LMFDB - Embedded newform 1792.3.g.d.127.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,3,Mod(127,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1792.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1792.g (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$48.8284633734$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1997017344.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 53x^{4} + 56x^{2} + 4 $ x^8 + 14x^6 + 53x^4 + 56*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.6
Root			$-0.277334i$ of defining polynomial
Character	$\chi$	$=$	1792.127
Dual form			1792.3.g.d.127.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.554669 q^{3} +4.57685i q^{5} -2.64575i q^{7} -8.69234 q^{9} +O(q^{10})$	$q+0.554669 q^{3} +4.57685i q^{5} -2.64575i q^{7} -8.69234 q^{9} +15.7367 q^{11} +8.57685i q^{13} +2.53864i q^{15} +28.3197 q^{17} -6.33599 q^{19} -1.46752i q^{21} -31.0647i q^{23} +4.05242 q^{25} -9.81339 q^{27} -0.846294i q^{29} -21.6354i q^{31} +8.72866 q^{33} +12.1092 q^{35} +33.6637i q^{37} +4.75731i q^{39} -66.9757 q^{41} +44.8781 q^{43} -39.7836i q^{45} +38.4528i q^{47} -7.00000 q^{49} +15.7081 q^{51} +14.8174i q^{53} +72.0246i q^{55} -3.51438 q^{57} +5.80942 q^{59} -52.6015i q^{61} +22.9978i q^{63} -39.2550 q^{65} +117.397 q^{67} -17.2306i q^{69} +81.2543i q^{71} +47.8054 q^{73} +2.24775 q^{75} -41.6354i q^{77} +57.4900i q^{79} +72.7879 q^{81} +102.855 q^{83} +129.615i q^{85} -0.469413i q^{87} +89.2955 q^{89} +22.6922 q^{91} -12.0005i q^{93} -28.9989i q^{95} -3.44452 q^{97} -136.789 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 40 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 + 40 * q^9	$ 8 q - 8 q^{3} + 40 q^{9} - 32 q^{11} - 16 q^{17} - 88 q^{19} - 104 q^{25} - 176 q^{27} - 56 q^{35} - 144 q^{41} + 224 q^{43} - 56 q^{49} - 16 q^{51} + 400 q^{57} - 232 q^{59} - 304 q^{65} + 368 q^{67} - 272 q^{73} + 664 q^{75} + 504 q^{81} + 424 q^{83} + 80 q^{89} - 56 q^{91} + 528 q^{97} - 544 q^{99}+O(q^{100}) $ 8 * q - 8 * q^3 + 40 * q^9 - 32 * q^11 - 16 * q^17 - 88 * q^19 - 104 * q^25 - 176 * q^27 - 56 * q^35 - 144 * q^41 + 224 * q^43 - 56 * q^49 - 16 * q^51 + 400 * q^57 - 232 * q^59 - 304 * q^65 + 368 * q^67 - 272 * q^73 + 664 * q^75 + 504 * q^81 + 424 * q^83 + 80 * q^89 - 56 * q^91 + 528 * q^97 - 544 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.554669	0.184890	0.0924448	−	0.995718i	$-0.470532\pi$
$3$	0.554669	0.184890	0.0924448	+	0.995718i	$0.470532\pi$
$4$	0	0
$5$	4.57685i	0.915371i	0.889114	+	0.457685i	$0.151321\pi$
$5$	4.57685i	0.915371i	−0.889114	+	0.457685i	$0.848679\pi$
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	0	0
$9$	−8.69234	−0.965816
$10$	0	0
$11$	15.7367	1.43061	0.715305	−	0.698812i	$-0.246288\pi$
$11$	15.7367	1.43061	0.715305	+	0.698812i	$0.246288\pi$
$12$	0	0
$13$	8.57685i	0.659758i	0.944023	+	0.329879i	$0.107008\pi$
$13$	8.57685i	0.659758i	−0.944023	+	0.329879i	$0.892992\pi$
$14$	0	0
$15$	2.53864i	0.169242i
$16$	0	0
$17$	28.3197	1.66587	0.832933	−	0.553374i	$-0.186660\pi$
$17$	28.3197	1.66587	0.832933	+	0.553374i	$0.186660\pi$
$18$	0	0
$19$	−6.33599	−0.333473	−0.166737	−	0.986001i	$-0.553323\pi$
$19$	−6.33599	−0.333473	−0.166737	+	0.986001i	$0.553323\pi$
$20$	0	0
$21$	− 1.46752i	− 0.0698817i
$22$	0	0
$23$	− 31.0647i	− 1.35064i	−0.737525	−	0.675320i	$-0.764005\pi$
$23$	− 31.0647i	− 1.35064i	0.737525	−	0.675320i	$-0.235995\pi$
$24$	0	0
$25$	4.05242	0.162097
$26$	0	0
$27$	−9.81339	−0.363459
$28$	0	0
$29$	− 0.846294i	− 0.0291826i	−0.999894	−	0.0145913i	$-0.995355\pi$
$29$	− 0.846294i	− 0.0291826i	0.999894	−	0.0145913i	$-0.00464471\pi$
$30$	0	0
$31$	− 21.6354i	− 0.697917i	−0.937138	−	0.348958i	$-0.886535\pi$
$31$	− 21.6354i	− 0.697917i	0.937138	−	0.348958i	$-0.113465\pi$
$32$	0	0
$33$	8.72866	0.264505
$34$	0	0
$35$	12.1092	0.345978
$36$	0	0
$37$	33.6637i	0.909830i	0.890535	+	0.454915i	$0.150330\pi$
$37$	33.6637i	0.909830i	−0.890535	+	0.454915i	$0.849670\pi$
$38$	0	0
$39$	4.75731i	0.121982i
$40$	0	0
$41$	−66.9757	−1.63355	−0.816777	−	0.576953i	$-0.804242\pi$
$41$	−66.9757	−1.63355	−0.816777	+	0.576953i	$0.804242\pi$
$42$	0	0
$43$	44.8781	1.04368	0.521839	−	0.853044i	$-0.325246\pi$
$43$	44.8781	1.04368	0.521839	+	0.853044i	$0.325246\pi$
$44$	0	0
$45$	− 39.7836i	− 0.884079i
$46$	0	0
$47$	38.4528i	0.818145i	0.912502	+	0.409073i	$0.134148\pi$
$47$	38.4528i	0.818145i	−0.912502	+	0.409073i	$0.865852\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	15.7081	0.308001
$52$	0	0
$53$	14.8174i	0.279574i	0.990182	+	0.139787i	$0.0446417\pi$
$53$	14.8174i	0.279574i	−0.990182	+	0.139787i	$0.955358\pi$
$54$	0	0
$55$	72.0246i	1.30954i
$56$	0	0
$57$	−3.51438	−0.0616557
$58$	0	0
$59$	5.80942	0.0984647	0.0492323	−	0.998787i	$-0.484323\pi$
$59$	5.80942	0.0984647	0.0492323	+	0.998787i	$0.484323\pi$
$60$	0	0
$61$	− 52.6015i	− 0.862319i	−0.902276	−	0.431160i	$-0.858105\pi$
$61$	− 52.6015i	− 0.862319i	0.902276	−	0.431160i	$-0.141895\pi$
$62$	0	0
$63$	22.9978i	0.365044i
$64$	0	0
$65$	−39.2550	−0.603923
$66$	0	0
$67$	117.397	1.75219	0.876095	−	0.482138i	$-0.160140\pi$
$67$	117.397	1.75219	0.876095	+	0.482138i	$0.160140\pi$
$68$	0	0
$69$	− 17.2306i	− 0.249719i
$70$	0	0
$71$	81.2543i	1.14443i	0.820105	+	0.572213i	$0.193915\pi$
$71$	81.2543i	1.14443i	−0.820105	+	0.572213i	$0.806085\pi$
$72$	0	0
$73$	47.8054	0.654869	0.327435	−	0.944874i	$-0.393816\pi$
$73$	47.8054	0.654869	0.327435	+	0.944874i	$0.393816\pi$
$74$	0	0
$75$	2.24775	0.0299700
$76$	0	0
$77$	− 41.6354i	− 0.540720i
$78$	0	0
$79$	57.4900i	0.727722i	0.931453	+	0.363861i	$0.118542\pi$
$79$	57.4900i	0.727722i	−0.931453	+	0.363861i	$0.881458\pi$
$80$	0	0
$81$	72.7879	0.898616
$82$	0	0
$83$	102.855	1.23921	0.619606	−	0.784913i	$-0.287292\pi$
$83$	102.855	1.23921	0.619606	+	0.784913i	$0.287292\pi$
$84$	0	0
$85$	129.615i	1.52488i
$86$	0	0
$87$	− 0.469413i	− 0.00539555i
$88$	0	0
$89$	89.2955	1.00332	0.501660	−	0.865065i	$-0.332723\pi$
$89$	89.2955	1.00332	0.501660	+	0.865065i	$0.332723\pi$
$90$	0	0
$91$	22.6922	0.249365
$92$	0	0
$93$	− 12.0005i	− 0.129038i
$94$	0	0
$95$	− 28.9989i	− 0.305252i
$96$	0	0
$97$	−3.44452	−0.0355106	−0.0177553	−	0.999842i	$-0.505652\pi$
$97$	−3.44452	−0.0355106	−0.0177553	+	0.999842i	$0.505652\pi$
$98$	0	0
$99$	−136.789	−1.38171
$100$	0	0
$101$	− 143.034i	− 1.41618i	−0.706123	−	0.708089i	$-0.749557\pi$
$101$	− 143.034i	− 1.41618i	0.706123	−	0.708089i	$-0.250443\pi$
$102$	0	0
$103$	173.424i	1.68373i	0.539688	+	0.841865i	$0.318542\pi$
$103$	173.424i	1.68373i	−0.539688	+	0.841865i	$0.681458\pi$
$104$	0	0
$105$	6.71660	0.0639676
$106$	0	0
$107$	95.5581	0.893067	0.446533	−	0.894767i	$-0.352658\pi$
$107$	95.5581	0.893067	0.446533	+	0.894767i	$0.352658\pi$
$108$	0	0
$109$	185.517i	1.70199i	0.525170	+	0.850997i	$0.324002\pi$
$109$	185.517i	1.70199i	−0.525170	+	0.850997i	$0.675998\pi$
$110$	0	0
$111$	18.6722i	0.168218i
$112$	0	0
$113$	−103.409	−0.915124	−0.457562	−	0.889178i	$-0.651277\pi$
$113$	−103.409	−0.915124	−0.457562	+	0.889178i	$0.651277\pi$
$114$	0	0
$115$	142.179	1.23634
$116$	0	0
$117$	− 74.5529i	− 0.637205i
$118$	0	0
$119$	− 74.9269i	− 0.629638i
$120$	0	0
$121$	126.644	1.04665
$122$	0	0
$123$	−37.1493	−0.302027
$124$	0	0
$125$	132.969i	1.06375i
$126$	0	0
$127$	147.970i	1.16512i	0.812787	+	0.582560i	$0.197949\pi$
$127$	147.970i	1.16512i	−0.812787	+	0.582560i	$0.802051\pi$
$128$	0	0
$129$	24.8925	0.192965
$130$	0	0
$131$	−259.412	−1.98025	−0.990123	−	0.140200i	$-0.955225\pi$
$131$	−259.412	−1.98025	−0.990123	+	0.140200i	$0.955225\pi$
$132$	0	0
$133$	16.7635i	0.126041i
$134$	0	0
$135$	− 44.9144i	− 0.332700i
$136$	0	0
$137$	−3.07777	−0.0224654	−0.0112327	−	0.999937i	$-0.503576\pi$
$137$	−3.07777	−0.0224654	−0.0112327	+	0.999937i	$0.503576\pi$
$138$	0	0
$139$	−90.3041	−0.649670	−0.324835	−	0.945771i	$-0.605309\pi$
$139$	−90.3041	−0.649670	−0.324835	+	0.945771i	$0.605309\pi$
$140$	0	0
$141$	21.3286i	0.151266i
$142$	0	0
$143$	134.971i	0.943856i
$144$	0	0
$145$	3.87336	0.0267129
$146$	0	0
$147$	−3.88268	−0.0264128
$148$	0	0
$149$	− 41.3766i	− 0.277695i	−0.990314	−	0.138848i	$-0.955660\pi$
$149$	− 41.3766i	− 0.277695i	0.990314	−	0.138848i	$-0.0443398\pi$
$150$	0	0
$151$	− 41.5182i	− 0.274955i	−0.990505	−	0.137477i	$-0.956101\pi$
$151$	− 41.5182i	− 0.274955i	0.990505	−	0.137477i	$-0.0438994\pi$
$152$	0	0
$153$	−246.165	−1.60892
$154$	0	0
$155$	99.0221	0.638853
$156$	0	0
$157$	− 26.9848i	− 0.171878i	−0.996300	−	0.0859389i	$-0.972611\pi$
$157$	− 26.9848i	− 0.171878i	0.996300	−	0.0859389i	$-0.0273890\pi$
$158$	0	0
$159$	8.21875i	0.0516902i
$160$	0	0
$161$	−82.1895	−0.510494
$162$	0	0
$163$	126.684	0.777200	0.388600	−	0.921407i	$-0.372959\pi$
$163$	126.684	0.777200	0.388600	+	0.921407i	$0.372959\pi$
$164$	0	0
$165$	39.9498i	0.242120i
$166$	0	0
$167$	246.243i	1.47451i	0.675615	+	0.737254i	$0.263878\pi$
$167$	246.243i	1.47451i	−0.675615	+	0.737254i	$0.736122\pi$
$168$	0	0
$169$	95.4376	0.564720
$170$	0	0
$171$	55.0746	0.322074
$172$	0	0
$173$	74.5776i	0.431084i	0.976495	+	0.215542i	$0.0691518\pi$
$173$	74.5776i	0.431084i	−0.976495	+	0.215542i	$0.930848\pi$
$174$	0	0
$175$	− 10.7217i	− 0.0612668i
$176$	0	0
$177$	3.22230	0.0182051
$178$	0	0
$179$	201.073	1.12331	0.561656	−	0.827371i	$-0.310165\pi$
$179$	201.073	1.12331	0.561656	+	0.827371i	$0.310165\pi$
$180$	0	0
$181$	− 252.696i	− 1.39611i	−0.716045	−	0.698054i	$-0.754050\pi$
$181$	− 252.696i	− 1.39611i	0.716045	−	0.698054i	$-0.245950\pi$
$182$	0	0
$183$	− 29.1764i	− 0.159434i
$184$	0	0
$185$	−154.074	−0.832831
$186$	0	0
$187$	445.659	2.38320
$188$	0	0
$189$	25.9638i	0.137375i
$190$	0	0
$191$	332.333i	1.73996i	0.493085	+	0.869981i	$0.335869\pi$
$191$	332.333i	1.73996i	−0.493085	+	0.869981i	$0.664131\pi$
$192$	0	0
$193$	−53.2218	−0.275761	−0.137880	−	0.990449i	$-0.544029\pi$
$193$	−53.2218	−0.275761	−0.137880	+	0.990449i	$0.544029\pi$
$194$	0	0
$195$	−21.7735	−0.111659
$196$	0	0
$197$	− 276.248i	− 1.40227i	−0.713026	−	0.701137i	$-0.752676\pi$
$197$	− 276.248i	− 1.40227i	0.713026	−	0.701137i	$-0.247324\pi$
$198$	0	0
$199$	58.5094i	0.294017i	0.989135	+	0.147008i	$0.0469644\pi$
$199$	58.5094i	0.294017i	−0.989135	+	0.147008i	$0.953036\pi$
$200$	0	0
$201$	65.1163	0.323962
$202$	0	0
$203$	−2.23908	−0.0110300
$204$	0	0
$205$	− 306.538i	− 1.49531i
$206$	0	0
$207$	270.025i	1.30447i
$208$	0	0
$209$	−99.7077	−0.477070
$210$	0	0
$211$	146.237	0.693068	0.346534	−	0.938037i	$-0.387359\pi$
$211$	146.237	0.693068	0.346534	+	0.938037i	$0.387359\pi$
$212$	0	0
$213$	45.0692i	0.211592i
$214$	0	0
$215$	205.401i	0.955351i
$216$	0	0
$217$	−57.2420	−0.263788
$218$	0	0
$219$	26.5162	0.121078
$220$	0	0
$221$	242.894i	1.09907i
$222$	0	0
$223$	− 422.290i	− 1.89368i	−0.321709	−	0.946839i	$-0.604257\pi$
$223$	− 422.290i	− 1.89368i	0.321709	−	0.946839i	$-0.395743\pi$
$224$	0	0
$225$	−35.2250	−0.156556
$226$	0	0
$227$	271.103	1.19429	0.597144	−	0.802134i	$-0.296302\pi$
$227$	271.103	1.19429	0.597144	+	0.802134i	$0.296302\pi$
$228$	0	0
$229$	400.572i	1.74922i	0.484826	+	0.874611i	$0.338883\pi$
$229$	400.572i	1.74922i	−0.484826	+	0.874611i	$0.661117\pi$
$230$	0	0
$231$	− 23.0939i	− 0.0999734i
$232$	0	0
$233$	257.611	1.10563	0.552813	−	0.833305i	$-0.313554\pi$
$233$	257.611	1.10563	0.552813	+	0.833305i	$0.313554\pi$
$234$	0	0
$235$	−175.993	−0.748906
$236$	0	0
$237$	31.8879i	0.134548i
$238$	0	0
$239$	− 152.650i	− 0.638704i	−0.947636	−	0.319352i	$-0.896535\pi$
$239$	− 152.650i	− 0.638704i	0.947636	−	0.319352i	$-0.103465\pi$
$240$	0	0
$241$	−47.1218	−0.195526	−0.0977630	−	0.995210i	$-0.531169\pi$
$241$	−47.1218	−0.195526	−0.0977630	+	0.995210i	$0.531169\pi$
$242$	0	0
$243$	128.694	0.529604
$244$	0	0
$245$	− 32.0380i	− 0.130767i
$246$	0	0
$247$	− 54.3429i	− 0.220012i
$248$	0	0
$249$	57.0502	0.229117
$250$	0	0
$251$	236.051	0.940443	0.470221	−	0.882549i	$-0.344174\pi$
$251$	236.051	0.940443	0.470221	+	0.882549i	$0.344174\pi$
$252$	0	0
$253$	− 488.857i	− 1.93224i
$254$	0	0
$255$	71.8935i	0.281935i
$256$	0	0
$257$	−27.5524	−0.107208	−0.0536039	−	0.998562i	$-0.517071\pi$
$257$	−27.5524	−0.107208	−0.0536039	+	0.998562i	$0.517071\pi$
$258$	0	0
$259$	89.0658	0.343883
$260$	0	0
$261$	7.35628i	0.0281850i
$262$	0	0
$263$	− 88.2675i	− 0.335618i	−0.985820	−	0.167809i	$-0.946331\pi$
$263$	− 88.2675i	− 0.335618i	0.985820	−	0.167809i	$-0.0536691\pi$
$264$	0	0
$265$	−67.8170	−0.255913
$266$	0	0
$267$	49.5294	0.185503
$268$	0	0
$269$	21.9135i	0.0814629i	0.999170	+	0.0407314i	$0.0129688\pi$
$269$	21.9135i	0.0814629i	−0.999170	+	0.0407314i	$0.987031\pi$
$270$	0	0
$271$	− 428.897i	− 1.58265i	−0.611399	−	0.791323i	$-0.709393\pi$
$271$	− 428.897i	− 1.58265i	0.611399	−	0.791323i	$-0.290607\pi$
$272$	0	0
$273$	12.5867	0.0461050
$274$	0	0
$275$	63.7717	0.231897
$276$	0	0
$277$	457.076i	1.65009i	0.565065	+	0.825046i	$0.308851\pi$
$277$	457.076i	1.65009i	−0.565065	+	0.825046i	$0.691149\pi$
$278$	0	0
$279$	188.063i	0.674059i
$280$	0	0
$281$	−95.5032	−0.339869	−0.169934	−	0.985455i	$-0.554356\pi$
$281$	−95.5032	−0.339869	−0.169934	+	0.985455i	$0.554356\pi$
$282$	0	0
$283$	−131.804	−0.465737	−0.232869	−	0.972508i	$-0.574811\pi$
$283$	−131.804	−0.465737	−0.232869	+	0.972508i	$0.574811\pi$
$284$	0	0
$285$	− 16.0848i	− 0.0564379i
$286$	0	0
$287$	177.201i	0.617426i
$288$	0	0
$289$	513.006	1.77511
$290$	0	0
$291$	−1.91057	−0.00656553
$292$	0	0
$293$	325.183i	1.10984i	0.831904	+	0.554920i	$0.187251\pi$
$293$	325.183i	1.10984i	−0.831904	+	0.554920i	$0.812749\pi$
$294$	0	0
$295$	26.5888i	0.0901317i
$296$	0	0
$297$	−154.430	−0.519968
$298$	0	0
$299$	266.438	0.891096
$300$	0	0
$301$	− 118.736i	− 0.394473i
$302$	0	0
$303$	− 79.3365i	− 0.261837i
$304$	0	0
$305$	240.749	0.789341
$306$	0	0
$307$	34.3658	0.111941	0.0559704	−	0.998432i	$-0.482175\pi$
$307$	34.3658	0.111941	0.0559704	+	0.998432i	$0.482175\pi$
$308$	0	0
$309$	96.1930i	0.311304i
$310$	0	0
$311$	− 195.190i	− 0.627620i	−0.949486	−	0.313810i	$-0.898394\pi$
$311$	− 195.190i	− 0.627620i	0.949486	−	0.313810i	$-0.101606\pi$
$312$	0	0
$313$	−19.8987	−0.0635740	−0.0317870	−	0.999495i	$-0.510120\pi$
$313$	−19.8987	−0.0635740	−0.0317870	+	0.999495i	$0.510120\pi$
$314$	0	0
$315$	−105.257	−0.334151
$316$	0	0
$317$	168.672i	0.532088i	0.963961	+	0.266044i	$0.0857166\pi$
$317$	168.672i	0.532088i	−0.963961	+	0.266044i	$0.914283\pi$
$318$	0	0
$319$	− 13.3179i	− 0.0417489i
$320$	0	0
$321$	53.0031	0.165119
$322$	0	0
$323$	−179.434	−0.555522
$324$	0	0
$325$	34.7570i	0.106945i
$326$	0	0
$327$	102.901i	0.314681i
$328$	0	0
$329$	101.737	0.309230
$330$	0	0
$331$	−603.602	−1.82357	−0.911786	−	0.410666i	$-0.865296\pi$
$331$	−603.602	−1.82357	−0.911786	+	0.410666i	$0.865296\pi$
$332$	0	0
$333$	− 292.616i	− 0.878728i
$334$	0	0
$335$	537.308i	1.60390i
$336$	0	0
$337$	189.903	0.563511	0.281756	−	0.959486i	$-0.409083\pi$
$337$	189.903	0.563511	0.281756	+	0.959486i	$0.409083\pi$
$338$	0	0
$339$	−57.3578	−0.169197
$340$	0	0
$341$	− 340.470i	− 0.998447i
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	0	0
$345$	78.8621	0.228586
$346$	0	0
$347$	−81.4988	−0.234867	−0.117433	−	0.993081i	$-0.537467\pi$
$347$	−81.4988	−0.234867	−0.117433	+	0.993081i	$0.537467\pi$
$348$	0	0
$349$	− 223.673i	− 0.640898i	−0.947266	−	0.320449i	$-0.896166\pi$
$349$	− 223.673i	− 0.640898i	0.947266	−	0.320449i	$-0.103834\pi$
$350$	0	0
$351$	− 84.1680i	− 0.239795i
$352$	0	0
$353$	−613.342	−1.73751	−0.868756	−	0.495241i	$-0.835080\pi$
$353$	−613.342	−1.73751	−0.868756	+	0.495241i	$0.835080\pi$
$354$	0	0
$355$	−371.889	−1.04757
$356$	0	0
$357$	− 41.5596i	− 0.116414i
$358$	0	0
$359$	79.0559i	0.220212i	0.993920	+	0.110106i	$0.0351190\pi$
$359$	79.0559i	0.220212i	−0.993920	+	0.110106i	$0.964881\pi$
$360$	0	0
$361$	−320.855	−0.888796
$362$	0	0
$363$	70.2455	0.193514
$364$	0	0
$365$	218.798i	0.599448i
$366$	0	0
$367$	− 0.705581i	− 0.00192257i	−1.00000		0.000961283i	$-0.999694\pi$
$367$	− 0.705581i	− 0.00192257i	1.00000		0.000961283i	$-0.000305986\pi$
$368$	0	0
$369$	582.176	1.57771
$370$	0	0
$371$	39.2031	0.105669
$372$	0	0
$373$	− 250.574i	− 0.671781i	−0.941901	−	0.335890i	$-0.890963\pi$
$373$	− 250.574i	− 0.671781i	0.941901	−	0.335890i	$-0.109037\pi$
$374$	0	0
$375$	73.7535i	0.196676i
$376$	0	0
$377$	7.25854	0.0192534
$378$	0	0
$379$	532.859	1.40596	0.702981	−	0.711209i	$-0.251852\pi$
$379$	532.859	1.40596	0.702981	+	0.711209i	$0.251852\pi$
$380$	0	0
$381$	82.0745i	0.215419i
$382$	0	0
$383$	− 441.226i	− 1.15203i	−0.817441	−	0.576013i	$-0.804608\pi$
$383$	− 441.226i	− 1.15203i	0.817441	−	0.576013i	$-0.195392\pi$
$384$	0	0
$385$	190.559	0.494959
$386$	0	0
$387$	−390.096	−1.00800
$388$	0	0
$389$	− 174.837i	− 0.449452i	−0.974422	−	0.224726i	$-0.927851\pi$
$389$	− 174.837i	− 0.449452i	0.974422	−	0.224726i	$-0.0721488\pi$
$390$	0	0
$391$	− 879.744i	− 2.24999i
$392$	0	0
$393$	−143.888	−0.366127
$394$	0	0
$395$	−263.123	−0.666135
$396$	0	0
$397$	129.750i	0.326826i	0.986558	+	0.163413i	$0.0522504\pi$
$397$	129.750i	0.326826i	−0.986558	+	0.163413i	$0.947750\pi$
$398$	0	0
$399$	9.29817i	0.0233037i
$400$	0	0
$401$	−135.495	−0.337892	−0.168946	−	0.985625i	$-0.554036\pi$
$401$	−135.495	−0.337892	−0.168946	+	0.985625i	$0.554036\pi$
$402$	0	0
$403$	185.564	0.460456
$404$	0	0
$405$	333.140i	0.822567i
$406$	0	0
$407$	529.756i	1.30161i
$408$	0	0
$409$	−243.575	−0.595538	−0.297769	−	0.954638i	$-0.596242\pi$
$409$	−243.575	−0.595538	−0.297769	+	0.954638i	$0.596242\pi$
$410$	0	0
$411$	−1.70714	−0.00415363
$412$	0	0
$413$	− 15.3703i	− 0.0372161i
$414$	0	0
$415$	470.750i	1.13434i
$416$	0	0
$417$	−50.0888	−0.120117
$418$	0	0
$419$	−499.348	−1.19176	−0.595881	−	0.803073i	$-0.703197\pi$
$419$	−499.348	−1.19176	−0.595881	+	0.803073i	$0.703197\pi$
$420$	0	0
$421$	− 537.031i	− 1.27561i	−0.770199	−	0.637804i	$-0.779843\pi$
$421$	− 537.031i	− 1.27561i	0.770199	−	0.637804i	$-0.220157\pi$
$422$	0	0
$423$	− 334.245i	− 0.790178i
$424$	0	0
$425$	114.763	0.270031
$426$	0	0
$427$	−139.170	−0.325926
$428$	0	0
$429$	74.8644i	0.174509i
$430$	0	0
$431$	559.209i	1.29747i	0.761015	+	0.648735i	$0.224702\pi$
$431$	559.209i	1.29747i	−0.761015	+	0.648735i	$0.775298\pi$
$432$	0	0
$433$	−812.706	−1.87692	−0.938459	−	0.345389i	$-0.887747\pi$
$433$	−812.706	−1.87692	−0.938459	+	0.345389i	$0.887747\pi$
$434$	0	0
$435$	2.14843	0.00493893
$436$	0	0
$437$	196.826i	0.450403i
$438$	0	0
$439$	− 346.809i	− 0.789998i	−0.918681	−	0.394999i	$-0.870745\pi$
$439$	− 346.809i	− 0.789998i	0.918681	−	0.394999i	$-0.129255\pi$
$440$	0	0
$441$	60.8464	0.137974
$442$	0	0
$443$	−369.535	−0.834166	−0.417083	−	0.908868i	$-0.636948\pi$
$443$	−369.535	−0.834166	−0.417083	+	0.908868i	$0.636948\pi$
$444$	0	0
$445$	408.692i	0.918409i
$446$	0	0
$447$	− 22.9503i	− 0.0513430i
$448$	0	0
$449$	315.180	0.701961	0.350980	−	0.936383i	$-0.385848\pi$
$449$	315.180	0.701961	0.350980	+	0.936383i	$0.385848\pi$
$450$	0	0
$451$	−1053.98	−2.33698
$452$	0	0
$453$	− 23.0288i	− 0.0508363i
$454$	0	0
$455$	103.859i	0.228261i
$456$	0	0
$457$	−781.559	−1.71019	−0.855097	−	0.518468i	$-0.826503\pi$
$457$	−781.559	−1.71019	−0.855097	+	0.518468i	$0.826503\pi$
$458$	0	0
$459$	−277.912	−0.605474
$460$	0	0
$461$	526.503i	1.14209i	0.820919	+	0.571044i	$0.193461\pi$
$461$	526.503i	1.14209i	−0.820919	+	0.571044i	$0.806539\pi$
$462$	0	0
$463$	− 754.257i	− 1.62906i	−0.580118	−	0.814532i	$-0.696994\pi$
$463$	− 754.257i	− 1.62906i	0.580118	−	0.814532i	$-0.303006\pi$
$464$	0	0
$465$	54.9245	0.118117
$466$	0	0
$467$	480.381	1.02865	0.514326	−	0.857594i	$-0.328042\pi$
$467$	480.381	1.02865	0.514326	+	0.857594i	$0.328042\pi$
$468$	0	0
$469$	− 310.603i	− 0.662266i
$470$	0	0
$471$	− 14.9676i	− 0.0317784i
$472$	0	0
$473$	706.234	1.49309
$474$	0	0
$475$	−25.6761	−0.0540549
$476$	0	0
$477$	− 128.798i	− 0.270017i
$478$	0	0
$479$	− 23.0908i	− 0.0482062i	−0.999709	−	0.0241031i	$-0.992327\pi$
$479$	− 23.0908i	− 0.0482062i	0.999709	−	0.0241031i	$-0.00767300\pi$
$480$	0	0
$481$	−288.729	−0.600267
$482$	0	0
$483$	−45.5880	−0.0943850
$484$	0	0
$485$	− 15.7651i	− 0.0325053i
$486$	0	0
$487$	− 517.867i	− 1.06338i	−0.846939	−	0.531691i	$-0.821557\pi$
$487$	− 517.867i	− 1.06338i	0.846939	−	0.531691i	$-0.178443\pi$
$488$	0	0
$489$	70.2674	0.143696
$490$	0	0
$491$	−385.683	−0.785505	−0.392753	−	0.919644i	$-0.628477\pi$
$491$	−385.683	−0.785505	−0.392753	+	0.919644i	$0.628477\pi$
$492$	0	0
$493$	− 23.9668i	− 0.0486142i
$494$	0	0
$495$	− 626.063i	− 1.26477i
$496$	0	0
$497$	214.979	0.432552
$498$	0	0
$499$	726.518	1.45595	0.727974	−	0.685605i	$-0.240462\pi$
$499$	726.518	1.45595	0.727974	+	0.685605i	$0.240462\pi$
$500$	0	0
$501$	136.583i	0.272621i
$502$	0	0
$503$	253.383i	0.503743i	0.967761	+	0.251872i	$0.0810461\pi$
$503$	253.383i	0.503743i	−0.967761	+	0.251872i	$0.918954\pi$
$504$	0	0
$505$	654.646	1.29633
$506$	0	0
$507$	52.9362	0.104411
$508$	0	0
$509$	− 338.344i	− 0.664723i	−0.943152	−	0.332361i	$-0.892155\pi$
$509$	− 338.344i	− 0.664723i	0.943152	−	0.332361i	$-0.107845\pi$
$510$	0	0
$511$	− 126.481i	− 0.247517i
$512$	0	0
$513$	62.1776	0.121204
$514$	0	0
$515$	−793.737	−1.54124
$516$	0	0
$517$	605.121i	1.17045i
$518$	0	0
$519$	41.3658i	0.0797030i
$520$	0	0
$521$	333.599	0.640305	0.320152	−	0.947366i	$-0.396266\pi$
$521$	333.599	0.640305	0.320152	+	0.947366i	$0.396266\pi$
$522$	0	0
$523$	38.7942	0.0741763	0.0370882	−	0.999312i	$-0.488192\pi$
$523$	38.7942	0.0741763	0.0370882	+	0.999312i	$0.488192\pi$
$524$	0	0
$525$	− 5.94699i	− 0.0113276i
$526$	0	0
$527$	− 612.709i	− 1.16264i
$528$	0	0
$529$	−436.017	−0.824229
$530$	0	0
$531$	−50.4974	−0.0950987
$532$	0	0
$533$	− 574.441i	− 1.07775i
$534$	0	0
$535$	437.355i	0.817487i
$536$	0	0
$537$	111.529	0.207689
$538$	0	0
$539$	−110.157	−0.204373
$540$	0	0
$541$	543.111i	1.00390i	0.864896	+	0.501951i	$0.167384\pi$
$541$	543.111i	1.00390i	−0.864896	+	0.501951i	$0.832616\pi$
$542$	0	0
$543$	− 140.162i	− 0.258126i
$544$	0	0
$545$	−849.086	−1.55796
$546$	0	0
$547$	−262.532	−0.479949	−0.239974	−	0.970779i	$-0.577139\pi$
$547$	−262.532	−0.479949	−0.239974	+	0.970779i	$0.577139\pi$
$548$	0	0
$549$	457.230i	0.832841i
$550$	0	0
$551$	5.36212i	0.00973161i
$552$	0	0
$553$	152.104	0.275053
$554$	0	0
$555$	−85.4599	−0.153982
$556$	0	0
$557$	− 652.276i	− 1.17105i	−0.810654	−	0.585526i	$-0.800888\pi$
$557$	− 652.276i	− 1.17105i	0.810654	−	0.585526i	$-0.199112\pi$
$558$	0	0
$559$	384.913i	0.688574i
$560$	0	0
$561$	247.193	0.440630
$562$	0	0
$563$	993.435	1.76454	0.882269	−	0.470746i	$-0.156015\pi$
$563$	993.435	1.76454	0.882269	+	0.470746i	$0.156015\pi$
$564$	0	0
$565$	− 473.288i	− 0.837678i
$566$	0	0
$567$	− 192.579i	− 0.339645i
$568$	0	0
$569$	450.590	0.791898	0.395949	−	0.918272i	$-0.370416\pi$
$569$	450.590	0.791898	0.395949	+	0.918272i	$0.370416\pi$
$570$	0	0
$571$	373.493	0.654103	0.327051	−	0.945007i	$-0.393945\pi$
$571$	373.493	0.654103	0.327051	+	0.945007i	$0.393945\pi$
$572$	0	0
$573$	184.335i	0.321701i
$574$	0	0
$575$	− 125.887i	− 0.218934i
$576$	0	0
$577$	370.687	0.642439	0.321219	−	0.947005i	$-0.395907\pi$
$577$	370.687	0.642439	0.321219	+	0.947005i	$0.395907\pi$
$578$	0	0
$579$	−29.5205	−0.0509853
$580$	0	0
$581$	− 272.128i	− 0.468378i
$582$	0	0
$583$	233.177i	0.399961i
$584$	0	0
$585$	341.218	0.583278
$586$	0	0
$587$	182.976	0.311714	0.155857	−	0.987780i	$-0.450186\pi$
$587$	182.976	0.311714	0.155857	+	0.987780i	$0.450186\pi$
$588$	0	0
$589$	137.082i	0.232737i
$590$	0	0
$591$	− 153.226i	− 0.259266i
$592$	0	0
$593$	333.716	0.562759	0.281379	−	0.959597i	$-0.409208\pi$
$593$	333.716	0.562759	0.281379	+	0.959597i	$0.409208\pi$
$594$	0	0
$595$	342.930	0.576352
$596$	0	0
$597$	32.4533i	0.0543607i
$598$	0	0
$599$	− 219.331i	− 0.366163i	−0.983098	−	0.183081i	$-0.941393\pi$
$599$	− 219.331i	− 0.366163i	0.983098	−	0.183081i	$-0.0586072\pi$
$600$	0	0
$601$	−268.193	−0.446245	−0.223122	−	0.974790i	$-0.571625\pi$
$601$	−268.193	−0.446245	−0.223122	+	0.974790i	$0.571625\pi$
$602$	0	0
$603$	−1020.45	−1.69229
$604$	0	0
$605$	579.631i	0.958068i
$606$	0	0
$607$	585.050i	0.963839i	0.876216	+	0.481919i	$0.160060\pi$
$607$	585.050i	0.963839i	−0.876216	+	0.481919i	$0.839940\pi$
$608$	0	0
$609$	−1.24195	−0.00203933
$610$	0	0
$611$	−329.804	−0.539778
$612$	0	0
$613$	− 29.3420i	− 0.0478662i	−0.999714	−	0.0239331i	$-0.992381\pi$
$613$	− 29.3420i	− 0.0478662i	0.999714	−	0.0239331i	$-0.00761887\pi$
$614$	0	0
$615$	− 170.027i	− 0.276467i
$616$	0	0
$617$	413.079	0.669497	0.334748	−	0.942308i	$-0.391349\pi$
$617$	413.079	0.669497	0.334748	+	0.942308i	$0.391349\pi$
$618$	0	0
$619$	−225.596	−0.364453	−0.182226	−	0.983257i	$-0.558330\pi$
$619$	−225.596	−0.364453	−0.182226	+	0.983257i	$0.558330\pi$
$620$	0	0
$621$	304.850i	0.490902i
$622$	0	0
$623$	− 236.254i	− 0.379219i
$624$	0	0
$625$	−507.267	−0.811628
$626$	0	0
$627$	−55.3047	−0.0882053
$628$	0	0
$629$	953.346i	1.51565i
$630$	0	0
$631$	− 914.619i	− 1.44948i	−0.689025	−	0.724738i	$-0.741961\pi$
$631$	− 914.619i	− 1.44948i	0.689025	−	0.724738i	$-0.258039\pi$
$632$	0	0
$633$	81.1132	0.128141
$634$	0	0
$635$	−677.239	−1.06652
$636$	0	0
$637$	− 60.0380i	− 0.0942511i
$638$	0	0
$639$	− 706.290i	− 1.10530i
$640$	0	0
$641$	−1025.61	−1.60002	−0.800009	−	0.599987i	$-0.795172\pi$
$641$	−1025.61	−1.60002	−0.800009	+	0.599987i	$0.795172\pi$
$642$	0	0
$643$	−864.377	−1.34429	−0.672144	−	0.740421i	$-0.734626\pi$
$643$	−864.377	−1.34429	−0.672144	+	0.740421i	$0.734626\pi$
$644$	0	0
$645$	113.929i	0.176634i
$646$	0	0
$647$	− 689.240i	− 1.06529i	−0.846340	−	0.532643i	$-0.821199\pi$
$647$	− 689.240i	− 1.06529i	0.846340	−	0.532643i	$-0.178801\pi$
$648$	0	0
$649$	91.4211	0.140865
$650$	0	0
$651$	−31.7503	−0.0487716
$652$	0	0
$653$	− 565.975i	− 0.866730i	−0.901218	−	0.433365i	$-0.857326\pi$
$653$	− 565.975i	− 0.866730i	0.901218	−	0.433365i	$-0.142674\pi$
$654$	0	0
$655$	− 1187.29i	− 1.81266i
$656$	0	0
$657$	−415.541	−0.632483
$658$	0	0
$659$	289.064	0.438640	0.219320	−	0.975653i	$-0.429616\pi$
$659$	289.064	0.438640	0.219320	+	0.975653i	$0.429616\pi$
$660$	0	0
$661$	− 283.787i	− 0.429329i	−0.976688	−	0.214665i	$-0.931134\pi$
$661$	− 283.787i	− 0.429329i	0.976688	−	0.214665i	$-0.0688658\pi$
$662$	0	0
$663$	134.726i	0.203206i
$664$	0	0
$665$	−76.7239	−0.115374
$666$	0	0
$667$	−26.2899	−0.0394151
$668$	0	0
$669$	− 234.231i	− 0.350121i
$670$	0	0
$671$	− 827.774i	− 1.23364i
$672$	0	0
$673$	987.489	1.46729	0.733647	−	0.679531i	$-0.237817\pi$
$673$	987.489	1.46729	0.733647	+	0.679531i	$0.237817\pi$
$674$	0	0
$675$	−39.7680	−0.0589155
$676$	0	0
$677$	− 787.301i	− 1.16293i	−0.813573	−	0.581463i	$-0.802481\pi$
$677$	− 787.301i	− 1.16293i	0.813573	−	0.581463i	$-0.197519\pi$
$678$	0	0
$679$	9.11336i	0.0134217i
$680$	0	0
$681$	150.373	0.220811
$682$	0	0
$683$	193.168	0.282823	0.141411	−	0.989951i	$-0.454836\pi$
$683$	193.168	0.282823	0.141411	+	0.989951i	$0.454836\pi$
$684$	0	0
$685$	− 14.0865i	− 0.0205642i
$686$	0	0
$687$	222.185i	0.323413i
$688$	0	0
$689$	−127.087	−0.184451
$690$	0	0
$691$	−967.147	−1.39963	−0.699817	−	0.714322i	$-0.746735\pi$
$691$	−967.147	−1.39963	−0.699817	+	0.714322i	$0.746735\pi$
$692$	0	0
$693$	361.909i	0.522236i
$694$	0	0
$695$	− 413.308i	− 0.594688i
$696$	0	0
$697$	−1896.73	−2.72128
$698$	0	0
$699$	142.889	0.204419
$700$	0	0
$701$	− 246.322i	− 0.351386i	−0.984445	−	0.175693i	$-0.943783\pi$
$701$	− 246.322i	− 0.351386i	0.984445	−	0.175693i	$-0.0562167\pi$
$702$	0	0
$703$	− 213.293i	− 0.303404i
$704$	0	0
$705$	−97.6178	−0.138465
$706$	0	0
$707$	−378.432	−0.535265
$708$	0	0
$709$	− 1191.27i	− 1.68021i	−0.542428	−	0.840103i	$-0.682495\pi$
$709$	− 1191.27i	− 1.68021i	0.542428	−	0.840103i	$-0.317505\pi$
$710$	0	0
$711$	− 499.723i	− 0.702845i
$712$	0	0
$713$	−672.098	−0.942635
$714$	0	0
$715$	−617.744	−0.863978
$716$	0	0
$717$	− 84.6703i	− 0.118090i
$718$	0	0
$719$	434.163i	0.603843i	0.953333	+	0.301922i	$0.0976281\pi$
$719$	434.163i	0.603843i	−0.953333	+	0.301922i	$0.902372\pi$
$720$	0	0
$721$	458.838	0.636390
$722$	0	0
$723$	−26.1370	−0.0361507
$724$	0	0
$725$	− 3.42954i	− 0.00473040i
$726$	0	0
$727$	− 931.815i	− 1.28173i	−0.767655	−	0.640863i	$-0.778577\pi$
$727$	− 931.815i	− 1.28173i	0.767655	−	0.640863i	$-0.221423\pi$
$728$	0	0
$729$	−583.709	−0.800698
$730$	0	0
$731$	1270.94	1.73863
$732$	0	0
$733$	− 1426.30i	− 1.94584i	−0.231150	−	0.972918i	$-0.574249\pi$
$733$	− 1426.30i	− 1.94584i	0.231150	−	0.972918i	$-0.425751\pi$
$734$	0	0
$735$	− 17.7705i	− 0.0241775i
$736$	0	0
$737$	1847.44	2.50670
$738$	0	0
$739$	−216.382	−0.292804	−0.146402	−	0.989225i	$-0.546769\pi$
$739$	−216.382	−0.292804	−0.146402	+	0.989225i	$0.546769\pi$
$740$	0	0
$741$	− 30.1423i	− 0.0406779i
$742$	0	0
$743$	− 81.5074i	− 0.109700i	−0.998495	−	0.0548502i	$-0.982532\pi$
$743$	− 81.5074i	− 0.109700i	0.998495	−	0.0548502i	$-0.0174681\pi$
$744$	0	0
$745$	189.375	0.254194
$746$	0	0
$747$	−894.048	−1.19685
$748$	0	0
$749$	− 252.823i	− 0.337547i
$750$	0	0
$751$	− 142.603i	− 0.189884i	−0.995483	−	0.0949422i	$-0.969733\pi$
$751$	− 142.603i	− 0.189884i	0.995483	−	0.0949422i	$-0.0302666\pi$
$752$	0	0
$753$	130.930	0.173878
$754$	0	0
$755$	190.023	0.251686
$756$	0	0
$757$	372.503i	0.492078i	0.969260	+	0.246039i	$0.0791292\pi$
$757$	372.503i	0.492078i	−0.969260	+	0.246039i	$0.920871\pi$
$758$	0	0
$759$	− 271.153i	− 0.357251i
$760$	0	0
$761$	−663.881	−0.872380	−0.436190	−	0.899855i	$-0.643672\pi$
$761$	−663.881	−0.872380	−0.436190	+	0.899855i	$0.643672\pi$
$762$	0	0
$763$	490.833	0.643294
$764$	0	0
$765$	− 1126.66i	− 1.47276i
$766$	0	0
$767$	49.8265i	0.0649628i
$768$	0	0
$769$	732.653	0.952734	0.476367	−	0.879247i	$-0.341953\pi$
$769$	732.653	0.952734	0.476367	+	0.879247i	$0.341953\pi$
$770$	0	0
$771$	−15.2824	−0.0198216
$772$	0	0
$773$	928.973i	1.20178i	0.799333	+	0.600888i	$0.205186\pi$
$773$	928.973i	1.20178i	−0.799333	+	0.600888i	$0.794814\pi$
$774$	0	0
$775$	− 87.6758i	− 0.113130i
$776$	0	0
$777$	49.4020	0.0635804
$778$	0	0
$779$	424.358	0.544747
$780$	0	0
$781$	1278.67i	1.63723i
$782$	0	0
$783$	8.30502i	0.0106067i
$784$	0	0
$785$	123.506	0.157332
$786$	0	0
$787$	−470.202	−0.597461	−0.298730	−	0.954338i	$-0.596563\pi$
$787$	−470.202	−0.597461	−0.298730	+	0.954338i	$0.596563\pi$
$788$	0	0
$789$	− 48.9592i	− 0.0620522i
$790$	0	0
$791$	273.595i	0.345884i
$792$	0	0
$793$	451.155	0.568922
$794$	0	0
$795$	−37.6160	−0.0473157
$796$	0	0
$797$	832.694i	1.04479i	0.852705	+	0.522393i	$0.174960\pi$
$797$	832.694i	1.04479i	−0.852705	+	0.522393i	$0.825040\pi$
$798$	0	0
$799$	1088.97i	1.36292i
$800$	0	0
$801$	−776.187	−0.969022
$802$	0	0
$803$	752.300	0.936862
$804$	0	0
$805$	− 376.169i	− 0.467291i
$806$	0	0
$807$	12.1547i	0.0150616i
$808$	0	0
$809$	−976.293	−1.20679	−0.603395	−	0.797442i	$-0.706186\pi$
$809$	−976.293	−1.20679	−0.603395	+	0.797442i	$0.706186\pi$
$810$	0	0
$811$	−564.470	−0.696018	−0.348009	−	0.937491i	$-0.613142\pi$
$811$	−564.470	−0.696018	−0.348009	+	0.937491i	$0.613142\pi$
$812$	0	0
$813$	− 237.896i	− 0.292615i
$814$	0	0
$815$	579.812i	0.711426i
$816$	0	0
$817$	−284.347	−0.348038
$818$	0	0
$819$	−197.249	−0.240841
$820$	0	0
$821$	− 779.033i	− 0.948883i	−0.880287	−	0.474441i	$-0.842650\pi$
$821$	− 779.033i	− 0.948883i	0.880287	−	0.474441i	$-0.157350\pi$
$822$	0	0
$823$	363.394i	0.441548i	0.975325	+	0.220774i	$0.0708582\pi$
$823$	363.394i	0.441548i	−0.975325	+	0.220774i	$0.929142\pi$
$824$	0	0
$825$	35.3722	0.0428754
$826$	0	0
$827$	620.477	0.750275	0.375137	−	0.926969i	$-0.377596\pi$
$827$	620.477	0.750275	0.375137	+	0.926969i	$0.377596\pi$
$828$	0	0
$829$	737.842i	0.890038i	0.895521	+	0.445019i	$0.146803\pi$
$829$	737.842i	0.890038i	−0.895521	+	0.445019i	$0.853197\pi$
$830$	0	0
$831$	253.525i	0.305085i
$832$	0	0
$833$	−198.238	−0.237981
$834$	0	0
$835$	−1127.02	−1.34972
$836$	0	0
$837$	212.317i	0.253664i
$838$	0	0
$839$	897.093i	1.06924i	0.845092	+	0.534620i	$0.179545\pi$
$839$	897.093i	1.06924i	−0.845092	+	0.534620i	$0.820455\pi$
$840$	0	0
$841$	840.284	0.999148
$842$	0	0
$843$	−52.9726	−0.0628382
$844$	0	0
$845$	436.804i	0.516928i
$846$	0	0
$847$	− 335.069i	− 0.395595i
$848$	0	0
$849$	−73.1074	−0.0861100
$850$	0	0
$851$	1045.75	1.22885
$852$	0	0
$853$	1470.98i	1.72448i	0.506501	+	0.862239i	$0.330939\pi$
$853$	1470.98i	1.72448i	−0.506501	+	0.862239i	$0.669061\pi$
$854$	0	0
$855$	252.068i	0.294817i
$856$	0	0
$857$	−851.862	−0.994005	−0.497002	−	0.867749i	$-0.665566\pi$
$857$	−851.862	−0.994005	−0.497002	+	0.867749i	$0.665566\pi$
$858$	0	0
$859$	−140.400	−0.163446	−0.0817229	−	0.996655i	$-0.526042\pi$
$859$	−140.400	−0.163446	−0.0817229	+	0.996655i	$0.526042\pi$
$860$	0	0
$861$	98.2879i	0.114156i
$862$	0	0
$863$	− 32.1489i	− 0.0372525i	−0.999827	−	0.0186263i	$-0.994071\pi$
$863$	− 32.1489i	− 0.0372525i	0.999827	−	0.0186263i	$-0.00592927\pi$
$864$	0	0
$865$	−341.330	−0.394602
$866$	0	0
$867$	284.549	0.328199
$868$	0	0
$869$	904.704i	1.04109i
$870$	0	0
$871$	1006.89i	1.15602i
$872$	0	0
$873$	29.9410	0.0342967
$874$	0	0
$875$	351.802	0.402059
$876$	0	0
$877$	− 1039.93i	− 1.18578i	−0.805285	−	0.592888i	$-0.797988\pi$
$877$	− 1039.93i	− 1.18578i	0.805285	−	0.592888i	$-0.202012\pi$
$878$	0	0
$879$	180.369i	0.205198i
$880$	0	0
$881$	−416.344	−0.472581	−0.236291	−	0.971682i	$-0.575932\pi$
$881$	−416.344	−0.472581	−0.236291	+	0.971682i	$0.575932\pi$
$882$	0	0
$883$	231.281	0.261926	0.130963	−	0.991387i	$-0.458193\pi$
$883$	231.281	0.261926	0.130963	+	0.991387i	$0.458193\pi$
$884$	0	0
$885$	14.7480i	0.0166644i
$886$	0	0
$887$	6.35223i	0.00716148i	0.999994	+	0.00358074i	$0.00113979\pi$
$887$	6.35223i	0.00716148i	−0.999994	+	0.00358074i	$0.998860\pi$
$888$	0	0
$889$	391.493	0.440374
$890$	0	0
$891$	1145.44	1.28557
$892$	0	0
$893$	− 243.637i	− 0.272830i
$894$	0	0
$895$	920.280i	1.02825i
$896$	0	0
$897$	147.785	0.164754
$898$	0	0
$899$	−18.3099	−0.0203670
$900$	0	0
$901$	419.624i	0.465732i
$902$	0	0
$903$	− 65.8593i	− 0.0729339i
$904$	0	0
$905$	1156.55	1.27796
$906$	0	0
$907$	−201.716	−0.222399	−0.111200	−	0.993798i	$-0.535469\pi$
$907$	−201.716	−0.222399	−0.111200	+	0.993798i	$0.535469\pi$
$908$	0	0
$909$	1243.30i	1.36777i
$910$	0	0
$911$	484.675i	0.532025i	0.963970	+	0.266013i	$0.0857063\pi$
$911$	484.675i	0.532025i	−0.963970	+	0.266013i	$0.914294\pi$
$912$	0	0
$913$	1618.59	1.77283
$914$	0	0
$915$	133.536	0.145941
$916$	0	0
$917$	686.340i	0.748463i
$918$	0	0
$919$	650.067i	0.707363i	0.935366	+	0.353681i	$0.115070\pi$
$919$	650.067i	0.707363i	−0.935366	+	0.353681i	$0.884930\pi$
$920$	0	0
$921$	19.0616	0.0206967
$922$	0	0
$923$	−696.906	−0.755044
$924$	0	0
$925$	136.419i	0.147480i
$926$	0	0
$927$	− 1507.46i	− 1.62617i
$928$	0	0
$929$	522.394	0.562318	0.281159	−	0.959661i	$-0.409281\pi$
$929$	522.394	0.562318	0.281159	+	0.959661i	$0.409281\pi$
$930$	0	0
$931$	44.3520	0.0476391
$932$	0	0
$933$	− 108.266i	− 0.116040i
$934$	0	0
$935$	2039.72i	2.18151i
$936$	0	0
$937$	−1579.17	−1.68535	−0.842676	−	0.538422i	$-0.819021\pi$
$937$	−1579.17	−1.68535	−0.842676	+	0.538422i	$0.819021\pi$
$938$	0	0
$939$	−11.0372	−0.0117542
$940$	0	0
$941$	− 1080.68i	− 1.14844i	−0.818703	−	0.574218i	$-0.805306\pi$
$941$	− 1080.68i	− 1.14844i	0.818703	−	0.574218i	$-0.194694\pi$
$942$	0	0
$943$	2080.58i	2.20634i
$944$	0	0
$945$	−118.832	−0.125749
$946$	0	0
$947$	538.713	0.568863	0.284431	−	0.958696i	$-0.408195\pi$
$947$	538.713	0.568863	0.284431	+	0.958696i	$0.408195\pi$
$948$	0	0
$949$	410.020i	0.432055i
$950$	0	0
$951$	93.5570i	0.0983775i
$952$	0	0
$953$	1099.01	1.15322	0.576608	−	0.817021i	$-0.304376\pi$
$953$	1099.01	1.15322	0.576608	+	0.817021i	$0.304376\pi$
$954$	0	0
$955$	−1521.04	−1.59271
$956$	0	0
$957$	− 7.38702i	− 0.00771893i
$958$	0	0
$959$	8.14300i	0.00849114i
$960$	0	0
$961$	492.908	0.512912
$962$	0	0
$963$	−830.624	−0.862538
$964$	0	0
$965$	− 243.588i	− 0.252423i
$966$	0	0
$967$	− 847.122i	− 0.876031i	−0.898967	−	0.438016i	$-0.855681\pi$
$967$	− 847.122i	− 0.876031i	0.898967	−	0.438016i	$-0.144319\pi$
$968$	0	0
$969$	−99.5262	−0.102710
$970$	0	0
$971$	−1059.01	−1.09064	−0.545322	−	0.838227i	$-0.683592\pi$
$971$	−1059.01	−1.09064	−0.545322	+	0.838227i	$0.683592\pi$
$972$	0	0
$973$	238.922i	0.245552i
$974$	0	0
$975$	19.2786i	0.0197729i
$976$	0	0
$977$	−1223.46	−1.25226	−0.626131	−	0.779718i	$-0.715362\pi$
$977$	−1223.46	−1.25226	−0.626131	+	0.779718i	$0.715362\pi$
$978$	0	0
$979$	1405.22	1.43536
$980$	0	0
$981$	− 1612.58i	− 1.64381i
$982$	0	0
$983$	− 256.802i	− 0.261244i	−0.991432	−	0.130622i	$-0.958303\pi$
$983$	− 256.802i	− 0.261244i	0.991432	−	0.130622i	$-0.0416974\pi$
$984$	0	0
$985$	1264.35	1.28360
$986$	0	0
$987$	56.4301	0.0571734
$988$	0	0
$989$	− 1394.13i	− 1.40963i
$990$	0	0
$991$	− 988.781i	− 0.997761i	−0.866671	−	0.498881i	$-0.833745\pi$
$991$	− 988.781i	− 0.997761i	0.866671	−	0.498881i	$-0.166255\pi$
$992$	0	0
$993$	−334.799	−0.337159
$994$	0	0
$995$	−267.789	−0.269134
$996$	0	0
$997$	581.918i	0.583669i	0.956469	+	0.291835i	$0.0942657\pi$
$997$	581.918i	0.583669i	−0.956469	+	0.291835i	$0.905734\pi$
$998$	0	0
$999$	− 330.355i	− 0.330686i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.3.g.d.127.6		8
4.3	odd	2		1792.3.g.f.127.4		8
8.3	odd	2	inner	1792.3.g.d.127.5		8
8.5	even	2		1792.3.g.f.127.3		8
16.3	odd	4		448.3.d.e.127.4		8
16.5	even	4		224.3.d.b.127.4	✓	8
16.11	odd	4		224.3.d.b.127.5	yes	8
16.13	even	4		448.3.d.e.127.5		8
48.5	odd	4		2016.3.m.c.127.6		8
48.11	even	4		2016.3.m.c.127.5		8
112.27	even	4		1568.3.d.n.1471.4		8
112.69	odd	4		1568.3.d.n.1471.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.d.b.127.4	✓	8	16.5	even	4
224.3.d.b.127.5	yes	8	16.11	odd	4
448.3.d.e.127.4		8	16.3	odd	4
448.3.d.e.127.5		8	16.13	even	4
1568.3.d.n.1471.4		8	112.27	even	4
1568.3.d.n.1471.5		8	112.69	odd	4
1792.3.g.d.127.5		8	8.3	odd	2	inner
1792.3.g.d.127.6		8	1.1	even	1	trivial
1792.3.g.f.127.3		8	8.5	even	2
1792.3.g.f.127.4		8	4.3	odd	2
2016.3.m.c.127.5		8	48.11	even	4
2016.3.m.c.127.6		8	48.5	odd	4

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:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1792.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.554669 q^{3} +4.57685i q^{5} -2.64575i q^{7} -8.69234 q^{9} +O(q^{10})\)	\(q+0.554669 q^{3} +4.57685i q^{5} -2.64575i q^{7} -8.69234 q^{9} +15.7367 q^{11} +8.57685i q^{13} +2.53864i q^{15} +28.3197 q^{17} -6.33599 q^{19} -1.46752i q^{21} -31.0647i q^{23} +4.05242 q^{25} -9.81339 q^{27} -0.846294i q^{29} -21.6354i q^{31} +8.72866 q^{33} +12.1092 q^{35} +33.6637i q^{37} +4.75731i q^{39} -66.9757 q^{41} +44.8781 q^{43} -39.7836i q^{45} +38.4528i q^{47} -7.00000 q^{49} +15.7081 q^{51} +14.8174i q^{53} +72.0246i q^{55} -3.51438 q^{57} +5.80942 q^{59} -52.6015i q^{61} +22.9978i q^{63} -39.2550 q^{65} +117.397 q^{67} -17.2306i q^{69} +81.2543i q^{71} +47.8054 q^{73} +2.24775 q^{75} -41.6354i q^{77} +57.4900i q^{79} +72.7879 q^{81} +102.855 q^{83} +129.615i q^{85} -0.469413i q^{87} +89.2955 q^{89} +22.6922 q^{91} -12.0005i q^{93} -28.9989i q^{95} -3.44452 q^{97} -136.789 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 40 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 + 40 * q^9	\( 8 q - 8 q^{3} + 40 q^{9} - 32 q^{11} - 16 q^{17} - 88 q^{19} - 104 q^{25} - 176 q^{27} - 56 q^{35} - 144 q^{41} + 224 q^{43} - 56 q^{49} - 16 q^{51} + 400 q^{57} - 232 q^{59} - 304 q^{65} + 368 q^{67} - 272 q^{73} + 664 q^{75} + 504 q^{81} + 424 q^{83} + 80 q^{89} - 56 q^{91} + 528 q^{97} - 544 q^{99}+O(q^{100}) \) 8 * q - 8 * q^3 + 40 * q^9 - 32 * q^11 - 16 * q^17 - 88 * q^19 - 104 * q^25 - 176 * q^27 - 56 * q^35 - 144 * q^41 + 224 * q^43 - 56 * q^49 - 16 * q^51 + 400 * q^57 - 232 * q^59 - 304 * q^65 + 368 * q^67 - 272 * q^73 + 664 * q^75 + 504 * q^81 + 424 * q^83 + 80 * q^89 - 56 * q^91 + 528 * q^97 - 544 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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