LMFDB - Embedded newform 1104.6.a.ba.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:	$0$
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} - 4608x^{6} - 3161x^{5} + 6284039x^{4} - 8279002x^{3} - 2677454576x^{2} + 13573192447x + 77901299860 $ x^8 - 4608x^6 - 3161x^5 + 6284039x^4 - 8279002x^3 - 2677454576x^2 + 13573192447x + 77901299860
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{18}\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			$-47.4070$ 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)

$f(q)$	$=$	$q-9.00000 q^{3} -92.8141 q^{5} +181.635 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} -92.8141 q^{5} +181.635 q^{7} +81.0000 q^{9} -24.0163 q^{11} -1168.42 q^{13} +835.327 q^{15} -1150.62 q^{17} -220.172 q^{19} -1634.71 q^{21} +529.000 q^{23} +5489.46 q^{25} -729.000 q^{27} -5390.73 q^{29} +6773.98 q^{31} +216.146 q^{33} -16858.3 q^{35} -6554.47 q^{37} +10515.8 q^{39} -6762.14 q^{41} +4754.96 q^{43} -7517.94 q^{45} +602.217 q^{47} +16184.1 q^{49} +10355.5 q^{51} +13067.9 q^{53} +2229.05 q^{55} +1981.55 q^{57} -21901.8 q^{59} -20035.3 q^{61} +14712.4 q^{63} +108446. q^{65} -55962.0 q^{67} -4761.00 q^{69} -37075.6 q^{71} -11810.8 q^{73} -49405.1 q^{75} -4362.19 q^{77} -87617.9 q^{79} +6561.00 q^{81} -31466.4 q^{83} +106793. q^{85} +48516.6 q^{87} -146765. q^{89} -212225. q^{91} -60965.8 q^{93} +20435.1 q^{95} -57166.0 q^{97} -1945.32 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 72 q^{3} + 16 q^{5} + 36 q^{7} + 648 q^{9}+O(q^{10}) $ 8 * q - 72 * q^3 + 16 * q^5 + 36 * q^7 + 648 * q^9	$ 8 q - 72 q^{3} + 16 q^{5} + 36 q^{7} + 648 q^{9} + 148 q^{11} + 696 q^{13} - 144 q^{15} + 572 q^{17} - 2456 q^{19} - 324 q^{21} + 4232 q^{23} + 11896 q^{25} - 5832 q^{27} + 6504 q^{29} - 3128 q^{31} - 1332 q^{33} - 6960 q^{35} + 3844 q^{37} - 6264 q^{39} + 6440 q^{41} - 7048 q^{43} + 1296 q^{45} - 30464 q^{47} + 46984 q^{49} - 5148 q^{51} + 63696 q^{53} - 32688 q^{55} + 22104 q^{57} - 54872 q^{59} + 12108 q^{61} + 2916 q^{63} + 124088 q^{65} - 139216 q^{67} - 38088 q^{69} - 76216 q^{71} + 13632 q^{73} - 107064 q^{75} - 78248 q^{77} - 126380 q^{79} + 52488 q^{81} - 238196 q^{83} + 117536 q^{85} - 58536 q^{87} + 123668 q^{89} - 248176 q^{91} + 28152 q^{93} - 183408 q^{95} + 18576 q^{97} + 11988 q^{99}+O(q^{100}) $ 8 * q - 72 * q^3 + 16 * q^5 + 36 * q^7 + 648 * q^9 + 148 * q^11 + 696 * q^13 - 144 * q^15 + 572 * q^17 - 2456 * q^19 - 324 * q^21 + 4232 * q^23 + 11896 * q^25 - 5832 * q^27 + 6504 * q^29 - 3128 * q^31 - 1332 * q^33 - 6960 * q^35 + 3844 * q^37 - 6264 * q^39 + 6440 * q^41 - 7048 * q^43 + 1296 * q^45 - 30464 * q^47 + 46984 * q^49 - 5148 * q^51 + 63696 * q^53 - 32688 * q^55 + 22104 * q^57 - 54872 * q^59 + 12108 * q^61 + 2916 * q^63 + 124088 * q^65 - 139216 * q^67 - 38088 * q^69 - 76216 * q^71 + 13632 * q^73 - 107064 * q^75 - 78248 * q^77 - 126380 * q^79 + 52488 * q^81 - 238196 * q^83 + 117536 * q^85 - 58536 * q^87 + 123668 * q^89 - 248176 * q^91 + 28152 * q^93 - 183408 * q^95 + 18576 * q^97 + 11988 * 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
$5$	−92.8141	−1.66031	−0.830155	−	0.557533i	$-0.811748\pi$
$5$	−92.8141	−1.66031	−0.830155	+	0.557533i	$0.811748\pi$
$6$	0	0
$7$	181.635	1.40105	0.700525	−	0.713628i	$-0.252949\pi$
$7$	181.635	1.40105	0.700525	+	0.713628i	$0.252949\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−24.0163	−0.0598445	−0.0299222	−	0.999552i	$-0.509526\pi$
$11$	−24.0163	−0.0598445	−0.0299222	+	0.999552i	$0.509526\pi$
$12$	0	0
$13$	−1168.42	−1.91752	−0.958760	−	0.284218i	$-0.908266\pi$
$13$	−1168.42	−1.91752	−0.958760	+	0.284218i	$0.908266\pi$
$14$	0	0
$15$	835.327	0.958580
$16$	0	0
$17$	−1150.62	−0.965623	−0.482812	−	0.875724i	$-0.660384\pi$
$17$	−1150.62	−0.965623	−0.482812	+	0.875724i	$0.660384\pi$
$18$	0	0
$19$	−220.172	−0.139919	−0.0699597	−	0.997550i	$-0.522287\pi$
$19$	−220.172	−0.139919	−0.0699597	+	0.997550i	$0.522287\pi$
$20$	0	0
$21$	−1634.71	−0.808897
$22$	0	0
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	5489.46	1.75663
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−5390.73	−1.19029	−0.595145	−	0.803618i	$-0.702905\pi$
$29$	−5390.73	−1.19029	−0.595145	+	0.803618i	$0.702905\pi$
$30$	0	0
$31$	6773.98	1.26602	0.633009	−	0.774145i	$-0.281820\pi$
$31$	6773.98	1.26602	0.633009	+	0.774145i	$0.281820\pi$
$32$	0	0
$33$	216.146	0.0345512
$34$	0	0
$35$	−16858.3	−2.32618
$36$	0	0
$37$	−6554.47	−0.787106	−0.393553	−	0.919302i	$-0.628754\pi$
$37$	−6554.47	−0.787106	−0.393553	+	0.919302i	$0.628754\pi$
$38$	0	0
$39$	10515.8	1.10708
$40$	0	0
$41$	−6762.14	−0.628238	−0.314119	−	0.949384i	$-0.601709\pi$
$41$	−6762.14	−0.628238	−0.314119	+	0.949384i	$0.601709\pi$
$42$	0	0
$43$	4754.96	0.392171	0.196086	−	0.980587i	$-0.437177\pi$
$43$	4754.96	0.392171	0.196086	+	0.980587i	$0.437177\pi$
$44$	0	0
$45$	−7517.94	−0.553436
$46$	0	0
$47$	602.217	0.0397656	0.0198828	−	0.999802i	$-0.493671\pi$
$47$	602.217	0.0397656	0.0198828	+	0.999802i	$0.493671\pi$
$48$	0	0
$49$	16184.1	0.962941
$50$	0	0
$51$	10355.5	0.557503
$52$	0	0
$53$	13067.9	0.639024	0.319512	−	0.947582i	$-0.396481\pi$
$53$	13067.9	0.639024	0.319512	+	0.947582i	$0.396481\pi$
$54$	0	0
$55$	2229.05	0.0993603
$56$	0	0
$57$	1981.55	0.0807825
$58$	0	0
$59$	−21901.8	−0.819126	−0.409563	−	0.912282i	$-0.634319\pi$
$59$	−21901.8	−0.819126	−0.409563	+	0.912282i	$0.634319\pi$
$60$	0	0
$61$	−20035.3	−0.689398	−0.344699	−	0.938713i	$-0.612019\pi$
$61$	−20035.3	−0.689398	−0.344699	+	0.938713i	$0.612019\pi$
$62$	0	0
$63$	14712.4	0.467017
$64$	0	0
$65$	108446.	3.18367
$66$	0	0
$67$	−55962.0	−1.52302	−0.761512	−	0.648151i	$-0.775542\pi$
$67$	−55962.0	−1.52302	−0.761512	+	0.648151i	$0.775542\pi$
$68$	0	0
$69$	−4761.00	−0.120386
$70$	0	0
$71$	−37075.6	−0.872854	−0.436427	−	0.899740i	$-0.643756\pi$
$71$	−37075.6	−0.872854	−0.436427	+	0.899740i	$0.643756\pi$
$72$	0	0
$73$	−11810.8	−0.259402	−0.129701	−	0.991553i	$-0.541402\pi$
$73$	−11810.8	−0.259402	−0.129701	+	0.991553i	$0.541402\pi$
$74$	0	0
$75$	−49405.1	−1.01419
$76$	0	0
$77$	−4362.19	−0.0838451
$78$	0	0
$79$	−87617.9	−1.57952	−0.789760	−	0.613416i	$-0.789795\pi$
$79$	−87617.9	−1.57952	−0.789760	+	0.613416i	$0.789795\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−31466.4	−0.501363	−0.250681	−	0.968070i	$-0.580655\pi$
$83$	−31466.4	−0.501363	−0.250681	+	0.968070i	$0.580655\pi$
$84$	0	0
$85$	106793.	1.60323
$86$	0	0
$87$	48516.6	0.687214
$88$	0	0
$89$	−146765.	−1.96402	−0.982010	−	0.188828i	$-0.939531\pi$
$89$	−146765.	−1.96402	−0.982010	+	0.188828i	$0.939531\pi$
$90$	0	0
$91$	−212225.	−2.68654
$92$	0	0
$93$	−60965.8	−0.730936
$94$	0	0
$95$	20435.1	0.232309
$96$	0	0
$97$	−57166.0	−0.616891	−0.308446	−	0.951242i	$-0.599809\pi$
$97$	−57166.0	−0.616891	−0.308446	+	0.951242i	$0.599809\pi$
$98$	0	0
$99$	−1945.32	−0.0199482
$100$	0	0
$101$	−178180.	−1.73803	−0.869014	−	0.494788i	$-0.835246\pi$
$101$	−178180.	−1.73803	−0.869014	+	0.494788i	$0.835246\pi$
$102$	0	0
$103$	27643.5	0.256743	0.128372	−	0.991726i	$-0.459025\pi$
$103$	27643.5	0.256743	0.128372	+	0.991726i	$0.459025\pi$
$104$	0	0
$105$	151724.	1.34302
$106$	0	0
$107$	−1692.34	−0.0142899	−0.00714494	−	0.999974i	$-0.502274\pi$
$107$	−1692.34	−0.0142899	−0.00714494	+	0.999974i	$0.502274\pi$
$108$	0	0
$109$	226922.	1.82941	0.914703	−	0.404127i	$-0.132425\pi$
$109$	226922.	1.82941	0.914703	+	0.404127i	$0.132425\pi$
$110$	0	0
$111$	58990.2	0.454436
$112$	0	0
$113$	19243.2	0.141769	0.0708844	−	0.997485i	$-0.477418\pi$
$113$	19243.2	0.141769	0.0708844	+	0.997485i	$0.477418\pi$
$114$	0	0
$115$	−49098.7	−0.346198
$116$	0	0
$117$	−94641.8	−0.639173
$118$	0	0
$119$	−208992.	−1.35289
$120$	0	0
$121$	−160474.	−0.996419
$122$	0	0
$123$	60859.2	0.362713
$124$	0	0
$125$	−219455.	−1.25623
$126$	0	0
$127$	160768.	0.884486	0.442243	−	0.896895i	$-0.354183\pi$
$127$	160768.	0.884486	0.442243	+	0.896895i	$0.354183\pi$
$128$	0	0
$129$	−42794.7	−0.226420
$130$	0	0
$131$	99896.1	0.508593	0.254296	−	0.967126i	$-0.418156\pi$
$131$	99896.1	0.508593	0.254296	+	0.967126i	$0.418156\pi$
$132$	0	0
$133$	−39990.8	−0.196034
$134$	0	0
$135$	67661.5	0.319527
$136$	0	0
$137$	175154.	0.797294	0.398647	−	0.917104i	$-0.369480\pi$
$137$	175154.	0.797294	0.398647	+	0.917104i	$0.369480\pi$
$138$	0	0
$139$	280729.	1.23240	0.616198	−	0.787591i	$-0.288672\pi$
$139$	280729.	1.23240	0.616198	+	0.787591i	$0.288672\pi$
$140$	0	0
$141$	−5419.95	−0.0229587
$142$	0	0
$143$	28061.0	0.114753
$144$	0	0
$145$	500336.	1.97625
$146$	0	0
$147$	−145657.	−0.555954
$148$	0	0
$149$	−154280.	−0.569304	−0.284652	−	0.958631i	$-0.591878\pi$
$149$	−154280.	−0.569304	−0.284652	+	0.958631i	$0.591878\pi$
$150$	0	0
$151$	−342089.	−1.22095	−0.610473	−	0.792037i	$-0.709021\pi$
$151$	−342089.	−1.22095	−0.610473	+	0.792037i	$0.709021\pi$
$152$	0	0
$153$	−93199.8	−0.321874
$154$	0	0
$155$	−628721.	−2.10198
$156$	0	0
$157$	443145.	1.43482	0.717409	−	0.696652i	$-0.245328\pi$
$157$	443145.	1.43482	0.717409	+	0.696652i	$0.245328\pi$
$158$	0	0
$159$	−117611.	−0.368941
$160$	0	0
$161$	96084.7	0.292139
$162$	0	0
$163$	14399.8	0.0424509	0.0212255	−	0.999775i	$-0.493243\pi$
$163$	14399.8	0.0424509	0.0212255	+	0.999775i	$0.493243\pi$
$164$	0	0
$165$	−20061.4	−0.0573657
$166$	0	0
$167$	42515.5	0.117966	0.0589828	−	0.998259i	$-0.481214\pi$
$167$	42515.5	0.117966	0.0589828	+	0.998259i	$0.481214\pi$
$168$	0	0
$169$	993907.	2.67688
$170$	0	0
$171$	−17833.9	−0.0466398
$172$	0	0
$173$	−101337.	−0.257425	−0.128713	−	0.991682i	$-0.541085\pi$
$173$	−101337.	−0.257425	−0.128713	+	0.991682i	$0.541085\pi$
$174$	0	0
$175$	997076.	2.46112
$176$	0	0
$177$	197117.	0.472922
$178$	0	0
$179$	408269.	0.952387	0.476194	−	0.879340i	$-0.342016\pi$
$179$	408269.	0.952387	0.476194	+	0.879340i	$0.342016\pi$
$180$	0	0
$181$	518848.	1.17718	0.588591	−	0.808431i	$-0.299683\pi$
$181$	518848.	1.17718	0.588591	+	0.808431i	$0.299683\pi$
$182$	0	0
$183$	180317.	0.398024
$184$	0	0
$185$	608347.	1.30684
$186$	0	0
$187$	27633.5	0.0577872
$188$	0	0
$189$	−132412.	−0.269632
$190$	0	0
$191$	917645.	1.82008	0.910041	−	0.414517i	$-0.136050\pi$
$191$	917645.	1.82008	0.910041	+	0.414517i	$0.136050\pi$
$192$	0	0
$193$	−25558.5	−0.0493904	−0.0246952	−	0.999695i	$-0.507862\pi$
$193$	−25558.5	−0.0493904	−0.0246952	+	0.999695i	$0.507862\pi$
$194$	0	0
$195$	−976011.	−1.83810
$196$	0	0
$197$	−827062.	−1.51835	−0.759176	−	0.650886i	$-0.774398\pi$
$197$	−827062.	−1.51835	−0.759176	+	0.650886i	$0.774398\pi$
$198$	0	0
$199$	898260.	1.60794	0.803969	−	0.594671i	$-0.202718\pi$
$199$	898260.	1.60794	0.803969	+	0.594671i	$0.202718\pi$
$200$	0	0
$201$	503658.	0.879318
$202$	0	0
$203$	−979144.	−1.66766
$204$	0	0
$205$	627622.	1.04307
$206$	0	0
$207$	42849.0	0.0695048
$208$	0	0
$209$	5287.71	0.00837340
$210$	0	0
$211$	711013.	1.09944	0.549720	−	0.835349i	$-0.314734\pi$
$211$	711013.	1.09944	0.549720	+	0.835349i	$0.314734\pi$
$212$	0	0
$213$	333680.	0.503943
$214$	0	0
$215$	−441328.	−0.651126
$216$	0	0
$217$	1.23039e6	1.77375
$218$	0	0
$219$	106297.	0.149766
$220$	0	0
$221$	1.34440e6	1.85160
$222$	0	0
$223$	−527293.	−0.710051	−0.355026	−	0.934857i	$-0.615528\pi$
$223$	−527293.	−0.710051	−0.355026	+	0.934857i	$0.615528\pi$
$224$	0	0
$225$	444646.	0.585542
$226$	0	0
$227$	−737069.	−0.949387	−0.474694	−	0.880151i	$-0.657441\pi$
$227$	−737069.	−0.949387	−0.474694	+	0.880151i	$0.657441\pi$
$228$	0	0
$229$	17678.6	0.0222772	0.0111386	−	0.999938i	$-0.496454\pi$
$229$	17678.6	0.0222772	0.0111386	+	0.999938i	$0.496454\pi$
$230$	0	0
$231$	39259.7	0.0484080
$232$	0	0
$233$	1.35902e6	1.63997	0.819986	−	0.572383i	$-0.193981\pi$
$233$	1.35902e6	1.63997	0.819986	+	0.572383i	$0.193981\pi$
$234$	0	0
$235$	−55894.2	−0.0660233
$236$	0	0
$237$	788561.	0.911936
$238$	0	0
$239$	1.13653e6	1.28702	0.643510	−	0.765438i	$-0.277477\pi$
$239$	1.13653e6	1.28702	0.643510	+	0.765438i	$0.277477\pi$
$240$	0	0
$241$	−1.45243e6	−1.61084	−0.805421	−	0.592704i	$-0.798061\pi$
$241$	−1.45243e6	−1.61084	−0.805421	+	0.592704i	$0.798061\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−1.50212e6	−1.59878
$246$	0	0
$247$	257253.	0.268298
$248$	0	0
$249$	283198.	0.289462
$250$	0	0
$251$	−217591.	−0.218000	−0.109000	−	0.994042i	$-0.534765\pi$
$251$	−217591.	−0.218000	−0.109000	+	0.994042i	$0.534765\pi$
$252$	0	0
$253$	−12704.6	−0.0124784
$254$	0	0
$255$	−961140.	−0.925627
$256$	0	0
$257$	−1.85937e6	−1.75604	−0.878018	−	0.478628i	$-0.841134\pi$
$257$	−1.85937e6	−1.75604	−0.878018	+	0.478628i	$0.841134\pi$
$258$	0	0
$259$	−1.19052e6	−1.10278
$260$	0	0
$261$	−436649.	−0.396763
$262$	0	0
$263$	−1.10186e6	−0.982280	−0.491140	−	0.871081i	$-0.663420\pi$
$263$	−1.10186e6	−0.982280	−0.491140	+	0.871081i	$0.663420\pi$
$264$	0	0
$265$	−1.21289e6	−1.06098
$266$	0	0
$267$	1.32088e6	1.13393
$268$	0	0
$269$	1.35947e6	1.14548	0.572741	−	0.819737i	$-0.305880\pi$
$269$	1.35947e6	1.14548	0.572741	+	0.819737i	$0.305880\pi$
$270$	0	0
$271$	−1.30244e6	−1.07730	−0.538649	−	0.842530i	$-0.681065\pi$
$271$	−1.30244e6	−1.07730	−0.538649	+	0.842530i	$0.681065\pi$
$272$	0	0
$273$	1.91003e6	1.55107
$274$	0	0
$275$	−131836.	−0.105124
$276$	0	0
$277$	2.42107e6	1.89586	0.947932	−	0.318472i	$-0.103170\pi$
$277$	2.42107e6	1.89586	0.947932	+	0.318472i	$0.103170\pi$
$278$	0	0
$279$	548693.	0.422006
$280$	0	0
$281$	−1.41435e6	−1.06854	−0.534271	−	0.845313i	$-0.679414\pi$
$281$	−1.41435e6	−1.06854	−0.534271	+	0.845313i	$0.679414\pi$
$282$	0	0
$283$	132703.	0.0984951	0.0492476	−	0.998787i	$-0.484318\pi$
$283$	132703.	0.0984951	0.0492476	+	0.998787i	$0.484318\pi$
$284$	0	0
$285$	−183915.	−0.134124
$286$	0	0
$287$	−1.22824e6	−0.880193
$288$	0	0
$289$	−95942.0	−0.0675716
$290$	0	0
$291$	514494.	0.356162
$292$	0	0
$293$	−1.29301e6	−0.879899	−0.439949	−	0.898023i	$-0.645004\pi$
$293$	−1.29301e6	−0.879899	−0.439949	+	0.898023i	$0.645004\pi$
$294$	0	0
$295$	2.03280e6	1.36000
$296$	0	0
$297$	17507.9	0.0115171
$298$	0	0
$299$	−618093.	−0.399830
$300$	0	0
$301$	863666.	0.549452
$302$	0	0
$303$	1.60362e6	1.00345
$304$	0	0
$305$	1.85955e6	1.14461
$306$	0	0
$307$	−276264.	−0.167293	−0.0836466	−	0.996495i	$-0.526657\pi$
$307$	−276264.	−0.167293	−0.0836466	+	0.996495i	$0.526657\pi$
$308$	0	0
$309$	−248791.	−0.148231
$310$	0	0
$311$	3.03455e6	1.77907	0.889535	−	0.456868i	$-0.151029\pi$
$311$	3.03455e6	1.77907	0.889535	+	0.456868i	$0.151029\pi$
$312$	0	0
$313$	754019.	0.435032	0.217516	−	0.976057i	$-0.430205\pi$
$313$	754019.	0.435032	0.217516	+	0.976057i	$0.430205\pi$
$314$	0	0
$315$	−1.36552e6	−0.775392
$316$	0	0
$317$	−1.89570e6	−1.05955	−0.529775	−	0.848138i	$-0.677724\pi$
$317$	−1.89570e6	−1.05955	−0.529775	+	0.848138i	$0.677724\pi$
$318$	0	0
$319$	129465.	0.0712322
$320$	0	0
$321$	15231.1	0.00825026
$322$	0	0
$323$	253333.	0.135109
$324$	0	0
$325$	−6.41398e6	−3.36837
$326$	0	0
$327$	−2.04230e6	−1.05621
$328$	0	0
$329$	109383.	0.0557137
$330$	0	0
$331$	3.77271e6	1.89271	0.946353	−	0.323134i	$-0.104736\pi$
$331$	3.77271e6	1.89271	0.946353	+	0.323134i	$0.104736\pi$
$332$	0	0
$333$	−530912.	−0.262369
$334$	0	0
$335$	5.19407e6	2.52869
$336$	0	0
$337$	1.40856e6	0.675619	0.337809	−	0.941215i	$-0.390314\pi$
$337$	1.40856e6	0.675619	0.337809	+	0.941215i	$0.390314\pi$
$338$	0	0
$339$	−173189.	−0.0818503
$340$	0	0
$341$	−162686.	−0.0757642
$342$	0	0
$343$	−113132.	−0.0519219
$344$	0	0
$345$	441888.	0.199878
$346$	0	0
$347$	−278032.	−0.123957	−0.0619785	−	0.998077i	$-0.519741\pi$
$347$	−278032.	−0.123957	−0.0619785	+	0.998077i	$0.519741\pi$
$348$	0	0
$349$	−242850.	−0.106727	−0.0533634	−	0.998575i	$-0.516994\pi$
$349$	−242850.	−0.106727	−0.0533634	+	0.998575i	$0.516994\pi$
$350$	0	0
$351$	851777.	0.369027
$352$	0	0
$353$	2.45325e6	1.04787	0.523933	−	0.851759i	$-0.324464\pi$
$353$	2.45325e6	1.04787	0.523933	+	0.851759i	$0.324464\pi$
$354$	0	0
$355$	3.44113e6	1.44921
$356$	0	0
$357$	1.88092e6	0.781089
$358$	0	0
$359$	990712.	0.405706	0.202853	−	0.979209i	$-0.434979\pi$
$359$	990712.	0.405706	0.202853	+	0.979209i	$0.434979\pi$
$360$	0	0
$361$	−2.42762e6	−0.980423
$362$	0	0
$363$	1.44427e6	0.575283
$364$	0	0
$365$	1.09621e6	0.430687
$366$	0	0
$367$	−2.19648e6	−0.851260	−0.425630	−	0.904897i	$-0.639947\pi$
$367$	−2.19648e6	−0.851260	−0.425630	+	0.904897i	$0.639947\pi$
$368$	0	0
$369$	−547733.	−0.209413
$370$	0	0
$371$	2.37359e6	0.895305
$372$	0	0
$373$	433728.	0.161415	0.0807077	−	0.996738i	$-0.474282\pi$
$373$	433728.	0.161415	0.0807077	+	0.996738i	$0.474282\pi$
$374$	0	0
$375$	1.97509e6	0.725287
$376$	0	0
$377$	6.29863e6	2.28240
$378$	0	0
$379$	22086.2	0.00789810	0.00394905	−	0.999992i	$-0.498743\pi$
$379$	22086.2	0.00789810	0.00394905	+	0.999992i	$0.498743\pi$
$380$	0	0
$381$	−1.44691e6	−0.510658
$382$	0	0
$383$	−345008.	−0.120180	−0.0600900	−	0.998193i	$-0.519139\pi$
$383$	−345008.	−0.120180	−0.0600900	+	0.998193i	$0.519139\pi$
$384$	0	0
$385$	404873.	0.139209
$386$	0	0
$387$	385152.	0.130724
$388$	0	0
$389$	−1.52089e6	−0.509592	−0.254796	−	0.966995i	$-0.582008\pi$
$389$	−1.52089e6	−0.509592	−0.254796	+	0.966995i	$0.582008\pi$
$390$	0	0
$391$	−608675.	−0.201346
$392$	0	0
$393$	−899065.	−0.293636
$394$	0	0
$395$	8.13217e6	2.62249
$396$	0	0
$397$	102791.	0.0327325	0.0163662	−	0.999866i	$-0.494790\pi$
$397$	102791.	0.0327325	0.0163662	+	0.999866i	$0.494790\pi$
$398$	0	0
$399$	359918.	0.113180
$400$	0	0
$401$	−926490.	−0.287726	−0.143863	−	0.989598i	$-0.545953\pi$
$401$	−926490.	−0.287726	−0.143863	+	0.989598i	$0.545953\pi$
$402$	0	0
$403$	−7.91484e6	−2.42761
$404$	0	0
$405$	−608953.	−0.184479
$406$	0	0
$407$	157414.	0.0471039
$408$	0	0
$409$	−391970.	−0.115863	−0.0579314	−	0.998321i	$-0.518450\pi$
$409$	−391970.	−0.115863	−0.0579314	+	0.998321i	$0.518450\pi$
$410$	0	0
$411$	−1.57639e6	−0.460318
$412$	0	0
$413$	−3.97813e6	−1.14764
$414$	0	0
$415$	2.92053e6	0.832417
$416$	0	0
$417$	−2.52656e6	−0.711525
$418$	0	0
$419$	4.19826e6	1.16825	0.584123	−	0.811665i	$-0.301439\pi$
$419$	4.19826e6	1.16825	0.584123	+	0.811665i	$0.301439\pi$
$420$	0	0
$421$	6.68650e6	1.83863	0.919313	−	0.393528i	$-0.128745\pi$
$421$	6.68650e6	1.83863	0.919313	+	0.393528i	$0.128745\pi$
$422$	0	0
$423$	48779.5	0.0132552
$424$	0	0
$425$	−6.31625e6	−1.69624
$426$	0	0
$427$	−3.63910e6	−0.965882
$428$	0	0
$429$	−252549.	−0.0662526
$430$	0	0
$431$	1.98861e6	0.515651	0.257826	−	0.966191i	$-0.416994\pi$
$431$	1.98861e6	0.515651	0.257826	+	0.966191i	$0.416994\pi$
$432$	0	0
$433$	1.34963e6	0.345935	0.172968	−	0.984928i	$-0.444664\pi$
$433$	1.34963e6	0.345935	0.172968	+	0.984928i	$0.444664\pi$
$434$	0	0
$435$	−4.50302e6	−1.14099
$436$	0	0
$437$	−116471.	−0.0291752
$438$	0	0
$439$	−109649.	−0.0271545	−0.0135773	−	0.999908i	$-0.504322\pi$
$439$	−109649.	−0.0271545	−0.0135773	+	0.999908i	$0.504322\pi$
$440$	0	0
$441$	1.31092e6	0.320980
$442$	0	0
$443$	−5.54743e6	−1.34302	−0.671510	−	0.740995i	$-0.734354\pi$
$443$	−5.54743e6	−1.34302	−0.671510	+	0.740995i	$0.734354\pi$
$444$	0	0
$445$	1.36218e7	3.26088
$446$	0	0
$447$	1.38852e6	0.328688
$448$	0	0
$449$	−4.34336e6	−1.01674	−0.508370	−	0.861139i	$-0.669752\pi$
$449$	−4.34336e6	−1.01674	−0.508370	+	0.861139i	$0.669752\pi$
$450$	0	0
$451$	162401.	0.0375966
$452$	0	0
$453$	3.07880e6	0.704913
$454$	0	0
$455$	1.96975e7	4.46049
$456$	0	0
$457$	−1.06256e6	−0.237993	−0.118996	−	0.992895i	$-0.537968\pi$
$457$	−1.06256e6	−0.237993	−0.118996	+	0.992895i	$0.537968\pi$
$458$	0	0
$459$	838798.	0.185834
$460$	0	0
$461$	1.06785e6	0.234022	0.117011	−	0.993131i	$-0.462669\pi$
$461$	1.06785e6	0.234022	0.117011	+	0.993131i	$0.462669\pi$
$462$	0	0
$463$	5.65391e6	1.22573	0.612867	−	0.790186i	$-0.290016\pi$
$463$	5.65391e6	1.22573	0.612867	+	0.790186i	$0.290016\pi$
$464$	0	0
$465$	5.65849e6	1.21358
$466$	0	0
$467$	6.53791e6	1.38722	0.693612	−	0.720349i	$-0.256018\pi$
$467$	6.53791e6	1.38722	0.693612	+	0.720349i	$0.256018\pi$
$468$	0	0
$469$	−1.01646e7	−2.13383
$470$	0	0
$471$	−3.98830e6	−0.828392
$472$	0	0
$473$	−114196.	−0.0234693
$474$	0	0
$475$	−1.20862e6	−0.245786
$476$	0	0
$477$	1.05850e6	0.213008
$478$	0	0
$479$	1.20709e6	0.240381	0.120191	−	0.992751i	$-0.461649\pi$
$479$	1.20709e6	0.240381	0.120191	+	0.992751i	$0.461649\pi$
$480$	0	0
$481$	7.65836e6	1.50929
$482$	0	0
$483$	−864763.	−0.168667
$484$	0	0
$485$	5.30581e6	1.02423
$486$	0	0
$487$	−2.10100e6	−0.401424	−0.200712	−	0.979650i	$-0.564326\pi$
$487$	−2.10100e6	−0.401424	−0.200712	+	0.979650i	$0.564326\pi$
$488$	0	0
$489$	−129598.	−0.0245091
$490$	0	0
$491$	2.53024e6	0.473650	0.236825	−	0.971552i	$-0.423893\pi$
$491$	2.53024e6	0.473650	0.236825	+	0.971552i	$0.423893\pi$
$492$	0	0
$493$	6.20266e6	1.14937
$494$	0	0
$495$	180553.	0.0331201
$496$	0	0
$497$	−6.73421e6	−1.22291
$498$	0	0
$499$	−4.12225e6	−0.741110	−0.370555	−	0.928811i	$-0.620832\pi$
$499$	−4.12225e6	−0.741110	−0.370555	+	0.928811i	$0.620832\pi$
$500$	0	0
$501$	−382639.	−0.0681075
$502$	0	0
$503$	9.64238e6	1.69928	0.849639	−	0.527365i	$-0.176820\pi$
$503$	9.64238e6	1.69928	0.849639	+	0.527365i	$0.176820\pi$
$504$	0	0
$505$	1.65377e7	2.88566
$506$	0	0
$507$	−8.94516e6	−1.54550
$508$	0	0
$509$	−5.06889e6	−0.867199	−0.433599	−	0.901106i	$-0.642757\pi$
$509$	−5.06889e6	−0.867199	−0.433599	+	0.901106i	$0.642757\pi$
$510$	0	0
$511$	−2.14526e6	−0.363435
$512$	0	0
$513$	160505.	0.0269275
$514$	0	0
$515$	−2.56570e6	−0.426273
$516$	0	0
$517$	−14463.0	−0.00237975
$518$	0	0
$519$	912030.	0.148625
$520$	0	0
$521$	−5.21898e6	−0.842347	−0.421174	−	0.906980i	$-0.638382\pi$
$521$	−5.21898e6	−0.842347	−0.421174	+	0.906980i	$0.638382\pi$
$522$	0	0
$523$	1.14382e7	1.82854	0.914271	−	0.405102i	$-0.132764\pi$
$523$	1.14382e7	1.82854	0.914271	+	0.405102i	$0.132764\pi$
$524$	0	0
$525$	−8.97368e6	−1.42093
$526$	0	0
$527$	−7.79425e6	−1.22250
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	−1.77405e6	−0.273042
$532$	0	0
$533$	7.90100e6	1.20466
$534$	0	0
$535$	157073.	0.0237256
$536$	0	0
$537$	−3.67442e6	−0.549861
$538$	0	0
$539$	−388683.	−0.0576267
$540$	0	0
$541$	−9.64880e6	−1.41736	−0.708680	−	0.705530i	$-0.750709\pi$
$541$	−9.64880e6	−1.41736	−0.708680	+	0.705530i	$0.750709\pi$
$542$	0	0
$543$	−4.66963e6	−0.679646
$544$	0	0
$545$	−2.10615e7	−3.03738
$546$	0	0
$547$	1.03888e7	1.48456	0.742281	−	0.670088i	$-0.233744\pi$
$547$	1.03888e7	1.48456	0.742281	+	0.670088i	$0.233744\pi$
$548$	0	0
$549$	−1.62286e6	−0.229799
$550$	0	0
$551$	1.18689e6	0.166545
$552$	0	0
$553$	−1.59144e7	−2.21299
$554$	0	0
$555$	−5.47513e6	−0.754504
$556$	0	0
$557$	7.11017e6	0.971051	0.485525	−	0.874223i	$-0.338628\pi$
$557$	7.11017e6	0.971051	0.485525	+	0.874223i	$0.338628\pi$
$558$	0	0
$559$	−5.55578e6	−0.751996
$560$	0	0
$561$	−248701.	−0.0333635
$562$	0	0
$563$	−2.20497e6	−0.293179	−0.146589	−	0.989197i	$-0.546830\pi$
$563$	−2.20497e6	−0.293179	−0.146589	+	0.989197i	$0.546830\pi$
$564$	0	0
$565$	−1.78604e6	−0.235380
$566$	0	0
$567$	1.19170e6	0.155672
$568$	0	0
$569$	−4.91284e6	−0.636139	−0.318070	−	0.948067i	$-0.603035\pi$
$569$	−4.91284e6	−0.636139	−0.318070	+	0.948067i	$0.603035\pi$
$570$	0	0
$571$	−166775.	−0.0214063	−0.0107031	−	0.999943i	$-0.503407\pi$
$571$	−166775.	−0.0214063	−0.0107031	+	0.999943i	$0.503407\pi$
$572$	0	0
$573$	−8.25880e6	−1.05083
$574$	0	0
$575$	2.90392e6	0.366282
$576$	0	0
$577$	1.54780e7	1.93542	0.967708	−	0.252075i	$-0.0811130\pi$
$577$	1.54780e7	1.93542	0.967708	+	0.252075i	$0.0811130\pi$
$578$	0	0
$579$	230027.	0.0285155
$580$	0	0
$581$	−5.71539e6	−0.702434
$582$	0	0
$583$	−313843.	−0.0382421
$584$	0	0
$585$	8.78410e6	1.06122
$586$	0	0
$587$	−6.69649e6	−0.802144	−0.401072	−	0.916047i	$-0.631362\pi$
$587$	−6.69649e6	−0.802144	−0.401072	+	0.916047i	$0.631362\pi$
$588$	0	0
$589$	−1.49144e6	−0.177140
$590$	0	0
$591$	7.44356e6	0.876621
$592$	0	0
$593$	−8.34847e6	−0.974923	−0.487462	−	0.873144i	$-0.662077\pi$
$593$	−8.34847e6	−0.974923	−0.487462	+	0.873144i	$0.662077\pi$
$594$	0	0
$595$	1.93974e7	2.24621
$596$	0	0
$597$	−8.08434e6	−0.928344
$598$	0	0
$599$	−6.44836e6	−0.734314	−0.367157	−	0.930159i	$-0.619669\pi$
$599$	−6.44836e6	−0.734314	−0.367157	+	0.930159i	$0.619669\pi$
$600$	0	0
$601$	−3.26437e6	−0.368649	−0.184324	−	0.982865i	$-0.559010\pi$
$601$	−3.26437e6	−0.368649	−0.184324	+	0.982865i	$0.559010\pi$
$602$	0	0
$603$	−4.53292e6	−0.507674
$604$	0	0
$605$	1.48943e7	1.65436
$606$	0	0
$607$	−9.96030e6	−1.09724	−0.548619	−	0.836073i	$-0.684846\pi$
$607$	−9.96030e6	−1.09724	−0.548619	+	0.836073i	$0.684846\pi$
$608$	0	0
$609$	8.81229e6	0.962821
$610$	0	0
$611$	−703641.	−0.0762514
$612$	0	0
$613$	1.55991e6	0.167667	0.0838334	−	0.996480i	$-0.473284\pi$
$613$	1.55991e6	0.167667	0.0838334	+	0.996480i	$0.473284\pi$
$614$	0	0
$615$	−5.64859e6	−0.602216
$616$	0	0
$617$	−8.75043e6	−0.925373	−0.462686	−	0.886522i	$-0.653114\pi$
$617$	−8.75043e6	−0.925373	−0.462686	+	0.886522i	$0.653114\pi$
$618$	0	0
$619$	−9.82960e6	−1.03112	−0.515560	−	0.856853i	$-0.672416\pi$
$619$	−9.82960e6	−1.03112	−0.515560	+	0.856853i	$0.672416\pi$
$620$	0	0
$621$	−385641.	−0.0401286
$622$	0	0
$623$	−2.66575e7	−2.75169
$624$	0	0
$625$	3.21396e6	0.329110
$626$	0	0
$627$	−47589.4	−0.00483438
$628$	0	0
$629$	7.54167e6	0.760048
$630$	0	0
$631$	130172.	0.0130150	0.00650750	−	0.999979i	$-0.497929\pi$
$631$	130172.	0.0130150	0.00650750	+	0.999979i	$0.497929\pi$
$632$	0	0
$633$	−6.39912e6	−0.634762
$634$	0	0
$635$	−1.49216e7	−1.46852
$636$	0	0
$637$	−1.89098e7	−1.84646
$638$	0	0
$639$	−3.00312e6	−0.290951
$640$	0	0
$641$	1.89935e7	1.82583	0.912914	−	0.408151i	$-0.133826\pi$
$641$	1.89935e7	1.82583	0.912914	+	0.408151i	$0.133826\pi$
$642$	0	0
$643$	1.07997e7	1.03011	0.515055	−	0.857157i	$-0.327772\pi$
$643$	1.07997e7	1.03011	0.515055	+	0.857157i	$0.327772\pi$
$644$	0	0
$645$	3.97195e6	0.375928
$646$	0	0
$647$	−7.72961e6	−0.725933	−0.362967	−	0.931802i	$-0.618236\pi$
$647$	−7.72961e6	−0.725933	−0.362967	+	0.931802i	$0.618236\pi$
$648$	0	0
$649$	526001.	0.0490201
$650$	0	0
$651$	−1.10735e7	−1.02408
$652$	0	0
$653$	−6.12939e6	−0.562515	−0.281258	−	0.959632i	$-0.590752\pi$
$653$	−6.12939e6	−0.562515	−0.281258	+	0.959632i	$0.590752\pi$
$654$	0	0
$655$	−9.27177e6	−0.844421
$656$	0	0
$657$	−956677.	−0.0864673
$658$	0	0
$659$	9.21169e6	0.826277	0.413139	−	0.910668i	$-0.364432\pi$
$659$	9.21169e6	0.826277	0.413139	+	0.910668i	$0.364432\pi$
$660$	0	0
$661$	1.15098e7	1.02462	0.512310	−	0.858800i	$-0.328790\pi$
$661$	1.15098e7	1.02462	0.512310	+	0.858800i	$0.328790\pi$
$662$	0	0
$663$	−1.20996e7	−1.06902
$664$	0	0
$665$	3.71171e6	0.325477
$666$	0	0
$667$	−2.85170e6	−0.248193
$668$	0	0
$669$	4.74564e6	0.409948
$670$	0	0
$671$	481172.	0.0412567
$672$	0	0
$673$	6.01149e6	0.511616	0.255808	−	0.966728i	$-0.417658\pi$
$673$	6.01149e6	0.511616	0.255808	+	0.966728i	$0.417658\pi$
$674$	0	0
$675$	−4.00181e6	−0.338063
$676$	0	0
$677$	−2.89719e6	−0.242943	−0.121472	−	0.992595i	$-0.538761\pi$
$677$	−2.89719e6	−0.242943	−0.121472	+	0.992595i	$0.538761\pi$
$678$	0	0
$679$	−1.03833e7	−0.864295
$680$	0	0
$681$	6.63362e6	0.548129
$682$	0	0
$683$	1.53709e7	1.26080	0.630401	−	0.776269i	$-0.282890\pi$
$683$	1.53709e7	1.26080	0.630401	+	0.776269i	$0.282890\pi$
$684$	0	0
$685$	−1.62568e7	−1.32375
$686$	0	0
$687$	−159108.	−0.0128617
$688$	0	0
$689$	−1.52688e7	−1.22534
$690$	0	0
$691$	−4.72378e6	−0.376353	−0.188176	−	0.982135i	$-0.560258\pi$
$691$	−4.72378e6	−0.376353	−0.188176	+	0.982135i	$0.560258\pi$
$692$	0	0
$693$	−353337.	−0.0279484
$694$	0	0
$695$	−2.60556e7	−2.04616
$696$	0	0
$697$	7.78062e6	0.606641
$698$	0	0
$699$	−1.22312e7	−0.946838
$700$	0	0
$701$	2.13344e7	1.63978	0.819890	−	0.572521i	$-0.194034\pi$
$701$	2.13344e7	1.63978	0.819890	+	0.572521i	$0.194034\pi$
$702$	0	0
$703$	1.44311e6	0.110131
$704$	0	0
$705$	503048.	0.0381185
$706$	0	0
$707$	−3.23638e7	−2.43506
$708$	0	0
$709$	1.18282e7	0.883694	0.441847	−	0.897090i	$-0.354323\pi$
$709$	1.18282e7	0.883694	0.441847	+	0.897090i	$0.354323\pi$
$710$	0	0
$711$	−7.09705e6	−0.526506
$712$	0	0
$713$	3.58344e6	0.263983
$714$	0	0
$715$	−2.60446e6	−0.190525
$716$	0	0
$717$	−1.02288e7	−0.743061
$718$	0	0
$719$	2.16577e7	1.56239	0.781195	−	0.624288i	$-0.214611\pi$
$719$	2.16577e7	1.56239	0.781195	+	0.624288i	$0.214611\pi$
$720$	0	0
$721$	5.02101e6	0.359710
$722$	0	0
$723$	1.30719e7	0.930020
$724$	0	0
$725$	−2.95922e7	−2.09089
$726$	0	0
$727$	−1.77127e7	−1.24293	−0.621466	−	0.783441i	$-0.713463\pi$
$727$	−1.77127e7	−1.24293	−0.621466	+	0.783441i	$0.713463\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−5.47113e6	−0.378690
$732$	0	0
$733$	−369850.	−0.0254253	−0.0127126	−	0.999919i	$-0.504047\pi$
$733$	−369850.	−0.0254253	−0.0127126	+	0.999919i	$0.504047\pi$
$734$	0	0
$735$	1.35191e7	0.923056
$736$	0	0
$737$	1.34400e6	0.0911445
$738$	0	0
$739$	2.79824e7	1.88484	0.942418	−	0.334439i	$-0.108547\pi$
$739$	2.79824e7	1.88484	0.942418	+	0.334439i	$0.108547\pi$
$740$	0	0
$741$	−2.31527e6	−0.154902
$742$	0	0
$743$	1.51857e7	1.00917	0.504584	−	0.863363i	$-0.331646\pi$
$743$	1.51857e7	1.00917	0.504584	+	0.863363i	$0.331646\pi$
$744$	0	0
$745$	1.43194e7	0.945220
$746$	0	0
$747$	−2.54878e6	−0.167121
$748$	0	0
$749$	−307388.	−0.0200208
$750$	0	0
$751$	−1.48086e7	−0.958107	−0.479053	−	0.877786i	$-0.659020\pi$
$751$	−1.48086e7	−0.958107	−0.479053	+	0.877786i	$0.659020\pi$
$752$	0	0
$753$	1.95832e6	0.125862
$754$	0	0
$755$	3.17507e7	2.02715
$756$	0	0
$757$	2.27527e7	1.44309	0.721544	−	0.692369i	$-0.243433\pi$
$757$	2.27527e7	1.44309	0.721544	+	0.692369i	$0.243433\pi$
$758$	0	0
$759$	114341.	0.00720443
$760$	0	0
$761$	1.54881e7	0.969474	0.484737	−	0.874660i	$-0.338915\pi$
$761$	1.54881e7	0.969474	0.484737	+	0.874660i	$0.338915\pi$
$762$	0	0
$763$	4.12169e7	2.56309
$764$	0	0
$765$	8.65026e6	0.534411
$766$	0	0
$767$	2.55905e7	1.57069
$768$	0	0
$769$	4.57454e6	0.278953	0.139477	−	0.990225i	$-0.455458\pi$
$769$	4.57454e6	0.278953	0.139477	+	0.990225i	$0.455458\pi$
$770$	0	0
$771$	1.67343e7	1.01385
$772$	0	0
$773$	2.90104e7	1.74625	0.873123	−	0.487500i	$-0.162091\pi$
$773$	2.90104e7	1.74625	0.873123	+	0.487500i	$0.162091\pi$
$774$	0	0
$775$	3.71855e7	2.22392
$776$	0	0
$777$	1.07147e7	0.636688
$778$	0	0
$779$	1.48883e6	0.0879027
$780$	0	0
$781$	890417.	0.0522355
$782$	0	0
$783$	3.92984e6	0.229071
$784$	0	0
$785$	−4.11301e7	−2.38224
$786$	0	0
$787$	−2.69458e7	−1.55079	−0.775396	−	0.631475i	$-0.782450\pi$
$787$	−2.69458e7	−1.55079	−0.775396	+	0.631475i	$0.782450\pi$
$788$	0	0
$789$	9.91670e6	0.567119
$790$	0	0
$791$	3.49523e6	0.198625
$792$	0	0
$793$	2.34095e7	1.32193
$794$	0	0
$795$	1.09160e7	0.612556
$796$	0	0
$797$	−317708.	−0.0177167	−0.00885835	−	0.999961i	$-0.502820\pi$
$797$	−317708.	−0.0177167	−0.00885835	+	0.999961i	$0.502820\pi$
$798$	0	0
$799$	−692919.	−0.0383986
$800$	0	0
$801$	−1.18879e7	−0.654673
$802$	0	0
$803$	283652.	0.0155238
$804$	0	0
$805$	−8.91802e6	−0.485041
$806$	0	0
$807$	−1.22352e7	−0.661344
$808$	0	0
$809$	1.13772e7	0.611175	0.305588	−	0.952164i	$-0.401147\pi$
$809$	1.13772e7	0.611175	0.305588	+	0.952164i	$0.401147\pi$
$810$	0	0
$811$	−1.46389e6	−0.0781551	−0.0390775	−	0.999236i	$-0.512442\pi$
$811$	−1.46389e6	−0.0781551	−0.0390775	+	0.999236i	$0.512442\pi$
$812$	0	0
$813$	1.17220e7	0.621979
$814$	0	0
$815$	−1.33650e6	−0.0704817
$816$	0	0
$817$	−1.04691e6	−0.0548724
$818$	0	0
$819$	−1.71902e7	−0.895513
$820$	0	0
$821$	4.62793e6	0.239623	0.119812	−	0.992797i	$-0.461771\pi$
$821$	4.62793e6	0.239623	0.119812	+	0.992797i	$0.461771\pi$
$822$	0	0
$823$	1.50684e7	0.775472	0.387736	−	0.921770i	$-0.373257\pi$
$823$	1.50684e7	0.775472	0.387736	+	0.921770i	$0.373257\pi$
$824$	0	0
$825$	1.18653e6	0.0606936
$826$	0	0
$827$	−3.46926e7	−1.76390	−0.881949	−	0.471344i	$-0.843769\pi$
$827$	−3.46926e7	−1.76390	−0.881949	+	0.471344i	$0.843769\pi$
$828$	0	0
$829$	−3.14158e7	−1.58768	−0.793838	−	0.608129i	$-0.791920\pi$
$829$	−3.14158e7	−1.58768	−0.793838	+	0.608129i	$0.791920\pi$
$830$	0	0
$831$	−2.17896e7	−1.09458
$832$	0	0
$833$	−1.86217e7	−0.929838
$834$	0	0
$835$	−3.94603e6	−0.195859
$836$	0	0
$837$	−4.93823e6	−0.243645
$838$	0	0
$839$	−2.56102e7	−1.25605	−0.628026	−	0.778192i	$-0.716137\pi$
$839$	−2.56102e7	−1.25605	−0.628026	+	0.778192i	$0.716137\pi$
$840$	0	0
$841$	8.54883e6	0.416790
$842$	0	0
$843$	1.27292e7	0.616923
$844$	0	0
$845$	−9.22486e7	−4.44445
$846$	0	0
$847$	−2.91477e7	−1.39603
$848$	0	0
$849$	−1.19433e6	−0.0568662
$850$	0	0
$851$	−3.46732e6	−0.164123
$852$	0	0
$853$	5.90055e6	0.277664	0.138832	−	0.990316i	$-0.455665\pi$
$853$	5.90055e6	0.277664	0.138832	+	0.990316i	$0.455665\pi$
$854$	0	0
$855$	1.65524e6	0.0774365
$856$	0	0
$857$	−832247.	−0.0387080	−0.0193540	−	0.999813i	$-0.506161\pi$
$857$	−832247.	−0.0387080	−0.0193540	+	0.999813i	$0.506161\pi$
$858$	0	0
$859$	−3.03686e7	−1.40424	−0.702122	−	0.712057i	$-0.747764\pi$
$859$	−3.03686e7	−1.40424	−0.702122	+	0.712057i	$0.747764\pi$
$860$	0	0
$861$	1.10541e7	0.508180
$862$	0	0
$863$	−2.83833e7	−1.29728	−0.648642	−	0.761093i	$-0.724663\pi$
$863$	−2.83833e7	−1.29728	−0.648642	+	0.761093i	$0.724663\pi$
$864$	0	0
$865$	9.40547e6	0.427406
$866$	0	0
$867$	863478.	0.0390125
$868$	0	0
$869$	2.10425e6	0.0945255
$870$	0	0
$871$	6.53870e7	2.92043
$872$	0	0
$873$	−4.63045e6	−0.205630
$874$	0	0
$875$	−3.98606e7	−1.76005
$876$	0	0
$877$	1.62138e7	0.711846	0.355923	−	0.934515i	$-0.384167\pi$
$877$	1.62138e7	0.711846	0.355923	+	0.934515i	$0.384167\pi$
$878$	0	0
$879$	1.16371e7	0.508010
$880$	0	0
$881$	2.94112e7	1.27665	0.638327	−	0.769765i	$-0.279627\pi$
$881$	2.94112e7	1.27665	0.638327	+	0.769765i	$0.279627\pi$
$882$	0	0
$883$	−1.07170e7	−0.462562	−0.231281	−	0.972887i	$-0.574292\pi$
$883$	−1.07170e7	−0.462562	−0.231281	+	0.972887i	$0.574292\pi$
$884$	0	0
$885$	−1.82952e7	−0.785197
$886$	0	0
$887$	2.28458e7	0.974984	0.487492	−	0.873128i	$-0.337912\pi$
$887$	2.28458e7	0.974984	0.487492	+	0.873128i	$0.337912\pi$
$888$	0	0
$889$	2.92011e7	1.23921
$890$	0	0
$891$	−157571.	−0.00664938
$892$	0	0
$893$	−132591.	−0.00556398
$894$	0	0
$895$	−3.78931e7	−1.58126
$896$	0	0
$897$	5.56284e6	0.230842
$898$	0	0
$899$	−3.65167e7	−1.50693
$900$	0	0
$901$	−1.50362e7	−0.617057
$902$	0	0
$903$	−7.77299e6	−0.317226
$904$	0	0
$905$	−4.81564e7	−1.95449
$906$	0	0
$907$	−3.80166e7	−1.53446	−0.767230	−	0.641373i	$-0.778365\pi$
$907$	−3.80166e7	−1.53446	−0.767230	+	0.641373i	$0.778365\pi$
$908$	0	0
$909$	−1.44326e7	−0.579343
$910$	0	0
$911$	−9.15349e6	−0.365419	−0.182709	−	0.983167i	$-0.558487\pi$
$911$	−9.15349e6	−0.365419	−0.182709	+	0.983167i	$0.558487\pi$
$912$	0	0
$913$	755706.	0.0300038
$914$	0	0
$915$	−1.67360e7	−0.660844
$916$	0	0
$917$	1.81446e7	0.712564
$918$	0	0
$919$	6.42128e6	0.250803	0.125401	−	0.992106i	$-0.459978\pi$
$919$	6.42128e6	0.250803	0.125401	+	0.992106i	$0.459978\pi$
$920$	0	0
$921$	2.48638e6	0.0965868
$922$	0	0
$923$	4.33197e7	1.67372
$924$	0	0
$925$	−3.59805e7	−1.38265
$926$	0	0
$927$	2.23912e6	0.0855811
$928$	0	0
$929$	−4.18903e7	−1.59248	−0.796241	−	0.604979i	$-0.793181\pi$
$929$	−4.18903e7	−1.59248	−0.796241	+	0.604979i	$0.793181\pi$
$930$	0	0
$931$	−3.56329e6	−0.134734
$932$	0	0
$933$	−2.73109e7	−1.02715
$934$	0	0
$935$	−2.56478e6	−0.0959446
$936$	0	0
$937$	2.24446e7	0.835145	0.417573	−	0.908644i	$-0.362881\pi$
$937$	2.24446e7	0.835145	0.417573	+	0.908644i	$0.362881\pi$
$938$	0	0
$939$	−6.78617e6	−0.251166
$940$	0	0
$941$	3.37595e7	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$941$	3.37595e7	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$942$	0	0
$943$	−3.57717e6	−0.130997
$944$	0	0
$945$	1.22897e7	0.447673
$946$	0	0
$947$	−3.42481e7	−1.24097	−0.620486	−	0.784218i	$-0.713064\pi$
$947$	−3.42481e7	−1.24097	−0.620486	+	0.784218i	$0.713064\pi$
$948$	0	0
$949$	1.38000e7	0.497408
$950$	0	0
$951$	1.70613e7	0.611732
$952$	0	0
$953$	−2.06697e7	−0.737227	−0.368613	−	0.929583i	$-0.620167\pi$
$953$	−2.06697e7	−0.737227	−0.368613	+	0.929583i	$0.620167\pi$
$954$	0	0
$955$	−8.51704e7	−3.02190
$956$	0	0
$957$	−1.16519e6	−0.0411260
$958$	0	0
$959$	3.18140e7	1.11705
$960$	0	0
$961$	1.72577e7	0.602801
$962$	0	0
$963$	−137080.	−0.00476329
$964$	0	0
$965$	2.37219e6	0.0820033
$966$	0	0
$967$	2.63298e7	0.905485	0.452742	−	0.891641i	$-0.350446\pi$
$967$	2.63298e7	0.905485	0.452742	+	0.891641i	$0.350446\pi$
$968$	0	0
$969$	−2.28000e6	−0.0780055
$970$	0	0
$971$	−4.17371e7	−1.42061	−0.710305	−	0.703894i	$-0.751443\pi$
$971$	−4.17371e7	−1.42061	−0.710305	+	0.703894i	$0.751443\pi$
$972$	0	0
$973$	5.09901e7	1.72665
$974$	0	0
$975$	5.77258e7	1.94473
$976$	0	0
$977$	−2.96368e7	−0.993333	−0.496666	−	0.867942i	$-0.665443\pi$
$977$	−2.96368e7	−0.993333	−0.496666	+	0.867942i	$0.665443\pi$
$978$	0	0
$979$	3.52474e6	0.117536
$980$	0	0
$981$	1.83807e7	0.609802
$982$	0	0
$983$	−2.15053e7	−0.709843	−0.354922	−	0.934896i	$-0.615492\pi$
$983$	−2.15053e7	−0.709843	−0.354922	+	0.934896i	$0.615492\pi$
$984$	0	0
$985$	7.67630e7	2.52093
$986$	0	0
$987$	−984451.	−0.0321663
$988$	0	0
$989$	2.51537e6	0.0817734
$990$	0	0
$991$	5.07020e7	1.63999	0.819995	−	0.572371i	$-0.193976\pi$
$991$	5.07020e7	1.63999	0.819995	+	0.572371i	$0.193976\pi$
$992$	0	0
$993$	−3.39544e7	−1.09275
$994$	0	0
$995$	−8.33712e7	−2.66967
$996$	0	0
$997$	−2.58074e7	−0.822254	−0.411127	−	0.911578i	$-0.634865\pi$
$997$	−2.58074e7	−0.822254	−0.411127	+	0.911578i	$0.634865\pi$
$998$	0	0
$999$	4.77821e6	0.151479

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.6.a.i.1.1	✓	8	4.3	odd	2
1104.6.a.ba.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)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -92.8141 q^{5} +181.635 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} -92.8141 q^{5} +181.635 q^{7} +81.0000 q^{9} -24.0163 q^{11} -1168.42 q^{13} +835.327 q^{15} -1150.62 q^{17} -220.172 q^{19} -1634.71 q^{21} +529.000 q^{23} +5489.46 q^{25} -729.000 q^{27} -5390.73 q^{29} +6773.98 q^{31} +216.146 q^{33} -16858.3 q^{35} -6554.47 q^{37} +10515.8 q^{39} -6762.14 q^{41} +4754.96 q^{43} -7517.94 q^{45} +602.217 q^{47} +16184.1 q^{49} +10355.5 q^{51} +13067.9 q^{53} +2229.05 q^{55} +1981.55 q^{57} -21901.8 q^{59} -20035.3 q^{61} +14712.4 q^{63} +108446. q^{65} -55962.0 q^{67} -4761.00 q^{69} -37075.6 q^{71} -11810.8 q^{73} -49405.1 q^{75} -4362.19 q^{77} -87617.9 q^{79} +6561.00 q^{81} -31466.4 q^{83} +106793. q^{85} +48516.6 q^{87} -146765. q^{89} -212225. q^{91} -60965.8 q^{93} +20435.1 q^{95} -57166.0 q^{97} -1945.32 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 72 q^{3} + 16 q^{5} + 36 q^{7} + 648 q^{9}+O(q^{10}) \) 8 * q - 72 * q^3 + 16 * q^5 + 36 * q^7 + 648 * q^9	\( 8 q - 72 q^{3} + 16 q^{5} + 36 q^{7} + 648 q^{9} + 148 q^{11} + 696 q^{13} - 144 q^{15} + 572 q^{17} - 2456 q^{19} - 324 q^{21} + 4232 q^{23} + 11896 q^{25} - 5832 q^{27} + 6504 q^{29} - 3128 q^{31} - 1332 q^{33} - 6960 q^{35} + 3844 q^{37} - 6264 q^{39} + 6440 q^{41} - 7048 q^{43} + 1296 q^{45} - 30464 q^{47} + 46984 q^{49} - 5148 q^{51} + 63696 q^{53} - 32688 q^{55} + 22104 q^{57} - 54872 q^{59} + 12108 q^{61} + 2916 q^{63} + 124088 q^{65} - 139216 q^{67} - 38088 q^{69} - 76216 q^{71} + 13632 q^{73} - 107064 q^{75} - 78248 q^{77} - 126380 q^{79} + 52488 q^{81} - 238196 q^{83} + 117536 q^{85} - 58536 q^{87} + 123668 q^{89} - 248176 q^{91} + 28152 q^{93} - 183408 q^{95} + 18576 q^{97} + 11988 q^{99}+O(q^{100}) \) 8 * q - 72 * q^3 + 16 * q^5 + 36 * q^7 + 648 * q^9 + 148 * q^11 + 696 * q^13 - 144 * q^15 + 572 * q^17 - 2456 * q^19 - 324 * q^21 + 4232 * q^23 + 11896 * q^25 - 5832 * q^27 + 6504 * q^29 - 3128 * q^31 - 1332 * q^33 - 6960 * q^35 + 3844 * q^37 - 6264 * q^39 + 6440 * q^41 - 7048 * q^43 + 1296 * q^45 - 30464 * q^47 + 46984 * q^49 - 5148 * q^51 + 63696 * q^53 - 32688 * q^55 + 22104 * q^57 - 54872 * q^59 + 12108 * q^61 + 2916 * q^63 + 124088 * q^65 - 139216 * q^67 - 38088 * q^69 - 76216 * q^71 + 13632 * q^73 - 107064 * q^75 - 78248 * q^77 - 126380 * q^79 + 52488 * q^81 - 238196 * q^83 + 117536 * q^85 - 58536 * q^87 + 123668 * q^89 - 248176 * q^91 + 28152 * q^93 - 183408 * q^95 + 18576 * q^97 + 11988 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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