LMFDB - Embedded newform 1104.6.a.x.1.5

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 3225x^{5} + 19410x^{4} + 2132445x^{3} - 10443621x^{2} - 341555347x - 181104660 $ x^7 - 3x^6 - 3225x^5 + 19410x^4 + 2132445x^3 - 10443621x^2 - 341555347x - 181104660
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 552)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-11.7085$ 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)

$f(q)$	$=$	$q-9.00000 q^{3} +9.41706 q^{5} -178.224 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} +9.41706 q^{5} -178.224 q^{7} +81.0000 q^{9} -7.24503 q^{11} -760.686 q^{13} -84.7536 q^{15} -1682.71 q^{17} -130.563 q^{19} +1604.02 q^{21} +529.000 q^{23} -3036.32 q^{25} -729.000 q^{27} -6244.17 q^{29} +6520.49 q^{31} +65.2053 q^{33} -1678.35 q^{35} -15756.4 q^{37} +6846.17 q^{39} +6572.02 q^{41} -17080.9 q^{43} +762.782 q^{45} -14550.4 q^{47} +14956.9 q^{49} +15144.4 q^{51} -2903.00 q^{53} -68.2269 q^{55} +1175.06 q^{57} +7693.49 q^{59} +8841.18 q^{61} -14436.2 q^{63} -7163.43 q^{65} +22723.3 q^{67} -4761.00 q^{69} +12633.5 q^{71} -69546.3 q^{73} +27326.9 q^{75} +1291.24 q^{77} -63402.8 q^{79} +6561.00 q^{81} -8475.45 q^{83} -15846.2 q^{85} +56197.6 q^{87} -22692.3 q^{89} +135573. q^{91} -58684.4 q^{93} -1229.52 q^{95} -156967. q^{97} -586.848 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 63 q^{3} - 104 q^{5} + 182 q^{7} + 567 q^{9}+O(q^{10}) $ 7 * q - 63 * q^3 - 104 * q^5 + 182 * q^7 + 567 * q^9	$ 7 q - 63 q^{3} - 104 q^{5} + 182 q^{7} + 567 q^{9} + 124 q^{11} + 294 q^{13} + 936 q^{15} + 428 q^{17} + 1826 q^{19} - 1638 q^{21} + 3703 q^{23} + 5501 q^{25} - 5103 q^{27} - 1682 q^{29} + 14420 q^{31} - 1116 q^{33} - 3312 q^{35} - 14218 q^{37} - 2646 q^{39} + 7294 q^{41} - 3630 q^{43} - 8424 q^{45} + 19808 q^{47} - 4325 q^{49} - 3852 q^{51} - 77412 q^{53} + 61136 q^{55} - 16434 q^{57} + 51076 q^{59} - 67186 q^{61} + 14742 q^{63} - 121800 q^{65} + 70870 q^{67} - 33327 q^{69} + 45784 q^{71} - 175522 q^{73} - 49509 q^{75} - 148632 q^{77} + 54538 q^{79} + 45927 q^{81} - 27612 q^{83} - 197552 q^{85} + 15138 q^{87} - 184360 q^{89} + 217412 q^{91} - 129780 q^{93} + 82608 q^{95} - 339766 q^{97} + 10044 q^{99}+O(q^{100}) $ 7 * q - 63 * q^3 - 104 * q^5 + 182 * q^7 + 567 * q^9 + 124 * q^11 + 294 * q^13 + 936 * q^15 + 428 * q^17 + 1826 * q^19 - 1638 * q^21 + 3703 * q^23 + 5501 * q^25 - 5103 * q^27 - 1682 * q^29 + 14420 * q^31 - 1116 * q^33 - 3312 * q^35 - 14218 * q^37 - 2646 * q^39 + 7294 * q^41 - 3630 * q^43 - 8424 * q^45 + 19808 * q^47 - 4325 * q^49 - 3852 * q^51 - 77412 * q^53 + 61136 * q^55 - 16434 * q^57 + 51076 * q^59 - 67186 * q^61 + 14742 * q^63 - 121800 * q^65 + 70870 * q^67 - 33327 * q^69 + 45784 * q^71 - 175522 * q^73 - 49509 * q^75 - 148632 * q^77 + 54538 * q^79 + 45927 * q^81 - 27612 * q^83 - 197552 * q^85 + 15138 * q^87 - 184360 * q^89 + 217412 * q^91 - 129780 * q^93 + 82608 * q^95 - 339766 * q^97 + 10044 * 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
$5$	9.41706	0.168458	0.0842288	−	0.996446i	$-0.473157\pi$
$5$	9.41706	0.168458	0.0842288	+	0.996446i	$0.473157\pi$
$6$	0	0
$7$	−178.224	−1.37474	−0.687372	−	0.726306i	$-0.741236\pi$
$7$	−178.224	−1.37474	−0.687372	+	0.726306i	$0.741236\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−7.24503	−0.0180534	−0.00902669	−	0.999959i	$-0.502873\pi$
$11$	−7.24503	−0.0180534	−0.00902669	+	0.999959i	$0.502873\pi$
$12$	0	0
$13$	−760.686	−1.24838	−0.624190	−	0.781272i	$-0.714571\pi$
$13$	−760.686	−1.24838	−0.624190	+	0.781272i	$0.714571\pi$
$14$	0	0
$15$	−84.7536	−0.0972590
$16$	0	0
$17$	−1682.71	−1.41217	−0.706084	−	0.708128i	$-0.749540\pi$
$17$	−1682.71	−1.41217	−0.706084	+	0.708128i	$0.749540\pi$
$18$	0	0
$19$	−130.563	−0.0829726	−0.0414863	−	0.999139i	$-0.513209\pi$
$19$	−130.563	−0.0829726	−0.0414863	+	0.999139i	$0.513209\pi$
$20$	0	0
$21$	1604.02	0.793709
$22$	0	0
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	−3036.32	−0.971622
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−6244.17	−1.37873	−0.689366	−	0.724413i	$-0.742111\pi$
$29$	−6244.17	−1.37873	−0.689366	+	0.724413i	$0.742111\pi$
$30$	0	0
$31$	6520.49	1.21864	0.609321	−	0.792924i	$-0.291442\pi$
$31$	6520.49	1.21864	0.609321	+	0.792924i	$0.291442\pi$
$32$	0	0
$33$	65.2053	0.0104231
$34$	0	0
$35$	−1678.35	−0.231586
$36$	0	0
$37$	−15756.4	−1.89214	−0.946068	−	0.323968i	$-0.894983\pi$
$37$	−15756.4	−1.89214	−0.946068	+	0.323968i	$0.894983\pi$
$38$	0	0
$39$	6846.17	0.720753
$40$	0	0
$41$	6572.02	0.610575	0.305288	−	0.952260i	$-0.401247\pi$
$41$	6572.02	0.610575	0.305288	+	0.952260i	$0.401247\pi$
$42$	0	0
$43$	−17080.9	−1.40877	−0.704384	−	0.709819i	$-0.748777\pi$
$43$	−17080.9	−1.40877	−0.704384	+	0.709819i	$0.748777\pi$
$44$	0	0
$45$	762.782	0.0561525
$46$	0	0
$47$	−14550.4	−0.960796	−0.480398	−	0.877051i	$-0.659508\pi$
$47$	−14550.4	−0.960796	−0.480398	+	0.877051i	$0.659508\pi$
$48$	0	0
$49$	14956.9	0.889920
$50$	0	0
$51$	15144.4	0.815316
$52$	0	0
$53$	−2903.00	−0.141957	−0.0709785	−	0.997478i	$-0.522612\pi$
$53$	−2903.00	−0.141957	−0.0709785	+	0.997478i	$0.522612\pi$
$54$	0	0
$55$	−68.2269	−0.00304123
$56$	0	0
$57$	1175.06	0.0479043
$58$	0	0
$59$	7693.49	0.287736	0.143868	−	0.989597i	$-0.454046\pi$
$59$	7693.49	0.287736	0.143868	+	0.989597i	$0.454046\pi$
$60$	0	0
$61$	8841.18	0.304219	0.152109	−	0.988364i	$-0.451393\pi$
$61$	8841.18	0.304219	0.152109	+	0.988364i	$0.451393\pi$
$62$	0	0
$63$	−14436.2	−0.458248
$64$	0	0
$65$	−7163.43	−0.210299
$66$	0	0
$67$	22723.3	0.618421	0.309210	−	0.950994i	$-0.399935\pi$
$67$	22723.3	0.618421	0.309210	+	0.950994i	$0.399935\pi$
$68$	0	0
$69$	−4761.00	−0.120386
$70$	0	0
$71$	12633.5	0.297424	0.148712	−	0.988881i	$-0.452487\pi$
$71$	12633.5	0.297424	0.148712	+	0.988881i	$0.452487\pi$
$72$	0	0
$73$	−69546.3	−1.52745	−0.763725	−	0.645542i	$-0.776631\pi$
$73$	−69546.3	−1.52745	−0.763725	+	0.645542i	$0.776631\pi$
$74$	0	0
$75$	27326.9	0.560966
$76$	0	0
$77$	1291.24	0.0248188
$78$	0	0
$79$	−63402.8	−1.14299	−0.571493	−	0.820607i	$-0.693635\pi$
$79$	−63402.8	−1.14299	−0.571493	+	0.820607i	$0.693635\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−8475.45	−0.135042	−0.0675208	−	0.997718i	$-0.521509\pi$
$83$	−8475.45	−0.135042	−0.0675208	+	0.997718i	$0.521509\pi$
$84$	0	0
$85$	−15846.2	−0.237891
$86$	0	0
$87$	56197.6	0.796012
$88$	0	0
$89$	−22692.3	−0.303671	−0.151836	−	0.988406i	$-0.548518\pi$
$89$	−22692.3	−0.303671	−0.151836	+	0.988406i	$0.548518\pi$
$90$	0	0
$91$	135573.	1.71620
$92$	0	0
$93$	−58684.4	−0.703583
$94$	0	0
$95$	−1229.52	−0.0139774
$96$	0	0
$97$	−156967.	−1.69386	−0.846932	−	0.531702i	$-0.821553\pi$
$97$	−156967.	−1.69386	−0.846932	+	0.531702i	$0.821553\pi$
$98$	0	0
$99$	−586.848	−0.00601780
$100$	0	0
$101$	186108.	1.81535	0.907677	−	0.419670i	$-0.137854\pi$
$101$	186108.	1.81535	0.907677	+	0.419670i	$0.137854\pi$
$102$	0	0
$103$	−154756.	−1.43733	−0.718664	−	0.695358i	$-0.755246\pi$
$103$	−154756.	−1.43733	−0.718664	+	0.695358i	$0.755246\pi$
$104$	0	0
$105$	15105.1	0.133706
$106$	0	0
$107$	−105156.	−0.887918	−0.443959	−	0.896047i	$-0.646426\pi$
$107$	−105156.	−0.887918	−0.443959	+	0.896047i	$0.646426\pi$
$108$	0	0
$109$	−35071.6	−0.282741	−0.141371	−	0.989957i	$-0.545151\pi$
$109$	−35071.6	−0.282741	−0.141371	+	0.989957i	$0.545151\pi$
$110$	0	0
$111$	141807.	1.09243
$112$	0	0
$113$	−84928.9	−0.625690	−0.312845	−	0.949804i	$-0.601282\pi$
$113$	−84928.9	−0.625690	−0.312845	+	0.949804i	$0.601282\pi$
$114$	0	0
$115$	4981.63	0.0351258
$116$	0	0
$117$	−61615.6	−0.416127
$118$	0	0
$119$	299900.	1.94137
$120$	0	0
$121$	−160999.	−0.999674
$122$	0	0
$123$	−59148.2	−0.352516
$124$	0	0
$125$	−58021.5	−0.332135
$126$	0	0
$127$	13815.6	0.0760080	0.0380040	−	0.999278i	$-0.487900\pi$
$127$	13815.6	0.0760080	0.0380040	+	0.999278i	$0.487900\pi$
$128$	0	0
$129$	153728.	0.813353
$130$	0	0
$131$	119937.	0.610627	0.305314	−	0.952252i	$-0.401239\pi$
$131$	119937.	0.610627	0.305314	+	0.952252i	$0.401239\pi$
$132$	0	0
$133$	23269.4	0.114066
$134$	0	0
$135$	−6865.04	−0.0324197
$136$	0	0
$137$	152139.	0.692531	0.346265	−	0.938137i	$-0.387450\pi$
$137$	152139.	0.692531	0.346265	+	0.938137i	$0.387450\pi$
$138$	0	0
$139$	−8594.70	−0.0377306	−0.0188653	−	0.999822i	$-0.506005\pi$
$139$	−8594.70	−0.0377306	−0.0188653	+	0.999822i	$0.506005\pi$
$140$	0	0
$141$	130954.	0.554716
$142$	0	0
$143$	5511.20	0.0225375
$144$	0	0
$145$	−58801.8	−0.232258
$146$	0	0
$147$	−134612.	−0.513796
$148$	0	0
$149$	130864.	0.482897	0.241449	−	0.970414i	$-0.422377\pi$
$149$	130864.	0.482897	0.241449	+	0.970414i	$0.422377\pi$
$150$	0	0
$151$	434643.	1.55128	0.775640	−	0.631176i	$-0.217427\pi$
$151$	434643.	1.55128	0.775640	+	0.631176i	$0.217427\pi$
$152$	0	0
$153$	−136299.	−0.470723
$154$	0	0
$155$	61403.8	0.205289
$156$	0	0
$157$	369905.	1.19768	0.598841	−	0.800868i	$-0.295628\pi$
$157$	369905.	1.19768	0.598841	+	0.800868i	$0.295628\pi$
$158$	0	0
$159$	26127.0	0.0819589
$160$	0	0
$161$	−94280.6	−0.286654
$162$	0	0
$163$	−392419.	−1.15686	−0.578431	−	0.815731i	$-0.696335\pi$
$163$	−392419.	−1.15686	−0.578431	+	0.815731i	$0.696335\pi$
$164$	0	0
$165$	614.043	0.00175585
$166$	0	0
$167$	654484.	1.81597	0.907983	−	0.419007i	$-0.137622\pi$
$167$	654484.	1.81597	0.907983	+	0.419007i	$0.137622\pi$
$168$	0	0
$169$	207350.	0.558454
$170$	0	0
$171$	−10575.6	−0.0276575
$172$	0	0
$173$	−434703.	−1.10428	−0.552138	−	0.833753i	$-0.686188\pi$
$173$	−434703.	−1.10428	−0.552138	+	0.833753i	$0.686188\pi$
$174$	0	0
$175$	541146.	1.33573
$176$	0	0
$177$	−69241.4	−0.166124
$178$	0	0
$179$	−105990.	−0.247247	−0.123624	−	0.992329i	$-0.539452\pi$
$179$	−105990.	−0.247247	−0.123624	+	0.992329i	$0.539452\pi$
$180$	0	0
$181$	−466514.	−1.05845	−0.529223	−	0.848483i	$-0.677516\pi$
$181$	−466514.	−1.05845	−0.529223	+	0.848483i	$0.677516\pi$
$182$	0	0
$183$	−79570.7	−0.175641
$184$	0	0
$185$	−148379.	−0.318745
$186$	0	0
$187$	12191.3	0.0254944
$188$	0	0
$189$	129925.	0.264570
$190$	0	0
$191$	−319385.	−0.633477	−0.316739	−	0.948513i	$-0.602588\pi$
$191$	−319385.	−0.633477	−0.316739	+	0.948513i	$0.602588\pi$
$192$	0	0
$193$	−113001.	−0.218369	−0.109184	−	0.994022i	$-0.534824\pi$
$193$	−113001.	−0.218369	−0.109184	+	0.994022i	$0.534824\pi$
$194$	0	0
$195$	64470.8	0.121416
$196$	0	0
$197$	−762710.	−1.40021	−0.700106	−	0.714039i	$-0.746864\pi$
$197$	−762710.	−1.40021	−0.700106	+	0.714039i	$0.746864\pi$
$198$	0	0
$199$	−680065.	−1.21736	−0.608678	−	0.793417i	$-0.708300\pi$
$199$	−680065.	−1.21736	−0.608678	+	0.793417i	$0.708300\pi$
$200$	0	0
$201$	−204510.	−0.357045
$202$	0	0
$203$	1.11286e6	1.89540
$204$	0	0
$205$	61889.1	0.102856
$206$	0	0
$207$	42849.0	0.0695048
$208$	0	0
$209$	945.931	0.00149794
$210$	0	0
$211$	−347666.	−0.537596	−0.268798	−	0.963197i	$-0.586626\pi$
$211$	−347666.	−0.537596	−0.268798	+	0.963197i	$0.586626\pi$
$212$	0	0
$213$	−113701.	−0.171718
$214$	0	0
$215$	−160852.	−0.237318
$216$	0	0
$217$	−1.16211e6	−1.67532
$218$	0	0
$219$	625917.	0.881873
$220$	0	0
$221$	1.28001e6	1.76292
$222$	0	0
$223$	1.09025e6	1.46813	0.734066	−	0.679078i	$-0.237620\pi$
$223$	1.09025e6	1.46813	0.734066	+	0.679078i	$0.237620\pi$
$224$	0	0
$225$	−245942.	−0.323874
$226$	0	0
$227$	481524.	0.620231	0.310116	−	0.950699i	$-0.399632\pi$
$227$	481524.	0.620231	0.310116	+	0.950699i	$0.399632\pi$
$228$	0	0
$229$	189444.	0.238722	0.119361	−	0.992851i	$-0.461915\pi$
$229$	189444.	0.238722	0.119361	+	0.992851i	$0.461915\pi$
$230$	0	0
$231$	−11621.2	−0.0143291
$232$	0	0
$233$	−810573.	−0.978143	−0.489071	−	0.872244i	$-0.662664\pi$
$233$	−810573.	−0.978143	−0.489071	+	0.872244i	$0.662664\pi$
$234$	0	0
$235$	−137022.	−0.161853
$236$	0	0
$237$	570625.	0.659903
$238$	0	0
$239$	551058.	0.624026	0.312013	−	0.950078i	$-0.398997\pi$
$239$	551058.	0.624026	0.312013	+	0.950078i	$0.398997\pi$
$240$	0	0
$241$	167872.	0.186181	0.0930906	−	0.995658i	$-0.470325\pi$
$241$	167872.	0.186181	0.0930906	+	0.995658i	$0.470325\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	140850.	0.149914
$246$	0	0
$247$	99317.1	0.103581
$248$	0	0
$249$	76279.0	0.0779663
$250$	0	0
$251$	−174379.	−0.174707	−0.0873534	−	0.996177i	$-0.527841\pi$
$251$	−174379.	−0.174707	−0.0873534	+	0.996177i	$0.527841\pi$
$252$	0	0
$253$	−3832.62	−0.00376439
$254$	0	0
$255$	142616.	0.137346
$256$	0	0
$257$	1.70962e6	1.61460	0.807302	−	0.590139i	$-0.200927\pi$
$257$	1.70962e6	1.61460	0.807302	+	0.590139i	$0.200927\pi$
$258$	0	0
$259$	2.80817e6	2.60120
$260$	0	0
$261$	−505778.	−0.459578
$262$	0	0
$263$	−1.47147e6	−1.31178	−0.655891	−	0.754855i	$-0.727707\pi$
$263$	−1.47147e6	−1.31178	−0.655891	+	0.754855i	$0.727707\pi$
$264$	0	0
$265$	−27337.7	−0.0239137
$266$	0	0
$267$	204231.	0.175325
$268$	0	0
$269$	990535.	0.834621	0.417310	−	0.908764i	$-0.362973\pi$
$269$	990535.	0.834621	0.417310	+	0.908764i	$0.362973\pi$
$270$	0	0
$271$	2.07133e6	1.71327	0.856635	−	0.515923i	$-0.172551\pi$
$271$	2.07133e6	1.71327	0.856635	+	0.515923i	$0.172551\pi$
$272$	0	0
$273$	−1.22015e6	−0.990850
$274$	0	0
$275$	21998.2	0.0175411
$276$	0	0
$277$	−675413.	−0.528895	−0.264448	−	0.964400i	$-0.585190\pi$
$277$	−675413.	−0.528895	−0.264448	+	0.964400i	$0.585190\pi$
$278$	0	0
$279$	528159.	0.406214
$280$	0	0
$281$	2.40160e6	1.81441	0.907205	−	0.420689i	$-0.138212\pi$
$281$	2.40160e6	1.81441	0.907205	+	0.420689i	$0.138212\pi$
$282$	0	0
$283$	1.88594e6	1.39979	0.699893	−	0.714247i	$-0.253231\pi$
$283$	1.88594e6	1.39979	0.699893	+	0.714247i	$0.253231\pi$
$284$	0	0
$285$	11065.6	0.00806984
$286$	0	0
$287$	−1.17129e6	−0.839385
$288$	0	0
$289$	1.41165e6	0.994221
$290$	0	0
$291$	1.41270e6	0.977952
$292$	0	0
$293$	−1.95649e6	−1.33140	−0.665699	−	0.746221i	$-0.731866\pi$
$293$	−1.95649e6	−1.33140	−0.665699	+	0.746221i	$0.731866\pi$
$294$	0	0
$295$	72450.1	0.0484712
$296$	0	0
$297$	5281.63	0.00347438
$298$	0	0
$299$	−402403.	−0.260305
$300$	0	0
$301$	3.04423e6	1.93670
$302$	0	0
$303$	−1.67497e6	−1.04809
$304$	0	0
$305$	83258.0	0.0512479
$306$	0	0
$307$	390236.	0.236310	0.118155	−	0.992995i	$-0.462302\pi$
$307$	390236.	0.236310	0.118155	+	0.992995i	$0.462302\pi$
$308$	0	0
$309$	1.39281e6	0.829842
$310$	0	0
$311$	−2.15294e6	−1.26221	−0.631104	−	0.775698i	$-0.717398\pi$
$311$	−2.15294e6	−1.26221	−0.631104	+	0.775698i	$0.717398\pi$
$312$	0	0
$313$	−909243.	−0.524589	−0.262294	−	0.964988i	$-0.584479\pi$
$313$	−909243.	−0.524589	−0.262294	+	0.964988i	$0.584479\pi$
$314$	0	0
$315$	−135946.	−0.0771953
$316$	0	0
$317$	1.77917e6	0.994421	0.497210	−	0.867630i	$-0.334358\pi$
$317$	1.77917e6	0.994421	0.497210	+	0.867630i	$0.334358\pi$
$318$	0	0
$319$	45239.3	0.0248908
$320$	0	0
$321$	946400.	0.512640
$322$	0	0
$323$	219699.	0.117171
$324$	0	0
$325$	2.30969e6	1.21295
$326$	0	0
$327$	315644.	0.163241
$328$	0	0
$329$	2.59324e6	1.32085
$330$	0	0
$331$	−1.67019e6	−0.837906	−0.418953	−	0.908008i	$-0.637603\pi$
$331$	−1.67019e6	−0.837906	−0.418953	+	0.908008i	$0.637603\pi$
$332$	0	0
$333$	−1.27627e6	−0.630712
$334$	0	0
$335$	213987.	0.104178
$336$	0	0
$337$	3.48561e6	1.67188	0.835938	−	0.548824i	$-0.184924\pi$
$337$	3.48561e6	1.67188	0.835938	+	0.548824i	$0.184924\pi$
$338$	0	0
$339$	764360.	0.361243
$340$	0	0
$341$	−47241.2	−0.0220006
$342$	0	0
$343$	329734.	0.151331
$344$	0	0
$345$	−44834.6	−0.0202799
$346$	0	0
$347$	512426.	0.228459	0.114229	−	0.993454i	$-0.463560\pi$
$347$	512426.	0.228459	0.114229	+	0.993454i	$0.463560\pi$
$348$	0	0
$349$	922966.	0.405623	0.202811	−	0.979218i	$-0.434992\pi$
$349$	922966.	0.405623	0.202811	+	0.979218i	$0.434992\pi$
$350$	0	0
$351$	554540.	0.240251
$352$	0	0
$353$	2.62524e6	1.12133	0.560663	−	0.828044i	$-0.310546\pi$
$353$	2.62524e6	1.12133	0.560663	+	0.828044i	$0.310546\pi$
$354$	0	0
$355$	118970.	0.0501034
$356$	0	0
$357$	−2.69910e6	−1.12085
$358$	0	0
$359$	4.64239e6	1.90110	0.950550	−	0.310570i	$-0.100520\pi$
$359$	4.64239e6	1.90110	0.950550	+	0.310570i	$0.100520\pi$
$360$	0	0
$361$	−2.45905e6	−0.993116
$362$	0	0
$363$	1.44899e6	0.577162
$364$	0	0
$365$	−654922.	−0.257310
$366$	0	0
$367$	4.83117e6	1.87235	0.936176	−	0.351531i	$-0.114339\pi$
$367$	4.83117e6	1.87235	0.936176	+	0.351531i	$0.114339\pi$
$368$	0	0
$369$	532334.	0.203525
$370$	0	0
$371$	517384.	0.195154
$372$	0	0
$373$	2.25580e6	0.839514	0.419757	−	0.907637i	$-0.362115\pi$
$373$	2.25580e6	0.839514	0.419757	+	0.907637i	$0.362115\pi$
$374$	0	0
$375$	522194.	0.191758
$376$	0	0
$377$	4.74986e6	1.72118
$378$	0	0
$379$	1.00342e6	0.358826	0.179413	−	0.983774i	$-0.442580\pi$
$379$	1.00342e6	0.358826	0.179413	+	0.983774i	$0.442580\pi$
$380$	0	0
$381$	−124340.	−0.0438832
$382$	0	0
$383$	−3.88205e6	−1.35227	−0.676135	−	0.736777i	$-0.736347\pi$
$383$	−3.88205e6	−1.35227	−0.676135	+	0.736777i	$0.736347\pi$
$384$	0	0
$385$	12159.7	0.00418091
$386$	0	0
$387$	−1.38355e6	−0.469589
$388$	0	0
$389$	−596166.	−0.199753	−0.0998765	−	0.995000i	$-0.531845\pi$
$389$	−596166.	−0.199753	−0.0998765	+	0.995000i	$0.531845\pi$
$390$	0	0
$391$	−890153.	−0.294458
$392$	0	0
$393$	−1.07944e6	−0.352546
$394$	0	0
$395$	−597068.	−0.192545
$396$	0	0
$397$	1.95305e6	0.621924	0.310962	−	0.950422i	$-0.399349\pi$
$397$	1.95305e6	0.621924	0.310962	+	0.950422i	$0.399349\pi$
$398$	0	0
$399$	−209425.	−0.0658561
$400$	0	0
$401$	3.85354e6	1.19674	0.598368	−	0.801221i	$-0.295816\pi$
$401$	3.85354e6	1.19674	0.598368	+	0.801221i	$0.295816\pi$
$402$	0	0
$403$	−4.96004e6	−1.52133
$404$	0	0
$405$	61785.4	0.0187175
$406$	0	0
$407$	114156.	0.0341595
$408$	0	0
$409$	−976672.	−0.288696	−0.144348	−	0.989527i	$-0.546108\pi$
$409$	−976672.	−0.288696	−0.144348	+	0.989527i	$0.546108\pi$
$410$	0	0
$411$	−1.36925e6	−0.399833
$412$	0	0
$413$	−1.37117e6	−0.395563
$414$	0	0
$415$	−79813.8	−0.0227488
$416$	0	0
$417$	77352.3	0.0217838
$418$	0	0
$419$	−5.47986e6	−1.52488	−0.762438	−	0.647061i	$-0.775998\pi$
$419$	−5.47986e6	−1.52488	−0.762438	+	0.647061i	$0.775998\pi$
$420$	0	0
$421$	−2.78411e6	−0.765562	−0.382781	−	0.923839i	$-0.625034\pi$
$421$	−2.78411e6	−0.765562	−0.382781	+	0.923839i	$0.625034\pi$
$422$	0	0
$423$	−1.17858e6	−0.320265
$424$	0	0
$425$	5.10924e6	1.37209
$426$	0	0
$427$	−1.57571e6	−0.418223
$428$	0	0
$429$	−49600.8	−0.0130120
$430$	0	0
$431$	−1.53410e6	−0.397797	−0.198899	−	0.980020i	$-0.563736\pi$
$431$	−1.53410e6	−0.397797	−0.198899	+	0.980020i	$0.563736\pi$
$432$	0	0
$433$	−3.49120e6	−0.894859	−0.447430	−	0.894319i	$-0.647660\pi$
$433$	−3.49120e6	−0.894859	−0.447430	+	0.894319i	$0.647660\pi$
$434$	0	0
$435$	529216.	0.134094
$436$	0	0
$437$	−69067.6	−0.0173010
$438$	0	0
$439$	−6.05397e6	−1.49927	−0.749634	−	0.661853i	$-0.769770\pi$
$439$	−6.05397e6	−1.49927	−0.749634	+	0.661853i	$0.769770\pi$
$440$	0	0
$441$	1.21151e6	0.296640
$442$	0	0
$443$	−2.60534e6	−0.630746	−0.315373	−	0.948968i	$-0.602130\pi$
$443$	−2.60534e6	−0.630746	−0.315373	+	0.948968i	$0.602130\pi$
$444$	0	0
$445$	−213695.	−0.0511557
$446$	0	0
$447$	−1.17778e6	−0.278801
$448$	0	0
$449$	−4.84447e6	−1.13405	−0.567023	−	0.823702i	$-0.691905\pi$
$449$	−4.84447e6	−1.13405	−0.567023	+	0.823702i	$0.691905\pi$
$450$	0	0
$451$	−47614.5	−0.0110230
$452$	0	0
$453$	−3.91178e6	−0.895631
$454$	0	0
$455$	1.27670e6	0.289107
$456$	0	0
$457$	2.56933e6	0.575478	0.287739	−	0.957709i	$-0.407096\pi$
$457$	2.56933e6	0.575478	0.287739	+	0.957709i	$0.407096\pi$
$458$	0	0
$459$	1.22669e6	0.271772
$460$	0	0
$461$	6.99997e6	1.53406	0.767032	−	0.641608i	$-0.221733\pi$
$461$	6.99997e6	1.53406	0.767032	+	0.641608i	$0.221733\pi$
$462$	0	0
$463$	3.62725e6	0.786366	0.393183	−	0.919460i	$-0.371374\pi$
$463$	3.62725e6	0.786366	0.393183	+	0.919460i	$0.371374\pi$
$464$	0	0
$465$	−552635.	−0.118524
$466$	0	0
$467$	−6.63513e6	−1.40785	−0.703926	−	0.710273i	$-0.748571\pi$
$467$	−6.63513e6	−1.40785	−0.703926	+	0.710273i	$0.748571\pi$
$468$	0	0
$469$	−4.04984e6	−0.850170
$470$	0	0
$471$	−3.32915e6	−0.691482
$472$	0	0
$473$	123752.	0.0254330
$474$	0	0
$475$	396430.	0.0806180
$476$	0	0
$477$	−235143.	−0.0473190
$478$	0	0
$479$	−2.78829e6	−0.555263	−0.277632	−	0.960688i	$-0.589549\pi$
$479$	−2.78829e6	−0.555263	−0.277632	+	0.960688i	$0.589549\pi$
$480$	0	0
$481$	1.19857e7	2.36211
$482$	0	0
$483$	848526.	0.165500
$484$	0	0
$485$	−1.47817e6	−0.285344
$486$	0	0
$487$	1.03474e7	1.97702	0.988508	−	0.151168i	$-0.0483034\pi$
$487$	1.03474e7	1.97702	0.988508	+	0.151168i	$0.0483034\pi$
$488$	0	0
$489$	3.53177e6	0.667914
$490$	0	0
$491$	18059.3	0.00338062	0.00169031	−	0.999999i	$-0.499462\pi$
$491$	18059.3	0.00338062	0.00169031	+	0.999999i	$0.499462\pi$
$492$	0	0
$493$	1.05071e7	1.94700
$494$	0	0
$495$	−5526.38	−0.00101374
$496$	0	0
$497$	−2.25159e6	−0.408882
$498$	0	0
$499$	−29657.5	−0.00533192	−0.00266596	−	0.999996i	$-0.500849\pi$
$499$	−29657.5	−0.00533192	−0.00266596	+	0.999996i	$0.500849\pi$
$500$	0	0
$501$	−5.89035e6	−1.04845
$502$	0	0
$503$	−1.77599e6	−0.312983	−0.156491	−	0.987679i	$-0.550018\pi$
$503$	−1.77599e6	−0.312983	−0.156491	+	0.987679i	$0.550018\pi$
$504$	0	0
$505$	1.75259e6	0.305810
$506$	0	0
$507$	−1.86615e6	−0.322424
$508$	0	0
$509$	−9.22074e6	−1.57751	−0.788754	−	0.614709i	$-0.789273\pi$
$509$	−9.22074e6	−1.57751	−0.788754	+	0.614709i	$0.789273\pi$
$510$	0	0
$511$	1.23948e7	2.09985
$512$	0	0
$513$	95180.1	0.0159681
$514$	0	0
$515$	−1.45735e6	−0.242129
$516$	0	0
$517$	105418.	0.0173456
$518$	0	0
$519$	3.91233e6	0.637554
$520$	0	0
$521$	1.29690e6	0.209321	0.104660	−	0.994508i	$-0.466624\pi$
$521$	1.29690e6	0.209321	0.104660	+	0.994508i	$0.466624\pi$
$522$	0	0
$523$	6.60238e6	1.05547	0.527736	−	0.849409i	$-0.323041\pi$
$523$	6.60238e6	1.05547	0.527736	+	0.849409i	$0.323041\pi$
$524$	0	0
$525$	−4.87031e6	−0.771185
$526$	0	0
$527$	−1.09721e7	−1.72093
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	623173.	0.0959118
$532$	0	0
$533$	−4.99924e6	−0.762230
$534$	0	0
$535$	−990257.	−0.149576
$536$	0	0
$537$	953908.	0.142748
$538$	0	0
$539$	−108363.	−0.0160661
$540$	0	0
$541$	−1.11340e7	−1.63553	−0.817764	−	0.575554i	$-0.804787\pi$
$541$	−1.11340e7	−1.63553	−0.817764	+	0.575554i	$0.804787\pi$
$542$	0	0
$543$	4.19863e6	0.611094
$544$	0	0
$545$	−330271.	−0.0476299
$546$	0	0
$547$	−2.63007e6	−0.375837	−0.187919	−	0.982185i	$-0.560174\pi$
$547$	−2.63007e6	−0.375837	−0.187919	+	0.982185i	$0.560174\pi$
$548$	0	0
$549$	716136.	0.101406
$550$	0	0
$551$	815256.	0.114397
$552$	0	0
$553$	1.12999e7	1.57131
$554$	0	0
$555$	1.33541e6	0.184027
$556$	0	0
$557$	1.09719e7	1.49845	0.749227	−	0.662314i	$-0.230425\pi$
$557$	1.09719e7	1.49845	0.749227	+	0.662314i	$0.230425\pi$
$558$	0	0
$559$	1.29932e7	1.75868
$560$	0	0
$561$	−109722.	−0.0147192
$562$	0	0
$563$	1.98368e6	0.263754	0.131877	−	0.991266i	$-0.457900\pi$
$563$	1.98368e6	0.263754	0.131877	+	0.991266i	$0.457900\pi$
$564$	0	0
$565$	−799781.	−0.105402
$566$	0	0
$567$	−1.16933e6	−0.152749
$568$	0	0
$569$	−1.72725e6	−0.223653	−0.111826	−	0.993728i	$-0.535670\pi$
$569$	−1.72725e6	−0.223653	−0.111826	+	0.993728i	$0.535670\pi$
$570$	0	0
$571$	−1.10546e7	−1.41890	−0.709451	−	0.704755i	$-0.751057\pi$
$571$	−1.10546e7	−1.41890	−0.709451	+	0.704755i	$0.751057\pi$
$572$	0	0
$573$	2.87446e6	0.365738
$574$	0	0
$575$	−1.60621e6	−0.202597
$576$	0	0
$577$	−5.31223e6	−0.664258	−0.332129	−	0.943234i	$-0.607767\pi$
$577$	−5.31223e6	−0.664258	−0.332129	+	0.943234i	$0.607767\pi$
$578$	0	0
$579$	1.01701e6	0.126075
$580$	0	0
$581$	1.51053e6	0.185648
$582$	0	0
$583$	21032.3	0.00256280
$584$	0	0
$585$	−580238.	−0.0700997
$586$	0	0
$587$	−953392.	−0.114203	−0.0571013	−	0.998368i	$-0.518186\pi$
$587$	−953392.	−0.114203	−0.0571013	+	0.998368i	$0.518186\pi$
$588$	0	0
$589$	−851332.	−0.101114
$590$	0	0
$591$	6.86439e6	0.808413
$592$	0	0
$593$	7.46913e6	0.872234	0.436117	−	0.899890i	$-0.356353\pi$
$593$	7.46913e6	0.872234	0.436117	+	0.899890i	$0.356353\pi$
$594$	0	0
$595$	2.82417e6	0.327038
$596$	0	0
$597$	6.12058e6	0.702841
$598$	0	0
$599$	8.60595e6	0.980013	0.490006	−	0.871719i	$-0.336994\pi$
$599$	8.60595e6	0.980013	0.490006	+	0.871719i	$0.336994\pi$
$600$	0	0
$601$	−8.59761e6	−0.970939	−0.485469	−	0.874254i	$-0.661351\pi$
$601$	−8.59761e6	−0.970939	−0.485469	+	0.874254i	$0.661351\pi$
$602$	0	0
$603$	1.84059e6	0.206140
$604$	0	0
$605$	−1.51613e6	−0.168403
$606$	0	0
$607$	−5.78141e6	−0.636886	−0.318443	−	0.947942i	$-0.603160\pi$
$607$	−5.78141e6	−0.636886	−0.318443	+	0.947942i	$0.603160\pi$
$608$	0	0
$609$	−1.00158e7	−1.09431
$610$	0	0
$611$	1.10683e7	1.19944
$612$	0	0
$613$	−1.09608e7	−1.17813	−0.589063	−	0.808087i	$-0.700503\pi$
$613$	−1.09608e7	−1.17813	−0.589063	+	0.808087i	$0.700503\pi$
$614$	0	0
$615$	−557002.	−0.0593839
$616$	0	0
$617$	4.00285e6	0.423308	0.211654	−	0.977345i	$-0.432115\pi$
$617$	4.00285e6	0.423308	0.211654	+	0.977345i	$0.432115\pi$
$618$	0	0
$619$	2.52071e6	0.264421	0.132211	−	0.991222i	$-0.457792\pi$
$619$	2.52071e6	0.264421	0.132211	+	0.991222i	$0.457792\pi$
$620$	0	0
$621$	−385641.	−0.0401286
$622$	0	0
$623$	4.04432e6	0.417470
$624$	0	0
$625$	8.94210e6	0.915671
$626$	0	0
$627$	−8513.38	−0.000864834 0
$628$	0	0
$629$	2.65134e7	2.67202
$630$	0	0
$631$	−8.18947e6	−0.818808	−0.409404	−	0.912353i	$-0.634263\pi$
$631$	−8.18947e6	−0.818808	−0.409404	+	0.912353i	$0.634263\pi$
$632$	0	0
$633$	3.12899e6	0.310381
$634$	0	0
$635$	130102.	0.0128041
$636$	0	0
$637$	−1.13775e7	−1.11096
$638$	0	0
$639$	1.02331e6	0.0991414
$640$	0	0
$641$	−1.48981e7	−1.43214	−0.716070	−	0.698028i	$-0.754061\pi$
$641$	−1.48981e7	−1.43214	−0.716070	+	0.698028i	$0.754061\pi$
$642$	0	0
$643$	1.85830e6	0.177251	0.0886255	−	0.996065i	$-0.471753\pi$
$643$	1.85830e6	0.177251	0.0886255	+	0.996065i	$0.471753\pi$
$644$	0	0
$645$	1.44767e6	0.137015
$646$	0	0
$647$	−1.71915e7	−1.61456	−0.807278	−	0.590171i	$-0.799060\pi$
$647$	−1.71915e7	−1.61456	−0.807278	+	0.590171i	$0.799060\pi$
$648$	0	0
$649$	−55739.6	−0.00519460
$650$	0	0
$651$	1.04590e7	0.967246
$652$	0	0
$653$	−1.94603e7	−1.78594	−0.892969	−	0.450118i	$-0.851382\pi$
$653$	−1.94603e7	−1.78594	−0.892969	+	0.450118i	$0.851382\pi$
$654$	0	0
$655$	1.12946e6	0.102865
$656$	0	0
$657$	−5.63325e6	−0.509150
$658$	0	0
$659$	−6.19056e6	−0.555286	−0.277643	−	0.960684i	$-0.589553\pi$
$659$	−6.19056e6	−0.555286	−0.277643	+	0.960684i	$0.589553\pi$
$660$	0	0
$661$	−9.08953e6	−0.809167	−0.404583	−	0.914501i	$-0.632583\pi$
$661$	−9.08953e6	−0.809167	−0.404583	+	0.914501i	$0.632583\pi$
$662$	0	0
$663$	−1.15201e7	−1.01782
$664$	0	0
$665$	219130.	0.0192153
$666$	0	0
$667$	−3.30317e6	−0.287486
$668$	0	0
$669$	−9.81228e6	−0.847627
$670$	0	0
$671$	−64054.7	−0.00549218
$672$	0	0
$673$	−8.23692e6	−0.701015	−0.350508	−	0.936560i	$-0.613991\pi$
$673$	−8.23692e6	−0.701015	−0.350508	+	0.936560i	$0.613991\pi$
$674$	0	0
$675$	2.21348e6	0.186989
$676$	0	0
$677$	6.57462e6	0.551314	0.275657	−	0.961256i	$-0.411105\pi$
$677$	6.57462e6	0.551314	0.275657	+	0.961256i	$0.411105\pi$
$678$	0	0
$679$	2.79753e7	2.32863
$680$	0	0
$681$	−4.33372e6	−0.358091
$682$	0	0
$683$	−1.28381e7	−1.05305	−0.526524	−	0.850160i	$-0.676505\pi$
$683$	−1.28381e7	−1.05305	−0.526524	+	0.850160i	$0.676505\pi$
$684$	0	0
$685$	1.43270e6	0.116662
$686$	0	0
$687$	−1.70500e6	−0.137826
$688$	0	0
$689$	2.20827e6	0.177216
$690$	0	0
$691$	−2.29348e7	−1.82726	−0.913629	−	0.406549i	$-0.866732\pi$
$691$	−2.29348e7	−1.82726	−0.913629	+	0.406549i	$0.866732\pi$
$692$	0	0
$693$	104591.	0.00827293
$694$	0	0
$695$	−80936.8	−0.00635600
$696$	0	0
$697$	−1.10588e7	−0.862235
$698$	0	0
$699$	7.29516e6	0.564731
$700$	0	0
$701$	8.19891e6	0.630175	0.315087	−	0.949063i	$-0.397966\pi$
$701$	8.19891e6	0.630175	0.315087	+	0.949063i	$0.397966\pi$
$702$	0	0
$703$	2.05719e6	0.156995
$704$	0	0
$705$	1.23320e6	0.0934461
$706$	0	0
$707$	−3.31689e7	−2.49565
$708$	0	0
$709$	−1.98568e7	−1.48352	−0.741759	−	0.670666i	$-0.766008\pi$
$709$	−1.98568e7	−1.48352	−0.741759	+	0.670666i	$0.766008\pi$
$710$	0	0
$711$	−5.13563e6	−0.380995
$712$	0	0
$713$	3.44934e6	0.254104
$714$	0	0
$715$	51899.3	0.00379661
$716$	0	0
$717$	−4.95952e6	−0.360281
$718$	0	0
$719$	9.16780e6	0.661368	0.330684	−	0.943742i	$-0.392721\pi$
$719$	9.16780e6	0.661368	0.330684	+	0.943742i	$0.392721\pi$
$720$	0	0
$721$	2.75814e7	1.97596
$722$	0	0
$723$	−1.51085e6	−0.107492
$724$	0	0
$725$	1.89593e7	1.33961
$726$	0	0
$727$	−3.67841e6	−0.258121	−0.129061	−	0.991637i	$-0.541196\pi$
$727$	−3.67841e6	−0.258121	−0.129061	+	0.991637i	$0.541196\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	2.87422e7	1.98942
$732$	0	0
$733$	6.10780e6	0.419880	0.209940	−	0.977714i	$-0.432673\pi$
$733$	6.10780e6	0.419880	0.209940	+	0.977714i	$0.432673\pi$
$734$	0	0
$735$	−1.26765e6	−0.0865528
$736$	0	0
$737$	−164631.	−0.0111646
$738$	0	0
$739$	1.88178e7	1.26753	0.633763	−	0.773527i	$-0.281509\pi$
$739$	1.88178e7	1.26753	0.633763	+	0.773527i	$0.281509\pi$
$740$	0	0
$741$	−893854.	−0.0598028
$742$	0	0
$743$	−1.34404e7	−0.893180	−0.446590	−	0.894739i	$-0.647362\pi$
$743$	−1.34404e7	−0.893180	−0.446590	+	0.894739i	$0.647362\pi$
$744$	0	0
$745$	1.23236e6	0.0813477
$746$	0	0
$747$	−686511.	−0.0450138
$748$	0	0
$749$	1.87413e7	1.22066
$750$	0	0
$751$	2.19820e7	1.42222	0.711110	−	0.703081i	$-0.248193\pi$
$751$	2.19820e7	1.42222	0.711110	+	0.703081i	$0.248193\pi$
$752$	0	0
$753$	1.56941e6	0.100867
$754$	0	0
$755$	4.09306e6	0.261325
$756$	0	0
$757$	−405077.	−0.0256920	−0.0128460	−	0.999917i	$-0.504089\pi$
$757$	−405077.	−0.0256920	−0.0128460	+	0.999917i	$0.504089\pi$
$758$	0	0
$759$	34493.6	0.00217337
$760$	0	0
$761$	−2.21836e7	−1.38858	−0.694290	−	0.719696i	$-0.744281\pi$
$761$	−2.21836e7	−1.38858	−0.694290	+	0.719696i	$0.744281\pi$
$762$	0	0
$763$	6.25061e6	0.388697
$764$	0	0
$765$	−1.28354e6	−0.0792968
$766$	0	0
$767$	−5.85233e6	−0.359203
$768$	0	0
$769$	−2.92667e7	−1.78467	−0.892334	−	0.451376i	$-0.850933\pi$
$769$	−2.92667e7	−1.78467	−0.892334	+	0.451376i	$0.850933\pi$
$770$	0	0
$771$	−1.53865e7	−0.932192
$772$	0	0
$773$	−1.76039e7	−1.05964	−0.529822	−	0.848109i	$-0.677741\pi$
$773$	−1.76039e7	−1.05964	−0.529822	+	0.848109i	$0.677741\pi$
$774$	0	0
$775$	−1.97983e7	−1.18406
$776$	0	0
$777$	−2.52735e7	−1.50180
$778$	0	0
$779$	−858060.	−0.0506610
$780$	0	0
$781$	−91529.8	−0.00536951
$782$	0	0
$783$	4.55200e6	0.265337
$784$	0	0
$785$	3.48342e6	0.201758
$786$	0	0
$787$	−1.11095e7	−0.639376	−0.319688	−	0.947523i	$-0.603578\pi$
$787$	−1.11095e7	−0.639376	−0.319688	+	0.947523i	$0.603578\pi$
$788$	0	0
$789$	1.32432e7	0.757358
$790$	0	0
$791$	1.51364e7	0.860164
$792$	0	0
$793$	−6.72536e6	−0.379781
$794$	0	0
$795$	246039.	0.0138066
$796$	0	0
$797$	−2.96296e7	−1.65227	−0.826134	−	0.563473i	$-0.809465\pi$
$797$	−2.96296e7	−1.65227	−0.826134	+	0.563473i	$0.809465\pi$
$798$	0	0
$799$	2.44841e7	1.35681
$800$	0	0
$801$	−1.83808e6	−0.101224
$802$	0	0
$803$	503865.	0.0275756
$804$	0	0
$805$	−887847.	−0.0482890
$806$	0	0
$807$	−8.91481e6	−0.481868
$808$	0	0
$809$	−1.46529e7	−0.787143	−0.393571	−	0.919294i	$-0.628761\pi$
$809$	−1.46529e7	−0.787143	−0.393571	+	0.919294i	$0.628761\pi$
$810$	0	0
$811$	1.11235e7	0.593866	0.296933	−	0.954898i	$-0.404036\pi$
$811$	1.11235e7	0.593866	0.296933	+	0.954898i	$0.404036\pi$
$812$	0	0
$813$	−1.86420e7	−0.989157
$814$	0	0
$815$	−3.69544e6	−0.194882
$816$	0	0
$817$	2.23013e6	0.116889
$818$	0	0
$819$	1.09814e7	0.572068
$820$	0	0
$821$	−3.01057e7	−1.55880	−0.779401	−	0.626526i	$-0.784476\pi$
$821$	−3.01057e7	−1.55880	−0.779401	+	0.626526i	$0.784476\pi$
$822$	0	0
$823$	8.75398e6	0.450512	0.225256	−	0.974300i	$-0.427678\pi$
$823$	8.75398e6	0.450512	0.225256	+	0.974300i	$0.427678\pi$
$824$	0	0
$825$	−197984.	−0.0101273
$826$	0	0
$827$	−1.27243e7	−0.646952	−0.323476	−	0.946236i	$-0.604851\pi$
$827$	−1.27243e7	−0.646952	−0.323476	+	0.946236i	$0.604851\pi$
$828$	0	0
$829$	−6.67806e6	−0.337492	−0.168746	−	0.985660i	$-0.553972\pi$
$829$	−6.67806e6	−0.337492	−0.168746	+	0.985660i	$0.553972\pi$
$830$	0	0
$831$	6.07871e6	0.305358
$832$	0	0
$833$	−2.51681e7	−1.25672
$834$	0	0
$835$	6.16332e6	0.305913
$836$	0	0
$837$	−4.75344e6	−0.234528
$838$	0	0
$839$	−2.87088e7	−1.40802	−0.704012	−	0.710188i	$-0.748610\pi$
$839$	−2.87088e7	−1.40802	−0.704012	+	0.710188i	$0.748610\pi$
$840$	0	0
$841$	1.84786e7	0.900904
$842$	0	0
$843$	−2.16144e7	−1.04755
$844$	0	0
$845$	1.95263e6	0.0940758
$846$	0	0
$847$	2.86938e7	1.37430
$848$	0	0
$849$	−1.69735e7	−0.808167
$850$	0	0
$851$	−8.33513e6	−0.394538
$852$	0	0
$853$	1.33163e7	0.626630	0.313315	−	0.949649i	$-0.398560\pi$
$853$	1.33163e7	0.626630	0.313315	+	0.949649i	$0.398560\pi$
$854$	0	0
$855$	−99590.8	−0.00465912
$856$	0	0
$857$	−3.30521e7	−1.53726	−0.768630	−	0.639693i	$-0.779061\pi$
$857$	−3.30521e7	−1.53726	−0.768630	+	0.639693i	$0.779061\pi$
$858$	0	0
$859$	−5.03878e6	−0.232993	−0.116496	−	0.993191i	$-0.537166\pi$
$859$	−5.03878e6	−0.232993	−0.116496	+	0.993191i	$0.537166\pi$
$860$	0	0
$861$	1.05416e7	0.484619
$862$	0	0
$863$	1.93073e7	0.882458	0.441229	−	0.897395i	$-0.354543\pi$
$863$	1.93073e7	0.882458	0.441229	+	0.897395i	$0.354543\pi$
$864$	0	0
$865$	−4.09363e6	−0.186024
$866$	0	0
$867$	−1.27049e7	−0.574014
$868$	0	0
$869$	459356.	0.0206348
$870$	0	0
$871$	−1.72853e7	−0.772025
$872$	0	0
$873$	−1.27143e7	−0.564621
$874$	0	0
$875$	1.03408e7	0.456600
$876$	0	0
$877$	3.18382e7	1.39782	0.698908	−	0.715211i	$-0.253670\pi$
$877$	3.18382e7	1.39782	0.698908	+	0.715211i	$0.253670\pi$
$878$	0	0
$879$	1.76084e7	0.768683
$880$	0	0
$881$	−1.11266e7	−0.482973	−0.241487	−	0.970404i	$-0.577635\pi$
$881$	−1.11266e7	−0.482973	−0.241487	+	0.970404i	$0.577635\pi$
$882$	0	0
$883$	−2.72511e7	−1.17620	−0.588101	−	0.808787i	$-0.700124\pi$
$883$	−2.72511e7	−1.17620	−0.588101	+	0.808787i	$0.700124\pi$
$884$	0	0
$885$	−652051.	−0.0279849
$886$	0	0
$887$	1.82135e7	0.777291	0.388646	−	0.921387i	$-0.372943\pi$
$887$	1.82135e7	0.777291	0.388646	+	0.921387i	$0.372943\pi$
$888$	0	0
$889$	−2.46227e6	−0.104491
$890$	0	0
$891$	−47534.7	−0.00200593
$892$	0	0
$893$	1.89974e6	0.0797198
$894$	0	0
$895$	−998112.	−0.0416507
$896$	0	0
$897$	3.62163e6	0.150287
$898$	0	0
$899$	−4.07151e7	−1.68018
$900$	0	0
$901$	4.88490e6	0.200467
$902$	0	0
$903$	−2.73981e7	−1.11815
$904$	0	0
$905$	−4.39319e6	−0.178303
$906$	0	0
$907$	−2.39287e7	−0.965829	−0.482914	−	0.875668i	$-0.660422\pi$
$907$	−2.39287e7	−0.965829	−0.482914	+	0.875668i	$0.660422\pi$
$908$	0	0
$909$	1.50747e7	0.605118
$910$	0	0
$911$	4.11928e7	1.64447	0.822234	−	0.569150i	$-0.192728\pi$
$911$	4.11928e7	1.64447	0.822234	+	0.569150i	$0.192728\pi$
$912$	0	0
$913$	61404.9	0.00243796
$914$	0	0
$915$	−749322.	−0.0295880
$916$	0	0
$917$	−2.13757e7	−0.839456
$918$	0	0
$919$	9.40442e6	0.367319	0.183659	−	0.982990i	$-0.441206\pi$
$919$	9.40442e6	0.367319	0.183659	+	0.982990i	$0.441206\pi$
$920$	0	0
$921$	−3.51213e6	−0.136433
$922$	0	0
$923$	−9.61009e6	−0.371299
$924$	0	0
$925$	4.78414e7	1.83844
$926$	0	0
$927$	−1.25353e7	−0.479109
$928$	0	0
$929$	4.29668e7	1.63341	0.816703	−	0.577058i	$-0.195799\pi$
$929$	4.29668e7	1.63341	0.816703	+	0.577058i	$0.195799\pi$
$930$	0	0
$931$	−1.95281e6	−0.0738390
$932$	0	0
$933$	1.93765e7	0.728736
$934$	0	0
$935$	114806.	0.00429473
$936$	0	0
$937$	173086.	0.00644041	0.00322020	−	0.999995i	$-0.498975\pi$
$937$	173086.	0.00644041	0.00322020	+	0.999995i	$0.498975\pi$
$938$	0	0
$939$	8.18319e6	0.302871
$940$	0	0
$941$	−3.50414e7	−1.29005	−0.645027	−	0.764160i	$-0.723154\pi$
$941$	−3.50414e7	−1.29005	−0.645027	+	0.764160i	$0.723154\pi$
$942$	0	0
$943$	3.47660e6	0.127314
$944$	0	0
$945$	1.22352e6	0.0445687
$946$	0	0
$947$	−1.16529e6	−0.0422241	−0.0211121	−	0.999777i	$-0.506721\pi$
$947$	−1.16529e6	−0.0422241	−0.0211121	+	0.999777i	$0.506721\pi$
$948$	0	0
$949$	5.29029e7	1.90684
$950$	0	0
$951$	−1.60126e7	−0.574129
$952$	0	0
$953$	−3.68644e7	−1.31485	−0.657424	−	0.753521i	$-0.728354\pi$
$953$	−3.68644e7	−1.31485	−0.657424	+	0.753521i	$0.728354\pi$
$954$	0	0
$955$	−3.00767e6	−0.106714
$956$	0	0
$957$	−407153.	−0.0143707
$958$	0	0
$959$	−2.71149e7	−0.952053
$960$	0	0
$961$	1.38876e7	0.485086
$962$	0	0
$963$	−8.51760e6	−0.295973
$964$	0	0
$965$	−1.06414e6	−0.0367859
$966$	0	0
$967$	5.38683e7	1.85254	0.926269	−	0.376864i	$-0.122997\pi$
$967$	5.38683e7	1.85254	0.926269	+	0.376864i	$0.122997\pi$
$968$	0	0
$969$	−1.97729e6	−0.0676489
$970$	0	0
$971$	5.31262e6	0.180826	0.0904129	−	0.995904i	$-0.471181\pi$
$971$	5.31262e6	0.180826	0.0904129	+	0.995904i	$0.471181\pi$
$972$	0	0
$973$	1.53178e6	0.0518699
$974$	0	0
$975$	−2.07872e7	−0.700299
$976$	0	0
$977$	−1.69318e6	−0.0567500	−0.0283750	−	0.999597i	$-0.509033\pi$
$977$	−1.69318e6	−0.0567500	−0.0283750	+	0.999597i	$0.509033\pi$
$978$	0	0
$979$	164406.	0.00548229
$980$	0	0
$981$	−2.84080e6	−0.0942471
$982$	0	0
$983$	−2.47933e7	−0.818371	−0.409185	−	0.912451i	$-0.634187\pi$
$983$	−2.47933e7	−0.818371	−0.409185	+	0.912451i	$0.634187\pi$
$984$	0	0
$985$	−7.18249e6	−0.235876
$986$	0	0
$987$	−2.33392e7	−0.762592
$988$	0	0
$989$	−9.03579e6	−0.293748
$990$	0	0
$991$	3.37438e6	0.109147	0.0545733	−	0.998510i	$-0.482620\pi$
$991$	3.37438e6	0.109147	0.0545733	+	0.998510i	$0.482620\pi$
$992$	0	0
$993$	1.50317e7	0.483765
$994$	0	0
$995$	−6.40421e6	−0.205073
$996$	0	0
$997$	3.96279e7	1.26259	0.631296	−	0.775542i	$-0.282523\pi$
$997$	3.96279e7	1.26259	0.631296	+	0.775542i	$0.282523\pi$
$998$	0	0
$999$	1.14864e7	0.364142

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.x.1.5		7
4.3	odd	2		552.6.a.g.1.5	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.6.a.g.1.5	✓	7	4.3	odd	2
1104.6.a.x.1.5		7	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)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +9.41706 q^{5} -178.224 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} +9.41706 q^{5} -178.224 q^{7} +81.0000 q^{9} -7.24503 q^{11} -760.686 q^{13} -84.7536 q^{15} -1682.71 q^{17} -130.563 q^{19} +1604.02 q^{21} +529.000 q^{23} -3036.32 q^{25} -729.000 q^{27} -6244.17 q^{29} +6520.49 q^{31} +65.2053 q^{33} -1678.35 q^{35} -15756.4 q^{37} +6846.17 q^{39} +6572.02 q^{41} -17080.9 q^{43} +762.782 q^{45} -14550.4 q^{47} +14956.9 q^{49} +15144.4 q^{51} -2903.00 q^{53} -68.2269 q^{55} +1175.06 q^{57} +7693.49 q^{59} +8841.18 q^{61} -14436.2 q^{63} -7163.43 q^{65} +22723.3 q^{67} -4761.00 q^{69} +12633.5 q^{71} -69546.3 q^{73} +27326.9 q^{75} +1291.24 q^{77} -63402.8 q^{79} +6561.00 q^{81} -8475.45 q^{83} -15846.2 q^{85} +56197.6 q^{87} -22692.3 q^{89} +135573. q^{91} -58684.4 q^{93} -1229.52 q^{95} -156967. q^{97} -586.848 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 63 q^{3} - 104 q^{5} + 182 q^{7} + 567 q^{9}+O(q^{10}) \) 7 * q - 63 * q^3 - 104 * q^5 + 182 * q^7 + 567 * q^9	\( 7 q - 63 q^{3} - 104 q^{5} + 182 q^{7} + 567 q^{9} + 124 q^{11} + 294 q^{13} + 936 q^{15} + 428 q^{17} + 1826 q^{19} - 1638 q^{21} + 3703 q^{23} + 5501 q^{25} - 5103 q^{27} - 1682 q^{29} + 14420 q^{31} - 1116 q^{33} - 3312 q^{35} - 14218 q^{37} - 2646 q^{39} + 7294 q^{41} - 3630 q^{43} - 8424 q^{45} + 19808 q^{47} - 4325 q^{49} - 3852 q^{51} - 77412 q^{53} + 61136 q^{55} - 16434 q^{57} + 51076 q^{59} - 67186 q^{61} + 14742 q^{63} - 121800 q^{65} + 70870 q^{67} - 33327 q^{69} + 45784 q^{71} - 175522 q^{73} - 49509 q^{75} - 148632 q^{77} + 54538 q^{79} + 45927 q^{81} - 27612 q^{83} - 197552 q^{85} + 15138 q^{87} - 184360 q^{89} + 217412 q^{91} - 129780 q^{93} + 82608 q^{95} - 339766 q^{97} + 10044 q^{99}+O(q^{100}) \) 7 * q - 63 * q^3 - 104 * q^5 + 182 * q^7 + 567 * q^9 + 124 * q^11 + 294 * q^13 + 936 * q^15 + 428 * q^17 + 1826 * q^19 - 1638 * q^21 + 3703 * q^23 + 5501 * q^25 - 5103 * q^27 - 1682 * q^29 + 14420 * q^31 - 1116 * q^33 - 3312 * q^35 - 14218 * q^37 - 2646 * q^39 + 7294 * q^41 - 3630 * q^43 - 8424 * q^45 + 19808 * q^47 - 4325 * q^49 - 3852 * q^51 - 77412 * q^53 + 61136 * q^55 - 16434 * q^57 + 51076 * q^59 - 67186 * q^61 + 14742 * q^63 - 121800 * q^65 + 70870 * q^67 - 33327 * q^69 + 45784 * q^71 - 175522 * q^73 - 49509 * q^75 - 148632 * q^77 + 54538 * q^79 + 45927 * q^81 - 27612 * q^83 - 197552 * q^85 + 15138 * q^87 - 184360 * q^89 + 217412 * q^91 - 129780 * q^93 + 82608 * q^95 - 339766 * q^97 + 10044 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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