LMFDB - Embedded newform 1104.6.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,6,Mod(1,1104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1104.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1104.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$177.063737074$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2 x^{7} - 11608 x^{6} + 58841 x^{5} + 38776784 x^{4} - 553865060 x^{3} - 41132174448 x^{2} + \cdots - 2675538259200 $ x^8 - 2x^7 - 11608x^6 + 58841x^5 + 38776784x^4 - 553865060x^3 - 41132174448x^2 + 992726803008*x - 2675538259200
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 552)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-58.2723$ of defining polynomial
Character	$\chi$	$=$	1104.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} -101.758 q^{5} -82.1684 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} -101.758 q^{5} -82.1684 q^{7} +81.0000 q^{9} +283.569 q^{11} -275.629 q^{13} +915.821 q^{15} -1670.45 q^{17} +398.288 q^{19} +739.515 q^{21} -529.000 q^{23} +7229.66 q^{25} -729.000 q^{27} -5088.40 q^{29} -8405.25 q^{31} -2552.12 q^{33} +8361.28 q^{35} -5530.27 q^{37} +2480.66 q^{39} +21268.6 q^{41} +21409.6 q^{43} -8242.39 q^{45} +19241.9 q^{47} -10055.4 q^{49} +15034.1 q^{51} +22216.7 q^{53} -28855.3 q^{55} -3584.59 q^{57} +22684.4 q^{59} +39650.0 q^{61} -6655.64 q^{63} +28047.4 q^{65} -47662.0 q^{67} +4761.00 q^{69} +15414.8 q^{71} +73753.4 q^{73} -65067.0 q^{75} -23300.4 q^{77} +7875.77 q^{79} +6561.00 q^{81} +54939.1 q^{83} +169981. q^{85} +45795.6 q^{87} -71374.7 q^{89} +22648.0 q^{91} +75647.2 q^{93} -40529.0 q^{95} +169350. q^{97} +22969.1 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 72 q^{3} - 4 q^{5} - 210 q^{7} + 648 q^{9}+O(q^{10}) $ 8 * q - 72 * q^3 - 4 * q^5 - 210 * q^7 + 648 * q^9	$ 8 q - 72 q^{3} - 4 q^{5} - 210 q^{7} + 648 q^{9} - 118 q^{11} - 12 q^{13} + 36 q^{15} + 310 q^{17} - 1352 q^{19} + 1890 q^{21} - 4232 q^{23} + 6916 q^{25} - 5832 q^{27} + 7440 q^{29} - 12896 q^{31} + 1062 q^{33} - 3840 q^{35} + 3262 q^{37} + 108 q^{39} + 9520 q^{41} + 5756 q^{43} - 324 q^{45} - 8068 q^{47} + 54880 q^{49} - 2790 q^{51} + 44568 q^{53} - 37356 q^{55} + 12168 q^{57} - 32116 q^{59} + 47322 q^{61} - 17010 q^{63} + 46912 q^{65} - 60148 q^{67} + 38088 q^{69} - 24200 q^{71} + 167340 q^{73} - 62244 q^{75} + 178316 q^{77} - 92366 q^{79} + 52488 q^{81} + 710 q^{83} + 227864 q^{85} - 66960 q^{87} + 103678 q^{89} - 2092 q^{91} + 116064 q^{93} - 82020 q^{95} + 255456 q^{97} - 9558 q^{99}+O(q^{100}) $ 8 * q - 72 * q^3 - 4 * q^5 - 210 * q^7 + 648 * q^9 - 118 * q^11 - 12 * q^13 + 36 * q^15 + 310 * q^17 - 1352 * q^19 + 1890 * q^21 - 4232 * q^23 + 6916 * q^25 - 5832 * q^27 + 7440 * q^29 - 12896 * q^31 + 1062 * q^33 - 3840 * q^35 + 3262 * q^37 + 108 * q^39 + 9520 * q^41 + 5756 * q^43 - 324 * q^45 - 8068 * q^47 + 54880 * q^49 - 2790 * q^51 + 44568 * q^53 - 37356 * q^55 + 12168 * q^57 - 32116 * q^59 + 47322 * q^61 - 17010 * q^63 + 46912 * q^65 - 60148 * q^67 + 38088 * q^69 - 24200 * q^71 + 167340 * q^73 - 62244 * q^75 + 178316 * q^77 - 92366 * q^79 + 52488 * q^81 + 710 * q^83 + 227864 * q^85 - 66960 * q^87 + 103678 * q^89 - 2092 * q^91 + 116064 * q^93 - 82020 * q^95 + 255456 * q^97 - 9558 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−101.758	−1.82030	−0.910150	−	0.414279i	$-0.864034\pi$
$5$	−101.758	−1.82030	−0.910150	+	0.414279i	$0.864034\pi$
$6$	0	0
$7$	−82.1684	−0.633811	−0.316905	−	0.948457i	$-0.602644\pi$
$7$	−82.1684	−0.633811	−0.316905	+	0.948457i	$0.602644\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	283.569	0.706605	0.353302	−	0.935509i	$-0.385059\pi$
$11$	283.569	0.706605	0.353302	+	0.935509i	$0.385059\pi$
$12$	0	0
$13$	−275.629	−0.452342	−0.226171	−	0.974088i	$-0.572621\pi$
$13$	−275.629	−0.452342	−0.226171	+	0.974088i	$0.572621\pi$
$14$	0	0
$15$	915.821	1.05095
$16$	0	0
$17$	−1670.45	−1.40188	−0.700941	−	0.713220i	$-0.747236\pi$
$17$	−1670.45	−1.40188	−0.700941	+	0.713220i	$0.747236\pi$
$18$	0	0
$19$	398.288	0.253112	0.126556	−	0.991959i	$-0.459608\pi$
$19$	398.288	0.253112	0.126556	+	0.991959i	$0.459608\pi$
$20$	0	0
$21$	739.515	0.365931
$22$	0	0
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	7229.66	2.31349
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−5088.40	−1.12354	−0.561768	−	0.827295i	$-0.689878\pi$
$29$	−5088.40	−1.12354	−0.561768	+	0.827295i	$0.689878\pi$
$30$	0	0
$31$	−8405.25	−1.57089	−0.785446	−	0.618930i	$-0.787566\pi$
$31$	−8405.25	−1.57089	−0.785446	+	0.618930i	$0.787566\pi$
$32$	0	0
$33$	−2552.12	−0.407958
$34$	0	0
$35$	8361.28	1.15373
$36$	0	0
$37$	−5530.27	−0.664113	−0.332056	−	0.943260i	$-0.607742\pi$
$37$	−5530.27	−0.664113	−0.332056	+	0.943260i	$0.607742\pi$
$38$	0	0
$39$	2480.66	0.261159
$40$	0	0
$41$	21268.6	1.97596	0.987980	−	0.154579i	$-0.0494023\pi$
$41$	21268.6	1.97596	0.987980	+	0.154579i	$0.0494023\pi$
$42$	0	0
$43$	21409.6	1.76579	0.882894	−	0.469573i	$-0.155592\pi$
$43$	21409.6	1.76579	0.882894	+	0.469573i	$0.155592\pi$
$44$	0	0
$45$	−8242.39	−0.606767
$46$	0	0
$47$	19241.9	1.27058	0.635292	−	0.772272i	$-0.280880\pi$
$47$	19241.9	1.27058	0.635292	+	0.772272i	$0.280880\pi$
$48$	0	0
$49$	−10055.4	−0.598284
$50$	0	0
$51$	15034.1	0.809376
$52$	0	0
$53$	22216.7	1.08640	0.543201	−	0.839603i	$-0.317212\pi$
$53$	22216.7	1.08640	0.543201	+	0.839603i	$0.317212\pi$
$54$	0	0
$55$	−28855.3	−1.28623
$56$	0	0
$57$	−3584.59	−0.146135
$58$	0	0
$59$	22684.4	0.848393	0.424196	−	0.905570i	$-0.360557\pi$
$59$	22684.4	0.848393	0.424196	+	0.905570i	$0.360557\pi$
$60$	0	0
$61$	39650.0	1.36433	0.682164	−	0.731200i	$-0.261039\pi$
$61$	39650.0	1.36433	0.682164	+	0.731200i	$0.261039\pi$
$62$	0	0
$63$	−6655.64	−0.211270
$64$	0	0
$65$	28047.4	0.823397
$66$	0	0
$67$	−47662.0	−1.29713	−0.648567	−	0.761157i	$-0.724631\pi$
$67$	−47662.0	−1.29713	−0.648567	+	0.761157i	$0.724631\pi$
$68$	0	0
$69$	4761.00	0.120386
$70$	0	0
$71$	15414.8	0.362905	0.181453	−	0.983400i	$-0.441920\pi$
$71$	15414.8	0.362905	0.181453	+	0.983400i	$0.441920\pi$
$72$	0	0
$73$	73753.4	1.61985	0.809926	−	0.586533i	$-0.199507\pi$
$73$	73753.4	1.61985	0.809926	+	0.586533i	$0.199507\pi$
$74$	0	0
$75$	−65067.0	−1.33570
$76$	0	0
$77$	−23300.4	−0.447853
$78$	0	0
$79$	7875.77	0.141979	0.0709897	−	0.997477i	$-0.477384\pi$
$79$	7875.77	0.141979	0.0709897	+	0.997477i	$0.477384\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	54939.1	0.875360	0.437680	−	0.899131i	$-0.355800\pi$
$83$	54939.1	0.875360	0.437680	+	0.899131i	$0.355800\pi$
$84$	0	0
$85$	169981.	2.55184
$86$	0	0
$87$	45795.6	0.648673
$88$	0	0
$89$	−71374.7	−0.955144	−0.477572	−	0.878593i	$-0.658483\pi$
$89$	−71374.7	−0.955144	−0.477572	+	0.878593i	$0.658483\pi$
$90$	0	0
$91$	22648.0	0.286699
$92$	0	0
$93$	75647.2	0.906955
$94$	0	0
$95$	−40529.0	−0.460741
$96$	0	0
$97$	169350.	1.82750	0.913749	−	0.406279i	$-0.133174\pi$
$97$	169350.	1.82750	0.913749	+	0.406279i	$0.133174\pi$
$98$	0	0
$99$	22969.1	0.235535
$100$	0	0
$101$	155408.	1.51590	0.757951	−	0.652311i	$-0.226200\pi$
$101$	155408.	1.51590	0.757951	+	0.652311i	$0.226200\pi$
$102$	0	0
$103$	−56273.2	−0.522647	−0.261324	−	0.965251i	$-0.584159\pi$
$103$	−56273.2	−0.522647	−0.261324	+	0.965251i	$0.584159\pi$
$104$	0	0
$105$	−75251.5	−0.666104
$106$	0	0
$107$	−141516.	−1.19494	−0.597468	−	0.801893i	$-0.703826\pi$
$107$	−141516.	−1.19494	−0.597468	+	0.801893i	$0.703826\pi$
$108$	0	0
$109$	−88254.9	−0.711496	−0.355748	−	0.934582i	$-0.615774\pi$
$109$	−88254.9	−0.711496	−0.355748	+	0.934582i	$0.615774\pi$
$110$	0	0
$111$	49772.4	0.383426
$112$	0	0
$113$	179737.	1.32416	0.662082	−	0.749431i	$-0.269673\pi$
$113$	179737.	1.32416	0.662082	+	0.749431i	$0.269673\pi$
$114$	0	0
$115$	53829.9	0.379559
$116$	0	0
$117$	−22325.9	−0.150781
$118$	0	0
$119$	137258.	0.888527
$120$	0	0
$121$	−80639.8	−0.500710
$122$	0	0
$123$	−191417.	−1.14082
$124$	0	0
$125$	−417682.	−2.39095
$126$	0	0
$127$	−321411.	−1.76828	−0.884142	−	0.467219i	$-0.845256\pi$
$127$	−321411.	−1.76828	−0.884142	+	0.467219i	$0.845256\pi$
$128$	0	0
$129$	−192687.	−1.01948
$130$	0	0
$131$	−335676.	−1.70900	−0.854499	−	0.519452i	$-0.826136\pi$
$131$	−335676.	−1.70900	−0.854499	+	0.519452i	$0.826136\pi$
$132$	0	0
$133$	−32726.7	−0.160425
$134$	0	0
$135$	74181.5	0.350317
$136$	0	0
$137$	−308715.	−1.40526	−0.702629	−	0.711556i	$-0.747991\pi$
$137$	−308715.	−1.40526	−0.702629	+	0.711556i	$0.747991\pi$
$138$	0	0
$139$	100545.	0.441390	0.220695	−	0.975343i	$-0.429167\pi$
$139$	100545.	0.441390	0.220695	+	0.975343i	$0.429167\pi$
$140$	0	0
$141$	−173177.	−0.733572
$142$	0	0
$143$	−78159.7	−0.319627
$144$	0	0
$145$	517785.	2.04517
$146$	0	0
$147$	90498.2	0.345419
$148$	0	0
$149$	15561.6	0.0574234	0.0287117	−	0.999588i	$-0.490860\pi$
$149$	15561.6	0.0574234	0.0287117	+	0.999588i	$0.490860\pi$
$150$	0	0
$151$	−264226.	−0.943046	−0.471523	−	0.881854i	$-0.656296\pi$
$151$	−264226.	−0.943046	−0.471523	+	0.881854i	$0.656296\pi$
$152$	0	0
$153$	−135306.	−0.467294
$154$	0	0
$155$	855300.	2.85949
$156$	0	0
$157$	−459736.	−1.48854	−0.744268	−	0.667881i	$-0.767202\pi$
$157$	−459736.	−1.48854	−0.744268	+	0.667881i	$0.767202\pi$
$158$	0	0
$159$	−199951.	−0.627234
$160$	0	0
$161$	43467.1	0.132159
$162$	0	0
$163$	170981.	0.504057	0.252029	−	0.967720i	$-0.418902\pi$
$163$	170981.	0.504057	0.252029	+	0.967720i	$0.418902\pi$
$164$	0	0
$165$	259698.	0.742607
$166$	0	0
$167$	−367430.	−1.01949	−0.509746	−	0.860325i	$-0.670261\pi$
$167$	−367430.	−1.01949	−0.509746	+	0.860325i	$0.670261\pi$
$168$	0	0
$169$	−295322.	−0.795387
$170$	0	0
$171$	32261.3	0.0843708
$172$	0	0
$173$	623193.	1.58310	0.791549	−	0.611106i	$-0.209275\pi$
$173$	623193.	1.58310	0.791549	+	0.611106i	$0.209275\pi$
$174$	0	0
$175$	−594049.	−1.46632
$176$	0	0
$177$	−204159.	−0.489820
$178$	0	0
$179$	37287.1	0.0869814	0.0434907	−	0.999054i	$-0.486152\pi$
$179$	37287.1	0.0869814	0.0434907	+	0.999054i	$0.486152\pi$
$180$	0	0
$181$	−14059.0	−0.0318976	−0.0159488	−	0.999873i	$-0.505077\pi$
$181$	−14059.0	−0.0318976	−0.0159488	+	0.999873i	$0.505077\pi$
$182$	0	0
$183$	−356850.	−0.787695
$184$	0	0
$185$	562748.	1.20888
$186$	0	0
$187$	−473687.	−0.990576
$188$	0	0
$189$	59900.7	0.121977
$190$	0	0
$191$	−182682.	−0.362336	−0.181168	−	0.983452i	$-0.557988\pi$
$191$	−182682.	−0.362336	−0.181168	+	0.983452i	$0.557988\pi$
$192$	0	0
$193$	268889.	0.519613	0.259807	−	0.965661i	$-0.416341\pi$
$193$	268889.	0.519613	0.259807	+	0.965661i	$0.416341\pi$
$194$	0	0
$195$	−252427.	−0.475389
$196$	0	0
$197$	443593.	0.814366	0.407183	−	0.913347i	$-0.366511\pi$
$197$	443593.	0.814366	0.407183	+	0.913347i	$0.366511\pi$
$198$	0	0
$199$	−806042.	−1.44286	−0.721431	−	0.692486i	$-0.756515\pi$
$199$	−806042.	−1.44286	−0.721431	+	0.692486i	$0.756515\pi$
$200$	0	0
$201$	428958.	0.748901
$202$	0	0
$203$	418106.	0.712108
$204$	0	0
$205$	−2.16424e6	−3.59684
$206$	0	0
$207$	−42849.0	−0.0695048
$208$	0	0
$209$	112942.	0.178850
$210$	0	0
$211$	143999.	0.222665	0.111332	−	0.993783i	$-0.464488\pi$
$211$	143999.	0.222665	0.111332	+	0.993783i	$0.464488\pi$
$212$	0	0
$213$	−138734.	−0.209523
$214$	0	0
$215$	−2.17860e6	−3.21426
$216$	0	0
$217$	690645.	0.995648
$218$	0	0
$219$	−663781.	−0.935222
$220$	0	0
$221$	460424.	0.634129
$222$	0	0
$223$	15033.2	0.0202437	0.0101219	−	0.999949i	$-0.496778\pi$
$223$	15033.2	0.0202437	0.0101219	+	0.999949i	$0.496778\pi$
$224$	0	0
$225$	585603.	0.771164
$226$	0	0
$227$	329228.	0.424065	0.212032	−	0.977263i	$-0.431992\pi$
$227$	329228.	0.424065	0.212032	+	0.977263i	$0.431992\pi$
$228$	0	0
$229$	753745.	0.949807	0.474904	−	0.880038i	$-0.342483\pi$
$229$	753745.	0.949807	0.474904	+	0.880038i	$0.342483\pi$
$230$	0	0
$231$	209703.	0.258568
$232$	0	0
$233$	−109148.	−0.131712	−0.0658561	−	0.997829i	$-0.520978\pi$
$233$	−109148.	−0.131712	−0.0658561	+	0.997829i	$0.520978\pi$
$234$	0	0
$235$	−1.95801e6	−2.31284
$236$	0	0
$237$	−70882.0	−0.0819719
$238$	0	0
$239$	283467.	0.321002	0.160501	−	0.987036i	$-0.448689\pi$
$239$	283467.	0.321002	0.160501	+	0.987036i	$0.448689\pi$
$240$	0	0
$241$	226769.	0.251502	0.125751	−	0.992062i	$-0.459866\pi$
$241$	226769.	0.251502	0.125751	+	0.992062i	$0.459866\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	1.02321e6	1.08906
$246$	0	0
$247$	−109780.	−0.114493
$248$	0	0
$249$	−494452.	−0.505389
$250$	0	0
$251$	597883.	0.599007	0.299504	−	0.954095i	$-0.403179\pi$
$251$	597883.	0.599007	0.299504	+	0.954095i	$0.403179\pi$
$252$	0	0
$253$	−150008.	−0.147337
$254$	0	0
$255$	−1.52983e6	−1.47331
$256$	0	0
$257$	18479.8	0.0174528	0.00872640	−	0.999962i	$-0.497222\pi$
$257$	18479.8	0.0174528	0.00872640	+	0.999962i	$0.497222\pi$
$258$	0	0
$259$	454413.	0.420922
$260$	0	0
$261$	−412161.	−0.374512
$262$	0	0
$263$	880095.	0.784585	0.392292	−	0.919841i	$-0.371682\pi$
$263$	880095.	0.784585	0.392292	+	0.919841i	$0.371682\pi$
$264$	0	0
$265$	−2.26073e6	−1.97758
$266$	0	0
$267$	642372.	0.551453
$268$	0	0
$269$	−303526.	−0.255750	−0.127875	−	0.991790i	$-0.540816\pi$
$269$	−303526.	−0.255750	−0.127875	+	0.991790i	$0.540816\pi$
$270$	0	0
$271$	885495.	0.732424	0.366212	−	0.930531i	$-0.380654\pi$
$271$	885495.	0.732424	0.366212	+	0.930531i	$0.380654\pi$
$272$	0	0
$273$	−203832.	−0.165526
$274$	0	0
$275$	2.05011e6	1.63472
$276$	0	0
$277$	1.83886e6	1.43995	0.719976	−	0.693999i	$-0.244153\pi$
$277$	1.83886e6	1.43995	0.719976	+	0.693999i	$0.244153\pi$
$278$	0	0
$279$	−680825.	−0.523631
$280$	0	0
$281$	83955.4	0.0634283	0.0317141	−	0.999497i	$-0.489903\pi$
$281$	83955.4	0.0634283	0.0317141	+	0.999497i	$0.489903\pi$
$282$	0	0
$283$	2.05022e6	1.52172	0.760861	−	0.648915i	$-0.224777\pi$
$283$	2.05022e6	1.52172	0.760861	+	0.648915i	$0.224777\pi$
$284$	0	0
$285$	364761.	0.266009
$286$	0	0
$287$	−1.74760e6	−1.25238
$288$	0	0
$289$	1.37055e6	0.965271
$290$	0	0
$291$	−1.52415e6	−1.05511
$292$	0	0
$293$	−629557.	−0.428417	−0.214208	−	0.976788i	$-0.568717\pi$
$293$	−629557.	−0.428417	−0.214208	+	0.976788i	$0.568717\pi$
$294$	0	0
$295$	−2.30831e6	−1.54433
$296$	0	0
$297$	−206722.	−0.135986
$298$	0	0
$299$	145808.	0.0943197
$300$	0	0
$301$	−1.75920e6	−1.11917
$302$	0	0
$303$	−1.39868e6	−0.875207
$304$	0	0
$305$	−4.03470e6	−2.48348
$306$	0	0
$307$	1.81407e6	1.09852	0.549259	−	0.835652i	$-0.314910\pi$
$307$	1.81407e6	1.09852	0.549259	+	0.835652i	$0.314910\pi$
$308$	0	0
$309$	506459.	0.301751
$310$	0	0
$311$	753118.	0.441532	0.220766	−	0.975327i	$-0.429144\pi$
$311$	753118.	0.441532	0.220766	+	0.975327i	$0.429144\pi$
$312$	0	0
$313$	−1.96405e6	−1.13316	−0.566582	−	0.824005i	$-0.691735\pi$
$313$	−1.96405e6	−1.13316	−0.566582	+	0.824005i	$0.691735\pi$
$314$	0	0
$315$	677263.	0.384575
$316$	0	0
$317$	1.47338e6	0.823507	0.411753	−	0.911295i	$-0.364917\pi$
$317$	1.47338e6	0.823507	0.411753	+	0.911295i	$0.364917\pi$
$318$	0	0
$319$	−1.44291e6	−0.793895
$320$	0	0
$321$	1.27364e6	0.689897
$322$	0	0
$323$	−665321.	−0.354834
$324$	0	0
$325$	−1.99270e6	−1.04649
$326$	0	0
$327$	794294.	0.410782
$328$	0	0
$329$	−1.58108e6	−0.805309
$330$	0	0
$331$	−1.27424e6	−0.639265	−0.319632	−	0.947542i	$-0.603559\pi$
$331$	−1.27424e6	−0.639265	−0.319632	+	0.947542i	$0.603559\pi$
$332$	0	0
$333$	−447952.	−0.221371
$334$	0	0
$335$	4.84998e6	2.36117
$336$	0	0
$337$	−4.00369e6	−1.92037	−0.960186	−	0.279360i	$-0.909878\pi$
$337$	−4.00369e6	−1.92037	−0.960186	+	0.279360i	$0.909878\pi$
$338$	0	0
$339$	−1.61764e6	−0.764507
$340$	0	0
$341$	−2.38346e6	−1.11000
$342$	0	0
$343$	2.20724e6	1.01301
$344$	0	0
$345$	−484469.	−0.219138
$346$	0	0
$347$	−34770.1	−0.0155018	−0.00775089	−	0.999970i	$-0.502467\pi$
$347$	−34770.1	−0.0155018	−0.00775089	+	0.999970i	$0.502467\pi$
$348$	0	0
$349$	1.18311e6	0.519949	0.259975	−	0.965615i	$-0.416286\pi$
$349$	1.18311e6	0.519949	0.259975	+	0.965615i	$0.416286\pi$
$350$	0	0
$351$	200934.	0.0870532
$352$	0	0
$353$	−294491.	−0.125787	−0.0628934	−	0.998020i	$-0.520033\pi$
$353$	−294491.	−0.125787	−0.0628934	+	0.998020i	$0.520033\pi$
$354$	0	0
$355$	−1.56858e6	−0.660596
$356$	0	0
$357$	−1.23532e6	−0.512991
$358$	0	0
$359$	2.46149e6	1.00800	0.504001	−	0.863703i	$-0.331861\pi$
$359$	2.46149e6	1.00800	0.504001	+	0.863703i	$0.331861\pi$
$360$	0	0
$361$	−2.31747e6	−0.935934
$362$	0	0
$363$	725759.	0.289085
$364$	0	0
$365$	−7.50499e6	−2.94862
$366$	0	0
$367$	−3.49011e6	−1.35261	−0.676307	−	0.736620i	$-0.736421\pi$
$367$	−3.49011e6	−1.35261	−0.676307	+	0.736620i	$0.736421\pi$
$368$	0	0
$369$	1.72275e6	0.658654
$370$	0	0
$371$	−1.82551e6	−0.688573
$372$	0	0
$373$	656455.	0.244305	0.122153	−	0.992511i	$-0.461020\pi$
$373$	656455.	0.244305	0.122153	+	0.992511i	$0.461020\pi$
$374$	0	0
$375$	3.75913e6	1.38041
$376$	0	0
$377$	1.40251e6	0.508222
$378$	0	0
$379$	−5.14308e6	−1.83919	−0.919593	−	0.392873i	$-0.871481\pi$
$379$	−5.14308e6	−1.83919	−0.919593	+	0.392873i	$0.871481\pi$
$380$	0	0
$381$	2.89270e6	1.02092
$382$	0	0
$383$	−3.84312e6	−1.33871	−0.669356	−	0.742942i	$-0.733430\pi$
$383$	−3.84312e6	−1.33871	−0.669356	+	0.742942i	$0.733430\pi$
$384$	0	0
$385$	2.37100e6	0.815228
$386$	0	0
$387$	1.73418e6	0.588596
$388$	0	0
$389$	3.51047e6	1.17623	0.588114	−	0.808778i	$-0.299871\pi$
$389$	3.51047e6	1.17623	0.588114	+	0.808778i	$0.299871\pi$
$390$	0	0
$391$	883668.	0.292312
$392$	0	0
$393$	3.02108e6	0.986691
$394$	0	0
$395$	−801422.	−0.258445
$396$	0	0
$397$	1.97798e6	0.629863	0.314932	−	0.949114i	$-0.398018\pi$
$397$	1.97798e6	0.629863	0.314932	+	0.949114i	$0.398018\pi$
$398$	0	0
$399$	294540.	0.0926216
$400$	0	0
$401$	−1.83375e6	−0.569481	−0.284741	−	0.958605i	$-0.591907\pi$
$401$	−1.83375e6	−0.569481	−0.284741	+	0.958605i	$0.591907\pi$
$402$	0	0
$403$	2.31673e6	0.710580
$404$	0	0
$405$	−667633.	−0.202256
$406$	0	0
$407$	−1.56821e6	−0.469265
$408$	0	0
$409$	−4.59674e6	−1.35876	−0.679378	−	0.733788i	$-0.737750\pi$
$409$	−4.59674e6	−1.35876	−0.679378	+	0.733788i	$0.737750\pi$
$410$	0	0
$411$	2.77843e6	0.811326
$412$	0	0
$413$	−1.86394e6	−0.537720
$414$	0	0
$415$	−5.59049e6	−1.59342
$416$	0	0
$417$	−904903.	−0.254837
$418$	0	0
$419$	5.19345e6	1.44518	0.722588	−	0.691279i	$-0.242953\pi$
$419$	5.19345e6	1.44518	0.722588	+	0.691279i	$0.242953\pi$
$420$	0	0
$421$	−1.75282e6	−0.481984	−0.240992	−	0.970527i	$-0.577473\pi$
$421$	−1.75282e6	−0.481984	−0.240992	+	0.970527i	$0.577473\pi$
$422$	0	0
$423$	1.55859e6	0.423528
$424$	0	0
$425$	−1.20768e7	−3.24324
$426$	0	0
$427$	−3.25798e6	−0.864725
$428$	0	0
$429$	703437.	0.184536
$430$	0	0
$431$	−1.81337e6	−0.470213	−0.235106	−	0.971970i	$-0.575544\pi$
$431$	−1.81337e6	−0.470213	−0.235106	+	0.971970i	$0.575544\pi$
$432$	0	0
$433$	439363.	0.112617	0.0563085	−	0.998413i	$-0.482067\pi$
$433$	439363.	0.112617	0.0563085	+	0.998413i	$0.482067\pi$
$434$	0	0
$435$	−4.66007e6	−1.18078
$436$	0	0
$437$	−210694.	−0.0527776
$438$	0	0
$439$	−102874.	−0.0254767	−0.0127384	−	0.999919i	$-0.504055\pi$
$439$	−102874.	−0.0254767	−0.0127384	+	0.999919i	$0.504055\pi$
$440$	0	0
$441$	−814484.	−0.199428
$442$	0	0
$443$	−1.10048e6	−0.266425	−0.133212	−	0.991088i	$-0.542529\pi$
$443$	−1.10048e6	−0.266425	−0.133212	+	0.991088i	$0.542529\pi$
$444$	0	0
$445$	7.26293e6	1.73865
$446$	0	0
$447$	−140054.	−0.0331534
$448$	0	0
$449$	−4.49808e6	−1.05296	−0.526479	−	0.850188i	$-0.676488\pi$
$449$	−4.49808e6	−1.05296	−0.526479	+	0.850188i	$0.676488\pi$
$450$	0	0
$451$	6.03109e6	1.39622
$452$	0	0
$453$	2.37803e6	0.544468
$454$	0	0
$455$	−2.30461e6	−0.521878
$456$	0	0
$457$	4.07230e6	0.912115	0.456057	−	0.889950i	$-0.349261\pi$
$457$	4.07230e6	0.912115	0.456057	+	0.889950i	$0.349261\pi$
$458$	0	0
$459$	1.21776e6	0.269792
$460$	0	0
$461$	−4.80711e6	−1.05349	−0.526747	−	0.850022i	$-0.676588\pi$
$461$	−4.80711e6	−1.05349	−0.526747	+	0.850022i	$0.676588\pi$
$462$	0	0
$463$	−6.60484e6	−1.43189	−0.715946	−	0.698156i	$-0.754004\pi$
$463$	−6.60484e6	−1.43189	−0.715946	+	0.698156i	$0.754004\pi$
$464$	0	0
$465$	−7.69770e6	−1.65093
$466$	0	0
$467$	6.40602e6	1.35924	0.679620	−	0.733565i	$-0.262145\pi$
$467$	6.40602e6	1.35924	0.679620	+	0.733565i	$0.262145\pi$
$468$	0	0
$469$	3.91631e6	0.822137
$470$	0	0
$471$	4.13763e6	0.859407
$472$	0	0
$473$	6.07110e6	1.24771
$474$	0	0
$475$	2.87949e6	0.585574
$476$	0	0
$477$	1.79955e6	0.362134
$478$	0	0
$479$	−6.67058e6	−1.32839	−0.664194	−	0.747561i	$-0.731225\pi$
$479$	−6.67058e6	−1.32839	−0.664194	+	0.747561i	$0.731225\pi$
$480$	0	0
$481$	1.52430e6	0.300406
$482$	0	0
$483$	−391204.	−0.0763018
$484$	0	0
$485$	−1.72327e7	−3.32659
$486$	0	0
$487$	−3.65966e6	−0.699227	−0.349614	−	0.936894i	$-0.613687\pi$
$487$	−3.65966e6	−0.699227	−0.349614	+	0.936894i	$0.613687\pi$
$488$	0	0
$489$	−1.53883e6	−0.291018
$490$	0	0
$491$	5.22897e6	0.978841	0.489421	−	0.872048i	$-0.337208\pi$
$491$	5.22897e6	0.978841	0.489421	+	0.872048i	$0.337208\pi$
$492$	0	0
$493$	8.49993e6	1.57506
$494$	0	0
$495$	−2.33728e6	−0.428744
$496$	0	0
$497$	−1.26661e6	−0.230013
$498$	0	0
$499$	−2.98181e6	−0.536078	−0.268039	−	0.963408i	$-0.586376\pi$
$499$	−2.98181e6	−0.536078	−0.268039	+	0.963408i	$0.586376\pi$
$500$	0	0
$501$	3.30687e6	0.588604
$502$	0	0
$503$	4.47229e6	0.788152	0.394076	−	0.919078i	$-0.371065\pi$
$503$	4.47229e6	0.788152	0.394076	+	0.919078i	$0.371065\pi$
$504$	0	0
$505$	−1.58140e7	−2.75940
$506$	0	0
$507$	2.65790e6	0.459217
$508$	0	0
$509$	−6.15814e6	−1.05355	−0.526775	−	0.850005i	$-0.676599\pi$
$509$	−6.15814e6	−1.05355	−0.526775	+	0.850005i	$0.676599\pi$
$510$	0	0
$511$	−6.06020e6	−1.02668
$512$	0	0
$513$	−290352.	−0.0487115
$514$	0	0
$515$	5.72624e6	0.951375
$516$	0	0
$517$	5.45640e6	0.897800
$518$	0	0
$519$	−5.60874e6	−0.914001
$520$	0	0
$521$	7.04783e6	1.13752	0.568762	−	0.822502i	$-0.307422\pi$
$521$	7.04783e6	1.13752	0.568762	+	0.822502i	$0.307422\pi$
$522$	0	0
$523$	6.65868e6	1.06447	0.532236	−	0.846596i	$-0.321352\pi$
$523$	6.65868e6	1.06447	0.532236	+	0.846596i	$0.321352\pi$
$524$	0	0
$525$	5.34644e6	0.846578
$526$	0	0
$527$	1.40405e7	2.20220
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	1.83744e6	0.282798
$532$	0	0
$533$	−5.86223e6	−0.893809
$534$	0	0
$535$	1.44003e7	2.17514
$536$	0	0
$537$	−335584.	−0.0502187
$538$	0	0
$539$	−2.85138e6	−0.422750
$540$	0	0
$541$	−4.46002e6	−0.655154	−0.327577	−	0.944824i	$-0.606232\pi$
$541$	−4.46002e6	−0.655154	−0.327577	+	0.944824i	$0.606232\pi$
$542$	0	0
$543$	126531.	0.0184161
$544$	0	0
$545$	8.98062e6	1.29514
$546$	0	0
$547$	6.35995e6	0.908835	0.454418	−	0.890789i	$-0.349847\pi$
$547$	6.35995e6	0.908835	0.454418	+	0.890789i	$0.349847\pi$
$548$	0	0
$549$	3.21165e6	0.454776
$550$	0	0
$551$	−2.02665e6	−0.284381
$552$	0	0
$553$	−647139.	−0.0899881
$554$	0	0
$555$	−5.06473e6	−0.697950
$556$	0	0
$557$	2.73196e6	0.373109	0.186555	−	0.982445i	$-0.440268\pi$
$557$	2.73196e6	0.373109	0.186555	+	0.982445i	$0.440268\pi$
$558$	0	0
$559$	−5.90112e6	−0.798739
$560$	0	0
$561$	4.26318e6	0.571909
$562$	0	0
$563$	−5.58783e6	−0.742972	−0.371486	−	0.928439i	$-0.621152\pi$
$563$	−5.58783e6	−0.742972	−0.371486	+	0.928439i	$0.621152\pi$
$564$	0	0
$565$	−1.82897e7	−2.41038
$566$	0	0
$567$	−539107.	−0.0704234
$568$	0	0
$569$	2.85322e6	0.369449	0.184724	−	0.982790i	$-0.440861\pi$
$569$	2.85322e6	0.369449	0.184724	+	0.982790i	$0.440861\pi$
$570$	0	0
$571$	−1.31199e7	−1.68399	−0.841997	−	0.539482i	$-0.818620\pi$
$571$	−1.31199e7	−1.68399	−0.841997	+	0.539482i	$0.818620\pi$
$572$	0	0
$573$	1.64414e6	0.209195
$574$	0	0
$575$	−3.82449e6	−0.482396
$576$	0	0
$577$	−1.50775e7	−1.88534	−0.942671	−	0.333723i	$-0.891695\pi$
$577$	−1.50775e7	−1.88534	−0.942671	+	0.333723i	$0.891695\pi$
$578$	0	0
$579$	−2.42000e6	−0.299999
$580$	0	0
$581$	−4.51426e6	−0.554812
$582$	0	0
$583$	6.29997e6	0.767656
$584$	0	0
$585$	2.27184e6	0.274466
$586$	0	0
$587$	−2.34907e6	−0.281385	−0.140693	−	0.990053i	$-0.544933\pi$
$587$	−2.34907e6	−0.281385	−0.140693	+	0.990053i	$0.544933\pi$
$588$	0	0
$589$	−3.34771e6	−0.397612
$590$	0	0
$591$	−3.99234e6	−0.470174
$592$	0	0
$593$	5.81448e6	0.679007	0.339504	−	0.940605i	$-0.389741\pi$
$593$	5.81448e6	0.679007	0.339504	+	0.940605i	$0.389741\pi$
$594$	0	0
$595$	−1.39671e7	−1.61739
$596$	0	0
$597$	7.25437e6	0.833037
$598$	0	0
$599$	−8.79862e6	−1.00195	−0.500977	−	0.865461i	$-0.667026\pi$
$599$	−8.79862e6	−1.00195	−0.500977	+	0.865461i	$0.667026\pi$
$600$	0	0
$601$	1.11765e7	1.26218	0.631090	−	0.775710i	$-0.282608\pi$
$601$	1.11765e7	1.26218	0.631090	+	0.775710i	$0.282608\pi$
$602$	0	0
$603$	−3.86062e6	−0.432378
$604$	0	0
$605$	8.20574e6	0.911442
$606$	0	0
$607$	−216664.	−0.0238680	−0.0119340	−	0.999929i	$-0.503799\pi$
$607$	−216664.	−0.0238680	−0.0119340	+	0.999929i	$0.503799\pi$
$608$	0	0
$609$	−3.76295e6	−0.411136
$610$	0	0
$611$	−5.30362e6	−0.574738
$612$	0	0
$613$	1.41844e7	1.52461	0.762305	−	0.647218i	$-0.224068\pi$
$613$	1.41844e7	1.52461	0.762305	+	0.647218i	$0.224068\pi$
$614$	0	0
$615$	1.94782e7	2.07664
$616$	0	0
$617$	1.44690e6	0.153012	0.0765058	−	0.997069i	$-0.475624\pi$
$617$	1.44690e6	0.153012	0.0765058	+	0.997069i	$0.475624\pi$
$618$	0	0
$619$	−3.34912e6	−0.351321	−0.175660	−	0.984451i	$-0.556206\pi$
$619$	−3.34912e6	−0.351321	−0.175660	+	0.984451i	$0.556206\pi$
$620$	0	0
$621$	385641.	0.0401286
$622$	0	0
$623$	5.86474e6	0.605381
$624$	0	0
$625$	1.99097e7	2.03875
$626$	0	0
$627$	−1.01648e6	−0.103259
$628$	0	0
$629$	9.23804e6	0.931007
$630$	0	0
$631$	−1.01130e7	−1.01113	−0.505567	−	0.862788i	$-0.668717\pi$
$631$	−1.01130e7	−1.01113	−0.505567	+	0.862788i	$0.668717\pi$
$632$	0	0
$633$	−1.29599e6	−0.128556
$634$	0	0
$635$	3.27061e7	3.21881
$636$	0	0
$637$	2.77155e6	0.270629
$638$	0	0
$639$	1.24860e6	0.120968
$640$	0	0
$641$	8.14562e6	0.783031	0.391516	−	0.920172i	$-0.371951\pi$
$641$	8.14562e6	0.783031	0.391516	+	0.920172i	$0.371951\pi$
$642$	0	0
$643$	7.03003e6	0.670547	0.335274	−	0.942121i	$-0.391171\pi$
$643$	7.03003e6	0.670547	0.335274	+	0.942121i	$0.391171\pi$
$644$	0	0
$645$	1.96074e7	1.85575
$646$	0	0
$647$	1.91669e7	1.80008	0.900041	−	0.435806i	$-0.143536\pi$
$647$	1.91669e7	1.80008	0.900041	+	0.435806i	$0.143536\pi$
$648$	0	0
$649$	6.43258e6	0.599478
$650$	0	0
$651$	−6.21581e6	−0.574838
$652$	0	0
$653$	−9.08374e6	−0.833646	−0.416823	−	0.908988i	$-0.636857\pi$
$653$	−9.08374e6	−0.833646	−0.416823	+	0.908988i	$0.636857\pi$
$654$	0	0
$655$	3.41577e7	3.11089
$656$	0	0
$657$	5.97403e6	0.539950
$658$	0	0
$659$	−1.72281e7	−1.54534	−0.772670	−	0.634807i	$-0.781079\pi$
$659$	−1.72281e7	−1.54534	−0.772670	+	0.634807i	$0.781079\pi$
$660$	0	0
$661$	−2.00417e7	−1.78415	−0.892073	−	0.451892i	$-0.850749\pi$
$661$	−2.00417e7	−1.78415	−0.892073	+	0.451892i	$0.850749\pi$
$662$	0	0
$663$	−4.14382e6	−0.366115
$664$	0	0
$665$	3.33020e6	0.292022
$666$	0	0
$667$	2.69177e6	0.234273
$668$	0	0
$669$	−135299.	−0.0116877
$670$	0	0
$671$	1.12435e7	0.964040
$672$	0	0
$673$	1.73726e7	1.47852	0.739260	−	0.673420i	$-0.235175\pi$
$673$	1.73726e7	1.47852	0.739260	+	0.673420i	$0.235175\pi$
$674$	0	0
$675$	−5.27042e6	−0.445232
$676$	0	0
$677$	1.60159e7	1.34301	0.671507	−	0.740999i	$-0.265647\pi$
$677$	1.60159e7	1.34301	0.671507	+	0.740999i	$0.265647\pi$
$678$	0	0
$679$	−1.39152e7	−1.15829
$680$	0	0
$681$	−2.96305e6	−0.244834
$682$	0	0
$683$	−8.40319e6	−0.689275	−0.344638	−	0.938736i	$-0.611998\pi$
$683$	−8.40319e6	−0.689275	−0.344638	+	0.938736i	$0.611998\pi$
$684$	0	0
$685$	3.14142e7	2.55799
$686$	0	0
$687$	−6.78370e6	−0.548372
$688$	0	0
$689$	−6.12357e6	−0.491425
$690$	0	0
$691$	−1.06831e7	−0.851145	−0.425572	−	0.904924i	$-0.639927\pi$
$691$	−1.06831e7	−0.851145	−0.425572	+	0.904924i	$0.639927\pi$
$692$	0	0
$693$	−1.88733e6	−0.149284
$694$	0	0
$695$	−1.02312e7	−0.803462
$696$	0	0
$697$	−3.55281e7	−2.77006
$698$	0	0
$699$	982332.	0.0760441
$700$	0	0
$701$	2.13669e6	0.164228	0.0821139	−	0.996623i	$-0.473833\pi$
$701$	2.13669e6	0.164228	0.0821139	+	0.996623i	$0.473833\pi$
$702$	0	0
$703$	−2.20264e6	−0.168095
$704$	0	0
$705$	1.76221e7	1.33532
$706$	0	0
$707$	−1.27697e7	−0.960795
$708$	0	0
$709$	−2.74210e6	−0.204865	−0.102432	−	0.994740i	$-0.532663\pi$
$709$	−2.74210e6	−0.204865	−0.102432	+	0.994740i	$0.532663\pi$
$710$	0	0
$711$	637938.	0.0473265
$712$	0	0
$713$	4.44638e6	0.327554
$714$	0	0
$715$	7.95337e6	0.581816
$716$	0	0
$717$	−2.55120e6	−0.185330
$718$	0	0
$719$	2.08967e7	1.50749	0.753746	−	0.657166i	$-0.228245\pi$
$719$	2.08967e7	1.50749	0.753746	+	0.657166i	$0.228245\pi$
$720$	0	0
$721$	4.62388e6	0.331259
$722$	0	0
$723$	−2.04092e6	−0.145205
$724$	0	0
$725$	−3.67874e7	−2.59929
$726$	0	0
$727$	−3.90380e6	−0.273938	−0.136969	−	0.990575i	$-0.543736\pi$
$727$	−3.90380e6	−0.273938	−0.136969	+	0.990575i	$0.543736\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−3.57637e7	−2.47542
$732$	0	0
$733$	8.10804e6	0.557386	0.278693	−	0.960380i	$-0.410099\pi$
$733$	8.10804e6	0.557386	0.278693	+	0.960380i	$0.410099\pi$
$734$	0	0
$735$	−9.20891e6	−0.628767
$736$	0	0
$737$	−1.35154e7	−0.916561
$738$	0	0
$739$	−7.40086e6	−0.498507	−0.249253	−	0.968438i	$-0.580185\pi$
$739$	−7.40086e6	−0.498507	−0.249253	+	0.968438i	$0.580185\pi$
$740$	0	0
$741$	988018.	0.0661027
$742$	0	0
$743$	2.62112e7	1.74187	0.870934	−	0.491400i	$-0.163515\pi$
$743$	2.62112e7	1.74187	0.870934	+	0.491400i	$0.163515\pi$
$744$	0	0
$745$	−1.58352e6	−0.104528
$746$	0	0
$747$	4.45007e6	0.291787
$748$	0	0
$749$	1.16281e7	0.757363
$750$	0	0
$751$	7.41278e6	0.479602	0.239801	−	0.970822i	$-0.422918\pi$
$751$	7.41278e6	0.479602	0.239801	+	0.970822i	$0.422918\pi$
$752$	0	0
$753$	−5.38095e6	−0.345837
$754$	0	0
$755$	2.68871e7	1.71663
$756$	0	0
$757$	−3.53896e6	−0.224459	−0.112229	−	0.993682i	$-0.535799\pi$
$757$	−3.53896e6	−0.224459	−0.112229	+	0.993682i	$0.535799\pi$
$758$	0	0
$759$	1.35007e6	0.0850652
$760$	0	0
$761$	−2.20882e6	−0.138260	−0.0691302	−	0.997608i	$-0.522022\pi$
$761$	−2.20882e6	−0.138260	−0.0691302	+	0.997608i	$0.522022\pi$
$762$	0	0
$763$	7.25176e6	0.450954
$764$	0	0
$765$	1.37685e7	0.850615
$766$	0	0
$767$	−6.25247e6	−0.383763
$768$	0	0
$769$	−2.05829e7	−1.25514	−0.627569	−	0.778561i	$-0.715950\pi$
$769$	−2.05829e7	−1.25514	−0.627569	+	0.778561i	$0.715950\pi$
$770$	0	0
$771$	−166318.	−0.0100764
$772$	0	0
$773$	−1.83310e7	−1.10341	−0.551706	−	0.834039i	$-0.686023\pi$
$773$	−1.83310e7	−1.10341	−0.551706	+	0.834039i	$0.686023\pi$
$774$	0	0
$775$	−6.07671e7	−3.63424
$776$	0	0
$777$	−4.08972e6	−0.243019
$778$	0	0
$779$	8.47101e6	0.500140
$780$	0	0
$781$	4.37116e6	0.256430
$782$	0	0
$783$	3.70945e6	0.216224
$784$	0	0
$785$	4.67818e7	2.70958
$786$	0	0
$787$	−1.82884e7	−1.05254	−0.526270	−	0.850318i	$-0.676410\pi$
$787$	−1.82884e7	−1.05254	−0.526270	+	0.850318i	$0.676410\pi$
$788$	0	0
$789$	−7.92085e6	−0.452980
$790$	0	0
$791$	−1.47687e7	−0.839270
$792$	0	0
$793$	−1.09287e7	−0.617142
$794$	0	0
$795$	2.03465e7	1.14175
$796$	0	0
$797$	2.18110e6	0.121627	0.0608136	−	0.998149i	$-0.480630\pi$
$797$	2.18110e6	0.121627	0.0608136	+	0.998149i	$0.480630\pi$
$798$	0	0
$799$	−3.21426e7	−1.78121
$800$	0	0
$801$	−5.78135e6	−0.318381
$802$	0	0
$803$	2.09142e7	1.14459
$804$	0	0
$805$	−4.42312e6	−0.240568
$806$	0	0
$807$	2.73174e6	0.147657
$808$	0	0
$809$	2.11274e7	1.13495	0.567473	−	0.823392i	$-0.307921\pi$
$809$	2.11274e7	1.13495	0.567473	+	0.823392i	$0.307921\pi$
$810$	0	0
$811$	1.89757e7	1.01309	0.506544	−	0.862214i	$-0.330923\pi$
$811$	1.89757e7	1.01309	0.506544	+	0.862214i	$0.330923\pi$
$812$	0	0
$813$	−7.96945e6	−0.422865
$814$	0	0
$815$	−1.73987e7	−0.917535
$816$	0	0
$817$	8.52721e6	0.446943
$818$	0	0
$819$	1.83449e6	0.0955663
$820$	0	0
$821$	−4.63603e6	−0.240042	−0.120021	−	0.992771i	$-0.538296\pi$
$821$	−4.63603e6	−0.240042	−0.120021	+	0.992771i	$0.538296\pi$
$822$	0	0
$823$	1.15241e7	0.593072	0.296536	−	0.955022i	$-0.404169\pi$
$823$	1.15241e7	0.593072	0.296536	+	0.955022i	$0.404169\pi$
$824$	0	0
$825$	−1.84509e7	−0.943808
$826$	0	0
$827$	−1.90387e7	−0.967997	−0.483999	−	0.875069i	$-0.660816\pi$
$827$	−1.90387e7	−0.967997	−0.483999	+	0.875069i	$0.660816\pi$
$828$	0	0
$829$	1.13938e7	0.575815	0.287907	−	0.957658i	$-0.407040\pi$
$829$	1.13938e7	0.575815	0.287907	+	0.957658i	$0.407040\pi$
$830$	0	0
$831$	−1.65497e7	−0.831357
$832$	0	0
$833$	1.67970e7	0.838723
$834$	0	0
$835$	3.73889e7	1.85578
$836$	0	0
$837$	6.12742e6	0.302318
$838$	0	0
$839$	6.96218e6	0.341460	0.170730	−	0.985318i	$-0.445387\pi$
$839$	6.96218e6	0.341460	0.170730	+	0.985318i	$0.445387\pi$
$840$	0	0
$841$	5.38071e6	0.262331
$842$	0	0
$843$	−755599.	−0.0366203
$844$	0	0
$845$	3.00513e7	1.44784
$846$	0	0
$847$	6.62604e6	0.317355
$848$	0	0
$849$	−1.84520e7	−0.878566
$850$	0	0
$851$	2.92551e6	0.138477
$852$	0	0
$853$	9.86525e6	0.464233	0.232116	−	0.972688i	$-0.425435\pi$
$853$	9.86525e6	0.464233	0.232116	+	0.972688i	$0.425435\pi$
$854$	0	0
$855$	−3.28285e6	−0.153580
$856$	0	0
$857$	2.49451e6	0.116020	0.0580100	−	0.998316i	$-0.481524\pi$
$857$	2.49451e6	0.116020	0.0580100	+	0.998316i	$0.481524\pi$
$858$	0	0
$859$	2.14899e6	0.0993694	0.0496847	−	0.998765i	$-0.484178\pi$
$859$	2.14899e6	0.0993694	0.0496847	+	0.998765i	$0.484178\pi$
$860$	0	0
$861$	1.57284e7	0.723065
$862$	0	0
$863$	−1.29718e7	−0.592891	−0.296445	−	0.955050i	$-0.595801\pi$
$863$	−1.29718e7	−0.592891	−0.296445	+	0.955050i	$0.595801\pi$
$864$	0	0
$865$	−6.34148e7	−2.88171
$866$	0	0
$867$	−1.23349e7	−0.557299
$868$	0	0
$869$	2.23332e6	0.100323
$870$	0	0
$871$	1.31370e7	0.586748
$872$	0	0
$873$	1.37174e7	0.609166
$874$	0	0
$875$	3.43202e7	1.51541
$876$	0	0
$877$	1.80873e7	0.794100	0.397050	−	0.917797i	$-0.370034\pi$
$877$	1.80873e7	0.794100	0.397050	+	0.917797i	$0.370034\pi$
$878$	0	0
$879$	5.66602e6	0.247346
$880$	0	0
$881$	1.05688e7	0.458758	0.229379	−	0.973337i	$-0.426330\pi$
$881$	1.05688e7	0.458758	0.229379	+	0.973337i	$0.426330\pi$
$882$	0	0
$883$	−3.99141e7	−1.72276	−0.861380	−	0.507962i	$-0.830399\pi$
$883$	−3.99141e7	−1.72276	−0.861380	+	0.507962i	$0.830399\pi$
$884$	0	0
$885$	2.07748e7	0.891619
$886$	0	0
$887$	4.09888e7	1.74927	0.874633	−	0.484786i	$-0.161103\pi$
$887$	4.09888e7	1.74927	0.874633	+	0.484786i	$0.161103\pi$
$888$	0	0
$889$	2.64098e7	1.12076
$890$	0	0
$891$	1.86049e6	0.0785116
$892$	0	0
$893$	7.66382e6	0.321601
$894$	0	0
$895$	−3.79426e6	−0.158332
$896$	0	0
$897$	−1.31227e6	−0.0544555
$898$	0	0
$899$	4.27693e7	1.76495
$900$	0	0
$901$	−3.71119e7	−1.52301
$902$	0	0
$903$	1.58328e7	0.646156
$904$	0	0
$905$	1.43061e6	0.0580632
$906$	0	0
$907$	−2.61391e7	−1.05505	−0.527523	−	0.849541i	$-0.676879\pi$
$907$	−2.61391e7	−1.05505	−0.527523	+	0.849541i	$0.676879\pi$
$908$	0	0
$909$	1.25881e7	0.505301
$910$	0	0
$911$	−2.61805e7	−1.04516	−0.522579	−	0.852591i	$-0.675030\pi$
$911$	−2.61805e7	−1.04516	−0.522579	+	0.852591i	$0.675030\pi$
$912$	0	0
$913$	1.55790e7	0.618533
$914$	0	0
$915$	3.63123e7	1.43384
$916$	0	0
$917$	2.75819e7	1.08318
$918$	0	0
$919$	−4.29697e7	−1.67831	−0.839157	−	0.543889i	$-0.816951\pi$
$919$	−4.29697e7	−1.67831	−0.839157	+	0.543889i	$0.816951\pi$
$920$	0	0
$921$	−1.63266e7	−0.634230
$922$	0	0
$923$	−4.24878e6	−0.164157
$924$	0	0
$925$	−3.99820e7	−1.53642
$926$	0	0
$927$	−4.55813e6	−0.174216
$928$	0	0
$929$	3.43391e7	1.30542	0.652709	−	0.757609i	$-0.273633\pi$
$929$	3.43391e7	1.30542	0.652709	+	0.757609i	$0.273633\pi$
$930$	0	0
$931$	−4.00493e6	−0.151433
$932$	0	0
$933$	−6.77806e6	−0.254918
$934$	0	0
$935$	4.82014e7	1.80314
$936$	0	0
$937$	2.85062e7	1.06070	0.530348	−	0.847780i	$-0.322061\pi$
$937$	2.85062e7	1.06070	0.530348	+	0.847780i	$0.322061\pi$
$938$	0	0
$939$	1.76765e7	0.654233
$940$	0	0
$941$	3.11610e7	1.14719	0.573597	−	0.819138i	$-0.305547\pi$
$941$	3.11610e7	1.14719	0.573597	+	0.819138i	$0.305547\pi$
$942$	0	0
$943$	−1.12511e7	−0.412016
$944$	0	0
$945$	−6.09537e6	−0.222035
$946$	0	0
$947$	−4.20029e7	−1.52196	−0.760982	−	0.648773i	$-0.775283\pi$
$947$	−4.20029e7	−1.52196	−0.760982	+	0.648773i	$0.775283\pi$
$948$	0	0
$949$	−2.03286e7	−0.732726
$950$	0	0
$951$	−1.32604e7	−0.475452
$952$	0	0
$953$	−4.29183e6	−0.153077	−0.0765385	−	0.997067i	$-0.524387\pi$
$953$	−4.29183e6	−0.153077	−0.0765385	+	0.997067i	$0.524387\pi$
$954$	0	0
$955$	1.85893e7	0.659560
$956$	0	0
$957$	1.29862e7	0.458355
$958$	0	0
$959$	2.53666e7	0.890668
$960$	0	0
$961$	4.20190e7	1.46770
$962$	0	0
$963$	−1.14628e7	−0.398312
$964$	0	0
$965$	−2.73616e7	−0.945852
$966$	0	0
$967$	−2.28602e7	−0.786165	−0.393083	−	0.919503i	$-0.628591\pi$
$967$	−2.28602e7	−0.786165	−0.393083	+	0.919503i	$0.628591\pi$
$968$	0	0
$969$	5.98789e6	0.204863
$970$	0	0
$971$	2.48677e7	0.846422	0.423211	−	0.906031i	$-0.360903\pi$
$971$	2.48677e7	0.846422	0.423211	+	0.906031i	$0.360903\pi$
$972$	0	0
$973$	−8.26160e6	−0.279758
$974$	0	0
$975$	1.79343e7	0.604190
$976$	0	0
$977$	−6.94717e6	−0.232848	−0.116424	−	0.993200i	$-0.537143\pi$
$977$	−6.94717e6	−0.232848	−0.116424	+	0.993200i	$0.537143\pi$
$978$	0	0
$979$	−2.02396e7	−0.674909
$980$	0	0
$981$	−7.14864e6	−0.237165
$982$	0	0
$983$	−2.84653e7	−0.939576	−0.469788	−	0.882779i	$-0.655670\pi$
$983$	−2.84653e7	−0.939576	−0.469788	+	0.882779i	$0.655670\pi$
$984$	0	0
$985$	−4.51391e7	−1.48239
$986$	0	0
$987$	1.42297e7	0.464946
$988$	0	0
$989$	−1.13257e7	−0.368192
$990$	0	0
$991$	−1.16285e7	−0.376132	−0.188066	−	0.982156i	$-0.560222\pi$
$991$	−1.16285e7	−0.376132	−0.188066	+	0.982156i	$0.560222\pi$
$992$	0	0
$993$	1.14681e7	0.369080
$994$	0	0
$995$	8.20211e7	2.62644
$996$	0	0
$997$	8.01605e6	0.255401	0.127701	−	0.991813i	$-0.459240\pi$
$997$	8.01605e6	0.255401	0.127701	+	0.991813i	$0.459240\pi$
$998$	0	0
$999$	4.03157e6	0.127809

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.z.1.1		8
4.3	odd	2		552.6.a.h.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.6.a.h.1.1	✓	8	4.3	odd	2
1104.6.a.z.1.1		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1104.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -101.758 q^{5} -82.1684 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} -101.758 q^{5} -82.1684 q^{7} +81.0000 q^{9} +283.569 q^{11} -275.629 q^{13} +915.821 q^{15} -1670.45 q^{17} +398.288 q^{19} +739.515 q^{21} -529.000 q^{23} +7229.66 q^{25} -729.000 q^{27} -5088.40 q^{29} -8405.25 q^{31} -2552.12 q^{33} +8361.28 q^{35} -5530.27 q^{37} +2480.66 q^{39} +21268.6 q^{41} +21409.6 q^{43} -8242.39 q^{45} +19241.9 q^{47} -10055.4 q^{49} +15034.1 q^{51} +22216.7 q^{53} -28855.3 q^{55} -3584.59 q^{57} +22684.4 q^{59} +39650.0 q^{61} -6655.64 q^{63} +28047.4 q^{65} -47662.0 q^{67} +4761.00 q^{69} +15414.8 q^{71} +73753.4 q^{73} -65067.0 q^{75} -23300.4 q^{77} +7875.77 q^{79} +6561.00 q^{81} +54939.1 q^{83} +169981. q^{85} +45795.6 q^{87} -71374.7 q^{89} +22648.0 q^{91} +75647.2 q^{93} -40529.0 q^{95} +169350. q^{97} +22969.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 72 q^{3} - 4 q^{5} - 210 q^{7} + 648 q^{9}+O(q^{10}) \) 8 * q - 72 * q^3 - 4 * q^5 - 210 * q^7 + 648 * q^9	\( 8 q - 72 q^{3} - 4 q^{5} - 210 q^{7} + 648 q^{9} - 118 q^{11} - 12 q^{13} + 36 q^{15} + 310 q^{17} - 1352 q^{19} + 1890 q^{21} - 4232 q^{23} + 6916 q^{25} - 5832 q^{27} + 7440 q^{29} - 12896 q^{31} + 1062 q^{33} - 3840 q^{35} + 3262 q^{37} + 108 q^{39} + 9520 q^{41} + 5756 q^{43} - 324 q^{45} - 8068 q^{47} + 54880 q^{49} - 2790 q^{51} + 44568 q^{53} - 37356 q^{55} + 12168 q^{57} - 32116 q^{59} + 47322 q^{61} - 17010 q^{63} + 46912 q^{65} - 60148 q^{67} + 38088 q^{69} - 24200 q^{71} + 167340 q^{73} - 62244 q^{75} + 178316 q^{77} - 92366 q^{79} + 52488 q^{81} + 710 q^{83} + 227864 q^{85} - 66960 q^{87} + 103678 q^{89} - 2092 q^{91} + 116064 q^{93} - 82020 q^{95} + 255456 q^{97} - 9558 q^{99}+O(q^{100}) \) 8 * q - 72 * q^3 - 4 * q^5 - 210 * q^7 + 648 * q^9 - 118 * q^11 - 12 * q^13 + 36 * q^15 + 310 * q^17 - 1352 * q^19 + 1890 * q^21 - 4232 * q^23 + 6916 * q^25 - 5832 * q^27 + 7440 * q^29 - 12896 * q^31 + 1062 * q^33 - 3840 * q^35 + 3262 * q^37 + 108 * q^39 + 9520 * q^41 + 5756 * q^43 - 324 * q^45 - 8068 * q^47 + 54880 * q^49 - 2790 * q^51 + 44568 * q^53 - 37356 * q^55 + 12168 * q^57 - 32116 * q^59 + 47322 * q^61 - 17010 * q^63 + 46912 * q^65 - 60148 * q^67 + 38088 * q^69 - 24200 * q^71 + 167340 * q^73 - 62244 * q^75 + 178316 * q^77 - 92366 * q^79 + 52488 * q^81 + 710 * q^83 + 227864 * q^85 - 66960 * q^87 + 103678 * q^89 - 2092 * q^91 + 116064 * q^93 - 82020 * q^95 + 255456 * q^97 - 9558 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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