LMFDB - Embedded newform 1840.3.k.c.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:	$16$
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} - 64 x^{14} - 16 x^{13} + 2252 x^{12} + 648 x^{11} - 30106 x^{10} + 12360 x^{9} + \cdots + 1535848276 $ x^16 - 64x^14 - 16x^13 + 2252x^12 + 648x^11 - 30106x^10 + 12360x^9 + 374528x^8 + 196544x^7 + 1261236x^6 - 4237944x^5 + 38013345x^4 - 8913096x^3 + 327235800x^2 + 72566336x + 1535848276
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 460)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.4
Root			$3.41677 + 2.23607i$ of defining polynomial
Character	$\chi$	$=$	1840.321
Dual form			1840.3.k.c.321.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.41677 q^{3} +2.23607i q^{5} +11.7428i q^{7} +2.67432 q^{9} +O(q^{10})$	$q-3.41677 q^{3} +2.23607i q^{5} +11.7428i q^{7} +2.67432 q^{9} +1.72427i q^{11} +16.3966 q^{13} -7.64013i q^{15} +6.55616i q^{17} +31.1218i q^{19} -40.1223i q^{21} +(-16.3910 + 16.1350i) q^{23} -5.00000 q^{25} +21.6134 q^{27} +53.4215 q^{29} +40.6914 q^{31} -5.89145i q^{33} -26.2576 q^{35} +48.8504i q^{37} -56.0233 q^{39} +71.5037 q^{41} -13.4681i q^{43} +5.97997i q^{45} +37.5216 q^{47} -88.8924 q^{49} -22.4009i q^{51} +36.8290i q^{53} -3.85560 q^{55} -106.336i q^{57} -16.3758 q^{59} -76.3705i q^{61} +31.4039i q^{63} +36.6639i q^{65} -16.5965i q^{67} +(56.0041 - 55.1294i) q^{69} +76.3909 q^{71} +77.4490 q^{73} +17.0839 q^{75} -20.2477 q^{77} -139.546i q^{79} -97.9169 q^{81} +71.7972i q^{83} -14.6600 q^{85} -182.529 q^{87} +96.3117i q^{89} +192.541i q^{91} -139.033 q^{93} -69.5904 q^{95} -111.760i q^{97} +4.61127i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 64 q^{9}+O(q^{10}) $ 16 * q + 64 * q^9	$ 16 q + 64 q^{9} - 12 q^{13} + 14 q^{23} - 80 q^{25} - 48 q^{27} + 90 q^{29} - 10 q^{31} - 30 q^{35} - 20 q^{39} + 186 q^{41} + 320 q^{47} + 2 q^{49} + 120 q^{55} + 90 q^{59} - 232 q^{69} + 238 q^{71} - 280 q^{73} + 324 q^{77} + 704 q^{81} - 30 q^{85} - 724 q^{87} - 380 q^{93} - 80 q^{95}+O(q^{100}) $ 16 * q + 64 * q^9 - 12 * q^13 + 14 * q^23 - 80 * q^25 - 48 * q^27 + 90 * q^29 - 10 * q^31 - 30 * q^35 - 20 * q^39 + 186 * q^41 + 320 * q^47 + 2 * q^49 + 120 * q^55 + 90 * q^59 - 232 * q^69 + 238 * q^71 - 280 * q^73 + 324 * q^77 + 704 * q^81 - 30 * q^85 - 724 * q^87 - 380 * q^93 - 80 * 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
$3$	−3.41677			−1.13892			−0.569462	−	0.822018i	$-0.692848\pi$
$3$	−3.41677			−1.13892			−0.569462	+	0.822018i	$0.692848\pi$
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$			11.7428i			1.67754i	0.544488	+	0.838769i	$0.316724\pi$
$7$			11.7428i			1.67754i	−0.544488	+	0.838769i	$0.683276\pi$
$8$		0			0
$9$	2.67432			0.297147
$10$		0			0
$11$			1.72427i			0.156752i	0.996924	+	0.0783761i	$0.0249735\pi$
$11$			1.72427i			0.156752i	−0.996924	+	0.0783761i	$0.975026\pi$
$12$		0			0
$13$	16.3966			1.26128			0.630638	−	0.776077i	$-0.282793\pi$
$13$	16.3966			1.26128			0.630638	+	0.776077i	$0.282793\pi$
$14$		0			0
$15$		−	7.64013i		−	0.509342i
$16$		0			0
$17$			6.55616i			0.385657i	0.981233	+	0.192828i	$0.0617660\pi$
$17$			6.55616i			0.385657i	−0.981233	+	0.192828i	$0.938234\pi$
$18$		0			0
$19$			31.1218i			1.63799i	0.573801	+	0.818994i	$0.305468\pi$
$19$			31.1218i			1.63799i	−0.573801	+	0.818994i	$0.694532\pi$
$20$		0			0
$21$		−	40.1223i		−	1.91059i
$22$		0			0
$23$	−16.3910	+	16.1350i	−0.712650	+	0.701520i
$23$	−16.3910	+	16.1350i	−0.712650	+	0.701520i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	21.6134			0.800496
$28$		0			0
$29$	53.4215			1.84212			0.921060	−	0.389420i	$-0.127325\pi$
$29$	53.4215			1.84212			0.921060	+	0.389420i	$0.127325\pi$
$30$		0			0
$31$	40.6914			1.31262			0.656312	−	0.754490i	$-0.272115\pi$
$31$	40.6914			1.31262			0.656312	+	0.754490i	$0.272115\pi$
$32$		0			0
$33$		−	5.89145i		−	0.178529i
$34$		0			0
$35$	−26.2576			−0.750217
$36$		0			0
$37$			48.8504i			1.32028i	0.751142	+	0.660141i	$0.229504\pi$
$37$			48.8504i			1.32028i	−0.751142	+	0.660141i	$0.770496\pi$
$38$		0			0
$39$	−56.0233			−1.43650
$40$		0			0
$41$	71.5037			1.74399			0.871997	−	0.489512i	$-0.162825\pi$
$41$	71.5037			1.74399			0.871997	+	0.489512i	$0.162825\pi$
$42$		0			0
$43$		−	13.4681i		−	0.313210i	−0.987661	−	0.156605i	$-0.949945\pi$
$43$		−	13.4681i		−	0.313210i	0.987661	−	0.156605i	$-0.0500550\pi$
$44$		0			0
$45$			5.97997i			0.132888i
$46$		0			0
$47$	37.5216			0.798331			0.399166	−	0.916879i	$-0.369300\pi$
$47$	37.5216			0.798331			0.399166	+	0.916879i	$0.369300\pi$
$48$		0			0
$49$	−88.8924			−1.81413
$50$		0			0
$51$		−	22.4009i		−	0.439233i
$52$		0			0
$53$			36.8290i			0.694888i	0.937701	+	0.347444i	$0.112950\pi$
$53$			36.8290i			0.694888i	−0.937701	+	0.347444i	$0.887050\pi$
$54$		0			0
$55$	−3.85560			−0.0701017
$56$		0			0
$57$		−	106.336i		−	1.86554i
$58$		0			0
$59$	−16.3758			−0.277557			−0.138778	−	0.990323i	$-0.544318\pi$
$59$	−16.3758			−0.277557			−0.138778	+	0.990323i	$0.544318\pi$
$60$		0			0
$61$		−	76.3705i		−	1.25197i	−0.779833	−	0.625987i	$-0.784696\pi$
$61$		−	76.3705i		−	1.25197i	0.779833	−	0.625987i	$-0.215304\pi$
$62$		0			0
$63$			31.4039i			0.498475i
$64$		0			0
$65$			36.6639i			0.564059i
$66$		0			0
$67$		−	16.5965i		−	0.247709i	−0.992300	−	0.123855i	$-0.960474\pi$
$67$		−	16.5965i		−	0.247709i	0.992300	−	0.123855i	$-0.0395256\pi$
$68$		0			0
$69$	56.0041	−	55.1294i	0.811654	−	0.798977i
$70$		0			0
$71$	76.3909			1.07593			0.537964	−	0.842968i	$-0.319193\pi$
$71$	76.3909			1.07593			0.537964	+	0.842968i	$0.319193\pi$
$72$		0			0
$73$	77.4490			1.06094			0.530472	−	0.847702i	$-0.322015\pi$
$73$	77.4490			1.06094			0.530472	+	0.847702i	$0.322015\pi$
$74$		0			0
$75$	17.0839			0.227785
$76$		0			0
$77$	−20.2477			−0.262958
$78$		0			0
$79$		−	139.546i		−	1.76641i	−0.468990	−	0.883203i	$-0.655382\pi$
$79$		−	139.546i		−	1.76641i	0.468990	−	0.883203i	$-0.344618\pi$
$80$		0			0
$81$	−97.9169			−1.20885
$82$		0			0
$83$			71.7972i			0.865027i	0.901628	+	0.432513i	$0.142373\pi$
$83$			71.7972i			0.865027i	−0.901628	+	0.432513i	$0.857627\pi$
$84$		0			0
$85$	−14.6600			−0.172471
$86$		0			0
$87$	−182.529			−2.09803
$88$		0			0
$89$			96.3117i			1.08215i	0.840973	+	0.541077i	$0.181983\pi$
$89$			96.3117i			1.08215i	−0.840973	+	0.541077i	$0.818017\pi$
$90$		0			0
$91$			192.541i			2.11584i
$92$		0			0
$93$	−139.033			−1.49498
$94$		0			0
$95$	−69.5904			−0.732531
$96$		0			0
$97$		−	111.760i		−	1.15217i	−0.817390	−	0.576084i	$-0.804580\pi$
$97$		−	111.760i		−	1.15217i	0.817390	−	0.576084i	$-0.195420\pi$
$98$		0			0
$99$			4.61127i			0.0465785i
$100$		0			0
$101$	−146.243			−1.44795			−0.723976	−	0.689826i	$-0.757687\pi$
$101$	−146.243			−1.44795			−0.723976	+	0.689826i	$0.757687\pi$
$102$		0			0
$103$		−	60.2344i		−	0.584800i	−0.956296	−	0.292400i	$-0.905546\pi$
$103$		−	60.2344i		−	0.584800i	0.956296	−	0.292400i	$-0.0944539\pi$
$104$		0			0
$105$	89.7162			0.854440
$106$		0			0
$107$			117.433i			1.09751i	0.835984	+	0.548754i	$0.184897\pi$
$107$			117.433i			1.09751i	−0.835984	+	0.548754i	$0.815103\pi$
$108$		0			0
$109$			97.9451i			0.898579i	0.893386	+	0.449290i	$0.148323\pi$
$109$			97.9451i			0.898579i	−0.893386	+	0.449290i	$0.851677\pi$
$110$		0			0
$111$		−	166.911i		−	1.50370i
$112$		0			0
$113$			110.416i			0.977137i	0.872526	+	0.488568i	$0.162481\pi$
$113$			110.416i			0.977137i	−0.872526	+	0.488568i	$0.837519\pi$
$114$		0			0
$115$	−36.0789	−	36.6513i	−0.313729	−	0.318707i
$116$		0			0
$117$	43.8498			0.374784
$118$		0			0
$119$	−76.9874			−0.646953
$120$		0			0
$121$	118.027			0.975429
$122$		0			0
$123$	−244.312			−1.98628
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	27.1726			0.213957			0.106979	−	0.994261i	$-0.465882\pi$
$127$	27.1726			0.213957			0.106979	+	0.994261i	$0.465882\pi$
$128$		0			0
$129$			46.0172i			0.356723i
$130$		0			0
$131$	−98.9559			−0.755388			−0.377694	−	0.925930i	$-0.623283\pi$
$131$	−98.9559			−0.755388			−0.377694	+	0.925930i	$0.623283\pi$
$132$		0			0
$133$	−365.456			−2.74779
$134$		0			0
$135$			48.3290i			0.357993i
$136$		0			0
$137$			201.031i			1.46738i	0.679483	+	0.733691i	$0.262204\pi$
$137$			201.031i			1.46738i	−0.679483	+	0.733691i	$0.737796\pi$
$138$		0			0
$139$	93.3805			0.671802			0.335901	−	0.941897i	$-0.390959\pi$
$139$	93.3805			0.671802			0.335901	+	0.941897i	$0.390959\pi$
$140$		0			0
$141$	−128.203			−0.909238
$142$		0			0
$143$			28.2722i			0.197708i
$144$		0			0
$145$			119.454i			0.823821i
$146$		0			0
$147$	303.725			2.06616
$148$		0			0
$149$			58.0226i			0.389414i	0.980861	+	0.194707i	$0.0623755\pi$
$149$			58.0226i			0.389414i	−0.980861	+	0.194707i	$0.937624\pi$
$150$		0			0
$151$	32.4550			0.214934			0.107467	−	0.994209i	$-0.465726\pi$
$151$	32.4550			0.214934			0.107467	+	0.994209i	$0.465726\pi$
$152$		0			0
$153$			17.5333i			0.114597i
$154$		0			0
$155$			90.9886i			0.587023i
$156$		0			0
$157$			45.1546i			0.287609i	0.989606	+	0.143805i	$0.0459337\pi$
$157$			45.1546i			0.287609i	−0.989606	+	0.143805i	$0.954066\pi$
$158$		0			0
$159$		−	125.836i		−	0.791424i
$160$		0			0
$161$	−189.469	−	192.475i	−1.17683	−	1.19550i
$162$		0			0
$163$	258.680			1.58699			0.793496	−	0.608576i	$-0.208259\pi$
$163$	258.680			1.58699			0.793496	+	0.608576i	$0.208259\pi$
$164$		0			0
$165$	13.1737			0.0798405
$166$		0			0
$167$	268.469			1.60760			0.803801	−	0.594899i	$-0.202808\pi$
$167$	268.469			1.60760			0.803801	+	0.594899i	$0.202808\pi$
$168$		0			0
$169$	99.8477			0.590815
$170$		0			0
$171$			83.2298i			0.486724i
$172$		0			0
$173$	−16.9111			−0.0977522			−0.0488761	−	0.998805i	$-0.515564\pi$
$173$	−16.9111			−0.0977522			−0.0488761	+	0.998805i	$0.515564\pi$
$174$		0			0
$175$		−	58.7138i		−	0.335507i
$176$		0			0
$177$	55.9525			0.316116
$178$		0			0
$179$	−169.690			−0.947988			−0.473994	−	0.880528i	$-0.657188\pi$
$179$	−169.690			−0.947988			−0.473994	+	0.880528i	$0.657188\pi$
$180$		0			0
$181$			307.427i			1.69849i	0.528000	+	0.849245i	$0.322942\pi$
$181$			307.427i			1.69849i	−0.528000	+	0.849245i	$0.677058\pi$
$182$		0			0
$183$			260.940i			1.42590i
$184$		0			0
$185$	−109.233			−0.590448
$186$		0			0
$187$	−11.3046			−0.0604525
$188$		0			0
$189$			253.801i			1.34286i
$190$		0			0
$191$		−	47.5368i		−	0.248884i	−0.992227	−	0.124442i	$-0.960286\pi$
$191$		−	47.5368i		−	0.248884i	0.992227	−	0.124442i	$-0.0397141\pi$
$192$		0			0
$193$	−299.912			−1.55395			−0.776975	−	0.629531i	$-0.783247\pi$
$193$	−299.912			−1.55395			−0.776975	+	0.629531i	$0.783247\pi$
$194$		0			0
$195$		−	125.272i		−	0.642421i
$196$		0			0
$197$	197.608			1.00308			0.501542	−	0.865133i	$-0.332766\pi$
$197$	197.608			1.00308			0.501542	+	0.865133i	$0.332766\pi$
$198$		0			0
$199$		−	316.197i		−	1.58893i	−0.607311	−	0.794464i	$-0.707752\pi$
$199$		−	316.197i		−	1.58893i	0.607311	−	0.794464i	$-0.292248\pi$
$200$		0			0
$201$			56.7065i			0.282122i
$202$		0			0
$203$			627.316i			3.09023i
$204$		0			0
$205$			159.887i			0.779938i
$206$		0			0
$207$	−43.8347	+	43.1501i	−0.211762	+	0.208455i
$208$		0			0
$209$	−53.6625			−0.256758
$210$		0			0
$211$	−312.629			−1.48165			−0.740826	−	0.671697i	$-0.765566\pi$
$211$	−312.629			−1.48165			−0.740826	+	0.671697i	$0.765566\pi$
$212$		0			0
$213$	−261.010			−1.22540
$214$		0			0
$215$	30.1155			0.140072
$216$		0			0
$217$			477.829i			2.20198i
$218$		0			0
$219$	−264.625			−1.20834
$220$		0			0
$221$			107.499i			0.486419i
$222$		0			0
$223$	−71.8360			−0.322135			−0.161067	−	0.986943i	$-0.551494\pi$
$223$	−71.8360			−0.322135			−0.161067	+	0.986943i	$0.551494\pi$
$224$		0			0
$225$	−13.3716			−0.0594294
$226$		0			0
$227$			95.9906i			0.422866i	0.977392	+	0.211433i	$0.0678130\pi$
$227$			95.9906i			0.422866i	−0.977392	+	0.211433i	$0.932187\pi$
$228$		0			0
$229$		−	287.272i		−	1.25446i	−0.778833	−	0.627231i	$-0.784188\pi$
$229$		−	287.272i		−	1.25446i	0.778833	−	0.627231i	$-0.215812\pi$
$230$		0			0
$231$	69.1819			0.299489
$232$		0			0
$233$	164.613			0.706493			0.353247	−	0.935530i	$-0.385078\pi$
$233$	164.613			0.706493			0.353247	+	0.935530i	$0.385078\pi$
$234$		0			0
$235$			83.9008i			0.357025i
$236$		0			0
$237$			476.797i			2.01180i
$238$		0			0
$239$	−290.498			−1.21547			−0.607737	−	0.794138i	$-0.707923\pi$
$239$	−290.498			−1.21547			−0.607737	+	0.794138i	$0.707923\pi$
$240$		0			0
$241$		−	11.6098i		−	0.0481733i	−0.999710	−	0.0240866i	$-0.992332\pi$
$241$		−	11.6098i		−	0.0481733i	0.999710	−	0.0240866i	$-0.00766775\pi$
$242$		0			0
$243$	140.039			0.576293
$244$		0			0
$245$		−	198.769i		−	0.811304i
$246$		0			0
$247$			510.291i			2.06595i
$248$		0			0
$249$		−	245.315i		−	0.985199i
$250$		0			0
$251$		−	319.118i		−	1.27139i	−0.771942	−	0.635693i	$-0.780714\pi$
$251$		−	319.118i		−	1.27139i	0.771942	−	0.635693i	$-0.219286\pi$
$252$		0			0
$253$	−27.8211	−	28.2625i	−0.109965	−	0.111709i
$254$		0			0
$255$	50.0899			0.196431
$256$		0			0
$257$	119.651			0.465568			0.232784	−	0.972528i	$-0.425216\pi$
$257$	119.651			0.465568			0.232784	+	0.972528i	$0.425216\pi$
$258$		0			0
$259$	−573.639			−2.21482
$260$		0			0
$261$	142.866			0.547381
$262$		0			0
$263$			7.65745i			0.0291158i	0.999894	+	0.0145579i	$0.00463408\pi$
$263$			7.65745i			0.0291158i	−0.999894	+	0.0145579i	$0.995366\pi$
$264$		0			0
$265$	−82.3523			−0.310763
$266$		0			0
$267$		−	329.075i		−	1.23249i
$268$		0			0
$269$	−99.2577			−0.368988			−0.184494	−	0.982834i	$-0.559065\pi$
$269$	−99.2577			−0.368988			−0.184494	+	0.982834i	$0.559065\pi$
$270$		0			0
$271$	−76.5517			−0.282479			−0.141239	−	0.989975i	$-0.545109\pi$
$271$	−76.5517			−0.282479			−0.141239	+	0.989975i	$0.545109\pi$
$272$		0			0
$273$		−	657.869i		−	2.40978i
$274$		0			0
$275$		−	8.62137i		−	0.0313504i
$276$		0			0
$277$	−156.882			−0.566361			−0.283180	−	0.959067i	$-0.591389\pi$
$277$	−156.882			−0.566361			−0.283180	+	0.959067i	$0.591389\pi$
$278$		0			0
$279$	108.822			0.390043
$280$		0			0
$281$		−	446.253i		−	1.58809i	−0.607860	−	0.794044i	$-0.707972\pi$
$281$		−	446.253i		−	1.58809i	0.607860	−	0.794044i	$-0.292028\pi$
$282$		0			0
$283$		−	285.334i		−	1.00825i	−0.863632	−	0.504123i	$-0.831816\pi$
$283$		−	285.334i		−	1.00825i	0.863632	−	0.504123i	$-0.168184\pi$
$284$		0			0
$285$	237.775			0.834297
$286$		0			0
$287$			839.651i			2.92561i
$288$		0			0
$289$	246.017			0.851269
$290$		0			0
$291$			381.860i			1.31223i
$292$		0			0
$293$		−	421.745i		−	1.43940i	−0.694283	−	0.719702i	$-0.744278\pi$
$293$		−	421.745i		−	1.43940i	0.694283	−	0.719702i	$-0.255722\pi$
$294$		0			0
$295$		−	36.6175i		−	0.124127i
$296$		0			0
$297$			37.2674i			0.125480i
$298$		0			0
$299$	−268.756	+	264.558i	−0.898848	+	0.884809i
$300$		0			0
$301$	158.152			0.525422
$302$		0			0
$303$	499.679			1.64911
$304$		0			0
$305$	170.770			0.559900
$306$		0			0
$307$	−66.6161			−0.216990			−0.108495	−	0.994097i	$-0.534603\pi$
$307$	−66.6161			−0.216990			−0.108495	+	0.994097i	$0.534603\pi$
$308$		0			0
$309$			205.807i			0.666043i
$310$		0			0
$311$	−162.073			−0.521135			−0.260568	−	0.965456i	$-0.583910\pi$
$311$	−162.073			−0.521135			−0.260568	+	0.965456i	$0.583910\pi$
$312$		0			0
$313$		−	221.123i		−	0.706464i	−0.935536	−	0.353232i	$-0.885083\pi$
$313$		−	221.123i		−	0.706464i	0.935536	−	0.353232i	$-0.114917\pi$
$314$		0			0
$315$	−70.2214			−0.222925
$316$		0			0
$317$	−191.646			−0.604563			−0.302281	−	0.953219i	$-0.597748\pi$
$317$	−191.646			−0.604563			−0.302281	+	0.953219i	$0.597748\pi$
$318$		0			0
$319$			92.1133i			0.288757i
$320$		0			0
$321$		−	401.243i		−	1.24998i
$322$		0			0
$323$	−204.039			−0.631701
$324$		0			0
$325$	−81.9829			−0.252255
$326$		0			0
$327$		−	334.656i		−	1.02341i
$328$		0			0
$329$			440.607i			1.33923i
$330$		0			0
$331$	−75.5419			−0.228223			−0.114112	−	0.993468i	$-0.536402\pi$
$331$	−75.5419			−0.228223			−0.114112	+	0.993468i	$0.536402\pi$
$332$		0			0
$333$			130.642i			0.392318i
$334$		0			0
$335$	37.1109			0.110779
$336$		0			0
$337$		−	117.380i		−	0.348310i	−0.984718	−	0.174155i	$-0.944281\pi$
$337$		−	117.380i		−	0.348310i	0.984718	−	0.174155i	$-0.0557194\pi$
$338$		0			0
$339$		−	377.268i		−	1.11288i
$340$		0			0
$341$			70.1631i			0.205757i
$342$		0			0
$343$		−	468.447i		−	1.36573i
$344$		0			0
$345$	123.273	+	125.229i	0.357314	+	0.362983i
$346$		0			0
$347$	−207.363			−0.597589			−0.298795	−	0.954317i	$-0.596585\pi$
$347$	−207.363			−0.597589			−0.298795	+	0.954317i	$0.596585\pi$
$348$		0			0
$349$	−469.884			−1.34637			−0.673187	−	0.739473i	$-0.735075\pi$
$349$	−469.884			−1.34637			−0.673187	+	0.739473i	$0.735075\pi$
$350$		0			0
$351$	354.386			1.00965
$352$		0			0
$353$	65.5764			0.185769			0.0928844	−	0.995677i	$-0.470391\pi$
$353$	65.5764			0.185769			0.0928844	+	0.995677i	$0.470391\pi$
$354$		0			0
$355$			170.815i			0.481170i
$356$		0			0
$357$	263.048			0.736830
$358$		0			0
$359$		−	70.9882i		−	0.197739i	−0.995100	−	0.0988694i	$-0.968477\pi$
$359$		−	70.9882i		−	0.197739i	0.995100	−	0.0988694i	$-0.0315226\pi$
$360$		0			0
$361$	−607.566			−1.68301
$362$		0			0
$363$	−403.271			−1.11094
$364$		0			0
$365$			173.181i			0.474469i
$366$		0			0
$367$			457.100i			1.24550i	0.782419	+	0.622752i	$0.213985\pi$
$367$			457.100i			1.24550i	−0.782419	+	0.622752i	$0.786015\pi$
$368$		0			0
$369$	191.224			0.518223
$370$		0			0
$371$	−432.475			−1.16570
$372$		0			0
$373$		−	157.057i		−	0.421065i	−0.977587	−	0.210533i	$-0.932480\pi$
$373$		−	157.057i		−	0.421065i	0.977587	−	0.210533i	$-0.0675198\pi$
$374$		0			0
$375$			38.2007i			0.101868i
$376$		0			0
$377$	875.930			2.32342
$378$		0			0
$379$		−	116.170i		−	0.306518i	−0.988186	−	0.153259i	$-0.951023\pi$
$379$		−	116.170i		−	0.306518i	0.988186	−	0.153259i	$-0.0489768\pi$
$380$		0			0
$381$	−92.8424			−0.243681
$382$		0			0
$383$		−	275.024i		−	0.718078i	−0.933323	−	0.359039i	$-0.883104\pi$
$383$		−	275.024i		−	0.718078i	0.933323	−	0.359039i	$-0.116896\pi$
$384$		0			0
$385$		−	45.2753i		−	0.117598i
$386$		0			0
$387$		−	36.0179i		−	0.0930696i
$388$		0			0
$389$			257.284i			0.661397i	0.943736	+	0.330699i	$0.107284\pi$
$389$			257.284i			0.661397i	−0.943736	+	0.330699i	$0.892716\pi$
$390$		0			0
$391$	−105.783	−	107.462i	−0.270546	−	0.274838i
$392$		0			0
$393$	338.110			0.860330
$394$		0			0
$395$	312.035			0.789961
$396$		0			0
$397$	636.084			1.60223			0.801114	−	0.598512i	$-0.204241\pi$
$397$	636.084			1.60223			0.801114	+	0.598512i	$0.204241\pi$
$398$		0			0
$399$	1248.68			3.12952
$400$		0			0
$401$		−	211.157i		−	0.526576i	−0.964717	−	0.263288i	$-0.915193\pi$
$401$		−	211.157i		−	0.526576i	0.964717	−	0.263288i	$-0.0848069\pi$
$402$		0			0
$403$	667.199			1.65558
$404$		0			0
$405$		−	218.949i		−	0.540614i
$406$		0			0
$407$	−84.2316			−0.206957
$408$		0			0
$409$	350.463			0.856878			0.428439	−	0.903571i	$-0.359064\pi$
$409$	350.463			0.856878			0.428439	+	0.903571i	$0.359064\pi$
$410$		0			0
$411$		−	686.878i		−	1.67124i
$412$		0			0
$413$		−	192.298i		−	0.465612i
$414$		0			0
$415$	−160.543			−0.386852
$416$		0			0
$417$	−319.060			−0.765132
$418$		0			0
$419$			209.980i			0.501145i	0.968098	+	0.250572i	$0.0806188\pi$
$419$			209.980i			0.501145i	−0.968098	+	0.250572i	$0.919381\pi$
$420$		0			0
$421$		−	283.864i		−	0.674260i	−0.941458	−	0.337130i	$-0.890544\pi$
$421$		−	283.864i		−	0.674260i	0.941458	−	0.337130i	$-0.109456\pi$
$422$		0			0
$423$	100.345			0.237222
$424$		0			0
$425$		−	32.7808i		−	0.0771313i
$426$		0			0
$427$	896.800			2.10023
$428$		0			0
$429$		−	96.5996i		−	0.225174i
$430$		0			0
$431$		−	651.479i		−	1.51155i	−0.654831	−	0.755776i	$-0.727260\pi$
$431$		−	651.479i		−	1.51155i	0.654831	−	0.755776i	$-0.272740\pi$
$432$		0			0
$433$			302.244i			0.698022i	0.937119	+	0.349011i	$0.113482\pi$
$433$			302.244i			0.698022i	−0.937119	+	0.349011i	$0.886518\pi$
$434$		0			0
$435$		−	408.147i		−	0.938270i
$436$		0			0
$437$	−502.149	−	510.116i	−1.14908	−	1.16731i
$438$		0			0
$439$	193.495			0.440763			0.220381	−	0.975414i	$-0.429270\pi$
$439$	193.495			0.440763			0.220381	+	0.975414i	$0.429270\pi$
$440$		0			0
$441$	−237.727			−0.539064
$442$		0			0
$443$	−309.781			−0.699280			−0.349640	−	0.936884i	$-0.613696\pi$
$443$	−309.781			−0.699280			−0.349640	+	0.936884i	$0.613696\pi$
$444$		0			0
$445$	−215.360			−0.483954
$446$		0			0
$447$		−	198.250i		−	0.443512i
$448$		0			0
$449$	388.848			0.866031			0.433015	−	0.901387i	$-0.357450\pi$
$449$	388.848			0.866031			0.433015	+	0.901387i	$0.357450\pi$
$450$		0			0
$451$			123.292i			0.273375i
$452$		0			0
$453$	−110.891			−0.244793
$454$		0			0
$455$	−430.535			−0.946231
$456$		0			0
$457$		−	745.529i		−	1.63135i	−0.578508	−	0.815677i	$-0.696365\pi$
$457$		−	745.529i		−	1.63135i	0.578508	−	0.815677i	$-0.303635\pi$
$458$		0			0
$459$			141.701i			0.308716i
$460$		0			0
$461$	128.523			0.278792			0.139396	−	0.990237i	$-0.455484\pi$
$461$	128.523			0.278792			0.139396	+	0.990237i	$0.455484\pi$
$462$		0			0
$463$	−42.0774			−0.0908799			−0.0454399	−	0.998967i	$-0.514469\pi$
$463$	−42.0774			−0.0908799			−0.0454399	+	0.998967i	$0.514469\pi$
$464$		0			0
$465$		−	310.887i		−	0.668575i
$466$		0			0
$467$		−	69.8471i		−	0.149565i	−0.997200	−	0.0747827i	$-0.976174\pi$
$467$		−	69.8471i		−	0.149565i	0.997200	−	0.0747827i	$-0.0238263\pi$
$468$		0			0
$469$	194.889			0.415541
$470$		0			0
$471$		−	154.283i		−	0.327565i
$472$		0			0
$473$	23.2226			0.0490964
$474$		0			0
$475$		−	155.609i		−	0.327598i
$476$		0			0
$477$			98.4928i			0.206484i
$478$		0			0
$479$			233.155i			0.486753i	0.969932	+	0.243376i	$0.0782550\pi$
$479$			233.155i			0.486753i	−0.969932	+	0.243376i	$0.921745\pi$
$480$		0			0
$481$			800.980i			1.66524i
$482$		0			0
$483$	647.372	+	657.643i	1.34031	+	1.36158i
$484$		0			0
$485$	249.904			0.515266
$486$		0			0
$487$	302.310			0.620759			0.310379	−	0.950613i	$-0.399544\pi$
$487$	302.310			0.620759			0.310379	+	0.950613i	$0.399544\pi$
$488$		0			0
$489$	−883.849			−1.80746
$490$		0			0
$491$	−240.848			−0.490525			−0.245262	−	0.969457i	$-0.578874\pi$
$491$	−240.848			−0.490525			−0.245262	+	0.969457i	$0.578874\pi$
$492$		0			0
$493$			350.240i			0.710426i
$494$		0			0
$495$	−10.3111			−0.0208305
$496$		0			0
$497$			897.040i			1.80491i
$498$		0			0
$499$	−122.950			−0.246393			−0.123196	−	0.992382i	$-0.539315\pi$
$499$	−122.950			−0.246393			−0.123196	+	0.992382i	$0.539315\pi$
$500$		0			0
$501$	−917.298			−1.83093
$502$		0			0
$503$		−	704.971i		−	1.40153i	−0.713390	−	0.700767i	$-0.752841\pi$
$503$		−	704.971i		−	1.40153i	0.713390	−	0.700767i	$-0.247159\pi$
$504$		0			0
$505$		−	327.009i		−	0.647543i
$506$		0			0
$507$	−341.157			−0.672893
$508$		0			0
$509$	554.316			1.08903			0.544515	−	0.838751i	$-0.316714\pi$
$509$	554.316			1.08903			0.544515	+	0.838751i	$0.316714\pi$
$510$		0			0
$511$			909.465i			1.77977i
$512$		0			0
$513$			672.647i			1.31120i
$514$		0			0
$515$	134.688			0.261531
$516$		0			0
$517$			64.6975i			0.125140i
$518$		0			0
$519$	57.7814			0.111332
$520$		0			0
$521$		−	738.767i		−	1.41798i	−0.705219	−	0.708989i	$-0.749151\pi$
$521$		−	738.767i		−	1.41798i	0.705219	−	0.708989i	$-0.250849\pi$
$522$		0			0
$523$		−	600.107i		−	1.14743i	−0.819054	−	0.573716i	$-0.805501\pi$
$523$		−	600.107i		−	1.14743i	0.819054	−	0.573716i	$-0.194499\pi$
$524$		0			0
$525$			200.612i			0.382117i
$526$		0			0
$527$			266.779i			0.506222i
$528$		0			0
$529$	8.32659	−	528.934i	0.0157402	−	0.999876i
$530$		0			0
$531$	−43.7943			−0.0824752
$532$		0			0
$533$	1172.42			2.19966
$534$		0			0
$535$	−262.589			−0.490820
$536$		0			0
$537$	579.792			1.07969
$538$		0			0
$539$		−	153.275i		−	0.284369i
$540$		0			0
$541$	293.461			0.542441			0.271221	−	0.962517i	$-0.412573\pi$
$541$	293.461			0.542441			0.271221	+	0.962517i	$0.412573\pi$
$542$		0			0
$543$		−	1050.41i		−	1.93445i
$544$		0			0
$545$	−219.012			−0.401857
$546$		0			0
$547$	1028.06			1.87946			0.939729	−	0.341920i	$-0.111077\pi$
$547$	1028.06			1.87946			0.939729	+	0.341920i	$0.111077\pi$
$548$		0			0
$549$		−	204.239i		−	0.372021i
$550$		0			0
$551$			1662.57i			3.01737i
$552$		0			0
$553$	1638.66			2.96321
$554$		0			0
$555$	373.224			0.672475
$556$		0			0
$557$			413.647i			0.742634i	0.928506	+	0.371317i	$0.121094\pi$
$557$			413.647i			0.742634i	−0.928506	+	0.371317i	$0.878906\pi$
$558$		0			0
$559$		−	220.830i		−	0.395045i
$560$		0			0
$561$	38.6253			0.0688508
$562$		0			0
$563$			424.924i			0.754750i	0.926061	+	0.377375i	$0.123173\pi$
$563$			424.924i			0.754750i	−0.926061	+	0.377375i	$0.876827\pi$
$564$		0			0
$565$	−246.899			−0.436989
$566$		0			0
$567$		−	1149.81i		−	2.02789i
$568$		0			0
$569$		−	745.878i		−	1.31086i	−0.755257	−	0.655429i	$-0.772488\pi$
$569$		−	745.878i		−	1.31086i	0.755257	−	0.655429i	$-0.227512\pi$
$570$		0			0
$571$		−	913.237i		−	1.59937i	−0.600423	−	0.799683i	$-0.705001\pi$
$571$		−	913.237i		−	1.59937i	0.600423	−	0.799683i	$-0.294999\pi$
$572$		0			0
$573$			162.422i			0.283460i
$574$		0			0
$575$	81.9548	−	80.6748i	0.142530	−	0.140304i
$576$		0			0
$577$	−180.241			−0.312376			−0.156188	−	0.987727i	$-0.549921\pi$
$577$	−180.241			−0.312376			−0.156188	+	0.987727i	$0.549921\pi$
$578$		0			0
$579$	1024.73			1.76983
$580$		0			0
$581$	−843.097			−1.45111
$582$		0			0
$583$	−63.5034			−0.108925
$584$		0			0
$585$			98.0511i			0.167609i
$586$		0			0
$587$	888.655			1.51389			0.756946	−	0.653477i	$-0.226690\pi$
$587$	888.655			1.51389			0.756946	+	0.653477i	$0.226690\pi$
$588$		0			0
$589$			1266.39i			2.15006i
$590$		0			0
$591$	−675.180			−1.14244
$592$		0			0
$593$	−331.765			−0.559469			−0.279735	−	0.960077i	$-0.590246\pi$
$593$	−331.765			−0.559469			−0.279735	+	0.960077i	$0.590246\pi$
$594$		0			0
$595$		−	172.149i		−	0.289326i
$596$		0			0
$597$			1080.37i			1.80967i
$598$		0			0
$599$	300.792			0.502156			0.251078	−	0.967967i	$-0.419215\pi$
$599$	300.792			0.502156			0.251078	+	0.967967i	$0.419215\pi$
$600$		0			0
$601$	153.905			0.256081			0.128040	−	0.991769i	$-0.459131\pi$
$601$	153.905			0.256081			0.128040	+	0.991769i	$0.459131\pi$
$602$		0			0
$603$		−	44.3844i		−	0.0736060i
$604$		0			0
$605$			263.916i			0.436225i
$606$		0			0
$607$	484.641			0.798420			0.399210	−	0.916859i	$-0.369284\pi$
$607$	484.641			0.798420			0.399210	+	0.916859i	$0.369284\pi$
$608$		0			0
$609$		−	2143.39i		−	3.51953i
$610$		0			0
$611$	615.225			1.00692
$612$		0			0
$613$			75.7339i			0.123546i	0.998090	+	0.0617732i	$0.0196755\pi$
$613$			75.7339i			0.123546i	−0.998090	+	0.0617732i	$0.980324\pi$
$614$		0			0
$615$		−	546.298i		−	0.888289i
$616$		0			0
$617$		−	1035.08i		−	1.67760i	−0.544437	−	0.838802i	$-0.683257\pi$
$617$		−	1035.08i		−	1.67760i	0.544437	−	0.838802i	$-0.316743\pi$
$618$		0			0
$619$			434.516i			0.701965i	0.936382	+	0.350983i	$0.114152\pi$
$619$			434.516i			0.701965i	−0.936382	+	0.350983i	$0.885848\pi$
$620$		0			0
$621$	−354.264	+	348.731i	−0.570473	+	0.561564i
$622$		0			0
$623$	−1130.97			−1.81535
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	183.353			0.292428
$628$		0			0
$629$	−320.271			−0.509175
$630$		0			0
$631$		−	130.479i		−	0.206782i	−0.994641	−	0.103391i	$-0.967031\pi$
$631$		−	130.479i		−	0.206782i	0.994641	−	0.103391i	$-0.0329692\pi$
$632$		0			0
$633$	1068.18			1.68749
$634$		0			0
$635$			60.7597i			0.0956846i
$636$		0			0
$637$	−1457.53			−2.28812
$638$		0			0
$639$	204.294			0.319709
$640$		0			0
$641$		−	318.505i		−	0.496888i	−0.968646	−	0.248444i	$-0.920081\pi$
$641$		−	318.505i		−	0.496888i	0.968646	−	0.248444i	$-0.0799191\pi$
$642$		0			0
$643$		−	231.689i		−	0.360324i	−0.983637	−	0.180162i	$-0.942338\pi$
$643$		−	231.689i		−	0.360324i	0.983637	−	0.180162i	$-0.0576623\pi$
$644$		0			0
$645$	−102.898			−0.159531
$646$		0			0
$647$	−823.423			−1.27268			−0.636339	−	0.771410i	$-0.719552\pi$
$647$	−823.423			−1.27268			−0.636339	+	0.771410i	$0.719552\pi$
$648$		0			0
$649$		−	28.2364i		−	0.0435076i
$650$		0			0
$651$		−	1632.63i		−	2.50788i
$652$		0			0
$653$	−558.550			−0.855361			−0.427680	−	0.903930i	$-0.640669\pi$
$653$	−558.550			−0.855361			−0.427680	+	0.903930i	$0.640669\pi$
$654$		0			0
$655$		−	221.272i		−	0.337820i
$656$		0			0
$657$	207.124			0.315257
$658$		0			0
$659$			193.919i			0.294262i	0.989117	+	0.147131i	$0.0470039\pi$
$659$			193.919i			0.294262i	−0.989117	+	0.147131i	$0.952996\pi$
$660$		0			0
$661$			244.241i			0.369502i	0.982785	+	0.184751i	$0.0591478\pi$
$661$			244.241i			0.369502i	−0.982785	+	0.184751i	$0.940852\pi$
$662$		0			0
$663$		−	367.298i		−	0.553994i
$664$		0			0
$665$		−	817.184i		−	1.22885i
$666$		0			0
$667$	−875.629	+	861.953i	−1.31279	+	1.29228i
$668$		0			0
$669$	245.447			0.366887
$670$		0			0
$671$	131.684			0.196250
$672$		0			0
$673$	−834.142			−1.23944			−0.619719	−	0.784824i	$-0.712754\pi$
$673$	−834.142			−1.23944			−0.619719	+	0.784824i	$0.712754\pi$
$674$		0			0
$675$	−108.067			−0.160099
$676$		0			0
$677$			53.0242i			0.0783224i	0.999233	+	0.0391612i	$0.0124686\pi$
$677$			53.0242i			0.0783224i	−0.999233	+	0.0391612i	$0.987531\pi$
$678$		0			0
$679$	1312.38			1.93281
$680$		0			0
$681$		−	327.978i		−	0.481612i
$682$		0			0
$683$	772.814			1.13150			0.565749	−	0.824577i	$-0.308587\pi$
$683$	772.814			1.13150			0.565749	+	0.824577i	$0.308587\pi$
$684$		0			0
$685$	−449.520			−0.656233
$686$		0			0
$687$			981.543i			1.42874i
$688$		0			0
$689$			603.870i			0.876445i
$690$		0			0
$691$	−264.260			−0.382431			−0.191216	−	0.981548i	$-0.561243\pi$
$691$	−264.260			−0.382431			−0.191216	+	0.981548i	$0.561243\pi$
$692$		0			0
$693$	−54.1490			−0.0781371
$694$		0			0
$695$			208.805i			0.300439i
$696$		0			0
$697$			468.790i			0.672582i
$698$		0			0
$699$	−562.445			−0.804642
$700$		0			0
$701$			506.422i			0.722427i	0.932483	+	0.361214i	$0.117638\pi$
$701$			506.422i			0.722427i	−0.932483	+	0.361214i	$0.882362\pi$
$702$		0			0
$703$	−1520.31			−2.16261
$704$		0			0
$705$		−	286.670i		−	0.406624i
$706$		0			0
$707$		−	1717.30i		−	2.42899i
$708$		0			0
$709$			202.075i			0.285013i	0.989794	+	0.142507i	$0.0455163\pi$
$709$			202.075i			0.285013i	−0.989794	+	0.142507i	$0.954484\pi$
$710$		0			0
$711$		−	373.192i		−	0.524883i
$712$		0			0
$713$	−666.970	+	656.553i	−0.935442	+	0.920832i
$714$		0			0
$715$	−63.2186			−0.0884176
$716$		0			0
$717$	992.566			1.38433
$718$		0			0
$719$	−1052.19			−1.46341			−0.731707	−	0.681619i	$-0.761276\pi$
$719$	−1052.19			−1.46341			−0.731707	+	0.681619i	$0.761276\pi$
$720$		0			0
$721$	707.318			0.981024
$722$		0			0
$723$			39.6679i			0.0548657i
$724$		0			0
$725$	−267.107			−0.368424
$726$		0			0
$727$			633.110i			0.870852i	0.900224	+	0.435426i	$0.143402\pi$
$727$			633.110i			0.870852i	−0.900224	+	0.435426i	$0.856598\pi$
$728$		0			0
$729$	402.770			0.552497
$730$		0			0
$731$	88.2987			0.120792
$732$		0			0
$733$		−	684.719i		−	0.934133i	−0.884222	−	0.467066i	$-0.845311\pi$
$733$		−	684.719i		−	0.934133i	0.884222	−	0.467066i	$-0.154689\pi$
$734$		0			0
$735$			679.150i			0.924013i
$736$		0			0
$737$	28.6169			0.0388289
$738$		0			0
$739$	406.342			0.549854			0.274927	−	0.961465i	$-0.411346\pi$
$739$	406.342			0.549854			0.274927	+	0.961465i	$0.411346\pi$
$740$		0			0
$741$		−	1743.55i		−	2.35296i
$742$		0			0
$743$			428.667i			0.576941i	0.957489	+	0.288470i	$0.0931467\pi$
$743$			428.667i			0.576941i	−0.957489	+	0.288470i	$0.906853\pi$
$744$		0			0
$745$	−129.743			−0.174151
$746$		0			0
$747$			192.009i			0.257040i
$748$		0			0
$749$	−1378.99			−1.84111
$750$		0			0
$751$		−	1018.01i		−	1.35554i	−0.735276	−	0.677768i	$-0.762947\pi$
$751$		−	1018.01i		−	1.35554i	0.735276	−	0.677768i	$-0.237053\pi$
$752$		0			0
$753$			1090.35i			1.44801i
$754$		0			0
$755$			72.5716i			0.0961213i
$756$		0			0
$757$		−	208.907i		−	0.275966i	−0.990435	−	0.137983i	$-0.955938\pi$
$757$		−	208.907i		−	0.275966i	0.990435	−	0.137983i	$-0.0440620\pi$
$758$		0			0
$759$	95.0583	+	96.5665i	0.125241	+	0.127229i
$760$		0			0
$761$	−301.393			−0.396048			−0.198024	−	0.980197i	$-0.563452\pi$
$761$	−301.393			−0.396048			−0.198024	+	0.980197i	$0.563452\pi$
$762$		0			0
$763$	−1150.15			−1.50740
$764$		0			0
$765$	−39.2056			−0.0512492
$766$		0			0
$767$	−268.508			−0.350075
$768$		0			0
$769$		−	367.042i		−	0.477298i	−0.971106	−	0.238649i	$-0.923296\pi$
$769$		−	367.042i		−	0.477298i	0.971106	−	0.238649i	$-0.0767045\pi$
$770$		0			0
$771$	−408.820			−0.530247
$772$		0			0
$773$			1011.85i			1.30899i	0.756065	+	0.654496i	$0.227119\pi$
$773$			1011.85i			1.30899i	−0.756065	+	0.654496i	$0.772881\pi$
$774$		0			0
$775$	−203.457			−0.262525
$776$		0			0
$777$	1959.99			2.52251
$778$		0			0
$779$			2225.32i			2.85664i
$780$		0			0
$781$			131.719i			0.168654i
$782$		0			0
$783$	1154.62			1.47461
$784$		0			0
$785$	−100.969			−0.128623
$786$		0			0
$787$		−	16.2497i		−	0.0206477i	−0.999947	−	0.0103238i	$-0.996714\pi$
$787$		−	16.2497i		−	0.0206477i	0.999947	−	0.0103238i	$-0.00328624\pi$
$788$		0			0
$789$		−	26.1638i		−	0.0331607i
$790$		0			0
$791$	−1296.59			−1.63918
$792$		0			0
$793$		−	1252.21i		−	1.57908i
$794$		0			0
$795$	281.379			0.353936
$796$		0			0
$797$			736.568i			0.924176i	0.886834	+	0.462088i	$0.152900\pi$
$797$			736.568i			0.924176i	−0.886834	+	0.462088i	$0.847100\pi$
$798$		0			0
$799$			245.997i			0.307882i
$800$		0			0
$801$			257.569i			0.321559i
$802$		0			0
$803$			133.543i			0.166305i
$804$		0			0
$805$	430.387	−	423.665i	0.534642	−	0.526292i
$806$		0			0
$807$	339.141			0.420249
$808$		0			0
$809$	−18.5942			−0.0229842			−0.0114921	−	0.999934i	$-0.503658\pi$
$809$	−18.5942			−0.0229842			−0.0114921	+	0.999934i	$0.503658\pi$
$810$		0			0
$811$	−577.893			−0.712568			−0.356284	−	0.934378i	$-0.615956\pi$
$811$	−577.893			−0.712568			−0.356284	+	0.934378i	$0.615956\pi$
$812$		0			0
$813$	261.560			0.321722
$814$		0			0
$815$			578.425i			0.709724i
$816$		0			0
$817$	419.150			0.513035
$818$		0			0
$819$			514.917i			0.628715i
$820$		0			0
$821$	916.480			1.11630			0.558149	−	0.829741i	$-0.311512\pi$
$821$	916.480			1.11630			0.558149	+	0.829741i	$0.311512\pi$
$822$		0			0
$823$	1230.37			1.49498			0.747491	−	0.664272i	$-0.231258\pi$
$823$	1230.37			1.49498			0.747491	+	0.664272i	$0.231258\pi$
$824$		0			0
$825$			29.4573i			0.0357058i
$826$		0			0
$827$			1140.04i			1.37852i	0.724514	+	0.689260i	$0.242064\pi$
$827$			1140.04i			1.37852i	−0.724514	+	0.689260i	$0.757936\pi$
$828$		0			0
$829$	−285.489			−0.344377			−0.172189	−	0.985064i	$-0.555084\pi$
$829$	−285.489			−0.344377			−0.172189	+	0.985064i	$0.555084\pi$
$830$		0			0
$831$	536.029			0.645042
$832$		0			0
$833$		−	582.793i		−	0.699631i
$834$		0			0
$835$			600.316i			0.718941i
$836$		0			0
$837$	879.478			1.05075
$838$		0			0
$839$		−	1052.44i		−	1.25440i	−0.778857	−	0.627201i	$-0.784200\pi$
$839$		−	1052.44i		−	1.25440i	0.778857	−	0.627201i	$-0.215800\pi$
$840$		0			0
$841$	2012.86			2.39341
$842$		0			0
$843$			1524.74i			1.80871i
$844$		0			0
$845$			223.266i			0.264221i
$846$		0			0
$847$			1385.96i			1.63632i
$848$		0			0
$849$			974.920i			1.14832i
$850$		0			0
$851$	−788.200	−	800.705i	−0.926204	−	0.940899i
$852$		0			0
$853$	622.997			0.730359			0.365180	−	0.930937i	$-0.381008\pi$
$853$	622.997			0.730359			0.365180	+	0.930937i	$0.381008\pi$
$854$		0			0
$855$	−186.107			−0.217669
$856$		0			0
$857$	−1490.05			−1.73868			−0.869341	−	0.494212i	$-0.835457\pi$
$857$	−1490.05			−1.73868			−0.869341	+	0.494212i	$0.835457\pi$
$858$		0			0
$859$	1258.53			1.46511			0.732555	−	0.680708i	$-0.238328\pi$
$859$	1258.53			1.46511			0.732555	+	0.680708i	$0.238328\pi$
$860$		0			0
$861$		−	2868.90i		−	3.33205i
$862$		0			0
$863$	−1190.24			−1.37919			−0.689596	−	0.724194i	$-0.742212\pi$
$863$	−1190.24			−1.37919			−0.689596	+	0.724194i	$0.742212\pi$
$864$		0			0
$865$		−	37.8144i		−	0.0437161i
$866$		0			0
$867$	−840.583			−0.969530
$868$		0			0
$869$	240.616			0.276888
$870$		0			0
$871$		−	272.126i		−	0.312429i
$872$		0			0
$873$		−	298.883i		−	0.342364i
$874$		0			0
$875$	131.288			0.150043
$876$		0			0
$877$	−85.1511			−0.0970936			−0.0485468	−	0.998821i	$-0.515459\pi$
$877$	−85.1511			−0.0970936			−0.0485468	+	0.998821i	$0.515459\pi$
$878$		0			0
$879$			1441.01i			1.63937i
$880$		0			0
$881$			489.208i			0.555288i	0.960684	+	0.277644i	$0.0895535\pi$
$881$			489.208i			0.555288i	−0.960684	+	0.277644i	$0.910447\pi$
$882$		0			0
$883$	256.997			0.291049			0.145525	−	0.989355i	$-0.453513\pi$
$883$	256.997			0.291049			0.145525	+	0.989355i	$0.453513\pi$
$884$		0			0
$885$			125.114i			0.141371i
$886$		0			0
$887$	−599.075			−0.675395			−0.337697	−	0.941255i	$-0.609648\pi$
$887$	−599.075			−0.675395			−0.337697	+	0.941255i	$0.609648\pi$
$888$		0			0
$889$			319.081i			0.358921i
$890$		0			0
$891$		−	168.836i		−	0.189490i
$892$		0			0
$893$			1167.74i			1.30766i
$894$		0			0
$895$		−	379.438i		−	0.423953i
$896$		0			0
$897$	918.276	−	903.934i	1.02372	−	1.00773i
$898$		0			0
$899$	2173.79			2.41801
$900$		0			0
$901$	−241.457			−0.267988
$902$		0			0
$903$	−540.369			−0.598416
$904$		0			0
$905$	−687.427			−0.759587
$906$		0			0
$907$		−	574.026i		−	0.632884i	−0.948612	−	0.316442i	$-0.897512\pi$
$907$		−	574.026i		−	0.632884i	0.948612	−	0.316442i	$-0.102488\pi$
$908$		0			0
$909$	−391.101			−0.430255
$910$		0			0
$911$		−	275.692i		−	0.302626i	−0.988486	−	0.151313i	$-0.951650\pi$
$911$		−	275.692i		−	0.302626i	0.988486	−	0.151313i	$-0.0483501\pi$
$912$		0			0
$913$	−123.798			−0.135595
$914$		0			0
$915$	−583.480			−0.637683
$916$		0			0
$917$		−	1162.02i		−	1.26719i
$918$		0			0
$919$		−	584.025i		−	0.635501i	−0.948174	−	0.317750i	$-0.897073\pi$
$919$		−	584.025i		−	0.635501i	0.948174	−	0.317750i	$-0.102927\pi$
$920$		0			0
$921$	227.612			0.247136
$922$		0			0
$923$	1252.55			1.35704
$924$		0			0
$925$		−	244.252i		−	0.264056i
$926$		0			0
$927$		−	161.086i		−	0.173772i
$928$		0			0
$929$	−751.145			−0.808552			−0.404276	−	0.914637i	$-0.632476\pi$
$929$	−751.145			−0.808552			−0.404276	+	0.914637i	$0.632476\pi$
$930$		0			0
$931$		−	2766.49i		−	2.97153i
$932$		0			0
$933$	553.767			0.593533
$934$		0			0
$935$		−	25.2779i		−	0.0270352i
$936$		0			0
$937$			746.497i			0.796688i	0.917236	+	0.398344i	$0.130415\pi$
$937$			746.497i			0.796688i	−0.917236	+	0.398344i	$0.869585\pi$
$938$		0			0
$939$			755.528i			0.804609i
$940$		0			0
$941$			1064.49i			1.13124i	0.824667	+	0.565618i	$0.191362\pi$
$941$			1064.49i			1.13124i	−0.824667	+	0.565618i	$0.808638\pi$
$942$		0			0
$943$	−1172.01	+	1153.71i	−1.24286	+	1.22345i
$944$		0			0
$945$	−567.516			−0.600546
$946$		0			0
$947$	−2.97936			−0.00314610			−0.00157305	−	0.999999i	$-0.500501\pi$
$947$	−2.97936			−0.00314610			−0.00157305	+	0.999999i	$0.500501\pi$
$948$		0			0
$949$	1269.90			1.33814
$950$		0			0
$951$	654.812			0.688551
$952$		0			0
$953$		−	41.1732i		−	0.0432038i	−0.999767	−	0.0216019i	$-0.993123\pi$
$953$		−	41.1732i		−	0.0432038i	0.999767	−	0.0216019i	$-0.00687664\pi$
$954$		0			0
$955$	106.296			0.111304
$956$		0			0
$957$		−	314.730i		−	0.328872i
$958$		0			0
$959$	−2360.66			−2.46159
$960$		0			0
$961$	694.786			0.722982
$962$		0			0
$963$			314.055i			0.326121i
$964$		0			0
$965$		−	670.624i		−	0.694948i
$966$		0			0
$967$	950.230			0.982658			0.491329	−	0.870974i	$-0.336511\pi$
$967$	950.230			0.982658			0.491329	+	0.870974i	$0.336511\pi$
$968$		0			0
$969$	697.156			0.719459
$970$		0			0
$971$			660.649i			0.680380i	0.940357	+	0.340190i	$0.110491\pi$
$971$			660.649i			0.680380i	−0.940357	+	0.340190i	$0.889509\pi$
$972$		0			0
$973$			1096.55i			1.12697i
$974$		0			0
$975$	280.117			0.287299
$976$		0			0
$977$		−	254.106i		−	0.260088i	−0.991508	−	0.130044i	$-0.958488\pi$
$977$		−	254.106i		−	0.260088i	0.991508	−	0.130044i	$-0.0415118\pi$
$978$		0			0
$979$	−166.068			−0.169630
$980$		0			0
$981$			261.937i			0.267010i
$982$		0			0
$983$		−	1209.95i		−	1.23087i	−0.788187	−	0.615436i	$-0.788980\pi$
$983$		−	1209.95i		−	1.23087i	0.788187	−	0.615436i	$-0.211020\pi$
$984$		0			0
$985$			441.864i			0.448593i
$986$		0			0
$987$		−	1505.45i		−	1.52528i
$988$		0			0
$989$	217.306	+	220.754i	0.219723	+	0.223209i
$990$		0			0
$991$	172.365			0.173930			0.0869652	−	0.996211i	$-0.472283\pi$
$991$	172.365			0.173930			0.0869652	+	0.996211i	$0.472283\pi$
$992$		0			0
$993$	258.110			0.259929
$994$		0			0
$995$	707.037			0.710590
$996$		0			0
$997$	−661.665			−0.663656			−0.331828	−	0.943340i	$-0.607665\pi$
$997$	−661.665			−0.663656			−0.331828	+	0.943340i	$0.607665\pi$
$998$		0			0
$999$			1055.82i			1.05688i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.k.c.321.4		16
4.3	odd	2		460.3.f.a.321.14	yes	16
12.11	even	2		4140.3.d.a.2161.1		16
20.3	even	4		2300.3.d.b.1149.32		32
20.7	even	4		2300.3.d.b.1149.1		32
20.19	odd	2		2300.3.f.e.1701.4		16
23.22	odd	2	inner	1840.3.k.c.321.3		16
92.91	even	2		460.3.f.a.321.13	✓	16
276.275	odd	2		4140.3.d.a.2161.16		16
460.183	odd	4		2300.3.d.b.1149.2		32
460.367	odd	4		2300.3.d.b.1149.31		32
460.459	even	2		2300.3.f.e.1701.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.3.f.a.321.13	✓	16	92.91	even	2
460.3.f.a.321.14	yes	16	4.3	odd	2
1840.3.k.c.321.3		16	23.22	odd	2	inner
1840.3.k.c.321.4		16	1.1	even	1	trivial
2300.3.d.b.1149.1		32	20.7	even	4
2300.3.d.b.1149.2		32	460.183	odd	4
2300.3.d.b.1149.31		32	460.367	odd	4
2300.3.d.b.1149.32		32	20.3	even	4
2300.3.f.e.1701.3		16	460.459	even	2
2300.3.f.e.1701.4		16	20.19	odd	2
4140.3.d.a.2161.1		16	12.11	even	2
4140.3.d.a.2161.16		16	276.275	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1840.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.41677 q^{3} +2.23607i q^{5} +11.7428i q^{7} +2.67432 q^{9} +O(q^{10})\)	\(q-3.41677 q^{3} +2.23607i q^{5} +11.7428i q^{7} +2.67432 q^{9} +1.72427i q^{11} +16.3966 q^{13} -7.64013i q^{15} +6.55616i q^{17} +31.1218i q^{19} -40.1223i q^{21} +(-16.3910 + 16.1350i) q^{23} -5.00000 q^{25} +21.6134 q^{27} +53.4215 q^{29} +40.6914 q^{31} -5.89145i q^{33} -26.2576 q^{35} +48.8504i q^{37} -56.0233 q^{39} +71.5037 q^{41} -13.4681i q^{43} +5.97997i q^{45} +37.5216 q^{47} -88.8924 q^{49} -22.4009i q^{51} +36.8290i q^{53} -3.85560 q^{55} -106.336i q^{57} -16.3758 q^{59} -76.3705i q^{61} +31.4039i q^{63} +36.6639i q^{65} -16.5965i q^{67} +(56.0041 - 55.1294i) q^{69} +76.3909 q^{71} +77.4490 q^{73} +17.0839 q^{75} -20.2477 q^{77} -139.546i q^{79} -97.9169 q^{81} +71.7972i q^{83} -14.6600 q^{85} -182.529 q^{87} +96.3117i q^{89} +192.541i q^{91} -139.033 q^{93} -69.5904 q^{95} -111.760i q^{97} +4.61127i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 64 q^{9}+O(q^{10}) \) 16 * q + 64 * q^9	\( 16 q + 64 q^{9} - 12 q^{13} + 14 q^{23} - 80 q^{25} - 48 q^{27} + 90 q^{29} - 10 q^{31} - 30 q^{35} - 20 q^{39} + 186 q^{41} + 320 q^{47} + 2 q^{49} + 120 q^{55} + 90 q^{59} - 232 q^{69} + 238 q^{71} - 280 q^{73} + 324 q^{77} + 704 q^{81} - 30 q^{85} - 724 q^{87} - 380 q^{93} - 80 q^{95}+O(q^{100}) \) 16 * q + 64 * q^9 - 12 * q^13 + 14 * q^23 - 80 * q^25 - 48 * q^27 + 90 * q^29 - 10 * q^31 - 30 * q^35 - 20 * q^39 + 186 * q^41 + 320 * q^47 + 2 * q^49 + 120 * q^55 + 90 * q^59 - 232 * q^69 + 238 * q^71 - 280 * q^73 + 324 * q^77 + 704 * q^81 - 30 * q^85 - 724 * q^87 - 380 * q^93 - 80 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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