LMFDB - Embedded newform 1840.3.k.c.321.7

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			321.7
Root			$0.613622 - 2.23607i$ of defining polynomial
Character	$\chi$	$=$	1840.321
Dual form			1840.3.k.c.321.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.613622 q^{3} -2.23607i q^{5} +2.35348i q^{7} -8.62347 q^{9} +O(q^{10})$	$q-0.613622 q^{3} -2.23607i q^{5} +2.35348i q^{7} -8.62347 q^{9} -0.167613i q^{11} -23.3530 q^{13} +1.37210i q^{15} +25.5778i q^{17} -24.3833i q^{19} -1.44415i q^{21} +(-9.84182 + 20.7879i) q^{23} -5.00000 q^{25} +10.8141 q^{27} +15.2749 q^{29} +39.4857 q^{31} +0.102851i q^{33} +5.26254 q^{35} +4.29869i q^{37} +14.3299 q^{39} -29.5894 q^{41} -50.9539i q^{43} +19.2827i q^{45} +66.5518 q^{47} +43.4611 q^{49} -15.6951i q^{51} -15.1926i q^{53} -0.374794 q^{55} +14.9621i q^{57} +58.1119 q^{59} -93.2439i q^{61} -20.2951i q^{63} +52.2190i q^{65} +97.0102i q^{67} +(6.03916 - 12.7559i) q^{69} +14.1710 q^{71} -64.9345 q^{73} +3.06811 q^{75} +0.394473 q^{77} -110.968i q^{79} +70.9754 q^{81} +4.10459i q^{83} +57.1936 q^{85} -9.37300 q^{87} +87.7504i q^{89} -54.9609i q^{91} -24.2293 q^{93} -54.5227 q^{95} -61.4252i q^{97} +1.44540i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.613622			−0.204541			−0.102270	−	0.994757i	$-0.532611\pi$
$3$	−0.613622			−0.204541			−0.102270	+	0.994757i	$0.532611\pi$
$4$		0			0
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$			2.35348i			0.336211i	0.985769	+	0.168106i	$0.0537650\pi$
$7$			2.35348i			0.336211i	−0.985769	+	0.168106i	$0.946235\pi$
$8$		0			0
$9$	−8.62347			−0.958163
$10$		0			0
$11$		−	0.167613i		−	0.0152375i	−0.999971	−	0.00761877i	$-0.997575\pi$
$11$		−	0.167613i		−	0.0152375i	0.999971	−	0.00761877i	$-0.00242515\pi$
$12$		0			0
$13$	−23.3530			−1.79639			−0.898194	−	0.439600i	$-0.855120\pi$
$13$	−23.3530			−1.79639			−0.898194	+	0.439600i	$0.855120\pi$
$14$		0			0
$15$			1.37210i			0.0914734i
$16$		0			0
$17$			25.5778i			1.50457i	0.658835	+	0.752287i	$0.271049\pi$
$17$			25.5778i			1.50457i	−0.658835	+	0.752287i	$0.728951\pi$
$18$		0			0
$19$		−	24.3833i		−	1.28333i	−0.766984	−	0.641666i	$-0.778244\pi$
$19$		−	24.3833i		−	1.28333i	0.766984	−	0.641666i	$-0.221756\pi$
$20$		0			0
$21$		−	1.44415i		−	0.0687688i
$22$		0			0
$23$	−9.84182	+	20.7879i	−0.427905	+	0.903824i
$23$	−9.84182	+	20.7879i	−0.427905	+	0.903824i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	10.8141			0.400524
$28$		0			0
$29$	15.2749			0.526720			0.263360	−	0.964698i	$-0.415169\pi$
$29$	15.2749			0.526720			0.263360	+	0.964698i	$0.415169\pi$
$30$		0			0
$31$	39.4857			1.27373			0.636867	−	0.770974i	$-0.280230\pi$
$31$	39.4857			1.27373			0.636867	+	0.770974i	$0.280230\pi$
$32$		0			0
$33$			0.102851i			0.00311670i
$34$		0			0
$35$	5.26254			0.150358
$36$		0			0
$37$			4.29869i			0.116181i	0.998311	+	0.0580904i	$0.0185012\pi$
$37$			4.29869i			0.116181i	−0.998311	+	0.0580904i	$0.981499\pi$
$38$		0			0
$39$	14.3299			0.367434
$40$		0			0
$41$	−29.5894			−0.721693			−0.360846	−	0.932625i	$-0.617512\pi$
$41$	−29.5894			−0.721693			−0.360846	+	0.932625i	$0.617512\pi$
$42$		0			0
$43$		−	50.9539i		−	1.18497i	−0.805580	−	0.592487i	$-0.798146\pi$
$43$		−	50.9539i		−	1.18497i	0.805580	−	0.592487i	$-0.201854\pi$
$44$		0			0
$45$			19.2827i			0.428504i
$46$		0			0
$47$	66.5518			1.41600			0.707998	−	0.706214i	$-0.249599\pi$
$47$	66.5518			1.41600			0.707998	+	0.706214i	$0.249599\pi$
$48$		0			0
$49$	43.4611			0.886962
$50$		0			0
$51$		−	15.6951i		−	0.307747i
$52$		0			0
$53$		−	15.1926i		−	0.286653i	−0.989675	−	0.143326i	$-0.954220\pi$
$53$		−	15.1926i		−	0.286653i	0.989675	−	0.143326i	$-0.0457799\pi$
$54$		0			0
$55$	−0.374794			−0.00681444
$56$		0			0
$57$			14.9621i			0.262493i
$58$		0			0
$59$	58.1119			0.984948			0.492474	−	0.870327i	$-0.336093\pi$
$59$	58.1119			0.984948			0.492474	+	0.870327i	$0.336093\pi$
$60$		0			0
$61$		−	93.2439i		−	1.52859i	−0.644867	−	0.764294i	$-0.723087\pi$
$61$		−	93.2439i		−	1.52859i	0.644867	−	0.764294i	$-0.276913\pi$
$62$		0			0
$63$		−	20.2951i		−	0.322145i
$64$		0			0
$65$			52.2190i			0.803369i
$66$		0			0
$67$			97.0102i			1.44791i	0.689845	+	0.723957i	$0.257679\pi$
$67$			97.0102i			1.44791i	−0.689845	+	0.723957i	$0.742321\pi$
$68$		0			0
$69$	6.03916	−	12.7559i	0.0875240	−	0.184869i
$70$		0			0
$71$	14.1710			0.199592			0.0997960	−	0.995008i	$-0.468181\pi$
$71$	14.1710			0.199592			0.0997960	+	0.995008i	$0.468181\pi$
$72$		0			0
$73$	−64.9345			−0.889513			−0.444757	−	0.895651i	$-0.646710\pi$
$73$	−64.9345			−0.889513			−0.444757	+	0.895651i	$0.646710\pi$
$74$		0			0
$75$	3.06811			0.0409081
$76$		0			0
$77$	0.394473			0.00512303
$78$		0			0
$79$		−	110.968i		−	1.40466i	−0.711851	−	0.702331i	$-0.752143\pi$
$79$		−	110.968i		−	1.40466i	0.711851	−	0.702331i	$-0.247857\pi$
$80$		0			0
$81$	70.9754			0.876240
$82$		0			0
$83$			4.10459i			0.0494529i	0.999694	+	0.0247265i	$0.00787148\pi$
$83$			4.10459i			0.0494529i	−0.999694	+	0.0247265i	$0.992129\pi$
$84$		0			0
$85$	57.1936			0.672866
$86$		0			0
$87$	−9.37300			−0.107736
$88$		0			0
$89$			87.7504i			0.985960i	0.870041	+	0.492980i	$0.164092\pi$
$89$			87.7504i			0.985960i	−0.870041	+	0.492980i	$0.835908\pi$
$90$		0			0
$91$		−	54.9609i		−	0.603965i
$92$		0			0
$93$	−24.2293			−0.260530
$94$		0			0
$95$	−54.5227			−0.573923
$96$		0			0
$97$		−	61.4252i		−	0.633249i	−0.948551	−	0.316625i	$-0.897450\pi$
$97$		−	61.4252i		−	0.633249i	0.948551	−	0.316625i	$-0.102550\pi$
$98$		0			0
$99$			1.44540i			0.0146000i
$100$		0			0
$101$	−37.9300			−0.375544			−0.187772	−	0.982213i	$-0.560127\pi$
$101$	−37.9300			−0.375544			−0.187772	+	0.982213i	$0.560127\pi$
$102$		0			0
$103$			121.156i			1.17627i	0.808763	+	0.588135i	$0.200138\pi$
$103$			121.156i			1.17627i	−0.808763	+	0.588135i	$0.799862\pi$
$104$		0			0
$105$	−3.22921			−0.0307544
$106$		0			0
$107$		−	62.2820i		−	0.582074i	−0.956712	−	0.291037i	$-0.906000\pi$
$107$		−	62.2820i		−	0.582074i	0.956712	−	0.291037i	$-0.0940004\pi$
$108$		0			0
$109$			0.133627i			0.00122594i	1.00000		0.000612968i	$0.000195114\pi$
$109$			0.133627i			0.00122594i	−1.00000		0.000612968i	$0.999805\pi$
$110$		0			0
$111$		−	2.63777i		−	0.0237637i
$112$		0			0
$113$		−	210.838i		−	1.86583i	−0.360101	−	0.932913i	$-0.617258\pi$
$113$		−	210.838i		−	1.86583i	0.360101	−	0.932913i	$-0.382742\pi$
$114$		0			0
$115$	46.4833	+	22.0070i	0.404202	+	0.191365i
$116$		0			0
$117$	201.384			1.72123
$118$		0			0
$119$	−60.1967			−0.505855
$120$		0			0
$121$	120.972			0.999768
$122$		0			0
$123$	18.1567			0.147616
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	30.3950			0.239331			0.119665	−	0.992814i	$-0.461818\pi$
$127$	30.3950			0.239331			0.119665	+	0.992814i	$0.461818\pi$
$128$		0			0
$129$			31.2664i			0.242375i
$130$		0			0
$131$	120.707			0.921429			0.460715	−	0.887548i	$-0.347593\pi$
$131$	120.707			0.921429			0.460715	+	0.887548i	$0.347593\pi$
$132$		0			0
$133$	57.3856			0.431470
$134$		0			0
$135$		−	24.1812i		−	0.179120i
$136$		0			0
$137$			43.8304i			0.319930i	0.987123	+	0.159965i	$0.0511382\pi$
$137$			43.8304i			0.319930i	−0.987123	+	0.159965i	$0.948862\pi$
$138$		0			0
$139$	−194.276			−1.39767			−0.698834	−	0.715284i	$-0.746297\pi$
$139$	−194.276			−1.39767			−0.698834	+	0.715284i	$0.746297\pi$
$140$		0			0
$141$	−40.8377			−0.289629
$142$		0			0
$143$			3.91427i			0.0273725i
$144$		0			0
$145$		−	34.1557i		−	0.235556i
$146$		0			0
$147$	−26.6687			−0.181420
$148$		0			0
$149$			198.743i			1.33384i	0.745127	+	0.666922i	$0.232389\pi$
$149$			198.743i			1.33384i	−0.745127	+	0.666922i	$0.767611\pi$
$150$		0			0
$151$	−150.390			−0.995963			−0.497981	−	0.867188i	$-0.665925\pi$
$151$	−150.390			−0.995963			−0.497981	+	0.867188i	$0.665925\pi$
$152$		0			0
$153$		−	220.569i		−	1.44163i
$154$		0			0
$155$		−	88.2928i		−	0.569631i
$156$		0			0
$157$		−	156.829i		−	0.998910i	−0.866340	−	0.499455i	$-0.833534\pi$
$157$		−	156.829i		−	0.998910i	0.866340	−	0.499455i	$-0.166466\pi$
$158$		0			0
$159$			9.32252i			0.0586322i
$160$		0			0
$161$	−48.9240	−	23.1625i	−0.303876	−	0.143867i
$162$		0			0
$163$	224.124			1.37499			0.687497	−	0.726187i	$-0.258709\pi$
$163$	224.124			1.37499			0.687497	+	0.726187i	$0.258709\pi$
$164$		0			0
$165$	0.229982			0.00139383
$166$		0			0
$167$	52.2597			0.312932			0.156466	−	0.987683i	$-0.449990\pi$
$167$	52.2597			0.312932			0.156466	+	0.987683i	$0.449990\pi$
$168$		0			0
$169$	376.364			2.22701
$170$		0			0
$171$			210.269i			1.22964i
$172$		0			0
$173$	165.785			0.958294			0.479147	−	0.877735i	$-0.340946\pi$
$173$	165.785			0.958294			0.479147	+	0.877735i	$0.340946\pi$
$174$		0			0
$175$		−	11.7674i		−	0.0672422i
$176$		0			0
$177$	−35.6588			−0.201462
$178$		0			0
$179$	−46.4803			−0.259667			−0.129833	−	0.991536i	$-0.541444\pi$
$179$	−46.4803			−0.259667			−0.129833	+	0.991536i	$0.541444\pi$
$180$		0			0
$181$		−	251.378i		−	1.38883i	−0.719576	−	0.694413i	$-0.755664\pi$
$181$		−	251.378i		−	1.38883i	0.719576	−	0.694413i	$-0.244336\pi$
$182$		0			0
$183$			57.2165i			0.312659i
$184$		0			0
$185$	9.61217			0.0519577
$186$		0			0
$187$	4.28716			0.0229260
$188$		0			0
$189$			25.4509i			0.134661i
$190$		0			0
$191$		−	6.75124i		−	0.0353468i	−0.999844	−	0.0176734i	$-0.994374\pi$
$191$		−	6.75124i		−	0.0353468i	0.999844	−	0.0176734i	$-0.00562591\pi$
$192$		0			0
$193$	316.647			1.64066			0.820330	−	0.571890i	$-0.193790\pi$
$193$	316.647			1.64066			0.820330	+	0.571890i	$0.193790\pi$
$194$		0			0
$195$		−	32.0427i		−	0.164322i
$196$		0			0
$197$	161.256			0.818559			0.409280	−	0.912409i	$-0.365780\pi$
$197$	161.256			0.818559			0.409280	+	0.912409i	$0.365780\pi$
$198$		0			0
$199$		−	264.616i		−	1.32973i	−0.746965	−	0.664864i	$-0.768490\pi$
$199$		−	264.616i		−	1.32973i	0.746965	−	0.664864i	$-0.231510\pi$
$200$		0			0
$201$		−	59.5276i		−	0.296157i
$202$		0			0
$203$			35.9491i			0.177089i
$204$		0			0
$205$			66.1639i			0.322751i
$206$		0			0
$207$	84.8706	−	179.264i	0.410003	−	0.866010i
$208$		0			0
$209$	−4.08696			−0.0195548
$210$		0			0
$211$	68.5753			0.325002			0.162501	−	0.986708i	$-0.448044\pi$
$211$	68.5753			0.325002			0.162501	+	0.986708i	$0.448044\pi$
$212$		0			0
$213$	−8.69566			−0.0408247
$214$		0			0
$215$	−113.936			−0.529936
$216$		0			0
$217$			92.9288i			0.428244i
$218$		0			0
$219$	39.8452			0.181942
$220$		0			0
$221$		−	597.318i		−	2.70280i
$222$		0			0
$223$	327.315			1.46778			0.733889	−	0.679269i	$-0.237703\pi$
$223$	327.315			1.46778			0.733889	+	0.679269i	$0.237703\pi$
$224$		0			0
$225$	43.1173			0.191633
$226$		0			0
$227$		−	67.7935i		−	0.298650i	−0.988788	−	0.149325i	$-0.952290\pi$
$227$		−	67.7935i		−	0.298650i	0.988788	−	0.149325i	$-0.0477100\pi$
$228$		0			0
$229$			334.659i			1.46139i	0.682703	+	0.730696i	$0.260804\pi$
$229$			334.659i			1.46139i	−0.682703	+	0.730696i	$0.739196\pi$
$230$		0			0
$231$	−0.242058			−0.00104787
$232$		0			0
$233$	−401.337			−1.72247			−0.861237	−	0.508203i	$-0.830310\pi$
$233$	−401.337			−1.72247			−0.861237	+	0.508203i	$0.830310\pi$
$234$		0			0
$235$		−	148.814i		−	0.633253i
$236$		0			0
$237$			68.0926i			0.287310i
$238$		0			0
$239$	421.040			1.76167			0.880837	−	0.473419i	$-0.156980\pi$
$239$	421.040			1.76167			0.880837	+	0.473419i	$0.156980\pi$
$240$		0			0
$241$			386.686i			1.60451i	0.596985	+	0.802253i	$0.296365\pi$
$241$			386.686i			1.60451i	−0.596985	+	0.802253i	$0.703635\pi$
$242$		0			0
$243$	−140.879			−0.579751
$244$		0			0
$245$		−	97.1821i		−	0.396661i
$246$		0			0
$247$			569.424i			2.30536i
$248$		0			0
$249$		−	2.51867i		−	0.0101151i
$250$		0			0
$251$		−	433.551i		−	1.72730i	−0.504095	−	0.863648i	$-0.668174\pi$
$251$		−	433.551i		−	1.72730i	0.504095	−	0.863648i	$-0.331826\pi$
$252$		0			0
$253$	3.48433	+	1.64962i	0.0137720	+	0.00652022i
$254$		0			0
$255$	−35.0953			−0.137628
$256$		0			0
$257$	−270.409			−1.05218			−0.526088	−	0.850430i	$-0.676342\pi$
$257$	−270.409			−1.05218			−0.526088	+	0.850430i	$0.676342\pi$
$258$		0			0
$259$	−10.1169			−0.0390613
$260$		0			0
$261$	−131.722			−0.504684
$262$		0			0
$263$			97.1155i			0.369260i	0.982808	+	0.184630i	$0.0591087\pi$
$263$			97.1155i			0.369260i	−0.982808	+	0.184630i	$0.940891\pi$
$264$		0			0
$265$	−33.9717			−0.128195
$266$		0			0
$267$		−	53.8456i		−	0.201669i
$268$		0			0
$269$	447.101			1.66208			0.831042	−	0.556209i	$-0.187745\pi$
$269$	447.101			1.66208			0.831042	+	0.556209i	$0.187745\pi$
$270$		0			0
$271$	12.1461			0.0448195			0.0224097	−	0.999749i	$-0.492866\pi$
$271$	12.1461			0.0448195			0.0224097	+	0.999749i	$0.492866\pi$
$272$		0			0
$273$			33.7252i			0.123535i
$274$		0			0
$275$			0.838065i			0.00304751i
$276$		0			0
$277$	−5.18232			−0.0187087			−0.00935437	−	0.999956i	$-0.502978\pi$
$277$	−5.18232			−0.0187087			−0.00935437	+	0.999956i	$0.502978\pi$
$278$		0			0
$279$	−340.504			−1.22044
$280$		0			0
$281$			362.929i			1.29156i	0.763523	+	0.645780i	$0.223468\pi$
$281$			362.929i			1.29156i	−0.763523	+	0.645780i	$0.776532\pi$
$282$		0			0
$283$			551.340i			1.94820i	0.226126	+	0.974098i	$0.427394\pi$
$283$			551.340i			1.94820i	−0.226126	+	0.974098i	$0.572606\pi$
$284$		0			0
$285$	33.4563			0.117391
$286$		0			0
$287$		−	69.6380i		−	0.242641i
$288$		0			0
$289$	−365.222			−1.26374
$290$		0			0
$291$			37.6918i			0.129525i
$292$		0			0
$293$		−	62.4364i		−	0.213094i	−0.994308	−	0.106547i	$-0.966021\pi$
$293$		−	62.4364i		−	0.213094i	0.994308	−	0.106547i	$-0.0339794\pi$
$294$		0			0
$295$		−	129.942i		−	0.440482i
$296$		0			0
$297$		−	1.81259i		−	0.00610300i
$298$		0			0
$299$	229.836	−	485.462i	0.768684	−	1.62362i
$300$		0			0
$301$	119.919			0.398401
$302$		0			0
$303$	23.2747			0.0768140
$304$		0			0
$305$	−208.500			−0.683606
$306$		0			0
$307$	73.2150			0.238485			0.119243	−	0.992865i	$-0.461953\pi$
$307$	73.2150			0.238485			0.119243	+	0.992865i	$0.461953\pi$
$308$		0			0
$309$		−	74.3439i		−	0.240595i
$310$		0			0
$311$	453.724			1.45892			0.729461	−	0.684023i	$-0.239771\pi$
$311$	453.724			1.45892			0.729461	+	0.684023i	$0.239771\pi$
$312$		0			0
$313$		−	88.0757i		−	0.281392i	−0.990053	−	0.140696i	$-0.955066\pi$
$313$		−	88.0757i		−	0.281392i	0.990053	−	0.140696i	$-0.0449340\pi$
$314$		0			0
$315$	−45.3813			−0.144068
$316$		0			0
$317$	20.6220			0.0650535			0.0325267	−	0.999471i	$-0.489645\pi$
$317$	20.6220			0.0650535			0.0325267	+	0.999471i	$0.489645\pi$
$318$		0			0
$319$		−	2.56027i		−	0.00802592i
$320$		0			0
$321$			38.2176i			0.119058i
$322$		0			0
$323$	623.670			1.93087
$324$		0			0
$325$	116.765			0.359277
$326$		0			0
$327$		−	0.0819965i		−	0.000250754i
$328$		0			0
$329$			156.628i			0.476074i
$330$		0			0
$331$	38.9486			0.117669			0.0588347	−	0.998268i	$-0.481262\pi$
$331$	38.9486			0.117669			0.0588347	+	0.998268i	$0.481262\pi$
$332$		0			0
$333$		−	37.0696i		−	0.111320i
$334$		0			0
$335$	216.921			0.647527
$336$		0			0
$337$			313.014i			0.928825i	0.885619	+	0.464412i	$0.153734\pi$
$337$			313.014i			0.928825i	−0.885619	+	0.464412i	$0.846266\pi$
$338$		0			0
$339$			129.375i			0.381637i
$340$		0			0
$341$		−	6.61832i		−	0.0194086i
$342$		0			0
$343$			217.605i			0.634418i
$344$		0			0
$345$	−28.5231	−	13.5040i	−0.0826758	−	0.0391419i
$346$		0			0
$347$	300.671			0.866488			0.433244	−	0.901277i	$-0.357369\pi$
$347$	300.671			0.866488			0.433244	+	0.901277i	$0.357369\pi$
$348$		0			0
$349$	7.08765			0.0203084			0.0101542	−	0.999948i	$-0.496768\pi$
$349$	7.08765			0.0203084			0.0101542	+	0.999948i	$0.496768\pi$
$350$		0			0
$351$	−252.543			−0.719496
$352$		0			0
$353$	−577.956			−1.63727			−0.818635	−	0.574314i	$-0.805269\pi$
$353$	−577.956			−1.63727			−0.818635	+	0.574314i	$0.805269\pi$
$354$		0			0
$355$		−	31.6874i		−	0.0892603i
$356$		0			0
$357$	36.9380			0.103468
$358$		0			0
$359$		−	204.788i		−	0.570440i	−0.958462	−	0.285220i	$-0.907933\pi$
$359$		−	204.788i		−	0.570440i	0.958462	−	0.285220i	$-0.0920667\pi$
$360$		0			0
$361$	−233.545			−0.646940
$362$		0			0
$363$	−74.2310			−0.204493
$364$		0			0
$365$			145.198i			0.397802i
$366$		0			0
$367$		−	160.597i		−	0.437593i	−0.975770	−	0.218797i	$-0.929787\pi$
$367$		−	160.597i		−	0.437593i	0.975770	−	0.218797i	$-0.0702132\pi$
$368$		0			0
$369$	255.163			0.691500
$370$		0			0
$371$	35.7555			0.0963759
$372$		0			0
$373$			3.71586i			0.00996208i	0.999988	+	0.00498104i	$0.00158552\pi$
$373$			3.71586i			0.00996208i	−0.999988	+	0.00498104i	$0.998414\pi$
$374$		0			0
$375$		−	6.86050i		−	0.0182947i
$376$		0			0
$377$	−356.715			−0.946193
$378$		0			0
$379$			207.746i			0.548142i	0.961709	+	0.274071i	$0.0883704\pi$
$379$			207.746i			0.548142i	−0.961709	+	0.274071i	$0.911630\pi$
$380$		0			0
$381$	−18.6510			−0.0489528
$382$		0			0
$383$		−	417.212i		−	1.08933i	−0.838655	−	0.544663i	$-0.816658\pi$
$383$		−	417.212i		−	1.08933i	0.838655	−	0.544663i	$-0.183342\pi$
$384$		0			0
$385$		−	0.882069i		−	0.00229109i
$386$		0			0
$387$			439.399i			1.13540i
$388$		0			0
$389$		−	235.791i		−	0.606148i	−0.952967	−	0.303074i	$-0.901987\pi$
$389$		−	235.791i		−	0.606148i	0.952967	−	0.303074i	$-0.0980129\pi$
$390$		0			0
$391$	−531.709	−	251.732i	−1.35987	−	0.643815i
$392$		0			0
$393$	−74.0686			−0.188470
$394$		0			0
$395$	−248.133			−0.628184
$396$		0			0
$397$	319.562			0.804942			0.402471	−	0.915433i	$-0.368151\pi$
$397$	319.562			0.804942			0.402471	+	0.915433i	$0.368151\pi$
$398$		0			0
$399$	−35.2130			−0.0882532
$400$		0			0
$401$		−	250.400i		−	0.624440i	−0.950010	−	0.312220i	$-0.898927\pi$
$401$		−	250.400i		−	0.624440i	0.950010	−	0.312220i	$-0.101073\pi$
$402$		0			0
$403$	−922.112			−2.28812
$404$		0			0
$405$		−	158.706i		−	0.391866i
$406$		0			0
$407$	0.720516			0.00177031
$408$		0			0
$409$	−389.521			−0.952375			−0.476188	−	0.879344i	$-0.657982\pi$
$409$	−389.521			−0.952375			−0.476188	+	0.879344i	$0.657982\pi$
$410$		0			0
$411$		−	26.8953i		−	0.0654387i
$412$		0			0
$413$			136.765i			0.331151i
$414$		0			0
$415$	9.17815			0.0221160
$416$		0			0
$417$	119.212			0.285880
$418$		0			0
$419$		−	589.258i		−	1.40634i	−0.711020	−	0.703171i	$-0.751767\pi$
$419$		−	589.258i		−	1.40634i	0.711020	−	0.703171i	$-0.248233\pi$
$420$		0			0
$421$		−	665.934i		−	1.58179i	−0.611952	−	0.790895i	$-0.709615\pi$
$421$		−	665.934i		−	1.58179i	0.611952	−	0.790895i	$-0.290385\pi$
$422$		0			0
$423$	−573.908			−1.35676
$424$		0			0
$425$		−	127.889i		−	0.300915i
$426$		0			0
$427$	219.448			0.513929
$428$		0			0
$429$		−	2.40188i		−	0.00559879i
$430$		0			0
$431$			171.958i			0.398975i	0.979900	+	0.199488i	$0.0639277\pi$
$431$			171.958i			0.398975i	−0.979900	+	0.199488i	$0.936072\pi$
$432$		0			0
$433$			513.166i			1.18514i	0.805519	+	0.592570i	$0.201887\pi$
$433$			513.166i			1.18514i	−0.805519	+	0.592570i	$0.798113\pi$
$434$		0			0
$435$			20.9587i			0.0481808i
$436$		0			0
$437$	506.879	+	239.976i	1.15991	+	0.549144i
$438$		0			0
$439$	−65.3345			−0.148826			−0.0744128	−	0.997228i	$-0.523708\pi$
$439$	−65.3345			−0.148826			−0.0744128	+	0.997228i	$0.523708\pi$
$440$		0			0
$441$	−374.786			−0.849854
$442$		0			0
$443$	159.582			0.360230			0.180115	−	0.983646i	$-0.442353\pi$
$443$	159.582			0.360230			0.180115	+	0.983646i	$0.442353\pi$
$444$		0			0
$445$	196.216			0.440935
$446$		0			0
$447$		−	121.953i		−	0.272825i
$448$		0			0
$449$	−546.515			−1.21718			−0.608592	−	0.793484i	$-0.708265\pi$
$449$	−546.515			−1.21718			−0.608592	+	0.793484i	$0.708265\pi$
$450$		0			0
$451$			4.95957i			0.0109968i
$452$		0			0
$453$	92.2828			0.203715
$454$		0			0
$455$	−122.896			−0.270102
$456$		0			0
$457$			253.821i			0.555408i	0.960667	+	0.277704i	$0.0895734\pi$
$457$			253.821i			0.555408i	−0.960667	+	0.277704i	$0.910427\pi$
$458$		0			0
$459$			276.602i			0.602618i
$460$		0			0
$461$	−380.491			−0.825360			−0.412680	−	0.910876i	$-0.635407\pi$
$461$	−380.491			−0.825360			−0.412680	+	0.910876i	$0.635407\pi$
$462$		0			0
$463$	245.745			0.530767			0.265384	−	0.964143i	$-0.414501\pi$
$463$	245.745			0.530767			0.265384	+	0.964143i	$0.414501\pi$
$464$		0			0
$465$			54.1784i			0.116513i
$466$		0			0
$467$			609.764i			1.30571i	0.757485	+	0.652853i	$0.226428\pi$
$467$			609.764i			1.30571i	−0.757485	+	0.652853i	$0.773572\pi$
$468$		0			0
$469$	−228.311			−0.486805
$470$		0			0
$471$			96.2337i			0.204318i
$472$		0			0
$473$	−8.54053			−0.0180561
$474$		0			0
$475$			121.916i			0.256666i
$476$		0			0
$477$			131.013i			0.274660i
$478$		0			0
$479$			476.721i			0.995241i	0.867395	+	0.497621i	$0.165793\pi$
$479$			476.721i			0.995241i	−0.867395	+	0.497621i	$0.834207\pi$
$480$		0			0
$481$		−	100.387i		−	0.208706i
$482$		0			0
$483$	30.0208	+	14.2130i	0.0621549	+	0.0294266i
$484$		0			0
$485$	−137.351			−0.283198
$486$		0			0
$487$	−561.472			−1.15292			−0.576460	−	0.817126i	$-0.695566\pi$
$487$	−561.472			−1.15292			−0.576460	+	0.817126i	$0.695566\pi$
$488$		0			0
$489$	−137.527			−0.281242
$490$		0			0
$491$	−55.5940			−0.113226			−0.0566130	−	0.998396i	$-0.518030\pi$
$491$	−55.5940			−0.113226			−0.0566130	+	0.998396i	$0.518030\pi$
$492$		0			0
$493$			390.697i			0.792489i
$494$		0			0
$495$	3.23202			0.00652934
$496$		0			0
$497$			33.3512i			0.0671051i
$498$		0			0
$499$	−761.749			−1.52655			−0.763275	−	0.646074i	$-0.776410\pi$
$499$	−761.749			−1.52655			−0.763275	+	0.646074i	$0.776410\pi$
$500$		0			0
$501$	−32.0677			−0.0640074
$502$		0			0
$503$		−	742.151i		−	1.47545i	−0.675101	−	0.737725i	$-0.735900\pi$
$503$		−	742.151i		−	1.47545i	0.675101	−	0.737725i	$-0.264100\pi$
$504$		0			0
$505$			84.8139i			0.167948i
$506$		0			0
$507$	−230.945			−0.455513
$508$		0			0
$509$	783.302			1.53890			0.769452	−	0.638704i	$-0.220529\pi$
$509$	783.302			1.53890			0.769452	+	0.638704i	$0.220529\pi$
$510$		0			0
$511$		−	152.822i		−	0.299064i
$512$		0			0
$513$		−	263.685i		−	0.514005i
$514$		0			0
$515$	270.913			0.526044
$516$		0			0
$517$		−	11.1549i		−	0.0215763i
$518$		0			0
$519$	−101.729			−0.196010
$520$		0			0
$521$		−	816.170i		−	1.56655i	−0.621678	−	0.783273i	$-0.713549\pi$
$521$		−	816.170i		−	1.56655i	0.621678	−	0.783273i	$-0.286451\pi$
$522$		0			0
$523$			533.298i			1.01969i	0.860266	+	0.509845i	$0.170297\pi$
$523$			533.298i			1.01969i	−0.860266	+	0.509845i	$0.829703\pi$
$524$		0			0
$525$			7.22073i			0.0137538i
$526$		0			0
$527$			1009.96i			1.91643i
$528$		0			0
$529$	−335.277	−	409.182i	−0.633794	−	0.773502i
$530$		0			0
$531$	−501.127			−0.943741
$532$		0			0
$533$	691.002			1.29644
$534$		0			0
$535$	−139.267			−0.260312
$536$		0			0
$537$	28.5213			0.0531124
$538$		0			0
$539$		−	7.28465i		−	0.0135151i
$540$		0			0
$541$	397.274			0.734332			0.367166	−	0.930155i	$-0.380328\pi$
$541$	397.274			0.734332			0.367166	+	0.930155i	$0.380328\pi$
$542$		0			0
$543$			154.251i			0.284071i
$544$		0			0
$545$	0.298799			0.000548255
$546$		0			0
$547$	123.740			0.226215			0.113108	−	0.993583i	$-0.463919\pi$
$547$	123.740			0.226215			0.113108	+	0.993583i	$0.463919\pi$
$548$		0			0
$549$			804.086i			1.46464i
$550$		0			0
$551$		−	372.452i		−	0.675956i
$552$		0			0
$553$	261.161			0.472263
$554$		0			0
$555$	−5.89824			−0.0106275
$556$		0			0
$557$			826.982i			1.48471i	0.670008	+	0.742354i	$0.266291\pi$
$557$			826.982i			1.48471i	−0.670008	+	0.742354i	$0.733709\pi$
$558$		0			0
$559$			1189.93i			2.12867i
$560$		0			0
$561$	−2.63070			−0.00468930
$562$		0			0
$563$			114.429i			0.203249i	0.994823	+	0.101624i	$0.0324039\pi$
$563$			114.429i			0.203249i	−0.994823	+	0.101624i	$0.967596\pi$
$564$		0			0
$565$	−471.449			−0.834423
$566$		0			0
$567$			167.039i			0.294602i
$568$		0			0
$569$		−	218.123i		−	0.383345i	−0.981459	−	0.191672i	$-0.938609\pi$
$569$		−	218.123i		−	0.383345i	0.981459	−	0.191672i	$-0.0613911\pi$
$570$		0			0
$571$		−	893.615i		−	1.56500i	−0.622651	−	0.782500i	$-0.713944\pi$
$571$		−	893.615i		−	1.56500i	0.622651	−	0.782500i	$-0.286056\pi$
$572$		0			0
$573$			4.14271i			0.00722985i
$574$		0			0
$575$	49.2091	−	103.940i	0.0855811	−	0.180765i
$576$		0			0
$577$	218.968			0.379495			0.189747	−	0.981833i	$-0.439233\pi$
$577$	218.968			0.379495			0.189747	+	0.981833i	$0.439233\pi$
$578$		0			0
$579$	−194.302			−0.335582
$580$		0			0
$581$	−9.66007			−0.0166266
$582$		0			0
$583$	−2.54648			−0.00436789
$584$		0			0
$585$		−	450.309i		−	0.769758i
$586$		0			0
$587$	383.491			0.653307			0.326653	−	0.945144i	$-0.394079\pi$
$587$	383.491			0.653307			0.326653	+	0.945144i	$0.394079\pi$
$588$		0			0
$589$		−	962.793i		−	1.63462i
$590$		0			0
$591$	−98.9503			−0.167429
$592$		0			0
$593$	453.370			0.764537			0.382269	−	0.924051i	$-0.375143\pi$
$593$	453.370			0.764537			0.382269	+	0.924051i	$0.375143\pi$
$594$		0			0
$595$			134.604i			0.226225i
$596$		0			0
$597$			162.374i			0.271983i
$598$		0			0
$599$	195.922			0.327082			0.163541	−	0.986537i	$-0.447708\pi$
$599$	195.922			0.327082			0.163541	+	0.986537i	$0.447708\pi$
$600$		0			0
$601$	−354.949			−0.590597			−0.295299	−	0.955405i	$-0.595419\pi$
$601$	−354.949			−0.590597			−0.295299	+	0.955405i	$0.595419\pi$
$602$		0			0
$603$		−	836.565i		−	1.38734i
$604$		0			0
$605$		−	270.501i		−	0.447110i
$606$		0			0
$607$	−476.386			−0.784820			−0.392410	−	0.919790i	$-0.628359\pi$
$607$	−476.386			−0.784820			−0.392410	+	0.919790i	$0.628359\pi$
$608$		0			0
$609$		−	22.0592i		−	0.0362219i
$610$		0			0
$611$	−1554.19			−2.54368
$612$		0			0
$613$		−	548.445i		−	0.894690i	−0.894361	−	0.447345i	$-0.852370\pi$
$613$		−	548.445i		−	0.894690i	0.894361	−	0.447345i	$-0.147630\pi$
$614$		0			0
$615$		−	40.5996i		−	0.0660157i
$616$		0			0
$617$			784.342i			1.27122i	0.772011	+	0.635609i	$0.219251\pi$
$617$			784.342i			1.27122i	−0.772011	+	0.635609i	$0.780749\pi$
$618$		0			0
$619$			551.280i			0.890597i	0.895382	+	0.445298i	$0.146902\pi$
$619$			551.280i			0.890597i	−0.895382	+	0.445298i	$0.853098\pi$
$620$		0			0
$621$	−106.431	+	224.804i	−0.171386	+	0.362003i
$622$		0			0
$623$	−206.519			−0.331491
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	2.50785			0.00399975
$628$		0			0
$629$	−109.951			−0.174803
$630$		0			0
$631$		−	636.749i		−	1.00911i	−0.863379	−	0.504555i	$-0.831656\pi$
$631$		−	636.749i		−	1.00911i	0.863379	−	0.504555i	$-0.168344\pi$
$632$		0			0
$633$	−42.0793			−0.0664760
$634$		0			0
$635$		−	67.9652i		−	0.107032i
$636$		0			0
$637$	−1014.95			−1.59333
$638$		0			0
$639$	−122.203			−0.191242
$640$		0			0
$641$			1157.64i			1.80599i	0.429655	+	0.902993i	$0.358635\pi$
$641$			1157.64i			1.80599i	−0.429655	+	0.902993i	$0.641365\pi$
$642$		0			0
$643$		−	709.420i		−	1.10330i	−0.834077	−	0.551648i	$-0.813999\pi$
$643$		−	709.420i		−	1.10330i	0.834077	−	0.551648i	$-0.186001\pi$
$644$		0			0
$645$	69.9138			0.108394
$646$		0			0
$647$	809.209			1.25071			0.625355	−	0.780341i	$-0.284954\pi$
$647$	809.209			1.25071			0.625355	+	0.780341i	$0.284954\pi$
$648$		0			0
$649$		−	9.74031i		−	0.0150082i
$650$		0			0
$651$		−	57.0232i		−	0.0875932i
$652$		0			0
$653$	313.648			0.480319			0.240159	−	0.970733i	$-0.422800\pi$
$653$	313.648			0.480319			0.240159	+	0.970733i	$0.422800\pi$
$654$		0			0
$655$		−	269.910i		−	0.412076i
$656$		0			0
$657$	559.960			0.852299
$658$		0			0
$659$			1138.57i			1.72772i	0.503731	+	0.863860i	$0.331960\pi$
$659$			1138.57i			1.72772i	−0.503731	+	0.863860i	$0.668040\pi$
$660$		0			0
$661$		−	295.689i		−	0.447336i	−0.974665	−	0.223668i	$-0.928197\pi$
$661$		−	295.689i		−	0.447336i	0.974665	−	0.223668i	$-0.0718031\pi$
$662$		0			0
$663$			366.528i			0.552832i
$664$		0			0
$665$		−	128.318i		−	0.192959i
$666$		0			0
$667$	−150.333	+	317.533i	−0.225386	+	0.476062i
$668$		0			0
$669$	−200.847			−0.300220
$670$		0			0
$671$	−15.6289			−0.0232919
$672$		0			0
$673$	−464.696			−0.690485			−0.345242	−	0.938514i	$-0.612203\pi$
$673$	−464.696			−0.690485			−0.345242	+	0.938514i	$0.612203\pi$
$674$		0			0
$675$	−54.0707			−0.0801048
$676$		0			0
$677$			15.9301i			0.0235304i	0.999931	+	0.0117652i	$0.00374506\pi$
$677$			15.9301i			0.0235304i	−0.999931	+	0.0117652i	$0.996255\pi$
$678$		0			0
$679$	144.563			0.212905
$680$		0			0
$681$			41.5996i			0.0610860i
$682$		0			0
$683$	−541.640			−0.793031			−0.396515	−	0.918028i	$-0.629781\pi$
$683$	−541.640			−0.793031			−0.396515	+	0.918028i	$0.629781\pi$
$684$		0			0
$685$	98.0079			0.143077
$686$		0			0
$687$		−	205.354i		−	0.298914i
$688$		0			0
$689$			354.793i			0.514940i
$690$		0			0
$691$	−1177.74			−1.70440			−0.852200	−	0.523216i	$-0.824732\pi$
$691$	−1177.74			−1.70440			−0.852200	+	0.523216i	$0.824732\pi$
$692$		0			0
$693$	−3.40173			−0.00490870
$694$		0			0
$695$			434.414i			0.625056i
$696$		0			0
$697$		−	756.831i		−	1.08584i
$698$		0			0
$699$	246.269			0.352316
$700$		0			0
$701$			146.183i			0.208535i	0.994549	+	0.104268i	$0.0332499\pi$
$701$			146.183i			0.208535i	−0.994549	+	0.104268i	$0.966750\pi$
$702$		0			0
$703$	104.816			0.149099
$704$		0			0
$705$			91.3158i			0.129526i
$706$		0			0
$707$		−	89.2673i		−	0.126262i
$708$		0			0
$709$			394.795i			0.556833i	0.960460	+	0.278417i	$0.0898096\pi$
$709$			394.795i			0.556833i	−0.960460	+	0.278417i	$0.910190\pi$
$710$		0			0
$711$			956.931i			1.34590i
$712$		0			0
$713$	−388.612	+	820.827i	−0.545037	+	1.15123i
$714$		0			0
$715$	8.75258			0.0122414
$716$		0			0
$717$	−258.360			−0.360334
$718$		0			0
$719$	916.745			1.27503			0.637514	−	0.770439i	$-0.279963\pi$
$719$	916.745			1.27503			0.637514	+	0.770439i	$0.279963\pi$
$720$		0			0
$721$	−285.138			−0.395475
$722$		0			0
$723$		−	237.279i		−	0.328187i
$724$		0			0
$725$	−76.3744			−0.105344
$726$		0			0
$727$			1076.69i			1.48100i	0.672057	+	0.740500i	$0.265411\pi$
$727$			1076.69i			1.48100i	−0.672057	+	0.740500i	$0.734589\pi$
$728$		0			0
$729$	−552.332			−0.757657
$730$		0			0
$731$	1303.29			1.78288
$732$		0			0
$733$			23.0078i			0.0313885i	0.999877	+	0.0156942i	$0.00499584\pi$
$733$			23.0078i			0.0313885i	−0.999877	+	0.0156942i	$0.995004\pi$
$734$		0			0
$735$			59.6330i			0.0811334i
$736$		0			0
$737$	16.2602			0.0220626
$738$		0			0
$739$	846.184			1.14504			0.572520	−	0.819891i	$-0.305966\pi$
$739$	846.184			1.14504			0.572520	+	0.819891i	$0.305966\pi$
$740$		0			0
$741$		−	349.411i		−	0.471540i
$742$		0			0
$743$		−	792.433i		−	1.06653i	−0.845948	−	0.533266i	$-0.820965\pi$
$743$		−	792.433i		−	1.06653i	0.845948	−	0.533266i	$-0.179035\pi$
$744$		0			0
$745$	444.402			0.596513
$746$		0			0
$747$		−	35.3958i		−	0.0473840i
$748$		0			0
$749$	146.579			0.195700
$750$		0			0
$751$		−	882.680i		−	1.17534i	−0.809101	−	0.587670i	$-0.800045\pi$
$751$		−	882.680i		−	1.17534i	0.809101	−	0.587670i	$-0.199955\pi$
$752$		0			0
$753$			266.037i			0.353302i
$754$		0			0
$755$			336.283i			0.445408i
$756$		0			0
$757$			490.893i			0.648472i	0.945976	+	0.324236i	$0.105107\pi$
$757$			490.893i			0.648472i	−0.945976	+	0.324236i	$0.894893\pi$
$758$		0			0
$759$	−2.13806	−	1.01224i	−0.00281694	−	0.00133365i
$760$		0			0
$761$	−852.432			−1.12015			−0.560073	−	0.828443i	$-0.689227\pi$
$761$	−852.432			−1.12015			−0.560073	+	0.828443i	$0.689227\pi$
$762$		0			0
$763$	−0.314488			−0.000412173
$764$		0			0
$765$	−493.207			−0.644715
$766$		0			0
$767$	−1357.09			−1.76935
$768$		0			0
$769$			323.724i			0.420968i	0.977597	+	0.210484i	$0.0675039\pi$
$769$			323.724i			0.420968i	−0.977597	+	0.210484i	$0.932496\pi$
$770$		0			0
$771$	165.929			0.215213
$772$		0			0
$773$		−	224.753i		−	0.290754i	−0.989376	−	0.145377i	$-0.953560\pi$
$773$		−	224.753i		−	0.290754i	0.989376	−	0.145377i	$-0.0464396\pi$
$774$		0			0
$775$	−197.429			−0.254747
$776$		0			0
$777$	6.20794			0.00798962
$778$		0			0
$779$			721.487i			0.926171i
$780$		0			0
$781$		−	2.37525i		−	0.00304129i
$782$		0			0
$783$	165.185			0.210964
$784$		0			0
$785$	−350.680			−0.446726
$786$		0			0
$787$		−	387.295i		−	0.492116i	−0.969255	−	0.246058i	$-0.920865\pi$
$787$		−	387.295i		−	0.492116i	0.969255	−	0.246058i	$-0.0791354\pi$
$788$		0			0
$789$		−	59.5922i		−	0.0755287i
$790$		0			0
$791$	496.204			0.627312
$792$		0			0
$793$			2177.53i			2.74594i
$794$		0			0
$795$	20.8458			0.0262211
$796$		0			0
$797$		−	143.007i		−	0.179432i	−0.995967	−	0.0897161i	$-0.971404\pi$
$797$		−	143.007i		−	0.179432i	0.995967	−	0.0897161i	$-0.0285960\pi$
$798$		0			0
$799$			1702.25i			2.13047i
$800$		0			0
$801$		−	756.713i		−	0.944710i
$802$		0			0
$803$			10.8839i			0.0135540i
$804$		0			0
$805$	−51.7930	+	109.397i	−0.0643391	+	0.135897i
$806$		0			0
$807$	−274.351			−0.339964
$808$		0			0
$809$	−757.216			−0.935990			−0.467995	−	0.883731i	$-0.655024\pi$
$809$	−757.216			−0.935990			−0.467995	+	0.883731i	$0.655024\pi$
$810$		0			0
$811$	−8.16598			−0.0100690			−0.00503451	−	0.999987i	$-0.501603\pi$
$811$	−8.16598			−0.0100690			−0.00503451	+	0.999987i	$0.501603\pi$
$812$		0			0
$813$	−7.45310			−0.00916740
$814$		0			0
$815$		−	501.157i		−	0.614916i
$816$		0			0
$817$	−1242.42			−1.52071
$818$		0			0
$819$			473.953i			0.578697i
$820$		0			0
$821$	98.3490			0.119792			0.0598959	−	0.998205i	$-0.480923\pi$
$821$	98.3490			0.119792			0.0598959	+	0.998205i	$0.480923\pi$
$822$		0			0
$823$	−786.402			−0.955531			−0.477765	−	0.878487i	$-0.658553\pi$
$823$	−786.402			−0.955531			−0.477765	+	0.878487i	$0.658553\pi$
$824$		0			0
$825$		−	0.514255i		−	0.000623339i
$826$		0			0
$827$		−	1566.53i		−	1.89423i	−0.320895	−	0.947115i	$-0.603984\pi$
$827$		−	1566.53i		−	1.89423i	0.320895	−	0.947115i	$-0.396016\pi$
$828$		0			0
$829$	376.874			0.454613			0.227307	−	0.973823i	$-0.427008\pi$
$829$	376.874			0.454613			0.227307	+	0.973823i	$0.427008\pi$
$830$		0			0
$831$	3.17999			0.00382670
$832$		0			0
$833$			1111.64i			1.33450i
$834$		0			0
$835$		−	116.856i		−	0.139948i
$836$		0			0
$837$	427.005			0.510161
$838$		0			0
$839$		−	511.907i		−	0.610139i	−0.952330	−	0.305070i	$-0.901320\pi$
$839$		−	511.907i		−	0.610139i	0.952330	−	0.305070i	$-0.0986797\pi$
$840$		0			0
$841$	−607.678			−0.722566
$842$		0			0
$843$		−	222.701i		−	0.264177i
$844$		0			0
$845$		−	841.576i		−	0.995948i
$846$		0			0
$847$			284.705i			0.336133i
$848$		0			0
$849$		−	338.314i		−	0.398485i
$850$		0			0
$851$	−89.3610	−	42.3070i	−0.105007	−	0.0497144i
$852$		0			0
$853$	−833.730			−0.977409			−0.488704	−	0.872449i	$-0.662530\pi$
$853$	−833.730			−0.977409			−0.488704	+	0.872449i	$0.662530\pi$
$854$		0			0
$855$	470.175			0.549912
$856$		0			0
$857$	76.9417			0.0897803			0.0448902	−	0.998992i	$-0.485706\pi$
$857$	76.9417			0.0897803			0.0448902	+	0.998992i	$0.485706\pi$
$858$		0			0
$859$	1173.16			1.36572			0.682862	−	0.730548i	$-0.260735\pi$
$859$	1173.16			1.36572			0.682862	+	0.730548i	$0.260735\pi$
$860$		0			0
$861$			42.7314i			0.0496300i
$862$		0			0
$863$	41.6403			0.0482506			0.0241253	−	0.999709i	$-0.492320\pi$
$863$	41.6403			0.0482506			0.0241253	+	0.999709i	$0.492320\pi$
$864$		0			0
$865$		−	370.706i		−	0.428562i
$866$		0			0
$867$	224.108			0.258487
$868$		0			0
$869$	−18.5997			−0.0214036
$870$		0			0
$871$		−	2265.48i		−	2.60101i
$872$		0			0
$873$			529.698i			0.606756i
$874$		0			0
$875$	−26.3127			−0.0300716
$876$		0			0
$877$	855.659			0.975665			0.487833	−	0.872937i	$-0.337788\pi$
$877$	855.659			0.975665			0.487833	+	0.872937i	$0.337788\pi$
$878$		0			0
$879$			38.3123i			0.0435863i
$880$		0			0
$881$		−	660.677i		−	0.749917i	−0.927042	−	0.374958i	$-0.877657\pi$
$881$		−	660.677i		−	0.749917i	0.927042	−	0.374958i	$-0.122343\pi$
$882$		0			0
$883$	249.768			0.282863			0.141431	−	0.989948i	$-0.454830\pi$
$883$	249.768			0.282863			0.141431	+	0.989948i	$0.454830\pi$
$884$		0			0
$885$			79.7354i			0.0900965i
$886$		0			0
$887$	1459.55			1.64549			0.822747	−	0.568408i	$-0.192440\pi$
$887$	1459.55			1.64549			0.822747	+	0.568408i	$0.192440\pi$
$888$		0			0
$889$			71.5339i			0.0804656i
$890$		0			0
$891$		−	11.8964i		−	0.0133517i
$892$		0			0
$893$		−	1622.75i		−	1.81719i
$894$		0			0
$895$			103.933i			0.116126i
$896$		0			0
$897$	−141.033	+	297.890i	−0.157227	+	0.332096i
$898$		0			0
$899$	603.140			0.670901
$900$		0			0
$901$	388.593			0.431291
$902$		0			0
$903$	−73.5848			−0.0814893
$904$		0			0
$905$	−562.097			−0.621102
$906$		0			0
$907$		−	504.127i		−	0.555818i	−0.960607	−	0.277909i	$-0.910359\pi$
$907$		−	504.127i		−	0.555818i	0.960607	−	0.277909i	$-0.0896414\pi$
$908$		0			0
$909$	327.088			0.359832
$910$		0			0
$911$			95.1985i			0.104499i	0.998634	+	0.0522494i	$0.0166391\pi$
$911$			95.1985i			0.104499i	−0.998634	+	0.0522494i	$0.983361\pi$
$912$		0			0
$913$	0.687983			0.000753541
$914$		0			0
$915$	127.940			0.139825
$916$		0			0
$917$			284.082i			0.309795i
$918$		0			0
$919$		−	897.170i		−	0.976246i	−0.872775	−	0.488123i	$-0.837682\pi$
$919$		−	897.170i		−	0.976246i	0.872775	−	0.488123i	$-0.162318\pi$
$920$		0			0
$921$	−44.9264			−0.0487800
$922$		0			0
$923$	−330.937			−0.358545
$924$		0			0
$925$		−	21.4935i		−	0.0232362i
$926$		0			0
$927$		−	1044.78i		−	1.12706i
$928$		0			0
$929$	−193.891			−0.208709			−0.104354	−	0.994540i	$-0.533278\pi$
$929$	−193.891			−0.208709			−0.104354	+	0.994540i	$0.533278\pi$
$930$		0			0
$931$		−	1059.73i		−	1.13827i
$932$		0			0
$933$	−278.415			−0.298409
$934$		0			0
$935$		−	9.58639i		−	0.0102528i
$936$		0			0
$937$		−	1049.54i		−	1.12011i	−0.828455	−	0.560056i	$-0.810780\pi$
$937$		−	1049.54i		−	1.12011i	0.828455	−	0.560056i	$-0.189220\pi$
$938$		0			0
$939$			54.0452i			0.0575561i
$940$		0			0
$941$		−	1194.28i		−	1.26917i	−0.772855	−	0.634583i	$-0.781172\pi$
$941$		−	1194.28i		−	1.26917i	0.772855	−	0.634583i	$-0.218828\pi$
$942$		0			0
$943$	291.214	−	615.103i	0.308816	−	0.652283i
$944$		0			0
$945$	56.9098			0.0602221
$946$		0			0
$947$	−273.098			−0.288382			−0.144191	−	0.989550i	$-0.546058\pi$
$947$	−273.098			−0.288382			−0.144191	+	0.989550i	$0.546058\pi$
$948$		0			0
$949$	1516.42			1.59791
$950$		0			0
$951$	−12.6541			−0.0133061
$952$		0			0
$953$		−	222.571i		−	0.233548i	−0.993159	−	0.116774i	$-0.962745\pi$
$953$		−	222.571i		−	0.233548i	0.993159	−	0.116774i	$-0.0372554\pi$
$954$		0			0
$955$	−15.0962			−0.0158076
$956$		0			0
$957$			1.57104i			0.00164163i
$958$		0			0
$959$	−103.154			−0.107564
$960$		0			0
$961$	598.124			0.622398
$962$		0			0
$963$			537.087i			0.557722i
$964$		0			0
$965$		−	708.045i		−	0.733726i
$966$		0			0
$967$	1285.39			1.32926			0.664629	−	0.747174i	$-0.268590\pi$
$967$	1285.39			1.32926			0.664629	+	0.747174i	$0.268590\pi$
$968$		0			0
$969$	−382.698			−0.394941
$970$		0			0
$971$		−	1108.87i		−	1.14198i	−0.820956	−	0.570991i	$-0.806559\pi$
$971$		−	1108.87i		−	1.14198i	0.820956	−	0.570991i	$-0.193441\pi$
$972$		0			0
$973$		−	457.224i		−	0.469911i
$974$		0			0
$975$	−71.6497			−0.0734868
$976$		0			0
$977$			520.016i			0.532258i	0.963937	+	0.266129i	$0.0857447\pi$
$977$			520.016i			0.532258i	−0.963937	+	0.266129i	$0.914255\pi$
$978$		0			0
$979$	14.7081			0.0150236
$980$		0			0
$981$		−	1.15233i		−	0.00117465i
$982$		0			0
$983$			1523.16i			1.54950i	0.632269	+	0.774749i	$0.282124\pi$
$983$			1523.16i			1.54950i	−0.632269	+	0.774749i	$0.717876\pi$
$984$		0			0
$985$		−	360.580i		−	0.366071i
$986$		0			0
$987$		−	96.1105i		−	0.0973764i
$988$		0			0
$989$	1059.23	+	501.479i	1.07101	+	0.507056i
$990$		0			0
$991$	−1084.41			−1.09426			−0.547131	−	0.837047i	$-0.684280\pi$
$991$	−1084.41			−1.09426			−0.547131	+	0.837047i	$0.684280\pi$
$992$		0			0
$993$	−23.8997			−0.0240682
$994$		0			0
$995$	−591.699			−0.594672
$996$		0			0
$997$	−805.761			−0.808185			−0.404093	−	0.914718i	$-0.632413\pi$
$997$	−805.761			−0.808185			−0.404093	+	0.914718i	$0.632413\pi$
$998$		0			0
$999$			46.4867i			0.0465332i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.k.c.321.7		16
4.3	odd	2		460.3.f.a.321.9	✓	16
12.11	even	2		4140.3.d.a.2161.11		16
20.3	even	4		2300.3.d.b.1149.20		32
20.7	even	4		2300.3.d.b.1149.13		32
20.19	odd	2		2300.3.f.e.1701.8		16
23.22	odd	2	inner	1840.3.k.c.321.8		16
92.91	even	2		460.3.f.a.321.10	yes	16
276.275	odd	2		4140.3.d.a.2161.6		16
460.183	odd	4		2300.3.d.b.1149.14		32
460.367	odd	4		2300.3.d.b.1149.19		32
460.459	even	2		2300.3.f.e.1701.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.3.f.a.321.9	✓	16	4.3	odd	2
460.3.f.a.321.10	yes	16	92.91	even	2
1840.3.k.c.321.7		16	1.1	even	1	trivial
1840.3.k.c.321.8		16	23.22	odd	2	inner
2300.3.d.b.1149.13		32	20.7	even	4
2300.3.d.b.1149.14		32	460.183	odd	4
2300.3.d.b.1149.19		32	460.367	odd	4
2300.3.d.b.1149.20		32	20.3	even	4
2300.3.f.e.1701.7		16	460.459	even	2
2300.3.f.e.1701.8		16	20.19	odd	2
4140.3.d.a.2161.6		16	276.275	odd	2
4140.3.d.a.2161.11		16	12.11	even	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)

\(f(q)\)	\(=\)	\(q-0.613622 q^{3} -2.23607i q^{5} +2.35348i q^{7} -8.62347 q^{9} +O(q^{10})\)	\(q-0.613622 q^{3} -2.23607i q^{5} +2.35348i q^{7} -8.62347 q^{9} -0.167613i q^{11} -23.3530 q^{13} +1.37210i q^{15} +25.5778i q^{17} -24.3833i q^{19} -1.44415i q^{21} +(-9.84182 + 20.7879i) q^{23} -5.00000 q^{25} +10.8141 q^{27} +15.2749 q^{29} +39.4857 q^{31} +0.102851i q^{33} +5.26254 q^{35} +4.29869i q^{37} +14.3299 q^{39} -29.5894 q^{41} -50.9539i q^{43} +19.2827i q^{45} +66.5518 q^{47} +43.4611 q^{49} -15.6951i q^{51} -15.1926i q^{53} -0.374794 q^{55} +14.9621i q^{57} +58.1119 q^{59} -93.2439i q^{61} -20.2951i q^{63} +52.2190i q^{65} +97.0102i q^{67} +(6.03916 - 12.7559i) q^{69} +14.1710 q^{71} -64.9345 q^{73} +3.06811 q^{75} +0.394473 q^{77} -110.968i q^{79} +70.9754 q^{81} +4.10459i q^{83} +57.1936 q^{85} -9.37300 q^{87} +87.7504i q^{89} -54.9609i q^{91} -24.2293 q^{93} -54.5227 q^{95} -61.4252i q^{97} +1.44540i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 64 q^{9}+O(q^{10}) \) 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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