LMFDB - Embedded newform 480.4.b.b.431.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,4,Mod(431,480)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(480, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("480.431");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	480.b (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:	$28.3209168028$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.7
Character	$\chi$	$=$	480.431
Dual form			480.4.b.b.431.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.31357 - 4.00253i) q^{3} +5.00000 q^{5} -30.8728i q^{7} +(-5.04045 + 26.5253i) q^{9} +O(q^{10})$	\(q+(-3.31357 - 4.00253i) q^{3} +5.00000 q^{5} -30.8728i q^{7} +(-5.04045 + 26.5253i) q^{9} +45.8002i q^{11} +76.8279i q^{13} +(-16.5679 - 20.0126i) q^{15} -13.8879i q^{17} -45.8869 q^{19} +(-123.569 + 102.299i) q^{21} -74.3371 q^{23} +25.0000 q^{25} +(122.870 - 67.7191i) q^{27} +85.8680 q^{29} +227.728i q^{31} +(183.317 - 151.762i) q^{33} -154.364i q^{35} -5.58615i q^{37} +(307.506 - 254.575i) q^{39} +61.0663i q^{41} -65.1565 q^{43} +(-25.2023 + 132.627i) q^{45} +529.167 q^{47} -610.132 q^{49} +(-55.5867 + 46.0186i) q^{51} +22.7454 q^{53} +229.001i q^{55} +(152.050 + 183.664i) q^{57} +384.963i q^{59} +588.386i q^{61} +(818.912 + 155.613i) q^{63} +384.139i q^{65} -369.398 q^{67} +(246.321 + 297.536i) q^{69} +102.090 q^{71} +897.637 q^{73} +(-82.8394 - 100.063i) q^{75} +1413.98 q^{77} +1020.54i q^{79} +(-678.188 - 267.400i) q^{81} -588.935i q^{83} -69.4395i q^{85} +(-284.530 - 343.689i) q^{87} -1260.99i q^{89} +2371.89 q^{91} +(911.486 - 754.593i) q^{93} -229.435 q^{95} +936.527 q^{97} +(-1214.87 - 230.854i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 120 q^{5}+O(q^{10}) $ 24 * q + 120 * q^5	$ 24 q + 120 q^{5} - 12 q^{19} - 4 q^{21} - 228 q^{23} + 600 q^{25} - 132 q^{27} + 116 q^{33} + 656 q^{39} + 924 q^{47} - 816 q^{49} + 700 q^{51} + 528 q^{53} - 172 q^{57} - 476 q^{63} - 1632 q^{67} + 980 q^{69} - 216 q^{71} - 216 q^{73} + 152 q^{81} - 252 q^{87} + 1800 q^{91} - 60 q^{95} + 792 q^{97} + 1328 q^{99}+O(q^{100}) $ 24 * q + 120 * q^5 - 12 * q^19 - 4 * q^21 - 228 * q^23 + 600 * q^25 - 132 * q^27 + 116 * q^33 + 656 * q^39 + 924 * q^47 - 816 * q^49 + 700 * q^51 + 528 * q^53 - 172 * q^57 - 476 * q^63 - 1632 * q^67 + 980 * q^69 - 216 * q^71 - 216 * q^73 + 152 * q^81 - 252 * q^87 + 1800 * q^91 - 60 * q^95 + 792 * q^97 + 1328 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$421$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$	−3.31357	−	4.00253i	−0.637698	−	0.770287i
$3$	−3.31357	−	4.00253i	−0.637698	−	0.770287i
$4$		0			0
$5$	5.00000			0.447214
$5$	5.00000			0.447214
$6$		0			0
$7$		−	30.8728i		−	1.66698i	−0.552537	−	0.833488i	$-0.686340\pi$
$7$		−	30.8728i		−	1.66698i	0.552537	−	0.833488i	$-0.313660\pi$
$8$		0			0
$9$	−5.04045	+	26.5253i	−0.186683	+	0.982420i
$10$		0			0
$11$			45.8002i			1.25539i	0.778459	+	0.627695i	$0.216001\pi$
$11$			45.8002i			1.25539i	−0.778459	+	0.627695i	$0.783999\pi$
$12$		0			0
$13$			76.8279i			1.63909i	0.573012	+	0.819547i	$0.305775\pi$
$13$			76.8279i			1.63909i	−0.573012	+	0.819547i	$0.694225\pi$
$14$		0			0
$15$	−16.5679	−	20.0126i	−0.285187	−	0.344483i
$16$		0			0
$17$		−	13.8879i		−	0.198136i	−0.995081	−	0.0990679i	$-0.968414\pi$
$17$		−	13.8879i		−	0.198136i	0.995081	−	0.0990679i	$-0.0315861\pi$
$18$		0			0
$19$	−45.8869			−0.554062			−0.277031	−	0.960861i	$-0.589350\pi$
$19$	−45.8869			−0.554062			−0.277031	+	0.960861i	$0.589350\pi$
$20$		0			0
$21$	−123.569	+	102.299i	−1.28405	+	1.06303i
$22$		0			0
$23$	−74.3371			−0.673928			−0.336964	−	0.941517i	$-0.609400\pi$
$23$	−74.3371			−0.673928			−0.336964	+	0.941517i	$0.609400\pi$
$24$		0			0
$25$	25.0000			0.200000
$26$		0			0
$27$	122.870	−	67.7191i	0.875793	−	0.482687i
$28$		0			0
$29$	85.8680			0.549838			0.274919	−	0.961467i	$-0.411349\pi$
$29$	85.8680			0.549838			0.274919	+	0.961467i	$0.411349\pi$
$30$		0			0
$31$			227.728i			1.31939i	0.751533	+	0.659695i	$0.229315\pi$
$31$			227.728i			1.31939i	−0.751533	+	0.659695i	$0.770685\pi$
$32$		0			0
$33$	183.317	−	151.762i	0.967010	−	0.800559i
$34$		0			0
$35$		−	154.364i		−	0.745494i
$36$		0			0
$37$		−	5.58615i		−	0.0248205i	−0.999923	−	0.0124102i	$-0.996050\pi$
$37$		−	5.58615i		−	0.0248205i	0.999923	−	0.0124102i	$-0.00395041\pi$
$38$		0			0
$39$	307.506	−	254.575i	1.26257	−	1.04525i
$40$		0			0
$41$			61.0663i			0.232609i	0.993214	+	0.116304i	$0.0371048\pi$
$41$			61.0663i			0.232609i	−0.993214	+	0.116304i	$0.962895\pi$
$42$		0			0
$43$	−65.1565			−0.231076			−0.115538	−	0.993303i	$-0.536859\pi$
$43$	−65.1565			−0.231076			−0.115538	+	0.993303i	$0.536859\pi$
$44$		0			0
$45$	−25.2023	+	132.627i	−0.0834874	+	0.439352i
$46$		0			0
$47$	529.167			1.64228			0.821138	−	0.570730i	$-0.193340\pi$
$47$	529.167			1.64228			0.821138	+	0.570730i	$0.193340\pi$
$48$		0			0
$49$	−610.132			−1.77881
$50$		0			0
$51$	−55.5867	+	46.0186i	−0.152621	+	0.126351i
$52$		0			0
$53$	22.7454			0.0589496			0.0294748	−	0.999566i	$-0.490617\pi$
$53$	22.7454			0.0589496			0.0294748	+	0.999566i	$0.490617\pi$
$54$		0			0
$55$			229.001i			0.561427i
$56$		0			0
$57$	152.050	+	183.664i	0.353324	+	0.426787i
$58$		0			0
$59$			384.963i			0.849456i	0.905321	+	0.424728i	$0.139630\pi$
$59$			384.963i			0.849456i	−0.905321	+	0.424728i	$0.860370\pi$
$60$		0			0
$61$			588.386i			1.23500i	0.786571	+	0.617500i	$0.211855\pi$
$61$			588.386i			1.23500i	−0.786571	+	0.617500i	$0.788145\pi$
$62$		0			0
$63$	818.912	+	155.613i	1.63767	+	0.311197i
$64$		0			0
$65$			384.139i			0.733025i
$66$		0			0
$67$	−369.398			−0.673570			−0.336785	−	0.941582i	$-0.609340\pi$
$67$	−369.398			−0.673570			−0.336785	+	0.941582i	$0.609340\pi$
$68$		0			0
$69$	246.321	+	297.536i	0.429762	+	0.519118i
$70$		0			0
$71$	102.090			0.170646			0.0853229	−	0.996353i	$-0.472808\pi$
$71$	102.090			0.170646			0.0853229	+	0.996353i	$0.472808\pi$
$72$		0			0
$73$	897.637			1.43918			0.719592	−	0.694397i	$-0.244329\pi$
$73$	897.637			1.43918			0.719592	+	0.694397i	$0.244329\pi$
$74$		0			0
$75$	−82.8394	−	100.063i	−0.127540	−	0.154057i
$76$		0			0
$77$	1413.98			2.09271
$78$		0			0
$79$			1020.54i			1.45341i	0.686951	+	0.726704i	$0.258949\pi$
$79$			1020.54i			1.45341i	−0.686951	+	0.726704i	$0.741051\pi$
$80$		0			0
$81$	−678.188	−	267.400i	−0.930299	−	0.366803i
$82$		0			0
$83$		−	588.935i		−	0.778844i	−0.921059	−	0.389422i	$-0.872675\pi$
$83$		−	588.935i		−	0.778844i	0.921059	−	0.389422i	$-0.127325\pi$
$84$		0			0
$85$		−	69.4395i		−	0.0886091i
$86$		0			0
$87$	−284.530	−	343.689i	−0.350630	−	0.423533i
$88$		0			0
$89$		−	1260.99i		−	1.50185i	−0.660387	−	0.750925i	$-0.729608\pi$
$89$		−	1260.99i		−	1.50185i	0.660387	−	0.750925i	$-0.270392\pi$
$90$		0			0
$91$	2371.89			2.73233
$92$		0			0
$93$	911.486	−	754.593i	1.01631	−	0.841372i
$94$		0			0
$95$	−229.435			−0.247784
$96$		0			0
$97$	936.527			0.980308			0.490154	−	0.871636i	$-0.336941\pi$
$97$	936.527			0.980308			0.490154	+	0.871636i	$0.336941\pi$
$98$		0			0
$99$	−1214.87	−	230.854i	−1.23332	−	0.234361i
$100$		0			0
$101$	1632.64			1.60845			0.804227	−	0.594322i	$-0.202579\pi$
$101$	1632.64			1.60845			0.804227	+	0.594322i	$0.202579\pi$
$102$		0			0
$103$			524.863i			0.502100i	0.967974	+	0.251050i	$0.0807759\pi$
$103$			524.863i			0.502100i	−0.967974	+	0.251050i	$0.919224\pi$
$104$		0			0
$105$	−617.847	+	511.497i	−0.574244	+	0.475400i
$106$		0			0
$107$			1358.24i			1.22716i	0.789633	+	0.613580i	$0.210271\pi$
$107$			1358.24i			1.22716i	−0.789633	+	0.613580i	$0.789729\pi$
$108$		0			0
$109$			188.470i			0.165616i	0.996566	+	0.0828080i	$0.0263888\pi$
$109$			188.470i			0.165616i	−0.996566	+	0.0828080i	$0.973611\pi$
$110$		0			0
$111$	−22.3587	+	18.5101i	−0.0191189	+	0.0158280i
$112$		0			0
$113$			663.242i			0.552146i	0.961137	+	0.276073i	$0.0890332\pi$
$113$			663.242i			0.552146i	−0.961137	+	0.276073i	$0.910967\pi$
$114$		0			0
$115$	−371.685			−0.301390
$116$		0			0
$117$	−2037.89	−	387.247i	−1.61028	−	0.305992i
$118$		0			0
$119$	−428.759			−0.330288
$120$		0			0
$121$	−766.661			−0.576004
$122$		0			0
$123$	244.420	−	202.348i	0.179175	−	0.148334i
$124$		0			0
$125$	125.000			0.0894427
$126$		0			0
$127$			1963.64i			1.37200i	0.727600	+	0.686002i	$0.240636\pi$
$127$			1963.64i			1.37200i	−0.727600	+	0.686002i	$0.759364\pi$
$128$		0			0
$129$	215.901	+	260.791i	0.147357	+	0.177995i
$130$		0			0
$131$		−	2280.27i		−	1.52082i	−0.649441	−	0.760412i	$-0.724997\pi$
$131$		−	2280.27i		−	1.52082i	0.649441	−	0.760412i	$-0.275003\pi$
$132$		0			0
$133$			1416.66i			0.923608i
$134$		0			0
$135$	614.352	−	338.596i	0.391666	−	0.215864i
$136$		0			0
$137$		−	1285.92i		−	0.801926i	−0.916094	−	0.400963i	$-0.868676\pi$
$137$		−	1285.92i		−	0.801926i	0.916094	−	0.400963i	$-0.131324\pi$
$138$		0			0
$139$	−3056.74			−1.86525			−0.932623	−	0.360853i	$-0.882486\pi$
$139$	−3056.74			−1.86525			−0.932623	+	0.360853i	$0.882486\pi$
$140$		0			0
$141$	−1753.43	−	2118.01i	−1.04727	−	1.26502i
$142$		0			0
$143$	−3518.74			−2.05770
$144$		0			0
$145$	429.340			0.245895
$146$		0			0
$147$	2021.72	+	2442.07i	1.13434	+	1.37019i
$148$		0			0
$149$	−213.268			−0.117259			−0.0586296	−	0.998280i	$-0.518673\pi$
$149$	−213.268			−0.117259			−0.0586296	+	0.998280i	$0.518673\pi$
$150$		0			0
$151$		−	1145.23i		−	0.617202i	−0.951192	−	0.308601i	$-0.900139\pi$
$151$		−	1145.23i		−	0.617202i	0.951192	−	0.308601i	$-0.0998608\pi$
$152$		0			0
$153$	368.381	+	70.0013i	0.194653	+	0.0369887i
$154$		0			0
$155$			1138.64i			0.590049i
$156$		0			0
$157$			1821.66i			0.926016i	0.886354	+	0.463008i	$0.153230\pi$
$157$			1821.66i			0.926016i	−0.886354	+	0.463008i	$0.846770\pi$
$158$		0			0
$159$	−75.3687	−	91.0393i	−0.0375920	−	0.0454081i
$160$		0			0
$161$			2295.00i			1.12342i
$162$		0			0
$163$	−45.0906			−0.0216673			−0.0108336	−	0.999941i	$-0.503449\pi$
$163$	−45.0906			−0.0216673			−0.0108336	+	0.999941i	$0.503449\pi$
$164$		0			0
$165$	916.583	−	758.812i	0.432460	−	0.358021i
$166$		0			0
$167$	−1273.78			−0.590226			−0.295113	−	0.955462i	$-0.595357\pi$
$167$	−1273.78			−0.590226			−0.295113	+	0.955462i	$0.595357\pi$
$168$		0			0
$169$	−3705.53			−1.68663
$170$		0			0
$171$	231.291	−	1217.17i	0.103434	−	0.544322i
$172$		0			0
$173$	−2464.79			−1.08321			−0.541603	−	0.840634i	$-0.682183\pi$
$173$	−2464.79			−1.08321			−0.541603	+	0.840634i	$0.682183\pi$
$174$		0			0
$175$		−	771.821i		−	0.333395i
$176$		0			0
$177$	1540.83	−	1275.60i	0.654325	−	0.541696i
$178$		0			0
$179$			940.520i			0.392725i	0.980531	+	0.196362i	$0.0629129\pi$
$179$			940.520i			0.392725i	−0.980531	+	0.196362i	$0.937087\pi$
$180$		0			0
$181$			72.3126i			0.0296959i	0.999890	+	0.0148479i	$0.00472642\pi$
$181$			72.3126i			0.0296959i	−0.999890	+	0.0148479i	$0.995274\pi$
$182$		0			0
$183$	2355.03	−	1949.66i	0.951305	−	0.787557i
$184$		0			0
$185$		−	27.9308i		−	0.0111001i
$186$		0			0
$187$	636.069			0.248738
$188$		0			0
$189$	−2090.68	−	3793.35i	−0.804628	−	1.45993i
$190$		0			0
$191$	2309.87			0.875060			0.437530	−	0.899204i	$-0.355853\pi$
$191$	2309.87			0.875060			0.437530	+	0.899204i	$0.355853\pi$
$192$		0			0
$193$	−73.8936			−0.0275595			−0.0137797	−	0.999905i	$-0.504386\pi$
$193$	−73.8936			−0.0275595			−0.0137797	+	0.999905i	$0.504386\pi$
$194$		0			0
$195$	1537.53	−	1272.87i	0.564640	−	0.467448i
$196$		0			0
$197$	−3741.28			−1.35307			−0.676536	−	0.736409i	$-0.736520\pi$
$197$	−3741.28			−1.35307			−0.676536	+	0.736409i	$0.736520\pi$
$198$		0			0
$199$		−	1758.41i		−	0.626382i	−0.949690	−	0.313191i	$-0.898602\pi$
$199$		−	1758.41i		−	0.626382i	0.949690	−	0.313191i	$-0.101398\pi$
$200$		0			0
$201$	1224.03	+	1478.53i	0.429534	+	0.518842i
$202$		0			0
$203$		−	2650.99i		−	0.916566i
$204$		0			0
$205$			305.332i			0.104026i
$206$		0			0
$207$	374.693	−	1971.82i	0.125811	−	0.662081i
$208$		0			0
$209$		−	2101.63i		−	0.695564i
$210$		0			0
$211$	−327.697			−0.106917			−0.0534587	−	0.998570i	$-0.517025\pi$
$211$	−327.697			−0.106917			−0.0534587	+	0.998570i	$0.517025\pi$
$212$		0			0
$213$	−338.283	−	408.618i	−0.108820	−	0.131446i
$214$		0			0
$215$	−325.783			−0.103340
$216$		0			0
$217$	7030.60			2.19939
$218$		0			0
$219$	−2974.39	−	3592.82i	−0.917764	−	1.10858i
$220$		0			0
$221$	1066.98			0.324763
$222$		0			0
$223$			724.529i			0.217570i	0.994065	+	0.108785i	$0.0346959\pi$
$223$			724.529i			0.217570i	−0.994065	+	0.108785i	$0.965304\pi$
$224$		0			0
$225$	−126.011	+	663.134i	−0.0373367	+	0.196484i
$226$		0			0
$227$			4049.18i			1.18394i	0.805961	+	0.591968i	$0.201649\pi$
$227$			4049.18i			1.18394i	−0.805961	+	0.591968i	$0.798351\pi$
$228$		0			0
$229$			5915.98i			1.70716i	0.520965	+	0.853578i	$0.325572\pi$
$229$			5915.98i			1.70716i	−0.520965	+	0.853578i	$0.674428\pi$
$230$		0			0
$231$	−4685.34	−	5659.50i	−1.33451	−	1.61198i
$232$		0			0
$233$		−	2260.11i		−	0.635471i	−0.948179	−	0.317736i	$-0.897078\pi$
$233$		−	2260.11i		−	0.635471i	0.948179	−	0.317736i	$-0.102922\pi$
$234$		0			0
$235$	2645.83			0.734448
$236$		0			0
$237$	4084.72	−	3381.62i	1.11954	−	0.926834i
$238$		0			0
$239$	928.002			0.251161			0.125581	−	0.992083i	$-0.459921\pi$
$239$	928.002			0.251161			0.125581	+	0.992083i	$0.459921\pi$
$240$		0			0
$241$	−5101.54			−1.36357			−0.681783	−	0.731554i	$-0.738795\pi$
$241$	−5101.54			−1.36357			−0.681783	+	0.731554i	$0.738795\pi$
$242$		0			0
$243$	1176.95	+	3600.51i	0.310705	+	0.950506i
$244$		0			0
$245$	−3050.66			−0.795508
$246$		0			0
$247$		−	3525.40i		−	0.908160i
$248$		0			0
$249$	−2357.23	+	1951.48i	−0.599933	+	0.496667i
$250$		0			0
$251$		−	396.163i		−	0.0996238i	−0.998759	−	0.0498119i	$-0.984138\pi$
$251$		−	396.163i		−	0.0996238i	0.998759	−	0.0498119i	$-0.0158622\pi$
$252$		0			0
$253$		−	3404.65i		−	0.846043i
$254$		0			0
$255$	−277.933	+	230.093i	−0.0682544	+	0.0565058i
$256$		0			0
$257$			4197.44i			1.01879i	0.860532	+	0.509396i	$0.170131\pi$
$257$			4197.44i			1.01879i	−0.860532	+	0.509396i	$0.829869\pi$
$258$		0			0
$259$	−172.460			−0.0413752
$260$		0			0
$261$	−432.814	+	2277.68i	−0.102646	+	0.540171i
$262$		0			0
$263$	4060.33			0.951981			0.475990	−	0.879450i	$-0.342090\pi$
$263$	4060.33			0.951981			0.475990	+	0.879450i	$0.342090\pi$
$264$		0			0
$265$	113.727			0.0263631
$266$		0			0
$267$	−5047.15	+	4178.38i	−1.15686	+	0.957726i
$268$		0			0
$269$	−166.812			−0.0378094			−0.0189047	−	0.999821i	$-0.506018\pi$
$269$	−166.812			−0.0378094			−0.0189047	+	0.999821i	$0.506018\pi$
$270$		0			0
$271$			6388.24i			1.43195i	0.698127	+	0.715974i	$0.254017\pi$
$271$			6388.24i			1.43195i	−0.698127	+	0.715974i	$0.745983\pi$
$272$		0			0
$273$	−7859.45	−	9493.57i	−1.74240	−	2.10468i
$274$		0			0
$275$			1145.01i			0.251078i
$276$		0			0
$277$		−	3245.23i		−	0.703924i	−0.936014	−	0.351962i	$-0.885515\pi$
$277$		−	3245.23i		−	0.703924i	0.936014	−	0.351962i	$-0.114485\pi$
$278$		0			0
$279$	−6040.56	−	1147.85i	−1.29620	−	0.246308i
$280$		0			0
$281$			6182.32i			1.31248i	0.754553	+	0.656239i	$0.227854\pi$
$281$			6182.32i			1.31248i	−0.754553	+	0.656239i	$0.772146\pi$
$282$		0			0
$283$	−5095.93			−1.07039			−0.535197	−	0.844727i	$-0.679763\pi$
$283$	−5095.93			−1.07039			−0.535197	+	0.844727i	$0.679763\pi$
$284$		0			0
$285$	760.248	+	918.318i	0.158011	+	0.190865i
$286$		0			0
$287$	1885.29			0.387753
$288$		0			0
$289$	4720.13			0.960742
$290$		0			0
$291$	−3103.25	−	3748.47i	−0.625140	−	0.755118i
$292$		0			0
$293$	−304.740			−0.0607614			−0.0303807	−	0.999538i	$-0.509672\pi$
$293$	−304.740			−0.0607614			−0.0303807	+	0.999538i	$0.509672\pi$
$294$		0			0
$295$			1924.82i			0.379888i
$296$		0			0
$297$	3101.55	+	5627.49i	0.605961	+	1.09946i
$298$		0			0
$299$		−	5711.16i		−	1.10463i
$300$		0			0
$301$			2011.57i			0.385199i
$302$		0			0
$303$	−5409.88	−	6534.69i	−1.02571	−	1.23897i
$304$		0			0
$305$			2941.93i			0.552309i
$306$		0			0
$307$	9307.57			1.73033			0.865165	−	0.501488i	$-0.167214\pi$
$307$	9307.57			1.73033			0.865165	+	0.501488i	$0.167214\pi$
$308$		0			0
$309$	2100.78	−	1739.17i	0.386761	−	0.320188i
$310$		0			0
$311$	−6574.29			−1.19869			−0.599347	−	0.800490i	$-0.704573\pi$
$311$	−6574.29			−1.19869			−0.599347	+	0.800490i	$0.704573\pi$
$312$		0			0
$313$	−1809.44			−0.326759			−0.163380	−	0.986563i	$-0.552240\pi$
$313$	−1809.44			−0.326759			−0.163380	+	0.986563i	$0.552240\pi$
$314$		0			0
$315$	4094.56	+	778.065i	0.732389	+	0.139171i
$316$		0			0
$317$	2060.25			0.365031			0.182516	−	0.983203i	$-0.441576\pi$
$317$	2060.25			0.365031			0.182516	+	0.983203i	$0.441576\pi$
$318$		0			0
$319$			3932.77i			0.690261i
$320$		0			0
$321$	5436.39	−	4500.63i	0.945264	−	0.782556i
$322$		0			0
$323$			637.273i			0.109780i
$324$		0			0
$325$			1920.70i			0.327819i
$326$		0			0
$327$	754.356	−	624.509i	0.127572	−	0.105613i
$328$		0			0
$329$		−	16336.9i		−	2.73763i
$330$		0			0
$331$	−474.742			−0.0788343			−0.0394172	−	0.999223i	$-0.512550\pi$
$331$	−474.742			−0.0788343			−0.0394172	+	0.999223i	$0.512550\pi$
$332$		0			0
$333$	148.175	+	28.1568i	0.0243841	+	0.00463358i
$334$		0			0
$335$	−1846.99			−0.301230
$336$		0			0
$337$	1554.93			0.251342			0.125671	−	0.992072i	$-0.459892\pi$
$337$	1554.93			0.251342			0.125671	+	0.992072i	$0.459892\pi$
$338$		0			0
$339$	2654.64	−	2197.70i	0.425311	−	0.352102i
$340$		0			0
$341$	−10430.0			−1.65635
$342$		0			0
$343$			8247.11i			1.29826i
$344$		0			0
$345$	1231.61	+	1487.68i	0.192196	+	0.232157i
$346$		0			0
$347$			227.214i			0.0351512i	0.999846	+	0.0175756i	$0.00559478\pi$
$347$			227.214i			0.0351512i	−0.999846	+	0.0175756i	$0.994405\pi$
$348$		0			0
$349$		−	11005.7i		−	1.68802i	−0.536327	−	0.844010i	$-0.680189\pi$
$349$		−	11005.7i		−	1.68802i	0.536327	−	0.844010i	$-0.319811\pi$
$350$		0			0
$351$	5202.72	+	9439.87i	0.791170	+	1.43551i
$352$		0			0
$353$			3366.22i			0.507551i	0.967263	+	0.253776i	$0.0816725\pi$
$353$			3366.22i			0.507551i	−0.967263	+	0.253776i	$0.918327\pi$
$354$		0			0
$355$	510.450			0.0763151
$356$		0			0
$357$	1420.72	+	1716.12i	0.210624	+	0.254416i
$358$		0			0
$359$	7235.65			1.06374			0.531870	−	0.846826i	$-0.321489\pi$
$359$	7235.65			1.06374			0.531870	+	0.846826i	$0.321489\pi$
$360$		0			0
$361$	−4753.39			−0.693015
$362$		0			0
$363$	2540.39	+	3068.58i	0.367316	+	0.443688i
$364$		0			0
$365$	4488.18			0.643623
$366$		0			0
$367$			8609.76i			1.22459i	0.790628	+	0.612297i	$0.209754\pi$
$367$			8609.76i			1.22459i	−0.790628	+	0.612297i	$0.790246\pi$
$368$		0			0
$369$	−1619.81	−	307.802i	−0.228519	−	0.0434242i
$370$		0			0
$371$		−	702.216i		−	0.0982675i
$372$		0			0
$373$			8108.93i			1.12564i	0.826579	+	0.562821i	$0.190284\pi$
$373$			8108.93i			1.12564i	−0.826579	+	0.562821i	$0.809716\pi$
$374$		0			0
$375$	−414.197	−	500.316i	−0.0570374	−	0.0688965i
$376$		0			0
$377$			6597.06i			0.901236i
$378$		0			0
$379$	699.156			0.0947578			0.0473789	−	0.998877i	$-0.484913\pi$
$379$	699.156			0.0947578			0.0473789	+	0.998877i	$0.484913\pi$
$380$		0			0
$381$	7859.51	−	6506.65i	1.05684	−	0.874924i
$382$		0			0
$383$	464.645			0.0619902			0.0309951	−	0.999520i	$-0.490132\pi$
$383$	464.645			0.0619902			0.0309951	+	0.999520i	$0.490132\pi$
$384$		0			0
$385$	7069.91			0.935886
$386$		0			0
$387$	328.419	−	1728.30i	0.0431381	−	0.227014i
$388$		0			0
$389$	−2826.36			−0.368386			−0.184193	−	0.982890i	$-0.558967\pi$
$389$	−2826.36			−0.368386			−0.184193	+	0.982890i	$0.558967\pi$
$390$		0			0
$391$			1032.39i			0.133529i
$392$		0			0
$393$	−9126.84	+	7555.84i	−1.17147	+	0.969826i
$394$		0			0
$395$			5102.68i			0.649983i
$396$		0			0
$397$			1121.59i			0.141791i	0.997484	+	0.0708957i	$0.0225857\pi$
$397$			1121.59i			0.141791i	−0.997484	+	0.0708957i	$0.977414\pi$
$398$		0			0
$399$	5670.22	−	4694.20i	0.711443	−	0.588983i
$400$		0			0
$401$		−	3167.14i		−	0.394412i	−0.980362	−	0.197206i	$-0.936813\pi$
$401$		−	3167.14i		−	0.394412i	0.980362	−	0.197206i	$-0.0631868\pi$
$402$		0			0
$403$	−17495.8			−2.16261
$404$		0			0
$405$	−3390.94	−	1337.00i	−0.416042	−	0.164039i
$406$		0			0
$407$	255.847			0.0311594
$408$		0			0
$409$	−6367.99			−0.769870			−0.384935	−	0.922944i	$-0.625776\pi$
$409$	−6367.99			−0.769870			−0.384935	+	0.922944i	$0.625776\pi$
$410$		0			0
$411$	−5146.94	+	4261.00i	−0.617713	+	0.511386i
$412$		0			0
$413$	11884.9			1.41602
$414$		0			0
$415$		−	2944.68i		−	0.348310i
$416$		0			0
$417$	10128.7	+	12234.7i	1.18946	+	1.43677i
$418$		0			0
$419$			5043.53i			0.588049i	0.955798	+	0.294025i	$0.0949948\pi$
$419$			5043.53i			0.588049i	−0.955798	+	0.294025i	$0.905005\pi$
$420$		0			0
$421$			7446.89i			0.862089i	0.902331	+	0.431044i	$0.141855\pi$
$421$			7446.89i			0.862089i	−0.902331	+	0.431044i	$0.858145\pi$
$422$		0			0
$423$	−2667.24	+	14036.3i	−0.306586	+	1.61340i
$424$		0			0
$425$		−	347.197i		−	0.0396272i
$426$		0			0
$427$	18165.1			2.05872
$428$		0			0
$429$	11659.6	+	14083.8i	1.31219	+	1.58502i
$430$		0			0
$431$	8372.86			0.935746			0.467873	−	0.883796i	$-0.345020\pi$
$431$	8372.86			0.935746			0.467873	+	0.883796i	$0.345020\pi$
$432$		0			0
$433$	−4694.85			−0.521063			−0.260531	−	0.965465i	$-0.583898\pi$
$433$	−4694.85			−0.521063			−0.260531	+	0.965465i	$0.583898\pi$
$434$		0			0
$435$	−1422.65	−	1718.45i	−0.156807	−	0.189410i
$436$		0			0
$437$	3411.10			0.373398
$438$		0			0
$439$			5414.00i			0.588602i	0.955713	+	0.294301i	$0.0950868\pi$
$439$			5414.00i			0.588602i	−0.955713	+	0.294301i	$0.904913\pi$
$440$		0			0
$441$	3075.34	−	16183.9i	0.332074	−	1.74754i
$442$		0			0
$443$		−	8147.10i		−	0.873771i	−0.899517	−	0.436886i	$-0.856081\pi$
$443$		−	8147.10i		−	0.873771i	0.899517	−	0.436886i	$-0.143919\pi$
$444$		0			0
$445$		−	6304.95i		−	0.671648i
$446$		0			0
$447$	706.680	+	853.612i	0.0747759	+	0.0903232i
$448$		0			0
$449$		−	2592.32i		−	0.272470i	−0.990677	−	0.136235i	$-0.956500\pi$
$449$		−	2592.32i		−	0.272470i	0.990677	−	0.136235i	$-0.0435002\pi$
$450$		0			0
$451$	−2796.85			−0.292015
$452$		0			0
$453$	−4583.82	+	3794.81i	−0.475423	+	0.393588i
$454$		0			0
$455$	11859.5			1.22194
$456$		0			0
$457$	−5102.14			−0.522249			−0.261125	−	0.965305i	$-0.584093\pi$
$457$	−5102.14			−0.522249			−0.261125	+	0.965305i	$0.584093\pi$
$458$		0			0
$459$	−940.476	−	1706.41i	−0.0956376	−	0.173526i
$460$		0			0
$461$	15229.7			1.53865			0.769326	−	0.638857i	$-0.220592\pi$
$461$	15229.7			1.53865			0.769326	+	0.638857i	$0.220592\pi$
$462$		0			0
$463$		−	14047.2i		−	1.41000i	−0.709206	−	0.705001i	$-0.750947\pi$
$463$		−	14047.2i		−	1.41000i	0.709206	−	0.705001i	$-0.249053\pi$
$464$		0			0
$465$	4557.43	−	3772.96i	0.454507	−	0.376273i
$466$		0			0
$467$		−	12290.1i		−	1.21781i	−0.793242	−	0.608906i	$-0.791608\pi$
$467$		−	12290.1i		−	1.21781i	0.793242	−	0.608906i	$-0.208392\pi$
$468$		0			0
$469$			11404.4i			1.12283i
$470$		0			0
$471$	7291.26	−	6036.22i	0.713298	−	0.590518i
$472$		0			0
$473$		−	2984.18i		−	0.290091i
$474$		0			0
$475$	−1147.17			−0.110812
$476$		0			0
$477$	−114.647	+	603.331i	−0.0110049	+	0.0579132i
$478$		0			0
$479$	−9853.78			−0.939939			−0.469970	−	0.882683i	$-0.655735\pi$
$479$	−9853.78			−0.939939			−0.469970	+	0.882683i	$0.655735\pi$
$480$		0			0
$481$	429.172			0.0406831
$482$		0			0
$483$	9185.78	−	7604.64i	0.865357	−	0.716404i
$484$		0			0
$485$	4682.63			0.438407
$486$		0			0
$487$		−	10068.4i		−	0.936845i	−0.883505	−	0.468423i	$-0.844823\pi$
$487$		−	10068.4i		−	0.936845i	0.883505	−	0.468423i	$-0.155177\pi$
$488$		0			0
$489$	149.411	+	180.476i	0.0138172	+	0.0166900i
$490$		0			0
$491$			15664.3i			1.43975i	0.694102	+	0.719877i	$0.255802\pi$
$491$			15664.3i			1.43975i	−0.694102	+	0.719877i	$0.744198\pi$
$492$		0			0
$493$		−	1192.53i		−	0.108943i
$494$		0			0
$495$	−6074.33	−	1154.27i	−0.551558	−	0.104809i
$496$		0			0
$497$		−	3151.81i		−	0.284463i
$498$		0			0
$499$	−12529.3			−1.12403			−0.562013	−	0.827128i	$-0.689973\pi$
$499$	−12529.3			−1.12403			−0.562013	+	0.827128i	$0.689973\pi$
$500$		0			0
$501$	4220.75	+	5098.32i	0.376386	+	0.454643i
$502$		0			0
$503$	21913.7			1.94251			0.971256	−	0.238036i	$-0.0765037\pi$
$503$	21913.7			1.94251			0.971256	+	0.238036i	$0.0765037\pi$
$504$		0			0
$505$	8163.21			0.719323
$506$		0			0
$507$	12278.5	+	14831.5i	1.07556	+	1.29919i
$508$		0			0
$509$	−3965.77			−0.345344			−0.172672	−	0.984979i	$-0.555240\pi$
$509$	−3965.77			−0.345344			−0.172672	+	0.984979i	$0.555240\pi$
$510$		0			0
$511$		−	27712.6i		−	2.39909i
$512$		0			0
$513$	−5638.14	+	3107.42i	−0.485244	+	0.267439i
$514$		0			0
$515$			2624.32i			0.224546i
$516$		0			0
$517$			24236.0i			2.06170i
$518$		0			0
$519$	8167.27	+	9865.40i	0.690758	+	0.834380i
$520$		0			0
$521$			5250.21i			0.441489i	0.975332	+	0.220745i	$0.0708488\pi$
$521$			5250.21i			0.441489i	−0.975332	+	0.220745i	$0.929151\pi$
$522$		0			0
$523$	−12458.4			−1.04162			−0.520809	−	0.853673i	$-0.674370\pi$
$523$	−12458.4			−1.04162			−0.520809	+	0.853673i	$0.674370\pi$
$524$		0			0
$525$	−3089.23	+	2557.49i	−0.256810	+	0.212605i
$526$		0			0
$527$	3162.66			0.261419
$528$		0			0
$529$	−6641.00			−0.545821
$530$		0			0
$531$	−10211.3	−	1940.39i	−0.834523	−	0.158579i
$532$		0			0
$533$	−4691.60			−0.381268
$534$		0			0
$535$			6791.20i			0.548802i
$536$		0			0
$537$	3764.46	−	3116.48i	0.302511	−	0.250440i
$538$		0			0
$539$		−	27944.2i		−	2.23310i
$540$		0			0
$541$		−	17111.4i		−	1.35984i	−0.733285	−	0.679922i	$-0.762014\pi$
$541$		−	17111.4i		−	1.35984i	0.733285	−	0.679922i	$-0.237986\pi$
$542$		0			0
$543$	289.433	−	239.613i	0.0228743	−	0.0189370i
$544$		0			0
$545$			942.349i			0.0740657i
$546$		0			0
$547$	8732.92			0.682619			0.341310	−	0.939951i	$-0.389130\pi$
$547$	8732.92			0.682619			0.341310	+	0.939951i	$0.389130\pi$
$548$		0			0
$549$	−15607.1	−	2965.73i	−1.21329	−	0.230554i
$550$		0			0
$551$	−3940.22			−0.304644
$552$		0			0
$553$	31506.8			2.42279
$554$		0			0
$555$	−111.794	+	92.5507i	−0.00855023	+	0.00707848i
$556$		0			0
$557$	−9911.76			−0.753994			−0.376997	−	0.926214i	$-0.623043\pi$
$557$	−9911.76			−0.753994			−0.376997	+	0.926214i	$0.623043\pi$
$558$		0			0
$559$		−	5005.84i		−	0.378756i
$560$		0			0
$561$	−2107.66	−	2545.88i	−0.158619	−	0.191599i
$562$		0			0
$563$		−	11569.0i		−	0.866028i	−0.901387	−	0.433014i	$-0.857450\pi$
$563$		−	11569.0i		−	0.866028i	0.901387	−	0.433014i	$-0.142550\pi$
$564$		0			0
$565$			3316.21i			0.246927i
$566$		0			0
$567$	−8255.38	+	20937.6i	−0.611452	+	1.55079i
$568$		0			0
$569$			16691.6i			1.22978i	0.788611	+	0.614892i	$0.210800\pi$
$569$			16691.6i			1.22978i	−0.788611	+	0.614892i	$0.789200\pi$
$570$		0			0
$571$	17494.0			1.28214			0.641068	−	0.767484i	$-0.278492\pi$
$571$	17494.0			1.28214			0.641068	+	0.767484i	$0.278492\pi$
$572$		0			0
$573$	−7653.93	−	9245.32i	−0.558023	−	0.674047i
$574$		0			0
$575$	−1858.43			−0.134786
$576$		0			0
$577$	−15697.4			−1.13257			−0.566285	−	0.824210i	$-0.691620\pi$
$577$	−15697.4			−1.13257			−0.566285	+	0.824210i	$0.691620\pi$
$578$		0			0
$579$	244.852	+	295.761i	0.0175746	+	0.0212287i
$580$		0			0
$581$	−18182.1			−1.29831
$582$		0			0
$583$			1041.75i			0.0740047i
$584$		0			0
$585$	−10189.4	−	1936.24i	−0.720139	−	0.136844i
$586$		0			0
$587$			20677.6i			1.45393i	0.686676	+	0.726964i	$0.259069\pi$
$587$			20677.6i			1.45393i	−0.686676	+	0.726964i	$0.740931\pi$
$588$		0			0
$589$		−	10449.7i		−	0.731024i
$590$		0			0
$591$	12397.0	+	14974.6i	0.862851	+	1.04225i
$592$		0			0
$593$		−	6.94591i		−	0.000481002i	−1.00000		0.000240501i	$-0.999923\pi$
$593$		−	6.94591i		−	0.000481002i	1.00000		0.000240501i	$-7.65539e-5\pi$
$594$		0			0
$595$	−2143.79			−0.147709
$596$		0			0
$597$	−7038.07	+	5826.61i	−0.482494	+	0.399443i
$598$		0			0
$599$	−19963.5			−1.36174			−0.680872	−	0.732402i	$-0.738399\pi$
$599$	−19963.5			−1.36174			−0.680872	+	0.732402i	$0.738399\pi$
$600$		0			0
$601$	−2588.32			−0.175674			−0.0878369	−	0.996135i	$-0.527995\pi$
$601$	−2588.32			−0.175674			−0.0878369	+	0.996135i	$0.527995\pi$
$602$		0			0
$603$	1861.94	−	9798.42i	0.125744	−	0.661729i
$604$		0			0
$605$	−3833.31			−0.257597
$606$		0			0
$607$		−	10641.9i		−	0.711599i	−0.934562	−	0.355800i	$-0.884209\pi$
$607$		−	10641.9i		−	0.711599i	0.934562	−	0.355800i	$-0.115791\pi$
$608$		0			0
$609$	−10610.7	+	8784.25i	−0.706019	+	0.584492i
$610$		0			0
$611$			40654.8i			2.69184i
$612$		0			0
$613$		−	18486.2i		−	1.21802i	−0.793161	−	0.609012i	$-0.791566\pi$
$613$		−	18486.2i		−	1.21802i	0.793161	−	0.609012i	$-0.208434\pi$
$614$		0			0
$615$	1222.10	−	1011.74i	0.0801297	−	0.0663370i
$616$		0			0
$617$			1293.82i			0.0844201i	0.999109	+	0.0422101i	$0.0134399\pi$
$617$			1293.82i			0.0844201i	−0.999109	+	0.0422101i	$0.986560\pi$
$618$		0			0
$619$	2590.13			0.168184			0.0840922	−	0.996458i	$-0.473201\pi$
$619$	2590.13			0.168184			0.0840922	+	0.996458i	$0.473201\pi$
$620$		0			0
$621$	−9133.82	+	5034.04i	−0.590222	+	0.325297i
$622$		0			0
$623$	−38930.3			−2.50355
$624$		0			0
$625$	625.000			0.0400000
$626$		0			0
$627$	−8411.84	+	6963.91i	−0.535784	+	0.443560i
$628$		0			0
$629$	−77.5799			−0.00491783
$630$		0			0
$631$			20155.5i			1.27160i	0.771854	+	0.635800i	$0.219330\pi$
$631$			20155.5i			1.27160i	−0.771854	+	0.635800i	$0.780670\pi$
$632$		0			0
$633$	1085.85	+	1311.62i	0.0681810	+	0.0823571i
$634$		0			0
$635$			9818.18i			0.613579i
$636$		0			0
$637$		−	46875.1i		−	2.91564i
$638$		0			0
$639$	−514.580	+	2707.97i	−0.0318568	+	0.167646i
$640$		0			0
$641$		−	22915.7i		−	1.41204i	−0.708193	−	0.706019i	$-0.750489\pi$
$641$		−	22915.7i		−	1.41204i	0.708193	−	0.706019i	$-0.249511\pi$
$642$		0			0
$643$	7306.88			0.448142			0.224071	−	0.974573i	$-0.428065\pi$
$643$	7306.88			0.448142			0.224071	+	0.974573i	$0.428065\pi$
$644$		0			0
$645$	1079.51	+	1303.95i	0.0659000	+	0.0796018i
$646$		0			0
$647$	−12065.5			−0.733146			−0.366573	−	0.930389i	$-0.619469\pi$
$647$	−12065.5			−0.733146			−0.366573	+	0.930389i	$0.619469\pi$
$648$		0			0
$649$	−17631.4			−1.06640
$650$		0			0
$651$	−23296.4	−	28140.2i	−1.40255	−	1.69416i
$652$		0			0
$653$	17110.6			1.02540			0.512701	−	0.858567i	$-0.328645\pi$
$653$	17110.6			1.02540			0.512701	+	0.858567i	$0.328645\pi$
$654$		0			0
$655$		−	11401.3i		−	0.680133i
$656$		0			0
$657$	−4524.50	+	23810.1i	−0.268672	+	1.41388i
$658$		0			0
$659$		−	6115.63i		−	0.361504i	−0.983529	−	0.180752i	$-0.942147\pi$
$659$		−	6115.63i		−	0.361504i	0.983529	−	0.180752i	$-0.0578531\pi$
$660$		0			0
$661$		−	6240.95i		−	0.367239i	−0.982997	−	0.183619i	$-0.941219\pi$
$661$		−	6240.95i		−	0.367239i	0.982997	−	0.183619i	$-0.0587814\pi$
$662$		0			0
$663$	−3535.51	−	4270.61i	−0.207101	−	0.250161i
$664$		0			0
$665$			7083.29i			0.413050i
$666$		0			0
$667$	−6383.18			−0.370551
$668$		0			0
$669$	2899.95	−	2400.78i	0.167591	−	0.138744i
$670$		0			0
$671$	−26948.2			−1.55041
$672$		0			0
$673$	22328.5			1.27890			0.639450	−	0.768833i	$-0.279162\pi$
$673$	22328.5			1.27890			0.639450	+	0.768833i	$0.279162\pi$
$674$		0			0
$675$	3071.76	−	1692.98i	0.175159	−	0.0965374i
$676$		0			0
$677$	10294.2			0.584398			0.292199	−	0.956358i	$-0.405613\pi$
$677$	10294.2			0.584398			0.292199	+	0.956358i	$0.405613\pi$
$678$		0			0
$679$		−	28913.2i		−	1.63415i
$680$		0			0
$681$	16207.0	−	13417.3i	0.911970	−	0.754993i
$682$		0			0
$683$			5988.48i			0.335495i	0.985830	+	0.167747i	$0.0536493\pi$
$683$			5988.48i			0.335495i	−0.985830	+	0.167747i	$0.946351\pi$
$684$		0			0
$685$		−	6429.61i		−	0.358632i
$686$		0			0
$687$	23678.9	−	19603.0i	1.31500	−	1.08865i
$688$		0			0
$689$			1747.48i			0.0966239i
$690$		0			0
$691$	17513.5			0.964176			0.482088	−	0.876123i	$-0.339879\pi$
$691$	17513.5			0.964176			0.482088	+	0.876123i	$0.339879\pi$
$692$		0			0
$693$	−7127.12	+	37506.4i	−0.390673	+	2.05592i
$694$		0			0
$695$	−15283.7			−0.834163
$696$		0			0
$697$	848.083			0.0460881
$698$		0			0
$699$	−9046.16	+	7489.05i	−0.489495	+	0.405239i
$700$		0			0
$701$	−11644.9			−0.627422			−0.313711	−	0.949519i	$-0.601572\pi$
$701$	−11644.9			−0.627422			−0.313711	+	0.949519i	$0.601572\pi$
$702$		0			0
$703$			256.331i			0.0137521i
$704$		0			0
$705$	−8767.17	−	10590.0i	−0.468356	−	0.565735i
$706$		0			0
$707$		−	50404.3i		−	2.68126i
$708$		0			0
$709$		−	6434.79i		−	0.340852i	−0.985371	−	0.170426i	$-0.945486\pi$
$709$		−	6434.79i		−	0.340852i	0.985371	−	0.170426i	$-0.0545143\pi$
$710$		0			0
$711$	−27070.0	−	5143.96i	−1.42786	−	0.271327i
$712$		0			0
$713$		−	16928.6i		−	0.889174i
$714$		0			0
$715$	−17593.7			−0.920233
$716$		0			0
$717$	−3075.00	−	3714.35i	−0.160165	−	0.193466i
$718$		0			0
$719$	14776.3			0.766432			0.383216	−	0.923659i	$-0.374817\pi$
$719$	14776.3			0.766432			0.383216	+	0.923659i	$0.374817\pi$
$720$		0			0
$721$	16204.0			0.836989
$722$		0			0
$723$	16904.3	+	20419.1i	0.869543	+	1.05034i
$724$		0			0
$725$	2146.70			0.109968
$726$		0			0
$727$		−	13213.0i		−	0.674060i	−0.941494	−	0.337030i	$-0.890578\pi$
$727$		−	13213.0i		−	0.674060i	0.941494	−	0.337030i	$-0.109422\pi$
$728$		0			0
$729$	10511.2	−	16641.3i	0.534026	−	0.845468i
$730$		0			0
$731$			904.887i			0.0457845i
$732$		0			0
$733$		−	7834.21i		−	0.394766i	−0.980326	−	0.197383i	$-0.936756\pi$
$733$		−	7834.21i		−	0.394766i	0.980326	−	0.197383i	$-0.0632442\pi$
$734$		0			0
$735$	10108.6	+	12210.3i	0.507293	+	0.612769i
$736$		0			0
$737$		−	16918.5i		−	0.845593i
$738$		0			0
$739$	7534.72			0.375060			0.187530	−	0.982259i	$-0.439952\pi$
$739$	7534.72			0.375060			0.187530	+	0.982259i	$0.439952\pi$
$740$		0			0
$741$	−14110.5	+	11681.7i	−0.699544	+	0.579131i
$742$		0			0
$743$	196.228			0.00968896			0.00484448	−	0.999988i	$-0.498458\pi$
$743$	196.228			0.00968896			0.00484448	+	0.999988i	$0.498458\pi$
$744$		0			0
$745$	−1066.34			−0.0524399
$746$		0			0
$747$	15621.7	+	2968.50i	0.765152	+	0.145397i
$748$		0			0
$749$	41932.7			2.04564
$750$		0			0
$751$			6242.17i			0.303302i	0.988434	+	0.151651i	$0.0484590\pi$
$751$			6242.17i			0.303302i	−0.988434	+	0.151651i	$0.951541\pi$
$752$		0			0
$753$	−1585.65	+	1312.71i	−0.0767389	+	0.0635298i
$754$		0			0
$755$		−	5726.15i		−	0.276021i
$756$		0			0
$757$		−	12861.5i		−	0.617517i	−0.951141	−	0.308758i	$-0.900087\pi$
$757$		−	12861.5i		−	0.617517i	0.951141	−	0.308758i	$-0.0999134\pi$
$758$		0			0
$759$	−13627.2	+	11281.6i	−0.651696	+	0.539519i
$760$		0			0
$761$			33111.2i			1.57724i	0.614879	+	0.788621i	$0.289205\pi$
$761$			33111.2i			1.57724i	−0.614879	+	0.788621i	$0.710795\pi$
$762$		0			0
$763$	5818.60			0.276078
$764$		0			0
$765$	1841.91	+	350.007i	0.0870513	+	0.0165418i
$766$		0			0
$767$	−29575.9			−1.39234
$768$		0			0
$769$	−31567.2			−1.48029			−0.740145	−	0.672448i	$-0.765243\pi$
$769$	−31567.2			−1.48029			−0.740145	+	0.672448i	$0.765243\pi$
$770$		0			0
$771$	16800.4	−	13908.5i	0.784761	−	0.649681i
$772$		0			0
$773$	−30856.5			−1.43575			−0.717873	−	0.696174i	$-0.754884\pi$
$773$	−30856.5			−1.43575			−0.717873	+	0.696174i	$0.754884\pi$
$774$		0			0
$775$			5693.19i			0.263878i
$776$		0			0
$777$	571.460	+	690.277i	0.0263848	+	0.0318707i
$778$		0			0
$779$		−	2802.15i		−	0.128880i
$780$		0			0
$781$			4675.75i			0.214227i
$782$		0			0
$783$	10550.6	−	5814.91i	0.481544	−	0.265400i
$784$		0			0
$785$			9108.32i			0.414127i
$786$		0			0
$787$	−22387.1			−1.01399			−0.506996	−	0.861948i	$-0.669244\pi$
$787$	−22387.1			−1.01399			−0.506996	+	0.861948i	$0.669244\pi$
$788$		0			0
$789$	−13454.2	−	16251.6i	−0.607076	−	0.733298i
$790$		0			0
$791$	20476.2			0.920415
$792$		0			0
$793$	−45204.4			−2.02428
$794$		0			0
$795$	−376.844	−	455.196i	−0.0168117	−	0.0203071i
$796$		0			0
$797$	22479.6			0.999083			0.499541	−	0.866290i	$-0.333502\pi$
$797$	22479.6			0.999083			0.499541	+	0.866290i	$0.333502\pi$
$798$		0			0
$799$		−	7349.02i		−	0.325394i
$800$		0			0
$801$	33448.2	+	6355.96i	1.47545	+	0.280371i
$802$		0			0
$803$			41112.0i			1.80674i
$804$		0			0
$805$			11475.0i			0.502410i
$806$		0			0
$807$	552.745	+	667.671i	0.0241110	+	0.0291241i
$808$		0			0
$809$			6314.08i			0.274402i	0.990543	+	0.137201i	$0.0438106\pi$
$809$			6314.08i			0.274402i	−0.990543	+	0.137201i	$0.956189\pi$
$810$		0			0
$811$	−33274.4			−1.44072			−0.720359	−	0.693601i	$-0.756023\pi$
$811$	−33274.4			−1.44072			−0.720359	+	0.693601i	$0.756023\pi$
$812$		0			0
$813$	25569.1	−	21167.9i	1.10301	−	0.913150i
$814$		0			0
$815$	−225.453			−0.00968990
$816$		0			0
$817$	2989.83			0.128031
$818$		0			0
$819$	−11955.4	+	62915.3i	−0.510081	+	2.68430i
$820$		0			0
$821$	40381.8			1.71661			0.858304	−	0.513141i	$-0.171518\pi$
$821$	40381.8			1.71661			0.858304	+	0.513141i	$0.171518\pi$
$822$		0			0
$823$			2147.64i			0.0909625i	0.998965	+	0.0454812i	$0.0144821\pi$
$823$			2147.64i			0.0909625i	−0.998965	+	0.0454812i	$0.985518\pi$
$824$		0			0
$825$	4582.92	−	3794.06i	0.193402	−	0.160112i
$826$		0			0
$827$		−	15402.2i		−	0.647625i	−0.946121	−	0.323812i	$-0.895035\pi$
$827$		−	15402.2i		−	0.647625i	0.946121	−	0.323812i	$-0.104965\pi$
$828$		0			0
$829$		−	19592.3i		−	0.820832i	−0.911898	−	0.410416i	$-0.865383\pi$
$829$		−	19592.3i		−	0.820832i	0.911898	−	0.410416i	$-0.134617\pi$
$830$		0			0
$831$	−12989.1	+	10753.3i	−0.542224	+	0.448891i
$832$		0			0
$833$			8473.45i			0.352446i
$834$		0			0
$835$	−6368.88			−0.263957
$836$		0			0
$837$	15421.5	+	27981.0i	0.636853	+	1.15551i
$838$		0			0
$839$	30637.0			1.26068			0.630338	−	0.776320i	$-0.282916\pi$
$839$	30637.0			1.26068			0.630338	+	0.776320i	$0.282916\pi$
$840$		0			0
$841$	−17015.7			−0.697679
$842$		0			0
$843$	24744.9	−	20485.6i	1.01098	−	0.836964i
$844$		0			0
$845$	−18527.6			−0.754284
$846$		0			0
$847$			23669.0i			0.960185i
$848$		0			0
$849$	16885.7	+	20396.6i	0.682587	+	0.824510i
$850$		0			0
$851$			415.258i			0.0167272i
$852$		0			0
$853$		−	39184.1i		−	1.57285i	−0.617687	−	0.786424i	$-0.711930\pi$
$853$		−	39184.1i		−	1.57285i	0.617687	−	0.786424i	$-0.288070\pi$
$854$		0			0
$855$	1156.45	−	6085.83i	0.0462572	−	0.243428i
$856$		0			0
$857$		−	40181.7i		−	1.60161i	−0.598924	−	0.800806i	$-0.704405\pi$
$857$		−	40181.7i		−	1.60161i	0.598924	−	0.800806i	$-0.295595\pi$
$858$		0			0
$859$	−17782.5			−0.706323			−0.353162	−	0.935562i	$-0.614893\pi$
$859$	−17782.5			−0.706323			−0.353162	+	0.935562i	$0.614893\pi$
$860$		0			0
$861$	−6247.05	−	7545.93i	−0.247269	−	0.298681i
$862$		0			0
$863$	40594.3			1.60121			0.800607	−	0.599190i	$-0.204511\pi$
$863$	40594.3			1.60121			0.800607	+	0.599190i	$0.204511\pi$
$864$		0			0
$865$	−12324.0			−0.484425
$866$		0			0
$867$	−15640.5	−	18892.4i	−0.612663	−	0.740047i
$868$		0			0
$869$	−46740.7			−1.82459
$870$		0			0
$871$		−	28380.1i		−	1.10404i
$872$		0			0
$873$	−4720.52	+	24841.7i	−0.183007	+	0.963074i
$874$		0			0
$875$		−	3859.10i		−	0.149099i
$876$		0			0
$877$			20000.2i			0.770079i	0.922900	+	0.385039i	$0.125812\pi$
$877$			20000.2i			0.770079i	−0.922900	+	0.385039i	$0.874188\pi$
$878$		0			0
$879$	1009.78	+	1219.73i	0.0387474	+	0.0468037i
$880$		0			0
$881$			23868.5i			0.912770i	0.889782	+	0.456385i	$0.150856\pi$
$881$			23868.5i			0.912770i	−0.889782	+	0.456385i	$0.849144\pi$
$882$		0			0
$883$	31165.4			1.18777			0.593885	−	0.804550i	$-0.297594\pi$
$883$	31165.4			1.18777			0.593885	+	0.804550i	$0.297594\pi$
$884$		0			0
$885$	7704.13	−	6378.02i	0.292623	−	0.242254i
$886$		0			0
$887$	45311.6			1.71523			0.857617	−	0.514288i	$-0.171944\pi$
$887$	45311.6			1.71523			0.857617	+	0.514288i	$0.171944\pi$
$888$		0			0
$889$	60623.0			2.28710
$890$		0			0
$891$	12247.0	−	31061.2i	0.460481	−	1.16789i
$892$		0			0
$893$	−24281.8			−0.909922
$894$		0			0
$895$			4702.60i			0.175632i
$896$		0			0
$897$	−22859.1	+	18924.4i	−0.850883	+	0.704421i
$898$		0			0
$899$			19554.5i			0.725450i
$900$		0			0
$901$		−	315.886i		−	0.0116800i
$902$		0			0
$903$	8051.35	−	6665.48i	0.296713	−	0.245640i
$904$		0			0
$905$			361.563i			0.0132804i
$906$		0			0
$907$	−2344.08			−0.0858146			−0.0429073	−	0.999079i	$-0.513662\pi$
$907$	−2344.08			−0.0858146			−0.0429073	+	0.999079i	$0.513662\pi$
$908$		0			0
$909$	−8229.26	+	43306.4i	−0.300272	+	1.58018i
$910$		0			0
$911$	35403.8			1.28757			0.643787	−	0.765205i	$-0.277362\pi$
$911$	35403.8			1.28757			0.643787	+	0.765205i	$0.277362\pi$
$912$		0			0
$913$	26973.4			0.977753
$914$		0			0
$915$	11775.1	−	9748.30i	0.425436	−	0.352206i
$916$		0			0
$917$	−70398.3			−2.53518
$918$		0			0
$919$			23879.3i			0.857132i	0.903510	+	0.428566i	$0.140981\pi$
$919$			23879.3i			0.857132i	−0.903510	+	0.428566i	$0.859019\pi$
$920$		0			0
$921$	−30841.3	−	37253.8i	−1.10343	−	1.33285i
$922$		0			0
$923$			7843.36i			0.279705i
$924$		0			0
$925$		−	139.654i		−	0.00496410i
$926$		0			0
$927$	−13922.2	−	2645.55i	−0.493273	−	0.0937338i
$928$		0			0
$929$		−	23794.9i		−	0.840350i	−0.907443	−	0.420175i	$-0.861969\pi$
$929$		−	23794.9i		−	0.840350i	0.907443	−	0.420175i	$-0.138031\pi$
$930$		0			0
$931$	27997.1			0.985571
$932$		0			0
$933$	21784.4	+	26313.8i	0.764404	+	0.923337i
$934$		0			0
$935$	3180.34			0.111239
$936$		0			0
$937$	30230.7			1.05400			0.526999	−	0.849866i	$-0.323317\pi$
$937$	30230.7			1.05400			0.526999	+	0.849866i	$0.323317\pi$
$938$		0			0
$939$	5995.72	+	7242.34i	0.208374	+	0.251698i
$940$		0			0
$941$	17271.7			0.598343			0.299172	−	0.954199i	$-0.403290\pi$
$941$	17271.7			0.598343			0.299172	+	0.954199i	$0.403290\pi$
$942$		0			0
$943$		−	4539.49i		−	0.156762i
$944$		0			0
$945$	−10453.4	−	18966.8i	−0.359841	−	0.652899i
$946$		0			0
$947$		−	46463.0i		−	1.59434i	−0.603752	−	0.797172i	$-0.706328\pi$
$947$		−	46463.0i		−	1.59434i	0.603752	−	0.797172i	$-0.293672\pi$
$948$		0			0
$949$			68963.6i			2.35896i
$950$		0			0
$951$	−6826.78	−	8246.19i	−0.232780	−	0.281179i
$952$		0			0
$953$		−	38846.0i		−	1.32040i	−0.751088	−	0.660202i	$-0.770471\pi$
$953$		−	38846.0i		−	1.32040i	0.751088	−	0.660202i	$-0.229529\pi$
$954$		0			0
$955$	11549.4			0.391339
$956$		0			0
$957$	15741.0	−	13031.5i	0.531699	−	0.440177i
$958$		0			0
$959$	−39700.1			−1.33679
$960$		0			0
$961$	−22068.9			−0.740791
$962$		0			0
$963$	−36027.8	−	6846.15i	−1.20559	−	0.229090i
$964$		0			0
$965$	−369.468			−0.0123250
$966$		0			0
$967$			9489.48i			0.315575i	0.987473	+	0.157787i	$0.0504361\pi$
$967$			9489.48i			0.315575i	−0.987473	+	0.157787i	$0.949564\pi$
$968$		0			0
$969$	2550.70	−	2111.65i	0.0845618	−	0.0700062i
$970$		0			0
$971$		−	14606.1i		−	0.482732i	−0.970434	−	0.241366i	$-0.922405\pi$
$971$		−	14606.1i		−	0.482732i	0.970434	−	0.241366i	$-0.0775954\pi$
$972$		0			0
$973$			94370.1i			3.10932i
$974$		0			0
$975$	7687.64	−	6364.37i	0.252515	−	0.209049i
$976$		0			0
$977$		−	30617.5i		−	1.00260i	−0.865273	−	0.501300i	$-0.832855\pi$
$977$		−	30617.5i		−	1.00260i	0.865273	−	0.501300i	$-0.167145\pi$
$978$		0			0
$979$	57753.6			1.88541
$980$		0			0
$981$	−4999.23	−	949.974i	−0.162704	−	0.0309178i
$982$		0			0
$983$	−52367.2			−1.69914			−0.849571	−	0.527475i	$-0.823139\pi$
$983$	−52367.2			−1.69914			−0.849571	+	0.527475i	$0.823139\pi$
$984$		0			0
$985$	−18706.4			−0.605113
$986$		0			0
$987$	−65388.8	+	54133.5i	−2.10876	+	1.74578i
$988$		0			0
$989$	4843.55			0.155729
$990$		0			0
$991$		−	19821.2i		−	0.635360i	−0.948198	−	0.317680i	$-0.897096\pi$
$991$		−	19821.2i		−	0.635360i	0.948198	−	0.317680i	$-0.102904\pi$
$992$		0			0
$993$	1573.09	+	1900.17i	0.0502725	+	0.0607250i
$994$		0			0
$995$		−	8792.03i		−	0.280127i
$996$		0			0
$997$			27175.1i			0.863233i	0.902057	+	0.431616i	$0.142057\pi$
$997$			27175.1i			0.863233i	−0.902057	+	0.431616i	$0.857943\pi$
$998$		0			0
$999$	−378.290	−	686.373i	−0.0119805	−	0.0217376i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.4.b.b.431.7		24
3.2	odd	2		480.4.b.a.431.8		24
4.3	odd	2		120.4.b.a.11.13	✓	24
8.3	odd	2		480.4.b.a.431.7		24
8.5	even	2		120.4.b.b.11.11	yes	24
12.11	even	2		120.4.b.b.11.12	yes	24
24.5	odd	2		120.4.b.a.11.14	yes	24
24.11	even	2	inner	480.4.b.b.431.8		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.b.a.11.13	✓	24	4.3	odd	2
120.4.b.a.11.14	yes	24	24.5	odd	2
120.4.b.b.11.11	yes	24	8.5	even	2
120.4.b.b.11.12	yes	24	12.11	even	2
480.4.b.a.431.7		24	8.3	odd	2
480.4.b.a.431.8		24	3.2	odd	2
480.4.b.b.431.7		24	1.1	even	1	trivial
480.4.b.b.431.8		24	24.11	even	2	inner

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:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	480.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.31357 - 4.00253i) q^{3} +5.00000 q^{5} -30.8728i q^{7} +(-5.04045 + 26.5253i) q^{9} +O(q^{10})\)	\(q+(-3.31357 - 4.00253i) q^{3} +5.00000 q^{5} -30.8728i q^{7} +(-5.04045 + 26.5253i) q^{9} +45.8002i q^{11} +76.8279i q^{13} +(-16.5679 - 20.0126i) q^{15} -13.8879i q^{17} -45.8869 q^{19} +(-123.569 + 102.299i) q^{21} -74.3371 q^{23} +25.0000 q^{25} +(122.870 - 67.7191i) q^{27} +85.8680 q^{29} +227.728i q^{31} +(183.317 - 151.762i) q^{33} -154.364i q^{35} -5.58615i q^{37} +(307.506 - 254.575i) q^{39} +61.0663i q^{41} -65.1565 q^{43} +(-25.2023 + 132.627i) q^{45} +529.167 q^{47} -610.132 q^{49} +(-55.5867 + 46.0186i) q^{51} +22.7454 q^{53} +229.001i q^{55} +(152.050 + 183.664i) q^{57} +384.963i q^{59} +588.386i q^{61} +(818.912 + 155.613i) q^{63} +384.139i q^{65} -369.398 q^{67} +(246.321 + 297.536i) q^{69} +102.090 q^{71} +897.637 q^{73} +(-82.8394 - 100.063i) q^{75} +1413.98 q^{77} +1020.54i q^{79} +(-678.188 - 267.400i) q^{81} -588.935i q^{83} -69.4395i q^{85} +(-284.530 - 343.689i) q^{87} -1260.99i q^{89} +2371.89 q^{91} +(911.486 - 754.593i) q^{93} -229.435 q^{95} +936.527 q^{97} +(-1214.87 - 230.854i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 120 q^{5}+O(q^{10}) \) 24 * q + 120 * q^5	\( 24 q + 120 q^{5} - 12 q^{19} - 4 q^{21} - 228 q^{23} + 600 q^{25} - 132 q^{27} + 116 q^{33} + 656 q^{39} + 924 q^{47} - 816 q^{49} + 700 q^{51} + 528 q^{53} - 172 q^{57} - 476 q^{63} - 1632 q^{67} + 980 q^{69} - 216 q^{71} - 216 q^{73} + 152 q^{81} - 252 q^{87} + 1800 q^{91} - 60 q^{95} + 792 q^{97} + 1328 q^{99}+O(q^{100}) \) 24 * q + 120 * q^5 - 12 * q^19 - 4 * q^21 - 228 * q^23 + 600 * q^25 - 132 * q^27 + 116 * q^33 + 656 * q^39 + 924 * q^47 - 816 * q^49 + 700 * q^51 + 528 * q^53 - 172 * q^57 - 476 * q^63 - 1632 * q^67 + 980 * q^69 - 216 * q^71 - 216 * q^73 + 152 * q^81 - 252 * q^87 + 1800 * q^91 - 60 * q^95 + 792 * q^97 + 1328 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more