LMFDB - Embedded newform 2070.3.c.b.91.5

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:	$32$
Twist minimal:	no (minimal twist has level 690)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.5
Character	$\chi$	$=$	2070.91
Dual form			2070.3.c.b.91.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} +6.64524i q^{7} -2.82843 q^{8} +O(q^{10})$	\(q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} +6.64524i q^{7} -2.82843 q^{8} +3.16228i q^{10} +9.54001i q^{11} +10.3241 q^{13} -9.39779i q^{14} +4.00000 q^{16} -15.5981i q^{17} -0.664113i q^{19} -4.47214i q^{20} -13.4916i q^{22} +(-16.5468 - 15.9751i) q^{23} -5.00000 q^{25} -14.6005 q^{26} +13.2905i q^{28} -48.3347 q^{29} -42.5781 q^{31} -5.65685 q^{32} +22.0590i q^{34} +14.8592 q^{35} +37.9575i q^{37} +0.939198i q^{38} +6.32456i q^{40} +32.8401 q^{41} +56.3079i q^{43} +19.0800i q^{44} +(23.4007 + 22.5922i) q^{46} -26.7091 q^{47} +4.84077 q^{49} +7.07107 q^{50} +20.6482 q^{52} -13.8360i q^{53} +21.3321 q^{55} -18.7956i q^{56} +68.3556 q^{58} +92.5057 q^{59} -114.986i q^{61} +60.2145 q^{62} +8.00000 q^{64} -23.0854i q^{65} -119.439i q^{67} -31.1962i q^{68} -21.0141 q^{70} -104.336 q^{71} +4.46718 q^{73} -53.6800i q^{74} -1.32823i q^{76} -63.3957 q^{77} -29.4634i q^{79} -8.94427i q^{80} -46.4429 q^{82} +136.169i q^{83} -34.8784 q^{85} -79.6313i q^{86} -26.9832i q^{88} +19.1748i q^{89} +68.6063i q^{91} +(-33.0935 - 31.9502i) q^{92} +37.7723 q^{94} -1.48500 q^{95} +3.15840i q^{97} -6.84588 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 64 q^{4}+O(q^{10}) $ 32 * q + 64 * q^4	$ 32 q + 64 q^{4} - 48 q^{13} + 128 q^{16} + 80 q^{23} - 160 q^{25} - 120 q^{29} + 248 q^{31} + 120 q^{35} - 72 q^{41} + 160 q^{46} - 400 q^{47} - 344 q^{49} - 96 q^{52} - 256 q^{58} - 120 q^{59} - 160 q^{62} + 256 q^{64} - 104 q^{71} + 16 q^{73} - 240 q^{77} + 64 q^{82} - 120 q^{85} + 160 q^{92} + 96 q^{94} + 160 q^{95} - 64 q^{98}+O(q^{100}) $ 32 * q + 64 * q^4 - 48 * q^13 + 128 * q^16 + 80 * q^23 - 160 * q^25 - 120 * q^29 + 248 * q^31 + 120 * q^35 - 72 * q^41 + 160 * q^46 - 400 * q^47 - 344 * q^49 - 96 * q^52 - 256 * q^58 - 120 * q^59 - 160 * q^62 + 256 * q^64 - 104 * q^71 + 16 * q^73 - 240 * q^77 + 64 * q^82 - 120 * q^85 + 160 * q^92 + 96 * q^94 + 160 * q^95 - 64 * 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
$7$			6.64524i			0.949320i	0.880169	+	0.474660i	$0.157429\pi$
$7$			6.64524i			0.949320i	−0.880169	+	0.474660i	$0.842571\pi$
$8$	−2.82843			−0.353553
$9$		0			0
$10$			3.16228i			0.316228i
$11$			9.54001i			0.867274i	0.901088	+	0.433637i	$0.142770\pi$
$11$			9.54001i			0.867274i	−0.901088	+	0.433637i	$0.857230\pi$
$12$		0			0
$13$	10.3241			0.794163			0.397082	−	0.917783i	$-0.370023\pi$
$13$	10.3241			0.794163			0.397082	+	0.917783i	$0.370023\pi$
$14$		−	9.39779i		−	0.671271i
$15$		0			0
$16$	4.00000			0.250000
$17$		−	15.5981i		−	0.917534i	−0.888557	−	0.458767i	$-0.848291\pi$
$17$		−	15.5981i		−	0.917534i	0.888557	−	0.458767i	$-0.151709\pi$
$18$		0			0
$19$		−	0.664113i		−	0.0349533i	−0.999847	−	0.0174767i	$-0.994437\pi$
$19$		−	0.664113i		−	0.0349533i	0.999847	−	0.0174767i	$-0.00556328\pi$
$20$		−	4.47214i		−	0.223607i
$21$		0			0
$22$		−	13.4916i		−	0.613255i
$23$	−16.5468	−	15.9751i	−0.719425	−	0.694570i
$23$	−16.5468	−	15.9751i	−0.719425	−	0.694570i
$24$		0			0
$25$	−5.00000			−0.200000
$26$	−14.6005			−0.561558
$27$		0			0
$28$			13.2905i			0.474660i
$29$	−48.3347			−1.66672			−0.833358	−	0.552734i	$-0.813584\pi$
$29$	−48.3347			−1.66672			−0.833358	+	0.552734i	$0.813584\pi$
$30$		0			0
$31$	−42.5781			−1.37349			−0.686743	−	0.726900i	$-0.740960\pi$
$31$	−42.5781			−1.37349			−0.686743	+	0.726900i	$0.740960\pi$
$32$	−5.65685			−0.176777
$33$		0			0
$34$			22.0590i			0.648795i
$35$	14.8592			0.424549
$36$		0			0
$37$			37.9575i			1.02588i	0.858425	+	0.512939i	$0.171443\pi$
$37$			37.9575i			1.02588i	−0.858425	+	0.512939i	$0.828557\pi$
$38$			0.939198i			0.0247157i
$39$		0			0
$40$			6.32456i			0.158114i
$41$	32.8401			0.800977			0.400489	−	0.916302i	$-0.368840\pi$
$41$	32.8401			0.800977			0.400489	+	0.916302i	$0.368840\pi$
$42$		0			0
$43$			56.3079i			1.30948i	0.755852	+	0.654742i	$0.227223\pi$
$43$			56.3079i			1.30948i	−0.755852	+	0.654742i	$0.772777\pi$
$44$			19.0800i			0.433637i
$45$		0			0
$46$	23.4007	+	22.5922i	0.508710	+	0.491135i
$47$	−26.7091			−0.568278			−0.284139	−	0.958783i	$-0.591708\pi$
$47$	−26.7091			−0.568278			−0.284139	+	0.958783i	$0.591708\pi$
$48$		0			0
$49$	4.84077			0.0987912
$50$	7.07107			0.141421
$51$		0			0
$52$	20.6482			0.397082
$53$		−	13.8360i		−	0.261056i	−0.991445	−	0.130528i	$-0.958333\pi$
$53$		−	13.8360i		−	0.261056i	0.991445	−	0.130528i	$-0.0416672\pi$
$54$		0			0
$55$	21.3321			0.387857
$56$		−	18.7956i		−	0.335635i
$57$		0			0
$58$	68.3556			1.17855
$59$	92.5057			1.56789			0.783947	−	0.620828i	$-0.213203\pi$
$59$	92.5057			1.56789			0.783947	+	0.620828i	$0.213203\pi$
$60$		0			0
$61$		−	114.986i		−	1.88502i	−0.334185	−	0.942508i	$-0.608461\pi$
$61$		−	114.986i		−	1.88502i	0.334185	−	0.942508i	$-0.391539\pi$
$62$	60.2145			0.971201
$63$		0			0
$64$	8.00000			0.125000
$65$		−	23.0854i		−	0.355161i
$66$		0			0
$67$		−	119.439i		−	1.78268i	−0.453337	−	0.891339i	$-0.649767\pi$
$67$		−	119.439i		−	1.78268i	0.453337	−	0.891339i	$-0.350233\pi$
$68$		−	31.1962i		−	0.458767i
$69$		0			0
$70$	−21.0141			−0.300201
$71$	−104.336			−1.46952			−0.734761	−	0.678326i	$-0.762706\pi$
$71$	−104.336			−1.46952			−0.734761	+	0.678326i	$0.762706\pi$
$72$		0			0
$73$	4.46718			0.0611943			0.0305972	−	0.999532i	$-0.490259\pi$
$73$	4.46718			0.0611943			0.0305972	+	0.999532i	$0.490259\pi$
$74$		−	53.6800i		−	0.725405i
$75$		0			0
$76$		−	1.32823i		−	0.0174767i
$77$	−63.3957			−0.823320
$78$		0			0
$79$		−	29.4634i		−	0.372954i	−0.982459	−	0.186477i	$-0.940293\pi$
$79$		−	29.4634i		−	0.372954i	0.982459	−	0.186477i	$-0.0597070\pi$
$80$		−	8.94427i		−	0.111803i
$81$		0			0
$82$	−46.4429			−0.566376
$83$			136.169i			1.64060i	0.571937	+	0.820298i	$0.306192\pi$
$83$			136.169i			1.64060i	−0.571937	+	0.820298i	$0.693808\pi$
$84$		0			0
$85$	−34.8784			−0.410334
$86$		−	79.6313i		−	0.925946i
$87$		0			0
$88$		−	26.9832i		−	0.306627i
$89$			19.1748i			0.215447i	0.994181	+	0.107724i	$0.0343562\pi$
$89$			19.1748i			0.215447i	−0.994181	+	0.107724i	$0.965644\pi$
$90$		0			0
$91$			68.6063i			0.753915i
$92$	−33.0935	−	31.9502i	−0.359712	−	0.347285i
$93$		0			0
$94$	37.7723			0.401833
$95$	−1.48500			−0.0156316
$96$		0			0
$97$			3.15840i			0.0325609i	0.999867	+	0.0162804i	$0.00518245\pi$
$97$			3.15840i			0.0325609i	−0.999867	+	0.0162804i	$0.994818\pi$
$98$	−6.84588			−0.0698559
$99$		0			0
$100$	−10.0000			−0.100000
$101$	134.694			1.33361			0.666803	−	0.745234i	$-0.267662\pi$
$101$	134.694			1.33361			0.666803	+	0.745234i	$0.267662\pi$
$102$		0			0
$103$		−	39.6863i		−	0.385303i	−0.981267	−	0.192652i	$-0.938291\pi$
$103$		−	39.6863i		−	0.385303i	0.981267	−	0.192652i	$-0.0617088\pi$
$104$	−29.2010			−0.280779
$105$		0			0
$106$			19.5670i			0.184594i
$107$		−	37.7240i		−	0.352561i	−0.984340	−	0.176281i	$-0.943593\pi$
$107$		−	37.7240i		−	0.352561i	0.984340	−	0.176281i	$-0.0564066\pi$
$108$		0			0
$109$		−	180.885i		−	1.65949i	−0.558142	−	0.829746i	$-0.688485\pi$
$109$		−	180.885i		−	1.65949i	0.558142	−	0.829746i	$-0.311515\pi$
$110$	−30.1682			−0.274256
$111$		0			0
$112$			26.5810i			0.237330i
$113$		−	168.729i		−	1.49317i	−0.665288	−	0.746587i	$-0.731691\pi$
$113$		−	168.729i		−	1.49317i	0.665288	−	0.746587i	$-0.268309\pi$
$114$		0			0
$115$	−35.7214	+	36.9997i	−0.310621	+	0.321737i
$116$	−96.6695			−0.833358
$117$		0			0
$118$	−130.823			−1.10867
$119$	103.653			0.871034
$120$		0			0
$121$	29.9882			0.247837
$122$			162.615i			1.33291i
$123$		0			0
$124$	−85.1561			−0.686743
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	−206.225			−1.62382			−0.811911	−	0.583781i	$-0.801573\pi$
$127$	−206.225			−1.62382			−0.811911	+	0.583781i	$0.801573\pi$
$128$	−11.3137			−0.0883883
$129$		0			0
$130$			32.6477i			0.251136i
$131$	−26.2454			−0.200347			−0.100173	−	0.994970i	$-0.531940\pi$
$131$	−26.2454			−0.200347			−0.100173	+	0.994970i	$0.531940\pi$
$132$		0			0
$133$	4.41319			0.0331819
$134$			168.913i			1.26054i
$135$		0			0
$136$			44.1180i			0.324397i
$137$			68.4377i			0.499545i	0.968305	+	0.249773i	$0.0803559\pi$
$137$			68.4377i			0.499545i	−0.968305	+	0.249773i	$0.919644\pi$
$138$		0			0
$139$	256.704			1.84679			0.923396	−	0.383849i	$-0.125402\pi$
$139$	256.704			1.84679			0.923396	+	0.383849i	$0.125402\pi$
$140$	29.7184			0.212274
$141$		0			0
$142$	147.553			1.03911
$143$			98.4922i			0.688757i
$144$		0			0
$145$			108.080i			0.745378i
$146$	−6.31755			−0.0432709
$147$		0			0
$148$			75.9150i			0.512939i
$149$			14.9048i			0.100032i	0.998748	+	0.0500161i	$0.0159273\pi$
$149$			14.9048i			0.100032i	−0.998748	+	0.0500161i	$0.984073\pi$
$150$		0			0
$151$	−179.587			−1.18931			−0.594657	−	0.803979i	$-0.702712\pi$
$151$	−179.587			−1.18931			−0.594657	+	0.803979i	$0.702712\pi$
$152$			1.87840i			0.0123579i
$153$		0			0
$154$	89.6550			0.582175
$155$			95.2075i			0.614242i
$156$		0			0
$157$		−	164.415i		−	1.04723i	−0.851956	−	0.523614i	$-0.824583\pi$
$157$		−	164.415i		−	1.04723i	0.851956	−	0.523614i	$-0.175417\pi$
$158$			41.6675i			0.263719i
$159$		0			0
$160$			12.6491i			0.0790569i
$161$	106.158	−	109.957i	0.659370	−	0.682965i
$162$		0			0
$163$	−283.644			−1.74015			−0.870074	−	0.492921i	$-0.835929\pi$
$163$	−283.644			−1.74015			−0.870074	+	0.492921i	$0.835929\pi$
$164$	65.6801			0.400489
$165$		0			0
$166$		−	192.573i		−	1.16008i
$167$	−213.320			−1.27737			−0.638684	−	0.769469i	$-0.720521\pi$
$167$	−213.320			−1.27737			−0.638684	+	0.769469i	$0.720521\pi$
$168$		0			0
$169$	−62.4126			−0.369305
$170$	49.3255			0.290150
$171$		0			0
$172$			112.616i			0.654742i
$173$	−158.006			−0.913327			−0.456664	−	0.889639i	$-0.650956\pi$
$173$	−158.006			−0.913327			−0.456664	+	0.889639i	$0.650956\pi$
$174$		0			0
$175$		−	33.2262i		−	0.189864i
$176$			38.1600i			0.216818i
$177$		0			0
$178$		−	27.1173i		−	0.152344i
$179$	−88.4216			−0.493975			−0.246988	−	0.969019i	$-0.579441\pi$
$179$	−88.4216			−0.493975			−0.246988	+	0.969019i	$0.579441\pi$
$180$		0			0
$181$		−	142.050i		−	0.784805i	−0.919794	−	0.392403i	$-0.871644\pi$
$181$		−	142.050i		−	0.784805i	0.919794	−	0.392403i	$-0.128356\pi$
$182$		−	97.0239i		−	0.533098i
$183$		0			0
$184$	46.8013	+	45.1845i	0.254355	+	0.245568i
$185$	84.8755			0.458787
$186$		0			0
$187$	148.806			0.795753
$188$	−53.4181			−0.284139
$189$		0			0
$190$	2.10011			0.0110532
$191$		−	58.6375i		−	0.307002i	−0.988148	−	0.153501i	$-0.950945\pi$
$191$		−	58.6375i		−	0.307002i	0.988148	−	0.153501i	$-0.0490549\pi$
$192$		0			0
$193$	−265.804			−1.37722			−0.688612	−	0.725130i	$-0.741780\pi$
$193$	−265.804			−1.37722			−0.688612	+	0.725130i	$0.741780\pi$
$194$		−	4.46666i		−	0.0230240i
$195$		0			0
$196$	9.68154			0.0493956
$197$	27.0148			0.137131			0.0685655	−	0.997647i	$-0.478158\pi$
$197$	27.0148			0.137131			0.0685655	+	0.997647i	$0.478158\pi$
$198$		0			0
$199$		−	4.41948i		−	0.0222085i	−0.999938	−	0.0111042i	$-0.996465\pi$
$199$		−	4.41948i		−	0.0222085i	0.999938	−	0.0111042i	$-0.00353466\pi$
$200$	14.1421			0.0707107
$201$		0			0
$202$	−190.486			−0.943002
$203$		−	321.196i		−	1.58225i
$204$		0			0
$205$		−	73.4326i		−	0.358208i
$206$			56.1248i			0.272451i
$207$		0			0
$208$	41.2965			0.198541
$209$	6.33564			0.0303141
$210$		0			0
$211$	−65.1177			−0.308615			−0.154307	−	0.988023i	$-0.549315\pi$
$211$	−65.1177			−0.308615			−0.154307	+	0.988023i	$0.549315\pi$
$212$		−	27.6719i		−	0.130528i
$213$		0			0
$214$			53.3498i			0.249298i
$215$	125.908			0.585619
$216$		0			0
$217$		−	282.942i		−	1.30388i
$218$			255.809i			1.17344i
$219$		0			0
$220$	42.6642			0.193928
$221$		−	161.036i		−	0.728672i
$222$		0			0
$223$	−269.495			−1.20850			−0.604248	−	0.796796i	$-0.706526\pi$
$223$	−269.495			−1.20850			−0.604248	+	0.796796i	$0.706526\pi$
$224$		−	37.5912i		−	0.167818i
$225$		0			0
$226$			238.618i			1.05583i
$227$			312.590i			1.37705i	0.725214	+	0.688523i	$0.241741\pi$
$227$			312.590i			1.37705i	−0.725214	+	0.688523i	$0.758259\pi$
$228$		0			0
$229$		−	213.240i		−	0.931179i	−0.885001	−	0.465589i	$-0.845842\pi$
$229$		−	213.240i		−	0.931179i	0.885001	−	0.465589i	$-0.154158\pi$
$230$	50.5178	−	52.3255i	0.219642	−	0.227502i
$231$		0			0
$232$	136.711			0.589273
$233$	241.296			1.03560			0.517802	−	0.855500i	$-0.326750\pi$
$233$	241.296			1.03560			0.517802	+	0.855500i	$0.326750\pi$
$234$		0			0
$235$			59.7233i			0.254142i
$236$	185.011			0.783947
$237$		0			0
$238$	−146.588			−0.615914
$239$	−49.0519			−0.205238			−0.102619	−	0.994721i	$-0.532722\pi$
$239$	−49.0519			−0.205238			−0.102619	+	0.994721i	$0.532722\pi$
$240$		0			0
$241$			125.719i			0.521654i	0.965386	+	0.260827i	$0.0839953\pi$
$241$			125.719i			0.521654i	−0.965386	+	0.260827i	$0.916005\pi$
$242$	−42.4098			−0.175247
$243$		0			0
$244$		−	229.972i		−	0.942508i
$245$		−	10.8243i		−	0.0441808i
$246$		0			0
$247$		−	6.85638i		−	0.0277586i
$248$	120.429			0.485601
$249$		0			0
$250$		−	15.8114i		−	0.0632456i
$251$			167.693i			0.668099i	0.942556	+	0.334049i	$0.108415\pi$
$251$			167.693i			0.668099i	−0.942556	+	0.334049i	$0.891585\pi$
$252$		0			0
$253$	152.403	−	157.856i	0.602382	−	0.623938i
$254$	291.647			1.14822
$255$		0			0
$256$	16.0000			0.0625000
$257$	−447.839			−1.74256			−0.871282	−	0.490783i	$-0.836711\pi$
$257$	−447.839			−1.74256			−0.871282	+	0.490783i	$0.836711\pi$
$258$		0			0
$259$	−252.237			−0.973887
$260$		−	46.1709i		−	0.177580i
$261$		0			0
$262$	37.1166			0.141666
$263$		−	206.991i		−	0.787039i	−0.919316	−	0.393519i	$-0.871257\pi$
$263$		−	206.991i		−	0.787039i	0.919316	−	0.393519i	$-0.128743\pi$
$264$		0			0
$265$	−30.9382			−0.116748
$266$	−6.24119			−0.0234631
$267$		0			0
$268$		−	238.879i		−	0.891339i
$269$	−60.4178			−0.224602			−0.112301	−	0.993674i	$-0.535822\pi$
$269$	−60.4178			−0.224602			−0.112301	+	0.993674i	$0.535822\pi$
$270$		0			0
$271$	−374.057			−1.38028			−0.690141	−	0.723675i	$-0.742452\pi$
$271$	−374.057			−1.38028			−0.690141	+	0.723675i	$0.742452\pi$
$272$		−	62.3923i		−	0.229384i
$273$		0			0
$274$		−	96.7855i		−	0.353232i
$275$		−	47.7000i		−	0.173455i
$276$		0			0
$277$	168.436			0.608074			0.304037	−	0.952660i	$-0.401665\pi$
$277$	168.436			0.608074			0.304037	+	0.952660i	$0.401665\pi$
$278$	−363.034			−1.30588
$279$		0			0
$280$	−42.0282			−0.150101
$281$		−	451.713i		−	1.60752i	−0.594953	−	0.803760i	$-0.702829\pi$
$281$		−	451.713i		−	1.60752i	0.594953	−	0.803760i	$-0.297171\pi$
$282$		0			0
$283$		−	272.170i		−	0.961733i	−0.876794	−	0.480866i	$-0.840322\pi$
$283$		−	272.170i		−	0.961733i	0.876794	−	0.480866i	$-0.159678\pi$
$284$	−208.672			−0.734761
$285$		0			0
$286$		−	139.289i		−	0.487024i
$287$			218.230i			0.760384i
$288$		0			0
$289$	45.6998			0.158131
$290$		−	152.848i		−	0.527062i
$291$		0			0
$292$	8.93437			0.0305972
$293$		−	97.2149i		−	0.331792i	−0.986143	−	0.165896i	$-0.946948\pi$
$293$		−	97.2149i		−	0.331792i	0.986143	−	0.165896i	$-0.0530515\pi$
$294$		0			0
$295$		−	206.849i		−	0.701183i
$296$		−	107.360i		−	0.362703i
$297$		0			0
$298$		−	21.0786i		−	0.0707334i
$299$	−170.831	−	164.929i	−0.571341	−	0.551602i
$300$		0			0
$301$	−374.179			−1.24312
$302$	253.974			0.840973
$303$		0			0
$304$		−	2.65645i		−	0.00873833i
$305$	−257.116			−0.843004
$306$		0			0
$307$	64.9944			0.211708			0.105854	−	0.994382i	$-0.466242\pi$
$307$	64.9944			0.211708			0.105854	+	0.994382i	$0.466242\pi$
$308$	−126.791			−0.411660
$309$		0			0
$310$		−	134.644i		−	0.434334i
$311$	277.605			0.892621			0.446310	−	0.894878i	$-0.352738\pi$
$311$	277.605			0.892621			0.446310	+	0.894878i	$0.352738\pi$
$312$		0			0
$313$			424.841i			1.35732i	0.734453	+	0.678660i	$0.237439\pi$
$313$			424.841i			1.35732i	−0.734453	+	0.678660i	$0.762561\pi$
$314$			232.518i			0.740502i
$315$		0			0
$316$		−	58.9268i		−	0.186477i
$317$	−222.185			−0.700900			−0.350450	−	0.936581i	$-0.613971\pi$
$317$	−222.185			−0.700900			−0.350450	+	0.936581i	$0.613971\pi$
$318$		0			0
$319$		−	461.114i		−	1.44550i
$320$		−	17.8885i		−	0.0559017i
$321$		0			0
$322$	−150.131	+	155.503i	−0.466245	+	0.482929i
$323$	−10.3589			−0.0320709
$324$		0			0
$325$	−51.6206			−0.158833
$326$	401.134			1.23047
$327$		0			0
$328$	−92.8857			−0.283188
$329$		−	177.488i		−	0.539478i
$330$		0			0
$331$	−228.815			−0.691284			−0.345642	−	0.938366i	$-0.612339\pi$
$331$	−228.815			−0.691284			−0.345642	+	0.938366i	$0.612339\pi$
$332$			272.339i			0.820298i
$333$		0			0
$334$	301.680			0.903235
$335$	−267.075			−0.797238
$336$		0			0
$337$		−	309.895i		−	0.919571i	−0.888030	−	0.459786i	$-0.847926\pi$
$337$		−	309.895i		−	0.919571i	0.888030	−	0.459786i	$-0.152074\pi$
$338$	88.2647			0.261138
$339$		0			0
$340$	−69.7567			−0.205167
$341$		−	406.195i		−	1.19119i
$342$		0			0
$343$			357.785i			1.04310i
$344$		−	159.263i		−	0.462973i
$345$		0			0
$346$	223.454			0.645820
$347$	347.850			1.00245			0.501224	−	0.865318i	$-0.332883\pi$
$347$	347.850			1.00245			0.501224	+	0.865318i	$0.332883\pi$
$348$		0			0
$349$	50.4420			0.144533			0.0722665	−	0.997385i	$-0.476977\pi$
$349$	50.4420			0.144533			0.0722665	+	0.997385i	$0.476977\pi$
$350$			46.9890i			0.134254i
$351$		0			0
$352$		−	53.9664i		−	0.153314i
$353$	−521.736			−1.47801			−0.739003	−	0.673702i	$-0.764703\pi$
$353$	−521.736			−1.47801			−0.739003	+	0.673702i	$0.764703\pi$
$354$		0			0
$355$			233.302i			0.657190i
$356$			38.3497i			0.107724i
$357$		0			0
$358$	125.047			0.349293
$359$		−	0.869077i		−	0.00242083i	−0.999999	−	0.00121041i	$-0.999615\pi$
$359$		−	0.869077i		−	0.00242083i	0.999999	−	0.00121041i	$-0.000385287\pi$
$360$		0			0
$361$	360.559			0.998778
$362$			200.889i			0.554941i
$363$		0			0
$364$			137.213i			0.376958i
$365$		−	9.98893i		−	0.0273669i
$366$		0			0
$367$		−	589.026i		−	1.60498i	−0.596668	−	0.802488i	$-0.703509\pi$
$367$		−	589.026i		−	1.60498i	0.596668	−	0.802488i	$-0.296491\pi$
$368$	−66.1871	−	63.9005i	−0.179856	−	0.173643i
$369$		0			0
$370$	−120.032			−0.324411
$371$	91.9433			0.247826
$372$		0			0
$373$			335.776i			0.900205i	0.892977	+	0.450102i	$0.148613\pi$
$373$			335.776i			0.900205i	−0.892977	+	0.450102i	$0.851387\pi$
$374$	−210.443			−0.562682
$375$		0			0
$376$	75.5447			0.200917
$377$	−499.014			−1.32364
$378$		0			0
$379$		−	530.492i		−	1.39971i	−0.714283	−	0.699857i	$-0.753247\pi$
$379$		−	530.492i		−	1.39971i	0.714283	−	0.699857i	$-0.246753\pi$
$380$	−2.97000			−0.00781580
$381$		0			0
$382$			82.9259i			0.217083i
$383$			617.105i			1.61124i	0.592433	+	0.805620i	$0.298167\pi$
$383$			617.105i			1.61124i	−0.592433	+	0.805620i	$0.701833\pi$
$384$		0			0
$385$			141.757i			0.368200i
$386$	375.904			0.973844
$387$		0			0
$388$			6.31681i			0.0162804i
$389$		−	543.176i		−	1.39634i	−0.715933	−	0.698169i	$-0.753998\pi$
$389$		−	543.176i		−	1.39634i	0.715933	−	0.698169i	$-0.246002\pi$
$390$		0			0
$391$	−249.181	+	258.098i	−0.637292	+	0.660097i
$392$	−13.6918			−0.0349280
$393$		0			0
$394$	−38.2047			−0.0969662
$395$	−65.8821			−0.166790
$396$		0			0
$397$	−6.47467			−0.0163090			−0.00815450	−	0.999967i	$-0.502596\pi$
$397$	−6.47467			−0.0163090			−0.00815450	+	0.999967i	$0.502596\pi$
$398$			6.25010i			0.0157038i
$399$		0			0
$400$	−20.0000			−0.0500000
$401$		−	132.267i		−	0.329843i	−0.986307	−	0.164921i	$-0.947263\pi$
$401$		−	132.267i		−	0.329843i	0.986307	−	0.164921i	$-0.0527371\pi$
$402$		0			0
$403$	−439.581			−1.09077
$404$	269.388			0.666803
$405$		0			0
$406$			454.240i			1.11882i
$407$	−362.115			−0.889717
$408$		0			0
$409$	−290.718			−0.710802			−0.355401	−	0.934714i	$-0.615656\pi$
$409$	−290.718			−0.710802			−0.355401	+	0.934714i	$0.615656\pi$
$410$			103.849i			0.253291i
$411$		0			0
$412$		−	79.3725i		−	0.192652i
$413$			614.723i			1.48843i
$414$		0			0
$415$	304.484			0.733696
$416$	−58.4020			−0.140390
$417$		0			0
$418$	−8.95995			−0.0214353
$419$		−	715.612i		−	1.70790i	−0.520352	−	0.853952i	$-0.674199\pi$
$419$		−	715.612i		−	1.70790i	0.520352	−	0.853952i	$-0.325801\pi$
$420$		0			0
$421$			236.715i			0.562268i	0.959669	+	0.281134i	$0.0907105\pi$
$421$			236.715i			0.562268i	−0.959669	+	0.281134i	$0.909289\pi$
$422$	92.0903			0.218223
$423$		0			0
$424$			39.1340i			0.0922972i
$425$			77.9904i			0.183507i
$426$		0			0
$427$	764.109			1.78948
$428$		−	75.4481i		−	0.176281i
$429$		0			0
$430$	−178.061			−0.414096
$431$		−	719.166i		−	1.66860i	−0.551312	−	0.834299i	$-0.685873\pi$
$431$		−	719.166i		−	1.66860i	0.551312	−	0.834299i	$-0.314127\pi$
$432$		0			0
$433$			344.163i			0.794834i	0.917638	+	0.397417i	$0.130093\pi$
$433$			344.163i			0.794834i	−0.917638	+	0.397417i	$0.869907\pi$
$434$			400.140i			0.921981i
$435$		0			0
$436$		−	361.769i		−	0.829746i
$437$	−10.6093	+	10.9889i	−0.0242775	+	0.0251463i
$438$		0			0
$439$	481.890			1.09770			0.548850	−	0.835921i	$-0.315066\pi$
$439$	481.890			1.09770			0.548850	+	0.835921i	$0.315066\pi$
$440$	−60.3363			−0.137128
$441$		0			0
$442$			227.740i			0.515249i
$443$	−499.957			−1.12857			−0.564286	−	0.825580i	$-0.690848\pi$
$443$	−499.957			−1.12857			−0.564286	+	0.825580i	$0.690848\pi$
$444$		0			0
$445$	42.8762			0.0963510
$446$	381.123			0.854536
$447$		0			0
$448$			53.1619i			0.118665i
$449$	312.769			0.696590			0.348295	−	0.937385i	$-0.386761\pi$
$449$	312.769			0.696590			0.348295	+	0.937385i	$0.386761\pi$
$450$		0			0
$451$			313.295i			0.694666i
$452$		−	337.457i		−	0.746587i
$453$		0			0
$454$		−	442.068i		−	0.973719i
$455$	153.408			0.337161
$456$		0			0
$457$			91.3525i			0.199896i	0.994993	+	0.0999481i	$0.0318677\pi$
$457$			91.3525i			0.199896i	−0.994993	+	0.0999481i	$0.968132\pi$
$458$			301.567i			0.658443i
$459$		0			0
$460$	−71.4429	+	73.9994i	−0.155311	+	0.160868i
$461$	32.8781			0.0713191			0.0356596	−	0.999364i	$-0.488647\pi$
$461$	32.8781			0.0713191			0.0356596	+	0.999364i	$0.488647\pi$
$462$		0			0
$463$	439.384			0.948995			0.474497	−	0.880257i	$-0.342630\pi$
$463$	439.384			0.948995			0.474497	+	0.880257i	$0.342630\pi$
$464$	−193.339			−0.416679
$465$		0			0
$466$	−341.244			−0.732283
$467$		−	175.531i		−	0.375870i	−0.982181	−	0.187935i	$-0.939821\pi$
$467$		−	175.531i		−	0.375870i	0.982181	−	0.187935i	$-0.0601794\pi$
$468$		0			0
$469$	793.704			1.69233
$470$		−	84.4615i		−	0.179705i
$471$		0			0
$472$	−261.646			−0.554334
$473$	−537.177			−1.13568
$474$		0			0
$475$			3.32056i			0.00699066i
$476$	207.306			0.435517
$477$		0			0
$478$	69.3698			0.145125
$479$			50.8178i			0.106091i	0.998592	+	0.0530457i	$0.0168929\pi$
$479$			50.8178i			0.106091i	−0.998592	+	0.0530457i	$0.983107\pi$
$480$		0			0
$481$			391.878i			0.814714i
$482$		−	177.793i		−	0.368865i
$483$		0			0
$484$	59.9765			0.123918
$485$	7.06241			0.0145617
$486$		0			0
$487$	14.3618			0.0294903			0.0147451	−	0.999891i	$-0.495306\pi$
$487$	14.3618			0.0294903			0.0147451	+	0.999891i	$0.495306\pi$
$488$			325.229i			0.666454i
$489$		0			0
$490$			15.3079i			0.0312405i
$491$	382.245			0.778503			0.389252	−	0.921131i	$-0.372734\pi$
$491$	382.245			0.778503			0.389252	+	0.921131i	$0.372734\pi$
$492$		0			0
$493$			753.929i			1.52927i
$494$			9.69639i			0.0196283i
$495$		0			0
$496$	−170.312			−0.343372
$497$		−	693.338i		−	1.39505i
$498$		0			0
$499$	−465.646			−0.933159			−0.466579	−	0.884479i	$-0.654514\pi$
$499$	−465.646			−0.933159			−0.466579	+	0.884479i	$0.654514\pi$
$500$			22.3607i			0.0447214i
$501$		0			0
$502$		−	237.153i		−	0.472417i
$503$			445.223i			0.885134i	0.896735	+	0.442567i	$0.145932\pi$
$503$			445.223i			0.885134i	−0.896735	+	0.442567i	$0.854068\pi$
$504$		0			0
$505$		−	301.185i		−	0.596407i
$506$	−215.530	+	223.243i	−0.425949	+	0.441191i
$507$		0			0
$508$	−412.451			−0.811911
$509$	−211.434			−0.415392			−0.207696	−	0.978193i	$-0.566596\pi$
$509$	−211.434			−0.415392			−0.207696	+	0.978193i	$0.566596\pi$
$510$		0			0
$511$			29.6855i			0.0580930i
$512$	−22.6274			−0.0441942
$513$		0			0
$514$	633.340			1.23218
$515$	−88.7412			−0.172313
$516$		0			0
$517$		−	254.805i		−	0.492853i
$518$	356.716			0.688642
$519$		0			0
$520$			65.2955i			0.125568i
$521$		−	685.288i		−	1.31533i	−0.753310	−	0.657666i	$-0.771544\pi$
$521$		−	685.288i		−	1.31533i	0.753310	−	0.657666i	$-0.228456\pi$
$522$		0			0
$523$		−	535.072i		−	1.02308i	−0.859259	−	0.511541i	$-0.829075\pi$
$523$		−	535.072i		−	1.02308i	0.859259	−	0.511541i	$-0.170925\pi$
$524$	−52.4908			−0.100173
$525$		0			0
$526$			292.730i			0.556520i
$527$			664.136i			1.26022i
$528$		0			0
$529$	18.5913	+	528.673i	0.0351443	+	0.999382i
$530$	43.7532			0.0825532
$531$		0			0
$532$	8.82638			0.0165909
$533$	339.045			0.636107
$534$		0			0
$535$	−84.3535			−0.157670
$536$			337.826i			0.630272i
$537$		0			0
$538$	85.4437			0.158817
$539$			46.1810i			0.0856790i
$540$		0			0
$541$	240.218			0.444026			0.222013	−	0.975044i	$-0.428737\pi$
$541$	240.218			0.444026			0.222013	+	0.975044i	$0.428737\pi$
$542$	528.996			0.976007
$543$		0			0
$544$			88.2361i			0.162199i
$545$	−404.470			−0.742147
$546$		0			0
$547$	560.977			1.02555			0.512776	−	0.858522i	$-0.328617\pi$
$547$	560.977			1.02555			0.512776	+	0.858522i	$0.328617\pi$
$548$			136.875i			0.249773i
$549$		0			0
$550$			67.4580i			0.122651i
$551$			32.0997i			0.0582572i
$552$		0			0
$553$	195.791			0.354053
$554$	−238.205			−0.429973
$555$		0			0
$556$	513.408			0.923396
$557$			243.211i			0.436645i	0.975877	+	0.218322i	$0.0700584\pi$
$557$			243.211i			0.436645i	−0.975877	+	0.218322i	$0.929942\pi$
$558$		0			0
$559$			581.329i			1.03994i
$560$	59.4368			0.106137
$561$		0			0
$562$			638.819i			1.13669i
$563$			403.682i			0.717020i	0.933526	+	0.358510i	$0.116715\pi$
$563$			403.682i			0.717020i	−0.933526	+	0.358510i	$0.883285\pi$
$564$		0			0
$565$	−377.289			−0.667768
$566$			384.907i			0.680048i
$567$		0			0
$568$	295.107			0.519554
$569$		−	404.655i		−	0.711169i	−0.934644	−	0.355584i	$-0.884282\pi$
$569$		−	404.655i		−	0.711169i	0.934644	−	0.355584i	$-0.115718\pi$
$570$		0			0
$571$			747.309i			1.30877i	0.756161	+	0.654386i	$0.227073\pi$
$571$			747.309i			1.30877i	−0.756161	+	0.654386i	$0.772927\pi$
$572$			196.984i			0.344378i
$573$		0			0
$574$		−	308.624i		−	0.537673i
$575$	82.7339	+	79.8756i	0.143885	+	0.138914i
$576$		0			0
$577$	952.285			1.65041			0.825203	−	0.564836i	$-0.191060\pi$
$577$	952.285			1.65041			0.825203	+	0.564836i	$0.191060\pi$
$578$	−64.6293			−0.111815
$579$		0			0
$580$			216.160i			0.372689i
$581$	−904.879			−1.55745
$582$		0			0
$583$	131.995			0.226407
$584$	−12.6351			−0.0216355
$585$		0			0
$586$			137.483i			0.234612i
$587$	−295.831			−0.503972			−0.251986	−	0.967731i	$-0.581084\pi$
$587$	−295.831			−0.503972			−0.251986	+	0.967731i	$0.581084\pi$
$588$		0			0
$589$			28.2767i			0.0480079i
$590$			292.529i			0.495812i
$591$		0			0
$592$			151.830i			0.256469i
$593$	−380.724			−0.642031			−0.321015	−	0.947074i	$-0.604024\pi$
$593$	−380.724			−0.642031			−0.321015	+	0.947074i	$0.604024\pi$
$594$		0			0
$595$		−	231.775i		−	0.389538i
$596$			29.8096i			0.0500161i
$597$		0			0
$598$	241.591	+	233.245i	0.403999	+	0.390042i
$599$	432.741			0.722440			0.361220	−	0.932481i	$-0.382360\pi$
$599$	432.741			0.722440			0.361220	+	0.932481i	$0.382360\pi$
$600$		0			0
$601$	−103.199			−0.171713			−0.0858563	−	0.996308i	$-0.527363\pi$
$601$	−103.199			−0.171713			−0.0858563	+	0.996308i	$0.527363\pi$
$602$	529.169			0.879019
$603$		0			0
$604$	−359.173			−0.594657
$605$		−	67.0557i		−	0.110836i
$606$		0			0
$607$	−220.607			−0.363439			−0.181719	−	0.983350i	$-0.558166\pi$
$607$	−220.607			−0.363439			−0.181719	+	0.983350i	$0.558166\pi$
$608$			3.75679i			0.00617893i
$609$		0			0
$610$	363.617			0.596094
$611$	−275.748			−0.451305
$612$		0			0
$613$			932.888i			1.52184i	0.648846	+	0.760920i	$0.275252\pi$
$613$			932.888i			1.52184i	−0.648846	+	0.760920i	$0.724748\pi$
$614$	−91.9159			−0.149700
$615$		0			0
$616$	179.310			0.291088
$617$		−	1218.37i		−	1.97467i	−0.158643	−	0.987336i	$-0.550712\pi$
$617$		−	1218.37i		−	1.97467i	0.158643	−	0.987336i	$-0.449288\pi$
$618$		0			0
$619$			338.996i			0.547652i	0.961779	+	0.273826i	$0.0882892\pi$
$619$			338.996i			0.547652i	−0.961779	+	0.273826i	$0.911711\pi$
$620$			190.415i			0.307121i
$621$		0			0
$622$	−392.593			−0.631178
$623$	−127.421			−0.204529
$624$		0			0
$625$	25.0000			0.0400000
$626$		−	600.816i		−	0.959770i
$627$		0			0
$628$		−	328.830i		−	0.523614i
$629$	592.064			0.941278
$630$		0			0
$631$		−	55.2614i		−	0.0875775i	−0.999041	−	0.0437887i	$-0.986057\pi$
$631$		−	55.2614i		−	0.0875775i	0.999041	−	0.0437887i	$-0.0139428\pi$
$632$			83.3351i			0.131859i
$633$		0			0
$634$	314.218			0.495611
$635$			461.134i			0.726195i
$636$		0			0
$637$	49.9767			0.0784563
$638$			652.113i			1.02212i
$639$		0			0
$640$			25.2982i			0.0395285i
$641$			1079.52i			1.68411i	0.539390	+	0.842056i	$0.318655\pi$
$641$			1079.52i			1.68411i	−0.539390	+	0.842056i	$0.681345\pi$
$642$		0			0
$643$		−	256.649i		−	0.399143i	−0.979883	−	0.199572i	$-0.936045\pi$
$643$		−	256.649i		−	0.399143i	0.979883	−	0.199572i	$-0.0639550\pi$
$644$	212.317	−	219.915i	0.329685	−	0.341482i
$645$		0			0
$646$	14.6497			0.0226775
$647$	961.900			1.48671			0.743354	−	0.668899i	$-0.233234\pi$
$647$	961.900			1.48671			0.743354	+	0.668899i	$0.233234\pi$
$648$		0			0
$649$			882.506i			1.35979i
$650$	73.0025			0.112312
$651$		0			0
$652$	−567.288			−0.870074
$653$	−860.220			−1.31734			−0.658668	−	0.752434i	$-0.728880\pi$
$653$	−860.220			−1.31734			−0.658668	+	0.752434i	$0.728880\pi$
$654$		0			0
$655$			58.6865i			0.0895977i
$656$	131.360			0.200244
$657$		0			0
$658$			251.006i			0.381468i
$659$			45.8016i			0.0695016i	0.999396	+	0.0347508i	$0.0110638\pi$
$659$			45.8016i			0.0695016i	−0.999396	+	0.0347508i	$0.988936\pi$
$660$		0			0
$661$		−	190.025i		−	0.287481i	−0.989615	−	0.143741i	$-0.954087\pi$
$661$		−	190.025i		−	0.287481i	0.989615	−	0.143741i	$-0.0459131\pi$
$662$	323.593			0.488812
$663$		0			0
$664$		−	385.145i		−	0.580038i
$665$		−	9.86819i		−	0.0148394i
$666$		0			0
$667$	799.784	+	772.153i	1.19908	+	1.15765i
$668$	−426.641			−0.638684
$669$		0			0
$670$	377.701			0.563732
$671$	1096.97			1.63482
$672$		0			0
$673$	−223.701			−0.332394			−0.166197	−	0.986093i	$-0.553149\pi$
$673$	−223.701			−0.332394			−0.166197	+	0.986093i	$0.553149\pi$
$674$			438.258i			0.650235i
$675$		0			0
$676$	−124.825			−0.184653
$677$		−	879.263i		−	1.29876i	−0.760463	−	0.649382i	$-0.775028\pi$
$677$		−	879.263i		−	1.29876i	0.760463	−	0.649382i	$-0.224972\pi$
$678$		0			0
$679$	−20.9884			−0.0309107
$680$	98.6509			0.145075
$681$		0			0
$682$			574.447i			0.842297i
$683$	−323.930			−0.474275			−0.237137	−	0.971476i	$-0.576209\pi$
$683$	−323.930			−0.474275			−0.237137	+	0.971476i	$0.576209\pi$
$684$		0			0
$685$	153.031			0.223403
$686$		−	505.984i		−	0.737586i
$687$		0			0
$688$			225.231i			0.327371i
$689$		−	142.844i		−	0.207321i
$690$		0			0
$691$	−574.754			−0.831772			−0.415886	−	0.909417i	$-0.636528\pi$
$691$	−574.754			−0.831772			−0.415886	+	0.909417i	$0.636528\pi$
$692$	−316.011			−0.456664
$693$		0			0
$694$	−491.934			−0.708838
$695$		−	574.008i		−	0.825911i
$696$		0			0
$697$		−	512.242i		−	0.734924i
$698$	−71.3357			−0.102200
$699$		0			0
$700$		−	66.4524i		−	0.0949320i
$701$			206.109i			0.294021i	0.989135	+	0.147011i	$0.0469651\pi$
$701$			206.109i			0.294021i	−0.989135	+	0.147011i	$0.953035\pi$
$702$		0			0
$703$	25.2081			0.0358578
$704$			76.3201i			0.108409i
$705$		0			0
$706$	737.847			1.04511
$707$			895.076i			1.26602i
$708$		0			0
$709$			759.295i			1.07094i	0.844555	+	0.535469i	$0.179865\pi$
$709$			759.295i			1.07094i	−0.844555	+	0.535469i	$0.820135\pi$
$710$		−	329.939i		−	0.464704i
$711$		0			0
$712$		−	54.2346i		−	0.0761722i
$713$	704.530	+	680.190i	0.988120	+	0.953983i
$714$		0			0
$715$	220.235			0.308021
$716$	−176.843			−0.246988
$717$		0			0
$718$			1.22906i			0.00171178i
$719$	1039.63			1.44594			0.722970	−	0.690880i	$-0.242777\pi$
$719$	1039.63			1.44594			0.722970	+	0.690880i	$0.242777\pi$
$720$		0			0
$721$	263.725			0.365776
$722$	−509.907			−0.706243
$723$		0			0
$724$		−	284.100i		−	0.392403i
$725$	241.674			0.333343
$726$		0			0
$727$		−	911.165i		−	1.25332i	−0.779292	−	0.626661i	$-0.784421\pi$
$727$		−	911.165i		−	1.25332i	0.779292	−	0.626661i	$-0.215579\pi$
$728$		−	194.048i		−	0.266549i
$729$		0			0
$730$			14.1265i			0.0193513i
$731$	878.295			1.20150
$732$		0			0
$733$			487.839i			0.665538i	0.943008	+	0.332769i	$0.107983\pi$
$733$			487.839i			0.665538i	−0.943008	+	0.332769i	$0.892017\pi$
$734$			833.009i			1.13489i
$735$		0			0
$736$	93.6027	+	90.3689i	0.127178	+	0.122784i
$737$	1139.45			1.54607
$738$		0			0
$739$	861.878			1.16628			0.583138	−	0.812373i	$-0.301825\pi$
$739$	861.878			1.16628			0.583138	+	0.812373i	$0.301825\pi$
$740$	169.751			0.229393
$741$		0			0
$742$	−130.028			−0.175239
$743$			999.092i			1.34467i	0.740246	+	0.672337i	$0.234709\pi$
$743$			999.092i			1.34467i	−0.740246	+	0.672337i	$0.765291\pi$
$744$		0			0
$745$	33.3281			0.0447358
$746$		−	474.860i		−	0.636541i
$747$		0			0
$748$	297.612			0.397877
$749$	250.685			0.334693
$750$		0			0
$751$		−	80.7630i		−	0.107541i	−0.998553	−	0.0537703i	$-0.982876\pi$
$751$		−	80.7630i		−	0.107541i	0.998553	−	0.0537703i	$-0.0171239\pi$
$752$	−106.836			−0.142070
$753$		0			0
$754$	705.712			0.935957
$755$			401.568i			0.531878i
$756$		0			0
$757$		−	75.6838i		−	0.0999786i	−0.998750	−	0.0499893i	$-0.984081\pi$
$757$		−	75.6838i		−	0.0999786i	0.998750	−	0.0499893i	$-0.0159187\pi$
$758$			750.229i			0.989747i
$759$		0			0
$760$	4.20022			0.00552660
$761$	−249.328			−0.327633			−0.163816	−	0.986491i	$-0.552380\pi$
$761$	−249.328			−0.327633			−0.163816	+	0.986491i	$0.552380\pi$
$762$		0			0
$763$	1202.02			1.57539
$764$		−	117.275i		−	0.153501i
$765$		0			0
$766$		−	872.718i		−	1.13932i
$767$	955.040			1.24516
$768$		0			0
$769$		−	1367.54i		−	1.77834i	−0.457581	−	0.889168i	$-0.651284\pi$
$769$		−	1367.54i		−	1.77834i	0.457581	−	0.889168i	$-0.348716\pi$
$770$		−	200.475i		−	0.260357i
$771$		0			0
$772$	−531.608			−0.688612
$773$			965.836i			1.24946i	0.780839	+	0.624732i	$0.214792\pi$
$773$			965.836i			1.24946i	−0.780839	+	0.624732i	$0.785208\pi$
$774$		0			0
$775$	212.890			0.274697
$776$		−	8.93332i		−	0.0115120i
$777$		0			0
$778$			768.166i			0.987360i
$779$		−	21.8095i		−	0.0279968i
$780$		0			0
$781$		−	995.367i		−	1.27448i
$782$	352.395	−	365.006i	0.450634	−	0.466759i
$783$		0			0
$784$	19.3631			0.0246978
$785$	−367.643			−0.468335
$786$		0			0
$787$		−	1.86804i		−	0.00237362i	−0.999999	−	0.00118681i	$-0.999622\pi$
$787$		−	1.86804i		−	0.00237362i	0.999999	−	0.00118681i	$-0.000377773\pi$
$788$	54.0296			0.0685655
$789$		0			0
$790$	93.1714			0.117939
$791$	1121.24			1.41750
$792$		0			0
$793$		−	1187.13i		−	1.49701i
$794$	9.15657			0.0115322
$795$		0			0
$796$		−	8.83897i		−	0.0111042i
$797$			20.0019i			0.0250965i	0.999921	+	0.0125483i	$0.00399434\pi$
$797$			20.0019i			0.0250965i	−0.999921	+	0.0125483i	$0.996006\pi$
$798$		0			0
$799$			416.610i			0.521415i
$800$	28.2843			0.0353553
$801$		0			0
$802$			187.054i			0.233234i
$803$			42.6170i			0.0530722i
$804$		0			0
$805$	−245.872	−	237.378i	−0.305431	−	0.294879i
$806$	621.662			0.771292
$807$		0			0
$808$	−380.973			−0.471501
$809$	−1217.58			−1.50504			−0.752521	−	0.658568i	$-0.771163\pi$
$809$	−1217.58			−1.50504			−0.752521	+	0.658568i	$0.771163\pi$
$810$		0			0
$811$	−357.718			−0.441082			−0.220541	−	0.975378i	$-0.570782\pi$
$811$	−357.718			−0.441082			−0.220541	+	0.975378i	$0.570782\pi$
$812$		−	642.392i		−	0.791123i
$813$		0			0
$814$	512.108			0.629125
$815$			634.248i			0.778218i
$816$		0			0
$817$	37.3948			0.0457708
$818$	411.138			0.502613
$819$		0			0
$820$		−	146.865i		−	0.179104i
$821$	−266.233			−0.324279			−0.162140	−	0.986768i	$-0.551839\pi$
$821$	−266.233			−0.324279			−0.162140	+	0.986768i	$0.551839\pi$
$822$		0			0
$823$	−861.623			−1.04693			−0.523465	−	0.852047i	$-0.675361\pi$
$823$	−861.623			−1.04693			−0.523465	+	0.852047i	$0.675361\pi$
$824$			112.250i			0.136225i
$825$		0			0
$826$		−	869.350i		−	1.05248i
$827$			400.784i			0.484624i	0.970198	+	0.242312i	$0.0779058\pi$
$827$			400.784i			0.484624i	−0.970198	+	0.242312i	$0.922094\pi$
$828$		0			0
$829$	−91.0241			−0.109800			−0.0548999	−	0.998492i	$-0.517484\pi$
$829$	−91.0241			−0.109800			−0.0548999	+	0.998492i	$0.517484\pi$
$830$	−430.605			−0.518802
$831$		0			0
$832$	82.5930			0.0992704
$833$		−	75.5067i		−	0.0906443i
$834$		0			0
$835$			476.999i			0.571256i
$836$	12.6713			0.0151570
$837$		0			0
$838$			1012.03i			1.20767i
$839$			1436.60i			1.71228i	0.516747	+	0.856138i	$0.327143\pi$
$839$			1436.60i			1.71228i	−0.516747	+	0.856138i	$0.672857\pi$
$840$		0			0
$841$	1495.25			1.77794
$842$		−	334.765i		−	0.397583i
$843$		0			0
$844$	−130.235			−0.154307
$845$			139.559i			0.165158i
$846$		0			0
$847$			199.279i			0.235276i
$848$		−	55.3439i		−	0.0652640i
$849$		0			0
$850$		−	110.295i		−	0.129759i
$851$	606.375	−	628.074i	0.712544	−	0.738042i
$852$		0			0
$853$	−298.556			−0.350007			−0.175004	−	0.984568i	$-0.555994\pi$
$853$	−298.556			−0.350007			−0.175004	+	0.984568i	$0.555994\pi$
$854$	−1080.61			−1.26536
$855$		0			0
$856$			106.700i			0.124649i
$857$	747.993			0.872804			0.436402	−	0.899752i	$-0.356252\pi$
$857$	747.993			0.872804			0.436402	+	0.899752i	$0.356252\pi$
$858$		0			0
$859$	−1105.06			−1.28645			−0.643225	−	0.765677i	$-0.722404\pi$
$859$	−1105.06			−1.28645			−0.643225	+	0.765677i	$0.722404\pi$
$860$	251.816			0.292810
$861$		0			0
$862$			1017.05i			1.17988i
$863$	115.567			0.133913			0.0669567	−	0.997756i	$-0.478671\pi$
$863$	115.567			0.133913			0.0669567	+	0.997756i	$0.478671\pi$
$864$		0			0
$865$			353.311i			0.408452i
$866$		−	486.720i		−	0.562032i
$867$		0			0
$868$		−	565.883i		−	0.651939i
$869$	281.081			0.323453
$870$		0			0
$871$		−	1233.11i		−	1.41574i
$872$			511.619i			0.586719i
$873$		0			0
$874$	15.0038	−	15.5407i	0.0171668	−	0.0177811i
$875$	−74.2961			−0.0849098
$876$		0			0
$877$	−1523.47			−1.73714			−0.868568	−	0.495570i	$-0.834959\pi$
$877$	−1523.47			−1.73714			−0.868568	+	0.495570i	$0.834959\pi$
$878$	−681.496			−0.776191
$879$		0			0
$880$	85.3284			0.0969641
$881$			1164.99i			1.32235i	0.750234	+	0.661173i	$0.229941\pi$
$881$			1164.99i			1.32235i	−0.750234	+	0.661173i	$0.770059\pi$
$882$		0			0
$883$	−1045.05			−1.18352			−0.591760	−	0.806114i	$-0.701567\pi$
$883$	−1045.05			−1.18352			−0.591760	+	0.806114i	$0.701567\pi$
$884$		−	322.073i		−	0.364336i
$885$		0			0
$886$	707.046			0.798021
$887$	1455.96			1.64144			0.820720	−	0.571330i	$-0.193572\pi$
$887$	1455.96			1.64144			0.820720	+	0.571330i	$0.193572\pi$
$888$		0			0
$889$		−	1370.42i		−	1.54153i
$890$	−60.6361			−0.0681305
$891$		0			0
$892$	−538.989			−0.604248
$893$			17.7378i			0.0198632i
$894$		0			0
$895$			197.717i			0.220913i
$896$		−	75.1823i		−	0.0839088i
$897$		0			0
$898$	−442.322			−0.492563
$899$	2058.00			2.28921
$900$		0			0
$901$	−215.815			−0.239528
$902$		−	443.065i		−	0.491203i
$903$		0			0
$904$			477.237i			0.527917i
$905$	−317.633			−0.350976
$906$		0			0
$907$			1339.28i			1.47660i	0.674470	+	0.738302i	$0.264372\pi$
$907$			1339.28i			1.47660i	−0.674470	+	0.738302i	$0.735628\pi$
$908$			625.179i			0.688523i
$909$		0			0
$910$	−216.952			−0.238409
$911$		−	772.255i		−	0.847700i	−0.905732	−	0.423850i	$-0.860678\pi$
$911$		−	772.255i		−	0.847700i	0.905732	−	0.423850i	$-0.139322\pi$
$912$		0			0
$913$	−1299.06			−1.42284
$914$		−	129.192i		−	0.141348i
$915$		0			0
$916$		−	426.480i		−	0.465589i
$917$		−	174.407i		−	0.190193i
$918$		0			0
$919$		−	913.892i		−	0.994441i	−0.867624	−	0.497221i	$-0.834354\pi$
$919$		−	913.892i		−	0.994441i	0.867624	−	0.497221i	$-0.165646\pi$
$920$	101.036	−	104.651i	0.109821	−	0.113751i
$921$		0			0
$922$	−46.4967			−0.0504302
$923$	−1077.18			−1.16704
$924$		0			0
$925$		−	189.787i		−	0.205176i
$926$	−621.383			−0.671040
$927$		0			0
$928$	273.423			0.294636
$929$	−536.691			−0.577709			−0.288854	−	0.957373i	$-0.593274\pi$
$929$	−536.691			−0.577709			−0.288854	+	0.957373i	$0.593274\pi$
$930$		0			0
$931$		−	3.21482i		−	0.00345308i
$932$	482.592			0.517802
$933$		0			0
$934$			248.239i			0.265780i
$935$		−	332.740i		−	0.355872i
$936$		0			0
$937$			636.687i			0.679495i	0.940517	+	0.339748i	$0.110342\pi$
$937$			636.687i			0.679495i	−0.940517	+	0.339748i	$0.889658\pi$
$938$	−1122.47			−1.19666
$939$		0			0
$940$			119.447i			0.127071i
$941$			452.052i			0.480395i	0.970724	+	0.240197i	$0.0772122\pi$
$941$			452.052i			0.480395i	−0.970724	+	0.240197i	$0.922788\pi$
$942$		0			0
$943$	−543.397	−	524.624i	−0.576243	−	0.556335i
$944$	370.023			0.391973
$945$		0			0
$946$	759.684			0.803048
$947$	138.936			0.146711			0.0733557	−	0.997306i	$-0.476629\pi$
$947$	138.936			0.146711			0.0733557	+	0.997306i	$0.476629\pi$
$948$		0			0
$949$	46.1197			0.0485983
$950$		−	4.69599i		−	0.00494315i
$951$		0			0
$952$	−293.175			−0.307957
$953$		−	902.261i		−	0.946759i	−0.880859	−	0.473380i	$-0.843034\pi$
$953$		−	902.261i		−	0.946759i	0.880859	−	0.473380i	$-0.156966\pi$
$954$		0			0
$955$	−131.117			−0.137296
$956$	−98.1038			−0.102619
$957$		0			0
$958$		−	71.8672i		−	0.0750180i
$959$	−454.785			−0.474228
$960$		0			0
$961$	851.892			0.886465
$962$		−	554.199i		−	0.576090i
$963$		0			0
$964$			251.437i			0.260827i
$965$			594.356i			0.615913i
$966$		0			0
$967$	−110.851			−0.114634			−0.0573172	−	0.998356i	$-0.518255\pi$
$967$	−110.851			−0.114634			−0.0573172	+	0.998356i	$0.518255\pi$
$968$	−84.8195			−0.0876235
$969$		0			0
$970$	−9.98775			−0.0102967
$971$			1126.21i			1.15985i	0.814672	+	0.579923i	$0.196917\pi$
$971$			1126.21i			1.15985i	−0.814672	+	0.579923i	$0.803083\pi$
$972$		0			0
$973$			1705.86i			1.75320i
$974$	−20.3106			−0.0208528
$975$		0			0
$976$		−	459.944i		−	0.471254i
$977$			1127.18i			1.15371i	0.816845	+	0.576857i	$0.195721\pi$
$977$			1127.18i			1.15371i	−0.816845	+	0.576857i	$0.804279\pi$
$978$		0			0
$979$	−182.928			−0.186852
$980$		−	21.6486i		−	0.0220904i
$981$		0			0
$982$	−540.576			−0.550485
$983$		−	843.799i		−	0.858392i	−0.903211	−	0.429196i	$-0.858797\pi$
$983$		−	843.799i		−	0.858392i	0.903211	−	0.429196i	$-0.141203\pi$
$984$		0			0
$985$		−	60.4069i		−	0.0613268i
$986$		−	1066.22i		−	1.08136i
$987$		0			0
$988$		−	13.7128i		−	0.0138793i
$989$	899.524	−	931.713i	0.909529	−	0.942076i
$990$		0			0
$991$	712.967			0.719442			0.359721	−	0.933060i	$-0.382872\pi$
$991$	712.967			0.719442			0.359721	+	0.933060i	$0.382872\pi$
$992$	240.858			0.242800
$993$		0			0
$994$			980.528i			0.986447i
$995$	−9.88227			−0.00993193
$996$		0			0
$997$	997.105			1.00011			0.500053	−	0.865995i	$-0.333314\pi$
$997$	997.105			1.00011			0.500053	+	0.865995i	$0.333314\pi$
$998$	658.523			0.659843
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.3.c.b.91.5		32
3.2	odd	2		690.3.c.a.91.31	yes	32
23.22	odd	2	inner	2070.3.c.b.91.12		32
69.68	even	2		690.3.c.a.91.26	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.c.a.91.26	✓	32	69.68	even	2
690.3.c.a.91.31	yes	32	3.2	odd	2
2070.3.c.b.91.5		32	1.1	even	1	trivial
2070.3.c.b.91.12		32	23.22	odd	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:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2070.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} +6.64524i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} +6.64524i q^{7} -2.82843 q^{8} +3.16228i q^{10} +9.54001i q^{11} +10.3241 q^{13} -9.39779i q^{14} +4.00000 q^{16} -15.5981i q^{17} -0.664113i q^{19} -4.47214i q^{20} -13.4916i q^{22} +(-16.5468 - 15.9751i) q^{23} -5.00000 q^{25} -14.6005 q^{26} +13.2905i q^{28} -48.3347 q^{29} -42.5781 q^{31} -5.65685 q^{32} +22.0590i q^{34} +14.8592 q^{35} +37.9575i q^{37} +0.939198i q^{38} +6.32456i q^{40} +32.8401 q^{41} +56.3079i q^{43} +19.0800i q^{44} +(23.4007 + 22.5922i) q^{46} -26.7091 q^{47} +4.84077 q^{49} +7.07107 q^{50} +20.6482 q^{52} -13.8360i q^{53} +21.3321 q^{55} -18.7956i q^{56} +68.3556 q^{58} +92.5057 q^{59} -114.986i q^{61} +60.2145 q^{62} +8.00000 q^{64} -23.0854i q^{65} -119.439i q^{67} -31.1962i q^{68} -21.0141 q^{70} -104.336 q^{71} +4.46718 q^{73} -53.6800i q^{74} -1.32823i q^{76} -63.3957 q^{77} -29.4634i q^{79} -8.94427i q^{80} -46.4429 q^{82} +136.169i q^{83} -34.8784 q^{85} -79.6313i q^{86} -26.9832i q^{88} +19.1748i q^{89} +68.6063i q^{91} +(-33.0935 - 31.9502i) q^{92} +37.7723 q^{94} -1.48500 q^{95} +3.15840i q^{97} -6.84588 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4}+O(q^{10}) \) 32 * q + 64 * q^4	\( 32 q + 64 q^{4} - 48 q^{13} + 128 q^{16} + 80 q^{23} - 160 q^{25} - 120 q^{29} + 248 q^{31} + 120 q^{35} - 72 q^{41} + 160 q^{46} - 400 q^{47} - 344 q^{49} - 96 q^{52} - 256 q^{58} - 120 q^{59} - 160 q^{62} + 256 q^{64} - 104 q^{71} + 16 q^{73} - 240 q^{77} + 64 q^{82} - 120 q^{85} + 160 q^{92} + 96 q^{94} + 160 q^{95} - 64 q^{98}+O(q^{100}) \) 32 * q + 64 * q^4 - 48 * q^13 + 128 * q^16 + 80 * q^23 - 160 * q^25 - 120 * q^29 + 248 * q^31 + 120 * q^35 - 72 * q^41 + 160 * q^46 - 400 * q^47 - 344 * q^49 - 96 * q^52 - 256 * q^58 - 120 * q^59 - 160 * q^62 + 256 * q^64 - 104 * q^71 + 16 * q^73 - 240 * q^77 + 64 * q^82 - 120 * q^85 + 160 * q^92 + 96 * q^94 + 160 * q^95 - 64 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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