LMFDB - Embedded newform 2070.3.c.a.91.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,3,Mod(91,2070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2070.91");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2070.c (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:	$56.4034147226$
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} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 $ x^16 + 78x^14 + 2165x^12 + 28310x^10 + 184804x^8 + 569634x^6 + 696037x^4 + 285578*x^2 + 529
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.16
Root			$2.98291i$ of defining polynomial
Character	$\chi$	$=$	2070.91
Dual form			2070.3.c.a.91.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +8.51262i q^{7} +2.82843 q^{8} +O(q^{10})$	\(q+1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +8.51262i q^{7} +2.82843 q^{8} +3.16228i q^{10} +7.57553i q^{11} -2.64076 q^{13} +12.0387i q^{14} +4.00000 q^{16} -7.56057i q^{17} +24.2676i q^{19} +4.47214i q^{20} +10.7134i q^{22} +(15.7366 - 16.7738i) q^{23} -5.00000 q^{25} -3.73460 q^{26} +17.0252i q^{28} +31.8513 q^{29} -56.5071 q^{31} +5.65685 q^{32} -10.6923i q^{34} -19.0348 q^{35} +39.9378i q^{37} +34.3195i q^{38} +6.32456i q^{40} +42.5710 q^{41} +20.5721i q^{43} +15.1511i q^{44} +(22.2549 - 23.7217i) q^{46} -84.3049 q^{47} -23.4647 q^{49} -7.07107 q^{50} -5.28152 q^{52} -11.9189i q^{53} -16.9394 q^{55} +24.0773i q^{56} +45.0446 q^{58} -67.6561 q^{59} -35.1621i q^{61} -79.9131 q^{62} +8.00000 q^{64} -5.90492i q^{65} +44.0660i q^{67} -15.1211i q^{68} -26.9193 q^{70} -8.86597 q^{71} -87.4150 q^{73} +56.4805i q^{74} +48.5352i q^{76} -64.4876 q^{77} -154.217i q^{79} +8.94427i q^{80} +60.2046 q^{82} +141.642i q^{83} +16.9059 q^{85} +29.0933i q^{86} +21.4268i q^{88} +63.7252i q^{89} -22.4798i q^{91} +(31.4732 - 33.5476i) q^{92} -119.225 q^{94} -54.2639 q^{95} +143.322i q^{97} -33.1841 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4}+O(q^{10}) $ 16 * q + 32 * q^4	$ 16 q + 32 q^{4} + 24 q^{13} + 64 q^{16} - 4 q^{23} - 80 q^{25} - 96 q^{26} + 108 q^{29} - 116 q^{31} - 60 q^{35} + 156 q^{41} - 124 q^{46} + 128 q^{47} - 28 q^{49} + 48 q^{52} + 160 q^{58} - 204 q^{59} - 64 q^{62} + 128 q^{64} - 120 q^{70} - 236 q^{71} - 112 q^{73} + 936 q^{77} - 64 q^{82} + 60 q^{85} - 8 q^{92} - 216 q^{94} + 160 q^{95} - 256 q^{98}+O(q^{100}) $ 16 * q + 32 * q^4 + 24 * q^13 + 64 * q^16 - 4 * q^23 - 80 * q^25 - 96 * q^26 + 108 * q^29 - 116 * q^31 - 60 * q^35 + 156 * q^41 - 124 * q^46 + 128 * q^47 - 28 * q^49 + 48 * q^52 + 160 * q^58 - 204 * q^59 - 64 * q^62 + 128 * q^64 - 120 * q^70 - 236 * q^71 - 112 * q^73 + 936 * q^77 - 64 * q^82 + 60 * q^85 - 8 * q^92 - 216 * q^94 + 160 * q^95 - 256 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$1891$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	1.41421			0.707107
$2$	1.41421			0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$			8.51262i			1.21609i	0.793903	+	0.608044i	$0.208046\pi$
$7$			8.51262i			1.21609i	−0.793903	+	0.608044i	$0.791954\pi$
$8$	2.82843			0.353553
$9$		0			0
$10$			3.16228i			0.316228i
$11$			7.57553i			0.688684i	0.938844	+	0.344342i	$0.111898\pi$
$11$			7.57553i			0.688684i	−0.938844	+	0.344342i	$0.888102\pi$
$12$		0			0
$13$	−2.64076			−0.203135			−0.101568	−	0.994829i	$-0.532386\pi$
$13$	−2.64076			−0.203135			−0.101568	+	0.994829i	$0.532386\pi$
$14$			12.0387i			0.859904i
$15$		0			0
$16$	4.00000			0.250000
$17$		−	7.56057i		−	0.444739i	−0.974962	−	0.222370i	$-0.928621\pi$
$17$		−	7.56057i		−	0.444739i	0.974962	−	0.222370i	$-0.0713792\pi$
$18$		0			0
$19$			24.2676i			1.27724i	0.769522	+	0.638620i	$0.220495\pi$
$19$			24.2676i			1.27724i	−0.769522	+	0.638620i	$0.779505\pi$
$20$			4.47214i			0.223607i
$21$		0			0
$22$			10.7134i			0.486973i
$23$	15.7366	−	16.7738i	0.684199	−	0.729295i
$23$	15.7366	−	16.7738i	0.684199	−	0.729295i
$24$		0			0
$25$	−5.00000			−0.200000
$26$	−3.73460			−0.143638
$27$		0			0
$28$			17.0252i			0.608044i
$29$	31.8513			1.09832			0.549161	−	0.835717i	$-0.314947\pi$
$29$	31.8513			1.09832			0.549161	+	0.835717i	$0.314947\pi$
$30$		0			0
$31$	−56.5071			−1.82281			−0.911405	−	0.411511i	$-0.865001\pi$
$31$	−56.5071			−1.82281			−0.911405	+	0.411511i	$0.865001\pi$
$32$	5.65685			0.176777
$33$		0			0
$34$		−	10.6923i		−	0.314478i
$35$	−19.0348			−0.543851
$36$		0			0
$37$			39.9378i			1.07940i	0.841858	+	0.539700i	$0.181462\pi$
$37$			39.9378i			1.07940i	−0.841858	+	0.539700i	$0.818538\pi$
$38$			34.3195i			0.903146i
$39$		0			0
$40$			6.32456i			0.158114i
$41$	42.5710			1.03832			0.519159	−	0.854678i	$-0.326245\pi$
$41$	42.5710			1.03832			0.519159	+	0.854678i	$0.326245\pi$
$42$		0			0
$43$			20.5721i			0.478420i	0.970968	+	0.239210i	$0.0768884\pi$
$43$			20.5721i			0.478420i	−0.970968	+	0.239210i	$0.923112\pi$
$44$			15.1511i			0.344342i
$45$		0			0
$46$	22.2549	−	23.7217i	0.483802	−	0.515690i
$47$	−84.3049			−1.79372			−0.896860	−	0.442314i	$-0.854158\pi$
$47$	−84.3049			−1.79372			−0.896860	+	0.442314i	$0.854158\pi$
$48$		0			0
$49$	−23.4647			−0.478871
$50$	−7.07107			−0.141421
$51$		0			0
$52$	−5.28152			−0.101568
$53$		−	11.9189i		−	0.224885i	−0.993658	−	0.112443i	$-0.964133\pi$
$53$		−	11.9189i		−	0.224885i	0.993658	−	0.112443i	$-0.0358674\pi$
$54$		0			0
$55$	−16.9394			−0.307989
$56$			24.0773i			0.429952i
$57$		0			0
$58$	45.0446			0.776631
$59$	−67.6561			−1.14671			−0.573357	−	0.819306i	$-0.694359\pi$
$59$	−67.6561			−1.14671			−0.573357	+	0.819306i	$0.694359\pi$
$60$		0			0
$61$		−	35.1621i		−	0.576428i	−0.957566	−	0.288214i	$-0.906939\pi$
$61$		−	35.1621i		−	0.576428i	0.957566	−	0.288214i	$-0.0930614\pi$
$62$	−79.9131			−1.28892
$63$		0			0
$64$	8.00000			0.125000
$65$		−	5.90492i		−	0.0908449i
$66$		0			0
$67$			44.0660i			0.657701i	0.944382	+	0.328850i	$0.106661\pi$
$67$			44.0660i			0.657701i	−0.944382	+	0.328850i	$0.893339\pi$
$68$		−	15.1211i		−	0.222370i
$69$		0			0
$70$	−26.9193			−0.384561
$71$	−8.86597			−0.124873			−0.0624364	−	0.998049i	$-0.519887\pi$
$71$	−8.86597			−0.124873			−0.0624364	+	0.998049i	$0.519887\pi$
$72$		0			0
$73$	−87.4150			−1.19747			−0.598733	−	0.800949i	$-0.704329\pi$
$73$	−87.4150			−1.19747			−0.598733	+	0.800949i	$0.704329\pi$
$74$			56.4805i			0.763250i
$75$		0			0
$76$			48.5352i			0.638620i
$77$	−64.4876			−0.837501
$78$		0			0
$79$		−	154.217i		−	1.95211i	−0.217517	−	0.976057i	$-0.569796\pi$
$79$		−	154.217i		−	1.95211i	0.217517	−	0.976057i	$-0.430204\pi$
$80$			8.94427i			0.111803i
$81$		0			0
$82$	60.2046			0.734202
$83$			141.642i			1.70653i	0.521476	+	0.853266i	$0.325382\pi$
$83$			141.642i			1.70653i	−0.521476	+	0.853266i	$0.674618\pi$
$84$		0			0
$85$	16.9059			0.198893
$86$			29.0933i			0.338294i
$87$		0			0
$88$			21.4268i			0.243487i
$89$			63.7252i			0.716013i	0.933719	+	0.358007i	$0.116543\pi$
$89$			63.7252i			0.716013i	−0.933719	+	0.358007i	$0.883457\pi$
$90$		0			0
$91$		−	22.4798i		−	0.247030i
$92$	31.4732	−	33.5476i	0.342100	−	0.364648i
$93$		0			0
$94$	−119.225			−1.26835
$95$	−54.2639			−0.571199
$96$		0			0
$97$			143.322i			1.47755i	0.673952	+	0.738775i	$0.264596\pi$
$97$			143.322i			1.47755i	−0.673952	+	0.738775i	$0.735404\pi$
$98$	−33.1841			−0.338613
$99$		0			0
$100$	−10.0000			−0.100000
$101$	−27.7102			−0.274359			−0.137179	−	0.990546i	$-0.543804\pi$
$101$	−27.7102			−0.274359			−0.137179	+	0.990546i	$0.543804\pi$
$102$		0			0
$103$		−	133.542i		−	1.29652i	−0.761418	−	0.648261i	$-0.775497\pi$
$103$		−	133.542i		−	1.29652i	0.761418	−	0.648261i	$-0.224503\pi$
$104$	−7.46919			−0.0718192
$105$		0			0
$106$		−	16.8559i		−	0.159018i
$107$		−	50.3091i		−	0.470179i	−0.971974	−	0.235089i	$-0.924462\pi$
$107$		−	50.3091i		−	0.470179i	0.971974	−	0.235089i	$-0.0755383\pi$
$108$		0			0
$109$			128.819i			1.18182i	0.806737	+	0.590911i	$0.201232\pi$
$109$			128.819i			1.18182i	−0.806737	+	0.590911i	$0.798768\pi$
$110$	−23.9559			−0.217781
$111$		0			0
$112$			34.0505i			0.304022i
$113$		−	86.3028i		−	0.763742i	−0.924216	−	0.381871i	$-0.875280\pi$
$113$		−	86.3028i		−	0.763742i	0.924216	−	0.381871i	$-0.124720\pi$
$114$		0			0
$115$	37.5073	+	35.1881i	0.326151	+	0.305983i
$116$	63.7027			0.549161
$117$		0			0
$118$	−95.6802			−0.810849
$119$	64.3602			0.540842
$120$		0			0
$121$	63.6114			0.525714
$122$		−	49.7267i		−	0.407596i
$123$		0			0
$124$	−113.014			−0.911405
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	−11.0135			−0.0867206			−0.0433603	−	0.999059i	$-0.513806\pi$
$127$	−11.0135			−0.0867206			−0.0433603	+	0.999059i	$0.513806\pi$
$128$	11.3137			0.0883883
$129$		0			0
$130$		−	8.35081i		−	0.0642370i
$131$	−3.63941			−0.0277818			−0.0138909	−	0.999904i	$-0.504422\pi$
$131$	−3.63941			−0.0277818			−0.0138909	+	0.999904i	$0.504422\pi$
$132$		0			0
$133$	−206.581			−1.55324
$134$			62.3187i			0.465065i
$135$		0			0
$136$		−	21.3845i		−	0.157239i
$137$			9.69785i			0.0707873i	0.999373	+	0.0353936i	$0.0112685\pi$
$137$			9.69785i			0.0707873i	−0.999373	+	0.0353936i	$0.988732\pi$
$138$		0			0
$139$	8.05485			0.0579486			0.0289743	−	0.999580i	$-0.490776\pi$
$139$	8.05485			0.0579486			0.0289743	+	0.999580i	$0.490776\pi$
$140$	−38.0696			−0.271926
$141$		0			0
$142$	−12.5384			−0.0882984
$143$		−	20.0051i		−	0.139896i
$144$		0			0
$145$			71.2217i			0.491184i
$146$	−123.624			−0.846737
$147$		0			0
$148$			79.8755i			0.539700i
$149$			144.062i			0.966859i	0.875383	+	0.483430i	$0.160609\pi$
$149$			144.062i			0.966859i	−0.875383	+	0.483430i	$0.839391\pi$
$150$		0			0
$151$	−109.956			−0.728188			−0.364094	−	0.931362i	$-0.618621\pi$
$151$	−109.956			−0.728188			−0.364094	+	0.931362i	$0.618621\pi$
$152$			68.6391i			0.451573i
$153$		0			0
$154$	−91.1992			−0.592202
$155$		−	126.354i		−	0.815185i
$156$		0			0
$157$			24.8208i			0.158094i	0.996871	+	0.0790471i	$0.0251877\pi$
$157$			24.8208i			0.158094i	−0.996871	+	0.0790471i	$0.974812\pi$
$158$		−	218.096i		−	1.38035i
$159$		0			0
$160$			12.6491i			0.0790569i
$161$	142.789	+	133.960i	0.886887	+	0.832047i
$162$		0			0
$163$	108.964			0.668489			0.334244	−	0.942486i	$-0.391519\pi$
$163$	108.964			0.668489			0.334244	+	0.942486i	$0.391519\pi$
$164$	85.1421			0.519159
$165$		0			0
$166$			200.312i			1.20670i
$167$	−72.1383			−0.431966			−0.215983	−	0.976397i	$-0.569296\pi$
$167$	−72.1383			−0.431966			−0.215983	+	0.976397i	$0.569296\pi$
$168$		0			0
$169$	−162.026			−0.958736
$170$	23.9086			0.140639
$171$		0			0
$172$			41.1441i			0.239210i
$173$	150.077			0.867497			0.433748	−	0.901034i	$-0.357191\pi$
$173$	150.077			0.867497			0.433748	+	0.901034i	$0.357191\pi$
$174$		0			0
$175$		−	42.5631i		−	0.243218i
$176$			30.3021i			0.172171i
$177$		0			0
$178$			90.1210i			0.506298i
$179$	−207.058			−1.15675			−0.578375	−	0.815771i	$-0.696313\pi$
$179$	−207.058			−1.15675			−0.578375	+	0.815771i	$0.696313\pi$
$180$		0			0
$181$			331.138i			1.82949i	0.404031	+	0.914746i	$0.367609\pi$
$181$			331.138i			1.82949i	−0.404031	+	0.914746i	$0.632391\pi$
$182$		−	31.7912i		−	0.174677i
$183$		0			0
$184$	44.5098	−	47.4434i	0.241901	−	0.257845i
$185$	−89.3036			−0.482722
$186$		0			0
$187$	57.2753			0.306285
$188$	−168.610			−0.896860
$189$		0			0
$190$	−76.7408			−0.403899
$191$		−	172.749i		−	0.904447i	−0.891905	−	0.452223i	$-0.850631\pi$
$191$		−	172.749i		−	0.904447i	0.891905	−	0.452223i	$-0.149369\pi$
$192$		0			0
$193$	136.450			0.706996			0.353498	−	0.935435i	$-0.384992\pi$
$193$	136.450			0.706996			0.353498	+	0.935435i	$0.384992\pi$
$194$			202.689i			1.04479i
$195$		0			0
$196$	−46.9293			−0.239435
$197$	271.090			1.37609			0.688046	−	0.725667i	$-0.258469\pi$
$197$	271.090			1.37609			0.688046	+	0.725667i	$0.258469\pi$
$198$		0			0
$199$			257.292i			1.29292i	0.762946	+	0.646462i	$0.223752\pi$
$199$			257.292i			1.29292i	−0.762946	+	0.646462i	$0.776248\pi$
$200$	−14.1421			−0.0707107
$201$		0			0
$202$	−39.1882			−0.194001
$203$			271.138i			1.33566i
$204$		0			0
$205$			95.1918i			0.464350i
$206$		−	188.857i		−	0.916780i
$207$		0			0
$208$	−10.5630			−0.0507838
$209$	−183.840			−0.879615
$210$		0			0
$211$	54.1944			0.256846			0.128423	−	0.991720i	$-0.459009\pi$
$211$	54.1944			0.256846			0.128423	+	0.991720i	$0.459009\pi$
$212$		−	23.8378i		−	0.112443i
$213$		0			0
$214$		−	71.1478i		−	0.332467i
$215$	−46.0005			−0.213956
$216$		0			0
$217$		−	481.023i		−	2.21670i
$218$			182.177i			0.835674i
$219$		0			0
$220$	−33.8788			−0.153994
$221$			19.9656i			0.0903423i
$222$		0			0
$223$	104.611			0.469105			0.234553	−	0.972103i	$-0.424637\pi$
$223$	104.611			0.469105			0.234553	+	0.972103i	$0.424637\pi$
$224$			48.1546i			0.214976i
$225$		0			0
$226$		−	122.051i		−	0.540047i
$227$		−	200.484i		−	0.883190i	−0.897215	−	0.441595i	$-0.854413\pi$
$227$		−	200.484i		−	0.883190i	0.897215	−	0.441595i	$-0.145587\pi$
$228$		0			0
$229$			22.0718i			0.0963834i	0.998838	+	0.0481917i	$0.0153458\pi$
$229$			22.0718i			0.0963834i	−0.998838	+	0.0481917i	$0.984654\pi$
$230$	53.0434	+	49.7634i	0.230623	+	0.216363i
$231$		0			0
$232$	90.0892			0.388315
$233$	338.632			1.45335			0.726677	−	0.686979i	$-0.241064\pi$
$233$	338.632			1.45335			0.726677	+	0.686979i	$0.241064\pi$
$234$		0			0
$235$		−	188.511i		−	0.802176i
$236$	−135.312			−0.573357
$237$		0			0
$238$	91.0191			0.382433
$239$	−149.374			−0.624997			−0.312499	−	0.949918i	$-0.601166\pi$
$239$	−149.374			−0.624997			−0.312499	+	0.949918i	$0.601166\pi$
$240$		0			0
$241$			133.030i			0.551991i	0.961159	+	0.275995i	$0.0890074\pi$
$241$			133.030i			0.551991i	−0.961159	+	0.275995i	$0.910993\pi$
$242$	89.9601			0.371736
$243$		0			0
$244$		−	70.3242i		−	0.288214i
$245$		−	52.4686i		−	0.214158i
$246$		0			0
$247$		−	64.0848i		−	0.259453i
$248$	−159.826			−0.644461
$249$		0			0
$250$		−	15.8114i		−	0.0632456i
$251$			376.920i			1.50167i	0.660489	+	0.750836i	$0.270349\pi$
$251$			376.920i			1.50167i	−0.660489	+	0.750836i	$0.729651\pi$
$252$		0			0
$253$	127.070	+	119.213i	0.502254	+	0.471197i
$254$	−15.5755			−0.0613208
$255$		0			0
$256$	16.0000			0.0625000
$257$	−226.085			−0.879708			−0.439854	−	0.898069i	$-0.644970\pi$
$257$	−226.085			−0.879708			−0.439854	+	0.898069i	$0.644970\pi$
$258$		0			0
$259$	−339.975			−1.31264
$260$		−	11.8098i		−	0.0454224i
$261$		0			0
$262$	−5.14690			−0.0196447
$263$			2.43654i			0.00926441i	0.999989	+	0.00463221i	$0.00147448\pi$
$263$			2.43654i			0.00926441i	−0.999989	+	0.00463221i	$0.998526\pi$
$264$		0			0
$265$	26.6515			0.100572
$266$	−292.149			−1.09830
$267$		0			0
$268$			88.1319i			0.328850i
$269$	311.337			1.15739			0.578694	−	0.815545i	$-0.303563\pi$
$269$	311.337			1.15739			0.578694	+	0.815545i	$0.303563\pi$
$270$		0			0
$271$	260.751			0.962182			0.481091	−	0.876671i	$-0.340241\pi$
$271$	260.751			0.962182			0.481091	+	0.876671i	$0.340241\pi$
$272$		−	30.2423i		−	0.111185i
$273$		0			0
$274$			13.7148i			0.0500541i
$275$		−	37.8776i		−	0.137737i
$276$		0			0
$277$	−182.267			−0.658004			−0.329002	−	0.944329i	$-0.606712\pi$
$277$	−182.267			−0.658004			−0.329002	+	0.944329i	$0.606712\pi$
$278$	11.3913			0.0409758
$279$		0			0
$280$	−53.8385			−0.192280
$281$		−	4.60502i		−	0.0163880i	−0.999966	−	0.00819398i	$-0.997392\pi$
$281$		−	4.60502i		−	0.0163880i	0.999966	−	0.00819398i	$-0.00260825\pi$
$282$		0			0
$283$			332.897i			1.17631i	0.808747	+	0.588157i	$0.200146\pi$
$283$			332.897i			1.17631i	−0.808747	+	0.588157i	$0.799854\pi$
$284$	−17.7319			−0.0624364
$285$		0			0
$286$		−	28.2915i		−	0.0989215i
$287$			362.391i			1.26269i
$288$		0			0
$289$	231.838			0.802207
$290$			100.723i			0.347320i
$291$		0			0
$292$	−174.830			−0.598733
$293$		−	336.114i		−	1.14715i	−0.819154	−	0.573574i	$-0.805557\pi$
$293$		−	336.114i		−	1.14715i	0.819154	−	0.573574i	$-0.194443\pi$
$294$		0			0
$295$		−	151.284i		−	0.512826i
$296$			112.961i			0.381625i
$297$		0			0
$298$			203.734i			0.683673i
$299$	−41.5565	+	44.2955i	−0.138985	+	0.148146i
$300$		0			0
$301$	−175.122			−0.581801
$302$	−155.502			−0.514907
$303$		0			0
$304$			97.0703i			0.319310i
$305$	78.6248			0.257786
$306$		0			0
$307$	−457.934			−1.49164			−0.745821	−	0.666147i	$-0.767943\pi$
$307$	−457.934			−1.49164			−0.745821	+	0.666147i	$0.767943\pi$
$308$	−128.975			−0.418750
$309$		0			0
$310$		−	178.691i		−	0.576423i
$311$	455.620			1.46502			0.732508	−	0.680758i	$-0.238350\pi$
$311$	455.620			1.46502			0.732508	+	0.680758i	$0.238350\pi$
$312$		0			0
$313$		−	589.353i		−	1.88292i	−0.337130	−	0.941458i	$-0.609456\pi$
$313$		−	589.353i		−	1.88292i	0.337130	−	0.941458i	$-0.390544\pi$
$314$			35.1019i			0.111789i
$315$		0			0
$316$		−	308.434i		−	0.976057i
$317$	386.251			1.21846			0.609229	−	0.792994i	$-0.291479\pi$
$317$	386.251			1.21846			0.609229	+	0.792994i	$0.291479\pi$
$318$		0			0
$319$			241.291i			0.756397i
$320$			17.8885i			0.0559017i
$321$		0			0
$322$	201.934	+	189.447i	0.627124	+	0.588346i
$323$	183.477			0.568039
$324$		0			0
$325$	13.2038			0.0406271
$326$	154.098			0.472693
$327$		0			0
$328$	120.409			0.367101
$329$		−	717.655i		−	2.18132i
$330$		0			0
$331$	220.712			0.666803			0.333402	−	0.942785i	$-0.391804\pi$
$331$	220.712			0.666803			0.333402	+	0.942785i	$0.391804\pi$
$332$			283.284i			0.853266i
$333$		0			0
$334$	−102.019			−0.305446
$335$	−98.5345			−0.294133
$336$		0			0
$337$			193.563i			0.574371i	0.957875	+	0.287185i	$0.0927196\pi$
$337$			193.563i			0.574371i	−0.957875	+	0.287185i	$0.907280\pi$
$338$	−229.140			−0.677929
$339$		0			0
$340$	33.8119			0.0994467
$341$		−	428.071i		−	1.25534i
$342$		0			0
$343$			217.373i			0.633739i
$344$			58.1866i			0.169147i
$345$		0			0
$346$	212.241			0.613413
$347$	262.429			0.756278			0.378139	−	0.925749i	$-0.376564\pi$
$347$	262.429			0.756278			0.378139	+	0.925749i	$0.376564\pi$
$348$		0			0
$349$	229.831			0.658543			0.329271	−	0.944235i	$-0.393197\pi$
$349$	229.831			0.658543			0.329271	+	0.944235i	$0.393197\pi$
$350$		−	60.1933i		−	0.171981i
$351$		0			0
$352$			42.8536i			0.121743i
$353$	−367.482			−1.04103			−0.520513	−	0.853854i	$-0.674259\pi$
$353$	−367.482			−1.04103			−0.520513	+	0.853854i	$0.674259\pi$
$354$		0			0
$355$		−	19.8249i		−	0.0558448i
$356$			127.450i			0.358007i
$357$		0			0
$358$	−292.824			−0.817945
$359$		−	239.971i		−	0.668442i	−0.942495	−	0.334221i	$-0.891527\pi$
$359$		−	239.971i		−	0.668442i	0.942495	−	0.334221i	$-0.108473\pi$
$360$		0			0
$361$	−227.915			−0.631344
$362$			468.300i			1.29365i
$363$		0			0
$364$		−	44.9595i		−	0.123515i
$365$		−	195.466i		−	0.535523i
$366$		0			0
$367$			389.113i			1.06025i	0.847918	+	0.530127i	$0.177856\pi$
$367$			389.113i			1.06025i	−0.847918	+	0.530127i	$0.822144\pi$
$368$	62.9463	−	67.0952i	0.171050	−	0.182324i
$369$		0			0
$370$	−126.294			−0.341336
$371$	101.461			0.273480
$372$		0			0
$373$		−	546.309i		−	1.46463i	−0.680964	−	0.732317i	$-0.738439\pi$
$373$		−	546.309i		−	1.46463i	0.680964	−	0.732317i	$-0.261561\pi$
$374$	80.9995			0.216576
$375$		0			0
$376$	−238.450			−0.634176
$377$	−84.1117			−0.223108
$378$		0			0
$379$		−	295.837i		−	0.780573i	−0.920693	−	0.390287i	$-0.872376\pi$
$379$		−	295.837i		−	0.780573i	0.920693	−	0.390287i	$-0.127624\pi$
$380$	−108.528			−0.285600
$381$		0			0
$382$		−	244.304i		−	0.639541i
$383$		−	566.033i		−	1.47789i	−0.673765	−	0.738946i	$-0.735324\pi$
$383$		−	566.033i		−	1.47789i	0.673765	−	0.738946i	$-0.264676\pi$
$384$		0			0
$385$		−	144.199i		−	0.374542i
$386$	192.970			0.499921
$387$		0			0
$388$			286.645i			0.738775i
$389$		−	196.043i		−	0.503966i	−0.967732	−	0.251983i	$-0.918917\pi$
$389$		−	196.043i		−	0.503966i	0.967732	−	0.251983i	$-0.0810826\pi$
$390$		0			0
$391$	−126.819	−	118.978i	−0.324346	−	0.304290i
$392$	−66.3681			−0.169306
$393$		0			0
$394$	383.380			0.973045
$395$	344.840			0.873012
$396$		0			0
$397$	−298.788			−0.752616			−0.376308	−	0.926495i	$-0.622806\pi$
$397$	−298.788			−0.752616			−0.376308	+	0.926495i	$0.622806\pi$
$398$			363.866i			0.914236i
$399$		0			0
$400$	−20.0000			−0.0500000
$401$			785.114i			1.95789i	0.204121	+	0.978946i	$0.434567\pi$
$401$			785.114i			1.95789i	−0.204121	+	0.978946i	$0.565433\pi$
$402$		0			0
$403$	149.222			0.370277
$404$	−55.4204			−0.137179
$405$		0			0
$406$			383.447i			0.944452i
$407$	−302.550			−0.743365
$408$		0			0
$409$	358.186			0.875761			0.437881	−	0.899033i	$-0.355729\pi$
$409$	358.186			0.875761			0.437881	+	0.899033i	$0.355729\pi$
$410$			134.621i			0.328345i
$411$		0			0
$412$		−	267.084i		−	0.648261i
$413$		−	575.930i		−	1.39450i
$414$		0			0
$415$	−316.722			−0.763184
$416$	−14.9384			−0.0359096
$417$		0			0
$418$	−259.988			−0.621982
$419$		−	195.271i		−	0.466042i	−0.972472	−	0.233021i	$-0.925139\pi$
$419$		−	195.271i		−	0.466042i	0.972472	−	0.233021i	$-0.0748610\pi$
$420$		0			0
$421$			768.042i			1.82433i	0.409826	+	0.912164i	$0.365590\pi$
$421$			768.042i			1.82433i	−0.409826	+	0.912164i	$0.634410\pi$
$422$	76.6425			0.181617
$423$		0			0
$424$		−	33.7118i		−	0.0795089i
$425$			37.8028i			0.0889479i
$426$		0			0
$427$	299.321			0.700987
$428$		−	100.618i		−	0.235089i
$429$		0			0
$430$	−65.0546			−0.151290
$431$			333.122i			0.772905i	0.922309	+	0.386453i	$0.126300\pi$
$431$			333.122i			0.772905i	−0.922309	+	0.386453i	$0.873700\pi$
$432$		0			0
$433$			7.49741i			0.0173150i	0.999963	+	0.00865752i	$0.00275581\pi$
$433$			7.49741i			0.0173150i	−0.999963	+	0.00865752i	$0.997244\pi$
$434$		−	680.270i		−	1.56744i
$435$		0			0
$436$			257.637i			0.590911i
$437$	407.059	+	381.889i	0.931485	+	0.873887i
$438$		0			0
$439$	−303.939			−0.692344			−0.346172	−	0.938171i	$-0.612519\pi$
$439$	−303.939			−0.692344			−0.346172	+	0.938171i	$0.612519\pi$
$440$	−47.9118			−0.108891
$441$		0			0
$442$			28.2357i			0.0638816i
$443$	−637.145			−1.43825			−0.719125	−	0.694881i	$-0.755457\pi$
$443$	−637.145			−1.43825			−0.719125	+	0.694881i	$0.755457\pi$
$444$		0			0
$445$	−142.494			−0.320211
$446$	147.942			0.331708
$447$		0			0
$448$			68.1009i			0.152011i
$449$	−88.9331			−0.198069			−0.0990346	−	0.995084i	$-0.531575\pi$
$449$	−88.9331			−0.198069			−0.0990346	+	0.995084i	$0.531575\pi$
$450$		0			0
$451$			322.498i			0.715073i
$452$		−	172.606i		−	0.381871i
$453$		0			0
$454$		−	283.527i		−	0.624509i
$455$	50.2663			0.110475
$456$		0			0
$457$		−	377.669i		−	0.826410i	−0.910638	−	0.413205i	$-0.864409\pi$
$457$		−	377.669i		−	0.826410i	0.910638	−	0.413205i	$-0.135591\pi$
$458$			31.2142i			0.0681534i
$459$		0			0
$460$	75.0147	+	70.3761i	0.163075	+	0.152992i
$461$	585.070			1.26913			0.634566	−	0.772869i	$-0.281179\pi$
$461$	585.070			1.26913			0.634566	+	0.772869i	$0.281179\pi$
$462$		0			0
$463$	−225.877			−0.487854			−0.243927	−	0.969794i	$-0.578436\pi$
$463$	−225.877			−0.487854			−0.243927	+	0.969794i	$0.578436\pi$
$464$	127.405			0.274580
$465$		0			0
$466$	478.897			1.02768
$467$		−	702.984i		−	1.50532i	−0.658410	−	0.752659i	$-0.728771\pi$
$467$		−	702.984i		−	1.50532i	0.658410	−	0.752659i	$-0.271229\pi$
$468$		0			0
$469$	−375.117			−0.799822
$470$		−	266.595i		−	0.567224i
$471$		0			0
$472$	−191.360			−0.405424
$473$	−155.844			−0.329480
$474$		0			0
$475$		−	121.338i		−	0.255448i
$476$	128.720			0.270421
$477$		0			0
$478$	−211.247			−0.441940
$479$		−	291.706i		−	0.608989i	−0.952514	−	0.304494i	$-0.901512\pi$
$479$		−	291.706i		−	0.608989i	0.952514	−	0.304494i	$-0.0984875\pi$
$480$		0			0
$481$		−	105.466i		−	0.219264i
$482$			188.132i			0.390316i
$483$		0			0
$484$	127.223			0.262857
$485$	−320.479			−0.660781
$486$		0			0
$487$	205.957			0.422909			0.211454	−	0.977388i	$-0.432180\pi$
$487$	205.957			0.422909			0.211454	+	0.977388i	$0.432180\pi$
$488$		−	99.4534i		−	0.203798i
$489$		0			0
$490$		−	74.2018i		−	0.151432i
$491$	192.478			0.392012			0.196006	−	0.980603i	$-0.437203\pi$
$491$	192.478			0.392012			0.196006	+	0.980603i	$0.437203\pi$
$492$		0			0
$493$		−	240.814i		−	0.488467i
$494$		−	90.6296i		−	0.183461i
$495$		0			0
$496$	−226.028			−0.455703
$497$		−	75.4726i		−	0.151856i
$498$		0			0
$499$	887.386			1.77833			0.889164	−	0.457589i	$-0.151287\pi$
$499$	887.386			1.77833			0.889164	+	0.457589i	$0.151287\pi$
$500$		−	22.3607i		−	0.0447214i
$501$		0			0
$502$			533.045i			1.06184i
$503$		−	330.810i		−	0.657673i	−0.944387	−	0.328837i	$-0.893343\pi$
$503$		−	330.810i		−	0.657673i	0.944387	−	0.328837i	$-0.106657\pi$
$504$		0			0
$505$		−	61.9619i		−	0.122697i
$506$	179.704	+	168.592i	0.355147	+	0.333187i
$507$		0			0
$508$	−22.0270			−0.0433603
$509$	407.928			0.801430			0.400715	−	0.916203i	$-0.368762\pi$
$509$	407.928			0.801430			0.400715	+	0.916203i	$0.368762\pi$
$510$		0			0
$511$		−	744.131i		−	1.45622i
$512$	22.6274			0.0441942
$513$		0			0
$514$	−319.733			−0.622048
$515$	298.609			0.579822
$516$		0			0
$517$		−	638.654i		−	1.23531i
$518$	−480.797			−0.928180
$519$		0			0
$520$		−	16.7016i		−	0.0321185i
$521$		−	458.780i		−	0.880576i	−0.897857	−	0.440288i	$-0.854876\pi$
$521$		−	458.780i		−	0.880576i	0.897857	−	0.440288i	$-0.145124\pi$
$522$		0			0
$523$			404.987i			0.774354i	0.922005	+	0.387177i	$0.126550\pi$
$523$			404.987i			0.774354i	−0.922005	+	0.387177i	$0.873450\pi$
$524$	−7.27882			−0.0138909
$525$		0			0
$526$			3.44579i			0.00655093i
$527$			427.226i			0.810675i
$528$		0			0
$529$	−33.7199	−	527.924i	−0.0637427	−	0.997966i
$530$	37.6909			0.0711149
$531$		0			0
$532$	−413.161			−0.776619
$533$	−112.420			−0.210919
$534$		0			0
$535$	112.495			0.210270
$536$			124.637i			0.232532i
$537$		0			0
$538$	440.298			0.818397
$539$		−	177.757i		−	0.329791i
$540$		0			0
$541$	1047.51			1.93625			0.968123	−	0.250477i	$-0.0805875\pi$
$541$	1047.51			1.93625			0.968123	+	0.250477i	$0.0805875\pi$
$542$	368.758			0.680366
$543$		0			0
$544$		−	42.7690i		−	0.0786195i
$545$	−288.047			−0.528527
$546$		0			0
$547$	123.784			0.226296			0.113148	−	0.993578i	$-0.463907\pi$
$547$	123.784			0.226296			0.113148	+	0.993578i	$0.463907\pi$
$548$			19.3957i			0.0353936i
$549$		0			0
$550$		−	53.5671i		−	0.0973946i
$551$			772.955i			1.40282i
$552$		0			0
$553$	1312.79			2.37394
$554$	−257.765			−0.465279
$555$		0			0
$556$	16.1097			0.0289743
$557$			246.292i			0.442176i	0.975254	+	0.221088i	$0.0709608\pi$
$557$			246.292i			0.442176i	−0.975254	+	0.221088i	$0.929039\pi$
$558$		0			0
$559$		−	54.3259i		−	0.0971840i
$560$	−76.1392			−0.135963
$561$		0			0
$562$		−	6.51248i		−	0.0115880i
$563$			574.776i			1.02092i	0.859902	+	0.510458i	$0.170524\pi$
$563$			574.776i			1.02092i	−0.859902	+	0.510458i	$0.829476\pi$
$564$		0			0
$565$	192.979			0.341556
$566$			470.787i			0.831780i
$567$		0			0
$568$	−25.0768			−0.0441492
$569$			794.332i			1.39601i	0.716091	+	0.698007i	$0.245930\pi$
$569$			794.332i			1.39601i	−0.716091	+	0.698007i	$0.754070\pi$
$570$		0			0
$571$			416.688i			0.729751i	0.931056	+	0.364876i	$0.118889\pi$
$571$			416.688i			0.729751i	−0.931056	+	0.364876i	$0.881111\pi$
$572$		−	40.0103i		−	0.0699480i
$573$		0			0
$574$			512.498i			0.892854i
$575$	−78.6829	+	83.8689i	−0.136840	+	0.145859i
$576$		0			0
$577$	−282.647			−0.489856			−0.244928	−	0.969541i	$-0.578764\pi$
$577$	−282.647			−0.489856			−0.244928	+	0.969541i	$0.578764\pi$
$578$	327.868			0.567246
$579$		0			0
$580$			142.443i			0.245592i
$581$	−1205.75			−2.07529
$582$		0			0
$583$	90.2920			0.154875
$584$	−247.247			−0.423368
$585$		0			0
$586$		−	475.337i		−	0.811156i
$587$	168.123			0.286410			0.143205	−	0.989693i	$-0.454259\pi$
$587$	168.123			0.286410			0.143205	+	0.989693i	$0.454259\pi$
$588$		0			0
$589$		−	1371.29i		−	2.32817i
$590$		−	213.947i		−	0.362623i
$591$		0			0
$592$			159.751i			0.269850i
$593$	419.364			0.707191			0.353595	−	0.935398i	$-0.384959\pi$
$593$	419.364			0.707191			0.353595	+	0.935398i	$0.384959\pi$
$594$		0			0
$595$			143.914i			0.241872i
$596$			288.124i			0.483430i
$597$		0			0
$598$	−58.7698	+	62.6433i	−0.0982773	+	0.104755i
$599$	519.774			0.867736			0.433868	−	0.900977i	$-0.357148\pi$
$599$	519.774			0.867736			0.433868	+	0.900977i	$0.357148\pi$
$600$		0			0
$601$	1089.19			1.81230			0.906151	−	0.422955i	$-0.139007\pi$
$601$	1089.19			1.81230			0.906151	+	0.422955i	$0.139007\pi$
$602$	−247.660			−0.411395
$603$		0			0
$604$	−219.913			−0.364094
$605$			142.239i			0.235107i
$606$		0			0
$607$	890.673			1.46734			0.733668	−	0.679508i	$-0.237807\pi$
$607$	890.673			1.46734			0.733668	+	0.679508i	$0.237807\pi$
$608$			137.278i			0.225786i
$609$		0			0
$610$	111.192			0.182282
$611$	222.629			0.364368
$612$		0			0
$613$			629.354i			1.02668i	0.858186	+	0.513339i	$0.171592\pi$
$613$			629.354i			1.02668i	−0.858186	+	0.513339i	$0.828408\pi$
$614$	−647.616			−1.05475
$615$		0			0
$616$	−182.398			−0.296101
$617$			157.187i			0.254760i	0.991854	+	0.127380i	$0.0406568\pi$
$617$			157.187i			0.254760i	−0.991854	+	0.127380i	$0.959343\pi$
$618$		0			0
$619$		−	462.293i		−	0.746838i	−0.927663	−	0.373419i	$-0.878185\pi$
$619$		−	462.293i		−	0.746838i	0.927663	−	0.373419i	$-0.121815\pi$
$620$		−	252.707i		−	0.407593i
$621$		0			0
$622$	644.344			1.03592
$623$	−542.468			−0.870735
$624$		0			0
$625$	25.0000			0.0400000
$626$		−	833.471i		−	1.33142i
$627$		0			0
$628$			49.6416i			0.0790471i
$629$	301.952			0.480051
$630$		0			0
$631$		−	411.630i		−	0.652345i	−0.945310	−	0.326173i	$-0.894241\pi$
$631$		−	411.630i		−	0.652345i	0.945310	−	0.326173i	$-0.105759\pi$
$632$		−	436.191i		−	0.690176i
$633$		0			0
$634$	546.242			0.861581
$635$		−	24.6270i		−	0.0387827i
$636$		0			0
$637$	61.9645			0.0972756
$638$			341.236i			0.534853i
$639$		0			0
$640$			25.2982i			0.0395285i
$641$		−	251.089i		−	0.391715i	−0.980632	−	0.195857i	$-0.937251\pi$
$641$		−	251.089i		−	0.391715i	0.980632	−	0.195857i	$-0.0627490\pi$
$642$		0			0
$643$		−	102.907i		−	0.160042i	−0.996793	−	0.0800211i	$-0.974501\pi$
$643$		−	102.907i		−	0.160042i	0.996793	−	0.0800211i	$-0.0254988\pi$
$644$	285.578	+	267.919i	0.443444	+	0.416023i
$645$		0			0
$646$	259.475			0.401664
$647$	437.368			0.675993			0.337997	−	0.941147i	$-0.390251\pi$
$647$	437.368			0.675993			0.337997	+	0.941147i	$0.390251\pi$
$648$		0			0
$649$		−	512.530i		−	0.789723i
$650$	18.6730			0.0287277
$651$		0			0
$652$	217.927			0.334244
$653$	94.4001			0.144564			0.0722819	−	0.997384i	$-0.476972\pi$
$653$	94.4001			0.144564			0.0722819	+	0.997384i	$0.476972\pi$
$654$		0			0
$655$		−	8.13797i		−	0.0124244i
$656$	170.284			0.259580
$657$		0			0
$658$		−	1014.92i		−	1.54243i
$659$		−	759.727i		−	1.15285i	−0.817151	−	0.576424i	$-0.804448\pi$
$659$		−	759.727i		−	1.15285i	0.817151	−	0.576424i	$-0.195552\pi$
$660$		0			0
$661$			581.754i			0.880112i	0.897970	+	0.440056i	$0.145041\pi$
$661$			581.754i			0.880112i	−0.897970	+	0.440056i	$0.854959\pi$
$662$	312.134			0.471501
$663$		0			0
$664$			400.625i			0.603350i
$665$		−	461.928i		−	0.694629i
$666$		0			0
$667$	501.231	−	534.267i	0.751471	−	0.801001i
$668$	−144.277			−0.215983
$669$		0			0
$670$	−139.349			−0.207983
$671$	266.371			0.396977
$672$		0			0
$673$	−831.173			−1.23503			−0.617513	−	0.786560i	$-0.711860\pi$
$673$	−831.173			−1.23503			−0.617513	+	0.786560i	$0.711860\pi$
$674$			273.739i			0.406142i
$675$		0			0
$676$	−324.053			−0.479368
$677$		−	106.706i		−	0.157616i	−0.996890	−	0.0788079i	$-0.974889\pi$
$677$		−	106.706i		−	0.157616i	0.996890	−	0.0788079i	$-0.0251114\pi$
$678$		0			0
$679$	−1220.05			−1.79683
$680$	47.8172			0.0703195
$681$		0			0
$682$		−	605.384i		−	0.887660i
$683$	−380.675			−0.557357			−0.278678	−	0.960385i	$-0.589896\pi$
$683$	−380.675			−0.557357			−0.278678	+	0.960385i	$0.589896\pi$
$684$		0			0
$685$	−21.6851			−0.0316570
$686$			307.411i			0.448121i
$687$		0			0
$688$			82.2883i			0.119605i
$689$			31.4750i			0.0456821i
$690$		0			0
$691$	−318.845			−0.461425			−0.230713	−	0.973022i	$-0.574106\pi$
$691$	−318.845			−0.461425			−0.230713	+	0.973022i	$0.574106\pi$
$692$	300.154			0.433748
$693$		0			0
$694$	371.130			0.534769
$695$			18.0112i			0.0259154i
$696$		0			0
$697$		−	321.861i		−	0.461781i
$698$	325.031			0.465660
$699$		0			0
$700$		−	85.1262i		−	0.121609i
$701$		−	112.754i		−	0.160848i	−0.996761	−	0.0804239i	$-0.974373\pi$
$701$		−	112.754i		−	0.160848i	0.996761	−	0.0804239i	$-0.0256274\pi$
$702$		0			0
$703$	−969.193			−1.37865
$704$			60.6042i			0.0860855i
$705$		0			0
$706$	−519.698			−0.736116
$707$		−	235.887i		−	0.333644i
$708$		0			0
$709$		−	465.481i		−	0.656531i	−0.944585	−	0.328266i	$-0.893536\pi$
$709$		−	465.481i		−	0.656531i	0.944585	−	0.328266i	$-0.106464\pi$
$710$		−	28.0367i		−	0.0394883i
$711$		0			0
$712$			180.242i			0.253149i
$713$	−889.229	+	947.838i	−1.24717	+	1.32937i
$714$		0			0
$715$	44.7328			0.0625634
$716$	−414.116			−0.578375
$717$		0			0
$718$		−	339.370i		−	0.472660i
$719$	347.913			0.483885			0.241943	−	0.970291i	$-0.422215\pi$
$719$	347.913			0.483885			0.241943	+	0.970291i	$0.422215\pi$
$720$		0			0
$721$	1136.79			1.57669
$722$	−322.321			−0.446428
$723$		0			0
$724$			662.276i			0.914746i
$725$	−159.257			−0.219664
$726$		0			0
$727$			88.3077i			0.121469i	0.998154	+	0.0607343i	$0.0193442\pi$
$727$			88.3077i			0.121469i	−0.998154	+	0.0607343i	$0.980656\pi$
$728$		−	63.5824i		−	0.0873385i
$729$		0			0
$730$		−	276.431i		−	0.378672i
$731$	155.536			0.212772
$732$		0			0
$733$		−	510.848i		−	0.696928i	−0.937322	−	0.348464i	$-0.886703\pi$
$733$		−	510.848i		−	0.696928i	0.937322	−	0.348464i	$-0.113297\pi$
$734$			550.289i			0.749712i
$735$		0			0
$736$	89.0196	−	94.8869i	0.120950	−	0.128922i
$737$	−333.823			−0.452948
$738$		0			0
$739$	−416.763			−0.563955			−0.281978	−	0.959421i	$-0.590990\pi$
$739$	−416.763			−0.563955			−0.281978	+	0.959421i	$0.590990\pi$
$740$	−178.607			−0.241361
$741$		0			0
$742$	143.488			0.193380
$743$		−	633.469i		−	0.852583i	−0.904586	−	0.426291i	$-0.859820\pi$
$743$		−	633.469i		−	0.852583i	0.904586	−	0.426291i	$-0.140180\pi$
$744$		0			0
$745$	−322.132			−0.432393
$746$		−	772.597i		−	1.03565i
$747$		0			0
$748$	114.551			0.153142
$749$	428.262			0.571779
$750$		0			0
$751$			1282.52i			1.70775i	0.520475	+	0.853877i	$0.325755\pi$
$751$			1282.52i			1.70775i	−0.520475	+	0.853877i	$0.674245\pi$
$752$	−337.219			−0.448430
$753$		0			0
$754$	−118.952			−0.157761
$755$		−	245.870i		−	0.325655i
$756$		0			0
$757$			1233.64i			1.62964i	0.579716	+	0.814819i	$0.303164\pi$
$757$			1233.64i			1.62964i	−0.579716	+	0.814819i	$0.696836\pi$
$758$		−	418.377i		−	0.551949i
$759$		0			0
$760$	−153.482			−0.201950
$761$	262.559			0.345018			0.172509	−	0.985008i	$-0.444813\pi$
$761$	262.559			0.345018			0.172509	+	0.985008i	$0.444813\pi$
$762$		0			0
$763$	−1096.58			−1.43720
$764$		−	345.499i		−	0.452223i
$765$		0			0
$766$		−	800.491i		−	1.04503i
$767$	178.663			0.232938
$768$		0			0
$769$		−	801.510i		−	1.04228i	−0.853473	−	0.521138i	$-0.825508\pi$
$769$		−	801.510i		−	1.04228i	0.853473	−	0.521138i	$-0.174492\pi$
$770$		−	203.928i		−	0.264841i
$771$		0			0
$772$	272.900			0.353498
$773$		−	1143.79i		−	1.47967i	−0.672786	−	0.739837i	$-0.734903\pi$
$773$		−	1143.79i		−	1.47967i	0.672786	−	0.739837i	$-0.265097\pi$
$774$		0			0
$775$	282.536			0.364562
$776$			405.377i			0.522393i
$777$		0			0
$778$		−	277.246i		−	0.356357i
$779$			1033.10i			1.32618i
$780$		0			0
$781$		−	67.1644i		−	0.0859979i
$782$	−179.350	−	168.260i	−0.229347	−	0.215166i
$783$		0			0
$784$	−93.8587			−0.119718
$785$	−55.5010			−0.0707019
$786$		0			0
$787$			111.467i			0.141635i	0.997489	+	0.0708175i	$0.0225608\pi$
$787$			111.467i			0.141635i	−0.997489	+	0.0708175i	$0.977439\pi$
$788$	542.181			0.688046
$789$		0			0
$790$	487.677			0.617312
$791$	734.663			0.928777
$792$		0			0
$793$			92.8546i			0.117093i
$794$	−422.551			−0.532180
$795$		0			0
$796$			514.584i			0.646462i
$797$		−	231.014i		−	0.289855i	−0.989442	−	0.144927i	$-0.953705\pi$
$797$		−	231.014i		−	0.289855i	0.989442	−	0.144927i	$-0.0462949\pi$
$798$		0			0
$799$			637.393i			0.797738i
$800$	−28.2843			−0.0353553
$801$		0			0
$802$			1110.32i			1.38444i
$803$		−	662.215i		−	0.824676i
$804$		0			0
$805$	−299.543	+	319.286i	−0.372103	+	0.396628i
$806$	211.031			0.261825
$807$		0			0
$808$	−78.3763			−0.0970004
$809$	165.201			0.204204			0.102102	−	0.994774i	$-0.467443\pi$
$809$	165.201			0.204204			0.102102	+	0.994774i	$0.467443\pi$
$810$		0			0
$811$	317.322			0.391273			0.195636	−	0.980677i	$-0.437323\pi$
$811$	317.322			0.391273			0.195636	+	0.980677i	$0.437323\pi$
$812$			542.276i			0.667828i
$813$		0			0
$814$	−427.870			−0.525638
$815$			243.650i			0.298957i
$816$		0			0
$817$	−499.234			−0.611058
$818$	506.552			0.619257
$819$		0			0
$820$			190.384i			0.232175i
$821$	1389.82			1.69284			0.846421	−	0.532514i	$-0.178753\pi$
$821$	1389.82			1.69284			0.846421	+	0.532514i	$0.178753\pi$
$822$		0			0
$823$	948.001			1.15188			0.575942	−	0.817490i	$-0.304635\pi$
$823$	948.001			1.15188			0.575942	+	0.817490i	$0.304635\pi$
$824$		−	377.713i		−	0.458390i
$825$		0			0
$826$		−	814.489i		−	0.986064i
$827$		−	1177.54i		−	1.42387i	−0.702245	−	0.711935i	$-0.747819\pi$
$827$		−	1177.54i		−	1.42387i	0.702245	−	0.711935i	$-0.252181\pi$
$828$		0			0
$829$	−1278.78			−1.54256			−0.771279	−	0.636497i	$-0.780383\pi$
$829$	−1278.78			−1.54256			−0.771279	+	0.636497i	$0.780383\pi$
$830$	−447.912			−0.539653
$831$		0			0
$832$	−21.1261			−0.0253919
$833$			177.406i			0.212973i
$834$		0			0
$835$		−	161.306i		−	0.193181i
$836$	−367.679			−0.439808
$837$		0			0
$838$		−	276.156i		−	0.329541i
$839$			1435.77i			1.71128i	0.517570	+	0.855641i	$0.326837\pi$
$839$			1435.77i			1.71128i	−0.517570	+	0.855641i	$0.673163\pi$
$840$		0			0
$841$	173.507			0.206311
$842$			1086.18i			1.28999i
$843$		0			0
$844$	108.389			0.128423
$845$		−	362.302i		−	0.428760i
$846$		0			0
$847$			541.500i			0.639315i
$848$		−	47.6757i		−	0.0562213i
$849$		0			0
$850$			53.4613i			0.0628956i
$851$	669.908	+	628.484i	0.787201	+	0.738524i
$852$		0			0
$853$	812.996			0.953102			0.476551	−	0.879147i	$-0.341887\pi$
$853$	812.996			0.953102			0.476551	+	0.879147i	$0.341887\pi$
$854$	423.304			0.495673
$855$		0			0
$856$		−	142.296i		−	0.166233i
$857$	958.841			1.11883			0.559417	−	0.828886i	$-0.311025\pi$
$857$	958.841			1.11883			0.559417	+	0.828886i	$0.311025\pi$
$858$		0			0
$859$	−383.654			−0.446628			−0.223314	−	0.974747i	$-0.571688\pi$
$859$	−383.654			−0.446628			−0.223314	+	0.974747i	$0.571688\pi$
$860$	−92.0011			−0.106978
$861$		0			0
$862$			471.106i			0.546526i
$863$	−1017.48			−1.17900			−0.589502	−	0.807767i	$-0.700676\pi$
$863$	−1017.48			−1.17900			−0.589502	+	0.807767i	$0.700676\pi$
$864$		0			0
$865$			335.582i			0.387956i
$866$			10.6029i			0.0122436i
$867$		0			0
$868$		−	962.047i		−	1.10835i
$869$	1168.27			1.34439
$870$		0			0
$871$		−	116.368i		−	0.133602i
$872$			364.354i			0.417837i
$873$		0			0
$874$	575.669	+	540.072i	0.658660	+	0.617932i
$875$	95.1740			0.108770
$876$		0			0
$877$	1443.83			1.64633			0.823163	−	0.567806i	$-0.192207\pi$
$877$	1443.83			1.64633			0.823163	+	0.567806i	$0.192207\pi$
$878$	−429.835			−0.489561
$879$		0			0
$880$	−67.7576			−0.0769972
$881$			894.988i			1.01588i	0.861393	+	0.507939i	$0.169592\pi$
$881$			894.988i			1.01588i	−0.861393	+	0.507939i	$0.830408\pi$
$882$		0			0
$883$	−870.115			−0.985407			−0.492704	−	0.870197i	$-0.663991\pi$
$883$	−870.115			−0.985407			−0.492704	+	0.870197i	$0.663991\pi$
$884$			39.9313i			0.0451711i
$885$		0			0
$886$	−901.058			−1.01700
$887$	529.330			0.596764			0.298382	−	0.954446i	$-0.403553\pi$
$887$	529.330			0.596764			0.298382	+	0.954446i	$0.403553\pi$
$888$		0			0
$889$		−	93.7539i		−	0.105460i
$890$	−201.517			−0.226423
$891$		0			0
$892$	209.221			0.234553
$893$		−	2045.87i		−	2.29101i
$894$		0			0
$895$		−	462.996i		−	0.517314i
$896$			96.3093i			0.107488i
$897$		0			0
$898$	−125.770			−0.140056
$899$	−1799.83			−2.00203
$900$		0			0
$901$	−90.1138			−0.100015
$902$			456.081i			0.505633i
$903$		0			0
$904$		−	244.101i		−	0.270023i
$905$	−740.447			−0.818173
$906$		0			0
$907$			691.482i			0.762384i	0.924496	+	0.381192i	$0.124486\pi$
$907$			691.482i			0.762384i	−0.924496	+	0.381192i	$0.875514\pi$
$908$		−	400.968i		−	0.441595i
$909$		0			0
$910$	71.0873			0.0781179
$911$			1794.49i			1.96980i	0.173115	+	0.984902i	$0.444617\pi$
$911$			1794.49i			1.96980i	−0.173115	+	0.984902i	$0.555383\pi$
$912$		0			0
$913$	−1073.01			−1.17526
$914$		−	534.105i		−	0.584360i
$915$		0			0
$916$			44.1436i			0.0481917i
$917$		−	30.9809i		−	0.0337851i
$918$		0			0
$919$			428.705i			0.466491i	0.972418	+	0.233245i	$0.0749345\pi$
$919$			428.705i			0.466491i	−0.972418	+	0.233245i	$0.925065\pi$
$920$	106.087	+	99.5269i	0.115312	+	0.108181i
$921$		0			0
$922$	827.413			0.897411
$923$	23.4129			0.0253661
$924$		0			0
$925$		−	199.689i		−	0.215880i
$926$	−319.438			−0.344965
$927$		0			0
$928$	180.178			0.194158
$929$	−1577.01			−1.69754			−0.848768	−	0.528766i	$-0.822655\pi$
$929$	−1577.01			−1.69754			−0.848768	+	0.528766i	$0.822655\pi$
$930$		0			0
$931$		−	569.431i		−	0.611633i
$932$	677.263			0.726677
$933$		0			0
$934$		−	994.169i		−	1.06442i
$935$			128.071i			0.136975i
$936$		0			0
$937$			141.317i			0.150818i	0.997153	+	0.0754091i	$0.0240263\pi$
$937$			141.317i			0.150818i	−0.997153	+	0.0754091i	$0.975974\pi$
$938$	−530.495			−0.565560
$939$		0			0
$940$		−	377.023i		−	0.401088i
$941$		−	1506.21i		−	1.60065i	−0.599566	−	0.800325i	$-0.704660\pi$
$941$		−	1506.21i		−	1.60065i	0.599566	−	0.800325i	$-0.295340\pi$
$942$		0			0
$943$	669.923	−	714.078i	0.710417	−	0.757240i
$944$	−270.624			−0.286678
$945$		0			0
$946$	−220.397			−0.232978
$947$	−1689.84			−1.78441			−0.892205	−	0.451632i	$-0.850842\pi$
$947$	−1689.84			−1.78441			−0.892205	+	0.451632i	$0.850842\pi$
$948$		0			0
$949$	230.842			0.243248
$950$		−	171.598i		−	0.180629i
$951$		0			0
$952$	182.038			0.191217
$953$		−	990.482i		−	1.03933i	−0.854370	−	0.519665i	$-0.826057\pi$
$953$		−	990.482i		−	1.03933i	0.854370	−	0.519665i	$-0.173943\pi$
$954$		0			0
$955$	386.279			0.404481
$956$	−298.749			−0.312499
$957$		0			0
$958$		−	412.534i		−	0.430620i
$959$	−82.5541			−0.0860836
$960$		0			0
$961$	2232.05			2.32264
$962$		−	149.151i		−	0.155043i
$963$		0			0
$964$			266.059i			0.275995i
$965$			305.112i			0.316178i
$966$		0			0
$967$	1528.09			1.58024			0.790118	−	0.612954i	$-0.210019\pi$
$967$	1528.09			1.58024			0.790118	+	0.612954i	$0.210019\pi$
$968$	179.920			0.185868
$969$		0			0
$970$	−453.225			−0.467243
$971$			616.239i			0.634643i	0.948318	+	0.317322i	$0.102783\pi$
$971$			616.239i			0.634643i	−0.948318	+	0.317322i	$0.897217\pi$
$972$		0			0
$973$			68.5679i			0.0704706i
$974$	291.267			0.299042
$975$		0			0
$976$		−	140.648i		−	0.144107i
$977$			1393.79i			1.42660i	0.700859	+	0.713299i	$0.252800\pi$
$977$			1393.79i			1.42660i	−0.700859	+	0.713299i	$0.747200\pi$
$978$		0			0
$979$	−482.752			−0.493107
$980$		−	104.937i		−	0.107079i
$981$		0			0
$982$	272.205			0.277195
$983$			1033.22i			1.05109i	0.850767	+	0.525543i	$0.176138\pi$
$983$			1033.22i			1.05109i	−0.850767	+	0.525543i	$0.823862\pi$
$984$		0			0
$985$			606.176i			0.615407i
$986$		−	340.563i		−	0.345398i
$987$		0			0
$988$		−	128.170i		−	0.129726i
$989$	345.071	+	323.734i	0.348909	+	0.327335i
$990$		0			0
$991$	−1122.80			−1.13300			−0.566501	−	0.824061i	$-0.691703\pi$
$991$	−1122.80			−1.13300			−0.566501	+	0.824061i	$0.691703\pi$
$992$	−319.652			−0.322230
$993$		0			0
$994$		−	106.734i		−	0.107379i
$995$	−575.323			−0.578214
$996$		0			0
$997$	−718.650			−0.720812			−0.360406	−	0.932795i	$-0.617362\pi$
$997$	−718.650			−0.720812			−0.360406	+	0.932795i	$0.617362\pi$
$998$	1254.95			1.25747
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.3.c.a.91.16		16
3.2	odd	2		230.3.d.a.91.3	✓	16
12.11	even	2		1840.3.k.d.321.9		16
15.2	even	4		1150.3.c.c.1149.25		32
15.8	even	4		1150.3.c.c.1149.8		32
15.14	odd	2		1150.3.d.b.551.13		16
23.22	odd	2	inner	2070.3.c.a.91.9		16
69.68	even	2		230.3.d.a.91.4	yes	16
276.275	odd	2		1840.3.k.d.321.10		16
345.68	odd	4		1150.3.c.c.1149.26		32
345.137	odd	4		1150.3.c.c.1149.7		32
345.344	even	2		1150.3.d.b.551.14		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.d.a.91.3	✓	16	3.2	odd	2
230.3.d.a.91.4	yes	16	69.68	even	2
1150.3.c.c.1149.7		32	345.137	odd	4
1150.3.c.c.1149.8		32	15.8	even	4
1150.3.c.c.1149.25		32	15.2	even	4
1150.3.c.c.1149.26		32	345.68	odd	4
1150.3.d.b.551.13		16	15.14	odd	2
1150.3.d.b.551.14		16	345.344	even	2
1840.3.k.d.321.9		16	12.11	even	2
1840.3.k.d.321.10		16	276.275	odd	2
2070.3.c.a.91.9		16	23.22	odd	2	inner
2070.3.c.a.91.16		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2070.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +8.51262i q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +8.51262i q^{7} +2.82843 q^{8} +3.16228i q^{10} +7.57553i q^{11} -2.64076 q^{13} +12.0387i q^{14} +4.00000 q^{16} -7.56057i q^{17} +24.2676i q^{19} +4.47214i q^{20} +10.7134i q^{22} +(15.7366 - 16.7738i) q^{23} -5.00000 q^{25} -3.73460 q^{26} +17.0252i q^{28} +31.8513 q^{29} -56.5071 q^{31} +5.65685 q^{32} -10.6923i q^{34} -19.0348 q^{35} +39.9378i q^{37} +34.3195i q^{38} +6.32456i q^{40} +42.5710 q^{41} +20.5721i q^{43} +15.1511i q^{44} +(22.2549 - 23.7217i) q^{46} -84.3049 q^{47} -23.4647 q^{49} -7.07107 q^{50} -5.28152 q^{52} -11.9189i q^{53} -16.9394 q^{55} +24.0773i q^{56} +45.0446 q^{58} -67.6561 q^{59} -35.1621i q^{61} -79.9131 q^{62} +8.00000 q^{64} -5.90492i q^{65} +44.0660i q^{67} -15.1211i q^{68} -26.9193 q^{70} -8.86597 q^{71} -87.4150 q^{73} +56.4805i q^{74} +48.5352i q^{76} -64.4876 q^{77} -154.217i q^{79} +8.94427i q^{80} +60.2046 q^{82} +141.642i q^{83} +16.9059 q^{85} +29.0933i q^{86} +21.4268i q^{88} +63.7252i q^{89} -22.4798i q^{91} +(31.4732 - 33.5476i) q^{92} -119.225 q^{94} -54.2639 q^{95} +143.322i q^{97} -33.1841 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4}+O(q^{10}) \) 16 * q + 32 * q^4	\( 16 q + 32 q^{4} + 24 q^{13} + 64 q^{16} - 4 q^{23} - 80 q^{25} - 96 q^{26} + 108 q^{29} - 116 q^{31} - 60 q^{35} + 156 q^{41} - 124 q^{46} + 128 q^{47} - 28 q^{49} + 48 q^{52} + 160 q^{58} - 204 q^{59} - 64 q^{62} + 128 q^{64} - 120 q^{70} - 236 q^{71} - 112 q^{73} + 936 q^{77} - 64 q^{82} + 60 q^{85} - 8 q^{92} - 216 q^{94} + 160 q^{95} - 256 q^{98}+O(q^{100}) \) 16 * q + 32 * q^4 + 24 * q^13 + 64 * q^16 - 4 * q^23 - 80 * q^25 - 96 * q^26 + 108 * q^29 - 116 * q^31 - 60 * q^35 + 156 * q^41 - 124 * q^46 + 128 * q^47 - 28 * q^49 + 48 * q^52 + 160 * q^58 - 204 * q^59 - 64 * q^62 + 128 * q^64 - 120 * q^70 - 236 * q^71 - 112 * q^73 + 936 * q^77 - 64 * q^82 + 60 * q^85 - 8 * q^92 - 216 * q^94 + 160 * q^95 - 256 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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