LMFDB - Embedded newform 224.3.d.b.127.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,3,Mod(127,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("224.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	224.d (of order $2$, degree $1$, 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:	$6.10355792167$
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^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.4
Root			$-0.277334i$ of defining polynomial
Character	$\chi$	$=$	224.127
Dual form			224.3.d.b.127.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.554669i q^{3} -4.57685 q^{5} +2.64575i q^{7} +8.69234 q^{9} +O(q^{10})$	$q-0.554669i q^{3} -4.57685 q^{5} +2.64575i q^{7} +8.69234 q^{9} +15.7367i q^{11} +8.57685 q^{13} +2.53864i q^{15} +28.3197 q^{17} +6.33599i q^{19} +1.46752 q^{21} +31.0647i q^{23} -4.05242 q^{25} -9.81339i q^{27} -0.846294 q^{29} -21.6354i q^{31} +8.72866 q^{33} -12.1092i q^{35} -33.6637 q^{37} -4.75731i q^{39} +66.9757 q^{41} +44.8781i q^{43} -39.7836 q^{45} +38.4528i q^{47} -7.00000 q^{49} -15.7081i q^{51} -14.8174 q^{53} -72.0246i q^{55} +3.51438 q^{57} +5.80942i q^{59} -52.6015 q^{61} +22.9978i q^{63} -39.2550 q^{65} -117.397i q^{67} +17.2306 q^{69} -81.2543i q^{71} -47.8054 q^{73} +2.24775i q^{75} -41.6354 q^{77} +57.4900i q^{79} +72.7879 q^{81} -102.855i q^{83} -129.615 q^{85} +0.469413i q^{87} -89.2955 q^{89} +22.6922i q^{91} -12.0005 q^{93} -28.9989i q^{95} -3.44452 q^{97} +136.789i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 40 q^{9}+O(q^{10}) $ 8 * q - 40 * q^9	$ 8 q - 40 q^{9} + 32 q^{13} - 16 q^{17} + 104 q^{25} - 80 q^{29} - 176 q^{37} + 144 q^{41} + 256 q^{45} - 56 q^{49} + 48 q^{53} - 400 q^{57} - 192 q^{61} - 304 q^{65} + 576 q^{69} + 272 q^{73} - 112 q^{77} + 504 q^{81} - 160 q^{85} - 80 q^{89} - 608 q^{93} + 528 q^{97}+O(q^{100}) $ 8 * q - 40 * q^9 + 32 * q^13 - 16 * q^17 + 104 * q^25 - 80 * q^29 - 176 * q^37 + 144 * q^41 + 256 * q^45 - 56 * q^49 + 48 * q^53 - 400 * q^57 - 192 * q^61 - 304 * q^65 + 576 * q^69 + 272 * q^73 - 112 * q^77 + 504 * q^81 - 160 * q^85 - 80 * q^89 - 608 * q^93 + 528 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 0.554669i	− 0.184890i	−0.995718	−	0.0924448i	$-0.970532\pi$
$3$	− 0.554669i	− 0.184890i	0.995718	−	0.0924448i	$-0.0294682\pi$
$4$	0	0
$5$	−4.57685	−0.915371	−0.457685	−	0.889114i	$-0.651321\pi$
$5$	−4.57685	−0.915371	−0.457685	+	0.889114i	$0.651321\pi$
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	0	0
$9$	8.69234	0.965816
$10$	0	0
$11$	15.7367i	1.43061i	0.698812	+	0.715305i	$0.253712\pi$
$11$	15.7367i	1.43061i	−0.698812	+	0.715305i	$0.746288\pi$
$12$	0	0
$13$	8.57685	0.659758	0.329879	−	0.944023i	$-0.392992\pi$
$13$	8.57685	0.659758	0.329879	+	0.944023i	$0.392992\pi$
$14$	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
$19$	6.33599i	0.333473i	0.986001	+	0.166737i	$0.0533230\pi$
$19$	6.33599i	0.333473i	−0.986001	+	0.166737i	$0.946677\pi$
$20$	0	0
$21$	1.46752	0.0698817
$22$	0	0
$23$	31.0647i	1.35064i	0.737525	+	0.675320i	$0.235995\pi$
$23$	31.0647i	1.35064i	−0.737525	+	0.675320i	$0.764005\pi$
$24$	0	0
$25$	−4.05242	−0.162097
$26$	0	0
$27$	− 9.81339i	− 0.363459i
$28$	0	0
$29$	−0.846294	−0.0291826	−0.0145913	−	0.999894i	$-0.504645\pi$
$29$	−0.846294	−0.0291826	−0.0145913	+	0.999894i	$0.504645\pi$
$30$	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
$35$	− 12.1092i	− 0.345978i
$36$	0	0
$37$	−33.6637	−0.909830	−0.454915	−	0.890535i	$-0.650330\pi$
$37$	−33.6637	−0.909830	−0.454915	+	0.890535i	$0.650330\pi$
$38$	0	0
$39$	− 4.75731i	− 0.121982i
$40$	0	0
$41$	66.9757	1.63355	0.816777	−	0.576953i	$-0.195758\pi$
$41$	66.9757	1.63355	0.816777	+	0.576953i	$0.195758\pi$
$42$	0	0
$43$	44.8781i	1.04368i	0.853044	+	0.521839i	$0.174754\pi$
$43$	44.8781i	1.04368i	−0.853044	+	0.521839i	$0.825246\pi$
$44$	0	0
$45$	−39.7836	−0.884079
$46$	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
$51$	− 15.7081i	− 0.308001i
$52$	0	0
$53$	−14.8174	−0.279574	−0.139787	−	0.990182i	$-0.544642\pi$
$53$	−14.8174	−0.279574	−0.139787	+	0.990182i	$0.544642\pi$
$54$	0	0
$55$	− 72.0246i	− 1.30954i
$56$	0	0
$57$	3.51438	0.0616557
$58$	0	0
$59$	5.80942i	0.0984647i	0.998787	+	0.0492323i	$0.0156775\pi$
$59$	5.80942i	0.0984647i	−0.998787	+	0.0492323i	$0.984323\pi$
$60$	0	0
$61$	−52.6015	−0.862319	−0.431160	−	0.902276i	$-0.641895\pi$
$61$	−52.6015	−0.862319	−0.431160	+	0.902276i	$0.641895\pi$
$62$	0	0
$63$	22.9978i	0.365044i
$64$	0	0
$65$	−39.2550	−0.603923
$66$	0	0
$67$	− 117.397i	− 1.75219i	−0.482138	−	0.876095i	$-0.660140\pi$
$67$	− 117.397i	− 1.75219i	0.482138	−	0.876095i	$-0.339860\pi$
$68$	0	0
$69$	17.2306	0.249719
$70$	0	0
$71$	− 81.2543i	− 1.14443i	−0.820105	−	0.572213i	$-0.806085\pi$
$71$	− 81.2543i	− 1.14443i	0.820105	−	0.572213i	$-0.193915\pi$
$72$	0	0
$73$	−47.8054	−0.654869	−0.327435	−	0.944874i	$-0.606184\pi$
$73$	−47.8054	−0.654869	−0.327435	+	0.944874i	$0.606184\pi$
$74$	0	0
$75$	2.24775i	0.0299700i
$76$	0	0
$77$	−41.6354	−0.540720
$78$	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
$83$	− 102.855i	− 1.23921i	−0.784913	−	0.619606i	$-0.787292\pi$
$83$	− 102.855i	− 1.23921i	0.784913	−	0.619606i	$-0.212708\pi$
$84$	0	0
$85$	−129.615	−1.52488
$86$	0	0
$87$	0.469413i	0.00539555i
$88$	0	0
$89$	−89.2955	−1.00332	−0.501660	−	0.865065i	$-0.667277\pi$
$89$	−89.2955	−1.00332	−0.501660	+	0.865065i	$0.667277\pi$
$90$	0	0
$91$	22.6922i	0.249365i
$92$	0	0
$93$	−12.0005	−0.129038
$94$	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
$99$	136.789i	1.38171i
$100$	0	0
$101$	143.034	1.41618	0.708089	−	0.706123i	$-0.249557\pi$
$101$	143.034	1.41618	0.708089	+	0.706123i	$0.249557\pi$
$102$	0	0
$103$	− 173.424i	− 1.68373i	−0.539688	−	0.841865i	$-0.681458\pi$
$103$	− 173.424i	− 1.68373i	0.539688	−	0.841865i	$-0.318542\pi$
$104$	0	0
$105$	−6.71660	−0.0639676
$106$	0	0
$107$	95.5581i	0.893067i	0.894767	+	0.446533i	$0.147342\pi$
$107$	95.5581i	0.893067i	−0.894767	+	0.446533i	$0.852658\pi$
$108$	0	0
$109$	185.517	1.70199	0.850997	−	0.525170i	$-0.175998\pi$
$109$	185.517	1.70199	0.850997	+	0.525170i	$0.175998\pi$
$110$	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
$115$	− 142.179i	− 1.23634i
$116$	0	0
$117$	74.5529	0.637205
$118$	0	0
$119$	74.9269i	0.629638i
$120$	0	0
$121$	−126.644	−1.04665
$122$	0	0
$123$	− 37.1493i	− 0.302027i
$124$	0	0
$125$	132.969	1.06375
$126$	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
$131$	259.412i	1.98025i	0.140200	+	0.990123i	$0.455225\pi$
$131$	259.412i	1.98025i	−0.140200	+	0.990123i	$0.544775\pi$
$132$	0	0
$133$	−16.7635	−0.126041
$134$	0	0
$135$	44.9144i	0.332700i
$136$	0	0
$137$	3.07777	0.0224654	0.0112327	−	0.999937i	$-0.496424\pi$
$137$	3.07777	0.0224654	0.0112327	+	0.999937i	$0.496424\pi$
$138$	0	0
$139$	− 90.3041i	− 0.649670i	−0.945771	−	0.324835i	$-0.894691\pi$
$139$	− 90.3041i	− 0.649670i	0.945771	−	0.324835i	$-0.105309\pi$
$140$	0	0
$141$	21.3286	0.151266
$142$	0	0
$143$	134.971i	0.943856i
$144$	0	0
$145$	3.87336	0.0267129
$146$	0	0
$147$	3.88268i	0.0264128i
$148$	0	0
$149$	41.3766	0.277695	0.138848	−	0.990314i	$-0.455660\pi$
$149$	41.3766	0.277695	0.138848	+	0.990314i	$0.455660\pi$
$150$	0	0
$151$	41.5182i	0.274955i	0.990505	+	0.137477i	$0.0438994\pi$
$151$	41.5182i	0.274955i	−0.990505	+	0.137477i	$0.956101\pi$
$152$	0	0
$153$	246.165	1.60892
$154$	0	0
$155$	99.0221i	0.638853i
$156$	0	0
$157$	−26.9848	−0.171878	−0.0859389	−	0.996300i	$-0.527389\pi$
$157$	−26.9848	−0.171878	−0.0859389	+	0.996300i	$0.527389\pi$
$158$	0	0
$159$	8.21875i	0.0516902i
$160$	0	0
$161$	−82.1895	−0.510494
$162$	0	0
$163$	− 126.684i	− 0.777200i	−0.921407	−	0.388600i	$-0.872959\pi$
$163$	− 126.684i	− 0.777200i	0.921407	−	0.388600i	$-0.127041\pi$
$164$	0	0
$165$	−39.9498	−0.242120
$166$	0	0
$167$	− 246.243i	− 1.47451i	−0.675615	−	0.737254i	$-0.736122\pi$
$167$	− 246.243i	− 1.47451i	0.675615	−	0.737254i	$-0.263878\pi$
$168$	0	0
$169$	−95.4376	−0.564720
$170$	0	0
$171$	55.0746i	0.322074i
$172$	0	0
$173$	74.5776	0.431084	0.215542	−	0.976495i	$-0.430848\pi$
$173$	74.5776	0.431084	0.215542	+	0.976495i	$0.430848\pi$
$174$	0	0
$175$	− 10.7217i	− 0.0612668i
$176$	0	0
$177$	3.22230	0.0182051
$178$	0	0
$179$	− 201.073i	− 1.12331i	−0.827371	−	0.561656i	$-0.810165\pi$
$179$	− 201.073i	− 1.12331i	0.827371	−	0.561656i	$-0.189835\pi$
$180$	0	0
$181$	252.696	1.39611	0.698054	−	0.716045i	$-0.254050\pi$
$181$	252.696	1.39611	0.698054	+	0.716045i	$0.254050\pi$
$182$	0	0
$183$	29.1764i	0.159434i
$184$	0	0
$185$	154.074	0.832831
$186$	0	0
$187$	445.659i	2.38320i
$188$	0	0
$189$	25.9638	0.137375
$190$	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
$195$	21.7735i	0.111659i
$196$	0	0
$197$	276.248	1.40227	0.701137	−	0.713026i	$-0.252676\pi$
$197$	276.248	1.40227	0.701137	+	0.713026i	$0.252676\pi$
$198$	0	0
$199$	− 58.5094i	− 0.294017i	−0.989135	−	0.147008i	$-0.953036\pi$
$199$	− 58.5094i	− 0.294017i	0.989135	−	0.147008i	$-0.0469644\pi$
$200$	0	0
$201$	−65.1163	−0.323962
$202$	0	0
$203$	− 2.23908i	− 0.0110300i
$204$	0	0
$205$	−306.538	−1.49531
$206$	0	0
$207$	270.025i	1.30447i
$208$	0	0
$209$	−99.7077	−0.477070
$210$	0	0
$211$	− 146.237i	− 0.693068i	−0.938037	−	0.346534i	$-0.887359\pi$
$211$	− 146.237i	− 0.693068i	0.938037	−	0.346534i	$-0.112641\pi$
$212$	0	0
$213$	−45.0692	−0.211592
$214$	0	0
$215$	− 205.401i	− 0.955351i
$216$	0	0
$217$	57.2420	0.263788
$218$	0	0
$219$	26.5162i	0.121078i
$220$	0	0
$221$	242.894	1.09907
$222$	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
$227$	− 271.103i	− 1.19429i	−0.802134	−	0.597144i	$-0.796302\pi$
$227$	− 271.103i	− 1.19429i	0.802134	−	0.597144i	$-0.203698\pi$
$228$	0	0
$229$	−400.572	−1.74922	−0.874611	−	0.484826i	$-0.838883\pi$
$229$	−400.572	−1.74922	−0.874611	+	0.484826i	$0.838883\pi$
$230$	0	0
$231$	23.0939i	0.0999734i
$232$	0	0
$233$	−257.611	−1.10563	−0.552813	−	0.833305i	$-0.686446\pi$
$233$	−257.611	−1.10563	−0.552813	+	0.833305i	$0.686446\pi$
$234$	0	0
$235$	− 175.993i	− 0.748906i
$236$	0	0
$237$	31.8879	0.134548
$238$	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
$243$	− 128.694i	− 0.529604i
$244$	0	0
$245$	32.0380	0.130767
$246$	0	0
$247$	54.3429i	0.220012i
$248$	0	0
$249$	−57.0502	−0.229117
$250$	0	0
$251$	236.051i	0.940443i	0.882549	+	0.470221i	$0.155826\pi$
$251$	236.051i	0.940443i	−0.882549	+	0.470221i	$0.844174\pi$
$252$	0	0
$253$	−488.857	−1.93224
$254$	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
$259$	− 89.0658i	− 0.343883i
$260$	0	0
$261$	−7.35628	−0.0281850
$262$	0	0
$263$	88.2675i	0.335618i	0.985820	+	0.167809i	$0.0536691\pi$
$263$	88.2675i	0.335618i	−0.985820	+	0.167809i	$0.946331\pi$
$264$	0	0
$265$	67.8170	0.255913
$266$	0	0
$267$	49.5294i	0.185503i
$268$	0	0
$269$	21.9135	0.0814629	0.0407314	−	0.999170i	$-0.487031\pi$
$269$	21.9135	0.0814629	0.0407314	+	0.999170i	$0.487031\pi$
$270$	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
$275$	− 63.7717i	− 0.231897i
$276$	0	0
$277$	−457.076	−1.65009	−0.825046	−	0.565065i	$-0.808851\pi$
$277$	−457.076	−1.65009	−0.825046	+	0.565065i	$0.808851\pi$
$278$	0	0
$279$	− 188.063i	− 0.674059i
$280$	0	0
$281$	95.5032	0.339869	0.169934	−	0.985455i	$-0.445644\pi$
$281$	95.5032	0.339869	0.169934	+	0.985455i	$0.445644\pi$
$282$	0	0
$283$	− 131.804i	− 0.465737i	−0.972508	−	0.232869i	$-0.925189\pi$
$283$	− 131.804i	− 0.465737i	0.972508	−	0.232869i	$-0.0748112\pi$
$284$	0	0
$285$	−16.0848	−0.0564379
$286$	0	0
$287$	177.201i	0.617426i
$288$	0	0
$289$	513.006	1.77511
$290$	0	0
$291$	1.91057i	0.00656553i
$292$	0	0
$293$	−325.183	−1.10984	−0.554920	−	0.831904i	$-0.687251\pi$
$293$	−325.183	−1.10984	−0.554920	+	0.831904i	$0.687251\pi$
$294$	0	0
$295$	− 26.5888i	− 0.0901317i
$296$	0	0
$297$	154.430	0.519968
$298$	0	0
$299$	266.438i	0.891096i
$300$	0	0
$301$	−118.736	−0.394473
$302$	0	0
$303$	− 79.3365i	− 0.261837i
$304$	0	0
$305$	240.749	0.789341
$306$	0	0
$307$	− 34.3658i	− 0.111941i	−0.998432	−	0.0559704i	$-0.982175\pi$
$307$	− 34.3658i	− 0.111941i	0.998432	−	0.0559704i	$-0.0178252\pi$
$308$	0	0
$309$	−96.1930	−0.311304
$310$	0	0
$311$	195.190i	0.627620i	0.949486	+	0.313810i	$0.101606\pi$
$311$	195.190i	0.627620i	−0.949486	+	0.313810i	$0.898394\pi$
$312$	0	0
$313$	19.8987	0.0635740	0.0317870	−	0.999495i	$-0.489880\pi$
$313$	19.8987	0.0635740	0.0317870	+	0.999495i	$0.489880\pi$
$314$	0	0
$315$	− 105.257i	− 0.334151i
$316$	0	0
$317$	168.672	0.532088	0.266044	−	0.963961i	$-0.414283\pi$
$317$	168.672	0.532088	0.266044	+	0.963961i	$0.414283\pi$
$318$	0	0
$319$	− 13.3179i	− 0.0417489i
$320$	0	0
$321$	53.0031	0.165119
$322$	0	0
$323$	179.434i	0.555522i
$324$	0	0
$325$	−34.7570	−0.106945
$326$	0	0
$327$	− 102.901i	− 0.314681i
$328$	0	0
$329$	−101.737	−0.309230
$330$	0	0
$331$	− 603.602i	− 1.82357i	−0.410666	−	0.911786i	$-0.634704\pi$
$331$	− 603.602i	− 1.82357i	0.410666	−	0.911786i	$-0.365296\pi$
$332$	0	0
$333$	−292.616	−0.878728
$334$	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
$339$	57.3578i	0.169197i
$340$	0	0
$341$	340.470	0.998447
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	0	0
$345$	−78.8621	−0.228586
$346$	0	0
$347$	− 81.4988i	− 0.234867i	−0.993081	−	0.117433i	$-0.962533\pi$
$347$	− 81.4988i	− 0.234867i	0.993081	−	0.117433i	$-0.0374667\pi$
$348$	0	0
$349$	−223.673	−0.640898	−0.320449	−	0.947266i	$-0.603834\pi$
$349$	−223.673	−0.640898	−0.320449	+	0.947266i	$0.603834\pi$
$350$	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
$355$	371.889i	1.04757i
$356$	0	0
$357$	41.5596	0.116414
$358$	0	0
$359$	− 79.0559i	− 0.220212i	−0.993920	−	0.110106i	$-0.964881\pi$
$359$	− 79.0559i	− 0.220212i	0.993920	−	0.110106i	$-0.0351190\pi$
$360$	0	0
$361$	320.855	0.888796
$362$	0	0
$363$	70.2455i	0.193514i
$364$	0	0
$365$	218.798	0.599448
$366$	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
$371$	− 39.2031i	− 0.105669i
$372$	0	0
$373$	250.574	0.671781	0.335890	−	0.941901i	$-0.390963\pi$
$373$	250.574	0.671781	0.335890	+	0.941901i	$0.390963\pi$
$374$	0	0
$375$	− 73.7535i	− 0.196676i
$376$	0	0
$377$	−7.25854	−0.0192534
$378$	0	0
$379$	532.859i	1.40596i	0.711209	+	0.702981i	$0.248148\pi$
$379$	532.859i	1.40596i	−0.711209	+	0.702981i	$0.751852\pi$
$380$	0	0
$381$	82.0745	0.215419
$382$	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
$387$	390.096i	1.00800i
$388$	0	0
$389$	174.837	0.449452	0.224726	−	0.974422i	$-0.427851\pi$
$389$	174.837	0.449452	0.224726	+	0.974422i	$0.427851\pi$
$390$	0	0
$391$	879.744i	2.24999i
$392$	0	0
$393$	143.888	0.366127
$394$	0	0
$395$	− 263.123i	− 0.666135i
$396$	0	0
$397$	129.750	0.326826	0.163413	−	0.986558i	$-0.447750\pi$
$397$	129.750	0.326826	0.163413	+	0.986558i	$0.447750\pi$
$398$	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
$403$	− 185.564i	− 0.460456i
$404$	0	0
$405$	−333.140	−0.822567
$406$	0	0
$407$	− 529.756i	− 1.30161i
$408$	0	0
$409$	243.575	0.595538	0.297769	−	0.954638i	$-0.403758\pi$
$409$	243.575	0.595538	0.297769	+	0.954638i	$0.403758\pi$
$410$	0	0
$411$	− 1.70714i	− 0.00415363i
$412$	0	0
$413$	−15.3703	−0.0372161
$414$	0	0
$415$	470.750i	1.13434i
$416$	0	0
$417$	−50.0888	−0.120117
$418$	0	0
$419$	499.348i	1.19176i	0.803073	+	0.595881i	$0.203197\pi$
$419$	499.348i	1.19176i	−0.803073	+	0.595881i	$0.796803\pi$
$420$	0	0
$421$	537.031	1.27561	0.637804	−	0.770199i	$-0.279843\pi$
$421$	537.031	1.27561	0.637804	+	0.770199i	$0.279843\pi$
$422$	0	0
$423$	334.245i	0.790178i
$424$	0	0
$425$	−114.763	−0.270031
$426$	0	0
$427$	− 139.170i	− 0.325926i
$428$	0	0
$429$	74.8644	0.174509
$430$	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
$435$	− 2.14843i	− 0.00493893i
$436$	0	0
$437$	−196.826	−0.450403
$438$	0	0
$439$	346.809i	0.789998i	0.918681	+	0.394999i	$0.129255\pi$
$439$	346.809i	0.789998i	−0.918681	+	0.394999i	$0.870745\pi$
$440$	0	0
$441$	−60.8464	−0.137974
$442$	0	0
$443$	− 369.535i	− 0.834166i	−0.908868	−	0.417083i	$-0.863052\pi$
$443$	− 369.535i	− 0.834166i	0.908868	−	0.417083i	$-0.136948\pi$
$444$	0	0
$445$	408.692	0.918409
$446$	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
$451$	1053.98i	2.33698i
$452$	0	0
$453$	23.0288	0.0508363
$454$	0	0
$455$	− 103.859i	− 0.228261i
$456$	0	0
$457$	781.559	1.71019	0.855097	−	0.518468i	$-0.173497\pi$
$457$	781.559	1.71019	0.855097	+	0.518468i	$0.173497\pi$
$458$	0	0
$459$	− 277.912i	− 0.605474i
$460$	0	0
$461$	526.503	1.14209	0.571044	−	0.820919i	$-0.306539\pi$
$461$	526.503	1.14209	0.571044	+	0.820919i	$0.306539\pi$
$462$	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
$467$	− 480.381i	− 1.02865i	−0.857594	−	0.514326i	$-0.828042\pi$
$467$	− 480.381i	− 1.02865i	0.857594	−	0.514326i	$-0.171958\pi$
$468$	0	0
$469$	310.603	0.662266
$470$	0	0
$471$	14.9676i	0.0317784i
$472$	0	0
$473$	−706.234	−1.49309
$474$	0	0
$475$	− 25.6761i	− 0.0540549i
$476$	0	0
$477$	−128.798	−0.270017
$478$	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
$483$	45.5880i	0.0943850i
$484$	0	0
$485$	15.7651	0.0325053
$486$	0	0
$487$	517.867i	1.06338i	0.846939	+	0.531691i	$0.178443\pi$
$487$	517.867i	1.06338i	−0.846939	+	0.531691i	$0.821557\pi$
$488$	0	0
$489$	−70.2674	−0.143696
$490$	0	0
$491$	− 385.683i	− 0.785505i	−0.919644	−	0.392753i	$-0.871523\pi$
$491$	− 385.683i	− 0.785505i	0.919644	−	0.392753i	$-0.128477\pi$
$492$	0	0
$493$	−23.9668	−0.0486142
$494$	0	0
$495$	− 626.063i	− 1.26477i
$496$	0	0
$497$	214.979	0.432552
$498$	0	0
$499$	− 726.518i	− 1.45595i	−0.685605	−	0.727974i	$-0.740462\pi$
$499$	− 726.518i	− 1.45595i	0.685605	−	0.727974i	$-0.259538\pi$
$500$	0	0
$501$	−136.583	−0.272621
$502$	0	0
$503$	− 253.383i	− 0.503743i	−0.967761	−	0.251872i	$-0.918954\pi$
$503$	− 253.383i	− 0.503743i	0.967761	−	0.251872i	$-0.0810461\pi$
$504$	0	0
$505$	−654.646	−1.29633
$506$	0	0
$507$	52.9362i	0.104411i
$508$	0	0
$509$	−338.344	−0.664723	−0.332361	−	0.943152i	$-0.607845\pi$
$509$	−338.344	−0.664723	−0.332361	+	0.943152i	$0.607845\pi$
$510$	0	0
$511$	− 126.481i	− 0.247517i
$512$	0	0
$513$	62.1776	0.121204
$514$	0	0
$515$	793.737i	1.54124i
$516$	0	0
$517$	−605.121	−1.17045
$518$	0	0
$519$	− 41.3658i	− 0.0797030i
$520$	0	0
$521$	−333.599	−0.640305	−0.320152	−	0.947366i	$-0.603734\pi$
$521$	−333.599	−0.640305	−0.320152	+	0.947366i	$0.603734\pi$
$522$	0	0
$523$	38.7942i	0.0741763i	0.999312	+	0.0370882i	$0.0118082\pi$
$523$	38.7942i	0.0741763i	−0.999312	+	0.0370882i	$0.988192\pi$
$524$	0	0
$525$	−5.94699	−0.0113276
$526$	0	0
$527$	− 612.709i	− 1.16264i
$528$	0	0
$529$	−436.017	−0.824229
$530$	0	0
$531$	50.4974i	0.0950987i
$532$	0	0
$533$	574.441	1.07775
$534$	0	0
$535$	− 437.355i	− 0.817487i
$536$	0	0
$537$	−111.529	−0.207689
$538$	0	0
$539$	− 110.157i	− 0.204373i
$540$	0	0
$541$	543.111	1.00390	0.501951	−	0.864896i	$-0.332616\pi$
$541$	543.111	1.00390	0.501951	+	0.864896i	$0.332616\pi$
$542$	0	0
$543$	− 140.162i	− 0.258126i
$544$	0	0
$545$	−849.086	−1.55796
$546$	0	0
$547$	262.532i	0.479949i	0.970779	+	0.239974i	$0.0771391\pi$
$547$	262.532i	0.479949i	−0.970779	+	0.239974i	$0.922861\pi$
$548$	0	0
$549$	−457.230	−0.832841
$550$	0	0
$551$	− 5.36212i	− 0.00973161i
$552$	0	0
$553$	−152.104	−0.275053
$554$	0	0
$555$	− 85.4599i	− 0.153982i
$556$	0	0
$557$	−652.276	−1.17105	−0.585526	−	0.810654i	$-0.699112\pi$
$557$	−652.276	−1.17105	−0.585526	+	0.810654i	$0.699112\pi$
$558$	0	0
$559$	384.913i	0.688574i
$560$	0	0
$561$	247.193	0.440630
$562$	0	0
$563$	− 993.435i	− 1.76454i	−0.470746	−	0.882269i	$-0.656015\pi$
$563$	− 993.435i	− 1.76454i	0.470746	−	0.882269i	$-0.343985\pi$
$564$	0	0
$565$	473.288	0.837678
$566$	0	0
$567$	192.579i	0.339645i
$568$	0	0
$569$	−450.590	−0.791898	−0.395949	−	0.918272i	$-0.629584\pi$
$569$	−450.590	−0.791898	−0.395949	+	0.918272i	$0.629584\pi$
$570$	0	0
$571$	373.493i	0.654103i	0.945007	+	0.327051i	$0.106055\pi$
$571$	373.493i	0.654103i	−0.945007	+	0.327051i	$0.893945\pi$
$572$	0	0
$573$	184.335	0.321701
$574$	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
$579$	29.5205i	0.0509853i
$580$	0	0
$581$	272.128	0.468378
$582$	0	0
$583$	− 233.177i	− 0.399961i
$584$	0	0
$585$	−341.218	−0.583278
$586$	0	0
$587$	182.976i	0.311714i	0.987780	+	0.155857i	$0.0498139\pi$
$587$	182.976i	0.311714i	−0.987780	+	0.155857i	$0.950186\pi$
$588$	0	0
$589$	137.082	0.232737
$590$	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
$595$	− 342.930i	− 0.576352i
$596$	0	0
$597$	−32.4533	−0.0543607
$598$	0	0
$599$	219.331i	0.366163i	0.983098	+	0.183081i	$0.0586072\pi$
$599$	219.331i	0.366163i	−0.983098	+	0.183081i	$0.941393\pi$
$600$	0	0
$601$	268.193	0.446245	0.223122	−	0.974790i	$-0.428375\pi$
$601$	268.193	0.446245	0.223122	+	0.974790i	$0.428375\pi$
$602$	0	0
$603$	− 1020.45i	− 1.69229i
$604$	0	0
$605$	579.631	0.958068
$606$	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
$611$	329.804i	0.539778i
$612$	0	0
$613$	29.3420	0.0478662	0.0239331	−	0.999714i	$-0.492381\pi$
$613$	29.3420	0.0478662	0.0239331	+	0.999714i	$0.492381\pi$
$614$	0	0
$615$	170.027i	0.276467i
$616$	0	0
$617$	−413.079	−0.669497	−0.334748	−	0.942308i	$-0.608651\pi$
$617$	−413.079	−0.669497	−0.334748	+	0.942308i	$0.608651\pi$
$618$	0	0
$619$	− 225.596i	− 0.364453i	−0.983257	−	0.182226i	$-0.941670\pi$
$619$	− 225.596i	− 0.364453i	0.983257	−	0.182226i	$-0.0583304\pi$
$620$	0	0
$621$	304.850	0.490902
$622$	0	0
$623$	− 236.254i	− 0.379219i
$624$	0	0
$625$	−507.267	−0.811628
$626$	0	0
$627$	55.3047i	0.0882053i
$628$	0	0
$629$	−953.346	−1.51565
$630$	0	0
$631$	914.619i	1.44948i	0.689025	+	0.724738i	$0.258039\pi$
$631$	914.619i	1.44948i	−0.689025	+	0.724738i	$0.741961\pi$
$632$	0	0
$633$	−81.1132	−0.128141
$634$	0	0
$635$	− 677.239i	− 1.06652i
$636$	0	0
$637$	−60.0380	−0.0942511
$638$	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
$643$	864.377i	1.34429i	0.740421	+	0.672144i	$0.234626\pi$
$643$	864.377i	1.34429i	−0.740421	+	0.672144i	$0.765374\pi$
$644$	0	0
$645$	−113.929	−0.176634
$646$	0	0
$647$	689.240i	1.06529i	0.846340	+	0.532643i	$0.178801\pi$
$647$	689.240i	1.06529i	−0.846340	+	0.532643i	$0.821199\pi$
$648$	0	0
$649$	−91.4211	−0.140865
$650$	0	0
$651$	− 31.7503i	− 0.0487716i
$652$	0	0
$653$	−565.975	−0.866730	−0.433365	−	0.901218i	$-0.642674\pi$
$653$	−565.975	−0.866730	−0.433365	+	0.901218i	$0.642674\pi$
$654$	0	0
$655$	− 1187.29i	− 1.81266i
$656$	0	0
$657$	−415.541	−0.632483
$658$	0	0
$659$	− 289.064i	− 0.438640i	−0.975653	−	0.219320i	$-0.929616\pi$
$659$	− 289.064i	− 0.438640i	0.975653	−	0.219320i	$-0.0703838\pi$
$660$	0	0
$661$	283.787	0.429329	0.214665	−	0.976688i	$-0.431134\pi$
$661$	283.787	0.429329	0.214665	+	0.976688i	$0.431134\pi$
$662$	0	0
$663$	− 134.726i	− 0.203206i
$664$	0	0
$665$	76.7239	0.115374
$666$	0	0
$667$	− 26.2899i	− 0.0394151i
$668$	0	0
$669$	−234.231	−0.350121
$670$	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
$675$	39.7680i	0.0589155i
$676$	0	0
$677$	787.301	1.16293	0.581463	−	0.813573i	$-0.302481\pi$
$677$	787.301	1.16293	0.581463	+	0.813573i	$0.302481\pi$
$678$	0	0
$679$	− 9.11336i	− 0.0134217i
$680$	0	0
$681$	−150.373	−0.220811
$682$	0	0
$683$	193.168i	0.282823i	0.989951	+	0.141411i	$0.0451640\pi$
$683$	193.168i	0.282823i	−0.989951	+	0.141411i	$0.954836\pi$
$684$	0	0
$685$	−14.0865	−0.0205642
$686$	0	0
$687$	222.185i	0.323413i
$688$	0	0
$689$	−127.087	−0.184451
$690$	0	0
$691$	967.147i	1.39963i	0.714322	+	0.699817i	$0.246735\pi$
$691$	967.147i	1.39963i	−0.714322	+	0.699817i	$0.753265\pi$
$692$	0	0
$693$	−361.909	−0.522236
$694$	0	0
$695$	413.308i	0.594688i
$696$	0	0
$697$	1896.73	2.72128
$698$	0	0
$699$	142.889i	0.204419i
$700$	0	0
$701$	−246.322	−0.351386	−0.175693	−	0.984445i	$-0.556217\pi$
$701$	−246.322	−0.351386	−0.175693	+	0.984445i	$0.556217\pi$
$702$	0	0
$703$	− 213.293i	− 0.303404i
$704$	0	0
$705$	−97.6178	−0.138465
$706$	0	0
$707$	378.432i	0.535265i
$708$	0	0
$709$	1191.27	1.68021	0.840103	−	0.542428i	$-0.182495\pi$
$709$	1191.27	1.68021	0.840103	+	0.542428i	$0.182495\pi$
$710$	0	0
$711$	499.723i	0.702845i
$712$	0	0
$713$	672.098	0.942635
$714$	0	0
$715$	− 617.744i	− 0.863978i
$716$	0	0
$717$	−84.6703	−0.118090
$718$	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
$723$	26.1370i	0.0361507i
$724$	0	0
$725$	3.42954	0.00473040
$726$	0	0
$727$	931.815i	1.28173i	0.767655	+	0.640863i	$0.221423\pi$
$727$	931.815i	1.28173i	−0.767655	+	0.640863i	$0.778577\pi$
$728$	0	0
$729$	583.709	0.800698
$730$	0	0
$731$	1270.94i	1.73863i
$732$	0	0
$733$	−1426.30	−1.94584	−0.972918	−	0.231150i	$-0.925751\pi$
$733$	−1426.30	−1.94584	−0.972918	+	0.231150i	$0.925751\pi$
$734$	0	0
$735$	− 17.7705i	− 0.0241775i
$736$	0	0
$737$	1847.44	2.50670
$738$	0	0
$739$	216.382i	0.292804i	0.989225	+	0.146402i	$0.0467693\pi$
$739$	216.382i	0.292804i	−0.989225	+	0.146402i	$0.953231\pi$
$740$	0	0
$741$	30.1423	0.0406779
$742$	0	0
$743$	81.5074i	0.109700i	0.998495	+	0.0548502i	$0.0174681\pi$
$743$	81.5074i	0.109700i	−0.998495	+	0.0548502i	$0.982532\pi$
$744$	0	0
$745$	−189.375	−0.254194
$746$	0	0
$747$	− 894.048i	− 1.19685i
$748$	0	0
$749$	−252.823	−0.337547
$750$	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
$755$	− 190.023i	− 0.251686i
$756$	0	0
$757$	−372.503	−0.492078	−0.246039	−	0.969260i	$-0.579129\pi$
$757$	−372.503	−0.492078	−0.246039	+	0.969260i	$0.579129\pi$
$758$	0	0
$759$	271.153i	0.357251i
$760$	0	0
$761$	663.881	0.872380	0.436190	−	0.899855i	$-0.356328\pi$
$761$	663.881	0.872380	0.436190	+	0.899855i	$0.356328\pi$
$762$	0	0
$763$	490.833i	0.643294i
$764$	0	0
$765$	−1126.66	−1.47276
$766$	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
$771$	15.2824i	0.0198216i
$772$	0	0
$773$	−928.973	−1.20178	−0.600888	−	0.799333i	$-0.705186\pi$
$773$	−928.973	−1.20178	−0.600888	+	0.799333i	$0.705186\pi$
$774$	0	0
$775$	87.6758i	0.113130i
$776$	0	0
$777$	−49.4020	−0.0635804
$778$	0	0
$779$	424.358i	0.544747i
$780$	0	0
$781$	1278.67	1.63723
$782$	0	0
$783$	8.30502i	0.0106067i
$784$	0	0
$785$	123.506	0.157332
$786$	0	0
$787$	470.202i	0.597461i	0.954338	+	0.298730i	$0.0965631\pi$
$787$	470.202i	0.597461i	−0.954338	+	0.298730i	$0.903437\pi$
$788$	0	0
$789$	48.9592	0.0620522
$790$	0	0
$791$	− 273.595i	− 0.345884i
$792$	0	0
$793$	−451.155	−0.568922
$794$	0	0
$795$	− 37.6160i	− 0.0473157i
$796$	0	0
$797$	832.694	1.04479	0.522393	−	0.852705i	$-0.325040\pi$
$797$	832.694	1.04479	0.522393	+	0.852705i	$0.325040\pi$
$798$	0	0
$799$	1088.97i	1.36292i
$800$	0	0
$801$	−776.187	−0.969022
$802$	0	0
$803$	− 752.300i	− 0.936862i
$804$	0	0
$805$	376.169	0.467291
$806$	0	0
$807$	− 12.1547i	− 0.0150616i
$808$	0	0
$809$	976.293	1.20679	0.603395	−	0.797442i	$-0.293814\pi$
$809$	976.293	1.20679	0.603395	+	0.797442i	$0.293814\pi$
$810$	0	0
$811$	− 564.470i	− 0.696018i	−0.937491	−	0.348009i	$-0.886858\pi$
$811$	− 564.470i	− 0.696018i	0.937491	−	0.348009i	$-0.113142\pi$
$812$	0	0
$813$	−237.896	−0.292615
$814$	0	0
$815$	579.812i	0.711426i
$816$	0	0
$817$	−284.347	−0.348038
$818$	0	0
$819$	197.249i	0.240841i
$820$	0	0
$821$	779.033	0.948883	0.474441	−	0.880287i	$-0.342650\pi$
$821$	779.033	0.948883	0.474441	+	0.880287i	$0.342650\pi$
$822$	0	0
$823$	− 363.394i	− 0.441548i	−0.975325	−	0.220774i	$-0.929142\pi$
$823$	− 363.394i	− 0.441548i	0.975325	−	0.220774i	$-0.0708582\pi$
$824$	0	0
$825$	−35.3722	−0.0428754
$826$	0	0
$827$	620.477i	0.750275i	0.926969	+	0.375137i	$0.122404\pi$
$827$	620.477i	0.750275i	−0.926969	+	0.375137i	$0.877596\pi$
$828$	0	0
$829$	737.842	0.890038	0.445019	−	0.895521i	$-0.353197\pi$
$829$	737.842	0.890038	0.445019	+	0.895521i	$0.353197\pi$
$830$	0	0
$831$	253.525i	0.305085i
$832$	0	0
$833$	−198.238	−0.237981
$834$	0	0
$835$	1127.02i	1.34972i
$836$	0	0
$837$	−212.317	−0.253664
$838$	0	0
$839$	− 897.093i	− 1.06924i	−0.845092	−	0.534620i	$-0.820455\pi$
$839$	− 897.093i	− 1.06924i	0.845092	−	0.534620i	$-0.179545\pi$
$840$	0	0
$841$	−840.284	−0.999148
$842$	0	0
$843$	− 52.9726i	− 0.0628382i
$844$	0	0
$845$	436.804	0.516928
$846$	0	0
$847$	− 335.069i	− 0.395595i
$848$	0	0
$849$	−73.1074	−0.0861100
$850$	0	0
$851$	− 1045.75i	− 1.22885i
$852$	0	0
$853$	−1470.98	−1.72448	−0.862239	−	0.506501i	$-0.830939\pi$
$853$	−1470.98	−1.72448	−0.862239	+	0.506501i	$0.830939\pi$
$854$	0	0
$855$	− 252.068i	− 0.294817i
$856$	0	0
$857$	851.862	0.994005	0.497002	−	0.867749i	$-0.334434\pi$
$857$	851.862	0.994005	0.497002	+	0.867749i	$0.334434\pi$
$858$	0	0
$859$	− 140.400i	− 0.163446i	−0.996655	−	0.0817229i	$-0.973958\pi$
$859$	− 140.400i	− 0.163446i	0.996655	−	0.0817229i	$-0.0260422\pi$
$860$	0	0
$861$	98.2879	0.114156
$862$	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
$867$	− 284.549i	− 0.328199i
$868$	0	0
$869$	−904.704	−1.04109
$870$	0	0
$871$	− 1006.89i	− 1.15602i
$872$	0	0
$873$	−29.9410	−0.0342967
$874$	0	0
$875$	351.802i	0.402059i
$876$	0	0
$877$	−1039.93	−1.18578	−0.592888	−	0.805285i	$-0.702012\pi$
$877$	−1039.93	−1.18578	−0.592888	+	0.805285i	$0.702012\pi$
$878$	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
$883$	− 231.281i	− 0.261926i	−0.991387	−	0.130963i	$-0.958193\pi$
$883$	− 231.281i	− 0.261926i	0.991387	−	0.130963i	$-0.0418069\pi$
$884$	0	0
$885$	−14.7480	−0.0166644
$886$	0	0
$887$	− 6.35223i	− 0.00716148i	−0.999994	−	0.00358074i	$-0.998860\pi$
$887$	− 6.35223i	− 0.00716148i	0.999994	−	0.00358074i	$-0.00113979\pi$
$888$	0	0
$889$	−391.493	−0.440374
$890$	0	0
$891$	1145.44i	1.28557i
$892$	0	0
$893$	−243.637	−0.272830
$894$	0	0
$895$	920.280i	1.02825i
$896$	0	0
$897$	147.785	0.164754
$898$	0	0
$899$	18.3099i	0.0203670i
$900$	0	0
$901$	−419.624	−0.465732
$902$	0	0
$903$	65.8593i	0.0729339i
$904$	0	0
$905$	−1156.55	−1.27796
$906$	0	0
$907$	− 201.716i	− 0.222399i	−0.993798	−	0.111200i	$-0.964531\pi$
$907$	− 201.716i	− 0.222399i	0.993798	−	0.111200i	$-0.0354693\pi$
$908$	0	0
$909$	1243.30	1.36777
$910$	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
$915$	− 133.536i	− 0.145941i
$916$	0	0
$917$	−686.340	−0.748463
$918$	0	0
$919$	− 650.067i	− 0.707363i	−0.935366	−	0.353681i	$-0.884930\pi$
$919$	− 650.067i	− 0.707363i	0.935366	−	0.353681i	$-0.115070\pi$
$920$	0	0
$921$	−19.0616	−0.0206967
$922$	0	0
$923$	− 696.906i	− 0.755044i
$924$	0	0
$925$	136.419	0.147480
$926$	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
$931$	− 44.3520i	− 0.0476391i
$932$	0	0
$933$	108.266	0.116040
$934$	0	0
$935$	− 2039.72i	− 2.18151i
$936$	0	0
$937$	1579.17	1.68535	0.842676	−	0.538422i	$-0.180979\pi$
$937$	1579.17	1.68535	0.842676	+	0.538422i	$0.180979\pi$
$938$	0	0
$939$	− 11.0372i	− 0.0117542i
$940$	0	0
$941$	−1080.68	−1.14844	−0.574218	−	0.818703i	$-0.694694\pi$
$941$	−1080.68	−1.14844	−0.574218	+	0.818703i	$0.694694\pi$
$942$	0	0
$943$	2080.58i	2.20634i
$944$	0	0
$945$	−118.832	−0.125749
$946$	0	0
$947$	− 538.713i	− 0.568863i	−0.958696	−	0.284431i	$-0.908195\pi$
$947$	− 538.713i	− 0.568863i	0.958696	−	0.284431i	$-0.0918048\pi$
$948$	0	0
$949$	−410.020	−0.432055
$950$	0	0
$951$	− 93.5570i	− 0.0983775i
$952$	0	0
$953$	−1099.01	−1.15322	−0.576608	−	0.817021i	$-0.695624\pi$
$953$	−1099.01	−1.15322	−0.576608	+	0.817021i	$0.695624\pi$
$954$	0	0
$955$	− 1521.04i	− 1.59271i
$956$	0	0
$957$	−7.38702	−0.00771893
$958$	0	0
$959$	8.14300i	0.00849114i
$960$	0	0
$961$	492.908	0.512912
$962$	0	0
$963$	830.624i	0.862538i
$964$	0	0
$965$	243.588	0.252423
$966$	0	0
$967$	847.122i	0.876031i	0.898967	+	0.438016i	$0.144319\pi$
$967$	847.122i	0.876031i	−0.898967	+	0.438016i	$0.855681\pi$
$968$	0	0
$969$	99.5262	0.102710
$970$	0	0
$971$	− 1059.01i	− 1.09064i	−0.838227	−	0.545322i	$-0.816408\pi$
$971$	− 1059.01i	− 1.09064i	0.838227	−	0.545322i	$-0.183592\pi$
$972$	0	0
$973$	238.922	0.245552
$974$	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
$979$	− 1405.22i	− 1.43536i
$980$	0	0
$981$	1612.58	1.64381
$982$	0	0
$983$	256.802i	0.261244i	0.991432	+	0.130622i	$0.0416974\pi$
$983$	256.802i	0.261244i	−0.991432	+	0.130622i	$0.958303\pi$
$984$	0	0
$985$	−1264.35	−1.28360
$986$	0	0
$987$	56.4301i	0.0571734i
$988$	0	0
$989$	−1394.13	−1.40963
$990$	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
$995$	267.789i	0.269134i
$996$	0	0
$997$	−581.918	−0.583669	−0.291835	−	0.956469i	$-0.594266\pi$
$997$	−581.918	−0.583669	−0.291835	+	0.956469i	$0.594266\pi$
$998$	0	0
$999$	330.355i	0.330686i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.d.b.127.4	✓	8	1.1	even	1	trivial
224.3.d.b.127.5	yes	8	4.3	odd	2	inner
448.3.d.e.127.4		8	8.3	odd	2
448.3.d.e.127.5		8	8.5	even	2
1568.3.d.n.1471.4		8	28.27	even	2
1568.3.d.n.1471.5		8	7.6	odd	2
1792.3.g.d.127.5		8	16.11	odd	4
1792.3.g.d.127.6		8	16.13	even	4
1792.3.g.f.127.3		8	16.5	even	4
1792.3.g.f.127.4		8	16.3	odd	4
2016.3.m.c.127.5		8	12.11	even	2
2016.3.m.c.127.6		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	224.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.554669i q^{3} -4.57685 q^{5} +2.64575i q^{7} +8.69234 q^{9} +O(q^{10})\)	\(q-0.554669i q^{3} -4.57685 q^{5} +2.64575i q^{7} +8.69234 q^{9} +15.7367i q^{11} +8.57685 q^{13} +2.53864i q^{15} +28.3197 q^{17} +6.33599i q^{19} +1.46752 q^{21} +31.0647i q^{23} -4.05242 q^{25} -9.81339i q^{27} -0.846294 q^{29} -21.6354i q^{31} +8.72866 q^{33} -12.1092i q^{35} -33.6637 q^{37} -4.75731i q^{39} +66.9757 q^{41} +44.8781i q^{43} -39.7836 q^{45} +38.4528i q^{47} -7.00000 q^{49} -15.7081i q^{51} -14.8174 q^{53} -72.0246i q^{55} +3.51438 q^{57} +5.80942i q^{59} -52.6015 q^{61} +22.9978i q^{63} -39.2550 q^{65} -117.397i q^{67} +17.2306 q^{69} -81.2543i q^{71} -47.8054 q^{73} +2.24775i q^{75} -41.6354 q^{77} +57.4900i q^{79} +72.7879 q^{81} -102.855i q^{83} -129.615 q^{85} +0.469413i q^{87} -89.2955 q^{89} +22.6922i q^{91} -12.0005 q^{93} -28.9989i q^{95} -3.44452 q^{97} +136.789i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 40 q^{9}+O(q^{10}) \) 8 * q - 40 * q^9	\( 8 q - 40 q^{9} + 32 q^{13} - 16 q^{17} + 104 q^{25} - 80 q^{29} - 176 q^{37} + 144 q^{41} + 256 q^{45} - 56 q^{49} + 48 q^{53} - 400 q^{57} - 192 q^{61} - 304 q^{65} + 576 q^{69} + 272 q^{73} - 112 q^{77} + 504 q^{81} - 160 q^{85} - 80 q^{89} - 608 q^{93} + 528 q^{97}+O(q^{100}) \) 8 * q - 40 * q^9 + 32 * q^13 - 16 * q^17 + 104 * q^25 - 80 * q^29 - 176 * q^37 + 144 * q^41 + 256 * q^45 - 56 * q^49 + 48 * q^53 - 400 * q^57 - 192 * q^61 - 304 * q^65 + 576 * q^69 + 272 * q^73 - 112 * q^77 + 504 * q^81 - 160 * q^85 - 80 * q^89 - 608 * q^93 + 528 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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