LMFDB - Embedded newform 1840.3.k.c.321.13

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			321.13
Root			$-4.53300 - 2.23607i$ of defining polynomial
Character	$\chi$	$=$	1840.321
Dual form			1840.3.k.c.321.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.53300 q^{3} -2.23607i q^{5} -5.73038i q^{7} +11.5481 q^{9} +O(q^{10})$	$q+4.53300 q^{3} -2.23607i q^{5} -5.73038i q^{7} +11.5481 q^{9} +14.4494i q^{11} -22.5859 q^{13} -10.1361i q^{15} -15.8960i q^{17} -21.5685i q^{19} -25.9758i q^{21} +(-14.6376 - 17.7410i) q^{23} -5.00000 q^{25} +11.5506 q^{27} -23.3223 q^{29} -52.1637 q^{31} +65.4993i q^{33} -12.8135 q^{35} +29.3528i q^{37} -102.382 q^{39} +69.3985 q^{41} -59.0128i q^{43} -25.8223i q^{45} -35.0771 q^{47} +16.1627 q^{49} -72.0564i q^{51} -55.7924i q^{53} +32.3099 q^{55} -97.7701i q^{57} -34.7849 q^{59} +35.4314i q^{61} -66.1750i q^{63} +50.5035i q^{65} -111.151i q^{67} +(-66.3521 - 80.4198i) q^{69} +94.6728 q^{71} +8.86831 q^{73} -22.6650 q^{75} +82.8008 q^{77} +86.9680i q^{79} -51.5742 q^{81} -153.870i q^{83} -35.5444 q^{85} -105.720 q^{87} +152.838i q^{89} +129.426i q^{91} -236.458 q^{93} -48.2286 q^{95} -14.2228i q^{97} +166.864i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	4.53300			1.51100			0.755500	−	0.655148i	$-0.227394\pi$
$3$	4.53300			1.51100			0.755500	+	0.655148i	$0.227394\pi$
$4$		0			0
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$		−	5.73038i		−	0.818626i	−0.912394	−	0.409313i	$-0.865768\pi$
$7$		−	5.73038i		−	0.818626i	0.912394	−	0.409313i	$-0.134232\pi$
$8$		0			0
$9$	11.5481			1.28312
$10$		0			0
$11$			14.4494i			1.31359i	0.754071	+	0.656793i	$0.228087\pi$
$11$			14.4494i			1.31359i	−0.754071	+	0.656793i	$0.771913\pi$
$12$		0			0
$13$	−22.5859			−1.73737			−0.868687	−	0.495361i	$-0.835036\pi$
$13$	−22.5859			−1.73737			−0.868687	+	0.495361i	$0.835036\pi$
$14$		0			0
$15$		−	10.1361i		−	0.675740i
$16$		0			0
$17$		−	15.8960i		−	0.935056i	−0.883978	−	0.467528i	$-0.845145\pi$
$17$		−	15.8960i		−	0.935056i	0.883978	−	0.467528i	$-0.154855\pi$
$18$		0			0
$19$		−	21.5685i		−	1.13518i	−0.823310	−	0.567592i	$-0.807875\pi$
$19$		−	21.5685i		−	1.13518i	0.823310	−	0.567592i	$-0.192125\pi$
$20$		0			0
$21$		−	25.9758i		−	1.23694i
$22$		0			0
$23$	−14.6376	−	17.7410i	−0.636416	−	0.771346i
$23$	−14.6376	−	17.7410i	−0.636416	−	0.771346i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	11.5506			0.427799
$28$		0			0
$29$	−23.3223			−0.804216			−0.402108	−	0.915592i	$-0.631722\pi$
$29$	−23.3223			−0.804216			−0.402108	+	0.915592i	$0.631722\pi$
$30$		0			0
$31$	−52.1637			−1.68270			−0.841349	−	0.540492i	$-0.818238\pi$
$31$	−52.1637			−1.68270			−0.841349	+	0.540492i	$0.818238\pi$
$32$		0			0
$33$			65.4993i			1.98483i
$34$		0			0
$35$	−12.8135			−0.366101
$36$		0			0
$37$			29.3528i			0.793319i	0.917966	+	0.396660i	$0.129831\pi$
$37$			29.3528i			0.793319i	−0.917966	+	0.396660i	$0.870169\pi$
$38$		0			0
$39$	−102.382			−2.62517
$40$		0			0
$41$	69.3985			1.69265			0.846323	−	0.532670i	$-0.178811\pi$
$41$	69.3985			1.69265			0.846323	+	0.532670i	$0.178811\pi$
$42$		0			0
$43$		−	59.0128i		−	1.37239i	−0.727418	−	0.686195i	$-0.759280\pi$
$43$		−	59.0128i		−	1.37239i	0.727418	−	0.686195i	$-0.240720\pi$
$44$		0			0
$45$		−	25.8223i		−	0.573830i
$46$		0			0
$47$	−35.0771			−0.746320			−0.373160	−	0.927767i	$-0.621726\pi$
$47$	−35.0771			−0.746320			−0.373160	+	0.927767i	$0.621726\pi$
$48$		0			0
$49$	16.1627			0.329852
$50$		0			0
$51$		−	72.0564i		−	1.41287i
$52$		0			0
$53$		−	55.7924i		−	1.05269i	−0.850272	−	0.526343i	$-0.823563\pi$
$53$		−	55.7924i		−	1.05269i	0.850272	−	0.526343i	$-0.176437\pi$
$54$		0			0
$55$	32.3099			0.587453
$56$		0			0
$57$		−	97.7701i		−	1.71526i
$58$		0			0
$59$	−34.7849			−0.589575			−0.294788	−	0.955563i	$-0.595249\pi$
$59$	−34.7849			−0.589575			−0.294788	+	0.955563i	$0.595249\pi$
$60$		0			0
$61$			35.4314i			0.580842i	0.956899	+	0.290421i	$0.0937954\pi$
$61$			35.4314i			0.580842i	−0.956899	+	0.290421i	$0.906205\pi$
$62$		0			0
$63$		−	66.1750i		−	1.05040i
$64$		0			0
$65$			50.5035i			0.776977i
$66$		0			0
$67$		−	111.151i		−	1.65898i	−0.558525	−	0.829488i	$-0.688632\pi$
$67$		−	111.151i		−	1.65898i	0.558525	−	0.829488i	$-0.311368\pi$
$68$		0			0
$69$	−66.3521	−	80.4198i	−0.961625	−	1.16550i
$70$		0			0
$71$	94.6728			1.33342			0.666710	−	0.745317i	$-0.267702\pi$
$71$	94.6728			1.33342			0.666710	+	0.745317i	$0.267702\pi$
$72$		0			0
$73$	8.86831			0.121484			0.0607418	−	0.998154i	$-0.480653\pi$
$73$	8.86831			0.121484			0.0607418	+	0.998154i	$0.480653\pi$
$74$		0			0
$75$	−22.6650			−0.302200
$76$		0			0
$77$	82.8008			1.07534
$78$		0			0
$79$			86.9680i			1.10086i	0.834881	+	0.550431i	$0.185536\pi$
$79$			86.9680i			1.10086i	−0.834881	+	0.550431i	$0.814464\pi$
$80$		0			0
$81$	−51.5742			−0.636719
$82$		0			0
$83$		−	153.870i		−	1.85386i	−0.375234	−	0.926930i	$-0.622438\pi$
$83$		−	153.870i		−	1.85386i	0.375234	−	0.926930i	$-0.377562\pi$
$84$		0			0
$85$	−35.5444			−0.418170
$86$		0			0
$87$	−105.720			−1.21517
$88$		0			0
$89$			152.838i			1.71728i	0.512578	+	0.858641i	$0.328690\pi$
$89$			152.838i			1.71728i	−0.512578	+	0.858641i	$0.671310\pi$
$90$		0			0
$91$			129.426i			1.42226i
$92$		0			0
$93$	−236.458			−2.54256
$94$		0			0
$95$	−48.2286			−0.507670
$96$		0			0
$97$		−	14.2228i		−	0.146627i	−0.997309	−	0.0733136i	$-0.976643\pi$
$97$		−	14.2228i		−	0.146627i	0.997309	−	0.0733136i	$-0.0233574\pi$
$98$		0			0
$99$			166.864i			1.68549i
$100$		0			0
$101$	133.799			1.32474			0.662369	−	0.749178i	$-0.269551\pi$
$101$	133.799			1.32474			0.662369	+	0.749178i	$0.269551\pi$
$102$		0			0
$103$			31.1112i			0.302050i	0.988530	+	0.151025i	$0.0482574\pi$
$103$			31.1112i			0.302050i	−0.988530	+	0.151025i	$0.951743\pi$
$104$		0			0
$105$	−58.0837			−0.553178
$106$		0			0
$107$			149.721i			1.39926i	0.714505	+	0.699630i	$0.246652\pi$
$107$			149.721i			1.39926i	−0.714505	+	0.699630i	$0.753348\pi$
$108$		0			0
$109$		−	157.255i		−	1.44270i	−0.692568	−	0.721352i	$-0.743521\pi$
$109$		−	157.255i		−	1.44270i	0.692568	−	0.721352i	$-0.256479\pi$
$110$		0			0
$111$			133.056i			1.19871i
$112$		0			0
$113$			34.7000i			0.307080i	0.988142	+	0.153540i	$0.0490674\pi$
$113$			34.7000i			0.307080i	−0.988142	+	0.153540i	$0.950933\pi$
$114$		0			0
$115$	−39.6700	+	32.7306i	−0.344957	+	0.284614i
$116$		0			0
$117$	−260.824			−2.22926
$118$		0			0
$119$	−91.0899			−0.765461
$120$		0			0
$121$	−87.7863			−0.725507
$122$		0			0
$123$	314.583			2.55759
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	−142.267			−1.12021			−0.560107	−	0.828420i	$-0.689240\pi$
$127$	−142.267			−1.12021			−0.560107	+	0.828420i	$0.689240\pi$
$128$		0			0
$129$		−	267.505i		−	2.07368i
$130$		0			0
$131$	−6.26810			−0.0478481			−0.0239240	−	0.999714i	$-0.507616\pi$
$131$	−6.26810			−0.0478481			−0.0239240	+	0.999714i	$0.507616\pi$
$132$		0			0
$133$	−123.596			−0.929291
$134$		0			0
$135$		−	25.8278i		−	0.191317i
$136$		0			0
$137$		−	76.6062i		−	0.559170i	−0.960121	−	0.279585i	$-0.909803\pi$
$137$		−	76.6062i		−	0.559170i	0.960121	−	0.279585i	$-0.0901968\pi$
$138$		0			0
$139$	71.9853			0.517880			0.258940	−	0.965893i	$-0.416627\pi$
$139$	71.9853			0.517880			0.258940	+	0.965893i	$0.416627\pi$
$140$		0			0
$141$	−159.004			−1.12769
$142$		0			0
$143$		−	326.353i		−	2.28219i
$144$		0			0
$145$			52.1502i			0.359657i
$146$		0			0
$147$	73.2657			0.498406
$148$		0			0
$149$		−	55.2401i		−	0.370739i	−0.982669	−	0.185370i	$-0.940652\pi$
$149$		−	55.2401i		−	0.370739i	0.982669	−	0.185370i	$-0.0593482\pi$
$150$		0			0
$151$	−107.049			−0.708934			−0.354467	−	0.935069i	$-0.615338\pi$
$151$	−107.049			−0.708934			−0.354467	+	0.935069i	$0.615338\pi$
$152$		0			0
$153$		−	183.568i		−	1.19979i
$154$		0			0
$155$			116.641i			0.752526i
$156$		0			0
$157$			184.159i			1.17298i	0.809955	+	0.586492i	$0.199492\pi$
$157$			184.159i			1.17298i	−0.809955	+	0.586492i	$0.800508\pi$
$158$		0			0
$159$		−	252.907i		−	1.59061i
$160$		0			0
$161$	−101.662	+	83.8788i	−0.631444	+	0.520986i
$162$		0			0
$163$	−149.086			−0.914640			−0.457320	−	0.889302i	$-0.651191\pi$
$163$	−149.086			−0.914640			−0.457320	+	0.889302i	$0.651191\pi$
$164$		0			0
$165$	146.461			0.887642
$166$		0			0
$167$	42.2896			0.253231			0.126616	−	0.991952i	$-0.459589\pi$
$167$	42.2896			0.253231			0.126616	+	0.991952i	$0.459589\pi$
$168$		0			0
$169$	341.121			2.01847
$170$		0			0
$171$		−	249.075i		−	1.45658i
$172$		0			0
$173$	134.739			0.778838			0.389419	−	0.921061i	$-0.372676\pi$
$173$	134.739			0.778838			0.389419	+	0.921061i	$0.372676\pi$
$174$		0			0
$175$			28.6519i			0.163725i
$176$		0			0
$177$	−157.680			−0.890848
$178$		0			0
$179$	−17.6581			−0.0986485			−0.0493242	−	0.998783i	$-0.515707\pi$
$179$	−17.6581			−0.0986485			−0.0493242	+	0.998783i	$0.515707\pi$
$180$		0			0
$181$			44.8426i			0.247749i	0.992298	+	0.123875i	$0.0395321\pi$
$181$			44.8426i			0.247749i	−0.992298	+	0.123875i	$0.960468\pi$
$182$		0			0
$183$			160.611i			0.877653i
$184$		0			0
$185$	65.6349			0.354783
$186$		0			0
$187$	229.688			1.22828
$188$		0			0
$189$		−	66.1891i		−	0.350207i
$190$		0			0
$191$		−	126.936i		−	0.664584i	−0.943177	−	0.332292i	$-0.892178\pi$
$191$		−	126.936i		−	0.664584i	0.943177	−	0.332292i	$-0.107822\pi$
$192$		0			0
$193$	−239.282			−1.23980			−0.619901	−	0.784680i	$-0.712827\pi$
$193$	−239.282			−1.23980			−0.619901	+	0.784680i	$0.712827\pi$
$194$		0			0
$195$			228.933i			1.17401i
$196$		0			0
$197$	−139.910			−0.710203			−0.355102	−	0.934828i	$-0.615554\pi$
$197$	−139.910			−0.710203			−0.355102	+	0.934828i	$0.615554\pi$
$198$		0			0
$199$		−	106.506i		−	0.535207i	−0.963529	−	0.267603i	$-0.913768\pi$
$199$		−	106.506i		−	0.535207i	0.963529	−	0.267603i	$-0.0862317\pi$
$200$		0			0
$201$		−	503.849i		−	2.50671i
$202$		0			0
$203$			133.646i			0.658352i
$204$		0			0
$205$		−	155.180i		−	0.756974i
$206$		0			0
$207$	−169.036	−	204.874i	−0.816599	−	0.989732i
$208$		0			0
$209$	311.653			1.49116
$210$		0			0
$211$	131.377			0.622638			0.311319	−	0.950305i	$-0.399229\pi$
$211$	131.377			0.622638			0.311319	+	0.950305i	$0.399229\pi$
$212$		0			0
$213$	429.152			2.01480
$214$		0			0
$215$	−131.957			−0.613751
$216$		0			0
$217$			298.918i			1.37750i
$218$		0			0
$219$	40.2001			0.183562
$220$		0			0
$221$			359.024i			1.62454i
$222$		0			0
$223$	152.786			0.685140			0.342570	−	0.939492i	$-0.388703\pi$
$223$	152.786			0.685140			0.342570	+	0.939492i	$0.388703\pi$
$224$		0			0
$225$	−57.7405			−0.256625
$226$		0			0
$227$			76.9755i			0.339099i	0.985522	+	0.169550i	$0.0542313\pi$
$227$			76.9755i			0.339099i	−0.985522	+	0.169550i	$0.945769\pi$
$228$		0			0
$229$		−	183.105i		−	0.799586i	−0.916605	−	0.399793i	$-0.869082\pi$
$229$		−	183.105i		−	0.799586i	0.916605	−	0.399793i	$-0.130918\pi$
$230$		0			0
$231$	375.336			1.62483
$232$		0			0
$233$	−98.8182			−0.424112			−0.212056	−	0.977257i	$-0.568016\pi$
$233$	−98.8182			−0.424112			−0.212056	+	0.977257i	$0.568016\pi$
$234$		0			0
$235$			78.4347i			0.333765i
$236$		0			0
$237$			394.226i			1.66340i
$238$		0			0
$239$	354.489			1.48322			0.741610	−	0.670832i	$-0.234063\pi$
$239$	354.489			1.48322			0.741610	+	0.670832i	$0.234063\pi$
$240$		0			0
$241$			371.085i			1.53977i	0.638182	+	0.769885i	$0.279687\pi$
$241$			371.085i			1.53977i	−0.638182	+	0.769885i	$0.720313\pi$
$242$		0			0
$243$	−337.741			−1.38988
$244$		0			0
$245$		−	36.1410i		−	0.147514i
$246$		0			0
$247$			487.143i			1.97224i
$248$		0			0
$249$		−	697.495i		−	2.80118i
$250$		0			0
$251$		−	357.639i		−	1.42486i	−0.701744	−	0.712429i	$-0.747595\pi$
$251$		−	357.639i		−	1.42486i	0.701744	−	0.712429i	$-0.252405\pi$
$252$		0			0
$253$	256.347	−	211.505i	1.01323	−	0.835987i
$254$		0			0
$255$	−161.123			−0.631855
$256$		0			0
$257$	127.799			0.497272			0.248636	−	0.968597i	$-0.420018\pi$
$257$	127.799			0.497272			0.248636	+	0.968597i	$0.420018\pi$
$258$		0			0
$259$	168.203			0.649432
$260$		0			0
$261$	−269.328			−1.03191
$262$		0			0
$263$		−	271.488i		−	1.03227i	−0.856506	−	0.516137i	$-0.827370\pi$
$263$		−	271.488i		−	1.03227i	0.856506	−	0.516137i	$-0.172630\pi$
$264$		0			0
$265$	−124.756			−0.470776
$266$		0			0
$267$			692.815i			2.59481i
$268$		0			0
$269$	238.833			0.887855			0.443928	−	0.896063i	$-0.353585\pi$
$269$	238.833			0.887855			0.443928	+	0.896063i	$0.353585\pi$
$270$		0			0
$271$	−115.745			−0.427103			−0.213551	−	0.976932i	$-0.568503\pi$
$271$	−115.745			−0.427103			−0.213551	+	0.976932i	$0.568503\pi$
$272$		0			0
$273$			586.687i			2.14904i
$274$		0			0
$275$		−	72.2472i		−	0.262717i
$276$		0			0
$277$	26.2892			0.0949067			0.0474534	−	0.998873i	$-0.484889\pi$
$277$	26.2892			0.0949067			0.0474534	+	0.998873i	$0.484889\pi$
$278$		0			0
$279$	−602.391			−2.15911
$280$		0			0
$281$			170.570i			0.607010i	0.952830	+	0.303505i	$0.0981570\pi$
$281$			170.570i			0.607010i	−0.952830	+	0.303505i	$0.901843\pi$
$282$		0			0
$283$			136.360i			0.481836i	0.970545	+	0.240918i	$0.0774485\pi$
$283$			136.360i			0.481836i	−0.970545	+	0.240918i	$0.922551\pi$
$284$		0			0
$285$	−218.621			−0.767090
$286$		0			0
$287$		−	397.680i		−	1.38564i
$288$		0			0
$289$	36.3186			0.125670
$290$		0			0
$291$		−	64.4721i		−	0.221554i
$292$		0			0
$293$		−	175.814i		−	0.600049i	−0.953931	−	0.300025i	$-0.903005\pi$
$293$		−	175.814i		−	0.600049i	0.953931	−	0.300025i	$-0.0969949\pi$
$294$		0			0
$295$			77.7815i			0.263666i
$296$		0			0
$297$			166.899i			0.561950i
$298$		0			0
$299$	330.602	+	400.695i	1.10569	+	1.34012i
$300$		0			0
$301$	−338.166			−1.12347
$302$		0			0
$303$	606.509			2.00168
$304$		0			0
$305$	79.2270			0.259761
$306$		0			0
$307$	45.6453			0.148682			0.0743409	−	0.997233i	$-0.476315\pi$
$307$	45.6453			0.148682			0.0743409	+	0.997233i	$0.476315\pi$
$308$		0			0
$309$			141.027i			0.456398i
$310$		0			0
$311$	−369.398			−1.18778			−0.593888	−	0.804548i	$-0.702408\pi$
$311$	−369.398			−1.18778			−0.593888	+	0.804548i	$0.702408\pi$
$312$		0			0
$313$		−	176.338i		−	0.563380i	−0.959505	−	0.281690i	$-0.909105\pi$
$313$		−	176.338i		−	0.563380i	0.959505	−	0.281690i	$-0.0908950\pi$
$314$		0			0
$315$	−147.972			−0.469752
$316$		0			0
$317$	−190.147			−0.599831			−0.299916	−	0.953966i	$-0.596959\pi$
$317$	−190.147			−0.599831			−0.299916	+	0.953966i	$0.596959\pi$
$318$		0			0
$319$		−	336.994i		−	1.05641i
$320$		0			0
$321$			678.685i			2.11428i
$322$		0			0
$323$	−342.852			−1.06146
$324$		0			0
$325$	112.929			0.347475
$326$		0			0
$327$		−	712.836i		−	2.17993i
$328$		0			0
$329$			201.005i			0.610957i
$330$		0			0
$331$	−153.515			−0.463792			−0.231896	−	0.972741i	$-0.574493\pi$
$331$	−153.515			−0.463792			−0.231896	+	0.972741i	$0.574493\pi$
$332$		0			0
$333$			338.969i			1.01793i
$334$		0			0
$335$	−248.542			−0.741916
$336$		0			0
$337$		−	545.713i		−	1.61933i	−0.586895	−	0.809663i	$-0.699650\pi$
$337$		−	545.713i		−	1.61933i	0.586895	−	0.809663i	$-0.300350\pi$
$338$		0			0
$339$			157.295i			0.463998i
$340$		0			0
$341$		−	753.736i		−	2.21037i
$342$		0			0
$343$		−	373.407i		−	1.08865i
$344$		0			0
$345$	−179.824	+	148.368i	−0.521229	+	0.430052i
$346$		0			0
$347$	−536.006			−1.54469			−0.772343	−	0.635206i	$-0.780915\pi$
$347$	−536.006			−1.54469			−0.772343	+	0.635206i	$0.780915\pi$
$348$		0			0
$349$	327.902			0.939546			0.469773	−	0.882787i	$-0.344336\pi$
$349$	327.902			0.939546			0.469773	+	0.882787i	$0.344336\pi$
$350$		0			0
$351$	−260.879			−0.743246
$352$		0			0
$353$	−136.555			−0.386841			−0.193420	−	0.981116i	$-0.561958\pi$
$353$	−136.555			−0.386841			−0.193420	+	0.981116i	$0.561958\pi$
$354$		0			0
$355$		−	211.695i		−	0.596323i
$356$		0			0
$357$	−412.911			−1.15661
$358$		0			0
$359$		−	199.326i		−	0.555226i	−0.960693	−	0.277613i	$-0.910457\pi$
$359$		−	199.326i		−	0.555226i	0.960693	−	0.277613i	$-0.0895432\pi$
$360$		0			0
$361$	−104.200			−0.288644
$362$		0			0
$363$	−397.936			−1.09624
$364$		0			0
$365$		−	19.8301i		−	0.0543291i
$366$		0			0
$367$			451.248i			1.22956i	0.788700	+	0.614779i	$0.210755\pi$
$367$			451.248i			1.22956i	−0.788700	+	0.614779i	$0.789245\pi$
$368$		0			0
$369$	801.421			2.17187
$370$		0			0
$371$	−319.711			−0.861756
$372$		0			0
$373$		−	566.051i		−	1.51756i	−0.651345	−	0.758782i	$-0.725795\pi$
$373$		−	566.051i		−	1.51756i	0.651345	−	0.758782i	$-0.274205\pi$
$374$		0			0
$375$			50.6805i			0.135148i
$376$		0			0
$377$	526.754			1.39723
$378$		0			0
$379$		−	548.345i		−	1.44682i	−0.690419	−	0.723410i	$-0.742574\pi$
$379$		−	548.345i		−	1.44682i	0.690419	−	0.723410i	$-0.257426\pi$
$380$		0			0
$381$	−644.897			−1.69264
$382$		0			0
$383$			275.822i			0.720161i	0.932921	+	0.360080i	$0.117251\pi$
$383$			275.822i			0.720161i	−0.932921	+	0.360080i	$0.882749\pi$
$384$		0			0
$385$		−	185.148i		−	0.480904i
$386$		0			0
$387$		−	681.485i		−	1.76094i
$388$		0			0
$389$		−	19.2474i		−	0.0494791i	−0.999694	−	0.0247395i	$-0.992124\pi$
$389$		−	19.2474i		−	0.0494791i	0.999694	−	0.0247395i	$-0.00787564\pi$
$390$		0			0
$391$	−282.010	+	232.678i	−0.721252	+	0.595085i
$392$		0			0
$393$	−28.4133			−0.0722984
$394$		0			0
$395$	194.466			0.492320
$396$		0			0
$397$	345.136			0.869360			0.434680	−	0.900585i	$-0.356861\pi$
$397$	345.136			0.869360			0.434680	+	0.900585i	$0.356861\pi$
$398$		0			0
$399$	−560.260			−1.40416
$400$		0			0
$401$			493.627i			1.23099i	0.788141	+	0.615495i	$0.211044\pi$
$401$			493.627i			1.23099i	−0.788141	+	0.615495i	$0.788956\pi$
$402$		0			0
$403$	1178.16			2.92348
$404$		0			0
$405$			115.323i			0.284749i
$406$		0			0
$407$	−424.132			−1.04209
$408$		0			0
$409$	−435.568			−1.06496			−0.532479	−	0.846443i	$-0.678740\pi$
$409$	−435.568			−1.06496			−0.532479	+	0.846443i	$0.678740\pi$
$410$		0			0
$411$		−	347.256i		−	0.844905i
$412$		0			0
$413$			199.331i			0.482642i
$414$		0			0
$415$	−344.065			−0.829071
$416$		0			0
$417$	326.309			0.782516
$418$		0			0
$419$		−	655.748i		−	1.56503i	−0.622631	−	0.782515i	$-0.713936\pi$
$419$		−	655.748i		−	1.56503i	0.622631	−	0.782515i	$-0.286064\pi$
$420$		0			0
$421$			136.328i			0.323818i	0.986806	+	0.161909i	$0.0517652\pi$
$421$			136.328i			0.323818i	−0.986806	+	0.161909i	$0.948235\pi$
$422$		0			0
$423$	−405.073			−0.957621
$424$		0			0
$425$			79.4798i			0.187011i
$426$		0			0
$427$	203.035			0.475493
$428$		0			0
$429$		−	1479.36i		−	3.44839i
$430$		0			0
$431$			246.087i			0.570969i	0.958383	+	0.285484i	$0.0921544\pi$
$431$			246.087i			0.570969i	−0.958383	+	0.285484i	$0.907846\pi$
$432$		0			0
$433$			582.571i			1.34543i	0.739902	+	0.672715i	$0.234872\pi$
$433$			582.571i			1.34543i	−0.739902	+	0.672715i	$0.765128\pi$
$434$		0			0
$435$			236.397i			0.543441i
$436$		0			0
$437$	−382.646	+	315.710i	−0.875620	+	0.722449i
$438$		0			0
$439$	708.320			1.61348			0.806742	−	0.590903i	$-0.201229\pi$
$439$	708.320			1.61348			0.806742	+	0.590903i	$0.201229\pi$
$440$		0			0
$441$	186.649			0.423240
$442$		0			0
$443$	−338.574			−0.764274			−0.382137	−	0.924106i	$-0.624812\pi$
$443$	−338.574			−0.764274			−0.382137	+	0.924106i	$0.624812\pi$
$444$		0			0
$445$	341.756			0.767991
$446$		0			0
$447$		−	250.404i		−	0.560187i
$448$		0			0
$449$	276.872			0.616641			0.308320	−	0.951283i	$-0.400233\pi$
$449$	276.872			0.616641			0.308320	+	0.951283i	$0.400233\pi$
$450$		0			0
$451$			1002.77i			2.22344i
$452$		0			0
$453$	−485.253			−1.07120
$454$		0			0
$455$	289.405			0.636054
$456$		0			0
$457$			641.187i			1.40304i	0.712652	+	0.701518i	$0.247494\pi$
$457$			641.187i			1.40304i	−0.712652	+	0.701518i	$0.752506\pi$
$458$		0			0
$459$		−	183.607i		−	0.400016i
$460$		0			0
$461$	204.088			0.442707			0.221354	−	0.975194i	$-0.428952\pi$
$461$	204.088			0.442707			0.221354	+	0.975194i	$0.428952\pi$
$462$		0			0
$463$	596.319			1.28795			0.643973	−	0.765048i	$-0.277285\pi$
$463$	596.319			1.28795			0.643973	+	0.765048i	$0.277285\pi$
$464$		0			0
$465$			528.736i			1.13707i
$466$		0			0
$467$		−	144.242i		−	0.308870i	−0.988003	−	0.154435i	$-0.950644\pi$
$467$		−	144.242i		−	0.308870i	0.988003	−	0.154435i	$-0.0493558\pi$
$468$		0			0
$469$	−636.940			−1.35808
$470$		0			0
$471$			834.791i			1.77238i
$472$		0			0
$473$	852.701			1.80275
$474$		0			0
$475$			107.843i			0.227037i
$476$		0			0
$477$		−	644.296i		−	1.35073i
$478$		0			0
$479$			153.818i			0.321124i	0.987026	+	0.160562i	$0.0513307\pi$
$479$			153.818i			0.321124i	−0.987026	+	0.160562i	$0.948669\pi$
$480$		0			0
$481$		−	662.959i		−	1.37829i
$482$		0			0
$483$	−460.836	+	380.223i	−0.954112	+	0.787211i
$484$		0			0
$485$	−31.8032			−0.0655736
$486$		0			0
$487$	612.573			1.25785			0.628925	−	0.777466i	$-0.283495\pi$
$487$	612.573			1.25785			0.628925	+	0.777466i	$0.283495\pi$
$488$		0			0
$489$	−675.808			−1.38202
$490$		0			0
$491$	612.246			1.24694			0.623468	−	0.781849i	$-0.285723\pi$
$491$	612.246			1.24694			0.623468	+	0.781849i	$0.285723\pi$
$492$		0			0
$493$			370.730i			0.751988i
$494$		0			0
$495$	373.118			0.753775
$496$		0			0
$497$		−	542.511i		−	1.09157i
$498$		0			0
$499$	639.084			1.28073			0.640364	−	0.768071i	$-0.278783\pi$
$499$	639.084			1.28073			0.640364	+	0.768071i	$0.278783\pi$
$500$		0			0
$501$	191.699			0.382632
$502$		0			0
$503$		−	797.548i		−	1.58558i	−0.609494	−	0.792791i	$-0.708627\pi$
$503$		−	797.548i		−	1.58558i	0.609494	−	0.792791i	$-0.291373\pi$
$504$		0			0
$505$		−	299.183i		−	0.592441i
$506$		0			0
$507$	1546.30			3.04991
$508$		0			0
$509$	−380.000			−0.746563			−0.373281	−	0.927718i	$-0.621767\pi$
$509$	−380.000			−0.746563			−0.373281	+	0.927718i	$0.621767\pi$
$510$		0			0
$511$		−	50.8188i		−	0.0994497i
$512$		0			0
$513$		−	249.128i		−	0.485630i
$514$		0			0
$515$	69.5667			0.135081
$516$		0			0
$517$		−	506.844i		−	0.980356i
$518$		0			0
$519$	610.772			1.17683
$520$		0			0
$521$			82.2165i			0.157805i	0.996882	+	0.0789026i	$0.0251416\pi$
$521$			82.2165i			0.157805i	−0.996882	+	0.0789026i	$0.974858\pi$
$522$		0			0
$523$			529.301i			1.01205i	0.862519	+	0.506024i	$0.168885\pi$
$523$			529.301i			1.01205i	−0.862519	+	0.506024i	$0.831115\pi$
$524$		0			0
$525$			129.879i			0.247389i
$526$		0			0
$527$			829.191i			1.57342i
$528$		0			0
$529$	−100.484	+	519.369i	−0.189950	+	0.981794i
$530$		0			0
$531$	−401.700			−0.756497
$532$		0			0
$533$	−1567.43			−2.94076
$534$		0			0
$535$	334.786			0.625768
$536$		0			0
$537$	−80.0441			−0.149058
$538$		0			0
$539$			233.542i			0.433288i
$540$		0			0
$541$	1077.12			1.99097			0.995487	−	0.0948980i	$-0.0302525\pi$
$541$	1077.12			1.99097			0.995487	+	0.0948980i	$0.0302525\pi$
$542$		0			0
$543$			203.272i			0.374349i
$544$		0			0
$545$	−351.632			−0.645197
$546$		0			0
$547$	−538.450			−0.984369			−0.492184	−	0.870491i	$-0.663801\pi$
$547$	−538.450			−0.984369			−0.492184	+	0.870491i	$0.663801\pi$
$548$		0			0
$549$			409.165i			0.745292i
$550$		0			0
$551$			503.027i			0.912934i
$552$		0			0
$553$	498.360			0.901193
$554$		0			0
$555$	297.523			0.536077
$556$		0			0
$557$		−	543.116i		−	0.975074i	−0.873102	−	0.487537i	$-0.837895\pi$
$557$		−	543.116i		−	0.975074i	0.873102	−	0.487537i	$-0.162105\pi$
$558$		0			0
$559$			1332.85i			2.38435i
$560$		0			0
$561$	1041.17			1.85593
$562$		0			0
$563$			263.998i			0.468912i	0.972127	+	0.234456i	$0.0753310\pi$
$563$			263.998i			0.468912i	−0.972127	+	0.234456i	$0.924669\pi$
$564$		0			0
$565$	77.5917			0.137330
$566$		0			0
$567$			295.540i			0.521235i
$568$		0			0
$569$		−	75.5139i		−	0.132713i	−0.997796	−	0.0663567i	$-0.978862\pi$
$569$		−	75.5139i		−	0.132713i	0.997796	−	0.0663567i	$-0.0211375\pi$
$570$		0			0
$571$			252.684i			0.442529i	0.975214	+	0.221264i	$0.0710184\pi$
$571$			252.684i			0.442529i	−0.975214	+	0.221264i	$0.928982\pi$
$572$		0			0
$573$		−	575.399i		−	1.00419i
$574$		0			0
$575$	73.1878	+	88.7048i	0.127283	+	0.154269i
$576$		0			0
$577$	443.113			0.767961			0.383980	−	0.923341i	$-0.374553\pi$
$577$	443.113			0.767961			0.383980	+	0.923341i	$0.374553\pi$
$578$		0			0
$579$	−1084.66			−1.87334
$580$		0			0
$581$	−881.736			−1.51762
$582$		0			0
$583$	806.168			1.38279
$584$		0			0
$585$			583.220i			0.996957i
$586$		0			0
$587$	−946.434			−1.61232			−0.806162	−	0.591695i	$-0.798459\pi$
$587$	−946.434			−1.61232			−0.806162	+	0.591695i	$0.798459\pi$
$588$		0			0
$589$			1125.09i			1.91017i
$590$		0			0
$591$	−634.212			−1.07312
$592$		0			0
$593$	−349.409			−0.589222			−0.294611	−	0.955617i	$-0.595190\pi$
$593$	−349.409			−0.589222			−0.294611	+	0.955617i	$0.595190\pi$
$594$		0			0
$595$			203.683i			0.342325i
$596$		0			0
$597$		−	482.793i		−	0.808698i
$598$		0			0
$599$	−448.083			−0.748052			−0.374026	−	0.927418i	$-0.622023\pi$
$599$	−448.083			−0.748052			−0.374026	+	0.927418i	$0.622023\pi$
$600$		0			0
$601$	−467.984			−0.778676			−0.389338	−	0.921095i	$-0.627296\pi$
$601$	−467.984			−0.778676			−0.389338	+	0.921095i	$0.627296\pi$
$602$		0			0
$603$		−	1283.59i		−	2.12867i
$604$		0			0
$605$			196.296i			0.324457i
$606$		0			0
$607$	−666.697			−1.09835			−0.549174	−	0.835708i	$-0.685058\pi$
$607$	−666.697			−1.09835			−0.549174	+	0.835708i	$0.685058\pi$
$608$		0			0
$609$			605.815i			0.994771i
$610$		0			0
$611$	792.246			1.29664
$612$		0			0
$613$			482.025i			0.786338i	0.919466	+	0.393169i	$0.128621\pi$
$613$			482.025i			0.786338i	−0.919466	+	0.393169i	$0.871379\pi$
$614$		0			0
$615$		−	703.430i		−	1.14379i
$616$		0			0
$617$			137.235i			0.222422i	0.993797	+	0.111211i	$0.0354730\pi$
$617$			137.235i			0.222422i	−0.993797	+	0.111211i	$0.964527\pi$
$618$		0			0
$619$		−	470.924i		−	0.760783i	−0.924826	−	0.380391i	$-0.875789\pi$
$619$		−	470.924i		−	0.760783i	0.924826	−	0.380391i	$-0.124211\pi$
$620$		0			0
$621$	−169.072	−	204.918i	−0.272258	−	0.329981i
$622$		0			0
$623$	875.820			1.40581
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	1412.72			2.25315
$628$		0			0
$629$	466.591			0.741798
$630$		0			0
$631$		−	791.573i		−	1.25447i	−0.778829	−	0.627237i	$-0.784186\pi$
$631$		−	791.573i		−	1.25447i	0.778829	−	0.627237i	$-0.215814\pi$
$632$		0			0
$633$	595.531			0.940807
$634$		0			0
$635$			318.119i			0.500975i
$636$		0			0
$637$	−365.049			−0.573076
$638$		0			0
$639$	1093.29			1.71094
$640$		0			0
$641$			587.063i			0.915855i	0.888990	+	0.457927i	$0.151408\pi$
$641$			587.063i			0.915855i	−0.888990	+	0.457927i	$0.848592\pi$
$642$		0			0
$643$		−	1071.84i		−	1.66693i	−0.552573	−	0.833464i	$-0.686354\pi$
$643$		−	1071.84i		−	1.66693i	0.552573	−	0.833464i	$-0.313646\pi$
$644$		0			0
$645$	−598.159			−0.927378
$646$		0			0
$647$	−276.415			−0.427226			−0.213613	−	0.976918i	$-0.568523\pi$
$647$	−276.415			−0.427226			−0.213613	+	0.976918i	$0.568523\pi$
$648$		0			0
$649$		−	502.623i		−	0.774457i
$650$		0			0
$651$			1354.99i			2.08140i
$652$		0			0
$653$	248.340			0.380307			0.190153	−	0.981754i	$-0.439101\pi$
$653$	248.340			0.380307			0.190153	+	0.981754i	$0.439101\pi$
$654$		0			0
$655$			14.0159i			0.0213983i
$656$		0			0
$657$	102.412			0.155878
$658$		0			0
$659$		−	211.399i		−	0.320787i	−0.987053	−	0.160394i	$-0.948724\pi$
$659$		−	211.399i		−	0.320787i	0.987053	−	0.160394i	$-0.0512763\pi$
$660$		0			0
$661$		−	843.142i		−	1.27556i	−0.770221	−	0.637778i	$-0.779854\pi$
$661$		−	843.142i		−	1.27556i	0.770221	−	0.637778i	$-0.220146\pi$
$662$		0			0
$663$			1627.46i			2.45469i
$664$		0			0
$665$			276.369i			0.415592i
$666$		0			0
$667$	341.381	+	413.760i	0.511816	+	0.620329i
$668$		0			0
$669$	692.580			1.03525
$670$		0			0
$671$	−511.964			−0.762986
$672$		0			0
$673$	1117.23			1.66008			0.830041	−	0.557703i	$-0.188317\pi$
$673$	1117.23			1.66008			0.830041	+	0.557703i	$0.188317\pi$
$674$		0			0
$675$	−57.7528			−0.0855597
$676$		0			0
$677$		−	503.627i		−	0.743911i	−0.928251	−	0.371955i	$-0.878687\pi$
$677$		−	503.627i		−	0.743911i	0.928251	−	0.371955i	$-0.121313\pi$
$678$		0			0
$679$	−81.5022			−0.120033
$680$		0			0
$681$			348.930i			0.512379i
$682$		0			0
$683$	−1117.09			−1.63557			−0.817783	−	0.575526i	$-0.804797\pi$
$683$	−1117.09			−1.63557			−0.817783	+	0.575526i	$0.804797\pi$
$684$		0			0
$685$	−171.297			−0.250068
$686$		0			0
$687$		−	830.016i		−	1.20818i
$688$		0			0
$689$			1260.12i			1.82891i
$690$		0			0
$691$	373.797			0.540951			0.270476	−	0.962727i	$-0.412819\pi$
$691$	373.797			0.540951			0.270476	+	0.962727i	$0.412819\pi$
$692$		0			0
$693$	956.192			1.37979
$694$		0			0
$695$		−	160.964i		−	0.231603i
$696$		0			0
$697$		−	1103.16i		−	1.58272i
$698$		0			0
$699$	−447.943			−0.640834
$700$		0			0
$701$		−	392.530i		−	0.559957i	−0.960006	−	0.279978i	$-0.909673\pi$
$701$		−	392.530i		−	0.559957i	0.960006	−	0.279978i	$-0.0903273\pi$
$702$		0			0
$703$	633.096			0.900564
$704$		0			0
$705$			355.545i			0.504318i
$706$		0			0
$707$		−	766.717i		−	1.08446i
$708$		0			0
$709$			1069.06i			1.50784i	0.656964	+	0.753922i	$0.271840\pi$
$709$			1069.06i			1.50784i	−0.656964	+	0.753922i	$0.728160\pi$
$710$		0			0
$711$			1004.32i			1.41254i
$712$		0			0
$713$	763.549	+	925.433i	1.07090	+	1.29794i
$714$		0			0
$715$	−729.748			−1.02063
$716$		0			0
$717$	1606.90			2.24115
$718$		0			0
$719$	520.733			0.724247			0.362123	−	0.932130i	$-0.382052\pi$
$719$	520.733			0.724247			0.362123	+	0.932130i	$0.382052\pi$
$720$		0			0
$721$	178.279			0.247266
$722$		0			0
$723$			1682.13i			2.32659i
$724$		0			0
$725$	116.611			0.160843
$726$		0			0
$727$		−	149.011i		−	0.204967i	−0.994735	−	0.102483i	$-0.967321\pi$
$727$		−	149.011i		−	0.204967i	0.994735	−	0.102483i	$-0.0326788\pi$
$728$		0			0
$729$	−1066.81			−1.46339
$730$		0			0
$731$	−938.064			−1.28326
$732$		0			0
$733$		−	161.168i		−	0.219875i	−0.993939	−	0.109937i	$-0.964935\pi$
$733$		−	161.168i		−	0.219875i	0.993939	−	0.109937i	$-0.0350650\pi$
$734$		0			0
$735$		−	163.827i		−	0.222894i
$736$		0			0
$737$	1606.07			2.17921
$738$		0			0
$739$	722.812			0.978095			0.489048	−	0.872257i	$-0.337344\pi$
$739$	722.812			0.978095			0.489048	+	0.872257i	$0.337344\pi$
$740$		0			0
$741$			2208.22i			2.98006i
$742$		0			0
$743$			585.251i			0.787687i	0.919178	+	0.393843i	$0.128855\pi$
$743$			585.251i			0.787687i	−0.919178	+	0.393843i	$0.871145\pi$
$744$		0			0
$745$	−123.521			−0.165800
$746$		0			0
$747$		−	1776.91i		−	2.37873i
$748$		0			0
$749$	857.958			1.14547
$750$		0			0
$751$		−	248.757i		−	0.331234i	−0.986190	−	0.165617i	$-0.947038\pi$
$751$		−	248.757i		−	0.331234i	0.986190	−	0.165617i	$-0.0529615\pi$
$752$		0			0
$753$		−	1621.18i		−	2.15296i
$754$		0			0
$755$			239.369i			0.317045i
$756$		0			0
$757$			147.079i			0.194292i	0.995270	+	0.0971459i	$0.0309714\pi$
$757$			147.079i			0.194292i	−0.995270	+	0.0971459i	$0.969029\pi$
$758$		0			0
$759$	1162.02	−	958.751i	1.53099	−	1.26318i
$760$		0			0
$761$	−715.969			−0.940826			−0.470413	−	0.882446i	$-0.655895\pi$
$761$	−715.969			−0.940826			−0.470413	+	0.882446i	$0.655895\pi$
$762$		0			0
$763$	−901.130			−1.18104
$764$		0			0
$765$	−410.471			−0.536563
$766$		0			0
$767$	785.648			1.02431
$768$		0			0
$769$			563.688i			0.733014i	0.930415	+	0.366507i	$0.119446\pi$
$769$			563.688i			0.733014i	−0.930415	+	0.366507i	$0.880554\pi$
$770$		0			0
$771$	579.313			0.751379
$772$		0			0
$773$		−	776.912i		−	1.00506i	−0.864560	−	0.502530i	$-0.832403\pi$
$773$		−	776.912i		−	1.00506i	0.864560	−	0.502530i	$-0.167597\pi$
$774$		0			0
$775$	260.818			0.336540
$776$		0			0
$777$	762.463			0.981291
$778$		0			0
$779$		−	1496.82i		−	1.92147i
$780$		0			0
$781$			1367.97i			1.75156i
$782$		0			0
$783$	−269.385			−0.344043
$784$		0			0
$785$	411.791			0.524575
$786$		0			0
$787$		−	700.182i		−	0.889685i	−0.895609	−	0.444843i	$-0.853260\pi$
$787$		−	700.182i		−	0.889685i	0.895609	−	0.444843i	$-0.146740\pi$
$788$		0			0
$789$		−	1230.66i		−	1.55977i
$790$		0			0
$791$	198.844			0.251384
$792$		0			0
$793$		−	800.249i		−	1.00914i
$794$		0			0
$795$	−565.517			−0.711342
$796$		0			0
$797$			973.433i			1.22137i	0.791873	+	0.610686i	$0.209106\pi$
$797$			973.433i			1.22137i	−0.791873	+	0.610686i	$0.790894\pi$
$798$		0			0
$799$			557.583i			0.697852i
$800$		0			0
$801$			1764.99i			2.20348i
$802$		0			0
$803$			128.142i			0.159579i
$804$		0			0
$805$	187.559	+	227.324i	0.232992	+	0.282390i
$806$		0			0
$807$	1082.63			1.34155
$808$		0			0
$809$	437.494			0.540784			0.270392	−	0.962750i	$-0.412847\pi$
$809$	437.494			0.540784			0.270392	+	0.962750i	$0.412847\pi$
$810$		0			0
$811$	−1277.85			−1.57564			−0.787821	−	0.615904i	$-0.788791\pi$
$811$	−1277.85			−1.57564			−0.787821	+	0.615904i	$0.788791\pi$
$812$		0			0
$813$	−524.672			−0.645352
$814$		0			0
$815$			333.367i			0.409039i
$816$		0			0
$817$	−1272.82			−1.55792
$818$		0			0
$819$			1494.62i			1.82493i
$820$		0			0
$821$	592.870			0.722131			0.361066	−	0.932540i	$-0.382413\pi$
$821$	592.870			0.722131			0.361066	+	0.932540i	$0.382413\pi$
$822$		0			0
$823$	107.071			0.130099			0.0650494	−	0.997882i	$-0.479279\pi$
$823$	107.071			0.130099			0.0650494	+	0.997882i	$0.479279\pi$
$824$		0			0
$825$		−	327.497i		−	0.396966i
$826$		0			0
$827$			700.568i			0.847119i	0.905868	+	0.423560i	$0.139220\pi$
$827$			700.568i			0.847119i	−0.905868	+	0.423560i	$0.860780\pi$
$828$		0			0
$829$	−1195.25			−1.44179			−0.720897	−	0.693042i	$-0.756270\pi$
$829$	−1195.25			−1.44179			−0.720897	+	0.693042i	$0.756270\pi$
$830$		0			0
$831$	119.169			0.143404
$832$		0			0
$833$		−	256.922i		−	0.308430i
$834$		0			0
$835$		−	94.5624i		−	0.113248i
$836$		0			0
$837$	−602.519			−0.719856
$838$		0			0
$839$			173.900i			0.207270i	0.994615	+	0.103635i	$0.0330474\pi$
$839$			173.900i			0.207270i	−0.994615	+	0.103635i	$0.966953\pi$
$840$		0			0
$841$	−297.071			−0.353236
$842$		0			0
$843$			773.194i			0.917193i
$844$		0			0
$845$		−	762.771i		−	0.902687i
$846$		0			0
$847$			503.049i			0.593919i
$848$		0			0
$849$			618.118i			0.728055i
$850$		0			0
$851$	520.747	−	429.654i	0.611924	−	0.504881i
$852$		0			0
$853$	1443.06			1.69174			0.845872	−	0.533386i	$-0.179081\pi$
$853$	1443.06			1.69174			0.845872	+	0.533386i	$0.179081\pi$
$854$		0			0
$855$	−556.949			−0.651403
$856$		0			0
$857$	368.637			0.430148			0.215074	−	0.976598i	$-0.431001\pi$
$857$	368.637			0.430148			0.215074	+	0.976598i	$0.431001\pi$
$858$		0			0
$859$	7.48252			0.00871073			0.00435537	−	0.999991i	$-0.498614\pi$
$859$	7.48252			0.00871073			0.00435537	+	0.999991i	$0.498614\pi$
$860$		0			0
$861$		−	1802.68i		−	2.09371i
$862$		0			0
$863$	304.740			0.353117			0.176558	−	0.984290i	$-0.443504\pi$
$863$	304.740			0.353117			0.176558	+	0.984290i	$0.443504\pi$
$864$		0			0
$865$		−	301.286i		−	0.348307i
$866$		0			0
$867$	164.632			0.189887
$868$		0			0
$869$	−1256.64			−1.44608
$870$		0			0
$871$			2510.45i			2.88226i
$872$		0			0
$873$		−	164.247i		−	0.188141i
$874$		0			0
$875$	64.0676			0.0732201
$876$		0			0
$877$	−188.312			−0.214723			−0.107361	−	0.994220i	$-0.534240\pi$
$877$	−188.312			−0.214723			−0.107361	+	0.994220i	$0.534240\pi$
$878$		0			0
$879$		−	796.967i		−	0.906674i
$880$		0			0
$881$		−	213.922i		−	0.242817i	−0.992603	−	0.121409i	$-0.961259\pi$
$881$		−	213.922i		−	0.242817i	0.992603	−	0.121409i	$-0.0387411\pi$
$882$		0			0
$883$	727.655			0.824071			0.412036	−	0.911168i	$-0.364818\pi$
$883$	727.655			0.824071			0.412036	+	0.911168i	$0.364818\pi$
$884$		0			0
$885$			352.584i			0.398400i
$886$		0			0
$887$	926.692			1.04475			0.522374	−	0.852716i	$-0.325046\pi$
$887$	926.692			1.04475			0.522374	+	0.852716i	$0.325046\pi$
$888$		0			0
$889$			815.245i			0.917036i
$890$		0			0
$891$		−	745.219i		−	0.836385i
$892$		0			0
$893$			756.560i			0.847211i
$894$		0			0
$895$			39.4847i			0.0441169i
$896$		0			0
$897$	1498.62	+	1816.35i	1.67070	+	2.02492i
$898$		0			0
$899$	1216.58			1.35325
$900$		0			0
$901$	−886.873			−0.984321
$902$		0			0
$903$	−1532.91			−1.69757
$904$		0			0
$905$	100.271			0.110797
$906$		0			0
$907$		−	509.932i		−	0.562218i	−0.959676	−	0.281109i	$-0.909298\pi$
$907$		−	509.932i		−	0.562218i	0.959676	−	0.281109i	$-0.0907023\pi$
$908$		0			0
$909$	1545.12			1.69980
$910$		0			0
$911$			241.322i			0.264898i	0.991190	+	0.132449i	$0.0422841\pi$
$911$			241.322i			0.264898i	−0.991190	+	0.132449i	$0.957716\pi$
$912$		0			0
$913$	2223.34			2.43520
$914$		0			0
$915$	359.136			0.392498
$916$		0			0
$917$			35.9186i			0.0391697i
$918$		0			0
$919$		−	489.612i		−	0.532766i	−0.963867	−	0.266383i	$-0.914171\pi$
$919$		−	489.612i		−	0.532766i	0.963867	−	0.266383i	$-0.0858286\pi$
$920$		0			0
$921$	206.910			0.224658
$922$		0			0
$923$	−2138.27			−2.31665
$924$		0			0
$925$		−	146.764i		−	0.158664i
$926$		0			0
$927$			359.275i			0.387568i
$928$		0			0
$929$	−337.438			−0.363227			−0.181613	−	0.983370i	$-0.558132\pi$
$929$	−337.438			−0.363227			−0.181613	+	0.983370i	$0.558132\pi$
$930$		0			0
$931$		−	348.606i		−	0.374442i
$932$		0			0
$933$	−1674.48			−1.79473
$934$		0			0
$935$		−	513.597i		−	0.549302i
$936$		0			0
$937$			578.923i			0.617847i	0.951087	+	0.308923i	$0.0999687\pi$
$937$			578.923i			0.617847i	−0.951087	+	0.308923i	$0.900031\pi$
$938$		0			0
$939$		−	799.340i		−	0.851267i
$940$		0			0
$941$			685.415i			0.728390i	0.931323	+	0.364195i	$0.118656\pi$
$941$			685.415i			0.728390i	−0.931323	+	0.364195i	$0.881344\pi$
$942$		0			0
$943$	−1015.82	−	1231.20i	−1.07723	−	1.30562i
$944$		0			0
$945$	−148.003			−0.156617
$946$		0			0
$947$	603.328			0.637094			0.318547	−	0.947907i	$-0.396805\pi$
$947$	603.328			0.637094			0.318547	+	0.947907i	$0.396805\pi$
$948$		0			0
$949$	−200.298			−0.211063
$950$		0			0
$951$	−861.935			−0.906346
$952$		0			0
$953$		−	898.707i		−	0.943030i	−0.881858	−	0.471515i	$-0.843707\pi$
$953$		−	898.707i		−	0.943030i	0.881858	−	0.471515i	$-0.156293\pi$
$954$		0			0
$955$	−283.837			−0.297211
$956$		0			0
$957$		−	1527.59i		−	1.59623i
$958$		0			0
$959$	−438.983			−0.457751
$960$		0			0
$961$	1760.05			1.83147
$962$		0			0
$963$			1728.99i			1.79542i
$964$		0			0
$965$			535.050i			0.554456i
$966$		0			0
$967$	−62.7154			−0.0648556			−0.0324278	−	0.999474i	$-0.510324\pi$
$967$	−62.7154			−0.0648556			−0.0324278	+	0.999474i	$0.510324\pi$
$968$		0			0
$969$	−1554.15			−1.60387
$970$		0			0
$971$		−	1127.41i		−	1.16108i	−0.814232	−	0.580540i	$-0.802842\pi$
$971$		−	1127.41i		−	1.16108i	0.814232	−	0.580540i	$-0.197158\pi$
$972$		0			0
$973$		−	412.503i		−	0.423950i
$974$		0			0
$975$	511.909			0.525035
$976$		0			0
$977$		−	709.445i		−	0.726147i	−0.931761	−	0.363073i	$-0.881727\pi$
$977$		−	709.445i		−	0.726147i	0.931761	−	0.363073i	$-0.118273\pi$
$978$		0			0
$979$	−2208.42			−2.25580
$980$		0			0
$981$		−	1815.99i		−	1.85117i
$982$		0			0
$983$		−	409.376i		−	0.416456i	−0.978080	−	0.208228i	$-0.933230\pi$
$983$		−	409.376i		−	0.416456i	0.978080	−	0.208228i	$-0.0667696\pi$
$984$		0			0
$985$			312.848i			0.317612i
$986$		0			0
$987$			911.156i			0.923157i
$988$		0			0
$989$	−1046.94	+	863.803i	−1.05859	+	0.873410i
$990$		0			0
$991$	−192.336			−0.194083			−0.0970414	−	0.995280i	$-0.530938\pi$
$991$	−192.336			−0.194083			−0.0970414	+	0.995280i	$0.530938\pi$
$992$		0			0
$993$	−695.884			−0.700790
$994$		0			0
$995$	−238.155			−0.239352
$996$		0			0
$997$	1426.03			1.43032			0.715160	−	0.698961i	$-0.246354\pi$
$997$	1426.03			1.43032			0.715160	+	0.698961i	$0.246354\pi$
$998$		0			0
$999$			339.041i			0.339381i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.k.c.321.13		16
4.3	odd	2		460.3.f.a.321.3	✓	16
12.11	even	2		4140.3.d.a.2161.14		16
20.3	even	4		2300.3.d.b.1149.23		32
20.7	even	4		2300.3.d.b.1149.10		32
20.19	odd	2		2300.3.f.e.1701.13		16
23.22	odd	2	inner	1840.3.k.c.321.14		16
92.91	even	2		460.3.f.a.321.4	yes	16
276.275	odd	2		4140.3.d.a.2161.3		16
460.183	odd	4		2300.3.d.b.1149.9		32
460.367	odd	4		2300.3.d.b.1149.24		32
460.459	even	2		2300.3.f.e.1701.14		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.3.f.a.321.3	✓	16	4.3	odd	2
460.3.f.a.321.4	yes	16	92.91	even	2
1840.3.k.c.321.13		16	1.1	even	1	trivial
1840.3.k.c.321.14		16	23.22	odd	2	inner
2300.3.d.b.1149.9		32	460.183	odd	4
2300.3.d.b.1149.10		32	20.7	even	4
2300.3.d.b.1149.23		32	20.3	even	4
2300.3.d.b.1149.24		32	460.367	odd	4
2300.3.f.e.1701.13		16	20.19	odd	2
2300.3.f.e.1701.14		16	460.459	even	2
4140.3.d.a.2161.3		16	276.275	odd	2
4140.3.d.a.2161.14		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)

\(f(q)\)	\(=\)	\(q+4.53300 q^{3} -2.23607i q^{5} -5.73038i q^{7} +11.5481 q^{9} +O(q^{10})\)	\(q+4.53300 q^{3} -2.23607i q^{5} -5.73038i q^{7} +11.5481 q^{9} +14.4494i q^{11} -22.5859 q^{13} -10.1361i q^{15} -15.8960i q^{17} -21.5685i q^{19} -25.9758i q^{21} +(-14.6376 - 17.7410i) q^{23} -5.00000 q^{25} +11.5506 q^{27} -23.3223 q^{29} -52.1637 q^{31} +65.4993i q^{33} -12.8135 q^{35} +29.3528i q^{37} -102.382 q^{39} +69.3985 q^{41} -59.0128i q^{43} -25.8223i q^{45} -35.0771 q^{47} +16.1627 q^{49} -72.0564i q^{51} -55.7924i q^{53} +32.3099 q^{55} -97.7701i q^{57} -34.7849 q^{59} +35.4314i q^{61} -66.1750i q^{63} +50.5035i q^{65} -111.151i q^{67} +(-66.3521 - 80.4198i) q^{69} +94.6728 q^{71} +8.86831 q^{73} -22.6650 q^{75} +82.8008 q^{77} +86.9680i q^{79} -51.5742 q^{81} -153.870i q^{83} -35.5444 q^{85} -105.720 q^{87} +152.838i q^{89} +129.426i q^{91} -236.458 q^{93} -48.2286 q^{95} -14.2228i q^{97} +166.864i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 64 q^{9}+O(q^{10}) \) 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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