LMFDB - Embedded newform 2070.3.c.b.91.1

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			91.1
Character	$\chi$	$=$	2070.91
Dual form			2070.3.c.b.91.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} -11.4203i q^{7} -2.82843 q^{8} +O(q^{10})$	\(q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} -11.4203i q^{7} -2.82843 q^{8} +3.16228i q^{10} +13.8723i q^{11} -2.98777 q^{13} +16.1508i q^{14} +4.00000 q^{16} +15.3412i q^{17} +2.52942i q^{19} -4.47214i q^{20} -19.6184i q^{22} +(20.9247 - 9.54768i) q^{23} -5.00000 q^{25} +4.22535 q^{26} -22.8407i q^{28} -11.6231 q^{29} +2.71207 q^{31} -5.65685 q^{32} -21.6958i q^{34} -25.5367 q^{35} -31.7707i q^{37} -3.57714i q^{38} +6.32456i q^{40} -36.3226 q^{41} +19.9718i q^{43} +27.7446i q^{44} +(-29.5920 + 13.5025i) q^{46} -52.2318 q^{47} -81.4243 q^{49} +7.07107 q^{50} -5.97554 q^{52} -59.4952i q^{53} +31.0194 q^{55} +32.3016i q^{56} +16.4376 q^{58} -81.6065 q^{59} -42.2136i q^{61} -3.83545 q^{62} +8.00000 q^{64} +6.68086i q^{65} +58.5139i q^{67} +30.6825i q^{68} +36.1143 q^{70} -100.588 q^{71} -16.7595 q^{73} +44.9306i q^{74} +5.05884i q^{76} +158.427 q^{77} +60.6495i q^{79} -8.94427i q^{80} +51.3679 q^{82} -44.1530i q^{83} +34.3041 q^{85} -28.2444i q^{86} -39.2368i q^{88} +106.245i q^{89} +34.1214i q^{91} +(41.8493 - 19.0954i) q^{92} +73.8669 q^{94} +5.65595 q^{95} +37.9918i q^{97} +115.151 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	11.4203i		−	1.63148i	−0.578420	−	0.815739i	$-0.696331\pi$
$7$		−	11.4203i		−	1.63148i	0.578420	−	0.815739i	$-0.303669\pi$
$8$	−2.82843			−0.353553
$9$		0			0
$10$			3.16228i			0.316228i
$11$			13.8723i			1.26112i	0.776141	+	0.630559i	$0.217175\pi$
$11$			13.8723i			1.26112i	−0.776141	+	0.630559i	$0.782825\pi$
$12$		0			0
$13$	−2.98777			−0.229829			−0.114914	−	0.993375i	$-0.536659\pi$
$13$	−2.98777			−0.229829			−0.114914	+	0.993375i	$0.536659\pi$
$14$			16.1508i			1.15363i
$15$		0			0
$16$	4.00000			0.250000
$17$			15.3412i			0.902426i	0.892416	+	0.451213i	$0.149009\pi$
$17$			15.3412i			0.902426i	−0.892416	+	0.451213i	$0.850991\pi$
$18$		0			0
$19$			2.52942i			0.133127i	0.997782	+	0.0665636i	$0.0212035\pi$
$19$			2.52942i			0.133127i	−0.997782	+	0.0665636i	$0.978796\pi$
$20$		−	4.47214i		−	0.223607i
$21$		0			0
$22$		−	19.6184i		−	0.891746i
$23$	20.9247	−	9.54768i	0.909768	−	0.415117i
$23$	20.9247	−	9.54768i	0.909768	−	0.415117i
$24$		0			0
$25$	−5.00000			−0.200000
$26$	4.22535			0.162513
$27$		0			0
$28$		−	22.8407i		−	0.815739i
$29$	−11.6231			−0.400798			−0.200399	−	0.979714i	$-0.564224\pi$
$29$	−11.6231			−0.400798			−0.200399	+	0.979714i	$0.564224\pi$
$30$		0			0
$31$	2.71207			0.0874861			0.0437431	−	0.999043i	$-0.486072\pi$
$31$	2.71207			0.0874861			0.0437431	+	0.999043i	$0.486072\pi$
$32$	−5.65685			−0.176777
$33$		0			0
$34$		−	21.6958i		−	0.638112i
$35$	−25.5367			−0.729619
$36$		0			0
$37$		−	31.7707i		−	0.858668i	−0.903146	−	0.429334i	$-0.858748\pi$
$37$		−	31.7707i		−	0.858668i	0.903146	−	0.429334i	$-0.141252\pi$
$38$		−	3.57714i		−	0.0941352i
$39$		0			0
$40$			6.32456i			0.158114i
$41$	−36.3226			−0.885917			−0.442959	−	0.896542i	$-0.646071\pi$
$41$	−36.3226			−0.885917			−0.442959	+	0.896542i	$0.646071\pi$
$42$		0			0
$43$			19.9718i			0.464461i	0.972661	+	0.232231i	$0.0746024\pi$
$43$			19.9718i			0.464461i	−0.972661	+	0.232231i	$0.925398\pi$
$44$			27.7446i			0.630559i
$45$		0			0
$46$	−29.5920	+	13.5025i	−0.643303	+	0.293532i
$47$	−52.2318			−1.11131			−0.555657	−	0.831411i	$-0.687533\pi$
$47$	−52.2318			−1.11131			−0.555657	+	0.831411i	$0.687533\pi$
$48$		0			0
$49$	−81.4243			−1.66172
$50$	7.07107			0.141421
$51$		0			0
$52$	−5.97554			−0.114914
$53$		−	59.4952i		−	1.12255i	−0.827629	−	0.561275i	$-0.810311\pi$
$53$		−	59.4952i		−	1.12255i	0.827629	−	0.561275i	$-0.189689\pi$
$54$		0			0
$55$	31.0194			0.563990
$56$			32.3016i			0.576815i
$57$		0			0
$58$	16.4376			0.283407
$59$	−81.6065			−1.38316			−0.691581	−	0.722299i	$-0.743085\pi$
$59$	−81.6065			−1.38316			−0.691581	+	0.722299i	$0.743085\pi$
$60$		0			0
$61$		−	42.2136i		−	0.692026i	−0.938230	−	0.346013i	$-0.887535\pi$
$61$		−	42.2136i		−	0.692026i	0.938230	−	0.346013i	$-0.112465\pi$
$62$	−3.83545			−0.0618620
$63$		0			0
$64$	8.00000			0.125000
$65$			6.68086i			0.102782i
$66$		0			0
$67$			58.5139i			0.873342i	0.899621	+	0.436671i	$0.143843\pi$
$67$			58.5139i			0.873342i	−0.899621	+	0.436671i	$0.856157\pi$
$68$			30.6825i			0.451213i
$69$		0			0
$70$	36.1143			0.515919
$71$	−100.588			−1.41673			−0.708367	−	0.705844i	$-0.750568\pi$
$71$	−100.588			−1.41673			−0.708367	+	0.705844i	$0.750568\pi$
$72$		0			0
$73$	−16.7595			−0.229582			−0.114791	−	0.993390i	$-0.536620\pi$
$73$	−16.7595			−0.229582			−0.114791	+	0.993390i	$0.536620\pi$
$74$			44.9306i			0.607170i
$75$		0			0
$76$			5.05884i			0.0665636i
$77$	158.427			2.05749
$78$		0			0
$79$			60.6495i			0.767715i	0.923392	+	0.383858i	$0.125405\pi$
$79$			60.6495i			0.767715i	−0.923392	+	0.383858i	$0.874595\pi$
$80$		−	8.94427i		−	0.111803i
$81$		0			0
$82$	51.3679			0.626438
$83$		−	44.1530i		−	0.531964i	−0.963978	−	0.265982i	$-0.914304\pi$
$83$		−	44.1530i		−	0.531964i	0.963978	−	0.265982i	$-0.0856962\pi$
$84$		0			0
$85$	34.3041			0.403577
$86$		−	28.2444i		−	0.328424i
$87$		0			0
$88$		−	39.2368i		−	0.445873i
$89$			106.245i			1.19376i	0.802330	+	0.596881i	$0.203594\pi$
$89$			106.245i			1.19376i	−0.802330	+	0.596881i	$0.796406\pi$
$90$		0			0
$91$			34.1214i			0.374960i
$92$	41.8493	−	19.0954i	0.454884	−	0.207558i
$93$		0			0
$94$	73.8669			0.785818
$95$	5.65595			0.0595363
$96$		0			0
$97$			37.9918i			0.391668i	0.980637	+	0.195834i	$0.0627414\pi$
$97$			37.9918i			0.391668i	−0.980637	+	0.195834i	$0.937259\pi$
$98$	115.151			1.17501
$99$		0			0
$100$	−10.0000			−0.100000
$101$	147.465			1.46005			0.730024	−	0.683422i	$-0.239509\pi$
$101$	147.465			1.46005			0.730024	+	0.683422i	$0.239509\pi$
$102$		0			0
$103$			74.8003i			0.726217i	0.931747	+	0.363108i	$0.118285\pi$
$103$			74.8003i			0.726217i	−0.931747	+	0.363108i	$0.881715\pi$
$104$	8.45069			0.0812567
$105$		0			0
$106$			84.1389i			0.793763i
$107$		−	36.8895i		−	0.344762i	−0.985030	−	0.172381i	$-0.944854\pi$
$107$		−	36.8895i		−	0.344762i	0.985030	−	0.172381i	$-0.0551460\pi$
$108$		0			0
$109$			89.1884i			0.818242i	0.912480	+	0.409121i	$0.134165\pi$
$109$			89.1884i			0.818242i	−0.912480	+	0.409121i	$0.865835\pi$
$110$	−43.8681			−0.398801
$111$		0			0
$112$		−	45.6814i		−	0.407869i
$113$			39.7138i			0.351450i	0.984439	+	0.175725i	$0.0562269\pi$
$113$			39.7138i			0.351450i	−0.984439	+	0.175725i	$0.943773\pi$
$114$		0			0
$115$	−21.3493	−	46.7890i	−0.185646	−	0.406861i
$116$	−23.2463			−0.200399
$117$		0			0
$118$	115.409			0.978043
$119$	175.202			1.47229
$120$		0			0
$121$	−71.4409			−0.590421
$122$			59.6990i			0.489336i
$123$		0			0
$124$	5.42414			0.0437431
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	−130.656			−1.02878			−0.514392	−	0.857555i	$-0.671982\pi$
$127$	−130.656			−1.02878			−0.514392	+	0.857555i	$0.671982\pi$
$128$	−11.3137			−0.0883883
$129$		0			0
$130$		−	9.44816i		−	0.0726782i
$131$	−2.98786			−0.0228081			−0.0114041	−	0.999935i	$-0.503630\pi$
$131$	−2.98786			−0.0228081			−0.0114041	+	0.999935i	$0.503630\pi$
$132$		0			0
$133$	28.8868			0.217194
$134$		−	82.7512i		−	0.617546i
$135$		0			0
$136$		−	43.3916i		−	0.319056i
$137$			120.177i			0.877207i	0.898681	+	0.438603i	$0.144527\pi$
$137$			120.177i			0.877207i	−0.898681	+	0.438603i	$0.855473\pi$
$138$		0			0
$139$	−127.272			−0.915628			−0.457814	−	0.889048i	$-0.651367\pi$
$139$	−127.272			−0.915628			−0.457814	+	0.889048i	$0.651367\pi$
$140$	−51.0733			−0.364810
$141$		0			0
$142$	142.253			1.00178
$143$		−	41.4473i		−	0.289841i
$144$		0			0
$145$			25.9901i			0.179242i
$146$	23.7015			0.162339
$147$		0			0
$148$		−	63.5414i		−	0.429334i
$149$			131.641i			0.883494i	0.897140	+	0.441747i	$0.145641\pi$
$149$			131.641i			0.883494i	−0.897140	+	0.441747i	$0.854359\pi$
$150$		0			0
$151$	149.135			0.987650			0.493825	−	0.869561i	$-0.335598\pi$
$151$	149.135			0.987650			0.493825	+	0.869561i	$0.335598\pi$
$152$		−	7.15427i		−	0.0470676i
$153$		0			0
$154$	−224.049			−1.45486
$155$		−	6.06437i		−	0.0391250i
$156$		0			0
$157$			228.188i			1.45343i	0.686941	+	0.726713i	$0.258953\pi$
$157$			228.188i			1.45343i	−0.686941	+	0.726713i	$0.741047\pi$
$158$		−	85.7713i		−	0.542857i
$159$		0			0
$160$			12.6491i			0.0790569i
$161$	−109.038	−	238.967i	−0.677253	−	1.48427i
$162$		0			0
$163$	260.536			1.59838			0.799191	−	0.601077i	$-0.205261\pi$
$163$	260.536			1.59838			0.799191	+	0.601077i	$0.205261\pi$
$164$	−72.6452			−0.442959
$165$		0			0
$166$			62.4418i			0.376155i
$167$	177.110			1.06054			0.530271	−	0.847828i	$-0.322090\pi$
$167$	177.110			1.06054			0.530271	+	0.847828i	$0.322090\pi$
$168$		0			0
$169$	−160.073			−0.947179
$170$	−48.5133			−0.285372
$171$		0			0
$172$			39.9437i			0.232231i
$173$	−7.60898			−0.0439825			−0.0219913	−	0.999758i	$-0.507001\pi$
$173$	−7.60898			−0.0439825			−0.0219913	+	0.999758i	$0.507001\pi$
$174$		0			0
$175$			57.1017i			0.326296i
$176$			55.4892i			0.315280i
$177$		0			0
$178$		−	150.253i		−	0.844118i
$179$	−319.384			−1.78427			−0.892133	−	0.451772i	$-0.850792\pi$
$179$	−319.384			−1.78427			−0.892133	+	0.451772i	$0.850792\pi$
$180$		0			0
$181$			115.161i			0.636248i	0.948049	+	0.318124i	$0.103053\pi$
$181$			115.161i			0.636248i	−0.948049	+	0.318124i	$0.896947\pi$
$182$		−	48.2549i		−	0.265137i
$183$		0			0
$184$	−59.1839	+	27.0049i	−0.321652	+	0.146766i
$185$	−71.0414			−0.384008
$186$		0			0
$187$	−212.818			−1.13807
$188$	−104.464			−0.555657
$189$		0			0
$190$	−7.99872			−0.0420985
$191$		−	104.681i		−	0.548068i	−0.961720	−	0.274034i	$-0.911642\pi$
$191$		−	104.681i		−	0.548068i	0.961720	−	0.274034i	$-0.0883581\pi$
$192$		0			0
$193$	58.0026			0.300532			0.150266	−	0.988646i	$-0.451987\pi$
$193$	58.0026			0.300532			0.150266	+	0.988646i	$0.451987\pi$
$194$		−	53.7285i		−	0.276951i
$195$		0			0
$196$	−162.849			−0.830860
$197$	−96.0748			−0.487689			−0.243845	−	0.969814i	$-0.578409\pi$
$197$	−96.0748			−0.487689			−0.243845	+	0.969814i	$0.578409\pi$
$198$		0			0
$199$			198.494i			0.997459i	0.866758	+	0.498729i	$0.166200\pi$
$199$			198.494i			0.997459i	−0.866758	+	0.498729i	$0.833800\pi$
$200$	14.1421			0.0707107
$201$		0			0
$202$	−208.547			−1.03241
$203$			132.740i			0.653893i
$204$		0			0
$205$			81.2198i			0.396194i
$206$		−	105.784i		−	0.513513i
$207$		0			0
$208$	−11.9511			−0.0574571
$209$	−35.0889			−0.167889
$210$		0			0
$211$	−154.074			−0.730207			−0.365103	−	0.930967i	$-0.618966\pi$
$211$	−154.074			−0.730207			−0.365103	+	0.930967i	$0.618966\pi$
$212$		−	118.990i		−	0.561275i
$213$		0			0
$214$			52.1697i			0.243784i
$215$	44.6584			0.207713
$216$		0			0
$217$		−	30.9728i		−	0.142732i
$218$		−	126.131i		−	0.578584i
$219$		0			0
$220$	62.0388			0.281995
$221$		−	45.8361i		−	0.207403i
$222$		0			0
$223$	−88.2708			−0.395833			−0.197916	−	0.980219i	$-0.563417\pi$
$223$	−88.2708			−0.395833			−0.197916	+	0.980219i	$0.563417\pi$
$224$			64.6032i			0.288407i
$225$		0			0
$226$		−	56.1638i		−	0.248512i
$227$			220.495i			0.971345i	0.874141	+	0.485673i	$0.161425\pi$
$227$			220.495i			0.971345i	−0.874141	+	0.485673i	$0.838575\pi$
$228$		0			0
$229$		−	138.113i		−	0.603115i	−0.953448	−	0.301557i	$-0.902494\pi$
$229$		−	138.113i		−	0.603115i	0.953448	−	0.301557i	$-0.0975065\pi$
$230$	30.1924	+	66.1696i	0.131271	+	0.287694i
$231$		0			0
$232$	32.8752			0.141703
$233$	−98.1877			−0.421406			−0.210703	−	0.977550i	$-0.567575\pi$
$233$	−98.1877			−0.421406			−0.210703	+	0.977550i	$0.567575\pi$
$234$		0			0
$235$			116.794i			0.496995i
$236$	−163.213			−0.691581
$237$		0			0
$238$	−247.773			−1.04106
$239$	421.219			1.76242			0.881211	−	0.472722i	$-0.156729\pi$
$239$	421.219			1.76242			0.881211	+	0.472722i	$0.156729\pi$
$240$		0			0
$241$			181.752i			0.754160i	0.926181	+	0.377080i	$0.123072\pi$
$241$			181.752i			0.754160i	−0.926181	+	0.377080i	$0.876928\pi$
$242$	101.033			0.417491
$243$		0			0
$244$		−	84.4272i		−	0.346013i
$245$			182.070i			0.743144i
$246$		0			0
$247$		−	7.55732i		−	0.0305964i
$248$	−7.67089			−0.0309310
$249$		0			0
$250$		−	15.8114i		−	0.0632456i
$251$		−	208.037i		−	0.828835i	−0.910087	−	0.414417i	$-0.863985\pi$
$251$		−	208.037i		−	0.828835i	0.910087	−	0.414417i	$-0.136015\pi$
$252$		0			0
$253$	132.448	+	290.273i	0.523511	+	1.14733i
$254$	184.775			0.727460
$255$		0			0
$256$	16.0000			0.0625000
$257$	463.102			1.80196			0.900978	−	0.433866i	$-0.142851\pi$
$257$	463.102			1.80196			0.900978	+	0.433866i	$0.142851\pi$
$258$		0			0
$259$	−362.832			−1.40090
$260$			13.3617i			0.0513912i
$261$		0			0
$262$	4.22548			0.0161278
$263$		−	34.0014i		−	0.129283i	−0.997909	−	0.0646415i	$-0.979410\pi$
$263$		−	34.0014i		−	0.129283i	0.997909	−	0.0646415i	$-0.0205904\pi$
$264$		0			0
$265$	−133.035			−0.502020
$266$	−40.8521			−0.153579
$267$		0			0
$268$			117.028i			0.436671i
$269$	−499.913			−1.85841			−0.929207	−	0.369560i	$-0.879509\pi$
$269$	−499.913			−1.85841			−0.929207	+	0.369560i	$0.879509\pi$
$270$		0			0
$271$	246.666			0.910207			0.455104	−	0.890439i	$-0.349602\pi$
$271$	246.666			0.910207			0.455104	+	0.890439i	$0.349602\pi$
$272$			61.3650i			0.225607i
$273$		0			0
$274$		−	169.956i		−	0.620279i
$275$		−	69.3615i		−	0.252224i
$276$		0			0
$277$	−350.841			−1.26657			−0.633287	−	0.773917i	$-0.718295\pi$
$277$	−350.841			−1.26657			−0.633287	+	0.773917i	$0.718295\pi$
$278$	179.990			0.647447
$279$		0			0
$280$	72.2286			0.257959
$281$			232.377i			0.826965i	0.910512	+	0.413482i	$0.135688\pi$
$281$			232.377i			0.826965i	−0.910512	+	0.413482i	$0.864312\pi$
$282$		0			0
$283$			462.787i			1.63529i	0.575723	+	0.817645i	$0.304721\pi$
$283$			462.787i			1.63529i	−0.575723	+	0.817645i	$0.695279\pi$
$284$	−201.176			−0.708367
$285$		0			0
$286$			58.6153i			0.204949i
$287$			414.817i			1.44535i
$288$		0			0
$289$	53.6463			0.185627
$290$		−	36.7556i		−	0.126743i
$291$		0			0
$292$	−33.5189			−0.114791
$293$			163.785i			0.558995i	0.960146	+	0.279497i	$0.0901678\pi$
$293$			163.785i			0.558995i	−0.960146	+	0.279497i	$0.909832\pi$
$294$		0			0
$295$			182.478i			0.618568i
$296$			89.8611i			0.303585i
$297$		0			0
$298$		−	186.168i		−	0.624725i
$299$	−62.5181	+	28.5263i	−0.209091	+	0.0954056i
$300$		0			0
$301$	228.085			0.757758
$302$	−210.909			−0.698374
$303$		0			0
$304$			10.1177i			0.0332818i
$305$	−94.3924			−0.309483
$306$		0			0
$307$	−359.655			−1.17152			−0.585758	−	0.810486i	$-0.699203\pi$
$307$	−359.655			−1.17152			−0.585758	+	0.810486i	$0.699203\pi$
$308$	316.853			1.02874
$309$		0			0
$310$			8.57632i			0.0276655i
$311$	−314.588			−1.01154			−0.505768	−	0.862669i	$-0.668791\pi$
$311$	−314.588			−1.01154			−0.505768	+	0.862669i	$0.668791\pi$
$312$		0			0
$313$			451.729i			1.44322i	0.692298	+	0.721612i	$0.256599\pi$
$313$			451.729i			1.44322i	−0.692298	+	0.721612i	$0.743401\pi$
$314$		−	322.706i		−	1.02773i
$315$		0			0
$316$			121.299i			0.383858i
$317$	−179.052			−0.564834			−0.282417	−	0.959292i	$-0.591136\pi$
$317$	−179.052			−0.564834			−0.282417	+	0.959292i	$0.591136\pi$
$318$		0			0
$319$		−	161.240i		−	0.505454i
$320$		−	17.8885i		−	0.0559017i
$321$		0			0
$322$	154.203	+	337.950i	0.478891	+	1.04954i
$323$	−38.8044			−0.120138
$324$		0			0
$325$	14.9389			0.0459657
$326$	−368.454			−1.13023
$327$		0			0
$328$	102.736			0.313219
$329$			596.505i			1.81309i
$330$		0			0
$331$	528.798			1.59758			0.798788	−	0.601612i	$-0.205475\pi$
$331$	528.798			1.59758			0.798788	+	0.601612i	$0.205475\pi$
$332$		−	88.3060i		−	0.265982i
$333$		0			0
$334$	−250.472			−0.749916
$335$	130.841			0.390570
$336$		0			0
$337$		−	493.496i		−	1.46438i	−0.681100	−	0.732191i	$-0.738498\pi$
$337$		−	493.496i		−	1.46438i	0.681100	−	0.732191i	$-0.261502\pi$
$338$	226.378			0.669757
$339$		0			0
$340$	68.6081			0.201789
$341$			37.6227i			0.110330i
$342$		0			0
$343$			370.297i			1.07958i
$344$		−	56.4889i		−	0.164212i
$345$		0			0
$346$	10.7607			0.0311003
$347$	−133.171			−0.383778			−0.191889	−	0.981417i	$-0.561461\pi$
$347$	−133.171			−0.383778			−0.191889	+	0.981417i	$0.561461\pi$
$348$		0			0
$349$	62.9714			0.180434			0.0902169	−	0.995922i	$-0.471244\pi$
$349$	62.9714			0.180434			0.0902169	+	0.995922i	$0.471244\pi$
$350$		−	80.7540i		−	0.230726i
$351$		0			0
$352$		−	78.4736i		−	0.222936i
$353$	103.337			0.292739			0.146370	−	0.989230i	$-0.453241\pi$
$353$	103.337			0.292739			0.146370	+	0.989230i	$0.453241\pi$
$354$		0			0
$355$			224.922i			0.633583i
$356$			212.490i			0.596881i
$357$		0			0
$358$	451.677			1.26167
$359$		−	608.721i		−	1.69560i	−0.530314	−	0.847801i	$-0.677926\pi$
$359$		−	608.721i		−	1.69560i	0.530314	−	0.847801i	$-0.322074\pi$
$360$		0			0
$361$	354.602			0.982277
$362$		−	162.862i		−	0.449895i
$363$		0			0
$364$			68.2428i			0.187480i
$365$			37.4753i			0.102672i
$366$		0			0
$367$		−	213.718i		−	0.582338i	−0.956672	−	0.291169i	$-0.905956\pi$
$367$		−	213.718i		−	0.582338i	0.956672	−	0.291169i	$-0.0940441\pi$
$368$	83.6987	−	38.1907i	0.227442	−	0.103779i
$369$		0			0
$370$	100.468			0.271535
$371$	−679.455			−1.83142
$372$		0			0
$373$			542.556i			1.45457i	0.686334	+	0.727287i	$0.259219\pi$
$373$			542.556i			1.45457i	−0.686334	+	0.727287i	$0.740781\pi$
$374$	300.971			0.804735
$375$		0			0
$376$	147.734			0.392909
$377$	34.7273			0.0921148
$378$		0			0
$379$		−	348.769i		−	0.920235i	−0.887858	−	0.460117i	$-0.847807\pi$
$379$		−	348.769i		−	0.920235i	0.887858	−	0.460117i	$-0.152193\pi$
$380$	11.3119			0.0297682
$381$		0			0
$382$			148.041i			0.387543i
$383$			672.210i			1.75512i	0.479470	+	0.877559i	$0.340829\pi$
$383$			672.210i			1.75512i	−0.479470	+	0.877559i	$0.659171\pi$
$384$		0			0
$385$		−	354.253i		−	0.920136i
$386$	−82.0281			−0.212508
$387$		0			0
$388$			75.9836i			0.195834i
$389$			464.622i			1.19440i	0.802092	+	0.597200i	$0.203720\pi$
$389$			464.622i			1.19440i	−0.802092	+	0.597200i	$0.796280\pi$
$390$		0			0
$391$	146.473	+	321.010i	0.374612	+	0.820999i
$392$	230.303			0.587507
$393$		0			0
$394$	135.870			0.344848
$395$	135.616			0.343333
$396$		0			0
$397$	−1.92370			−0.00484560			−0.00242280	−	0.999997i	$-0.500771\pi$
$397$	−1.92370			−0.00484560			−0.00242280	+	0.999997i	$0.500771\pi$
$398$		−	280.713i		−	0.705310i
$399$		0			0
$400$	−20.0000			−0.0500000
$401$			417.281i			1.04060i	0.853983	+	0.520301i	$0.174180\pi$
$401$			417.281i			1.04060i	−0.853983	+	0.520301i	$0.825820\pi$
$402$		0			0
$403$	−8.10305			−0.0201068
$404$	294.930			0.730024
$405$		0			0
$406$		−	187.723i		−	0.462372i
$407$	440.733			1.08288
$408$		0			0
$409$	−812.820			−1.98733			−0.993667	−	0.112361i	$-0.964159\pi$
$409$	−812.820			−1.98733			−0.993667	+	0.112361i	$0.964159\pi$
$410$		−	114.862i		−	0.280152i
$411$		0			0
$412$			149.601i			0.363108i
$413$			931.974i			2.25660i
$414$		0			0
$415$	−98.7291			−0.237901
$416$	16.9014			0.0406283
$417$		0			0
$418$	49.6232			0.118716
$419$		−	268.155i		−	0.639988i	−0.947420	−	0.319994i	$-0.896319\pi$
$419$		−	268.155i		−	0.639988i	0.947420	−	0.319994i	$-0.103681\pi$
$420$		0			0
$421$		−	174.098i		−	0.413533i	−0.978390	−	0.206767i	$-0.933706\pi$
$421$		−	174.098i		−	0.413533i	0.978390	−	0.206767i	$-0.0662941\pi$
$422$	217.893			0.516334
$423$		0			0
$424$			168.278i			0.396881i
$425$		−	76.7062i		−	0.180485i
$426$		0			0
$427$	−482.094			−1.12903
$428$		−	73.7791i		−	0.172381i
$429$		0			0
$430$	−63.1565			−0.146876
$431$		−	291.962i		−	0.677407i	−0.940893	−	0.338704i	$-0.890012\pi$
$431$		−	291.962i		−	0.677407i	0.940893	−	0.338704i	$-0.109988\pi$
$432$		0			0
$433$		−	439.111i		−	1.01411i	−0.861913	−	0.507056i	$-0.830734\pi$
$433$		−	439.111i		−	1.01411i	0.861913	−	0.507056i	$-0.169266\pi$
$434$			43.8021i			0.100927i
$435$		0			0
$436$			178.377i			0.409121i
$437$	24.1501	+	52.9272i	0.0552633	+	0.121115i
$438$		0			0
$439$	−62.0644			−0.141377			−0.0706884	−	0.997498i	$-0.522520\pi$
$439$	−62.0644			−0.141377			−0.0706884	+	0.997498i	$0.522520\pi$
$440$	−87.7362			−0.199400
$441$		0			0
$442$			64.8221i			0.146656i
$443$	261.415			0.590102			0.295051	−	0.955482i	$-0.404663\pi$
$443$	261.415			0.590102			0.295051	+	0.955482i	$0.404663\pi$
$444$		0			0
$445$	237.571			0.533867
$446$	124.834			0.279896
$447$		0			0
$448$		−	91.3628i		−	0.203935i
$449$	−431.325			−0.960634			−0.480317	−	0.877095i	$-0.659478\pi$
$449$	−431.325			−0.960634			−0.480317	+	0.877095i	$0.659478\pi$
$450$		0			0
$451$		−	503.878i		−	1.11725i
$452$			79.4276i			0.175725i
$453$		0			0
$454$		−	311.828i		−	0.686845i
$455$	76.2977			0.167687
$456$		0			0
$457$		−	734.574i		−	1.60738i	−0.595046	−	0.803692i	$-0.702866\pi$
$457$		−	734.574i		−	1.60738i	0.595046	−	0.803692i	$-0.297134\pi$
$458$			195.322i			0.426466i
$459$		0			0
$460$	−42.6985	−	93.5780i	−0.0928229	−	0.203430i
$461$	−626.021			−1.35796			−0.678982	−	0.734155i	$-0.737578\pi$
$461$	−626.021			−1.35796			−0.678982	+	0.734155i	$0.737578\pi$
$462$		0			0
$463$	−320.725			−0.692712			−0.346356	−	0.938103i	$-0.612581\pi$
$463$	−320.725			−0.692712			−0.346356	+	0.938103i	$0.612581\pi$
$464$	−46.4925			−0.100199
$465$		0			0
$466$	138.858			0.297979
$467$			148.416i			0.317807i	0.987294	+	0.158904i	$0.0507959\pi$
$467$			148.416i			0.317807i	−0.987294	+	0.158904i	$0.949204\pi$
$468$		0			0
$469$	668.249			1.42484
$470$		−	165.171i		−	0.351429i
$471$		0			0
$472$	230.818			0.489021
$473$	−277.055			−0.585741
$474$		0			0
$475$		−	12.6471i		−	0.0266255i
$476$	350.405			0.736144
$477$		0			0
$478$	−595.694			−1.24622
$479$		−	749.170i		−	1.56403i	−0.623260	−	0.782014i	$-0.714192\pi$
$479$		−	749.170i		−	1.56403i	0.623260	−	0.782014i	$-0.285808\pi$
$480$		0			0
$481$			94.9236i			0.197346i
$482$		−	257.037i		−	0.533271i
$483$		0			0
$484$	−142.882			−0.295210
$485$	84.9523			0.175159
$486$		0			0
$487$	289.122			0.593679			0.296839	−	0.954927i	$-0.404067\pi$
$487$	289.122			0.593679			0.296839	+	0.954927i	$0.404067\pi$
$488$			119.398i			0.244668i
$489$		0			0
$490$		−	257.486i		−	0.525482i
$491$	−768.805			−1.56579			−0.782897	−	0.622152i	$-0.786259\pi$
$491$	−768.805			−1.56579			−0.782897	+	0.622152i	$0.786259\pi$
$492$		0			0
$493$		−	178.313i		−	0.361690i
$494$			10.6877i			0.0216350i
$495$		0			0
$496$	10.8483			0.0218715
$497$			1148.75i			2.31137i
$498$		0			0
$499$	201.586			0.403981			0.201990	−	0.979388i	$-0.435259\pi$
$499$	201.586			0.403981			0.201990	+	0.979388i	$0.435259\pi$
$500$			22.3607i			0.0447214i
$501$		0			0
$502$			294.209i			0.586075i
$503$			385.180i			0.765766i	0.923797	+	0.382883i	$0.125069\pi$
$503$			385.180i			0.765766i	−0.923797	+	0.382883i	$0.874931\pi$
$504$		0			0
$505$		−	329.741i		−	0.652953i
$506$	−187.310	−	410.509i	−0.370178	−	0.811282i
$507$		0			0
$508$	−261.311			−0.514392
$509$	−563.072			−1.10623			−0.553116	−	0.833104i	$-0.686561\pi$
$509$	−563.072			−1.10623			−0.553116	+	0.833104i	$0.686561\pi$
$510$		0			0
$511$			191.399i			0.374557i
$512$	−22.6274			−0.0441942
$513$		0			0
$514$	−654.926			−1.27417
$515$	167.259			0.324774
$516$		0			0
$517$		−	724.576i		−	1.40150i
$518$	513.122			0.990584
$519$		0			0
$520$		−	18.8963i		−	0.0363391i
$521$		−	687.431i		−	1.31945i	−0.751509	−	0.659723i	$-0.770674\pi$
$521$		−	687.431i		−	1.31945i	0.751509	−	0.659723i	$-0.229326\pi$
$522$		0			0
$523$		−	580.256i		−	1.10948i	−0.832025	−	0.554738i	$-0.812819\pi$
$523$		−	580.256i		−	1.10948i	0.832025	−	0.554738i	$-0.187181\pi$
$524$	−5.97573			−0.0114041
$525$		0			0
$526$			48.0853i			0.0914169i
$527$			41.6065i			0.0789498i
$528$		0			0
$529$	346.684	−	399.564i	0.655357	−	0.755320i
$530$	188.140			0.354982
$531$		0			0
$532$	57.7737			0.108597
$533$	108.524			0.203609
$534$		0			0
$535$	−82.4875			−0.154182
$536$		−	165.502i		−	0.308773i
$537$		0			0
$538$	706.984			1.31410
$539$		−	1129.54i		−	2.09563i
$540$		0			0
$541$	148.687			0.274838			0.137419	−	0.990513i	$-0.456119\pi$
$541$	148.687			0.274838			0.137419	+	0.990513i	$0.456119\pi$
$542$	−348.839			−0.643614
$543$		0			0
$544$		−	86.7832i		−	0.159528i
$545$	199.431			0.365929
$546$		0			0
$547$	−648.445			−1.18546			−0.592729	−	0.805402i	$-0.701949\pi$
$547$	−648.445			−1.18546			−0.592729	+	0.805402i	$0.701949\pi$
$548$			240.355i			0.438603i
$549$		0			0
$550$			98.0920i			0.178349i
$551$		−	29.3998i		−	0.0533571i
$552$		0			0
$553$	692.638			1.25251
$554$	496.164			0.895603
$555$		0			0
$556$	−254.545			−0.457814
$557$		−	697.203i		−	1.25171i	−0.779939	−	0.625856i	$-0.784750\pi$
$557$		−	697.203i		−	1.25171i	0.779939	−	0.625856i	$-0.215250\pi$
$558$		0			0
$559$		−	59.6713i		−	0.106746i
$560$	−102.147			−0.182405
$561$		0			0
$562$		−	328.631i		−	0.584752i
$563$			890.248i			1.58126i	0.612295	+	0.790629i	$0.290246\pi$
$563$			890.248i			1.58126i	−0.612295	+	0.790629i	$0.709754\pi$
$564$		0			0
$565$	88.8028			0.157173
$566$		−	654.480i		−	1.15632i
$567$		0			0
$568$	284.506			0.500891
$569$			548.456i			0.963894i	0.876200	+	0.481947i	$0.160070\pi$
$569$			548.456i			0.963894i	−0.876200	+	0.481947i	$0.839930\pi$
$570$		0			0
$571$			479.794i			0.840270i	0.907462	+	0.420135i	$0.138017\pi$
$571$			479.794i			0.840270i	−0.907462	+	0.420135i	$0.861983\pi$
$572$		−	82.8946i		−	0.144921i
$573$		0			0
$574$		−	586.639i		−	1.02202i
$575$	−104.623	+	47.7384i	−0.181954	+	0.0830233i
$576$		0			0
$577$	12.3741			0.0214456			0.0107228	−	0.999943i	$-0.496587\pi$
$577$	12.3741			0.0214456			0.0107228	+	0.999943i	$0.496587\pi$
$578$	−75.8673			−0.131258
$579$		0			0
$580$			51.9802i			0.0896211i
$581$	−504.243			−0.867887
$582$		0			0
$583$	825.335			1.41567
$584$	47.4029			0.0811694
$585$		0			0
$586$		−	231.628i		−	0.395269i
$587$	−613.203			−1.04464			−0.522320	−	0.852750i	$-0.674933\pi$
$587$	−613.203			−1.04464			−0.522320	+	0.852750i	$0.674933\pi$
$588$		0			0
$589$			6.85996i			0.0116468i
$590$		−	258.062i		−	0.437394i
$591$		0			0
$592$		−	127.083i		−	0.214667i
$593$	208.251			0.351182			0.175591	−	0.984463i	$-0.443816\pi$
$593$	208.251			0.351182			0.175591	+	0.984463i	$0.443816\pi$
$594$		0			0
$595$		−	391.764i		−	0.658427i
$596$			263.281i			0.441747i
$597$		0			0
$598$	88.4140	−	40.3423i	0.147849	−	0.0674620i
$599$	−925.513			−1.54510			−0.772548	−	0.634956i	$-0.781018\pi$
$599$	−925.513			−1.54510			−0.772548	+	0.634956i	$0.781018\pi$
$600$		0			0
$601$	−922.104			−1.53428			−0.767142	−	0.641478i	$-0.778322\pi$
$601$	−922.104			−1.53428			−0.767142	+	0.641478i	$0.778322\pi$
$602$	−322.561			−0.535816
$603$		0			0
$604$	298.270			0.493825
$605$			159.747i			0.264044i
$606$		0			0
$607$	−73.7934			−0.121571			−0.0607853	−	0.998151i	$-0.519360\pi$
$607$	−73.7934			−0.121571			−0.0607853	+	0.998151i	$0.519360\pi$
$608$		−	14.3085i		−	0.0235338i
$609$		0			0
$610$	133.491			0.218838
$611$	156.057			0.255412
$612$		0			0
$613$		−	808.004i		−	1.31811i	−0.752093	−	0.659057i	$-0.770956\pi$
$613$		−	808.004i		−	1.31811i	0.752093	−	0.659057i	$-0.229044\pi$
$614$	508.629			0.828387
$615$		0			0
$616$	−448.098			−0.727432
$617$		−	484.018i		−	0.784470i	−0.919865	−	0.392235i	$-0.871702\pi$
$617$		−	484.018i		−	0.784470i	0.919865	−	0.392235i	$-0.128298\pi$
$618$		0			0
$619$		−	454.285i		−	0.733902i	−0.930240	−	0.366951i	$-0.880402\pi$
$619$		−	454.285i		−	0.733902i	0.930240	−	0.366951i	$-0.119598\pi$
$620$		−	12.1287i		−	0.0195625i
$621$		0			0
$622$	444.895			0.715265
$623$	1213.35			1.94760
$624$		0			0
$625$	25.0000			0.0400000
$626$		−	638.841i		−	1.02051i
$627$		0			0
$628$			456.376i			0.726713i
$629$	487.402			0.774884
$630$		0			0
$631$		−	385.780i		−	0.611380i	−0.952131	−	0.305690i	$-0.901113\pi$
$631$		−	385.780i		−	0.611380i	0.952131	−	0.305690i	$-0.0988870\pi$
$632$		−	171.543i		−	0.271428i
$633$		0			0
$634$	253.218			0.399398
$635$			292.155i			0.460086i
$636$		0			0
$637$	243.277			0.381911
$638$			228.027i			0.357410i
$639$		0			0
$640$			25.2982i			0.0395285i
$641$			1009.69i			1.57518i	0.616199	+	0.787591i	$0.288672\pi$
$641$			1009.69i			1.57518i	−0.616199	+	0.787591i	$0.711328\pi$
$642$		0			0
$643$		−	849.482i		−	1.32112i	−0.750772	−	0.660561i	$-0.770318\pi$
$643$		−	849.482i		−	1.32112i	0.750772	−	0.660561i	$-0.229682\pi$
$644$	−218.076	−	477.934i	−0.338627	−	0.742133i
$645$		0			0
$646$	54.8777			0.0849501
$647$	788.852			1.21924			0.609622	−	0.792692i	$-0.291321\pi$
$647$	788.852			1.21924			0.609622	+	0.792692i	$0.291321\pi$
$648$		0			0
$649$		−	1132.07i		−	1.74433i
$650$	−21.1267			−0.0325027
$651$		0			0
$652$	521.073			0.799191
$653$	912.671			1.39766			0.698830	−	0.715288i	$-0.253705\pi$
$653$	912.671			1.39766			0.698830	+	0.715288i	$0.253705\pi$
$654$		0			0
$655$			6.68107i			0.0102001i
$656$	−145.290			−0.221479
$657$		0			0
$658$		−	843.586i		−	1.28205i
$659$		−	845.536i		−	1.28306i	−0.767098	−	0.641530i	$-0.778300\pi$
$659$		−	845.536i		−	1.28306i	0.767098	−	0.641530i	$-0.221700\pi$
$660$		0			0
$661$			594.404i			0.899250i	0.893217	+	0.449625i	$0.148442\pi$
$661$			594.404i			0.899250i	−0.893217	+	0.449625i	$0.851558\pi$
$662$	−747.833			−1.12966
$663$		0			0
$664$			124.884i			0.188078i
$665$		−	64.5929i		−	0.0971322i
$666$		0			0
$667$	−243.210	+	110.974i	−0.364633	+	0.166378i
$668$	354.221			0.530271
$669$		0			0
$670$	−185.037			−0.276175
$671$	585.600			0.872727
$672$		0			0
$673$	−49.6946			−0.0738404			−0.0369202	−	0.999318i	$-0.511755\pi$
$673$	−49.6946			−0.0738404			−0.0369202	+	0.999318i	$0.511755\pi$
$674$			697.909i			1.03547i
$675$		0			0
$676$	−320.146			−0.473589
$677$		−	256.435i		−	0.378782i	−0.981902	−	0.189391i	$-0.939349\pi$
$677$		−	256.435i		−	0.378782i	0.981902	−	0.189391i	$-0.0606513\pi$
$678$		0			0
$679$	433.880			0.638998
$680$	−97.0265			−0.142686
$681$		0			0
$682$		−	53.2065i		−	0.0780154i
$683$	1344.91			1.96912			0.984560	−	0.175045i	$-0.0560072\pi$
$683$	1344.91			1.96912			0.984560	+	0.175045i	$0.0560072\pi$
$684$		0			0
$685$	268.725			0.392299
$686$		−	523.679i		−	0.763380i
$687$		0			0
$688$			79.8873i			0.116115i
$689$			177.758i			0.257994i
$690$		0			0
$691$	559.144			0.809180			0.404590	−	0.914498i	$-0.367414\pi$
$691$	559.144			0.809180			0.404590	+	0.914498i	$0.367414\pi$
$692$	−15.2180			−0.0219913
$693$		0			0
$694$	188.332			0.271372
$695$			284.589i			0.409481i
$696$		0			0
$697$		−	557.234i		−	0.799475i
$698$	−89.0550			−0.127586
$699$		0			0
$700$			114.203i			0.163148i
$701$			1187.16i			1.69352i	0.531977	+	0.846758i	$0.321449\pi$
$701$			1187.16i			1.69352i	−0.531977	+	0.846758i	$0.678551\pi$
$702$		0			0
$703$	80.3614			0.114312
$704$			110.978i			0.157640i
$705$		0			0
$706$	−146.140			−0.206998
$707$		−	1684.10i		−	2.38203i
$708$		0			0
$709$			1093.87i			1.54284i	0.636325	+	0.771421i	$0.280454\pi$
$709$			1093.87i			1.54284i	−0.636325	+	0.771421i	$0.719546\pi$
$710$		−	318.088i		−	0.448011i
$711$		0			0
$712$		−	300.506i		−	0.422059i
$713$	56.7492	−	25.8940i	0.0795921	−	0.0363169i
$714$		0			0
$715$	−92.6789			−0.129621
$716$	−638.767			−0.892133
$717$		0			0
$718$			860.862i			1.19897i
$719$	1191.45			1.65709			0.828545	−	0.559923i	$-0.189169\pi$
$719$	1191.45			1.65709			0.828545	+	0.559923i	$0.189169\pi$
$720$		0			0
$721$	854.245			1.18481
$722$	−501.483			−0.694575
$723$		0			0
$724$			230.322i			0.318124i
$725$	58.1157			0.0801596
$726$		0			0
$727$			1123.33i			1.54516i	0.634920	+	0.772578i	$0.281033\pi$
$727$			1123.33i			1.54516i	−0.634920	+	0.772578i	$0.718967\pi$
$728$		−	96.5098i		−	0.132568i
$729$		0			0
$730$		−	52.9981i		−	0.0726001i
$731$	−306.393			−0.419142
$732$		0			0
$733$			81.6381i			0.111375i	0.998448	+	0.0556877i	$0.0177351\pi$
$733$			81.6381i			0.111375i	−0.998448	+	0.0556877i	$0.982265\pi$
$734$			302.243i			0.411775i
$735$		0			0
$736$	−118.368	+	54.0098i	−0.160826	+	0.0733829i
$737$	−811.723			−1.10139
$738$		0			0
$739$	−706.438			−0.955938			−0.477969	−	0.878377i	$-0.658627\pi$
$739$	−706.438			−0.955938			−0.477969	+	0.878377i	$0.658627\pi$
$740$	−142.083			−0.192004
$741$		0			0
$742$	960.895			1.29501
$743$		−	1028.94i		−	1.38484i	−0.721494	−	0.692420i	$-0.756544\pi$
$743$		−	1028.94i		−	1.38484i	0.721494	−	0.692420i	$-0.243456\pi$
$744$		0			0
$745$	294.357			0.395111
$746$		−	767.290i		−	1.02854i
$747$		0			0
$748$	−425.637			−0.569033
$749$	−421.291			−0.562472
$750$		0			0
$751$			650.518i			0.866202i	0.901345	+	0.433101i	$0.142581\pi$
$751$			650.518i			0.866202i	−0.901345	+	0.433101i	$0.857419\pi$
$752$	−208.927			−0.277829
$753$		0			0
$754$	−49.1118			−0.0651350
$755$		−	333.476i		−	0.441690i
$756$		0			0
$757$			372.692i			0.492328i	0.969228	+	0.246164i	$0.0791702\pi$
$757$			372.692i			0.492328i	−0.969228	+	0.246164i	$0.920830\pi$
$758$			493.234i			0.650704i
$759$		0			0
$760$	−15.9974			−0.0210493
$761$	127.577			0.167644			0.0838219	−	0.996481i	$-0.473287\pi$
$761$	127.577			0.167644			0.0838219	+	0.996481i	$0.473287\pi$
$762$		0			0
$763$	1018.56			1.33494
$764$		−	209.362i		−	0.274034i
$765$		0			0
$766$		−	950.648i		−	1.24106i
$767$	243.822			0.317890
$768$		0			0
$769$			1274.35i			1.65715i	0.559878	+	0.828575i	$0.310848\pi$
$769$			1274.35i			1.65715i	−0.559878	+	0.828575i	$0.689152\pi$
$770$			500.989i			0.650635i
$771$		0			0
$772$	116.005			0.150266
$773$		−	1009.21i		−	1.30557i	−0.757543	−	0.652785i	$-0.773601\pi$
$773$		−	1009.21i		−	1.30557i	0.757543	−	0.652785i	$-0.226399\pi$
$774$		0			0
$775$	−13.5604			−0.0174972
$776$		−	107.457i		−	0.138476i
$777$		0			0
$778$		−	657.074i		−	0.844569i
$779$		−	91.8751i		−	0.117940i
$780$		0			0
$781$		−	1395.39i		−	1.78667i
$782$	−207.145	−	453.977i	−0.264891	−	0.580534i
$783$		0			0
$784$	−325.697			−0.415430
$785$	510.243			0.649992
$786$		0			0
$787$			618.658i			0.786096i	0.919518	+	0.393048i	$0.128579\pi$
$787$			618.658i			0.786096i	−0.919518	+	0.393048i	$0.871421\pi$
$788$	−192.150			−0.243845
$789$		0			0
$790$	−191.791			−0.242773
$791$	453.545			0.573382
$792$		0			0
$793$			126.125i			0.159047i
$794$	2.72053			0.00342636
$795$		0			0
$796$			396.989i			0.498729i
$797$		−	849.216i		−	1.06552i	−0.846268	−	0.532758i	$-0.821156\pi$
$797$		−	849.216i		−	1.06552i	0.846268	−	0.532758i	$-0.178844\pi$
$798$		0			0
$799$		−	801.301i		−	1.00288i
$800$	28.2843			0.0353553
$801$		0			0
$802$		−	590.125i		−	0.735817i
$803$		−	232.492i		−	0.289530i
$804$		0			0
$805$	−534.346	+	243.816i	−0.663784	+	0.302877i
$806$	11.4594			0.0142177
$807$		0			0
$808$	−417.093			−0.516205
$809$	−194.891			−0.240904			−0.120452	−	0.992719i	$-0.538434\pi$
$809$	−194.891			−0.240904			−0.120452	+	0.992719i	$0.538434\pi$
$810$		0			0
$811$	−224.475			−0.276788			−0.138394	−	0.990377i	$-0.544194\pi$
$811$	−224.475			−0.276788			−0.138394	+	0.990377i	$0.544194\pi$
$812$			265.480i			0.326946i
$813$		0			0
$814$	−623.291			−0.765713
$815$		−	582.577i		−	0.714818i
$816$		0			0
$817$	−50.5171			−0.0618324
$818$	1149.50			1.40526
$819$		0			0
$820$			162.440i			0.198097i
$821$	472.157			0.575100			0.287550	−	0.957766i	$-0.407159\pi$
$821$	472.157			0.575100			0.287550	+	0.957766i	$0.407159\pi$
$822$		0			0
$823$	1081.17			1.31370			0.656849	−	0.754022i	$-0.271889\pi$
$823$	1081.17			1.31370			0.656849	+	0.754022i	$0.271889\pi$
$824$		−	211.567i		−	0.256756i
$825$		0			0
$826$		−	1318.01i		−	1.59565i
$827$		−	64.5413i		−	0.0780427i	−0.999238	−	0.0390214i	$-0.987576\pi$
$827$		−	64.5413i		−	0.0780427i	0.999238	−	0.0390214i	$-0.0124240\pi$
$828$		0			0
$829$	−548.619			−0.661784			−0.330892	−	0.943669i	$-0.607350\pi$
$829$	−548.619			−0.661784			−0.330892	+	0.943669i	$0.607350\pi$
$830$	139.624			0.168222
$831$		0			0
$832$	−23.9022			−0.0287286
$833$		−	1249.15i		−	1.49958i
$834$		0			0
$835$		−	396.031i		−	0.474289i
$836$	−70.1777			−0.0839447
$837$		0			0
$838$			379.229i			0.452540i
$839$		−	927.859i		−	1.10591i	−0.833211	−	0.552955i	$-0.813500\pi$
$839$		−	927.859i		−	1.10591i	0.833211	−	0.552955i	$-0.186500\pi$
$840$		0			0
$841$	−705.903			−0.839361
$842$			246.211i			0.292412i
$843$		0			0
$844$	−308.147			−0.365103
$845$			357.935i			0.423591i
$846$		0			0
$847$			815.880i			0.963259i
$848$		−	237.981i		−	0.280638i
$849$		0			0
$850$			108.479i			0.127622i
$851$	−303.337	−	664.791i	−0.356447	−	0.781189i
$852$		0			0
$853$	−348.485			−0.408540			−0.204270	−	0.978915i	$-0.565482\pi$
$853$	−348.485			−0.408540			−0.204270	+	0.978915i	$0.565482\pi$
$854$	681.783			0.798341
$855$		0			0
$856$			104.339i			0.121892i
$857$	−1026.47			−1.19774			−0.598872	−	0.800845i	$-0.704384\pi$
$857$	−1026.47			−1.19774			−0.598872	+	0.800845i	$0.704384\pi$
$858$		0			0
$859$	1188.51			1.38360			0.691801	−	0.722088i	$-0.256817\pi$
$859$	1188.51			1.38360			0.691801	+	0.722088i	$0.256817\pi$
$860$	89.3167			0.103857
$861$		0			0
$862$			412.897i			0.478999i
$863$	1055.95			1.22359			0.611793	−	0.791018i	$-0.290449\pi$
$863$	1055.95			1.22359			0.611793	+	0.791018i	$0.290449\pi$
$864$		0			0
$865$			17.0142i			0.0196696i
$866$			620.996i			0.717086i
$867$		0			0
$868$		−	61.9456i		−	0.0713659i
$869$	−841.348			−0.968180
$870$		0			0
$871$		−	174.826i		−	0.200719i
$872$		−	252.263i		−	0.289292i
$873$		0			0
$874$	−34.1534	−	74.8504i	−0.0390771	−	0.0856412i
$875$	127.683			0.145924
$876$		0			0
$877$	−921.525			−1.05077			−0.525385	−	0.850865i	$-0.676079\pi$
$877$	−921.525			−1.05077			−0.525385	+	0.850865i	$0.676079\pi$
$878$	87.7724			0.0999685
$879$		0			0
$880$	124.078			0.140997
$881$		−	1210.60i		−	1.37412i	−0.726600	−	0.687061i	$-0.758901\pi$
$881$		−	1210.60i		−	1.37412i	0.726600	−	0.687061i	$-0.241099\pi$
$882$		0			0
$883$	−386.146			−0.437311			−0.218656	−	0.975802i	$-0.570167\pi$
$883$	−386.146			−0.437311			−0.218656	+	0.975802i	$0.570167\pi$
$884$		−	91.6722i		−	0.103702i
$885$		0			0
$886$	−369.697			−0.417265
$887$	493.959			0.556887			0.278444	−	0.960453i	$-0.410181\pi$
$887$	493.959			0.556887			0.278444	+	0.960453i	$0.410181\pi$
$888$		0			0
$889$			1492.13i			1.67844i
$890$	−335.976			−0.377501
$891$		0			0
$892$	−176.542			−0.197916
$893$		−	132.116i		−	0.147946i
$894$		0			0
$895$			714.164i			0.797948i
$896$			129.206i			0.144204i
$897$		0			0
$898$	609.985			0.679271
$899$	−31.5228			−0.0350643
$900$		0			0
$901$	912.730			1.01302
$902$			712.592i			0.790013i
$903$		0			0
$904$		−	112.328i		−	0.124256i
$905$	257.508			0.284539
$906$		0			0
$907$		−	332.986i		−	0.367129i	−0.983008	−	0.183565i	$-0.941236\pi$
$907$		−	332.986i		−	0.367129i	0.983008	−	0.183565i	$-0.0587637\pi$
$908$			440.991i			0.485673i
$909$		0			0
$910$	−107.901			−0.118573
$911$		−	310.817i		−	0.341182i	−0.985342	−	0.170591i	$-0.945432\pi$
$911$		−	310.817i		−	0.341182i	0.985342	−	0.170591i	$-0.0545677\pi$
$912$		0			0
$913$	612.504			0.670870
$914$			1038.84i			1.13659i
$915$		0			0
$916$		−	276.227i		−	0.301557i
$917$			34.1225i			0.0372110i
$918$		0			0
$919$		−	487.215i		−	0.530158i	−0.964227	−	0.265079i	$-0.914602\pi$
$919$		−	487.215i		−	0.530158i	0.964227	−	0.265079i	$-0.0853980\pi$
$920$	60.3848	+	132.339i	0.0656357	+	0.143847i
$921$		0			0
$922$	885.327			0.960225
$923$	300.534			0.325606
$924$		0			0
$925$			158.854i			0.171734i
$926$	453.574			0.489821
$927$		0			0
$928$	65.7504			0.0708517
$929$	−1224.58			−1.31817			−0.659085	−	0.752068i	$-0.729056\pi$
$929$	−1224.58			−1.31817			−0.659085	+	0.752068i	$0.729056\pi$
$930$		0			0
$931$		−	205.956i		−	0.221220i
$932$	−196.375			−0.210703
$933$		0			0
$934$		−	209.892i		−	0.224724i
$935$			475.877i			0.508959i
$936$		0			0
$937$		−	412.886i		−	0.440647i	−0.975427	−	0.220323i	$-0.929289\pi$
$937$		−	412.886i		−	0.440647i	0.975427	−	0.220323i	$-0.0707112\pi$
$938$	−945.047			−1.00751
$939$		0			0
$940$			233.588i			0.248498i
$941$			729.608i			0.775354i	0.921795	+	0.387677i	$0.126722\pi$
$941$			729.608i			0.775354i	−0.921795	+	0.387677i	$0.873278\pi$
$942$		0			0
$943$	−760.038	+	346.797i	−0.805979	+	0.367759i
$944$	−326.426			−0.345790
$945$		0			0
$946$	391.815			0.414181
$947$	−1691.14			−1.78579			−0.892894	−	0.450268i	$-0.851329\pi$
$947$	−1691.14			−1.78579			−0.892894	+	0.450268i	$0.851329\pi$
$948$		0			0
$949$	50.0734			0.0527644
$950$			17.8857i			0.0188270i
$951$		0			0
$952$	−495.547			−0.520532
$953$		−	133.087i		−	0.139651i	−0.997559	−	0.0698255i	$-0.977756\pi$
$953$		−	133.087i		−	0.139651i	0.997559	−	0.0698255i	$-0.0222443\pi$
$954$		0			0
$955$	−234.074			−0.245104
$956$	842.438			0.881211
$957$		0			0
$958$			1059.49i			1.10594i
$959$	1372.47			1.43114
$960$		0			0
$961$	−953.645			−0.992346
$962$		−	134.242i		−	0.139545i
$963$		0			0
$964$			363.505i			0.377080i
$965$		−	129.698i		−	0.134402i
$966$		0			0
$967$	310.427			0.321020			0.160510	−	0.987034i	$-0.448686\pi$
$967$	310.427			0.321020			0.160510	+	0.987034i	$0.448686\pi$
$968$	202.065			0.208745
$969$		0			0
$970$	−120.141			−0.123856
$971$		−	1571.54i		−	1.61847i	−0.587483	−	0.809237i	$-0.699881\pi$
$971$		−	1571.54i		−	1.61847i	0.587483	−	0.809237i	$-0.300119\pi$
$972$		0			0
$973$			1453.49i			1.49383i
$974$	−408.880			−0.419794
$975$		0			0
$976$		−	168.854i		−	0.173006i
$977$		−	283.092i		−	0.289757i	−0.989449	−	0.144878i	$-0.953721\pi$
$977$		−	283.092i		−	0.289757i	0.989449	−	0.144878i	$-0.0462791\pi$
$978$		0			0
$979$	−1473.86			−1.50548
$980$			364.141i			0.371572i
$981$		0			0
$982$	1087.25			1.10718
$983$		−	1310.77i		−	1.33344i	−0.745310	−	0.666718i	$-0.767698\pi$
$983$		−	1310.77i		−	1.33344i	0.745310	−	0.666718i	$-0.232302\pi$
$984$		0			0
$985$			214.830i			0.218101i
$986$			252.173i			0.255754i
$987$		0			0
$988$		−	15.1146i		−	0.0152982i
$989$	190.685	+	417.904i	0.192806	+	0.422552i
$990$		0			0
$991$	−999.520			−1.00860			−0.504299	−	0.863529i	$-0.668249\pi$
$991$	−999.520			−1.00860			−0.504299	+	0.863529i	$0.668249\pi$
$992$	−15.3418			−0.0154655
$993$		0			0
$994$		−	1624.58i		−	1.63439i
$995$	443.847			0.446077
$996$		0			0
$997$	−806.118			−0.808543			−0.404272	−	0.914639i	$-0.632475\pi$
$997$	−806.118			−0.808543			−0.404272	+	0.914639i	$0.632475\pi$
$998$	−285.086			−0.285657
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.3.c.b.91.1		32
3.2	odd	2		690.3.c.a.91.29	yes	32
23.22	odd	2	inner	2070.3.c.b.91.16		32
69.68	even	2		690.3.c.a.91.28	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.c.a.91.28	✓	32	69.68	even	2
690.3.c.a.91.29	yes	32	3.2	odd	2
2070.3.c.b.91.1		32	1.1	even	1	trivial
2070.3.c.b.91.16		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)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} -11.4203i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} -11.4203i q^{7} -2.82843 q^{8} +3.16228i q^{10} +13.8723i q^{11} -2.98777 q^{13} +16.1508i q^{14} +4.00000 q^{16} +15.3412i q^{17} +2.52942i q^{19} -4.47214i q^{20} -19.6184i q^{22} +(20.9247 - 9.54768i) q^{23} -5.00000 q^{25} +4.22535 q^{26} -22.8407i q^{28} -11.6231 q^{29} +2.71207 q^{31} -5.65685 q^{32} -21.6958i q^{34} -25.5367 q^{35} -31.7707i q^{37} -3.57714i q^{38} +6.32456i q^{40} -36.3226 q^{41} +19.9718i q^{43} +27.7446i q^{44} +(-29.5920 + 13.5025i) q^{46} -52.2318 q^{47} -81.4243 q^{49} +7.07107 q^{50} -5.97554 q^{52} -59.4952i q^{53} +31.0194 q^{55} +32.3016i q^{56} +16.4376 q^{58} -81.6065 q^{59} -42.2136i q^{61} -3.83545 q^{62} +8.00000 q^{64} +6.68086i q^{65} +58.5139i q^{67} +30.6825i q^{68} +36.1143 q^{70} -100.588 q^{71} -16.7595 q^{73} +44.9306i q^{74} +5.05884i q^{76} +158.427 q^{77} +60.6495i q^{79} -8.94427i q^{80} +51.3679 q^{82} -44.1530i q^{83} +34.3041 q^{85} -28.2444i q^{86} -39.2368i q^{88} +106.245i q^{89} +34.1214i q^{91} +(41.8493 - 19.0954i) q^{92} +73.8669 q^{94} +5.65595 q^{95} +37.9918i q^{97} +115.151 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4}+O(q^{10}) \) 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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