LMFDB - Embedded newform 7728.2.a.bx.1.2

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3864)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.167449$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -0.167449 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.167449 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.80451 q^{11} -3.63706 q^{13} -0.167449 q^{15} +4.13941 q^{17} -5.80451 q^{19} -1.00000 q^{21} +1.00000 q^{23} -4.97196 q^{25} +1.00000 q^{27} +5.00000 q^{29} -0.195488 q^{31} +1.80451 q^{33} +0.167449 q^{35} -3.33490 q^{37} -3.63706 q^{39} -4.94392 q^{41} -1.30216 q^{43} -0.167449 q^{45} +3.13941 q^{47} +1.00000 q^{49} +4.13941 q^{51} +3.30686 q^{53} -0.302164 q^{55} -5.80451 q^{57} +4.91588 q^{59} +8.77647 q^{61} -1.00000 q^{63} +0.609023 q^{65} -13.2508 q^{67} +1.00000 q^{69} -8.77647 q^{71} -12.4182 q^{73} -4.97196 q^{75} -1.80451 q^{77} -1.52569 q^{79} +1.00000 q^{81} -14.7531 q^{83} -0.693141 q^{85} +5.00000 q^{87} +8.77647 q^{89} +3.63706 q^{91} -0.195488 q^{93} +0.971961 q^{95} -14.2741 q^{97} +1.80451 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9} - 6 q^{11} - 6 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{25} + 3 q^{27} + 15 q^{29} - 12 q^{31} - 6 q^{33} - 9 q^{37} + 9 q^{41} + 6 q^{43} - 3 q^{47} + 3 q^{49} - 3 q^{53} + 9 q^{55} - 6 q^{57} - 21 q^{59} + 3 q^{61} - 3 q^{63} - 21 q^{65} - 3 q^{67} + 3 q^{69} - 3 q^{71} - 3 q^{75} + 6 q^{77} - 18 q^{79} + 3 q^{81} - 6 q^{83} - 15 q^{85} + 15 q^{87} + 3 q^{89} - 12 q^{93} - 9 q^{95} - 21 q^{97} - 6 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9 - 6 * q^11 - 6 * q^19 - 3 * q^21 + 3 * q^23 - 3 * q^25 + 3 * q^27 + 15 * q^29 - 12 * q^31 - 6 * q^33 - 9 * q^37 + 9 * q^41 + 6 * q^43 - 3 * q^47 + 3 * q^49 - 3 * q^53 + 9 * q^55 - 6 * q^57 - 21 * q^59 + 3 * q^61 - 3 * q^63 - 21 * q^65 - 3 * q^67 + 3 * q^69 - 3 * q^71 - 3 * q^75 + 6 * q^77 - 18 * q^79 + 3 * q^81 - 6 * q^83 - 15 * q^85 + 15 * q^87 + 3 * q^89 - 12 * q^93 - 9 * q^95 - 21 * q^97 - 6 * 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
$5$	−0.167449	−0.0748856	−0.0374428	−	0.999299i	$-0.511921\pi$
$5$	−0.167449	−0.0748856	−0.0374428	+	0.999299i	$0.511921\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.80451	0.544081	0.272040	−	0.962286i	$-0.412302\pi$
$11$	1.80451	0.544081	0.272040	+	0.962286i	$0.412302\pi$
$12$	0	0
$13$	−3.63706	−1.00874	−0.504370	−	0.863488i	$-0.668275\pi$
$13$	−3.63706	−1.00874	−0.504370	+	0.863488i	$0.668275\pi$
$14$	0	0
$15$	−0.167449	−0.0432352
$16$	0	0
$17$	4.13941	1.00395	0.501977	−	0.864881i	$-0.332606\pi$
$17$	4.13941	1.00395	0.501977	+	0.864881i	$0.332606\pi$
$18$	0	0
$19$	−5.80451	−1.33165	−0.665823	−	0.746110i	$-0.731919\pi$
$19$	−5.80451	−1.33165	−0.665823	+	0.746110i	$0.731919\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.97196	−0.994392
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	−0.195488	−0.0351108	−0.0175554	−	0.999846i	$-0.505588\pi$
$31$	−0.195488	−0.0351108	−0.0175554	+	0.999846i	$0.505588\pi$
$32$	0	0
$33$	1.80451	0.314125
$34$	0	0
$35$	0.167449	0.0283041
$36$	0	0
$37$	−3.33490	−0.548254	−0.274127	−	0.961694i	$-0.588389\pi$
$37$	−3.33490	−0.548254	−0.274127	+	0.961694i	$0.588389\pi$
$38$	0	0
$39$	−3.63706	−0.582396
$40$	0	0
$41$	−4.94392	−0.772111	−0.386055	−	0.922476i	$-0.626163\pi$
$41$	−4.94392	−0.772111	−0.386055	+	0.922476i	$0.626163\pi$
$42$	0	0
$43$	−1.30216	−0.198578	−0.0992891	−	0.995059i	$-0.531657\pi$
$43$	−1.30216	−0.198578	−0.0992891	+	0.995059i	$0.531657\pi$
$44$	0	0
$45$	−0.167449	−0.0249619
$46$	0	0
$47$	3.13941	0.457930	0.228965	−	0.973435i	$-0.426466\pi$
$47$	3.13941	0.457930	0.228965	+	0.973435i	$0.426466\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.13941	0.579633
$52$	0	0
$53$	3.30686	0.454232	0.227116	−	0.973868i	$-0.427070\pi$
$53$	3.30686	0.454232	0.227116	+	0.973868i	$0.427070\pi$
$54$	0	0
$55$	−0.302164	−0.0407438
$56$	0	0
$57$	−5.80451	−0.768826
$58$	0	0
$59$	4.91588	0.639993	0.319997	−	0.947419i	$-0.396318\pi$
$59$	4.91588	0.639993	0.319997	+	0.947419i	$0.396318\pi$
$60$	0	0
$61$	8.77647	1.12371	0.561856	−	0.827235i	$-0.310087\pi$
$61$	8.77647	1.12371	0.561856	+	0.827235i	$0.310087\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0.609023	0.0755400
$66$	0	0
$67$	−13.2508	−1.61884	−0.809420	−	0.587230i	$-0.800218\pi$
$67$	−13.2508	−1.61884	−0.809420	+	0.587230i	$0.800218\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−8.77647	−1.04158	−0.520788	−	0.853686i	$-0.674362\pi$
$71$	−8.77647	−1.04158	−0.520788	+	0.853686i	$0.674362\pi$
$72$	0	0
$73$	−12.4182	−1.45344	−0.726722	−	0.686932i	$-0.758957\pi$
$73$	−12.4182	−1.45344	−0.726722	+	0.686932i	$0.758957\pi$
$74$	0	0
$75$	−4.97196	−0.574113
$76$	0	0
$77$	−1.80451	−0.205643
$78$	0	0
$79$	−1.52569	−0.171654	−0.0858269	−	0.996310i	$-0.527353\pi$
$79$	−1.52569	−0.171654	−0.0858269	+	0.996310i	$0.527353\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.7531	−1.61937	−0.809683	−	0.586867i	$-0.800361\pi$
$83$	−14.7531	−1.61937	−0.809683	+	0.586867i	$0.800361\pi$
$84$	0	0
$85$	−0.693141	−0.0751817
$86$	0	0
$87$	5.00000	0.536056
$88$	0	0
$89$	8.77647	0.930304	0.465152	−	0.885231i	$-0.346000\pi$
$89$	8.77647	0.930304	0.465152	+	0.885231i	$0.346000\pi$
$90$	0	0
$91$	3.63706	0.381268
$92$	0	0
$93$	−0.195488	−0.0202712
$94$	0	0
$95$	0.971961	0.0997211
$96$	0	0
$97$	−14.2741	−1.44932	−0.724659	−	0.689108i	$-0.758003\pi$
$97$	−14.2741	−1.44932	−0.724659	+	0.689108i	$0.758003\pi$
$98$	0	0
$99$	1.80451	0.181360
$100$	0	0
$101$	8.63706	0.859420	0.429710	−	0.902967i	$-0.358616\pi$
$101$	8.63706	0.859420	0.429710	+	0.902967i	$0.358616\pi$
$102$	0	0
$103$	−8.13471	−0.801537	−0.400769	−	0.916179i	$-0.631257\pi$
$103$	−8.13471	−0.801537	−0.400769	+	0.916179i	$0.631257\pi$
$104$	0	0
$105$	0.167449	0.0163414
$106$	0	0
$107$	−7.16745	−0.692903	−0.346452	−	0.938068i	$-0.612614\pi$
$107$	−7.16745	−0.692903	−0.346452	+	0.938068i	$0.612614\pi$
$108$	0	0
$109$	2.44627	0.234310	0.117155	−	0.993114i	$-0.462623\pi$
$109$	2.44627	0.234310	0.117155	+	0.993114i	$0.462623\pi$
$110$	0	0
$111$	−3.33490	−0.316535
$112$	0	0
$113$	5.10668	0.480396	0.240198	−	0.970724i	$-0.422788\pi$
$113$	5.10668	0.480396	0.240198	+	0.970724i	$0.422788\pi$
$114$	0	0
$115$	−0.167449	−0.0156147
$116$	0	0
$117$	−3.63706	−0.336247
$118$	0	0
$119$	−4.13941	−0.379459
$120$	0	0
$121$	−7.74374	−0.703976
$122$	0	0
$123$	−4.94392	−0.445778
$124$	0	0
$125$	1.66980	0.149351
$126$	0	0
$127$	8.97666	0.796549	0.398275	−	0.917266i	$-0.369609\pi$
$127$	8.97666	0.796549	0.398275	+	0.917266i	$0.369609\pi$
$128$	0	0
$129$	−1.30216	−0.114649
$130$	0	0
$131$	−9.86059	−0.861524	−0.430762	−	0.902466i	$-0.641755\pi$
$131$	−9.86059	−0.861524	−0.430762	+	0.902466i	$0.641755\pi$
$132$	0	0
$133$	5.80451	0.503315
$134$	0	0
$135$	−0.167449	−0.0144117
$136$	0	0
$137$	−13.0000	−1.11066	−0.555332	−	0.831628i	$-0.687409\pi$
$137$	−13.0000	−1.11066	−0.555332	+	0.831628i	$0.687409\pi$
$138$	0	0
$139$	14.3902	1.22056	0.610280	−	0.792186i	$-0.291057\pi$
$139$	14.3902	1.22056	0.610280	+	0.792186i	$0.291057\pi$
$140$	0	0
$141$	3.13941	0.264386
$142$	0	0
$143$	−6.56312	−0.548836
$144$	0	0
$145$	−0.837246	−0.0695295
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−2.33490	−0.191282	−0.0956412	−	0.995416i	$-0.530490\pi$
$149$	−2.33490	−0.191282	−0.0956412	+	0.995416i	$0.530490\pi$
$150$	0	0
$151$	−11.8092	−0.961020	−0.480510	−	0.876989i	$-0.659548\pi$
$151$	−11.8092	−0.961020	−0.480510	+	0.876989i	$0.659548\pi$
$152$	0	0
$153$	4.13941	0.334651
$154$	0	0
$155$	0.0327344	0.00262929
$156$	0	0
$157$	2.72588	0.217549	0.108774	−	0.994066i	$-0.465307\pi$
$157$	2.72588	0.217549	0.108774	+	0.994066i	$0.465307\pi$
$158$	0	0
$159$	3.30686	0.262251
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−14.6371	−1.14646	−0.573232	−	0.819393i	$-0.694310\pi$
$163$	−14.6371	−1.14646	−0.573232	+	0.819393i	$0.694310\pi$
$164$	0	0
$165$	−0.302164	−0.0235234
$166$	0	0
$167$	−1.20018	−0.0928730	−0.0464365	−	0.998921i	$-0.514787\pi$
$167$	−1.20018	−0.0928730	−0.0464365	+	0.998921i	$0.514787\pi$
$168$	0	0
$169$	0.228223	0.0175556
$170$	0	0
$171$	−5.80451	−0.443882
$172$	0	0
$173$	19.0880	1.45124	0.725618	−	0.688098i	$-0.241554\pi$
$173$	19.0880	1.45124	0.725618	+	0.688098i	$0.241554\pi$
$174$	0	0
$175$	4.97196	0.375845
$176$	0	0
$177$	4.91588	0.369500
$178$	0	0
$179$	−7.49765	−0.560401	−0.280200	−	0.959942i	$-0.590401\pi$
$179$	−7.49765	−0.560401	−0.280200	+	0.959942i	$0.590401\pi$
$180$	0	0
$181$	16.4088	1.21966	0.609830	−	0.792532i	$-0.291238\pi$
$181$	16.4088	1.21966	0.609830	+	0.792532i	$0.291238\pi$
$182$	0	0
$183$	8.77647	0.648776
$184$	0	0
$185$	0.558426	0.0410563
$186$	0	0
$187$	7.46961	0.546232
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	1.55294	0.112367	0.0561836	−	0.998420i	$-0.482107\pi$
$191$	1.55294	0.112367	0.0561836	+	0.998420i	$0.482107\pi$
$192$	0	0
$193$	−13.1394	−0.945795	−0.472898	−	0.881117i	$-0.656792\pi$
$193$	−13.1394	−0.945795	−0.472898	+	0.881117i	$0.656792\pi$
$194$	0	0
$195$	0.609023	0.0436131
$196$	0	0
$197$	−13.8598	−0.987470	−0.493735	−	0.869612i	$-0.664369\pi$
$197$	−13.8598	−0.987470	−0.493735	+	0.869612i	$0.664369\pi$
$198$	0	0
$199$	−19.0600	−1.35113	−0.675563	−	0.737302i	$-0.736100\pi$
$199$	−19.0600	−1.35113	−0.675563	+	0.737302i	$0.736100\pi$
$200$	0	0
$201$	−13.2508	−0.934638
$202$	0	0
$203$	−5.00000	−0.350931
$204$	0	0
$205$	0.827856	0.0578199
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−10.4743	−0.724523
$210$	0	0
$211$	7.46961	0.514229	0.257115	−	0.966381i	$-0.417228\pi$
$211$	7.46961	0.514229	0.257115	+	0.966381i	$0.417228\pi$
$212$	0	0
$213$	−8.77647	−0.601354
$214$	0	0
$215$	0.218046	0.0148706
$216$	0	0
$217$	0.195488	0.0132706
$218$	0	0
$219$	−12.4182	−0.839146
$220$	0	0
$221$	−15.0553	−1.01273
$222$	0	0
$223$	−7.64176	−0.511730	−0.255865	−	0.966713i	$-0.582360\pi$
$223$	−7.64176	−0.511730	−0.255865	+	0.966713i	$0.582360\pi$
$224$	0	0
$225$	−4.97196	−0.331464
$226$	0	0
$227$	−20.1114	−1.33484	−0.667419	−	0.744682i	$-0.732601\pi$
$227$	−20.1114	−1.33484	−0.667419	+	0.744682i	$0.732601\pi$
$228$	0	0
$229$	7.30686	0.482851	0.241425	−	0.970419i	$-0.422385\pi$
$229$	7.30686	0.482851	0.241425	+	0.970419i	$0.422385\pi$
$230$	0	0
$231$	−1.80451	−0.118728
$232$	0	0
$233$	2.30216	0.150820	0.0754099	−	0.997153i	$-0.475973\pi$
$233$	2.30216	0.150820	0.0754099	+	0.997153i	$0.475973\pi$
$234$	0	0
$235$	−0.525692	−0.0342923
$236$	0	0
$237$	−1.52569	−0.0991043
$238$	0	0
$239$	−11.9767	−0.774705	−0.387353	−	0.921932i	$-0.626610\pi$
$239$	−11.9767	−0.774705	−0.387353	+	0.921932i	$0.626610\pi$
$240$	0	0
$241$	30.1620	1.94290	0.971452	−	0.237238i	$-0.0762421\pi$
$241$	30.1620	1.94290	0.971452	+	0.237238i	$0.0762421\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.167449	−0.0106979
$246$	0	0
$247$	21.1114	1.34328
$248$	0	0
$249$	−14.7531	−0.934942
$250$	0	0
$251$	−18.8606	−1.19047	−0.595235	−	0.803552i	$-0.702941\pi$
$251$	−18.8606	−1.19047	−0.595235	+	0.803552i	$0.702941\pi$
$252$	0	0
$253$	1.80451	0.113449
$254$	0	0
$255$	−0.693141	−0.0434062
$256$	0	0
$257$	−0.613718	−0.0382827	−0.0191413	−	0.999817i	$-0.506093\pi$
$257$	−0.613718	−0.0382827	−0.0191413	+	0.999817i	$0.506093\pi$
$258$	0	0
$259$	3.33490	0.207221
$260$	0	0
$261$	5.00000	0.309492
$262$	0	0
$263$	−19.8831	−1.22605	−0.613024	−	0.790065i	$-0.710047\pi$
$263$	−19.8831	−1.22605	−0.613024	+	0.790065i	$0.710047\pi$
$264$	0	0
$265$	−0.553731	−0.0340154
$266$	0	0
$267$	8.77647	0.537111
$268$	0	0
$269$	13.1020	0.798842	0.399421	−	0.916768i	$-0.369211\pi$
$269$	13.1020	0.798842	0.399421	+	0.916768i	$0.369211\pi$
$270$	0	0
$271$	4.53039	0.275201	0.137601	−	0.990488i	$-0.456061\pi$
$271$	4.53039	0.275201	0.137601	+	0.990488i	$0.456061\pi$
$272$	0	0
$273$	3.63706	0.220125
$274$	0	0
$275$	−8.97196	−0.541030
$276$	0	0
$277$	1.22353	0.0735147	0.0367573	−	0.999324i	$-0.488297\pi$
$277$	1.22353	0.0735147	0.0367573	+	0.999324i	$0.488297\pi$
$278$	0	0
$279$	−0.195488	−0.0117036
$280$	0	0
$281$	16.3575	0.975804	0.487902	−	0.872898i	$-0.337762\pi$
$281$	16.3575	0.975804	0.487902	+	0.872898i	$0.337762\pi$
$282$	0	0
$283$	−0.637062	−0.0378694	−0.0189347	−	0.999821i	$-0.506027\pi$
$283$	−0.637062	−0.0378694	−0.0189347	+	0.999821i	$0.506027\pi$
$284$	0	0
$285$	0.971961	0.0575740
$286$	0	0
$287$	4.94392	0.291830
$288$	0	0
$289$	0.134715	0.00792440
$290$	0	0
$291$	−14.2741	−0.836764
$292$	0	0
$293$	13.4969	0.788495	0.394248	−	0.919004i	$-0.371005\pi$
$293$	13.4969	0.788495	0.394248	+	0.919004i	$0.371005\pi$
$294$	0	0
$295$	−0.823161	−0.0479263
$296$	0	0
$297$	1.80451	0.104708
$298$	0	0
$299$	−3.63706	−0.210337
$300$	0	0
$301$	1.30216	0.0750555
$302$	0	0
$303$	8.63706	0.496186
$304$	0	0
$305$	−1.46961	−0.0841498
$306$	0	0
$307$	−18.6184	−1.06261	−0.531304	−	0.847181i	$-0.678298\pi$
$307$	−18.6184	−1.06261	−0.531304	+	0.847181i	$0.678298\pi$
$308$	0	0
$309$	−8.13471	−0.462768
$310$	0	0
$311$	−17.9945	−1.02038	−0.510188	−	0.860063i	$-0.670424\pi$
$311$	−17.9945	−1.02038	−0.510188	+	0.860063i	$0.670424\pi$
$312$	0	0
$313$	33.3847	1.88702	0.943508	−	0.331351i	$-0.107504\pi$
$313$	33.3847	1.88702	0.943508	+	0.331351i	$0.107504\pi$
$314$	0	0
$315$	0.167449	0.00943469
$316$	0	0
$317$	−3.77647	−0.212108	−0.106054	−	0.994360i	$-0.533822\pi$
$317$	−3.77647	−0.212108	−0.106054	+	0.994360i	$0.533822\pi$
$318$	0	0
$319$	9.02256	0.505166
$320$	0	0
$321$	−7.16745	−0.400048
$322$	0	0
$323$	−24.0273	−1.33691
$324$	0	0
$325$	18.0833	1.00308
$326$	0	0
$327$	2.44627	0.135279
$328$	0	0
$329$	−3.13941	−0.173081
$330$	0	0
$331$	7.60902	0.418230	0.209115	−	0.977891i	$-0.432942\pi$
$331$	7.60902	0.418230	0.209115	+	0.977891i	$0.432942\pi$
$332$	0	0
$333$	−3.33490	−0.182751
$334$	0	0
$335$	2.21883	0.121228
$336$	0	0
$337$	−22.3855	−1.21942	−0.609708	−	0.792626i	$-0.708713\pi$
$337$	−22.3855	−1.21942	−0.609708	+	0.792626i	$0.708713\pi$
$338$	0	0
$339$	5.10668	0.277357
$340$	0	0
$341$	−0.352761	−0.0191031
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−0.167449	−0.00901516
$346$	0	0
$347$	−25.4922	−1.36849	−0.684246	−	0.729252i	$-0.739868\pi$
$347$	−25.4922	−1.36849	−0.684246	+	0.729252i	$0.739868\pi$
$348$	0	0
$349$	10.1900	0.545459	0.272729	−	0.962091i	$-0.412074\pi$
$349$	10.1900	0.545459	0.272729	+	0.962091i	$0.412074\pi$
$350$	0	0
$351$	−3.63706	−0.194132
$352$	0	0
$353$	19.2134	1.02262	0.511312	−	0.859395i	$-0.329160\pi$
$353$	19.2134	1.02262	0.511312	+	0.859395i	$0.329160\pi$
$354$	0	0
$355$	1.46961	0.0779990
$356$	0	0
$357$	−4.13941	−0.219081
$358$	0	0
$359$	−1.11607	−0.0589037	−0.0294519	−	0.999566i	$-0.509376\pi$
$359$	−1.11607	−0.0589037	−0.0294519	+	0.999566i	$0.509376\pi$
$360$	0	0
$361$	14.6924	0.773282
$362$	0	0
$363$	−7.74374	−0.406441
$364$	0	0
$365$	2.07942	0.108842
$366$	0	0
$367$	−5.69314	−0.297180	−0.148590	−	0.988899i	$-0.547473\pi$
$367$	−5.69314	−0.297180	−0.148590	+	0.988899i	$0.547473\pi$
$368$	0	0
$369$	−4.94392	−0.257370
$370$	0	0
$371$	−3.30686	−0.171684
$372$	0	0
$373$	−28.2967	−1.46515	−0.732573	−	0.680688i	$-0.761681\pi$
$373$	−28.2967	−1.46515	−0.732573	+	0.680688i	$0.761681\pi$
$374$	0	0
$375$	1.66980	0.0862279
$376$	0	0
$377$	−18.1853	−0.936591
$378$	0	0
$379$	−13.3622	−0.686368	−0.343184	−	0.939268i	$-0.611505\pi$
$379$	−13.3622	−0.686368	−0.343184	+	0.939268i	$0.611505\pi$
$380$	0	0
$381$	8.97666	0.459888
$382$	0	0
$383$	2.75313	0.140678	0.0703391	−	0.997523i	$-0.477592\pi$
$383$	2.75313	0.140678	0.0703391	+	0.997523i	$0.477592\pi$
$384$	0	0
$385$	0.302164	0.0153997
$386$	0	0
$387$	−1.30216	−0.0661927
$388$	0	0
$389$	6.92136	0.350927	0.175464	−	0.984486i	$-0.443858\pi$
$389$	6.92136	0.350927	0.175464	+	0.984486i	$0.443858\pi$
$390$	0	0
$391$	4.13941	0.209339
$392$	0	0
$393$	−9.86059	−0.497401
$394$	0	0
$395$	0.255476	0.0128544
$396$	0	0
$397$	−2.60433	−0.130707	−0.0653537	−	0.997862i	$-0.520818\pi$
$397$	−2.60433	−0.130707	−0.0653537	+	0.997862i	$0.520818\pi$
$398$	0	0
$399$	5.80451	0.290589
$400$	0	0
$401$	−2.13941	−0.106837	−0.0534185	−	0.998572i	$-0.517012\pi$
$401$	−2.13941	−0.106837	−0.0534185	+	0.998572i	$0.517012\pi$
$402$	0	0
$403$	0.711004	0.0354176
$404$	0	0
$405$	−0.167449	−0.00832062
$406$	0	0
$407$	−6.01786	−0.298294
$408$	0	0
$409$	−34.2500	−1.69355	−0.846777	−	0.531949i	$-0.821460\pi$
$409$	−34.2500	−1.69355	−0.846777	+	0.531949i	$0.821460\pi$
$410$	0	0
$411$	−13.0000	−0.641243
$412$	0	0
$413$	−4.91588	−0.241895
$414$	0	0
$415$	2.47040	0.121267
$416$	0	0
$417$	14.3902	0.704691
$418$	0	0
$419$	−3.91119	−0.191074	−0.0955370	−	0.995426i	$-0.530457\pi$
$419$	−3.91119	−0.191074	−0.0955370	+	0.995426i	$0.530457\pi$
$420$	0	0
$421$	−31.5249	−1.53643	−0.768215	−	0.640192i	$-0.778855\pi$
$421$	−31.5249	−1.53643	−0.768215	+	0.640192i	$0.778855\pi$
$422$	0	0
$423$	3.13941	0.152643
$424$	0	0
$425$	−20.5810	−0.998324
$426$	0	0
$427$	−8.77647	−0.424723
$428$	0	0
$429$	−6.56312	−0.316870
$430$	0	0
$431$	19.5716	0.942730	0.471365	−	0.881938i	$-0.343761\pi$
$431$	19.5716	0.942730	0.471365	+	0.881938i	$0.343761\pi$
$432$	0	0
$433$	−16.0880	−0.773141	−0.386571	−	0.922260i	$-0.626340\pi$
$433$	−16.0880	−0.773141	−0.386571	+	0.922260i	$0.626340\pi$
$434$	0	0
$435$	−0.837246	−0.0401429
$436$	0	0
$437$	−5.80451	−0.277667
$438$	0	0
$439$	−20.0833	−0.958525	−0.479263	−	0.877672i	$-0.659096\pi$
$439$	−20.0833	−0.958525	−0.479263	+	0.877672i	$0.659096\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	11.0833	0.526585	0.263292	−	0.964716i	$-0.415192\pi$
$443$	11.0833	0.526585	0.263292	+	0.964716i	$0.415192\pi$
$444$	0	0
$445$	−1.46961	−0.0696663
$446$	0	0
$447$	−2.33490	−0.110437
$448$	0	0
$449$	−30.8598	−1.45636	−0.728182	−	0.685384i	$-0.759634\pi$
$449$	−30.8598	−1.45636	−0.728182	+	0.685384i	$0.759634\pi$
$450$	0	0
$451$	−8.92136	−0.420091
$452$	0	0
$453$	−11.8092	−0.554845
$454$	0	0
$455$	−0.609023	−0.0285514
$456$	0	0
$457$	6.91588	0.323511	0.161756	−	0.986831i	$-0.448284\pi$
$457$	6.91588	0.323511	0.161756	+	0.986831i	$0.448284\pi$
$458$	0	0
$459$	4.13941	0.193211
$460$	0	0
$461$	−30.4688	−1.41907	−0.709537	−	0.704668i	$-0.751096\pi$
$461$	−30.4688	−1.41907	−0.709537	+	0.704668i	$0.751096\pi$
$462$	0	0
$463$	29.9392	1.39139	0.695697	−	0.718335i	$-0.255096\pi$
$463$	29.9392	1.39139	0.695697	+	0.718335i	$0.255096\pi$
$464$	0	0
$465$	0.0327344	0.00151802
$466$	0	0
$467$	6.21805	0.287737	0.143868	−	0.989597i	$-0.454046\pi$
$467$	6.21805	0.287737	0.143868	+	0.989597i	$0.454046\pi$
$468$	0	0
$469$	13.2508	0.611864
$470$	0	0
$471$	2.72588	0.125602
$472$	0	0
$473$	−2.34977	−0.108043
$474$	0	0
$475$	28.8598	1.32418
$476$	0	0
$477$	3.30686	0.151411
$478$	0	0
$479$	−43.6457	−1.99422	−0.997111	−	0.0759622i	$-0.975797\pi$
$479$	−43.6457	−1.99422	−0.997111	+	0.0759622i	$0.975797\pi$
$480$	0	0
$481$	12.1292	0.553045
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	2.39019	0.108533
$486$	0	0
$487$	26.1059	1.18297	0.591485	−	0.806316i	$-0.298542\pi$
$487$	26.1059	1.18297	0.591485	+	0.806316i	$0.298542\pi$
$488$	0	0
$489$	−14.6371	−0.661911
$490$	0	0
$491$	4.99921	0.225611	0.112806	−	0.993617i	$-0.464016\pi$
$491$	4.99921	0.225611	0.112806	+	0.993617i	$0.464016\pi$
$492$	0	0
$493$	20.6970	0.932148
$494$	0	0
$495$	−0.302164	−0.0135813
$496$	0	0
$497$	8.77647	0.393679
$498$	0	0
$499$	27.2508	1.21991	0.609956	−	0.792435i	$-0.291187\pi$
$499$	27.2508	1.21991	0.609956	+	0.792435i	$0.291187\pi$
$500$	0	0
$501$	−1.20018	−0.0536202
$502$	0	0
$503$	39.4174	1.75754	0.878768	−	0.477248i	$-0.158366\pi$
$503$	39.4174	1.75754	0.878768	+	0.477248i	$0.158366\pi$
$504$	0	0
$505$	−1.44627	−0.0643581
$506$	0	0
$507$	0.228223	0.0101357
$508$	0	0
$509$	−36.7710	−1.62985	−0.814923	−	0.579570i	$-0.803221\pi$
$509$	−36.7710	−1.62985	−0.814923	+	0.579570i	$0.803221\pi$
$510$	0	0
$511$	12.4182	0.549350
$512$	0	0
$513$	−5.80451	−0.256275
$514$	0	0
$515$	1.36215	0.0600236
$516$	0	0
$517$	5.66510	0.249151
$518$	0	0
$519$	19.0880	0.837871
$520$	0	0
$521$	21.8878	0.958924	0.479462	−	0.877563i	$-0.340832\pi$
$521$	21.8878	0.958924	0.479462	+	0.877563i	$0.340832\pi$
$522$	0	0
$523$	17.2180	0.752893	0.376446	−	0.926438i	$-0.377146\pi$
$523$	17.2180	0.752893	0.376446	+	0.926438i	$0.377146\pi$
$524$	0	0
$525$	4.97196	0.216994
$526$	0	0
$527$	−0.809207	−0.0352496
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	4.91588	0.213331
$532$	0	0
$533$	17.9814	0.778859
$534$	0	0
$535$	1.20018	0.0518885
$536$	0	0
$537$	−7.49765	−0.323548
$538$	0	0
$539$	1.80451	0.0777258
$540$	0	0
$541$	−23.6745	−1.01785	−0.508923	−	0.860812i	$-0.669956\pi$
$541$	−23.6745	−1.01785	−0.508923	+	0.860812i	$0.669956\pi$
$542$	0	0
$543$	16.4088	0.704171
$544$	0	0
$545$	−0.409626	−0.0175464
$546$	0	0
$547$	39.9206	1.70688	0.853440	−	0.521191i	$-0.174512\pi$
$547$	39.9206	1.70688	0.853440	+	0.521191i	$0.174512\pi$
$548$	0	0
$549$	8.77647	0.374571
$550$	0	0
$551$	−29.0226	−1.23640
$552$	0	0
$553$	1.52569	0.0648790
$554$	0	0
$555$	0.558426	0.0237039
$556$	0	0
$557$	23.2835	0.986554	0.493277	−	0.869872i	$-0.335799\pi$
$557$	23.2835	0.986554	0.493277	+	0.869872i	$0.335799\pi$
$558$	0	0
$559$	4.73605	0.200314
$560$	0	0
$561$	7.46961	0.315367
$562$	0	0
$563$	32.4735	1.36860	0.684298	−	0.729203i	$-0.260109\pi$
$563$	32.4735	1.36860	0.684298	+	0.729203i	$0.260109\pi$
$564$	0	0
$565$	−0.855109	−0.0359747
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−14.1441	−0.592952	−0.296476	−	0.955040i	$-0.595811\pi$
$569$	−14.1441	−0.592952	−0.296476	+	0.955040i	$0.595811\pi$
$570$	0	0
$571$	−32.3715	−1.35471	−0.677353	−	0.735658i	$-0.736873\pi$
$571$	−32.3715	−1.35471	−0.677353	+	0.735658i	$0.736873\pi$
$572$	0	0
$573$	1.55294	0.0648752
$574$	0	0
$575$	−4.97196	−0.207345
$576$	0	0
$577$	−13.7757	−0.573489	−0.286745	−	0.958007i	$-0.592573\pi$
$577$	−13.7757	−0.573489	−0.286745	+	0.958007i	$0.592573\pi$
$578$	0	0
$579$	−13.1394	−0.546055
$580$	0	0
$581$	14.7531	0.612063
$582$	0	0
$583$	5.96727	0.247139
$584$	0	0
$585$	0.609023	0.0251800
$586$	0	0
$587$	22.1721	0.915142	0.457571	−	0.889173i	$-0.348719\pi$
$587$	22.1721	0.915142	0.457571	+	0.889173i	$0.348719\pi$
$588$	0	0
$589$	1.13471	0.0467551
$590$	0	0
$591$	−13.8598	−0.570116
$592$	0	0
$593$	26.3668	1.08276	0.541378	−	0.840779i	$-0.317903\pi$
$593$	26.3668	1.08276	0.541378	+	0.840779i	$0.317903\pi$
$594$	0	0
$595$	0.693141	0.0284160
$596$	0	0
$597$	−19.0600	−0.780073
$598$	0	0
$599$	1.89332	0.0773591	0.0386796	−	0.999252i	$-0.487685\pi$
$599$	1.89332	0.0773591	0.0386796	+	0.999252i	$0.487685\pi$
$600$	0	0
$601$	−2.96257	−0.120846	−0.0604229	−	0.998173i	$-0.519245\pi$
$601$	−2.96257	−0.120846	−0.0604229	+	0.998173i	$0.519245\pi$
$602$	0	0
$603$	−13.2508	−0.539614
$604$	0	0
$605$	1.29668	0.0527176
$606$	0	0
$607$	1.67058	0.0678069	0.0339034	−	0.999425i	$-0.489206\pi$
$607$	1.67058	0.0678069	0.0339034	+	0.999425i	$0.489206\pi$
$608$	0	0
$609$	−5.00000	−0.202610
$610$	0	0
$611$	−11.4182	−0.461932
$612$	0	0
$613$	−14.1059	−0.569732	−0.284866	−	0.958567i	$-0.591949\pi$
$613$	−14.1059	−0.569732	−0.284866	+	0.958567i	$0.591949\pi$
$614$	0	0
$615$	0.827856	0.0333824
$616$	0	0
$617$	12.4510	0.501257	0.250628	−	0.968083i	$-0.419363\pi$
$617$	12.4510	0.501257	0.250628	+	0.968083i	$0.419363\pi$
$618$	0	0
$619$	−15.9945	−0.642874	−0.321437	−	0.946931i	$-0.604166\pi$
$619$	−15.9945	−0.642874	−0.321437	+	0.946931i	$0.604166\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−8.77647	−0.351622
$624$	0	0
$625$	24.5802	0.983208
$626$	0	0
$627$	−10.4743	−0.418304
$628$	0	0
$629$	−13.8045	−0.550422
$630$	0	0
$631$	−44.6970	−1.77936	−0.889681	−	0.456583i	$-0.849073\pi$
$631$	−44.6970	−1.77936	−0.889681	+	0.456583i	$0.849073\pi$
$632$	0	0
$633$	7.46961	0.296890
$634$	0	0
$635$	−1.50313	−0.0596500
$636$	0	0
$637$	−3.63706	−0.144106
$638$	0	0
$639$	−8.77647	−0.347192
$640$	0	0
$641$	−6.37233	−0.251692	−0.125846	−	0.992050i	$-0.540164\pi$
$641$	−6.37233	−0.251692	−0.125846	+	0.992050i	$0.540164\pi$
$642$	0	0
$643$	−2.11607	−0.0834495	−0.0417247	−	0.999129i	$-0.513285\pi$
$643$	−2.11607	−0.0834495	−0.0417247	+	0.999129i	$0.513285\pi$
$644$	0	0
$645$	0.218046	0.00858557
$646$	0	0
$647$	−6.98135	−0.274465	−0.137233	−	0.990539i	$-0.543821\pi$
$647$	−6.98135	−0.274465	−0.137233	+	0.990539i	$0.543821\pi$
$648$	0	0
$649$	8.87077	0.348208
$650$	0	0
$651$	0.195488	0.00766180
$652$	0	0
$653$	47.5522	1.86086	0.930430	−	0.366470i	$-0.119434\pi$
$653$	47.5522	1.86086	0.930430	+	0.366470i	$0.119434\pi$
$654$	0	0
$655$	1.65115	0.0645157
$656$	0	0
$657$	−12.4182	−0.484481
$658$	0	0
$659$	39.5708	1.54146	0.770730	−	0.637162i	$-0.219892\pi$
$659$	39.5708	1.54146	0.770730	+	0.637162i	$0.219892\pi$
$660$	0	0
$661$	32.4088	1.26056	0.630279	−	0.776369i	$-0.282941\pi$
$661$	32.4088	1.26056	0.630279	+	0.776369i	$0.282941\pi$
$662$	0	0
$663$	−15.0553	−0.584699
$664$	0	0
$665$	−0.971961	−0.0376910
$666$	0	0
$667$	5.00000	0.193601
$668$	0	0
$669$	−7.64176	−0.295447
$670$	0	0
$671$	15.8372	0.611390
$672$	0	0
$673$	−8.05138	−0.310358	−0.155179	−	0.987886i	$-0.549595\pi$
$673$	−8.05138	−0.310358	−0.155179	+	0.987886i	$0.549595\pi$
$674$	0	0
$675$	−4.97196	−0.191371
$676$	0	0
$677$	−15.0647	−0.578983	−0.289491	−	0.957181i	$-0.593486\pi$
$677$	−15.0647	−0.578983	−0.289491	+	0.957181i	$0.593486\pi$
$678$	0	0
$679$	14.2741	0.547791
$680$	0	0
$681$	−20.1114	−0.770669
$682$	0	0
$683$	−33.3481	−1.27603	−0.638014	−	0.770025i	$-0.720244\pi$
$683$	−33.3481	−1.27603	−0.638014	+	0.770025i	$0.720244\pi$
$684$	0	0
$685$	2.17684	0.0831728
$686$	0	0
$687$	7.30686	0.278774
$688$	0	0
$689$	−12.0273	−0.458202
$690$	0	0
$691$	24.1853	0.920053	0.460026	−	0.887905i	$-0.347840\pi$
$691$	24.1853	0.920053	0.460026	+	0.887905i	$0.347840\pi$
$692$	0	0
$693$	−1.80451	−0.0685477
$694$	0	0
$695$	−2.40963	−0.0914023
$696$	0	0
$697$	−20.4649	−0.775164
$698$	0	0
$699$	2.30216	0.0870758
$700$	0	0
$701$	8.69314	0.328335	0.164168	−	0.986432i	$-0.447506\pi$
$701$	8.69314	0.328335	0.164168	+	0.986432i	$0.447506\pi$
$702$	0	0
$703$	19.3575	0.730080
$704$	0	0
$705$	−0.525692	−0.0197987
$706$	0	0
$707$	−8.63706	−0.324830
$708$	0	0
$709$	−14.0973	−0.529435	−0.264717	−	0.964326i	$-0.585279\pi$
$709$	−14.0973	−0.529435	−0.264717	+	0.964326i	$0.585279\pi$
$710$	0	0
$711$	−1.52569	−0.0572179
$712$	0	0
$713$	−0.195488	−0.00732110
$714$	0	0
$715$	1.09899	0.0410999
$716$	0	0
$717$	−11.9767	−0.447276
$718$	0	0
$719$	0.0607736	0.00226647	0.00113324	−	0.999999i	$-0.499639\pi$
$719$	0.0607736	0.00226647	0.00113324	+	0.999999i	$0.499639\pi$
$720$	0	0
$721$	8.13471	0.302953
$722$	0	0
$723$	30.1620	1.12174
$724$	0	0
$725$	−24.8598	−0.923270
$726$	0	0
$727$	−0.743738	−0.0275837	−0.0137919	−	0.999905i	$-0.504390\pi$
$727$	−0.743738	−0.0275837	−0.0137919	+	0.999905i	$0.504390\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.39019	−0.199363
$732$	0	0
$733$	−36.6043	−1.35201	−0.676006	−	0.736896i	$-0.736291\pi$
$733$	−36.6043	−1.35201	−0.676006	+	0.736896i	$0.736291\pi$
$734$	0	0
$735$	−0.167449	−0.00617646
$736$	0	0
$737$	−23.9112	−0.880780
$738$	0	0
$739$	13.8606	0.509870	0.254935	−	0.966958i	$-0.417946\pi$
$739$	13.8606	0.509870	0.254935	+	0.966958i	$0.417946\pi$
$740$	0	0
$741$	21.1114	0.775546
$742$	0	0
$743$	5.96727	0.218918	0.109459	−	0.993991i	$-0.465088\pi$
$743$	5.96727	0.218918	0.109459	+	0.993991i	$0.465088\pi$
$744$	0	0
$745$	0.390977	0.0143243
$746$	0	0
$747$	−14.7531	−0.539789
$748$	0	0
$749$	7.16745	0.261893
$750$	0	0
$751$	5.19627	0.189615	0.0948074	−	0.995496i	$-0.469776\pi$
$751$	5.19627	0.189615	0.0948074	+	0.995496i	$0.469776\pi$
$752$	0	0
$753$	−18.8606	−0.687318
$754$	0	0
$755$	1.97744	0.0719665
$756$	0	0
$757$	16.4088	0.596389	0.298195	−	0.954505i	$-0.403616\pi$
$757$	16.4088	0.596389	0.298195	+	0.954505i	$0.403616\pi$
$758$	0	0
$759$	1.80451	0.0654996
$760$	0	0
$761$	20.2882	0.735447	0.367724	−	0.929935i	$-0.380137\pi$
$761$	20.2882	0.735447	0.367724	+	0.929935i	$0.380137\pi$
$762$	0	0
$763$	−2.44627	−0.0885609
$764$	0	0
$765$	−0.693141	−0.0250606
$766$	0	0
$767$	−17.8794	−0.645587
$768$	0	0
$769$	−9.92136	−0.357774	−0.178887	−	0.983870i	$-0.557250\pi$
$769$	−9.92136	−0.357774	−0.178887	+	0.983870i	$0.557250\pi$
$770$	0	0
$771$	−0.613718	−0.0221025
$772$	0	0
$773$	−45.7055	−1.64391	−0.821957	−	0.569550i	$-0.807117\pi$
$773$	−45.7055	−1.64391	−0.821957	+	0.569550i	$0.807117\pi$
$774$	0	0
$775$	0.971961	0.0349139
$776$	0	0
$777$	3.33490	0.119639
$778$	0	0
$779$	28.6970	1.02818
$780$	0	0
$781$	−15.8372	−0.566701
$782$	0	0
$783$	5.00000	0.178685
$784$	0	0
$785$	−0.456446	−0.0162912
$786$	0	0
$787$	37.8084	1.34772	0.673862	−	0.738857i	$-0.264634\pi$
$787$	37.8084	1.34772	0.673862	+	0.738857i	$0.264634\pi$
$788$	0	0
$789$	−19.8831	−0.707859
$790$	0	0
$791$	−5.10668	−0.181572
$792$	0	0
$793$	−31.9206	−1.13353
$794$	0	0
$795$	−0.553731	−0.0196388
$796$	0	0
$797$	10.9151	0.386633	0.193316	−	0.981136i	$-0.438076\pi$
$797$	10.9151	0.386633	0.193316	+	0.981136i	$0.438076\pi$
$798$	0	0
$799$	12.9953	0.459741
$800$	0	0
$801$	8.77647	0.310101
$802$	0	0
$803$	−22.4088	−0.790791
$804$	0	0
$805$	0.167449	0.00590181
$806$	0	0
$807$	13.1020	0.461212
$808$	0	0
$809$	14.2733	0.501824	0.250912	−	0.968010i	$-0.419270\pi$
$809$	14.2733	0.501824	0.250912	+	0.968010i	$0.419270\pi$
$810$	0	0
$811$	43.7804	1.53734	0.768669	−	0.639647i	$-0.220920\pi$
$811$	43.7804	1.53734	0.768669	+	0.639647i	$0.220920\pi$
$812$	0	0
$813$	4.53039	0.158888
$814$	0	0
$815$	2.45096	0.0858535
$816$	0	0
$817$	7.55843	0.264436
$818$	0	0
$819$	3.63706	0.127089
$820$	0	0
$821$	31.1620	1.08756	0.543780	−	0.839228i	$-0.316993\pi$
$821$	31.1620	1.08756	0.543780	+	0.839228i	$0.316993\pi$
$822$	0	0
$823$	−15.7859	−0.550261	−0.275130	−	0.961407i	$-0.588721\pi$
$823$	−15.7859	−0.550261	−0.275130	+	0.961407i	$0.588721\pi$
$824$	0	0
$825$	−8.97196	−0.312364
$826$	0	0
$827$	−40.3661	−1.40367	−0.701833	−	0.712342i	$-0.747635\pi$
$827$	−40.3661	−1.40367	−0.701833	+	0.712342i	$0.747635\pi$
$828$	0	0
$829$	9.42293	0.327272	0.163636	−	0.986521i	$-0.447678\pi$
$829$	9.42293	0.327272	0.163636	+	0.986521i	$0.447678\pi$
$830$	0	0
$831$	1.22353	0.0424437
$832$	0	0
$833$	4.13941	0.143422
$834$	0	0
$835$	0.200970	0.00695485
$836$	0	0
$837$	−0.195488	−0.00675707
$838$	0	0
$839$	−5.83725	−0.201524	−0.100762	−	0.994911i	$-0.532128\pi$
$839$	−5.83725	−0.201524	−0.100762	+	0.994911i	$0.532128\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	16.3575	0.563381
$844$	0	0
$845$	−0.0382157	−0.00131466
$846$	0	0
$847$	7.74374	0.266078
$848$	0	0
$849$	−0.637062	−0.0218639
$850$	0	0
$851$	−3.33490	−0.114319
$852$	0	0
$853$	11.9120	0.407858	0.203929	−	0.978986i	$-0.434629\pi$
$853$	11.9120	0.407858	0.203929	+	0.978986i	$0.434629\pi$
$854$	0	0
$855$	0.971961	0.0332404
$856$	0	0
$857$	17.6970	0.604520	0.302260	−	0.953226i	$-0.402259\pi$
$857$	17.6970	0.604520	0.302260	+	0.953226i	$0.402259\pi$
$858$	0	0
$859$	22.6924	0.774253	0.387127	−	0.922027i	$-0.373468\pi$
$859$	22.6924	0.774253	0.387127	+	0.922027i	$0.373468\pi$
$860$	0	0
$861$	4.94392	0.168488
$862$	0	0
$863$	19.8412	0.675401	0.337700	−	0.941254i	$-0.390351\pi$
$863$	19.8412	0.675401	0.337700	+	0.941254i	$0.390351\pi$
$864$	0	0
$865$	−3.19627	−0.108677
$866$	0	0
$867$	0.134715	0.00457515
$868$	0	0
$869$	−2.75313	−0.0933935
$870$	0	0
$871$	48.1939	1.63299
$872$	0	0
$873$	−14.2741	−0.483106
$874$	0	0
$875$	−1.66980	−0.0564494
$876$	0	0
$877$	−33.0965	−1.11759	−0.558795	−	0.829306i	$-0.688736\pi$
$877$	−33.0965	−1.11759	−0.558795	+	0.829306i	$0.688736\pi$
$878$	0	0
$879$	13.4969	0.455238
$880$	0	0
$881$	−46.2134	−1.55697	−0.778484	−	0.627665i	$-0.784011\pi$
$881$	−46.2134	−1.55697	−0.778484	+	0.627665i	$0.784011\pi$
$882$	0	0
$883$	10.7671	0.362341	0.181171	−	0.983452i	$-0.442011\pi$
$883$	10.7671	0.362341	0.181171	+	0.983452i	$0.442011\pi$
$884$	0	0
$885$	−0.823161	−0.0276702
$886$	0	0
$887$	−35.3153	−1.18577	−0.592886	−	0.805286i	$-0.702012\pi$
$887$	−35.3153	−1.18577	−0.592886	+	0.805286i	$0.702012\pi$
$888$	0	0
$889$	−8.97666	−0.301067
$890$	0	0
$891$	1.80451	0.0604534
$892$	0	0
$893$	−18.2227	−0.609801
$894$	0	0
$895$	1.25548	0.0419659
$896$	0	0
$897$	−3.63706	−0.121438
$898$	0	0
$899$	−0.977442	−0.0325995
$900$	0	0
$901$	13.6884	0.456028
$902$	0	0
$903$	1.30216	0.0433333
$904$	0	0
$905$	−2.74765	−0.0913349
$906$	0	0
$907$	26.7157	0.887080	0.443540	−	0.896255i	$-0.353722\pi$
$907$	26.7157	0.887080	0.443540	+	0.896255i	$0.353722\pi$
$908$	0	0
$909$	8.63706	0.286473
$910$	0	0
$911$	14.2453	0.471968	0.235984	−	0.971757i	$-0.424169\pi$
$911$	14.2453	0.471968	0.235984	+	0.971757i	$0.424169\pi$
$912$	0	0
$913$	−26.6222	−0.881066
$914$	0	0
$915$	−1.46961	−0.0485839
$916$	0	0
$917$	9.86059	0.325625
$918$	0	0
$919$	9.20018	0.303486	0.151743	−	0.988420i	$-0.451511\pi$
$919$	9.20018	0.303486	0.151743	+	0.988420i	$0.451511\pi$
$920$	0	0
$921$	−18.6184	−0.613498
$922$	0	0
$923$	31.9206	1.05068
$924$	0	0
$925$	16.5810	0.545179
$926$	0	0
$927$	−8.13471	−0.267179
$928$	0	0
$929$	−27.4416	−0.900329	−0.450164	−	0.892946i	$-0.648635\pi$
$929$	−27.4416	−0.900329	−0.450164	+	0.892946i	$0.648635\pi$
$930$	0	0
$931$	−5.80451	−0.190235
$932$	0	0
$933$	−17.9945	−0.589114
$934$	0	0
$935$	−1.25078	−0.0409049
$936$	0	0
$937$	20.8785	0.682069	0.341035	−	0.940051i	$-0.389223\pi$
$937$	20.8785	0.682069	0.341035	+	0.940051i	$0.389223\pi$
$938$	0	0
$939$	33.3847	1.08947
$940$	0	0
$941$	24.2469	0.790425	0.395213	−	0.918590i	$-0.370671\pi$
$941$	24.2469	0.790425	0.395213	+	0.918590i	$0.370671\pi$
$942$	0	0
$943$	−4.94392	−0.160996
$944$	0	0
$945$	0.167449	0.00544712
$946$	0	0
$947$	−4.41353	−0.143421	−0.0717103	−	0.997426i	$-0.522846\pi$
$947$	−4.41353	−0.143421	−0.0717103	+	0.997426i	$0.522846\pi$
$948$	0	0
$949$	45.1659	1.46615
$950$	0	0
$951$	−3.77647	−0.122461
$952$	0	0
$953$	−8.21883	−0.266234	−0.133117	−	0.991100i	$-0.542499\pi$
$953$	−8.21883	−0.266234	−0.133117	+	0.991100i	$0.542499\pi$
$954$	0	0
$955$	−0.260039	−0.00841468
$956$	0	0
$957$	9.02256	0.291658
$958$	0	0
$959$	13.0000	0.419792
$960$	0	0
$961$	−30.9618	−0.998767
$962$	0	0
$963$	−7.16745	−0.230968
$964$	0	0
$965$	2.20018	0.0708264
$966$	0	0
$967$	−5.48748	−0.176465	−0.0882327	−	0.996100i	$-0.528122\pi$
$967$	−5.48748	−0.176465	−0.0882327	+	0.996100i	$0.528122\pi$
$968$	0	0
$969$	−24.0273	−0.771867
$970$	0	0
$971$	−51.2686	−1.64529	−0.822645	−	0.568556i	$-0.807502\pi$
$971$	−51.2686	−1.64529	−0.822645	+	0.568556i	$0.807502\pi$
$972$	0	0
$973$	−14.3902	−0.461328
$974$	0	0
$975$	18.0833	0.579130
$976$	0	0
$977$	3.25548	0.104152	0.0520759	−	0.998643i	$-0.483416\pi$
$977$	3.25548	0.104152	0.0520759	+	0.998643i	$0.483416\pi$
$978$	0	0
$979$	15.8372	0.506161
$980$	0	0
$981$	2.44627	0.0781034
$982$	0	0
$983$	9.03195	0.288074	0.144037	−	0.989572i	$-0.453992\pi$
$983$	9.03195	0.288074	0.144037	+	0.989572i	$0.453992\pi$
$984$	0	0
$985$	2.32081	0.0739472
$986$	0	0
$987$	−3.13941	−0.0999285
$988$	0	0
$989$	−1.30216	−0.0414064
$990$	0	0
$991$	−42.4510	−1.34850	−0.674250	−	0.738503i	$-0.735533\pi$
$991$	−42.4510	−1.34850	−0.674250	+	0.738503i	$0.735533\pi$
$992$	0	0
$993$	7.60902	0.241465
$994$	0	0
$995$	3.19158	0.101180
$996$	0	0
$997$	40.6222	1.28652	0.643259	−	0.765649i	$-0.277582\pi$
$997$	40.6222	1.28652	0.643259	+	0.765649i	$0.277582\pi$
$998$	0	0
$999$	−3.33490	−0.105512

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bx.1.2		3
4.3	odd	2		3864.2.a.m.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.m.1.2	✓	3	4.3	odd	2
7728.2.a.bx.1.2		3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.167449 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.167449 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.80451 q^{11} -3.63706 q^{13} -0.167449 q^{15} +4.13941 q^{17} -5.80451 q^{19} -1.00000 q^{21} +1.00000 q^{23} -4.97196 q^{25} +1.00000 q^{27} +5.00000 q^{29} -0.195488 q^{31} +1.80451 q^{33} +0.167449 q^{35} -3.33490 q^{37} -3.63706 q^{39} -4.94392 q^{41} -1.30216 q^{43} -0.167449 q^{45} +3.13941 q^{47} +1.00000 q^{49} +4.13941 q^{51} +3.30686 q^{53} -0.302164 q^{55} -5.80451 q^{57} +4.91588 q^{59} +8.77647 q^{61} -1.00000 q^{63} +0.609023 q^{65} -13.2508 q^{67} +1.00000 q^{69} -8.77647 q^{71} -12.4182 q^{73} -4.97196 q^{75} -1.80451 q^{77} -1.52569 q^{79} +1.00000 q^{81} -14.7531 q^{83} -0.693141 q^{85} +5.00000 q^{87} +8.77647 q^{89} +3.63706 q^{91} -0.195488 q^{93} +0.971961 q^{95} -14.2741 q^{97} +1.80451 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9} - 6 q^{11} - 6 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{25} + 3 q^{27} + 15 q^{29} - 12 q^{31} - 6 q^{33} - 9 q^{37} + 9 q^{41} + 6 q^{43} - 3 q^{47} + 3 q^{49} - 3 q^{53} + 9 q^{55} - 6 q^{57} - 21 q^{59} + 3 q^{61} - 3 q^{63} - 21 q^{65} - 3 q^{67} + 3 q^{69} - 3 q^{71} - 3 q^{75} + 6 q^{77} - 18 q^{79} + 3 q^{81} - 6 q^{83} - 15 q^{85} + 15 q^{87} + 3 q^{89} - 12 q^{93} - 9 q^{95} - 21 q^{97} - 6 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9 - 6 * q^11 - 6 * q^19 - 3 * q^21 + 3 * q^23 - 3 * q^25 + 3 * q^27 + 15 * q^29 - 12 * q^31 - 6 * q^33 - 9 * q^37 + 9 * q^41 + 6 * q^43 - 3 * q^47 + 3 * q^49 - 3 * q^53 + 9 * q^55 - 6 * q^57 - 21 * q^59 + 3 * q^61 - 3 * q^63 - 21 * q^65 - 3 * q^67 + 3 * q^69 - 3 * q^71 - 3 * q^75 + 6 * q^77 - 18 * q^79 + 3 * q^81 - 6 * q^83 - 15 * q^85 + 15 * q^87 + 3 * q^89 - 12 * q^93 - 9 * q^95 - 21 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more