LMFDB - Embedded newform 1840.3.k.b.321.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,3,Mod(321,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1840.321");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1840.k (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:	$50.1363686423$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2 x^{9} - 10 x^{8} + 34 x^{7} + 346 x^{6} - 968 x^{5} + 165 x^{4} + 6972 x^{3} + 19344 x^{2} + \cdots + 225444 $ x^10 - 2x^9 - 10x^8 + 34x^7 + 346x^6 - 968x^5 + 165x^4 + 6972x^3 + 19344x^2 - 89080*x + 225444
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 5 $
Twist minimal:	no (minimal twist has level 115)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.4
Root			$-2.67869 + 2.23607i$ of defining polynomial
Character	$\chi$	$=$	1840.321
Dual form			1840.3.k.b.321.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.23330 q^{3} +2.23607i q^{5} +0.521669i q^{7} -7.47898 q^{9} +O(q^{10})$	$q-1.23330 q^{3} +2.23607i q^{5} +0.521669i q^{7} -7.47898 q^{9} -10.5914i q^{11} +14.3029 q^{13} -2.75774i q^{15} +33.4342i q^{17} -23.1053i q^{19} -0.643373i q^{21} +(-22.5772 - 4.39000i) q^{23} -5.00000 q^{25} +20.3235 q^{27} +11.7613 q^{29} +17.8286 q^{31} +13.0624i q^{33} -1.16649 q^{35} -57.2590i q^{37} -17.6398 q^{39} -33.7426 q^{41} +53.7180i q^{43} -16.7235i q^{45} -14.0973 q^{47} +48.7279 q^{49} -41.2343i q^{51} +85.7672i q^{53} +23.6831 q^{55} +28.4958i q^{57} +70.7106 q^{59} -31.8109i q^{61} -3.90155i q^{63} +31.9824i q^{65} -39.4556i q^{67} +(27.8443 + 5.41418i) q^{69} -68.9236 q^{71} +30.3374 q^{73} +6.16649 q^{75} +5.52521 q^{77} +49.0126i q^{79} +42.2459 q^{81} +96.7624i q^{83} -74.7611 q^{85} -14.5052 q^{87} -56.7850i q^{89} +7.46140i q^{91} -21.9879 q^{93} +51.6651 q^{95} +80.0997i q^{97} +79.2129i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{3} - 16 q^{9}+O(q^{10}) $ 10 * q + 2 * q^3 - 16 * q^9	$ 10 q + 2 q^{3} - 16 q^{9} - 2 q^{13} - 44 q^{23} - 50 q^{25} - 40 q^{27} - 46 q^{29} - 16 q^{31} + 60 q^{35} - 72 q^{39} - 84 q^{41} - 112 q^{47} + 50 q^{49} + 10 q^{55} + 262 q^{59} + 124 q^{69} - 236 q^{71} + 168 q^{73} - 10 q^{75} + 300 q^{77} - 258 q^{81} - 540 q^{87} + 100 q^{93} + 90 q^{95}+O(q^{100}) $ 10 * q + 2 * q^3 - 16 * q^9 - 2 * q^13 - 44 * q^23 - 50 * q^25 - 40 * q^27 - 46 * q^29 - 16 * q^31 + 60 * q^35 - 72 * q^39 - 84 * q^41 - 112 * q^47 + 50 * q^49 + 10 * q^55 + 262 * q^59 + 124 * q^69 - 236 * q^71 + 168 * q^73 - 10 * q^75 + 300 * q^77 - 258 * q^81 - 540 * q^87 + 100 * q^93 + 90 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	−1.23330			−0.411099			−0.205550	−	0.978647i	$-0.565898\pi$
$3$	−1.23330			−0.411099			−0.205550	+	0.978647i	$0.565898\pi$
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$			0.521669i			0.0745241i	0.999306	+	0.0372620i	$0.0118636\pi$
$7$			0.521669i			0.0745241i	−0.999306	+	0.0372620i	$0.988136\pi$
$8$		0			0
$9$	−7.47898			−0.830998
$10$		0			0
$11$		−	10.5914i		−	0.962856i	−0.876486	−	0.481428i	$-0.840118\pi$
$11$		−	10.5914i		−	0.962856i	0.876486	−	0.481428i	$-0.159882\pi$
$12$		0			0
$13$	14.3029			1.10023			0.550113	−	0.835090i	$-0.314585\pi$
$13$	14.3029			1.10023			0.550113	+	0.835090i	$0.314585\pi$
$14$		0			0
$15$		−	2.75774i		−	0.183849i
$16$		0			0
$17$			33.4342i			1.96672i	0.181676	+	0.983358i	$0.441848\pi$
$17$			33.4342i			1.96672i	−0.181676	+	0.983358i	$0.558152\pi$
$18$		0			0
$19$		−	23.1053i		−	1.21607i	−0.793910	−	0.608035i	$-0.791958\pi$
$19$		−	23.1053i		−	1.21607i	0.793910	−	0.608035i	$-0.208042\pi$
$20$		0			0
$21$		−	0.643373i		−	0.0306368i
$22$		0			0
$23$	−22.5772	−	4.39000i	−0.981615	−	0.190870i
$23$	−22.5772	−	4.39000i	−0.981615	−	0.190870i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	20.3235			0.752721
$28$		0			0
$29$	11.7613			0.405563			0.202782	−	0.979224i	$-0.435002\pi$
$29$	11.7613			0.405563			0.202782	+	0.979224i	$0.435002\pi$
$30$		0			0
$31$	17.8286			0.575115			0.287558	−	0.957763i	$-0.407157\pi$
$31$	17.8286			0.575115			0.287558	+	0.957763i	$0.407157\pi$
$32$		0			0
$33$			13.0624i			0.395829i
$34$		0			0
$35$	−1.16649			−0.0333282
$36$		0			0
$37$		−	57.2590i		−	1.54754i	−0.633466	−	0.773771i	$-0.718368\pi$
$37$		−	57.2590i		−	1.54754i	0.633466	−	0.773771i	$-0.281632\pi$
$38$		0			0
$39$	−17.6398			−0.452302
$40$		0			0
$41$	−33.7426			−0.822991			−0.411496	−	0.911412i	$-0.634993\pi$
$41$	−33.7426			−0.822991			−0.411496	+	0.911412i	$0.634993\pi$
$42$		0			0
$43$			53.7180i			1.24926i	0.780922	+	0.624628i	$0.214749\pi$
$43$			53.7180i			1.24926i	−0.780922	+	0.624628i	$0.785251\pi$
$44$		0			0
$45$		−	16.7235i		−	0.371633i
$46$		0			0
$47$	−14.0973			−0.299942			−0.149971	−	0.988690i	$-0.547918\pi$
$47$	−14.0973			−0.299942			−0.149971	+	0.988690i	$0.547918\pi$
$48$		0			0
$49$	48.7279			0.994446
$50$		0			0
$51$		−	41.2343i		−	0.808516i
$52$		0			0
$53$			85.7672i			1.61825i	0.587638	+	0.809124i	$0.300058\pi$
$53$			85.7672i			1.61825i	−0.587638	+	0.809124i	$0.699942\pi$
$54$		0			0
$55$	23.6831			0.430602
$56$		0			0
$57$			28.4958i			0.499926i
$58$		0			0
$59$	70.7106			1.19848			0.599242	−	0.800568i	$-0.295469\pi$
$59$	70.7106			1.19848			0.599242	+	0.800568i	$0.295469\pi$
$60$		0			0
$61$		−	31.8109i		−	0.521490i	−0.965408	−	0.260745i	$-0.916032\pi$
$61$		−	31.8109i		−	0.521490i	0.965408	−	0.260745i	$-0.0839682\pi$
$62$		0			0
$63$		−	3.90155i		−	0.0619293i
$64$		0			0
$65$			31.9824i			0.492036i
$66$		0			0
$67$		−	39.4556i		−	0.588890i	−0.955668	−	0.294445i	$-0.904865\pi$
$67$		−	39.4556i		−	0.588890i	0.955668	−	0.294445i	$-0.0951348\pi$
$68$		0			0
$69$	27.8443	+	5.41418i	0.403541	+	0.0784664i
$70$		0			0
$71$	−68.9236			−0.970754			−0.485377	−	0.874305i	$-0.661318\pi$
$71$	−68.9236			−0.970754			−0.485377	+	0.874305i	$0.661318\pi$
$72$		0			0
$73$	30.3374			0.415580			0.207790	−	0.978173i	$-0.433373\pi$
$73$	30.3374			0.415580			0.207790	+	0.978173i	$0.433373\pi$
$74$		0			0
$75$	6.16649			0.0822198
$76$		0			0
$77$	5.52521			0.0717559
$78$		0			0
$79$			49.0126i			0.620412i	0.950669	+	0.310206i	$0.100398\pi$
$79$			49.0126i			0.620412i	−0.950669	+	0.310206i	$0.899602\pi$
$80$		0			0
$81$	42.2459			0.521554
$82$		0			0
$83$			96.7624i			1.16581i	0.812540	+	0.582906i	$0.198084\pi$
$83$			96.7624i			1.16581i	−0.812540	+	0.582906i	$0.801916\pi$
$84$		0			0
$85$	−74.7611			−0.879543
$86$		0			0
$87$	−14.5052			−0.166727
$88$		0			0
$89$		−	56.7850i		−	0.638034i	−0.947749	−	0.319017i	$-0.896647\pi$
$89$		−	56.7850i		−	0.638034i	0.947749	−	0.319017i	$-0.103353\pi$
$90$		0			0
$91$			7.46140i			0.0819934i
$92$		0			0
$93$	−21.9879			−0.236429
$94$		0			0
$95$	51.6651			0.543843
$96$		0			0
$97$			80.0997i			0.825770i	0.910783	+	0.412885i	$0.135479\pi$
$97$			80.0997i			0.825770i	−0.910783	+	0.412885i	$0.864521\pi$
$98$		0			0
$99$			79.2129i			0.800131i
$100$		0			0
$101$	−20.8975			−0.206906			−0.103453	−	0.994634i	$-0.532989\pi$
$101$	−20.8975			−0.206906			−0.103453	+	0.994634i	$0.532989\pi$
$102$		0			0
$103$			8.78760i			0.0853165i	0.999090	+	0.0426582i	$0.0135827\pi$
$103$			8.78760i			0.0853165i	−0.999090	+	0.0426582i	$0.986417\pi$
$104$		0			0
$105$	1.43862			0.0137012
$106$		0			0
$107$			198.899i			1.85887i	0.368992	+	0.929433i	$0.379703\pi$
$107$			198.899i			1.85887i	−0.368992	+	0.929433i	$0.620297\pi$
$108$		0			0
$109$			142.315i			1.30564i	0.757512	+	0.652821i	$0.226415\pi$
$109$			142.315i			1.30564i	−0.757512	+	0.652821i	$0.773585\pi$
$110$		0			0
$111$			70.6174i			0.636193i
$112$		0			0
$113$			22.9994i			0.203534i	0.994808	+	0.101767i	$0.0324497\pi$
$113$			22.9994i			0.203534i	−0.994808	+	0.101767i	$0.967550\pi$
$114$		0			0
$115$	9.81635	−	50.4841i	0.0853595	−	0.438992i
$116$		0			0
$117$	−106.971			−0.914286
$118$		0			0
$119$	−17.4416			−0.146568
$120$		0			0
$121$	8.82202			0.0729093
$122$		0			0
$123$	41.6147			0.338331
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	59.8833			0.471522			0.235761	−	0.971811i	$-0.424242\pi$
$127$	59.8833			0.471522			0.235761	+	0.971811i	$0.424242\pi$
$128$		0			0
$129$		−	66.2503i		−	0.513568i
$130$		0			0
$131$	−158.280			−1.20824			−0.604121	−	0.796892i	$-0.706476\pi$
$131$	−158.280			−1.20824			−0.604121	+	0.796892i	$0.706476\pi$
$132$		0			0
$133$	12.0533			0.0906266
$134$		0			0
$135$			45.4447i			0.336627i
$136$		0			0
$137$			92.0533i			0.671922i	0.941876	+	0.335961i	$0.109061\pi$
$137$			92.0533i			0.671922i	−0.941876	+	0.335961i	$0.890939\pi$
$138$		0			0
$139$	−3.03747			−0.0218523			−0.0109262	−	0.999940i	$-0.503478\pi$
$139$	−3.03747			−0.0218523			−0.0109262	+	0.999940i	$0.503478\pi$
$140$		0			0
$141$	17.3861			0.123306
$142$		0			0
$143$		−	151.488i		−	1.05936i
$144$		0			0
$145$			26.2991i			0.181373i
$146$		0			0
$147$	−60.0959			−0.408816
$148$		0			0
$149$			127.915i			0.858493i	0.903187	+	0.429247i	$0.141221\pi$
$149$			127.915i			0.858493i	−0.903187	+	0.429247i	$0.858779\pi$
$150$		0			0
$151$	−45.2103			−0.299406			−0.149703	−	0.988731i	$-0.547832\pi$
$151$	−45.2103			−0.299406			−0.149703	+	0.988731i	$0.547832\pi$
$152$		0			0
$153$		−	250.054i		−	1.63434i
$154$		0			0
$155$			39.8659i			0.257199i
$156$		0			0
$157$			19.3773i			0.123422i	0.998094	+	0.0617110i	$0.0196557\pi$
$157$			19.3773i			0.123422i	−0.998094	+	0.0617110i	$0.980344\pi$
$158$		0			0
$159$		−	105.776i		−	0.665260i
$160$		0			0
$161$	2.29013	−	11.7778i	0.0142244	−	0.0731540i
$162$		0			0
$163$	−293.153			−1.79849			−0.899243	−	0.437450i	$-0.855882\pi$
$163$	−293.153			−1.79849			−0.899243	+	0.437450i	$0.855882\pi$
$164$		0			0
$165$	−29.2083			−0.177020
$166$		0			0
$167$	158.029			0.946282			0.473141	−	0.880987i	$-0.343120\pi$
$167$	158.029			0.946282			0.473141	+	0.880987i	$0.343120\pi$
$168$		0			0
$169$	35.5742			0.210498
$170$		0			0
$171$			172.804i			1.01055i
$172$		0			0
$173$	−114.157			−0.659868			−0.329934	−	0.944004i	$-0.607027\pi$
$173$	−114.157			−0.659868			−0.329934	+	0.944004i	$0.607027\pi$
$174$		0			0
$175$		−	2.60834i		−	0.0149048i
$176$		0			0
$177$	−87.2071			−0.492696
$178$		0			0
$179$	172.889			0.965862			0.482931	−	0.875659i	$-0.339572\pi$
$179$	172.889			0.965862			0.482931	+	0.875659i	$0.339572\pi$
$180$		0			0
$181$			173.085i			0.956270i	0.878286	+	0.478135i	$0.158687\pi$
$181$			173.085i			0.956270i	−0.878286	+	0.478135i	$0.841313\pi$
$182$		0			0
$183$			39.2323i			0.214384i
$184$		0			0
$185$	128.035			0.692082
$186$		0			0
$187$	354.115			1.89366
$188$		0			0
$189$			10.6021i			0.0560959i
$190$		0			0
$191$			213.863i			1.11970i	0.828594	+	0.559850i	$0.189141\pi$
$191$			213.863i			1.11970i	−0.828594	+	0.559850i	$0.810859\pi$
$192$		0			0
$193$	232.186			1.20303			0.601517	−	0.798860i	$-0.294563\pi$
$193$	232.186			1.20303			0.601517	+	0.798860i	$0.294563\pi$
$194$		0			0
$195$		−	39.4438i		−	0.202276i
$196$		0			0
$197$	301.664			1.53129			0.765646	−	0.643262i	$-0.222420\pi$
$197$	301.664			1.53129			0.765646	+	0.643262i	$0.222420\pi$
$198$		0			0
$199$			349.041i			1.75397i	0.480515	+	0.876986i	$0.340450\pi$
$199$			349.041i			1.75397i	−0.480515	+	0.876986i	$0.659550\pi$
$200$		0			0
$201$			48.6605i			0.242092i
$202$		0			0
$203$			6.13552i			0.0302242i
$204$		0			0
$205$		−	75.4508i		−	0.368053i
$206$		0			0
$207$	168.854	+	32.8327i	0.815720	+	0.158612i
$208$		0			0
$209$	−244.718			−1.17090
$210$		0			0
$211$	178.918			0.847952			0.423976	−	0.905673i	$-0.360634\pi$
$211$	178.918			0.847952			0.423976	+	0.905673i	$0.360634\pi$
$212$		0			0
$213$	85.0032			0.399076
$214$		0			0
$215$	−120.117			−0.558684
$216$		0			0
$217$			9.30060i			0.0428599i
$218$		0			0
$219$	−37.4150			−0.170845
$220$		0			0
$221$			478.207i			2.16383i
$222$		0			0
$223$	−131.985			−0.591862			−0.295931	−	0.955209i	$-0.595630\pi$
$223$	−131.985			−0.591862			−0.295931	+	0.955209i	$0.595630\pi$
$224$		0			0
$225$	37.3949			0.166200
$226$		0			0
$227$			91.7919i			0.404369i	0.979347	+	0.202185i	$0.0648041\pi$
$227$			91.7919i			0.404369i	−0.979347	+	0.202185i	$0.935196\pi$
$228$		0			0
$229$			304.840i			1.33118i	0.746318	+	0.665589i	$0.231820\pi$
$229$			304.840i			1.33118i	−0.746318	+	0.665589i	$0.768180\pi$
$230$		0			0
$231$	−6.81422			−0.0294988
$232$		0			0
$233$	239.908			1.02965			0.514824	−	0.857296i	$-0.327857\pi$
$233$	239.908			1.02965			0.514824	+	0.857296i	$0.327857\pi$
$234$		0			0
$235$		−	31.5224i		−	0.134138i
$236$		0			0
$237$		−	60.4471i		−	0.255051i
$238$		0			0
$239$	324.673			1.35846			0.679232	−	0.733923i	$-0.262313\pi$
$239$	324.673			1.35846			0.679232	+	0.733923i	$0.262313\pi$
$240$		0			0
$241$		−	133.840i		−	0.555353i	−0.960675	−	0.277676i	$-0.910436\pi$
$241$		−	133.840i		−	0.555353i	0.960675	−	0.277676i	$-0.0895643\pi$
$242$		0			0
$243$	−235.013			−0.967132
$244$		0			0
$245$			108.959i			0.444730i
$246$		0			0
$247$		−	330.475i		−	1.33795i
$248$		0			0
$249$		−	119.337i		−	0.479264i
$250$		0			0
$251$			134.518i			0.535929i	0.963429	+	0.267964i	$0.0863509\pi$
$251$			134.518i			0.535929i	−0.963429	+	0.267964i	$0.913649\pi$
$252$		0			0
$253$	−46.4963	+	239.124i	−0.183780	+	0.945154i
$254$		0			0
$255$	92.2027			0.361579
$256$		0			0
$257$	186.968			0.727503			0.363752	−	0.931496i	$-0.381496\pi$
$257$	186.968			0.727503			0.363752	+	0.931496i	$0.381496\pi$
$258$		0			0
$259$	29.8703			0.115329
$260$		0			0
$261$	−87.9628			−0.337022
$262$		0			0
$263$			188.537i			0.716871i	0.933555	+	0.358435i	$0.116690\pi$
$263$			188.537i			0.716871i	−0.933555	+	0.358435i	$0.883310\pi$
$264$		0			0
$265$	−191.781			−0.723703
$266$		0			0
$267$			70.0328i			0.262295i
$268$		0			0
$269$	−238.007			−0.884786			−0.442393	−	0.896821i	$-0.645870\pi$
$269$	−238.007			−0.884786			−0.442393	+	0.896821i	$0.645870\pi$
$270$		0			0
$271$	−134.114			−0.494885			−0.247442	−	0.968903i	$-0.579590\pi$
$271$	−134.114			−0.494885			−0.247442	+	0.968903i	$0.579590\pi$
$272$		0			0
$273$		−	9.20212i		−	0.0337074i
$274$		0			0
$275$			52.9571i			0.192571i
$276$		0			0
$277$	389.169			1.40494			0.702471	−	0.711713i	$-0.252080\pi$
$277$	389.169			1.40494			0.702471	+	0.711713i	$0.252080\pi$
$278$		0			0
$279$	−133.339			−0.477919
$280$		0			0
$281$		−	76.2923i		−	0.271503i	−0.990743	−	0.135751i	$-0.956655\pi$
$281$		−	76.2923i		−	0.271503i	0.990743	−	0.135751i	$-0.0433449\pi$
$282$		0			0
$283$			263.833i			0.932271i	0.884713	+	0.466136i	$0.154354\pi$
$283$			263.833i			0.932271i	−0.884713	+	0.466136i	$0.845646\pi$
$284$		0			0
$285$	−63.7185			−0.223574
$286$		0			0
$287$		−	17.6025i		−	0.0613327i
$288$		0			0
$289$	−828.845			−2.86798
$290$		0			0
$291$		−	98.7868i		−	0.339473i
$292$		0			0
$293$		−	66.4738i		−	0.226873i	−0.993545	−	0.113437i	$-0.963814\pi$
$293$		−	66.4738i		−	0.226873i	0.993545	−	0.113437i	$-0.0361859\pi$
$294$		0			0
$295$			158.114i			0.535978i
$296$		0			0
$297$		−	215.254i		−	0.724762i
$298$		0			0
$299$	−322.920	−	62.7900i	−1.08000	−	0.210000i
$300$		0			0
$301$	−28.0230			−0.0930997
$302$		0			0
$303$	25.7728			0.0850589
$304$		0			0
$305$	71.1314			0.233218
$306$		0			0
$307$	−76.2175			−0.248266			−0.124133	−	0.992266i	$-0.539615\pi$
$307$	−76.2175			−0.248266			−0.124133	+	0.992266i	$0.539615\pi$
$308$		0			0
$309$		−	10.8377i		−	0.0350735i
$310$		0			0
$311$	−378.675			−1.21761			−0.608803	−	0.793321i	$-0.708350\pi$
$311$	−378.675			−1.21761			−0.608803	+	0.793321i	$0.708350\pi$
$312$		0			0
$313$			226.302i			0.723010i	0.932370	+	0.361505i	$0.117737\pi$
$313$			226.302i			0.723010i	−0.932370	+	0.361505i	$0.882263\pi$
$314$		0			0
$315$	8.72413			0.0276956
$316$		0			0
$317$	431.012			1.35966			0.679830	−	0.733370i	$-0.262054\pi$
$317$	431.012			1.35966			0.679830	+	0.733370i	$0.262054\pi$
$318$		0			0
$319$		−	124.569i		−	0.390499i
$320$		0			0
$321$		−	245.301i		−	0.764178i
$322$		0			0
$323$	772.508			2.39167
$324$		0			0
$325$	−71.5147			−0.220045
$326$		0			0
$327$		−	175.517i		−	0.536748i
$328$		0			0
$329$		−	7.35410i		−	0.0223529i
$330$		0			0
$331$	−385.002			−1.16315			−0.581574	−	0.813493i	$-0.697563\pi$
$331$	−385.002			−1.16315			−0.581574	+	0.813493i	$0.697563\pi$
$332$		0			0
$333$			428.239i			1.28600i
$334$		0			0
$335$	88.2255			0.263360
$336$		0			0
$337$		−	203.770i		−	0.604658i	−0.953204	−	0.302329i	$-0.902236\pi$
$337$		−	203.770i		−	0.604658i	0.953204	−	0.302329i	$-0.0977642\pi$
$338$		0			0
$339$		−	28.3651i		−	0.0836728i
$340$		0			0
$341$		−	188.830i		−	0.553753i
$342$		0			0
$343$			50.9816i			0.148634i
$344$		0			0
$345$	−12.1065	+	62.2618i	−0.0350912	+	0.180469i
$346$		0			0
$347$	−55.5063			−0.159961			−0.0799803	−	0.996796i	$-0.525486\pi$
$347$	−55.5063			−0.159961			−0.0799803	+	0.996796i	$0.525486\pi$
$348$		0			0
$349$	588.423			1.68603			0.843013	−	0.537893i	$-0.180780\pi$
$349$	588.423			1.68603			0.843013	+	0.537893i	$0.180780\pi$
$350$		0			0
$351$	290.686			0.828164
$352$		0			0
$353$	−447.976			−1.26905			−0.634527	−	0.772900i	$-0.718805\pi$
$353$	−447.976			−1.26905			−0.634527	+	0.772900i	$0.718805\pi$
$354$		0			0
$355$		−	154.118i		−	0.434135i
$356$		0			0
$357$	21.5106			0.0602539
$358$		0			0
$359$		−	563.019i		−	1.56830i	−0.620573	−	0.784149i	$-0.713100\pi$
$359$		−	563.019i		−	1.56830i	0.620573	−	0.784149i	$-0.286900\pi$
$360$		0			0
$361$	−172.857			−0.478828
$362$		0			0
$363$	−10.8802			−0.0299729
$364$		0			0
$365$			67.8364i			0.185853i
$366$		0			0
$367$			306.018i			0.833836i	0.908944	+	0.416918i	$0.136890\pi$
$367$			306.018i			0.833836i	−0.908944	+	0.416918i	$0.863110\pi$
$368$		0			0
$369$	252.360			0.683904
$370$		0			0
$371$	−44.7420			−0.120598
$372$		0			0
$373$		−	80.0977i		−	0.214739i	−0.994219	−	0.107370i	$-0.965757\pi$
$373$		−	80.0977i		−	0.214739i	0.994219	−	0.107370i	$-0.0342428\pi$
$374$		0			0
$375$			13.7887i			0.0367698i
$376$		0			0
$377$	168.222			0.446212
$378$		0			0
$379$			5.99870i			0.0158277i	0.999969	+	0.00791385i	$0.00251908\pi$
$379$			5.99870i			0.0158277i	−0.999969	+	0.00791385i	$0.997481\pi$
$380$		0			0
$381$	−73.8539			−0.193842
$382$		0			0
$383$		−	490.051i		−	1.27951i	−0.768580	−	0.639753i	$-0.779037\pi$
$383$		−	490.051i		−	1.27951i	0.768580	−	0.639753i	$-0.220963\pi$
$384$		0			0
$385$			12.3547i			0.0320902i
$386$		0			0
$387$		−	401.756i		−	1.03813i
$388$		0			0
$389$		−	536.332i		−	1.37874i	−0.724407	−	0.689372i	$-0.757886\pi$
$389$		−	536.332i		−	1.37874i	0.724407	−	0.689372i	$-0.242114\pi$
$390$		0			0
$391$	146.776	−	754.849i	0.375387	−	1.93056i
$392$		0			0
$393$	195.206			0.496707
$394$		0			0
$395$	−109.595			−0.277457
$396$		0			0
$397$	97.6151			0.245882			0.122941	−	0.992414i	$-0.460767\pi$
$397$	97.6151			0.245882			0.122941	+	0.992414i	$0.460767\pi$
$398$		0			0
$399$	−14.8653			−0.0372565
$400$		0			0
$401$			756.313i			1.88607i	0.332696	+	0.943034i	$0.392042\pi$
$401$			756.313i			1.88607i	−0.332696	+	0.943034i	$0.607958\pi$
$402$		0			0
$403$	255.001			0.632757
$404$		0			0
$405$			94.4647i			0.233246i
$406$		0			0
$407$	−606.454			−1.49006
$408$		0			0
$409$	−425.609			−1.04061			−0.520304	−	0.853981i	$-0.674181\pi$
$409$	−425.609			−1.04061			−0.520304	+	0.853981i	$0.674181\pi$
$410$		0			0
$411$		−	113.529i		−	0.276226i
$412$		0			0
$413$			36.8875i			0.0893159i
$414$		0			0
$415$	−216.367			−0.521367
$416$		0			0
$417$	3.74611			0.00898347
$418$		0			0
$419$		−	663.073i		−	1.58251i	−0.611484	−	0.791257i	$-0.709427\pi$
$419$		−	663.073i		−	1.58251i	0.611484	−	0.791257i	$-0.290573\pi$
$420$		0			0
$421$		−	163.431i		−	0.388197i	−0.980982	−	0.194098i	$-0.937822\pi$
$421$		−	163.431i		−	0.388197i	0.980982	−	0.194098i	$-0.0621781\pi$
$422$		0			0
$423$	105.433			0.249251
$424$		0			0
$425$		−	167.171i		−	0.393343i
$426$		0			0
$427$	16.5948			0.0388636
$428$		0			0
$429$			186.830i			0.435502i
$430$		0			0
$431$			213.125i			0.494490i	0.968953	+	0.247245i	$0.0795252\pi$
$431$			213.125i			0.494490i	−0.968953	+	0.247245i	$0.920475\pi$
$432$		0			0
$433$		−	769.179i		−	1.77640i	−0.459461	−	0.888198i	$-0.651957\pi$
$433$		−	769.179i		−	1.77640i	0.459461	−	0.888198i	$-0.348043\pi$
$434$		0			0
$435$		−	32.4347i		−	0.0745625i
$436$		0			0
$437$	−101.433	+	521.653i	−0.232111	+	1.19371i
$438$		0			0
$439$	615.630			1.40235			0.701173	−	0.712991i	$-0.252660\pi$
$439$	615.630			1.40235			0.701173	+	0.712991i	$0.252660\pi$
$440$		0			0
$441$	−364.435			−0.826382
$442$		0			0
$443$	34.9484			0.0788904			0.0394452	−	0.999222i	$-0.487441\pi$
$443$	34.9484			0.0788904			0.0394452	+	0.999222i	$0.487441\pi$
$444$		0			0
$445$	126.975			0.285337
$446$		0			0
$447$		−	157.758i		−	0.352926i
$448$		0			0
$449$	259.106			0.577074			0.288537	−	0.957469i	$-0.406831\pi$
$449$	259.106			0.577074			0.288537	+	0.957469i	$0.406831\pi$
$450$		0			0
$451$			357.382i			0.792422i
$452$		0			0
$453$	55.7578			0.123086
$454$		0			0
$455$	−16.6842			−0.0366686
$456$		0			0
$457$		−	362.602i		−	0.793440i	−0.917940	−	0.396720i	$-0.870148\pi$
$457$		−	362.602i		−	0.793440i	0.917940	−	0.396720i	$-0.129852\pi$
$458$		0			0
$459$			679.499i			1.48039i
$460$		0			0
$461$	−703.013			−1.52497			−0.762487	−	0.647003i	$-0.776022\pi$
$461$	−703.013			−1.52497			−0.762487	+	0.647003i	$0.776022\pi$
$462$		0			0
$463$	−775.327			−1.67457			−0.837286	−	0.546765i	$-0.815859\pi$
$463$	−775.327			−1.67457			−0.837286	+	0.546765i	$0.815859\pi$
$464$		0			0
$465$		−	49.1665i		−	0.105734i
$466$		0			0
$467$		−	154.199i		−	0.330190i	−0.986278	−	0.165095i	$-0.947207\pi$
$467$		−	154.199i		−	0.330190i	0.986278	−	0.165095i	$-0.0527931\pi$
$468$		0			0
$469$	20.5828			0.0438865
$470$		0			0
$471$		−	23.8979i		−	0.0507387i
$472$		0			0
$473$	568.949			1.20285
$474$		0			0
$475$			115.527i			0.243214i
$476$		0			0
$477$		−	641.451i		−	1.34476i
$478$		0			0
$479$			496.640i			1.03683i	0.855130	+	0.518413i	$0.173477\pi$
$479$			496.640i			1.03683i	−0.855130	+	0.518413i	$0.826523\pi$
$480$		0			0
$481$		−	818.973i		−	1.70265i
$482$		0			0
$483$	−2.82441	+	14.5255i	−0.00584764	+	0.0300735i
$484$		0			0
$485$	−179.108			−0.369296
$486$		0			0
$487$	321.989			0.661168			0.330584	−	0.943777i	$-0.392754\pi$
$487$	321.989			0.661168			0.330584	+	0.943777i	$0.392754\pi$
$488$		0			0
$489$	361.545			0.739356
$490$		0			0
$491$	−247.376			−0.503820			−0.251910	−	0.967751i	$-0.581059\pi$
$491$	−247.376			−0.503820			−0.251910	+	0.967751i	$0.581059\pi$
$492$		0			0
$493$			393.231i			0.797628i
$494$		0			0
$495$	−177.125			−0.357829
$496$		0			0
$497$		−	35.9553i		−	0.0723446i
$498$		0			0
$499$	463.163			0.928183			0.464091	−	0.885787i	$-0.346381\pi$
$499$	463.163			0.928183			0.464091	+	0.885787i	$0.346381\pi$
$500$		0			0
$501$	−194.897			−0.389016
$502$		0			0
$503$			117.024i			0.232652i	0.993211	+	0.116326i	$0.0371118\pi$
$503$			117.024i			0.232652i	−0.993211	+	0.116326i	$0.962888\pi$
$504$		0			0
$505$		−	46.7282i		−	0.0925312i
$506$		0			0
$507$	−43.8736			−0.0865357
$508$		0			0
$509$	−547.729			−1.07609			−0.538044	−	0.842917i	$-0.680837\pi$
$509$	−547.729			−1.07609			−0.538044	+	0.842917i	$0.680837\pi$
$510$		0			0
$511$			15.8261i			0.0309707i
$512$		0			0
$513$		−	469.581i		−	0.915363i
$514$		0			0
$515$	−19.6497			−0.0381547
$516$		0			0
$517$			149.310i			0.288800i
$518$		0			0
$519$	140.790			0.271271
$520$		0			0
$521$			23.6219i			0.0453395i	0.999743	+	0.0226698i	$0.00721663\pi$
$521$			23.6219i			0.0453395i	−0.999743	+	0.0226698i	$0.992783\pi$
$522$		0			0
$523$		−	151.625i		−	0.289914i	−0.989438	−	0.144957i	$-0.953696\pi$
$523$		−	151.625i		−	0.289914i	0.989438	−	0.144957i	$-0.0463044\pi$
$524$		0			0
$525$			3.21686i			0.00612736i
$526$		0			0
$527$			596.084i			1.13109i
$528$		0			0
$529$	490.456	+	198.228i	0.927138	+	0.374721i
$530$		0			0
$531$	−528.843			−0.995937
$532$		0			0
$533$	−482.619			−0.905477
$534$		0			0
$535$	−444.751			−0.831310
$536$		0			0
$537$	−213.224			−0.397065
$538$		0			0
$539$		−	516.097i		−	0.957508i
$540$		0			0
$541$	243.075			0.449308			0.224654	−	0.974439i	$-0.427875\pi$
$541$	243.075			0.449308			0.224654	+	0.974439i	$0.427875\pi$
$542$		0			0
$543$		−	213.465i		−	0.393122i
$544$		0			0
$545$	−318.226			−0.583901
$546$		0			0
$547$	−66.6228			−0.121797			−0.0608983	−	0.998144i	$-0.519397\pi$
$547$	−66.6228			−0.121797			−0.0608983	+	0.998144i	$0.519397\pi$
$548$		0			0
$549$			237.913i			0.433357i
$550$		0			0
$551$		−	271.750i		−	0.493194i
$552$		0			0
$553$	−25.5683			−0.0462357
$554$		0			0
$555$	−157.905			−0.284514
$556$		0			0
$557$			548.308i			0.984395i	0.870483	+	0.492198i	$0.163806\pi$
$557$			548.308i			0.984395i	−0.870483	+	0.492198i	$0.836194\pi$
$558$		0			0
$559$			768.326i			1.37446i
$560$		0			0
$561$	−436.729			−0.778484
$562$		0			0
$563$		−	249.788i		−	0.443673i	−0.975084	−	0.221837i	$-0.928795\pi$
$563$		−	249.788i		−	0.443673i	0.975084	−	0.221837i	$-0.0712052\pi$
$564$		0			0
$565$	−51.4282			−0.0910234
$566$		0			0
$567$			22.0384i			0.0388684i
$568$		0			0
$569$		−	771.295i		−	1.35553i	−0.735280	−	0.677763i	$-0.762949\pi$
$569$		−	771.295i		−	1.35553i	0.735280	−	0.677763i	$-0.237051\pi$
$570$		0			0
$571$			34.8847i			0.0610940i	0.999533	+	0.0305470i	$0.00972492\pi$
$571$			34.8847i			0.0610940i	−0.999533	+	0.0305470i	$0.990275\pi$
$572$		0			0
$573$		−	263.756i		−	0.460308i
$574$		0			0
$575$	112.886	+	21.9500i	0.196323	+	0.0381739i
$576$		0			0
$577$	472.936			0.819646			0.409823	−	0.912165i	$-0.365590\pi$
$577$	472.936			0.819646			0.409823	+	0.912165i	$0.365590\pi$
$578$		0			0
$579$	−286.354			−0.494567
$580$		0			0
$581$	−50.4779			−0.0868810
$582$		0			0
$583$	908.395			1.55814
$584$		0			0
$585$		−	239.195i		−	0.408881i
$586$		0			0
$587$	737.792			1.25689			0.628443	−	0.777855i	$-0.283692\pi$
$587$	737.792			1.25689			0.628443	+	0.777855i	$0.283692\pi$
$588$		0			0
$589$		−	411.935i		−	0.699381i
$590$		0			0
$591$	−372.042			−0.629513
$592$		0			0
$593$	−438.218			−0.738986			−0.369493	−	0.929234i	$-0.620469\pi$
$593$	−438.218			−0.738986			−0.369493	+	0.929234i	$0.620469\pi$
$594$		0			0
$595$		−	39.0005i		−	0.0655471i
$596$		0			0
$597$		−	430.471i		−	0.721057i
$598$		0			0
$599$	−721.525			−1.20455			−0.602275	−	0.798289i	$-0.705739\pi$
$599$	−721.525			−1.20455			−0.602275	+	0.798289i	$0.705739\pi$
$600$		0			0
$601$	−152.892			−0.254395			−0.127198	−	0.991877i	$-0.540598\pi$
$601$	−152.892			−0.254395			−0.127198	+	0.991877i	$0.540598\pi$
$602$		0			0
$603$			295.088i			0.489366i
$604$		0			0
$605$			19.7266i			0.0326060i
$606$		0			0
$607$	694.250			1.14374			0.571870	−	0.820344i	$-0.306218\pi$
$607$	694.250			1.14374			0.571870	+	0.820344i	$0.306218\pi$
$608$		0			0
$609$		−	7.56692i		−	0.0124252i
$610$		0			0
$611$	−201.632			−0.330004
$612$		0			0
$613$		−	916.314i		−	1.49480i	−0.664373	−	0.747401i	$-0.731301\pi$
$613$		−	916.314i		−	1.49480i	0.664373	−	0.747401i	$-0.268699\pi$
$614$		0			0
$615$			93.0533i			0.151306i
$616$		0			0
$617$		−	414.587i		−	0.671941i	−0.941873	−	0.335970i	$-0.890936\pi$
$617$		−	414.587i		−	0.671941i	0.941873	−	0.335970i	$-0.109064\pi$
$618$		0			0
$619$		−	904.936i		−	1.46193i	−0.682414	−	0.730966i	$-0.739070\pi$
$619$		−	904.936i		−	1.46193i	0.682414	−	0.730966i	$-0.260930\pi$
$620$		0			0
$621$	−458.846	−	89.2201i	−0.738883	−	0.143672i
$622$		0			0
$623$	29.6230			0.0475489
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	301.810			0.481356
$628$		0			0
$629$	1914.41			3.04358
$630$		0			0
$631$		−	190.276i		−	0.301547i	−0.988568	−	0.150773i	$-0.951824\pi$
$631$		−	190.276i		−	0.301547i	0.988568	−	0.150773i	$-0.0481763\pi$
$632$		0			0
$633$	−220.659			−0.348592
$634$		0			0
$635$			133.903i			0.210871i
$636$		0			0
$637$	696.952			1.09412
$638$		0			0
$639$	515.478			0.806694
$640$		0			0
$641$		−	915.056i		−	1.42755i	−0.700378	−	0.713773i	$-0.746985\pi$
$641$		−	915.056i		−	1.42755i	0.700378	−	0.713773i	$-0.253015\pi$
$642$		0			0
$643$		−	998.723i		−	1.55322i	−0.629979	−	0.776612i	$-0.716936\pi$
$643$		−	998.723i		−	1.55322i	0.629979	−	0.776612i	$-0.283064\pi$
$644$		0			0
$645$	148.140			0.229675
$646$		0			0
$647$	484.292			0.748519			0.374260	−	0.927324i	$-0.377897\pi$
$647$	484.292			0.748519			0.374260	+	0.927324i	$0.377897\pi$
$648$		0			0
$649$		−	748.925i		−	1.15397i
$650$		0			0
$651$		−	11.4704i		−	0.0176197i
$652$		0			0
$653$	161.111			0.246724			0.123362	−	0.992362i	$-0.460632\pi$
$653$	161.111			0.246724			0.123362	+	0.992362i	$0.460632\pi$
$654$		0			0
$655$		−	353.924i		−	0.540343i
$656$		0			0
$657$	−226.892			−0.345346
$658$		0			0
$659$			388.234i			0.589126i	0.955632	+	0.294563i	$0.0951741\pi$
$659$			388.234i			0.589126i	−0.955632	+	0.294563i	$0.904826\pi$
$660$		0			0
$661$		−	44.1368i		−	0.0667728i	−0.999443	−	0.0333864i	$-0.989371\pi$
$661$		−	44.1368i		−	0.0667728i	0.999443	−	0.0333864i	$-0.0106292\pi$
$662$		0			0
$663$		−	589.772i		−	0.889550i
$664$		0			0
$665$			26.9521i			0.0405294i
$666$		0			0
$667$	−265.538	−	51.6323i	−0.398107	−	0.0774098i
$668$		0			0
$669$	162.777			0.243314
$670$		0			0
$671$	−336.922			−0.502120
$672$		0			0
$673$	−549.707			−0.816801			−0.408401	−	0.912803i	$-0.633913\pi$
$673$	−549.707			−0.816801			−0.408401	+	0.912803i	$0.633913\pi$
$674$		0			0
$675$	−101.617			−0.150544
$676$		0			0
$677$			767.490i			1.13366i	0.823834	+	0.566832i	$0.191831\pi$
$677$			767.490i			1.13366i	−0.823834	+	0.566832i	$0.808169\pi$
$678$		0			0
$679$	−41.7855			−0.0615398
$680$		0			0
$681$		−	113.207i		−	0.166236i
$682$		0			0
$683$	791.512			1.15888			0.579438	−	0.815017i	$-0.303272\pi$
$683$	791.512			1.15888			0.579438	+	0.815017i	$0.303272\pi$
$684$		0			0
$685$	−205.837			−0.300493
$686$		0			0
$687$		−	375.958i		−	0.547246i
$688$		0			0
$689$			1226.72i			1.78044i
$690$		0			0
$691$	464.599			0.672357			0.336179	−	0.941798i	$-0.390865\pi$
$691$	464.599			0.672357			0.336179	+	0.941798i	$0.390865\pi$
$692$		0			0
$693$	−41.3229			−0.0596290
$694$		0			0
$695$		−	6.79200i		−	0.00977266i
$696$		0			0
$697$		−	1128.16i		−	1.61859i
$698$		0			0
$699$	−295.878			−0.423287
$700$		0			0
$701$			646.813i			0.922700i	0.887218	+	0.461350i	$0.152635\pi$
$701$			646.813i			0.922700i	−0.887218	+	0.461350i	$0.847365\pi$
$702$		0			0
$703$	−1322.99			−1.88192
$704$		0			0
$705$			38.8765i			0.0551440i
$706$		0			0
$707$		−	10.9016i		−	0.0154195i
$708$		0			0
$709$			469.683i			0.662458i	0.943550	+	0.331229i	$0.107463\pi$
$709$			469.683i			0.662458i	−0.943550	+	0.331229i	$0.892537\pi$
$710$		0			0
$711$		−	366.564i		−	0.515561i
$712$		0			0
$713$	−402.518	−	78.2675i	−0.564542	−	0.109772i
$714$		0			0
$715$	338.738			0.473760
$716$		0			0
$717$	−400.418			−0.558464
$718$		0			0
$719$	−661.292			−0.919738			−0.459869	−	0.887987i	$-0.652104\pi$
$719$	−661.292			−0.919738			−0.459869	+	0.887987i	$0.652104\pi$
$720$		0			0
$721$	−4.58422			−0.00635813
$722$		0			0
$723$			165.065i			0.228305i
$724$		0			0
$725$	−58.8067			−0.0811127
$726$		0			0
$727$			20.5037i			0.0282032i	0.999901	+	0.0141016i	$0.00448882\pi$
$727$			20.5037i			0.0282032i	−0.999901	+	0.0141016i	$0.995511\pi$
$728$		0			0
$729$	−90.3722			−0.123967
$730$		0			0
$731$	−1796.02			−2.45693
$732$		0			0
$733$			90.9287i			0.124050i	0.998075	+	0.0620250i	$0.0197559\pi$
$733$			90.9287i			0.124050i	−0.998075	+	0.0620250i	$0.980244\pi$
$734$		0			0
$735$		−	134.379i		−	0.182828i
$736$		0			0
$737$	−417.891			−0.567016
$738$		0			0
$739$	679.537			0.919537			0.459768	−	0.888039i	$-0.347932\pi$
$739$	679.537			0.919537			0.459768	+	0.888039i	$0.347932\pi$
$740$		0			0
$741$			407.573i			0.550031i
$742$		0			0
$743$			748.573i			1.00750i	0.863849	+	0.503750i	$0.168047\pi$
$743$			748.573i			1.00750i	−0.863849	+	0.503750i	$0.831953\pi$
$744$		0			0
$745$	−286.028			−0.383930
$746$		0			0
$747$		−	723.683i		−	0.968786i
$748$		0			0
$749$	−103.759			−0.138530
$750$		0			0
$751$			765.774i			1.01967i	0.860272	+	0.509836i	$0.170294\pi$
$751$			765.774i			1.01967i	−0.860272	+	0.509836i	$0.829706\pi$
$752$		0			0
$753$		−	165.901i		−	0.220320i
$754$		0			0
$755$		−	101.093i		−	0.133899i
$756$		0			0
$757$			665.274i			0.878830i	0.898284	+	0.439415i	$0.144814\pi$
$757$			665.274i			0.878830i	−0.898284	+	0.439415i	$0.855186\pi$
$758$		0			0
$759$	57.3438	−	294.911i	0.0755518	−	0.388552i
$760$		0			0
$761$	656.887			0.863189			0.431594	−	0.902068i	$-0.357951\pi$
$761$	656.887			0.863189			0.431594	+	0.902068i	$0.357951\pi$
$762$		0			0
$763$	−74.2413			−0.0973018
$764$		0			0
$765$	559.137			0.730898
$766$		0			0
$767$	1011.37			1.31860
$768$		0			0
$769$		−	790.399i		−	1.02783i	−0.857842	−	0.513914i	$-0.828195\pi$
$769$		−	790.399i		−	1.02783i	0.857842	−	0.513914i	$-0.171805\pi$
$770$		0			0
$771$	−230.588			−0.299076
$772$		0			0
$773$			783.714i			1.01386i	0.861987	+	0.506930i	$0.169220\pi$
$773$			783.714i			1.01386i	−0.861987	+	0.506930i	$0.830780\pi$
$774$		0			0
$775$	−89.1428			−0.115023
$776$		0			0
$777$	−36.8389			−0.0474117
$778$		0			0
$779$			779.635i			1.00082i
$780$		0			0
$781$			729.998i			0.934696i
$782$		0			0
$783$	239.031			0.305276
$784$		0			0
$785$	−43.3289			−0.0551960
$786$		0			0
$787$			131.079i			0.166556i	0.996526	+	0.0832779i	$0.0265389\pi$
$787$			131.079i			0.166556i	−0.996526	+	0.0832779i	$0.973461\pi$
$788$		0			0
$789$		−	232.522i		−	0.294705i
$790$		0			0
$791$	−11.9981			−0.0151682
$792$		0			0
$793$		−	454.990i		−	0.573758i
$794$		0			0
$795$	236.523			0.297514
$796$		0			0
$797$			627.924i			0.787859i	0.919141	+	0.393930i	$0.128885\pi$
$797$			627.924i			0.787859i	−0.919141	+	0.393930i	$0.871115\pi$
$798$		0			0
$799$		−	471.330i		−	0.589900i
$800$		0			0
$801$			424.694i			0.530204i
$802$		0			0
$803$		−	321.315i		−	0.400144i
$804$		0			0
$805$	26.3359	+	5.12088i	0.0327155	+	0.00636134i
$806$		0			0
$807$	293.534			0.363735
$808$		0			0
$809$	125.748			0.155436			0.0777179	−	0.996975i	$-0.475237\pi$
$809$	125.748			0.155436			0.0777179	+	0.996975i	$0.475237\pi$
$810$		0			0
$811$	450.426			0.555396			0.277698	−	0.960668i	$-0.410429\pi$
$811$	450.426			0.555396			0.277698	+	0.960668i	$0.410429\pi$
$812$		0			0
$813$	165.402			0.203447
$814$		0			0
$815$		−	655.510i		−	0.804307i
$816$		0			0
$817$	1241.17			1.51918
$818$		0			0
$819$		−	55.8036i		−	0.0681363i
$820$		0			0
$821$	−47.5865			−0.0579616			−0.0289808	−	0.999580i	$-0.509226\pi$
$821$	−47.5865			−0.0579616			−0.0289808	+	0.999580i	$0.509226\pi$
$822$		0			0
$823$	−600.613			−0.729784			−0.364892	−	0.931050i	$-0.618894\pi$
$823$	−600.613			−0.729784			−0.364892	+	0.931050i	$0.618894\pi$
$824$		0			0
$825$		−	65.3118i		−	0.0791658i
$826$		0			0
$827$		−	1073.18i		−	1.29768i	−0.760925	−	0.648840i	$-0.775255\pi$
$827$		−	1073.18i		−	1.29768i	0.760925	−	0.648840i	$-0.224745\pi$
$828$		0			0
$829$	972.327			1.17289			0.586446	−	0.809988i	$-0.300527\pi$
$829$	972.327			1.17289			0.586446	+	0.809988i	$0.300527\pi$
$830$		0			0
$831$	−479.961			−0.577570
$832$		0			0
$833$			1629.18i			1.95579i
$834$		0			0
$835$			353.364i			0.423190i
$836$		0			0
$837$	362.338			0.432901
$838$		0			0
$839$		−	1524.09i		−	1.81656i	−0.418364	−	0.908279i	$-0.637396\pi$
$839$		−	1524.09i		−	1.81656i	0.418364	−	0.908279i	$-0.362604\pi$
$840$		0			0
$841$	−702.671			−0.835518
$842$		0			0
$843$			94.0911i			0.111615i
$844$		0			0
$845$			79.5464i			0.0941378i
$846$		0			0
$847$			4.60217i			0.00543350i
$848$		0			0
$849$		−	325.384i		−	0.383256i
$850$		0			0
$851$	−251.367	+	1292.75i	−0.295379	+	1.51909i
$852$		0			0
$853$	−485.965			−0.569713			−0.284857	−	0.958570i	$-0.591946\pi$
$853$	−485.965			−0.569713			−0.284857	+	0.958570i	$0.591946\pi$
$854$		0			0
$855$	−386.402			−0.451933
$856$		0			0
$857$	−1253.82			−1.46304			−0.731518	−	0.681822i	$-0.761188\pi$
$857$	−1253.82			−1.46304			−0.731518	+	0.681822i	$0.761188\pi$
$858$		0			0
$859$	−992.552			−1.15547			−0.577737	−	0.816223i	$-0.696064\pi$
$859$	−992.552			−1.15547			−0.577737	+	0.816223i	$0.696064\pi$
$860$		0			0
$861$			21.7091i			0.0252138i
$862$		0			0
$863$	1170.45			1.35626			0.678128	−	0.734944i	$-0.262792\pi$
$863$	1170.45			1.35626			0.678128	+	0.734944i	$0.262792\pi$
$864$		0			0
$865$		−	255.263i		−	0.295102i
$866$		0			0
$867$	1022.21			1.17902
$868$		0			0
$869$	519.112			0.597367
$870$		0			0
$871$		−	564.332i		−	0.647912i
$872$		0			0
$873$		−	599.064i		−	0.686213i
$874$		0			0
$875$	5.83243			0.00666564
$876$		0			0
$877$	382.998			0.436714			0.218357	−	0.975869i	$-0.429930\pi$
$877$	382.998			0.436714			0.218357	+	0.975869i	$0.429930\pi$
$878$		0			0
$879$			81.9820i			0.0932673i
$880$		0			0
$881$		−	514.592i		−	0.584100i	−0.956403	−	0.292050i	$-0.905663\pi$
$881$		−	514.592i		−	0.584100i	0.956403	−	0.292050i	$-0.0943373\pi$
$882$		0			0
$883$	28.1064			0.0318306			0.0159153	−	0.999873i	$-0.494934\pi$
$883$	28.1064			0.0318306			0.0159153	+	0.999873i	$0.494934\pi$
$884$		0			0
$885$		−	195.001i		−	0.220340i
$886$		0			0
$887$	651.934			0.734988			0.367494	−	0.930026i	$-0.380216\pi$
$887$	651.934			0.734988			0.367494	+	0.930026i	$0.380216\pi$
$888$		0			0
$889$			31.2392i			0.0351397i
$890$		0			0
$891$		−	447.444i		−	0.502182i
$892$		0			0
$893$			325.722i			0.364750i
$894$		0			0
$895$			386.592i			0.431946i
$896$		0			0
$897$	398.256	+	77.4387i	0.443987	+	0.0863308i
$898$		0			0
$899$	209.688			0.233246
$900$		0			0
$901$	−2867.56			−3.18264
$902$		0			0
$903$	34.5607			0.0382732
$904$		0			0
$905$	−387.030			−0.427657
$906$		0			0
$907$			249.920i			0.275546i	0.990464	+	0.137773i	$0.0439944\pi$
$907$			249.920i			0.275546i	−0.990464	+	0.137773i	$0.956006\pi$
$908$		0			0
$909$	156.292			0.171938
$910$		0			0
$911$			1000.23i			1.09795i	0.835839	+	0.548975i	$0.184982\pi$
$911$			1000.23i			1.09795i	−0.835839	+	0.548975i	$0.815018\pi$
$912$		0			0
$913$	1024.85			1.12251
$914$		0			0
$915$	−87.7261			−0.0958755
$916$		0			0
$917$		−	82.5696i		−	0.0900432i
$918$		0			0
$919$			759.452i			0.826389i	0.910643	+	0.413195i	$0.135587\pi$
$919$			759.452i			0.826389i	−0.910643	+	0.413195i	$0.864413\pi$
$920$		0			0
$921$	93.9989			0.102062
$922$		0			0
$923$	−985.810			−1.06805
$924$		0			0
$925$			286.295i			0.309508i
$926$		0			0
$927$		−	65.7223i		−	0.0708978i
$928$		0			0
$929$	1097.98			1.18190			0.590950	−	0.806708i	$-0.298753\pi$
$929$	1097.98			1.18190			0.590950	+	0.806708i	$0.298753\pi$
$930$		0			0
$931$		−	1125.87i		−	1.20932i
$932$		0			0
$933$	467.019			0.500557
$934$		0			0
$935$			791.826i			0.846872i
$936$		0			0
$937$		−	429.342i		−	0.458209i	−0.973402	−	0.229105i	$-0.926420\pi$
$937$		−	429.342i		−	0.458209i	0.973402	−	0.229105i	$-0.0735798\pi$
$938$		0			0
$939$		−	279.098i		−	0.297229i
$940$		0			0
$941$			1813.97i			1.92770i	0.266442	+	0.963851i	$0.414152\pi$
$941$			1813.97i			1.92770i	−0.266442	+	0.963851i	$0.585848\pi$
$942$		0			0
$943$	761.813	+	148.130i	0.807861	+	0.157084i
$944$		0			0
$945$	−23.7071			−0.0250868
$946$		0			0
$947$	−1020.51			−1.07762			−0.538810	−	0.842427i	$-0.681126\pi$
$947$	−1020.51			−1.07762			−0.538810	+	0.842427i	$0.681126\pi$
$948$		0			0
$949$	433.914			0.457233
$950$		0			0
$951$	−531.566			−0.558955
$952$		0			0
$953$			1222.97i			1.28328i	0.767006	+	0.641640i	$0.221746\pi$
$953$			1222.97i			1.28328i	−0.767006	+	0.641640i	$0.778254\pi$
$954$		0			0
$955$	−478.211			−0.500745
$956$		0			0
$957$			153.631i			0.160534i
$958$		0			0
$959$	−48.0213			−0.0500744
$960$		0			0
$961$	−643.142			−0.669243
$962$		0			0
$963$		−	1487.56i		−	1.54471i
$964$		0			0
$965$			519.183i			0.538014i
$966$		0			0
$967$	−604.850			−0.625491			−0.312745	−	0.949837i	$-0.601249\pi$
$967$	−604.850			−0.625491			−0.312745	+	0.949837i	$0.601249\pi$
$968$		0			0
$969$	−952.733			−0.983212
$970$		0			0
$971$		−	1403.19i		−	1.44510i	−0.691321	−	0.722548i	$-0.742971\pi$
$971$		−	1403.19i		−	1.44510i	0.691321	−	0.722548i	$-0.257029\pi$
$972$		0			0
$973$		−	1.58456i		−	0.00162853i
$974$		0			0
$975$	88.1989			0.0904604
$976$		0			0
$977$		−	1557.63i		−	1.59430i	−0.603780	−	0.797151i	$-0.706339\pi$
$977$		−	1557.63i		−	1.59430i	0.603780	−	0.797151i	$-0.293661\pi$
$978$		0			0
$979$	−601.433			−0.614334
$980$		0			0
$981$		−	1064.37i		−	1.08499i
$982$		0			0
$983$			1074.26i			1.09284i	0.837512	+	0.546418i	$0.184009\pi$
$983$			1074.26i			1.09284i	−0.837512	+	0.546418i	$0.815991\pi$
$984$		0			0
$985$			674.542i			0.684814i
$986$		0			0
$987$			9.06979i			0.00918925i
$988$		0			0
$989$	235.822	−	1212.80i	0.238445	−	1.22629i
$990$		0			0
$991$	974.643			0.983494			0.491747	−	0.870738i	$-0.336358\pi$
$991$	974.643			0.983494			0.491747	+	0.870738i	$0.336358\pi$
$992$		0			0
$993$	474.822			0.478169
$994$		0			0
$995$	−780.479			−0.784401
$996$		0			0
$997$	−1575.15			−1.57989			−0.789945	−	0.613178i	$-0.789891\pi$
$997$	−1575.15			−1.57989			−0.789945	+	0.613178i	$0.789891\pi$
$998$		0			0
$999$		−	1163.70i		−	1.16487i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.k.b.321.4		10
4.3	odd	2		115.3.d.b.91.4	yes	10
12.11	even	2		1035.3.g.b.91.7		10
20.3	even	4		575.3.c.d.574.15		20
20.7	even	4		575.3.c.d.574.6		20
20.19	odd	2		575.3.d.g.551.8		10
23.22	odd	2	inner	1840.3.k.b.321.3		10
92.91	even	2		115.3.d.b.91.3	✓	10
276.275	odd	2		1035.3.g.b.91.8		10
460.183	odd	4		575.3.c.d.574.16		20
460.367	odd	4		575.3.c.d.574.5		20
460.459	even	2		575.3.d.g.551.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.3.d.b.91.3	✓	10	92.91	even	2
115.3.d.b.91.4	yes	10	4.3	odd	2
575.3.c.d.574.5		20	460.367	odd	4
575.3.c.d.574.6		20	20.7	even	4
575.3.c.d.574.15		20	20.3	even	4
575.3.c.d.574.16		20	460.183	odd	4
575.3.d.g.551.7		10	460.459	even	2
575.3.d.g.551.8		10	20.19	odd	2
1035.3.g.b.91.7		10	12.11	even	2
1035.3.g.b.91.8		10	276.275	odd	2
1840.3.k.b.321.3		10	23.22	odd	2	inner
1840.3.k.b.321.4		10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1840.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.23330 q^{3} +2.23607i q^{5} +0.521669i q^{7} -7.47898 q^{9} +O(q^{10})\)	\(q-1.23330 q^{3} +2.23607i q^{5} +0.521669i q^{7} -7.47898 q^{9} -10.5914i q^{11} +14.3029 q^{13} -2.75774i q^{15} +33.4342i q^{17} -23.1053i q^{19} -0.643373i q^{21} +(-22.5772 - 4.39000i) q^{23} -5.00000 q^{25} +20.3235 q^{27} +11.7613 q^{29} +17.8286 q^{31} +13.0624i q^{33} -1.16649 q^{35} -57.2590i q^{37} -17.6398 q^{39} -33.7426 q^{41} +53.7180i q^{43} -16.7235i q^{45} -14.0973 q^{47} +48.7279 q^{49} -41.2343i q^{51} +85.7672i q^{53} +23.6831 q^{55} +28.4958i q^{57} +70.7106 q^{59} -31.8109i q^{61} -3.90155i q^{63} +31.9824i q^{65} -39.4556i q^{67} +(27.8443 + 5.41418i) q^{69} -68.9236 q^{71} +30.3374 q^{73} +6.16649 q^{75} +5.52521 q^{77} +49.0126i q^{79} +42.2459 q^{81} +96.7624i q^{83} -74.7611 q^{85} -14.5052 q^{87} -56.7850i q^{89} +7.46140i q^{91} -21.9879 q^{93} +51.6651 q^{95} +80.0997i q^{97} +79.2129i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} - 16 q^{9}+O(q^{10}) \) 10 * q + 2 * q^3 - 16 * q^9	\( 10 q + 2 q^{3} - 16 q^{9} - 2 q^{13} - 44 q^{23} - 50 q^{25} - 40 q^{27} - 46 q^{29} - 16 q^{31} + 60 q^{35} - 72 q^{39} - 84 q^{41} - 112 q^{47} + 50 q^{49} + 10 q^{55} + 262 q^{59} + 124 q^{69} - 236 q^{71} + 168 q^{73} - 10 q^{75} + 300 q^{77} - 258 q^{81} - 540 q^{87} + 100 q^{93} + 90 q^{95}+O(q^{100}) \) 10 * q + 2 * q^3 - 16 * q^9 - 2 * q^13 - 44 * q^23 - 50 * q^25 - 40 * q^27 - 46 * q^29 - 16 * q^31 + 60 * q^35 - 72 * q^39 - 84 * q^41 - 112 * q^47 + 50 * q^49 + 10 * q^55 + 262 * q^59 + 124 * q^69 - 236 * q^71 + 168 * q^73 - 10 * q^75 + 300 * q^77 - 258 * q^81 - 540 * q^87 + 100 * q^93 + 90 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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