LMFDB - Embedded newform 7728.2.a.bo.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1932)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.618034 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.618034 q^{5} +1.00000 q^{7} +1.00000 q^{9} -0.236068 q^{11} +0.381966 q^{13} -0.618034 q^{15} -2.23607 q^{17} -3.47214 q^{19} +1.00000 q^{21} -1.00000 q^{23} -4.61803 q^{25} +1.00000 q^{27} +5.47214 q^{29} +1.76393 q^{31} -0.236068 q^{33} -0.618034 q^{35} -5.47214 q^{37} +0.381966 q^{39} +7.47214 q^{41} +12.5623 q^{43} -0.618034 q^{45} +0.763932 q^{47} +1.00000 q^{49} -2.23607 q^{51} -7.09017 q^{53} +0.145898 q^{55} -3.47214 q^{57} +3.38197 q^{59} +5.32624 q^{61} +1.00000 q^{63} -0.236068 q^{65} +3.38197 q^{67} -1.00000 q^{69} +10.8541 q^{71} +1.94427 q^{73} -4.61803 q^{75} -0.236068 q^{77} +8.23607 q^{79} +1.00000 q^{81} +7.94427 q^{83} +1.38197 q^{85} +5.47214 q^{87} -2.85410 q^{89} +0.381966 q^{91} +1.76393 q^{93} +2.14590 q^{95} -0.236068 q^{97} -0.236068 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{11} + 3 q^{13} + q^{15} + 2 q^{19} + 2 q^{21} - 2 q^{23} - 7 q^{25} + 2 q^{27} + 2 q^{29} + 8 q^{31} + 4 q^{33} + q^{35} - 2 q^{37} + 3 q^{39} + 6 q^{41} + 5 q^{43} + q^{45} + 6 q^{47} + 2 q^{49} - 3 q^{53} + 7 q^{55} + 2 q^{57} + 9 q^{59} - 5 q^{61} + 2 q^{63} + 4 q^{65} + 9 q^{67} - 2 q^{69} + 15 q^{71} - 14 q^{73} - 7 q^{75} + 4 q^{77} + 12 q^{79} + 2 q^{81} - 2 q^{83} + 5 q^{85} + 2 q^{87} + q^{89} + 3 q^{91} + 8 q^{93} + 11 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9 + 4 * q^11 + 3 * q^13 + q^15 + 2 * q^19 + 2 * q^21 - 2 * q^23 - 7 * q^25 + 2 * q^27 + 2 * q^29 + 8 * q^31 + 4 * q^33 + q^35 - 2 * q^37 + 3 * q^39 + 6 * q^41 + 5 * q^43 + q^45 + 6 * q^47 + 2 * q^49 - 3 * q^53 + 7 * q^55 + 2 * q^57 + 9 * q^59 - 5 * q^61 + 2 * q^63 + 4 * q^65 + 9 * q^67 - 2 * q^69 + 15 * q^71 - 14 * q^73 - 7 * q^75 + 4 * q^77 + 12 * q^79 + 2 * q^81 - 2 * q^83 + 5 * q^85 + 2 * q^87 + q^89 + 3 * q^91 + 8 * q^93 + 11 * q^95 + 4 * 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
$5$	−0.618034	−0.276393	−0.138197	−	0.990405i	$-0.544131\pi$
$5$	−0.618034	−0.276393	−0.138197	+	0.990405i	$0.544131\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.236068	−0.0711772	−0.0355886	−	0.999367i	$-0.511331\pi$
$11$	−0.236068	−0.0711772	−0.0355886	+	0.999367i	$0.511331\pi$
$12$	0	0
$13$	0.381966	0.105938	0.0529692	−	0.998596i	$-0.483131\pi$
$13$	0.381966	0.105938	0.0529692	+	0.998596i	$0.483131\pi$
$14$	0	0
$15$	−0.618034	−0.159576
$16$	0	0
$17$	−2.23607	−0.542326	−0.271163	−	0.962533i	$-0.587408\pi$
$17$	−2.23607	−0.542326	−0.271163	+	0.962533i	$0.587408\pi$
$18$	0	0
$19$	−3.47214	−0.796563	−0.398281	−	0.917263i	$-0.630393\pi$
$19$	−3.47214	−0.796563	−0.398281	+	0.917263i	$0.630393\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.61803	−0.923607
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	5.47214	1.01615	0.508075	−	0.861313i	$-0.330357\pi$
$29$	5.47214	1.01615	0.508075	+	0.861313i	$0.330357\pi$
$30$	0	0
$31$	1.76393	0.316812	0.158406	−	0.987374i	$-0.449365\pi$
$31$	1.76393	0.316812	0.158406	+	0.987374i	$0.449365\pi$
$32$	0	0
$33$	−0.236068	−0.0410942
$34$	0	0
$35$	−0.618034	−0.104467
$36$	0	0
$37$	−5.47214	−0.899614	−0.449807	−	0.893126i	$-0.648507\pi$
$37$	−5.47214	−0.899614	−0.449807	+	0.893126i	$0.648507\pi$
$38$	0	0
$39$	0.381966	0.0611635
$40$	0	0
$41$	7.47214	1.16695	0.583476	−	0.812131i	$-0.301692\pi$
$41$	7.47214	1.16695	0.583476	+	0.812131i	$0.301692\pi$
$42$	0	0
$43$	12.5623	1.91573	0.957867	−	0.287213i	$-0.0927286\pi$
$43$	12.5623	1.91573	0.957867	+	0.287213i	$0.0927286\pi$
$44$	0	0
$45$	−0.618034	−0.0921311
$46$	0	0
$47$	0.763932	0.111431	0.0557155	−	0.998447i	$-0.482256\pi$
$47$	0.763932	0.111431	0.0557155	+	0.998447i	$0.482256\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.23607	−0.313112
$52$	0	0
$53$	−7.09017	−0.973910	−0.486955	−	0.873427i	$-0.661892\pi$
$53$	−7.09017	−0.973910	−0.486955	+	0.873427i	$0.661892\pi$
$54$	0	0
$55$	0.145898	0.0196729
$56$	0	0
$57$	−3.47214	−0.459896
$58$	0	0
$59$	3.38197	0.440294	0.220147	−	0.975467i	$-0.429346\pi$
$59$	3.38197	0.440294	0.220147	+	0.975467i	$0.429346\pi$
$60$	0	0
$61$	5.32624	0.681955	0.340977	−	0.940071i	$-0.389242\pi$
$61$	5.32624	0.681955	0.340977	+	0.940071i	$0.389242\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−0.236068	−0.0292806
$66$	0	0
$67$	3.38197	0.413173	0.206586	−	0.978428i	$-0.433765\pi$
$67$	3.38197	0.413173	0.206586	+	0.978428i	$0.433765\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	10.8541	1.28814	0.644072	−	0.764964i	$-0.277244\pi$
$71$	10.8541	1.28814	0.644072	+	0.764964i	$0.277244\pi$
$72$	0	0
$73$	1.94427	0.227560	0.113780	−	0.993506i	$-0.463704\pi$
$73$	1.94427	0.227560	0.113780	+	0.993506i	$0.463704\pi$
$74$	0	0
$75$	−4.61803	−0.533245
$76$	0	0
$77$	−0.236068	−0.0269024
$78$	0	0
$79$	8.23607	0.926630	0.463315	−	0.886194i	$-0.346660\pi$
$79$	8.23607	0.926630	0.463315	+	0.886194i	$0.346660\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.94427	0.871997	0.435999	−	0.899947i	$-0.356395\pi$
$83$	7.94427	0.871997	0.435999	+	0.899947i	$0.356395\pi$
$84$	0	0
$85$	1.38197	0.149895
$86$	0	0
$87$	5.47214	0.586675
$88$	0	0
$89$	−2.85410	−0.302534	−0.151267	−	0.988493i	$-0.548335\pi$
$89$	−2.85410	−0.302534	−0.151267	+	0.988493i	$0.548335\pi$
$90$	0	0
$91$	0.381966	0.0400409
$92$	0	0
$93$	1.76393	0.182911
$94$	0	0
$95$	2.14590	0.220164
$96$	0	0
$97$	−0.236068	−0.0239691	−0.0119845	−	0.999928i	$-0.503815\pi$
$97$	−0.236068	−0.0239691	−0.0119845	+	0.999928i	$0.503815\pi$
$98$	0	0
$99$	−0.236068	−0.0237257
$100$	0	0
$101$	−1.38197	−0.137511	−0.0687554	−	0.997634i	$-0.521903\pi$
$101$	−1.38197	−0.137511	−0.0687554	+	0.997634i	$0.521903\pi$
$102$	0	0
$103$	8.94427	0.881305	0.440653	−	0.897678i	$-0.354747\pi$
$103$	8.94427	0.881305	0.440653	+	0.897678i	$0.354747\pi$
$104$	0	0
$105$	−0.618034	−0.0603139
$106$	0	0
$107$	10.3820	1.00366	0.501831	−	0.864966i	$-0.332660\pi$
$107$	10.3820	1.00366	0.501831	+	0.864966i	$0.332660\pi$
$108$	0	0
$109$	−6.61803	−0.633893	−0.316946	−	0.948443i	$-0.602658\pi$
$109$	−6.61803	−0.633893	−0.316946	+	0.948443i	$0.602658\pi$
$110$	0	0
$111$	−5.47214	−0.519392
$112$	0	0
$113$	7.56231	0.711402	0.355701	−	0.934600i	$-0.384242\pi$
$113$	7.56231	0.711402	0.355701	+	0.934600i	$0.384242\pi$
$114$	0	0
$115$	0.618034	0.0576320
$116$	0	0
$117$	0.381966	0.0353128
$118$	0	0
$119$	−2.23607	−0.204980
$120$	0	0
$121$	−10.9443	−0.994934
$122$	0	0
$123$	7.47214	0.673740
$124$	0	0
$125$	5.94427	0.531672
$126$	0	0
$127$	7.14590	0.634096	0.317048	−	0.948410i	$-0.397308\pi$
$127$	7.14590	0.634096	0.317048	+	0.948410i	$0.397308\pi$
$128$	0	0
$129$	12.5623	1.10605
$130$	0	0
$131$	−0.0557281	−0.00486899	−0.00243449	−	0.999997i	$-0.500775\pi$
$131$	−0.0557281	−0.00486899	−0.00243449	+	0.999997i	$0.500775\pi$
$132$	0	0
$133$	−3.47214	−0.301072
$134$	0	0
$135$	−0.618034	−0.0531919
$136$	0	0
$137$	−0.236068	−0.0201686	−0.0100843	−	0.999949i	$-0.503210\pi$
$137$	−0.236068	−0.0201686	−0.0100843	+	0.999949i	$0.503210\pi$
$138$	0	0
$139$	12.0344	1.02075	0.510374	−	0.859953i	$-0.329507\pi$
$139$	12.0344	1.02075	0.510374	+	0.859953i	$0.329507\pi$
$140$	0	0
$141$	0.763932	0.0643347
$142$	0	0
$143$	−0.0901699	−0.00754039
$144$	0	0
$145$	−3.38197	−0.280857
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	14.6525	1.20038	0.600189	−	0.799858i	$-0.295092\pi$
$149$	14.6525	1.20038	0.600189	+	0.799858i	$0.295092\pi$
$150$	0	0
$151$	2.29180	0.186504	0.0932519	−	0.995643i	$-0.470274\pi$
$151$	2.29180	0.186504	0.0932519	+	0.995643i	$0.470274\pi$
$152$	0	0
$153$	−2.23607	−0.180775
$154$	0	0
$155$	−1.09017	−0.0875646
$156$	0	0
$157$	−17.2361	−1.37559	−0.687794	−	0.725906i	$-0.741421\pi$
$157$	−17.2361	−1.37559	−0.687794	+	0.725906i	$0.741421\pi$
$158$	0	0
$159$	−7.09017	−0.562287
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−3.56231	−0.279021	−0.139511	−	0.990221i	$-0.544553\pi$
$163$	−3.56231	−0.279021	−0.139511	+	0.990221i	$0.544553\pi$
$164$	0	0
$165$	0.145898	0.0113581
$166$	0	0
$167$	14.8885	1.15211	0.576055	−	0.817411i	$-0.304591\pi$
$167$	14.8885	1.15211	0.576055	+	0.817411i	$0.304591\pi$
$168$	0	0
$169$	−12.8541	−0.988777
$170$	0	0
$171$	−3.47214	−0.265521
$172$	0	0
$173$	19.9443	1.51633	0.758167	−	0.652060i	$-0.226095\pi$
$173$	19.9443	1.51633	0.758167	+	0.652060i	$0.226095\pi$
$174$	0	0
$175$	−4.61803	−0.349091
$176$	0	0
$177$	3.38197	0.254204
$178$	0	0
$179$	2.79837	0.209160	0.104580	−	0.994516i	$-0.466650\pi$
$179$	2.79837	0.209160	0.104580	+	0.994516i	$0.466650\pi$
$180$	0	0
$181$	−4.23607	−0.314864	−0.157432	−	0.987530i	$-0.550322\pi$
$181$	−4.23607	−0.314864	−0.157432	+	0.987530i	$0.550322\pi$
$182$	0	0
$183$	5.32624	0.393727
$184$	0	0
$185$	3.38197	0.248647
$186$	0	0
$187$	0.527864	0.0386012
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−21.1246	−1.52852	−0.764262	−	0.644906i	$-0.776896\pi$
$191$	−21.1246	−1.52852	−0.764262	+	0.644906i	$0.776896\pi$
$192$	0	0
$193$	−4.29180	−0.308930	−0.154465	−	0.987998i	$-0.549365\pi$
$193$	−4.29180	−0.308930	−0.154465	+	0.987998i	$0.549365\pi$
$194$	0	0
$195$	−0.236068	−0.0169052
$196$	0	0
$197$	−6.38197	−0.454696	−0.227348	−	0.973814i	$-0.573006\pi$
$197$	−6.38197	−0.454696	−0.227348	+	0.973814i	$0.573006\pi$
$198$	0	0
$199$	5.67376	0.402202	0.201101	−	0.979570i	$-0.435548\pi$
$199$	5.67376	0.402202	0.201101	+	0.979570i	$0.435548\pi$
$200$	0	0
$201$	3.38197	0.238545
$202$	0	0
$203$	5.47214	0.384069
$204$	0	0
$205$	−4.61803	−0.322537
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	0.819660	0.0566971
$210$	0	0
$211$	2.23607	0.153937	0.0769686	−	0.997034i	$-0.475476\pi$
$211$	2.23607	0.153937	0.0769686	+	0.997034i	$0.475476\pi$
$212$	0	0
$213$	10.8541	0.743711
$214$	0	0
$215$	−7.76393	−0.529496
$216$	0	0
$217$	1.76393	0.119744
$218$	0	0
$219$	1.94427	0.131382
$220$	0	0
$221$	−0.854102	−0.0574531
$222$	0	0
$223$	−26.2148	−1.75547	−0.877736	−	0.479145i	$-0.840947\pi$
$223$	−26.2148	−1.75547	−0.877736	+	0.479145i	$0.840947\pi$
$224$	0	0
$225$	−4.61803	−0.307869
$226$	0	0
$227$	−1.09017	−0.0723571	−0.0361786	−	0.999345i	$-0.511519\pi$
$227$	−1.09017	−0.0723571	−0.0361786	+	0.999345i	$0.511519\pi$
$228$	0	0
$229$	14.6180	0.965987	0.482993	−	0.875624i	$-0.339549\pi$
$229$	14.6180	0.965987	0.482993	+	0.875624i	$0.339549\pi$
$230$	0	0
$231$	−0.236068	−0.0155321
$232$	0	0
$233$	−9.09017	−0.595517	−0.297758	−	0.954641i	$-0.596239\pi$
$233$	−9.09017	−0.595517	−0.297758	+	0.954641i	$0.596239\pi$
$234$	0	0
$235$	−0.472136	−0.0307988
$236$	0	0
$237$	8.23607	0.534990
$238$	0	0
$239$	22.0344	1.42529	0.712645	−	0.701525i	$-0.247497\pi$
$239$	22.0344	1.42529	0.712645	+	0.701525i	$0.247497\pi$
$240$	0	0
$241$	−24.7082	−1.59160	−0.795798	−	0.605563i	$-0.792948\pi$
$241$	−24.7082	−1.59160	−0.795798	+	0.605563i	$0.792948\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.618034	−0.0394847
$246$	0	0
$247$	−1.32624	−0.0843865
$248$	0	0
$249$	7.94427	0.503448
$250$	0	0
$251$	−22.1803	−1.40001	−0.700005	−	0.714138i	$-0.746819\pi$
$251$	−22.1803	−1.40001	−0.700005	+	0.714138i	$0.746819\pi$
$252$	0	0
$253$	0.236068	0.0148415
$254$	0	0
$255$	1.38197	0.0865421
$256$	0	0
$257$	25.7082	1.60363	0.801817	−	0.597570i	$-0.203867\pi$
$257$	25.7082	1.60363	0.801817	+	0.597570i	$0.203867\pi$
$258$	0	0
$259$	−5.47214	−0.340022
$260$	0	0
$261$	5.47214	0.338717
$262$	0	0
$263$	7.94427	0.489865	0.244932	−	0.969540i	$-0.421234\pi$
$263$	7.94427	0.489865	0.244932	+	0.969540i	$0.421234\pi$
$264$	0	0
$265$	4.38197	0.269182
$266$	0	0
$267$	−2.85410	−0.174668
$268$	0	0
$269$	11.9098	0.726155	0.363078	−	0.931759i	$-0.381726\pi$
$269$	11.9098	0.726155	0.363078	+	0.931759i	$0.381726\pi$
$270$	0	0
$271$	13.3607	0.811603	0.405802	−	0.913961i	$-0.366992\pi$
$271$	13.3607	0.811603	0.405802	+	0.913961i	$0.366992\pi$
$272$	0	0
$273$	0.381966	0.0231176
$274$	0	0
$275$	1.09017	0.0657397
$276$	0	0
$277$	6.38197	0.383455	0.191728	−	0.981448i	$-0.438591\pi$
$277$	6.38197	0.383455	0.191728	+	0.981448i	$0.438591\pi$
$278$	0	0
$279$	1.76393	0.105604
$280$	0	0
$281$	−8.65248	−0.516163	−0.258082	−	0.966123i	$-0.583090\pi$
$281$	−8.65248	−0.516163	−0.258082	+	0.966123i	$0.583090\pi$
$282$	0	0
$283$	19.0902	1.13479	0.567396	−	0.823445i	$-0.307951\pi$
$283$	19.0902	1.13479	0.567396	+	0.823445i	$0.307951\pi$
$284$	0	0
$285$	2.14590	0.127112
$286$	0	0
$287$	7.47214	0.441066
$288$	0	0
$289$	−12.0000	−0.705882
$290$	0	0
$291$	−0.236068	−0.0138385
$292$	0	0
$293$	7.05573	0.412200	0.206100	−	0.978531i	$-0.433923\pi$
$293$	7.05573	0.412200	0.206100	+	0.978531i	$0.433923\pi$
$294$	0	0
$295$	−2.09017	−0.121694
$296$	0	0
$297$	−0.236068	−0.0136981
$298$	0	0
$299$	−0.381966	−0.0220897
$300$	0	0
$301$	12.5623	0.724079
$302$	0	0
$303$	−1.38197	−0.0793919
$304$	0	0
$305$	−3.29180	−0.188488
$306$	0	0
$307$	−5.76393	−0.328965	−0.164482	−	0.986380i	$-0.552595\pi$
$307$	−5.76393	−0.328965	−0.164482	+	0.986380i	$0.552595\pi$
$308$	0	0
$309$	8.94427	0.508822
$310$	0	0
$311$	15.9787	0.906070	0.453035	−	0.891493i	$-0.350341\pi$
$311$	15.9787	0.906070	0.453035	+	0.891493i	$0.350341\pi$
$312$	0	0
$313$	6.47214	0.365827	0.182913	−	0.983129i	$-0.441447\pi$
$313$	6.47214	0.365827	0.182913	+	0.983129i	$0.441447\pi$
$314$	0	0
$315$	−0.618034	−0.0348223
$316$	0	0
$317$	34.0344	1.91156	0.955782	−	0.294075i	$-0.0950115\pi$
$317$	34.0344	1.91156	0.955782	+	0.294075i	$0.0950115\pi$
$318$	0	0
$319$	−1.29180	−0.0723267
$320$	0	0
$321$	10.3820	0.579465
$322$	0	0
$323$	7.76393	0.431997
$324$	0	0
$325$	−1.76393	−0.0978453
$326$	0	0
$327$	−6.61803	−0.365978
$328$	0	0
$329$	0.763932	0.0421169
$330$	0	0
$331$	−1.05573	−0.0580281	−0.0290140	−	0.999579i	$-0.509237\pi$
$331$	−1.05573	−0.0580281	−0.0290140	+	0.999579i	$0.509237\pi$
$332$	0	0
$333$	−5.47214	−0.299871
$334$	0	0
$335$	−2.09017	−0.114198
$336$	0	0
$337$	−14.5623	−0.793259	−0.396630	−	0.917979i	$-0.629820\pi$
$337$	−14.5623	−0.793259	−0.396630	+	0.917979i	$0.629820\pi$
$338$	0	0
$339$	7.56231	0.410728
$340$	0	0
$341$	−0.416408	−0.0225498
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.618034	0.0332738
$346$	0	0
$347$	−19.1803	−1.02965	−0.514827	−	0.857294i	$-0.672144\pi$
$347$	−19.1803	−1.02965	−0.514827	+	0.857294i	$0.672144\pi$
$348$	0	0
$349$	23.0902	1.23599	0.617994	−	0.786183i	$-0.287945\pi$
$349$	23.0902	1.23599	0.617994	+	0.786183i	$0.287945\pi$
$350$	0	0
$351$	0.381966	0.0203878
$352$	0	0
$353$	8.05573	0.428763	0.214382	−	0.976750i	$-0.431226\pi$
$353$	8.05573	0.428763	0.214382	+	0.976750i	$0.431226\pi$
$354$	0	0
$355$	−6.70820	−0.356034
$356$	0	0
$357$	−2.23607	−0.118345
$358$	0	0
$359$	−8.32624	−0.439442	−0.219721	−	0.975563i	$-0.570515\pi$
$359$	−8.32624	−0.439442	−0.219721	+	0.975563i	$0.570515\pi$
$360$	0	0
$361$	−6.94427	−0.365488
$362$	0	0
$363$	−10.9443	−0.574425
$364$	0	0
$365$	−1.20163	−0.0628960
$366$	0	0
$367$	11.7984	0.615870	0.307935	−	0.951407i	$-0.400362\pi$
$367$	11.7984	0.615870	0.307935	+	0.951407i	$0.400362\pi$
$368$	0	0
$369$	7.47214	0.388984
$370$	0	0
$371$	−7.09017	−0.368103
$372$	0	0
$373$	25.9443	1.34334	0.671672	−	0.740849i	$-0.265577\pi$
$373$	25.9443	1.34334	0.671672	+	0.740849i	$0.265577\pi$
$374$	0	0
$375$	5.94427	0.306961
$376$	0	0
$377$	2.09017	0.107649
$378$	0	0
$379$	−26.8328	−1.37831	−0.689155	−	0.724614i	$-0.742018\pi$
$379$	−26.8328	−1.37831	−0.689155	+	0.724614i	$0.742018\pi$
$380$	0	0
$381$	7.14590	0.366095
$382$	0	0
$383$	−0.527864	−0.0269726	−0.0134863	−	0.999909i	$-0.504293\pi$
$383$	−0.527864	−0.0269726	−0.0134863	+	0.999909i	$0.504293\pi$
$384$	0	0
$385$	0.145898	0.00743565
$386$	0	0
$387$	12.5623	0.638578
$388$	0	0
$389$	−0.416408	−0.0211127	−0.0105564	−	0.999944i	$-0.503360\pi$
$389$	−0.416408	−0.0211127	−0.0105564	+	0.999944i	$0.503360\pi$
$390$	0	0
$391$	2.23607	0.113083
$392$	0	0
$393$	−0.0557281	−0.00281111
$394$	0	0
$395$	−5.09017	−0.256114
$396$	0	0
$397$	28.1803	1.41433	0.707165	−	0.707048i	$-0.249974\pi$
$397$	28.1803	1.41433	0.707165	+	0.707048i	$0.249974\pi$
$398$	0	0
$399$	−3.47214	−0.173824
$400$	0	0
$401$	−11.3607	−0.567325	−0.283663	−	0.958924i	$-0.591550\pi$
$401$	−11.3607	−0.567325	−0.283663	+	0.958924i	$0.591550\pi$
$402$	0	0
$403$	0.673762	0.0335625
$404$	0	0
$405$	−0.618034	−0.0307104
$406$	0	0
$407$	1.29180	0.0640320
$408$	0	0
$409$	−4.23607	−0.209460	−0.104730	−	0.994501i	$-0.533398\pi$
$409$	−4.23607	−0.209460	−0.104730	+	0.994501i	$0.533398\pi$
$410$	0	0
$411$	−0.236068	−0.0116444
$412$	0	0
$413$	3.38197	0.166416
$414$	0	0
$415$	−4.90983	−0.241014
$416$	0	0
$417$	12.0344	0.589329
$418$	0	0
$419$	−40.2148	−1.96462	−0.982310	−	0.187260i	$-0.940039\pi$
$419$	−40.2148	−1.96462	−0.982310	+	0.187260i	$0.940039\pi$
$420$	0	0
$421$	17.5066	0.853218	0.426609	−	0.904436i	$-0.359708\pi$
$421$	17.5066	0.853218	0.426609	+	0.904436i	$0.359708\pi$
$422$	0	0
$423$	0.763932	0.0371436
$424$	0	0
$425$	10.3262	0.500896
$426$	0	0
$427$	5.32624	0.257755
$428$	0	0
$429$	−0.0901699	−0.00435345
$430$	0	0
$431$	22.5623	1.08679	0.543394	−	0.839478i	$-0.317139\pi$
$431$	22.5623	1.08679	0.543394	+	0.839478i	$0.317139\pi$
$432$	0	0
$433$	−23.7082	−1.13934	−0.569672	−	0.821872i	$-0.692930\pi$
$433$	−23.7082	−1.13934	−0.569672	+	0.821872i	$0.692930\pi$
$434$	0	0
$435$	−3.38197	−0.162153
$436$	0	0
$437$	3.47214	0.166095
$438$	0	0
$439$	−6.81966	−0.325485	−0.162742	−	0.986669i	$-0.552034\pi$
$439$	−6.81966	−0.325485	−0.162742	+	0.986669i	$0.552034\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	13.8885	0.659865	0.329932	−	0.944005i	$-0.392974\pi$
$443$	13.8885	0.659865	0.329932	+	0.944005i	$0.392974\pi$
$444$	0	0
$445$	1.76393	0.0836184
$446$	0	0
$447$	14.6525	0.693038
$448$	0	0
$449$	9.09017	0.428992	0.214496	−	0.976725i	$-0.431189\pi$
$449$	9.09017	0.428992	0.214496	+	0.976725i	$0.431189\pi$
$450$	0	0
$451$	−1.76393	−0.0830603
$452$	0	0
$453$	2.29180	0.107678
$454$	0	0
$455$	−0.236068	−0.0110670
$456$	0	0
$457$	−3.85410	−0.180287	−0.0901436	−	0.995929i	$-0.528733\pi$
$457$	−3.85410	−0.180287	−0.0901436	+	0.995929i	$0.528733\pi$
$458$	0	0
$459$	−2.23607	−0.104371
$460$	0	0
$461$	15.5623	0.724809	0.362404	−	0.932021i	$-0.381956\pi$
$461$	15.5623	0.724809	0.362404	+	0.932021i	$0.381956\pi$
$462$	0	0
$463$	36.8885	1.71436	0.857178	−	0.515020i	$-0.172216\pi$
$463$	36.8885	1.71436	0.857178	+	0.515020i	$0.172216\pi$
$464$	0	0
$465$	−1.09017	−0.0505554
$466$	0	0
$467$	−14.4164	−0.667112	−0.333556	−	0.942730i	$-0.608249\pi$
$467$	−14.4164	−0.667112	−0.333556	+	0.942730i	$0.608249\pi$
$468$	0	0
$469$	3.38197	0.156165
$470$	0	0
$471$	−17.2361	−0.794196
$472$	0	0
$473$	−2.96556	−0.136357
$474$	0	0
$475$	16.0344	0.735711
$476$	0	0
$477$	−7.09017	−0.324637
$478$	0	0
$479$	−7.18034	−0.328078	−0.164039	−	0.986454i	$-0.552452\pi$
$479$	−7.18034	−0.328078	−0.164039	+	0.986454i	$0.552452\pi$
$480$	0	0
$481$	−2.09017	−0.0953035
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	0.145898	0.00662489
$486$	0	0
$487$	−20.5279	−0.930206	−0.465103	−	0.885256i	$-0.653983\pi$
$487$	−20.5279	−0.930206	−0.465103	+	0.885256i	$0.653983\pi$
$488$	0	0
$489$	−3.56231	−0.161093
$490$	0	0
$491$	0.854102	0.0385451	0.0192725	−	0.999814i	$-0.493865\pi$
$491$	0.854102	0.0385451	0.0192725	+	0.999814i	$0.493865\pi$
$492$	0	0
$493$	−12.2361	−0.551085
$494$	0	0
$495$	0.145898	0.00655763
$496$	0	0
$497$	10.8541	0.486873
$498$	0	0
$499$	1.27051	0.0568758	0.0284379	−	0.999596i	$-0.490947\pi$
$499$	1.27051	0.0568758	0.0284379	+	0.999596i	$0.490947\pi$
$500$	0	0
$501$	14.8885	0.665171
$502$	0	0
$503$	4.43769	0.197867	0.0989335	−	0.995094i	$-0.468457\pi$
$503$	4.43769	0.197867	0.0989335	+	0.995094i	$0.468457\pi$
$504$	0	0
$505$	0.854102	0.0380070
$506$	0	0
$507$	−12.8541	−0.570871
$508$	0	0
$509$	2.29180	0.101582	0.0507910	−	0.998709i	$-0.483826\pi$
$509$	2.29180	0.101582	0.0507910	+	0.998709i	$0.483826\pi$
$510$	0	0
$511$	1.94427	0.0860095
$512$	0	0
$513$	−3.47214	−0.153299
$514$	0	0
$515$	−5.52786	−0.243587
$516$	0	0
$517$	−0.180340	−0.00793134
$518$	0	0
$519$	19.9443	0.875456
$520$	0	0
$521$	8.47214	0.371171	0.185586	−	0.982628i	$-0.440582\pi$
$521$	8.47214	0.371171	0.185586	+	0.982628i	$0.440582\pi$
$522$	0	0
$523$	17.4164	0.761566	0.380783	−	0.924664i	$-0.375654\pi$
$523$	17.4164	0.761566	0.380783	+	0.924664i	$0.375654\pi$
$524$	0	0
$525$	−4.61803	−0.201548
$526$	0	0
$527$	−3.94427	−0.171815
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	3.38197	0.146765
$532$	0	0
$533$	2.85410	0.123625
$534$	0	0
$535$	−6.41641	−0.277406
$536$	0	0
$537$	2.79837	0.120759
$538$	0	0
$539$	−0.236068	−0.0101682
$540$	0	0
$541$	−2.29180	−0.0985320	−0.0492660	−	0.998786i	$-0.515688\pi$
$541$	−2.29180	−0.0985320	−0.0492660	+	0.998786i	$0.515688\pi$
$542$	0	0
$543$	−4.23607	−0.181787
$544$	0	0
$545$	4.09017	0.175204
$546$	0	0
$547$	−3.09017	−0.132126	−0.0660631	−	0.997815i	$-0.521044\pi$
$547$	−3.09017	−0.132126	−0.0660631	+	0.997815i	$0.521044\pi$
$548$	0	0
$549$	5.32624	0.227318
$550$	0	0
$551$	−19.0000	−0.809427
$552$	0	0
$553$	8.23607	0.350233
$554$	0	0
$555$	3.38197	0.143556
$556$	0	0
$557$	4.65248	0.197132	0.0985659	−	0.995131i	$-0.468574\pi$
$557$	4.65248	0.197132	0.0985659	+	0.995131i	$0.468574\pi$
$558$	0	0
$559$	4.79837	0.202950
$560$	0	0
$561$	0.527864	0.0222864
$562$	0	0
$563$	37.5066	1.58071	0.790357	−	0.612647i	$-0.209895\pi$
$563$	37.5066	1.58071	0.790357	+	0.612647i	$0.209895\pi$
$564$	0	0
$565$	−4.67376	−0.196627
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	26.4721	1.10977	0.554885	−	0.831927i	$-0.312762\pi$
$569$	26.4721	1.10977	0.554885	+	0.831927i	$0.312762\pi$
$570$	0	0
$571$	−3.29180	−0.137757	−0.0688787	−	0.997625i	$-0.521942\pi$
$571$	−3.29180	−0.137757	−0.0688787	+	0.997625i	$0.521942\pi$
$572$	0	0
$573$	−21.1246	−0.882493
$574$	0	0
$575$	4.61803	0.192585
$576$	0	0
$577$	−30.9443	−1.28823	−0.644113	−	0.764930i	$-0.722774\pi$
$577$	−30.9443	−1.28823	−0.644113	+	0.764930i	$0.722774\pi$
$578$	0	0
$579$	−4.29180	−0.178361
$580$	0	0
$581$	7.94427	0.329584
$582$	0	0
$583$	1.67376	0.0693201
$584$	0	0
$585$	−0.236068	−0.00976021
$586$	0	0
$587$	17.0344	0.703087	0.351543	−	0.936172i	$-0.385657\pi$
$587$	17.0344	0.703087	0.351543	+	0.936172i	$0.385657\pi$
$588$	0	0
$589$	−6.12461	−0.252360
$590$	0	0
$591$	−6.38197	−0.262519
$592$	0	0
$593$	2.76393	0.113501	0.0567505	−	0.998388i	$-0.481926\pi$
$593$	2.76393	0.113501	0.0567505	+	0.998388i	$0.481926\pi$
$594$	0	0
$595$	1.38197	0.0566551
$596$	0	0
$597$	5.67376	0.232212
$598$	0	0
$599$	−37.3951	−1.52792	−0.763962	−	0.645262i	$-0.776748\pi$
$599$	−37.3951	−1.52792	−0.763962	+	0.645262i	$0.776748\pi$
$600$	0	0
$601$	34.6869	1.41491	0.707454	−	0.706759i	$-0.249843\pi$
$601$	34.6869	1.41491	0.707454	+	0.706759i	$0.249843\pi$
$602$	0	0
$603$	3.38197	0.137724
$604$	0	0
$605$	6.76393	0.274993
$606$	0	0
$607$	−40.6312	−1.64917	−0.824585	−	0.565739i	$-0.808591\pi$
$607$	−40.6312	−1.64917	−0.824585	+	0.565739i	$0.808591\pi$
$608$	0	0
$609$	5.47214	0.221742
$610$	0	0
$611$	0.291796	0.0118048
$612$	0	0
$613$	−17.7639	−0.717478	−0.358739	−	0.933438i	$-0.616793\pi$
$613$	−17.7639	−0.717478	−0.358739	+	0.933438i	$0.616793\pi$
$614$	0	0
$615$	−4.61803	−0.186217
$616$	0	0
$617$	−15.9098	−0.640506	−0.320253	−	0.947332i	$-0.603768\pi$
$617$	−15.9098	−0.640506	−0.320253	+	0.947332i	$0.603768\pi$
$618$	0	0
$619$	−7.03444	−0.282738	−0.141369	−	0.989957i	$-0.545150\pi$
$619$	−7.03444	−0.282738	−0.141369	+	0.989957i	$0.545150\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−2.85410	−0.114347
$624$	0	0
$625$	19.4164	0.776656
$626$	0	0
$627$	0.819660	0.0327341
$628$	0	0
$629$	12.2361	0.487884
$630$	0	0
$631$	9.47214	0.377080	0.188540	−	0.982066i	$-0.439625\pi$
$631$	9.47214	0.377080	0.188540	+	0.982066i	$0.439625\pi$
$632$	0	0
$633$	2.23607	0.0888757
$634$	0	0
$635$	−4.41641	−0.175260
$636$	0	0
$637$	0.381966	0.0151340
$638$	0	0
$639$	10.8541	0.429382
$640$	0	0
$641$	−34.8541	−1.37665	−0.688327	−	0.725400i	$-0.741655\pi$
$641$	−34.8541	−1.37665	−0.688327	+	0.725400i	$0.741655\pi$
$642$	0	0
$643$	−19.9787	−0.787884	−0.393942	−	0.919135i	$-0.628889\pi$
$643$	−19.9787	−0.787884	−0.393942	+	0.919135i	$0.628889\pi$
$644$	0	0
$645$	−7.76393	−0.305705
$646$	0	0
$647$	30.9098	1.21519	0.607595	−	0.794247i	$-0.292134\pi$
$647$	30.9098	1.21519	0.607595	+	0.794247i	$0.292134\pi$
$648$	0	0
$649$	−0.798374	−0.0313389
$650$	0	0
$651$	1.76393	0.0691339
$652$	0	0
$653$	−10.0344	−0.392678	−0.196339	−	0.980536i	$-0.562905\pi$
$653$	−10.0344	−0.392678	−0.196339	+	0.980536i	$0.562905\pi$
$654$	0	0
$655$	0.0344419	0.00134575
$656$	0	0
$657$	1.94427	0.0758533
$658$	0	0
$659$	41.0689	1.59982	0.799908	−	0.600122i	$-0.204881\pi$
$659$	41.0689	1.59982	0.799908	+	0.600122i	$0.204881\pi$
$660$	0	0
$661$	−8.81966	−0.343045	−0.171523	−	0.985180i	$-0.554869\pi$
$661$	−8.81966	−0.343045	−0.171523	+	0.985180i	$0.554869\pi$
$662$	0	0
$663$	−0.854102	−0.0331706
$664$	0	0
$665$	2.14590	0.0832144
$666$	0	0
$667$	−5.47214	−0.211882
$668$	0	0
$669$	−26.2148	−1.01352
$670$	0	0
$671$	−1.25735	−0.0485396
$672$	0	0
$673$	−34.4164	−1.32666	−0.663328	−	0.748329i	$-0.730856\pi$
$673$	−34.4164	−1.32666	−0.663328	+	0.748329i	$0.730856\pi$
$674$	0	0
$675$	−4.61803	−0.177748
$676$	0	0
$677$	−21.9787	−0.844711	−0.422355	−	0.906430i	$-0.638797\pi$
$677$	−21.9787	−0.844711	−0.422355	+	0.906430i	$0.638797\pi$
$678$	0	0
$679$	−0.236068	−0.00905946
$680$	0	0
$681$	−1.09017	−0.0417754
$682$	0	0
$683$	15.7639	0.603190	0.301595	−	0.953436i	$-0.402481\pi$
$683$	15.7639	0.603190	0.301595	+	0.953436i	$0.402481\pi$
$684$	0	0
$685$	0.145898	0.00557448
$686$	0	0
$687$	14.6180	0.557713
$688$	0	0
$689$	−2.70820	−0.103174
$690$	0	0
$691$	−2.56231	−0.0974747	−0.0487374	−	0.998812i	$-0.515520\pi$
$691$	−2.56231	−0.0974747	−0.0487374	+	0.998812i	$0.515520\pi$
$692$	0	0
$693$	−0.236068	−0.00896748
$694$	0	0
$695$	−7.43769	−0.282128
$696$	0	0
$697$	−16.7082	−0.632868
$698$	0	0
$699$	−9.09017	−0.343822
$700$	0	0
$701$	−46.2148	−1.74551	−0.872754	−	0.488160i	$-0.837668\pi$
$701$	−46.2148	−1.74551	−0.872754	+	0.488160i	$0.837668\pi$
$702$	0	0
$703$	19.0000	0.716599
$704$	0	0
$705$	−0.472136	−0.0177817
$706$	0	0
$707$	−1.38197	−0.0519742
$708$	0	0
$709$	8.27051	0.310606	0.155303	−	0.987867i	$-0.450365\pi$
$709$	8.27051	0.310606	0.155303	+	0.987867i	$0.450365\pi$
$710$	0	0
$711$	8.23607	0.308877
$712$	0	0
$713$	−1.76393	−0.0660598
$714$	0	0
$715$	0.0557281	0.00208411
$716$	0	0
$717$	22.0344	0.822891
$718$	0	0
$719$	1.76393	0.0657836	0.0328918	−	0.999459i	$-0.489528\pi$
$719$	1.76393	0.0657836	0.0328918	+	0.999459i	$0.489528\pi$
$720$	0	0
$721$	8.94427	0.333102
$722$	0	0
$723$	−24.7082	−0.918908
$724$	0	0
$725$	−25.2705	−0.938523
$726$	0	0
$727$	−45.4721	−1.68647	−0.843234	−	0.537547i	$-0.819351\pi$
$727$	−45.4721	−1.68647	−0.843234	+	0.537547i	$0.819351\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−28.0902	−1.03895
$732$	0	0
$733$	−41.5967	−1.53641	−0.768205	−	0.640203i	$-0.778850\pi$
$733$	−41.5967	−1.53641	−0.768205	+	0.640203i	$0.778850\pi$
$734$	0	0
$735$	−0.618034	−0.0227965
$736$	0	0
$737$	−0.798374	−0.0294085
$738$	0	0
$739$	−24.5967	−0.904806	−0.452403	−	0.891814i	$-0.649433\pi$
$739$	−24.5967	−0.904806	−0.452403	+	0.891814i	$0.649433\pi$
$740$	0	0
$741$	−1.32624	−0.0487206
$742$	0	0
$743$	29.2016	1.07130	0.535652	−	0.844439i	$-0.320066\pi$
$743$	29.2016	1.07130	0.535652	+	0.844439i	$0.320066\pi$
$744$	0	0
$745$	−9.05573	−0.331776
$746$	0	0
$747$	7.94427	0.290666
$748$	0	0
$749$	10.3820	0.379349
$750$	0	0
$751$	38.0344	1.38790	0.693948	−	0.720025i	$-0.255870\pi$
$751$	38.0344	1.38790	0.693948	+	0.720025i	$0.255870\pi$
$752$	0	0
$753$	−22.1803	−0.808297
$754$	0	0
$755$	−1.41641	−0.0515484
$756$	0	0
$757$	35.0000	1.27210	0.636048	−	0.771649i	$-0.280568\pi$
$757$	35.0000	1.27210	0.636048	+	0.771649i	$0.280568\pi$
$758$	0	0
$759$	0.236068	0.00856872
$760$	0	0
$761$	10.8328	0.392689	0.196345	−	0.980535i	$-0.437093\pi$
$761$	10.8328	0.392689	0.196345	+	0.980535i	$0.437093\pi$
$762$	0	0
$763$	−6.61803	−0.239589
$764$	0	0
$765$	1.38197	0.0499651
$766$	0	0
$767$	1.29180	0.0466441
$768$	0	0
$769$	30.1803	1.08833	0.544165	−	0.838978i	$-0.316846\pi$
$769$	30.1803	1.08833	0.544165	+	0.838978i	$0.316846\pi$
$770$	0	0
$771$	25.7082	0.925858
$772$	0	0
$773$	6.52786	0.234791	0.117395	−	0.993085i	$-0.462545\pi$
$773$	6.52786	0.234791	0.117395	+	0.993085i	$0.462545\pi$
$774$	0	0
$775$	−8.14590	−0.292609
$776$	0	0
$777$	−5.47214	−0.196312
$778$	0	0
$779$	−25.9443	−0.929550
$780$	0	0
$781$	−2.56231	−0.0916865
$782$	0	0
$783$	5.47214	0.195558
$784$	0	0
$785$	10.6525	0.380203
$786$	0	0
$787$	17.4508	0.622056	0.311028	−	0.950401i	$-0.399327\pi$
$787$	17.4508	0.622056	0.311028	+	0.950401i	$0.399327\pi$
$788$	0	0
$789$	7.94427	0.282824
$790$	0	0
$791$	7.56231	0.268885
$792$	0	0
$793$	2.03444	0.0722451
$794$	0	0
$795$	4.38197	0.155412
$796$	0	0
$797$	28.5410	1.01097	0.505487	−	0.862834i	$-0.331313\pi$
$797$	28.5410	1.01097	0.505487	+	0.862834i	$0.331313\pi$
$798$	0	0
$799$	−1.70820	−0.0604319
$800$	0	0
$801$	−2.85410	−0.100845
$802$	0	0
$803$	−0.458980	−0.0161971
$804$	0	0
$805$	0.618034	0.0217828
$806$	0	0
$807$	11.9098	0.419246
$808$	0	0
$809$	16.3820	0.575959	0.287980	−	0.957637i	$-0.407016\pi$
$809$	16.3820	0.575959	0.287980	+	0.957637i	$0.407016\pi$
$810$	0	0
$811$	−5.58359	−0.196066	−0.0980332	−	0.995183i	$-0.531255\pi$
$811$	−5.58359	−0.196066	−0.0980332	+	0.995183i	$0.531255\pi$
$812$	0	0
$813$	13.3607	0.468579
$814$	0	0
$815$	2.20163	0.0771196
$816$	0	0
$817$	−43.6180	−1.52600
$818$	0	0
$819$	0.381966	0.0133470
$820$	0	0
$821$	41.1246	1.43526	0.717629	−	0.696425i	$-0.245227\pi$
$821$	41.1246	1.43526	0.717629	+	0.696425i	$0.245227\pi$
$822$	0	0
$823$	−49.2705	−1.71746	−0.858731	−	0.512427i	$-0.828747\pi$
$823$	−49.2705	−1.71746	−0.858731	+	0.512427i	$0.828747\pi$
$824$	0	0
$825$	1.09017	0.0379548
$826$	0	0
$827$	23.5623	0.819342	0.409671	−	0.912233i	$-0.365644\pi$
$827$	23.5623	0.819342	0.409671	+	0.912233i	$0.365644\pi$
$828$	0	0
$829$	−20.1246	−0.698957	−0.349478	−	0.936944i	$-0.613641\pi$
$829$	−20.1246	−0.698957	−0.349478	+	0.936944i	$0.613641\pi$
$830$	0	0
$831$	6.38197	0.221388
$832$	0	0
$833$	−2.23607	−0.0774752
$834$	0	0
$835$	−9.20163	−0.318435
$836$	0	0
$837$	1.76393	0.0609704
$838$	0	0
$839$	34.9230	1.20568	0.602838	−	0.797864i	$-0.294037\pi$
$839$	34.9230	1.20568	0.602838	+	0.797864i	$0.294037\pi$
$840$	0	0
$841$	0.944272	0.0325611
$842$	0	0
$843$	−8.65248	−0.298007
$844$	0	0
$845$	7.94427	0.273291
$846$	0	0
$847$	−10.9443	−0.376050
$848$	0	0
$849$	19.0902	0.655173
$850$	0	0
$851$	5.47214	0.187582
$852$	0	0
$853$	−28.6525	−0.981042	−0.490521	−	0.871429i	$-0.663194\pi$
$853$	−28.6525	−0.981042	−0.490521	+	0.871429i	$0.663194\pi$
$854$	0	0
$855$	2.14590	0.0733882
$856$	0	0
$857$	0.875388	0.0299027	0.0149513	−	0.999888i	$-0.495241\pi$
$857$	0.875388	0.0299027	0.0149513	+	0.999888i	$0.495241\pi$
$858$	0	0
$859$	14.9443	0.509892	0.254946	−	0.966955i	$-0.417942\pi$
$859$	14.9443	0.509892	0.254946	+	0.966955i	$0.417942\pi$
$860$	0	0
$861$	7.47214	0.254650
$862$	0	0
$863$	−3.12461	−0.106363	−0.0531815	−	0.998585i	$-0.516936\pi$
$863$	−3.12461	−0.106363	−0.0531815	+	0.998585i	$0.516936\pi$
$864$	0	0
$865$	−12.3262	−0.419105
$866$	0	0
$867$	−12.0000	−0.407541
$868$	0	0
$869$	−1.94427	−0.0659549
$870$	0	0
$871$	1.29180	0.0437708
$872$	0	0
$873$	−0.236068	−0.00798969
$874$	0	0
$875$	5.94427	0.200953
$876$	0	0
$877$	−47.1935	−1.59361	−0.796806	−	0.604236i	$-0.793478\pi$
$877$	−47.1935	−1.59361	−0.796806	+	0.604236i	$0.793478\pi$
$878$	0	0
$879$	7.05573	0.237984
$880$	0	0
$881$	−35.7082	−1.20304	−0.601520	−	0.798858i	$-0.705438\pi$
$881$	−35.7082	−1.20304	−0.601520	+	0.798858i	$0.705438\pi$
$882$	0	0
$883$	32.7426	1.10188	0.550939	−	0.834546i	$-0.314270\pi$
$883$	32.7426	1.10188	0.550939	+	0.834546i	$0.314270\pi$
$884$	0	0
$885$	−2.09017	−0.0702603
$886$	0	0
$887$	20.5623	0.690415	0.345207	−	0.938526i	$-0.387809\pi$
$887$	20.5623	0.690415	0.345207	+	0.938526i	$0.387809\pi$
$888$	0	0
$889$	7.14590	0.239666
$890$	0	0
$891$	−0.236068	−0.00790857
$892$	0	0
$893$	−2.65248	−0.0887617
$894$	0	0
$895$	−1.72949	−0.0578105
$896$	0	0
$897$	−0.381966	−0.0127535
$898$	0	0
$899$	9.65248	0.321928
$900$	0	0
$901$	15.8541	0.528177
$902$	0	0
$903$	12.5623	0.418047
$904$	0	0
$905$	2.61803	0.0870264
$906$	0	0
$907$	−17.9656	−0.596537	−0.298268	−	0.954482i	$-0.596409\pi$
$907$	−17.9656	−0.596537	−0.298268	+	0.954482i	$0.596409\pi$
$908$	0	0
$909$	−1.38197	−0.0458369
$910$	0	0
$911$	−32.5410	−1.07813	−0.539066	−	0.842264i	$-0.681223\pi$
$911$	−32.5410	−1.07813	−0.539066	+	0.842264i	$0.681223\pi$
$912$	0	0
$913$	−1.87539	−0.0620663
$914$	0	0
$915$	−3.29180	−0.108823
$916$	0	0
$917$	−0.0557281	−0.00184030
$918$	0	0
$919$	6.41641	0.211658	0.105829	−	0.994384i	$-0.466250\pi$
$919$	6.41641	0.211658	0.105829	+	0.994384i	$0.466250\pi$
$920$	0	0
$921$	−5.76393	−0.189928
$922$	0	0
$923$	4.14590	0.136464
$924$	0	0
$925$	25.2705	0.830889
$926$	0	0
$927$	8.94427	0.293768
$928$	0	0
$929$	−30.5066	−1.00089	−0.500444	−	0.865769i	$-0.666830\pi$
$929$	−30.5066	−1.00089	−0.500444	+	0.865769i	$0.666830\pi$
$930$	0	0
$931$	−3.47214	−0.113795
$932$	0	0
$933$	15.9787	0.523120
$934$	0	0
$935$	−0.326238	−0.0106691
$936$	0	0
$937$	31.4721	1.02815	0.514075	−	0.857745i	$-0.328135\pi$
$937$	31.4721	1.02815	0.514075	+	0.857745i	$0.328135\pi$
$938$	0	0
$939$	6.47214	0.211210
$940$	0	0
$941$	28.0000	0.912774	0.456387	−	0.889781i	$-0.349143\pi$
$941$	28.0000	0.912774	0.456387	+	0.889781i	$0.349143\pi$
$942$	0	0
$943$	−7.47214	−0.243326
$944$	0	0
$945$	−0.618034	−0.0201046
$946$	0	0
$947$	−24.1115	−0.783517	−0.391759	−	0.920068i	$-0.628133\pi$
$947$	−24.1115	−0.783517	−0.391759	+	0.920068i	$0.628133\pi$
$948$	0	0
$949$	0.742646	0.0241073
$950$	0	0
$951$	34.0344	1.10364
$952$	0	0
$953$	−13.4377	−0.435290	−0.217645	−	0.976028i	$-0.569837\pi$
$953$	−13.4377	−0.435290	−0.217645	+	0.976028i	$0.569837\pi$
$954$	0	0
$955$	13.0557	0.422473
$956$	0	0
$957$	−1.29180	−0.0417578
$958$	0	0
$959$	−0.236068	−0.00762303
$960$	0	0
$961$	−27.8885	−0.899630
$962$	0	0
$963$	10.3820	0.334554
$964$	0	0
$965$	2.65248	0.0853862
$966$	0	0
$967$	−16.4721	−0.529708	−0.264854	−	0.964288i	$-0.585324\pi$
$967$	−16.4721	−0.529708	−0.264854	+	0.964288i	$0.585324\pi$
$968$	0	0
$969$	7.76393	0.249413
$970$	0	0
$971$	29.3262	0.941124	0.470562	−	0.882367i	$-0.344051\pi$
$971$	29.3262	0.941124	0.470562	+	0.882367i	$0.344051\pi$
$972$	0	0
$973$	12.0344	0.385806
$974$	0	0
$975$	−1.76393	−0.0564910
$976$	0	0
$977$	−45.9098	−1.46879	−0.734393	−	0.678725i	$-0.762533\pi$
$977$	−45.9098	−1.46879	−0.734393	+	0.678725i	$0.762533\pi$
$978$	0	0
$979$	0.673762	0.0215335
$980$	0	0
$981$	−6.61803	−0.211298
$982$	0	0
$983$	−22.7771	−0.726476	−0.363238	−	0.931696i	$-0.618329\pi$
$983$	−22.7771	−0.726476	−0.363238	+	0.931696i	$0.618329\pi$
$984$	0	0
$985$	3.94427	0.125675
$986$	0	0
$987$	0.763932	0.0243162
$988$	0	0
$989$	−12.5623	−0.399458
$990$	0	0
$991$	10.1591	0.322713	0.161356	−	0.986896i	$-0.448413\pi$
$991$	10.1591	0.322713	0.161356	+	0.986896i	$0.448413\pi$
$992$	0	0
$993$	−1.05573	−0.0335025
$994$	0	0
$995$	−3.50658	−0.111166
$996$	0	0
$997$	−12.1246	−0.383990	−0.191995	−	0.981396i	$-0.561496\pi$
$997$	−12.1246	−0.383990	−0.191995	+	0.981396i	$0.561496\pi$
$998$	0	0
$999$	−5.47214	−0.173131

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bo.1.1		2
4.3	odd	2		1932.2.a.d.1.1	✓	2
12.11	even	2		5796.2.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.a.d.1.1	✓	2	4.3	odd	2
5796.2.a.i.1.2		2	12.11	even	2
7728.2.a.bo.1.1		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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.618034 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.618034 q^{5} +1.00000 q^{7} +1.00000 q^{9} -0.236068 q^{11} +0.381966 q^{13} -0.618034 q^{15} -2.23607 q^{17} -3.47214 q^{19} +1.00000 q^{21} -1.00000 q^{23} -4.61803 q^{25} +1.00000 q^{27} +5.47214 q^{29} +1.76393 q^{31} -0.236068 q^{33} -0.618034 q^{35} -5.47214 q^{37} +0.381966 q^{39} +7.47214 q^{41} +12.5623 q^{43} -0.618034 q^{45} +0.763932 q^{47} +1.00000 q^{49} -2.23607 q^{51} -7.09017 q^{53} +0.145898 q^{55} -3.47214 q^{57} +3.38197 q^{59} +5.32624 q^{61} +1.00000 q^{63} -0.236068 q^{65} +3.38197 q^{67} -1.00000 q^{69} +10.8541 q^{71} +1.94427 q^{73} -4.61803 q^{75} -0.236068 q^{77} +8.23607 q^{79} +1.00000 q^{81} +7.94427 q^{83} +1.38197 q^{85} +5.47214 q^{87} -2.85410 q^{89} +0.381966 q^{91} +1.76393 q^{93} +2.14590 q^{95} -0.236068 q^{97} -0.236068 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{11} + 3 q^{13} + q^{15} + 2 q^{19} + 2 q^{21} - 2 q^{23} - 7 q^{25} + 2 q^{27} + 2 q^{29} + 8 q^{31} + 4 q^{33} + q^{35} - 2 q^{37} + 3 q^{39} + 6 q^{41} + 5 q^{43} + q^{45} + 6 q^{47} + 2 q^{49} - 3 q^{53} + 7 q^{55} + 2 q^{57} + 9 q^{59} - 5 q^{61} + 2 q^{63} + 4 q^{65} + 9 q^{67} - 2 q^{69} + 15 q^{71} - 14 q^{73} - 7 q^{75} + 4 q^{77} + 12 q^{79} + 2 q^{81} - 2 q^{83} + 5 q^{85} + 2 q^{87} + q^{89} + 3 q^{91} + 8 q^{93} + 11 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9 + 4 * q^11 + 3 * q^13 + q^15 + 2 * q^19 + 2 * q^21 - 2 * q^23 - 7 * q^25 + 2 * q^27 + 2 * q^29 + 8 * q^31 + 4 * q^33 + q^35 - 2 * q^37 + 3 * q^39 + 6 * q^41 + 5 * q^43 + q^45 + 6 * q^47 + 2 * q^49 - 3 * q^53 + 7 * q^55 + 2 * q^57 + 9 * q^59 - 5 * q^61 + 2 * q^63 + 4 * q^65 + 9 * q^67 - 2 * q^69 + 15 * q^71 - 14 * q^73 - 7 * q^75 + 4 * q^77 + 12 * q^79 + 2 * q^81 - 2 * q^83 + 5 * q^85 + 2 * q^87 + q^89 + 3 * q^91 + 8 * q^93 + 11 * q^95 + 4 * q^97 + 4 * q^99

Introduction

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