LMFDB - Embedded newform 1104.6.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1104.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$177.063737074$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.5333.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 11x + 8 $ x^3 - x^2 - 11*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.20733$ 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)

$f(q)$	$=$	$q-9.00000 q^{3} -59.5955 q^{5} -96.8268 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} -59.5955 q^{5} -96.8268 q^{7} +81.0000 q^{9} +731.078 q^{11} -631.879 q^{13} +536.360 q^{15} +75.7854 q^{17} -2.35318 q^{19} +871.441 q^{21} -529.000 q^{23} +426.625 q^{25} -729.000 q^{27} -2425.82 q^{29} +349.683 q^{31} -6579.70 q^{33} +5770.44 q^{35} +8314.43 q^{37} +5686.91 q^{39} +4198.56 q^{41} -9372.62 q^{43} -4827.24 q^{45} +2436.94 q^{47} -7431.57 q^{49} -682.069 q^{51} -34979.0 q^{53} -43569.0 q^{55} +21.1787 q^{57} -15086.9 q^{59} +987.986 q^{61} -7842.97 q^{63} +37657.1 q^{65} -44337.8 q^{67} +4761.00 q^{69} +33934.4 q^{71} -42483.0 q^{73} -3839.62 q^{75} -70788.0 q^{77} -51187.8 q^{79} +6561.00 q^{81} +88496.5 q^{83} -4516.47 q^{85} +21832.4 q^{87} -34083.9 q^{89} +61182.8 q^{91} -3147.15 q^{93} +140.239 q^{95} -156803. q^{97} +59217.3 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 27 q^{3} - 56 q^{5} + 114 q^{7} + 243 q^{9}+O(q^{10}) $ 3 * q - 27 * q^3 - 56 * q^5 + 114 * q^7 + 243 * q^9	$ 3 q - 27 q^{3} - 56 q^{5} + 114 q^{7} + 243 q^{9} + 376 q^{11} - 858 q^{13} + 504 q^{15} - 2548 q^{17} + 2846 q^{19} - 1026 q^{21} - 1587 q^{23} + 753 q^{25} - 2187 q^{27} - 16370 q^{29} + 14756 q^{31} - 3384 q^{33} + 18520 q^{35} + 15874 q^{37} + 7722 q^{39} + 12606 q^{41} - 3154 q^{43} - 4536 q^{45} - 29928 q^{47} + 4471 q^{49} + 22932 q^{51} - 44084 q^{53} - 38360 q^{55} - 25614 q^{57} + 29300 q^{59} + 54010 q^{61} + 9234 q^{63} - 51216 q^{65} - 43390 q^{67} + 14283 q^{69} - 23424 q^{71} - 91402 q^{73} - 6777 q^{75} - 97208 q^{77} + 49398 q^{79} + 19683 q^{81} + 103936 q^{83} + 5888 q^{85} + 147330 q^{87} + 96112 q^{89} - 129228 q^{91} - 132804 q^{93} + 55928 q^{95} - 135318 q^{97} + 30456 q^{99}+O(q^{100}) $ 3 * q - 27 * q^3 - 56 * q^5 + 114 * q^7 + 243 * q^9 + 376 * q^11 - 858 * q^13 + 504 * q^15 - 2548 * q^17 + 2846 * q^19 - 1026 * q^21 - 1587 * q^23 + 753 * q^25 - 2187 * q^27 - 16370 * q^29 + 14756 * q^31 - 3384 * q^33 + 18520 * q^35 + 15874 * q^37 + 7722 * q^39 + 12606 * q^41 - 3154 * q^43 - 4536 * q^45 - 29928 * q^47 + 4471 * q^49 + 22932 * q^51 - 44084 * q^53 - 38360 * q^55 - 25614 * q^57 + 29300 * q^59 + 54010 * q^61 + 9234 * q^63 - 51216 * q^65 - 43390 * q^67 + 14283 * q^69 - 23424 * q^71 - 91402 * q^73 - 6777 * q^75 - 97208 * q^77 + 49398 * q^79 + 19683 * q^81 + 103936 * q^83 + 5888 * q^85 + 147330 * q^87 + 96112 * q^89 - 129228 * q^91 - 132804 * q^93 + 55928 * q^95 - 135318 * q^97 + 30456 * 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
$5$	−59.5955	−1.06608	−0.533038	−	0.846091i	$-0.678950\pi$
$5$	−59.5955	−1.06608	−0.533038	+	0.846091i	$0.678950\pi$
$6$	0	0
$7$	−96.8268	−0.746879	−0.373440	−	0.927654i	$-0.621822\pi$
$7$	−96.8268	−0.746879	−0.373440	+	0.927654i	$0.621822\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	731.078	1.82172	0.910861	−	0.412713i	$-0.135419\pi$
$11$	731.078	1.82172	0.910861	+	0.412713i	$0.135419\pi$
$12$	0	0
$13$	−631.879	−1.03699	−0.518496	−	0.855080i	$-0.673508\pi$
$13$	−631.879	−1.03699	−0.518496	+	0.855080i	$0.673508\pi$
$14$	0	0
$15$	536.360	0.615500
$16$	0	0
$17$	75.7854	0.0636009	0.0318005	−	0.999494i	$-0.489876\pi$
$17$	75.7854	0.0636009	0.0318005	+	0.999494i	$0.489876\pi$
$18$	0	0
$19$	−2.35318	−0.00149545	−0.000747725	−	1.00000i	$-0.500238\pi$
$19$	−2.35318	−0.00149545	−0.000747725		1.00000i	$0.500238\pi$
$20$	0	0
$21$	871.441	0.431211
$22$	0	0
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	426.625	0.136520
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−2425.82	−0.535628	−0.267814	−	0.963471i	$-0.586301\pi$
$29$	−2425.82	−0.535628	−0.267814	+	0.963471i	$0.586301\pi$
$30$	0	0
$31$	349.683	0.0653537	0.0326769	−	0.999466i	$-0.489597\pi$
$31$	349.683	0.0653537	0.0326769	+	0.999466i	$0.489597\pi$
$32$	0	0
$33$	−6579.70	−1.05177
$34$	0	0
$35$	5770.44	0.796231
$36$	0	0
$37$	8314.43	0.998454	0.499227	−	0.866471i	$-0.333617\pi$
$37$	8314.43	0.998454	0.499227	+	0.866471i	$0.333617\pi$
$38$	0	0
$39$	5686.91	0.598707
$40$	0	0
$41$	4198.56	0.390068	0.195034	−	0.980796i	$-0.437518\pi$
$41$	4198.56	0.390068	0.195034	+	0.980796i	$0.437518\pi$
$42$	0	0
$43$	−9372.62	−0.773019	−0.386509	−	0.922285i	$-0.626319\pi$
$43$	−9372.62	−0.773019	−0.386509	+	0.922285i	$0.626319\pi$
$44$	0	0
$45$	−4827.24	−0.355359
$46$	0	0
$47$	2436.94	0.160916	0.0804581	−	0.996758i	$-0.474362\pi$
$47$	2436.94	0.160916	0.0804581	+	0.996758i	$0.474362\pi$
$48$	0	0
$49$	−7431.57	−0.442171
$50$	0	0
$51$	−682.069	−0.0367200
$52$	0	0
$53$	−34979.0	−1.71048	−0.855241	−	0.518231i	$-0.826591\pi$
$53$	−34979.0	−1.71048	−0.855241	+	0.518231i	$0.826591\pi$
$54$	0	0
$55$	−43569.0	−1.94210
$56$	0	0
$57$	21.1787	0.000863399 0
$58$	0	0
$59$	−15086.9	−0.564249	−0.282125	−	0.959378i	$-0.591039\pi$
$59$	−15086.9	−0.564249	−0.282125	+	0.959378i	$0.591039\pi$
$60$	0	0
$61$	987.986	0.0339959	0.0169979	−	0.999856i	$-0.494589\pi$
$61$	987.986	0.0339959	0.0169979	+	0.999856i	$0.494589\pi$
$62$	0	0
$63$	−7842.97	−0.248960
$64$	0	0
$65$	37657.1	1.10551
$66$	0	0
$67$	−44337.8	−1.20667	−0.603333	−	0.797490i	$-0.706161\pi$
$67$	−44337.8	−1.20667	−0.603333	+	0.797490i	$0.706161\pi$
$68$	0	0
$69$	4761.00	0.120386
$70$	0	0
$71$	33934.4	0.798904	0.399452	−	0.916754i	$-0.369200\pi$
$71$	33934.4	0.798904	0.399452	+	0.916754i	$0.369200\pi$
$72$	0	0
$73$	−42483.0	−0.933058	−0.466529	−	0.884506i	$-0.654496\pi$
$73$	−42483.0	−0.933058	−0.466529	+	0.884506i	$0.654496\pi$
$74$	0	0
$75$	−3839.62	−0.0788198
$76$	0	0
$77$	−70788.0	−1.36061
$78$	0	0
$79$	−51187.8	−0.922782	−0.461391	−	0.887197i	$-0.652649\pi$
$79$	−51187.8	−0.922782	−0.461391	+	0.887197i	$0.652649\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	88496.5	1.41004	0.705019	−	0.709188i	$-0.250938\pi$
$83$	88496.5	1.41004	0.705019	+	0.709188i	$0.250938\pi$
$84$	0	0
$85$	−4516.47	−0.0678035
$86$	0	0
$87$	21832.4	0.309245
$88$	0	0
$89$	−34083.9	−0.456115	−0.228058	−	0.973648i	$-0.573237\pi$
$89$	−34083.9	−0.456115	−0.228058	+	0.973648i	$0.573237\pi$
$90$	0	0
$91$	61182.8	0.774508
$92$	0	0
$93$	−3147.15	−0.0377320
$94$	0	0
$95$	140.239	0.00159427
$96$	0	0
$97$	−156803.	−1.69209	−0.846047	−	0.533108i	$-0.821024\pi$
$97$	−156803.	−1.69209	−0.846047	+	0.533108i	$0.821024\pi$
$98$	0	0
$99$	59217.3	0.607241
$100$	0	0
$101$	−148400.	−1.44754	−0.723772	−	0.690039i	$-0.757593\pi$
$101$	−148400.	−1.44754	−0.723772	+	0.690039i	$0.757593\pi$
$102$	0	0
$103$	−155116.	−1.44067	−0.720335	−	0.693626i	$-0.756012\pi$
$103$	−155116.	−1.44067	−0.720335	+	0.693626i	$0.756012\pi$
$104$	0	0
$105$	−51934.0	−0.459704
$106$	0	0
$107$	177527.	1.49901	0.749504	−	0.662000i	$-0.230292\pi$
$107$	177527.	1.49901	0.749504	+	0.662000i	$0.230292\pi$
$108$	0	0
$109$	−169314.	−1.36498	−0.682492	−	0.730893i	$-0.739104\pi$
$109$	−169314.	−1.36498	−0.682492	+	0.730893i	$0.739104\pi$
$110$	0	0
$111$	−74829.8	−0.576458
$112$	0	0
$113$	81710.2	0.601977	0.300989	−	0.953628i	$-0.402683\pi$
$113$	81710.2	0.601977	0.300989	+	0.953628i	$0.402683\pi$
$114$	0	0
$115$	31526.0	0.222292
$116$	0	0
$117$	−51182.2	−0.345664
$118$	0	0
$119$	−7338.06	−0.0475022
$120$	0	0
$121$	373424.	2.31867
$122$	0	0
$123$	−37787.0	−0.225206
$124$	0	0
$125$	160811.	0.920536
$126$	0	0
$127$	317307.	1.74570	0.872851	−	0.487986i	$-0.162268\pi$
$127$	317307.	1.74570	0.872851	+	0.487986i	$0.162268\pi$
$128$	0	0
$129$	84353.6	0.446303
$130$	0	0
$131$	154610.	0.787151	0.393576	−	0.919292i	$-0.371238\pi$
$131$	154610.	0.787151	0.393576	+	0.919292i	$0.371238\pi$
$132$	0	0
$133$	227.851	0.00111692
$134$	0	0
$135$	43445.1	0.205167
$136$	0	0
$137$	217676.	0.990852	0.495426	−	0.868650i	$-0.335012\pi$
$137$	217676.	0.990852	0.495426	+	0.868650i	$0.335012\pi$
$138$	0	0
$139$	−236914.	−1.04005	−0.520023	−	0.854152i	$-0.674077\pi$
$139$	−236914.	−1.04005	−0.520023	+	0.854152i	$0.674077\pi$
$140$	0	0
$141$	−21932.4	−0.0929050
$142$	0	0
$143$	−461953.	−1.88911
$144$	0	0
$145$	144568.	0.571021
$146$	0	0
$147$	66884.1	0.255288
$148$	0	0
$149$	434197.	1.60221	0.801107	−	0.598521i	$-0.204245\pi$
$149$	434197.	1.60221	0.801107	+	0.598521i	$0.204245\pi$
$150$	0	0
$151$	135760.	0.484541	0.242270	−	0.970209i	$-0.422108\pi$
$151$	135760.	0.484541	0.242270	+	0.970209i	$0.422108\pi$
$152$	0	0
$153$	6138.62	0.0212003
$154$	0	0
$155$	−20839.5	−0.0696721
$156$	0	0
$157$	−355399.	−1.15071	−0.575356	−	0.817903i	$-0.695136\pi$
$157$	−355399.	−1.15071	−0.575356	+	0.817903i	$0.695136\pi$
$158$	0	0
$159$	314811.	0.987547
$160$	0	0
$161$	51221.4	0.155735
$162$	0	0
$163$	514344.	1.51630	0.758148	−	0.652082i	$-0.226104\pi$
$163$	514344.	1.51630	0.758148	+	0.652082i	$0.226104\pi$
$164$	0	0
$165$	392121.	1.12127
$166$	0	0
$167$	−178285.	−0.494680	−0.247340	−	0.968929i	$-0.579556\pi$
$167$	−178285.	−0.494680	−0.247340	+	0.968929i	$0.579556\pi$
$168$	0	0
$169$	27977.5	0.0753515
$170$	0	0
$171$	−190.608	−0.000498483 0
$172$	0	0
$173$	−535921.	−1.36140	−0.680699	−	0.732563i	$-0.738324\pi$
$173$	−535921.	−1.36140	−0.680699	+	0.732563i	$0.738324\pi$
$174$	0	0
$175$	−41308.7	−0.101964
$176$	0	0
$177$	135782.	0.325769
$178$	0	0
$179$	58891.2	0.137378	0.0686891	−	0.997638i	$-0.478118\pi$
$179$	58891.2	0.137378	0.0686891	+	0.997638i	$0.478118\pi$
$180$	0	0
$181$	829741.	1.88255	0.941274	−	0.337644i	$-0.109630\pi$
$181$	829741.	1.88255	0.941274	+	0.337644i	$0.109630\pi$
$182$	0	0
$183$	−8891.88	−0.0196275
$184$	0	0
$185$	−495502.	−1.06443
$186$	0	0
$187$	55405.1	0.115863
$188$	0	0
$189$	70586.7	0.143737
$190$	0	0
$191$	−149335.	−0.296195	−0.148097	−	0.988973i	$-0.547315\pi$
$191$	−149335.	−0.296195	−0.148097	+	0.988973i	$0.547315\pi$
$192$	0	0
$193$	913030.	1.76438	0.882189	−	0.470895i	$-0.156069\pi$
$193$	913030.	1.76438	0.882189	+	0.470895i	$0.156069\pi$
$194$	0	0
$195$	−338914.	−0.638268
$196$	0	0
$197$	−919205.	−1.68751	−0.843756	−	0.536727i	$-0.819661\pi$
$197$	−919205.	−1.68751	−0.843756	+	0.536727i	$0.819661\pi$
$198$	0	0
$199$	−493730.	−0.883806	−0.441903	−	0.897063i	$-0.645696\pi$
$199$	−493730.	−0.883806	−0.441903	+	0.897063i	$0.645696\pi$
$200$	0	0
$201$	399040.	0.696669
$202$	0	0
$203$	234884.	0.400050
$204$	0	0
$205$	−250215.	−0.415843
$206$	0	0
$207$	−42849.0	−0.0695048
$208$	0	0
$209$	−1720.36	−0.00272430
$210$	0	0
$211$	133387.	0.206257	0.103129	−	0.994668i	$-0.467115\pi$
$211$	133387.	0.206257	0.103129	+	0.994668i	$0.467115\pi$
$212$	0	0
$213$	−305410.	−0.461247
$214$	0	0
$215$	558566.	0.824098
$216$	0	0
$217$	−33858.7	−0.0488114
$218$	0	0
$219$	382347.	0.538701
$220$	0	0
$221$	−47887.2	−0.0659536
$222$	0	0
$223$	−362735.	−0.488457	−0.244229	−	0.969718i	$-0.578535\pi$
$223$	−362735.	−0.488457	−0.244229	+	0.969718i	$0.578535\pi$
$224$	0	0
$225$	34556.6	0.0455067
$226$	0	0
$227$	190962.	0.245970	0.122985	−	0.992409i	$-0.460753\pi$
$227$	190962.	0.245970	0.122985	+	0.992409i	$0.460753\pi$
$228$	0	0
$229$	745968.	0.940008	0.470004	−	0.882664i	$-0.344252\pi$
$229$	745968.	0.940008	0.470004	+	0.882664i	$0.344252\pi$
$230$	0	0
$231$	637092.	0.785547
$232$	0	0
$233$	636804.	0.768450	0.384225	−	0.923239i	$-0.374469\pi$
$233$	636804.	0.768450	0.384225	+	0.923239i	$0.374469\pi$
$234$	0	0
$235$	−145231.	−0.171549
$236$	0	0
$237$	460691.	0.532768
$238$	0	0
$239$	−1.44028e6	−1.63099	−0.815494	−	0.578765i	$-0.803535\pi$
$239$	−1.44028e6	−1.63099	−0.815494	+	0.578765i	$0.803535\pi$
$240$	0	0
$241$	1.13226e6	1.25575	0.627876	−	0.778313i	$-0.283924\pi$
$241$	1.13226e6	1.25575	0.627876	+	0.778313i	$0.283924\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	442888.	0.471388
$246$	0	0
$247$	1486.93	0.00155077
$248$	0	0
$249$	−796469.	−0.814086
$250$	0	0
$251$	622529.	0.623700	0.311850	−	0.950131i	$-0.399051\pi$
$251$	622529.	0.623700	0.311850	+	0.950131i	$0.399051\pi$
$252$	0	0
$253$	−386740.	−0.379855
$254$	0	0
$255$	40648.2	0.0391463
$256$	0	0
$257$	−2.10304e6	−1.98616	−0.993082	−	0.117419i	$-0.962538\pi$
$257$	−2.10304e6	−1.98616	−0.993082	+	0.117419i	$0.962538\pi$
$258$	0	0
$259$	−805059.	−0.745725
$260$	0	0
$261$	−196491.	−0.178543
$262$	0	0
$263$	−838283.	−0.747310	−0.373655	−	0.927568i	$-0.621896\pi$
$263$	−838283.	−0.747310	−0.373655	+	0.927568i	$0.621896\pi$
$264$	0	0
$265$	2.08459e6	1.82350
$266$	0	0
$267$	306755.	0.263338
$268$	0	0
$269$	897532.	0.756257	0.378128	−	0.925753i	$-0.376568\pi$
$269$	897532.	0.756257	0.378128	+	0.925753i	$0.376568\pi$
$270$	0	0
$271$	1.00088e6	0.827864	0.413932	−	0.910308i	$-0.364155\pi$
$271$	1.00088e6	0.827864	0.413932	+	0.910308i	$0.364155\pi$
$272$	0	0
$273$	−550645.	−0.447162
$274$	0	0
$275$	311896.	0.248701
$276$	0	0
$277$	2.09974e6	1.64424	0.822120	−	0.569314i	$-0.192791\pi$
$277$	2.09974e6	1.64424	0.822120	+	0.569314i	$0.192791\pi$
$278$	0	0
$279$	28324.3	0.0217846
$280$	0	0
$281$	503227.	0.380188	0.190094	−	0.981766i	$-0.439121\pi$
$281$	503227.	0.380188	0.190094	+	0.981766i	$0.439121\pi$
$282$	0	0
$283$	1.61771e6	1.20070	0.600349	−	0.799738i	$-0.295028\pi$
$283$	1.61771e6	1.20070	0.600349	+	0.799738i	$0.295028\pi$
$284$	0	0
$285$	−1262.15	−0.000920449 0
$286$	0	0
$287$	−406533.	−0.291334
$288$	0	0
$289$	−1.41411e6	−0.995955
$290$	0	0
$291$	1.41123e6	0.976932
$292$	0	0
$293$	1.04521e6	0.711270	0.355635	−	0.934625i	$-0.384265\pi$
$293$	1.04521e6	0.711270	0.355635	+	0.934625i	$0.384265\pi$
$294$	0	0
$295$	899113.	0.601533
$296$	0	0
$297$	−532956.	−0.350591
$298$	0	0
$299$	334264.	0.216228
$300$	0	0
$301$	907521.	0.577352
$302$	0	0
$303$	1.33560e6	0.835740
$304$	0	0
$305$	−58879.6	−0.0362422
$306$	0	0
$307$	697539.	0.422399	0.211199	−	0.977443i	$-0.432263\pi$
$307$	697539.	0.422399	0.211199	+	0.977443i	$0.432263\pi$
$308$	0	0
$309$	1.39605e6	0.831771
$310$	0	0
$311$	2.68619e6	1.57484	0.787419	−	0.616419i	$-0.211417\pi$
$311$	2.68619e6	1.57484	0.787419	+	0.616419i	$0.211417\pi$
$312$	0	0
$313$	1.62222e6	0.935944	0.467972	−	0.883743i	$-0.344985\pi$
$313$	1.62222e6	0.935944	0.467972	+	0.883743i	$0.344985\pi$
$314$	0	0
$315$	467406.	0.265410
$316$	0	0
$317$	648171.	0.362277	0.181139	−	0.983458i	$-0.442022\pi$
$317$	648171.	0.362277	0.181139	+	0.983458i	$0.442022\pi$
$318$	0	0
$319$	−1.77346e6	−0.975766
$320$	0	0
$321$	−1.59774e6	−0.865452
$322$	0	0
$323$	−178.337	−9.51120e−5 0
$324$	0	0
$325$	−269575.	−0.141570
$326$	0	0
$327$	1.52383e6	0.788074
$328$	0	0
$329$	−235961.	−0.120185
$330$	0	0
$331$	1.66550e6	0.835552	0.417776	−	0.908550i	$-0.362810\pi$
$331$	1.66550e6	0.835552	0.417776	+	0.908550i	$0.362810\pi$
$332$	0	0
$333$	673468.	0.332818
$334$	0	0
$335$	2.64233e6	1.28640
$336$	0	0
$337$	−2.11943e6	−1.01659	−0.508294	−	0.861184i	$-0.669724\pi$
$337$	−2.11943e6	−1.01659	−0.508294	+	0.861184i	$0.669724\pi$
$338$	0	0
$339$	−735392.	−0.347552
$340$	0	0
$341$	255646.	0.119056
$342$	0	0
$343$	2.34694e6	1.07713
$344$	0	0
$345$	−283734.	−0.128341
$346$	0	0
$347$	791717.	0.352977	0.176488	−	0.984303i	$-0.443526\pi$
$347$	791717.	0.352977	0.176488	+	0.984303i	$0.443526\pi$
$348$	0	0
$349$	743781.	0.326875	0.163437	−	0.986554i	$-0.447742\pi$
$349$	743781.	0.326875	0.163437	+	0.986554i	$0.447742\pi$
$350$	0	0
$351$	460639.	0.199569
$352$	0	0
$353$	−646287.	−0.276051	−0.138025	−	0.990429i	$-0.544076\pi$
$353$	−646287.	−0.276051	−0.138025	+	0.990429i	$0.544076\pi$
$354$	0	0
$355$	−2.02234e6	−0.851693
$356$	0	0
$357$	66042.5	0.0274254
$358$	0	0
$359$	−1.89862e6	−0.777503	−0.388752	−	0.921343i	$-0.627094\pi$
$359$	−1.89862e6	−0.777503	−0.388752	+	0.921343i	$0.627094\pi$
$360$	0	0
$361$	−2.47609e6	−0.999998
$362$	0	0
$363$	−3.36082e6	−1.33869
$364$	0	0
$365$	2.53180e6	0.994711
$366$	0	0
$367$	−1.19082e6	−0.461509	−0.230754	−	0.973012i	$-0.574119\pi$
$367$	−1.19082e6	−0.461509	−0.230754	+	0.973012i	$0.574119\pi$
$368$	0	0
$369$	340083.	0.130023
$370$	0	0
$371$	3.38691e6	1.27752
$372$	0	0
$373$	−254301.	−0.0946402	−0.0473201	−	0.998880i	$-0.515068\pi$
$373$	−254301.	−0.0946402	−0.0473201	+	0.998880i	$0.515068\pi$
$374$	0	0
$375$	−1.44730e6	−0.531472
$376$	0	0
$377$	1.53282e6	0.555442
$378$	0	0
$379$	5.16696e6	1.84772	0.923861	−	0.382728i	$-0.125015\pi$
$379$	5.16696e6	1.84772	0.923861	+	0.382728i	$0.125015\pi$
$380$	0	0
$381$	−2.85576e6	−1.00788
$382$	0	0
$383$	−5.10331e6	−1.77769	−0.888843	−	0.458212i	$-0.848490\pi$
$383$	−5.10331e6	−1.77769	−0.888843	+	0.458212i	$0.848490\pi$
$384$	0	0
$385$	4.21865e6	1.45051
$386$	0	0
$387$	−759183.	−0.257673
$388$	0	0
$389$	−1.00400e6	−0.336403	−0.168201	−	0.985753i	$-0.553796\pi$
$389$	−1.00400e6	−0.336403	−0.168201	+	0.985753i	$0.553796\pi$
$390$	0	0
$391$	−40090.5	−0.0132617
$392$	0	0
$393$	−1.39149e6	−0.454462
$394$	0	0
$395$	3.05057e6	0.983757
$396$	0	0
$397$	3.62003e6	1.15275	0.576377	−	0.817184i	$-0.304466\pi$
$397$	3.62003e6	1.15275	0.576377	+	0.817184i	$0.304466\pi$
$398$	0	0
$399$	−2050.66	−0.000644855 0
$400$	0	0
$401$	1.18825e6	0.369016	0.184508	−	0.982831i	$-0.440931\pi$
$401$	1.18825e6	0.369016	0.184508	+	0.982831i	$0.440931\pi$
$402$	0	0
$403$	−220957.	−0.0677713
$404$	0	0
$405$	−391006.	−0.118453
$406$	0	0
$407$	6.07850e6	1.81891
$408$	0	0
$409$	−64602.3	−0.0190959	−0.00954794	−	0.999954i	$-0.503039\pi$
$409$	−64602.3	−0.0190959	−0.00954794	+	0.999954i	$0.503039\pi$
$410$	0	0
$411$	−1.95908e6	−0.572068
$412$	0	0
$413$	1.46082e6	0.421426
$414$	0	0
$415$	−5.27399e6	−1.50321
$416$	0	0
$417$	2.13222e6	0.600471
$418$	0	0
$419$	2.92057e6	0.812704	0.406352	−	0.913717i	$-0.366801\pi$
$419$	2.92057e6	0.812704	0.406352	+	0.913717i	$0.366801\pi$
$420$	0	0
$421$	−2.99592e6	−0.823807	−0.411904	−	0.911227i	$-0.635136\pi$
$421$	−2.99592e6	−0.823807	−0.411904	+	0.911227i	$0.635136\pi$
$422$	0	0
$423$	197392.	0.0536387
$424$	0	0
$425$	32331.9	0.00868280
$426$	0	0
$427$	−95663.6	−0.0253908
$428$	0	0
$429$	4.15757e6	1.09068
$430$	0	0
$431$	−5.82982e6	−1.51169	−0.755844	−	0.654752i	$-0.772773\pi$
$431$	−5.82982e6	−1.51169	−0.755844	+	0.654752i	$0.772773\pi$
$432$	0	0
$433$	−2.98624e6	−0.765429	−0.382714	−	0.923867i	$-0.625011\pi$
$433$	−2.98624e6	−0.765429	−0.382714	+	0.923867i	$0.625011\pi$
$434$	0	0
$435$	−1.30111e6	−0.329679
$436$	0	0
$437$	1244.83	0.000311823 0
$438$	0	0
$439$	5.68591e6	1.40812	0.704058	−	0.710142i	$-0.251369\pi$
$439$	5.68591e6	1.40812	0.704058	+	0.710142i	$0.251369\pi$
$440$	0	0
$441$	−601957.	−0.147390
$442$	0	0
$443$	68926.8	0.0166870	0.00834352	−	0.999965i	$-0.497344\pi$
$443$	68926.8	0.0166870	0.00834352	+	0.999965i	$0.497344\pi$
$444$	0	0
$445$	2.03125e6	0.486254
$446$	0	0
$447$	−3.90777e6	−0.925039
$448$	0	0
$449$	4.63362e6	1.08469	0.542343	−	0.840157i	$-0.317537\pi$
$449$	4.63362e6	1.08469	0.542343	+	0.840157i	$0.317537\pi$
$450$	0	0
$451$	3.06948e6	0.710596
$452$	0	0
$453$	−1.22184e6	−0.279750
$454$	0	0
$455$	−3.64622e6	−0.825685
$456$	0	0
$457$	−4.33421e6	−0.970777	−0.485388	−	0.874299i	$-0.661322\pi$
$457$	−4.33421e6	−0.970777	−0.485388	+	0.874299i	$0.661322\pi$
$458$	0	0
$459$	−55247.6	−0.0122400
$460$	0	0
$461$	2.08958e6	0.457938	0.228969	−	0.973434i	$-0.426465\pi$
$461$	2.08958e6	0.457938	0.228969	+	0.973434i	$0.426465\pi$
$462$	0	0
$463$	1.24932e6	0.270846	0.135423	−	0.990788i	$-0.456761\pi$
$463$	1.24932e6	0.270846	0.135423	+	0.990788i	$0.456761\pi$
$464$	0	0
$465$	187556.	0.0402252
$466$	0	0
$467$	157302.	0.0333765	0.0166883	−	0.999861i	$-0.494688\pi$
$467$	157302.	0.0333765	0.0166883	+	0.999861i	$0.494688\pi$
$468$	0	0
$469$	4.29309e6	0.901234
$470$	0	0
$471$	3.19859e6	0.664364
$472$	0	0
$473$	−6.85212e6	−1.40823
$474$	0	0
$475$	−1003.93	−0.000204159 0
$476$	0	0
$477$	−2.83330e6	−0.570160
$478$	0	0
$479$	−2.74855e6	−0.547350	−0.273675	−	0.961822i	$-0.588239\pi$
$479$	−2.74855e6	−0.547350	−0.273675	+	0.961822i	$0.588239\pi$
$480$	0	0
$481$	−5.25371e6	−1.03539
$482$	0	0
$483$	−460992.	−0.0899137
$484$	0	0
$485$	9.34475e6	1.80390
$486$	0	0
$487$	1.01838e7	1.94576	0.972878	−	0.231319i	$-0.0743041\pi$
$487$	1.01838e7	1.94576	0.972878	+	0.231319i	$0.0743041\pi$
$488$	0	0
$489$	−4.62909e6	−0.875434
$490$	0	0
$491$	8.81893e6	1.65087	0.825434	−	0.564499i	$-0.190931\pi$
$491$	8.81893e6	1.65087	0.825434	+	0.564499i	$0.190931\pi$
$492$	0	0
$493$	−183842.	−0.0340664
$494$	0	0
$495$	−3.52909e6	−0.647365
$496$	0	0
$497$	−3.28576e6	−0.596685
$498$	0	0
$499$	7.27705e6	1.30829	0.654145	−	0.756369i	$-0.273029\pi$
$499$	7.27705e6	1.30829	0.654145	+	0.756369i	$0.273029\pi$
$500$	0	0
$501$	1.60457e6	0.285604
$502$	0	0
$503$	2.82184e6	0.497293	0.248646	−	0.968594i	$-0.420014\pi$
$503$	2.82184e6	0.497293	0.248646	+	0.968594i	$0.420014\pi$
$504$	0	0
$505$	8.84400e6	1.54319
$506$	0	0
$507$	−251797.	−0.0435042
$508$	0	0
$509$	−4.78063e6	−0.817881	−0.408941	−	0.912561i	$-0.634102\pi$
$509$	−4.78063e6	−0.817881	−0.408941	+	0.912561i	$0.634102\pi$
$510$	0	0
$511$	4.11350e6	0.696882
$512$	0	0
$513$	1715.47	0.000287800 0
$514$	0	0
$515$	9.24424e6	1.53587
$516$	0	0
$517$	1.78159e6	0.293145
$518$	0	0
$519$	4.82329e6	0.786004
$520$	0	0
$521$	7.11976e6	1.14914	0.574568	−	0.818457i	$-0.305170\pi$
$521$	7.11976e6	1.14914	0.574568	+	0.818457i	$0.305170\pi$
$522$	0	0
$523$	−4.97980e6	−0.796082	−0.398041	−	0.917368i	$-0.630310\pi$
$523$	−4.97980e6	−0.796082	−0.398041	+	0.917368i	$0.630310\pi$
$524$	0	0
$525$	371779.	0.0588689
$526$	0	0
$527$	26500.9	0.00415656
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	−1.22204e6	−0.188083
$532$	0	0
$533$	−2.65298e6	−0.404498
$534$	0	0
$535$	−1.05798e7	−1.59806
$536$	0	0
$537$	−530021.	−0.0793153
$538$	0	0
$539$	−5.43306e6	−0.805513
$540$	0	0
$541$	2.34995e6	0.345196	0.172598	−	0.984992i	$-0.444784\pi$
$541$	2.34995e6	0.345196	0.172598	+	0.984992i	$0.444784\pi$
$542$	0	0
$543$	−7.46767e6	−1.08689
$544$	0	0
$545$	1.00904e7	1.45518
$546$	0	0
$547$	−1.27753e7	−1.82559	−0.912795	−	0.408419i	$-0.866080\pi$
$547$	−1.27753e7	−1.82559	−0.912795	+	0.408419i	$0.866080\pi$
$548$	0	0
$549$	80026.9	0.0113320
$550$	0	0
$551$	5708.40	0.000801005 0
$552$	0	0
$553$	4.95636e6	0.689207
$554$	0	0
$555$	4.45952e6	0.614548
$556$	0	0
$557$	−6.13111e6	−0.837339	−0.418670	−	0.908139i	$-0.637503\pi$
$557$	−6.13111e6	−0.837339	−0.418670	+	0.908139i	$0.637503\pi$
$558$	0	0
$559$	5.92236e6	0.801614
$560$	0	0
$561$	−498646.	−0.0668937
$562$	0	0
$563$	−1.30613e7	−1.73667	−0.868334	−	0.495979i	$-0.834809\pi$
$563$	−1.30613e7	−1.73667	−0.868334	+	0.495979i	$0.834809\pi$
$564$	0	0
$565$	−4.86956e6	−0.641754
$566$	0	0
$567$	−635281.	−0.0829866
$568$	0	0
$569$	−5.03952e6	−0.652542	−0.326271	−	0.945276i	$-0.605792\pi$
$569$	−5.03952e6	−0.652542	−0.326271	+	0.945276i	$0.605792\pi$
$570$	0	0
$571$	7.67025e6	0.984508	0.492254	−	0.870452i	$-0.336173\pi$
$571$	7.67025e6	0.984508	0.492254	+	0.870452i	$0.336173\pi$
$572$	0	0
$573$	1.34401e6	0.171008
$574$	0	0
$575$	−225685.	−0.0284664
$576$	0	0
$577$	1.25985e7	1.57536	0.787681	−	0.616083i	$-0.211281\pi$
$577$	1.25985e7	1.57536	0.787681	+	0.616083i	$0.211281\pi$
$578$	0	0
$579$	−8.21727e6	−1.01866
$580$	0	0
$581$	−8.56883e6	−1.05313
$582$	0	0
$583$	−2.55724e7	−3.11602
$584$	0	0
$585$	3.05023e6	0.368504
$586$	0	0
$587$	−5.11295e6	−0.612457	−0.306229	−	0.951958i	$-0.599067\pi$
$587$	−5.11295e6	−0.612457	−0.306229	+	0.951958i	$0.599067\pi$
$588$	0	0
$589$	−822.869	−9.77333e−5 0
$590$	0	0
$591$	8.27285e6	0.974286
$592$	0	0
$593$	5.45932e6	0.637532	0.318766	−	0.947833i	$-0.396732\pi$
$593$	5.45932e6	0.637532	0.318766	+	0.947833i	$0.396732\pi$
$594$	0	0
$595$	437315.	0.0506410
$596$	0	0
$597$	4.44357e6	0.510265
$598$	0	0
$599$	22496.6	0.00256183	0.00128091	−	0.999999i	$-0.499592\pi$
$599$	22496.6	0.00256183	0.00128091	+	0.999999i	$0.499592\pi$
$600$	0	0
$601$	1.00912e6	0.113962	0.0569808	−	0.998375i	$-0.481853\pi$
$601$	1.00912e6	0.113962	0.0569808	+	0.998375i	$0.481853\pi$
$602$	0	0
$603$	−3.59136e6	−0.402222
$604$	0	0
$605$	−2.22544e7	−2.47188
$606$	0	0
$607$	68503.9	0.00754646	0.00377323	−	0.999993i	$-0.498799\pi$
$607$	68503.9	0.00754646	0.00377323	+	0.999993i	$0.498799\pi$
$608$	0	0
$609$	−2.11396e6	−0.230969
$610$	0	0
$611$	−1.53985e6	−0.166869
$612$	0	0
$613$	7.67884e6	0.825362	0.412681	−	0.910876i	$-0.364593\pi$
$613$	7.67884e6	0.825362	0.412681	+	0.910876i	$0.364593\pi$
$614$	0	0
$615$	2.25194e6	0.240087
$616$	0	0
$617$	−1.85528e6	−0.196199	−0.0980995	−	0.995177i	$-0.531276\pi$
$617$	−1.85528e6	−0.196199	−0.0980995	+	0.995177i	$0.531276\pi$
$618$	0	0
$619$	2.88941e6	0.303098	0.151549	−	0.988450i	$-0.451574\pi$
$619$	2.88941e6	0.303098	0.151549	+	0.988450i	$0.451574\pi$
$620$	0	0
$621$	385641.	0.0401286
$622$	0	0
$623$	3.30024e6	0.340663
$624$	0	0
$625$	−1.09168e7	−1.11788
$626$	0	0
$627$	15483.3	0.00157287
$628$	0	0
$629$	630112.	0.0635026
$630$	0	0
$631$	1.22946e7	1.22925	0.614625	−	0.788819i	$-0.289307\pi$
$631$	1.22946e7	1.22925	0.614625	+	0.788819i	$0.289307\pi$
$632$	0	0
$633$	−1.20049e6	−0.119083
$634$	0	0
$635$	−1.89101e7	−1.86105
$636$	0	0
$637$	4.69585e6	0.458528
$638$	0	0
$639$	2.74869e6	0.266301
$640$	0	0
$641$	−7.87250e6	−0.756776	−0.378388	−	0.925647i	$-0.623522\pi$
$641$	−7.87250e6	−0.756776	−0.378388	+	0.925647i	$0.623522\pi$
$642$	0	0
$643$	−3.14575e6	−0.300052	−0.150026	−	0.988682i	$-0.547936\pi$
$643$	−3.14575e6	−0.300052	−0.150026	+	0.988682i	$0.547936\pi$
$644$	0	0
$645$	−5.02710e6	−0.475793
$646$	0	0
$647$	−4.14059e6	−0.388867	−0.194434	−	0.980916i	$-0.562287\pi$
$647$	−4.14059e6	−0.388867	−0.194434	+	0.980916i	$0.562287\pi$
$648$	0	0
$649$	−1.10297e7	−1.02791
$650$	0	0
$651$	304728.	0.0281813
$652$	0	0
$653$	1.09894e7	1.00854	0.504268	−	0.863547i	$-0.331762\pi$
$653$	1.09894e7	1.00854	0.504268	+	0.863547i	$0.331762\pi$
$654$	0	0
$655$	−9.21404e6	−0.839164
$656$	0	0
$657$	−3.44113e6	−0.311019
$658$	0	0
$659$	−1.78286e7	−1.59920	−0.799602	−	0.600530i	$-0.794956\pi$
$659$	−1.78286e7	−1.59920	−0.799602	+	0.600530i	$0.794956\pi$
$660$	0	0
$661$	4.39680e6	0.391411	0.195705	−	0.980663i	$-0.437300\pi$
$661$	4.39680e6	0.391411	0.195705	+	0.980663i	$0.437300\pi$
$662$	0	0
$663$	430985.	0.0380783
$664$	0	0
$665$	−13578.9	−0.00119072
$666$	0	0
$667$	1.28326e6	0.111686
$668$	0	0
$669$	3.26461e6	0.282011
$670$	0	0
$671$	722295.	0.0619311
$672$	0	0
$673$	−1.98954e7	−1.69322	−0.846612	−	0.532210i	$-0.821362\pi$
$673$	−1.98954e7	−1.69322	−0.846612	+	0.532210i	$0.821362\pi$
$674$	0	0
$675$	−311010.	−0.0262733
$676$	0	0
$677$	−796059.	−0.0667535	−0.0333767	−	0.999443i	$-0.510626\pi$
$677$	−796059.	−0.0667535	−0.0333767	+	0.999443i	$0.510626\pi$
$678$	0	0
$679$	1.51827e7	1.26379
$680$	0	0
$681$	−1.71865e6	−0.142011
$682$	0	0
$683$	9.97449e6	0.818162	0.409081	−	0.912498i	$-0.365849\pi$
$683$	9.97449e6	0.818162	0.409081	+	0.912498i	$0.365849\pi$
$684$	0	0
$685$	−1.29725e7	−1.05632
$686$	0	0
$687$	−6.71371e6	−0.542714
$688$	0	0
$689$	2.21025e7	1.77375
$690$	0	0
$691$	6.67183e6	0.531557	0.265779	−	0.964034i	$-0.414371\pi$
$691$	6.67183e6	0.531557	0.265779	+	0.964034i	$0.414371\pi$
$692$	0	0
$693$	−5.73383e6	−0.453536
$694$	0	0
$695$	1.41190e7	1.10877
$696$	0	0
$697$	318190.	0.0248087
$698$	0	0
$699$	−5.73123e6	−0.443665
$700$	0	0
$701$	5.64132e6	0.433597	0.216798	−	0.976216i	$-0.430439\pi$
$701$	5.64132e6	0.433597	0.216798	+	0.976216i	$0.430439\pi$
$702$	0	0
$703$	−19565.4	−0.00149314
$704$	0	0
$705$	1.30708e6	0.0990439
$706$	0	0
$707$	1.43691e7	1.08114
$708$	0	0
$709$	−1.86950e6	−0.139672	−0.0698361	−	0.997558i	$-0.522248\pi$
$709$	−1.86950e6	−0.139672	−0.0698361	+	0.997558i	$0.522248\pi$
$710$	0	0
$711$	−4.14622e6	−0.307594
$712$	0	0
$713$	−184982.	−0.0136272
$714$	0	0
$715$	2.75303e7	2.01394
$716$	0	0
$717$	1.29625e7	0.941652
$718$	0	0
$719$	−1.87348e7	−1.35153	−0.675766	−	0.737116i	$-0.736187\pi$
$719$	−1.87348e7	−1.35153	−0.675766	+	0.737116i	$0.736187\pi$
$720$	0	0
$721$	1.50194e7	1.07601
$722$	0	0
$723$	−1.01904e7	−0.725009
$724$	0	0
$725$	−1.03492e6	−0.0731240
$726$	0	0
$727$	−7.88165e6	−0.553071	−0.276536	−	0.961004i	$-0.589186\pi$
$727$	−7.88165e6	−0.553071	−0.276536	+	0.961004i	$0.589186\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−710308.	−0.0491647
$732$	0	0
$733$	196627.	0.0135171	0.00675854	−	0.999977i	$-0.497849\pi$
$733$	196627.	0.0135171	0.00675854	+	0.999977i	$0.497849\pi$
$734$	0	0
$735$	−3.98599e6	−0.272156
$736$	0	0
$737$	−3.24144e7	−2.19821
$738$	0	0
$739$	1.89161e7	1.27415	0.637074	−	0.770803i	$-0.280145\pi$
$739$	1.89161e7	1.27415	0.637074	+	0.770803i	$0.280145\pi$
$740$	0	0
$741$	−13382.3	−0.000895337 0
$742$	0	0
$743$	5.29482e6	0.351867	0.175934	−	0.984402i	$-0.443706\pi$
$743$	5.29482e6	0.351867	0.175934	+	0.984402i	$0.443706\pi$
$744$	0	0
$745$	−2.58762e7	−1.70808
$746$	0	0
$747$	7.16822e6	0.470013
$748$	0	0
$749$	−1.71893e7	−1.11958
$750$	0	0
$751$	1.23104e7	0.796478	0.398239	−	0.917282i	$-0.369621\pi$
$751$	1.23104e7	0.796478	0.398239	+	0.917282i	$0.369621\pi$
$752$	0	0
$753$	−5.60276e6	−0.360093
$754$	0	0
$755$	−8.09070e6	−0.516558
$756$	0	0
$757$	−7.33051e6	−0.464938	−0.232469	−	0.972604i	$-0.574680\pi$
$757$	−7.33051e6	−0.464938	−0.232469	+	0.972604i	$0.574680\pi$
$758$	0	0
$759$	3.48066e6	0.219310
$760$	0	0
$761$	3.11913e6	0.195242	0.0976208	−	0.995224i	$-0.468877\pi$
$761$	3.11913e6	0.195242	0.0976208	+	0.995224i	$0.468877\pi$
$762$	0	0
$763$	1.63942e7	1.01948
$764$	0	0
$765$	−365834.	−0.0226012
$766$	0	0
$767$	9.53311e6	0.585122
$768$	0	0
$769$	−7.07903e6	−0.431676	−0.215838	−	0.976429i	$-0.569248\pi$
$769$	−7.07903e6	−0.431676	−0.215838	+	0.976429i	$0.569248\pi$
$770$	0	0
$771$	1.89274e7	1.14671
$772$	0	0
$773$	1.91702e7	1.15393	0.576964	−	0.816770i	$-0.304237\pi$
$773$	1.91702e7	1.15393	0.576964	+	0.816770i	$0.304237\pi$
$774$	0	0
$775$	149184.	0.00892209
$776$	0	0
$777$	7.24553e6	0.430544
$778$	0	0
$779$	−9879.99	−0.000583328 0
$780$	0	0
$781$	2.48087e7	1.45538
$782$	0	0
$783$	1.76842e6	0.103082
$784$	0	0
$785$	2.11802e7	1.22675
$786$	0	0
$787$	−8.38766e6	−0.482730	−0.241365	−	0.970434i	$-0.577595\pi$
$787$	−8.38766e6	−0.482730	−0.241365	+	0.970434i	$0.577595\pi$
$788$	0	0
$789$	7.54454e6	0.431460
$790$	0	0
$791$	−7.91173e6	−0.449604
$792$	0	0
$793$	−624287.	−0.0352534
$794$	0	0
$795$	−1.87613e7	−1.05280
$796$	0	0
$797$	−1.36638e7	−0.761950	−0.380975	−	0.924585i	$-0.624412\pi$
$797$	−1.36638e7	−0.761950	−0.380975	+	0.924585i	$0.624412\pi$
$798$	0	0
$799$	184684.	0.0102344
$800$	0	0
$801$	−2.76080e6	−0.152038
$802$	0	0
$803$	−3.10584e7	−1.69977
$804$	0	0
$805$	−3.05256e6	−0.166026
$806$	0	0
$807$	−8.07779e6	−0.436625
$808$	0	0
$809$	5.16198e6	0.277297	0.138648	−	0.990342i	$-0.455724\pi$
$809$	5.16198e6	0.277297	0.138648	+	0.990342i	$0.455724\pi$
$810$	0	0
$811$	−1.20275e7	−0.642131	−0.321065	−	0.947057i	$-0.604041\pi$
$811$	−1.20275e7	−0.642131	−0.321065	+	0.947057i	$0.604041\pi$
$812$	0	0
$813$	−9.00793e6	−0.477968
$814$	0	0
$815$	−3.06526e7	−1.61649
$816$	0	0
$817$	22055.5	0.00115601
$818$	0	0
$819$	4.95580e6	0.258169
$820$	0	0
$821$	−4.25894e6	−0.220518	−0.110259	−	0.993903i	$-0.535168\pi$
$821$	−4.25894e6	−0.220518	−0.110259	+	0.993903i	$0.535168\pi$
$822$	0	0
$823$	−1.63473e7	−0.841290	−0.420645	−	0.907225i	$-0.638196\pi$
$823$	−1.63473e7	−0.841290	−0.420645	+	0.907225i	$0.638196\pi$
$824$	0	0
$825$	−2.80707e6	−0.143588
$826$	0	0
$827$	1.62566e7	0.826542	0.413271	−	0.910608i	$-0.364386\pi$
$827$	1.62566e7	0.826542	0.413271	+	0.910608i	$0.364386\pi$
$828$	0	0
$829$	3.87989e7	1.96080	0.980400	−	0.197018i	$-0.0631256\pi$
$829$	3.87989e7	1.96080	0.980400	+	0.197018i	$0.0631256\pi$
$830$	0	0
$831$	−1.88976e7	−0.949302
$832$	0	0
$833$	−563205.	−0.0281225
$834$	0	0
$835$	1.06250e7	0.527367
$836$	0	0
$837$	−254919.	−0.0125773
$838$	0	0
$839$	2.96779e7	1.45555	0.727776	−	0.685815i	$-0.240554\pi$
$839$	2.96779e7	1.45555	0.727776	+	0.685815i	$0.240554\pi$
$840$	0	0
$841$	−1.46265e7	−0.713102
$842$	0	0
$843$	−4.52904e6	−0.219502
$844$	0	0
$845$	−1.66733e6	−0.0803305
$846$	0	0
$847$	−3.61575e7	−1.73177
$848$	0	0
$849$	−1.45594e7	−0.693224
$850$	0	0
$851$	−4.39833e6	−0.208192
$852$	0	0
$853$	−3.20704e7	−1.50915	−0.754573	−	0.656216i	$-0.772156\pi$
$853$	−3.20704e7	−1.50915	−0.754573	+	0.656216i	$0.772156\pi$
$854$	0	0
$855$	11359.4	0.000531422 0
$856$	0	0
$857$	−1.10540e7	−0.514123	−0.257061	−	0.966395i	$-0.582754\pi$
$857$	−1.10540e7	−0.514123	−0.257061	+	0.966395i	$0.582754\pi$
$858$	0	0
$859$	−1.49699e6	−0.0692206	−0.0346103	−	0.999401i	$-0.511019\pi$
$859$	−1.49699e6	−0.0692206	−0.0346103	+	0.999401i	$0.511019\pi$
$860$	0	0
$861$	3.65880e6	0.168202
$862$	0	0
$863$	−7.26672e6	−0.332132	−0.166066	−	0.986115i	$-0.553107\pi$
$863$	−7.26672e6	−0.332132	−0.166066	+	0.986115i	$0.553107\pi$
$864$	0	0
$865$	3.19385e7	1.45136
$866$	0	0
$867$	1.27270e7	0.575015
$868$	0	0
$869$	−3.74223e7	−1.68105
$870$	0	0
$871$	2.80161e7	1.25130
$872$	0	0
$873$	−1.27010e7	−0.564032
$874$	0	0
$875$	−1.55708e7	−0.687529
$876$	0	0
$877$	−3.36708e6	−0.147827	−0.0739137	−	0.997265i	$-0.523549\pi$
$877$	−3.36708e6	−0.147827	−0.0739137	+	0.997265i	$0.523549\pi$
$878$	0	0
$879$	−9.40688e6	−0.410652
$880$	0	0
$881$	−1.40164e7	−0.608409	−0.304204	−	0.952607i	$-0.598391\pi$
$881$	−1.40164e7	−0.608409	−0.304204	+	0.952607i	$0.598391\pi$
$882$	0	0
$883$	4.00921e7	1.73044	0.865220	−	0.501392i	$-0.167179\pi$
$883$	4.00921e7	1.73044	0.865220	+	0.501392i	$0.167179\pi$
$884$	0	0
$885$	−8.09202e6	−0.347295
$886$	0	0
$887$	−3.57779e7	−1.52688	−0.763442	−	0.645877i	$-0.776492\pi$
$887$	−3.57779e7	−1.52688	−0.763442	+	0.645877i	$0.776492\pi$
$888$	0	0
$889$	−3.07238e7	−1.30383
$890$	0	0
$891$	4.79660e6	0.202414
$892$	0	0
$893$	−5734.56	−0.000240642 0
$894$	0	0
$895$	−3.50965e6	−0.146456
$896$	0	0
$897$	−3.00837e6	−0.124839
$898$	0	0
$899$	−848268.	−0.0350053
$900$	0	0
$901$	−2.65090e6	−0.108788
$902$	0	0
$903$	−8.16769e6	−0.333334
$904$	0	0
$905$	−4.94488e7	−2.00694
$906$	0	0
$907$	−2.64880e6	−0.106913	−0.0534565	−	0.998570i	$-0.517024\pi$
$907$	−2.64880e6	−0.106913	−0.0534565	+	0.998570i	$0.517024\pi$
$908$	0	0
$909$	−1.20204e7	−0.482515
$910$	0	0
$911$	3.49769e7	1.39632	0.698161	−	0.715941i	$-0.254002\pi$
$911$	3.49769e7	1.39632	0.698161	+	0.715941i	$0.254002\pi$
$912$	0	0
$913$	6.46979e7	2.56870
$914$	0	0
$915$	529916.	0.0209245
$916$	0	0
$917$	−1.49704e7	−0.587907
$918$	0	0
$919$	2.60166e7	1.01616	0.508079	−	0.861310i	$-0.330356\pi$
$919$	2.60166e7	1.01616	0.508079	+	0.861310i	$0.330356\pi$
$920$	0	0
$921$	−6.27785e6	−0.243872
$922$	0	0
$923$	−2.14424e7	−0.828457
$924$	0	0
$925$	3.54714e6	0.136309
$926$	0	0
$927$	−1.25644e7	−0.480223
$928$	0	0
$929$	4.76233e7	1.81042	0.905212	−	0.424960i	$-0.139712\pi$
$929$	4.76233e7	1.81042	0.905212	+	0.424960i	$0.139712\pi$
$930$	0	0
$931$	17487.9	0.000661245 0
$932$	0	0
$933$	−2.41757e7	−0.909233
$934$	0	0
$935$	−3.30189e6	−0.123519
$936$	0	0
$937$	−5.81079e6	−0.216215	−0.108108	−	0.994139i	$-0.534479\pi$
$937$	−5.81079e6	−0.216215	−0.108108	+	0.994139i	$0.534479\pi$
$938$	0	0
$939$	−1.46000e7	−0.540367
$940$	0	0
$941$	−6.47870e6	−0.238514	−0.119257	−	0.992863i	$-0.538051\pi$
$941$	−6.47870e6	−0.238514	−0.119257	+	0.992863i	$0.538051\pi$
$942$	0	0
$943$	−2.22104e6	−0.0813349
$944$	0	0
$945$	−4.20665e6	−0.153235
$946$	0	0
$947$	4.16257e6	0.150830	0.0754149	−	0.997152i	$-0.475972\pi$
$947$	4.16257e6	0.150830	0.0754149	+	0.997152i	$0.475972\pi$
$948$	0	0
$949$	2.68441e7	0.967573
$950$	0	0
$951$	−5.83354e6	−0.209161
$952$	0	0
$953$	−2.93375e7	−1.04638	−0.523191	−	0.852216i	$-0.675259\pi$
$953$	−2.93375e7	−1.04638	−0.523191	+	0.852216i	$0.675259\pi$
$954$	0	0
$955$	8.89968e6	0.315766
$956$	0	0
$957$	1.59612e7	0.563359
$958$	0	0
$959$	−2.10768e7	−0.740047
$960$	0	0
$961$	−2.85069e7	−0.995729
$962$	0	0
$963$	1.43796e7	0.499669
$964$	0	0
$965$	−5.44125e7	−1.88096
$966$	0	0
$967$	−4.34493e6	−0.149423	−0.0747113	−	0.997205i	$-0.523804\pi$
$967$	−4.34493e6	−0.149423	−0.0747113	+	0.997205i	$0.523804\pi$
$968$	0	0
$969$	1605.03	5.49129e−5 0
$970$	0	0
$971$	1.84656e7	0.628514	0.314257	−	0.949338i	$-0.398245\pi$
$971$	1.84656e7	0.628514	0.314257	+	0.949338i	$0.398245\pi$
$972$	0	0
$973$	2.29396e7	0.776790
$974$	0	0
$975$	2.42618e6	0.0817355
$976$	0	0
$977$	4.05887e7	1.36041	0.680203	−	0.733023i	$-0.261891\pi$
$977$	4.05887e7	1.36041	0.680203	+	0.733023i	$0.261891\pi$
$978$	0	0
$979$	−2.49180e7	−0.830915
$980$	0	0
$981$	−1.37145e7	−0.454995
$982$	0	0
$983$	−5.29637e7	−1.74822	−0.874108	−	0.485732i	$-0.838553\pi$
$983$	−5.29637e7	−1.74822	−0.874108	+	0.485732i	$0.838553\pi$
$984$	0	0
$985$	5.47805e7	1.79902
$986$	0	0
$987$	2.12365e6	0.0693888
$988$	0	0
$989$	4.95812e6	0.161186
$990$	0	0
$991$	−1.26953e7	−0.410636	−0.205318	−	0.978695i	$-0.565823\pi$
$991$	−1.26953e7	−0.410636	−0.205318	+	0.978695i	$0.565823\pi$
$992$	0	0
$993$	−1.49895e7	−0.482406
$994$	0	0
$995$	2.94241e7	0.942205
$996$	0	0
$997$	4.65074e7	1.48178	0.740890	−	0.671626i	$-0.234404\pi$
$997$	4.65074e7	1.48178	0.740890	+	0.671626i	$0.234404\pi$
$998$	0	0
$999$	−6.06122e6	−0.192153

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.i.1.1		3
4.3	odd	2		69.6.a.b.1.3	✓	3
12.11	even	2		207.6.a.c.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.b.1.3	✓	3	4.3	odd	2
207.6.a.c.1.1		3	12.11	even	2
1104.6.a.i.1.1		3	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)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -59.5955 q^{5} -96.8268 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} -59.5955 q^{5} -96.8268 q^{7} +81.0000 q^{9} +731.078 q^{11} -631.879 q^{13} +536.360 q^{15} +75.7854 q^{17} -2.35318 q^{19} +871.441 q^{21} -529.000 q^{23} +426.625 q^{25} -729.000 q^{27} -2425.82 q^{29} +349.683 q^{31} -6579.70 q^{33} +5770.44 q^{35} +8314.43 q^{37} +5686.91 q^{39} +4198.56 q^{41} -9372.62 q^{43} -4827.24 q^{45} +2436.94 q^{47} -7431.57 q^{49} -682.069 q^{51} -34979.0 q^{53} -43569.0 q^{55} +21.1787 q^{57} -15086.9 q^{59} +987.986 q^{61} -7842.97 q^{63} +37657.1 q^{65} -44337.8 q^{67} +4761.00 q^{69} +33934.4 q^{71} -42483.0 q^{73} -3839.62 q^{75} -70788.0 q^{77} -51187.8 q^{79} +6561.00 q^{81} +88496.5 q^{83} -4516.47 q^{85} +21832.4 q^{87} -34083.9 q^{89} +61182.8 q^{91} -3147.15 q^{93} +140.239 q^{95} -156803. q^{97} +59217.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 27 q^{3} - 56 q^{5} + 114 q^{7} + 243 q^{9}+O(q^{10}) \) 3 * q - 27 * q^3 - 56 * q^5 + 114 * q^7 + 243 * q^9	\( 3 q - 27 q^{3} - 56 q^{5} + 114 q^{7} + 243 q^{9} + 376 q^{11} - 858 q^{13} + 504 q^{15} - 2548 q^{17} + 2846 q^{19} - 1026 q^{21} - 1587 q^{23} + 753 q^{25} - 2187 q^{27} - 16370 q^{29} + 14756 q^{31} - 3384 q^{33} + 18520 q^{35} + 15874 q^{37} + 7722 q^{39} + 12606 q^{41} - 3154 q^{43} - 4536 q^{45} - 29928 q^{47} + 4471 q^{49} + 22932 q^{51} - 44084 q^{53} - 38360 q^{55} - 25614 q^{57} + 29300 q^{59} + 54010 q^{61} + 9234 q^{63} - 51216 q^{65} - 43390 q^{67} + 14283 q^{69} - 23424 q^{71} - 91402 q^{73} - 6777 q^{75} - 97208 q^{77} + 49398 q^{79} + 19683 q^{81} + 103936 q^{83} + 5888 q^{85} + 147330 q^{87} + 96112 q^{89} - 129228 q^{91} - 132804 q^{93} + 55928 q^{95} - 135318 q^{97} + 30456 q^{99}+O(q^{100}) \) 3 * q - 27 * q^3 - 56 * q^5 + 114 * q^7 + 243 * q^9 + 376 * q^11 - 858 * q^13 + 504 * q^15 - 2548 * q^17 + 2846 * q^19 - 1026 * q^21 - 1587 * q^23 + 753 * q^25 - 2187 * q^27 - 16370 * q^29 + 14756 * q^31 - 3384 * q^33 + 18520 * q^35 + 15874 * q^37 + 7722 * q^39 + 12606 * q^41 - 3154 * q^43 - 4536 * q^45 - 29928 * q^47 + 4471 * q^49 + 22932 * q^51 - 44084 * q^53 - 38360 * q^55 - 25614 * q^57 + 29300 * q^59 + 54010 * q^61 + 9234 * q^63 - 51216 * q^65 - 43390 * q^67 + 14283 * q^69 - 23424 * q^71 - 91402 * q^73 - 6777 * q^75 - 97208 * q^77 + 49398 * q^79 + 19683 * q^81 + 103936 * q^83 + 5888 * q^85 + 147330 * q^87 + 96112 * q^89 - 129228 * q^91 - 132804 * q^93 + 55928 * q^95 - 135318 * q^97 + 30456 * q^99

Introduction

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