LMFDB - Embedded newform 2151.2.b.a.2150.17

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			2150.17
Character	$\chi$	$=$	2151.2150
Dual form			2151.2.b.a.2150.64

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.79525i q^{2} -1.22293 q^{4} -1.07382i q^{5} -2.61327i q^{7} -1.39503i q^{8} +O(q^{10})$	\(q-1.79525i q^{2} -1.22293 q^{4} -1.07382i q^{5} -2.61327i q^{7} -1.39503i q^{8} -1.92778 q^{10} -3.92260i q^{11} +1.27874i q^{13} -4.69148 q^{14} -4.95030 q^{16} -4.77577i q^{17} -3.64178i q^{19} +1.31321i q^{20} -7.04207 q^{22} +2.95613 q^{23} +3.84691 q^{25} +2.29567 q^{26} +3.19586i q^{28} +2.37382i q^{29} +2.28399 q^{31} +6.09698i q^{32} -8.57371 q^{34} -2.80619 q^{35} +11.4949i q^{37} -6.53792 q^{38} -1.49801 q^{40} +9.63344 q^{41} +1.14834i q^{43} +4.79709i q^{44} -5.30700i q^{46} -9.16990 q^{47} +0.170824 q^{49} -6.90617i q^{50} -1.56382i q^{52} -2.40064 q^{53} -4.21218 q^{55} -3.64559 q^{56} +4.26160 q^{58} +8.14794 q^{59} -14.4217 q^{61} -4.10035i q^{62} +1.04503 q^{64} +1.37314 q^{65} -9.77833 q^{67} +5.84045i q^{68} +5.03782i q^{70} -10.3481i q^{71} -4.89967i q^{73} +20.6362 q^{74} +4.45366i q^{76} -10.2508 q^{77} +5.60994i q^{79} +5.31574i q^{80} -17.2945i q^{82} +2.40419i q^{83} -5.12832 q^{85} +2.06156 q^{86} -5.47215 q^{88} +14.0514 q^{89} +3.34170 q^{91} -3.61515 q^{92} +16.4623i q^{94} -3.91063 q^{95} +16.5363i q^{97} -0.306672i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	1.79525i		−	1.26944i	−0.772744	−	0.634718i	$-0.781116\pi$
$2$		−	1.79525i		−	1.26944i	0.772744	−	0.634718i	$-0.218884\pi$
$3$		0			0
$3$		0			0
$4$	−1.22293			−0.611467
$5$		−	1.07382i		−	0.480228i	−0.970745	−	0.240114i	$-0.922815\pi$
$5$		−	1.07382i		−	0.480228i	0.970745	−	0.240114i	$-0.0771848\pi$
$6$		0			0
$7$		−	2.61327i		−	0.987723i	−0.869541	−	0.493861i	$-0.835585\pi$
$7$		−	2.61327i		−	0.987723i	0.869541	−	0.493861i	$-0.164415\pi$
$8$		−	1.39503i		−	0.493217i
$9$		0			0
$10$	−1.92778			−0.609619
$11$		−	3.92260i		−	1.18271i	−0.806412	−	0.591355i	$-0.798593\pi$
$11$		−	3.92260i		−	1.18271i	0.806412	−	0.591355i	$-0.201407\pi$
$12$		0			0
$13$			1.27874i			0.354659i	0.984152	+	0.177330i	$0.0567459\pi$
$13$			1.27874i			0.354659i	−0.984152	+	0.177330i	$0.943254\pi$
$14$	−4.69148			−1.25385
$15$		0			0
$16$	−4.95030			−1.23758
$17$		−	4.77577i		−	1.15829i	−0.815223	−	0.579147i	$-0.803386\pi$
$17$		−	4.77577i		−	1.15829i	0.815223	−	0.579147i	$-0.196614\pi$
$18$		0			0
$19$		−	3.64178i		−	0.835482i	−0.908566	−	0.417741i	$-0.862822\pi$
$19$		−	3.64178i		−	0.835482i	0.908566	−	0.417741i	$-0.137178\pi$
$20$			1.31321i			0.293644i
$21$		0			0
$22$	−7.04207			−1.50137
$23$	2.95613			0.616396			0.308198	−	0.951322i	$-0.400274\pi$
$23$	2.95613			0.616396			0.308198	+	0.951322i	$0.400274\pi$
$24$		0			0
$25$	3.84691			0.769381
$26$	2.29567			0.450217
$27$		0			0
$28$			3.19586i			0.603960i
$29$			2.37382i			0.440807i	0.975409	+	0.220403i	$0.0707374\pi$
$29$			2.37382i			0.440807i	−0.975409	+	0.220403i	$0.929263\pi$
$30$		0			0
$31$	2.28399			0.410217			0.205109	−	0.978739i	$-0.434245\pi$
$31$	2.28399			0.410217			0.205109	+	0.978739i	$0.434245\pi$
$32$			6.09698i			1.07780i
$33$		0			0
$34$	−8.57371			−1.47038
$35$	−2.80619			−0.474332
$36$		0			0
$37$			11.4949i			1.88975i	0.327435	+	0.944874i	$0.393816\pi$
$37$			11.4949i			1.88975i	−0.327435	+	0.944874i	$0.606184\pi$
$38$	−6.53792			−1.06059
$39$		0			0
$40$	−1.49801			−0.236857
$41$	9.63344			1.50449			0.752245	−	0.658883i	$-0.228971\pi$
$41$	9.63344			1.50449			0.752245	+	0.658883i	$0.228971\pi$
$42$		0			0
$43$			1.14834i			0.175120i	0.996159	+	0.0875602i	$0.0279070\pi$
$43$			1.14834i			0.175120i	−0.996159	+	0.0875602i	$0.972093\pi$
$44$			4.79709i			0.723188i
$45$		0			0
$46$		−	5.30700i		−	0.782475i
$47$	−9.16990			−1.33757			−0.668784	−	0.743457i	$-0.733185\pi$
$47$	−9.16990			−1.33757			−0.668784	+	0.743457i	$0.733185\pi$
$48$		0			0
$49$	0.170824			0.0244034
$50$		−	6.90617i		−	0.976680i
$51$		0			0
$52$		−	1.56382i		−	0.216863i
$53$	−2.40064			−0.329754			−0.164877	−	0.986314i	$-0.552723\pi$
$53$	−2.40064			−0.329754			−0.164877	+	0.986314i	$0.552723\pi$
$54$		0			0
$55$	−4.21218			−0.567970
$56$	−3.64559			−0.487162
$57$		0			0
$58$	4.26160			0.559576
$59$	8.14794			1.06077			0.530386	−	0.847756i	$-0.322047\pi$
$59$	8.14794			1.06077			0.530386	+	0.847756i	$0.322047\pi$
$60$		0			0
$61$	−14.4217			−1.84650			−0.923252	−	0.384194i	$-0.874479\pi$
$61$	−14.4217			−1.84650			−0.923252	+	0.384194i	$0.874479\pi$
$62$		−	4.10035i		−	0.520745i
$63$		0			0
$64$	1.04503			0.130629
$65$	1.37314			0.170317
$66$		0			0
$67$	−9.77833			−1.19461			−0.597307	−	0.802013i	$-0.703762\pi$
$67$	−9.77833			−1.19461			−0.597307	+	0.802013i	$0.703762\pi$
$68$			5.84045i			0.708259i
$69$		0			0
$70$			5.03782i			0.602134i
$71$		−	10.3481i		−	1.22809i	−0.789271	−	0.614045i	$-0.789541\pi$
$71$		−	10.3481i		−	1.22809i	0.789271	−	0.614045i	$-0.210459\pi$
$72$		0			0
$73$		−	4.89967i		−	0.573463i	−0.958011	−	0.286731i	$-0.907431\pi$
$73$		−	4.89967i		−	0.573463i	0.958011	−	0.286731i	$-0.0925687\pi$
$74$	20.6362			2.39891
$75$		0			0
$76$			4.45366i			0.510870i
$77$	−10.2508			−1.16819
$78$		0			0
$79$			5.60994i			0.631168i	0.948898	+	0.315584i	$0.102200\pi$
$79$			5.60994i			0.631168i	−0.948898	+	0.315584i	$0.897800\pi$
$80$			5.31574i			0.594318i
$81$		0			0
$82$		−	17.2945i		−	1.90985i
$83$			2.40419i			0.263894i	0.991257	+	0.131947i	$0.0421229\pi$
$83$			2.40419i			0.263894i	−0.991257	+	0.131947i	$0.957877\pi$
$84$		0			0
$85$	−5.12832			−0.556245
$86$	2.06156			0.222304
$87$		0			0
$88$	−5.47215			−0.583333
$89$	14.0514			1.48945			0.744723	−	0.667373i	$-0.232581\pi$
$89$	14.0514			1.48945			0.744723	+	0.667373i	$0.232581\pi$
$90$		0			0
$91$	3.34170			0.350305
$92$	−3.61515			−0.376906
$93$		0			0
$94$			16.4623i			1.69796i
$95$	−3.91063			−0.401222
$96$		0			0
$97$			16.5363i			1.67901i	0.543354	+	0.839503i	$0.317154\pi$
$97$			16.5363i			1.67901i	−0.543354	+	0.839503i	$0.682846\pi$
$98$		−	0.306672i		−	0.0309786i
$99$		0			0
$100$	−4.70451			−0.470451
$101$			4.61643i			0.459352i	0.973267	+	0.229676i	$0.0737666\pi$
$101$			4.61643i			0.459352i	−0.973267	+	0.229676i	$0.926233\pi$
$102$		0			0
$103$		−	14.1455i		−	1.39380i	−0.717169	−	0.696899i	$-0.754562\pi$
$103$		−	14.1455i		−	1.39380i	0.717169	−	0.696899i	$-0.245438\pi$
$104$	1.78388			0.174924
$105$		0			0
$106$			4.30976i			0.418601i
$107$	11.3005			1.09246			0.546232	−	0.837634i	$-0.316062\pi$
$107$	11.3005			1.09246			0.546232	+	0.837634i	$0.316062\pi$
$108$		0			0
$109$	−14.4669			−1.38568			−0.692840	−	0.721092i	$-0.743641\pi$
$109$	−14.4669			−1.38568			−0.692840	+	0.721092i	$0.743641\pi$
$110$			7.56193i			0.721002i
$111$		0			0
$112$			12.9365i			1.22238i
$113$		−	4.29507i		−	0.404047i	−0.979381	−	0.202023i	$-0.935248\pi$
$113$		−	4.29507i		−	0.404047i	0.979381	−	0.202023i	$-0.0647517\pi$
$114$		0			0
$115$		−	3.17436i		−	0.296011i
$116$		−	2.90302i		−	0.269539i
$117$		0			0
$118$		−	14.6276i		−	1.34658i
$119$	−12.4804			−1.14407
$120$		0			0
$121$	−4.38682			−0.398801
$122$			25.8905i			2.34402i
$123$		0			0
$124$	−2.79318			−0.250835
$125$		−	9.50001i		−	0.849706i
$126$		0			0
$127$	4.53361			0.402293			0.201146	−	0.979561i	$-0.435533\pi$
$127$	4.53361			0.402293			0.201146	+	0.979561i	$0.435533\pi$
$128$			10.3179i			0.911980i
$129$		0			0
$130$		−	2.46514i		−	0.216207i
$131$	−21.6995			−1.89590			−0.947949	−	0.318423i	$-0.896847\pi$
$131$	−21.6995			−1.89590			−0.947949	+	0.318423i	$0.896847\pi$
$132$		0			0
$133$	−9.51696			−0.825225
$134$			17.5546i			1.51648i
$135$		0			0
$136$	−6.66233			−0.571290
$137$	2.64629			0.226088			0.113044	−	0.993590i	$-0.463940\pi$
$137$	2.64629			0.226088			0.113044	+	0.993590i	$0.463940\pi$
$138$		0			0
$139$		−	11.7192i		−	0.994007i	−0.867748	−	0.497004i	$-0.834434\pi$
$139$		−	11.7192i		−	0.994007i	0.867748	−	0.497004i	$-0.165566\pi$
$140$	3.43178			0.290039
$141$		0			0
$142$	−18.5774			−1.55898
$143$	5.01600			0.419459
$144$		0			0
$145$	2.54906			0.211688
$146$	−8.79615			−0.727974
$147$		0			0
$148$		−	14.0575i		−	1.15552i
$149$	−17.1712			−1.40672			−0.703361	−	0.710832i	$-0.748318\pi$
$149$	−17.1712			−1.40672			−0.703361	+	0.710832i	$0.748318\pi$
$150$		0			0
$151$			15.3977i			1.25305i	0.779402	+	0.626524i	$0.215523\pi$
$151$			15.3977i			1.25305i	−0.779402	+	0.626524i	$0.784477\pi$
$152$	−5.08039			−0.412074
$153$		0			0
$154$			18.4028i			1.48294i
$155$		−	2.45260i		−	0.196998i
$156$		0			0
$157$	16.5981			1.32467			0.662336	−	0.749207i	$-0.269565\pi$
$157$	16.5981			1.32467			0.662336	+	0.749207i	$0.269565\pi$
$158$	10.0713			0.801227
$159$		0			0
$160$	6.54708			0.517592
$161$		−	7.72516i		−	0.608828i
$162$		0			0
$163$	5.74423			0.449923			0.224961	−	0.974368i	$-0.427774\pi$
$163$	5.74423			0.449923			0.224961	+	0.974368i	$0.427774\pi$
$164$	−11.7811			−0.919947
$165$		0			0
$166$	4.31613			0.334997
$167$	6.11765			0.473398			0.236699	−	0.971583i	$-0.423935\pi$
$167$	6.11765			0.473398			0.236699	+	0.971583i	$0.423935\pi$
$168$		0			0
$169$	11.3648			0.874217
$170$			9.20664i			0.706117i
$171$		0			0
$172$		−	1.40435i		−	0.107080i
$173$	14.8858			1.13175			0.565874	−	0.824491i	$-0.308539\pi$
$173$	14.8858			1.13175			0.565874	+	0.824491i	$0.308539\pi$
$174$		0			0
$175$		−	10.0530i		−	0.759935i
$176$			19.4181i			1.46369i
$177$		0			0
$178$		−	25.2258i		−	1.89076i
$179$	−25.6167			−1.91468			−0.957341	−	0.288961i	$-0.906690\pi$
$179$	−25.6167			−1.91468			−0.957341	+	0.288961i	$0.906690\pi$
$180$		0			0
$181$			6.04885i			0.449608i	0.974404	+	0.224804i	$0.0721741\pi$
$181$			6.04885i			0.449608i	−0.974404	+	0.224804i	$0.927826\pi$
$182$		−	5.99919i		−	0.444690i
$183$		0			0
$184$		−	4.12389i		−	0.304017i
$185$	12.3435			0.907510
$186$		0			0
$187$	−18.7334			−1.36992
$188$	11.2142			0.817879
$189$		0			0
$190$			7.02057i			0.509326i
$191$	−4.61890			−0.334212			−0.167106	−	0.985939i	$-0.553442\pi$
$191$	−4.61890			−0.334212			−0.167106	+	0.985939i	$0.553442\pi$
$192$		0			0
$193$	14.7620			1.06259			0.531297	−	0.847185i	$-0.321705\pi$
$193$	14.7620			1.06259			0.531297	+	0.847185i	$0.321705\pi$
$194$	29.6868			2.13139
$195$		0			0
$196$	−0.208906			−0.0149219
$197$			9.27949i			0.661136i	0.943782	+	0.330568i	$0.107240\pi$
$197$			9.27949i			0.661136i	−0.943782	+	0.330568i	$0.892760\pi$
$198$		0			0
$199$			5.40294i			0.383004i	0.981492	+	0.191502i	$0.0613358\pi$
$199$			5.40294i			0.383004i	−0.981492	+	0.191502i	$0.938664\pi$
$200$		−	5.36655i		−	0.379472i
$201$		0			0
$202$	8.28766			0.583118
$203$	6.20342			0.435395
$204$		0			0
$205$		−	10.3446i		−	0.722499i
$206$	−25.3948			−1.76934
$207$		0			0
$208$		−	6.33016i		−	0.438917i
$209$	−14.2853			−0.988133
$210$		0			0
$211$	2.89138			0.199051			0.0995255	−	0.995035i	$-0.468268\pi$
$211$	2.89138			0.199051			0.0995255	+	0.995035i	$0.468268\pi$
$212$	2.93583			0.201634
$213$		0			0
$214$		−	20.2873i		−	1.38681i
$215$	1.23311			0.0840977
$216$		0			0
$217$		−	5.96869i		−	0.405181i
$218$			25.9718i			1.75903i
$219$		0			0
$220$	5.15122			0.347295
$221$	6.10697			0.410799
$222$		0			0
$223$			11.2281i			0.751890i	0.926642	+	0.375945i	$0.122682\pi$
$223$			11.2281i			0.751890i	−0.926642	+	0.375945i	$0.877318\pi$
$224$	15.9331			1.06457
$225$		0			0
$226$	−7.71075			−0.512911
$227$	23.2132			1.54071			0.770357	−	0.637613i	$-0.220078\pi$
$227$	23.2132			1.54071			0.770357	+	0.637613i	$0.220078\pi$
$228$		0			0
$229$			5.42020i			0.358177i	0.983833	+	0.179088i	$0.0573148\pi$
$229$			5.42020i			0.358177i	−0.983833	+	0.179088i	$0.942685\pi$
$230$	−5.69878			−0.375766
$231$		0			0
$232$	3.31155			0.217414
$233$	12.9286			0.846983			0.423492	−	0.905900i	$-0.360804\pi$
$233$	12.9286			0.846983			0.423492	+	0.905900i	$0.360804\pi$
$234$		0			0
$235$			9.84684i			0.642337i
$236$	−9.96440			−0.648627
$237$		0			0
$238$			22.4054i			1.45233i
$239$	−1.97380	+	15.3331i	−0.127675	+	0.991816i
$239$	−1.97380	+	15.3331i	−0.127675	+	0.991816i
$240$		0			0
$241$	−18.7987			−1.21093			−0.605465	−	0.795872i	$-0.707013\pi$
$241$	−18.7987			−1.21093			−0.605465	+	0.795872i	$0.707013\pi$
$242$			7.87544i			0.506253i
$243$		0			0
$244$	17.6367			1.12908
$245$		−	0.183435i		−	0.0117192i
$246$		0			0
$247$	4.65690			0.296312
$248$		−	3.18624i		−	0.202326i
$249$		0			0
$250$	−17.0549			−1.07865
$251$			0.0802468i			0.00506513i	0.999997	+	0.00253257i	$0.000806141\pi$
$251$			0.0802468i			0.00506513i	−0.999997	+	0.00253257i	$0.999194\pi$
$252$		0			0
$253$		−	11.5957i		−	0.729017i
$254$		−	8.13897i		−	0.510685i
$255$		0			0
$256$	20.6133			1.28833
$257$		−	5.92846i		−	0.369807i	−0.982757	−	0.184904i	$-0.940803\pi$
$257$		−	5.92846i		−	0.369807i	0.982757	−	0.184904i	$-0.0591973\pi$
$258$		0			0
$259$	30.0392			1.86655
$260$	−1.67926			−0.104143
$261$		0			0
$262$			38.9562i			2.40672i
$263$		−	17.7619i		−	1.09525i	−0.836725	−	0.547623i	$-0.815533\pi$
$263$		−	17.7619i		−	1.09525i	0.836725	−	0.547623i	$-0.184467\pi$
$264$		0			0
$265$			2.57786i			0.158357i
$266$			17.0854i			1.04757i
$267$		0			0
$268$	11.9583			0.730467
$269$		−	11.8284i		−	0.721191i	−0.932722	−	0.360595i	$-0.882574\pi$
$269$		−	11.8284i		−	0.721191i	0.932722	−	0.360595i	$-0.117426\pi$
$270$		0			0
$271$	−0.800591			−0.0486324			−0.0243162	−	0.999704i	$-0.507741\pi$
$271$	−0.800591			−0.0486324			−0.0243162	+	0.999704i	$0.507741\pi$
$272$			23.6415i			1.43347i
$273$		0			0
$274$		−	4.75076i		−	0.287004i
$275$		−	15.0899i		−	0.909954i
$276$		0			0
$277$		−	25.3724i		−	1.52448i	−0.647296	−	0.762239i	$-0.724100\pi$
$277$		−	25.3724i		−	1.52448i	0.647296	−	0.762239i	$-0.275900\pi$
$278$	−21.0389			−1.26183
$279$		0			0
$280$			3.91471i			0.233949i
$281$	−1.72706			−0.103028			−0.0515139	−	0.998672i	$-0.516405\pi$
$281$	−1.72706			−0.103028			−0.0515139	+	0.998672i	$0.516405\pi$
$282$		0			0
$283$	23.0086			1.36772			0.683860	−	0.729614i	$-0.260300\pi$
$283$	23.0086			1.36772			0.683860	+	0.729614i	$0.260300\pi$
$284$			12.6550i			0.750937i
$285$		0			0
$286$		−	9.00499i		−	0.532476i
$287$		−	25.1748i		−	1.48602i
$288$		0			0
$289$	−5.80794			−0.341644
$290$		−	4.57621i		−	0.268724i
$291$		0			0
$292$			5.99197i			0.350654i
$293$			5.82436i			0.340263i	0.985421	+	0.170131i	$0.0544192\pi$
$293$			5.82436i			0.340263i	−0.985421	+	0.170131i	$0.945581\pi$
$294$		0			0
$295$		−	8.74944i		−	0.509412i
$296$	16.0357			0.932056
$297$		0			0
$298$			30.8267i			1.78574i
$299$			3.78013i			0.218610i
$300$		0			0
$301$	3.00092			0.172970
$302$	27.6428			1.59066
$303$		0			0
$304$			18.0279i			1.03397i
$305$			15.4863i			0.886743i
$306$		0			0
$307$	26.3633			1.50463			0.752315	−	0.658803i	$-0.228937\pi$
$307$	26.3633			1.50463			0.752315	+	0.658803i	$0.228937\pi$
$308$	12.5361			0.714309
$309$		0			0
$310$	−4.40305			−0.250076
$311$		−	24.9192i		−	1.41304i	−0.707695	−	0.706518i	$-0.750265\pi$
$311$		−	24.9192i		−	1.41304i	0.707695	−	0.706518i	$-0.249735\pi$
$312$		0			0
$313$			11.0398i			0.624004i	0.950081	+	0.312002i	$0.101000\pi$
$313$			11.0398i			0.624004i	−0.950081	+	0.312002i	$0.899000\pi$
$314$		−	29.7978i		−	1.68159i
$315$		0			0
$316$		−	6.86059i		−	0.385938i
$317$	−11.8752			−0.666976			−0.333488	−	0.942754i	$-0.608226\pi$
$317$	−11.8752			−0.666976			−0.333488	+	0.942754i	$0.608226\pi$
$318$		0			0
$319$	9.31154			0.521346
$320$		−	1.12218i		−	0.0627317i
$321$		0			0
$322$	−13.8686			−0.772868
$323$	−17.3923			−0.967734
$324$		0			0
$325$			4.91920i			0.272868i
$326$		−	10.3123i		−	0.571148i
$327$		0			0
$328$		−	13.4389i		−	0.742041i
$329$			23.9634i			1.32115i
$330$		0			0
$331$		−	29.8109i		−	1.63856i	−0.573396	−	0.819278i	$-0.694375\pi$
$331$		−	29.8109i		−	1.63856i	0.573396	−	0.819278i	$-0.305625\pi$
$332$		−	2.94017i		−	0.161363i
$333$		0			0
$334$		−	10.9827i		−	0.600948i
$335$			10.5002i			0.573687i
$336$		0			0
$337$	33.8810			1.84562			0.922808	−	0.385260i	$-0.125888\pi$
$337$	33.8810			1.84562			0.922808	+	0.385260i	$0.125888\pi$
$338$		−	20.4027i		−	1.10976i
$339$		0			0
$340$	6.27161			0.340126
$341$		−	8.95920i		−	0.485168i
$342$		0			0
$343$		−	18.7393i		−	1.01183i
$344$	1.60197			0.0863724
$345$		0			0
$346$		−	26.7238i		−	1.43668i
$347$			7.19789i			0.386403i	0.981159	+	0.193201i	$0.0618871\pi$
$347$			7.19789i			0.386403i	−0.981159	+	0.193201i	$0.938113\pi$
$348$		0			0
$349$	13.6948			0.733067			0.366533	−	0.930405i	$-0.380545\pi$
$349$	13.6948			0.733067			0.366533	+	0.930405i	$0.380545\pi$
$350$	−18.0477			−0.964689
$351$		0			0
$352$	23.9161			1.27473
$353$	−19.5182			−1.03885			−0.519424	−	0.854517i	$-0.673853\pi$
$353$	−19.5182			−1.03885			−0.519424	+	0.854517i	$0.673853\pi$
$354$		0			0
$355$	−11.1120			−0.589763
$356$	−17.1840			−0.910748
$357$		0			0
$358$			45.9885i			2.43057i
$359$		−	17.7350i		−	0.936018i	−0.883724	−	0.468009i	$-0.844972\pi$
$359$		−	17.7350i		−	0.936018i	0.883724	−	0.468009i	$-0.155028\pi$
$360$		0			0
$361$	5.73741			0.301969
$362$	10.8592			0.570748
$363$		0			0
$364$	−4.08668			−0.214200
$365$	−5.26137			−0.275393
$366$		0			0
$367$	−20.2592			−1.05752			−0.528761	−	0.848770i	$-0.677343\pi$
$367$	−20.2592			−1.05752			−0.528761	+	0.848770i	$0.677343\pi$
$368$	−14.6337			−0.762836
$369$		0			0
$370$		−	22.1596i		−	1.15203i
$371$			6.27352i			0.325705i
$372$		0			0
$373$	−0.0530776			−0.00274826			−0.00137413	−	0.999999i	$-0.500437\pi$
$373$	−0.0530776			−0.00274826			−0.00137413	+	0.999999i	$0.500437\pi$
$374$			33.6313i			1.73903i
$375$		0			0
$376$			12.7923i			0.659711i
$377$	−3.03550			−0.156336
$378$		0			0
$379$			22.9250i			1.17758i	0.808288	+	0.588788i	$0.200395\pi$
$379$			22.9250i			1.17758i	−0.808288	+	0.588788i	$0.799605\pi$
$380$	4.78244			0.245334
$381$		0			0
$382$			8.29210i			0.424261i
$383$			22.0284i			1.12560i	0.826594	+	0.562799i	$0.190275\pi$
$383$			22.0284i			1.12560i	−0.826594	+	0.562799i	$0.809725\pi$
$384$		0			0
$385$			11.0076i			0.560997i
$386$		−	26.5016i		−	1.34890i
$387$		0			0
$388$		−	20.2228i		−	1.02666i
$389$		−	20.6876i		−	1.04890i	−0.851440	−	0.524452i	$-0.824270\pi$
$389$		−	20.6876i		−	1.04890i	0.851440	−	0.524452i	$-0.175730\pi$
$390$		0			0
$391$		−	14.1178i		−	0.713967i
$392$		−	0.238304i		−	0.0120362i
$393$		0			0
$394$	16.6590			0.839270
$395$	6.02408			0.303104
$396$		0			0
$397$		−	25.4526i		−	1.27743i	−0.769444	−	0.638714i	$-0.779467\pi$
$397$		−	25.4526i		−	1.27743i	0.769444	−	0.638714i	$-0.220533\pi$
$398$	9.69964			0.486199
$399$		0			0
$400$	−19.0433			−0.952167
$401$		−	11.0109i		−	0.549860i	−0.961464	−	0.274930i	$-0.911345\pi$
$401$		−	11.0109i		−	0.549860i	0.961464	−	0.274930i	$-0.0886547\pi$
$402$		0			0
$403$			2.92064i			0.145487i
$404$		−	5.64559i		−	0.280879i
$405$		0			0
$406$		−	11.1367i		−	0.552706i
$407$	45.0899			2.23502
$408$		0			0
$409$	0.0630477			0.00311751			0.00155875	−	0.999999i	$-0.499504\pi$
$409$	0.0630477			0.00311751			0.00155875	+	0.999999i	$0.499504\pi$
$410$	−18.5712			−0.917166
$411$		0			0
$412$			17.2990i			0.852262i
$413$		−	21.2928i		−	1.04775i
$414$		0			0
$415$	2.58167			0.126729
$416$	−7.79647			−0.382253
$417$		0			0
$418$			25.6457i			1.25437i
$419$			34.4935i			1.68512i	0.538603	+	0.842559i	$0.318952\pi$
$419$			34.4935i			1.68512i	−0.538603	+	0.842559i	$0.681048\pi$
$420$		0			0
$421$	−32.4922			−1.58357			−0.791787	−	0.610798i	$-0.790849\pi$
$421$	−32.4922			−1.58357			−0.791787	+	0.610798i	$0.790849\pi$
$422$		−	5.19076i		−	0.252682i
$423$		0			0
$424$			3.34897i			0.162640i
$425$		−	18.3719i		−	0.891169i
$426$		0			0
$427$			37.6877i			1.82383i
$428$	−13.8198			−0.668006
$429$		0			0
$430$		−	2.21375i		−	0.106757i
$431$		−	29.5987i		−	1.42572i	−0.701306	−	0.712860i	$-0.747399\pi$
$431$		−	29.5987i		−	1.42572i	0.701306	−	0.712860i	$-0.252601\pi$
$432$		0			0
$433$			27.2157i			1.30790i	0.756536	+	0.653952i	$0.226890\pi$
$433$			27.2157i			1.30790i	−0.756536	+	0.653952i	$0.773110\pi$
$434$	−10.7153			−0.514351
$435$		0			0
$436$	17.6921			0.847298
$437$		−	10.7656i		−	0.514988i
$438$		0			0
$439$	26.4835			1.26399			0.631994	−	0.774973i	$-0.282237\pi$
$439$	26.4835			1.26399			0.631994	+	0.774973i	$0.282237\pi$
$440$			5.87611i			0.280133i
$441$		0			0
$442$		−	10.9636i		−	0.521484i
$443$		−	16.9216i		−	0.803970i	−0.915646	−	0.401985i	$-0.868320\pi$
$443$		−	16.9216i		−	0.803970i	0.915646	−	0.401985i	$-0.131680\pi$
$444$		0			0
$445$		−	15.0887i		−	0.715274i
$446$	20.1573			0.954476
$447$		0			0
$448$		−	2.73095i		−	0.129025i
$449$	2.68280			0.126609			0.0633046	−	0.997994i	$-0.479836\pi$
$449$	2.68280			0.126609			0.0633046	+	0.997994i	$0.479836\pi$
$450$		0			0
$451$		−	37.7882i		−	1.77938i
$452$			5.25260i			0.247061i
$453$		0			0
$454$		−	41.6736i		−	1.95584i
$455$		−	3.58839i		−	0.168226i
$456$		0			0
$457$	2.78882			0.130456			0.0652278	−	0.997870i	$-0.479223\pi$
$457$	2.78882			0.130456			0.0652278	+	0.997870i	$0.479223\pi$
$458$	9.73063			0.454682
$459$		0			0
$460$			3.88203i			0.181001i
$461$	2.78866			0.129881			0.0649405	−	0.997889i	$-0.479314\pi$
$461$	2.78866			0.129881			0.0649405	+	0.997889i	$0.479314\pi$
$462$		0			0
$463$		−	32.6731i		−	1.51845i	−0.650830	−	0.759223i	$-0.725579\pi$
$463$		−	32.6731i		−	1.51845i	0.650830	−	0.759223i	$-0.274421\pi$
$464$		−	11.7511i		−	0.545532i
$465$		0			0
$466$		−	23.2102i		−	1.07519i
$467$	−9.13298			−0.422624			−0.211312	−	0.977419i	$-0.567774\pi$
$467$	−9.13298			−0.422624			−0.211312	+	0.977419i	$0.567774\pi$
$468$		0			0
$469$			25.5534i			1.17995i
$470$	17.6776			0.815406
$471$		0			0
$472$		−	11.3666i		−	0.523191i
$473$	4.50449			0.207116
$474$		0			0
$475$		−	14.0096i		−	0.642804i
$476$	15.2627			0.699563
$477$		0			0
$478$	27.5268	+	3.54347i	1.25905	+	0.162075i
$479$		−	29.7697i		−	1.36021i	−0.733114	−	0.680106i	$-0.761934\pi$
$479$		−	29.7697i		−	1.36021i	0.733114	−	0.680106i	$-0.238066\pi$
$480$		0			0
$481$	−14.6990			−0.670216
$482$			33.7484i			1.53720i
$483$		0			0
$484$	5.36479			0.243854
$485$	17.7570			0.806306
$486$		0			0
$487$	6.74934			0.305842			0.152921	−	0.988238i	$-0.451132\pi$
$487$	6.74934			0.305842			0.152921	+	0.988238i	$0.451132\pi$
$488$			20.1186i			0.910728i
$489$		0			0
$490$	−0.329311			−0.0148768
$491$	−26.0243			−1.17446			−0.587230	−	0.809420i	$-0.699782\pi$
$491$	−26.0243			−1.17446			−0.587230	+	0.809420i	$0.699782\pi$
$492$		0			0
$493$	11.3368			0.510584
$494$		−	8.36032i		−	0.376148i
$495$		0			0
$496$	−11.3065			−0.507675
$497$	−27.0423			−1.21301
$498$		0			0
$499$			10.7300i			0.480339i	0.970731	+	0.240169i	$0.0772030\pi$
$499$			10.7300i			0.480339i	−0.970731	+	0.240169i	$0.922797\pi$
$500$			11.6179i			0.519568i
$501$		0			0
$502$	0.144063			0.00642986
$503$		−	1.84910i		−	0.0824471i	−0.999150	−	0.0412235i	$-0.986874\pi$
$503$		−	1.84910i		−	0.0824471i	0.999150	−	0.0412235i	$-0.0131256\pi$
$504$		0			0
$505$	4.95723			0.220594
$506$	−20.8173			−0.925440
$507$		0			0
$508$	−5.54430			−0.245989
$509$			22.5771i			1.00071i	0.865820	+	0.500356i	$0.166798\pi$
$509$			22.5771i			1.00071i	−0.865820	+	0.500356i	$0.833202\pi$
$510$		0			0
$511$	−12.8042			−0.566422
$512$		−	16.3703i		−	0.723471i
$513$		0			0
$514$	−10.6431			−0.469447
$515$	−15.1898			−0.669341
$516$		0			0
$517$			35.9699i			1.58195i
$518$		−	53.9280i		−	2.36946i
$519$		0			0
$520$		−	1.91557i		−	0.0840034i
$521$	−19.1504			−0.838995			−0.419498	−	0.907756i	$-0.637794\pi$
$521$	−19.1504			−0.838995			−0.419498	+	0.907756i	$0.637794\pi$
$522$		0			0
$523$	23.1340			1.01158			0.505790	−	0.862657i	$-0.331201\pi$
$523$	23.1340			1.01158			0.505790	+	0.862657i	$0.331201\pi$
$524$	26.5371			1.15928
$525$		0			0
$526$	−31.8871			−1.39035
$527$		−	10.9078i		−	0.475152i
$528$		0			0
$529$	−14.2613			−0.620056
$530$	4.62792			0.201024
$531$		0			0
$532$	11.6386			0.504598
$533$			12.3187i			0.533581i
$534$		0			0
$535$		−	12.1348i		−	0.524631i
$536$			13.6411i			0.589204i
$537$		0			0
$538$	−21.2350			−0.915505
$539$		−	0.670074i		−	0.0288621i
$540$		0			0
$541$			35.1335i			1.51051i	0.655434	+	0.755253i	$0.272486\pi$
$541$			35.1335i			1.51051i	−0.655434	+	0.755253i	$0.727514\pi$
$542$			1.43726i			0.0617358i
$543$		0			0
$544$	29.1178			1.24841
$545$			15.5349i			0.665442i
$546$		0			0
$547$		−	20.6882i		−	0.884561i	−0.896877	−	0.442281i	$-0.854169\pi$
$547$		−	20.6882i		−	0.884561i	0.896877	−	0.442281i	$-0.145831\pi$
$548$	−3.23624			−0.138245
$549$		0			0
$550$	−27.0902			−1.15513
$551$	8.64493			0.368286
$552$		0			0
$553$	14.6603			0.623419
$554$	−45.5498			−1.93523
$555$		0			0
$556$			14.3318i			0.607803i
$557$	−21.5610			−0.913569			−0.456785	−	0.889577i	$-0.650999\pi$
$557$	−21.5610			−0.913569			−0.456785	+	0.889577i	$0.650999\pi$
$558$		0			0
$559$	−1.46843			−0.0621080
$560$	13.8915			0.587022
$561$		0			0
$562$			3.10051i			0.130787i
$563$		−	6.85121i		−	0.288744i	−0.989523	−	0.144372i	$-0.953884\pi$
$563$		−	6.85121i		−	0.288744i	0.989523	−	0.144372i	$-0.0461162\pi$
$564$		0			0
$565$	−4.61215			−0.194035
$566$		−	41.3063i		−	1.73623i
$567$		0			0
$568$	−14.4359			−0.605715
$569$		−	35.7165i		−	1.49731i	−0.662958	−	0.748657i	$-0.730699\pi$
$569$		−	35.7165i		−	1.49731i	0.662958	−	0.748657i	$-0.269301\pi$
$570$		0			0
$571$	6.14654			0.257225			0.128612	−	0.991695i	$-0.458948\pi$
$571$	6.14654			0.257225			0.128612	+	0.991695i	$0.458948\pi$
$572$	−6.13424			−0.256485
$573$		0			0
$574$	−45.1951			−1.88641
$575$	11.3720			0.474243
$576$		0			0
$577$	−23.4275			−0.975301			−0.487651	−	0.873039i	$-0.662146\pi$
$577$	−23.4275			−0.975301			−0.487651	+	0.873039i	$0.662146\pi$
$578$			10.4267i			0.433695i
$579$		0			0
$580$	−3.11733			−0.129440
$581$	6.28279			0.260654
$582$		0			0
$583$			9.41677i			0.390003i
$584$	−6.83518			−0.282842
$585$		0			0
$586$	10.4562			0.431942
$587$		−	14.0498i		−	0.579895i	−0.957042	−	0.289948i	$-0.906362\pi$
$587$		−	14.0498i		−	0.579895i	0.957042	−	0.289948i	$-0.0936379\pi$
$588$		0			0
$589$		−	8.31781i		−	0.342729i
$590$	−15.7075			−0.646666
$591$		0			0
$592$		−	56.9031i		−	2.33870i
$593$	−3.68854			−0.151470			−0.0757351	−	0.997128i	$-0.524130\pi$
$593$	−3.68854			−0.151470			−0.0757351	+	0.997128i	$0.524130\pi$
$594$		0			0
$595$			13.4017i			0.549416i
$596$	20.9993			0.860165
$597$		0			0
$598$	6.78629			0.277512
$599$		−	27.2153i		−	1.11199i	−0.831187	−	0.555993i	$-0.812338\pi$
$599$		−	27.2153i		−	1.11199i	0.831187	−	0.555993i	$-0.187662\pi$
$600$		0			0
$601$			12.7582i			0.520419i	0.965552	+	0.260209i	$0.0837916\pi$
$601$			12.7582i			0.520419i	−0.965552	+	0.260209i	$0.916208\pi$
$602$		−	5.38742i		−	0.219575i
$603$		0			0
$604$		−	18.8304i		−	0.766198i
$605$			4.71066i			0.191516i
$606$		0			0
$607$			48.6254i			1.97365i	0.161804	+	0.986823i	$0.448269\pi$
$607$			48.6254i			1.97365i	−0.161804	+	0.986823i	$0.551731\pi$
$608$	22.2039			0.900487
$609$		0			0
$610$	27.8018			1.12566
$611$		−	11.7259i		−	0.474381i
$612$		0			0
$613$	−14.9216			−0.602678			−0.301339	−	0.953517i	$-0.597434\pi$
$613$	−14.9216			−0.602678			−0.301339	+	0.953517i	$0.597434\pi$
$614$		−	47.3287i		−	1.91003i
$615$		0			0
$616$			14.3002i			0.576171i
$617$	5.23042			0.210569			0.105284	−	0.994442i	$-0.466425\pi$
$617$	5.23042			0.210569			0.105284	+	0.994442i	$0.466425\pi$
$618$		0			0
$619$			12.8934i			0.518228i	0.965847	+	0.259114i	$0.0834305\pi$
$619$			12.8934i			0.518228i	−0.965847	+	0.259114i	$0.916569\pi$
$620$			2.99937i			0.120458i
$621$		0			0
$622$	−44.7362			−1.79376
$623$		−	36.7201i		−	1.47116i
$624$		0			0
$625$	9.03321			0.361328
$626$	19.8192			0.792133
$627$		0			0
$628$	−20.2984			−0.809993
$629$	54.8969			2.18888
$630$		0			0
$631$	−18.1260			−0.721583			−0.360792	−	0.932646i	$-0.617493\pi$
$631$	−18.1260			−0.721583			−0.360792	+	0.932646i	$0.617493\pi$
$632$	7.82603			0.311303
$633$		0			0
$634$			21.3189i			0.846683i
$635$		−	4.86829i		−	0.193192i
$636$		0			0
$637$			0.218440i			0.00865490i
$638$		−	16.7166i		−	0.661816i
$639$		0			0
$640$	11.0796			0.437958
$641$		−	43.3995i		−	1.71418i	−0.515170	−	0.857088i	$-0.672271\pi$
$641$		−	43.3995i		−	1.71418i	0.515170	−	0.857088i	$-0.327729\pi$
$642$		0			0
$643$	27.6639			1.09096			0.545478	−	0.838125i	$-0.316348\pi$
$643$	27.6639			1.09096			0.545478	+	0.838125i	$0.316348\pi$
$644$			9.44737i			0.372279i
$645$		0			0
$646$			31.2236i			1.22848i
$647$			10.1282i			0.398180i	0.979981	+	0.199090i	$0.0637987\pi$
$647$			10.1282i			0.398180i	−0.979981	+	0.199090i	$0.936201\pi$
$648$		0			0
$649$		−	31.9611i		−	1.25458i
$650$	8.83121			0.346389
$651$		0			0
$652$	−7.02482			−0.275113
$653$	−47.5775			−1.86185			−0.930926	−	0.365209i	$-0.880998\pi$
$653$	−47.5775			−1.86185			−0.930926	+	0.365209i	$0.880998\pi$
$654$		0			0
$655$			23.3015i			0.910463i
$656$	−47.6884			−1.86192
$657$		0			0
$658$	43.0204			1.67711
$659$	10.1081			0.393758			0.196879	−	0.980428i	$-0.436919\pi$
$659$	10.1081			0.393758			0.196879	+	0.980428i	$0.436919\pi$
$660$		0			0
$661$	−19.4644			−0.757078			−0.378539	−	0.925585i	$-0.623573\pi$
$661$	−19.4644			−0.757078			−0.378539	+	0.925585i	$0.623573\pi$
$662$	−53.5181			−2.08004
$663$		0			0
$664$	3.35391			0.130157
$665$			10.2195i			0.396296i
$666$		0			0
$667$			7.01731i			0.271712i
$668$	−7.48148			−0.289467
$669$		0			0
$670$	18.8505			0.728258
$671$			56.5705i			2.18388i
$672$		0			0
$673$		−	16.3324i		−	0.629567i	−0.949164	−	0.314783i	$-0.898068\pi$
$673$		−	16.3324i		−	0.629567i	0.949164	−	0.314783i	$-0.101932\pi$
$674$		−	60.8250i		−	2.34289i
$675$		0			0
$676$	−13.8984			−0.534555
$677$	22.9229			0.880997			0.440499	−	0.897753i	$-0.354802\pi$
$677$	22.9229			0.880997			0.440499	+	0.897753i	$0.354802\pi$
$678$		0			0
$679$	43.2138			1.65839
$680$			7.15416i			0.274350i
$681$		0			0
$682$	−16.0840			−0.615890
$683$	−3.07206			−0.117549			−0.0587745	−	0.998271i	$-0.518719\pi$
$683$	−3.07206			−0.117549			−0.0587745	+	0.998271i	$0.518719\pi$
$684$		0			0
$685$		−	2.84165i		−	0.108574i
$686$	−33.6418			−1.28445
$687$		0			0
$688$		−	5.68463i		−	0.216725i
$689$		−	3.06980i		−	0.116950i
$690$		0			0
$691$	30.5149			1.16084			0.580421	−	0.814317i	$-0.302888\pi$
$691$	30.5149			1.16084			0.580421	+	0.814317i	$0.302888\pi$
$692$	−18.2044			−0.692027
$693$		0			0
$694$	12.9220			0.490514
$695$	−12.5843			−0.477350
$696$		0			0
$697$		−	46.0071i		−	1.74264i
$698$		−	24.5857i		−	0.930581i
$699$		0			0
$700$			12.2942i			0.464676i
$701$	41.2701			1.55875			0.779376	−	0.626557i	$-0.215536\pi$
$701$	41.2701			1.55875			0.779376	+	0.626557i	$0.215536\pi$
$702$		0			0
$703$	41.8619			1.57885
$704$		−	4.09924i		−	0.154496i
$705$		0			0
$706$			35.0401i			1.31875i
$707$	12.0640			0.453712
$708$		0			0
$709$		−	31.1056i		−	1.16819i	−0.811684	−	0.584097i	$-0.801449\pi$
$709$		−	31.1056i		−	1.16819i	0.811684	−	0.584097i	$-0.198551\pi$
$710$			19.9488i			0.748667i
$711$		0			0
$712$		−	19.6021i		−	0.734621i
$713$	6.75178			0.252856
$714$		0			0
$715$		−	5.38629i		−	0.201436i
$716$	31.3275			1.17077
$717$		0			0
$718$	−31.8388			−1.18821
$719$		−	1.36054i		−	0.0507396i	−0.999678	−	0.0253698i	$-0.991924\pi$
$719$		−	1.36054i		−	0.0507396i	0.999678	−	0.0253698i	$-0.00807633\pi$
$720$		0			0
$721$	−36.9660			−1.37669
$722$		−	10.3001i		−	0.383331i
$723$		0			0
$724$		−	7.39735i		−	0.274920i
$725$			9.13185i			0.339148i
$726$		0			0
$727$	−10.2819			−0.381333			−0.190667	−	0.981655i	$-0.561065\pi$
$727$	−10.2819			−0.381333			−0.190667	+	0.981655i	$0.561065\pi$
$728$		−	4.66177i		−	0.172777i
$729$		0			0
$730$			9.44550i			0.349594i
$731$	5.48421			0.202841
$732$		0			0
$733$	43.9066			1.62173			0.810864	−	0.585235i	$-0.198998\pi$
$733$	43.9066			1.62173			0.810864	+	0.585235i	$0.198998\pi$
$734$			36.3704i			1.34246i
$735$		0			0
$736$			18.0235i			0.664354i
$737$			38.3565i			1.41288i
$738$		0			0
$739$	38.5550			1.41827			0.709134	−	0.705073i	$-0.249086\pi$
$739$	38.5550			1.41827			0.709134	+	0.705073i	$0.249086\pi$
$740$	−15.0953			−0.554912
$741$		0			0
$742$	11.2626			0.413462
$743$	−0.720577			−0.0264354			−0.0132177	−	0.999913i	$-0.504207\pi$
$743$	−0.720577			−0.0264354			−0.0132177	+	0.999913i	$0.504207\pi$
$744$		0			0
$745$			18.4389i			0.675548i
$746$			0.0952878i			0.00348873i
$747$		0			0
$748$	22.9098			0.837664
$749$		−	29.5313i		−	1.07905i
$750$		0			0
$751$	−23.7928			−0.868211			−0.434105	−	0.900862i	$-0.642935\pi$
$751$	−23.7928			−0.868211			−0.434105	+	0.900862i	$0.642935\pi$
$752$	45.3938			1.65534
$753$		0			0
$754$			5.44949i			0.198459i
$755$	16.5344			0.601749
$756$		0			0
$757$	2.15248			0.0782332			0.0391166	−	0.999235i	$-0.487546\pi$
$757$	2.15248			0.0782332			0.0391166	+	0.999235i	$0.487546\pi$
$758$	41.1561			1.49486
$759$		0			0
$760$			5.45544i			0.197890i
$761$			40.9000i			1.48262i	0.671161	+	0.741312i	$0.265796\pi$
$761$			40.9000i			1.48262i	−0.671161	+	0.741312i	$0.734204\pi$
$762$		0			0
$763$			37.8060i			1.36867i
$764$	5.64862			0.204360
$765$		0			0
$766$	39.5465			1.42887
$767$			10.4191i			0.376212i
$768$		0			0
$769$		−	2.39300i		−	0.0862939i	−0.999069	−	0.0431469i	$-0.986262\pi$
$769$		−	2.39300i		−	0.0862939i	0.999069	−	0.0431469i	$-0.0137384\pi$
$770$	19.7614			0.712150
$771$		0			0
$772$	−18.0530			−0.649742
$773$	−9.79379			−0.352258			−0.176129	−	0.984367i	$-0.556358\pi$
$773$	−9.79379			−0.352258			−0.176129	+	0.984367i	$0.556358\pi$
$774$		0			0
$775$	8.78631			0.315614
$776$	23.0686			0.828115
$777$		0			0
$778$	−37.1395			−1.33152
$779$		−	35.0829i		−	1.25698i
$780$		0			0
$781$	−40.5914			−1.45247
$782$	−25.3450			−0.906336
$783$		0			0
$784$	−0.845630			−0.0302011
$785$		−	17.8234i		−	0.636144i
$786$		0			0
$787$		−	13.9995i		−	0.499029i	−0.968371	−	0.249515i	$-0.919729\pi$
$787$		−	13.9995i		−	0.499029i	0.968371	−	0.249515i	$-0.0802711\pi$
$788$		−	11.3482i		−	0.404263i
$789$		0			0
$790$		−	10.8148i		−	0.384772i
$791$	−11.2242			−0.399086
$792$		0			0
$793$		−	18.4416i		−	0.654880i
$794$	−45.6938			−1.62161
$795$		0			0
$796$		−	6.60744i		−	0.234194i
$797$			50.2460i			1.77980i	0.456153	+	0.889901i	$0.349227\pi$
$797$			50.2460i			1.77980i	−0.456153	+	0.889901i	$0.650773\pi$
$798$		0			0
$799$			43.7933i			1.54930i
$800$			23.4545i			0.829243i
$801$		0			0
$802$	−19.7674			−0.698012
$803$	−19.2195			−0.678240
$804$		0			0
$805$	−8.29546			−0.292376
$806$	5.24329			0.184687
$807$		0			0
$808$	6.44005			0.226560
$809$	26.5304			0.932758			0.466379	−	0.884585i	$-0.345558\pi$
$809$	26.5304			0.932758			0.466379	+	0.884585i	$0.345558\pi$
$810$		0			0
$811$		−	32.6705i		−	1.14721i	−0.819130	−	0.573607i	$-0.805544\pi$
$811$		−	32.6705i		−	1.14721i	0.819130	−	0.573607i	$-0.194456\pi$
$812$	−7.58638			−0.266230
$813$		0			0
$814$		−	80.9477i		−	2.83722i
$815$		−	6.16828i		−	0.216066i
$816$		0			0
$817$	4.18201			0.146310
$818$		−	0.113187i		−	0.00395748i
$819$		0			0
$820$			12.6508i			0.441784i
$821$	−2.46303			−0.0859603			−0.0429802	−	0.999076i	$-0.513685\pi$
$821$	−2.46303			−0.0859603			−0.0429802	+	0.999076i	$0.513685\pi$
$822$		0			0
$823$			33.3189i			1.16142i	0.814109	+	0.580712i	$0.197226\pi$
$823$			33.3189i			1.16142i	−0.814109	+	0.580712i	$0.802774\pi$
$824$	−19.7334			−0.687445
$825$		0			0
$826$	−38.2259			−1.33005
$827$			46.7108i			1.62430i	0.583452	+	0.812148i	$0.301702\pi$
$827$			46.7108i			1.62430i	−0.583452	+	0.812148i	$0.698298\pi$
$828$		0			0
$829$		−	14.5179i		−	0.504226i	−0.967698	−	0.252113i	$-0.918874\pi$
$829$		−	14.5179i		−	0.504226i	0.967698	−	0.252113i	$-0.0811255\pi$
$830$		−	4.63476i		−	0.160875i
$831$		0			0
$832$			1.33633i			0.0463288i
$833$		−	0.815815i		−	0.0282663i
$834$		0			0
$835$		−	6.56927i		−	0.227339i
$836$	17.4700			0.604211
$837$		0			0
$838$	61.9246			2.13915
$839$			22.5246i			0.777634i	0.921315	+	0.388817i	$0.127116\pi$
$839$			22.5246i			0.777634i	−0.921315	+	0.388817i	$0.872884\pi$
$840$		0			0
$841$	23.3650			0.805689
$842$			58.3318i			2.01024i
$843$		0			0
$844$	−3.53597			−0.121713
$845$		−	12.2038i		−	0.419823i
$846$		0			0
$847$			11.4639i			0.393905i
$848$	11.8839			0.408095
$849$		0			0
$850$	−32.9822			−1.13128
$851$			33.9804i			1.16483i
$852$		0			0
$853$	−3.34971			−0.114692			−0.0573459	−	0.998354i	$-0.518264\pi$
$853$	−3.34971			−0.114692			−0.0573459	+	0.998354i	$0.518264\pi$
$854$	67.6589			2.31524
$855$		0			0
$856$		−	15.7646i		−	0.538822i
$857$	−4.71834			−0.161175			−0.0805877	−	0.996748i	$-0.525680\pi$
$857$	−4.71834			−0.161175			−0.0805877	+	0.996748i	$0.525680\pi$
$858$		0			0
$859$	−10.5025			−0.358340			−0.179170	−	0.983818i	$-0.557341\pi$
$859$	−10.5025			−0.358340			−0.179170	+	0.983818i	$0.557341\pi$
$860$	−1.50802			−0.0514230
$861$		0			0
$862$	−53.1372			−1.80986
$863$	−20.2184			−0.688244			−0.344122	−	0.938925i	$-0.611823\pi$
$863$	−20.2184			−0.688244			−0.344122	+	0.938925i	$0.611823\pi$
$864$		0			0
$865$		−	15.9847i		−	0.543498i
$866$	48.8591			1.66030
$867$		0			0
$868$			7.29932i			0.247755i
$869$	22.0056			0.746488
$870$		0			0
$871$		−	12.5040i		−	0.423681i
$872$			20.1818i			0.683441i
$873$		0			0
$874$	−19.3270			−0.653744
$875$	−24.8261			−0.839274
$876$		0			0
$877$	4.30591			0.145400			0.0727001	−	0.997354i	$-0.476838\pi$
$877$	4.30591			0.145400			0.0727001	+	0.997354i	$0.476838\pi$
$878$		−	47.5446i		−	1.60455i
$879$		0			0
$880$	20.8516			0.702906
$881$	−18.6070			−0.626887			−0.313443	−	0.949607i	$-0.601483\pi$
$881$	−18.6070			−0.626887			−0.313443	+	0.949607i	$0.601483\pi$
$882$		0			0
$883$	20.1933			0.679559			0.339779	−	0.940505i	$-0.389648\pi$
$883$	20.1933			0.679559			0.339779	+	0.940505i	$0.389648\pi$
$884$	−7.46843			−0.251190
$885$		0			0
$886$	−30.3786			−1.02059
$887$			24.2216i			0.813281i	0.913588	+	0.406641i	$0.133300\pi$
$887$			24.2216i			0.813281i	−0.913588	+	0.406641i	$0.866700\pi$
$888$		0			0
$889$		−	11.8475i		−	0.397354i
$890$	−27.0881			−0.907994
$891$		0			0
$892$		−	13.7312i		−	0.459756i
$893$			33.3948i			1.11751i
$894$		0			0
$895$			27.5078i			0.919484i
$896$	26.9634			0.900783
$897$		0			0
$898$		−	4.81631i		−	0.160722i
$899$			5.42179i			0.180827i
$900$		0			0
$901$			11.4649i			0.381951i
$902$	−67.8393			−2.25880
$903$		0			0
$904$	−5.99175			−0.199283
$905$	6.49539			0.215914
$906$		0			0
$907$		−	7.98815i		−	0.265242i	−0.991167	−	0.132621i	$-0.957661\pi$
$907$		−	7.98815i		−	0.265242i	0.991167	−	0.132621i	$-0.0423393\pi$
$908$	−28.3882			−0.942096
$909$		0			0
$910$	−6.44207			−0.213552
$911$	11.6980			0.387572			0.193786	−	0.981044i	$-0.437923\pi$
$911$	11.6980			0.387572			0.193786	+	0.981044i	$0.437923\pi$
$912$		0			0
$913$	9.43068			0.312110
$914$		−	5.00664i		−	0.165605i
$915$		0			0
$916$		−	6.62855i		−	0.219013i
$917$			56.7067i			1.87262i
$918$		0			0
$919$	56.5221			1.86449			0.932246	−	0.361826i	$-0.117846\pi$
$919$	56.5221			1.86449			0.932246	+	0.361826i	$0.117846\pi$
$920$	−4.42832			−0.145998
$921$		0			0
$922$		−	5.00636i		−	0.164876i
$923$	13.2325			0.435553
$924$		0			0
$925$			44.2197i			1.45394i
$926$	−58.6564			−1.92757
$927$		0			0
$928$	−14.4731			−0.475104
$929$	−52.1932			−1.71240			−0.856202	−	0.516641i	$-0.827182\pi$
$929$	−52.1932			−1.71240			−0.856202	+	0.516641i	$0.827182\pi$
$930$		0			0
$931$		−	0.622104i		−	0.0203886i
$932$	−15.8109			−0.517903
$933$		0			0
$934$			16.3960i			0.536494i
$935$			20.1164i			0.657876i
$936$		0			0
$937$	−47.6473			−1.55657			−0.778284	−	0.627912i	$-0.783910\pi$
$937$	−47.6473			−1.55657			−0.778284	+	0.627912i	$0.783910\pi$
$938$	45.8748			1.49787
$939$		0			0
$940$		−	12.0420i		−	0.392768i
$941$	−30.5145			−0.994745			−0.497372	−	0.867537i	$-0.665702\pi$
$941$	−30.5145			−0.994745			−0.497372	+	0.867537i	$0.665702\pi$
$942$		0			0
$943$	28.4777			0.927362
$944$	−40.3347			−1.31278
$945$		0			0
$946$		−	8.08669i		−	0.262921i
$947$	11.9564			0.388532			0.194266	−	0.980949i	$-0.437768\pi$
$947$	11.9564			0.388532			0.194266	+	0.980949i	$0.437768\pi$
$948$		0			0
$949$	6.26541			0.203384
$950$	−25.1508			−0.815999
$951$		0			0
$952$			17.4105i			0.564277i
$953$	−47.1480			−1.52727			−0.763636	−	0.645647i	$-0.776588\pi$
$953$	−47.1480			−1.52727			−0.763636	+	0.645647i	$0.776588\pi$
$954$		0			0
$955$			4.95988i			0.160498i
$956$	2.41383	−	18.7514i	0.0780688	−	0.606463i
$957$		0			0
$958$	−53.4441			−1.72670
$959$		−	6.91547i		−	0.223312i
$960$		0			0
$961$	−25.7834			−0.831722
$962$			26.3884i			0.850797i
$963$		0			0
$964$	22.9896			0.740444
$965$		−	15.8518i		−	0.510288i
$966$		0			0
$967$	−16.5755			−0.533032			−0.266516	−	0.963831i	$-0.585872\pi$
$967$	−16.5755			−0.533032			−0.266516	+	0.963831i	$0.585872\pi$
$968$			6.11974i			0.196696i
$969$		0			0
$970$		−	31.8784i		−	1.02355i
$971$			3.79596i			0.121818i	0.998143	+	0.0609091i	$0.0194000\pi$
$971$			3.79596i			0.121818i	−0.998143	+	0.0609091i	$0.980600\pi$
$972$		0			0
$973$	−30.6254			−0.981804
$974$		−	12.1168i		−	0.388246i
$975$		0			0
$976$	71.3916			2.28519
$977$	53.1350			1.69994			0.849970	−	0.526831i	$-0.176620\pi$
$977$	53.1350			1.69994			0.849970	+	0.526831i	$0.176620\pi$
$978$		0			0
$979$		−	55.1181i		−	1.76158i
$980$			0.224328i			0.00716591i
$981$		0			0
$982$			46.7202i			1.49090i
$983$		−	4.62332i		−	0.147461i	−0.997278	−	0.0737305i	$-0.976510\pi$
$983$		−	4.62332i		−	0.147461i	0.997278	−	0.0737305i	$-0.0234905\pi$
$984$		0			0
$985$	9.96452			0.317496
$986$		−	20.3524i		−	0.648153i
$987$		0			0
$988$	−5.69509			−0.181185
$989$			3.39465i			0.107943i
$990$		0			0
$991$			13.0796i			0.415486i	0.978183	+	0.207743i	$0.0666118\pi$
$991$			13.0796i			0.415486i	−0.978183	+	0.207743i	$0.933388\pi$
$992$			13.9255i			0.442134i
$993$		0			0
$994$			48.5478i			1.53984i
$995$	5.80179			0.183929
$996$		0			0
$997$			8.04000i			0.254630i	0.991862	+	0.127315i	$0.0406358\pi$
$997$			8.04000i			0.254630i	−0.991862	+	0.127315i	$0.959364\pi$
$998$	19.2630			0.609759
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.b.a.2150.17	✓	80
3.2	odd	2	inner	2151.2.b.a.2150.63	yes	80
239.238	odd	2	inner	2151.2.b.a.2150.18	yes	80
717.716	even	2	inner	2151.2.b.a.2150.64	yes	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.b.a.2150.17	✓	80	1.1	even	1	trivial
2151.2.b.a.2150.18	yes	80	239.238	odd	2	inner
2151.2.b.a.2150.63	yes	80	3.2	odd	2	inner
2151.2.b.a.2150.64	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)

\(f(q)\)	\(=\)	\(q-1.79525i q^{2} -1.22293 q^{4} -1.07382i q^{5} -2.61327i q^{7} -1.39503i q^{8} +O(q^{10})\)	\(q-1.79525i q^{2} -1.22293 q^{4} -1.07382i q^{5} -2.61327i q^{7} -1.39503i q^{8} -1.92778 q^{10} -3.92260i q^{11} +1.27874i q^{13} -4.69148 q^{14} -4.95030 q^{16} -4.77577i q^{17} -3.64178i q^{19} +1.31321i q^{20} -7.04207 q^{22} +2.95613 q^{23} +3.84691 q^{25} +2.29567 q^{26} +3.19586i q^{28} +2.37382i q^{29} +2.28399 q^{31} +6.09698i q^{32} -8.57371 q^{34} -2.80619 q^{35} +11.4949i q^{37} -6.53792 q^{38} -1.49801 q^{40} +9.63344 q^{41} +1.14834i q^{43} +4.79709i q^{44} -5.30700i q^{46} -9.16990 q^{47} +0.170824 q^{49} -6.90617i q^{50} -1.56382i q^{52} -2.40064 q^{53} -4.21218 q^{55} -3.64559 q^{56} +4.26160 q^{58} +8.14794 q^{59} -14.4217 q^{61} -4.10035i q^{62} +1.04503 q^{64} +1.37314 q^{65} -9.77833 q^{67} +5.84045i q^{68} +5.03782i q^{70} -10.3481i q^{71} -4.89967i q^{73} +20.6362 q^{74} +4.45366i q^{76} -10.2508 q^{77} +5.60994i q^{79} +5.31574i q^{80} -17.2945i q^{82} +2.40419i q^{83} -5.12832 q^{85} +2.06156 q^{86} -5.47215 q^{88} +14.0514 q^{89} +3.34170 q^{91} -3.61515 q^{92} +16.4623i q^{94} -3.91063 q^{95} +16.5363i q^{97} -0.306672i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 80 q^{4}+O(q^{10}) \) 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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