LMFDB - Embedded newform 240.10.a.m.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{4729}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1182 $ x^2 - x - 1182
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{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			$-33.8839$ 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)

$f(q)$	$=$	$q-81.0000 q^{3} -625.000 q^{5} +7861.50 q^{7} +6561.00 q^{9} +O(q^{10})$	$q-81.0000 q^{3} -625.000 q^{5} +7861.50 q^{7} +6561.00 q^{9} +49373.3 q^{11} +24250.7 q^{13} +50625.0 q^{15} +268222. q^{17} +168364. q^{19} -636781. q^{21} +2.12200e6 q^{23} +390625. q^{25} -531441. q^{27} +389624. q^{29} -90532.2 q^{31} -3.99924e6 q^{33} -4.91344e6 q^{35} -3.31991e6 q^{37} -1.96431e6 q^{39} +2.32694e7 q^{41} -1.91140e7 q^{43} -4.10062e6 q^{45} -6.28153e7 q^{47} +2.14495e7 q^{49} -2.17260e7 q^{51} -180207. q^{53} -3.08583e7 q^{55} -1.36375e7 q^{57} -3.84564e7 q^{59} -553620. q^{61} +5.15793e7 q^{63} -1.51567e7 q^{65} +2.39163e8 q^{67} -1.71882e8 q^{69} -1.28653e8 q^{71} -2.39376e8 q^{73} -3.16406e7 q^{75} +3.88148e8 q^{77} +5.28027e8 q^{79} +4.30467e7 q^{81} -2.12210e8 q^{83} -1.67639e8 q^{85} -3.15595e7 q^{87} -2.07724e8 q^{89} +1.90647e8 q^{91} +7.33311e6 q^{93} -1.05228e8 q^{95} +1.70780e9 q^{97} +3.23938e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 162 q^{3} - 1250 q^{5} + 11872 q^{7} + 13122 q^{9}+O(q^{10}) $ 2 * q - 162 * q^3 - 1250 * q^5 + 11872 * q^7 + 13122 * q^9	$ 2 q - 162 q^{3} - 1250 q^{5} + 11872 q^{7} + 13122 q^{9} - 35488 q^{11} + 143676 q^{13} + 101250 q^{15} + 385156 q^{17} + 403296 q^{19} - 961632 q^{21} - 223704 q^{23} + 781250 q^{25} - 1062882 q^{27} - 74572 q^{29} + 5027128 q^{31} + 2874528 q^{33} - 7420000 q^{35} + 5373628 q^{37} - 11637756 q^{39} + 14211332 q^{41} - 27748920 q^{43} - 8201250 q^{45} - 95966440 q^{47} - 2819950 q^{49} - 31197636 q^{51} - 64305596 q^{53} + 22180000 q^{55} - 32666976 q^{57} - 187863136 q^{59} + 154080060 q^{61} + 77892192 q^{63} - 89797500 q^{65} - 33592376 q^{67} + 18120024 q^{69} + 228270976 q^{71} - 33122316 q^{73} - 63281250 q^{75} + 47811456 q^{77} + 932406760 q^{79} + 86093442 q^{81} - 207040152 q^{83} - 240722500 q^{85} + 6040332 q^{87} + 224518164 q^{89} + 669602528 q^{91} - 407197368 q^{93} - 252060000 q^{95} + 387134596 q^{97} - 232836768 q^{99}+O(q^{100}) $ 2 * q - 162 * q^3 - 1250 * q^5 + 11872 * q^7 + 13122 * q^9 - 35488 * q^11 + 143676 * q^13 + 101250 * q^15 + 385156 * q^17 + 403296 * q^19 - 961632 * q^21 - 223704 * q^23 + 781250 * q^25 - 1062882 * q^27 - 74572 * q^29 + 5027128 * q^31 + 2874528 * q^33 - 7420000 * q^35 + 5373628 * q^37 - 11637756 * q^39 + 14211332 * q^41 - 27748920 * q^43 - 8201250 * q^45 - 95966440 * q^47 - 2819950 * q^49 - 31197636 * q^51 - 64305596 * q^53 + 22180000 * q^55 - 32666976 * q^57 - 187863136 * q^59 + 154080060 * q^61 + 77892192 * q^63 - 89797500 * q^65 - 33592376 * q^67 + 18120024 * q^69 + 228270976 * q^71 - 33122316 * q^73 - 63281250 * q^75 + 47811456 * q^77 + 932406760 * q^79 + 86093442 * q^81 - 207040152 * q^83 - 240722500 * q^85 + 6040332 * q^87 + 224518164 * q^89 + 669602528 * q^91 - 407197368 * q^93 - 252060000 * q^95 + 387134596 * q^97 - 232836768 * 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
$7$	7861.50	1.23755	0.618777	−	0.785567i	$-0.287629\pi$
$7$	7861.50	1.23755	0.618777	+	0.785567i	$0.287629\pi$
$8$	0	0
$9$	6561.00	0.333333
$10$	0	0
$11$	49373.3	1.01678	0.508388	−	0.861128i	$-0.330242\pi$
$11$	49373.3	1.01678	0.508388	+	0.861128i	$0.330242\pi$
$12$	0	0
$13$	24250.7	0.235494	0.117747	−	0.993044i	$-0.462433\pi$
$13$	24250.7	0.235494	0.117747	+	0.993044i	$0.462433\pi$
$14$	0	0
$15$	50625.0	0.258199
$16$	0	0
$17$	268222.	0.778888	0.389444	−	0.921050i	$-0.372667\pi$
$17$	268222.	0.778888	0.389444	+	0.921050i	$0.372667\pi$
$18$	0	0
$19$	168364.	0.296387	0.148193	−	0.988958i	$-0.452654\pi$
$19$	168364.	0.296387	0.148193	+	0.988958i	$0.452654\pi$
$20$	0	0
$21$	−636781.	−0.714502
$22$	0	0
$23$	2.12200e6	1.58114	0.790569	−	0.612373i	$-0.209785\pi$
$23$	2.12200e6	1.58114	0.790569	+	0.612373i	$0.209785\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	0	0
$27$	−531441.	−0.192450
$28$	0	0
$29$	389624.	0.102295	0.0511475	−	0.998691i	$-0.483712\pi$
$29$	389624.	0.102295	0.0511475	+	0.998691i	$0.483712\pi$
$30$	0	0
$31$	−90532.2	−0.0176066	−0.00880330	−	0.999961i	$-0.502802\pi$
$31$	−90532.2	−0.0176066	−0.00880330	+	0.999961i	$0.502802\pi$
$32$	0	0
$33$	−3.99924e6	−0.587036
$34$	0	0
$35$	−4.91344e6	−0.553451
$36$	0	0
$37$	−3.31991e6	−0.291218	−0.145609	−	0.989342i	$-0.546514\pi$
$37$	−3.31991e6	−0.291218	−0.145609	+	0.989342i	$0.546514\pi$
$38$	0	0
$39$	−1.96431e6	−0.135963
$40$	0	0
$41$	2.32694e7	1.28605	0.643024	−	0.765846i	$-0.277679\pi$
$41$	2.32694e7	1.28605	0.643024	+	0.765846i	$0.277679\pi$
$42$	0	0
$43$	−1.91140e7	−0.852597	−0.426298	−	0.904583i	$-0.640183\pi$
$43$	−1.91140e7	−0.852597	−0.426298	+	0.904583i	$0.640183\pi$
$44$	0	0
$45$	−4.10062e6	−0.149071
$46$	0	0
$47$	−6.28153e7	−1.87770	−0.938848	−	0.344332i	$-0.888105\pi$
$47$	−6.28153e7	−1.87770	−0.938848	+	0.344332i	$0.888105\pi$
$48$	0	0
$49$	2.14495e7	0.531539
$50$	0	0
$51$	−2.17260e7	−0.449691
$52$	0	0
$53$	−180207.	−0.00313711	−0.00156856	−	0.999999i	$-0.500499\pi$
$53$	−180207.	−0.00313711	−0.00156856	+	0.999999i	$0.500499\pi$
$54$	0	0
$55$	−3.08583e7	−0.454716
$56$	0	0
$57$	−1.36375e7	−0.171119
$58$	0	0
$59$	−3.84564e7	−0.413175	−0.206588	−	0.978428i	$-0.566236\pi$
$59$	−3.84564e7	−0.413175	−0.206588	+	0.978428i	$0.566236\pi$
$60$	0	0
$61$	−553620.	−0.00511950	−0.00255975	−	0.999997i	$-0.500815\pi$
$61$	−553620.	−0.00511950	−0.00255975	+	0.999997i	$0.500815\pi$
$62$	0	0
$63$	5.15793e7	0.412518
$64$	0	0
$65$	−1.51567e7	−0.105316
$66$	0	0
$67$	2.39163e8	1.44996	0.724982	−	0.688768i	$-0.241848\pi$
$67$	2.39163e8	1.44996	0.724982	+	0.688768i	$0.241848\pi$
$68$	0	0
$69$	−1.71882e8	−0.912871
$70$	0	0
$71$	−1.28653e8	−0.600838	−0.300419	−	0.953807i	$-0.597127\pi$
$71$	−1.28653e8	−0.600838	−0.300419	+	0.953807i	$0.597127\pi$
$72$	0	0
$73$	−2.39376e8	−0.986569	−0.493284	−	0.869868i	$-0.664204\pi$
$73$	−2.39376e8	−0.986569	−0.493284	+	0.869868i	$0.664204\pi$
$74$	0	0
$75$	−3.16406e7	−0.115470
$76$	0	0
$77$	3.88148e8	1.25831
$78$	0	0
$79$	5.28027e8	1.52523	0.762613	−	0.646855i	$-0.223916\pi$
$79$	5.28027e8	1.52523	0.762613	+	0.646855i	$0.223916\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	−2.12210e8	−0.490812	−0.245406	−	0.969420i	$-0.578921\pi$
$83$	−2.12210e8	−0.490812	−0.245406	+	0.969420i	$0.578921\pi$
$84$	0	0
$85$	−1.67639e8	−0.348329
$86$	0	0
$87$	−3.15595e7	−0.0590601
$88$	0	0
$89$	−2.07724e8	−0.350939	−0.175469	−	0.984485i	$-0.556144\pi$
$89$	−2.07724e8	−0.350939	−0.175469	+	0.984485i	$0.556144\pi$
$90$	0	0
$91$	1.90647e8	0.291436
$92$	0	0
$93$	7.33311e6	0.0101652
$94$	0	0
$95$	−1.05228e8	−0.132548
$96$	0	0
$97$	1.70780e9	1.95868	0.979341	−	0.202214i	$-0.0648136\pi$
$97$	1.70780e9	1.95868	0.979341	+	0.202214i	$0.0648136\pi$
$98$	0	0
$99$	3.23938e8	0.338925
$100$	0	0
$101$	−1.81384e8	−0.173441	−0.0867206	−	0.996233i	$-0.527639\pi$
$101$	−1.81384e8	−0.173441	−0.0867206	+	0.996233i	$0.527639\pi$
$102$	0	0
$103$	1.70453e9	1.49223	0.746116	−	0.665816i	$-0.231917\pi$
$103$	1.70453e9	1.49223	0.746116	+	0.665816i	$0.231917\pi$
$104$	0	0
$105$	3.97988e8	0.319535
$106$	0	0
$107$	−1.73298e9	−1.27811	−0.639053	−	0.769163i	$-0.720674\pi$
$107$	−1.73298e9	−1.27811	−0.639053	+	0.769163i	$0.720674\pi$
$108$	0	0
$109$	−1.17238e9	−0.795515	−0.397757	−	0.917491i	$-0.630211\pi$
$109$	−1.17238e9	−0.795515	−0.397757	+	0.917491i	$0.630211\pi$
$110$	0	0
$111$	2.68913e8	0.168135
$112$	0	0
$113$	−1.22180e9	−0.704935	−0.352467	−	0.935824i	$-0.614657\pi$
$113$	−1.22180e9	−0.704935	−0.352467	+	0.935824i	$0.614657\pi$
$114$	0	0
$115$	−1.32625e9	−0.707107
$116$	0	0
$117$	1.59109e8	0.0784980
$118$	0	0
$119$	2.10863e9	0.963916
$120$	0	0
$121$	7.97750e7	0.0338324
$122$	0	0
$123$	−1.88482e9	−0.742501
$124$	0	0
$125$	−2.44141e8	−0.0894427
$126$	0	0
$127$	9.50324e8	0.324157	0.162078	−	0.986778i	$-0.448180\pi$
$127$	9.50324e8	0.324157	0.162078	+	0.986778i	$0.448180\pi$
$128$	0	0
$129$	1.54823e9	0.492247
$130$	0	0
$131$	4.03050e9	1.19574	0.597872	−	0.801592i	$-0.296013\pi$
$131$	4.03050e9	1.19574	0.597872	+	0.801592i	$0.296013\pi$
$132$	0	0
$133$	1.32360e9	0.366795
$134$	0	0
$135$	3.32151e8	0.0860663
$136$	0	0
$137$	5.51871e9	1.33843	0.669214	−	0.743070i	$-0.266631\pi$
$137$	5.51871e9	1.33843	0.669214	+	0.743070i	$0.266631\pi$
$138$	0	0
$139$	3.97776e9	0.903798	0.451899	−	0.892069i	$-0.350747\pi$
$139$	3.97776e9	0.903798	0.451899	+	0.892069i	$0.350747\pi$
$140$	0	0
$141$	5.08804e9	1.08409
$142$	0	0
$143$	1.19734e9	0.239445
$144$	0	0
$145$	−2.43515e8	−0.0457478
$146$	0	0
$147$	−1.73741e9	−0.306884
$148$	0	0
$149$	−2.26538e9	−0.376533	−0.188266	−	0.982118i	$-0.560287\pi$
$149$	−2.26538e9	−0.376533	−0.188266	+	0.982118i	$0.560287\pi$
$150$	0	0
$151$	8.49158e9	1.32921	0.664603	−	0.747197i	$-0.268601\pi$
$151$	8.49158e9	1.32921	0.664603	+	0.747197i	$0.268601\pi$
$152$	0	0
$153$	1.75981e9	0.259629
$154$	0	0
$155$	5.65826e7	0.00787391
$156$	0	0
$157$	7.28379e9	0.956774	0.478387	−	0.878149i	$-0.341222\pi$
$157$	7.28379e9	0.956774	0.478387	+	0.878149i	$0.341222\pi$
$158$	0	0
$159$	1.45968e7	0.00181121
$160$	0	0
$161$	1.66821e10	1.95674
$162$	0	0
$163$	−1.38419e10	−1.53586	−0.767929	−	0.640535i	$-0.778712\pi$
$163$	−1.38419e10	−1.53586	−0.767929	+	0.640535i	$0.778712\pi$
$164$	0	0
$165$	2.49952e9	0.262530
$166$	0	0
$167$	6.92038e9	0.688503	0.344252	−	0.938877i	$-0.388133\pi$
$167$	6.92038e9	0.688503	0.344252	+	0.938877i	$0.388133\pi$
$168$	0	0
$169$	−1.00164e10	−0.944543
$170$	0	0
$171$	1.10464e9	0.0987957
$172$	0	0
$173$	3.26987e9	0.277539	0.138769	−	0.990325i	$-0.455685\pi$
$173$	3.26987e9	0.277539	0.138769	+	0.990325i	$0.455685\pi$
$174$	0	0
$175$	3.07090e9	0.247511
$176$	0	0
$177$	3.11497e9	0.238547
$178$	0	0
$179$	1.12431e10	0.818556	0.409278	−	0.912410i	$-0.365781\pi$
$179$	1.12431e10	0.818556	0.409278	+	0.912410i	$0.365781\pi$
$180$	0	0
$181$	3.75039e9	0.259731	0.129865	−	0.991532i	$-0.458546\pi$
$181$	3.75039e9	0.259731	0.129865	+	0.991532i	$0.458546\pi$
$182$	0	0
$183$	4.48432e7	0.00295574
$184$	0	0
$185$	2.07494e9	0.130237
$186$	0	0
$187$	1.32430e10	0.791954
$188$	0	0
$189$	−4.17792e9	−0.238167
$190$	0	0
$191$	−1.72426e10	−0.937460	−0.468730	−	0.883342i	$-0.655288\pi$
$191$	−1.72426e10	−0.937460	−0.468730	+	0.883342i	$0.655288\pi$
$192$	0	0
$193$	−6.01060e9	−0.311824	−0.155912	−	0.987771i	$-0.549832\pi$
$193$	−6.01060e9	−0.311824	−0.155912	+	0.987771i	$0.549832\pi$
$194$	0	0
$195$	1.22769e9	0.0608043
$196$	0	0
$197$	−1.19077e10	−0.563289	−0.281645	−	0.959519i	$-0.590880\pi$
$197$	−1.19077e10	−0.563289	−0.281645	+	0.959519i	$0.590880\pi$
$198$	0	0
$199$	6.10946e9	0.276162	0.138081	−	0.990421i	$-0.455907\pi$
$199$	6.10946e9	0.276162	0.138081	+	0.990421i	$0.455907\pi$
$200$	0	0
$201$	−1.93722e10	−0.837137
$202$	0	0
$203$	3.06303e9	0.126596
$204$	0	0
$205$	−1.45434e10	−0.575139
$206$	0	0
$207$	1.39224e10	0.527046
$208$	0	0
$209$	8.31271e9	0.301359
$210$	0	0
$211$	3.67892e10	1.27776	0.638881	−	0.769306i	$-0.279398\pi$
$211$	3.67892e10	1.27776	0.638881	+	0.769306i	$0.279398\pi$
$212$	0	0
$213$	1.04209e10	0.346894
$214$	0	0
$215$	1.19463e10	0.381293
$216$	0	0
$217$	−7.11719e8	−0.0217891
$218$	0	0
$219$	1.93894e10	0.569596
$220$	0	0
$221$	6.50459e9	0.183423
$222$	0	0
$223$	5.01307e9	0.135747	0.0678737	−	0.997694i	$-0.478379\pi$
$223$	5.01307e9	0.135747	0.0678737	+	0.997694i	$0.478379\pi$
$224$	0	0
$225$	2.56289e9	0.0666667
$226$	0	0
$227$	2.73858e10	0.684555	0.342278	−	0.939599i	$-0.388802\pi$
$227$	2.73858e10	0.684555	0.342278	+	0.939599i	$0.388802\pi$
$228$	0	0
$229$	5.36868e10	1.29005	0.645027	−	0.764159i	$-0.276846\pi$
$229$	5.36868e10	1.29005	0.645027	+	0.764159i	$0.276846\pi$
$230$	0	0
$231$	−3.14400e10	−0.726488
$232$	0	0
$233$	7.25874e10	1.61347	0.806733	−	0.590916i	$-0.201234\pi$
$233$	7.25874e10	1.61347	0.806733	+	0.590916i	$0.201234\pi$
$234$	0	0
$235$	3.92596e10	0.839731
$236$	0	0
$237$	−4.27702e10	−0.880590
$238$	0	0
$239$	−1.73313e10	−0.343589	−0.171795	−	0.985133i	$-0.554957\pi$
$239$	−1.73313e10	−0.343589	−0.171795	+	0.985133i	$0.554957\pi$
$240$	0	0
$241$	4.06448e10	0.776119	0.388059	−	0.921634i	$-0.373146\pi$
$241$	4.06448e10	0.776119	0.388059	+	0.921634i	$0.373146\pi$
$242$	0	0
$243$	−3.48678e9	−0.0641500
$244$	0	0
$245$	−1.34059e10	−0.237711
$246$	0	0
$247$	4.08296e9	0.0697973
$248$	0	0
$249$	1.71890e10	0.283371
$250$	0	0
$251$	−1.00920e11	−1.60490	−0.802448	−	0.596723i	$-0.796469\pi$
$251$	−1.00920e11	−1.60490	−0.802448	+	0.596723i	$0.796469\pi$
$252$	0	0
$253$	1.04770e11	1.60766
$254$	0	0
$255$	1.35788e10	0.201108
$256$	0	0
$257$	5.03660e10	0.720176	0.360088	−	0.932918i	$-0.382747\pi$
$257$	5.03660e10	0.720176	0.360088	+	0.932918i	$0.382747\pi$
$258$	0	0
$259$	−2.60995e10	−0.360398
$260$	0	0
$261$	2.55632e9	0.0340984
$262$	0	0
$263$	5.58677e10	0.720046	0.360023	−	0.932943i	$-0.382769\pi$
$263$	5.58677e10	0.720046	0.360023	+	0.932943i	$0.382769\pi$
$264$	0	0
$265$	1.12629e8	0.00140296
$266$	0	0
$267$	1.68256e10	0.202614
$268$	0	0
$269$	−1.05175e11	−1.22469	−0.612345	−	0.790590i	$-0.709774\pi$
$269$	−1.05175e11	−1.22469	−0.612345	+	0.790590i	$0.709774\pi$
$270$	0	0
$271$	5.44180e10	0.612887	0.306444	−	0.951889i	$-0.400861\pi$
$271$	5.44180e10	0.612887	0.306444	+	0.951889i	$0.400861\pi$
$272$	0	0
$273$	−1.54424e10	−0.168261
$274$	0	0
$275$	1.92864e10	0.203355
$276$	0	0
$277$	1.64528e11	1.67912	0.839560	−	0.543268i	$-0.182813\pi$
$277$	1.64528e11	1.67912	0.839560	+	0.543268i	$0.182813\pi$
$278$	0	0
$279$	−5.93982e8	−0.00586887
$280$	0	0
$281$	−6.74255e10	−0.645128	−0.322564	−	0.946548i	$-0.604545\pi$
$281$	−6.74255e10	−0.645128	−0.322564	+	0.946548i	$0.604545\pi$
$282$	0	0
$283$	−1.41917e11	−1.31521	−0.657603	−	0.753364i	$-0.728430\pi$
$283$	−1.41917e11	−1.31521	−0.657603	+	0.753364i	$0.728430\pi$
$284$	0	0
$285$	8.52345e9	0.0765268
$286$	0	0
$287$	1.82932e11	1.59155
$288$	0	0
$289$	−4.66446e10	−0.393333
$290$	0	0
$291$	−1.38332e11	−1.13085
$292$	0	0
$293$	2.79097e10	0.221233	0.110617	−	0.993863i	$-0.464717\pi$
$293$	2.79097e10	0.221233	0.110617	+	0.993863i	$0.464717\pi$
$294$	0	0
$295$	2.40352e10	0.184778
$296$	0	0
$297$	−2.62390e10	−0.195679
$298$	0	0
$299$	5.14600e10	0.372349
$300$	0	0
$301$	−1.50265e11	−1.05513
$302$	0	0
$303$	1.46921e10	0.100136
$304$	0	0
$305$	3.46012e8	0.00228951
$306$	0	0
$307$	−1.59767e11	−1.02651	−0.513257	−	0.858235i	$-0.671561\pi$
$307$	−1.59767e11	−1.02651	−0.513257	+	0.858235i	$0.671561\pi$
$308$	0	0
$309$	−1.38067e11	−0.861540
$310$	0	0
$311$	7.84292e10	0.475396	0.237698	−	0.971339i	$-0.423607\pi$
$311$	7.84292e10	0.475396	0.237698	+	0.971339i	$0.423607\pi$
$312$	0	0
$313$	−1.26229e11	−0.743378	−0.371689	−	0.928357i	$-0.621221\pi$
$313$	−1.26229e11	−0.743378	−0.371689	+	0.928357i	$0.621221\pi$
$314$	0	0
$315$	−3.22370e10	−0.184484
$316$	0	0
$317$	−9.59499e10	−0.533676	−0.266838	−	0.963741i	$-0.585979\pi$
$317$	−9.59499e10	−0.533676	−0.266838	+	0.963741i	$0.585979\pi$
$318$	0	0
$319$	1.92370e10	0.104011
$320$	0	0
$321$	1.40372e11	0.737915
$322$	0	0
$323$	4.51591e10	0.230852
$324$	0	0
$325$	9.47294e9	0.0470988
$326$	0	0
$327$	9.49626e10	0.459291
$328$	0	0
$329$	−4.93822e11	−2.32375
$330$	0	0
$331$	−7.73728e10	−0.354293	−0.177146	−	0.984185i	$-0.556687\pi$
$331$	−7.73728e10	−0.354293	−0.177146	+	0.984185i	$0.556687\pi$
$332$	0	0
$333$	−2.17819e10	−0.0970727
$334$	0	0
$335$	−1.49477e11	−0.648443
$336$	0	0
$337$	1.73809e11	0.734071	0.367035	−	0.930207i	$-0.380373\pi$
$337$	1.73809e11	0.734071	0.367035	+	0.930207i	$0.380373\pi$
$338$	0	0
$339$	9.89662e10	0.406994
$340$	0	0
$341$	−4.46987e9	−0.0179020
$342$	0	0
$343$	−1.48614e11	−0.579746
$344$	0	0
$345$	1.07426e11	0.408248
$346$	0	0
$347$	−4.11335e11	−1.52305	−0.761523	−	0.648138i	$-0.775548\pi$
$347$	−4.11335e11	−1.52305	−0.761523	+	0.648138i	$0.775548\pi$
$348$	0	0
$349$	2.97923e11	1.07495	0.537477	−	0.843278i	$-0.319377\pi$
$349$	2.97923e11	1.07495	0.537477	+	0.843278i	$0.319377\pi$
$350$	0	0
$351$	−1.28878e10	−0.0453208
$352$	0	0
$353$	5.69499e9	0.0195212	0.00976061	−	0.999952i	$-0.496893\pi$
$353$	5.69499e9	0.0195212	0.00976061	+	0.999952i	$0.496893\pi$
$354$	0	0
$355$	8.04082e10	0.268703
$356$	0	0
$357$	−1.70799e11	−0.556517
$358$	0	0
$359$	−6.19565e11	−1.96862	−0.984310	−	0.176446i	$-0.943540\pi$
$359$	−6.19565e11	−1.96862	−0.984310	+	0.176446i	$0.943540\pi$
$360$	0	0
$361$	−2.94341e11	−0.912155
$362$	0	0
$363$	−6.46178e9	−0.0195331
$364$	0	0
$365$	1.49610e11	0.441207
$366$	0	0
$367$	4.06833e9	0.0117063	0.00585314	−	0.999983i	$-0.498137\pi$
$367$	4.06833e9	0.0117063	0.00585314	+	0.999983i	$0.498137\pi$
$368$	0	0
$369$	1.52670e11	0.428683
$370$	0	0
$371$	−1.41670e9	−0.00388235
$372$	0	0
$373$	−6.75397e11	−1.80663	−0.903315	−	0.428978i	$-0.858874\pi$
$373$	−6.75397e11	−1.80663	−0.903315	+	0.428978i	$0.858874\pi$
$374$	0	0
$375$	1.97754e10	0.0516398
$376$	0	0
$377$	9.44867e9	0.0240899
$378$	0	0
$379$	3.67146e11	0.914033	0.457017	−	0.889458i	$-0.348918\pi$
$379$	3.67146e11	0.914033	0.457017	+	0.889458i	$0.348918\pi$
$380$	0	0
$381$	−7.69763e10	−0.187152
$382$	0	0
$383$	−2.22057e11	−0.527314	−0.263657	−	0.964616i	$-0.584929\pi$
$383$	−2.22057e11	−0.527314	−0.263657	+	0.964616i	$0.584929\pi$
$384$	0	0
$385$	−2.42593e11	−0.562735
$386$	0	0
$387$	−1.25407e11	−0.284199
$388$	0	0
$389$	−7.23043e11	−1.60100	−0.800499	−	0.599334i	$-0.795432\pi$
$389$	−7.23043e11	−1.60100	−0.800499	+	0.599334i	$0.795432\pi$
$390$	0	0
$391$	5.69168e11	1.23153
$392$	0	0
$393$	−3.26470e11	−0.690363
$394$	0	0
$395$	−3.30017e11	−0.682102
$396$	0	0
$397$	6.06161e11	1.22470	0.612352	−	0.790585i	$-0.290224\pi$
$397$	6.06161e11	1.22470	0.612352	+	0.790585i	$0.290224\pi$
$398$	0	0
$399$	−1.07211e11	−0.211769
$400$	0	0
$401$	6.72510e11	1.29882	0.649411	−	0.760438i	$-0.275016\pi$
$401$	6.72510e11	1.29882	0.649411	+	0.760438i	$0.275016\pi$
$402$	0	0
$403$	−2.19547e9	−0.00414625
$404$	0	0
$405$	−2.69042e10	−0.0496904
$406$	0	0
$407$	−1.63915e11	−0.296103
$408$	0	0
$409$	−3.02482e10	−0.0534497	−0.0267248	−	0.999643i	$-0.508508\pi$
$409$	−3.02482e10	−0.0534497	−0.0267248	+	0.999643i	$0.508508\pi$
$410$	0	0
$411$	−4.47015e11	−0.772742
$412$	0	0
$413$	−3.02325e11	−0.511326
$414$	0	0
$415$	1.32632e11	0.219498
$416$	0	0
$417$	−3.22198e11	−0.521808
$418$	0	0
$419$	1.61938e11	0.256676	0.128338	−	0.991731i	$-0.459036\pi$
$419$	1.61938e11	0.256676	0.128338	+	0.991731i	$0.459036\pi$
$420$	0	0
$421$	−8.06547e11	−1.25130	−0.625648	−	0.780105i	$-0.715165\pi$
$421$	−8.06547e11	−1.25130	−0.625648	+	0.780105i	$0.715165\pi$
$422$	0	0
$423$	−4.12131e11	−0.625899
$424$	0	0
$425$	1.04774e11	0.155778
$426$	0	0
$427$	−4.35228e9	−0.00633565
$428$	0	0
$429$	−9.69844e10	−0.138243
$430$	0	0
$431$	−8.54653e11	−1.19300	−0.596502	−	0.802611i	$-0.703443\pi$
$431$	−8.54653e11	−1.19300	−0.596502	+	0.802611i	$0.703443\pi$
$432$	0	0
$433$	8.21884e11	1.12361	0.561804	−	0.827270i	$-0.310107\pi$
$433$	8.21884e11	1.12361	0.561804	+	0.827270i	$0.310107\pi$
$434$	0	0
$435$	1.97247e10	0.0264125
$436$	0	0
$437$	3.57269e11	0.468629
$438$	0	0
$439$	−9.97807e11	−1.28220	−0.641100	−	0.767457i	$-0.721522\pi$
$439$	−9.97807e11	−1.28220	−0.641100	+	0.767457i	$0.721522\pi$
$440$	0	0
$441$	1.40730e11	0.177180
$442$	0	0
$443$	2.22586e11	0.274588	0.137294	−	0.990530i	$-0.456160\pi$
$443$	2.22586e11	0.274588	0.137294	+	0.990530i	$0.456160\pi$
$444$	0	0
$445$	1.29827e11	0.156945
$446$	0	0
$447$	1.83496e11	0.217391
$448$	0	0
$449$	1.28886e12	1.49657	0.748287	−	0.663375i	$-0.230877\pi$
$449$	1.28886e12	1.49657	0.748287	+	0.663375i	$0.230877\pi$
$450$	0	0
$451$	1.14889e12	1.30762
$452$	0	0
$453$	−6.87818e11	−0.767417
$454$	0	0
$455$	−1.19154e11	−0.130334
$456$	0	0
$457$	8.77644e11	0.941230	0.470615	−	0.882339i	$-0.344032\pi$
$457$	8.77644e11	0.941230	0.470615	+	0.882339i	$0.344032\pi$
$458$	0	0
$459$	−1.42544e11	−0.149897
$460$	0	0
$461$	−3.31142e11	−0.341476	−0.170738	−	0.985316i	$-0.554615\pi$
$461$	−3.31142e11	−0.341476	−0.170738	+	0.985316i	$0.554615\pi$
$462$	0	0
$463$	1.05840e12	1.07037	0.535186	−	0.844735i	$-0.320242\pi$
$463$	1.05840e12	1.07037	0.535186	+	0.844735i	$0.320242\pi$
$464$	0	0
$465$	−4.58319e9	−0.00454600
$466$	0	0
$467$	1.49420e12	1.45373	0.726864	−	0.686782i	$-0.240977\pi$
$467$	1.49420e12	1.45373	0.726864	+	0.686782i	$0.240977\pi$
$468$	0	0
$469$	1.88018e12	1.79441
$470$	0	0
$471$	−5.89987e11	−0.552394
$472$	0	0
$473$	−9.43722e11	−0.866900
$474$	0	0
$475$	6.57674e10	0.0592774
$476$	0	0
$477$	−1.18234e9	−0.00104570
$478$	0	0
$479$	−1.04497e12	−0.906970	−0.453485	−	0.891264i	$-0.649819\pi$
$479$	−1.04497e12	−0.906970	−0.453485	+	0.891264i	$0.649819\pi$
$480$	0	0
$481$	−8.05102e10	−0.0685801
$482$	0	0
$483$	−1.35125e12	−1.12973
$484$	0	0
$485$	−1.06737e12	−0.875950
$486$	0	0
$487$	2.30343e12	1.85564	0.927820	−	0.373027i	$-0.121680\pi$
$487$	2.30343e12	1.85564	0.927820	+	0.373027i	$0.121680\pi$
$488$	0	0
$489$	1.12119e12	0.886728
$490$	0	0
$491$	5.37950e11	0.417711	0.208855	−	0.977947i	$-0.433026\pi$
$491$	5.37950e11	0.417711	0.208855	+	0.977947i	$0.433026\pi$
$492$	0	0
$493$	1.04506e11	0.0796764
$494$	0	0
$495$	−2.02461e11	−0.151572
$496$	0	0
$497$	−1.01141e12	−0.743570
$498$	0	0
$499$	1.01572e12	0.733364	0.366682	−	0.930346i	$-0.380494\pi$
$499$	1.01572e12	0.733364	0.366682	+	0.930346i	$0.380494\pi$
$500$	0	0
$501$	−5.60551e11	−0.397508
$502$	0	0
$503$	1.07349e12	0.747723	0.373861	−	0.927485i	$-0.378034\pi$
$503$	1.07349e12	0.747723	0.373861	+	0.927485i	$0.378034\pi$
$504$	0	0
$505$	1.13365e11	0.0775653
$506$	0	0
$507$	8.11329e11	0.545332
$508$	0	0
$509$	−1.48831e12	−0.982799	−0.491400	−	0.870934i	$-0.663515\pi$
$509$	−1.48831e12	−0.982799	−0.491400	+	0.870934i	$0.663515\pi$
$510$	0	0
$511$	−1.88185e12	−1.22093
$512$	0	0
$513$	−8.94758e10	−0.0570397
$514$	0	0
$515$	−1.06533e12	−0.667346
$516$	0	0
$517$	−3.10140e12	−1.90919
$518$	0	0
$519$	−2.64860e11	−0.160237
$520$	0	0
$521$	−9.98544e11	−0.593742	−0.296871	−	0.954918i	$-0.595943\pi$
$521$	−9.98544e11	−0.593742	−0.296871	+	0.954918i	$0.595943\pi$
$522$	0	0
$523$	1.59973e12	0.934954	0.467477	−	0.884005i	$-0.345163\pi$
$523$	1.59973e12	0.934954	0.467477	+	0.884005i	$0.345163\pi$
$524$	0	0
$525$	−2.48743e11	−0.142900
$526$	0	0
$527$	−2.42828e10	−0.0137136
$528$	0	0
$529$	2.70173e12	1.50000
$530$	0	0
$531$	−2.52312e11	−0.137725
$532$	0	0
$533$	5.64300e11	0.302857
$534$	0	0
$535$	1.08311e12	0.571586
$536$	0	0
$537$	−9.10693e11	−0.472593
$538$	0	0
$539$	1.05903e12	0.540456
$540$	0	0
$541$	5.11810e11	0.256875	0.128437	−	0.991718i	$-0.459004\pi$
$541$	5.11810e11	0.256875	0.128437	+	0.991718i	$0.459004\pi$
$542$	0	0
$543$	−3.03782e11	−0.149956
$544$	0	0
$545$	7.32736e11	0.355765
$546$	0	0
$547$	−1.40910e12	−0.672977	−0.336489	−	0.941688i	$-0.609239\pi$
$547$	−1.40910e12	−0.672977	−0.336489	+	0.941688i	$0.609239\pi$
$548$	0	0
$549$	−3.63230e9	−0.00170650
$550$	0	0
$551$	6.55988e10	0.0303189
$552$	0	0
$553$	4.15108e12	1.88755
$554$	0	0
$555$	−1.68070e11	−0.0751922
$556$	0	0
$557$	2.36835e12	1.04255	0.521276	−	0.853388i	$-0.325456\pi$
$557$	2.36835e12	1.04255	0.521276	+	0.853388i	$0.325456\pi$
$558$	0	0
$559$	−4.63529e11	−0.200781
$560$	0	0
$561$	−1.07269e12	−0.457235
$562$	0	0
$563$	2.36245e12	0.991005	0.495502	−	0.868607i	$-0.334984\pi$
$563$	2.36245e12	0.991005	0.495502	+	0.868607i	$0.334984\pi$
$564$	0	0
$565$	7.63628e11	0.315256
$566$	0	0
$567$	3.38412e11	0.137506
$568$	0	0
$569$	2.25679e11	0.0902582	0.0451291	−	0.998981i	$-0.485630\pi$
$569$	2.25679e11	0.0902582	0.0451291	+	0.998981i	$0.485630\pi$
$570$	0	0
$571$	9.15389e11	0.360366	0.180183	−	0.983633i	$-0.442331\pi$
$571$	9.15389e11	0.360366	0.180183	+	0.983633i	$0.442331\pi$
$572$	0	0
$573$	1.39665e12	0.541243
$574$	0	0
$575$	8.28906e11	0.316228
$576$	0	0
$577$	2.22404e12	0.835318	0.417659	−	0.908604i	$-0.362851\pi$
$577$	2.22404e12	0.835318	0.417659	+	0.908604i	$0.362851\pi$
$578$	0	0
$579$	4.86859e11	0.180032
$580$	0	0
$581$	−1.66829e12	−0.607407
$582$	0	0
$583$	−8.89741e9	−0.00318974
$584$	0	0
$585$	−9.94432e10	−0.0351054
$586$	0	0
$587$	−5.60186e12	−1.94742	−0.973712	−	0.227783i	$-0.926852\pi$
$587$	−5.60186e12	−1.94742	−0.973712	+	0.227783i	$0.926852\pi$
$588$	0	0
$589$	−1.52424e10	−0.00521837
$590$	0	0
$591$	9.64527e11	0.325215
$592$	0	0
$593$	2.78326e12	0.924289	0.462145	−	0.886805i	$-0.347080\pi$
$593$	2.78326e12	0.924289	0.462145	+	0.886805i	$0.347080\pi$
$594$	0	0
$595$	−1.31789e12	−0.431076
$596$	0	0
$597$	−4.94867e11	−0.159442
$598$	0	0
$599$	1.36720e12	0.433922	0.216961	−	0.976180i	$-0.430386\pi$
$599$	1.36720e12	0.433922	0.216961	+	0.976180i	$0.430386\pi$
$600$	0	0
$601$	4.74171e12	1.48252	0.741259	−	0.671219i	$-0.234229\pi$
$601$	4.74171e12	1.48252	0.741259	+	0.671219i	$0.234229\pi$
$602$	0	0
$603$	1.56915e12	0.483321
$604$	0	0
$605$	−4.98594e10	−0.0151303
$606$	0	0
$607$	5.74836e12	1.71868	0.859339	−	0.511406i	$-0.170875\pi$
$607$	5.74836e12	1.71868	0.859339	+	0.511406i	$0.170875\pi$
$608$	0	0
$609$	−2.48105e11	−0.0730900
$610$	0	0
$611$	−1.52332e12	−0.442186
$612$	0	0
$613$	2.99054e12	0.855416	0.427708	−	0.903917i	$-0.359321\pi$
$613$	2.99054e12	0.855416	0.427708	+	0.903917i	$0.359321\pi$
$614$	0	0
$615$	1.17801e12	0.332056
$616$	0	0
$617$	6.27174e12	1.74223	0.871114	−	0.491081i	$-0.163398\pi$
$617$	6.27174e12	1.74223	0.871114	+	0.491081i	$0.163398\pi$
$618$	0	0
$619$	2.67240e12	0.731632	0.365816	−	0.930687i	$-0.380790\pi$
$619$	2.67240e12	0.731632	0.365816	+	0.930687i	$0.380790\pi$
$620$	0	0
$621$	−1.12772e12	−0.304290
$622$	0	0
$623$	−1.63302e12	−0.434305
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	0	0
$627$	−6.73329e11	−0.173990
$628$	0	0
$629$	−8.90474e11	−0.226826
$630$	0	0
$631$	−2.73608e12	−0.687064	−0.343532	−	0.939141i	$-0.611623\pi$
$631$	−2.73608e12	−0.687064	−0.343532	+	0.939141i	$0.611623\pi$
$632$	0	0
$633$	−2.97993e12	−0.737716
$634$	0	0
$635$	−5.93953e11	−0.144967
$636$	0	0
$637$	5.20167e11	0.125174
$638$	0	0
$639$	−8.44093e11	−0.200279
$640$	0	0
$641$	−2.58542e12	−0.604882	−0.302441	−	0.953168i	$-0.597802\pi$
$641$	−2.58542e12	−0.604882	−0.302441	+	0.953168i	$0.597802\pi$
$642$	0	0
$643$	4.53865e12	1.04707	0.523537	−	0.852003i	$-0.324612\pi$
$643$	4.53865e12	1.04707	0.523537	+	0.852003i	$0.324612\pi$
$644$	0	0
$645$	−9.67647e11	−0.220140
$646$	0	0
$647$	−2.68597e12	−0.602604	−0.301302	−	0.953529i	$-0.597421\pi$
$647$	−2.68597e12	−0.602604	−0.301302	+	0.953529i	$0.597421\pi$
$648$	0	0
$649$	−1.89872e12	−0.420106
$650$	0	0
$651$	5.76492e10	0.0125799
$652$	0	0
$653$	−6.39120e12	−1.37554	−0.687770	−	0.725929i	$-0.741410\pi$
$653$	−6.39120e12	−1.37554	−0.687770	+	0.725929i	$0.741410\pi$
$654$	0	0
$655$	−2.51906e12	−0.534753
$656$	0	0
$657$	−1.57054e12	−0.328856
$658$	0	0
$659$	4.23105e12	0.873904	0.436952	−	0.899485i	$-0.356058\pi$
$659$	4.23105e12	0.873904	0.436952	+	0.899485i	$0.356058\pi$
$660$	0	0
$661$	−3.77872e12	−0.769907	−0.384953	−	0.922936i	$-0.625782\pi$
$661$	−3.77872e12	−0.769907	−0.384953	+	0.922936i	$0.625782\pi$
$662$	0	0
$663$	−5.26872e11	−0.105900
$664$	0	0
$665$	−8.27248e11	−0.164036
$666$	0	0
$667$	8.26782e11	0.161743
$668$	0	0
$669$	−4.06058e11	−0.0783738
$670$	0	0
$671$	−2.73340e10	−0.00520538
$672$	0	0
$673$	2.75736e12	0.518114	0.259057	−	0.965862i	$-0.416588\pi$
$673$	2.75736e12	0.518114	0.259057	+	0.965862i	$0.416588\pi$
$674$	0	0
$675$	−2.07594e11	−0.0384900
$676$	0	0
$677$	−2.40567e12	−0.440136	−0.220068	−	0.975485i	$-0.570628\pi$
$677$	−2.40567e12	−0.440136	−0.220068	+	0.975485i	$0.570628\pi$
$678$	0	0
$679$	1.34259e13	2.42397
$680$	0	0
$681$	−2.21825e12	−0.395228
$682$	0	0
$683$	7.88192e11	0.138592	0.0692961	−	0.997596i	$-0.477925\pi$
$683$	7.88192e11	0.138592	0.0692961	+	0.997596i	$0.477925\pi$
$684$	0	0
$685$	−3.44919e12	−0.598563
$686$	0	0
$687$	−4.34863e12	−0.744813
$688$	0	0
$689$	−4.37015e9	−0.000738771 0
$690$	0	0
$691$	−3.54288e12	−0.591161	−0.295580	−	0.955318i	$-0.595513\pi$
$691$	−3.54288e12	−0.591161	−0.295580	+	0.955318i	$0.595513\pi$
$692$	0	0
$693$	2.54664e12	0.419438
$694$	0	0
$695$	−2.48610e12	−0.404191
$696$	0	0
$697$	6.24137e12	1.00169
$698$	0	0
$699$	−5.87958e12	−0.931535
$700$	0	0
$701$	−3.66998e12	−0.574027	−0.287014	−	0.957926i	$-0.592663\pi$
$701$	−3.66998e12	−0.574027	−0.287014	+	0.957926i	$0.592663\pi$
$702$	0	0
$703$	−5.58955e11	−0.0863133
$704$	0	0
$705$	−3.18003e12	−0.484819
$706$	0	0
$707$	−1.42595e12	−0.214643
$708$	0	0
$709$	6.68373e12	0.993369	0.496685	−	0.867931i	$-0.334551\pi$
$709$	6.68373e12	0.993369	0.496685	+	0.867931i	$0.334551\pi$
$710$	0	0
$711$	3.46438e12	0.508409
$712$	0	0
$713$	−1.92109e11	−0.0278385
$714$	0	0
$715$	−7.48337e11	−0.107083
$716$	0	0
$717$	1.40383e12	0.198371
$718$	0	0
$719$	1.23056e13	1.71721	0.858604	−	0.512640i	$-0.171332\pi$
$719$	1.23056e13	1.71721	0.858604	+	0.512640i	$0.171332\pi$
$720$	0	0
$721$	1.34001e13	1.84672
$722$	0	0
$723$	−3.29223e12	−0.448092
$724$	0	0
$725$	1.52197e11	0.0204590
$726$	0	0
$727$	1.07672e13	1.42954	0.714770	−	0.699360i	$-0.246531\pi$
$727$	1.07672e13	1.42954	0.714770	+	0.699360i	$0.246531\pi$
$728$	0	0
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	−5.12681e12	−0.664078
$732$	0	0
$733$	−1.06410e13	−1.36149	−0.680744	−	0.732521i	$-0.738343\pi$
$733$	−1.06410e13	−1.36149	−0.680744	+	0.732521i	$0.738343\pi$
$734$	0	0
$735$	1.08588e12	0.137243
$736$	0	0
$737$	1.18083e13	1.47429
$738$	0	0
$739$	4.24704e12	0.523825	0.261912	−	0.965092i	$-0.415647\pi$
$739$	4.24704e12	0.523825	0.261912	+	0.965092i	$0.415647\pi$
$740$	0	0
$741$	−3.30720e11	−0.0402975
$742$	0	0
$743$	−1.10732e13	−1.33298	−0.666488	−	0.745516i	$-0.732203\pi$
$743$	−1.10732e13	−1.33298	−0.666488	+	0.745516i	$0.732203\pi$
$744$	0	0
$745$	1.41586e12	0.168391
$746$	0	0
$747$	−1.39231e12	−0.163604
$748$	0	0
$749$	−1.36238e13	−1.58172
$750$	0	0
$751$	−1.45365e13	−1.66755	−0.833777	−	0.552102i	$-0.813826\pi$
$751$	−1.45365e13	−1.66755	−0.833777	+	0.552102i	$0.813826\pi$
$752$	0	0
$753$	8.17454e12	0.926587
$754$	0	0
$755$	−5.30724e12	−0.594439
$756$	0	0
$757$	−5.75556e12	−0.637025	−0.318512	−	0.947919i	$-0.603183\pi$
$757$	−5.75556e12	−0.637025	−0.318512	+	0.947919i	$0.603183\pi$
$758$	0	0
$759$	−8.48638e12	−0.928184
$760$	0	0
$761$	−1.44303e13	−1.55971	−0.779854	−	0.625961i	$-0.784707\pi$
$761$	−1.44303e13	−1.55971	−0.779854	+	0.625961i	$0.784707\pi$
$762$	0	0
$763$	−9.21664e12	−0.984492
$764$	0	0
$765$	−1.09988e12	−0.116110
$766$	0	0
$767$	−9.32595e11	−0.0973003
$768$	0	0
$769$	2.65690e12	0.273973	0.136986	−	0.990573i	$-0.456258\pi$
$769$	2.65690e12	0.273973	0.136986	+	0.990573i	$0.456258\pi$
$770$	0	0
$771$	−4.07965e12	−0.415794
$772$	0	0
$773$	−4.35569e12	−0.438783	−0.219391	−	0.975637i	$-0.570407\pi$
$773$	−4.35569e12	−0.438783	−0.219391	+	0.975637i	$0.570407\pi$
$774$	0	0
$775$	−3.53641e10	−0.00352132
$776$	0	0
$777$	2.11406e12	0.208076
$778$	0	0
$779$	3.91774e12	0.381168
$780$	0	0
$781$	−6.35203e12	−0.610918
$782$	0	0
$783$	−2.07062e11	−0.0196867
$784$	0	0
$785$	−4.55237e12	−0.427882
$786$	0	0
$787$	2.02013e13	1.87713	0.938564	−	0.345106i	$-0.112157\pi$
$787$	2.02013e13	1.87713	0.938564	+	0.345106i	$0.112157\pi$
$788$	0	0
$789$	−4.52529e12	−0.415719
$790$	0	0
$791$	−9.60521e12	−0.872394
$792$	0	0
$793$	−1.34257e10	−0.00120561
$794$	0	0
$795$	−9.12298e9	−0.000809999 0
$796$	0	0
$797$	−2.07284e12	−0.181971	−0.0909857	−	0.995852i	$-0.529002\pi$
$797$	−2.07284e12	−0.181971	−0.0909857	+	0.995852i	$0.529002\pi$
$798$	0	0
$799$	−1.68485e13	−1.46251
$800$	0	0
$801$	−1.36288e12	−0.116980
$802$	0	0
$803$	−1.18188e13	−1.00312
$804$	0	0
$805$	−1.04263e13	−0.875082
$806$	0	0
$807$	8.51916e12	0.707075
$808$	0	0
$809$	1.41673e13	1.16284	0.581418	−	0.813605i	$-0.302498\pi$
$809$	1.41673e13	1.16284	0.581418	+	0.813605i	$0.302498\pi$
$810$	0	0
$811$	−2.04580e12	−0.166062	−0.0830310	−	0.996547i	$-0.526460\pi$
$811$	−2.04580e12	−0.166062	−0.0830310	+	0.996547i	$0.526460\pi$
$812$	0	0
$813$	−4.40786e12	−0.353851
$814$	0	0
$815$	8.65117e12	0.686856
$816$	0	0
$817$	−3.21812e12	−0.252699
$818$	0	0
$819$	1.25084e12	0.0971455
$820$	0	0
$821$	−1.27265e13	−0.977607	−0.488804	−	0.872394i	$-0.662567\pi$
$821$	−1.27265e13	−0.977607	−0.488804	+	0.872394i	$0.662567\pi$
$822$	0	0
$823$	−6.94698e12	−0.527833	−0.263917	−	0.964546i	$-0.585014\pi$
$823$	−6.94698e12	−0.527833	−0.263917	+	0.964546i	$0.585014\pi$
$824$	0	0
$825$	−1.56220e12	−0.117407
$826$	0	0
$827$	−5.09129e11	−0.0378489	−0.0189244	−	0.999821i	$-0.506024\pi$
$827$	−5.09129e11	−0.0378489	−0.0189244	+	0.999821i	$0.506024\pi$
$828$	0	0
$829$	−1.87663e13	−1.38001	−0.690006	−	0.723804i	$-0.742392\pi$
$829$	−1.87663e13	−1.38001	−0.690006	+	0.723804i	$0.742392\pi$
$830$	0	0
$831$	−1.33268e13	−0.969440
$832$	0	0
$833$	5.75324e12	0.414009
$834$	0	0
$835$	−4.32524e12	−0.307908
$836$	0	0
$837$	4.81125e10	0.00338839
$838$	0	0
$839$	−6.68721e12	−0.465925	−0.232962	−	0.972486i	$-0.574842\pi$
$839$	−6.68721e12	−0.465925	−0.232962	+	0.972486i	$0.574842\pi$
$840$	0	0
$841$	−1.43553e13	−0.989536
$842$	0	0
$843$	5.46147e12	0.372465
$844$	0	0
$845$	6.26025e12	0.422412
$846$	0	0
$847$	6.27151e11	0.0418694
$848$	0	0
$849$	1.14952e13	0.759335
$850$	0	0
$851$	−7.04484e12	−0.460456
$852$	0	0
$853$	9.37029e12	0.606014	0.303007	−	0.952988i	$-0.402009\pi$
$853$	9.37029e12	0.606014	0.303007	+	0.952988i	$0.402009\pi$
$854$	0	0
$855$	−6.90399e11	−0.0441828
$856$	0	0
$857$	8.81996e12	0.558538	0.279269	−	0.960213i	$-0.409908\pi$
$857$	8.81996e12	0.558538	0.279269	+	0.960213i	$0.409908\pi$
$858$	0	0
$859$	−1.98932e13	−1.24663	−0.623313	−	0.781973i	$-0.714214\pi$
$859$	−1.98932e13	−1.24663	−0.623313	+	0.781973i	$0.714214\pi$
$860$	0	0
$861$	−1.48175e13	−0.918884
$862$	0	0
$863$	1.20953e13	0.742278	0.371139	−	0.928577i	$-0.378967\pi$
$863$	1.20953e13	0.742278	0.371139	+	0.928577i	$0.378967\pi$
$864$	0	0
$865$	−2.04367e12	−0.124119
$866$	0	0
$867$	3.77821e12	0.227091
$868$	0	0
$869$	2.60704e13	1.55081
$870$	0	0
$871$	5.79987e12	0.341458
$872$	0	0
$873$	1.12049e13	0.652894
$874$	0	0
$875$	−1.91931e12	−0.110690
$876$	0	0
$877$	−2.52693e13	−1.44243	−0.721216	−	0.692710i	$-0.756416\pi$
$877$	−2.52693e13	−1.44243	−0.721216	+	0.692710i	$0.756416\pi$
$878$	0	0
$879$	−2.26068e12	−0.127729
$880$	0	0
$881$	4.23113e12	0.236628	0.118314	−	0.992976i	$-0.462251\pi$
$881$	4.23113e12	0.236628	0.118314	+	0.992976i	$0.462251\pi$
$882$	0	0
$883$	−5.41348e12	−0.299677	−0.149839	−	0.988710i	$-0.547875\pi$
$883$	−5.41348e12	−0.299677	−0.149839	+	0.988710i	$0.547875\pi$
$884$	0	0
$885$	−1.94685e12	−0.106681
$886$	0	0
$887$	6.28957e12	0.341165	0.170583	−	0.985343i	$-0.445435\pi$
$887$	6.28957e12	0.341165	0.170583	+	0.985343i	$0.445435\pi$
$888$	0	0
$889$	7.47097e12	0.401161
$890$	0	0
$891$	2.12536e12	0.112975
$892$	0	0
$893$	−1.05759e13	−0.556525
$894$	0	0
$895$	−7.02695e12	−0.366069
$896$	0	0
$897$	−4.16826e12	−0.214976
$898$	0	0
$899$	−3.52735e10	−0.00180107
$900$	0	0
$901$	−4.83356e10	−0.00244346
$902$	0	0
$903$	1.21714e13	0.609182
$904$	0	0
$905$	−2.34400e12	−0.116155
$906$	0	0
$907$	1.27094e13	0.623580	0.311790	−	0.950151i	$-0.399072\pi$
$907$	1.27094e13	0.623580	0.311790	+	0.950151i	$0.399072\pi$
$908$	0	0
$909$	−1.19006e12	−0.0578138
$910$	0	0
$911$	−9.10912e12	−0.438171	−0.219086	−	0.975706i	$-0.570307\pi$
$911$	−9.10912e12	−0.438171	−0.219086	+	0.975706i	$0.570307\pi$
$912$	0	0
$913$	−1.04775e13	−0.499046
$914$	0	0
$915$	−2.80270e10	−0.00132185
$916$	0	0
$917$	3.16858e13	1.47980
$918$	0	0
$919$	6.78526e12	0.313795	0.156898	−	0.987615i	$-0.449851\pi$
$919$	6.78526e12	0.313795	0.156898	+	0.987615i	$0.449851\pi$
$920$	0	0
$921$	1.29411e13	0.592658
$922$	0	0
$923$	−3.11993e12	−0.141494
$924$	0	0
$925$	−1.29684e12	−0.0582436
$926$	0	0
$927$	1.11834e13	0.497410
$928$	0	0
$929$	3.16131e12	0.139250	0.0696252	−	0.997573i	$-0.477820\pi$
$929$	3.16131e12	0.139250	0.0696252	+	0.997573i	$0.477820\pi$
$930$	0	0
$931$	3.61134e12	0.157541
$932$	0	0
$933$	−6.35276e12	−0.274470
$934$	0	0
$935$	−8.27689e12	−0.354173
$936$	0	0
$937$	3.75670e13	1.59213	0.796065	−	0.605211i	$-0.206911\pi$
$937$	3.75670e13	1.59213	0.796065	+	0.605211i	$0.206911\pi$
$938$	0	0
$939$	1.02246e13	0.429190
$940$	0	0
$941$	−3.13713e13	−1.30431	−0.652153	−	0.758087i	$-0.726134\pi$
$941$	−3.13713e13	−1.30431	−0.652153	+	0.758087i	$0.726134\pi$
$942$	0	0
$943$	4.93776e13	2.03342
$944$	0	0
$945$	2.61120e12	0.106512
$946$	0	0
$947$	1.78574e13	0.721513	0.360757	−	0.932660i	$-0.382519\pi$
$947$	1.78574e13	0.721513	0.360757	+	0.932660i	$0.382519\pi$
$948$	0	0
$949$	−5.80504e12	−0.232331
$950$	0	0
$951$	7.77194e12	0.308118
$952$	0	0
$953$	8.76602e12	0.344258	0.172129	−	0.985074i	$-0.444935\pi$
$953$	8.76602e12	0.344258	0.172129	+	0.985074i	$0.444935\pi$
$954$	0	0
$955$	1.07766e13	0.419245
$956$	0	0
$957$	−1.55820e12	−0.0600509
$958$	0	0
$959$	4.33853e13	1.65638
$960$	0	0
$961$	−2.64314e13	−0.999690
$962$	0	0
$963$	−1.13701e13	−0.426035
$964$	0	0
$965$	3.75663e12	0.139452
$966$	0	0
$967$	−2.92145e13	−1.07443	−0.537217	−	0.843444i	$-0.680524\pi$
$967$	−2.92145e13	−1.07443	−0.537217	+	0.843444i	$0.680524\pi$
$968$	0	0
$969$	−3.65789e12	−0.133283
$970$	0	0
$971$	−1.10119e13	−0.397536	−0.198768	−	0.980047i	$-0.563694\pi$
$971$	−1.10119e13	−0.397536	−0.198768	+	0.980047i	$0.563694\pi$
$972$	0	0
$973$	3.12711e13	1.11850
$974$	0	0
$975$	−7.67308e11	−0.0271925
$976$	0	0
$977$	8.77989e12	0.308293	0.154146	−	0.988048i	$-0.450737\pi$
$977$	8.77989e12	0.308293	0.154146	+	0.988048i	$0.450737\pi$
$978$	0	0
$979$	−1.02560e13	−0.356826
$980$	0	0
$981$	−7.69197e12	−0.265172
$982$	0	0
$983$	5.68473e13	1.94186	0.970932	−	0.239355i	$-0.0769359\pi$
$983$	5.68473e13	1.94186	0.970932	+	0.239355i	$0.0769359\pi$
$984$	0	0
$985$	7.44234e12	0.251911
$986$	0	0
$987$	3.99996e13	1.34162
$988$	0	0
$989$	−4.05599e13	−1.34807
$990$	0	0
$991$	−4.32479e13	−1.42440	−0.712202	−	0.701974i	$-0.752302\pi$
$991$	−4.32479e13	−1.42440	−0.712202	+	0.701974i	$0.752302\pi$
$992$	0	0
$993$	6.26720e12	0.204551
$994$	0	0
$995$	−3.81841e12	−0.123503
$996$	0	0
$997$	5.30139e13	1.69927	0.849633	−	0.527374i	$-0.176823\pi$
$997$	5.30139e13	1.69927	0.849633	+	0.527374i	$0.176823\pi$
$998$	0	0
$999$	1.76434e12	0.0560450

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.10.a.m.1.2		2
4.3	odd	2		15.10.a.c.1.2	✓	2
12.11	even	2		45.10.a.e.1.1		2
20.3	even	4		75.10.b.e.49.1		4
20.7	even	4		75.10.b.e.49.4		4
20.19	odd	2		75.10.a.g.1.1		2
60.23	odd	4		225.10.b.g.199.4		4
60.47	odd	4		225.10.b.g.199.1		4
60.59	even	2		225.10.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.10.a.c.1.2	✓	2	4.3	odd	2
45.10.a.e.1.1		2	12.11	even	2
75.10.a.g.1.1		2	20.19	odd	2
75.10.b.e.49.1		4	20.3	even	4
75.10.b.e.49.4		4	20.7	even	4
225.10.a.j.1.2		2	60.59	even	2
225.10.b.g.199.1		4	60.47	odd	4
225.10.b.g.199.4		4	60.23	odd	4
240.10.a.m.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)

\(f(q)\)	\(=\)	\(q-81.0000 q^{3} -625.000 q^{5} +7861.50 q^{7} +6561.00 q^{9} +O(q^{10})\)	\(q-81.0000 q^{3} -625.000 q^{5} +7861.50 q^{7} +6561.00 q^{9} +49373.3 q^{11} +24250.7 q^{13} +50625.0 q^{15} +268222. q^{17} +168364. q^{19} -636781. q^{21} +2.12200e6 q^{23} +390625. q^{25} -531441. q^{27} +389624. q^{29} -90532.2 q^{31} -3.99924e6 q^{33} -4.91344e6 q^{35} -3.31991e6 q^{37} -1.96431e6 q^{39} +2.32694e7 q^{41} -1.91140e7 q^{43} -4.10062e6 q^{45} -6.28153e7 q^{47} +2.14495e7 q^{49} -2.17260e7 q^{51} -180207. q^{53} -3.08583e7 q^{55} -1.36375e7 q^{57} -3.84564e7 q^{59} -553620. q^{61} +5.15793e7 q^{63} -1.51567e7 q^{65} +2.39163e8 q^{67} -1.71882e8 q^{69} -1.28653e8 q^{71} -2.39376e8 q^{73} -3.16406e7 q^{75} +3.88148e8 q^{77} +5.28027e8 q^{79} +4.30467e7 q^{81} -2.12210e8 q^{83} -1.67639e8 q^{85} -3.15595e7 q^{87} -2.07724e8 q^{89} +1.90647e8 q^{91} +7.33311e6 q^{93} -1.05228e8 q^{95} +1.70780e9 q^{97} +3.23938e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 162 q^{3} - 1250 q^{5} + 11872 q^{7} + 13122 q^{9}+O(q^{10}) \) 2 * q - 162 * q^3 - 1250 * q^5 + 11872 * q^7 + 13122 * q^9	\( 2 q - 162 q^{3} - 1250 q^{5} + 11872 q^{7} + 13122 q^{9} - 35488 q^{11} + 143676 q^{13} + 101250 q^{15} + 385156 q^{17} + 403296 q^{19} - 961632 q^{21} - 223704 q^{23} + 781250 q^{25} - 1062882 q^{27} - 74572 q^{29} + 5027128 q^{31} + 2874528 q^{33} - 7420000 q^{35} + 5373628 q^{37} - 11637756 q^{39} + 14211332 q^{41} - 27748920 q^{43} - 8201250 q^{45} - 95966440 q^{47} - 2819950 q^{49} - 31197636 q^{51} - 64305596 q^{53} + 22180000 q^{55} - 32666976 q^{57} - 187863136 q^{59} + 154080060 q^{61} + 77892192 q^{63} - 89797500 q^{65} - 33592376 q^{67} + 18120024 q^{69} + 228270976 q^{71} - 33122316 q^{73} - 63281250 q^{75} + 47811456 q^{77} + 932406760 q^{79} + 86093442 q^{81} - 207040152 q^{83} - 240722500 q^{85} + 6040332 q^{87} + 224518164 q^{89} + 669602528 q^{91} - 407197368 q^{93} - 252060000 q^{95} + 387134596 q^{97} - 232836768 q^{99}+O(q^{100}) \) 2 * q - 162 * q^3 - 1250 * q^5 + 11872 * q^7 + 13122 * q^9 - 35488 * q^11 + 143676 * q^13 + 101250 * q^15 + 385156 * q^17 + 403296 * q^19 - 961632 * q^21 - 223704 * q^23 + 781250 * q^25 - 1062882 * q^27 - 74572 * q^29 + 5027128 * q^31 + 2874528 * q^33 - 7420000 * q^35 + 5373628 * q^37 - 11637756 * q^39 + 14211332 * q^41 - 27748920 * q^43 - 8201250 * q^45 - 95966440 * q^47 - 2819950 * q^49 - 31197636 * q^51 - 64305596 * q^53 + 22180000 * q^55 - 32666976 * q^57 - 187863136 * q^59 + 154080060 * q^61 + 77892192 * q^63 - 89797500 * q^65 - 33592376 * q^67 + 18120024 * q^69 + 228270976 * q^71 - 33122316 * q^73 - 63281250 * q^75 + 47811456 * q^77 + 932406760 * q^79 + 86093442 * q^81 - 207040152 * q^83 - 240722500 * q^85 + 6040332 * q^87 + 224518164 * q^89 + 669602528 * q^91 - 407197368 * q^93 - 252060000 * q^95 + 387134596 * q^97 - 232836768 * q^99

Introduction

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