LMFDB - Embedded newform 99.4.d.a.98.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,4,Mod(98,99)]

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, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("99.98");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 99 = 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	99.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:	$5.84118909057$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			98.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	99.98
Dual form			99.4.d.a.98.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{2} +1.00000 q^{4} -19.7990i q^{5} +16.9706i q^{7} +21.0000 q^{8} +O(q^{10})$	\(q-3.00000 q^{2} +1.00000 q^{4} -19.7990i q^{5} +16.9706i q^{7} +21.0000 q^{8} +59.3970i q^{10} +(-33.0000 + 15.5563i) q^{11} -29.6985i q^{13} -50.9117i q^{14} -71.0000 q^{16} -126.000 q^{17} +89.0955i q^{19} -19.7990i q^{20} +(99.0000 - 46.6690i) q^{22} +120.208i q^{23} -267.000 q^{25} +89.0955i q^{26} +16.9706i q^{28} +24.0000 q^{29} -70.0000 q^{31} +45.0000 q^{32} +378.000 q^{34} +336.000 q^{35} +182.000 q^{37} -267.286i q^{38} -415.779i q^{40} -294.000 q^{41} +4.24264i q^{43} +(-33.0000 + 15.5563i) q^{44} -360.624i q^{46} -108.894i q^{47} +55.0000 q^{49} +801.000 q^{50} -29.6985i q^{52} -147.078i q^{53} +(308.000 + 653.367i) q^{55} +356.382i q^{56} -72.0000 q^{58} +514.774i q^{59} -326.683i q^{61} +210.000 q^{62} +433.000 q^{64} -588.000 q^{65} -880.000 q^{67} -126.000 q^{68} -1008.00 q^{70} -337.997i q^{71} -178.191i q^{73} -546.000 q^{74} +89.0955i q^{76} +(-264.000 - 560.029i) q^{77} -772.161i q^{79} +1405.73i q^{80} +882.000 q^{82} -1218.00 q^{83} +2494.67i q^{85} -12.7279i q^{86} +(-693.000 + 326.683i) q^{88} -1534.42i q^{89} +504.000 q^{91} +120.208i q^{92} +326.683i q^{94} +1764.00 q^{95} -196.000 q^{97} -165.000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{2} + 2 q^{4} + 42 q^{8}+O(q^{10}) $ 2 * q - 6 * q^2 + 2 * q^4 + 42 * q^8	$ 2 q - 6 q^{2} + 2 q^{4} + 42 q^{8} - 66 q^{11} - 142 q^{16} - 252 q^{17} + 198 q^{22} - 534 q^{25} + 48 q^{29} - 140 q^{31} + 90 q^{32} + 756 q^{34} + 672 q^{35} + 364 q^{37} - 588 q^{41} - 66 q^{44} + 110 q^{49} + 1602 q^{50} + 616 q^{55} - 144 q^{58} + 420 q^{62} + 866 q^{64} - 1176 q^{65} - 1760 q^{67} - 252 q^{68} - 2016 q^{70} - 1092 q^{74} - 528 q^{77} + 1764 q^{82} - 2436 q^{83} - 1386 q^{88} + 1008 q^{91} + 3528 q^{95} - 392 q^{97} - 330 q^{98}+O(q^{100}) $ 2 * q - 6 * q^2 + 2 * q^4 + 42 * q^8 - 66 * q^11 - 142 * q^16 - 252 * q^17 + 198 * q^22 - 534 * q^25 + 48 * q^29 - 140 * q^31 + 90 * q^32 + 756 * q^34 + 672 * q^35 + 364 * q^37 - 588 * q^41 - 66 * q^44 + 110 * q^49 + 1602 * q^50 + 616 * q^55 - 144 * q^58 + 420 * q^62 + 866 * q^64 - 1176 * q^65 - 1760 * q^67 - 252 * q^68 - 2016 * q^70 - 1092 * q^74 - 528 * q^77 + 1764 * q^82 - 2436 * q^83 - 1386 * q^88 + 1008 * q^91 + 3528 * q^95 - 392 * q^97 - 330 * q^98

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 $

$2$	−3.00000			−1.06066			−0.530330	−	0.847791i	$-0.677932\pi$
$2$	−3.00000			−1.06066			−0.530330	+	0.847791i	$0.677932\pi$
$3$		0			0
$3$		0			0
$4$	1.00000			0.125000
$5$		−	19.7990i		−	1.77088i	−0.464758	−	0.885438i	$-0.653859\pi$
$5$		−	19.7990i		−	1.77088i	0.464758	−	0.885438i	$-0.346141\pi$
$6$		0			0
$7$			16.9706i			0.916324i	0.888869	+	0.458162i	$0.151492\pi$
$7$			16.9706i			0.916324i	−0.888869	+	0.458162i	$0.848508\pi$
$8$	21.0000			0.928078
$9$		0			0
$10$			59.3970i			1.87830i
$11$	−33.0000	+	15.5563i	−0.904534	+	0.426401i
$11$	−33.0000	+	15.5563i	−0.904534	+	0.426401i
$12$		0			0
$13$		−	29.6985i		−	0.633606i	−0.948491	−	0.316803i	$-0.897391\pi$
$13$		−	29.6985i		−	0.633606i	0.948491	−	0.316803i	$-0.102609\pi$
$14$		−	50.9117i		−	0.971909i
$15$		0			0
$16$	−71.0000			−1.10938
$17$	−126.000			−1.79762			−0.898808	−	0.438342i	$-0.855566\pi$
$17$	−126.000			−1.79762			−0.898808	+	0.438342i	$0.855566\pi$
$18$		0			0
$19$			89.0955i			1.07578i	0.843014	+	0.537892i	$0.180779\pi$
$19$			89.0955i			1.07578i	−0.843014	+	0.537892i	$0.819221\pi$
$20$		−	19.7990i		−	0.221359i
$21$		0			0
$22$	99.0000	−	46.6690i	0.959403	−	0.452267i
$23$			120.208i			1.08979i	0.838505	+	0.544894i	$0.183430\pi$
$23$			120.208i			1.08979i	−0.838505	+	0.544894i	$0.816570\pi$
$24$		0			0
$25$	−267.000			−2.13600
$26$			89.0955i			0.672041i
$27$		0			0
$28$			16.9706i			0.114541i
$29$	24.0000			0.153679			0.0768395	−	0.997043i	$-0.475517\pi$
$29$	24.0000			0.153679			0.0768395	+	0.997043i	$0.475517\pi$
$30$		0			0
$31$	−70.0000			−0.405560			−0.202780	−	0.979224i	$-0.564998\pi$
$31$	−70.0000			−0.405560			−0.202780	+	0.979224i	$0.564998\pi$
$32$	45.0000			0.248592
$33$		0			0
$34$	378.000			1.90666
$35$	336.000			1.62270
$36$		0			0
$37$	182.000			0.808665			0.404333	−	0.914612i	$-0.367504\pi$
$37$	182.000			0.808665			0.404333	+	0.914612i	$0.367504\pi$
$38$		−	267.286i		−	1.14104i
$39$		0			0
$40$		−	415.779i		−	1.64351i
$41$	−294.000			−1.11988			−0.559940	−	0.828533i	$-0.689176\pi$
$41$	−294.000			−1.11988			−0.559940	+	0.828533i	$0.689176\pi$
$42$		0			0
$43$			4.24264i			0.0150464i	0.999972	+	0.00752322i	$0.00239474\pi$
$43$			4.24264i			0.0150464i	−0.999972	+	0.00752322i	$0.997605\pi$
$44$	−33.0000	+	15.5563i	−0.113067	+	0.0533002i
$45$		0			0
$46$		−	360.624i		−	1.15590i
$47$		−	108.894i		−	0.337955i	−0.985620	−	0.168978i	$-0.945953\pi$
$47$		−	108.894i		−	0.337955i	0.985620	−	0.168978i	$-0.0540465\pi$
$48$		0			0
$49$	55.0000			0.160350
$50$	801.000			2.26557
$51$		0			0
$52$		−	29.6985i		−	0.0792007i
$53$		−	147.078i		−	0.381184i	−0.981669	−	0.190592i	$-0.938959\pi$
$53$		−	147.078i		−	0.381184i	0.981669	−	0.190592i	$-0.0610407\pi$
$54$		0			0
$55$	308.000	+	653.367i	0.755104	+	1.60182i
$56$			356.382i			0.850420i
$57$		0			0
$58$	−72.0000			−0.163001
$59$			514.774i			1.13590i	0.823065	+	0.567948i	$0.192262\pi$
$59$			514.774i			1.13590i	−0.823065	+	0.567948i	$0.807738\pi$
$60$		0			0
$61$		−	326.683i		−	0.685697i	−0.939391	−	0.342848i	$-0.888608\pi$
$61$		−	326.683i		−	0.685697i	0.939391	−	0.342848i	$-0.111392\pi$
$62$	210.000			0.430162
$63$		0			0
$64$	433.000			0.845703
$65$	−588.000			−1.12204
$66$		0			0
$67$	−880.000			−1.60461			−0.802307	−	0.596912i	$-0.796394\pi$
$67$	−880.000			−1.60461			−0.802307	+	0.596912i	$0.796394\pi$
$68$	−126.000			−0.224702
$69$		0			0
$70$	−1008.00			−1.72113
$71$		−	337.997i		−	0.564970i	−0.959272	−	0.282485i	$-0.908841\pi$
$71$		−	337.997i		−	0.564970i	0.959272	−	0.282485i	$-0.0911587\pi$
$72$		0			0
$73$		−	178.191i		−	0.285694i	−0.989745	−	0.142847i	$-0.954374\pi$
$73$		−	178.191i		−	0.285694i	0.989745	−	0.142847i	$-0.0456257\pi$
$74$	−546.000			−0.857719
$75$		0			0
$76$			89.0955i			0.134473i
$77$	−264.000	−	560.029i	−0.390722	−	0.828846i
$78$		0			0
$79$		−	772.161i		−	1.09968i	−0.835269	−	0.549841i	$-0.814688\pi$
$79$		−	772.161i		−	1.09968i	0.835269	−	0.549841i	$-0.185312\pi$
$80$			1405.73i			1.96456i
$81$		0			0
$82$	882.000			1.18781
$83$	−1218.00			−1.61076			−0.805379	−	0.592761i	$-0.798038\pi$
$83$	−1218.00			−1.61076			−0.805379	+	0.592761i	$0.798038\pi$
$84$		0			0
$85$			2494.67i			3.18336i
$86$		−	12.7279i		−	0.0159592i
$87$		0			0
$88$	−693.000	+	326.683i	−0.839478	+	0.395734i
$89$		−	1534.42i		−	1.82751i	−0.406266	−	0.913755i	$-0.633169\pi$
$89$		−	1534.42i		−	1.82751i	0.406266	−	0.913755i	$-0.366831\pi$
$90$		0			0
$91$	504.000			0.580589
$92$			120.208i			0.136224i
$93$		0			0
$94$			326.683i			0.358455i
$95$	1764.00			1.90508
$96$		0			0
$97$	−196.000			−0.205163			−0.102581	−	0.994725i	$-0.532710\pi$
$97$	−196.000			−0.205163			−0.102581	+	0.994725i	$0.532710\pi$
$98$	−165.000			−0.170077
$99$		0			0
$100$	−267.000			−0.267000
$101$	546.000			0.537911			0.268956	−	0.963153i	$-0.413322\pi$
$101$	546.000			0.537911			0.268956	+	0.963153i	$0.413322\pi$
$102$		0			0
$103$	−826.000			−0.790177			−0.395088	−	0.918643i	$-0.629286\pi$
$103$	−826.000			−0.790177			−0.395088	+	0.918643i	$0.629286\pi$
$104$		−	623.668i		−	0.588036i
$105$		0			0
$106$			441.235i			0.404307i
$107$	−108.000			−0.0975771			−0.0487886	−	0.998809i	$-0.515536\pi$
$107$	−108.000			−0.0975771			−0.0487886	+	0.998809i	$0.515536\pi$
$108$		0			0
$109$			1183.70i			1.04016i	0.854117	+	0.520081i	$0.174098\pi$
$109$			1183.70i			1.04016i	−0.854117	+	0.520081i	$0.825902\pi$
$110$	−924.000	−	1960.10i	−0.800909	−	1.69898i
$111$		0			0
$112$		−	1204.91i		−	1.01655i
$113$			1622.10i			1.35039i	0.737637	+	0.675197i	$0.235942\pi$
$113$			1622.10i			1.35039i	−0.737637	+	0.675197i	$0.764058\pi$
$114$		0			0
$115$	2380.00			1.92988
$116$	24.0000			0.0192099
$117$		0			0
$118$		−	1544.32i		−	1.20480i
$119$		−	2138.29i		−	1.64720i
$120$		0			0
$121$	847.000	−	1026.72i	0.636364	−	0.771389i
$122$			980.050i			0.727291i
$123$		0			0
$124$	−70.0000			−0.0506950
$125$			2811.46i			2.01171i
$126$		0			0
$127$		−	178.191i		−	0.124503i	−0.998060	−	0.0622515i	$-0.980172\pi$
$127$		−	178.191i		−	0.124503i	0.998060	−	0.0622515i	$-0.0198281\pi$
$128$	−1659.00			−1.14560
$129$		0			0
$130$	1764.00			1.19010
$131$	588.000			0.392166			0.196083	−	0.980587i	$-0.437178\pi$
$131$	588.000			0.392166			0.196083	+	0.980587i	$0.437178\pi$
$132$		0			0
$133$	−1512.00			−0.985767
$134$	2640.00			1.70195
$135$		0			0
$136$	−2646.00			−1.66833
$137$		−	337.997i		−	0.210781i	−0.994431	−	0.105391i	$-0.966391\pi$
$137$		−	337.997i		−	0.210781i	0.994431	−	0.105391i	$-0.0336093\pi$
$138$		0			0
$139$		−	2880.75i		−	1.75786i	−0.476952	−	0.878929i	$-0.658259\pi$
$139$		−	2880.75i		−	1.75786i	0.476952	−	0.878929i	$-0.341741\pi$
$140$	336.000			0.202837
$141$		0			0
$142$			1013.99i			0.599241i
$143$	462.000	+	980.050i	0.270170	+	0.573118i
$144$		0			0
$145$		−	475.176i		−	0.272146i
$146$			534.573i			0.303024i
$147$		0			0
$148$	182.000			0.101083
$149$	2082.00			1.14473			0.572363	−	0.820001i	$-0.306027\pi$
$149$	2082.00			1.14473			0.572363	+	0.820001i	$0.306027\pi$
$150$		0			0
$151$		−	1637.66i		−	0.882588i	−0.897363	−	0.441294i	$-0.854520\pi$
$151$		−	1637.66i		−	0.882588i	0.897363	−	0.441294i	$-0.145480\pi$
$152$			1871.00i			0.998411i
$153$		0			0
$154$	792.000	+	1680.09i	0.414423	+	0.879124i
$155$			1385.93i			0.718197i
$156$		0			0
$157$	−2338.00			−1.18849			−0.594244	−	0.804285i	$-0.702549\pi$
$157$	−2338.00			−1.18849			−0.594244	+	0.804285i	$0.702549\pi$
$158$			2316.48i			1.16639i
$159$		0			0
$160$		−	890.955i		−	0.440226i
$161$	−2040.00			−0.998600
$162$		0			0
$163$	128.000			0.0615076			0.0307538	−	0.999527i	$-0.490209\pi$
$163$	128.000			0.0615076			0.0307538	+	0.999527i	$0.490209\pi$
$164$	−294.000			−0.139985
$165$		0			0
$166$	3654.00			1.70847
$167$	−1176.00			−0.544920			−0.272460	−	0.962167i	$-0.587837\pi$
$167$	−1176.00			−0.544920			−0.272460	+	0.962167i	$0.587837\pi$
$168$		0			0
$169$	1315.00			0.598543
$170$		−	7484.02i		−	3.37646i
$171$		0			0
$172$			4.24264i			0.00188080i
$173$	−1092.00			−0.479903			−0.239952	−	0.970785i	$-0.577132\pi$
$173$	−1092.00			−0.479903			−0.239952	+	0.970785i	$0.577132\pi$
$174$		0			0
$175$		−	4531.14i		−	1.95727i
$176$	2343.00	−	1104.50i	1.00347	−	0.473039i
$177$		0			0
$178$			4603.27i			1.93837i
$179$			387.495i			0.161803i	0.996722	+	0.0809014i	$0.0257799\pi$
$179$			387.495i			0.161803i	−0.996722	+	0.0809014i	$0.974220\pi$
$180$		0			0
$181$	−1330.00			−0.546177			−0.273089	−	0.961989i	$-0.588045\pi$
$181$	−1330.00			−0.546177			−0.273089	+	0.961989i	$0.588045\pi$
$182$	−1512.00			−0.615807
$183$		0			0
$184$			2524.37i			1.01141i
$185$		−	3603.42i		−	1.43205i
$186$		0			0
$187$	4158.00	−	1960.10i	1.62601	−	0.766506i
$188$		−	108.894i		−	0.0422444i
$189$		0			0
$190$	−5292.00			−2.02064
$191$			3404.01i			1.28956i	0.764369	+	0.644779i	$0.223051\pi$
$191$			3404.01i			1.28956i	−0.764369	+	0.644779i	$0.776949\pi$
$192$		0			0
$193$			3419.57i			1.27537i	0.770298	+	0.637684i	$0.220107\pi$
$193$			3419.57i			1.27537i	−0.770298	+	0.637684i	$0.779893\pi$
$194$	588.000			0.217608
$195$		0			0
$196$	55.0000			0.0200437
$197$	3798.00			1.37359			0.686793	−	0.726853i	$-0.259018\pi$
$197$	3798.00			1.37359			0.686793	+	0.726853i	$0.259018\pi$
$198$		0			0
$199$	560.000			0.199484			0.0997421	−	0.995013i	$-0.468198\pi$
$199$	560.000			0.199484			0.0997421	+	0.995013i	$0.468198\pi$
$200$	−5607.00			−1.98237
$201$		0			0
$202$	−1638.00			−0.570541
$203$			407.294i			0.140820i
$204$		0			0
$205$			5820.90i			1.98317i
$206$	2478.00			0.838109
$207$		0			0
$208$			2108.59i			0.702907i
$209$	−1386.00	−	2940.15i	−0.458716	−	0.973083i
$210$		0			0
$211$			1904.95i			0.621525i	0.950488	+	0.310763i	$0.100584\pi$
$211$			1904.95i			0.621525i	−0.950488	+	0.310763i	$0.899416\pi$
$212$		−	147.078i		−	0.0476480i
$213$		0			0
$214$	324.000			0.103496
$215$	84.0000			0.0266454
$216$		0			0
$217$		−	1187.94i		−	0.371625i
$218$		−	3551.09i		−	1.10326i
$219$		0			0
$220$	308.000	+	653.367i	0.0943880	+	0.200227i
$221$			3742.01i			1.13898i
$222$		0			0
$223$	−2968.00			−0.891264			−0.445632	−	0.895216i	$-0.647021\pi$
$223$	−2968.00			−0.891264			−0.445632	+	0.895216i	$0.647021\pi$
$224$			763.675i			0.227791i
$225$		0			0
$226$		−	4866.31i		−	1.43231i
$227$	3444.00			1.00699			0.503494	−	0.863999i	$-0.332048\pi$
$227$	3444.00			1.00699			0.503494	+	0.863999i	$0.332048\pi$
$228$		0			0
$229$	−574.000			−0.165638			−0.0828188	−	0.996565i	$-0.526392\pi$
$229$	−574.000			−0.165638			−0.0828188	+	0.996565i	$0.526392\pi$
$230$	−7140.00			−2.04695
$231$		0			0
$232$	504.000			0.142626
$233$	−5310.00			−1.49300			−0.746501	−	0.665384i	$-0.768268\pi$
$233$	−5310.00			−1.49300			−0.746501	+	0.665384i	$0.768268\pi$
$234$		0			0
$235$	−2156.00			−0.598476
$236$			514.774i			0.141987i
$237$		0			0
$238$			6414.87i			1.74712i
$239$	1236.00			0.334520			0.167260	−	0.985913i	$-0.446508\pi$
$239$	1236.00			0.334520			0.167260	+	0.985913i	$0.446508\pi$
$240$		0			0
$241$		−	3445.02i		−	0.920803i	−0.887711	−	0.460401i	$-0.847705\pi$
$241$		−	3445.02i		−	0.920803i	0.887711	−	0.460401i	$-0.152295\pi$
$242$	−2541.00	+	3080.16i	−0.674966	+	0.818182i
$243$		0			0
$244$		−	326.683i		−	0.0857121i
$245$		−	1088.94i		−	0.283960i
$246$		0			0
$247$	2646.00			0.681623
$248$	−1470.00			−0.376392
$249$		0			0
$250$		−	8434.37i		−	2.13375i
$251$			5682.31i			1.42894i	0.699665	+	0.714471i	$0.253332\pi$
$251$			5682.31i			1.42894i	−0.699665	+	0.714471i	$0.746668\pi$
$252$		0			0
$253$	−1870.00	−	3966.87i	−0.464687	−	0.985751i
$254$			534.573i			0.132055i
$255$		0			0
$256$	1513.00			0.369385
$257$			1494.82i			0.362819i	0.983408	+	0.181410i	$0.0580660\pi$
$257$			1494.82i			0.362819i	−0.983408	+	0.181410i	$0.941934\pi$
$258$		0			0
$259$			3088.64i			0.741000i
$260$	−588.000			−0.140255
$261$		0			0
$262$	−1764.00			−0.415955
$263$	4056.00			0.950965			0.475482	−	0.879725i	$-0.342274\pi$
$263$	4056.00			0.950965			0.475482	+	0.879725i	$0.342274\pi$
$264$		0			0
$265$	−2912.00			−0.675029
$266$	4536.00			1.04556
$267$		0			0
$268$	−880.000			−0.200577
$269$		−	6256.48i		−	1.41808i	−0.705167	−	0.709042i	$-0.749128\pi$
$269$		−	6256.48i		−	1.41808i	0.705167	−	0.709042i	$-0.250872\pi$
$270$		0			0
$271$		−	1603.72i		−	0.359479i	−0.983714	−	0.179740i	$-0.942474\pi$
$271$		−	1603.72i		−	0.359479i	0.983714	−	0.179740i	$-0.0575256\pi$
$272$	8946.00			1.99423
$273$		0			0
$274$			1013.99i			0.223567i
$275$	8811.00	−	4153.55i	1.93208	−	0.910793i
$276$		0			0
$277$			5019.04i			1.08868i	0.838864	+	0.544341i	$0.183220\pi$
$277$			5019.04i			1.08868i	−0.838864	+	0.544341i	$0.816780\pi$
$278$			8642.26i			1.86449i
$279$		0			0
$280$	7056.00			1.50599
$281$	−2874.00			−0.610137			−0.305068	−	0.952330i	$-0.598679\pi$
$281$	−2874.00			−0.610137			−0.305068	+	0.952330i	$0.598679\pi$
$282$		0			0
$283$		−	2167.99i		−	0.455384i	−0.973733	−	0.227692i	$-0.926882\pi$
$283$		−	2167.99i		−	0.455384i	0.973733	−	0.227692i	$-0.0731179\pi$
$284$		−	337.997i		−	0.0706212i
$285$		0			0
$286$	−1386.00	−	2940.15i	−0.286559	−	0.607884i
$287$		−	4989.35i		−	1.02617i
$288$		0			0
$289$	10963.0			2.23143
$290$			1425.53i			0.288655i
$291$		0			0
$292$		−	178.191i		−	0.0357118i
$293$	−9240.00			−1.84234			−0.921172	−	0.389157i	$-0.872766\pi$
$293$	−9240.00			−1.84234			−0.921172	+	0.389157i	$0.872766\pi$
$294$		0			0
$295$	10192.0			2.01153
$296$	3822.00			0.750504
$297$		0			0
$298$	−6246.00			−1.21416
$299$	3570.00			0.690496
$300$		0			0
$301$	−72.0000			−0.0137874
$302$			4912.98i			0.936126i
$303$		0			0
$304$		−	6325.78i		−	1.19345i
$305$	−6468.00			−1.21428
$306$		0			0
$307$		−	3831.10i		−	0.712224i	−0.934443	−	0.356112i	$-0.884102\pi$
$307$		−	3831.10i		−	0.712224i	0.934443	−	0.356112i	$-0.115898\pi$
$308$	−264.000	−	560.029i	−0.0488402	−	0.103606i
$309$		0			0
$310$		−	4157.79i		−	0.761763i
$311$		−	287.085i		−	0.0523444i	−0.999657	−	0.0261722i	$-0.991668\pi$
$311$		−	287.085i		−	0.0523444i	0.999657	−	0.0261722i	$-0.00833183\pi$
$312$		0			0
$313$	−5992.00			−1.08207			−0.541035	−	0.841000i	$-0.681967\pi$
$313$	−5992.00			−1.08207			−0.541035	+	0.841000i	$0.681967\pi$
$314$	7014.00			1.26058
$315$		0			0
$316$		−	772.161i		−	0.137460i
$317$		−	6307.39i		−	1.11753i	−0.829324	−	0.558767i	$-0.811275\pi$
$317$		−	6307.39i		−	1.11753i	0.829324	−	0.558767i	$-0.188725\pi$
$318$		0			0
$319$	−792.000	+	373.352i	−0.139008	+	0.0655289i
$320$		−	8572.96i		−	1.49763i
$321$		0			0
$322$	6120.00			1.05917
$323$		−	11226.0i		−	1.93385i
$324$		0			0
$325$			7929.50i			1.35338i
$326$	−384.000			−0.0652386
$327$		0			0
$328$	−6174.00			−1.03934
$329$	1848.00			0.309676
$330$		0			0
$331$	−1708.00			−0.283626			−0.141813	−	0.989893i	$-0.545293\pi$
$331$	−1708.00			−0.283626			−0.141813	+	0.989893i	$0.545293\pi$
$332$	−1218.00			−0.201345
$333$		0			0
$334$	3528.00			0.577975
$335$			17423.1i			2.84157i
$336$		0			0
$337$		−	6864.59i		−	1.10961i	−0.831981	−	0.554804i	$-0.812793\pi$
$337$		−	6864.59i		−	1.10961i	0.831981	−	0.554804i	$-0.187207\pi$
$338$	−3945.00			−0.634851
$339$		0			0
$340$			2494.67i			0.397919i
$341$	2310.00	−	1088.94i	0.366843	−	0.172932i
$342$		0			0
$343$			6754.28i			1.06326i
$344$			89.0955i			0.0139643i
$345$		0			0
$346$	3276.00			0.509014
$347$	−2922.00			−0.452050			−0.226025	−	0.974122i	$-0.572573\pi$
$347$	−2922.00			−0.452050			−0.226025	+	0.974122i	$0.572573\pi$
$348$		0			0
$349$			8048.29i			1.23443i	0.786796	+	0.617214i	$0.211739\pi$
$349$			8048.29i			1.23443i	−0.786796	+	0.617214i	$0.788261\pi$
$350$			13593.4i			2.07600i
$351$		0			0
$352$	−1485.00	+	700.036i	−0.224860	+	0.106000i
$353$			8087.89i			1.21948i	0.792603	+	0.609738i	$0.208725\pi$
$353$			8087.89i			1.21948i	−0.792603	+	0.609738i	$0.791275\pi$
$354$		0			0
$355$	−6692.00			−1.00049
$356$		−	1534.42i		−	0.228439i
$357$		0			0
$358$		−	1162.48i		−	0.171618i
$359$	−5436.00			−0.799167			−0.399584	−	0.916697i	$-0.630845\pi$
$359$	−5436.00			−0.799167			−0.399584	+	0.916697i	$0.630845\pi$
$360$		0			0
$361$	−1079.00			−0.157312
$362$	3990.00			0.579309
$363$		0			0
$364$	504.000			0.0725736
$365$	−3528.00			−0.505929
$366$		0			0
$367$	−11536.0			−1.64080			−0.820401	−	0.571789i	$-0.806250\pi$
$367$	−11536.0			−1.64080			−0.820401	+	0.571789i	$0.806250\pi$
$368$		−	8534.78i		−	1.20898i
$369$		0			0
$370$			10810.2i			1.51891i
$371$	2496.00			0.349288
$372$		0			0
$373$			10305.4i			1.43054i	0.698847	+	0.715271i	$0.253697\pi$
$373$			10305.4i			1.43054i	−0.698847	+	0.715271i	$0.746303\pi$
$374$	−12474.0	+	5880.30i	−1.72464	+	0.813003i
$375$		0			0
$376$		−	2286.78i		−	0.313649i
$377$		−	712.764i		−	0.0973719i
$378$		0			0
$379$	−3724.00			−0.504720			−0.252360	−	0.967633i	$-0.581207\pi$
$379$	−3724.00			−0.504720			−0.252360	+	0.967633i	$0.581207\pi$
$380$	1764.00			0.238135
$381$		0			0
$382$		−	10212.0i		−	1.36778i
$383$		−	5098.24i		−	0.680177i	−0.940393	−	0.340089i	$-0.889543\pi$
$383$		−	5098.24i		−	0.680177i	0.940393	−	0.340089i	$-0.110457\pi$
$384$		0			0
$385$	−11088.0	+	5226.93i	−1.46778	+	0.691920i
$386$		−	10258.7i		−	1.35273i
$387$		0			0
$388$	−196.000			−0.0256453
$389$		−	70.7107i		−	0.00921638i	−0.999989	−	0.00460819i	$-0.998533\pi$
$389$		−	70.7107i		−	0.00921638i	0.999989	−	0.00460819i	$-0.00146684\pi$
$390$		0			0
$391$		−	15146.2i		−	1.95902i
$392$	1155.00			0.148817
$393$		0			0
$394$	−11394.0			−1.45691
$395$	−15288.0			−1.94740
$396$		0			0
$397$	8498.00			1.07431			0.537157	−	0.843482i	$-0.319498\pi$
$397$	8498.00			1.07431			0.537157	+	0.843482i	$0.319498\pi$
$398$	−1680.00			−0.211585
$399$		0			0
$400$	18957.0			2.36963
$401$			2614.88i			0.325638i	0.986656	+	0.162819i	$0.0520587\pi$
$401$			2614.88i			0.325638i	−0.986656	+	0.162819i	$0.947941\pi$
$402$		0			0
$403$			2078.89i			0.256965i
$404$	546.000			0.0672389
$405$		0			0
$406$		−	1221.88i		−	0.149362i
$407$	−6006.00	+	2831.26i	−0.731465	+	0.344816i
$408$		0			0
$409$		−	475.176i		−	0.0574473i	−0.999587	−	0.0287236i	$-0.990856\pi$
$409$		−	475.176i		−	0.0574473i	0.999587	−	0.0287236i	$-0.00914427\pi$
$410$		−	17462.7i		−	2.10347i
$411$		0			0
$412$	−826.000			−0.0987721
$413$	−8736.00			−1.04085
$414$		0			0
$415$			24115.2i			2.85245i
$416$		−	1336.43i		−	0.157510i
$417$		0			0
$418$	4158.00	+	8820.45i	0.486542	+	1.03211i
$419$			12809.9i			1.49357i	0.665064	+	0.746786i	$0.268404\pi$
$419$			12809.9i			1.49357i	−0.665064	+	0.746786i	$0.731596\pi$
$420$		0			0
$421$	6554.00			0.758723			0.379362	−	0.925249i	$-0.376144\pi$
$421$	6554.00			0.758723			0.379362	+	0.925249i	$0.376144\pi$
$422$		−	5714.84i		−	0.659227i
$423$		0			0
$424$		−	3088.64i		−	0.353768i
$425$	33642.0			3.83971
$426$		0			0
$427$	5544.00			0.628321
$428$	−108.000			−0.0121971
$429$		0			0
$430$	−252.000			−0.0282617
$431$	10692.0			1.19493			0.597466	−	0.801894i	$-0.296174\pi$
$431$	10692.0			1.19493			0.597466	+	0.801894i	$0.296174\pi$
$432$		0			0
$433$	7616.00			0.845269			0.422635	−	0.906300i	$-0.361105\pi$
$433$	7616.00			0.845269			0.422635	+	0.906300i	$0.361105\pi$
$434$			3563.82i			0.394168i
$435$		0			0
$436$			1183.70i			0.130020i
$437$	−10710.0			−1.17238
$438$		0			0
$439$		−	13423.7i		−	1.45941i	−0.683765	−	0.729703i	$-0.739658\pi$
$439$		−	13423.7i		−	1.45941i	0.683765	−	0.729703i	$-0.260342\pi$
$440$	6468.00	+	13720.7i	0.700795	+	1.48661i
$441$		0			0
$442$		−	11226.0i		−	1.20807i
$443$		−	18322.6i		−	1.96508i	−0.186050	−	0.982540i	$-0.559569\pi$
$443$		−	18322.6i		−	1.96508i	0.186050	−	0.982540i	$-0.440431\pi$
$444$		0			0
$445$	−30380.0			−3.23629
$446$	8904.00			0.945329
$447$		0			0
$448$			7348.25i			0.774938i
$449$			654.781i			0.0688219i	0.999408	+	0.0344109i	$0.0109555\pi$
$449$			654.781i			0.0688219i	−0.999408	+	0.0344109i	$0.989044\pi$
$450$		0			0
$451$	9702.00	−	4573.57i	1.01297	−	0.477518i
$452$			1622.10i			0.168799i
$453$		0			0
$454$	−10332.0			−1.06807
$455$		−	9978.69i		−	1.02815i
$456$		0			0
$457$			6024.55i			0.616666i	0.951278	+	0.308333i	$0.0997712\pi$
$457$			6024.55i			0.616666i	−0.951278	+	0.308333i	$0.900229\pi$
$458$	1722.00			0.175685
$459$		0			0
$460$	2380.00			0.241235
$461$	4452.00			0.449784			0.224892	−	0.974384i	$-0.427797\pi$
$461$	4452.00			0.449784			0.224892	+	0.974384i	$0.427797\pi$
$462$		0			0
$463$	14978.0			1.50343			0.751713	−	0.659490i	$-0.229228\pi$
$463$	14978.0			1.50343			0.751713	+	0.659490i	$0.229228\pi$
$464$	−1704.00			−0.170488
$465$		0			0
$466$	15930.0			1.58357
$467$			9246.13i			0.916188i	0.888904	+	0.458094i	$0.151468\pi$
$467$			9246.13i			0.916188i	−0.888904	+	0.458094i	$0.848532\pi$
$468$		0			0
$469$		−	14934.1i		−	1.47035i
$470$	6468.00			0.634780
$471$		0			0
$472$			10810.2i			1.05420i
$473$	−66.0000	−	140.007i	−0.00641582	−	0.0136100i
$474$		0			0
$475$		−	23788.5i		−	2.29787i
$476$		−	2138.29i		−	0.205900i
$477$		0			0
$478$	−3708.00			−0.354812
$479$	−5124.00			−0.488771			−0.244386	−	0.969678i	$-0.578586\pi$
$479$	−5124.00			−0.488771			−0.244386	+	0.969678i	$0.578586\pi$
$480$		0			0
$481$		−	5405.12i		−	0.512375i
$482$			10335.1i			0.976659i
$483$		0			0
$484$	847.000	−	1026.72i	0.0795455	−	0.0964237i
$485$			3880.60i			0.363318i
$486$		0			0
$487$	5096.00			0.474172			0.237086	−	0.971489i	$-0.423808\pi$
$487$	5096.00			0.474172			0.237086	+	0.971489i	$0.423808\pi$
$488$		−	6860.35i		−	0.636380i
$489$		0			0
$490$			3266.83i			0.301185i
$491$	−13164.0			−1.20995			−0.604973	−	0.796246i	$-0.706816\pi$
$491$	−13164.0			−1.20995			−0.604973	+	0.796246i	$0.706816\pi$
$492$		0			0
$493$	−3024.00			−0.276256
$494$	−7938.00			−0.722971
$495$		0			0
$496$	4970.00			0.449919
$497$	5736.00			0.517696
$498$		0			0
$499$	−4228.00			−0.379301			−0.189651	−	0.981852i	$-0.560736\pi$
$499$	−4228.00			−0.379301			−0.189651	+	0.981852i	$0.560736\pi$
$500$			2811.46i			0.251464i
$501$		0			0
$502$		−	17046.9i		−	1.51562i
$503$	−10332.0			−0.915867			−0.457934	−	0.888986i	$-0.651410\pi$
$503$	−10332.0			−0.915867			−0.457934	+	0.888986i	$0.651410\pi$
$504$		0			0
$505$		−	10810.2i		−	0.952574i
$506$	5610.00	+	11900.6i	0.492875	+	1.04555i
$507$		0			0
$508$		−	178.191i		−	0.0155629i
$509$			5504.12i			0.479304i	0.970859	+	0.239652i	$0.0770334\pi$
$509$			5504.12i			0.479304i	−0.970859	+	0.239652i	$0.922967\pi$
$510$		0			0
$511$	3024.00			0.261788
$512$	8733.00			0.753804
$513$		0			0
$514$		−	4484.47i		−	0.384828i
$515$			16354.0i			1.39930i
$516$		0			0
$517$	1694.00	+	3593.52i	0.144105	+	0.305692i
$518$		−	9265.93i		−	0.785949i
$519$		0			0
$520$	−12348.0			−1.04134
$521$		−	643.467i		−	0.0541090i	−0.999634	−	0.0270545i	$-0.991387\pi$
$521$		−	643.467i		−	0.0541090i	0.999634	−	0.0270545i	$-0.00861277\pi$
$522$		0			0
$523$			11136.9i			0.931136i	0.885012	+	0.465568i	$0.154150\pi$
$523$			11136.9i			0.931136i	−0.885012	+	0.465568i	$0.845850\pi$
$524$	588.000			0.0490208
$525$		0			0
$526$	−12168.0			−1.00865
$527$	8820.00			0.729042
$528$		0			0
$529$	−2283.00			−0.187639
$530$	8736.00			0.715977
$531$		0			0
$532$	−1512.00			−0.123221
$533$			8731.35i			0.709563i
$534$		0			0
$535$			2138.29i			0.172797i
$536$	−18480.0			−1.48921
$537$		0			0
$538$			18769.4i			1.50410i
$539$	−1815.00	+	855.599i	−0.145042	+	0.0683734i
$540$		0			0
$541$		−	16304.5i		−	1.29572i	−0.761760	−	0.647859i	$-0.775664\pi$
$541$		−	16304.5i		−	1.29572i	0.761760	−	0.647859i	$-0.224336\pi$
$542$			4811.15i			0.381286i
$543$		0			0
$544$	−5670.00			−0.446874
$545$	23436.0			1.84200
$546$		0			0
$547$			5019.04i			0.392320i	0.980572	+	0.196160i	$0.0628471\pi$
$547$			5019.04i			0.392320i	−0.980572	+	0.196160i	$0.937153\pi$
$548$		−	337.997i		−	0.0263477i
$549$		0			0
$550$	−26433.0	+	12460.6i	−2.04929	+	0.966042i
$551$			2138.29i			0.165325i
$552$		0			0
$553$	13104.0			1.00767
$554$		−	15057.1i		−	1.15472i
$555$		0			0
$556$		−	2880.75i		−	0.219732i
$557$	−4932.00			−0.375181			−0.187590	−	0.982247i	$-0.560068\pi$
$557$	−4932.00			−0.375181			−0.187590	+	0.982247i	$0.560068\pi$
$558$		0			0
$559$	126.000			0.00953351
$560$	−23856.0			−1.80018
$561$		0			0
$562$	8622.00			0.647148
$563$	17598.0			1.31735			0.658674	−	0.752428i	$-0.271118\pi$
$563$	17598.0			1.31735			0.658674	+	0.752428i	$0.271118\pi$
$564$		0			0
$565$	32116.0			2.39138
$566$			6503.97i			0.483007i
$567$		0			0
$568$		−	7097.94i		−	0.524336i
$569$	18546.0			1.36641			0.683206	−	0.730225i	$-0.260585\pi$
$569$	18546.0			1.36641			0.683206	+	0.730225i	$0.260585\pi$
$570$		0			0
$571$			20105.9i			1.47356i	0.676131	+	0.736782i	$0.263655\pi$
$571$			20105.9i			1.47356i	−0.676131	+	0.736782i	$0.736345\pi$
$572$	462.000	+	980.050i	0.0337713	+	0.0716398i
$573$		0			0
$574$			14968.0i			1.08842i
$575$		−	32095.6i		−	2.32779i
$576$		0			0
$577$	−3220.00			−0.232323			−0.116161	−	0.993230i	$-0.537059\pi$
$577$	−3220.00			−0.232323			−0.116161	+	0.993230i	$0.537059\pi$
$578$	−32889.0			−2.36679
$579$		0			0
$580$		−	475.176i		−	0.0340183i
$581$		−	20670.1i		−	1.47598i
$582$		0			0
$583$	2288.00	+	4853.58i	0.162537	+	0.344794i
$584$		−	3742.01i		−	0.265146i
$585$		0			0
$586$	27720.0			1.95410
$587$		−	18017.1i		−	1.26686i	−0.773802	−	0.633428i	$-0.781647\pi$
$587$		−	18017.1i		−	1.26686i	0.773802	−	0.633428i	$-0.218353\pi$
$588$		0			0
$589$		−	6236.68i		−	0.436295i
$590$	−30576.0			−2.13355
$591$		0			0
$592$	−12922.0			−0.897113
$593$	6426.00			0.444999			0.222499	−	0.974933i	$-0.428578\pi$
$593$	6426.00			0.444999			0.222499	+	0.974933i	$0.428578\pi$
$594$		0			0
$595$	−42336.0			−2.91699
$596$	2082.00			0.143091
$597$		0			0
$598$	−10710.0			−0.732382
$599$			20968.5i			1.43030i	0.698969	+	0.715152i	$0.253642\pi$
$599$			20968.5i			1.43030i	−0.698969	+	0.715152i	$0.746358\pi$
$600$		0			0
$601$			9028.34i			0.612768i	0.951908	+	0.306384i	$0.0991192\pi$
$601$			9028.34i			0.612768i	−0.951908	+	0.306384i	$0.900881\pi$
$602$	216.000			0.0146238
$603$		0			0
$604$		−	1637.66i		−	0.110324i
$605$	−20328.0	−	16769.7i	−1.36603	−	1.12692i
$606$		0			0
$607$			16809.3i			1.12400i	0.827136	+	0.562002i	$0.189969\pi$
$607$			16809.3i			1.12400i	−0.827136	+	0.562002i	$0.810031\pi$
$608$			4009.30i			0.267432i
$609$		0			0
$610$	19404.0			1.28794
$611$	−3234.00			−0.214130
$612$		0			0
$613$		−	5791.20i		−	0.381573i	−0.981632	−	0.190787i	$-0.938896\pi$
$613$		−	5791.20i		−	0.381573i	0.981632	−	0.190787i	$-0.0611039\pi$
$614$			11493.3i			0.755427i
$615$		0			0
$616$	−5544.00	−	11760.6i	−0.362620	−	0.769234i
$617$		−	57.9828i		−	0.00378330i	−0.999998	−	0.00189165i	$-0.999398\pi$
$617$		−	57.9828i		−	0.00378330i	0.999998	−	0.00189165i	$-0.000602132\pi$
$618$		0			0
$619$	15680.0			1.01815			0.509073	−	0.860723i	$-0.329988\pi$
$619$	15680.0			1.01815			0.509073	+	0.860723i	$0.329988\pi$
$620$			1385.93i			0.0897746i
$621$		0			0
$622$			861.256i			0.0555196i
$623$	26040.0			1.67459
$624$		0			0
$625$	22289.0			1.42650
$626$	17976.0			1.14771
$627$		0			0
$628$	−2338.00			−0.148561
$629$	−22932.0			−1.45367
$630$		0			0
$631$	−24334.0			−1.53522			−0.767608	−	0.640920i	$-0.778553\pi$
$631$	−24334.0			−1.53522			−0.767608	+	0.640920i	$0.778553\pi$
$632$		−	16215.4i		−	1.02059i
$633$		0			0
$634$			18922.2i			1.18532i
$635$	−3528.00			−0.220479
$636$		0			0
$637$		−	1633.42i		−	0.101599i
$638$	2376.00	−	1120.06i	0.147440	−	0.0695039i
$639$		0			0
$640$			32846.5i			2.02871i
$641$			22215.9i			1.36892i	0.729053	+	0.684458i	$0.239961\pi$
$641$			22215.9i			1.36892i	−0.729053	+	0.684458i	$0.760039\pi$
$642$		0			0
$643$	5096.00			0.312545			0.156273	−	0.987714i	$-0.450052\pi$
$643$	5096.00			0.312545			0.156273	+	0.987714i	$0.450052\pi$
$644$	−2040.00			−0.124825
$645$		0			0
$646$			33678.1i			2.05116i
$647$		−	13829.6i		−	0.840336i	−0.907446	−	0.420168i	$-0.861971\pi$
$647$		−	13829.6i		−	0.840336i	0.907446	−	0.420168i	$-0.138029\pi$
$648$		0			0
$649$	−8008.00	−	16987.5i	−0.484347	−	1.02746i
$650$		−	23788.5i		−	1.43548i
$651$		0			0
$652$	128.000			0.00768845
$653$			1813.02i			0.108651i	0.998523	+	0.0543254i	$0.0173008\pi$
$653$			1813.02i			0.108651i	−0.998523	+	0.0543254i	$0.982699\pi$
$654$		0			0
$655$		−	11641.8i		−	0.694478i
$656$	20874.0			1.24237
$657$		0			0
$658$	−5544.00			−0.328461
$659$	16566.0			0.979241			0.489620	−	0.871936i	$-0.337135\pi$
$659$	16566.0			0.979241			0.489620	+	0.871936i	$0.337135\pi$
$660$		0			0
$661$	−6118.00			−0.360004			−0.180002	−	0.983666i	$-0.557610\pi$
$661$	−6118.00			−0.360004			−0.180002	+	0.983666i	$0.557610\pi$
$662$	5124.00			0.300831
$663$		0			0
$664$	−25578.0			−1.49491
$665$			29936.1i			1.74567i
$666$		0			0
$667$			2885.00i			0.167477i
$668$	−1176.00			−0.0681150
$669$		0			0
$670$		−	52269.3i		−	3.01394i
$671$	5082.00	+	10780.5i	0.292382	+	0.620236i
$672$		0			0
$673$		−	20933.2i		−	1.19898i	−0.800381	−	0.599491i	$-0.795370\pi$
$673$		−	20933.2i		−	1.19898i	0.800381	−	0.599491i	$-0.204630\pi$
$674$			20593.8i			1.17692i
$675$		0			0
$676$	1315.00			0.0748179
$677$	−12558.0			−0.712915			−0.356457	−	0.934312i	$-0.616015\pi$
$677$	−12558.0			−0.712915			−0.356457	+	0.934312i	$0.616015\pi$
$678$		0			0
$679$		−	3326.23i		−	0.187996i
$680$			52388.1i			2.95440i
$681$		0			0
$682$	−6930.00	+	3266.83i	−0.389096	+	0.183422i
$683$		−	2209.00i		−	0.123756i	−0.998084	−	0.0618778i	$-0.980291\pi$
$683$		−	2209.00i		−	0.123756i	0.998084	−	0.0618778i	$-0.0197089\pi$
$684$		0			0
$685$	−6692.00			−0.373267
$686$		−	20262.9i		−	1.12775i
$687$		0			0
$688$		−	301.227i		−	0.0166921i
$689$	−4368.00			−0.241520
$690$		0			0
$691$	2828.00			0.155691			0.0778453	−	0.996965i	$-0.475196\pi$
$691$	2828.00			0.155691			0.0778453	+	0.996965i	$0.475196\pi$
$692$	−1092.00			−0.0599879
$693$		0			0
$694$	8766.00			0.479471
$695$	−57036.0			−3.11295
$696$		0			0
$697$	37044.0			2.01312
$698$		−	24144.9i		−	1.30931i
$699$		0			0
$700$		−	4531.14i		−	0.244659i
$701$	8424.00			0.453880			0.226940	−	0.973909i	$-0.427128\pi$
$701$	8424.00			0.453880			0.226940	+	0.973909i	$0.427128\pi$
$702$		0			0
$703$			16215.4i			0.869949i
$704$	−14289.0	+	6735.90i	−0.764967	+	0.360609i
$705$		0			0
$706$		−	24263.7i		−	1.29345i
$707$			9265.93i			0.492901i
$708$		0			0
$709$	−8890.00			−0.470904			−0.235452	−	0.971886i	$-0.575657\pi$
$709$	−8890.00			−0.470904			−0.235452	+	0.971886i	$0.575657\pi$
$710$	20076.0			1.06118
$711$		0			0
$712$		−	32222.9i		−	1.69607i
$713$		−	8414.57i		−	0.441975i
$714$		0			0
$715$	19404.0	−	9147.13i	1.01492	−	0.478438i
$716$			387.495i			0.0202253i
$717$		0			0
$718$	16308.0			0.847645
$719$			23946.9i			1.24210i	0.783772	+	0.621049i	$0.213293\pi$
$719$			23946.9i			1.24210i	−0.783772	+	0.621049i	$0.786707\pi$
$720$		0			0
$721$		−	14017.7i		−	0.724058i
$722$	3237.00			0.166854
$723$		0			0
$724$	−1330.00			−0.0682722
$725$	−6408.00			−0.328258
$726$		0			0
$727$	−10024.0			−0.511375			−0.255687	−	0.966759i	$-0.582302\pi$
$727$	−10024.0			−0.511375			−0.255687	+	0.966759i	$0.582302\pi$
$728$	10584.0			0.538831
$729$		0			0
$730$	10584.0			0.536618
$731$		−	534.573i		−	0.0270477i
$732$		0			0
$733$			6206.98i			0.312770i	0.987696	+	0.156385i	$0.0499840\pi$
$733$			6206.98i			0.312770i	−0.987696	+	0.156385i	$0.950016\pi$
$734$	34608.0			1.74033
$735$		0			0
$736$			5409.37i			0.270913i
$737$	29040.0	−	13689.6i	1.45143	−	0.684210i
$738$		0			0
$739$			37331.0i			1.85824i	0.369772	+	0.929122i	$0.379436\pi$
$739$			37331.0i			1.85824i	−0.369772	+	0.929122i	$0.620564\pi$
$740$		−	3603.42i		−	0.179006i
$741$		0			0
$742$	−7488.00			−0.370476
$743$	14340.0			0.708053			0.354027	−	0.935235i	$-0.384812\pi$
$743$	14340.0			0.708053			0.354027	+	0.935235i	$0.384812\pi$
$744$		0			0
$745$		−	41221.5i		−	2.02717i
$746$		−	30916.1i		−	1.51732i
$747$		0			0
$748$	4158.00	−	1960.10i	0.203251	−	0.0958133i
$749$		−	1832.82i		−	0.0894123i
$750$		0			0
$751$	−33334.0			−1.61967			−0.809837	−	0.586655i	$-0.800444\pi$
$751$	−33334.0			−1.61967			−0.809837	+	0.586655i	$0.800444\pi$
$752$			7731.51i			0.374919i
$753$		0			0
$754$			2138.29i			0.103278i
$755$	−32424.0			−1.56295
$756$		0			0
$757$	−4354.00			−0.209047			−0.104524	−	0.994522i	$-0.533332\pi$
$757$	−4354.00			−0.209047			−0.104524	+	0.994522i	$0.533332\pi$
$758$	11172.0			0.535337
$759$		0			0
$760$	37044.0			1.76806
$761$	22638.0			1.07835			0.539177	−	0.842193i	$-0.318736\pi$
$761$	22638.0			1.07835			0.539177	+	0.842193i	$0.318736\pi$
$762$		0			0
$763$	−20088.0			−0.953125
$764$			3404.01i			0.161195i
$765$		0			0
$766$			15294.7i			0.721437i
$767$	15288.0			0.719710
$768$		0			0
$769$		−	27025.6i		−	1.26732i	−0.773612	−	0.633660i	$-0.781552\pi$
$769$		−	27025.6i		−	1.26732i	0.773612	−	0.633660i	$-0.218448\pi$
$770$	33264.0	−	15680.8i	1.55682	−	0.733892i
$771$		0			0
$772$			3419.57i			0.159421i
$773$		−	20511.8i		−	0.954407i	−0.878793	−	0.477203i	$-0.841650\pi$
$773$		−	20511.8i		−	0.954407i	0.878793	−	0.477203i	$-0.158350\pi$
$774$		0			0
$775$	18690.0			0.866277
$776$	−4116.00			−0.190407
$777$		0			0
$778$			212.132i			0.00977545i
$779$		−	26194.1i		−	1.20475i
$780$		0			0
$781$	5258.00	+	11153.9i	0.240904	+	0.511035i
$782$			45438.7i			2.07786i
$783$		0			0
$784$	−3905.00			−0.177888
$785$			46290.0i			2.10467i
$786$		0			0
$787$			37746.8i			1.70969i	0.518882	+	0.854846i	$0.326348\pi$
$787$			37746.8i			1.70969i	−0.518882	+	0.854846i	$0.673652\pi$
$788$	3798.00			0.171698
$789$		0			0
$790$	45864.0			2.06553
$791$	−27528.0			−1.23740
$792$		0			0
$793$	−9702.00			−0.434462
$794$	−25494.0			−1.13948
$795$		0			0
$796$	560.000			0.0249355
$797$		−	19264.4i		−	0.856187i	−0.903735	−	0.428093i	$-0.859185\pi$
$797$		−	19264.4i		−	0.856187i	0.903735	−	0.428093i	$-0.140815\pi$
$798$		0			0
$799$			13720.7i			0.607514i
$800$	−12015.0			−0.530993
$801$		0			0
$802$		−	7844.64i		−	0.345391i
$803$	2772.00	+	5880.30i	0.121820	+	0.258420i
$804$		0			0
$805$			40389.9i			1.76840i
$806$		−	6236.68i		−	0.272553i
$807$		0			0
$808$	11466.0			0.499223
$809$	27486.0			1.19451			0.597254	−	0.802052i	$-0.296259\pi$
$809$	27486.0			1.19451			0.597254	+	0.802052i	$0.296259\pi$
$810$		0			0
$811$		−	40894.8i		−	1.77067i	−0.464956	−	0.885334i	$-0.653930\pi$
$811$		−	40894.8i		−	1.77067i	0.464956	−	0.885334i	$-0.346070\pi$
$812$			407.294i			0.0176025i
$813$		0			0
$814$	18018.0	−	8493.77i	0.775836	−	0.365733i
$815$		−	2534.27i		−	0.108922i
$816$		0			0
$817$	−378.000			−0.0161867
$818$			1425.53i			0.0609320i
$819$		0			0
$820$			5820.90i			0.247896i
$821$	−25770.0			−1.09547			−0.547734	−	0.836653i	$-0.684509\pi$
$821$	−25770.0			−1.09547			−0.547734	+	0.836653i	$0.684509\pi$
$822$		0			0
$823$	−42208.0			−1.78770			−0.893851	−	0.448365i	$-0.852007\pi$
$823$	−42208.0			−1.78770			−0.893851	+	0.448365i	$0.852007\pi$
$824$	−17346.0			−0.733345
$825$		0			0
$826$	26208.0			1.10399
$827$	8082.00			0.339829			0.169915	−	0.985459i	$-0.445651\pi$
$827$	8082.00			0.339829			0.169915	+	0.985459i	$0.445651\pi$
$828$		0			0
$829$	−11914.0			−0.499144			−0.249572	−	0.968356i	$-0.580290\pi$
$829$	−11914.0			−0.499144			−0.249572	+	0.968356i	$0.580290\pi$
$830$		−	72345.5i		−	3.02548i
$831$		0			0
$832$		−	12859.4i		−	0.535843i
$833$	−6930.00			−0.288248
$834$		0			0
$835$			23283.6i			0.964985i
$836$	−1386.00	−	2940.15i	−0.0573395	−	0.121635i
$837$		0			0
$838$		−	38429.8i		−	1.58417i
$839$		−	27193.9i		−	1.11900i	−0.828832	−	0.559498i	$-0.810994\pi$
$839$		−	27193.9i		−	1.11900i	0.828832	−	0.559498i	$-0.189006\pi$
$840$		0			0
$841$	−23813.0			−0.976383
$842$	−19662.0			−0.804747
$843$		0			0
$844$			1904.95i			0.0776907i
$845$		−	26035.7i		−	1.05995i
$846$		0			0
$847$	17424.0	+	14374.1i	0.706843	+	0.583115i
$848$			10442.6i			0.422876i
$849$		0			0
$850$	−100926.			−4.07263
$851$			21877.9i			0.881274i
$852$		0			0
$853$			5197.23i			0.208617i	0.994545	+	0.104308i	$0.0332629\pi$
$853$			5197.23i			0.208617i	−0.994545	+	0.104308i	$0.966737\pi$
$854$	−16632.0			−0.666435
$855$		0			0
$856$	−2268.00			−0.0905592
$857$	−24402.0			−0.972645			−0.486322	−	0.873779i	$-0.661662\pi$
$857$	−24402.0			−0.972645			−0.486322	+	0.873779i	$0.661662\pi$
$858$		0			0
$859$	22232.0			0.883057			0.441529	−	0.897247i	$-0.354436\pi$
$859$	22232.0			0.883057			0.441529	+	0.897247i	$0.354436\pi$
$860$	84.0000			0.00333067
$861$		0			0
$862$	−32076.0			−1.26742
$863$			21223.1i			0.837130i	0.908187	+	0.418565i	$0.137467\pi$
$863$			21223.1i			0.837130i	−0.908187	+	0.418565i	$0.862533\pi$
$864$		0			0
$865$			21620.5i			0.849848i
$866$	−22848.0			−0.896543
$867$		0			0
$868$		−	1187.94i		−	0.0464531i
$869$	12012.0	+	25481.3i	0.468906	+	0.994700i
$870$		0			0
$871$			26134.7i			1.01669i
$872$			24857.6i			0.965350i
$873$		0			0
$874$	32130.0			1.24349
$875$	−47712.0			−1.84338
$876$		0			0
$877$			34183.0i			1.31616i	0.752946	+	0.658082i	$0.228632\pi$
$877$			34183.0i			1.31616i	−0.752946	+	0.658082i	$0.771368\pi$
$878$			40271.1i			1.54793i
$879$		0			0
$880$	−21868.0	−	46389.0i	−0.837693	−	1.77702i
$881$			28758.0i			1.09975i	0.835246	+	0.549877i	$0.185325\pi$
$881$			28758.0i			1.09975i	−0.835246	+	0.549877i	$0.814675\pi$
$882$		0			0
$883$	8120.00			0.309467			0.154734	−	0.987956i	$-0.450548\pi$
$883$	8120.00			0.309467			0.154734	+	0.987956i	$0.450548\pi$
$884$			3742.01i			0.142373i
$885$		0			0
$886$			54967.7i			2.08428i
$887$	−9996.00			−0.378391			−0.189196	−	0.981939i	$-0.560588\pi$
$887$	−9996.00			−0.378391			−0.189196	+	0.981939i	$0.560588\pi$
$888$		0			0
$889$	3024.00			0.114085
$890$	91140.0			3.43261
$891$		0			0
$892$	−2968.00			−0.111408
$893$	9702.00			0.363567
$894$		0			0
$895$	7672.00			0.286533
$896$		−	28154.2i		−	1.04974i
$897$		0			0
$898$		−	1964.34i		−	0.0729966i
$899$	−1680.00			−0.0623261
$900$		0			0
$901$			18531.9i			0.685223i
$902$	−29106.0	+	13720.7i	−1.07442	+	0.506485i
$903$		0			0
$904$			34064.2i			1.25327i
$905$			26332.7i			0.967212i
$906$		0			0
$907$	−37744.0			−1.38177			−0.690887	−	0.722963i	$-0.742780\pi$
$907$	−37744.0			−1.38177			−0.690887	+	0.722963i	$0.742780\pi$
$908$	3444.00			0.125874
$909$		0			0
$910$			29936.1i			1.09052i
$911$			1800.29i			0.0654735i	0.999464	+	0.0327368i	$0.0104223\pi$
$911$			1800.29i			0.0654735i	−0.999464	+	0.0327368i	$0.989578\pi$
$912$		0			0
$913$	40194.0	−	18947.6i	1.45698	−	0.686829i
$914$		−	18073.6i		−	0.654074i
$915$		0			0
$916$	−574.000			−0.0207047
$917$			9978.69i			0.359352i
$918$		0			0
$919$		−	25277.7i		−	0.907326i	−0.891173	−	0.453663i	$-0.850117\pi$
$919$		−	25277.7i		−	0.907326i	0.891173	−	0.453663i	$-0.149883\pi$
$920$	49980.0			1.79108
$921$		0			0
$922$	−13356.0			−0.477068
$923$	−10038.0			−0.357968
$924$		0			0
$925$	−48594.0			−1.72731
$926$	−44934.0			−1.59463
$927$		0			0
$928$	1080.00			0.0382034
$929$		−	39667.3i		−	1.40091i	−0.713699	−	0.700453i	$-0.752981\pi$
$929$		−	39667.3i		−	1.40091i	0.713699	−	0.700453i	$-0.247019\pi$
$930$		0			0
$931$			4900.25i			0.172502i
$932$	−5310.00			−0.186625
$933$		0			0
$934$		−	27738.4i		−	0.971764i
$935$	−38808.0	−	82324.2i	−1.35739	−	2.87945i
$936$		0			0
$937$		−	2851.05i		−	0.0994022i	−0.998764	−	0.0497011i	$-0.984173\pi$
$937$		−	2851.05i		−	0.0994022i	0.998764	−	0.0497011i	$-0.0158269\pi$
$938$			44802.3i			1.55954i
$939$		0			0
$940$	−2156.00			−0.0748095
$941$	3990.00			0.138226			0.0691128	−	0.997609i	$-0.477983\pi$
$941$	3990.00			0.138226			0.0691128	+	0.997609i	$0.477983\pi$
$942$		0			0
$943$		−	35341.2i		−	1.22043i
$944$		−	36548.9i		−	1.26013i
$945$		0			0
$946$	198.000	+	420.021i	0.00680501	+	0.0144356i
$947$		−	40774.6i		−	1.39915i	−0.714558	−	0.699576i	$-0.753372\pi$
$947$		−	40774.6i		−	1.39915i	0.714558	−	0.699576i	$-0.246628\pi$
$948$		0			0
$949$	−5292.00			−0.181017
$950$			71365.5i			2.43726i
$951$		0			0
$952$		−	44904.1i		−	1.52873i
$953$	−11322.0			−0.384843			−0.192422	−	0.981312i	$-0.561634\pi$
$953$	−11322.0			−0.384843			−0.192422	+	0.981312i	$0.561634\pi$
$954$		0			0
$955$	67396.0			2.28365
$956$	1236.00			0.0418150
$957$		0			0
$958$	15372.0			0.518420
$959$	5736.00			0.193144
$960$		0			0
$961$	−24891.0			−0.835521
$962$			16215.4i			0.543456i
$963$		0			0
$964$		−	3445.02i		−	0.115100i
$965$	67704.0			2.25852
$966$		0			0
$967$		−	2791.66i		−	0.0928373i	−0.998922	−	0.0464186i	$-0.985219\pi$
$967$		−	2791.66i		−	0.0928373i	0.998922	−	0.0464186i	$-0.0147808\pi$
$968$	17787.0	−	21561.1i	0.590595	−	0.715909i
$969$		0			0
$970$		−	11641.8i		−	0.385357i
$971$			6216.88i			0.205468i	0.994709	+	0.102734i	$0.0327590\pi$
$971$			6216.88i			0.205468i	−0.994709	+	0.102734i	$0.967241\pi$
$972$		0			0
$973$	48888.0			1.61077
$974$	−15288.0			−0.502935
$975$		0			0
$976$			23194.5i			0.760695i
$977$		−	56468.1i		−	1.84911i	−0.381055	−	0.924553i	$-0.624439\pi$
$977$		−	56468.1i		−	1.84911i	0.381055	−	0.924553i	$-0.375561\pi$
$978$		0			0
$979$	23870.0	+	50635.9i	0.779253	+	1.65304i
$980$		−	1088.94i		−	0.0354950i
$981$		0			0
$982$	39492.0			1.28334
$983$		−	2781.76i		−	0.0902587i	−0.998981	−	0.0451294i	$-0.985630\pi$
$983$		−	2781.76i		−	0.0902587i	0.998981	−	0.0451294i	$-0.0143700\pi$
$984$		0			0
$985$		−	75196.6i		−	2.43245i
$986$	9072.00			0.293014
$987$		0			0
$988$	2646.00			0.0852029
$989$	−510.000			−0.0163974
$990$		0			0
$991$	−7882.00			−0.252654			−0.126327	−	0.991989i	$-0.540319\pi$
$991$	−7882.00			−0.252654			−0.126327	+	0.991989i	$0.540319\pi$
$992$	−3150.00			−0.100819
$993$		0			0
$994$	−17208.0			−0.549099
$995$		−	11087.4i		−	0.353262i
$996$		0			0
$997$			5910.00i			0.187735i	0.995585	+	0.0938674i	$0.0299230\pi$
$997$			5910.00i			0.187735i	−0.995585	+	0.0938674i	$0.970077\pi$
$998$	12684.0			0.402310
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.4.d.a.98.1	✓	2
3.2	odd	2		99.4.d.b.98.2	yes	2
4.3	odd	2		1584.4.b.b.593.1		2
11.10	odd	2		99.4.d.b.98.1	yes	2
12.11	even	2		1584.4.b.a.593.2		2
33.32	even	2	inner	99.4.d.a.98.2	yes	2
44.43	even	2		1584.4.b.a.593.1		2
132.131	odd	2		1584.4.b.b.593.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.4.d.a.98.1	✓	2	1.1	even	1	trivial
99.4.d.a.98.2	yes	2	33.32	even	2	inner
99.4.d.b.98.1	yes	2	11.10	odd	2
99.4.d.b.98.2	yes	2	3.2	odd	2
1584.4.b.a.593.1		2	44.43	even	2
1584.4.b.a.593.2		2	12.11	even	2
1584.4.b.b.593.1		2	4.3	odd	2
1584.4.b.b.593.2		2	132.131	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:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	99.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{2} +1.00000 q^{4} -19.7990i q^{5} +16.9706i q^{7} +21.0000 q^{8} +O(q^{10})\)	\(q-3.00000 q^{2} +1.00000 q^{4} -19.7990i q^{5} +16.9706i q^{7} +21.0000 q^{8} +59.3970i q^{10} +(-33.0000 + 15.5563i) q^{11} -29.6985i q^{13} -50.9117i q^{14} -71.0000 q^{16} -126.000 q^{17} +89.0955i q^{19} -19.7990i q^{20} +(99.0000 - 46.6690i) q^{22} +120.208i q^{23} -267.000 q^{25} +89.0955i q^{26} +16.9706i q^{28} +24.0000 q^{29} -70.0000 q^{31} +45.0000 q^{32} +378.000 q^{34} +336.000 q^{35} +182.000 q^{37} -267.286i q^{38} -415.779i q^{40} -294.000 q^{41} +4.24264i q^{43} +(-33.0000 + 15.5563i) q^{44} -360.624i q^{46} -108.894i q^{47} +55.0000 q^{49} +801.000 q^{50} -29.6985i q^{52} -147.078i q^{53} +(308.000 + 653.367i) q^{55} +356.382i q^{56} -72.0000 q^{58} +514.774i q^{59} -326.683i q^{61} +210.000 q^{62} +433.000 q^{64} -588.000 q^{65} -880.000 q^{67} -126.000 q^{68} -1008.00 q^{70} -337.997i q^{71} -178.191i q^{73} -546.000 q^{74} +89.0955i q^{76} +(-264.000 - 560.029i) q^{77} -772.161i q^{79} +1405.73i q^{80} +882.000 q^{82} -1218.00 q^{83} +2494.67i q^{85} -12.7279i q^{86} +(-693.000 + 326.683i) q^{88} -1534.42i q^{89} +504.000 q^{91} +120.208i q^{92} +326.683i q^{94} +1764.00 q^{95} -196.000 q^{97} -165.000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{2} + 2 q^{4} + 42 q^{8}+O(q^{10}) \) 2 * q - 6 * q^2 + 2 * q^4 + 42 * q^8	\( 2 q - 6 q^{2} + 2 q^{4} + 42 q^{8} - 66 q^{11} - 142 q^{16} - 252 q^{17} + 198 q^{22} - 534 q^{25} + 48 q^{29} - 140 q^{31} + 90 q^{32} + 756 q^{34} + 672 q^{35} + 364 q^{37} - 588 q^{41} - 66 q^{44} + 110 q^{49} + 1602 q^{50} + 616 q^{55} - 144 q^{58} + 420 q^{62} + 866 q^{64} - 1176 q^{65} - 1760 q^{67} - 252 q^{68} - 2016 q^{70} - 1092 q^{74} - 528 q^{77} + 1764 q^{82} - 2436 q^{83} - 1386 q^{88} + 1008 q^{91} + 3528 q^{95} - 392 q^{97} - 330 q^{98}+O(q^{100}) \) 2 * q - 6 * q^2 + 2 * q^4 + 42 * q^8 - 66 * q^11 - 142 * q^16 - 252 * q^17 + 198 * q^22 - 534 * q^25 + 48 * q^29 - 140 * q^31 + 90 * q^32 + 756 * q^34 + 672 * q^35 + 364 * q^37 - 588 * q^41 - 66 * q^44 + 110 * q^49 + 1602 * q^50 + 616 * q^55 - 144 * q^58 + 420 * q^62 + 866 * q^64 - 1176 * q^65 - 1760 * q^67 - 252 * q^68 - 2016 * q^70 - 1092 * q^74 - 528 * q^77 + 1764 * q^82 - 2436 * q^83 - 1386 * q^88 + 1008 * q^91 + 3528 * q^95 - 392 * q^97 - 330 * q^98

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