LMFDB - Embedded newform 4560.2.d.i.2431.11

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:	$12$
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} - 4 x^{11} + 8 x^{10} + 4 x^{9} + 36 x^{8} - 128 x^{7} + 232 x^{6} + 104 x^{5} + 324 x^{4} + \cdots + 64 $ x^12 - 4x^11 + 8x^10 + 4x^9 + 36x^8 - 128x^7 + 232x^6 + 104x^5 + 324x^4 - 784x^3 + 800x^2 - 320*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.11
Root			$2.28268 + 2.28268i$ of defining polynomial
Character	$\chi$	$=$	4560.2431
Dual form			4560.2.d.i.2431.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +3.67104i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +3.67104i q^{7} +1.00000 q^{9} +4.75021i q^{11} -4.23639i q^{13} +1.00000 q^{15} +1.85589 q^{17} +(-1.32070 - 4.15400i) q^{19} -3.67104i q^{21} -4.55656i q^{23} +1.00000 q^{25} -1.00000 q^{27} -4.50254i q^{29} -2.95714 q^{31} -4.75021i q^{33} -3.67104i q^{35} +2.49374i q^{37} +4.23639i q^{39} -4.10196i q^{41} -0.970359i q^{43} -1.00000 q^{45} +4.14411i q^{47} -6.47651 q^{49} -1.85589 q^{51} -14.2232i q^{53} -4.75021i q^{55} +(1.32070 + 4.15400i) q^{57} +0.965936 q^{59} +3.72057 q^{61} +3.67104i q^{63} +4.23639i q^{65} +4.81585 q^{67} +4.55656i q^{69} -4.62810 q^{71} -0.257348 q^{73} -1.00000 q^{75} -17.4382 q^{77} +13.6652 q^{79} +1.00000 q^{81} -1.93878i q^{83} -1.85589 q^{85} +4.50254i q^{87} +2.54529i q^{89} +15.5520 q^{91} +2.95714 q^{93} +(1.32070 + 4.15400i) q^{95} +11.9942i q^{97} +4.75021i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} - 12 q^{5} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 - 12 * q^5 + 12 * q^9	$ 12 q - 12 q^{3} - 12 q^{5} + 12 q^{9} + 12 q^{15} - 8 q^{17} - 8 q^{19} + 12 q^{25} - 12 q^{27} - 12 q^{45} - 20 q^{49} + 8 q^{51} + 8 q^{57} - 8 q^{59} + 8 q^{67} - 32 q^{71} - 24 q^{73} - 12 q^{75} + 56 q^{77} - 16 q^{79} + 12 q^{81} + 8 q^{85} + 16 q^{91} + 8 q^{95}+O(q^{100}) $ 12 * q - 12 * q^3 - 12 * q^5 + 12 * q^9 + 12 * q^15 - 8 * q^17 - 8 * q^19 + 12 * q^25 - 12 * q^27 - 12 * q^45 - 20 * q^49 + 8 * q^51 + 8 * q^57 - 8 * q^59 + 8 * q^67 - 32 * q^71 - 24 * q^73 - 12 * q^75 + 56 * q^77 - 16 * q^79 + 12 * q^81 + 8 * q^85 + 16 * q^91 + 8 * 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
$7$			3.67104i			1.38752i	0.720206	+	0.693761i	$0.244047\pi$
$7$			3.67104i			1.38752i	−0.720206	+	0.693761i	$0.755953\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			4.75021i			1.43224i	0.697976	+	0.716121i	$0.254084\pi$
$11$			4.75021i			1.43224i	−0.697976	+	0.716121i	$0.745916\pi$
$12$		0			0
$13$		−	4.23639i		−	1.17496i	−0.809237	−	0.587482i	$-0.800119\pi$
$13$		−	4.23639i		−	1.17496i	0.809237	−	0.587482i	$-0.199881\pi$
$14$		0			0
$15$	1.00000			0.258199
$16$		0			0
$17$	1.85589			0.450119			0.225059	−	0.974345i	$-0.427742\pi$
$17$	1.85589			0.450119			0.225059	+	0.974345i	$0.427742\pi$
$18$		0			0
$19$	−1.32070	−	4.15400i	−0.302989	−	0.952994i
$19$	−1.32070	−	4.15400i	−0.302989	−	0.952994i
$20$		0			0
$21$		−	3.67104i		−	0.801086i
$22$		0			0
$23$		−	4.55656i		−	0.950109i	−0.879956	−	0.475055i	$-0.842428\pi$
$23$		−	4.55656i		−	0.950109i	0.879956	−	0.475055i	$-0.157572\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$		−	4.50254i		−	0.836100i	−0.908424	−	0.418050i	$-0.862714\pi$
$29$		−	4.50254i		−	0.836100i	0.908424	−	0.418050i	$-0.137286\pi$
$30$		0			0
$31$	−2.95714			−0.531119			−0.265559	−	0.964095i	$-0.585557\pi$
$31$	−2.95714			−0.531119			−0.265559	+	0.964095i	$0.585557\pi$
$32$		0			0
$33$		−	4.75021i		−	0.826905i
$34$		0			0
$35$		−	3.67104i		−	0.620518i
$36$		0			0
$37$			2.49374i			0.409969i	0.978765	+	0.204984i	$0.0657143\pi$
$37$			2.49374i			0.409969i	−0.978765	+	0.204984i	$0.934286\pi$
$38$		0			0
$39$			4.23639i			0.678366i
$40$		0			0
$41$		−	4.10196i		−	0.640618i	−0.947313	−	0.320309i	$-0.896213\pi$
$41$		−	4.10196i		−	0.640618i	0.947313	−	0.320309i	$-0.103787\pi$
$42$		0			0
$43$		−	0.970359i		−	0.147978i	−0.997259	−	0.0739892i	$-0.976427\pi$
$43$		−	0.970359i		−	0.147978i	0.997259	−	0.0739892i	$-0.0235730\pi$
$44$		0			0
$45$	−1.00000			−0.149071
$46$		0			0
$47$			4.14411i			0.604481i	0.953232	+	0.302240i	$0.0977346\pi$
$47$			4.14411i			0.604481i	−0.953232	+	0.302240i	$0.902265\pi$
$48$		0			0
$49$	−6.47651			−0.925216
$50$		0			0
$51$	−1.85589			−0.259876
$52$		0			0
$53$		−	14.2232i		−	1.95370i	−0.213917	−	0.976852i	$-0.568622\pi$
$53$		−	14.2232i		−	1.95370i	0.213917	−	0.976852i	$-0.431378\pi$
$54$		0			0
$55$		−	4.75021i		−	0.640518i
$56$		0			0
$57$	1.32070	+	4.15400i	0.174931	+	0.550211i
$58$		0			0
$59$	0.965936			0.125754			0.0628771	−	0.998021i	$-0.479972\pi$
$59$	0.965936			0.125754			0.0628771	+	0.998021i	$0.479972\pi$
$60$		0			0
$61$	3.72057			0.476370			0.238185	−	0.971220i	$-0.423448\pi$
$61$	3.72057			0.476370			0.238185	+	0.971220i	$0.423448\pi$
$62$		0			0
$63$			3.67104i			0.462507i
$64$		0			0
$65$			4.23639i			0.525460i
$66$		0			0
$67$	4.81585			0.588350			0.294175	−	0.955752i	$-0.404955\pi$
$67$	4.81585			0.588350			0.294175	+	0.955752i	$0.404955\pi$
$68$		0			0
$69$			4.55656i			0.548546i
$70$		0			0
$71$	−4.62810			−0.549255			−0.274627	−	0.961551i	$-0.588555\pi$
$71$	−4.62810			−0.549255			−0.274627	+	0.961551i	$0.588555\pi$
$72$		0			0
$73$	−0.257348			−0.0301203			−0.0150601	−	0.999887i	$-0.504794\pi$
$73$	−0.257348			−0.0301203			−0.0150601	+	0.999887i	$0.504794\pi$
$74$		0			0
$75$	−1.00000			−0.115470
$76$		0			0
$77$	−17.4382			−1.98727
$78$		0			0
$79$	13.6652			1.53745			0.768727	−	0.639577i	$-0.220890\pi$
$79$	13.6652			1.53745			0.768727	+	0.639577i	$0.220890\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	1.93878i		−	0.212809i	−0.994323	−	0.106404i	$-0.966066\pi$
$83$		−	1.93878i		−	0.212809i	0.994323	−	0.106404i	$-0.0339338\pi$
$84$		0			0
$85$	−1.85589			−0.201299
$86$		0			0
$87$			4.50254i			0.482722i
$88$		0			0
$89$			2.54529i			0.269800i	0.990859	+	0.134900i	$0.0430713\pi$
$89$			2.54529i			0.269800i	−0.990859	+	0.134900i	$0.956929\pi$
$90$		0			0
$91$	15.5520			1.63029
$92$		0			0
$93$	2.95714			0.306641
$94$		0			0
$95$	1.32070	+	4.15400i	0.135501	+	0.426192i
$96$		0			0
$97$			11.9942i			1.21782i	0.793238	+	0.608911i	$0.208393\pi$
$97$			11.9942i			1.21782i	−0.793238	+	0.608911i	$0.791607\pi$
$98$		0			0
$99$			4.75021i			0.477414i
$100$		0			0
$101$	−5.19000			−0.516424			−0.258212	−	0.966088i	$-0.583133\pi$
$101$	−5.19000			−0.516424			−0.258212	+	0.966088i	$0.583133\pi$
$102$		0			0
$103$	4.30115			0.423805			0.211903	−	0.977291i	$-0.432034\pi$
$103$	4.30115			0.423805			0.211903	+	0.977291i	$0.432034\pi$
$104$		0			0
$105$			3.67104i			0.358257i
$106$		0			0
$107$	13.5432			1.30927			0.654634	−	0.755946i	$-0.272823\pi$
$107$	13.5432			1.30927			0.654634	+	0.755946i	$0.272823\pi$
$108$		0			0
$109$			0.211359i			0.0202445i	0.999949	+	0.0101222i	$0.00322206\pi$
$109$			0.211359i			0.0202445i	−0.999949	+	0.0101222i	$0.996778\pi$
$110$		0			0
$111$		−	2.49374i		−	0.236696i
$112$		0			0
$113$			9.46216i			0.890125i	0.895499	+	0.445063i	$0.146819\pi$
$113$			9.46216i			0.890125i	−0.895499	+	0.445063i	$0.853181\pi$
$114$		0			0
$115$			4.55656i			0.424902i
$116$		0			0
$117$		−	4.23639i		−	0.391655i
$118$		0			0
$119$			6.81303i			0.624549i
$120$		0			0
$121$	−11.5645			−1.05132
$122$		0			0
$123$			4.10196i			0.369861i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	12.7610			1.13236			0.566178	−	0.824283i	$-0.308421\pi$
$127$	12.7610			1.13236			0.566178	+	0.824283i	$0.308421\pi$
$128$		0			0
$129$			0.970359i			0.0854354i
$130$		0			0
$131$			9.93718i			0.868215i	0.900861	+	0.434108i	$0.142936\pi$
$131$			9.93718i			0.868215i	−0.900861	+	0.434108i	$0.857064\pi$
$132$		0			0
$133$	15.2495	−	4.84833i	1.32230	−	0.420404i
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$	−5.52444			−0.471985			−0.235992	−	0.971755i	$-0.575834\pi$
$137$	−5.52444			−0.471985			−0.235992	+	0.971755i	$0.575834\pi$
$138$		0			0
$139$		−	7.58975i		−	0.643754i	−0.946782	−	0.321877i	$-0.895686\pi$
$139$		−	7.58975i		−	0.643754i	0.946782	−	0.321877i	$-0.104314\pi$
$140$		0			0
$141$		−	4.14411i		−	0.348997i
$142$		0			0
$143$	20.1238			1.68283
$144$		0			0
$145$			4.50254i			0.373915i
$146$		0			0
$147$	6.47651			0.534174
$148$		0			0
$149$	−7.01554			−0.574736			−0.287368	−	0.957820i	$-0.592780\pi$
$149$	−7.01554			−0.574736			−0.287368	+	0.957820i	$0.592780\pi$
$150$		0			0
$151$	−12.0275			−0.978786			−0.489393	−	0.872063i	$-0.662782\pi$
$151$	−12.0275			−0.978786			−0.489393	+	0.872063i	$0.662782\pi$
$152$		0			0
$153$	1.85589			0.150040
$154$		0			0
$155$	2.95714			0.237523
$156$		0			0
$157$	18.9064			1.50890			0.754449	−	0.656358i	$-0.227904\pi$
$157$	18.9064			1.50890			0.754449	+	0.656358i	$0.227904\pi$
$158$		0			0
$159$			14.2232i			1.12797i
$160$		0			0
$161$	16.7273			1.31830
$162$		0			0
$163$		−	12.9569i		−	1.01486i	−0.861693	−	0.507430i	$-0.830596\pi$
$163$		−	12.9569i		−	1.01486i	0.861693	−	0.507430i	$-0.169404\pi$
$164$		0			0
$165$			4.75021i			0.369803i
$166$		0			0
$167$	16.6384			1.28752			0.643761	−	0.765227i	$-0.277373\pi$
$167$	16.6384			1.28752			0.643761	+	0.765227i	$0.277373\pi$
$168$		0			0
$169$	−4.94704			−0.380541
$170$		0			0
$171$	−1.32070	−	4.15400i	−0.100996	−	0.317665i
$172$		0			0
$173$		−	3.21916i		−	0.244748i	−0.992484	−	0.122374i	$-0.960949\pi$
$173$		−	3.21916i		−	0.244748i	0.992484	−	0.122374i	$-0.0390508\pi$
$174$		0			0
$175$			3.67104i			0.277504i
$176$		0			0
$177$	−0.965936			−0.0726042
$178$		0			0
$179$	−21.1329			−1.57955			−0.789774	−	0.613398i	$-0.789802\pi$
$179$	−21.1329			−1.57955			−0.789774	+	0.613398i	$0.789802\pi$
$180$		0			0
$181$		−	14.4107i		−	1.07114i	−0.844491	−	0.535569i	$-0.820097\pi$
$181$		−	14.4107i		−	1.07114i	0.844491	−	0.535569i	$-0.179903\pi$
$182$		0			0
$183$	−3.72057			−0.275032
$184$		0			0
$185$		−	2.49374i		−	0.183344i
$186$		0			0
$187$			8.81585i			0.644679i
$188$		0			0
$189$		−	3.67104i		−	0.267029i
$190$		0			0
$191$			25.3382i			1.83341i	0.399570	+	0.916703i	$0.369160\pi$
$191$			25.3382i			1.83341i	−0.399570	+	0.916703i	$0.630840\pi$
$192$		0			0
$193$		−	25.4026i		−	1.82852i	−0.405129	−	0.914259i	$-0.632773\pi$
$193$		−	25.4026i		−	1.82852i	0.405129	−	0.914259i	$-0.367227\pi$
$194$		0			0
$195$		−	4.23639i		−	0.303375i
$196$		0			0
$197$	−8.14170			−0.580072			−0.290036	−	0.957016i	$-0.593667\pi$
$197$	−8.14170			−0.580072			−0.290036	+	0.957016i	$0.593667\pi$
$198$		0			0
$199$		−	22.1822i		−	1.57245i	−0.617940	−	0.786226i	$-0.712032\pi$
$199$		−	22.1822i		−	1.57245i	0.617940	−	0.786226i	$-0.287968\pi$
$200$		0			0
$201$	−4.81585			−0.339684
$202$		0			0
$203$	16.5290			1.16011
$204$		0			0
$205$			4.10196i			0.286493i
$206$		0			0
$207$		−	4.55656i		−	0.316703i
$208$		0			0
$209$	19.7324	−	6.27359i	1.36492	−	0.433953i
$210$		0			0
$211$	22.9871			1.58250			0.791249	−	0.611494i	$-0.209431\pi$
$211$	22.9871			1.58250			0.791249	+	0.611494i	$0.209431\pi$
$212$		0			0
$213$	4.62810			0.317112
$214$		0			0
$215$			0.970359i			0.0661780i
$216$		0			0
$217$		−	10.8558i		−	0.736938i
$218$		0			0
$219$	0.257348			0.0173900
$220$		0			0
$221$		−	7.86227i		−	0.528874i
$222$		0			0
$223$	17.0446			1.14139			0.570695	−	0.821162i	$-0.306674\pi$
$223$	17.0446			1.14139			0.570695	+	0.821162i	$0.306674\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	−11.2066			−0.743811			−0.371905	−	0.928271i	$-0.621295\pi$
$227$	−11.2066			−0.743811			−0.371905	+	0.928271i	$0.621295\pi$
$228$		0			0
$229$	25.6197			1.69300			0.846500	−	0.532389i	$-0.178706\pi$
$229$	25.6197			1.69300			0.846500	+	0.532389i	$0.178706\pi$
$230$		0			0
$231$	17.4382			1.14735
$232$		0			0
$233$	−8.01873			−0.525324			−0.262662	−	0.964888i	$-0.584600\pi$
$233$	−8.01873			−0.525324			−0.262662	+	0.964888i	$0.584600\pi$
$234$		0			0
$235$		−	4.14411i		−	0.270332i
$236$		0			0
$237$	−13.6652			−0.887650
$238$		0			0
$239$			4.99829i			0.323313i	0.986847	+	0.161656i	$0.0516836\pi$
$239$			4.99829i			0.323313i	−0.986847	+	0.161656i	$0.948316\pi$
$240$		0			0
$241$		−	17.1735i		−	1.10624i	−0.833102	−	0.553120i	$-0.813437\pi$
$241$		−	17.1735i		−	1.10624i	0.833102	−	0.553120i	$-0.186563\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	6.47651			0.413769
$246$		0			0
$247$	−17.5980	+	5.59500i	−1.11973	+	0.356001i
$248$		0			0
$249$			1.93878i			0.122865i
$250$		0			0
$251$			20.4291i			1.28947i	0.764405	+	0.644737i	$0.223033\pi$
$251$			20.4291i			1.28947i	−0.764405	+	0.644737i	$0.776967\pi$
$252$		0			0
$253$	21.6446			1.36079
$254$		0			0
$255$	1.85589			0.116220
$256$		0			0
$257$		−	8.02208i		−	0.500404i	−0.968194	−	0.250202i	$-0.919503\pi$
$257$		−	8.02208i		−	0.500404i	0.968194	−	0.250202i	$-0.0804970\pi$
$258$		0			0
$259$	−9.15462			−0.568840
$260$		0			0
$261$		−	4.50254i		−	0.278700i
$262$		0			0
$263$			14.6158i			0.901252i	0.892713	+	0.450626i	$0.148799\pi$
$263$			14.6158i			0.901252i	−0.892713	+	0.450626i	$0.851201\pi$
$264$		0			0
$265$			14.2232i			0.873723i
$266$		0			0
$267$		−	2.54529i		−	0.155769i
$268$		0			0
$269$		−	31.2383i		−	1.90463i	−0.305109	−	0.952317i	$-0.598693\pi$
$269$		−	31.2383i		−	1.90463i	0.305109	−	0.952317i	$-0.401307\pi$
$270$		0			0
$271$		−	20.9879i		−	1.27493i	−0.770481	−	0.637463i	$-0.779984\pi$
$271$		−	20.9879i		−	1.27493i	0.770481	−	0.637463i	$-0.220016\pi$
$272$		0			0
$273$	−15.5520			−0.941247
$274$		0			0
$275$			4.75021i			0.286448i
$276$		0			0
$277$	25.8982			1.55607			0.778037	−	0.628219i	$-0.216216\pi$
$277$	25.8982			1.55607			0.778037	+	0.628219i	$0.216216\pi$
$278$		0			0
$279$	−2.95714			−0.177040
$280$		0			0
$281$			25.5991i			1.52712i	0.645739	+	0.763558i	$0.276549\pi$
$281$			25.5991i			1.52712i	−0.645739	+	0.763558i	$0.723451\pi$
$282$		0			0
$283$		−	5.39716i		−	0.320828i	−0.987050	−	0.160414i	$-0.948717\pi$
$283$		−	5.39716i		−	0.320828i	0.987050	−	0.160414i	$-0.0512829\pi$
$284$		0			0
$285$	−1.32070	−	4.15400i	−0.0782314	−	0.246062i
$286$		0			0
$287$	15.0584			0.888871
$288$		0			0
$289$	−13.5557			−0.797393
$290$		0			0
$291$		−	11.9942i		−	0.703110i
$292$		0			0
$293$			6.16284i			0.360037i	0.983663	+	0.180019i	$0.0576158\pi$
$293$			6.16284i			0.360037i	−0.983663	+	0.180019i	$0.942384\pi$
$294$		0			0
$295$	−0.965936			−0.0562390
$296$		0			0
$297$		−	4.75021i		−	0.275635i
$298$		0			0
$299$	−19.3034			−1.11634
$300$		0			0
$301$	3.56223			0.205323
$302$		0			0
$303$	5.19000			0.298157
$304$		0			0
$305$	−3.72057			−0.213039
$306$		0			0
$307$	18.4548			1.05327			0.526637	−	0.850090i	$-0.323453\pi$
$307$	18.4548			1.05327			0.526637	+	0.850090i	$0.323453\pi$
$308$		0			0
$309$	−4.30115			−0.244684
$310$		0			0
$311$		−	10.1604i		−	0.576144i	−0.957609	−	0.288072i	$-0.906986\pi$
$311$		−	10.1604i		−	0.576144i	0.957609	−	0.288072i	$-0.0930142\pi$
$312$		0			0
$313$	22.2080			1.25527			0.627635	−	0.778508i	$-0.284023\pi$
$313$	22.2080			1.25527			0.627635	+	0.778508i	$0.284023\pi$
$314$		0			0
$315$		−	3.67104i		−	0.206839i
$316$		0			0
$317$		−	10.5114i		−	0.590379i	−0.955439	−	0.295190i	$-0.904617\pi$
$317$		−	10.5114i		−	0.590379i	0.955439	−	0.295190i	$-0.0953828\pi$
$318$		0			0
$319$	21.3880			1.19750
$320$		0			0
$321$	−13.5432			−0.755906
$322$		0			0
$323$	−2.45107	−	7.70936i	−0.136381	−	0.428961i
$324$		0			0
$325$		−	4.23639i		−	0.234993i
$326$		0			0
$327$		−	0.211359i		−	0.0116882i
$328$		0			0
$329$	−15.2132			−0.838730
$330$		0			0
$331$	−2.22476			−0.122284			−0.0611419	−	0.998129i	$-0.519474\pi$
$331$	−2.22476			−0.122284			−0.0611419	+	0.998129i	$0.519474\pi$
$332$		0			0
$333$			2.49374i			0.136656i
$334$		0			0
$335$	−4.81585			−0.263118
$336$		0			0
$337$			8.01169i			0.436424i	0.975901	+	0.218212i	$0.0700225\pi$
$337$			8.01169i			0.436424i	−0.975901	+	0.218212i	$0.929977\pi$
$338$		0			0
$339$		−	9.46216i		−	0.513914i
$340$		0			0
$341$		−	14.0470i		−	0.760690i
$342$		0			0
$343$			1.92175i			0.103765i
$344$		0			0
$345$		−	4.55656i		−	0.245317i
$346$		0			0
$347$		−	18.4295i		−	0.989347i	−0.869079	−	0.494674i	$-0.835288\pi$
$347$		−	18.4295i		−	0.989347i	0.869079	−	0.494674i	$-0.164712\pi$
$348$		0			0
$349$	4.68557			0.250813			0.125406	−	0.992105i	$-0.459977\pi$
$349$	4.68557			0.250813			0.125406	+	0.992105i	$0.459977\pi$
$350$		0			0
$351$			4.23639i			0.226122i
$352$		0			0
$353$	22.7068			1.20856			0.604279	−	0.796773i	$-0.293461\pi$
$353$	22.7068			1.20856			0.604279	+	0.796773i	$0.293461\pi$
$354$		0			0
$355$	4.62810			0.245634
$356$		0			0
$357$		−	6.81303i		−	0.360584i
$358$		0			0
$359$		−	14.2843i		−	0.753898i	−0.926234	−	0.376949i	$-0.876973\pi$
$359$		−	14.2843i		−	0.753898i	0.926234	−	0.376949i	$-0.123027\pi$
$360$		0			0
$361$	−15.5115	+	10.9724i	−0.816395	+	0.577493i
$362$		0			0
$363$	11.5645			0.606978
$364$		0			0
$365$	0.257348			0.0134702
$366$		0			0
$367$			20.2311i			1.05606i	0.849227	+	0.528029i	$0.177069\pi$
$367$			20.2311i			1.05606i	−0.849227	+	0.528029i	$0.822931\pi$
$368$		0			0
$369$		−	4.10196i		−	0.213539i
$370$		0			0
$371$	52.2138			2.71081
$372$		0			0
$373$			4.80411i			0.248747i	0.992235	+	0.124374i	$0.0396921\pi$
$373$			4.80411i			0.248747i	−0.992235	+	0.124374i	$0.960308\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	−19.0745			−0.982387
$378$		0			0
$379$	−24.5183			−1.25942			−0.629710	−	0.776830i	$-0.716826\pi$
$379$	−24.5183			−1.25942			−0.629710	+	0.776830i	$0.716826\pi$
$380$		0			0
$381$	−12.7610			−0.653767
$382$		0			0
$383$	33.1067			1.69168			0.845838	−	0.533440i	$-0.179101\pi$
$383$	33.1067			1.69168			0.845838	+	0.533440i	$0.179101\pi$
$384$		0			0
$385$	17.4382			0.888732
$386$		0			0
$387$		−	0.970359i		−	0.0493261i
$388$		0			0
$389$	25.1305			1.27417			0.637084	−	0.770795i	$-0.280141\pi$
$389$	25.1305			1.27417			0.637084	+	0.770795i	$0.280141\pi$
$390$		0			0
$391$		−	8.45647i		−	0.427662i
$392$		0			0
$393$		−	9.93718i		−	0.501264i
$394$		0			0
$395$	−13.6652			−0.687571
$396$		0			0
$397$	17.5127			0.878937			0.439469	−	0.898258i	$-0.355167\pi$
$397$	17.5127			0.878937			0.439469	+	0.898258i	$0.355167\pi$
$398$		0			0
$399$	−15.2495	+	4.84833i	−0.763430	+	0.242720i
$400$		0			0
$401$			14.5969i			0.728936i	0.931216	+	0.364468i	$0.118749\pi$
$401$			14.5969i			0.728936i	−0.931216	+	0.364468i	$0.881251\pi$
$402$		0			0
$403$			12.5276i			0.624045i
$404$		0			0
$405$	−1.00000			−0.0496904
$406$		0			0
$407$	−11.8458			−0.587174
$408$		0			0
$409$			10.0011i			0.494520i	0.968949	+	0.247260i	$0.0795302\pi$
$409$			10.0011i			0.494520i	−0.968949	+	0.247260i	$0.920470\pi$
$410$		0			0
$411$	5.52444			0.272500
$412$		0			0
$413$			3.54599i			0.174487i
$414$		0			0
$415$			1.93878i			0.0951710i
$416$		0			0
$417$			7.58975i			0.371671i
$418$		0			0
$419$		−	9.54428i		−	0.466269i	−0.972445	−	0.233134i	$-0.925102\pi$
$419$		−	9.54428i		−	0.466269i	0.972445	−	0.233134i	$-0.0748982\pi$
$420$		0			0
$421$		−	1.09787i		−	0.0535067i	−0.999642	−	0.0267534i	$-0.991483\pi$
$421$		−	1.09787i		−	0.0535067i	0.999642	−	0.0267534i	$-0.00851688\pi$
$422$		0			0
$423$			4.14411i			0.201494i
$424$		0			0
$425$	1.85589			0.0900238
$426$		0			0
$427$			13.6583i			0.660973i
$428$		0			0
$429$	−20.1238			−0.971584
$430$		0			0
$431$	−20.1670			−0.971409			−0.485705	−	0.874123i	$-0.661437\pi$
$431$	−20.1670			−0.971409			−0.485705	+	0.874123i	$0.661437\pi$
$432$		0			0
$433$			17.8221i			0.856476i	0.903666	+	0.428238i	$0.140865\pi$
$433$			17.8221i			0.856476i	−0.903666	+	0.428238i	$0.859135\pi$
$434$		0			0
$435$		−	4.50254i		−	0.215880i
$436$		0			0
$437$	−18.9280	+	6.01785i	−0.905449	+	0.287873i
$438$		0			0
$439$	−39.4813			−1.88434			−0.942169	−	0.335138i	$-0.891217\pi$
$439$	−39.4813			−1.88434			−0.942169	+	0.335138i	$0.891217\pi$
$440$		0			0
$441$	−6.47651			−0.308405
$442$		0			0
$443$			30.8331i			1.46493i	0.680807	+	0.732463i	$0.261629\pi$
$443$			30.8331i			1.46493i	−0.680807	+	0.732463i	$0.738371\pi$
$444$		0			0
$445$		−	2.54529i		−	0.120658i
$446$		0			0
$447$	7.01554			0.331824
$448$		0			0
$449$			16.0931i			0.759481i	0.925093	+	0.379740i	$0.123987\pi$
$449$			16.0931i			0.759481i	−0.925093	+	0.379740i	$0.876013\pi$
$450$		0			0
$451$	19.4851			0.917520
$452$		0			0
$453$	12.0275			0.565102
$454$		0			0
$455$	−15.5520			−0.729087
$456$		0			0
$457$	40.6370			1.90092			0.950459	−	0.310851i	$-0.100614\pi$
$457$	40.6370			1.90092			0.950459	+	0.310851i	$0.100614\pi$
$458$		0			0
$459$	−1.85589			−0.0866254
$460$		0			0
$461$	−14.9980			−0.698524			−0.349262	−	0.937025i	$-0.613568\pi$
$461$	−14.9980			−0.698524			−0.349262	+	0.937025i	$0.613568\pi$
$462$		0			0
$463$			16.5006i			0.766848i	0.923572	+	0.383424i	$0.125255\pi$
$463$			16.5006i			0.766848i	−0.923572	+	0.383424i	$0.874745\pi$
$464$		0			0
$465$	−2.95714			−0.137134
$466$		0			0
$467$		−	35.3662i		−	1.63655i	−0.574824	−	0.818277i	$-0.694930\pi$
$467$		−	35.3662i		−	1.63655i	0.574824	−	0.818277i	$-0.305070\pi$
$468$		0			0
$469$			17.6792i			0.816348i
$470$		0			0
$471$	−18.9064			−0.871163
$472$		0			0
$473$	4.60941			0.211941
$474$		0			0
$475$	−1.32070	−	4.15400i	−0.0605978	−	0.190599i
$476$		0			0
$477$		−	14.2232i		−	0.651235i
$478$		0			0
$479$		−	13.7819i		−	0.629709i	−0.949140	−	0.314855i	$-0.898044\pi$
$479$		−	13.7819i		−	0.629709i	0.949140	−	0.314855i	$-0.101956\pi$
$480$		0			0
$481$	10.5645			0.481699
$482$		0			0
$483$	−16.7273			−0.761119
$484$		0			0
$485$		−	11.9942i		−	0.544627i
$486$		0			0
$487$	17.0496			0.772592			0.386296	−	0.922375i	$-0.373754\pi$
$487$	17.0496			0.772592			0.386296	+	0.922375i	$0.373754\pi$
$488$		0			0
$489$			12.9569i			0.585929i
$490$		0			0
$491$			13.1111i			0.591694i	0.955235	+	0.295847i	$0.0956018\pi$
$491$			13.1111i			0.591694i	−0.955235	+	0.295847i	$0.904398\pi$
$492$		0			0
$493$		−	8.35620i		−	0.376344i
$494$		0			0
$495$		−	4.75021i		−	0.213506i
$496$		0			0
$497$		−	16.9899i		−	0.762103i
$498$		0			0
$499$			11.3756i			0.509240i	0.967041	+	0.254620i	$0.0819504\pi$
$499$			11.3756i			0.509240i	−0.967041	+	0.254620i	$0.918050\pi$
$500$		0			0
$501$	−16.6384			−0.743351
$502$		0			0
$503$		−	18.7559i		−	0.836284i	−0.908382	−	0.418142i	$-0.862681\pi$
$503$		−	18.7559i		−	0.836284i	0.908382	−	0.418142i	$-0.137319\pi$
$504$		0			0
$505$	5.19000			0.230952
$506$		0			0
$507$	4.94704			0.219706
$508$		0			0
$509$			1.90374i			0.0843817i	0.999110	+	0.0421909i	$0.0134338\pi$
$509$			1.90374i			0.0843817i	−0.999110	+	0.0421909i	$0.986566\pi$
$510$		0			0
$511$		−	0.944733i		−	0.0417925i
$512$		0			0
$513$	1.32070	+	4.15400i	0.0583102	+	0.183404i
$514$		0			0
$515$	−4.30115			−0.189532
$516$		0			0
$517$	−19.6854			−0.865763
$518$		0			0
$519$			3.21916i			0.141306i
$520$		0			0
$521$		−	34.5249i		−	1.51256i	−0.654247	−	0.756281i	$-0.727014\pi$
$521$		−	34.5249i		−	1.51256i	0.654247	−	0.756281i	$-0.272986\pi$
$522$		0			0
$523$	11.7011			0.511654			0.255827	−	0.966723i	$-0.417652\pi$
$523$	11.7011			0.511654			0.255827	+	0.966723i	$0.417652\pi$
$524$		0			0
$525$		−	3.67104i		−	0.160217i
$526$		0			0
$527$	−5.48812			−0.239066
$528$		0			0
$529$	2.23772			0.0972921
$530$		0			0
$531$	0.965936			0.0419181
$532$		0			0
$533$	−17.3775			−0.752703
$534$		0			0
$535$	−13.5432			−0.585522
$536$		0			0
$537$	21.1329			0.911953
$538$		0			0
$539$		−	30.7648i		−	1.32513i
$540$		0			0
$541$	32.4900			1.39685			0.698427	−	0.715681i	$-0.253884\pi$
$541$	32.4900			1.39685			0.698427	+	0.715681i	$0.253884\pi$
$542$		0			0
$543$			14.4107i			0.618422i
$544$		0			0
$545$		−	0.211359i		−	0.00905361i
$546$		0			0
$547$	−15.9866			−0.683536			−0.341768	−	0.939784i	$-0.611026\pi$
$547$	−15.9866			−0.683536			−0.341768	+	0.939784i	$0.611026\pi$
$548$		0			0
$549$	3.72057			0.158790
$550$		0			0
$551$	−18.7036	+	5.94649i	−0.796798	+	0.253329i
$552$		0			0
$553$			50.1654i			2.13325i
$554$		0			0
$555$			2.49374i			0.105853i
$556$		0			0
$557$	−28.9926			−1.22845			−0.614227	−	0.789129i	$-0.710532\pi$
$557$	−28.9926			−1.22845			−0.614227	+	0.789129i	$0.710532\pi$
$558$		0			0
$559$	−4.11083			−0.173869
$560$		0			0
$561$		−	8.81585i		−	0.372205i
$562$		0			0
$563$	−14.0604			−0.592577			−0.296289	−	0.955098i	$-0.595749\pi$
$563$	−14.0604			−0.592577			−0.296289	+	0.955098i	$0.595749\pi$
$564$		0			0
$565$		−	9.46216i		−	0.398076i
$566$		0			0
$567$			3.67104i			0.154169i
$568$		0			0
$569$			28.8612i			1.20992i	0.796254	+	0.604962i	$0.206812\pi$
$569$			28.8612i			1.20992i	−0.796254	+	0.604962i	$0.793188\pi$
$570$		0			0
$571$		−	14.1218i		−	0.590980i	−0.955346	−	0.295490i	$-0.904517\pi$
$571$		−	14.1218i		−	0.590980i	0.955346	−	0.295490i	$-0.0954828\pi$
$572$		0			0
$573$		−	25.3382i		−	1.05852i
$574$		0			0
$575$		−	4.55656i		−	0.190022i
$576$		0			0
$577$	−8.64479			−0.359887			−0.179944	−	0.983677i	$-0.557592\pi$
$577$	−8.64479			−0.359887			−0.179944	+	0.983677i	$0.557592\pi$
$578$		0			0
$579$			25.4026i			1.05570i
$580$		0			0
$581$	7.11734			0.295277
$582$		0			0
$583$	67.5631			2.79818
$584$		0			0
$585$			4.23639i			0.175153i
$586$		0			0
$587$			8.50637i			0.351095i	0.984471	+	0.175548i	$0.0561696\pi$
$587$			8.50637i			0.351095i	−0.984471	+	0.175548i	$0.943830\pi$
$588$		0			0
$589$	3.90549	+	12.2840i	0.160923	+	0.506153i
$590$		0			0
$591$	8.14170			0.334905
$592$		0			0
$593$	18.5456			0.761576			0.380788	−	0.924662i	$-0.375653\pi$
$593$	18.5456			0.761576			0.380788	+	0.924662i	$0.375653\pi$
$594$		0			0
$595$		−	6.81303i		−	0.279307i
$596$		0			0
$597$			22.1822i			0.907855i
$598$		0			0
$599$	−15.4524			−0.631369			−0.315685	−	0.948864i	$-0.602234\pi$
$599$	−15.4524			−0.631369			−0.315685	+	0.948864i	$0.602234\pi$
$600$		0			0
$601$			30.2318i			1.23318i	0.787284	+	0.616591i	$0.211487\pi$
$601$			30.2318i			1.23318i	−0.787284	+	0.616591i	$0.788513\pi$
$602$		0			0
$603$	4.81585			0.196117
$604$		0			0
$605$	11.5645			0.470163
$606$		0			0
$607$	−38.5892			−1.56629			−0.783144	−	0.621840i	$-0.786385\pi$
$607$	−38.5892			−1.56629			−0.783144	+	0.621840i	$0.786385\pi$
$608$		0			0
$609$	−16.5290			−0.669788
$610$		0			0
$611$	17.5561			0.710244
$612$		0			0
$613$	−1.00748			−0.0406917			−0.0203459	−	0.999793i	$-0.506477\pi$
$613$	−1.00748			−0.0406917			−0.0203459	+	0.999793i	$0.506477\pi$
$614$		0			0
$615$		−	4.10196i		−	0.165407i
$616$		0			0
$617$	−32.5675			−1.31112			−0.655560	−	0.755143i	$-0.727567\pi$
$617$	−32.5675			−1.31112			−0.655560	+	0.755143i	$0.727567\pi$
$618$		0			0
$619$			0.372972i			0.0149910i	0.999972	+	0.00749551i	$0.00238592\pi$
$619$			0.372972i			0.0149910i	−0.999972	+	0.00749551i	$0.997614\pi$
$620$		0			0
$621$			4.55656i			0.182849i
$622$		0			0
$623$	−9.34384			−0.374353
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−19.7324	+	6.27359i	−0.788036	+	0.250543i
$628$		0			0
$629$			4.62810i			0.184535i
$630$		0			0
$631$			2.00099i			0.0796582i	0.999207	+	0.0398291i	$0.0126814\pi$
$631$			2.00099i			0.0796582i	−0.999207	+	0.0398291i	$0.987319\pi$
$632$		0			0
$633$	−22.9871			−0.913656
$634$		0			0
$635$	−12.7610			−0.506405
$636$		0			0
$637$			27.4371i			1.08710i
$638$		0			0
$639$	−4.62810			−0.183085
$640$		0			0
$641$			3.91416i			0.154600i	0.997008	+	0.0772999i	$0.0246299\pi$
$641$			3.91416i			0.154600i	−0.997008	+	0.0772999i	$0.975370\pi$
$642$		0			0
$643$		−	19.3955i		−	0.764883i	−0.923980	−	0.382441i	$-0.875083\pi$
$643$		−	19.3955i		−	0.764883i	0.923980	−	0.382441i	$-0.124917\pi$
$644$		0			0
$645$		−	0.970359i		−	0.0382079i
$646$		0			0
$647$			26.8783i			1.05670i	0.849028	+	0.528348i	$0.177188\pi$
$647$			26.8783i			1.05670i	−0.849028	+	0.528348i	$0.822812\pi$
$648$		0			0
$649$			4.58840i			0.180110i
$650$		0			0
$651$			10.8558i			0.425472i
$652$		0			0
$653$	29.1647			1.14130			0.570652	−	0.821192i	$-0.306691\pi$
$653$	29.1647			1.14130			0.570652	+	0.821192i	$0.306691\pi$
$654$		0			0
$655$		−	9.93718i		−	0.388278i
$656$		0			0
$657$	−0.257348			−0.0100401
$658$		0			0
$659$	24.2670			0.945310			0.472655	−	0.881248i	$-0.343296\pi$
$659$	24.2670			0.945310			0.472655	+	0.881248i	$0.343296\pi$
$660$		0			0
$661$		−	15.9625i		−	0.620869i	−0.950595	−	0.310434i	$-0.899526\pi$
$661$		−	15.9625i		−	0.620869i	0.950595	−	0.310434i	$-0.100474\pi$
$662$		0			0
$663$			7.86227i			0.305345i
$664$		0			0
$665$	−15.2495	+	4.84833i	−0.591350	+	0.188010i
$666$		0			0
$667$	−20.5161			−0.794386
$668$		0			0
$669$	−17.0446			−0.658982
$670$		0			0
$671$			17.6735i			0.682277i
$672$		0			0
$673$		−	6.67633i		−	0.257354i	−0.991687	−	0.128677i	$-0.958927\pi$
$673$		−	6.67633i		−	0.257354i	0.991687	−	0.128677i	$-0.0410730\pi$
$674$		0			0
$675$	−1.00000			−0.0384900
$676$		0			0
$677$		−	24.6669i		−	0.948026i	−0.880518	−	0.474013i	$-0.842805\pi$
$677$		−	24.6669i		−	0.948026i	0.880518	−	0.474013i	$-0.157195\pi$
$678$		0			0
$679$	−44.0310			−1.68975
$680$		0			0
$681$	11.2066			0.429439
$682$		0			0
$683$	17.6600			0.675740			0.337870	−	0.941193i	$-0.390294\pi$
$683$	17.6600			0.675740			0.337870	+	0.941193i	$0.390294\pi$
$684$		0			0
$685$	5.52444			0.211078
$686$		0			0
$687$	−25.6197			−0.977453
$688$		0			0
$689$	−60.2550			−2.29553
$690$		0			0
$691$			24.8253i			0.944399i	0.881492	+	0.472200i	$0.156540\pi$
$691$			24.8253i			0.944399i	−0.881492	+	0.472200i	$0.843460\pi$
$692$		0			0
$693$	−17.4382			−0.662422
$694$		0			0
$695$			7.58975i			0.287895i
$696$		0			0
$697$		−	7.61277i		−	0.288354i
$698$		0			0
$699$	8.01873			0.303296
$700$		0			0
$701$	−9.16194			−0.346042			−0.173021	−	0.984918i	$-0.555353\pi$
$701$	−9.16194			−0.346042			−0.173021	+	0.984918i	$0.555353\pi$
$702$		0			0
$703$	10.3590	−	3.29348i	0.390698	−	0.124216i
$704$		0			0
$705$			4.14411i			0.156076i
$706$		0			0
$707$		−	19.0527i		−	0.716549i
$708$		0			0
$709$	−5.92530			−0.222529			−0.111265	−	0.993791i	$-0.535490\pi$
$709$	−5.92530			−0.222529			−0.111265	+	0.993791i	$0.535490\pi$
$710$		0			0
$711$	13.6652			0.512485
$712$		0			0
$713$			13.4744i			0.504621i
$714$		0			0
$715$	−20.1238			−0.752586
$716$		0			0
$717$		−	4.99829i		−	0.186665i
$718$		0			0
$719$		−	25.2599i		−	0.942037i	−0.882123	−	0.471018i	$-0.843887\pi$
$719$		−	25.2599i		−	0.942037i	0.882123	−	0.471018i	$-0.156113\pi$
$720$		0			0
$721$			15.7897i			0.588039i
$722$		0			0
$723$			17.1735i			0.638688i
$724$		0			0
$725$		−	4.50254i		−	0.167220i
$726$		0			0
$727$		−	31.1485i		−	1.15524i	−0.816308	−	0.577618i	$-0.803983\pi$
$727$		−	31.1485i		−	1.15524i	0.816308	−	0.577618i	$-0.196017\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		−	1.80088i		−	0.0666079i
$732$		0			0
$733$	−7.84149			−0.289632			−0.144816	−	0.989459i	$-0.546259\pi$
$733$	−7.84149			−0.289632			−0.144816	+	0.989459i	$0.546259\pi$
$734$		0			0
$735$	−6.47651			−0.238890
$736$		0			0
$737$			22.8763i			0.842659i
$738$		0			0
$739$		−	17.8528i		−	0.656725i	−0.944552	−	0.328363i	$-0.893503\pi$
$739$		−	17.8528i		−	0.656725i	0.944552	−	0.328363i	$-0.106497\pi$
$740$		0			0
$741$	17.5980	−	5.59500i	0.646479	−	0.205537i
$742$		0			0
$743$	15.4024			0.565060			0.282530	−	0.959259i	$-0.408826\pi$
$743$	15.4024			0.565060			0.282530	+	0.959259i	$0.408826\pi$
$744$		0			0
$745$	7.01554			0.257030
$746$		0			0
$747$		−	1.93878i		−	0.0709363i
$748$		0			0
$749$			49.7175i			1.81664i
$750$		0			0
$751$	42.4989			1.55081			0.775403	−	0.631467i	$-0.217547\pi$
$751$	42.4989			1.55081			0.775403	+	0.631467i	$0.217547\pi$
$752$		0			0
$753$		−	20.4291i		−	0.744478i
$754$		0			0
$755$	12.0275			0.437726
$756$		0			0
$757$	22.8445			0.830298			0.415149	−	0.909753i	$-0.363729\pi$
$757$	22.8445			0.830298			0.415149	+	0.909753i	$0.363729\pi$
$758$		0			0
$759$	−21.6446			−0.785650
$760$		0			0
$761$	−25.4880			−0.923938			−0.461969	−	0.886896i	$-0.652857\pi$
$761$	−25.4880			−0.923938			−0.461969	+	0.886896i	$0.652857\pi$
$762$		0			0
$763$	−0.775905			−0.0280897
$764$		0			0
$765$	−1.85589			−0.0670997
$766$		0			0
$767$		−	4.09209i		−	0.147757i
$768$		0			0
$769$	−3.22104			−0.116154			−0.0580768	−	0.998312i	$-0.518497\pi$
$769$	−3.22104			−0.116154			−0.0580768	+	0.998312i	$0.518497\pi$
$770$		0			0
$771$			8.02208i			0.288908i
$772$		0			0
$773$		−	30.1514i		−	1.08447i	−0.840227	−	0.542235i	$-0.817578\pi$
$773$		−	30.1514i		−	1.08447i	0.840227	−	0.542235i	$-0.182422\pi$
$774$		0			0
$775$	−2.95714			−0.106224
$776$		0			0
$777$	9.15462			0.328420
$778$		0			0
$779$	−17.0395	+	5.41745i	−0.610505	+	0.194100i
$780$		0			0
$781$		−	21.9845i		−	0.786666i
$782$		0			0
$783$			4.50254i			0.160907i
$784$		0			0
$785$	−18.9064			−0.674800
$786$		0			0
$787$	41.9138			1.49406			0.747032	−	0.664788i	$-0.231478\pi$
$787$	41.9138			1.49406			0.747032	+	0.664788i	$0.231478\pi$
$788$		0			0
$789$		−	14.6158i		−	0.520338i
$790$		0			0
$791$	−34.7360			−1.23507
$792$		0			0
$793$		−	15.7618i		−	0.559718i
$794$		0			0
$795$		−	14.2232i		−	0.504444i
$796$		0			0
$797$		−	27.4723i		−	0.973119i	−0.873647	−	0.486559i	$-0.838252\pi$
$797$		−	27.4723i		−	0.973119i	0.873647	−	0.486559i	$-0.161748\pi$
$798$		0			0
$799$			7.69101i			0.272088i
$800$		0			0
$801$			2.54529i			0.0899333i
$802$		0			0
$803$		−	1.22246i		−	0.0431395i
$804$		0			0
$805$	−16.7273			−0.589560
$806$		0			0
$807$			31.2383i			1.09964i
$808$		0			0
$809$	−25.6609			−0.902188			−0.451094	−	0.892476i	$-0.648966\pi$
$809$	−25.6609			−0.902188			−0.451094	+	0.892476i	$0.648966\pi$
$810$		0			0
$811$	−13.9923			−0.491337			−0.245668	−	0.969354i	$-0.579007\pi$
$811$	−13.9923			−0.491337			−0.245668	+	0.969354i	$0.579007\pi$
$812$		0			0
$813$			20.9879i			0.736078i
$814$		0			0
$815$			12.9569i			0.453859i
$816$		0			0
$817$	−4.03088	+	1.28155i	−0.141023	+	0.0448358i
$818$		0			0
$819$	15.5520			0.543429
$820$		0			0
$821$	31.5722			1.10188			0.550939	−	0.834546i	$-0.314270\pi$
$821$	31.5722			1.10188			0.550939	+	0.834546i	$0.314270\pi$
$822$		0			0
$823$		−	48.7352i		−	1.69880i	−0.527747	−	0.849402i	$-0.676963\pi$
$823$		−	48.7352i		−	1.69880i	0.527747	−	0.849402i	$-0.323037\pi$
$824$		0			0
$825$		−	4.75021i		−	0.165381i
$826$		0			0
$827$	−25.4882			−0.886311			−0.443155	−	0.896445i	$-0.646141\pi$
$827$	−25.4882			−0.886311			−0.443155	+	0.896445i	$0.646141\pi$
$828$		0			0
$829$			2.83035i			0.0983021i	0.998791	+	0.0491510i	$0.0156516\pi$
$829$			2.83035i			0.0983021i	−0.998791	+	0.0491510i	$0.984348\pi$
$830$		0			0
$831$	−25.8982			−0.898399
$832$		0			0
$833$	−12.0197			−0.416457
$834$		0			0
$835$	−16.6384			−0.575797
$836$		0			0
$837$	2.95714			0.102214
$838$		0			0
$839$	−28.5615			−0.986053			−0.493027	−	0.870014i	$-0.664110\pi$
$839$	−28.5615			−0.986053			−0.493027	+	0.870014i	$0.664110\pi$
$840$		0			0
$841$	8.72718			0.300937
$842$		0			0
$843$		−	25.5991i		−	0.881681i
$844$		0			0
$845$	4.94704			0.170183
$846$		0			0
$847$		−	42.4536i		−	1.45872i
$848$		0			0
$849$			5.39716i			0.185230i
$850$		0			0
$851$	11.3629			0.389515
$852$		0			0
$853$	−44.8509			−1.53567			−0.767833	−	0.640650i	$-0.778665\pi$
$853$	−44.8509			−1.53567			−0.767833	+	0.640650i	$0.778665\pi$
$854$		0			0
$855$	1.32070	+	4.15400i	0.0451669	+	0.142064i
$856$		0			0
$857$			0.507014i			0.0173193i	0.999963	+	0.00865963i	$0.00275648\pi$
$857$			0.507014i			0.0173193i	−0.999963	+	0.00865963i	$0.997244\pi$
$858$		0			0
$859$			38.9738i			1.32977i	0.746946	+	0.664885i	$0.231520\pi$
$859$			38.9738i			1.32977i	−0.746946	+	0.664885i	$0.768480\pi$
$860$		0			0
$861$	−15.0584			−0.513190
$862$		0			0
$863$	2.64721			0.0901121			0.0450561	−	0.998984i	$-0.485653\pi$
$863$	2.64721			0.0901121			0.0450561	+	0.998984i	$0.485653\pi$
$864$		0			0
$865$			3.21916i			0.109455i
$866$		0			0
$867$	13.5557			0.460375
$868$		0			0
$869$			64.9125i			2.20201i
$870$		0			0
$871$		−	20.4018i		−	0.691290i
$872$		0			0
$873$			11.9942i			0.405941i
$874$		0			0
$875$		−	3.67104i		−	0.124104i
$876$		0			0
$877$		−	47.7471i		−	1.61231i	−0.591708	−	0.806153i	$-0.701546\pi$
$877$		−	47.7471i		−	1.61231i	0.591708	−	0.806153i	$-0.298454\pi$
$878$		0			0
$879$		−	6.16284i		−	0.207867i
$880$		0			0
$881$	8.62405			0.290552			0.145276	−	0.989391i	$-0.453593\pi$
$881$	8.62405			0.290552			0.145276	+	0.989391i	$0.453593\pi$
$882$		0			0
$883$			19.6878i			0.662546i	0.943535	+	0.331273i	$0.107478\pi$
$883$			19.6878i			0.662546i	−0.943535	+	0.331273i	$0.892522\pi$
$884$		0			0
$885$	0.965936			0.0324696
$886$		0			0
$887$	20.7575			0.696968			0.348484	−	0.937315i	$-0.386697\pi$
$887$	20.7575			0.696968			0.348484	+	0.937315i	$0.386697\pi$
$888$		0			0
$889$			46.8462i			1.57117i
$890$		0			0
$891$			4.75021i			0.159138i
$892$		0			0
$893$	17.2147	−	5.47312i	0.576067	−	0.183151i
$894$		0			0
$895$	21.1329			0.706395
$896$		0			0
$897$	19.3034			0.644522
$898$		0			0
$899$			13.3146i			0.444068i
$900$		0			0
$901$		−	26.3966i		−	0.879399i
$902$		0			0
$903$	−3.56223			−0.118543
$904$		0			0
$905$			14.4107i			0.479027i
$906$		0			0
$907$	−35.4444			−1.17691			−0.588456	−	0.808529i	$-0.700264\pi$
$907$	−35.4444			−1.17691			−0.588456	+	0.808529i	$0.700264\pi$
$908$		0			0
$909$	−5.19000			−0.172141
$910$		0			0
$911$	−22.6192			−0.749407			−0.374704	−	0.927145i	$-0.622255\pi$
$911$	−22.6192			−0.749407			−0.374704	+	0.927145i	$0.622255\pi$
$912$		0			0
$913$	9.20961			0.304794
$914$		0			0
$915$	3.72057			0.122998
$916$		0			0
$917$	−36.4797			−1.20467
$918$		0			0
$919$		−	59.1693i		−	1.95181i	−0.218185	−	0.975907i	$-0.570014\pi$
$919$		−	59.1693i		−	1.95181i	0.218185	−	0.975907i	$-0.429986\pi$
$920$		0			0
$921$	−18.4548			−0.608108
$922$		0			0
$923$			19.6065i			0.645355i
$924$		0			0
$925$			2.49374i			0.0819937i
$926$		0			0
$927$	4.30115			0.141268
$928$		0			0
$929$	28.1666			0.924118			0.462059	−	0.886849i	$-0.347111\pi$
$929$	28.1666			0.924118			0.462059	+	0.886849i	$0.347111\pi$
$930$		0			0
$931$	8.55352	+	26.9035i	0.280330	+	0.881725i
$932$		0			0
$933$			10.1604i			0.332637i
$934$		0			0
$935$		−	8.81585i		−	0.288309i
$936$		0			0
$937$	43.6505			1.42600			0.712999	−	0.701165i	$-0.247336\pi$
$937$	43.6505			1.42600			0.712999	+	0.701165i	$0.247336\pi$
$938$		0			0
$939$	−22.2080			−0.724731
$940$		0			0
$941$		−	47.1932i		−	1.53846i	−0.638975	−	0.769228i	$-0.720641\pi$
$941$		−	47.1932i		−	1.53846i	0.638975	−	0.769228i	$-0.279359\pi$
$942$		0			0
$943$	−18.6908			−0.608657
$944$		0			0
$945$			3.67104i			0.119419i
$946$		0			0
$947$			48.7018i			1.58260i	0.611431	+	0.791298i	$0.290594\pi$
$947$			48.7018i			1.58260i	−0.611431	+	0.791298i	$0.709406\pi$
$948$		0			0
$949$			1.09023i			0.0353903i
$950$		0			0
$951$			10.5114i			0.340856i
$952$		0			0
$953$		−	35.7917i		−	1.15941i	−0.814827	−	0.579704i	$-0.803168\pi$
$953$		−	35.7917i		−	1.15941i	0.814827	−	0.579704i	$-0.196832\pi$
$954$		0			0
$955$		−	25.3382i		−	0.819924i
$956$		0			0
$957$	−21.3880			−0.691375
$958$		0			0
$959$		−	20.2804i		−	0.654889i
$960$		0			0
$961$	−22.2553			−0.717913
$962$		0			0
$963$	13.5432			0.436422
$964$		0			0
$965$			25.4026i			0.817738i
$966$		0			0
$967$		−	27.9652i		−	0.899299i	−0.893205	−	0.449650i	$-0.851549\pi$
$967$		−	27.9652i		−	0.899299i	0.893205	−	0.449650i	$-0.148451\pi$
$968$		0			0
$969$	2.45107	+	7.70936i	0.0787396	+	0.247660i
$970$		0			0
$971$	18.6661			0.599025			0.299512	−	0.954092i	$-0.403176\pi$
$971$	18.6661			0.599025			0.299512	+	0.954092i	$0.403176\pi$
$972$		0			0
$973$	27.8622			0.893222
$974$		0			0
$975$			4.23639i			0.135673i
$976$		0			0
$977$		−	61.0269i		−	1.95242i	−0.216817	−	0.976212i	$-0.569568\pi$
$977$		−	61.0269i		−	1.95242i	0.216817	−	0.976212i	$-0.430432\pi$
$978$		0			0
$979$	−12.0906			−0.386418
$980$		0			0
$981$			0.211359i			0.00674816i
$982$		0			0
$983$	−6.58414			−0.210001			−0.105001	−	0.994472i	$-0.533484\pi$
$983$	−6.58414			−0.210001			−0.105001	+	0.994472i	$0.533484\pi$
$984$		0			0
$985$	8.14170			0.259416
$986$		0			0
$987$	15.2132			0.484241
$988$		0			0
$989$	−4.42151			−0.140596
$990$		0			0
$991$	−59.0557			−1.87597			−0.937983	−	0.346681i	$-0.887309\pi$
$991$	−59.0557			−1.87597			−0.937983	+	0.346681i	$0.887309\pi$
$992$		0			0
$993$	2.22476			0.0706006
$994$		0			0
$995$			22.1822i			0.703222i
$996$		0			0
$997$	2.86565			0.0907561			0.0453780	−	0.998970i	$-0.485551\pi$
$997$	2.86565			0.0907561			0.0453780	+	0.998970i	$0.485551\pi$
$998$		0			0
$999$		−	2.49374i		−	0.0788985i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.d.i.2431.11	yes	12
4.3	odd	2		4560.2.d.k.2431.2	yes	12
19.18	odd	2		4560.2.d.k.2431.11	yes	12
76.75	even	2	inner	4560.2.d.i.2431.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4560.2.d.i.2431.2	✓	12	76.75	even	2	inner
4560.2.d.i.2431.11	yes	12	1.1	even	1	trivial
4560.2.d.k.2431.2	yes	12	4.3	odd	2
4560.2.d.k.2431.11	yes	12	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +3.67104i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +3.67104i q^{7} +1.00000 q^{9} +4.75021i q^{11} -4.23639i q^{13} +1.00000 q^{15} +1.85589 q^{17} +(-1.32070 - 4.15400i) q^{19} -3.67104i q^{21} -4.55656i q^{23} +1.00000 q^{25} -1.00000 q^{27} -4.50254i q^{29} -2.95714 q^{31} -4.75021i q^{33} -3.67104i q^{35} +2.49374i q^{37} +4.23639i q^{39} -4.10196i q^{41} -0.970359i q^{43} -1.00000 q^{45} +4.14411i q^{47} -6.47651 q^{49} -1.85589 q^{51} -14.2232i q^{53} -4.75021i q^{55} +(1.32070 + 4.15400i) q^{57} +0.965936 q^{59} +3.72057 q^{61} +3.67104i q^{63} +4.23639i q^{65} +4.81585 q^{67} +4.55656i q^{69} -4.62810 q^{71} -0.257348 q^{73} -1.00000 q^{75} -17.4382 q^{77} +13.6652 q^{79} +1.00000 q^{81} -1.93878i q^{83} -1.85589 q^{85} +4.50254i q^{87} +2.54529i q^{89} +15.5520 q^{91} +2.95714 q^{93} +(1.32070 + 4.15400i) q^{95} +11.9942i q^{97} +4.75021i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} - 12 q^{5} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 - 12 * q^5 + 12 * q^9	\( 12 q - 12 q^{3} - 12 q^{5} + 12 q^{9} + 12 q^{15} - 8 q^{17} - 8 q^{19} + 12 q^{25} - 12 q^{27} - 12 q^{45} - 20 q^{49} + 8 q^{51} + 8 q^{57} - 8 q^{59} + 8 q^{67} - 32 q^{71} - 24 q^{73} - 12 q^{75} + 56 q^{77} - 16 q^{79} + 12 q^{81} + 8 q^{85} + 16 q^{91} + 8 q^{95}+O(q^{100}) \) 12 * q - 12 * q^3 - 12 * q^5 + 12 * q^9 + 12 * q^15 - 8 * q^17 - 8 * q^19 + 12 * q^25 - 12 * q^27 - 12 * q^45 - 20 * q^49 + 8 * q^51 + 8 * q^57 - 8 * q^59 + 8 * q^67 - 32 * q^71 - 24 * q^73 - 12 * q^75 + 56 * q^77 - 16 * q^79 + 12 * q^81 + 8 * q^85 + 16 * q^91 + 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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