LMFDB - Embedded newform 1530.2.d.f.919.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1530,2,Mod(919,1530)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1530, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1530.919");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1530.d (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:	$12.2171115093$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 510)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			919.4
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	1530.919
Dual form			1530.2.d.f.919.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.00000 q^{4} +2.23607i q^{5} -3.23607i q^{7} -1.00000i q^{8} -2.23607 q^{10} -2.00000 q^{11} +0.763932i q^{13} +3.23607 q^{14} +1.00000 q^{16} +1.00000i q^{17} -2.47214 q^{19} -2.23607i q^{20} -2.00000i q^{22} -4.00000i q^{23} -5.00000 q^{25} -0.763932 q^{26} +3.23607i q^{28} -4.00000 q^{29} -6.47214 q^{31} +1.00000i q^{32} -1.00000 q^{34} +7.23607 q^{35} -2.00000i q^{37} -2.47214i q^{38} +2.23607 q^{40} -5.70820 q^{41} -10.1803i q^{43} +2.00000 q^{44} +4.00000 q^{46} +1.52786i q^{47} -3.47214 q^{49} -5.00000i q^{50} -0.763932i q^{52} -6.94427i q^{53} -4.47214i q^{55} -3.23607 q^{56} -4.00000i q^{58} -1.70820 q^{59} -4.47214 q^{61} -6.47214i q^{62} -1.00000 q^{64} -1.70820 q^{65} -11.7082i q^{67} -1.00000i q^{68} +7.23607i q^{70} -6.76393 q^{71} +13.2361i q^{73} +2.00000 q^{74} +2.47214 q^{76} +6.47214i q^{77} +16.9443 q^{79} +2.23607i q^{80} -5.70820i q^{82} -1.52786i q^{83} -2.23607 q^{85} +10.1803 q^{86} +2.00000i q^{88} +3.52786 q^{89} +2.47214 q^{91} +4.00000i q^{92} -1.52786 q^{94} -5.52786i q^{95} -11.7082i q^{97} -3.47214i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 8 q^{11} + 4 q^{14} + 4 q^{16} + 8 q^{19} - 20 q^{25} - 12 q^{26} - 16 q^{29} - 8 q^{31} - 4 q^{34} + 20 q^{35} + 4 q^{41} + 8 q^{44} + 16 q^{46} + 4 q^{49} - 4 q^{56} + 20 q^{59} - 4 q^{64}+ \cdots - 24 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 - 8 * q^11 + 4 * q^14 + 4 * q^16 + 8 * q^19 - 20 * q^25 - 12 * q^26 - 16 * q^29 - 8 * q^31 - 4 * q^34 + 20 * q^35 + 4 * q^41 + 8 * q^44 + 16 * q^46 + 4 * q^49 - 4 * q^56 + 20 * q^59 - 4 * q^64 + 20 * q^65 - 36 * q^71 + 8 * q^74 - 8 * q^76 + 32 * q^79 - 4 * q^86 + 32 * q^89 - 8 * q^91 - 24 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1530\mathbb{Z}\right)^\times$.

$n$	$307$	$1261$	$1361$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	2.23607i	1.00000i
$5$	2.23607i	1.00000i
$6$	0	0
$7$	− 3.23607i	− 1.22312i	−0.791199	−	0.611559i	$-0.790543\pi$
$7$	− 3.23607i	− 1.22312i	0.791199	−	0.611559i	$-0.209457\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	−2.23607	−0.707107
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	0.763932i	0.211877i	0.994373	+	0.105938i	$0.0337846\pi$
$13$	0.763932i	0.211877i	−0.994373	+	0.105938i	$0.966215\pi$
$14$	3.23607	0.864876
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000i	0.242536i
$17$	1.00000i	0.242536i
$18$	0	0
$19$	−2.47214	−0.567147	−0.283573	−	0.958951i	$-0.591520\pi$
$19$	−2.47214	−0.567147	−0.283573	+	0.958951i	$0.591520\pi$
$20$	− 2.23607i	− 0.500000i
$21$	0	0
$22$	− 2.00000i	− 0.426401i
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	−0.763932	−0.149819
$27$	0	0
$28$	3.23607i	0.611559i
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	−6.47214	−1.16243	−0.581215	−	0.813750i	$-0.697422\pi$
$31$	−6.47214	−1.16243	−0.581215	+	0.813750i	$0.697422\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−1.00000	−0.171499
$35$	7.23607	1.22312
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	− 2.47214i	− 0.401033i
$39$	0	0
$40$	2.23607	0.353553
$41$	−5.70820	−0.891472	−0.445736	−	0.895165i	$-0.647058\pi$
$41$	−5.70820	−0.891472	−0.445736	+	0.895165i	$0.647058\pi$
$42$	0	0
$43$	− 10.1803i	− 1.55249i	−0.630433	−	0.776244i	$-0.717123\pi$
$43$	− 10.1803i	− 1.55249i	0.630433	−	0.776244i	$-0.282877\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	4.00000	0.589768
$47$	1.52786i	0.222862i	0.993772	+	0.111431i	$0.0355434\pi$
$47$	1.52786i	0.222862i	−0.993772	+	0.111431i	$0.964457\pi$
$48$	0	0
$49$	−3.47214	−0.496019
$50$	− 5.00000i	− 0.707107i
$51$	0	0
$52$	− 0.763932i	− 0.105938i
$53$	− 6.94427i	− 0.953869i	−0.878939	−	0.476935i	$-0.841748\pi$
$53$	− 6.94427i	− 0.953869i	0.878939	−	0.476935i	$-0.158252\pi$
$54$	0	0
$55$	− 4.47214i	− 0.603023i
$56$	−3.23607	−0.432438
$57$	0	0
$58$	− 4.00000i	− 0.525226i
$59$	−1.70820	−0.222389	−0.111195	−	0.993799i	$-0.535468\pi$
$59$	−1.70820	−0.222389	−0.111195	+	0.993799i	$0.535468\pi$
$60$	0	0
$61$	−4.47214	−0.572598	−0.286299	−	0.958140i	$-0.592425\pi$
$61$	−4.47214	−0.572598	−0.286299	+	0.958140i	$0.592425\pi$
$62$	− 6.47214i	− 0.821962i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	−1.70820	−0.211877
$66$	0	0
$67$	− 11.7082i	− 1.43038i	−0.698928	−	0.715192i	$-0.746339\pi$
$67$	− 11.7082i	− 1.43038i	0.698928	−	0.715192i	$-0.253661\pi$
$68$	− 1.00000i	− 0.121268i
$69$	0	0
$70$	7.23607i	0.864876i
$71$	−6.76393	−0.802731	−0.401366	−	0.915918i	$-0.631464\pi$
$71$	−6.76393	−0.802731	−0.401366	+	0.915918i	$0.631464\pi$
$72$	0	0
$73$	13.2361i	1.54916i	0.632473	+	0.774582i	$0.282040\pi$
$73$	13.2361i	1.54916i	−0.632473	+	0.774582i	$0.717960\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	2.47214	0.283573
$77$	6.47214i	0.737568i
$78$	0	0
$79$	16.9443	1.90638	0.953190	−	0.302373i	$-0.0977787\pi$
$79$	16.9443	1.90638	0.953190	+	0.302373i	$0.0977787\pi$
$80$	2.23607i	0.250000i
$81$	0	0
$82$	− 5.70820i	− 0.630366i
$83$	− 1.52786i	− 0.167705i	−0.996478	−	0.0838524i	$-0.973278\pi$
$83$	− 1.52786i	− 0.167705i	0.996478	−	0.0838524i	$-0.0267224\pi$
$84$	0	0
$85$	−2.23607	−0.242536
$86$	10.1803	1.09777
$87$	0	0
$88$	2.00000i	0.213201i
$89$	3.52786	0.373953	0.186976	−	0.982364i	$-0.440131\pi$
$89$	3.52786	0.373953	0.186976	+	0.982364i	$0.440131\pi$
$90$	0	0
$91$	2.47214	0.259150
$92$	4.00000i	0.417029i
$93$	0	0
$94$	−1.52786	−0.157587
$95$	− 5.52786i	− 0.567147i
$96$	0	0
$97$	− 11.7082i	− 1.18879i	−0.804174	−	0.594394i	$-0.797392\pi$
$97$	− 11.7082i	− 1.18879i	0.804174	−	0.594394i	$-0.202608\pi$
$98$	− 3.47214i	− 0.350739i
$99$	0	0
$100$	5.00000	0.500000
$101$	−11.7082	−1.16501	−0.582505	−	0.812827i	$-0.697927\pi$
$101$	−11.7082	−1.16501	−0.582505	+	0.812827i	$0.697927\pi$
$102$	0	0
$103$	− 6.94427i	− 0.684239i	−0.939656	−	0.342120i	$-0.888855\pi$
$103$	− 6.94427i	− 0.684239i	0.939656	−	0.342120i	$-0.111145\pi$
$104$	0.763932	0.0749097
$105$	0	0
$106$	6.94427	0.674487
$107$	8.00000i	0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$	8.00000i	0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$	0	0
$109$	16.4721	1.57774	0.788872	−	0.614557i	$-0.210665\pi$
$109$	16.4721	1.57774	0.788872	+	0.614557i	$0.210665\pi$
$110$	4.47214	0.426401
$111$	0	0
$112$	− 3.23607i	− 0.305780i
$113$	10.0000i	0.940721i	0.882474	+	0.470360i	$0.155876\pi$
$113$	10.0000i	0.940721i	−0.882474	+	0.470360i	$0.844124\pi$
$114$	0	0
$115$	8.94427	0.834058
$116$	4.00000	0.371391
$117$	0	0
$118$	− 1.70820i	− 0.157253i
$119$	3.23607	0.296650
$120$	0	0
$121$	−7.00000	−0.636364
$122$	− 4.47214i	− 0.404888i
$123$	0	0
$124$	6.47214	0.581215
$125$	− 11.1803i	− 1.00000i
$126$	0	0
$127$	− 14.0000i	− 1.24230i	−0.783692	−	0.621150i	$-0.786666\pi$
$127$	− 14.0000i	− 1.24230i	0.783692	−	0.621150i	$-0.213334\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	− 1.70820i	− 0.149819i
$131$	−3.52786	−0.308231	−0.154115	−	0.988053i	$-0.549253\pi$
$131$	−3.52786	−0.308231	−0.154115	+	0.988053i	$0.549253\pi$
$132$	0	0
$133$	8.00000i	0.693688i
$134$	11.7082	1.01143
$135$	0	0
$136$	1.00000	0.0857493
$137$	− 2.00000i	− 0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$	− 2.00000i	− 0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$	0	0
$139$	−10.4721	−0.888235	−0.444117	−	0.895969i	$-0.646483\pi$
$139$	−10.4721	−0.888235	−0.444117	+	0.895969i	$0.646483\pi$
$140$	−7.23607	−0.611559
$141$	0	0
$142$	− 6.76393i	− 0.567617i
$143$	− 1.52786i	− 0.127766i
$144$	0	0
$145$	− 8.94427i	− 0.742781i
$146$	−13.2361	−1.09542
$147$	0	0
$148$	2.00000i	0.164399i
$149$	−10.1803	−0.834006	−0.417003	−	0.908905i	$-0.636920\pi$
$149$	−10.1803	−0.834006	−0.417003	+	0.908905i	$0.636920\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	2.47214i	0.200517i
$153$	0	0
$154$	−6.47214	−0.521540
$155$	− 14.4721i	− 1.16243i
$156$	0	0
$157$	12.1803i	0.972097i	0.873932	+	0.486048i	$0.161562\pi$
$157$	12.1803i	0.972097i	−0.873932	+	0.486048i	$0.838438\pi$
$158$	16.9443i	1.34801i
$159$	0	0
$160$	−2.23607	−0.176777
$161$	−12.9443	−1.02015
$162$	0	0
$163$	− 13.5279i	− 1.05958i	−0.848128	−	0.529792i	$-0.822270\pi$
$163$	− 13.5279i	− 1.05958i	0.848128	−	0.529792i	$-0.177730\pi$
$164$	5.70820	0.445736
$165$	0	0
$166$	1.52786	0.118585
$167$	21.8885i	1.69379i	0.531763	+	0.846893i	$0.321530\pi$
$167$	21.8885i	1.69379i	−0.531763	+	0.846893i	$0.678470\pi$
$168$	0	0
$169$	12.4164	0.955108
$170$	− 2.23607i	− 0.171499i
$171$	0	0
$172$	10.1803i	0.776244i
$173$	− 19.8885i	− 1.51210i	−0.654515	−	0.756049i	$-0.727127\pi$
$173$	− 19.8885i	− 1.51210i	0.654515	−	0.756049i	$-0.272873\pi$
$174$	0	0
$175$	16.1803i	1.22312i
$176$	−2.00000	−0.150756
$177$	0	0
$178$	3.52786i	0.264425i
$179$	5.70820	0.426651	0.213326	−	0.976981i	$-0.431570\pi$
$179$	5.70820	0.426651	0.213326	+	0.976981i	$0.431570\pi$
$180$	0	0
$181$	−8.47214	−0.629729	−0.314864	−	0.949137i	$-0.601959\pi$
$181$	−8.47214	−0.629729	−0.314864	+	0.949137i	$0.601959\pi$
$182$	2.47214i	0.183247i
$183$	0	0
$184$	−4.00000	−0.294884
$185$	4.47214	0.328798
$186$	0	0
$187$	− 2.00000i	− 0.146254i
$188$	− 1.52786i	− 0.111431i
$189$	0	0
$190$	5.52786	0.401033
$191$	−14.4721	−1.04717	−0.523584	−	0.851974i	$-0.675405\pi$
$191$	−14.4721	−1.04717	−0.523584	+	0.851974i	$0.675405\pi$
$192$	0	0
$193$	25.5967i	1.84249i	0.388978	+	0.921247i	$0.372828\pi$
$193$	25.5967i	1.84249i	−0.388978	+	0.921247i	$0.627172\pi$
$194$	11.7082	0.840600
$195$	0	0
$196$	3.47214	0.248010
$197$	− 19.8885i	− 1.41700i	−0.705711	−	0.708500i	$-0.749372\pi$
$197$	− 19.8885i	− 1.41700i	0.705711	−	0.708500i	$-0.250628\pi$
$198$	0	0
$199$	−8.94427	−0.634043	−0.317021	−	0.948418i	$-0.602683\pi$
$199$	−8.94427	−0.634043	−0.317021	+	0.948418i	$0.602683\pi$
$200$	5.00000i	0.353553i
$201$	0	0
$202$	− 11.7082i	− 0.823786i
$203$	12.9443i	0.908510i
$204$	0	0
$205$	− 12.7639i	− 0.891472i
$206$	6.94427	0.483830
$207$	0	0
$208$	0.763932i	0.0529692i
$209$	4.94427	0.342002
$210$	0	0
$211$	−7.05573	−0.485736	−0.242868	−	0.970059i	$-0.578088\pi$
$211$	−7.05573	−0.485736	−0.242868	+	0.970059i	$0.578088\pi$
$212$	6.94427i	0.476935i
$213$	0	0
$214$	−8.00000	−0.546869
$215$	22.7639	1.55249
$216$	0	0
$217$	20.9443i	1.42179i
$218$	16.4721i	1.11563i
$219$	0	0
$220$	4.47214i	0.301511i
$221$	−0.763932	−0.0513876
$222$	0	0
$223$	26.9443i	1.80432i	0.431400	+	0.902161i	$0.358020\pi$
$223$	26.9443i	1.80432i	−0.431400	+	0.902161i	$0.641980\pi$
$224$	3.23607	0.216219
$225$	0	0
$226$	−10.0000	−0.665190
$227$	− 25.8885i	− 1.71828i	−0.511738	−	0.859142i	$-0.670998\pi$
$227$	− 25.8885i	− 1.71828i	0.511738	−	0.859142i	$-0.329002\pi$
$228$	0	0
$229$	−5.41641	−0.357926	−0.178963	−	0.983856i	$-0.557274\pi$
$229$	−5.41641	−0.357926	−0.178963	+	0.983856i	$0.557274\pi$
$230$	8.94427i	0.589768i
$231$	0	0
$232$	4.00000i	0.262613i
$233$	21.4164i	1.40304i	0.712652	+	0.701518i	$0.247494\pi$
$233$	21.4164i	1.40304i	−0.712652	+	0.701518i	$0.752506\pi$
$234$	0	0
$235$	−3.41641	−0.222862
$236$	1.70820	0.111195
$237$	0	0
$238$	3.23607i	0.209763i
$239$	13.8885	0.898375	0.449188	−	0.893437i	$-0.351713\pi$
$239$	13.8885	0.898375	0.449188	+	0.893437i	$0.351713\pi$
$240$	0	0
$241$	−17.4164	−1.12189	−0.560945	−	0.827853i	$-0.689562\pi$
$241$	−17.4164	−1.12189	−0.560945	+	0.827853i	$0.689562\pi$
$242$	− 7.00000i	− 0.449977i
$243$	0	0
$244$	4.47214	0.286299
$245$	− 7.76393i	− 0.496019i
$246$	0	0
$247$	− 1.88854i	− 0.120165i
$248$	6.47214i	0.410981i
$249$	0	0
$250$	11.1803	0.707107
$251$	−12.1803	−0.768816	−0.384408	−	0.923163i	$-0.625594\pi$
$251$	−12.1803	−0.768816	−0.384408	+	0.923163i	$0.625594\pi$
$252$	0	0
$253$	8.00000i	0.502956i
$254$	14.0000	0.878438
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	0	0
$259$	−6.47214	−0.402159
$260$	1.70820	0.105938
$261$	0	0
$262$	− 3.52786i	− 0.217952i
$263$	4.00000i	0.246651i	0.992366	+	0.123325i	$0.0393559\pi$
$263$	4.00000i	0.246651i	−0.992366	+	0.123325i	$0.960644\pi$
$264$	0	0
$265$	15.5279	0.953869
$266$	−8.00000	−0.490511
$267$	0	0
$268$	11.7082i	0.715192i
$269$	−24.9443	−1.52088	−0.760440	−	0.649409i	$-0.775016\pi$
$269$	−24.9443	−1.52088	−0.760440	+	0.649409i	$0.775016\pi$
$270$	0	0
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	1.00000i	0.0606339i
$273$	0	0
$274$	2.00000	0.120824
$275$	10.0000	0.603023
$276$	0	0
$277$	18.3607i	1.10319i	0.834113	+	0.551593i	$0.185980\pi$
$277$	18.3607i	1.10319i	−0.834113	+	0.551593i	$0.814020\pi$
$278$	− 10.4721i	− 0.628077i
$279$	0	0
$280$	− 7.23607i	− 0.432438i
$281$	25.4164	1.51622	0.758108	−	0.652129i	$-0.226124\pi$
$281$	25.4164	1.51622	0.758108	+	0.652129i	$0.226124\pi$
$282$	0	0
$283$	9.88854i	0.587813i	0.955834	+	0.293906i	$0.0949554\pi$
$283$	9.88854i	0.587813i	−0.955834	+	0.293906i	$0.905045\pi$
$284$	6.76393	0.401366
$285$	0	0
$286$	1.52786	0.0903445
$287$	18.4721i	1.09038i
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	8.94427	0.525226
$291$	0	0
$292$	− 13.2361i	− 0.774582i
$293$	− 2.58359i	− 0.150935i	−0.997148	−	0.0754675i	$-0.975955\pi$
$293$	− 2.58359i	− 0.150935i	0.997148	−	0.0754675i	$-0.0240449\pi$
$294$	0	0
$295$	− 3.81966i	− 0.222389i
$296$	−2.00000	−0.116248
$297$	0	0
$298$	− 10.1803i	− 0.589731i
$299$	3.05573	0.176717
$300$	0	0
$301$	−32.9443	−1.89888
$302$	− 4.00000i	− 0.230174i
$303$	0	0
$304$	−2.47214	−0.141787
$305$	− 10.0000i	− 0.572598i
$306$	0	0
$307$	18.1803i	1.03761i	0.854894	+	0.518803i	$0.173622\pi$
$307$	18.1803i	1.03761i	−0.854894	+	0.518803i	$0.826378\pi$
$308$	− 6.47214i	− 0.368784i
$309$	0	0
$310$	14.4721	0.821962
$311$	29.5967	1.67828	0.839139	−	0.543917i	$-0.183059\pi$
$311$	29.5967	1.67828	0.839139	+	0.543917i	$0.183059\pi$
$312$	0	0
$313$	− 8.29180i	− 0.468680i	−0.972155	−	0.234340i	$-0.924707\pi$
$313$	− 8.29180i	− 0.468680i	0.972155	−	0.234340i	$-0.0752929\pi$
$314$	−12.1803	−0.687376
$315$	0	0
$316$	−16.9443	−0.953190
$317$	− 23.8885i	− 1.34171i	−0.741587	−	0.670857i	$-0.765926\pi$
$317$	− 23.8885i	− 1.34171i	0.741587	−	0.670857i	$-0.234074\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	− 2.23607i	− 0.125000i
$321$	0	0
$322$	− 12.9443i	− 0.721356i
$323$	− 2.47214i	− 0.137553i
$324$	0	0
$325$	− 3.81966i	− 0.211877i
$326$	13.5279	0.749239
$327$	0	0
$328$	5.70820i	0.315183i
$329$	4.94427	0.272587
$330$	0	0
$331$	23.4164	1.28708	0.643541	−	0.765412i	$-0.277465\pi$
$331$	23.4164	1.28708	0.643541	+	0.765412i	$0.277465\pi$
$332$	1.52786i	0.0838524i
$333$	0	0
$334$	−21.8885	−1.19769
$335$	26.1803	1.43038
$336$	0	0
$337$	− 2.18034i	− 0.118771i	−0.998235	−	0.0593853i	$-0.981086\pi$
$337$	− 2.18034i	− 0.118771i	0.998235	−	0.0593853i	$-0.0189141\pi$
$338$	12.4164i	0.675364i
$339$	0	0
$340$	2.23607	0.121268
$341$	12.9443	0.700972
$342$	0	0
$343$	− 11.4164i	− 0.616428i
$344$	−10.1803	−0.548887
$345$	0	0
$346$	19.8885	1.06921
$347$	30.8328i	1.65519i	0.561324	+	0.827596i	$0.310292\pi$
$347$	30.8328i	1.65519i	−0.561324	+	0.827596i	$0.689708\pi$
$348$	0	0
$349$	19.5279	1.04530	0.522651	−	0.852547i	$-0.324943\pi$
$349$	19.5279	1.04530	0.522651	+	0.852547i	$0.324943\pi$
$350$	−16.1803	−0.864876
$351$	0	0
$352$	− 2.00000i	− 0.106600i
$353$	− 2.94427i	− 0.156708i	−0.996926	−	0.0783539i	$-0.975034\pi$
$353$	− 2.94427i	− 0.156708i	0.996926	−	0.0783539i	$-0.0249664\pi$
$354$	0	0
$355$	− 15.1246i	− 0.802731i
$356$	−3.52786	−0.186976
$357$	0	0
$358$	5.70820i	0.301688i
$359$	−18.4721	−0.974922	−0.487461	−	0.873145i	$-0.662077\pi$
$359$	−18.4721	−0.974922	−0.487461	+	0.873145i	$0.662077\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	− 8.47214i	− 0.445286i
$363$	0	0
$364$	−2.47214	−0.129575
$365$	−29.5967	−1.54916
$366$	0	0
$367$	28.1803i	1.47100i	0.677524	+	0.735501i	$0.263053\pi$
$367$	28.1803i	1.47100i	−0.677524	+	0.735501i	$0.736947\pi$
$368$	− 4.00000i	− 0.208514i
$369$	0	0
$370$	4.47214i	0.232495i
$371$	−22.4721	−1.16670
$372$	0	0
$373$	3.23607i	0.167557i	0.996484	+	0.0837786i	$0.0266989\pi$
$373$	3.23607i	0.167557i	−0.996484	+	0.0837786i	$0.973301\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	1.52786	0.0787936
$377$	− 3.05573i	− 0.157378i
$378$	0	0
$379$	10.4721	0.537917	0.268959	−	0.963152i	$-0.413320\pi$
$379$	10.4721	0.537917	0.268959	+	0.963152i	$0.413320\pi$
$380$	5.52786i	0.283573i
$381$	0	0
$382$	− 14.4721i	− 0.740459i
$383$	17.8885i	0.914062i	0.889451	+	0.457031i	$0.151087\pi$
$383$	17.8885i	0.914062i	−0.889451	+	0.457031i	$0.848913\pi$
$384$	0	0
$385$	−14.4721	−0.737568
$386$	−25.5967	−1.30284
$387$	0	0
$388$	11.7082i	0.594394i
$389$	−2.18034	−0.110548	−0.0552738	−	0.998471i	$-0.517603\pi$
$389$	−2.18034	−0.110548	−0.0552738	+	0.998471i	$0.517603\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	3.47214i	0.175369i
$393$	0	0
$394$	19.8885	1.00197
$395$	37.8885i	1.90638i
$396$	0	0
$397$	31.8885i	1.60044i	0.599706	+	0.800220i	$0.295284\pi$
$397$	31.8885i	1.60044i	−0.599706	+	0.800220i	$0.704716\pi$
$398$	− 8.94427i	− 0.448336i
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−22.0689	−1.10207	−0.551034	−	0.834483i	$-0.685766\pi$
$401$	−22.0689	−1.10207	−0.551034	+	0.834483i	$0.685766\pi$
$402$	0	0
$403$	− 4.94427i	− 0.246292i
$404$	11.7082	0.582505
$405$	0	0
$406$	−12.9443	−0.642413
$407$	4.00000i	0.198273i
$408$	0	0
$409$	18.3607	0.907877	0.453939	−	0.891033i	$-0.350019\pi$
$409$	18.3607	0.907877	0.453939	+	0.891033i	$0.350019\pi$
$410$	12.7639	0.630366
$411$	0	0
$412$	6.94427i	0.342120i
$413$	5.52786i	0.272008i
$414$	0	0
$415$	3.41641	0.167705
$416$	−0.763932	−0.0374548
$417$	0	0
$418$	4.94427i	0.241832i
$419$	−38.3607	−1.87404	−0.937021	−	0.349273i	$-0.886429\pi$
$419$	−38.3607	−1.87404	−0.937021	+	0.349273i	$0.886429\pi$
$420$	0	0
$421$	−10.5836	−0.515813	−0.257906	−	0.966170i	$-0.583033\pi$
$421$	−10.5836	−0.515813	−0.257906	+	0.966170i	$0.583033\pi$
$422$	− 7.05573i	− 0.343467i
$423$	0	0
$424$	−6.94427	−0.337244
$425$	− 5.00000i	− 0.242536i
$426$	0	0
$427$	14.4721i	0.700356i
$428$	− 8.00000i	− 0.386695i
$429$	0	0
$430$	22.7639i	1.09777i
$431$	17.5967	0.847606	0.423803	−	0.905755i	$-0.360695\pi$
$431$	17.5967	0.847606	0.423803	+	0.905755i	$0.360695\pi$
$432$	0	0
$433$	− 16.3607i	− 0.786244i	−0.919486	−	0.393122i	$-0.871395\pi$
$433$	− 16.3607i	− 0.786244i	0.919486	−	0.393122i	$-0.128605\pi$
$434$	−20.9443	−1.00536
$435$	0	0
$436$	−16.4721	−0.788872
$437$	9.88854i	0.473033i
$438$	0	0
$439$	−22.4721	−1.07254	−0.536268	−	0.844048i	$-0.680166\pi$
$439$	−22.4721	−1.07254	−0.536268	+	0.844048i	$0.680166\pi$
$440$	−4.47214	−0.213201
$441$	0	0
$442$	− 0.763932i	− 0.0363365i
$443$	− 4.00000i	− 0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$	− 4.00000i	− 0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	0	0
$445$	7.88854i	0.373953i
$446$	−26.9443	−1.27585
$447$	0	0
$448$	3.23607i	0.152890i
$449$	−31.5967	−1.49114	−0.745571	−	0.666426i	$-0.767823\pi$
$449$	−31.5967	−1.49114	−0.745571	+	0.666426i	$0.767823\pi$
$450$	0	0
$451$	11.4164	0.537578
$452$	− 10.0000i	− 0.470360i
$453$	0	0
$454$	25.8885	1.21501
$455$	5.52786i	0.259150i
$456$	0	0
$457$	− 28.3607i	− 1.32666i	−0.748328	−	0.663328i	$-0.769143\pi$
$457$	− 28.3607i	− 1.32666i	0.748328	−	0.663328i	$-0.230857\pi$
$458$	− 5.41641i	− 0.253092i
$459$	0	0
$460$	−8.94427	−0.417029
$461$	−28.0689	−1.30730	−0.653649	−	0.756798i	$-0.726763\pi$
$461$	−28.0689	−1.30730	−0.653649	+	0.756798i	$0.726763\pi$
$462$	0	0
$463$	− 1.05573i	− 0.0490638i	−0.999699	−	0.0245319i	$-0.992190\pi$
$463$	− 1.05573i	− 0.0490638i	0.999699	−	0.0245319i	$-0.00780954\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	−21.4164	−0.992096
$467$	− 16.9443i	− 0.784087i	−0.919947	−	0.392044i	$-0.871768\pi$
$467$	− 16.9443i	− 0.784087i	0.919947	−	0.392044i	$-0.128232\pi$
$468$	0	0
$469$	−37.8885	−1.74953
$470$	− 3.41641i	− 0.157587i
$471$	0	0
$472$	1.70820i	0.0786265i
$473$	20.3607i	0.936185i
$474$	0	0
$475$	12.3607	0.567147
$476$	−3.23607	−0.148325
$477$	0	0
$478$	13.8885i	0.635247i
$479$	13.5967	0.621251	0.310626	−	0.950532i	$-0.399461\pi$
$479$	13.5967	0.621251	0.310626	+	0.950532i	$0.399461\pi$
$480$	0	0
$481$	1.52786	0.0696646
$482$	− 17.4164i	− 0.793296i
$483$	0	0
$484$	7.00000	0.318182
$485$	26.1803	1.18879
$486$	0	0
$487$	29.7082i	1.34621i	0.739548	+	0.673104i	$0.235039\pi$
$487$	29.7082i	1.34621i	−0.739548	+	0.673104i	$0.764961\pi$
$488$	4.47214i	0.202444i
$489$	0	0
$490$	7.76393	0.350739
$491$	−25.7082	−1.16020	−0.580098	−	0.814547i	$-0.696986\pi$
$491$	−25.7082	−1.16020	−0.580098	+	0.814547i	$0.696986\pi$
$492$	0	0
$493$	− 4.00000i	− 0.180151i
$494$	1.88854	0.0849696
$495$	0	0
$496$	−6.47214	−0.290607
$497$	21.8885i	0.981835i
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	11.1803i	0.500000i
$501$	0	0
$502$	− 12.1803i	− 0.543635i
$503$	0.583592i	0.0260211i	0.999915	+	0.0130105i	$0.00414150\pi$
$503$	0.583592i	0.0260211i	−0.999915	+	0.0130105i	$0.995858\pi$
$504$	0	0
$505$	− 26.1803i	− 1.16501i
$506$	−8.00000	−0.355643
$507$	0	0
$508$	14.0000i	0.621150i
$509$	32.6525	1.44730	0.723648	−	0.690169i	$-0.242464\pi$
$509$	32.6525	1.44730	0.723648	+	0.690169i	$0.242464\pi$
$510$	0	0
$511$	42.8328	1.89481
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	−18.0000	−0.793946
$515$	15.5279	0.684239
$516$	0	0
$517$	− 3.05573i	− 0.134391i
$518$	− 6.47214i	− 0.284369i
$519$	0	0
$520$	1.70820i	0.0749097i
$521$	−28.1803	−1.23460	−0.617302	−	0.786727i	$-0.711774\pi$
$521$	−28.1803	−1.23460	−0.617302	+	0.786727i	$0.711774\pi$
$522$	0	0
$523$	− 27.7082i	− 1.21160i	−0.795619	−	0.605798i	$-0.792854\pi$
$523$	− 27.7082i	− 1.21160i	0.795619	−	0.605798i	$-0.207146\pi$
$524$	3.52786	0.154115
$525$	0	0
$526$	−4.00000	−0.174408
$527$	− 6.47214i	− 0.281931i
$528$	0	0
$529$	7.00000	0.304348
$530$	15.5279i	0.674487i
$531$	0	0
$532$	− 8.00000i	− 0.346844i
$533$	− 4.36068i	− 0.188882i
$534$	0	0
$535$	−17.8885	−0.773389
$536$	−11.7082	−0.505717
$537$	0	0
$538$	− 24.9443i	− 1.07542i
$539$	6.94427	0.299111
$540$	0	0
$541$	31.3050	1.34590	0.672952	−	0.739686i	$-0.265026\pi$
$541$	31.3050	1.34590	0.672952	+	0.739686i	$0.265026\pi$
$542$	4.00000i	0.171815i
$543$	0	0
$544$	−1.00000	−0.0428746
$545$	36.8328i	1.57774i
$546$	0	0
$547$	16.9443i	0.724485i	0.932084	+	0.362242i	$0.117989\pi$
$547$	16.9443i	0.724485i	−0.932084	+	0.362242i	$0.882011\pi$
$548$	2.00000i	0.0854358i
$549$	0	0
$550$	10.0000i	0.426401i
$551$	9.88854	0.421266
$552$	0	0
$553$	− 54.8328i	− 2.33173i
$554$	−18.3607	−0.780071
$555$	0	0
$556$	10.4721	0.444117
$557$	− 42.9443i	− 1.81961i	−0.415039	−	0.909804i	$-0.636232\pi$
$557$	− 42.9443i	− 1.81961i	0.415039	−	0.909804i	$-0.363768\pi$
$558$	0	0
$559$	7.77709	0.328936
$560$	7.23607	0.305780
$561$	0	0
$562$	25.4164i	1.07213i
$563$	− 37.8885i	− 1.59681i	−0.602120	−	0.798406i	$-0.705677\pi$
$563$	− 37.8885i	− 1.59681i	0.602120	−	0.798406i	$-0.294323\pi$
$564$	0	0
$565$	−22.3607	−0.940721
$566$	−9.88854	−0.415646
$567$	0	0
$568$	6.76393i	0.283808i
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−8.36068	−0.349884	−0.174942	−	0.984579i	$-0.555974\pi$
$571$	−8.36068	−0.349884	−0.174942	+	0.984579i	$0.555974\pi$
$572$	1.52786i	0.0638832i
$573$	0	0
$574$	−18.4721	−0.771012
$575$	20.0000i	0.834058i
$576$	0	0
$577$	8.58359i	0.357340i	0.983909	+	0.178670i	$0.0571794\pi$
$577$	8.58359i	0.357340i	−0.983909	+	0.178670i	$0.942821\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	0	0
$580$	8.94427i	0.371391i
$581$	−4.94427	−0.205123
$582$	0	0
$583$	13.8885i	0.575205i
$584$	13.2361	0.547712
$585$	0	0
$586$	2.58359	0.106727
$587$	− 14.4721i	− 0.597329i	−0.954358	−	0.298664i	$-0.903459\pi$
$587$	− 14.4721i	− 0.597329i	0.954358	−	0.298664i	$-0.0965411\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	3.81966	0.157253
$591$	0	0
$592$	− 2.00000i	− 0.0821995i
$593$	− 13.0557i	− 0.536134i	−0.963400	−	0.268067i	$-0.913615\pi$
$593$	− 13.0557i	− 0.536134i	0.963400	−	0.268067i	$-0.0863849\pi$
$594$	0	0
$595$	7.23607i	0.296650i
$596$	10.1803	0.417003
$597$	0	0
$598$	3.05573i	0.124958i
$599$	34.4721	1.40849	0.704247	−	0.709955i	$-0.251285\pi$
$599$	34.4721	1.40849	0.704247	+	0.709955i	$0.251285\pi$
$600$	0	0
$601$	−19.5279	−0.796558	−0.398279	−	0.917264i	$-0.630392\pi$
$601$	−19.5279	−0.796558	−0.398279	+	0.917264i	$0.630392\pi$
$602$	− 32.9443i	− 1.34271i
$603$	0	0
$604$	4.00000	0.162758
$605$	− 15.6525i	− 0.636364i
$606$	0	0
$607$	− 9.12461i	− 0.370357i	−0.982705	−	0.185178i	$-0.940714\pi$
$607$	− 9.12461i	− 0.370357i	0.982705	−	0.185178i	$-0.0592863\pi$
$608$	− 2.47214i	− 0.100258i
$609$	0	0
$610$	10.0000	0.404888
$611$	−1.16718	−0.0472192
$612$	0	0
$613$	− 19.8197i	− 0.800509i	−0.916404	−	0.400254i	$-0.868922\pi$
$613$	− 19.8197i	− 0.800509i	0.916404	−	0.400254i	$-0.131078\pi$
$614$	−18.1803	−0.733699
$615$	0	0
$616$	6.47214	0.260770
$617$	− 11.5279i	− 0.464094i	−0.972705	−	0.232047i	$-0.925458\pi$
$617$	− 11.5279i	− 0.464094i	0.972705	−	0.232047i	$-0.0745424\pi$
$618$	0	0
$619$	31.0557	1.24824	0.624118	−	0.781330i	$-0.285459\pi$
$619$	31.0557	1.24824	0.624118	+	0.781330i	$0.285459\pi$
$620$	14.4721i	0.581215i
$621$	0	0
$622$	29.5967i	1.18672i
$623$	− 11.4164i	− 0.457389i
$624$	0	0
$625$	25.0000	1.00000
$626$	8.29180	0.331407
$627$	0	0
$628$	− 12.1803i	− 0.486048i
$629$	2.00000	0.0797452
$630$	0	0
$631$	25.8885	1.03061	0.515303	−	0.857008i	$-0.327679\pi$
$631$	25.8885	1.03061	0.515303	+	0.857008i	$0.327679\pi$
$632$	− 16.9443i	− 0.674007i
$633$	0	0
$634$	23.8885	0.948735
$635$	31.3050	1.24230
$636$	0	0
$637$	− 2.65248i	− 0.105095i
$638$	8.00000i	0.316723i
$639$	0	0
$640$	2.23607	0.0883883
$641$	1.70820	0.0674700	0.0337350	−	0.999431i	$-0.489260\pi$
$641$	1.70820	0.0674700	0.0337350	+	0.999431i	$0.489260\pi$
$642$	0	0
$643$	− 0.583592i	− 0.0230146i	−0.999934	−	0.0115073i	$-0.996337\pi$
$643$	− 0.583592i	− 0.0230146i	0.999934	−	0.0115073i	$-0.00366297\pi$
$644$	12.9443	0.510076
$645$	0	0
$646$	2.47214	0.0972649
$647$	− 3.05573i	− 0.120133i	−0.998194	−	0.0600665i	$-0.980869\pi$
$647$	− 3.05573i	− 0.120133i	0.998194	−	0.0600665i	$-0.0191313\pi$
$648$	0	0
$649$	3.41641	0.134106
$650$	3.81966	0.149819
$651$	0	0
$652$	13.5279i	0.529792i
$653$	12.4721i	0.488072i	0.969766	+	0.244036i	$0.0784716\pi$
$653$	12.4721i	0.488072i	−0.969766	+	0.244036i	$0.921528\pi$
$654$	0	0
$655$	− 7.88854i	− 0.308231i
$656$	−5.70820	−0.222868
$657$	0	0
$658$	4.94427i	0.192748i
$659$	41.7082	1.62472	0.812360	−	0.583156i	$-0.198182\pi$
$659$	41.7082	1.62472	0.812360	+	0.583156i	$0.198182\pi$
$660$	0	0
$661$	−13.4164	−0.521838	−0.260919	−	0.965361i	$-0.584026\pi$
$661$	−13.4164	−0.521838	−0.260919	+	0.965361i	$0.584026\pi$
$662$	23.4164i	0.910105i
$663$	0	0
$664$	−1.52786	−0.0592926
$665$	−17.8885	−0.693688
$666$	0	0
$667$	16.0000i	0.619522i
$668$	− 21.8885i	− 0.846893i
$669$	0	0
$670$	26.1803i	1.01143i
$671$	8.94427	0.345290
$672$	0	0
$673$	− 18.1803i	− 0.700801i	−0.936600	−	0.350400i	$-0.886046\pi$
$673$	− 18.1803i	− 0.700801i	0.936600	−	0.350400i	$-0.113954\pi$
$674$	2.18034	0.0839836
$675$	0	0
$676$	−12.4164	−0.477554
$677$	− 26.9443i	− 1.03555i	−0.855516	−	0.517776i	$-0.826760\pi$
$677$	− 26.9443i	− 1.03555i	0.855516	−	0.517776i	$-0.173240\pi$
$678$	0	0
$679$	−37.8885	−1.45403
$680$	2.23607i	0.0857493i
$681$	0	0
$682$	12.9443i	0.495662i
$683$	15.0557i	0.576091i	0.957617	+	0.288046i	$0.0930055\pi$
$683$	15.0557i	0.576091i	−0.957617	+	0.288046i	$0.906994\pi$
$684$	0	0
$685$	4.47214	0.170872
$686$	11.4164	0.435880
$687$	0	0
$688$	− 10.1803i	− 0.388122i
$689$	5.30495	0.202103
$690$	0	0
$691$	12.9443	0.492423	0.246212	−	0.969216i	$-0.420814\pi$
$691$	12.9443	0.492423	0.246212	+	0.969216i	$0.420814\pi$
$692$	19.8885i	0.756049i
$693$	0	0
$694$	−30.8328	−1.17040
$695$	− 23.4164i	− 0.888235i
$696$	0	0
$697$	− 5.70820i	− 0.216214i
$698$	19.5279i	0.739141i
$699$	0	0
$700$	− 16.1803i	− 0.611559i
$701$	37.5967	1.42001	0.710005	−	0.704197i	$-0.248693\pi$
$701$	37.5967	1.42001	0.710005	+	0.704197i	$0.248693\pi$
$702$	0	0
$703$	4.94427i	0.186477i
$704$	2.00000	0.0753778
$705$	0	0
$706$	2.94427	0.110809
$707$	37.8885i	1.42495i
$708$	0	0
$709$	48.4721	1.82041	0.910205	−	0.414159i	$-0.135924\pi$
$709$	48.4721	1.82041	0.910205	+	0.414159i	$0.135924\pi$
$710$	15.1246	0.567617
$711$	0	0
$712$	− 3.52786i	− 0.132212i
$713$	25.8885i	0.969534i
$714$	0	0
$715$	3.41641	0.127766
$716$	−5.70820	−0.213326
$717$	0	0
$718$	− 18.4721i	− 0.689374i
$719$	30.5410	1.13899	0.569494	−	0.821996i	$-0.307139\pi$
$719$	30.5410	1.13899	0.569494	+	0.821996i	$0.307139\pi$
$720$	0	0
$721$	−22.4721	−0.836906
$722$	− 12.8885i	− 0.479662i
$723$	0	0
$724$	8.47214	0.314864
$725$	20.0000	0.742781
$726$	0	0
$727$	− 11.8885i	− 0.440922i	−0.975396	−	0.220461i	$-0.929244\pi$
$727$	− 11.8885i	− 0.440922i	0.975396	−	0.220461i	$-0.0707561\pi$
$728$	− 2.47214i	− 0.0916235i
$729$	0	0
$730$	− 29.5967i	− 1.09542i
$731$	10.1803	0.376533
$732$	0	0
$733$	− 28.5410i	− 1.05419i	−0.849807	−	0.527093i	$-0.823282\pi$
$733$	− 28.5410i	− 1.05419i	0.849807	−	0.527093i	$-0.176718\pi$
$734$	−28.1803	−1.04016
$735$	0	0
$736$	4.00000	0.147442
$737$	23.4164i	0.862554i
$738$	0	0
$739$	−42.4721	−1.56236	−0.781181	−	0.624304i	$-0.785383\pi$
$739$	−42.4721	−1.56236	−0.781181	+	0.624304i	$0.785383\pi$
$740$	−4.47214	−0.164399
$741$	0	0
$742$	− 22.4721i	− 0.824978i
$743$	− 42.4721i	− 1.55815i	−0.626930	−	0.779076i	$-0.715689\pi$
$743$	− 42.4721i	− 1.55815i	0.626930	−	0.779076i	$-0.284311\pi$
$744$	0	0
$745$	− 22.7639i	− 0.834006i
$746$	−3.23607	−0.118481
$747$	0	0
$748$	2.00000i	0.0731272i
$749$	25.8885	0.945947
$750$	0	0
$751$	−5.88854	−0.214876	−0.107438	−	0.994212i	$-0.534265\pi$
$751$	−5.88854	−0.214876	−0.107438	+	0.994212i	$0.534265\pi$
$752$	1.52786i	0.0557155i
$753$	0	0
$754$	3.05573	0.111283
$755$	− 8.94427i	− 0.325515i
$756$	0	0
$757$	− 11.5967i	− 0.421491i	−0.977541	−	0.210745i	$-0.932411\pi$
$757$	− 11.5967i	− 0.421491i	0.977541	−	0.210745i	$-0.0675891\pi$
$758$	10.4721i	0.380365i
$759$	0	0
$760$	−5.52786	−0.200517
$761$	0.832816	0.0301895	0.0150948	−	0.999886i	$-0.495195\pi$
$761$	0.832816	0.0301895	0.0150948	+	0.999886i	$0.495195\pi$
$762$	0	0
$763$	− 53.3050i	− 1.92977i
$764$	14.4721	0.523584
$765$	0	0
$766$	−17.8885	−0.646339
$767$	− 1.30495i	− 0.0471191i
$768$	0	0
$769$	−10.3607	−0.373616	−0.186808	−	0.982396i	$-0.559814\pi$
$769$	−10.3607	−0.373616	−0.186808	+	0.982396i	$0.559814\pi$
$770$	− 14.4721i	− 0.521540i
$771$	0	0
$772$	− 25.5967i	− 0.921247i
$773$	− 6.58359i	− 0.236795i	−0.992966	−	0.118398i	$-0.962224\pi$
$773$	− 6.58359i	− 0.236795i	0.992966	−	0.118398i	$-0.0377758\pi$
$774$	0	0
$775$	32.3607	1.16243
$776$	−11.7082	−0.420300
$777$	0	0
$778$	− 2.18034i	− 0.0781690i
$779$	14.1115	0.505595
$780$	0	0
$781$	13.5279	0.484065
$782$	4.00000i	0.143040i
$783$	0	0
$784$	−3.47214	−0.124005
$785$	−27.2361	−0.972097
$786$	0	0
$787$	− 37.5279i	− 1.33772i	−0.743387	−	0.668862i	$-0.766782\pi$
$787$	− 37.5279i	− 1.33772i	0.743387	−	0.668862i	$-0.233218\pi$
$788$	19.8885i	0.708500i
$789$	0	0
$790$	−37.8885	−1.34801
$791$	32.3607	1.15061
$792$	0	0
$793$	− 3.41641i	− 0.121320i
$794$	−31.8885	−1.13168
$795$	0	0
$796$	8.94427	0.317021
$797$	46.0000i	1.62940i	0.579880	+	0.814702i	$0.303099\pi$
$797$	46.0000i	1.62940i	−0.579880	+	0.814702i	$0.696901\pi$
$798$	0	0
$799$	−1.52786	−0.0540519
$800$	− 5.00000i	− 0.176777i
$801$	0	0
$802$	− 22.0689i	− 0.779279i
$803$	− 26.4721i	− 0.934181i
$804$	0	0
$805$	− 28.9443i	− 1.02015i
$806$	4.94427	0.174155
$807$	0	0
$808$	11.7082i	0.411893i
$809$	6.87539	0.241726	0.120863	−	0.992669i	$-0.461434\pi$
$809$	6.87539	0.241726	0.120863	+	0.992669i	$0.461434\pi$
$810$	0	0
$811$	−38.2492	−1.34311	−0.671556	−	0.740954i	$-0.734374\pi$
$811$	−38.2492	−1.34311	−0.671556	+	0.740954i	$0.734374\pi$
$812$	− 12.9443i	− 0.454255i
$813$	0	0
$814$	−4.00000	−0.140200
$815$	30.2492	1.05958
$816$	0	0
$817$	25.1672i	0.880488i
$818$	18.3607i	0.641966i
$819$	0	0
$820$	12.7639i	0.445736i
$821$	11.4164	0.398435	0.199218	−	0.979955i	$-0.436160\pi$
$821$	11.4164	0.398435	0.199218	+	0.979955i	$0.436160\pi$
$822$	0	0
$823$	15.8197i	0.551439i	0.961238	+	0.275719i	$0.0889160\pi$
$823$	15.8197i	0.551439i	−0.961238	+	0.275719i	$0.911084\pi$
$824$	−6.94427	−0.241915
$825$	0	0
$826$	−5.52786	−0.192339
$827$	− 40.9443i	− 1.42377i	−0.702295	−	0.711886i	$-0.747841\pi$
$827$	− 40.9443i	− 1.42377i	0.702295	−	0.711886i	$-0.252159\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	3.41641i	0.118585i
$831$	0	0
$832$	− 0.763932i	− 0.0264846i
$833$	− 3.47214i	− 0.120302i
$834$	0	0
$835$	−48.9443	−1.69379
$836$	−4.94427	−0.171001
$837$	0	0
$838$	− 38.3607i	− 1.32515i
$839$	36.0689	1.24524	0.622618	−	0.782526i	$-0.286069\pi$
$839$	36.0689	1.24524	0.622618	+	0.782526i	$0.286069\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	− 10.5836i	− 0.364735i
$843$	0	0
$844$	7.05573	0.242868
$845$	27.7639i	0.955108i
$846$	0	0
$847$	22.6525i	0.778348i
$848$	− 6.94427i	− 0.238467i
$849$	0	0
$850$	5.00000	0.171499
$851$	−8.00000	−0.274236
$852$	0	0
$853$	38.3607i	1.31344i	0.754132	+	0.656722i	$0.228058\pi$
$853$	38.3607i	1.31344i	−0.754132	+	0.656722i	$0.771942\pi$
$854$	−14.4721	−0.495226
$855$	0	0
$856$	8.00000	0.273434
$857$	5.63932i	0.192636i	0.995351	+	0.0963178i	$0.0307065\pi$
$857$	5.63932i	0.192636i	−0.995351	+	0.0963178i	$0.969294\pi$
$858$	0	0
$859$	−24.9443	−0.851088	−0.425544	−	0.904938i	$-0.639917\pi$
$859$	−24.9443	−0.851088	−0.425544	+	0.904938i	$0.639917\pi$
$860$	−22.7639	−0.776244
$861$	0	0
$862$	17.5967i	0.599348i
$863$	− 41.3050i	− 1.40604i	−0.711172	−	0.703018i	$-0.751835\pi$
$863$	− 41.3050i	− 1.40604i	0.711172	−	0.703018i	$-0.248165\pi$
$864$	0	0
$865$	44.4721	1.51210
$866$	16.3607	0.555959
$867$	0	0
$868$	− 20.9443i	− 0.710895i
$869$	−33.8885	−1.14959
$870$	0	0
$871$	8.94427	0.303065
$872$	− 16.4721i	− 0.557817i
$873$	0	0
$874$	−9.88854	−0.334485
$875$	−36.1803	−1.22312
$876$	0	0
$877$	31.5279i	1.06462i	0.846549	+	0.532310i	$0.178676\pi$
$877$	31.5279i	1.06462i	−0.846549	+	0.532310i	$0.821324\pi$
$878$	− 22.4721i	− 0.758398i
$879$	0	0
$880$	− 4.47214i	− 0.150756i
$881$	−33.1246	−1.11600	−0.557998	−	0.829842i	$-0.688430\pi$
$881$	−33.1246	−1.11600	−0.557998	+	0.829842i	$0.688430\pi$
$882$	0	0
$883$	− 22.7639i	− 0.766067i	−0.923734	−	0.383034i	$-0.874879\pi$
$883$	− 22.7639i	− 0.766067i	0.923734	−	0.383034i	$-0.125121\pi$
$884$	0.763932	0.0256938
$885$	0	0
$886$	4.00000	0.134383
$887$	− 8.94427i	− 0.300319i	−0.988662	−	0.150160i	$-0.952021\pi$
$887$	− 8.94427i	− 0.300319i	0.988662	−	0.150160i	$-0.0479788\pi$
$888$	0	0
$889$	−45.3050	−1.51948
$890$	−7.88854	−0.264425
$891$	0	0
$892$	− 26.9443i	− 0.902161i
$893$	− 3.77709i	− 0.126395i
$894$	0	0
$895$	12.7639i	0.426651i
$896$	−3.23607	−0.108109
$897$	0	0
$898$	− 31.5967i	− 1.05440i
$899$	25.8885	0.863431
$900$	0	0
$901$	6.94427	0.231347
$902$	11.4164i	0.380125i
$903$	0	0
$904$	10.0000	0.332595
$905$	− 18.9443i	− 0.629729i
$906$	0	0
$907$	40.3607i	1.34015i	0.742291	+	0.670077i	$0.233739\pi$
$907$	40.3607i	1.34015i	−0.742291	+	0.670077i	$0.766261\pi$
$908$	25.8885i	0.859142i
$909$	0	0
$910$	−5.52786	−0.183247
$911$	−54.5410	−1.80702	−0.903512	−	0.428562i	$-0.859020\pi$
$911$	−54.5410	−1.80702	−0.903512	+	0.428562i	$0.859020\pi$
$912$	0	0
$913$	3.05573i	0.101130i
$914$	28.3607	0.938088
$915$	0	0
$916$	5.41641	0.178963
$917$	11.4164i	0.377003i
$918$	0	0
$919$	36.7214	1.21133	0.605663	−	0.795721i	$-0.292908\pi$
$919$	36.7214	1.21133	0.605663	+	0.795721i	$0.292908\pi$
$920$	− 8.94427i	− 0.294884i
$921$	0	0
$922$	− 28.0689i	− 0.924399i
$923$	− 5.16718i	− 0.170080i
$924$	0	0
$925$	10.0000i	0.328798i
$926$	1.05573	0.0346934
$927$	0	0
$928$	− 4.00000i	− 0.131306i
$929$	−34.0689	−1.11776	−0.558882	−	0.829247i	$-0.688769\pi$
$929$	−34.0689	−1.11776	−0.558882	+	0.829247i	$0.688769\pi$
$930$	0	0
$931$	8.58359	0.281316
$932$	− 21.4164i	− 0.701518i
$933$	0	0
$934$	16.9443	0.554434
$935$	4.47214	0.146254
$936$	0	0
$937$	− 26.4721i	− 0.864807i	−0.901680	−	0.432403i	$-0.857666\pi$
$937$	− 26.4721i	− 0.864807i	0.901680	−	0.432403i	$-0.142334\pi$
$938$	− 37.8885i	− 1.23710i
$939$	0	0
$940$	3.41641	0.111431
$941$	49.3050	1.60730	0.803648	−	0.595105i	$-0.202890\pi$
$941$	49.3050	1.60730	0.803648	+	0.595105i	$0.202890\pi$
$942$	0	0
$943$	22.8328i	0.743539i
$944$	−1.70820	−0.0555973
$945$	0	0
$946$	−20.3607	−0.661983
$947$	− 40.7214i	− 1.32327i	−0.749828	−	0.661633i	$-0.769864\pi$
$947$	− 40.7214i	− 1.32327i	0.749828	−	0.661633i	$-0.230136\pi$
$948$	0	0
$949$	−10.1115	−0.328232
$950$	12.3607i	0.401033i
$951$	0	0
$952$	− 3.23607i	− 0.104882i
$953$	− 9.05573i	− 0.293344i	−0.989185	−	0.146672i	$-0.953144\pi$
$953$	− 9.05573i	− 0.293344i	0.989185	−	0.146672i	$-0.0468562\pi$
$954$	0	0
$955$	− 32.3607i	− 1.04717i
$956$	−13.8885	−0.449188
$957$	0	0
$958$	13.5967i	0.439291i
$959$	−6.47214	−0.208996
$960$	0	0
$961$	10.8885	0.351243
$962$	1.52786i	0.0492603i
$963$	0	0
$964$	17.4164	0.560945
$965$	−57.2361	−1.84249
$966$	0	0
$967$	− 8.47214i	− 0.272446i	−0.990678	−	0.136223i	$-0.956504\pi$
$967$	− 8.47214i	− 0.272446i	0.990678	−	0.136223i	$-0.0434963\pi$
$968$	7.00000i	0.224989i
$969$	0	0
$970$	26.1803i	0.840600i
$971$	27.2361	0.874047	0.437024	−	0.899450i	$-0.356033\pi$
$971$	27.2361	0.874047	0.437024	+	0.899450i	$0.356033\pi$
$972$	0	0
$973$	33.8885i	1.08642i
$974$	−29.7082	−0.951912
$975$	0	0
$976$	−4.47214	−0.143150
$977$	9.05573i	0.289718i	0.989452	+	0.144859i	$0.0462729\pi$
$977$	9.05573i	0.289718i	−0.989452	+	0.144859i	$0.953727\pi$
$978$	0	0
$979$	−7.05573	−0.225502
$980$	7.76393i	0.248010i
$981$	0	0
$982$	− 25.7082i	− 0.820382i
$983$	15.4164i	0.491707i	0.969307	+	0.245854i	$0.0790682\pi$
$983$	15.4164i	0.491707i	−0.969307	+	0.245854i	$0.920932\pi$
$984$	0	0
$985$	44.4721	1.41700
$986$	4.00000	0.127386
$987$	0	0
$988$	1.88854i	0.0600826i
$989$	−40.7214	−1.29486
$990$	0	0
$991$	42.8328	1.36063	0.680315	−	0.732920i	$-0.261843\pi$
$991$	42.8328	1.36063	0.680315	+	0.732920i	$0.261843\pi$
$992$	− 6.47214i	− 0.205491i
$993$	0	0
$994$	−21.8885	−0.694262
$995$	− 20.0000i	− 0.634043i
$996$	0	0
$997$	− 58.9443i	− 1.86678i	−0.358859	−	0.933392i	$-0.616834\pi$
$997$	− 58.9443i	− 1.86678i	0.358859	−	0.933392i	$-0.383166\pi$
$998$	16.0000i	0.506471i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1530.2.d.f.919.4		4
3.2	odd	2		510.2.d.b.409.1	✓	4
5.2	odd	4		7650.2.a.cx.1.2		2
5.3	odd	4		7650.2.a.da.1.1		2
5.4	even	2	inner	1530.2.d.f.919.1		4
12.11	even	2		4080.2.m.m.2449.3		4
15.2	even	4		2550.2.a.bk.1.2		2
15.8	even	4		2550.2.a.bh.1.1		2
15.14	odd	2		510.2.d.b.409.4	yes	4
60.59	even	2		4080.2.m.m.2449.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.d.b.409.1	✓	4	3.2	odd	2
510.2.d.b.409.4	yes	4	15.14	odd	2
1530.2.d.f.919.1		4	5.4	even	2	inner
1530.2.d.f.919.4		4	1.1	even	1	trivial
2550.2.a.bh.1.1		2	15.8	even	4
2550.2.a.bk.1.2		2	15.2	even	4
4080.2.m.m.2449.2		4	60.59	even	2
4080.2.m.m.2449.3		4	12.11	even	2
7650.2.a.cx.1.2		2	5.2	odd	4
7650.2.a.da.1.1		2	5.3	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:	\( N \)	\(=\)	\( 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1530.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +2.23607i q^{5} -3.23607i q^{7} -1.00000i q^{8} -2.23607 q^{10} -2.00000 q^{11} +0.763932i q^{13} +3.23607 q^{14} +1.00000 q^{16} +1.00000i q^{17} -2.47214 q^{19} -2.23607i q^{20} -2.00000i q^{22} -4.00000i q^{23} -5.00000 q^{25} -0.763932 q^{26} +3.23607i q^{28} -4.00000 q^{29} -6.47214 q^{31} +1.00000i q^{32} -1.00000 q^{34} +7.23607 q^{35} -2.00000i q^{37} -2.47214i q^{38} +2.23607 q^{40} -5.70820 q^{41} -10.1803i q^{43} +2.00000 q^{44} +4.00000 q^{46} +1.52786i q^{47} -3.47214 q^{49} -5.00000i q^{50} -0.763932i q^{52} -6.94427i q^{53} -4.47214i q^{55} -3.23607 q^{56} -4.00000i q^{58} -1.70820 q^{59} -4.47214 q^{61} -6.47214i q^{62} -1.00000 q^{64} -1.70820 q^{65} -11.7082i q^{67} -1.00000i q^{68} +7.23607i q^{70} -6.76393 q^{71} +13.2361i q^{73} +2.00000 q^{74} +2.47214 q^{76} +6.47214i q^{77} +16.9443 q^{79} +2.23607i q^{80} -5.70820i q^{82} -1.52786i q^{83} -2.23607 q^{85} +10.1803 q^{86} +2.00000i q^{88} +3.52786 q^{89} +2.47214 q^{91} +4.00000i q^{92} -1.52786 q^{94} -5.52786i q^{95} -11.7082i q^{97} -3.47214i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 8 q^{11} + 4 q^{14} + 4 q^{16} + 8 q^{19} - 20 q^{25} - 12 q^{26} - 16 q^{29} - 8 q^{31} - 4 q^{34} + 20 q^{35} + 4 q^{41} + 8 q^{44} + 16 q^{46} + 4 q^{49} - 4 q^{56} + 20 q^{59} - 4 q^{64}+ \cdots - 24 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 - 8 * q^11 + 4 * q^14 + 4 * q^16 + 8 * q^19 - 20 * q^25 - 12 * q^26 - 16 * q^29 - 8 * q^31 - 4 * q^34 + 20 * q^35 + 4 * q^41 + 8 * q^44 + 16 * q^46 + 4 * q^49 - 4 * q^56 + 20 * q^59 - 4 * q^64 + 20 * q^65 - 36 * q^71 + 8 * q^74 - 8 * q^76 + 32 * q^79 - 4 * q^86 + 32 * q^89 - 8 * q^91 - 24 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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