382795	+	0.923833i	$0.625038\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−42.9893	−1.57713	−0.788563	−	0.614954i	$-0.789174\pi$
$743$	−42.9893	−1.57713	−0.788563	+	0.614954i	$0.789174\pi$
$744$	0	0
$745$	23.3470	0.855369
$746$	0	0
$747$	0	0
$748$	0	0
$749$	12.6137	0.460896
$750$	0	0
$751$	2.38993	0.0872096	0.0436048	−	0.999049i	$-0.486116\pi$
$751$	2.38993	0.0872096	0.0436048	+	0.999049i	$0.486116\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	46.1343	1.67900
$756$	0	0
$757$	22.9891	0.835552	0.417776	−	0.908550i	$-0.362810\pi$
$757$	22.9891	0.835552	0.417776	+	0.908550i	$0.362810\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−7.01746	−0.254383	−0.127191	−	0.991878i	$-0.540596\pi$
$761$	−7.01746	−0.254383	−0.127191	+	0.991878i	$0.540596\pi$
$762$	0	0
$763$	−81.1372	−2.93737
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−0.134105	−0.00484225
$768$	0	0
$769$	12.0764	0.435485	0.217743	−	0.976006i	$-0.430131\pi$
$769$	12.0764	0.435485	0.217743	+	0.976006i	$0.430131\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−4.39208	−0.157972	−0.0789861	−	0.996876i	$-0.525168\pi$
$773$	−4.39208	−0.157972	−0.0789861	+	0.996876i	$0.525168\pi$
$774$	0	0
$775$	−20.6275	−0.740963
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.384653	0.0137816
$780$	0	0
$781$	−30.1460	−1.07871
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.31280	0.118239
$786$	0	0
$787$	−4.19545	−0.149552	−0.0747759	−	0.997200i	$-0.523824\pi$
$787$	−4.19545	−0.149552	−0.0747759	+	0.997200i	$0.523824\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−79.8106	−2.83774
$792$	0	0
$793$	−17.3412	−0.615804
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−11.0553	−0.391599	−0.195799	−	0.980644i	$-0.562730\pi$
$797$	−11.0553	−0.391599	−0.195799	+	0.980644i	$0.562730\pi$
$798$	0	0
$799$	−6.99709	−0.247539
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−23.4447	−0.827345
$804$	0	0
$805$	−3.13119	−0.110360
$806$	0	0
$807$	0	0
$808$	0	0
$809$	15.9423	0.560500	0.280250	−	0.959927i	$-0.409583\pi$
$809$	15.9423	0.560500	0.280250	+	0.959927i	$0.409583\pi$
$810$	0	0
$811$	−19.7246	−0.692623	−0.346311	−	0.938120i	$-0.612566\pi$
$811$	−19.7246	−0.692623	−0.346311	+	0.938120i	$0.612566\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−6.97672	−0.244384
$816$	0	0
$817$	0.245635	0.00859369
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−4.48981	−0.156695	−0.0783477	−	0.996926i	$-0.524964\pi$
$821$	−4.48981	−0.156695	−0.0783477	+	0.996926i	$0.524964\pi$
$822$	0	0
$823$	−20.3858	−0.710605	−0.355303	−	0.934751i	$-0.615622\pi$
$823$	−20.3858	−0.710605	−0.355303	+	0.934751i	$0.615622\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−42.6488	−1.48304	−0.741521	−	0.670929i	$-0.765895\pi$
$827$	−42.6488	−1.48304	−0.741521	+	0.670929i	$0.765895\pi$
$828$	0	0
$829$	−23.9301	−0.831128	−0.415564	−	0.909564i	$-0.636416\pi$
$829$	−23.9301	−0.831128	−0.415564	+	0.909564i	$0.636416\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−15.4796	−0.536337
$834$	0	0
$835$	−65.4038	−2.26339
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−43.5721	−1.50428	−0.752138	−	0.659005i	$-0.770977\pi$
$839$	−43.5721	−1.50428	−0.752138	+	0.659005i	$0.770977\pi$
$840$	0	0
$841$	−5.48545	−0.189153
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−40.0781	−1.37873
$846$	0	0
$847$	−23.8201	−0.818468
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−0.915369	−0.0313785
$852$	0	0
$853$	−19.8783	−0.680620	−0.340310	−	0.940313i	$-0.610532\pi$
$853$	−19.8783	−0.680620	−0.340310	+	0.940313i	$0.610532\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−12.5858	−0.429923	−0.214962	−	0.976623i	$-0.568963\pi$
$857$	−12.5858	−0.429923	−0.214962	+	0.976623i	$0.568963\pi$
$858$	0	0
$859$	11.5043	0.392522	0.196261	−	0.980552i	$-0.437120\pi$
$859$	11.5043	0.392522	0.196261	+	0.980552i	$0.437120\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−26.2321	−0.892951	−0.446476	−	0.894796i	$-0.647321\pi$
$863$	−26.2321	−0.892951	−0.446476	+	0.894796i	$0.647321\pi$
$864$	0	0
$865$	59.4091	2.01997
$866$	0	0
$867$	0	0
$868$	0	0
$869$	15.2929	0.518776
$870$	0	0
$871$	16.4200	0.556371
$872$	0	0
$873$	0	0
$874$	0	0
$875$	37.1530	1.25600
$876$	0	0
$877$	−37.5109	−1.26665	−0.633326	−	0.773885i	$-0.718311\pi$
$877$	−37.5109	−1.26665	−0.633326	+	0.773885i	$0.718311\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	10.2318	0.344718	0.172359	−	0.985034i	$-0.444861\pi$
$881$	10.2318	0.344718	0.172359	+	0.985034i	$0.444861\pi$
$882$	0	0
$883$	−21.5673	−0.725797	−0.362898	−	0.931829i	$-0.618213\pi$
$883$	−21.5673	−0.725797	−0.362898	+	0.931829i	$0.618213\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	49.6073	1.66565	0.832825	−	0.553537i	$-0.186722\pi$
$887$	49.6073	1.66565	0.832825	+	0.553537i	$0.186722\pi$
$888$	0	0
$889$	59.5148	1.99606
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3.11082	−0.104100
$894$	0	0
$895$	−11.3525	−0.379473
$896$	0	0
$897$	0	0
$898$	0	0
$899$	13.8162	0.460796
$900$	0	0
$901$	−4.10791	−0.136854
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−51.4610	−1.71062
$906$	0	0
$907$	−16.7054	−0.554693	−0.277346	−	0.960770i	$-0.589455\pi$
$907$	−16.7054	−0.554693	−0.277346	+	0.960770i	$0.589455\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−44.7004	−1.48099	−0.740495	−	0.672062i	$-0.765409\pi$
$911$	−44.7004	−1.48099	−0.740495	+	0.672062i	$0.765409\pi$
$912$	0	0
$913$	28.0701	0.928984
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−58.9755	−1.94754
$918$	0	0
$919$	−43.1560	−1.42359	−0.711793	−	0.702390i	$-0.752116\pi$
$919$	−43.1560	−1.42359	−0.711793	+	0.702390i	$0.752116\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−15.3249	−0.504426
$924$	0	0
$925$	35.1072	1.15432
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−30.6915	−1.00696	−0.503478	−	0.864008i	$-0.667947\pi$
$929$	−30.6915	−1.00696	−0.503478	+	0.864008i	$0.667947\pi$
$930$	0	0
$931$	−6.88206	−0.225550
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.55250	−0.279697
$936$	0	0
$937$	37.0072	1.20897	0.604487	−	0.796615i	$-0.293378\pi$
$937$	37.0072	1.20897	0.604487	+	0.796615i	$0.293378\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−39.4350	−1.28554	−0.642772	−	0.766058i	$-0.722216\pi$
$941$	−39.4350	−1.28554	−0.642772	+	0.766058i	$0.722216\pi$
$942$	0	0
$943$	0.163320	0.00531842
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.4360	−0.924045	−0.462022	−	0.886868i	$-0.652876\pi$
$947$	−28.4360	−0.924045	−0.462022	+	0.886868i	$0.652876\pi$
$948$	0	0
$949$	−11.9183	−0.386884
$950$	0	0
$951$	0	0
$952$	0	0
$953$	22.9113	0.742169	0.371084	−	0.928599i	$-0.378986\pi$
$953$	22.9113	0.742169	0.371084	+	0.928599i	$0.378986\pi$
$954$	0	0
$955$	73.7248	2.38568
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−53.9301	−1.74150
$960$	0	0
$961$	−22.8822	−0.738135
$962$	0	0
$963$	0	0
$964$	0	0
$965$	51.6925	1.66404
$966$	0	0
$967$	16.8660	0.542374	0.271187	−	0.962527i	$-0.412584\pi$
$967$	16.8660	0.542374	0.271187	+	0.962527i	$0.412584\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−6.77544	−0.217434	−0.108717	−	0.994073i	$-0.534674\pi$
$971$	−6.77544	−0.217434	−0.108717	+	0.994073i	$0.534674\pi$
$972$	0	0
$973$	94.2861	3.02267
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.16203	0.197141	0.0985703	−	0.995130i	$-0.468573\pi$
$977$	6.16203	0.197141	0.0985703	+	0.995130i	$0.468573\pi$
$978$	0	0
$979$	−16.8412	−0.538247
$980$	0	0
$981$	0	0
$982$	0	0
$983$	34.0113	1.08479	0.542396	−	0.840123i	$-0.317517\pi$
$983$	34.0113	1.08479	0.542396	+	0.840123i	$0.317517\pi$
$984$	0	0
$985$	5.26309	0.167696
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.104294	0.00331637
$990$	0	0
$991$	57.0431	1.81203	0.906017	−	0.423241i	$-0.139107\pi$
$991$	57.0431	1.81203	0.906017	+	0.423241i	$0.139107\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−74.5994	−2.36496
$996$	0	0
$997$	−8.63336	−0.273421	−0.136711	−	0.990611i	$-0.543653\pi$
$997$	−8.63336	−0.273421	−0.136711	+	0.990611i	$0.543653\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4896.2.a.bj.1.4	yes	4
3.2	odd	2		4896.2.a.bh.1.1	yes	4
4.3	odd	2		4896.2.a.bi.1.4	yes	4
8.3	odd	2		9792.2.a.dm.1.1		4
8.5	even	2		9792.2.a.dn.1.1		4
12.11	even	2		4896.2.a.bg.1.1	✓	4
24.5	odd	2		9792.2.a.dp.1.4		4
24.11	even	2		9792.2.a.do.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4896.2.a.bg.1.1	✓	4	12.11	even	2
4896.2.a.bh.1.1	yes	4	3.2	odd	2
4896.2.a.bi.1.4	yes	4	4.3	odd	2
4896.2.a.bj.1.4	yes	4	1.1	even	1	trivial
9792.2.a.dm.1.1		4	8.3	odd	2
9792.2.a.dn.1.1		4	8.5	even	2
9792.2.a.do.1.4		4	24.11	even	2
9792.2.a.dp.1.4		4	24.5	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:	\( N \)	\(=\)	\( 4896 = 2^{5} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4896.a (trivial)

\(f(q)\)	\(=\)	\(q+3.49854 q^{5} +4.74127 q^{7} +2.44459 q^{11} +1.24272 q^{13} -1.00000 q^{17} -0.444588 q^{19} -0.188768 q^{23} +7.23981 q^{25} -4.84918 q^{29} -2.84918 q^{31} +16.5875 q^{35} +4.84918 q^{37} -0.865188 q^{41} -0.552500 q^{43} +6.99709 q^{47} +15.4796 q^{49} +4.10791 q^{53} +8.55250 q^{55} -0.107912 q^{59} -13.9542 q^{61} +4.34772 q^{65} +13.2129 q^{67} -12.3317 q^{71} -9.59045 q^{73} +11.5904 q^{77} +6.25582 q^{79} +11.4825 q^{83} -3.49854 q^{85} -6.88918 q^{89} +5.89209 q^{91} -1.55541 q^{95} +14.1021 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5} + 4 q^{7} + 2 q^{13} - 4 q^{17} + 8 q^{19} + 2 q^{25} + 8 q^{31} + 8 q^{35} - 2 q^{41} + 12 q^{43} + 4 q^{47} + 8 q^{49} + 12 q^{53} + 20 q^{55} + 4 q^{59} - 8 q^{61} - 14 q^{65} + 20 q^{67}+ \cdots + 4 q^{97}+O(q^{100}) \) 4 * q + 2 * q^5 + 4 * q^7 + 2 * q^13 - 4 * q^17 + 8 * q^19 + 2 * q^25 + 8 * q^31 + 8 * q^35 - 2 * q^41 + 12 * q^43 + 4 * q^47 + 8 * q^49 + 12 * q^53 + 20 * q^55 + 4 * q^59 - 8 * q^61 - 14 * q^65 + 20 * q^67 - 4 * q^73 + 12 * q^77 + 16 * q^79 + 16 * q^83 - 2 * q^85 - 8 * q^89 + 28 * q^91 - 16 * q^95 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more