LMFDB - Embedded newform 325.6.a.j.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(1,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, 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("325.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.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:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 5 x^{10} - 257 x^{9} + 1165 x^{8} + 22234 x^{7} - 90282 x^{6} - 751180 x^{5} + 2564400 x^{4} + \cdots + 44115200 $ x^11 - 5x^10 - 257x^9 + 1165x^8 + 22234x^7 - 90282x^6 - 751180x^5 + 2564400x^4 + 9571584x^3 - 23191072x^2 - 33154496x + 44115200
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 5^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.89241$ of defining polynomial
Character	$\chi$	$=$	325.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.89241 q^{2} +3.45730 q^{3} -23.6340 q^{4} -9.99993 q^{6} +148.288 q^{7} +160.916 q^{8} -231.047 q^{9} -712.658 q^{11} -81.7096 q^{12} -169.000 q^{13} -428.910 q^{14} +290.850 q^{16} +1131.92 q^{17} +668.283 q^{18} +1400.39 q^{19} +512.676 q^{21} +2061.30 q^{22} -897.571 q^{23} +556.336 q^{24} +488.818 q^{26} -1638.92 q^{27} -3504.63 q^{28} +3238.07 q^{29} -7976.53 q^{31} -5990.58 q^{32} -2463.87 q^{33} -3273.98 q^{34} +5460.56 q^{36} -4978.35 q^{37} -4050.50 q^{38} -584.284 q^{39} -15559.9 q^{41} -1482.87 q^{42} +2308.61 q^{43} +16842.9 q^{44} +2596.15 q^{46} +7602.52 q^{47} +1005.56 q^{48} +5182.35 q^{49} +3913.39 q^{51} +3994.14 q^{52} +14138.1 q^{53} +4740.44 q^{54} +23862.0 q^{56} +4841.56 q^{57} -9365.83 q^{58} +49808.7 q^{59} -2516.69 q^{61} +23071.4 q^{62} -34261.5 q^{63} +8020.03 q^{64} +7126.54 q^{66} +38549.3 q^{67} -26751.8 q^{68} -3103.17 q^{69} +68008.5 q^{71} -37179.2 q^{72} +21305.2 q^{73} +14399.4 q^{74} -33096.7 q^{76} -105679. q^{77} +1689.99 q^{78} +3984.60 q^{79} +50478.2 q^{81} +45005.8 q^{82} +13876.6 q^{83} -12116.6 q^{84} -6677.44 q^{86} +11195.0 q^{87} -114678. q^{88} +89289.8 q^{89} -25060.7 q^{91} +21213.1 q^{92} -27577.2 q^{93} -21989.6 q^{94} -20711.2 q^{96} +147549. q^{97} -14989.5 q^{98} +164658. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 5 q^{2} - 11 q^{3} + 187 q^{4} + 351 q^{6} - 208 q^{7} - 165 q^{8} + 1372 q^{9} + 1276 q^{11} - 1533 q^{12} - 1859 q^{13} + 578 q^{14} + 5707 q^{16} - 2218 q^{17} + 6776 q^{18} + 3520 q^{19} + 1706 q^{21}+ \cdots + 426698 q^{99}+O(q^{100}) $ 11 * q - 5 * q^2 - 11 * q^3 + 187 * q^4 + 351 * q^6 - 208 * q^7 - 165 * q^8 + 1372 * q^9 + 1276 * q^11 - 1533 * q^12 - 1859 * q^13 + 578 * q^14 + 5707 * q^16 - 2218 * q^17 + 6776 * q^18 + 3520 * q^19 + 1706 * q^21 + 6785 * q^22 + 1215 * q^23 + 7723 * q^24 + 845 * q^26 - 161 * q^27 - 12746 * q^28 + 18425 * q^29 + 9068 * q^31 - 21165 * q^32 - 19748 * q^33 - 14455 * q^34 + 37684 * q^36 + 2280 * q^37 + 23783 * q^38 + 1859 * q^39 - 10010 * q^41 + 44184 * q^42 - 6763 * q^43 + 60651 * q^44 - 9858 * q^46 - 4110 * q^47 - 63289 * q^48 + 73313 * q^49 + 36606 * q^51 - 31603 * q^52 - 15633 * q^53 + 48013 * q^54 - 1262 * q^56 + 205974 * q^57 - 77652 * q^58 + 131642 * q^59 - 43365 * q^61 + 115940 * q^62 - 218466 * q^63 + 180071 * q^64 - 58607 * q^66 + 100904 * q^67 - 371277 * q^68 + 59345 * q^69 + 68420 * q^71 + 210708 * q^72 - 124436 * q^73 + 371396 * q^74 + 22517 * q^76 + 150978 * q^77 - 59319 * q^78 + 52239 * q^79 + 107507 * q^81 + 351137 * q^82 - 211538 * q^83 + 852716 * q^84 + 202246 * q^86 + 28273 * q^87 + 28107 * q^88 + 327712 * q^89 + 35152 * q^91 + 413252 * q^92 - 243338 * q^93 + 649974 * q^94 + 899663 * q^96 + 110796 * q^97 - 27143 * q^98 + 426698 * q^99

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 $

$2$	−2.89241	−0.511311	−0.255656	−	0.966768i	$-0.582291\pi$
$2$	−2.89241	−0.511311	−0.255656	+	0.966768i	$0.582291\pi$
$3$	3.45730	0.221786	0.110893	−	0.993832i	$-0.464629\pi$
$3$	3.45730	0.221786	0.110893	+	0.993832i	$0.464629\pi$
$4$	−23.6340	−0.738561
$5$	0	0
$5$	0	0
$6$	−9.99993	−0.113402
$7$	148.288	1.14383	0.571914	−	0.820313i	$-0.306201\pi$
$7$	148.288	1.14383	0.571914	+	0.820313i	$0.306201\pi$
$8$	160.916	0.888945
$9$	−231.047	−0.950811
$10$	0	0
$11$	−712.658	−1.77582	−0.887912	−	0.460014i	$-0.847844\pi$
$11$	−712.658	−1.77582	−0.887912	+	0.460014i	$0.847844\pi$
$12$	−81.7096	−0.163802
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	−428.910	−0.584852
$15$	0	0
$16$	290.850	0.284033
$17$	1131.92	0.949935	0.474968	−	0.880003i	$-0.342460\pi$
$17$	1131.92	0.949935	0.474968	+	0.880003i	$0.342460\pi$
$18$	668.283	0.486160
$19$	1400.39	0.889947	0.444974	−	0.895544i	$-0.353213\pi$
$19$	1400.39	0.889947	0.444974	+	0.895544i	$0.353213\pi$
$20$	0	0
$21$	512.676	0.253685
$22$	2061.30	0.907998
$23$	−897.571	−0.353793	−0.176896	−	0.984229i	$-0.556606\pi$
$23$	−897.571	−0.353793	−0.176896	+	0.984229i	$0.556606\pi$
$24$	556.336	0.197156
$25$	0	0
$26$	488.818	0.141812
$27$	−1638.92	−0.432662
$28$	−3504.63	−0.844787
$29$	3238.07	0.714975	0.357488	−	0.933918i	$-0.383633\pi$
$29$	3238.07	0.714975	0.357488	+	0.933918i	$0.383633\pi$
$30$	0	0
$31$	−7976.53	−1.49077	−0.745383	−	0.666636i	$-0.767733\pi$
$31$	−7976.53	−1.49077	−0.745383	+	0.666636i	$0.767733\pi$
$32$	−5990.58	−1.03417
$33$	−2463.87	−0.393852
$34$	−3273.98	−0.485712
$35$	0	0
$36$	5460.56	0.702232
$37$	−4978.35	−0.597835	−0.298917	−	0.954279i	$-0.596626\pi$
$37$	−4978.35	−0.597835	−0.298917	+	0.954279i	$0.596626\pi$
$38$	−4050.50	−0.455040
$39$	−584.284	−0.0615123
$40$	0	0
$41$	−15559.9	−1.44560	−0.722800	−	0.691057i	$-0.757145\pi$
$41$	−15559.9	−1.44560	−0.722800	+	0.691057i	$0.757145\pi$
$42$	−1482.87	−0.129712
$43$	2308.61	0.190405	0.0952026	−	0.995458i	$-0.469650\pi$
$43$	2308.61	0.190405	0.0952026	+	0.995458i	$0.469650\pi$
$44$	16842.9	1.31155
$45$	0	0
$46$	2596.15	0.180898
$47$	7602.52	0.502011	0.251005	−	0.967986i	$-0.419239\pi$
$47$	7602.52	0.502011	0.251005	+	0.967986i	$0.419239\pi$
$48$	1005.56	0.0629946
$49$	5182.35	0.308344
$50$	0	0
$51$	3913.39	0.210682
$52$	3994.14	0.204840
$53$	14138.1	0.691357	0.345678	−	0.938353i	$-0.387649\pi$
$53$	14138.1	0.691357	0.345678	+	0.938353i	$0.387649\pi$
$54$	4740.44	0.221225
$55$	0	0
$56$	23862.0	1.01680
$57$	4841.56	0.197378
$58$	−9365.83	−0.365575
$59$	49808.7	1.86284	0.931419	−	0.363948i	$-0.118572\pi$
$59$	49808.7	1.86284	0.931419	+	0.363948i	$0.118572\pi$
$60$	0	0
$61$	−2516.69	−0.0865973	−0.0432986	−	0.999062i	$-0.513787\pi$
$61$	−2516.69	−0.0865973	−0.0432986	+	0.999062i	$0.513787\pi$
$62$	23071.4	0.762245
$63$	−34261.5	−1.08757
$64$	8020.03	0.244752
$65$	0	0
$66$	7126.54	0.201381
$67$	38549.3	1.04913	0.524566	−	0.851370i	$-0.324228\pi$
$67$	38549.3	1.04913	0.524566	+	0.851370i	$0.324228\pi$
$68$	−26751.8	−0.701585
$69$	−3103.17	−0.0784663
$70$	0	0
$71$	68008.5	1.60110	0.800548	−	0.599269i	$-0.204542\pi$
$71$	68008.5	1.60110	0.800548	+	0.599269i	$0.204542\pi$
$72$	−37179.2	−0.845219
$73$	21305.2	0.467927	0.233963	−	0.972245i	$-0.424830\pi$
$73$	21305.2	0.467927	0.233963	+	0.972245i	$0.424830\pi$
$74$	14399.4	0.305680
$75$	0	0
$76$	−33096.7	−0.657280
$77$	−105679.	−2.03124
$78$	1689.99	0.0314519
$79$	3984.60	0.0718318	0.0359159	−	0.999355i	$-0.488565\pi$
$79$	3984.60	0.0718318	0.0359159	+	0.999355i	$0.488565\pi$
$80$	0	0
$81$	50478.2	0.854853
$82$	45005.8	0.739152
$83$	13876.6	0.221100	0.110550	−	0.993871i	$-0.464739\pi$
$83$	13876.6	0.221100	0.110550	+	0.993871i	$0.464739\pi$
$84$	−12116.6	−0.187362
$85$	0	0
$86$	−6677.44	−0.0973563
$87$	11195.0	0.158571
$88$	−114678.	−1.57861
$89$	89289.8	1.19489	0.597443	−	0.801911i	$-0.296183\pi$
$89$	89289.8	1.19489	0.597443	+	0.801911i	$0.296183\pi$
$90$	0	0
$91$	−25060.7	−0.317241
$92$	21213.1	0.261298
$93$	−27577.2	−0.330631
$94$	−21989.6	−0.256684
$95$	0	0
$96$	−20711.2	−0.229365
$97$	147549.	1.59223	0.796116	−	0.605145i	$-0.206885\pi$
$97$	147549.	1.59223	0.796116	+	0.605145i	$0.206885\pi$
$98$	−14989.5	−0.157660
$99$	164658.	1.68847
$100$	0	0
$101$	−10084.2	−0.0983642	−0.0491821	−	0.998790i	$-0.515661\pi$
$101$	−10084.2	−0.0983642	−0.0491821	+	0.998790i	$0.515661\pi$
$102$	−11319.1	−0.107724
$103$	−175080.	−1.62608	−0.813041	−	0.582206i	$-0.802190\pi$
$103$	−175080.	−1.62608	−0.813041	+	0.582206i	$0.802190\pi$
$104$	−27194.9	−0.246549
$105$	0	0
$106$	−40893.3	−0.353498
$107$	225117.	1.90086	0.950428	−	0.310946i	$-0.100646\pi$
$107$	225117.	1.90086	0.950428	+	0.310946i	$0.100646\pi$
$108$	38734.2	0.319547
$109$	25083.5	0.202219	0.101109	−	0.994875i	$-0.467761\pi$
$109$	25083.5	0.202219	0.101109	+	0.994875i	$0.467761\pi$
$110$	0	0
$111$	−17211.6	−0.132591
$112$	43129.6	0.324885
$113$	39258.9	0.289229	0.144615	−	0.989488i	$-0.453806\pi$
$113$	39258.9	0.289229	0.144615	+	0.989488i	$0.453806\pi$
$114$	−14003.8	−0.100921
$115$	0	0
$116$	−76528.3	−0.528053
$117$	39047.0	0.263708
$118$	−144067.	−0.952490
$119$	167850.	1.08656
$120$	0	0
$121$	346831.	2.15355
$122$	7279.29	0.0442782
$123$	−53795.4	−0.320614
$124$	188517.	1.10102
$125$	0	0
$126$	99098.4	0.556084
$127$	267258.	1.47035	0.735177	−	0.677875i	$-0.237099\pi$
$127$	267258.	1.47035	0.735177	+	0.677875i	$0.237099\pi$
$128$	168501.	0.909031
$129$	7981.55	0.0422292
$130$	0	0
$131$	−157114.	−0.799903	−0.399952	−	0.916536i	$-0.630973\pi$
$131$	−157114.	−0.799903	−0.399952	+	0.916536i	$0.630973\pi$
$132$	58231.1	0.290884
$133$	207661.	1.01795
$134$	−111501.	−0.536433
$135$	0	0
$136$	182145.	0.844441
$137$	−132383.	−0.602603	−0.301302	−	0.953529i	$-0.597421\pi$
$137$	−132383.	−0.602603	−0.301302	+	0.953529i	$0.597421\pi$
$138$	8975.65	0.0401207
$139$	−285040.	−1.25132	−0.625661	−	0.780095i	$-0.715171\pi$
$139$	−285040.	−1.25132	−0.625661	+	0.780095i	$0.715171\pi$
$140$	0	0
$141$	26284.2	0.111339
$142$	−196709.	−0.818658
$143$	120439.	0.492525
$144$	−67200.1	−0.270062
$145$	0	0
$146$	−61623.4	−0.239256
$147$	17916.9	0.0683864
$148$	117658.	0.441537
$149$	−156686.	−0.578180	−0.289090	−	0.957302i	$-0.593353\pi$
$149$	−156686.	−0.578180	−0.289090	+	0.957302i	$0.593353\pi$
$150$	0	0
$151$	422557.	1.50814	0.754072	−	0.656792i	$-0.228087\pi$
$151$	422557.	1.50814	0.754072	+	0.656792i	$0.228087\pi$
$152$	225345.	0.791115
$153$	−261527.	−0.903209
$154$	305666.	1.03859
$155$	0	0
$156$	13808.9	0.0454306
$157$	129540.	0.419427	0.209713	−	0.977763i	$-0.432747\pi$
$157$	129540.	0.419427	0.209713	+	0.977763i	$0.432747\pi$
$158$	−11525.1	−0.0367284
$159$	48879.7	0.153333
$160$	0	0
$161$	−133099.	−0.404679
$162$	−146004.	−0.437096
$163$	−616763.	−1.81823	−0.909116	−	0.416542i	$-0.863242\pi$
$163$	−616763.	−1.81823	−0.909116	+	0.416542i	$0.863242\pi$
$164$	367743.	1.06766
$165$	0	0
$166$	−40136.9	−0.113051
$167$	−512670.	−1.42248	−0.711241	−	0.702948i	$-0.751867\pi$
$167$	−512670.	−1.42248	−0.711241	+	0.702948i	$0.751867\pi$
$168$	82498.0	0.225512
$169$	28561.0	0.0769231
$170$	0	0
$171$	−323556.	−0.846172
$172$	−54561.5	−0.140626
$173$	462817.	1.17569	0.587846	−	0.808973i	$-0.299976\pi$
$173$	462817.	1.17569	0.587846	+	0.808973i	$0.299976\pi$
$174$	−32380.5	−0.0810793
$175$	0	0
$176$	−207277.	−0.504393
$177$	172204.	0.413151
$178$	−258263.	−0.610959
$179$	413926.	0.965584	0.482792	−	0.875735i	$-0.339623\pi$
$179$	413926.	0.965584	0.482792	+	0.875735i	$0.339623\pi$
$180$	0	0
$181$	−167089.	−0.379099	−0.189549	−	0.981871i	$-0.560703\pi$
$181$	−167089.	−0.379099	−0.189549	+	0.981871i	$0.560703\pi$
$182$	72485.8	0.162209
$183$	−8700.93	−0.0192061
$184$	−144434.	−0.314503
$185$	0	0
$186$	79764.7	0.169055
$187$	−806673.	−1.68692
$188$	−179678.	−0.370766
$189$	−243033.	−0.494892
$190$	0	0
$191$	30723.8	0.0609385	0.0304692	−	0.999536i	$-0.490300\pi$
$191$	30723.8	0.0609385	0.0304692	+	0.999536i	$0.490300\pi$
$192$	27727.6	0.0542825
$193$	1.01259e6	1.95677	0.978387	−	0.206784i	$-0.0662996\pi$
$193$	1.01259e6	1.95677	0.978387	+	0.206784i	$0.0662996\pi$
$194$	−426772.	−0.814125
$195$	0	0
$196$	−122479.	−0.227731
$197$	422275.	0.775229	0.387614	−	0.921822i	$-0.373299\pi$
$197$	422275.	0.775229	0.387614	+	0.921822i	$0.373299\pi$
$198$	−476258.	−0.863335
$199$	553506.	0.990808	0.495404	−	0.868663i	$-0.335020\pi$
$199$	553506.	0.990808	0.495404	+	0.868663i	$0.335020\pi$
$200$	0	0
$201$	133277.	0.232683
$202$	29167.6	0.0502947
$203$	480167.	0.817809
$204$	−92488.9	−0.155602
$205$	0	0
$206$	506402.	0.831434
$207$	207381.	0.336390
$208$	−49153.7	−0.0787767
$209$	−997998.	−1.58039
$210$	0	0
$211$	−758171.	−1.17236	−0.586180	−	0.810181i	$-0.699369\pi$
$211$	−758171.	−1.17236	−0.586180	+	0.810181i	$0.699369\pi$
$212$	−334140.	−0.510609
$213$	235126.	0.355100
$214$	−651132.	−0.971928
$215$	0	0
$216$	−263729.	−0.384613
$217$	−1.18282e6	−1.70518
$218$	−72551.8	−0.103397
$219$	73658.4	0.103780
$220$	0	0
$221$	−191295.	−0.263465
$222$	49783.2	0.0677954
$223$	142765.	0.192247	0.0961236	−	0.995369i	$-0.469356\pi$
$223$	142765.	0.192247	0.0961236	+	0.995369i	$0.469356\pi$
$224$	−888332.	−1.18292
$225$	0	0
$226$	−113553.	−0.147886
$227$	−578969.	−0.745746	−0.372873	−	0.927882i	$-0.621627\pi$
$227$	−578969.	−0.745746	−0.372873	+	0.927882i	$0.621627\pi$
$228$	−114425.	−0.145775
$229$	−242404.	−0.305457	−0.152729	−	0.988268i	$-0.548806\pi$
$229$	−242404.	−0.305457	−0.152729	+	0.988268i	$0.548806\pi$
$230$	0	0
$231$	−365363.	−0.450500
$232$	521058.	0.635574
$233$	540278.	0.651970	0.325985	−	0.945375i	$-0.394304\pi$
$233$	540278.	0.651970	0.325985	+	0.945375i	$0.394304\pi$
$234$	−112940.	−0.134837
$235$	0	0
$236$	−1.17718e6	−1.37582
$237$	13776.0	0.0159313
$238$	−485493.	−0.555572
$239$	−223034.	−0.252567	−0.126284	−	0.991994i	$-0.540305\pi$
$239$	−223034.	−0.252567	−0.126284	+	0.991994i	$0.540305\pi$
$240$	0	0
$241$	1.31596e6	1.45948	0.729742	−	0.683723i	$-0.239640\pi$
$241$	1.31596e6	1.45948	0.729742	+	0.683723i	$0.239640\pi$
$242$	−1.00318e6	−1.10113
$243$	572776.	0.622256
$244$	59479.2	0.0639574
$245$	0	0
$246$	155598.	0.163933
$247$	−236666.	−0.246827
$248$	−1.28355e6	−1.32521
$249$	47975.6	0.0490368
$250$	0	0
$251$	475886.	0.476781	0.238390	−	0.971169i	$-0.423380\pi$
$251$	475886.	0.476781	0.238390	+	0.971169i	$0.423380\pi$
$252$	809735.	0.803233
$253$	639661.	0.628274
$254$	−773021.	−0.751808
$255$	0	0
$256$	−744016.	−0.709549
$257$	184678.	0.174415	0.0872074	−	0.996190i	$-0.472206\pi$
$257$	184678.	0.174415	0.0872074	+	0.996190i	$0.472206\pi$
$258$	−23085.9	−0.0215923
$259$	−738230.	−0.683821
$260$	0	0
$261$	−748146.	−0.679806
$262$	454439.	0.408999
$263$	−541222.	−0.482487	−0.241244	−	0.970465i	$-0.577555\pi$
$263$	−541222.	−0.482487	−0.241244	+	0.970465i	$0.577555\pi$
$264$	−396477.	−0.350113
$265$	0	0
$266$	−600641.	−0.520488
$267$	308702.	0.265009
$268$	−911073.	−0.774848
$269$	198419.	0.167187	0.0835934	−	0.996500i	$-0.473360\pi$
$269$	198419.	0.167187	0.0835934	+	0.996500i	$0.473360\pi$
$270$	0	0
$271$	−258403.	−0.213734	−0.106867	−	0.994273i	$-0.534082\pi$
$271$	−258403.	−0.213734	−0.106867	+	0.994273i	$0.534082\pi$
$272$	329219.	0.269813
$273$	−86642.3	−0.0703596
$274$	382907.	0.308118
$275$	0	0
$276$	73340.2	0.0579521
$277$	−2.19835e6	−1.72146	−0.860729	−	0.509063i	$-0.829992\pi$
$277$	−2.19835e6	−1.72146	−0.860729	+	0.509063i	$0.829992\pi$
$278$	824454.	0.639815
$279$	1.84295e6	1.41744
$280$	0	0
$281$	−842.387	−0.000636423 0	−0.000318211	−	1.00000i	$-0.500101\pi$
$281$	−842.387	−0.000636423 0	−0.000318211		1.00000i	$0.500101\pi$
$282$	−76024.7	−0.0569288
$283$	1.93205e6	1.43401	0.717006	−	0.697067i	$-0.245512\pi$
$283$	1.93205e6	1.43401	0.717006	+	0.697067i	$0.245512\pi$
$284$	−1.60731e6	−1.18251
$285$	0	0
$286$	−348360.	−0.251833
$287$	−2.30735e6	−1.65352
$288$	1.38411e6	0.983305
$289$	−138611.	−0.0976230
$290$	0	0
$291$	510120.	0.353134
$292$	−503526.	−0.345593
$293$	1.01321e6	0.689493	0.344747	−	0.938696i	$-0.387965\pi$
$293$	1.01321e6	0.689493	0.344747	+	0.938696i	$0.387965\pi$
$294$	−51823.1	−0.0349667
$295$	0	0
$296$	−801098.	−0.531443
$297$	1.16799e6	0.768332
$298$	453199.	0.295630
$299$	151689.	0.0981245
$300$	0	0
$301$	342339.	0.217791
$302$	−1.22221e6	−0.771131
$303$	−34864.0	−0.0218158
$304$	407303.	0.252775
$305$	0	0
$306$	756444.	0.461821
$307$	386274.	0.233910	0.116955	−	0.993137i	$-0.462687\pi$
$307$	386274.	0.233910	0.116955	+	0.993137i	$0.462687\pi$
$308$	2.49761e6	1.50019
$309$	−605303.	−0.360642
$310$	0	0
$311$	1.37666e6	0.807099	0.403550	−	0.914958i	$-0.367776\pi$
$311$	1.37666e6	0.807099	0.403550	+	0.914958i	$0.367776\pi$
$312$	−94020.8	−0.0546811
$313$	−541373.	−0.312346	−0.156173	−	0.987730i	$-0.549916\pi$
$313$	−541373.	−0.312346	−0.156173	+	0.987730i	$0.549916\pi$
$314$	−374684.	−0.214458
$315$	0	0
$316$	−94171.8	−0.0530522
$317$	1.91853e6	1.07231	0.536154	−	0.844120i	$-0.319876\pi$
$317$	1.91853e6	1.07231	0.536154	+	0.844120i	$0.319876\pi$
$318$	−141380.	−0.0784009
$319$	−2.30764e6	−1.26967
$320$	0	0
$321$	778297.	0.421583
$322$	384977.	0.206917
$323$	1.58513e6	0.845392
$324$	−1.19300e6	−0.631361
$325$	0	0
$326$	1.78393e6	0.929683
$327$	86721.2	0.0448493
$328$	−2.50385e6	−1.28506
$329$	1.12736e6	0.574215
$330$	0	0
$331$	−2.40144e6	−1.20477	−0.602383	−	0.798207i	$-0.705782\pi$
$331$	−2.40144e6	−1.20477	−0.602383	+	0.798207i	$0.705782\pi$
$332$	−327959.	−0.163296
$333$	1.15023e6	0.568428
$334$	1.48285e6	0.727331
$335$	0	0
$336$	149112.	0.0720550
$337$	3.84325e6	1.84342	0.921709	−	0.387882i	$-0.126793\pi$
$337$	3.84325e6	1.84342	0.921709	+	0.387882i	$0.126793\pi$
$338$	−82610.2	−0.0393316
$339$	135730.	0.0641469
$340$	0	0
$341$	5.68454e6	2.64734
$342$	935856.	0.432657
$343$	−1.72380e6	−0.791136
$344$	371493.	0.169260
$345$	0	0
$346$	−1.33866e6	−0.601144
$347$	−1.02696e6	−0.457858	−0.228929	−	0.973443i	$-0.573522\pi$
$347$	−1.02696e6	−0.457858	−0.228929	+	0.973443i	$0.573522\pi$
$348$	−264581.	−0.117115
$349$	1.24908e6	0.548944	0.274472	−	0.961595i	$-0.411497\pi$
$349$	1.24908e6	0.548944	0.274472	+	0.961595i	$0.411497\pi$
$350$	0	0
$351$	276978.	0.119999
$352$	4.26924e6	1.83651
$353$	−3.13090e6	−1.33731	−0.668654	−	0.743573i	$-0.733129\pi$
$353$	−3.13090e6	−1.33731	−0.668654	+	0.743573i	$0.733129\pi$
$354$	−498084.	−0.211249
$355$	0	0
$356$	−2.11027e6	−0.882497
$357$	580309.	0.240984
$358$	−1.19724e6	−0.493714
$359$	−2.63535e6	−1.07920	−0.539600	−	0.841921i	$-0.681425\pi$
$359$	−2.63535e6	−1.07920	−0.539600	+	0.841921i	$0.681425\pi$
$360$	0	0
$361$	−515013.	−0.207994
$362$	483291.	0.193837
$363$	1.19910e6	0.477626
$364$	592283.	0.234302
$365$	0	0
$366$	25166.7	0.00982027
$367$	−2.23239e6	−0.865175	−0.432588	−	0.901592i	$-0.642399\pi$
$367$	−2.23239e6	−0.865175	−0.432588	+	0.901592i	$0.642399\pi$
$368$	−261059.	−0.100489
$369$	3.59508e6	1.37449
$370$	0	0
$371$	2.09651e6	0.790794
$372$	651759.	0.244191
$373$	−3.72332e6	−1.38566	−0.692832	−	0.721099i	$-0.743637\pi$
$373$	−3.72332e6	−1.38566	−0.692832	+	0.721099i	$0.743637\pi$
$374$	2.33323e6	0.862539
$375$	0	0
$376$	1.22337e6	0.446260
$377$	−547233.	−0.198298
$378$	702951.	0.253044
$379$	−2.11153e6	−0.755089	−0.377545	−	0.925991i	$-0.623232\pi$
$379$	−2.11153e6	−0.755089	−0.377545	+	0.925991i	$0.623232\pi$
$380$	0	0
$381$	923992.	0.326104
$382$	−88865.9	−0.0311585
$383$	−414336.	−0.144330	−0.0721648	−	0.997393i	$-0.522991\pi$
$383$	−414336.	−0.144330	−0.0721648	+	0.997393i	$0.522991\pi$
$384$	582560.	0.201610
$385$	0	0
$386$	−2.92883e6	−1.00052
$387$	−533397.	−0.181039
$388$	−3.48716e6	−1.17596
$389$	4.10572e6	1.37567	0.687837	−	0.725865i	$-0.258560\pi$
$389$	4.10572e6	1.37567	0.687837	+	0.725865i	$0.258560\pi$
$390$	0	0
$391$	−1.01598e6	−0.336080
$392$	833924.	0.274101
$393$	−543191.	−0.177407
$394$	−1.22139e6	−0.396383
$395$	0	0
$396$	−3.89151e6	−1.24704
$397$	−5.78249e6	−1.84136	−0.920681	−	0.390317i	$-0.872365\pi$
$397$	−5.78249e6	−1.84136	−0.920681	+	0.390317i	$0.872365\pi$
$398$	−1.60097e6	−0.506611
$399$	717945.	0.225766
$400$	0	0
$401$	−3.48722e6	−1.08298	−0.541488	−	0.840708i	$-0.682139\pi$
$401$	−3.48722e6	−1.08298	−0.541488	+	0.840708i	$0.682139\pi$
$402$	−385491.	−0.118973
$403$	1.34803e6	0.413464
$404$	238329.	0.0726479
$405$	0	0
$406$	−1.38884e6	−0.418155
$407$	3.54786e6	1.06165
$408$	629729.	0.187285
$409$	−3.23381e6	−0.955886	−0.477943	−	0.878391i	$-0.658618\pi$
$409$	−3.23381e6	−0.955886	−0.477943	+	0.878391i	$0.658618\pi$
$410$	0	0
$411$	−457688.	−0.133649
$412$	4.13782e6	1.20096
$413$	7.38603e6	2.13077
$414$	−599832.	−0.172000
$415$	0	0
$416$	1.01241e6	0.286829
$417$	−985470.	−0.277526
$418$	2.88662e6	0.808070
$419$	3.63720e6	1.01212	0.506060	−	0.862498i	$-0.331101\pi$
$419$	3.63720e6	1.01212	0.506060	+	0.862498i	$0.331101\pi$
$420$	0	0
$421$	2.11596e6	0.581839	0.290920	−	0.956747i	$-0.406039\pi$
$421$	2.11596e6	0.581839	0.290920	+	0.956747i	$0.406039\pi$
$422$	2.19294e6	0.599441
$423$	−1.75654e6	−0.477318
$424$	2.27505e6	0.614578
$425$	0	0
$426$	−680081.	−0.181567
$427$	−373194.	−0.0990525
$428$	−5.32041e6	−1.40390
$429$	416395.	0.109235
$430$	0	0
$431$	6.14425e6	1.59322	0.796609	−	0.604495i	$-0.206625\pi$
$431$	6.14425e6	1.59322	0.796609	+	0.604495i	$0.206625\pi$
$432$	−476681.	−0.122890
$433$	−6.08667e6	−1.56013	−0.780064	−	0.625700i	$-0.784813\pi$
$433$	−6.08667e6	−1.56013	−0.780064	+	0.625700i	$0.784813\pi$
$434$	3.42121e6	0.871878
$435$	0	0
$436$	−592822.	−0.149351
$437$	−1.25695e6	−0.314857
$438$	−213050.	−0.0530636
$439$	5.46357e6	1.35305	0.676527	−	0.736418i	$-0.263484\pi$
$439$	5.46357e6	1.35305	0.676527	+	0.736418i	$0.263484\pi$
$440$	0	0
$441$	−1.19737e6	−0.293177
$442$	553303.	0.134712
$443$	−4.25637e6	−1.03046	−0.515229	−	0.857052i	$-0.672293\pi$
$443$	−4.25637e6	−1.03046	−0.515229	+	0.857052i	$0.672293\pi$
$444$	406779.	0.0979268
$445$	0	0
$446$	−412935.	−0.0982981
$447$	−541709.	−0.128232
$448$	1.18927e6	0.279954
$449$	3.07736e6	0.720382	0.360191	−	0.932879i	$-0.382712\pi$
$449$	3.07736e6	0.720382	0.360191	+	0.932879i	$0.382712\pi$
$450$	0	0
$451$	1.10889e7	2.56713
$452$	−927843.	−0.213613
$453$	1.46091e6	0.334485
$454$	1.67462e6	0.381308
$455$	0	0
$456$	779086.	0.175458
$457$	−1.92830e6	−0.431901	−0.215951	−	0.976404i	$-0.569285\pi$
$457$	−1.92830e6	−0.431901	−0.215951	+	0.976404i	$0.569285\pi$
$458$	701132.	0.156184
$459$	−1.85513e6	−0.411001
$460$	0	0
$461$	1.53125e6	0.335578	0.167789	−	0.985823i	$-0.446337\pi$
$461$	1.53125e6	0.335578	0.167789	+	0.985823i	$0.446337\pi$
$462$	1.05678e6	0.230346
$463$	−2.13296e6	−0.462412	−0.231206	−	0.972905i	$-0.574267\pi$
$463$	−2.13296e6	−0.462412	−0.231206	+	0.972905i	$0.574267\pi$
$464$	941792.	0.203077
$465$	0	0
$466$	−1.56271e6	−0.333359
$467$	−2.47557e6	−0.525270	−0.262635	−	0.964895i	$-0.584592\pi$
$467$	−2.47557e6	−0.525270	−0.262635	+	0.964895i	$0.584592\pi$
$468$	−922834.	−0.194764
$469$	5.71641e6	1.20003
$470$	0	0
$471$	447860.	0.0930230
$472$	8.01503e6	1.65596
$473$	−1.64525e6	−0.338126
$474$	−39845.7	−0.00814584
$475$	0	0
$476$	−3.96697e6	−0.802493
$477$	−3.26657e6	−0.657349
$478$	645108.	0.129140
$479$	6.52504e6	1.29940	0.649702	−	0.760189i	$-0.274894\pi$
$479$	6.52504e6	1.29940	0.649702	+	0.760189i	$0.274894\pi$
$480$	0	0
$481$	841341.	0.165810
$482$	−3.80629e6	−0.746250
$483$	−460163.	−0.0897520
$484$	−8.19699e6	−1.59053
$485$	0	0
$486$	−1.65671e6	−0.318167
$487$	5.50061e6	1.05097	0.525483	−	0.850804i	$-0.323885\pi$
$487$	5.50061e6	1.05097	0.525483	+	0.850804i	$0.323885\pi$
$488$	−404976.	−0.0769803
$489$	−2.13233e6	−0.403258
$490$	0	0
$491$	8.54562e6	1.59970	0.799852	−	0.600197i	$-0.204911\pi$
$491$	8.54562e6	1.59970	0.799852	+	0.600197i	$0.204911\pi$
$492$	1.27140e6	0.236793
$493$	3.66524e6	0.679180
$494$	684534.	0.126205
$495$	0	0
$496$	−2.31997e6	−0.423427
$497$	1.00848e7	1.83138
$498$	−138765.	−0.0250730
$499$	2.98189e6	0.536094	0.268047	−	0.963406i	$-0.413622\pi$
$499$	2.98189e6	0.536094	0.268047	+	0.963406i	$0.413622\pi$
$500$	0	0
$501$	−1.77245e6	−0.315486
$502$	−1.37646e6	−0.243783
$503$	4.24939e6	0.748871	0.374436	−	0.927253i	$-0.377836\pi$
$503$	4.24939e6	0.748871	0.374436	+	0.927253i	$0.377836\pi$
$504$	−5.51324e6	−0.966786
$505$	0	0
$506$	−1.85016e6	−0.321243
$507$	98743.9	0.0170604
$508$	−6.31637e6	−1.08595
$509$	−1.22747e6	−0.209999	−0.105000	−	0.994472i	$-0.533484\pi$
$509$	−1.22747e6	−0.209999	−0.105000	+	0.994472i	$0.533484\pi$
$510$	0	0
$511$	3.15930e6	0.535228
$512$	−3.24004e6	−0.546230
$513$	−2.29513e6	−0.385047
$514$	−534166.	−0.0891802
$515$	0	0
$516$	−188635.	−0.0311888
$517$	−5.41800e6	−0.891483
$518$	2.13527e6	0.349645
$519$	1.60010e6	0.260752
$520$	0	0
$521$	−7.91981e6	−1.27826	−0.639131	−	0.769097i	$-0.720706\pi$
$521$	−7.91981e6	−1.27826	−0.639131	+	0.769097i	$0.720706\pi$
$522$	2.16395e6	0.347592
$523$	4.76548e6	0.761821	0.380910	−	0.924612i	$-0.375611\pi$
$523$	4.76548e6	0.761821	0.380910	+	0.924612i	$0.375611\pi$
$524$	3.71323e6	0.590777
$525$	0	0
$526$	1.56544e6	0.246701
$527$	−9.02880e6	−1.41613
$528$	−716618.	−0.111867
$529$	−5.63071e6	−0.874831
$530$	0	0
$531$	−1.15082e7	−1.77121
$532$	−4.90784e6	−0.751816
$533$	2.62963e6	0.400938
$534$	−892892.	−0.135502
$535$	0	0
$536$	6.20322e6	0.932621
$537$	1.43107e6	0.214153
$538$	−573909.	−0.0854845
$539$	−3.69324e6	−0.547565
$540$	0	0
$541$	−5.85407e6	−0.859934	−0.429967	−	0.902845i	$-0.641475\pi$
$541$	−5.85407e6	−0.859934	−0.429967	+	0.902845i	$0.641475\pi$
$542$	747408.	0.109285
$543$	−577678.	−0.0840787
$544$	−6.78087e6	−0.982399
$545$	0	0
$546$	250605.	0.0359756
$547$	587042.	0.0838882	0.0419441	−	0.999120i	$-0.486645\pi$
$547$	587042.	0.0838882	0.0419441	+	0.999120i	$0.486645\pi$
$548$	3.12874e6	0.445059
$549$	581473.	0.0823377
$550$	0	0
$551$	4.53455e6	0.636290
$552$	−499351.	−0.0697522
$553$	590868.	0.0821633
$554$	6.35852e6	0.880201
$555$	0	0
$556$	6.73663e6	0.924178
$557$	−3.23702e6	−0.442086	−0.221043	−	0.975264i	$-0.570946\pi$
$557$	−3.23702e6	−0.442086	−0.221043	+	0.975264i	$0.570946\pi$
$558$	−5.33058e6	−0.724751
$559$	−390155.	−0.0528089
$560$	0	0
$561$	−2.78891e6	−0.374134
$562$	2436.53	0.000325410 0
$563$	1.09630e7	1.45767	0.728835	−	0.684689i	$-0.240062\pi$
$563$	1.09630e7	1.45767	0.728835	+	0.684689i	$0.240062\pi$
$564$	−621200.	−0.0822306
$565$	0	0
$566$	−5.58829e6	−0.733226
$567$	7.48531e6	0.977805
$568$	1.09437e7	1.42329
$569$	4.35413e6	0.563795	0.281897	−	0.959445i	$-0.409036\pi$
$569$	4.35413e6	0.563795	0.281897	+	0.959445i	$0.409036\pi$
$570$	0	0
$571$	4.59961e6	0.590379	0.295189	−	0.955439i	$-0.404617\pi$
$571$	4.59961e6	0.590379	0.295189	+	0.955439i	$0.404617\pi$
$572$	−2.84646e6	−0.363760
$573$	106221.	0.0135153
$574$	6.67382e6	0.845463
$575$	0	0
$576$	−1.85300e6	−0.232713
$577$	−7.57061e6	−0.946654	−0.473327	−	0.880887i	$-0.656947\pi$
$577$	−7.57061e6	−0.946654	−0.473327	+	0.880887i	$0.656947\pi$
$578$	400919.	0.0499157
$579$	3.50083e6	0.433985
$580$	0	0
$581$	2.05773e6	0.252900
$582$	−1.47548e6	−0.180561
$583$	−1.00757e7	−1.22773
$584$	3.42835e6	0.415962
$585$	0	0
$586$	−2.93062e6	−0.352546
$587$	1.48215e7	1.77541	0.887703	−	0.460416i	$-0.152300\pi$
$587$	1.48215e7	1.77541	0.887703	+	0.460416i	$0.152300\pi$
$588$	−423448.	−0.0505076
$589$	−1.11702e7	−1.32670
$590$	0	0
$591$	1.45993e6	0.171935
$592$	−1.44795e6	−0.169805
$593$	1.00021e6	0.116803	0.0584017	−	0.998293i	$-0.481400\pi$
$593$	1.00021e6	0.116803	0.0584017	+	0.998293i	$0.481400\pi$
$594$	−3.37831e6	−0.392856
$595$	0	0
$596$	3.70310e6	0.427021
$597$	1.91363e6	0.219747
$598$	−438749.	−0.0501721
$599$	−8.35745e6	−0.951715	−0.475857	−	0.879522i	$-0.657862\pi$
$599$	−8.35745e6	−0.951715	−0.475857	+	0.879522i	$0.657862\pi$
$600$	0	0
$601$	−6.04104e6	−0.682222	−0.341111	−	0.940023i	$-0.610803\pi$
$601$	−6.04104e6	−0.682222	−0.341111	+	0.940023i	$0.610803\pi$
$602$	−990185.	−0.111359
$603$	−8.90671e6	−0.997526
$604$	−9.98669e6	−1.11386
$605$	0	0
$606$	100841.	0.0111547
$607$	−137224.	−0.0151168	−0.00755838	−	0.999971i	$-0.502406\pi$
$607$	−137224.	−0.0151168	−0.00755838	+	0.999971i	$0.502406\pi$
$608$	−8.38914e6	−0.920361
$609$	1.66008e6	0.181378
$610$	0	0
$611$	−1.28483e6	−0.139233
$612$	6.18092e6	0.667075
$613$	1.09895e7	1.18121	0.590606	−	0.806960i	$-0.298889\pi$
$613$	1.09895e7	1.18121	0.590606	+	0.806960i	$0.298889\pi$
$614$	−1.11726e6	−0.119601
$615$	0	0
$616$	−1.70054e7	−1.80566
$617$	3.42268e6	0.361954	0.180977	−	0.983487i	$-0.442074\pi$
$617$	3.42268e6	0.361954	0.180977	+	0.983487i	$0.442074\pi$
$618$	1.75078e6	0.184400
$619$	−1.22543e7	−1.28547	−0.642736	−	0.766088i	$-0.722201\pi$
$619$	−1.22543e7	−1.28547	−0.642736	+	0.766088i	$0.722201\pi$
$620$	0	0
$621$	1.47105e6	0.153073
$622$	−3.98188e6	−0.412679
$623$	1.32406e7	1.36675
$624$	−169939.	−0.0174715
$625$	0	0
$626$	1.56587e6	0.159706
$627$	−3.45038e6	−0.350508
$628$	−3.06155e6	−0.309772
$629$	−5.63510e6	−0.567904
$630$	0	0
$631$	7.99788e6	0.799653	0.399827	−	0.916591i	$-0.369070\pi$
$631$	7.99788e6	0.799653	0.399827	+	0.916591i	$0.369070\pi$
$632$	641187.	0.0638546
$633$	−2.62122e6	−0.260013
$634$	−5.54917e6	−0.548283
$635$	0	0
$636$	−1.15522e6	−0.113246
$637$	−875816.	−0.0855194
$638$	6.67463e6	0.649196
$639$	−1.57132e7	−1.52234
$640$	0	0
$641$	8.18872e6	0.787175	0.393587	−	0.919287i	$-0.371234\pi$
$641$	8.18872e6	0.787175	0.393587	+	0.919287i	$0.371234\pi$
$642$	−2.25116e6	−0.215560
$643$	−1.64743e7	−1.57137	−0.785686	−	0.618626i	$-0.787690\pi$
$643$	−1.64743e7	−1.57137	−0.785686	+	0.618626i	$0.787690\pi$
$644$	3.14566e6	0.298880
$645$	0	0
$646$	−4.58485e6	−0.432258
$647$	−3.65606e6	−0.343362	−0.171681	−	0.985153i	$-0.554920\pi$
$647$	−3.65606e6	−0.343362	−0.171681	+	0.985153i	$0.554920\pi$
$648$	8.12277e6	0.759917
$649$	−3.54966e7	−3.30807
$650$	0	0
$651$	−4.08938e6	−0.378185
$652$	1.45766e7	1.34288
$653$	1.22553e7	1.12471	0.562355	−	0.826896i	$-0.309896\pi$
$653$	1.22553e7	1.12471	0.562355	+	0.826896i	$0.309896\pi$
$654$	−250833.	−0.0229319
$655$	0	0
$656$	−4.52561e6	−0.410599
$657$	−4.92250e6	−0.444910
$658$	−3.26080e6	−0.293602
$659$	708926.	0.0635898	0.0317949	−	0.999494i	$-0.489878\pi$
$659$	708926.	0.0635898	0.0317949	+	0.999494i	$0.489878\pi$
$660$	0	0
$661$	1.69284e7	1.50700	0.753500	−	0.657448i	$-0.228364\pi$
$661$	1.69284e7	1.50700	0.753500	+	0.657448i	$0.228364\pi$
$662$	6.94597e6	0.616010
$663$	−661363.	−0.0584327
$664$	2.23297e6	0.196546
$665$	0	0
$666$	−3.32695e6	−0.290644
$667$	−2.90639e6	−0.252953
$668$	1.21164e7	1.05059
$669$	493582.	0.0426377
$670$	0	0
$671$	1.79354e6	0.153781
$672$	−3.07123e6	−0.262355
$673$	−1.00330e7	−0.853873	−0.426937	−	0.904281i	$-0.640407\pi$
$673$	−1.00330e7	−0.853873	−0.426937	+	0.904281i	$0.640407\pi$
$674$	−1.11163e7	−0.942560
$675$	0	0
$676$	−675009.	−0.0568124
$677$	−1.54623e7	−1.29659	−0.648296	−	0.761388i	$-0.724518\pi$
$677$	−1.54623e7	−1.29659	−0.648296	+	0.761388i	$0.724518\pi$
$678$	−392586.	−0.0327990
$679$	2.18797e7	1.82124
$680$	0	0
$681$	−2.00167e6	−0.165396
$682$	−1.64420e7	−1.35361
$683$	9.19066e6	0.753868	0.376934	−	0.926240i	$-0.376978\pi$
$683$	9.19066e6	0.753868	0.376934	+	0.926240i	$0.376978\pi$
$684$	7.64689e6	0.624949
$685$	0	0
$686$	4.98593e6	0.404516
$687$	−838062.	−0.0677461
$688$	671459.	0.0540814
$689$	−2.38934e6	−0.191748
$690$	0	0
$691$	−1.38192e7	−1.10100	−0.550502	−	0.834834i	$-0.685564\pi$
$691$	−1.38192e7	−1.10100	−0.550502	+	0.834834i	$0.685564\pi$
$692$	−1.09382e7	−0.868320
$693$	2.44168e7	1.93132
$694$	2.97040e6	0.234108
$695$	0	0
$696$	1.80145e6	0.140961
$697$	−1.76126e7	−1.37323
$698$	−3.61286e6	−0.280681
$699$	1.86790e6	0.144598
$700$	0	0
$701$	1.58946e7	1.22167	0.610834	−	0.791758i	$-0.290834\pi$
$701$	1.58946e7	1.22167	0.610834	+	0.791758i	$0.290834\pi$
$702$	−801134.	−0.0613568
$703$	−6.97162e6	−0.532041
$704$	−5.71554e6	−0.434636
$705$	0	0
$706$	9.05584e6	0.683781
$707$	−1.49536e6	−0.112512
$708$	−4.06985e6	−0.305137
$709$	1.20549e7	0.900635	0.450317	−	0.892869i	$-0.351311\pi$
$709$	1.20549e7	0.900635	0.450317	+	0.892869i	$0.351311\pi$
$710$	0	0
$711$	−920630.	−0.0682985
$712$	1.43682e7	1.06219
$713$	7.15950e6	0.527423
$714$	−1.67849e6	−0.123218
$715$	0	0
$716$	−9.78271e6	−0.713143
$717$	−771097.	−0.0560159
$718$	7.62252e6	0.551807
$719$	1.21480e7	0.876357	0.438178	−	0.898888i	$-0.355624\pi$
$719$	1.21480e7	0.876357	0.438178	+	0.898888i	$0.355624\pi$
$720$	0	0
$721$	−2.59622e7	−1.85996
$722$	1.48963e6	0.106350
$723$	4.54966e6	0.323693
$724$	3.94898e6	0.279988
$725$	0	0
$726$	−3.46829e6	−0.244216
$727$	1.37550e7	0.965215	0.482607	−	0.875837i	$-0.339690\pi$
$727$	1.37550e7	0.965215	0.482607	+	0.875837i	$0.339690\pi$
$728$	−4.03267e6	−0.282010
$729$	−1.02859e7	−0.716845
$730$	0	0
$731$	2.61316e6	0.180873
$732$	205637.	0.0141848
$733$	−5.10914e6	−0.351227	−0.175614	−	0.984459i	$-0.556191\pi$
$733$	−5.10914e6	−0.351227	−0.175614	+	0.984459i	$0.556191\pi$
$734$	6.45698e6	0.442374
$735$	0	0
$736$	5.37697e6	0.365884
$737$	−2.74725e7	−1.86307
$738$	−1.03985e7	−0.702794
$739$	4.07246e6	0.274312	0.137156	−	0.990549i	$-0.456204\pi$
$739$	4.07246e6	0.274312	0.137156	+	0.990549i	$0.456204\pi$
$740$	0	0
$741$	−818224.	−0.0547427
$742$	−6.06398e6	−0.404342
$743$	−1.86637e7	−1.24029	−0.620147	−	0.784485i	$-0.712927\pi$
$743$	−1.86637e7	−1.24029	−0.620147	+	0.784485i	$0.712927\pi$
$744$	−4.43763e6	−0.293913
$745$	0	0
$746$	1.07694e7	0.708505
$747$	−3.20615e6	−0.210224
$748$	1.90649e7	1.24589
$749$	3.33822e7	2.17425
$750$	0	0
$751$	−2.20898e7	−1.42919	−0.714597	−	0.699536i	$-0.753390\pi$
$751$	−2.20898e7	−1.42919	−0.714597	+	0.699536i	$0.753390\pi$
$752$	2.21119e6	0.142588
$753$	1.64528e6	0.105743
$754$	1.58282e6	0.101392
$755$	0	0
$756$	5.74382e6	0.365508
$757$	2.27912e7	1.44553	0.722765	−	0.691094i	$-0.242871\pi$
$757$	2.27912e7	1.44553	0.722765	+	0.691094i	$0.242871\pi$
$758$	6.10740e6	0.386085
$759$	2.21150e6	0.139342
$760$	0	0
$761$	2.05216e7	1.28455	0.642274	−	0.766475i	$-0.277991\pi$
$761$	2.05216e7	1.28455	0.642274	+	0.766475i	$0.277991\pi$
$762$	−2.67257e6	−0.166740
$763$	3.71958e6	0.231304
$764$	−726125.	−0.0450068
$765$	0	0
$766$	1.19843e6	0.0737973
$767$	−8.41767e6	−0.516658
$768$	−2.57229e6	−0.157368
$769$	−2.11516e7	−1.28982	−0.644908	−	0.764260i	$-0.723104\pi$
$769$	−2.11516e7	−1.28982	−0.644908	+	0.764260i	$0.723104\pi$
$770$	0	0
$771$	638488.	0.0386827
$772$	−2.39315e7	−1.44520
$773$	6.75134e6	0.406389	0.203194	−	0.979138i	$-0.434868\pi$
$773$	6.75134e6	0.406389	0.203194	+	0.979138i	$0.434868\pi$
$774$	1.54280e6	0.0925675
$775$	0	0
$776$	2.37430e7	1.41541
$777$	−2.55228e6	−0.151662
$778$	−1.18754e7	−0.703398
$779$	−2.17900e7	−1.28651
$780$	0	0
$781$	−4.84668e7	−2.84326
$782$	2.93863e6	0.171842
$783$	−5.30694e6	−0.309343
$784$	1.50729e6	0.0875801
$785$	0	0
$786$	1.57113e6	0.0907103
$787$	1.78318e7	1.02626	0.513132	−	0.858310i	$-0.328485\pi$
$787$	1.78318e7	1.02626	0.513132	+	0.858310i	$0.328485\pi$
$788$	−9.98003e6	−0.572554
$789$	−1.87117e6	−0.107009
$790$	0	0
$791$	5.82163e6	0.330829
$792$	2.64961e7	1.50096
$793$	425320.	0.0240178
$794$	1.67254e7	0.941508
$795$	0	0
$796$	−1.30815e7	−0.731772
$797$	−1.22612e7	−0.683732	−0.341866	−	0.939749i	$-0.611059\pi$
$797$	−1.22612e7	−0.683732	−0.341866	+	0.939749i	$0.611059\pi$
$798$	−2.07659e6	−0.115437
$799$	8.60546e6	0.476878
$800$	0	0
$801$	−2.06301e7	−1.13611
$802$	1.00865e7	0.553738
$803$	−1.51833e7	−0.830956
$804$	−3.14985e6	−0.171850
$805$	0	0
$806$	−3.89907e6	−0.211409
$807$	685993.	0.0370797
$808$	−1.62271e6	−0.0874404
$809$	−1.86542e7	−1.00208	−0.501042	−	0.865423i	$-0.667050\pi$
$809$	−1.86542e7	−1.00208	−0.501042	+	0.865423i	$0.667050\pi$
$810$	0	0
$811$	2.65131e6	0.141550	0.0707748	−	0.997492i	$-0.477453\pi$
$811$	2.65131e6	0.141550	0.0707748	+	0.997492i	$0.477453\pi$
$812$	−1.13482e7	−0.604002
$813$	−893377.	−0.0474033
$814$	−1.02619e7	−0.542833
$815$	0	0
$816$	1.13821e6	0.0598408
$817$	3.23295e6	0.169451
$818$	9.35351e6	0.488755
$819$	5.79020e6	0.301636
$820$	0	0
$821$	2.04948e7	1.06117	0.530587	−	0.847630i	$-0.321972\pi$
$821$	2.04948e7	1.06117	0.530587	+	0.847630i	$0.321972\pi$
$822$	1.32382e6	0.0683361
$823$	−1.26841e7	−0.652768	−0.326384	−	0.945237i	$-0.605830\pi$
$823$	−1.26841e7	−0.652768	−0.326384	+	0.945237i	$0.605830\pi$
$824$	−2.81732e7	−1.44550
$825$	0	0
$826$	−2.13635e7	−1.08949
$827$	−9.90644e6	−0.503679	−0.251839	−	0.967769i	$-0.581035\pi$
$827$	−9.90644e6	−0.503679	−0.251839	+	0.967769i	$0.581035\pi$
$828$	−4.90124e6	−0.248445
$829$	8.05866e6	0.407265	0.203632	−	0.979047i	$-0.434725\pi$
$829$	8.05866e6	0.407265	0.203632	+	0.979047i	$0.434725\pi$
$830$	0	0
$831$	−7.60034e6	−0.381795
$832$	−1.35538e6	−0.0678819
$833$	5.86601e6	0.292907
$834$	2.85039e6	0.141902
$835$	0	0
$836$	2.35866e7	1.16721
$837$	1.30729e7	0.644998
$838$	−1.05203e7	−0.517508
$839$	2.15392e7	1.05639	0.528196	−	0.849122i	$-0.322869\pi$
$839$	2.15392e7	1.05639	0.528196	+	0.849122i	$0.322869\pi$
$840$	0	0
$841$	−1.00261e7	−0.488811
$842$	−6.12024e6	−0.297501
$843$	−2912.38	−0.000141150 0
$844$	1.79186e7	0.865860
$845$	0	0
$846$	5.08064e6	0.244058
$847$	5.14309e7	2.46329
$848$	4.11207e6	0.196368
$849$	6.67968e6	0.318044
$850$	0	0
$851$	4.46842e6	0.211510
$852$	−5.55695e6	−0.262263
$853$	−1.55629e7	−0.732351	−0.366175	−	0.930546i	$-0.619333\pi$
$853$	−1.55629e7	−0.732351	−0.366175	+	0.930546i	$0.619333\pi$
$854$	1.07943e6	0.0506466
$855$	0	0
$856$	3.62250e7	1.68976
$857$	3.66500e7	1.70460	0.852299	−	0.523055i	$-0.175208\pi$
$857$	3.66500e7	1.70460	0.852299	+	0.523055i	$0.175208\pi$
$858$	−1.20438e6	−0.0558531
$859$	3.16288e7	1.46251	0.731257	−	0.682102i	$-0.238934\pi$
$859$	3.16288e7	1.46251	0.731257	+	0.682102i	$0.238934\pi$
$860$	0	0
$861$	−7.97721e6	−0.366727
$862$	−1.77717e7	−0.814630
$863$	−1.63149e7	−0.745691	−0.372845	−	0.927894i	$-0.621618\pi$
$863$	−1.63149e7	−0.745691	−0.372845	+	0.927894i	$0.621618\pi$
$864$	9.81810e6	0.447448
$865$	0	0
$866$	1.76052e7	0.797710
$867$	−479219.	−0.0216514
$868$	2.79548e7	1.25938
$869$	−2.83966e6	−0.127561
$870$	0	0
$871$	−6.51484e6	−0.290977
$872$	4.03634e6	0.179762
$873$	−3.40907e7	−1.51391
$874$	3.63561e6	0.160990
$875$	0	0
$876$	−1.74084e6	−0.0766475
$877$	2.64601e7	1.16170	0.580848	−	0.814012i	$-0.302721\pi$
$877$	2.64601e7	1.16170	0.580848	+	0.814012i	$0.302721\pi$
$878$	−1.58029e7	−0.691831
$879$	3.50297e6	0.152920
$880$	0	0
$881$	1.51155e7	0.656119	0.328059	−	0.944657i	$-0.393605\pi$
$881$	1.51155e7	0.656119	0.328059	+	0.944657i	$0.393605\pi$
$882$	3.46328e6	0.149905
$883$	2.08156e7	0.898437	0.449218	−	0.893422i	$-0.351703\pi$
$883$	2.08156e7	0.898437	0.449218	+	0.893422i	$0.351703\pi$
$884$	4.52105e6	0.194585
$885$	0	0
$886$	1.23112e7	0.526885
$887$	1.01116e7	0.431529	0.215764	−	0.976445i	$-0.430776\pi$
$887$	1.01116e7	0.431529	0.215764	+	0.976445i	$0.430776\pi$
$888$	−2.76964e6	−0.117866
$889$	3.96312e7	1.68183
$890$	0	0
$891$	−3.59737e7	−1.51807
$892$	−3.37410e6	−0.141986
$893$	1.06465e7	0.446763
$894$	1.56684e6	0.0655665
$895$	0	0
$896$	2.49867e7	1.03978
$897$	524436.	0.0217626
$898$	−8.90100e6	−0.368339
$899$	−2.58285e7	−1.06586
$900$	0	0
$901$	1.60032e7	0.656744
$902$	−3.20737e7	−1.31260
$903$	1.18357e6	0.0483030
$904$	6.31740e6	0.257109
$905$	0	0
$906$	−4.22554e6	−0.171026
$907$	−3.04365e6	−0.122850	−0.0614251	−	0.998112i	$-0.519565\pi$
$907$	−3.04365e6	−0.122850	−0.0614251	+	0.998112i	$0.519565\pi$
$908$	1.36833e7	0.550779
$909$	2.32992e6	0.0935257
$910$	0	0
$911$	−2.90597e7	−1.16010	−0.580050	−	0.814581i	$-0.696967\pi$
$911$	−2.90597e7	−1.16010	−0.580050	+	0.814581i	$0.696967\pi$
$912$	1.40817e6	0.0560618
$913$	−9.88928e6	−0.392634
$914$	5.57744e6	0.220836
$915$	0	0
$916$	5.72896e6	0.225599
$917$	−2.32982e7	−0.914952
$918$	5.36581e6	0.210149
$919$	3.38583e7	1.32244	0.661220	−	0.750192i	$-0.270039\pi$
$919$	3.38583e7	1.32244	0.661220	+	0.750192i	$0.270039\pi$
$920$	0	0
$921$	1.33546e6	0.0518780
$922$	−4.42900e6	−0.171585
$923$	−1.14934e7	−0.444064
$924$	8.63497e6	0.332722
$925$	0	0
$926$	6.16939e6	0.236437
$927$	4.04516e7	1.54610
$928$	−1.93979e7	−0.739409
$929$	3.78136e7	1.43750	0.718751	−	0.695267i	$-0.244714\pi$
$929$	3.78136e7	1.43750	0.718751	+	0.695267i	$0.244714\pi$
$930$	0	0
$931$	7.25729e6	0.274410
$932$	−1.27689e7	−0.481519
$933$	4.75954e6	0.179003
$934$	7.16037e6	0.268577
$935$	0	0
$936$	6.28329e6	0.234422
$937$	2.00563e7	0.746281	0.373140	−	0.927775i	$-0.378281\pi$
$937$	2.00563e7	0.746281	0.373140	+	0.927775i	$0.378281\pi$
$938$	−1.65342e7	−0.613587
$939$	−1.87169e6	−0.0692739
$940$	0	0
$941$	−2.54644e7	−0.937473	−0.468737	−	0.883338i	$-0.655291\pi$
$941$	−2.54644e7	−0.937473	−0.468737	+	0.883338i	$0.655291\pi$
$942$	−1.29540e6	−0.0475637
$943$	1.39662e7	0.511443
$944$	1.44869e7	0.529108
$945$	0	0
$946$	4.75874e6	0.172888
$947$	−1.42435e7	−0.516110	−0.258055	−	0.966130i	$-0.583082\pi$
$947$	−1.42435e7	−0.516110	−0.258055	+	0.966130i	$0.583082\pi$
$948$	−325580.	−0.0117662
$949$	−3.60058e6	−0.129780
$950$	0	0
$951$	6.63292e6	0.237823
$952$	2.70099e7	0.965896
$953$	3.95389e7	1.41024	0.705119	−	0.709089i	$-0.250894\pi$
$953$	3.95389e7	1.41024	0.705119	+	0.709089i	$0.250894\pi$
$954$	9.44827e6	0.336110
$955$	0	0
$956$	5.27119e6	0.186536
$957$	−7.97819e6	−0.281595
$958$	−1.88731e7	−0.664399
$959$	−1.96308e7	−0.689275
$960$	0	0
$961$	3.49958e7	1.22238
$962$	−2.43351e6	−0.0847803
$963$	−5.20127e7	−1.80735
$964$	−3.11013e7	−1.07792
$965$	0	0
$966$	1.33098e6	0.0458912
$967$	−4.28774e7	−1.47456	−0.737279	−	0.675588i	$-0.763890\pi$
$967$	−4.28774e7	−1.47456	−0.737279	+	0.675588i	$0.763890\pi$
$968$	5.58108e7	1.91439
$969$	5.48027e6	0.187496
$970$	0	0
$971$	1.56895e7	0.534023	0.267011	−	0.963693i	$-0.413964\pi$
$971$	1.56895e7	0.534023	0.267011	+	0.963693i	$0.413964\pi$
$972$	−1.35370e7	−0.459574
$973$	−4.22681e7	−1.43130
$974$	−1.59100e7	−0.537370
$975$	0	0
$976$	−731978.	−0.0245965
$977$	2.22874e7	0.747004	0.373502	−	0.927629i	$-0.378157\pi$
$977$	2.22874e7	0.747004	0.373502	+	0.927629i	$0.378157\pi$
$978$	6.16759e6	0.206190
$979$	−6.36331e7	−2.12191
$980$	0	0
$981$	−5.79547e6	−0.192272
$982$	−2.47174e7	−0.817946
$983$	−2.46890e6	−0.0814929	−0.0407465	−	0.999170i	$-0.512974\pi$
$983$	−2.46890e6	−0.0814929	−0.0407465	+	0.999170i	$0.512974\pi$
$984$	−8.65656e6	−0.285008
$985$	0	0
$986$	−1.06014e7	−0.347272
$987$	3.89763e6	0.127353
$988$	5.59334e6	0.182297
$989$	−2.07214e6	−0.0673640
$990$	0	0
$991$	1.73304e7	0.560563	0.280281	−	0.959918i	$-0.409572\pi$
$991$	1.73304e7	0.560563	0.280281	+	0.959918i	$0.409572\pi$
$992$	4.77840e7	1.54171
$993$	−8.30251e6	−0.267200
$994$	−2.91695e7	−0.936405
$995$	0	0
$996$	−1.13385e6	−0.0362166
$997$	5.90647e7	1.88187	0.940936	−	0.338586i	$-0.109948\pi$
$997$	5.90647e7	1.88187	0.940936	+	0.338586i	$0.109948\pi$
$998$	−8.62487e6	−0.274111
$999$	8.15913e6	0.258661

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.a.j.1.5	✓	11
5.2	odd	4		325.6.b.i.274.9		22
5.3	odd	4		325.6.b.i.274.14		22
5.4	even	2		325.6.a.k.1.7	yes	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.j.1.5	✓	11	1.1	even	1	trivial
325.6.a.k.1.7	yes	11	5.4	even	2
325.6.b.i.274.9		22	5.2	odd	4
325.6.b.i.274.14		22	5.3	odd	4

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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.a (trivial)

\(f(q)\)	\(=\)	\(q-2.89241 q^{2} +3.45730 q^{3} -23.6340 q^{4} -9.99993 q^{6} +148.288 q^{7} +160.916 q^{8} -231.047 q^{9} -712.658 q^{11} -81.7096 q^{12} -169.000 q^{13} -428.910 q^{14} +290.850 q^{16} +1131.92 q^{17} +668.283 q^{18} +1400.39 q^{19} +512.676 q^{21} +2061.30 q^{22} -897.571 q^{23} +556.336 q^{24} +488.818 q^{26} -1638.92 q^{27} -3504.63 q^{28} +3238.07 q^{29} -7976.53 q^{31} -5990.58 q^{32} -2463.87 q^{33} -3273.98 q^{34} +5460.56 q^{36} -4978.35 q^{37} -4050.50 q^{38} -584.284 q^{39} -15559.9 q^{41} -1482.87 q^{42} +2308.61 q^{43} +16842.9 q^{44} +2596.15 q^{46} +7602.52 q^{47} +1005.56 q^{48} +5182.35 q^{49} +3913.39 q^{51} +3994.14 q^{52} +14138.1 q^{53} +4740.44 q^{54} +23862.0 q^{56} +4841.56 q^{57} -9365.83 q^{58} +49808.7 q^{59} -2516.69 q^{61} +23071.4 q^{62} -34261.5 q^{63} +8020.03 q^{64} +7126.54 q^{66} +38549.3 q^{67} -26751.8 q^{68} -3103.17 q^{69} +68008.5 q^{71} -37179.2 q^{72} +21305.2 q^{73} +14399.4 q^{74} -33096.7 q^{76} -105679. q^{77} +1689.99 q^{78} +3984.60 q^{79} +50478.2 q^{81} +45005.8 q^{82} +13876.6 q^{83} -12116.6 q^{84} -6677.44 q^{86} +11195.0 q^{87} -114678. q^{88} +89289.8 q^{89} -25060.7 q^{91} +21213.1 q^{92} -27577.2 q^{93} -21989.6 q^{94} -20711.2 q^{96} +147549. q^{97} -14989.5 q^{98} +164658. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 5 q^{2} - 11 q^{3} + 187 q^{4} + 351 q^{6} - 208 q^{7} - 165 q^{8} + 1372 q^{9} + 1276 q^{11} - 1533 q^{12} - 1859 q^{13} + 578 q^{14} + 5707 q^{16} - 2218 q^{17} + 6776 q^{18} + 3520 q^{19} + 1706 q^{21}+ \cdots + 426698 q^{99}+O(q^{100}) \) 11 * q - 5 * q^2 - 11 * q^3 + 187 * q^4 + 351 * q^6 - 208 * q^7 - 165 * q^8 + 1372 * q^9 + 1276 * q^11 - 1533 * q^12 - 1859 * q^13 + 578 * q^14 + 5707 * q^16 - 2218 * q^17 + 6776 * q^18 + 3520 * q^19 + 1706 * q^21 + 6785 * q^22 + 1215 * q^23 + 7723 * q^24 + 845 * q^26 - 161 * q^27 - 12746 * q^28 + 18425 * q^29 + 9068 * q^31 - 21165 * q^32 - 19748 * q^33 - 14455 * q^34 + 37684 * q^36 + 2280 * q^37 + 23783 * q^38 + 1859 * q^39 - 10010 * q^41 + 44184 * q^42 - 6763 * q^43 + 60651 * q^44 - 9858 * q^46 - 4110 * q^47 - 63289 * q^48 + 73313 * q^49 + 36606 * q^51 - 31603 * q^52 - 15633 * q^53 + 48013 * q^54 - 1262 * q^56 + 205974 * q^57 - 77652 * q^58 + 131642 * q^59 - 43365 * q^61 + 115940 * q^62 - 218466 * q^63 + 180071 * q^64 - 58607 * q^66 + 100904 * q^67 - 371277 * q^68 + 59345 * q^69 + 68420 * q^71 + 210708 * q^72 - 124436 * q^73 + 371396 * q^74 + 22517 * q^76 + 150978 * q^77 - 59319 * q^78 + 52239 * q^79 + 107507 * q^81 + 351137 * q^82 - 211538 * q^83 + 852716 * q^84 + 202246 * q^86 + 28273 * q^87 + 28107 * q^88 + 327712 * q^89 + 35152 * q^91 + 413252 * q^92 - 243338 * q^93 + 649974 * q^94 + 899663 * q^96 + 110796 * q^97 - 27143 * q^98 + 426698 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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