LMFDB - Embedded newform 2070.3.c.b.91.7

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			91.7
Character	$\chi$	$=$	2070.91
Dual form			2070.3.c.b.91.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} +7.96402i q^{7} -2.82843 q^{8} +O(q^{10})$	\(q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} +7.96402i q^{7} -2.82843 q^{8} +3.16228i q^{10} -13.6667i q^{11} -14.0318 q^{13} -11.2628i q^{14} +4.00000 q^{16} +17.2668i q^{17} -26.4661i q^{19} -4.47214i q^{20} +19.3277i q^{22} +(11.7666 - 19.7623i) q^{23} -5.00000 q^{25} +19.8440 q^{26} +15.9280i q^{28} +54.4984 q^{29} -22.1628 q^{31} -5.65685 q^{32} -24.4189i q^{34} +17.8081 q^{35} +32.2493i q^{37} +37.4287i q^{38} +6.32456i q^{40} -9.59193 q^{41} +65.3706i q^{43} -27.3335i q^{44} +(-16.6405 + 27.9481i) q^{46} -82.5547 q^{47} -14.4257 q^{49} +7.07107 q^{50} -28.0636 q^{52} -21.1000i q^{53} -30.5598 q^{55} -22.5257i q^{56} -77.0724 q^{58} +105.537 q^{59} +33.4803i q^{61} +31.3429 q^{62} +8.00000 q^{64} +31.3760i q^{65} -34.9712i q^{67} +34.5336i q^{68} -25.1845 q^{70} -65.1127 q^{71} -126.274 q^{73} -45.6075i q^{74} -52.9321i q^{76} +108.842 q^{77} +45.0370i q^{79} -8.94427i q^{80} +13.5650 q^{82} -10.3487i q^{83} +38.6097 q^{85} -92.4480i q^{86} +38.6554i q^{88} +20.8576i q^{89} -111.750i q^{91} +(23.5332 - 39.5245i) q^{92} +116.750 q^{94} -59.1799 q^{95} +146.441i q^{97} +20.4010 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$			7.96402i			1.13772i	0.822435	+	0.568859i	$0.192615\pi$
$7$			7.96402i			1.13772i	−0.822435	+	0.568859i	$0.807385\pi$
$8$	−2.82843			−0.353553
$9$		0			0
$10$			3.16228i			0.316228i
$11$		−	13.6667i		−	1.24243i	−0.783640	−	0.621216i	$-0.786639\pi$
$11$		−	13.6667i		−	1.24243i	0.783640	−	0.621216i	$-0.213361\pi$
$12$		0			0
$13$	−14.0318			−1.07937			−0.539684	−	0.841867i	$-0.681456\pi$
$13$	−14.0318			−1.07937			−0.539684	+	0.841867i	$0.681456\pi$
$14$		−	11.2628i		−	0.804488i
$15$		0			0
$16$	4.00000			0.250000
$17$			17.2668i			1.01569i	0.861447	+	0.507847i	$0.169558\pi$
$17$			17.2668i			1.01569i	−0.861447	+	0.507847i	$0.830442\pi$
$18$		0			0
$19$		−	26.4661i		−	1.39295i	−0.717581	−	0.696475i	$-0.754750\pi$
$19$		−	26.4661i		−	1.39295i	0.717581	−	0.696475i	$-0.245250\pi$
$20$		−	4.47214i		−	0.223607i
$21$		0			0
$22$			19.3277i			0.878532i
$23$	11.7666	−	19.7623i	0.511592	−	0.859229i
$23$	11.7666	−	19.7623i	0.511592	−	0.859229i
$24$		0			0
$25$	−5.00000			−0.200000
$26$	19.8440			0.763229
$27$		0			0
$28$			15.9280i			0.568859i
$29$	54.4984			1.87926			0.939628	−	0.342197i	$-0.111171\pi$
$29$	54.4984			1.87926			0.939628	+	0.342197i	$0.111171\pi$
$30$		0			0
$31$	−22.1628			−0.714929			−0.357465	−	0.933927i	$-0.616359\pi$
$31$	−22.1628			−0.714929			−0.357465	+	0.933927i	$0.616359\pi$
$32$	−5.65685			−0.176777
$33$		0			0
$34$		−	24.4189i		−	0.718204i
$35$	17.8081			0.508803
$36$		0			0
$37$			32.2493i			0.871604i	0.900043	+	0.435802i	$0.143535\pi$
$37$			32.2493i			0.871604i	−0.900043	+	0.435802i	$0.856465\pi$
$38$			37.4287i			0.984965i
$39$		0			0
$40$			6.32456i			0.158114i
$41$	−9.59193			−0.233949			−0.116975	−	0.993135i	$-0.537320\pi$
$41$	−9.59193			−0.233949			−0.116975	+	0.993135i	$0.537320\pi$
$42$		0			0
$43$			65.3706i			1.52025i	0.649778	+	0.760124i	$0.274862\pi$
$43$			65.3706i			1.52025i	−0.649778	+	0.760124i	$0.725138\pi$
$44$		−	27.3335i		−	0.621216i
$45$		0			0
$46$	−16.6405	+	27.9481i	−0.361750	+	0.607566i
$47$	−82.5547			−1.75648			−0.878242	−	0.478217i	$-0.841283\pi$
$47$	−82.5547			−1.75648			−0.878242	+	0.478217i	$0.841283\pi$
$48$		0			0
$49$	−14.4257			−0.294402
$50$	7.07107			0.141421
$51$		0			0
$52$	−28.0636			−0.539684
$53$		−	21.1000i		−	0.398113i	−0.979988	−	0.199056i	$-0.936212\pi$
$53$		−	21.1000i		−	0.398113i	0.979988	−	0.199056i	$-0.0637877\pi$
$54$		0			0
$55$	−30.5598			−0.555632
$56$		−	22.5257i		−	0.402244i
$57$		0			0
$58$	−77.0724			−1.32883
$59$	105.537			1.78877			0.894385	−	0.447299i	$-0.147614\pi$
$59$	105.537			1.78877			0.894385	+	0.447299i	$0.147614\pi$
$60$		0			0
$61$			33.4803i			0.548858i	0.961607	+	0.274429i	$0.0884888\pi$
$61$			33.4803i			0.548858i	−0.961607	+	0.274429i	$0.911511\pi$
$62$	31.3429			0.505531
$63$		0			0
$64$	8.00000			0.125000
$65$			31.3760i			0.482708i
$66$		0			0
$67$		−	34.9712i		−	0.521958i	−0.965344	−	0.260979i	$-0.915955\pi$
$67$		−	34.9712i		−	0.521958i	0.965344	−	0.260979i	$-0.0840453\pi$
$68$			34.5336i			0.507847i
$69$		0			0
$70$	−25.1845			−0.359778
$71$	−65.1127			−0.917080			−0.458540	−	0.888674i	$-0.651627\pi$
$71$	−65.1127			−0.917080			−0.458540	+	0.888674i	$0.651627\pi$
$72$		0			0
$73$	−126.274			−1.72978			−0.864891	−	0.501959i	$-0.832613\pi$
$73$	−126.274			−1.72978			−0.864891	+	0.501959i	$0.832613\pi$
$74$		−	45.6075i		−	0.616317i
$75$		0			0
$76$		−	52.9321i		−	0.696475i
$77$	108.842			1.41354
$78$		0			0
$79$			45.0370i			0.570089i	0.958514	+	0.285045i	$0.0920084\pi$
$79$			45.0370i			0.570089i	−0.958514	+	0.285045i	$0.907992\pi$
$80$		−	8.94427i		−	0.111803i
$81$		0			0
$82$	13.5650			0.165427
$83$		−	10.3487i		−	0.124684i	−0.998055	−	0.0623418i	$-0.980143\pi$
$83$		−	10.3487i		−	0.124684i	0.998055	−	0.0623418i	$-0.0198569\pi$
$84$		0			0
$85$	38.6097			0.454232
$86$		−	92.4480i		−	1.07498i
$87$		0			0
$88$			38.6554i			0.439266i
$89$			20.8576i			0.234356i	0.993111	+	0.117178i	$0.0373847\pi$
$89$			20.8576i			0.234356i	−0.993111	+	0.117178i	$0.962615\pi$
$90$		0			0
$91$		−	111.750i		−	1.22802i
$92$	23.5332	−	39.5245i	0.255796	−	0.429614i
$93$		0			0
$94$	116.750			1.24202
$95$	−59.1799			−0.622947
$96$		0			0
$97$			146.441i			1.50970i	0.655895	+	0.754852i	$0.272291\pi$
$97$			146.441i			1.50970i	−0.655895	+	0.754852i	$0.727709\pi$
$98$	20.4010			0.208173
$99$		0			0
$100$	−10.0000			−0.100000
$101$	−110.767			−1.09671			−0.548354	−	0.836246i	$-0.684745\pi$
$101$	−110.767			−1.09671			−0.548354	+	0.836246i	$0.684745\pi$
$102$		0			0
$103$		−	75.0106i		−	0.728259i	−0.931348	−	0.364129i	$-0.881367\pi$
$103$		−	75.0106i		−	0.728259i	0.931348	−	0.364129i	$-0.118633\pi$
$104$	39.6879			0.381615
$105$		0			0
$106$			29.8399i			0.281508i
$107$		−	1.13721i		−	0.0106281i	−0.999986	−	0.00531405i	$-0.998308\pi$
$107$		−	1.13721i		−	0.0106281i	0.999986	−	0.00531405i	$-0.00169152\pi$
$108$		0			0
$109$		−	74.1091i		−	0.679900i	−0.940443	−	0.339950i	$-0.889590\pi$
$109$		−	74.1091i		−	0.679900i	0.940443	−	0.339950i	$-0.110410\pi$
$110$	43.2180			0.392891
$111$		0			0
$112$			31.8561i			0.284429i
$113$			121.550i			1.07566i	0.843053	+	0.537830i	$0.180756\pi$
$113$			121.550i			1.07566i	−0.843053	+	0.537830i	$0.819244\pi$
$114$		0			0
$115$	−44.1897	−	26.3110i	−0.384259	−	0.228791i
$116$	108.997			0.939628
$117$		0			0
$118$	−149.252			−1.26485
$119$	−137.513			−1.15557
$120$		0			0
$121$	−65.7799			−0.543635
$122$		−	47.3484i		−	0.388101i
$123$		0			0
$124$	−44.3256			−0.357465
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	−46.6213			−0.367097			−0.183549	−	0.983011i	$-0.558758\pi$
$127$	−46.6213			−0.367097			−0.183549	+	0.983011i	$0.558758\pi$
$128$	−11.3137			−0.0883883
$129$		0			0
$130$		−	44.3724i		−	0.341326i
$131$	−172.666			−1.31806			−0.659030	−	0.752116i	$-0.729033\pi$
$131$	−172.666			−1.31806			−0.659030	+	0.752116i	$0.729033\pi$
$132$		0			0
$133$	210.776			1.58478
$134$			49.4567i			0.369080i
$135$		0			0
$136$		−	48.8379i		−	0.359102i
$137$		−	269.189i		−	1.96488i	−0.186572	−	0.982441i	$-0.559738\pi$
$137$		−	269.189i		−	1.96488i	0.186572	−	0.982441i	$-0.440262\pi$
$138$		0			0
$139$	−112.626			−0.810260			−0.405130	−	0.914259i	$-0.632774\pi$
$139$	−112.626			−0.810260			−0.405130	+	0.914259i	$0.632774\pi$
$140$	35.6162			0.254401
$141$		0			0
$142$	92.0832			0.648474
$143$			191.769i			1.34104i
$144$		0			0
$145$		−	121.862i		−	0.840429i
$146$	178.579			1.22314
$147$		0			0
$148$			64.4987i			0.435802i
$149$		−	53.3780i		−	0.358241i	−0.983827	−	0.179121i	$-0.942675\pi$
$149$		−	53.3780i		−	0.358241i	0.983827	−	0.179121i	$-0.0573253\pi$
$150$		0			0
$151$	39.7896			0.263507			0.131753	−	0.991283i	$-0.457939\pi$
$151$	39.7896			0.263507			0.131753	+	0.991283i	$0.457939\pi$
$152$			74.8573i			0.492483i
$153$		0			0
$154$	−153.926			−0.999521
$155$			49.5575i			0.319726i
$156$		0			0
$157$			263.251i			1.67676i	0.545086	+	0.838380i	$0.316497\pi$
$157$			263.251i			1.67676i	−0.545086	+	0.838380i	$0.683503\pi$
$158$		−	63.6920i		−	0.403114i
$159$		0			0
$160$			12.6491i			0.0790569i
$161$	157.387	+	93.7096i	0.977559	+	0.582047i
$162$		0			0
$163$	239.678			1.47042			0.735209	−	0.677840i	$-0.237084\pi$
$163$	239.678			1.47042			0.735209	+	0.677840i	$0.237084\pi$
$164$	−19.1839			−0.116975
$165$		0			0
$166$			14.6353i			0.0881646i
$167$	−316.884			−1.89751			−0.948753	−	0.316017i	$-0.897654\pi$
$167$	−316.884			−1.89751			−0.948753	+	0.316017i	$0.897654\pi$
$168$		0			0
$169$	27.8913			0.165037
$170$	−54.6024			−0.321191
$171$		0			0
$172$			130.741i			0.760124i
$173$	−115.891			−0.669890			−0.334945	−	0.942238i	$-0.608718\pi$
$173$	−115.891			−0.669890			−0.334945	+	0.942238i	$0.608718\pi$
$174$		0			0
$175$		−	39.8201i		−	0.227544i
$176$		−	54.6670i		−	0.310608i
$177$		0			0
$178$		−	29.4972i		−	0.165714i
$179$	62.1264			0.347075			0.173537	−	0.984827i	$-0.444480\pi$
$179$	62.1264			0.347075			0.173537	+	0.984827i	$0.444480\pi$
$180$		0			0
$181$			34.2525i			0.189240i	0.995513	+	0.0946202i	$0.0301637\pi$
$181$			34.2525i			0.189240i	−0.995513	+	0.0946202i	$0.969836\pi$
$182$			158.038i			0.868339i
$183$		0			0
$184$	−33.2810	+	55.8961i	−0.180875	+	0.303783i
$185$	72.1117			0.389793
$186$		0			0
$187$	235.981			1.26193
$188$	−165.109			−0.878242
$189$		0			0
$190$	83.6931			0.440490
$191$			365.807i			1.91522i	0.288065	+	0.957611i	$0.406988\pi$
$191$			365.807i			1.91522i	−0.288065	+	0.957611i	$0.593012\pi$
$192$		0			0
$193$	−223.842			−1.15980			−0.579901	−	0.814687i	$-0.696909\pi$
$193$	−223.842			−1.15980			−0.579901	+	0.814687i	$0.696909\pi$
$194$		−	207.099i		−	1.06752i
$195$		0			0
$196$	−28.8513			−0.147201
$197$	−89.6193			−0.454920			−0.227460	−	0.973787i	$-0.573042\pi$
$197$	−89.6193			−0.454920			−0.227460	+	0.973787i	$0.573042\pi$
$198$		0			0
$199$			5.90005i			0.0296485i	0.999890	+	0.0148243i	$0.00471888\pi$
$199$			5.90005i			0.0296485i	−0.999890	+	0.0148243i	$0.995281\pi$
$200$	14.1421			0.0707107
$201$		0			0
$202$	156.649			0.775489
$203$			434.027i			2.13806i
$204$		0			0
$205$			21.4482i			0.104625i
$206$			106.081i			0.514957i
$207$		0			0
$208$	−56.1272			−0.269842
$209$	−361.705			−1.73065
$210$		0			0
$211$	−231.698			−1.09810			−0.549048	−	0.835791i	$-0.685009\pi$
$211$	−231.698			−1.09810			−0.549048	+	0.835791i	$0.685009\pi$
$212$		−	42.1999i		−	0.199056i
$213$		0			0
$214$			1.60825i			0.00751521i
$215$	146.173			0.679875
$216$		0			0
$217$		−	176.505i		−	0.813387i
$218$			104.806i			0.480762i
$219$		0			0
$220$	−61.1195			−0.277816
$221$		−	242.284i		−	1.09631i
$222$		0			0
$223$	130.000			0.582962			0.291481	−	0.956577i	$-0.405852\pi$
$223$	130.000			0.582962			0.291481	+	0.956577i	$0.405852\pi$
$224$		−	45.0513i		−	0.201122i
$225$		0			0
$226$		−	171.897i		−	0.760606i
$227$		−	167.115i		−	0.736190i	−0.929788	−	0.368095i	$-0.880010\pi$
$227$		−	167.115i		−	0.736190i	0.929788	−	0.368095i	$-0.119990\pi$
$228$		0			0
$229$		−	19.2126i		−	0.0838976i	−0.999120	−	0.0419488i	$-0.986643\pi$
$229$		−	19.2126i		−	0.0838976i	0.999120	−	0.0419488i	$-0.0133566\pi$
$230$	62.4937	+	37.2093i	0.271712	+	0.161780i
$231$		0			0
$232$	−154.145			−0.664417
$233$	−101.698			−0.436470			−0.218235	−	0.975896i	$-0.570030\pi$
$233$	−101.698			−0.436470			−0.218235	+	0.975896i	$0.570030\pi$
$234$		0			0
$235$			184.598i			0.785523i
$236$	211.075			0.894385
$237$		0			0
$238$	194.473			0.817113
$239$	65.7260			0.275004			0.137502	−	0.990501i	$-0.456093\pi$
$239$	65.7260			0.275004			0.137502	+	0.990501i	$0.456093\pi$
$240$		0			0
$241$			191.368i			0.794059i	0.917806	+	0.397029i	$0.129959\pi$
$241$			191.368i			0.794059i	−0.917806	+	0.397029i	$0.870041\pi$
$242$	93.0268			0.384408
$243$		0			0
$244$			66.9607i			0.274429i
$245$			32.2568i			0.131660i
$246$		0			0
$247$			371.366i			1.50351i
$248$	62.6859			0.252766
$249$		0			0
$250$		−	15.8114i		−	0.0632456i
$251$			269.641i			1.07427i	0.843498	+	0.537133i	$0.180493\pi$
$251$			269.641i			1.07427i	−0.843498	+	0.537133i	$0.819507\pi$
$252$		0			0
$253$	−270.086	−	160.811i	−1.06753	−	0.635618i
$254$	65.9325			0.259577
$255$		0			0
$256$	16.0000			0.0625000
$257$	−130.359			−0.507234			−0.253617	−	0.967305i	$-0.581620\pi$
$257$	−130.359			−0.507234			−0.253617	+	0.967305i	$0.581620\pi$
$258$		0			0
$259$	−256.834			−0.991639
$260$			62.7521i			0.241354i
$261$		0			0
$262$	244.187			0.932010
$263$		−	179.446i		−	0.682306i	−0.940008	−	0.341153i	$-0.889183\pi$
$263$		−	179.446i		−	0.682306i	0.940008	−	0.341153i	$-0.110817\pi$
$264$		0			0
$265$	−47.1810			−0.178041
$266$	−298.083			−1.12061
$267$		0			0
$268$		−	69.9424i		−	0.260979i
$269$	−483.252			−1.79648			−0.898239	−	0.439508i	$-0.855153\pi$
$269$	−483.252			−1.79648			−0.898239	+	0.439508i	$0.855153\pi$
$270$		0			0
$271$	25.8751			0.0954799			0.0477400	−	0.998860i	$-0.484798\pi$
$271$	25.8751			0.0954799			0.0477400	+	0.998860i	$0.484798\pi$
$272$			69.0672i			0.253923i
$273$		0			0
$274$			380.691i			1.38938i
$275$			68.3337i			0.248486i
$276$		0			0
$277$	−322.130			−1.16293			−0.581463	−	0.813573i	$-0.697519\pi$
$277$	−322.130			−1.16293			−0.581463	+	0.813573i	$0.697519\pi$
$278$	159.277			0.572940
$279$		0			0
$280$	−50.3689			−0.179889
$281$			152.134i			0.541401i	0.962664	+	0.270701i	$0.0872553\pi$
$281$			152.134i			0.541401i	−0.962664	+	0.270701i	$0.912745\pi$
$282$		0			0
$283$			256.411i			0.906047i	0.891499	+	0.453024i	$0.149655\pi$
$283$			256.411i			0.906047i	−0.891499	+	0.453024i	$0.850345\pi$
$284$	−130.225			−0.458540
$285$		0			0
$286$		−	271.202i		−	0.948260i
$287$		−	76.3903i		−	0.266168i
$288$		0			0
$289$	−9.14209			−0.0316335
$290$			172.339i			0.594273i
$291$		0			0
$292$	−252.548			−0.864891
$293$		−	371.098i		−	1.26655i	−0.773929	−	0.633273i	$-0.781711\pi$
$293$		−	371.098i		−	1.26655i	0.773929	−	0.633273i	$-0.218289\pi$
$294$		0			0
$295$		−	235.989i		−	0.799962i
$296$		−	91.2149i		−	0.308158i
$297$		0			0
$298$			75.4878i			0.253315i
$299$	−165.107	+	277.300i	−0.552196	+	0.927425i
$300$		0			0
$301$	−520.613			−1.72961
$302$	−56.2709			−0.186328
$303$		0			0
$304$		−	105.864i		−	0.348238i
$305$	74.8643			0.245457
$306$		0			0
$307$	526.653			1.71548			0.857742	−	0.514081i	$-0.171867\pi$
$307$	526.653			1.71548			0.857742	+	0.514081i	$0.171867\pi$
$308$	217.685			0.706768
$309$		0			0
$310$		−	70.0849i		−	0.226080i
$311$	−537.154			−1.72718			−0.863592	−	0.504192i	$-0.831790\pi$
$311$	−537.154			−1.72718			−0.863592	+	0.504192i	$0.831790\pi$
$312$		0			0
$313$			296.354i			0.946819i	0.880842	+	0.473409i	$0.156977\pi$
$313$			296.354i			0.946819i	−0.880842	+	0.473409i	$0.843023\pi$
$314$		−	372.294i		−	1.18565i
$315$		0			0
$316$			90.0741i			0.285045i
$317$	144.472			0.455748			0.227874	−	0.973691i	$-0.426823\pi$
$317$	144.472			0.455748			0.227874	+	0.973691i	$0.426823\pi$
$318$		0			0
$319$		−	744.816i		−	2.33485i
$320$		−	17.8885i		−	0.0559017i
$321$		0			0
$322$	−222.579	−	132.525i	−0.691239	−	0.411570i
$323$	456.984			1.41481
$324$		0			0
$325$	70.1590			0.215874
$326$	−338.956			−1.03974
$327$		0			0
$328$	27.1301			0.0827136
$329$		−	657.468i		−	1.99838i
$330$		0			0
$331$	−481.196			−1.45376			−0.726882	−	0.686763i	$-0.759031\pi$
$331$	−481.196			−1.45376			−0.726882	+	0.686763i	$0.759031\pi$
$332$		−	20.6975i		−	0.0623418i
$333$		0			0
$334$	448.141			1.34174
$335$	−78.1980			−0.233427
$336$		0			0
$337$		−	227.981i		−	0.676501i	−0.941056	−	0.338251i	$-0.890165\pi$
$337$		−	227.981i		−	0.676501i	0.941056	−	0.338251i	$-0.109835\pi$
$338$	−39.4443			−0.116699
$339$		0			0
$340$	77.2194			0.227116
$341$			302.893i			0.888250i
$342$		0			0
$343$			275.351i			0.802772i
$344$		−	184.896i		−	0.537489i
$345$		0			0
$346$	163.895			0.473684
$347$	−284.411			−0.819628			−0.409814	−	0.912169i	$-0.634406\pi$
$347$	−284.411			−0.819628			−0.409814	+	0.912169i	$0.634406\pi$
$348$		0			0
$349$	−426.741			−1.22275			−0.611377	−	0.791340i	$-0.709384\pi$
$349$	−426.741			−1.22275			−0.611377	+	0.791340i	$0.709384\pi$
$350$			56.3142i			0.160898i
$351$		0			0
$352$			77.3108i			0.219633i
$353$	−159.245			−0.451118			−0.225559	−	0.974230i	$-0.572421\pi$
$353$	−159.245			−0.451118			−0.225559	+	0.974230i	$0.572421\pi$
$354$		0			0
$355$			145.596i			0.410131i
$356$			41.7153i			0.117178i
$357$		0			0
$358$	−87.8600			−0.245419
$359$		−	660.255i		−	1.83915i	−0.392914	−	0.919575i	$-0.628533\pi$
$359$		−	660.255i		−	1.83915i	0.392914	−	0.919575i	$-0.371467\pi$
$360$		0			0
$361$	−339.453			−0.940312
$362$		−	48.4404i		−	0.133813i
$363$		0			0
$364$		−	223.499i		−	0.614009i
$365$			282.358i			0.773582i
$366$		0			0
$367$			153.906i			0.419363i	0.977770	+	0.209681i	$0.0672427\pi$
$367$			153.906i			0.419363i	−0.977770	+	0.209681i	$0.932757\pi$
$368$	47.0665	−	79.0490i	0.127898	−	0.214807i
$369$		0			0
$370$	−101.981			−0.275625
$371$	168.041			0.452940
$372$		0			0
$373$			114.077i			0.305836i	0.988239	+	0.152918i	$0.0488671\pi$
$373$			114.077i			0.305836i	−0.988239	+	0.152918i	$0.951133\pi$
$374$	−333.727			−0.892319
$375$		0			0
$376$	233.500			0.621011
$377$	−764.711			−2.02841
$378$		0			0
$379$			568.948i			1.50118i	0.660767	+	0.750591i	$0.270231\pi$
$379$			568.948i			1.50118i	−0.660767	+	0.750591i	$0.729769\pi$
$380$	−118.360			−0.311473
$381$		0			0
$382$		−	517.330i		−	1.35427i
$383$			598.817i			1.56349i	0.623597	+	0.781746i	$0.285671\pi$
$383$			598.817i			1.56349i	−0.623597	+	0.781746i	$0.714329\pi$
$384$		0			0
$385$		−	243.379i		−	0.632153i
$386$	316.560			0.820103
$387$		0			0
$388$			292.883i			0.754852i
$389$			296.143i			0.761293i	0.924721	+	0.380647i	$0.124299\pi$
$389$			296.143i			0.761293i	−0.924721	+	0.380647i	$0.875701\pi$
$390$		0			0
$391$	341.231	+	203.172i	0.872713	+	0.519621i
$392$	40.8020			0.104087
$393$		0			0
$394$	126.741			0.321677
$395$	100.706			0.254952
$396$		0			0
$397$	−706.135			−1.77868			−0.889338	−	0.457249i	$-0.848835\pi$
$397$	−706.135			−1.77868			−0.889338	+	0.457249i	$0.848835\pi$
$398$		−	8.34394i		−	0.0209647i
$399$		0			0
$400$	−20.0000			−0.0500000
$401$			501.908i			1.25164i	0.779967	+	0.625821i	$0.215236\pi$
$401$			501.908i			1.25164i	−0.779967	+	0.625821i	$0.784764\pi$
$402$		0			0
$403$	310.984			0.771672
$404$	−221.535			−0.548354
$405$		0			0
$406$		−	613.807i		−	1.51184i
$407$	440.743			1.08291
$408$		0			0
$409$	−351.306			−0.858939			−0.429470	−	0.903081i	$-0.641299\pi$
$409$	−351.306			−0.858939			−0.429470	+	0.903081i	$0.641299\pi$
$410$		−	30.3323i		−	0.0739813i
$411$		0			0
$412$		−	150.021i		−	0.364129i
$413$			840.502i			2.03511i
$414$		0			0
$415$	−23.1405			−0.0557602
$416$	79.3758			0.190807
$417$		0			0
$418$	511.528			1.22375
$419$			497.695i			1.18782i	0.804533	+	0.593908i	$0.202416\pi$
$419$			497.695i			1.18782i	−0.804533	+	0.593908i	$0.797584\pi$
$420$		0			0
$421$			323.572i			0.768581i	0.923212	+	0.384290i	$0.125554\pi$
$421$			323.572i			0.768581i	−0.923212	+	0.384290i	$0.874446\pi$
$422$	327.671			0.776470
$423$		0			0
$424$			59.6797i			0.140754i
$425$		−	86.3340i		−	0.203139i
$426$		0			0
$427$	−266.638			−0.624446
$428$		−	2.27442i		−	0.00531405i
$429$		0			0
$430$	−206.720			−0.480744
$431$		−	539.731i		−	1.25228i	−0.779712	−	0.626138i	$-0.784635\pi$
$431$		−	539.731i		−	1.25228i	0.779712	−	0.626138i	$-0.215365\pi$
$432$		0			0
$433$		−	421.843i		−	0.974234i	−0.873337	−	0.487117i	$-0.838049\pi$
$433$		−	421.843i		−	0.974234i	0.873337	−	0.487117i	$-0.161951\pi$
$434$			249.616i			0.575152i
$435$		0			0
$436$		−	148.218i		−	0.339950i
$437$	−523.029	−	311.416i	−1.19686	−	0.712623i
$438$		0			0
$439$	324.408			0.738970			0.369485	−	0.929237i	$-0.379534\pi$
$439$	324.408			0.738970			0.369485	+	0.929237i	$0.379534\pi$
$440$	86.4361			0.196446
$441$		0			0
$442$			342.641i			0.775207i
$443$	−221.797			−0.500671			−0.250335	−	0.968159i	$-0.580541\pi$
$443$	−221.797			−0.500671			−0.250335	+	0.968159i	$0.580541\pi$
$444$		0			0
$445$	46.6391			0.104807
$446$	−183.848			−0.412216
$447$		0			0
$448$			63.7122i			0.142215i
$449$	523.920			1.16686			0.583430	−	0.812164i	$-0.301710\pi$
$449$	523.920			1.16686			0.583430	+	0.812164i	$0.301710\pi$
$450$		0			0
$451$			131.090i			0.290666i
$452$			243.099i			0.537830i
$453$		0			0
$454$			236.337i			0.520565i
$455$	−249.880			−0.549186
$456$		0			0
$457$			379.414i			0.830227i	0.909770	+	0.415114i	$0.136258\pi$
$457$			379.414i			0.830227i	−0.909770	+	0.415114i	$0.863742\pi$
$458$			27.1707i			0.0593246i
$459$		0			0
$460$	−88.3795	−	52.6219i	−0.192129	−	0.114395i
$461$	449.296			0.974611			0.487306	−	0.873231i	$-0.337980\pi$
$461$	449.296			0.974611			0.487306	+	0.873231i	$0.337980\pi$
$462$		0			0
$463$	581.451			1.25583			0.627917	−	0.778280i	$-0.283908\pi$
$463$	581.451			1.25583			0.627917	+	0.778280i	$0.283908\pi$
$464$	217.994			0.469814
$465$		0			0
$466$	143.822			0.308631
$467$			622.103i			1.33213i	0.745895	+	0.666064i	$0.232022\pi$
$467$			622.103i			1.33213i	−0.745895	+	0.666064i	$0.767978\pi$
$468$		0			0
$469$	278.511			0.593841
$470$		−	261.061i		−	0.555449i
$471$		0			0
$472$	−298.505			−0.632425
$473$	893.404			1.88880
$474$		0			0
$475$			132.330i			0.278590i
$476$	−275.026			−0.577786
$477$		0			0
$478$	−92.9506			−0.194457
$479$			269.556i			0.562747i	0.959598	+	0.281373i	$0.0907900\pi$
$479$			269.556i			0.562747i	−0.959598	+	0.281373i	$0.909210\pi$
$480$		0			0
$481$		−	452.516i		−	0.940782i
$482$		−	270.635i		−	0.561484i
$483$		0			0
$484$	−131.560			−0.271818
$485$	327.453			0.675160
$486$		0			0
$487$	321.930			0.661048			0.330524	−	0.943798i	$-0.392775\pi$
$487$	321.930			0.661048			0.330524	+	0.943798i	$0.392775\pi$
$488$		−	94.6967i		−	0.194051i
$489$		0			0
$490$		−	45.6180i		−	0.0930979i
$491$	420.200			0.855805			0.427903	−	0.903825i	$-0.359253\pi$
$491$	420.200			0.855805			0.427903	+	0.903825i	$0.359253\pi$
$492$		0			0
$493$			941.013i			1.90875i
$494$		−	525.192i		−	1.06314i
$495$		0			0
$496$	−88.6512			−0.178732
$497$		−	518.559i		−	1.04338i
$498$		0			0
$499$	−259.403			−0.519845			−0.259922	−	0.965629i	$-0.583697\pi$
$499$	−259.403			−0.519845			−0.259922	+	0.965629i	$0.583697\pi$
$500$			22.3607i			0.0447214i
$501$		0			0
$502$		−	381.330i		−	0.759621i
$503$		−	687.217i		−	1.36624i	−0.730307	−	0.683119i	$-0.760623\pi$
$503$		−	687.217i		−	1.36624i	0.730307	−	0.683119i	$-0.239377\pi$
$504$		0			0
$505$			247.684i			0.490463i
$506$	381.959	+	227.422i	0.754859	+	0.449450i
$507$		0			0
$508$	−93.2427			−0.183549
$509$	537.856			1.05669			0.528345	−	0.849030i	$-0.322813\pi$
$509$	537.856			1.05669			0.528345	+	0.849030i	$0.322813\pi$
$510$		0			0
$511$		−	1005.65i		−	1.96800i
$512$	−22.6274			−0.0441942
$513$		0			0
$514$	184.356			0.358669
$515$	−167.729			−0.325687
$516$		0			0
$517$			1128.25i			2.18231i
$518$	363.219			0.701195
$519$		0			0
$520$		−	88.7449i		−	0.170663i
$521$		−	295.782i		−	0.567720i	−0.958866	−	0.283860i	$-0.908385\pi$
$521$		−	295.782i		−	0.567720i	0.958866	−	0.283860i	$-0.0916151\pi$
$522$		0			0
$523$			889.701i			1.70115i	0.525855	+	0.850575i	$0.323746\pi$
$523$			889.701i			1.70115i	−0.525855	+	0.850575i	$0.676254\pi$
$524$	−345.332			−0.659030
$525$		0			0
$526$			253.776i			0.482463i
$527$		−	382.680i		−	0.726149i
$528$		0			0
$529$	−252.094	−	465.070i	−0.476547	−	0.879149i
$530$	66.7240			0.125894
$531$		0			0
$532$	421.553			0.792392
$533$	134.592			0.252518
$534$		0			0
$535$	−2.54287			−0.00475303
$536$			98.9135i			0.184540i
$537$		0			0
$538$	683.422			1.27030
$539$			197.152i			0.365774i
$540$		0			0
$541$	704.031			1.30135			0.650676	−	0.759356i	$-0.274486\pi$
$541$	704.031			1.30135			0.650676	+	0.759356i	$0.274486\pi$
$542$	−36.5929			−0.0675145
$543$		0			0
$544$		−	97.6757i		−	0.179551i
$545$	−165.713			−0.304061
$546$		0			0
$547$	469.176			0.857726			0.428863	−	0.903370i	$-0.358914\pi$
$547$	469.176			0.857726			0.428863	+	0.903370i	$0.358914\pi$
$548$		−	538.378i		−	0.982441i
$549$		0			0
$550$		−	96.6385i		−	0.175706i
$551$		−	1442.36i		−	2.61771i
$552$		0			0
$553$	−358.676			−0.648600
$554$	455.561			0.822313
$555$		0			0
$556$	−225.252			−0.405130
$557$		−	280.095i		−	0.502864i	−0.967875	−	0.251432i	$-0.919099\pi$
$557$		−	280.095i		−	0.502864i	0.967875	−	0.251432i	$-0.0809015\pi$
$558$		0			0
$559$		−	917.268i		−	1.64091i
$560$	71.2324			0.127201
$561$		0			0
$562$		−	215.150i		−	0.382828i
$563$			30.3039i			0.0538257i	0.999638	+	0.0269129i	$0.00856767\pi$
$563$			30.3039i			0.0538257i	−0.999638	+	0.0269129i	$0.991432\pi$
$564$		0			0
$565$	271.793			0.481050
$566$		−	362.620i		−	0.640672i
$567$		0			0
$568$	184.166			0.324237
$569$			253.630i			0.445746i	0.974847	+	0.222873i	$0.0715436\pi$
$569$			253.630i			0.445746i	−0.974847	+	0.222873i	$0.928456\pi$
$570$		0			0
$571$		−	329.218i		−	0.576564i	−0.957546	−	0.288282i	$-0.906916\pi$
$571$		−	329.218i		−	0.576564i	0.957546	−	0.288282i	$-0.0930841\pi$
$572$			383.538i			0.670521i
$573$		0			0
$574$			108.032i			0.188209i
$575$	−58.8331	+	98.8113i	−0.102318	+	0.171846i
$576$		0			0
$577$	172.230			0.298492			0.149246	−	0.988800i	$-0.452315\pi$
$577$	172.230			0.298492			0.149246	+	0.988800i	$0.452315\pi$
$578$	12.9289			0.0223683
$579$		0			0
$580$		−	243.724i		−	0.420214i
$581$	82.4175			0.141855
$582$		0			0
$583$	−288.368			−0.494628
$584$	357.157			0.611570
$585$		0			0
$586$			524.811i			0.895583i
$587$	627.853			1.06960			0.534798	−	0.844980i	$-0.320388\pi$
$587$	627.853			1.06960			0.534798	+	0.844980i	$0.320388\pi$
$588$		0			0
$589$			586.562i			0.995861i
$590$			333.739i			0.565659i
$591$		0			0
$592$			128.997i			0.217901i
$593$	916.147			1.54494			0.772468	−	0.635054i	$-0.219022\pi$
$593$	916.147			1.54494			0.772468	+	0.635054i	$0.219022\pi$
$594$		0			0
$595$			307.489i			0.516788i
$596$		−	106.756i		−	0.179121i
$597$		0			0
$598$	233.496	−	392.161i	0.390462	−	0.655788i
$599$	−767.429			−1.28118			−0.640592	−	0.767881i	$-0.721311\pi$
$599$	−767.429			−1.28118			−0.640592	+	0.767881i	$0.721311\pi$
$600$		0			0
$601$	−486.951			−0.810234			−0.405117	−	0.914265i	$-0.632769\pi$
$601$	−486.951			−0.810234			−0.405117	+	0.914265i	$0.632769\pi$
$602$	736.258			1.22302
$603$		0			0
$604$	79.5791			0.131753
$605$			147.088i			0.243121i
$606$		0			0
$607$	−856.769			−1.41148			−0.705740	−	0.708471i	$-0.749385\pi$
$607$	−856.769			−1.41148			−0.705740	+	0.708471i	$0.749385\pi$
$608$			149.715i			0.246241i
$609$		0			0
$610$	−105.874			−0.173564
$611$	1158.39			1.89589
$612$		0			0
$613$		−	996.963i		−	1.62637i	−0.582008	−	0.813183i	$-0.697733\pi$
$613$		−	996.963i		−	1.62637i	0.582008	−	0.813183i	$-0.302267\pi$
$614$	−744.800			−1.21303
$615$		0			0
$616$	−307.852			−0.499760
$617$			403.163i			0.653425i	0.945124	+	0.326712i	$0.105941\pi$
$617$			403.163i			0.653425i	−0.945124	+	0.326712i	$0.894059\pi$
$618$		0			0
$619$		−	6.17899i		−	0.00998222i	−0.999988	−	0.00499111i	$-0.998411\pi$
$619$		−	6.17899i		−	0.00998222i	0.999988	−	0.00499111i	$-0.00158873\pi$
$620$			99.1151i			0.159863i
$621$		0			0
$622$	759.651			1.22130
$623$	−166.111			−0.266630
$624$		0			0
$625$	25.0000			0.0400000
$626$		−	419.108i		−	0.669502i
$627$		0			0
$628$			526.503i			0.838380i
$629$	−556.843			−0.885282
$630$		0			0
$631$		−	478.508i		−	0.758333i	−0.925328	−	0.379166i	$-0.876211\pi$
$631$		−	478.508i		−	0.758333i	0.925328	−	0.379166i	$-0.123789\pi$
$632$		−	127.384i		−	0.201557i
$633$		0			0
$634$	−204.314			−0.322262
$635$			104.248i			0.164171i
$636$		0			0
$637$	202.418			0.317768
$638$			1053.33i			1.65099i
$639$		0			0
$640$			25.2982i			0.0395285i
$641$			431.496i			0.673160i	0.941655	+	0.336580i	$0.109270\pi$
$641$			431.496i			0.673160i	−0.941655	+	0.336580i	$0.890730\pi$
$642$		0			0
$643$		−	155.797i		−	0.242297i	−0.992634	−	0.121149i	$-0.961342\pi$
$643$		−	155.797i		−	0.242297i	0.992634	−	0.121149i	$-0.0386578\pi$
$644$	314.774	+	187.419i	0.488780	+	0.291024i
$645$		0			0
$646$	−646.273			−1.00042
$647$	797.916			1.23326			0.616628	−	0.787255i	$-0.288498\pi$
$647$	797.916			1.23326			0.616628	+	0.787255i	$0.288498\pi$
$648$		0			0
$649$		−	1442.35i		−	2.22242i
$650$	−99.2198			−0.152646
$651$		0			0
$652$	479.356			0.735209
$653$	−579.230			−0.887028			−0.443514	−	0.896267i	$-0.646268\pi$
$653$	−579.230			−0.887028			−0.443514	+	0.896267i	$0.646268\pi$
$654$		0			0
$655$			386.093i			0.589455i
$656$	−38.3677			−0.0584873
$657$		0			0
$658$			929.800i			1.41307i
$659$		−	991.515i		−	1.50458i	−0.658835	−	0.752288i	$-0.728950\pi$
$659$		−	991.515i		−	1.50458i	0.658835	−	0.752288i	$-0.271050\pi$
$660$		0			0
$661$		−	985.923i		−	1.49156i	−0.666191	−	0.745781i	$-0.732077\pi$
$661$		−	985.923i		−	1.49156i	0.666191	−	0.745781i	$-0.267923\pi$
$662$	680.513			1.02797
$663$		0			0
$664$			29.2706i			0.0440823i
$665$		−	471.310i		−	0.708737i
$666$		0			0
$667$	641.262	−	1077.01i	0.961412	−	1.61471i
$668$	−633.767			−0.948753
$669$		0			0
$670$	110.589			0.165058
$671$	457.567			0.681918
$672$		0			0
$673$	809.473			1.20278			0.601392	−	0.798954i	$-0.294613\pi$
$673$	809.473			1.20278			0.601392	+	0.798954i	$0.294613\pi$
$674$			322.414i			0.478358i
$675$		0			0
$676$	55.7826			0.0825186
$677$		−	1104.73i		−	1.63181i	−0.578188	−	0.815903i	$-0.696240\pi$
$677$		−	1104.73i		−	1.63181i	0.578188	−	0.815903i	$-0.303760\pi$
$678$		0			0
$679$	−1166.26			−1.71762
$680$	−109.205			−0.160595
$681$		0			0
$682$		−	428.356i		−	0.628088i
$683$	503.585			0.737313			0.368657	−	0.929566i	$-0.379818\pi$
$683$	503.585			0.737313			0.368657	+	0.929566i	$0.379818\pi$
$684$		0			0
$685$	−601.925			−0.878722
$686$		−	389.405i		−	0.567645i
$687$		0			0
$688$			261.483i			0.380062i
$689$			296.071i			0.429710i
$690$		0			0
$691$	693.803			1.00406			0.502028	−	0.864851i	$-0.332587\pi$
$691$	693.803			1.00406			0.502028	+	0.864851i	$0.332587\pi$
$692$	−231.782			−0.334945
$693$		0			0
$694$	402.218			0.579565
$695$			251.840i			0.362359i
$696$		0			0
$697$		−	165.622i		−	0.237621i
$698$	603.503			0.864617
$699$		0			0
$700$		−	79.6402i		−	0.113772i
$701$		−	456.823i		−	0.651673i	−0.945426	−	0.325836i	$-0.894354\pi$
$701$		−	456.823i		−	0.651673i	0.945426	−	0.325836i	$-0.105646\pi$
$702$		0			0
$703$	853.513			1.21410
$704$		−	109.334i		−	0.155304i
$705$		0			0
$706$	225.206			0.318989
$707$		−	882.155i		−	1.24774i
$708$		0			0
$709$		−	54.3734i		−	0.0766903i	−0.999265	−	0.0383452i	$-0.987791\pi$
$709$		−	54.3734i		−	0.0766903i	0.999265	−	0.0383452i	$-0.0122086\pi$
$710$		−	205.904i		−	0.290006i
$711$		0			0
$712$		−	58.9943i		−	0.0828572i
$713$	−260.781	+	437.987i	−0.365752	+	0.614287i
$714$		0			0
$715$	428.808			0.599732
$716$	124.253			0.173537
$717$		0			0
$718$			933.742i			1.30048i
$719$	−154.273			−0.214566			−0.107283	−	0.994229i	$-0.534215\pi$
$719$	−154.273			−0.214566			−0.107283	+	0.994229i	$0.534215\pi$
$720$		0			0
$721$	597.386			0.828553
$722$	480.059			0.664901
$723$		0			0
$724$			68.5051i			0.0946202i
$725$	−272.492			−0.375851
$726$		0			0
$727$		−	772.570i		−	1.06268i	−0.847158	−	0.531341i	$-0.821688\pi$
$727$		−	772.570i		−	1.06268i	0.847158	−	0.531341i	$-0.178312\pi$
$728$			316.075i			0.434170i
$729$		0			0
$730$		−	399.314i		−	0.547005i
$731$	−1128.74			−1.54411
$732$		0			0
$733$			590.416i			0.805479i	0.915315	+	0.402739i	$0.131942\pi$
$733$			590.416i			0.805479i	−0.915315	+	0.402739i	$0.868058\pi$
$734$		−	217.656i		−	0.296534i
$735$		0			0
$736$	−66.5620	+	111.792i	−0.0904375	+	0.151892i
$737$	−477.942			−0.648497
$738$		0			0
$739$	99.2703			0.134331			0.0671653	−	0.997742i	$-0.478605\pi$
$739$	99.2703			0.134331			0.0671653	+	0.997742i	$0.478605\pi$
$740$	144.223			0.194897
$741$		0			0
$742$	−237.645			−0.320277
$743$		−	5.78824i		−	0.00779036i	−0.999992	−	0.00389518i	$-0.998760\pi$
$743$		−	5.78824i		−	0.00779036i	0.999992	−	0.00389518i	$-0.00123988\pi$
$744$		0			0
$745$	−119.357			−0.160210
$746$		−	161.329i		−	0.216259i
$747$		0			0
$748$	471.962			0.630965
$749$	9.05675			0.0120918
$750$		0			0
$751$			4.93292i			0.00656847i	0.999995	+	0.00328423i	$0.00104541\pi$
$751$			4.93292i			0.00656847i	−0.999995	+	0.00328423i	$0.998955\pi$
$752$	−330.219			−0.439121
$753$		0			0
$754$	1081.46			1.43430
$755$		−	88.9721i		−	0.117844i
$756$		0			0
$757$			841.391i			1.11148i	0.831356	+	0.555740i	$0.187565\pi$
$757$			841.391i			1.11148i	−0.831356	+	0.555740i	$0.812435\pi$
$758$		−	804.614i		−	1.06150i
$759$		0			0
$760$	167.386			0.220245
$761$	−122.636			−0.161151			−0.0805754	−	0.996749i	$-0.525676\pi$
$761$	−122.636			−0.161151			−0.0805754	+	0.996749i	$0.525676\pi$
$762$		0			0
$763$	590.207			0.773535
$764$			731.615i			0.957611i
$765$		0			0
$766$		−	846.855i		−	1.10556i
$767$	−1480.88			−1.93074
$768$		0			0
$769$			495.138i			0.643873i	0.946761	+	0.321936i	$0.104334\pi$
$769$			495.138i			0.643873i	−0.946761	+	0.321936i	$0.895666\pi$
$770$			344.189i			0.446999i
$771$		0			0
$772$	−447.683			−0.579901
$773$			314.784i			0.407224i	0.979052	+	0.203612i	$0.0652680\pi$
$773$			314.784i			0.407224i	−0.979052	+	0.203612i	$0.934732\pi$
$774$		0			0
$775$	110.814			0.142986
$776$		−	414.199i		−	0.533761i
$777$		0			0
$778$		−	418.810i		−	0.538316i
$779$			253.861i			0.325880i
$780$		0			0
$781$			889.878i			1.13941i
$782$	−482.573	−	287.328i	−0.617101	−	0.367427i
$783$		0			0
$784$	−57.7027			−0.0736004
$785$	588.648			0.749870
$786$		0			0
$787$		−	870.883i		−	1.10659i	−0.832987	−	0.553293i	$-0.813371\pi$
$787$		−	870.883i		−	1.10659i	0.832987	−	0.553293i	$-0.186629\pi$
$788$	−179.239			−0.227460
$789$		0			0
$790$	−142.420			−0.180278
$791$	−968.024			−1.22380
$792$		0			0
$793$		−	469.789i		−	0.592420i
$794$	998.625			1.25771
$795$		0			0
$796$			11.8001i			0.0148243i
$797$			614.333i			0.770807i	0.922748	+	0.385403i	$0.125938\pi$
$797$			614.333i			0.770807i	−0.922748	+	0.385403i	$0.874062\pi$
$798$		0			0
$799$		−	1425.46i		−	1.78405i
$800$	28.2843			0.0353553
$801$		0			0
$802$		−	709.806i		−	0.885045i
$803$			1725.76i			2.14914i
$804$		0			0
$805$	209.541	−	351.928i	0.260299	−	0.437178i
$806$	−439.798			−0.545655
$807$		0			0
$808$	313.298			0.387745
$809$	380.557			0.470405			0.235202	−	0.971946i	$-0.424425\pi$
$809$	380.557			0.470405			0.235202	+	0.971946i	$0.424425\pi$
$810$		0			0
$811$	1032.12			1.27265			0.636324	−	0.771422i	$-0.280454\pi$
$811$	1032.12			1.27265			0.636324	+	0.771422i	$0.280454\pi$
$812$			868.054i			1.06903i
$813$		0			0
$814$	−623.305			−0.765731
$815$		−	535.937i		−	0.657591i
$816$		0			0
$817$	1730.10			2.11763
$818$	496.822			0.607362
$819$		0			0
$820$			42.8964i			0.0523127i
$821$	−1218.00			−1.48355			−0.741777	−	0.670647i	$-0.766016\pi$
$821$	−1218.00			−1.48355			−0.741777	+	0.670647i	$0.766016\pi$
$822$		0			0
$823$	361.478			0.439220			0.219610	−	0.975588i	$-0.429522\pi$
$823$	361.478			0.439220			0.219610	+	0.975588i	$0.429522\pi$
$824$			212.162i			0.257478i
$825$		0			0
$826$		−	1188.65i		−	1.43904i
$827$		−	1564.84i		−	1.89219i	−0.323893	−	0.946094i	$-0.604992\pi$
$827$		−	1564.84i		−	1.89219i	0.323893	−	0.946094i	$-0.395008\pi$
$828$		0			0
$829$	817.899			0.986609			0.493304	−	0.869857i	$-0.335789\pi$
$829$	817.899			0.986609			0.493304	+	0.869857i	$0.335789\pi$
$830$	32.7256			0.0394284
$831$		0			0
$832$	−112.254			−0.134921
$833$		−	249.085i		−	0.299022i
$834$		0			0
$835$			708.573i			0.848591i
$836$	−723.410			−0.865323
$837$		0			0
$838$		−	703.847i		−	0.839913i
$839$		−	281.562i		−	0.335592i	−0.985822	−	0.167796i	$-0.946335\pi$
$839$		−	281.562i		−	0.335592i	0.985822	−	0.167796i	$-0.0536650\pi$
$840$		0			0
$841$	2129.08			2.53160
$842$		−	457.600i		−	0.543469i
$843$		0			0
$844$	−463.396			−0.549048
$845$		−	62.3668i		−	0.0738069i
$846$		0			0
$847$		−	523.873i		−	0.618504i
$848$		−	84.3999i		−	0.0995282i
$849$		0			0
$850$			122.095i			0.143641i
$851$	637.320	+	379.466i	0.748907	+	0.445905i
$852$		0			0
$853$	−856.416			−1.00400			−0.502002	−	0.864866i	$-0.667403\pi$
$853$	−856.416			−1.00400			−0.502002	+	0.864866i	$0.667403\pi$
$854$	377.083			0.441550
$855$		0			0
$856$			3.21651i			0.00375760i
$857$	261.702			0.305369			0.152685	−	0.988275i	$-0.451208\pi$
$857$	261.702			0.305369			0.152685	+	0.988275i	$0.451208\pi$
$858$		0			0
$859$	−816.256			−0.950240			−0.475120	−	0.879921i	$-0.657595\pi$
$859$	−816.256			−0.950240			−0.475120	+	0.879921i	$0.657595\pi$
$860$	292.346			0.339938
$861$		0			0
$862$			763.295i			0.885493i
$863$	915.224			1.06051			0.530257	−	0.847837i	$-0.322095\pi$
$863$	915.224			1.06051			0.530257	+	0.847837i	$0.322095\pi$
$864$		0			0
$865$			259.140i			0.299584i
$866$			596.576i			0.688887i
$867$		0			0
$868$		−	353.010i		−	0.406694i
$869$	615.510			0.708297
$870$		0			0
$871$			490.709i			0.563385i
$872$			209.612i			0.240381i
$873$		0			0
$874$	739.675	+	440.409i	0.846310	+	0.503900i
$875$	−89.0405			−0.101761
$876$		0			0
$877$	−891.151			−1.01614			−0.508068	−	0.861317i	$-0.669640\pi$
$877$	−891.151			−1.01614			−0.508068	+	0.861317i	$0.669640\pi$
$878$	−458.782			−0.522531
$879$		0			0
$880$	−122.239			−0.138908
$881$		−	330.926i		−	0.375625i	−0.982205	−	0.187813i	$-0.939860\pi$
$881$		−	330.926i		−	0.375625i	0.982205	−	0.187813i	$-0.0601398\pi$
$882$		0			0
$883$	−369.742			−0.418734			−0.209367	−	0.977837i	$-0.567140\pi$
$883$	−369.742			−0.418734			−0.209367	+	0.977837i	$0.567140\pi$
$884$		−	484.568i		−	0.548154i
$885$		0			0
$886$	313.668			0.354028
$887$	−876.383			−0.988031			−0.494015	−	0.869453i	$-0.664471\pi$
$887$	−876.383			−0.988031			−0.494015	+	0.869453i	$0.664471\pi$
$888$		0			0
$889$		−	371.293i		−	0.417653i
$890$	−65.9577			−0.0741097
$891$		0			0
$892$	260.001			0.291481
$893$			2184.90i			2.44670i
$894$		0			0
$895$		−	138.919i		−	0.155217i
$896$		−	90.1026i		−	0.100561i
$897$		0			0
$898$	−740.935			−0.825094
$899$	−1207.84			−1.34354
$900$		0			0
$901$	364.329			0.404361
$902$		−	185.390i		−	0.205532i
$903$		0			0
$904$		−	343.794i		−	0.380303i
$905$	76.5910			0.0846309
$906$		0			0
$907$		−	1391.06i		−	1.53369i	−0.641832	−	0.766845i	$-0.721825\pi$
$907$		−	1391.06i		−	1.53369i	0.641832	−	0.766845i	$-0.278175\pi$
$908$		−	334.230i		−	0.368095i
$909$		0			0
$910$	353.383			0.388333
$911$			633.904i			0.695833i	0.937525	+	0.347917i	$0.113111\pi$
$911$			633.904i			0.695833i	−0.937525	+	0.347917i	$0.886889\pi$
$912$		0			0
$913$	−141.433			−0.154911
$914$		−	536.572i		−	0.587059i
$915$		0			0
$916$		−	38.4251i		−	0.0419488i
$917$		−	1375.12i		−	1.49958i
$918$		0			0
$919$			89.6435i			0.0975446i	0.998810	+	0.0487723i	$0.0155309\pi$
$919$			89.6435i			0.0975446i	−0.998810	+	0.0487723i	$0.984469\pi$
$920$	124.987	+	74.4186i	0.135856	+	0.0808898i
$921$		0			0
$922$	−635.400			−0.689154
$923$	913.648			0.989868
$924$		0			0
$925$		−	161.247i		−	0.174321i
$926$	−822.296			−0.888008
$927$		0			0
$928$	−308.290			−0.332209
$929$	−129.658			−0.139568			−0.0697838	−	0.997562i	$-0.522231\pi$
$929$	−129.658			−0.139568			−0.0697838	+	0.997562i	$0.522231\pi$
$930$		0			0
$931$			381.791i			0.410087i
$932$	−203.395			−0.218235
$933$		0			0
$934$		−	879.787i		−	0.941956i
$935$		−	527.669i		−	0.564352i
$936$		0			0
$937$		−	171.827i		−	0.183380i	−0.995788	−	0.0916900i	$-0.970773\pi$
$937$		−	171.827i		−	0.183380i	0.995788	−	0.0916900i	$-0.0292269\pi$
$938$	−393.875			−0.419909
$939$		0			0
$940$			369.196i			0.392762i
$941$		−	6.11599i		−	0.00649946i	−0.999995	−	0.00324973i	$-0.998966\pi$
$941$		−	6.11599i		−	0.00649946i	0.999995	−	0.00324973i	$-0.00103442\pi$
$942$		0			0
$943$	−112.864	+	189.558i	−0.119687	+	0.201016i
$944$	422.150			0.447192
$945$		0			0
$946$	−1263.46			−1.33559
$947$	−131.841			−0.139219			−0.0696097	−	0.997574i	$-0.522175\pi$
$947$	−131.841			−0.139219			−0.0696097	+	0.997574i	$0.522175\pi$
$948$		0			0
$949$	1771.85			1.86707
$950$		−	187.143i		−	0.196993i
$951$		0			0
$952$	388.946			0.408557
$953$			609.278i			0.639326i	0.947531	+	0.319663i	$0.103570\pi$
$953$			609.278i			0.639326i	−0.947531	+	0.319663i	$0.896430\pi$
$954$		0			0
$955$	817.970			0.856513
$956$	131.452			0.137502
$957$		0			0
$958$		−	381.209i		−	0.397922i
$959$	2143.83			2.23548
$960$		0			0
$961$	−469.810			−0.488876
$962$			639.954i			0.665233i
$963$		0			0
$964$			382.736i			0.397029i
$965$			500.525i			0.518679i
$966$		0			0
$967$	443.855			0.459002			0.229501	−	0.973308i	$-0.426291\pi$
$967$	443.855			0.459002			0.229501	+	0.973308i	$0.426291\pi$
$968$	186.054			0.192204
$969$		0			0
$970$	−463.088			−0.477410
$971$		−	1070.02i		−	1.10198i	−0.834511	−	0.550991i	$-0.814250\pi$
$971$		−	1070.02i		−	1.10198i	0.834511	−	0.550991i	$-0.185750\pi$
$972$		0			0
$973$		−	896.957i		−	0.921847i
$974$	−455.278			−0.467431
$975$		0			0
$976$			133.921i			0.137215i
$977$			1707.50i			1.74770i	0.486195	+	0.873850i	$0.338384\pi$
$977$			1707.50i			1.74770i	−0.486195	+	0.873850i	$0.661616\pi$
$978$		0			0
$979$	285.056			0.291171
$980$			64.5136i			0.0658302i
$981$		0			0
$982$	−594.253			−0.605146
$983$			1086.75i			1.10555i	0.833332	+	0.552773i	$0.186431\pi$
$983$			1086.75i			1.10555i	−0.833332	+	0.552773i	$0.813569\pi$
$984$		0			0
$985$			200.395i			0.203446i
$986$		−	1330.79i		−	1.34969i
$987$		0			0
$988$			742.733i			0.751754i
$989$	1291.87	+	769.191i	1.30624	+	0.777746i
$990$		0			0
$991$	374.649			0.378051			0.189026	−	0.981972i	$-0.439467\pi$
$991$	374.649			0.378051			0.189026	+	0.981972i	$0.439467\pi$
$992$	125.372			0.126383
$993$		0			0
$994$			733.353i			0.737780i
$995$	13.1929			0.0132592
$996$		0			0
$997$	−761.430			−0.763721			−0.381860	−	0.924220i	$-0.624716\pi$
$997$	−761.430			−0.763721			−0.381860	+	0.924220i	$0.624716\pi$
$998$	366.851			0.367586
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.3.c.b.91.7		32
3.2	odd	2		690.3.c.a.91.23	yes	32
23.22	odd	2	inner	2070.3.c.b.91.10		32
69.68	even	2		690.3.c.a.91.18	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.c.a.91.18	✓	32	69.68	even	2
690.3.c.a.91.23	yes	32	3.2	odd	2
2070.3.c.b.91.7		32	1.1	even	1	trivial
2070.3.c.b.91.10		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)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} +7.96402i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} +7.96402i q^{7} -2.82843 q^{8} +3.16228i q^{10} -13.6667i q^{11} -14.0318 q^{13} -11.2628i q^{14} +4.00000 q^{16} +17.2668i q^{17} -26.4661i q^{19} -4.47214i q^{20} +19.3277i q^{22} +(11.7666 - 19.7623i) q^{23} -5.00000 q^{25} +19.8440 q^{26} +15.9280i q^{28} +54.4984 q^{29} -22.1628 q^{31} -5.65685 q^{32} -24.4189i q^{34} +17.8081 q^{35} +32.2493i q^{37} +37.4287i q^{38} +6.32456i q^{40} -9.59193 q^{41} +65.3706i q^{43} -27.3335i q^{44} +(-16.6405 + 27.9481i) q^{46} -82.5547 q^{47} -14.4257 q^{49} +7.07107 q^{50} -28.0636 q^{52} -21.1000i q^{53} -30.5598 q^{55} -22.5257i q^{56} -77.0724 q^{58} +105.537 q^{59} +33.4803i q^{61} +31.3429 q^{62} +8.00000 q^{64} +31.3760i q^{65} -34.9712i q^{67} +34.5336i q^{68} -25.1845 q^{70} -65.1127 q^{71} -126.274 q^{73} -45.6075i q^{74} -52.9321i q^{76} +108.842 q^{77} +45.0370i q^{79} -8.94427i q^{80} +13.5650 q^{82} -10.3487i q^{83} +38.6097 q^{85} -92.4480i q^{86} +38.6554i q^{88} +20.8576i q^{89} -111.750i q^{91} +(23.5332 - 39.5245i) q^{92} +116.750 q^{94} -59.1799 q^{95} +146.441i q^{97} +20.4010 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4}+O(q^{10}) \) 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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