LMFDB - Embedded newform 261.6.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [261,6,Mod(1,261)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(261, 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("261.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 261 = 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	261.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:	$41.8601769712$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.3257317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 34x^{2} - 27x + 10 $ x^4 - 34x^2 - 27x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.10057$ of defining polynomial
Character	$\chi$	$=$	261.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.16235 q^{2} -14.6748 q^{4} -32.5670 q^{5} -220.793 q^{7} +194.277 q^{8} +O(q^{10})$	\(q-4.16235 q^{2} -14.6748 q^{4} -32.5670 q^{5} -220.793 q^{7} +194.277 q^{8} +135.555 q^{10} +85.7296 q^{11} +1034.02 q^{13} +919.018 q^{14} -339.054 q^{16} +313.020 q^{17} +458.534 q^{19} +477.916 q^{20} -356.836 q^{22} +3448.84 q^{23} -2064.39 q^{25} -4303.94 q^{26} +3240.10 q^{28} +841.000 q^{29} -7983.23 q^{31} -4805.60 q^{32} -1302.90 q^{34} +7190.58 q^{35} +152.624 q^{37} -1908.58 q^{38} -6327.03 q^{40} +18492.2 q^{41} +2072.84 q^{43} -1258.07 q^{44} -14355.3 q^{46} -15845.6 q^{47} +31942.6 q^{49} +8592.70 q^{50} -15174.0 q^{52} -9240.52 q^{53} -2791.96 q^{55} -42895.0 q^{56} -3500.54 q^{58} +14323.2 q^{59} -19580.2 q^{61} +33229.0 q^{62} +30852.3 q^{64} -33674.9 q^{65} -9193.70 q^{67} -4593.53 q^{68} -29929.7 q^{70} +19374.7 q^{71} -56912.4 q^{73} -635.276 q^{74} -6728.91 q^{76} -18928.5 q^{77} +51573.6 q^{79} +11042.0 q^{80} -76971.0 q^{82} -19978.1 q^{83} -10194.2 q^{85} -8627.87 q^{86} +16655.3 q^{88} -130663. q^{89} -228304. q^{91} -50611.1 q^{92} +65954.9 q^{94} -14933.1 q^{95} +43603.5 q^{97} -132956. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{4} + 68 q^{5} - 208 q^{7} + 504 q^{8}+O(q^{10}) $ 4 * q + 10 * q^4 + 68 * q^5 - 208 * q^7 + 504 * q^8	$ 4 q + 10 q^{4} + 68 q^{5} - 208 q^{7} + 504 q^{8} - 788 q^{10} + 124 q^{11} - 460 q^{13} - 768 q^{14} - 414 q^{16} - 184 q^{17} - 2392 q^{19} - 2822 q^{20} + 5538 q^{22} + 1192 q^{23} + 1824 q^{25} - 4724 q^{26} + 44 q^{28} + 3364 q^{29} - 19212 q^{31} - 6552 q^{32} - 7612 q^{34} + 22944 q^{35} - 10928 q^{37} + 456 q^{38} - 20 q^{40} + 1120 q^{41} - 21420 q^{43} + 1932 q^{44} - 7588 q^{46} - 23772 q^{47} + 10452 q^{49} - 43240 q^{50} - 29062 q^{52} - 8860 q^{53} - 52652 q^{55} - 34304 q^{56} + 10840 q^{59} + 49448 q^{61} - 18518 q^{62} - 20734 q^{64} - 97836 q^{65} - 7840 q^{67} - 20724 q^{68} - 77496 q^{70} + 48744 q^{71} - 74992 q^{73} + 35920 q^{74} - 140792 q^{76} - 128656 q^{77} - 106076 q^{79} - 58638 q^{80} - 234132 q^{82} - 62888 q^{83} + 23848 q^{85} + 216014 q^{86} - 39426 q^{88} - 107568 q^{89} - 268896 q^{91} + 26268 q^{92} + 30542 q^{94} - 147352 q^{95} - 49520 q^{97} - 242304 q^{98}+O(q^{100}) $ 4 * q + 10 * q^4 + 68 * q^5 - 208 * q^7 + 504 * q^8 - 788 * q^10 + 124 * q^11 - 460 * q^13 - 768 * q^14 - 414 * q^16 - 184 * q^17 - 2392 * q^19 - 2822 * q^20 + 5538 * q^22 + 1192 * q^23 + 1824 * q^25 - 4724 * q^26 + 44 * q^28 + 3364 * q^29 - 19212 * q^31 - 6552 * q^32 - 7612 * q^34 + 22944 * q^35 - 10928 * q^37 + 456 * q^38 - 20 * q^40 + 1120 * q^41 - 21420 * q^43 + 1932 * q^44 - 7588 * q^46 - 23772 * q^47 + 10452 * q^49 - 43240 * q^50 - 29062 * q^52 - 8860 * q^53 - 52652 * q^55 - 34304 * q^56 + 10840 * q^59 + 49448 * q^61 - 18518 * q^62 - 20734 * q^64 - 97836 * q^65 - 7840 * q^67 - 20724 * q^68 - 77496 * q^70 + 48744 * q^71 - 74992 * q^73 + 35920 * q^74 - 140792 * q^76 - 128656 * q^77 - 106076 * q^79 - 58638 * q^80 - 234132 * q^82 - 62888 * q^83 + 23848 * q^85 + 216014 * q^86 - 39426 * q^88 - 107568 * q^89 - 268896 * q^91 + 26268 * q^92 + 30542 * q^94 - 147352 * q^95 - 49520 * q^97 - 242304 * 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 $

$2$	−4.16235	−0.735806	−0.367903	−	0.929864i	$-0.619924\pi$
$2$	−4.16235	−0.735806	−0.367903	+	0.929864i	$0.619924\pi$
$3$	0	0
$3$	0	0
$4$	−14.6748	−0.458589
$5$	−32.5670	−0.582577	−0.291289	−	0.956635i	$-0.594084\pi$
$5$	−32.5670	−0.582577	−0.291289	+	0.956635i	$0.594084\pi$
$6$	0	0
$7$	−220.793	−1.70310	−0.851551	−	0.524272i	$-0.824337\pi$
$7$	−220.793	−1.70310	−0.851551	+	0.524272i	$0.824337\pi$
$8$	194.277	1.07324
$9$	0	0
$10$	135.555	0.428664
$11$	85.7296	0.213623	0.106812	−	0.994279i	$-0.465936\pi$
$11$	85.7296	0.213623	0.106812	+	0.994279i	$0.465936\pi$
$12$	0	0
$13$	1034.02	1.69695	0.848475	−	0.529235i	$-0.177521\pi$
$13$	1034.02	1.69695	0.848475	+	0.529235i	$0.177521\pi$
$14$	919.018	1.25315
$15$	0	0
$16$	−339.054	−0.331108
$17$	313.020	0.262694	0.131347	−	0.991336i	$-0.458070\pi$
$17$	313.020	0.262694	0.131347	+	0.991336i	$0.458070\pi$
$18$	0	0
$19$	458.534	0.291398	0.145699	−	0.989329i	$-0.453457\pi$
$19$	458.534	0.291398	0.145699	+	0.989329i	$0.453457\pi$
$20$	477.916	0.267163
$21$	0	0
$22$	−356.836	−0.157186
$23$	3448.84	1.35942	0.679709	−	0.733482i	$-0.262106\pi$
$23$	3448.84	1.35942	0.679709	+	0.733482i	$0.262106\pi$
$24$	0	0
$25$	−2064.39	−0.660604
$26$	−4303.94	−1.24863
$27$	0	0
$28$	3240.10	0.781023
$29$	841.000	0.185695
$29$	841.000	0.185695
$30$	0	0
$31$	−7983.23	−1.49202	−0.746009	−	0.665936i	$-0.768033\pi$
$31$	−7983.23	−1.49202	−0.746009	+	0.665936i	$0.768033\pi$
$32$	−4805.60	−0.829608
$33$	0	0
$34$	−1302.90	−0.193292
$35$	7190.58	0.992188
$36$	0	0
$37$	152.624	0.0183282	0.00916409	−	0.999958i	$-0.497083\pi$
$37$	152.624	0.0183282	0.00916409	+	0.999958i	$0.497083\pi$
$38$	−1908.58	−0.214413
$39$	0	0
$40$	−6327.03	−0.625244
$41$	18492.2	1.71802	0.859011	−	0.511957i	$-0.171079\pi$
$41$	18492.2	1.71802	0.859011	+	0.511957i	$0.171079\pi$
$42$	0	0
$43$	2072.84	0.170960	0.0854799	−	0.996340i	$-0.472758\pi$
$43$	2072.84	0.170960	0.0854799	+	0.996340i	$0.472758\pi$
$44$	−1258.07	−0.0979653
$45$	0	0
$46$	−14355.3	−1.00027
$47$	−15845.6	−1.04632	−0.523159	−	0.852235i	$-0.675247\pi$
$47$	−15845.6	−1.04632	−0.523159	+	0.852235i	$0.675247\pi$
$48$	0	0
$49$	31942.6	1.90055
$50$	8592.70	0.486077
$51$	0	0
$52$	−15174.0	−0.778203
$53$	−9240.52	−0.451863	−0.225931	−	0.974143i	$-0.572543\pi$
$53$	−9240.52	−0.451863	−0.225931	+	0.974143i	$0.572543\pi$
$54$	0	0
$55$	−2791.96	−0.124452
$56$	−42895.0	−1.82783
$57$	0	0
$58$	−3500.54	−0.136636
$59$	14323.2	0.535686	0.267843	−	0.963463i	$-0.413689\pi$
$59$	14323.2	0.535686	0.267843	+	0.963463i	$0.413689\pi$
$60$	0	0
$61$	−19580.2	−0.673739	−0.336870	−	0.941551i	$-0.609368\pi$
$61$	−19580.2	−0.673739	−0.336870	+	0.941551i	$0.609368\pi$
$62$	33229.0	1.09784
$63$	0	0
$64$	30852.3	0.941539
$65$	−33674.9	−0.988604
$66$	0	0
$67$	−9193.70	−0.250209	−0.125105	−	0.992144i	$-0.539927\pi$
$67$	−9193.70	−0.250209	−0.125105	+	0.992144i	$0.539927\pi$
$68$	−4593.53	−0.120469
$69$	0	0
$70$	−29929.7	−0.730058
$71$	19374.7	0.456131	0.228066	−	0.973646i	$-0.426760\pi$
$71$	19374.7	0.456131	0.228066	+	0.973646i	$0.426760\pi$
$72$	0	0
$73$	−56912.4	−1.24997	−0.624985	−	0.780637i	$-0.714895\pi$
$73$	−56912.4	−1.24997	−0.624985	+	0.780637i	$0.714895\pi$
$74$	−635.276	−0.0134860
$75$	0	0
$76$	−6728.91	−0.133632
$77$	−18928.5	−0.363822
$78$	0	0
$79$	51573.6	0.929736	0.464868	−	0.885380i	$-0.346102\pi$
$79$	51573.6	0.929736	0.464868	+	0.885380i	$0.346102\pi$
$80$	11042.0	0.192896
$81$	0	0
$82$	−76971.0	−1.26413
$83$	−19978.1	−0.318317	−0.159158	−	0.987253i	$-0.550878\pi$
$83$	−19978.1	−0.318317	−0.159158	+	0.987253i	$0.550878\pi$
$84$	0	0
$85$	−10194.2	−0.153040
$86$	−8627.87	−0.125793
$87$	0	0
$88$	16655.3	0.229269
$89$	−130663.	−1.74855	−0.874274	−	0.485432i	$-0.838662\pi$
$89$	−130663.	−1.74855	−0.874274	+	0.485432i	$0.838662\pi$
$90$	0	0
$91$	−228304.	−2.89008
$92$	−50611.1	−0.623414
$93$	0	0
$94$	65954.9	0.769888
$95$	−14933.1	−0.169762
$96$	0	0
$97$	43603.5	0.470535	0.235268	−	0.971931i	$-0.424403\pi$
$97$	43603.5	0.470535	0.235268	+	0.971931i	$0.424403\pi$
$98$	−132956.	−1.39844
$99$	0	0
$100$	30294.6	0.302946
$101$	56686.0	0.552933	0.276467	−	0.961024i	$-0.410836\pi$
$101$	56686.0	0.552933	0.276467	+	0.961024i	$0.410836\pi$
$102$	0	0
$103$	−111450.	−1.03511	−0.517556	−	0.855649i	$-0.673158\pi$
$103$	−111450.	−1.03511	−0.517556	+	0.855649i	$0.673158\pi$
$104$	200886.	1.82123
$105$	0	0
$106$	38462.3	0.332484
$107$	−189434.	−1.59955	−0.799775	−	0.600300i	$-0.795048\pi$
$107$	−189434.	−1.59955	−0.799775	+	0.600300i	$0.795048\pi$
$108$	0	0
$109$	100232.	0.808054	0.404027	−	0.914747i	$-0.367610\pi$
$109$	100232.	0.808054	0.404027	+	0.914747i	$0.367610\pi$
$110$	11621.1	0.0915727
$111$	0	0
$112$	74860.8	0.563910
$113$	103444.	0.762092	0.381046	−	0.924556i	$-0.375564\pi$
$113$	103444.	0.762092	0.381046	+	0.924556i	$0.375564\pi$
$114$	0	0
$115$	−112318.	−0.791966
$116$	−12341.5	−0.0851578
$117$	0	0
$118$	−59618.2	−0.394161
$119$	−69112.8	−0.447395
$120$	0	0
$121$	−153701.	−0.954365
$122$	81499.5	0.495742
$123$	0	0
$124$	117153.	0.684223
$125$	169003.	0.967430
$126$	0	0
$127$	−247538.	−1.36186	−0.680930	−	0.732349i	$-0.738424\pi$
$127$	−247538.	−1.36186	−0.680930	+	0.732349i	$0.738424\pi$
$128$	25361.1	0.136818
$129$	0	0
$130$	140167.	0.727422
$131$	−198458.	−1.01040	−0.505198	−	0.863004i	$-0.668580\pi$
$131$	−198458.	−1.01040	−0.505198	+	0.863004i	$0.668580\pi$
$132$	0	0
$133$	−101241.	−0.496281
$134$	38267.4	0.184106
$135$	0	0
$136$	60812.7	0.281934
$137$	−141442.	−0.643839	−0.321920	−	0.946767i	$-0.604328\pi$
$137$	−141442.	−0.643839	−0.321920	+	0.946767i	$0.604328\pi$
$138$	0	0
$139$	−325330.	−1.42819	−0.714097	−	0.700047i	$-0.753162\pi$
$139$	−325330.	−1.42819	−0.714097	+	0.700047i	$0.753162\pi$
$140$	−105521.	−0.455006
$141$	0	0
$142$	−80644.4	−0.335624
$143$	88645.8	0.362508
$144$	0	0
$145$	−27388.9	−0.108182
$146$	236889.	0.919736
$147$	0	0
$148$	−2239.74	−0.00840510
$149$	391462.	1.44452	0.722261	−	0.691621i	$-0.243103\pi$
$149$	391462.	1.44452	0.722261	+	0.691621i	$0.243103\pi$
$150$	0	0
$151$	−216585.	−0.773013	−0.386507	−	0.922287i	$-0.626318\pi$
$151$	−216585.	−0.773013	−0.386507	+	0.922287i	$0.626318\pi$
$152$	89082.6	0.312740
$153$	0	0
$154$	78787.0	0.267703
$155$	259990.	0.869216
$156$	0	0
$157$	564396.	1.82740	0.913702	−	0.406385i	$-0.133211\pi$
$157$	564396.	1.82740	0.913702	+	0.406385i	$0.133211\pi$
$158$	−214667.	−0.684106
$159$	0	0
$160$	156504.	0.483311
$161$	−761480.	−2.31523
$162$	0	0
$163$	476986.	1.40617	0.703083	−	0.711108i	$-0.251806\pi$
$163$	476986.	1.40617	0.703083	+	0.711108i	$0.251806\pi$
$164$	−271370.	−0.787866
$165$	0	0
$166$	83155.9	0.234219
$167$	203397.	0.564357	0.282178	−	0.959362i	$-0.408943\pi$
$167$	203397.	0.564357	0.282178	+	0.959362i	$0.408943\pi$
$168$	0	0
$169$	697898.	1.87964
$170$	42431.6	0.112608
$171$	0	0
$172$	−30418.6	−0.0784003
$173$	95409.0	0.242367	0.121184	−	0.992630i	$-0.461331\pi$
$173$	95409.0	0.242367	0.121184	+	0.992630i	$0.461331\pi$
$174$	0	0
$175$	455803.	1.12508
$176$	−29067.0	−0.0707323
$177$	0	0
$178$	543865.	1.28659
$179$	327011.	0.762833	0.381416	−	0.924403i	$-0.375436\pi$
$179$	327011.	0.762833	0.381416	+	0.924403i	$0.375436\pi$
$180$	0	0
$181$	108581.	0.246353	0.123177	−	0.992385i	$-0.460692\pi$
$181$	108581.	0.246353	0.123177	+	0.992385i	$0.460692\pi$
$182$	950280.	2.12654
$183$	0	0
$184$	670030.	1.45898
$185$	−4970.52	−0.0106776
$186$	0	0
$187$	26835.1	0.0561176
$188$	232532.	0.479830
$189$	0	0
$190$	62156.7	0.124912
$191$	315738.	0.626244	0.313122	−	0.949713i	$-0.398625\pi$
$191$	315738.	0.626244	0.313122	+	0.949713i	$0.398625\pi$
$192$	0	0
$193$	41432.1	0.0800651	0.0400326	−	0.999198i	$-0.487254\pi$
$193$	41432.1	0.0800651	0.0400326	+	0.999198i	$0.487254\pi$
$194$	−181493.	−0.346223
$195$	0	0
$196$	−468753.	−0.871573
$197$	−354739.	−0.651243	−0.325622	−	0.945500i	$-0.605574\pi$
$197$	−354739.	−0.651243	−0.325622	+	0.945500i	$0.605574\pi$
$198$	0	0
$199$	−949129.	−1.69900	−0.849498	−	0.527591i	$-0.823095\pi$
$199$	−949129.	−1.69900	−0.849498	+	0.527591i	$0.823095\pi$
$200$	−401063.	−0.708986
$201$	0	0
$202$	−235947.	−0.406852
$203$	−185687.	−0.316258
$204$	0	0
$205$	−602236.	−1.00088
$206$	463894.	0.761642
$207$	0	0
$208$	−350588.	−0.561873
$209$	39309.9	0.0622495
$210$	0	0
$211$	−380574.	−0.588481	−0.294241	−	0.955731i	$-0.595067\pi$
$211$	−380574.	−0.588481	−0.294241	+	0.955731i	$0.595067\pi$
$212$	135603.	0.207219
$213$	0	0
$214$	788489.	1.17696
$215$	−67506.2	−0.0995973
$216$	0	0
$217$	1.76264e6	2.54106
$218$	−417201.	−0.594571
$219$	0	0
$220$	40971.5	0.0570723
$221$	323668.	0.445779
$222$	0	0
$223$	−68362.6	−0.0920569	−0.0460284	−	0.998940i	$-0.514656\pi$
$223$	−68362.6	−0.0920569	−0.0460284	+	0.998940i	$0.514656\pi$
$224$	1.06104e6	1.41291
$225$	0	0
$226$	−430569.	−0.560753
$227$	1.07914e6	1.39000	0.694998	−	0.719012i	$-0.255405\pi$
$227$	1.07914e6	1.39000	0.694998	+	0.719012i	$0.255405\pi$
$228$	0	0
$229$	−724221.	−0.912604	−0.456302	−	0.889825i	$-0.650826\pi$
$229$	−724221.	−0.912604	−0.456302	+	0.889825i	$0.650826\pi$
$230$	467509.	0.582734
$231$	0	0
$232$	163387.	0.199296
$233$	−1.37311e6	−1.65697	−0.828486	−	0.560009i	$-0.810797\pi$
$233$	−1.37311e6	−1.65697	−0.828486	+	0.560009i	$0.810797\pi$
$234$	0	0
$235$	516044.	0.609561
$236$	−210191.	−0.245660
$237$	0	0
$238$	287672.	0.329196
$239$	−1.11446e6	−1.26204	−0.631018	−	0.775769i	$-0.717362\pi$
$239$	−1.11446e6	−1.26204	−0.631018	+	0.775769i	$0.717362\pi$
$240$	0	0
$241$	−1.47058e6	−1.63097	−0.815486	−	0.578776i	$-0.803530\pi$
$241$	−1.47058e6	−1.63097	−0.815486	+	0.578776i	$0.803530\pi$
$242$	639759.	0.702228
$243$	0	0
$244$	287336.	0.308969
$245$	−1.04028e6	−1.10722
$246$	0	0
$247$	474132.	0.494489
$248$	−1.55096e6	−1.60129
$249$	0	0
$250$	−703450.	−0.711841
$251$	−375105.	−0.375810	−0.187905	−	0.982187i	$-0.560170\pi$
$251$	−375105.	−0.375810	−0.187905	+	0.982187i	$0.560170\pi$
$252$	0	0
$253$	295667.	0.290404
$254$	1.03034e6	1.00207
$255$	0	0
$256$	−1.09284e6	−1.04221
$257$	471074.	0.444894	0.222447	−	0.974945i	$-0.428596\pi$
$257$	471074.	0.444894	0.222447	+	0.974945i	$0.428596\pi$
$258$	0	0
$259$	−33698.4	−0.0312147
$260$	494173.	0.453363
$261$	0	0
$262$	826053.	0.743455
$263$	−1.11373e6	−0.992865	−0.496433	−	0.868075i	$-0.665357\pi$
$263$	−1.11373e6	−0.992865	−0.496433	+	0.868075i	$0.665357\pi$
$264$	0	0
$265$	300936.	0.263245
$266$	421401.	0.365167
$267$	0	0
$268$	134916.	0.114743
$269$	381568.	0.321508	0.160754	−	0.986995i	$-0.448607\pi$
$269$	381568.	0.321508	0.160754	+	0.986995i	$0.448607\pi$
$270$	0	0
$271$	1.08834e6	0.900208	0.450104	−	0.892976i	$-0.351387\pi$
$271$	1.08834e6	0.900208	0.450104	+	0.892976i	$0.351387\pi$
$272$	−106131.	−0.0869800
$273$	0	0
$274$	588732.	0.473741
$275$	−176979.	−0.141120
$276$	0	0
$277$	−2.18958e6	−1.71459	−0.857295	−	0.514825i	$-0.827857\pi$
$277$	−2.18958e6	−1.71459	−0.857295	+	0.514825i	$0.827857\pi$
$278$	1.35414e6	1.05087
$279$	0	0
$280$	1.39696e6	1.06485
$281$	1.02712e6	0.775992	0.387996	−	0.921661i	$-0.373167\pi$
$281$	1.02712e6	0.775992	0.387996	+	0.921661i	$0.373167\pi$
$282$	0	0
$283$	889156.	0.659952	0.329976	−	0.943989i	$-0.392959\pi$
$283$	889156.	0.659952	0.329976	+	0.943989i	$0.392959\pi$
$284$	−284321.	−0.209177
$285$	0	0
$286$	−368975.	−0.266736
$287$	−4.08295e6	−2.92597
$288$	0	0
$289$	−1.32188e6	−0.930992
$290$	114002.	0.0796009
$291$	0	0
$292$	835180.	0.573222
$293$	482000.	0.328003	0.164001	−	0.986460i	$-0.447560\pi$
$293$	482000.	0.328003	0.164001	+	0.986460i	$0.447560\pi$
$294$	0	0
$295$	−466465.	−0.312079
$296$	29651.4	0.0196705
$297$	0	0
$298$	−1.62940e6	−1.06289
$299$	3.56616e6	2.30687
$300$	0	0
$301$	−457668.	−0.291162
$302$	901505.	0.568788
$303$	0	0
$304$	−155468.	−0.0964842
$305$	637668.	0.392505
$306$	0	0
$307$	229158.	0.138768	0.0693839	−	0.997590i	$-0.477897\pi$
$307$	229158.	0.138768	0.0693839	+	0.997590i	$0.477897\pi$
$308$	277773.	0.166845
$309$	0	0
$310$	−1.08217e6	−0.639575
$311$	−2.49699e6	−1.46392	−0.731958	−	0.681350i	$-0.761393\pi$
$311$	−2.49699e6	−1.46392	−0.731958	+	0.681350i	$0.761393\pi$
$312$	0	0
$313$	−2.78111e6	−1.60457	−0.802283	−	0.596944i	$-0.796381\pi$
$313$	−2.78111e6	−1.60457	−0.802283	+	0.596944i	$0.796381\pi$
$314$	−2.34921e6	−1.34462
$315$	0	0
$316$	−756834.	−0.426366
$317$	−1.94375e6	−1.08641	−0.543203	−	0.839602i	$-0.682788\pi$
$317$	−1.94375e6	−1.08641	−0.543203	+	0.839602i	$0.682788\pi$
$318$	0	0
$319$	72098.6	0.0396689
$320$	−1.00477e6	−0.548519
$321$	0	0
$322$	3.16955e6	1.70356
$323$	143530.	0.0765487
$324$	0	0
$325$	−2.13461e6	−1.12101
$326$	−1.98538e6	−1.03467
$327$	0	0
$328$	3.59261e6	1.84385
$329$	3.49860e6	1.78199
$330$	0	0
$331$	−60445.1	−0.0303243	−0.0151622	−	0.999885i	$-0.504826\pi$
$331$	−60445.1	−0.0303243	−0.0151622	+	0.999885i	$0.504826\pi$
$332$	293176.	0.145976
$333$	0	0
$334$	−846611.	−0.415258
$335$	299412.	0.145766
$336$	0	0
$337$	−1.72999e6	−0.829791	−0.414895	−	0.909869i	$-0.636182\pi$
$337$	−1.72999e6	−0.829791	−0.414895	+	0.909869i	$0.636182\pi$
$338$	−2.90489e6	−1.38305
$339$	0	0
$340$	149598.	0.0701822
$341$	−684398.	−0.318730
$342$	0	0
$343$	−3.34184e6	−1.53373
$344$	402705.	0.183481
$345$	0	0
$346$	−397126.	−0.178335
$347$	−1.17667e6	−0.524605	−0.262303	−	0.964986i	$-0.584482\pi$
$347$	−1.17667e6	−0.524605	−0.262303	+	0.964986i	$0.584482\pi$
$348$	0	0
$349$	1.20893e6	0.531297	0.265648	−	0.964070i	$-0.414414\pi$
$349$	1.20893e6	0.531297	0.265648	+	0.964070i	$0.414414\pi$
$350$	−1.89721e6	−0.827838
$351$	0	0
$352$	−411982.	−0.177224
$353$	−1.00723e6	−0.430223	−0.215111	−	0.976589i	$-0.569011\pi$
$353$	−1.00723e6	−0.430223	−0.215111	+	0.976589i	$0.569011\pi$
$354$	0	0
$355$	−630978.	−0.265732
$356$	1.91746e6	0.801865
$357$	0	0
$358$	−1.36113e6	−0.561297
$359$	−1.10145e6	−0.451055	−0.225527	−	0.974237i	$-0.572411\pi$
$359$	−1.10145e6	−0.451055	−0.225527	+	0.974237i	$0.572411\pi$
$360$	0	0
$361$	−2.26585e6	−0.915087
$362$	−451953.	−0.181268
$363$	0	0
$364$	3.35032e6	1.32536
$365$	1.85347e6	0.728204
$366$	0	0
$367$	2.99288e6	1.15991	0.579955	−	0.814648i	$-0.303070\pi$
$367$	2.99288e6	1.15991	0.579955	+	0.814648i	$0.303070\pi$
$368$	−1.16934e6	−0.450114
$369$	0	0
$370$	20689.0	0.00785663
$371$	2.04024e6	0.769568
$372$	0	0
$373$	4.23178e6	1.57489	0.787446	−	0.616384i	$-0.211403\pi$
$373$	4.23178e6	1.57489	0.787446	+	0.616384i	$0.211403\pi$
$374$	−111697.	−0.0412917
$375$	0	0
$376$	−3.07844e6	−1.12295
$377$	869608.	0.315116
$378$	0	0
$379$	1.06268e6	0.380018	0.190009	−	0.981782i	$-0.439148\pi$
$379$	1.06268e6	0.380018	0.190009	+	0.981782i	$0.439148\pi$
$380$	219141.	0.0778510
$381$	0	0
$382$	−1.31421e6	−0.460794
$383$	3.75736e6	1.30884	0.654418	−	0.756133i	$-0.272913\pi$
$383$	3.75736e6	1.30884	0.654418	+	0.756133i	$0.272913\pi$
$384$	0	0
$385$	616445.	0.211955
$386$	−172455.	−0.0589125
$387$	0	0
$388$	−639875.	−0.215782
$389$	3.88561e6	1.30192	0.650961	−	0.759112i	$-0.274366\pi$
$389$	3.88561e6	1.30192	0.650961	+	0.759112i	$0.274366\pi$
$390$	0	0
$391$	1.07956e6	0.357111
$392$	6.20571e6	2.03975
$393$	0	0
$394$	1.47655e6	0.479189
$395$	−1.67960e6	−0.541643
$396$	0	0
$397$	719537.	0.229127	0.114564	−	0.993416i	$-0.463453\pi$
$397$	719537.	0.229127	0.114564	+	0.993416i	$0.463453\pi$
$398$	3.95061e6	1.25013
$399$	0	0
$400$	699939.	0.218731
$401$	1.57519e6	0.489185	0.244592	−	0.969626i	$-0.421346\pi$
$401$	1.57519e6	0.489185	0.244592	+	0.969626i	$0.421346\pi$
$402$	0	0
$403$	−8.25479e6	−2.53188
$404$	−831859.	−0.253569
$405$	0	0
$406$	772894.	0.232705
$407$	13084.4	0.00391533
$408$	0	0
$409$	−3.35355e6	−0.991279	−0.495639	−	0.868528i	$-0.665066\pi$
$409$	−3.35355e6	−0.991279	−0.495639	+	0.868528i	$0.665066\pi$
$410$	2.50672e6	0.736454
$411$	0	0
$412$	1.63551e6	0.474691
$413$	−3.16247e6	−0.912328
$414$	0	0
$415$	650628.	0.185444
$416$	−4.96907e6	−1.40780
$417$	0	0
$418$	−163622.	−0.0458036
$419$	−6.79853e6	−1.89182	−0.945911	−	0.324427i	$-0.894829\pi$
$419$	−6.79853e6	−1.89182	−0.945911	+	0.324427i	$0.894829\pi$
$420$	0	0
$421$	−1.77259e6	−0.487420	−0.243710	−	0.969848i	$-0.578365\pi$
$421$	−1.77259e6	−0.487420	−0.243710	+	0.969848i	$0.578365\pi$
$422$	1.58408e6	0.433008
$423$	0	0
$424$	−1.79522e6	−0.484957
$425$	−646196.	−0.173537
$426$	0	0
$427$	4.32317e6	1.14745
$428$	2.77991e6	0.733536
$429$	0	0
$430$	280984.	0.0732843
$431$	−1.30221e6	−0.337666	−0.168833	−	0.985645i	$-0.554000\pi$
$431$	−1.30221e6	−0.337666	−0.168833	+	0.985645i	$0.554000\pi$
$432$	0	0
$433$	2.15386e6	0.552074	0.276037	−	0.961147i	$-0.410979\pi$
$433$	2.15386e6	0.552074	0.276037	+	0.961147i	$0.410979\pi$
$434$	−7.33673e6	−1.86973
$435$	0	0
$436$	−1.47089e6	−0.370564
$437$	1.58141e6	0.396133
$438$	0	0
$439$	−5.81916e6	−1.44112	−0.720558	−	0.693394i	$-0.756114\pi$
$439$	−5.81916e6	−1.44112	−0.720558	+	0.693394i	$0.756114\pi$
$440$	−542413.	−0.133567
$441$	0	0
$442$	−1.34722e6	−0.328007
$443$	−4.34332e6	−1.05151	−0.525754	−	0.850637i	$-0.676217\pi$
$443$	−4.34332e6	−1.05151	−0.525754	+	0.850637i	$0.676217\pi$
$444$	0	0
$445$	4.25531e6	1.01866
$446$	284549.	0.0677361
$447$	0	0
$448$	−6.81198e6	−1.60354
$449$	−2.68108e6	−0.627615	−0.313807	−	0.949487i	$-0.601605\pi$
$449$	−2.68108e6	−0.627615	−0.313807	+	0.949487i	$0.601605\pi$
$450$	0	0
$451$	1.58533e6	0.367010
$452$	−1.51802e6	−0.349487
$453$	0	0
$454$	−4.49176e6	−1.02277
$455$	7.43518e6	1.68369
$456$	0	0
$457$	−4.29834e6	−0.962743	−0.481371	−	0.876517i	$-0.659861\pi$
$457$	−4.29834e6	−0.962743	−0.481371	+	0.876517i	$0.659861\pi$
$458$	3.01446e6	0.671500
$459$	0	0
$460$	1.64826e6	0.363187
$461$	−2.28441e6	−0.500636	−0.250318	−	0.968164i	$-0.580535\pi$
$461$	−2.28441e6	−0.500636	−0.250318	+	0.968164i	$0.580535\pi$
$462$	0	0
$463$	1.51960e6	0.329440	0.164720	−	0.986340i	$-0.447328\pi$
$463$	1.51960e6	0.329440	0.164720	+	0.986340i	$0.447328\pi$
$464$	−285144.	−0.0614851
$465$	0	0
$466$	5.71536e6	1.21921
$467$	4.71846e6	1.00117	0.500585	−	0.865687i	$-0.333118\pi$
$467$	4.71846e6	1.00117	0.500585	+	0.865687i	$0.333118\pi$
$468$	0	0
$469$	2.02991e6	0.426132
$470$	−2.14796e6	−0.448519
$471$	0	0
$472$	2.78267e6	0.574919
$473$	177703.	0.0365210
$474$	0	0
$475$	−946591.	−0.192499
$476$	1.01422e6	0.205170
$477$	0	0
$478$	4.63879e6	0.928614
$479$	−5.85157e6	−1.16529	−0.582644	−	0.812727i	$-0.697982\pi$
$479$	−5.85157e6	−1.16529	−0.582644	+	0.812727i	$0.697982\pi$
$480$	0	0
$481$	157816.	0.0311020
$482$	6.12108e6	1.20008
$483$	0	0
$484$	2.25554e6	0.437661
$485$	−1.42004e6	−0.274123
$486$	0	0
$487$	1.83575e6	0.350745	0.175372	−	0.984502i	$-0.443887\pi$
$487$	1.83575e6	0.350745	0.175372	+	0.984502i	$0.443887\pi$
$488$	−3.80398e6	−0.723083
$489$	0	0
$490$	4.32999e6	0.814699
$491$	5.75635e6	1.07757	0.538783	−	0.842445i	$-0.318884\pi$
$491$	5.75635e6	1.07757	0.538783	+	0.842445i	$0.318884\pi$
$492$	0	0
$493$	263250.	0.0487811
$494$	−1.97350e6	−0.363848
$495$	0	0
$496$	2.70675e6	0.494019
$497$	−4.27781e6	−0.776838
$498$	0	0
$499$	4.10166e6	0.737409	0.368705	−	0.929547i	$-0.379801\pi$
$499$	4.10166e6	0.737409	0.368705	+	0.929547i	$0.379801\pi$
$500$	−2.48009e6	−0.443652
$501$	0	0
$502$	1.56132e6	0.276523
$503$	1.12157e6	0.197654	0.0988270	−	0.995105i	$-0.468491\pi$
$503$	1.12157e6	0.197654	0.0988270	+	0.995105i	$0.468491\pi$
$504$	0	0
$505$	−1.84610e6	−0.322126
$506$	−1.23067e6	−0.213681
$507$	0	0
$508$	3.63258e6	0.624534
$509$	2.68902e6	0.460045	0.230023	−	0.973185i	$-0.426120\pi$
$509$	2.68902e6	0.460045	0.230023	+	0.973185i	$0.426120\pi$
$510$	0	0
$511$	1.25659e7	2.12883
$512$	3.73721e6	0.630047
$513$	0	0
$514$	−1.96078e6	−0.327356
$515$	3.62960e6	0.603033
$516$	0	0
$517$	−1.35844e6	−0.223518
$518$	140264.	0.0229680
$519$	0	0
$520$	−6.54225e6	−1.06101
$521$	−2.88911e6	−0.466305	−0.233152	−	0.972440i	$-0.574904\pi$
$521$	−2.88911e6	−0.466305	−0.233152	+	0.972440i	$0.574904\pi$
$522$	0	0
$523$	−1.40417e6	−0.224474	−0.112237	−	0.993681i	$-0.535802\pi$
$523$	−1.40417e6	−0.224474	−0.112237	+	0.993681i	$0.535802\pi$
$524$	2.91235e6	0.463356
$525$	0	0
$526$	4.63573e6	0.730557
$527$	−2.49891e6	−0.391945
$528$	0	0
$529$	5.45814e6	0.848019
$530$	−1.25260e6	−0.193697
$531$	0	0
$532$	1.48570e6	0.227589
$533$	1.91212e7	2.91540
$534$	0	0
$535$	6.16930e6	0.931861
$536$	−1.78612e6	−0.268534
$537$	0	0
$538$	−1.58822e6	−0.236568
$539$	2.73843e6	0.406003
$540$	0	0
$541$	−211154.	−0.0310174	−0.0155087	−	0.999880i	$-0.504937\pi$
$541$	−211154.	−0.0310174	−0.0155087	+	0.999880i	$0.504937\pi$
$542$	−4.53007e6	−0.662379
$543$	0	0
$544$	−1.50425e6	−0.217933
$545$	−3.26426e6	−0.470754
$546$	0	0
$547$	−4.61571e6	−0.659584	−0.329792	−	0.944054i	$-0.606979\pi$
$547$	−4.61571e6	−0.659584	−0.329792	+	0.944054i	$0.606979\pi$
$548$	2.07564e6	0.295257
$549$	0	0
$550$	736649.	0.103837
$551$	385627.	0.0541113
$552$	0	0
$553$	−1.13871e7	−1.58343
$554$	9.11378e6	1.26161
$555$	0	0
$556$	4.77416e6	0.654953
$557$	−1.01209e7	−1.38223	−0.691114	−	0.722746i	$-0.742880\pi$
$557$	−1.01209e7	−1.38223	−0.691114	+	0.722746i	$0.742880\pi$
$558$	0	0
$559$	2.14335e6	0.290110
$560$	−2.43800e6	−0.328521
$561$	0	0
$562$	−4.27525e6	−0.570980
$563$	−3.84774e6	−0.511604	−0.255802	−	0.966729i	$-0.582340\pi$
$563$	−3.84774e6	−0.511604	−0.255802	+	0.966729i	$0.582340\pi$
$564$	0	0
$565$	−3.36885e6	−0.443978
$566$	−3.70098e6	−0.485597
$567$	0	0
$568$	3.76406e6	0.489538
$569$	−4.38443e6	−0.567718	−0.283859	−	0.958866i	$-0.591615\pi$
$569$	−4.38443e6	−0.567718	−0.283859	+	0.958866i	$0.591615\pi$
$570$	0	0
$571$	2.61241e6	0.335313	0.167656	−	0.985845i	$-0.446380\pi$
$571$	2.61241e6	0.335313	0.167656	+	0.985845i	$0.446380\pi$
$572$	−1.30086e6	−0.166242
$573$	0	0
$574$	1.69947e7	2.15295
$575$	−7.11974e6	−0.898037
$576$	0	0
$577$	−7.06124e6	−0.882961	−0.441480	−	0.897271i	$-0.645547\pi$
$577$	−7.06124e6	−0.882961	−0.441480	+	0.897271i	$0.645547\pi$
$578$	5.50211e6	0.685030
$579$	0	0
$580$	401928.	0.0496110
$581$	4.41103e6	0.542125
$582$	0	0
$583$	−792186.	−0.0965285
$584$	−1.10568e7	−1.34152
$585$	0	0
$586$	−2.00625e6	−0.241347
$587$	8.18957e6	0.980993	0.490496	−	0.871443i	$-0.336815\pi$
$587$	8.18957e6	0.980993	0.490496	+	0.871443i	$0.336815\pi$
$588$	0	0
$589$	−3.66058e6	−0.434772
$590$	1.94159e6	0.229629
$591$	0	0
$592$	−51747.9	−0.00606860
$593$	9.78828e6	1.14306	0.571531	−	0.820580i	$-0.306350\pi$
$593$	9.78828e6	1.14306	0.571531	+	0.820580i	$0.306350\pi$
$594$	0	0
$595$	2.25080e6	0.260642
$596$	−5.74465e6	−0.662442
$597$	0	0
$598$	−1.48436e7	−1.69741
$599$	−4.44247e6	−0.505892	−0.252946	−	0.967480i	$-0.581400\pi$
$599$	−4.44247e6	−0.505892	−0.252946	+	0.967480i	$0.581400\pi$
$600$	0	0
$601$	248973.	0.0281168	0.0140584	−	0.999901i	$-0.495525\pi$
$601$	248973.	0.0281168	0.0140584	+	0.999901i	$0.495525\pi$
$602$	1.90498e6	0.214239
$603$	0	0
$604$	3.17836e6	0.354495
$605$	5.00560e6	0.555991
$606$	0	0
$607$	−5.61487e6	−0.618540	−0.309270	−	0.950974i	$-0.600085\pi$
$607$	−5.61487e6	−0.618540	−0.309270	+	0.950974i	$0.600085\pi$
$608$	−2.20353e6	−0.241747
$609$	0	0
$610$	−2.65420e6	−0.288808
$611$	−1.63846e7	−1.77555
$612$	0	0
$613$	7.11213e6	0.764449	0.382225	−	0.924069i	$-0.375158\pi$
$613$	7.11213e6	0.764449	0.382225	+	0.924069i	$0.375158\pi$
$614$	−953834.	−0.102106
$615$	0	0
$616$	−3.67737e6	−0.390468
$617$	−872669.	−0.0922862	−0.0461431	−	0.998935i	$-0.514693\pi$
$617$	−872669.	−0.0922862	−0.0461431	+	0.998935i	$0.514693\pi$
$618$	0	0
$619$	−9.40244e6	−0.986312	−0.493156	−	0.869941i	$-0.664157\pi$
$619$	−9.40244e6	−0.986312	−0.493156	+	0.869941i	$0.664157\pi$
$620$	−3.81531e6	−0.398613
$621$	0	0
$622$	1.03934e7	1.07716
$623$	2.88495e7	2.97796
$624$	0	0
$625$	947282.	0.0970017
$626$	1.15760e7	1.18065
$627$	0	0
$628$	−8.28242e6	−0.838027
$629$	47774.5	0.00481470
$630$	0	0
$631$	−1.90990e7	−1.90957	−0.954786	−	0.297293i	$-0.903916\pi$
$631$	−1.90990e7	−1.90957	−0.954786	+	0.297293i	$0.903916\pi$
$632$	1.00196e7	0.997829
$633$	0	0
$634$	8.09056e6	0.799384
$635$	8.06158e6	0.793388
$636$	0	0
$637$	3.30292e7	3.22515
$638$	−300099.	−0.0291886
$639$	0	0
$640$	−825936.	−0.0797070
$641$	4.06757e6	0.391012	0.195506	−	0.980703i	$-0.437365\pi$
$641$	4.06757e6	0.391012	0.195506	+	0.980703i	$0.437365\pi$
$642$	0	0
$643$	−1.00899e6	−0.0962410	−0.0481205	−	0.998842i	$-0.515323\pi$
$643$	−1.00899e6	−0.0962410	−0.0481205	+	0.998842i	$0.515323\pi$
$644$	1.11746e7	1.06174
$645$	0	0
$646$	−597424.	−0.0563250
$647$	−1.99986e7	−1.87819	−0.939095	−	0.343657i	$-0.888334\pi$
$647$	−1.99986e7	−1.87819	−0.939095	+	0.343657i	$0.888334\pi$
$648$	0	0
$649$	1.22792e6	0.114435
$650$	8.88500e6	0.824848
$651$	0	0
$652$	−6.99970e6	−0.644852
$653$	1.16599e7	1.07007	0.535035	−	0.844830i	$-0.320298\pi$
$653$	1.16599e7	1.07007	0.535035	+	0.844830i	$0.320298\pi$
$654$	0	0
$655$	6.46320e6	0.588633
$656$	−6.26986e6	−0.568850
$657$	0	0
$658$	−1.45624e7	−1.31120
$659$	−1.40204e7	−1.25761	−0.628806	−	0.777562i	$-0.716456\pi$
$659$	−1.40204e7	−1.25761	−0.628806	+	0.777562i	$0.716456\pi$
$660$	0	0
$661$	5.39932e6	0.480657	0.240328	−	0.970692i	$-0.422745\pi$
$661$	5.39932e6	0.480657	0.240328	+	0.970692i	$0.422745\pi$
$662$	251594.	0.0223128
$663$	0	0
$664$	−3.88129e6	−0.341630
$665$	3.29712e6	0.289122
$666$	0	0
$667$	2.90047e6	0.252438
$668$	−2.98482e6	−0.258808
$669$	0	0
$670$	−1.24626e6	−0.107256
$671$	−1.67860e6	−0.143926
$672$	0	0
$673$	2.37236e6	0.201903	0.100951	−	0.994891i	$-0.467811\pi$
$673$	2.37236e6	0.201903	0.100951	+	0.994891i	$0.467811\pi$
$674$	7.20082e6	0.610565
$675$	0	0
$676$	−1.02415e7	−0.861983
$677$	−1.48279e7	−1.24339	−0.621695	−	0.783260i	$-0.713556\pi$
$677$	−1.48279e7	−1.24339	−0.621695	+	0.783260i	$0.713556\pi$
$678$	0	0
$679$	−9.62736e6	−0.801369
$680$	−1.98049e6	−0.164248
$681$	0	0
$682$	2.84871e6	0.234524
$683$	1.45710e7	1.19519	0.597594	−	0.801799i	$-0.296123\pi$
$683$	1.45710e7	1.19519	0.597594	+	0.801799i	$0.296123\pi$
$684$	0	0
$685$	4.60635e6	0.375086
$686$	1.39099e7	1.12853
$687$	0	0
$688$	−702804.	−0.0566061
$689$	−9.55485e6	−0.766789
$690$	0	0
$691$	−2.39101e6	−0.190496	−0.0952480	−	0.995454i	$-0.530364\pi$
$691$	−2.39101e6	−0.190496	−0.0952480	+	0.995454i	$0.530364\pi$
$692$	−1.40011e6	−0.111147
$693$	0	0
$694$	4.89773e6	0.386008
$695$	1.05950e7	0.832033
$696$	0	0
$697$	5.78844e6	0.451315
$698$	−5.03198e6	−0.390932
$699$	0	0
$700$	−6.68883e6	−0.515947
$701$	2.67685e6	0.205745	0.102872	−	0.994695i	$-0.467197\pi$
$701$	2.67685e6	0.205745	0.102872	+	0.994695i	$0.467197\pi$
$702$	0	0
$703$	69983.4	0.00534080
$704$	2.64496e6	0.201135
$705$	0	0
$706$	4.19246e6	0.316561
$707$	−1.25159e7	−0.941701
$708$	0	0
$709$	1.61177e7	1.20417	0.602085	−	0.798432i	$-0.294337\pi$
$709$	1.61177e7	1.20417	0.602085	+	0.798432i	$0.294337\pi$
$710$	2.62635e6	0.195527
$711$	0	0
$712$	−2.53848e7	−1.87661
$713$	−2.75329e7	−2.02828
$714$	0	0
$715$	−2.88693e6	−0.211189
$716$	−4.79883e6	−0.349826
$717$	0	0
$718$	4.58463e6	0.331889
$719$	−2.74012e7	−1.97673	−0.988363	−	0.152111i	$-0.951393\pi$
$719$	−2.74012e7	−1.97673	−0.988363	+	0.152111i	$0.951393\pi$
$720$	0	0
$721$	2.46074e7	1.76290
$722$	9.43124e6	0.673327
$723$	0	0
$724$	−1.59341e6	−0.112975
$725$	−1.73615e6	−0.122671
$726$	0	0
$727$	−1.44905e7	−1.01683	−0.508413	−	0.861113i	$-0.669768\pi$
$727$	−1.44905e7	−1.01683	−0.508413	+	0.861113i	$0.669768\pi$
$728$	−4.43542e7	−3.10175
$729$	0	0
$730$	−7.71478e6	−0.535817
$731$	648840.	0.0449101
$732$	0	0
$733$	9.24243e6	0.635370	0.317685	−	0.948196i	$-0.397095\pi$
$733$	9.24243e6	0.635370	0.317685	+	0.948196i	$0.397095\pi$
$734$	−1.24574e7	−0.853470
$735$	0	0
$736$	−1.65737e7	−1.12778
$737$	−788172.	−0.0534505
$738$	0	0
$739$	−270924.	−0.0182489	−0.00912445	−	0.999958i	$-0.502904\pi$
$739$	−270924.	−0.0182489	−0.00912445	+	0.999958i	$0.502904\pi$
$740$	72941.6	0.00489662
$741$	0	0
$742$	−8.49221e6	−0.566253
$743$	−1.56128e7	−1.03755	−0.518776	−	0.854910i	$-0.673612\pi$
$743$	−1.56128e7	−1.03755	−0.518776	+	0.854910i	$0.673612\pi$
$744$	0	0
$745$	−1.27488e7	−0.841545
$746$	−1.76141e7	−1.15882
$747$	0	0
$748$	−393801.	−0.0257349
$749$	4.18257e7	2.72420
$750$	0	0
$751$	2.81796e6	0.182320	0.0911600	−	0.995836i	$-0.470943\pi$
$751$	2.81796e6	0.182320	0.0911600	+	0.995836i	$0.470943\pi$
$752$	5.37251e6	0.346444
$753$	0	0
$754$	−3.61961e6	−0.231864
$755$	7.05355e6	0.450340
$756$	0	0
$757$	1.47622e6	0.0936293	0.0468146	−	0.998904i	$-0.485093\pi$
$757$	1.47622e6	0.0936293	0.0468146	+	0.998904i	$0.485093\pi$
$758$	−4.42325e6	−0.279620
$759$	0	0
$760$	−2.90116e6	−0.182195
$761$	1.59710e7	0.999700	0.499850	−	0.866112i	$-0.333388\pi$
$761$	1.59710e7	0.999700	0.499850	+	0.866112i	$0.333388\pi$
$762$	0	0
$763$	−2.21305e7	−1.37620
$764$	−4.63340e6	−0.287188
$765$	0	0
$766$	−1.56394e7	−0.963051
$767$	1.48104e7	0.909033
$768$	0	0
$769$	−2.69038e7	−1.64058	−0.820291	−	0.571947i	$-0.806188\pi$
$769$	−2.69038e7	−1.64058	−0.820291	+	0.571947i	$0.806188\pi$
$770$	−2.56586e6	−0.155958
$771$	0	0
$772$	−608009.	−0.0367170
$773$	−2.10506e7	−1.26711	−0.633556	−	0.773697i	$-0.718405\pi$
$773$	−2.10506e7	−1.26711	−0.633556	+	0.773697i	$0.718405\pi$
$774$	0	0
$775$	1.64805e7	0.985633
$776$	8.47116e6	0.504997
$777$	0	0
$778$	−1.61732e7	−0.957962
$779$	8.47930e6	0.500629
$780$	0	0
$781$	1.66099e6	0.0974403
$782$	−4.49349e6	−0.262765
$783$	0	0
$784$	−1.08303e7	−0.629288
$785$	−1.83807e7	−1.06460
$786$	0	0
$787$	−1.46428e7	−0.842728	−0.421364	−	0.906892i	$-0.638448\pi$
$787$	−1.46428e7	−0.842728	−0.421364	+	0.906892i	$0.638448\pi$
$788$	5.20574e6	0.298653
$789$	0	0
$790$	6.99108e6	0.398544
$791$	−2.28396e7	−1.29792
$792$	0	0
$793$	−2.02462e7	−1.14330
$794$	−2.99496e6	−0.168593
$795$	0	0
$796$	1.39283e7	0.779141
$797$	1.53622e7	0.856660	0.428330	−	0.903622i	$-0.359102\pi$
$797$	1.53622e7	0.856660	0.428330	+	0.903622i	$0.359102\pi$
$798$	0	0
$799$	−4.96000e6	−0.274862
$800$	9.92063e6	0.548042
$801$	0	0
$802$	−6.55651e6	−0.359945
$803$	−4.87907e6	−0.267023
$804$	0	0
$805$	2.47991e7	1.34880
$806$	3.43593e7	1.86298
$807$	0	0
$808$	1.10128e7	0.593430
$809$	7.26215e6	0.390116	0.195058	−	0.980792i	$-0.437510\pi$
$809$	7.26215e6	0.390116	0.195058	+	0.980792i	$0.437510\pi$
$810$	0	0
$811$	4.17728e6	0.223019	0.111510	−	0.993763i	$-0.464431\pi$
$811$	4.17728e6	0.223019	0.111510	+	0.993763i	$0.464431\pi$
$812$	2.72493e6	0.145032
$813$	0	0
$814$	−54461.9	−0.00288092
$815$	−1.55340e7	−0.819200
$816$	0	0
$817$	950466.	0.0498174
$818$	1.39586e7	0.729390
$819$	0	0
$820$	8.83772e6	0.458993
$821$	−2.70285e7	−1.39947	−0.699736	−	0.714402i	$-0.746699\pi$
$821$	−2.70285e7	−1.39947	−0.699736	+	0.714402i	$0.746699\pi$
$822$	0	0
$823$	−6.95032e6	−0.357689	−0.178844	−	0.983877i	$-0.557236\pi$
$823$	−6.95032e6	−0.357689	−0.178844	+	0.983877i	$0.557236\pi$
$824$	−2.16522e7	−1.11092
$825$	0	0
$826$	1.31633e7	0.671297
$827$	370060.	0.0188152	0.00940759	−	0.999956i	$-0.497005\pi$
$827$	370060.	0.0188152	0.00940759	+	0.999956i	$0.497005\pi$
$828$	0	0
$829$	1.38451e7	0.699698	0.349849	−	0.936806i	$-0.386233\pi$
$829$	1.38451e7	0.699698	0.349849	+	0.936806i	$0.386233\pi$
$830$	−2.70814e6	−0.136451
$831$	0	0
$832$	3.19018e7	1.59774
$833$	9.99869e6	0.499264
$834$	0	0
$835$	−6.62405e6	−0.328781
$836$	−576867.	−0.0285469
$837$	0	0
$838$	2.82979e7	1.39201
$839$	−1.86174e7	−0.913089	−0.456544	−	0.889701i	$-0.650913\pi$
$839$	−1.86174e7	−0.913089	−0.456544	+	0.889701i	$0.650913\pi$
$840$	0	0
$841$	707281.	0.0344828
$842$	7.37815e6	0.358647
$843$	0	0
$844$	5.58486e6	0.269871
$845$	−2.27285e7	−1.09504
$846$	0	0
$847$	3.39362e7	1.62538
$848$	3.13304e6	0.149615
$849$	0	0
$850$	2.68969e6	0.127690
$851$	526376.	0.0249157
$852$	0	0
$853$	2.15389e6	0.101356	0.0506782	−	0.998715i	$-0.483862\pi$
$853$	2.15389e6	0.101356	0.0506782	+	0.998715i	$0.483862\pi$
$854$	−1.79945e7	−0.844298
$855$	0	0
$856$	−3.68026e7	−1.71670
$857$	3.33513e7	1.55118	0.775588	−	0.631239i	$-0.217453\pi$
$857$	3.33513e7	1.55118	0.775588	+	0.631239i	$0.217453\pi$
$858$	0	0
$859$	−3.11061e7	−1.43834	−0.719171	−	0.694833i	$-0.755478\pi$
$859$	−3.11061e7	−1.43834	−0.719171	+	0.694833i	$0.755478\pi$
$860$	990643.	0.0456742
$861$	0	0
$862$	5.42025e6	0.248457
$863$	1.59387e7	0.728494	0.364247	−	0.931302i	$-0.381326\pi$
$863$	1.59387e7	0.728494	0.364247	+	0.931302i	$0.381326\pi$
$864$	0	0
$865$	−3.10719e6	−0.141198
$866$	−8.96511e6	−0.406220
$867$	0	0
$868$	−2.58665e7	−1.16530
$869$	4.42138e6	0.198613
$870$	0	0
$871$	−9.50644e6	−0.424593
$872$	1.94728e7	0.867235
$873$	0	0
$874$	−6.58238e6	−0.291477
$875$	−3.73147e7	−1.64763
$876$	0	0
$877$	666578.	0.0292652	0.0146326	−	0.999893i	$-0.495342\pi$
$877$	666578.	0.0292652	0.0146326	+	0.999893i	$0.495342\pi$
$878$	2.42214e7	1.06038
$879$	0	0
$880$	946625.	0.0412070
$881$	2.30910e7	1.00231	0.501155	−	0.865358i	$-0.332909\pi$
$881$	2.30910e7	1.00231	0.501155	+	0.865358i	$0.332909\pi$
$882$	0	0
$883$	3.85060e6	0.166198	0.0830992	−	0.996541i	$-0.473518\pi$
$883$	3.85060e6	0.166198	0.0830992	+	0.996541i	$0.473518\pi$
$884$	−4.74978e6	−0.204429
$885$	0	0
$886$	1.80784e7	0.773706
$887$	4.14426e7	1.76863	0.884316	−	0.466889i	$-0.154625\pi$
$887$	4.14426e7	1.76863	0.884316	+	0.466889i	$0.154625\pi$
$888$	0	0
$889$	5.46547e7	2.31939
$890$	−1.77121e7	−0.749540
$891$	0	0
$892$	1.00321e6	0.0422163
$893$	−7.26574e6	−0.304896
$894$	0	0
$895$	−1.06498e7	−0.444409
$896$	−5.59955e6	−0.233015
$897$	0	0
$898$	1.11596e7	0.461803
$899$	−6.71389e6	−0.277061
$900$	0	0
$901$	−2.89247e6	−0.118702
$902$	−6.59869e6	−0.270048
$903$	0	0
$904$	2.00967e7	0.817907
$905$	−3.53617e6	−0.143520
$906$	0	0
$907$	−4.63163e7	−1.86946	−0.934729	−	0.355363i	$-0.884357\pi$
$907$	−4.63163e7	−1.86946	−0.934729	+	0.355363i	$0.884357\pi$
$908$	−1.58362e7	−0.637436
$909$	0	0
$910$	−3.09478e7	−1.23887
$911$	−2.69515e6	−0.107594	−0.0537969	−	0.998552i	$-0.517132\pi$
$911$	−2.69515e6	−0.107594	−0.0537969	+	0.998552i	$0.517132\pi$
$912$	0	0
$913$	−1.71272e6	−0.0679999
$914$	1.78912e7	0.708392
$915$	0	0
$916$	1.06278e7	0.418510
$917$	4.38183e7	1.72081
$918$	0	0
$919$	1.39792e7	0.546000	0.273000	−	0.962014i	$-0.411984\pi$
$919$	1.39792e7	0.546000	0.273000	+	0.962014i	$0.411984\pi$
$920$	−2.18209e7	−0.849969
$921$	0	0
$922$	9.50852e6	0.368371
$923$	2.00338e7	0.774032
$924$	0	0
$925$	−315076.	−0.0121077
$926$	−6.32510e6	−0.242404
$927$	0	0
$928$	−4.04151e6	−0.154054
$929$	7.25580e6	0.275833	0.137916	−	0.990444i	$-0.455959\pi$
$929$	7.25580e6	0.275833	0.137916	+	0.990444i	$0.455959\pi$
$930$	0	0
$931$	1.46468e7	0.553819
$932$	2.01502e7	0.759869
$933$	0	0
$934$	−1.96399e7	−0.736667
$935$	−873940.	−0.0326928
$936$	0	0
$937$	3.95850e6	0.147293	0.0736464	−	0.997284i	$-0.476536\pi$
$937$	3.95850e6	0.147293	0.0736464	+	0.997284i	$0.476536\pi$
$938$	−8.44918e6	−0.313550
$939$	0	0
$940$	−7.57287e6	−0.279538
$941$	4.07672e7	1.50085	0.750425	−	0.660956i	$-0.229849\pi$
$941$	4.07672e7	1.50085	0.750425	+	0.660956i	$0.229849\pi$
$942$	0	0
$943$	6.37766e7	2.33551
$944$	−4.85635e6	−0.177370
$945$	0	0
$946$	−739664.	−0.0268724
$947$	2.67878e7	0.970650	0.485325	−	0.874334i	$-0.338701\pi$
$947$	2.67878e7	0.970650	0.485325	+	0.874334i	$0.338701\pi$
$948$	0	0
$949$	−5.88484e7	−2.12114
$950$	3.94004e6	0.141642
$951$	0	0
$952$	−1.34270e7	−0.480162
$953$	1.81430e7	0.647107	0.323553	−	0.946210i	$-0.395123\pi$
$953$	1.81430e7	0.647107	0.323553	+	0.946210i	$0.395123\pi$
$954$	0	0
$955$	−1.02827e7	−0.364835
$956$	1.63546e7	0.578755
$957$	0	0
$958$	2.43563e7	0.857427
$959$	3.12294e7	1.09652
$960$	0	0
$961$	3.51028e7	1.22612
$962$	−656886.	−0.0228851
$963$	0	0
$964$	2.15806e7	0.747946
$965$	−1.34932e6	−0.0466441
$966$	0	0
$967$	−1.40529e7	−0.483282	−0.241641	−	0.970366i	$-0.577686\pi$
$967$	−1.40529e7	−0.483282	−0.241641	+	0.970366i	$0.577686\pi$
$968$	−2.98607e7	−1.02426
$969$	0	0
$970$	5.91069e6	0.201702
$971$	−1.77622e7	−0.604573	−0.302287	−	0.953217i	$-0.597750\pi$
$971$	−1.77622e7	−0.604573	−0.302287	+	0.953217i	$0.597750\pi$
$972$	0	0
$973$	7.18306e7	2.43236
$974$	−7.64103e6	−0.258080
$975$	0	0
$976$	6.63873e6	0.223080
$977$	1.66625e7	0.558474	0.279237	−	0.960222i	$-0.409918\pi$
$977$	1.66625e7	0.558474	0.279237	+	0.960222i	$0.409918\pi$
$978$	0	0
$979$	−1.12017e7	−0.373531
$980$	1.52659e7	0.507758
$981$	0	0
$982$	−2.39600e7	−0.792880
$983$	2.28571e7	0.754461	0.377231	−	0.926119i	$-0.376876\pi$
$983$	2.28571e7	0.754461	0.377231	+	0.926119i	$0.376876\pi$
$984$	0	0
$985$	1.15528e7	0.379399
$986$	−1.09574e6	−0.0358934
$987$	0	0
$988$	−6.95781e6	−0.226767
$989$	7.14888e6	0.232406
$990$	0	0
$991$	−2.52864e7	−0.817904	−0.408952	−	0.912556i	$-0.634106\pi$
$991$	−2.52864e7	−0.817904	−0.408952	+	0.912556i	$0.634106\pi$
$992$	3.83642e7	1.23779
$993$	0	0
$994$	1.78057e7	0.571602
$995$	3.09103e7	0.989796
$996$	0	0
$997$	4.44014e7	1.41468	0.707340	−	0.706873i	$-0.249895\pi$
$997$	4.44014e7	1.41468	0.707340	+	0.706873i	$0.249895\pi$
$998$	−1.70726e7	−0.542590
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	261.6.a.a.1.2		4
3.2	odd	2		29.6.a.a.1.3	✓	4
12.11	even	2		464.6.a.i.1.3		4
15.14	odd	2		725.6.a.a.1.2		4
87.86	odd	2		841.6.a.a.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.a.a.1.3	✓	4	3.2	odd	2
261.6.a.a.1.2		4	1.1	even	1	trivial
464.6.a.i.1.3		4	12.11	even	2
725.6.a.a.1.2		4	15.14	odd	2
841.6.a.a.1.2		4	87.86	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:	\( N \)	\(=\)	\( 261 = 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	261.a (trivial)

\(f(q)\)	\(=\)	\(q-4.16235 q^{2} -14.6748 q^{4} -32.5670 q^{5} -220.793 q^{7} +194.277 q^{8} +O(q^{10})\)	\(q-4.16235 q^{2} -14.6748 q^{4} -32.5670 q^{5} -220.793 q^{7} +194.277 q^{8} +135.555 q^{10} +85.7296 q^{11} +1034.02 q^{13} +919.018 q^{14} -339.054 q^{16} +313.020 q^{17} +458.534 q^{19} +477.916 q^{20} -356.836 q^{22} +3448.84 q^{23} -2064.39 q^{25} -4303.94 q^{26} +3240.10 q^{28} +841.000 q^{29} -7983.23 q^{31} -4805.60 q^{32} -1302.90 q^{34} +7190.58 q^{35} +152.624 q^{37} -1908.58 q^{38} -6327.03 q^{40} +18492.2 q^{41} +2072.84 q^{43} -1258.07 q^{44} -14355.3 q^{46} -15845.6 q^{47} +31942.6 q^{49} +8592.70 q^{50} -15174.0 q^{52} -9240.52 q^{53} -2791.96 q^{55} -42895.0 q^{56} -3500.54 q^{58} +14323.2 q^{59} -19580.2 q^{61} +33229.0 q^{62} +30852.3 q^{64} -33674.9 q^{65} -9193.70 q^{67} -4593.53 q^{68} -29929.7 q^{70} +19374.7 q^{71} -56912.4 q^{73} -635.276 q^{74} -6728.91 q^{76} -18928.5 q^{77} +51573.6 q^{79} +11042.0 q^{80} -76971.0 q^{82} -19978.1 q^{83} -10194.2 q^{85} -8627.87 q^{86} +16655.3 q^{88} -130663. q^{89} -228304. q^{91} -50611.1 q^{92} +65954.9 q^{94} -14933.1 q^{95} +43603.5 q^{97} -132956. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{4} + 68 q^{5} - 208 q^{7} + 504 q^{8}+O(q^{10}) \) 4 * q + 10 * q^4 + 68 * q^5 - 208 * q^7 + 504 * q^8	\( 4 q + 10 q^{4} + 68 q^{5} - 208 q^{7} + 504 q^{8} - 788 q^{10} + 124 q^{11} - 460 q^{13} - 768 q^{14} - 414 q^{16} - 184 q^{17} - 2392 q^{19} - 2822 q^{20} + 5538 q^{22} + 1192 q^{23} + 1824 q^{25} - 4724 q^{26} + 44 q^{28} + 3364 q^{29} - 19212 q^{31} - 6552 q^{32} - 7612 q^{34} + 22944 q^{35} - 10928 q^{37} + 456 q^{38} - 20 q^{40} + 1120 q^{41} - 21420 q^{43} + 1932 q^{44} - 7588 q^{46} - 23772 q^{47} + 10452 q^{49} - 43240 q^{50} - 29062 q^{52} - 8860 q^{53} - 52652 q^{55} - 34304 q^{56} + 10840 q^{59} + 49448 q^{61} - 18518 q^{62} - 20734 q^{64} - 97836 q^{65} - 7840 q^{67} - 20724 q^{68} - 77496 q^{70} + 48744 q^{71} - 74992 q^{73} + 35920 q^{74} - 140792 q^{76} - 128656 q^{77} - 106076 q^{79} - 58638 q^{80} - 234132 q^{82} - 62888 q^{83} + 23848 q^{85} + 216014 q^{86} - 39426 q^{88} - 107568 q^{89} - 268896 q^{91} + 26268 q^{92} + 30542 q^{94} - 147352 q^{95} - 49520 q^{97} - 242304 q^{98}+O(q^{100}) \) 4 * q + 10 * q^4 + 68 * q^5 - 208 * q^7 + 504 * q^8 - 788 * q^10 + 124 * q^11 - 460 * q^13 - 768 * q^14 - 414 * q^16 - 184 * q^17 - 2392 * q^19 - 2822 * q^20 + 5538 * q^22 + 1192 * q^23 + 1824 * q^25 - 4724 * q^26 + 44 * q^28 + 3364 * q^29 - 19212 * q^31 - 6552 * q^32 - 7612 * q^34 + 22944 * q^35 - 10928 * q^37 + 456 * q^38 - 20 * q^40 + 1120 * q^41 - 21420 * q^43 + 1932 * q^44 - 7588 * q^46 - 23772 * q^47 + 10452 * q^49 - 43240 * q^50 - 29062 * q^52 - 8860 * q^53 - 52652 * q^55 - 34304 * q^56 + 10840 * q^59 + 49448 * q^61 - 18518 * q^62 - 20734 * q^64 - 97836 * q^65 - 7840 * q^67 - 20724 * q^68 - 77496 * q^70 + 48744 * q^71 - 74992 * q^73 + 35920 * q^74 - 140792 * q^76 - 128656 * q^77 - 106076 * q^79 - 58638 * q^80 - 234132 * q^82 - 62888 * q^83 + 23848 * q^85 + 216014 * q^86 - 39426 * q^88 - 107568 * q^89 - 268896 * q^91 + 26268 * q^92 + 30542 * q^94 - 147352 * q^95 - 49520 * q^97 - 242304 * q^98

Introduction

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