LMFDB - Embedded newform 240.10.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,10,Mod(1,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("240.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	240.a (trivial)

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:	yes
Analytic conductor:	$123.608600679$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{241}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 60 $ x^2 - x - 60
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-7.26209$ of defining polynomial
Character	$\chi$	$=$	240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+81.0000 q^{3} +625.000 q^{5} -1839.88 q^{7} +6561.00 q^{9} +O(q^{10})$	$q+81.0000 q^{3} +625.000 q^{5} -1839.88 q^{7} +6561.00 q^{9} -44385.9 q^{11} +136584. q^{13} +50625.0 q^{15} -253591. q^{17} -85435.7 q^{19} -149030. q^{21} +979409. q^{23} +390625. q^{25} +531441. q^{27} +2.58640e6 q^{29} -8.94787e6 q^{31} -3.59526e6 q^{33} -1.14992e6 q^{35} +1.56064e7 q^{37} +1.10633e7 q^{39} +2.44893e7 q^{41} -1.27592e7 q^{43} +4.10062e6 q^{45} +6.16764e7 q^{47} -3.69685e7 q^{49} -2.05408e7 q^{51} +5.70418e6 q^{53} -2.77412e7 q^{55} -6.92029e6 q^{57} -8.35095e7 q^{59} +1.48622e8 q^{61} -1.20714e7 q^{63} +8.53649e7 q^{65} +1.68003e8 q^{67} +7.93321e7 q^{69} -2.10986e8 q^{71} -1.43534e8 q^{73} +3.16406e7 q^{75} +8.16646e7 q^{77} +4.55960e8 q^{79} +4.30467e7 q^{81} +3.55106e8 q^{83} -1.58494e8 q^{85} +2.09498e8 q^{87} -4.24540e8 q^{89} -2.51297e8 q^{91} -7.24777e8 q^{93} -5.33973e7 q^{95} +1.19905e9 q^{97} -2.91216e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 162 q^{3} + 1250 q^{5} - 14112 q^{7} + 13122 q^{9}+O(q^{10}) $ 2 * q + 162 * q^3 + 1250 * q^5 - 14112 * q^7 + 13122 * q^9	$ 2 q + 162 q^{3} + 1250 q^{5} - 14112 q^{7} + 13122 q^{9} + 21512 q^{11} + 24284 q^{13} + 101250 q^{15} - 156956 q^{17} + 95896 q^{19} - 1143072 q^{21} + 735264 q^{23} + 781250 q^{25} + 1062882 q^{27} - 2678212 q^{29} - 10782432 q^{31} + 1742472 q^{33} - 8820000 q^{35} + 21968332 q^{37} + 1967004 q^{39} + 26060372 q^{41} + 7191160 q^{43} + 8201250 q^{45} + 31580240 q^{47} + 73282930 q^{49} - 12713436 q^{51} + 3131116 q^{53} + 13445000 q^{55} + 7767576 q^{57} + 35494664 q^{59} + 341497340 q^{61} - 92588832 q^{63} + 15177500 q^{65} + 288195816 q^{67} + 59556384 q^{69} - 210286064 q^{71} - 232663084 q^{73} + 63281250 q^{75} - 727042176 q^{77} + 24755040 q^{79} + 86093442 q^{81} + 372082152 q^{83} - 98097500 q^{85} - 216935172 q^{87} - 427639116 q^{89} + 1126859328 q^{91} - 873376992 q^{93} + 59935000 q^{95} + 1771658884 q^{97} + 141140232 q^{99}+O(q^{100}) $ 2 * q + 162 * q^3 + 1250 * q^5 - 14112 * q^7 + 13122 * q^9 + 21512 * q^11 + 24284 * q^13 + 101250 * q^15 - 156956 * q^17 + 95896 * q^19 - 1143072 * q^21 + 735264 * q^23 + 781250 * q^25 + 1062882 * q^27 - 2678212 * q^29 - 10782432 * q^31 + 1742472 * q^33 - 8820000 * q^35 + 21968332 * q^37 + 1967004 * q^39 + 26060372 * q^41 + 7191160 * q^43 + 8201250 * q^45 + 31580240 * q^47 + 73282930 * q^49 - 12713436 * q^51 + 3131116 * q^53 + 13445000 * q^55 + 7767576 * q^57 + 35494664 * q^59 + 341497340 * q^61 - 92588832 * q^63 + 15177500 * q^65 + 288195816 * q^67 + 59556384 * q^69 - 210286064 * q^71 - 232663084 * q^73 + 63281250 * q^75 - 727042176 * q^77 + 24755040 * q^79 + 86093442 * q^81 + 372082152 * q^83 - 98097500 * q^85 - 216935172 * q^87 - 427639116 * q^89 + 1126859328 * q^91 - 873376992 * q^93 + 59935000 * q^95 + 1771658884 * q^97 + 141140232 * q^99

Display 100 coefficients 10 coefficients

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
$3$	81.0000	0.577350
$3$	81.0000	0.577350
$4$	0	0
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	−1839.88	−0.289633	−0.144816	−	0.989459i	$-0.546259\pi$
$7$	−1839.88	−0.289633	−0.144816	+	0.989459i	$0.546259\pi$
$8$	0	0
$9$	6561.00	0.333333
$10$	0	0
$11$	−44385.9	−0.914066	−0.457033	−	0.889450i	$-0.651088\pi$
$11$	−44385.9	−0.914066	−0.457033	+	0.889450i	$0.651088\pi$
$12$	0	0
$13$	136584.	1.32634	0.663169	−	0.748470i	$-0.269211\pi$
$13$	136584.	1.32634	0.663169	+	0.748470i	$0.269211\pi$
$14$	0	0
$15$	50625.0	0.258199
$16$	0	0
$17$	−253591.	−0.736399	−0.368199	−	0.929747i	$-0.620026\pi$
$17$	−253591.	−0.736399	−0.368199	+	0.929747i	$0.620026\pi$
$18$	0	0
$19$	−85435.7	−0.150400	−0.0752001	−	0.997168i	$-0.523960\pi$
$19$	−85435.7	−0.150400	−0.0752001	+	0.997168i	$0.523960\pi$
$20$	0	0
$21$	−149030.	−0.167220
$22$	0	0
$23$	979409.	0.729775	0.364887	−	0.931052i	$-0.381108\pi$
$23$	979409.	0.729775	0.364887	+	0.931052i	$0.381108\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	0	0
$27$	531441.	0.192450
$28$	0	0
$29$	2.58640e6	0.679054	0.339527	−	0.940596i	$-0.389733\pi$
$29$	2.58640e6	0.679054	0.339527	+	0.940596i	$0.389733\pi$
$30$	0	0
$31$	−8.94787e6	−1.74017	−0.870085	−	0.492901i	$-0.835936\pi$
$31$	−8.94787e6	−1.74017	−0.870085	+	0.492901i	$0.835936\pi$
$32$	0	0
$33$	−3.59526e6	−0.527736
$34$	0	0
$35$	−1.14992e6	−0.129528
$36$	0	0
$37$	1.56064e7	1.36897	0.684486	−	0.729026i	$-0.260026\pi$
$37$	1.56064e7	1.36897	0.684486	+	0.729026i	$0.260026\pi$
$38$	0	0
$39$	1.10633e7	0.765761
$40$	0	0
$41$	2.44893e7	1.35347	0.676735	−	0.736227i	$-0.263394\pi$
$41$	2.44893e7	1.35347	0.676735	+	0.736227i	$0.263394\pi$
$42$	0	0
$43$	−1.27592e7	−0.569135	−0.284568	−	0.958656i	$-0.591850\pi$
$43$	−1.27592e7	−0.569135	−0.284568	+	0.958656i	$0.591850\pi$
$44$	0	0
$45$	4.10062e6	0.149071
$46$	0	0
$47$	6.16764e7	1.84365	0.921825	−	0.387607i	$-0.126698\pi$
$47$	6.16764e7	1.84365	0.921825	+	0.387607i	$0.126698\pi$
$48$	0	0
$49$	−3.69685e7	−0.916113
$50$	0	0
$51$	−2.05408e7	−0.425160
$52$	0	0
$53$	5.70418e6	0.0993006	0.0496503	−	0.998767i	$-0.484189\pi$
$53$	5.70418e6	0.0993006	0.0496503	+	0.998767i	$0.484189\pi$
$54$	0	0
$55$	−2.77412e7	−0.408783
$56$	0	0
$57$	−6.92029e6	−0.0868336
$58$	0	0
$59$	−8.35095e7	−0.897226	−0.448613	−	0.893726i	$-0.648082\pi$
$59$	−8.35095e7	−0.897226	−0.448613	+	0.893726i	$0.648082\pi$
$60$	0	0
$61$	1.48622e8	1.37435	0.687177	−	0.726490i	$-0.258850\pi$
$61$	1.48622e8	1.37435	0.687177	+	0.726490i	$0.258850\pi$
$62$	0	0
$63$	−1.20714e7	−0.0965443
$64$	0	0
$65$	8.53649e7	0.593156
$66$	0	0
$67$	1.68003e8	1.01854	0.509272	−	0.860606i	$-0.329915\pi$
$67$	1.68003e8	1.01854	0.509272	+	0.860606i	$0.329915\pi$
$68$	0	0
$69$	7.93321e7	0.421336
$70$	0	0
$71$	−2.10986e8	−0.985349	−0.492675	−	0.870214i	$-0.663981\pi$
$71$	−2.10986e8	−0.985349	−0.492675	+	0.870214i	$0.663981\pi$
$72$	0	0
$73$	−1.43534e8	−0.591566	−0.295783	−	0.955255i	$-0.595580\pi$
$73$	−1.43534e8	−0.591566	−0.295783	+	0.955255i	$0.595580\pi$
$74$	0	0
$75$	3.16406e7	0.115470
$76$	0	0
$77$	8.16646e7	0.264744
$78$	0	0
$79$	4.55960e8	1.31706	0.658529	−	0.752555i	$-0.271179\pi$
$79$	4.55960e8	1.31706	0.658529	+	0.752555i	$0.271179\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	3.55106e8	0.821309	0.410655	−	0.911791i	$-0.365300\pi$
$83$	3.55106e8	0.821309	0.410655	+	0.911791i	$0.365300\pi$
$84$	0	0
$85$	−1.58494e8	−0.329328
$86$	0	0
$87$	2.09498e8	0.392052
$88$	0	0
$89$	−4.24540e8	−0.717239	−0.358620	−	0.933484i	$-0.616752\pi$
$89$	−4.24540e8	−0.717239	−0.358620	+	0.933484i	$0.616752\pi$
$90$	0	0
$91$	−2.51297e8	−0.384151
$92$	0	0
$93$	−7.24777e8	−1.00469
$94$	0	0
$95$	−5.33973e7	−0.0672610
$96$	0	0
$97$	1.19905e9	1.37520	0.687598	−	0.726092i	$-0.258665\pi$
$97$	1.19905e9	1.37520	0.687598	+	0.726092i	$0.258665\pi$
$98$	0	0
$99$	−2.91216e8	−0.304689
$100$	0	0
$101$	−1.77085e9	−1.69331	−0.846654	−	0.532144i	$-0.821387\pi$
$101$	−1.77085e9	−1.69331	−0.846654	+	0.532144i	$0.821387\pi$
$102$	0	0
$103$	−3.03322e8	−0.265544	−0.132772	−	0.991147i	$-0.542388\pi$
$103$	−3.03322e8	−0.265544	−0.132772	+	0.991147i	$0.542388\pi$
$104$	0	0
$105$	−9.31438e7	−0.0747829
$106$	0	0
$107$	1.95414e8	0.144121	0.0720607	−	0.997400i	$-0.477042\pi$
$107$	1.95414e8	0.144121	0.0720607	+	0.997400i	$0.477042\pi$
$108$	0	0
$109$	2.31494e9	1.57080	0.785401	−	0.618988i	$-0.212457\pi$
$109$	2.31494e9	1.57080	0.785401	+	0.618988i	$0.212457\pi$
$110$	0	0
$111$	1.26412e9	0.790377
$112$	0	0
$113$	1.31945e9	0.761270	0.380635	−	0.924725i	$-0.375706\pi$
$113$	1.31945e9	0.761270	0.380635	+	0.924725i	$0.375706\pi$
$114$	0	0
$115$	6.12130e8	0.326365
$116$	0	0
$117$	8.96126e8	0.442113
$118$	0	0
$119$	4.66576e8	0.213285
$120$	0	0
$121$	−3.87842e8	−0.164483
$122$	0	0
$123$	1.98363e9	0.781426
$124$	0	0
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	−3.18960e9	−1.08798	−0.543988	−	0.839093i	$-0.683086\pi$
$127$	−3.18960e9	−1.08798	−0.543988	+	0.839093i	$0.683086\pi$
$128$	0	0
$129$	−1.03350e9	−0.328590
$130$	0	0
$131$	5.40115e9	1.60238	0.801190	−	0.598410i	$-0.204201\pi$
$131$	5.40115e9	1.60238	0.801190	+	0.598410i	$0.204201\pi$
$132$	0	0
$133$	1.57191e8	0.0435608
$134$	0	0
$135$	3.32151e8	0.0860663
$136$	0	0
$137$	1.19903e7	0.00290795	0.00145398	−	0.999999i	$-0.499537\pi$
$137$	1.19903e7	0.00290795	0.00145398	+	0.999999i	$0.499537\pi$
$138$	0	0
$139$	−9.15482e8	−0.208010	−0.104005	−	0.994577i	$-0.533166\pi$
$139$	−9.15482e8	−0.208010	−0.104005	+	0.994577i	$0.533166\pi$
$140$	0	0
$141$	4.99578e9	1.06443
$142$	0	0
$143$	−6.06239e9	−1.21236
$144$	0	0
$145$	1.61650e9	0.303682
$146$	0	0
$147$	−2.99445e9	−0.528918
$148$	0	0
$149$	5.00462e9	0.831827	0.415913	−	0.909404i	$-0.363462\pi$
$149$	5.00462e9	0.831827	0.415913	+	0.909404i	$0.363462\pi$
$150$	0	0
$151$	−3.20554e9	−0.501770	−0.250885	−	0.968017i	$-0.580722\pi$
$151$	−3.20554e9	−0.501770	−0.250885	+	0.968017i	$0.580722\pi$
$152$	0	0
$153$	−1.66381e9	−0.245466
$154$	0	0
$155$	−5.59242e9	−0.778228
$156$	0	0
$157$	−4.63430e8	−0.0608745	−0.0304373	−	0.999537i	$-0.509690\pi$
$157$	−4.63430e8	−0.0608745	−0.0304373	+	0.999537i	$0.509690\pi$
$158$	0	0
$159$	4.62038e8	0.0573312
$160$	0	0
$161$	−1.80199e9	−0.211367
$162$	0	0
$163$	1.27948e10	1.41968	0.709840	−	0.704363i	$-0.248767\pi$
$163$	1.27948e10	1.41968	0.709840	+	0.704363i	$0.248767\pi$
$164$	0	0
$165$	−2.24703e9	−0.236011
$166$	0	0
$167$	1.85699e10	1.84750	0.923752	−	0.382992i	$-0.125106\pi$
$167$	1.85699e10	1.84750	0.923752	+	0.382992i	$0.125106\pi$
$168$	0	0
$169$	8.05063e9	0.759171
$170$	0	0
$171$	−5.60544e8	−0.0501334
$172$	0	0
$173$	4.90746e9	0.416533	0.208266	−	0.978072i	$-0.433218\pi$
$173$	4.90746e9	0.416533	0.208266	+	0.978072i	$0.433218\pi$
$174$	0	0
$175$	−7.18702e8	−0.0579266
$176$	0	0
$177$	−6.76427e9	−0.518014
$178$	0	0
$179$	1.28930e10	0.938678	0.469339	−	0.883018i	$-0.344492\pi$
$179$	1.28930e10	0.938678	0.469339	+	0.883018i	$0.344492\pi$
$180$	0	0
$181$	2.66233e10	1.84378	0.921889	−	0.387453i	$-0.126645\pi$
$181$	2.66233e10	1.84378	0.921889	+	0.387453i	$0.126645\pi$
$182$	0	0
$183$	1.20384e10	0.793483
$184$	0	0
$185$	9.75400e9	0.612223
$186$	0	0
$187$	1.12558e10	0.673117
$188$	0	0
$189$	−9.77786e8	−0.0557399
$190$	0	0
$191$	2.72802e10	1.48319	0.741595	−	0.670848i	$-0.234070\pi$
$191$	2.72802e10	1.48319	0.741595	+	0.670848i	$0.234070\pi$
$192$	0	0
$193$	−9.65442e9	−0.500862	−0.250431	−	0.968134i	$-0.580572\pi$
$193$	−9.65442e9	−0.500862	−0.250431	+	0.968134i	$0.580572\pi$
$194$	0	0
$195$	6.91455e9	0.342459
$196$	0	0
$197$	1.21091e10	0.572812	0.286406	−	0.958108i	$-0.407539\pi$
$197$	1.21091e10	0.572812	0.286406	+	0.958108i	$0.407539\pi$
$198$	0	0
$199$	−1.89741e10	−0.857673	−0.428837	−	0.903382i	$-0.641076\pi$
$199$	−1.89741e10	−0.857673	−0.428837	+	0.903382i	$0.641076\pi$
$200$	0	0
$201$	1.36082e10	0.588056
$202$	0	0
$203$	−4.75866e9	−0.196676
$204$	0	0
$205$	1.53058e10	0.605290
$206$	0	0
$207$	6.42590e9	0.243258
$208$	0	0
$209$	3.79214e9	0.137476
$210$	0	0
$211$	8.91928e9	0.309784	0.154892	−	0.987931i	$-0.450497\pi$
$211$	8.91928e9	0.309784	0.154892	+	0.987931i	$0.450497\pi$
$212$	0	0
$213$	−1.70898e10	−0.568892
$214$	0	0
$215$	−7.97450e9	−0.254525
$216$	0	0
$217$	1.64630e10	0.504010
$218$	0	0
$219$	−1.16263e10	−0.341541
$220$	0	0
$221$	−3.46364e10	−0.976714
$222$	0	0
$223$	4.42755e10	1.19892	0.599462	−	0.800403i	$-0.295381\pi$
$223$	4.42755e10	1.19892	0.599462	+	0.800403i	$0.295381\pi$
$224$	0	0
$225$	2.56289e9	0.0666667
$226$	0	0
$227$	5.40045e10	1.34994	0.674968	−	0.737847i	$-0.264157\pi$
$227$	5.40045e10	1.34994	0.674968	+	0.737847i	$0.264157\pi$
$228$	0	0
$229$	5.80250e9	0.139430	0.0697149	−	0.997567i	$-0.477791\pi$
$229$	5.80250e9	0.139430	0.0697149	+	0.997567i	$0.477791\pi$
$230$	0	0
$231$	6.61483e9	0.152850
$232$	0	0
$233$	5.41865e10	1.20445	0.602226	−	0.798326i	$-0.294281\pi$
$233$	5.41865e10	1.20445	0.602226	+	0.798326i	$0.294281\pi$
$234$	0	0
$235$	3.85477e10	0.824505
$236$	0	0
$237$	3.69328e10	0.760404
$238$	0	0
$239$	−7.91761e10	−1.56965	−0.784826	−	0.619716i	$-0.787248\pi$
$239$	−7.91761e10	−1.56965	−0.784826	+	0.619716i	$0.787248\pi$
$240$	0	0
$241$	−6.14920e10	−1.17420	−0.587100	−	0.809514i	$-0.699730\pi$
$241$	−6.14920e10	−1.17420	−0.587100	+	0.809514i	$0.699730\pi$
$242$	0	0
$243$	3.48678e9	0.0641500
$244$	0	0
$245$	−2.31053e10	−0.409698
$246$	0	0
$247$	−1.16691e10	−0.199481
$248$	0	0
$249$	2.87636e10	0.474183
$250$	0	0
$251$	−2.89319e10	−0.460093	−0.230046	−	0.973180i	$-0.573888\pi$
$251$	−2.89319e10	−0.460093	−0.230046	+	0.973180i	$0.573888\pi$
$252$	0	0
$253$	−4.34719e10	−0.667062
$254$	0	0
$255$	−1.28380e10	−0.190137
$256$	0	0
$257$	1.22388e11	1.75001	0.875005	−	0.484114i	$-0.160858\pi$
$257$	1.22388e11	1.75001	0.875005	+	0.484114i	$0.160858\pi$
$258$	0	0
$259$	−2.87139e10	−0.396499
$260$	0	0
$261$	1.69694e10	0.226351
$262$	0	0
$263$	6.24892e10	0.805386	0.402693	−	0.915335i	$-0.368074\pi$
$263$	6.24892e10	0.805386	0.402693	+	0.915335i	$0.368074\pi$
$264$	0	0
$265$	3.56511e9	0.0444086
$266$	0	0
$267$	−3.43878e10	−0.414098
$268$	0	0
$269$	1.27214e11	1.48132	0.740659	−	0.671881i	$-0.234513\pi$
$269$	1.27214e11	1.48132	0.740659	+	0.671881i	$0.234513\pi$
$270$	0	0
$271$	−1.54116e10	−0.173574	−0.0867871	−	0.996227i	$-0.527660\pi$
$271$	−1.54116e10	−0.173574	−0.0867871	+	0.996227i	$0.527660\pi$
$272$	0	0
$273$	−2.03551e10	−0.221790
$274$	0	0
$275$	−1.73382e10	−0.182813
$276$	0	0
$277$	−9.20867e10	−0.939806	−0.469903	−	0.882718i	$-0.655711\pi$
$277$	−9.20867e10	−0.939806	−0.469903	+	0.882718i	$0.655711\pi$
$278$	0	0
$279$	−5.87070e10	−0.580057
$280$	0	0
$281$	−7.78782e10	−0.745139	−0.372570	−	0.928004i	$-0.621523\pi$
$281$	−7.78782e10	−0.745139	−0.372570	+	0.928004i	$0.621523\pi$
$282$	0	0
$283$	−2.92463e10	−0.271039	−0.135520	−	0.990775i	$-0.543270\pi$
$283$	−2.92463e10	−0.271039	−0.135520	+	0.990775i	$0.543270\pi$
$284$	0	0
$285$	−4.32518e9	−0.0388331
$286$	0	0
$287$	−4.50572e10	−0.392009
$288$	0	0
$289$	−5.42796e10	−0.457717
$290$	0	0
$291$	9.71231e10	0.793970
$292$	0	0
$293$	−2.27403e11	−1.80257	−0.901285	−	0.433227i	$-0.857375\pi$
$293$	−2.27403e11	−1.80257	−0.901285	+	0.433227i	$0.857375\pi$
$294$	0	0
$295$	−5.21935e10	−0.401252
$296$	0	0
$297$	−2.35885e10	−0.175912
$298$	0	0
$299$	1.33771e11	0.967927
$300$	0	0
$301$	2.34754e10	0.164840
$302$	0	0
$303$	−1.43439e11	−0.977632
$304$	0	0
$305$	9.28887e10	0.614630
$306$	0	0
$307$	6.02579e10	0.387160	0.193580	−	0.981084i	$-0.437990\pi$
$307$	6.02579e10	0.387160	0.193580	+	0.981084i	$0.437990\pi$
$308$	0	0
$309$	−2.45691e10	−0.153312
$310$	0	0
$311$	−1.36816e11	−0.829308	−0.414654	−	0.909979i	$-0.636097\pi$
$311$	−1.36816e11	−0.829308	−0.414654	+	0.909979i	$0.636097\pi$
$312$	0	0
$313$	2.25535e11	1.32820	0.664101	−	0.747643i	$-0.268815\pi$
$313$	2.25535e11	1.32820	0.664101	+	0.747643i	$0.268815\pi$
$314$	0	0
$315$	−7.54465e9	−0.0431759
$316$	0	0
$317$	−1.28386e11	−0.714089	−0.357045	−	0.934087i	$-0.616216\pi$
$317$	−1.28386e11	−0.714089	−0.357045	+	0.934087i	$0.616216\pi$
$318$	0	0
$319$	−1.14800e11	−0.620701
$320$	0	0
$321$	1.58285e10	0.0832086
$322$	0	0
$323$	2.16657e10	0.110755
$324$	0	0
$325$	5.33530e10	0.265268
$326$	0	0
$327$	1.87510e11	0.906902
$328$	0	0
$329$	−1.13477e11	−0.533981
$330$	0	0
$331$	−1.15018e10	−0.0526673	−0.0263337	−	0.999653i	$-0.508383\pi$
$331$	−1.15018e10	−0.0526673	−0.0263337	+	0.999653i	$0.508383\pi$
$332$	0	0
$333$	1.02394e11	0.456324
$334$	0	0
$335$	1.05002e11	0.455506
$336$	0	0
$337$	−2.79631e11	−1.18100	−0.590502	−	0.807036i	$-0.701070\pi$
$337$	−2.79631e11	−1.18100	−0.590502	+	0.807036i	$0.701070\pi$
$338$	0	0
$339$	1.06875e11	0.439519
$340$	0	0
$341$	3.97159e11	1.59063
$342$	0	0
$343$	1.42263e11	0.554969
$344$	0	0
$345$	4.95826e10	0.188427
$346$	0	0
$347$	−9.07088e10	−0.335867	−0.167933	−	0.985798i	$-0.553709\pi$
$347$	−9.07088e10	−0.335867	−0.167933	+	0.985798i	$0.553709\pi$
$348$	0	0
$349$	−3.98825e11	−1.43902	−0.719511	−	0.694481i	$-0.755634\pi$
$349$	−3.98825e11	−1.43902	−0.719511	+	0.694481i	$0.755634\pi$
$350$	0	0
$351$	7.25862e10	0.255254
$352$	0	0
$353$	−4.56192e11	−1.56373	−0.781865	−	0.623448i	$-0.785731\pi$
$353$	−4.56192e11	−1.56373	−0.781865	+	0.623448i	$0.785731\pi$
$354$	0	0
$355$	−1.31866e11	−0.440662
$356$	0	0
$357$	3.77926e10	0.123140
$358$	0	0
$359$	−1.89176e11	−0.601094	−0.300547	−	0.953767i	$-0.597169\pi$
$359$	−1.89176e11	−0.601094	−0.300547	+	0.953767i	$0.597169\pi$
$360$	0	0
$361$	−3.15388e11	−0.977380
$362$	0	0
$363$	−3.14152e10	−0.0949643
$364$	0	0
$365$	−8.97090e10	−0.264556
$366$	0	0
$367$	3.17920e10	0.0914787	0.0457394	−	0.998953i	$-0.485436\pi$
$367$	3.17920e10	0.0914787	0.0457394	+	0.998953i	$0.485436\pi$
$368$	0	0
$369$	1.60674e11	0.451156
$370$	0	0
$371$	−1.04950e10	−0.0287607
$372$	0	0
$373$	−4.46388e11	−1.19405	−0.597025	−	0.802222i	$-0.703651\pi$
$373$	−4.46388e11	−1.19405	−0.597025	+	0.802222i	$0.703651\pi$
$374$	0	0
$375$	1.97754e10	0.0516398
$376$	0	0
$377$	3.53260e11	0.900655
$378$	0	0
$379$	−3.53467e11	−0.879978	−0.439989	−	0.898003i	$-0.645018\pi$
$379$	−3.53467e11	−0.879978	−0.439989	+	0.898003i	$0.645018\pi$
$380$	0	0
$381$	−2.58357e11	−0.628143
$382$	0	0
$383$	−3.14974e11	−0.747963	−0.373981	−	0.927436i	$-0.622008\pi$
$383$	−3.14974e11	−0.747963	−0.373981	+	0.927436i	$0.622008\pi$
$384$	0	0
$385$	5.10403e10	0.118397
$386$	0	0
$387$	−8.37131e10	−0.189712
$388$	0	0
$389$	−4.20729e11	−0.931600	−0.465800	−	0.884890i	$-0.654233\pi$
$389$	−4.20729e11	−0.931600	−0.465800	+	0.884890i	$0.654233\pi$
$390$	0	0
$391$	−2.48369e11	−0.537405
$392$	0	0
$393$	4.37493e11	0.925134
$394$	0	0
$395$	2.84975e11	0.589007
$396$	0	0
$397$	5.63093e11	1.13769	0.568844	−	0.822446i	$-0.307391\pi$
$397$	5.63093e11	1.13769	0.568844	+	0.822446i	$0.307391\pi$
$398$	0	0
$399$	1.27325e10	0.0251498
$400$	0	0
$401$	2.75897e11	0.532841	0.266420	−	0.963857i	$-0.414159\pi$
$401$	2.75897e11	0.532841	0.266420	+	0.963857i	$0.414159\pi$
$402$	0	0
$403$	−1.22213e12	−2.30805
$404$	0	0
$405$	2.69042e10	0.0496904
$406$	0	0
$407$	−6.92703e11	−1.25133
$408$	0	0
$409$	−6.54552e10	−0.115662	−0.0578308	−	0.998326i	$-0.518418\pi$
$409$	−6.54552e10	−0.115662	−0.0578308	+	0.998326i	$0.518418\pi$
$410$	0	0
$411$	9.71214e8	0.00167891
$412$	0	0
$413$	1.53647e11	0.259866
$414$	0	0
$415$	2.21941e11	0.367301
$416$	0	0
$417$	−7.41541e10	−0.120094
$418$	0	0
$419$	−2.70250e10	−0.0428354	−0.0214177	−	0.999771i	$-0.506818\pi$
$419$	−2.70250e10	−0.0428354	−0.0214177	+	0.999771i	$0.506818\pi$
$420$	0	0
$421$	−4.29698e11	−0.666644	−0.333322	−	0.942813i	$-0.608170\pi$
$421$	−4.29698e11	−0.666644	−0.333322	+	0.942813i	$0.608170\pi$
$422$	0	0
$423$	4.04659e11	0.614550
$424$	0	0
$425$	−9.90589e10	−0.147280
$426$	0	0
$427$	−2.73446e11	−0.398058
$428$	0	0
$429$	−4.91054e11	−0.699957
$430$	0	0
$431$	9.44700e11	1.31870	0.659350	−	0.751836i	$-0.270831\pi$
$431$	9.44700e11	1.31870	0.659350	+	0.751836i	$0.270831\pi$
$432$	0	0
$433$	2.10762e11	0.288136	0.144068	−	0.989568i	$-0.453982\pi$
$433$	2.10762e11	0.288136	0.144068	+	0.989568i	$0.453982\pi$
$434$	0	0
$435$	1.30936e11	0.175331
$436$	0	0
$437$	−8.36765e10	−0.109758
$438$	0	0
$439$	−7.69079e11	−0.988282	−0.494141	−	0.869382i	$-0.664517\pi$
$439$	−7.69079e11	−0.988282	−0.494141	+	0.869382i	$0.664517\pi$
$440$	0	0
$441$	−2.42550e11	−0.305371
$442$	0	0
$443$	6.49894e11	0.801726	0.400863	−	0.916138i	$-0.368710\pi$
$443$	6.49894e11	0.801726	0.400863	+	0.916138i	$0.368710\pi$
$444$	0	0
$445$	−2.65338e11	−0.320759
$446$	0	0
$447$	4.05374e11	0.480255
$448$	0	0
$449$	−4.51858e11	−0.524679	−0.262339	−	0.964976i	$-0.584494\pi$
$449$	−4.51858e11	−0.524679	−0.262339	+	0.964976i	$0.584494\pi$
$450$	0	0
$451$	−1.08698e12	−1.23716
$452$	0	0
$453$	−2.59649e11	−0.289697
$454$	0	0
$455$	−1.57061e11	−0.171797
$456$	0	0
$457$	−6.18434e11	−0.663240	−0.331620	−	0.943413i	$-0.607595\pi$
$457$	−6.18434e11	−0.663240	−0.331620	+	0.943413i	$0.607595\pi$
$458$	0	0
$459$	−1.34768e11	−0.141720
$460$	0	0
$461$	−2.76450e11	−0.285078	−0.142539	−	0.989789i	$-0.545527\pi$
$461$	−2.76450e11	−0.285078	−0.142539	+	0.989789i	$0.545527\pi$
$462$	0	0
$463$	−6.09969e11	−0.616870	−0.308435	−	0.951245i	$-0.599805\pi$
$463$	−6.09969e11	−0.616870	−0.308435	+	0.951245i	$0.599805\pi$
$464$	0	0
$465$	−4.52986e11	−0.449310
$466$	0	0
$467$	−2.83260e11	−0.275587	−0.137794	−	0.990461i	$-0.544001\pi$
$467$	−2.83260e11	−0.275587	−0.137794	+	0.990461i	$0.544001\pi$
$468$	0	0
$469$	−3.09104e11	−0.295004
$470$	0	0
$471$	−3.75378e10	−0.0351459
$472$	0	0
$473$	5.66328e11	0.520227
$474$	0	0
$475$	−3.33733e10	−0.0300800
$476$	0	0
$477$	3.74251e10	0.0331002
$478$	0	0
$479$	1.39149e12	1.20773	0.603865	−	0.797087i	$-0.293627\pi$
$479$	1.39149e12	1.20773	0.603865	+	0.797087i	$0.293627\pi$
$480$	0	0
$481$	2.13158e12	1.81572
$482$	0	0
$483$	−1.45961e11	−0.122033
$484$	0	0
$485$	7.49406e11	0.615006
$486$	0	0
$487$	1.41532e11	0.114018	0.0570092	−	0.998374i	$-0.481844\pi$
$487$	1.41532e11	0.114018	0.0570092	+	0.998374i	$0.481844\pi$
$488$	0	0
$489$	1.03638e12	0.819652
$490$	0	0
$491$	4.66246e11	0.362034	0.181017	−	0.983480i	$-0.442061\pi$
$491$	4.66246e11	0.362034	0.181017	+	0.983480i	$0.442061\pi$
$492$	0	0
$493$	−6.55887e11	−0.500055
$494$	0	0
$495$	−1.82010e11	−0.136261
$496$	0	0
$497$	3.88188e11	0.285389
$498$	0	0
$499$	2.21254e12	1.59749	0.798746	−	0.601668i	$-0.205497\pi$
$499$	2.21254e12	1.59749	0.798746	+	0.601668i	$0.205497\pi$
$500$	0	0
$501$	1.50416e12	1.06666
$502$	0	0
$503$	−3.39574e11	−0.236526	−0.118263	−	0.992982i	$-0.537733\pi$
$503$	−3.39574e11	−0.236526	−0.118263	+	0.992982i	$0.537733\pi$
$504$	0	0
$505$	−1.10678e12	−0.757270
$506$	0	0
$507$	6.52101e11	0.438308
$508$	0	0
$509$	−5.66691e11	−0.374211	−0.187106	−	0.982340i	$-0.559911\pi$
$509$	−5.66691e11	−0.374211	−0.187106	+	0.982340i	$0.559911\pi$
$510$	0	0
$511$	2.64086e11	0.171337
$512$	0	0
$513$	−4.54040e10	−0.0289445
$514$	0	0
$515$	−1.89576e11	−0.118755
$516$	0	0
$517$	−2.73756e12	−1.68522
$518$	0	0
$519$	3.97504e11	0.240485
$520$	0	0
$521$	4.42970e11	0.263393	0.131697	−	0.991290i	$-0.457958\pi$
$521$	4.42970e11	0.263393	0.131697	+	0.991290i	$0.457958\pi$
$522$	0	0
$523$	1.44683e11	0.0845591	0.0422796	−	0.999106i	$-0.486538\pi$
$523$	1.44683e11	0.0845591	0.0422796	+	0.999106i	$0.486538\pi$
$524$	0	0
$525$	−5.82149e10	−0.0334439
$526$	0	0
$527$	2.26910e12	1.28146
$528$	0	0
$529$	−8.41911e11	−0.467429
$530$	0	0
$531$	−5.47906e11	−0.299075
$532$	0	0
$533$	3.34484e12	1.79516
$534$	0	0
$535$	1.22134e11	0.0644531
$536$	0	0
$537$	1.04434e12	0.541946
$538$	0	0
$539$	1.64088e12	0.837388
$540$	0	0
$541$	−3.10308e11	−0.155742	−0.0778710	−	0.996963i	$-0.524812\pi$
$541$	−3.10308e11	−0.155742	−0.0778710	+	0.996963i	$0.524812\pi$
$542$	0	0
$543$	2.15649e12	1.06451
$544$	0	0
$545$	1.44684e12	0.702484
$546$	0	0
$547$	−1.68502e12	−0.804750	−0.402375	−	0.915475i	$-0.631815\pi$
$547$	−1.68502e12	−0.804750	−0.402375	+	0.915475i	$0.631815\pi$
$548$	0	0
$549$	9.75108e11	0.458118
$550$	0	0
$551$	−2.20971e11	−0.102130
$552$	0	0
$553$	−8.38911e11	−0.381463
$554$	0	0
$555$	7.90074e11	0.353467
$556$	0	0
$557$	−2.31328e12	−1.01831	−0.509156	−	0.860674i	$-0.670042\pi$
$557$	−2.31328e12	−1.01831	−0.509156	+	0.860674i	$0.670042\pi$
$558$	0	0
$559$	−1.74270e12	−0.754865
$560$	0	0
$561$	9.11723e11	0.388624
$562$	0	0
$563$	−7.03648e11	−0.295167	−0.147583	−	0.989050i	$-0.547150\pi$
$563$	−7.03648e11	−0.295167	−0.147583	+	0.989050i	$0.547150\pi$
$564$	0	0
$565$	8.24653e11	0.340450
$566$	0	0
$567$	−7.92007e10	−0.0321814
$568$	0	0
$569$	3.21997e12	1.28779	0.643896	−	0.765113i	$-0.277317\pi$
$569$	3.21997e12	1.28779	0.643896	+	0.765113i	$0.277317\pi$
$570$	0	0
$571$	1.21116e12	0.476801	0.238401	−	0.971167i	$-0.423377\pi$
$571$	1.21116e12	0.476801	0.238401	+	0.971167i	$0.423377\pi$
$572$	0	0
$573$	2.20969e12	0.856320
$574$	0	0
$575$	3.82582e11	0.145955
$576$	0	0
$577$	7.30673e11	0.274430	0.137215	−	0.990541i	$-0.456185\pi$
$577$	7.30673e11	0.274430	0.137215	+	0.990541i	$0.456185\pi$
$578$	0	0
$579$	−7.82008e11	−0.289173
$580$	0	0
$581$	−6.53352e11	−0.237878
$582$	0	0
$583$	−2.53185e11	−0.0907673
$584$	0	0
$585$	5.60079e11	0.197719
$586$	0	0
$587$	−4.55331e12	−1.58291	−0.791453	−	0.611230i	$-0.790675\pi$
$587$	−4.55331e12	−1.58291	−0.791453	+	0.611230i	$0.790675\pi$
$588$	0	0
$589$	7.64467e11	0.261722
$590$	0	0
$591$	9.80833e11	0.330713
$592$	0	0
$593$	3.00074e12	0.996510	0.498255	−	0.867030i	$-0.333974\pi$
$593$	3.00074e12	0.996510	0.498255	+	0.867030i	$0.333974\pi$
$594$	0	0
$595$	2.91610e11	0.0953841
$596$	0	0
$597$	−1.53690e12	−0.495178
$598$	0	0
$599$	4.03514e12	1.28067	0.640336	−	0.768095i	$-0.278795\pi$
$599$	4.03514e12	1.28067	0.640336	+	0.768095i	$0.278795\pi$
$600$	0	0
$601$	2.04760e12	0.640192	0.320096	−	0.947385i	$-0.396285\pi$
$601$	2.04760e12	0.640192	0.320096	+	0.947385i	$0.396285\pi$
$602$	0	0
$603$	1.10227e12	0.339514
$604$	0	0
$605$	−2.42401e11	−0.0735590
$606$	0	0
$607$	−3.15792e12	−0.944175	−0.472088	−	0.881552i	$-0.656499\pi$
$607$	−3.15792e12	−0.944175	−0.472088	+	0.881552i	$0.656499\pi$
$608$	0	0
$609$	−3.85451e11	−0.113551
$610$	0	0
$611$	8.42399e12	2.44530
$612$	0	0
$613$	−2.89302e12	−0.827520	−0.413760	−	0.910386i	$-0.635785\pi$
$613$	−2.89302e12	−0.827520	−0.413760	+	0.910386i	$0.635785\pi$
$614$	0	0
$615$	1.23977e12	0.349464
$616$	0	0
$617$	6.03603e12	1.67675	0.838375	−	0.545094i	$-0.183506\pi$
$617$	6.03603e12	1.67675	0.838375	+	0.545094i	$0.183506\pi$
$618$	0	0
$619$	−4.05606e12	−1.11044	−0.555222	−	0.831702i	$-0.687367\pi$
$619$	−4.05606e12	−1.11044	−0.555222	+	0.831702i	$0.687367\pi$
$620$	0	0
$621$	5.20498e11	0.140445
$622$	0	0
$623$	7.81102e11	0.207736
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	0	0
$627$	3.07163e11	0.0793716
$628$	0	0
$629$	−3.95764e12	−1.00811
$630$	0	0
$631$	5.34498e12	1.34219	0.671095	−	0.741372i	$-0.265824\pi$
$631$	5.34498e12	1.34219	0.671095	+	0.741372i	$0.265824\pi$
$632$	0	0
$633$	7.22461e11	0.178854
$634$	0	0
$635$	−1.99350e12	−0.486558
$636$	0	0
$637$	−5.04929e12	−1.21507
$638$	0	0
$639$	−1.38428e12	−0.328450
$640$	0	0
$641$	4.81739e12	1.12707	0.563534	−	0.826093i	$-0.309441\pi$
$641$	4.81739e12	1.12707	0.563534	+	0.826093i	$0.309441\pi$
$642$	0	0
$643$	5.85505e11	0.135077	0.0675385	−	0.997717i	$-0.478485\pi$
$643$	5.85505e11	0.135077	0.0675385	+	0.997717i	$0.478485\pi$
$644$	0	0
$645$	−6.45935e11	−0.146950
$646$	0	0
$647$	−4.54903e12	−1.02059	−0.510293	−	0.860001i	$-0.670463\pi$
$647$	−4.54903e12	−1.02059	−0.510293	+	0.860001i	$0.670463\pi$
$648$	0	0
$649$	3.70664e12	0.820124
$650$	0	0
$651$	1.33350e12	0.290991
$652$	0	0
$653$	−7.04350e12	−1.51593	−0.757965	−	0.652295i	$-0.773806\pi$
$653$	−7.04350e12	−1.51593	−0.757965	+	0.652295i	$0.773806\pi$
$654$	0	0
$655$	3.37572e12	0.716606
$656$	0	0
$657$	−9.41729e11	−0.197189
$658$	0	0
$659$	−5.23559e12	−1.08139	−0.540694	−	0.841219i	$-0.681838\pi$
$659$	−5.23559e12	−1.08139	−0.540694	+	0.841219i	$0.681838\pi$
$660$	0	0
$661$	−5.79017e12	−1.17974	−0.589868	−	0.807500i	$-0.700820\pi$
$661$	−5.79017e12	−1.17974	−0.589868	+	0.807500i	$0.700820\pi$
$662$	0	0
$663$	−2.80555e12	−0.563906
$664$	0	0
$665$	9.82445e10	0.0194810
$666$	0	0
$667$	2.53314e12	0.495557
$668$	0	0
$669$	3.58632e12	0.692199
$670$	0	0
$671$	−6.59671e12	−1.25625
$672$	0	0
$673$	−5.92328e10	−0.0111300	−0.00556499	−	0.999985i	$-0.501771\pi$
$673$	−5.92328e10	−0.0111300	−0.00556499	+	0.999985i	$0.501771\pi$
$674$	0	0
$675$	2.07594e11	0.0384900
$676$	0	0
$677$	4.57177e12	0.836440	0.418220	−	0.908346i	$-0.362654\pi$
$677$	4.57177e12	0.836440	0.418220	+	0.908346i	$0.362654\pi$
$678$	0	0
$679$	−2.20611e12	−0.398302
$680$	0	0
$681$	4.37436e12	0.779386
$682$	0	0
$683$	9.71286e12	1.70787	0.853934	−	0.520382i	$-0.174210\pi$
$683$	9.71286e12	1.70787	0.853934	+	0.520382i	$0.174210\pi$
$684$	0	0
$685$	7.49393e9	0.00130048
$686$	0	0
$687$	4.70003e11	0.0804998
$688$	0	0
$689$	7.79098e11	0.131706
$690$	0	0
$691$	−2.23726e12	−0.373306	−0.186653	−	0.982426i	$-0.559764\pi$
$691$	−2.23726e12	−0.373306	−0.186653	+	0.982426i	$0.559764\pi$
$692$	0	0
$693$	5.35801e11	0.0882478
$694$	0	0
$695$	−5.72176e11	−0.0930247
$696$	0	0
$697$	−6.21025e12	−0.996693
$698$	0	0
$699$	4.38911e12	0.695391
$700$	0	0
$701$	−2.29477e10	−0.00358929	−0.00179464	−	0.999998i	$-0.500571\pi$
$701$	−2.29477e10	−0.00358929	−0.00179464	+	0.999998i	$0.500571\pi$
$702$	0	0
$703$	−1.33334e12	−0.205894
$704$	0	0
$705$	3.12237e12	0.476028
$706$	0	0
$707$	3.25815e12	0.490438
$708$	0	0
$709$	−9.57637e12	−1.42329	−0.711644	−	0.702540i	$-0.752049\pi$
$709$	−9.57637e12	−1.42329	−0.711644	+	0.702540i	$0.752049\pi$
$710$	0	0
$711$	2.99156e12	0.439020
$712$	0	0
$713$	−8.76362e12	−1.26993
$714$	0	0
$715$	−3.78899e12	−0.542184
$716$	0	0
$717$	−6.41326e12	−0.906239
$718$	0	0
$719$	5.46390e12	0.762469	0.381235	−	0.924478i	$-0.375499\pi$
$719$	5.46390e12	0.762469	0.381235	+	0.924478i	$0.375499\pi$
$720$	0	0
$721$	5.58075e11	0.0769102
$722$	0	0
$723$	−4.98085e12	−0.677925
$724$	0	0
$725$	1.01031e12	0.135811
$726$	0	0
$727$	−6.59842e12	−0.876062	−0.438031	−	0.898960i	$-0.644324\pi$
$727$	−6.59842e12	−0.876062	−0.438031	+	0.898960i	$0.644324\pi$
$728$	0	0
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	3.23561e12	0.419111
$732$	0	0
$733$	6.91041e11	0.0884170	0.0442085	−	0.999022i	$-0.485923\pi$
$733$	6.91041e11	0.0884170	0.0442085	+	0.999022i	$0.485923\pi$
$734$	0	0
$735$	−1.87153e12	−0.236539
$736$	0	0
$737$	−7.45694e12	−0.931016
$738$	0	0
$739$	−7.60189e12	−0.937608	−0.468804	−	0.883302i	$-0.655315\pi$
$739$	−7.60189e12	−0.937608	−0.468804	+	0.883302i	$0.655315\pi$
$740$	0	0
$741$	−9.45200e11	−0.115171
$742$	0	0
$743$	2.23002e12	0.268448	0.134224	−	0.990951i	$-0.457146\pi$
$743$	2.23002e12	0.268448	0.134224	+	0.990951i	$0.457146\pi$
$744$	0	0
$745$	3.12789e12	0.372004
$746$	0	0
$747$	2.32985e12	0.273770
$748$	0	0
$749$	−3.59538e11	−0.0417423
$750$	0	0
$751$	−1.65708e12	−0.190092	−0.0950459	−	0.995473i	$-0.530300\pi$
$751$	−1.65708e12	−0.190092	−0.0950459	+	0.995473i	$0.530300\pi$
$752$	0	0
$753$	−2.34348e12	−0.265635
$754$	0	0
$755$	−2.00346e12	−0.224398
$756$	0	0
$757$	−1.28420e13	−1.42135	−0.710675	−	0.703521i	$-0.751610\pi$
$757$	−1.28420e13	−1.42135	−0.710675	+	0.703521i	$0.751610\pi$
$758$	0	0
$759$	−3.52122e12	−0.385129
$760$	0	0
$761$	1.61852e13	1.74939	0.874694	−	0.484676i	$-0.161063\pi$
$761$	1.61852e13	1.74939	0.874694	+	0.484676i	$0.161063\pi$
$762$	0	0
$763$	−4.25921e12	−0.454955
$764$	0	0
$765$	−1.03988e12	−0.109776
$766$	0	0
$767$	−1.14060e13	−1.19002
$768$	0	0
$769$	5.30462e12	0.546998	0.273499	−	0.961872i	$-0.411819\pi$
$769$	5.30462e12	0.546998	0.273499	+	0.961872i	$0.411819\pi$
$770$	0	0
$771$	9.91345e12	1.01037
$772$	0	0
$773$	−1.26188e13	−1.27119	−0.635596	−	0.772022i	$-0.719245\pi$
$773$	−1.26188e13	−1.27119	−0.635596	+	0.772022i	$0.719245\pi$
$774$	0	0
$775$	−3.49526e12	−0.348034
$776$	0	0
$777$	−2.32582e12	−0.228919
$778$	0	0
$779$	−2.09226e12	−0.203562
$780$	0	0
$781$	9.36478e12	0.900675
$782$	0	0
$783$	1.37452e12	0.130684
$784$	0	0
$785$	−2.89644e11	−0.0272239
$786$	0	0
$787$	−1.22887e13	−1.14188	−0.570941	−	0.820991i	$-0.693421\pi$
$787$	−1.22887e13	−1.14188	−0.570941	+	0.820991i	$0.693421\pi$
$788$	0	0
$789$	5.06162e12	0.464990
$790$	0	0
$791$	−2.42762e12	−0.220489
$792$	0	0
$793$	2.02993e13	1.82286
$794$	0	0
$795$	2.88774e11	0.0256393
$796$	0	0
$797$	−3.56495e12	−0.312962	−0.156481	−	0.987681i	$-0.550015\pi$
$797$	−3.56495e12	−0.312962	−0.156481	+	0.987681i	$0.550015\pi$
$798$	0	0
$799$	−1.56405e13	−1.35766
$800$	0	0
$801$	−2.78541e12	−0.239080
$802$	0	0
$803$	6.37090e12	0.540730
$804$	0	0
$805$	−1.12624e12	−0.0945260
$806$	0	0
$807$	1.03043e13	0.855239
$808$	0	0
$809$	−3.11624e12	−0.255778	−0.127889	−	0.991789i	$-0.540820\pi$
$809$	−3.11624e12	−0.255778	−0.127889	+	0.991789i	$0.540820\pi$
$810$	0	0
$811$	2.30256e12	0.186904	0.0934518	−	0.995624i	$-0.470210\pi$
$811$	2.30256e12	0.186904	0.0934518	+	0.995624i	$0.470210\pi$
$812$	0	0
$813$	−1.24834e12	−0.100213
$814$	0	0
$815$	7.99677e12	0.634900
$816$	0	0
$817$	1.09009e12	0.0855980
$818$	0	0
$819$	−1.64876e12	−0.128050
$820$	0	0
$821$	−6.28720e12	−0.482962	−0.241481	−	0.970406i	$-0.577633\pi$
$821$	−6.28720e12	−0.482962	−0.241481	+	0.970406i	$0.577633\pi$
$822$	0	0
$823$	−3.76056e12	−0.285728	−0.142864	−	0.989742i	$-0.545631\pi$
$823$	−3.76056e12	−0.285728	−0.142864	+	0.989742i	$0.545631\pi$
$824$	0	0
$825$	−1.40440e12	−0.105547
$826$	0	0
$827$	−1.75054e13	−1.30136	−0.650679	−	0.759353i	$-0.725516\pi$
$827$	−1.75054e13	−1.30136	−0.650679	+	0.759353i	$0.725516\pi$
$828$	0	0
$829$	−2.42570e12	−0.178378	−0.0891891	−	0.996015i	$-0.528428\pi$
$829$	−2.42570e12	−0.178378	−0.0891891	+	0.996015i	$0.528428\pi$
$830$	0	0
$831$	−7.45902e12	−0.542597
$832$	0	0
$833$	9.37486e12	0.674625
$834$	0	0
$835$	1.16062e13	0.826229
$836$	0	0
$837$	−4.75526e12	−0.334896
$838$	0	0
$839$	−9.69596e11	−0.0675557	−0.0337778	−	0.999429i	$-0.510754\pi$
$839$	−9.69596e11	−0.0675557	−0.0337778	+	0.999429i	$0.510754\pi$
$840$	0	0
$841$	−7.81769e12	−0.538885
$842$	0	0
$843$	−6.30813e12	−0.430206
$844$	0	0
$845$	5.03164e12	0.339512
$846$	0	0
$847$	7.13582e11	0.0476397
$848$	0	0
$849$	−2.36895e12	−0.156485
$850$	0	0
$851$	1.52850e13	0.999042
$852$	0	0
$853$	−2.11898e13	−1.37043	−0.685215	−	0.728340i	$-0.740292\pi$
$853$	−2.11898e13	−1.37043	−0.685215	+	0.728340i	$0.740292\pi$
$854$	0	0
$855$	−3.50340e11	−0.0224203
$856$	0	0
$857$	8.34904e12	0.528717	0.264358	−	0.964425i	$-0.414840\pi$
$857$	8.34904e12	0.528717	0.264358	+	0.964425i	$0.414840\pi$
$858$	0	0
$859$	8.24621e12	0.516755	0.258378	−	0.966044i	$-0.416812\pi$
$859$	8.24621e12	0.516755	0.258378	+	0.966044i	$0.416812\pi$
$860$	0	0
$861$	−3.64964e12	−0.226326
$862$	0	0
$863$	−4.35663e12	−0.267364	−0.133682	−	0.991024i	$-0.542680\pi$
$863$	−4.35663e12	−0.267364	−0.133682	+	0.991024i	$0.542680\pi$
$864$	0	0
$865$	3.06716e12	0.186279
$866$	0	0
$867$	−4.39665e12	−0.264263
$868$	0	0
$869$	−2.02382e13	−1.20388
$870$	0	0
$871$	2.29464e13	1.35093
$872$	0	0
$873$	7.86697e12	0.458399
$874$	0	0
$875$	−4.49189e11	−0.0259055
$876$	0	0
$877$	−1.21873e12	−0.0695680	−0.0347840	−	0.999395i	$-0.511074\pi$
$877$	−1.21873e12	−0.0695680	−0.0347840	+	0.999395i	$0.511074\pi$
$878$	0	0
$879$	−1.84197e13	−1.04071
$880$	0	0
$881$	3.98577e12	0.222905	0.111453	−	0.993770i	$-0.464450\pi$
$881$	3.98577e12	0.222905	0.111453	+	0.993770i	$0.464450\pi$
$882$	0	0
$883$	−2.19963e13	−1.21766	−0.608831	−	0.793300i	$-0.708361\pi$
$883$	−2.19963e13	−1.21766	−0.608831	+	0.793300i	$0.708361\pi$
$884$	0	0
$885$	−4.22767e12	−0.231663
$886$	0	0
$887$	−1.39860e13	−0.758640	−0.379320	−	0.925266i	$-0.623842\pi$
$887$	−1.39860e13	−0.758640	−0.379320	+	0.925266i	$0.623842\pi$
$888$	0	0
$889$	5.86847e12	0.315113
$890$	0	0
$891$	−1.91067e12	−0.101563
$892$	0	0
$893$	−5.26936e12	−0.277285
$894$	0	0
$895$	8.05815e12	0.419790
$896$	0	0
$897$	1.08355e13	0.558833
$898$	0	0
$899$	−2.31428e13	−1.18167
$900$	0	0
$901$	−1.44653e12	−0.0731248
$902$	0	0
$903$	1.90150e12	0.0951705
$904$	0	0
$905$	1.66396e13	0.824563
$906$	0	0
$907$	8.00749e12	0.392884	0.196442	−	0.980515i	$-0.437061\pi$
$907$	8.00749e12	0.392884	0.196442	+	0.980515i	$0.437061\pi$
$908$	0	0
$909$	−1.16186e13	−0.564436
$910$	0	0
$911$	4.09248e13	1.96858	0.984291	−	0.176555i	$-0.0564953\pi$
$911$	4.09248e13	1.96858	0.984291	+	0.176555i	$0.0564953\pi$
$912$	0	0
$913$	−1.57617e13	−0.750731
$914$	0	0
$915$	7.52398e12	0.354857
$916$	0	0
$917$	−9.93745e12	−0.464102
$918$	0	0
$919$	−8.12730e12	−0.375860	−0.187930	−	0.982182i	$-0.560178\pi$
$919$	−8.12730e12	−0.375860	−0.187930	+	0.982182i	$0.560178\pi$
$920$	0	0
$921$	4.88089e12	0.223527
$922$	0	0
$923$	−2.88172e13	−1.30691
$924$	0	0
$925$	6.09625e12	0.273795
$926$	0	0
$927$	−1.99010e12	−0.0885147
$928$	0	0
$929$	−4.43436e13	−1.95326	−0.976631	−	0.214923i	$-0.931050\pi$
$929$	−4.43436e13	−1.95326	−0.976631	+	0.214923i	$0.931050\pi$
$930$	0	0
$931$	3.15843e12	0.137783
$932$	0	0
$933$	−1.10821e13	−0.478801
$934$	0	0
$935$	7.03490e12	0.301027
$936$	0	0
$937$	−3.56882e13	−1.51250	−0.756251	−	0.654281i	$-0.772971\pi$
$937$	−3.56882e13	−1.51250	−0.756251	+	0.654281i	$0.772971\pi$
$938$	0	0
$939$	1.82683e13	0.766838
$940$	0	0
$941$	3.76351e13	1.56473	0.782366	−	0.622818i	$-0.214012\pi$
$941$	3.76351e13	1.56473	0.782366	+	0.622818i	$0.214012\pi$
$942$	0	0
$943$	2.39850e13	0.987727
$944$	0	0
$945$	−6.11116e11	−0.0249276
$946$	0	0
$947$	1.01014e13	0.408139	0.204070	−	0.978956i	$-0.434583\pi$
$947$	1.01014e13	0.408139	0.204070	+	0.978956i	$0.434583\pi$
$948$	0	0
$949$	−1.96045e13	−0.784616
$950$	0	0
$951$	−1.03993e13	−0.412280
$952$	0	0
$953$	1.49469e13	0.586993	0.293497	−	0.955960i	$-0.405181\pi$
$953$	1.49469e13	0.586993	0.293497	+	0.955960i	$0.405181\pi$
$954$	0	0
$955$	1.70501e13	0.663303
$956$	0	0
$957$	−9.29876e12	−0.358362
$958$	0	0
$959$	−2.20607e10	−0.000842238 0
$960$	0	0
$961$	5.36247e13	2.02819
$962$	0	0
$963$	1.28211e12	0.0480405
$964$	0	0
$965$	−6.03401e12	−0.223992
$966$	0	0
$967$	−2.42942e13	−0.893475	−0.446738	−	0.894665i	$-0.647414\pi$
$967$	−2.42942e13	−0.893475	−0.446738	+	0.894665i	$0.647414\pi$
$968$	0	0
$969$	1.75492e12	0.0639441
$970$	0	0
$971$	1.07032e13	0.386392	0.193196	−	0.981160i	$-0.438115\pi$
$971$	1.07032e13	0.386392	0.193196	+	0.981160i	$0.438115\pi$
$972$	0	0
$973$	1.68437e12	0.0602464
$974$	0	0
$975$	4.32160e12	0.153152
$976$	0	0
$977$	−1.53038e13	−0.537372	−0.268686	−	0.963228i	$-0.586589\pi$
$977$	−1.53038e13	−0.537372	−0.268686	+	0.963228i	$0.586589\pi$
$978$	0	0
$979$	1.88436e13	0.655604
$980$	0	0
$981$	1.51883e13	0.523600
$982$	0	0
$983$	−3.27251e13	−1.11787	−0.558934	−	0.829212i	$-0.688790\pi$
$983$	−3.27251e13	−1.11787	−0.558934	+	0.829212i	$0.688790\pi$
$984$	0	0
$985$	7.56816e12	0.256169
$986$	0	0
$987$	−9.19163e12	−0.308294
$988$	0	0
$989$	−1.24965e13	−0.415340
$990$	0	0
$991$	−3.05488e13	−1.00615	−0.503075	−	0.864243i	$-0.667798\pi$
$991$	−3.05488e13	−1.00615	−0.503075	+	0.864243i	$0.667798\pi$
$992$	0	0
$993$	−9.31649e11	−0.0304075
$994$	0	0
$995$	−1.18588e13	−0.383563
$996$	0	0
$997$	4.87857e13	1.56374	0.781870	−	0.623441i	$-0.214266\pi$
$997$	4.87857e13	1.56374	0.781870	+	0.623441i	$0.214266\pi$
$998$	0	0
$999$	8.29388e12	0.263459

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.10.a.r.1.2		2
4.3	odd	2		15.10.a.d.1.1	✓	2
12.11	even	2		45.10.a.d.1.2		2
20.3	even	4		75.10.b.f.49.3		4
20.7	even	4		75.10.b.f.49.2		4
20.19	odd	2		75.10.a.f.1.2		2
60.23	odd	4		225.10.b.i.199.2		4
60.47	odd	4		225.10.b.i.199.3		4
60.59	even	2		225.10.a.k.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.10.a.d.1.1	✓	2	4.3	odd	2
45.10.a.d.1.2		2	12.11	even	2
75.10.a.f.1.2		2	20.19	odd	2
75.10.b.f.49.2		4	20.7	even	4
75.10.b.f.49.3		4	20.3	even	4
225.10.a.k.1.1		2	60.59	even	2
225.10.b.i.199.2		4	60.23	odd	4
225.10.b.i.199.3		4	60.47	odd	4
240.10.a.r.1.2		2	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:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	240.a (trivial)

\(f(q)\)	\(=\)	\(q+81.0000 q^{3} +625.000 q^{5} -1839.88 q^{7} +6561.00 q^{9} +O(q^{10})\)	\(q+81.0000 q^{3} +625.000 q^{5} -1839.88 q^{7} +6561.00 q^{9} -44385.9 q^{11} +136584. q^{13} +50625.0 q^{15} -253591. q^{17} -85435.7 q^{19} -149030. q^{21} +979409. q^{23} +390625. q^{25} +531441. q^{27} +2.58640e6 q^{29} -8.94787e6 q^{31} -3.59526e6 q^{33} -1.14992e6 q^{35} +1.56064e7 q^{37} +1.10633e7 q^{39} +2.44893e7 q^{41} -1.27592e7 q^{43} +4.10062e6 q^{45} +6.16764e7 q^{47} -3.69685e7 q^{49} -2.05408e7 q^{51} +5.70418e6 q^{53} -2.77412e7 q^{55} -6.92029e6 q^{57} -8.35095e7 q^{59} +1.48622e8 q^{61} -1.20714e7 q^{63} +8.53649e7 q^{65} +1.68003e8 q^{67} +7.93321e7 q^{69} -2.10986e8 q^{71} -1.43534e8 q^{73} +3.16406e7 q^{75} +8.16646e7 q^{77} +4.55960e8 q^{79} +4.30467e7 q^{81} +3.55106e8 q^{83} -1.58494e8 q^{85} +2.09498e8 q^{87} -4.24540e8 q^{89} -2.51297e8 q^{91} -7.24777e8 q^{93} -5.33973e7 q^{95} +1.19905e9 q^{97} -2.91216e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 162 q^{3} + 1250 q^{5} - 14112 q^{7} + 13122 q^{9}+O(q^{10}) \) 2 * q + 162 * q^3 + 1250 * q^5 - 14112 * q^7 + 13122 * q^9	\( 2 q + 162 q^{3} + 1250 q^{5} - 14112 q^{7} + 13122 q^{9} + 21512 q^{11} + 24284 q^{13} + 101250 q^{15} - 156956 q^{17} + 95896 q^{19} - 1143072 q^{21} + 735264 q^{23} + 781250 q^{25} + 1062882 q^{27} - 2678212 q^{29} - 10782432 q^{31} + 1742472 q^{33} - 8820000 q^{35} + 21968332 q^{37} + 1967004 q^{39} + 26060372 q^{41} + 7191160 q^{43} + 8201250 q^{45} + 31580240 q^{47} + 73282930 q^{49} - 12713436 q^{51} + 3131116 q^{53} + 13445000 q^{55} + 7767576 q^{57} + 35494664 q^{59} + 341497340 q^{61} - 92588832 q^{63} + 15177500 q^{65} + 288195816 q^{67} + 59556384 q^{69} - 210286064 q^{71} - 232663084 q^{73} + 63281250 q^{75} - 727042176 q^{77} + 24755040 q^{79} + 86093442 q^{81} + 372082152 q^{83} - 98097500 q^{85} - 216935172 q^{87} - 427639116 q^{89} + 1126859328 q^{91} - 873376992 q^{93} + 59935000 q^{95} + 1771658884 q^{97} + 141140232 q^{99}+O(q^{100}) \) 2 * q + 162 * q^3 + 1250 * q^5 - 14112 * q^7 + 13122 * q^9 + 21512 * q^11 + 24284 * q^13 + 101250 * q^15 - 156956 * q^17 + 95896 * q^19 - 1143072 * q^21 + 735264 * q^23 + 781250 * q^25 + 1062882 * q^27 - 2678212 * q^29 - 10782432 * q^31 + 1742472 * q^33 - 8820000 * q^35 + 21968332 * q^37 + 1967004 * q^39 + 26060372 * q^41 + 7191160 * q^43 + 8201250 * q^45 + 31580240 * q^47 + 73282930 * q^49 - 12713436 * q^51 + 3131116 * q^53 + 13445000 * q^55 + 7767576 * q^57 + 35494664 * q^59 + 341497340 * q^61 - 92588832 * q^63 + 15177500 * q^65 + 288195816 * q^67 + 59556384 * q^69 - 210286064 * q^71 - 232663084 * q^73 + 63281250 * q^75 - 727042176 * q^77 + 24755040 * q^79 + 86093442 * q^81 + 372082152 * q^83 - 98097500 * q^85 - 216935172 * q^87 - 427639116 * q^89 + 1126859328 * q^91 - 873376992 * q^93 + 59935000 * q^95 + 1771658884 * q^97 + 141140232 * q^99

Introduction

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