LMFDB - Embedded newform 4560.2.d.f.2431.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4560,2,Mod(2431,4560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("4560.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4560.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:	$36.4117833217$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.792772608.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 22x^{4} + 62x^{2} + 12 $ x^6 + 22x^4 + 62x^2 + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.5
Root			$-0.457038i$ of defining polynomial
Character	$\chi$	$=$	4560.2431
Dual form			4560.2.d.f.2431.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +2.36627i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} +2.36627i q^{7} +1.00000 q^{9} -0.457038i q^{11} -2.36627i q^{13} -1.00000 q^{15} +7.27334 q^{17} +(-3.91852 - 1.90924i) q^{19} -2.36627i q^{21} -3.24311i q^{23} +1.00000 q^{25} -1.00000 q^{27} +6.94326i q^{29} +0.918523 q^{31} +0.457038i q^{33} +2.36627i q^{35} -2.78607i q^{37} +2.36627i q^{39} -9.84766i q^{41} +2.78607i q^{43} +1.00000 q^{45} -13.5478i q^{47} +1.40075 q^{49} -7.27334 q^{51} +3.24311i q^{53} -0.457038i q^{55} +(3.91852 + 1.90924i) q^{57} +6.87259 q^{59} -3.95407 q^{61} +2.36627i q^{63} -2.36627i q^{65} -2.51777 q^{67} +3.24311i q^{69} -6.87259 q^{71} -0.517774 q^{73} -1.00000 q^{75} +1.08148 q^{77} +6.35482 q^{79} +1.00000 q^{81} +11.7941i q^{83} +7.27334 q^{85} -6.94326i q^{87} -14.1604i q^{89} +5.59925 q^{91} -0.918523 q^{93} +(-3.91852 - 1.90924i) q^{95} -12.6710i q^{97} -0.457038i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{5} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 6 * q^5 + 6 * q^9	$ 6 q - 6 q^{3} + 6 q^{5} + 6 q^{9} - 6 q^{15} - 4 q^{17} - 6 q^{19} + 6 q^{25} - 6 q^{27} - 12 q^{31} + 6 q^{45} - 14 q^{49} + 4 q^{51} + 6 q^{57} + 16 q^{59} - 16 q^{61} - 20 q^{67} - 16 q^{71} - 8 q^{73} - 6 q^{75} + 24 q^{77} + 8 q^{79} + 6 q^{81} - 4 q^{85} + 56 q^{91} + 12 q^{93} - 6 q^{95}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^5 + 6 * q^9 - 6 * q^15 - 4 * q^17 - 6 * q^19 + 6 * q^25 - 6 * q^27 - 12 * q^31 + 6 * q^45 - 14 * q^49 + 4 * q^51 + 6 * q^57 + 16 * q^59 - 16 * q^61 - 20 * q^67 - 16 * q^71 - 8 * q^73 - 6 * q^75 + 24 * q^77 + 8 * q^79 + 6 * q^81 - 4 * q^85 + 56 * q^91 + 12 * q^93 - 6 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times$.

$n$	$1141$	$1711$	$1921$	$2737$	$3041$
$\chi(n)$	$1$	$-1$	$-1$	$1$	$1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$			2.36627i			0.894367i	0.894442	+	0.447184i	$0.147573\pi$
$7$			2.36627i			0.894367i	−0.894442	+	0.447184i	$0.852427\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	0.457038i		−	0.137802i	−0.997623	−	0.0689011i	$-0.978051\pi$
$11$		−	0.457038i		−	0.137802i	0.997623	−	0.0689011i	$-0.0219493\pi$
$12$		0			0
$13$		−	2.36627i		−	0.656286i	−0.944628	−	0.328143i	$-0.893577\pi$
$13$		−	2.36627i		−	0.656286i	0.944628	−	0.328143i	$-0.106423\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$	7.27334			1.76404			0.882022	−	0.471208i	$-0.156182\pi$
$17$	7.27334			1.76404			0.882022	+	0.471208i	$0.156182\pi$
$18$		0			0
$19$	−3.91852	−	1.90924i	−0.898971	−	0.438009i
$19$	−3.91852	−	1.90924i	−0.898971	−	0.438009i
$20$		0			0
$21$		−	2.36627i		−	0.516363i
$22$		0			0
$23$		−	3.24311i		−	0.676236i	−0.941104	−	0.338118i	$-0.890210\pi$
$23$		−	3.24311i		−	0.676236i	0.941104	−	0.338118i	$-0.109790\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$			6.94326i			1.28933i	0.764465	+	0.644666i	$0.223003\pi$
$29$			6.94326i			1.28933i	−0.764465	+	0.644666i	$0.776997\pi$
$30$		0			0
$31$	0.918523			0.164972			0.0824858	−	0.996592i	$-0.473714\pi$
$31$	0.918523			0.164972			0.0824858	+	0.996592i	$0.473714\pi$
$32$		0			0
$33$			0.457038i			0.0795601i
$34$		0			0
$35$			2.36627i			0.399973i
$36$		0			0
$37$		−	2.78607i		−	0.458028i	−0.973423	−	0.229014i	$-0.926450\pi$
$37$		−	2.78607i		−	0.458028i	0.973423	−	0.229014i	$-0.0735501\pi$
$38$		0			0
$39$			2.36627i			0.378907i
$40$		0			0
$41$		−	9.84766i		−	1.53795i	−0.639282	−	0.768973i	$-0.720768\pi$
$41$		−	9.84766i		−	1.53795i	0.639282	−	0.768973i	$-0.279232\pi$
$42$		0			0
$43$			2.78607i			0.424872i	0.977175	+	0.212436i	$0.0681397\pi$
$43$			2.78607i			0.424872i	−0.977175	+	0.212436i	$0.931860\pi$
$44$		0			0
$45$	1.00000			0.149071
$46$		0			0
$47$		−	13.5478i		−	1.97615i	−0.153971	−	0.988075i	$-0.549206\pi$
$47$		−	13.5478i		−	1.97615i	0.153971	−	0.988075i	$-0.450794\pi$
$48$		0			0
$49$	1.40075			0.200107
$50$		0			0
$51$	−7.27334			−1.01847
$52$		0			0
$53$			3.24311i			0.445476i	0.974878	+	0.222738i	$0.0714994\pi$
$53$			3.24311i			0.445476i	−0.974878	+	0.222738i	$0.928501\pi$
$54$		0			0
$55$		−	0.457038i		−	0.0616270i
$56$		0			0
$57$	3.91852	+	1.90924i	0.519021	+	0.252884i
$58$		0			0
$59$	6.87259			0.894735			0.447368	−	0.894350i	$-0.352362\pi$
$59$	6.87259			0.894735			0.447368	+	0.894350i	$0.352362\pi$
$60$		0			0
$61$	−3.95407			−0.506267			−0.253133	−	0.967431i	$-0.581461\pi$
$61$	−3.95407			−0.506267			−0.253133	+	0.967431i	$0.581461\pi$
$62$		0			0
$63$			2.36627i			0.298122i
$64$		0			0
$65$		−	2.36627i		−	0.293500i
$66$		0			0
$67$	−2.51777			−0.307595			−0.153798	−	0.988102i	$-0.549150\pi$
$67$	−2.51777			−0.307595			−0.153798	+	0.988102i	$0.549150\pi$
$68$		0			0
$69$			3.24311i			0.390425i
$70$		0			0
$71$	−6.87259			−0.815627			−0.407813	−	0.913065i	$-0.633709\pi$
$71$	−6.87259			−0.815627			−0.407813	+	0.913065i	$0.633709\pi$
$72$		0			0
$73$	−0.517774			−0.0606009			−0.0303004	−	0.999541i	$-0.509646\pi$
$73$	−0.517774			−0.0606009			−0.0303004	+	0.999541i	$0.509646\pi$
$74$		0			0
$75$	−1.00000			−0.115470
$76$		0			0
$77$	1.08148			0.123246
$78$		0			0
$79$	6.35482			0.714973			0.357487	−	0.933918i	$-0.383634\pi$
$79$	6.35482			0.714973			0.357487	+	0.933918i	$0.383634\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			11.7941i			1.29457i	0.762246	+	0.647287i	$0.224096\pi$
$83$			11.7941i			1.29457i	−0.762246	+	0.647287i	$0.775904\pi$
$84$		0			0
$85$	7.27334			0.788905
$86$		0			0
$87$		−	6.94326i		−	0.744396i
$88$		0			0
$89$		−	14.1604i		−	1.50100i	−0.660871	−	0.750500i	$-0.729813\pi$
$89$		−	14.1604i		−	1.50100i	0.660871	−	0.750500i	$-0.270187\pi$
$90$		0			0
$91$	5.59925			0.586961
$92$		0			0
$93$	−0.918523			−0.0952464
$94$		0			0
$95$	−3.91852	−	1.90924i	−0.402032	−	0.195883i
$96$		0			0
$97$		−	12.6710i		−	1.28654i	−0.765639	−	0.643271i	$-0.777577\pi$
$97$		−	12.6710i		−	1.28654i	0.765639	−	0.643271i	$-0.222423\pi$
$98$		0			0
$99$		−	0.457038i		−	0.0459341i
$100$		0			0
$101$	3.83705			0.381800			0.190900	−	0.981609i	$-0.438859\pi$
$101$	3.83705			0.381800			0.190900	+	0.981609i	$0.438859\pi$
$102$		0			0
$103$	−10.5467			−1.03920			−0.519598	−	0.854411i	$-0.673918\pi$
$103$	−10.5467			−1.03920			−0.519598	+	0.854411i	$0.673918\pi$
$104$		0			0
$105$		−	2.36627i		−	0.230925i
$106$		0			0
$107$	2.35482			0.227649			0.113824	−	0.993501i	$-0.463690\pi$
$107$	2.35482			0.227649			0.113824	+	0.993501i	$0.463690\pi$
$108$		0			0
$109$		−	9.39062i		−	0.899458i	−0.893165	−	0.449729i	$-0.851521\pi$
$109$		−	9.39062i		−	0.899458i	0.893165	−	0.449729i	$-0.148479\pi$
$110$		0			0
$111$			2.78607i			0.264442i
$112$		0			0
$113$			14.4619i			1.36046i	0.732999	+	0.680230i	$0.238120\pi$
$113$			14.4619i			1.36046i	−0.732999	+	0.680230i	$0.761880\pi$
$114$		0			0
$115$		−	3.24311i		−	0.302422i
$116$		0			0
$117$		−	2.36627i		−	0.218762i
$118$		0			0
$119$			17.2107i			1.57770i
$120$		0			0
$121$	10.7911			0.981011
$122$		0			0
$123$			9.84766i			0.887933i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−2.51777			−0.223416			−0.111708	−	0.993741i	$-0.535632\pi$
$127$	−2.51777			−0.223416			−0.111708	+	0.993741i	$0.535632\pi$
$128$		0			0
$129$		−	2.78607i		−	0.245300i
$130$		0			0
$131$			11.1815i			0.976935i	0.872582	+	0.488468i	$0.162444\pi$
$131$			11.1815i			0.976935i	−0.872582	+	0.488468i	$0.837556\pi$
$132$		0			0
$133$	4.51777	−	9.27230i	0.391741	−	0.804010i
$134$		0			0
$135$	−1.00000			−0.0860663
$136$		0			0
$137$	10.5178			0.898594			0.449297	−	0.893382i	$-0.351674\pi$
$137$	10.5178			0.898594			0.449297	+	0.893382i	$0.351674\pi$
$138$		0			0
$139$			13.8865i			1.17784i	0.808191	+	0.588920i	$0.200447\pi$
$139$			13.8865i			1.17784i	−0.808191	+	0.588920i	$0.799553\pi$
$140$		0			0
$141$			13.5478i			1.14093i
$142$		0			0
$143$	−1.08148			−0.0904377
$144$		0			0
$145$			6.94326i			0.576607i
$146$		0			0
$147$	−1.40075			−0.115532
$148$		0			0
$149$	20.5467			1.68325			0.841625	−	0.540062i	$-0.181599\pi$
$149$	20.5467			1.68325			0.841625	+	0.540062i	$0.181599\pi$
$150$		0			0
$151$	21.9830			1.78895			0.894475	−	0.447118i	$-0.147550\pi$
$151$	21.9830			1.78895			0.894475	+	0.447118i	$0.147550\pi$
$152$		0			0
$153$	7.27334			0.588015
$154$		0			0
$155$	0.918523			0.0737775
$156$		0			0
$157$	4.19186			0.334547			0.167274	−	0.985911i	$-0.446504\pi$
$157$	4.19186			0.334547			0.167274	+	0.985911i	$0.446504\pi$
$158$		0			0
$159$		−	3.24311i		−	0.257195i
$160$		0			0
$161$	7.67409			0.604803
$162$		0			0
$163$		−	14.4991i		−	1.13566i	−0.823146	−	0.567829i	$-0.807783\pi$
$163$		−	14.4991i		−	1.13566i	0.823146	−	0.567829i	$-0.192217\pi$
$164$		0			0
$165$			0.457038i			0.0355804i
$166$		0			0
$167$	−8.14594			−0.630351			−0.315176	−	0.949033i	$-0.602063\pi$
$167$	−8.14594			−0.630351			−0.315176	+	0.949033i	$0.602063\pi$
$168$		0			0
$169$	7.40075			0.569288
$170$		0			0
$171$	−3.91852	−	1.90924i	−0.299657	−	0.146003i
$172$		0			0
$173$			6.98711i			0.531220i	0.964081	+	0.265610i	$0.0855733\pi$
$173$			6.98711i			0.531220i	−0.964081	+	0.265610i	$0.914427\pi$
$174$		0			0
$175$			2.36627i			0.178873i
$176$		0			0
$177$	−6.87259			−0.516576
$178$		0			0
$179$	−7.67409			−0.573588			−0.286794	−	0.957992i	$-0.592590\pi$
$179$	−7.67409			−0.573588			−0.286794	+	0.957992i	$0.592590\pi$
$180$		0			0
$181$			0.914076i			0.0679428i	0.999423	+	0.0339714i	$0.0108155\pi$
$181$			0.914076i			0.0679428i	−0.999423	+	0.0339714i	$0.989184\pi$
$182$		0			0
$183$	3.95407			0.292293
$184$		0			0
$185$		−	2.78607i		−	0.204836i
$186$		0			0
$187$		−	3.32419i		−	0.243089i
$188$		0			0
$189$		−	2.36627i		−	0.172121i
$190$		0			0
$191$		−	8.93358i		−	0.646411i	−0.946329	−	0.323206i	$-0.895239\pi$
$191$		−	8.93358i		−	0.646411i	0.946329	−	0.323206i	$-0.104761\pi$
$192$		0			0
$193$		−	1.45220i		−	0.104531i	−0.998633	−	0.0522657i	$-0.983356\pi$
$193$		−	1.45220i		−	0.104531i	0.998633	−	0.0522657i	$-0.0166443\pi$
$194$		0			0
$195$			2.36627i			0.169452i
$196$		0			0
$197$	−14.1459			−1.00786			−0.503928	−	0.863746i	$-0.668112\pi$
$197$	−14.1459			−1.00786			−0.503928	+	0.863746i	$0.668112\pi$
$198$		0			0
$199$		−	15.0372i		−	1.06596i	−0.846127	−	0.532981i	$-0.821072\pi$
$199$		−	15.0372i		−	1.06596i	0.846127	−	0.532981i	$-0.178928\pi$
$200$		0			0
$201$	2.51777			0.177590
$202$		0			0
$203$	−16.4297			−1.15314
$204$		0			0
$205$		−	9.84766i		−	0.687790i
$206$		0			0
$207$		−	3.24311i		−	0.225412i
$208$		0			0
$209$	−0.872594	+	1.79091i	−0.0603586	+	0.123880i
$210$		0			0
$211$	21.8660			1.50531			0.752657	−	0.658413i	$-0.228772\pi$
$211$	21.8660			1.50531			0.752657	+	0.658413i	$0.228772\pi$
$212$		0			0
$213$	6.87259			0.470902
$214$		0			0
$215$			2.78607i			0.190009i
$216$		0			0
$217$			2.17348i			0.147545i
$218$		0			0
$219$	0.517774			0.0349879
$220$		0			0
$221$		−	17.2107i		−	1.15772i
$222$		0			0
$223$	18.5467			1.24198			0.620989	−	0.783819i	$-0.286731\pi$
$223$	18.5467			1.24198			0.620989	+	0.783819i	$0.286731\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	11.1104			0.737422			0.368711	−	0.929544i	$-0.379799\pi$
$227$	11.1104			0.737422			0.368711	+	0.929544i	$0.379799\pi$
$228$		0			0
$229$	8.40075			0.555137			0.277569	−	0.960706i	$-0.410471\pi$
$229$	8.40075			0.555137			0.277569	+	0.960706i	$0.410471\pi$
$230$		0			0
$231$	−1.08148			−0.0711560
$232$		0			0
$233$	−10.7096			−0.701612			−0.350806	−	0.936448i	$-0.614092\pi$
$233$	−10.7096			−0.701612			−0.350806	+	0.936448i	$0.614092\pi$
$234$		0			0
$235$		−	13.5478i		−	0.883761i
$236$		0			0
$237$	−6.35482			−0.412790
$238$		0			0
$239$			12.0956i			0.782400i	0.920306	+	0.391200i	$0.127940\pi$
$239$			12.0956i			0.782400i	−0.920306	+	0.391200i	$0.872060\pi$
$240$		0			0
$241$		−	2.97887i		−	0.191886i	−0.995387	−	0.0959430i	$-0.969413\pi$
$241$		−	2.97887i		−	0.191886i	0.995387	−	0.0959430i	$-0.0305867\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	1.40075			0.0894905
$246$		0			0
$247$	−4.51777	+	9.27230i	−0.287459	+	0.589982i
$248$		0			0
$249$		−	11.7941i		−	0.747423i
$250$		0			0
$251$		−	15.9141i		−	1.00449i	−0.864726	−	0.502244i	$-0.832508\pi$
$251$		−	15.9141i		−	1.00449i	0.864726	−	0.502244i	$-0.167492\pi$
$252$		0			0
$253$	−1.48223			−0.0931867
$254$		0			0
$255$	−7.27334			−0.455474
$256$		0			0
$257$		−	3.47975i		−	0.217061i	−0.994093	−	0.108531i	$-0.965385\pi$
$257$		−	3.47975i		−	0.217061i	0.994093	−	0.108531i	$-0.0346145\pi$
$258$		0			0
$259$	6.59261			0.409645
$260$		0			0
$261$			6.94326i			0.429777i
$262$		0			0
$263$			2.32904i			0.143614i	0.997419	+	0.0718072i	$0.0228766\pi$
$263$			2.32904i			0.143614i	−0.997419	+	0.0718072i	$0.977123\pi$
$264$		0			0
$265$			3.24311i			0.199223i
$266$		0			0
$267$			14.1604i			0.866603i
$268$		0			0
$269$		−	13.2463i		−	0.807643i	−0.914838	−	0.403821i	$-0.867682\pi$
$269$		−	13.2463i		−	0.807643i	0.914838	−	0.403821i	$-0.132318\pi$
$270$		0			0
$271$		−	23.5138i		−	1.42836i	−0.699962	−	0.714180i	$-0.746800\pi$
$271$		−	23.5138i		−	1.42836i	0.699962	−	0.714180i	$-0.253200\pi$
$272$		0			0
$273$	−5.59925			−0.338882
$274$		0			0
$275$		−	0.457038i		−	0.0275604i
$276$		0			0
$277$	−0.162955			−0.00979101			−0.00489550	−	0.999988i	$-0.501558\pi$
$277$	−0.162955			−0.00979101			−0.00489550	+	0.999988i	$0.501558\pi$
$278$		0			0
$279$	0.918523			0.0549905
$280$		0			0
$281$			11.2560i			0.671477i	0.941955	+	0.335739i	$0.108986\pi$
$281$			11.2560i			0.671477i	−0.941955	+	0.335739i	$0.891014\pi$
$282$		0			0
$283$			0.375955i			0.0223482i	0.999938	+	0.0111741i	$0.00355690\pi$
$283$			0.375955i			0.0223482i	−0.999938	+	0.0111741i	$0.996443\pi$
$284$		0			0
$285$	3.91852	+	1.90924i	0.232113	+	0.113093i
$286$		0			0
$287$	23.3023			1.37549
$288$		0			0
$289$	35.9015			2.11185
$290$		0			0
$291$			12.6710i			0.742785i
$292$		0			0
$293$		−	31.2528i		−	1.82581i	−0.408174	−	0.912904i	$-0.633834\pi$
$293$		−	31.2528i		−	1.82581i	0.408174	−	0.912904i	$-0.366166\pi$
$294$		0			0
$295$	6.87259			0.400138
$296$		0			0
$297$			0.457038i			0.0265200i
$298$		0			0
$299$	−7.67409			−0.443804
$300$		0			0
$301$	−6.59261			−0.379992
$302$		0			0
$303$	−3.83705			−0.220432
$304$		0			0
$305$	−3.95407			−0.226409
$306$		0			0
$307$	−2.87259			−0.163948			−0.0819738	−	0.996634i	$-0.526122\pi$
$307$	−2.87259			−0.163948			−0.0819738	+	0.996634i	$0.526122\pi$
$308$		0			0
$309$	10.5467			0.599980
$310$		0			0
$311$			5.68386i			0.322302i	0.986930	+	0.161151i	$0.0515207\pi$
$311$			5.68386i			0.322302i	−0.986930	+	0.161151i	$0.948479\pi$
$312$		0			0
$313$	−15.0645			−0.851494			−0.425747	−	0.904842i	$-0.639989\pi$
$313$	−15.0645			−0.851494			−0.425747	+	0.904842i	$0.639989\pi$
$314$		0			0
$315$			2.36627i			0.133324i
$316$		0			0
$317$			18.2059i			1.02254i	0.859419	+	0.511272i	$0.170826\pi$
$317$			18.2059i			1.02254i	−0.859419	+	0.511272i	$0.829174\pi$
$318$		0			0
$319$	3.17334			0.177673
$320$		0			0
$321$	−2.35482			−0.131433
$322$		0			0
$323$	−28.5008	−	13.8865i	−1.58582	−	0.772667i
$324$		0			0
$325$		−	2.36627i		−	0.131257i
$326$		0			0
$327$			9.39062i			0.519302i
$328$		0			0
$329$	32.0578			1.76740
$330$		0			0
$331$	18.0748			0.993483			0.496741	−	0.867899i	$-0.334530\pi$
$331$	18.0748			0.993483			0.496741	+	0.867899i	$0.334530\pi$
$332$		0			0
$333$		−	2.78607i		−	0.152676i
$334$		0			0
$335$	−2.51777			−0.137561
$336$		0			0
$337$			32.6984i			1.78119i	0.454792	+	0.890597i	$0.349713\pi$
$337$			32.6984i			1.78119i	−0.454792	+	0.890597i	$0.650287\pi$
$338$		0			0
$339$		−	14.4619i		−	0.785462i
$340$		0			0
$341$		−	0.419800i		−	0.0227334i
$342$		0			0
$343$			19.8785i			1.07334i
$344$		0			0
$345$			3.24311i			0.174603i
$346$		0			0
$347$		−	9.80381i		−	0.526296i	−0.964755	−	0.263148i	$-0.915239\pi$
$347$		−	9.80381i		−	0.526296i	0.964755	−	0.263148i	$-0.0847608\pi$
$348$		0			0
$349$	5.15632			0.276011			0.138006	−	0.990431i	$-0.455931\pi$
$349$	5.15632			0.276011			0.138006	+	0.990431i	$0.455931\pi$
$350$		0			0
$351$			2.36627i			0.126302i
$352$		0			0
$353$	7.74519			0.412235			0.206117	−	0.978527i	$-0.433917\pi$
$353$	7.74519			0.412235			0.206117	+	0.978527i	$0.433917\pi$
$354$		0			0
$355$	−6.87259			−0.364759
$356$		0			0
$357$		−	17.2107i		−	0.910888i
$358$		0			0
$359$		−	24.6483i		−	1.30089i	−0.759555	−	0.650443i	$-0.774583\pi$
$359$		−	24.6483i		−	1.30089i	0.759555	−	0.650443i	$-0.225417\pi$
$360$		0			0
$361$	11.7096	+	14.9628i	0.616297	+	0.787514i
$362$		0			0
$363$	−10.7911			−0.566387
$364$		0			0
$365$	−0.517774			−0.0271015
$366$		0			0
$367$		−	33.7002i		−	1.75913i	−0.475775	−	0.879567i	$-0.657832\pi$
$367$		−	33.7002i		−	1.75913i	0.475775	−	0.879567i	$-0.342168\pi$
$368$		0			0
$369$		−	9.84766i		−	0.512649i
$370$		0			0
$371$	−7.67409			−0.398419
$372$		0			0
$373$			16.4894i			0.853791i	0.904301	+	0.426895i	$0.140393\pi$
$373$			16.4894i			0.853791i	−0.904301	+	0.426895i	$0.859607\pi$
$374$		0			0
$375$	−1.00000			−0.0516398
$376$		0			0
$377$	16.4297			0.846170
$378$		0			0
$379$	18.0748			0.928442			0.464221	−	0.885719i	$-0.346334\pi$
$379$	18.0748			0.928442			0.464221	+	0.885719i	$0.346334\pi$
$380$		0			0
$381$	2.51777			0.128989
$382$		0			0
$383$	9.41928			0.481303			0.240651	−	0.970612i	$-0.422639\pi$
$383$	9.41928			0.481303			0.240651	+	0.970612i	$0.422639\pi$
$384$		0			0
$385$	1.08148			0.0551172
$386$		0			0
$387$			2.78607i			0.141624i
$388$		0			0
$389$	24.8726			1.26109			0.630545	−	0.776152i	$-0.282831\pi$
$389$	24.8726			1.26109			0.630545	+	0.776152i	$0.282831\pi$
$390$		0			0
$391$		−	23.5883i		−	1.19291i
$392$		0			0
$393$		−	11.1815i		−	0.564034i
$394$		0			0
$395$	6.35482			0.319746
$396$		0			0
$397$	−17.6741			−0.887037			−0.443519	−	0.896265i	$-0.646270\pi$
$397$	−17.6741			−0.887037			−0.443519	+	0.896265i	$0.646270\pi$
$398$		0			0
$399$	−4.51777	+	9.27230i	−0.226172	+	0.464195i
$400$		0			0
$401$		−	28.5412i		−	1.42528i	−0.701530	−	0.712640i	$-0.747499\pi$
$401$		−	28.5412i		−	1.42528i	0.701530	−	0.712640i	$-0.252501\pi$
$402$		0			0
$403$		−	2.17348i		−	0.108269i
$404$		0			0
$405$	1.00000			0.0496904
$406$		0			0
$407$	−1.27334			−0.0631172
$408$		0			0
$409$		−	26.4927i		−	1.30998i	−0.755639	−	0.654989i	$-0.772673\pi$
$409$		−	26.4927i		−	1.30998i	0.755639	−	0.654989i	$-0.227327\pi$
$410$		0			0
$411$	−10.5178			−0.518804
$412$		0			0
$413$			16.2624i			0.800222i
$414$		0			0
$415$			11.7941i			0.578951i
$416$		0			0
$417$		−	13.8865i		−	0.680026i
$418$		0			0
$419$			8.33062i			0.406978i	0.979077	+	0.203489i	$0.0652280\pi$
$419$			8.33062i			0.406978i	−0.979077	+	0.203489i	$0.934772\pi$
$420$		0			0
$421$		−	4.80702i		−	0.234280i	−0.993115	−	0.117140i	$-0.962627\pi$
$421$		−	4.80702i		−	0.234280i	0.993115	−	0.117140i	$-0.0373726\pi$
$422$		0			0
$423$		−	13.5478i		−	0.658717i
$424$		0			0
$425$	7.27334			0.352809
$426$		0			0
$427$		−	9.35641i		−	0.452789i
$428$		0			0
$429$	1.08148			0.0522142
$430$		0			0
$431$	14.5467			0.700689			0.350345	−	0.936621i	$-0.386064\pi$
$431$	14.5467			0.700689			0.350345	+	0.936621i	$0.386064\pi$
$432$		0			0
$433$		−	28.6223i		−	1.37550i	−0.725948	−	0.687750i	$-0.758599\pi$
$433$		−	28.6223i		−	1.37550i	0.725948	−	0.687750i	$-0.241401\pi$
$434$		0			0
$435$		−	6.94326i		−	0.332904i
$436$		0			0
$437$	−6.19186	+	12.7082i	−0.296197	+	0.607916i
$438$		0			0
$439$	16.0000			0.763638			0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000			0.763638			0.381819	+	0.924237i	$0.375298\pi$
$440$		0			0
$441$	1.40075			0.0667023
$442$		0			0
$443$			1.72608i			0.0820085i	0.999159	+	0.0410042i	$0.0130557\pi$
$443$			1.72608i			0.0820085i	−0.999159	+	0.0410042i	$0.986944\pi$
$444$		0			0
$445$		−	14.1604i		−	0.671268i
$446$		0			0
$447$	−20.5467			−0.971825
$448$		0			0
$449$		−	20.9920i		−	0.990671i	−0.868702	−	0.495336i	$-0.835045\pi$
$449$		−	20.9920i		−	0.990671i	0.868702	−	0.495336i	$-0.164955\pi$
$450$		0			0
$451$	−4.50075			−0.211932
$452$		0			0
$453$	−21.9830			−1.03285
$454$		0			0
$455$	5.59925			0.262497
$456$		0			0
$457$	32.9015			1.53907			0.769534	−	0.638606i	$-0.220489\pi$
$457$	32.9015			1.53907			0.769534	+	0.638606i	$0.220489\pi$
$458$		0			0
$459$	−7.27334			−0.339491
$460$		0			0
$461$	−35.0934			−1.63446			−0.817231	−	0.576311i	$-0.804492\pi$
$461$	−35.0934			−1.63446			−0.817231	+	0.576311i	$0.804492\pi$
$462$		0			0
$463$			22.3192i			1.03726i	0.854998	+	0.518631i	$0.173558\pi$
$463$			22.3192i			1.03726i	−0.854998	+	0.518631i	$0.826442\pi$
$464$		0			0
$465$	−0.918523			−0.0425955
$466$		0			0
$467$			16.3645i			0.757259i	0.925548	+	0.378630i	$0.123605\pi$
$467$			16.3645i			0.757259i	−0.925548	+	0.378630i	$0.876395\pi$
$468$		0			0
$469$		−	5.95774i		−	0.275103i
$470$		0			0
$471$	−4.19186			−0.193151
$472$		0			0
$473$	1.27334			0.0585483
$474$		0			0
$475$	−3.91852	−	1.90924i	−0.179794	−	0.0876017i
$476$		0			0
$477$			3.24311i			0.148492i
$478$		0			0
$479$			18.6563i			0.852429i	0.904622	+	0.426214i	$0.140153\pi$
$479$			18.6563i			0.852429i	−0.904622	+	0.426214i	$0.859847\pi$
$480$		0			0
$481$	−6.59261			−0.300597
$482$		0			0
$483$	−7.67409			−0.349183
$484$		0			0
$485$		−	12.6710i		−	0.575359i
$486$		0			0
$487$	1.39037			0.0630036			0.0315018	−	0.999504i	$-0.489971\pi$
$487$	1.39037			0.0630036			0.0315018	+	0.999504i	$0.489971\pi$
$488$		0			0
$489$			14.4991i			0.655673i
$490$		0			0
$491$			32.2107i			1.45365i	0.686823	+	0.726825i	$0.259005\pi$
$491$			32.2107i			1.45365i	−0.686823	+	0.726825i	$0.740995\pi$
$492$		0			0
$493$			50.5007i			2.27444i
$494$		0			0
$495$		−	0.457038i		−	0.0205423i
$496$		0			0
$497$		−	16.2624i		−	0.729470i
$498$		0			0
$499$		−	30.0000i		−	1.34298i	−0.741011	−	0.671492i	$-0.765654\pi$
$499$		−	30.0000i		−	1.34298i	0.741011	−	0.671492i	$-0.234346\pi$
$500$		0			0
$501$	8.14594			0.363934
$502$		0			0
$503$			18.2804i			0.815081i	0.913187	+	0.407540i	$0.133613\pi$
$503$			18.2804i			0.815081i	−0.913187	+	0.407540i	$0.866387\pi$
$504$		0			0
$505$	3.83705			0.170746
$506$		0			0
$507$	−7.40075			−0.328679
$508$		0			0
$509$			35.8670i			1.58978i	0.606755	+	0.794889i	$0.292471\pi$
$509$			35.8670i			1.58978i	−0.606755	+	0.794889i	$0.707529\pi$
$510$		0			0
$511$		−	1.22520i		−	0.0541995i
$512$		0			0
$513$	3.91852	+	1.90924i	0.173007	+	0.0842948i
$514$		0			0
$515$	−10.5467			−0.464742
$516$		0			0
$517$	−6.19186			−0.272318
$518$		0			0
$519$		−	6.98711i		−	0.306700i
$520$		0			0
$521$			10.5786i			0.463456i	0.972781	+	0.231728i	$0.0744379\pi$
$521$			10.5786i			0.463456i	−0.972781	+	0.231728i	$0.925562\pi$
$522$		0			0
$523$	41.1223			1.79815			0.899076	−	0.437793i	$-0.144240\pi$
$523$	41.1223			1.79815			0.899076	+	0.437793i	$0.144240\pi$
$524$		0			0
$525$		−	2.36627i		−	0.103273i
$526$		0			0
$527$	6.68073			0.291017
$528$		0			0
$529$	12.4822			0.542705
$530$		0			0
$531$	6.87259			0.298245
$532$		0			0
$533$	−23.3023			−1.00933
$534$		0			0
$535$	2.35482			0.101808
$536$		0			0
$537$	7.67409			0.331161
$538$		0			0
$539$		−	0.640195i		−	0.0275752i
$540$		0			0
$541$	−15.6741			−0.673882			−0.336941	−	0.941526i	$-0.609392\pi$
$541$	−15.6741			−0.673882			−0.336941	+	0.941526i	$0.609392\pi$
$542$		0			0
$543$		−	0.914076i		−	0.0392268i
$544$		0			0
$545$		−	9.39062i		−	0.402250i
$546$		0			0
$547$	39.5822			1.69241			0.846207	−	0.532854i	$-0.178881\pi$
$547$	39.5822			1.69241			0.846207	+	0.532854i	$0.178881\pi$
$548$		0			0
$549$	−3.95407			−0.168756
$550$		0			0
$551$	13.2563	−	27.2073i	0.564738	−	1.15907i
$552$		0			0
$553$			15.0372i			0.639449i
$554$		0			0
$555$			2.78607i			0.118262i
$556$		0			0
$557$	0.262961			0.0111420			0.00557101	−	0.999984i	$-0.498227\pi$
$557$	0.262961			0.0111420			0.00557101	+	0.999984i	$0.498227\pi$
$558$		0			0
$559$	6.59261			0.278838
$560$		0			0
$561$			3.32419i			0.140348i
$562$		0			0
$563$	−32.5637			−1.37240			−0.686198	−	0.727415i	$-0.740722\pi$
$563$	−32.5637			−1.37240			−0.686198	+	0.727415i	$0.740722\pi$
$564$		0			0
$565$			14.4619i			0.608416i
$566$		0			0
$567$			2.36627i			0.0993742i
$568$		0			0
$569$			7.54622i			0.316354i	0.987411	+	0.158177i	$0.0505617\pi$
$569$			7.54622i			0.316354i	−0.987411	+	0.158177i	$0.949438\pi$
$570$		0			0
$571$		−	12.3695i		−	0.517647i	−0.965925	−	0.258824i	$-0.916665\pi$
$571$		−	12.3695i		−	0.517647i	0.965925	−	0.258824i	$-0.0833348\pi$
$572$		0			0
$573$			8.93358i			0.373206i
$574$		0			0
$575$		−	3.24311i		−	0.135247i
$576$		0			0
$577$	−1.31927			−0.0549220			−0.0274610	−	0.999623i	$-0.508742\pi$
$577$	−1.31927			−0.0549220			−0.0274610	+	0.999623i	$0.508742\pi$
$578$		0			0
$579$			1.45220i			0.0603513i
$580$		0			0
$581$	−27.9081			−1.15782
$582$		0			0
$583$	1.48223			0.0613875
$584$		0			0
$585$		−	2.36627i		−	0.0978334i
$586$		0			0
$587$		−	14.6985i		−	0.606673i	−0.952883	−	0.303337i	$-0.901899\pi$
$587$		−	14.6985i		−	0.606673i	0.952883	−	0.303337i	$-0.0981006\pi$
$588$		0			0
$589$	−3.59925	−	1.75368i	−0.148305	−	0.0722590i
$590$		0			0
$591$	14.1459			0.581886
$592$		0			0
$593$	−19.9371			−0.818717			−0.409358	−	0.912374i	$-0.634247\pi$
$593$	−19.9371			−0.818717			−0.409358	+	0.912374i	$0.634247\pi$
$594$		0			0
$595$			17.2107i			0.705571i
$596$		0			0
$597$			15.0372i			0.615433i
$598$		0			0
$599$	−29.5111			−1.20579			−0.602896	−	0.797820i	$-0.705987\pi$
$599$	−29.5111			−1.20579			−0.602896	+	0.797820i	$0.705987\pi$
$600$		0			0
$601$			38.4633i			1.56895i	0.620159	+	0.784476i	$0.287068\pi$
$601$			38.4633i			1.56895i	−0.620159	+	0.784476i	$0.712932\pi$
$602$		0			0
$603$	−2.51777			−0.102532
$604$		0			0
$605$	10.7911			0.438721
$606$		0			0
$607$	−2.51777			−0.102193			−0.0510967	−	0.998694i	$-0.516272\pi$
$607$	−2.51777			−0.102193			−0.0510967	+	0.998694i	$0.516272\pi$
$608$		0			0
$609$	16.4297			0.665763
$610$		0			0
$611$	−32.0578			−1.29692
$612$		0			0
$613$	−12.9355			−0.522462			−0.261231	−	0.965276i	$-0.584128\pi$
$613$	−12.9355			−0.522462			−0.261231	+	0.965276i	$0.584128\pi$
$614$		0			0
$615$			9.84766i			0.397096i
$616$		0			0
$617$	11.5993			0.466968			0.233484	−	0.972361i	$-0.424987\pi$
$617$	11.5993			0.466968			0.233484	+	0.972361i	$0.424987\pi$
$618$		0			0
$619$			7.08918i			0.284938i	0.989799	+	0.142469i	$0.0455042\pi$
$619$			7.08918i			0.284938i	−0.989799	+	0.142469i	$0.954496\pi$
$620$		0			0
$621$			3.24311i			0.130142i
$622$		0			0
$623$	33.5074			1.34245
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	0.872594	−	1.79091i	0.0348480	−	0.0715222i
$628$		0			0
$629$		−	20.2641i		−	0.807981i
$630$		0			0
$631$			48.6936i			1.93846i	0.246157	+	0.969230i	$0.420832\pi$
$631$			48.6936i			1.93846i	−0.246157	+	0.969230i	$0.579168\pi$
$632$		0			0
$633$	−21.8660			−0.869094
$634$		0			0
$635$	−2.51777			−0.0999148
$636$		0			0
$637$		−	3.31455i		−	0.131327i
$638$		0			0
$639$	−6.87259			−0.271876
$640$		0			0
$641$			15.3111i			0.604753i	0.953189	+	0.302377i	$0.0977800\pi$
$641$			15.3111i			0.604753i	−0.953189	+	0.302377i	$0.902220\pi$
$642$		0			0
$643$		−	21.5673i		−	0.850532i	−0.905068	−	0.425266i	$-0.860181\pi$
$643$		−	21.5673i		−	0.850532i	0.905068	−	0.425266i	$-0.139819\pi$
$644$		0			0
$645$		−	2.78607i		−	0.109702i
$646$		0			0
$647$			40.7179i			1.60079i	0.599476	+	0.800393i	$0.295376\pi$
$647$			40.7179i			1.60079i	−0.599476	+	0.800393i	$0.704624\pi$
$648$		0			0
$649$		−	3.14104i		−	0.123297i
$650$		0			0
$651$		−	2.17348i		−	0.0851852i
$652$		0			0
$653$	−35.0393			−1.37119			−0.685597	−	0.727982i	$-0.740459\pi$
$653$	−35.0393			−1.37119			−0.685597	+	0.727982i	$0.740459\pi$
$654$		0			0
$655$			11.1815i			0.436899i
$656$		0			0
$657$	−0.517774			−0.0202003
$658$		0			0
$659$	−42.4548			−1.65381			−0.826903	−	0.562345i	$-0.809899\pi$
$659$	−42.4548			−1.65381			−0.826903	+	0.562345i	$0.809899\pi$
$660$		0			0
$661$			34.8070i			1.35384i	0.736058	+	0.676918i	$0.236685\pi$
$661$			34.8070i			1.35384i	−0.736058	+	0.676918i	$0.763315\pi$
$662$		0			0
$663$			17.2107i			0.668409i
$664$		0			0
$665$	4.51777	−	9.27230i	0.175192	−	0.359564i
$666$		0			0
$667$	22.5178			0.871892
$668$		0			0
$669$	−18.5467			−0.717056
$670$		0			0
$671$			1.80716i			0.0697647i
$672$		0			0
$673$		−	5.60278i		−	0.215971i	−0.994152	−	0.107986i	$-0.965560\pi$
$673$		−	5.60278i		−	0.215971i	0.994152	−	0.107986i	$-0.0344401\pi$
$674$		0			0
$675$	−1.00000			−0.0384900
$676$		0			0
$677$			20.1962i			0.776203i	0.921617	+	0.388101i	$0.126869\pi$
$677$			20.1962i			0.776203i	−0.921617	+	0.388101i	$0.873131\pi$
$678$		0			0
$679$	29.9830			1.15064
$680$		0			0
$681$	−11.1104			−0.425751
$682$		0			0
$683$	−34.4126			−1.31676			−0.658382	−	0.752684i	$-0.728759\pi$
$683$	−34.4126			−1.31676			−0.658382	+	0.752684i	$0.728759\pi$
$684$		0			0
$685$	10.5178			0.401863
$686$		0			0
$687$	−8.40075			−0.320509
$688$		0			0
$689$	7.67409			0.292360
$690$		0			0
$691$		−	11.5299i		−	0.438618i	−0.975655	−	0.219309i	$-0.929620\pi$
$691$		−	11.5299i		−	0.438618i	0.975655	−	0.219309i	$-0.0703802\pi$
$692$		0			0
$693$	1.08148			0.0410819
$694$		0			0
$695$			13.8865i			0.526746i
$696$		0			0
$697$		−	71.6254i		−	2.71300i
$698$		0			0
$699$	10.7096			0.405076
$700$		0			0
$701$	2.61778			0.0988722			0.0494361	−	0.998777i	$-0.484258\pi$
$701$	2.61778			0.0988722			0.0494361	+	0.998777i	$0.484258\pi$
$702$		0			0
$703$	−5.31927	+	10.9173i	−0.200620	+	0.411753i
$704$		0			0
$705$			13.5478i			0.510240i
$706$		0			0
$707$			9.07950i			0.341470i
$708$		0			0
$709$	−18.8845			−0.709222			−0.354611	−	0.935014i	$-0.615387\pi$
$709$	−18.8845			−0.709222			−0.354611	+	0.935014i	$0.615387\pi$
$710$		0			0
$711$	6.35482			0.238324
$712$		0			0
$713$		−	2.97887i		−	0.111560i
$714$		0			0
$715$	−1.08148			−0.0404450
$716$		0			0
$717$		−	12.0956i		−	0.451719i
$718$		0			0
$719$		−	34.6076i		−	1.29065i	−0.763910	−	0.645323i	$-0.776723\pi$
$719$		−	34.6076i		−	1.29065i	0.763910	−	0.645323i	$-0.223277\pi$
$720$		0			0
$721$		−	24.9563i		−	0.929423i
$722$		0			0
$723$			2.97887i			0.110785i
$724$		0			0
$725$			6.94326i			0.257866i
$726$		0			0
$727$			32.3663i			1.20040i	0.799850	+	0.600199i	$0.204912\pi$
$727$			32.3663i			1.20040i	−0.799850	+	0.600199i	$0.795088\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$			20.2641i			0.749494i
$732$		0			0
$733$	−52.2497			−1.92989			−0.964943	−	0.262459i	$-0.915466\pi$
$733$	−52.2497			−1.92989			−0.964943	+	0.262459i	$0.915466\pi$
$734$		0			0
$735$	−1.40075			−0.0516674
$736$		0			0
$737$			1.15072i			0.0423873i
$738$		0			0
$739$			2.13927i			0.0786944i	0.999226	+	0.0393472i	$0.0125278\pi$
$739$			2.13927i			0.0786944i	−0.999226	+	0.0393472i	$0.987472\pi$
$740$		0			0
$741$	4.51777	−	9.27230i	0.165965	−	0.340626i
$742$		0			0
$743$	−41.4771			−1.52165			−0.760824	−	0.648959i	$-0.775205\pi$
$743$	−41.4771			−1.52165			−0.760824	+	0.648959i	$0.775205\pi$
$744$		0			0
$745$	20.5467			0.752772
$746$		0			0
$747$			11.7941i			0.431525i
$748$		0			0
$749$			5.57215i			0.203602i
$750$		0			0
$751$	−34.6008			−1.26260			−0.631300	−	0.775539i	$-0.717478\pi$
$751$	−34.6008			−1.26260			−0.631300	+	0.775539i	$0.717478\pi$
$752$		0			0
$753$			15.9141i			0.579941i
$754$		0			0
$755$	21.9830			0.800043
$756$		0			0
$757$	−27.5111			−0.999909			−0.499955	−	0.866052i	$-0.666650\pi$
$757$	−27.5111			−0.999909			−0.499955	+	0.866052i	$0.666650\pi$
$758$		0			0
$759$	1.48223			0.0538014
$760$		0			0
$761$	−32.9304			−1.19373			−0.596863	−	0.802343i	$-0.703587\pi$
$761$	−32.9304			−1.19373			−0.596863	+	0.802343i	$0.703587\pi$
$762$		0			0
$763$	22.2208			0.804446
$764$		0			0
$765$	7.27334			0.262968
$766$		0			0
$767$		−	16.2624i		−	0.587203i
$768$		0			0
$769$	24.0578			0.867547			0.433774	−	0.901022i	$-0.357182\pi$
$769$	24.0578			0.867547			0.433774	+	0.901022i	$0.357182\pi$
$770$		0			0
$771$			3.47975i			0.125320i
$772$		0			0
$773$			0.886479i			0.0318844i	0.999873	+	0.0159422i	$0.00507478\pi$
$773$			0.886479i			0.0318844i	−0.999873	+	0.0159422i	$0.994925\pi$
$774$		0			0
$775$	0.918523			0.0329943
$776$		0			0
$777$	−6.59261			−0.236509
$778$		0			0
$779$	−18.8015	+	38.5883i	−0.673634	+	1.38257i
$780$		0			0
$781$			3.14104i			0.112395i
$782$		0			0
$783$		−	6.94326i		−	0.248132i
$784$		0			0
$785$	4.19186			0.149614
$786$		0			0
$787$	47.1934			1.68226			0.841131	−	0.540831i	$-0.181890\pi$
$787$	47.1934			1.68226			0.841131	+	0.540831i	$0.181890\pi$
$788$		0			0
$789$		−	2.32904i		−	0.0829159i
$790$		0			0
$791$	−34.2208			−1.21675
$792$		0			0
$793$			9.35641i			0.332256i
$794$		0			0
$795$		−	3.24311i		−	0.115021i
$796$		0			0
$797$			4.63047i			0.164020i	0.996632	+	0.0820099i	$0.0261339\pi$
$797$			4.63047i			0.164020i	−0.996632	+	0.0820099i	$0.973866\pi$
$798$		0			0
$799$		−	98.5378i		−	3.48602i
$800$		0			0
$801$		−	14.1604i		−	0.500333i
$802$		0			0
$803$			0.236643i			0.00835093i
$804$		0			0
$805$	7.67409			0.270476
$806$		0			0
$807$			13.2463i			0.466293i
$808$		0			0
$809$	−2.23405			−0.0785451			−0.0392725	−	0.999229i	$-0.512504\pi$
$809$	−2.23405			−0.0785451			−0.0392725	+	0.999229i	$0.512504\pi$
$810$		0			0
$811$	37.4193			1.31397			0.656984	−	0.753904i	$-0.271832\pi$
$811$	37.4193			1.31397			0.656984	+	0.753904i	$0.271832\pi$
$812$		0			0
$813$			23.5138i			0.824664i
$814$		0			0
$815$		−	14.4991i		−	0.507882i
$816$		0			0
$817$	5.31927	−	10.9173i	0.186098	−	0.381948i
$818$		0			0
$819$	5.59925			0.195654
$820$		0			0
$821$	16.7807			0.585652			0.292826	−	0.956166i	$-0.405404\pi$
$821$	16.7807			0.585652			0.292826	+	0.956166i	$0.405404\pi$
$822$		0			0
$823$		−	45.9075i		−	1.60023i	−0.599844	−	0.800117i	$-0.704771\pi$
$823$		−	45.9075i		−	1.60023i	0.599844	−	0.800117i	$-0.295229\pi$
$824$		0			0
$825$			0.457038i			0.0159120i
$826$		0			0
$827$	−5.59925			−0.194705			−0.0973525	−	0.995250i	$-0.531037\pi$
$827$	−5.59925			−0.194705			−0.0973525	+	0.995250i	$0.531037\pi$
$828$		0			0
$829$		−	26.8590i		−	0.932851i	−0.884561	−	0.466425i	$-0.845542\pi$
$829$		−	26.8590i		−	0.932851i	0.884561	−	0.466425i	$-0.154458\pi$
$830$		0			0
$831$	0.162955			0.00565284
$832$		0			0
$833$	10.1881			0.352998
$834$		0			0
$835$	−8.14594			−0.281902
$836$		0			0
$837$	−0.918523			−0.0317488
$838$		0			0
$839$	−35.5822			−1.22844			−0.614218	−	0.789137i	$-0.710528\pi$
$839$	−35.5822			−1.22844			−0.614218	+	0.789137i	$0.710528\pi$
$840$		0			0
$841$	−19.2089			−0.662375
$842$		0			0
$843$		−	11.2560i		−	0.387678i
$844$		0			0
$845$	7.40075			0.254593
$846$		0			0
$847$			25.5347i			0.877384i
$848$		0			0
$849$		−	0.375955i		−	0.0129027i
$850$		0			0
$851$	−9.03555			−0.309735
$852$		0			0
$853$	−35.1852			−1.20472			−0.602360	−	0.798225i	$-0.705773\pi$
$853$	−35.1852			−1.20472			−0.602360	+	0.798225i	$0.705773\pi$
$854$		0			0
$855$	−3.91852	−	1.90924i	−0.134011	−	0.0652945i
$856$		0			0
$857$			6.89942i			0.235680i	0.993033	+	0.117840i	$0.0375969\pi$
$857$			6.89942i			0.235680i	−0.993033	+	0.117840i	$0.962403\pi$
$858$		0			0
$859$			0.677434i			0.0231137i	0.999933	+	0.0115569i	$0.00367875\pi$
$859$			0.677434i			0.0231137i	−0.999933	+	0.0115569i	$0.996321\pi$
$860$		0			0
$861$	−23.3023			−0.794139
$862$		0			0
$863$	−35.7741			−1.21776			−0.608882	−	0.793261i	$-0.708382\pi$
$863$	−35.7741			−1.21776			−0.608882	+	0.793261i	$0.708382\pi$
$864$		0			0
$865$			6.98711i			0.237569i
$866$		0			0
$867$	−35.9015			−1.21928
$868$		0			0
$869$		−	2.90439i		−	0.0985248i
$870$		0			0
$871$			5.95774i			0.201870i
$872$		0			0
$873$		−	12.6710i		−	0.428847i
$874$		0			0
$875$			2.36627i			0.0799947i
$876$		0			0
$877$		−	2.31279i		−	0.0780973i	−0.999237	−	0.0390487i	$-0.987567\pi$
$877$		−	2.31279i		−	0.0780973i	0.999237	−	0.0390487i	$-0.0124327\pi$
$878$		0			0
$879$			31.2528i			1.05413i
$880$		0			0
$881$	−48.4548			−1.63248			−0.816242	−	0.577710i	$-0.803947\pi$
$881$	−48.4548			−1.63248			−0.816242	+	0.577710i	$0.803947\pi$
$882$		0			0
$883$		−	52.2105i		−	1.75702i	−0.477719	−	0.878512i	$-0.658536\pi$
$883$		−	52.2105i		−	1.75702i	0.477719	−	0.878512i	$-0.341464\pi$
$884$		0			0
$885$	−6.87259			−0.231020
$886$		0			0
$887$	16.7096			0.561055			0.280527	−	0.959846i	$-0.409491\pi$
$887$	16.7096			0.561055			0.280527	+	0.959846i	$0.409491\pi$
$888$		0			0
$889$		−	5.95774i		−	0.199816i
$890$		0			0
$891$		−	0.457038i		−	0.0153114i
$892$		0			0
$893$	−25.8660	+	53.0874i	−0.865571	+	1.77650i
$894$		0			0
$895$	−7.67409			−0.256517
$896$		0			0
$897$	7.67409			0.256230
$898$		0			0
$899$			6.37754i			0.212703i
$900$		0			0
$901$			23.5883i			0.785839i
$902$		0			0
$903$	6.59261			0.219388
$904$		0			0
$905$			0.914076i			0.0303849i
$906$		0			0
$907$	−8.97260			−0.297930			−0.148965	−	0.988842i	$-0.547594\pi$
$907$	−8.97260			−0.297930			−0.148965	+	0.988842i	$0.547594\pi$
$908$		0			0
$909$	3.83705			0.127267
$910$		0			0
$911$	5.12741			0.169879			0.0849393	−	0.996386i	$-0.472930\pi$
$911$	5.12741			0.169879			0.0849393	+	0.996386i	$0.472930\pi$
$912$		0			0
$913$	5.39037			0.178395
$914$		0			0
$915$	3.95407			0.130718
$916$		0			0
$917$	−26.4586			−0.873739
$918$		0			0
$919$			10.6158i			0.350183i	0.984552	+	0.175092i	$0.0560222\pi$
$919$			10.6158i			0.350183i	−0.984552	+	0.175092i	$0.943978\pi$
$920$		0			0
$921$	2.87259			0.0946552
$922$		0			0
$923$			16.2624i			0.535285i
$924$		0			0
$925$		−	2.78607i		−	0.0916055i
$926$		0			0
$927$	−10.5467			−0.346399
$928$		0			0
$929$	27.4193			0.899597			0.449799	−	0.893130i	$-0.351496\pi$
$929$	27.4193			0.899597			0.449799	+	0.893130i	$0.351496\pi$
$930$		0			0
$931$	−5.48886	−	2.67436i	−0.179890	−	0.0876486i
$932$		0			0
$933$		−	5.68386i		−	0.186081i
$934$		0			0
$935$		−	3.32419i		−	0.108713i
$936$		0			0
$937$	−6.23405			−0.203658			−0.101829	−	0.994802i	$-0.532469\pi$
$937$	−6.23405			−0.203658			−0.101829	+	0.994802i	$0.532469\pi$
$938$		0			0
$939$	15.0645			0.491610
$940$		0			0
$941$			31.4246i			1.02441i	0.858862	+	0.512206i	$0.171172\pi$
$941$			31.4246i			1.02441i	−0.858862	+	0.512206i	$0.828828\pi$
$942$		0			0
$943$	−31.9371			−1.04001
$944$		0			0
$945$		−	2.36627i		−	0.0769749i
$946$		0			0
$947$		−	16.4522i		−	0.534625i	−0.963610	−	0.267312i	$-0.913864\pi$
$947$		−	16.4522i		−	0.534625i	0.963610	−	0.267312i	$-0.0861355\pi$
$948$		0			0
$949$			1.22520i			0.0397715i
$950$		0			0
$951$		−	18.2059i		−	0.590366i
$952$		0			0
$953$			33.5542i			1.08693i	0.839432	+	0.543464i	$0.182888\pi$
$953$			33.5542i			1.08693i	−0.839432	+	0.543464i	$0.817112\pi$
$954$		0			0
$955$		−	8.93358i		−	0.289084i
$956$		0			0
$957$	−3.17334			−0.102579
$958$		0			0
$959$			24.8879i			0.803673i
$960$		0			0
$961$	−30.1563			−0.972784
$962$		0			0
$963$	2.35482			0.0758830
$964$		0			0
$965$		−	1.45220i		−	0.0467479i
$966$		0			0
$967$			12.9076i			0.415081i	0.978226	+	0.207540i	$0.0665458\pi$
$967$			12.9076i			0.415081i	−0.978226	+	0.207540i	$0.933454\pi$
$968$		0			0
$969$	28.5008	+	13.8865i	0.915576	+	0.446099i
$970$		0			0
$971$	24.3837			0.782511			0.391256	−	0.920282i	$-0.372041\pi$
$971$	24.3837			0.782511			0.391256	+	0.920282i	$0.372041\pi$
$972$		0			0
$973$	−32.8593			−1.05342
$974$		0			0
$975$			2.36627i			0.0757814i
$976$		0			0
$977$			58.7340i			1.87907i	0.342456	+	0.939534i	$0.388741\pi$
$977$			58.7340i			1.87907i	−0.342456	+	0.939534i	$0.611259\pi$
$978$		0			0
$979$	−6.47185			−0.206841
$980$		0			0
$981$		−	9.39062i		−	0.299819i
$982$		0			0
$983$	59.8778			1.90981			0.954903	−	0.296917i	$-0.0959584\pi$
$983$	59.8778			1.90981			0.954903	+	0.296917i	$0.0959584\pi$
$984$		0			0
$985$	−14.1459			−0.450727
$986$		0			0
$987$	−32.0578			−1.02041
$988$		0			0
$989$	9.03555			0.287314
$990$		0			0
$991$	−45.5822			−1.44797			−0.723984	−	0.689817i	$-0.757691\pi$
$991$	−45.5822			−1.44797			−0.723984	+	0.689817i	$0.757691\pi$
$992$		0			0
$993$	−18.0748			−0.573588
$994$		0			0
$995$		−	15.0372i		−	0.476713i
$996$		0			0
$997$	−44.5756			−1.41172			−0.705862	−	0.708350i	$-0.749440\pi$
$997$	−44.5756			−1.41172			−0.705862	+	0.708350i	$0.749440\pi$
$998$		0			0
$999$			2.78607i			0.0881475i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.d.f.2431.5	yes	6
4.3	odd	2		4560.2.d.h.2431.2	yes	6
19.18	odd	2		4560.2.d.h.2431.5	yes	6
76.75	even	2	inner	4560.2.d.f.2431.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4560.2.d.f.2431.2	✓	6	76.75	even	2	inner
4560.2.d.f.2431.5	yes	6	1.1	even	1	trivial
4560.2.d.h.2431.2	yes	6	4.3	odd	2
4560.2.d.h.2431.5	yes	6	19.18	odd	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:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +2.36627i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} +2.36627i q^{7} +1.00000 q^{9} -0.457038i q^{11} -2.36627i q^{13} -1.00000 q^{15} +7.27334 q^{17} +(-3.91852 - 1.90924i) q^{19} -2.36627i q^{21} -3.24311i q^{23} +1.00000 q^{25} -1.00000 q^{27} +6.94326i q^{29} +0.918523 q^{31} +0.457038i q^{33} +2.36627i q^{35} -2.78607i q^{37} +2.36627i q^{39} -9.84766i q^{41} +2.78607i q^{43} +1.00000 q^{45} -13.5478i q^{47} +1.40075 q^{49} -7.27334 q^{51} +3.24311i q^{53} -0.457038i q^{55} +(3.91852 + 1.90924i) q^{57} +6.87259 q^{59} -3.95407 q^{61} +2.36627i q^{63} -2.36627i q^{65} -2.51777 q^{67} +3.24311i q^{69} -6.87259 q^{71} -0.517774 q^{73} -1.00000 q^{75} +1.08148 q^{77} +6.35482 q^{79} +1.00000 q^{81} +11.7941i q^{83} +7.27334 q^{85} -6.94326i q^{87} -14.1604i q^{89} +5.59925 q^{91} -0.918523 q^{93} +(-3.91852 - 1.90924i) q^{95} -12.6710i q^{97} -0.457038i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{5} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 6 * q^5 + 6 * q^9	\( 6 q - 6 q^{3} + 6 q^{5} + 6 q^{9} - 6 q^{15} - 4 q^{17} - 6 q^{19} + 6 q^{25} - 6 q^{27} - 12 q^{31} + 6 q^{45} - 14 q^{49} + 4 q^{51} + 6 q^{57} + 16 q^{59} - 16 q^{61} - 20 q^{67} - 16 q^{71} - 8 q^{73} - 6 q^{75} + 24 q^{77} + 8 q^{79} + 6 q^{81} - 4 q^{85} + 56 q^{91} + 12 q^{93} - 6 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^5 + 6 * q^9 - 6 * q^15 - 4 * q^17 - 6 * q^19 + 6 * q^25 - 6 * q^27 - 12 * q^31 + 6 * q^45 - 14 * q^49 + 4 * q^51 + 6 * q^57 + 16 * q^59 - 16 * q^61 - 20 * q^67 - 16 * q^71 - 8 * q^73 - 6 * q^75 + 24 * q^77 + 8 * q^79 + 6 * q^81 - 4 * q^85 + 56 * q^91 + 12 * q^93 - 6 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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