LMFDB - Embedded newform 2240.2.l.a.1569.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1569,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("2240.1569");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.l (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:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 3x^{10} + 11x^{8} + 22x^{6} + 99x^{4} + 243x^{2} + 729 $ x^12 + 3x^10 + 11x^8 + 22x^6 + 99x^4 + 243*x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1569.3
Root			$1.02957 - 1.39283i$ of defining polynomial
Character	$\chi$	$=$	2240.1569
Dual form			2240.2.l.a.1569.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.05914 q^{3} +(1.27280 - 1.83847i) q^{5} -1.00000i q^{7} +1.24006 q^{9} +O(q^{10})$	$q-2.05914 q^{3} +(1.27280 - 1.83847i) q^{5} -1.00000i q^{7} +1.24006 q^{9} +5.33127i q^{11} -0.785667 q^{13} +(-2.62088 + 3.78567i) q^{15} -1.61780i q^{17} +4.54561i q^{19} +2.05914i q^{21} -7.09122i q^{23} +(-1.75994 - 4.68002i) q^{25} +3.62397 q^{27} -7.74225i q^{29} -4.11828 q^{31} -10.9778i q^{33} +(-1.83847 - 1.27280i) q^{35} +0.480119 q^{37} +1.61780 q^{39} +2.00000 q^{41} +2.00617 q^{43} +(1.57835 - 2.27981i) q^{45} -2.24006i q^{47} -1.00000 q^{49} +3.33127i q^{51} -5.57133 q^{53} +(9.80139 + 6.78567i) q^{55} -9.36004i q^{57} +5.02573i q^{59} +4.55962i q^{61} -1.24006i q^{63} +(-1.00000 + 1.44442i) q^{65} -12.5956 q^{67} +14.6018i q^{69} -13.4783 q^{71} +2.00617i q^{73} +(3.62397 + 9.63682i) q^{75} +5.33127 q^{77} +6.61876 q^{79} -11.1824 q^{81} -8.91870 q^{83} +(-2.97427 - 2.05914i) q^{85} +15.9424i q^{87} +0.428666 q^{89} +0.785667i q^{91} +8.48012 q^{93} +(8.35696 + 5.78567i) q^{95} -13.0900i q^{97} +6.61110i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{5} + 24 q^{9}+O(q^{10}) $ 12 * q - 4 * q^5 + 24 * q^9	$ 12 q - 4 q^{5} + 24 q^{9} + 20 q^{13} - 12 q^{25} + 24 q^{37} + 24 q^{41} - 48 q^{45} - 12 q^{49} - 8 q^{53} - 12 q^{65} - 4 q^{77} + 20 q^{81} - 56 q^{85} + 64 q^{89} + 120 q^{93}+O(q^{100}) $ 12 * q - 4 * q^5 + 24 * q^9 + 20 * q^13 - 12 * q^25 + 24 * q^37 + 24 * q^41 - 48 * q^45 - 12 * q^49 - 8 * q^53 - 12 * q^65 - 4 * q^77 + 20 * q^81 - 56 * q^85 + 64 * q^89 + 120 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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			0
$2$		0			0
$3$	−2.05914			−1.18885			−0.594423	−	0.804153i	$-0.702619\pi$
$3$	−2.05914			−1.18885			−0.594423	+	0.804153i	$0.702619\pi$
$4$		0			0
$5$	1.27280	−	1.83847i	0.569215	−	0.822189i
$5$	1.27280	−	1.83847i	0.569215	−	0.822189i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.24006			0.413353
$10$		0			0
$11$			5.33127i			1.60744i	0.595008	+	0.803720i	$0.297149\pi$
$11$			5.33127i			1.60744i	−0.595008	+	0.803720i	$0.702851\pi$
$12$		0			0
$13$	−0.785667			−0.217905			−0.108952	−	0.994047i	$-0.534750\pi$
$13$	−0.785667			−0.217905			−0.108952	+	0.994047i	$0.534750\pi$
$14$		0			0
$15$	−2.62088	+	3.78567i	−0.676709	+	0.977455i
$16$		0			0
$17$		−	1.61780i		−	0.392374i	−0.980567	−	0.196187i	$-0.937144\pi$
$17$		−	1.61780i		−	0.392374i	0.980567	−	0.196187i	$-0.0628559\pi$
$18$		0			0
$19$			4.54561i			1.04283i	0.853302	+	0.521417i	$0.174596\pi$
$19$			4.54561i			1.04283i	−0.853302	+	0.521417i	$0.825404\pi$
$20$		0			0
$21$			2.05914i			0.449341i
$22$		0			0
$23$		−	7.09122i		−	1.47862i	−0.673365	−	0.739310i	$-0.735152\pi$
$23$		−	7.09122i		−	1.47862i	0.673365	−	0.739310i	$-0.264848\pi$
$24$		0			0
$25$	−1.75994	−	4.68002i	−0.351988	−	0.936004i
$26$		0			0
$27$	3.62397			0.697432
$28$		0			0
$29$		−	7.74225i		−	1.43770i	−0.695166	−	0.718849i	$-0.744669\pi$
$29$		−	7.74225i		−	1.43770i	0.695166	−	0.718849i	$-0.255331\pi$
$30$		0			0
$31$	−4.11828			−0.739665			−0.369833	−	0.929098i	$-0.620585\pi$
$31$	−4.11828			−0.739665			−0.369833	+	0.929098i	$0.620585\pi$
$32$		0			0
$33$		−	10.9778i		−	1.91100i
$34$		0			0
$35$	−1.83847	−	1.27280i	−0.310758	−	0.215143i
$36$		0			0
$37$	0.480119			0.0789310			0.0394655	−	0.999221i	$-0.487434\pi$
$37$	0.480119			0.0789310			0.0394655	+	0.999221i	$0.487434\pi$
$38$		0			0
$39$	1.61780			0.259055
$40$		0			0
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$	2.00617			0.305937			0.152969	−	0.988231i	$-0.451117\pi$
$43$	2.00617			0.305937			0.152969	+	0.988231i	$0.451117\pi$
$44$		0			0
$45$	1.57835	−	2.27981i	0.235287	−	0.339854i
$46$		0			0
$47$		−	2.24006i		−	0.326746i	−0.986564	−	0.163373i	$-0.947763\pi$
$47$		−	2.24006i		−	0.326746i	0.986564	−	0.163373i	$-0.0522374\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$			3.33127i			0.466472i
$52$		0			0
$53$	−5.57133			−0.765282			−0.382641	−	0.923897i	$-0.624985\pi$
$53$	−5.57133			−0.765282			−0.382641	+	0.923897i	$0.624985\pi$
$54$		0			0
$55$	9.80139	+	6.78567i	1.32162	+	0.914979i
$56$		0			0
$57$		−	9.36004i		−	1.23977i
$58$		0			0
$59$			5.02573i			0.654294i	0.944973	+	0.327147i	$0.106087\pi$
$59$			5.02573i			0.654294i	−0.944973	+	0.327147i	$0.893913\pi$
$60$		0			0
$61$			4.55962i			0.583800i	0.956449	+	0.291900i	$0.0942874\pi$
$61$			4.55962i			0.583800i	−0.956449	+	0.291900i	$0.905713\pi$
$62$		0			0
$63$		−	1.24006i		−	0.156233i
$64$		0			0
$65$	−1.00000	+	1.44442i	−0.124035	+	0.179159i
$66$		0			0
$67$	−12.5956			−1.53880			−0.769401	−	0.638766i	$-0.779445\pi$
$67$	−12.5956			−1.53880			−0.769401	+	0.638766i	$0.779445\pi$
$68$		0			0
$69$			14.6018i			1.75785i
$70$		0			0
$71$	−13.4783			−1.59958			−0.799791	−	0.600278i	$-0.795057\pi$
$71$	−13.4783			−1.59958			−0.799791	+	0.600278i	$0.795057\pi$
$72$		0			0
$73$			2.00617i			0.234804i	0.993084	+	0.117402i	$0.0374566\pi$
$73$			2.00617i			0.234804i	−0.993084	+	0.117402i	$0.962543\pi$
$74$		0			0
$75$	3.62397	+	9.63682i	0.418459	+	1.11276i
$76$		0			0
$77$	5.33127			0.607555
$78$		0			0
$79$	6.61876			0.744669			0.372335	−	0.928099i	$-0.378557\pi$
$79$	6.61876			0.744669			0.372335	+	0.928099i	$0.378557\pi$
$80$		0			0
$81$	−11.1824			−1.24249
$82$		0			0
$83$	−8.91870			−0.978955			−0.489477	−	0.872016i	$-0.662812\pi$
$83$	−8.91870			−0.978955			−0.489477	+	0.872016i	$0.662812\pi$
$84$		0			0
$85$	−2.97427	−	2.05914i	−0.322605	−	0.223345i
$86$		0			0
$87$			15.9424i			1.70920i
$88$		0			0
$89$	0.428666			0.0454385			0.0227193	−	0.999742i	$-0.492768\pi$
$89$	0.428666			0.0454385			0.0227193	+	0.999742i	$0.492768\pi$
$90$		0			0
$91$			0.785667i			0.0823603i
$92$		0			0
$93$	8.48012			0.879347
$94$		0			0
$95$	8.35696	+	5.78567i	0.857406	+	0.593597i
$96$		0			0
$97$		−	13.0900i		−	1.32908i	−0.747251	−	0.664542i	$-0.768627\pi$
$97$		−	13.0900i		−	1.32908i	0.747251	−	0.664542i	$-0.231373\pi$
$98$		0			0
$99$			6.61110i			0.664440i
$100$		0			0
$101$			11.9135i			1.18544i	0.805409	+	0.592719i	$0.201945\pi$
$101$			11.9135i			1.18544i	−0.805409	+	0.592719i	$0.798055\pi$
$102$		0			0
$103$		−	11.3313i		−	1.11650i	−0.829672	−	0.558252i	$-0.811472\pi$
$103$		−	11.3313i		−	1.11650i	0.829672	−	0.558252i	$-0.188528\pi$
$104$		0			0
$105$	3.78567	+	2.62088i	0.369443	+	0.255772i
$106$		0			0
$107$	−17.5966			−1.70113			−0.850564	−	0.525871i	$-0.823739\pi$
$107$	−17.5966			−1.70113			−0.850564	+	0.525871i	$0.823739\pi$
$108$		0			0
$109$		−	3.72991i		−	0.357261i	−0.983916	−	0.178630i	$-0.942833\pi$
$109$		−	3.72991i		−	0.357261i	0.983916	−	0.178630i	$-0.0571667\pi$
$110$		0			0
$111$	−0.988632			−0.0938368
$112$		0			0
$113$		−	2.11211i		−	0.198691i	−0.995053	−	0.0993455i	$-0.968325\pi$
$113$		−	2.11211i		−	0.198691i	0.995053	−	0.0993455i	$-0.0316749\pi$
$114$		0			0
$115$	−13.0370	−	9.02573i	−1.21571	−	0.841653i
$116$		0			0
$117$	−0.974274			−0.0900716
$118$		0			0
$119$	−1.61780			−0.148303
$120$		0			0
$121$	−17.4225			−1.58386
$122$		0			0
$123$	−4.11828			−0.371333
$124$		0			0
$125$	−10.8441	−	2.72115i	−0.969929	−	0.243387i
$126$		0			0
$127$		−	10.2339i		−	0.908110i	−0.890974	−	0.454055i	$-0.849977\pi$
$127$		−	10.2339i		−	0.908110i	0.890974	−	0.454055i	$-0.150023\pi$
$128$		0			0
$129$	−4.13098			−0.363712
$130$		0			0
$131$		−	12.1684i		−	1.06316i	−0.847009	−	0.531579i	$-0.821599\pi$
$131$		−	12.1684i		−	1.06316i	0.847009	−	0.531579i	$-0.178401\pi$
$132$		0			0
$133$	4.54561			0.394154
$134$		0			0
$135$	4.61260	−	6.66255i	0.396989	−	0.573421i
$136$		0			0
$137$			18.7201i			1.59937i	0.600423	+	0.799683i	$0.294999\pi$
$137$			18.7201i			1.59937i	−0.600423	+	0.799683i	$0.705001\pi$
$138$		0			0
$139$			9.15670i			0.776662i	0.921520	+	0.388331i	$0.126948\pi$
$139$			9.15670i			0.776662i	−0.921520	+	0.388331i	$0.873052\pi$
$140$		0			0
$141$			4.61260i			0.388451i
$142$		0			0
$143$		−	4.18861i		−	0.350269i
$144$		0			0
$145$	−14.2339	−	9.85436i	−1.18206	−	0.818360i
$146$		0			0
$147$	2.05914			0.169835
$148$		0			0
$149$		−	18.7201i		−	1.53361i	−0.641880	−	0.766805i	$-0.721845\pi$
$149$		−	18.7201i		−	1.53361i	0.641880	−	0.766805i	$-0.278155\pi$
$150$		0			0
$151$	10.0952			0.821532			0.410766	−	0.911741i	$-0.365261\pi$
$151$	10.0952			0.821532			0.410766	+	0.911741i	$0.365261\pi$
$152$		0			0
$153$		−	2.00617i		−	0.162189i
$154$		0			0
$155$	−5.24176	+	7.57133i	−0.421029	+	0.608144i
$156$		0			0
$157$	−21.6883			−1.73091			−0.865456	−	0.500984i	$-0.832971\pi$
$157$	−21.6883			−1.73091			−0.865456	+	0.500984i	$0.832971\pi$
$158$		0			0
$159$	11.4722			0.909801
$160$		0			0
$161$	−7.09122			−0.558866
$162$		0			0
$163$	−11.7130			−0.917430			−0.458715	−	0.888584i	$-0.651690\pi$
$163$	−11.7130			−0.917430			−0.458715	+	0.888584i	$0.651690\pi$
$164$		0			0
$165$	−20.1824	−	13.9726i	−1.57120	−	1.08777i
$166$		0			0
$167$			18.9541i			1.46671i	0.679846	+	0.733355i	$0.262046\pi$
$167$			18.9541i			1.46671i	−0.679846	+	0.733355i	$0.737954\pi$
$168$		0			0
$169$	−12.3827			−0.952517
$170$		0			0
$171$			5.63682i			0.431059i
$172$		0			0
$173$	16.7857			1.27619			0.638095	−	0.769957i	$-0.279723\pi$
$173$	16.7857			1.27619			0.638095	+	0.769957i	$0.279723\pi$
$174$		0			0
$175$	−4.68002	+	1.75994i	−0.353776	+	0.133039i
$176$		0			0
$177$		−	10.3487i		−	0.777854i
$178$		0			0
$179$		−	24.6625i		−	1.84337i	−0.387944	−	0.921683i	$-0.626815\pi$
$179$		−	24.6625i		−	1.84337i	0.387944	−	0.921683i	$-0.373185\pi$
$180$		0			0
$181$			25.6326i			1.90526i	0.304135	+	0.952629i	$0.401633\pi$
$181$			25.6326i			1.90526i	−0.304135	+	0.952629i	$0.598367\pi$
$182$		0			0
$183$		−	9.38890i		−	0.694048i
$184$		0			0
$185$	0.611097	−	0.882683i	0.0449287	−	0.0648962i
$186$		0			0
$187$	8.62493			0.630717
$188$		0			0
$189$		−	3.62397i		−	0.263605i
$190$		0			0
$191$	−25.3389			−1.83345			−0.916727	−	0.399513i	$-0.869179\pi$
$191$	−25.3389			−1.83345			−0.916727	+	0.399513i	$0.869179\pi$
$192$		0			0
$193$		−	5.34771i		−	0.384937i	−0.981303	−	0.192468i	$-0.938351\pi$
$193$		−	5.34771i		−	0.384937i	0.981303	−	0.192468i	$-0.0616493\pi$
$194$		0			0
$195$	2.05914	−	2.97427i	0.147458	−	0.212992i
$196$		0			0
$197$	16.6111			1.18349			0.591746	−	0.806125i	$-0.298439\pi$
$197$	16.6111			1.18349			0.591746	+	0.806125i	$0.298439\pi$
$198$		0			0
$199$	−11.4722			−0.813240			−0.406620	−	0.913597i	$-0.633293\pi$
$199$	−11.4722			−0.813240			−0.406620	+	0.913597i	$0.633293\pi$
$200$		0			0
$201$	25.9362			1.82940
$202$		0			0
$203$	−7.74225			−0.543399
$204$		0			0
$205$	2.54561	−	3.67694i	0.177793	−	0.256809i
$206$		0			0
$207$		−	8.79353i		−	0.611192i
$208$		0			0
$209$	−24.2339			−1.67629
$210$		0			0
$211$		−	10.9026i		−	0.750567i	−0.926910	−	0.375283i	$-0.877545\pi$
$211$		−	10.9026i		−	0.750567i	0.926910	−	0.375283i	$-0.122455\pi$
$212$		0			0
$213$	27.7538			1.90166
$214$		0			0
$215$	2.55346	−	3.68828i	0.174144	−	0.251538i
$216$		0			0
$217$			4.11828i			0.279567i
$218$		0			0
$219$		−	4.13098i		−	0.279146i
$220$		0			0
$221$			1.27105i			0.0855001i
$222$		0			0
$223$			20.6564i			1.38325i	0.722255	+	0.691627i	$0.243106\pi$
$223$			20.6564i			1.38325i	−0.722255	+	0.691627i	$0.756894\pi$
$224$		0			0
$225$	−2.18243	−	5.80351i	−0.145495	−	0.386900i
$226$		0			0
$227$	7.06010			0.468596			0.234298	−	0.972165i	$-0.424721\pi$
$227$	7.06010			0.468596			0.234298	+	0.972165i	$0.424721\pi$
$228$		0			0
$229$		−	26.5153i		−	1.75218i	−0.482147	−	0.876090i	$-0.660143\pi$
$229$		−	26.5153i		−	1.75218i	0.482147	−	0.876090i	$-0.339857\pi$
$230$		0			0
$231$	−10.9778			−0.722289
$232$		0			0
$233$			14.7078i			0.963537i	0.876298	+	0.481769i	$0.160005\pi$
$233$			14.7078i			0.963537i	−0.876298	+	0.481769i	$0.839995\pi$
$234$		0			0
$235$	−4.11828	−	2.85116i	−0.268647	−	0.185989i
$236$		0			0
$237$	−13.6290			−0.885296
$238$		0			0
$239$	−1.85860			−0.120223			−0.0601114	−	0.998192i	$-0.519146\pi$
$239$	−1.85860			−0.120223			−0.0601114	+	0.998192i	$0.519146\pi$
$240$		0			0
$241$	−18.2339			−1.17455			−0.587274	−	0.809388i	$-0.699799\pi$
$241$	−18.2339			−1.17455			−0.587274	+	0.809388i	$0.699799\pi$
$242$		0			0
$243$	12.1543			0.779699
$244$		0			0
$245$	−1.27280	+	1.83847i	−0.0813165	+	0.117456i
$246$		0			0
$247$		−	3.57133i		−	0.227239i
$248$		0			0
$249$	18.3649			1.16383
$250$		0			0
$251$			5.88306i			0.371335i	0.982613	+	0.185668i	$0.0594448\pi$
$251$			5.88306i			0.371335i	−0.982613	+	0.185668i	$0.940555\pi$
$252$		0			0
$253$	37.8052			2.37679
$254$		0			0
$255$	6.12445	+	4.24006i	0.383528	+	0.265523i
$256$		0			0
$257$		−	1.12348i		−	0.0700809i	−0.999386	−	0.0350405i	$-0.988844\pi$
$257$		−	1.12348i		−	0.0700809i	0.999386	−	0.0350405i	$-0.0111560\pi$
$258$		0			0
$259$		−	0.480119i		−	0.0298331i
$260$		0			0
$261$		−	9.60084i		−	0.594277i
$262$		0			0
$263$		−	15.0398i		−	0.927392i	−0.885994	−	0.463696i	$-0.846523\pi$
$263$		−	15.0398i		−	0.927392i	0.885994	−	0.463696i	$-0.153477\pi$
$264$		0			0
$265$	−7.09122	+	10.2427i	−0.435610	+	0.629206i
$266$		0			0
$267$	−0.882683			−0.0540193
$268$		0			0
$269$			17.3961i			1.06066i	0.847792	+	0.530328i	$0.177931\pi$
$269$			17.3961i			1.06066i	−0.847792	+	0.530328i	$0.822069\pi$
$270$		0			0
$271$	−12.2489			−0.744067			−0.372034	−	0.928219i	$-0.621339\pi$
$271$	−12.2489			−0.744067			−0.372034	+	0.928219i	$0.621339\pi$
$272$		0			0
$273$		−	1.61780i		−	0.0979136i
$274$		0			0
$275$	24.9505	−	9.38273i	1.50457	−	0.565800i
$276$		0			0
$277$	−30.8964			−1.85639			−0.928193	−	0.372098i	$-0.878638\pi$
$277$	−30.8964			−1.85639			−0.928193	+	0.372098i	$0.878638\pi$
$278$		0			0
$279$	−5.10691			−0.305743
$280$		0			0
$281$	17.5652			1.04785			0.523925	−	0.851765i	$-0.324467\pi$
$281$	17.5652			1.04785			0.523925	+	0.851765i	$0.324467\pi$
$282$		0			0
$283$	−6.17742			−0.367210			−0.183605	−	0.983000i	$-0.558777\pi$
$283$	−6.17742			−0.367210			−0.183605	+	0.983000i	$0.558777\pi$
$284$		0			0
$285$	−17.2082	−	11.9135i	−1.01932	−	0.705695i
$286$		0			0
$287$		−	2.00000i		−	0.118056i
$288$		0			0
$289$	14.3827			0.846043
$290$		0			0
$291$			26.9541i			1.58008i
$292$		0			0
$293$	−3.31724			−0.193795			−0.0968976	−	0.995294i	$-0.530892\pi$
$293$	−3.31724			−0.193795			−0.0968976	+	0.995294i	$0.530892\pi$
$294$		0			0
$295$	9.23964	+	6.39676i	0.537953	+	0.372434i
$296$		0			0
$297$			19.3204i			1.12108i
$298$		0			0
$299$			5.57133i			0.322199i
$300$		0			0
$301$		−	2.00617i		−	0.115634i
$302$		0			0
$303$		−	24.5316i		−	1.40930i
$304$		0			0
$305$	8.38273	+	5.80351i	0.479994	+	0.332308i
$306$		0			0
$307$	−18.4263			−1.05165			−0.525823	−	0.850594i	$-0.676242\pi$
$307$	−18.4263			−1.05165			−0.525823	+	0.850594i	$0.676242\pi$
$308$		0			0
$309$			23.3327i			1.32735i
$310$		0			0
$311$	−3.34155			−0.189482			−0.0947408	−	0.995502i	$-0.530202\pi$
$311$	−3.34155			−0.189482			−0.0947408	+	0.995502i	$0.530202\pi$
$312$		0			0
$313$		−	2.50048i		−	0.141336i	−0.997500	−	0.0706678i	$-0.977487\pi$
$313$		−	2.50048i		−	0.141336i	0.997500	−	0.0706678i	$-0.0225130\pi$
$314$		0			0
$315$	−2.27981	−	1.57835i	−0.128453	−	0.0889301i
$316$		0			0
$317$	−10.9088			−0.612698			−0.306349	−	0.951919i	$-0.599108\pi$
$317$	−10.9088			−0.612698			−0.306349	+	0.951919i	$0.599108\pi$
$318$		0			0
$319$	41.2760			2.31101
$320$		0			0
$321$	36.2339			2.02238
$322$		0			0
$323$	7.35388			0.409181
$324$		0			0
$325$	1.38273	+	3.67694i	0.0766999	+	0.203960i
$326$		0			0
$327$			7.68042i			0.424728i
$328$		0			0
$329$	−2.24006			−0.123498
$330$		0			0
$331$			1.75376i			0.0963956i	0.998838	+	0.0481978i	$0.0153478\pi$
$331$			1.75376i			0.0963956i	−0.998838	+	0.0481978i	$0.984652\pi$
$332$		0			0
$333$	0.595376			0.0326264
$334$		0			0
$335$	−16.0318	+	23.1567i	−0.875910	+	1.26519i
$336$		0			0
$337$		−	2.11211i		−	0.115054i	−0.998344	−	0.0575271i	$-0.981678\pi$
$337$		−	2.11211i		−	0.115054i	0.998344	−	0.0575271i	$-0.0183216\pi$
$338$		0			0
$339$			4.34914i			0.236213i
$340$		0			0
$341$		−	21.9557i		−	1.18897i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	26.8450	+	18.5852i	1.44529	+	1.00060i
$346$		0			0
$347$	28.1861			1.51311			0.756554	−	0.653931i	$-0.226881\pi$
$347$	28.1861			1.51311			0.756554	+	0.653931i	$0.226881\pi$
$348$		0			0
$349$			10.1481i			0.543217i	0.962408	+	0.271609i	$0.0875556\pi$
$349$			10.1481i			0.543217i	−0.962408	+	0.271609i	$0.912444\pi$
$350$		0			0
$351$	−2.84723			−0.151974
$352$		0			0
$353$		−	21.2206i		−	1.12946i	−0.825277	−	0.564729i	$-0.808981\pi$
$353$		−	21.2206i		−	1.12946i	0.825277	−	0.564729i	$-0.191019\pi$
$354$		0			0
$355$	−17.1553	+	24.7795i	−0.910507	+	1.31516i
$356$		0			0
$357$	3.33127			0.176310
$358$		0			0
$359$	−9.01330			−0.475704			−0.237852	−	0.971301i	$-0.576443\pi$
$359$	−9.01330			−0.475704			−0.237852	+	0.971301i	$0.576443\pi$
$360$		0			0
$361$	−1.66255			−0.0875026
$362$		0			0
$363$	35.8754			1.88297
$364$		0			0
$365$	3.68828	+	2.55346i	0.193053	+	0.133654i
$366$		0			0
$367$			9.14884i			0.477566i	0.971073	+	0.238783i	$0.0767484\pi$
$367$			9.14884i			0.477566i	−0.971073	+	0.238783i	$0.923252\pi$
$368$		0			0
$369$	2.48012			0.129110
$370$		0			0
$371$			5.57133i			0.289249i
$372$		0			0
$373$	31.2736			1.61929			0.809644	−	0.586921i	$-0.199660\pi$
$373$	31.2736			1.61929			0.809644	+	0.586921i	$0.199660\pi$
$374$		0			0
$375$	22.3296	+	5.60324i	1.15310	+	0.289350i
$376$		0			0
$377$			6.08283i			0.313282i
$378$		0			0
$379$			2.61110i			0.134123i	0.997749	+	0.0670615i	$0.0213624\pi$
$379$			2.61110i			0.134123i	−0.997749	+	0.0670615i	$0.978638\pi$
$380$		0			0
$381$			21.0730i			1.07960i
$382$		0			0
$383$		−	33.3251i		−	1.70283i	−0.524490	−	0.851417i	$-0.675744\pi$
$383$		−	33.3251i		−	1.70283i	0.524490	−	0.851417i	$-0.324256\pi$
$384$		0			0
$385$	6.78567	−	9.80139i	0.345830	−	0.499525i
$386$		0			0
$387$	2.48777			0.126460
$388$		0			0
$389$		−	23.4386i		−	1.18839i	−0.804323	−	0.594193i	$-0.797472\pi$
$389$		−	23.4386i		−	1.18839i	0.804323	−	0.594193i	$-0.202528\pi$
$390$		0			0
$391$	−11.4722			−0.580172
$392$		0			0
$393$			25.0564i			1.26393i
$394$		0			0
$395$	8.42439	−	12.1684i	0.423877	−	0.612258i
$396$		0			0
$397$	−19.3453			−0.970913			−0.485457	−	0.874261i	$-0.661347\pi$
$397$	−19.3453			−0.970913			−0.485457	+	0.874261i	$0.661347\pi$
$398$		0			0
$399$	−9.36004			−0.468588
$400$		0			0
$401$	−13.3827			−0.668302			−0.334151	−	0.942520i	$-0.608449\pi$
$401$	−13.3827			−0.668302			−0.334151	+	0.942520i	$0.608449\pi$
$402$		0			0
$403$	3.23560			0.161177
$404$		0			0
$405$	−14.2330	+	20.5586i	−0.707245	+	1.02156i
$406$		0			0
$407$			2.55964i			0.126877i
$408$		0			0
$409$	24.7655			1.22457			0.612286	−	0.790636i	$-0.290250\pi$
$409$	24.7655			1.22457			0.612286	+	0.790636i	$0.290250\pi$
$410$		0			0
$411$		−	38.5473i		−	1.90140i
$412$		0			0
$413$	5.02573			0.247300
$414$		0			0
$415$	−11.3518	+	16.3968i	−0.557236	+	0.804885i
$416$		0			0
$417$		−	18.8549i		−	0.923330i
$418$		0			0
$419$		−	20.0655i		−	0.980263i	−0.871648	−	0.490132i	$-0.836949\pi$
$419$		−	20.0655i		−	0.980263i	0.871648	−	0.490132i	$-0.163051\pi$
$420$		0			0
$421$			15.0961i			0.735740i	0.929877	+	0.367870i	$0.119913\pi$
$421$			15.0961i			0.735740i	−0.929877	+	0.367870i	$0.880087\pi$
$422$		0			0
$423$		−	2.77781i		−	0.135062i
$424$		0			0
$425$	−7.57133	+	2.84723i	−0.367264	+	0.138111i
$426$		0			0
$427$	4.55962			0.220656
$428$		0			0
$429$			8.62493i			0.416416i
$430$		0			0
$431$	33.0395			1.59146			0.795728	−	0.605654i	$-0.207089\pi$
$431$	33.0395			1.59146			0.795728	+	0.605654i	$0.207089\pi$
$432$		0			0
$433$			10.2427i			0.492234i	0.969240	+	0.246117i	$0.0791548\pi$
$433$			10.2427i			0.492234i	−0.969240	+	0.246117i	$0.920845\pi$
$434$		0			0
$435$	29.3096	+	20.2915i	1.40529	+	0.972904i
$436$		0			0
$437$	32.2339			1.54196
$438$		0			0
$439$	−14.6018			−0.696906			−0.348453	−	0.937326i	$-0.613293\pi$
$439$	−14.6018			−0.696906			−0.348453	+	0.937326i	$0.613293\pi$
$440$		0			0
$441$	−1.24006			−0.0590504
$442$		0			0
$443$	32.3044			1.53483			0.767413	−	0.641153i	$-0.221543\pi$
$443$	32.3044			1.53483			0.767413	+	0.641153i	$0.221543\pi$
$444$		0			0
$445$	0.545608	−	0.788089i	0.0258643	−	0.0373590i
$446$		0			0
$447$			38.5473i			1.82322i
$448$		0			0
$449$	5.94237			0.280438			0.140219	−	0.990121i	$-0.455219\pi$
$449$	5.94237			0.280438			0.140219	+	0.990121i	$0.455219\pi$
$450$		0			0
$451$			10.6625i			0.502080i
$452$		0			0
$453$	−20.7874			−0.976675
$454$		0			0
$455$	1.44442	+	1.00000i	0.0677157	+	0.0468807i
$456$		0			0
$457$		−	30.8341i		−	1.44236i	−0.692748	−	0.721180i	$-0.743600\pi$
$457$		−	30.8341i		−	1.44236i	0.692748	−	0.721180i	$-0.256400\pi$
$458$		0			0
$459$		−	5.86285i		−	0.273654i
$460$		0			0
$461$			11.0308i			0.513756i	0.966444	+	0.256878i	$0.0826939\pi$
$461$			11.0308i			0.513756i	−0.966444	+	0.256878i	$0.917306\pi$
$462$		0			0
$463$		−	23.0398i		−	1.07075i	−0.844615	−	0.535374i	$-0.820171\pi$
$463$		−	23.0398i		−	1.07075i	0.844615	−	0.535374i	$-0.179829\pi$
$464$		0			0
$465$	10.7935	−	15.5904i	0.500538	−	0.722989i
$466$		0			0
$467$	32.3573			1.49732			0.748659	−	0.662955i	$-0.230698\pi$
$467$	32.3573			1.49732			0.748659	+	0.662955i	$0.230698\pi$
$468$		0			0
$469$			12.5956i			0.581613i
$470$		0			0
$471$	44.6592			2.05779
$472$		0			0
$473$			10.6954i			0.491776i
$474$		0			0
$475$	21.2735	−	8.00000i	0.976097	−	0.367065i
$476$		0			0
$477$	−6.90878			−0.316332
$478$		0			0
$479$	24.6037			1.12417			0.562087	−	0.827078i	$-0.309999\pi$
$479$	24.6037			1.12417			0.562087	+	0.827078i	$0.309999\pi$
$480$		0			0
$481$	−0.377213			−0.0171994
$482$		0			0
$483$	14.6018			0.664405
$484$		0			0
$485$	−24.0655	−	16.6609i	−1.09276	−	0.756535i
$486$		0			0
$487$		−	2.23388i		−	0.101227i	−0.998718	−	0.0506135i	$-0.983882\pi$
$487$		−	2.23388i		−	0.101227i	0.998718	−	0.0506135i	$-0.0161177\pi$
$488$		0			0
$489$	24.1186			1.09068
$490$		0			0
$491$		−	39.1646i		−	1.76747i	−0.467986	−	0.883736i	$-0.655020\pi$
$491$		−	39.1646i		−	1.76747i	0.467986	−	0.883736i	$-0.344980\pi$
$492$		0			0
$493$	−12.5254			−0.564115
$494$		0			0
$495$	12.1543	+	8.41463i	0.546295	+	0.378209i
$496$		0			0
$497$			13.4783i			0.604586i
$498$		0			0
$499$			5.09739i			0.228191i	0.993470	+	0.114095i	$0.0363969\pi$
$499$			5.09739i			0.228191i	−0.993470	+	0.114095i	$0.963603\pi$
$500$		0			0
$501$		−	39.0291i		−	1.74369i
$502$		0			0
$503$		−	16.9026i		−	0.753650i	−0.926284	−	0.376825i	$-0.877016\pi$
$503$		−	16.9026i		−	0.753650i	0.926284	−	0.376825i	$-0.122984\pi$
$504$		0			0
$505$	21.9026	+	15.1635i	0.974653	+	0.674769i
$506$		0			0
$507$	25.4978			1.13240
$508$		0			0
$509$			5.33636i			0.236530i	0.992982	+	0.118265i	$0.0377332\pi$
$509$			5.33636i			0.236530i	−0.992982	+	0.118265i	$0.962267\pi$
$510$		0			0
$511$	2.00617			0.0887476
$512$		0			0
$513$			16.4731i			0.727306i
$514$		0			0
$515$	−20.8322	−	14.4225i	−0.917977	−	0.635531i
$516$		0			0
$517$	11.9424			0.525225
$518$		0			0
$519$	−34.5640			−1.51719
$520$		0			0
$521$	22.7421			0.996348			0.498174	−	0.867077i	$-0.334004\pi$
$521$	22.7421			0.996348			0.498174	+	0.867077i	$0.334004\pi$
$522$		0			0
$523$	−35.8754			−1.56872			−0.784360	−	0.620306i	$-0.787008\pi$
$523$	−35.8754			−1.56872			−0.784360	+	0.620306i	$0.787008\pi$
$524$		0			0
$525$	9.63682	−	3.62397i	0.420585	−	0.158163i
$526$		0			0
$527$			6.66255i			0.290225i
$528$		0			0
$529$	−27.2853			−1.18632
$530$		0			0
$531$			6.23220i			0.270454i
$532$		0			0
$533$	−1.57133			−0.0680620
$534$		0			0
$535$	−22.3970	+	32.3508i	−0.968308	+	1.39865i
$536$		0			0
$537$			50.7837i			2.19148i
$538$		0			0
$539$		−	5.33127i		−	0.229634i
$540$		0			0
$541$			3.62397i			0.155806i	0.996961	+	0.0779032i	$0.0248225\pi$
$541$			3.62397i			0.155806i	−0.996961	+	0.0779032i	$0.975177\pi$
$542$		0			0
$543$		−	52.7812i		−	2.26506i
$544$		0			0
$545$	−6.85733	−	4.74745i	−0.293736	−	0.203358i
$546$		0			0
$547$	−12.7016			−0.543081			−0.271540	−	0.962427i	$-0.587533\pi$
$547$	−12.7016			−0.543081			−0.271540	+	0.962427i	$0.587533\pi$
$548$		0			0
$549$			5.65420i			0.241315i
$550$		0			0
$551$	35.1932			1.49928
$552$		0			0
$553$		−	6.61876i		−	0.281458i
$554$		0			0
$555$	−1.25833	+	1.81757i	−0.0534133	+	0.0771515i
$556$		0			0
$557$	−22.3134			−0.945449			−0.472725	−	0.881210i	$-0.656730\pi$
$557$	−22.3134			−0.945449			−0.472725	+	0.881210i	$0.656730\pi$
$558$		0			0
$559$	−1.57618			−0.0666653
$560$		0			0
$561$	−17.7599			−0.749825
$562$		0			0
$563$	22.7438			0.958536			0.479268	−	0.877669i	$-0.340902\pi$
$563$	22.7438			0.958536			0.479268	+	0.877669i	$0.340902\pi$
$564$		0			0
$565$	−3.88306	−	2.68831i	−0.163361	−	0.113098i
$566$		0			0
$567$			11.1824i			0.469618i
$568$		0			0
$569$	11.5199			0.482939			0.241469	−	0.970408i	$-0.422371\pi$
$569$	11.5199			0.482939			0.241469	+	0.970408i	$0.422371\pi$
$570$		0			0
$571$			13.6509i			0.571271i	0.958338	+	0.285635i	$0.0922046\pi$
$571$			13.6509i			0.571271i	−0.958338	+	0.285635i	$0.907795\pi$
$572$		0			0
$573$	52.1763			2.17969
$574$		0			0
$575$	−33.1870	+	12.4801i	−1.38400	+	0.520457i
$576$		0			0
$577$			41.7060i			1.73624i	0.496350	+	0.868122i	$0.334673\pi$
$577$			41.7060i			1.73624i	−0.496350	+	0.868122i	$0.665327\pi$
$578$		0			0
$579$			11.0117i			0.457630i
$580$		0			0
$581$			8.91870i			0.370010i
$582$		0			0
$583$		−	29.7023i		−	1.23014i
$584$		0			0
$585$	−1.24006	+	1.79117i	−0.0512701	+	0.0740559i
$586$		0			0
$587$	−17.1553			−0.708074			−0.354037	−	0.935232i	$-0.615191\pi$
$587$	−17.1553			−0.708074			−0.354037	+	0.935232i	$0.615191\pi$
$588$		0			0
$589$		−	18.7201i		−	0.771348i
$590$		0			0
$591$	−34.2046			−1.40699
$592$		0			0
$593$			31.7041i			1.30193i	0.759107	+	0.650966i	$0.225636\pi$
$593$			31.7041i			1.30193i	−0.759107	+	0.650966i	$0.774364\pi$
$594$		0			0
$595$	−2.05914	+	2.97427i	−0.0844165	+	0.121933i
$596$		0			0
$597$	23.6228			0.966816
$598$		0			0
$599$	18.5725			0.758853			0.379426	−	0.925222i	$-0.376121\pi$
$599$	18.5725			0.758853			0.379426	+	0.925222i	$0.376121\pi$
$600$		0			0
$601$	12.4287			0.506976			0.253488	−	0.967339i	$-0.418422\pi$
$601$	12.4287			0.506976			0.253488	+	0.967339i	$0.418422\pi$
$602$		0			0
$603$	−15.6193			−0.636069
$604$		0			0
$605$	−22.1754	+	32.0307i	−0.901559	+	1.30223i
$606$		0			0
$607$		−	18.4739i		−	0.749834i	−0.927058	−	0.374917i	$-0.877671\pi$
$607$		−	18.4739i		−	0.749834i	0.927058	−	0.374917i	$-0.122329\pi$
$608$		0			0
$609$	15.9424			0.646017
$610$		0			0
$611$			1.75994i			0.0711996i
$612$		0			0
$613$	−13.3251			−0.538196			−0.269098	−	0.963113i	$-0.586725\pi$
$613$	−13.3251			−0.538196			−0.269098	+	0.963113i	$0.586725\pi$
$614$		0			0
$615$	−5.24176	+	7.57133i	−0.211368	+	0.305306i
$616$		0			0
$617$			29.5504i			1.18965i	0.803854	+	0.594826i	$0.202779\pi$
$617$			29.5504i			1.18965i	−0.803854	+	0.594826i	$0.797221\pi$
$618$		0			0
$619$		−	23.3391i		−	0.938079i	−0.883177	−	0.469040i	$-0.844600\pi$
$619$		−	23.3391i		−	0.938079i	0.883177	−	0.469040i	$-0.155400\pi$
$620$		0			0
$621$		−	25.6983i		−	1.03124i
$622$		0			0
$623$		−	0.428666i		−	0.0171741i
$624$		0			0
$625$	−18.8052	+	16.4731i	−0.752209	+	0.658925i
$626$		0			0
$627$	49.9010			1.99285
$628$		0			0
$629$		−	0.776735i		−	0.0309705i
$630$		0			0
$631$	2.60643			0.103760			0.0518802	−	0.998653i	$-0.483479\pi$
$631$	2.60643			0.103760			0.0518802	+	0.998653i	$0.483479\pi$
$632$		0			0
$633$			22.4500i			0.892307i
$634$		0			0
$635$	−18.8147	−	13.0257i	−0.746638	−	0.516910i
$636$		0			0
$637$	0.785667			0.0311293
$638$		0			0
$639$	−16.7139			−0.661193
$640$		0			0
$641$	15.5199			0.612998			0.306499	−	0.951871i	$-0.400842\pi$
$641$	15.5199			0.612998			0.306499	+	0.951871i	$0.400842\pi$
$642$		0			0
$643$	−24.0148			−0.947052			−0.473526	−	0.880780i	$-0.657019\pi$
$643$	−24.0148			−0.947052			−0.473526	+	0.880780i	$0.657019\pi$
$644$		0			0
$645$	−5.25792	+	7.59468i	−0.207031	+	0.299040i
$646$		0			0
$647$		−	6.90878i		−	0.271612i	−0.990735	−	0.135806i	$-0.956638\pi$
$647$		−	6.90878i		−	0.271612i	0.990735	−	0.135806i	$-0.0433624\pi$
$648$		0			0
$649$	−26.7935			−1.05174
$650$		0			0
$651$		−	8.48012i		−	0.332362i
$652$		0			0
$653$	−34.6502			−1.35597			−0.677984	−	0.735077i	$-0.737146\pi$
$653$	−34.6502			−1.35597			−0.677984	+	0.735077i	$0.737146\pi$
$654$		0			0
$655$	−22.3712	−	15.4880i	−0.874116	−	0.605165i
$656$		0			0
$657$			2.48777i			0.0970570i
$658$		0			0
$659$		−	19.5137i		−	0.760146i	−0.924956	−	0.380073i	$-0.875899\pi$
$659$		−	19.5137i		−	0.760146i	0.924956	−	0.380073i	$-0.124101\pi$
$660$		0			0
$661$		−	34.6459i		−	1.34757i	−0.738927	−	0.673785i	$-0.764667\pi$
$661$		−	34.6459i		−	1.34757i	0.738927	−	0.673785i	$-0.235333\pi$
$662$		0			0
$663$		−	2.61727i		−	0.101646i
$664$		0			0
$665$	5.78567	−	8.35696i	0.224359	−	0.324069i
$666$		0			0
$667$	−54.9019			−2.12581
$668$		0			0
$669$		−	42.5344i		−	1.64447i
$670$		0			0
$671$	−24.3086			−0.938423
$672$		0			0
$673$		−	45.6767i		−	1.76071i	−0.474316	−	0.880354i	$-0.657305\pi$
$673$		−	45.6767i		−	1.76071i	0.474316	−	0.880354i	$-0.342695\pi$
$674$		0			0
$675$	−6.37796	−	16.9602i	−0.245488	−	0.652800i
$676$		0			0
$677$	28.3055			1.08787			0.543935	−	0.839127i	$-0.316934\pi$
$677$	28.3055			1.08787			0.543935	+	0.839127i	$0.316934\pi$
$678$		0			0
$679$	−13.0900			−0.502346
$680$		0			0
$681$	−14.5377			−0.557088
$682$		0			0
$683$	−1.12348			−0.0429889			−0.0214944	−	0.999769i	$-0.506842\pi$
$683$	−1.12348			−0.0429889			−0.0214944	+	0.999769i	$0.506842\pi$
$684$		0			0
$685$	34.4163	+	23.8270i	1.31498	+	0.910383i
$686$		0			0
$687$			54.5987i			2.08307i
$688$		0			0
$689$	4.37721			0.166759
$690$		0			0
$691$		−	29.5058i		−	1.12246i	−0.827661	−	0.561228i	$-0.810329\pi$
$691$		−	29.5058i		−	1.12246i	0.827661	−	0.561228i	$-0.189671\pi$
$692$		0			0
$693$	6.61110			0.251135
$694$		0			0
$695$	16.8343	+	11.6547i	0.638562	+	0.442088i
$696$		0			0
$697$		−	3.23560i		−	0.122557i
$698$		0			0
$699$		−	30.2853i		−	1.14550i
$700$		0			0
$701$		−	16.7555i		−	0.632848i	−0.948618	−	0.316424i	$-0.897518\pi$
$701$		−	16.7555i		−	0.632848i	0.948618	−	0.316424i	$-0.102482\pi$
$702$		0			0
$703$			2.18243i			0.0823119i
$704$		0			0
$705$	8.48012	+	5.87093i	0.319380	+	0.221112i
$706$		0			0
$707$	11.9135			0.448053
$708$		0			0
$709$			14.9902i			0.562968i	0.959566	+	0.281484i	$0.0908266\pi$
$709$			14.9902i			0.562968i	−0.959566	+	0.281484i	$0.909173\pi$
$710$		0			0
$711$	8.20766			0.307811
$712$		0			0
$713$			29.2036i			1.09368i
$714$		0			0
$715$	−7.70063	−	5.33127i	−0.287987	−	0.199378i
$716$		0			0
$717$	3.82711			0.142926
$718$		0			0
$719$	33.5338			1.25060			0.625300	−	0.780385i	$-0.284977\pi$
$719$	33.5338			1.25060			0.625300	+	0.780385i	$0.284977\pi$
$720$		0			0
$721$	−11.3313			−0.421999
$722$		0			0
$723$	37.5461			1.39636
$724$		0			0
$725$	−36.2339	+	13.6259i	−1.34569	+	0.506053i
$726$		0			0
$727$		−	17.1941i		−	0.637695i	−0.947806	−	0.318847i	$-0.896704\pi$
$727$		−	17.1941i		−	0.637695i	0.947806	−	0.318847i	$-0.103296\pi$
$728$		0			0
$729$	8.51988			0.315551
$730$		0			0
$731$		−	3.24557i		−	0.120042i
$732$		0			0
$733$	39.0991			1.44416			0.722079	−	0.691811i	$-0.243187\pi$
$733$	39.0991			1.44416			0.722079	+	0.691811i	$0.243187\pi$
$734$		0			0
$735$	2.62088	−	3.78567i	0.0966727	−	0.139636i
$736$		0			0
$737$		−	67.1508i		−	2.47353i
$738$		0			0
$739$			25.2160i			0.927586i	0.885944	+	0.463793i	$0.153512\pi$
$739$			25.2160i			0.927586i	−0.885944	+	0.463793i	$0.846488\pi$
$740$		0			0
$741$			7.35388i			0.270151i
$742$		0			0
$743$			15.0912i			0.553643i	0.960921	+	0.276821i	$0.0892811\pi$
$743$			15.0912i			0.553643i	−0.960921	+	0.276821i	$0.910719\pi$
$744$		0			0
$745$	−34.4163	−	23.8270i	−1.26092	−	0.872954i
$746$		0			0
$747$	−11.0597			−0.404654
$748$		0			0
$749$			17.5966i			0.642966i
$750$		0			0
$751$	20.8738			0.761697			0.380848	−	0.924638i	$-0.375632\pi$
$751$	20.8738			0.761697			0.380848	+	0.924638i	$0.375632\pi$
$752$		0			0
$753$		−	12.1140i		−	0.441460i
$754$		0			0
$755$	12.8492	−	18.5596i	0.467629	−	0.675455i
$756$		0			0
$757$	−43.5075			−1.58131			−0.790654	−	0.612263i	$-0.790259\pi$
$757$	−43.5075			−1.58131			−0.790654	+	0.612263i	$0.790259\pi$
$758$		0			0
$759$	−77.8463			−2.82564
$760$		0			0
$761$	−6.85733			−0.248578			−0.124289	−	0.992246i	$-0.539665\pi$
$761$	−6.85733			−0.248578			−0.124289	+	0.992246i	$0.539665\pi$
$762$		0			0
$763$	−3.72991			−0.135032
$764$		0			0
$765$	−3.68828	−	2.55346i	−0.133350	−	0.0923204i
$766$		0			0
$767$		−	3.94855i		−	0.142574i
$768$		0			0
$769$	−49.2613			−1.77641			−0.888204	−	0.459450i	$-0.848047\pi$
$769$	−49.2613			−1.77641			−0.888204	+	0.459450i	$0.848047\pi$
$770$		0			0
$771$			2.31341i			0.0833154i
$772$		0			0
$773$	46.3446			1.66690			0.833451	−	0.552594i	$-0.186362\pi$
$773$	46.3446			1.66690			0.833451	+	0.552594i	$0.186362\pi$
$774$		0			0
$775$	7.24793	+	19.2736i	0.260353	+	0.692330i
$776$		0			0
$777$			0.988632i			0.0354670i
$778$		0			0
$779$			9.09122i			0.325727i
$780$		0			0
$781$		−	71.8567i		−	2.57123i
$782$		0			0
$783$		−	28.0576i		−	1.00270i
$784$		0			0
$785$	−27.6049	+	39.8732i	−0.985262	+	1.42314i
$786$		0			0
$787$	11.1784			0.398466			0.199233	−	0.979952i	$-0.436155\pi$
$787$	11.1784			0.398466			0.199233	+	0.979952i	$0.436155\pi$
$788$		0			0
$789$			30.9690i			1.10253i
$790$		0			0
$791$	−2.11211			−0.0750981
$792$		0			0
$793$		−	3.58235i		−	0.127213i
$794$		0			0
$795$	14.6018	−	21.0912i	0.517873	−	0.748028i
$796$		0			0
$797$	13.8488			0.490550			0.245275	−	0.969454i	$-0.421122\pi$
$797$	13.8488			0.490550			0.245275	+	0.969454i	$0.421122\pi$
$798$		0			0
$799$	−3.62397			−0.128207
$800$		0			0
$801$	0.531571			0.0187821
$802$		0			0
$803$	−10.6954			−0.377433
$804$		0			0
$805$	−9.02573	+	13.0370i	−0.318115	+	0.459493i
$806$		0			0
$807$		−	35.8209i		−	1.26096i
$808$		0			0
$809$	−7.09739			−0.249531			−0.124765	−	0.992186i	$-0.539818\pi$
$809$	−7.09739			−0.249531			−0.124765	+	0.992186i	$0.539818\pi$
$810$		0			0
$811$			46.2479i			1.62398i	0.583668	+	0.811992i	$0.301617\pi$
$811$			46.2479i			1.62398i	−0.583668	+	0.811992i	$0.698383\pi$
$812$		0			0
$813$	25.2222			0.884581
$814$		0			0
$815$	−14.9083	+	21.5339i	−0.522215	+	0.754300i
$816$		0			0
$817$			9.11925i			0.319042i
$818$		0			0
$819$			0.974274i			0.0340439i
$820$		0			0
$821$		−	21.0857i		−	0.735897i	−0.929846	−	0.367948i	$-0.880060\pi$
$821$		−	21.0857i		−	0.735897i	0.929846	−	0.367948i	$-0.119940\pi$
$822$		0			0
$823$		−	8.37721i		−	0.292011i	−0.989284	−	0.146006i	$-0.953358\pi$
$823$		−	8.37721i		−	0.292011i	0.989284	−	0.146006i	$-0.0466418\pi$
$824$		0			0
$825$	−51.3766	+	19.3204i	−1.78870	+	0.672648i
$826$		0			0
$827$	−26.6099			−0.925317			−0.462659	−	0.886537i	$-0.653104\pi$
$827$	−26.6099			−0.925317			−0.462659	+	0.886537i	$0.653104\pi$
$828$		0			0
$829$			9.15950i			0.318123i	0.987269	+	0.159061i	$0.0508468\pi$
$829$			9.15950i			0.318123i	−0.987269	+	0.159061i	$0.949153\pi$
$830$		0			0
$831$	63.6201			2.20696
$832$		0			0
$833$			1.61780i			0.0560534i
$834$		0			0
$835$	34.8465	+	24.1248i	1.20591	+	0.834873i
$836$		0			0
$837$	−14.9245			−0.515866
$838$		0			0
$839$	−13.7191			−0.473637			−0.236818	−	0.971554i	$-0.576105\pi$
$839$	−13.7191			−0.473637			−0.236818	+	0.971554i	$0.576105\pi$
$840$		0			0
$841$	−30.9424			−1.06698
$842$		0			0
$843$	−36.1691			−1.24573
$844$		0			0
$845$	−15.7608	+	22.7653i	−0.542187	+	0.783149i
$846$		0			0
$847$			17.4225i			0.598644i
$848$		0			0
$849$	12.7202			0.436555
$850$		0			0
$851$		−	3.40462i		−	0.116709i
$852$		0			0
$853$	20.7000			0.708753			0.354377	−	0.935103i	$-0.384693\pi$
$853$	20.7000			0.708753			0.354377	+	0.935103i	$0.384693\pi$
$854$		0			0
$855$	10.3631	+	7.17457i	0.354411	+	0.245365i
$856$		0			0
$857$		−	3.37045i		−	0.115132i	−0.998342	−	0.0575662i	$-0.981666\pi$
$857$		−	3.37045i		−	0.115132i	0.998342	−	0.0575662i	$-0.0183340\pi$
$858$		0			0
$859$		−	3.45439i		−	0.117862i	−0.998262	−	0.0589312i	$-0.981231\pi$
$859$		−	3.45439i		−	0.117862i	0.998262	−	0.0589312i	$-0.0187692\pi$
$860$		0			0
$861$			4.11828i			0.140351i
$862$		0			0
$863$			49.8052i			1.69539i	0.530484	+	0.847695i	$0.322010\pi$
$863$			49.8052i			1.69539i	−0.530484	+	0.847695i	$0.677990\pi$
$864$		0			0
$865$	21.3649	−	30.8599i	0.726427	−	1.04927i
$866$		0			0
$867$	−29.6161			−1.00581
$868$		0			0
$869$			35.2864i			1.19701i
$870$		0			0
$871$	9.89598			0.335313
$872$		0			0
$873$		−	16.2323i		−	0.549381i
$874$		0			0
$875$	−2.72115	+	10.8441i	−0.0919918	+	0.366599i
$876$		0			0
$877$	−44.4678			−1.50157			−0.750785	−	0.660547i	$-0.770325\pi$
$877$	−44.4678			−1.50157			−0.750785	+	0.660547i	$0.770325\pi$
$878$		0			0
$879$	6.83066			0.230392
$880$		0			0
$881$	22.4801			0.757374			0.378687	−	0.925525i	$-0.376376\pi$
$881$	22.4801			0.757374			0.378687	+	0.925525i	$0.376376\pi$
$882$		0			0
$883$	9.36004			0.314991			0.157495	−	0.987520i	$-0.449658\pi$
$883$	9.36004			0.314991			0.157495	+	0.987520i	$0.449658\pi$
$884$		0			0
$885$	−19.0257	−	13.1718i	−0.639543	−	0.442766i
$886$		0			0
$887$		−	32.3368i		−	1.08576i	−0.839809	−	0.542882i	$-0.817333\pi$
$887$		−	32.3368i		−	1.08576i	0.839809	−	0.542882i	$-0.182667\pi$
$888$		0			0
$889$	−10.2339			−0.343233
$890$		0			0
$891$		−	59.6166i		−	1.99723i
$892$		0			0
$893$	10.1824			0.340742
$894$		0			0
$895$	−45.3413	−	31.3906i	−1.51559	−	1.04927i
$896$		0			0
$897$		−	11.4722i		−	0.383044i
$898$		0			0
$899$			31.8847i			1.06342i
$900$		0			0
$901$			9.01330i			0.300276i
$902$		0			0
$903$			4.13098i			0.137470i
$904$		0			0
$905$	47.1248	+	32.6253i	1.56648	+	1.08450i
$906$		0			0
$907$	3.66559			0.121714			0.0608569	−	0.998147i	$-0.480617\pi$
$907$	3.66559			0.121714			0.0608569	+	0.998147i	$0.480617\pi$
$908$		0			0
$909$			14.7734i			0.490004i
$910$		0			0
$911$	−7.45983			−0.247155			−0.123578	−	0.992335i	$-0.539437\pi$
$911$	−7.45983			−0.247155			−0.123578	+	0.992335i	$0.539437\pi$
$912$		0			0
$913$		−	47.5481i		−	1.57361i
$914$		0			0
$915$	−17.2612	−	11.9502i	−0.570638	−	0.395062i
$916$		0			0
$917$	−12.1684			−0.401836
$918$		0			0
$919$	45.5292			1.50187			0.750934	−	0.660377i	$-0.229603\pi$
$919$	45.5292			1.50187			0.750934	+	0.660377i	$0.229603\pi$
$920$		0			0
$921$	37.9424			1.25024
$922$		0			0
$923$	10.5895			0.348557
$924$		0			0
$925$	−0.844980	−	2.24697i	−0.0277828	−	0.0738798i
$926$		0			0
$927$		−	14.0515i		−	0.461510i
$928$		0			0
$929$	26.2339			0.860706			0.430353	−	0.902661i	$-0.358389\pi$
$929$	26.2339			0.860706			0.430353	+	0.902661i	$0.358389\pi$
$930$		0			0
$931$		−	4.54561i		−	0.148976i
$932$		0			0
$933$	6.88071			0.225264
$934$		0			0
$935$	10.9778	−	15.8567i	0.359014	−	0.518569i
$936$		0			0
$937$		−	52.9890i		−	1.73108i	−0.500844	−	0.865538i	$-0.666977\pi$
$937$		−	52.9890i		−	1.73108i	0.500844	−	0.865538i	$-0.333023\pi$
$938$		0			0
$939$			5.14884i			0.168026i
$940$		0			0
$941$			32.8806i			1.07188i	0.844258	+	0.535938i	$0.180042\pi$
$941$			32.8806i			1.07188i	−0.844258	+	0.535938i	$0.819958\pi$
$942$		0			0
$943$		−	14.1824i		−	0.461843i
$944$		0			0
$945$	−6.66255	−	4.61260i	−0.216733	−	0.150048i
$946$		0			0
$947$	−31.7168			−1.03066			−0.515329	−	0.856992i	$-0.672330\pi$
$947$	−31.7168			−1.03066			−0.515329	+	0.856992i	$0.672330\pi$
$948$		0			0
$949$		−	1.57618i		−	0.0511649i
$950$		0			0
$951$	22.4627			0.728404
$952$		0			0
$953$			33.7230i			1.09239i	0.837657	+	0.546197i	$0.183925\pi$
$953$			33.7230i			1.09239i	−0.837657	+	0.546197i	$0.816075\pi$
$954$		0			0
$955$	−32.2514	+	46.5847i	−1.04363	+	1.50745i
$956$		0			0
$957$	−84.9932			−2.74744
$958$		0			0
$959$	18.7201			0.604503
$960$		0			0
$961$	−14.0398			−0.452896
$962$		0			0
$963$	−21.8208			−0.703167
$964$		0			0
$965$	−9.83161	−	6.80659i	−0.316491	−	0.219112i
$966$		0			0
$967$			24.6345i			0.792191i	0.918209	+	0.396096i	$0.129635\pi$
$967$			24.6345i			0.792191i	−0.918209	+	0.396096i	$0.870365\pi$
$968$		0			0
$969$	−15.1427			−0.486453
$970$		0			0
$971$			40.5052i			1.29987i	0.759988	+	0.649937i	$0.225205\pi$
$971$			40.5052i			1.29987i	−0.759988	+	0.649937i	$0.774795\pi$
$972$		0			0
$973$	9.15670			0.293550
$974$		0			0
$975$	−2.84723	−	7.57133i	−0.0911843	−	0.242477i
$976$		0			0
$977$			39.2055i			1.25430i	0.778900	+	0.627148i	$0.215778\pi$
$977$			39.2055i			1.25430i	−0.778900	+	0.627148i	$0.784222\pi$
$978$		0			0
$979$			2.28534i			0.0730397i
$980$		0			0
$981$		−	4.62531i		−	0.147675i
$982$		0			0
$983$			24.1763i			0.771103i	0.922686	+	0.385551i	$0.125989\pi$
$983$			24.1763i			0.771103i	−0.922686	+	0.385551i	$0.874011\pi$
$984$		0			0
$985$	21.1427	−	30.5390i	0.673661	−	0.973053i
$986$		0			0
$987$	4.61260			0.146821
$988$		0			0
$989$		−	14.2262i		−	0.452365i
$990$		0			0
$991$	53.3774			1.69559			0.847794	−	0.530326i	$-0.177930\pi$
$991$	53.3774			1.69559			0.847794	+	0.530326i	$0.177930\pi$
$992$		0			0
$993$		−	3.61125i		−	0.114599i
$994$		0			0
$995$	−14.6018	+	21.0912i	−0.462908	+	0.668636i
$996$		0			0
$997$	54.9557			1.74047			0.870233	−	0.492640i	$-0.163968\pi$
$997$	54.9557			1.74047			0.870233	+	0.492640i	$0.163968\pi$
$998$		0			0
$999$	1.73993			0.0550490

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.l.a.1569.3	✓	12
4.3	odd	2	inner	2240.2.l.a.1569.9	yes	12
5.4	even	2		2240.2.l.b.1569.9	yes	12
8.3	odd	2		2240.2.l.b.1569.4	yes	12
8.5	even	2		2240.2.l.b.1569.10	yes	12
20.19	odd	2		2240.2.l.b.1569.3	yes	12
40.19	odd	2	inner	2240.2.l.a.1569.10	yes	12
40.29	even	2	inner	2240.2.l.a.1569.4	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.l.a.1569.3	✓	12	1.1	even	1	trivial
2240.2.l.a.1569.4	yes	12	40.29	even	2	inner
2240.2.l.a.1569.9	yes	12	4.3	odd	2	inner
2240.2.l.a.1569.10	yes	12	40.19	odd	2	inner
2240.2.l.b.1569.3	yes	12	20.19	odd	2
2240.2.l.b.1569.4	yes	12	8.3	odd	2
2240.2.l.b.1569.9	yes	12	5.4	even	2
2240.2.l.b.1569.10	yes	12	8.5	even	2

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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.05914 q^{3} +(1.27280 - 1.83847i) q^{5} -1.00000i q^{7} +1.24006 q^{9} +O(q^{10})\)	\(q-2.05914 q^{3} +(1.27280 - 1.83847i) q^{5} -1.00000i q^{7} +1.24006 q^{9} +5.33127i q^{11} -0.785667 q^{13} +(-2.62088 + 3.78567i) q^{15} -1.61780i q^{17} +4.54561i q^{19} +2.05914i q^{21} -7.09122i q^{23} +(-1.75994 - 4.68002i) q^{25} +3.62397 q^{27} -7.74225i q^{29} -4.11828 q^{31} -10.9778i q^{33} +(-1.83847 - 1.27280i) q^{35} +0.480119 q^{37} +1.61780 q^{39} +2.00000 q^{41} +2.00617 q^{43} +(1.57835 - 2.27981i) q^{45} -2.24006i q^{47} -1.00000 q^{49} +3.33127i q^{51} -5.57133 q^{53} +(9.80139 + 6.78567i) q^{55} -9.36004i q^{57} +5.02573i q^{59} +4.55962i q^{61} -1.24006i q^{63} +(-1.00000 + 1.44442i) q^{65} -12.5956 q^{67} +14.6018i q^{69} -13.4783 q^{71} +2.00617i q^{73} +(3.62397 + 9.63682i) q^{75} +5.33127 q^{77} +6.61876 q^{79} -11.1824 q^{81} -8.91870 q^{83} +(-2.97427 - 2.05914i) q^{85} +15.9424i q^{87} +0.428666 q^{89} +0.785667i q^{91} +8.48012 q^{93} +(8.35696 + 5.78567i) q^{95} -13.0900i q^{97} +6.61110i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{5} + 24 q^{9}+O(q^{10}) \) 12 * q - 4 * q^5 + 24 * q^9	\( 12 q - 4 q^{5} + 24 q^{9} + 20 q^{13} - 12 q^{25} + 24 q^{37} + 24 q^{41} - 48 q^{45} - 12 q^{49} - 8 q^{53} - 12 q^{65} - 4 q^{77} + 20 q^{81} - 56 q^{85} + 64 q^{89} + 120 q^{93}+O(q^{100}) \) 12 * q - 4 * q^5 + 24 * q^9 + 20 * q^13 - 12 * q^25 + 24 * q^37 + 24 * q^41 - 48 * q^45 - 12 * q^49 - 8 * q^53 - 12 * q^65 - 4 * q^77 + 20 * q^81 - 56 * q^85 + 64 * q^89 + 120 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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