LMFDB - Embedded newform 99.3.b.a.89.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,3,Mod(89,99)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("99.89");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 99 = 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	99.b (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:	$2.69755461717$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.65306824704.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + x^{4} - 40x^{3} + 36x^{2} - 12x + 27 $ x^8 - 4x^7 - 2x^6 + 20x^5 + x^4 - 40x^3 + 36x^2 - 12x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			89.2
Root			$-1.75726 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	99.89
Dual form			99.3.b.a.89.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.48514i q^{2} -2.17590 q^{4} -4.68911i q^{5} -3.30128 q^{7} -4.53313i q^{8} +O(q^{10})$	\(q-2.48514i q^{2} -2.17590 q^{4} -4.68911i q^{5} -3.30128 q^{7} -4.53313i q^{8} -11.6531 q^{10} -3.31662i q^{11} +1.00848 q^{13} +8.20412i q^{14} -19.9691 q^{16} +6.45594i q^{17} +25.1291 q^{19} +10.2030i q^{20} -8.24226 q^{22} +24.4236i q^{23} +3.01224 q^{25} -2.50622i q^{26} +7.18325 q^{28} -50.8695i q^{29} +48.4055 q^{31} +31.4933i q^{32} +16.0439 q^{34} +15.4800i q^{35} +65.8199 q^{37} -62.4493i q^{38} -21.2564 q^{40} +6.41379i q^{41} -48.5582 q^{43} +7.21665i q^{44} +60.6959 q^{46} +56.8721i q^{47} -38.1016 q^{49} -7.48582i q^{50} -2.19436 q^{52} +67.3272i q^{53} -15.5520 q^{55} +14.9651i q^{56} -126.418 q^{58} +0.307118i q^{59} -86.8383 q^{61} -120.294i q^{62} -1.61108 q^{64} -4.72888i q^{65} +29.6749 q^{67} -14.0475i q^{68} +38.4700 q^{70} +40.9307i q^{71} -61.0447 q^{73} -163.571i q^{74} -54.6785 q^{76} +10.9491i q^{77} +85.4268 q^{79} +93.6371i q^{80} +15.9391 q^{82} -64.7194i q^{83} +30.2726 q^{85} +120.674i q^{86} -15.0347 q^{88} -64.1065i q^{89} -3.32928 q^{91} -53.1433i q^{92} +141.335 q^{94} -117.833i q^{95} -86.8082 q^{97} +94.6876i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} + 16 q^{7}+O(q^{10}) $ 8 * q - 16 * q^4 + 16 * q^7	$ 8 q - 16 q^{4} + 16 q^{7} - 48 q^{10} - 8 q^{13} + 104 q^{16} + 40 q^{19} - 112 q^{25} - 32 q^{28} - 56 q^{31} - 216 q^{34} + 136 q^{37} + 432 q^{40} - 104 q^{43} + 24 q^{46} - 96 q^{49} + 280 q^{52} - 432 q^{58} - 8 q^{61} - 592 q^{64} + 112 q^{67} + 168 q^{70} + 448 q^{73} - 344 q^{76} + 448 q^{79} + 504 q^{82} + 48 q^{85} - 264 q^{88} - 544 q^{91} + 360 q^{94} - 152 q^{97}+O(q^{100}) $ 8 * q - 16 * q^4 + 16 * q^7 - 48 * q^10 - 8 * q^13 + 104 * q^16 + 40 * q^19 - 112 * q^25 - 32 * q^28 - 56 * q^31 - 216 * q^34 + 136 * q^37 + 432 * q^40 - 104 * q^43 + 24 * q^46 - 96 * q^49 + 280 * q^52 - 432 * q^58 - 8 * q^61 - 592 * q^64 + 112 * q^67 + 168 * q^70 + 448 * q^73 - 344 * q^76 + 448 * q^79 + 504 * q^82 + 48 * q^85 - 264 * q^88 - 544 * q^91 + 360 * q^94 - 152 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/99\mathbb{Z}\right)^\times$.

$n$	$46$	$56$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.48514i	− 1.24257i	−0.783585	−	0.621284i	$-0.786611\pi$
$2$	− 2.48514i	− 1.24257i	0.783585	−	0.621284i	$-0.213389\pi$
$3$	0	0
$3$	0	0
$4$	−2.17590	−0.543976
$5$	− 4.68911i	− 0.937822i	−0.883245	−	0.468911i	$-0.844646\pi$
$5$	− 4.68911i	− 0.937822i	0.883245	−	0.468911i	$-0.155354\pi$
$6$	0	0
$7$	−3.30128	−0.471611	−0.235805	−	0.971800i	$-0.575773\pi$
$7$	−3.30128	−0.471611	−0.235805	+	0.971800i	$0.575773\pi$
$8$	− 4.53313i	− 0.566641i
$9$	0	0
$10$	−11.6531	−1.16531
$11$	− 3.31662i	− 0.301511i
$11$	− 3.31662i	− 0.301511i
$12$	0	0
$13$	1.00848	0.0775756	0.0387878	−	0.999247i	$-0.487650\pi$
$13$	1.00848	0.0775756	0.0387878	+	0.999247i	$0.487650\pi$
$14$	8.20412i	0.586009i
$15$	0	0
$16$	−19.9691	−1.24807
$17$	6.45594i	0.379761i	0.981807	+	0.189881i	$0.0608101\pi$
$17$	6.45594i	0.379761i	−0.981807	+	0.189881i	$0.939190\pi$
$18$	0	0
$19$	25.1291	1.32259	0.661293	−	0.750128i	$-0.270008\pi$
$19$	25.1291	1.32259	0.661293	+	0.750128i	$0.270008\pi$
$20$	10.2030i	0.510152i
$21$	0	0
$22$	−8.24226	−0.374648
$23$	24.4236i	1.06189i	0.847405	+	0.530947i	$0.178164\pi$
$23$	24.4236i	1.06189i	−0.847405	+	0.530947i	$0.821836\pi$
$24$	0	0
$25$	3.01224	0.120489
$26$	− 2.50622i	− 0.0963929i
$27$	0	0
$28$	7.18325	0.256545
$29$	− 50.8695i	− 1.75412i	−0.480379	−	0.877061i	$-0.659501\pi$
$29$	− 50.8695i	− 1.75412i	0.480379	−	0.877061i	$-0.340499\pi$
$30$	0	0
$31$	48.4055	1.56147	0.780734	−	0.624864i	$-0.214846\pi$
$31$	48.4055	1.56147	0.780734	+	0.624864i	$0.214846\pi$
$32$	31.4933i	0.984166i
$33$	0	0
$34$	16.0439	0.471879
$35$	15.4800i	0.442287i
$36$	0	0
$37$	65.8199	1.77892	0.889458	−	0.457017i	$-0.151082\pi$
$37$	65.8199	1.77892	0.889458	+	0.457017i	$0.151082\pi$
$38$	− 62.4493i	− 1.64340i
$39$	0	0
$40$	−21.2564	−0.531409
$41$	6.41379i	0.156434i	0.996936	+	0.0782169i	$0.0249227\pi$
$41$	6.41379i	0.156434i	−0.996936	+	0.0782169i	$0.975077\pi$
$42$	0	0
$43$	−48.5582	−1.12926	−0.564631	−	0.825344i	$-0.690981\pi$
$43$	−48.5582	−1.12926	−0.564631	+	0.825344i	$0.690981\pi$
$44$	7.21665i	0.164015i
$45$	0	0
$46$	60.6959	1.31948
$47$	56.8721i	1.21004i	0.796209	+	0.605022i	$0.206836\pi$
$47$	56.8721i	1.21004i	−0.796209	+	0.605022i	$0.793164\pi$
$48$	0	0
$49$	−38.1016	−0.777583
$50$	− 7.48582i	− 0.149716i
$51$	0	0
$52$	−2.19436	−0.0421992
$53$	67.3272i	1.27033i	0.772379	+	0.635163i	$0.219067\pi$
$53$	67.3272i	1.27033i	−0.772379	+	0.635163i	$0.780933\pi$
$54$	0	0
$55$	−15.5520	−0.282764
$56$	14.9651i	0.267234i
$57$	0	0
$58$	−126.418	−2.17962
$59$	0.307118i	0.00520539i	0.999997	+	0.00260270i	$0.000828465\pi$
$59$	0.307118i	0.00520539i	−0.999997	+	0.00260270i	$0.999172\pi$
$60$	0	0
$61$	−86.8383	−1.42358	−0.711790	−	0.702393i	$-0.752115\pi$
$61$	−86.8383	−1.42358	−0.711790	+	0.702393i	$0.752115\pi$
$62$	− 120.294i	− 1.94023i
$63$	0	0
$64$	−1.61108	−0.0251731
$65$	− 4.72888i	− 0.0727521i
$66$	0	0
$67$	29.6749	0.442909	0.221455	−	0.975171i	$-0.428920\pi$
$67$	29.6749	0.442909	0.221455	+	0.975171i	$0.428920\pi$
$68$	− 14.0475i	− 0.206581i
$69$	0	0
$70$	38.4700	0.549572
$71$	40.9307i	0.576489i	0.957557	+	0.288244i	$0.0930716\pi$
$71$	40.9307i	0.576489i	−0.957557	+	0.288244i	$0.906928\pi$
$72$	0	0
$73$	−61.0447	−0.836229	−0.418115	−	0.908394i	$-0.637309\pi$
$73$	−61.0447	−0.836229	−0.418115	+	0.908394i	$0.637309\pi$
$74$	− 163.571i	− 2.21042i
$75$	0	0
$76$	−54.6785	−0.719454
$77$	10.9491i	0.142196i
$78$	0	0
$79$	85.4268	1.08135	0.540676	−	0.841231i	$-0.318169\pi$
$79$	85.4268	1.08135	0.540676	+	0.841231i	$0.318169\pi$
$80$	93.6371i	1.17046i
$81$	0	0
$82$	15.9391	0.194380
$83$	− 64.7194i	− 0.779751i	−0.920867	−	0.389876i	$-0.872518\pi$
$83$	− 64.7194i	− 0.779751i	0.920867	−	0.389876i	$-0.127482\pi$
$84$	0	0
$85$	30.2726	0.356149
$86$	120.674i	1.40318i
$87$	0	0
$88$	−15.0347	−0.170849
$89$	− 64.1065i	− 0.720297i	−0.932895	−	0.360149i	$-0.882726\pi$
$89$	− 64.1065i	− 0.720297i	0.932895	−	0.360149i	$-0.117274\pi$
$90$	0	0
$91$	−3.32928	−0.0365855
$92$	− 53.1433i	− 0.577644i
$93$	0	0
$94$	141.335	1.50356
$95$	− 117.833i	− 1.24035i
$96$	0	0
$97$	−86.8082	−0.894930	−0.447465	−	0.894302i	$-0.647673\pi$
$97$	−86.8082	−0.894930	−0.447465	+	0.894302i	$0.647673\pi$
$98$	94.6876i	0.966200i
$99$	0	0
$100$	−6.55433	−0.0655433
$101$	− 114.951i	− 1.13813i	−0.822292	−	0.569065i	$-0.807305\pi$
$101$	− 114.951i	− 1.13813i	0.822292	−	0.569065i	$-0.192695\pi$
$102$	0	0
$103$	28.6367	0.278026	0.139013	−	0.990291i	$-0.455607\pi$
$103$	28.6367	0.278026	0.139013	+	0.990291i	$0.455607\pi$
$104$	− 4.57158i	− 0.0439575i
$105$	0	0
$106$	167.317	1.57847
$107$	119.479i	1.11662i	0.829631	+	0.558312i	$0.188551\pi$
$107$	119.479i	1.11662i	−0.829631	+	0.558312i	$0.811449\pi$
$108$	0	0
$109$	−76.7577	−0.704199	−0.352100	−	0.935962i	$-0.614532\pi$
$109$	−76.7577	−0.704199	−0.352100	+	0.935962i	$0.614532\pi$
$110$	38.6489i	0.351354i
$111$	0	0
$112$	65.9234	0.588601
$113$	84.4614i	0.747446i	0.927540	+	0.373723i	$0.121919\pi$
$113$	84.4614i	0.747446i	−0.927540	+	0.373723i	$0.878081\pi$
$114$	0	0
$115$	114.525	0.995867
$116$	110.687i	0.954199i
$117$	0	0
$118$	0.763231	0.00646806
$119$	− 21.3129i	− 0.179100i
$120$	0	0
$121$	−11.0000	−0.0909091
$122$	215.805i	1.76889i
$123$	0	0
$124$	−105.326	−0.849400
$125$	− 131.352i	− 1.05082i
$126$	0	0
$127$	−99.8661	−0.786347	−0.393174	−	0.919464i	$-0.628623\pi$
$127$	−99.8661	−0.786347	−0.393174	+	0.919464i	$0.628623\pi$
$128$	129.977i	1.01545i
$129$	0	0
$130$	−11.7519	−0.0903994
$131$	165.140i	1.26061i	0.776347	+	0.630305i	$0.217070\pi$
$131$	165.140i	1.26061i	−0.776347	+	0.630305i	$0.782930\pi$
$132$	0	0
$133$	−82.9582	−0.623746
$134$	− 73.7462i	− 0.550345i
$135$	0	0
$136$	29.2656	0.215189
$137$	23.5961i	0.172234i	0.996285	+	0.0861170i	$0.0274459\pi$
$137$	23.5961i	0.172234i	−0.996285	+	0.0861170i	$0.972554\pi$
$138$	0	0
$139$	272.764	1.96233	0.981167	−	0.193162i	$-0.0618743\pi$
$139$	272.764	1.96233	0.981167	+	0.193162i	$0.0618743\pi$
$140$	− 33.6831i	− 0.240593i
$141$	0	0
$142$	101.718	0.716327
$143$	− 3.34476i	− 0.0233899i
$144$	0	0
$145$	−238.533	−1.64505
$146$	151.704i	1.03907i
$147$	0	0
$148$	−143.218	−0.967687
$149$	− 34.7567i	− 0.233266i	−0.993175	−	0.116633i	$-0.962790\pi$
$149$	− 34.7567i	− 0.233266i	0.993175	−	0.116633i	$-0.0372102\pi$
$150$	0	0
$151$	123.990	0.821124	0.410562	−	0.911833i	$-0.365333\pi$
$151$	123.990	0.821124	0.410562	+	0.911833i	$0.365333\pi$
$152$	− 113.914i	− 0.749432i
$153$	0	0
$154$	27.2100	0.176688
$155$	− 226.979i	− 1.46438i
$156$	0	0
$157$	157.296	1.00188	0.500942	−	0.865481i	$-0.332987\pi$
$157$	157.296	1.00188	0.500942	+	0.865481i	$0.332987\pi$
$158$	− 212.297i	− 1.34365i
$159$	0	0
$160$	147.676	0.922973
$161$	− 80.6289i	− 0.500800i
$162$	0	0
$163$	−300.686	−1.84470	−0.922350	−	0.386356i	$-0.873734\pi$
$163$	−300.686	−1.84470	−0.922350	+	0.386356i	$0.873734\pi$
$164$	− 13.9558i	− 0.0850962i
$165$	0	0
$166$	−160.836	−0.968894
$167$	− 90.7043i	− 0.543139i	−0.962419	−	0.271570i	$-0.912457\pi$
$167$	− 90.7043i	− 0.543139i	0.962419	−	0.271570i	$-0.0875427\pi$
$168$	0	0
$169$	−167.983	−0.993982
$170$	− 75.2316i	− 0.442539i
$171$	0	0
$172$	105.658	0.614291
$173$	− 194.822i	− 1.12614i	−0.826410	−	0.563069i	$-0.809620\pi$
$173$	− 194.822i	− 1.12614i	0.826410	−	0.563069i	$-0.190380\pi$
$174$	0	0
$175$	−9.94422	−0.0568241
$176$	66.2299i	0.376306i
$177$	0	0
$178$	−159.313	−0.895018
$179$	− 27.8164i	− 0.155399i	−0.996977	−	0.0776994i	$-0.975243\pi$
$179$	− 27.8164i	− 0.155399i	0.996977	−	0.0776994i	$-0.0247574\pi$
$180$	0	0
$181$	194.094	1.07234	0.536171	−	0.844109i	$-0.319870\pi$
$181$	194.094	1.07234	0.536171	+	0.844109i	$0.319870\pi$
$182$	8.27371i	0.0454599i
$183$	0	0
$184$	110.715	0.601713
$185$	− 308.637i	− 1.66831i
$186$	0	0
$187$	21.4119	0.114502
$188$	− 123.748i	− 0.658235i
$189$	0	0
$190$	−292.832	−1.54122
$191$	297.244i	1.55625i	0.628108	+	0.778126i	$0.283829\pi$
$191$	297.244i	1.55625i	−0.628108	+	0.778126i	$0.716171\pi$
$192$	0	0
$193$	−197.890	−1.02533	−0.512667	−	0.858587i	$-0.671343\pi$
$193$	−197.890	−1.02533	−0.512667	+	0.858587i	$0.671343\pi$
$194$	215.730i	1.11201i
$195$	0	0
$196$	82.9053	0.422986
$197$	249.339i	1.26568i	0.774282	+	0.632841i	$0.218111\pi$
$197$	249.339i	1.26568i	−0.774282	+	0.632841i	$0.781889\pi$
$198$	0	0
$199$	101.803	0.511571	0.255786	−	0.966733i	$-0.417666\pi$
$199$	101.803	0.511571	0.255786	+	0.966733i	$0.417666\pi$
$200$	− 13.6549i	− 0.0682743i
$201$	0	0
$202$	−285.669	−1.41420
$203$	167.934i	0.827263i
$204$	0	0
$205$	30.0750	0.146707
$206$	− 71.1661i	− 0.345466i
$207$	0	0
$208$	−20.1384	−0.0968194
$209$	− 83.3439i	− 0.398775i
$210$	0	0
$211$	−37.9452	−0.179835	−0.0899175	−	0.995949i	$-0.528660\pi$
$211$	−37.9452	−0.179835	−0.0899175	+	0.995949i	$0.528660\pi$
$212$	− 146.497i	− 0.691026i
$213$	0	0
$214$	296.921	1.38748
$215$	227.695i	1.05905i
$216$	0	0
$217$	−159.800	−0.736405
$218$	190.753i	0.875016i
$219$	0	0
$220$	33.8397	0.153817
$221$	6.51071i	0.0294602i
$222$	0	0
$223$	119.504	0.535893	0.267946	−	0.963434i	$-0.413655\pi$
$223$	119.504	0.535893	0.267946	+	0.963434i	$0.413655\pi$
$224$	− 103.968i	− 0.464143i
$225$	0	0
$226$	209.898	0.928753
$227$	233.484i	1.02856i	0.857621	+	0.514282i	$0.171941\pi$
$227$	233.484i	1.02856i	−0.857621	+	0.514282i	$0.828059\pi$
$228$	0	0
$229$	−170.303	−0.743683	−0.371841	−	0.928296i	$-0.621274\pi$
$229$	−170.303	−0.743683	−0.371841	+	0.928296i	$0.621274\pi$
$230$	− 284.610i	− 1.23743i
$231$	0	0
$232$	−230.598	−0.993958
$233$	149.406i	0.641228i	0.947210	+	0.320614i	$0.103889\pi$
$233$	149.406i	0.641228i	−0.947210	+	0.320614i	$0.896111\pi$
$234$	0	0
$235$	266.680	1.13481
$236$	− 0.668259i	− 0.00283161i
$237$	0	0
$238$	−52.9653	−0.222543
$239$	267.637i	1.11982i	0.828554	+	0.559909i	$0.189164\pi$
$239$	267.637i	1.11982i	−0.828554	+	0.559909i	$0.810836\pi$
$240$	0	0
$241$	202.338	0.839578	0.419789	−	0.907622i	$-0.362104\pi$
$241$	202.338	0.839578	0.419789	+	0.907622i	$0.362104\pi$
$242$	27.3365i	0.112961i
$243$	0	0
$244$	188.952	0.774392
$245$	178.663i	0.729235i
$246$	0	0
$247$	25.3423	0.102600
$248$	− 219.428i	− 0.884792i
$249$	0	0
$250$	−326.429	−1.30572
$251$	− 169.400i	− 0.674901i	−0.941343	−	0.337450i	$-0.890435\pi$
$251$	− 169.400i	− 0.674901i	0.941343	−	0.337450i	$-0.109565\pi$
$252$	0	0
$253$	81.0038	0.320173
$254$	248.181i	0.977090i
$255$	0	0
$256$	316.566	1.23659
$257$	− 208.654i	− 0.811885i	−0.913899	−	0.405942i	$-0.866943\pi$
$257$	− 208.654i	− 0.811885i	0.913899	−	0.405942i	$-0.133057\pi$
$258$	0	0
$259$	−217.290	−0.838956
$260$	10.2896i	0.0395754i
$261$	0	0
$262$	410.395	1.56639
$263$	− 117.552i	− 0.446967i	−0.974708	−	0.223483i	$-0.928257\pi$
$263$	− 117.552i	− 0.446967i	0.974708	−	0.223483i	$-0.0717429\pi$
$264$	0	0
$265$	315.705	1.19134
$266$	206.162i	0.775047i
$267$	0	0
$268$	−64.5697	−0.240932
$269$	467.823i	1.73912i	0.493830	+	0.869559i	$0.335597\pi$
$269$	467.823i	1.73912i	−0.493830	+	0.869559i	$0.664403\pi$
$270$	0	0
$271$	−116.293	−0.429126	−0.214563	−	0.976710i	$-0.568833\pi$
$271$	−116.293	−0.429126	−0.214563	+	0.976710i	$0.568833\pi$
$272$	− 128.919i	− 0.473967i
$273$	0	0
$274$	58.6394	0.214013
$275$	− 9.99046i	− 0.0363289i
$276$	0	0
$277$	290.493	1.04871	0.524356	−	0.851499i	$-0.324306\pi$
$277$	290.493	1.04871	0.524356	+	0.851499i	$0.324306\pi$
$278$	− 677.857i	− 2.43833i
$279$	0	0
$280$	70.1731	0.250618
$281$	− 26.7812i	− 0.0953066i	−0.998864	−	0.0476533i	$-0.984826\pi$
$281$	− 26.7812i	− 0.0953066i	0.998864	−	0.0476533i	$-0.0151743\pi$
$282$	0	0
$283$	23.7154	0.0838001	0.0419001	−	0.999122i	$-0.486659\pi$
$283$	23.7154	0.0838001	0.0419001	+	0.999122i	$0.486659\pi$
$284$	− 89.0612i	− 0.313596i
$285$	0	0
$286$	−8.31218	−0.0290636
$287$	− 21.1737i	− 0.0737759i
$288$	0	0
$289$	247.321	0.855781
$290$	592.787i	2.04409i
$291$	0	0
$292$	132.827	0.454888
$293$	− 379.831i	− 1.29635i	−0.761491	−	0.648175i	$-0.775532\pi$
$293$	− 379.831i	− 1.29635i	0.761491	−	0.648175i	$-0.224468\pi$
$294$	0	0
$295$	1.44011	0.00488173
$296$	− 298.370i	− 1.00801i
$297$	0	0
$298$	−86.3750	−0.289849
$299$	24.6307i	0.0823770i
$300$	0	0
$301$	160.304	0.532572
$302$	− 308.131i	− 1.02030i
$303$	0	0
$304$	−501.805	−1.65067
$305$	407.195i	1.33506i
$306$	0	0
$307$	41.2092	0.134232	0.0671160	−	0.997745i	$-0.478620\pi$
$307$	41.2092	0.134232	0.0671160	+	0.997745i	$0.478620\pi$
$308$	− 23.8242i	− 0.0773512i
$309$	0	0
$310$	−564.073	−1.81959
$311$	− 520.132i	− 1.67245i	−0.548387	−	0.836224i	$-0.684758\pi$
$311$	− 520.132i	− 1.67245i	0.548387	−	0.836224i	$-0.315242\pi$
$312$	0	0
$313$	−558.820	−1.78537	−0.892684	−	0.450683i	$-0.851181\pi$
$313$	−558.820	−1.78537	−0.892684	+	0.450683i	$0.851181\pi$
$314$	− 390.902i	− 1.24491i
$315$	0	0
$316$	−185.880	−0.588229
$317$	434.973i	1.37215i	0.727529	+	0.686077i	$0.240669\pi$
$317$	434.973i	1.37215i	−0.727529	+	0.686077i	$0.759331\pi$
$318$	0	0
$319$	−168.715	−0.528888
$320$	7.55451i	0.0236079i
$321$	0	0
$322$	−200.374	−0.622279
$323$	162.232i	0.502267i
$324$	0	0
$325$	3.03779	0.00934704
$326$	747.246i	2.29216i
$327$	0	0
$328$	29.0745	0.0886419
$329$	− 187.750i	− 0.570670i
$330$	0	0
$331$	185.292	0.559795	0.279897	−	0.960030i	$-0.409700\pi$
$331$	185.292	0.559795	0.279897	+	0.960030i	$0.409700\pi$
$332$	140.823i	0.424166i
$333$	0	0
$334$	−225.412	−0.674888
$335$	− 139.149i	− 0.415370i
$336$	0	0
$337$	−366.174	−1.08657	−0.543285	−	0.839549i	$-0.682820\pi$
$337$	−366.174	−1.08657	−0.543285	+	0.839549i	$0.682820\pi$
$338$	417.461i	1.23509i
$339$	0	0
$340$	−65.8703	−0.193736
$341$	− 160.543i	− 0.470800i
$342$	0	0
$343$	287.546	0.838327
$344$	220.121i	0.639886i
$345$	0	0
$346$	−484.159	−1.39930
$347$	232.794i	0.670876i	0.942062	+	0.335438i	$0.108884\pi$
$347$	232.794i	0.670876i	−0.942062	+	0.335438i	$0.891116\pi$
$348$	0	0
$349$	−190.183	−0.544938	−0.272469	−	0.962165i	$-0.587840\pi$
$349$	−190.183	−0.544938	−0.272469	+	0.962165i	$0.587840\pi$
$350$	24.7127i	0.0706078i
$351$	0	0
$352$	104.451	0.296737
$353$	− 650.001i	− 1.84136i	−0.390316	−	0.920681i	$-0.627634\pi$
$353$	− 650.001i	− 1.84136i	0.390316	−	0.920681i	$-0.372366\pi$
$354$	0	0
$355$	191.929	0.540644
$356$	139.489i	0.391824i
$357$	0	0
$358$	−69.1275	−0.193094
$359$	− 318.134i	− 0.886167i	−0.896480	−	0.443083i	$-0.853885\pi$
$359$	− 318.134i	− 0.886167i	0.896480	−	0.443083i	$-0.146115\pi$
$360$	0	0
$361$	270.473	0.749233
$362$	− 482.350i	− 1.33246i
$363$	0	0
$364$	7.24418	0.0199016
$365$	286.246i	0.784234i
$366$	0	0
$367$	−10.0024	−0.0272546	−0.0136273	−	0.999907i	$-0.504338\pi$
$367$	−10.0024	−0.0272546	−0.0136273	+	0.999907i	$0.504338\pi$
$368$	− 487.715i	− 1.32531i
$369$	0	0
$370$	−767.004	−2.07299
$371$	− 222.266i	− 0.599099i
$372$	0	0
$373$	373.434	1.00116	0.500582	−	0.865689i	$-0.333119\pi$
$373$	373.434	1.00116	0.500582	+	0.865689i	$0.333119\pi$
$374$	− 53.2116i	− 0.142277i
$375$	0	0
$376$	257.809	0.685661
$377$	− 51.3010i	− 0.136077i
$378$	0	0
$379$	−273.614	−0.721937	−0.360969	−	0.932578i	$-0.617554\pi$
$379$	−273.614	−0.721937	−0.360969	+	0.932578i	$0.617554\pi$
$380$	256.394i	0.674720i
$381$	0	0
$382$	738.692	1.93375
$383$	− 129.063i	− 0.336978i	−0.985703	−	0.168489i	$-0.946111\pi$
$383$	− 129.063i	− 0.336978i	0.985703	−	0.168489i	$-0.0538888\pi$
$384$	0	0
$385$	51.3415	0.133355
$386$	491.783i	1.27405i
$387$	0	0
$388$	188.886	0.486820
$389$	311.684i	0.801244i	0.916243	+	0.400622i	$0.131206\pi$
$389$	311.684i	0.801244i	−0.916243	+	0.400622i	$0.868794\pi$
$390$	0	0
$391$	−157.677	−0.403266
$392$	172.719i	0.440611i
$393$	0	0
$394$	619.642	1.57270
$395$	− 400.576i	− 1.01412i
$396$	0	0
$397$	449.573	1.13243	0.566213	−	0.824259i	$-0.308408\pi$
$397$	449.573	1.13243	0.566213	+	0.824259i	$0.308408\pi$
$398$	− 252.994i	− 0.635662i
$399$	0	0
$400$	−60.1515	−0.150379
$401$	− 348.583i	− 0.869284i	−0.900603	−	0.434642i	$-0.856875\pi$
$401$	− 348.583i	− 0.869284i	0.900603	−	0.434642i	$-0.143125\pi$
$402$	0	0
$403$	48.8161	0.121132
$404$	250.122i	0.619115i
$405$	0	0
$406$	417.340	1.02793
$407$	− 218.300i	− 0.536363i
$408$	0	0
$409$	−532.578	−1.30215	−0.651073	−	0.759015i	$-0.725681\pi$
$409$	−532.578	−1.30215	−0.651073	+	0.759015i	$0.725681\pi$
$410$	− 74.7404i	− 0.182294i
$411$	0	0
$412$	−62.3106	−0.151239
$413$	− 1.01388i	− 0.00245492i
$414$	0	0
$415$	−303.476	−0.731268
$416$	31.7604i	0.0763472i
$417$	0	0
$418$	−207.121	−0.495505
$419$	267.800i	0.639141i	0.947563	+	0.319570i	$0.103539\pi$
$419$	267.800i	0.639141i	−0.947563	+	0.319570i	$0.896461\pi$
$420$	0	0
$421$	371.455	0.882315	0.441158	−	0.897430i	$-0.354568\pi$
$421$	371.455	0.882315	0.441158	+	0.897430i	$0.354568\pi$
$422$	94.2990i	0.223457i
$423$	0	0
$424$	305.203	0.719819
$425$	19.4468i	0.0457572i
$426$	0	0
$427$	286.677	0.671375
$428$	− 259.974i	− 0.607416i
$429$	0	0
$430$	565.853	1.31594
$431$	285.915i	0.663376i	0.943389	+	0.331688i	$0.107618\pi$
$431$	285.915i	0.663376i	−0.943389	+	0.331688i	$0.892382\pi$
$432$	0	0
$433$	228.990	0.528845	0.264423	−	0.964407i	$-0.414819\pi$
$433$	228.990	0.528845	0.264423	+	0.964407i	$0.414819\pi$
$434$	397.124i	0.915033i
$435$	0	0
$436$	167.017	0.383067
$437$	613.743i	1.40445i
$438$	0	0
$439$	−815.871	−1.85848	−0.929238	−	0.369481i	$-0.879536\pi$
$439$	−815.871	−1.85848	−0.929238	+	0.369481i	$0.879536\pi$
$440$	70.4994i	0.160226i
$441$	0	0
$442$	16.1800	0.0366063
$443$	− 434.432i	− 0.980660i	−0.871537	−	0.490330i	$-0.836876\pi$
$443$	− 434.432i	− 0.980660i	0.871537	−	0.490330i	$-0.163124\pi$
$444$	0	0
$445$	−300.602	−0.675511
$446$	− 296.984i	− 0.665883i
$447$	0	0
$448$	5.31860	0.0118719
$449$	73.1022i	0.162811i	0.996681	+	0.0814056i	$0.0259409\pi$
$449$	73.1022i	0.162811i	−0.996681	+	0.0814056i	$0.974059\pi$
$450$	0	0
$451$	21.2721	0.0471666
$452$	− 183.780i	− 0.406592i
$453$	0	0
$454$	580.239	1.27806
$455$	15.6114i	0.0343107i
$456$	0	0
$457$	−49.1411	−0.107530	−0.0537649	−	0.998554i	$-0.517122\pi$
$457$	−49.1411	−0.107530	−0.0537649	+	0.998554i	$0.517122\pi$
$458$	423.227i	0.924077i
$459$	0	0
$460$	−249.195	−0.541728
$461$	657.645i	1.42656i	0.700878	+	0.713281i	$0.252792\pi$
$461$	657.645i	1.42656i	−0.700878	+	0.713281i	$0.747208\pi$
$462$	0	0
$463$	−195.703	−0.422685	−0.211343	−	0.977412i	$-0.567784\pi$
$463$	−195.703	−0.422685	−0.211343	+	0.977412i	$0.567784\pi$
$464$	1015.82i	2.18926i
$465$	0	0
$466$	371.295	0.796770
$467$	− 13.5937i	− 0.0291086i	−0.999894	−	0.0145543i	$-0.995367\pi$
$467$	− 13.5937i	− 0.0291086i	0.999894	−	0.0145543i	$-0.00463295\pi$
$468$	0	0
$469$	−97.9651	−0.208881
$470$	− 662.735i	− 1.41007i
$471$	0	0
$472$	1.39221	0.00294959
$473$	161.049i	0.340485i
$474$	0	0
$475$	75.6949	0.159358
$476$	46.3747i	0.0974258i
$477$	0	0
$478$	665.113	1.39145
$479$	369.614i	0.771637i	0.922575	+	0.385818i	$0.126081\pi$
$479$	369.614i	0.771637i	−0.922575	+	0.385818i	$0.873919\pi$
$480$	0	0
$481$	66.3782	0.138000
$482$	− 502.838i	− 1.04323i
$483$	0	0
$484$	23.9349	0.0494523
$485$	407.053i	0.839285i
$486$	0	0
$487$	−485.893	−0.997727	−0.498863	−	0.866681i	$-0.666249\pi$
$487$	−485.893	−0.997727	−0.498863	+	0.866681i	$0.666249\pi$
$488$	393.650i	0.806659i
$489$	0	0
$490$	444.001	0.906124
$491$	− 538.550i	− 1.09684i	−0.836202	−	0.548421i	$-0.815229\pi$
$491$	− 538.550i	− 1.09684i	0.836202	−	0.548421i	$-0.184771\pi$
$492$	0	0
$493$	328.411	0.666148
$494$	− 62.9790i	− 0.127488i
$495$	0	0
$496$	−966.612	−1.94881
$497$	− 135.124i	− 0.271878i
$498$	0	0
$499$	−949.252	−1.90231	−0.951154	−	0.308716i	$-0.900101\pi$
$499$	−949.252	−1.90231	−0.951154	+	0.308716i	$0.900101\pi$
$500$	285.810i	0.571620i
$501$	0	0
$502$	−420.982	−0.838610
$503$	207.815i	0.413151i	0.978431	+	0.206575i	$0.0662319\pi$
$503$	207.815i	0.413151i	−0.978431	+	0.206575i	$0.933768\pi$
$504$	0	0
$505$	−539.019	−1.06736
$506$	− 201.305i	− 0.397837i
$507$	0	0
$508$	217.299	0.427754
$509$	− 473.501i	− 0.930258i	−0.885243	−	0.465129i	$-0.846008\pi$
$509$	− 473.501i	− 0.930258i	0.885243	−	0.465129i	$-0.153992\pi$
$510$	0	0
$511$	201.526	0.394375
$512$	− 266.802i	− 0.521098i
$513$	0	0
$514$	−518.535	−1.00882
$515$	− 134.281i	− 0.260739i
$516$	0	0
$517$	188.623	0.364842
$518$	539.994i	1.04246i
$519$	0	0
$520$	−21.4367	−0.0412243
$521$	− 756.599i	− 1.45221i	−0.687586	−	0.726103i	$-0.741330\pi$
$521$	− 756.599i	− 1.45221i	0.687586	−	0.726103i	$-0.258670\pi$
$522$	0	0
$523$	−782.107	−1.49543	−0.747713	−	0.664022i	$-0.768848\pi$
$523$	−782.107	−1.49543	−0.747713	+	0.664022i	$0.768848\pi$
$524$	− 359.329i	− 0.685741i
$525$	0	0
$526$	−292.134	−0.555387
$527$	312.503i	0.592985i
$528$	0	0
$529$	−67.5098	−0.127618
$530$	− 784.570i	− 1.48032i
$531$	0	0
$532$	180.509	0.339302
$533$	6.46819i	0.0121354i
$534$	0	0
$535$	560.249	1.04719
$536$	− 134.520i	− 0.250971i
$537$	0	0
$538$	1162.60	2.16097
$539$	126.369i	0.234450i
$540$	0	0
$541$	−55.2149	−0.102061	−0.0510304	−	0.998697i	$-0.516251\pi$
$541$	−55.2149	−0.102061	−0.0510304	+	0.998697i	$0.516251\pi$
$542$	289.005i	0.533219i
$543$	0	0
$544$	−203.319	−0.373748
$545$	359.926i	0.660414i
$546$	0	0
$547$	−358.392	−0.655195	−0.327597	−	0.944817i	$-0.606239\pi$
$547$	−358.392	−0.655195	−0.327597	+	0.944817i	$0.606239\pi$
$548$	− 51.3427i	− 0.0936911i
$549$	0	0
$550$	−24.8276	−0.0451412
$551$	− 1278.31i	− 2.31998i
$552$	0	0
$553$	−282.017	−0.509977
$554$	− 721.916i	− 1.30310i
$555$	0	0
$556$	−593.509	−1.06746
$557$	644.969i	1.15793i	0.815351	+	0.578967i	$0.196544\pi$
$557$	644.969i	1.15793i	−0.815351	+	0.578967i	$0.803456\pi$
$558$	0	0
$559$	−48.9701	−0.0876031
$560$	− 309.122i	− 0.552004i
$561$	0	0
$562$	−66.5548	−0.118425
$563$	− 730.584i	− 1.29766i	−0.760932	−	0.648831i	$-0.775258\pi$
$563$	− 730.584i	− 1.29766i	0.760932	−	0.648831i	$-0.224742\pi$
$564$	0	0
$565$	396.049	0.700972
$566$	− 58.9361i	− 0.104127i
$567$	0	0
$568$	185.544	0.326662
$569$	− 1037.85i	− 1.82399i	−0.410200	−	0.911996i	$-0.634541\pi$
$569$	− 1037.85i	− 1.82399i	0.410200	−	0.911996i	$-0.365459\pi$
$570$	0	0
$571$	−161.245	−0.282391	−0.141196	−	0.989982i	$-0.545095\pi$
$571$	−161.245	−0.282391	−0.141196	+	0.989982i	$0.545095\pi$
$572$	7.27786i	0.0127235i
$573$	0	0
$574$	−52.6195	−0.0916716
$575$	73.5695i	0.127947i
$576$	0	0
$577$	988.690	1.71350	0.856750	−	0.515732i	$-0.172480\pi$
$577$	988.690	1.71350	0.856750	+	0.515732i	$0.172480\pi$
$578$	− 614.626i	− 1.06337i
$579$	0	0
$580$	519.024	0.894869
$581$	213.656i	0.367739i
$582$	0	0
$583$	223.299	0.383017
$584$	276.724i	0.473842i
$585$	0	0
$586$	−943.931	−1.61080
$587$	517.799i	0.882110i	0.897480	+	0.441055i	$0.145396\pi$
$587$	517.799i	0.882110i	−0.897480	+	0.441055i	$0.854604\pi$
$588$	0	0
$589$	1216.39	2.06517
$590$	− 3.57887i	− 0.00606589i
$591$	0	0
$592$	−1314.36	−2.22020
$593$	396.829i	0.669188i	0.942362	+	0.334594i	$0.108599\pi$
$593$	396.829i	0.669188i	−0.942362	+	0.334594i	$0.891401\pi$
$594$	0	0
$595$	−99.9383	−0.167964
$596$	75.6271i	0.126891i
$597$	0	0
$598$	61.2107	0.102359
$599$	865.942i	1.44565i	0.691034	+	0.722823i	$0.257156\pi$
$599$	865.942i	1.44565i	−0.691034	+	0.722823i	$0.742844\pi$
$600$	0	0
$601$	412.367	0.686134	0.343067	−	0.939311i	$-0.388534\pi$
$601$	412.367	0.686134	0.343067	+	0.939311i	$0.388534\pi$
$602$	− 398.378i	− 0.661757i
$603$	0	0
$604$	−269.789	−0.446671
$605$	51.5802i	0.0852566i
$606$	0	0
$607$	−248.291	−0.409045	−0.204523	−	0.978862i	$-0.565564\pi$
$607$	−248.291	−0.409045	−0.204523	+	0.978862i	$0.565564\pi$
$608$	791.399i	1.30164i
$609$	0	0
$610$	1011.93	1.65891
$611$	57.3545i	0.0938699i
$612$	0	0
$613$	59.7914	0.0975390	0.0487695	−	0.998810i	$-0.484470\pi$
$613$	59.7914	0.0975390	0.0487695	+	0.998810i	$0.484470\pi$
$614$	− 102.410i	− 0.166792i
$615$	0	0
$616$	49.6337	0.0805741
$617$	− 615.297i	− 0.997240i	−0.866821	−	0.498620i	$-0.833840\pi$
$617$	− 615.297i	− 0.997240i	0.866821	−	0.498620i	$-0.166160\pi$
$618$	0	0
$619$	−308.597	−0.498541	−0.249270	−	0.968434i	$-0.580191\pi$
$619$	−308.597	−0.498541	−0.249270	+	0.968434i	$0.580191\pi$
$620$	493.883i	0.796586i
$621$	0	0
$622$	−1292.60	−2.07813
$623$	211.633i	0.339700i
$624$	0	0
$625$	−540.621	−0.864993
$626$	1388.74i	2.21844i
$627$	0	0
$628$	−342.260	−0.545001
$629$	424.930i	0.675564i
$630$	0	0
$631$	−473.277	−0.750042	−0.375021	−	0.927016i	$-0.622365\pi$
$631$	−473.277	−0.750042	−0.375021	+	0.927016i	$0.622365\pi$
$632$	− 387.251i	− 0.612739i
$633$	0	0
$634$	1080.97	1.70500
$635$	468.283i	0.737454i
$636$	0	0
$637$	−38.4248	−0.0603214
$638$	419.280i	0.657179i
$639$	0	0
$640$	609.476	0.952307
$641$	280.129i	0.437019i	0.975835	+	0.218509i	$0.0701194\pi$
$641$	280.129i	0.437019i	−0.975835	+	0.218509i	$0.929881\pi$
$642$	0	0
$643$	−1239.37	−1.92748	−0.963740	−	0.266843i	$-0.914019\pi$
$643$	−1239.37	−1.92748	−0.963740	+	0.266843i	$0.914019\pi$
$644$	175.441i	0.272423i
$645$	0	0
$646$	403.169	0.624101
$647$	19.9618i	0.0308529i	0.999881	+	0.0154264i	$0.00491059\pi$
$647$	19.9618i	0.0308529i	−0.999881	+	0.0154264i	$0.995089\pi$
$648$	0	0
$649$	1.01860	0.00156949
$650$	− 7.54931i	− 0.0116143i
$651$	0	0
$652$	654.263	1.00347
$653$	− 420.634i	− 0.644156i	−0.946713	−	0.322078i	$-0.895619\pi$
$653$	− 420.634i	− 0.644156i	0.946713	−	0.322078i	$-0.104381\pi$
$654$	0	0
$655$	774.360	1.18223
$656$	− 128.077i	− 0.195240i
$657$	0	0
$658$	−466.585	−0.709097
$659$	− 380.713i	− 0.577714i	−0.957372	−	0.288857i	$-0.906725\pi$
$659$	− 380.713i	− 0.577714i	0.957372	−	0.288857i	$-0.0932751\pi$
$660$	0	0
$661$	339.630	0.513812	0.256906	−	0.966436i	$-0.417297\pi$
$661$	339.630	0.513812	0.256906	+	0.966436i	$0.417297\pi$
$662$	− 460.476i	− 0.695583i
$663$	0	0
$664$	−293.381	−0.441839
$665$	389.000i	0.584963i
$666$	0	0
$667$	1242.41	1.86269
$668$	197.364i	0.295455i
$669$	0	0
$670$	−345.804	−0.516126
$671$	288.010i	0.429225i
$672$	0	0
$673$	−219.920	−0.326775	−0.163388	−	0.986562i	$-0.552242\pi$
$673$	−219.920	−0.326775	−0.163388	+	0.986562i	$0.552242\pi$
$674$	909.992i	1.35014i
$675$	0	0
$676$	365.515	0.540702
$677$	336.004i	0.496313i	0.968720	+	0.248157i	$0.0798248\pi$
$677$	336.004i	0.496313i	−0.968720	+	0.248157i	$0.920175\pi$
$678$	0	0
$679$	286.578	0.422058
$680$	− 137.230i	− 0.201809i
$681$	0	0
$682$	−398.971	−0.585001
$683$	231.915i	0.339553i	0.985483	+	0.169777i	$0.0543046\pi$
$683$	231.915i	0.339553i	−0.985483	+	0.169777i	$0.945695\pi$
$684$	0	0
$685$	110.645	0.161525
$686$	− 714.592i	− 1.04168i
$687$	0	0
$688$	969.662	1.40939
$689$	67.8983i	0.0985462i
$690$	0	0
$691$	404.654	0.585606	0.292803	−	0.956173i	$-0.405412\pi$
$691$	404.654	0.585606	0.292803	+	0.956173i	$0.405412\pi$
$692$	423.914i	0.612592i
$693$	0	0
$694$	578.525	0.833609
$695$	− 1279.02i	− 1.84032i
$696$	0	0
$697$	−41.4071	−0.0594075
$698$	472.632i	0.677123i
$699$	0	0
$700$	21.6377	0.0309109
$701$	− 144.543i	− 0.206196i	−0.994671	−	0.103098i	$-0.967125\pi$
$701$	− 144.543i	− 0.206196i	0.994671	−	0.103098i	$-0.0328755\pi$
$702$	0	0
$703$	1654.00	2.35277
$704$	5.34333i	0.00758996i
$705$	0	0
$706$	−1615.34	−2.28802
$707$	379.485i	0.536755i
$708$	0	0
$709$	1033.34	1.45746	0.728730	−	0.684802i	$-0.240111\pi$
$709$	1033.34	1.45746	0.728730	+	0.684802i	$0.240111\pi$
$710$	− 476.969i	− 0.671787i
$711$	0	0
$712$	−290.603	−0.408150
$713$	1182.23i	1.65811i
$714$	0	0
$715$	−15.6839	−0.0219356
$716$	60.5258i	0.0845332i
$717$	0	0
$718$	−790.606	−1.10112
$719$	− 1286.83i	− 1.78974i	−0.446324	−	0.894872i	$-0.647267\pi$
$719$	− 1286.83i	− 1.78974i	0.446324	−	0.894872i	$-0.352733\pi$
$720$	0	0
$721$	−94.5376	−0.131120
$722$	− 672.162i	− 0.930973i
$723$	0	0
$724$	−422.329	−0.583328
$725$	− 153.231i	− 0.211353i
$726$	0	0
$727$	−835.384	−1.14908	−0.574542	−	0.818475i	$-0.694820\pi$
$727$	−835.384	−1.14908	−0.574542	+	0.818475i	$0.694820\pi$
$728$	15.0921i	0.0207308i
$729$	0	0
$730$	711.359	0.974465
$731$	− 313.489i	− 0.428850i
$732$	0	0
$733$	1300.03	1.77358	0.886790	−	0.462173i	$-0.152930\pi$
$733$	1300.03	1.77358	0.886790	+	0.462173i	$0.152930\pi$
$734$	24.8574i	0.0338657i
$735$	0	0
$736$	−769.178	−1.04508
$737$	− 98.4206i	− 0.133542i
$738$	0	0
$739$	993.426	1.34428	0.672142	−	0.740422i	$-0.265374\pi$
$739$	993.426	1.34428	0.672142	+	0.740422i	$0.265374\pi$
$740$	671.563i	0.907518i
$741$	0	0
$742$	−552.361	−0.744421
$743$	762.477i	1.02621i	0.858325	+	0.513107i	$0.171506\pi$
$743$	762.477i	1.02621i	−0.858325	+	0.513107i	$0.828494\pi$
$744$	0	0
$745$	−162.978	−0.218762
$746$	− 928.035i	− 1.24401i
$747$	0	0
$748$	−46.5903	−0.0622865
$749$	− 394.432i	− 0.526612i
$750$	0	0
$751$	−647.654	−0.862389	−0.431194	−	0.902259i	$-0.641908\pi$
$751$	−647.654	−0.862389	−0.431194	+	0.902259i	$0.641908\pi$
$752$	− 1135.68i	− 1.51022i
$753$	0	0
$754$	−127.490	−0.169085
$755$	− 581.401i	− 0.770068i
$756$	0	0
$757$	−43.6130	−0.0576130	−0.0288065	−	0.999585i	$-0.509171\pi$
$757$	−43.6130	−0.0576130	−0.0288065	+	0.999585i	$0.509171\pi$
$758$	679.969i	0.897057i
$759$	0	0
$760$	−534.154	−0.702834
$761$	130.629i	0.171655i	0.996310	+	0.0858274i	$0.0273534\pi$
$761$	130.629i	0.171655i	−0.996310	+	0.0858274i	$0.972647\pi$
$762$	0	0
$763$	253.398	0.332108
$764$	− 646.774i	− 0.846563i
$765$	0	0
$766$	−320.739	−0.418719
$767$	0.309723i	0 0.000403811i
$768$	0	0
$769$	−1261.92	−1.64099	−0.820495	−	0.571654i	$-0.806302\pi$
$769$	−1261.92	−1.64099	−0.820495	+	0.571654i	$0.806302\pi$
$770$	− 127.591i	− 0.165702i
$771$	0	0
$772$	430.588	0.557757
$773$	436.493i	0.564674i	0.959315	+	0.282337i	$0.0911097\pi$
$773$	436.493i	0.564674i	−0.959315	+	0.282337i	$0.908890\pi$
$774$	0	0
$775$	145.809	0.188140
$776$	393.513i	0.507104i
$777$	0	0
$778$	774.577	0.995600
$779$	161.173i	0.206897i
$780$	0	0
$781$	135.752	0.173818
$782$	391.849i	0.501086i
$783$	0	0
$784$	760.853	0.970475
$785$	− 737.578i	− 0.939590i
$786$	0	0
$787$	38.8895	0.0494148	0.0247074	−	0.999695i	$-0.492135\pi$
$787$	38.8895	0.0494148	0.0247074	+	0.999695i	$0.492135\pi$
$788$	− 542.538i	− 0.688500i
$789$	0	0
$790$	−995.485	−1.26011
$791$	− 278.830i	− 0.352504i
$792$	0	0
$793$	−87.5749	−0.110435
$794$	− 1117.25i	− 1.40712i
$795$	0	0
$796$	−221.513	−0.278282
$797$	− 864.722i	− 1.08497i	−0.840065	−	0.542486i	$-0.817483\pi$
$797$	− 864.722i	− 1.08497i	0.840065	−	0.542486i	$-0.182517\pi$
$798$	0	0
$799$	−367.163	−0.459528
$800$	94.8653i	0.118582i
$801$	0	0
$802$	−866.276	−1.08014
$803$	202.462i	0.252133i
$804$	0	0
$805$	−378.078	−0.469662
$806$	− 121.315i	− 0.150514i
$807$	0	0
$808$	−521.089	−0.644912
$809$	− 267.814i	− 0.331043i	−0.986206	−	0.165521i	$-0.947069\pi$
$809$	− 267.814i	− 0.331043i	0.986206	−	0.165521i	$-0.0529307\pi$
$810$	0	0
$811$	62.9645	0.0776381	0.0388190	−	0.999246i	$-0.487640\pi$
$811$	62.9645	0.0776381	0.0388190	+	0.999246i	$0.487640\pi$
$812$	− 365.409i	− 0.450011i
$813$	0	0
$814$	−542.505	−0.666468
$815$	1409.95i	1.73000i
$816$	0	0
$817$	−1220.23	−1.49354
$818$	1323.53i	1.61800i
$819$	0	0
$820$	−65.4402	−0.0798051
$821$	205.138i	0.249863i	0.992165	+	0.124932i	$0.0398712\pi$
$821$	205.138i	0.249863i	−0.992165	+	0.124932i	$0.960129\pi$
$822$	0	0
$823$	92.1457	0.111963	0.0559816	−	0.998432i	$-0.482171\pi$
$823$	92.1457	0.111963	0.0559816	+	0.998432i	$0.482171\pi$
$824$	− 129.814i	− 0.157541i
$825$	0	0
$826$	−2.51963	−0.00305041
$827$	152.560i	0.184475i	0.995737	+	0.0922373i	$0.0294018\pi$
$827$	152.560i	0.184475i	−0.995737	+	0.0922373i	$0.970598\pi$
$828$	0	0
$829$	−527.763	−0.636626	−0.318313	−	0.947986i	$-0.603116\pi$
$829$	−527.763	−0.636626	−0.318313	+	0.947986i	$0.603116\pi$
$830$	754.180i	0.908651i
$831$	0	0
$832$	−1.62474	−0.00195281
$833$	− 245.982i	− 0.295296i
$834$	0	0
$835$	−425.322	−0.509368
$836$	181.348i	0.216924i
$837$	0	0
$838$	665.519	0.794176
$839$	548.065i	0.653237i	0.945156	+	0.326618i	$0.105909\pi$
$839$	548.065i	0.653237i	−0.945156	+	0.326618i	$0.894091\pi$
$840$	0	0
$841$	−1746.71	−2.07694
$842$	− 923.116i	− 1.09634i
$843$	0	0
$844$	82.5650	0.0978259
$845$	787.691i	0.932178i
$846$	0	0
$847$	36.3140	0.0428737
$848$	− 1344.46i	− 1.58545i
$849$	0	0
$850$	48.3280	0.0568565
$851$	1607.56i	1.88902i
$852$	0	0
$853$	−88.0263	−0.103196	−0.0515980	−	0.998668i	$-0.516431\pi$
$853$	−88.0263	−0.103196	−0.0515980	+	0.998668i	$0.516431\pi$
$854$	− 712.432i	− 0.834230i
$855$	0	0
$856$	541.613	0.632725
$857$	1341.27i	1.56508i	0.622601	+	0.782539i	$0.286076\pi$
$857$	1341.27i	1.56508i	−0.622601	+	0.782539i	$0.713924\pi$
$858$	0	0
$859$	1230.21	1.43214	0.716069	−	0.698030i	$-0.245940\pi$
$859$	1230.21	1.43214	0.716069	+	0.698030i	$0.245940\pi$
$860$	− 495.442i	− 0.576095i
$861$	0	0
$862$	710.538	0.824290
$863$	− 218.670i	− 0.253383i	−0.991942	−	0.126692i	$-0.959564\pi$
$863$	− 218.670i	− 0.253383i	0.991942	−	0.126692i	$-0.0404358\pi$
$864$	0	0
$865$	−913.542	−1.05612
$866$	− 569.071i	− 0.657126i
$867$	0	0
$868$	347.709	0.400586
$869$	− 283.329i	− 0.326040i
$870$	0	0
$871$	29.9266	0.0343589
$872$	347.953i	0.399029i
$873$	0	0
$874$	1525.23	1.74512
$875$	433.631i	0.495578i
$876$	0	0
$877$	547.663	0.624474	0.312237	−	0.950004i	$-0.398922\pi$
$877$	547.663	0.624474	0.312237	+	0.950004i	$0.398922\pi$
$878$	2027.55i	2.30928i
$879$	0	0
$880$	310.559	0.352908
$881$	235.271i	0.267050i	0.991045	+	0.133525i	$0.0426297\pi$
$881$	235.271i	0.267050i	−0.991045	+	0.133525i	$0.957370\pi$
$882$	0	0
$883$	−246.902	−0.279617	−0.139808	−	0.990179i	$-0.544649\pi$
$883$	−246.902	−0.279617	−0.139808	+	0.990179i	$0.544649\pi$
$884$	− 14.1667i	− 0.0160256i
$885$	0	0
$886$	−1079.62	−1.21854
$887$	− 667.037i	− 0.752014i	−0.926617	−	0.376007i	$-0.877297\pi$
$887$	− 667.037i	− 0.752014i	0.926617	−	0.376007i	$-0.122703\pi$
$888$	0	0
$889$	329.686	0.370850
$890$	747.038i	0.839368i
$891$	0	0
$892$	−260.029	−0.291513
$893$	1429.15i	1.60039i
$894$	0	0
$895$	−130.434	−0.145736
$896$	− 429.090i	− 0.478895i
$897$	0	0
$898$	181.669	0.202304
$899$	− 2462.36i	− 2.73900i
$900$	0	0
$901$	−434.661	−0.482420
$902$	− 52.8641i	− 0.0586077i
$903$	0	0
$904$	382.875	0.423534
$905$	− 910.128i	− 1.00567i
$906$	0	0
$907$	−115.024	−0.126819	−0.0634093	−	0.997988i	$-0.520197\pi$
$907$	−115.024	−0.126819	−0.0634093	+	0.997988i	$0.520197\pi$
$908$	− 508.038i	− 0.559513i
$909$	0	0
$910$	38.7963	0.0426333
$911$	− 545.185i	− 0.598447i	−0.954183	−	0.299223i	$-0.903272\pi$
$911$	− 545.185i	− 0.598447i	0.954183	−	0.299223i	$-0.0967276\pi$
$912$	0	0
$913$	−214.650	−0.235104
$914$	122.122i	0.133613i
$915$	0	0
$916$	370.564	0.404545
$917$	− 545.173i	− 0.594518i
$918$	0	0
$919$	−116.321	−0.126573	−0.0632866	−	0.997995i	$-0.520158\pi$
$919$	−116.321	−0.126573	−0.0632866	+	0.997995i	$0.520158\pi$
$920$	− 519.156i	− 0.564300i
$921$	0	0
$922$	1634.34	1.77260
$923$	41.2779i	0.0447214i
$924$	0	0
$925$	198.265	0.214341
$926$	486.349i	0.525215i
$927$	0	0
$928$	1602.05	1.72635
$929$	− 25.3182i	− 0.0272531i	−0.999907	−	0.0136266i	$-0.995662\pi$
$929$	− 25.3182i	− 0.0272531i	0.999907	−	0.0136266i	$-0.00433760\pi$
$930$	0	0
$931$	−957.459	−1.02842
$932$	− 325.093i	− 0.348812i
$933$	0	0
$934$	−33.7823	−0.0361695
$935$	− 100.403i	− 0.107383i
$936$	0	0
$937$	1360.70	1.45219	0.726093	−	0.687597i	$-0.241334\pi$
$937$	1360.70	1.45219	0.726093	+	0.687597i	$0.241334\pi$
$938$	243.457i	0.259549i
$939$	0	0
$940$	−580.269	−0.617307
$941$	− 908.864i	− 0.965850i	−0.875662	−	0.482925i	$-0.839574\pi$
$941$	− 908.864i	− 0.965850i	0.875662	−	0.482925i	$-0.160426\pi$
$942$	0	0
$943$	−156.647	−0.166116
$944$	− 6.13286i	− 0.00649668i
$945$	0	0
$946$	400.230	0.423076
$947$	1593.80i	1.68300i	0.540256	+	0.841501i	$0.318327\pi$
$947$	1593.80i	1.68300i	−0.540256	+	0.841501i	$0.681673\pi$
$948$	0	0
$949$	−61.5625	−0.0648709
$950$	− 188.112i	− 0.198013i
$951$	0	0
$952$	−96.6140	−0.101485
$953$	163.175i	0.171222i	0.996329	+	0.0856110i	$0.0272842\pi$
$953$	163.175i	0.171222i	−0.996329	+	0.0856110i	$0.972716\pi$
$954$	0	0
$955$	1393.81	1.45949
$956$	− 582.351i	− 0.609154i
$957$	0	0
$958$	918.541	0.958811
$959$	− 77.8971i	− 0.0812274i
$960$	0	0
$961$	1382.09	1.43818
$962$	− 164.959i	− 0.171475i
$963$	0	0
$964$	−440.269	−0.456710
$965$	927.926i	0.961582i
$966$	0	0
$967$	751.225	0.776862	0.388431	−	0.921478i	$-0.373017\pi$
$967$	751.225	0.776862	0.388431	+	0.921478i	$0.373017\pi$
$968$	49.8644i	0.0515129i
$969$	0	0
$970$	1011.58	1.04287
$971$	− 134.041i	− 0.138044i	−0.997615	−	0.0690221i	$-0.978012\pi$
$971$	− 134.041i	− 0.138044i	0.997615	−	0.0690221i	$-0.0219879\pi$
$972$	0	0
$973$	−900.470	−0.925458
$974$	1207.51i	1.23974i
$975$	0	0
$976$	1734.08	1.77672
$977$	− 1081.18i	− 1.10664i	−0.832970	−	0.553319i	$-0.813361\pi$
$977$	− 1081.18i	− 1.10664i	0.832970	−	0.553319i	$-0.186639\pi$
$978$	0	0
$979$	−212.617	−0.217178
$980$	− 388.752i	− 0.396686i
$981$	0	0
$982$	−1338.37	−1.36290
$983$	1114.11i	1.13337i	0.823933	+	0.566687i	$0.191775\pi$
$983$	1114.11i	1.13337i	−0.823933	+	0.566687i	$0.808225\pi$
$984$	0	0
$985$	1169.18	1.18698
$986$	− 816.146i	− 0.827734i
$987$	0	0
$988$	−55.1423	−0.0558121
$989$	− 1185.96i	− 1.19916i
$990$	0	0
$991$	−1199.41	−1.21031	−0.605153	−	0.796109i	$-0.706888\pi$
$991$	−1199.41	−1.21031	−0.605153	+	0.796109i	$0.706888\pi$
$992$	1524.45i	1.53674i
$993$	0	0
$994$	−335.800	−0.337827
$995$	− 477.364i	− 0.479763i
$996$	0	0
$997$	−995.763	−0.998760	−0.499380	−	0.866383i	$-0.666439\pi$
$997$	−995.763	−0.998760	−0.499380	+	0.866383i	$0.666439\pi$
$998$	2359.02i	2.36375i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.3.b.a.89.2	✓	8
3.2	odd	2	inner	99.3.b.a.89.7	yes	8
4.3	odd	2		1584.3.i.b.881.4		8
11.10	odd	2		1089.3.b.g.485.7		8
12.11	even	2		1584.3.i.b.881.5		8
33.32	even	2		1089.3.b.g.485.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.3.b.a.89.2	✓	8	1.1	even	1	trivial
99.3.b.a.89.7	yes	8	3.2	odd	2	inner
1089.3.b.g.485.2		8	33.32	even	2
1089.3.b.g.485.7		8	11.10	odd	2
1584.3.i.b.881.4		8	4.3	odd	2
1584.3.i.b.881.5		8	12.11	even	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:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	99.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.48514i q^{2} -2.17590 q^{4} -4.68911i q^{5} -3.30128 q^{7} -4.53313i q^{8} +O(q^{10})\)	\(q-2.48514i q^{2} -2.17590 q^{4} -4.68911i q^{5} -3.30128 q^{7} -4.53313i q^{8} -11.6531 q^{10} -3.31662i q^{11} +1.00848 q^{13} +8.20412i q^{14} -19.9691 q^{16} +6.45594i q^{17} +25.1291 q^{19} +10.2030i q^{20} -8.24226 q^{22} +24.4236i q^{23} +3.01224 q^{25} -2.50622i q^{26} +7.18325 q^{28} -50.8695i q^{29} +48.4055 q^{31} +31.4933i q^{32} +16.0439 q^{34} +15.4800i q^{35} +65.8199 q^{37} -62.4493i q^{38} -21.2564 q^{40} +6.41379i q^{41} -48.5582 q^{43} +7.21665i q^{44} +60.6959 q^{46} +56.8721i q^{47} -38.1016 q^{49} -7.48582i q^{50} -2.19436 q^{52} +67.3272i q^{53} -15.5520 q^{55} +14.9651i q^{56} -126.418 q^{58} +0.307118i q^{59} -86.8383 q^{61} -120.294i q^{62} -1.61108 q^{64} -4.72888i q^{65} +29.6749 q^{67} -14.0475i q^{68} +38.4700 q^{70} +40.9307i q^{71} -61.0447 q^{73} -163.571i q^{74} -54.6785 q^{76} +10.9491i q^{77} +85.4268 q^{79} +93.6371i q^{80} +15.9391 q^{82} -64.7194i q^{83} +30.2726 q^{85} +120.674i q^{86} -15.0347 q^{88} -64.1065i q^{89} -3.32928 q^{91} -53.1433i q^{92} +141.335 q^{94} -117.833i q^{95} -86.8082 q^{97} +94.6876i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} + 16 q^{7}+O(q^{10}) \) 8 * q - 16 * q^4 + 16 * q^7	\( 8 q - 16 q^{4} + 16 q^{7} - 48 q^{10} - 8 q^{13} + 104 q^{16} + 40 q^{19} - 112 q^{25} - 32 q^{28} - 56 q^{31} - 216 q^{34} + 136 q^{37} + 432 q^{40} - 104 q^{43} + 24 q^{46} - 96 q^{49} + 280 q^{52} - 432 q^{58} - 8 q^{61} - 592 q^{64} + 112 q^{67} + 168 q^{70} + 448 q^{73} - 344 q^{76} + 448 q^{79} + 504 q^{82} + 48 q^{85} - 264 q^{88} - 544 q^{91} + 360 q^{94} - 152 q^{97}+O(q^{100}) \) 8 * q - 16 * q^4 + 16 * q^7 - 48 * q^10 - 8 * q^13 + 104 * q^16 + 40 * q^19 - 112 * q^25 - 32 * q^28 - 56 * q^31 - 216 * q^34 + 136 * q^37 + 432 * q^40 - 104 * q^43 + 24 * q^46 - 96 * q^49 + 280 * q^52 - 432 * q^58 - 8 * q^61 - 592 * q^64 + 112 * q^67 + 168 * q^70 + 448 * q^73 - 344 * q^76 + 448 * q^79 + 504 * q^82 + 48 * q^85 - 264 * q^88 - 544 * q^91 + 360 * q^94 - 152 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

Properties

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