LMFDB - Embedded newform 342.6.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,6,Mod(1,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("342.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	342.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:	$54.8512663760$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1441}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 360 $ x^2 - x - 360
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$19.4803$ of defining polynomial
Character	$\chi$	$=$	342.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000 q^{2} +16.0000 q^{4} -34.4408 q^{5} -18.9210 q^{7} +64.0000 q^{8} +O(q^{10})$	\(q+4.00000 q^{2} +16.0000 q^{4} -34.4408 q^{5} -18.9210 q^{7} +64.0000 q^{8} -137.763 q^{10} -349.480 q^{11} +711.599 q^{13} -75.6840 q^{14} +256.000 q^{16} -221.803 q^{17} +361.000 q^{19} -551.052 q^{20} -1397.92 q^{22} +662.468 q^{23} -1938.83 q^{25} +2846.39 q^{26} -302.736 q^{28} +7219.28 q^{29} +5407.76 q^{31} +1024.00 q^{32} -887.210 q^{34} +651.654 q^{35} +1979.40 q^{37} +1444.00 q^{38} -2204.21 q^{40} +3111.11 q^{41} +318.049 q^{43} -5591.68 q^{44} +2649.87 q^{46} +27240.5 q^{47} -16449.0 q^{49} -7755.34 q^{50} +11385.6 q^{52} +1114.63 q^{53} +12036.4 q^{55} -1210.94 q^{56} +28877.1 q^{58} +37904.9 q^{59} +37469.2 q^{61} +21631.0 q^{62} +4096.00 q^{64} -24508.0 q^{65} -54955.3 q^{67} -3548.84 q^{68} +2606.62 q^{70} +7177.04 q^{71} +64746.1 q^{73} +7917.59 q^{74} +5776.00 q^{76} +6612.52 q^{77} +36104.4 q^{79} -8816.83 q^{80} +12444.4 q^{82} +51782.7 q^{83} +7639.05 q^{85} +1272.20 q^{86} -22366.7 q^{88} -145254. q^{89} -13464.2 q^{91} +10599.5 q^{92} +108962. q^{94} -12433.1 q^{95} +39512.8 q^{97} -65796.0 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} + 32 q^{4} + 45 q^{5} + 114 q^{7} + 128 q^{8}+O(q^{10}) $ 2 * q + 8 * q^2 + 32 * q^4 + 45 * q^5 + 114 * q^7 + 128 * q^8	$ 2 q + 8 q^{2} + 32 q^{4} + 45 q^{5} + 114 q^{7} + 128 q^{8} + 180 q^{10} - 661 q^{11} + 1613 q^{13} + 456 q^{14} + 512 q^{16} - 64 q^{17} + 722 q^{19} + 720 q^{20} - 2644 q^{22} + 3185 q^{23} + 1247 q^{25} + 6452 q^{26} + 1824 q^{28} + 2481 q^{29} - 1180 q^{31} + 2048 q^{32} - 256 q^{34} + 11211 q^{35} + 10488 q^{37} + 2888 q^{38} + 2880 q^{40} - 16630 q^{41} + 11303 q^{43} - 10576 q^{44} + 12740 q^{46} + 12155 q^{47} - 15588 q^{49} + 4988 q^{50} + 25808 q^{52} - 20585 q^{53} - 12711 q^{55} + 7296 q^{56} + 9924 q^{58} + 78581 q^{59} + 43621 q^{61} - 4720 q^{62} + 8192 q^{64} + 47100 q^{65} + 7805 q^{67} - 1024 q^{68} + 44844 q^{70} + 62488 q^{71} + 16218 q^{73} + 41952 q^{74} + 11552 q^{76} - 34795 q^{77} + 67122 q^{79} + 11520 q^{80} - 66520 q^{82} + 10714 q^{83} + 20175 q^{85} + 45212 q^{86} - 42304 q^{88} - 128188 q^{89} + 106351 q^{91} + 50960 q^{92} + 48620 q^{94} + 16245 q^{95} + 178558 q^{97} - 62352 q^{98}+O(q^{100}) $ 2 * q + 8 * q^2 + 32 * q^4 + 45 * q^5 + 114 * q^7 + 128 * q^8 + 180 * q^10 - 661 * q^11 + 1613 * q^13 + 456 * q^14 + 512 * q^16 - 64 * q^17 + 722 * q^19 + 720 * q^20 - 2644 * q^22 + 3185 * q^23 + 1247 * q^25 + 6452 * q^26 + 1824 * q^28 + 2481 * q^29 - 1180 * q^31 + 2048 * q^32 - 256 * q^34 + 11211 * q^35 + 10488 * q^37 + 2888 * q^38 + 2880 * q^40 - 16630 * q^41 + 11303 * q^43 - 10576 * q^44 + 12740 * q^46 + 12155 * q^47 - 15588 * q^49 + 4988 * q^50 + 25808 * q^52 - 20585 * q^53 - 12711 * q^55 + 7296 * q^56 + 9924 * q^58 + 78581 * q^59 + 43621 * q^61 - 4720 * q^62 + 8192 * q^64 + 47100 * q^65 + 7805 * q^67 - 1024 * q^68 + 44844 * q^70 + 62488 * q^71 + 16218 * q^73 + 41952 * q^74 + 11552 * q^76 - 34795 * q^77 + 67122 * q^79 + 11520 * q^80 - 66520 * q^82 + 10714 * q^83 + 20175 * q^85 + 45212 * q^86 - 42304 * q^88 - 128188 * q^89 + 106351 * q^91 + 50960 * q^92 + 48620 * q^94 + 16245 * q^95 + 178558 * q^97 - 62352 * q^98

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	−34.4408	−0.616095	−0.308048	−	0.951371i	$-0.599676\pi$
$5$	−34.4408	−0.616095	−0.308048	+	0.951371i	$0.599676\pi$
$6$	0	0
$7$	−18.9210	−0.145948	−0.0729742	−	0.997334i	$-0.523249\pi$
$7$	−18.9210	−0.145948	−0.0729742	+	0.997334i	$0.523249\pi$
$8$	64.0000	0.353553
$9$	0	0
$10$	−137.763	−0.435645
$11$	−349.480	−0.870845	−0.435423	−	0.900226i	$-0.643401\pi$
$11$	−349.480	−0.870845	−0.435423	+	0.900226i	$0.643401\pi$
$12$	0	0
$13$	711.599	1.16782	0.583911	−	0.811818i	$-0.301522\pi$
$13$	711.599	1.16782	0.583911	+	0.811818i	$0.301522\pi$
$14$	−75.6840	−0.103201
$15$	0	0
$16$	256.000	0.250000
$17$	−221.803	−0.186142	−0.0930710	−	0.995659i	$-0.529668\pi$
$17$	−221.803	−0.186142	−0.0930710	+	0.995659i	$0.529668\pi$
$18$	0	0
$19$	361.000	0.229416
$19$	361.000	0.229416
$20$	−551.052	−0.308048
$21$	0	0
$22$	−1397.92	−0.615780
$23$	662.468	0.261123	0.130561	−	0.991440i	$-0.458322\pi$
$23$	662.468	0.261123	0.130561	+	0.991440i	$0.458322\pi$
$24$	0	0
$25$	−1938.83	−0.620427
$26$	2846.39	0.825775
$27$	0	0
$28$	−302.736	−0.0729742
$29$	7219.28	1.59404	0.797019	−	0.603954i	$-0.206409\pi$
$29$	7219.28	1.59404	0.797019	+	0.603954i	$0.206409\pi$
$30$	0	0
$31$	5407.76	1.01068	0.505339	−	0.862921i	$-0.331367\pi$
$31$	5407.76	1.01068	0.505339	+	0.862921i	$0.331367\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	−887.210	−0.131622
$35$	651.654	0.0899181
$36$	0	0
$37$	1979.40	0.237700	0.118850	−	0.992912i	$-0.462079\pi$
$37$	1979.40	0.237700	0.118850	+	0.992912i	$0.462079\pi$
$38$	1444.00	0.162221
$39$	0	0
$40$	−2204.21	−0.217822
$41$	3111.11	0.289039	0.144519	−	0.989502i	$-0.453836\pi$
$41$	3111.11	0.289039	0.144519	+	0.989502i	$0.453836\pi$
$42$	0	0
$43$	318.049	0.0262315	0.0131157	−	0.999914i	$-0.495825\pi$
$43$	318.049	0.0262315	0.0131157	+	0.999914i	$0.495825\pi$
$44$	−5591.68	−0.435423
$45$	0	0
$46$	2649.87	0.184642
$47$	27240.5	1.79875	0.899374	−	0.437181i	$-0.144023\pi$
$47$	27240.5	1.79875	0.899374	+	0.437181i	$0.144023\pi$
$48$	0	0
$49$	−16449.0	−0.978699
$50$	−7755.34	−0.438708
$51$	0	0
$52$	11385.6	0.583911
$53$	1114.63	0.0545057	0.0272528	−	0.999629i	$-0.491324\pi$
$53$	1114.63	0.0545057	0.0272528	+	0.999629i	$0.491324\pi$
$54$	0	0
$55$	12036.4	0.536523
$56$	−1210.94	−0.0516005
$57$	0	0
$58$	28877.1	1.12716
$59$	37904.9	1.41764	0.708820	−	0.705390i	$-0.249228\pi$
$59$	37904.9	1.41764	0.708820	+	0.705390i	$0.249228\pi$
$60$	0	0
$61$	37469.2	1.28929	0.644644	−	0.764483i	$-0.277006\pi$
$61$	37469.2	1.28929	0.644644	+	0.764483i	$0.277006\pi$
$62$	21631.0	0.714658
$63$	0	0
$64$	4096.00	0.125000
$65$	−24508.0	−0.719490
$66$	0	0
$67$	−54955.3	−1.49562	−0.747812	−	0.663911i	$-0.768895\pi$
$67$	−54955.3	−1.49562	−0.747812	+	0.663911i	$0.768895\pi$
$68$	−3548.84	−0.0930710
$69$	0	0
$70$	2606.62	0.0635817
$71$	7177.04	0.168966	0.0844830	−	0.996425i	$-0.473076\pi$
$71$	7177.04	0.168966	0.0844830	+	0.996425i	$0.473076\pi$
$72$	0	0
$73$	64746.1	1.42202	0.711011	−	0.703181i	$-0.248238\pi$
$73$	64746.1	1.42202	0.711011	+	0.703181i	$0.248238\pi$
$74$	7917.59	0.168079
$75$	0	0
$76$	5776.00	0.114708
$77$	6612.52	0.127098
$78$	0	0
$79$	36104.4	0.650866	0.325433	−	0.945565i	$-0.394490\pi$
$79$	36104.4	0.650866	0.325433	+	0.945565i	$0.394490\pi$
$80$	−8816.83	−0.154024
$81$	0	0
$82$	12444.4	0.204381
$83$	51782.7	0.825067	0.412534	−	0.910942i	$-0.364644\pi$
$83$	51782.7	0.825067	0.412534	+	0.910942i	$0.364644\pi$
$84$	0	0
$85$	7639.05	0.114681
$86$	1272.20	0.0185485
$87$	0	0
$88$	−22366.7	−0.307890
$89$	−145254.	−1.94380	−0.971900	−	0.235392i	$-0.924363\pi$
$89$	−145254.	−1.94380	−0.971900	+	0.235392i	$0.924363\pi$
$90$	0	0
$91$	−13464.2	−0.170442
$92$	10599.5	0.130561
$93$	0	0
$94$	108962.	1.27191
$95$	−12433.1	−0.141342
$96$	0	0
$97$	39512.8	0.426391	0.213196	−	0.977010i	$-0.431613\pi$
$97$	39512.8	0.426391	0.213196	+	0.977010i	$0.431613\pi$
$98$	−65796.0	−0.692045
$99$	0	0
$100$	−31021.3	−0.310213
$101$	16722.9	0.163120	0.0815602	−	0.996668i	$-0.474010\pi$
$101$	16722.9	0.163120	0.0815602	+	0.996668i	$0.474010\pi$
$102$	0	0
$103$	92664.4	0.860636	0.430318	−	0.902677i	$-0.358401\pi$
$103$	92664.4	0.860636	0.430318	+	0.902677i	$0.358401\pi$
$104$	45542.3	0.412888
$105$	0	0
$106$	4458.53	0.0385413
$107$	47156.3	0.398180	0.199090	−	0.979981i	$-0.436201\pi$
$107$	47156.3	0.398180	0.199090	+	0.979981i	$0.436201\pi$
$108$	0	0
$109$	52450.1	0.422844	0.211422	−	0.977395i	$-0.432191\pi$
$109$	52450.1	0.422844	0.211422	+	0.977395i	$0.432191\pi$
$110$	48145.5	0.379379
$111$	0	0
$112$	−4843.78	−0.0364871
$113$	−150596.	−1.10947	−0.554737	−	0.832026i	$-0.687181\pi$
$113$	−150596.	−1.10947	−0.554737	+	0.832026i	$0.687181\pi$
$114$	0	0
$115$	−22815.9	−0.160877
$116$	115508.	0.797019
$117$	0	0
$118$	151620.	1.00242
$119$	4196.73	0.0271671
$120$	0	0
$121$	−38914.6	−0.241629
$122$	149877.	0.911664
$123$	0	0
$124$	86524.2	0.505339
$125$	174402.	0.998337
$126$	0	0
$127$	352240.	1.93789	0.968945	−	0.247276i	$-0.0795353\pi$
$127$	352240.	1.93789	0.968945	+	0.247276i	$0.0795353\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	−98032.0	−0.508756
$131$	−19070.2	−0.0970907	−0.0485453	−	0.998821i	$-0.515459\pi$
$131$	−19070.2	−0.0970907	−0.0485453	+	0.998821i	$0.515459\pi$
$132$	0	0
$133$	−6830.49	−0.0334829
$134$	−219821.	−1.05757
$135$	0	0
$136$	−14195.4	−0.0658111
$137$	266677.	1.21390	0.606952	−	0.794738i	$-0.292392\pi$
$137$	266677.	1.21390	0.606952	+	0.794738i	$0.292392\pi$
$138$	0	0
$139$	−294888.	−1.29456	−0.647278	−	0.762254i	$-0.724093\pi$
$139$	−294888.	−1.29456	−0.647278	+	0.762254i	$0.724093\pi$
$140$	10426.5	0.0449590
$141$	0	0
$142$	28708.2	0.119477
$143$	−248690.	−1.01699
$144$	0	0
$145$	−248637.	−0.982079
$146$	258984.	1.00552
$147$	0	0
$148$	31670.3	0.118850
$149$	103990.	0.383729	0.191864	−	0.981421i	$-0.438547\pi$
$149$	103990.	0.383729	0.191864	+	0.981421i	$0.438547\pi$
$150$	0	0
$151$	−477343.	−1.70368	−0.851840	−	0.523802i	$-0.824513\pi$
$151$	−477343.	−1.70368	−0.851840	+	0.523802i	$0.824513\pi$
$152$	23104.0	0.0811107
$153$	0	0
$154$	26450.1	0.0898722
$155$	−186247.	−0.622674
$156$	0	0
$157$	434955.	1.40830	0.704151	−	0.710051i	$-0.251328\pi$
$157$	434955.	1.40830	0.704151	+	0.710051i	$0.251328\pi$
$158$	144417.	0.460232
$159$	0	0
$160$	−35267.3	−0.108911
$161$	−12534.6	−0.0381105
$162$	0	0
$163$	−255083.	−0.751990	−0.375995	−	0.926622i	$-0.622699\pi$
$163$	−255083.	−0.751990	−0.375995	+	0.926622i	$0.622699\pi$
$164$	49777.8	0.144519
$165$	0	0
$166$	207131.	0.583411
$167$	−510501.	−1.41646	−0.708232	−	0.705980i	$-0.750507\pi$
$167$	−510501.	−1.41646	−0.708232	+	0.705980i	$0.750507\pi$
$168$	0	0
$169$	135080.	0.363809
$170$	30556.2	0.0810918
$171$	0	0
$172$	5088.78	0.0131157
$173$	−773453.	−1.96480	−0.982401	−	0.186783i	$-0.940194\pi$
$173$	−773453.	−1.96480	−0.982401	+	0.186783i	$0.940194\pi$
$174$	0	0
$175$	36684.7	0.0905503
$176$	−89466.9	−0.217711
$177$	0	0
$178$	−581014.	−1.37447
$179$	477664.	1.11427	0.557135	−	0.830422i	$-0.311901\pi$
$179$	477664.	1.11427	0.557135	+	0.830422i	$0.311901\pi$
$180$	0	0
$181$	729114.	1.65424	0.827121	−	0.562024i	$-0.189977\pi$
$181$	729114.	1.65424	0.827121	+	0.562024i	$0.189977\pi$
$182$	−53856.7	−0.120521
$183$	0	0
$184$	42397.9	0.0923209
$185$	−68171.9	−0.146446
$186$	0	0
$187$	77515.6	0.162101
$188$	435848.	0.899374
$189$	0	0
$190$	−49732.5	−0.0999438
$191$	285584.	0.566435	0.283218	−	0.959056i	$-0.408598\pi$
$191$	285584.	0.566435	0.283218	+	0.959056i	$0.408598\pi$
$192$	0	0
$193$	−53690.2	−0.103753	−0.0518766	−	0.998654i	$-0.516520\pi$
$193$	−53690.2	−0.103753	−0.0518766	+	0.998654i	$0.516520\pi$
$194$	158051.	0.301504
$195$	0	0
$196$	−263184.	−0.489350
$197$	−14538.9	−0.0266910	−0.0133455	−	0.999911i	$-0.504248\pi$
$197$	−14538.9	−0.0266910	−0.0133455	+	0.999911i	$0.504248\pi$
$198$	0	0
$199$	−143700.	−0.257232	−0.128616	−	0.991695i	$-0.541053\pi$
$199$	−143700.	−0.257232	−0.128616	+	0.991695i	$0.541053\pi$
$200$	−124085.	−0.219354
$201$	0	0
$202$	66891.6	0.115343
$203$	−136596.	−0.232647
$204$	0	0
$205$	−107149.	−0.178075
$206$	370658.	0.608562
$207$	0	0
$208$	182169.	0.291956
$209$	−126162.	−0.199786
$210$	0	0
$211$	83653.6	0.129354	0.0646768	−	0.997906i	$-0.479398\pi$
$211$	83653.6	0.129354	0.0646768	+	0.997906i	$0.479398\pi$
$212$	17834.1	0.0272528
$213$	0	0
$214$	188625.	0.281556
$215$	−10953.8	−0.0161611
$216$	0	0
$217$	−102320.	−0.147507
$218$	209800.	0.298996
$219$	0	0
$220$	192582.	0.268262
$221$	−157834.	−0.217381
$222$	0	0
$223$	−665414.	−0.896045	−0.448022	−	0.894022i	$-0.647872\pi$
$223$	−665414.	−0.896045	−0.448022	+	0.894022i	$0.647872\pi$
$224$	−19375.1	−0.0258003
$225$	0	0
$226$	−602384.	−0.784517
$227$	−1.15382e6	−1.48619	−0.743096	−	0.669185i	$-0.766643\pi$
$227$	−1.15382e6	−1.48619	−0.743096	+	0.669185i	$0.766643\pi$
$228$	0	0
$229$	−433691.	−0.546502	−0.273251	−	0.961943i	$-0.588099\pi$
$229$	−433691.	−0.546502	−0.273251	+	0.961943i	$0.588099\pi$
$230$	−91263.5	−0.113757
$231$	0	0
$232$	462034.	0.563578
$233$	−387173.	−0.467214	−0.233607	−	0.972331i	$-0.575053\pi$
$233$	−387173.	−0.467214	−0.233607	+	0.972331i	$0.575053\pi$
$234$	0	0
$235$	−938183.	−1.10820
$236$	606479.	0.708820
$237$	0	0
$238$	16786.9	0.0192100
$239$	622463.	0.704886	0.352443	−	0.935833i	$-0.385351\pi$
$239$	622463.	0.704886	0.352443	+	0.935833i	$0.385351\pi$
$240$	0	0
$241$	−371454.	−0.411967	−0.205984	−	0.978555i	$-0.566039\pi$
$241$	−371454.	−0.411967	−0.205984	+	0.978555i	$0.566039\pi$
$242$	−155658.	−0.170857
$243$	0	0
$244$	599507.	0.644644
$245$	566516.	0.602972
$246$	0	0
$247$	256887.	0.267917
$248$	346097.	0.357329
$249$	0	0
$250$	697609.	0.705931
$251$	−376098.	−0.376805	−0.188403	−	0.982092i	$-0.560331\pi$
$251$	−376098.	−0.376805	−0.188403	+	0.982092i	$0.560331\pi$
$252$	0	0
$253$	−231519.	−0.227398
$254$	1.40896e6	1.37030
$255$	0	0
$256$	65536.0	0.0625000
$257$	−1.64382e6	−1.55247	−0.776233	−	0.630446i	$-0.782872\pi$
$257$	−1.64382e6	−1.55247	−0.776233	+	0.630446i	$0.782872\pi$
$258$	0	0
$259$	−37452.2	−0.0346919
$260$	−392128.	−0.359745
$261$	0	0
$262$	−76280.9	−0.0686535
$263$	−775299.	−0.691162	−0.345581	−	0.938389i	$-0.612318\pi$
$263$	−775299.	−0.691162	−0.345581	+	0.938389i	$0.612318\pi$
$264$	0	0
$265$	−38388.8	−0.0335807
$266$	−27321.9	−0.0236760
$267$	0	0
$268$	−879284.	−0.747812
$269$	−334851.	−0.282144	−0.141072	−	0.989999i	$-0.545055\pi$
$269$	−334851.	−0.282144	−0.141072	+	0.989999i	$0.545055\pi$
$270$	0	0
$271$	893377.	0.738944	0.369472	−	0.929242i	$-0.379539\pi$
$271$	893377.	0.738944	0.369472	+	0.929242i	$0.379539\pi$
$272$	−56781.4	−0.0465355
$273$	0	0
$274$	1.06671e6	0.858360
$275$	677584.	0.540296
$276$	0	0
$277$	−1.13462e6	−0.888483	−0.444242	−	0.895907i	$-0.646527\pi$
$277$	−1.13462e6	−0.888483	−0.444242	+	0.895907i	$0.646527\pi$
$278$	−1.17955e6	−0.915389
$279$	0	0
$280$	41705.9	0.0317908
$281$	1.98111e6	1.49672	0.748362	−	0.663290i	$-0.230841\pi$
$281$	1.98111e6	1.49672	0.748362	+	0.663290i	$0.230841\pi$
$282$	0	0
$283$	1.32192e6	0.981161	0.490580	−	0.871396i	$-0.336785\pi$
$283$	1.32192e6	0.981161	0.490580	+	0.871396i	$0.336785\pi$
$284$	114833.	0.0844830
$285$	0	0
$286$	−994759.	−0.719122
$287$	−58865.4	−0.0421847
$288$	0	0
$289$	−1.37066e6	−0.965351
$290$	−994550.	−0.694435
$291$	0	0
$292$	1.03594e6	0.711011
$293$	−1.35808e6	−0.924179	−0.462089	−	0.886833i	$-0.652900\pi$
$293$	−1.35808e6	−0.924179	−0.462089	+	0.886833i	$0.652900\pi$
$294$	0	0
$295$	−1.30547e6	−0.873401
$296$	126681.	0.0840395
$297$	0	0
$298$	415959.	0.271337
$299$	471411.	0.304945
$300$	0	0
$301$	−6017.81	−0.00382844
$302$	−1.90937e6	−1.20468
$303$	0	0
$304$	92416.0	0.0573539
$305$	−1.29047e6	−0.794324
$306$	0	0
$307$	−1.23842e6	−0.749930	−0.374965	−	0.927039i	$-0.622345\pi$
$307$	−1.23842e6	−0.749930	−0.374965	+	0.927039i	$0.622345\pi$
$308$	105800.	0.0635492
$309$	0	0
$310$	−744989.	−0.440297
$311$	1.78565e6	1.04688	0.523439	−	0.852063i	$-0.324649\pi$
$311$	1.78565e6	1.04688	0.523439	+	0.852063i	$0.324649\pi$
$312$	0	0
$313$	−3.01608e6	−1.74013	−0.870066	−	0.492935i	$-0.835924\pi$
$313$	−3.01608e6	−1.74013	−0.870066	+	0.492935i	$0.835924\pi$
$314$	1.73982e6	0.995819
$315$	0	0
$316$	577670.	0.325433
$317$	−515839.	−0.288314	−0.144157	−	0.989555i	$-0.546047\pi$
$317$	−515839.	−0.288314	−0.144157	+	0.989555i	$0.546047\pi$
$318$	0	0
$319$	−2.52300e6	−1.38816
$320$	−141069.	−0.0770119
$321$	0	0
$322$	−50138.2	−0.0269482
$323$	−80070.7	−0.0427039
$324$	0	0
$325$	−1.37967e6	−0.724548
$326$	−1.02033e6	−0.531737
$327$	0	0
$328$	199111.	0.102191
$329$	−515417.	−0.262524
$330$	0	0
$331$	−2.25449e6	−1.13104	−0.565521	−	0.824734i	$-0.691325\pi$
$331$	−2.25449e6	−1.13104	−0.565521	+	0.824734i	$0.691325\pi$
$332$	828523.	0.412534
$333$	0	0
$334$	−2.04200e6	−1.00159
$335$	1.89270e6	0.921446
$336$	0	0
$337$	2.40261e6	1.15242	0.576208	−	0.817303i	$-0.304532\pi$
$337$	2.40261e6	1.15242	0.576208	+	0.817303i	$0.304532\pi$
$338$	540319.	0.257252
$339$	0	0
$340$	122225.	0.0573406
$341$	−1.88991e6	−0.880145
$342$	0	0
$343$	629237.	0.288788
$344$	20355.1	0.00927423
$345$	0	0
$346$	−3.09381e6	−1.38933
$347$	684693.	0.305262	0.152631	−	0.988283i	$-0.451225\pi$
$347$	684693.	0.305262	0.152631	+	0.988283i	$0.451225\pi$
$348$	0	0
$349$	2.19857e6	0.966220	0.483110	−	0.875560i	$-0.339507\pi$
$349$	2.19857e6	0.966220	0.483110	+	0.875560i	$0.339507\pi$
$350$	146739.	0.0640287
$351$	0	0
$352$	−357868.	−0.153945
$353$	−2.03446e6	−0.868987	−0.434493	−	0.900675i	$-0.643073\pi$
$353$	−2.03446e6	−0.868987	−0.434493	+	0.900675i	$0.643073\pi$
$354$	0	0
$355$	−247183.	−0.104099
$356$	−2.32406e6	−0.971900
$357$	0	0
$358$	1.91066e6	0.787907
$359$	−2.30592e6	−0.944298	−0.472149	−	0.881519i	$-0.656522\pi$
$359$	−2.30592e6	−0.944298	−0.472149	+	0.881519i	$0.656522\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	2.91646e6	1.16973
$363$	0	0
$364$	−215427.	−0.0852209
$365$	−2.22990e6	−0.876101
$366$	0	0
$367$	2.64475e6	1.02499	0.512495	−	0.858690i	$-0.328721\pi$
$367$	2.64475e6	1.02499	0.512495	+	0.858690i	$0.328721\pi$
$368$	169592.	0.0652807
$369$	0	0
$370$	−272688.	−0.103553
$371$	−21090.0	−0.00795502
$372$	0	0
$373$	173405.	0.0645342	0.0322671	−	0.999479i	$-0.489727\pi$
$373$	173405.	0.0645342	0.0322671	+	0.999479i	$0.489727\pi$
$374$	310062.	0.114623
$375$	0	0
$376$	1.74339e6	0.635953
$377$	5.13723e6	1.86155
$378$	0	0
$379$	2.23408e6	0.798915	0.399458	−	0.916752i	$-0.369198\pi$
$379$	2.23408e6	0.798915	0.399458	+	0.916752i	$0.369198\pi$
$380$	−198930.	−0.0706709
$381$	0	0
$382$	1.14234e6	0.400530
$383$	4.44976e6	1.55003	0.775014	−	0.631944i	$-0.217743\pi$
$383$	4.44976e6	1.55003	0.775014	+	0.631944i	$0.217743\pi$
$384$	0	0
$385$	−227740.	−0.0783047
$386$	−214761.	−0.0733646
$387$	0	0
$388$	632204.	0.213196
$389$	−2.43083e6	−0.814480	−0.407240	−	0.913321i	$-0.633509\pi$
$389$	−2.43083e6	−0.814480	−0.407240	+	0.913321i	$0.633509\pi$
$390$	0	0
$391$	−146937.	−0.0486059
$392$	−1.05274e6	−0.346022
$393$	0	0
$394$	−58155.5	−0.0188734
$395$	−1.24346e6	−0.400996
$396$	0	0
$397$	2.61205e6	0.831775	0.415887	−	0.909416i	$-0.363471\pi$
$397$	2.61205e6	0.831775	0.415887	+	0.909416i	$0.363471\pi$
$398$	−574800.	−0.181890
$399$	0	0
$400$	−496342.	−0.155107
$401$	−1.42660e6	−0.443038	−0.221519	−	0.975156i	$-0.571101\pi$
$401$	−1.42660e6	−0.443038	−0.221519	+	0.975156i	$0.571101\pi$
$402$	0	0
$403$	3.84816e6	1.18029
$404$	267566.	0.0815602
$405$	0	0
$406$	−546384.	−0.164507
$407$	−691760.	−0.207000
$408$	0	0
$409$	4.73321e6	1.39910	0.699548	−	0.714586i	$-0.253385\pi$
$409$	4.73321e6	1.39910	0.699548	+	0.714586i	$0.253385\pi$
$410$	−428596.	−0.125918
$411$	0	0
$412$	1.48263e6	0.430318
$413$	−717200.	−0.206902
$414$	0	0
$415$	−1.78344e6	−0.508320
$416$	728677.	0.206444
$417$	0	0
$418$	−504649.	−0.141270
$419$	357759.	0.0995531	0.0497766	−	0.998760i	$-0.484149\pi$
$419$	357759.	0.0995531	0.0497766	+	0.998760i	$0.484149\pi$
$420$	0	0
$421$	652504.	0.179423	0.0897115	−	0.995968i	$-0.471405\pi$
$421$	652504.	0.179423	0.0897115	+	0.995968i	$0.471405\pi$
$422$	334614.	0.0914668
$423$	0	0
$424$	71336.4	0.0192707
$425$	430038.	0.115487
$426$	0	0
$427$	−708955.	−0.188169
$428$	754500.	0.199090
$429$	0	0
$430$	−43815.4	−0.0114276
$431$	6.25148e6	1.62103	0.810513	−	0.585721i	$-0.199188\pi$
$431$	6.25148e6	1.62103	0.810513	+	0.585721i	$0.199188\pi$
$432$	0	0
$433$	4.45832e6	1.14275	0.571375	−	0.820689i	$-0.306410\pi$
$433$	4.45832e6	1.14275	0.571375	+	0.820689i	$0.306410\pi$
$434$	−409281.	−0.104303
$435$	0	0
$436$	839201.	0.211422
$437$	239151.	0.0599057
$438$	0	0
$439$	5.00652e6	1.23986	0.619932	−	0.784655i	$-0.287160\pi$
$439$	5.00652e6	1.23986	0.619932	+	0.784655i	$0.287160\pi$
$440$	770327.	0.189690
$441$	0	0
$442$	−631338.	−0.153711
$443$	−715908.	−0.173320	−0.0866599	−	0.996238i	$-0.527619\pi$
$443$	−715908.	−0.173320	−0.0866599	+	0.996238i	$0.527619\pi$
$444$	0	0
$445$	5.00264e6	1.19757
$446$	−2.66166e6	−0.633599
$447$	0	0
$448$	−77500.5	−0.0182435
$449$	4.83183e6	1.13109	0.565544	−	0.824718i	$-0.308666\pi$
$449$	4.83183e6	1.13109	0.565544	+	0.824718i	$0.308666\pi$
$450$	0	0
$451$	−1.08727e6	−0.251708
$452$	−2.40954e6	−0.554737
$453$	0	0
$454$	−4.61530e6	−1.05090
$455$	463716.	0.105008
$456$	0	0
$457$	−6.44410e6	−1.44335	−0.721675	−	0.692232i	$-0.756628\pi$
$457$	−6.44410e6	−1.44335	−0.721675	+	0.692232i	$0.756628\pi$
$458$	−1.73477e6	−0.386436
$459$	0	0
$460$	−365054.	−0.0804383
$461$	4.10108e6	0.898764	0.449382	−	0.893340i	$-0.351644\pi$
$461$	4.10108e6	0.898764	0.449382	+	0.893340i	$0.351644\pi$
$462$	0	0
$463$	8.98683e6	1.94829	0.974146	−	0.225919i	$-0.0725386\pi$
$463$	8.98683e6	1.94829	0.974146	+	0.225919i	$0.0725386\pi$
$464$	1.84814e6	0.398510
$465$	0	0
$466$	−1.54869e6	−0.330370
$467$	8.84409e6	1.87655	0.938276	−	0.345886i	$-0.112422\pi$
$467$	8.84409e6	1.87655	0.938276	+	0.345886i	$0.112422\pi$
$468$	0	0
$469$	1.03981e6	0.218284
$470$	−3.75273e6	−0.783615
$471$	0	0
$472$	2.42592e6	0.501211
$473$	−111152.	−0.0228436
$474$	0	0
$475$	−699919.	−0.142336
$476$	67147.7	0.0135836
$477$	0	0
$478$	2.48985e6	0.498430
$479$	−2.89320e6	−0.576155	−0.288077	−	0.957607i	$-0.593016\pi$
$479$	−2.89320e6	−0.576155	−0.288077	+	0.957607i	$0.593016\pi$
$480$	0	0
$481$	1.40854e6	0.277591
$482$	−1.48582e6	−0.291305
$483$	0	0
$484$	−622633.	−0.120814
$485$	−1.36085e6	−0.262697
$486$	0	0
$487$	−3.18166e6	−0.607900	−0.303950	−	0.952688i	$-0.598306\pi$
$487$	−3.18166e6	−0.607900	−0.303950	+	0.952688i	$0.598306\pi$
$488$	2.39803e6	0.455832
$489$	0	0
$490$	2.26606e6	0.426365
$491$	4.45509e6	0.833975	0.416987	−	0.908912i	$-0.363086\pi$
$491$	4.45509e6	0.833975	0.416987	+	0.908912i	$0.363086\pi$
$492$	0	0
$493$	−1.60125e6	−0.296717
$494$	1.02755e6	0.189446
$495$	0	0
$496$	1.38439e6	0.252670
$497$	−135797.	−0.0246603
$498$	0	0
$499$	9.31472e6	1.67463	0.837315	−	0.546721i	$-0.184124\pi$
$499$	9.31472e6	1.67463	0.837315	+	0.546721i	$0.184124\pi$
$500$	2.79044e6	0.499168
$501$	0	0
$502$	−1.50439e6	−0.266442
$503$	2.04835e6	0.360980	0.180490	−	0.983577i	$-0.442232\pi$
$503$	2.04835e6	0.360980	0.180490	+	0.983577i	$0.442232\pi$
$504$	0	0
$505$	−575949.	−0.100498
$506$	−926077.	−0.160794
$507$	0	0
$508$	5.63584e6	0.968945
$509$	166211.	0.0284358	0.0142179	−	0.999899i	$-0.495474\pi$
$509$	166211.	0.0284358	0.0142179	+	0.999899i	$0.495474\pi$
$510$	0	0
$511$	−1.22506e6	−0.207542
$512$	262144.	0.0441942
$513$	0	0
$514$	−6.57529e6	−1.09776
$515$	−3.19143e6	−0.530234
$516$	0	0
$517$	−9.52001e6	−1.56643
$518$	−149809.	−0.0245309
$519$	0	0
$520$	−1.56851e6	−0.254378
$521$	−2.64602e6	−0.427069	−0.213535	−	0.976935i	$-0.568498\pi$
$521$	−2.64602e6	−0.427069	−0.213535	+	0.976935i	$0.568498\pi$
$522$	0	0
$523$	−7.02287e6	−1.12269	−0.561346	−	0.827581i	$-0.689716\pi$
$523$	−7.02287e6	−1.12269	−0.561346	+	0.827581i	$0.689716\pi$
$524$	−305124.	−0.0485453
$525$	0	0
$526$	−3.10120e6	−0.488726
$527$	−1.19945e6	−0.188130
$528$	0	0
$529$	−5.99748e6	−0.931815
$530$	−153555.	−0.0237451
$531$	0	0
$532$	−109288.	−0.0167414
$533$	2.21386e6	0.337546
$534$	0	0
$535$	−1.62410e6	−0.245317
$536$	−3.51714e6	−0.528783
$537$	0	0
$538$	−1.33940e6	−0.199506
$539$	5.74860e6	0.852295
$540$	0	0
$541$	4.35066e6	0.639090	0.319545	−	0.947571i	$-0.396470\pi$
$541$	4.35066e6	0.639090	0.319545	+	0.947571i	$0.396470\pi$
$542$	3.57351e6	0.522512
$543$	0	0
$544$	−227126.	−0.0329056
$545$	−1.80642e6	−0.260512
$546$	0	0
$547$	9.77794e6	1.39727	0.698633	−	0.715480i	$-0.253792\pi$
$547$	9.77794e6	1.39727	0.698633	+	0.715480i	$0.253792\pi$
$548$	4.26684e6	0.606952
$549$	0	0
$550$	2.71034e6	0.382047
$551$	2.60616e6	0.365698
$552$	0	0
$553$	−683131.	−0.0949929
$554$	−4.53846e6	−0.628252
$555$	0	0
$556$	−4.71822e6	−0.647278
$557$	−6.69323e6	−0.914108	−0.457054	−	0.889439i	$-0.651095\pi$
$557$	−6.69323e6	−0.914108	−0.457054	+	0.889439i	$0.651095\pi$
$558$	0	0
$559$	226323.	0.0306337
$560$	166823.	0.0224795
$561$	0	0
$562$	7.92442e6	1.05834
$563$	−7.54034e6	−1.00258	−0.501291	−	0.865279i	$-0.667141\pi$
$563$	−7.54034e6	−1.00258	−0.501291	+	0.865279i	$0.667141\pi$
$564$	0	0
$565$	5.18664e6	0.683542
$566$	5.28769e6	0.693785
$567$	0	0
$568$	459331.	0.0597385
$569$	−1.34726e6	−0.174450	−0.0872252	−	0.996189i	$-0.527800\pi$
$569$	−1.34726e6	−0.174450	−0.0872252	+	0.996189i	$0.527800\pi$
$570$	0	0
$571$	2.05762e6	0.264104	0.132052	−	0.991243i	$-0.457843\pi$
$571$	2.05762e6	0.264104	0.132052	+	0.991243i	$0.457843\pi$
$572$	−3.97904e6	−0.508496
$573$	0	0
$574$	−235462.	−0.0298291
$575$	−1.28441e6	−0.162008
$576$	0	0
$577$	7.88561e6	0.986043	0.493021	−	0.870017i	$-0.335892\pi$
$577$	7.88561e6	0.986043	0.493021	+	0.870017i	$0.335892\pi$
$578$	−5.48264e6	−0.682606
$579$	0	0
$580$	−3.97820e6	−0.491040
$581$	−979781.	−0.120417
$582$	0	0
$583$	−389542.	−0.0474660
$584$	4.14375e6	0.502761
$585$	0	0
$586$	−5.43232e6	−0.653493
$587$	−1.90439e6	−0.228119	−0.114059	−	0.993474i	$-0.536385\pi$
$587$	−1.90439e6	−0.228119	−0.114059	+	0.993474i	$0.536385\pi$
$588$	0	0
$589$	1.95220e6	0.231866
$590$	−5.22190e6	−0.617588
$591$	0	0
$592$	506726.	0.0594249
$593$	1.35835e7	1.58626	0.793132	−	0.609050i	$-0.208449\pi$
$593$	1.35835e7	1.58626	0.793132	+	0.609050i	$0.208449\pi$
$594$	0	0
$595$	−144539.	−0.0167375
$596$	1.66384e6	0.191864
$597$	0	0
$598$	1.88564e6	0.215629
$599$	1.04361e7	1.18842	0.594212	−	0.804309i	$-0.297464\pi$
$599$	1.04361e7	1.18842	0.594212	+	0.804309i	$0.297464\pi$
$600$	0	0
$601$	−1.17196e7	−1.32351	−0.661753	−	0.749722i	$-0.730187\pi$
$601$	−1.17196e7	−1.32351	−0.661753	+	0.749722i	$0.730187\pi$
$602$	−24071.2	−0.00270712
$603$	0	0
$604$	−7.63749e6	−0.851840
$605$	1.34025e6	0.148866
$606$	0	0
$607$	−7.53524e6	−0.830091	−0.415045	−	0.909801i	$-0.636234\pi$
$607$	−7.53524e6	−0.830091	−0.415045	+	0.909801i	$0.636234\pi$
$608$	369664.	0.0405554
$609$	0	0
$610$	−5.16187e6	−0.561672
$611$	1.93843e7	2.10062
$612$	0	0
$613$	−4.37292e6	−0.470025	−0.235012	−	0.971992i	$-0.575513\pi$
$613$	−4.37292e6	−0.470025	−0.235012	+	0.971992i	$0.575513\pi$
$614$	−4.95366e6	−0.530280
$615$	0	0
$616$	423201.	0.0449361
$617$	7.50595e6	0.793767	0.396883	−	0.917869i	$-0.370092\pi$
$617$	7.50595e6	0.793767	0.396883	+	0.917869i	$0.370092\pi$
$618$	0	0
$619$	−1.30877e7	−1.37289	−0.686447	−	0.727180i	$-0.740831\pi$
$619$	−1.30877e7	−1.37289	−0.686447	+	0.727180i	$0.740831\pi$
$620$	−2.97996e6	−0.311337
$621$	0	0
$622$	7.14261e6	0.740254
$623$	2.74834e6	0.283695
$624$	0	0
$625$	52309.5	0.00535649
$626$	−1.20643e7	−1.23046
$627$	0	0
$628$	6.95929e6	0.704151
$629$	−439035.	−0.0442459
$630$	0	0
$631$	8.92096e6	0.891945	0.445972	−	0.895047i	$-0.352858\pi$
$631$	8.92096e6	0.891945	0.445972	+	0.895047i	$0.352858\pi$
$632$	2.31068e6	0.230116
$633$	0	0
$634$	−2.06336e6	−0.203869
$635$	−1.21314e7	−1.19392
$636$	0	0
$637$	−1.17051e7	−1.14295
$638$	−1.00920e7	−0.981578
$639$	0	0
$640$	−564277.	−0.0544556
$641$	−1.45438e7	−1.39808	−0.699041	−	0.715082i	$-0.746389\pi$
$641$	−1.45438e7	−1.39808	−0.699041	+	0.715082i	$0.746389\pi$
$642$	0	0
$643$	1.45757e7	1.39028	0.695140	−	0.718874i	$-0.255342\pi$
$643$	1.45757e7	1.39028	0.695140	+	0.718874i	$0.255342\pi$
$644$	−200553.	−0.0190552
$645$	0	0
$646$	−320283.	−0.0301962
$647$	2.24522e6	0.210862	0.105431	−	0.994427i	$-0.466378\pi$
$647$	2.24522e6	0.210862	0.105431	+	0.994427i	$0.466378\pi$
$648$	0	0
$649$	−1.32470e7	−1.23454
$650$	−5.51869e6	−0.512333
$651$	0	0
$652$	−4.08132e6	−0.375995
$653$	−1.54461e7	−1.41754	−0.708770	−	0.705440i	$-0.750749\pi$
$653$	−1.54461e7	−1.41754	−0.708770	+	0.705440i	$0.750749\pi$
$654$	0	0
$655$	656793.	0.0598171
$656$	796445.	0.0722597
$657$	0	0
$658$	−2.06167e6	−0.185633
$659$	−8.68940e6	−0.779429	−0.389714	−	0.920936i	$-0.627426\pi$
$659$	−8.68940e6	−0.779429	−0.389714	+	0.920936i	$0.627426\pi$
$660$	0	0
$661$	−2.03442e7	−1.81108	−0.905538	−	0.424266i	$-0.860532\pi$
$661$	−2.03442e7	−1.81108	−0.905538	+	0.424266i	$0.860532\pi$
$662$	−9.01797e6	−0.799767
$663$	0	0
$664$	3.31409e6	0.291705
$665$	235247.	0.0206286
$666$	0	0
$667$	4.78254e6	0.416240
$668$	−8.16802e6	−0.708232
$669$	0	0
$670$	7.57080e6	0.651561
$671$	−1.30947e7	−1.12277
$672$	0	0
$673$	1.71139e7	1.45650	0.728251	−	0.685310i	$-0.240333\pi$
$673$	1.71139e7	1.45650	0.728251	+	0.685310i	$0.240333\pi$
$674$	9.61045e6	0.814881
$675$	0	0
$676$	2.16128e6	0.181905
$677$	−2.25299e6	−0.188924	−0.0944621	−	0.995528i	$-0.530113\pi$
$677$	−2.25299e6	−0.188924	−0.0944621	+	0.995528i	$0.530113\pi$
$678$	0	0
$679$	−747622.	−0.0622311
$680$	488899.	0.0405459
$681$	0	0
$682$	−7.55962e6	−0.622356
$683$	−5.33481e6	−0.437590	−0.218795	−	0.975771i	$-0.570213\pi$
$683$	−5.33481e6	−0.437590	−0.218795	+	0.975771i	$0.570213\pi$
$684$	0	0
$685$	−9.18457e6	−0.747881
$686$	2.51695e6	0.204204
$687$	0	0
$688$	81420.5	0.00655787
$689$	793171.	0.0636530
$690$	0	0
$691$	8.08495e6	0.644143	0.322071	−	0.946715i	$-0.395621\pi$
$691$	8.08495e6	0.644143	0.322071	+	0.946715i	$0.395621\pi$
$692$	−1.23753e7	−0.982401
$693$	0	0
$694$	2.73877e6	0.215853
$695$	1.01562e7	0.797569
$696$	0	0
$697$	−690053.	−0.0538022
$698$	8.79427e6	0.683221
$699$	0	0
$700$	586955.	0.0452751
$701$	−7.88116e6	−0.605752	−0.302876	−	0.953030i	$-0.597947\pi$
$701$	−7.88116e6	−0.605752	−0.302876	+	0.953030i	$0.597947\pi$
$702$	0	0
$703$	714562.	0.0545320
$704$	−1.43147e6	−0.108856
$705$	0	0
$706$	−8.13786e6	−0.614467
$707$	−316414.	−0.0238071
$708$	0	0
$709$	−2.07701e7	−1.55175	−0.775876	−	0.630885i	$-0.782692\pi$
$709$	−2.07701e7	−1.55175	−0.775876	+	0.630885i	$0.782692\pi$
$710$	−988731.	−0.0736092
$711$	0	0
$712$	−9.29623e6	−0.687237
$713$	3.58247e6	0.263911
$714$	0	0
$715$	8.56506e6	0.626564
$716$	7.64263e6	0.557135
$717$	0	0
$718$	−9.22370e6	−0.667719
$719$	−1.03826e7	−0.749004	−0.374502	−	0.927226i	$-0.622186\pi$
$719$	−1.03826e7	−0.749004	−0.374502	+	0.927226i	$0.622186\pi$
$720$	0	0
$721$	−1.75330e6	−0.125608
$722$	521284.	0.0372161
$723$	0	0
$724$	1.16658e7	0.827121
$725$	−1.39970e7	−0.988985
$726$	0	0
$727$	−6.52486e6	−0.457863	−0.228931	−	0.973443i	$-0.573523\pi$
$727$	−6.52486e6	−0.457863	−0.228931	+	0.973443i	$0.573523\pi$
$728$	−861707.	−0.0602603
$729$	0	0
$730$	−8.91962e6	−0.619497
$731$	−70544.1	−0.00488278
$732$	0	0
$733$	−2.46101e6	−0.169182	−0.0845908	−	0.996416i	$-0.526958\pi$
$733$	−2.46101e6	−0.169182	−0.0845908	+	0.996416i	$0.526958\pi$
$734$	1.05790e7	0.724778
$735$	0	0
$736$	678367.	0.0461605
$737$	1.92058e7	1.30246
$738$	0	0
$739$	−7.33586e6	−0.494128	−0.247064	−	0.968999i	$-0.579466\pi$
$739$	−7.33586e6	−0.494128	−0.247064	+	0.968999i	$0.579466\pi$
$740$	−1.09075e6	−0.0732228
$741$	0	0
$742$	−84359.9	−0.00562505
$743$	−2.44394e7	−1.62412	−0.812060	−	0.583574i	$-0.801654\pi$
$743$	−2.44394e7	−1.62412	−0.812060	+	0.583574i	$0.801654\pi$
$744$	0	0
$745$	−3.58148e6	−0.236414
$746$	693621.	0.0456326
$747$	0	0
$748$	1.24025e6	0.0810504
$749$	−892244.	−0.0581138
$750$	0	0
$751$	−2.11169e6	−0.136625	−0.0683126	−	0.997664i	$-0.521762\pi$
$751$	−2.11169e6	−0.136625	−0.0683126	+	0.997664i	$0.521762\pi$
$752$	6.97356e6	0.449687
$753$	0	0
$754$	2.05489e7	1.31632
$755$	1.64401e7	1.04963
$756$	0	0
$757$	−1.47627e7	−0.936325	−0.468162	−	0.883642i	$-0.655084\pi$
$757$	−1.47627e7	−0.936325	−0.468162	+	0.883642i	$0.655084\pi$
$758$	8.93632e6	0.564919
$759$	0	0
$760$	−795719.	−0.0499719
$761$	−2.67074e7	−1.67174	−0.835871	−	0.548926i	$-0.815037\pi$
$761$	−2.67074e7	−1.67174	−0.835871	+	0.548926i	$0.815037\pi$
$762$	0	0
$763$	−992408.	−0.0617133
$764$	4.56934e6	0.283218
$765$	0	0
$766$	1.77990e7	1.09604
$767$	2.69731e7	1.65555
$768$	0	0
$769$	3.05104e6	0.186051	0.0930256	−	0.995664i	$-0.470346\pi$
$769$	3.05104e6	0.186051	0.0930256	+	0.995664i	$0.470346\pi$
$770$	−910961.	−0.0553698
$771$	0	0
$772$	−859043.	−0.0518766
$773$	2.18723e7	1.31657	0.658287	−	0.752767i	$-0.271281\pi$
$773$	2.18723e7	1.31657	0.658287	+	0.752767i	$0.271281\pi$
$774$	0	0
$775$	−1.04847e7	−0.627052
$776$	2.52882e6	0.150752
$777$	0	0
$778$	−9.72331e6	−0.575924
$779$	1.12311e6	0.0663100
$780$	0	0
$781$	−2.50823e6	−0.147143
$782$	−587748.	−0.0343696
$783$	0	0
$784$	−4.21094e6	−0.244675
$785$	−1.49802e7	−0.867647
$786$	0	0
$787$	−1.26837e7	−0.729975	−0.364987	−	0.931013i	$-0.618927\pi$
$787$	−1.26837e7	−0.729975	−0.364987	+	0.931013i	$0.618927\pi$
$788$	−232622.	−0.0133455
$789$	0	0
$790$	−4.97385e6	−0.283547
$791$	2.84943e6	0.161926
$792$	0	0
$793$	2.66630e7	1.50566
$794$	1.04482e7	0.588154
$795$	0	0
$796$	−2.29920e6	−0.128616
$797$	−7.28989e6	−0.406514	−0.203257	−	0.979125i	$-0.565153\pi$
$797$	−7.28989e6	−0.406514	−0.203257	+	0.979125i	$0.565153\pi$
$798$	0	0
$799$	−6.04201e6	−0.334822
$800$	−1.98537e6	−0.109677
$801$	0	0
$802$	−5.70640e6	−0.313275
$803$	−2.26275e7	−1.23836
$804$	0	0
$805$	431700.	0.0234797
$806$	1.53926e7	0.834593
$807$	0	0
$808$	1.07027e6	0.0576717
$809$	2.72506e7	1.46388	0.731939	−	0.681370i	$-0.238615\pi$
$809$	2.72506e7	1.46388	0.731939	+	0.681370i	$0.238615\pi$
$810$	0	0
$811$	1.45954e7	0.779229	0.389615	−	0.920978i	$-0.372608\pi$
$811$	1.45954e7	0.779229	0.389615	+	0.920978i	$0.372608\pi$
$812$	−2.18554e6	−0.116324
$813$	0	0
$814$	−2.76704e6	−0.146371
$815$	8.78524e6	0.463297
$816$	0	0
$817$	114816.	0.00601791
$818$	1.89328e7	0.989310
$819$	0	0
$820$	−1.71439e6	−0.0890377
$821$	−4.16570e6	−0.215690	−0.107845	−	0.994168i	$-0.534395\pi$
$821$	−4.16570e6	−0.215690	−0.107845	+	0.994168i	$0.534395\pi$
$822$	0	0
$823$	−9.74487e6	−0.501506	−0.250753	−	0.968051i	$-0.580678\pi$
$823$	−9.74487e6	−0.501506	−0.250753	+	0.968051i	$0.580678\pi$
$824$	5.93052e6	0.304281
$825$	0	0
$826$	−2.86880e6	−0.146302
$827$	−1.04970e6	−0.0533707	−0.0266854	−	0.999644i	$-0.508495\pi$
$827$	−1.04970e6	−0.0533707	−0.0266854	+	0.999644i	$0.508495\pi$
$828$	0	0
$829$	1.87001e7	0.945056	0.472528	−	0.881316i	$-0.343342\pi$
$829$	1.87001e7	0.945056	0.472528	+	0.881316i	$0.343342\pi$
$830$	−7.13374e6	−0.359436
$831$	0	0
$832$	2.91471e6	0.145978
$833$	3.64843e6	0.182177
$834$	0	0
$835$	1.75820e7	0.872676
$836$	−2.01860e6	−0.0998928
$837$	0	0
$838$	1.43103e6	0.0703947
$839$	−1.96094e7	−0.961742	−0.480871	−	0.876791i	$-0.659680\pi$
$839$	−1.96094e7	−0.961742	−0.480871	+	0.876791i	$0.659680\pi$
$840$	0	0
$841$	3.16068e7	1.54096
$842$	2.61002e6	0.126871
$843$	0	0
$844$	1.33846e6	0.0646768
$845$	−4.65225e6	−0.224141
$846$	0	0
$847$	736303.	0.0352653
$848$	285346.	0.0136264
$849$	0	0
$850$	1.72015e6	0.0816620
$851$	1.31129e6	0.0620688
$852$	0	0
$853$	−5.00288e6	−0.235422	−0.117711	−	0.993048i	$-0.537556\pi$
$853$	−5.00288e6	−0.235422	−0.117711	+	0.993048i	$0.537556\pi$
$854$	−2.83582e6	−0.133056
$855$	0	0
$856$	3.01800e6	0.140778
$857$	−1.36218e7	−0.633553	−0.316776	−	0.948500i	$-0.602600\pi$
$857$	−1.36218e7	−0.633553	−0.316776	+	0.948500i	$0.602600\pi$
$858$	0	0
$859$	1.88600e7	0.872084	0.436042	−	0.899926i	$-0.356380\pi$
$859$	1.88600e7	0.872084	0.436042	+	0.899926i	$0.356380\pi$
$860$	−175262.	−0.00808054
$861$	0	0
$862$	2.50059e7	1.14624
$863$	3.62744e7	1.65796	0.828979	−	0.559280i	$-0.188922\pi$
$863$	3.62744e7	1.65796	0.828979	+	0.559280i	$0.188922\pi$
$864$	0	0
$865$	2.66383e7	1.21050
$866$	1.78333e7	0.808047
$867$	0	0
$868$	−1.63712e6	−0.0737535
$869$	−1.26178e7	−0.566804
$870$	0	0
$871$	−3.91061e7	−1.74662
$872$	3.35680e6	0.149498
$873$	0	0
$874$	956603.	0.0423597
$875$	−3.29987e6	−0.145706
$876$	0	0
$877$	−1.47531e7	−0.647715	−0.323857	−	0.946106i	$-0.604980\pi$
$877$	−1.47531e7	−0.647715	−0.323857	+	0.946106i	$0.604980\pi$
$878$	2.00261e7	0.876717
$879$	0	0
$880$	3.08131e6	0.134131
$881$	−1.41746e7	−0.615277	−0.307638	−	0.951503i	$-0.599539\pi$
$881$	−1.41746e7	−0.615277	−0.307638	+	0.951503i	$0.599539\pi$
$882$	0	0
$883$	2.05570e7	0.887276	0.443638	−	0.896206i	$-0.353688\pi$
$883$	2.05570e7	0.887276	0.443638	+	0.896206i	$0.353688\pi$
$884$	−2.52535e6	−0.108690
$885$	0	0
$886$	−2.86363e6	−0.122556
$887$	9.48962e6	0.404986	0.202493	−	0.979284i	$-0.435096\pi$
$887$	9.48962e6	0.404986	0.202493	+	0.979284i	$0.435096\pi$
$888$	0	0
$889$	−6.66473e6	−0.282832
$890$	2.00106e7	0.846807
$891$	0	0
$892$	−1.06466e7	−0.448022
$893$	9.83381e6	0.412661
$894$	0	0
$895$	−1.64511e7	−0.686496
$896$	−310002.	−0.0129001
$897$	0	0
$898$	1.93273e7	0.799800
$899$	3.90401e7	1.61106
$900$	0	0
$901$	−247228.	−0.0101458
$902$	−4.34909e6	−0.177984
$903$	0	0
$904$	−9.63814e6	−0.392258
$905$	−2.51112e7	−1.01917
$906$	0	0
$907$	1.30672e7	0.527428	0.263714	−	0.964601i	$-0.415053\pi$
$907$	1.30672e7	0.527428	0.263714	+	0.964601i	$0.415053\pi$
$908$	−1.84612e7	−0.743096
$909$	0	0
$910$	1.85486e6	0.0742521
$911$	4.60549e7	1.83857	0.919284	−	0.393594i	$-0.128768\pi$
$911$	4.60549e7	1.83857	0.919284	+	0.393594i	$0.128768\pi$
$912$	0	0
$913$	−1.80970e7	−0.718506
$914$	−2.57764e7	−1.02060
$915$	0	0
$916$	−6.93906e6	−0.273251
$917$	360828.	0.0141702
$918$	0	0
$919$	−6.67836e6	−0.260844	−0.130422	−	0.991459i	$-0.541633\pi$
$919$	−6.67836e6	−0.260844	−0.130422	+	0.991459i	$0.541633\pi$
$920$	−1.46022e6	−0.0568785
$921$	0	0
$922$	1.64043e7	0.635522
$923$	5.10717e6	0.197322
$924$	0	0
$925$	−3.83772e6	−0.147475
$926$	3.59473e7	1.37765
$927$	0	0
$928$	7.39254e6	0.281789
$929$	84737.5	0.00322134	0.00161067	−	0.999999i	$-0.499487\pi$
$929$	84737.5	0.00322134	0.00161067	+	0.999999i	$0.499487\pi$
$930$	0	0
$931$	−5.93809e6	−0.224529
$932$	−6.19477e6	−0.233607
$933$	0	0
$934$	3.53763e7	1.32692
$935$	−2.66970e6	−0.0998695
$936$	0	0
$937$	−3.12820e7	−1.16398	−0.581989	−	0.813196i	$-0.697725\pi$
$937$	−3.12820e7	−1.16398	−0.581989	+	0.813196i	$0.697725\pi$
$938$	4.15924e6	0.154350
$939$	0	0
$940$	−1.50109e7	−0.554100
$941$	−2.88841e7	−1.06337	−0.531685	−	0.846942i	$-0.678441\pi$
$941$	−2.88841e7	−1.06337	−0.531685	+	0.846942i	$0.678441\pi$
$942$	0	0
$943$	2.06101e6	0.0754746
$944$	9.70367e6	0.354410
$945$	0	0
$946$	−444607.	−0.0161528
$947$	−1.99925e7	−0.724422	−0.362211	−	0.932096i	$-0.617978\pi$
$947$	−1.99925e7	−0.724422	−0.362211	+	0.932096i	$0.617978\pi$
$948$	0	0
$949$	4.60732e7	1.66067
$950$	−2.79968e6	−0.100647
$951$	0	0
$952$	268591.	0.00960502
$953$	−5.22187e7	−1.86249	−0.931244	−	0.364396i	$-0.881276\pi$
$953$	−5.22187e7	−1.86249	−0.931244	+	0.364396i	$0.881276\pi$
$954$	0	0
$955$	−9.83573e6	−0.348978
$956$	9.95941e6	0.352443
$957$	0	0
$958$	−1.15728e7	−0.407403
$959$	−5.04580e6	−0.177167
$960$	0	0
$961$	614716.	0.0214717
$962$	5.63414e6	0.196286
$963$	0	0
$964$	−5.94327e6	−0.205984
$965$	1.84913e6	0.0639218
$966$	0	0
$967$	2.66303e7	0.915818	0.457909	−	0.888999i	$-0.348598\pi$
$967$	2.66303e7	0.915818	0.457909	+	0.888999i	$0.348598\pi$
$968$	−2.49053e6	−0.0854287
$969$	0	0
$970$	−5.44340e6	−0.185755
$971$	−1.04206e7	−0.354686	−0.177343	−	0.984149i	$-0.556750\pi$
$971$	−1.04206e7	−0.354686	−0.177343	+	0.984149i	$0.556750\pi$
$972$	0	0
$973$	5.57959e6	0.188938
$974$	−1.27267e7	−0.429850
$975$	0	0
$976$	9.59212e6	0.322322
$977$	−1.82795e7	−0.612673	−0.306336	−	0.951923i	$-0.599103\pi$
$977$	−1.82795e7	−0.612673	−0.306336	+	0.951923i	$0.599103\pi$
$978$	0	0
$979$	5.07633e7	1.69275
$980$	9.06425e6	0.301486
$981$	0	0
$982$	1.78204e7	0.589709
$983$	−2.25159e7	−0.743199	−0.371600	−	0.928393i	$-0.621191\pi$
$983$	−2.25159e7	−0.743199	−0.371600	+	0.928393i	$0.621191\pi$
$984$	0	0
$985$	500730.	0.0164442
$986$	−6.40502e6	−0.209811
$987$	0	0
$988$	4.11019e6	0.133958
$989$	210697.	0.00684964
$990$	0	0
$991$	3.42068e7	1.10644	0.553221	−	0.833034i	$-0.313398\pi$
$991$	3.42068e7	1.10644	0.553221	+	0.833034i	$0.313398\pi$
$992$	5.53755e6	0.178664
$993$	0	0
$994$	−543187.	−0.0174375
$995$	4.94914e6	0.158479
$996$	0	0
$997$	−1.12682e7	−0.359018	−0.179509	−	0.983756i	$-0.557451\pi$
$997$	−1.12682e7	−0.359018	−0.179509	+	0.983756i	$0.557451\pi$
$998$	3.72589e7	1.18414
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.6.a.i.1.1		2
3.2	odd	2		38.6.a.c.1.2	✓	2
12.11	even	2		304.6.a.f.1.1		2
15.14	odd	2		950.6.a.d.1.1		2
57.56	even	2		722.6.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.a.c.1.2	✓	2	3.2	odd	2
304.6.a.f.1.1		2	12.11	even	2
342.6.a.i.1.1		2	1.1	even	1	trivial
722.6.a.c.1.1		2	57.56	even	2
950.6.a.d.1.1		2	15.14	odd	2

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:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	342.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +16.0000 q^{4} -34.4408 q^{5} -18.9210 q^{7} +64.0000 q^{8} +O(q^{10})\)	\(q+4.00000 q^{2} +16.0000 q^{4} -34.4408 q^{5} -18.9210 q^{7} +64.0000 q^{8} -137.763 q^{10} -349.480 q^{11} +711.599 q^{13} -75.6840 q^{14} +256.000 q^{16} -221.803 q^{17} +361.000 q^{19} -551.052 q^{20} -1397.92 q^{22} +662.468 q^{23} -1938.83 q^{25} +2846.39 q^{26} -302.736 q^{28} +7219.28 q^{29} +5407.76 q^{31} +1024.00 q^{32} -887.210 q^{34} +651.654 q^{35} +1979.40 q^{37} +1444.00 q^{38} -2204.21 q^{40} +3111.11 q^{41} +318.049 q^{43} -5591.68 q^{44} +2649.87 q^{46} +27240.5 q^{47} -16449.0 q^{49} -7755.34 q^{50} +11385.6 q^{52} +1114.63 q^{53} +12036.4 q^{55} -1210.94 q^{56} +28877.1 q^{58} +37904.9 q^{59} +37469.2 q^{61} +21631.0 q^{62} +4096.00 q^{64} -24508.0 q^{65} -54955.3 q^{67} -3548.84 q^{68} +2606.62 q^{70} +7177.04 q^{71} +64746.1 q^{73} +7917.59 q^{74} +5776.00 q^{76} +6612.52 q^{77} +36104.4 q^{79} -8816.83 q^{80} +12444.4 q^{82} +51782.7 q^{83} +7639.05 q^{85} +1272.20 q^{86} -22366.7 q^{88} -145254. q^{89} -13464.2 q^{91} +10599.5 q^{92} +108962. q^{94} -12433.1 q^{95} +39512.8 q^{97} -65796.0 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} + 32 q^{4} + 45 q^{5} + 114 q^{7} + 128 q^{8}+O(q^{10}) \) 2 * q + 8 * q^2 + 32 * q^4 + 45 * q^5 + 114 * q^7 + 128 * q^8	\( 2 q + 8 q^{2} + 32 q^{4} + 45 q^{5} + 114 q^{7} + 128 q^{8} + 180 q^{10} - 661 q^{11} + 1613 q^{13} + 456 q^{14} + 512 q^{16} - 64 q^{17} + 722 q^{19} + 720 q^{20} - 2644 q^{22} + 3185 q^{23} + 1247 q^{25} + 6452 q^{26} + 1824 q^{28} + 2481 q^{29} - 1180 q^{31} + 2048 q^{32} - 256 q^{34} + 11211 q^{35} + 10488 q^{37} + 2888 q^{38} + 2880 q^{40} - 16630 q^{41} + 11303 q^{43} - 10576 q^{44} + 12740 q^{46} + 12155 q^{47} - 15588 q^{49} + 4988 q^{50} + 25808 q^{52} - 20585 q^{53} - 12711 q^{55} + 7296 q^{56} + 9924 q^{58} + 78581 q^{59} + 43621 q^{61} - 4720 q^{62} + 8192 q^{64} + 47100 q^{65} + 7805 q^{67} - 1024 q^{68} + 44844 q^{70} + 62488 q^{71} + 16218 q^{73} + 41952 q^{74} + 11552 q^{76} - 34795 q^{77} + 67122 q^{79} + 11520 q^{80} - 66520 q^{82} + 10714 q^{83} + 20175 q^{85} + 45212 q^{86} - 42304 q^{88} - 128188 q^{89} + 106351 q^{91} + 50960 q^{92} + 48620 q^{94} + 16245 q^{95} + 178558 q^{97} - 62352 q^{98}+O(q^{100}) \) 2 * q + 8 * q^2 + 32 * q^4 + 45 * q^5 + 114 * q^7 + 128 * q^8 + 180 * q^10 - 661 * q^11 + 1613 * q^13 + 456 * q^14 + 512 * q^16 - 64 * q^17 + 722 * q^19 + 720 * q^20 - 2644 * q^22 + 3185 * q^23 + 1247 * q^25 + 6452 * q^26 + 1824 * q^28 + 2481 * q^29 - 1180 * q^31 + 2048 * q^32 - 256 * q^34 + 11211 * q^35 + 10488 * q^37 + 2888 * q^38 + 2880 * q^40 - 16630 * q^41 + 11303 * q^43 - 10576 * q^44 + 12740 * q^46 + 12155 * q^47 - 15588 * q^49 + 4988 * q^50 + 25808 * q^52 - 20585 * q^53 - 12711 * q^55 + 7296 * q^56 + 9924 * q^58 + 78581 * q^59 + 43621 * q^61 - 4720 * q^62 + 8192 * q^64 + 47100 * q^65 + 7805 * q^67 - 1024 * q^68 + 44844 * q^70 + 62488 * q^71 + 16218 * q^73 + 41952 * q^74 + 11552 * q^76 - 34795 * q^77 + 67122 * q^79 + 11520 * q^80 - 66520 * q^82 + 10714 * q^83 + 20175 * q^85 + 45212 * q^86 - 42304 * q^88 - 128188 * q^89 + 106351 * q^91 + 50960 * q^92 + 48620 * q^94 + 16245 * q^95 + 178558 * q^97 - 62352 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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