LMFDB - Embedded newform 1104.6.a.h.1.2

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:	$2$
Coefficient field:	$\Q(\sqrt{29}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 7 $ x^2 - x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.19258$ 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)

$f(q)$	$=$	$q+9.00000 q^{3} +52.3852 q^{5} +118.237 q^{7} +81.0000 q^{9} +O(q^{10})$	$q+9.00000 q^{3} +52.3852 q^{5} +118.237 q^{7} +81.0000 q^{9} -741.598 q^{11} -542.502 q^{13} +471.466 q^{15} -834.993 q^{17} +1309.79 q^{19} +1064.13 q^{21} -529.000 q^{23} -380.795 q^{25} +729.000 q^{27} +5536.45 q^{29} +7267.14 q^{31} -6674.38 q^{33} +6193.85 q^{35} -10446.7 q^{37} -4882.52 q^{39} -4630.45 q^{41} +9201.23 q^{43} +4243.20 q^{45} -15272.3 q^{47} -2827.06 q^{49} -7514.94 q^{51} -36202.8 q^{53} -38848.7 q^{55} +11788.1 q^{57} -6436.34 q^{59} -2637.86 q^{61} +9577.18 q^{63} -28419.1 q^{65} -21068.5 q^{67} -4761.00 q^{69} -74668.2 q^{71} +80742.0 q^{73} -3427.15 q^{75} -87684.2 q^{77} -39107.7 q^{79} +6561.00 q^{81} -101465. q^{83} -43741.2 q^{85} +49828.1 q^{87} +22407.6 q^{89} -64143.7 q^{91} +65404.3 q^{93} +68613.4 q^{95} -29724.8 q^{97} -60069.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 18 q^{3} + 94 q^{5} + 118 q^{7} + 162 q^{9}+O(q^{10}) $ 2 * q + 18 * q^3 + 94 * q^5 + 118 * q^7 + 162 * q^9	$ 2 q + 18 q^{3} + 94 q^{5} + 118 q^{7} + 162 q^{9} - 320 q^{11} - 288 q^{13} + 846 q^{15} - 1810 q^{17} - 730 q^{19} + 1062 q^{21} - 1058 q^{23} - 1774 q^{25} + 1458 q^{27} + 8208 q^{29} - 1772 q^{31} - 2880 q^{33} + 6184 q^{35} - 23112 q^{37} - 2592 q^{39} + 5516 q^{41} - 10322 q^{43} + 7614 q^{45} - 42952 q^{47} - 19634 q^{49} - 16290 q^{51} - 25350 q^{53} - 21304 q^{55} - 6570 q^{57} - 18344 q^{59} + 37224 q^{61} + 9558 q^{63} - 17828 q^{65} + 7482 q^{67} - 9522 q^{69} - 126848 q^{71} + 137660 q^{73} - 15966 q^{75} - 87784 q^{77} - 62286 q^{79} + 13122 q^{81} - 83120 q^{83} - 84316 q^{85} + 73872 q^{87} + 69770 q^{89} - 64204 q^{91} - 15948 q^{93} - 16272 q^{95} - 170104 q^{97} - 25920 q^{99}+O(q^{100}) $ 2 * q + 18 * q^3 + 94 * q^5 + 118 * q^7 + 162 * q^9 - 320 * q^11 - 288 * q^13 + 846 * q^15 - 1810 * q^17 - 730 * q^19 + 1062 * q^21 - 1058 * q^23 - 1774 * q^25 + 1458 * q^27 + 8208 * q^29 - 1772 * q^31 - 2880 * q^33 + 6184 * q^35 - 23112 * q^37 - 2592 * q^39 + 5516 * q^41 - 10322 * q^43 + 7614 * q^45 - 42952 * q^47 - 19634 * q^49 - 16290 * q^51 - 25350 * q^53 - 21304 * q^55 - 6570 * q^57 - 18344 * q^59 + 37224 * q^61 + 9558 * q^63 - 17828 * q^65 + 7482 * q^67 - 9522 * q^69 - 126848 * q^71 + 137660 * q^73 - 15966 * q^75 - 87784 * q^77 - 62286 * q^79 + 13122 * q^81 - 83120 * q^83 - 84316 * q^85 + 73872 * q^87 + 69770 * q^89 - 64204 * q^91 - 15948 * q^93 - 16272 * q^95 - 170104 * q^97 - 25920 * 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
$5$	52.3852	0.937094	0.468547	−	0.883438i	$-0.344778\pi$
$5$	52.3852	0.937094	0.468547	+	0.883438i	$0.344778\pi$
$6$	0	0
$7$	118.237	0.912027	0.456013	−	0.889973i	$-0.349277\pi$
$7$	118.237	0.912027	0.456013	+	0.889973i	$0.349277\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−741.598	−1.84794	−0.923968	−	0.382471i	$-0.875073\pi$
$11$	−741.598	−1.84794	−0.923968	+	0.382471i	$0.875073\pi$
$12$	0	0
$13$	−542.502	−0.890314	−0.445157	−	0.895453i	$-0.646852\pi$
$13$	−542.502	−0.890314	−0.445157	+	0.895453i	$0.646852\pi$
$14$	0	0
$15$	471.466	0.541032
$16$	0	0
$17$	−834.993	−0.700746	−0.350373	−	0.936610i	$-0.613945\pi$
$17$	−834.993	−0.700746	−0.350373	+	0.936610i	$0.613945\pi$
$18$	0	0
$19$	1309.79	0.832370	0.416185	−	0.909280i	$-0.363367\pi$
$19$	1309.79	0.832370	0.416185	+	0.909280i	$0.363367\pi$
$20$	0	0
$21$	1064.13	0.526559
$22$	0	0
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	−380.795	−0.121854
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	5536.45	1.22247	0.611233	−	0.791451i	$-0.290674\pi$
$29$	5536.45	1.22247	0.611233	+	0.791451i	$0.290674\pi$
$30$	0	0
$31$	7267.14	1.35819	0.679093	−	0.734052i	$-0.262373\pi$
$31$	7267.14	1.35819	0.679093	+	0.734052i	$0.262373\pi$
$32$	0	0
$33$	−6674.38	−1.06691
$34$	0	0
$35$	6193.85	0.854655
$36$	0	0
$37$	−10446.7	−1.25451	−0.627253	−	0.778815i	$-0.715821\pi$
$37$	−10446.7	−1.25451	−0.627253	+	0.778815i	$0.715821\pi$
$38$	0	0
$39$	−4882.52	−0.514023
$40$	0	0
$41$	−4630.45	−0.430193	−0.215096	−	0.976593i	$-0.569007\pi$
$41$	−4630.45	−0.430193	−0.215096	+	0.976593i	$0.569007\pi$
$42$	0	0
$43$	9201.23	0.758883	0.379442	−	0.925216i	$-0.376116\pi$
$43$	9201.23	0.758883	0.379442	+	0.925216i	$0.376116\pi$
$44$	0	0
$45$	4243.20	0.312365
$46$	0	0
$47$	−15272.3	−1.00846	−0.504231	−	0.863569i	$-0.668224\pi$
$47$	−15272.3	−1.00846	−0.504231	+	0.863569i	$0.668224\pi$
$48$	0	0
$49$	−2827.06	−0.168207
$50$	0	0
$51$	−7514.94	−0.404576
$52$	0	0
$53$	−36202.8	−1.77032	−0.885161	−	0.465285i	$-0.845952\pi$
$53$	−36202.8	−1.77032	−0.885161	+	0.465285i	$0.845952\pi$
$54$	0	0
$55$	−38848.7	−1.73169
$56$	0	0
$57$	11788.1	0.480569
$58$	0	0
$59$	−6436.34	−0.240718	−0.120359	−	0.992730i	$-0.538405\pi$
$59$	−6436.34	−0.240718	−0.120359	+	0.992730i	$0.538405\pi$
$60$	0	0
$61$	−2637.86	−0.0907668	−0.0453834	−	0.998970i	$-0.514451\pi$
$61$	−2637.86	−0.0907668	−0.0453834	+	0.998970i	$0.514451\pi$
$62$	0	0
$63$	9577.18	0.304009
$64$	0	0
$65$	−28419.1	−0.834308
$66$	0	0
$67$	−21068.5	−0.573384	−0.286692	−	0.958023i	$-0.592556\pi$
$67$	−21068.5	−0.573384	−0.286692	+	0.958023i	$0.592556\pi$
$68$	0	0
$69$	−4761.00	−0.120386
$70$	0	0
$71$	−74668.2	−1.75788	−0.878941	−	0.476930i	$-0.841750\pi$
$71$	−74668.2	−1.75788	−0.878941	+	0.476930i	$0.841750\pi$
$72$	0	0
$73$	80742.0	1.77334	0.886671	−	0.462402i	$-0.153012\pi$
$73$	80742.0	1.77334	0.886671	+	0.462402i	$0.153012\pi$
$74$	0	0
$75$	−3427.15	−0.0703526
$76$	0	0
$77$	−87684.2	−1.68537
$78$	0	0
$79$	−39107.7	−0.705008	−0.352504	−	0.935810i	$-0.614670\pi$
$79$	−39107.7	−0.705008	−0.352504	+	0.935810i	$0.614670\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−101465.	−1.61666	−0.808331	−	0.588728i	$-0.799629\pi$
$83$	−101465.	−1.61666	−0.808331	+	0.588728i	$0.799629\pi$
$84$	0	0
$85$	−43741.2	−0.656665
$86$	0	0
$87$	49828.1	0.705791
$88$	0	0
$89$	22407.6	0.299861	0.149930	−	0.988697i	$-0.452095\pi$
$89$	22407.6	0.299861	0.149930	+	0.988697i	$0.452095\pi$
$90$	0	0
$91$	−64143.7	−0.811990
$92$	0	0
$93$	65404.3	0.784149
$94$	0	0
$95$	68613.4	0.780009
$96$	0	0
$97$	−29724.8	−0.320767	−0.160384	−	0.987055i	$-0.551273\pi$
$97$	−29724.8	−0.320767	−0.160384	+	0.987055i	$0.551273\pi$
$98$	0	0
$99$	−60069.4	−0.615978
$100$	0	0
$101$	96859.7	0.944800	0.472400	−	0.881384i	$-0.343388\pi$
$101$	96859.7	0.944800	0.472400	+	0.881384i	$0.343388\pi$
$102$	0	0
$103$	−2081.11	−0.0193287	−0.00966434	−	0.999953i	$-0.503076\pi$
$103$	−2081.11	−0.0193287	−0.00966434	+	0.999953i	$0.503076\pi$
$104$	0	0
$105$	55744.7	0.493435
$106$	0	0
$107$	76907.3	0.649393	0.324697	−	0.945818i	$-0.394738\pi$
$107$	76907.3	0.649393	0.324697	+	0.945818i	$0.394738\pi$
$108$	0	0
$109$	−27250.0	−0.219685	−0.109842	−	0.993949i	$-0.535035\pi$
$109$	−27250.0	−0.219685	−0.109842	+	0.993949i	$0.535035\pi$
$110$	0	0
$111$	−94019.9	−0.724290
$112$	0	0
$113$	−120738.	−0.889505	−0.444753	−	0.895653i	$-0.646708\pi$
$113$	−120738.	−0.889505	−0.444753	+	0.895653i	$0.646708\pi$
$114$	0	0
$115$	−27711.8	−0.195398
$116$	0	0
$117$	−43942.7	−0.296771
$118$	0	0
$119$	−98726.9	−0.639099
$120$	0	0
$121$	388916.	2.41486
$122$	0	0
$123$	−41674.0	−0.248372
$124$	0	0
$125$	−183652.	−1.05128
$126$	0	0
$127$	140823.	0.774755	0.387377	−	0.921921i	$-0.373381\pi$
$127$	140823.	0.774755	0.387377	+	0.921921i	$0.373381\pi$
$128$	0	0
$129$	82811.1	0.438141
$130$	0	0
$131$	137026.	0.697630	0.348815	−	0.937192i	$-0.386584\pi$
$131$	137026.	0.697630	0.348815	+	0.937192i	$0.386584\pi$
$132$	0	0
$133$	154865.	0.759144
$134$	0	0
$135$	38188.8	0.180344
$136$	0	0
$137$	73627.5	0.335149	0.167575	−	0.985859i	$-0.446406\pi$
$137$	73627.5	0.335149	0.167575	+	0.985859i	$0.446406\pi$
$138$	0	0
$139$	−214252.	−0.940562	−0.470281	−	0.882517i	$-0.655848\pi$
$139$	−214252.	−0.940562	−0.470281	+	0.882517i	$0.655848\pi$
$140$	0	0
$141$	−137451.	−0.582236
$142$	0	0
$143$	402318.	1.64524
$144$	0	0
$145$	290028.	1.14557
$146$	0	0
$147$	−25443.5	−0.0971144
$148$	0	0
$149$	132470.	0.488823	0.244412	−	0.969672i	$-0.421405\pi$
$149$	132470.	0.488823	0.244412	+	0.969672i	$0.421405\pi$
$150$	0	0
$151$	−171660.	−0.612669	−0.306334	−	0.951924i	$-0.599103\pi$
$151$	−171660.	−0.612669	−0.306334	+	0.951924i	$0.599103\pi$
$152$	0	0
$153$	−67634.4	−0.233582
$154$	0	0
$155$	380690.	1.27275
$156$	0	0
$157$	−359852.	−1.16513	−0.582565	−	0.812784i	$-0.697951\pi$
$157$	−359852.	−1.16513	−0.582565	+	0.812784i	$0.697951\pi$
$158$	0	0
$159$	−325825.	−1.02210
$160$	0	0
$161$	−62547.3	−0.190171
$162$	0	0
$163$	−454656.	−1.34034	−0.670168	−	0.742210i	$-0.733778\pi$
$163$	−454656.	−1.34034	−0.670168	+	0.742210i	$0.733778\pi$
$164$	0	0
$165$	−349639.	−0.999791
$166$	0	0
$167$	359837.	0.998424	0.499212	−	0.866480i	$-0.333623\pi$
$167$	359837.	0.998424	0.499212	+	0.866480i	$0.333623\pi$
$168$	0	0
$169$	−76984.4	−0.207341
$170$	0	0
$171$	106093.	0.277457
$172$	0	0
$173$	−308025.	−0.782476	−0.391238	−	0.920289i	$-0.627953\pi$
$173$	−308025.	−0.782476	−0.391238	+	0.920289i	$0.627953\pi$
$174$	0	0
$175$	−45023.9	−0.111134
$176$	0	0
$177$	−57927.0	−0.138979
$178$	0	0
$179$	−628893.	−1.46705	−0.733524	−	0.679663i	$-0.762126\pi$
$179$	−628893.	−1.46705	−0.733524	+	0.679663i	$0.762126\pi$
$180$	0	0
$181$	−667015.	−1.51335	−0.756675	−	0.653791i	$-0.773177\pi$
$181$	−667015.	−1.51335	−0.756675	+	0.653791i	$0.773177\pi$
$182$	0	0
$183$	−23740.7	−0.0524043
$184$	0	0
$185$	−547250.	−1.17559
$186$	0	0
$187$	619229.	1.29493
$188$	0	0
$189$	86194.6	0.175520
$190$	0	0
$191$	−573327.	−1.13715	−0.568577	−	0.822630i	$-0.692506\pi$
$191$	−573327.	−1.13715	−0.568577	+	0.822630i	$0.692506\pi$
$192$	0	0
$193$	21198.3	0.0409646	0.0204823	−	0.999790i	$-0.493480\pi$
$193$	21198.3	0.0409646	0.0204823	+	0.999790i	$0.493480\pi$
$194$	0	0
$195$	−255772.	−0.481688
$196$	0	0
$197$	−342207.	−0.628237	−0.314118	−	0.949384i	$-0.601709\pi$
$197$	−342207.	−0.628237	−0.314118	+	0.949384i	$0.601709\pi$
$198$	0	0
$199$	32335.5	0.0578825	0.0289412	−	0.999581i	$-0.490786\pi$
$199$	32335.5	0.0578825	0.0289412	+	0.999581i	$0.490786\pi$
$200$	0	0
$201$	−189616.	−0.331043
$202$	0	0
$203$	654613.	1.11492
$204$	0	0
$205$	−242567.	−0.403131
$206$	0	0
$207$	−42849.0	−0.0695048
$208$	0	0
$209$	−971335.	−1.53817
$210$	0	0
$211$	1.03399e6	1.59886	0.799430	−	0.600759i	$-0.205135\pi$
$211$	1.03399e6	1.59886	0.799430	+	0.600759i	$0.205135\pi$
$212$	0	0
$213$	−672014.	−1.01491
$214$	0	0
$215$	482008.	0.711145
$216$	0	0
$217$	859243.	1.23870
$218$	0	0
$219$	726678.	1.02384
$220$	0	0
$221$	452985.	0.623884
$222$	0	0
$223$	−990187.	−1.33338	−0.666692	−	0.745333i	$-0.732290\pi$
$223$	−990187.	−1.33338	−0.666692	+	0.745333i	$0.732290\pi$
$224$	0	0
$225$	−30844.4	−0.0406181
$226$	0	0
$227$	26685.3	0.0343723	0.0171861	−	0.999852i	$-0.494529\pi$
$227$	26685.3	0.0343723	0.0171861	+	0.999852i	$0.494529\pi$
$228$	0	0
$229$	150586.	0.189756	0.0948778	−	0.995489i	$-0.469754\pi$
$229$	150586.	0.189756	0.0948778	+	0.995489i	$0.469754\pi$
$230$	0	0
$231$	−789157.	−0.973047
$232$	0	0
$233$	1.29421e6	1.56176	0.780881	−	0.624680i	$-0.214770\pi$
$233$	1.29421e6	1.56176	0.780881	+	0.624680i	$0.214770\pi$
$234$	0	0
$235$	−800041.	−0.945024
$236$	0	0
$237$	−351969.	−0.407037
$238$	0	0
$239$	136898.	0.155025	0.0775123	−	0.996991i	$-0.475302\pi$
$239$	136898.	0.155025	0.0775123	+	0.996991i	$0.475302\pi$
$240$	0	0
$241$	−284796.	−0.315858	−0.157929	−	0.987450i	$-0.550482\pi$
$241$	−284796.	−0.315858	−0.157929	+	0.987450i	$0.550482\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	−148096.	−0.157626
$246$	0	0
$247$	−710562.	−0.741071
$248$	0	0
$249$	−913181.	−0.933380
$250$	0	0
$251$	−1.26148e6	−1.26385	−0.631927	−	0.775028i	$-0.717736\pi$
$251$	−1.26148e6	−1.26385	−0.631927	+	0.775028i	$0.717736\pi$
$252$	0	0
$253$	392305.	0.385321
$254$	0	0
$255$	−393671.	−0.379126
$256$	0	0
$257$	552720.	0.522002	0.261001	−	0.965338i	$-0.415947\pi$
$257$	552720.	0.522002	0.261001	+	0.965338i	$0.415947\pi$
$258$	0	0
$259$	−1.23518e6	−1.14414
$260$	0	0
$261$	448453.	0.407489
$262$	0	0
$263$	651514.	0.580811	0.290405	−	0.956904i	$-0.406210\pi$
$263$	651514.	0.580811	0.290405	+	0.956904i	$0.406210\pi$
$264$	0	0
$265$	−1.89649e6	−1.65896
$266$	0	0
$267$	201668.	0.173125
$268$	0	0
$269$	1.62659e6	1.37056	0.685280	−	0.728280i	$-0.259680\pi$
$269$	1.62659e6	1.37056	0.685280	+	0.728280i	$0.259680\pi$
$270$	0	0
$271$	−526548.	−0.435527	−0.217763	−	0.976002i	$-0.569876\pi$
$271$	−526548.	−0.435527	−0.217763	+	0.976002i	$0.569876\pi$
$272$	0	0
$273$	−577294.	−0.468803
$274$	0	0
$275$	282396.	0.225179
$276$	0	0
$277$	102974.	0.0806358	0.0403179	−	0.999187i	$-0.487163\pi$
$277$	102974.	0.0806358	0.0403179	+	0.999187i	$0.487163\pi$
$278$	0	0
$279$	588638.	0.452729
$280$	0	0
$281$	−1.79209e6	−1.35392	−0.676960	−	0.736020i	$-0.736703\pi$
$281$	−1.79209e6	−1.35392	−0.676960	+	0.736020i	$0.736703\pi$
$282$	0	0
$283$	1.06565e6	0.790946	0.395473	−	0.918478i	$-0.370581\pi$
$283$	1.06565e6	0.790946	0.395473	+	0.918478i	$0.370581\pi$
$284$	0	0
$285$	617520.	0.450339
$286$	0	0
$287$	−547489.	−0.392347
$288$	0	0
$289$	−722644.	−0.508955
$290$	0	0
$291$	−267523.	−0.185195
$292$	0	0
$293$	1.85858e6	1.26477	0.632387	−	0.774653i	$-0.282075\pi$
$293$	1.85858e6	1.26477	0.632387	+	0.774653i	$0.282075\pi$
$294$	0	0
$295$	−337169.	−0.225576
$296$	0	0
$297$	−540625.	−0.355635
$298$	0	0
$299$	286984.	0.185643
$300$	0	0
$301$	1.08792e6	0.692122
$302$	0	0
$303$	871738.	0.545480
$304$	0	0
$305$	−138185.	−0.0850571
$306$	0	0
$307$	2.13805e6	1.29471	0.647355	−	0.762189i	$-0.275875\pi$
$307$	2.13805e6	1.29471	0.647355	+	0.762189i	$0.275875\pi$
$308$	0	0
$309$	−18730.0	−0.0111594
$310$	0	0
$311$	−1.35485e6	−0.794311	−0.397155	−	0.917751i	$-0.630003\pi$
$311$	−1.35485e6	−0.794311	−0.397155	+	0.917751i	$0.630003\pi$
$312$	0	0
$313$	−523789.	−0.302201	−0.151100	−	0.988518i	$-0.548282\pi$
$313$	−523789.	−0.302201	−0.151100	+	0.988518i	$0.548282\pi$
$314$	0	0
$315$	501702.	0.284885
$316$	0	0
$317$	2.59918e6	1.45274	0.726371	−	0.687302i	$-0.241205\pi$
$317$	2.59918e6	1.45274	0.726371	+	0.687302i	$0.241205\pi$
$318$	0	0
$319$	−4.10582e6	−2.25904
$320$	0	0
$321$	692165.	0.374928
$322$	0	0
$323$	−1.09366e6	−0.583280
$324$	0	0
$325$	206582.	0.108489
$326$	0	0
$327$	−245250.	−0.126835
$328$	0	0
$329$	−1.80575e6	−0.919744
$330$	0	0
$331$	1.02887e6	0.516168	0.258084	−	0.966122i	$-0.416909\pi$
$331$	1.02887e6	0.516168	0.258084	+	0.966122i	$0.416909\pi$
$332$	0	0
$333$	−846179.	−0.418169
$334$	0	0
$335$	−1.10367e6	−0.537315
$336$	0	0
$337$	1.13501e6	0.544406	0.272203	−	0.962240i	$-0.412248\pi$
$337$	1.13501e6	0.544406	0.272203	+	0.962240i	$0.412248\pi$
$338$	0	0
$339$	−1.08664e6	−0.513556
$340$	0	0
$341$	−5.38929e6	−2.50984
$342$	0	0
$343$	−2.32147e6	−1.06544
$344$	0	0
$345$	−249406.	−0.112813
$346$	0	0
$347$	−1.08736e6	−0.484787	−0.242393	−	0.970178i	$-0.577932\pi$
$347$	−1.08736e6	−0.484787	−0.242393	+	0.970178i	$0.577932\pi$
$348$	0	0
$349$	147242.	0.0647097	0.0323548	−	0.999476i	$-0.489699\pi$
$349$	147242.	0.0647097	0.0323548	+	0.999476i	$0.489699\pi$
$350$	0	0
$351$	−395484.	−0.171341
$352$	0	0
$353$	1.58645e6	0.677623	0.338812	−	0.940854i	$-0.389975\pi$
$353$	1.58645e6	0.677623	0.338812	+	0.940854i	$0.389975\pi$
$354$	0	0
$355$	−3.91151e6	−1.64730
$356$	0	0
$357$	−888542.	−0.368984
$358$	0	0
$359$	1.85640e6	0.760211	0.380106	−	0.924943i	$-0.375888\pi$
$359$	1.85640e6	0.760211	0.380106	+	0.924943i	$0.375888\pi$
$360$	0	0
$361$	−760559.	−0.307160
$362$	0	0
$363$	3.50025e6	1.39422
$364$	0	0
$365$	4.22968e6	1.66179
$366$	0	0
$367$	2.01535e6	0.781061	0.390531	−	0.920590i	$-0.372292\pi$
$367$	2.01535e6	0.781061	0.390531	+	0.920590i	$0.372292\pi$
$368$	0	0
$369$	−375066.	−0.143398
$370$	0	0
$371$	−4.28050e6	−1.61458
$372$	0	0
$373$	−1.33526e6	−0.496928	−0.248464	−	0.968641i	$-0.579926\pi$
$373$	−1.33526e6	−0.496928	−0.248464	+	0.968641i	$0.579926\pi$
$374$	0	0
$375$	−1.65286e6	−0.606959
$376$	0	0
$377$	−3.00354e6	−1.08838
$378$	0	0
$379$	2.84308e6	1.01669	0.508347	−	0.861152i	$-0.330257\pi$
$379$	2.84308e6	1.01669	0.508347	+	0.861152i	$0.330257\pi$
$380$	0	0
$381$	1.26741e6	0.447305
$382$	0	0
$383$	−3.29481e6	−1.14771	−0.573857	−	0.818956i	$-0.694553\pi$
$383$	−3.29481e6	−1.14771	−0.573857	+	0.818956i	$0.694553\pi$
$384$	0	0
$385$	−4.59335e6	−1.57935
$386$	0	0
$387$	745300.	0.252961
$388$	0	0
$389$	−4.92017e6	−1.64857	−0.824283	−	0.566177i	$-0.808422\pi$
$389$	−4.92017e6	−1.64857	−0.824283	+	0.566177i	$0.808422\pi$
$390$	0	0
$391$	441711.	0.146116
$392$	0	0
$393$	1.23323e6	0.402777
$394$	0	0
$395$	−2.04866e6	−0.660659
$396$	0	0
$397$	−4.13215e6	−1.31583	−0.657915	−	0.753092i	$-0.728561\pi$
$397$	−4.13215e6	−1.31583	−0.657915	+	0.753092i	$0.728561\pi$
$398$	0	0
$399$	1.39378e6	0.438292
$400$	0	0
$401$	−3.07921e6	−0.956266	−0.478133	−	0.878287i	$-0.658686\pi$
$401$	−3.07921e6	−0.956266	−0.478133	+	0.878287i	$0.658686\pi$
$402$	0	0
$403$	−3.94244e6	−1.20921
$404$	0	0
$405$	343699.	0.104122
$406$	0	0
$407$	7.74722e6	2.31825
$408$	0	0
$409$	−483481.	−0.142913	−0.0714564	−	0.997444i	$-0.522765\pi$
$409$	−483481.	−0.142913	−0.0714564	+	0.997444i	$0.522765\pi$
$410$	0	0
$411$	662647.	0.193499
$412$	0	0
$413$	−761012.	−0.219541
$414$	0	0
$415$	−5.31524e6	−1.51496
$416$	0	0
$417$	−1.92827e6	−0.543034
$418$	0	0
$419$	−1.51752e6	−0.422279	−0.211140	−	0.977456i	$-0.567718\pi$
$419$	−1.51752e6	−0.422279	−0.211140	+	0.977456i	$0.567718\pi$
$420$	0	0
$421$	3.76256e6	1.03461	0.517306	−	0.855800i	$-0.326935\pi$
$421$	3.76256e6	1.03461	0.517306	+	0.855800i	$0.326935\pi$
$422$	0	0
$423$	−1.23706e6	−0.336154
$424$	0	0
$425$	317961.	0.0853888
$426$	0	0
$427$	−311892.	−0.0827818
$428$	0	0
$429$	3.62087e6	0.949881
$430$	0	0
$431$	−2.80235e6	−0.726658	−0.363329	−	0.931661i	$-0.618360\pi$
$431$	−2.80235e6	−0.726658	−0.363329	+	0.931661i	$0.618360\pi$
$432$	0	0
$433$	3.06756e6	0.786272	0.393136	−	0.919480i	$-0.371390\pi$
$433$	3.06756e6	0.786272	0.393136	+	0.919480i	$0.371390\pi$
$434$	0	0
$435$	2.61025e6	0.661393
$436$	0	0
$437$	−692877.	−0.173561
$438$	0	0
$439$	4.01436e6	0.994158	0.497079	−	0.867705i	$-0.334406\pi$
$439$	4.01436e6	0.994158	0.497079	+	0.867705i	$0.334406\pi$
$440$	0	0
$441$	−228992.	−0.0560690
$442$	0	0
$443$	−4.74625e6	−1.14906	−0.574528	−	0.818485i	$-0.694814\pi$
$443$	−4.74625e6	−1.14906	−0.574528	+	0.818485i	$0.694814\pi$
$444$	0	0
$445$	1.17382e6	0.280998
$446$	0	0
$447$	1.19223e6	0.282222
$448$	0	0
$449$	1.57070e6	0.367687	0.183843	−	0.982956i	$-0.441146\pi$
$449$	1.57070e6	0.367687	0.183843	+	0.982956i	$0.441146\pi$
$450$	0	0
$451$	3.43393e6	0.794968
$452$	0	0
$453$	−1.54494e6	−0.353725
$454$	0	0
$455$	−3.36018e6	−0.760911
$456$	0	0
$457$	4.94573e6	1.10775	0.553873	−	0.832601i	$-0.313149\pi$
$457$	4.94573e6	1.10775	0.553873	+	0.832601i	$0.313149\pi$
$458$	0	0
$459$	−608710.	−0.134859
$460$	0	0
$461$	4.86491e6	1.06616	0.533080	−	0.846065i	$-0.321035\pi$
$461$	4.86491e6	1.06616	0.533080	+	0.846065i	$0.321035\pi$
$462$	0	0
$463$	2.04379e6	0.443082	0.221541	−	0.975151i	$-0.428891\pi$
$463$	2.04379e6	0.443082	0.221541	+	0.975151i	$0.428891\pi$
$464$	0	0
$465$	3.42621e6	0.734822
$466$	0	0
$467$	6.33570e6	1.34432	0.672159	−	0.740407i	$-0.265367\pi$
$467$	6.33570e6	1.34432	0.672159	+	0.740407i	$0.265367\pi$
$468$	0	0
$469$	−2.49107e6	−0.522942
$470$	0	0
$471$	−3.23866e6	−0.672688
$472$	0	0
$473$	−6.82362e6	−1.40237
$474$	0	0
$475$	−498759.	−0.101428
$476$	0	0
$477$	−2.93243e6	−0.590107
$478$	0	0
$479$	−2.39044e6	−0.476036	−0.238018	−	0.971261i	$-0.576498\pi$
$479$	−2.39044e6	−0.476036	−0.238018	+	0.971261i	$0.576498\pi$
$480$	0	0
$481$	5.66733e6	1.11690
$482$	0	0
$483$	−562925.	−0.109795
$484$	0	0
$485$	−1.55714e6	−0.300589
$486$	0	0
$487$	8.69235e6	1.66079	0.830395	−	0.557175i	$-0.188115\pi$
$487$	8.69235e6	1.66079	0.830395	+	0.557175i	$0.188115\pi$
$488$	0	0
$489$	−4.09190e6	−0.773843
$490$	0	0
$491$	−3.02753e6	−0.566742	−0.283371	−	0.959010i	$-0.591453\pi$
$491$	−3.02753e6	−0.566742	−0.283371	+	0.959010i	$0.591453\pi$
$492$	0	0
$493$	−4.62290e6	−0.856638
$494$	0	0
$495$	−3.14675e6	−0.577230
$496$	0	0
$497$	−8.82853e6	−1.60324
$498$	0	0
$499$	−3.24808e6	−0.583950	−0.291975	−	0.956426i	$-0.594312\pi$
$499$	−3.24808e6	−0.583950	−0.291975	+	0.956426i	$0.594312\pi$
$500$	0	0
$501$	3.23853e6	0.576440
$502$	0	0
$503$	1.14587e6	0.201937	0.100969	−	0.994890i	$-0.467806\pi$
$503$	1.14587e6	0.201937	0.100969	+	0.994890i	$0.467806\pi$
$504$	0	0
$505$	5.07401e6	0.885367
$506$	0	0
$507$	−692859.	−0.119709
$508$	0	0
$509$	−7.47756e6	−1.27928	−0.639640	−	0.768675i	$-0.720917\pi$
$509$	−7.47756e6	−1.27928	−0.639640	+	0.768675i	$0.720917\pi$
$510$	0	0
$511$	9.54667e6	1.61733
$512$	0	0
$513$	954834.	0.160190
$514$	0	0
$515$	−109019.	−0.0181128
$516$	0	0
$517$	1.13259e7	1.86357
$518$	0	0
$519$	−2.77223e6	−0.451763
$520$	0	0
$521$	−568216.	−0.0917106	−0.0458553	−	0.998948i	$-0.514601\pi$
$521$	−568216.	−0.0917106	−0.0458553	+	0.998948i	$0.514601\pi$
$522$	0	0
$523$	−6.11824e6	−0.978076	−0.489038	−	0.872262i	$-0.662652\pi$
$523$	−6.11824e6	−0.978076	−0.489038	+	0.872262i	$0.662652\pi$
$524$	0	0
$525$	−405215.	−0.0641634
$526$	0	0
$527$	−6.06801e6	−0.951743
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	−521343.	−0.0802394
$532$	0	0
$533$	2.51203e6	0.383007
$534$	0	0
$535$	4.02880e6	0.608543
$536$	0	0
$537$	−5.66004e6	−0.847001
$538$	0	0
$539$	2.09654e6	0.310836
$540$	0	0
$541$	−4.31434e6	−0.633755	−0.316878	−	0.948466i	$-0.602634\pi$
$541$	−4.31434e6	−0.633755	−0.316878	+	0.948466i	$0.602634\pi$
$542$	0	0
$543$	−6.00314e6	−0.873733
$544$	0	0
$545$	−1.42749e6	−0.205865
$546$	0	0
$547$	−7.34052e6	−1.04896	−0.524480	−	0.851423i	$-0.675740\pi$
$547$	−7.34052e6	−1.04896	−0.524480	+	0.851423i	$0.675740\pi$
$548$	0	0
$549$	−213667.	−0.0302556
$550$	0	0
$551$	7.25157e6	1.01754
$552$	0	0
$553$	−4.62396e6	−0.642986
$554$	0	0
$555$	−4.92525e6	−0.678728
$556$	0	0
$557$	8.10617e6	1.10708	0.553538	−	0.832824i	$-0.313277\pi$
$557$	8.10617e6	1.10708	0.553538	+	0.832824i	$0.313277\pi$
$558$	0	0
$559$	−4.99169e6	−0.675644
$560$	0	0
$561$	5.57306e6	0.747630
$562$	0	0
$563$	−1.15177e7	−1.53142	−0.765709	−	0.643187i	$-0.777612\pi$
$563$	−1.15177e7	−1.53142	−0.765709	+	0.643187i	$0.777612\pi$
$564$	0	0
$565$	−6.32489e6	−0.833550
$566$	0	0
$567$	775752.	0.101336
$568$	0	0
$569$	248635.	0.0321945	0.0160972	−	0.999870i	$-0.494876\pi$
$569$	248635.	0.0321945	0.0160972	+	0.999870i	$0.494876\pi$
$570$	0	0
$571$	−2.55456e6	−0.327888	−0.163944	−	0.986470i	$-0.552422\pi$
$571$	−2.55456e6	−0.327888	−0.163944	+	0.986470i	$0.552422\pi$
$572$	0	0
$573$	−5.15994e6	−0.656536
$574$	0	0
$575$	201440.	0.0254084
$576$	0	0
$577$	3.96849e6	0.496233	0.248116	−	0.968730i	$-0.420188\pi$
$577$	3.96849e6	0.496233	0.248116	+	0.968730i	$0.420188\pi$
$578$	0	0
$579$	190785.	0.0236509
$580$	0	0
$581$	−1.19968e7	−1.47444
$582$	0	0
$583$	2.68479e7	3.27144
$584$	0	0
$585$	−2.30194e6	−0.278103
$586$	0	0
$587$	−7.38560e6	−0.884689	−0.442344	−	0.896845i	$-0.645853\pi$
$587$	−7.38560e6	−0.884689	−0.442344	+	0.896845i	$0.645853\pi$
$588$	0	0
$589$	9.51840e6	1.13051
$590$	0	0
$591$	−3.07986e6	−0.362713
$592$	0	0
$593$	−1.34366e7	−1.56910	−0.784552	−	0.620063i	$-0.787107\pi$
$593$	−1.34366e7	−1.56910	−0.784552	+	0.620063i	$0.787107\pi$
$594$	0	0
$595$	−5.17182e6	−0.598896
$596$	0	0
$597$	291020.	0.0334185
$598$	0	0
$599$	−1.65576e6	−0.188552	−0.0942760	−	0.995546i	$-0.530054\pi$
$599$	−1.65576e6	−0.188552	−0.0942760	+	0.995546i	$0.530054\pi$
$600$	0	0
$601$	−8.29015e6	−0.936216	−0.468108	−	0.883671i	$-0.655064\pi$
$601$	−8.29015e6	−0.936216	−0.468108	+	0.883671i	$0.655064\pi$
$602$	0	0
$603$	−1.70654e6	−0.191128
$604$	0	0
$605$	2.03734e7	2.26296
$606$	0	0
$607$	1.24477e7	1.37125	0.685624	−	0.727956i	$-0.259530\pi$
$607$	1.24477e7	1.37125	0.685624	+	0.727956i	$0.259530\pi$
$608$	0	0
$609$	5.89151e6	0.643700
$610$	0	0
$611$	8.28525e6	0.897848
$612$	0	0
$613$	−9.98753e6	−1.07351	−0.536756	−	0.843737i	$-0.680351\pi$
$613$	−9.98753e6	−1.07351	−0.536756	+	0.843737i	$0.680351\pi$
$614$	0	0
$615$	−2.18310e6	−0.232748
$616$	0	0
$617$	−1.14994e7	−1.21608	−0.608042	−	0.793905i	$-0.708045\pi$
$617$	−1.14994e7	−1.21608	−0.608042	+	0.793905i	$0.708045\pi$
$618$	0	0
$619$	5.70831e6	0.598799	0.299400	−	0.954128i	$-0.403214\pi$
$619$	5.70831e6	0.598799	0.299400	+	0.954128i	$0.403214\pi$
$620$	0	0
$621$	−385641.	−0.0401286
$622$	0	0
$623$	2.64940e6	0.273481
$624$	0	0
$625$	−8.43064e6	−0.863297
$626$	0	0
$627$	−8.74201e6	−0.888060
$628$	0	0
$629$	8.72288e6	0.879090
$630$	0	0
$631$	−6.75550e6	−0.675436	−0.337718	−	0.941247i	$-0.609655\pi$
$631$	−6.75550e6	−0.675436	−0.337718	+	0.941247i	$0.609655\pi$
$632$	0	0
$633$	9.30592e6	0.923102
$634$	0	0
$635$	7.37703e6	0.726018
$636$	0	0
$637$	1.53368e6	0.149757
$638$	0	0
$639$	−6.04813e6	−0.585961
$640$	0	0
$641$	355815.	0.0342042	0.0171021	−	0.999854i	$-0.494556\pi$
$641$	355815.	0.0342042	0.0171021	+	0.999854i	$0.494556\pi$
$642$	0	0
$643$	−1.12269e6	−0.107086	−0.0535432	−	0.998566i	$-0.517051\pi$
$643$	−1.12269e6	−0.107086	−0.0535432	+	0.998566i	$0.517051\pi$
$644$	0	0
$645$	4.33807e6	0.410580
$646$	0	0
$647$	1.91791e7	1.80123	0.900613	−	0.434621i	$-0.143118\pi$
$647$	1.91791e7	1.80123	0.900613	+	0.434621i	$0.143118\pi$
$648$	0	0
$649$	4.77317e6	0.444831
$650$	0	0
$651$	7.73319e6	0.715165
$652$	0	0
$653$	−9.62631e6	−0.883440	−0.441720	−	0.897153i	$-0.645632\pi$
$653$	−9.62631e6	−0.883440	−0.441720	+	0.897153i	$0.645632\pi$
$654$	0	0
$655$	7.17813e6	0.653745
$656$	0	0
$657$	6.54010e6	0.591114
$658$	0	0
$659$	6.82930e6	0.612579	0.306290	−	0.951938i	$-0.400912\pi$
$659$	6.82930e6	0.612579	0.306290	+	0.951938i	$0.400912\pi$
$660$	0	0
$661$	9.20374e6	0.819333	0.409667	−	0.912235i	$-0.365645\pi$
$661$	9.20374e6	0.819333	0.409667	+	0.912235i	$0.365645\pi$
$662$	0	0
$663$	4.07687e6	0.360199
$664$	0	0
$665$	8.11263e6	0.711389
$666$	0	0
$667$	−2.92878e6	−0.254902
$668$	0	0
$669$	−8.91169e6	−0.769830
$670$	0	0
$671$	1.95623e6	0.167731
$672$	0	0
$673$	−9.25649e6	−0.787787	−0.393893	−	0.919156i	$-0.628872\pi$
$673$	−9.25649e6	−0.787787	−0.393893	+	0.919156i	$0.628872\pi$
$674$	0	0
$675$	−277599.	−0.0234509
$676$	0	0
$677$	−4.89629e6	−0.410578	−0.205289	−	0.978701i	$-0.565813\pi$
$677$	−4.89629e6	−0.410578	−0.205289	+	0.978701i	$0.565813\pi$
$678$	0	0
$679$	−3.51457e6	−0.292548
$680$	0	0
$681$	240168.	0.0198448
$682$	0	0
$683$	1.01443e6	0.0832086	0.0416043	−	0.999134i	$-0.486753\pi$
$683$	1.01443e6	0.0832086	0.0416043	+	0.999134i	$0.486753\pi$
$684$	0	0
$685$	3.85699e6	0.314067
$686$	0	0
$687$	1.35527e6	0.109555
$688$	0	0
$689$	1.96401e7	1.57614
$690$	0	0
$691$	9.64799e6	0.768673	0.384336	−	0.923193i	$-0.374430\pi$
$691$	9.64799e6	0.768673	0.384336	+	0.923193i	$0.374430\pi$
$692$	0	0
$693$	−7.10242e6	−0.561789
$694$	0	0
$695$	−1.12236e7	−0.881396
$696$	0	0
$697$	3.86639e6	0.301456
$698$	0	0
$699$	1.16479e7	0.901684
$700$	0	0
$701$	−1.47825e7	−1.13620	−0.568098	−	0.822961i	$-0.692321\pi$
$701$	−1.47825e7	−1.13620	−0.568098	+	0.822961i	$0.692321\pi$
$702$	0	0
$703$	−1.36829e7	−1.04421
$704$	0	0
$705$	−7.20037e6	−0.545610
$706$	0	0
$707$	1.14524e7	0.861683
$708$	0	0
$709$	−585189.	−0.0437201	−0.0218600	−	0.999761i	$-0.506959\pi$
$709$	−585189.	−0.0437201	−0.0218600	+	0.999761i	$0.506959\pi$
$710$	0	0
$711$	−3.16772e6	−0.235003
$712$	0	0
$713$	−3.84432e6	−0.283201
$714$	0	0
$715$	2.10755e7	1.54175
$716$	0	0
$717$	1.23208e6	0.0895035
$718$	0	0
$719$	3.42448e6	0.247043	0.123521	−	0.992342i	$-0.460581\pi$
$719$	3.42448e6	0.247043	0.123521	+	0.992342i	$0.460581\pi$
$720$	0	0
$721$	−246064.	−0.0176283
$722$	0	0
$723$	−2.56317e6	−0.182361
$724$	0	0
$725$	−2.10825e6	−0.148963
$726$	0	0
$727$	1.91984e7	1.34719	0.673595	−	0.739100i	$-0.264749\pi$
$727$	1.91984e7	1.34719	0.673595	+	0.739100i	$0.264749\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−7.68297e6	−0.531784
$732$	0	0
$733$	2.66362e7	1.83110	0.915552	−	0.402200i	$-0.131754\pi$
$733$	2.66362e7	1.83110	0.915552	+	0.402200i	$0.131754\pi$
$734$	0	0
$735$	−1.33286e6	−0.0910053
$736$	0	0
$737$	1.56243e7	1.05958
$738$	0	0
$739$	6.81746e6	0.459210	0.229605	−	0.973284i	$-0.426257\pi$
$739$	6.81746e6	0.459210	0.229605	+	0.973284i	$0.426257\pi$
$740$	0	0
$741$	−6.39506e6	−0.427857
$742$	0	0
$743$	377932.	0.0251155	0.0125577	−	0.999921i	$-0.496003\pi$
$743$	377932.	0.0251155	0.0125577	+	0.999921i	$0.496003\pi$
$744$	0	0
$745$	6.93946e6	0.458073
$746$	0	0
$747$	−8.21863e6	−0.538887
$748$	0	0
$749$	9.09327e6	0.592264
$750$	0	0
$751$	8.02610e6	0.519283	0.259642	−	0.965705i	$-0.416396\pi$
$751$	8.02610e6	0.519283	0.259642	+	0.965705i	$0.416396\pi$
$752$	0	0
$753$	−1.13533e7	−0.729686
$754$	0	0
$755$	−8.99242e6	−0.574129
$756$	0	0
$757$	−4.55894e6	−0.289150	−0.144575	−	0.989494i	$-0.546182\pi$
$757$	−4.55894e6	−0.289150	−0.144575	+	0.989494i	$0.546182\pi$
$758$	0	0
$759$	3.53075e6	0.222465
$760$	0	0
$761$	2.77860e7	1.73926	0.869629	−	0.493705i	$-0.164358\pi$
$761$	2.77860e7	1.73926	0.869629	+	0.493705i	$0.164358\pi$
$762$	0	0
$763$	−3.22195e6	−0.200358
$764$	0	0
$765$	−3.54304e6	−0.218888
$766$	0	0
$767$	3.49173e6	0.214315
$768$	0	0
$769$	1.47056e7	0.896742	0.448371	−	0.893847i	$-0.352004\pi$
$769$	1.47056e7	0.896742	0.448371	+	0.893847i	$0.352004\pi$
$770$	0	0
$771$	4.97448e6	0.301378
$772$	0	0
$773$	1.46941e7	0.884495	0.442248	−	0.896893i	$-0.354181\pi$
$773$	1.46941e7	0.884495	0.442248	+	0.896893i	$0.354181\pi$
$774$	0	0
$775$	−2.76729e6	−0.165501
$776$	0	0
$777$	−1.11166e7	−0.660572
$778$	0	0
$779$	−6.06489e6	−0.358080
$780$	0	0
$781$	5.53738e7	3.24845
$782$	0	0
$783$	4.03607e6	0.235264
$784$	0	0
$785$	−1.88509e7	−1.09184
$786$	0	0
$787$	−1.02770e7	−0.591465	−0.295732	−	0.955271i	$-0.595564\pi$
$787$	−1.02770e7	−0.591465	−0.295732	+	0.955271i	$0.595564\pi$
$788$	0	0
$789$	5.86363e6	0.335331
$790$	0	0
$791$	−1.42757e7	−0.811253
$792$	0	0
$793$	1.43105e6	0.0808110
$794$	0	0
$795$	−1.70684e7	−0.957800
$796$	0	0
$797$	2.13182e7	1.18879	0.594395	−	0.804173i	$-0.297392\pi$
$797$	2.13182e7	1.18879	0.594395	+	0.804173i	$0.297392\pi$
$798$	0	0
$799$	1.27523e7	0.706675
$800$	0	0
$801$	1.81501e6	0.0999536
$802$	0	0
$803$	−5.98781e7	−3.27702
$804$	0	0
$805$	−3.27655e6	−0.178208
$806$	0	0
$807$	1.46393e7	0.791293
$808$	0	0
$809$	2.64965e7	1.42337	0.711685	−	0.702499i	$-0.247932\pi$
$809$	2.64965e7	1.42337	0.711685	+	0.702499i	$0.247932\pi$
$810$	0	0
$811$	−1.59727e7	−0.852758	−0.426379	−	0.904545i	$-0.640211\pi$
$811$	−1.59727e7	−0.852758	−0.426379	+	0.904545i	$0.640211\pi$
$812$	0	0
$813$	−4.73893e6	−0.251452
$814$	0	0
$815$	−2.38172e7	−1.25602
$816$	0	0
$817$	1.20517e7	0.631672
$818$	0	0
$819$	−5.19564e6	−0.270663
$820$	0	0
$821$	2.19797e7	1.13806	0.569029	−	0.822317i	$-0.307319\pi$
$821$	2.19797e7	1.13806	0.569029	+	0.822317i	$0.307319\pi$
$822$	0	0
$823$	3.47354e7	1.78761	0.893806	−	0.448454i	$-0.148025\pi$
$823$	3.47354e7	1.78761	0.893806	+	0.448454i	$0.148025\pi$
$824$	0	0
$825$	2.54157e6	0.130007
$826$	0	0
$827$	1.25679e7	0.638997	0.319499	−	0.947587i	$-0.396485\pi$
$827$	1.25679e7	0.638997	0.319499	+	0.947587i	$0.396485\pi$
$828$	0	0
$829$	−3.35386e7	−1.69495	−0.847477	−	0.530831i	$-0.821880\pi$
$829$	−3.35386e7	−1.69495	−0.847477	+	0.530831i	$0.821880\pi$
$830$	0	0
$831$	926766.	0.0465551
$832$	0	0
$833$	2.36057e6	0.117870
$834$	0	0
$835$	1.88501e7	0.935617
$836$	0	0
$837$	5.29774e6	0.261383
$838$	0	0
$839$	−1.15158e6	−0.0564791	−0.0282395	−	0.999601i	$-0.508990\pi$
$839$	−1.15158e6	−0.0564791	−0.0282395	+	0.999601i	$0.508990\pi$
$840$	0	0
$841$	1.01412e7	0.494422
$842$	0	0
$843$	−1.61288e7	−0.781686
$844$	0	0
$845$	−4.03284e6	−0.194298
$846$	0	0
$847$	4.59842e7	2.20242
$848$	0	0
$849$	9.59081e6	0.456653
$850$	0	0
$851$	5.52628e6	0.261583
$852$	0	0
$853$	−1.19102e7	−0.560465	−0.280232	−	0.959932i	$-0.590411\pi$
$853$	−1.19102e7	−0.560465	−0.280232	+	0.959932i	$0.590411\pi$
$854$	0	0
$855$	5.55768e6	0.260003
$856$	0	0
$857$	−1.59627e7	−0.742430	−0.371215	−	0.928547i	$-0.621059\pi$
$857$	−1.59627e7	−0.742430	−0.371215	+	0.928547i	$0.621059\pi$
$858$	0	0
$859$	−1.22935e7	−0.568452	−0.284226	−	0.958757i	$-0.591737\pi$
$859$	−1.22935e7	−0.568452	−0.284226	+	0.958757i	$0.591737\pi$
$860$	0	0
$861$	−4.92740e6	−0.226522
$862$	0	0
$863$	2.14243e7	0.979219	0.489609	−	0.871942i	$-0.337139\pi$
$863$	2.14243e7	0.979219	0.489609	+	0.871942i	$0.337139\pi$
$864$	0	0
$865$	−1.61360e7	−0.733254
$866$	0	0
$867$	−6.50380e6	−0.293846
$868$	0	0
$869$	2.90022e7	1.30281
$870$	0	0
$871$	1.14297e7	0.510492
$872$	0	0
$873$	−2.40771e6	−0.106922
$874$	0	0
$875$	−2.17144e7	−0.958799
$876$	0	0
$877$	−3.70233e6	−0.162546	−0.0812730	−	0.996692i	$-0.525899\pi$
$877$	−3.70233e6	−0.162546	−0.0812730	+	0.996692i	$0.525899\pi$
$878$	0	0
$879$	1.67272e7	0.730217
$880$	0	0
$881$	5.90856e6	0.256473	0.128237	−	0.991744i	$-0.459068\pi$
$881$	5.90856e6	0.256473	0.128237	+	0.991744i	$0.459068\pi$
$882$	0	0
$883$	−1.83087e7	−0.790233	−0.395117	−	0.918631i	$-0.629296\pi$
$883$	−1.83087e7	−0.790233	−0.395117	+	0.918631i	$0.629296\pi$
$884$	0	0
$885$	−3.03452e6	−0.130236
$886$	0	0
$887$	−2.01751e7	−0.861005	−0.430503	−	0.902589i	$-0.641664\pi$
$887$	−2.01751e7	−0.861005	−0.430503	+	0.902589i	$0.641664\pi$
$888$	0	0
$889$	1.66505e7	0.706597
$890$	0	0
$891$	−4.86562e6	−0.205326
$892$	0	0
$893$	−2.00034e7	−0.839413
$894$	0	0
$895$	−3.29447e7	−1.37476
$896$	0	0
$897$	2.58285e6	0.107181
$898$	0	0
$899$	4.02342e7	1.66034
$900$	0	0
$901$	3.02291e7	1.24055
$902$	0	0
$903$	9.79132e6	0.399597
$904$	0	0
$905$	−3.49417e7	−1.41815
$906$	0	0
$907$	3.53066e7	1.42508	0.712538	−	0.701634i	$-0.247546\pi$
$907$	3.53066e7	1.42508	0.712538	+	0.701634i	$0.247546\pi$
$908$	0	0
$909$	7.84564e6	0.314933
$910$	0	0
$911$	1.91557e7	0.764720	0.382360	−	0.924014i	$-0.375111\pi$
$911$	1.91557e7	0.764720	0.382360	+	0.924014i	$0.375111\pi$
$912$	0	0
$913$	7.52459e7	2.98749
$914$	0	0
$915$	−1.24366e6	−0.0491077
$916$	0	0
$917$	1.62015e7	0.636257
$918$	0	0
$919$	3.36053e7	1.31256	0.656279	−	0.754518i	$-0.272129\pi$
$919$	3.36053e7	1.31256	0.656279	+	0.754518i	$0.272129\pi$
$920$	0	0
$921$	1.92425e7	0.747501
$922$	0	0
$923$	4.05077e7	1.56507
$924$	0	0
$925$	3.97803e6	0.152867
$926$	0	0
$927$	−168570.	−0.00644289
$928$	0	0
$929$	−2.51698e7	−0.956842	−0.478421	−	0.878131i	$-0.658791\pi$
$929$	−2.51698e7	−0.956842	−0.478421	+	0.878131i	$0.658791\pi$
$930$	0	0
$931$	−3.70284e6	−0.140011
$932$	0	0
$933$	−1.21937e7	−0.458596
$934$	0	0
$935$	3.24384e7	1.21347
$936$	0	0
$937$	−2.32062e7	−0.863484	−0.431742	−	0.901997i	$-0.642101\pi$
$937$	−2.32062e7	−0.863484	−0.431742	+	0.901997i	$0.642101\pi$
$938$	0	0
$939$	−4.71410e6	−0.174476
$940$	0	0
$941$	−1.44758e7	−0.532930	−0.266465	−	0.963845i	$-0.585856\pi$
$941$	−1.44758e7	−0.532930	−0.266465	+	0.963845i	$0.585856\pi$
$942$	0	0
$943$	2.44951e6	0.0897014
$944$	0	0
$945$	4.51532e6	0.164478
$946$	0	0
$947$	3.57094e7	1.29392	0.646960	−	0.762524i	$-0.276040\pi$
$947$	3.57094e7	1.29392	0.646960	+	0.762524i	$0.276040\pi$
$948$	0	0
$949$	−4.38027e7	−1.57883
$950$	0	0
$951$	2.33926e7	0.838741
$952$	0	0
$953$	−6.09472e6	−0.217381	−0.108691	−	0.994076i	$-0.534666\pi$
$953$	−6.09472e6	−0.217381	−0.108691	+	0.994076i	$0.534666\pi$
$954$	0	0
$955$	−3.00338e7	−1.06562
$956$	0	0
$957$	−3.69524e7	−1.30426
$958$	0	0
$959$	8.70548e6	0.305665
$960$	0	0
$961$	2.41822e7	0.844669
$962$	0	0
$963$	6.22949e6	0.216464
$964$	0	0
$965$	1.11048e6	0.0383877
$966$	0	0
$967$	−1.48405e7	−0.510366	−0.255183	−	0.966893i	$-0.582136\pi$
$967$	−1.48405e7	−0.510366	−0.255183	+	0.966893i	$0.582136\pi$
$968$	0	0
$969$	−9.84296e6	−0.336757
$970$	0	0
$971$	−1.73779e7	−0.591493	−0.295746	−	0.955267i	$-0.595568\pi$
$971$	−1.73779e7	−0.591493	−0.295746	+	0.955267i	$0.595568\pi$
$972$	0	0
$973$	−2.53325e7	−0.857818
$974$	0	0
$975$	1.85924e6	0.0626359
$976$	0	0
$977$	−1.30124e7	−0.436136	−0.218068	−	0.975934i	$-0.569976\pi$
$977$	−1.30124e7	−0.436136	−0.218068	+	0.975934i	$0.569976\pi$
$978$	0	0
$979$	−1.66174e7	−0.554123
$980$	0	0
$981$	−2.20725e6	−0.0732282
$982$	0	0
$983$	−1.94435e7	−0.641787	−0.320893	−	0.947115i	$-0.603983\pi$
$983$	−1.94435e7	−0.641787	−0.320893	+	0.947115i	$0.603983\pi$
$984$	0	0
$985$	−1.79266e7	−0.588717
$986$	0	0
$987$	−1.62517e7	−0.531015
$988$	0	0
$989$	−4.86745e6	−0.158238
$990$	0	0
$991$	3.56706e6	0.115379	0.0576894	−	0.998335i	$-0.481627\pi$
$991$	3.56706e6	0.115379	0.0576894	+	0.998335i	$0.481627\pi$
$992$	0	0
$993$	9.25984e6	0.298010
$994$	0	0
$995$	1.69390e6	0.0542414
$996$	0	0
$997$	3.36336e7	1.07161	0.535804	−	0.844342i	$-0.320009\pi$
$997$	3.36336e7	1.07161	0.535804	+	0.844342i	$0.320009\pi$
$998$	0	0
$999$	−7.61561e6	−0.241430

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.h.1.2		2
4.3	odd	2		69.6.a.a.1.2	✓	2
12.11	even	2		207.6.a.a.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.a.1.2	✓	2	4.3	odd	2
207.6.a.a.1.1		2	12.11	even	2
1104.6.a.h.1.2		2	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)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} +52.3852 q^{5} +118.237 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q+9.00000 q^{3} +52.3852 q^{5} +118.237 q^{7} +81.0000 q^{9} -741.598 q^{11} -542.502 q^{13} +471.466 q^{15} -834.993 q^{17} +1309.79 q^{19} +1064.13 q^{21} -529.000 q^{23} -380.795 q^{25} +729.000 q^{27} +5536.45 q^{29} +7267.14 q^{31} -6674.38 q^{33} +6193.85 q^{35} -10446.7 q^{37} -4882.52 q^{39} -4630.45 q^{41} +9201.23 q^{43} +4243.20 q^{45} -15272.3 q^{47} -2827.06 q^{49} -7514.94 q^{51} -36202.8 q^{53} -38848.7 q^{55} +11788.1 q^{57} -6436.34 q^{59} -2637.86 q^{61} +9577.18 q^{63} -28419.1 q^{65} -21068.5 q^{67} -4761.00 q^{69} -74668.2 q^{71} +80742.0 q^{73} -3427.15 q^{75} -87684.2 q^{77} -39107.7 q^{79} +6561.00 q^{81} -101465. q^{83} -43741.2 q^{85} +49828.1 q^{87} +22407.6 q^{89} -64143.7 q^{91} +65404.3 q^{93} +68613.4 q^{95} -29724.8 q^{97} -60069.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 18 q^{3} + 94 q^{5} + 118 q^{7} + 162 q^{9}+O(q^{10}) \) 2 * q + 18 * q^3 + 94 * q^5 + 118 * q^7 + 162 * q^9	\( 2 q + 18 q^{3} + 94 q^{5} + 118 q^{7} + 162 q^{9} - 320 q^{11} - 288 q^{13} + 846 q^{15} - 1810 q^{17} - 730 q^{19} + 1062 q^{21} - 1058 q^{23} - 1774 q^{25} + 1458 q^{27} + 8208 q^{29} - 1772 q^{31} - 2880 q^{33} + 6184 q^{35} - 23112 q^{37} - 2592 q^{39} + 5516 q^{41} - 10322 q^{43} + 7614 q^{45} - 42952 q^{47} - 19634 q^{49} - 16290 q^{51} - 25350 q^{53} - 21304 q^{55} - 6570 q^{57} - 18344 q^{59} + 37224 q^{61} + 9558 q^{63} - 17828 q^{65} + 7482 q^{67} - 9522 q^{69} - 126848 q^{71} + 137660 q^{73} - 15966 q^{75} - 87784 q^{77} - 62286 q^{79} + 13122 q^{81} - 83120 q^{83} - 84316 q^{85} + 73872 q^{87} + 69770 q^{89} - 64204 q^{91} - 15948 q^{93} - 16272 q^{95} - 170104 q^{97} - 25920 q^{99}+O(q^{100}) \) 2 * q + 18 * q^3 + 94 * q^5 + 118 * q^7 + 162 * q^9 - 320 * q^11 - 288 * q^13 + 846 * q^15 - 1810 * q^17 - 730 * q^19 + 1062 * q^21 - 1058 * q^23 - 1774 * q^25 + 1458 * q^27 + 8208 * q^29 - 1772 * q^31 - 2880 * q^33 + 6184 * q^35 - 23112 * q^37 - 2592 * q^39 + 5516 * q^41 - 10322 * q^43 + 7614 * q^45 - 42952 * q^47 - 19634 * q^49 - 16290 * q^51 - 25350 * q^53 - 21304 * q^55 - 6570 * q^57 - 18344 * q^59 + 37224 * q^61 + 9558 * q^63 - 17828 * q^65 + 7482 * q^67 - 9522 * q^69 - 126848 * q^71 + 137660 * q^73 - 15966 * q^75 - 87784 * q^77 - 62286 * q^79 + 13122 * q^81 - 83120 * q^83 - 84316 * q^85 + 73872 * q^87 + 69770 * q^89 - 64204 * q^91 - 15948 * q^93 - 16272 * q^95 - 170104 * q^97 - 25920 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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