LMFDB - Embedded newform 765.2.g.b.271.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [765,2,Mod(271,765)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(765, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("765.271");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 765 = 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	765.g (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:	$6.10855575463$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 85)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			271.4
Root			$1.45161 + 1.45161i$ of defining polynomial
Character	$\chi$	$=$	765.271
Dual form			765.2.g.b.271.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.311108 q^{2} -1.90321 q^{4} +1.00000i q^{5} -1.59210i q^{7} -1.21432 q^{8} +0.311108i q^{10} -1.31111i q^{11} +3.52543 q^{13} -0.495316i q^{14} +3.42864 q^{16} +(-4.11753 - 0.214320i) q^{17} -4.42864 q^{19} -1.90321i q^{20} -0.407896i q^{22} -4.96989i q^{23} -1.00000 q^{25} +1.09679 q^{26} +3.03011i q^{28} -8.42864i q^{29} -7.73975i q^{31} +3.49532 q^{32} +(-1.28100 - 0.0666765i) q^{34} +1.59210 q^{35} -7.05086i q^{37} -1.37778 q^{38} -1.21432i q^{40} -3.67307i q^{41} +2.47457 q^{43} +2.49532i q^{44} -1.54617i q^{46} -3.33185 q^{47} +4.46520 q^{49} -0.311108 q^{50} -6.70964 q^{52} +9.18421 q^{53} +1.31111 q^{55} +1.93332i q^{56} -2.62222i q^{58} -1.37778 q^{59} +15.4193i q^{61} -2.40790i q^{62} -5.76986 q^{64} +3.52543i q^{65} -9.13828 q^{67} +(7.83654 + 0.407896i) q^{68} +0.495316 q^{70} -10.5970i q^{71} -5.57136i q^{73} -2.19358i q^{74} +8.42864 q^{76} -2.08742 q^{77} +7.87310i q^{79} +3.42864i q^{80} -1.14272i q^{82} -7.19850 q^{83} +(0.214320 - 4.11753i) q^{85} +0.769859 q^{86} +1.59210i q^{88} -11.6271 q^{89} -5.61285i q^{91} +9.45875i q^{92} -1.03657 q^{94} -4.42864i q^{95} +15.4795i q^{97} +1.38916 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 2 q^{4} + 6 q^{8} + 8 q^{13} - 6 q^{16} + 2 q^{17} - 6 q^{25} + 20 q^{26} - 6 q^{32} + 6 q^{34} - 4 q^{35} - 8 q^{38} + 28 q^{43} + 20 q^{47} - 14 q^{49} - 2 q^{50} + 28 q^{53} + 8 q^{55}+ \cdots - 86 q^{98}+O(q^{100}) $ 6 * q + 2 * q^2 + 2 * q^4 + 6 * q^8 + 8 * q^13 - 6 * q^16 + 2 * q^17 - 6 * q^25 + 20 * q^26 - 6 * q^32 + 6 * q^34 - 4 * q^35 - 8 * q^38 + 28 * q^43 + 20 * q^47 - 14 * q^49 - 2 * q^50 + 28 * q^53 + 8 * q^55 - 8 * q^59 - 22 * q^64 + 12 * q^67 + 34 * q^68 - 24 * q^70 + 24 * q^76 + 28 * q^77 - 4 * q^83 - 12 * q^85 - 8 * q^86 - 4 * q^89 + 8 * q^94 - 86 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/765\mathbb{Z}\right)^\times$.

$n$	$307$	$496$	$596$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	0.311108			0.219986			0.109993	−	0.993932i	$-0.464917\pi$
$2$	0.311108			0.219986			0.109993	+	0.993932i	$0.464917\pi$
$3$		0			0
$3$		0			0
$4$	−1.90321			−0.951606
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$		−	1.59210i		−	0.601759i	−0.953662	−	0.300879i	$-0.902720\pi$
$7$		−	1.59210i		−	0.601759i	0.953662	−	0.300879i	$-0.0972802\pi$
$8$	−1.21432			−0.429327
$9$		0			0
$10$			0.311108i			0.0983809i
$11$		−	1.31111i		−	0.395314i	−0.980271	−	0.197657i	$-0.936667\pi$
$11$		−	1.31111i		−	0.395314i	0.980271	−	0.197657i	$-0.0633332\pi$
$12$		0			0
$13$	3.52543			0.977778			0.488889	−	0.872346i	$-0.337402\pi$
$13$	3.52543			0.977778			0.488889	+	0.872346i	$0.337402\pi$
$14$		−	0.495316i		−	0.132379i
$15$		0			0
$16$	3.42864			0.857160
$17$	−4.11753	−	0.214320i	−0.998648	−	0.0519802i
$17$	−4.11753	−	0.214320i	−0.998648	−	0.0519802i
$18$		0			0
$19$	−4.42864			−1.01600			−0.508000	−	0.861357i	$-0.669615\pi$
$19$	−4.42864			−1.01600			−0.508000	+	0.861357i	$0.669615\pi$
$20$		−	1.90321i		−	0.425571i
$21$		0			0
$22$		−	0.407896i		−	0.0869637i
$23$		−	4.96989i		−	1.03629i	−0.855292	−	0.518147i	$-0.826622\pi$
$23$		−	4.96989i		−	1.03629i	0.855292	−	0.518147i	$-0.173378\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$	1.09679			0.215098
$27$		0			0
$28$			3.03011i			0.572637i
$29$		−	8.42864i		−	1.56516i	−0.622551	−	0.782580i	$-0.713904\pi$
$29$		−	8.42864i		−	1.56516i	0.622551	−	0.782580i	$-0.286096\pi$
$30$		0			0
$31$		−	7.73975i		−	1.39010i	−0.718962	−	0.695050i	$-0.755382\pi$
$31$		−	7.73975i		−	1.39010i	0.718962	−	0.695050i	$-0.244618\pi$
$32$	3.49532			0.617890
$33$		0			0
$34$	−1.28100	−	0.0666765i	−0.219689	−	0.0114349i
$35$	1.59210			0.269115
$36$		0			0
$37$		−	7.05086i		−	1.15915i	−0.814918	−	0.579577i	$-0.803218\pi$
$37$		−	7.05086i		−	1.15915i	0.814918	−	0.579577i	$-0.196782\pi$
$38$	−1.37778			−0.223506
$39$		0			0
$40$		−	1.21432i		−	0.192001i
$41$		−	3.67307i		−	0.573637i	−0.957985	−	0.286819i	$-0.907402\pi$
$41$		−	3.67307i		−	0.573637i	0.957985	−	0.286819i	$-0.0925977\pi$
$42$		0			0
$43$	2.47457			0.377369			0.188684	−	0.982038i	$-0.439578\pi$
$43$	2.47457			0.377369			0.188684	+	0.982038i	$0.439578\pi$
$44$			2.49532i			0.376183i
$45$		0			0
$46$		−	1.54617i		−	0.227970i
$47$	−3.33185			−0.486000			−0.243000	−	0.970026i	$-0.578132\pi$
$47$	−3.33185			−0.486000			−0.243000	+	0.970026i	$0.578132\pi$
$48$		0			0
$49$	4.46520			0.637886
$50$	−0.311108			−0.0439973
$51$		0			0
$52$	−6.70964			−0.930459
$53$	9.18421			1.26155			0.630774	−	0.775967i	$-0.282737\pi$
$53$	9.18421			1.26155			0.630774	+	0.775967i	$0.282737\pi$
$54$		0			0
$55$	1.31111			0.176790
$56$			1.93332i			0.258351i
$57$		0			0
$58$		−	2.62222i		−	0.344314i
$59$	−1.37778			−0.179372			−0.0896861	−	0.995970i	$-0.528586\pi$
$59$	−1.37778			−0.179372			−0.0896861	+	0.995970i	$0.528586\pi$
$60$		0			0
$61$			15.4193i			1.97424i	0.159996	+	0.987118i	$0.448852\pi$
$61$			15.4193i			1.97424i	−0.159996	+	0.987118i	$0.551148\pi$
$62$		−	2.40790i		−	0.305803i
$63$		0			0
$64$	−5.76986			−0.721232
$65$			3.52543i			0.437275i
$66$		0			0
$67$	−9.13828			−1.11642			−0.558209	−	0.829700i	$-0.688511\pi$
$67$	−9.13828			−1.11642			−0.558209	+	0.829700i	$0.688511\pi$
$68$	7.83654	+	0.407896i	0.950320	+	0.0494646i
$69$		0			0
$70$	0.495316			0.0592016
$71$		−	10.5970i		−	1.25764i	−0.777553	−	0.628818i	$-0.783539\pi$
$71$		−	10.5970i		−	1.25764i	0.777553	−	0.628818i	$-0.216461\pi$
$72$		0			0
$73$		−	5.57136i		−	0.652078i	−0.945356	−	0.326039i	$-0.894286\pi$
$73$		−	5.57136i		−	0.652078i	0.945356	−	0.326039i	$-0.105714\pi$
$74$		−	2.19358i		−	0.254998i
$75$		0			0
$76$	8.42864			0.966831
$77$	−2.08742			−0.237884
$78$		0			0
$79$			7.87310i			0.885793i	0.896573	+	0.442897i	$0.146049\pi$
$79$			7.87310i			0.885793i	−0.896573	+	0.442897i	$0.853951\pi$
$80$			3.42864i			0.383334i
$81$		0			0
$82$		−	1.14272i		−	0.126192i
$83$	−7.19850			−0.790138			−0.395069	−	0.918651i	$-0.629279\pi$
$83$	−7.19850			−0.790138			−0.395069	+	0.918651i	$0.629279\pi$
$84$		0			0
$85$	0.214320	−	4.11753i	0.0232462	−	0.446609i
$86$	0.769859			0.0830160
$87$		0			0
$88$			1.59210i			0.169719i
$89$	−11.6271			−1.23247			−0.616237	−	0.787561i	$-0.711344\pi$
$89$	−11.6271			−1.23247			−0.616237	+	0.787561i	$0.711344\pi$
$90$		0			0
$91$		−	5.61285i		−	0.588386i
$92$			9.45875i			0.986143i
$93$		0			0
$94$	−1.03657			−0.106914
$95$		−	4.42864i		−	0.454369i
$96$		0			0
$97$			15.4795i			1.57170i	0.618414	+	0.785852i	$0.287775\pi$
$97$			15.4795i			1.57170i	−0.618414	+	0.785852i	$0.712225\pi$
$98$	1.38916			0.140326
$99$		0			0
$100$	1.90321			0.190321
$101$	14.3827			1.43113			0.715566	−	0.698545i	$-0.246169\pi$
$101$	14.3827			1.43113			0.715566	+	0.698545i	$0.246169\pi$
$102$		0			0
$103$	−9.39207			−0.925429			−0.462714	−	0.886507i	$-0.653124\pi$
$103$	−9.39207			−0.925429			−0.462714	+	0.886507i	$0.653124\pi$
$104$	−4.28100			−0.419786
$105$		0			0
$106$	2.85728			0.277523
$107$			6.77631i			0.655091i	0.944835	+	0.327545i	$0.106221\pi$
$107$			6.77631i			0.655091i	−0.944835	+	0.327545i	$0.893779\pi$
$108$		0			0
$109$		−	2.85728i		−	0.273678i	−0.990593	−	0.136839i	$-0.956306\pi$
$109$		−	2.85728i		−	0.273678i	0.990593	−	0.136839i	$-0.0436942\pi$
$110$	0.407896			0.0388913
$111$		0			0
$112$		−	5.45875i		−	0.515803i
$113$			5.86665i			0.551888i	0.961174	+	0.275944i	$0.0889904\pi$
$113$			5.86665i			0.551888i	−0.961174	+	0.275944i	$0.911010\pi$
$114$		0			0
$115$	4.96989			0.463444
$116$			16.0415i			1.48941i
$117$		0			0
$118$	−0.428639			−0.0394595
$119$	−0.341219	+	6.55554i	−0.0312795	+	0.600945i
$120$		0			0
$121$	9.28100			0.843727
$122$			4.79706i			0.434305i
$123$		0			0
$124$			14.7304i			1.32283i
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	0.280996			0.0249344			0.0124672	−	0.999922i	$-0.496031\pi$
$127$	0.280996			0.0249344			0.0124672	+	0.999922i	$0.496031\pi$
$128$	−8.78568			−0.776552
$129$		0			0
$130$			1.09679i			0.0961947i
$131$			13.4128i			1.17188i	0.810353	+	0.585942i	$0.199275\pi$
$131$			13.4128i			1.17188i	−0.810353	+	0.585942i	$0.800725\pi$
$132$		0			0
$133$			7.05086i			0.611387i
$134$	−2.84299			−0.245597
$135$		0			0
$136$	5.00000	+	0.260253i	0.428746	+	0.0223165i
$137$	−10.2810			−0.878365			−0.439182	−	0.898398i	$-0.644732\pi$
$137$	−10.2810			−0.878365			−0.439182	+	0.898398i	$0.644732\pi$
$138$		0			0
$139$		−	18.3017i		−	1.55233i	−0.630528	−	0.776167i	$-0.717162\pi$
$139$		−	18.3017i		−	1.55233i	0.630528	−	0.776167i	$-0.282838\pi$
$140$	−3.03011			−0.256091
$141$		0			0
$142$		−	3.29682i		−	0.276663i
$143$		−	4.62222i		−	0.386529i
$144$		0			0
$145$	8.42864			0.699960
$146$		−	1.73329i		−	0.143448i
$147$		0			0
$148$			13.4193i			1.10306i
$149$	4.91750			0.402857			0.201429	−	0.979503i	$-0.435442\pi$
$149$	4.91750			0.402857			0.201429	+	0.979503i	$0.435442\pi$
$150$		0			0
$151$	12.7239			1.03546			0.517729	−	0.855545i	$-0.326777\pi$
$151$	12.7239			1.03546			0.517729	+	0.855545i	$0.326777\pi$
$152$	5.37778			0.436196
$153$		0			0
$154$	−0.649413			−0.0523312
$155$	7.73975			0.621671
$156$		0			0
$157$	−9.18421			−0.732980			−0.366490	−	0.930422i	$-0.619441\pi$
$157$	−9.18421			−0.732980			−0.366490	+	0.930422i	$0.619441\pi$
$158$			2.44938i			0.194862i
$159$		0			0
$160$			3.49532i			0.276329i
$161$	−7.91258			−0.623599
$162$		0			0
$163$		−	5.07160i		−	0.397238i	−0.980077	−	0.198619i	$-0.936354\pi$
$163$		−	5.07160i		−	0.397238i	0.980077	−	0.198619i	$-0.0636457\pi$
$164$			6.99063i			0.545877i
$165$		0			0
$166$	−2.23951			−0.173820
$167$		−	19.8272i		−	1.53427i	−0.641484	−	0.767136i	$-0.721681\pi$
$167$		−	19.8272i		−	1.53427i	0.641484	−	0.767136i	$-0.278319\pi$
$168$		0			0
$169$	−0.571361			−0.0439508
$170$	0.0666765	−	1.28100i	0.00511386	−	0.0982479i
$171$		0			0
$172$	−4.70964			−0.359106
$173$		−	2.48886i		−	0.189225i	−0.995514	−	0.0946124i	$-0.969839\pi$
$173$		−	2.48886i		−	0.189225i	0.995514	−	0.0946124i	$-0.0301612\pi$
$174$		0			0
$175$			1.59210i			0.120352i
$176$		−	4.49532i		−	0.338847i
$177$		0			0
$178$	−3.61729			−0.271128
$179$	11.6128			0.867985			0.433992	−	0.900916i	$-0.357104\pi$
$179$	11.6128			0.867985			0.433992	+	0.900916i	$0.357104\pi$
$180$		0			0
$181$		−	3.86665i		−	0.287406i	−0.989621	−	0.143703i	$-0.954099\pi$
$181$		−	3.86665i		−	0.287406i	0.989621	−	0.143703i	$-0.0459009\pi$
$182$		−	1.74620i		−	0.129437i
$183$		0			0
$184$			6.03503i			0.444909i
$185$	7.05086			0.518389
$186$		0			0
$187$	−0.280996	+	5.39853i	−0.0205485	+	0.394779i
$188$	6.34122			0.462481
$189$		0			0
$190$		−	1.37778i		−	0.0999550i
$191$	14.9175			1.07939			0.539696	−	0.841860i	$-0.318539\pi$
$191$	14.9175			1.07939			0.539696	+	0.841860i	$0.318539\pi$
$192$		0			0
$193$		−	17.7462i		−	1.27740i	−0.769456	−	0.638700i	$-0.779473\pi$
$193$		−	17.7462i		−	1.27740i	0.769456	−	0.638700i	$-0.220527\pi$
$194$			4.81579i			0.345754i
$195$		0			0
$196$	−8.49823			−0.607016
$197$			9.41927i			0.671095i	0.942023	+	0.335548i	$0.108921\pi$
$197$			9.41927i			0.671095i	−0.942023	+	0.335548i	$0.891079\pi$
$198$		0			0
$199$			7.21924i			0.511758i	0.966709	+	0.255879i	$0.0823649\pi$
$199$			7.21924i			0.511758i	−0.966709	+	0.255879i	$0.917635\pi$
$200$	1.21432			0.0858654
$201$		0			0
$202$	4.47457			0.314830
$203$	−13.4193			−0.941848
$204$		0			0
$205$	3.67307			0.256538
$206$	−2.92195			−0.203582
$207$		0			0
$208$	12.0874			0.838112
$209$			5.80642i			0.401639i
$210$		0			0
$211$			14.8825i			1.02455i	0.858821	+	0.512276i	$0.171197\pi$
$211$			14.8825i			1.02455i	−0.858821	+	0.512276i	$0.828803\pi$
$212$	−17.4795			−1.20050
$213$		0			0
$214$			2.10816i			0.144111i
$215$			2.47457i			0.168764i
$216$		0			0
$217$	−12.3225			−0.836505
$218$		−	0.888922i		−	0.0602054i
$219$		0			0
$220$	−2.49532			−0.168234
$221$	−14.5161	−	0.755569i	−0.976456	−	0.0508251i
$222$		0			0
$223$	−23.4652			−1.57135			−0.785673	−	0.618642i	$-0.787683\pi$
$223$	−23.4652			−1.57135			−0.785673	+	0.618642i	$0.787683\pi$
$224$		−	5.56491i		−	0.371821i
$225$		0			0
$226$			1.82516i			0.121408i
$227$		−	4.34767i		−	0.288565i	−0.989537	−	0.144283i	$-0.953913\pi$
$227$		−	4.34767i		−	0.288565i	0.989537	−	0.144283i	$-0.0460874\pi$
$228$		0			0
$229$	6.47457			0.427852			0.213926	−	0.976850i	$-0.431375\pi$
$229$	6.47457			0.427852			0.213926	+	0.976850i	$0.431375\pi$
$230$	1.54617			0.101952
$231$		0			0
$232$			10.2351i			0.671965i
$233$			18.5303i			1.21396i	0.794716	+	0.606982i	$0.207620\pi$
$233$			18.5303i			1.21396i	−0.794716	+	0.606982i	$0.792380\pi$
$234$		0			0
$235$		−	3.33185i		−	0.217346i
$236$	2.62222			0.170692
$237$		0			0
$238$	−0.106156	+	2.03948i	−0.00688107	+	0.132200i
$239$	21.4193			1.38550			0.692749	−	0.721179i	$-0.256399\pi$
$239$	21.4193			1.38550			0.692749	+	0.721179i	$0.256399\pi$
$240$		0			0
$241$		−	12.2351i		−	0.788130i	−0.919083	−	0.394065i	$-0.871069\pi$
$241$		−	12.2351i		−	0.788130i	0.919083	−	0.394065i	$-0.128931\pi$
$242$	2.88739			0.185608
$243$		0			0
$244$		−	29.3461i		−	1.87869i
$245$			4.46520i			0.285271i
$246$		0			0
$247$	−15.6128			−0.993422
$248$			9.39853i			0.596807i
$249$		0			0
$250$		−	0.311108i		−	0.0196762i
$251$	4.52051			0.285332			0.142666	−	0.989771i	$-0.454433\pi$
$251$	4.52051			0.285332			0.142666	+	0.989771i	$0.454433\pi$
$252$		0			0
$253$	−6.51606			−0.409661
$254$	0.0874201			0.00548523
$255$		0			0
$256$	8.80642			0.550401
$257$	6.18913			0.386067			0.193034	−	0.981192i	$-0.438167\pi$
$257$	6.18913			0.386067			0.193034	+	0.981192i	$0.438167\pi$
$258$		0			0
$259$	−11.2257			−0.697531
$260$		−	6.70964i		−	0.416114i
$261$		0			0
$262$			4.17283i			0.257798i
$263$	−8.18913			−0.504963			−0.252482	−	0.967602i	$-0.581247\pi$
$263$	−8.18913			−0.504963			−0.252482	+	0.967602i	$0.581247\pi$
$264$		0			0
$265$			9.18421i			0.564181i
$266$			2.19358i			0.134497i
$267$		0			0
$268$	17.3921			1.06239
$269$		−	3.89829i		−	0.237683i	−0.992913	−	0.118841i	$-0.962082\pi$
$269$		−	3.89829i		−	0.237683i	0.992913	−	0.118841i	$-0.0379180\pi$
$270$		0			0
$271$	8.85728			0.538041			0.269021	−	0.963134i	$-0.413300\pi$
$271$	8.85728			0.538041			0.269021	+	0.963134i	$0.413300\pi$
$272$	−14.1175	−	0.734825i	−0.856001	−	0.0445553i
$273$		0			0
$274$	−3.19850			−0.193228
$275$			1.31111i			0.0790628i
$276$		0			0
$277$		−	1.14272i		−	0.0686595i	−0.999411	−	0.0343297i	$-0.989070\pi$
$277$		−	1.14272i		−	0.0686595i	0.999411	−	0.0343297i	$-0.0109296\pi$
$278$		−	5.69381i		−	0.341492i
$279$		0			0
$280$	−1.93332			−0.115538
$281$	14.6953			0.876651			0.438325	−	0.898816i	$-0.355572\pi$
$281$	14.6953			0.876651			0.438325	+	0.898816i	$0.355572\pi$
$282$		0			0
$283$			1.97926i			0.117655i	0.998268	+	0.0588273i	$0.0187361\pi$
$283$			1.97926i			0.117655i	−0.998268	+	0.0588273i	$0.981264\pi$
$284$			20.1684i			1.19677i
$285$		0			0
$286$		−	1.43801i		−	0.0850312i
$287$	−5.84791			−0.345191
$288$		0			0
$289$	16.9081	+	1.76494i	0.994596	+	0.103820i
$290$	2.62222			0.153982
$291$		0			0
$292$			10.6035i			0.620522i
$293$	−2.94914			−0.172291			−0.0861454	−	0.996283i	$-0.527455\pi$
$293$	−2.94914			−0.172291			−0.0861454	+	0.996283i	$0.527455\pi$
$294$		0			0
$295$		−	1.37778i		−	0.0802177i
$296$			8.56199i			0.497656i
$297$		0			0
$298$	1.52987			0.0886232
$299$		−	17.5210i		−	1.01326i
$300$		0			0
$301$		−	3.93978i		−	0.227085i
$302$	3.95851			0.227787
$303$		0			0
$304$	−15.1842			−0.870874
$305$	−15.4193			−0.882905
$306$		0			0
$307$	−5.68736			−0.324595			−0.162297	−	0.986742i	$-0.551890\pi$
$307$	−5.68736			−0.324595			−0.162297	+	0.986742i	$0.551890\pi$
$308$	3.97280			0.226371
$309$		0			0
$310$	2.40790			0.136759
$311$		−	23.7210i		−	1.34510i	−0.740054	−	0.672548i	$-0.765200\pi$
$311$		−	23.7210i		−	1.34510i	0.740054	−	0.672548i	$-0.234800\pi$
$312$		0			0
$313$			30.8988i			1.74650i	0.487271	+	0.873251i	$0.337992\pi$
$313$			30.8988i			1.74650i	−0.487271	+	0.873251i	$0.662008\pi$
$314$	−2.85728			−0.161246
$315$		0			0
$316$		−	14.9842i		−	0.842926i
$317$			13.7047i			0.769733i	0.922972	+	0.384867i	$0.125753\pi$
$317$			13.7047i			0.769733i	−0.922972	+	0.384867i	$0.874247\pi$
$318$		0			0
$319$	−11.0509			−0.618729
$320$		−	5.76986i		−	0.322545i
$321$		0			0
$322$	−2.46167			−0.137183
$323$	18.2351	+	0.949145i	1.01463	+	0.0528118i
$324$		0			0
$325$	−3.52543			−0.195556
$326$		−	1.57781i		−	0.0873870i
$327$		0			0
$328$			4.46028i			0.246278i
$329$			5.30465i			0.292455i
$330$		0			0
$331$	24.8988			1.36856			0.684280	−	0.729219i	$-0.260117\pi$
$331$	24.8988			1.36856			0.684280	+	0.729219i	$0.260117\pi$
$332$	13.7003			0.751900
$333$		0			0
$334$		−	6.16839i		−	0.337519i
$335$		−	9.13828i		−	0.499277i
$336$		0			0
$337$		−	5.28592i		−	0.287942i	−0.989582	−	0.143971i	$-0.954013\pi$
$337$		−	5.28592i		−	0.287942i	0.989582	−	0.143971i	$-0.0459873\pi$
$338$	−0.177755			−0.00966858
$339$		0			0
$340$	−0.407896	+	7.83654i	−0.0221213	+	0.424996i
$341$	−10.1476			−0.549526
$342$		0			0
$343$		−	18.2538i		−	0.985613i
$344$	−3.00492			−0.162015
$345$		0			0
$346$		−	0.774305i		−	0.0416269i
$347$		−	31.6019i		−	1.69648i	−0.529611	−	0.848241i	$-0.677662\pi$
$347$		−	31.6019i		−	1.69648i	0.529611	−	0.848241i	$-0.322338\pi$
$348$		0			0
$349$	26.5116			1.41913			0.709567	−	0.704638i	$-0.248891\pi$
$349$	26.5116			1.41913			0.709567	+	0.704638i	$0.248891\pi$
$350$			0.495316i			0.0264758i
$351$		0			0
$352$		−	4.58274i		−	0.244261i
$353$	3.71456			0.197706			0.0988530	−	0.995102i	$-0.468483\pi$
$353$	3.71456			0.197706			0.0988530	+	0.995102i	$0.468483\pi$
$354$		0			0
$355$	10.5970			0.562432
$356$	22.1289			1.17283
$357$		0			0
$358$	3.61285			0.190945
$359$	−10.6637			−0.562809			−0.281404	−	0.959589i	$-0.590800\pi$
$359$	−10.6637			−0.562809			−0.281404	+	0.959589i	$0.590800\pi$
$360$		0			0
$361$	0.612848			0.0322551
$362$		−	1.20294i		−	0.0632253i
$363$		0			0
$364$			10.6824i			0.559912i
$365$	5.57136			0.291618
$366$		0			0
$367$			10.8780i			0.567828i	0.958850	+	0.283914i	$0.0916330\pi$
$367$			10.8780i			0.567828i	−0.958850	+	0.283914i	$0.908367\pi$
$368$		−	17.0400i		−	0.888269i
$369$		0			0
$370$	2.19358			0.114039
$371$		−	14.6222i		−	0.759148i
$372$		0			0
$373$	−16.5575			−0.857317			−0.428659	−	0.903467i	$-0.641014\pi$
$373$	−16.5575			−0.857317			−0.428659	+	0.903467i	$0.641014\pi$
$374$	−0.0874201	+	1.67952i	−0.00452039	+	0.0868461i
$375$		0			0
$376$	4.04593			0.208653
$377$		−	29.7146i		−	1.53038i
$378$		0			0
$379$		−	8.16839i		−	0.419582i	−0.977746	−	0.209791i	$-0.932722\pi$
$379$		−	8.16839i		−	0.419582i	0.977746	−	0.209791i	$-0.0672783\pi$
$380$			8.42864i			0.432380i
$381$		0			0
$382$	4.64095			0.237452
$383$	−36.7926			−1.88001			−0.940007	−	0.341154i	$-0.889182\pi$
$383$	−36.7926			−1.88001			−0.940007	+	0.341154i	$0.889182\pi$
$384$		0			0
$385$		−	2.08742i		−	0.106385i
$386$		−	5.52098i		−	0.281011i
$387$		0			0
$388$		−	29.4608i		−	1.49564i
$389$	−12.5348			−0.635539			−0.317770	−	0.948168i	$-0.602934\pi$
$389$	−12.5348			−0.635539			−0.317770	+	0.948168i	$0.602934\pi$
$390$		0			0
$391$	−1.06515	+	20.4637i	−0.0538667	+	1.03489i
$392$	−5.42219			−0.273862
$393$		0			0
$394$			2.93041i			0.147632i
$395$	−7.87310			−0.396139
$396$		0			0
$397$		−	32.1017i		−	1.61114i	−0.592502	−	0.805569i	$-0.701860\pi$
$397$		−	32.1017i		−	1.61114i	0.592502	−	0.805569i	$-0.298140\pi$
$398$			2.24596i			0.112580i
$399$		0			0
$400$	−3.42864			−0.171432
$401$			13.0321i			0.650793i	0.945578	+	0.325396i	$0.105498\pi$
$401$			13.0321i			0.650793i	−0.945578	+	0.325396i	$0.894502\pi$
$402$		0			0
$403$		−	27.2859i		−	1.35921i
$404$	−27.3733			−1.36187
$405$		0			0
$406$	−4.17484			−0.207194
$407$	−9.24443			−0.458229
$408$		0			0
$409$	3.12399			0.154471			0.0772356	−	0.997013i	$-0.475391\pi$
$409$	3.12399			0.154471			0.0772356	+	0.997013i	$0.475391\pi$
$410$	1.14272			0.0564350
$411$		0			0
$412$	17.8751			0.880643
$413$			2.19358i			0.107939i
$414$		0			0
$415$		−	7.19850i		−	0.353360i
$416$	12.3225			0.604159
$417$		0			0
$418$			1.80642i			0.0883551i
$419$		−	13.1590i		−	0.642860i	−0.946933	−	0.321430i	$-0.895836\pi$
$419$		−	13.1590i		−	0.642860i	0.946933	−	0.321430i	$-0.104164\pi$
$420$		0			0
$421$	−27.6686			−1.34849			−0.674243	−	0.738509i	$-0.735530\pi$
$421$	−27.6686			−1.34849			−0.674243	+	0.738509i	$0.735530\pi$
$422$			4.63005i			0.225387i
$423$		0			0
$424$	−11.1526			−0.541616
$425$	4.11753	+	0.214320i	0.199730	+	0.0103960i
$426$		0			0
$427$	24.5491			1.18801
$428$		−	12.8968i		−	0.623388i
$429$		0			0
$430$			0.769859i			0.0371259i
$431$		−	17.6795i		−	0.851593i	−0.904819	−	0.425796i	$-0.859994\pi$
$431$		−	17.6795i		−	0.851593i	0.904819	−	0.425796i	$-0.140006\pi$
$432$		0			0
$433$	−24.9447			−1.19877			−0.599383	−	0.800462i	$-0.704587\pi$
$433$	−24.9447			−1.19877			−0.599383	+	0.800462i	$0.704587\pi$
$434$	−3.83362			−0.184020
$435$		0			0
$436$			5.43801i			0.260433i
$437$			22.0098i			1.05287i
$438$		0			0
$439$		−	11.9748i		−	0.571527i	−0.958300	−	0.285763i	$-0.907753\pi$
$439$		−	11.9748i		−	0.571527i	0.958300	−	0.285763i	$-0.0922471\pi$
$440$	−1.59210			−0.0759006
$441$		0			0
$442$	−4.51606	−	0.235063i	−0.214807	−	0.0111808i
$443$	30.6909			1.45817			0.729084	−	0.684424i	$-0.239946\pi$
$443$	30.6909			1.45817			0.729084	+	0.684424i	$0.239946\pi$
$444$		0			0
$445$		−	11.6271i		−	0.551179i
$446$	−7.30021			−0.345675
$447$		0			0
$448$			9.18622i			0.434008i
$449$			13.6543i			0.644388i	0.946674	+	0.322194i	$0.104420\pi$
$449$			13.6543i			0.644388i	−0.946674	+	0.322194i	$0.895580\pi$
$450$		0			0
$451$	−4.81579			−0.226767
$452$		−	11.1655i		−	0.525180i
$453$		0			0
$454$		−	1.35260i		−	0.0634804i
$455$	5.61285			0.263134
$456$		0			0
$457$	−8.20787			−0.383948			−0.191974	−	0.981400i	$-0.561489\pi$
$457$	−8.20787			−0.383948			−0.191974	+	0.981400i	$0.561489\pi$
$458$	2.01429			0.0941216
$459$		0			0
$460$	−9.45875			−0.441017
$461$	−5.18421			−0.241453			−0.120726	−	0.992686i	$-0.538522\pi$
$461$	−5.18421			−0.241453			−0.120726	+	0.992686i	$0.538522\pi$
$462$		0			0
$463$	23.1985			1.07813			0.539063	−	0.842266i	$-0.318779\pi$
$463$	23.1985			1.07813			0.539063	+	0.842266i	$0.318779\pi$
$464$		−	28.8988i		−	1.34159i
$465$		0			0
$466$			5.76494i			0.267056i
$467$	35.2400			1.63071			0.815356	−	0.578960i	$-0.196541\pi$
$467$	35.2400			1.63071			0.815356	+	0.578960i	$0.196541\pi$
$468$		0			0
$469$			14.5491i			0.671814i
$470$		−	1.03657i		−	0.0478132i
$471$		0			0
$472$	1.67307			0.0770093
$473$		−	3.24443i		−	0.149179i
$474$		0			0
$475$	4.42864			0.203200
$476$	0.649413	−	12.4766i	0.0297658	−	0.571863i
$477$		0			0
$478$	6.66370			0.304791
$479$			11.4445i			0.522911i	0.965216	+	0.261455i	$0.0842024\pi$
$479$			11.4445i			0.522911i	−0.965216	+	0.261455i	$0.915798\pi$
$480$		0			0
$481$		−	24.8573i		−	1.13339i
$482$		−	3.80642i		−	0.173378i
$483$		0			0
$484$	−17.6637			−0.802896
$485$	−15.4795			−0.702888
$486$		0			0
$487$			29.7540i			1.34828i	0.738602	+	0.674142i	$0.235486\pi$
$487$			29.7540i			1.34828i	−0.738602	+	0.674142i	$0.764514\pi$
$488$		−	18.7239i		−	0.847592i
$489$		0			0
$490$			1.38916i			0.0627559i
$491$	−25.4193			−1.14716			−0.573578	−	0.819151i	$-0.694445\pi$
$491$	−25.4193			−1.14716			−0.573578	+	0.819151i	$0.694445\pi$
$492$		0			0
$493$	−1.80642	+	34.7052i	−0.0813572	+	1.56304i
$494$	−4.85728			−0.218539
$495$		0			0
$496$		−	26.5368i		−	1.19154i
$497$	−16.8716			−0.756793
$498$		0			0
$499$		−	14.8035i		−	0.662696i	−0.943509	−	0.331348i	$-0.892497\pi$
$499$		−	14.8035i		−	0.662696i	0.943509	−	0.331348i	$-0.107503\pi$
$500$			1.90321i			0.0851142i
$501$		0			0
$502$	1.40636			0.0627691
$503$			4.34767i			0.193853i	0.995292	+	0.0969266i	$0.0309012\pi$
$503$			4.34767i			0.193853i	−0.995292	+	0.0969266i	$0.969099\pi$
$504$		0			0
$505$			14.3827i			0.640022i
$506$	−2.02720			−0.0901199
$507$		0			0
$508$	−0.534795			−0.0237277
$509$	15.9813			0.708357			0.354179	−	0.935178i	$-0.384761\pi$
$509$	15.9813			0.708357			0.354179	+	0.935178i	$0.384761\pi$
$510$		0			0
$511$	−8.87019			−0.392394
$512$	20.3111			0.897633
$513$		0			0
$514$	1.92549			0.0849296
$515$		−	9.39207i		−	0.413864i
$516$		0			0
$517$			4.36842i			0.192123i
$518$	−3.49240			−0.153447
$519$		0			0
$520$		−	4.28100i		−	0.187734i
$521$			27.9081i			1.22268i	0.791369	+	0.611339i	$0.209369\pi$
$521$			27.9081i			1.22268i	−0.791369	+	0.611339i	$0.790631\pi$
$522$		0			0
$523$	15.8938			0.694989			0.347495	−	0.937682i	$-0.387032\pi$
$523$	15.8938			0.694989			0.347495	+	0.937682i	$0.387032\pi$
$524$		−	25.5274i		−	1.11517i
$525$		0			0
$526$	−2.54770			−0.111085
$527$	−1.65878	+	31.8687i	−0.0722576	+	1.38822i
$528$		0			0
$529$	−1.69979			−0.0739040
$530$			2.85728i			0.124112i
$531$		0			0
$532$		−	13.4193i		−	0.581799i
$533$		−	12.9491i		−	0.560890i
$534$		0			0
$535$	−6.77631			−0.292966
$536$	11.0968			0.479308
$537$		0			0
$538$		−	1.21279i		−	0.0522870i
$539$		−	5.85436i		−	0.252165i
$540$		0			0
$541$			24.2449i			1.04237i	0.853444	+	0.521185i	$0.174510\pi$
$541$			24.2449i			1.04237i	−0.853444	+	0.521185i	$0.825490\pi$
$542$	2.75557			0.118362
$543$		0			0
$544$	−14.3921	−	0.749115i	−0.617055	−	0.0321181i
$545$	2.85728			0.122392
$546$		0			0
$547$		−	11.7255i		−	0.501344i	−0.968072	−	0.250672i	$-0.919348\pi$
$547$		−	11.7255i		−	0.501344i	0.968072	−	0.250672i	$-0.0806516\pi$
$548$	19.5669			0.835857
$549$		0			0
$550$			0.407896i			0.0173927i
$551$			37.3274i			1.59020i
$552$		0			0
$553$	12.5348			0.533034
$554$		−	0.355509i		−	0.0151041i
$555$		0			0
$556$			34.8321i			1.47721i
$557$	31.4336			1.33188			0.665941	−	0.746004i	$-0.268030\pi$
$557$	31.4336			1.33188			0.665941	+	0.746004i	$0.268030\pi$
$558$		0			0
$559$	8.72393			0.368983
$560$	5.45875			0.230674
$561$		0			0
$562$	4.57184			0.192851
$563$	−11.9353			−0.503014			−0.251507	−	0.967855i	$-0.580926\pi$
$563$	−11.9353			−0.503014			−0.251507	+	0.967855i	$0.580926\pi$
$564$		0			0
$565$	−5.86665			−0.246812
$566$			0.615762i			0.0258824i
$567$		0			0
$568$			12.8682i			0.539937i
$569$	−1.14272			−0.0479054			−0.0239527	−	0.999713i	$-0.507625\pi$
$569$	−1.14272			−0.0479054			−0.0239527	+	0.999713i	$0.507625\pi$
$570$		0			0
$571$		−	26.7402i		−	1.11904i	−0.828816	−	0.559522i	$-0.810985\pi$
$571$		−	26.7402i		−	1.11904i	0.828816	−	0.559522i	$-0.189015\pi$
$572$			8.79706i			0.367823i
$573$		0			0
$574$	−1.81933			−0.0759374
$575$			4.96989i			0.207259i
$576$		0			0
$577$	27.9956			1.16547			0.582735	−	0.812662i	$-0.301983\pi$
$577$	27.9956			1.16547			0.582735	+	0.812662i	$0.301983\pi$
$578$	5.26025	+	0.549086i	0.218798	+	0.0228389i
$579$		0			0
$580$	−16.0415			−0.666087
$581$			11.4608i			0.475472i
$582$		0			0
$583$		−	12.0415i		−	0.498707i
$584$			6.76541i			0.279955i
$585$		0			0
$586$	−0.917502			−0.0379017
$587$	−6.60793			−0.272738			−0.136369	−	0.990658i	$-0.543543\pi$
$587$	−6.60793			−0.272738			−0.136369	+	0.990658i	$0.543543\pi$
$588$		0			0
$589$			34.2766i			1.41234i
$590$		−	0.428639i		−	0.0176468i
$591$		0			0
$592$		−	24.1748i		−	0.993580i
$593$	4.82564			0.198165			0.0990826	−	0.995079i	$-0.468409\pi$
$593$	4.82564			0.198165			0.0990826	+	0.995079i	$0.468409\pi$
$594$		0			0
$595$	−6.55554	−	0.341219i	−0.268751	−	0.0139886i
$596$	−9.35905			−0.383362
$597$		0			0
$598$		−	5.45091i		−	0.222904i
$599$	39.0005			1.59352			0.796758	−	0.604298i	$-0.206546\pi$
$599$	39.0005			1.59352			0.796758	+	0.604298i	$0.206546\pi$
$600$		0			0
$601$			7.19405i			0.293452i	0.989177	+	0.146726i	$0.0468735\pi$
$601$			7.19405i			0.293452i	−0.989177	+	0.146726i	$0.953127\pi$
$602$		−	1.22570i		−	0.0499556i
$603$		0			0
$604$	−24.2163			−0.985348
$605$			9.28100i			0.377326i
$606$		0			0
$607$		−	7.23353i		−	0.293600i	−0.989166	−	0.146800i	$-0.953103\pi$
$607$		−	7.23353i		−	0.293600i	0.989166	−	0.146800i	$-0.0468974\pi$
$608$	−15.4795			−0.627776
$609$		0			0
$610$	−4.79706			−0.194227
$611$	−11.7462			−0.475200
$612$		0			0
$613$	14.7654			0.596369			0.298185	−	0.954508i	$-0.403619\pi$
$613$	14.7654			0.596369			0.298185	+	0.954508i	$0.403619\pi$
$614$	−1.76938			−0.0714065
$615$		0			0
$616$	2.53480			0.102130
$617$			10.5205i			0.423540i	0.977320	+	0.211770i	$0.0679227\pi$
$617$			10.5205i			0.423540i	−0.977320	+	0.211770i	$0.932077\pi$
$618$		0			0
$619$			7.94623i			0.319386i	0.987167	+	0.159693i	$0.0510504\pi$
$619$			7.94623i			0.319386i	−0.987167	+	0.159693i	$0.948950\pi$
$620$	−14.7304			−0.591586
$621$		0			0
$622$		−	7.37979i		−	0.295903i
$623$			18.5116i			0.741652i
$624$		0			0
$625$	1.00000			0.0400000
$626$			9.61285i			0.384207i
$627$		0			0
$628$	17.4795			0.697508
$629$	−1.51114	+	29.0321i	−0.0602530	+	1.15759i
$630$		0			0
$631$	−14.2636			−0.567827			−0.283913	−	0.958850i	$-0.591633\pi$
$631$	−14.2636			−0.567827			−0.283913	+	0.958850i	$0.591633\pi$
$632$		−	9.56046i		−	0.380295i
$633$		0			0
$634$			4.26364i			0.169331i
$635$			0.280996i			0.0111510i
$636$		0			0
$637$	15.7418			0.623711
$638$	−3.43801			−0.136112
$639$		0			0
$640$		−	8.78568i		−	0.347285i
$641$		−	25.4509i		−	1.00525i	−0.864504	−	0.502625i	$-0.832368\pi$
$641$		−	25.4509i		−	1.00525i	0.864504	−	0.502625i	$-0.167632\pi$
$642$		0			0
$643$		−	16.1126i		−	0.635419i	−0.948188	−	0.317710i	$-0.897086\pi$
$643$		−	16.1126i		−	0.635419i	0.948188	−	0.317710i	$-0.102914\pi$
$644$	15.0593			0.593420
$645$		0			0
$646$	5.67307	+	0.295286i	0.223204	+	0.0116179i
$647$	−40.4558			−1.59048			−0.795242	−	0.606293i	$-0.792656\pi$
$647$	−40.4558			−1.59048			−0.795242	+	0.606293i	$0.792656\pi$
$648$		0			0
$649$			1.80642i			0.0709083i
$650$	−1.09679			−0.0430196
$651$		0			0
$652$			9.65233i			0.378014i
$653$			6.87601i			0.269079i	0.990908	+	0.134540i	$0.0429556\pi$
$653$			6.87601i			0.269079i	−0.990908	+	0.134540i	$0.957044\pi$
$654$		0			0
$655$	−13.4128			−0.524082
$656$		−	12.5936i		−	0.491699i
$657$		0			0
$658$			1.65032i			0.0643361i
$659$	−16.0286			−0.624385			−0.312192	−	0.950019i	$-0.601063\pi$
$659$	−16.0286			−0.624385			−0.312192	+	0.950019i	$0.601063\pi$
$660$		0			0
$661$	−31.1655			−1.21220			−0.606098	−	0.795390i	$-0.707266\pi$
$661$	−31.1655			−1.21220			−0.606098	+	0.795390i	$0.707266\pi$
$662$	7.74620			0.301065
$663$		0			0
$664$	8.74128			0.339227
$665$	−7.05086			−0.273420
$666$		0			0
$667$	−41.8894			−1.62196
$668$			37.7353i			1.46002i
$669$		0			0
$670$		−	2.84299i		−	0.109834i
$671$	20.2163			0.780443
$672$		0			0
$673$		−	19.8479i		−	0.765081i	−0.923939	−	0.382540i	$-0.875049\pi$
$673$		−	19.8479i		−	0.765081i	0.923939	−	0.382540i	$-0.124951\pi$
$674$		−	1.64449i		−	0.0633434i
$675$		0			0
$676$	1.08742			0.0418239
$677$			28.8385i			1.10836i	0.832398	+	0.554178i	$0.186967\pi$
$677$			28.8385i			1.10836i	−0.832398	+	0.554178i	$0.813033\pi$
$678$		0			0
$679$	24.6450			0.945787
$680$	−0.260253	+	5.00000i	−0.00998024	+	0.191741i
$681$		0			0
$682$	−3.15701			−0.120888
$683$		−	19.2050i		−	0.734857i	−0.930052	−	0.367429i	$-0.880238\pi$
$683$		−	19.2050i		−	0.734857i	0.930052	−	0.367429i	$-0.119762\pi$
$684$		0			0
$685$		−	10.2810i		−	0.392817i
$686$		−	5.67890i		−	0.216821i
$687$		0			0
$688$	8.48442			0.323465
$689$	32.3783			1.23351
$690$		0			0
$691$			34.5555i			1.31455i	0.753649	+	0.657277i	$0.228292\pi$
$691$			34.5555i			1.31455i	−0.753649	+	0.657277i	$0.771708\pi$
$692$			4.73683i			0.180067i
$693$		0			0
$694$		−	9.83161i		−	0.373203i
$695$	18.3017			0.694225
$696$		0			0
$697$	−0.787212	+	15.1240i	−0.0298178	+	0.572862i
$698$	8.24797			0.312190
$699$		0			0
$700$		−	3.03011i		−	0.114527i
$701$	20.9131			0.789875			0.394938	−	0.918708i	$-0.370766\pi$
$701$	20.9131			0.789875			0.394938	+	0.918708i	$0.370766\pi$
$702$		0			0
$703$			31.2257i			1.17770i
$704$			7.56491i			0.285113i
$705$		0			0
$706$	1.15563			0.0434926
$707$		−	22.8988i		−	0.861197i
$708$		0			0
$709$			32.9906i			1.23899i	0.785001	+	0.619495i	$0.212662\pi$
$709$			32.9906i			1.23899i	−0.785001	+	0.619495i	$0.787338\pi$
$710$	3.29682			0.123727
$711$		0			0
$712$	14.1191			0.529134
$713$	−38.4657			−1.44055
$714$		0			0
$715$	4.62222			0.172861
$716$	−22.1017			−0.825980
$717$		0			0
$718$	−3.31756			−0.123810
$719$		−	10.5141i		−	0.392108i	−0.980593	−	0.196054i	$-0.937187\pi$
$719$		−	10.5141i		−	0.392108i	0.980593	−	0.196054i	$-0.0628128\pi$
$720$		0			0
$721$			14.9532i			0.556885i
$722$	0.190662			0.00709569
$723$		0			0
$724$			7.35905i			0.273497i
$725$			8.42864i			0.313032i
$726$		0			0
$727$	43.2815			1.60522			0.802610	−	0.596503i	$-0.203444\pi$
$727$	43.2815			1.60522			0.802610	+	0.596503i	$0.203444\pi$
$728$			6.81579i			0.252610i
$729$		0			0
$730$	1.73329			0.0641521
$731$	−10.1891	−	0.530350i	−0.376859	−	0.0196157i
$732$		0			0
$733$	−22.8859			−0.845308			−0.422654	−	0.906291i	$-0.638902\pi$
$733$	−22.8859			−0.845308			−0.422654	+	0.906291i	$0.638902\pi$
$734$			3.38424i			0.124914i
$735$		0			0
$736$		−	17.3713i		−	0.640316i
$737$			11.9813i			0.441336i
$738$		0			0
$739$	0.815792			0.0300094			0.0150047	−	0.999887i	$-0.495224\pi$
$739$	0.815792			0.0300094			0.0150047	+	0.999887i	$0.495224\pi$
$740$	−13.4193			−0.493302
$741$		0			0
$742$		−	4.54909i		−	0.167002i
$743$			30.1639i			1.10661i	0.832980	+	0.553304i	$0.186633\pi$
$743$			30.1639i			1.10661i	−0.832980	+	0.553304i	$0.813367\pi$
$744$		0			0
$745$			4.91750i			0.180163i
$746$	−5.15118			−0.188598
$747$		0			0
$748$	0.534795	−	10.2745i	0.0195541	−	0.375674i
$749$	10.7886			0.394207
$750$		0			0
$751$		−	2.17728i		−	0.0794500i	−0.999211	−	0.0397250i	$-0.987352\pi$
$751$		−	2.17728i		−	0.0794500i	0.999211	−	0.0397250i	$-0.0126482\pi$
$752$	−11.4237			−0.416580
$753$		0			0
$754$		−	9.24443i		−	0.336662i
$755$			12.7239i			0.463071i
$756$		0			0
$757$	29.3230			1.06576			0.532881	−	0.846190i	$-0.321110\pi$
$757$	29.3230			1.06576			0.532881	+	0.846190i	$0.321110\pi$
$758$		−	2.54125i		−	0.0923023i
$759$		0			0
$760$			5.37778i			0.195073i
$761$	−49.2212			−1.78427			−0.892134	−	0.451770i	$-0.850793\pi$
$761$	−49.2212			−1.78427			−0.892134	+	0.451770i	$0.850793\pi$
$762$		0			0
$763$	−4.54909			−0.164688
$764$	−28.3912			−1.02716
$765$		0			0
$766$	−11.4465			−0.413578
$767$	−4.85728			−0.175386
$768$		0			0
$769$	37.0178			1.33490			0.667449	−	0.744656i	$-0.267386\pi$
$769$	37.0178			1.33490			0.667449	+	0.744656i	$0.267386\pi$
$770$		−	0.649413i		−	0.0234032i
$771$		0			0
$772$			33.7748i			1.21558i
$773$	4.20787			0.151346			0.0756732	−	0.997133i	$-0.475889\pi$
$773$	4.20787			0.151346			0.0756732	+	0.997133i	$0.475889\pi$
$774$		0			0
$775$			7.73975i			0.278020i
$776$		−	18.7971i		−	0.674775i
$777$		0			0
$778$	−3.89967			−0.139810
$779$			16.2667i			0.582815i
$780$		0			0
$781$	−13.8938			−0.497161
$782$	−0.331375	+	6.36641i	−0.0118499	+	0.227662i
$783$		0			0
$784$	15.3096			0.546771
$785$		−	9.18421i		−	0.327798i
$786$		0			0
$787$		−	23.8557i		−	0.850366i	−0.905108	−	0.425183i	$-0.860210\pi$
$787$		−	23.8557i		−	0.850366i	0.905108	−	0.425183i	$-0.139790\pi$
$788$		−	17.9269i		−	0.638618i
$789$		0			0
$790$	−2.44938			−0.0871451
$791$	9.34031			0.332103
$792$		0			0
$793$			54.3595i			1.93036i
$794$		−	9.98709i		−	0.354429i
$795$		0			0
$796$		−	13.7397i		−	0.486992i
$797$	34.4415			1.21998			0.609991	−	0.792408i	$-0.291173\pi$
$797$	34.4415			1.21998			0.609991	+	0.792408i	$0.291173\pi$
$798$		0			0
$799$	13.7190	+	0.714082i	0.485343	+	0.0252624i
$800$	−3.49532			−0.123578
$801$		0			0
$802$			4.05439i			0.143166i
$803$	−7.30465			−0.257776
$804$		0			0
$805$		−	7.91258i		−	0.278882i
$806$		−	8.48886i		−	0.299007i
$807$		0			0
$808$	−17.4652			−0.614424
$809$		−	40.2578i		−	1.41539i	−0.706518	−	0.707695i	$-0.749735\pi$
$809$		−	40.2578i		−	1.41539i	0.706518	−	0.707695i	$-0.250265\pi$
$810$		0			0
$811$		−	20.4953i		−	0.719688i	−0.933013	−	0.359844i	$-0.882830\pi$
$811$		−	20.4953i		−	0.719688i	0.933013	−	0.359844i	$-0.117170\pi$
$812$	25.5397			0.896268
$813$		0			0
$814$	−2.87601			−0.100804
$815$	5.07160			0.177650
$816$		0			0
$817$	−10.9590			−0.383407
$818$	0.971896			0.0339816
$819$		0			0
$820$	−6.99063			−0.244123
$821$			33.6958i			1.17599i	0.808864	+	0.587996i	$0.200083\pi$
$821$			33.6958i			1.17599i	−0.808864	+	0.587996i	$0.799917\pi$
$822$		0			0
$823$			16.6113i			0.579034i	0.957173	+	0.289517i	$0.0934947\pi$
$823$			16.6113i			0.579034i	−0.957173	+	0.289517i	$0.906505\pi$
$824$	11.4050			0.397311
$825$		0			0
$826$			0.682439i			0.0237451i
$827$			0.316030i			0.0109894i	0.999985	+	0.00549472i	$0.00174903\pi$
$827$			0.316030i			0.0109894i	−0.999985	+	0.00549472i	$0.998251\pi$
$828$		0			0
$829$	18.2034			0.632231			0.316115	−	0.948721i	$-0.397621\pi$
$829$	18.2034			0.632231			0.316115	+	0.948721i	$0.397621\pi$
$830$		−	2.23951i		−	0.0777345i
$831$		0			0
$832$	−20.3412			−0.705205
$833$	−18.3856	−	0.956981i	−0.637024	−	0.0331574i
$834$		0			0
$835$	19.8272			0.686147
$836$		−	11.0509i		−	0.382202i
$837$		0			0
$838$		−	4.09387i		−	0.141421i
$839$			2.63851i			0.0910916i	0.998962	+	0.0455458i	$0.0145027\pi$
$839$			2.63851i			0.0910916i	−0.998962	+	0.0455458i	$0.985497\pi$
$840$		0			0
$841$	−42.0420			−1.44972
$842$	−8.60793			−0.296649
$843$		0			0
$844$		−	28.3245i		−	0.974969i
$845$		−	0.571361i		−	0.0196554i
$846$		0			0
$847$		−	14.7763i		−	0.507720i
$848$	31.4893			1.08135
$849$		0			0
$850$	1.28100	+	0.0666765i	0.0439378	+	0.00228699i
$851$	−35.0420			−1.20122
$852$		0			0
$853$			4.68244i			0.160324i	0.996782	+	0.0801618i	$0.0255437\pi$
$853$			4.68244i			0.160324i	−0.996782	+	0.0801618i	$0.974456\pi$
$854$	7.63741			0.261347
$855$		0			0
$856$		−	8.22861i		−	0.281248i
$857$			47.3689i			1.61809i	0.587746	+	0.809045i	$0.300015\pi$
$857$			47.3689i			1.61809i	−0.587746	+	0.809045i	$0.699985\pi$
$858$		0			0
$859$	16.1748			0.551878			0.275939	−	0.961175i	$-0.411011\pi$
$859$	16.1748			0.551878			0.275939	+	0.961175i	$0.411011\pi$
$860$		−	4.70964i		−	0.160597i
$861$		0			0
$862$		−	5.50024i		−	0.187339i
$863$	32.5674			1.10861			0.554303	−	0.832315i	$-0.312985\pi$
$863$	32.5674			1.10861			0.554303	+	0.832315i	$0.312985\pi$
$864$		0			0
$865$	2.48886			0.0846239
$866$	−7.76049			−0.263712
$867$		0			0
$868$	23.4523			0.796023
$869$	10.3225			0.350166
$870$		0			0
$871$	−32.2163			−1.09161
$872$			3.46965i			0.117497i
$873$		0			0
$874$			6.84743i			0.231618i
$875$	−1.59210			−0.0538229
$876$		0			0
$877$		−	13.2543i		−	0.447565i	−0.974639	−	0.223783i	$-0.928159\pi$
$877$		−	13.2543i		−	0.447565i	0.974639	−	0.223783i	$-0.0718405\pi$
$878$		−	3.72546i		−	0.125728i
$879$		0			0
$880$	4.49532			0.151537
$881$			2.78721i			0.0939035i	0.998897	+	0.0469518i	$0.0149507\pi$
$881$			2.78721i			0.0939035i	−0.998897	+	0.0469518i	$0.985049\pi$
$882$		0			0
$883$	−27.1985			−0.915302			−0.457651	−	0.889132i	$-0.651309\pi$
$883$	−27.1985			−0.915302			−0.457651	+	0.889132i	$0.651309\pi$
$884$	27.6271	+	1.43801i	0.929201	+	0.0483654i
$885$		0			0
$886$	9.54818			0.320777
$887$			48.2657i			1.62060i	0.586014	+	0.810301i	$0.300696\pi$
$887$			48.2657i			1.62060i	−0.586014	+	0.810301i	$0.699304\pi$
$888$		0			0
$889$		−	0.447375i		−	0.0150045i
$890$		−	3.61729i		−	0.121252i
$891$		0			0
$892$	44.6593			1.49530
$893$	14.7556			0.493776
$894$		0			0
$895$			11.6128i			0.388175i
$896$			13.9877i			0.467297i
$897$		0			0
$898$			4.24797i			0.141757i
$899$	−65.2355			−2.17573
$900$		0			0
$901$	−37.8163	−	1.96836i	−1.25984	−	0.0655755i
$902$	−1.49823			−0.0498856
$903$		0			0
$904$		−	7.12399i		−	0.236940i
$905$	3.86665			0.128532
$906$		0			0
$907$			8.93825i			0.296790i	0.988928	+	0.148395i	$0.0474106\pi$
$907$			8.93825i			0.296790i	−0.988928	+	0.148395i	$0.952589\pi$
$908$			8.27454i			0.274600i
$909$		0			0
$910$	1.74620			0.0578860
$911$			51.6795i			1.71222i	0.516794	+	0.856110i	$0.327125\pi$
$911$			51.6795i			1.71222i	−0.516794	+	0.856110i	$0.672875\pi$
$912$		0			0
$913$			9.43801i			0.312352i
$914$	−2.55353			−0.0844633
$915$		0			0
$916$	−12.3225			−0.407146
$917$	21.3546			0.705191
$918$		0			0
$919$	18.9719			0.625825			0.312913	−	0.949782i	$-0.398695\pi$
$919$	18.9719			0.625825			0.312913	+	0.949782i	$0.398695\pi$
$920$	−6.03503			−0.198969
$921$		0			0
$922$	−1.61285			−0.0531163
$923$		−	37.3590i		−	1.22969i
$924$		0			0
$925$			7.05086i			0.231831i
$926$	7.21723			0.237173
$927$		0			0
$928$		−	29.4608i		−	0.967097i
$929$			50.3912i			1.65328i	0.562731	+	0.826640i	$0.309751\pi$
$929$			50.3912i			1.65328i	−0.562731	+	0.826640i	$0.690249\pi$
$930$		0			0
$931$	−19.7748			−0.648092
$932$		−	35.2672i		−	1.15521i
$933$		0			0
$934$	10.9634			0.358735
$935$	−5.39853	−	0.280996i	−0.176551	−	0.00918956i
$936$		0			0
$937$	58.1530			1.89978			0.949889	−	0.312589i	$-0.101196\pi$
$937$	58.1530			1.89978			0.949889	+	0.312589i	$0.101196\pi$
$938$			4.52633i			0.147790i
$939$		0			0
$940$			6.34122i			0.206828i
$941$		−	9.25734i		−	0.301781i	−0.988551	−	0.150890i	$-0.951786\pi$
$941$		−	9.25734i		−	0.301781i	0.988551	−	0.150890i	$-0.0482140\pi$
$942$		0			0
$943$	−18.2548			−0.594457
$944$	−4.72393			−0.153751
$945$		0			0
$946$		−	1.00937i		−	0.0328174i
$947$		−	43.9516i		−	1.42824i	−0.700025	−	0.714118i	$-0.746828\pi$
$947$		−	43.9516i		−	1.42824i	0.700025	−	0.714118i	$-0.253172\pi$
$948$		0			0
$949$		−	19.6414i		−	0.637588i
$950$	1.37778			0.0447012
$951$		0			0
$952$	0.414349	−	7.96052i	0.0134291	−	0.258002i
$953$	38.1891			1.23707			0.618534	−	0.785758i	$-0.287727\pi$
$953$	38.1891			1.23707			0.618534	+	0.785758i	$0.287727\pi$
$954$		0			0
$955$			14.9175i			0.482719i
$956$	−40.7654			−1.31845
$957$		0			0
$958$			3.56046i			0.115033i
$959$			16.3684i			0.528564i
$960$		0			0
$961$	−28.9037			−0.932377
$962$		−	7.73329i		−	0.249331i
$963$		0			0
$964$			23.2859i			0.749989i
$965$	17.7462			0.571270
$966$		0			0
$967$	−7.36043			−0.236696			−0.118348	−	0.992972i	$-0.537760\pi$
$967$	−7.36043			−0.236696			−0.118348	+	0.992972i	$0.537760\pi$
$968$	−11.2701			−0.362235
$969$		0			0
$970$	−4.81579			−0.154626
$971$	35.2257			1.13045			0.565223	−	0.824938i	$-0.308790\pi$
$971$	35.2257			1.13045			0.565223	+	0.824938i	$0.308790\pi$
$972$		0			0
$973$	−29.1383			−0.934130
$974$			9.25671i			0.296604i
$975$		0			0
$976$			52.8671i			1.69224i
$977$	46.9273			1.50134			0.750669	−	0.660678i	$-0.229731\pi$
$977$	46.9273			1.50134			0.750669	+	0.660678i	$0.229731\pi$
$978$		0			0
$979$			15.2444i			0.487214i
$980$		−	8.49823i		−	0.271466i
$981$		0			0
$982$	−7.90813			−0.252359
$983$		−	8.21432i		−	0.261996i	−0.991383	−	0.130998i	$-0.958182\pi$
$983$		−	8.21432i		−	0.261996i	0.991383	−	0.130998i	$-0.0418182\pi$
$984$		0			0
$985$	−9.41927			−0.300123
$986$	−0.561993	+	10.7971i	−0.0178975	+	0.343848i
$987$		0			0
$988$	29.7146			0.945346
$989$		−	12.2983i		−	0.391065i
$990$		0			0
$991$			17.7683i			0.564430i	0.959351	+	0.282215i	$0.0910691\pi$
$991$			17.7683i			0.564430i	−0.959351	+	0.282215i	$0.908931\pi$
$992$		−	27.0529i		−	0.858929i
$993$		0			0
$994$	−5.24888			−0.166484
$995$	−7.21924			−0.228865
$996$		0			0
$997$		−	11.6958i		−	0.370410i	−0.982700	−	0.185205i	$-0.940705\pi$
$997$		−	11.6958i		−	0.370410i	0.982700	−	0.185205i	$-0.0592950\pi$
$998$		−	4.60549i		−	0.145784i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	765.2.g.b.271.4		6
3.2	odd	2		85.2.d.a.16.3	✓	6
12.11	even	2		1360.2.c.f.1121.6		6
15.2	even	4		425.2.c.a.424.3		6
15.8	even	4		425.2.c.b.424.4		6
15.14	odd	2		425.2.d.c.101.4		6
17.16	even	2	inner	765.2.g.b.271.3		6
51.38	odd	4		1445.2.a.j.1.2		3
51.47	odd	4		1445.2.a.k.1.2		3
51.50	odd	2		85.2.d.a.16.4	yes	6
204.203	even	2		1360.2.c.f.1121.1		6
255.89	odd	4		7225.2.a.q.1.2		3
255.149	odd	4		7225.2.a.r.1.2		3
255.152	even	4		425.2.c.b.424.3		6
255.203	even	4		425.2.c.a.424.4		6
255.254	odd	2		425.2.d.c.101.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.d.a.16.3	✓	6	3.2	odd	2
85.2.d.a.16.4	yes	6	51.50	odd	2
425.2.c.a.424.3		6	15.2	even	4
425.2.c.a.424.4		6	255.203	even	4
425.2.c.b.424.3		6	255.152	even	4
425.2.c.b.424.4		6	15.8	even	4
425.2.d.c.101.3		6	255.254	odd	2
425.2.d.c.101.4		6	15.14	odd	2
765.2.g.b.271.3		6	17.16	even	2	inner
765.2.g.b.271.4		6	1.1	even	1	trivial
1360.2.c.f.1121.1		6	204.203	even	2
1360.2.c.f.1121.6		6	12.11	even	2
1445.2.a.j.1.2		3	51.38	odd	4
1445.2.a.k.1.2		3	51.47	odd	4
7225.2.a.q.1.2		3	255.89	odd	4
7225.2.a.r.1.2		3	255.149	odd	4

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:	\( N \)	\(=\)	\( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	765.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.311108 q^{2} -1.90321 q^{4} +1.00000i q^{5} -1.59210i q^{7} -1.21432 q^{8} +0.311108i q^{10} -1.31111i q^{11} +3.52543 q^{13} -0.495316i q^{14} +3.42864 q^{16} +(-4.11753 - 0.214320i) q^{17} -4.42864 q^{19} -1.90321i q^{20} -0.407896i q^{22} -4.96989i q^{23} -1.00000 q^{25} +1.09679 q^{26} +3.03011i q^{28} -8.42864i q^{29} -7.73975i q^{31} +3.49532 q^{32} +(-1.28100 - 0.0666765i) q^{34} +1.59210 q^{35} -7.05086i q^{37} -1.37778 q^{38} -1.21432i q^{40} -3.67307i q^{41} +2.47457 q^{43} +2.49532i q^{44} -1.54617i q^{46} -3.33185 q^{47} +4.46520 q^{49} -0.311108 q^{50} -6.70964 q^{52} +9.18421 q^{53} +1.31111 q^{55} +1.93332i q^{56} -2.62222i q^{58} -1.37778 q^{59} +15.4193i q^{61} -2.40790i q^{62} -5.76986 q^{64} +3.52543i q^{65} -9.13828 q^{67} +(7.83654 + 0.407896i) q^{68} +0.495316 q^{70} -10.5970i q^{71} -5.57136i q^{73} -2.19358i q^{74} +8.42864 q^{76} -2.08742 q^{77} +7.87310i q^{79} +3.42864i q^{80} -1.14272i q^{82} -7.19850 q^{83} +(0.214320 - 4.11753i) q^{85} +0.769859 q^{86} +1.59210i q^{88} -11.6271 q^{89} -5.61285i q^{91} +9.45875i q^{92} -1.03657 q^{94} -4.42864i q^{95} +15.4795i q^{97} +1.38916 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 2 q^{4} + 6 q^{8} + 8 q^{13} - 6 q^{16} + 2 q^{17} - 6 q^{25} + 20 q^{26} - 6 q^{32} + 6 q^{34} - 4 q^{35} - 8 q^{38} + 28 q^{43} + 20 q^{47} - 14 q^{49} - 2 q^{50} + 28 q^{53} + 8 q^{55}+ \cdots - 86 q^{98}+O(q^{100}) \) 6 * q + 2 * q^2 + 2 * q^4 + 6 * q^8 + 8 * q^13 - 6 * q^16 + 2 * q^17 - 6 * q^25 + 20 * q^26 - 6 * q^32 + 6 * q^34 - 4 * q^35 - 8 * q^38 + 28 * q^43 + 20 * q^47 - 14 * q^49 - 2 * q^50 + 28 * q^53 + 8 * q^55 - 8 * q^59 - 22 * q^64 + 12 * q^67 + 34 * q^68 - 24 * q^70 + 24 * q^76 + 28 * q^77 - 4 * q^83 - 12 * q^85 - 8 * q^86 - 4 * q^89 + 8 * q^94 - 86 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more