LMFDB - Embedded newform 912.6.a.r.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [912,6,Mod(1,912)] 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("912.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(912, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	912.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,36,0,-8] 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:	$146.270043669$
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} - x^{3} - 90x^{2} + 118x + 1412 $ x^4 - x^3 - 90x^2 + 118x + 1412
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 57)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$5.82184$ of defining polynomial
Character	$\chi$	$=$	912.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} +23.6174 q^{5} +250.612 q^{7} +81.0000 q^{9} -414.956 q^{11} -920.892 q^{13} +212.557 q^{15} -238.401 q^{17} +361.000 q^{19} +2255.50 q^{21} -1138.97 q^{23} -2567.22 q^{25} +729.000 q^{27} +7031.41 q^{29} -495.673 q^{31} -3734.61 q^{33} +5918.80 q^{35} +2981.99 q^{37} -8288.03 q^{39} +9339.27 q^{41} +11537.4 q^{43} +1913.01 q^{45} +19955.6 q^{47} +45999.2 q^{49} -2145.61 q^{51} -20089.6 q^{53} -9800.20 q^{55} +3249.00 q^{57} +48844.4 q^{59} +24559.4 q^{61} +20299.5 q^{63} -21749.1 q^{65} +59482.1 q^{67} -10250.7 q^{69} +4635.76 q^{71} -26670.8 q^{73} -23105.0 q^{75} -103993. q^{77} +68240.6 q^{79} +6561.00 q^{81} -2260.79 q^{83} -5630.41 q^{85} +63282.7 q^{87} +749.444 q^{89} -230786. q^{91} -4461.06 q^{93} +8525.89 q^{95} +88106.3 q^{97} -33611.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 36 q^{3} - 8 q^{5} + 142 q^{7} + 324 q^{9} + 714 q^{11} - 74 q^{13} - 72 q^{15} - 3690 q^{17} + 1444 q^{19} + 1278 q^{21} - 862 q^{23} + 3282 q^{25} + 2916 q^{27} + 992 q^{29} + 7382 q^{31} + 6426 q^{33}+ \cdots + 57834 q^{99}+O(q^{100}) $ 4 * q + 36 * q^3 - 8 * q^5 + 142 * q^7 + 324 * q^9 + 714 * q^11 - 74 * q^13 - 72 * q^15 - 3690 * q^17 + 1444 * q^19 + 1278 * q^21 - 862 * q^23 + 3282 * q^25 + 2916 * q^27 + 992 * q^29 + 7382 * q^31 + 6426 * q^33 + 10242 * q^35 + 19114 * q^37 - 666 * q^39 - 5792 * q^41 + 18634 * q^43 - 648 * q^45 + 35424 * q^47 + 15910 * q^49 - 33210 * q^51 - 20256 * q^53 - 23538 * q^55 + 12996 * q^57 + 76944 * q^59 + 11562 * q^61 + 11502 * q^63 - 157464 * q^65 + 110044 * q^67 - 7758 * q^69 + 71184 * q^71 - 138830 * q^73 + 29538 * q^75 - 76122 * q^77 + 59470 * q^79 + 26244 * q^81 + 191600 * q^83 + 23682 * q^85 + 8928 * q^87 + 158920 * q^89 - 267196 * q^91 + 66438 * q^93 - 2888 * q^95 + 47460 * q^97 + 57834 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	23.6174	0.422482	0.211241	−	0.977434i	$-0.432250\pi$
$5$	23.6174	0.422482	0.211241	+	0.977434i	$0.432250\pi$
$6$	0	0
$7$	250.612	1.93311	0.966554	−	0.256464i	$-0.0825575\pi$
$7$	250.612	1.93311	0.966554	+	0.256464i	$0.0825575\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−414.956	−1.03400	−0.517000	−	0.855985i	$-0.672951\pi$
$11$	−414.956	−1.03400	−0.517000	+	0.855985i	$0.672951\pi$
$12$	0	0
$13$	−920.892	−1.51130	−0.755650	−	0.654976i	$-0.772679\pi$
$13$	−920.892	−1.51130	−0.755650	+	0.654976i	$0.772679\pi$
$14$	0	0
$15$	212.557	0.243920
$16$	0	0
$17$	−238.401	−0.200071	−0.100036	−	0.994984i	$-0.531896\pi$
$17$	−238.401	−0.200071	−0.100036	+	0.994984i	$0.531896\pi$
$18$	0	0
$19$	361.000	0.229416
$19$	361.000	0.229416
$20$	0	0
$21$	2255.50	1.11608
$22$	0	0
$23$	−1138.97	−0.448945	−0.224472	−	0.974480i	$-0.572066\pi$
$23$	−1138.97	−0.448945	−0.224472	+	0.974480i	$0.572066\pi$
$24$	0	0
$25$	−2567.22	−0.821509
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	7031.41	1.55256	0.776278	−	0.630391i	$-0.217105\pi$
$29$	7031.41	1.55256	0.776278	+	0.630391i	$0.217105\pi$
$30$	0	0
$31$	−495.673	−0.0926384	−0.0463192	−	0.998927i	$-0.514749\pi$
$31$	−495.673	−0.0926384	−0.0463192	+	0.998927i	$0.514749\pi$
$32$	0	0
$33$	−3734.61	−0.596980
$34$	0	0
$35$	5918.80	0.816702
$36$	0	0
$37$	2981.99	0.358098	0.179049	−	0.983840i	$-0.442698\pi$
$37$	2981.99	0.358098	0.179049	+	0.983840i	$0.442698\pi$
$38$	0	0
$39$	−8288.03	−0.872549
$40$	0	0
$41$	9339.27	0.867668	0.433834	−	0.900993i	$-0.357160\pi$
$41$	9339.27	0.867668	0.433834	+	0.900993i	$0.357160\pi$
$42$	0	0
$43$	11537.4	0.951563	0.475781	−	0.879564i	$-0.342165\pi$
$43$	11537.4	0.951563	0.475781	+	0.879564i	$0.342165\pi$
$44$	0	0
$45$	1913.01	0.140827
$46$	0	0
$47$	19955.6	1.31771	0.658857	−	0.752268i	$-0.271040\pi$
$47$	19955.6	1.31771	0.658857	+	0.752268i	$0.271040\pi$
$48$	0	0
$49$	45999.2	2.73690
$50$	0	0
$51$	−2145.61	−0.115511
$52$	0	0
$53$	−20089.6	−0.982385	−0.491193	−	0.871051i	$-0.663439\pi$
$53$	−20089.6	−0.982385	−0.491193	+	0.871051i	$0.663439\pi$
$54$	0	0
$55$	−9800.20	−0.436846
$56$	0	0
$57$	3249.00	0.132453
$58$	0	0
$59$	48844.4	1.82677	0.913387	−	0.407093i	$-0.133457\pi$
$59$	48844.4	1.82677	0.913387	+	0.407093i	$0.133457\pi$
$60$	0	0
$61$	24559.4	0.845069	0.422535	−	0.906347i	$-0.361140\pi$
$61$	24559.4	0.845069	0.422535	+	0.906347i	$0.361140\pi$
$62$	0	0
$63$	20299.5	0.644369
$64$	0	0
$65$	−21749.1	−0.638496
$66$	0	0
$67$	59482.1	1.61882	0.809411	−	0.587243i	$-0.199787\pi$
$67$	59482.1	1.61882	0.809411	+	0.587243i	$0.199787\pi$
$68$	0	0
$69$	−10250.7	−0.259199
$70$	0	0
$71$	4635.76	0.109138	0.0545689	−	0.998510i	$-0.482622\pi$
$71$	4635.76	0.109138	0.0545689	+	0.998510i	$0.482622\pi$
$72$	0	0
$73$	−26670.8	−0.585772	−0.292886	−	0.956147i	$-0.594616\pi$
$73$	−26670.8	−0.585772	−0.292886	+	0.956147i	$0.594616\pi$
$74$	0	0
$75$	−23105.0	−0.474299
$76$	0	0
$77$	−103993.	−1.99883
$78$	0	0
$79$	68240.6	1.23020	0.615100	−	0.788449i	$-0.289116\pi$
$79$	68240.6	1.23020	0.615100	+	0.788449i	$0.289116\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−2260.79	−0.0360217	−0.0180109	−	0.999838i	$-0.505733\pi$
$83$	−2260.79	−0.0360217	−0.0180109	+	0.999838i	$0.505733\pi$
$84$	0	0
$85$	−5630.41	−0.0845265
$86$	0	0
$87$	63282.7	0.896368
$88$	0	0
$89$	749.444	0.0100291	0.00501457	−	0.999987i	$-0.498404\pi$
$89$	749.444	0.0100291	0.00501457	+	0.999987i	$0.498404\pi$
$90$	0	0
$91$	−230786.	−2.92150
$92$	0	0
$93$	−4461.06	−0.0534848
$94$	0	0
$95$	8525.89	0.0969239
$96$	0	0
$97$	88106.3	0.950775	0.475387	−	0.879777i	$-0.342308\pi$
$97$	88106.3	0.950775	0.475387	+	0.879777i	$0.342308\pi$
$98$	0	0
$99$	−33611.5	−0.344667
$100$	0	0
$101$	−125404.	−1.22323	−0.611613	−	0.791157i	$-0.709479\pi$
$101$	−125404.	−1.22323	−0.611613	+	0.791157i	$0.709479\pi$
$102$	0	0
$103$	110218.	1.02367	0.511835	−	0.859084i	$-0.328966\pi$
$103$	110218.	1.02367	0.511835	+	0.859084i	$0.328966\pi$
$104$	0	0
$105$	53269.2	0.471523
$106$	0	0
$107$	−160783.	−1.35763	−0.678813	−	0.734311i	$-0.737505\pi$
$107$	−160783.	−1.35763	−0.678813	+	0.734311i	$0.737505\pi$
$108$	0	0
$109$	−111776.	−0.901122	−0.450561	−	0.892746i	$-0.648776\pi$
$109$	−111776.	−0.901122	−0.450561	+	0.892746i	$0.648776\pi$
$110$	0	0
$111$	26837.9	0.206748
$112$	0	0
$113$	116128.	0.855544	0.427772	−	0.903887i	$-0.359299\pi$
$113$	116128.	0.855544	0.427772	+	0.903887i	$0.359299\pi$
$114$	0	0
$115$	−26899.6	−0.189671
$116$	0	0
$117$	−74592.3	−0.503766
$118$	0	0
$119$	−59746.0	−0.386760
$120$	0	0
$121$	11137.7	0.0691561
$122$	0	0
$123$	84053.5	0.500948
$124$	0	0
$125$	−134436.	−0.769554
$126$	0	0
$127$	6062.47	0.0333534	0.0166767	−	0.999861i	$-0.494691\pi$
$127$	6062.47	0.0333534	0.0166767	+	0.999861i	$0.494691\pi$
$128$	0	0
$129$	103837.	0.549385
$130$	0	0
$131$	−115145.	−0.586228	−0.293114	−	0.956078i	$-0.594692\pi$
$131$	−115145.	−0.586228	−0.293114	+	0.956078i	$0.594692\pi$
$132$	0	0
$133$	90470.8	0.443485
$134$	0	0
$135$	17217.1	0.0813066
$136$	0	0
$137$	326643.	1.48687	0.743434	−	0.668810i	$-0.233196\pi$
$137$	326643.	1.48687	0.743434	+	0.668810i	$0.233196\pi$
$138$	0	0
$139$	228617.	1.00362	0.501812	−	0.864977i	$-0.332667\pi$
$139$	228617.	1.00362	0.501812	+	0.864977i	$0.332667\pi$
$140$	0	0
$141$	179601.	0.760782
$142$	0	0
$143$	382130.	1.56268
$144$	0	0
$145$	166064.	0.655926
$146$	0	0
$147$	413992.	1.58015
$148$	0	0
$149$	225286.	0.831321	0.415661	−	0.909520i	$-0.363550\pi$
$149$	225286.	0.831321	0.415661	+	0.909520i	$0.363550\pi$
$150$	0	0
$151$	−102524.	−0.365918	−0.182959	−	0.983121i	$-0.558568\pi$
$151$	−102524.	−0.365918	−0.182959	+	0.983121i	$0.558568\pi$
$152$	0	0
$153$	−19310.5	−0.0666905
$154$	0	0
$155$	−11706.5	−0.0391380
$156$	0	0
$157$	441599.	1.42981	0.714906	−	0.699220i	$-0.246469\pi$
$157$	441599.	1.42981	0.714906	+	0.699220i	$0.246469\pi$
$158$	0	0
$159$	−180806.	−0.567180
$160$	0	0
$161$	−285439.	−0.867859
$162$	0	0
$163$	451041.	1.32968	0.664839	−	0.746986i	$-0.268500\pi$
$163$	451041.	1.32968	0.664839	+	0.746986i	$0.268500\pi$
$164$	0	0
$165$	−88201.8	−0.252213
$166$	0	0
$167$	−447874.	−1.24270	−0.621348	−	0.783535i	$-0.713415\pi$
$167$	−447874.	−1.24270	−0.621348	+	0.783535i	$0.713415\pi$
$168$	0	0
$169$	476750.	1.28403
$170$	0	0
$171$	29241.0	0.0764719
$172$	0	0
$173$	−381949.	−0.970266	−0.485133	−	0.874440i	$-0.661229\pi$
$173$	−381949.	−0.970266	−0.485133	+	0.874440i	$0.661229\pi$
$174$	0	0
$175$	−643374.	−1.58807
$176$	0	0
$177$	439600.	1.05469
$178$	0	0
$179$	−193497.	−0.451379	−0.225690	−	0.974199i	$-0.572464\pi$
$179$	−193497.	−0.451379	−0.225690	+	0.974199i	$0.572464\pi$
$180$	0	0
$181$	52655.8	0.119468	0.0597338	−	0.998214i	$-0.480975\pi$
$181$	52655.8	0.119468	0.0597338	+	0.998214i	$0.480975\pi$
$182$	0	0
$183$	221034.	0.487901
$184$	0	0
$185$	70426.9	0.151290
$186$	0	0
$187$	98925.8	0.206874
$188$	0	0
$189$	182696.	0.372027
$190$	0	0
$191$	303888.	0.602739	0.301370	−	0.953507i	$-0.402556\pi$
$191$	303888.	0.602739	0.301370	+	0.953507i	$0.402556\pi$
$192$	0	0
$193$	134586.	0.260080	0.130040	−	0.991509i	$-0.458489\pi$
$193$	134586.	0.260080	0.130040	+	0.991509i	$0.458489\pi$
$194$	0	0
$195$	−195742.	−0.368636
$196$	0	0
$197$	−651096.	−1.19531	−0.597654	−	0.801754i	$-0.703900\pi$
$197$	−651096.	−1.19531	−0.597654	+	0.801754i	$0.703900\pi$
$198$	0	0
$199$	291770.	0.522286	0.261143	−	0.965300i	$-0.415901\pi$
$199$	291770.	0.522286	0.261143	+	0.965300i	$0.415901\pi$
$200$	0	0
$201$	535339.	0.934627
$202$	0	0
$203$	1.76215e6	3.00126
$204$	0	0
$205$	220570.	0.366574
$206$	0	0
$207$	−92256.7	−0.149648
$208$	0	0
$209$	−149799.	−0.237216
$210$	0	0
$211$	−1.05072e6	−1.62472	−0.812361	−	0.583154i	$-0.801818\pi$
$211$	−1.05072e6	−1.62472	−0.812361	+	0.583154i	$0.801818\pi$
$212$	0	0
$213$	41721.8	0.0630107
$214$	0	0
$215$	272484.	0.402018
$216$	0	0
$217$	−124221.	−0.179080
$218$	0	0
$219$	−240037.	−0.338196
$220$	0	0
$221$	219541.	0.302368
$222$	0	0
$223$	81993.8	0.110413	0.0552063	−	0.998475i	$-0.482418\pi$
$223$	81993.8	0.110413	0.0552063	+	0.998475i	$0.482418\pi$
$224$	0	0
$225$	−207945.	−0.273836
$226$	0	0
$227$	−911528.	−1.17410	−0.587051	−	0.809550i	$-0.699711\pi$
$227$	−911528.	−1.17410	−0.587051	+	0.809550i	$0.699711\pi$
$228$	0	0
$229$	−660815.	−0.832705	−0.416353	−	0.909203i	$-0.636692\pi$
$229$	−660815.	−0.832705	−0.416353	+	0.909203i	$0.636692\pi$
$230$	0	0
$231$	−935935.	−1.15403
$232$	0	0
$233$	866250.	1.04533	0.522665	−	0.852538i	$-0.324938\pi$
$233$	866250.	1.04533	0.522665	+	0.852538i	$0.324938\pi$
$234$	0	0
$235$	471301.	0.556710
$236$	0	0
$237$	614166.	0.710256
$238$	0	0
$239$	385761.	0.436841	0.218421	−	0.975855i	$-0.429910\pi$
$239$	385761.	0.436841	0.218421	+	0.975855i	$0.429910\pi$
$240$	0	0
$241$	−1.27179e6	−1.41050	−0.705252	−	0.708957i	$-0.749166\pi$
$241$	−1.27179e6	−1.41050	−0.705252	+	0.708957i	$0.749166\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	1.08638e6	1.15629
$246$	0	0
$247$	−332442.	−0.346716
$248$	0	0
$249$	−20347.1	−0.0207972
$250$	0	0
$251$	1.13475e6	1.13689	0.568444	−	0.822722i	$-0.307546\pi$
$251$	1.13475e6	1.13689	0.568444	+	0.822722i	$0.307546\pi$
$252$	0	0
$253$	472623.	0.464209
$254$	0	0
$255$	−50673.7	−0.0488014
$256$	0	0
$257$	−129194.	−0.122014	−0.0610071	−	0.998137i	$-0.519431\pi$
$257$	−129194.	−0.122014	−0.0610071	+	0.998137i	$0.519431\pi$
$258$	0	0
$259$	747320.	0.692241
$260$	0	0
$261$	569544.	0.517519
$262$	0	0
$263$	−913758.	−0.814595	−0.407298	−	0.913295i	$-0.633529\pi$
$263$	−913758.	−0.814595	−0.407298	+	0.913295i	$0.633529\pi$
$264$	0	0
$265$	−474465.	−0.415040
$266$	0	0
$267$	6744.99	0.00579033
$268$	0	0
$269$	−460736.	−0.388214	−0.194107	−	0.980980i	$-0.562181\pi$
$269$	−460736.	−0.388214	−0.194107	+	0.980980i	$0.562181\pi$
$270$	0	0
$271$	726289.	0.600739	0.300370	−	0.953823i	$-0.402890\pi$
$271$	726289.	0.600739	0.300370	+	0.953823i	$0.402890\pi$
$272$	0	0
$273$	−2.07708e6	−1.68673
$274$	0	0
$275$	1.06528e6	0.849441
$276$	0	0
$277$	−588940.	−0.461181	−0.230590	−	0.973051i	$-0.574066\pi$
$277$	−588940.	−0.461181	−0.230590	+	0.973051i	$0.574066\pi$
$278$	0	0
$279$	−40149.5	−0.0308795
$280$	0	0
$281$	111634.	0.0843396	0.0421698	−	0.999110i	$-0.486573\pi$
$281$	111634.	0.0843396	0.0421698	+	0.999110i	$0.486573\pi$
$282$	0	0
$283$	686287.	0.509377	0.254689	−	0.967023i	$-0.418027\pi$
$283$	686287.	0.509377	0.254689	+	0.967023i	$0.418027\pi$
$284$	0	0
$285$	76733.0	0.0559590
$286$	0	0
$287$	2.34053e6	1.67730
$288$	0	0
$289$	−1.36302e6	−0.959971
$290$	0	0
$291$	792957.	0.548930
$292$	0	0
$293$	−1.87734e6	−1.27754	−0.638769	−	0.769399i	$-0.720556\pi$
$293$	−1.87734e6	−1.27754	−0.638769	+	0.769399i	$0.720556\pi$
$294$	0	0
$295$	1.15358e6	0.771778
$296$	0	0
$297$	−302503.	−0.198993
$298$	0	0
$299$	1.04887e6	0.678490
$300$	0	0
$301$	2.89141e6	1.83947
$302$	0	0
$303$	−1.12863e6	−0.706230
$304$	0	0
$305$	580029.	0.357026
$306$	0	0
$307$	−396709.	−0.240229	−0.120115	−	0.992760i	$-0.538326\pi$
$307$	−396709.	−0.240229	−0.120115	+	0.992760i	$0.538326\pi$
$308$	0	0
$309$	991964.	0.591016
$310$	0	0
$311$	1.18936e6	0.697290	0.348645	−	0.937255i	$-0.386642\pi$
$311$	1.18936e6	0.697290	0.348645	+	0.937255i	$0.386642\pi$
$312$	0	0
$313$	−778041.	−0.448892	−0.224446	−	0.974487i	$-0.572057\pi$
$313$	−778041.	−0.448892	−0.224446	+	0.974487i	$0.572057\pi$
$314$	0	0
$315$	479423.	0.272234
$316$	0	0
$317$	3.52584e6	1.97067	0.985335	−	0.170630i	$-0.0545802\pi$
$317$	3.52584e6	1.97067	0.985335	+	0.170630i	$0.0545802\pi$
$318$	0	0
$319$	−2.91773e6	−1.60534
$320$	0	0
$321$	−1.44705e6	−0.783826
$322$	0	0
$323$	−86062.7	−0.0458995
$324$	0	0
$325$	2.36413e6	1.24155
$326$	0	0
$327$	−1.00599e6	−0.520263
$328$	0	0
$329$	5.00112e6	2.54728
$330$	0	0
$331$	−2.21135e6	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$331$	−2.21135e6	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$332$	0	0
$333$	241541.	0.119366
$334$	0	0
$335$	1.40481e6	0.683922
$336$	0	0
$337$	3.52355e6	1.69008	0.845038	−	0.534706i	$-0.179578\pi$
$337$	3.52355e6	1.69008	0.845038	+	0.534706i	$0.179578\pi$
$338$	0	0
$339$	1.04516e6	0.493949
$340$	0	0
$341$	205683.	0.0957881
$342$	0	0
$343$	7.31589e6	3.35762
$344$	0	0
$345$	−242096.	−0.109507
$346$	0	0
$347$	−3.53103e6	−1.57426	−0.787131	−	0.616785i	$-0.788435\pi$
$347$	−3.53103e6	−1.57426	−0.787131	+	0.616785i	$0.788435\pi$
$348$	0	0
$349$	−1.25322e6	−0.550760	−0.275380	−	0.961335i	$-0.588804\pi$
$349$	−1.25322e6	−0.550760	−0.275380	+	0.961335i	$0.588804\pi$
$350$	0	0
$351$	−671330.	−0.290850
$352$	0	0
$353$	964634.	0.412027	0.206014	−	0.978549i	$-0.433951\pi$
$353$	964634.	0.412027	0.206014	+	0.978549i	$0.433951\pi$
$354$	0	0
$355$	109485.	0.0461087
$356$	0	0
$357$	−537714.	−0.223296
$358$	0	0
$359$	1.26957e6	0.519900	0.259950	−	0.965622i	$-0.416294\pi$
$359$	1.26957e6	0.519900	0.259950	+	0.965622i	$0.416294\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	0	0
$363$	100239.	0.0399273
$364$	0	0
$365$	−629896.	−0.247478
$366$	0	0
$367$	1.26919e6	0.491881	0.245941	−	0.969285i	$-0.420903\pi$
$367$	1.26919e6	0.491881	0.245941	+	0.969285i	$0.420903\pi$
$368$	0	0
$369$	756481.	0.289223
$370$	0	0
$371$	−5.03469e6	−1.89906
$372$	0	0
$373$	3.47150e6	1.29195	0.645975	−	0.763359i	$-0.276451\pi$
$373$	3.47150e6	1.29195	0.645975	+	0.763359i	$0.276451\pi$
$374$	0	0
$375$	−1.20992e6	−0.444302
$376$	0	0
$377$	−6.47517e6	−2.34638
$378$	0	0
$379$	2.84273e6	1.01657	0.508286	−	0.861188i	$-0.330279\pi$
$379$	2.84273e6	1.01657	0.508286	+	0.861188i	$0.330279\pi$
$380$	0	0
$381$	54562.2	0.0192566
$382$	0	0
$383$	1.59297e6	0.554894	0.277447	−	0.960741i	$-0.410512\pi$
$383$	1.59297e6	0.554894	0.277447	+	0.960741i	$0.410512\pi$
$384$	0	0
$385$	−2.45604e6	−0.844470
$386$	0	0
$387$	934531.	0.317188
$388$	0	0
$389$	−5.24604e6	−1.75775	−0.878876	−	0.477050i	$-0.841706\pi$
$389$	−5.24604e6	−1.75775	−0.878876	+	0.477050i	$0.841706\pi$
$390$	0	0
$391$	271532.	0.0898211
$392$	0	0
$393$	−1.03630e6	−0.338459
$394$	0	0
$395$	1.61167e6	0.519736
$396$	0	0
$397$	−361755.	−0.115196	−0.0575982	−	0.998340i	$-0.518344\pi$
$397$	−361755.	−0.115196	−0.0575982	+	0.998340i	$0.518344\pi$
$398$	0	0
$399$	814237.	0.256046
$400$	0	0
$401$	−3.07851e6	−0.956049	−0.478025	−	0.878346i	$-0.658647\pi$
$401$	−3.07851e6	−0.956049	−0.478025	+	0.878346i	$0.658647\pi$
$402$	0	0
$403$	456461.	0.140004
$404$	0	0
$405$	154954.	0.0469424
$406$	0	0
$407$	−1.23739e6	−0.370273
$408$	0	0
$409$	4.13583e6	1.22252	0.611258	−	0.791432i	$-0.290664\pi$
$409$	4.13583e6	1.22252	0.611258	+	0.791432i	$0.290664\pi$
$410$	0	0
$411$	2.93979e6	0.858443
$412$	0	0
$413$	1.22410e7	3.53135
$414$	0	0
$415$	−53394.0	−0.0152185
$416$	0	0
$417$	2.05755e6	0.579443
$418$	0	0
$419$	−3.19560e6	−0.889238	−0.444619	−	0.895720i	$-0.646661\pi$
$419$	−3.19560e6	−0.889238	−0.444619	+	0.895720i	$0.646661\pi$
$420$	0	0
$421$	996574.	0.274034	0.137017	−	0.990569i	$-0.456249\pi$
$421$	996574.	0.274034	0.137017	+	0.990569i	$0.456249\pi$
$422$	0	0
$423$	1.61641e6	0.439238
$424$	0	0
$425$	612026.	0.164361
$426$	0	0
$427$	6.15486e6	1.63361
$428$	0	0
$429$	3.43917e6	0.902216
$430$	0	0
$431$	2.51955e6	0.653325	0.326662	−	0.945141i	$-0.394076\pi$
$431$	2.51955e6	0.653325	0.326662	+	0.945141i	$0.394076\pi$
$432$	0	0
$433$	−3.82101e6	−0.979395	−0.489698	−	0.871892i	$-0.662893\pi$
$433$	−3.82101e6	−0.979395	−0.489698	+	0.871892i	$0.662893\pi$
$434$	0	0
$435$	1.49457e6	0.378699
$436$	0	0
$437$	−411169.	−0.102995
$438$	0	0
$439$	−619538.	−0.153429	−0.0767144	−	0.997053i	$-0.524443\pi$
$439$	−619538.	−0.153429	−0.0767144	+	0.997053i	$0.524443\pi$
$440$	0	0
$441$	3.72593e6	0.912302
$442$	0	0
$443$	1.08715e6	0.263198	0.131599	−	0.991303i	$-0.457989\pi$
$443$	1.08715e6	0.263198	0.131599	+	0.991303i	$0.457989\pi$
$444$	0	0
$445$	17699.9	0.00423713
$446$	0	0
$447$	2.02758e6	0.479963
$448$	0	0
$449$	−4.56352e6	−1.06828	−0.534139	−	0.845397i	$-0.679364\pi$
$449$	−4.56352e6	−1.06828	−0.534139	+	0.845397i	$0.679364\pi$
$450$	0	0
$451$	−3.87539e6	−0.897169
$452$	0	0
$453$	−922717.	−0.211263
$454$	0	0
$455$	−5.45058e6	−1.23428
$456$	0	0
$457$	605485.	0.135617	0.0678083	−	0.997698i	$-0.478399\pi$
$457$	605485.	0.135617	0.0678083	+	0.997698i	$0.478399\pi$
$458$	0	0
$459$	−173794.	−0.0385038
$460$	0	0
$461$	−7.26112e6	−1.59130	−0.795649	−	0.605759i	$-0.792870\pi$
$461$	−7.26112e6	−1.59130	−0.795649	+	0.605759i	$0.792870\pi$
$462$	0	0
$463$	−1.84692e6	−0.400400	−0.200200	−	0.979755i	$-0.564159\pi$
$463$	−1.84692e6	−0.400400	−0.200200	+	0.979755i	$0.564159\pi$
$464$	0	0
$465$	−105359.	−0.0225963
$466$	0	0
$467$	−6.63166e6	−1.40712	−0.703559	−	0.710637i	$-0.748407\pi$
$467$	−6.63166e6	−1.40712	−0.703559	+	0.710637i	$0.748407\pi$
$468$	0	0
$469$	1.49069e7	3.12936
$470$	0	0
$471$	3.97439e6	0.825503
$472$	0	0
$473$	−4.78752e6	−0.983916
$474$	0	0
$475$	−926765.	−0.188467
$476$	0	0
$477$	−1.62726e6	−0.327462
$478$	0	0
$479$	2.39693e6	0.477328	0.238664	−	0.971102i	$-0.423291\pi$
$479$	2.39693e6	0.477328	0.238664	+	0.971102i	$0.423291\pi$
$480$	0	0
$481$	−2.74609e6	−0.541193
$482$	0	0
$483$	−2.56895e6	−0.501059
$484$	0	0
$485$	2.08084e6	0.401685
$486$	0	0
$487$	2.17359e6	0.415294	0.207647	−	0.978204i	$-0.433420\pi$
$487$	2.17359e6	0.415294	0.207647	+	0.978204i	$0.433420\pi$
$488$	0	0
$489$	4.05937e6	0.767690
$490$	0	0
$491$	−9.09470e6	−1.70249	−0.851245	−	0.524768i	$-0.824152\pi$
$491$	−9.09470e6	−1.70249	−0.851245	+	0.524768i	$0.824152\pi$
$492$	0	0
$493$	−1.67629e6	−0.310622
$494$	0	0
$495$	−793816.	−0.145615
$496$	0	0
$497$	1.16177e6	0.210975
$498$	0	0
$499$	−8.69614e6	−1.56342	−0.781709	−	0.623644i	$-0.785652\pi$
$499$	−8.69614e6	−1.56342	−0.781709	+	0.623644i	$0.785652\pi$
$500$	0	0
$501$	−4.03087e6	−0.717470
$502$	0	0
$503$	2.72206e6	0.479708	0.239854	−	0.970809i	$-0.422900\pi$
$503$	2.72206e6	0.479708	0.239854	+	0.970809i	$0.422900\pi$
$504$	0	0
$505$	−2.96171e6	−0.516790
$506$	0	0
$507$	4.29075e6	0.741332
$508$	0	0
$509$	7.17949e6	1.22828	0.614142	−	0.789195i	$-0.289502\pi$
$509$	7.17949e6	1.22828	0.614142	+	0.789195i	$0.289502\pi$
$510$	0	0
$511$	−6.68401e6	−1.13236
$512$	0	0
$513$	263169.	0.0441511
$514$	0	0
$515$	2.60307e6	0.432482
$516$	0	0
$517$	−8.28072e6	−1.36252
$518$	0	0
$519$	−3.43755e6	−0.560183
$520$	0	0
$521$	−2.71363e6	−0.437982	−0.218991	−	0.975727i	$-0.570277\pi$
$521$	−2.71363e6	−0.437982	−0.218991	+	0.975727i	$0.570277\pi$
$522$	0	0
$523$	8.12813e6	1.29938	0.649690	−	0.760199i	$-0.274899\pi$
$523$	8.12813e6	1.29938	0.649690	+	0.760199i	$0.274899\pi$
$524$	0	0
$525$	−5.79037e6	−0.916870
$526$	0	0
$527$	118169.	0.0185343
$528$	0	0
$529$	−5.13909e6	−0.798448
$530$	0	0
$531$	3.95640e6	0.608925
$532$	0	0
$533$	−8.60047e6	−1.31131
$534$	0	0
$535$	−3.79728e6	−0.573572
$536$	0	0
$537$	−1.74147e6	−0.260604
$538$	0	0
$539$	−1.90876e7	−2.82996
$540$	0	0
$541$	6.55456e6	0.962831	0.481416	−	0.876492i	$-0.340123\pi$
$541$	6.55456e6	0.962831	0.481416	+	0.876492i	$0.340123\pi$
$542$	0	0
$543$	473903.	0.0689747
$544$	0	0
$545$	−2.63987e6	−0.380707
$546$	0	0
$547$	−4.13403e6	−0.590752	−0.295376	−	0.955381i	$-0.595445\pi$
$547$	−4.13403e6	−0.590752	−0.295376	+	0.955381i	$0.595445\pi$
$548$	0	0
$549$	1.98931e6	0.281690
$550$	0	0
$551$	2.53834e6	0.356181
$552$	0	0
$553$	1.71019e7	2.37811
$554$	0	0
$555$	633842.	0.0873471
$556$	0	0
$557$	−1.13647e7	−1.55211	−0.776053	−	0.630668i	$-0.782781\pi$
$557$	−1.13647e7	−1.55211	−0.776053	+	0.630668i	$0.782781\pi$
$558$	0	0
$559$	−1.06247e7	−1.43810
$560$	0	0
$561$	890333.	0.119439
$562$	0	0
$563$	−1.00795e7	−1.34019	−0.670095	−	0.742276i	$-0.733746\pi$
$563$	−1.00795e7	−1.34019	−0.670095	+	0.742276i	$0.733746\pi$
$564$	0	0
$565$	2.74266e6	0.361452
$566$	0	0
$567$	1.64426e6	0.214790
$568$	0	0
$569$	1.25833e7	1.62934	0.814671	−	0.579924i	$-0.196918\pi$
$569$	1.25833e7	1.62934	0.814671	+	0.579924i	$0.196918\pi$
$570$	0	0
$571$	1.32744e7	1.70382	0.851909	−	0.523690i	$-0.175445\pi$
$571$	1.32744e7	1.70382	0.851909	+	0.523690i	$0.175445\pi$
$572$	0	0
$573$	2.73499e6	0.347992
$574$	0	0
$575$	2.92399e6	0.368813
$576$	0	0
$577$	−7.77405e6	−0.972093	−0.486046	−	0.873933i	$-0.661561\pi$
$577$	−7.77405e6	−0.972093	−0.486046	+	0.873933i	$0.661561\pi$
$578$	0	0
$579$	1.21127e6	0.150157
$580$	0	0
$581$	−566580.	−0.0696339
$582$	0	0
$583$	8.33631e6	1.01579
$584$	0	0
$585$	−1.76168e6	−0.212832
$586$	0	0
$587$	9.72112e6	1.16445	0.582225	−	0.813027i	$-0.302182\pi$
$587$	9.72112e6	1.16445	0.582225	+	0.813027i	$0.302182\pi$
$588$	0	0
$589$	−178938.	−0.0212527
$590$	0	0
$591$	−5.85986e6	−0.690111
$592$	0	0
$593$	−9.51207e6	−1.11081	−0.555403	−	0.831581i	$-0.687436\pi$
$593$	−9.51207e6	−1.11081	−0.555403	+	0.831581i	$0.687436\pi$
$594$	0	0
$595$	−1.41105e6	−0.163399
$596$	0	0
$597$	2.62593e6	0.301542
$598$	0	0
$599$	−2.79130e6	−0.317863	−0.158931	−	0.987290i	$-0.550805\pi$
$599$	−2.79130e6	−0.317863	−0.158931	+	0.987290i	$0.550805\pi$
$600$	0	0
$601$	−1.71913e6	−0.194143	−0.0970716	−	0.995277i	$-0.530948\pi$
$601$	−1.71913e6	−0.194143	−0.0970716	+	0.995277i	$0.530948\pi$
$602$	0	0
$603$	4.81805e6	0.539607
$604$	0	0
$605$	263043.	0.0292172
$606$	0	0
$607$	1.11303e7	1.22612	0.613061	−	0.790035i	$-0.289938\pi$
$607$	1.11303e7	1.22612	0.613061	+	0.790035i	$0.289938\pi$
$608$	0	0
$609$	1.58594e7	1.73278
$610$	0	0
$611$	−1.83770e7	−1.99146
$612$	0	0
$613$	998180.	0.107290	0.0536448	−	0.998560i	$-0.482916\pi$
$613$	998180.	0.107290	0.0536448	+	0.998560i	$0.482916\pi$
$614$	0	0
$615$	1.98513e6	0.211641
$616$	0	0
$617$	−5.14322e6	−0.543904	−0.271952	−	0.962311i	$-0.587669\pi$
$617$	−5.14322e6	−0.543904	−0.271952	+	0.962311i	$0.587669\pi$
$618$	0	0
$619$	1.51122e7	1.58526	0.792632	−	0.609701i	$-0.208710\pi$
$619$	1.51122e7	1.58526	0.792632	+	0.609701i	$0.208710\pi$
$620$	0	0
$621$	−830310.	−0.0863995
$622$	0	0
$623$	187819.	0.0193874
$624$	0	0
$625$	4.84753e6	0.496387
$626$	0	0
$627$	−1.34819e6	−0.136957
$628$	0	0
$629$	−710908.	−0.0716451
$630$	0	0
$631$	3.02455e6	0.302404	0.151202	−	0.988503i	$-0.451686\pi$
$631$	3.02455e6	0.302404	0.151202	+	0.988503i	$0.451686\pi$
$632$	0	0
$633$	−9.45644e6	−0.938034
$634$	0	0
$635$	143180.	0.0140912
$636$	0	0
$637$	−4.23603e7	−4.13628
$638$	0	0
$639$	375496.	0.0363792
$640$	0	0
$641$	−6.09294e6	−0.585709	−0.292855	−	0.956157i	$-0.594605\pi$
$641$	−6.09294e6	−0.585709	−0.292855	+	0.956157i	$0.594605\pi$
$642$	0	0
$643$	−7.43059e6	−0.708755	−0.354377	−	0.935103i	$-0.615307\pi$
$643$	−7.43059e6	−0.708755	−0.354377	+	0.935103i	$0.615307\pi$
$644$	0	0
$645$	2.45236e6	0.232105
$646$	0	0
$647$	1.75431e6	0.164758	0.0823789	−	0.996601i	$-0.473748\pi$
$647$	1.75431e6	0.164758	0.0823789	+	0.996601i	$0.473748\pi$
$648$	0	0
$649$	−2.02683e7	−1.88888
$650$	0	0
$651$	−1.11799e6	−0.103392
$652$	0	0
$653$	−6.02593e6	−0.553021	−0.276510	−	0.961011i	$-0.589178\pi$
$653$	−6.02593e6	−0.553021	−0.276510	+	0.961011i	$0.589178\pi$
$654$	0	0
$655$	−2.71943e6	−0.247670
$656$	0	0
$657$	−2.16033e6	−0.195257
$658$	0	0
$659$	1.15725e7	1.03804	0.519018	−	0.854763i	$-0.326298\pi$
$659$	1.15725e7	1.03804	0.519018	+	0.854763i	$0.326298\pi$
$660$	0	0
$661$	−8.34628e6	−0.743001	−0.371500	−	0.928433i	$-0.621157\pi$
$661$	−8.34628e6	−0.743001	−0.371500	+	0.928433i	$0.621157\pi$
$662$	0	0
$663$	1.97587e6	0.174572
$664$	0	0
$665$	2.13669e6	0.187364
$666$	0	0
$667$	−8.00857e6	−0.697012
$668$	0	0
$669$	737944.	0.0637468
$670$	0	0
$671$	−1.01911e7	−0.873802
$672$	0	0
$673$	−5.33266e6	−0.453843	−0.226922	−	0.973913i	$-0.572866\pi$
$673$	−5.33266e6	−0.453843	−0.226922	+	0.973913i	$0.572866\pi$
$674$	0	0
$675$	−1.87150e6	−0.158100
$676$	0	0
$677$	2.14247e7	1.79656	0.898282	−	0.439418i	$-0.144815\pi$
$677$	2.14247e7	1.79656	0.898282	+	0.439418i	$0.144815\pi$
$678$	0	0
$679$	2.20805e7	1.83795
$680$	0	0
$681$	−8.20376e6	−0.677868
$682$	0	0
$683$	−2.01614e7	−1.65375	−0.826874	−	0.562388i	$-0.809883\pi$
$683$	−2.01614e7	−1.65375	−0.826874	+	0.562388i	$0.809883\pi$
$684$	0	0
$685$	7.71448e6	0.628174
$686$	0	0
$687$	−5.94734e6	−0.480763
$688$	0	0
$689$	1.85004e7	1.48468
$690$	0	0
$691$	−1.54453e7	−1.23056	−0.615279	−	0.788310i	$-0.710957\pi$
$691$	−1.54453e7	−1.23056	−0.615279	+	0.788310i	$0.710957\pi$
$692$	0	0
$693$	−8.42342e6	−0.666278
$694$	0	0
$695$	5.39934e6	0.424013
$696$	0	0
$697$	−2.22649e6	−0.173596
$698$	0	0
$699$	7.79625e6	0.603521
$700$	0	0
$701$	−3.36415e6	−0.258571	−0.129285	−	0.991607i	$-0.541268\pi$
$701$	−3.36415e6	−0.258571	−0.129285	+	0.991607i	$0.541268\pi$
$702$	0	0
$703$	1.07650e6	0.0821532
$704$	0	0
$705$	4.24171e6	0.321417
$706$	0	0
$707$	−3.14276e7	−2.36463
$708$	0	0
$709$	9.46957e6	0.707481	0.353740	−	0.935344i	$-0.384910\pi$
$709$	9.46957e6	0.707481	0.353740	+	0.935344i	$0.384910\pi$
$710$	0	0
$711$	5.52749e6	0.410066
$712$	0	0
$713$	564557.	0.0415895
$714$	0	0
$715$	9.02493e6	0.660205
$716$	0	0
$717$	3.47185e6	0.252210
$718$	0	0
$719$	1.17104e7	0.844790	0.422395	−	0.906412i	$-0.361189\pi$
$719$	1.17104e7	0.844790	0.422395	+	0.906412i	$0.361189\pi$
$720$	0	0
$721$	2.76219e7	1.97886
$722$	0	0
$723$	−1.14461e7	−0.814355
$724$	0	0
$725$	−1.80511e7	−1.27544
$726$	0	0
$727$	−1.55978e7	−1.09453	−0.547265	−	0.836959i	$-0.684331\pi$
$727$	−1.55978e7	−1.09453	−0.547265	+	0.836959i	$0.684331\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−2.75053e6	−0.190381
$732$	0	0
$733$	172411.	0.0118523	0.00592617	−	0.999982i	$-0.498114\pi$
$733$	172411.	0.0118523	0.00592617	+	0.999982i	$0.498114\pi$
$734$	0	0
$735$	9.77744e6	0.667585
$736$	0	0
$737$	−2.46825e7	−1.67386
$738$	0	0
$739$	1.29022e6	0.0869064	0.0434532	−	0.999055i	$-0.486164\pi$
$739$	1.29022e6	0.0869064	0.0434532	+	0.999055i	$0.486164\pi$
$740$	0	0
$741$	−2.99198e6	−0.200176
$742$	0	0
$743$	3.69707e6	0.245689	0.122844	−	0.992426i	$-0.460798\pi$
$743$	3.69707e6	0.245689	0.122844	+	0.992426i	$0.460798\pi$
$744$	0	0
$745$	5.32068e6	0.351218
$746$	0	0
$747$	−183124.	−0.0120072
$748$	0	0
$749$	−4.02940e7	−2.62444
$750$	0	0
$751$	188742.	0.0122115	0.00610575	−	0.999981i	$-0.498056\pi$
$751$	188742.	0.0122115	0.00610575	+	0.999981i	$0.498056\pi$
$752$	0	0
$753$	1.02128e7	0.656382
$754$	0	0
$755$	−2.42136e6	−0.154594
$756$	0	0
$757$	−2.13322e7	−1.35299	−0.676497	−	0.736446i	$-0.736503\pi$
$757$	−2.13322e7	−1.35299	−0.676497	+	0.736446i	$0.736503\pi$
$758$	0	0
$759$	4.25361e6	0.268011
$760$	0	0
$761$	−2.83469e7	−1.77437	−0.887185	−	0.461414i	$-0.847342\pi$
$761$	−2.83469e7	−1.77437	−0.887185	+	0.461414i	$0.847342\pi$
$762$	0	0
$763$	−2.80124e7	−1.74197
$764$	0	0
$765$	−456063.	−0.0281755
$766$	0	0
$767$	−4.49804e7	−2.76080
$768$	0	0
$769$	2.37153e7	1.44615	0.723073	−	0.690772i	$-0.242729\pi$
$769$	2.37153e7	1.44615	0.723073	+	0.690772i	$0.242729\pi$
$770$	0	0
$771$	−1.16275e6	−0.0704450
$772$	0	0
$773$	92234.1	0.00555192	0.00277596	−	0.999996i	$-0.499116\pi$
$773$	92234.1	0.00555192	0.00277596	+	0.999996i	$0.499116\pi$
$774$	0	0
$775$	1.27250e6	0.0761033
$776$	0	0
$777$	6.72588e6	0.399666
$778$	0	0
$779$	3.37148e6	0.199057
$780$	0	0
$781$	−1.92364e6	−0.112848
$782$	0	0
$783$	5.12589e6	0.298789
$784$	0	0
$785$	1.04294e7	0.604070
$786$	0	0
$787$	−350387.	−0.0201656	−0.0100828	−	0.999949i	$-0.503210\pi$
$787$	−350387.	−0.0201656	−0.0100828	+	0.999949i	$0.503210\pi$
$788$	0	0
$789$	−8.22382e6	−0.470307
$790$	0	0
$791$	2.91031e7	1.65386
$792$	0	0
$793$	−2.26165e7	−1.27715
$794$	0	0
$795$	−4.27018e6	−0.239623
$796$	0	0
$797$	9.25461e6	0.516074	0.258037	−	0.966135i	$-0.416924\pi$
$797$	9.25461e6	0.516074	0.258037	+	0.966135i	$0.416924\pi$
$798$	0	0
$799$	−4.75744e6	−0.263637
$800$	0	0
$801$	60704.9	0.00334305
$802$	0	0
$803$	1.10672e7	0.605688
$804$	0	0
$805$	−6.74134e6	−0.366654
$806$	0	0
$807$	−4.14663e6	−0.224136
$808$	0	0
$809$	−2.94050e6	−0.157961	−0.0789805	−	0.996876i	$-0.525166\pi$
$809$	−2.94050e6	−0.157961	−0.0789805	+	0.996876i	$0.525166\pi$
$810$	0	0
$811$	−2.12110e7	−1.13242	−0.566211	−	0.824260i	$-0.691591\pi$
$811$	−2.12110e7	−1.13242	−0.566211	+	0.824260i	$0.691591\pi$
$812$	0	0
$813$	6.53660e6	0.346837
$814$	0	0
$815$	1.06524e7	0.561765
$816$	0	0
$817$	4.16501e6	0.218303
$818$	0	0
$819$	−1.86937e7	−0.973835
$820$	0	0
$821$	−28537.6	−0.00147761	−0.000738804	−	1.00000i	$-0.500235\pi$
$821$	−28537.6	−0.00147761	−0.000738804		1.00000i	$0.500235\pi$
$822$	0	0
$823$	1.60864e7	0.827863	0.413931	−	0.910308i	$-0.364155\pi$
$823$	1.60864e7	0.827863	0.413931	+	0.910308i	$0.364155\pi$
$824$	0	0
$825$	9.58754e6	0.490425
$826$	0	0
$827$	1.43894e7	0.731607	0.365803	−	0.930692i	$-0.380794\pi$
$827$	1.43894e7	0.731607	0.365803	+	0.930692i	$0.380794\pi$
$828$	0	0
$829$	2.29442e7	1.15954	0.579771	−	0.814779i	$-0.303142\pi$
$829$	2.29442e7	1.15954	0.579771	+	0.814779i	$0.303142\pi$
$830$	0	0
$831$	−5.30046e6	−0.266263
$832$	0	0
$833$	−1.09662e7	−0.547577
$834$	0	0
$835$	−1.05776e7	−0.525016
$836$	0	0
$837$	−361345.	−0.0178283
$838$	0	0
$839$	3.34833e7	1.64219	0.821095	−	0.570792i	$-0.193364\pi$
$839$	3.34833e7	1.64219	0.821095	+	0.570792i	$0.193364\pi$
$840$	0	0
$841$	2.89295e7	1.41043
$842$	0	0
$843$	1.00471e6	0.0486935
$844$	0	0
$845$	1.12596e7	0.542477
$846$	0	0
$847$	2.79123e6	0.133686
$848$	0	0
$849$	6.17658e6	0.294089
$850$	0	0
$851$	−3.39640e6	−0.160766
$852$	0	0
$853$	−9.55702e6	−0.449728	−0.224864	−	0.974390i	$-0.572194\pi$
$853$	−9.55702e6	−0.449728	−0.224864	+	0.974390i	$0.572194\pi$
$854$	0	0
$855$	690597.	0.0323080
$856$	0	0
$857$	1.69429e7	0.788019	0.394009	−	0.919106i	$-0.371088\pi$
$857$	1.69429e7	0.788019	0.394009	+	0.919106i	$0.371088\pi$
$858$	0	0
$859$	6.40368e6	0.296106	0.148053	−	0.988979i	$-0.452699\pi$
$859$	6.40368e6	0.296106	0.148053	+	0.988979i	$0.452699\pi$
$860$	0	0
$861$	2.10648e7	0.968387
$862$	0	0
$863$	−3.12340e6	−0.142758	−0.0713791	−	0.997449i	$-0.522740\pi$
$863$	−3.12340e6	−0.142758	−0.0713791	+	0.997449i	$0.522740\pi$
$864$	0	0
$865$	−9.02067e6	−0.409919
$866$	0	0
$867$	−1.22672e7	−0.554240
$868$	0	0
$869$	−2.83169e7	−1.27203
$870$	0	0
$871$	−5.47766e7	−2.44652
$872$	0	0
$873$	7.13661e6	0.316925
$874$	0	0
$875$	−3.36911e7	−1.48763
$876$	0	0
$877$	3.31907e7	1.45719	0.728597	−	0.684943i	$-0.240173\pi$
$877$	3.31907e7	1.45719	0.728597	+	0.684943i	$0.240173\pi$
$878$	0	0
$879$	−1.68961e7	−0.737587
$880$	0	0
$881$	3.73467e7	1.62111	0.810555	−	0.585662i	$-0.199165\pi$
$881$	3.73467e7	1.62111	0.810555	+	0.585662i	$0.199165\pi$
$882$	0	0
$883$	3.95144e7	1.70551	0.852753	−	0.522314i	$-0.174931\pi$
$883$	3.95144e7	1.70551	0.852753	+	0.522314i	$0.174931\pi$
$884$	0	0
$885$	1.03822e7	0.445586
$886$	0	0
$887$	−3.39325e7	−1.44813	−0.724064	−	0.689732i	$-0.757728\pi$
$887$	−3.39325e7	−1.44813	−0.724064	+	0.689732i	$0.757728\pi$
$888$	0	0
$889$	1.51932e6	0.0644757
$890$	0	0
$891$	−2.72253e6	−0.114889
$892$	0	0
$893$	7.20399e6	0.302304
$894$	0	0
$895$	−4.56990e6	−0.190699
$896$	0	0
$897$	9.43983e6	0.391726
$898$	0	0
$899$	−3.48528e6	−0.143826
$900$	0	0
$901$	4.78938e6	0.196547
$902$	0	0
$903$	2.60227e7	1.06202
$904$	0	0
$905$	1.24360e6	0.0504729
$906$	0	0
$907$	1.00269e7	0.404716	0.202358	−	0.979312i	$-0.435140\pi$
$907$	1.00269e7	0.404716	0.202358	+	0.979312i	$0.435140\pi$
$908$	0	0
$909$	−1.01577e7	−0.407742
$910$	0	0
$911$	6.57126e6	0.262333	0.131166	−	0.991360i	$-0.458128\pi$
$911$	6.57126e6	0.262333	0.131166	+	0.991360i	$0.458128\pi$
$912$	0	0
$913$	938128.	0.0372465
$914$	0	0
$915$	5.22026e6	0.206129
$916$	0	0
$917$	−2.88566e7	−1.13324
$918$	0	0
$919$	7.60638e6	0.297091	0.148545	−	0.988906i	$-0.452541\pi$
$919$	7.60638e6	0.297091	0.148545	+	0.988906i	$0.452541\pi$
$920$	0	0
$921$	−3.57038e6	−0.138696
$922$	0	0
$923$	−4.26903e6	−0.164940
$924$	0	0
$925$	−7.65541e6	−0.294181
$926$	0	0
$927$	8.92767e6	0.341223
$928$	0	0
$929$	−3.40362e7	−1.29390	−0.646951	−	0.762532i	$-0.723956\pi$
$929$	−3.40362e7	−1.29390	−0.646951	+	0.762532i	$0.723956\pi$
$930$	0	0
$931$	1.66057e7	0.627889
$932$	0	0
$933$	1.07043e7	0.402580
$934$	0	0
$935$	2.33637e6	0.0874004
$936$	0	0
$937$	1.77553e6	0.0660663	0.0330332	−	0.999454i	$-0.489483\pi$
$937$	1.77553e6	0.0660663	0.0330332	+	0.999454i	$0.489483\pi$
$938$	0	0
$939$	−7.00237e6	−0.259168
$940$	0	0
$941$	1.16181e7	0.427721	0.213860	−	0.976864i	$-0.431396\pi$
$941$	1.16181e7	0.427721	0.213860	+	0.976864i	$0.431396\pi$
$942$	0	0
$943$	−1.06372e7	−0.389535
$944$	0	0
$945$	4.31481e6	0.157174
$946$	0	0
$947$	−3.38853e7	−1.22782	−0.613912	−	0.789374i	$-0.710405\pi$
$947$	−3.38853e7	−1.22782	−0.613912	+	0.789374i	$0.710405\pi$
$948$	0	0
$949$	2.45609e7	0.885277
$950$	0	0
$951$	3.17325e7	1.13777
$952$	0	0
$953$	−2.44407e7	−0.871729	−0.435864	−	0.900012i	$-0.643557\pi$
$953$	−2.44407e7	−0.871729	−0.435864	+	0.900012i	$0.643557\pi$
$954$	0	0
$955$	7.17705e6	0.254646
$956$	0	0
$957$	−2.62595e7	−0.926845
$958$	0	0
$959$	8.18606e7	2.87427
$960$	0	0
$961$	−2.83835e7	−0.991418
$962$	0	0
$963$	−1.30234e7	−0.452542
$964$	0	0
$965$	3.17858e6	0.109879
$966$	0	0
$967$	3.67340e7	1.26329	0.631644	−	0.775258i	$-0.282380\pi$
$967$	3.67340e7	1.26329	0.631644	+	0.775258i	$0.282380\pi$
$968$	0	0
$969$	−774564.	−0.0265001
$970$	0	0
$971$	3.43205e7	1.16817	0.584085	−	0.811693i	$-0.301454\pi$
$971$	3.43205e7	1.16817	0.584085	+	0.811693i	$0.301454\pi$
$972$	0	0
$973$	5.72940e7	1.94011
$974$	0	0
$975$	2.12772e7	0.716807
$976$	0	0
$977$	−8.40409e6	−0.281679	−0.140839	−	0.990032i	$-0.544980\pi$
$977$	−8.40409e6	−0.281679	−0.140839	+	0.990032i	$0.544980\pi$
$978$	0	0
$979$	−310986.	−0.0103701
$980$	0	0
$981$	−9.05388e6	−0.300374
$982$	0	0
$983$	−1.87755e7	−0.619738	−0.309869	−	0.950779i	$-0.600285\pi$
$983$	−1.87755e7	−0.619738	−0.309869	+	0.950779i	$0.600285\pi$
$984$	0	0
$985$	−1.53772e7	−0.504995
$986$	0	0
$987$	4.50100e7	1.47067
$988$	0	0
$989$	−1.31408e7	−0.427199
$990$	0	0
$991$	1.98114e6	0.0640814	0.0320407	−	0.999487i	$-0.489799\pi$
$991$	1.98114e6	0.0640814	0.0320407	+	0.999487i	$0.489799\pi$
$992$	0	0
$993$	−1.99022e7	−0.640512
$994$	0	0
$995$	6.89086e6	0.220656
$996$	0	0
$997$	1.55752e6	0.0496244	0.0248122	−	0.999692i	$-0.492101\pi$
$997$	1.55752e6	0.0496244	0.0248122	+	0.999692i	$0.492101\pi$
$998$	0	0
$999$	2.17387e6	0.0689159

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.6.a.r.1.3		4
4.3	odd	2		57.6.a.f.1.3	✓	4
12.11	even	2		171.6.a.j.1.2		4
76.75	even	2		1083.6.a.g.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.6.a.f.1.3	✓	4	4.3	odd	2
171.6.a.j.1.2		4	12.11	even	2
912.6.a.r.1.3		4	1.1	even	1	trivial
1083.6.a.g.1.2		4	76.75	even	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:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	912.a (trivial)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} +23.6174 q^{5} +250.612 q^{7} +81.0000 q^{9} -414.956 q^{11} -920.892 q^{13} +212.557 q^{15} -238.401 q^{17} +361.000 q^{19} +2255.50 q^{21} -1138.97 q^{23} -2567.22 q^{25} +729.000 q^{27} +7031.41 q^{29} -495.673 q^{31} -3734.61 q^{33} +5918.80 q^{35} +2981.99 q^{37} -8288.03 q^{39} +9339.27 q^{41} +11537.4 q^{43} +1913.01 q^{45} +19955.6 q^{47} +45999.2 q^{49} -2145.61 q^{51} -20089.6 q^{53} -9800.20 q^{55} +3249.00 q^{57} +48844.4 q^{59} +24559.4 q^{61} +20299.5 q^{63} -21749.1 q^{65} +59482.1 q^{67} -10250.7 q^{69} +4635.76 q^{71} -26670.8 q^{73} -23105.0 q^{75} -103993. q^{77} +68240.6 q^{79} +6561.00 q^{81} -2260.79 q^{83} -5630.41 q^{85} +63282.7 q^{87} +749.444 q^{89} -230786. q^{91} -4461.06 q^{93} +8525.89 q^{95} +88106.3 q^{97} -33611.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 36 q^{3} - 8 q^{5} + 142 q^{7} + 324 q^{9} + 714 q^{11} - 74 q^{13} - 72 q^{15} - 3690 q^{17} + 1444 q^{19} + 1278 q^{21} - 862 q^{23} + 3282 q^{25} + 2916 q^{27} + 992 q^{29} + 7382 q^{31} + 6426 q^{33}+ \cdots + 57834 q^{99}+O(q^{100}) \) 4 * q + 36 * q^3 - 8 * q^5 + 142 * q^7 + 324 * q^9 + 714 * q^11 - 74 * q^13 - 72 * q^15 - 3690 * q^17 + 1444 * q^19 + 1278 * q^21 - 862 * q^23 + 3282 * q^25 + 2916 * q^27 + 992 * q^29 + 7382 * q^31 + 6426 * q^33 + 10242 * q^35 + 19114 * q^37 - 666 * q^39 - 5792 * q^41 + 18634 * q^43 - 648 * q^45 + 35424 * q^47 + 15910 * q^49 - 33210 * q^51 - 20256 * q^53 - 23538 * q^55 + 12996 * q^57 + 76944 * q^59 + 11562 * q^61 + 11502 * q^63 - 157464 * q^65 + 110044 * q^67 - 7758 * q^69 + 71184 * q^71 - 138830 * q^73 + 29538 * q^75 - 76122 * q^77 + 59470 * q^79 + 26244 * q^81 + 191600 * q^83 + 23682 * q^85 + 8928 * q^87 + 158920 * q^89 - 267196 * q^91 + 66438 * q^93 - 2888 * q^95 + 47460 * q^97 + 57834 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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