LMFDB - Embedded newform 558.6.a.k.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [558,6,Mod(1,558)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(558, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("558.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 558 = 2 \cdot 3^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	558.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,16,0,64,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$89.4941714556$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 221x^{2} - 140x + 2412 $ x^4 - 2x^3 - 221x^2 - 140*x + 2412
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 62)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$15.8744$ of defining polynomial
Character	$\chi$	$=$	558.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.00000 q^{2} +16.0000 q^{4} -4.15476 q^{5} -174.927 q^{7} +64.0000 q^{8} -16.6190 q^{10} +235.351 q^{11} +1029.68 q^{13} -699.709 q^{14} +256.000 q^{16} -1455.25 q^{17} +1049.81 q^{19} -66.4761 q^{20} +941.406 q^{22} +59.4958 q^{23} -3107.74 q^{25} +4118.72 q^{26} -2798.84 q^{28} -4193.48 q^{29} +961.000 q^{31} +1024.00 q^{32} -5820.99 q^{34} +726.781 q^{35} +7798.04 q^{37} +4199.25 q^{38} -265.905 q^{40} +4010.52 q^{41} -1999.94 q^{43} +3765.62 q^{44} +237.983 q^{46} -13311.7 q^{47} +13792.6 q^{49} -12431.0 q^{50} +16474.9 q^{52} +31868.1 q^{53} -977.829 q^{55} -11195.3 q^{56} -16773.9 q^{58} +35467.1 q^{59} +45806.4 q^{61} +3844.00 q^{62} +4096.00 q^{64} -4278.07 q^{65} +55514.2 q^{67} -23284.0 q^{68} +2907.12 q^{70} +47049.1 q^{71} +79212.2 q^{73} +31192.2 q^{74} +16797.0 q^{76} -41169.4 q^{77} +814.146 q^{79} -1063.62 q^{80} +16042.1 q^{82} -47660.7 q^{83} +6046.20 q^{85} -7999.76 q^{86} +15062.5 q^{88} +46071.5 q^{89} -180119. q^{91} +951.933 q^{92} -53246.8 q^{94} -4361.72 q^{95} +54266.6 q^{97} +55170.3 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{2} + 64 q^{4} - 2 q^{5} + 146 q^{7} + 256 q^{8} - 8 q^{10} + 270 q^{11} + 1038 q^{13} + 584 q^{14} + 1024 q^{16} - 2020 q^{17} + 6214 q^{19} - 32 q^{20} + 1080 q^{22} - 8292 q^{23} + 9974 q^{25}+ \cdots + 86520 q^{98}+O(q^{100}) $ 4 * q + 16 * q^2 + 64 * q^4 - 2 * q^5 + 146 * q^7 + 256 * q^8 - 8 * q^10 + 270 * q^11 + 1038 * q^13 + 584 * q^14 + 1024 * q^16 - 2020 * q^17 + 6214 * q^19 - 32 * q^20 + 1080 * q^22 - 8292 * q^23 + 9974 * q^25 + 4152 * q^26 + 2336 * q^28 - 1894 * q^29 + 3844 * q^31 + 4096 * q^32 - 8080 * q^34 + 9146 * q^35 - 5470 * q^37 + 24856 * q^38 - 128 * q^40 + 16878 * q^41 + 1530 * q^43 + 4320 * q^44 - 33168 * q^46 + 25460 * q^47 + 21630 * q^49 + 39896 * q^50 + 16608 * q^52 + 7790 * q^53 + 5760 * q^55 + 9344 * q^56 - 7576 * q^58 + 35802 * q^59 + 48366 * q^61 + 15376 * q^62 + 16384 * q^64 - 12096 * q^65 + 54696 * q^67 - 32320 * q^68 + 36584 * q^70 + 35342 * q^71 + 59336 * q^73 - 21880 * q^74 + 99424 * q^76 + 86976 * q^77 - 3652 * q^79 - 512 * q^80 + 67512 * q^82 + 80074 * q^83 + 81860 * q^85 + 6120 * q^86 + 17280 * q^88 + 12060 * q^89 - 74952 * q^91 - 132672 * q^92 + 101840 * q^94 + 286138 * q^95 + 232670 * q^97 + 86520 * 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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	−4.15476	−0.0743226	−0.0371613	−	0.999309i	$-0.511832\pi$
$5$	−4.15476	−0.0743226	−0.0371613	+	0.999309i	$0.511832\pi$
$6$	0	0
$7$	−174.927	−1.34931	−0.674656	−	0.738132i	$-0.735708\pi$
$7$	−174.927	−1.34931	−0.674656	+	0.738132i	$0.735708\pi$
$8$	64.0000	0.353553
$9$	0	0
$10$	−16.6190	−0.0525540
$11$	235.351	0.586456	0.293228	−	0.956043i	$-0.405271\pi$
$11$	235.351	0.586456	0.293228	+	0.956043i	$0.405271\pi$
$12$	0	0
$13$	1029.68	1.68983	0.844916	−	0.534899i	$-0.179650\pi$
$13$	1029.68	1.68983	0.844916	+	0.534899i	$0.179650\pi$
$14$	−699.709	−0.954108
$15$	0	0
$16$	256.000	0.250000
$17$	−1455.25	−1.22128	−0.610639	−	0.791909i	$-0.709087\pi$
$17$	−1455.25	−1.22128	−0.610639	+	0.791909i	$0.709087\pi$
$18$	0	0
$19$	1049.81	0.667157	0.333579	−	0.942722i	$-0.391744\pi$
$19$	1049.81	0.667157	0.333579	+	0.942722i	$0.391744\pi$
$20$	−66.4761	−0.0371613
$21$	0	0
$22$	941.406	0.414687
$23$	59.4958	0.0234513	0.0117256	−	0.999931i	$-0.496268\pi$
$23$	59.4958	0.0234513	0.0117256	+	0.999931i	$0.496268\pi$
$24$	0	0
$25$	−3107.74	−0.994476
$26$	4118.72	1.19489
$27$	0	0
$28$	−2798.84	−0.674656
$29$	−4193.48	−0.925933	−0.462966	−	0.886376i	$-0.653215\pi$
$29$	−4193.48	−0.925933	−0.462966	+	0.886376i	$0.653215\pi$
$30$	0	0
$31$	961.000	0.179605
$31$	961.000	0.179605
$32$	1024.00	0.176777
$33$	0	0
$34$	−5820.99	−0.863574
$35$	726.781	0.100284
$36$	0	0
$37$	7798.04	0.936443	0.468221	−	0.883611i	$-0.344895\pi$
$37$	7798.04	0.936443	0.468221	+	0.883611i	$0.344895\pi$
$38$	4199.25	0.471751
$39$	0	0
$40$	−265.905	−0.0262770
$41$	4010.52	0.372599	0.186299	−	0.982493i	$-0.440351\pi$
$41$	4010.52	0.372599	0.186299	+	0.982493i	$0.440351\pi$
$42$	0	0
$43$	−1999.94	−0.164947	−0.0824737	−	0.996593i	$-0.526282\pi$
$43$	−1999.94	−0.164947	−0.0824737	+	0.996593i	$0.526282\pi$
$44$	3765.62	0.293228
$45$	0	0
$46$	237.983	0.0165826
$47$	−13311.7	−0.879000	−0.439500	−	0.898243i	$-0.644844\pi$
$47$	−13311.7	−0.879000	−0.439500	+	0.898243i	$0.644844\pi$
$48$	0	0
$49$	13792.6	0.820645
$50$	−12431.0	−0.703201
$51$	0	0
$52$	16474.9	0.844916
$53$	31868.1	1.55835	0.779177	−	0.626804i	$-0.215637\pi$
$53$	31868.1	1.55835	0.779177	+	0.626804i	$0.215637\pi$
$54$	0	0
$55$	−977.829	−0.0435869
$56$	−11195.3	−0.477054
$57$	0	0
$58$	−16773.9	−0.654733
$59$	35467.1	1.32646	0.663231	−	0.748414i	$-0.269185\pi$
$59$	35467.1	1.32646	0.663231	+	0.748414i	$0.269185\pi$
$60$	0	0
$61$	45806.4	1.57617	0.788083	−	0.615569i	$-0.211074\pi$
$61$	45806.4	1.57617	0.788083	+	0.615569i	$0.211074\pi$
$62$	3844.00	0.127000
$63$	0	0
$64$	4096.00	0.125000
$65$	−4278.07	−0.125593
$66$	0	0
$67$	55514.2	1.51083	0.755417	−	0.655244i	$-0.227434\pi$
$67$	55514.2	1.51083	0.755417	+	0.655244i	$0.227434\pi$
$68$	−23284.0	−0.610639
$69$	0	0
$70$	2907.12	0.0709118
$71$	47049.1	1.10766	0.553829	−	0.832630i	$-0.313166\pi$
$71$	47049.1	1.10766	0.553829	+	0.832630i	$0.313166\pi$
$72$	0	0
$73$	79212.2	1.73974	0.869871	−	0.493280i	$-0.164202\pi$
$73$	79212.2	1.73974	0.869871	+	0.493280i	$0.164202\pi$
$74$	31192.2	0.662165
$75$	0	0
$76$	16797.0	0.333579
$77$	−41169.4	−0.791312
$78$	0	0
$79$	814.146	0.0146769	0.00733845	−	0.999973i	$-0.497664\pi$
$79$	814.146	0.0146769	0.00733845	+	0.999973i	$0.497664\pi$
$80$	−1063.62	−0.0185806
$81$	0	0
$82$	16042.1	0.263467
$83$	−47660.7	−0.759390	−0.379695	−	0.925112i	$-0.623971\pi$
$83$	−47660.7	−0.759390	−0.379695	+	0.925112i	$0.623971\pi$
$84$	0	0
$85$	6046.20	0.0907686
$86$	−7999.76	−0.116635
$87$	0	0
$88$	15062.5	0.207343
$89$	46071.5	0.616535	0.308267	−	0.951300i	$-0.400251\pi$
$89$	46071.5	0.616535	0.308267	+	0.951300i	$0.400251\pi$
$90$	0	0
$91$	−180119.	−2.28011
$92$	951.933	0.0117256
$93$	0	0
$94$	−53246.8	−0.621547
$95$	−4361.72	−0.0495848
$96$	0	0
$97$	54266.6	0.585603	0.292802	−	0.956173i	$-0.405412\pi$
$97$	54266.6	0.585603	0.292802	+	0.956173i	$0.405412\pi$
$98$	55170.3	0.580283
$99$	0	0
$100$	−49723.8	−0.497238
$101$	−82647.7	−0.806171	−0.403086	−	0.915162i	$-0.632062\pi$
$101$	−82647.7	−0.806171	−0.403086	+	0.915162i	$0.632062\pi$
$102$	0	0
$103$	84118.3	0.781263	0.390631	−	0.920547i	$-0.372257\pi$
$103$	84118.3	0.781263	0.390631	+	0.920547i	$0.372257\pi$
$104$	65899.4	0.597446
$105$	0	0
$106$	127472.	1.10192
$107$	108238.	0.913942	0.456971	−	0.889482i	$-0.348934\pi$
$107$	108238.	0.913942	0.456971	+	0.889482i	$0.348934\pi$
$108$	0	0
$109$	−76250.3	−0.614717	−0.307359	−	0.951594i	$-0.599445\pi$
$109$	−76250.3	−0.614717	−0.307359	+	0.951594i	$0.599445\pi$
$110$	−3911.31	−0.0308206
$111$	0	0
$112$	−44781.4	−0.337328
$113$	−113394.	−0.835400	−0.417700	−	0.908585i	$-0.637164\pi$
$113$	−113394.	−0.835400	−0.417700	+	0.908585i	$0.637164\pi$
$114$	0	0
$115$	−247.191	−0.00174296
$116$	−67095.7	−0.462966
$117$	0	0
$118$	141868.	0.937951
$119$	254563.	1.64789
$120$	0	0
$121$	−105661.	−0.656070
$122$	183226.	1.11452
$123$	0	0
$124$	15376.0	0.0898027
$125$	25895.5	0.148235
$126$	0	0
$127$	127074.	0.699110	0.349555	−	0.936916i	$-0.386333\pi$
$127$	127074.	0.699110	0.349555	+	0.936916i	$0.386333\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	−17112.3	−0.0888074
$131$	−9469.80	−0.0482128	−0.0241064	−	0.999709i	$-0.507674\pi$
$131$	−9469.80	−0.0482128	−0.0241064	+	0.999709i	$0.507674\pi$
$132$	0	0
$133$	−183641.	−0.900204
$134$	222057.	1.06832
$135$	0	0
$136$	−93135.9	−0.431787
$137$	−66791.2	−0.304031	−0.152016	−	0.988378i	$-0.548576\pi$
$137$	−66791.2	−0.304031	−0.152016	+	0.988378i	$0.548576\pi$
$138$	0	0
$139$	264672.	1.16190	0.580952	−	0.813938i	$-0.302680\pi$
$139$	264672.	1.16190	0.580952	+	0.813938i	$0.302680\pi$
$140$	11628.5	0.0501422
$141$	0	0
$142$	188197.	0.783233
$143$	242336.	0.991011
$144$	0	0
$145$	17422.9	0.0688177
$146$	316849.	1.23018
$147$	0	0
$148$	124769.	0.468221
$149$	−272372.	−1.00507	−0.502535	−	0.864557i	$-0.667599\pi$
$149$	−272372.	−1.00507	−0.502535	+	0.864557i	$0.667599\pi$
$150$	0	0
$151$	223300.	0.796978	0.398489	−	0.917173i	$-0.369535\pi$
$151$	223300.	0.796978	0.398489	+	0.917173i	$0.369535\pi$
$152$	67188.1	0.235876
$153$	0	0
$154$	−164678.	−0.559542
$155$	−3992.72	−0.0133487
$156$	0	0
$157$	−7971.22	−0.0258093	−0.0129046	−	0.999917i	$-0.504108\pi$
$157$	−7971.22	−0.0258093	−0.0129046	+	0.999917i	$0.504108\pi$
$158$	3256.58	0.0103781
$159$	0	0
$160$	−4254.47	−0.0131385
$161$	−10407.4	−0.0316431
$162$	0	0
$163$	−192360.	−0.567083	−0.283541	−	0.958960i	$-0.591509\pi$
$163$	−192360.	−0.567083	−0.283541	+	0.958960i	$0.591509\pi$
$164$	64168.3	0.186299
$165$	0	0
$166$	−190643.	−0.536970
$167$	−23682.7	−0.0657114	−0.0328557	−	0.999460i	$-0.510460\pi$
$167$	−23682.7	−0.0657114	−0.0328557	+	0.999460i	$0.510460\pi$
$168$	0	0
$169$	688946.	1.85553
$170$	24184.8	0.0641831
$171$	0	0
$172$	−31999.0	−0.0824737
$173$	−712817.	−1.81077	−0.905385	−	0.424593i	$-0.860417\pi$
$173$	−712817.	−1.81077	−0.905385	+	0.424593i	$0.860417\pi$
$174$	0	0
$175$	543628.	1.34186
$176$	60250.0	0.146614
$177$	0	0
$178$	184286.	0.435956
$179$	591690.	1.38026	0.690131	−	0.723684i	$-0.257553\pi$
$179$	591690.	1.38026	0.690131	+	0.723684i	$0.257553\pi$
$180$	0	0
$181$	649491.	1.47359	0.736794	−	0.676117i	$-0.236339\pi$
$181$	649491.	1.47359	0.736794	+	0.676117i	$0.236339\pi$
$182$	−720476.	−1.61228
$183$	0	0
$184$	3807.73	0.00829129
$185$	−32399.0	−0.0695988
$186$	0	0
$187$	−342495.	−0.716226
$188$	−212987.	−0.439500
$189$	0	0
$190$	−17446.9	−0.0350618
$191$	−662573.	−1.31417	−0.657083	−	0.753818i	$-0.728210\pi$
$191$	−662573.	−1.31417	−0.657083	+	0.753818i	$0.728210\pi$
$192$	0	0
$193$	944863.	1.82589	0.912947	−	0.408078i	$-0.133801\pi$
$193$	944863.	1.82589	0.912947	+	0.408078i	$0.133801\pi$
$194$	217066.	0.414084
$195$	0	0
$196$	220681.	0.410322
$197$	443922.	0.814969	0.407485	−	0.913212i	$-0.366406\pi$
$197$	443922.	0.814969	0.407485	+	0.913212i	$0.366406\pi$
$198$	0	0
$199$	−24623.0	−0.0440765	−0.0220383	−	0.999757i	$-0.507016\pi$
$199$	−24623.0	−0.0440765	−0.0220383	+	0.999757i	$0.507016\pi$
$200$	−198895.	−0.351600
$201$	0	0
$202$	−330591.	−0.570049
$203$	733554.	1.24937
$204$	0	0
$205$	−16662.7	−0.0276925
$206$	336473.	0.552436
$207$	0	0
$208$	263598.	0.422458
$209$	247075.	0.391258
$210$	0	0
$211$	−389799.	−0.602747	−0.301373	−	0.953506i	$-0.597445\pi$
$211$	−389799.	−0.602747	−0.301373	+	0.953506i	$0.597445\pi$
$212$	509889.	0.779177
$213$	0	0
$214$	432950.	0.646255
$215$	8309.27	0.0122593
$216$	0	0
$217$	−168105.	−0.242344
$218$	−305001.	−0.434671
$219$	0	0
$220$	−15645.3	−0.0217935
$221$	−1.49844e6	−2.06375
$222$	0	0
$223$	−467711.	−0.629819	−0.314910	−	0.949122i	$-0.601974\pi$
$223$	−467711.	−0.629819	−0.314910	+	0.949122i	$0.601974\pi$
$224$	−179126.	−0.238527
$225$	0	0
$226$	−453577.	−0.590717
$227$	898140.	1.15686	0.578428	−	0.815733i	$-0.303666\pi$
$227$	898140.	1.15686	0.578428	+	0.815733i	$0.303666\pi$
$228$	0	0
$229$	−630277.	−0.794223	−0.397112	−	0.917770i	$-0.629987\pi$
$229$	−630277.	−0.794223	−0.397112	+	0.917770i	$0.629987\pi$
$230$	−988.763	−0.00123246
$231$	0	0
$232$	−268383.	−0.327367
$233$	473923.	0.571898	0.285949	−	0.958245i	$-0.407691\pi$
$233$	473923.	0.571898	0.285949	+	0.958245i	$0.407691\pi$
$234$	0	0
$235$	55306.9	0.0653296
$236$	567473.	0.663231
$237$	0	0
$238$	1.01825e6	1.16523
$239$	−123418.	−0.139761	−0.0698803	−	0.997555i	$-0.522262\pi$
$239$	−123418.	−0.139761	−0.0698803	+	0.997555i	$0.522262\pi$
$240$	0	0
$241$	−1.80023e6	−1.99657	−0.998287	−	0.0584992i	$-0.981368\pi$
$241$	−1.80023e6	−1.99657	−0.998287	+	0.0584992i	$0.981368\pi$
$242$	−422643.	−0.463911
$243$	0	0
$244$	732903.	0.788083
$245$	−57304.8	−0.0609924
$246$	0	0
$247$	1.08097e6	1.12738
$248$	61504.0	0.0635001
$249$	0	0
$250$	103582.	0.104818
$251$	−885581.	−0.887246	−0.443623	−	0.896213i	$-0.646307\pi$
$251$	−885581.	−0.887246	−0.443623	+	0.896213i	$0.646307\pi$
$252$	0	0
$253$	14002.4	0.0137531
$254$	508294.	0.494346
$255$	0	0
$256$	65536.0	0.0625000
$257$	−786523.	−0.742812	−0.371406	−	0.928471i	$-0.621124\pi$
$257$	−786523.	−0.742812	−0.371406	+	0.928471i	$0.621124\pi$
$258$	0	0
$259$	−1.36409e6	−1.26355
$260$	−68449.1	−0.0627963
$261$	0	0
$262$	−37879.2	−0.0340916
$263$	−1.48145e6	−1.32068	−0.660341	−	0.750966i	$-0.729588\pi$
$263$	−1.48145e6	−1.32068	−0.660341	+	0.750966i	$0.729588\pi$
$264$	0	0
$265$	−132404.	−0.115821
$266$	−734564.	−0.636540
$267$	0	0
$268$	888227.	0.755417
$269$	1.16750e6	0.983727	0.491863	−	0.870672i	$-0.336316\pi$
$269$	1.16750e6	0.983727	0.491863	+	0.870672i	$0.336316\pi$
$270$	0	0
$271$	1.20110e6	0.993477	0.496738	−	0.867900i	$-0.334531\pi$
$271$	1.20110e6	0.993477	0.496738	+	0.867900i	$0.334531\pi$
$272$	−372543.	−0.305320
$273$	0	0
$274$	−267165.	−0.214982
$275$	−731411.	−0.583216
$276$	0	0
$277$	−1.84842e6	−1.44745	−0.723723	−	0.690091i	$-0.757571\pi$
$277$	−1.84842e6	−1.44745	−0.723723	+	0.690091i	$0.757571\pi$
$278$	1.05869e6	0.821590
$279$	0	0
$280$	46514.0	0.0354559
$281$	−982107.	−0.741981	−0.370991	−	0.928637i	$-0.620982\pi$
$281$	−982107.	−0.741981	−0.370991	+	0.928637i	$0.620982\pi$
$282$	0	0
$283$	−76041.8	−0.0564399	−0.0282200	−	0.999602i	$-0.508984\pi$
$283$	−76041.8	−0.0564399	−0.0282200	+	0.999602i	$0.508984\pi$
$284$	752786.	0.553829
$285$	0	0
$286$	969346.	0.700751
$287$	−701550.	−0.502752
$288$	0	0
$289$	697890.	0.491521
$290$	69691.6	0.0486615
$291$	0	0
$292$	1.26739e6	0.869871
$293$	2.66036e6	1.81039	0.905194	−	0.424998i	$-0.139725\pi$
$293$	2.66036e6	1.81039	0.905194	+	0.424998i	$0.139725\pi$
$294$	0	0
$295$	−147357.	−0.0985861
$296$	499075.	0.331082
$297$	0	0
$298$	−1.08949e6	−0.710692
$299$	61261.6	0.0396287
$300$	0	0
$301$	349844.	0.222566
$302$	893200.	0.563549
$303$	0	0
$304$	268752.	0.166789
$305$	−190315.	−0.117145
$306$	0	0
$307$	951120.	0.575956	0.287978	−	0.957637i	$-0.407017\pi$
$307$	951120.	0.575956	0.287978	+	0.957637i	$0.407017\pi$
$308$	−658710.	−0.395656
$309$	0	0
$310$	−15970.9	−0.00943898
$311$	−699861.	−0.410309	−0.205154	−	0.978730i	$-0.565770\pi$
$311$	−699861.	−0.410309	−0.205154	+	0.978730i	$0.565770\pi$
$312$	0	0
$313$	3.03373e6	1.75031	0.875156	−	0.483841i	$-0.160759\pi$
$313$	3.03373e6	1.75031	0.875156	+	0.483841i	$0.160759\pi$
$314$	−31884.9	−0.0182499
$315$	0	0
$316$	13026.3	0.00733845
$317$	−2.04461e6	−1.14278	−0.571390	−	0.820679i	$-0.693596\pi$
$317$	−2.04461e6	−1.14278	−0.571390	+	0.820679i	$0.693596\pi$
$318$	0	0
$319$	−986941.	−0.543019
$320$	−17017.9	−0.00929032
$321$	0	0
$322$	−41629.8	−0.0223751
$323$	−1.52774e6	−0.814785
$324$	0	0
$325$	−3.19997e6	−1.68050
$326$	−769441.	−0.400988
$327$	0	0
$328$	256673.	0.131733
$329$	2.32858e6	1.18605
$330$	0	0
$331$	3.17478e6	1.59273	0.796367	−	0.604814i	$-0.206753\pi$
$331$	3.17478e6	1.59273	0.796367	+	0.604814i	$0.206753\pi$
$332$	−762571.	−0.379695
$333$	0	0
$334$	−94730.9	−0.0464649
$335$	−230648.	−0.112289
$336$	0	0
$337$	288974.	0.138607	0.0693033	−	0.997596i	$-0.477922\pi$
$337$	288974.	0.138607	0.0693033	+	0.997596i	$0.477922\pi$
$338$	2.75578e6	1.31206
$339$	0	0
$340$	96739.3	0.0453843
$341$	226173.	0.105331
$342$	0	0
$343$	527306.	0.242007
$344$	−127996.	−0.0583177
$345$	0	0
$346$	−2.85127e6	−1.28041
$347$	−3.73614e6	−1.66571	−0.832856	−	0.553490i	$-0.813296\pi$
$347$	−3.73614e6	−1.66571	−0.832856	+	0.553490i	$0.813296\pi$
$348$	0	0
$349$	−3.49319e6	−1.53518	−0.767588	−	0.640943i	$-0.778543\pi$
$349$	−3.49319e6	−1.53518	−0.767588	+	0.640943i	$0.778543\pi$
$350$	2.17451e6	0.948838
$351$	0	0
$352$	241000.	0.103672
$353$	2.86521e6	1.22383	0.611914	−	0.790925i	$-0.290400\pi$
$353$	2.86521e6	1.22383	0.611914	+	0.790925i	$0.290400\pi$
$354$	0	0
$355$	−195478.	−0.0823240
$356$	737144.	0.308267
$357$	0	0
$358$	2.36676e6	0.975993
$359$	4.21678e6	1.72681	0.863406	−	0.504510i	$-0.168327\pi$
$359$	4.21678e6	1.72681	0.863406	+	0.504510i	$0.168327\pi$
$360$	0	0
$361$	−1.37399e6	−0.554901
$362$	2.59796e6	1.04198
$363$	0	0
$364$	−2.88190e6	−1.14006
$365$	−329107.	−0.129302
$366$	0	0
$367$	−2.80326e6	−1.08642	−0.543211	−	0.839596i	$-0.682792\pi$
$367$	−2.80326e6	−1.08642	−0.543211	+	0.839596i	$0.682792\pi$
$368$	15230.9	0.00586282
$369$	0	0
$370$	−129596.	−0.0492138
$371$	−5.57460e6	−2.10271
$372$	0	0
$373$	−3.03545e6	−1.12967	−0.564834	−	0.825204i	$-0.691060\pi$
$373$	−3.03545e6	−1.12967	−0.564834	+	0.825204i	$0.691060\pi$
$374$	−1.36998e6	−0.506448
$375$	0	0
$376$	−851949.	−0.310774
$377$	−4.31794e6	−1.56467
$378$	0	0
$379$	−1.35861e6	−0.485845	−0.242922	−	0.970046i	$-0.578106\pi$
$379$	−1.35861e6	−0.485845	−0.242922	+	0.970046i	$0.578106\pi$
$380$	−69787.6	−0.0247924
$381$	0	0
$382$	−2.65029e6	−0.929256
$383$	4.81834e6	1.67842	0.839209	−	0.543809i	$-0.183018\pi$
$383$	4.81834e6	1.67842	0.839209	+	0.543809i	$0.183018\pi$
$384$	0	0
$385$	171049.	0.0588124
$386$	3.77945e6	1.29110
$387$	0	0
$388$	868266.	0.292802
$389$	2.84397e6	0.952909	0.476455	−	0.879199i	$-0.341922\pi$
$389$	2.84397e6	0.952909	0.476455	+	0.879199i	$0.341922\pi$
$390$	0	0
$391$	−86581.2	−0.0286406
$392$	882725.	0.290142
$393$	0	0
$394$	1.77569e6	0.576270
$395$	−3382.58	−0.00109083
$396$	0	0
$397$	−4.23047e6	−1.34714	−0.673569	−	0.739124i	$-0.735240\pi$
$397$	−4.23047e6	−1.34714	−0.673569	+	0.739124i	$0.735240\pi$
$398$	−98491.8	−0.0311668
$399$	0	0
$400$	−795581.	−0.248619
$401$	−938762.	−0.291538	−0.145769	−	0.989319i	$-0.546566\pi$
$401$	−938762.	−0.291538	−0.145769	+	0.989319i	$0.546566\pi$
$402$	0	0
$403$	989521.	0.303503
$404$	−1.32236e6	−0.403086
$405$	0	0
$406$	2.93422e6	0.883440
$407$	1.83528e6	0.549182
$408$	0	0
$409$	1.18216e6	0.349437	0.174718	−	0.984618i	$-0.444098\pi$
$409$	1.18216e6	0.349437	0.174718	+	0.984618i	$0.444098\pi$
$410$	−66651.0	−0.0195815
$411$	0	0
$412$	1.34589e6	0.390631
$413$	−6.20416e6	−1.78981
$414$	0	0
$415$	198019.	0.0564399
$416$	1.05439e6	0.298723
$417$	0	0
$418$	988301.	0.276661
$419$	−5.55949e6	−1.54703	−0.773517	−	0.633775i	$-0.781505\pi$
$419$	−5.55949e6	−1.54703	−0.773517	+	0.633775i	$0.781505\pi$
$420$	0	0
$421$	−539989.	−0.148484	−0.0742420	−	0.997240i	$-0.523654\pi$
$421$	−539989.	−0.148484	−0.0742420	+	0.997240i	$0.523654\pi$
$422$	−1.55920e6	−0.426206
$423$	0	0
$424$	2.03956e6	0.550961
$425$	4.52253e6	1.21453
$426$	0	0
$427$	−8.01279e6	−2.12674
$428$	1.73180e6	0.456971
$429$	0	0
$430$	33237.1	0.00866865
$431$	−4.82032e6	−1.24992	−0.624961	−	0.780656i	$-0.714885\pi$
$431$	−4.82032e6	−1.24992	−0.624961	+	0.780656i	$0.714885\pi$
$432$	0	0
$433$	684866.	0.175544	0.0877720	−	0.996141i	$-0.472025\pi$
$433$	684866.	0.175544	0.0877720	+	0.996141i	$0.472025\pi$
$434$	−672421.	−0.171363
$435$	0	0
$436$	−1.22001e6	−0.307359
$437$	62459.5	0.0156457
$438$	0	0
$439$	3.94888e6	0.977940	0.488970	−	0.872301i	$-0.337373\pi$
$439$	3.94888e6	0.977940	0.488970	+	0.872301i	$0.337373\pi$
$440$	−62581.0	−0.0154103
$441$	0	0
$442$	−5.99375e6	−1.45930
$443$	−3.73154e6	−0.903397	−0.451699	−	0.892171i	$-0.649182\pi$
$443$	−3.73154e6	−0.903397	−0.451699	+	0.892171i	$0.649182\pi$
$444$	0	0
$445$	−191416.	−0.0458224
$446$	−1.87085e6	−0.445349
$447$	0	0
$448$	−716502.	−0.168664
$449$	5.80405e6	1.35867	0.679337	−	0.733826i	$-0.262267\pi$
$449$	5.80405e6	1.35867	0.679337	+	0.733826i	$0.262267\pi$
$450$	0	0
$451$	943882.	0.218513
$452$	−1.81431e6	−0.417700
$453$	0	0
$454$	3.59256e6	0.818021
$455$	748351.	0.169464
$456$	0	0
$457$	1.86959e6	0.418750	0.209375	−	0.977835i	$-0.432857\pi$
$457$	1.86959e6	0.418750	0.209375	+	0.977835i	$0.432857\pi$
$458$	−2.52111e6	−0.561600
$459$	0	0
$460$	−3955.05	−0.000871481 0
$461$	−3.30270e6	−0.723797	−0.361898	−	0.932218i	$-0.617871\pi$
$461$	−3.30270e6	−0.723797	−0.361898	+	0.932218i	$0.617871\pi$
$462$	0	0
$463$	1.42250e6	0.308389	0.154195	−	0.988041i	$-0.450722\pi$
$463$	1.42250e6	0.308389	0.154195	+	0.988041i	$0.450722\pi$
$464$	−1.07353e6	−0.231483
$465$	0	0
$466$	1.89569e6	0.404393
$467$	6.32630e6	1.34232	0.671162	−	0.741311i	$-0.265795\pi$
$467$	6.32630e6	1.34232	0.671162	+	0.741311i	$0.265795\pi$
$468$	0	0
$469$	−9.71094e6	−2.03859
$470$	221228.	0.0461950
$471$	0	0
$472$	2.26989e6	0.468975
$473$	−470689.	−0.0967344
$474$	0	0
$475$	−3.26255e6	−0.663472
$476$	4.07300e6	0.823943
$477$	0	0
$478$	−493673.	−0.0988257
$479$	7.24181e6	1.44214	0.721072	−	0.692860i	$-0.243650\pi$
$479$	7.24181e6	1.44214	0.721072	+	0.692860i	$0.243650\pi$
$480$	0	0
$481$	8.02948e6	1.58243
$482$	−7.20092e6	−1.41179
$483$	0	0
$484$	−1.69057e6	−0.328035
$485$	−225465.	−0.0435235
$486$	0	0
$487$	4.99256e6	0.953896	0.476948	−	0.878931i	$-0.341743\pi$
$487$	4.99256e6	0.953896	0.476948	+	0.878931i	$0.341743\pi$
$488$	2.93161e6	0.557259
$489$	0	0
$490$	−229219.	−0.0431282
$491$	−4.15456e6	−0.777716	−0.388858	−	0.921298i	$-0.627130\pi$
$491$	−4.15456e6	−0.777716	−0.388858	+	0.921298i	$0.627130\pi$
$492$	0	0
$493$	6.10255e6	1.13082
$494$	4.32388e6	0.797180
$495$	0	0
$496$	246016.	0.0449013
$497$	−8.23018e6	−1.49458
$498$	0	0
$499$	8.96959e6	1.61258	0.806290	−	0.591520i	$-0.201472\pi$
$499$	8.96959e6	1.61258	0.806290	+	0.591520i	$0.201472\pi$
$500$	414328.	0.0741173
$501$	0	0
$502$	−3.54233e6	−0.627378
$503$	−3.70803e6	−0.653467	−0.326734	−	0.945116i	$-0.605948\pi$
$503$	−3.70803e6	−0.653467	−0.326734	+	0.945116i	$0.605948\pi$
$504$	0	0
$505$	343381.	0.0599167
$506$	56009.7	0.00972494
$507$	0	0
$508$	2.03318e6	0.349555
$509$	−5.11101e6	−0.874405	−0.437202	−	0.899363i	$-0.644031\pi$
$509$	−5.11101e6	−0.874405	−0.437202	+	0.899363i	$0.644031\pi$
$510$	0	0
$511$	−1.38564e7	−2.34746
$512$	262144.	0.0441942
$513$	0	0
$514$	−3.14609e6	−0.525247
$515$	−349491.	−0.0580655
$516$	0	0
$517$	−3.13293e6	−0.515495
$518$	−5.45636e6	−0.893468
$519$	0	0
$520$	−273796.	−0.0444037
$521$	−6.62277e6	−1.06892	−0.534460	−	0.845194i	$-0.679485\pi$
$521$	−6.62277e6	−1.06892	−0.534460	+	0.845194i	$0.679485\pi$
$522$	0	0
$523$	−5.90383e6	−0.943799	−0.471899	−	0.881652i	$-0.656431\pi$
$523$	−5.90383e6	−0.943799	−0.471899	+	0.881652i	$0.656431\pi$
$524$	−151517.	−0.0241064
$525$	0	0
$526$	−5.92580e6	−0.933863
$527$	−1.39849e6	−0.219348
$528$	0	0
$529$	−6.43280e6	−0.999450
$530$	−529617.	−0.0818977
$531$	0	0
$532$	−2.93826e6	−0.450102
$533$	4.12955e6	0.629629
$534$	0	0
$535$	−449701.	−0.0679265
$536$	3.55291e6	0.534160
$537$	0	0
$538$	4.66998e6	0.695600
$539$	3.24610e6	0.481272
$540$	0	0
$541$	−4.62041e6	−0.678715	−0.339357	−	0.940658i	$-0.610210\pi$
$541$	−4.62041e6	−0.678715	−0.339357	+	0.940658i	$0.610210\pi$
$542$	4.80442e6	0.702494
$543$	0	0
$544$	−1.49017e6	−0.215894
$545$	316802.	0.0456874
$546$	0	0
$547$	7.61421e6	1.08807	0.544035	−	0.839063i	$-0.316896\pi$
$547$	7.61421e6	1.08807	0.544035	+	0.839063i	$0.316896\pi$
$548$	−1.06866e6	−0.152016
$549$	0	0
$550$	−2.92564e6	−0.412396
$551$	−4.40237e6	−0.617743
$552$	0	0
$553$	−142416.	−0.0198037
$554$	−7.39370e6	−1.02350
$555$	0	0
$556$	4.23474e6	0.580952
$557$	8.83369e6	1.20644	0.603218	−	0.797577i	$-0.293885\pi$
$557$	8.83369e6	1.20644	0.603218	+	0.797577i	$0.293885\pi$
$558$	0	0
$559$	−2.05930e6	−0.278733
$560$	186056.	0.0250711
$561$	0	0
$562$	−3.92843e6	−0.524660
$563$	−9.26342e6	−1.23169	−0.615843	−	0.787869i	$-0.711185\pi$
$563$	−9.26342e6	−1.23169	−0.615843	+	0.787869i	$0.711185\pi$
$564$	0	0
$565$	471125.	0.0620891
$566$	−304167.	−0.0399091
$567$	0	0
$568$	3.01115e6	0.391616
$569$	−1.18171e6	−0.153013	−0.0765067	−	0.997069i	$-0.524377\pi$
$569$	−1.18171e6	−0.153013	−0.0765067	+	0.997069i	$0.524377\pi$
$570$	0	0
$571$	−1.38342e7	−1.77568	−0.887840	−	0.460151i	$-0.847795\pi$
$571$	−1.38342e7	−1.77568	−0.887840	+	0.460151i	$0.847795\pi$
$572$	3.87738e6	0.495506
$573$	0	0
$574$	−2.80620e6	−0.355499
$575$	−184897.	−0.0233218
$576$	0	0
$577$	−7.78469e6	−0.973423	−0.486712	−	0.873563i	$-0.661804\pi$
$577$	−7.78469e6	−0.973423	−0.486712	+	0.873563i	$0.661804\pi$
$578$	2.79156e6	0.347558
$579$	0	0
$580$	278766.	0.0344089
$581$	8.33716e6	1.02466
$582$	0	0
$583$	7.50020e6	0.913906
$584$	5.06958e6	0.615092
$585$	0	0
$586$	1.06414e7	1.28014
$587$	−691004.	−0.0827723	−0.0413862	−	0.999143i	$-0.513177\pi$
$587$	−691004.	−0.0827723	−0.0413862	+	0.999143i	$0.513177\pi$
$588$	0	0
$589$	1.00887e6	0.119825
$590$	−589428.	−0.0697109
$591$	0	0
$592$	1.99630e6	0.234111
$593$	−1.66505e7	−1.94442	−0.972209	−	0.234115i	$-0.924781\pi$
$593$	−1.66505e7	−1.94442	−0.972209	+	0.234115i	$0.924781\pi$
$594$	0	0
$595$	−1.05765e6	−0.122475
$596$	−4.35795e6	−0.502535
$597$	0	0
$598$	245046.	0.0280218
$599$	−5.35814e6	−0.610165	−0.305082	−	0.952326i	$-0.598684\pi$
$599$	−5.35814e6	−0.610165	−0.305082	+	0.952326i	$0.598684\pi$
$600$	0	0
$601$	326874.	0.0369143	0.0184571	−	0.999830i	$-0.494125\pi$
$601$	326874.	0.0369143	0.0184571	+	0.999830i	$0.494125\pi$
$602$	1.39938e6	0.157378
$603$	0	0
$604$	3.57280e6	0.398489
$605$	438995.	0.0487608
$606$	0	0
$607$	1.72684e7	1.90230	0.951152	−	0.308724i	$-0.0999018\pi$
$607$	1.72684e7	1.90230	0.951152	+	0.308724i	$0.0999018\pi$
$608$	1.07501e6	0.117938
$609$	0	0
$610$	−761258.	−0.0828338
$611$	−1.37068e7	−1.48536
$612$	0	0
$613$	−8.78767e6	−0.944545	−0.472273	−	0.881453i	$-0.656566\pi$
$613$	−8.78767e6	−0.944545	−0.472273	+	0.881453i	$0.656566\pi$
$614$	3.80448e6	0.407263
$615$	0	0
$616$	−2.63484e6	−0.279771
$617$	−3.55288e6	−0.375723	−0.187861	−	0.982196i	$-0.560156\pi$
$617$	−3.55288e6	−0.375723	−0.187861	+	0.982196i	$0.560156\pi$
$618$	0	0
$619$	1.34718e7	1.41319	0.706595	−	0.707619i	$-0.250231\pi$
$619$	1.34718e7	1.41319	0.706595	+	0.707619i	$0.250231\pi$
$620$	−63883.6	−0.00667437
$621$	0	0
$622$	−2.79944e6	−0.290132
$623$	−8.05917e6	−0.831898
$624$	0	0
$625$	9.60409e6	0.983459
$626$	1.21349e7	1.23766
$627$	0	0
$628$	−127540.	−0.0129046
$629$	−1.13481e7	−1.14366
$630$	0	0
$631$	−1.19893e6	−0.119872	−0.0599362	−	0.998202i	$-0.519090\pi$
$631$	−1.19893e6	−0.119872	−0.0599362	+	0.998202i	$0.519090\pi$
$632$	52105.3	0.00518907
$633$	0	0
$634$	−8.17844e6	−0.808067
$635$	−527960.	−0.0519597
$636$	0	0
$637$	1.42019e7	1.38675
$638$	−3.94777e6	−0.383972
$639$	0	0
$640$	−68071.6	−0.00656925
$641$	1.61440e6	0.155191	0.0775953	−	0.996985i	$-0.475276\pi$
$641$	1.61440e6	0.155191	0.0775953	+	0.996985i	$0.475276\pi$
$642$	0	0
$643$	5.51394e6	0.525938	0.262969	−	0.964804i	$-0.415298\pi$
$643$	5.51394e6	0.525938	0.262969	+	0.964804i	$0.415298\pi$
$644$	−166519.	−0.0158216
$645$	0	0
$646$	−6.11096e6	−0.576140
$647$	1.67629e7	1.57430	0.787150	−	0.616761i	$-0.211556\pi$
$647$	1.67629e7	1.57430	0.787150	+	0.616761i	$0.211556\pi$
$648$	0	0
$649$	8.34722e6	0.777912
$650$	−1.27999e7	−1.18829
$651$	0	0
$652$	−3.07776e6	−0.283541
$653$	7.92672e6	0.727462	0.363731	−	0.931504i	$-0.381503\pi$
$653$	7.92672e6	0.727462	0.363731	+	0.931504i	$0.381503\pi$
$654$	0	0
$655$	39344.7	0.00358330
$656$	1.02669e6	0.0931496
$657$	0	0
$658$	9.31432e6	0.838661
$659$	1.79313e7	1.60841	0.804206	−	0.594351i	$-0.202591\pi$
$659$	1.79313e7	1.60841	0.804206	+	0.594351i	$0.202591\pi$
$660$	0	0
$661$	1.63759e6	0.145781	0.0728904	−	0.997340i	$-0.476778\pi$
$661$	1.63759e6	0.145781	0.0728904	+	0.997340i	$0.476778\pi$
$662$	1.26991e7	1.12623
$663$	0	0
$664$	−3.05028e6	−0.268485
$665$	762984.	0.0669055
$666$	0	0
$667$	−249494.	−0.0217143
$668$	−378924.	−0.0328557
$669$	0	0
$670$	−922592.	−0.0794004
$671$	1.07806e7	0.924351
$672$	0	0
$673$	6.68369e6	0.568825	0.284413	−	0.958702i	$-0.408201\pi$
$673$	6.68369e6	0.568825	0.284413	+	0.958702i	$0.408201\pi$
$674$	1.15590e6	0.0980097
$675$	0	0
$676$	1.10231e7	0.927765
$677$	−435721.	−0.0365373	−0.0182687	−	0.999833i	$-0.505815\pi$
$677$	−435721.	−0.0365373	−0.0182687	+	0.999833i	$0.505815\pi$
$678$	0	0
$679$	−9.49271e6	−0.790162
$680$	386957.	0.0320915
$681$	0	0
$682$	904691.	0.0744799
$683$	7.88577e6	0.646834	0.323417	−	0.946257i	$-0.395168\pi$
$683$	7.88577e6	0.646834	0.323417	+	0.946257i	$0.395168\pi$
$684$	0	0
$685$	277501.	0.0225964
$686$	2.10922e6	0.171125
$687$	0	0
$688$	−511984.	−0.0412369
$689$	3.28139e7	2.63336
$690$	0	0
$691$	−2.42470e6	−0.193180	−0.0965900	−	0.995324i	$-0.530794\pi$
$691$	−2.42470e6	−0.193180	−0.0965900	+	0.995324i	$0.530794\pi$
$692$	−1.14051e7	−0.905385
$693$	0	0
$694$	−1.49446e7	−1.17784
$695$	−1.09965e6	−0.0863557
$696$	0	0
$697$	−5.83630e6	−0.455047
$698$	−1.39727e7	−1.08553
$699$	0	0
$700$	8.69805e6	0.670930
$701$	1.34755e7	1.03574	0.517871	−	0.855459i	$-0.326725\pi$
$701$	1.34755e7	1.03574	0.517871	+	0.855459i	$0.326725\pi$
$702$	0	0
$703$	8.18649e6	0.624754
$704$	964000.	0.0733070
$705$	0	0
$706$	1.14608e7	0.865376
$707$	1.44573e7	1.08778
$708$	0	0
$709$	1.62749e7	1.21591	0.607956	−	0.793971i	$-0.291990\pi$
$709$	1.62749e7	1.21591	0.607956	+	0.793971i	$0.291990\pi$
$710$	−781911.	−0.0582119
$711$	0	0
$712$	2.94858e6	0.217978
$713$	57175.5	0.00421198
$714$	0	0
$715$	−1.00685e6	−0.0736545
$716$	9.46704e6	0.690131
$717$	0	0
$718$	1.68671e7	1.22104
$719$	5.81254e6	0.419319	0.209659	−	0.977775i	$-0.432765\pi$
$719$	5.81254e6	0.419319	0.209659	+	0.977775i	$0.432765\pi$
$720$	0	0
$721$	−1.47146e7	−1.05417
$722$	−5.49596e6	−0.392374
$723$	0	0
$724$	1.03918e7	0.736794
$725$	1.30322e7	0.920818
$726$	0	0
$727$	−6.21696e6	−0.436256	−0.218128	−	0.975920i	$-0.569995\pi$
$727$	−6.21696e6	−0.436256	−0.218128	+	0.975920i	$0.569995\pi$
$728$	−1.15276e7	−0.806141
$729$	0	0
$730$	−1.31643e6	−0.0914304
$731$	2.91041e6	0.201447
$732$	0	0
$733$	1.81331e7	1.24656	0.623279	−	0.781999i	$-0.285800\pi$
$733$	1.81331e7	1.24656	0.623279	+	0.781999i	$0.285800\pi$
$734$	−1.12131e7	−0.768217
$735$	0	0
$736$	60923.7	0.00414564
$737$	1.30653e7	0.886037
$738$	0	0
$739$	−2.73725e7	−1.84375	−0.921876	−	0.387485i	$-0.873344\pi$
$739$	−2.73725e7	−1.84375	−0.921876	+	0.387485i	$0.873344\pi$
$740$	−518384.	−0.0347994
$741$	0	0
$742$	−2.22984e7	−1.48684
$743$	2.90995e7	1.93381	0.966904	−	0.255142i	$-0.0821221\pi$
$743$	2.90995e7	1.93381	0.966904	+	0.255142i	$0.0821221\pi$
$744$	0	0
$745$	1.13164e6	0.0746994
$746$	−1.21418e7	−0.798796
$747$	0	0
$748$	−5.47992e6	−0.358113
$749$	−1.89337e7	−1.23319
$750$	0	0
$751$	4.40114e6	0.284751	0.142376	−	0.989813i	$-0.454526\pi$
$751$	4.40114e6	0.284751	0.142376	+	0.989813i	$0.454526\pi$
$752$	−3.40780e6	−0.219750
$753$	0	0
$754$	−1.72717e7	−1.10639
$755$	−927758.	−0.0592335
$756$	0	0
$757$	6.88173e6	0.436474	0.218237	−	0.975896i	$-0.429969\pi$
$757$	6.88173e6	0.436474	0.218237	+	0.975896i	$0.429969\pi$
$758$	−5.43445e6	−0.343544
$759$	0	0
$760$	−279150.	−0.0175309
$761$	−1.81095e7	−1.13356	−0.566781	−	0.823869i	$-0.691811\pi$
$761$	−1.81095e7	−1.13356	−0.566781	+	0.823869i	$0.691811\pi$
$762$	0	0
$763$	1.33383e7	0.829446
$764$	−1.06012e7	−0.657083
$765$	0	0
$766$	1.92733e7	1.18682
$767$	3.65197e7	2.24150
$768$	0	0
$769$	−1.21359e7	−0.740044	−0.370022	−	0.929023i	$-0.620650\pi$
$769$	−1.21359e7	−0.740044	−0.370022	+	0.929023i	$0.620650\pi$
$770$	684196.	0.0415866
$771$	0	0
$772$	1.51178e7	0.912947
$773$	6.85275e6	0.412493	0.206246	−	0.978500i	$-0.433875\pi$
$773$	6.85275e6	0.412493	0.206246	+	0.978500i	$0.433875\pi$
$774$	0	0
$775$	−2.98654e6	−0.178613
$776$	3.47306e6	0.207042
$777$	0	0
$778$	1.13759e7	0.673808
$779$	4.21030e6	0.248582
$780$	0	0
$781$	1.10731e7	0.649593
$782$	−346325.	−0.0202519
$783$	0	0
$784$	3.53090e6	0.205161
$785$	33118.5	0.00191821
$786$	0	0
$787$	−9.20801e6	−0.529943	−0.264971	−	0.964256i	$-0.585363\pi$
$787$	−9.20801e6	−0.529943	−0.264971	+	0.964256i	$0.585363\pi$
$788$	7.10275e6	0.407485
$789$	0	0
$790$	−13530.3	−0.000771330 0
$791$	1.98357e7	1.12722
$792$	0	0
$793$	4.71659e7	2.66345
$794$	−1.69219e7	−0.952570
$795$	0	0
$796$	−393967.	−0.0220383
$797$	1.24590e7	0.694762	0.347381	−	0.937724i	$-0.387071\pi$
$797$	1.24590e7	0.694762	0.347381	+	0.937724i	$0.387071\pi$
$798$	0	0
$799$	1.93718e7	1.07350
$800$	−3.18232e6	−0.175800
$801$	0	0
$802$	−3.75505e6	−0.206148
$803$	1.86427e7	1.02028
$804$	0	0
$805$	43240.4	0.00235180
$806$	3.95809e6	0.214609
$807$	0	0
$808$	−5.28945e6	−0.285025
$809$	−1.39177e7	−0.747645	−0.373823	−	0.927500i	$-0.621953\pi$
$809$	−1.39177e7	−0.747645	−0.373823	+	0.927500i	$0.621953\pi$
$810$	0	0
$811$	2.23695e7	1.19427	0.597137	−	0.802140i	$-0.296305\pi$
$811$	2.23695e7	1.19427	0.597137	+	0.802140i	$0.296305\pi$
$812$	1.17369e7	0.624686
$813$	0	0
$814$	7.34112e6	0.388330
$815$	799211.	0.0421471
$816$	0	0
$817$	−2.09956e6	−0.110046
$818$	4.72865e6	0.247089
$819$	0	0
$820$	−266604.	−0.0138462
$821$	−5.80930e6	−0.300792	−0.150396	−	0.988626i	$-0.548055\pi$
$821$	−5.80930e6	−0.300792	−0.150396	+	0.988626i	$0.548055\pi$
$822$	0	0
$823$	−2.13414e7	−1.09830	−0.549152	−	0.835722i	$-0.685049\pi$
$823$	−2.13414e7	−1.09830	−0.549152	+	0.835722i	$0.685049\pi$
$824$	5.38357e6	0.276218
$825$	0	0
$826$	−2.48166e7	−1.26559
$827$	−2.57909e7	−1.31130	−0.655651	−	0.755064i	$-0.727606\pi$
$827$	−2.57909e7	−1.31130	−0.655651	+	0.755064i	$0.727606\pi$
$828$	0	0
$829$	−2.36243e7	−1.19391	−0.596956	−	0.802274i	$-0.703623\pi$
$829$	−2.36243e7	−1.19391	−0.596956	+	0.802274i	$0.703623\pi$
$830$	792075.	0.0399090
$831$	0	0
$832$	4.21756e6	0.211229
$833$	−2.00716e7	−1.00224
$834$	0	0
$835$	98396.0	0.00488384
$836$	3.95320e6	0.195629
$837$	0	0
$838$	−2.22380e7	−1.09392
$839$	−2.79824e7	−1.37240	−0.686198	−	0.727415i	$-0.740722\pi$
$839$	−2.79824e7	−1.37240	−0.686198	+	0.727415i	$0.740722\pi$
$840$	0	0
$841$	−2.92588e6	−0.142648
$842$	−2.15996e6	−0.104994
$843$	0	0
$844$	−6.23679e6	−0.301373
$845$	−2.86240e6	−0.137908
$846$	0	0
$847$	1.84829e7	0.885243
$848$	8.15823e6	0.389589
$849$	0	0
$850$	1.80901e7	0.858804
$851$	463951.	0.0219608
$852$	0	0
$853$	−2.04470e7	−0.962182	−0.481091	−	0.876671i	$-0.659759\pi$
$853$	−2.04470e7	−0.962182	−0.481091	+	0.876671i	$0.659759\pi$
$854$	−3.20512e7	−1.50383
$855$	0	0
$856$	6.92721e6	0.323127
$857$	−3.08472e7	−1.43471	−0.717355	−	0.696708i	$-0.754648\pi$
$857$	−3.08472e7	−1.43471	−0.717355	+	0.696708i	$0.754648\pi$
$858$	0	0
$859$	−1.90536e7	−0.881038	−0.440519	−	0.897743i	$-0.645206\pi$
$859$	−1.90536e7	−0.881038	−0.440519	+	0.897743i	$0.645206\pi$
$860$	132948.	0.00612966
$861$	0	0
$862$	−1.92813e7	−0.883828
$863$	1.60269e7	0.732525	0.366263	−	0.930512i	$-0.380637\pi$
$863$	1.60269e7	0.732525	0.366263	+	0.930512i	$0.380637\pi$
$864$	0	0
$865$	2.96158e6	0.134581
$866$	2.73946e6	0.124128
$867$	0	0
$868$	−2.68968e6	−0.121172
$869$	191610.	0.00860736
$870$	0	0
$871$	5.71618e7	2.55305
$872$	−4.88002e6	−0.217335
$873$	0	0
$874$	249838.	0.0110632
$875$	−4.52983e6	−0.200015
$876$	0	0
$877$	6.46053e6	0.283641	0.141821	−	0.989892i	$-0.454704\pi$
$877$	6.46053e6	0.283641	0.141821	+	0.989892i	$0.454704\pi$
$878$	1.57955e7	0.691508
$879$	0	0
$880$	−250324.	−0.0108967
$881$	−4.43941e7	−1.92702	−0.963508	−	0.267679i	$-0.913743\pi$
$881$	−4.43941e7	−1.92702	−0.963508	+	0.267679i	$0.913743\pi$
$882$	0	0
$883$	3.31650e7	1.43146	0.715729	−	0.698378i	$-0.246095\pi$
$883$	3.31650e7	1.43146	0.715729	+	0.698378i	$0.246095\pi$
$884$	−2.39750e7	−1.03188
$885$	0	0
$886$	−1.49262e7	−0.638798
$887$	3.31641e7	1.41533	0.707667	−	0.706546i	$-0.249748\pi$
$887$	3.31641e7	1.41533	0.707667	+	0.706546i	$0.249748\pi$
$888$	0	0
$889$	−2.22286e7	−0.943318
$890$	−765664.	−0.0324014
$891$	0	0
$892$	−7.48338e6	−0.314910
$893$	−1.39748e7	−0.586431
$894$	0	0
$895$	−2.45833e6	−0.102585
$896$	−2.86601e6	−0.119264
$897$	0	0
$898$	2.32162e7	0.960728
$899$	−4.02993e6	−0.166302
$900$	0	0
$901$	−4.63759e7	−1.90318
$902$	3.77553e6	0.154512
$903$	0	0
$904$	−7.25723e6	−0.295359
$905$	−2.69848e6	−0.109521
$906$	0	0
$907$	2.70053e7	1.09001	0.545005	−	0.838433i	$-0.316528\pi$
$907$	2.70053e7	1.09001	0.545005	+	0.838433i	$0.316528\pi$
$908$	1.43702e7	0.578428
$909$	0	0
$910$	2.99340e6	0.119829
$911$	−1.82765e7	−0.729621	−0.364811	−	0.931082i	$-0.618866\pi$
$911$	−1.82765e7	−0.729621	−0.364811	+	0.931082i	$0.618866\pi$
$912$	0	0
$913$	−1.12170e7	−0.445349
$914$	7.47835e6	0.296101
$915$	0	0
$916$	−1.00844e7	−0.397112
$917$	1.65653e6	0.0650542
$918$	0	0
$919$	2.05975e7	0.804498	0.402249	−	0.915530i	$-0.368229\pi$
$919$	2.05975e7	0.804498	0.402249	+	0.915530i	$0.368229\pi$
$920$	−15820.2	−0.000616230 0
$921$	0	0
$922$	−1.32108e7	−0.511801
$923$	4.84455e7	1.87176
$924$	0	0
$925$	−2.42343e7	−0.931270
$926$	5.68999e6	0.218064
$927$	0	0
$928$	−4.29412e6	−0.163683
$929$	2.79827e7	1.06378	0.531888	−	0.846815i	$-0.321483\pi$
$929$	2.79827e7	1.06378	0.531888	+	0.846815i	$0.321483\pi$
$930$	0	0
$931$	1.44796e7	0.547499
$932$	7.58277e6	0.285949
$933$	0	0
$934$	2.53052e7	0.949166
$935$	1.42298e6	0.0532317
$936$	0	0
$937$	−1.18863e7	−0.442281	−0.221141	−	0.975242i	$-0.570978\pi$
$937$	−1.18863e7	−0.442281	−0.221141	+	0.975242i	$0.570978\pi$
$938$	−3.88438e7	−1.44150
$939$	0	0
$940$	884911.	0.0326648
$941$	2.99706e7	1.10337	0.551686	−	0.834052i	$-0.313985\pi$
$941$	2.99706e7	1.10337	0.551686	+	0.834052i	$0.313985\pi$
$942$	0	0
$943$	238609.	0.00873792
$944$	9.07957e6	0.331616
$945$	0	0
$946$	−1.88275e6	−0.0684015
$947$	−2.88301e6	−0.104465	−0.0522325	−	0.998635i	$-0.516634\pi$
$947$	−2.88301e6	−0.104465	−0.0522325	+	0.998635i	$0.516634\pi$
$948$	0	0
$949$	8.15631e7	2.93987
$950$	−1.30502e7	−0.469146
$951$	0	0
$952$	1.62920e7	0.582616
$953$	1.21859e7	0.434637	0.217319	−	0.976101i	$-0.430269\pi$
$953$	1.21859e7	0.434637	0.217319	+	0.976101i	$0.430269\pi$
$954$	0	0
$955$	2.75283e6	0.0976722
$956$	−1.97469e6	−0.0698803
$957$	0	0
$958$	2.89672e7	1.01975
$959$	1.16836e7	0.410233
$960$	0	0
$961$	923521.	0.0322581
$962$	3.21179e7	1.11895
$963$	0	0
$964$	−2.88037e7	−0.998287
$965$	−3.92568e6	−0.135705
$966$	0	0
$967$	4.36736e7	1.50194	0.750970	−	0.660337i	$-0.229586\pi$
$967$	4.36736e7	1.50194	0.750970	+	0.660337i	$0.229586\pi$
$968$	−6.76228e6	−0.231956
$969$	0	0
$970$	−901859.	−0.0307758
$971$	−2.72990e6	−0.0929178	−0.0464589	−	0.998920i	$-0.514794\pi$
$971$	−2.72990e6	−0.0929178	−0.0464589	+	0.998920i	$0.514794\pi$
$972$	0	0
$973$	−4.62983e7	−1.56777
$974$	1.99702e7	0.674506
$975$	0	0
$976$	1.17264e7	0.394041
$977$	5.61565e6	0.188219	0.0941096	−	0.995562i	$-0.470000\pi$
$977$	5.61565e6	0.188219	0.0941096	+	0.995562i	$0.470000\pi$
$978$	0	0
$979$	1.08430e7	0.361570
$980$	−916877.	−0.0304962
$981$	0	0
$982$	−1.66182e7	−0.549928
$983$	−7.21208e6	−0.238055	−0.119027	−	0.992891i	$-0.537978\pi$
$983$	−7.21208e6	−0.238055	−0.119027	+	0.992891i	$0.537978\pi$
$984$	0	0
$985$	−1.84439e6	−0.0605706
$986$	2.44102e7	0.799612
$987$	0	0
$988$	1.72955e7	0.563692
$989$	−118988.	−0.00386823
$990$	0	0
$991$	−2.05082e7	−0.663351	−0.331675	−	0.943394i	$-0.607614\pi$
$991$	−2.05082e7	−0.663351	−0.331675	+	0.943394i	$0.607614\pi$
$992$	984064.	0.0317500
$993$	0	0
$994$	−3.29207e7	−1.05683
$995$	102302.	0.00327588
$996$	0	0
$997$	−3.56424e7	−1.13561	−0.567805	−	0.823163i	$-0.692207\pi$
$997$	−3.56424e7	−1.13561	−0.567805	+	0.823163i	$0.692207\pi$
$998$	3.58784e7	1.14027
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	558.6.a.k.1.3		4
3.2	odd	2		62.6.a.c.1.1	✓	4
12.11	even	2		496.6.a.d.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
62.6.a.c.1.1	✓	4	3.2	odd	2
496.6.a.d.1.4		4	12.11	even	2
558.6.a.k.1.3		4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 558 = 2 \cdot 3^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	558.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +16.0000 q^{4} -4.15476 q^{5} -174.927 q^{7} +64.0000 q^{8} -16.6190 q^{10} +235.351 q^{11} +1029.68 q^{13} -699.709 q^{14} +256.000 q^{16} -1455.25 q^{17} +1049.81 q^{19} -66.4761 q^{20} +941.406 q^{22} +59.4958 q^{23} -3107.74 q^{25} +4118.72 q^{26} -2798.84 q^{28} -4193.48 q^{29} +961.000 q^{31} +1024.00 q^{32} -5820.99 q^{34} +726.781 q^{35} +7798.04 q^{37} +4199.25 q^{38} -265.905 q^{40} +4010.52 q^{41} -1999.94 q^{43} +3765.62 q^{44} +237.983 q^{46} -13311.7 q^{47} +13792.6 q^{49} -12431.0 q^{50} +16474.9 q^{52} +31868.1 q^{53} -977.829 q^{55} -11195.3 q^{56} -16773.9 q^{58} +35467.1 q^{59} +45806.4 q^{61} +3844.00 q^{62} +4096.00 q^{64} -4278.07 q^{65} +55514.2 q^{67} -23284.0 q^{68} +2907.12 q^{70} +47049.1 q^{71} +79212.2 q^{73} +31192.2 q^{74} +16797.0 q^{76} -41169.4 q^{77} +814.146 q^{79} -1063.62 q^{80} +16042.1 q^{82} -47660.7 q^{83} +6046.20 q^{85} -7999.76 q^{86} +15062.5 q^{88} +46071.5 q^{89} -180119. q^{91} +951.933 q^{92} -53246.8 q^{94} -4361.72 q^{95} +54266.6 q^{97} +55170.3 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{2} + 64 q^{4} - 2 q^{5} + 146 q^{7} + 256 q^{8} - 8 q^{10} + 270 q^{11} + 1038 q^{13} + 584 q^{14} + 1024 q^{16} - 2020 q^{17} + 6214 q^{19} - 32 q^{20} + 1080 q^{22} - 8292 q^{23} + 9974 q^{25}+ \cdots + 86520 q^{98}+O(q^{100}) \) 4 * q + 16 * q^2 + 64 * q^4 - 2 * q^5 + 146 * q^7 + 256 * q^8 - 8 * q^10 + 270 * q^11 + 1038 * q^13 + 584 * q^14 + 1024 * q^16 - 2020 * q^17 + 6214 * q^19 - 32 * q^20 + 1080 * q^22 - 8292 * q^23 + 9974 * q^25 + 4152 * q^26 + 2336 * q^28 - 1894 * q^29 + 3844 * q^31 + 4096 * q^32 - 8080 * q^34 + 9146 * q^35 - 5470 * q^37 + 24856 * q^38 - 128 * q^40 + 16878 * q^41 + 1530 * q^43 + 4320 * q^44 - 33168 * q^46 + 25460 * q^47 + 21630 * q^49 + 39896 * q^50 + 16608 * q^52 + 7790 * q^53 + 5760 * q^55 + 9344 * q^56 - 7576 * q^58 + 35802 * q^59 + 48366 * q^61 + 15376 * q^62 + 16384 * q^64 - 12096 * q^65 + 54696 * q^67 - 32320 * q^68 + 36584 * q^70 + 35342 * q^71 + 59336 * q^73 - 21880 * q^74 + 99424 * q^76 + 86976 * q^77 - 3652 * q^79 - 512 * q^80 + 67512 * q^82 + 80074 * q^83 + 81860 * q^85 + 6120 * q^86 + 17280 * q^88 + 12060 * q^89 - 74952 * q^91 - 132672 * q^92 + 101840 * q^94 + 286138 * q^95 + 232670 * q^97 + 86520 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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