LMFDB - Embedded newform 1840.3.k.e.321.10

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			321.10
Character	$\chi$	$=$	1840.321
Dual form			1840.3.k.e.321.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.03006 q^{3} +2.23607i q^{5} +6.21370i q^{7} +0.181242 q^{9} +O(q^{10})$	$q-3.03006 q^{3} +2.23607i q^{5} +6.21370i q^{7} +0.181242 q^{9} -20.6025i q^{11} -10.8920 q^{13} -6.77541i q^{15} +10.5685i q^{17} -8.71830i q^{19} -18.8279i q^{21} +(21.1485 - 9.04113i) q^{23} -5.00000 q^{25} +26.7213 q^{27} +5.68403 q^{29} +16.3448 q^{31} +62.4268i q^{33} -13.8943 q^{35} +34.1967i q^{37} +33.0034 q^{39} +45.6218 q^{41} -21.0039i q^{43} +0.405270i q^{45} -28.3001 q^{47} +10.3899 q^{49} -32.0233i q^{51} +62.5365i q^{53} +46.0687 q^{55} +26.4169i q^{57} -92.5030 q^{59} -60.0362i q^{61} +1.12619i q^{63} -24.3553i q^{65} -6.11342i q^{67} +(-64.0811 + 27.3951i) q^{69} +80.3085 q^{71} -129.608 q^{73} +15.1503 q^{75} +128.018 q^{77} +45.3382i q^{79} -82.5983 q^{81} -94.7894i q^{83} -23.6320 q^{85} -17.2229 q^{87} +10.9318i q^{89} -67.6796i q^{91} -49.5258 q^{93} +19.4947 q^{95} +54.0146i q^{97} -3.73405i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.03006			−1.01002			−0.505009	−	0.863114i	$-0.668511\pi$
$3$	−3.03006			−1.01002			−0.505009	+	0.863114i	$0.668511\pi$
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$			6.21370i			0.887672i	0.896108	+	0.443836i	$0.146383\pi$
$7$			6.21370i			0.887672i	−0.896108	+	0.443836i	$0.853617\pi$
$8$		0			0
$9$	0.181242			0.0201380
$10$		0			0
$11$		−	20.6025i		−	1.87296i	−0.350724	−	0.936479i	$-0.614065\pi$
$11$		−	20.6025i		−	1.87296i	0.350724	−	0.936479i	$-0.385935\pi$
$12$		0			0
$13$	−10.8920			−0.837846			−0.418923	−	0.908022i	$-0.637592\pi$
$13$	−10.8920			−0.837846			−0.418923	+	0.908022i	$0.637592\pi$
$14$		0			0
$15$		−	6.77541i		−	0.451694i
$16$		0			0
$17$			10.5685i			0.621679i	0.950462	+	0.310839i	$0.100610\pi$
$17$			10.5685i			0.621679i	−0.950462	+	0.310839i	$0.899390\pi$
$18$		0			0
$19$		−	8.71830i		−	0.458858i	−0.973325	−	0.229429i	$-0.926314\pi$
$19$		−	8.71830i		−	0.458858i	0.973325	−	0.229429i	$-0.0736858\pi$
$20$		0			0
$21$		−	18.8279i		−	0.896565i
$22$		0			0
$23$	21.1485	−	9.04113i	0.919499	−	0.393093i
$23$	21.1485	−	9.04113i	0.919499	−	0.393093i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	26.7213			0.989679
$28$		0			0
$29$	5.68403			0.196001			0.0980005	−	0.995186i	$-0.468755\pi$
$29$	5.68403			0.196001			0.0980005	+	0.995186i	$0.468755\pi$
$30$		0			0
$31$	16.3448			0.527253			0.263626	−	0.964625i	$-0.415081\pi$
$31$	16.3448			0.527253			0.263626	+	0.964625i	$0.415081\pi$
$32$		0			0
$33$			62.4268i			1.89172i
$34$		0			0
$35$	−13.8943			−0.396979
$36$		0			0
$37$			34.1967i			0.924234i	0.886819	+	0.462117i	$0.152910\pi$
$37$			34.1967i			0.924234i	−0.886819	+	0.462117i	$0.847090\pi$
$38$		0			0
$39$	33.0034			0.846240
$40$		0			0
$41$	45.6218			1.11273			0.556363	−	0.830939i	$-0.312196\pi$
$41$	45.6218			1.11273			0.556363	+	0.830939i	$0.312196\pi$
$42$		0			0
$43$		−	21.0039i		−	0.488464i	−0.969717	−	0.244232i	$-0.921464\pi$
$43$		−	21.0039i		−	0.488464i	0.969717	−	0.244232i	$-0.0785358\pi$
$44$		0			0
$45$			0.405270i			0.00900600i
$46$		0			0
$47$	−28.3001			−0.602130			−0.301065	−	0.953604i	$-0.597342\pi$
$47$	−28.3001			−0.602130			−0.301065	+	0.953604i	$0.597342\pi$
$48$		0			0
$49$	10.3899			0.212039
$50$		0			0
$51$		−	32.0233i		−	0.627907i
$52$		0			0
$53$			62.5365i			1.17993i	0.807428	+	0.589967i	$0.200859\pi$
$53$			62.5365i			1.17993i	−0.807428	+	0.589967i	$0.799141\pi$
$54$		0			0
$55$	46.0687			0.837612
$56$		0			0
$57$			26.4169i			0.463455i
$58$		0			0
$59$	−92.5030			−1.56785			−0.783924	−	0.620857i	$-0.786785\pi$
$59$	−92.5030			−1.56785			−0.783924	+	0.620857i	$0.786785\pi$
$60$		0			0
$61$		−	60.0362i		−	0.984199i	−0.870539	−	0.492100i	$-0.836230\pi$
$61$		−	60.0362i		−	0.984199i	0.870539	−	0.492100i	$-0.163770\pi$
$62$		0			0
$63$			1.12619i			0.0178760i
$64$		0			0
$65$		−	24.3553i		−	0.374696i
$66$		0			0
$67$		−	6.11342i		−	0.0912451i	−0.998959	−	0.0456225i	$-0.985473\pi$
$67$		−	6.11342i		−	0.0912451i	0.998959	−	0.0456225i	$-0.0145271\pi$
$68$		0			0
$69$	−64.0811	+	27.3951i	−0.928711	+	0.397031i
$70$		0			0
$71$	80.3085			1.13111			0.565553	−	0.824712i	$-0.308663\pi$
$71$	80.3085			1.13111			0.565553	+	0.824712i	$0.308663\pi$
$72$		0			0
$73$	−129.608			−1.77545			−0.887727	−	0.460369i	$-0.847717\pi$
$73$	−129.608			−1.77545			−0.887727	+	0.460369i	$0.847717\pi$
$74$		0			0
$75$	15.1503			0.202004
$76$		0			0
$77$	128.018			1.66257
$78$		0			0
$79$			45.3382i			0.573901i	0.957945	+	0.286951i	$0.0926416\pi$
$79$			45.3382i			0.573901i	−0.957945	+	0.286951i	$0.907358\pi$
$80$		0			0
$81$	−82.5983			−1.01973
$82$		0			0
$83$		−	94.7894i		−	1.14204i	−0.820936	−	0.571020i	$-0.806548\pi$
$83$		−	94.7894i		−	1.14204i	0.820936	−	0.571020i	$-0.193452\pi$
$84$		0			0
$85$	−23.6320			−0.278023
$86$		0			0
$87$	−17.2229			−0.197965
$88$		0			0
$89$			10.9318i			0.122829i	0.998112	+	0.0614146i	$0.0195612\pi$
$89$			10.9318i			0.122829i	−0.998112	+	0.0614146i	$0.980439\pi$
$90$		0			0
$91$		−	67.6796i		−	0.743732i
$92$		0			0
$93$	−49.5258			−0.532535
$94$		0			0
$95$	19.4947			0.205207
$96$		0			0
$97$			54.0146i			0.556852i	0.960458	+	0.278426i	$0.0898127\pi$
$97$			54.0146i			0.556852i	−0.960458	+	0.278426i	$0.910187\pi$
$98$		0			0
$99$		−	3.73405i		−	0.0377177i
$100$		0			0
$101$	14.3259			0.141840			0.0709201	−	0.997482i	$-0.477406\pi$
$101$	14.3259			0.141840			0.0709201	+	0.997482i	$0.477406\pi$
$102$		0			0
$103$			173.154i			1.68111i	0.541726	+	0.840555i	$0.317771\pi$
$103$			173.154i			1.68111i	−0.541726	+	0.840555i	$0.682229\pi$
$104$		0			0
$105$	42.1004			0.400956
$106$		0			0
$107$			17.9872i			0.168104i	0.996461	+	0.0840522i	$0.0267863\pi$
$107$			17.9872i			0.168104i	−0.996461	+	0.0840522i	$0.973214\pi$
$108$		0			0
$109$		−	12.4904i		−	0.114591i	−0.998357	−	0.0572954i	$-0.981752\pi$
$109$		−	12.4904i		−	0.114591i	0.998357	−	0.0572954i	$-0.0182477\pi$
$110$		0			0
$111$		−	103.618i		−	0.933494i
$112$		0			0
$113$		−	37.5305i		−	0.332128i	−0.986115	−	0.166064i	$-0.946894\pi$
$113$		−	37.5305i		−	0.332128i	0.986115	−	0.166064i	$-0.0531058\pi$
$114$		0			0
$115$	20.2166	+	47.2894i	0.175796	+	0.411212i
$116$		0			0
$117$	−1.97409			−0.0168726
$118$		0			0
$119$	−65.6698			−0.551847
$120$		0			0
$121$	−303.464			−2.50797
$122$		0			0
$123$	−138.237			−1.12387
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	−43.0292			−0.338813			−0.169406	−	0.985546i	$-0.554185\pi$
$127$	−43.0292			−0.338813			−0.169406	+	0.985546i	$0.554185\pi$
$128$		0			0
$129$			63.6431i			0.493358i
$130$		0			0
$131$	10.1368			0.0773800			0.0386900	−	0.999251i	$-0.487682\pi$
$131$	10.1368			0.0773800			0.0386900	+	0.999251i	$0.487682\pi$
$132$		0			0
$133$	54.1729			0.407315
$134$		0			0
$135$			59.7507i			0.442598i
$136$		0			0
$137$			168.598i			1.23064i	0.788277	+	0.615321i	$0.210974\pi$
$137$			168.598i			1.23064i	−0.788277	+	0.615321i	$0.789026\pi$
$138$		0			0
$139$	−260.818			−1.87639			−0.938193	−	0.346113i	$-0.887501\pi$
$139$	−260.818			−1.87639			−0.938193	+	0.346113i	$0.887501\pi$
$140$		0			0
$141$	85.7509			0.608163
$142$		0			0
$143$			224.403i			1.56925i
$144$		0			0
$145$			12.7099i			0.0876543i
$146$		0			0
$147$	−31.4820			−0.214163
$148$		0			0
$149$			68.9129i			0.462503i	0.972894	+	0.231251i	$0.0742820\pi$
$149$			68.9129i			0.462503i	−0.972894	+	0.231251i	$0.925718\pi$
$150$		0			0
$151$	−179.465			−1.18851			−0.594256	−	0.804276i	$-0.702553\pi$
$151$	−179.465			−1.18851			−0.594256	+	0.804276i	$0.702553\pi$
$152$		0			0
$153$			1.91547i			0.0125194i
$154$		0			0
$155$			36.5482i			0.235795i
$156$		0			0
$157$			217.297i			1.38406i	0.721869	+	0.692029i	$0.243283\pi$
$157$			217.297i			1.38406i	−0.721869	+	0.692029i	$0.756717\pi$
$158$		0			0
$159$		−	189.489i		−	1.19176i
$160$		0			0
$161$	56.1789	+	131.410i	0.348937	+	0.816213i
$162$		0			0
$163$	191.093			1.17235			0.586173	−	0.810186i	$-0.300634\pi$
$163$	191.093			1.17235			0.586173	+	0.810186i	$0.300634\pi$
$164$		0			0
$165$	−139.591			−0.846004
$166$		0			0
$167$	−19.4590			−0.116521			−0.0582606	−	0.998301i	$-0.518555\pi$
$167$	−19.4590			−0.116521			−0.0582606	+	0.998301i	$0.518555\pi$
$168$		0			0
$169$	−50.3643			−0.298014
$170$		0			0
$171$		−	1.58012i		−	0.00924050i
$172$		0			0
$173$	−92.4652			−0.534481			−0.267240	−	0.963630i	$-0.586112\pi$
$173$	−92.4652			−0.534481			−0.267240	+	0.963630i	$0.586112\pi$
$174$		0			0
$175$		−	31.0685i		−	0.177534i
$176$		0			0
$177$	280.289			1.58356
$178$		0			0
$179$	154.892			0.865320			0.432660	−	0.901557i	$-0.357575\pi$
$179$	154.892			0.865320			0.432660	+	0.901557i	$0.357575\pi$
$180$		0			0
$181$			220.070i			1.21586i	0.793991	+	0.607929i	$0.208000\pi$
$181$			220.070i			1.21586i	−0.793991	+	0.607929i	$0.792000\pi$
$182$		0			0
$183$			181.913i			0.994060i
$184$		0			0
$185$	−76.4661			−0.413330
$186$		0			0
$187$	217.739			1.16438
$188$		0			0
$189$			166.038i			0.878510i
$190$		0			0
$191$			302.866i			1.58568i	0.609428	+	0.792842i	$0.291399\pi$
$191$			302.866i			1.58568i	−0.609428	+	0.792842i	$0.708601\pi$
$192$		0			0
$193$	227.009			1.17621			0.588106	−	0.808784i	$-0.299874\pi$
$193$	227.009			1.17621			0.588106	+	0.808784i	$0.299874\pi$
$194$		0			0
$195$			73.7978i			0.378450i
$196$		0			0
$197$	245.467			1.24602			0.623011	−	0.782213i	$-0.285909\pi$
$197$	245.467			1.24602			0.623011	+	0.782213i	$0.285909\pi$
$198$		0			0
$199$			162.469i			0.816429i	0.912886	+	0.408215i	$0.133848\pi$
$199$			162.469i			0.816429i	−0.912886	+	0.408215i	$0.866152\pi$
$200$		0			0
$201$			18.5240i			0.0921593i
$202$		0			0
$203$			35.3188i			0.173984i
$204$		0			0
$205$			102.013i			0.497626i
$206$		0			0
$207$	3.83300	−	1.63864i	0.0185169	−	0.00791611i
$208$		0			0
$209$	−179.619			−0.859421
$210$		0			0
$211$	137.845			0.653292			0.326646	−	0.945147i	$-0.394081\pi$
$211$	137.845			0.653292			0.326646	+	0.945147i	$0.394081\pi$
$212$		0			0
$213$	−243.339			−1.14244
$214$		0			0
$215$	46.9662			0.218448
$216$		0			0
$217$			101.562i			0.468027i
$218$		0			0
$219$	392.720			1.79324
$220$		0			0
$221$		−	115.113i		−	0.520871i
$222$		0			0
$223$	76.6673			0.343799			0.171900	−	0.985114i	$-0.445010\pi$
$223$	76.6673			0.343799			0.171900	+	0.985114i	$0.445010\pi$
$224$		0			0
$225$	−0.906212			−0.00402761
$226$		0			0
$227$			148.845i			0.655705i	0.944729	+	0.327853i	$0.106325\pi$
$227$			148.845i			0.655705i	−0.944729	+	0.327853i	$0.893675\pi$
$228$		0			0
$229$		−	119.430i		−	0.521528i	−0.965403	−	0.260764i	$-0.916025\pi$
$229$		−	119.430i		−	0.521528i	0.965403	−	0.260764i	$-0.0839745\pi$
$230$		0			0
$231$	−387.902			−1.67923
$232$		0			0
$233$	−339.069			−1.45523			−0.727615	−	0.685986i	$-0.759371\pi$
$233$	−339.069			−1.45523			−0.727615	+	0.685986i	$0.759371\pi$
$234$		0			0
$235$		−	63.2810i		−	0.269281i
$236$		0			0
$237$		−	137.377i		−	0.579651i
$238$		0			0
$239$	−179.802			−0.752310			−0.376155	−	0.926557i	$-0.622754\pi$
$239$	−179.802			−0.752310			−0.376155	+	0.926557i	$0.622754\pi$
$240$		0			0
$241$			28.6351i			0.118818i	0.998234	+	0.0594089i	$0.0189216\pi$
$241$			28.6351i			0.118818i	−0.998234	+	0.0594089i	$0.981078\pi$
$242$		0			0
$243$	9.78561			0.0402700
$244$		0			0
$245$			23.2325i			0.0948267i
$246$		0			0
$247$			94.9597i			0.384452i
$248$		0			0
$249$			287.217i			1.15348i
$250$		0			0
$251$			466.392i			1.85814i	0.369910	+	0.929068i	$0.379389\pi$
$251$			466.392i			1.85814i	−0.369910	+	0.929068i	$0.620611\pi$
$252$		0			0
$253$	−186.270	−	435.712i	−0.736246	−	1.72218i
$254$		0			0
$255$	71.6062			0.280809
$256$		0			0
$257$	45.9172			0.178666			0.0893331	−	0.996002i	$-0.471526\pi$
$257$	45.9172			0.178666			0.0893331	+	0.996002i	$0.471526\pi$
$258$		0			0
$259$	−212.488			−0.820417
$260$		0			0
$261$	1.03019			0.00394707
$262$		0			0
$263$			119.146i			0.453025i	0.974008	+	0.226512i	$0.0727324\pi$
$263$			119.146i			0.453025i	−0.974008	+	0.226512i	$0.927268\pi$
$264$		0			0
$265$	−139.836			−0.527682
$266$		0			0
$267$		−	33.1240i		−	0.124060i
$268$		0			0
$269$	354.317			1.31716			0.658581	−	0.752510i	$-0.271157\pi$
$269$	354.317			1.31716			0.658581	+	0.752510i	$0.271157\pi$
$270$		0			0
$271$	−246.190			−0.908450			−0.454225	−	0.890887i	$-0.650084\pi$
$271$	−246.190			−0.908450			−0.454225	+	0.890887i	$0.650084\pi$
$272$		0			0
$273$			205.073i			0.751184i
$274$		0			0
$275$			103.013i			0.374591i
$276$		0			0
$277$	42.1468			0.152154			0.0760772	−	0.997102i	$-0.475760\pi$
$277$	42.1468			0.152154			0.0760772	+	0.997102i	$0.475760\pi$
$278$		0			0
$279$	2.96238			0.0106178
$280$		0			0
$281$			1.41091i			0.00502102i	0.999997	+	0.00251051i	$0.000799120\pi$
$281$			1.41091i			0.00502102i	−0.999997	+	0.00251051i	$0.999201\pi$
$282$		0			0
$283$		−	446.108i		−	1.57635i	−0.615449	−	0.788176i	$-0.711025\pi$
$283$		−	446.108i		−	1.57635i	0.615449	−	0.788176i	$-0.288975\pi$
$284$		0			0
$285$	−59.0701			−0.207263
$286$		0			0
$287$			283.480i			0.987736i
$288$		0			0
$289$	177.306			0.613515
$290$		0			0
$291$		−	163.667i		−	0.562431i
$292$		0			0
$293$			89.1768i			0.304358i	0.988353	+	0.152179i	$0.0486290\pi$
$293$			89.1768i			0.304358i	−0.988353	+	0.152179i	$0.951371\pi$
$294$		0			0
$295$		−	206.843i		−	0.701163i
$296$		0			0
$297$		−	550.527i		−	1.85363i
$298$		0			0
$299$	−230.349	+	98.4760i	−0.770399	+	0.329351i
$300$		0			0
$301$	130.512			0.433596
$302$		0			0
$303$	−43.4082			−0.143261
$304$		0			0
$305$	134.245			0.440147
$306$		0			0
$307$	−465.016			−1.51471			−0.757356	−	0.653002i	$-0.773509\pi$
$307$	−465.016			−1.51471			−0.757356	+	0.653002i	$0.773509\pi$
$308$		0			0
$309$		−	524.667i		−	1.69795i
$310$		0			0
$311$	−539.001			−1.73312			−0.866560	−	0.499072i	$-0.833674\pi$
$311$	−539.001			−1.73312			−0.866560	+	0.499072i	$0.833674\pi$
$312$		0			0
$313$			243.724i			0.778670i	0.921096	+	0.389335i	$0.127295\pi$
$313$			243.724i			0.778670i	−0.921096	+	0.389335i	$0.872705\pi$
$314$		0			0
$315$	−2.51823			−0.00799437
$316$		0			0
$317$	117.946			0.372071			0.186035	−	0.982543i	$-0.440436\pi$
$317$	117.946			0.372071			0.186035	+	0.982543i	$0.440436\pi$
$318$		0			0
$319$		−	117.105i		−	0.367101i
$320$		0			0
$321$		−	54.5022i		−	0.169789i
$322$		0			0
$323$	92.1397			0.285262
$324$		0			0
$325$	54.4600			0.167569
$326$		0			0
$327$			37.8466i			0.115739i
$328$		0			0
$329$		−	175.848i		−	0.534494i
$330$		0			0
$331$	−301.031			−0.909460			−0.454730	−	0.890629i	$-0.650264\pi$
$331$	−301.031			−0.909460			−0.454730	+	0.890629i	$0.650264\pi$
$332$		0			0
$333$			6.19788i			0.0186123i
$334$		0			0
$335$	13.6700			0.0408060
$336$		0			0
$337$			131.650i			0.390652i	0.980738	+	0.195326i	$0.0625765\pi$
$337$			131.650i			0.390652i	−0.980738	+	0.195326i	$0.937423\pi$
$338$		0			0
$339$			113.719i			0.335456i
$340$		0			0
$341$		−	336.745i		−	0.987522i
$342$		0			0
$343$			369.031i			1.07589i
$344$		0			0
$345$	−61.2574	−	143.290i	−0.177558	−	0.415332i
$346$		0			0
$347$	516.437			1.48829			0.744146	−	0.668017i	$-0.232857\pi$
$347$	516.437			1.48829			0.744146	+	0.668017i	$0.232857\pi$
$348$		0			0
$349$	−347.734			−0.996373			−0.498186	−	0.867070i	$-0.666000\pi$
$349$	−347.734			−0.996373			−0.498186	+	0.867070i	$0.666000\pi$
$350$		0			0
$351$	−291.049			−0.829199
$352$		0			0
$353$	−379.715			−1.07568			−0.537840	−	0.843047i	$-0.680759\pi$
$353$	−379.715			−1.07568			−0.537840	+	0.843047i	$0.680759\pi$
$354$		0			0
$355$			179.575i			0.505846i
$356$		0			0
$357$	198.983			0.557376
$358$		0			0
$359$			95.0757i			0.264835i	0.991194	+	0.132417i	$0.0422739\pi$
$359$			95.0757i			0.264835i	−0.991194	+	0.132417i	$0.957726\pi$
$360$		0			0
$361$	284.991			0.789450
$362$		0			0
$363$	919.514			2.53310
$364$		0			0
$365$		−	289.813i		−	0.794008i
$366$		0			0
$367$		−	327.970i		−	0.893651i	−0.894621	−	0.446825i	$-0.852555\pi$
$367$		−	327.970i		−	0.893651i	0.894621	−	0.446825i	$-0.147445\pi$
$368$		0			0
$369$	8.26860			0.0224081
$370$		0			0
$371$	−388.583			−1.04739
$372$		0			0
$373$		−	675.193i		−	1.81017i	−0.425232	−	0.905084i	$-0.639808\pi$
$373$		−	675.193i		−	1.81017i	0.425232	−	0.905084i	$-0.360192\pi$
$374$		0			0
$375$			33.8771i			0.0903388i
$376$		0			0
$377$	−61.9104			−0.164219
$378$		0			0
$379$			591.703i			1.56122i	0.625018	+	0.780610i	$0.285092\pi$
$379$			591.703i			1.56122i	−0.625018	+	0.780610i	$0.714908\pi$
$380$		0			0
$381$	130.381			0.342207
$382$		0			0
$383$			309.211i			0.807340i	0.914905	+	0.403670i	$0.132266\pi$
$383$			309.211i			0.807340i	−0.914905	+	0.403670i	$0.867734\pi$
$384$		0			0
$385$			286.257i			0.743525i
$386$		0			0
$387$		−	3.80680i		−	0.00983670i
$388$		0			0
$389$			76.6822i			0.197126i	0.995131	+	0.0985632i	$0.0314247\pi$
$389$			76.6822i			0.197126i	−0.995131	+	0.0985632i	$0.968575\pi$
$390$		0			0
$391$	95.5515	+	223.508i	0.244377	+	0.571633i
$392$		0			0
$393$	−30.7150			−0.0781552
$394$		0			0
$395$	−101.379			−0.256657
$396$		0			0
$397$	582.338			1.46685			0.733423	−	0.679773i	$-0.237922\pi$
$397$	582.338			1.46685			0.733423	+	0.679773i	$0.237922\pi$
$398$		0			0
$399$	−164.147			−0.411396
$400$		0			0
$401$			396.912i			0.989806i	0.868948	+	0.494903i	$0.164796\pi$
$401$			396.912i			0.989806i	−0.868948	+	0.494903i	$0.835204\pi$
$402$		0			0
$403$	−178.028			−0.441757
$404$		0			0
$405$		−	184.695i		−	0.456038i
$406$		0			0
$407$	704.538			1.73105
$408$		0			0
$409$	−173.058			−0.423126			−0.211563	−	0.977364i	$-0.567855\pi$
$409$	−173.058			−0.423126			−0.211563	+	0.977364i	$0.567855\pi$
$410$		0			0
$411$		−	510.861i		−	1.24297i
$412$		0			0
$413$		−	574.786i		−	1.39173i
$414$		0			0
$415$	211.956			0.510736
$416$		0			0
$417$	790.292			1.89518
$418$		0			0
$419$		−	716.510i		−	1.71005i	−0.518590	−	0.855023i	$-0.673543\pi$
$419$		−	716.510i		−	1.71005i	0.518590	−	0.855023i	$-0.326457\pi$
$420$		0			0
$421$			218.283i			0.518487i	0.965812	+	0.259243i	$0.0834732\pi$
$421$			218.283i			0.518487i	−0.965812	+	0.259243i	$0.916527\pi$
$422$		0			0
$423$	−5.12918			−0.0121257
$424$		0			0
$425$		−	52.8427i		−	0.124336i
$426$		0			0
$427$	373.047			0.873646
$428$		0			0
$429$		−	679.953i		−	1.58497i
$430$		0			0
$431$			178.804i			0.414857i	0.978250	+	0.207429i	$0.0665095\pi$
$431$			178.804i			0.414857i	−0.978250	+	0.207429i	$0.933490\pi$
$432$		0			0
$433$		−	219.392i		−	0.506679i	−0.967377	−	0.253340i	$-0.918471\pi$
$433$		−	219.392i		−	0.506679i	0.967377	−	0.253340i	$-0.0815290\pi$
$434$		0			0
$435$		−	38.5116i		−	0.0885325i
$436$		0			0
$437$	−78.8233	−	184.379i	−0.180374	−	0.421919i
$438$		0			0
$439$	−113.568			−0.258697			−0.129348	−	0.991599i	$-0.541289\pi$
$439$	−113.568			−0.258697			−0.129348	+	0.991599i	$0.541289\pi$
$440$		0			0
$441$	1.88309			0.00427005
$442$		0			0
$443$	−81.1026			−0.183076			−0.0915379	−	0.995802i	$-0.529178\pi$
$443$	−81.1026			−0.183076			−0.0915379	+	0.995802i	$0.529178\pi$
$444$		0			0
$445$	−24.4442			−0.0549309
$446$		0			0
$447$		−	208.810i		−	0.467137i
$448$		0			0
$449$	402.574			0.896602			0.448301	−	0.893883i	$-0.352029\pi$
$449$	402.574			0.896602			0.448301	+	0.893883i	$0.352029\pi$
$450$		0			0
$451$		−	939.924i		−	2.08409i
$452$		0			0
$453$	543.790			1.20042
$454$		0			0
$455$	151.336			0.332607
$456$		0			0
$457$			440.952i			0.964884i	0.875928	+	0.482442i	$0.160250\pi$
$457$			440.952i			0.964884i	−0.875928	+	0.482442i	$0.839750\pi$
$458$		0			0
$459$			282.405i			0.615262i
$460$		0			0
$461$	−389.075			−0.843981			−0.421991	−	0.906600i	$-0.638668\pi$
$461$	−389.075			−0.843981			−0.421991	+	0.906600i	$0.638668\pi$
$462$		0			0
$463$	677.231			1.46270			0.731350	−	0.682002i	$-0.238890\pi$
$463$	677.231			1.46270			0.731350	+	0.682002i	$0.238890\pi$
$464$		0			0
$465$		−	110.743i		−	0.238157i
$466$		0			0
$467$			751.553i			1.60932i	0.593735	+	0.804661i	$0.297653\pi$
$467$			751.553i			1.60932i	−0.593735	+	0.804661i	$0.702347\pi$
$468$		0			0
$469$	37.9870			0.0809957
$470$		0			0
$471$		−	658.423i		−	1.39793i
$472$		0			0
$473$	−432.734			−0.914872
$474$		0			0
$475$			43.5915i			0.0917716i
$476$		0			0
$477$			11.3343i			0.0237615i
$478$		0			0
$479$		−	736.665i		−	1.53792i	−0.639295	−	0.768962i	$-0.720774\pi$
$479$		−	736.665i		−	1.53792i	0.639295	−	0.768962i	$-0.279226\pi$
$480$		0			0
$481$		−	372.470i		−	0.774366i
$482$		0			0
$483$	−170.225	−	398.181i	−0.352433	−	0.824391i
$484$		0			0
$485$	−120.780			−0.249032
$486$		0			0
$487$	413.305			0.848676			0.424338	−	0.905504i	$-0.360507\pi$
$487$	413.305			0.848676			0.424338	+	0.905504i	$0.360507\pi$
$488$		0			0
$489$	−579.021			−1.18409
$490$		0			0
$491$	−810.534			−1.65078			−0.825391	−	0.564561i	$-0.809045\pi$
$491$	−810.534			−1.65078			−0.825391	+	0.564561i	$0.809045\pi$
$492$		0			0
$493$			60.0719i			0.121850i
$494$		0			0
$495$	8.34959			0.0168679
$496$		0			0
$497$			499.013i			1.00405i
$498$		0			0
$499$	786.846			1.57684			0.788422	−	0.615134i	$-0.210898\pi$
$499$	786.846			1.57684			0.788422	+	0.615134i	$0.210898\pi$
$500$		0			0
$501$	58.9620			0.117689
$502$		0			0
$503$		−	39.6272i		−	0.0787818i	−0.999224	−	0.0393909i	$-0.987458\pi$
$503$		−	39.6272i		−	0.0787818i	0.999224	−	0.0393909i	$-0.0125418\pi$
$504$		0			0
$505$			32.0336i			0.0634328i
$506$		0			0
$507$	152.607			0.301000
$508$		0			0
$509$	276.695			0.543605			0.271803	−	0.962353i	$-0.412380\pi$
$509$	276.695			0.543605			0.271803	+	0.962353i	$0.412380\pi$
$510$		0			0
$511$		−	805.347i		−	1.57602i
$512$		0			0
$513$		−	232.965i		−	0.454122i
$514$		0			0
$515$	−387.185			−0.751815
$516$		0			0
$517$			583.054i			1.12776i
$518$		0			0
$519$	280.175			0.539836
$520$		0			0
$521$			103.084i			0.197859i	0.995094	+	0.0989293i	$0.0315418\pi$
$521$			103.084i			0.197859i	−0.995094	+	0.0989293i	$0.968458\pi$
$522$		0			0
$523$		−	173.383i		−	0.331516i	−0.986166	−	0.165758i	$-0.946993\pi$
$523$		−	173.383i		−	0.331516i	0.986166	−	0.165758i	$-0.0530071\pi$
$524$		0			0
$525$			94.1393i			0.179313i
$526$		0			0
$527$			172.741i			0.327782i
$528$		0			0
$529$	365.516	−	382.412i	0.690956	−	0.722896i
$530$		0			0
$531$	−16.7655			−0.0315734
$532$		0			0
$533$	−496.912			−0.932294
$534$		0			0
$535$	−40.2205			−0.0751786
$536$		0			0
$537$	−469.332			−0.873989
$538$		0			0
$539$		−	214.058i		−	0.397140i
$540$		0			0
$541$	92.2168			0.170456			0.0852281	−	0.996361i	$-0.472838\pi$
$541$	92.2168			0.170456			0.0852281	+	0.996361i	$0.472838\pi$
$542$		0			0
$543$		−	666.826i		−	1.22804i
$544$		0			0
$545$	27.9294			0.0512466
$546$		0			0
$547$	138.930			0.253986			0.126993	−	0.991904i	$-0.459467\pi$
$547$	138.930			0.253986			0.126993	+	0.991904i	$0.459467\pi$
$548$		0			0
$549$		−	10.8811i		−	0.0198198i
$550$		0			0
$551$		−	49.5550i		−	0.0899366i
$552$		0			0
$553$	−281.718			−0.509436
$554$		0			0
$555$	231.697			0.417471
$556$		0			0
$557$			294.970i			0.529569i	0.964308	+	0.264785i	$0.0853008\pi$
$557$			294.970i			0.529569i	−0.964308	+	0.264785i	$0.914699\pi$
$558$		0			0
$559$			228.775i			0.409258i
$560$		0			0
$561$	−659.760			−1.17604
$562$		0			0
$563$		−	123.008i		−	0.218486i	−0.994015	−	0.109243i	$-0.965157\pi$
$563$		−	123.008i		−	0.218486i	0.994015	−	0.109243i	$-0.0348427\pi$
$564$		0			0
$565$	83.9207			0.148532
$566$		0			0
$567$		−	513.241i		−	0.905188i
$568$		0			0
$569$			928.439i			1.63170i	0.578262	+	0.815851i	$0.303731\pi$
$569$			928.439i			1.63170i	−0.578262	+	0.815851i	$0.696269\pi$
$570$		0			0
$571$		−	500.255i		−	0.876103i	−0.898950	−	0.438051i	$-0.855669\pi$
$571$		−	500.255i		−	0.876103i	0.898950	−	0.438051i	$-0.144331\pi$
$572$		0			0
$573$		−	917.700i		−	1.60157i
$574$		0			0
$575$	−105.742	+	45.2057i	−0.183900	+	0.0786185i
$576$		0			0
$577$	−2.41022			−0.00417716			−0.00208858	−	0.999998i	$-0.500665\pi$
$577$	−2.41022			−0.00417716			−0.00208858	+	0.999998i	$0.500665\pi$
$578$		0			0
$579$	−687.850			−1.18800
$580$		0			0
$581$	588.993			1.01376
$582$		0			0
$583$	1288.41			2.20997
$584$		0			0
$585$		−	4.41420i		−	0.00754565i
$586$		0			0
$587$	−616.864			−1.05088			−0.525438	−	0.850832i	$-0.676098\pi$
$587$	−616.864			−1.05088			−0.525438	+	0.850832i	$0.676098\pi$
$588$		0			0
$589$		−	142.499i		−	0.241934i
$590$		0			0
$591$	−743.777			−1.25851
$592$		0			0
$593$	−105.330			−0.177622			−0.0888108	−	0.996049i	$-0.528307\pi$
$593$	−105.330			−0.177622			−0.0888108	+	0.996049i	$0.528307\pi$
$594$		0			0
$595$		−	146.842i		−	0.246793i
$596$		0			0
$597$		−	492.292i		−	0.824609i
$598$		0			0
$599$	361.338			0.603235			0.301617	−	0.953429i	$-0.402474\pi$
$599$	361.338			0.603235			0.301617	+	0.953429i	$0.402474\pi$
$600$		0			0
$601$	809.360			1.34669			0.673345	−	0.739329i	$-0.264857\pi$
$601$	809.360			1.34669			0.673345	+	0.739329i	$0.264857\pi$
$602$		0			0
$603$		−	1.10801i		−	0.00183750i
$604$		0			0
$605$		−	678.567i		−	1.12160i
$606$		0			0
$607$	775.706			1.27793			0.638967	−	0.769234i	$-0.279362\pi$
$607$	775.706			1.27793			0.638967	+	0.769234i	$0.279362\pi$
$608$		0			0
$609$		−	107.018i		−	0.175728i
$610$		0			0
$611$	308.245			0.504492
$612$		0			0
$613$			957.447i			1.56190i	0.624591	+	0.780952i	$0.285266\pi$
$613$			957.447i			1.56190i	−0.624591	+	0.780952i	$0.714734\pi$
$614$		0			0
$615$		−	309.106i		−	0.502612i
$616$		0			0
$617$			833.518i			1.35092i	0.737396	+	0.675461i	$0.236055\pi$
$617$			833.518i			1.35092i	−0.737396	+	0.675461i	$0.763945\pi$
$618$		0			0
$619$		−	937.486i		−	1.51452i	−0.653115	−	0.757259i	$-0.726538\pi$
$619$		−	937.486i		−	1.51452i	0.653115	−	0.757259i	$-0.273462\pi$
$620$		0			0
$621$	565.115	−	241.591i	0.910009	−	0.389036i
$622$		0			0
$623$	−67.9269			−0.109032
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	544.256			0.868032
$628$		0			0
$629$	−361.409			−0.574577
$630$		0			0
$631$		−	179.539i		−	0.284531i	−0.989829	−	0.142265i	$-0.954561\pi$
$631$		−	179.539i		−	0.284531i	0.989829	−	0.142265i	$-0.0454387\pi$
$632$		0			0
$633$	−417.677			−0.659838
$634$		0			0
$635$		−	96.2162i		−	0.151522i
$636$		0			0
$637$	−113.167			−0.177656
$638$		0			0
$639$	14.5553			0.0227783
$640$		0			0
$641$			559.271i			0.872497i	0.899826	+	0.436249i	$0.143693\pi$
$641$			559.271i			0.872497i	−0.899826	+	0.436249i	$0.856307\pi$
$642$		0			0
$643$		−	192.835i		−	0.299898i	−0.988694	−	0.149949i	$-0.952089\pi$
$643$		−	192.835i		−	0.299898i	0.988694	−	0.149949i	$-0.0479110\pi$
$644$		0			0
$645$	−142.310			−0.220636
$646$		0			0
$647$	−635.856			−0.982777			−0.491388	−	0.870941i	$-0.663510\pi$
$647$	−635.856			−0.982777			−0.491388	+	0.870941i	$0.663510\pi$
$648$		0			0
$649$			1905.80i			2.93651i
$650$		0			0
$651$		−	307.738i		−	0.472716i
$652$		0			0
$653$	−361.501			−0.553600			−0.276800	−	0.960928i	$-0.589274\pi$
$653$	−361.501			−0.553600			−0.276800	+	0.960928i	$0.589274\pi$
$654$		0			0
$655$			22.6665i			0.0346054i
$656$		0			0
$657$	−23.4905			−0.0357542
$658$		0			0
$659$			1107.34i			1.68033i	0.542331	+	0.840165i	$0.317542\pi$
$659$			1107.34i			1.68033i	−0.542331	+	0.840165i	$0.682458\pi$
$660$		0			0
$661$			754.882i			1.14203i	0.820939	+	0.571015i	$0.193450\pi$
$661$			754.882i			1.14203i	−0.820939	+	0.571015i	$0.806550\pi$
$662$		0			0
$663$			348.797i			0.526090i
$664$		0			0
$665$			121.134i			0.182157i
$666$		0			0
$667$	120.208	−	51.3900i	0.180223	−	0.0770465i
$668$		0			0
$669$	−232.306			−0.347244
$670$		0			0
$671$	−1236.90			−1.84336
$672$		0			0
$673$	366.666			0.544823			0.272411	−	0.962181i	$-0.412179\pi$
$673$	366.666			0.544823			0.272411	+	0.962181i	$0.412179\pi$
$674$		0			0
$675$	−133.607			−0.197936
$676$		0			0
$677$		−	947.890i		−	1.40013i	−0.714078	−	0.700066i	$-0.753154\pi$
$677$		−	947.890i		−	1.40013i	0.714078	−	0.700066i	$-0.246846\pi$
$678$		0			0
$679$	−335.631			−0.494302
$680$		0			0
$681$		−	451.009i		−	0.662275i
$682$		0			0
$683$	20.8596			0.0305411			0.0152706	−	0.999883i	$-0.495139\pi$
$683$	20.8596			0.0305411			0.0152706	+	0.999883i	$0.495139\pi$
$684$		0			0
$685$	−376.997			−0.550360
$686$		0			0
$687$			361.880i			0.526753i
$688$		0			0
$689$		−	681.147i		−	0.988603i
$690$		0			0
$691$	−727.549			−1.05289			−0.526446	−	0.850209i	$-0.676476\pi$
$691$	−727.549			−1.05289			−0.526446	+	0.850209i	$0.676476\pi$
$692$		0			0
$693$	23.2023			0.0334809
$694$		0			0
$695$		−	583.206i		−	0.839145i
$696$		0			0
$697$			482.156i			0.691758i
$698$		0			0
$699$	1027.40			1.46981
$700$		0			0
$701$		−	561.470i		−	0.800956i	−0.916306	−	0.400478i	$-0.868844\pi$
$701$		−	561.470i		−	0.800956i	0.916306	−	0.400478i	$-0.131156\pi$
$702$		0			0
$703$	298.137			0.424092
$704$		0			0
$705$			191.745i			0.271979i
$706$		0			0
$707$			89.0166i			0.125907i
$708$		0			0
$709$			424.005i			0.598033i	0.954248	+	0.299016i	$0.0966585\pi$
$709$			424.005i			0.598033i	−0.954248	+	0.299016i	$0.903341\pi$
$710$		0			0
$711$			8.21720i			0.0115572i
$712$		0			0
$713$	345.668	−	147.776i	0.484808	−	0.207259i
$714$		0			0
$715$	−501.780			−0.701790
$716$		0			0
$717$	544.811			0.759848
$718$		0			0
$719$	700.701			0.974549			0.487275	−	0.873249i	$-0.337991\pi$
$719$	700.701			0.974549			0.487275	+	0.873249i	$0.337991\pi$
$720$		0			0
$721$	−1075.93			−1.49227
$722$		0			0
$723$		−	86.7659i		−	0.120008i
$724$		0			0
$725$	−28.4201			−0.0392002
$726$		0			0
$727$		−	529.531i		−	0.728379i	−0.931325	−	0.364189i	$-0.881346\pi$
$727$		−	529.531i		−	0.728379i	0.931325	−	0.364189i	$-0.118654\pi$
$728$		0			0
$729$	713.734			0.979059
$730$		0			0
$731$	221.981			0.303668
$732$		0			0
$733$			1190.95i			1.62476i	0.583128	+	0.812380i	$0.301829\pi$
$733$			1190.95i			1.62476i	−0.583128	+	0.812380i	$0.698171\pi$
$734$		0			0
$735$		−	70.3959i		−	0.0957767i
$736$		0			0
$737$	−125.952			−0.170898
$738$		0			0
$739$	74.6864			0.101064			0.0505321	−	0.998722i	$-0.483908\pi$
$739$	74.6864			0.101064			0.0505321	+	0.998722i	$0.483908\pi$
$740$		0			0
$741$		−	287.733i		−	0.388304i
$742$		0			0
$743$		−	636.616i		−	0.856818i	−0.903585	−	0.428409i	$-0.859074\pi$
$743$		−	636.616i		−	0.856818i	0.903585	−	0.428409i	$-0.140926\pi$
$744$		0			0
$745$	−154.094			−0.206838
$746$		0			0
$747$		−	17.1799i		−	0.0229985i
$748$		0			0
$749$	−111.767			−0.149222
$750$		0			0
$751$		−	335.655i		−	0.446944i	−0.974710	−	0.223472i	$-0.928261\pi$
$751$		−	335.655i		−	0.446944i	0.974710	−	0.223472i	$-0.0717391\pi$
$752$		0			0
$753$		−	1413.19i		−	1.87675i
$754$		0			0
$755$		−	401.297i		−	0.531519i
$756$		0			0
$757$		−	930.224i		−	1.22883i	−0.788983	−	0.614415i	$-0.789392\pi$
$757$		−	930.224i		−	1.22883i	0.788983	−	0.614415i	$-0.210608\pi$
$758$		0			0
$759$	564.409	+	1320.23i	0.743622	+	1.73944i
$760$		0			0
$761$	12.9551			0.0170237			0.00851186	−	0.999964i	$-0.497291\pi$
$761$	12.9551			0.0170237			0.00851186	+	0.999964i	$0.497291\pi$
$762$		0			0
$763$	77.6116			0.101719
$764$		0			0
$765$	−4.28311			−0.00559884
$766$		0			0
$767$	1007.54			1.31362
$768$		0			0
$769$		−	260.342i		−	0.338546i	−0.985569	−	0.169273i	$-0.945858\pi$
$769$		−	260.342i		−	0.338546i	0.985569	−	0.169273i	$-0.0541420\pi$
$770$		0			0
$771$	−139.132			−0.180456
$772$		0			0
$773$			1193.19i			1.54359i	0.635874	+	0.771793i	$0.280640\pi$
$773$			1193.19i			1.54359i	−0.635874	+	0.771793i	$0.719360\pi$
$774$		0			0
$775$	−81.7241			−0.105451
$776$		0			0
$777$	643.850			0.828636
$778$		0			0
$779$		−	397.744i		−	0.510583i
$780$		0			0
$781$		−	1654.56i		−	2.11851i
$782$		0			0
$783$	151.885			0.193978
$784$		0			0
$785$	−485.891			−0.618970
$786$		0			0
$787$			1417.92i			1.80168i	0.434156	+	0.900838i	$0.357047\pi$
$787$			1417.92i			1.80168i	−0.434156	+	0.900838i	$0.642953\pi$
$788$		0			0
$789$		−	361.018i		−	0.457564i
$790$		0			0
$791$	233.203			0.294821
$792$		0			0
$793$			653.914i			0.824608i
$794$		0			0
$795$	423.710			0.532969
$796$		0			0
$797$		−	424.377i		−	0.532468i	−0.963908	−	0.266234i	$-0.914221\pi$
$797$		−	424.377i		−	0.532468i	0.963908	−	0.266234i	$-0.0857794\pi$
$798$		0			0
$799$		−	299.091i		−	0.374331i
$800$		0			0
$801$			1.98130i			0.00247354i
$802$		0			0
$803$			2670.26i			3.32535i
$804$		0			0
$805$	−293.842	+	125.620i	−0.365022	+	0.156049i
$806$		0			0
$807$	−1073.60			−1.33036
$808$		0			0
$809$	1605.74			1.98485			0.992424	−	0.122858i	$-0.0392060\pi$
$809$	1605.74			1.98485			0.992424	+	0.122858i	$0.0392060\pi$
$810$		0			0
$811$	565.162			0.696870			0.348435	−	0.937333i	$-0.386713\pi$
$811$	565.162			0.696870			0.348435	+	0.937333i	$0.386713\pi$
$812$		0			0
$813$	745.970			0.917552
$814$		0			0
$815$			427.296i			0.524290i
$816$		0			0
$817$	−183.119			−0.224135
$818$		0			0
$819$		−	12.2664i		−	0.0149773i
$820$		0			0
$821$	−1042.03			−1.26922			−0.634608	−	0.772834i	$-0.718838\pi$
$821$	−1042.03			−1.26922			−0.634608	+	0.772834i	$0.718838\pi$
$822$		0			0
$823$	780.083			0.947853			0.473926	−	0.880564i	$-0.342836\pi$
$823$	780.083			0.947853			0.473926	+	0.880564i	$0.342836\pi$
$824$		0			0
$825$		−	312.134i		−	0.378344i
$826$		0			0
$827$		−	646.765i		−	0.782062i	−0.920377	−	0.391031i	$-0.872118\pi$
$827$		−	646.765i		−	0.782062i	0.920377	−	0.391031i	$-0.127882\pi$
$828$		0			0
$829$	−274.585			−0.331224			−0.165612	−	0.986191i	$-0.552960\pi$
$829$	−274.585			−0.331224			−0.165612	+	0.986191i	$0.552960\pi$
$830$		0			0
$831$	−127.707			−0.153679
$832$		0			0
$833$			109.806i			0.131820i
$834$		0			0
$835$		−	43.5118i		−	0.0521099i
$836$		0			0
$837$	436.756			0.521811
$838$		0			0
$839$			586.898i			0.699521i	0.936839	+	0.349760i	$0.113737\pi$
$839$			586.898i			0.699521i	−0.936839	+	0.349760i	$0.886263\pi$
$840$		0			0
$841$	−808.692			−0.961584
$842$		0			0
$843$		−	4.27512i		−	0.00507132i
$844$		0			0
$845$		−	112.618i		−	0.133276i
$846$		0			0
$847$		−	1885.64i		−	2.22625i
$848$		0			0
$849$			1351.73i			1.59215i
$850$		0			0
$851$	309.177	+	723.207i	0.363310	+	0.849832i
$852$		0			0
$853$	880.617			1.03238			0.516188	−	0.856475i	$-0.327350\pi$
$853$	880.617			1.03238			0.516188	+	0.856475i	$0.327350\pi$
$854$		0			0
$855$	3.53327			0.00413248
$856$		0			0
$857$	−1268.18			−1.47979			−0.739896	−	0.672721i	$-0.765125\pi$
$857$	−1268.18			−1.47979			−0.739896	+	0.672721i	$0.765125\pi$
$858$		0			0
$859$	396.640			0.461746			0.230873	−	0.972984i	$-0.425842\pi$
$859$	396.640			0.461746			0.230873	+	0.972984i	$0.425842\pi$
$860$		0			0
$861$		−	858.961i		−	0.997632i
$862$		0			0
$863$	−108.795			−0.126066			−0.0630329	−	0.998011i	$-0.520077\pi$
$863$	−108.795			−0.126066			−0.0630329	+	0.998011i	$0.520077\pi$
$864$		0			0
$865$		−	206.758i		−	0.239027i
$866$		0			0
$867$	−537.247			−0.619662
$868$		0			0
$869$	934.082			1.07489
$870$		0			0
$871$			66.5874i			0.0764493i
$872$		0			0
$873$			9.78974i			0.0112139i
$874$		0			0
$875$	69.4713			0.0793958
$876$		0			0
$877$	53.9219			0.0614845			0.0307422	−	0.999527i	$-0.490213\pi$
$877$	53.9219			0.0614845			0.0307422	+	0.999527i	$0.490213\pi$
$878$		0			0
$879$		−	270.211i		−	0.307407i
$880$		0			0
$881$			487.899i			0.553801i	0.960899	+	0.276901i	$0.0893073\pi$
$881$			487.899i			0.553801i	−0.960899	+	0.276901i	$0.910693\pi$
$882$		0			0
$883$	−1227.86			−1.39055			−0.695277	−	0.718742i	$-0.744718\pi$
$883$	−1227.86			−1.39055			−0.695277	+	0.718742i	$0.744718\pi$
$884$		0			0
$885$			626.746i			0.708188i
$886$		0			0
$887$	−880.193			−0.992326			−0.496163	−	0.868230i	$-0.665258\pi$
$887$	−880.193			−0.992326			−0.496163	+	0.868230i	$0.665258\pi$
$888$		0			0
$889$		−	267.371i		−	0.300754i
$890$		0			0
$891$			1701.73i			1.90992i
$892$		0			0
$893$			246.729i			0.276292i
$894$		0			0
$895$			346.350i			0.386983i
$896$		0			0
$897$	697.971	−	298.388i	0.778117	−	0.332651i
$898$		0			0
$899$	92.9045			0.103342
$900$		0			0
$901$	−660.919			−0.733540
$902$		0			0
$903$	−395.459			−0.437940
$904$		0			0
$905$	−492.092			−0.543749
$906$		0			0
$907$			256.796i			0.283127i	0.989929	+	0.141563i	$0.0452129\pi$
$907$			256.796i			0.283127i	−0.989929	+	0.141563i	$0.954787\pi$
$908$		0			0
$909$	2.59645			0.00285638
$910$		0			0
$911$		−	1649.93i		−	1.81112i	−0.424215	−	0.905562i	$-0.639450\pi$
$911$		−	1649.93i		−	1.81112i	0.424215	−	0.905562i	$-0.360550\pi$
$912$		0			0
$913$	−1952.90			−2.13899
$914$		0			0
$915$	−406.770			−0.444557
$916$		0			0
$917$			62.9869i			0.0686880i
$918$		0			0
$919$		−	1670.53i		−	1.81777i	−0.417046	−	0.908885i	$-0.636935\pi$
$919$		−	1670.53i		−	1.81777i	0.417046	−	0.908885i	$-0.363065\pi$
$920$		0			0
$921$	1409.03			1.52989
$922$		0			0
$923$	−874.721			−0.947693
$924$		0			0
$925$		−	170.983i		−	0.184847i
$926$		0			0
$927$			31.3829i			0.0338543i
$928$		0			0
$929$	308.143			0.331694			0.165847	−	0.986152i	$-0.446964\pi$
$929$	308.143			0.331694			0.165847	+	0.986152i	$0.446964\pi$
$930$		0			0
$931$		−	90.5823i		−	0.0972957i
$932$		0			0
$933$	1633.20			1.75048
$934$		0			0
$935$			486.878i			0.520726i
$936$		0			0
$937$			752.662i			0.803268i	0.915800	+	0.401634i	$0.131558\pi$
$937$			752.662i			0.803268i	−0.915800	+	0.401634i	$0.868442\pi$
$938$		0			0
$939$		−	738.496i		−	0.786471i
$940$		0			0
$941$		−	921.779i		−	0.979574i	−0.871842	−	0.489787i	$-0.837075\pi$
$941$		−	921.779i		−	0.979574i	0.871842	−	0.489787i	$-0.162925\pi$
$942$		0			0
$943$	964.831	−	412.473i	1.02315	−	0.437405i
$944$		0			0
$945$	−371.273			−0.392882
$946$		0			0
$947$	67.7919			0.0715860			0.0357930	−	0.999359i	$-0.488604\pi$
$947$	67.7919			0.0715860			0.0357930	+	0.999359i	$0.488604\pi$
$948$		0			0
$949$	1411.69			1.48756
$950$		0			0
$951$	−357.384			−0.375798
$952$		0			0
$953$			820.240i			0.860692i	0.902664	+	0.430346i	$0.141609\pi$
$953$			820.240i			0.860692i	−0.902664	+	0.430346i	$0.858391\pi$
$954$		0			0
$955$	−677.228			−0.709139
$956$		0			0
$957$			354.836i			0.370779i
$958$		0			0
$959$	−1047.62			−1.09241
$960$		0			0
$961$	−693.847			−0.722005
$962$		0			0
$963$			3.26004i			0.00338529i
$964$		0			0
$965$			507.608i			0.526018i
$966$		0			0
$967$	−1628.31			−1.68388			−0.841939	−	0.539573i	$-0.818586\pi$
$967$	−1628.31			−1.68388			−0.841939	+	0.539573i	$0.818586\pi$
$968$		0			0
$969$	−279.188			−0.288120
$970$		0			0
$971$			1796.85i			1.85052i	0.379338	+	0.925258i	$0.376152\pi$
$971$			1796.85i			1.85052i	−0.379338	+	0.925258i	$0.623848\pi$
$972$		0			0
$973$		−	1620.64i		−	1.66561i
$974$		0			0
$975$	−165.017			−0.169248
$976$		0			0
$977$			609.955i			0.624315i	0.950030	+	0.312157i	$0.101052\pi$
$977$			609.955i			0.624315i	−0.950030	+	0.312157i	$0.898948\pi$
$978$		0			0
$979$	225.223			0.230054
$980$		0			0
$981$		−	2.26379i		−	0.00230763i
$982$		0			0
$983$			1586.54i			1.61398i	0.590567	+	0.806988i	$0.298904\pi$
$983$			1586.54i			1.61398i	−0.590567	+	0.806988i	$0.701096\pi$
$984$		0			0
$985$			548.880i			0.557238i
$986$		0			0
$987$			532.831i			0.539849i
$988$		0			0
$989$	−189.899	−	444.201i	−0.192012	−	0.449142i
$990$		0			0
$991$	−678.849			−0.685014			−0.342507	−	0.939515i	$-0.611276\pi$
$991$	−678.849			−0.685014			−0.342507	+	0.939515i	$0.611276\pi$
$992$		0			0
$993$	912.142			0.918572
$994$		0			0
$995$	−363.293			−0.365118
$996$		0			0
$997$	1461.82			1.46622			0.733108	−	0.680113i	$-0.238069\pi$
$997$	1461.82			1.46622			0.733108	+	0.680113i	$0.238069\pi$
$998$		0			0
$999$			913.781i			0.914695i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.k.e.321.10		48
4.3	odd	2		920.3.k.a.321.40	yes	48
23.22	odd	2	inner	1840.3.k.e.321.9		48
92.91	even	2		920.3.k.a.321.39	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.3.k.a.321.39	✓	48	92.91	even	2
920.3.k.a.321.40	yes	48	4.3	odd	2
1840.3.k.e.321.9		48	23.22	odd	2	inner
1840.3.k.e.321.10		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)

\(f(q)\)	\(=\)	\(q-3.03006 q^{3} +2.23607i q^{5} +6.21370i q^{7} +0.181242 q^{9} +O(q^{10})\)	\(q-3.03006 q^{3} +2.23607i q^{5} +6.21370i q^{7} +0.181242 q^{9} -20.6025i q^{11} -10.8920 q^{13} -6.77541i q^{15} +10.5685i q^{17} -8.71830i q^{19} -18.8279i q^{21} +(21.1485 - 9.04113i) q^{23} -5.00000 q^{25} +26.7213 q^{27} +5.68403 q^{29} +16.3448 q^{31} +62.4268i q^{33} -13.8943 q^{35} +34.1967i q^{37} +33.0034 q^{39} +45.6218 q^{41} -21.0039i q^{43} +0.405270i q^{45} -28.3001 q^{47} +10.3899 q^{49} -32.0233i q^{51} +62.5365i q^{53} +46.0687 q^{55} +26.4169i q^{57} -92.5030 q^{59} -60.0362i q^{61} +1.12619i q^{63} -24.3553i q^{65} -6.11342i q^{67} +(-64.0811 + 27.3951i) q^{69} +80.3085 q^{71} -129.608 q^{73} +15.1503 q^{75} +128.018 q^{77} +45.3382i q^{79} -82.5983 q^{81} -94.7894i q^{83} -23.6320 q^{85} -17.2229 q^{87} +10.9318i q^{89} -67.6796i q^{91} -49.5258 q^{93} +19.4947 q^{95} +54.0146i q^{97} -3.73405i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 128 q^{9}+O(q^{10}) \) 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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