LMFDB - Embedded newform 225.6.a.v.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,6,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("225.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	225.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:	$36.0863594579$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{5}, \sqrt{14})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 29x^{2} + 30x + 155 $ x^4 - 2x^3 - 29x^2 + 30*x + 155
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.12362$ of defining polynomial
Character	$\chi$	$=$	225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.47214 q^{2} -12.0000 q^{4} -224.499 q^{7} -196.774 q^{8} +O(q^{10})$	$q+4.47214 q^{2} -12.0000 q^{4} -224.499 q^{7} -196.774 q^{8} +501.996 q^{11} -224.499 q^{13} -1003.99 q^{14} -496.000 q^{16} +1668.11 q^{17} +484.000 q^{19} +2244.99 q^{22} +2262.90 q^{23} -1003.99 q^{26} +2693.99 q^{28} +5521.96 q^{29} +3608.00 q^{31} +4078.59 q^{32} +7460.00 q^{34} -7408.48 q^{37} +2164.51 q^{38} -11043.9 q^{41} -12572.0 q^{43} -6023.95 q^{44} +10120.0 q^{46} +9543.54 q^{47} +33593.0 q^{49} +2693.99 q^{52} +4771.77 q^{53} +44175.6 q^{56} +24694.9 q^{58} -5521.96 q^{59} +21362.0 q^{61} +16135.5 q^{62} +34112.0 q^{64} +34572.9 q^{67} -20017.3 q^{68} -33131.7 q^{71} +2693.99 q^{73} -33131.7 q^{74} -5808.00 q^{76} -112698. q^{77} +99616.0 q^{79} -49389.9 q^{82} -57118.1 q^{83} -56223.6 q^{86} -98779.8 q^{88} +120479. q^{89} +50400.0 q^{91} -27154.8 q^{92} +42680.0 q^{94} -64206.8 q^{97} +150232. q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 48 q^{4}+O(q^{10}) $ 4 * q - 48 * q^4	$ 4 q - 48 q^{4} - 1984 q^{16} + 1936 q^{19} + 14432 q^{31} + 29840 q^{34} + 40480 q^{46} + 134372 q^{49} + 85448 q^{61} + 136448 q^{64} - 23232 q^{76} + 398464 q^{79} + 201600 q^{91} + 170720 q^{94}+O(q^{100}) $ 4 * q - 48 * q^4 - 1984 * q^16 + 1936 * q^19 + 14432 * q^31 + 29840 * q^34 + 40480 * q^46 + 134372 * q^49 + 85448 * q^61 + 136448 * q^64 - 23232 * q^76 + 398464 * q^79 + 201600 * q^91 + 170720 * q^94

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 $

$2$	4.47214	0.790569	0.395285	−	0.918559i	$-0.370646\pi$
$2$	4.47214	0.790569	0.395285	+	0.918559i	$0.370646\pi$
$3$	0	0
$3$	0	0
$4$	−12.0000	−0.375000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−224.499	−1.73169	−0.865845	−	0.500312i	$-0.833219\pi$
$7$	−224.499	−1.73169	−0.865845	+	0.500312i	$0.833219\pi$
$8$	−196.774	−1.08703
$9$	0	0
$10$	0	0
$11$	501.996	1.25089	0.625444	−	0.780269i	$-0.284918\pi$
$11$	501.996	1.25089	0.625444	+	0.780269i	$0.284918\pi$
$12$	0	0
$13$	−224.499	−0.368432	−0.184216	−	0.982886i	$-0.558975\pi$
$13$	−224.499	−0.368432	−0.184216	+	0.982886i	$0.558975\pi$
$14$	−1003.99	−1.36902
$15$	0	0
$16$	−496.000	−0.484375
$17$	1668.11	1.39991	0.699957	−	0.714185i	$-0.253202\pi$
$17$	1668.11	1.39991	0.699957	+	0.714185i	$0.253202\pi$
$18$	0	0
$19$	484.000	0.307582	0.153791	−	0.988103i	$-0.450852\pi$
$19$	484.000	0.307582	0.153791	+	0.988103i	$0.450852\pi$
$20$	0	0
$21$	0	0
$22$	2244.99	0.988914
$23$	2262.90	0.891961	0.445981	−	0.895043i	$-0.352855\pi$
$23$	2262.90	0.891961	0.445981	+	0.895043i	$0.352855\pi$
$24$	0	0
$25$	0	0
$26$	−1003.99	−0.291271
$27$	0	0
$28$	2693.99	0.649384
$29$	5521.96	1.21926	0.609632	−	0.792684i	$-0.291317\pi$
$29$	5521.96	1.21926	0.609632	+	0.792684i	$0.291317\pi$
$30$	0	0
$31$	3608.00	0.674314	0.337157	−	0.941448i	$-0.390535\pi$
$31$	3608.00	0.674314	0.337157	+	0.941448i	$0.390535\pi$
$32$	4078.59	0.704101
$33$	0	0
$34$	7460.00	1.10673
$35$	0	0
$36$	0	0
$37$	−7408.48	−0.889662	−0.444831	−	0.895615i	$-0.646736\pi$
$37$	−7408.48	−0.889662	−0.444831	+	0.895615i	$0.646736\pi$
$38$	2164.51	0.243165
$39$	0	0
$40$	0	0
$41$	−11043.9	−1.02604	−0.513019	−	0.858377i	$-0.671473\pi$
$41$	−11043.9	−1.02604	−0.513019	+	0.858377i	$0.671473\pi$
$42$	0	0
$43$	−12572.0	−1.03689	−0.518444	−	0.855111i	$-0.673489\pi$
$43$	−12572.0	−1.03689	−0.518444	+	0.855111i	$0.673489\pi$
$44$	−6023.95	−0.469083
$45$	0	0
$46$	10120.0	0.705157
$47$	9543.54	0.630180	0.315090	−	0.949062i	$-0.397965\pi$
$47$	9543.54	0.630180	0.315090	+	0.949062i	$0.397965\pi$
$48$	0	0
$49$	33593.0	1.99875
$50$	0	0
$51$	0	0
$52$	2693.99	0.138162
$53$	4771.77	0.233340	0.116670	−	0.993171i	$-0.462778\pi$
$53$	4771.77	0.233340	0.116670	+	0.993171i	$0.462778\pi$
$54$	0	0
$55$	0	0
$56$	44175.6	1.88240
$57$	0	0
$58$	24694.9	0.963913
$59$	−5521.96	−0.206520	−0.103260	−	0.994654i	$-0.532927\pi$
$59$	−5521.96	−0.206520	−0.103260	+	0.994654i	$0.532927\pi$
$60$	0	0
$61$	21362.0	0.735051	0.367525	−	0.930013i	$-0.380205\pi$
$61$	21362.0	0.735051	0.367525	+	0.930013i	$0.380205\pi$
$62$	16135.5	0.533092
$63$	0	0
$64$	34112.0	1.04102
$65$	0	0
$66$	0	0
$67$	34572.9	0.940912	0.470456	−	0.882423i	$-0.344089\pi$
$67$	34572.9	0.940912	0.470456	+	0.882423i	$0.344089\pi$
$68$	−20017.3	−0.524968
$69$	0	0
$70$	0	0
$71$	−33131.7	−0.780007	−0.390003	−	0.920813i	$-0.627526\pi$
$71$	−33131.7	−0.780007	−0.390003	+	0.920813i	$0.627526\pi$
$72$	0	0
$73$	2693.99	0.0591683	0.0295842	−	0.999562i	$-0.490582\pi$
$73$	2693.99	0.0591683	0.0295842	+	0.999562i	$0.490582\pi$
$74$	−33131.7	−0.703339
$75$	0	0
$76$	−5808.00	−0.115343
$77$	−112698.	−2.16615
$78$	0	0
$79$	99616.0	1.79581	0.897907	−	0.440185i	$-0.145087\pi$
$79$	99616.0	1.79581	0.897907	+	0.440185i	$0.145087\pi$
$80$	0	0
$81$	0	0
$82$	−49389.9	−0.811154
$83$	−57118.1	−0.910078	−0.455039	−	0.890471i	$-0.650375\pi$
$83$	−57118.1	−0.910078	−0.455039	+	0.890471i	$0.650375\pi$
$84$	0	0
$85$	0	0
$86$	−56223.6	−0.819733
$87$	0	0
$88$	−98779.8	−1.35976
$89$	120479.	1.61227	0.806133	−	0.591735i	$-0.201557\pi$
$89$	120479.	1.61227	0.806133	+	0.591735i	$0.201557\pi$
$90$	0	0
$91$	50400.0	0.638009
$92$	−27154.8	−0.334485
$93$	0	0
$94$	42680.0	0.498201
$95$	0	0
$96$	0	0
$97$	−64206.8	−0.692870	−0.346435	−	0.938074i	$-0.612608\pi$
$97$	−64206.8	−0.692870	−0.346435	+	0.938074i	$0.612608\pi$
$98$	150232.	1.58015
$99$	0	0
$100$	0	0
$101$	60741.5	0.592492	0.296246	−	0.955112i	$-0.404265\pi$
$101$	60741.5	0.592492	0.296246	+	0.955112i	$0.404265\pi$
$102$	0	0
$103$	150639.	1.39909	0.699544	−	0.714590i	$-0.253387\pi$
$103$	150639.	1.39909	0.699544	+	0.714590i	$0.253387\pi$
$104$	44175.6	0.400497
$105$	0	0
$106$	21340.0	0.184472
$107$	−1914.07	−0.0161622	−0.00808108	−	0.999967i	$-0.502572\pi$
$107$	−1914.07	−0.0161622	−0.00808108	+	0.999967i	$0.502572\pi$
$108$	0	0
$109$	−1694.00	−0.0136567	−0.00682837	−	0.999977i	$-0.502174\pi$
$109$	−1694.00	−0.0136567	−0.00682837	+	0.999977i	$0.502174\pi$
$110$	0	0
$111$	0	0
$112$	111352.	0.838787
$113$	−148712.	−1.09559	−0.547797	−	0.836611i	$-0.684533\pi$
$113$	−148712.	−1.09559	−0.547797	+	0.836611i	$0.684533\pi$
$114$	0	0
$115$	0	0
$116$	−66263.5	−0.457224
$117$	0	0
$118$	−24694.9	−0.163269
$119$	−374489.	−2.42422
$120$	0	0
$121$	90949.0	0.564722
$122$	95533.8	0.581109
$123$	0	0
$124$	−43296.0	−0.252868
$125$	0	0
$126$	0	0
$127$	173089.	0.952271	0.476135	−	0.879372i	$-0.342037\pi$
$127$	173089.	0.952271	0.476135	+	0.879372i	$0.342037\pi$
$128$	22038.7	0.118894
$129$	0	0
$130$	0	0
$131$	270576.	1.37756	0.688780	−	0.724970i	$-0.258146\pi$
$131$	270576.	1.37756	0.688780	+	0.724970i	$0.258146\pi$
$132$	0	0
$133$	−108658.	−0.532637
$134$	154615.	0.743856
$135$	0	0
$136$	−328240.	−1.52175
$137$	319020.	1.45217	0.726083	−	0.687607i	$-0.241339\pi$
$137$	319020.	1.45217	0.726083	+	0.687607i	$0.241339\pi$
$138$	0	0
$139$	−226556.	−0.994577	−0.497289	−	0.867585i	$-0.665671\pi$
$139$	−226556.	−0.994577	−0.497289	+	0.867585i	$0.665671\pi$
$140$	0	0
$141$	0	0
$142$	−148170.	−0.616649
$143$	−112698.	−0.460867
$144$	0	0
$145$	0	0
$146$	12047.9	0.0467767
$147$	0	0
$148$	88901.8	0.333623
$149$	160137.	0.590915	0.295458	−	0.955356i	$-0.404528\pi$
$149$	160137.	0.590915	0.295458	+	0.955356i	$0.404528\pi$
$150$	0	0
$151$	8888.00	0.0317221	0.0158610	−	0.999874i	$-0.494951\pi$
$151$	8888.00	0.0317221	0.0158610	+	0.999874i	$0.494951\pi$
$152$	−95238.6	−0.334352
$153$	0	0
$154$	−504000.	−1.71249
$155$	0	0
$156$	0	0
$157$	−506246.	−1.63913	−0.819564	−	0.572988i	$-0.805784\pi$
$157$	−506246.	−1.63913	−0.819564	+	0.572988i	$0.805784\pi$
$158$	445496.	1.41972
$159$	0	0
$160$	0	0
$161$	−508020.	−1.54460
$162$	0	0
$163$	−222254.	−0.655211	−0.327606	−	0.944815i	$-0.606242\pi$
$163$	−222254.	−0.655211	−0.327606	+	0.944815i	$0.606242\pi$
$164$	132527.	0.384764
$165$	0	0
$166$	−255440.	−0.719480
$167$	97555.2	0.270682	0.135341	−	0.990799i	$-0.456787\pi$
$167$	97555.2	0.270682	0.135341	+	0.990799i	$0.456787\pi$
$168$	0	0
$169$	−320893.	−0.864258
$170$	0	0
$171$	0	0
$172$	150864.	0.388833
$173$	276669.	0.702821	0.351411	−	0.936221i	$-0.385702\pi$
$173$	276669.	0.702821	0.351411	+	0.936221i	$0.385702\pi$
$174$	0	0
$175$	0	0
$176$	−248990.	−0.605899
$177$	0	0
$178$	538799.	1.27461
$179$	28613.8	0.0667487	0.0333743	−	0.999443i	$-0.489375\pi$
$179$	28613.8	0.0667487	0.0333743	+	0.999443i	$0.489375\pi$
$180$	0	0
$181$	−81862.0	−0.185732	−0.0928658	−	0.995679i	$-0.529603\pi$
$181$	−81862.0	−0.185732	−0.0928658	+	0.995679i	$0.529603\pi$
$182$	225396.	0.504391
$183$	0	0
$184$	−445280.	−0.969591
$185$	0	0
$186$	0	0
$187$	837383.	1.75114
$188$	−114522.	−0.236318
$189$	0	0
$190$	0	0
$191$	486936.	0.965803	0.482902	−	0.875675i	$-0.339583\pi$
$191$	486936.	0.965803	0.482902	+	0.875675i	$0.339583\pi$
$192$	0	0
$193$	−541044.	−1.04554	−0.522768	−	0.852475i	$-0.675101\pi$
$193$	−541044.	−1.04554	−0.522768	+	0.852475i	$0.675101\pi$
$194$	−287142.	−0.547762
$195$	0	0
$196$	−403116.	−0.749531
$197$	−812976.	−1.49249	−0.746246	−	0.665670i	$-0.768146\pi$
$197$	−812976.	−1.49249	−0.746246	+	0.665670i	$0.768146\pi$
$198$	0	0
$199$	−146144.	−0.261606	−0.130803	−	0.991408i	$-0.541756\pi$
$199$	−146144.	−0.261606	−0.130803	+	0.991408i	$0.541756\pi$
$200$	0	0
$201$	0	0
$202$	271644.	0.468406
$203$	−1.23968e6	−2.11139
$204$	0	0
$205$	0	0
$206$	673679.	1.10608
$207$	0	0
$208$	111352.	0.178459
$209$	242966.	0.384751
$210$	0	0
$211$	846692.	1.30924	0.654620	−	0.755958i	$-0.272829\pi$
$211$	846692.	1.30924	0.654620	+	0.755958i	$0.272829\pi$
$212$	−57261.2	−0.0875026
$213$	0	0
$214$	−8560.00	−0.0127773
$215$	0	0
$216$	0	0
$217$	−809994.	−1.16770
$218$	−7575.80	−0.0107966
$219$	0	0
$220$	0	0
$221$	−374489.	−0.515773
$222$	0	0
$223$	812463.	1.09406	0.547031	−	0.837113i	$-0.315758\pi$
$223$	812463.	1.09406	0.547031	+	0.837113i	$0.315758\pi$
$224$	−915641.	−1.21928
$225$	0	0
$226$	−665060.	−0.866143
$227$	−355034.	−0.457304	−0.228652	−	0.973508i	$-0.573432\pi$
$227$	−355034.	−0.457304	−0.228652	+	0.973508i	$0.573432\pi$
$228$	0	0
$229$	411094.	0.518027	0.259014	−	0.965874i	$-0.416603\pi$
$229$	411094.	0.518027	0.259014	+	0.965874i	$0.416603\pi$
$230$	0	0
$231$	0	0
$232$	−1.08658e6	−1.32538
$233$	738193.	0.890800	0.445400	−	0.895332i	$-0.353061\pi$
$233$	738193.	0.890800	0.445400	+	0.895332i	$0.353061\pi$
$234$	0	0
$235$	0	0
$236$	66263.5	0.0774451
$237$	0	0
$238$	−1.67477e6	−1.91651
$239$	−121483.	−0.137569	−0.0687845	−	0.997632i	$-0.521912\pi$
$239$	−121483.	−0.137569	−0.0687845	+	0.997632i	$0.521912\pi$
$240$	0	0
$241$	1.12224e6	1.24464	0.622320	−	0.782763i	$-0.286190\pi$
$241$	1.12224e6	1.24464	0.622320	+	0.782763i	$0.286190\pi$
$242$	406736.	0.446452
$243$	0	0
$244$	−256344.	−0.275644
$245$	0	0
$246$	0	0
$247$	−108658.	−0.113323
$248$	−709961.	−0.733002
$249$	0	0
$250$	0	0
$251$	−16565.9	−0.0165970	−0.00829851	−	0.999966i	$-0.502642\pi$
$251$	−16565.9	−0.0165970	−0.00829851	+	0.999966i	$0.502642\pi$
$252$	0	0
$253$	1.13597e6	1.11574
$254$	774078.	0.752836
$255$	0	0
$256$	−993024.	−0.947021
$257$	−1.10779e6	−1.04622	−0.523111	−	0.852264i	$-0.675229\pi$
$257$	−1.10779e6	−1.04622	−0.523111	+	0.852264i	$0.675229\pi$
$258$	0	0
$259$	1.66320e6	1.54062
$260$	0	0
$261$	0	0
$262$	1.21005e6	1.08906
$263$	1.82582e6	1.62768	0.813840	−	0.581090i	$-0.197373\pi$
$263$	1.82582e6	1.62768	0.813840	+	0.581090i	$0.197373\pi$
$264$	0	0
$265$	0	0
$266$	−485932.	−0.421087
$267$	0	0
$268$	−414875.	−0.352842
$269$	1.34184e6	1.13062	0.565312	−	0.824877i	$-0.308756\pi$
$269$	1.34184e6	1.13062	0.565312	+	0.824877i	$0.308756\pi$
$270$	0	0
$271$	724592.	0.599336	0.299668	−	0.954044i	$-0.403124\pi$
$271$	724592.	0.599336	0.299668	+	0.954044i	$0.403124\pi$
$272$	−827381.	−0.678084
$273$	0	0
$274$	1.42670e6	1.14804
$275$	0	0
$276$	0	0
$277$	2.00007e6	1.56619	0.783096	−	0.621901i	$-0.213639\pi$
$277$	2.00007e6	1.56619	0.783096	+	0.621901i	$0.213639\pi$
$278$	−1.01319e6	−0.786282
$279$	0	0
$280$	0	0
$281$	−2.20878e6	−1.66873	−0.834367	−	0.551209i	$-0.814167\pi$
$281$	−2.20878e6	−1.66873	−0.834367	+	0.551209i	$0.814167\pi$
$282$	0	0
$283$	442264.	0.328258	0.164129	−	0.986439i	$-0.447519\pi$
$283$	442264.	0.328258	0.164129	+	0.986439i	$0.447519\pi$
$284$	397581.	0.292503
$285$	0	0
$286$	−504000.	−0.364347
$287$	2.47935e6	1.77678
$288$	0	0
$289$	1.36272e6	0.959761
$290$	0	0
$291$	0	0
$292$	−32327.9	−0.0221881
$293$	−1.01093e6	−0.687943	−0.343971	−	0.938980i	$-0.611772\pi$
$293$	−1.01093e6	−0.687943	−0.343971	+	0.938980i	$0.611772\pi$
$294$	0	0
$295$	0	0
$296$	1.45780e6	0.967092
$297$	0	0
$298$	716153.	0.467160
$299$	−508020.	−0.328627
$300$	0	0
$301$	2.82240e6	1.79557
$302$	39748.3	0.0250785
$303$	0	0
$304$	−240064.	−0.148985
$305$	0	0
$306$	0	0
$307$	−2.37341e6	−1.43723	−0.718615	−	0.695408i	$-0.755224\pi$
$307$	−2.37341e6	−1.43723	−0.718615	+	0.695408i	$0.755224\pi$
$308$	1.35237e6	0.812307
$309$	0	0
$310$	0	0
$311$	2.57323e6	1.50861	0.754307	−	0.656522i	$-0.227973\pi$
$311$	2.57323e6	1.50861	0.754307	+	0.656522i	$0.227973\pi$
$312$	0	0
$313$	−1.53603e6	−0.886212	−0.443106	−	0.896469i	$-0.646123\pi$
$313$	−1.53603e6	−0.886212	−0.443106	+	0.896469i	$0.646123\pi$
$314$	−2.26400e6	−1.29584
$315$	0	0
$316$	−1.19539e6	−0.673430
$317$	2.26708e6	1.26712	0.633562	−	0.773692i	$-0.281592\pi$
$317$	2.26708e6	1.26712	0.633562	+	0.773692i	$0.281592\pi$
$318$	0	0
$319$	2.77200e6	1.52516
$320$	0	0
$321$	0	0
$322$	−2.27193e6	−1.22111
$323$	807364.	0.430589
$324$	0	0
$325$	0	0
$326$	−993952.	−0.517990
$327$	0	0
$328$	2.17315e6	1.11534
$329$	−2.14252e6	−1.09128
$330$	0	0
$331$	1.65981e6	0.832701	0.416350	−	0.909204i	$-0.363309\pi$
$331$	1.65981e6	0.832701	0.416350	+	0.909204i	$0.363309\pi$
$332$	685417.	0.341279
$333$	0	0
$334$	436280.	0.213993
$335$	0	0
$336$	0	0
$337$	822566.	0.394544	0.197272	−	0.980349i	$-0.436792\pi$
$337$	822566.	0.394544	0.197272	+	0.980349i	$0.436792\pi$
$338$	−1.43508e6	−0.683256
$339$	0	0
$340$	0	0
$341$	1.81120e6	0.843492
$342$	0	0
$343$	−3.76845e6	−1.72953
$344$	2.47384e6	1.12713
$345$	0	0
$346$	1.23730e6	0.555629
$347$	977323.	0.435727	0.217863	−	0.975979i	$-0.430091\pi$
$347$	977323.	0.435727	0.217863	+	0.975979i	$0.430091\pi$
$348$	0	0
$349$	774994.	0.340592	0.170296	−	0.985393i	$-0.445528\pi$
$349$	774994.	0.340592	0.170296	+	0.985393i	$0.445528\pi$
$350$	0	0
$351$	0	0
$352$	2.04743e6	0.880752
$353$	2.76069e6	1.17918	0.589591	−	0.807702i	$-0.299289\pi$
$353$	2.76069e6	1.17918	0.589591	+	0.807702i	$0.299289\pi$
$354$	0	0
$355$	0	0
$356$	−1.44575e6	−0.604599
$357$	0	0
$358$	127965.	0.0527694
$359$	−1.12648e6	−0.461304	−0.230652	−	0.973036i	$-0.574086\pi$
$359$	−1.12648e6	−0.461304	−0.230652	+	0.973036i	$0.574086\pi$
$360$	0	0
$361$	−2.24184e6	−0.905393
$362$	−366098.	−0.146834
$363$	0	0
$364$	−604800.	−0.239254
$365$	0	0
$366$	0	0
$367$	−264236.	−0.102406	−0.0512031	−	0.998688i	$-0.516306\pi$
$367$	−264236.	−0.102406	−0.0512031	+	0.998688i	$0.516306\pi$
$368$	−1.12240e6	−0.432044
$369$	0	0
$370$	0	0
$371$	−1.07126e6	−0.404073
$372$	0	0
$373$	3.07227e6	1.14337	0.571687	−	0.820472i	$-0.306289\pi$
$373$	3.07227e6	1.14337	0.571687	+	0.820472i	$0.306289\pi$
$374$	3.74489e6	1.38440
$375$	0	0
$376$	−1.87792e6	−0.685027
$377$	−1.23968e6	−0.449216
$378$	0	0
$379$	−1.52304e6	−0.544643	−0.272322	−	0.962206i	$-0.587791\pi$
$379$	−1.52304e6	−0.544643	−0.272322	+	0.962206i	$0.587791\pi$
$380$	0	0
$381$	0	0
$382$	2.17764e6	0.763534
$383$	−2.40625e6	−0.838193	−0.419096	−	0.907942i	$-0.637653\pi$
$383$	−2.40625e6	−0.838193	−0.419096	+	0.907942i	$0.637653\pi$
$384$	0	0
$385$	0	0
$386$	−2.41962e6	−0.826569
$387$	0	0
$388$	770482.	0.259826
$389$	1.26453e6	0.423696	0.211848	−	0.977303i	$-0.432052\pi$
$389$	1.26453e6	0.423696	0.211848	+	0.977303i	$0.432052\pi$
$390$	0	0
$391$	3.77476e6	1.24867
$392$	−6.61023e6	−2.17271
$393$	0	0
$394$	−3.63574e6	−1.17992
$395$	0	0
$396$	0	0
$397$	1.56319e6	0.497778	0.248889	−	0.968532i	$-0.419935\pi$
$397$	1.56319e6	0.497778	0.248889	+	0.968532i	$0.419935\pi$
$398$	−653576.	−0.206818
$399$	0	0
$400$	0	0
$401$	1.36844e6	0.424977	0.212488	−	0.977164i	$-0.431843\pi$
$401$	1.36844e6	0.424977	0.212488	+	0.977164i	$0.431843\pi$
$402$	0	0
$403$	−809994.	−0.248439
$404$	−728898.	−0.222184
$405$	0	0
$406$	−5.54400e6	−1.66920
$407$	−3.71903e6	−1.11287
$408$	0	0
$409$	−4.31427e6	−1.27526	−0.637630	−	0.770343i	$-0.720085\pi$
$409$	−4.31427e6	−1.27526	−0.637630	+	0.770343i	$0.720085\pi$
$410$	0	0
$411$	0	0
$412$	−1.80767e6	−0.524658
$413$	1.23968e6	0.357629
$414$	0	0
$415$	0	0
$416$	−915641.	−0.259413
$417$	0	0
$418$	1.08658e6	0.304172
$419$	−689241.	−0.191794	−0.0958972	−	0.995391i	$-0.530572\pi$
$419$	−689241.	−0.191794	−0.0958972	+	0.995391i	$0.530572\pi$
$420$	0	0
$421$	−1.03556e6	−0.284755	−0.142377	−	0.989812i	$-0.545475\pi$
$421$	−1.03556e6	−0.284755	−0.142377	+	0.989812i	$0.545475\pi$
$422$	3.78652e6	1.03505
$423$	0	0
$424$	−938960.	−0.253649
$425$	0	0
$426$	0	0
$427$	−4.79576e6	−1.27288
$428$	22968.9	0.00606081
$429$	0	0
$430$	0	0
$431$	−2.78307e6	−0.721656	−0.360828	−	0.932632i	$-0.617506\pi$
$431$	−2.78307e6	−0.721656	−0.360828	+	0.932632i	$0.617506\pi$
$432$	0	0
$433$	−400058.	−0.102542	−0.0512712	−	0.998685i	$-0.516327\pi$
$433$	−400058.	−0.102542	−0.0512712	+	0.998685i	$0.516327\pi$
$434$	−3.62240e6	−0.923150
$435$	0	0
$436$	20328.0	0.00512128
$437$	1.09524e6	0.274351
$438$	0	0
$439$	−3.10006e6	−0.767731	−0.383866	−	0.923389i	$-0.625407\pi$
$439$	−3.10006e6	−0.767731	−0.383866	+	0.923389i	$0.625407\pi$
$440$	0	0
$441$	0	0
$442$	−1.67477e6	−0.407754
$443$	219797.	0.0532122	0.0266061	−	0.999646i	$-0.491530\pi$
$443$	219797.	0.0532122	0.0266061	+	0.999646i	$0.491530\pi$
$444$	0	0
$445$	0	0
$446$	3.63345e6	0.864931
$447$	0	0
$448$	−7.65813e6	−1.80272
$449$	2.47283e6	0.578867	0.289434	−	0.957198i	$-0.406533\pi$
$449$	2.47283e6	0.578867	0.289434	+	0.957198i	$0.406533\pi$
$450$	0	0
$451$	−5.54400e6	−1.28346
$452$	1.78454e6	0.410848
$453$	0	0
$454$	−1.58776e6	−0.361531
$455$	0	0
$456$	0	0
$457$	2.89649e6	0.648757	0.324378	−	0.945927i	$-0.394845\pi$
$457$	2.89649e6	0.648757	0.324378	+	0.945927i	$0.394845\pi$
$458$	1.83847e6	0.409536
$459$	0	0
$460$	0	0
$461$	7.05154e6	1.54537	0.772683	−	0.634792i	$-0.218914\pi$
$461$	7.05154e6	1.54537	0.772683	+	0.634792i	$0.218914\pi$
$462$	0	0
$463$	2.89178e6	0.626920	0.313460	−	0.949601i	$-0.398512\pi$
$463$	2.89178e6	0.626920	0.313460	+	0.949601i	$0.398512\pi$
$464$	−2.73889e6	−0.590581
$465$	0	0
$466$	3.30130e6	0.704239
$467$	−7.71866e6	−1.63776	−0.818879	−	0.573967i	$-0.805404\pi$
$467$	−7.71866e6	−1.63776	−0.818879	+	0.573967i	$0.805404\pi$
$468$	0	0
$469$	−7.76160e6	−1.62937
$470$	0	0
$471$	0	0
$472$	1.08658e6	0.224494
$473$	−6.31108e6	−1.29703
$474$	0	0
$475$	0	0
$476$	4.49387e6	0.909082
$477$	0	0
$478$	−543289.	−0.108758
$479$	−209834.	−0.0417867	−0.0208933	−	0.999782i	$-0.506651\pi$
$479$	−209834.	−0.0417867	−0.0208933	+	0.999782i	$0.506651\pi$
$480$	0	0
$481$	1.66320e6	0.327779
$482$	5.01882e6	0.983975
$483$	0	0
$484$	−1.09139e6	−0.211771
$485$	0	0
$486$	0	0
$487$	777891.	0.148626	0.0743132	−	0.997235i	$-0.476324\pi$
$487$	777891.	0.148626	0.0743132	+	0.997235i	$0.476324\pi$
$488$	−4.20349e6	−0.799024
$489$	0	0
$490$	0	0
$491$	−9.93400e6	−1.85960	−0.929802	−	0.368061i	$-0.880022\pi$
$491$	−9.93400e6	−1.85960	−0.929802	+	0.368061i	$0.880022\pi$
$492$	0	0
$493$	9.21121e6	1.70687
$494$	−485932.	−0.0895897
$495$	0	0
$496$	−1.78957e6	−0.326621
$497$	7.43806e6	1.35073
$498$	0	0
$499$	5.10332e6	0.917489	0.458745	−	0.888568i	$-0.348299\pi$
$499$	5.10332e6	0.917489	0.458745	+	0.888568i	$0.348299\pi$
$500$	0	0
$501$	0	0
$502$	−74084.8	−0.0131211
$503$	4.00824e6	0.706373	0.353186	−	0.935553i	$-0.385098\pi$
$503$	4.00824e6	0.706373	0.353186	+	0.935553i	$0.385098\pi$
$504$	0	0
$505$	0	0
$506$	5.08020e6	0.882073
$507$	0	0
$508$	−2.07707e6	−0.357101
$509$	−1.96029e6	−0.335372	−0.167686	−	0.985840i	$-0.553629\pi$
$509$	−1.96029e6	−0.335372	−0.167686	+	0.985840i	$0.553629\pi$
$510$	0	0
$511$	−604800.	−0.102461
$512$	−5.14618e6	−0.867580
$513$	0	0
$514$	−4.95418e6	−0.827111
$515$	0	0
$516$	0	0
$517$	4.79082e6	0.788285
$518$	7.43806e6	1.21797
$519$	0	0
$520$	0	0
$521$	−9.65338e6	−1.55806	−0.779032	−	0.626984i	$-0.784289\pi$
$521$	−9.65338e6	−1.55806	−0.779032	+	0.626984i	$0.784289\pi$
$522$	0	0
$523$	462020.	0.0738595	0.0369298	−	0.999318i	$-0.488242\pi$
$523$	462020.	0.0738595	0.0369298	+	0.999318i	$0.488242\pi$
$524$	−3.24691e6	−0.516585
$525$	0	0
$526$	8.16532e6	1.28679
$527$	6.01853e6	0.943982
$528$	0	0
$529$	−1.31562e6	−0.204405
$530$	0	0
$531$	0	0
$532$	1.30389e6	0.199739
$533$	2.47935e6	0.378025
$534$	0	0
$535$	0	0
$536$	−6.80305e6	−1.02280
$537$	0	0
$538$	6.00087e6	0.893837
$539$	1.68636e7	2.50021
$540$	0	0
$541$	−1.08238e7	−1.58996	−0.794978	−	0.606639i	$-0.792517\pi$
$541$	−1.08238e7	−1.58996	−0.794978	+	0.606639i	$0.792517\pi$
$542$	3.24047e6	0.473817
$543$	0	0
$544$	6.80352e6	0.985681
$545$	0	0
$546$	0	0
$547$	2.02229e6	0.288985	0.144493	−	0.989506i	$-0.453845\pi$
$547$	2.02229e6	0.288985	0.144493	+	0.989506i	$0.453845\pi$
$548$	−3.82824e6	−0.544562
$549$	0	0
$550$	0	0
$551$	2.67263e6	0.375024
$552$	0	0
$553$	−2.23637e7	−3.10979
$554$	8.94457e6	1.23818
$555$	0	0
$556$	2.71867e6	0.372967
$557$	−3.92593e6	−0.536173	−0.268086	−	0.963395i	$-0.586391\pi$
$557$	−3.92593e6	−0.536173	−0.268086	+	0.963395i	$0.586391\pi$
$558$	0	0
$559$	2.82240e6	0.382023
$560$	0	0
$561$	0	0
$562$	−9.87798e6	−1.31925
$563$	1.12906e7	1.50123	0.750616	−	0.660739i	$-0.229757\pi$
$563$	1.12906e7	1.50123	0.750616	+	0.660739i	$0.229757\pi$
$564$	0	0
$565$	0	0
$566$	1.97786e6	0.259511
$567$	0	0
$568$	6.51946e6	0.847893
$569$	−1.10770e7	−1.43431	−0.717155	−	0.696913i	$-0.754556\pi$
$569$	−1.10770e7	−1.43431	−0.717155	+	0.696913i	$0.754556\pi$
$570$	0	0
$571$	3.28605e6	0.421778	0.210889	−	0.977510i	$-0.432364\pi$
$571$	3.28605e6	0.421778	0.210889	+	0.977510i	$0.432364\pi$
$572$	1.35237e6	0.172825
$573$	0	0
$574$	1.10880e7	1.40467
$575$	0	0
$576$	0	0
$577$	−1.42539e7	−1.78236	−0.891179	−	0.453653i	$-0.850121\pi$
$577$	−1.42539e7	−1.78236	−0.891179	+	0.453653i	$0.850121\pi$
$578$	6.09428e6	0.758757
$579$	0	0
$580$	0	0
$581$	1.28230e7	1.57597
$582$	0	0
$583$	2.39541e6	0.291883
$584$	−530108.	−0.0643179
$585$	0	0
$586$	−4.52102e6	−0.543867
$587$	6.12026e6	0.733119	0.366560	−	0.930395i	$-0.380536\pi$
$587$	6.12026e6	0.733119	0.366560	+	0.930395i	$0.380536\pi$
$588$	0	0
$589$	1.74627e6	0.207407
$590$	0	0
$591$	0	0
$592$	3.67461e6	0.430930
$593$	−1.38377e7	−1.61595	−0.807976	−	0.589215i	$-0.799437\pi$
$593$	−1.38377e7	−1.61595	−0.807976	+	0.589215i	$0.799437\pi$
$594$	0	0
$595$	0	0
$596$	−1.92164e6	−0.221593
$597$	0	0
$598$	−2.27193e6	−0.259802
$599$	−6.37133e6	−0.725543	−0.362772	−	0.931878i	$-0.618169\pi$
$599$	−6.37133e6	−0.725543	−0.362772	+	0.931878i	$0.618169\pi$
$600$	0	0
$601$	2.06116e6	0.232769	0.116384	−	0.993204i	$-0.462870\pi$
$601$	2.06116e6	0.232769	0.116384	+	0.993204i	$0.462870\pi$
$602$	1.26222e7	1.41952
$603$	0	0
$604$	−106656.	−0.0118958
$605$	0	0
$606$	0	0
$607$	8.89040e6	0.979377	0.489688	−	0.871898i	$-0.337111\pi$
$607$	8.89040e6	0.979377	0.489688	+	0.871898i	$0.337111\pi$
$608$	1.97404e6	0.216569
$609$	0	0
$610$	0	0
$611$	−2.14252e6	−0.232178
$612$	0	0
$613$	9.20605e6	0.989514	0.494757	−	0.869031i	$-0.335257\pi$
$613$	9.20605e6	0.989514	0.494757	+	0.869031i	$0.335257\pi$
$614$	−1.06142e7	−1.13623
$615$	0	0
$616$	2.21760e7	2.35468
$617$	−1.07622e7	−1.13812	−0.569061	−	0.822296i	$-0.692693\pi$
$617$	−1.07622e7	−1.13812	−0.569061	+	0.822296i	$0.692693\pi$
$618$	0	0
$619$	1.40448e7	1.47329	0.736644	−	0.676281i	$-0.236409\pi$
$619$	1.40448e7	1.47329	0.736644	+	0.676281i	$0.236409\pi$
$620$	0	0
$621$	0	0
$622$	1.15078e7	1.19266
$623$	−2.70475e7	−2.79194
$624$	0	0
$625$	0	0
$626$	−6.86931e6	−0.700612
$627$	0	0
$628$	6.07495e6	0.614673
$629$	−1.23581e7	−1.24545
$630$	0	0
$631$	−1.00120e7	−1.00104	−0.500518	−	0.865726i	$-0.666857\pi$
$631$	−1.00120e7	−1.00104	−0.500518	+	0.865726i	$0.666857\pi$
$632$	−1.96018e7	−1.95211
$633$	0	0
$634$	1.01387e7	1.00175
$635$	0	0
$636$	0	0
$637$	−7.54161e6	−0.736403
$638$	1.23968e7	1.20575
$639$	0	0
$640$	0	0
$641$	9.60720e6	0.923532	0.461766	−	0.887002i	$-0.347216\pi$
$641$	9.60720e6	0.923532	0.461766	+	0.887002i	$0.347216\pi$
$642$	0	0
$643$	2.21761e6	0.211523	0.105761	−	0.994392i	$-0.466272\pi$
$643$	2.21761e6	0.211523	0.105761	+	0.994392i	$0.466272\pi$
$644$	6.09624e6	0.579225
$645$	0	0
$646$	3.61064e6	0.340410
$647$	−5.71481e6	−0.536712	−0.268356	−	0.963320i	$-0.586480\pi$
$647$	−5.71481e6	−0.536712	−0.268356	+	0.963320i	$0.586480\pi$
$648$	0	0
$649$	−2.77200e6	−0.258334
$650$	0	0
$651$	0	0
$652$	2.66705e6	0.245704
$653$	−4.71603e6	−0.432807	−0.216403	−	0.976304i	$-0.569433\pi$
$653$	−4.71603e6	−0.432807	−0.216403	+	0.976304i	$0.569433\pi$
$654$	0	0
$655$	0	0
$656$	5.47778e6	0.496987
$657$	0	0
$658$	−9.58164e6	−0.862730
$659$	2.01496e7	1.80740	0.903698	−	0.428171i	$-0.140842\pi$
$659$	2.01496e7	1.80740	0.903698	+	0.428171i	$0.140842\pi$
$660$	0	0
$661$	−564742.	−0.0502743	−0.0251372	−	0.999684i	$-0.508002\pi$
$661$	−564742.	−0.0502743	−0.0251372	+	0.999684i	$0.508002\pi$
$662$	7.42290e6	0.658308
$663$	0	0
$664$	1.12394e7	0.989285
$665$	0	0
$666$	0	0
$667$	1.24956e7	1.08754
$668$	−1.17066e6	−0.101506
$669$	0	0
$670$	0	0
$671$	1.07236e7	0.919466
$672$	0	0
$673$	−1.07198e7	−0.912328	−0.456164	−	0.889896i	$-0.650777\pi$
$673$	−1.07198e7	−0.912328	−0.456164	+	0.889896i	$0.650777\pi$
$674$	3.67863e6	0.311915
$675$	0	0
$676$	3.85072e6	0.324097
$677$	−3.36926e6	−0.282529	−0.141264	−	0.989972i	$-0.545117\pi$
$677$	−3.36926e6	−0.282529	−0.141264	+	0.989972i	$0.545117\pi$
$678$	0	0
$679$	1.44144e7	1.19984
$680$	0	0
$681$	0	0
$682$	8.09994e6	0.666839
$683$	1.55817e7	1.27810	0.639049	−	0.769166i	$-0.279328\pi$
$683$	1.55817e7	1.27810	0.639049	+	0.769166i	$0.279328\pi$
$684$	0	0
$685$	0	0
$686$	−1.68530e7	−1.36731
$687$	0	0
$688$	6.23570e6	0.502243
$689$	−1.07126e6	−0.0859699
$690$	0	0
$691$	9.73771e6	0.775821	0.387911	−	0.921697i	$-0.373197\pi$
$691$	9.73771e6	0.775821	0.387911	+	0.921697i	$0.373197\pi$
$692$	−3.32002e6	−0.263558
$693$	0	0
$694$	4.37072e6	0.344472
$695$	0	0
$696$	0	0
$697$	−1.84224e7	−1.43637
$698$	3.46588e6	0.269262
$699$	0	0
$700$	0	0
$701$	−1.72451e7	−1.32547	−0.662735	−	0.748854i	$-0.730604\pi$
$701$	−1.72451e7	−1.32547	−0.662735	+	0.748854i	$0.730604\pi$
$702$	0	0
$703$	−3.58571e6	−0.273644
$704$	1.71241e7	1.30219
$705$	0	0
$706$	1.23462e7	0.932225
$707$	−1.36364e7	−1.02601
$708$	0	0
$709$	−1.81083e7	−1.35289	−0.676443	−	0.736495i	$-0.736480\pi$
$709$	−1.81083e7	−1.35289	−0.676443	+	0.736495i	$0.736480\pi$
$710$	0	0
$711$	0	0
$712$	−2.37071e7	−1.75259
$713$	8.16455e6	0.601462
$714$	0	0
$715$	0	0
$716$	−343365.	−0.0250307
$717$	0	0
$718$	−5.03777e6	−0.364693
$719$	−1.77044e7	−1.27720	−0.638600	−	0.769539i	$-0.720486\pi$
$719$	−1.77044e7	−1.27720	−0.638600	+	0.769539i	$0.720486\pi$
$720$	0	0
$721$	−3.38184e7	−2.42279
$722$	−1.00258e7	−0.715776
$723$	0	0
$724$	982344.	0.0696494
$725$	0	0
$726$	0	0
$727$	−7.80607e6	−0.547768	−0.273884	−	0.961763i	$-0.588308\pi$
$727$	−7.80607e6	−0.547768	−0.273884	+	0.961763i	$0.588308\pi$
$728$	−9.91741e6	−0.693537
$729$	0	0
$730$	0	0
$731$	−2.09714e7	−1.45156
$732$	0	0
$733$	2.11490e7	1.45388	0.726941	−	0.686700i	$-0.240941\pi$
$733$	2.11490e7	1.45388	0.726941	+	0.686700i	$0.240941\pi$
$734$	−1.18170e6	−0.0809593
$735$	0	0
$736$	9.22944e6	0.628031
$737$	1.73555e7	1.17698
$738$	0	0
$739$	−9.20256e6	−0.619865	−0.309933	−	0.950759i	$-0.600307\pi$
$739$	−9.20256e6	−0.619865	−0.309933	+	0.950759i	$0.600307\pi$
$740$	0	0
$741$	0	0
$742$	−4.79082e6	−0.319448
$743$	−4.78756e6	−0.318157	−0.159079	−	0.987266i	$-0.550852\pi$
$743$	−4.78756e6	−0.318157	−0.159079	+	0.987266i	$0.550852\pi$
$744$	0	0
$745$	0	0
$746$	1.37396e7	0.903916
$747$	0	0
$748$	−1.00486e7	−0.656676
$749$	429709.	0.0279878
$750$	0	0
$751$	7.03369e6	0.455075	0.227538	−	0.973769i	$-0.426933\pi$
$751$	7.03369e6	0.455075	0.227538	+	0.973769i	$0.426933\pi$
$752$	−4.73359e6	−0.305244
$753$	0	0
$754$	−5.54400e6	−0.355136
$755$	0	0
$756$	0	0
$757$	1.99807e7	1.26727	0.633637	−	0.773631i	$-0.281561\pi$
$757$	1.99807e7	1.26727	0.633637	+	0.773631i	$0.281561\pi$
$758$	−6.81122e6	−0.430578
$759$	0	0
$760$	0	0
$761$	1.15851e7	0.725165	0.362582	−	0.931952i	$-0.381895\pi$
$761$	1.15851e7	0.725165	0.362582	+	0.931952i	$0.381895\pi$
$762$	0	0
$763$	380302.	0.0236492
$764$	−5.84323e6	−0.362176
$765$	0	0
$766$	−1.07611e7	−0.662650
$767$	1.23968e6	0.0760886
$768$	0	0
$769$	−1.88308e7	−1.14830	−0.574148	−	0.818752i	$-0.694667\pi$
$769$	−1.88308e7	−1.14830	−0.574148	+	0.818752i	$0.694667\pi$
$770$	0	0
$771$	0	0
$772$	6.49252e6	0.392076
$773$	−9.80638e6	−0.590283	−0.295141	−	0.955454i	$-0.595367\pi$
$773$	−9.80638e6	−0.590283	−0.295141	+	0.955454i	$0.595367\pi$
$774$	0	0
$775$	0	0
$776$	1.26342e7	0.753173
$777$	0	0
$778$	5.65514e6	0.334961
$779$	−5.34525e6	−0.315591
$780$	0	0
$781$	−1.66320e7	−0.975701
$782$	1.68812e7	0.987160
$783$	0	0
$784$	−1.66621e7	−0.968145
$785$	0	0
$786$	0	0
$787$	3.21012e7	1.84750	0.923749	−	0.382998i	$-0.125108\pi$
$787$	3.21012e7	1.84750	0.923749	+	0.382998i	$0.125108\pi$
$788$	9.75571e6	0.559685
$789$	0	0
$790$	0	0
$791$	3.33857e7	1.89723
$792$	0	0
$793$	−4.79576e6	−0.270816
$794$	6.99080e6	0.393528
$795$	0	0
$796$	1.75373e6	0.0981024
$797$	3.06076e7	1.70680	0.853400	−	0.521256i	$-0.174536\pi$
$797$	3.06076e7	1.70680	0.853400	+	0.521256i	$0.174536\pi$
$798$	0	0
$799$	1.59196e7	0.882198
$800$	0	0
$801$	0	0
$802$	6.11985e6	0.335974
$803$	1.35237e6	0.0740130
$804$	0	0
$805$	0	0
$806$	−3.62240e6	−0.196408
$807$	0	0
$808$	−1.19524e7	−0.644058
$809$	1.30208e7	0.699464	0.349732	−	0.936850i	$-0.386273\pi$
$809$	1.30208e7	0.699464	0.349732	+	0.936850i	$0.386273\pi$
$810$	0	0
$811$	−2.58466e7	−1.37991	−0.689956	−	0.723851i	$-0.742370\pi$
$811$	−2.58466e7	−1.37991	−0.689956	+	0.723851i	$0.742370\pi$
$812$	1.48761e7	0.791771
$813$	0	0
$814$	−1.66320e7	−0.879799
$815$	0	0
$816$	0	0
$817$	−6.08483e6	−0.318929
$818$	−1.92940e7	−1.00818
$819$	0	0
$820$	0	0
$821$	−1.88575e7	−0.976396	−0.488198	−	0.872733i	$-0.662346\pi$
$821$	−1.88575e7	−0.976396	−0.488198	+	0.872733i	$0.662346\pi$
$822$	0	0
$823$	1.90126e7	0.978459	0.489230	−	0.872155i	$-0.337278\pi$
$823$	1.90126e7	0.978459	0.489230	+	0.872155i	$0.337278\pi$
$824$	−2.96419e7	−1.52085
$825$	0	0
$826$	5.54400e6	0.282731
$827$	823839.	0.0418869	0.0209435	−	0.999781i	$-0.493333\pi$
$827$	823839.	0.0418869	0.0209435	+	0.999781i	$0.493333\pi$
$828$	0	0
$829$	−3.60900e7	−1.82390	−0.911948	−	0.410305i	$-0.865422\pi$
$829$	−3.60900e7	−1.82390	−0.911948	+	0.410305i	$0.865422\pi$
$830$	0	0
$831$	0	0
$832$	−7.65813e6	−0.383543
$833$	5.60367e7	2.79808
$834$	0	0
$835$	0	0
$836$	−2.91559e6	−0.144282
$837$	0	0
$838$	−3.08238e6	−0.151627
$839$	−1.23692e7	−0.606647	−0.303324	−	0.952888i	$-0.598096\pi$
$839$	−1.23692e7	−0.606647	−0.303324	+	0.952888i	$0.598096\pi$
$840$	0	0
$841$	9.98085e6	0.486606
$842$	−4.63117e6	−0.225118
$843$	0	0
$844$	−1.01603e7	−0.490965
$845$	0	0
$846$	0	0
$847$	−2.04180e7	−0.977923
$848$	−2.36680e6	−0.113024
$849$	0	0
$850$	0	0
$851$	−1.67647e7	−0.793544
$852$	0	0
$853$	−9.97159e6	−0.469237	−0.234618	−	0.972088i	$-0.575384\pi$
$853$	−9.97159e6	−0.469237	−0.234618	+	0.972088i	$0.575384\pi$
$854$	−2.14473e7	−1.00630
$855$	0	0
$856$	376640.	0.0175688
$857$	−1.48619e7	−0.691231	−0.345616	−	0.938376i	$-0.612330\pi$
$857$	−1.48619e7	−0.691231	−0.345616	+	0.938376i	$0.612330\pi$
$858$	0	0
$859$	−1.41452e7	−0.654071	−0.327036	−	0.945012i	$-0.606050\pi$
$859$	−1.41452e7	−0.654071	−0.327036	+	0.945012i	$0.606050\pi$
$860$	0	0
$861$	0	0
$862$	−1.24462e7	−0.570519
$863$	−1.72325e6	−0.0787627	−0.0393814	−	0.999224i	$-0.512539\pi$
$863$	−1.72325e6	−0.0787627	−0.0393814	+	0.999224i	$0.512539\pi$
$864$	0	0
$865$	0	0
$866$	−1.78911e6	−0.0810668
$867$	0	0
$868$	9.71993e6	0.437889
$869$	5.00068e7	2.24636
$870$	0	0
$871$	−7.76160e6	−0.346662
$872$	333335.	0.0148453
$873$	0	0
$874$	4.89808e6	0.216894
$875$	0	0
$876$	0	0
$877$	−8.53435e6	−0.374689	−0.187345	−	0.982294i	$-0.559988\pi$
$877$	−8.53435e6	−0.374689	−0.187345	+	0.982294i	$0.559988\pi$
$878$	−1.38639e7	−0.606945
$879$	0	0
$880$	0	0
$881$	−7.09019e6	−0.307764	−0.153882	−	0.988089i	$-0.549178\pi$
$881$	−7.09019e6	−0.307764	−0.153882	+	0.988089i	$0.549178\pi$
$882$	0	0
$883$	−7.54183e6	−0.325518	−0.162759	−	0.986666i	$-0.552039\pi$
$883$	−7.54183e6	−0.325518	−0.162759	+	0.986666i	$0.552039\pi$
$884$	4.49387e6	0.193415
$885$	0	0
$886$	982960.	0.0420680
$887$	4.11968e7	1.75815	0.879073	−	0.476687i	$-0.158163\pi$
$887$	4.11968e7	1.75815	0.879073	+	0.476687i	$0.158163\pi$
$888$	0	0
$889$	−3.88584e7	−1.64904
$890$	0	0
$891$	0	0
$892$	−9.74956e6	−0.410273
$893$	4.61907e6	0.193832
$894$	0	0
$895$	0	0
$896$	−4.94767e6	−0.205888
$897$	0	0
$898$	1.10588e7	0.457635
$899$	1.99232e7	0.822167
$900$	0	0
$901$	7.95982e6	0.326656
$902$	−2.47935e7	−1.01466
$903$	0	0
$904$	2.92626e7	1.19095
$905$	0	0
$906$	0	0
$907$	−3.99367e7	−1.61196	−0.805979	−	0.591945i	$-0.798360\pi$
$907$	−3.99367e7	−1.61196	−0.805979	+	0.591945i	$0.798360\pi$
$908$	4.26041e6	0.171489
$909$	0	0
$910$	0	0
$911$	9.30901e6	0.371627	0.185814	−	0.982585i	$-0.440508\pi$
$911$	9.30901e6	0.371627	0.185814	+	0.982585i	$0.440508\pi$
$912$	0	0
$913$	−2.86731e7	−1.13841
$914$	1.29535e7	0.512887
$915$	0	0
$916$	−4.93313e6	−0.194260
$917$	−6.07441e7	−2.38551
$918$	0	0
$919$	2.88035e7	1.12501	0.562506	−	0.826793i	$-0.309837\pi$
$919$	2.88035e7	1.12501	0.562506	+	0.826793i	$0.309837\pi$
$920$	0	0
$921$	0	0
$922$	3.15354e7	1.22172
$923$	7.43806e6	0.287379
$924$	0	0
$925$	0	0
$926$	1.29324e7	0.495624
$927$	0	0
$928$	2.25218e7	0.858485
$929$	2.03981e7	0.775444	0.387722	−	0.921776i	$-0.373262\pi$
$929$	2.03981e7	0.775444	0.387722	+	0.921776i	$0.373262\pi$
$930$	0	0
$931$	1.62590e7	0.614780
$932$	−8.85832e6	−0.334050
$933$	0	0
$934$	−3.45189e7	−1.29476
$935$	0	0
$936$	0	0
$937$	4.91357e7	1.82830	0.914152	−	0.405371i	$-0.132858\pi$
$937$	4.91357e7	1.82830	0.914152	+	0.405371i	$0.132858\pi$
$938$	−3.47109e7	−1.28813
$939$	0	0
$940$	0	0
$941$	−3.80960e7	−1.40251	−0.701254	−	0.712912i	$-0.747376\pi$
$941$	−3.80960e7	−1.40251	−0.701254	+	0.712912i	$0.747376\pi$
$942$	0	0
$943$	−2.49913e7	−0.915186
$944$	2.73889e6	0.100033
$945$	0	0
$946$	−2.82240e7	−1.02539
$947$	−1.45947e6	−0.0528836	−0.0264418	−	0.999650i	$-0.508418\pi$
$947$	−1.45947e6	−0.0528836	−0.0264418	+	0.999650i	$0.508418\pi$
$948$	0	0
$949$	−604800.	−0.0217995
$950$	0	0
$951$	0	0
$952$	7.36897e7	2.63520
$953$	−1.94841e7	−0.694940	−0.347470	−	0.937691i	$-0.612959\pi$
$953$	−1.94841e7	−0.694940	−0.347470	+	0.937691i	$0.612959\pi$
$954$	0	0
$955$	0	0
$956$	1.45780e6	0.0515884
$957$	0	0
$958$	−938408.	−0.0330353
$959$	−7.16198e7	−2.51470
$960$	0	0
$961$	−1.56115e7	−0.545300
$962$	7.43806e6	0.259132
$963$	0	0
$964$	−1.34669e7	−0.466740
$965$	0	0
$966$	0	0
$967$	−1.65207e7	−0.568149	−0.284074	−	0.958802i	$-0.591686\pi$
$967$	−1.65207e7	−0.568149	−0.284074	+	0.958802i	$0.591686\pi$
$968$	−1.78964e7	−0.613871
$969$	0	0
$970$	0	0
$971$	1.02422e7	0.348615	0.174308	−	0.984691i	$-0.444231\pi$
$971$	1.02422e7	0.348615	0.174308	+	0.984691i	$0.444231\pi$
$972$	0	0
$973$	5.08617e7	1.72230
$974$	3.47883e6	0.117500
$975$	0	0
$976$	−1.05956e7	−0.356040
$977$	−1.95090e7	−0.653881	−0.326941	−	0.945045i	$-0.606018\pi$
$977$	−1.95090e7	−0.653881	−0.326941	+	0.945045i	$0.606018\pi$
$978$	0	0
$979$	6.04800e7	2.01676
$980$	0	0
$981$	0	0
$982$	−4.44262e7	−1.47015
$983$	1.91895e7	0.633403	0.316701	−	0.948525i	$-0.397425\pi$
$983$	1.91895e7	0.633403	0.316701	+	0.948525i	$0.397425\pi$
$984$	0	0
$985$	0	0
$986$	4.11938e7	1.34940
$987$	0	0
$988$	1.30389e6	0.0424961
$989$	−2.84491e7	−0.924864
$990$	0	0
$991$	5.38361e7	1.74136	0.870682	−	0.491846i	$-0.163678\pi$
$991$	5.38361e7	1.74136	0.870682	+	0.491846i	$0.163678\pi$
$992$	1.47155e7	0.474785
$993$	0	0
$994$	3.32640e7	1.06785
$995$	0	0
$996$	0	0
$997$	3.25008e6	0.103551	0.0517757	−	0.998659i	$-0.483512\pi$
$997$	3.25008e6	0.103551	0.0517757	+	0.998659i	$0.483512\pi$
$998$	2.28227e7	0.725339
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.6.a.v.1.3		4
3.2	odd	2	inner	225.6.a.v.1.1		4
5.2	odd	4		45.6.b.d.19.4	yes	4
5.3	odd	4		45.6.b.d.19.2	yes	4
5.4	even	2	inner	225.6.a.v.1.2		4
15.2	even	4		45.6.b.d.19.1	✓	4
15.8	even	4		45.6.b.d.19.3	yes	4
15.14	odd	2	inner	225.6.a.v.1.4		4
20.3	even	4		720.6.f.k.289.4		4
20.7	even	4		720.6.f.k.289.3		4
60.23	odd	4		720.6.f.k.289.1		4
60.47	odd	4		720.6.f.k.289.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.6.b.d.19.1	✓	4	15.2	even	4
45.6.b.d.19.2	yes	4	5.3	odd	4
45.6.b.d.19.3	yes	4	15.8	even	4
45.6.b.d.19.4	yes	4	5.2	odd	4
225.6.a.v.1.1		4	3.2	odd	2	inner
225.6.a.v.1.2		4	5.4	even	2	inner
225.6.a.v.1.3		4	1.1	even	1	trivial
225.6.a.v.1.4		4	15.14	odd	2	inner
720.6.f.k.289.1		4	60.23	odd	4
720.6.f.k.289.2		4	60.47	odd	4
720.6.f.k.289.3		4	20.7	even	4
720.6.f.k.289.4		4	20.3	even	4

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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q+4.47214 q^{2} -12.0000 q^{4} -224.499 q^{7} -196.774 q^{8} +O(q^{10})\)	\(q+4.47214 q^{2} -12.0000 q^{4} -224.499 q^{7} -196.774 q^{8} +501.996 q^{11} -224.499 q^{13} -1003.99 q^{14} -496.000 q^{16} +1668.11 q^{17} +484.000 q^{19} +2244.99 q^{22} +2262.90 q^{23} -1003.99 q^{26} +2693.99 q^{28} +5521.96 q^{29} +3608.00 q^{31} +4078.59 q^{32} +7460.00 q^{34} -7408.48 q^{37} +2164.51 q^{38} -11043.9 q^{41} -12572.0 q^{43} -6023.95 q^{44} +10120.0 q^{46} +9543.54 q^{47} +33593.0 q^{49} +2693.99 q^{52} +4771.77 q^{53} +44175.6 q^{56} +24694.9 q^{58} -5521.96 q^{59} +21362.0 q^{61} +16135.5 q^{62} +34112.0 q^{64} +34572.9 q^{67} -20017.3 q^{68} -33131.7 q^{71} +2693.99 q^{73} -33131.7 q^{74} -5808.00 q^{76} -112698. q^{77} +99616.0 q^{79} -49389.9 q^{82} -57118.1 q^{83} -56223.6 q^{86} -98779.8 q^{88} +120479. q^{89} +50400.0 q^{91} -27154.8 q^{92} +42680.0 q^{94} -64206.8 q^{97} +150232. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 48 q^{4}+O(q^{10}) \) 4 * q - 48 * q^4	\( 4 q - 48 q^{4} - 1984 q^{16} + 1936 q^{19} + 14432 q^{31} + 29840 q^{34} + 40480 q^{46} + 134372 q^{49} + 85448 q^{61} + 136448 q^{64} - 23232 q^{76} + 398464 q^{79} + 201600 q^{91} + 170720 q^{94}+O(q^{100}) \) 4 * q - 48 * q^4 - 1984 * q^16 + 1936 * q^19 + 14432 * q^31 + 29840 * q^34 + 40480 * q^46 + 134372 * q^49 + 85448 * q^61 + 136448 * q^64 - 23232 * q^76 + 398464 * q^79 + 201600 * q^91 + 170720 * q^94

Introduction

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