LMFDB - Embedded newform 2070.3.c.a.91.8

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			91.8
Root			$6.02373i$ of defining polynomial
Character	$\chi$	$=$	2070.91
Dual form			2070.3.c.a.91.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +7.10180i q^{7} -2.82843 q^{8} +O(q^{10})$	\(q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +7.10180i q^{7} -2.82843 q^{8} -3.16228i q^{10} -11.2644i q^{11} +20.0597 q^{13} -10.0435i q^{14} +4.00000 q^{16} -1.63128i q^{17} +29.4164i q^{19} +4.47214i q^{20} +15.9302i q^{22} +(-20.0280 - 11.3084i) q^{23} -5.00000 q^{25} -28.3688 q^{26} +14.2036i q^{28} +50.3233 q^{29} +11.1316 q^{31} -5.65685 q^{32} +2.30698i q^{34} -15.8801 q^{35} +40.5429i q^{37} -41.6011i q^{38} -6.32456i q^{40} +7.24039 q^{41} -71.7020i q^{43} -22.5287i q^{44} +(28.3239 + 15.9924i) q^{46} +6.40666 q^{47} -1.43550 q^{49} +7.07107 q^{50} +40.1195 q^{52} +20.4148i q^{53} +25.1879 q^{55} -20.0869i q^{56} -71.1679 q^{58} +65.8889 q^{59} -37.7281i q^{61} -15.7425 q^{62} +8.00000 q^{64} +44.8549i q^{65} -124.242i q^{67} -3.26256i q^{68} +22.4578 q^{70} -43.5656 q^{71} +48.1194 q^{73} -57.3363i q^{74} +58.8328i q^{76} +79.9972 q^{77} +101.026i q^{79} +8.94427i q^{80} -10.2395 q^{82} -102.409i q^{83} +3.64765 q^{85} +101.402i q^{86} +31.8604i q^{88} +9.63875i q^{89} +142.460i q^{91} +(-40.0560 - 22.6167i) q^{92} -9.06039 q^{94} -65.7771 q^{95} +143.631i q^{97} +2.03010 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$			7.10180i			1.01454i	0.861787	+	0.507271i	$0.169346\pi$
$7$			7.10180i			1.01454i	−0.861787	+	0.507271i	$0.830654\pi$
$8$	−2.82843			−0.353553
$9$		0			0
$10$		−	3.16228i		−	0.316228i
$11$		−	11.2644i		−	1.02403i	−0.858975	−	0.512017i	$-0.828899\pi$
$11$		−	11.2644i		−	1.02403i	0.858975	−	0.512017i	$-0.171101\pi$
$12$		0			0
$13$	20.0597			1.54306			0.771528	−	0.636195i	$-0.219493\pi$
$13$	20.0597			1.54306			0.771528	+	0.636195i	$0.219493\pi$
$14$		−	10.0435i		−	0.717390i
$15$		0			0
$16$	4.00000			0.250000
$17$		−	1.63128i		−	0.0959575i	−0.998848	−	0.0479788i	$-0.984722\pi$
$17$		−	1.63128i		−	0.0959575i	0.998848	−	0.0479788i	$-0.0152780\pi$
$18$		0			0
$19$			29.4164i			1.54823i	0.633044	+	0.774116i	$0.281805\pi$
$19$			29.4164i			1.54823i	−0.633044	+	0.774116i	$0.718195\pi$
$20$			4.47214i			0.223607i
$21$		0			0
$22$			15.9302i			0.724101i
$23$	−20.0280	−	11.3084i	−0.870783	−	0.491668i
$23$	−20.0280	−	11.3084i	−0.870783	−	0.491668i
$24$		0			0
$25$	−5.00000			−0.200000
$26$	−28.3688			−1.09111
$27$		0			0
$28$			14.2036i			0.507271i
$29$	50.3233			1.73529			0.867644	−	0.497187i	$-0.165634\pi$
$29$	50.3233			1.73529			0.867644	+	0.497187i	$0.165634\pi$
$30$		0			0
$31$	11.1316			0.359084			0.179542	−	0.983750i	$-0.442538\pi$
$31$	11.1316			0.359084			0.179542	+	0.983750i	$0.442538\pi$
$32$	−5.65685			−0.176777
$33$		0			0
$34$			2.30698i			0.0678522i
$35$	−15.8801			−0.453717
$36$		0			0
$37$			40.5429i			1.09575i	0.836559	+	0.547877i	$0.184564\pi$
$37$			40.5429i			1.09575i	−0.836559	+	0.547877i	$0.815436\pi$
$38$		−	41.6011i		−	1.09477i
$39$		0			0
$40$		−	6.32456i		−	0.158114i
$41$	7.24039			0.176595			0.0882975	−	0.996094i	$-0.471857\pi$
$41$	7.24039			0.176595			0.0882975	+	0.996094i	$0.471857\pi$
$42$		0			0
$43$		−	71.7020i		−	1.66749i	−0.552150	−	0.833745i	$-0.686192\pi$
$43$		−	71.7020i		−	1.66749i	0.552150	−	0.833745i	$-0.313808\pi$
$44$		−	22.5287i		−	0.512017i
$45$		0			0
$46$	28.3239	+	15.9924i	0.615736	+	0.347662i
$47$	6.40666			0.136312			0.0681560	−	0.997675i	$-0.478288\pi$
$47$	6.40666			0.136312			0.0681560	+	0.997675i	$0.478288\pi$
$48$		0			0
$49$	−1.43550			−0.0292959
$50$	7.07107			0.141421
$51$		0			0
$52$	40.1195			0.771528
$53$			20.4148i			0.385184i	0.981279	+	0.192592i	$0.0616894\pi$
$53$			20.4148i			0.385184i	−0.981279	+	0.192592i	$0.938311\pi$
$54$		0			0
$55$	25.1879			0.457962
$56$		−	20.0869i		−	0.358695i
$57$		0			0
$58$	−71.1679			−1.22703
$59$	65.8889			1.11676			0.558381	−	0.829585i	$-0.311423\pi$
$59$	65.8889			1.11676			0.558381	+	0.829585i	$0.311423\pi$
$60$		0			0
$61$		−	37.7281i		−	0.618493i	−0.950982	−	0.309247i	$-0.899923\pi$
$61$		−	37.7281i		−	0.618493i	0.950982	−	0.309247i	$-0.100077\pi$
$62$	−15.7425			−0.253911
$63$		0			0
$64$	8.00000			0.125000
$65$			44.8549i			0.690076i
$66$		0			0
$67$		−	124.242i		−	1.85437i	−0.374609	−	0.927183i	$-0.622223\pi$
$67$		−	124.242i		−	1.85437i	0.374609	−	0.927183i	$-0.377777\pi$
$68$		−	3.26256i		−	0.0479788i
$69$		0			0
$70$	22.4578			0.320826
$71$	−43.5656			−0.613600			−0.306800	−	0.951774i	$-0.599258\pi$
$71$	−43.5656			−0.613600			−0.306800	+	0.951774i	$0.599258\pi$
$72$		0			0
$73$	48.1194			0.659169			0.329585	−	0.944126i	$-0.393091\pi$
$73$	48.1194			0.659169			0.329585	+	0.944126i	$0.393091\pi$
$74$		−	57.3363i		−	0.774815i
$75$		0			0
$76$			58.8328i			0.774116i
$77$	79.9972			1.03893
$78$		0			0
$79$			101.026i			1.27882i	0.768868	+	0.639408i	$0.220821\pi$
$79$			101.026i			1.27882i	−0.768868	+	0.639408i	$0.779179\pi$
$80$			8.94427i			0.111803i
$81$		0			0
$82$	−10.2395			−0.124871
$83$		−	102.409i		−	1.23384i	−0.787025	−	0.616921i	$-0.788380\pi$
$83$		−	102.409i		−	1.23384i	0.787025	−	0.616921i	$-0.211620\pi$
$84$		0			0
$85$	3.64765			0.0429135
$86$			101.402i			1.17909i
$87$		0			0
$88$			31.8604i			0.362050i
$89$			9.63875i			0.108301i	0.998533	+	0.0541503i	$0.0172450\pi$
$89$			9.63875i			0.108301i	−0.998533	+	0.0541503i	$0.982755\pi$
$90$		0			0
$91$			142.460i			1.56550i
$92$	−40.0560	−	22.6167i	−0.435391	−	0.245834i
$93$		0			0
$94$	−9.06039			−0.0963871
$95$	−65.7771			−0.692390
$96$		0			0
$97$			143.631i			1.48074i	0.672202	+	0.740368i	$0.265349\pi$
$97$			143.631i			1.48074i	−0.672202	+	0.740368i	$0.734651\pi$
$98$	2.03010			0.0207154
$99$		0			0
$100$	−10.0000			−0.100000
$101$	−103.099			−1.02078			−0.510391	−	0.859943i	$-0.670499\pi$
$101$	−103.099			−1.02078			−0.510391	+	0.859943i	$0.670499\pi$
$102$		0			0
$103$			98.8637i			0.959841i	0.877312	+	0.479921i	$0.159335\pi$
$103$			98.8637i			0.959841i	−0.877312	+	0.479921i	$0.840665\pi$
$104$	−56.7375			−0.545553
$105$		0			0
$106$		−	28.8708i		−	0.272366i
$107$		−	22.5494i		−	0.210742i	−0.994433	−	0.105371i	$-0.966397\pi$
$107$		−	22.5494i		−	0.210742i	0.994433	−	0.105371i	$-0.0336029\pi$
$108$		0			0
$109$		−	30.2389i		−	0.277421i	−0.990333	−	0.138711i	$-0.955704\pi$
$109$		−	30.2389i		−	0.277421i	0.990333	−	0.138711i	$-0.0442958\pi$
$110$	−35.6211			−0.323828
$111$		0			0
$112$			28.4072i			0.253636i
$113$			213.437i			1.88882i	0.328771	+	0.944410i	$0.393365\pi$
$113$			213.437i			1.88882i	−0.328771	+	0.944410i	$0.606635\pi$
$114$		0			0
$115$	25.2863	−	44.7840i	0.219881	−	0.389426i
$116$	100.647			0.867644
$117$		0			0
$118$	−93.1810			−0.789670
$119$	11.5850			0.0973530
$120$		0			0
$121$	−5.88598			−0.0486445
$122$			53.3556i			0.437341i
$123$		0			0
$124$	22.2632			0.179542
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	−29.9509			−0.235834			−0.117917	−	0.993023i	$-0.537622\pi$
$127$	−29.9509			−0.235834			−0.117917	+	0.993023i	$0.537622\pi$
$128$	−11.3137			−0.0883883
$129$		0			0
$130$		−	63.4345i		−	0.487957i
$131$	116.486			0.889208			0.444604	−	0.895727i	$-0.353344\pi$
$131$	116.486			0.889208			0.444604	+	0.895727i	$0.353344\pi$
$132$		0			0
$133$	−208.909			−1.57075
$134$			175.705i			1.31123i
$135$		0			0
$136$			4.61395i			0.0339261i
$137$			94.8211i			0.692125i	0.938212	+	0.346062i	$0.112481\pi$
$137$			94.8211i			0.692125i	−0.938212	+	0.346062i	$0.887519\pi$
$138$		0			0
$139$	89.2774			0.642284			0.321142	−	0.947031i	$-0.395933\pi$
$139$	89.2774			0.642284			0.321142	+	0.947031i	$0.395933\pi$
$140$	−31.7602			−0.226859
$141$		0			0
$142$	61.6111			0.433881
$143$		−	225.960i		−	1.58014i
$144$		0			0
$145$			112.526i			0.776044i
$146$	−68.0510			−0.466103
$147$		0			0
$148$			81.0857i			0.547877i
$149$			182.441i			1.22443i	0.790690	+	0.612217i	$0.209722\pi$
$149$			182.441i			1.22443i	−0.790690	+	0.612217i	$0.790278\pi$
$150$		0			0
$151$	29.7608			0.197092			0.0985458	−	0.995133i	$-0.468581\pi$
$151$	29.7608			0.197092			0.0985458	+	0.995133i	$0.468581\pi$
$152$		−	83.2022i		−	0.547383i
$153$		0			0
$154$	−113.133			−0.734631
$155$			24.8910i			0.160587i
$156$		0			0
$157$		−	64.1093i		−	0.408340i	−0.978935	−	0.204170i	$-0.934551\pi$
$157$		−	64.1093i		−	0.408340i	0.978935	−	0.204170i	$-0.0654495\pi$
$158$		−	142.873i		−	0.904260i
$159$		0			0
$160$		−	12.6491i		−	0.0790569i
$161$	80.3097	−	142.235i	0.498818	−	0.883446i
$162$		0			0
$163$	−75.3328			−0.462164			−0.231082	−	0.972934i	$-0.574227\pi$
$163$	−75.3328			−0.462164			−0.231082	+	0.972934i	$0.574227\pi$
$164$	14.4808			0.0882975
$165$		0			0
$166$			144.828i			0.872458i
$167$	−272.459			−1.63149			−0.815745	−	0.578412i	$-0.803673\pi$
$167$	−272.459			−1.63149			−0.815745	+	0.578412i	$0.803673\pi$
$168$		0			0
$169$	233.393			1.38102
$170$	−5.15855			−0.0303444
$171$		0			0
$172$		−	143.404i		−	0.833745i
$173$	−261.815			−1.51338			−0.756691	−	0.653773i	$-0.773185\pi$
$173$	−261.815			−1.51338			−0.756691	+	0.653773i	$0.773185\pi$
$174$		0			0
$175$		−	35.5090i		−	0.202908i
$176$		−	45.0575i		−	0.256008i
$177$		0			0
$178$		−	13.6312i		−	0.0765800i
$179$	184.406			1.03020			0.515101	−	0.857130i	$-0.327754\pi$
$179$	184.406			1.03020			0.515101	+	0.857130i	$0.327754\pi$
$180$		0			0
$181$		−	191.867i		−	1.06004i	−0.847986	−	0.530019i	$-0.822185\pi$
$181$		−	191.867i		−	1.06004i	0.847986	−	0.530019i	$-0.177815\pi$
$182$		−	201.469i		−	1.10697i
$183$		0			0
$184$	56.6477	+	31.9849i	0.307868	+	0.173831i
$185$	−90.6566			−0.490036
$186$		0			0
$187$	−18.3753			−0.0982637
$188$	12.8133			0.0681560
$189$		0			0
$190$	93.0228			0.489594
$191$		−	32.6185i		−	0.170778i	−0.996348	−	0.0853888i	$-0.972787\pi$
$191$		−	32.6185i		−	0.170778i	0.996348	−	0.0853888i	$-0.0272132\pi$
$192$		0			0
$193$	316.748			1.64118			0.820592	−	0.571515i	$-0.193644\pi$
$193$	316.748			1.64118			0.820592	+	0.571515i	$0.193644\pi$
$194$		−	203.125i		−	1.04704i
$195$		0			0
$196$	−2.87100			−0.0146480
$197$	194.946			0.989573			0.494787	−	0.869015i	$-0.335246\pi$
$197$	194.946			0.989573			0.494787	+	0.869015i	$0.335246\pi$
$198$		0			0
$199$			74.0815i			0.372269i	0.982524	+	0.186134i	$0.0595960\pi$
$199$			74.0815i			0.372269i	−0.982524	+	0.186134i	$0.940404\pi$
$200$	14.1421			0.0707107
$201$		0			0
$202$	145.804			0.721802
$203$			357.386i			1.76052i
$204$		0			0
$205$			16.1900i			0.0789757i
$206$		−	139.814i		−	0.678710i
$207$		0			0
$208$	80.2390			0.385764
$209$	331.357			1.58544
$210$		0			0
$211$	−4.75017			−0.0225126			−0.0112563	−	0.999937i	$-0.503583\pi$
$211$	−4.75017			−0.0225126			−0.0112563	+	0.999937i	$0.503583\pi$
$212$			40.8295i			0.192592i
$213$		0			0
$214$			31.8896i			0.149017i
$215$	160.331			0.745724
$216$		0			0
$217$			79.0544i			0.364306i
$218$			42.7643i			0.196167i
$219$		0			0
$220$	50.3758			0.228981
$221$		−	32.7230i		−	0.148068i
$222$		0			0
$223$	211.977			0.950571			0.475286	−	0.879832i	$-0.342345\pi$
$223$	211.977			0.950571			0.475286	+	0.879832i	$0.342345\pi$
$224$		−	40.1738i		−	0.179347i
$225$		0			0
$226$		−	301.845i		−	1.33560i
$227$			389.941i			1.71780i	0.512141	+	0.858901i	$0.328852\pi$
$227$			389.941i			1.71780i	−0.512141	+	0.858901i	$0.671148\pi$
$228$		0			0
$229$			156.000i			0.681224i	0.940204	+	0.340612i	$0.110634\pi$
$229$			156.000i			0.681224i	−0.940204	+	0.340612i	$0.889366\pi$
$230$	−35.7602	+	63.3341i	−0.155479	+	0.275366i
$231$		0			0
$232$	−142.336			−0.613517
$233$	46.4968			0.199557			0.0997785	−	0.995010i	$-0.468187\pi$
$233$	46.4968			0.199557			0.0997785	+	0.995010i	$0.468187\pi$
$234$		0			0
$235$			14.3257i			0.0609606i
$236$	131.778			0.558381
$237$		0			0
$238$	−16.3837			−0.0688389
$239$	454.735			1.90266			0.951328	−	0.308181i	$-0.0997201\pi$
$239$	454.735			1.90266			0.951328	+	0.308181i	$0.0997201\pi$
$240$		0			0
$241$		−	149.357i		−	0.619739i	−0.950779	−	0.309870i	$-0.899715\pi$
$241$		−	149.357i		−	0.619739i	0.950779	−	0.309870i	$-0.100285\pi$
$242$	8.32404			0.0343968
$243$		0			0
$244$		−	75.4562i		−	0.309247i
$245$		−	3.20988i		−	0.0131015i
$246$		0			0
$247$			590.085i			2.38901i
$248$	−31.4849			−0.126955
$249$		0			0
$250$			15.8114i			0.0632456i
$251$			364.513i			1.45224i	0.687567	+	0.726121i	$0.258679\pi$
$251$			364.513i			1.45224i	−0.687567	+	0.726121i	$0.741321\pi$
$252$		0			0
$253$	−127.382	+	225.603i	−0.503485	+	0.891711i
$254$	42.3570			0.166760
$255$		0			0
$256$	16.0000			0.0625000
$257$	−59.7088			−0.232330			−0.116165	−	0.993230i	$-0.537060\pi$
$257$	−59.7088			−0.232330			−0.116165	+	0.993230i	$0.537060\pi$
$258$		0			0
$259$	−287.927			−1.11169
$260$			89.7099i			0.345038i
$261$		0			0
$262$	−164.736			−0.628765
$263$			282.085i			1.07257i	0.844038	+	0.536284i	$0.180172\pi$
$263$			282.085i			1.07257i	−0.844038	+	0.536284i	$0.819828\pi$
$264$		0			0
$265$	−45.6488			−0.172260
$266$	295.442			1.11069
$267$		0			0
$268$		−	248.485i		−	0.927183i
$269$	−14.7823			−0.0549529			−0.0274764	−	0.999622i	$-0.508747\pi$
$269$	−14.7823			−0.0549529			−0.0274764	+	0.999622i	$0.508747\pi$
$270$		0			0
$271$	−34.7150			−0.128100			−0.0640499	−	0.997947i	$-0.520402\pi$
$271$	−34.7150			−0.128100			−0.0640499	+	0.997947i	$0.520402\pi$
$272$		−	6.52511i		−	0.0239894i
$273$		0			0
$274$		−	134.097i		−	0.489406i
$275$			56.3218i			0.204807i
$276$		0			0
$277$	−191.042			−0.689682			−0.344841	−	0.938661i	$-0.612067\pi$
$277$	−191.042			−0.689682			−0.344841	+	0.938661i	$0.612067\pi$
$278$	−126.257			−0.454163
$279$		0			0
$280$	44.9157			0.160413
$281$			471.349i			1.67740i	0.544596	+	0.838698i	$0.316683\pi$
$281$			471.349i			1.67740i	−0.544596	+	0.838698i	$0.683317\pi$
$282$		0			0
$283$		−	23.4603i		−	0.0828984i	−0.999141	−	0.0414492i	$-0.986803\pi$
$283$		−	23.4603i		−	0.0828984i	0.999141	−	0.0414492i	$-0.0131975\pi$
$284$	−87.1312			−0.306800
$285$		0			0
$286$			319.556i			1.11733i
$287$			51.4198i			0.179163i
$288$		0			0
$289$	286.339			0.990792
$290$		−	159.136i		−	0.548746i
$291$		0			0
$292$	96.2387			0.329585
$293$			289.288i			0.987331i	0.869652	+	0.493666i	$0.164343\pi$
$293$			289.288i			0.987331i	−0.869652	+	0.493666i	$0.835657\pi$
$294$		0			0
$295$			147.332i			0.499431i
$296$		−	114.673i		−	0.387407i
$297$		0			0
$298$		−	258.010i		−	0.865805i
$299$	−401.756	−	226.843i	−1.34367	−	0.758672i
$300$		0			0
$301$	509.213			1.69174
$302$	−42.0882			−0.139365
$303$		0			0
$304$			117.666i			0.387058i
$305$	84.3625			0.276599
$306$		0			0
$307$	−563.775			−1.83640			−0.918200	−	0.396117i	$-0.870358\pi$
$307$	−563.775			−1.83640			−0.918200	+	0.396117i	$0.870358\pi$
$308$	159.994			0.519463
$309$		0			0
$310$		−	35.2012i		−	0.113552i
$311$	−76.9428			−0.247404			−0.123702	−	0.992319i	$-0.539477\pi$
$311$	−76.9428			−0.247404			−0.123702	+	0.992319i	$0.539477\pi$
$312$		0			0
$313$			436.773i			1.39544i	0.716370	+	0.697721i	$0.245802\pi$
$313$			436.773i			1.39544i	−0.716370	+	0.697721i	$0.754198\pi$
$314$			90.6643i			0.288740i
$315$		0			0
$316$			202.053i			0.639408i
$317$	−95.6774			−0.301822			−0.150911	−	0.988547i	$-0.548221\pi$
$317$	−95.6774			−0.301822			−0.150911	+	0.988547i	$0.548221\pi$
$318$		0			0
$319$		−	566.860i		−	1.77699i
$320$			17.8885i			0.0559017i
$321$		0			0
$322$	−113.575	+	201.150i	−0.352718	+	0.624691i
$323$	47.9863			0.148565
$324$		0			0
$325$	−100.299			−0.308611
$326$	106.537			0.326799
$327$		0			0
$328$	−20.4789			−0.0624357
$329$			45.4988i			0.138294i
$330$		0			0
$331$	515.137			1.55631			0.778153	−	0.628074i	$-0.216157\pi$
$331$	515.137			1.55631			0.778153	+	0.628074i	$0.216157\pi$
$332$		−	204.818i		−	0.616921i
$333$		0			0
$334$	385.315			1.15364
$335$	277.815			0.829297
$336$		0			0
$337$			251.793i			0.747161i	0.927598	+	0.373580i	$0.121870\pi$
$337$			251.793i			0.747161i	−0.927598	+	0.373580i	$0.878130\pi$
$338$	−330.068			−0.976532
$339$		0			0
$340$	7.29530			0.0214568
$341$		−	125.391i		−	0.367714i
$342$		0			0
$343$			337.793i			0.984820i
$344$			202.804i			0.589547i
$345$		0			0
$346$	370.262			1.07012
$347$	167.899			0.483858			0.241929	−	0.970294i	$-0.422220\pi$
$347$	167.899			0.483858			0.241929	+	0.970294i	$0.422220\pi$
$348$		0			0
$349$	131.699			0.377360			0.188680	−	0.982039i	$-0.439579\pi$
$349$	131.699			0.377360			0.188680	+	0.982039i	$0.439579\pi$
$350$			50.2173i			0.143478i
$351$		0			0
$352$			63.7209i			0.181025i
$353$	−232.683			−0.659159			−0.329580	−	0.944128i	$-0.606907\pi$
$353$	−232.683			−0.659159			−0.329580	+	0.944128i	$0.606907\pi$
$354$		0			0
$355$		−	97.4157i		−	0.274410i
$356$			19.2775i			0.0541503i
$357$		0			0
$358$	−260.790			−0.728463
$359$		−	205.862i		−	0.573432i	−0.958016	−	0.286716i	$-0.907436\pi$
$359$		−	205.862i		−	0.573432i	0.958016	−	0.286716i	$-0.0925636\pi$
$360$		0			0
$361$	−504.325			−1.39702
$362$			271.341i			0.749560i
$363$		0			0
$364$			284.920i			0.782748i
$365$			107.598i			0.294789i
$366$		0			0
$367$		−	158.406i		−	0.431623i	−0.976435	−	0.215811i	$-0.930760\pi$
$367$		−	158.406i		−	0.431623i	0.976435	−	0.215811i	$-0.0692396\pi$
$368$	−80.1120	−	45.2335i	−0.217696	−	0.122917i
$369$		0			0
$370$	128.208			0.346508
$371$	−144.981			−0.390786
$372$		0			0
$373$		−	67.1037i		−	0.179903i	−0.995946	−	0.0899513i	$-0.971329\pi$
$373$		−	67.1037i		−	0.179903i	0.995946	−	0.0899513i	$-0.0286712\pi$
$374$	25.9866			0.0694829
$375$		0			0
$376$	−18.1208			−0.0481936
$377$	1009.47			2.67765
$378$		0			0
$379$			10.5032i			0.0277128i	0.999904	+	0.0138564i	$0.00441078\pi$
$379$			10.5032i			0.0277128i	−0.999904	+	0.0138564i	$0.995589\pi$
$380$	−131.554			−0.346195
$381$		0			0
$382$			46.1296i			0.120758i
$383$			360.978i			0.942501i	0.882000	+	0.471250i	$0.156197\pi$
$383$			360.978i			0.942501i	−0.882000	+	0.471250i	$0.843803\pi$
$384$		0			0
$385$			178.879i			0.464621i
$386$	−447.950			−1.16049
$387$		0			0
$388$			287.263i			0.740368i
$389$		−	47.2280i		−	0.121409i	−0.998156	−	0.0607044i	$-0.980665\pi$
$389$		−	47.2280i		−	0.121409i	0.998156	−	0.0607044i	$-0.0193347\pi$
$390$		0			0
$391$	−18.4471	+	32.6712i	−0.0471793	+	0.0835582i
$392$	4.06021			0.0103577
$393$		0			0
$394$	−275.695			−0.699734
$395$	−225.902			−0.571904
$396$		0			0
$397$	4.85826			0.0122374			0.00611872	−	0.999981i	$-0.498052\pi$
$397$	4.85826			0.0122374			0.00611872	+	0.999981i	$0.498052\pi$
$398$		−	104.767i		−	0.263234i
$399$		0			0
$400$	−20.0000			−0.0500000
$401$		−	297.502i		−	0.741900i	−0.928653	−	0.370950i	$-0.879032\pi$
$401$		−	297.502i		−	0.741900i	0.928653	−	0.370950i	$-0.120968\pi$
$402$		0			0
$403$	223.297			0.554087
$404$	−206.198			−0.510391
$405$		0			0
$406$		−	505.420i		−	1.24488i
$407$	456.690			1.12209
$408$		0			0
$409$	238.943			0.584213			0.292106	−	0.956386i	$-0.405644\pi$
$409$	238.943			0.584213			0.292106	+	0.956386i	$0.405644\pi$
$410$		−	22.8961i		−	0.0558442i
$411$		0			0
$412$			197.727i			0.479921i
$413$			467.930i			1.13300i
$414$		0			0
$415$	228.993			0.551791
$416$	−113.475			−0.272776
$417$		0			0
$418$	−468.610			−1.12108
$419$			197.316i			0.470920i	0.971884	+	0.235460i	$0.0756597\pi$
$419$			197.316i			0.470920i	−0.971884	+	0.235460i	$0.924340\pi$
$420$		0			0
$421$		−	459.256i		−	1.09087i	−0.838153	−	0.545435i	$-0.816365\pi$
$421$		−	459.256i		−	1.09087i	0.838153	−	0.545435i	$-0.183635\pi$
$422$	6.71775			0.0159188
$423$		0			0
$424$		−	57.7417i		−	0.136183i
$425$			8.15639i			0.0191915i
$426$		0			0
$427$	267.937			0.627487
$428$		−	45.0987i		−	0.105371i
$429$		0			0
$430$	−226.742			−0.527306
$431$		−	475.283i		−	1.10275i	−0.834259	−	0.551373i	$-0.814104\pi$
$431$		−	475.283i		−	1.10275i	0.834259	−	0.551373i	$-0.185896\pi$
$432$		0			0
$433$			694.309i			1.60349i	0.597669	+	0.801743i	$0.296094\pi$
$433$			694.309i			1.60349i	−0.597669	+	0.801743i	$0.703906\pi$
$434$		−	111.800i		−	0.257603i
$435$		0			0
$436$		−	60.4779i		−	0.138711i
$437$	332.651	−	589.152i	0.761216	−	1.34817i
$438$		0			0
$439$	−692.132			−1.57661			−0.788305	−	0.615284i	$-0.789041\pi$
$439$	−692.132			−1.57661			−0.788305	+	0.615284i	$0.789041\pi$
$440$	−71.2421			−0.161914
$441$		0			0
$442$			46.2773i			0.104700i
$443$	282.065			0.636716			0.318358	−	0.947971i	$-0.396869\pi$
$443$	282.065			0.636716			0.318358	+	0.947971i	$0.396869\pi$
$444$		0			0
$445$	−21.5529			−0.0484335
$446$	−299.781			−0.672155
$447$		0			0
$448$			56.8144i			0.126818i
$449$	4.51574			0.0100573			0.00502866	−	0.999987i	$-0.498399\pi$
$449$	4.51574			0.0100573			0.00502866	+	0.999987i	$0.498399\pi$
$450$		0			0
$451$		−	81.5584i		−	0.180839i
$452$			426.873i			0.944410i
$453$		0			0
$454$		−	551.460i		−	1.21467i
$455$	−318.551			−0.700111
$456$		0			0
$457$		−	399.040i		−	0.873174i	−0.899662	−	0.436587i	$-0.856187\pi$
$457$		−	399.040i		−	0.873174i	0.899662	−	0.436587i	$-0.143813\pi$
$458$		−	220.618i		−	0.481698i
$459$		0			0
$460$	50.5726	−	89.5679i	0.109940	−	0.194713i
$461$	−44.2537			−0.0959950			−0.0479975	−	0.998847i	$-0.515284\pi$
$461$	−44.2537			−0.0959950			−0.0479975	+	0.998847i	$0.515284\pi$
$462$		0			0
$463$	−668.258			−1.44332			−0.721661	−	0.692247i	$-0.756621\pi$
$463$	−668.258			−1.44332			−0.721661	+	0.692247i	$0.756621\pi$
$464$	201.293			0.433822
$465$		0			0
$466$	−65.7564			−0.141108
$467$		−	670.150i		−	1.43501i	−0.696554	−	0.717505i	$-0.745284\pi$
$467$		−	670.150i		−	1.43501i	0.696554	−	0.717505i	$-0.254716\pi$
$468$		0			0
$469$	882.345			1.88133
$470$		−	20.2597i		−	0.0431056i
$471$		0			0
$472$	−186.362			−0.394835
$473$	−807.678			−1.70756
$474$		0			0
$475$		−	147.082i		−	0.309646i
$476$	23.1700			0.0486765
$477$		0			0
$478$	−643.092			−1.34538
$479$		−	310.492i		−	0.648209i	−0.946021	−	0.324105i	$-0.894937\pi$
$479$		−	310.492i		−	0.648209i	0.946021	−	0.324105i	$-0.105063\pi$
$480$		0			0
$481$			813.279i			1.69081i
$482$			211.223i			0.438222i
$483$		0			0
$484$	−11.7720			−0.0243222
$485$	−321.169			−0.662205
$486$		0			0
$487$	829.644			1.70358			0.851790	−	0.523883i	$-0.175517\pi$
$487$	829.644			1.70358			0.851790	+	0.523883i	$0.175517\pi$
$488$			106.711i			0.218670i
$489$		0			0
$490$			4.53945i			0.00926419i
$491$	123.794			0.252126			0.126063	−	0.992022i	$-0.459766\pi$
$491$	123.794			0.252126			0.126063	+	0.992022i	$0.459766\pi$
$492$		0			0
$493$		−	82.0913i		−	0.166514i
$494$		−	834.507i		−	1.68928i
$495$		0			0
$496$	44.5264			0.0897710
$497$		−	309.394i		−	0.622523i
$498$		0			0
$499$	757.919			1.51887			0.759437	−	0.650580i	$-0.225474\pi$
$499$	757.919			1.51887			0.759437	+	0.650580i	$0.225474\pi$
$500$		−	22.3607i		−	0.0447214i
$501$		0			0
$502$		−	515.499i		−	1.02689i
$503$			242.915i			0.482933i	0.970409	+	0.241467i	$0.0776284\pi$
$503$			242.915i			0.482933i	−0.970409	+	0.241467i	$0.922372\pi$
$504$		0			0
$505$		−	230.536i		−	0.456507i
$506$	180.145	−	319.050i	0.356017	−	0.630535i
$507$		0			0
$508$	−59.9018			−0.117917
$509$	−822.585			−1.61608			−0.808040	−	0.589127i	$-0.799472\pi$
$509$	−822.585			−1.61608			−0.808040	+	0.589127i	$0.799472\pi$
$510$		0			0
$511$			341.734i			0.668755i
$512$	−22.6274			−0.0441942
$513$		0			0
$514$	84.4409			0.164282
$515$	−221.066			−0.429254
$516$		0			0
$517$		−	72.1670i		−	0.139588i
$518$	407.191			0.786082
$519$		0			0
$520$		−	126.869i		−	0.243979i
$521$			95.3538i			0.183021i	0.995804	+	0.0915103i	$0.0291694\pi$
$521$			95.3538i			0.183021i	−0.995804	+	0.0915103i	$0.970831\pi$
$522$		0			0
$523$			277.463i			0.530522i	0.964177	+	0.265261i	$0.0854581\pi$
$523$			277.463i			0.530522i	−0.964177	+	0.265261i	$0.914542\pi$
$524$	232.972			0.444604
$525$		0			0
$526$		−	398.929i		−	0.758420i
$527$		−	18.1588i		−	0.0344568i
$528$		0			0
$529$	273.242	+	452.968i	0.516525	+	0.856272i
$530$	64.5571			0.121806
$531$		0			0
$532$	−417.819			−0.785373
$533$	145.240			0.272496
$534$		0			0
$535$	50.4219			0.0942465
$536$			351.411i			0.655617i
$537$		0			0
$538$	20.9054			0.0388575
$539$			16.1700i			0.0300000i
$540$		0			0
$541$	−666.108			−1.23125			−0.615627	−	0.788038i	$-0.711097\pi$
$541$	−666.108			−1.23125			−0.615627	+	0.788038i	$0.711097\pi$
$542$	49.0945			0.0905802
$543$		0			0
$544$			9.22790i			0.0169631i
$545$	67.6163			0.124067
$546$		0			0
$547$	349.611			0.639143			0.319571	−	0.947562i	$-0.396461\pi$
$547$	349.611			0.639143			0.319571	+	0.947562i	$0.396461\pi$
$548$			189.642i			0.346062i
$549$		0			0
$550$		−	79.6511i		−	0.144820i
$551$			1480.33i			2.68663i
$552$		0			0
$553$	−717.469			−1.29741
$554$	270.174			0.487679
$555$		0			0
$556$	178.555			0.321142
$557$			8.96150i			0.0160889i	0.999968	+	0.00804443i	$0.00256065\pi$
$557$			8.96150i			0.0160889i	−0.999968	+	0.00804443i	$0.997439\pi$
$558$		0			0
$559$		−	1438.32i		−	2.57303i
$560$	−63.5204			−0.113429
$561$		0			0
$562$		−	666.587i		−	1.18610i
$563$		−	732.683i		−	1.30139i	−0.759339	−	0.650696i	$-0.774477\pi$
$563$		−	732.683i		−	1.30139i	0.759339	−	0.650696i	$-0.225523\pi$
$564$		0			0
$565$	−477.259			−0.844706
$566$			33.1778i			0.0586181i
$567$		0			0
$568$	123.222			0.216940
$569$			42.5363i			0.0747563i	0.999301	+	0.0373781i	$0.0119006\pi$
$569$			42.5363i			0.0747563i	−0.999301	+	0.0373781i	$0.988099\pi$
$570$		0			0
$571$			459.356i			0.804476i	0.915535	+	0.402238i	$0.131768\pi$
$571$			459.356i			0.804476i	−0.915535	+	0.402238i	$0.868232\pi$
$572$		−	451.921i		−	0.790071i
$573$		0			0
$574$		−	72.7186i		−	0.126687i
$575$	100.140	+	56.5418i	0.174157	+	0.0983336i
$576$		0			0
$577$	831.608			1.44126			0.720630	−	0.693319i	$-0.243852\pi$
$577$	831.608			1.44126			0.720630	+	0.693319i	$0.243852\pi$
$578$	−404.944			−0.700596
$579$		0			0
$580$			225.053i			0.388022i
$581$	727.287			1.25179
$582$		0			0
$583$	229.959			0.394441
$584$	−136.102			−0.233052
$585$		0			0
$586$		−	409.115i		−	0.698149i
$587$	166.970			0.284447			0.142223	−	0.989835i	$-0.454575\pi$
$587$	166.970			0.284447			0.142223	+	0.989835i	$0.454575\pi$
$588$		0			0
$589$			327.452i			0.555945i
$590$		−	208.359i		−	0.353151i
$591$		0			0
$592$			162.171i			0.273938i
$593$	−900.895			−1.51922			−0.759608	−	0.650381i	$-0.774609\pi$
$593$	−900.895			−1.51922			−0.759608	+	0.650381i	$0.774609\pi$
$594$		0			0
$595$			25.9049i			0.0435376i
$596$			364.881i			0.612217i
$597$		0			0
$598$	568.169	+	320.804i	0.950116	+	0.536462i
$599$	129.221			0.215727			0.107864	−	0.994166i	$-0.465599\pi$
$599$	129.221			0.215727			0.107864	+	0.994166i	$0.465599\pi$
$600$		0			0
$601$	580.916			0.966582			0.483291	−	0.875460i	$-0.339441\pi$
$601$	580.916			0.966582			0.483291	+	0.875460i	$0.339441\pi$
$602$	−720.136			−1.19624
$603$		0			0
$604$	59.5217			0.0985458
$605$		−	13.1615i		−	0.0217545i
$606$		0			0
$607$	−1063.36			−1.75183			−0.875915	−	0.482466i	$-0.839741\pi$
$607$	−1063.36			−1.75183			−0.875915	+	0.482466i	$0.839741\pi$
$608$		−	166.404i		−	0.273691i
$609$		0			0
$610$	−119.307			−0.195585
$611$	128.516			0.210337
$612$		0			0
$613$		−	442.412i		−	0.721715i	−0.932621	−	0.360858i	$-0.882484\pi$
$613$		−	442.412i		−	0.721715i	0.932621	−	0.360858i	$-0.117516\pi$
$614$	797.298			1.29853
$615$		0			0
$616$	−226.266			−0.367316
$617$		−	936.724i		−	1.51819i	−0.650979	−	0.759095i	$-0.725642\pi$
$617$		−	936.724i		−	1.51819i	0.650979	−	0.759095i	$-0.274358\pi$
$618$		0			0
$619$		−	401.856i		−	0.649202i	−0.945851	−	0.324601i	$-0.894770\pi$
$619$		−	401.856i		−	0.649202i	0.945851	−	0.324601i	$-0.105230\pi$
$620$			49.7821i			0.0802937i
$621$		0			0
$622$	108.814			0.174941
$623$	−68.4524			−0.109875
$624$		0			0
$625$	25.0000			0.0400000
$626$		−	617.691i		−	0.986726i
$627$		0			0
$628$		−	128.219i		−	0.204170i
$629$	66.1367			0.105146
$630$		0			0
$631$		−	1033.92i		−	1.63854i	−0.573411	−	0.819268i	$-0.694380\pi$
$631$		−	1033.92i		−	1.63854i	0.573411	−	0.819268i	$-0.305620\pi$
$632$		−	285.746i		−	0.452130i
$633$		0			0
$634$	135.308			0.213420
$635$		−	66.9722i		−	0.105468i
$636$		0			0
$637$	−28.7958			−0.0452053
$638$			801.662i			1.25652i
$639$		0			0
$640$		−	25.2982i		−	0.0395285i
$641$		−	188.175i		−	0.293564i	−0.989169	−	0.146782i	$-0.953108\pi$
$641$		−	188.175i		−	0.293564i	0.989169	−	0.146782i	$-0.0468916\pi$
$642$		0			0
$643$		−	1055.82i		−	1.64203i	−0.570909	−	0.821013i	$-0.693409\pi$
$643$		−	1055.82i		−	1.64203i	0.570909	−	0.821013i	$-0.306591\pi$
$644$	160.619	−	284.470i	0.249409	−	0.441723i
$645$		0			0
$646$	−67.8629			−0.105051
$647$	443.636			0.685681			0.342841	−	0.939394i	$-0.388611\pi$
$647$	443.636			0.685681			0.342841	+	0.939394i	$0.388611\pi$
$648$		0			0
$649$		−	742.197i		−	1.14360i
$650$	141.844			0.218221
$651$		0			0
$652$	−150.666			−0.231082
$653$	−117.465			−0.179886			−0.0899429	−	0.995947i	$-0.528668\pi$
$653$	−117.465			−0.179886			−0.0899429	+	0.995947i	$0.528668\pi$
$654$		0			0
$655$			260.471i			0.397666i
$656$	28.9616			0.0441487
$657$		0			0
$658$		−	64.3451i		−	0.0977888i
$659$		−	664.743i		−	1.00871i	−0.863495	−	0.504357i	$-0.831730\pi$
$659$		−	664.743i		−	1.00871i	0.863495	−	0.504357i	$-0.168270\pi$
$660$		0			0
$661$			426.231i			0.644828i	0.946599	+	0.322414i	$0.104494\pi$
$661$			426.231i			0.644828i	−0.946599	+	0.322414i	$0.895506\pi$
$662$	−728.514			−1.10047
$663$		0			0
$664$			289.656i			0.436229i
$665$		−	467.135i		−	0.702459i
$666$		0			0
$667$	−1007.88	−	569.075i	−1.51106	−	0.853185i
$668$	−544.918			−0.815745
$669$		0			0
$670$	−392.889			−0.586402
$671$	−424.983			−0.633358
$672$		0			0
$673$	−824.444			−1.22503			−0.612514	−	0.790460i	$-0.709842\pi$
$673$	−824.444			−1.22503			−0.612514	+	0.790460i	$0.709842\pi$
$674$		−	356.089i		−	0.528322i
$675$		0			0
$676$	466.786			0.690512
$677$			530.039i			0.782924i	0.920194	+	0.391462i	$0.128031\pi$
$677$			530.039i			0.782924i	−0.920194	+	0.391462i	$0.871969\pi$
$678$		0			0
$679$	−1020.04			−1.50227
$680$	−10.3171			−0.0151722
$681$		0			0
$682$			177.329i			0.260013i
$683$	−414.954			−0.607546			−0.303773	−	0.952744i	$-0.598246\pi$
$683$	−414.954			−0.607546			−0.303773	+	0.952744i	$0.598246\pi$
$684$		0			0
$685$	−212.026			−0.309527
$686$		−	477.712i		−	0.696373i
$687$		0			0
$688$		−	286.808i		−	0.416872i
$689$			409.515i			0.594361i
$690$		0			0
$691$	−363.154			−0.525548			−0.262774	−	0.964857i	$-0.584637\pi$
$691$	−363.154			−0.525548			−0.262774	+	0.964857i	$0.584637\pi$
$692$	−523.630			−0.756691
$693$		0			0
$694$	−237.445			−0.342139
$695$			199.630i			0.287238i
$696$		0			0
$697$		−	11.8111i		−	0.0169456i
$698$	−186.250			−0.266834
$699$		0			0
$700$		−	71.0180i		−	0.101454i
$701$		−	928.839i		−	1.32502i	−0.749053	−	0.662510i	$-0.769491\pi$
$701$		−	928.839i		−	1.32502i	0.749053	−	0.662510i	$-0.230509\pi$
$702$		0			0
$703$	−1192.63			−1.69648
$704$		−	90.1149i		−	0.128004i
$705$		0			0
$706$	329.064			0.466096
$707$		−	732.188i		−	1.03563i
$708$		0			0
$709$		−	44.8088i		−	0.0632000i	−0.999501	−	0.0316000i	$-0.989940\pi$
$709$		−	44.8088i		−	0.0632000i	0.999501	−	0.0316000i	$-0.0100603\pi$
$710$			137.767i			0.194037i
$711$		0			0
$712$		−	27.2625i		−	0.0382900i
$713$	−222.944	−	125.880i	−0.312684	−	0.176550i
$714$		0			0
$715$	505.263			0.706661
$716$	368.812			0.515101
$717$		0			0
$718$			291.133i			0.405478i
$719$	−1308.36			−1.81969			−0.909844	−	0.414950i	$-0.863799\pi$
$719$	−1308.36			−1.81969			−0.909844	+	0.414950i	$0.863799\pi$
$720$		0			0
$721$	−702.110			−0.973800
$722$	713.223			0.987843
$723$		0			0
$724$		−	383.734i		−	0.530019i
$725$	−251.617			−0.347057
$726$		0			0
$727$		−	515.858i		−	0.709570i	−0.934948	−	0.354785i	$-0.884554\pi$
$727$		−	515.858i		−	0.709570i	0.934948	−	0.354785i	$-0.115446\pi$
$728$		−	402.938i		−	0.553487i
$729$		0			0
$730$		−	152.167i		−	0.208448i
$731$	−116.966			−0.160008
$732$		0			0
$733$			405.638i			0.553394i	0.960957	+	0.276697i	$0.0892398\pi$
$733$			405.638i			0.553394i	−0.960957	+	0.276697i	$0.910760\pi$
$734$			224.019i			0.305203i
$735$		0			0
$736$	113.295	+	63.9698i	0.153934	+	0.0869155i
$737$	−1399.51			−1.89893
$738$		0			0
$739$	601.397			0.813798			0.406899	−	0.913473i	$-0.366610\pi$
$739$	601.397			0.813798			0.406899	+	0.913473i	$0.366610\pi$
$740$	−181.313			−0.245018
$741$		0			0
$742$	205.035			0.276327
$743$			775.124i			1.04324i	0.853179	+	0.521618i	$0.174671\pi$
$743$			775.124i			1.04324i	−0.853179	+	0.521618i	$0.825329\pi$
$744$		0			0
$745$	−407.950			−0.547583
$746$			94.8990i			0.127210i
$747$		0			0
$748$	−36.7506			−0.0491319
$749$	160.141			0.213806
$750$		0			0
$751$			533.250i			0.710054i	0.934856	+	0.355027i	$0.115528\pi$
$751$			533.250i			0.710054i	−0.934856	+	0.355027i	$0.884472\pi$
$752$	25.6267			0.0340780
$753$		0			0
$754$	−1427.61			−1.89338
$755$			66.5473i			0.0881421i
$756$		0			0
$757$		−	794.660i		−	1.04975i	−0.851180	−	0.524875i	$-0.824112\pi$
$757$		−	794.660i		−	1.04975i	0.851180	−	0.524875i	$-0.175888\pi$
$758$		−	14.8537i		−	0.0195959i
$759$		0			0
$760$	186.046			0.244797
$761$	1322.01			1.73720			0.868599	−	0.495515i	$-0.165021\pi$
$761$	1322.01			1.73720			0.868599	+	0.495515i	$0.165021\pi$
$762$		0			0
$763$	214.751			0.281456
$764$		−	65.2371i		−	0.0853888i
$765$		0			0
$766$		−	510.500i		−	0.666449i
$767$	1321.71			1.72323
$768$		0			0
$769$			1443.85i			1.87757i	0.344500	+	0.938786i	$0.388048\pi$
$769$			1443.85i			1.87757i	−0.344500	+	0.938786i	$0.611952\pi$
$770$		−	252.973i		−	0.328537i
$771$		0			0
$772$	633.497			0.820592
$773$		−	64.8298i		−	0.0838678i	−0.999120	−	0.0419339i	$-0.986648\pi$
$773$		−	64.8298i		−	0.0838678i	0.999120	−	0.0419339i	$-0.0133519\pi$
$774$		0			0
$775$	−55.6580			−0.0718168
$776$		−	406.251i		−	0.523519i
$777$		0			0
$778$			66.7905i			0.0858490i
$779$			212.986i			0.273410i
$780$		0			0
$781$			490.739i			0.628347i
$782$	26.0881	−	46.2041i	0.0333608	−	0.0590845i
$783$		0			0
$784$	−5.74200			−0.00732398
$785$	143.353			0.182615
$786$		0			0
$787$		−	1247.04i		−	1.58455i	−0.610163	−	0.792276i	$-0.708896\pi$
$787$		−	1247.04i		−	1.58455i	0.610163	−	0.792276i	$-0.291104\pi$
$788$	389.892			0.494787
$789$		0			0
$790$	319.474			0.404397
$791$	−1515.78			−1.91629
$792$		0			0
$793$		−	756.815i		−	0.954370i
$794$	−6.87062			−0.00865317
$795$		0			0
$796$			148.163i			0.186134i
$797$			965.218i			1.21106i	0.795821	+	0.605532i	$0.207040\pi$
$797$			965.218i			1.21106i	−0.795821	+	0.605532i	$0.792960\pi$
$798$		0			0
$799$		−	10.4511i		−	0.0130802i
$800$	28.2843			0.0353553
$801$		0			0
$802$			420.731i			0.524602i
$803$		−	542.034i		−	0.675011i
$804$		0			0
$805$	318.047	+	179.578i	0.395089	+	0.223078i
$806$	−315.790			−0.391799
$807$		0			0
$808$	291.608			0.360901
$809$	1310.48			1.61988			0.809939	−	0.586514i	$-0.199500\pi$
$809$	1310.48			1.61988			0.809939	+	0.586514i	$0.199500\pi$
$810$		0			0
$811$	−1174.00			−1.44760			−0.723799	−	0.690010i	$-0.757606\pi$
$811$	−1174.00			−1.44760			−0.723799	+	0.690010i	$0.757606\pi$
$812$			714.772i			0.880261i
$813$		0			0
$814$	−645.857			−0.793436
$815$		−	168.449i		−	0.206686i
$816$		0			0
$817$	2109.22			2.58166
$818$	−337.917			−0.413101
$819$		0			0
$820$			32.3800i			0.0394878i
$821$	1238.00			1.50791			0.753956	−	0.656925i	$-0.228143\pi$
$821$	1238.00			1.50791			0.753956	+	0.656925i	$0.228143\pi$
$822$		0			0
$823$	−937.653			−1.13931			−0.569656	−	0.821883i	$-0.692923\pi$
$823$	−937.653			−1.13931			−0.569656	+	0.821883i	$0.692923\pi$
$824$		−	279.629i		−	0.339355i
$825$		0			0
$826$		−	661.753i		−	0.801153i
$827$		−	199.863i		−	0.241672i	−0.992672	−	0.120836i	$-0.961443\pi$
$827$		−	199.863i		−	0.241672i	0.992672	−	0.120836i	$-0.0385575\pi$
$828$		0			0
$829$	−892.787			−1.07695			−0.538473	−	0.842643i	$-0.680998\pi$
$829$	−892.787			−1.07695			−0.538473	+	0.842643i	$0.680998\pi$
$830$	−323.845			−0.390175
$831$		0			0
$832$	160.478			0.192882
$833$			2.34170i			0.00281117i
$834$		0			0
$835$		−	609.236i		−	0.729624i
$836$	662.714			0.792721
$837$		0			0
$838$		−	279.046i		−	0.332991i
$839$			515.108i			0.613955i	0.951717	+	0.306977i	$0.0993176\pi$
$839$			515.108i			0.613955i	−0.951717	+	0.306977i	$0.900682\pi$
$840$		0			0
$841$	1691.44			2.01122
$842$			649.486i			0.771361i
$843$		0			0
$844$	−9.50033			−0.0112563
$845$			521.883i			0.617613i
$846$		0			0
$847$		−	41.8010i		−	0.0493519i
$848$			81.6590i			0.0962960i
$849$		0			0
$850$		−	11.5349i		−	0.0135704i
$851$	458.474	−	811.993i	0.538747	−	0.954163i
$852$		0			0
$853$	−483.197			−0.566467			−0.283234	−	0.959051i	$-0.591407\pi$
$853$	−483.197			−0.566467			−0.283234	+	0.959051i	$0.591407\pi$
$854$	−378.920			−0.443701
$855$		0			0
$856$			63.7792i			0.0745084i
$857$	−920.413			−1.07399			−0.536997	−	0.843584i	$-0.680441\pi$
$857$	−920.413			−1.07399			−0.536997	+	0.843584i	$0.680441\pi$
$858$		0			0
$859$	1173.30			1.36589			0.682947	−	0.730468i	$-0.260698\pi$
$859$	1173.30			1.36589			0.682947	+	0.730468i	$0.260698\pi$
$860$	320.661			0.372862
$861$		0			0
$862$			672.152i			0.779759i
$863$	−866.353			−1.00388			−0.501942	−	0.864901i	$-0.667381\pi$
$863$	−866.353			−1.00388			−0.501942	+	0.864901i	$0.667381\pi$
$864$		0			0
$865$		−	585.436i		−	0.676805i
$866$		−	981.901i		−	1.13384i
$867$		0			0
$868$			158.109i			0.182153i
$869$	1138.00			1.30955
$870$		0			0
$871$		−	2492.27i		−	2.86139i
$872$			85.5286i			0.0980833i
$873$		0			0
$874$	−470.440	+	833.186i	−0.538261	+	0.953303i
$875$	79.4005			0.0907434
$876$		0			0
$877$	281.590			0.321084			0.160542	−	0.987029i	$-0.448676\pi$
$877$	281.590			0.321084			0.160542	+	0.987029i	$0.448676\pi$
$878$	978.823			1.11483
$879$		0			0
$880$	100.752			0.114490
$881$		−	919.123i		−	1.04327i	−0.853168	−	0.521636i	$-0.825322\pi$
$881$		−	919.123i		−	1.04327i	0.853168	−	0.521636i	$-0.174678\pi$
$882$		0			0
$883$	−821.800			−0.930691			−0.465346	−	0.885129i	$-0.654070\pi$
$883$	−821.800			−0.930691			−0.465346	+	0.885129i	$0.654070\pi$
$884$		−	65.4460i		−	0.0740340i
$885$		0			0
$886$	−398.901			−0.450226
$887$	−1280.91			−1.44409			−0.722047	−	0.691844i	$-0.756799\pi$
$887$	−1280.91			−1.44409			−0.722047	+	0.691844i	$0.756799\pi$
$888$		0			0
$889$		−	212.705i		−	0.239263i
$890$	30.4804			0.0342476
$891$		0			0
$892$	423.955			0.475286
$893$			188.461i			0.211043i
$894$		0			0
$895$			412.345i			0.460720i
$896$		−	80.3476i		−	0.0896737i
$897$		0			0
$898$	−6.38621			−0.00711160
$899$	560.180			0.623114
$900$		0			0
$901$	33.3022			0.0369613
$902$			115.341i			0.127873i
$903$		0			0
$904$		−	603.690i		−	0.667798i
$905$	429.027			0.474063
$906$		0			0
$907$			1726.39i			1.90341i	0.307021	+	0.951703i	$0.400668\pi$
$907$			1726.39i			1.90341i	−0.307021	+	0.951703i	$0.599332\pi$
$908$			779.882i			0.858901i
$909$		0			0
$910$	450.499			0.495053
$911$			791.171i			0.868464i	0.900801	+	0.434232i	$0.142980\pi$
$911$			791.171i			0.868464i	−0.900801	+	0.434232i	$0.857020\pi$
$912$		0			0
$913$	−1153.57			−1.26350
$914$			564.328i			0.617427i
$915$		0			0
$916$			312.001i			0.340612i
$917$			827.262i			0.902139i
$918$		0			0
$919$		−	1272.36i		−	1.38451i	−0.721655	−	0.692253i	$-0.756618\pi$
$919$		−	1272.36i		−	1.38451i	0.721655	−	0.692253i	$-0.243382\pi$
$920$	−71.5204	+	126.668i	−0.0777396	+	0.137683i
$921$		0			0
$922$	62.5842			0.0678787
$923$	−873.915			−0.946820
$924$		0			0
$925$		−	202.714i		−	0.219151i
$926$	945.060			1.02058
$927$		0			0
$928$	−284.672			−0.306758
$929$	−672.653			−0.724062			−0.362031	−	0.932166i	$-0.617917\pi$
$929$	−672.653			−0.724062			−0.362031	+	0.932166i	$0.617917\pi$
$930$		0			0
$931$		−	42.2273i		−	0.0453569i
$932$	92.9936			0.0997785
$933$		0			0
$934$			947.735i			1.01471i
$935$		−	41.0885i		−	0.0439449i
$936$		0			0
$937$		−	1599.57i		−	1.70711i	−0.520999	−	0.853557i	$-0.674440\pi$
$937$		−	1599.57i		−	1.70711i	0.520999	−	0.853557i	$-0.325560\pi$
$938$	−1247.82			−1.33030
$939$		0			0
$940$			28.6515i			0.0304803i
$941$		−	1628.06i		−	1.73014i	−0.501651	−	0.865070i	$-0.667274\pi$
$941$		−	1628.06i		−	1.73014i	0.501651	−	0.865070i	$-0.332726\pi$
$942$		0			0
$943$	−145.011	−	81.8770i	−0.153776	−	0.0868261i
$944$	263.556			0.279190
$945$		0			0
$946$	1142.23			1.20743
$947$	1829.88			1.93229			0.966147	−	0.257992i	$-0.0830608\pi$
$947$	1829.88			1.93229			0.966147	+	0.257992i	$0.0830608\pi$
$948$		0			0
$949$	965.262			1.01714
$950$			208.005i			0.218953i
$951$		0			0
$952$	−32.7673			−0.0344195
$953$		−	655.714i		−	0.688053i	−0.938960	−	0.344026i	$-0.888209\pi$
$953$		−	655.714i		−	0.688053i	0.938960	−	0.344026i	$-0.111791\pi$
$954$		0			0
$955$	72.9373			0.0763741
$956$	909.469			0.951328
$957$		0			0
$958$			439.102i			0.458353i
$959$	−673.400			−0.702190
$960$		0			0
$961$	−837.087			−0.871059
$962$		−	1150.15i		−	1.19558i
$963$		0			0
$964$		−	298.714i		−	0.309870i
$965$			708.271i			0.733960i
$966$		0			0
$967$	143.641			0.148543			0.0742716	−	0.997238i	$-0.476337\pi$
$967$	143.641			0.148543			0.0742716	+	0.997238i	$0.476337\pi$
$968$	16.6481			0.0171984
$969$		0			0
$970$	454.202			0.468250
$971$		−	1107.60i		−	1.14068i	−0.821408	−	0.570341i	$-0.806811\pi$
$971$		−	1107.60i		−	1.14068i	0.821408	−	0.570341i	$-0.193189\pi$
$972$		0			0
$973$			634.030i			0.651624i
$974$	−1173.29			−1.20461
$975$		0			0
$976$		−	150.912i		−	0.154623i
$977$			626.322i			0.641066i	0.947237	+	0.320533i	$0.103862\pi$
$977$			626.322i			0.641066i	−0.947237	+	0.320533i	$0.896138\pi$
$978$		0			0
$979$	108.574			0.110903
$980$		−	6.41976i		−	0.00655077i
$981$		0			0
$982$	−175.071			−0.178280
$983$		−	202.538i		−	0.206041i	−0.994679	−	0.103020i	$-0.967149\pi$
$983$		−	202.538i		−	0.206041i	0.994679	−	0.103020i	$-0.0328507\pi$
$984$		0			0
$985$			435.912i			0.442551i
$986$			116.095i			0.117743i
$987$		0			0
$988$			1180.17i			1.19450i
$989$	−810.833	+	1436.05i	−0.819851	+	1.45202i
$990$		0			0
$991$	354.302			0.357520			0.178760	−	0.983893i	$-0.442791\pi$
$991$	354.302			0.357520			0.178760	+	0.983893i	$0.442791\pi$
$992$	−62.9699			−0.0634777
$993$		0			0
$994$			437.549i			0.440190i
$995$	−165.651			−0.166484
$996$		0			0
$997$	184.596			0.185152			0.0925758	−	0.995706i	$-0.470490\pi$
$997$	184.596			0.185152			0.0925758	+	0.995706i	$0.470490\pi$
$998$	−1071.86			−1.07401
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.3.c.a.91.8		16
3.2	odd	2		230.3.d.a.91.9	✓	16
12.11	even	2		1840.3.k.d.321.13		16
15.2	even	4		1150.3.c.c.1149.10		32
15.8	even	4		1150.3.c.c.1149.23		32
15.14	odd	2		1150.3.d.b.551.7		16
23.22	odd	2	inner	2070.3.c.a.91.1		16
69.68	even	2		230.3.d.a.91.10	yes	16
276.275	odd	2		1840.3.k.d.321.14		16
345.68	odd	4		1150.3.c.c.1149.9		32
345.137	odd	4		1150.3.c.c.1149.24		32
345.344	even	2		1150.3.d.b.551.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.3.d.a.91.9	✓	16	3.2	odd	2
230.3.d.a.91.10	yes	16	69.68	even	2
1150.3.c.c.1149.9		32	345.68	odd	4
1150.3.c.c.1149.10		32	15.2	even	4
1150.3.c.c.1149.23		32	15.8	even	4
1150.3.c.c.1149.24		32	345.137	odd	4
1150.3.d.b.551.7		16	15.14	odd	2
1150.3.d.b.551.8		16	345.344	even	2
1840.3.k.d.321.13		16	12.11	even	2
1840.3.k.d.321.14		16	276.275	odd	2
2070.3.c.a.91.1		16	23.22	odd	2	inner
2070.3.c.a.91.8		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)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +7.10180i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +7.10180i q^{7} -2.82843 q^{8} -3.16228i q^{10} -11.2644i q^{11} +20.0597 q^{13} -10.0435i q^{14} +4.00000 q^{16} -1.63128i q^{17} +29.4164i q^{19} +4.47214i q^{20} +15.9302i q^{22} +(-20.0280 - 11.3084i) q^{23} -5.00000 q^{25} -28.3688 q^{26} +14.2036i q^{28} +50.3233 q^{29} +11.1316 q^{31} -5.65685 q^{32} +2.30698i q^{34} -15.8801 q^{35} +40.5429i q^{37} -41.6011i q^{38} -6.32456i q^{40} +7.24039 q^{41} -71.7020i q^{43} -22.5287i q^{44} +(28.3239 + 15.9924i) q^{46} +6.40666 q^{47} -1.43550 q^{49} +7.07107 q^{50} +40.1195 q^{52} +20.4148i q^{53} +25.1879 q^{55} -20.0869i q^{56} -71.1679 q^{58} +65.8889 q^{59} -37.7281i q^{61} -15.7425 q^{62} +8.00000 q^{64} +44.8549i q^{65} -124.242i q^{67} -3.26256i q^{68} +22.4578 q^{70} -43.5656 q^{71} +48.1194 q^{73} -57.3363i q^{74} +58.8328i q^{76} +79.9972 q^{77} +101.026i q^{79} +8.94427i q^{80} -10.2395 q^{82} -102.409i q^{83} +3.64765 q^{85} +101.402i q^{86} +31.8604i q^{88} +9.63875i q^{89} +142.460i q^{91} +(-40.0560 - 22.6167i) q^{92} -9.06039 q^{94} -65.7771 q^{95} +143.631i q^{97} +2.03010 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4}+O(q^{10}) \) 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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