LMFDB - Embedded newform 7744.2.a.cp.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7744,2,Mod(1,7744)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7744, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7744.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7744 = 2^{6} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7744.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:	$61.8361513253$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 352)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	7744.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803 q^{3} -3.23607 q^{5} -4.47214 q^{7} -0.381966 q^{9} +O(q^{10})$	$q+1.61803 q^{3} -3.23607 q^{5} -4.47214 q^{7} -0.381966 q^{9} -1.23607 q^{13} -5.23607 q^{15} +1.85410 q^{17} -5.38197 q^{19} -7.23607 q^{21} -5.23607 q^{23} +5.47214 q^{25} -5.47214 q^{27} +8.47214 q^{29} -0.472136 q^{31} +14.4721 q^{35} -9.23607 q^{37} -2.00000 q^{39} -7.32624 q^{41} +4.09017 q^{43} +1.23607 q^{45} -6.47214 q^{47} +13.0000 q^{49} +3.00000 q^{51} -6.47214 q^{53} -8.70820 q^{57} -9.09017 q^{59} -2.47214 q^{61} +1.70820 q^{63} +4.00000 q^{65} -1.38197 q^{67} -8.47214 q^{69} -3.70820 q^{71} -4.09017 q^{73} +8.85410 q^{75} +6.00000 q^{79} -7.70820 q^{81} -2.38197 q^{83} -6.00000 q^{85} +13.7082 q^{87} +6.38197 q^{89} +5.52786 q^{91} -0.763932 q^{93} +17.4164 q^{95} +5.61803 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 2 q^{5} - 3 q^{9}+O(q^{10}) $ 2 * q + q^3 - 2 * q^5 - 3 * q^9	$ 2 q + q^{3} - 2 q^{5} - 3 q^{9} + 2 q^{13} - 6 q^{15} - 3 q^{17} - 13 q^{19} - 10 q^{21} - 6 q^{23} + 2 q^{25} - 2 q^{27} + 8 q^{29} + 8 q^{31} + 20 q^{35} - 14 q^{37} - 4 q^{39} + q^{41} - 3 q^{43} - 2 q^{45} - 4 q^{47} + 26 q^{49} + 6 q^{51} - 4 q^{53} - 4 q^{57} - 7 q^{59} + 4 q^{61} - 10 q^{63} + 8 q^{65} - 5 q^{67} - 8 q^{69} + 6 q^{71} + 3 q^{73} + 11 q^{75} + 12 q^{79} - 2 q^{81} - 7 q^{83} - 12 q^{85} + 14 q^{87} + 15 q^{89} + 20 q^{91} - 6 q^{93} + 8 q^{95} + 9 q^{97}+O(q^{100}) $ 2 * q + q^3 - 2 * q^5 - 3 * q^9 + 2 * q^13 - 6 * q^15 - 3 * q^17 - 13 * q^19 - 10 * q^21 - 6 * q^23 + 2 * q^25 - 2 * q^27 + 8 * q^29 + 8 * q^31 + 20 * q^35 - 14 * q^37 - 4 * q^39 + q^41 - 3 * q^43 - 2 * q^45 - 4 * q^47 + 26 * q^49 + 6 * q^51 - 4 * q^53 - 4 * q^57 - 7 * q^59 + 4 * q^61 - 10 * q^63 + 8 * q^65 - 5 * q^67 - 8 * q^69 + 6 * q^71 + 3 * q^73 + 11 * q^75 + 12 * q^79 - 2 * q^81 - 7 * q^83 - 12 * q^85 + 14 * q^87 + 15 * q^89 + 20 * q^91 - 6 * q^93 + 8 * q^95 + 9 * q^97

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
$3$	1.61803	0.934172	0.467086	−	0.884212i	$-0.345304\pi$
$3$	1.61803	0.934172	0.467086	+	0.884212i	$0.345304\pi$
$4$	0	0
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	0	0
$7$	−4.47214	−1.69031	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	−4.47214	−1.69031	−0.845154	+	0.534522i	$0.820491\pi$
$8$	0	0
$9$	−0.381966	−0.127322
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.23607	−0.342824	−0.171412	−	0.985199i	$-0.554833\pi$
$13$	−1.23607	−0.342824	−0.171412	+	0.985199i	$0.554833\pi$
$14$	0	0
$15$	−5.23607	−1.35195
$16$	0	0
$17$	1.85410	0.449686	0.224843	−	0.974395i	$-0.427813\pi$
$17$	1.85410	0.449686	0.224843	+	0.974395i	$0.427813\pi$
$18$	0	0
$19$	−5.38197	−1.23471	−0.617354	−	0.786686i	$-0.711795\pi$
$19$	−5.38197	−1.23471	−0.617354	+	0.786686i	$0.711795\pi$
$20$	0	0
$21$	−7.23607	−1.57904
$22$	0	0
$23$	−5.23607	−1.09180	−0.545898	−	0.837852i	$-0.683811\pi$
$23$	−5.23607	−1.09180	−0.545898	+	0.837852i	$0.683811\pi$
$24$	0	0
$25$	5.47214	1.09443
$26$	0	0
$27$	−5.47214	−1.05311
$28$	0	0
$29$	8.47214	1.57324	0.786618	−	0.617440i	$-0.211830\pi$
$29$	8.47214	1.57324	0.786618	+	0.617440i	$0.211830\pi$
$30$	0	0
$31$	−0.472136	−0.0847981	−0.0423991	−	0.999101i	$-0.513500\pi$
$31$	−0.472136	−0.0847981	−0.0423991	+	0.999101i	$0.513500\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	14.4721	2.44624
$36$	0	0
$37$	−9.23607	−1.51840	−0.759200	−	0.650857i	$-0.774410\pi$
$37$	−9.23607	−1.51840	−0.759200	+	0.650857i	$0.774410\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−7.32624	−1.14417	−0.572083	−	0.820196i	$-0.693865\pi$
$41$	−7.32624	−1.14417	−0.572083	+	0.820196i	$0.693865\pi$
$42$	0	0
$43$	4.09017	0.623745	0.311873	−	0.950124i	$-0.399044\pi$
$43$	4.09017	0.623745	0.311873	+	0.950124i	$0.399044\pi$
$44$	0	0
$45$	1.23607	0.184262
$46$	0	0
$47$	−6.47214	−0.944058	−0.472029	−	0.881583i	$-0.656478\pi$
$47$	−6.47214	−0.944058	−0.472029	+	0.881583i	$0.656478\pi$
$48$	0	0
$49$	13.0000	1.85714
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	−6.47214	−0.889016	−0.444508	−	0.895775i	$-0.646622\pi$
$53$	−6.47214	−0.889016	−0.444508	+	0.895775i	$0.646622\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−8.70820	−1.15343
$58$	0	0
$59$	−9.09017	−1.18344	−0.591720	−	0.806144i	$-0.701551\pi$
$59$	−9.09017	−1.18344	−0.591720	+	0.806144i	$0.701551\pi$
$60$	0	0
$61$	−2.47214	−0.316525	−0.158262	−	0.987397i	$-0.550589\pi$
$61$	−2.47214	−0.316525	−0.158262	+	0.987397i	$0.550589\pi$
$62$	0	0
$63$	1.70820	0.215213
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	−1.38197	−0.168834	−0.0844170	−	0.996431i	$-0.526903\pi$
$67$	−1.38197	−0.168834	−0.0844170	+	0.996431i	$0.526903\pi$
$68$	0	0
$69$	−8.47214	−1.01993
$70$	0	0
$71$	−3.70820	−0.440083	−0.220041	−	0.975491i	$-0.570619\pi$
$71$	−3.70820	−0.440083	−0.220041	+	0.975491i	$0.570619\pi$
$72$	0	0
$73$	−4.09017	−0.478718	−0.239359	−	0.970931i	$-0.576937\pi$
$73$	−4.09017	−0.478718	−0.239359	+	0.970931i	$0.576937\pi$
$74$	0	0
$75$	8.85410	1.02238
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	−2.38197	−0.261455	−0.130727	−	0.991418i	$-0.541731\pi$
$83$	−2.38197	−0.261455	−0.130727	+	0.991418i	$0.541731\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	0	0
$87$	13.7082	1.46967
$88$	0	0
$89$	6.38197	0.676487	0.338244	−	0.941059i	$-0.390167\pi$
$89$	6.38197	0.676487	0.338244	+	0.941059i	$0.390167\pi$
$90$	0	0
$91$	5.52786	0.579478
$92$	0	0
$93$	−0.763932	−0.0792161
$94$	0	0
$95$	17.4164	1.78689
$96$	0	0
$97$	5.61803	0.570425	0.285212	−	0.958464i	$-0.407936\pi$
$97$	5.61803	0.570425	0.285212	+	0.958464i	$0.407936\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.29180	0.228042	0.114021	−	0.993478i	$-0.463627\pi$
$101$	2.29180	0.228042	0.114021	+	0.993478i	$0.463627\pi$
$102$	0	0
$103$	17.2361	1.69832	0.849160	−	0.528135i	$-0.177109\pi$
$103$	17.2361	1.69832	0.849160	+	0.528135i	$0.177109\pi$
$104$	0	0
$105$	23.4164	2.28521
$106$	0	0
$107$	−3.32624	−0.321560	−0.160780	−	0.986990i	$-0.551401\pi$
$107$	−3.32624	−0.321560	−0.160780	+	0.986990i	$0.551401\pi$
$108$	0	0
$109$	−2.94427	−0.282010	−0.141005	−	0.990009i	$-0.545033\pi$
$109$	−2.94427	−0.282010	−0.141005	+	0.990009i	$0.545033\pi$
$110$	0	0
$111$	−14.9443	−1.41845
$112$	0	0
$113$	19.0902	1.79585	0.897926	−	0.440146i	$-0.145073\pi$
$113$	19.0902	1.79585	0.897926	+	0.440146i	$0.145073\pi$
$114$	0	0
$115$	16.9443	1.58006
$116$	0	0
$117$	0.472136	0.0436490
$118$	0	0
$119$	−8.29180	−0.760108
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−11.8541	−1.06885
$124$	0	0
$125$	−1.52786	−0.136656
$126$	0	0
$127$	20.4721	1.81661	0.908304	−	0.418310i	$-0.137378\pi$
$127$	20.4721	1.81661	0.908304	+	0.418310i	$0.137378\pi$
$128$	0	0
$129$	6.61803	0.582685
$130$	0	0
$131$	−1.09017	−0.0952486	−0.0476243	−	0.998865i	$-0.515165\pi$
$131$	−1.09017	−0.0952486	−0.0476243	+	0.998865i	$0.515165\pi$
$132$	0	0
$133$	24.0689	2.08704
$134$	0	0
$135$	17.7082	1.52408
$136$	0	0
$137$	−9.79837	−0.837132	−0.418566	−	0.908186i	$-0.637467\pi$
$137$	−9.79837	−0.837132	−0.418566	+	0.908186i	$0.637467\pi$
$138$	0	0
$139$	16.9443	1.43719	0.718597	−	0.695427i	$-0.244785\pi$
$139$	16.9443	1.43719	0.718597	+	0.695427i	$0.244785\pi$
$140$	0	0
$141$	−10.4721	−0.881913
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−27.4164	−2.27681
$146$	0	0
$147$	21.0344	1.73489
$148$	0	0
$149$	4.29180	0.351598	0.175799	−	0.984426i	$-0.443749\pi$
$149$	4.29180	0.351598	0.175799	+	0.984426i	$0.443749\pi$
$150$	0	0
$151$	−12.9443	−1.05339	−0.526695	−	0.850054i	$-0.676569\pi$
$151$	−12.9443	−1.05339	−0.526695	+	0.850054i	$0.676569\pi$
$152$	0	0
$153$	−0.708204	−0.0572549
$154$	0	0
$155$	1.52786	0.122721
$156$	0	0
$157$	−9.70820	−0.774799	−0.387400	−	0.921912i	$-0.626627\pi$
$157$	−9.70820	−0.774799	−0.387400	+	0.921912i	$0.626627\pi$
$158$	0	0
$159$	−10.4721	−0.830494
$160$	0	0
$161$	23.4164	1.84547
$162$	0	0
$163$	−21.0902	−1.65191	−0.825955	−	0.563736i	$-0.809363\pi$
$163$	−21.0902	−1.65191	−0.825955	+	0.563736i	$0.809363\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.6525	0.979078	0.489539	−	0.871981i	$-0.337165\pi$
$167$	12.6525	0.979078	0.489539	+	0.871981i	$0.337165\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	0	0
$171$	2.05573	0.157205
$172$	0	0
$173$	8.94427	0.680020	0.340010	−	0.940422i	$-0.389569\pi$
$173$	8.94427	0.680020	0.340010	+	0.940422i	$0.389569\pi$
$174$	0	0
$175$	−24.4721	−1.84992
$176$	0	0
$177$	−14.7082	−1.10554
$178$	0	0
$179$	−9.67376	−0.723051	−0.361525	−	0.932362i	$-0.617744\pi$
$179$	−9.67376	−0.723051	−0.361525	+	0.932362i	$0.617744\pi$
$180$	0	0
$181$	−0.763932	−0.0567826	−0.0283913	−	0.999597i	$-0.509038\pi$
$181$	−0.763932	−0.0567826	−0.0283913	+	0.999597i	$0.509038\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	0	0
$185$	29.8885	2.19745
$186$	0	0
$187$	0	0
$188$	0	0
$189$	24.4721	1.78009
$190$	0	0
$191$	5.70820	0.413031	0.206516	−	0.978443i	$-0.433788\pi$
$191$	5.70820	0.413031	0.206516	+	0.978443i	$0.433788\pi$
$192$	0	0
$193$	17.4164	1.25366	0.626830	−	0.779156i	$-0.284352\pi$
$193$	17.4164	1.25366	0.626830	+	0.779156i	$0.284352\pi$
$194$	0	0
$195$	6.47214	0.463479
$196$	0	0
$197$	8.00000	0.569976	0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000	0.569976	0.284988	+	0.958531i	$0.408010\pi$
$198$	0	0
$199$	22.3607	1.58511	0.792553	−	0.609803i	$-0.208751\pi$
$199$	22.3607	1.58511	0.792553	+	0.609803i	$0.208751\pi$
$200$	0	0
$201$	−2.23607	−0.157720
$202$	0	0
$203$	−37.8885	−2.65925
$204$	0	0
$205$	23.7082	1.65585
$206$	0	0
$207$	2.00000	0.139010
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0.0901699	0.00620755	0.00310378	−	0.999995i	$-0.499012\pi$
$211$	0.0901699	0.00620755	0.00310378	+	0.999995i	$0.499012\pi$
$212$	0	0
$213$	−6.00000	−0.411113
$214$	0	0
$215$	−13.2361	−0.902692
$216$	0	0
$217$	2.11146	0.143335
$218$	0	0
$219$	−6.61803	−0.447205
$220$	0	0
$221$	−2.29180	−0.154163
$222$	0	0
$223$	−2.29180	−0.153470	−0.0767350	−	0.997052i	$-0.524450\pi$
$223$	−2.29180	−0.153470	−0.0767350	+	0.997052i	$0.524450\pi$
$224$	0	0
$225$	−2.09017	−0.139345
$226$	0	0
$227$	−23.8541	−1.58325	−0.791626	−	0.611006i	$-0.790765\pi$
$227$	−23.8541	−1.58325	−0.791626	+	0.611006i	$0.790765\pi$
$228$	0	0
$229$	16.1803	1.06923	0.534613	−	0.845097i	$-0.320457\pi$
$229$	16.1803	1.06923	0.534613	+	0.845097i	$0.320457\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.90983	0.518190	0.259095	−	0.965852i	$-0.416576\pi$
$233$	7.90983	0.518190	0.259095	+	0.965852i	$0.416576\pi$
$234$	0	0
$235$	20.9443	1.36625
$236$	0	0
$237$	9.70820	0.630616
$238$	0	0
$239$	14.2918	0.924459	0.462230	−	0.886760i	$-0.347050\pi$
$239$	14.2918	0.924459	0.462230	+	0.886760i	$0.347050\pi$
$240$	0	0
$241$	28.3262	1.82465	0.912327	−	0.409463i	$-0.134284\pi$
$241$	28.3262	1.82465	0.912327	+	0.409463i	$0.134284\pi$
$242$	0	0
$243$	3.94427	0.253025
$244$	0	0
$245$	−42.0689	−2.68768
$246$	0	0
$247$	6.65248	0.423287
$248$	0	0
$249$	−3.85410	−0.244244
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−9.70820	−0.607951
$256$	0	0
$257$	−18.5623	−1.15788	−0.578942	−	0.815368i	$-0.696534\pi$
$257$	−18.5623	−1.15788	−0.578942	+	0.815368i	$0.696534\pi$
$258$	0	0
$259$	41.3050	2.56656
$260$	0	0
$261$	−3.23607	−0.200308
$262$	0	0
$263$	−0.180340	−0.0111202	−0.00556012	−	0.999985i	$-0.501770\pi$
$263$	−0.180340	−0.0111202	−0.00556012	+	0.999985i	$0.501770\pi$
$264$	0	0
$265$	20.9443	1.28660
$266$	0	0
$267$	10.3262	0.631955
$268$	0	0
$269$	−14.1803	−0.864591	−0.432295	−	0.901732i	$-0.642296\pi$
$269$	−14.1803	−0.864591	−0.432295	+	0.901732i	$0.642296\pi$
$270$	0	0
$271$	−8.47214	−0.514646	−0.257323	−	0.966326i	$-0.582840\pi$
$271$	−8.47214	−0.514646	−0.257323	+	0.966326i	$0.582840\pi$
$272$	0	0
$273$	8.94427	0.541332
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−10.1803	−0.611677	−0.305839	−	0.952083i	$-0.598937\pi$
$277$	−10.1803	−0.611677	−0.305839	+	0.952083i	$0.598937\pi$
$278$	0	0
$279$	0.180340	0.0107967
$280$	0	0
$281$	24.2705	1.44786	0.723929	−	0.689875i	$-0.242334\pi$
$281$	24.2705	1.44786	0.723929	+	0.689875i	$0.242334\pi$
$282$	0	0
$283$	8.94427	0.531682	0.265841	−	0.964017i	$-0.414350\pi$
$283$	8.94427	0.531682	0.265841	+	0.964017i	$0.414350\pi$
$284$	0	0
$285$	28.1803	1.66926
$286$	0	0
$287$	32.7639	1.93399
$288$	0	0
$289$	−13.5623	−0.797783
$290$	0	0
$291$	9.09017	0.532875
$292$	0	0
$293$	−13.5279	−0.790306	−0.395153	−	0.918615i	$-0.629308\pi$
$293$	−13.5279	−0.790306	−0.395153	+	0.918615i	$0.629308\pi$
$294$	0	0
$295$	29.4164	1.71269
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.47214	0.374293
$300$	0	0
$301$	−18.2918	−1.05432
$302$	0	0
$303$	3.70820	0.213031
$304$	0	0
$305$	8.00000	0.458079
$306$	0	0
$307$	−4.85410	−0.277038	−0.138519	−	0.990360i	$-0.544234\pi$
$307$	−4.85410	−0.277038	−0.138519	+	0.990360i	$0.544234\pi$
$308$	0	0
$309$	27.8885	1.58652
$310$	0	0
$311$	−8.76393	−0.496957	−0.248478	−	0.968637i	$-0.579931\pi$
$311$	−8.76393	−0.496957	−0.248478	+	0.968637i	$0.579931\pi$
$312$	0	0
$313$	15.3820	0.869440	0.434720	−	0.900566i	$-0.356847\pi$
$313$	15.3820	0.869440	0.434720	+	0.900566i	$0.356847\pi$
$314$	0	0
$315$	−5.52786	−0.311460
$316$	0	0
$317$	1.81966	0.102202	0.0511011	−	0.998693i	$-0.483727\pi$
$317$	1.81966	0.102202	0.0511011	+	0.998693i	$0.483727\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−5.38197	−0.300392
$322$	0	0
$323$	−9.97871	−0.555230
$324$	0	0
$325$	−6.76393	−0.375195
$326$	0	0
$327$	−4.76393	−0.263446
$328$	0	0
$329$	28.9443	1.59575
$330$	0	0
$331$	−29.7984	−1.63787	−0.818933	−	0.573889i	$-0.805434\pi$
$331$	−29.7984	−1.63787	−0.818933	+	0.573889i	$0.805434\pi$
$332$	0	0
$333$	3.52786	0.193326
$334$	0	0
$335$	4.47214	0.244339
$336$	0	0
$337$	−21.8541	−1.19047	−0.595234	−	0.803552i	$-0.702941\pi$
$337$	−21.8541	−1.19047	−0.595234	+	0.803552i	$0.702941\pi$
$338$	0	0
$339$	30.8885	1.67764
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−26.8328	−1.44884
$344$	0	0
$345$	27.4164	1.47605
$346$	0	0
$347$	−14.3262	−0.769073	−0.384536	−	0.923110i	$-0.625639\pi$
$347$	−14.3262	−0.769073	−0.384536	+	0.923110i	$0.625639\pi$
$348$	0	0
$349$	20.7639	1.11147	0.555734	−	0.831360i	$-0.312437\pi$
$349$	20.7639	1.11147	0.555734	+	0.831360i	$0.312437\pi$
$350$	0	0
$351$	6.76393	0.361032
$352$	0	0
$353$	26.5066	1.41080	0.705401	−	0.708808i	$-0.250767\pi$
$353$	26.5066	1.41080	0.705401	+	0.708808i	$0.250767\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	0	0
$357$	−13.4164	−0.710072
$358$	0	0
$359$	−5.70820	−0.301267	−0.150634	−	0.988590i	$-0.548131\pi$
$359$	−5.70820	−0.301267	−0.150634	+	0.988590i	$0.548131\pi$
$360$	0	0
$361$	9.96556	0.524503
$362$	0	0
$363$	0	0
$364$	0	0
$365$	13.2361	0.692807
$366$	0	0
$367$	−20.6525	−1.07805	−0.539025	−	0.842290i	$-0.681207\pi$
$367$	−20.6525	−1.07805	−0.539025	+	0.842290i	$0.681207\pi$
$368$	0	0
$369$	2.79837	0.145678
$370$	0	0
$371$	28.9443	1.50271
$372$	0	0
$373$	−16.6525	−0.862233	−0.431116	−	0.902296i	$-0.641880\pi$
$373$	−16.6525	−0.862233	−0.431116	+	0.902296i	$0.641880\pi$
$374$	0	0
$375$	−2.47214	−0.127661
$376$	0	0
$377$	−10.4721	−0.539342
$378$	0	0
$379$	1.03444	0.0531357	0.0265679	−	0.999647i	$-0.491542\pi$
$379$	1.03444	0.0531357	0.0265679	+	0.999647i	$0.491542\pi$
$380$	0	0
$381$	33.1246	1.69703
$382$	0	0
$383$	−7.41641	−0.378961	−0.189480	−	0.981885i	$-0.560680\pi$
$383$	−7.41641	−0.378961	−0.189480	+	0.981885i	$0.560680\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.56231	−0.0794165
$388$	0	0
$389$	3.81966	0.193664	0.0968322	−	0.995301i	$-0.469129\pi$
$389$	3.81966	0.193664	0.0968322	+	0.995301i	$0.469129\pi$
$390$	0	0
$391$	−9.70820	−0.490965
$392$	0	0
$393$	−1.76393	−0.0889786
$394$	0	0
$395$	−19.4164	−0.976946
$396$	0	0
$397$	−8.76393	−0.439849	−0.219925	−	0.975517i	$-0.570581\pi$
$397$	−8.76393	−0.439849	−0.219925	+	0.975517i	$0.570581\pi$
$398$	0	0
$399$	38.9443	1.94965
$400$	0	0
$401$	12.6738	0.632897	0.316449	−	0.948610i	$-0.397509\pi$
$401$	12.6738	0.632897	0.316449	+	0.948610i	$0.397509\pi$
$402$	0	0
$403$	0.583592	0.0290708
$404$	0	0
$405$	24.9443	1.23949
$406$	0	0
$407$	0	0
$408$	0	0
$409$	12.4721	0.616707	0.308354	−	0.951272i	$-0.400222\pi$
$409$	12.4721	0.616707	0.308354	+	0.951272i	$0.400222\pi$
$410$	0	0
$411$	−15.8541	−0.782025
$412$	0	0
$413$	40.6525	2.00038
$414$	0	0
$415$	7.70820	0.378381
$416$	0	0
$417$	27.4164	1.34259
$418$	0	0
$419$	26.3820	1.28884	0.644422	−	0.764670i	$-0.277098\pi$
$419$	26.3820	1.28884	0.644422	+	0.764670i	$0.277098\pi$
$420$	0	0
$421$	−21.5279	−1.04920	−0.524602	−	0.851348i	$-0.675786\pi$
$421$	−21.5279	−1.04920	−0.524602	+	0.851348i	$0.675786\pi$
$422$	0	0
$423$	2.47214	0.120199
$424$	0	0
$425$	10.1459	0.492148
$426$	0	0
$427$	11.0557	0.535024
$428$	0	0
$429$	0	0
$430$	0	0
$431$	29.8885	1.43968	0.719840	−	0.694140i	$-0.244215\pi$
$431$	29.8885	1.43968	0.719840	+	0.694140i	$0.244215\pi$
$432$	0	0
$433$	4.32624	0.207906	0.103953	−	0.994582i	$-0.466851\pi$
$433$	4.32624	0.207906	0.103953	+	0.994582i	$0.466851\pi$
$434$	0	0
$435$	−44.3607	−2.12693
$436$	0	0
$437$	28.1803	1.34805
$438$	0	0
$439$	−6.00000	−0.286364	−0.143182	−	0.989696i	$-0.545733\pi$
$439$	−6.00000	−0.286364	−0.143182	+	0.989696i	$0.545733\pi$
$440$	0	0
$441$	−4.96556	−0.236455
$442$	0	0
$443$	−34.7426	−1.65067	−0.825336	−	0.564641i	$-0.809015\pi$
$443$	−34.7426	−1.65067	−0.825336	+	0.564641i	$0.809015\pi$
$444$	0	0
$445$	−20.6525	−0.979021
$446$	0	0
$447$	6.94427	0.328453
$448$	0	0
$449$	11.7426	0.554170	0.277085	−	0.960845i	$-0.410632\pi$
$449$	11.7426	0.554170	0.277085	+	0.960845i	$0.410632\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−20.9443	−0.984048
$454$	0	0
$455$	−17.8885	−0.838628
$456$	0	0
$457$	9.61803	0.449913	0.224956	−	0.974369i	$-0.427776\pi$
$457$	9.61803	0.449913	0.224956	+	0.974369i	$0.427776\pi$
$458$	0	0
$459$	−10.1459	−0.473570
$460$	0	0
$461$	30.6525	1.42763	0.713814	−	0.700335i	$-0.246966\pi$
$461$	30.6525	1.42763	0.713814	+	0.700335i	$0.246966\pi$
$462$	0	0
$463$	−25.5279	−1.18638	−0.593190	−	0.805062i	$-0.702132\pi$
$463$	−25.5279	−1.18638	−0.593190	+	0.805062i	$0.702132\pi$
$464$	0	0
$465$	2.47214	0.114643
$466$	0	0
$467$	25.5279	1.18129	0.590644	−	0.806932i	$-0.298874\pi$
$467$	25.5279	1.18129	0.590644	+	0.806932i	$0.298874\pi$
$468$	0	0
$469$	6.18034	0.285382
$470$	0	0
$471$	−15.7082	−0.723796
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−29.4508	−1.35130
$476$	0	0
$477$	2.47214	0.113191
$478$	0	0
$479$	−39.4164	−1.80098	−0.900491	−	0.434875i	$-0.856793\pi$
$479$	−39.4164	−1.80098	−0.900491	+	0.434875i	$0.856793\pi$
$480$	0	0
$481$	11.4164	0.520543
$482$	0	0
$483$	37.8885	1.72399
$484$	0	0
$485$	−18.1803	−0.825527
$486$	0	0
$487$	−33.1246	−1.50102	−0.750510	−	0.660859i	$-0.770192\pi$
$487$	−33.1246	−1.50102	−0.750510	+	0.660859i	$0.770192\pi$
$488$	0	0
$489$	−34.1246	−1.54317
$490$	0	0
$491$	−8.27051	−0.373243	−0.186621	−	0.982432i	$-0.559754\pi$
$491$	−8.27051	−0.373243	−0.186621	+	0.982432i	$0.559754\pi$
$492$	0	0
$493$	15.7082	0.707462
$494$	0	0
$495$	0	0
$496$	0	0
$497$	16.5836	0.743876
$498$	0	0
$499$	−41.6180	−1.86308	−0.931540	−	0.363640i	$-0.881534\pi$
$499$	−41.6180	−1.86308	−0.931540	+	0.363640i	$0.881534\pi$
$500$	0	0
$501$	20.4721	0.914628
$502$	0	0
$503$	−10.2918	−0.458889	−0.229444	−	0.973322i	$-0.573691\pi$
$503$	−10.2918	−0.458889	−0.229444	+	0.973322i	$0.573691\pi$
$504$	0	0
$505$	−7.41641	−0.330026
$506$	0	0
$507$	−18.5623	−0.824381
$508$	0	0
$509$	13.2361	0.586678	0.293339	−	0.956008i	$-0.405233\pi$
$509$	13.2361	0.586678	0.293339	+	0.956008i	$0.405233\pi$
$510$	0	0
$511$	18.2918	0.809181
$512$	0	0
$513$	29.4508	1.30029
$514$	0	0
$515$	−55.7771	−2.45783
$516$	0	0
$517$	0	0
$518$	0	0
$519$	14.4721	0.635256
$520$	0	0
$521$	−10.1459	−0.444500	−0.222250	−	0.974990i	$-0.571340\pi$
$521$	−10.1459	−0.444500	−0.222250	+	0.974990i	$0.571340\pi$
$522$	0	0
$523$	−24.7984	−1.08436	−0.542179	−	0.840263i	$-0.682400\pi$
$523$	−24.7984	−1.08436	−0.542179	+	0.840263i	$0.682400\pi$
$524$	0	0
$525$	−39.5967	−1.72814
$526$	0	0
$527$	−0.875388	−0.0381325
$528$	0	0
$529$	4.41641	0.192018
$530$	0	0
$531$	3.47214	0.150678
$532$	0	0
$533$	9.05573	0.392247
$534$	0	0
$535$	10.7639	0.465365
$536$	0	0
$537$	−15.6525	−0.675454
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−24.6525	−1.05989	−0.529946	−	0.848031i	$-0.677788\pi$
$541$	−24.6525	−1.05989	−0.529946	+	0.848031i	$0.677788\pi$
$542$	0	0
$543$	−1.23607	−0.0530448
$544$	0	0
$545$	9.52786	0.408129
$546$	0	0
$547$	−20.5066	−0.876798	−0.438399	−	0.898780i	$-0.644454\pi$
$547$	−20.5066	−0.876798	−0.438399	+	0.898780i	$0.644454\pi$
$548$	0	0
$549$	0.944272	0.0403005
$550$	0	0
$551$	−45.5967	−1.94249
$552$	0	0
$553$	−26.8328	−1.14105
$554$	0	0
$555$	48.3607	2.05280
$556$	0	0
$557$	−0.180340	−0.00764125	−0.00382062	−	0.999993i	$-0.501216\pi$
$557$	−0.180340	−0.00764125	−0.00382062	+	0.999993i	$0.501216\pi$
$558$	0	0
$559$	−5.05573	−0.213835
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−17.4377	−0.734911	−0.367456	−	0.930041i	$-0.619771\pi$
$563$	−17.4377	−0.734911	−0.367456	+	0.930041i	$0.619771\pi$
$564$	0	0
$565$	−61.7771	−2.59898
$566$	0	0
$567$	34.4721	1.44769
$568$	0	0
$569$	−13.5623	−0.568561	−0.284281	−	0.958741i	$-0.591755\pi$
$569$	−13.5623	−0.568561	−0.284281	+	0.958741i	$0.591755\pi$
$570$	0	0
$571$	22.4721	0.940430	0.470215	−	0.882552i	$-0.344176\pi$
$571$	22.4721	0.940430	0.470215	+	0.882552i	$0.344176\pi$
$572$	0	0
$573$	9.23607	0.385842
$574$	0	0
$575$	−28.6525	−1.19489
$576$	0	0
$577$	33.9230	1.41223	0.706116	−	0.708096i	$-0.250446\pi$
$577$	33.9230	1.41223	0.706116	+	0.708096i	$0.250446\pi$
$578$	0	0
$579$	28.1803	1.17113
$580$	0	0
$581$	10.6525	0.441939
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−1.52786	−0.0631694
$586$	0	0
$587$	−1.96556	−0.0811273	−0.0405636	−	0.999177i	$-0.512915\pi$
$587$	−1.96556	−0.0811273	−0.0405636	+	0.999177i	$0.512915\pi$
$588$	0	0
$589$	2.54102	0.104701
$590$	0	0
$591$	12.9443	0.532456
$592$	0	0
$593$	−34.7984	−1.42900	−0.714499	−	0.699636i	$-0.753345\pi$
$593$	−34.7984	−1.42900	−0.714499	+	0.699636i	$0.753345\pi$
$594$	0	0
$595$	26.8328	1.10004
$596$	0	0
$597$	36.1803	1.48076
$598$	0	0
$599$	−12.8328	−0.524335	−0.262167	−	0.965022i	$-0.584437\pi$
$599$	−12.8328	−0.524335	−0.262167	+	0.965022i	$0.584437\pi$
$600$	0	0
$601$	3.32624	0.135680	0.0678400	−	0.997696i	$-0.478389\pi$
$601$	3.32624	0.135680	0.0678400	+	0.997696i	$0.478389\pi$
$602$	0	0
$603$	0.527864	0.0214963
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−20.2918	−0.823619	−0.411809	−	0.911270i	$-0.635103\pi$
$607$	−20.2918	−0.823619	−0.411809	+	0.911270i	$0.635103\pi$
$608$	0	0
$609$	−61.3050	−2.48420
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	−6.76393	−0.273193	−0.136596	−	0.990627i	$-0.543616\pi$
$613$	−6.76393	−0.273193	−0.136596	+	0.990627i	$0.543616\pi$
$614$	0	0
$615$	38.3607	1.54685
$616$	0	0
$617$	−4.90983	−0.197662	−0.0988312	−	0.995104i	$-0.531510\pi$
$617$	−4.90983	−0.197662	−0.0988312	+	0.995104i	$0.531510\pi$
$618$	0	0
$619$	−15.8541	−0.637230	−0.318615	−	0.947884i	$-0.603218\pi$
$619$	−15.8541	−0.637230	−0.318615	+	0.947884i	$0.603218\pi$
$620$	0	0
$621$	28.6525	1.14978
$622$	0	0
$623$	−28.5410	−1.14347
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−17.1246	−0.682803
$630$	0	0
$631$	6.18034	0.246035	0.123018	−	0.992404i	$-0.460743\pi$
$631$	6.18034	0.246035	0.123018	+	0.992404i	$0.460743\pi$
$632$	0	0
$633$	0.145898	0.00579893
$634$	0	0
$635$	−66.2492	−2.62902
$636$	0	0
$637$	−16.0689	−0.636672
$638$	0	0
$639$	1.41641	0.0560322
$640$	0	0
$641$	−32.8541	−1.29766	−0.648830	−	0.760934i	$-0.724741\pi$
$641$	−32.8541	−1.29766	−0.648830	+	0.760934i	$0.724741\pi$
$642$	0	0
$643$	−35.5066	−1.40024	−0.700121	−	0.714024i	$-0.746871\pi$
$643$	−35.5066	−1.40024	−0.700121	+	0.714024i	$0.746871\pi$
$644$	0	0
$645$	−21.4164	−0.843270
$646$	0	0
$647$	23.5279	0.924976	0.462488	−	0.886626i	$-0.346957\pi$
$647$	23.5279	0.924976	0.462488	+	0.886626i	$0.346957\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	3.41641	0.133900
$652$	0	0
$653$	45.0132	1.76150	0.880750	−	0.473581i	$-0.157039\pi$
$653$	45.0132	1.76150	0.880750	+	0.473581i	$0.157039\pi$
$654$	0	0
$655$	3.52786	0.137845
$656$	0	0
$657$	1.56231	0.0609514
$658$	0	0
$659$	−11.4934	−0.447720	−0.223860	−	0.974621i	$-0.571866\pi$
$659$	−11.4934	−0.447720	−0.223860	+	0.974621i	$0.571866\pi$
$660$	0	0
$661$	21.4164	0.833002	0.416501	−	0.909135i	$-0.363256\pi$
$661$	21.4164	0.833002	0.416501	+	0.909135i	$0.363256\pi$
$662$	0	0
$663$	−3.70820	−0.144015
$664$	0	0
$665$	−77.8885	−3.02039
$666$	0	0
$667$	−44.3607	−1.71765
$668$	0	0
$669$	−3.70820	−0.143367
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−44.3951	−1.71131	−0.855653	−	0.517550i	$-0.826844\pi$
$673$	−44.3951	−1.71131	−0.855653	+	0.517550i	$0.826844\pi$
$674$	0	0
$675$	−29.9443	−1.15256
$676$	0	0
$677$	12.4721	0.479343	0.239672	−	0.970854i	$-0.422960\pi$
$677$	12.4721	0.479343	0.239672	+	0.970854i	$0.422960\pi$
$678$	0	0
$679$	−25.1246	−0.964194
$680$	0	0
$681$	−38.5967	−1.47903
$682$	0	0
$683$	27.4164	1.04906	0.524530	−	0.851392i	$-0.324241\pi$
$683$	27.4164	1.04906	0.524530	+	0.851392i	$0.324241\pi$
$684$	0	0
$685$	31.7082	1.21151
$686$	0	0
$687$	26.1803	0.998842
$688$	0	0
$689$	8.00000	0.304776
$690$	0	0
$691$	38.9787	1.48282	0.741410	−	0.671052i	$-0.234157\pi$
$691$	38.9787	1.48282	0.741410	+	0.671052i	$0.234157\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−54.8328	−2.07993
$696$	0	0
$697$	−13.5836	−0.514515
$698$	0	0
$699$	12.7984	0.484079
$700$	0	0
$701$	−6.83282	−0.258072	−0.129036	−	0.991640i	$-0.541188\pi$
$701$	−6.83282	−0.258072	−0.129036	+	0.991640i	$0.541188\pi$
$702$	0	0
$703$	49.7082	1.87478
$704$	0	0
$705$	33.8885	1.27632
$706$	0	0
$707$	−10.2492	−0.385462
$708$	0	0
$709$	52.4721	1.97063	0.985316	−	0.170739i	$-0.0546156\pi$
$709$	52.4721	1.97063	0.985316	+	0.170739i	$0.0546156\pi$
$710$	0	0
$711$	−2.29180	−0.0859491
$712$	0	0
$713$	2.47214	0.0925822
$714$	0	0
$715$	0	0
$716$	0	0
$717$	23.1246	0.863604
$718$	0	0
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	0	0
$721$	−77.0820	−2.87069
$722$	0	0
$723$	45.8328	1.70454
$724$	0	0
$725$	46.3607	1.72179
$726$	0	0
$727$	6.87539	0.254994	0.127497	−	0.991839i	$-0.459306\pi$
$727$	6.87539	0.254994	0.127497	+	0.991839i	$0.459306\pi$
$728$	0	0
$729$	29.5066	1.09284
$730$	0	0
$731$	7.58359	0.280489
$732$	0	0
$733$	−10.3607	−0.382680	−0.191340	−	0.981524i	$-0.561283\pi$
$733$	−10.3607	−0.382680	−0.191340	+	0.981524i	$0.561283\pi$
$734$	0	0
$735$	−68.0689	−2.51076
$736$	0	0
$737$	0	0
$738$	0	0
$739$	25.3820	0.933691	0.466845	−	0.884339i	$-0.345390\pi$
$739$	25.3820	0.933691	0.466845	+	0.884339i	$0.345390\pi$
$740$	0	0
$741$	10.7639	0.395423
$742$	0	0
$743$	3.23607	0.118720	0.0593599	−	0.998237i	$-0.481094\pi$
$743$	3.23607	0.118720	0.0593599	+	0.998237i	$0.481094\pi$
$744$	0	0
$745$	−13.8885	−0.508837
$746$	0	0
$747$	0.909830	0.0332889
$748$	0	0
$749$	14.8754	0.543535
$750$	0	0
$751$	−34.3607	−1.25384	−0.626920	−	0.779084i	$-0.715685\pi$
$751$	−34.3607	−1.25384	−0.626920	+	0.779084i	$0.715685\pi$
$752$	0	0
$753$	−6.47214	−0.235858
$754$	0	0
$755$	41.8885	1.52448
$756$	0	0
$757$	−34.2918	−1.24636	−0.623178	−	0.782080i	$-0.714159\pi$
$757$	−34.2918	−1.24636	−0.623178	+	0.782080i	$0.714159\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	38.7984	1.40644	0.703220	−	0.710972i	$-0.251745\pi$
$761$	38.7984	1.40644	0.703220	+	0.710972i	$0.251745\pi$
$762$	0	0
$763$	13.1672	0.476684
$764$	0	0
$765$	2.29180	0.0828601
$766$	0	0
$767$	11.2361	0.405711
$768$	0	0
$769$	−18.5836	−0.670141	−0.335071	−	0.942193i	$-0.608760\pi$
$769$	−18.5836	−0.670141	−0.335071	+	0.942193i	$0.608760\pi$
$770$	0	0
$771$	−30.0344	−1.08166
$772$	0	0
$773$	24.8328	0.893174	0.446587	−	0.894740i	$-0.352639\pi$
$773$	24.8328	0.893174	0.446587	+	0.894740i	$0.352639\pi$
$774$	0	0
$775$	−2.58359	−0.0928054
$776$	0	0
$777$	66.8328	2.39761
$778$	0	0
$779$	39.4296	1.41271
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−46.3607	−1.65680
$784$	0	0
$785$	31.4164	1.12130
$786$	0	0
$787$	−1.45085	−0.0517172	−0.0258586	−	0.999666i	$-0.508232\pi$
$787$	−1.45085	−0.0517172	−0.0258586	+	0.999666i	$0.508232\pi$
$788$	0	0
$789$	−0.291796	−0.0103882
$790$	0	0
$791$	−85.3738	−3.03554
$792$	0	0
$793$	3.05573	0.108512
$794$	0	0
$795$	33.8885	1.20190
$796$	0	0
$797$	26.6525	0.944079	0.472040	−	0.881577i	$-0.343518\pi$
$797$	26.6525	0.944079	0.472040	+	0.881577i	$0.343518\pi$
$798$	0	0
$799$	−12.0000	−0.424529
$800$	0	0
$801$	−2.43769	−0.0861317
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−75.7771	−2.67079
$806$	0	0
$807$	−22.9443	−0.807677
$808$	0	0
$809$	21.8541	0.768349	0.384175	−	0.923260i	$-0.374486\pi$
$809$	21.8541	0.768349	0.384175	+	0.923260i	$0.374486\pi$
$810$	0	0
$811$	−20.8541	−0.732287	−0.366143	−	0.930558i	$-0.619322\pi$
$811$	−20.8541	−0.732287	−0.366143	+	0.930558i	$0.619322\pi$
$812$	0	0
$813$	−13.7082	−0.480768
$814$	0	0
$815$	68.2492	2.39067
$816$	0	0
$817$	−22.0132	−0.770143
$818$	0	0
$819$	−2.11146	−0.0737802
$820$	0	0
$821$	−30.5410	−1.06589	−0.532944	−	0.846150i	$-0.678915\pi$
$821$	−30.5410	−1.06589	−0.532944	+	0.846150i	$0.678915\pi$
$822$	0	0
$823$	2.87539	0.100230	0.0501149	−	0.998743i	$-0.484041\pi$
$823$	2.87539	0.100230	0.0501149	+	0.998743i	$0.484041\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	27.8541	0.968582	0.484291	−	0.874907i	$-0.339078\pi$
$827$	27.8541	0.968582	0.484291	+	0.874907i	$0.339078\pi$
$828$	0	0
$829$	−2.83282	−0.0983878	−0.0491939	−	0.998789i	$-0.515665\pi$
$829$	−2.83282	−0.0983878	−0.0491939	+	0.998789i	$0.515665\pi$
$830$	0	0
$831$	−16.4721	−0.571412
$832$	0	0
$833$	24.1033	0.835131
$834$	0	0
$835$	−40.9443	−1.41693
$836$	0	0
$837$	2.58359	0.0893020
$838$	0	0
$839$	−25.4164	−0.877472	−0.438736	−	0.898616i	$-0.644574\pi$
$839$	−25.4164	−0.877472	−0.438736	+	0.898616i	$0.644574\pi$
$840$	0	0
$841$	42.7771	1.47507
$842$	0	0
$843$	39.2705	1.35255
$844$	0	0
$845$	37.1246	1.27713
$846$	0	0
$847$	0	0
$848$	0	0
$849$	14.4721	0.496682
$850$	0	0
$851$	48.3607	1.65778
$852$	0	0
$853$	16.3607	0.560179	0.280090	−	0.959974i	$-0.409636\pi$
$853$	16.3607	0.560179	0.280090	+	0.959974i	$0.409636\pi$
$854$	0	0
$855$	−6.65248	−0.227510
$856$	0	0
$857$	−10.7426	−0.366962	−0.183481	−	0.983023i	$-0.558737\pi$
$857$	−10.7426	−0.366962	−0.183481	+	0.983023i	$0.558737\pi$
$858$	0	0
$859$	44.5066	1.51854	0.759272	−	0.650773i	$-0.225555\pi$
$859$	44.5066	1.51854	0.759272	+	0.650773i	$0.225555\pi$
$860$	0	0
$861$	53.0132	1.80668
$862$	0	0
$863$	2.18034	0.0742196	0.0371098	−	0.999311i	$-0.488185\pi$
$863$	2.18034	0.0742196	0.0371098	+	0.999311i	$0.488185\pi$
$864$	0	0
$865$	−28.9443	−0.984135
$866$	0	0
$867$	−21.9443	−0.745267
$868$	0	0
$869$	0	0
$870$	0	0
$871$	1.70820	0.0578803
$872$	0	0
$873$	−2.14590	−0.0726276
$874$	0	0
$875$	6.83282	0.230991
$876$	0	0
$877$	−12.5410	−0.423480	−0.211740	−	0.977326i	$-0.567913\pi$
$877$	−12.5410	−0.423480	−0.211740	+	0.977326i	$0.567913\pi$
$878$	0	0
$879$	−21.8885	−0.738282
$880$	0	0
$881$	−29.6180	−0.997857	−0.498928	−	0.866643i	$-0.666273\pi$
$881$	−29.6180	−0.997857	−0.498928	+	0.866643i	$0.666273\pi$
$882$	0	0
$883$	−22.2016	−0.747144	−0.373572	−	0.927601i	$-0.621867\pi$
$883$	−22.2016	−0.747144	−0.373572	+	0.927601i	$0.621867\pi$
$884$	0	0
$885$	47.5967	1.59995
$886$	0	0
$887$	8.47214	0.284466	0.142233	−	0.989833i	$-0.454572\pi$
$887$	8.47214	0.284466	0.142233	+	0.989833i	$0.454572\pi$
$888$	0	0
$889$	−91.5542	−3.07063
$890$	0	0
$891$	0	0
$892$	0	0
$893$	34.8328	1.16564
$894$	0	0
$895$	31.3050	1.04641
$896$	0	0
$897$	10.4721	0.349654
$898$	0	0
$899$	−4.00000	−0.133407
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	−29.5967	−0.984918
$904$	0	0
$905$	2.47214	0.0821766
$906$	0	0
$907$	3.85410	0.127973	0.0639867	−	0.997951i	$-0.479618\pi$
$907$	3.85410	0.127973	0.0639867	+	0.997951i	$0.479618\pi$
$908$	0	0
$909$	−0.875388	−0.0290348
$910$	0	0
$911$	29.5967	0.980584	0.490292	−	0.871558i	$-0.336890\pi$
$911$	29.5967	0.980584	0.490292	+	0.871558i	$0.336890\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	12.9443	0.427924
$916$	0	0
$917$	4.87539	0.161000
$918$	0	0
$919$	−43.4164	−1.43218	−0.716088	−	0.698010i	$-0.754069\pi$
$919$	−43.4164	−1.43218	−0.716088	+	0.698010i	$0.754069\pi$
$920$	0	0
$921$	−7.85410	−0.258801
$922$	0	0
$923$	4.58359	0.150871
$924$	0	0
$925$	−50.5410	−1.66178
$926$	0	0
$927$	−6.58359	−0.216234
$928$	0	0
$929$	−27.4508	−0.900633	−0.450317	−	0.892869i	$-0.648689\pi$
$929$	−27.4508	−0.900633	−0.450317	+	0.892869i	$0.648689\pi$
$930$	0	0
$931$	−69.9656	−2.29303
$932$	0	0
$933$	−14.1803	−0.464243
$934$	0	0
$935$	0	0
$936$	0	0
$937$	43.4508	1.41948	0.709739	−	0.704465i	$-0.248813\pi$
$937$	43.4508	1.41948	0.709739	+	0.704465i	$0.248813\pi$
$938$	0	0
$939$	24.8885	0.812207
$940$	0	0
$941$	30.6525	0.999242	0.499621	−	0.866244i	$-0.333473\pi$
$941$	30.6525	0.999242	0.499621	+	0.866244i	$0.333473\pi$
$942$	0	0
$943$	38.3607	1.24920
$944$	0	0
$945$	−79.1935	−2.57616
$946$	0	0
$947$	−24.7984	−0.805839	−0.402919	−	0.915235i	$-0.632005\pi$
$947$	−24.7984	−0.805839	−0.402919	+	0.915235i	$0.632005\pi$
$948$	0	0
$949$	5.05573	0.164116
$950$	0	0
$951$	2.94427	0.0954746
$952$	0	0
$953$	4.67376	0.151398	0.0756990	−	0.997131i	$-0.475881\pi$
$953$	4.67376	0.151398	0.0756990	+	0.997131i	$0.475881\pi$
$954$	0	0
$955$	−18.4721	−0.597744
$956$	0	0
$957$	0	0
$958$	0	0
$959$	43.8197	1.41501
$960$	0	0
$961$	−30.7771	−0.992809
$962$	0	0
$963$	1.27051	0.0409416
$964$	0	0
$965$	−56.3607	−1.81431
$966$	0	0
$967$	10.3607	0.333177	0.166589	−	0.986027i	$-0.446725\pi$
$967$	10.3607	0.333177	0.166589	+	0.986027i	$0.446725\pi$
$968$	0	0
$969$	−16.1459	−0.518681
$970$	0	0
$971$	35.7771	1.14814	0.574071	−	0.818806i	$-0.305363\pi$
$971$	35.7771	1.14814	0.574071	+	0.818806i	$0.305363\pi$
$972$	0	0
$973$	−75.7771	−2.42930
$974$	0	0
$975$	−10.9443	−0.350497
$976$	0	0
$977$	−23.5279	−0.752723	−0.376362	−	0.926473i	$-0.622825\pi$
$977$	−23.5279	−0.752723	−0.376362	+	0.926473i	$0.622825\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	1.12461	0.0359061
$982$	0	0
$983$	26.5836	0.847885	0.423942	−	0.905689i	$-0.360646\pi$
$983$	26.5836	0.847885	0.423942	+	0.905689i	$0.360646\pi$
$984$	0	0
$985$	−25.8885	−0.824878
$986$	0	0
$987$	46.8328	1.49070
$988$	0	0
$989$	−21.4164	−0.681002
$990$	0	0
$991$	−11.7082	−0.371923	−0.185962	−	0.982557i	$-0.559540\pi$
$991$	−11.7082	−0.371923	−0.185962	+	0.982557i	$0.559540\pi$
$992$	0	0
$993$	−48.2148	−1.53005
$994$	0	0
$995$	−72.3607	−2.29399
$996$	0	0
$997$	27.1246	0.859045	0.429523	−	0.903056i	$-0.358682\pi$
$997$	27.1246	0.859045	0.429523	+	0.903056i	$0.358682\pi$
$998$	0	0
$999$	50.5410	1.59905

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7744.2.a.cp.1.2		2
4.3	odd	2		7744.2.a.ca.1.1		2
8.3	odd	2		3872.2.a.y.1.2		2
8.5	even	2		3872.2.a.n.1.1		2
11.5	even	5		704.2.m.f.641.1		4
11.9	even	5		704.2.m.f.257.1		4
11.10	odd	2		7744.2.a.co.1.2		2
44.27	odd	10		704.2.m.c.641.1		4
44.31	odd	10		704.2.m.c.257.1		4
44.43	even	2		7744.2.a.bz.1.1		2
88.5	even	10		352.2.m.a.289.1	yes	4
88.21	odd	2		3872.2.a.o.1.1		2
88.27	odd	10		352.2.m.b.289.1	yes	4
88.43	even	2		3872.2.a.z.1.2		2
88.53	even	10		352.2.m.a.257.1	✓	4
88.75	odd	10		352.2.m.b.257.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.m.a.257.1	✓	4	88.53	even	10
352.2.m.a.289.1	yes	4	88.5	even	10
352.2.m.b.257.1	yes	4	88.75	odd	10
352.2.m.b.289.1	yes	4	88.27	odd	10
704.2.m.c.257.1		4	44.31	odd	10
704.2.m.c.641.1		4	44.27	odd	10
704.2.m.f.257.1		4	11.9	even	5
704.2.m.f.641.1		4	11.5	even	5
3872.2.a.n.1.1		2	8.5	even	2
3872.2.a.o.1.1		2	88.21	odd	2
3872.2.a.y.1.2		2	8.3	odd	2
3872.2.a.z.1.2		2	88.43	even	2
7744.2.a.bz.1.1		2	44.43	even	2
7744.2.a.ca.1.1		2	4.3	odd	2
7744.2.a.co.1.2		2	11.10	odd	2
7744.2.a.cp.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7744 = 2^{6} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7744.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{3} -3.23607 q^{5} -4.47214 q^{7} -0.381966 q^{9} +O(q^{10})\)	\(q+1.61803 q^{3} -3.23607 q^{5} -4.47214 q^{7} -0.381966 q^{9} -1.23607 q^{13} -5.23607 q^{15} +1.85410 q^{17} -5.38197 q^{19} -7.23607 q^{21} -5.23607 q^{23} +5.47214 q^{25} -5.47214 q^{27} +8.47214 q^{29} -0.472136 q^{31} +14.4721 q^{35} -9.23607 q^{37} -2.00000 q^{39} -7.32624 q^{41} +4.09017 q^{43} +1.23607 q^{45} -6.47214 q^{47} +13.0000 q^{49} +3.00000 q^{51} -6.47214 q^{53} -8.70820 q^{57} -9.09017 q^{59} -2.47214 q^{61} +1.70820 q^{63} +4.00000 q^{65} -1.38197 q^{67} -8.47214 q^{69} -3.70820 q^{71} -4.09017 q^{73} +8.85410 q^{75} +6.00000 q^{79} -7.70820 q^{81} -2.38197 q^{83} -6.00000 q^{85} +13.7082 q^{87} +6.38197 q^{89} +5.52786 q^{91} -0.763932 q^{93} +17.4164 q^{95} +5.61803 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 2 q^{5} - 3 q^{9}+O(q^{10}) \) 2 * q + q^3 - 2 * q^5 - 3 * q^9	\( 2 q + q^{3} - 2 q^{5} - 3 q^{9} + 2 q^{13} - 6 q^{15} - 3 q^{17} - 13 q^{19} - 10 q^{21} - 6 q^{23} + 2 q^{25} - 2 q^{27} + 8 q^{29} + 8 q^{31} + 20 q^{35} - 14 q^{37} - 4 q^{39} + q^{41} - 3 q^{43} - 2 q^{45} - 4 q^{47} + 26 q^{49} + 6 q^{51} - 4 q^{53} - 4 q^{57} - 7 q^{59} + 4 q^{61} - 10 q^{63} + 8 q^{65} - 5 q^{67} - 8 q^{69} + 6 q^{71} + 3 q^{73} + 11 q^{75} + 12 q^{79} - 2 q^{81} - 7 q^{83} - 12 q^{85} + 14 q^{87} + 15 q^{89} + 20 q^{91} - 6 q^{93} + 8 q^{95} + 9 q^{97}+O(q^{100}) \) 2 * q + q^3 - 2 * q^5 - 3 * q^9 + 2 * q^13 - 6 * q^15 - 3 * q^17 - 13 * q^19 - 10 * q^21 - 6 * q^23 + 2 * q^25 - 2 * q^27 + 8 * q^29 + 8 * q^31 + 20 * q^35 - 14 * q^37 - 4 * q^39 + q^41 - 3 * q^43 - 2 * q^45 - 4 * q^47 + 26 * q^49 + 6 * q^51 - 4 * q^53 - 4 * q^57 - 7 * q^59 + 4 * q^61 - 10 * q^63 + 8 * q^65 - 5 * q^67 - 8 * q^69 + 6 * q^71 + 3 * q^73 + 11 * q^75 + 12 * q^79 - 2 * q^81 - 7 * q^83 - 12 * q^85 + 14 * q^87 + 15 * q^89 + 20 * q^91 - 6 * q^93 + 8 * q^95 + 9 * q^97

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