LMFDB - Embedded newform 1840.3.k.e.321.12

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:	$48$
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.12
Character	$\chi$	$=$	1840.321
Dual form			1840.3.k.e.321.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.78603 q^{3} +2.23607i q^{5} +9.63233i q^{7} -1.23801 q^{9} +O(q^{10})$	$q-2.78603 q^{3} +2.23607i q^{5} +9.63233i q^{7} -1.23801 q^{9} +8.11353i q^{11} -3.04367 q^{13} -6.22976i q^{15} -28.9225i q^{17} -30.9204i q^{19} -26.8360i q^{21} +(20.0182 + 11.3256i) q^{23} -5.00000 q^{25} +28.5235 q^{27} +31.5158 q^{29} +51.7423 q^{31} -22.6046i q^{33} -21.5385 q^{35} -15.3465i q^{37} +8.47978 q^{39} -18.7569 q^{41} +29.9739i q^{43} -2.76828i q^{45} +66.5766 q^{47} -43.7817 q^{49} +80.5790i q^{51} -35.4563i q^{53} -18.1424 q^{55} +86.1453i q^{57} -61.9018 q^{59} -13.6548i q^{61} -11.9249i q^{63} -6.80586i q^{65} +18.8677i q^{67} +(-55.7715 - 31.5536i) q^{69} -132.162 q^{71} +47.8907 q^{73} +13.9302 q^{75} -78.1521 q^{77} +61.2230i q^{79} -68.3252 q^{81} +90.3862i q^{83} +64.6726 q^{85} -87.8042 q^{87} +5.60548i q^{89} -29.3176i q^{91} -144.156 q^{93} +69.1401 q^{95} -72.1236i q^{97} -10.0446i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 128 q^{9}+O(q^{10}) $ 48 * q + 128 * q^9	$ 48 q + 128 q^{9} - 8 q^{23} - 240 q^{25} + 72 q^{29} - 32 q^{31} + 40 q^{35} + 96 q^{39} - 104 q^{41} - 128 q^{47} - 344 q^{49} - 80 q^{55} - 248 q^{59} + 292 q^{69} - 208 q^{71} + 224 q^{73} - 288 q^{77} + 184 q^{81} + 48 q^{87} - 672 q^{93}+O(q^{100}) $ 48 * q + 128 * q^9 - 8 * q^23 - 240 * q^25 + 72 * q^29 - 32 * q^31 + 40 * q^35 + 96 * q^39 - 104 * q^41 - 128 * q^47 - 344 * q^49 - 80 * q^55 - 248 * q^59 + 292 * q^69 - 208 * q^71 + 224 * q^73 - 288 * q^77 + 184 * q^81 + 48 * q^87 - 672 * q^93

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
$3$	−2.78603			−0.928678			−0.464339	−	0.885658i	$-0.653708\pi$
$3$	−2.78603			−0.928678			−0.464339	+	0.885658i	$0.653708\pi$
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$			9.63233i			1.37605i	0.725689	+	0.688023i	$0.241521\pi$
$7$			9.63233i			1.37605i	−0.725689	+	0.688023i	$0.758479\pi$
$8$		0			0
$9$	−1.23801			−0.137557
$10$		0			0
$11$			8.11353i			0.737593i	0.929510	+	0.368797i	$0.120230\pi$
$11$			8.11353i			0.737593i	−0.929510	+	0.368797i	$0.879770\pi$
$12$		0			0
$13$	−3.04367			−0.234129			−0.117064	−	0.993124i	$-0.537348\pi$
$13$	−3.04367			−0.234129			−0.117064	+	0.993124i	$0.537348\pi$
$14$		0			0
$15$		−	6.22976i		−	0.415317i
$16$		0			0
$17$		−	28.9225i		−	1.70132i	−0.525715	−	0.850661i	$-0.676202\pi$
$17$		−	28.9225i		−	1.70132i	0.525715	−	0.850661i	$-0.323798\pi$
$18$		0			0
$19$		−	30.9204i		−	1.62739i	−0.581292	−	0.813695i	$-0.697453\pi$
$19$		−	30.9204i		−	1.62739i	0.581292	−	0.813695i	$-0.302547\pi$
$20$		0			0
$21$		−	26.8360i		−	1.27790i
$22$		0			0
$23$	20.0182	+	11.3256i	0.870358	+	0.492419i
$23$	20.0182	+	11.3256i	0.870358	+	0.492419i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	28.5235			1.05642
$28$		0			0
$29$	31.5158			1.08675			0.543376	−	0.839489i	$-0.317146\pi$
$29$	31.5158			1.08675			0.543376	+	0.839489i	$0.317146\pi$
$30$		0			0
$31$	51.7423			1.66911			0.834553	−	0.550928i	$-0.185726\pi$
$31$	51.7423			1.66911			0.834553	+	0.550928i	$0.185726\pi$
$32$		0			0
$33$		−	22.6046i		−	0.684987i
$34$		0			0
$35$	−21.5385			−0.615387
$36$		0			0
$37$		−	15.3465i		−	0.414771i	−0.978259	−	0.207386i	$-0.933504\pi$
$37$		−	15.3465i		−	0.414771i	0.978259	−	0.207386i	$-0.0664955\pi$
$38$		0			0
$39$	8.47978			0.217430
$40$		0			0
$41$	−18.7569			−0.457485			−0.228743	−	0.973487i	$-0.573461\pi$
$41$	−18.7569			−0.457485			−0.228743	+	0.973487i	$0.573461\pi$
$42$		0			0
$43$			29.9739i			0.697069i	0.937296	+	0.348534i	$0.113321\pi$
$43$			29.9739i			0.697069i	−0.937296	+	0.348534i	$0.886679\pi$
$44$		0			0
$45$		−	2.76828i		−	0.0615173i
$46$		0			0
$47$	66.5766			1.41652			0.708262	−	0.705950i	$-0.249480\pi$
$47$	66.5766			1.41652			0.708262	+	0.705950i	$0.249480\pi$
$48$		0			0
$49$	−43.7817			−0.893504
$50$		0			0
$51$			80.5790i			1.57998i
$52$		0			0
$53$		−	35.4563i		−	0.668988i	−0.942398	−	0.334494i	$-0.891435\pi$
$53$		−	35.4563i		−	0.668988i	0.942398	−	0.334494i	$-0.108565\pi$
$54$		0			0
$55$	−18.1424			−0.329862
$56$		0			0
$57$			86.1453i			1.51132i
$58$		0			0
$59$	−61.9018			−1.04918			−0.524592	−	0.851354i	$-0.675782\pi$
$59$	−61.9018			−1.04918			−0.524592	+	0.851354i	$0.675782\pi$
$60$		0			0
$61$		−	13.6548i		−	0.223848i	−0.993717	−	0.111924i	$-0.964299\pi$
$61$		−	13.6548i		−	0.223848i	0.993717	−	0.111924i	$-0.0357014\pi$
$62$		0			0
$63$		−	11.9249i		−	0.189285i
$64$		0			0
$65$		−	6.80586i		−	0.104706i
$66$		0			0
$67$			18.8677i			0.281607i	0.990038	+	0.140803i	$0.0449686\pi$
$67$			18.8677i			0.281607i	−0.990038	+	0.140803i	$0.955031\pi$
$68$		0			0
$69$	−55.7715	−	31.5536i	−0.808283	−	0.457299i
$70$		0			0
$71$	−132.162			−1.86144			−0.930718	−	0.365739i	$-0.880816\pi$
$71$	−132.162			−1.86144			−0.930718	+	0.365739i	$0.880816\pi$
$72$		0			0
$73$	47.8907			0.656037			0.328019	−	0.944671i	$-0.393619\pi$
$73$	47.8907			0.656037			0.328019	+	0.944671i	$0.393619\pi$
$74$		0			0
$75$	13.9302			0.185736
$76$		0			0
$77$	−78.1521			−1.01496
$78$		0			0
$79$			61.2230i			0.774975i	0.921875	+	0.387487i	$0.126657\pi$
$79$			61.2230i			0.774975i	−0.921875	+	0.387487i	$0.873343\pi$
$80$		0			0
$81$	−68.3252			−0.843521
$82$		0			0
$83$			90.3862i			1.08899i	0.838764	+	0.544495i	$0.183279\pi$
$83$			90.3862i			1.08899i	−0.838764	+	0.544495i	$0.816721\pi$
$84$		0			0
$85$	64.6726			0.760854
$86$		0			0
$87$	−87.8042			−1.00924
$88$		0			0
$89$			5.60548i			0.0629829i	0.999504	+	0.0314915i	$0.0100257\pi$
$89$			5.60548i			0.0629829i	−0.999504	+	0.0314915i	$0.989974\pi$
$90$		0			0
$91$		−	29.3176i		−	0.322172i
$92$		0			0
$93$	−144.156			−1.55006
$94$		0			0
$95$	69.1401			0.727791
$96$		0			0
$97$		−	72.1236i		−	0.743543i	−0.928324	−	0.371771i	$-0.878751\pi$
$97$		−	72.1236i		−	0.743543i	0.928324	−	0.371771i	$-0.121249\pi$
$98$		0			0
$99$		−	10.0446i		−	0.101461i
$100$		0			0
$101$	156.882			1.55328			0.776642	−	0.629942i	$-0.216921\pi$
$101$	156.882			1.55328			0.776642	+	0.629942i	$0.216921\pi$
$102$		0			0
$103$			193.115i			1.87490i	0.348116	+	0.937451i	$0.386822\pi$
$103$			193.115i			1.87490i	−0.348116	+	0.937451i	$0.613178\pi$
$104$		0			0
$105$	60.0071			0.571496
$106$		0			0
$107$		−	169.687i		−	1.58586i	−0.609312	−	0.792931i	$-0.708554\pi$
$107$		−	169.687i		−	1.58586i	0.609312	−	0.792931i	$-0.291446\pi$
$108$		0			0
$109$			82.1626i			0.753785i	0.926257	+	0.376893i	$0.123007\pi$
$109$			82.1626i			0.753785i	−0.926257	+	0.376893i	$0.876993\pi$
$110$		0			0
$111$			42.7560i			0.385189i
$112$		0			0
$113$			179.589i			1.58928i	0.607078	+	0.794642i	$0.292341\pi$
$113$			179.589i			1.58928i	−0.607078	+	0.794642i	$0.707659\pi$
$114$		0			0
$115$	−25.3249	+	44.7621i	−0.220217	+	0.389236i
$116$		0			0
$117$	3.76811			0.0322060
$118$		0			0
$119$	278.591			2.34110
$120$		0			0
$121$	55.1707			0.455956
$122$		0			0
$123$	52.2573			0.424856
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	120.300			0.947248			0.473624	−	0.880727i	$-0.342946\pi$
$127$	120.300			0.947248			0.473624	+	0.880727i	$0.342946\pi$
$128$		0			0
$129$		−	83.5084i		−	0.647352i
$130$		0			0
$131$	123.006			0.938981			0.469490	−	0.882938i	$-0.344438\pi$
$131$	123.006			0.938981			0.469490	+	0.882938i	$0.344438\pi$
$132$		0			0
$133$	297.835			2.23936
$134$		0			0
$135$			63.7804i			0.472447i
$136$		0			0
$137$		−	32.1467i		−	0.234647i	−0.993094	−	0.117324i	$-0.962569\pi$
$137$		−	32.1467i		−	0.234647i	0.993094	−	0.117324i	$-0.0374315\pi$
$138$		0			0
$139$	−112.568			−0.809841			−0.404920	−	0.914352i	$-0.632701\pi$
$139$	−112.568			−0.809841			−0.404920	+	0.914352i	$0.632701\pi$
$140$		0			0
$141$	−185.485			−1.31550
$142$		0			0
$143$		−	24.6949i		−	0.172692i
$144$		0			0
$145$			70.4715i			0.486011i
$146$		0			0
$147$	121.977			0.829778
$148$		0			0
$149$			169.519i			1.13771i	0.822437	+	0.568856i	$0.192614\pi$
$149$			169.519i			1.13771i	−0.822437	+	0.568856i	$0.807386\pi$
$150$		0			0
$151$	−92.2806			−0.611130			−0.305565	−	0.952171i	$-0.598845\pi$
$151$	−92.2806			−0.611130			−0.305565	+	0.952171i	$0.598845\pi$
$152$		0			0
$153$			35.8064i			0.234029i
$154$		0			0
$155$			115.699i			0.746447i
$156$		0			0
$157$		−	162.079i		−	1.03235i	−0.856483	−	0.516175i	$-0.827355\pi$
$157$		−	162.079i		−	1.03235i	0.856483	−	0.516175i	$-0.172645\pi$
$158$		0			0
$159$			98.7826i			0.621274i
$160$		0			0
$161$	−109.092	+	192.822i	−0.677592	+	1.19765i
$162$		0			0
$163$	−201.386			−1.23550			−0.617749	−	0.786376i	$-0.711955\pi$
$163$	−201.386			−1.23550			−0.617749	+	0.786376i	$0.711955\pi$
$164$		0			0
$165$	50.5453			0.306335
$166$		0			0
$167$	243.693			1.45924			0.729618	−	0.683854i	$-0.239698\pi$
$167$	243.693			1.45924			0.729618	+	0.683854i	$0.239698\pi$
$168$		0			0
$169$	−159.736			−0.945184
$170$		0			0
$171$			38.2799i			0.223859i
$172$		0			0
$173$	−147.862			−0.854691			−0.427345	−	0.904088i	$-0.640551\pi$
$173$	−147.862			−0.854691			−0.427345	+	0.904088i	$0.640551\pi$
$174$		0			0
$175$		−	48.1616i		−	0.275209i
$176$		0			0
$177$	172.461			0.974354
$178$		0			0
$179$	−67.0918			−0.374815			−0.187407	−	0.982282i	$-0.560008\pi$
$179$	−67.0918			−0.374815			−0.187407	+	0.982282i	$0.560008\pi$
$180$		0			0
$181$		−	91.9825i		−	0.508190i	−0.967179	−	0.254095i	$-0.918222\pi$
$181$		−	91.9825i		−	0.508190i	0.967179	−	0.254095i	$-0.0817777\pi$
$182$		0			0
$183$			38.0426i			0.207883i
$184$		0			0
$185$	34.3159			0.185491
$186$		0			0
$187$	234.663			1.25488
$188$		0			0
$189$			274.747i			1.45369i
$190$		0			0
$191$			30.0619i			0.157392i	0.996899	+	0.0786960i	$0.0250757\pi$
$191$			30.0619i			0.157392i	−0.996899	+	0.0786960i	$0.974924\pi$
$192$		0			0
$193$	95.0675			0.492578			0.246289	−	0.969196i	$-0.420789\pi$
$193$	95.0675			0.492578			0.246289	+	0.969196i	$0.420789\pi$
$194$		0			0
$195$			18.9614i			0.0972377i
$196$		0			0
$197$	−155.375			−0.788706			−0.394353	−	0.918959i	$-0.629031\pi$
$197$	−155.375			−0.788706			−0.394353	+	0.918959i	$0.629031\pi$
$198$		0			0
$199$			281.782i			1.41599i	0.706217	+	0.707995i	$0.250400\pi$
$199$			281.782i			1.41599i	−0.706217	+	0.707995i	$0.749600\pi$
$200$		0			0
$201$		−	52.5660i		−	0.261522i
$202$		0			0
$203$			303.571i			1.49542i
$204$		0			0
$205$		−	41.9417i		−	0.204594i
$206$		0			0
$207$	−24.7828	−	14.0213i	−0.119724	−	0.0677357i
$208$		0			0
$209$	250.874			1.20035
$210$		0			0
$211$	−236.061			−1.11877			−0.559387	−	0.828907i	$-0.688963\pi$
$211$	−236.061			−1.11877			−0.559387	+	0.828907i	$0.688963\pi$
$212$		0			0
$213$	368.208			1.72867
$214$		0			0
$215$	−67.0238			−0.311739
$216$		0			0
$217$			498.398i			2.29677i
$218$		0			0
$219$	−133.425			−0.609248
$220$		0			0
$221$			88.0305i			0.398328i
$222$		0			0
$223$	229.564			1.02944			0.514718	−	0.857359i	$-0.327897\pi$
$223$	229.564			1.02944			0.514718	+	0.857359i	$0.327897\pi$
$224$		0			0
$225$	6.19006			0.0275114
$226$		0			0
$227$		−	121.801i		−	0.536568i	−0.963340	−	0.268284i	$-0.913544\pi$
$227$		−	121.801i		−	0.536568i	0.963340	−	0.268284i	$-0.0864565\pi$
$228$		0			0
$229$			50.1047i			0.218798i	0.993998	+	0.109399i	$0.0348926\pi$
$229$			50.1047i			0.218798i	−0.993998	+	0.109399i	$0.965107\pi$
$230$		0			0
$231$	217.735			0.942574
$232$		0			0
$233$	282.243			1.21135			0.605673	−	0.795714i	$-0.292904\pi$
$233$	282.243			1.21135			0.605673	+	0.795714i	$0.292904\pi$
$234$		0			0
$235$			148.870i			0.633489i
$236$		0			0
$237$		−	170.569i		−	0.719702i
$238$		0			0
$239$	347.476			1.45388			0.726938	−	0.686703i	$-0.240943\pi$
$239$	347.476			1.45388			0.726938	+	0.686703i	$0.240943\pi$
$240$		0			0
$241$		−	47.0721i		−	0.195320i	−0.995220	−	0.0976600i	$-0.968864\pi$
$241$		−	47.0721i		−	0.195320i	0.995220	−	0.0976600i	$-0.0311358\pi$
$242$		0			0
$243$	−66.3547			−0.273065
$244$		0			0
$245$		−	97.8988i		−	0.399587i
$246$		0			0
$247$			94.1116i			0.381019i
$248$		0			0
$249$		−	251.819i		−	1.01132i
$250$		0			0
$251$		−	432.673i		−	1.72380i	−0.507082	−	0.861898i	$-0.669276\pi$
$251$		−	432.673i		−	1.72380i	0.507082	−	0.861898i	$-0.330724\pi$
$252$		0			0
$253$	−91.8909	+	162.418i	−0.363205	+	0.641970i
$254$		0			0
$255$	−180.180			−0.706589
$256$		0			0
$257$	352.087			1.36999			0.684994	−	0.728548i	$-0.259805\pi$
$257$	352.087			1.36999			0.684994	+	0.728548i	$0.259805\pi$
$258$		0			0
$259$	147.823			0.570745
$260$		0			0
$261$	−39.0170			−0.149490
$262$		0			0
$263$			76.2781i			0.290031i	0.989429	+	0.145015i	$0.0463232\pi$
$263$			76.2781i			0.290031i	−0.989429	+	0.145015i	$0.953677\pi$
$264$		0			0
$265$	79.2828			0.299180
$266$		0			0
$267$		−	15.6171i		−	0.0584909i
$268$		0			0
$269$	128.551			0.477887			0.238943	−	0.971034i	$-0.423199\pi$
$269$	128.551			0.477887			0.238943	+	0.971034i	$0.423199\pi$
$270$		0			0
$271$	−154.204			−0.569018			−0.284509	−	0.958673i	$-0.591831\pi$
$271$	−154.204			−0.569018			−0.284509	+	0.958673i	$0.591831\pi$
$272$		0			0
$273$			81.6800i			0.299194i
$274$		0			0
$275$		−	40.5676i		−	0.147519i
$276$		0			0
$277$	364.310			1.31520			0.657599	−	0.753368i	$-0.271572\pi$
$277$	364.310			1.31520			0.657599	+	0.753368i	$0.271572\pi$
$278$		0			0
$279$	−64.0576			−0.229597
$280$		0			0
$281$			226.459i			0.805903i	0.915221	+	0.402952i	$0.132016\pi$
$281$			226.459i			0.805903i	−0.915221	+	0.402952i	$0.867984\pi$
$282$		0			0
$283$		−	340.113i		−	1.20181i	−0.799319	−	0.600907i	$-0.794806\pi$
$283$		−	340.113i		−	1.20181i	0.799319	−	0.600907i	$-0.205194\pi$
$284$		0			0
$285$	−192.627			−0.675884
$286$		0			0
$287$		−	180.672i		−	0.629521i
$288$		0			0
$289$	−547.509			−1.89449
$290$		0			0
$291$			200.939i			0.690512i
$292$		0			0
$293$			155.630i			0.531161i	0.964089	+	0.265580i	$0.0855636\pi$
$293$			155.630i			0.531161i	−0.964089	+	0.265580i	$0.914436\pi$
$294$		0			0
$295$		−	138.417i		−	0.469209i
$296$		0			0
$297$			231.426i			0.779211i
$298$		0			0
$299$	−60.9290	−	34.4715i	−0.203776	−	0.115289i
$300$		0			0
$301$	−288.719			−0.959199
$302$		0			0
$303$	−437.078			−1.44250
$304$		0			0
$305$	30.5330			0.100108
$306$		0			0
$307$	434.604			1.41565			0.707825	−	0.706388i	$-0.249677\pi$
$307$	434.604			1.41565			0.707825	+	0.706388i	$0.249677\pi$
$308$		0			0
$309$		−	538.025i		−	1.74118i
$310$		0			0
$311$	−218.370			−0.702154			−0.351077	−	0.936347i	$-0.614184\pi$
$311$	−218.370			−0.702154			−0.351077	+	0.936347i	$0.614184\pi$
$312$		0			0
$313$			17.5030i			0.0559201i	0.999609	+	0.0279600i	$0.00890112\pi$
$313$			17.5030i			0.0559201i	−0.999609	+	0.0279600i	$0.991099\pi$
$314$		0			0
$315$	26.6650			0.0846507
$316$		0			0
$317$	394.049			1.24306			0.621528	−	0.783392i	$-0.286512\pi$
$317$	394.049			1.24306			0.621528	+	0.783392i	$0.286512\pi$
$318$		0			0
$319$			255.705i			0.801582i
$320$		0			0
$321$			472.754i			1.47275i
$322$		0			0
$323$	−894.295			−2.76871
$324$		0			0
$325$	15.2184			0.0468257
$326$		0			0
$327$		−	228.908i		−	0.700024i
$328$		0			0
$329$			641.288i			1.94920i
$330$		0			0
$331$	449.864			1.35910			0.679552	−	0.733627i	$-0.262174\pi$
$331$	449.864			1.35910			0.679552	+	0.733627i	$0.262174\pi$
$332$		0			0
$333$			18.9992i			0.0570547i
$334$		0			0
$335$	−42.1894			−0.125938
$336$		0			0
$337$			395.652i			1.17404i	0.809572	+	0.587021i	$0.199699\pi$
$337$			395.652i			1.17404i	−0.809572	+	0.587021i	$0.800301\pi$
$338$		0			0
$339$		−	500.341i		−	1.47593i
$340$		0			0
$341$			419.812i			1.23112i
$342$		0			0
$343$			50.2644i			0.146543i
$344$		0			0
$345$	70.5560	−	124.709i	0.204510	−	0.361475i
$346$		0			0
$347$	39.5081			0.113856			0.0569281	−	0.998378i	$-0.481869\pi$
$347$	39.5081			0.113856			0.0569281	+	0.998378i	$0.481869\pi$
$348$		0			0
$349$	−108.091			−0.309716			−0.154858	−	0.987937i	$-0.549492\pi$
$349$	−108.091			−0.309716			−0.154858	+	0.987937i	$0.549492\pi$
$350$		0			0
$351$	−86.8161			−0.247339
$352$		0			0
$353$	473.564			1.34154			0.670770	−	0.741665i	$-0.265964\pi$
$353$	473.564			1.34154			0.670770	+	0.741665i	$0.265964\pi$
$354$		0			0
$355$		−	295.523i		−	0.832459i
$356$		0			0
$357$	−776.163			−2.17413
$358$		0			0
$359$			168.522i			0.469421i	0.972065	+	0.234710i	$0.0754142\pi$
$359$			168.522i			0.469421i	−0.972065	+	0.234710i	$0.924586\pi$
$360$		0			0
$361$	−595.072			−1.64840
$362$		0			0
$363$	−153.707			−0.423437
$364$		0			0
$365$			107.087i			0.293389i
$366$		0			0
$367$			246.612i			0.671967i	0.941868	+	0.335983i	$0.109069\pi$
$367$			246.612i			0.671967i	−0.941868	+	0.335983i	$0.890931\pi$
$368$		0			0
$369$	23.2213			0.0629302
$370$		0			0
$371$	341.527			0.920558
$372$		0			0
$373$		−	556.664i		−	1.49240i	−0.665723	−	0.746199i	$-0.731877\pi$
$373$		−	556.664i		−	1.49240i	0.665723	−	0.746199i	$-0.268123\pi$
$374$		0			0
$375$			31.1488i			0.0830635i
$376$		0			0
$377$	−95.9239			−0.254440
$378$		0			0
$379$		−	686.540i		−	1.81145i	−0.423863	−	0.905726i	$-0.639326\pi$
$379$		−	686.540i		−	1.81145i	0.423863	−	0.905726i	$-0.360674\pi$
$380$		0			0
$381$	−335.161			−0.879688
$382$		0			0
$383$		−	505.974i		−	1.32108i	−0.750790	−	0.660541i	$-0.770327\pi$
$383$		−	505.974i		−	1.32108i	0.750790	−	0.660541i	$-0.229673\pi$
$384$		0			0
$385$		−	174.753i		−	0.453905i
$386$		0			0
$387$		−	37.1081i		−	0.0958866i
$388$		0			0
$389$			110.496i			0.284052i	0.989863	+	0.142026i	$0.0453616\pi$
$389$			110.496i			0.284052i	−0.989863	+	0.142026i	$0.954638\pi$
$390$		0			0
$391$	327.565	−	578.977i	0.837763	−	1.48076i
$392$		0			0
$393$	−342.700			−0.872011
$394$		0			0
$395$	−136.899			−0.346579
$396$		0			0
$397$	335.166			0.844246			0.422123	−	0.906538i	$-0.361285\pi$
$397$	335.166			0.844246			0.422123	+	0.906538i	$0.361285\pi$
$398$		0			0
$399$	−829.780			−2.07965
$400$		0			0
$401$			376.016i			0.937695i	0.883279	+	0.468847i	$0.155331\pi$
$401$			376.016i			0.937695i	−0.883279	+	0.468847i	$0.844669\pi$
$402$		0			0
$403$	−157.487			−0.390786
$404$		0			0
$405$		−	152.780i		−	0.377234i
$406$		0			0
$407$	124.515			0.305933
$408$		0			0
$409$	461.636			1.12870			0.564348	−	0.825537i	$-0.309128\pi$
$409$	461.636			1.12870			0.564348	+	0.825537i	$0.309128\pi$
$410$		0			0
$411$			89.5618i			0.217912i
$412$		0			0
$413$		−	596.259i		−	1.44373i
$414$		0			0
$415$	−202.110			−0.487011
$416$		0			0
$417$	313.618			0.752081
$418$		0			0
$419$			158.778i			0.378945i	0.981886	+	0.189473i	$0.0606779\pi$
$419$			158.778i			0.378945i	−0.981886	+	0.189473i	$0.939322\pi$
$420$		0			0
$421$			26.0495i			0.0618754i	0.999521	+	0.0309377i	$0.00984935\pi$
$421$			26.0495i			0.0618754i	−0.999521	+	0.0309377i	$0.990151\pi$
$422$		0			0
$423$	−82.4227			−0.194853
$424$		0			0
$425$			144.612i			0.340264i
$426$		0			0
$427$	131.527			0.308026
$428$		0			0
$429$			68.8009i			0.160375i
$430$		0			0
$431$			603.914i			1.40119i	0.713558	+	0.700596i	$0.247083\pi$
$431$			603.914i			1.40119i	−0.713558	+	0.700596i	$0.752917\pi$
$432$		0			0
$433$		−	520.189i		−	1.20136i	−0.799490	−	0.600680i	$-0.794897\pi$
$433$		−	520.189i		−	1.20136i	0.799490	−	0.600680i	$-0.205103\pi$
$434$		0			0
$435$		−	196.336i		−	0.451347i
$436$		0			0
$437$	350.193	−	618.972i	0.801358	−	1.41641i
$438$		0			0
$439$	−781.158			−1.77940			−0.889701	−	0.456544i	$-0.849087\pi$
$439$	−781.158			−1.77940			−0.889701	+	0.456544i	$0.849087\pi$
$440$		0			0
$441$	54.2023			0.122908
$442$		0			0
$443$	133.602			0.301585			0.150792	−	0.988565i	$-0.451817\pi$
$443$	133.602			0.301585			0.150792	+	0.988565i	$0.451817\pi$
$444$		0			0
$445$	−12.5342			−0.0281668
$446$		0			0
$447$		−	472.286i		−	1.05657i
$448$		0			0
$449$	450.890			1.00421			0.502104	−	0.864807i	$-0.332559\pi$
$449$	450.890			1.00421			0.502104	+	0.864807i	$0.332559\pi$
$450$		0			0
$451$		−	152.184i		−	0.337438i
$452$		0			0
$453$	257.097			0.567543
$454$		0			0
$455$	65.5563			0.144080
$456$		0			0
$457$		−	449.952i		−	0.984579i	−0.870432	−	0.492289i	$-0.836160\pi$
$457$		−	449.952i		−	0.984579i	0.870432	−	0.492289i	$-0.163840\pi$
$458$		0			0
$459$		−	824.969i		−	1.79732i
$460$		0			0
$461$	−118.990			−0.258112			−0.129056	−	0.991637i	$-0.541195\pi$
$461$	−118.990			−0.258112			−0.129056	+	0.991637i	$0.541195\pi$
$462$		0			0
$463$	−470.126			−1.01539			−0.507696	−	0.861537i	$-0.669503\pi$
$463$	−470.126			−1.01539			−0.507696	+	0.861537i	$0.669503\pi$
$464$		0			0
$465$		−	322.342i		−	0.693209i
$466$		0			0
$467$			625.533i			1.33947i	0.742600	+	0.669735i	$0.233592\pi$
$467$			625.533i			1.33947i	−0.742600	+	0.669735i	$0.766408\pi$
$468$		0			0
$469$	−181.739			−0.387504
$470$		0			0
$471$			451.558i			0.958722i
$472$		0			0
$473$	−243.194			−0.514153
$474$		0			0
$475$			154.602i			0.325478i
$476$		0			0
$477$			43.8954i			0.0920239i
$478$		0			0
$479$			502.989i			1.05008i	0.851077	+	0.525041i	$0.175950\pi$
$479$			502.989i			1.05008i	−0.851077	+	0.525041i	$0.824050\pi$
$480$		0			0
$481$			46.7099i			0.0971099i
$482$		0			0
$483$	303.935	−	537.209i	0.629265	−	1.11223i
$484$		0			0
$485$	161.273			0.332522
$486$		0			0
$487$	−17.9953			−0.0369514			−0.0184757	−	0.999829i	$-0.505881\pi$
$487$	−17.9953			−0.0369514			−0.0184757	+	0.999829i	$0.505881\pi$
$488$		0			0
$489$	561.069			1.14738
$490$		0			0
$491$	−309.184			−0.629703			−0.314851	−	0.949141i	$-0.601955\pi$
$491$	−309.184			−0.629703			−0.314851	+	0.949141i	$0.601955\pi$
$492$		0			0
$493$		−	911.516i		−	1.84892i
$494$		0			0
$495$	22.4605			0.0453748
$496$		0			0
$497$		−	1273.03i		−	2.56142i
$498$		0			0
$499$	657.133			1.31690			0.658450	−	0.752625i	$-0.271213\pi$
$499$	657.133			1.31690			0.658450	+	0.752625i	$0.271213\pi$
$500$		0			0
$501$	−678.936			−1.35516
$502$		0			0
$503$			523.861i			1.04147i	0.853717	+	0.520737i	$0.174343\pi$
$503$			523.861i			1.04147i	−0.853717	+	0.520737i	$0.825657\pi$
$504$		0			0
$505$			350.798i			0.694650i
$506$		0			0
$507$	445.030			0.877771
$508$		0			0
$509$	−955.951			−1.87810			−0.939048	−	0.343787i	$-0.888290\pi$
$509$	−955.951			−1.87810			−0.939048	+	0.343787i	$0.888290\pi$
$510$		0			0
$511$			461.299i			0.902738i
$512$		0			0
$513$		−	881.957i		−	1.71921i
$514$		0			0
$515$	−431.818			−0.838482
$516$		0			0
$517$			540.171i			1.04482i
$518$		0			0
$519$	411.947			0.793733
$520$		0			0
$521$			364.505i			0.699626i	0.936820	+	0.349813i	$0.113755\pi$
$521$			364.505i			0.699626i	−0.936820	+	0.349813i	$0.886245\pi$
$522$		0			0
$523$			656.426i			1.25512i	0.778570	+	0.627558i	$0.215946\pi$
$523$			656.426i			1.25512i	−0.778570	+	0.627558i	$0.784054\pi$
$524$		0			0
$525$			134.180i			0.255581i
$526$		0			0
$527$		−	1496.51i		−	2.83969i
$528$		0			0
$529$	272.460	+	453.439i	0.515047	+	0.857162i
$530$		0			0
$531$	76.6352			0.144322
$532$		0			0
$533$	57.0898			0.107110
$534$		0			0
$535$	379.432			0.709219
$536$		0			0
$537$	186.920			0.348082
$538$		0			0
$539$		−	355.224i		−	0.659043i
$540$		0			0
$541$	−841.289			−1.55506			−0.777532	−	0.628844i	$-0.783529\pi$
$541$	−841.289			−1.55506			−0.777532	+	0.628844i	$0.783529\pi$
$542$		0			0
$543$			256.266i			0.471945i
$544$		0			0
$545$	−183.721			−0.337103
$546$		0			0
$547$	893.917			1.63422			0.817109	−	0.576483i	$-0.195575\pi$
$547$	893.917			1.63422			0.817109	+	0.576483i	$0.195575\pi$
$548$		0			0
$549$			16.9048i			0.0307919i
$550$		0			0
$551$		−	974.483i		−	1.76857i
$552$		0			0
$553$	−589.720			−1.06640
$554$		0			0
$555$	−95.6053			−0.172262
$556$		0			0
$557$		−	674.886i		−	1.21165i	−0.795600	−	0.605823i	$-0.792844\pi$
$557$		−	674.886i		−	1.21165i	0.795600	−	0.605823i	$-0.207156\pi$
$558$		0			0
$559$		−	91.2309i		−	0.163204i
$560$		0			0
$561$	−653.780			−1.16538
$562$		0			0
$563$			642.543i			1.14128i	0.821199	+	0.570642i	$0.193306\pi$
$563$			642.543i			1.14128i	−0.821199	+	0.570642i	$0.806694\pi$
$564$		0			0
$565$	−401.573			−0.710749
$566$		0			0
$567$		−	658.131i		−	1.16072i
$568$		0			0
$569$			931.185i			1.63653i	0.574842	+	0.818265i	$0.305064\pi$
$569$			931.185i			1.63653i	−0.574842	+	0.818265i	$0.694936\pi$
$570$		0			0
$571$			567.864i			0.994508i	0.867605	+	0.497254i	$0.165658\pi$
$571$			567.864i			0.994508i	−0.867605	+	0.497254i	$0.834342\pi$
$572$		0			0
$573$		−	83.7534i		−	0.146167i
$574$		0			0
$575$	−100.091	−	56.6282i	−0.174072	−	0.0984838i
$576$		0			0
$577$	173.079			0.299963			0.149982	−	0.988689i	$-0.452079\pi$
$577$	173.079			0.299963			0.149982	+	0.988689i	$0.452079\pi$
$578$		0			0
$579$	−264.861			−0.457446
$580$		0			0
$581$	−870.630			−1.49850
$582$		0			0
$583$	287.676			0.493441
$584$		0			0
$585$			8.42574i			0.0144030i
$586$		0			0
$587$	−596.324			−1.01588			−0.507942	−	0.861391i	$-0.669594\pi$
$587$	−596.324			−1.01588			−0.507942	+	0.861391i	$0.669594\pi$
$588$		0			0
$589$		−	1599.89i		−	2.71629i
$590$		0			0
$591$	432.880			0.732454
$592$		0			0
$593$	205.466			0.346486			0.173243	−	0.984879i	$-0.444575\pi$
$593$	205.466			0.346486			0.173243	+	0.984879i	$0.444575\pi$
$594$		0			0
$595$			622.947i			1.04697i
$596$		0			0
$597$		−	785.055i		−	1.31500i
$598$		0			0
$599$	928.662			1.55035			0.775177	−	0.631744i	$-0.217661\pi$
$599$	928.662			1.55035			0.775177	+	0.631744i	$0.217661\pi$
$600$		0			0
$601$	−0.675554			−0.00112405			−0.000562025	−	1.00000i	$-0.500179\pi$
$601$	−0.675554			−0.00112405			−0.000562025		1.00000i	$0.500179\pi$
$602$		0			0
$603$		−	23.3584i		−	0.0387370i
$604$		0			0
$605$			123.365i			0.203910i
$606$		0			0
$607$	−767.471			−1.26437			−0.632184	−	0.774818i	$-0.717841\pi$
$607$	−767.471			−1.26437			−0.632184	+	0.774818i	$0.717841\pi$
$608$		0			0
$609$		−	845.759i		−	1.38877i
$610$		0			0
$611$	−202.638			−0.331649
$612$		0			0
$613$			111.752i			0.182304i	0.995837	+	0.0911521i	$0.0290550\pi$
$613$			111.752i			0.182304i	−0.995837	+	0.0911521i	$0.970945\pi$
$614$		0			0
$615$			116.851i			0.190002i
$616$		0			0
$617$		−	347.855i		−	0.563785i	−0.959446	−	0.281892i	$-0.909038\pi$
$617$		−	347.855i		−	0.563785i	0.959446	−	0.281892i	$-0.0909621\pi$
$618$		0			0
$619$			939.046i			1.51704i	0.651652	+	0.758518i	$0.274076\pi$
$619$			939.046i			1.51704i	−0.651652	+	0.758518i	$0.725924\pi$
$620$		0			0
$621$	570.989	+	323.046i	0.919467	+	0.520204i
$622$		0			0
$623$	−53.9938			−0.0866674
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−698.942			−1.11474
$628$		0			0
$629$	−443.860			−0.705659
$630$		0			0
$631$			554.612i			0.878941i	0.898257	+	0.439471i	$0.144834\pi$
$631$			554.612i			0.878941i	−0.898257	+	0.439471i	$0.855166\pi$
$632$		0			0
$633$	657.675			1.03898
$634$		0			0
$635$			269.000i			0.423622i
$636$		0			0
$637$	133.257			0.209195
$638$		0			0
$639$	163.618			0.256053
$640$		0			0
$641$			21.9441i			0.0342342i	0.999853	+	0.0171171i	$0.00544881\pi$
$641$			21.9441i			0.0342342i	−0.999853	+	0.0171171i	$0.994551\pi$
$642$		0			0
$643$		−	443.017i		−	0.688985i	−0.938789	−	0.344492i	$-0.888051\pi$
$643$		−	443.017i		−	0.688985i	0.938789	−	0.344492i	$-0.111949\pi$
$644$		0			0
$645$	186.731			0.289505
$646$		0			0
$647$	578.891			0.894732			0.447366	−	0.894351i	$-0.352362\pi$
$647$	578.891			0.894732			0.447366	+	0.894351i	$0.352362\pi$
$648$		0			0
$649$		−	502.242i		−	0.773871i
$650$		0			0
$651$		−	1388.56i		−	2.13296i
$652$		0			0
$653$	−489.328			−0.749353			−0.374677	−	0.927156i	$-0.622246\pi$
$653$	−489.328			−0.749353			−0.374677	+	0.927156i	$0.622246\pi$
$654$		0			0
$655$			275.051i			0.419925i
$656$		0			0
$657$	−59.2893			−0.0902425
$658$		0			0
$659$			708.863i			1.07567i	0.843052	+	0.537833i	$0.180757\pi$
$659$			708.863i			1.07567i	−0.843052	+	0.537833i	$0.819243\pi$
$660$		0			0
$661$		−	596.780i		−	0.902844i	−0.892311	−	0.451422i	$-0.850917\pi$
$661$		−	596.780i		−	0.902844i	0.892311	−	0.451422i	$-0.149083\pi$
$662$		0			0
$663$		−	245.256i		−	0.369919i
$664$		0			0
$665$			665.980i			1.00147i
$666$		0			0
$667$	630.891	+	356.937i	0.945864	+	0.535138i
$668$		0			0
$669$	−639.574			−0.956015
$670$		0			0
$671$	110.788			0.165109
$672$		0			0
$673$	318.498			0.473252			0.236626	−	0.971601i	$-0.423958\pi$
$673$	318.498			0.473252			0.236626	+	0.971601i	$0.423958\pi$
$674$		0			0
$675$	−142.617			−0.211285
$676$		0			0
$677$			932.322i			1.37714i	0.725171	+	0.688569i	$0.241761\pi$
$677$			932.322i			1.37714i	−0.725171	+	0.688569i	$0.758239\pi$
$678$		0			0
$679$	694.718			1.02315
$680$		0			0
$681$			339.341i			0.498299i
$682$		0			0
$683$	−1178.48			−1.72544			−0.862721	−	0.505680i	$-0.831242\pi$
$683$	−1178.48			−1.72544			−0.862721	+	0.505680i	$0.831242\pi$
$684$		0			0
$685$	71.8822			0.104937
$686$		0			0
$687$		−	139.593i		−	0.203193i
$688$		0			0
$689$			107.918i			0.156629i
$690$		0			0
$691$	271.488			0.392892			0.196446	−	0.980515i	$-0.437060\pi$
$691$	271.488			0.392892			0.196446	+	0.980515i	$0.437060\pi$
$692$		0			0
$693$	96.7533			0.139615
$694$		0			0
$695$		−	251.709i		−	0.362172i
$696$		0			0
$697$			542.495i			0.778329i
$698$		0			0
$699$	−786.340			−1.12495
$700$		0			0
$701$			67.5200i			0.0963196i	0.998840	+	0.0481598i	$0.0153357\pi$
$701$			67.5200i			0.0963196i	−0.998840	+	0.0481598i	$0.984664\pi$
$702$		0			0
$703$	−474.521			−0.674995
$704$		0			0
$705$		−	414.757i		−	0.588307i
$706$		0			0
$707$			1511.14i			2.13739i
$708$		0			0
$709$		−	1351.19i		−	1.90576i	−0.303340	−	0.952882i	$-0.598102\pi$
$709$		−	1351.19i		−	1.90576i	0.303340	−	0.952882i	$-0.401898\pi$
$710$		0			0
$711$		−	75.7949i		−	0.106603i
$712$		0			0
$713$	1035.79	+	586.014i	1.45272	+	0.821900i
$714$		0			0
$715$	55.2195			0.0772301
$716$		0			0
$717$	−968.081			−1.35018
$718$		0			0
$719$	480.778			0.668677			0.334338	−	0.942453i	$-0.391487\pi$
$719$	480.778			0.668677			0.334338	+	0.942453i	$0.391487\pi$
$720$		0			0
$721$	−1860.15			−2.57995
$722$		0			0
$723$			131.145i			0.181389i
$724$		0			0
$725$	−157.579			−0.217351
$726$		0			0
$727$			123.371i			0.169699i	0.996394	+	0.0848496i	$0.0270410\pi$
$727$			123.371i			0.169699i	−0.996394	+	0.0848496i	$0.972959\pi$
$728$		0			0
$729$	799.793			1.09711
$730$		0			0
$731$	866.920			1.18594
$732$		0			0
$733$		−	285.404i		−	0.389364i	−0.980866	−	0.194682i	$-0.937633\pi$
$733$		−	285.404i		−	0.389364i	0.980866	−	0.194682i	$-0.0623675\pi$
$734$		0			0
$735$			272.750i			0.371088i
$736$		0			0
$737$	−153.083			−0.207711
$738$		0			0
$739$	−201.545			−0.272727			−0.136363	−	0.990659i	$-0.543541\pi$
$739$	−201.545			−0.272727			−0.136363	+	0.990659i	$0.543541\pi$
$740$		0			0
$741$		−	262.198i		−	0.353844i
$742$		0			0
$743$		−	641.580i		−	0.863499i	−0.901994	−	0.431749i	$-0.857896\pi$
$743$		−	641.580i		−	0.863499i	0.901994	−	0.431749i	$-0.142104\pi$
$744$		0			0
$745$	−379.056			−0.508800
$746$		0			0
$747$		−	111.899i		−	0.149798i
$748$		0			0
$749$	1634.48			2.18222
$750$		0			0
$751$			329.589i			0.438867i	0.975627	+	0.219434i	$0.0704209\pi$
$751$			329.589i			0.438867i	−0.975627	+	0.219434i	$0.929579\pi$
$752$		0			0
$753$			1205.44i			1.60085i
$754$		0			0
$755$		−	206.346i		−	0.273306i
$756$		0			0
$757$			325.521i			0.430015i	0.976612	+	0.215007i	$0.0689776\pi$
$757$			325.521i			0.430015i	−0.976612	+	0.215007i	$0.931022\pi$
$758$		0			0
$759$	256.011	−	452.504i	0.337301	−	0.596184i
$760$		0			0
$761$	135.928			0.178617			0.0893086	−	0.996004i	$-0.471534\pi$
$761$	135.928			0.178617			0.0893086	+	0.996004i	$0.471534\pi$
$762$		0			0
$763$	−791.417			−1.03724
$764$		0			0
$765$	−80.0655			−0.104661
$766$		0			0
$767$	188.409			0.245644
$768$		0			0
$769$		−	855.993i		−	1.11312i	−0.830806	−	0.556562i	$-0.812120\pi$
$769$		−	855.993i		−	1.11312i	0.830806	−	0.556562i	$-0.187880\pi$
$770$		0			0
$771$	−980.927			−1.27228
$772$		0			0
$773$			1076.96i			1.39322i	0.717448	+	0.696612i	$0.245310\pi$
$773$			1076.96i			1.39322i	−0.717448	+	0.696612i	$0.754690\pi$
$774$		0			0
$775$	−258.711			−0.333821
$776$		0			0
$777$	−411.840			−0.530038
$778$		0			0
$779$			579.971i			0.744507i
$780$		0			0
$781$		−	1072.30i		−	1.37298i
$782$		0			0
$783$	898.940			1.14807
$784$		0			0
$785$	362.420			0.461681
$786$		0			0
$787$		−	126.777i		−	0.161089i	−0.996751	−	0.0805445i	$-0.974334\pi$
$787$		−	126.777i		−	0.161089i	0.996751	−	0.0805445i	$-0.0256659\pi$
$788$		0			0
$789$		−	212.513i		−	0.269345i
$790$		0			0
$791$	−1729.86			−2.18693
$792$		0			0
$793$			41.5606i			0.0524093i
$794$		0			0
$795$	−220.885			−0.277842
$796$		0			0
$797$		−	745.819i		−	0.935783i	−0.883786	−	0.467891i	$-0.845014\pi$
$797$		−	745.819i		−	0.935783i	0.883786	−	0.467891i	$-0.154986\pi$
$798$		0			0
$799$		−	1925.56i		−	2.40996i
$800$		0			0
$801$		−	6.93965i		−	0.00866374i
$802$		0			0
$803$			388.563i			0.483889i
$804$		0			0
$805$	−431.164	−	243.938i	−0.535607	−	0.303028i
$806$		0			0
$807$	−358.149			−0.443803
$808$		0			0
$809$	−841.111			−1.03969			−0.519846	−	0.854260i	$-0.674011\pi$
$809$	−841.111			−1.03969			−0.519846	+	0.854260i	$0.674011\pi$
$810$		0			0
$811$	−856.401			−1.05598			−0.527991	−	0.849250i	$-0.677054\pi$
$811$	−856.401			−1.05598			−0.527991	+	0.849250i	$0.677054\pi$
$812$		0			0
$813$	429.617			0.528434
$814$		0			0
$815$		−	450.313i		−	0.552531i
$816$		0			0
$817$	926.807			1.13440
$818$		0			0
$819$			36.2956i			0.0443170i
$820$		0			0
$821$	1073.97			1.30813			0.654064	−	0.756439i	$-0.273063\pi$
$821$	1073.97			1.30813			0.654064	+	0.756439i	$0.273063\pi$
$822$		0			0
$823$	751.313			0.912895			0.456448	−	0.889750i	$-0.349122\pi$
$823$	751.313			0.912895			0.456448	+	0.889750i	$0.349122\pi$
$824$		0			0
$825$			113.023i			0.136997i
$826$		0			0
$827$			184.561i			0.223169i	0.993755	+	0.111585i	$0.0355926\pi$
$827$			184.561i			0.223169i	−0.993755	+	0.111585i	$0.964407\pi$
$828$		0			0
$829$	−198.894			−0.239920			−0.119960	−	0.992779i	$-0.538277\pi$
$829$	−198.894			−0.239920			−0.119960	+	0.992779i	$0.538277\pi$
$830$		0			0
$831$	−1014.98			−1.22140
$832$		0			0
$833$			1266.27i			1.52014i
$834$		0			0
$835$			544.913i			0.652591i
$836$		0			0
$837$	1475.87			1.76328
$838$		0			0
$839$			396.084i			0.472091i	0.971742	+	0.236045i	$0.0758514\pi$
$839$			396.084i			0.472091i	−0.971742	+	0.236045i	$0.924149\pi$
$840$		0			0
$841$	152.248			0.181032
$842$		0			0
$843$		−	630.922i		−	0.748425i
$844$		0			0
$845$		−	357.181i		−	0.422699i
$846$		0			0
$847$			531.422i			0.627417i
$848$		0			0
$849$			947.568i			1.11610i
$850$		0			0
$851$	173.809	−	307.211i	0.204241	−	0.361000i
$852$		0			0
$853$	1097.11			1.28618			0.643088	−	0.765792i	$-0.277653\pi$
$853$	1097.11			1.28618			0.643088	+	0.765792i	$0.277653\pi$
$854$		0			0
$855$	−85.5964			−0.100113
$856$		0			0
$857$	−1335.36			−1.55818			−0.779092	−	0.626909i	$-0.784320\pi$
$857$	−1335.36			−1.55818			−0.779092	+	0.626909i	$0.784320\pi$
$858$		0			0
$859$	535.472			0.623366			0.311683	−	0.950186i	$-0.399107\pi$
$859$	535.472			0.623366			0.311683	+	0.950186i	$0.399107\pi$
$860$		0			0
$861$			503.360i			0.584622i
$862$		0			0
$863$	1423.19			1.64912			0.824560	−	0.565774i	$-0.191423\pi$
$863$	1423.19			1.64912			0.824560	+	0.565774i	$0.191423\pi$
$864$		0			0
$865$		−	330.628i		−	0.382229i
$866$		0			0
$867$	1525.38			1.75938
$868$		0			0
$869$	−496.735			−0.571616
$870$		0			0
$871$		−	57.4270i		−	0.0659323i
$872$		0			0
$873$			89.2900i			0.102279i
$874$		0			0
$875$	107.693			0.123077
$876$		0			0
$877$	−316.587			−0.360989			−0.180495	−	0.983576i	$-0.557770\pi$
$877$	−316.587			−0.360989			−0.180495	+	0.983576i	$0.557770\pi$
$878$		0			0
$879$		−	433.591i		−	0.493277i
$880$		0			0
$881$		−	124.297i		−	0.141087i	−0.997509	−	0.0705433i	$-0.977527\pi$
$881$		−	124.297i		−	0.141087i	0.997509	−	0.0705433i	$-0.0224733\pi$
$882$		0			0
$883$	607.679			0.688198			0.344099	−	0.938933i	$-0.388184\pi$
$883$	607.679			0.688198			0.344099	+	0.938933i	$0.388184\pi$
$884$		0			0
$885$			385.634i			0.435744i
$886$		0			0
$887$	−749.974			−0.845517			−0.422759	−	0.906242i	$-0.638938\pi$
$887$	−749.974			−0.845517			−0.422759	+	0.906242i	$0.638938\pi$
$888$		0			0
$889$			1158.77i			1.30346i
$890$		0			0
$891$		−	554.358i		−	0.622176i
$892$		0			0
$893$		−	2058.58i		−	2.30524i
$894$		0			0
$895$		−	150.022i		−	0.167622i
$896$		0			0
$897$	169.750	+	96.0389i	0.189242	+	0.107067i
$898$		0			0
$899$	1630.70			1.81391
$900$		0			0
$901$	−1025.48			−1.13816
$902$		0			0
$903$	804.381			0.890787
$904$		0			0
$905$	205.679			0.227270
$906$		0			0
$907$		−	1315.64i		−	1.45054i	−0.688465	−	0.725270i	$-0.741715\pi$
$907$		−	1315.64i		−	1.45054i	0.688465	−	0.725270i	$-0.258285\pi$
$908$		0			0
$909$	−194.222			−0.213665
$910$		0			0
$911$			250.666i			0.275155i	0.990491	+	0.137577i	$0.0439315\pi$
$911$			250.666i			0.275155i	−0.990491	+	0.137577i	$0.956068\pi$
$912$		0			0
$913$	−733.351			−0.803232
$914$		0			0
$915$	−85.0658			−0.0929681
$916$		0			0
$917$			1184.84i			1.29208i
$918$		0			0
$919$		−	954.979i		−	1.03915i	−0.854425	−	0.519575i	$-0.826090\pi$
$919$		−	954.979i		−	1.03915i	0.854425	−	0.519575i	$-0.173910\pi$
$920$		0			0
$921$	−1210.82			−1.31468
$922$		0			0
$923$	402.258			0.435815
$924$		0			0
$925$			76.7327i			0.0829543i
$926$		0			0
$927$		−	239.079i		−	0.257906i
$928$		0			0
$929$	1000.76			1.07725			0.538624	−	0.842546i	$-0.318944\pi$
$929$	1000.76			1.07725			0.538624	+	0.842546i	$0.318944\pi$
$930$		0			0
$931$			1353.75i			1.45408i
$932$		0			0
$933$	608.386			0.652075
$934$		0			0
$935$			524.723i			0.561201i
$936$		0			0
$937$		−	287.822i		−	0.307174i	−0.988135	−	0.153587i	$-0.950917\pi$
$937$		−	287.822i		−	0.307174i	0.988135	−	0.153587i	$-0.0490825\pi$
$938$		0			0
$939$		−	48.7639i		−	0.0519318i
$940$		0			0
$941$			978.106i			1.03943i	0.854339	+	0.519716i	$0.173962\pi$
$941$			978.106i			1.03943i	−0.854339	+	0.519716i	$0.826038\pi$
$942$		0			0
$943$	−375.480	−	212.434i	−0.398176	−	0.225274i
$944$		0			0
$945$	−614.353			−0.650109
$946$		0			0
$947$	1092.46			1.15360			0.576799	−	0.816886i	$-0.304302\pi$
$947$	1092.46			1.15360			0.576799	+	0.816886i	$0.304302\pi$
$948$		0			0
$949$	−145.764			−0.153597
$950$		0			0
$951$	−1097.83			−1.15440
$952$		0			0
$953$			1112.86i			1.16775i	0.811844	+	0.583874i	$0.198464\pi$
$953$			1112.86i			1.16775i	−0.811844	+	0.583874i	$0.801536\pi$
$954$		0			0
$955$	−67.2204			−0.0703879
$956$		0			0
$957$		−	712.402i		−	0.744411i
$958$		0			0
$959$	309.647			0.322886
$960$		0			0
$961$	1716.26			1.78591
$962$		0			0
$963$			210.075i			0.218146i
$964$		0			0
$965$			212.577i			0.220287i
$966$		0			0
$967$	−108.550			−0.112254			−0.0561270	−	0.998424i	$-0.517875\pi$
$967$	−108.550			−0.112254			−0.0561270	+	0.998424i	$0.517875\pi$
$968$		0			0
$969$	2491.54			2.57124
$970$		0			0
$971$			1446.92i			1.49013i	0.666990	+	0.745067i	$0.267582\pi$
$971$			1446.92i			1.49013i	−0.666990	+	0.745067i	$0.732418\pi$
$972$		0			0
$973$		−	1084.29i		−	1.11438i
$974$		0			0
$975$	−42.3989			−0.0434860
$976$		0			0
$977$			1250.06i			1.27949i	0.768588	+	0.639744i	$0.220959\pi$
$977$			1250.06i			1.27949i	−0.768588	+	0.639744i	$0.779041\pi$
$978$		0			0
$979$	−45.4802			−0.0464558
$980$		0			0
$981$		−	101.718i		−	0.103688i
$982$		0			0
$983$			501.398i			0.510070i	0.966932	+	0.255035i	$0.0820869\pi$
$983$			501.398i			0.510070i	−0.966932	+	0.255035i	$0.917913\pi$
$984$		0			0
$985$		−	347.429i		−	0.352720i
$986$		0			0
$987$		−	1786.65i		−	1.81018i
$988$		0			0
$989$	−339.474	+	600.026i	−0.343250	+	0.606699i
$990$		0			0
$991$	−496.193			−0.500699			−0.250350	−	0.968155i	$-0.580546\pi$
$991$	−496.193			−0.500699			−0.250350	+	0.968155i	$0.580546\pi$
$992$		0			0
$993$	−1253.34			−1.26217
$994$		0			0
$995$	−630.084			−0.633250
$996$		0			0
$997$	−213.066			−0.213707			−0.106854	−	0.994275i	$-0.534078\pi$
$997$	−213.066			−0.213707			−0.106854	+	0.994275i	$0.534078\pi$
$998$		0			0
$999$		−	437.736i		−	0.438175i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.k.e.321.12		48
4.3	odd	2		920.3.k.a.321.38	yes	48
23.22	odd	2	inner	1840.3.k.e.321.11		48
92.91	even	2		920.3.k.a.321.37	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.3.k.a.321.37	✓	48	92.91	even	2
920.3.k.a.321.38	yes	48	4.3	odd	2
1840.3.k.e.321.11		48	23.22	odd	2	inner
1840.3.k.e.321.12		48	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)

\(f(q)\)	\(=\)	\(q-2.78603 q^{3} +2.23607i q^{5} +9.63233i q^{7} -1.23801 q^{9} +O(q^{10})\)	\(q-2.78603 q^{3} +2.23607i q^{5} +9.63233i q^{7} -1.23801 q^{9} +8.11353i q^{11} -3.04367 q^{13} -6.22976i q^{15} -28.9225i q^{17} -30.9204i q^{19} -26.8360i q^{21} +(20.0182 + 11.3256i) q^{23} -5.00000 q^{25} +28.5235 q^{27} +31.5158 q^{29} +51.7423 q^{31} -22.6046i q^{33} -21.5385 q^{35} -15.3465i q^{37} +8.47978 q^{39} -18.7569 q^{41} +29.9739i q^{43} -2.76828i q^{45} +66.5766 q^{47} -43.7817 q^{49} +80.5790i q^{51} -35.4563i q^{53} -18.1424 q^{55} +86.1453i q^{57} -61.9018 q^{59} -13.6548i q^{61} -11.9249i q^{63} -6.80586i q^{65} +18.8677i q^{67} +(-55.7715 - 31.5536i) q^{69} -132.162 q^{71} +47.8907 q^{73} +13.9302 q^{75} -78.1521 q^{77} +61.2230i q^{79} -68.3252 q^{81} +90.3862i q^{83} +64.6726 q^{85} -87.8042 q^{87} +5.60548i q^{89} -29.3176i q^{91} -144.156 q^{93} +69.1401 q^{95} -72.1236i q^{97} -10.0446i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 128 q^{9}+O(q^{10}) \) 48 * q + 128 * q^9	\( 48 q + 128 q^{9} - 8 q^{23} - 240 q^{25} + 72 q^{29} - 32 q^{31} + 40 q^{35} + 96 q^{39} - 104 q^{41} - 128 q^{47} - 344 q^{49} - 80 q^{55} - 248 q^{59} + 292 q^{69} - 208 q^{71} + 224 q^{73} - 288 q^{77} + 184 q^{81} + 48 q^{87} - 672 q^{93}+O(q^{100}) \) 48 * q + 128 * q^9 - 8 * q^23 - 240 * q^25 + 72 * q^29 - 32 * q^31 + 40 * q^35 + 96 * q^39 - 104 * q^41 - 128 * q^47 - 344 * q^49 - 80 * q^55 - 248 * q^59 + 292 * q^69 - 208 * q^71 + 224 * q^73 - 288 * q^77 + 184 * q^81 + 48 * q^87 - 672 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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