LMFDB - Embedded newform 1104.6.a.x.1.6

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			1.6
Root			$-24.4425$ 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)

$f(q)$	$=$	$q-9.00000 q^{3} +34.8849 q^{5} +162.182 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} +34.8849 q^{5} +162.182 q^{7} +81.0000 q^{9} -438.677 q^{11} -538.859 q^{13} -313.964 q^{15} -1148.19 q^{17} -1847.30 q^{19} -1459.64 q^{21} +529.000 q^{23} -1908.04 q^{25} -729.000 q^{27} +466.567 q^{29} -810.795 q^{31} +3948.09 q^{33} +5657.71 q^{35} +5998.77 q^{37} +4849.73 q^{39} +231.761 q^{41} +10196.2 q^{43} +2825.68 q^{45} +19757.8 q^{47} +9496.00 q^{49} +10333.7 q^{51} -22937.4 q^{53} -15303.2 q^{55} +16625.7 q^{57} +32707.2 q^{59} -8509.30 q^{61} +13136.7 q^{63} -18798.0 q^{65} +2607.60 q^{67} -4761.00 q^{69} +32888.6 q^{71} +52349.2 q^{73} +17172.4 q^{75} -71145.5 q^{77} +44529.5 q^{79} +6561.00 q^{81} -486.060 q^{83} -40054.6 q^{85} -4199.10 q^{87} -7220.56 q^{89} -87393.2 q^{91} +7297.15 q^{93} -64442.9 q^{95} +29469.1 q^{97} -35532.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	34.8849	0.624040	0.312020	−	0.950075i	$-0.398994\pi$
$5$	34.8849	0.624040	0.312020	+	0.950075i	$0.398994\pi$
$6$	0	0
$7$	162.182	1.25100	0.625500	−	0.780224i	$-0.284895\pi$
$7$	162.182	1.25100	0.625500	+	0.780224i	$0.284895\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−438.677	−1.09311	−0.546554	−	0.837424i	$-0.684061\pi$
$11$	−438.677	−1.09311	−0.546554	+	0.837424i	$0.684061\pi$
$12$	0	0
$13$	−538.859	−0.884335	−0.442167	−	0.896933i	$-0.645790\pi$
$13$	−538.859	−0.884335	−0.442167	+	0.896933i	$0.645790\pi$
$14$	0	0
$15$	−313.964	−0.360290
$16$	0	0
$17$	−1148.19	−0.963589	−0.481795	−	0.876284i	$-0.660015\pi$
$17$	−1148.19	−0.963589	−0.481795	+	0.876284i	$0.660015\pi$
$18$	0	0
$19$	−1847.30	−1.17396	−0.586980	−	0.809602i	$-0.699683\pi$
$19$	−1847.30	−1.17396	−0.586980	+	0.809602i	$0.699683\pi$
$20$	0	0
$21$	−1459.64	−0.722266
$22$	0	0
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	−1908.04	−0.610574
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	466.567	0.103019	0.0515097	−	0.998672i	$-0.483597\pi$
$29$	466.567	0.103019	0.0515097	+	0.998672i	$0.483597\pi$
$30$	0	0
$31$	−810.795	−0.151533	−0.0757664	−	0.997126i	$-0.524140\pi$
$31$	−810.795	−0.151533	−0.0757664	+	0.997126i	$0.524140\pi$
$32$	0	0
$33$	3948.09	0.631106
$34$	0	0
$35$	5657.71	0.780675
$36$	0	0
$37$	5998.77	0.720374	0.360187	−	0.932880i	$-0.382713\pi$
$37$	5998.77	0.720374	0.360187	+	0.932880i	$0.382713\pi$
$38$	0	0
$39$	4849.73	0.510571
$40$	0	0
$41$	231.761	0.0215318	0.0107659	−	0.999942i	$-0.496573\pi$
$41$	231.761	0.0215318	0.0107659	+	0.999942i	$0.496573\pi$
$42$	0	0
$43$	10196.2	0.840945	0.420472	−	0.907305i	$-0.361864\pi$
$43$	10196.2	0.840945	0.420472	+	0.907305i	$0.361864\pi$
$44$	0	0
$45$	2825.68	0.208013
$46$	0	0
$47$	19757.8	1.30465	0.652324	−	0.757941i	$-0.273794\pi$
$47$	19757.8	1.30465	0.652324	+	0.757941i	$0.273794\pi$
$48$	0	0
$49$	9496.00	0.565003
$50$	0	0
$51$	10333.7	0.556328
$52$	0	0
$53$	−22937.4	−1.12164	−0.560820	−	0.827938i	$-0.689514\pi$
$53$	−22937.4	−1.12164	−0.560820	+	0.827938i	$0.689514\pi$
$54$	0	0
$55$	−15303.2	−0.682143
$56$	0	0
$57$	16625.7	0.677786
$58$	0	0
$59$	32707.2	1.22324	0.611622	−	0.791150i	$-0.290517\pi$
$59$	32707.2	1.22324	0.611622	+	0.791150i	$0.290517\pi$
$60$	0	0
$61$	−8509.30	−0.292799	−0.146399	−	0.989226i	$-0.546768\pi$
$61$	−8509.30	−0.292799	−0.146399	+	0.989226i	$0.546768\pi$
$62$	0	0
$63$	13136.7	0.417000
$64$	0	0
$65$	−18798.0	−0.551861
$66$	0	0
$67$	2607.60	0.0709667	0.0354834	−	0.999370i	$-0.488703\pi$
$67$	2607.60	0.0709667	0.0354834	+	0.999370i	$0.488703\pi$
$68$	0	0
$69$	−4761.00	−0.120386
$70$	0	0
$71$	32888.6	0.774282	0.387141	−	0.922020i	$-0.373463\pi$
$71$	32888.6	0.774282	0.387141	+	0.922020i	$0.373463\pi$
$72$	0	0
$73$	52349.2	1.14975	0.574875	−	0.818241i	$-0.305051\pi$
$73$	52349.2	1.14975	0.574875	+	0.818241i	$0.305051\pi$
$74$	0	0
$75$	17172.4	0.352515
$76$	0	0
$77$	−71145.5	−1.36748
$78$	0	0
$79$	44529.5	0.802750	0.401375	−	0.915914i	$-0.368532\pi$
$79$	44529.5	0.802750	0.401375	+	0.915914i	$0.368532\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−486.060	−0.00774452	−0.00387226	−	0.999993i	$-0.501233\pi$
$83$	−486.060	−0.00774452	−0.00387226	+	0.999993i	$0.501233\pi$
$84$	0	0
$85$	−40054.6	−0.601319
$86$	0	0
$87$	−4199.10	−0.0594782
$88$	0	0
$89$	−7220.56	−0.0966264	−0.0483132	−	0.998832i	$-0.515385\pi$
$89$	−7220.56	−0.0966264	−0.0483132	+	0.998832i	$0.515385\pi$
$90$	0	0
$91$	−87393.2	−1.10630
$92$	0	0
$93$	7297.15	0.0874875
$94$	0	0
$95$	−64442.9	−0.732598
$96$	0	0
$97$	29469.1	0.318008	0.159004	−	0.987278i	$-0.449172\pi$
$97$	29469.1	0.318008	0.159004	+	0.987278i	$0.449172\pi$
$98$	0	0
$99$	−35532.8	−0.364369
$100$	0	0
$101$	−23125.9	−0.225578	−0.112789	−	0.993619i	$-0.535978\pi$
$101$	−23125.9	−0.225578	−0.112789	+	0.993619i	$0.535978\pi$
$102$	0	0
$103$	71621.9	0.665201	0.332601	−	0.943068i	$-0.392074\pi$
$103$	71621.9	0.665201	0.332601	+	0.943068i	$0.392074\pi$
$104$	0	0
$105$	−50919.4	−0.450723
$106$	0	0
$107$	−80217.5	−0.677344	−0.338672	−	0.940904i	$-0.609978\pi$
$107$	−80217.5	−0.677344	−0.338672	+	0.940904i	$0.609978\pi$
$108$	0	0
$109$	31669.8	0.255317	0.127658	−	0.991818i	$-0.459254\pi$
$109$	31669.8	0.255317	0.127658	+	0.991818i	$0.459254\pi$
$110$	0	0
$111$	−53988.9	−0.415908
$112$	0	0
$113$	50413.7	0.371409	0.185704	−	0.982606i	$-0.440543\pi$
$113$	50413.7	0.371409	0.185704	+	0.982606i	$0.440543\pi$
$114$	0	0
$115$	18454.1	0.130121
$116$	0	0
$117$	−43647.6	−0.294778
$118$	0	0
$119$	−186216.	−1.20545
$120$	0	0
$121$	31386.3	0.194884
$122$	0	0
$123$	−2085.85	−0.0124314
$124$	0	0
$125$	−175577.	−1.00506
$126$	0	0
$127$	151472.	0.833341	0.416670	−	0.909058i	$-0.363197\pi$
$127$	151472.	0.833341	0.416670	+	0.909058i	$0.363197\pi$
$128$	0	0
$129$	−91765.8	−0.485520
$130$	0	0
$131$	3135.12	0.0159616	0.00798078	−	0.999968i	$-0.497460\pi$
$131$	3135.12	0.0159616	0.00798078	+	0.999968i	$0.497460\pi$
$132$	0	0
$133$	−299599.	−1.46862
$134$	0	0
$135$	−25431.1	−0.120097
$136$	0	0
$137$	−316.101	−0.00143888	−0.000719441	−	1.00000i	$-0.500229\pi$
$137$	−316.101	−0.00143888	−0.000719441		1.00000i	$0.500229\pi$
$138$	0	0
$139$	303932.	1.33426	0.667129	−	0.744942i	$-0.267523\pi$
$139$	303932.	1.33426	0.667129	+	0.744942i	$0.267523\pi$
$140$	0	0
$141$	−177820.	−0.753238
$142$	0	0
$143$	236385.	0.966673
$144$	0	0
$145$	16276.1	0.0642882
$146$	0	0
$147$	−85464.0	−0.326204
$148$	0	0
$149$	−151482.	−0.558980	−0.279490	−	0.960149i	$-0.590165\pi$
$149$	−151482.	−0.558980	−0.279490	+	0.960149i	$0.590165\pi$
$150$	0	0
$151$	32295.1	0.115264	0.0576321	−	0.998338i	$-0.481645\pi$
$151$	32295.1	0.115264	0.0576321	+	0.998338i	$0.481645\pi$
$152$	0	0
$153$	−93003.5	−0.321196
$154$	0	0
$155$	−28284.5	−0.0945626
$156$	0	0
$157$	−291272.	−0.943082	−0.471541	−	0.881844i	$-0.656302\pi$
$157$	−291272.	−0.943082	−0.471541	+	0.881844i	$0.656302\pi$
$158$	0	0
$159$	206436.	0.647579
$160$	0	0
$161$	85794.3	0.260852
$162$	0	0
$163$	159420.	0.469974	0.234987	−	0.971999i	$-0.424495\pi$
$163$	159420.	0.469974	0.234987	+	0.971999i	$0.424495\pi$
$164$	0	0
$165$	137729.	0.393836
$166$	0	0
$167$	492433.	1.36633	0.683165	−	0.730264i	$-0.260603\pi$
$167$	492433.	1.36633	0.683165	+	0.730264i	$0.260603\pi$
$168$	0	0
$169$	−80924.2	−0.217952
$170$	0	0
$171$	−149631.	−0.391320
$172$	0	0
$173$	−92962.7	−0.236153	−0.118077	−	0.993004i	$-0.537673\pi$
$173$	−92962.7	−0.236153	−0.118077	+	0.993004i	$0.537673\pi$
$174$	0	0
$175$	−309450.	−0.763828
$176$	0	0
$177$	−294365.	−0.706240
$178$	0	0
$179$	400588.	0.934469	0.467235	−	0.884133i	$-0.345250\pi$
$179$	400588.	0.934469	0.467235	+	0.884133i	$0.345250\pi$
$180$	0	0
$181$	404617.	0.918010	0.459005	−	0.888434i	$-0.348206\pi$
$181$	404617.	0.918010	0.459005	+	0.888434i	$0.348206\pi$
$182$	0	0
$183$	76583.7	0.169047
$184$	0	0
$185$	209267.	0.449542
$186$	0	0
$187$	503685.	1.05331
$188$	0	0
$189$	−118231.	−0.240755
$190$	0	0
$191$	−7439.23	−0.0147552	−0.00737759	−	0.999973i	$-0.502348\pi$
$191$	−7439.23	−0.0147552	−0.00737759	+	0.999973i	$0.502348\pi$
$192$	0	0
$193$	601203.	1.16179	0.580895	−	0.813978i	$-0.302703\pi$
$193$	601203.	1.16179	0.580895	+	0.813978i	$0.302703\pi$
$194$	0	0
$195$	169182.	0.318617
$196$	0	0
$197$	−257103.	−0.471999	−0.235999	−	0.971753i	$-0.575836\pi$
$197$	−257103.	−0.471999	−0.235999	+	0.971753i	$0.575836\pi$
$198$	0	0
$199$	−140342.	−0.251220	−0.125610	−	0.992080i	$-0.540089\pi$
$199$	−140342.	−0.251220	−0.125610	+	0.992080i	$0.540089\pi$
$200$	0	0
$201$	−23468.4	−0.0409726
$202$	0	0
$203$	75668.7	0.128877
$204$	0	0
$205$	8084.97	0.0134367
$206$	0	0
$207$	42849.0	0.0695048
$208$	0	0
$209$	810367.	1.28326
$210$	0	0
$211$	717281.	1.10913	0.554566	−	0.832140i	$-0.312884\pi$
$211$	717281.	1.10913	0.554566	+	0.832140i	$0.312884\pi$
$212$	0	0
$213$	−295997.	−0.447032
$214$	0	0
$215$	355694.	0.524783
$216$	0	0
$217$	−131496.	−0.189568
$218$	0	0
$219$	−471143.	−0.663808
$220$	0	0
$221$	618713.	0.852135
$222$	0	0
$223$	844640.	1.13739	0.568695	−	0.822548i	$-0.307448\pi$
$223$	844640.	1.13739	0.568695	+	0.822548i	$0.307448\pi$
$224$	0	0
$225$	−154551.	−0.203525
$226$	0	0
$227$	−75943.8	−0.0978200	−0.0489100	−	0.998803i	$-0.515575\pi$
$227$	−75943.8	−0.0978200	−0.0489100	+	0.998803i	$0.515575\pi$
$228$	0	0
$229$	502745.	0.633518	0.316759	−	0.948506i	$-0.397405\pi$
$229$	502745.	0.633518	0.316759	+	0.948506i	$0.397405\pi$
$230$	0	0
$231$	640309.	0.789514
$232$	0	0
$233$	1.27741e6	1.54149	0.770743	−	0.637146i	$-0.219885\pi$
$233$	1.27741e6	1.54149	0.770743	+	0.637146i	$0.219885\pi$
$234$	0	0
$235$	689248.	0.814153
$236$	0	0
$237$	−400766.	−0.463468
$238$	0	0
$239$	−1.08556e6	−1.22930	−0.614652	−	0.788798i	$-0.710704\pi$
$239$	−1.08556e6	−1.22930	−0.614652	+	0.788798i	$0.710704\pi$
$240$	0	0
$241$	−684231.	−0.758858	−0.379429	−	0.925221i	$-0.623879\pi$
$241$	−684231.	−0.758858	−0.379429	+	0.925221i	$0.623879\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	331267.	0.352585
$246$	0	0
$247$	995433.	1.03817
$248$	0	0
$249$	4374.54	0.00447130
$250$	0	0
$251$	754058.	0.755476	0.377738	−	0.925913i	$-0.376702\pi$
$251$	754058.	0.755476	0.377738	+	0.925913i	$0.376702\pi$
$252$	0	0
$253$	−232060.	−0.227929
$254$	0	0
$255$	360491.	0.347171
$256$	0	0
$257$	1.24451e6	1.17535	0.587674	−	0.809098i	$-0.300044\pi$
$257$	1.24451e6	1.17535	0.587674	+	0.809098i	$0.300044\pi$
$258$	0	0
$259$	972893.	0.901188
$260$	0	0
$261$	37791.9	0.0343398
$262$	0	0
$263$	1.15263e6	1.02755	0.513774	−	0.857926i	$-0.328247\pi$
$263$	1.15263e6	1.02755	0.513774	+	0.857926i	$0.328247\pi$
$264$	0	0
$265$	−800168.	−0.699949
$266$	0	0
$267$	64985.1	0.0557873
$268$	0	0
$269$	−1.44329e6	−1.21611	−0.608055	−	0.793895i	$-0.708050\pi$
$269$	−1.44329e6	−1.21611	−0.608055	+	0.793895i	$0.708050\pi$
$270$	0	0
$271$	−1.84788e6	−1.52845	−0.764224	−	0.644951i	$-0.776878\pi$
$271$	−1.84788e6	−1.52845	−0.764224	+	0.644951i	$0.776878\pi$
$272$	0	0
$273$	786539.	0.638724
$274$	0	0
$275$	837014.	0.667423
$276$	0	0
$277$	−2.24909e6	−1.76119	−0.880597	−	0.473865i	$-0.842858\pi$
$277$	−2.24909e6	−1.76119	−0.880597	+	0.473865i	$0.842858\pi$
$278$	0	0
$279$	−65674.4	−0.0505109
$280$	0	0
$281$	531673.	0.401679	0.200840	−	0.979624i	$-0.435633\pi$
$281$	531673.	0.401679	0.200840	+	0.979624i	$0.435633\pi$
$282$	0	0
$283$	−212034.	−0.157376	−0.0786880	−	0.996899i	$-0.525073\pi$
$283$	−212034.	−0.157376	−0.0786880	+	0.996899i	$0.525073\pi$
$284$	0	0
$285$	579986.	0.422966
$286$	0	0
$287$	37587.5	0.0269363
$288$	0	0
$289$	−101514.	−0.0714959
$290$	0	0
$291$	−265222.	−0.183602
$292$	0	0
$293$	−960498.	−0.653623	−0.326812	−	0.945090i	$-0.605974\pi$
$293$	−960498.	−0.653623	−0.326812	+	0.945090i	$0.605974\pi$
$294$	0	0
$295$	1.14099e6	0.763354
$296$	0	0
$297$	319795.	0.210369
$298$	0	0
$299$	−285056.	−0.184397
$300$	0	0
$301$	1.65364e6	1.05202
$302$	0	0
$303$	208133.	0.130237
$304$	0	0
$305$	−296846.	−0.182718
$306$	0	0
$307$	2.76773e6	1.67601	0.838006	−	0.545660i	$-0.183721\pi$
$307$	2.76773e6	1.67601	0.838006	+	0.545660i	$0.183721\pi$
$308$	0	0
$309$	−644597.	−0.384054
$310$	0	0
$311$	−1.75000e6	−1.02597	−0.512987	−	0.858396i	$-0.671461\pi$
$311$	−1.75000e6	−1.02597	−0.512987	+	0.858396i	$0.671461\pi$
$312$	0	0
$313$	421041.	0.242920	0.121460	−	0.992596i	$-0.461242\pi$
$313$	421041.	0.242920	0.121460	+	0.992596i	$0.461242\pi$
$314$	0	0
$315$	458274.	0.260225
$316$	0	0
$317$	2.16209e6	1.20844	0.604221	−	0.796817i	$-0.293485\pi$
$317$	2.16209e6	1.20844	0.604221	+	0.796817i	$0.293485\pi$
$318$	0	0
$319$	−204672.	−0.112611
$320$	0	0
$321$	721957.	0.391065
$322$	0	0
$323$	2.12105e6	1.13121
$324$	0	0
$325$	1.02817e6	0.539951
$326$	0	0
$327$	−285028.	−0.147407
$328$	0	0
$329$	3.20435e6	1.63211
$330$	0	0
$331$	−1.13572e6	−0.569774	−0.284887	−	0.958561i	$-0.591956\pi$
$331$	−1.13572e6	−0.569774	−0.284887	+	0.958561i	$0.591956\pi$
$332$	0	0
$333$	485901.	0.240125
$334$	0	0
$335$	90966.1	0.0442861
$336$	0	0
$337$	285708.	0.137040	0.0685200	−	0.997650i	$-0.478172\pi$
$337$	285708.	0.137040	0.0685200	+	0.997650i	$0.478172\pi$
$338$	0	0
$339$	−453723.	−0.214433
$340$	0	0
$341$	355677.	0.165642
$342$	0	0
$343$	−1.18571e6	−0.544182
$344$	0	0
$345$	−166087.	−0.0751256
$346$	0	0
$347$	2.31052e6	1.03011	0.515057	−	0.857156i	$-0.327771\pi$
$347$	2.31052e6	1.03011	0.515057	+	0.857156i	$0.327771\pi$
$348$	0	0
$349$	4.28233e6	1.88199	0.940993	−	0.338425i	$-0.109894\pi$
$349$	4.28233e6	1.88199	0.940993	+	0.338425i	$0.109894\pi$
$350$	0	0
$351$	392828.	0.170190
$352$	0	0
$353$	−26591.3	−0.0113580	−0.00567901	−	0.999984i	$-0.501808\pi$
$353$	−26591.3	−0.0113580	−0.00567901	+	0.999984i	$0.501808\pi$
$354$	0	0
$355$	1.14732e6	0.483183
$356$	0	0
$357$	1.67594e6	0.695967
$358$	0	0
$359$	−2.09512e6	−0.857972	−0.428986	−	0.903311i	$-0.641129\pi$
$359$	−2.09512e6	−0.857972	−0.428986	+	0.903311i	$0.641129\pi$
$360$	0	0
$361$	936414.	0.378181
$362$	0	0
$363$	−282476.	−0.112516
$364$	0	0
$365$	1.82620e6	0.717490
$366$	0	0
$367$	30767.9	0.0119243	0.00596215	−	0.999982i	$-0.498102\pi$
$367$	30767.9	0.0119243	0.00596215	+	0.999982i	$0.498102\pi$
$368$	0	0
$369$	18772.7	0.00717728
$370$	0	0
$371$	−3.72003e6	−1.40317
$372$	0	0
$373$	−1.13535e6	−0.422531	−0.211265	−	0.977429i	$-0.567758\pi$
$373$	−1.13535e6	−0.422531	−0.211265	+	0.977429i	$0.567758\pi$
$374$	0	0
$375$	1.58020e6	0.580273
$376$	0	0
$377$	−251414.	−0.0911036
$378$	0	0
$379$	−3.78379e6	−1.35310	−0.676549	−	0.736397i	$-0.736526\pi$
$379$	−3.78379e6	−1.35310	−0.676549	+	0.736397i	$0.736526\pi$
$380$	0	0
$381$	−1.36325e6	−0.481129
$382$	0	0
$383$	1.67884e6	0.584807	0.292404	−	0.956295i	$-0.405545\pi$
$383$	1.67884e6	0.584807	0.292404	+	0.956295i	$0.405545\pi$
$384$	0	0
$385$	−2.48190e6	−0.853362
$386$	0	0
$387$	825893.	0.280315
$388$	0	0
$389$	2.24460e6	0.752082	0.376041	−	0.926603i	$-0.377285\pi$
$389$	2.24460e6	0.752082	0.376041	+	0.926603i	$0.377285\pi$
$390$	0	0
$391$	−607393.	−0.200922
$392$	0	0
$393$	−28216.0	−0.00921541
$394$	0	0
$395$	1.55341e6	0.500948
$396$	0	0
$397$	3.73999e6	1.19095	0.595476	−	0.803373i	$-0.296964\pi$
$397$	3.73999e6	1.19095	0.595476	+	0.803373i	$0.296964\pi$
$398$	0	0
$399$	2.69639e6	0.847911
$400$	0	0
$401$	2.60609e6	0.809337	0.404668	−	0.914464i	$-0.367387\pi$
$401$	2.60609e6	0.809337	0.404668	+	0.914464i	$0.367387\pi$
$402$	0	0
$403$	436904.	0.134006
$404$	0	0
$405$	228880.	0.0693378
$406$	0	0
$407$	−2.63152e6	−0.787446
$408$	0	0
$409$	5.56109e6	1.64381	0.821906	−	0.569624i	$-0.192911\pi$
$409$	5.56109e6	1.64381	0.821906	+	0.569624i	$0.192911\pi$
$410$	0	0
$411$	2844.91	0.000830739 0
$412$	0	0
$413$	5.30452e6	1.53028
$414$	0	0
$415$	−16956.2	−0.00483289
$416$	0	0
$417$	−2.73539e6	−0.770334
$418$	0	0
$419$	−5.41356e6	−1.50643	−0.753213	−	0.657776i	$-0.771497\pi$
$419$	−5.41356e6	−1.50643	−0.753213	+	0.657776i	$0.771497\pi$
$420$	0	0
$421$	2.43193e6	0.668723	0.334362	−	0.942445i	$-0.391479\pi$
$421$	2.43193e6	0.668723	0.334362	+	0.942445i	$0.391479\pi$
$422$	0	0
$423$	1.60038e6	0.434882
$424$	0	0
$425$	2.19080e6	0.588342
$426$	0	0
$427$	−1.38006e6	−0.366292
$428$	0	0
$429$	−2.12746e6	−0.558109
$430$	0	0
$431$	−3.85512e6	−0.999642	−0.499821	−	0.866129i	$-0.666601\pi$
$431$	−3.85512e6	−0.999642	−0.499821	+	0.866129i	$0.666601\pi$
$432$	0	0
$433$	−1.99195e6	−0.510574	−0.255287	−	0.966865i	$-0.582170\pi$
$433$	−1.99195e6	−0.510574	−0.255287	+	0.966865i	$0.582170\pi$
$434$	0	0
$435$	−146485.	−0.0371168
$436$	0	0
$437$	−977221.	−0.244787
$438$	0	0
$439$	1.57771e6	0.390719	0.195360	−	0.980732i	$-0.437413\pi$
$439$	1.57771e6	0.390719	0.195360	+	0.980732i	$0.437413\pi$
$440$	0	0
$441$	769176.	0.188334
$442$	0	0
$443$	−3.34306e6	−0.809347	−0.404674	−	0.914461i	$-0.632615\pi$
$443$	−3.34306e6	−0.809347	−0.404674	+	0.914461i	$0.632615\pi$
$444$	0	0
$445$	−251889.	−0.0602988
$446$	0	0
$447$	1.36334e6	0.322727
$448$	0	0
$449$	6.67466e6	1.56247	0.781237	−	0.624234i	$-0.214589\pi$
$449$	6.67466e6	1.56247	0.781237	+	0.624234i	$0.214589\pi$
$450$	0	0
$451$	−101668.	−0.0235366
$452$	0	0
$453$	−290656.	−0.0665478
$454$	0	0
$455$	−3.04870e6	−0.690378
$456$	0	0
$457$	347285.	0.0777848	0.0388924	−	0.999243i	$-0.487617\pi$
$457$	347285.	0.0777848	0.0388924	+	0.999243i	$0.487617\pi$
$458$	0	0
$459$	837031.	0.185443
$460$	0	0
$461$	−5.19079e6	−1.13758	−0.568789	−	0.822483i	$-0.692588\pi$
$461$	−5.19079e6	−1.13758	−0.568789	+	0.822483i	$0.692588\pi$
$462$	0	0
$463$	−2.10613e6	−0.456597	−0.228299	−	0.973591i	$-0.573316\pi$
$463$	−2.10613e6	−0.456597	−0.228299	+	0.973591i	$0.573316\pi$
$464$	0	0
$465$	254561.	0.0545957
$466$	0	0
$467$	−3.30903e6	−0.702114	−0.351057	−	0.936354i	$-0.614178\pi$
$467$	−3.30903e6	−0.702114	−0.351057	+	0.936354i	$0.614178\pi$
$468$	0	0
$469$	422906.	0.0887794
$470$	0	0
$471$	2.62145e6	0.544489
$472$	0	0
$473$	−4.47284e6	−0.919243
$474$	0	0
$475$	3.52472e6	0.716789
$476$	0	0
$477$	−1.85793e6	−0.373880
$478$	0	0
$479$	−9.29734e6	−1.85148	−0.925742	−	0.378156i	$-0.876558\pi$
$479$	−9.29734e6	−1.85148	−0.925742	+	0.378156i	$0.876558\pi$
$480$	0	0
$481$	−3.23249e6	−0.637052
$482$	0	0
$483$	−772148.	−0.150603
$484$	0	0
$485$	1.02803e6	0.198450
$486$	0	0
$487$	7.69688e6	1.47059	0.735296	−	0.677746i	$-0.237043\pi$
$487$	7.69688e6	1.47059	0.735296	+	0.677746i	$0.237043\pi$
$488$	0	0
$489$	−1.43478e6	−0.271339
$490$	0	0
$491$	−1.39665e6	−0.261448	−0.130724	−	0.991419i	$-0.541730\pi$
$491$	−1.39665e6	−0.261448	−0.130724	+	0.991419i	$0.541730\pi$
$492$	0	0
$493$	−535708.	−0.0992683
$494$	0	0
$495$	−1.23956e6	−0.227381
$496$	0	0
$497$	5.33394e6	0.968628
$498$	0	0
$499$	1.03987e7	1.86950	0.934752	−	0.355301i	$-0.115622\pi$
$499$	1.03987e7	1.86950	0.934752	+	0.355301i	$0.115622\pi$
$500$	0	0
$501$	−4.43189e6	−0.788851
$502$	0	0
$503$	1.02872e7	1.81292	0.906461	−	0.422290i	$-0.138774\pi$
$503$	1.02872e7	1.81292	0.906461	+	0.422290i	$0.138774\pi$
$504$	0	0
$505$	−806746.	−0.140770
$506$	0	0
$507$	728317.	0.125835
$508$	0	0
$509$	5.31249e6	0.908874	0.454437	−	0.890779i	$-0.349841\pi$
$509$	5.31249e6	0.908874	0.454437	+	0.890779i	$0.349841\pi$
$510$	0	0
$511$	8.49010e6	1.43834
$512$	0	0
$513$	1.34668e6	0.225929
$514$	0	0
$515$	2.49853e6	0.415112
$516$	0	0
$517$	−8.66727e6	−1.42612
$518$	0	0
$519$	836665.	0.136343
$520$	0	0
$521$	−7.20082e6	−1.16222	−0.581109	−	0.813826i	$-0.697381\pi$
$521$	−7.20082e6	−1.16222	−0.581109	+	0.813826i	$0.697381\pi$
$522$	0	0
$523$	−4.62813e6	−0.739863	−0.369931	−	0.929059i	$-0.620619\pi$
$523$	−4.62813e6	−0.739863	−0.369931	+	0.929059i	$0.620619\pi$
$524$	0	0
$525$	2.78505e6	0.440996
$526$	0	0
$527$	930947.	0.146015
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	2.64928e6	0.407748
$532$	0	0
$533$	−124887.	−0.0190413
$534$	0	0
$535$	−2.79838e6	−0.422690
$536$	0	0
$537$	−3.60529e6	−0.539516
$538$	0	0
$539$	−4.16567e6	−0.617609
$540$	0	0
$541$	5.42154e6	0.796396	0.398198	−	0.917299i	$-0.369636\pi$
$541$	5.42154e6	0.796396	0.398198	+	0.917299i	$0.369636\pi$
$542$	0	0
$543$	−3.64155e6	−0.530013
$544$	0	0
$545$	1.10480e6	0.159328
$546$	0	0
$547$	−2.18699e6	−0.312520	−0.156260	−	0.987716i	$-0.549944\pi$
$547$	−2.18699e6	−0.312520	−0.156260	+	0.987716i	$0.549944\pi$
$548$	0	0
$549$	−689253.	−0.0975996
$550$	0	0
$551$	−861888.	−0.120941
$552$	0	0
$553$	7.22189e6	1.00424
$554$	0	0
$555$	−1.88340e6	−0.259543
$556$	0	0
$557$	−4.32655e6	−0.590886	−0.295443	−	0.955360i	$-0.595467\pi$
$557$	−4.32655e6	−0.590886	−0.295443	+	0.955360i	$0.595467\pi$
$558$	0	0
$559$	−5.49431e6	−0.743676
$560$	0	0
$561$	−4.53316e6	−0.608127
$562$	0	0
$563$	−1.41778e7	−1.88511	−0.942556	−	0.334049i	$-0.891585\pi$
$563$	−1.41778e7	−1.88511	−0.942556	+	0.334049i	$0.891585\pi$
$564$	0	0
$565$	1.75868e6	0.231774
$566$	0	0
$567$	1.06408e6	0.139000
$568$	0	0
$569$	−1.47247e6	−0.190662	−0.0953311	−	0.995446i	$-0.530391\pi$
$569$	−1.47247e6	−0.190662	−0.0953311	+	0.995446i	$0.530391\pi$
$570$	0	0
$571$	1.17767e7	1.51159	0.755797	−	0.654806i	$-0.227250\pi$
$571$	1.17767e7	1.51159	0.755797	+	0.654806i	$0.227250\pi$
$572$	0	0
$573$	66953.1	0.00851891
$574$	0	0
$575$	−1.00935e6	−0.127313
$576$	0	0
$577$	1.74435e6	0.218119	0.109060	−	0.994035i	$-0.465216\pi$
$577$	1.74435e6	0.218119	0.109060	+	0.994035i	$0.465216\pi$
$578$	0	0
$579$	−5.41083e6	−0.670760
$580$	0	0
$581$	−78830.2	−0.00968840
$582$	0	0
$583$	1.00621e7	1.22607
$584$	0	0
$585$	−1.52264e6	−0.183954
$586$	0	0
$587$	−1.41933e7	−1.70016	−0.850078	−	0.526657i	$-0.823445\pi$
$587$	−1.41933e7	−1.70016	−0.850078	+	0.526657i	$0.823445\pi$
$588$	0	0
$589$	1.49778e6	0.177893
$590$	0	0
$591$	2.31392e6	0.272509
$592$	0	0
$593$	1.17484e7	1.37196	0.685981	−	0.727620i	$-0.259373\pi$
$593$	1.17484e7	1.37196	0.685981	+	0.727620i	$0.259373\pi$
$594$	0	0
$595$	−6.49613e6	−0.752250
$596$	0	0
$597$	1.26307e6	0.145042
$598$	0	0
$599$	2.00229e6	0.228013	0.114007	−	0.993480i	$-0.463632\pi$
$599$	2.00229e6	0.228013	0.114007	+	0.993480i	$0.463632\pi$
$600$	0	0
$601$	1.28358e7	1.44956	0.724781	−	0.688980i	$-0.241941\pi$
$601$	1.28358e7	1.44956	0.724781	+	0.688980i	$0.241941\pi$
$602$	0	0
$603$	211216.	0.0236556
$604$	0	0
$605$	1.09491e6	0.121616
$606$	0	0
$607$	6.99644e6	0.770735	0.385368	−	0.922763i	$-0.374075\pi$
$607$	6.99644e6	0.770735	0.385368	+	0.922763i	$0.374075\pi$
$608$	0	0
$609$	−681019.	−0.0744073
$610$	0	0
$611$	−1.06466e7	−1.15374
$612$	0	0
$613$	−4.68651e6	−0.503731	−0.251865	−	0.967762i	$-0.581044\pi$
$613$	−4.68651e6	−0.503731	−0.251865	+	0.967762i	$0.581044\pi$
$614$	0	0
$615$	−72764.7	−0.00775770
$616$	0	0
$617$	−359289.	−0.0379954	−0.0189977	−	0.999820i	$-0.506048\pi$
$617$	−359289.	−0.0379954	−0.0189977	+	0.999820i	$0.506048\pi$
$618$	0	0
$619$	−4.65577e6	−0.488388	−0.244194	−	0.969726i	$-0.578523\pi$
$619$	−4.65577e6	−0.488388	−0.244194	+	0.969726i	$0.578523\pi$
$620$	0	0
$621$	−385641.	−0.0401286
$622$	0	0
$623$	−1.17105e6	−0.120880
$624$	0	0
$625$	−162366.	−0.0166263
$626$	0	0
$627$	−7.29330e6	−0.740893
$628$	0	0
$629$	−6.88774e6	−0.694145
$630$	0	0
$631$	1.19418e7	1.19398	0.596990	−	0.802249i	$-0.296363\pi$
$631$	1.19418e7	1.19398	0.596990	+	0.802249i	$0.296363\pi$
$632$	0	0
$633$	−6.45553e6	−0.640358
$634$	0	0
$635$	5.28408e6	0.520038
$636$	0	0
$637$	−5.11700e6	−0.499651
$638$	0	0
$639$	2.66398e6	0.258094
$640$	0	0
$641$	−4.55939e6	−0.438291	−0.219145	−	0.975692i	$-0.570327\pi$
$641$	−4.55939e6	−0.438291	−0.219145	+	0.975692i	$0.570327\pi$
$642$	0	0
$643$	1.50236e7	1.43300	0.716499	−	0.697588i	$-0.245743\pi$
$643$	1.50236e7	1.43300	0.716499	+	0.697588i	$0.245743\pi$
$644$	0	0
$645$	−3.20124e6	−0.302984
$646$	0	0
$647$	8.55319e6	0.803281	0.401640	−	0.915798i	$-0.368440\pi$
$647$	8.55319e6	0.803281	0.401640	+	0.915798i	$0.368440\pi$
$648$	0	0
$649$	−1.43479e7	−1.33714
$650$	0	0
$651$	1.18347e6	0.109447
$652$	0	0
$653$	−6.03849e6	−0.554173	−0.277086	−	0.960845i	$-0.589369\pi$
$653$	−6.03849e6	−0.554173	−0.277086	+	0.960845i	$0.589369\pi$
$654$	0	0
$655$	109368.	0.00996066
$656$	0	0
$657$	4.24029e6	0.383250
$658$	0	0
$659$	1.04911e7	0.941035	0.470517	−	0.882391i	$-0.344067\pi$
$659$	1.04911e7	0.941035	0.470517	+	0.882391i	$0.344067\pi$
$660$	0	0
$661$	−6.06871e6	−0.540248	−0.270124	−	0.962826i	$-0.587065\pi$
$661$	−6.06871e6	−0.540248	−0.270124	+	0.962826i	$0.587065\pi$
$662$	0	0
$663$	−5.56842e6	−0.491981
$664$	0	0
$665$	−1.04515e7	−0.916481
$666$	0	0
$667$	246814.	0.0214810
$668$	0	0
$669$	−7.60176e6	−0.656673
$670$	0	0
$671$	3.73283e6	0.320061
$672$	0	0
$673$	7.60617e6	0.647334	0.323667	−	0.946171i	$-0.395084\pi$
$673$	7.60617e6	0.647334	0.323667	+	0.946171i	$0.395084\pi$
$674$	0	0
$675$	1.39096e6	0.117505
$676$	0	0
$677$	−1.00615e7	−0.843704	−0.421852	−	0.906665i	$-0.638620\pi$
$677$	−1.00615e7	−0.843704	−0.421852	+	0.906665i	$0.638620\pi$
$678$	0	0
$679$	4.77936e6	0.397828
$680$	0	0
$681$	683494.	0.0564764
$682$	0	0
$683$	1.68072e6	0.137861	0.0689307	−	0.997621i	$-0.478041\pi$
$683$	1.68072e6	0.137861	0.0689307	+	0.997621i	$0.478041\pi$
$684$	0	0
$685$	−11027.2	−0.000897920 0
$686$	0	0
$687$	−4.52470e6	−0.365762
$688$	0	0
$689$	1.23600e7	0.991905
$690$	0	0
$691$	8.22232e6	0.655088	0.327544	−	0.944836i	$-0.393779\pi$
$691$	8.22232e6	0.655088	0.327544	+	0.944836i	$0.393779\pi$
$692$	0	0
$693$	−5.76278e6	−0.455826
$694$	0	0
$695$	1.06027e7	0.832631
$696$	0	0
$697$	−266106.	−0.0207478
$698$	0	0
$699$	−1.14967e7	−0.889978
$700$	0	0
$701$	1.06358e7	0.817479	0.408740	−	0.912651i	$-0.365968\pi$
$701$	1.06358e7	0.817479	0.408740	+	0.912651i	$0.365968\pi$
$702$	0	0
$703$	−1.10815e7	−0.845690
$704$	0	0
$705$	−6.20323e6	−0.470051
$706$	0	0
$707$	−3.75061e6	−0.282198
$708$	0	0
$709$	−7.34963e6	−0.549098	−0.274549	−	0.961573i	$-0.588529\pi$
$709$	−7.34963e6	−0.549098	−0.274549	+	0.961573i	$0.588529\pi$
$710$	0	0
$711$	3.60689e6	0.267583
$712$	0	0
$713$	−428910.	−0.0315968
$714$	0	0
$715$	8.24627e6	0.603243
$716$	0	0
$717$	9.77005e6	0.709739
$718$	0	0
$719$	−1.31051e7	−0.945407	−0.472703	−	0.881222i	$-0.656722\pi$
$719$	−1.31051e7	−0.945407	−0.472703	+	0.881222i	$0.656722\pi$
$720$	0	0
$721$	1.16158e7	0.832167
$722$	0	0
$723$	6.15808e6	0.438127
$724$	0	0
$725$	−890229.	−0.0629009
$726$	0	0
$727$	−5.88175e6	−0.412734	−0.206367	−	0.978475i	$-0.566164\pi$
$727$	−5.88175e6	−0.412734	−0.206367	+	0.978475i	$0.566164\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.17072e7	−0.810325
$732$	0	0
$733$	2.48757e7	1.71008	0.855040	−	0.518563i	$-0.173533\pi$
$733$	2.48757e7	1.71008	0.855040	+	0.518563i	$0.173533\pi$
$734$	0	0
$735$	−2.98140e6	−0.203565
$736$	0	0
$737$	−1.14390e6	−0.0775742
$738$	0	0
$739$	−6.20759e6	−0.418131	−0.209065	−	0.977902i	$-0.567042\pi$
$739$	−6.20759e6	−0.418131	−0.209065	+	0.977902i	$0.567042\pi$
$740$	0	0
$741$	−8.95890e6	−0.599390
$742$	0	0
$743$	1.81468e7	1.20595	0.602974	−	0.797760i	$-0.293982\pi$
$743$	1.81468e7	1.20595	0.602974	+	0.797760i	$0.293982\pi$
$744$	0	0
$745$	−5.28445e6	−0.348826
$746$	0	0
$747$	−39370.9	−0.00258151
$748$	0	0
$749$	−1.30098e7	−0.847358
$750$	0	0
$751$	1.05337e7	0.681524	0.340762	−	0.940150i	$-0.389315\pi$
$751$	1.05337e7	0.681524	0.340762	+	0.940150i	$0.389315\pi$
$752$	0	0
$753$	−6.78652e6	−0.436174
$754$	0	0
$755$	1.12661e6	0.0719295
$756$	0	0
$757$	−2.24620e7	−1.42465	−0.712327	−	0.701848i	$-0.752359\pi$
$757$	−2.24620e7	−1.42465	−0.712327	+	0.701848i	$0.752359\pi$
$758$	0	0
$759$	2.08854e6	0.131595
$760$	0	0
$761$	−1.06142e6	−0.0664393	−0.0332196	−	0.999448i	$-0.510576\pi$
$761$	−1.06142e6	−0.0664393	−0.0332196	+	0.999448i	$0.510576\pi$
$762$	0	0
$763$	5.13627e6	0.319401
$764$	0	0
$765$	−3.24442e6	−0.200440
$766$	0	0
$767$	−1.76246e7	−1.08176
$768$	0	0
$769$	−5.82945e6	−0.355477	−0.177739	−	0.984078i	$-0.556878\pi$
$769$	−5.82945e6	−0.355477	−0.177739	+	0.984078i	$0.556878\pi$
$770$	0	0
$771$	−1.12006e7	−0.678587
$772$	0	0
$773$	2.17921e7	1.31175	0.655874	−	0.754870i	$-0.272300\pi$
$773$	2.17921e7	1.31175	0.655874	+	0.754870i	$0.272300\pi$
$774$	0	0
$775$	1.54703e6	0.0925219
$776$	0	0
$777$	−8.75603e6	−0.520301
$778$	0	0
$779$	−428132.	−0.0252775
$780$	0	0
$781$	−1.44275e7	−0.846374
$782$	0	0
$783$	−340127.	−0.0198261
$784$	0	0
$785$	−1.01610e7	−0.588521
$786$	0	0
$787$	2.78945e7	1.60539	0.802696	−	0.596388i	$-0.203398\pi$
$787$	2.78945e7	1.60539	0.802696	+	0.596388i	$0.203398\pi$
$788$	0	0
$789$	−1.03737e7	−0.593255
$790$	0	0
$791$	8.17619e6	0.464633
$792$	0	0
$793$	4.58531e6	0.258932
$794$	0	0
$795$	7.20151e6	0.404116
$796$	0	0
$797$	−2.34028e7	−1.30503	−0.652517	−	0.757774i	$-0.726287\pi$
$797$	−2.34028e7	−1.30503	−0.652517	+	0.757774i	$0.726287\pi$
$798$	0	0
$799$	−2.26857e7	−1.25714
$800$	0	0
$801$	−584865.	−0.0322088
$802$	0	0
$803$	−2.29644e7	−1.25680
$804$	0	0
$805$	2.99293e6	0.162782
$806$	0	0
$807$	1.29896e7	0.702122
$808$	0	0
$809$	−1.11325e6	−0.0598029	−0.0299014	−	0.999553i	$-0.509519\pi$
$809$	−1.11325e6	−0.0598029	−0.0299014	+	0.999553i	$0.509519\pi$
$810$	0	0
$811$	−3.64357e7	−1.94525	−0.972623	−	0.232388i	$-0.925346\pi$
$811$	−3.64357e7	−1.94525	−0.972623	+	0.232388i	$0.925346\pi$
$812$	0	0
$813$	1.66309e7	0.882450
$814$	0	0
$815$	5.56135e6	0.293283
$816$	0	0
$817$	−1.88354e7	−0.987235
$818$	0	0
$819$	−7.07885e6	−0.368768
$820$	0	0
$821$	2.08077e7	1.07737	0.538687	−	0.842506i	$-0.318921\pi$
$821$	2.08077e7	1.07737	0.538687	+	0.842506i	$0.318921\pi$
$822$	0	0
$823$	2.37823e7	1.22392	0.611962	−	0.790887i	$-0.290380\pi$
$823$	2.37823e7	1.22392	0.611962	+	0.790887i	$0.290380\pi$
$824$	0	0
$825$	−7.53312e6	−0.385337
$826$	0	0
$827$	8.79713e6	0.447278	0.223639	−	0.974672i	$-0.428206\pi$
$827$	8.79713e6	0.447278	0.223639	+	0.974672i	$0.428206\pi$
$828$	0	0
$829$	1.24086e6	0.0627098	0.0313549	−	0.999508i	$-0.490018\pi$
$829$	1.24086e6	0.0627098	0.0313549	+	0.999508i	$0.490018\pi$
$830$	0	0
$831$	2.02418e7	1.01683
$832$	0	0
$833$	−1.09032e7	−0.544431
$834$	0	0
$835$	1.71785e7	0.852645
$836$	0	0
$837$	591069.	0.0291625
$838$	0	0
$839$	−7.46029e6	−0.365890	−0.182945	−	0.983123i	$-0.558563\pi$
$839$	−7.46029e6	−0.365890	−0.182945	+	0.983123i	$0.558563\pi$
$840$	0	0
$841$	−2.02935e7	−0.989387
$842$	0	0
$843$	−4.78506e6	−0.231910
$844$	0	0
$845$	−2.82303e6	−0.136011
$846$	0	0
$847$	5.09029e6	0.243800
$848$	0	0
$849$	1.90830e6	0.0908611
$850$	0	0
$851$	3.17335e6	0.150208
$852$	0	0
$853$	3.65353e7	1.71925	0.859627	−	0.510922i	$-0.170696\pi$
$853$	3.65353e7	1.71925	0.859627	+	0.510922i	$0.170696\pi$
$854$	0	0
$855$	−5.21987e6	−0.244199
$856$	0	0
$857$	−3.70881e7	−1.72498	−0.862488	−	0.506078i	$-0.831095\pi$
$857$	−3.70881e7	−1.72498	−0.862488	+	0.506078i	$0.831095\pi$
$858$	0	0
$859$	3.64383e6	0.168491	0.0842453	−	0.996445i	$-0.473152\pi$
$859$	3.64383e6	0.168491	0.0842453	+	0.996445i	$0.473152\pi$
$860$	0	0
$861$	−338287.	−0.0155517
$862$	0	0
$863$	−1.15452e7	−0.527684	−0.263842	−	0.964566i	$-0.584990\pi$
$863$	−1.15452e7	−0.527684	−0.263842	+	0.964566i	$0.584990\pi$
$864$	0	0
$865$	−3.24300e6	−0.147369
$866$	0	0
$867$	913626.	0.0412782
$868$	0	0
$869$	−1.95341e7	−0.877492
$870$	0	0
$871$	−1.40513e6	−0.0627583
$872$	0	0
$873$	2.38700e6	0.106003
$874$	0	0
$875$	−2.84755e7	−1.25733
$876$	0	0
$877$	−3.85078e7	−1.69063	−0.845317	−	0.534265i	$-0.820588\pi$
$877$	−3.85078e7	−1.69063	−0.845317	+	0.534265i	$0.820588\pi$
$878$	0	0
$879$	8.64448e6	0.377369
$880$	0	0
$881$	2.74476e7	1.19142	0.595710	−	0.803199i	$-0.296871\pi$
$881$	2.74476e7	1.19142	0.595710	+	0.803199i	$0.296871\pi$
$882$	0	0
$883$	−1.96630e7	−0.848687	−0.424343	−	0.905501i	$-0.639495\pi$
$883$	−1.96630e7	−0.848687	−0.424343	+	0.905501i	$0.639495\pi$
$884$	0	0
$885$	−1.02689e7	−0.440722
$886$	0	0
$887$	−8.92998e6	−0.381102	−0.190551	−	0.981677i	$-0.561028\pi$
$887$	−8.92998e6	−0.381102	−0.190551	+	0.981677i	$0.561028\pi$
$888$	0	0
$889$	2.45660e7	1.04251
$890$	0	0
$891$	−2.87816e6	−0.121456
$892$	0	0
$893$	−3.64985e7	−1.53160
$894$	0	0
$895$	1.39745e7	0.583147
$896$	0	0
$897$	2.56551e6	0.106461
$898$	0	0
$899$	−378290.	−0.0156108
$900$	0	0
$901$	2.63365e7	1.08080
$902$	0	0
$903$	−1.48828e7	−0.607385
$904$	0	0
$905$	1.41150e7	0.572875
$906$	0	0
$907$	9.01620e6	0.363919	0.181960	−	0.983306i	$-0.441756\pi$
$907$	9.01620e6	0.363919	0.181960	+	0.983306i	$0.441756\pi$
$908$	0	0
$909$	−1.87320e6	−0.0751925
$910$	0	0
$911$	7.05120e6	0.281493	0.140746	−	0.990046i	$-0.455050\pi$
$911$	7.05120e6	0.281493	0.140746	+	0.990046i	$0.455050\pi$
$912$	0	0
$913$	213223.	0.00846560
$914$	0	0
$915$	2.67162e6	0.105492
$916$	0	0
$917$	508459.	0.0199679
$918$	0	0
$919$	1.05682e7	0.412773	0.206386	−	0.978471i	$-0.433830\pi$
$919$	1.05682e7	0.412773	0.206386	+	0.978471i	$0.433830\pi$
$920$	0	0
$921$	−2.49095e7	−0.967647
$922$	0	0
$923$	−1.77223e7	−0.684725
$924$	0	0
$925$	−1.14459e7	−0.439841
$926$	0	0
$927$	5.80138e6	0.221734
$928$	0	0
$929$	−4.30045e7	−1.63484	−0.817419	−	0.576043i	$-0.804596\pi$
$929$	−4.30045e7	−1.63484	−0.817419	+	0.576043i	$0.804596\pi$
$930$	0	0
$931$	−1.75420e7	−0.663290
$932$	0	0
$933$	1.57500e7	0.592347
$934$	0	0
$935$	1.75710e7	0.657306
$936$	0	0
$937$	−3.25358e6	−0.121063	−0.0605317	−	0.998166i	$-0.519280\pi$
$937$	−3.25358e6	−0.121063	−0.0605317	+	0.998166i	$0.519280\pi$
$938$	0	0
$939$	−3.78937e6	−0.140250
$940$	0	0
$941$	−4.58567e7	−1.68822	−0.844110	−	0.536171i	$-0.819870\pi$
$941$	−4.58567e7	−1.68822	−0.844110	+	0.536171i	$0.819870\pi$
$942$	0	0
$943$	122602.	0.00448970
$944$	0	0
$945$	−4.12447e6	−0.150241
$946$	0	0
$947$	−2.29161e7	−0.830358	−0.415179	−	0.909740i	$-0.636281\pi$
$947$	−2.29161e7	−0.830358	−0.415179	+	0.909740i	$0.636281\pi$
$948$	0	0
$949$	−2.82089e7	−1.01676
$950$	0	0
$951$	−1.94588e7	−0.697694
$952$	0	0
$953$	2.17297e7	0.775037	0.387518	−	0.921862i	$-0.373332\pi$
$953$	2.17297e7	0.775037	0.387518	+	0.921862i	$0.373332\pi$
$954$	0	0
$955$	−259517.	−0.00920783
$956$	0	0
$957$	1.84205e6	0.0650161
$958$	0	0
$959$	−51266.0	−0.00180004
$960$	0	0
$961$	−2.79718e7	−0.977038
$962$	0	0
$963$	−6.49761e6	−0.225781
$964$	0	0
$965$	2.09729e7	0.725004
$966$	0	0
$967$	−5.03192e7	−1.73048	−0.865242	−	0.501355i	$-0.832835\pi$
$967$	−5.03192e7	−1.73048	−0.865242	+	0.501355i	$0.832835\pi$
$968$	0	0
$969$	−1.90895e7	−0.653107
$970$	0	0
$971$	−4.15430e7	−1.41400	−0.707001	−	0.707213i	$-0.749952\pi$
$971$	−4.15430e7	−1.41400	−0.707001	+	0.707213i	$0.749952\pi$
$972$	0	0
$973$	4.92923e7	1.66916
$974$	0	0
$975$	−9.25349e6	−0.311741
$976$	0	0
$977$	2.22784e7	0.746704	0.373352	−	0.927690i	$-0.378208\pi$
$977$	2.22784e7	0.746704	0.373352	+	0.927690i	$0.378208\pi$
$978$	0	0
$979$	3.16749e6	0.105623
$980$	0	0
$981$	2.56525e6	0.0851055
$982$	0	0
$983$	1.65096e7	0.544945	0.272472	−	0.962164i	$-0.412159\pi$
$983$	1.65096e7	0.544945	0.272472	+	0.962164i	$0.412159\pi$
$984$	0	0
$985$	−8.96900e6	−0.294546
$986$	0	0
$987$	−2.88392e7	−0.942302
$988$	0	0
$989$	5.39379e6	0.175349
$990$	0	0
$991$	2.10673e7	0.681436	0.340718	−	0.940166i	$-0.389330\pi$
$991$	2.10673e7	0.681436	0.340718	+	0.940166i	$0.389330\pi$
$992$	0	0
$993$	1.02215e7	0.328959
$994$	0	0
$995$	−4.89580e6	−0.156771
$996$	0	0
$997$	−7.05514e6	−0.224785	−0.112393	−	0.993664i	$-0.535851\pi$
$997$	−7.05514e6	−0.224785	−0.112393	+	0.993664i	$0.535851\pi$
$998$	0	0
$999$	−4.37310e6	−0.138636

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.x.1.6		7
4.3	odd	2		552.6.a.g.1.6	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.6.a.g.1.6	✓	7	4.3	odd	2
1104.6.a.x.1.6		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)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +34.8849 q^{5} +162.182 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} +34.8849 q^{5} +162.182 q^{7} +81.0000 q^{9} -438.677 q^{11} -538.859 q^{13} -313.964 q^{15} -1148.19 q^{17} -1847.30 q^{19} -1459.64 q^{21} +529.000 q^{23} -1908.04 q^{25} -729.000 q^{27} +466.567 q^{29} -810.795 q^{31} +3948.09 q^{33} +5657.71 q^{35} +5998.77 q^{37} +4849.73 q^{39} +231.761 q^{41} +10196.2 q^{43} +2825.68 q^{45} +19757.8 q^{47} +9496.00 q^{49} +10333.7 q^{51} -22937.4 q^{53} -15303.2 q^{55} +16625.7 q^{57} +32707.2 q^{59} -8509.30 q^{61} +13136.7 q^{63} -18798.0 q^{65} +2607.60 q^{67} -4761.00 q^{69} +32888.6 q^{71} +52349.2 q^{73} +17172.4 q^{75} -71145.5 q^{77} +44529.5 q^{79} +6561.00 q^{81} -486.060 q^{83} -40054.6 q^{85} -4199.10 q^{87} -7220.56 q^{89} -87393.2 q^{91} +7297.15 q^{93} -64442.9 q^{95} +29469.1 q^{97} -35532.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 63 q^{3} - 104 q^{5} + 182 q^{7} + 567 q^{9}+O(q^{10}) \) 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

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Database

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