LMFDB - Embedded newform 567.6.a.g.1.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [567,6,Mod(1,567)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("567.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(567, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 567 = 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	567.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$90.9376258340$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 4 x^{14} - 352 x^{13} + 1351 x^{12} + 47844 x^{11} - 176175 x^{10} - 3166121 x^{9} + \cdots - 9588032000 $ x^15 - 4x^14 - 352x^13 + 1351x^12 + 47844x^11 - 176175x^10 - 3166121x^9 + 11061548x^8 + 105332420x^7 - 333083360x^6 - 1647385344x^5 + 4024862784x^4 + 11037752704x^3 - 11783300224x^2 - 30569499136x - 9588032000
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{21} $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$-10.6447$ of defining polynomial
Character	$\chi$	$=$	567.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+10.6447 q^{2} +81.3091 q^{4} -19.3688 q^{5} -49.0000 q^{7} +524.880 q^{8} -206.175 q^{10} -685.239 q^{11} -21.0192 q^{13} -521.589 q^{14} +2985.28 q^{16} -566.915 q^{17} -2762.91 q^{19} -1574.86 q^{20} -7294.15 q^{22} -1605.57 q^{23} -2749.85 q^{25} -223.742 q^{26} -3984.15 q^{28} -2389.23 q^{29} +3764.94 q^{31} +14981.2 q^{32} -6034.62 q^{34} +949.072 q^{35} +11576.9 q^{37} -29410.2 q^{38} -10166.3 q^{40} +9575.52 q^{41} +6566.83 q^{43} -55716.2 q^{44} -17090.8 q^{46} -17632.2 q^{47} +2401.00 q^{49} -29271.2 q^{50} -1709.05 q^{52} -1608.88 q^{53} +13272.3 q^{55} -25719.1 q^{56} -25432.6 q^{58} -30522.4 q^{59} -7951.28 q^{61} +40076.6 q^{62} +63941.1 q^{64} +407.117 q^{65} +53819.2 q^{67} -46095.4 q^{68} +10102.6 q^{70} +74215.5 q^{71} -90409.2 q^{73} +123232. q^{74} -224649. q^{76} +33576.7 q^{77} -61306.6 q^{79} -57821.4 q^{80} +101928. q^{82} +5876.99 q^{83} +10980.5 q^{85} +69901.8 q^{86} -359668. q^{88} -94191.4 q^{89} +1029.94 q^{91} -130547. q^{92} -187689. q^{94} +53514.2 q^{95} +41205.4 q^{97} +25557.9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 4 q^{2} + 240 q^{4} - 71 q^{5} - 735 q^{7} + 21 q^{8} - 134 q^{11} - 165 q^{13} + 196 q^{14} + 3840 q^{16} - 1092 q^{17} - 447 q^{19} + 1214 q^{20} + 96 q^{22} + 3564 q^{23} + 3576 q^{25} - 865 q^{26}+ \cdots - 9604 q^{98}+O(q^{100}) $ 15 * q - 4 * q^2 + 240 * q^4 - 71 * q^5 - 735 * q^7 + 21 * q^8 - 134 * q^11 - 165 * q^13 + 196 * q^14 + 3840 * q^16 - 1092 * q^17 - 447 * q^19 + 1214 * q^20 + 96 * q^22 + 3564 * q^23 + 3576 * q^25 - 865 * q^26 - 11760 * q^28 - 4073 * q^29 + 2217 * q^31 - 12107 * q^32 + 15654 * q^34 + 3479 * q^35 + 5007 * q^37 - 48685 * q^38 + 7485 * q^40 - 22294 * q^41 + 3153 * q^43 - 82480 * q^44 - 18246 * q^46 - 74313 * q^47 + 36015 * q^49 - 111323 * q^50 - 33501 * q^52 - 27774 * q^53 + 38688 * q^55 - 1029 * q^56 + 3249 * q^58 - 90567 * q^59 + 3714 * q^61 - 178704 * q^62 + 47217 * q^64 - 79607 * q^65 + 11016 * q^67 - 241743 * q^68 + 53793 * q^71 - 120081 * q^73 - 95016 * q^74 - 122748 * q^76 + 6566 * q^77 - 11013 * q^79 - 296449 * q^80 + 81018 * q^82 - 7014 * q^83 + 36387 * q^85 - 170519 * q^86 + 108012 * q^88 - 76160 * q^89 + 8085 * q^91 - 153651 * q^92 + 266412 * q^94 + 64264 * q^95 + 81033 * q^97 - 9604 * q^98

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 $

$2$	10.6447	1.88173	0.940865	−	0.338781i	$-0.110014\pi$
$2$	10.6447	1.88173	0.940865	+	0.338781i	$0.110014\pi$
$3$	0	0
$3$	0	0
$4$	81.3091	2.54091
$5$	−19.3688	−0.346480	−0.173240	−	0.984880i	$-0.555424\pi$
$5$	−19.3688	−0.346480	−0.173240	+	0.984880i	$0.555424\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	524.880	2.89958
$9$	0	0
$10$	−206.175	−0.651982
$11$	−685.239	−1.70750	−0.853749	−	0.520684i	$-0.825677\pi$
$11$	−685.239	−1.70750	−0.853749	+	0.520684i	$0.825677\pi$
$12$	0	0
$13$	−21.0192	−0.0344951	−0.0172475	−	0.999851i	$-0.505490\pi$
$13$	−21.0192	−0.0344951	−0.0172475	+	0.999851i	$0.505490\pi$
$14$	−521.589	−0.711227
$15$	0	0
$16$	2985.28	2.91532
$17$	−566.915	−0.475768	−0.237884	−	0.971294i	$-0.576454\pi$
$17$	−566.915	−0.475768	−0.237884	+	0.971294i	$0.576454\pi$
$18$	0	0
$19$	−2762.91	−1.75583	−0.877914	−	0.478818i	$-0.841065\pi$
$19$	−2762.91	−1.75583	−0.877914	+	0.478818i	$0.841065\pi$
$20$	−1574.86	−0.880375
$21$	0	0
$22$	−7294.15	−3.21305
$23$	−1605.57	−0.632862	−0.316431	−	0.948615i	$-0.602485\pi$
$23$	−1605.57	−0.632862	−0.316431	+	0.948615i	$0.602485\pi$
$24$	0	0
$25$	−2749.85	−0.879952
$26$	−223.742	−0.0649105
$27$	0	0
$28$	−3984.15	−0.960374
$29$	−2389.23	−0.527549	−0.263774	−	0.964584i	$-0.584967\pi$
$29$	−2389.23	−0.527549	−0.263774	+	0.964584i	$0.584967\pi$
$30$	0	0
$31$	3764.94	0.703646	0.351823	−	0.936067i	$-0.385562\pi$
$31$	3764.94	0.703646	0.351823	+	0.936067i	$0.385562\pi$
$32$	14981.2	2.58626
$33$	0	0
$34$	−6034.62	−0.895268
$35$	949.072	0.130957
$36$	0	0
$37$	11576.9	1.39023	0.695115	−	0.718898i	$-0.255353\pi$
$37$	11576.9	1.39023	0.695115	+	0.718898i	$0.255353\pi$
$38$	−29410.2	−3.30400
$39$	0	0
$40$	−10166.3	−1.00465
$41$	9575.52	0.889616	0.444808	−	0.895626i	$-0.353272\pi$
$41$	9575.52	0.889616	0.444808	+	0.895626i	$0.353272\pi$
$42$	0	0
$43$	6566.83	0.541607	0.270804	−	0.962635i	$-0.412711\pi$
$43$	6566.83	0.541607	0.270804	+	0.962635i	$0.412711\pi$
$44$	−55716.2	−4.33860
$45$	0	0
$46$	−17090.8	−1.19088
$47$	−17632.2	−1.16429	−0.582146	−	0.813084i	$-0.697787\pi$
$47$	−17632.2	−1.16429	−0.582146	+	0.813084i	$0.697787\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	−29271.2	−1.65583
$51$	0	0
$52$	−1709.05	−0.0876489
$53$	−1608.88	−0.0786745	−0.0393373	−	0.999226i	$-0.512525\pi$
$53$	−1608.88	−0.0786745	−0.0393373	+	0.999226i	$0.512525\pi$
$54$	0	0
$55$	13272.3	0.591614
$56$	−25719.1	−1.09594
$57$	0	0
$58$	−25432.6	−0.992705
$59$	−30522.4	−1.14153	−0.570767	−	0.821112i	$-0.693354\pi$
$59$	−30522.4	−1.14153	−0.570767	+	0.821112i	$0.693354\pi$
$60$	0	0
$61$	−7951.28	−0.273598	−0.136799	−	0.990599i	$-0.543681\pi$
$61$	−7951.28	−0.273598	−0.136799	+	0.990599i	$0.543681\pi$
$62$	40076.6	1.32407
$63$	0	0
$64$	63941.1	1.95133
$65$	407.117	0.0119519
$66$	0	0
$67$	53819.2	1.46470	0.732352	−	0.680926i	$-0.238422\pi$
$67$	53819.2	1.46470	0.732352	+	0.680926i	$0.238422\pi$
$68$	−46095.4	−1.20888
$69$	0	0
$70$	10102.6	0.246426
$71$	74215.5	1.74722	0.873612	−	0.486623i	$-0.161771\pi$
$71$	74215.5	1.74722	0.873612	+	0.486623i	$0.161771\pi$
$72$	0	0
$73$	−90409.2	−1.98566	−0.992831	−	0.119526i	$-0.961862\pi$
$73$	−90409.2	−1.98566	−0.992831	+	0.119526i	$0.961862\pi$
$74$	123232.	2.61604
$75$	0	0
$76$	−224649.	−4.46140
$77$	33576.7	0.645374
$78$	0	0
$79$	−61306.6	−1.10520	−0.552598	−	0.833448i	$-0.686364\pi$
$79$	−61306.6	−1.10520	−0.552598	+	0.833448i	$0.686364\pi$
$80$	−57821.4	−1.01010
$81$	0	0
$82$	101928.	1.67402
$83$	5876.99	0.0936397	0.0468198	−	0.998903i	$-0.485091\pi$
$83$	5876.99	0.0936397	0.0468198	+	0.998903i	$0.485091\pi$
$84$	0	0
$85$	10980.5	0.164844
$86$	69901.8	1.01916
$87$	0	0
$88$	−359668.	−4.95103
$89$	−94191.4	−1.26048	−0.630240	−	0.776400i	$-0.717044\pi$
$89$	−94191.4	−1.26048	−0.630240	+	0.776400i	$0.717044\pi$
$90$	0	0
$91$	1029.94	0.0130379
$92$	−130547.	−1.60805
$93$	0	0
$94$	−187689.	−2.19088
$95$	53514.2	0.608359
$96$	0	0
$97$	41205.4	0.444657	0.222328	−	0.974972i	$-0.428634\pi$
$97$	41205.4	0.444657	0.222328	+	0.974972i	$0.428634\pi$
$98$	25557.9	0.268819
$99$	0	0
$100$	−223588.	−2.23588
$101$	−99492.1	−0.970477	−0.485238	−	0.874382i	$-0.661267\pi$
$101$	−99492.1	−0.970477	−0.485238	+	0.874382i	$0.661267\pi$
$102$	0	0
$103$	105242.	0.977453	0.488727	−	0.872437i	$-0.337462\pi$
$103$	105242.	0.977453	0.488727	+	0.872437i	$0.337462\pi$
$104$	−11032.5	−0.100021
$105$	0	0
$106$	−17126.0	−0.148044
$107$	99883.5	0.843401	0.421701	−	0.906735i	$-0.361433\pi$
$107$	99883.5	0.843401	0.421701	+	0.906735i	$0.361433\pi$
$108$	0	0
$109$	−130287.	−1.05035	−0.525177	−	0.850993i	$-0.676001\pi$
$109$	−130287.	−1.05035	−0.525177	+	0.850993i	$0.676001\pi$
$110$	141279.	1.11326
$111$	0	0
$112$	−146279.	−1.10189
$113$	23802.1	0.175355	0.0876776	−	0.996149i	$-0.472055\pi$
$113$	23802.1	0.175355	0.0876776	+	0.996149i	$0.472055\pi$
$114$	0	0
$115$	31098.0	0.219274
$116$	−194266.	−1.34045
$117$	0	0
$118$	−324901.	−2.14806
$119$	27778.8	0.179824
$120$	0	0
$121$	308501.	1.91555
$122$	−84638.8	−0.514837
$123$	0	0
$124$	306124.	1.78790
$125$	113789.	0.651366
$126$	0	0
$127$	4095.45	0.0225316	0.0112658	−	0.999937i	$-0.496414\pi$
$127$	4095.45	0.0225316	0.0112658	+	0.999937i	$0.496414\pi$
$128$	201234.	1.08562
$129$	0	0
$130$	4333.63	0.0224902
$131$	39066.0	0.198894	0.0994468	−	0.995043i	$-0.468293\pi$
$131$	39066.0	0.198894	0.0994468	+	0.995043i	$0.468293\pi$
$132$	0	0
$133$	135382.	0.663641
$134$	572888.	2.75618
$135$	0	0
$136$	−297562.	−1.37953
$137$	150737.	0.686150	0.343075	−	0.939308i	$-0.388532\pi$
$137$	150737.	0.686150	0.343075	+	0.939308i	$0.388532\pi$
$138$	0	0
$139$	−158030.	−0.693749	−0.346874	−	0.937912i	$-0.612757\pi$
$139$	−158030.	−0.693749	−0.346874	+	0.937912i	$0.612757\pi$
$140$	77168.3	0.332750
$141$	0	0
$142$	790000.	3.28781
$143$	14403.2	0.0589003
$144$	0	0
$145$	46276.5	0.182785
$146$	−962376.	−3.73648
$147$	0	0
$148$	941305.	3.53245
$149$	39751.3	0.146685	0.0733425	−	0.997307i	$-0.476633\pi$
$149$	39751.3	0.146685	0.0733425	+	0.997307i	$0.476633\pi$
$150$	0	0
$151$	−284764.	−1.01635	−0.508175	−	0.861254i	$-0.669680\pi$
$151$	−284764.	−1.01635	−0.508175	+	0.861254i	$0.669680\pi$
$152$	−1.45019e6	−5.09116
$153$	0	0
$154$	357413.	1.21442
$155$	−72922.5	−0.243799
$156$	0	0
$157$	62827.6	0.203424	0.101712	−	0.994814i	$-0.467568\pi$
$157$	62827.6	0.203424	0.101712	+	0.994814i	$0.467568\pi$
$158$	−652589.	−2.07968
$159$	0	0
$160$	−290169.	−0.896087
$161$	78672.9	0.239199
$162$	0	0
$163$	357419.	1.05368	0.526839	−	0.849965i	$-0.323377\pi$
$163$	357419.	1.05368	0.526839	+	0.849965i	$0.323377\pi$
$164$	778577.	2.26043
$165$	0	0
$166$	62558.7	0.176205
$167$	−103132.	−0.286156	−0.143078	−	0.989711i	$-0.545700\pi$
$167$	−103132.	−0.286156	−0.143078	+	0.989711i	$0.545700\pi$
$168$	0	0
$169$	−370851.	−0.998810
$170$	116884.	0.310192
$171$	0	0
$172$	533943.	1.37618
$173$	−672604.	−1.70861	−0.854307	−	0.519768i	$-0.826018\pi$
$173$	−672604.	−1.70861	−0.854307	+	0.519768i	$0.826018\pi$
$174$	0	0
$175$	134743.	0.332590
$176$	−2.04563e6	−4.97790
$177$	0	0
$178$	−1.00264e6	−2.37188
$179$	165363.	0.385750	0.192875	−	0.981223i	$-0.438219\pi$
$179$	165363.	0.385750	0.192875	+	0.981223i	$0.438219\pi$
$180$	0	0
$181$	−184682.	−0.419013	−0.209506	−	0.977807i	$-0.567186\pi$
$181$	−184682.	−0.419013	−0.209506	+	0.977807i	$0.567186\pi$
$182$	10963.4	0.0245339
$183$	0	0
$184$	−842730.	−1.83503
$185$	−224230.	−0.481687
$186$	0	0
$187$	388472.	0.812374
$188$	−1.43366e6	−2.95836
$189$	0	0
$190$	569642.	1.14477
$191$	420240.	0.833517	0.416758	−	0.909017i	$-0.363166\pi$
$191$	420240.	0.833517	0.416758	+	0.909017i	$0.363166\pi$
$192$	0	0
$193$	−50779.7	−0.0981288	−0.0490644	−	0.998796i	$-0.515624\pi$
$193$	−50779.7	−0.0981288	−0.0490644	+	0.998796i	$0.515624\pi$
$194$	438618.	0.836724
$195$	0	0
$196$	195223.	0.362987
$197$	−516023.	−0.947334	−0.473667	−	0.880704i	$-0.657070\pi$
$197$	−516023.	−0.947334	−0.473667	+	0.880704i	$0.657070\pi$
$198$	0	0
$199$	141678.	0.253612	0.126806	−	0.991928i	$-0.459527\pi$
$199$	141678.	0.253612	0.126806	+	0.991928i	$0.459527\pi$
$200$	−1.44334e6	−2.55149
$201$	0	0
$202$	−1.05906e6	−1.82618
$203$	117072.	0.199395
$204$	0	0
$205$	−185467.	−0.308234
$206$	1.12027e6	1.83930
$207$	0	0
$208$	−62748.2	−0.100564
$209$	1.89325e6	2.99807
$210$	0	0
$211$	885007.	1.36849	0.684244	−	0.729254i	$-0.260133\pi$
$211$	885007.	1.36849	0.684244	+	0.729254i	$0.260133\pi$
$212$	−130817.	−0.199905
$213$	0	0
$214$	1.06323e6	1.58705
$215$	−127192.	−0.187656
$216$	0	0
$217$	−184482.	−0.265953
$218$	−1.38687e6	−1.97648
$219$	0	0
$220$	1.07916e6	1.50324
$221$	11916.1	0.0164117
$222$	0	0
$223$	959379.	1.29190	0.645949	−	0.763381i	$-0.276462\pi$
$223$	959379.	1.29190	0.645949	+	0.763381i	$0.276462\pi$
$224$	−734080.	−0.977514
$225$	0	0
$226$	253365.	0.329971
$227$	−217441.	−0.280077	−0.140038	−	0.990146i	$-0.544723\pi$
$227$	−217441.	−0.280077	−0.140038	+	0.990146i	$0.544723\pi$
$228$	0	0
$229$	222942.	0.280933	0.140467	−	0.990085i	$-0.455140\pi$
$229$	222942.	0.280933	0.140467	+	0.990085i	$0.455140\pi$
$230$	331028.	0.412615
$231$	0	0
$232$	−1.25406e6	−1.52967
$233$	−795096.	−0.959466	−0.479733	−	0.877415i	$-0.659266\pi$
$233$	−795096.	−0.959466	−0.479733	+	0.877415i	$0.659266\pi$
$234$	0	0
$235$	341515.	0.403404
$236$	−2.48175e6	−2.90054
$237$	0	0
$238$	295697.	0.338379
$239$	−264010.	−0.298969	−0.149484	−	0.988764i	$-0.547761\pi$
$239$	−264010.	−0.298969	−0.149484	+	0.988764i	$0.547761\pi$
$240$	0	0
$241$	507101.	0.562409	0.281204	−	0.959648i	$-0.409266\pi$
$241$	507101.	0.562409	0.281204	+	0.959648i	$0.409266\pi$
$242$	3.28390e6	3.60455
$243$	0	0
$244$	−646512.	−0.695187
$245$	−46504.5	−0.0494972
$246$	0	0
$247$	58074.0	0.0605675
$248$	1.97614e6	2.04028
$249$	0	0
$250$	1.21125e6	1.22569
$251$	−836948.	−0.838521	−0.419261	−	0.907866i	$-0.637711\pi$
$251$	−836948.	−0.838521	−0.419261	+	0.907866i	$0.637711\pi$
$252$	0	0
$253$	1.10020e6	1.08061
$254$	43594.7	0.0423984
$255$	0	0
$256$	95953.1	0.0915080
$257$	−1.27571e6	−1.20482	−0.602408	−	0.798188i	$-0.705792\pi$
$257$	−1.27571e6	−1.20482	−0.602408	+	0.798188i	$0.705792\pi$
$258$	0	0
$259$	−567267.	−0.525458
$260$	33102.3	0.0303686
$261$	0	0
$262$	415845.	0.374264
$263$	1.22321e6	1.09047	0.545233	−	0.838285i	$-0.316441\pi$
$263$	1.22321e6	1.09047	0.545233	+	0.838285i	$0.316441\pi$
$264$	0	0
$265$	31162.1	0.0272592
$266$	1.44110e6	1.24879
$267$	0	0
$268$	4.37599e6	3.72168
$269$	−1.86618e6	−1.57244	−0.786220	−	0.617947i	$-0.787965\pi$
$269$	−1.86618e6	−1.57244	−0.786220	+	0.617947i	$0.787965\pi$
$270$	0	0
$271$	1.75496e6	1.45159	0.725796	−	0.687910i	$-0.241471\pi$
$271$	1.75496e6	1.45159	0.725796	+	0.687910i	$0.241471\pi$
$272$	−1.69240e6	−1.38701
$273$	0	0
$274$	1.60455e6	1.29115
$275$	1.88430e6	1.50252
$276$	0	0
$277$	1.45044e6	1.13580	0.567899	−	0.823098i	$-0.307756\pi$
$277$	1.45044e6	1.13580	0.567899	+	0.823098i	$0.307756\pi$
$278$	−1.68218e6	−1.30545
$279$	0	0
$280$	498149.	0.379721
$281$	868650.	0.656265	0.328132	−	0.944632i	$-0.393581\pi$
$281$	868650.	0.656265	0.328132	+	0.944632i	$0.393581\pi$
$282$	0	0
$283$	1.37681e6	1.02190	0.510949	−	0.859611i	$-0.329294\pi$
$283$	1.37681e6	1.02190	0.510949	+	0.859611i	$0.329294\pi$
$284$	6.03440e6	4.43954
$285$	0	0
$286$	153317.	0.110835
$287$	−469200.	−0.336243
$288$	0	0
$289$	−1.09846e6	−0.773645
$290$	492599.	0.343952
$291$	0	0
$292$	−7.35109e6	−5.04539
$293$	412109.	0.280442	0.140221	−	0.990120i	$-0.455219\pi$
$293$	412109.	0.280442	0.140221	+	0.990120i	$0.455219\pi$
$294$	0	0
$295$	591184.	0.395519
$296$	6.07647e6	4.03108
$297$	0	0
$298$	423140.	0.276022
$299$	33747.7	0.0218306
$300$	0	0
$301$	−321775.	−0.204708
$302$	−3.03122e6	−1.91250
$303$	0	0
$304$	−8.24805e6	−5.11879
$305$	154007.	0.0947961
$306$	0	0
$307$	2.34033e6	1.41720	0.708600	−	0.705611i	$-0.249327\pi$
$307$	2.34033e6	1.41720	0.708600	+	0.705611i	$0.249327\pi$
$308$	2.73009e6	1.63984
$309$	0	0
$310$	−776237.	−0.458765
$311$	−2.08176e6	−1.22048	−0.610238	−	0.792218i	$-0.708926\pi$
$311$	−2.08176e6	−1.22048	−0.610238	+	0.792218i	$0.708926\pi$
$312$	0	0
$313$	−2.57363e6	−1.48486	−0.742430	−	0.669923i	$-0.766327\pi$
$313$	−2.57363e6	−1.48486	−0.742430	+	0.669923i	$0.766327\pi$
$314$	668780.	0.382789
$315$	0	0
$316$	−4.98479e6	−2.80821
$317$	−1.81437e6	−1.01409	−0.507046	−	0.861919i	$-0.669263\pi$
$317$	−1.81437e6	−1.01409	−0.507046	+	0.861919i	$0.669263\pi$
$318$	0	0
$319$	1.63719e6	0.900789
$320$	−1.23846e6	−0.676097
$321$	0	0
$322$	837447.	0.450109
$323$	1.56633e6	0.835367
$324$	0	0
$325$	57799.6	0.0303540
$326$	3.80460e6	1.98274
$327$	0	0
$328$	5.02600e6	2.57951
$329$	863978.	0.440061
$330$	0	0
$331$	−3.49213e6	−1.75194	−0.875972	−	0.482361i	$-0.839779\pi$
$331$	−3.49213e6	−1.75194	−0.875972	+	0.482361i	$0.839779\pi$
$332$	477853.	0.237930
$333$	0	0
$334$	−1.09781e6	−0.538468
$335$	−1.04241e6	−0.507491
$336$	0	0
$337$	139126.	0.0667320	0.0333660	−	0.999443i	$-0.489377\pi$
$337$	139126.	0.0667320	0.0333660	+	0.999443i	$0.489377\pi$
$338$	−3.94759e6	−1.87949
$339$	0	0
$340$	892813.	0.418854
$341$	−2.57989e6	−1.20147
$342$	0	0
$343$	−117649.	−0.0539949
$344$	3.44680e6	1.57043
$345$	0	0
$346$	−7.15965e6	−3.21515
$347$	−726928.	−0.324091	−0.162046	−	0.986783i	$-0.551809\pi$
$347$	−726928.	−0.324091	−0.162046	+	0.986783i	$0.551809\pi$
$348$	0	0
$349$	2.73453e6	1.20176	0.600882	−	0.799338i	$-0.294816\pi$
$349$	2.73453e6	1.20176	0.600882	+	0.799338i	$0.294816\pi$
$350$	1.43429e6	0.625846
$351$	0	0
$352$	−1.02657e7	−4.41603
$353$	−60289.6	−0.0257517	−0.0128758	−	0.999917i	$-0.504099\pi$
$353$	−60289.6	−0.0257517	−0.0128758	+	0.999917i	$0.504099\pi$
$354$	0	0
$355$	−1.43747e6	−0.605378
$356$	−7.65862e6	−3.20277
$357$	0	0
$358$	1.76024e6	0.725878
$359$	638473.	0.261461	0.130730	−	0.991418i	$-0.458268\pi$
$359$	638473.	0.261461	0.130730	+	0.991418i	$0.458268\pi$
$360$	0	0
$361$	5.15755e6	2.08293
$362$	−1.96588e6	−0.788469
$363$	0	0
$364$	83743.5	0.0331282
$365$	1.75112e6	0.687992
$366$	0	0
$367$	1.52868e6	0.592451	0.296225	−	0.955118i	$-0.404272\pi$
$367$	1.52868e6	0.592451	0.296225	+	0.955118i	$0.404272\pi$
$368$	−4.79308e6	−1.84499
$369$	0	0
$370$	−2.38686e6	−0.906406
$371$	78835.2	0.0297362
$372$	0	0
$373$	858054.	0.319332	0.159666	−	0.987171i	$-0.448958\pi$
$373$	858054.	0.319332	0.159666	+	0.987171i	$0.448958\pi$
$374$	4.13516e6	1.52867
$375$	0	0
$376$	−9.25478e6	−3.37595
$377$	50219.6	0.0181978
$378$	0	0
$379$	477174.	0.170639	0.0853196	−	0.996354i	$-0.472809\pi$
$379$	477174.	0.170639	0.0853196	+	0.996354i	$0.472809\pi$
$380$	4.35120e6	1.54579
$381$	0	0
$382$	4.47332e6	1.56845
$383$	−3.56461e6	−1.24170	−0.620848	−	0.783931i	$-0.713212\pi$
$383$	−3.56461e6	−1.24170	−0.620848	+	0.783931i	$0.713212\pi$
$384$	0	0
$385$	−650341.	−0.223609
$386$	−540533.	−0.184652
$387$	0	0
$388$	3.35038e6	1.12983
$389$	3.42320e6	1.14698	0.573492	−	0.819211i	$-0.305588\pi$
$389$	3.42320e6	1.14698	0.573492	+	0.819211i	$0.305588\pi$
$390$	0	0
$391$	910221.	0.301096
$392$	1.26024e6	0.414225
$393$	0	0
$394$	−5.49290e6	−1.78263
$395$	1.18744e6	0.382929
$396$	0	0
$397$	−3.41868e6	−1.08863	−0.544317	−	0.838880i	$-0.683211\pi$
$397$	−3.41868e6	−1.08863	−0.544317	+	0.838880i	$0.683211\pi$
$398$	1.50812e6	0.477230
$399$	0	0
$400$	−8.20908e6	−2.56534
$401$	1.76030e6	0.546672	0.273336	−	0.961919i	$-0.411873\pi$
$401$	1.76030e6	0.546672	0.273336	+	0.961919i	$0.411873\pi$
$402$	0	0
$403$	−79136.0	−0.0242723
$404$	−8.08962e6	−2.46589
$405$	0	0
$406$	1.24620e6	0.375207
$407$	−7.93292e6	−2.37382
$408$	0	0
$409$	−361851.	−0.106960	−0.0534800	−	0.998569i	$-0.517031\pi$
$409$	−361851.	−0.106960	−0.0534800	+	0.998569i	$0.517031\pi$
$410$	−1.97423e6	−0.580014
$411$	0	0
$412$	8.55713e6	2.48362
$413$	1.49560e6	0.431460
$414$	0	0
$415$	−113830.	−0.0324443
$416$	−314893.	−0.0892133
$417$	0	0
$418$	2.01530e7	5.64157
$419$	−1.79426e6	−0.499288	−0.249644	−	0.968338i	$-0.580314\pi$
$419$	−1.79426e6	−0.499288	−0.249644	+	0.968338i	$0.580314\pi$
$420$	0	0
$421$	824279.	0.226657	0.113328	−	0.993558i	$-0.463849\pi$
$421$	824279.	0.226657	0.113328	+	0.993558i	$0.463849\pi$
$422$	9.42062e6	2.57512
$423$	0	0
$424$	−844469.	−0.228123
$425$	1.55893e6	0.418653
$426$	0	0
$427$	389613.	0.103410
$428$	8.12144e6	2.14301
$429$	0	0
$430$	−1.35392e6	−0.353118
$431$	−3.61199e6	−0.936598	−0.468299	−	0.883570i	$-0.655133\pi$
$431$	−3.61199e6	−0.936598	−0.468299	+	0.883570i	$0.655133\pi$
$432$	0	0
$433$	772471.	0.197999	0.0989994	−	0.995087i	$-0.468436\pi$
$433$	772471.	0.197999	0.0989994	+	0.995087i	$0.468436\pi$
$434$	−1.96375e6	−0.500452
$435$	0	0
$436$	−1.05935e7	−2.66886
$437$	4.43603e6	1.11120
$438$	0	0
$439$	1.84395e6	0.456655	0.228327	−	0.973584i	$-0.426674\pi$
$439$	1.84395e6	0.456655	0.228327	+	0.973584i	$0.426674\pi$
$440$	6.96635e6	1.71543
$441$	0	0
$442$	126843.	0.0308824
$443$	3.16424e6	0.766055	0.383028	−	0.923737i	$-0.374881\pi$
$443$	3.16424e6	0.766055	0.383028	+	0.923737i	$0.374881\pi$
$444$	0	0
$445$	1.82438e6	0.436731
$446$	1.02123e7	2.43100
$447$	0	0
$448$	−3.13312e6	−0.737533
$449$	−7.11111e6	−1.66465	−0.832323	−	0.554291i	$-0.812989\pi$
$449$	−7.11111e6	−1.66465	−0.832323	+	0.554291i	$0.812989\pi$
$450$	0	0
$451$	−6.56152e6	−1.51902
$452$	1.93533e6	0.445562
$453$	0	0
$454$	−2.31459e6	−0.527029
$455$	−19948.7	−0.00451738
$456$	0	0
$457$	2.13856e6	0.478995	0.239497	−	0.970897i	$-0.423017\pi$
$457$	2.13856e6	0.478995	0.239497	+	0.970897i	$0.423017\pi$
$458$	2.37314e6	0.528641
$459$	0	0
$460$	2.52855e6	0.557156
$461$	−6.59006e6	−1.44423	−0.722116	−	0.691772i	$-0.756830\pi$
$461$	−6.59006e6	−1.44423	−0.722116	+	0.691772i	$0.756830\pi$
$462$	0	0
$463$	−2.67862e6	−0.580708	−0.290354	−	0.956919i	$-0.593773\pi$
$463$	−2.67862e6	−0.580708	−0.290354	+	0.956919i	$0.593773\pi$
$464$	−7.13252e6	−1.53797
$465$	0	0
$466$	−8.46354e6	−1.80546
$467$	8.29780e6	1.76064	0.880320	−	0.474380i	$-0.157328\pi$
$467$	8.29780e6	1.76064	0.880320	+	0.474380i	$0.157328\pi$
$468$	0	0
$469$	−2.63714e6	−0.553606
$470$	3.63532e6	0.759097
$471$	0	0
$472$	−1.60206e7	−3.30997
$473$	−4.49985e6	−0.924794
$474$	0	0
$475$	7.59757e6	1.54504
$476$	2.25867e6	0.456915
$477$	0	0
$478$	−2.81030e6	−0.562579
$479$	−5.75377e6	−1.14581	−0.572907	−	0.819621i	$-0.694184\pi$
$479$	−5.75377e6	−1.14581	−0.572907	+	0.819621i	$0.694184\pi$
$480$	0	0
$481$	−243336.	−0.0479561
$482$	5.39793e6	1.05830
$483$	0	0
$484$	2.50840e7	4.86724
$485$	−798100.	−0.154065
$486$	0	0
$487$	−2.79860e6	−0.534711	−0.267356	−	0.963598i	$-0.586150\pi$
$487$	−2.79860e6	−0.534711	−0.267356	+	0.963598i	$0.586150\pi$
$488$	−4.17347e6	−0.793318
$489$	0	0
$490$	−495026.	−0.0931403
$491$	−4.35393e6	−0.815038	−0.407519	−	0.913197i	$-0.633606\pi$
$491$	−4.35393e6	−0.815038	−0.407519	+	0.913197i	$0.633606\pi$
$492$	0	0
$493$	1.35449e6	0.250991
$494$	618179.	0.113972
$495$	0	0
$496$	1.12394e7	2.05135
$497$	−3.63656e6	−0.660389
$498$	0	0
$499$	−427566.	−0.0768691	−0.0384345	−	0.999261i	$-0.512237\pi$
$499$	−427566.	−0.0768691	−0.0384345	+	0.999261i	$0.512237\pi$
$500$	9.25208e6	1.65506
$501$	0	0
$502$	−8.90904e6	−1.57787
$503$	2.86088e6	0.504173	0.252087	−	0.967705i	$-0.418883\pi$
$503$	2.86088e6	0.504173	0.252087	+	0.967705i	$0.418883\pi$
$504$	0	0
$505$	1.92705e6	0.336251
$506$	1.17113e7	2.03342
$507$	0	0
$508$	332997.	0.0572508
$509$	−9.08804e6	−1.55480	−0.777402	−	0.629004i	$-0.783463\pi$
$509$	−9.08804e6	−1.55480	−0.777402	+	0.629004i	$0.783463\pi$
$510$	0	0
$511$	4.43005e6	0.750510
$512$	−5.41810e6	−0.913423
$513$	0	0
$514$	−1.35796e7	−2.26714
$515$	−2.03841e6	−0.338668
$516$	0	0
$517$	1.20823e7	1.98803
$518$	−6.03837e6	−0.988770
$519$	0	0
$520$	213687.	0.0346554
$521$	7.13852e6	1.15216	0.576081	−	0.817392i	$-0.304581\pi$
$521$	7.13852e6	1.15216	0.576081	+	0.817392i	$0.304581\pi$
$522$	0	0
$523$	3.52396e6	0.563349	0.281674	−	0.959510i	$-0.409110\pi$
$523$	3.52396e6	0.563349	0.281674	+	0.959510i	$0.409110\pi$
$524$	3.17642e6	0.505371
$525$	0	0
$526$	1.30207e7	2.05196
$527$	−2.13440e6	−0.334773
$528$	0	0
$529$	−3.85849e6	−0.599485
$530$	331711.	0.0512944
$531$	0	0
$532$	1.10078e7	1.68625
$533$	−201270.	−0.0306874
$534$	0	0
$535$	−1.93463e6	−0.292222
$536$	2.82486e7	4.24703
$537$	0	0
$538$	−1.98649e7	−2.95891
$539$	−1.64526e6	−0.243928
$540$	0	0
$541$	−264828.	−0.0389018	−0.0194509	−	0.999811i	$-0.506192\pi$
$541$	−264828.	−0.0389018	−0.0194509	+	0.999811i	$0.506192\pi$
$542$	1.86810e7	2.73151
$543$	0	0
$544$	−8.49307e6	−1.23046
$545$	2.52351e6	0.363927
$546$	0	0
$547$	2.75562e6	0.393778	0.196889	−	0.980426i	$-0.436916\pi$
$547$	2.75562e6	0.393778	0.196889	+	0.980426i	$0.436916\pi$
$548$	1.22563e7	1.74345
$549$	0	0
$550$	2.00578e7	2.82733
$551$	6.60121e6	0.926285
$552$	0	0
$553$	3.00402e6	0.417725
$554$	1.54395e7	2.13727
$555$	0	0
$556$	−1.28493e7	−1.76275
$557$	−1.82948e6	−0.249856	−0.124928	−	0.992166i	$-0.539870\pi$
$557$	−1.82948e6	−0.249856	−0.124928	+	0.992166i	$0.539870\pi$
$558$	0	0
$559$	−138029.	−0.0186828
$560$	2.83325e6	0.381781
$561$	0	0
$562$	9.24650e6	1.23491
$563$	1.34471e7	1.78796	0.893982	−	0.448103i	$-0.147900\pi$
$563$	1.34471e7	1.78796	0.893982	+	0.448103i	$0.147900\pi$
$564$	0	0
$565$	−461018.	−0.0607571
$566$	1.46557e7	1.92294
$567$	0	0
$568$	3.89542e7	5.06621
$569$	−1.69125e6	−0.218991	−0.109495	−	0.993987i	$-0.534923\pi$
$569$	−1.69125e6	−0.218991	−0.109495	+	0.993987i	$0.534923\pi$
$570$	0	0
$571$	3.79607e6	0.487241	0.243621	−	0.969871i	$-0.421665\pi$
$571$	3.79607e6	0.487241	0.243621	+	0.969871i	$0.421665\pi$
$572$	1.17111e6	0.149660
$573$	0	0
$574$	−4.99449e6	−0.632719
$575$	4.41507e6	0.556888
$576$	0	0
$577$	6.69060e6	0.836615	0.418308	−	0.908305i	$-0.362623\pi$
$577$	6.69060e6	0.836615	0.418308	+	0.908305i	$0.362623\pi$
$578$	−1.16928e7	−1.45579
$579$	0	0
$580$	3.76271e6	0.464441
$581$	−287973.	−0.0353925
$582$	0	0
$583$	1.10247e6	0.134337
$584$	−4.74539e7	−5.75758
$585$	0	0
$586$	4.38676e6	0.527716
$587$	−6.49588e6	−0.778113	−0.389057	−	0.921214i	$-0.627199\pi$
$587$	−6.49588e6	−0.778113	−0.389057	+	0.921214i	$0.627199\pi$
$588$	0	0
$589$	−1.04022e7	−1.23548
$590$	6.29296e6	0.744260
$591$	0	0
$592$	3.45602e7	4.05296
$593$	7.76664e6	0.906977	0.453488	−	0.891262i	$-0.350179\pi$
$593$	7.76664e6	0.906977	0.453488	+	0.891262i	$0.350179\pi$
$594$	0	0
$595$	−538043.	−0.0623053
$596$	3.23214e6	0.372714
$597$	0	0
$598$	359234.	0.0410794
$599$	−9.51892e6	−1.08398	−0.541989	−	0.840385i	$-0.682329\pi$
$599$	−9.51892e6	−1.08398	−0.541989	+	0.840385i	$0.682329\pi$
$600$	0	0
$601$	1.14167e6	0.128931	0.0644653	−	0.997920i	$-0.479466\pi$
$601$	1.14167e6	0.128931	0.0644653	+	0.997920i	$0.479466\pi$
$602$	−3.42519e6	−0.385206
$603$	0	0
$604$	−2.31539e7	−2.58245
$605$	−5.97531e6	−0.663700
$606$	0	0
$607$	−5.85639e6	−0.645147	−0.322573	−	0.946544i	$-0.604548\pi$
$607$	−5.85639e6	−0.645147	−0.322573	+	0.946544i	$0.604548\pi$
$608$	−4.13917e7	−4.54103
$609$	0	0
$610$	1.63935e6	0.178381
$611$	370614.	0.0401623
$612$	0	0
$613$	9.01709e6	0.969204	0.484602	−	0.874735i	$-0.338965\pi$
$613$	9.01709e6	0.969204	0.484602	+	0.874735i	$0.338965\pi$
$614$	2.49120e7	2.66679
$615$	0	0
$616$	1.76237e7	1.87131
$617$	−1.00317e6	−0.106087	−0.0530433	−	0.998592i	$-0.516892\pi$
$617$	−1.00317e6	−0.106087	−0.0530433	+	0.998592i	$0.516892\pi$
$618$	0	0
$619$	1.72821e6	0.181288	0.0906442	−	0.995883i	$-0.471107\pi$
$619$	1.72821e6	0.181288	0.0906442	+	0.995883i	$0.471107\pi$
$620$	−5.92927e6	−0.619472
$621$	0	0
$622$	−2.21597e7	−2.29661
$623$	4.61538e6	0.476417
$624$	0	0
$625$	6.38932e6	0.654266
$626$	−2.73955e7	−2.79411
$627$	0	0
$628$	5.10846e6	0.516881
$629$	−6.56310e6	−0.661428
$630$	0	0
$631$	−1.39498e7	−1.39475	−0.697374	−	0.716708i	$-0.745648\pi$
$631$	−1.39498e7	−1.39475	−0.697374	+	0.716708i	$0.745648\pi$
$632$	−3.21786e7	−3.20460
$633$	0	0
$634$	−1.93134e7	−1.90825
$635$	−79324.0	−0.00780675
$636$	0	0
$637$	−50467.0	−0.00492787
$638$	1.74274e7	1.69504
$639$	0	0
$640$	−3.89766e6	−0.376144
$641$	−4.94165e6	−0.475037	−0.237518	−	0.971383i	$-0.576334\pi$
$641$	−4.94165e6	−0.475037	−0.237518	+	0.971383i	$0.576334\pi$
$642$	0	0
$643$	−3.11381e6	−0.297005	−0.148503	−	0.988912i	$-0.547445\pi$
$643$	−3.11381e6	−0.297005	−0.148503	+	0.988912i	$0.547445\pi$
$644$	6.39682e6	0.607784
$645$	0	0
$646$	1.66731e7	1.57194
$647$	−5.08098e6	−0.477185	−0.238593	−	0.971120i	$-0.576686\pi$
$647$	−5.08098e6	−0.477185	−0.238593	+	0.971120i	$0.576686\pi$
$648$	0	0
$649$	2.09152e7	1.94917
$650$	615258.	0.0571181
$651$	0	0
$652$	2.90614e7	2.67730
$653$	−1.32555e7	−1.21650	−0.608251	−	0.793745i	$-0.708129\pi$
$653$	−1.32555e7	−1.21650	−0.608251	+	0.793745i	$0.708129\pi$
$654$	0	0
$655$	−756663.	−0.0689127
$656$	2.85856e7	2.59351
$657$	0	0
$658$	9.19676e6	0.828076
$659$	−8.82994e6	−0.792035	−0.396017	−	0.918243i	$-0.629608\pi$
$659$	−8.82994e6	−0.792035	−0.396017	+	0.918243i	$0.629608\pi$
$660$	0	0
$661$	−3.83176e6	−0.341110	−0.170555	−	0.985348i	$-0.554556\pi$
$661$	−3.83176e6	−0.341110	−0.170555	+	0.985348i	$0.554556\pi$
$662$	−3.71726e7	−3.29669
$663$	0	0
$664$	3.08471e6	0.271516
$665$	−2.62220e6	−0.229938
$666$	0	0
$667$	3.83607e6	0.333866
$668$	−8.38558e6	−0.727096
$669$	0	0
$670$	−1.10962e7	−0.954961
$671$	5.44853e6	0.467168
$672$	0	0
$673$	1.35950e7	1.15702	0.578510	−	0.815676i	$-0.303635\pi$
$673$	1.35950e7	1.15702	0.578510	+	0.815676i	$0.303635\pi$
$674$	1.48095e6	0.125572
$675$	0	0
$676$	−3.01536e7	−2.53789
$677$	−1.36433e7	−1.14406	−0.572028	−	0.820234i	$-0.693843\pi$
$677$	−1.36433e7	−1.14406	−0.572028	+	0.820234i	$0.693843\pi$
$678$	0	0
$679$	−2.01907e6	−0.168064
$680$	5.76343e6	0.477979
$681$	0	0
$682$	−2.74621e7	−2.26085
$683$	2.54456e6	0.208718	0.104359	−	0.994540i	$-0.466721\pi$
$683$	2.54456e6	0.208718	0.104359	+	0.994540i	$0.466721\pi$
$684$	0	0
$685$	−2.91960e6	−0.237737
$686$	−1.25234e6	−0.101604
$687$	0	0
$688$	1.96038e7	1.57896
$689$	33817.4	0.00271389
$690$	0	0
$691$	4.89444e6	0.389949	0.194974	−	0.980808i	$-0.437538\pi$
$691$	4.89444e6	0.389949	0.194974	+	0.980808i	$0.437538\pi$
$692$	−5.46888e7	−4.34144
$693$	0	0
$694$	−7.73791e6	−0.609853
$695$	3.06085e6	0.240370
$696$	0	0
$697$	−5.42850e6	−0.423251
$698$	2.91082e7	2.26140
$699$	0	0
$700$	1.09558e7	0.845082
$701$	−5.48199e6	−0.421350	−0.210675	−	0.977556i	$-0.567566\pi$
$701$	−5.48199e6	−0.421350	−0.210675	+	0.977556i	$0.567566\pi$
$702$	0	0
$703$	−3.19858e7	−2.44101
$704$	−4.38150e7	−3.33189
$705$	0	0
$706$	−641764.	−0.0484578
$707$	4.87511e6	0.366806
$708$	0	0
$709$	−1.11203e7	−0.830805	−0.415402	−	0.909638i	$-0.636359\pi$
$709$	−1.11203e7	−0.830805	−0.415402	+	0.909638i	$0.636359\pi$
$710$	−1.53014e7	−1.13916
$711$	0	0
$712$	−4.94391e7	−3.65486
$713$	−6.04488e6	−0.445311
$714$	0	0
$715$	−278972.	−0.0204078
$716$	1.34455e7	0.980156
$717$	0	0
$718$	6.79634e6	0.491999
$719$	−5.36650e6	−0.387141	−0.193570	−	0.981086i	$-0.562007\pi$
$719$	−5.36650e6	−0.387141	−0.193570	+	0.981086i	$0.562007\pi$
$720$	0	0
$721$	−5.15686e6	−0.369443
$722$	5.49004e7	3.91952
$723$	0	0
$724$	−1.50163e7	−1.06467
$725$	6.57002e6	0.464217
$726$	0	0
$727$	−2.64839e7	−1.85843	−0.929215	−	0.369539i	$-0.879516\pi$
$727$	−2.64839e7	−1.85843	−0.929215	+	0.369539i	$0.879516\pi$
$728$	540595.	0.0378045
$729$	0	0
$730$	1.86401e7	1.29462
$731$	−3.72283e6	−0.257680
$732$	0	0
$733$	−9.78546e6	−0.672700	−0.336350	−	0.941737i	$-0.609193\pi$
$733$	−9.78546e6	−0.672700	−0.336350	+	0.941737i	$0.609193\pi$
$734$	1.62723e7	1.11483
$735$	0	0
$736$	−2.40534e7	−1.63675
$737$	−3.68790e7	−2.50098
$738$	0	0
$739$	5.97965e6	0.402777	0.201388	−	0.979511i	$-0.435455\pi$
$739$	5.97965e6	0.402777	0.201388	+	0.979511i	$0.435455\pi$
$740$	−1.82320e7	−1.22392
$741$	0	0
$742$	839175.	0.0559555
$743$	−9.68440e6	−0.643578	−0.321789	−	0.946811i	$-0.604284\pi$
$743$	−9.68440e6	−0.643578	−0.321789	+	0.946811i	$0.604284\pi$
$744$	0	0
$745$	−769936.	−0.0508234
$746$	9.13371e6	0.600897
$747$	0	0
$748$	3.15863e7	2.06417
$749$	−4.89429e6	−0.318776
$750$	0	0
$751$	−2.47557e7	−1.60168	−0.800839	−	0.598879i	$-0.795613\pi$
$751$	−2.47557e7	−1.60168	−0.800839	+	0.598879i	$0.795613\pi$
$752$	−5.26371e7	−3.39428
$753$	0	0
$754$	534572.	0.0342434
$755$	5.51555e6	0.352145
$756$	0	0
$757$	2.48859e7	1.57839	0.789193	−	0.614146i	$-0.210499\pi$
$757$	2.48859e7	1.57839	0.789193	+	0.614146i	$0.210499\pi$
$758$	5.07936e6	0.321097
$759$	0	0
$760$	2.80885e7	1.76399
$761$	431590.	0.0270153	0.0135076	−	0.999909i	$-0.495700\pi$
$761$	431590.	0.0270153	0.0135076	+	0.999909i	$0.495700\pi$
$762$	0	0
$763$	6.38408e6	0.396997
$764$	3.41694e7	2.11789
$765$	0	0
$766$	−3.79441e7	−2.33654
$767$	641556.	0.0393774
$768$	0	0
$769$	−1.30074e7	−0.793186	−0.396593	−	0.917995i	$-0.629808\pi$
$769$	−1.30074e7	−0.793186	−0.396593	+	0.917995i	$0.629808\pi$
$770$	−6.92267e6	−0.420772
$771$	0	0
$772$	−4.12885e6	−0.249337
$773$	−1.81937e7	−1.09515	−0.547574	−	0.836757i	$-0.684449\pi$
$773$	−1.81937e7	−1.09515	−0.547574	+	0.836757i	$0.684449\pi$
$774$	0	0
$775$	−1.03530e7	−0.619175
$776$	2.16279e7	1.28932
$777$	0	0
$778$	3.64388e7	2.15832
$779$	−2.64563e7	−1.56201
$780$	0	0
$781$	−5.08554e7	−2.98338
$782$	9.68900e6	0.566581
$783$	0	0
$784$	7.16766e6	0.416474
$785$	−1.21690e6	−0.0704823
$786$	0	0
$787$	−1.27059e7	−0.731252	−0.365626	−	0.930762i	$-0.619145\pi$
$787$	−1.27059e7	−0.731252	−0.365626	+	0.930762i	$0.619145\pi$
$788$	−4.19574e7	−2.40709
$789$	0	0
$790$	1.26399e7	0.720568
$791$	−1.16630e6	−0.0662780
$792$	0	0
$793$	167129.	0.00943778
$794$	−3.63907e7	−2.04851
$795$	0	0
$796$	1.15197e7	0.644406
$797$	2.89051e7	1.61186	0.805932	−	0.592008i	$-0.201665\pi$
$797$	2.89051e7	1.61186	0.805932	+	0.592008i	$0.201665\pi$
$798$	0	0
$799$	9.99595e6	0.553933
$800$	−4.11961e7	−2.27578
$801$	0	0
$802$	1.87379e7	1.02869
$803$	6.19519e7	3.39052
$804$	0	0
$805$	−1.52380e6	−0.0828778
$806$	−842377.	−0.0456740
$807$	0	0
$808$	−5.22214e7	−2.81397
$809$	−5.93719e6	−0.318941	−0.159470	−	0.987203i	$-0.550979\pi$
$809$	−5.93719e6	−0.318941	−0.159470	+	0.987203i	$0.550979\pi$
$810$	0	0
$811$	−2.80244e7	−1.49618	−0.748091	−	0.663596i	$-0.769029\pi$
$811$	−2.80244e7	−1.49618	−0.748091	+	0.663596i	$0.769029\pi$
$812$	9.51904e6	0.506644
$813$	0	0
$814$	−8.44434e7	−4.46688
$815$	−6.92278e6	−0.365079
$816$	0	0
$817$	−1.81435e7	−0.950970
$818$	−3.85179e6	−0.201270
$819$	0	0
$820$	−1.50801e7	−0.783196
$821$	8.31794e6	0.430683	0.215342	−	0.976539i	$-0.430914\pi$
$821$	8.31794e6	0.430683	0.215342	+	0.976539i	$0.430914\pi$
$822$	0	0
$823$	1.67972e6	0.0864444	0.0432222	−	0.999065i	$-0.486238\pi$
$823$	1.67972e6	0.0864444	0.0432222	+	0.999065i	$0.486238\pi$
$824$	5.52394e7	2.83420
$825$	0	0
$826$	1.59202e7	0.811891
$827$	−6.72497e6	−0.341921	−0.170961	−	0.985278i	$-0.554687\pi$
$827$	−6.72497e6	−0.341921	−0.170961	+	0.985278i	$0.554687\pi$
$828$	0	0
$829$	2.56662e6	0.129711	0.0648554	−	0.997895i	$-0.479341\pi$
$829$	2.56662e6	0.129711	0.0648554	+	0.997895i	$0.479341\pi$
$830$	−1.21169e6	−0.0610514
$831$	0	0
$832$	−1.34399e6	−0.0673113
$833$	−1.36116e6	−0.0679669
$834$	0	0
$835$	1.99755e6	0.0991472
$836$	1.53939e8	7.61784
$837$	0	0
$838$	−1.90994e7	−0.939526
$839$	4.46741e6	0.219104	0.109552	−	0.993981i	$-0.465058\pi$
$839$	4.46741e6	0.219104	0.109552	+	0.993981i	$0.465058\pi$
$840$	0	0
$841$	−1.48027e7	−0.721692
$842$	8.77418e6	0.426507
$843$	0	0
$844$	7.19592e7	3.47720
$845$	7.18295e6	0.346068
$846$	0	0
$847$	−1.51166e7	−0.724010
$848$	−4.80296e6	−0.229361
$849$	0	0
$850$	1.65943e7	0.787792
$851$	−1.85875e7	−0.879825
$852$	0	0
$853$	2.96386e7	1.39472	0.697358	−	0.716723i	$-0.254359\pi$
$853$	2.96386e7	1.39472	0.697358	+	0.716723i	$0.254359\pi$
$854$	4.14730e6	0.194590
$855$	0	0
$856$	5.24268e7	2.44551
$857$	−5.73764e6	−0.266858	−0.133429	−	0.991058i	$-0.542599\pi$
$857$	−5.73764e6	−0.266858	−0.133429	+	0.991058i	$0.542599\pi$
$858$	0	0
$859$	1.05190e7	0.486398	0.243199	−	0.969976i	$-0.421803\pi$
$859$	1.05190e7	0.486398	0.243199	+	0.969976i	$0.421803\pi$
$860$	−1.03419e7	−0.476817
$861$	0	0
$862$	−3.84485e7	−1.76243
$863$	2.63871e7	1.20605	0.603024	−	0.797723i	$-0.293962\pi$
$863$	2.63871e7	1.20605	0.603024	+	0.797723i	$0.293962\pi$
$864$	0	0
$865$	1.30275e7	0.592001
$866$	8.22271e6	0.372581
$867$	0	0
$868$	−1.50001e7	−0.675763
$869$	4.20097e7	1.88712
$870$	0	0
$871$	−1.13123e6	−0.0505251
$872$	−6.83852e7	−3.04558
$873$	0	0
$874$	4.72201e7	2.09097
$875$	−5.57566e6	−0.246193
$876$	0	0
$877$	1.20657e7	0.529730	0.264865	−	0.964286i	$-0.414673\pi$
$877$	1.20657e7	0.529730	0.264865	+	0.964286i	$0.414673\pi$
$878$	1.96283e7	0.859301
$879$	0	0
$880$	3.96215e7	1.72474
$881$	2.01221e7	0.873441	0.436720	−	0.899597i	$-0.356140\pi$
$881$	2.01221e7	0.873441	0.436720	+	0.899597i	$0.356140\pi$
$882$	0	0
$883$	1.08017e7	0.466219	0.233109	−	0.972451i	$-0.425110\pi$
$883$	1.08017e7	0.466219	0.233109	+	0.972451i	$0.425110\pi$
$884$	968886.	0.0417006
$885$	0	0
$886$	3.36823e7	1.44151
$887$	−2.39674e7	−1.02285	−0.511424	−	0.859328i	$-0.670882\pi$
$887$	−2.39674e7	−1.02285	−0.511424	+	0.859328i	$0.670882\pi$
$888$	0	0
$889$	−200677.	−0.00851614
$890$	1.94199e7	0.821811
$891$	0	0
$892$	7.80063e7	3.28259
$893$	4.87161e7	2.04430
$894$	0	0
$895$	−3.20289e6	−0.133655
$896$	−9.86046e6	−0.410324
$897$	0	0
$898$	−7.56955e7	−3.13242
$899$	−8.99531e6	−0.371208
$900$	0	0
$901$	912098.	0.0374309
$902$	−6.98452e7	−2.85838
$903$	0	0
$904$	1.24932e7	0.508456
$905$	3.57707e6	0.145180
$906$	0	0
$907$	−2.10207e7	−0.848455	−0.424227	−	0.905556i	$-0.639454\pi$
$907$	−2.10207e7	−0.848455	−0.424227	+	0.905556i	$0.639454\pi$
$908$	−1.76799e7	−0.711650
$909$	0	0
$910$	−212348.	−0.00850049
$911$	−2.33252e7	−0.931170	−0.465585	−	0.885003i	$-0.654156\pi$
$911$	−2.33252e7	−0.931170	−0.465585	+	0.885003i	$0.654156\pi$
$912$	0	0
$913$	−4.02714e6	−0.159890
$914$	2.27643e7	0.901339
$915$	0	0
$916$	1.81272e7	0.713826
$917$	−1.91424e6	−0.0751747
$918$	0	0
$919$	−2.47403e7	−0.966311	−0.483155	−	0.875535i	$-0.660509\pi$
$919$	−2.47403e7	−0.966311	−0.483155	+	0.875535i	$0.660509\pi$
$920$	1.63227e7	0.635803
$921$	0	0
$922$	−7.01490e7	−2.71766
$923$	−1.55995e6	−0.0602707
$924$	0	0
$925$	−3.18346e7	−1.22334
$926$	−2.85130e7	−1.09274
$927$	0	0
$928$	−3.57935e7	−1.36438
$929$	−1.35330e7	−0.514463	−0.257231	−	0.966350i	$-0.582810\pi$
$929$	−1.35330e7	−0.514463	−0.257231	+	0.966350i	$0.582810\pi$
$930$	0	0
$931$	−6.63374e6	−0.250833
$932$	−6.46485e7	−2.43792
$933$	0	0
$934$	8.83274e7	3.31305
$935$	−7.52425e6	−0.281471
$936$	0	0
$937$	−2.06413e7	−0.768048	−0.384024	−	0.923323i	$-0.625462\pi$
$937$	−2.06413e7	−0.768048	−0.384024	+	0.923323i	$0.625462\pi$
$938$	−2.80715e7	−1.04174
$939$	0	0
$940$	2.77683e7	1.02501
$941$	2.01077e7	0.740266	0.370133	−	0.928979i	$-0.379312\pi$
$941$	2.01077e7	0.740266	0.370133	+	0.928979i	$0.379312\pi$
$942$	0	0
$943$	−1.53741e7	−0.563004
$944$	−9.11181e7	−3.32793
$945$	0	0
$946$	−4.78994e7	−1.74021
$947$	−2.30754e7	−0.836131	−0.418065	−	0.908417i	$-0.637292\pi$
$947$	−2.30754e7	−0.836131	−0.418065	+	0.908417i	$0.637292\pi$
$948$	0	0
$949$	1.90033e6	0.0684956
$950$	8.08737e7	2.90736
$951$	0	0
$952$	1.45805e7	0.521412
$953$	3.80597e7	1.35748	0.678739	−	0.734379i	$-0.262527\pi$
$953$	3.80597e7	1.35748	0.678739	+	0.734379i	$0.262527\pi$
$954$	0	0
$955$	−8.13956e6	−0.288797
$956$	−2.14664e7	−0.759653
$957$	0	0
$958$	−6.12470e7	−2.15611
$959$	−7.38612e6	−0.259340
$960$	0	0
$961$	−1.44543e7	−0.504882
$962$	−2.59024e6	−0.0902406
$963$	0	0
$964$	4.12320e7	1.42903
$965$	983543.	0.0339997
$966$	0	0
$967$	4.28212e7	1.47263	0.736313	−	0.676641i	$-0.236565\pi$
$967$	4.28212e7	1.47263	0.736313	+	0.676641i	$0.236565\pi$
$968$	1.61926e8	5.55429
$969$	0	0
$970$	−8.49552e6	−0.289908
$971$	−3.09516e7	−1.05350	−0.526750	−	0.850020i	$-0.676590\pi$
$971$	−3.09516e7	−1.05350	−0.526750	+	0.850020i	$0.676590\pi$
$972$	0	0
$973$	7.74346e6	0.262212
$974$	−2.97902e7	−1.00618
$975$	0	0
$976$	−2.37368e7	−0.797623
$977$	−4.11114e7	−1.37793	−0.688963	−	0.724797i	$-0.741934\pi$
$977$	−4.11114e7	−1.37793	−0.688963	+	0.724797i	$0.741934\pi$
$978$	0	0
$979$	6.45436e7	2.15227
$980$	−3.78124e6	−0.125768
$981$	0	0
$982$	−4.63462e7	−1.53368
$983$	−3.46577e7	−1.14397	−0.571986	−	0.820263i	$-0.693827\pi$
$983$	−3.46577e7	−1.14397	−0.571986	+	0.820263i	$0.693827\pi$
$984$	0	0
$985$	9.99475e6	0.328232
$986$	1.44181e7	0.472297
$987$	0	0
$988$	4.72195e6	0.153897
$989$	−1.05435e7	−0.342763
$990$	0	0
$991$	3.20084e7	1.03533	0.517666	−	0.855582i	$-0.326801\pi$
$991$	3.20084e7	1.03533	0.517666	+	0.855582i	$0.326801\pi$
$992$	5.64034e7	1.81981
$993$	0	0
$994$	−3.87100e7	−1.24267
$995$	−2.74414e6	−0.0878715
$996$	0	0
$997$	−2.77643e7	−0.884604	−0.442302	−	0.896866i	$-0.645838\pi$
$997$	−2.77643e7	−0.884604	−0.442302	+	0.896866i	$0.645838\pi$
$998$	−4.55130e6	−0.144647
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.6.a.g.1.15		15
3.2	odd	2		567.6.a.h.1.1		15
9.2	odd	6		189.6.f.a.64.15		30
9.4	even	3		63.6.f.b.43.1	yes	30
9.5	odd	6		189.6.f.a.127.15		30
9.7	even	3		63.6.f.b.22.1	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.6.f.b.22.1	✓	30	9.7	even	3
63.6.f.b.43.1	yes	30	9.4	even	3
189.6.f.a.64.15		30	9.2	odd	6
189.6.f.a.127.15		30	9.5	odd	6
567.6.a.g.1.15		15	1.1	even	1	trivial
567.6.a.h.1.1		15	3.2	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 567 = 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	567.a (trivial)

\(f(q)\)	\(=\)	\(q+10.6447 q^{2} +81.3091 q^{4} -19.3688 q^{5} -49.0000 q^{7} +524.880 q^{8} -206.175 q^{10} -685.239 q^{11} -21.0192 q^{13} -521.589 q^{14} +2985.28 q^{16} -566.915 q^{17} -2762.91 q^{19} -1574.86 q^{20} -7294.15 q^{22} -1605.57 q^{23} -2749.85 q^{25} -223.742 q^{26} -3984.15 q^{28} -2389.23 q^{29} +3764.94 q^{31} +14981.2 q^{32} -6034.62 q^{34} +949.072 q^{35} +11576.9 q^{37} -29410.2 q^{38} -10166.3 q^{40} +9575.52 q^{41} +6566.83 q^{43} -55716.2 q^{44} -17090.8 q^{46} -17632.2 q^{47} +2401.00 q^{49} -29271.2 q^{50} -1709.05 q^{52} -1608.88 q^{53} +13272.3 q^{55} -25719.1 q^{56} -25432.6 q^{58} -30522.4 q^{59} -7951.28 q^{61} +40076.6 q^{62} +63941.1 q^{64} +407.117 q^{65} +53819.2 q^{67} -46095.4 q^{68} +10102.6 q^{70} +74215.5 q^{71} -90409.2 q^{73} +123232. q^{74} -224649. q^{76} +33576.7 q^{77} -61306.6 q^{79} -57821.4 q^{80} +101928. q^{82} +5876.99 q^{83} +10980.5 q^{85} +69901.8 q^{86} -359668. q^{88} -94191.4 q^{89} +1029.94 q^{91} -130547. q^{92} -187689. q^{94} +53514.2 q^{95} +41205.4 q^{97} +25557.9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 4 q^{2} + 240 q^{4} - 71 q^{5} - 735 q^{7} + 21 q^{8} - 134 q^{11} - 165 q^{13} + 196 q^{14} + 3840 q^{16} - 1092 q^{17} - 447 q^{19} + 1214 q^{20} + 96 q^{22} + 3564 q^{23} + 3576 q^{25} - 865 q^{26}+ \cdots - 9604 q^{98}+O(q^{100}) \) 15 * q - 4 * q^2 + 240 * q^4 - 71 * q^5 - 735 * q^7 + 21 * q^8 - 134 * q^11 - 165 * q^13 + 196 * q^14 + 3840 * q^16 - 1092 * q^17 - 447 * q^19 + 1214 * q^20 + 96 * q^22 + 3564 * q^23 + 3576 * q^25 - 865 * q^26 - 11760 * q^28 - 4073 * q^29 + 2217 * q^31 - 12107 * q^32 + 15654 * q^34 + 3479 * q^35 + 5007 * q^37 - 48685 * q^38 + 7485 * q^40 - 22294 * q^41 + 3153 * q^43 - 82480 * q^44 - 18246 * q^46 - 74313 * q^47 + 36015 * q^49 - 111323 * q^50 - 33501 * q^52 - 27774 * q^53 + 38688 * q^55 - 1029 * q^56 + 3249 * q^58 - 90567 * q^59 + 3714 * q^61 - 178704 * q^62 + 47217 * q^64 - 79607 * q^65 + 11016 * q^67 - 241743 * q^68 + 53793 * q^71 - 120081 * q^73 - 95016 * q^74 - 122748 * q^76 + 6566 * q^77 - 11013 * q^79 - 296449 * q^80 + 81018 * q^82 - 7014 * q^83 + 36387 * q^85 - 170519 * q^86 + 108012 * q^88 - 76160 * q^89 + 8085 * q^91 - 153651 * q^92 + 266412 * q^94 + 64264 * q^95 + 81033 * q^97 - 9604 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more