LMFDB - Embedded newform 4320.2.d.g.3889.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4320,2,Mod(3889,4320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("4320.3889");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4320 = 2^{5} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4320.d (of order $2$, degree $1$, not 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:	$34.4953736732$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 3 x^{15} - 3 x^{14} + 36 x^{13} - 78 x^{12} - 96 x^{11} + 1194 x^{10} + 1456 x^{9} + \cdots + 45658 $ x^16 - 3x^15 - 3x^14 + 36x^13 - 78x^12 - 96x^11 + 1194x^10 + 1456x^9 + 2243x^8 + 23019x^7 + 49749x^6 + 37798x^5 + 78784x^4 + 244612x^3 + 302532x^2 + 157882*x + 45658
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3889.12
Root			$-1.59571 + 0.665253i$ of defining polynomial
Character	$\chi$	$=$	4320.3889
Dual form			4320.2.d.g.3889.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.463409 + 2.18752i) q^{5} -3.14971i q^{7} -2.42180i q^{11} +3.12253 q^{13} -7.15192i q^{17} +2.34624i q^{19} -1.28934i q^{23} +(-4.57050 + 2.02743i) q^{25} -4.21495i q^{29} -3.05999 q^{31} +(6.89007 - 1.45960i) q^{35} -1.37346 q^{37} -11.0754 q^{41} -9.70199 q^{43} -0.627861i q^{47} -2.92070 q^{49} +9.54695 q^{53} +(5.29774 - 1.12228i) q^{55} -10.1066i q^{59} +12.7235i q^{61} +(1.44701 + 6.83059i) q^{65} -10.5125 q^{67} +7.39000 q^{71} +5.73160i q^{73} -7.62797 q^{77} -11.2071 q^{79} -3.12640 q^{83} +(15.6450 - 3.31426i) q^{85} -15.9470 q^{89} -9.83507i q^{91} +(-5.13246 + 1.08727i) q^{95} -14.6589i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 6 q^{5} - 22 q^{25} + 2 q^{35} - 44 q^{49} + 96 q^{53} - 34 q^{55} - 12 q^{77} - 4 q^{79} - 64 q^{83}+O(q^{100}) $ 16 * q - 6 * q^5 - 22 * q^25 + 2 * q^35 - 44 * q^49 + 96 * q^53 - 34 * q^55 - 12 * q^77 - 4 * q^79 - 64 * q^83

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4320\mathbb{Z}\right)^\times$.

$n$	$2081$	$2431$	$3457$	$3781$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.463409	+	2.18752i	0.207243	+	0.978290i
$5$	0.463409	+	2.18752i	0.207243	+	0.978290i
$6$		0			0
$7$		−	3.14971i		−	1.19048i	−0.803548	−	0.595240i	$-0.797057\pi$
$7$		−	3.14971i		−	1.19048i	0.803548	−	0.595240i	$-0.202943\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	2.42180i		−	0.730199i	−0.930968	−	0.365100i	$-0.881035\pi$
$11$		−	2.42180i		−	0.730199i	0.930968	−	0.365100i	$-0.118965\pi$
$12$		0			0
$13$	3.12253			0.866033			0.433016	−	0.901386i	$-0.357449\pi$
$13$	3.12253			0.866033			0.433016	+	0.901386i	$0.357449\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	7.15192i		−	1.73460i	−0.497790	−	0.867298i	$-0.665855\pi$
$17$		−	7.15192i		−	1.73460i	0.497790	−	0.867298i	$-0.334145\pi$
$18$		0			0
$19$			2.34624i			0.538265i	0.963103	+	0.269133i	$0.0867370\pi$
$19$			2.34624i			0.538265i	−0.963103	+	0.269133i	$0.913263\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.28934i		−	0.268846i	−0.990924	−	0.134423i	$-0.957082\pi$
$23$		−	1.28934i		−	0.268846i	0.990924	−	0.134423i	$-0.0429182\pi$
$24$		0			0
$25$	−4.57050	+	2.02743i	−0.914101	+	0.405487i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	4.21495i		−	0.782697i	−0.920242	−	0.391349i	$-0.872009\pi$
$29$		−	4.21495i		−	0.782697i	0.920242	−	0.391349i	$-0.127991\pi$
$30$		0			0
$31$	−3.05999			−0.549590			−0.274795	−	0.961503i	$-0.588610\pi$
$31$	−3.05999			−0.549590			−0.274795	+	0.961503i	$0.588610\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.89007	−	1.45960i	1.16463	−	0.246718i
$36$		0			0
$37$	−1.37346			−0.225795			−0.112898	−	0.993607i	$-0.536013\pi$
$37$	−1.37346			−0.225795			−0.112898	+	0.993607i	$0.536013\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−11.0754			−1.72969			−0.864847	−	0.502035i	$-0.832585\pi$
$41$	−11.0754			−1.72969			−0.864847	+	0.502035i	$0.832585\pi$
$42$		0			0
$43$	−9.70199			−1.47954			−0.739770	−	0.672860i	$-0.765066\pi$
$43$	−9.70199			−1.47954			−0.739770	+	0.672860i	$0.765066\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	0.627861i		−	0.0915829i	−0.998951	−	0.0457915i	$-0.985419\pi$
$47$		−	0.627861i		−	0.0915829i	0.998951	−	0.0457915i	$-0.0145810\pi$
$48$		0			0
$49$	−2.92070			−0.417243
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.54695			1.31137			0.655687	−	0.755033i	$-0.272379\pi$
$53$	9.54695			1.31137			0.655687	+	0.755033i	$0.272379\pi$
$54$		0			0
$55$	5.29774	−	1.12228i	0.714347	−	0.151328i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	10.1066i		−	1.31577i	−0.753117	−	0.657887i	$-0.771451\pi$
$59$		−	10.1066i		−	1.31577i	0.753117	−	0.657887i	$-0.228549\pi$
$60$		0			0
$61$			12.7235i			1.62908i	0.580106	+	0.814541i	$0.303011\pi$
$61$			12.7235i			1.62908i	−0.580106	+	0.814541i	$0.696989\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.44701	+	6.83059i	0.179479	+	0.847231i
$66$		0			0
$67$	−10.5125			−1.28431			−0.642155	−	0.766575i	$-0.721959\pi$
$67$	−10.5125			−1.28431			−0.642155	+	0.766575i	$0.721959\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	7.39000			0.877031			0.438516	−	0.898724i	$-0.355504\pi$
$71$	7.39000			0.877031			0.438516	+	0.898724i	$0.355504\pi$
$72$		0			0
$73$			5.73160i			0.670833i	0.942070	+	0.335417i	$0.108877\pi$
$73$			5.73160i			0.670833i	−0.942070	+	0.335417i	$0.891123\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−7.62797			−0.869288
$78$		0			0
$79$	−11.2071			−1.26090			−0.630449	−	0.776230i	$-0.717129\pi$
$79$	−11.2071			−1.26090			−0.630449	+	0.776230i	$0.717129\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.12640			−0.343167			−0.171584	−	0.985170i	$-0.554888\pi$
$83$	−3.12640			−0.343167			−0.171584	+	0.985170i	$0.554888\pi$
$84$		0			0
$85$	15.6450	−	3.31426i	1.69694	−	0.359482i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−15.9470			−1.69038			−0.845192	−	0.534464i	$-0.820514\pi$
$89$	−15.9470			−1.69038			−0.845192	+	0.534464i	$0.820514\pi$
$90$		0			0
$91$		−	9.83507i		−	1.03100i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.13246	+	1.08727i	−0.526579	+	0.111551i
$96$		0			0
$97$		−	14.6589i		−	1.48839i	−0.667963	−	0.744194i	$-0.732834\pi$
$97$		−	14.6589i		−	1.48839i	0.667963	−	0.744194i	$-0.267166\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	6.22901i		−	0.619810i	−0.950768	−	0.309905i	$-0.899703\pi$
$101$		−	6.22901i		−	0.619810i	0.950768	−	0.309905i	$-0.100297\pi$
$102$		0			0
$103$			10.6041i			1.04485i	0.852686	+	0.522424i	$0.174972\pi$
$103$			10.6041i			1.04485i	−0.852686	+	0.522424i	$0.825028\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.62797			−0.157382			−0.0786910	−	0.996899i	$-0.525074\pi$
$107$	−1.62797			−0.157382			−0.0786910	+	0.996899i	$0.525074\pi$
$108$		0			0
$109$			1.56551i			0.149949i	0.997185	+	0.0749743i	$0.0238875\pi$
$109$			1.56551i			0.149949i	−0.997185	+	0.0749743i	$0.976113\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			8.57935i			0.807077i	0.914963	+	0.403538i	$0.132220\pi$
$113$			8.57935i			0.807077i	−0.914963	+	0.403538i	$0.867780\pi$
$114$		0			0
$115$	2.82046	−	0.597492i	0.263010	−	0.0557164i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−22.5265			−2.06500
$120$		0			0
$121$	5.13490			0.466809
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−6.55307	−	9.05855i	−0.586124	−	0.810221i
$126$		0			0
$127$		−	10.7524i		−	0.954124i	−0.878870	−	0.477062i	$-0.841702\pi$
$127$		−	10.7524i		−	0.954124i	0.878870	−	0.477062i	$-0.158298\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	12.2083i		−	1.06664i	−0.845913	−	0.533321i	$-0.820944\pi$
$131$		−	12.2083i		−	1.06664i	0.845913	−	0.533321i	$-0.179056\pi$
$132$		0			0
$133$	7.39000			0.640794
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	7.44288i		−	0.635888i	−0.948110	−	0.317944i	$-0.897008\pi$
$137$		−	7.44288i		−	0.635888i	0.948110	−	0.317944i	$-0.102992\pi$
$138$		0			0
$139$			10.1260i			0.858878i	0.903096	+	0.429439i	$0.141289\pi$
$139$			10.1260i			0.858878i	−0.903096	+	0.429439i	$0.858711\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	7.56213i		−	0.632377i
$144$		0			0
$145$	9.22031	−	1.95325i	0.765705	−	0.162208i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	11.6233i		−	0.952217i	−0.879386	−	0.476109i	$-0.842047\pi$
$149$		−	11.6233i		−	0.952217i	0.879386	−	0.476109i	$-0.157953\pi$
$150$		0			0
$151$	10.0529			0.818094			0.409047	−	0.912513i	$-0.365861\pi$
$151$	10.0529			0.818094			0.409047	+	0.912513i	$0.365861\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.41803	−	6.69379i	−0.113899	−	0.537658i
$156$		0			0
$157$	13.3874			1.06843			0.534217	−	0.845347i	$-0.320606\pi$
$157$	13.3874			1.06843			0.534217	+	0.845347i	$0.320606\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−4.06106			−0.320056
$162$		0			0
$163$	1.37346			0.107578			0.0537888	−	0.998552i	$-0.482870\pi$
$163$	1.37346			0.107578			0.0537888	+	0.998552i	$0.482870\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			14.3227i			1.10832i	0.832410	+	0.554161i	$0.186961\pi$
$167$			14.3227i			1.10832i	−0.832410	+	0.554161i	$0.813039\pi$
$168$		0			0
$169$	−3.24983			−0.249987
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.96953			−0.301798			−0.150899	−	0.988549i	$-0.548217\pi$
$173$	−3.96953			−0.301798			−0.150899	+	0.988549i	$0.548217\pi$
$174$		0			0
$175$	6.38584	+	14.3958i	0.482724	+	1.08822i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	11.3491i		−	0.848273i	−0.905598	−	0.424137i	$-0.860578\pi$
$179$		−	11.3491i		−	0.848273i	0.905598	−	0.424137i	$-0.139422\pi$
$180$		0			0
$181$		−	15.5324i		−	1.15452i	−0.816561	−	0.577259i	$-0.804122\pi$
$181$		−	15.5324i		−	1.15452i	0.816561	−	0.577259i	$-0.195878\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.636473	−	3.00447i	−0.0467944	−	0.220893i
$186$		0			0
$187$	−17.3205			−1.26660
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−27.0225			−1.95528			−0.977639	−	0.210288i	$-0.932560\pi$
$191$	−27.0225			−1.95528			−0.977639	+	0.210288i	$0.932560\pi$
$192$		0			0
$193$			8.40105i			0.604721i	0.953194	+	0.302360i	$0.0977746\pi$
$193$			8.40105i			0.604721i	−0.953194	+	0.302360i	$0.902225\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	10.4003			0.740989			0.370494	−	0.928835i	$-0.379188\pi$
$197$	10.4003			0.740989			0.370494	+	0.928835i	$0.379188\pi$
$198$		0			0
$199$	13.5659			0.961665			0.480832	−	0.876813i	$-0.340335\pi$
$199$	13.5659			0.961665			0.480832	+	0.876813i	$0.340335\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−13.2759			−0.931786
$204$		0			0
$205$	−5.13246	−	24.2278i	−0.358466	−	1.69214i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5.68213			0.393041
$210$		0			0
$211$		−	24.5868i		−	1.69262i	−0.532688	−	0.846312i	$-0.678818\pi$
$211$		−	24.5868i		−	1.69262i	0.532688	−	0.846312i	$-0.321182\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.49599	−	21.2233i	−0.306624	−	1.44742i
$216$		0			0
$217$			9.63809i			0.654276i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	22.3321i		−	1.50222i
$222$		0			0
$223$		−	10.6041i		−	0.710100i	−0.934847	−	0.355050i	$-0.884464\pi$
$223$		−	10.6041i		−	0.710100i	0.934847	−	0.355050i	$-0.115536\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.5958			0.836011			0.418006	−	0.908444i	$-0.362729\pi$
$227$	12.5958			0.836011			0.418006	+	0.908444i	$0.362729\pi$
$228$		0			0
$229$			9.75550i			0.644661i	0.946627	+	0.322331i	$0.104466\pi$
$229$			9.75550i			0.644661i	−0.946627	+	0.322331i	$0.895534\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	1.25572i		−	0.0822651i	−0.999154	−	0.0411325i	$-0.986903\pi$
$233$		−	1.25572i		−	0.0822651i	0.999154	−	0.0411325i	$-0.0130966\pi$
$234$		0			0
$235$	1.37346	−	0.290956i	0.0895946	−	0.0189799i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2.89401			0.187198			0.0935990	−	0.995610i	$-0.470163\pi$
$239$	2.89401			0.187198			0.0935990	+	0.995610i	$0.470163\pi$
$240$		0			0
$241$	22.4166			1.44398			0.721990	−	0.691903i	$-0.243228\pi$
$241$	22.4166			1.44398			0.721990	+	0.691903i	$0.243228\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.35348	−	6.38910i	−0.0864706	−	0.408185i
$246$		0			0
$247$			7.32621i			0.466155i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			12.2787i			0.775023i	0.921865	+	0.387512i	$0.126665\pi$
$251$			12.2787i			0.775023i	−0.921865	+	0.387512i	$0.873335\pi$
$252$		0			0
$253$	−3.12253			−0.196312
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	16.1354i		−	1.00650i	−0.864141	−	0.503250i	$-0.832138\pi$
$257$		−	16.1354i		−	1.00650i	0.864141	−	0.503250i	$-0.167862\pi$
$258$		0			0
$259$			4.32600i			0.268805i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	5.20982i		−	0.321251i	−0.987015	−	0.160626i	$-0.948649\pi$
$263$		−	5.20982i		−	0.321251i	0.987015	−	0.160626i	$-0.0513512\pi$
$264$		0			0
$265$	4.42414	+	20.8842i	0.271773	+	1.28290i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	18.1171i		−	1.10462i	−0.833639	−	0.552310i	$-0.813747\pi$
$269$		−	18.1171i		−	1.10462i	0.833639	−	0.552310i	$-0.186253\pi$
$270$		0			0
$271$	−25.1014			−1.52480			−0.762402	−	0.647104i	$-0.775980\pi$
$271$	−25.1014			−1.52480			−0.762402	+	0.647104i	$0.775980\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.91003	+	11.0688i	0.296086	+	0.667476i
$276$		0			0
$277$	−23.9000			−1.43601			−0.718005	−	0.696038i	$-0.754944\pi$
$277$	−23.9000			−1.43601			−0.718005	+	0.696038i	$0.754944\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	18.8411			1.12396			0.561982	−	0.827150i	$-0.310039\pi$
$281$	18.8411			1.12396			0.561982	+	0.827150i	$0.310039\pi$
$282$		0			0
$283$	30.4794			1.81181			0.905907	−	0.423477i	$-0.139190\pi$
$283$	30.4794			1.81181			0.905907	+	0.423477i	$0.139190\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			34.8845i			2.05917i
$288$		0			0
$289$	−34.1499			−2.00882
$290$		0			0
$291$		0			0
$292$		0			0
$293$	23.6233			1.38009			0.690043	−	0.723768i	$-0.257592\pi$
$293$	23.6233			1.38009			0.690043	+	0.723768i	$0.257592\pi$
$294$		0			0
$295$	22.1085	−	4.68351i	1.28721	−	0.272684i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	4.02601i		−	0.232830i
$300$		0			0
$301$			30.5585i			1.76136i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−27.8330	+	5.89620i	−1.59371	+	0.337615i
$306$		0			0
$307$	9.13906			0.521594			0.260797	−	0.965394i	$-0.416015\pi$
$307$	9.13906			0.521594			0.260797	+	0.965394i	$0.416015\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−29.5409			−1.67511			−0.837555	−	0.546353i	$-0.816016\pi$
$311$	−29.5409			−1.67511			−0.837555	+	0.546353i	$0.816016\pi$
$312$		0			0
$313$		−	11.6116i		−	0.656325i	−0.944621	−	0.328163i	$-0.893571\pi$
$313$		−	11.6116i		−	0.656325i	0.944621	−	0.328163i	$-0.106429\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	23.8869			1.34162			0.670812	−	0.741628i	$-0.265946\pi$
$317$	23.8869			1.34162			0.670812	+	0.741628i	$0.265946\pi$
$318$		0			0
$319$	−10.2078			−0.571525
$320$		0			0
$321$		0			0
$322$		0			0
$323$	16.7801			0.933672
$324$		0			0
$325$	−14.2715	+	6.33071i	−0.791642	+	0.351165i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.97758			−0.109028
$330$		0			0
$331$		−	10.1260i		−	0.556577i	−0.960498	−	0.278288i	$-0.910233\pi$
$331$		−	10.1260i		−	0.556577i	0.960498	−	0.278288i	$-0.0897671\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4.87159	−	22.9964i	−0.266164	−	1.25643i
$336$		0			0
$337$		−	1.55511i		−	0.0847123i	−0.999103	−	0.0423561i	$-0.986514\pi$
$337$		−	1.55511i		−	0.0847123i	0.999103	−	0.0423561i	$-0.0134864\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			7.41068i			0.401311i
$342$		0			0
$343$		−	12.8486i		−	0.693760i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	15.3552			0.824310			0.412155	−	0.911114i	$-0.364776\pi$
$347$	15.3552			0.824310			0.412155	+	0.911114i	$0.364776\pi$
$348$		0			0
$349$			3.28995i			0.176107i	0.996116	+	0.0880536i	$0.0280647\pi$
$349$			3.28995i			0.176107i	−0.996116	+	0.0880536i	$0.971935\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	4.75322i		−	0.252989i	−0.991967	−	0.126494i	$-0.959627\pi$
$353$		−	4.75322i		−	0.252989i	0.991967	−	0.126494i	$-0.0403725\pi$
$354$		0			0
$355$	3.42459	+	16.1658i	0.181758	+	0.857990i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	7.76560			0.409853			0.204926	−	0.978777i	$-0.434304\pi$
$359$	7.76560			0.409853			0.204926	+	0.978777i	$0.434304\pi$
$360$		0			0
$361$	13.4951			0.710271
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−12.5380	+	2.65607i	−0.656269	+	0.139025i
$366$		0			0
$367$		−	16.8214i		−	0.878068i	−0.898471	−	0.439034i	$-0.855321\pi$
$367$		−	16.8214i		−	0.878068i	0.898471	−	0.439034i	$-0.144679\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	30.0702i		−	1.56117i
$372$		0			0
$373$	5.99745			0.310536			0.155268	−	0.987872i	$-0.450376\pi$
$373$	5.99745			0.310536			0.155268	+	0.987872i	$0.450376\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	13.1613i		−	0.677842i
$378$		0			0
$379$			22.3073i			1.14585i	0.819608	+	0.572925i	$0.194192\pi$
$379$			22.3073i			1.14585i	−0.819608	+	0.572925i	$0.805808\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	38.3484i		−	1.95951i	−0.200198	−	0.979756i	$-0.564158\pi$
$383$		−	38.3484i		−	1.95951i	0.200198	−	0.979756i	$-0.435842\pi$
$384$		0			0
$385$	−3.53487	−	16.6864i	−0.180154	−	0.850415i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			17.2044i			0.872298i	0.899874	+	0.436149i	$0.143658\pi$
$389$			17.2044i			0.872298i	−0.899874	+	0.436149i	$0.856342\pi$
$390$		0			0
$391$	−9.22127			−0.466340
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−5.19347	−	24.5158i	−0.261312	−	1.23352i
$396$		0			0
$397$	−12.0140			−0.602965			−0.301482	−	0.953472i	$-0.597481\pi$
$397$	−12.0140			−0.602965			−0.301482	+	0.953472i	$0.597481\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−16.7576			−0.836833			−0.418417	−	0.908255i	$-0.637415\pi$
$401$	−16.7576			−0.836833			−0.418417	+	0.908255i	$0.637415\pi$
$402$		0			0
$403$	−9.55490			−0.475963
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3.32624i			0.164876i
$408$		0			0
$409$	−33.7162			−1.66716			−0.833580	−	0.552399i	$-0.813712\pi$
$409$	−33.7162			−1.66716			−0.833580	+	0.552399i	$0.813712\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−31.8331			−1.56640
$414$		0			0
$415$	−1.44880	−	6.83908i	−0.0711189	−	0.335717i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			25.1125i			1.22683i	0.789762	+	0.613413i	$0.210204\pi$
$419$			25.1125i			1.22683i	−0.789762	+	0.613413i	$0.789796\pi$
$420$		0			0
$421$		−	19.7185i		−	0.961023i	−0.876988	−	0.480511i	$-0.840451\pi$
$421$		−	19.7185i		−	0.961023i	0.876988	−	0.480511i	$-0.159549\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	14.5000	+	32.6879i	0.703355	+	1.58560i
$426$		0			0
$427$	40.0755			1.93939
$428$		0			0
$429$		0			0
$430$		0			0
$431$	21.2536			1.02375			0.511874	−	0.859061i	$-0.328951\pi$
$431$	21.2536			1.02375			0.511874	+	0.859061i	$0.328951\pi$
$432$		0			0
$433$		−	33.7002i		−	1.61953i	−0.586757	−	0.809763i	$-0.699596\pi$
$433$		−	33.7002i		−	1.61953i	0.586757	−	0.809763i	$-0.300404\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.02511			0.144711
$438$		0			0
$439$	−25.1313			−1.19945			−0.599725	−	0.800206i	$-0.704723\pi$
$439$	−25.1313			−1.19945			−0.599725	+	0.800206i	$0.704723\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	20.5894			0.978230			0.489115	−	0.872219i	$-0.337320\pi$
$443$	20.5894			0.978230			0.489115	+	0.872219i	$0.337320\pi$
$444$		0			0
$445$	−7.39000	−	34.8845i	−0.350319	−	1.65368i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−9.09787			−0.429355			−0.214677	−	0.976685i	$-0.568870\pi$
$449$	−9.09787			−0.429355			−0.214677	+	0.976685i	$0.568870\pi$
$450$		0			0
$451$			26.8225i			1.26302i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	21.5144	−	4.55765i	1.00861	−	0.213666i
$456$		0			0
$457$			7.10221i			0.332228i	0.986107	+	0.166114i	$0.0531219\pi$
$457$			7.10221i			0.332228i	−0.986107	+	0.166114i	$0.946878\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	2.20090i		−	0.102506i	−0.998686	−	0.0512530i	$-0.983679\pi$
$461$		−	2.20090i		−	0.102506i	0.998686	−	0.0512530i	$-0.0163215\pi$
$462$		0			0
$463$			28.7916i			1.33806i	0.743236	+	0.669029i	$0.233290\pi$
$463$			28.7916i			1.33806i	−0.743236	+	0.669029i	$0.766710\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−37.9504			−1.75613			−0.878067	−	0.478538i	$-0.841167\pi$
$467$	−37.9504			−1.75613			−0.878067	+	0.478538i	$0.841167\pi$
$468$		0			0
$469$			33.1114i			1.52894i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			23.4963i			1.08036i
$474$		0			0
$475$	−4.75685	−	10.7235i	−0.218259	−	0.492029i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.49599			0.205427			0.102713	−	0.994711i	$-0.467248\pi$
$479$	4.49599			0.205427			0.102713	+	0.994711i	$0.467248\pi$
$480$		0			0
$481$	−4.28866			−0.195546
$482$		0			0
$483$		0			0
$484$		0			0
$485$	32.0667	−	6.79307i	1.45607	−	0.308457i
$486$		0			0
$487$			38.9154i			1.76342i	0.471789	+	0.881712i	$0.343609\pi$
$487$			38.9154i			1.76342i	−0.471789	+	0.881712i	$0.656391\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	4.19054i		−	0.189117i	−0.995519	−	0.0945583i	$-0.969856\pi$
$491$		−	4.19054i		−	0.189117i	0.995519	−	0.0945583i	$-0.0301439\pi$
$492$		0			0
$493$	−30.1450			−1.35766
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	23.2764i		−	1.04409i
$498$		0			0
$499$			37.4693i			1.67735i	0.544629	+	0.838677i	$0.316670\pi$
$499$			37.4693i			1.67735i	−0.544629	+	0.838677i	$0.683330\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			15.8317i			0.705900i	0.935642	+	0.352950i	$0.114821\pi$
$503$			15.8317i			0.705900i	−0.935642	+	0.352950i	$0.885179\pi$
$504$		0			0
$505$	13.6261	−	2.88658i	0.606354	−	0.128451i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			15.8382i			0.702018i	0.936372	+	0.351009i	$0.114161\pi$
$509$			15.8382i			0.702018i	−0.936372	+	0.351009i	$0.885839\pi$
$510$		0			0
$511$	18.0529			0.798614
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−23.1966	+	4.91401i	−1.02216	+	0.216537i
$516$		0			0
$517$	−1.52055			−0.0668738
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−2.08348			−0.0912788			−0.0456394	−	0.998958i	$-0.514533\pi$
$521$	−2.08348			−0.0912788			−0.0456394	+	0.998958i	$0.514533\pi$
$522$		0			0
$523$	−13.8224			−0.604410			−0.302205	−	0.953243i	$-0.597723\pi$
$523$	−13.8224			−0.604410			−0.302205	+	0.953243i	$0.597723\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			21.8848i			0.953317i
$528$		0			0
$529$	21.3376			0.927722
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−34.5834			−1.49797
$534$		0			0
$535$	−0.754416	−	3.56122i	−0.0326163	−	0.153965i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			7.07335i			0.304671i
$540$		0			0
$541$		−	6.68097i		−	0.287238i	−0.989633	−	0.143619i	$-0.954126\pi$
$541$		−	6.68097i		−	0.287238i	0.989633	−	0.143619i	$-0.0458739\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.42459	+	0.725471i	−0.146693	+	0.0310758i
$546$		0			0
$547$	18.9416			0.809883			0.404941	−	0.914343i	$-0.367292\pi$
$547$	18.9416			0.809883			0.404941	+	0.914343i	$0.367292\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	9.88931			0.421299
$552$		0			0
$553$			35.2992i			1.50108i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−33.0653			−1.40102			−0.700511	−	0.713642i	$-0.747044\pi$
$557$	−33.0653			−1.40102			−0.700511	+	0.713642i	$0.747044\pi$
$558$		0			0
$559$	−30.2947			−1.28133
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.95023			−0.124337			−0.0621686	−	0.998066i	$-0.519802\pi$
$563$	−2.95023			−0.124337			−0.0621686	+	0.998066i	$0.519802\pi$
$564$		0			0
$565$	−18.7675	+	3.97574i	−0.789555	+	0.167261i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.78811			−0.116884			−0.0584419	−	0.998291i	$-0.518613\pi$
$569$	−2.78811			−0.116884			−0.0584419	+	0.998291i	$0.518613\pi$
$570$		0			0
$571$		−	15.0427i		−	0.629516i	−0.949172	−	0.314758i	$-0.898077\pi$
$571$		−	15.0427i		−	0.629516i	0.949172	−	0.314758i	$-0.101923\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.61406	+	5.89295i	0.109014	+	0.245753i
$576$		0			0
$577$		−	23.1614i		−	0.964220i	−0.876111	−	0.482110i	$-0.839871\pi$
$577$		−	23.1614i		−	0.964220i	0.876111	−	0.482110i	$-0.160129\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			9.84728i			0.408534i
$582$		0			0
$583$		−	23.1208i		−	0.957565i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−14.0705			−0.580751			−0.290376	−	0.956913i	$-0.593780\pi$
$587$	−14.0705			−0.580751			−0.290376	+	0.956913i	$0.593780\pi$
$588$		0			0
$589$		−	7.17948i		−	0.295825i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	1.42743i		−	0.0586174i	−0.999570	−	0.0293087i	$-0.990669\pi$
$593$		−	1.42743i		−	0.0586174i	0.999570	−	0.0293087i	$-0.00933058\pi$
$594$		0			0
$595$	−10.4390	−	49.2772i	−0.427956	−	2.02017i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	26.6469			1.08876			0.544381	−	0.838838i	$-0.316765\pi$
$599$	26.6469			1.08876			0.544381	+	0.838838i	$0.316765\pi$
$600$		0			0
$601$	−19.8748			−0.810710			−0.405355	−	0.914159i	$-0.632852\pi$
$601$	−19.8748			−0.810710			−0.405355	+	0.914159i	$0.632852\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.37955	+	11.2327i	0.0967427	+	0.456674i
$606$		0			0
$607$			20.7546i			0.842403i	0.906967	+	0.421202i	$0.138391\pi$
$607$			20.7546i			0.842403i	−0.906967	+	0.421202i	$0.861609\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	1.96051i		−	0.0793138i
$612$		0			0
$613$	−19.4040			−0.783719			−0.391860	−	0.920025i	$-0.628168\pi$
$613$	−19.4040			−0.783719			−0.391860	+	0.920025i	$0.628168\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	38.5389i		−	1.55152i	−0.631029	−	0.775759i	$-0.717367\pi$
$617$		−	38.5389i		−	1.55152i	0.631029	−	0.775759i	$-0.282633\pi$
$618$		0			0
$619$			14.4520i			0.580876i	0.956894	+	0.290438i	$0.0938010\pi$
$619$			14.4520i			0.580876i	−0.956894	+	0.290438i	$0.906199\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			50.2286i			2.01237i
$624$		0			0
$625$	16.7790	−	18.5328i	0.671161	−	0.741311i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			9.82287i			0.391663i
$630$		0			0
$631$	40.5922			1.61595			0.807974	−	0.589218i	$-0.200564\pi$
$631$	40.5922			1.61595			0.807974	+	0.589218i	$0.200564\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	23.5212	−	4.98277i	0.933410	−	0.197735i
$636$		0			0
$637$	−9.11997			−0.361346
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−11.8860			−0.469468			−0.234734	−	0.972060i	$-0.575422\pi$
$641$	−11.8860			−0.469468			−0.234734	+	0.972060i	$0.575422\pi$
$642$		0			0
$643$	29.5630			1.16585			0.582925	−	0.812526i	$-0.301908\pi$
$643$	29.5630			1.16585			0.582925	+	0.812526i	$0.301908\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			13.5816i			0.533949i	0.963704	+	0.266974i	$0.0860239\pi$
$647$			13.5816i			0.533949i	−0.963704	+	0.266974i	$0.913976\pi$
$648$		0			0
$649$	−24.4763			−0.960777
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2.80243			−0.109667			−0.0548337	−	0.998496i	$-0.517463\pi$
$653$	−2.80243			−0.109667			−0.0548337	+	0.998496i	$0.517463\pi$
$654$		0			0
$655$	26.7058	−	5.65741i	1.04348	−	0.221053i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	0.0288607i		−	0.00112425i	−1.00000		0.000562127i	$-0.999821\pi$
$659$		−	0.0288607i		−	0.00112425i	1.00000		0.000562127i	$-0.000178931\pi$
$660$		0			0
$661$			2.35497i			0.0915977i	0.998951	+	0.0457989i	$0.0145833\pi$
$661$			2.35497i			0.0915977i	−0.998951	+	0.0457989i	$0.985417\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3.42459	+	16.1658i	0.132800	+	0.626882i
$666$		0			0
$667$	−5.43452			−0.210425
$668$		0			0
$669$		0			0
$670$		0			0
$671$	30.8138			1.18956
$672$		0			0
$673$		−	39.6306i		−	1.52765i	−0.645425	−	0.763823i	$-0.723320\pi$
$673$		−	39.6306i		−	1.52765i	0.645425	−	0.763823i	$-0.276680\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2.57428			−0.0989377			−0.0494688	−	0.998776i	$-0.515753\pi$
$677$	−2.57428			−0.0989377			−0.0494688	+	0.998776i	$0.515753\pi$
$678$		0			0
$679$	−46.1714			−1.77190
$680$		0			0
$681$		0			0
$682$		0			0
$683$	4.26099			0.163042			0.0815211	−	0.996672i	$-0.474022\pi$
$683$	4.26099			0.163042			0.0815211	+	0.996672i	$0.474022\pi$
$684$		0			0
$685$	16.2815	−	3.44909i	0.622082	−	0.131783i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	29.8106			1.13569
$690$		0			0
$691$		−	32.7768i		−	1.24689i	−0.781868	−	0.623443i	$-0.785733\pi$
$691$		−	32.7768i		−	1.24689i	0.781868	−	0.623443i	$-0.214267\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−22.1509	+	4.69249i	−0.840231	+	0.177996i
$696$		0			0
$697$			79.2107i			3.00032i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			33.0385i			1.24785i	0.781486	+	0.623923i	$0.214462\pi$
$701$			33.0385i			1.24785i	−0.781486	+	0.623923i	$0.785538\pi$
$702$		0			0
$703$		−	3.22247i		−	0.121538i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−19.6196			−0.737872
$708$		0			0
$709$			26.3511i			0.989637i	0.868996	+	0.494818i	$0.164765\pi$
$709$			26.3511i			0.989637i	−0.868996	+	0.494818i	$0.835235\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			3.94537i			0.147755i
$714$		0			0
$715$	16.5423	−	3.50435i	0.618648	−	0.131055i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−14.7800			−0.551201			−0.275600	−	0.961272i	$-0.588877\pi$
$719$	−14.7800			−0.551201			−0.275600	+	0.961272i	$0.588877\pi$
$720$		0			0
$721$	33.3998			1.24387
$722$		0			0
$723$		0			0
$724$		0			0
$725$	8.54554	+	19.2645i	0.317373	+	0.715465i
$726$		0			0
$727$		−	10.7322i		−	0.398035i	−0.979996	−	0.199018i	$-0.936225\pi$
$727$		−	10.7322i		−	0.398035i	0.979996	−	0.199018i	$-0.0637751\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			69.3878i			2.56640i
$732$		0			0
$733$	20.7774			0.767432			0.383716	−	0.923451i	$-0.374644\pi$
$733$	20.7774			0.767432			0.383716	+	0.923451i	$0.374644\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			25.4592i			0.937802i
$738$		0			0
$739$		−	14.7167i		−	0.541363i	−0.962669	−	0.270682i	$-0.912751\pi$
$739$		−	14.7167i		−	0.541363i	0.962669	−	0.270682i	$-0.0872490\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	10.2244i		−	0.375098i	−0.982255	−	0.187549i	$-0.939946\pi$
$743$		−	10.2244i		−	0.375098i	0.982255	−	0.187549i	$-0.0600543\pi$
$744$		0			0
$745$	25.4262	−	5.38633i	0.931544	−	0.197340i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			5.12765i			0.187360i
$750$		0			0
$751$	45.7533			1.66956			0.834780	−	0.550584i	$-0.185595\pi$
$751$	45.7533			1.66956			0.834780	+	0.550584i	$0.185595\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.65860	+	21.9910i	0.169544	+	0.800333i
$756$		0			0
$757$	54.8143			1.99226			0.996130	−	0.0878903i	$-0.0280125\pi$
$757$	54.8143			1.99226			0.996130	+	0.0878903i	$0.0280125\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	23.3179			0.845275			0.422637	−	0.906299i	$-0.361104\pi$
$761$	23.3179			0.845275			0.422637	+	0.906299i	$0.361104\pi$
$762$		0			0
$763$	4.93091			0.178511
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	31.5583i		−	1.13950i
$768$		0			0
$769$	−9.56833			−0.345043			−0.172521	−	0.985006i	$-0.555191\pi$
$769$	−9.56833			−0.345043			−0.172521	+	0.985006i	$0.555191\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	22.8294			0.821117			0.410559	−	0.911834i	$-0.365334\pi$
$773$	22.8294			0.821117			0.410559	+	0.911834i	$0.365334\pi$
$774$		0			0
$775$	13.9857	−	6.20392i	0.502381	−	0.222852i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	25.9857i		−	0.931034i
$780$		0			0
$781$		−	17.8971i		−	0.640408i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	6.20386	+	29.2853i	0.221425	+	1.04524i
$786$		0			0
$787$	23.6303			0.842328			0.421164	−	0.906985i	$-0.361622\pi$
$787$	23.6303			0.842328			0.421164	+	0.906985i	$0.361622\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	27.0225			0.960809
$792$		0			0
$793$			39.7296i			1.41084i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	15.5592			0.551134			0.275567	−	0.961282i	$-0.411134\pi$
$797$	15.5592			0.551134			0.275567	+	0.961282i	$0.411134\pi$
$798$		0			0
$799$	−4.49041			−0.158859
$800$		0			0
$801$		0			0
$802$		0			0
$803$	13.8808			0.489842
$804$		0			0
$805$	−1.88193	−	8.88366i	−0.0663293	−	0.313108i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	12.3484			0.434146			0.217073	−	0.976155i	$-0.430349\pi$
$809$	12.3484			0.434146			0.217073	+	0.976155i	$0.430349\pi$
$810$		0			0
$811$			36.9485i			1.29744i	0.761028	+	0.648719i	$0.224695\pi$
$811$			36.9485i			1.29744i	−0.761028	+	0.648719i	$0.775305\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.636473	+	3.00447i	0.0222947	+	0.105242i
$816$		0			0
$817$		−	22.7632i		−	0.796384i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			10.1482i			0.354175i	0.984195	+	0.177087i	$0.0566675\pi$
$821$			10.1482i			0.354175i	−0.984195	+	0.177087i	$0.943332\pi$
$822$		0			0
$823$		−	21.0044i		−	0.732166i	−0.930582	−	0.366083i	$-0.880699\pi$
$823$		−	21.0044i		−	0.732166i	0.930582	−	0.366083i	$-0.119301\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.54049			0.0535682			0.0267841	−	0.999641i	$-0.491473\pi$
$827$	1.54049			0.0535682			0.0267841	+	0.999641i	$0.491473\pi$
$828$		0			0
$829$			6.68097i			0.232040i	0.993247	+	0.116020i	$0.0370136\pi$
$829$			6.68097i			0.232040i	−0.993247	+	0.116020i	$0.962986\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			20.8886i			0.723748i
$834$		0			0
$835$	−31.3312	+	6.63725i	−1.08426	+	0.229691i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	33.1670			1.14505			0.572527	−	0.819886i	$-0.305963\pi$
$839$	33.1670			1.14505			0.572527	+	0.819886i	$0.305963\pi$
$840$		0			0
$841$	11.2342			0.387385
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.50600	−	7.10907i	−0.0518079	−	0.244560i
$846$		0			0
$847$		−	16.1735i		−	0.555727i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			1.77086i			0.0607043i
$852$		0			0
$853$	−41.5549			−1.42281			−0.711406	−	0.702781i	$-0.751941\pi$
$853$	−41.5549			−1.42281			−0.711406	+	0.702781i	$0.751941\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	40.4225i		−	1.38081i	−0.723425	−	0.690403i	$-0.757433\pi$
$857$		−	40.4225i		−	1.38081i	0.723425	−	0.690403i	$-0.242567\pi$
$858$		0			0
$859$			27.9251i			0.952794i	0.879230	+	0.476397i	$0.158057\pi$
$859$			27.9251i			0.952794i	−0.879230	+	0.476397i	$0.841943\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	38.4008i		−	1.30718i	−0.756849	−	0.653590i	$-0.773262\pi$
$863$		−	38.4008i		−	1.30718i	0.756849	−	0.653590i	$-0.226738\pi$
$864$		0			0
$865$	−1.83952	−	8.68344i	−0.0625454	−	0.295246i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			27.1414i			0.920708i
$870$		0			0
$871$	−32.8256			−1.11225
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−28.5318	+	20.6403i	−0.964553	+	0.697769i
$876$		0			0
$877$	21.1530			0.714288			0.357144	−	0.934049i	$-0.383751\pi$
$877$	21.1530			0.714288			0.357144	+	0.934049i	$0.383751\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	33.9776			1.14473			0.572367	−	0.819998i	$-0.306025\pi$
$881$	33.9776			1.14473			0.572367	+	0.819998i	$0.306025\pi$
$882$		0			0
$883$	−14.5736			−0.490440			−0.245220	−	0.969467i	$-0.578860\pi$
$883$	−14.5736			−0.490440			−0.245220	+	0.969467i	$0.578860\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			24.3962i			0.819146i	0.912277	+	0.409573i	$0.134322\pi$
$887$			24.3962i			0.819146i	−0.912277	+	0.409573i	$0.865678\pi$
$888$		0			0
$889$	−33.8671			−1.13587
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.47311			0.0492959
$894$		0			0
$895$	24.8264	−	5.25928i	0.829857	−	0.175798i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			12.8977i			0.430163i
$900$		0			0
$901$		−	68.2790i		−	2.27470i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	33.9776	−	7.19787i	1.12945	−	0.239265i
$906$		0			0
$907$	−24.7326			−0.821233			−0.410616	−	0.911808i	$-0.634686\pi$
$907$	−24.7326			−0.821233			−0.410616	+	0.911808i	$0.634686\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	29.3757			0.973260			0.486630	−	0.873608i	$-0.338226\pi$
$911$	29.3757			0.973260			0.486630	+	0.873608i	$0.338226\pi$
$912$		0			0
$913$			7.57152i			0.250581i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−38.4526			−1.26982
$918$		0			0
$919$	−18.4626			−0.609026			−0.304513	−	0.952508i	$-0.598494\pi$
$919$	−18.4626			−0.609026			−0.304513	+	0.952508i	$0.598494\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	23.0755			0.759538
$924$		0			0
$925$	6.27740	−	2.78460i	0.206400	−	0.0915570i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	45.4688			1.49178			0.745892	−	0.666067i	$-0.232023\pi$
$929$	45.4688			1.49178			0.745892	+	0.666067i	$0.232023\pi$
$930$		0			0
$931$		−	6.85268i		−	0.224588i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−8.02647	−	37.8890i	−0.262494	−	1.23910i
$936$		0			0
$937$			5.96656i			0.194919i	0.995239	+	0.0974595i	$0.0310716\pi$
$937$			5.96656i			0.194919i	−0.995239	+	0.0974595i	$0.968928\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			11.2275i			0.366006i	0.983112	+	0.183003i	$0.0585818\pi$
$941$			11.2275i			0.366006i	−0.983112	+	0.183003i	$0.941418\pi$
$942$		0			0
$943$			14.2800i			0.465022i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	33.1571			1.07746			0.538731	−	0.842478i	$-0.318904\pi$
$947$	33.1571			1.07746			0.538731	+	0.842478i	$0.318904\pi$
$948$		0			0
$949$			17.8971i			0.580964i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	9.07923i		−	0.294105i	−0.989129	−	0.147053i	$-0.953021\pi$
$953$		−	9.07923i		−	0.294105i	0.989129	−	0.147053i	$-0.0469786\pi$
$954$		0			0
$955$	−12.5225	−	59.1123i	−0.405217	−	1.91283i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−23.4429			−0.757012
$960$		0			0
$961$	−21.6365			−0.697950
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−18.3775	+	3.89312i	−0.591592	+	0.125324i
$966$		0			0
$967$			41.4918i			1.33429i	0.744929	+	0.667144i	$0.232483\pi$
$967$			41.4918i			1.33429i	−0.744929	+	0.667144i	$0.767517\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			50.8245i			1.63104i	0.578731	+	0.815519i	$0.303548\pi$
$971$			50.8245i			1.63104i	−0.578731	+	0.815519i	$0.696452\pi$
$972$		0			0
$973$	31.8941			1.02248
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	8.25072i		−	0.263964i	−0.991252	−	0.131982i	$-0.957866\pi$
$977$		−	8.25072i		−	0.263964i	0.991252	−	0.131982i	$-0.0421341\pi$
$978$		0			0
$979$			38.6205i			1.23432i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	16.5792i		−	0.528796i	−0.964414	−	0.264398i	$-0.914827\pi$
$983$		−	16.5792i		−	0.528796i	0.964414	−	0.264398i	$-0.0851732\pi$
$984$		0			0
$985$	4.81958	+	22.7508i	0.153564	+	0.724902i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			12.5092i			0.397769i
$990$		0			0
$991$	15.3390			0.487260			0.243630	−	0.969868i	$-0.421662\pi$
$991$	15.3390			0.487260			0.243630	+	0.969868i	$0.421662\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	6.28658	+	29.6758i	0.199298	+	0.940786i
$996$		0			0
$997$	40.0534			1.26850			0.634252	−	0.773126i	$-0.281308\pi$
$997$	40.0534			1.26850			0.634252	+	0.773126i	$0.281308\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4320.2.d.g.3889.12		16
3.2	odd	2		4320.2.d.h.3889.6		16
4.3	odd	2		1080.2.d.g.109.16	yes	16
5.4	even	2		4320.2.d.h.3889.7		16
8.3	odd	2		1080.2.d.h.109.3	yes	16
8.5	even	2		4320.2.d.h.3889.5		16
12.11	even	2		1080.2.d.h.109.1	yes	16
15.14	odd	2	inner	4320.2.d.g.3889.9		16
20.19	odd	2		1080.2.d.h.109.2	yes	16
24.5	odd	2	inner	4320.2.d.g.3889.11		16
24.11	even	2		1080.2.d.g.109.14	yes	16
40.19	odd	2		1080.2.d.g.109.13	✓	16
40.29	even	2	inner	4320.2.d.g.3889.10		16
60.59	even	2		1080.2.d.g.109.15	yes	16
120.29	odd	2		4320.2.d.h.3889.8		16
120.59	even	2		1080.2.d.h.109.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.d.g.109.13	✓	16	40.19	odd	2
1080.2.d.g.109.14	yes	16	24.11	even	2
1080.2.d.g.109.15	yes	16	60.59	even	2
1080.2.d.g.109.16	yes	16	4.3	odd	2
1080.2.d.h.109.1	yes	16	12.11	even	2
1080.2.d.h.109.2	yes	16	20.19	odd	2
1080.2.d.h.109.3	yes	16	8.3	odd	2
1080.2.d.h.109.4	yes	16	120.59	even	2
4320.2.d.g.3889.9		16	15.14	odd	2	inner
4320.2.d.g.3889.10		16	40.29	even	2	inner
4320.2.d.g.3889.11		16	24.5	odd	2	inner
4320.2.d.g.3889.12		16	1.1	even	1	trivial
4320.2.d.h.3889.5		16	8.5	even	2
4320.2.d.h.3889.6		16	3.2	odd	2
4320.2.d.h.3889.7		16	5.4	even	2
4320.2.d.h.3889.8		16	120.29	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:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.463409 + 2.18752i) q^{5} -3.14971i q^{7} -2.42180i q^{11} +3.12253 q^{13} -7.15192i q^{17} +2.34624i q^{19} -1.28934i q^{23} +(-4.57050 + 2.02743i) q^{25} -4.21495i q^{29} -3.05999 q^{31} +(6.89007 - 1.45960i) q^{35} -1.37346 q^{37} -11.0754 q^{41} -9.70199 q^{43} -0.627861i q^{47} -2.92070 q^{49} +9.54695 q^{53} +(5.29774 - 1.12228i) q^{55} -10.1066i q^{59} +12.7235i q^{61} +(1.44701 + 6.83059i) q^{65} -10.5125 q^{67} +7.39000 q^{71} +5.73160i q^{73} -7.62797 q^{77} -11.2071 q^{79} -3.12640 q^{83} +(15.6450 - 3.31426i) q^{85} -15.9470 q^{89} -9.83507i q^{91} +(-5.13246 + 1.08727i) q^{95} -14.6589i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{5} - 22 q^{25} + 2 q^{35} - 44 q^{49} + 96 q^{53} - 34 q^{55} - 12 q^{77} - 4 q^{79} - 64 q^{83}+O(q^{100}) \) 16 * q - 6 * q^5 - 22 * q^25 + 2 * q^35 - 44 * q^49 + 96 * q^53 - 34 * q^55 - 12 * q^77 - 4 * q^79 - 64 * q^83

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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