LMFDB - Embedded newform 240.10.a.r.1.1

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			1.1
Root			$8.26209$ 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)

$f(q)$	$=$	$q+81.0000 q^{3} +625.000 q^{5} -12272.1 q^{7} +6561.00 q^{9} +O(q^{10})$	$q+81.0000 q^{3} +625.000 q^{5} -12272.1 q^{7} +6561.00 q^{9} +65897.9 q^{11} -112300. q^{13} +50625.0 q^{15} +96634.7 q^{17} +181332. q^{19} -994042. q^{21} -244145. q^{23} +390625. q^{25} +531441. q^{27} -5.26461e6 q^{29} -1.83457e6 q^{31} +5.33773e6 q^{33} -7.67008e6 q^{35} +6.36194e6 q^{37} -9.09628e6 q^{39} +1.57111e6 q^{41} +1.99504e7 q^{43} +4.10062e6 q^{45} -3.00961e7 q^{47} +1.10251e8 q^{49} +7.82741e6 q^{51} -2.57306e6 q^{53} +4.11862e7 q^{55} +1.46879e7 q^{57} +1.19004e8 q^{59} +1.92875e8 q^{61} -8.05174e7 q^{63} -7.01874e7 q^{65} +1.20193e8 q^{67} -1.97757e7 q^{69} +699549. q^{71} -8.91287e7 q^{73} +3.16406e7 q^{75} -8.08707e8 q^{77} -4.31205e8 q^{79} +4.30467e7 q^{81} +1.69761e7 q^{83} +6.03967e7 q^{85} -4.26433e8 q^{87} -3.09863e6 q^{89} +1.37816e9 q^{91} -1.48600e8 q^{93} +1.13332e8 q^{95} +5.72609e8 q^{97} +4.32356e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−12272.1	−1.93187	−0.965936	−	0.258780i	$-0.916680\pi$
$7$	−12272.1	−1.93187	−0.965936	+	0.258780i	$0.916680\pi$
$8$	0	0
$9$	6561.00	0.333333
$10$	0	0
$11$	65897.9	1.35708	0.678538	−	0.734565i	$-0.262614\pi$
$11$	65897.9	1.35708	0.678538	+	0.734565i	$0.262614\pi$
$12$	0	0
$13$	−112300.	−1.09052	−0.545260	−	0.838267i	$-0.683569\pi$
$13$	−112300.	−1.09052	−0.545260	+	0.838267i	$0.683569\pi$
$14$	0	0
$15$	50625.0	0.258199
$16$	0	0
$17$	96634.7	0.280616	0.140308	−	0.990108i	$-0.455191\pi$
$17$	96634.7	0.280616	0.140308	+	0.990108i	$0.455191\pi$
$18$	0	0
$19$	181332.	0.319214	0.159607	−	0.987181i	$-0.448977\pi$
$19$	181332.	0.319214	0.159607	+	0.987181i	$0.448977\pi$
$20$	0	0
$21$	−994042.	−1.11537
$22$	0	0
$23$	−244145.	−0.181916	−0.0909582	−	0.995855i	$-0.528993\pi$
$23$	−244145.	−0.181916	−0.0909582	+	0.995855i	$0.528993\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	0	0
$27$	531441.	0.192450
$28$	0	0
$29$	−5.26461e6	−1.38221	−0.691107	−	0.722752i	$-0.742877\pi$
$29$	−5.26461e6	−1.38221	−0.691107	+	0.722752i	$0.742877\pi$
$30$	0	0
$31$	−1.83457e6	−0.356784	−0.178392	−	0.983959i	$-0.557090\pi$
$31$	−1.83457e6	−0.356784	−0.178392	+	0.983959i	$0.557090\pi$
$32$	0	0
$33$	5.33773e6	0.783508
$34$	0	0
$35$	−7.67008e6	−0.863960
$36$	0	0
$37$	6.36194e6	0.558061	0.279030	−	0.960282i	$-0.409987\pi$
$37$	6.36194e6	0.558061	0.279030	+	0.960282i	$0.409987\pi$
$38$	0	0
$39$	−9.09628e6	−0.629612
$40$	0	0
$41$	1.57111e6	0.0868319	0.0434159	−	0.999057i	$-0.486176\pi$
$41$	1.57111e6	0.0868319	0.0434159	+	0.999057i	$0.486176\pi$
$42$	0	0
$43$	1.99504e7	0.889903	0.444952	−	0.895555i	$-0.353221\pi$
$43$	1.99504e7	0.889903	0.444952	+	0.895555i	$0.353221\pi$
$44$	0	0
$45$	4.10062e6	0.149071
$46$	0	0
$47$	−3.00961e7	−0.899643	−0.449821	−	0.893119i	$-0.648512\pi$
$47$	−3.00961e7	−0.899643	−0.449821	+	0.893119i	$0.648512\pi$
$48$	0	0
$49$	1.10251e8	2.73213
$50$	0	0
$51$	7.82741e6	0.162014
$52$	0	0
$53$	−2.57306e6	−0.0447929	−0.0223964	−	0.999749i	$-0.507130\pi$
$53$	−2.57306e6	−0.0447929	−0.0223964	+	0.999749i	$0.507130\pi$
$54$	0	0
$55$	4.11862e7	0.606903
$56$	0	0
$57$	1.46879e7	0.184299
$58$	0	0
$59$	1.19004e8	1.27858	0.639290	−	0.768965i	$-0.279228\pi$
$59$	1.19004e8	1.27858	0.639290	+	0.768965i	$0.279228\pi$
$60$	0	0
$61$	1.92875e8	1.78358	0.891790	−	0.452449i	$-0.149450\pi$
$61$	1.92875e8	1.78358	0.891790	+	0.452449i	$0.149450\pi$
$62$	0	0
$63$	−8.05174e7	−0.643958
$64$	0	0
$65$	−7.01874e7	−0.487696
$66$	0	0
$67$	1.20193e8	0.728691	0.364345	−	0.931264i	$-0.381293\pi$
$67$	1.20193e8	0.728691	0.364345	+	0.931264i	$0.381293\pi$
$68$	0	0
$69$	−1.97757e7	−0.105030
$70$	0	0
$71$	699549.	0.00326705	0.00163352	−	0.999999i	$-0.499480\pi$
$71$	699549.	0.00326705	0.00163352	+	0.999999i	$0.499480\pi$
$72$	0	0
$73$	−8.91287e7	−0.367337	−0.183669	−	0.982988i	$-0.558797\pi$
$73$	−8.91287e7	−0.367337	−0.183669	+	0.982988i	$0.558797\pi$
$74$	0	0
$75$	3.16406e7	0.115470
$76$	0	0
$77$	−8.08707e8	−2.62170
$78$	0	0
$79$	−4.31205e8	−1.24555	−0.622776	−	0.782400i	$-0.713995\pi$
$79$	−4.31205e8	−1.24555	−0.622776	+	0.782400i	$0.713995\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	1.69761e7	0.0392633	0.0196316	−	0.999807i	$-0.493751\pi$
$83$	1.69761e7	0.0392633	0.0196316	+	0.999807i	$0.493751\pi$
$84$	0	0
$85$	6.03967e7	0.125495
$86$	0	0
$87$	−4.26433e8	−0.798022
$88$	0	0
$89$	−3.09863e6	−0.00523498	−0.00261749	−	0.999997i	$-0.500833\pi$
$89$	−3.09863e6	−0.00523498	−0.00261749	+	0.999997i	$0.500833\pi$
$90$	0	0
$91$	1.37816e9	2.10675
$92$	0	0
$93$	−1.48600e8	−0.205989
$94$	0	0
$95$	1.13332e8	0.142757
$96$	0	0
$97$	5.72609e8	0.656727	0.328364	−	0.944551i	$-0.393503\pi$
$97$	5.72609e8	0.656727	0.328364	+	0.944551i	$0.393503\pi$
$98$	0	0
$99$	4.32356e8	0.452359
$100$	0	0
$101$	1.78445e9	1.70631	0.853156	−	0.521655i	$-0.174685\pi$
$101$	1.78445e9	1.70631	0.853156	+	0.521655i	$0.174685\pi$
$102$	0	0
$103$	−1.16632e9	−1.02106	−0.510528	−	0.859861i	$-0.670550\pi$
$103$	−1.16632e9	−1.02106	−0.510528	+	0.859861i	$0.670550\pi$
$104$	0	0
$105$	−6.21276e8	−0.498807
$106$	0	0
$107$	1.72229e9	1.27022	0.635109	−	0.772422i	$-0.280955\pi$
$107$	1.72229e9	1.27022	0.635109	+	0.772422i	$0.280955\pi$
$108$	0	0
$109$	2.08468e9	1.41456	0.707279	−	0.706934i	$-0.249922\pi$
$109$	2.08468e9	1.41456	0.707279	+	0.706934i	$0.249922\pi$
$110$	0	0
$111$	5.15317e8	0.322197
$112$	0	0
$113$	−1.81995e9	−1.05004	−0.525020	−	0.851090i	$-0.675942\pi$
$113$	−1.81995e9	−1.05004	−0.525020	+	0.851090i	$0.675942\pi$
$114$	0	0
$115$	−1.52590e8	−0.0813555
$116$	0	0
$117$	−7.36799e8	−0.363507
$118$	0	0
$119$	−1.18591e9	−0.542115
$120$	0	0
$121$	1.98458e9	0.841656
$122$	0	0
$123$	1.27260e8	0.0501324
$124$	0	0
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	3.74913e9	1.27883	0.639417	−	0.768860i	$-0.279176\pi$
$127$	3.74913e9	1.27883	0.639417	+	0.768860i	$0.279176\pi$
$128$	0	0
$129$	1.61598e9	0.513786
$130$	0	0
$131$	1.96598e9	0.583256	0.291628	−	0.956532i	$-0.405803\pi$
$131$	1.96598e9	0.583256	0.291628	+	0.956532i	$0.405803\pi$
$132$	0	0
$133$	−2.22532e9	−0.616682
$134$	0	0
$135$	3.32151e8	0.0860663
$136$	0	0
$137$	−3.63223e8	−0.0880909	−0.0440455	−	0.999030i	$-0.514025\pi$
$137$	−3.63223e8	−0.0880909	−0.0440455	+	0.999030i	$0.514025\pi$
$138$	0	0
$139$	−4.55580e9	−1.03514	−0.517569	−	0.855642i	$-0.673163\pi$
$139$	−4.55580e9	−1.03514	−0.517569	+	0.855642i	$0.673163\pi$
$140$	0	0
$141$	−2.43779e9	−0.519409
$142$	0	0
$143$	−7.40032e9	−1.47992
$144$	0	0
$145$	−3.29038e9	−0.618145
$146$	0	0
$147$	8.93036e9	1.57740
$148$	0	0
$149$	8.79830e9	1.46238	0.731191	−	0.682173i	$-0.238965\pi$
$149$	8.79830e9	1.46238	0.731191	+	0.682173i	$0.238965\pi$
$150$	0	0
$151$	2.70702e8	0.0423736	0.0211868	−	0.999776i	$-0.493256\pi$
$151$	2.70702e8	0.0423736	0.0211868	+	0.999776i	$0.493256\pi$
$152$	0	0
$153$	6.34020e8	0.0935388
$154$	0	0
$155$	−1.14660e9	−0.159559
$156$	0	0
$157$	9.73240e9	1.27841	0.639207	−	0.769035i	$-0.279263\pi$
$157$	9.73240e9	1.27841	0.639207	+	0.769035i	$0.279263\pi$
$158$	0	0
$159$	−2.08418e8	−0.0258612
$160$	0	0
$161$	2.99617e9	0.351439
$162$	0	0
$163$	−7.76823e9	−0.861941	−0.430971	−	0.902366i	$-0.641829\pi$
$163$	−7.76823e9	−0.861941	−0.430971	+	0.902366i	$0.641829\pi$
$164$	0	0
$165$	3.33608e9	0.350396
$166$	0	0
$167$	8.41748e9	0.837448	0.418724	−	0.908114i	$-0.362478\pi$
$167$	8.41748e9	0.837448	0.418724	+	0.908114i	$0.362478\pi$
$168$	0	0
$169$	2.00674e9	0.189235
$170$	0	0
$171$	1.18972e9	0.106405
$172$	0	0
$173$	3.59838e8	0.0305422	0.0152711	−	0.999883i	$-0.495139\pi$
$173$	3.59838e8	0.0305422	0.0152711	+	0.999883i	$0.495139\pi$
$174$	0	0
$175$	−4.79380e9	−0.386375
$176$	0	0
$177$	9.63934e9	0.738189
$178$	0	0
$179$	8.65338e9	0.630009	0.315005	−	0.949090i	$-0.397994\pi$
$179$	8.65338e9	0.630009	0.315005	+	0.949090i	$0.397994\pi$
$180$	0	0
$181$	6.09713e9	0.422252	0.211126	−	0.977459i	$-0.432287\pi$
$181$	6.09713e9	0.422252	0.211126	+	0.977459i	$0.432287\pi$
$182$	0	0
$183$	1.56229e10	1.02975
$184$	0	0
$185$	3.97621e9	0.249572
$186$	0	0
$187$	6.36802e9	0.380818
$188$	0	0
$189$	−6.52191e9	−0.371789
$190$	0	0
$191$	2.96238e10	1.61061	0.805306	−	0.592859i	$-0.202001\pi$
$191$	2.96238e10	1.61061	0.805306	+	0.592859i	$0.202001\pi$
$192$	0	0
$193$	−4.63438e8	−0.0240427	−0.0120214	−	0.999928i	$-0.503827\pi$
$193$	−4.63438e8	−0.0240427	−0.0120214	+	0.999928i	$0.503827\pi$
$194$	0	0
$195$	−5.68518e9	−0.281571
$196$	0	0
$197$	1.23414e10	0.583803	0.291902	−	0.956448i	$-0.405712\pi$
$197$	1.23414e10	0.583803	0.291902	+	0.956448i	$0.405712\pi$
$198$	0	0
$199$	1.00354e10	0.453624	0.226812	−	0.973939i	$-0.427170\pi$
$199$	1.00354e10	0.453624	0.226812	+	0.973939i	$0.427170\pi$
$200$	0	0
$201$	9.73565e9	0.420710
$202$	0	0
$203$	6.46080e10	2.67026
$204$	0	0
$205$	9.81943e8	0.0388324
$206$	0	0
$207$	−1.60183e9	−0.0606388
$208$	0	0
$209$	1.19494e10	0.433198
$210$	0	0
$211$	3.56268e10	1.23739	0.618693	−	0.785633i	$-0.287662\pi$
$211$	3.56268e10	1.23739	0.618693	+	0.785633i	$0.287662\pi$
$212$	0	0
$213$	5.66635e7	0.00188623
$214$	0	0
$215$	1.24690e10	0.397977
$216$	0	0
$217$	2.25140e10	0.689262
$218$	0	0
$219$	−7.21943e9	−0.212082
$220$	0	0
$221$	−1.08521e10	−0.306018
$222$	0	0
$223$	3.41646e10	0.925133	0.462566	−	0.886585i	$-0.346929\pi$
$223$	3.41646e10	0.925133	0.462566	+	0.886585i	$0.346929\pi$
$224$	0	0
$225$	2.56289e9	0.0666667
$226$	0	0
$227$	−3.64936e10	−0.912221	−0.456110	−	0.889923i	$-0.650758\pi$
$227$	−3.64936e10	−0.912221	−0.456110	+	0.889923i	$0.650758\pi$
$228$	0	0
$229$	3.87446e10	0.931005	0.465502	−	0.885047i	$-0.345874\pi$
$229$	3.87446e10	0.931005	0.465502	+	0.885047i	$0.345874\pi$
$230$	0	0
$231$	−6.55052e10	−1.51364
$232$	0	0
$233$	−2.68717e10	−0.597302	−0.298651	−	0.954362i	$-0.596537\pi$
$233$	−2.68717e10	−0.597302	−0.298651	+	0.954362i	$0.596537\pi$
$234$	0	0
$235$	−1.88101e10	−0.402332
$236$	0	0
$237$	−3.49276e10	−0.719120
$238$	0	0
$239$	−1.34711e10	−0.267063	−0.133531	−	0.991045i	$-0.542632\pi$
$239$	−1.34711e10	−0.267063	−0.133531	+	0.991045i	$0.542632\pi$
$240$	0	0
$241$	−1.85860e10	−0.354902	−0.177451	−	0.984130i	$-0.556785\pi$
$241$	−1.85860e10	−0.354902	−0.177451	+	0.984130i	$0.556785\pi$
$242$	0	0
$243$	3.48678e9	0.0641500
$244$	0	0
$245$	6.89071e10	1.22185
$246$	0	0
$247$	−2.03635e10	−0.348110
$248$	0	0
$249$	1.37506e9	0.0226687
$250$	0	0
$251$	−4.30327e10	−0.684332	−0.342166	−	0.939640i	$-0.611161\pi$
$251$	−4.30327e10	−0.684332	−0.342166	+	0.939640i	$0.611161\pi$
$252$	0	0
$253$	−1.60886e10	−0.246875
$254$	0	0
$255$	4.89213e9	0.0724548
$256$	0	0
$257$	−6.26858e8	−0.00896335	−0.00448168	−	0.999990i	$-0.501427\pi$
$257$	−6.26858e8	−0.00896335	−0.00448168	+	0.999990i	$0.501427\pi$
$258$	0	0
$259$	−7.80745e10	−1.07810
$260$	0	0
$261$	−3.45411e10	−0.460738
$262$	0	0
$263$	−6.21446e10	−0.800944	−0.400472	−	0.916309i	$-0.631154\pi$
$263$	−6.21446e10	−0.800944	−0.400472	+	0.916309i	$0.631154\pi$
$264$	0	0
$265$	−1.60816e9	−0.0200320
$266$	0	0
$267$	−2.50989e8	−0.00302242
$268$	0	0
$269$	−5.44876e10	−0.634472	−0.317236	−	0.948347i	$-0.602755\pi$
$269$	−5.44876e10	−0.634472	−0.317236	+	0.948347i	$0.602755\pi$
$270$	0	0
$271$	9.74930e10	1.09802	0.549012	−	0.835815i	$-0.315004\pi$
$271$	9.74930e10	1.09802	0.549012	+	0.835815i	$0.315004\pi$
$272$	0	0
$273$	1.11631e11	1.21633
$274$	0	0
$275$	2.57414e10	0.271415
$276$	0	0
$277$	1.43195e11	1.46140	0.730702	−	0.682697i	$-0.239193\pi$
$277$	1.43195e11	1.46140	0.730702	+	0.682697i	$0.239193\pi$
$278$	0	0
$279$	−1.20366e10	−0.118928
$280$	0	0
$281$	6.51280e10	0.623146	0.311573	−	0.950222i	$-0.399144\pi$
$281$	6.51280e10	0.623146	0.311573	+	0.950222i	$0.399144\pi$
$282$	0	0
$283$	1.65431e11	1.53312	0.766561	−	0.642171i	$-0.221966\pi$
$283$	1.65431e11	1.53312	0.766561	+	0.642171i	$0.221966\pi$
$284$	0	0
$285$	9.17992e9	0.0824208
$286$	0	0
$287$	−1.92808e10	−0.167748
$288$	0	0
$289$	−1.09250e11	−0.921254
$290$	0	0
$291$	4.63813e10	0.379162
$292$	0	0
$293$	5.64261e10	0.447277	0.223638	−	0.974672i	$-0.428207\pi$
$293$	5.64261e10	0.447277	0.223638	+	0.974672i	$0.428207\pi$
$294$	0	0
$295$	7.43776e10	0.571799
$296$	0	0
$297$	3.50208e10	0.261169
$298$	0	0
$299$	2.74174e10	0.198384
$300$	0	0
$301$	−2.44833e11	−1.71918
$302$	0	0
$303$	1.44541e11	0.985140
$304$	0	0
$305$	1.20547e11	0.797641
$306$	0	0
$307$	−1.79360e10	−0.115240	−0.0576200	−	0.998339i	$-0.518351\pi$
$307$	−1.79360e10	−0.115240	−0.0576200	+	0.998339i	$0.518351\pi$
$308$	0	0
$309$	−9.44717e10	−0.589507
$310$	0	0
$311$	−3.19953e11	−1.93938	−0.969692	−	0.244330i	$-0.921432\pi$
$311$	−3.19953e11	−1.93938	−0.969692	+	0.244330i	$0.921432\pi$
$312$	0	0
$313$	−3.16987e11	−1.86677	−0.933387	−	0.358872i	$-0.883162\pi$
$313$	−3.16987e11	−1.86677	−0.933387	+	0.358872i	$0.883162\pi$
$314$	0	0
$315$	−5.03234e10	−0.287987
$316$	0	0
$317$	−3.08157e11	−1.71398	−0.856990	−	0.515334i	$-0.827668\pi$
$317$	−3.08157e11	−1.71398	−0.856990	+	0.515334i	$0.827668\pi$
$318$	0	0
$319$	−3.46927e11	−1.87577
$320$	0	0
$321$	1.39505e11	0.733361
$322$	0	0
$323$	1.75229e10	0.0895768
$324$	0	0
$325$	−4.38671e10	−0.218104
$326$	0	0
$327$	1.68859e11	0.816696
$328$	0	0
$329$	3.69343e11	1.73800
$330$	0	0
$331$	1.79184e11	0.820490	0.410245	−	0.911975i	$-0.365443\pi$
$331$	1.79184e11	0.820490	0.410245	+	0.911975i	$0.365443\pi$
$332$	0	0
$333$	4.17407e10	0.186020
$334$	0	0
$335$	7.51207e10	0.325880
$336$	0	0
$337$	−9.73351e10	−0.411088	−0.205544	−	0.978648i	$-0.565896\pi$
$337$	−9.73351e10	−0.411088	−0.205544	+	0.978648i	$0.565896\pi$
$338$	0	0
$339$	−1.47416e11	−0.606241
$340$	0	0
$341$	−1.20894e11	−0.484183
$342$	0	0
$343$	−8.57794e11	−3.34626
$344$	0	0
$345$	−1.23598e10	−0.0469706
$346$	0	0
$347$	3.47750e11	1.28761	0.643806	−	0.765189i	$-0.277354\pi$
$347$	3.47750e11	1.28761	0.643806	+	0.765189i	$0.277354\pi$
$348$	0	0
$349$	−1.10131e11	−0.397372	−0.198686	−	0.980063i	$-0.563667\pi$
$349$	−1.10131e11	−0.397372	−0.198686	+	0.980063i	$0.563667\pi$
$350$	0	0
$351$	−5.96807e10	−0.209871
$352$	0	0
$353$	−3.40052e11	−1.16563	−0.582814	−	0.812606i	$-0.698048\pi$
$353$	−3.40052e11	−1.16563	−0.582814	+	0.812606i	$0.698048\pi$
$354$	0	0
$355$	4.37218e8	0.00146107
$356$	0	0
$357$	−9.60589e10	−0.312990
$358$	0	0
$359$	2.75063e11	0.873993	0.436996	−	0.899463i	$-0.356042\pi$
$359$	2.75063e11	0.873993	0.436996	+	0.899463i	$0.356042\pi$
$360$	0	0
$361$	−2.89807e11	−0.898102
$362$	0	0
$363$	1.60751e11	0.485930
$364$	0	0
$365$	−5.57054e10	−0.164278
$366$	0	0
$367$	2.12218e11	0.610639	0.305320	−	0.952250i	$-0.401237\pi$
$367$	2.12218e11	0.610639	0.305320	+	0.952250i	$0.401237\pi$
$368$	0	0
$369$	1.03081e10	0.0289440
$370$	0	0
$371$	3.15769e10	0.0865341
$372$	0	0
$373$	6.08324e11	1.62721	0.813607	−	0.581415i	$-0.197501\pi$
$373$	6.08324e11	1.62721	0.813607	+	0.581415i	$0.197501\pi$
$374$	0	0
$375$	1.97754e10	0.0516398
$376$	0	0
$377$	5.91215e11	1.50733
$378$	0	0
$379$	−4.64136e11	−1.15550	−0.577749	−	0.816214i	$-0.696069\pi$
$379$	−4.64136e11	−1.15550	−0.577749	+	0.816214i	$0.696069\pi$
$380$	0	0
$381$	3.03680e11	0.738335
$382$	0	0
$383$	−5.22749e11	−1.24136	−0.620681	−	0.784063i	$-0.713144\pi$
$383$	−5.22749e11	−1.24136	−0.620681	+	0.784063i	$0.713144\pi$
$384$	0	0
$385$	−5.05442e11	−1.17246
$386$	0	0
$387$	1.30894e11	0.296634
$388$	0	0
$389$	−1.18899e11	−0.263271	−0.131636	−	0.991298i	$-0.542023\pi$
$389$	−1.18899e11	−0.263271	−0.131636	+	0.991298i	$0.542023\pi$
$390$	0	0
$391$	−2.35928e10	−0.0510487
$392$	0	0
$393$	1.59245e11	0.336743
$394$	0	0
$395$	−2.69503e11	−0.557028
$396$	0	0
$397$	−2.76954e11	−0.559564	−0.279782	−	0.960064i	$-0.590262\pi$
$397$	−2.76954e11	−0.559564	−0.279782	+	0.960064i	$0.590262\pi$
$398$	0	0
$399$	−1.80251e11	−0.356041
$400$	0	0
$401$	−1.53482e11	−0.296421	−0.148210	−	0.988956i	$-0.547351\pi$
$401$	−1.53482e11	−0.296421	−0.148210	+	0.988956i	$0.547351\pi$
$402$	0	0
$403$	2.06021e11	0.389080
$404$	0	0
$405$	2.69042e10	0.0496904
$406$	0	0
$407$	4.19238e11	0.757331
$408$	0	0
$409$	−1.52364e11	−0.269232	−0.134616	−	0.990898i	$-0.542980\pi$
$409$	−1.52364e11	−0.269232	−0.134616	+	0.990898i	$0.542980\pi$
$410$	0	0
$411$	−2.94211e10	−0.0508593
$412$	0	0
$413$	−1.46043e12	−2.47006
$414$	0	0
$415$	1.06101e10	0.0175591
$416$	0	0
$417$	−3.69020e11	−0.597637
$418$	0	0
$419$	9.09824e11	1.44210	0.721049	−	0.692885i	$-0.243660\pi$
$419$	9.09824e11	1.44210	0.721049	+	0.692885i	$0.243660\pi$
$420$	0	0
$421$	9.88529e11	1.53363	0.766814	−	0.641869i	$-0.221841\pi$
$421$	9.88529e11	1.53363	0.766814	+	0.641869i	$0.221841\pi$
$422$	0	0
$423$	−1.97461e11	−0.299881
$424$	0	0
$425$	3.77479e10	0.0561233
$426$	0	0
$427$	−2.36699e12	−3.44565
$428$	0	0
$429$	−5.99426e11	−0.854432
$430$	0	0
$431$	−2.03278e11	−0.283755	−0.141878	−	0.989884i	$-0.545314\pi$
$431$	−2.03278e11	−0.283755	−0.141878	+	0.989884i	$0.545314\pi$
$432$	0	0
$433$	−2.10847e11	−0.288251	−0.144126	−	0.989559i	$-0.546037\pi$
$433$	−2.10847e11	−0.288251	−0.144126	+	0.989559i	$0.546037\pi$
$434$	0	0
$435$	−2.66521e11	−0.356886
$436$	0	0
$437$	−4.42712e10	−0.0580704
$438$	0	0
$439$	−2.11413e11	−0.271670	−0.135835	−	0.990732i	$-0.543372\pi$
$439$	−2.11413e11	−0.271670	−0.135835	+	0.990732i	$0.543372\pi$
$440$	0	0
$441$	7.23359e11	0.910711
$442$	0	0
$443$	−1.45517e12	−1.79514	−0.897569	−	0.440874i	$-0.854669\pi$
$443$	−1.45517e12	−1.79514	−0.897569	+	0.440874i	$0.854669\pi$
$444$	0	0
$445$	−1.93665e9	−0.00234115
$446$	0	0
$447$	7.12663e11	0.844306
$448$	0	0
$449$	−2.19470e11	−0.254840	−0.127420	−	0.991849i	$-0.540670\pi$
$449$	−2.19470e11	−0.254840	−0.127420	+	0.991849i	$0.540670\pi$
$450$	0	0
$451$	1.03533e11	0.117837
$452$	0	0
$453$	2.19269e10	0.0244644
$454$	0	0
$455$	8.61348e11	0.942166
$456$	0	0
$457$	7.40816e11	0.794488	0.397244	−	0.917713i	$-0.369967\pi$
$457$	7.40816e11	0.794488	0.397244	+	0.917713i	$0.369967\pi$
$458$	0	0
$459$	5.13556e10	0.0540046
$460$	0	0
$461$	2.80170e11	0.288913	0.144457	−	0.989511i	$-0.453857\pi$
$461$	2.80170e11	0.288913	0.144457	+	0.989511i	$0.453857\pi$
$462$	0	0
$463$	5.07092e10	0.0512828	0.0256414	−	0.999671i	$-0.491837\pi$
$463$	5.07092e10	0.0512828	0.0256414	+	0.999671i	$0.491837\pi$
$464$	0	0
$465$	−9.28749e10	−0.0921213
$466$	0	0
$467$	−1.77228e12	−1.72427	−0.862135	−	0.506678i	$-0.830873\pi$
$467$	−1.77228e12	−1.72427	−0.862135	+	0.506678i	$0.830873\pi$
$468$	0	0
$469$	−1.47503e12	−1.40774
$470$	0	0
$471$	7.88324e11	0.738092
$472$	0	0
$473$	1.31469e12	1.20767
$474$	0	0
$475$	7.08327e10	0.0638429
$476$	0	0
$477$	−1.68819e10	−0.0149310
$478$	0	0
$479$	7.79555e11	0.676608	0.338304	−	0.941037i	$-0.390147\pi$
$479$	7.79555e11	0.676608	0.338304	+	0.941037i	$0.390147\pi$
$480$	0	0
$481$	−7.14444e11	−0.608577
$482$	0	0
$483$	2.42690e11	0.202904
$484$	0	0
$485$	3.57880e11	0.293697
$486$	0	0
$487$	1.33958e12	1.07916	0.539582	−	0.841933i	$-0.318582\pi$
$487$	1.33958e12	1.07916	0.539582	+	0.841933i	$0.318582\pi$
$488$	0	0
$489$	−6.29226e11	−0.497642
$490$	0	0
$491$	9.76349e11	0.758121	0.379061	−	0.925372i	$-0.376247\pi$
$491$	9.76349e11	0.758121	0.379061	+	0.925372i	$0.376247\pi$
$492$	0	0
$493$	−5.08744e11	−0.387872
$494$	0	0
$495$	2.70222e11	0.202301
$496$	0	0
$497$	−8.58495e9	−0.00631152
$498$	0	0
$499$	−1.32057e12	−0.953472	−0.476736	−	0.879046i	$-0.658180\pi$
$499$	−1.32057e12	−0.953472	−0.476736	+	0.879046i	$0.658180\pi$
$500$	0	0
$501$	6.81815e11	0.483501
$502$	0	0
$503$	1.16421e12	0.810917	0.405459	−	0.914113i	$-0.367112\pi$
$503$	1.16421e12	0.810917	0.405459	+	0.914113i	$0.367112\pi$
$504$	0	0
$505$	1.11528e12	0.763086
$506$	0	0
$507$	1.62546e11	0.109255
$508$	0	0
$509$	8.32285e11	0.549594	0.274797	−	0.961502i	$-0.411389\pi$
$509$	8.32285e11	0.549594	0.274797	+	0.961502i	$0.411389\pi$
$510$	0	0
$511$	1.09380e12	0.709649
$512$	0	0
$513$	9.63671e10	0.0614329
$514$	0	0
$515$	−7.28949e11	−0.456630
$516$	0	0
$517$	−1.98327e12	−1.22088
$518$	0	0
$519$	2.91469e10	0.0176335
$520$	0	0
$521$	2.43459e12	1.44763	0.723813	−	0.689997i	$-0.242388\pi$
$521$	2.43459e12	1.44763	0.723813	+	0.689997i	$0.242388\pi$
$522$	0	0
$523$	1.07631e12	0.629043	0.314522	−	0.949250i	$-0.398156\pi$
$523$	1.07631e12	0.629043	0.314522	+	0.949250i	$0.398156\pi$
$524$	0	0
$525$	−3.88298e11	−0.223073
$526$	0	0
$527$	−1.77283e11	−0.100119
$528$	0	0
$529$	−1.74155e12	−0.966906
$530$	0	0
$531$	7.80787e11	0.426194
$532$	0	0
$533$	−1.76435e11	−0.0946919
$534$	0	0
$535$	1.07643e12	0.568059
$536$	0	0
$537$	7.00923e11	0.363736
$538$	0	0
$539$	7.26533e12	3.70771
$540$	0	0
$541$	−3.12532e12	−1.56858	−0.784291	−	0.620393i	$-0.786973\pi$
$541$	−3.12532e12	−1.56858	−0.784291	+	0.620393i	$0.786973\pi$
$542$	0	0
$543$	4.93867e11	0.243787
$544$	0	0
$545$	1.30293e12	0.632610
$546$	0	0
$547$	−1.59061e12	−0.759663	−0.379832	−	0.925056i	$-0.624018\pi$
$547$	−1.59061e12	−0.759663	−0.379832	+	0.925056i	$0.624018\pi$
$548$	0	0
$549$	1.26546e12	0.594527
$550$	0	0
$551$	−9.54641e11	−0.441223
$552$	0	0
$553$	5.29180e12	2.40625
$554$	0	0
$555$	3.22073e11	0.144091
$556$	0	0
$557$	3.76617e12	1.65787	0.828937	−	0.559343i	$-0.188946\pi$
$557$	3.76617e12	1.65787	0.828937	+	0.559343i	$0.188946\pi$
$558$	0	0
$559$	−2.24042e12	−0.970458
$560$	0	0
$561$	5.15810e11	0.219865
$562$	0	0
$563$	2.32486e12	0.975234	0.487617	−	0.873058i	$-0.337866\pi$
$563$	2.32486e12	0.975234	0.487617	+	0.873058i	$0.337866\pi$
$564$	0	0
$565$	−1.13747e12	−0.469592
$566$	0	0
$567$	−5.28275e11	−0.214653
$568$	0	0
$569$	3.30244e11	0.132078	0.0660388	−	0.997817i	$-0.478964\pi$
$569$	3.30244e11	0.132078	0.0660388	+	0.997817i	$0.478964\pi$
$570$	0	0
$571$	−3.83190e12	−1.50852	−0.754261	−	0.656575i	$-0.772005\pi$
$571$	−3.83190e12	−1.50852	−0.754261	+	0.656575i	$0.772005\pi$
$572$	0	0
$573$	2.39953e12	0.929887
$574$	0	0
$575$	−9.53690e10	−0.0363833
$576$	0	0
$577$	−1.66616e12	−0.625786	−0.312893	−	0.949788i	$-0.601298\pi$
$577$	−1.66616e12	−0.625786	−0.312893	+	0.949788i	$0.601298\pi$
$578$	0	0
$579$	−3.75384e10	−0.0138811
$580$	0	0
$581$	−2.08333e11	−0.0758517
$582$	0	0
$583$	−1.69559e11	−0.0607874
$584$	0	0
$585$	−4.60499e11	−0.162565
$586$	0	0
$587$	−2.00482e12	−0.696953	−0.348476	−	0.937318i	$-0.613301\pi$
$587$	−2.00482e12	−0.696953	−0.348476	+	0.937318i	$0.613301\pi$
$588$	0	0
$589$	−3.32665e11	−0.113891
$590$	0	0
$591$	9.99654e11	0.337059
$592$	0	0
$593$	1.46848e12	0.487666	0.243833	−	0.969817i	$-0.421595\pi$
$593$	1.46848e12	0.487666	0.243833	+	0.969817i	$0.421595\pi$
$594$	0	0
$595$	−7.41195e11	−0.242441
$596$	0	0
$597$	8.12868e11	0.261900
$598$	0	0
$599$	−5.67302e12	−1.80050	−0.900251	−	0.435370i	$-0.856617\pi$
$599$	−5.67302e12	−1.80050	−0.900251	+	0.435370i	$0.856617\pi$
$600$	0	0
$601$	2.23747e12	0.699555	0.349778	−	0.936833i	$-0.386257\pi$
$601$	2.23747e12	0.699555	0.349778	+	0.936833i	$0.386257\pi$
$602$	0	0
$603$	7.88587e11	0.242897
$604$	0	0
$605$	1.24036e12	0.376400
$606$	0	0
$607$	−7.87986e11	−0.235597	−0.117798	−	0.993038i	$-0.537584\pi$
$607$	−7.87986e11	−0.235597	−0.117798	+	0.993038i	$0.537584\pi$
$608$	0	0
$609$	5.23324e12	1.54168
$610$	0	0
$611$	3.37979e12	0.981079
$612$	0	0
$613$	−4.35930e12	−1.24694	−0.623468	−	0.781848i	$-0.714277\pi$
$613$	−4.35930e12	−1.24694	−0.623468	+	0.781848i	$0.714277\pi$
$614$	0	0
$615$	7.95374e10	0.0224199
$616$	0	0
$617$	−2.04791e12	−0.568889	−0.284444	−	0.958693i	$-0.591809\pi$
$617$	−2.04791e12	−0.568889	−0.284444	+	0.958693i	$0.591809\pi$
$618$	0	0
$619$	6.31079e12	1.72773	0.863865	−	0.503723i	$-0.168037\pi$
$619$	6.31079e12	1.72773	0.863865	+	0.503723i	$0.168037\pi$
$620$	0	0
$621$	−1.29748e11	−0.0350098
$622$	0	0
$623$	3.80268e10	0.0101133
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	0	0
$627$	9.67899e11	0.250107
$628$	0	0
$629$	6.14784e11	0.156601
$630$	0	0
$631$	−4.94106e12	−1.24076	−0.620380	−	0.784301i	$-0.713022\pi$
$631$	−4.94106e12	−1.24076	−0.620380	+	0.784301i	$0.713022\pi$
$632$	0	0
$633$	2.88577e12	0.714405
$634$	0	0
$635$	2.34321e12	0.571912
$636$	0	0
$637$	−1.23812e13	−2.97945
$638$	0	0
$639$	4.58974e9	0.00108902
$640$	0	0
$641$	−6.55819e11	−0.153434	−0.0767172	−	0.997053i	$-0.524444\pi$
$641$	−6.55819e11	−0.153434	−0.0767172	+	0.997053i	$0.524444\pi$
$642$	0	0
$643$	−5.76324e11	−0.132959	−0.0664794	−	0.997788i	$-0.521177\pi$
$643$	−5.76324e11	−0.132959	−0.0664794	+	0.997788i	$0.521177\pi$
$644$	0	0
$645$	1.00999e12	0.229772
$646$	0	0
$647$	3.52372e12	0.790556	0.395278	−	0.918562i	$-0.370648\pi$
$647$	3.52372e12	0.790556	0.395278	+	0.918562i	$0.370648\pi$
$648$	0	0
$649$	7.84212e12	1.73513
$650$	0	0
$651$	1.82364e12	0.397945
$652$	0	0
$653$	−4.37841e11	−0.0942338	−0.0471169	−	0.998889i	$-0.515003\pi$
$653$	−4.37841e11	−0.0942338	−0.0471169	+	0.998889i	$0.515003\pi$
$654$	0	0
$655$	1.22874e12	0.260840
$656$	0	0
$657$	−5.84774e11	−0.122446
$658$	0	0
$659$	−1.98477e12	−0.409945	−0.204972	−	0.978768i	$-0.565710\pi$
$659$	−1.98477e12	−0.409945	−0.204972	+	0.978768i	$0.565710\pi$
$660$	0	0
$661$	2.80945e12	0.572420	0.286210	−	0.958167i	$-0.407604\pi$
$661$	2.80945e12	0.572420	0.286210	+	0.958167i	$0.407604\pi$
$662$	0	0
$663$	−8.79016e11	−0.176679
$664$	0	0
$665$	−1.39083e12	−0.275788
$666$	0	0
$667$	1.28533e12	0.251447
$668$	0	0
$669$	2.76733e12	0.534126
$670$	0	0
$671$	1.27101e13	2.42045
$672$	0	0
$673$	−2.08224e12	−0.391258	−0.195629	−	0.980678i	$-0.562675\pi$
$673$	−2.08224e12	−0.391258	−0.195629	+	0.980678i	$0.562675\pi$
$674$	0	0
$675$	2.07594e11	0.0384900
$676$	0	0
$677$	−6.44805e12	−1.17972	−0.589861	−	0.807505i	$-0.700817\pi$
$677$	−6.44805e12	−1.17972	−0.589861	+	0.807505i	$0.700817\pi$
$678$	0	0
$679$	−7.02712e12	−1.26871
$680$	0	0
$681$	−2.95598e12	−0.526671
$682$	0	0
$683$	4.02024e12	0.706902	0.353451	−	0.935453i	$-0.385008\pi$
$683$	4.02024e12	0.706902	0.353451	+	0.935453i	$0.385008\pi$
$684$	0	0
$685$	−2.27015e11	−0.0393955
$686$	0	0
$687$	3.13832e12	0.537516
$688$	0	0
$689$	2.88954e11	0.0488476
$690$	0	0
$691$	8.43146e12	1.40686	0.703431	−	0.710763i	$-0.251650\pi$
$691$	8.43146e12	1.40686	0.703431	+	0.710763i	$0.251650\pi$
$692$	0	0
$693$	−5.30592e12	−0.873900
$694$	0	0
$695$	−2.84737e12	−0.462927
$696$	0	0
$697$	1.51824e11	0.0243664
$698$	0	0
$699$	−2.17661e12	−0.344853
$700$	0	0
$701$	6.55956e12	1.02599	0.512995	−	0.858391i	$-0.328536\pi$
$701$	6.55956e12	1.02599	0.512995	+	0.858391i	$0.328536\pi$
$702$	0	0
$703$	1.15362e12	0.178141
$704$	0	0
$705$	−1.52362e12	−0.232287
$706$	0	0
$707$	−2.18990e13	−3.29638
$708$	0	0
$709$	−5.30959e12	−0.789138	−0.394569	−	0.918866i	$-0.629106\pi$
$709$	−5.30959e12	−0.789138	−0.394569	+	0.918866i	$0.629106\pi$
$710$	0	0
$711$	−2.82914e12	−0.415184
$712$	0	0
$713$	4.47899e11	0.0649049
$714$	0	0
$715$	−4.62520e12	−0.661840
$716$	0	0
$717$	−1.09116e12	−0.154189
$718$	0	0
$719$	−1.02053e13	−1.42411	−0.712056	−	0.702123i	$-0.752236\pi$
$719$	−1.02053e13	−1.42411	−0.712056	+	0.702123i	$0.752236\pi$
$720$	0	0
$721$	1.43132e13	1.97255
$722$	0	0
$723$	−1.50546e12	−0.204903
$724$	0	0
$725$	−2.05649e12	−0.276443
$726$	0	0
$727$	7.45813e12	0.990206	0.495103	−	0.868834i	$-0.335130\pi$
$727$	7.45813e12	0.990206	0.495103	+	0.868834i	$0.335130\pi$
$728$	0	0
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	1.92790e12	0.249721
$732$	0	0
$733$	−1.06610e13	−1.36405	−0.682024	−	0.731329i	$-0.738900\pi$
$733$	−1.06610e13	−1.36405	−0.682024	+	0.731329i	$0.738900\pi$
$734$	0	0
$735$	5.58148e12	0.705433
$736$	0	0
$737$	7.92047e12	0.988889
$738$	0	0
$739$	−2.33179e12	−0.287601	−0.143800	−	0.989607i	$-0.545932\pi$
$739$	−2.33179e12	−0.287601	−0.143800	+	0.989607i	$0.545932\pi$
$740$	0	0
$741$	−1.64944e12	−0.200981
$742$	0	0
$743$	7.80869e12	0.940001	0.470001	−	0.882666i	$-0.344254\pi$
$743$	7.80869e12	0.940001	0.470001	+	0.882666i	$0.344254\pi$
$744$	0	0
$745$	5.49894e12	0.653997
$746$	0	0
$747$	1.11380e11	0.0130878
$748$	0	0
$749$	−2.11361e13	−2.45390
$750$	0	0
$751$	−4.20664e12	−0.482565	−0.241283	−	0.970455i	$-0.577568\pi$
$751$	−4.20664e12	−0.482565	−0.241283	+	0.970455i	$0.577568\pi$
$752$	0	0
$753$	−3.48565e12	−0.395099
$754$	0	0
$755$	1.69189e11	0.0189501
$756$	0	0
$757$	−1.54834e13	−1.71370	−0.856848	−	0.515570i	$-0.827580\pi$
$757$	−1.54834e13	−1.71370	−0.856848	+	0.515570i	$0.827580\pi$
$758$	0	0
$759$	−1.30318e12	−0.142533
$760$	0	0
$761$	9.77615e12	1.05666	0.528332	−	0.849038i	$-0.322818\pi$
$761$	9.77615e12	1.05666	0.528332	+	0.849038i	$0.322818\pi$
$762$	0	0
$763$	−2.55835e13	−2.73275
$764$	0	0
$765$	3.96263e11	0.0418318
$766$	0	0
$767$	−1.33641e13	−1.39432
$768$	0	0
$769$	3.35411e12	0.345867	0.172933	−	0.984934i	$-0.444675\pi$
$769$	3.35411e12	0.345867	0.172933	+	0.984934i	$0.444675\pi$
$770$	0	0
$771$	−5.07755e10	−0.00517499
$772$	0	0
$773$	−6.04474e12	−0.608933	−0.304467	−	0.952523i	$-0.598478\pi$
$773$	−6.04474e12	−0.608933	−0.304467	+	0.952523i	$0.598478\pi$
$774$	0	0
$775$	−7.16627e11	−0.0713568
$776$	0	0
$777$	−6.32403e12	−0.622443
$778$	0	0
$779$	2.84892e11	0.0277180
$780$	0	0
$781$	4.60988e10	0.00443364
$782$	0	0
$783$	−2.79783e12	−0.266007
$784$	0	0
$785$	6.08275e12	0.571724
$786$	0	0
$787$	−4.16503e12	−0.387019	−0.193509	−	0.981098i	$-0.561987\pi$
$787$	−4.16503e12	−0.387019	−0.193509	+	0.981098i	$0.561987\pi$
$788$	0	0
$789$	−5.03371e12	−0.462425
$790$	0	0
$791$	2.23346e13	2.02854
$792$	0	0
$793$	−2.16599e13	−1.94503
$794$	0	0
$795$	−1.30261e11	−0.0115655
$796$	0	0
$797$	2.14334e13	1.88160	0.940802	−	0.338957i	$-0.110074\pi$
$797$	2.14334e13	1.88160	0.940802	+	0.338957i	$0.110074\pi$
$798$	0	0
$799$	−2.90833e12	−0.252454
$800$	0	0
$801$	−2.03301e10	−0.00174499
$802$	0	0
$803$	−5.87339e12	−0.498504
$804$	0	0
$805$	1.87261e12	0.157168
$806$	0	0
$807$	−4.41350e12	−0.366313
$808$	0	0
$809$	−1.82139e11	−0.0149498	−0.00747490	−	0.999972i	$-0.502379\pi$
$809$	−1.82139e11	−0.0149498	−0.00747490	+	0.999972i	$0.502379\pi$
$810$	0	0
$811$	1.32191e13	1.07302	0.536511	−	0.843894i	$-0.319742\pi$
$811$	1.32191e13	1.07302	0.536511	+	0.843894i	$0.319742\pi$
$812$	0	0
$813$	7.89693e12	0.633944
$814$	0	0
$815$	−4.85514e12	−0.385472
$816$	0	0
$817$	3.61763e12	0.284070
$818$	0	0
$819$	9.04209e12	0.702249
$820$	0	0
$821$	−1.15652e13	−0.888398	−0.444199	−	0.895928i	$-0.646512\pi$
$821$	−1.15652e13	−0.888398	−0.444199	+	0.895928i	$0.646512\pi$
$822$	0	0
$823$	−9.35914e12	−0.711110	−0.355555	−	0.934655i	$-0.615708\pi$
$823$	−9.35914e12	−0.711110	−0.355555	+	0.934655i	$0.615708\pi$
$824$	0	0
$825$	2.08505e12	0.156702
$826$	0	0
$827$	2.25652e13	1.67751	0.838753	−	0.544512i	$-0.183285\pi$
$827$	2.25652e13	1.67751	0.838753	+	0.544512i	$0.183285\pi$
$828$	0	0
$829$	1.22484e13	0.900705	0.450353	−	0.892851i	$-0.351298\pi$
$829$	1.22484e13	0.900705	0.450353	+	0.892851i	$0.351298\pi$
$830$	0	0
$831$	1.15988e13	0.843742
$832$	0	0
$833$	1.06541e13	0.766681
$834$	0	0
$835$	5.26092e12	0.374518
$836$	0	0
$837$	−9.74963e11	−0.0686631
$838$	0	0
$839$	−4.67277e12	−0.325571	−0.162786	−	0.986661i	$-0.552048\pi$
$839$	−4.67277e12	−0.325571	−0.162786	+	0.986661i	$0.552048\pi$
$840$	0	0
$841$	1.32090e13	0.910516
$842$	0	0
$843$	5.27537e12	0.359773
$844$	0	0
$845$	1.25421e12	0.0846285
$846$	0	0
$847$	−2.43550e13	−1.62597
$848$	0	0
$849$	1.33999e13	0.885149
$850$	0	0
$851$	−1.55323e12	−0.101520
$852$	0	0
$853$	4.04678e12	0.261721	0.130861	−	0.991401i	$-0.458226\pi$
$853$	4.04678e12	0.261721	0.130861	+	0.991401i	$0.458226\pi$
$854$	0	0
$855$	7.43573e11	0.0475857
$856$	0	0
$857$	−2.35372e13	−1.49053	−0.745267	−	0.666766i	$-0.767678\pi$
$857$	−2.35372e13	−1.49053	−0.745267	+	0.666766i	$0.767678\pi$
$858$	0	0
$859$	−2.19727e13	−1.37694	−0.688470	−	0.725265i	$-0.741717\pi$
$859$	−2.19727e13	−1.37694	−0.688470	+	0.725265i	$0.741717\pi$
$860$	0	0
$861$	−1.56175e12	−0.0968494
$862$	0	0
$863$	2.06939e13	1.26997	0.634984	−	0.772525i	$-0.281007\pi$
$863$	2.06939e13	1.26997	0.634984	+	0.772525i	$0.281007\pi$
$864$	0	0
$865$	2.24899e11	0.0136589
$866$	0	0
$867$	−8.84922e12	−0.531887
$868$	0	0
$869$	−2.84155e13	−1.69031
$870$	0	0
$871$	−1.34977e13	−0.794652
$872$	0	0
$873$	3.75689e12	0.218909
$874$	0	0
$875$	−2.99612e12	−0.172792
$876$	0	0
$877$	−8.18344e12	−0.467130	−0.233565	−	0.972341i	$-0.575039\pi$
$877$	−8.18344e12	−0.467130	−0.233565	+	0.972341i	$0.575039\pi$
$878$	0	0
$879$	4.57052e12	0.258235
$880$	0	0
$881$	2.43747e13	1.36316	0.681581	−	0.731743i	$-0.261293\pi$
$881$	2.43747e13	1.36316	0.681581	+	0.731743i	$0.261293\pi$
$882$	0	0
$883$	1.17064e13	0.648039	0.324019	−	0.946050i	$-0.394966\pi$
$883$	1.17064e13	0.648039	0.324019	+	0.946050i	$0.394966\pi$
$884$	0	0
$885$	6.02459e12	0.330128
$886$	0	0
$887$	−2.83247e13	−1.53642	−0.768210	−	0.640198i	$-0.778852\pi$
$887$	−2.83247e13	−1.53642	−0.768210	+	0.640198i	$0.778852\pi$
$888$	0	0
$889$	−4.60098e13	−2.47055
$890$	0	0
$891$	2.83669e12	0.150786
$892$	0	0
$893$	−5.45738e12	−0.287179
$894$	0	0
$895$	5.40836e12	0.281749
$896$	0	0
$897$	2.22081e12	0.114537
$898$	0	0
$899$	9.65827e12	0.493152
$900$	0	0
$901$	−2.48647e11	−0.0125696
$902$	0	0
$903$	−1.98315e13	−0.992569
$904$	0	0
$905$	3.81070e12	0.188837
$906$	0	0
$907$	−5.36313e12	−0.263139	−0.131570	−	0.991307i	$-0.542002\pi$
$907$	−5.36313e12	−0.263139	−0.131570	+	0.991307i	$0.542002\pi$
$908$	0	0
$909$	1.17078e13	0.568771
$910$	0	0
$911$	−3.43721e13	−1.65338	−0.826691	−	0.562656i	$-0.809780\pi$
$911$	−3.43721e13	−1.65338	−0.826691	+	0.562656i	$0.809780\pi$
$912$	0	0
$913$	1.11869e12	0.0532833
$914$	0	0
$915$	9.76432e12	0.460519
$916$	0	0
$917$	−2.41268e13	−1.12678
$918$	0	0
$919$	2.98037e13	1.37832	0.689160	−	0.724609i	$-0.257980\pi$
$919$	2.98037e13	1.37832	0.689160	+	0.724609i	$0.257980\pi$
$920$	0	0
$921$	−1.45282e12	−0.0665339
$922$	0	0
$923$	−7.85592e10	−0.00356278
$924$	0	0
$925$	2.48513e12	0.111612
$926$	0	0
$927$	−7.65221e12	−0.340352
$928$	0	0
$929$	−2.03150e13	−0.894843	−0.447421	−	0.894323i	$-0.647658\pi$
$929$	−2.03150e13	−0.894843	−0.447421	+	0.894323i	$0.647658\pi$
$930$	0	0
$931$	1.99921e13	0.872136
$932$	0	0
$933$	−2.59162e13	−1.11970
$934$	0	0
$935$	3.98001e12	0.170307
$936$	0	0
$937$	2.73553e13	1.15935	0.579674	−	0.814848i	$-0.303180\pi$
$937$	2.73553e13	1.15935	0.579674	+	0.814848i	$0.303180\pi$
$938$	0	0
$939$	−2.56759e13	−1.07778
$940$	0	0
$941$	−4.24123e13	−1.76335	−0.881675	−	0.471857i	$-0.843584\pi$
$941$	−4.24123e13	−1.76335	−0.881675	+	0.471857i	$0.843584\pi$
$942$	0	0
$943$	−3.83578e11	−0.0157961
$944$	0	0
$945$	−4.07619e12	−0.166269
$946$	0	0
$947$	−9.23741e12	−0.373229	−0.186614	−	0.982433i	$-0.559752\pi$
$947$	−9.23741e12	−0.373229	−0.186614	+	0.982433i	$0.559752\pi$
$948$	0	0
$949$	1.00091e13	0.400589
$950$	0	0
$951$	−2.49607e13	−0.989566
$952$	0	0
$953$	1.11343e13	0.437265	0.218633	−	0.975807i	$-0.429840\pi$
$953$	1.11343e13	0.437265	0.218633	+	0.975807i	$0.429840\pi$
$954$	0	0
$955$	1.85149e13	0.720288
$956$	0	0
$957$	−2.81011e13	−1.08298
$958$	0	0
$959$	4.45752e12	0.170180
$960$	0	0
$961$	−2.30740e13	−0.872705
$962$	0	0
$963$	1.12999e13	0.423406
$964$	0	0
$965$	−2.89649e11	−0.0107522
$966$	0	0
$967$	5.48630e12	0.201772	0.100886	−	0.994898i	$-0.467832\pi$
$967$	5.48630e12	0.201772	0.100886	+	0.994898i	$0.467832\pi$
$968$	0	0
$969$	1.41936e12	0.0517172
$970$	0	0
$971$	2.07887e13	0.750484	0.375242	−	0.926927i	$-0.377560\pi$
$971$	2.07887e13	0.750484	0.375242	+	0.926927i	$0.377560\pi$
$972$	0	0
$973$	5.59093e13	1.99975
$974$	0	0
$975$	−3.55324e12	−0.125922
$976$	0	0
$977$	2.25372e13	0.791362	0.395681	−	0.918388i	$-0.370509\pi$
$977$	2.25372e13	0.791362	0.395681	+	0.918388i	$0.370509\pi$
$978$	0	0
$979$	−2.04193e11	−0.00710427
$980$	0	0
$981$	1.36776e13	0.471519
$982$	0	0
$983$	2.20727e13	0.753989	0.376994	−	0.926216i	$-0.376958\pi$
$983$	2.20727e13	0.753989	0.376994	+	0.926216i	$0.376958\pi$
$984$	0	0
$985$	7.71338e12	0.261085
$986$	0	0
$987$	2.99168e13	1.00343
$988$	0	0
$989$	−4.87077e12	−0.161888
$990$	0	0
$991$	5.27788e13	1.73831	0.869157	−	0.494536i	$-0.164662\pi$
$991$	5.27788e13	1.73831	0.869157	+	0.494536i	$0.164662\pi$
$992$	0	0
$993$	1.45139e13	0.473710
$994$	0	0
$995$	6.27213e12	0.202867
$996$	0	0
$997$	−4.19157e13	−1.34353	−0.671767	−	0.740763i	$-0.734464\pi$
$997$	−4.19157e13	−1.34353	−0.671767	+	0.740763i	$0.734464\pi$
$998$	0	0
$999$	3.38099e12	0.107399

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.10.a.r.1.1		2
4.3	odd	2		15.10.a.d.1.2	✓	2
12.11	even	2		45.10.a.d.1.1		2
20.3	even	4		75.10.b.f.49.1		4
20.7	even	4		75.10.b.f.49.4		4
20.19	odd	2		75.10.a.f.1.1		2
60.23	odd	4		225.10.b.i.199.4		4
60.47	odd	4		225.10.b.i.199.1		4
60.59	even	2		225.10.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.10.a.d.1.2	✓	2	4.3	odd	2
45.10.a.d.1.1		2	12.11	even	2
75.10.a.f.1.1		2	20.19	odd	2
75.10.b.f.49.1		4	20.3	even	4
75.10.b.f.49.4		4	20.7	even	4
225.10.a.k.1.2		2	60.59	even	2
225.10.b.i.199.1		4	60.47	odd	4
225.10.b.i.199.4		4	60.23	odd	4
240.10.a.r.1.1		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)

\(f(q)\)	\(=\)	\(q+81.0000 q^{3} +625.000 q^{5} -12272.1 q^{7} +6561.00 q^{9} +O(q^{10})\)	\(q+81.0000 q^{3} +625.000 q^{5} -12272.1 q^{7} +6561.00 q^{9} +65897.9 q^{11} -112300. q^{13} +50625.0 q^{15} +96634.7 q^{17} +181332. q^{19} -994042. q^{21} -244145. q^{23} +390625. q^{25} +531441. q^{27} -5.26461e6 q^{29} -1.83457e6 q^{31} +5.33773e6 q^{33} -7.67008e6 q^{35} +6.36194e6 q^{37} -9.09628e6 q^{39} +1.57111e6 q^{41} +1.99504e7 q^{43} +4.10062e6 q^{45} -3.00961e7 q^{47} +1.10251e8 q^{49} +7.82741e6 q^{51} -2.57306e6 q^{53} +4.11862e7 q^{55} +1.46879e7 q^{57} +1.19004e8 q^{59} +1.92875e8 q^{61} -8.05174e7 q^{63} -7.01874e7 q^{65} +1.20193e8 q^{67} -1.97757e7 q^{69} +699549. q^{71} -8.91287e7 q^{73} +3.16406e7 q^{75} -8.08707e8 q^{77} -4.31205e8 q^{79} +4.30467e7 q^{81} +1.69761e7 q^{83} +6.03967e7 q^{85} -4.26433e8 q^{87} -3.09863e6 q^{89} +1.37816e9 q^{91} -1.48600e8 q^{93} +1.13332e8 q^{95} +5.72609e8 q^{97} +4.32356e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 162 q^{3} + 1250 q^{5} - 14112 q^{7} + 13122 q^{9}+O(q^{10}) \) 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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