LMFDB - Embedded newform 2151.2.b.a.2150.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(2150,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2151.2150");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.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:	$17.1758214748$
Analytic rank:	$0$
Dimension:	$80$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2150.1
Character	$\chi$	$=$	2151.2150
Dual form			2151.2.b.a.2150.80

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.75120i q^{2} -5.56908 q^{4} -1.21173i q^{5} -4.92033i q^{7} +9.81922i q^{8} +O(q^{10})$	\(q-2.75120i q^{2} -5.56908 q^{4} -1.21173i q^{5} -4.92033i q^{7} +9.81922i q^{8} -3.33371 q^{10} +4.34592i q^{11} +4.32722i q^{13} -13.5368 q^{14} +15.8765 q^{16} -2.49653i q^{17} +6.32761i q^{19} +6.74823i q^{20} +11.9565 q^{22} +4.80088 q^{23} +3.53170 q^{25} +11.9050 q^{26} +27.4017i q^{28} +5.82671i q^{29} -6.52141 q^{31} -24.0408i q^{32} -6.86844 q^{34} -5.96212 q^{35} +10.0969i q^{37} +17.4085 q^{38} +11.8983 q^{40} +0.699322 q^{41} +7.75381i q^{43} -24.2028i q^{44} -13.2082i q^{46} -0.961776 q^{47} -17.2096 q^{49} -9.71641i q^{50} -24.0986i q^{52} -5.76437 q^{53} +5.26610 q^{55} +48.3138 q^{56} +16.0304 q^{58} -0.200805 q^{59} -5.19148 q^{61} +17.9417i q^{62} -34.3880 q^{64} +5.24343 q^{65} +6.85991 q^{67} +13.9034i q^{68} +16.4030i q^{70} +3.70205i q^{71} +3.35993i q^{73} +27.7785 q^{74} -35.2389i q^{76} +21.3834 q^{77} +2.99293i q^{79} -19.2380i q^{80} -1.92397i q^{82} -7.32661i q^{83} -3.02513 q^{85} +21.3322 q^{86} -42.6736 q^{88} -6.84486 q^{89} +21.2913 q^{91} -26.7365 q^{92} +2.64603i q^{94} +7.66736 q^{95} +3.10836i q^{97} +47.3470i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 80 q^{4}+O(q^{10}) $ 80 * q - 80 * q^4	$ 80 q - 80 q^{4} + 16 q^{10} + 56 q^{16} + 40 q^{22} - 64 q^{25} - 8 q^{31} + 32 q^{34} - 24 q^{40} - 104 q^{49} - 24 q^{55} + 56 q^{58} + 40 q^{61} - 80 q^{64} - 8 q^{67} - 8 q^{85} - 120 q^{88} + 32 q^{91}+O(q^{100}) $ 80 * q - 80 * q^4 + 16 * q^10 + 56 * q^16 + 40 * q^22 - 64 * q^25 - 8 * q^31 + 32 * q^34 - 24 * q^40 - 104 * q^49 - 24 * q^55 + 56 * q^58 + 40 * q^61 - 80 * q^64 - 8 * q^67 - 8 * q^85 - 120 * q^88 + 32 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2151\mathbb{Z}\right)^\times$.

$n$	$479$	$1441$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	2.75120i		−	1.94539i	−0.232090	−	0.972694i	$-0.574556\pi$
$2$		−	2.75120i		−	1.94539i	0.232090	−	0.972694i	$-0.425444\pi$
$3$		0			0
$3$		0			0
$4$	−5.56908			−2.78454
$5$		−	1.21173i		−	0.541903i	−0.962593	−	0.270952i	$-0.912662\pi$
$5$		−	1.21173i		−	0.541903i	0.962593	−	0.270952i	$-0.0873383\pi$
$6$		0			0
$7$		−	4.92033i		−	1.85971i	−0.367928	−	0.929854i	$-0.619933\pi$
$7$		−	4.92033i		−	1.85971i	0.367928	−	0.929854i	$-0.380067\pi$
$8$			9.81922i			3.47162i
$9$		0			0
$10$	−3.33371			−1.05421
$11$			4.34592i			1.31035i	0.755479	+	0.655173i	$0.227404\pi$
$11$			4.34592i			1.31035i	−0.755479	+	0.655173i	$0.772596\pi$
$12$		0			0
$13$			4.32722i			1.20016i	0.799942	+	0.600078i	$0.204864\pi$
$13$			4.32722i			1.20016i	−0.799942	+	0.600078i	$0.795136\pi$
$14$	−13.5368			−3.61786
$15$		0			0
$16$	15.8765			3.96911
$17$		−	2.49653i		−	0.605498i	−0.953070	−	0.302749i	$-0.902096\pi$
$17$		−	2.49653i		−	0.605498i	0.953070	−	0.302749i	$-0.0979043\pi$
$18$		0			0
$19$			6.32761i			1.45165i	0.687878	+	0.725826i	$0.258542\pi$
$19$			6.32761i			1.45165i	−0.687878	+	0.725826i	$0.741458\pi$
$20$			6.74823i			1.50895i
$21$		0			0
$22$	11.9565			2.54913
$23$	4.80088			1.00105			0.500526	−	0.865721i	$-0.333140\pi$
$23$	4.80088			1.00105			0.500526	+	0.865721i	$0.333140\pi$
$24$		0			0
$25$	3.53170			0.706341
$26$	11.9050			2.33477
$27$		0			0
$28$			27.4017i			5.17843i
$29$			5.82671i			1.08199i	0.841025	+	0.540997i	$0.181953\pi$
$29$			5.82671i			1.08199i	−0.841025	+	0.540997i	$0.818047\pi$
$30$		0			0
$31$	−6.52141			−1.17128			−0.585640	−	0.810572i	$-0.699157\pi$
$31$	−6.52141			−1.17128			−0.585640	+	0.810572i	$0.699157\pi$
$32$		−	24.0408i		−	4.24985i
$33$		0			0
$34$	−6.86844			−1.17793
$35$	−5.96212			−1.00778
$36$		0			0
$37$			10.0969i			1.65992i	0.557824	+	0.829959i	$0.311636\pi$
$37$			10.0969i			1.65992i	−0.557824	+	0.829959i	$0.688364\pi$
$38$	17.4085			2.82403
$39$		0			0
$40$	11.8983			1.88128
$41$	0.699322			0.109216			0.0546079	−	0.998508i	$-0.482609\pi$
$41$	0.699322			0.109216			0.0546079	+	0.998508i	$0.482609\pi$
$42$		0			0
$43$			7.75381i			1.18244i	0.806509	+	0.591222i	$0.201354\pi$
$43$			7.75381i			1.18244i	−0.806509	+	0.591222i	$0.798646\pi$
$44$		−	24.2028i		−	3.64871i
$45$		0			0
$46$		−	13.2082i		−	1.94744i
$47$	−0.961776			−0.140289			−0.0701447	−	0.997537i	$-0.522346\pi$
$47$	−0.961776			−0.140289			−0.0701447	+	0.997537i	$0.522346\pi$
$48$		0			0
$49$	−17.2096			−2.45852
$50$		−	9.71641i		−	1.37411i
$51$		0			0
$52$		−	24.0986i		−	3.34188i
$53$	−5.76437			−0.791798			−0.395899	−	0.918294i	$-0.629567\pi$
$53$	−5.76437			−0.791798			−0.395899	+	0.918294i	$0.629567\pi$
$54$		0			0
$55$	5.26610			0.710080
$56$	48.3138			6.45620
$57$		0			0
$58$	16.0304			2.10490
$59$	−0.200805			−0.0261426			−0.0130713	−	0.999915i	$-0.504161\pi$
$59$	−0.200805			−0.0261426			−0.0130713	+	0.999915i	$0.504161\pi$
$60$		0			0
$61$	−5.19148			−0.664701			−0.332351	−	0.943156i	$-0.607842\pi$
$61$	−5.19148			−0.664701			−0.332351	+	0.943156i	$0.607842\pi$
$62$			17.9417i			2.27859i
$63$		0			0
$64$	−34.3880			−4.29849
$65$	5.24343			0.650368
$66$		0			0
$67$	6.85991			0.838071			0.419036	−	0.907970i	$-0.362368\pi$
$67$	6.85991			0.838071			0.419036	+	0.907970i	$0.362368\pi$
$68$			13.9034i			1.68603i
$69$		0			0
$70$			16.4030i			1.96053i
$71$			3.70205i			0.439352i	0.975573	+	0.219676i	$0.0705000\pi$
$71$			3.70205i			0.439352i	−0.975573	+	0.219676i	$0.929500\pi$
$72$		0			0
$73$			3.35993i			0.393250i	0.980479	+	0.196625i	$0.0629981\pi$
$73$			3.35993i			0.393250i	−0.980479	+	0.196625i	$0.937002\pi$
$74$	27.7785			3.22919
$75$		0			0
$76$		−	35.2389i		−	4.04218i
$77$	21.3834			2.43686
$78$		0			0
$79$			2.99293i			0.336730i	0.985725	+	0.168365i	$0.0538488\pi$
$79$			2.99293i			0.336730i	−0.985725	+	0.168365i	$0.946151\pi$
$80$		−	19.2380i		−	2.15087i
$81$		0			0
$82$		−	1.92397i		−	0.212467i
$83$		−	7.32661i		−	0.804200i	−0.915596	−	0.402100i	$-0.868280\pi$
$83$		−	7.32661i		−	0.804200i	0.915596	−	0.402100i	$-0.131720\pi$
$84$		0			0
$85$	−3.02513			−0.328121
$86$	21.3322			2.30031
$87$		0			0
$88$	−42.6736			−4.54902
$89$	−6.84486			−0.725554			−0.362777	−	0.931876i	$-0.618171\pi$
$89$	−6.84486			−0.725554			−0.362777	+	0.931876i	$0.618171\pi$
$90$		0			0
$91$	21.2913			2.23194
$92$	−26.7365			−2.78747
$93$		0			0
$94$			2.64603i			0.272917i
$95$	7.66736			0.786655
$96$		0			0
$97$			3.10836i			0.315607i	0.987471	+	0.157803i	$0.0504412\pi$
$97$			3.10836i			0.315607i	−0.987471	+	0.157803i	$0.949559\pi$
$98$			47.3470i			4.78277i
$99$		0			0
$100$	−19.6683			−1.96683
$101$		−	10.2400i		−	1.01891i	−0.860496	−	0.509457i	$-0.829846\pi$
$101$		−	10.2400i		−	1.01891i	0.860496	−	0.509457i	$-0.170154\pi$
$102$		0			0
$103$			16.6721i			1.64276i	0.570385	+	0.821378i	$0.306794\pi$
$103$			16.6721i			1.64276i	−0.570385	+	0.821378i	$0.693206\pi$
$104$	−42.4900			−4.16648
$105$		0			0
$106$			15.8589i			1.54035i
$107$	−0.173468			−0.0167698			−0.00838491	−	0.999965i	$-0.502669\pi$
$107$	−0.173468			−0.0167698			−0.00838491	+	0.999965i	$0.502669\pi$
$108$		0			0
$109$	−1.42164			−0.136168			−0.0680842	−	0.997680i	$-0.521689\pi$
$109$	−1.42164			−0.136168			−0.0680842	+	0.997680i	$0.521689\pi$
$110$		−	14.4881i		−	1.38138i
$111$		0			0
$112$		−	78.1173i		−	7.38139i
$113$		−	16.0178i		−	1.50683i	−0.657548	−	0.753413i	$-0.728406\pi$
$113$		−	16.0178i		−	1.50683i	0.657548	−	0.753413i	$-0.271594\pi$
$114$		0			0
$115$		−	5.81738i		−	0.542474i
$116$		−	32.4494i		−	3.01285i
$117$		0			0
$118$			0.552453i			0.0508574i
$119$	−12.2837			−1.12605
$120$		0			0
$121$	−7.88705			−0.717004
$122$			14.2828i			1.29310i
$123$		0			0
$124$	36.3182			3.26147
$125$		−	10.3381i		−	0.924672i
$126$		0			0
$127$	−11.8386			−1.05051			−0.525254	−	0.850945i	$-0.676030\pi$
$127$	−11.8386			−1.05051			−0.525254	+	0.850945i	$0.676030\pi$
$128$			46.5264i			4.11239i
$129$		0			0
$130$		−	14.4257i		−	1.26522i
$131$	−22.3723			−1.95468			−0.977340	−	0.211673i	$-0.932109\pi$
$131$	−22.3723			−1.95468			−0.977340	+	0.211673i	$0.932109\pi$
$132$		0			0
$133$	31.1339			2.69965
$134$		−	18.8730i		−	1.63037i
$135$		0			0
$136$	24.5140			2.10206
$137$	16.7876			1.43426			0.717131	−	0.696938i	$-0.245455\pi$
$137$	16.7876			1.43426			0.717131	+	0.696938i	$0.245455\pi$
$138$		0			0
$139$			0.615143i			0.0521758i	0.999660	+	0.0260879i	$0.00830498\pi$
$139$			0.615143i			0.0521758i	−0.999660	+	0.0260879i	$0.991695\pi$
$140$	33.2035			2.80621
$141$		0			0
$142$	10.1851			0.854711
$143$	−18.8058			−1.57262
$144$		0			0
$145$	7.06042			0.586336
$146$	9.24382			0.765024
$147$		0			0
$148$		−	56.2303i		−	4.62210i
$149$	19.1514			1.56894			0.784472	−	0.620164i	$-0.212934\pi$
$149$	19.1514			1.56894			0.784472	+	0.620164i	$0.212934\pi$
$150$		0			0
$151$			15.7130i			1.27871i	0.768913	+	0.639354i	$0.220798\pi$
$151$			15.7130i			1.27871i	−0.768913	+	0.639354i	$0.779202\pi$
$152$	−62.1322			−5.03959
$153$		0			0
$154$		−	58.8298i		−	4.74064i
$155$			7.90220i			0.634720i
$156$		0			0
$157$	−14.8157			−1.18242			−0.591210	−	0.806518i	$-0.701349\pi$
$157$	−14.8157			−1.18242			−0.591210	+	0.806518i	$0.701349\pi$
$158$	8.23412			0.655072
$159$		0			0
$160$	−29.1310			−2.30301
$161$		−	23.6219i		−	1.86167i
$162$		0			0
$163$	13.1029			1.02630			0.513150	−	0.858299i	$-0.328479\pi$
$163$	13.1029			1.02630			0.513150	+	0.858299i	$0.328479\pi$
$164$	−3.89458			−0.304115
$165$		0			0
$166$	−20.1569			−1.56448
$167$	−17.1571			−1.32766			−0.663829	−	0.747885i	$-0.731070\pi$
$167$	−17.1571			−1.32766			−0.663829	+	0.747885i	$0.731070\pi$
$168$		0			0
$169$	−5.72485			−0.440373
$170$			8.32272i			0.638323i
$171$		0			0
$172$		−	43.1815i		−	3.29256i
$173$	−10.3555			−0.787313			−0.393657	−	0.919258i	$-0.628790\pi$
$173$	−10.3555			−0.787313			−0.393657	+	0.919258i	$0.628790\pi$
$174$		0			0
$175$		−	17.3771i		−	1.31359i
$176$			68.9978i			5.20091i
$177$		0			0
$178$			18.8316i			1.41149i
$179$	−5.24717			−0.392192			−0.196096	−	0.980585i	$-0.562826\pi$
$179$	−5.24717			−0.392192			−0.196096	+	0.980585i	$0.562826\pi$
$180$		0			0
$181$			0.243166i			0.0180744i	0.999959	+	0.00903720i	$0.00287667\pi$
$181$			0.243166i			0.0180744i	−0.999959	+	0.00903720i	$0.997123\pi$
$182$		−	58.5766i		−	4.34199i
$183$		0			0
$184$			47.1409i			3.47527i
$185$	12.2347			0.899515
$186$		0			0
$187$	10.8497			0.793411
$188$	5.35620			0.390641
$189$		0			0
$190$		−	21.0944i		−	1.53035i
$191$	22.4120			1.62168			0.810838	−	0.585270i	$-0.199011\pi$
$191$	22.4120			1.62168			0.810838	+	0.585270i	$0.199011\pi$
$192$		0			0
$193$	−16.5627			−1.19221			−0.596103	−	0.802908i	$-0.703285\pi$
$193$	−16.5627			−1.19221			−0.596103	+	0.802908i	$0.703285\pi$
$194$	8.55172			0.613977
$195$		0			0
$196$	95.8416			6.84583
$197$			10.3899i			0.740246i	0.928983	+	0.370123i	$0.120685\pi$
$197$			10.3899i			0.740246i	−0.928983	+	0.370123i	$0.879315\pi$
$198$		0			0
$199$		−	4.86472i		−	0.344851i	−0.985023	−	0.172426i	$-0.944840\pi$
$199$		−	4.86472i		−	0.344851i	0.985023	−	0.172426i	$-0.0551604\pi$
$200$			34.6786i			2.45215i
$201$		0			0
$202$	−28.1721			−1.98218
$203$	28.6693			2.01219
$204$		0			0
$205$		−	0.847391i		−	0.0591843i
$206$	45.8683			3.19580
$207$		0			0
$208$			68.7009i			4.76355i
$209$	−27.4993			−1.90217
$210$		0			0
$211$	−0.619761			−0.0426661			−0.0213331	−	0.999772i	$-0.506791\pi$
$211$	−0.619761			−0.0426661			−0.0213331	+	0.999772i	$0.506791\pi$
$212$	32.1022			2.20479
$213$		0			0
$214$			0.477245i			0.0326238i
$215$	9.39554			0.640771
$216$		0			0
$217$			32.0874i			2.17824i
$218$			3.91121i			0.264900i
$219$		0			0
$220$	−29.3273			−1.97724
$221$	10.8030			0.726691
$222$		0			0
$223$		−	22.2904i		−	1.49268i	−0.665566	−	0.746339i	$-0.731810\pi$
$223$		−	22.2904i		−	1.49268i	0.665566	−	0.746339i	$-0.268190\pi$
$224$	−118.288			−7.90348
$225$		0			0
$226$	−44.0680			−2.93136
$227$	14.2918			0.948583			0.474292	−	0.880368i	$-0.342704\pi$
$227$	14.2918			0.948583			0.474292	+	0.880368i	$0.342704\pi$
$228$		0			0
$229$		−	5.93777i		−	0.392379i	−0.980566	−	0.196189i	$-0.937143\pi$
$229$		−	5.93777i		−	0.392379i	0.980566	−	0.196189i	$-0.0628568\pi$
$230$	−16.0048			−1.05532
$231$		0			0
$232$	−57.2138			−3.75627
$233$	16.0579			1.05199			0.525994	−	0.850488i	$-0.323693\pi$
$233$	16.0579			1.05199			0.525994	+	0.850488i	$0.323693\pi$
$234$		0			0
$235$			1.16541i			0.0760233i
$236$	1.11830			0.0727949
$237$		0			0
$238$			33.7950i			2.19060i
$239$	−15.0293	−	3.62222i	−0.972164	−	0.234302i
$239$	−15.0293	−	3.62222i	−0.972164	−	0.234302i
$240$		0			0
$241$	−20.3891			−1.31338			−0.656690	−	0.754160i	$-0.728044\pi$
$241$	−20.3891			−1.31338			−0.656690	+	0.754160i	$0.728044\pi$
$242$			21.6988i			1.39485i
$243$		0			0
$244$	28.9118			1.85089
$245$			20.8534i			1.33228i
$246$		0			0
$247$	−27.3810			−1.74221
$248$		−	64.0352i		−	4.06624i
$249$		0			0
$250$	−28.4422			−1.79885
$251$			12.0463i			0.760353i	0.924914	+	0.380176i	$0.124137\pi$
$251$			12.0463i			0.760353i	−0.924914	+	0.380176i	$0.875863\pi$
$252$		0			0
$253$			20.8643i			1.31172i
$254$			32.5704i			2.04365i
$255$		0			0
$256$	59.2274			3.70171
$257$			17.3295i			1.08098i	0.841350	+	0.540491i	$0.181761\pi$
$257$			17.3295i			1.08098i	−0.841350	+	0.540491i	$0.818239\pi$
$258$		0			0
$259$	49.6800			3.08696
$260$	−29.2011			−1.81097
$261$		0			0
$262$			61.5507i			3.80261i
$263$			15.7142i			0.968979i	0.874797	+	0.484490i	$0.160995\pi$
$263$			15.7142i			0.968979i	−0.874797	+	0.484490i	$0.839005\pi$
$264$		0			0
$265$			6.98488i			0.429078i
$266$		−	85.6554i		−	5.25187i
$267$		0			0
$268$	−38.2034			−2.33364
$269$			9.02473i			0.550247i	0.961409	+	0.275124i	$0.0887188\pi$
$269$			9.02473i			0.550247i	−0.961409	+	0.275124i	$0.911281\pi$
$270$		0			0
$271$	13.0973			0.795606			0.397803	−	0.917471i	$-0.369773\pi$
$271$	13.0973			0.795606			0.397803	+	0.917471i	$0.369773\pi$
$272$		−	39.6361i		−	2.40329i
$273$		0			0
$274$		−	46.1860i		−	2.79020i
$275$			15.3485i			0.925550i
$276$		0			0
$277$			2.95021i			0.177261i	0.996065	+	0.0886304i	$0.0282490\pi$
$277$			2.95021i			0.177261i	−0.996065	+	0.0886304i	$0.971751\pi$
$278$	1.69238			0.101502
$279$		0			0
$280$		−	58.5434i		−	3.49864i
$281$	−11.8135			−0.704735			−0.352367	−	0.935862i	$-0.614623\pi$
$281$	−11.8135			−0.704735			−0.352367	+	0.935862i	$0.614623\pi$
$282$		0			0
$283$	15.7602			0.936847			0.468424	−	0.883504i	$-0.344822\pi$
$283$	15.7602			0.936847			0.468424	+	0.883504i	$0.344822\pi$
$284$		−	20.6170i		−	1.22339i
$285$		0			0
$286$			51.7384i			3.05935i
$287$		−	3.44089i		−	0.203109i
$288$		0			0
$289$	10.7673			0.633372
$290$		−	19.4246i		−	1.14065i
$291$		0			0
$292$		−	18.7117i		−	1.09502i
$293$		−	12.2076i		−	0.713177i	−0.934262	−	0.356588i	$-0.883940\pi$
$293$		−	12.2076i		−	0.713177i	0.934262	−	0.356588i	$-0.116060\pi$
$294$		0			0
$295$			0.243322i			0.0141667i
$296$	−99.1436			−5.76260
$297$		0			0
$298$		−	52.6892i		−	3.05221i
$299$			20.7745i			1.20142i
$300$		0			0
$301$	38.1513			2.19900
$302$	43.2296			2.48758
$303$		0			0
$304$			100.460i			5.76177i
$305$			6.29069i			0.360204i
$306$		0			0
$307$	21.6488			1.23556			0.617780	−	0.786351i	$-0.288032\pi$
$307$	21.6488			1.23556			0.617780	+	0.786351i	$0.288032\pi$
$308$	−119.086			−6.78553
$309$		0			0
$310$	21.7405			1.23478
$311$			21.3397i			1.21006i	0.796202	+	0.605031i	$0.206839\pi$
$311$			21.3397i			1.21006i	−0.796202	+	0.605031i	$0.793161\pi$
$312$		0			0
$313$			29.8633i			1.68797i	0.536366	+	0.843986i	$0.319797\pi$
$313$			29.8633i			1.68797i	−0.536366	+	0.843986i	$0.680203\pi$
$314$			40.7608i			2.30026i
$315$		0			0
$316$		−	16.6678i		−	0.937639i
$317$	20.4143			1.14658			0.573291	−	0.819352i	$-0.305666\pi$
$317$	20.4143			1.14658			0.573291	+	0.819352i	$0.305666\pi$
$318$		0			0
$319$	−25.3224			−1.41778
$320$			41.6690i			2.32937i
$321$		0			0
$322$	−64.9884			−3.62166
$323$	15.7971			0.878972
$324$		0			0
$325$			15.2825i			0.847719i
$326$		−	36.0487i		−	1.99655i
$327$		0			0
$328$			6.86680i			0.379155i
$329$			4.73225i			0.260897i
$330$		0			0
$331$			8.51689i			0.468131i	0.972221	+	0.234065i	$0.0752030\pi$
$331$			8.51689i			0.468131i	−0.972221	+	0.234065i	$0.924797\pi$
$332$			40.8025i			2.23933i
$333$		0			0
$334$			47.2026i			2.58281i
$335$		−	8.31237i		−	0.454154i
$336$		0			0
$337$	27.0995			1.47620			0.738102	−	0.674689i	$-0.235722\pi$
$337$	27.0995			1.47620			0.738102	+	0.674689i	$0.235722\pi$
$338$			15.7502i			0.856697i
$339$		0			0
$340$	16.8472			0.913666
$341$		−	28.3415i		−	1.53478i
$342$		0			0
$343$			50.2346i			2.71241i
$344$	−76.1364			−4.10500
$345$		0			0
$346$			28.4900i			1.53163i
$347$			15.2451i			0.818401i	0.912445	+	0.409200i	$0.134192\pi$
$347$			15.2451i			0.818401i	−0.912445	+	0.409200i	$0.865808\pi$
$348$		0			0
$349$	−12.2341			−0.654876			−0.327438	−	0.944873i	$-0.606185\pi$
$349$	−12.2341			−0.654876			−0.327438	+	0.944873i	$0.606185\pi$
$350$	−47.8079			−2.55544
$351$		0			0
$352$	104.479			5.56877
$353$	−2.84927			−0.151651			−0.0758257	−	0.997121i	$-0.524159\pi$
$353$	−2.84927			−0.151651			−0.0758257	+	0.997121i	$0.524159\pi$
$354$		0			0
$355$	4.48589			0.238086
$356$	38.1196			2.02033
$357$		0			0
$358$			14.4360i			0.762966i
$359$		−	30.1564i		−	1.59159i	−0.605565	−	0.795796i	$-0.707053\pi$
$359$		−	30.1564i		−	1.59159i	0.605565	−	0.795796i	$-0.292947\pi$
$360$		0			0
$361$	−21.0386			−1.10729
$362$	0.668997			0.0351617
$363$		0			0
$364$	−118.573			−6.21492
$365$	4.07133			0.213103
$366$		0			0
$367$	−32.8208			−1.71323			−0.856615	−	0.515957i	$-0.827437\pi$
$367$	−32.8208			−1.71323			−0.856615	+	0.515957i	$0.827437\pi$
$368$	76.2209			3.97329
$369$		0			0
$370$		−	33.6601i		−	1.74991i
$371$			28.3626i			1.47251i
$372$		0			0
$373$	−5.25094			−0.271883			−0.135942	−	0.990717i	$-0.543406\pi$
$373$	−5.25094			−0.271883			−0.135942	+	0.990717i	$0.543406\pi$
$374$		−	29.8497i		−	1.54349i
$375$		0			0
$376$		−	9.44389i		−	0.487031i
$377$	−25.2135			−1.29856
$378$		0			0
$379$			8.85980i			0.455097i	0.973767	+	0.227549i	$0.0730711\pi$
$379$			8.85980i			0.455097i	−0.973767	+	0.227549i	$0.926929\pi$
$380$	−42.7001			−2.19047
$381$		0			0
$382$		−	61.6598i		−	3.15479i
$383$			18.0512i			0.922371i	0.887304	+	0.461185i	$0.152576\pi$
$383$			18.0512i			0.922371i	−0.887304	+	0.461185i	$0.847424\pi$
$384$		0			0
$385$		−	25.9109i		−	1.32054i
$386$			45.5671i			2.31931i
$387$		0			0
$388$		−	17.3107i		−	0.878818i
$389$		−	36.6136i		−	1.85638i	−0.372102	−	0.928192i	$-0.621363\pi$
$389$		−	36.6136i		−	1.85638i	0.372102	−	0.928192i	$-0.378637\pi$
$390$		0			0
$391$		−	11.9855i		−	0.606135i
$392$		−	168.985i		−	8.53503i
$393$		0			0
$394$	28.5845			1.44007
$395$	3.62662			0.182475
$396$		0			0
$397$			2.79185i			0.140119i	0.997543	+	0.0700596i	$0.0223189\pi$
$397$			2.79185i			0.140119i	−0.997543	+	0.0700596i	$0.977681\pi$
$398$	−13.3838			−0.670869
$399$		0			0
$400$	56.0709			2.80355
$401$		−	16.7221i		−	0.835063i	−0.908663	−	0.417531i	$-0.862895\pi$
$401$		−	16.7221i		−	0.835063i	0.908663	−	0.417531i	$-0.137105\pi$
$402$		0			0
$403$		−	28.2196i		−	1.40572i
$404$			57.0271i			2.83720i
$405$		0			0
$406$		−	78.8749i		−	3.91450i
$407$	−43.8803			−2.17507
$408$		0			0
$409$	7.50283			0.370991			0.185495	−	0.982645i	$-0.440611\pi$
$409$	7.50283			0.370991			0.185495	+	0.982645i	$0.440611\pi$
$410$	−2.33134			−0.115137
$411$		0			0
$412$		−	92.8485i		−	4.57431i
$413$			0.988025i			0.0486175i
$414$		0			0
$415$	−8.87789			−0.435799
$416$	104.030			5.10048
$417$		0			0
$418$			75.6559i			3.70045i
$419$			10.2128i			0.498929i	0.968384	+	0.249465i	$0.0802546\pi$
$419$			10.2128i			0.498929i	−0.968384	+	0.249465i	$0.919745\pi$
$420$		0			0
$421$	3.35723			0.163621			0.0818107	−	0.996648i	$-0.473930\pi$
$421$	3.35723			0.163621			0.0818107	+	0.996648i	$0.473930\pi$
$422$			1.70508i			0.0830022i
$423$		0			0
$424$		−	56.6017i		−	2.74882i
$425$		−	8.81701i		−	0.427688i
$426$		0			0
$427$			25.5438i			1.23615i
$428$	0.966059			0.0466962
$429$		0			0
$430$		−	25.8490i		−	1.24655i
$431$			17.1322i			0.825228i	0.910906	+	0.412614i	$0.135384\pi$
$431$			17.1322i			0.825228i	−0.910906	+	0.412614i	$0.864616\pi$
$432$		0			0
$433$		−	7.53265i		−	0.361996i	−0.983483	−	0.180998i	$-0.942067\pi$
$433$		−	7.53265i		−	0.361996i	0.983483	−	0.180998i	$-0.0579327\pi$
$434$	88.2788			4.23752
$435$		0			0
$436$	7.91722			0.379166
$437$			30.3781i			1.45318i
$438$		0			0
$439$	19.1180			0.912452			0.456226	−	0.889864i	$-0.349201\pi$
$439$	19.1180			0.912452			0.456226	+	0.889864i	$0.349201\pi$
$440$			51.7090i			2.46513i
$441$		0			0
$442$		−	29.7213i		−	1.41370i
$443$		−	22.6904i		−	1.07805i	−0.842288	−	0.539027i	$-0.818792\pi$
$443$		−	22.6904i		−	1.07805i	0.842288	−	0.539027i	$-0.181208\pi$
$444$		0			0
$445$			8.29414i			0.393180i
$446$	−61.3254			−2.90384
$447$		0			0
$448$			169.200i			7.99395i
$449$	25.9805			1.22610			0.613049	−	0.790045i	$-0.289943\pi$
$449$	25.9805			1.22610			0.613049	+	0.790045i	$0.289943\pi$
$450$		0			0
$451$			3.03920i			0.143110i
$452$			89.2042i			4.19581i
$453$		0			0
$454$		−	39.3197i		−	1.84536i
$455$		−	25.7994i		−	1.20949i
$456$		0			0
$457$	−38.4930			−1.80062			−0.900312	−	0.435245i	$-0.856662\pi$
$457$	−38.4930			−1.80062			−0.900312	+	0.435245i	$0.856662\pi$
$458$	−16.3360			−0.763330
$459$		0			0
$460$			32.3974i			1.51054i
$461$	17.4108			0.810901			0.405450	−	0.914117i	$-0.367115\pi$
$461$	17.4108			0.810901			0.405450	+	0.914117i	$0.367115\pi$
$462$		0			0
$463$			38.3441i			1.78200i	0.454003	+	0.891000i	$0.349995\pi$
$463$			38.3441i			1.78200i	−0.454003	+	0.891000i	$0.650005\pi$
$464$			92.5075i			4.29455i
$465$		0			0
$466$		−	44.1784i		−	2.04653i
$467$	5.73300			0.265292			0.132646	−	0.991164i	$-0.457653\pi$
$467$	5.73300			0.265292			0.132646	+	0.991164i	$0.457653\pi$
$468$		0			0
$469$		−	33.7530i		−	1.55857i
$470$	3.20628			0.147895
$471$		0			0
$472$		−	1.97175i		−	0.0907570i
$473$	−33.6975			−1.54941
$474$		0			0
$475$			22.3472i			1.02536i
$476$	68.4091			3.13553
$477$		0			0
$478$	−9.96545	+	41.3485i	−0.455809	+	1.89124i
$479$		−	11.6662i		−	0.533044i	−0.963829	−	0.266522i	$-0.914125\pi$
$479$		−	11.6662i		−	0.533044i	0.963829	−	0.266522i	$-0.0858745\pi$
$480$		0			0
$481$	−43.6915			−1.99216
$482$			56.0945i			2.55504i
$483$		0			0
$484$	43.9236			1.99653
$485$	3.76650			0.171028
$486$		0			0
$487$	19.6203			0.889080			0.444540	−	0.895759i	$-0.353367\pi$
$487$	19.6203			0.889080			0.444540	+	0.895759i	$0.353367\pi$
$488$		−	50.9763i		−	2.30759i
$489$		0			0
$490$	57.3719			2.59180
$491$	21.8862			0.987709			0.493855	−	0.869544i	$-0.335587\pi$
$491$	21.8862			0.987709			0.493855	+	0.869544i	$0.335587\pi$
$492$		0			0
$493$	14.5466			0.655145
$494$			75.3304i			3.38927i
$495$		0			0
$496$	−103.537			−4.64894
$497$	18.2153			0.817067
$498$		0			0
$499$			1.29864i			0.0581349i	0.999577	+	0.0290675i	$0.00925377\pi$
$499$			1.29864i			0.0581349i	−0.999577	+	0.0290675i	$0.990746\pi$
$500$			57.5739i			2.57478i
$501$		0			0
$502$	33.1416			1.47918
$503$		−	35.9826i		−	1.60439i	−0.597065	−	0.802193i	$-0.703667\pi$
$503$		−	35.9826i		−	1.60439i	0.597065	−	0.802193i	$-0.296333\pi$
$504$		0			0
$505$	−12.4081			−0.552152
$506$	57.4016			2.55181
$507$		0			0
$508$	65.9302			2.92518
$509$		−	3.55717i		−	0.157669i	−0.996888	−	0.0788344i	$-0.974880\pi$
$509$		−	3.55717i		−	0.157669i	0.996888	−	0.0788344i	$-0.0251198\pi$
$510$		0			0
$511$	16.5319			0.731330
$512$		−	69.8933i		−	3.08888i
$513$		0			0
$514$	47.6767			2.10293
$515$	20.2022			0.890214
$516$		0			0
$517$		−	4.17980i		−	0.183827i
$518$		−	136.679i		−	6.00535i
$519$		0			0
$520$			51.4865i			2.25783i
$521$	18.7018			0.819341			0.409671	−	0.912234i	$-0.365644\pi$
$521$	18.7018			0.819341			0.409671	+	0.912234i	$0.365644\pi$
$522$		0			0
$523$	15.8207			0.691789			0.345895	−	0.938273i	$-0.387575\pi$
$523$	15.8207			0.691789			0.345895	+	0.938273i	$0.387575\pi$
$524$	124.593			5.44288
$525$		0			0
$526$	43.2328			1.88504
$527$			16.2809i			0.709207i
$528$		0			0
$529$	0.0484447			0.00210629
$530$	19.2168			0.834723
$531$		0			0
$532$	−173.387			−7.51728
$533$			3.02612i			0.131076i
$534$		0			0
$535$			0.210197i			0.00908762i
$536$			67.3590i			2.90947i
$537$		0			0
$538$	24.8288			1.07044
$539$		−	74.7916i		−	3.22150i
$540$		0			0
$541$		−	7.22020i		−	0.310420i	−0.987881	−	0.155210i	$-0.950395\pi$
$541$		−	7.22020i		−	0.310420i	0.987881	−	0.155210i	$-0.0496055\pi$
$542$		−	36.0333i		−	1.54776i
$543$		0			0
$544$	−60.0185			−2.57327
$545$			1.72265i			0.0737900i
$546$		0			0
$547$			13.4082i			0.573291i	0.958037	+	0.286646i	$0.0925403\pi$
$547$			13.4082i			0.573291i	−0.958037	+	0.286646i	$0.907460\pi$
$548$	−93.4914			−3.99376
$549$		0			0
$550$	42.2268			1.80056
$551$	−36.8691			−1.57068
$552$		0			0
$553$	14.7262			0.626220
$554$	8.11660			0.344841
$555$		0			0
$556$		−	3.42578i		−	0.145285i
$557$	9.24598			0.391765			0.195882	−	0.980627i	$-0.437243\pi$
$557$	9.24598			0.391765			0.195882	+	0.980627i	$0.437243\pi$
$558$		0			0
$559$	−33.5525			−1.41912
$560$	−94.6573			−4.00000
$561$		0			0
$562$			32.5013i			1.37098i
$563$		−	12.3807i		−	0.521783i	−0.965368	−	0.260891i	$-0.915984\pi$
$563$		−	12.3807i		−	0.521783i	0.965368	−	0.260891i	$-0.0840165\pi$
$564$		0			0
$565$	−19.4093			−0.816554
$566$		−	43.3594i		−	1.82253i
$567$		0			0
$568$	−36.3512			−1.52526
$569$		−	2.26459i		−	0.0949364i	−0.998873	−	0.0474682i	$-0.984885\pi$
$569$		−	2.26459i		−	0.0949364i	0.998873	−	0.0474682i	$-0.0151153\pi$
$570$		0			0
$571$	2.63120			0.110112			0.0550561	−	0.998483i	$-0.482466\pi$
$571$	2.63120			0.110112			0.0550561	+	0.998483i	$0.482466\pi$
$572$	104.731			4.37901
$573$		0			0
$574$	−9.46656			−0.395127
$575$	16.9553			0.707084
$576$		0			0
$577$	46.7548			1.94643			0.973214	−	0.229900i	$-0.0738401\pi$
$577$	46.7548			1.94643			0.973214	+	0.229900i	$0.0738401\pi$
$578$		−	29.6230i		−	1.23216i
$579$		0			0
$580$	−39.3200			−1.63267
$581$	−36.0493			−1.49558
$582$		0			0
$583$		−	25.0515i		−	1.03753i
$584$	−32.9919			−1.36521
$585$		0			0
$586$	−33.5856			−1.38741
$587$			30.6653i			1.26569i	0.774277	+	0.632847i	$0.218114\pi$
$587$			30.6653i			1.26569i	−0.774277	+	0.632847i	$0.781886\pi$
$588$		0			0
$589$		−	41.2649i		−	1.70029i
$590$	0.669425			0.0275598
$591$		0			0
$592$			160.303i			6.58840i
$593$	1.90746			0.0783302			0.0391651	−	0.999233i	$-0.487530\pi$
$593$	1.90746			0.0783302			0.0391651	+	0.999233i	$0.487530\pi$
$594$		0			0
$595$			14.8846i			0.610210i
$596$	−106.656			−4.36878
$597$		0			0
$598$	57.1546			2.33723
$599$		−	25.4428i		−	1.03957i	−0.854299	−	0.519783i	$-0.826013\pi$
$599$		−	25.4428i		−	1.03957i	0.854299	−	0.519783i	$-0.173987\pi$
$600$		0			0
$601$			23.7731i			0.969726i	0.874590	+	0.484863i	$0.161130\pi$
$601$			23.7731i			0.969726i	−0.874590	+	0.484863i	$0.838870\pi$
$602$		−	104.962i		−	4.27792i
$603$		0			0
$604$		−	87.5070i		−	3.56061i
$605$			9.55699i			0.388547i
$606$		0			0
$607$		−	6.85005i		−	0.278035i	−0.990290	−	0.139017i	$-0.955606\pi$
$607$		−	6.85005i		−	0.278035i	0.990290	−	0.139017i	$-0.0443944\pi$
$608$	152.121			6.16930
$609$		0			0
$610$	17.3069			0.700736
$611$		−	4.16182i		−	0.168369i
$612$		0			0
$613$	−9.67687			−0.390845			−0.195423	−	0.980719i	$-0.562608\pi$
$613$	−9.67687			−0.390845			−0.195423	+	0.980719i	$0.562608\pi$
$614$		−	59.5600i		−	2.40365i
$615$		0			0
$616$			209.968i			8.45985i
$617$	−37.4955			−1.50951			−0.754757	−	0.656005i	$-0.772245\pi$
$617$	−37.4955			−1.50951			−0.754757	+	0.656005i	$0.772245\pi$
$618$		0			0
$619$			8.45381i			0.339787i	0.985462	+	0.169894i	$0.0543424\pi$
$619$			8.45381i			0.339787i	−0.985462	+	0.169894i	$0.945658\pi$
$620$		−	44.0079i		−	1.76740i
$621$		0			0
$622$	58.7096			2.35404
$623$			33.6790i			1.34932i
$624$		0			0
$625$	5.13146			0.205259
$626$	82.1597			3.28376
$627$		0			0
$628$	82.5096			3.29249
$629$	25.2072			1.00508
$630$		0			0
$631$	−5.80793			−0.231210			−0.115605	−	0.993295i	$-0.536881\pi$
$631$	−5.80793			−0.231210			−0.115605	+	0.993295i	$0.536881\pi$
$632$	−29.3882			−1.16900
$633$		0			0
$634$		−	56.1638i		−	2.23055i
$635$			14.3452i			0.569274i
$636$		0			0
$637$		−	74.4698i		−	2.95060i
$638$			69.6670i			2.75814i
$639$		0			0
$640$	56.3776			2.22852
$641$		−	7.17829i		−	0.283525i	−0.989901	−	0.141763i	$-0.954723\pi$
$641$		−	7.17829i		−	0.283525i	0.989901	−	0.141763i	$-0.0452770\pi$
$642$		0			0
$643$	6.21921			0.245262			0.122631	−	0.992452i	$-0.460867\pi$
$643$	6.21921			0.245262			0.122631	+	0.992452i	$0.460867\pi$
$644$			131.552i			5.18388i
$645$		0			0
$646$		−	43.4608i		−	1.70994i
$647$			18.1619i			0.714019i	0.934101	+	0.357010i	$0.116204\pi$
$647$			18.1619i			0.714019i	−0.934101	+	0.357010i	$0.883796\pi$
$648$		0			0
$649$		−	0.872682i		−	0.0342558i
$650$	42.0451			1.64914
$651$		0			0
$652$	−72.9711			−2.85777
$653$	−4.33066			−0.169472			−0.0847359	−	0.996403i	$-0.527005\pi$
$653$	−4.33066			−0.169472			−0.0847359	+	0.996403i	$0.527005\pi$
$654$		0			0
$655$			27.1093i			1.05925i
$656$	11.1027			0.433490
$657$		0			0
$658$	13.0193			0.507547
$659$	−24.0715			−0.937692			−0.468846	−	0.883280i	$-0.655330\pi$
$659$	−24.0715			−0.937692			−0.468846	+	0.883280i	$0.655330\pi$
$660$		0			0
$661$	−21.0013			−0.816857			−0.408428	−	0.912790i	$-0.633923\pi$
$661$	−21.0013			−0.816857			−0.408428	+	0.912790i	$0.633923\pi$
$662$	23.4316			0.910696
$663$		0			0
$664$	71.9417			2.79188
$665$		−	37.7259i		−	1.46295i
$666$		0			0
$667$			27.9733i			1.08313i
$668$	95.5493			3.69691
$669$		0			0
$670$	−22.8690			−0.883505
$671$		−	22.5618i		−	0.870988i
$672$		0			0
$673$		−	20.2940i		−	0.782277i	−0.920332	−	0.391138i	$-0.872081\pi$
$673$		−	20.2940i		−	0.782277i	0.920332	−	0.391138i	$-0.127919\pi$
$674$		−	74.5561i		−	2.87179i
$675$		0			0
$676$	31.8821			1.22624
$677$	−32.2460			−1.23931			−0.619656	−	0.784873i	$-0.712728\pi$
$677$	−32.2460			−1.23931			−0.619656	+	0.784873i	$0.712728\pi$
$678$		0			0
$679$	15.2942			0.586936
$680$		−	29.7044i		−	1.13911i
$681$		0			0
$682$	−77.9731			−2.98574
$683$	−45.1184			−1.72641			−0.863203	−	0.504857i	$-0.831545\pi$
$683$	−45.1184			−1.72641			−0.863203	+	0.504857i	$0.831545\pi$
$684$		0			0
$685$		−	20.3421i		−	0.777231i
$686$	138.205			5.27670
$687$		0			0
$688$			123.103i			4.69326i
$689$		−	24.9437i		−	0.950280i
$690$		0			0
$691$	−41.9994			−1.59773			−0.798866	−	0.601509i	$-0.794567\pi$
$691$	−41.9994			−1.59773			−0.798866	+	0.601509i	$0.794567\pi$
$692$	57.6705			2.19230
$693$		0			0
$694$	41.9423			1.59211
$695$	0.745389			0.0282742
$696$		0			0
$697$		−	1.74588i		−	0.0661299i
$698$			33.6584i			1.27399i
$699$		0			0
$700$			96.7746i			3.65774i
$701$	−0.149454			−0.00564481			−0.00282241	−	0.999996i	$-0.500898\pi$
$701$	−0.149454			−0.00564481			−0.00282241	+	0.999996i	$0.500898\pi$
$702$		0			0
$703$	−63.8891			−2.40962
$704$		−	149.447i		−	5.63251i
$705$		0			0
$706$			7.83890i			0.295021i
$707$	−50.3839			−1.89488
$708$		0			0
$709$		−	39.6852i		−	1.49041i	−0.666836	−	0.745204i	$-0.732352\pi$
$709$		−	39.6852i		−	1.49041i	0.666836	−	0.745204i	$-0.267648\pi$
$710$		−	12.3416i		−	0.463170i
$711$		0			0
$712$		−	67.2113i		−	2.51885i
$713$	−31.3085			−1.17251
$714$		0			0
$715$			22.7876i			0.852206i
$716$	29.2219			1.09207
$717$		0			0
$718$	−82.9661			−3.09627
$719$			17.0396i			0.635471i	0.948179	+	0.317736i	$0.102922\pi$
$719$			17.0396i			0.635471i	−0.948179	+	0.317736i	$0.897078\pi$
$720$		0			0
$721$	82.0324			3.05505
$722$			57.8813i			2.15412i
$723$		0			0
$724$		−	1.35421i		−	0.0503288i
$725$			20.5782i			0.764256i
$726$		0			0
$727$	−10.8353			−0.401858			−0.200929	−	0.979606i	$-0.564396\pi$
$727$	−10.8353			−0.401858			−0.200929	+	0.979606i	$0.564396\pi$
$728$			209.064i			7.74844i
$729$		0			0
$730$		−	11.2010i		−	0.414569i
$731$	19.3576			0.715968
$732$		0			0
$733$	41.4074			1.52942			0.764708	−	0.644377i	$-0.222883\pi$
$733$	41.4074			1.52942			0.764708	+	0.644377i	$0.222883\pi$
$734$			90.2963i			3.33290i
$735$		0			0
$736$		−	115.417i		−	4.25432i
$737$			29.8126i			1.09816i
$738$		0			0
$739$	−14.8075			−0.544703			−0.272352	−	0.962198i	$-0.587801\pi$
$739$	−14.8075			−0.544703			−0.272352	+	0.962198i	$0.587801\pi$
$740$	−68.1361			−2.50473
$741$		0			0
$742$	78.0311			2.86461
$743$	−6.37843			−0.234002			−0.117001	−	0.993132i	$-0.537328\pi$
$743$	−6.37843			−0.234002			−0.117001	+	0.993132i	$0.537328\pi$
$744$		0			0
$745$		−	23.2064i		−	0.850216i
$746$			14.4464i			0.528919i
$747$		0			0
$748$	−60.4230			−2.20928
$749$			0.853521i			0.0311870i
$750$		0			0
$751$	44.4848			1.62327			0.811637	−	0.584161i	$-0.198577\pi$
$751$	44.4848			1.62327			0.811637	+	0.584161i	$0.198577\pi$
$752$	−15.2696			−0.556824
$753$		0			0
$754$			69.3672i			2.52620i
$755$	19.0400			0.692936
$756$		0			0
$757$	−35.6892			−1.29715			−0.648574	−	0.761152i	$-0.724634\pi$
$757$	−35.6892			−1.29715			−0.648574	+	0.761152i	$0.724634\pi$
$758$	24.3750			0.885341
$759$		0			0
$760$			75.2876i			2.73097i
$761$			12.6027i			0.456847i	0.973562	+	0.228424i	$0.0733571\pi$
$761$			12.6027i			0.456847i	−0.973562	+	0.228424i	$0.926643\pi$
$762$		0			0
$763$			6.99493i			0.253233i
$764$	−124.814			−4.51562
$765$		0			0
$766$	49.6622			1.79437
$767$		−	0.868927i		−	0.0313751i
$768$		0			0
$769$		−	47.3832i		−	1.70868i	−0.519713	−	0.854341i	$-0.673961\pi$
$769$		−	47.3832i		−	1.70868i	0.519713	−	0.854341i	$-0.326039\pi$
$770$	−71.2860			−2.56897
$771$		0			0
$772$	92.2387			3.31975
$773$	−34.9390			−1.25667			−0.628333	−	0.777944i	$-0.716263\pi$
$773$	−34.9390			−1.25667			−0.628333	+	0.777944i	$0.716263\pi$
$774$		0			0
$775$	−23.0317			−0.827323
$776$	−30.5217			−1.09567
$777$		0			0
$778$	−100.731			−3.61139
$779$			4.42503i			0.158543i
$780$		0			0
$781$	−16.0888			−0.575703
$782$	−32.9746			−1.17917
$783$		0			0
$784$	−273.228			−9.75813
$785$			17.9526i			0.640757i
$786$		0			0
$787$		−	5.19381i		−	0.185139i	−0.995706	−	0.0925696i	$-0.970492\pi$
$787$		−	5.19381i		−	0.185139i	0.995706	−	0.0925696i	$-0.0295081\pi$
$788$		−	57.8619i		−	2.06124i
$789$		0			0
$790$		−	9.97755i		−	0.354985i
$791$	−78.8127			−2.80226
$792$		0			0
$793$		−	22.4647i		−	0.797745i
$794$	7.68094			0.272586
$795$		0			0
$796$			27.0920i			0.960251i
$797$		−	2.23081i		−	0.0790193i	−0.999219	−	0.0395097i	$-0.987420\pi$
$797$		−	2.23081i		−	0.0790193i	0.999219	−	0.0395097i	$-0.0125796\pi$
$798$		0			0
$799$			2.40110i			0.0849449i
$800$		−	84.9049i		−	3.00184i
$801$		0			0
$802$	−46.0058			−1.62452
$803$	−14.6020			−0.515293
$804$		0			0
$805$	−28.6234			−1.00884
$806$	−77.6376			−2.73467
$807$		0			0
$808$	100.548			3.53728
$809$	28.5848			1.00499			0.502494	−	0.864581i	$-0.332416\pi$
$809$	28.5848			1.00499			0.502494	+	0.864581i	$0.332416\pi$
$810$		0			0
$811$			45.6642i			1.60349i	0.597668	+	0.801744i	$0.296094\pi$
$811$			45.6642i			1.60349i	−0.597668	+	0.801744i	$0.703906\pi$
$812$	−159.662			−5.60303
$813$		0			0
$814$			120.723i			4.23135i
$815$		−	15.8772i		−	0.556155i
$816$		0			0
$817$	−49.0630			−1.71650
$818$		−	20.6417i		−	0.721722i
$819$		0			0
$820$			4.71918i			0.164801i
$821$	0.294928			0.0102931			0.00514653	−	0.999987i	$-0.498362\pi$
$821$	0.294928			0.0102931			0.00514653	+	0.999987i	$0.498362\pi$
$822$		0			0
$823$		−	23.8472i		−	0.831262i	−0.909533	−	0.415631i	$-0.863561\pi$
$823$		−	23.8472i		−	0.831262i	0.909533	−	0.415631i	$-0.136439\pi$
$824$	−163.708			−5.70302
$825$		0			0
$826$	2.71825			0.0945800
$827$			11.2552i			0.391381i	0.980666	+	0.195691i	$0.0626948\pi$
$827$			11.2552i			0.391381i	−0.980666	+	0.195691i	$0.937305\pi$
$828$		0			0
$829$		−	38.9689i		−	1.35345i	−0.736238	−	0.676723i	$-0.763399\pi$
$829$		−	38.9689i		−	1.35345i	0.736238	−	0.676723i	$-0.236601\pi$
$830$			24.4248i			0.847798i
$831$		0			0
$832$		−	148.804i		−	5.15886i
$833$			42.9643i			1.48863i
$834$		0			0
$835$			20.7898i			0.719462i
$836$	153.146			5.29665
$837$		0			0
$838$	28.0975			0.970611
$839$		−	43.7282i		−	1.50967i	−0.655917	−	0.754833i	$-0.727718\pi$
$839$		−	43.7282i		−	1.50967i	0.655917	−	0.754833i	$-0.272282\pi$
$840$		0			0
$841$	−4.95059			−0.170710
$842$		−	9.23640i		−	0.318307i
$843$		0			0
$844$	3.45150			0.118805
$845$			6.93698i			0.238639i
$846$		0			0
$847$			38.8068i			1.33342i
$848$	−91.5178			−3.14273
$849$		0			0
$850$	−24.2573			−0.832019
$851$			48.4739i			1.66167i
$852$		0			0
$853$	17.5755			0.601773			0.300887	−	0.953660i	$-0.402717\pi$
$853$	17.5755			0.601773			0.300887	+	0.953660i	$0.402717\pi$
$854$	70.2760			2.40479
$855$		0			0
$856$		−	1.70333i		−	0.0582185i
$857$	38.2990			1.30827			0.654134	−	0.756379i	$-0.273033\pi$
$857$	38.2990			1.30827			0.654134	+	0.756379i	$0.273033\pi$
$858$		0			0
$859$	0.580912			0.0198205			0.00991023	−	0.999951i	$-0.496845\pi$
$859$	0.580912			0.0198205			0.00991023	+	0.999951i	$0.496845\pi$
$860$	−52.3245			−1.78425
$861$		0			0
$862$	47.1340			1.60539
$863$	−4.46688			−0.152054			−0.0760272	−	0.997106i	$-0.524224\pi$
$863$	−4.46688			−0.152054			−0.0760272	+	0.997106i	$0.524224\pi$
$864$		0			0
$865$			12.5481i			0.426648i
$866$	−20.7238			−0.704223
$867$		0			0
$868$		−	178.697i		−	6.06539i
$869$	−13.0070			−0.441233
$870$		0			0
$871$			29.6844i			1.00582i
$872$		−	13.9594i		−	0.472725i
$873$		0			0
$874$	83.5760			2.82700
$875$	−50.8670			−1.71962
$876$		0			0
$877$	−1.51988			−0.0513227			−0.0256613	−	0.999671i	$-0.508169\pi$
$877$	−1.51988			−0.0513227			−0.0256613	+	0.999671i	$0.508169\pi$
$878$		−	52.5973i		−	1.77507i
$879$		0			0
$880$	83.6069			2.81839
$881$	14.9657			0.504207			0.252104	−	0.967700i	$-0.418878\pi$
$881$	14.9657			0.504207			0.252104	+	0.967700i	$0.418878\pi$
$882$		0			0
$883$	−2.52962			−0.0851286			−0.0425643	−	0.999094i	$-0.513553\pi$
$883$	−2.52962			−0.0851286			−0.0425643	+	0.999094i	$0.513553\pi$
$884$	−60.1630			−2.02350
$885$		0			0
$886$	−62.4258			−2.09724
$887$			2.68809i			0.0902571i	0.998981	+	0.0451286i	$0.0143697\pi$
$887$			2.68809i			0.0902571i	−0.998981	+	0.0451286i	$0.985630\pi$
$888$		0			0
$889$			58.2499i			1.95364i
$890$	22.8188			0.764888
$891$		0			0
$892$			124.137i			4.15642i
$893$		−	6.08574i		−	0.203651i
$894$		0			0
$895$			6.35817i			0.212530i
$896$	228.925			7.64786
$897$		0			0
$898$		−	71.4776i		−	2.38524i
$899$		−	37.9984i		−	1.26732i
$900$		0			0
$901$			14.3909i			0.479432i
$902$	8.36143			0.278405
$903$		0			0
$904$	157.282			5.23113
$905$	0.294652			0.00979457
$906$		0			0
$907$		−	10.9107i		−	0.362285i	−0.983457	−	0.181143i	$-0.942020\pi$
$907$		−	10.9107i		−	0.362285i	0.983457	−	0.181143i	$-0.0579795\pi$
$908$	−79.5924			−2.64137
$909$		0			0
$910$	−70.9792			−2.35294
$911$	−24.1050			−0.798636			−0.399318	−	0.916813i	$-0.630753\pi$
$911$	−24.1050			−0.798636			−0.399318	+	0.916813i	$0.630753\pi$
$912$		0			0
$913$	31.8409			1.05378
$914$			105.902i			3.50291i
$915$		0			0
$916$			33.0679i			1.09259i
$917$			110.079i			3.63514i
$918$		0			0
$919$	−28.7910			−0.949729			−0.474864	−	0.880059i	$-0.657503\pi$
$919$	−28.7910			−0.949729			−0.474864	+	0.880059i	$0.657503\pi$
$920$	57.1222			1.88326
$921$		0			0
$922$		−	47.9005i		−	1.57752i
$923$	−16.0196			−0.527291
$924$		0			0
$925$			35.6592i			1.17247i
$926$	105.492			3.46668
$927$		0			0
$928$	140.079			4.59831
$929$	−44.3659			−1.45560			−0.727798	−	0.685791i	$-0.759456\pi$
$929$	−44.3659			−1.45560			−0.727798	+	0.685791i	$0.759456\pi$
$930$		0			0
$931$		−	108.896i		−	3.56891i
$932$	−89.4277			−2.92930
$933$		0			0
$934$		−	15.7726i		−	0.516095i
$935$		−	13.1470i		−	0.429952i
$936$		0			0
$937$	10.8119			0.353210			0.176605	−	0.984282i	$-0.443488\pi$
$937$	10.8119			0.353210			0.176605	+	0.984282i	$0.443488\pi$
$938$	−92.8611			−3.03202
$939$		0			0
$940$		−	6.49028i		−	0.211690i
$941$	1.80218			0.0587493			0.0293747	−	0.999568i	$-0.490648\pi$
$941$	1.80218			0.0587493			0.0293747	+	0.999568i	$0.490648\pi$
$942$		0			0
$943$	3.35736			0.109331
$944$	−3.18807			−0.103763
$945$		0			0
$946$			92.7083i			3.01421i
$947$	39.6302			1.28781			0.643904	−	0.765106i	$-0.277314\pi$
$947$	39.6302			1.28781			0.643904	+	0.765106i	$0.277314\pi$
$948$		0			0
$949$	−14.5392			−0.471961
$950$	61.4816			1.99473
$951$		0			0
$952$		−	120.617i		−	3.90922i
$953$	24.7542			0.801866			0.400933	−	0.916107i	$-0.368686\pi$
$953$	24.7542			0.801866			0.400933	+	0.916107i	$0.368686\pi$
$954$		0			0
$955$		−	27.1574i		−	0.878792i
$956$	83.6992	+	20.1724i	2.70703	+	0.652423i
$957$		0			0
$958$	−32.0961			−1.03698
$959$		−	82.6005i		−	2.66731i
$960$		0			0
$961$	11.5287			0.371895
$962$			120.204i			3.87553i
$963$		0			0
$964$	113.549			3.65716
$965$			20.0695i			0.646061i
$966$		0			0
$967$	−39.1159			−1.25788			−0.628941	−	0.777453i	$-0.716511\pi$
$967$	−39.1159			−1.25788			−0.628941	+	0.777453i	$0.716511\pi$
$968$		−	77.4447i		−	2.48917i
$969$		0			0
$970$		−	10.3624i		−	0.332716i
$971$			42.1586i			1.35293i	0.736473	+	0.676467i	$0.236490\pi$
$971$			42.1586i			1.35293i	−0.736473	+	0.676467i	$0.763510\pi$
$972$		0			0
$973$	3.02671			0.0970318
$974$		−	53.9793i		−	1.72961i
$975$		0			0
$976$	−82.4223			−2.63827
$977$	−37.2644			−1.19219			−0.596096	−	0.802913i	$-0.703282\pi$
$977$	−37.2644			−1.19219			−0.596096	+	0.802913i	$0.703282\pi$
$978$		0			0
$979$		−	29.7473i		−	0.950726i
$980$		−	116.134i		−	3.70978i
$981$		0			0
$982$		−	60.2131i		−	1.92148i
$983$			0.117083i			0.00373436i	0.999998	+	0.00186718i	$0.000594342\pi$
$983$			0.117083i			0.00373436i	−0.999998	+	0.00186718i	$0.999406\pi$
$984$		0			0
$985$	12.5897			0.401142
$986$		−	40.0205i		−	1.27451i
$987$		0			0
$988$	152.487			4.85125
$989$			37.2251i			1.18369i
$990$		0			0
$991$		−	16.6340i		−	0.528397i	−0.964468	−	0.264199i	$-0.914893\pi$
$991$		−	16.6340i		−	0.528397i	0.964468	−	0.264199i	$-0.0851074\pi$
$992$			156.780i			4.97776i
$993$		0			0
$994$		−	50.1138i		−	1.58951i
$995$	−5.89474			−0.186876
$996$		0			0
$997$		−	6.60464i		−	0.209171i	−0.994516	−	0.104586i	$-0.966648\pi$
$997$		−	6.60464i		−	0.209171i	0.994516	−	0.104586i	$-0.0333516\pi$
$998$	3.57280			0.113095
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.b.a.2150.1	✓	80
3.2	odd	2	inner	2151.2.b.a.2150.79	yes	80
239.238	odd	2	inner	2151.2.b.a.2150.2	yes	80
717.716	even	2	inner	2151.2.b.a.2150.80	yes	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.b.a.2150.1	✓	80	1.1	even	1	trivial
2151.2.b.a.2150.2	yes	80	239.238	odd	2	inner
2151.2.b.a.2150.79	yes	80	3.2	odd	2	inner
2151.2.b.a.2150.80	yes	80	717.716	even	2	inner

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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.75120i q^{2} -5.56908 q^{4} -1.21173i q^{5} -4.92033i q^{7} +9.81922i q^{8} +O(q^{10})\)	\(q-2.75120i q^{2} -5.56908 q^{4} -1.21173i q^{5} -4.92033i q^{7} +9.81922i q^{8} -3.33371 q^{10} +4.34592i q^{11} +4.32722i q^{13} -13.5368 q^{14} +15.8765 q^{16} -2.49653i q^{17} +6.32761i q^{19} +6.74823i q^{20} +11.9565 q^{22} +4.80088 q^{23} +3.53170 q^{25} +11.9050 q^{26} +27.4017i q^{28} +5.82671i q^{29} -6.52141 q^{31} -24.0408i q^{32} -6.86844 q^{34} -5.96212 q^{35} +10.0969i q^{37} +17.4085 q^{38} +11.8983 q^{40} +0.699322 q^{41} +7.75381i q^{43} -24.2028i q^{44} -13.2082i q^{46} -0.961776 q^{47} -17.2096 q^{49} -9.71641i q^{50} -24.0986i q^{52} -5.76437 q^{53} +5.26610 q^{55} +48.3138 q^{56} +16.0304 q^{58} -0.200805 q^{59} -5.19148 q^{61} +17.9417i q^{62} -34.3880 q^{64} +5.24343 q^{65} +6.85991 q^{67} +13.9034i q^{68} +16.4030i q^{70} +3.70205i q^{71} +3.35993i q^{73} +27.7785 q^{74} -35.2389i q^{76} +21.3834 q^{77} +2.99293i q^{79} -19.2380i q^{80} -1.92397i q^{82} -7.32661i q^{83} -3.02513 q^{85} +21.3322 q^{86} -42.6736 q^{88} -6.84486 q^{89} +21.2913 q^{91} -26.7365 q^{92} +2.64603i q^{94} +7.66736 q^{95} +3.10836i q^{97} +47.3470i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 80 q^{4}+O(q^{10}) \) 80 * q - 80 * q^4	\( 80 q - 80 q^{4} + 16 q^{10} + 56 q^{16} + 40 q^{22} - 64 q^{25} - 8 q^{31} + 32 q^{34} - 24 q^{40} - 104 q^{49} - 24 q^{55} + 56 q^{58} + 40 q^{61} - 80 q^{64} - 8 q^{67} - 8 q^{85} - 120 q^{88} + 32 q^{91}+O(q^{100}) \) 80 * q - 80 * q^4 + 16 * q^10 + 56 * q^16 + 40 * q^22 - 64 * q^25 - 8 * q^31 + 32 * q^34 - 24 * q^40 - 104 * q^49 - 24 * q^55 + 56 * q^58 + 40 * q^61 - 80 * q^64 - 8 * q^67 - 8 * q^85 - 120 * q^88 + 32 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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