LMFDB - Embedded newform 6030.2.d.j.2411.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6030,2,Mod(2411,6030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6030, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6030.2411");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6030.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$48.1497924188$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 36x^{14} + 519x^{12} + 3876x^{10} + 16111x^{8} + 36772x^{6} + 41293x^{4} + 16036x^{2} + 4 $ x^16 + 36x^14 + 519x^12 + 3876x^10 + 16111x^8 + 36772x^6 + 41293x^4 + 16036*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{67}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2411.1
Root			$-3.35739i$ of defining polynomial
Character	$\chi$	$=$	6030.2411
Dual form			6030.2.d.j.2411.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -5.18179i q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -5.18179i q^{7} +1.00000 q^{8} -1.00000 q^{10} -5.96177 q^{11} +5.86888i q^{13} -5.18179i q^{14} +1.00000 q^{16} +4.42499i q^{17} +5.41782 q^{19} -1.00000 q^{20} -5.96177 q^{22} +2.69899i q^{23} +1.00000 q^{25} +5.86888i q^{26} -5.18179i q^{28} +6.15001i q^{29} -3.36640i q^{31} +1.00000 q^{32} +4.42499i q^{34} +5.18179i q^{35} +4.86463 q^{37} +5.41782 q^{38} -1.00000 q^{40} +4.68168 q^{41} -12.5329i q^{43} -5.96177 q^{44} +2.69899i q^{46} +1.66891i q^{47} -19.8510 q^{49} +1.00000 q^{50} +5.86888i q^{52} +0.782473 q^{53} +5.96177 q^{55} -5.18179i q^{56} +6.15001i q^{58} +5.16305i q^{59} +14.3511i q^{61} -3.36640i q^{62} +1.00000 q^{64} -5.86888i q^{65} +(7.18582 + 3.91969i) q^{67} +4.42499i q^{68} +5.18179i q^{70} -5.99641i q^{71} +4.25788 q^{73} +4.86463 q^{74} +5.41782 q^{76} +30.8927i q^{77} -1.83790i q^{79} -1.00000 q^{80} +4.68168 q^{82} +9.73260i q^{83} -4.42499i q^{85} -12.5329i q^{86} -5.96177 q^{88} -15.2721i q^{89} +30.4113 q^{91} +2.69899i q^{92} +1.66891i q^{94} -5.41782 q^{95} +6.02723i q^{97} -19.8510 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{2} + 16 q^{4} - 16 q^{5} + 16 q^{8}+O(q^{10}) $ 16 * q + 16 * q^2 + 16 * q^4 - 16 * q^5 + 16 * q^8	$ 16 q + 16 q^{2} + 16 q^{4} - 16 q^{5} + 16 q^{8} - 16 q^{10} - 20 q^{11} + 16 q^{16} - 8 q^{19} - 16 q^{20} - 20 q^{22} + 16 q^{25} + 16 q^{32} + 32 q^{37} - 8 q^{38} - 16 q^{40} - 8 q^{41} - 20 q^{44} - 88 q^{49} + 16 q^{50} + 8 q^{53} + 20 q^{55} + 16 q^{64} + 4 q^{67} + 16 q^{73} + 32 q^{74} - 8 q^{76} - 16 q^{80} - 8 q^{82} - 20 q^{88} + 40 q^{91} + 8 q^{95} - 88 q^{98}+O(q^{100}) $ 16 * q + 16 * q^2 + 16 * q^4 - 16 * q^5 + 16 * q^8 - 16 * q^10 - 20 * q^11 + 16 * q^16 - 8 * q^19 - 16 * q^20 - 20 * q^22 + 16 * q^25 + 16 * q^32 + 32 * q^37 - 8 * q^38 - 16 * q^40 - 8 * q^41 - 20 * q^44 - 88 * q^49 + 16 * q^50 + 8 * q^53 + 20 * q^55 + 16 * q^64 + 4 * q^67 + 16 * q^73 + 32 * q^74 - 8 * q^76 - 16 * q^80 - 8 * q^82 - 20 * q^88 + 40 * q^91 + 8 * q^95 - 88 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6030\mathbb{Z}\right)^\times$.

$n$	$1207$	$3151$	$4691$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$		−	5.18179i		−	1.95853i	−0.202573	−	0.979267i	$-0.564930\pi$
$7$		−	5.18179i		−	1.95853i	0.202573	−	0.979267i	$-0.435070\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$	−1.00000			−0.316228
$11$	−5.96177			−1.79754			−0.898771	−	0.438419i	$-0.855539\pi$
$11$	−5.96177			−1.79754			−0.898771	+	0.438419i	$0.855539\pi$
$12$		0			0
$13$			5.86888i			1.62773i	0.581051	+	0.813867i	$0.302641\pi$
$13$			5.86888i			1.62773i	−0.581051	+	0.813867i	$0.697359\pi$
$14$		−	5.18179i		−	1.38489i
$15$		0			0
$16$	1.00000			0.250000
$17$			4.42499i			1.07322i	0.843831	+	0.536609i	$0.180295\pi$
$17$			4.42499i			1.07322i	−0.843831	+	0.536609i	$0.819705\pi$
$18$		0			0
$19$	5.41782			1.24293			0.621466	−	0.783441i	$-0.286537\pi$
$19$	5.41782			1.24293			0.621466	+	0.783441i	$0.286537\pi$
$20$	−1.00000			−0.223607
$21$		0			0
$22$	−5.96177			−1.27105
$23$			2.69899i			0.562779i	0.959594	+	0.281389i	$0.0907952\pi$
$23$			2.69899i			0.562779i	−0.959594	+	0.281389i	$0.909205\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$			5.86888i			1.15098i
$27$		0			0
$28$		−	5.18179i		−	0.979267i
$29$			6.15001i			1.14203i	0.820941	+	0.571014i	$0.193450\pi$
$29$			6.15001i			1.14203i	−0.820941	+	0.571014i	$0.806550\pi$
$30$		0			0
$31$		−	3.36640i		−	0.604623i	−0.953209	−	0.302312i	$-0.902242\pi$
$31$		−	3.36640i		−	0.604623i	0.953209	−	0.302312i	$-0.0977583\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$			4.42499i			0.758879i
$35$			5.18179i			0.875883i
$36$		0			0
$37$	4.86463			0.799740			0.399870	−	0.916572i	$-0.369055\pi$
$37$	4.86463			0.799740			0.399870	+	0.916572i	$0.369055\pi$
$38$	5.41782			0.878886
$39$		0			0
$40$	−1.00000			−0.158114
$41$	4.68168			0.731156			0.365578	−	0.930781i	$-0.380871\pi$
$41$	4.68168			0.731156			0.365578	+	0.930781i	$0.380871\pi$
$42$		0			0
$43$		−	12.5329i		−	1.91125i	−0.294587	−	0.955625i	$-0.595182\pi$
$43$		−	12.5329i		−	1.91125i	0.294587	−	0.955625i	$-0.404818\pi$
$44$	−5.96177			−0.898771
$45$		0			0
$46$			2.69899i			0.397945i
$47$			1.66891i			0.243436i	0.992565	+	0.121718i	$0.0388404\pi$
$47$			1.66891i			0.243436i	−0.992565	+	0.121718i	$0.961160\pi$
$48$		0			0
$49$	−19.8510			−2.83586
$50$	1.00000			0.141421
$51$		0			0
$52$			5.86888i			0.813867i
$53$	0.782473			0.107481			0.0537405	−	0.998555i	$-0.482886\pi$
$53$	0.782473			0.107481			0.0537405	+	0.998555i	$0.482886\pi$
$54$		0			0
$55$	5.96177			0.803885
$56$		−	5.18179i		−	0.692446i
$57$		0			0
$58$			6.15001i			0.807535i
$59$			5.16305i			0.672172i	0.941831	+	0.336086i	$0.109103\pi$
$59$			5.16305i			0.672172i	−0.941831	+	0.336086i	$0.890897\pi$
$60$		0			0
$61$			14.3511i			1.83747i	0.394878	+	0.918733i	$0.370787\pi$
$61$			14.3511i			1.83747i	−0.394878	+	0.918733i	$0.629213\pi$
$62$		−	3.36640i		−	0.427533i
$63$		0			0
$64$	1.00000			0.125000
$65$		−	5.86888i		−	0.727945i
$66$		0			0
$67$	7.18582	+	3.91969i	0.877888	+	0.478867i
$67$	7.18582	+	3.91969i	0.877888	+	0.478867i
$68$			4.42499i			0.536609i
$69$		0			0
$70$			5.18179i			0.619343i
$71$		−	5.99641i		−	0.711643i	−0.934554	−	0.355822i	$-0.884201\pi$
$71$		−	5.99641i		−	0.711643i	0.934554	−	0.355822i	$-0.115799\pi$
$72$		0			0
$73$	4.25788			0.498347			0.249173	−	0.968459i	$-0.419841\pi$
$73$	4.25788			0.498347			0.249173	+	0.968459i	$0.419841\pi$
$74$	4.86463			0.565501
$75$		0			0
$76$	5.41782			0.621466
$77$			30.8927i			3.52055i
$78$		0			0
$79$		−	1.83790i		−	0.206780i	−0.994641	−	0.103390i	$-0.967031\pi$
$79$		−	1.83790i		−	0.206780i	0.994641	−	0.103390i	$-0.0329689\pi$
$80$	−1.00000			−0.111803
$81$		0			0
$82$	4.68168			0.517005
$83$			9.73260i			1.06829i	0.845392	+	0.534146i	$0.179367\pi$
$83$			9.73260i			1.06829i	−0.845392	+	0.534146i	$0.820633\pi$
$84$		0			0
$85$		−	4.42499i		−	0.479957i
$86$		−	12.5329i		−	1.35146i
$87$		0			0
$88$	−5.96177			−0.635527
$89$		−	15.2721i		−	1.61884i	−0.587232	−	0.809419i	$-0.699782\pi$
$89$		−	15.2721i		−	1.61884i	0.587232	−	0.809419i	$-0.300218\pi$
$90$		0			0
$91$	30.4113			3.18797
$92$			2.69899i			0.281389i
$93$		0			0
$94$			1.66891i			0.172135i
$95$	−5.41782			−0.555856
$96$		0			0
$97$			6.02723i			0.611972i	0.952036	+	0.305986i	$0.0989861\pi$
$97$			6.02723i			0.611972i	−0.952036	+	0.305986i	$0.901014\pi$
$98$	−19.8510			−2.00525
$99$		0			0
$100$	1.00000			0.100000
$101$	−7.13224			−0.709684			−0.354842	−	0.934926i	$-0.615465\pi$
$101$	−7.13224			−0.709684			−0.354842	+	0.934926i	$0.615465\pi$
$102$		0			0
$103$	5.56927			0.548756			0.274378	−	0.961622i	$-0.411528\pi$
$103$	5.56927			0.548756			0.274378	+	0.961622i	$0.411528\pi$
$104$			5.86888i			0.575491i
$105$		0			0
$106$	0.782473			0.0760005
$107$		−	5.45340i		−	0.527200i	−0.964632	−	0.263600i	$-0.915090\pi$
$107$		−	5.45340i		−	0.527200i	0.964632	−	0.263600i	$-0.0849099\pi$
$108$		0			0
$109$			12.3864i			1.18640i	0.805054	+	0.593202i	$0.202136\pi$
$109$			12.3864i			1.18640i	−0.805054	+	0.593202i	$0.797864\pi$
$110$	5.96177			0.568433
$111$		0			0
$112$		−	5.18179i		−	0.489634i
$113$	15.1850			1.42849			0.714245	−	0.699896i	$-0.246770\pi$
$113$	15.1850			1.42849			0.714245	+	0.699896i	$0.246770\pi$
$114$		0			0
$115$		−	2.69899i		−	0.251682i
$116$			6.15001i			0.571014i
$117$		0			0
$118$			5.16305i			0.475297i
$119$	22.9294			2.10193
$120$		0			0
$121$	24.5427			2.23116
$122$			14.3511i			1.29929i
$123$		0			0
$124$		−	3.36640i		−	0.302312i
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	17.8168			1.58098			0.790492	−	0.612472i	$-0.209825\pi$
$127$	17.8168			1.58098			0.790492	+	0.612472i	$0.209825\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$		−	5.86888i		−	0.514735i
$131$		−	10.6190i		−	0.927788i	−0.885891	−	0.463894i	$-0.846452\pi$
$131$		−	10.6190i		−	0.927788i	0.885891	−	0.463894i	$-0.153548\pi$
$132$		0			0
$133$		−	28.0740i		−	2.43433i
$134$	7.18582	+	3.91969i	0.620760	+	0.338610i
$135$		0			0
$136$			4.42499i			0.379440i
$137$	9.82609			0.839500			0.419750	−	0.907640i	$-0.362118\pi$
$137$	9.82609			0.839500			0.419750	+	0.907640i	$0.362118\pi$
$138$		0			0
$139$			3.96165i			0.336023i	0.985785	+	0.168012i	$0.0537346\pi$
$139$			3.96165i			0.336023i	−0.985785	+	0.168012i	$0.946265\pi$
$140$			5.18179i			0.437942i
$141$		0			0
$142$		−	5.99641i		−	0.503208i
$143$		−	34.9889i		−	2.92592i
$144$		0			0
$145$		−	6.15001i		−	0.510730i
$146$	4.25788			0.352384
$147$		0			0
$148$	4.86463			0.399870
$149$			13.5127i			1.10700i	0.832849	+	0.553501i	$0.186708\pi$
$149$			13.5127i			1.10700i	−0.832849	+	0.553501i	$0.813292\pi$
$150$		0			0
$151$	13.0171			1.05931			0.529657	−	0.848212i	$-0.322321\pi$
$151$	13.0171			1.05931			0.529657	+	0.848212i	$0.322321\pi$
$152$	5.41782			0.439443
$153$		0			0
$154$			30.8927i			2.48940i
$155$			3.36640i			0.270396i
$156$		0			0
$157$	−11.0167			−0.879228			−0.439614	−	0.898187i	$-0.644885\pi$
$157$	−11.0167			−0.879228			−0.439614	+	0.898187i	$0.644885\pi$
$158$		−	1.83790i		−	0.146215i
$159$		0			0
$160$	−1.00000			−0.0790569
$161$	13.9856			1.10222
$162$		0			0
$163$	−19.8732			−1.55659			−0.778296	−	0.627898i	$-0.783916\pi$
$163$	−19.8732			−1.55659			−0.778296	+	0.627898i	$0.783916\pi$
$164$	4.68168			0.365578
$165$		0			0
$166$			9.73260i			0.755396i
$167$			16.4348i			1.27176i	0.771787	+	0.635881i	$0.219363\pi$
$167$			16.4348i			1.27176i	−0.771787	+	0.635881i	$0.780637\pi$
$168$		0			0
$169$	−21.4437			−1.64952
$170$		−	4.42499i		−	0.339381i
$171$		0			0
$172$		−	12.5329i		−	0.955625i
$173$		−	13.4243i		−	1.02063i	−0.859987	−	0.510316i	$-0.829528\pi$
$173$		−	13.4243i		−	1.02063i	0.859987	−	0.510316i	$-0.170472\pi$
$174$		0			0
$175$		−	5.18179i		−	0.391707i
$176$	−5.96177			−0.449385
$177$		0			0
$178$		−	15.2721i		−	1.14469i
$179$	10.2736			0.767886			0.383943	−	0.923357i	$-0.374566\pi$
$179$	10.2736			0.767886			0.383943	+	0.923357i	$0.374566\pi$
$180$		0			0
$181$	−4.13822			−0.307591			−0.153796	−	0.988103i	$-0.549150\pi$
$181$	−4.13822			−0.307591			−0.153796	+	0.988103i	$0.549150\pi$
$182$	30.4113			2.25424
$183$		0			0
$184$			2.69899i			0.198972i
$185$	−4.86463			−0.357655
$186$		0			0
$187$		−	26.3808i		−	1.92915i
$188$			1.66891i			0.121718i
$189$		0			0
$190$	−5.41782			−0.393050
$191$	−0.859197			−0.0621693			−0.0310847	−	0.999517i	$-0.509896\pi$
$191$	−0.859197			−0.0621693			−0.0310847	+	0.999517i	$0.509896\pi$
$192$		0			0
$193$	15.8120			1.13817			0.569087	−	0.822277i	$-0.307297\pi$
$193$	15.8120			1.13817			0.569087	+	0.822277i	$0.307297\pi$
$194$			6.02723i			0.432730i
$195$		0			0
$196$	−19.8510			−1.41793
$197$	5.28463			0.376514			0.188257	−	0.982120i	$-0.439716\pi$
$197$	5.28463			0.376514			0.188257	+	0.982120i	$0.439716\pi$
$198$		0			0
$199$	1.20124			0.0851538			0.0425769	−	0.999093i	$-0.486443\pi$
$199$	1.20124			0.0851538			0.0425769	+	0.999093i	$0.486443\pi$
$200$	1.00000			0.0707107
$201$		0			0
$202$	−7.13224			−0.501823
$203$	31.8681			2.23670
$204$		0			0
$205$	−4.68168			−0.326983
$206$	5.56927			0.388029
$207$		0			0
$208$			5.86888i			0.406933i
$209$	−32.2998			−2.23422
$210$		0			0
$211$	1.22192			0.0841204			0.0420602	−	0.999115i	$-0.486608\pi$
$211$	1.22192			0.0841204			0.0420602	+	0.999115i	$0.486608\pi$
$212$	0.782473			0.0537405
$213$		0			0
$214$		−	5.45340i		−	0.372787i
$215$			12.5329i			0.854737i
$216$		0			0
$217$	−17.4440			−1.18418
$218$			12.3864i			0.838914i
$219$		0			0
$220$	5.96177			0.401943
$221$	−25.9697			−1.74691
$222$		0			0
$223$	7.88456			0.527989			0.263995	−	0.964524i	$-0.414960\pi$
$223$	7.88456			0.527989			0.263995	+	0.964524i	$0.414960\pi$
$224$		−	5.18179i		−	0.346223i
$225$		0			0
$226$	15.1850			1.01009
$227$			24.3653i			1.61718i	0.588372	+	0.808590i	$0.299769\pi$
$227$			24.3653i			1.61718i	−0.588372	+	0.808590i	$0.700231\pi$
$228$		0			0
$229$			9.99033i			0.660180i	0.943949	+	0.330090i	$0.107079\pi$
$229$			9.99033i			0.660180i	−0.943949	+	0.330090i	$0.892921\pi$
$230$		−	2.69899i		−	0.177966i
$231$		0			0
$232$			6.15001i			0.403768i
$233$	1.79171			0.117379			0.0586895	−	0.998276i	$-0.481308\pi$
$233$	1.79171			0.117379			0.0586895	+	0.998276i	$0.481308\pi$
$234$		0			0
$235$		−	1.66891i		−	0.108868i
$236$			5.16305i			0.336086i
$237$		0			0
$238$	22.9294			1.48629
$239$	28.9040			1.86965			0.934823	−	0.355114i	$-0.115558\pi$
$239$	28.9040			1.86965			0.934823	+	0.355114i	$0.115558\pi$
$240$		0			0
$241$	19.1263			1.23203			0.616015	−	0.787734i	$-0.288746\pi$
$241$	19.1263			1.23203			0.616015	+	0.787734i	$0.288746\pi$
$242$	24.5427			1.57767
$243$		0			0
$244$			14.3511i			0.918733i
$245$	19.8510			1.26823
$246$		0			0
$247$			31.7965i			2.02316i
$248$		−	3.36640i		−	0.213767i
$249$		0			0
$250$	−1.00000			−0.0632456
$251$	−14.0260			−0.885316			−0.442658	−	0.896690i	$-0.645964\pi$
$251$	−14.0260			−0.885316			−0.442658	+	0.896690i	$0.645964\pi$
$252$		0			0
$253$		−	16.0908i		−	1.01162i
$254$	17.8168			1.11792
$255$		0			0
$256$	1.00000			0.0625000
$257$			12.9070i			0.805116i	0.915395	+	0.402558i	$0.131879\pi$
$257$			12.9070i			0.805116i	−0.915395	+	0.402558i	$0.868121\pi$
$258$		0			0
$259$		−	25.2075i		−	1.56632i
$260$		−	5.86888i		−	0.363972i
$261$		0			0
$262$		−	10.6190i		−	0.656045i
$263$		−	20.5436i		−	1.26677i	−0.773837	−	0.633385i	$-0.781665\pi$
$263$		−	20.5436i		−	1.26677i	0.773837	−	0.633385i	$-0.218335\pi$
$264$		0			0
$265$	−0.782473			−0.0480669
$266$		−	28.0740i		−	1.72133i
$267$		0			0
$268$	7.18582	+	3.91969i	0.438944	+	0.239433i
$269$			22.0530i			1.34460i	0.740281	+	0.672298i	$0.234693\pi$
$269$			22.0530i			1.34460i	−0.740281	+	0.672298i	$0.765307\pi$
$270$		0			0
$271$		−	32.6182i		−	1.98141i	−0.136010	−	0.990707i	$-0.543428\pi$
$271$		−	32.6182i		−	1.98141i	0.136010	−	0.990707i	$-0.456572\pi$
$272$			4.42499i			0.268304i
$273$		0			0
$274$	9.82609			0.593616
$275$	−5.96177			−0.359508
$276$		0			0
$277$	−23.8168			−1.43101			−0.715505	−	0.698608i	$-0.753803\pi$
$277$	−23.8168			−1.43101			−0.715505	+	0.698608i	$0.753803\pi$
$278$			3.96165i			0.237604i
$279$		0			0
$280$			5.18179i			0.309671i
$281$	31.9169			1.90400			0.952002	−	0.306091i	$-0.0990210\pi$
$281$	31.9169			1.90400			0.952002	+	0.306091i	$0.0990210\pi$
$282$		0			0
$283$	−12.2435			−0.727799			−0.363900	−	0.931438i	$-0.618555\pi$
$283$	−12.2435			−0.727799			−0.363900	+	0.931438i	$0.618555\pi$
$284$		−	5.99641i		−	0.355822i
$285$		0			0
$286$		−	34.9889i		−	2.06894i
$287$		−	24.2595i		−	1.43199i
$288$		0			0
$289$	−2.58052			−0.151795
$290$		−	6.15001i		−	0.361141i
$291$		0			0
$292$	4.25788			0.249173
$293$			23.5765i			1.37735i	0.725069	+	0.688677i	$0.241808\pi$
$293$			23.5765i			1.37735i	−0.725069	+	0.688677i	$0.758192\pi$
$294$		0			0
$295$		−	5.16305i		−	0.300604i
$296$	4.86463			0.282751
$297$		0			0
$298$			13.5127i			0.782768i
$299$	−15.8400			−0.916054
$300$		0			0
$301$	−64.9429			−3.74325
$302$	13.0171			0.749048
$303$		0			0
$304$	5.41782			0.310733
$305$		−	14.3511i		−	0.821740i
$306$		0			0
$307$	−4.74518			−0.270822			−0.135411	−	0.990790i	$-0.543235\pi$
$307$	−4.74518			−0.270822			−0.135411	+	0.990790i	$0.543235\pi$
$308$			30.8927i			1.76027i
$309$		0			0
$310$			3.36640i			0.191199i
$311$	−11.3216			−0.641990			−0.320995	−	0.947081i	$-0.604017\pi$
$311$	−11.3216			−0.641990			−0.320995	+	0.947081i	$0.604017\pi$
$312$		0			0
$313$			11.3053i			0.639013i	0.947584	+	0.319506i	$0.103517\pi$
$313$			11.3053i			0.639013i	−0.947584	+	0.319506i	$0.896483\pi$
$314$	−11.0167			−0.621708
$315$		0			0
$316$		−	1.83790i		−	0.103390i
$317$			6.85114i			0.384798i	0.981317	+	0.192399i	$0.0616269\pi$
$317$			6.85114i			0.384798i	−0.981317	+	0.192399i	$0.938373\pi$
$318$		0			0
$319$		−	36.6649i		−	2.05284i
$320$	−1.00000			−0.0559017
$321$		0			0
$322$	13.9856			0.779388
$323$			23.9738i			1.33394i
$324$		0			0
$325$			5.86888i			0.325547i
$326$	−19.8732			−1.10068
$327$		0			0
$328$	4.68168			0.258503
$329$	8.64797			0.476778
$330$		0			0
$331$			1.28951i			0.0708779i	0.999372	+	0.0354389i	$0.0112829\pi$
$331$			1.28951i			0.0708779i	−0.999372	+	0.0354389i	$0.988717\pi$
$332$			9.73260i			0.534146i
$333$		0			0
$334$			16.4348i			0.899271i
$335$	−7.18582	−	3.91969i	−0.392603	−	0.214156i
$336$		0			0
$337$		−	10.3466i		−	0.563618i	−0.959471	−	0.281809i	$-0.909066\pi$
$337$		−	10.3466i		−	0.563618i	0.959471	−	0.281809i	$-0.0909344\pi$
$338$	−21.4437			−1.16638
$339$		0			0
$340$		−	4.42499i		−	0.239979i
$341$			20.0697i			1.08684i
$342$		0			0
$343$			66.5912i			3.59559i
$344$		−	12.5329i		−	0.675729i
$345$		0			0
$346$		−	13.4243i		−	0.721696i
$347$	−29.3425			−1.57519			−0.787593	−	0.616195i	$-0.788673\pi$
$347$	−29.3425			−1.57519			−0.787593	+	0.616195i	$0.788673\pi$
$348$		0			0
$349$	26.5886			1.42325			0.711627	−	0.702558i	$-0.247959\pi$
$349$	26.5886			1.42325			0.711627	+	0.702558i	$0.247959\pi$
$350$		−	5.18179i		−	0.276979i
$351$		0			0
$352$	−5.96177			−0.317764
$353$	−9.43485			−0.502166			−0.251083	−	0.967966i	$-0.580787\pi$
$353$	−9.43485			−0.502166			−0.251083	+	0.967966i	$0.580787\pi$
$354$		0			0
$355$			5.99641i			0.318256i
$356$		−	15.2721i		−	0.809419i
$357$		0			0
$358$	10.2736			0.542977
$359$		−	4.07390i		−	0.215012i	−0.994204	−	0.107506i	$-0.965713\pi$
$359$		−	4.07390i		−	0.215012i	0.994204	−	0.107506i	$-0.0342865\pi$
$360$		0			0
$361$	10.3528			0.544882
$362$	−4.13822			−0.217500
$363$		0			0
$364$	30.4113			1.59399
$365$	−4.25788			−0.222868
$366$		0			0
$367$			14.4846i			0.756088i	0.925788	+	0.378044i	$0.123403\pi$
$367$			14.4846i			0.756088i	−0.925788	+	0.378044i	$0.876597\pi$
$368$			2.69899i			0.140695i
$369$		0			0
$370$	−4.86463			−0.252900
$371$		−	4.05462i		−	0.210505i
$372$		0			0
$373$			9.64695i			0.499500i	0.968310	+	0.249750i	$0.0803485\pi$
$373$			9.64695i			0.499500i	−0.968310	+	0.249750i	$0.919652\pi$
$374$		−	26.3808i		−	1.36412i
$375$		0			0
$376$			1.66891i			0.0860677i
$377$	−36.0936			−1.85892
$378$		0			0
$379$			9.96973i			0.512111i	0.966662	+	0.256055i	$0.0824229\pi$
$379$			9.96973i			0.512111i	−0.966662	+	0.256055i	$0.917577\pi$
$380$	−5.41782			−0.277928
$381$		0			0
$382$	−0.859197			−0.0439603
$383$	−15.7905			−0.806856			−0.403428	−	0.915011i	$-0.632181\pi$
$383$	−15.7905			−0.806856			−0.403428	+	0.915011i	$0.632181\pi$
$384$		0			0
$385$		−	30.8927i		−	1.57444i
$386$	15.8120			0.804811
$387$		0			0
$388$			6.02723i			0.305986i
$389$		−	26.1647i		−	1.32660i	−0.748353	−	0.663300i	$-0.769155\pi$
$389$		−	26.1647i		−	1.32660i	0.748353	−	0.663300i	$-0.230845\pi$
$390$		0			0
$391$	−11.9430			−0.603984
$392$	−19.8510			−1.00263
$393$		0			0
$394$	5.28463			0.266236
$395$			1.83790i			0.0924747i
$396$		0			0
$397$	−3.16108			−0.158650			−0.0793250	−	0.996849i	$-0.525277\pi$
$397$	−3.16108			−0.158650			−0.0793250	+	0.996849i	$0.525277\pi$
$398$	1.20124			0.0602129
$399$		0			0
$400$	1.00000			0.0500000
$401$	−12.5387			−0.626154			−0.313077	−	0.949728i	$-0.601360\pi$
$401$	−12.5387			−0.626154			−0.313077	+	0.949728i	$0.601360\pi$
$402$		0			0
$403$	19.7570			0.984166
$404$	−7.13224			−0.354842
$405$		0			0
$406$	31.8681			1.58159
$407$	−29.0018			−1.43757
$408$		0			0
$409$			16.0800i			0.795105i	0.917579	+	0.397552i	$0.130140\pi$
$409$			16.0800i			0.795105i	−0.917579	+	0.397552i	$0.869860\pi$
$410$	−4.68168			−0.231212
$411$		0			0
$412$	5.56927			0.274378
$413$	26.7539			1.31647
$414$		0			0
$415$		−	9.73260i		−	0.477754i
$416$			5.86888i			0.287745i
$417$		0			0
$418$	−32.2998			−1.57983
$419$			20.2260i			0.988105i	0.869432	+	0.494053i	$0.164485\pi$
$419$			20.2260i			0.988105i	−0.869432	+	0.494053i	$0.835515\pi$
$420$		0			0
$421$	−7.95185			−0.387550			−0.193775	−	0.981046i	$-0.562073\pi$
$421$	−7.95185			−0.387550			−0.193775	+	0.981046i	$0.562073\pi$
$422$	1.22192			0.0594821
$423$		0			0
$424$	0.782473			0.0380003
$425$			4.42499i			0.214643i
$426$		0			0
$427$	74.3643			3.59874
$428$		−	5.45340i		−	0.263600i
$429$		0			0
$430$			12.5329i			0.604390i
$431$		−	4.51238i		−	0.217354i	−0.994077	−	0.108677i	$-0.965339\pi$
$431$		−	4.51238i		−	0.217354i	0.994077	−	0.108677i	$-0.0346613\pi$
$432$		0			0
$433$			11.6103i			0.557954i	0.960298	+	0.278977i	$0.0899953\pi$
$433$			11.6103i			0.557954i	−0.960298	+	0.278977i	$0.910005\pi$
$434$	−17.4440			−0.837339
$435$		0			0
$436$			12.3864i			0.593202i
$437$			14.6226i			0.699496i
$438$		0			0
$439$	−4.45507			−0.212629			−0.106314	−	0.994333i	$-0.533905\pi$
$439$	−4.45507			−0.212629			−0.106314	+	0.994333i	$0.533905\pi$
$440$	5.96177			0.284216
$441$		0			0
$442$	−25.9697			−1.23525
$443$	18.4945			0.878698			0.439349	−	0.898316i	$-0.355209\pi$
$443$	18.4945			0.878698			0.439349	+	0.898316i	$0.355209\pi$
$444$		0			0
$445$			15.2721i			0.723966i
$446$	7.88456			0.373345
$447$		0			0
$448$		−	5.18179i		−	0.244817i
$449$			25.7910i			1.21715i	0.793496	+	0.608576i	$0.208259\pi$
$449$			25.7910i			1.21715i	−0.793496	+	0.608576i	$0.791741\pi$
$450$		0			0
$451$	−27.9111			−1.31428
$452$	15.1850			0.714245
$453$		0			0
$454$			24.3653i			1.14352i
$455$	−30.4113			−1.42570
$456$		0			0
$457$	−21.6487			−1.01268			−0.506342	−	0.862333i	$-0.669003\pi$
$457$	−21.6487			−1.01268			−0.506342	+	0.862333i	$0.669003\pi$
$458$			9.99033i			0.466818i
$459$		0			0
$460$		−	2.69899i		−	0.125841i
$461$		−	12.5907i		−	0.586409i	−0.956050	−	0.293205i	$-0.905278\pi$
$461$		−	12.5907i		−	0.586409i	0.956050	−	0.293205i	$-0.0947217\pi$
$462$		0			0
$463$		−	26.2704i		−	1.22089i	−0.792058	−	0.610445i	$-0.790991\pi$
$463$		−	26.2704i		−	1.22089i	0.792058	−	0.610445i	$-0.209009\pi$
$464$			6.15001i			0.285507i
$465$		0			0
$466$	1.79171			0.0829994
$467$			27.7577i			1.28448i	0.766506	+	0.642238i	$0.221994\pi$
$467$			27.7577i			1.28448i	−0.766506	+	0.642238i	$0.778006\pi$
$468$		0			0
$469$	20.3110	−	37.2354i	0.937877	−	1.71937i
$470$		−	1.66891i		−	0.0769813i
$471$		0			0
$472$			5.16305i			0.237649i
$473$			74.7183i			3.43555i
$474$		0			0
$475$	5.41782			0.248587
$476$	22.9294			1.05097
$477$		0			0
$478$	28.9040			1.32204
$479$			37.3128i			1.70486i	0.522839	+	0.852432i	$0.324873\pi$
$479$			37.3128i			1.70486i	−0.522839	+	0.852432i	$0.675127\pi$
$480$		0			0
$481$			28.5499i			1.30176i
$482$	19.1263			0.871177
$483$		0			0
$484$	24.5427			1.11558
$485$		−	6.02723i		−	0.273682i
$486$		0			0
$487$		−	4.14488i		−	0.187822i	−0.995581	−	0.0939112i	$-0.970063\pi$
$487$		−	4.14488i		−	0.187822i	0.995581	−	0.0939112i	$-0.0299370\pi$
$488$			14.3511i			0.649643i
$489$		0			0
$490$	19.8510			0.896777
$491$		−	22.1959i		−	1.00169i	−0.865538	−	0.500843i	$-0.833023\pi$
$491$		−	22.1959i		−	1.00169i	0.865538	−	0.500843i	$-0.166977\pi$
$492$		0			0
$493$	−27.2137			−1.22564
$494$			31.7965i			1.43059i
$495$		0			0
$496$		−	3.36640i		−	0.151156i
$497$	−31.0722			−1.39378
$498$		0			0
$499$			13.8677i			0.620803i	0.950606	+	0.310401i	$0.100463\pi$
$499$			13.8677i			0.620803i	−0.950606	+	0.310401i	$0.899537\pi$
$500$	−1.00000			−0.0447214
$501$		0			0
$502$	−14.0260			−0.626013
$503$	−0.215477			−0.00960766			−0.00480383	−	0.999988i	$-0.501529\pi$
$503$	−0.215477			−0.00960766			−0.00480383	+	0.999988i	$0.501529\pi$
$504$		0			0
$505$	7.13224			0.317381
$506$		−	16.0908i		−	0.715322i
$507$		0			0
$508$	17.8168			0.790492
$509$		−	13.7251i		−	0.608353i	−0.952616	−	0.304177i	$-0.901619\pi$
$509$		−	13.7251i		−	0.608353i	0.952616	−	0.304177i	$-0.0983813\pi$
$510$		0			0
$511$		−	22.0635i		−	0.976030i
$512$	1.00000			0.0441942
$513$		0			0
$514$			12.9070i			0.569303i
$515$	−5.56927			−0.245411
$516$		0			0
$517$		−	9.94968i		−	0.437587i
$518$		−	25.2075i		−	1.10755i
$519$		0			0
$520$		−	5.86888i		−	0.257367i
$521$	−14.4768			−0.634241			−0.317120	−	0.948385i	$-0.602716\pi$
$521$	−14.4768			−0.634241			−0.317120	+	0.948385i	$0.602716\pi$
$522$		0			0
$523$	−24.3372			−1.06419			−0.532096	−	0.846684i	$-0.678596\pi$
$523$	−24.3372			−1.06419			−0.532096	+	0.846684i	$0.678596\pi$
$524$		−	10.6190i		−	0.463894i
$525$		0			0
$526$		−	20.5436i		−	0.895742i
$527$	14.8963			0.648892
$528$		0			0
$529$	15.7154			0.683280
$530$	−0.782473			−0.0339885
$531$		0			0
$532$		−	28.0740i		−	1.21716i
$533$			27.4762i			1.19013i
$534$		0			0
$535$			5.45340i			0.235771i
$536$	7.18582	+	3.91969i	0.310380	+	0.169305i
$537$		0			0
$538$			22.0530i			0.950773i
$539$	118.347			5.09757
$540$		0			0
$541$			34.5420i			1.48508i	0.669804	+	0.742538i	$0.266378\pi$
$541$			34.5420i			1.48508i	−0.669804	+	0.742538i	$0.733622\pi$
$542$		−	32.6182i		−	1.40107i
$543$		0			0
$544$			4.42499i			0.189720i
$545$		−	12.3864i		−	0.530576i
$546$		0			0
$547$			11.6844i			0.499590i	0.968299	+	0.249795i	$0.0803633\pi$
$547$			11.6844i			0.499590i	−0.968299	+	0.249795i	$0.919637\pi$
$548$	9.82609			0.419750
$549$		0			0
$550$	−5.96177			−0.254211
$551$			33.3196i			1.41946i
$552$		0			0
$553$	−9.52361			−0.404985
$554$	−23.8168			−1.01188
$555$		0			0
$556$			3.96165i			0.168012i
$557$		−	6.92022i		−	0.293219i	−0.989194	−	0.146610i	$-0.953164\pi$
$557$		−	6.92022i		−	0.293219i	0.989194	−	0.146610i	$-0.0468361\pi$
$558$		0			0
$559$	73.5540			3.11100
$560$			5.18179i			0.218971i
$561$		0			0
$562$	31.9169			1.34633
$563$	−4.90662			−0.206789			−0.103395	−	0.994640i	$-0.532970\pi$
$563$	−4.90662			−0.206789			−0.103395	+	0.994640i	$0.532970\pi$
$564$		0			0
$565$	−15.1850			−0.638840
$566$	−12.2435			−0.514632
$567$		0			0
$568$		−	5.99641i		−	0.251604i
$569$		−	18.2226i		−	0.763932i	−0.924176	−	0.381966i	$-0.875247\pi$
$569$		−	18.2226i		−	0.763932i	0.924176	−	0.381966i	$-0.124753\pi$
$570$		0			0
$571$	1.34666			0.0563560			0.0281780	−	0.999603i	$-0.491029\pi$
$571$	1.34666			0.0563560			0.0281780	+	0.999603i	$0.491029\pi$
$572$		−	34.9889i		−	1.46296i
$573$		0			0
$574$		−	24.2595i		−	1.01257i
$575$			2.69899i			0.112556i
$576$		0			0
$577$		−	28.1416i		−	1.17155i	−0.810474	−	0.585775i	$-0.800790\pi$
$577$		−	28.1416i		−	1.17155i	0.810474	−	0.585775i	$-0.199210\pi$
$578$	−2.58052			−0.107335
$579$		0			0
$580$		−	6.15001i		−	0.255365i
$581$	50.4323			2.09229
$582$		0			0
$583$	−4.66493			−0.193202
$584$	4.25788			0.176192
$585$		0			0
$586$			23.5765i			0.973936i
$587$	−11.7589			−0.485342			−0.242671	−	0.970109i	$-0.578024\pi$
$587$	−11.7589			−0.485342			−0.242671	+	0.970109i	$0.578024\pi$
$588$		0			0
$589$		−	18.2385i		−	0.751506i
$590$		−	5.16305i		−	0.212559i
$591$		0			0
$592$	4.86463			0.199935
$593$	−28.9293			−1.18798			−0.593992	−	0.804471i	$-0.702449\pi$
$593$	−28.9293			−1.18798			−0.593992	+	0.804471i	$0.702449\pi$
$594$		0			0
$595$	−22.9294			−0.940013
$596$			13.5127i			0.553501i
$597$		0			0
$598$	−15.8400			−0.647748
$599$	−13.5068			−0.551873			−0.275937	−	0.961176i	$-0.588988\pi$
$599$	−13.5068			−0.551873			−0.275937	+	0.961176i	$0.588988\pi$
$600$		0			0
$601$	11.0658			0.451382			0.225691	−	0.974199i	$-0.427536\pi$
$601$	11.0658			0.451382			0.225691	+	0.974199i	$0.427536\pi$
$602$	−64.9429			−2.64688
$603$		0			0
$604$	13.0171			0.529657
$605$	−24.5427			−0.997804
$606$		0			0
$607$	−24.9641			−1.01326			−0.506632	−	0.862163i	$-0.669110\pi$
$607$	−24.9641			−1.01326			−0.506632	+	0.862163i	$0.669110\pi$
$608$	5.41782			0.219722
$609$		0			0
$610$		−	14.3511i		−	0.581058i
$611$	−9.79465			−0.396249
$612$		0			0
$613$	−22.2307			−0.897890			−0.448945	−	0.893559i	$-0.648200\pi$
$613$	−22.2307			−0.897890			−0.448945	+	0.893559i	$0.648200\pi$
$614$	−4.74518			−0.191500
$615$		0			0
$616$			30.8927i			1.24470i
$617$			36.8553i			1.48374i	0.670544	+	0.741870i	$0.266061\pi$
$617$			36.8553i			1.48374i	−0.670544	+	0.741870i	$0.733939\pi$
$618$		0			0
$619$	−34.2387			−1.37617			−0.688085	−	0.725630i	$-0.741548\pi$
$619$	−34.2387			−1.37617			−0.688085	+	0.725630i	$0.741548\pi$
$620$			3.36640i			0.135198i
$621$		0			0
$622$	−11.3216			−0.453955
$623$	−79.1368			−3.17055
$624$		0			0
$625$	1.00000			0.0400000
$626$			11.3053i			0.451850i
$627$		0			0
$628$	−11.0167			−0.439614
$629$			21.5259i			0.858295i
$630$		0			0
$631$			25.5129i			1.01565i	0.861459	+	0.507827i	$0.169551\pi$
$631$			25.5129i			1.01565i	−0.861459	+	0.507827i	$0.830449\pi$
$632$		−	1.83790i		−	0.0731076i
$633$		0			0
$634$			6.85114i			0.272094i
$635$	−17.8168			−0.707037
$636$		0			0
$637$		−	116.503i		−	4.61602i
$638$		−	36.6649i		−	1.45158i
$639$		0			0
$640$	−1.00000			−0.0395285
$641$	−10.8327			−0.427868			−0.213934	−	0.976848i	$-0.568628\pi$
$641$	−10.8327			−0.427868			−0.213934	+	0.976848i	$0.568628\pi$
$642$		0			0
$643$	−3.69806			−0.145837			−0.0729186	−	0.997338i	$-0.523231\pi$
$643$	−3.69806			−0.145837			−0.0729186	+	0.997338i	$0.523231\pi$
$644$	13.9856			0.551111
$645$		0			0
$646$			23.9738i			0.943236i
$647$	36.3659			1.42969			0.714846	−	0.699282i	$-0.246497\pi$
$647$	36.3659			1.42969			0.714846	+	0.699282i	$0.246497\pi$
$648$		0			0
$649$		−	30.7809i		−	1.20826i
$650$			5.86888i			0.230196i
$651$		0			0
$652$	−19.8732			−0.778296
$653$	45.8717			1.79510			0.897550	−	0.440913i	$-0.145345\pi$
$653$	45.8717			1.79510			0.897550	+	0.440913i	$0.145345\pi$
$654$		0			0
$655$			10.6190i			0.414919i
$656$	4.68168			0.182789
$657$		0			0
$658$	8.64797			0.337133
$659$		−	9.40431i		−	0.366340i	−0.983081	−	0.183170i	$-0.941364\pi$
$659$		−	9.40431i		−	0.366340i	0.983081	−	0.183170i	$-0.0586359\pi$
$660$		0			0
$661$		−	24.5089i		−	0.953287i	−0.879097	−	0.476643i	$-0.841853\pi$
$661$		−	24.5089i		−	0.953287i	0.879097	−	0.476643i	$-0.158147\pi$
$662$			1.28951i			0.0501182i
$663$		0			0
$664$			9.73260i			0.377698i
$665$			28.0740i			1.08866i
$666$		0			0
$667$	−16.5988			−0.642709
$668$			16.4348i			0.635881i
$669$		0			0
$670$	−7.18582	−	3.91969i	−0.277612	−	0.151431i
$671$		−	85.5578i		−	3.30292i
$672$		0			0
$673$		−	36.8972i		−	1.42228i	−0.703050	−	0.711141i	$-0.748179\pi$
$673$		−	36.8972i		−	1.42228i	0.703050	−	0.711141i	$-0.251821\pi$
$674$		−	10.3466i		−	0.398538i
$675$		0			0
$676$	−21.4437			−0.824758
$677$	47.5385			1.82705			0.913526	−	0.406780i	$-0.133348\pi$
$677$	47.5385			1.82705			0.913526	+	0.406780i	$0.133348\pi$
$678$		0			0
$679$	31.2319			1.19857
$680$		−	4.42499i		−	0.169691i
$681$		0			0
$682$			20.0697i			0.768509i
$683$	−41.4401			−1.58566			−0.792831	−	0.609442i	$-0.791394\pi$
$683$	−41.4401			−1.58566			−0.792831	+	0.609442i	$0.791394\pi$
$684$		0			0
$685$	−9.82609			−0.375436
$686$			66.5912i			2.54247i
$687$		0			0
$688$		−	12.5329i		−	0.477812i
$689$			4.59224i			0.174950i
$690$		0			0
$691$	−46.4741			−1.76796			−0.883980	−	0.467526i	$-0.845146\pi$
$691$	−46.4741			−1.76796			−0.883980	+	0.467526i	$0.845146\pi$
$692$		−	13.4243i		−	0.510316i
$693$		0			0
$694$	−29.3425			−1.11383
$695$		−	3.96165i		−	0.150274i
$696$		0			0
$697$			20.7164i			0.784689i
$698$	26.5886			1.00639
$699$		0			0
$700$		−	5.18179i		−	0.195853i
$701$	21.1480			0.798749			0.399374	−	0.916788i	$-0.369227\pi$
$701$	21.1480			0.798749			0.399374	+	0.916788i	$0.369227\pi$
$702$		0			0
$703$	26.3557			0.994023
$704$	−5.96177			−0.224693
$705$		0			0
$706$	−9.43485			−0.355085
$707$			36.9578i			1.38994i
$708$		0			0
$709$	9.40658			0.353272			0.176636	−	0.984276i	$-0.443479\pi$
$709$	9.40658			0.353272			0.176636	+	0.984276i	$0.443479\pi$
$710$			5.99641i			0.225041i
$711$		0			0
$712$		−	15.2721i		−	0.572345i
$713$	9.08588			0.340269
$714$		0			0
$715$			34.9889i			1.30851i
$716$	10.2736			0.383943
$717$		0			0
$718$		−	4.07390i		−	0.152037i
$719$		−	8.88470i		−	0.331343i	−0.986181	−	0.165672i	$-0.947021\pi$
$719$		−	8.88470i		−	0.331343i	0.986181	−	0.165672i	$-0.0529792\pi$
$720$		0			0
$721$		−	28.8588i		−	1.07476i
$722$	10.3528			0.385290
$723$		0			0
$724$	−4.13822			−0.153796
$725$			6.15001i			0.228405i
$726$		0			0
$727$			33.7450i			1.25153i	0.780010	+	0.625767i	$0.215214\pi$
$727$			33.7450i			1.25153i	−0.780010	+	0.625767i	$0.784786\pi$
$728$	30.4113			1.12712
$729$		0			0
$730$	−4.25788			−0.157591
$731$	55.4579			2.05119
$732$		0			0
$733$		−	23.6757i		−	0.874483i	−0.899344	−	0.437242i	$-0.855955\pi$
$733$		−	23.6757i		−	0.874483i	0.899344	−	0.437242i	$-0.144045\pi$
$734$			14.4846i			0.534635i
$735$		0			0
$736$			2.69899i			0.0994861i
$737$	−42.8402	−	23.3683i	−1.57804	−	0.860783i
$738$		0			0
$739$		−	45.7882i		−	1.68435i	−0.539206	−	0.842174i	$-0.681276\pi$
$739$		−	45.7882i		−	1.68435i	0.539206	−	0.842174i	$-0.318724\pi$
$740$	−4.86463			−0.178827
$741$		0			0
$742$		−	4.05462i		−	0.148850i
$743$			1.62856i			0.0597462i	0.999554	+	0.0298731i	$0.00951032\pi$
$743$			1.62856i			0.0597462i	−0.999554	+	0.0298731i	$0.990490\pi$
$744$		0			0
$745$		−	13.5127i		−	0.495066i
$746$			9.64695i			0.353200i
$747$		0			0
$748$		−	26.3808i		−	0.964576i
$749$	−28.2584			−1.03254
$750$		0			0
$751$	−42.6152			−1.55505			−0.777525	−	0.628852i	$-0.783525\pi$
$751$	−42.6152			−1.55505			−0.777525	+	0.628852i	$0.783525\pi$
$752$			1.66891i			0.0608590i
$753$		0			0
$754$	−36.0936			−1.31445
$755$	−13.0171			−0.473740
$756$		0			0
$757$			0.881929i			0.0320542i	0.999872	+	0.0160271i	$0.00510181\pi$
$757$			0.881929i			0.0320542i	−0.999872	+	0.0160271i	$0.994898\pi$
$758$			9.96973i			0.362117i
$759$		0			0
$760$	−5.41782			−0.196525
$761$			14.6083i			0.529551i	0.964310	+	0.264775i	$0.0852978\pi$
$761$			14.6083i			0.529551i	−0.964310	+	0.264775i	$0.914702\pi$
$762$		0			0
$763$	64.1839			2.32361
$764$	−0.859197			−0.0310847
$765$		0			0
$766$	−15.7905			−0.570533
$767$	−30.3013			−1.09412
$768$		0			0
$769$			17.5810i			0.633989i	0.948427	+	0.316994i	$0.102674\pi$
$769$			17.5810i			0.633989i	−0.948427	+	0.316994i	$0.897326\pi$
$770$		−	30.8927i		−	1.11329i
$771$		0			0
$772$	15.8120			0.569087
$773$		−	30.3276i		−	1.09081i	−0.838174	−	0.545404i	$-0.816376\pi$
$773$		−	30.3276i		−	1.09081i	0.838174	−	0.545404i	$-0.183624\pi$
$774$		0			0
$775$		−	3.36640i		−	0.120925i
$776$			6.02723i			0.216365i
$777$		0			0
$778$		−	26.1647i		−	0.938048i
$779$	25.3645			0.908778
$780$		0			0
$781$			35.7492i			1.27921i
$782$	−11.9430			−0.427081
$783$		0			0
$784$	−19.8510			−0.708964
$785$	11.0167			0.393203
$786$		0			0
$787$			25.2682i			0.900714i	0.892848	+	0.450357i	$0.148703\pi$
$787$			25.2682i			0.900714i	−0.892848	+	0.450357i	$0.851297\pi$
$788$	5.28463			0.188257
$789$		0			0
$790$			1.83790i			0.0653895i
$791$		−	78.6858i		−	2.79775i
$792$		0			0
$793$	−84.2247			−2.99091
$794$	−3.16108			−0.112183
$795$		0			0
$796$	1.20124			0.0425769
$797$		−	13.6166i		−	0.482325i	−0.970485	−	0.241162i	$-0.922471\pi$
$797$		−	13.6166i		−	0.482325i	0.970485	−	0.241162i	$-0.0775286\pi$
$798$		0			0
$799$	−7.38492			−0.261260
$800$	1.00000			0.0353553
$801$		0			0
$802$	−12.5387			−0.442758
$803$	−25.3845			−0.895799
$804$		0			0
$805$	−13.9856			−0.492928
$806$	19.7570			0.695910
$807$		0			0
$808$	−7.13224			−0.250911
$809$	−14.0789			−0.494989			−0.247494	−	0.968889i	$-0.579607\pi$
$809$	−14.0789			−0.494989			−0.247494	+	0.968889i	$0.579607\pi$
$810$		0			0
$811$		−	55.2858i		−	1.94135i	−0.240401	−	0.970674i	$-0.577279\pi$
$811$		−	55.2858i		−	1.94135i	0.240401	−	0.970674i	$-0.422721\pi$
$812$	31.8681			1.11835
$813$		0			0
$814$	−29.0018			−1.01651
$815$	19.8732			0.696129
$816$		0			0
$817$		−	67.9010i		−	2.37555i
$818$			16.0800i			0.562224i
$819$		0			0
$820$	−4.68168			−0.163491
$821$		−	19.5601i		−	0.682654i	−0.939945	−	0.341327i	$-0.889124\pi$
$821$		−	19.5601i		−	0.682654i	0.939945	−	0.341327i	$-0.110876\pi$
$822$		0			0
$823$	−4.67161			−0.162842			−0.0814211	−	0.996680i	$-0.525946\pi$
$823$	−4.67161			−0.162842			−0.0814211	+	0.996680i	$0.525946\pi$
$824$	5.56927			0.194015
$825$		0			0
$826$	26.7539			0.930886
$827$			17.8383i			0.620300i	0.950688	+	0.310150i	$0.100379\pi$
$827$			17.8383i			0.620300i	−0.950688	+	0.310150i	$0.899621\pi$
$828$		0			0
$829$	54.8126			1.90372			0.951859	−	0.306535i	$-0.0991695\pi$
$829$	54.8126			1.90372			0.951859	+	0.306535i	$0.0991695\pi$
$830$		−	9.73260i		−	0.337823i
$831$		0			0
$832$			5.86888i			0.203467i
$833$		−	87.8404i		−	3.04349i
$834$		0			0
$835$		−	16.4348i		−	0.568749i
$836$	−32.2998			−1.11711
$837$		0			0
$838$			20.2260i			0.698696i
$839$			3.28845i			0.113530i	0.998388	+	0.0567650i	$0.0180786\pi$
$839$			3.28845i			0.113530i	−0.998388	+	0.0567650i	$0.981921\pi$
$840$		0			0
$841$	−8.82257			−0.304226
$842$	−7.95185			−0.274039
$843$		0			0
$844$	1.22192			0.0420602
$845$	21.4437			0.737686
$846$		0			0
$847$		−	127.175i		−	4.36980i
$848$	0.782473			0.0268702
$849$		0			0
$850$			4.42499i			0.151776i
$851$			13.1296i			0.450076i
$852$		0			0
$853$	54.1544			1.85421			0.927106	−	0.374799i	$-0.122288\pi$
$853$	54.1544			1.85421			0.927106	+	0.374799i	$0.122288\pi$
$854$	74.3643			2.54470
$855$		0			0
$856$		−	5.45340i		−	0.186393i
$857$	−44.0908			−1.50611			−0.753057	−	0.657955i	$-0.771422\pi$
$857$	−44.0908			−1.50611			−0.753057	+	0.657955i	$0.771422\pi$
$858$		0			0
$859$	36.7718			1.25464			0.627319	−	0.778762i	$-0.284152\pi$
$859$	36.7718			1.25464			0.627319	+	0.778762i	$0.284152\pi$
$860$			12.5329i			0.427368i
$861$		0			0
$862$		−	4.51238i		−	0.153692i
$863$		−	33.9821i		−	1.15676i	−0.815766	−	0.578382i	$-0.803684\pi$
$863$		−	33.9821i		−	1.15676i	0.815766	−	0.578382i	$-0.196316\pi$
$864$		0			0
$865$			13.4243i			0.456440i
$866$			11.6103i			0.394533i
$867$		0			0
$868$	−17.4440			−0.592088
$869$			10.9571i			0.371695i
$870$		0			0
$871$	−23.0042	+	42.1727i	−0.779467	+	1.42897i
$872$			12.3864i			0.419457i
$873$		0			0
$874$			14.6226i			0.494618i
$875$			5.18179i			0.175177i
$876$		0			0
$877$	−41.5537			−1.40317			−0.701585	−	0.712586i	$-0.747524\pi$
$877$	−41.5537			−1.40317			−0.701585	+	0.712586i	$0.747524\pi$
$878$	−4.45507			−0.150351
$879$		0			0
$880$	5.96177			0.200971
$881$			30.5570i			1.02949i	0.857343	+	0.514746i	$0.172114\pi$
$881$			30.5570i			1.02949i	−0.857343	+	0.514746i	$0.827886\pi$
$882$		0			0
$883$			0.532739i			0.0179281i	0.999960	+	0.00896405i	$0.00285338\pi$
$883$			0.532739i			0.0179281i	−0.999960	+	0.00896405i	$0.997147\pi$
$884$	−25.9697			−0.873456
$885$		0			0
$886$	18.4945			0.621334
$887$		−	39.5021i		−	1.32635i	−0.748463	−	0.663176i	$-0.769208\pi$
$887$		−	39.5021i		−	1.32635i	0.748463	−	0.663176i	$-0.230792\pi$
$888$		0			0
$889$		−	92.3229i		−	3.09641i
$890$			15.2721i			0.511921i
$891$		0			0
$892$	7.88456			0.263995
$893$			9.04187i			0.302575i
$894$		0			0
$895$	−10.2736			−0.343409
$896$		−	5.18179i		−	0.173112i
$897$		0			0
$898$			25.7910i			0.860656i
$899$	20.7034			0.690496
$900$		0			0
$901$			3.46243i			0.115350i
$902$	−27.9111			−0.929339
$903$		0			0
$904$	15.1850			0.505047
$905$	4.13822			0.137559
$906$		0			0
$907$	−43.0522			−1.42952			−0.714762	−	0.699368i	$-0.753465\pi$
$907$	−43.0522			−1.42952			−0.714762	+	0.699368i	$0.753465\pi$
$908$			24.3653i			0.808590i
$909$		0			0
$910$	−30.4113			−1.00813
$911$			13.5803i			0.449937i	0.974366	+	0.224968i	$0.0722279\pi$
$911$			13.5803i			0.449937i	−0.974366	+	0.224968i	$0.927772\pi$
$912$		0			0
$913$		−	58.0235i		−	1.92030i
$914$	−21.6487			−0.716075
$915$		0			0
$916$			9.99033i			0.330090i
$917$	−55.0256			−1.81710
$918$		0			0
$919$			15.9188i			0.525113i	0.964917	+	0.262556i	$0.0845656\pi$
$919$			15.9188i			0.525113i	−0.964917	+	0.262556i	$0.915434\pi$
$920$		−	2.69899i		−	0.0889831i
$921$		0			0
$922$		−	12.5907i		−	0.414654i
$923$	35.1922			1.15837
$924$		0			0
$925$	4.86463			0.159948
$926$		−	26.2704i		−	0.863300i
$927$		0			0
$928$			6.15001i			0.201884i
$929$	25.0663			0.822400			0.411200	−	0.911545i	$-0.365110\pi$
$929$	25.0663			0.822400			0.411200	+	0.911545i	$0.365110\pi$
$930$		0			0
$931$	−107.549			−3.52478
$932$	1.79171			0.0586895
$933$		0			0
$934$			27.7577i			0.908261i
$935$			26.3808i			0.862743i
$936$		0			0
$937$		−	7.95598i		−	0.259911i	−0.991520	−	0.129955i	$-0.958517\pi$
$937$		−	7.95598i		−	0.259911i	0.991520	−	0.129955i	$-0.0414834\pi$
$938$	20.3110	−	37.2354i	0.663179	−	1.21578i
$939$		0			0
$940$		−	1.66891i		−	0.0544340i
$941$	−30.8051			−1.00422			−0.502109	−	0.864804i	$-0.667442\pi$
$941$	−30.8051			−1.00422			−0.502109	+	0.864804i	$0.667442\pi$
$942$		0			0
$943$			12.6358i			0.411479i
$944$			5.16305i			0.168043i
$945$		0			0
$946$			74.7183i			2.42930i
$947$			10.4343i			0.339069i	0.985524	+	0.169535i	$0.0542265\pi$
$947$			10.4343i			0.339069i	−0.985524	+	0.169535i	$0.945774\pi$
$948$		0			0
$949$			24.9890i			0.811176i
$950$	5.41782			0.175777
$951$		0			0
$952$	22.9294			0.743145
$953$		−	60.1012i		−	1.94687i	−0.228967	−	0.973434i	$-0.573535\pi$
$953$		−	60.1012i		−	1.94687i	0.228967	−	0.973434i	$-0.426465\pi$
$954$		0			0
$955$	0.859197			0.0278030
$956$	28.9040			0.934823
$957$		0			0
$958$			37.3128i			1.20552i
$959$		−	50.9168i		−	1.64419i
$960$		0			0
$961$	19.6673			0.634431
$962$			28.5499i			0.920486i
$963$		0			0
$964$	19.1263			0.616015
$965$	−15.8120			−0.509007
$966$		0			0
$967$	14.8432			0.477326			0.238663	−	0.971102i	$-0.423291\pi$
$967$	14.8432			0.477326			0.238663	+	0.971102i	$0.423291\pi$
$968$	24.5427			0.788833
$969$		0			0
$970$		−	6.02723i		−	0.193523i
$971$			24.7327i			0.793709i	0.917882	+	0.396855i	$0.129898\pi$
$971$			24.7327i			0.793709i	−0.917882	+	0.396855i	$0.870102\pi$
$972$		0			0
$973$	20.5285			0.658113
$974$		−	4.14488i		−	0.132811i
$975$		0			0
$976$			14.3511i			0.459367i
$977$			16.6964i			0.534164i	0.963674	+	0.267082i	$0.0860595\pi$
$977$			16.6964i			0.534164i	−0.963674	+	0.267082i	$0.913941\pi$
$978$		0			0
$979$			91.0487i			2.90993i
$980$	19.8510			0.634117
$981$		0			0
$982$		−	22.1959i		−	0.708299i
$983$	−19.2387			−0.613619			−0.306810	−	0.951771i	$-0.599261\pi$
$983$	−19.2387			−0.613619			−0.306810	+	0.951771i	$0.599261\pi$
$984$		0			0
$985$	−5.28463			−0.168382
$986$	−27.2137			−0.866661
$987$		0			0
$988$			31.7965i			1.01158i
$989$	33.8262			1.07561
$990$		0			0
$991$			3.22867i			0.102562i	0.998684	+	0.0512810i	$0.0163304\pi$
$991$			3.22867i			0.102562i	−0.998684	+	0.0512810i	$0.983670\pi$
$992$		−	3.36640i		−	0.106883i
$993$		0			0
$994$	−31.0722			−0.985550
$995$	−1.20124			−0.0380820
$996$		0			0
$997$	53.7152			1.70118			0.850588	−	0.525832i	$-0.176246\pi$
$997$	53.7152			1.70118			0.850588	+	0.525832i	$0.176246\pi$
$998$			13.8677i			0.438974i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6030.2.d.j.2411.1	yes	16
3.2	odd	2		6030.2.d.i.2411.1	✓	16
67.66	odd	2		6030.2.d.i.2411.16	yes	16
201.200	even	2	inner	6030.2.d.j.2411.16	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6030.2.d.i.2411.1	✓	16	3.2	odd	2
6030.2.d.i.2411.16	yes	16	67.66	odd	2
6030.2.d.j.2411.1	yes	16	1.1	even	1	trivial
6030.2.d.j.2411.16	yes	16	201.200	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:	\( N \)	\(=\)	\( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6030.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -5.18179i q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -5.18179i q^{7} +1.00000 q^{8} -1.00000 q^{10} -5.96177 q^{11} +5.86888i q^{13} -5.18179i q^{14} +1.00000 q^{16} +4.42499i q^{17} +5.41782 q^{19} -1.00000 q^{20} -5.96177 q^{22} +2.69899i q^{23} +1.00000 q^{25} +5.86888i q^{26} -5.18179i q^{28} +6.15001i q^{29} -3.36640i q^{31} +1.00000 q^{32} +4.42499i q^{34} +5.18179i q^{35} +4.86463 q^{37} +5.41782 q^{38} -1.00000 q^{40} +4.68168 q^{41} -12.5329i q^{43} -5.96177 q^{44} +2.69899i q^{46} +1.66891i q^{47} -19.8510 q^{49} +1.00000 q^{50} +5.86888i q^{52} +0.782473 q^{53} +5.96177 q^{55} -5.18179i q^{56} +6.15001i q^{58} +5.16305i q^{59} +14.3511i q^{61} -3.36640i q^{62} +1.00000 q^{64} -5.86888i q^{65} +(7.18582 + 3.91969i) q^{67} +4.42499i q^{68} +5.18179i q^{70} -5.99641i q^{71} +4.25788 q^{73} +4.86463 q^{74} +5.41782 q^{76} +30.8927i q^{77} -1.83790i q^{79} -1.00000 q^{80} +4.68168 q^{82} +9.73260i q^{83} -4.42499i q^{85} -12.5329i q^{86} -5.96177 q^{88} -15.2721i q^{89} +30.4113 q^{91} +2.69899i q^{92} +1.66891i q^{94} -5.41782 q^{95} +6.02723i q^{97} -19.8510 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} + 16 q^{4} - 16 q^{5} + 16 q^{8}+O(q^{10}) \) 16 * q + 16 * q^2 + 16 * q^4 - 16 * q^5 + 16 * q^8	\( 16 q + 16 q^{2} + 16 q^{4} - 16 q^{5} + 16 q^{8} - 16 q^{10} - 20 q^{11} + 16 q^{16} - 8 q^{19} - 16 q^{20} - 20 q^{22} + 16 q^{25} + 16 q^{32} + 32 q^{37} - 8 q^{38} - 16 q^{40} - 8 q^{41} - 20 q^{44} - 88 q^{49} + 16 q^{50} + 8 q^{53} + 20 q^{55} + 16 q^{64} + 4 q^{67} + 16 q^{73} + 32 q^{74} - 8 q^{76} - 16 q^{80} - 8 q^{82} - 20 q^{88} + 40 q^{91} + 8 q^{95} - 88 q^{98}+O(q^{100}) \) 16 * q + 16 * q^2 + 16 * q^4 - 16 * q^5 + 16 * q^8 - 16 * q^10 - 20 * q^11 + 16 * q^16 - 8 * q^19 - 16 * q^20 - 20 * q^22 + 16 * q^25 + 16 * q^32 + 32 * q^37 - 8 * q^38 - 16 * q^40 - 8 * q^41 - 20 * q^44 - 88 * q^49 + 16 * q^50 + 8 * q^53 + 20 * q^55 + 16 * q^64 + 4 * q^67 + 16 * q^73 + 32 * q^74 - 8 * q^76 - 16 * q^80 - 8 * q^82 - 20 * q^88 + 40 * q^91 + 8 * q^95 - 88 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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