LMFDB - Embedded newform 1104.6.a.s.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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 3740x^{4} - 50049x^{3} + 3200252x^{2} + 86063268x + 576646848 $ x^6 - 2x^5 - 3740x^4 - 50049x^3 + 3200252x^2 + 86063268*x + 576646848
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 276)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-36.8883$ 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)

$f(q)$	$=$	$q-9.00000 q^{3} +84.9274 q^{5} -141.499 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} +84.9274 q^{5} -141.499 q^{7} +81.0000 q^{9} -576.669 q^{11} +357.346 q^{13} -764.346 q^{15} -373.676 q^{17} +265.506 q^{19} +1273.49 q^{21} -529.000 q^{23} +4087.66 q^{25} -729.000 q^{27} -8156.72 q^{29} -607.988 q^{31} +5190.02 q^{33} -12017.1 q^{35} -1073.36 q^{37} -3216.12 q^{39} -12732.7 q^{41} +19839.1 q^{43} +6879.12 q^{45} +2202.56 q^{47} +3215.00 q^{49} +3363.08 q^{51} +11191.2 q^{53} -48975.0 q^{55} -2389.55 q^{57} -3730.24 q^{59} +41151.8 q^{61} -11461.4 q^{63} +30348.5 q^{65} +51973.5 q^{67} +4761.00 q^{69} +73477.5 q^{71} +39717.2 q^{73} -36788.9 q^{75} +81598.2 q^{77} -53492.5 q^{79} +6561.00 q^{81} -93683.1 q^{83} -31735.3 q^{85} +73410.5 q^{87} -19076.4 q^{89} -50564.2 q^{91} +5471.90 q^{93} +22548.7 q^{95} -5830.26 q^{97} -46710.2 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 54 q^{3} + 50 q^{5} + 154 q^{7} + 486 q^{9}+O(q^{10}) $ 6 * q - 54 * q^3 + 50 * q^5 + 154 * q^7 + 486 * q^9	$ 6 q - 54 q^{3} + 50 q^{5} + 154 q^{7} + 486 q^{9} - 68 q^{11} + 1076 q^{13} - 450 q^{15} + 1650 q^{17} + 226 q^{19} - 1386 q^{21} - 3174 q^{23} + 14994 q^{25} - 4374 q^{27} + 12760 q^{29} - 10980 q^{31} + 612 q^{33} - 8492 q^{35} + 18012 q^{37} - 9684 q^{39} + 15396 q^{41} + 7146 q^{43} + 4050 q^{45} - 23588 q^{47} + 56182 q^{49} - 14850 q^{51} + 9310 q^{53} - 52048 q^{55} - 2034 q^{57} - 5260 q^{59} + 16932 q^{61} + 12474 q^{63} + 30852 q^{65} - 24858 q^{67} + 28566 q^{69} - 101832 q^{71} + 33100 q^{73} - 134946 q^{75} + 182232 q^{77} - 100570 q^{79} + 39366 q^{81} - 222524 q^{83} + 23824 q^{85} - 114840 q^{87} + 28374 q^{89} - 18204 q^{91} + 98820 q^{93} - 481996 q^{95} + 47632 q^{97} - 5508 q^{99}+O(q^{100}) $ 6 * q - 54 * q^3 + 50 * q^5 + 154 * q^7 + 486 * q^9 - 68 * q^11 + 1076 * q^13 - 450 * q^15 + 1650 * q^17 + 226 * q^19 - 1386 * q^21 - 3174 * q^23 + 14994 * q^25 - 4374 * q^27 + 12760 * q^29 - 10980 * q^31 + 612 * q^33 - 8492 * q^35 + 18012 * q^37 - 9684 * q^39 + 15396 * q^41 + 7146 * q^43 + 4050 * q^45 - 23588 * q^47 + 56182 * q^49 - 14850 * q^51 + 9310 * q^53 - 52048 * q^55 - 2034 * q^57 - 5260 * q^59 + 16932 * q^61 + 12474 * q^63 + 30852 * q^65 - 24858 * q^67 + 28566 * q^69 - 101832 * q^71 + 33100 * q^73 - 134946 * q^75 + 182232 * q^77 - 100570 * q^79 + 39366 * q^81 - 222524 * q^83 + 23824 * q^85 - 114840 * q^87 + 28374 * q^89 - 18204 * q^91 + 98820 * q^93 - 481996 * q^95 + 47632 * q^97 - 5508 * 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
$5$	84.9274	1.51923	0.759614	−	0.650375i	$-0.225388\pi$
$5$	84.9274	1.51923	0.759614	+	0.650375i	$0.225388\pi$
$6$	0	0
$7$	−141.499	−1.09146	−0.545731	−	0.837960i	$-0.683748\pi$
$7$	−141.499	−1.09146	−0.545731	+	0.837960i	$0.683748\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−576.669	−1.43696	−0.718481	−	0.695547i	$-0.755162\pi$
$11$	−576.669	−1.43696	−0.718481	+	0.695547i	$0.755162\pi$
$12$	0	0
$13$	357.346	0.586450	0.293225	−	0.956043i	$-0.405272\pi$
$13$	357.346	0.586450	0.293225	+	0.956043i	$0.405272\pi$
$14$	0	0
$15$	−764.346	−0.877126
$16$	0	0
$17$	−373.676	−0.313598	−0.156799	−	0.987631i	$-0.550117\pi$
$17$	−373.676	−0.313598	−0.156799	+	0.987631i	$0.550117\pi$
$18$	0	0
$19$	265.506	0.168729	0.0843646	−	0.996435i	$-0.473114\pi$
$19$	265.506	0.168729	0.0843646	+	0.996435i	$0.473114\pi$
$20$	0	0
$21$	1273.49	0.630156
$22$	0	0
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	4087.66	1.30805
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−8156.72	−1.80103	−0.900514	−	0.434826i	$-0.856810\pi$
$29$	−8156.72	−1.80103	−0.900514	+	0.434826i	$0.856810\pi$
$30$	0	0
$31$	−607.988	−0.113629	−0.0568147	−	0.998385i	$-0.518094\pi$
$31$	−607.988	−0.113629	−0.0568147	+	0.998385i	$0.518094\pi$
$32$	0	0
$33$	5190.02	0.829630
$34$	0	0
$35$	−12017.1	−1.65818
$36$	0	0
$37$	−1073.36	−0.128897	−0.0644484	−	0.997921i	$-0.520529\pi$
$37$	−1073.36	−0.128897	−0.0644484	+	0.997921i	$0.520529\pi$
$38$	0	0
$39$	−3216.12	−0.338587
$40$	0	0
$41$	−12732.7	−1.18293	−0.591466	−	0.806330i	$-0.701450\pi$
$41$	−12732.7	−1.18293	−0.591466	+	0.806330i	$0.701450\pi$
$42$	0	0
$43$	19839.1	1.63625	0.818125	−	0.575040i	$-0.195014\pi$
$43$	19839.1	1.63625	0.818125	+	0.575040i	$0.195014\pi$
$44$	0	0
$45$	6879.12	0.506409
$46$	0	0
$47$	2202.56	0.145440	0.0727198	−	0.997352i	$-0.476832\pi$
$47$	2202.56	0.145440	0.0727198	+	0.997352i	$0.476832\pi$
$48$	0	0
$49$	3215.00	0.191290
$50$	0	0
$51$	3363.08	0.181056
$52$	0	0
$53$	11191.2	0.547253	0.273627	−	0.961836i	$-0.411777\pi$
$53$	11191.2	0.547253	0.273627	+	0.961836i	$0.411777\pi$
$54$	0	0
$55$	−48975.0	−2.18307
$56$	0	0
$57$	−2389.55	−0.0974158
$58$	0	0
$59$	−3730.24	−0.139510	−0.0697551	−	0.997564i	$-0.522222\pi$
$59$	−3730.24	−0.139510	−0.0697551	+	0.997564i	$0.522222\pi$
$60$	0	0
$61$	41151.8	1.41600	0.708002	−	0.706210i	$-0.249597\pi$
$61$	41151.8	1.41600	0.708002	+	0.706210i	$0.249597\pi$
$62$	0	0
$63$	−11461.4	−0.363821
$64$	0	0
$65$	30348.5	0.890951
$66$	0	0
$67$	51973.5	1.41447	0.707236	−	0.706977i	$-0.249942\pi$
$67$	51973.5	1.41447	0.707236	+	0.706977i	$0.249942\pi$
$68$	0	0
$69$	4761.00	0.120386
$70$	0	0
$71$	73477.5	1.72985	0.864925	−	0.501901i	$-0.167366\pi$
$71$	73477.5	1.72985	0.864925	+	0.501901i	$0.167366\pi$
$72$	0	0
$73$	39717.2	0.872312	0.436156	−	0.899871i	$-0.356340\pi$
$73$	39717.2	0.872312	0.436156	+	0.899871i	$0.356340\pi$
$74$	0	0
$75$	−36788.9	−0.755203
$76$	0	0
$77$	81598.2	1.56839
$78$	0	0
$79$	−53492.5	−0.964330	−0.482165	−	0.876081i	$-0.660149\pi$
$79$	−53492.5	−0.964330	−0.482165	+	0.876081i	$0.660149\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−93683.1	−1.49268	−0.746338	−	0.665567i	$-0.768190\pi$
$83$	−93683.1	−1.49268	−0.746338	+	0.665567i	$0.768190\pi$
$84$	0	0
$85$	−31735.3	−0.476426
$86$	0	0
$87$	73410.5	1.03982
$88$	0	0
$89$	−19076.4	−0.255283	−0.127641	−	0.991820i	$-0.540741\pi$
$89$	−19076.4	−0.255283	−0.127641	+	0.991820i	$0.540741\pi$
$90$	0	0
$91$	−50564.2	−0.640088
$92$	0	0
$93$	5471.90	0.0656040
$94$	0	0
$95$	22548.7	0.256338
$96$	0	0
$97$	−5830.26	−0.0629156	−0.0314578	−	0.999505i	$-0.510015\pi$
$97$	−5830.26	−0.0629156	−0.0314578	+	0.999505i	$0.510015\pi$
$98$	0	0
$99$	−46710.2	−0.478987
$100$	0	0
$101$	−96025.2	−0.936659	−0.468330	−	0.883554i	$-0.655144\pi$
$101$	−96025.2	−0.936659	−0.468330	+	0.883554i	$0.655144\pi$
$102$	0	0
$103$	−54533.6	−0.506490	−0.253245	−	0.967402i	$-0.581498\pi$
$103$	−54533.6	−0.506490	−0.253245	+	0.967402i	$0.581498\pi$
$104$	0	0
$105$	108154.	0.957350
$106$	0	0
$107$	−55054.0	−0.464867	−0.232434	−	0.972612i	$-0.574669\pi$
$107$	−55054.0	−0.464867	−0.232434	+	0.972612i	$0.574669\pi$
$108$	0	0
$109$	−52304.2	−0.421668	−0.210834	−	0.977522i	$-0.567618\pi$
$109$	−52304.2	−0.421668	−0.210834	+	0.977522i	$0.567618\pi$
$110$	0	0
$111$	9660.26	0.0744186
$112$	0	0
$113$	209253.	1.54161	0.770806	−	0.637070i	$-0.219854\pi$
$113$	209253.	1.54161	0.770806	+	0.637070i	$0.219854\pi$
$114$	0	0
$115$	−44926.6	−0.316781
$116$	0	0
$117$	28945.1	0.195483
$118$	0	0
$119$	52874.8	0.342280
$120$	0	0
$121$	171496.	1.06486
$122$	0	0
$123$	114594.	0.682966
$124$	0	0
$125$	81756.0	0.467999
$126$	0	0
$127$	291320.	1.60273	0.801365	−	0.598175i	$-0.204107\pi$
$127$	291320.	1.60273	0.801365	+	0.598175i	$0.204107\pi$
$128$	0	0
$129$	−178551.	−0.944690
$130$	0	0
$131$	−324674.	−1.65298	−0.826492	−	0.562948i	$-0.809667\pi$
$131$	−324674.	−1.65298	−0.826492	+	0.562948i	$0.809667\pi$
$132$	0	0
$133$	−37568.8	−0.184161
$134$	0	0
$135$	−61912.1	−0.292375
$136$	0	0
$137$	133662.	0.608423	0.304212	−	0.952604i	$-0.401607\pi$
$137$	133662.	0.608423	0.304212	+	0.952604i	$0.401607\pi$
$138$	0	0
$139$	−276052.	−1.21187	−0.605933	−	0.795516i	$-0.707200\pi$
$139$	−276052.	−1.21187	−0.605933	+	0.795516i	$0.707200\pi$
$140$	0	0
$141$	−19823.0	−0.0839696
$142$	0	0
$143$	−206071.	−0.842706
$144$	0	0
$145$	−692729.	−2.73617
$146$	0	0
$147$	−28935.0	−0.110441
$148$	0	0
$149$	509785.	1.88114	0.940571	−	0.339596i	$-0.110290\pi$
$149$	509785.	1.88114	0.940571	+	0.339596i	$0.110290\pi$
$150$	0	0
$151$	365952.	1.30611	0.653057	−	0.757308i	$-0.273486\pi$
$151$	365952.	1.30611	0.653057	+	0.757308i	$0.273486\pi$
$152$	0	0
$153$	−30267.7	−0.104533
$154$	0	0
$155$	−51634.9	−0.172629
$156$	0	0
$157$	−373627.	−1.20973	−0.604865	−	0.796328i	$-0.706773\pi$
$157$	−373627.	−1.20973	−0.604865	+	0.796328i	$0.706773\pi$
$158$	0	0
$159$	−100721.	−0.315957
$160$	0	0
$161$	74853.0	0.227586
$162$	0	0
$163$	203047.	0.598586	0.299293	−	0.954161i	$-0.403249\pi$
$163$	203047.	0.598586	0.299293	+	0.954161i	$0.403249\pi$
$164$	0	0
$165$	440775.	1.26040
$166$	0	0
$167$	485401.	1.34682	0.673410	−	0.739269i	$-0.264829\pi$
$167$	485401.	1.34682	0.673410	+	0.739269i	$0.264829\pi$
$168$	0	0
$169$	−243597.	−0.656076
$170$	0	0
$171$	21506.0	0.0562430
$172$	0	0
$173$	−329700.	−0.837537	−0.418768	−	0.908093i	$-0.637538\pi$
$173$	−329700.	−0.837537	−0.418768	+	0.908093i	$0.637538\pi$
$174$	0	0
$175$	−578400.	−1.42769
$176$	0	0
$177$	33572.1	0.0805463
$178$	0	0
$179$	471816.	1.10063	0.550313	−	0.834958i	$-0.314508\pi$
$179$	471816.	1.10063	0.550313	+	0.834958i	$0.314508\pi$
$180$	0	0
$181$	−372549.	−0.845253	−0.422626	−	0.906304i	$-0.638892\pi$
$181$	−372549.	−0.845253	−0.422626	+	0.906304i	$0.638892\pi$
$182$	0	0
$183$	−370367.	−0.817531
$184$	0	0
$185$	−91157.9	−0.195823
$186$	0	0
$187$	215487.	0.450628
$188$	0	0
$189$	103153.	0.210052
$190$	0	0
$191$	−307499.	−0.609902	−0.304951	−	0.952368i	$-0.598640\pi$
$191$	−307499.	−0.609902	−0.304951	+	0.952368i	$0.598640\pi$
$192$	0	0
$193$	−212342.	−0.410338	−0.205169	−	0.978727i	$-0.565774\pi$
$193$	−212342.	−0.410338	−0.205169	+	0.978727i	$0.565774\pi$
$194$	0	0
$195$	−273136.	−0.514391
$196$	0	0
$197$	316598.	0.581222	0.290611	−	0.956841i	$-0.406141\pi$
$197$	316598.	0.581222	0.290611	+	0.956841i	$0.406141\pi$
$198$	0	0
$199$	46727.9	0.0836457	0.0418228	−	0.999125i	$-0.486683\pi$
$199$	46727.9	0.0836457	0.0418228	+	0.999125i	$0.486683\pi$
$200$	0	0
$201$	−467761.	−0.816646
$202$	0	0
$203$	1.15417e6	1.96575
$204$	0	0
$205$	−1.08135e6	−1.79714
$206$	0	0
$207$	−42849.0	−0.0695048
$208$	0	0
$209$	−153109.	−0.242457
$210$	0	0
$211$	553869.	0.856448	0.428224	−	0.903673i	$-0.359139\pi$
$211$	553869.	0.856448	0.428224	+	0.903673i	$0.359139\pi$
$212$	0	0
$213$	−661298.	−0.998730
$214$	0	0
$215$	1.68488e6	2.48584
$216$	0	0
$217$	86029.8	0.124022
$218$	0	0
$219$	−357455.	−0.503630
$220$	0	0
$221$	−133532.	−0.183909
$222$	0	0
$223$	606152.	0.816243	0.408121	−	0.912928i	$-0.366184\pi$
$223$	606152.	0.816243	0.408121	+	0.912928i	$0.366184\pi$
$224$	0	0
$225$	331100.	0.436017
$226$	0	0
$227$	221836.	0.285738	0.142869	−	0.989742i	$-0.454367\pi$
$227$	221836.	0.285738	0.142869	+	0.989742i	$0.454367\pi$
$228$	0	0
$229$	674179.	0.849546	0.424773	−	0.905300i	$-0.360354\pi$
$229$	674179.	0.849546	0.424773	+	0.905300i	$0.360354\pi$
$230$	0	0
$231$	−734384.	−0.905510
$232$	0	0
$233$	430042.	0.518944	0.259472	−	0.965751i	$-0.416451\pi$
$233$	430042.	0.518944	0.259472	+	0.965751i	$0.416451\pi$
$234$	0	0
$235$	187057.	0.220956
$236$	0	0
$237$	481433.	0.556756
$238$	0	0
$239$	1.57421e6	1.78265	0.891327	−	0.453361i	$-0.149775\pi$
$239$	1.57421e6	1.78265	0.891327	+	0.453361i	$0.149775\pi$
$240$	0	0
$241$	−30250.6	−0.0335499	−0.0167750	−	0.999859i	$-0.505340\pi$
$241$	−30250.6	−0.0335499	−0.0167750	+	0.999859i	$0.505340\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	273042.	0.290612
$246$	0	0
$247$	94877.5	0.0989512
$248$	0	0
$249$	843148.	0.861797
$250$	0	0
$251$	−86847.2	−0.0870105	−0.0435052	−	0.999053i	$-0.513853\pi$
$251$	−86847.2	−0.0870105	−0.0435052	+	0.999053i	$0.513853\pi$
$252$	0	0
$253$	305058.	0.299627
$254$	0	0
$255$	285618.	0.275065
$256$	0	0
$257$	1.58402e6	1.49599	0.747994	−	0.663705i	$-0.231017\pi$
$257$	1.58402e6	1.49599	0.747994	+	0.663705i	$0.231017\pi$
$258$	0	0
$259$	151880.	0.140686
$260$	0	0
$261$	−660695.	−0.600343
$262$	0	0
$263$	−828208.	−0.738329	−0.369164	−	0.929364i	$-0.620356\pi$
$263$	−828208.	−0.738329	−0.369164	+	0.929364i	$0.620356\pi$
$264$	0	0
$265$	950442.	0.831402
$266$	0	0
$267$	171688.	0.147388
$268$	0	0
$269$	953511.	0.803424	0.401712	−	0.915766i	$-0.368415\pi$
$269$	953511.	0.803424	0.401712	+	0.915766i	$0.368415\pi$
$270$	0	0
$271$	1.71958e6	1.42233	0.711165	−	0.703025i	$-0.248168\pi$
$271$	1.71958e6	1.42233	0.711165	+	0.703025i	$0.248168\pi$
$272$	0	0
$273$	455078.	0.369555
$274$	0	0
$275$	−2.35723e6	−1.87962
$276$	0	0
$277$	608283.	0.476328	0.238164	−	0.971225i	$-0.423454\pi$
$277$	608283.	0.476328	0.238164	+	0.971225i	$0.423454\pi$
$278$	0	0
$279$	−49247.1	−0.0378765
$280$	0	0
$281$	486264.	0.367372	0.183686	−	0.982985i	$-0.441197\pi$
$281$	486264.	0.367372	0.183686	+	0.982985i	$0.441197\pi$
$282$	0	0
$283$	−250286.	−0.185768	−0.0928839	−	0.995677i	$-0.529609\pi$
$283$	−250286.	−0.185768	−0.0928839	+	0.995677i	$0.529609\pi$
$284$	0	0
$285$	−202938.	−0.147997
$286$	0	0
$287$	1.80166e6	1.29112
$288$	0	0
$289$	−1.28022e6	−0.901657
$290$	0	0
$291$	52472.3	0.0363244
$292$	0	0
$293$	2.80763e6	1.91060	0.955302	−	0.295632i	$-0.0955302\pi$
$293$	2.80763e6	1.91060	0.955302	+	0.295632i	$0.0955302\pi$
$294$	0	0
$295$	−316799.	−0.211948
$296$	0	0
$297$	420392.	0.276543
$298$	0	0
$299$	−189036.	−0.122283
$300$	0	0
$301$	−2.80721e6	−1.78591
$302$	0	0
$303$	864227.	0.540780
$304$	0	0
$305$	3.49492e6	2.15123
$306$	0	0
$307$	1.93302e6	1.17055	0.585275	−	0.810835i	$-0.300986\pi$
$307$	1.93302e6	1.17055	0.585275	+	0.810835i	$0.300986\pi$
$308$	0	0
$309$	490802.	0.292422
$310$	0	0
$311$	−802489.	−0.470477	−0.235238	−	0.971938i	$-0.575587\pi$
$311$	−802489.	−0.470477	−0.235238	+	0.971938i	$0.575587\pi$
$312$	0	0
$313$	180021.	0.103863	0.0519316	−	0.998651i	$-0.483462\pi$
$313$	180021.	0.103863	0.0519316	+	0.998651i	$0.483462\pi$
$314$	0	0
$315$	−973389.	−0.552726
$316$	0	0
$317$	−684759.	−0.382727	−0.191364	−	0.981519i	$-0.561291\pi$
$317$	−684759.	−0.382727	−0.191364	+	0.981519i	$0.561291\pi$
$318$	0	0
$319$	4.70373e6	2.58801
$320$	0	0
$321$	495486.	0.268391
$322$	0	0
$323$	−99213.1	−0.0529130
$324$	0	0
$325$	1.46071e6	0.767106
$326$	0	0
$327$	470738.	0.243450
$328$	0	0
$329$	−311660.	−0.158742
$330$	0	0
$331$	860601.	0.431750	0.215875	−	0.976421i	$-0.430740\pi$
$331$	860601.	0.431750	0.215875	+	0.976421i	$0.430740\pi$
$332$	0	0
$333$	−86942.4	−0.0429656
$334$	0	0
$335$	4.41397e6	2.14891
$336$	0	0
$337$	3.09883e6	1.48636	0.743179	−	0.669093i	$-0.233317\pi$
$337$	3.09883e6	1.48636	0.743179	+	0.669093i	$0.233317\pi$
$338$	0	0
$339$	−1.88327e6	−0.890050
$340$	0	0
$341$	350608.	0.163281
$342$	0	0
$343$	1.92326e6	0.882677
$344$	0	0
$345$	404339.	0.182893
$346$	0	0
$347$	−3.03621e6	−1.35365	−0.676827	−	0.736142i	$-0.736645\pi$
$347$	−3.03621e6	−1.35365	−0.676827	+	0.736142i	$0.736645\pi$
$348$	0	0
$349$	3.85392e6	1.69371	0.846855	−	0.531823i	$-0.178493\pi$
$349$	3.85392e6	1.69371	0.846855	+	0.531823i	$0.178493\pi$
$350$	0	0
$351$	−260505.	−0.112862
$352$	0	0
$353$	−872688.	−0.372754	−0.186377	−	0.982478i	$-0.559675\pi$
$353$	−872688.	−0.372754	−0.186377	+	0.982478i	$0.559675\pi$
$354$	0	0
$355$	6.24025e6	2.62804
$356$	0	0
$357$	−475873.	−0.197615
$358$	0	0
$359$	−3.03864e6	−1.24435	−0.622177	−	0.782877i	$-0.713752\pi$
$359$	−3.03864e6	−1.24435	−0.622177	+	0.782877i	$0.713752\pi$
$360$	0	0
$361$	−2.40561e6	−0.971530
$362$	0	0
$363$	−1.54347e6	−0.614796
$364$	0	0
$365$	3.37308e6	1.32524
$366$	0	0
$367$	−395588.	−0.153312	−0.0766562	−	0.997058i	$-0.524424\pi$
$367$	−395588.	−0.153312	−0.0766562	+	0.997058i	$0.524424\pi$
$368$	0	0
$369$	−1.03135e6	−0.394310
$370$	0	0
$371$	−1.58355e6	−0.597306
$372$	0	0
$373$	2.52214e6	0.938636	0.469318	−	0.883029i	$-0.344500\pi$
$373$	2.52214e6	0.938636	0.469318	+	0.883029i	$0.344500\pi$
$374$	0	0
$375$	−735804.	−0.270199
$376$	0	0
$377$	−2.91477e6	−1.05621
$378$	0	0
$379$	692972.	0.247810	0.123905	−	0.992294i	$-0.460458\pi$
$379$	692972.	0.247810	0.123905	+	0.992294i	$0.460458\pi$
$380$	0	0
$381$	−2.62188e6	−0.925337
$382$	0	0
$383$	−1.49822e6	−0.521891	−0.260945	−	0.965354i	$-0.584034\pi$
$383$	−1.49822e6	−0.521891	−0.260945	+	0.965354i	$0.584034\pi$
$384$	0	0
$385$	6.92992e6	2.38274
$386$	0	0
$387$	1.60696e6	0.545417
$388$	0	0
$389$	77144.8	0.0258484	0.0129242	−	0.999916i	$-0.495886\pi$
$389$	77144.8	0.0258484	0.0129242	+	0.999916i	$0.495886\pi$
$390$	0	0
$391$	197675.	0.0653896
$392$	0	0
$393$	2.92206e6	0.954351
$394$	0	0
$395$	−4.54298e6	−1.46504
$396$	0	0
$397$	1.50574e6	0.479484	0.239742	−	0.970837i	$-0.422937\pi$
$397$	1.50574e6	0.479484	0.239742	+	0.970837i	$0.422937\pi$
$398$	0	0
$399$	338120.	0.106326
$400$	0	0
$401$	−1.94873e6	−0.605188	−0.302594	−	0.953119i	$-0.597853\pi$
$401$	−1.94873e6	−0.605188	−0.302594	+	0.953119i	$0.597853\pi$
$402$	0	0
$403$	−217262.	−0.0666380
$404$	0	0
$405$	557208.	0.168803
$406$	0	0
$407$	618975.	0.185220
$408$	0	0
$409$	42322.8	0.0125102	0.00625512	−	0.999980i	$-0.498009\pi$
$409$	42322.8	0.0125102	0.00625512	+	0.999980i	$0.498009\pi$
$410$	0	0
$411$	−1.20296e6	−0.351273
$412$	0	0
$413$	527825.	0.152270
$414$	0	0
$415$	−7.95626e6	−2.26772
$416$	0	0
$417$	2.48447e6	0.699671
$418$	0	0
$419$	−2.38045e6	−0.662405	−0.331203	−	0.943560i	$-0.607454\pi$
$419$	−2.38045e6	−0.662405	−0.331203	+	0.943560i	$0.607454\pi$
$420$	0	0
$421$	5.81066e6	1.59779	0.798896	−	0.601469i	$-0.205418\pi$
$421$	5.81066e6	1.59779	0.798896	+	0.601469i	$0.205418\pi$
$422$	0	0
$423$	178407.	0.0484799
$424$	0	0
$425$	−1.52746e6	−0.410202
$426$	0	0
$427$	−5.82295e6	−1.54552
$428$	0	0
$429$	1.85464e6	0.486536
$430$	0	0
$431$	1.40086e6	0.363248	0.181624	−	0.983368i	$-0.441865\pi$
$431$	1.40086e6	0.363248	0.181624	+	0.983368i	$0.441865\pi$
$432$	0	0
$433$	−6.46718e6	−1.65766	−0.828830	−	0.559501i	$-0.810993\pi$
$433$	−6.46718e6	−1.65766	−0.828830	+	0.559501i	$0.810993\pi$
$434$	0	0
$435$	6.23456e6	1.57973
$436$	0	0
$437$	−140453.	−0.0351825
$438$	0	0
$439$	−2.11762e6	−0.524429	−0.262215	−	0.965010i	$-0.584453\pi$
$439$	−2.11762e6	−0.524429	−0.262215	+	0.965010i	$0.584453\pi$
$440$	0	0
$441$	260415.	0.0637632
$442$	0	0
$443$	3.41798e6	0.827485	0.413743	−	0.910394i	$-0.364221\pi$
$443$	3.41798e6	0.827485	0.413743	+	0.910394i	$0.364221\pi$
$444$	0	0
$445$	−1.62011e6	−0.387833
$446$	0	0
$447$	−4.58807e6	−1.08608
$448$	0	0
$449$	−6.00867e6	−1.40657	−0.703287	−	0.710906i	$-0.748285\pi$
$449$	−6.00867e6	−1.40657	−0.703287	+	0.710906i	$0.748285\pi$
$450$	0	0
$451$	7.34253e6	1.69983
$452$	0	0
$453$	−3.29356e6	−0.754086
$454$	0	0
$455$	−4.29428e6	−0.972439
$456$	0	0
$457$	5.14764e6	1.15297	0.576484	−	0.817108i	$-0.304424\pi$
$457$	5.14764e6	1.15297	0.576484	+	0.817108i	$0.304424\pi$
$458$	0	0
$459$	272410.	0.0603519
$460$	0	0
$461$	4.46375e6	0.978244	0.489122	−	0.872215i	$-0.337317\pi$
$461$	4.46375e6	0.978244	0.489122	+	0.872215i	$0.337317\pi$
$462$	0	0
$463$	651674.	0.141279	0.0706396	−	0.997502i	$-0.477496\pi$
$463$	651674.	0.141279	0.0706396	+	0.997502i	$0.477496\pi$
$464$	0	0
$465$	464714.	0.0996674
$466$	0	0
$467$	−3.05986e6	−0.649246	−0.324623	−	0.945844i	$-0.605237\pi$
$467$	−3.05986e6	−0.649246	−0.324623	+	0.945844i	$0.605237\pi$
$468$	0	0
$469$	−7.35420e6	−1.54384
$470$	0	0
$471$	3.36264e6	0.698438
$472$	0	0
$473$	−1.14406e7	−2.35123
$474$	0	0
$475$	1.08530e6	0.220706
$476$	0	0
$477$	906490.	0.182418
$478$	0	0
$479$	4.45781e6	0.887734	0.443867	−	0.896093i	$-0.353606\pi$
$479$	4.45781e6	0.887734	0.443867	+	0.896093i	$0.353606\pi$
$480$	0	0
$481$	−383562.	−0.0755915
$482$	0	0
$483$	−673677.	−0.131397
$484$	0	0
$485$	−495149.	−0.0955831
$486$	0	0
$487$	−1.29604e6	−0.247626	−0.123813	−	0.992306i	$-0.539512\pi$
$487$	−1.29604e6	−0.247626	−0.123813	+	0.992306i	$0.539512\pi$
$488$	0	0
$489$	−1.82742e6	−0.345594
$490$	0	0
$491$	2.47737e6	0.463753	0.231876	−	0.972745i	$-0.425513\pi$
$491$	2.47737e6	0.463753	0.231876	+	0.972745i	$0.425513\pi$
$492$	0	0
$493$	3.04797e6	0.564798
$494$	0	0
$495$	−3.96698e6	−0.727690
$496$	0	0
$497$	−1.03970e7	−1.88807
$498$	0	0
$499$	3.58134e6	0.643865	0.321932	−	0.946763i	$-0.395668\pi$
$499$	3.58134e6	0.643865	0.321932	+	0.946763i	$0.395668\pi$
$500$	0	0
$501$	−4.36861e6	−0.777587
$502$	0	0
$503$	7.42820e6	1.30907	0.654536	−	0.756031i	$-0.272864\pi$
$503$	7.42820e6	1.30907	0.654536	+	0.756031i	$0.272864\pi$
$504$	0	0
$505$	−8.15517e6	−1.42300
$506$	0	0
$507$	2.19237e6	0.378786
$508$	0	0
$509$	−3.26043e6	−0.557802	−0.278901	−	0.960320i	$-0.589970\pi$
$509$	−3.26043e6	−0.557802	−0.278901	+	0.960320i	$0.589970\pi$
$510$	0	0
$511$	−5.61995e6	−0.952095
$512$	0	0
$513$	−193554.	−0.0324719
$514$	0	0
$515$	−4.63139e6	−0.769473
$516$	0	0
$517$	−1.27015e6	−0.208991
$518$	0	0
$519$	2.96730e6	0.483552
$520$	0	0
$521$	1.51959e6	0.245263	0.122631	−	0.992452i	$-0.460867\pi$
$521$	1.51959e6	0.245263	0.122631	+	0.992452i	$0.460867\pi$
$522$	0	0
$523$	−9.77044e6	−1.56192	−0.780962	−	0.624578i	$-0.785271\pi$
$523$	−9.77044e6	−1.56192	−0.780962	+	0.624578i	$0.785271\pi$
$524$	0	0
$525$	5.20560e6	0.824276
$526$	0	0
$527$	227191.	0.0356339
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	−302149.	−0.0465034
$532$	0	0
$533$	−4.54997e6	−0.693730
$534$	0	0
$535$	−4.67559e6	−0.706239
$536$	0	0
$537$	−4.24634e6	−0.635447
$538$	0	0
$539$	−1.85399e6	−0.274876
$540$	0	0
$541$	6.94915e6	1.02080	0.510398	−	0.859938i	$-0.329498\pi$
$541$	6.94915e6	1.02080	0.510398	+	0.859938i	$0.329498\pi$
$542$	0	0
$543$	3.35294e6	0.488007
$544$	0	0
$545$	−4.44206e6	−0.640609
$546$	0	0
$547$	1.25448e7	1.79265	0.896326	−	0.443395i	$-0.146226\pi$
$547$	1.25448e7	1.79265	0.896326	+	0.443395i	$0.146226\pi$
$548$	0	0
$549$	3.33330e6	0.472001
$550$	0	0
$551$	−2.16566e6	−0.303886
$552$	0	0
$553$	7.56915e6	1.05253
$554$	0	0
$555$	820421.	0.113059
$556$	0	0
$557$	−1.09702e7	−1.49822	−0.749111	−	0.662445i	$-0.769519\pi$
$557$	−1.09702e7	−1.49822	−0.749111	+	0.662445i	$0.769519\pi$
$558$	0	0
$559$	7.08941e6	0.959579
$560$	0	0
$561$	−1.93939e6	−0.260170
$562$	0	0
$563$	−575210.	−0.0764814	−0.0382407	−	0.999269i	$-0.512175\pi$
$563$	−575210.	−0.0764814	−0.0382407	+	0.999269i	$0.512175\pi$
$564$	0	0
$565$	1.77713e7	2.34206
$566$	0	0
$567$	−928376.	−0.121274
$568$	0	0
$569$	5.39848e6	0.699022	0.349511	−	0.936932i	$-0.386348\pi$
$569$	5.39848e6	0.699022	0.349511	+	0.936932i	$0.386348\pi$
$570$	0	0
$571$	1.53619e7	1.97177	0.985883	−	0.167436i	$-0.0535488\pi$
$571$	1.53619e7	1.97177	0.985883	+	0.167436i	$0.0535488\pi$
$572$	0	0
$573$	2.76749e6	0.352127
$574$	0	0
$575$	−2.16237e6	−0.272747
$576$	0	0
$577$	112946.	0.0141232	0.00706158	−	0.999975i	$-0.497752\pi$
$577$	112946.	0.0141232	0.00706158	+	0.999975i	$0.497752\pi$
$578$	0	0
$579$	1.91107e6	0.236909
$580$	0	0
$581$	1.32561e7	1.62920
$582$	0	0
$583$	−6.45364e6	−0.786382
$584$	0	0
$585$	2.45823e6	0.296984
$586$	0	0
$587$	7.17937e6	0.859985	0.429993	−	0.902832i	$-0.358516\pi$
$587$	7.17937e6	0.859985	0.429993	+	0.902832i	$0.358516\pi$
$588$	0	0
$589$	−161424.	−0.0191726
$590$	0	0
$591$	−2.84938e6	−0.335569
$592$	0	0
$593$	6.77778e6	0.791499	0.395750	−	0.918358i	$-0.370485\pi$
$593$	6.77778e6	0.791499	0.395750	+	0.918358i	$0.370485\pi$
$594$	0	0
$595$	4.49052e6	0.520001
$596$	0	0
$597$	−420551.	−0.0482929
$598$	0	0
$599$	1.56267e7	1.77950	0.889752	−	0.456445i	$-0.150877\pi$
$599$	1.56267e7	1.77950	0.889752	+	0.456445i	$0.150877\pi$
$600$	0	0
$601$	−8.59894e6	−0.971088	−0.485544	−	0.874212i	$-0.661378\pi$
$601$	−8.59894e6	−0.971088	−0.485544	+	0.874212i	$0.661378\pi$
$602$	0	0
$603$	4.20985e6	0.471491
$604$	0	0
$605$	1.45647e7	1.61776
$606$	0	0
$607$	7.03090e6	0.774532	0.387266	−	0.921968i	$-0.373420\pi$
$607$	7.03090e6	0.774532	0.387266	+	0.921968i	$0.373420\pi$
$608$	0	0
$609$	−1.03875e7	−1.13493
$610$	0	0
$611$	787076.	0.0852930
$612$	0	0
$613$	6.02161e6	0.647234	0.323617	−	0.946188i	$-0.395101\pi$
$613$	6.02161e6	0.647234	0.323617	+	0.946188i	$0.395101\pi$
$614$	0	0
$615$	9.73216e6	1.03758
$616$	0	0
$617$	1.28453e6	0.135841	0.0679205	−	0.997691i	$-0.478364\pi$
$617$	1.28453e6	0.135841	0.0679205	+	0.997691i	$0.478364\pi$
$618$	0	0
$619$	1.07456e7	1.12720	0.563602	−	0.826047i	$-0.309415\pi$
$619$	1.07456e7	1.12720	0.563602	+	0.826047i	$0.309415\pi$
$620$	0	0
$621$	385641.	0.0401286
$622$	0	0
$623$	2.69930e6	0.278632
$624$	0	0
$625$	−5.83061e6	−0.597054
$626$	0	0
$627$	1.37798e6	0.139983
$628$	0	0
$629$	401090.	0.0404217
$630$	0	0
$631$	629466.	0.0629359	0.0314680	−	0.999505i	$-0.489982\pi$
$631$	629466.	0.0629359	0.0314680	+	0.999505i	$0.489982\pi$
$632$	0	0
$633$	−4.98482e6	−0.494471
$634$	0	0
$635$	2.47410e7	2.43491
$636$	0	0
$637$	1.14887e6	0.112182
$638$	0	0
$639$	5.95168e6	0.576617
$640$	0	0
$641$	−6.24205e6	−0.600043	−0.300021	−	0.953932i	$-0.596994\pi$
$641$	−6.24205e6	−0.600043	−0.300021	+	0.953932i	$0.596994\pi$
$642$	0	0
$643$	−8.24908e6	−0.786825	−0.393412	−	0.919362i	$-0.628705\pi$
$643$	−8.24908e6	−0.786825	−0.393412	+	0.919362i	$0.628705\pi$
$644$	0	0
$645$	−1.51639e7	−1.43520
$646$	0	0
$647$	−1.67040e7	−1.56877	−0.784387	−	0.620271i	$-0.787023\pi$
$647$	−1.67040e7	−1.56877	−0.784387	+	0.620271i	$0.787023\pi$
$648$	0	0
$649$	2.15111e6	0.200471
$650$	0	0
$651$	−774269.	−0.0716043
$652$	0	0
$653$	2.36323e6	0.216882	0.108441	−	0.994103i	$-0.465414\pi$
$653$	2.36323e6	0.216882	0.108441	+	0.994103i	$0.465414\pi$
$654$	0	0
$655$	−2.75737e7	−2.51126
$656$	0	0
$657$	3.21710e6	0.290771
$658$	0	0
$659$	1.13305e7	1.01634	0.508168	−	0.861258i	$-0.330323\pi$
$659$	1.13305e7	1.01634	0.508168	+	0.861258i	$0.330323\pi$
$660$	0	0
$661$	−1.22951e7	−1.09453	−0.547265	−	0.836959i	$-0.684331\pi$
$661$	−1.22951e7	−1.09453	−0.547265	+	0.836959i	$0.684331\pi$
$662$	0	0
$663$	1.20179e6	0.106180
$664$	0	0
$665$	−3.19062e6	−0.279783
$666$	0	0
$667$	4.31491e6	0.375540
$668$	0	0
$669$	−5.45537e6	−0.471258
$670$	0	0
$671$	−2.37310e7	−2.03474
$672$	0	0
$673$	1.54803e7	1.31748	0.658738	−	0.752372i	$-0.271090\pi$
$673$	1.54803e7	1.31748	0.658738	+	0.752372i	$0.271090\pi$
$674$	0	0
$675$	−2.97990e6	−0.251734
$676$	0	0
$677$	1.25056e7	1.04866	0.524329	−	0.851516i	$-0.324316\pi$
$677$	1.25056e7	1.04866	0.524329	+	0.851516i	$0.324316\pi$
$678$	0	0
$679$	824977.	0.0686700
$680$	0	0
$681$	−1.99652e6	−0.164971
$682$	0	0
$683$	−1.30704e7	−1.07210	−0.536051	−	0.844186i	$-0.680084\pi$
$683$	−1.30704e7	−1.07210	−0.536051	+	0.844186i	$0.680084\pi$
$684$	0	0
$685$	1.13515e7	0.924333
$686$	0	0
$687$	−6.06761e6	−0.490485
$688$	0	0
$689$	3.99915e6	0.320937
$690$	0	0
$691$	−592109.	−0.0471744	−0.0235872	−	0.999722i	$-0.507509\pi$
$691$	−592109.	−0.0471744	−0.0235872	+	0.999722i	$0.507509\pi$
$692$	0	0
$693$	6.60945e6	0.522796
$694$	0	0
$695$	−2.34444e7	−1.84110
$696$	0	0
$697$	4.75789e6	0.370964
$698$	0	0
$699$	−3.87038e6	−0.299613
$700$	0	0
$701$	−9.05049e6	−0.695628	−0.347814	−	0.937564i	$-0.613076\pi$
$701$	−9.05049e6	−0.695628	−0.347814	+	0.937564i	$0.613076\pi$
$702$	0	0
$703$	−284984.	−0.0217486
$704$	0	0
$705$	−1.68352e6	−0.127569
$706$	0	0
$707$	1.35875e7	1.02233
$708$	0	0
$709$	9.82451e6	0.733999	0.366999	−	0.930221i	$-0.380385\pi$
$709$	9.82451e6	0.733999	0.366999	+	0.930221i	$0.380385\pi$
$710$	0	0
$711$	−4.33290e6	−0.321443
$712$	0	0
$713$	321626.	0.0236934
$714$	0	0
$715$	−1.75010e7	−1.28026
$716$	0	0
$717$	−1.41679e7	−1.02922
$718$	0	0
$719$	−9.49042e6	−0.684641	−0.342321	−	0.939583i	$-0.611213\pi$
$719$	−9.49042e6	−0.684641	−0.342321	+	0.939583i	$0.611213\pi$
$720$	0	0
$721$	7.71645e6	0.552815
$722$	0	0
$723$	272255.	0.0193700
$724$	0	0
$725$	−3.33419e7	−2.35584
$726$	0	0
$727$	−2.14150e7	−1.50274	−0.751368	−	0.659883i	$-0.770606\pi$
$727$	−2.14150e7	−1.50274	−0.751368	+	0.659883i	$0.770606\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−7.41338e6	−0.513124
$732$	0	0
$733$	−8.08977e6	−0.556130	−0.278065	−	0.960562i	$-0.589693\pi$
$733$	−8.08977e6	−0.556130	−0.278065	+	0.960562i	$0.589693\pi$
$734$	0	0
$735$	−2.45738e6	−0.167785
$736$	0	0
$737$	−2.99715e7	−2.03254
$738$	0	0
$739$	−8.24782e6	−0.555556	−0.277778	−	0.960645i	$-0.589598\pi$
$739$	−8.24782e6	−0.555556	−0.277778	+	0.960645i	$0.589598\pi$
$740$	0	0
$741$	−853898.	−0.0571295
$742$	0	0
$743$	5.08256e6	0.337762	0.168881	−	0.985636i	$-0.445985\pi$
$743$	5.08256e6	0.337762	0.168881	+	0.985636i	$0.445985\pi$
$744$	0	0
$745$	4.32947e7	2.85788
$746$	0	0
$747$	−7.58833e6	−0.497559
$748$	0	0
$749$	7.79009e6	0.507385
$750$	0	0
$751$	−2.76604e7	−1.78961	−0.894805	−	0.446458i	$-0.852685\pi$
$751$	−2.76604e7	−1.78961	−0.894805	+	0.446458i	$0.852685\pi$
$752$	0	0
$753$	781625.	0.0502355
$754$	0	0
$755$	3.10793e7	1.98428
$756$	0	0
$757$	−2.99448e7	−1.89925	−0.949625	−	0.313387i	$-0.898536\pi$
$757$	−2.99448e7	−1.89925	−0.949625	+	0.313387i	$0.898536\pi$
$758$	0	0
$759$	−2.74552e6	−0.172990
$760$	0	0
$761$	8.12735e6	0.508730	0.254365	−	0.967108i	$-0.418134\pi$
$761$	8.12735e6	0.508730	0.254365	+	0.967108i	$0.418134\pi$
$762$	0	0
$763$	7.40100e6	0.460234
$764$	0	0
$765$	−2.57056e6	−0.158809
$766$	0	0
$767$	−1.33299e6	−0.0818158
$768$	0	0
$769$	2.28845e7	1.39549	0.697743	−	0.716348i	$-0.254188\pi$
$769$	2.28845e7	1.39549	0.697743	+	0.716348i	$0.254188\pi$
$770$	0	0
$771$	−1.42562e7	−0.863709
$772$	0	0
$773$	2.03489e7	1.22487	0.612437	−	0.790519i	$-0.290189\pi$
$773$	2.03489e7	1.22487	0.612437	+	0.790519i	$0.290189\pi$
$774$	0	0
$775$	−2.48525e6	−0.148633
$776$	0	0
$777$	−1.36692e6	−0.0812251
$778$	0	0
$779$	−3.38059e6	−0.199595
$780$	0	0
$781$	−4.23722e7	−2.48573
$782$	0	0
$783$	5.94625e6	0.346608
$784$	0	0
$785$	−3.17311e7	−1.83786
$786$	0	0
$787$	1.52343e7	0.876772	0.438386	−	0.898787i	$-0.355550\pi$
$787$	1.52343e7	0.876772	0.438386	+	0.898787i	$0.355550\pi$
$788$	0	0
$789$	7.45387e6	0.426274
$790$	0	0
$791$	−2.96091e7	−1.68261
$792$	0	0
$793$	1.47055e7	0.830416
$794$	0	0
$795$	−8.55398e6	−0.480010
$796$	0	0
$797$	−2.90264e7	−1.61863	−0.809316	−	0.587374i	$-0.800162\pi$
$797$	−2.90264e7	−1.61863	−0.809316	+	0.587374i	$0.800162\pi$
$798$	0	0
$799$	−823042.	−0.0456095
$800$	0	0
$801$	−1.54519e6	−0.0850943
$802$	0	0
$803$	−2.29037e7	−1.25348
$804$	0	0
$805$	6.35707e6	0.345754
$806$	0	0
$807$	−8.58160e6	−0.463857
$808$	0	0
$809$	4.92029e6	0.264313	0.132157	−	0.991229i	$-0.457810\pi$
$809$	4.92029e6	0.264313	0.132157	+	0.991229i	$0.457810\pi$
$810$	0	0
$811$	−2.79391e7	−1.49163	−0.745813	−	0.666155i	$-0.767939\pi$
$811$	−2.79391e7	−1.49163	−0.745813	+	0.666155i	$0.767939\pi$
$812$	0	0
$813$	−1.54763e7	−0.821182
$814$	0	0
$815$	1.72442e7	0.909388
$816$	0	0
$817$	5.26738e6	0.276083
$818$	0	0
$819$	−4.09570e6	−0.213363
$820$	0	0
$821$	1.55694e7	0.806144	0.403072	−	0.915168i	$-0.367942\pi$
$821$	1.55694e7	0.806144	0.403072	+	0.915168i	$0.367942\pi$
$822$	0	0
$823$	−1.04383e7	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$823$	−1.04383e7	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$824$	0	0
$825$	2.12150e7	1.08520
$826$	0	0
$827$	−2.74975e6	−0.139807	−0.0699037	−	0.997554i	$-0.522269\pi$
$827$	−2.74975e6	−0.139807	−0.0699037	+	0.997554i	$0.522269\pi$
$828$	0	0
$829$	−956425.	−0.0483353	−0.0241677	−	0.999708i	$-0.507694\pi$
$829$	−956425.	−0.0483353	−0.0241677	+	0.999708i	$0.507694\pi$
$830$	0	0
$831$	−5.47455e6	−0.275008
$832$	0	0
$833$	−1.20137e6	−0.0599879
$834$	0	0
$835$	4.12239e7	2.04613
$836$	0	0
$837$	443224.	0.0218680
$838$	0	0
$839$	2.59060e7	1.27056	0.635280	−	0.772282i	$-0.280885\pi$
$839$	2.59060e7	1.27056	0.635280	+	0.772282i	$0.280885\pi$
$840$	0	0
$841$	4.60210e7	2.24371
$842$	0	0
$843$	−4.37637e6	−0.212102
$844$	0	0
$845$	−2.06880e7	−0.996729
$846$	0	0
$847$	−2.42666e7	−1.16225
$848$	0	0
$849$	2.25257e6	0.107253
$850$	0	0
$851$	567809.	0.0268768
$852$	0	0
$853$	−1.93551e7	−0.910800	−0.455400	−	0.890287i	$-0.650504\pi$
$853$	−1.93551e7	−0.910800	−0.455400	+	0.890287i	$0.650504\pi$
$854$	0	0
$855$	1.82645e6	0.0854459
$856$	0	0
$857$	−9.20011e6	−0.427899	−0.213949	−	0.976845i	$-0.568633\pi$
$857$	−9.20011e6	−0.427899	−0.213949	+	0.976845i	$0.568633\pi$
$858$	0	0
$859$	−7.33876e6	−0.339344	−0.169672	−	0.985501i	$-0.554271\pi$
$859$	−7.33876e6	−0.339344	−0.169672	+	0.985501i	$0.554271\pi$
$860$	0	0
$861$	−1.62149e7	−0.745431
$862$	0	0
$863$	−3.96363e7	−1.81162	−0.905808	−	0.423688i	$-0.860735\pi$
$863$	−3.96363e7	−1.81162	−0.905808	+	0.423688i	$0.860735\pi$
$864$	0	0
$865$	−2.80006e7	−1.27241
$866$	0	0
$867$	1.15220e7	0.520572
$868$	0	0
$869$	3.08475e7	1.38570
$870$	0	0
$871$	1.85725e7	0.829517
$872$	0	0
$873$	−472251.	−0.0209719
$874$	0	0
$875$	−1.15684e7	−0.510803
$876$	0	0
$877$	−3.22540e7	−1.41607	−0.708036	−	0.706177i	$-0.750418\pi$
$877$	−3.22540e7	−1.41607	−0.708036	+	0.706177i	$0.750418\pi$
$878$	0	0
$879$	−2.52687e7	−1.10309
$880$	0	0
$881$	−2.94978e7	−1.28041	−0.640206	−	0.768203i	$-0.721151\pi$
$881$	−2.94978e7	−1.28041	−0.640206	+	0.768203i	$0.721151\pi$
$882$	0	0
$883$	3.60339e7	1.55528	0.777642	−	0.628708i	$-0.216416\pi$
$883$	3.60339e7	1.55528	0.777642	+	0.628708i	$0.216416\pi$
$884$	0	0
$885$	2.85119e6	0.122368
$886$	0	0
$887$	−4.06205e7	−1.73355	−0.866774	−	0.498701i	$-0.833810\pi$
$887$	−4.06205e7	−1.73355	−0.866774	+	0.498701i	$0.833810\pi$
$888$	0	0
$889$	−4.12215e7	−1.74932
$890$	0	0
$891$	−3.78353e6	−0.159662
$892$	0	0
$893$	584792.	0.0245399
$894$	0	0
$895$	4.00701e7	1.67210
$896$	0	0
$897$	1.70133e6	0.0706003
$898$	0	0
$899$	4.95919e6	0.204650
$900$	0	0
$901$	−4.18189e6	−0.171617
$902$	0	0
$903$	2.52649e7	1.03109
$904$	0	0
$905$	−3.16396e7	−1.28413
$906$	0	0
$907$	3.48140e7	1.40519	0.702595	−	0.711590i	$-0.252024\pi$
$907$	3.48140e7	1.40519	0.702595	+	0.711590i	$0.252024\pi$
$908$	0	0
$909$	−7.77804e6	−0.312220
$910$	0	0
$911$	−4.11574e7	−1.64305	−0.821527	−	0.570170i	$-0.806877\pi$
$911$	−4.11574e7	−1.64305	−0.821527	+	0.570170i	$0.806877\pi$
$912$	0	0
$913$	5.40241e7	2.14492
$914$	0	0
$915$	−3.14543e7	−1.24201
$916$	0	0
$917$	4.59410e7	1.80417
$918$	0	0
$919$	1.10526e7	0.431694	0.215847	−	0.976427i	$-0.430749\pi$
$919$	1.10526e7	0.431694	0.215847	+	0.976427i	$0.430749\pi$
$920$	0	0
$921$	−1.73972e7	−0.675818
$922$	0	0
$923$	2.62569e7	1.01447
$924$	0	0
$925$	−4.38754e6	−0.168604
$926$	0	0
$927$	−4.41722e6	−0.168830
$928$	0	0
$929$	3.41103e7	1.29672	0.648360	−	0.761334i	$-0.275455\pi$
$929$	3.41103e7	1.29672	0.648360	+	0.761334i	$0.275455\pi$
$930$	0	0
$931$	853602.	0.0322761
$932$	0	0
$933$	7.22240e6	0.271630
$934$	0	0
$935$	1.83008e7	0.684606
$936$	0	0
$937$	6.87290e6	0.255735	0.127868	−	0.991791i	$-0.459187\pi$
$937$	6.87290e6	0.255735	0.127868	+	0.991791i	$0.459187\pi$
$938$	0	0
$939$	−1.62019e6	−0.0599655
$940$	0	0
$941$	2.88168e7	1.06089	0.530447	−	0.847718i	$-0.322024\pi$
$941$	2.88168e7	1.06089	0.530447	+	0.847718i	$0.322024\pi$
$942$	0	0
$943$	6.73558e6	0.246658
$944$	0	0
$945$	8.76050e6	0.319117
$946$	0	0
$947$	3.06941e7	1.11219	0.556097	−	0.831117i	$-0.312298\pi$
$947$	3.06941e7	1.11219	0.556097	+	0.831117i	$0.312298\pi$
$948$	0	0
$949$	1.41928e7	0.511567
$950$	0	0
$951$	6.16283e6	0.220968
$952$	0	0
$953$	3.38123e7	1.20599	0.602993	−	0.797746i	$-0.293975\pi$
$953$	3.38123e7	1.20599	0.602993	+	0.797746i	$0.293975\pi$
$954$	0	0
$955$	−2.61151e7	−0.926579
$956$	0	0
$957$	−4.23336e7	−1.49419
$958$	0	0
$959$	−1.89130e7	−0.664071
$960$	0	0
$961$	−2.82595e7	−0.987088
$962$	0	0
$963$	−4.45937e6	−0.154956
$964$	0	0
$965$	−1.80336e7	−0.623397
$966$	0	0
$967$	3.43624e7	1.18173	0.590864	−	0.806771i	$-0.298787\pi$
$967$	3.43624e7	1.18173	0.590864	+	0.806771i	$0.298787\pi$
$968$	0	0
$969$	892918.	0.0305494
$970$	0	0
$971$	−3.87651e7	−1.31945	−0.659725	−	0.751507i	$-0.729327\pi$
$971$	−3.87651e7	−1.31945	−0.659725	+	0.751507i	$0.729327\pi$
$972$	0	0
$973$	3.90612e7	1.32271
$974$	0	0
$975$	−1.31464e7	−0.442889
$976$	0	0
$977$	−1.48309e7	−0.497085	−0.248542	−	0.968621i	$-0.579952\pi$
$977$	−1.48309e7	−0.497085	−0.248542	+	0.968621i	$0.579952\pi$
$978$	0	0
$979$	1.10008e7	0.366832
$980$	0	0
$981$	−4.23664e6	−0.140556
$982$	0	0
$983$	1.54388e7	0.509602	0.254801	−	0.966994i	$-0.417990\pi$
$983$	1.54388e7	0.509602	0.254801	+	0.966994i	$0.417990\pi$
$984$	0	0
$985$	2.68878e7	0.883009
$986$	0	0
$987$	2.80494e6	0.0916496
$988$	0	0
$989$	−1.04949e7	−0.341182
$990$	0	0
$991$	−2.60341e7	−0.842090	−0.421045	−	0.907040i	$-0.638337\pi$
$991$	−2.60341e7	−0.842090	−0.421045	+	0.907040i	$0.638337\pi$
$992$	0	0
$993$	−7.74541e6	−0.249271
$994$	0	0
$995$	3.96848e6	0.127077
$996$	0	0
$997$	4.19388e7	1.33622	0.668110	−	0.744063i	$-0.267104\pi$
$997$	4.19388e7	1.33622	0.668110	+	0.744063i	$0.267104\pi$
$998$	0	0
$999$	782481.	0.0248062

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.s.1.5		6
4.3	odd	2		276.6.a.d.1.5	✓	6
12.11	even	2		828.6.a.e.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.6.a.d.1.5	✓	6	4.3	odd	2
828.6.a.e.1.2		6	12.11	even	2
1104.6.a.s.1.5		6	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)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +84.9274 q^{5} -141.499 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} +84.9274 q^{5} -141.499 q^{7} +81.0000 q^{9} -576.669 q^{11} +357.346 q^{13} -764.346 q^{15} -373.676 q^{17} +265.506 q^{19} +1273.49 q^{21} -529.000 q^{23} +4087.66 q^{25} -729.000 q^{27} -8156.72 q^{29} -607.988 q^{31} +5190.02 q^{33} -12017.1 q^{35} -1073.36 q^{37} -3216.12 q^{39} -12732.7 q^{41} +19839.1 q^{43} +6879.12 q^{45} +2202.56 q^{47} +3215.00 q^{49} +3363.08 q^{51} +11191.2 q^{53} -48975.0 q^{55} -2389.55 q^{57} -3730.24 q^{59} +41151.8 q^{61} -11461.4 q^{63} +30348.5 q^{65} +51973.5 q^{67} +4761.00 q^{69} +73477.5 q^{71} +39717.2 q^{73} -36788.9 q^{75} +81598.2 q^{77} -53492.5 q^{79} +6561.00 q^{81} -93683.1 q^{83} -31735.3 q^{85} +73410.5 q^{87} -19076.4 q^{89} -50564.2 q^{91} +5471.90 q^{93} +22548.7 q^{95} -5830.26 q^{97} -46710.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 54 q^{3} + 50 q^{5} + 154 q^{7} + 486 q^{9}+O(q^{10}) \) 6 * q - 54 * q^3 + 50 * q^5 + 154 * q^7 + 486 * q^9	\( 6 q - 54 q^{3} + 50 q^{5} + 154 q^{7} + 486 q^{9} - 68 q^{11} + 1076 q^{13} - 450 q^{15} + 1650 q^{17} + 226 q^{19} - 1386 q^{21} - 3174 q^{23} + 14994 q^{25} - 4374 q^{27} + 12760 q^{29} - 10980 q^{31} + 612 q^{33} - 8492 q^{35} + 18012 q^{37} - 9684 q^{39} + 15396 q^{41} + 7146 q^{43} + 4050 q^{45} - 23588 q^{47} + 56182 q^{49} - 14850 q^{51} + 9310 q^{53} - 52048 q^{55} - 2034 q^{57} - 5260 q^{59} + 16932 q^{61} + 12474 q^{63} + 30852 q^{65} - 24858 q^{67} + 28566 q^{69} - 101832 q^{71} + 33100 q^{73} - 134946 q^{75} + 182232 q^{77} - 100570 q^{79} + 39366 q^{81} - 222524 q^{83} + 23824 q^{85} - 114840 q^{87} + 28374 q^{89} - 18204 q^{91} + 98820 q^{93} - 481996 q^{95} + 47632 q^{97} - 5508 q^{99}+O(q^{100}) \) 6 * q - 54 * q^3 + 50 * q^5 + 154 * q^7 + 486 * q^9 - 68 * q^11 + 1076 * q^13 - 450 * q^15 + 1650 * q^17 + 226 * q^19 - 1386 * q^21 - 3174 * q^23 + 14994 * q^25 - 4374 * q^27 + 12760 * q^29 - 10980 * q^31 + 612 * q^33 - 8492 * q^35 + 18012 * q^37 - 9684 * q^39 + 15396 * q^41 + 7146 * q^43 + 4050 * q^45 - 23588 * q^47 + 56182 * q^49 - 14850 * q^51 + 9310 * q^53 - 52048 * q^55 - 2034 * q^57 - 5260 * q^59 + 16932 * q^61 + 12474 * q^63 + 30852 * q^65 - 24858 * q^67 + 28566 * q^69 - 101832 * q^71 + 33100 * q^73 - 134946 * q^75 + 182232 * q^77 - 100570 * q^79 + 39366 * q^81 - 222524 * q^83 + 23824 * q^85 - 114840 * q^87 + 28374 * q^89 - 18204 * q^91 + 98820 * q^93 - 481996 * q^95 + 47632 * q^97 - 5508 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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