LMFDB - Embedded newform 650.6.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,6,Mod(1,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, 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("650.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 650 = 2 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	650.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:	$104.249482878$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4 x^{8} - 1585 x^{7} + 2848 x^{6} + 753030 x^{5} - 686440 x^{4} - 130455000 x^{3} + \cdots - 22087917000 $ x^9 - 4x^8 - 1585x^7 + 2848x^6 + 753030x^5 - 686440x^4 - 130455000x^3 + 180669100x^2 + 6894601500x - 22087917000
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{5} $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-28.2730$ of defining polynomial
Character	$\chi$	$=$	650.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000 q^{2} -30.2730 q^{3} +16.0000 q^{4} +121.092 q^{6} -48.6525 q^{7} -64.0000 q^{8} +673.456 q^{9} -183.365 q^{11} -484.368 q^{12} -169.000 q^{13} +194.610 q^{14} +256.000 q^{16} +21.6273 q^{17} -2693.82 q^{18} +1858.66 q^{19} +1472.86 q^{21} +733.459 q^{22} +4771.82 q^{23} +1937.47 q^{24} +676.000 q^{26} -13031.2 q^{27} -778.439 q^{28} +4875.43 q^{29} +5155.25 q^{31} -1024.00 q^{32} +5551.01 q^{33} -86.5092 q^{34} +10775.3 q^{36} -1178.68 q^{37} -7434.65 q^{38} +5116.14 q^{39} +952.823 q^{41} -5891.43 q^{42} +8187.16 q^{43} -2933.84 q^{44} -19087.3 q^{46} -12733.5 q^{47} -7749.89 q^{48} -14439.9 q^{49} -654.724 q^{51} -2704.00 q^{52} -5475.33 q^{53} +52124.8 q^{54} +3113.76 q^{56} -56267.3 q^{57} -19501.7 q^{58} +10648.6 q^{59} -27963.6 q^{61} -20621.0 q^{62} -32765.3 q^{63} +4096.00 q^{64} -22204.0 q^{66} +21481.1 q^{67} +346.037 q^{68} -144457. q^{69} -72723.3 q^{71} -43101.2 q^{72} -78686.6 q^{73} +4714.73 q^{74} +29738.6 q^{76} +8921.15 q^{77} -20464.6 q^{78} +88076.0 q^{79} +230844. q^{81} -3811.29 q^{82} +118577. q^{83} +23565.7 q^{84} -32748.6 q^{86} -147594. q^{87} +11735.4 q^{88} -70518.1 q^{89} +8222.27 q^{91} +76349.1 q^{92} -156065. q^{93} +50933.8 q^{94} +30999.6 q^{96} -11261.1 q^{97} +57759.8 q^{98} -123488. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 36 q^{2} - 14 q^{3} + 144 q^{4} + 56 q^{6} - 164 q^{7} - 576 q^{8} + 1019 q^{9} + 326 q^{11} - 224 q^{12} - 1521 q^{13} + 656 q^{14} + 2304 q^{16} - 2144 q^{17} - 4076 q^{18} + 5558 q^{19} - 236 q^{21}+ \cdots + 231774 q^{99}+O(q^{100}) $ 9 * q - 36 * q^2 - 14 * q^3 + 144 * q^4 + 56 * q^6 - 164 * q^7 - 576 * q^8 + 1019 * q^9 + 326 * q^11 - 224 * q^12 - 1521 * q^13 + 656 * q^14 + 2304 * q^16 - 2144 * q^17 - 4076 * q^18 + 5558 * q^19 - 236 * q^21 - 1304 * q^22 - 660 * q^23 + 896 * q^24 + 6084 * q^26 - 1796 * q^27 - 2624 * q^28 + 6742 * q^29 + 3100 * q^31 - 9216 * q^32 + 5672 * q^33 + 8576 * q^34 + 16304 * q^36 - 4236 * q^37 - 22232 * q^38 + 2366 * q^39 + 15494 * q^41 + 944 * q^42 - 22630 * q^43 + 5216 * q^44 + 2640 * q^46 - 11620 * q^47 - 3584 * q^48 + 35575 * q^49 + 40698 * q^51 - 24336 * q^52 - 80830 * q^53 + 7184 * q^54 + 10496 * q^56 + 33508 * q^57 - 26968 * q^58 + 57474 * q^59 + 69486 * q^61 - 12400 * q^62 - 72522 * q^63 + 36864 * q^64 - 22688 * q^66 + 16450 * q^67 - 34304 * q^68 - 32980 * q^69 - 7228 * q^71 - 65216 * q^72 - 115278 * q^73 + 16944 * q^74 + 88928 * q^76 - 5568 * q^77 - 9464 * q^78 + 81836 * q^79 + 254885 * q^81 - 61976 * q^82 - 108670 * q^83 - 3776 * q^84 + 90520 * q^86 - 333048 * q^87 - 20864 * q^88 + 164994 * q^89 + 27716 * q^91 - 10560 * q^92 - 316844 * q^93 + 46480 * q^94 + 14336 * q^96 - 155190 * q^97 - 142300 * q^98 + 231774 * 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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	−30.2730	−1.94202	−0.971008	−	0.239048i	$-0.923165\pi$
$3$	−30.2730	−1.94202	−0.971008	+	0.239048i	$0.923165\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	121.092	1.37321
$7$	−48.6525	−0.375284	−0.187642	−	0.982238i	$-0.560084\pi$
$7$	−48.6525	−0.375284	−0.187642	+	0.982238i	$0.560084\pi$
$8$	−64.0000	−0.353553
$9$	673.456	2.77142
$10$	0	0
$11$	−183.365	−0.456914	−0.228457	−	0.973554i	$-0.573368\pi$
$11$	−183.365	−0.456914	−0.228457	+	0.973554i	$0.573368\pi$
$12$	−484.368	−0.971008
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	194.610	0.265366
$15$	0	0
$16$	256.000	0.250000
$17$	21.6273	0.0181501	0.00907507	−	0.999959i	$-0.497111\pi$
$17$	21.6273	0.0181501	0.00907507	+	0.999959i	$0.497111\pi$
$18$	−2693.82	−1.95969
$19$	1858.66	1.18118	0.590590	−	0.806971i	$-0.298895\pi$
$19$	1858.66	1.18118	0.590590	+	0.806971i	$0.298895\pi$
$20$	0	0
$21$	1472.86	0.728807
$22$	733.459	0.323087
$23$	4771.82	1.88089	0.940447	−	0.339940i	$-0.110407\pi$
$23$	4771.82	1.88089	0.940447	+	0.339940i	$0.110407\pi$
$24$	1937.47	0.686606
$25$	0	0
$26$	676.000	0.196116
$27$	−13031.2	−3.44013
$28$	−778.439	−0.187642
$29$	4875.43	1.07651	0.538255	−	0.842782i	$-0.319084\pi$
$29$	4875.43	1.07651	0.538255	+	0.842782i	$0.319084\pi$
$30$	0	0
$31$	5155.25	0.963486	0.481743	−	0.876312i	$-0.340004\pi$
$31$	5155.25	0.963486	0.481743	+	0.876312i	$0.340004\pi$
$32$	−1024.00	−0.176777
$33$	5551.01	0.887334
$34$	−86.5092	−0.0128341
$35$	0	0
$36$	10775.3	1.38571
$37$	−1178.68	−0.141544	−0.0707721	−	0.997493i	$-0.522546\pi$
$37$	−1178.68	−0.141544	−0.0707721	+	0.997493i	$0.522546\pi$
$38$	−7434.65	−0.835221
$39$	5116.14	0.538618
$40$	0	0
$41$	952.823	0.0885222	0.0442611	−	0.999020i	$-0.485907\pi$
$41$	952.823	0.0885222	0.0442611	+	0.999020i	$0.485907\pi$
$42$	−5891.43	−0.515344
$43$	8187.16	0.675246	0.337623	−	0.941281i	$-0.390377\pi$
$43$	8187.16	0.675246	0.337623	+	0.941281i	$0.390377\pi$
$44$	−2933.84	−0.228457
$45$	0	0
$46$	−19087.3	−1.32999
$47$	−12733.5	−0.840817	−0.420409	−	0.907335i	$-0.638113\pi$
$47$	−12733.5	−0.840817	−0.420409	+	0.907335i	$0.638113\pi$
$48$	−7749.89	−0.485504
$49$	−14439.9	−0.859162
$50$	0	0
$51$	−654.724	−0.0352479
$52$	−2704.00	−0.138675
$53$	−5475.33	−0.267745	−0.133872	−	0.990999i	$-0.542741\pi$
$53$	−5475.33	−0.267745	−0.133872	+	0.990999i	$0.542741\pi$
$54$	52124.8	2.43254
$55$	0	0
$56$	3113.76	0.132683
$57$	−56267.3	−2.29387
$58$	−19501.7	−0.761208
$59$	10648.6	0.398256	0.199128	−	0.979974i	$-0.436189\pi$
$59$	10648.6	0.398256	0.199128	+	0.979974i	$0.436189\pi$
$60$	0	0
$61$	−27963.6	−0.962208	−0.481104	−	0.876664i	$-0.659764\pi$
$61$	−27963.6	−0.962208	−0.481104	+	0.876664i	$0.659764\pi$
$62$	−20621.0	−0.681288
$63$	−32765.3	−1.04007
$64$	4096.00	0.125000
$65$	0	0
$66$	−22204.0	−0.627440
$67$	21481.1	0.584614	0.292307	−	0.956325i	$-0.405577\pi$
$67$	21481.1	0.584614	0.292307	+	0.956325i	$0.405577\pi$
$68$	346.037	0.00907507
$69$	−144457.	−3.65273
$70$	0	0
$71$	−72723.3	−1.71210	−0.856048	−	0.516897i	$-0.827087\pi$
$71$	−72723.3	−1.71210	−0.856048	+	0.516897i	$0.827087\pi$
$72$	−43101.2	−0.979846
$73$	−78686.6	−1.72820	−0.864100	−	0.503321i	$-0.832112\pi$
$73$	−78686.6	−1.72820	−0.864100	+	0.503321i	$0.832112\pi$
$74$	4714.73	0.100087
$75$	0	0
$76$	29738.6	0.590590
$77$	8921.15	0.171472
$78$	−20464.6	−0.380861
$79$	88076.0	1.58778	0.793889	−	0.608063i	$-0.208053\pi$
$79$	88076.0	1.58778	0.793889	+	0.608063i	$0.208053\pi$
$80$	0	0
$81$	230844.	3.90936
$82$	−3811.29	−0.0625947
$83$	118577.	1.88931	0.944657	−	0.328060i	$-0.106395\pi$
$83$	118577.	1.88931	0.944657	+	0.328060i	$0.106395\pi$
$84$	23565.7	0.364403
$85$	0	0
$86$	−32748.6	−0.477471
$87$	−147594.	−2.09060
$88$	11735.4	0.161543
$89$	−70518.1	−0.943682	−0.471841	−	0.881684i	$-0.656410\pi$
$89$	−70518.1	−0.943682	−0.471841	+	0.881684i	$0.656410\pi$
$90$	0	0
$91$	8222.27	0.104085
$92$	76349.1	0.940447
$93$	−156065.	−1.87110
$94$	50933.8	0.594548
$95$	0	0
$96$	30999.6	0.343303
$97$	−11261.1	−0.121522	−0.0607608	−	0.998152i	$-0.519353\pi$
$97$	−11261.1	−0.121522	−0.0607608	+	0.998152i	$0.519353\pi$
$98$	57759.8	0.607519
$99$	−123488.	−1.26630
$100$	0	0
$101$	40669.3	0.396701	0.198350	−	0.980131i	$-0.436442\pi$
$101$	40669.3	0.396701	0.198350	+	0.980131i	$0.436442\pi$
$102$	2618.90	0.0249240
$103$	−105122.	−0.976340	−0.488170	−	0.872749i	$-0.662335\pi$
$103$	−105122.	−0.976340	−0.488170	+	0.872749i	$0.662335\pi$
$104$	10816.0	0.0980581
$105$	0	0
$106$	21901.3	0.189324
$107$	−68533.0	−0.578682	−0.289341	−	0.957226i	$-0.593436\pi$
$107$	−68533.0	−0.578682	−0.289341	+	0.957226i	$0.593436\pi$
$108$	−208499.	−1.72007
$109$	159560.	1.28635	0.643175	−	0.765720i	$-0.277617\pi$
$109$	159560.	1.28635	0.643175	+	0.765720i	$0.277617\pi$
$110$	0	0
$111$	35682.3	0.274881
$112$	−12455.0	−0.0938209
$113$	−100116.	−0.737579	−0.368790	−	0.929513i	$-0.620228\pi$
$113$	−100116.	−0.737579	−0.368790	+	0.929513i	$0.620228\pi$
$114$	225069.	1.62201
$115$	0	0
$116$	78006.9	0.538255
$117$	−113814.	−0.768655
$118$	−42594.4	−0.281609
$119$	−1052.22	−0.00681145
$120$	0	0
$121$	−127428.	−0.791230
$122$	111854.	0.680383
$123$	−28844.8	−0.171912
$124$	82484.0	0.481743
$125$	0	0
$126$	131061.	0.735441
$127$	259872.	1.42972	0.714858	−	0.699270i	$-0.246491\pi$
$127$	259872.	1.42972	0.714858	+	0.699270i	$0.246491\pi$
$128$	−16384.0	−0.0883883
$129$	−247850.	−1.31134
$130$	0	0
$131$	−171651.	−0.873912	−0.436956	−	0.899483i	$-0.643943\pi$
$131$	−171651.	−0.873912	−0.436956	+	0.899483i	$0.643943\pi$
$132$	88816.1	0.443667
$133$	−90428.5	−0.443278
$134$	−85924.3	−0.413384
$135$	0	0
$136$	−1384.15	−0.00641705
$137$	218907.	0.996457	0.498228	−	0.867046i	$-0.333984\pi$
$137$	218907.	0.996457	0.498228	+	0.867046i	$0.333984\pi$
$138$	577829.	2.58287
$139$	−331271.	−1.45428	−0.727138	−	0.686492i	$-0.759150\pi$
$139$	−331271.	−1.45428	−0.727138	+	0.686492i	$0.759150\pi$
$140$	0	0
$141$	385480.	1.63288
$142$	290893.	1.21063
$143$	30988.7	0.126725
$144$	172405.	0.692856
$145$	0	0
$146$	314747.	1.22202
$147$	437141.	1.66851
$148$	−18858.9	−0.0707721
$149$	−56849.5	−0.209778	−0.104889	−	0.994484i	$-0.533449\pi$
$149$	−56849.5	−0.209778	−0.104889	+	0.994484i	$0.533449\pi$
$150$	0	0
$151$	67592.0	0.241242	0.120621	−	0.992699i	$-0.461511\pi$
$151$	67592.0	0.241242	0.120621	+	0.992699i	$0.461511\pi$
$152$	−118954.	−0.417611
$153$	14565.0	0.0503017
$154$	−35684.6	−0.121249
$155$	0	0
$156$	81858.3	0.269309
$157$	−222975.	−0.721950	−0.360975	−	0.932575i	$-0.617556\pi$
$157$	−222975.	−0.721950	−0.360975	+	0.932575i	$0.617556\pi$
$158$	−352304.	−1.12273
$159$	165755.	0.519964
$160$	0	0
$161$	−232161.	−0.705869
$162$	−923376.	−2.76434
$163$	606609.	1.78830	0.894149	−	0.447769i	$-0.147781\pi$
$163$	606609.	1.78830	0.894149	+	0.447769i	$0.147781\pi$
$164$	15245.2	0.0442611
$165$	0	0
$166$	−474307.	−1.33595
$167$	347642.	0.964587	0.482293	−	0.876010i	$-0.339804\pi$
$167$	347642.	0.964587	0.482293	+	0.876010i	$0.339804\pi$
$168$	−94262.8	−0.257672
$169$	28561.0	0.0769231
$170$	0	0
$171$	1.25173e6	3.27355
$172$	130995.	0.337623
$173$	−36365.7	−0.0923798	−0.0461899	−	0.998933i	$-0.514708\pi$
$173$	−36365.7	−0.0923798	−0.0461899	+	0.998933i	$0.514708\pi$
$174$	590376.	1.47828
$175$	0	0
$176$	−46941.4	−0.114228
$177$	−322365.	−0.773419
$178$	282072.	0.667284
$179$	−186888.	−0.435962	−0.217981	−	0.975953i	$-0.569947\pi$
$179$	−186888.	−0.435962	−0.217981	+	0.975953i	$0.569947\pi$
$180$	0	0
$181$	256584.	0.582148	0.291074	−	0.956701i	$-0.405987\pi$
$181$	256584.	0.582148	0.291074	+	0.956701i	$0.405987\pi$
$182$	−32889.1	−0.0735992
$183$	846543.	1.86862
$184$	−305396.	−0.664996
$185$	0	0
$186$	624260.	1.32307
$187$	−3965.69	−0.00829305
$188$	−203735.	−0.420409
$189$	634000.	1.29103
$190$	0	0
$191$	−374629.	−0.743050	−0.371525	−	0.928423i	$-0.621165\pi$
$191$	−374629.	−0.743050	−0.371525	+	0.928423i	$0.621165\pi$
$192$	−123998.	−0.242752
$193$	−30106.5	−0.0581791	−0.0290896	−	0.999577i	$-0.509261\pi$
$193$	−30106.5	−0.0581791	−0.0290896	+	0.999577i	$0.509261\pi$
$194$	45044.6	0.0859287
$195$	0	0
$196$	−231039.	−0.429581
$197$	813922.	1.49423	0.747114	−	0.664696i	$-0.231439\pi$
$197$	813922.	1.49423	0.747114	+	0.664696i	$0.231439\pi$
$198$	493953.	0.895411
$199$	856690.	1.53353	0.766763	−	0.641931i	$-0.221866\pi$
$199$	856690.	1.53353	0.766763	+	0.641931i	$0.221866\pi$
$200$	0	0
$201$	−650297.	−1.13533
$202$	−162677.	−0.280510
$203$	−237202.	−0.403997
$204$	−10475.6	−0.0176239
$205$	0	0
$206$	420489.	0.690377
$207$	3.21361e6	5.21275
$208$	−43264.0	−0.0693375
$209$	−340813.	−0.539698
$210$	0	0
$211$	93103.0	0.143965	0.0719826	−	0.997406i	$-0.477067\pi$
$211$	93103.0	0.143965	0.0719826	+	0.997406i	$0.477067\pi$
$212$	−87605.4	−0.133872
$213$	2.20156e6	3.32492
$214$	274132.	0.409190
$215$	0	0
$216$	833997.	1.21627
$217$	−250816.	−0.361581
$218$	−638242.	−0.909586
$219$	2.38208e6	3.35619
$220$	0	0
$221$	−3655.01	−0.00503394
$222$	−142729.	−0.194370
$223$	540262.	0.727515	0.363758	−	0.931494i	$-0.381494\pi$
$223$	540262.	0.727515	0.363758	+	0.931494i	$0.381494\pi$
$224$	49820.1	0.0663414
$225$	0	0
$226$	400465.	0.521547
$227$	−297798.	−0.383581	−0.191791	−	0.981436i	$-0.561429\pi$
$227$	−297798.	−0.383581	−0.191791	+	0.981436i	$0.561429\pi$
$228$	−900277.	−1.14694
$229$	835401.	1.05270	0.526352	−	0.850267i	$-0.323559\pi$
$229$	835401.	1.05270	0.526352	+	0.850267i	$0.323559\pi$
$230$	0	0
$231$	−270070.	−0.333002
$232$	−312028.	−0.380604
$233$	−72054.4	−0.0869502	−0.0434751	−	0.999055i	$-0.513843\pi$
$233$	−72054.4	−0.0869502	−0.0434751	+	0.999055i	$0.513843\pi$
$234$	455256.	0.543521
$235$	0	0
$236$	170377.	0.199128
$237$	−2.66633e6	−3.08349
$238$	4208.89	0.00481643
$239$	−1.13067e6	−1.28038	−0.640191	−	0.768216i	$-0.721145\pi$
$239$	−1.13067e6	−1.28038	−0.640191	+	0.768216i	$0.721145\pi$
$240$	0	0
$241$	−1.09237e6	−1.21152	−0.605758	−	0.795649i	$-0.707130\pi$
$241$	−1.09237e6	−1.21152	−0.605758	+	0.795649i	$0.707130\pi$
$242$	509713.	0.559484
$243$	−3.82177e6	−4.15191
$244$	−447418.	−0.481104
$245$	0	0
$246$	115379.	0.121560
$247$	−314114.	−0.327601
$248$	−329936.	−0.340644
$249$	−3.58968e6	−3.66908
$250$	0	0
$251$	1.90077e6	1.90435	0.952173	−	0.305559i	$-0.0988434\pi$
$251$	1.90077e6	1.90435	0.952173	+	0.305559i	$0.0988434\pi$
$252$	−524245.	−0.520035
$253$	−874984.	−0.859407
$254$	−1.03949e6	−1.01096
$255$	0	0
$256$	65536.0	0.0625000
$257$	−1.63127e6	−1.54062	−0.770308	−	0.637672i	$-0.779897\pi$
$257$	−1.63127e6	−1.54062	−0.770308	+	0.637672i	$0.779897\pi$
$258$	991401.	0.927257
$259$	57345.8	0.0531193
$260$	0	0
$261$	3.28339e6	2.98347
$262$	686603.	0.617949
$263$	−430511.	−0.383791	−0.191896	−	0.981415i	$-0.561464\pi$
$263$	−430511.	−0.383791	−0.191896	+	0.981415i	$0.561464\pi$
$264$	−355265.	−0.313720
$265$	0	0
$266$	361714.	0.313445
$267$	2.13480e6	1.83264
$268$	343697.	0.292307
$269$	29531.5	0.0248832	0.0124416	−	0.999923i	$-0.496040\pi$
$269$	29531.5	0.0248832	0.0124416	+	0.999923i	$0.496040\pi$
$270$	0	0
$271$	−281575.	−0.232901	−0.116450	−	0.993197i	$-0.537152\pi$
$271$	−281575.	−0.232901	−0.116450	+	0.993197i	$0.537152\pi$
$272$	5536.59	0.00453754
$273$	−248913.	−0.202135
$274$	−875628.	−0.704601
$275$	0	0
$276$	−2.31132e6	−1.82636
$277$	−2.07708e6	−1.62650	−0.813250	−	0.581915i	$-0.802304\pi$
$277$	−2.07708e6	−1.62650	−0.813250	+	0.581915i	$0.802304\pi$
$278$	1.32508e6	1.02833
$279$	3.47183e6	2.67023
$280$	0	0
$281$	−33435.8	−0.0252607	−0.0126304	−	0.999920i	$-0.504020\pi$
$281$	−33435.8	−0.0252607	−0.0126304	+	0.999920i	$0.504020\pi$
$282$	−1.54192e6	−1.15462
$283$	2.25537e6	1.67399	0.836994	−	0.547213i	$-0.184311\pi$
$283$	2.25537e6	1.67399	0.836994	+	0.547213i	$0.184311\pi$
$284$	−1.16357e6	−0.856048
$285$	0	0
$286$	−123955.	−0.0896082
$287$	−46357.2	−0.0332210
$288$	−689619.	−0.489923
$289$	−1.41939e6	−0.999671
$290$	0	0
$291$	340909.	0.235997
$292$	−1.25899e6	−0.864100
$293$	1.49777e6	1.01924	0.509619	−	0.860400i	$-0.329786\pi$
$293$	1.49777e6	1.01924	0.509619	+	0.860400i	$0.329786\pi$
$294$	−1.74856e6	−1.17981
$295$	0	0
$296$	75435.6	0.0500435
$297$	2.38946e6	1.57184
$298$	227398.	0.148336
$299$	−806437.	−0.521666
$300$	0	0
$301$	−398326.	−0.253409
$302$	−270368.	−0.170584
$303$	−1.23118e6	−0.770399
$304$	475817.	0.295295
$305$	0	0
$306$	−58260.1	−0.0355687
$307$	752586.	0.455733	0.227866	−	0.973692i	$-0.426825\pi$
$307$	752586.	0.455733	0.227866	+	0.973692i	$0.426825\pi$
$308$	142738.	0.0857362
$309$	3.18237e6	1.89607
$310$	0	0
$311$	2.24396e6	1.31557	0.657786	−	0.753205i	$-0.271493\pi$
$311$	2.24396e6	1.31557	0.657786	+	0.753205i	$0.271493\pi$
$312$	−327433.	−0.190430
$313$	−2.20149e6	−1.27015	−0.635075	−	0.772450i	$-0.719031\pi$
$313$	−2.20149e6	−1.27015	−0.635075	+	0.772450i	$0.719031\pi$
$314$	891900.	0.510496
$315$	0	0
$316$	1.40922e6	0.793889
$317$	2.27670e6	1.27250	0.636250	−	0.771483i	$-0.280485\pi$
$317$	2.27670e6	1.27250	0.636250	+	0.771483i	$0.280485\pi$
$318$	−663020.	−0.367670
$319$	−893983.	−0.491872
$320$	0	0
$321$	2.07470e6	1.12381
$322$	928643.	0.499125
$323$	40197.9	0.0214386
$324$	3.69351e6	1.95468
$325$	0	0
$326$	−2.42644e6	−1.26452
$327$	−4.83038e6	−2.49811
$328$	−60980.6	−0.0312973
$329$	619514.	0.315545
$330$	0	0
$331$	−426696.	−0.214066	−0.107033	−	0.994255i	$-0.534135\pi$
$331$	−426696.	−0.214066	−0.107033	+	0.994255i	$0.534135\pi$
$332$	1.89723e6	0.944657
$333$	−793790.	−0.392279
$334$	−1.39057e6	−0.682066
$335$	0	0
$336$	377051.	0.182202
$337$	2.06754e6	0.991697	0.495849	−	0.868409i	$-0.334857\pi$
$337$	2.06754e6	0.991697	0.495849	+	0.868409i	$0.334857\pi$
$338$	−114244.	−0.0543928
$339$	3.03082e6	1.43239
$340$	0	0
$341$	−945292.	−0.440230
$342$	−5.00691e6	−2.31475
$343$	1.52024e6	0.697713
$344$	−523978.	−0.238736
$345$	0	0
$346$	145463.	0.0653224
$347$	−984235.	−0.438808	−0.219404	−	0.975634i	$-0.570411\pi$
$347$	−984235.	−0.438808	−0.219404	+	0.975634i	$0.570411\pi$
$348$	−2.36150e6	−1.04530
$349$	437783.	0.192396	0.0961979	−	0.995362i	$-0.469332\pi$
$349$	437783.	0.192396	0.0961979	+	0.995362i	$0.469332\pi$
$350$	0	0
$351$	2.20227e6	0.954121
$352$	187766.	0.0807717
$353$	2.74405e6	1.17208	0.586038	−	0.810284i	$-0.300687\pi$
$353$	2.74405e6	1.17208	0.586038	+	0.810284i	$0.300687\pi$
$354$	1.28946e6	0.546890
$355$	0	0
$356$	−1.12829e6	−0.471841
$357$	31853.9	0.0132279
$358$	747551.	0.308272
$359$	−3.66684e6	−1.50161	−0.750803	−	0.660526i	$-0.770333\pi$
$359$	−3.66684e6	−1.50161	−0.750803	+	0.660526i	$0.770333\pi$
$360$	0	0
$361$	978526.	0.395188
$362$	−1.02634e6	−0.411641
$363$	3.85764e6	1.53658
$364$	131556.	0.0520425
$365$	0	0
$366$	−3.38617e6	−1.32132
$367$	−3.98978e6	−1.54627	−0.773133	−	0.634244i	$-0.781311\pi$
$367$	−3.98978e6	−1.54627	−0.773133	+	0.634244i	$0.781311\pi$
$368$	1.22159e6	0.470224
$369$	641684.	0.245333
$370$	0	0
$371$	266388.	0.100480
$372$	−2.49704e6	−0.935552
$373$	3.40693e6	1.26792	0.633960	−	0.773366i	$-0.281428\pi$
$373$	3.40693e6	1.26792	0.633960	+	0.773366i	$0.281428\pi$
$374$	15862.8	0.00586407
$375$	0	0
$376$	814941.	0.297274
$377$	−823948.	−0.298570
$378$	−2.53600e6	−0.912893
$379$	3.86571e6	1.38239	0.691196	−	0.722667i	$-0.257084\pi$
$379$	3.86571e6	1.38239	0.691196	+	0.722667i	$0.257084\pi$
$380$	0	0
$381$	−7.86710e6	−2.77653
$382$	1.49852e6	0.525416
$383$	−3.95015e6	−1.37599	−0.687997	−	0.725714i	$-0.741510\pi$
$383$	−3.95015e6	−1.37599	−0.687997	+	0.725714i	$0.741510\pi$
$384$	495993.	0.171652
$385$	0	0
$386$	120426.	0.0411388
$387$	5.51369e6	1.87139
$388$	−180178.	−0.0607608
$389$	−413950.	−0.138699	−0.0693497	−	0.997592i	$-0.522092\pi$
$389$	−413950.	−0.138699	−0.0693497	+	0.997592i	$0.522092\pi$
$390$	0	0
$391$	103202.	0.0341385
$392$	924156.	0.303760
$393$	5.19639e6	1.69715
$394$	−3.25569e6	−1.05658
$395$	0	0
$396$	−1.97581e6	−0.633151
$397$	−896816.	−0.285579	−0.142790	−	0.989753i	$-0.545607\pi$
$397$	−896816.	−0.285579	−0.142790	+	0.989753i	$0.545607\pi$
$398$	−3.42676e6	−1.08437
$399$	2.73754e6	0.860853
$400$	0	0
$401$	−226670.	−0.0703936	−0.0351968	−	0.999380i	$-0.511206\pi$
$401$	−226670.	−0.0703936	−0.0351968	+	0.999380i	$0.511206\pi$
$402$	2.60119e6	0.802799
$403$	−871237.	−0.267223
$404$	650708.	0.198350
$405$	0	0
$406$	948807.	0.285669
$407$	216129.	0.0646735
$408$	41902.3	0.0124620
$409$	−3.06994e6	−0.907448	−0.453724	−	0.891142i	$-0.649905\pi$
$409$	−3.06994e6	−0.907448	−0.453724	+	0.891142i	$0.649905\pi$
$410$	0	0
$411$	−6.62698e6	−1.93513
$412$	−1.68195e6	−0.488170
$413$	−518080.	−0.149459
$414$	−1.28544e7	−3.68597
$415$	0	0
$416$	173056.	0.0490290
$417$	1.00286e7	2.82422
$418$	1.36325e6	0.381624
$419$	−2.75158e6	−0.765679	−0.382839	−	0.923815i	$-0.625054\pi$
$419$	−2.75158e6	−0.765679	−0.382839	+	0.923815i	$0.625054\pi$
$420$	0	0
$421$	2.91415e6	0.801321	0.400660	−	0.916227i	$-0.368781\pi$
$421$	2.91415e6	0.801321	0.400660	+	0.916227i	$0.368781\pi$
$422$	−372412.	−0.101799
$423$	−8.57542e6	−2.33026
$424$	350421.	0.0946621
$425$	0	0
$426$	−8.80622e6	−2.35107
$427$	1.36050e6	0.361101
$428$	−1.09653e6	−0.289341
$429$	−938120.	−0.246102
$430$	0	0
$431$	−1.28558e6	−0.333353	−0.166677	−	0.986012i	$-0.553304\pi$
$431$	−1.28558e6	−0.333353	−0.166677	+	0.986012i	$0.553304\pi$
$432$	−3.33599e6	−0.860033
$433$	4.82256e6	1.23611	0.618056	−	0.786134i	$-0.287920\pi$
$433$	4.82256e6	1.23611	0.618056	+	0.786134i	$0.287920\pi$
$434$	1.00326e6	0.255676
$435$	0	0
$436$	2.55297e6	0.643175
$437$	8.86920e6	2.22168
$438$	−9.52833e6	−2.37318
$439$	686841.	0.170096	0.0850482	−	0.996377i	$-0.472896\pi$
$439$	686841.	0.170096	0.0850482	+	0.996377i	$0.472896\pi$
$440$	0	0
$441$	−9.72466e6	−2.38110
$442$	14620.1	0.00355954
$443$	3.59352e6	0.869983	0.434991	−	0.900435i	$-0.356751\pi$
$443$	3.59352e6	0.869983	0.434991	+	0.900435i	$0.356751\pi$
$444$	570916.	0.137441
$445$	0	0
$446$	−2.16105e6	−0.514431
$447$	1.72100e6	0.407393
$448$	−199280.	−0.0469105
$449$	−3.77675e6	−0.884102	−0.442051	−	0.896990i	$-0.645749\pi$
$449$	−3.77675e6	−0.884102	−0.442051	+	0.896990i	$0.645749\pi$
$450$	0	0
$451$	−174714.	−0.0404470
$452$	−1.60186e6	−0.368790
$453$	−2.04621e6	−0.468495
$454$	1.19119e6	0.271233
$455$	0	0
$456$	3.60111e6	0.811006
$457$	4.86264e6	1.08913	0.544567	−	0.838717i	$-0.316694\pi$
$457$	4.86264e6	1.08913	0.544567	+	0.838717i	$0.316694\pi$
$458$	−3.34161e6	−0.744375
$459$	−281830.	−0.0624389
$460$	0	0
$461$	−2.32471e6	−0.509468	−0.254734	−	0.967011i	$-0.581988\pi$
$461$	−2.32471e6	−0.509468	−0.254734	+	0.967011i	$0.581988\pi$
$462$	1.08028e6	0.235468
$463$	4.48469e6	0.972254	0.486127	−	0.873888i	$-0.338409\pi$
$463$	4.48469e6	0.972254	0.486127	+	0.873888i	$0.338409\pi$
$464$	1.24811e6	0.269128
$465$	0	0
$466$	288217.	0.0614831
$467$	−373349.	−0.0792179	−0.0396090	−	0.999215i	$-0.512611\pi$
$467$	−373349.	−0.0792179	−0.0396090	+	0.999215i	$0.512611\pi$
$468$	−1.82102e6	−0.384327
$469$	−1.04511e6	−0.219396
$470$	0	0
$471$	6.75013e6	1.40204
$472$	−681510.	−0.140805
$473$	−1.50124e6	−0.308529
$474$	1.06653e7	2.18036
$475$	0	0
$476$	−16835.5	−0.00340573
$477$	−3.68740e6	−0.742034
$478$	4.52266e6	0.905367
$479$	5.08290e6	1.01221	0.506107	−	0.862470i	$-0.331084\pi$
$479$	5.08290e6	1.01221	0.506107	+	0.862470i	$0.331084\pi$
$480$	0	0
$481$	199197.	0.0392573
$482$	4.36950e6	0.856671
$483$	7.02821e6	1.37081
$484$	−2.03885e6	−0.395615
$485$	0	0
$486$	1.52871e7	2.93585
$487$	5.05870e6	0.966532	0.483266	−	0.875474i	$-0.339450\pi$
$487$	5.05870e6	0.966532	0.483266	+	0.875474i	$0.339450\pi$
$488$	1.78967e6	0.340192
$489$	−1.83639e7	−3.47290
$490$	0	0
$491$	−2.93879e6	−0.550129	−0.275064	−	0.961426i	$-0.588699\pi$
$491$	−2.93879e6	−0.550129	−0.275064	+	0.961426i	$0.588699\pi$
$492$	−461517.	−0.0859558
$493$	105442.	0.0195388
$494$	1.25646e6	0.231649
$495$	0	0
$496$	1.31974e6	0.240872
$497$	3.53817e6	0.642521
$498$	1.43587e7	2.59443
$499$	944824.	0.169863	0.0849317	−	0.996387i	$-0.472933\pi$
$499$	944824.	0.169863	0.0849317	+	0.996387i	$0.472933\pi$
$500$	0	0
$501$	−1.05242e7	−1.87324
$502$	−7.60309e6	−1.34658
$503$	−5.43804e6	−0.958346	−0.479173	−	0.877720i	$-0.659063\pi$
$503$	−5.43804e6	−0.958346	−0.479173	+	0.877720i	$0.659063\pi$
$504$	2.09698e6	0.367720
$505$	0	0
$506$	3.49994e6	0.607692
$507$	−864628.	−0.149386
$508$	4.15795e6	0.714858
$509$	22475.6	0.00384518	0.00192259	−	0.999998i	$-0.499388\pi$
$509$	22475.6	0.00384518	0.00192259	+	0.999998i	$0.499388\pi$
$510$	0	0
$511$	3.82830e6	0.648565
$512$	−262144.	−0.0441942
$513$	−2.42206e7	−4.06342
$514$	6.52510e6	1.08938
$515$	0	0
$516$	−3.96560e6	−0.655669
$517$	2.33487e6	0.384181
$518$	−229383.	−0.0375610
$519$	1.10090e6	0.179403
$520$	0	0
$521$	4.79883e6	0.774534	0.387267	−	0.921968i	$-0.373419\pi$
$521$	4.79883e6	0.774534	0.387267	+	0.921968i	$0.373419\pi$
$522$	−1.31336e7	−2.10963
$523$	−828786.	−0.132491	−0.0662457	−	0.997803i	$-0.521102\pi$
$523$	−828786.	−0.132491	−0.0662457	+	0.997803i	$0.521102\pi$
$524$	−2.74641e6	−0.436956
$525$	0	0
$526$	1.72205e6	0.271381
$527$	111494.	0.0174874
$528$	1.42106e6	0.221833
$529$	1.63339e7	2.53776
$530$	0	0
$531$	7.17135e6	1.10374
$532$	−1.44686e6	−0.221639
$533$	−161027.	−0.0245517
$534$	−8.53918e6	−1.29588
$535$	0	0
$536$	−1.37479e6	−0.206692
$537$	5.65766e6	0.846645
$538$	−118126.	−0.0175950
$539$	2.64778e6	0.392563
$540$	0	0
$541$	−1.09983e7	−1.61559	−0.807795	−	0.589463i	$-0.799339\pi$
$541$	−1.09983e7	−1.61559	−0.807795	+	0.589463i	$0.799339\pi$
$542$	1.12630e6	0.164686
$543$	−7.76758e6	−1.13054
$544$	−22146.4	−0.00320852
$545$	0	0
$546$	995651.	0.142931
$547$	−6.04060e6	−0.863201	−0.431600	−	0.902065i	$-0.642051\pi$
$547$	−6.04060e6	−0.863201	−0.431600	+	0.902065i	$0.642051\pi$
$548$	3.50251e6	0.498228
$549$	−1.88323e7	−2.66668
$550$	0	0
$551$	9.06178e6	1.27155
$552$	9.24527e6	1.29143
$553$	−4.28511e6	−0.595867
$554$	8.30832e6	1.15011
$555$	0	0
$556$	−5.30034e6	−0.727138
$557$	−5.15197e6	−0.703616	−0.351808	−	0.936072i	$-0.614433\pi$
$557$	−5.15197e6	−0.703616	−0.351808	+	0.936072i	$0.614433\pi$
$558$	−1.38873e7	−1.88814
$559$	−1.38363e6	−0.187280
$560$	0	0
$561$	120053.	0.0161052
$562$	133743.	0.0178620
$563$	−614271.	−0.0816750	−0.0408375	−	0.999166i	$-0.513003\pi$
$563$	−614271.	−0.0816750	−0.0408375	+	0.999166i	$0.513003\pi$
$564$	6.16768e6	0.816440
$565$	0	0
$566$	−9.02149e6	−1.18369
$567$	−1.12311e7	−1.46712
$568$	4.65429e6	0.605317
$569$	−1.06927e7	−1.38454	−0.692270	−	0.721639i	$-0.743389\pi$
$569$	−1.06927e7	−1.38454	−0.692270	+	0.721639i	$0.743389\pi$
$570$	0	0
$571$	−1.74824e6	−0.224393	−0.112197	−	0.993686i	$-0.535789\pi$
$571$	−1.74824e6	−0.224393	−0.112197	+	0.993686i	$0.535789\pi$
$572$	495819.	0.0633626
$573$	1.13412e7	1.44301
$574$	185429.	0.0234908
$575$	0	0
$576$	2.75848e6	0.346428
$577$	1.02190e7	1.27782	0.638909	−	0.769282i	$-0.279386\pi$
$577$	1.02190e7	1.27782	0.638909	+	0.769282i	$0.279386\pi$
$578$	5.67756e6	0.706874
$579$	911415.	0.112985
$580$	0	0
$581$	−5.76905e6	−0.709029
$582$	−1.36364e6	−0.166875
$583$	1.00398e6	0.122336
$584$	5.03595e6	0.611011
$585$	0	0
$586$	−5.99107e6	−0.720710
$587$	−1.18011e7	−1.41360	−0.706802	−	0.707412i	$-0.749863\pi$
$587$	−1.18011e7	−1.41360	−0.706802	+	0.707412i	$0.749863\pi$
$588$	6.99425e6	0.834253
$589$	9.58187e6	1.13805
$590$	0	0
$591$	−2.46399e7	−2.90181
$592$	−301743.	−0.0353861
$593$	3.55381e6	0.415009	0.207504	−	0.978234i	$-0.433466\pi$
$593$	3.55381e6	0.415009	0.207504	+	0.978234i	$0.433466\pi$
$594$	−9.55786e6	−1.11146
$595$	0	0
$596$	−909591.	−0.104889
$597$	−2.59346e7	−2.97813
$598$	3.22575e6	0.368874
$599$	1.10074e7	1.25348	0.626742	−	0.779227i	$-0.284388\pi$
$599$	1.10074e7	1.25348	0.626742	+	0.779227i	$0.284388\pi$
$600$	0	0
$601$	1.18989e7	1.34375	0.671877	−	0.740663i	$-0.265489\pi$
$601$	1.18989e7	1.34375	0.671877	+	0.740663i	$0.265489\pi$
$602$	1.59330e6	0.179187
$603$	1.44666e7	1.62021
$604$	1.08147e6	0.120621
$605$	0	0
$606$	4.92473e6	0.544754
$607$	8.30894e6	0.915322	0.457661	−	0.889127i	$-0.348687\pi$
$607$	8.30894e6	0.915322	0.457661	+	0.889127i	$0.348687\pi$
$608$	−1.90327e6	−0.208805
$609$	7.18081e6	0.784568
$610$	0	0
$611$	2.15195e6	0.233201
$612$	233041.	0.0251509
$613$	−7.13046e6	−0.766419	−0.383210	−	0.923661i	$-0.625181\pi$
$613$	−7.13046e6	−0.766419	−0.383210	+	0.923661i	$0.625181\pi$
$614$	−3.01034e6	−0.322252
$615$	0	0
$616$	−570954.	−0.0606246
$617$	−3.60632e6	−0.381374	−0.190687	−	0.981651i	$-0.561072\pi$
$617$	−3.60632e6	−0.381374	−0.190687	+	0.981651i	$0.561072\pi$
$618$	−1.27295e7	−1.34072
$619$	−1.83897e6	−0.192907	−0.0964535	−	0.995337i	$-0.530750\pi$
$619$	−1.83897e6	−0.192907	−0.0964535	+	0.995337i	$0.530750\pi$
$620$	0	0
$621$	−6.21825e7	−6.47052
$622$	−8.97584e6	−0.930249
$623$	3.43088e6	0.354148
$624$	1.30973e6	0.134655
$625$	0	0
$626$	8.80595e6	0.898132
$627$	1.03174e7	1.04810
$628$	−3.56760e6	−0.360975
$629$	−25491.7	−0.00256905
$630$	0	0
$631$	−1.06969e7	−1.06951	−0.534753	−	0.845009i	$-0.679595\pi$
$631$	−1.06969e7	−1.06951	−0.534753	+	0.845009i	$0.679595\pi$
$632$	−5.63686e6	−0.561364
$633$	−2.81851e6	−0.279583
$634$	−9.10681e6	−0.899794
$635$	0	0
$636$	2.65208e6	0.259982
$637$	2.44035e6	0.238289
$638$	3.57593e6	0.347806
$639$	−4.89760e7	−4.74494
$640$	0	0
$641$	785551.	0.0755143	0.0377571	−	0.999287i	$-0.487979\pi$
$641$	785551.	0.0755143	0.0377571	+	0.999287i	$0.487979\pi$
$642$	−8.29880e6	−0.794654
$643$	1.56345e7	1.49127	0.745634	−	0.666356i	$-0.232147\pi$
$643$	1.56345e7	1.49127	0.745634	+	0.666356i	$0.232147\pi$
$644$	−3.71457e6	−0.352934
$645$	0	0
$646$	−160791.	−0.0151594
$647$	7.19435e6	0.675664	0.337832	−	0.941206i	$-0.390306\pi$
$647$	7.19435e6	0.675664	0.337832	+	0.941206i	$0.390306\pi$
$648$	−1.47740e7	−1.38217
$649$	−1.95258e6	−0.181969
$650$	0	0
$651$	7.59295e6	0.702195
$652$	9.70575e6	0.894149
$653$	1.14576e7	1.05150	0.525752	−	0.850638i	$-0.323784\pi$
$653$	1.14576e7	1.05150	0.525752	+	0.850638i	$0.323784\pi$
$654$	1.93215e7	1.76643
$655$	0	0
$656$	243923.	0.0221306
$657$	−5.29920e7	−4.78957
$658$	−2.47806e6	−0.223124
$659$	−5.39523e6	−0.483945	−0.241973	−	0.970283i	$-0.577794\pi$
$659$	−5.39523e6	−0.483945	−0.241973	+	0.970283i	$0.577794\pi$
$660$	0	0
$661$	1.58289e7	1.40912	0.704559	−	0.709645i	$-0.251145\pi$
$661$	1.58289e7	1.40912	0.704559	+	0.709645i	$0.251145\pi$
$662$	1.70678e6	0.151368
$663$	110648.	0.00977600
$664$	−7.58891e6	−0.667973
$665$	0	0
$666$	3.17516e6	0.277383
$667$	2.32647e7	2.02480
$668$	5.56227e6	0.482293
$669$	−1.63554e7	−1.41285
$670$	0	0
$671$	5.12754e6	0.439646
$672$	−1.50821e6	−0.128836
$673$	4.13517e6	0.351930	0.175965	−	0.984396i	$-0.443695\pi$
$673$	4.13517e6	0.351930	0.175965	+	0.984396i	$0.443695\pi$
$674$	−8.27016e6	−0.701236
$675$	0	0
$676$	456976.	0.0384615
$677$	−4.55672e6	−0.382103	−0.191052	−	0.981580i	$-0.561190\pi$
$677$	−4.55672e6	−0.382103	−0.191052	+	0.981580i	$0.561190\pi$
$678$	−1.21233e7	−1.01285
$679$	547882.	0.0456051
$680$	0	0
$681$	9.01525e6	0.744921
$682$	3.78117e6	0.311290
$683$	9.28853e6	0.761895	0.380947	−	0.924597i	$-0.375598\pi$
$683$	9.28853e6	0.761895	0.380947	+	0.924597i	$0.375598\pi$
$684$	2.00276e7	1.63678
$685$	0	0
$686$	−6.08096e6	−0.493358
$687$	−2.52901e7	−2.04437
$688$	2.09591e6	0.168812
$689$	925332.	0.0742590
$690$	0	0
$691$	−1.76099e6	−0.140301	−0.0701507	−	0.997536i	$-0.522348\pi$
$691$	−1.76099e6	−0.140301	−0.0701507	+	0.997536i	$0.522348\pi$
$692$	−581852.	−0.0461899
$693$	6.00800e6	0.475222
$694$	3.93694e6	0.310284
$695$	0	0
$696$	9.44602e6	0.739138
$697$	20607.0	0.00160669
$698$	−1.75113e6	−0.136044
$699$	2.18130e6	0.168859
$700$	0	0
$701$	1.20471e7	0.925950	0.462975	−	0.886371i	$-0.346782\pi$
$701$	1.20471e7	0.925950	0.462975	+	0.886371i	$0.346782\pi$
$702$	−8.80909e6	−0.674665
$703$	−2.19077e6	−0.167189
$704$	−751062.	−0.0571142
$705$	0	0
$706$	−1.09762e7	−0.828783
$707$	−1.97866e6	−0.148875
$708$	−5.15784e6	−0.386709
$709$	3.45858e6	0.258394	0.129197	−	0.991619i	$-0.458760\pi$
$709$	3.45858e6	0.258394	0.129197	+	0.991619i	$0.458760\pi$
$710$	0	0
$711$	5.93153e7	4.40041
$712$	4.51316e6	0.333642
$713$	2.45999e7	1.81222
$714$	−127416.	−0.00935357
$715$	0	0
$716$	−2.99021e6	−0.217981
$717$	3.42287e7	2.48652
$718$	1.46674e7	1.06180
$719$	−1.66383e7	−1.20029	−0.600146	−	0.799891i	$-0.704891\pi$
$719$	−1.66383e7	−1.20029	−0.600146	+	0.799891i	$0.704891\pi$
$720$	0	0
$721$	5.11445e6	0.366405
$722$	−3.91410e6	−0.279440
$723$	3.30695e7	2.35278
$724$	4.10535e6	0.291074
$725$	0	0
$726$	−1.54306e7	−1.08653
$727$	5.70304e6	0.400194	0.200097	−	0.979776i	$-0.435874\pi$
$727$	5.70304e6	0.400194	0.200097	+	0.979776i	$0.435874\pi$
$728$	−526225.	−0.0367996
$729$	5.96013e7	4.15372
$730$	0	0
$731$	177066.	0.0122558
$732$	1.35447e7	0.934311
$733$	−1.80882e7	−1.24347	−0.621735	−	0.783228i	$-0.713572\pi$
$733$	−1.80882e7	−1.24347	−0.621735	+	0.783228i	$0.713572\pi$
$734$	1.59591e7	1.09337
$735$	0	0
$736$	−4.88634e6	−0.332498
$737$	−3.93887e6	−0.267118
$738$	−2.56674e6	−0.173476
$739$	8.58937e6	0.578563	0.289281	−	0.957244i	$-0.406584\pi$
$739$	8.58937e6	0.578563	0.289281	+	0.957244i	$0.406584\pi$
$740$	0	0
$741$	9.50918e6	0.636205
$742$	−1.06555e6	−0.0710503
$743$	−3.77059e6	−0.250575	−0.125287	−	0.992120i	$-0.539985\pi$
$743$	−3.77059e6	−0.250575	−0.125287	+	0.992120i	$0.539985\pi$
$744$	9.98816e6	0.661536
$745$	0	0
$746$	−1.36277e7	−0.896554
$747$	7.98562e7	5.23609
$748$	−63451.0	−0.00414653
$749$	3.33430e6	0.217170
$750$	0	0
$751$	69637.4	0.00450549	0.00225275	−	0.999997i	$-0.499283\pi$
$751$	69637.4	0.00450549	0.00225275	+	0.999997i	$0.499283\pi$
$752$	−3.25976e6	−0.210204
$753$	−5.75421e7	−3.69827
$754$	3.29579e6	0.211121
$755$	0	0
$756$	1.01440e7	0.645513
$757$	1.98883e7	1.26141	0.630706	−	0.776022i	$-0.282765\pi$
$757$	1.98883e7	1.26141	0.630706	+	0.776022i	$0.282765\pi$
$758$	−1.54628e7	−0.977499
$759$	2.64884e7	1.66898
$760$	0	0
$761$	−8.37595e6	−0.524291	−0.262146	−	0.965028i	$-0.584430\pi$
$761$	−8.37595e6	−0.524291	−0.262146	+	0.965028i	$0.584430\pi$
$762$	3.14684e7	1.96330
$763$	−7.76300e6	−0.482746
$764$	−5.99407e6	−0.371525
$765$	0	0
$766$	1.58006e7	0.972974
$767$	−1.79961e6	−0.110456
$768$	−1.98397e6	−0.121376
$769$	1.14325e7	0.697148	0.348574	−	0.937281i	$-0.386666\pi$
$769$	1.14325e7	0.697148	0.348574	+	0.937281i	$0.386666\pi$
$770$	0	0
$771$	4.93836e7	2.99190
$772$	−481704.	−0.0290896
$773$	1.04075e7	0.626467	0.313234	−	0.949676i	$-0.398588\pi$
$773$	1.04075e7	0.626467	0.313234	+	0.949676i	$0.398588\pi$
$774$	−2.20548e7	−1.32328
$775$	0	0
$776$	720713.	0.0429644
$777$	−1.73603e6	−0.103158
$778$	1.65580e6	0.0980752
$779$	1.77098e6	0.104561
$780$	0	0
$781$	1.33349e7	0.782280
$782$	−412806.	−0.0241396
$783$	−6.35327e7	−3.70334
$784$	−3.69662e6	−0.214791
$785$	0	0
$786$	−2.07856e7	−1.20007
$787$	−2.60331e7	−1.49827	−0.749134	−	0.662418i	$-0.769530\pi$
$787$	−2.60331e7	−1.49827	−0.749134	+	0.662418i	$0.769530\pi$
$788$	1.30227e7	0.747114
$789$	1.30329e7	0.745329
$790$	0	0
$791$	4.87090e6	0.276801
$792$	7.90324e6	0.447705
$793$	4.72585e6	0.266868
$794$	3.58726e6	0.201935
$795$	0	0
$796$	1.37070e7	0.766763
$797$	1.37335e7	0.765837	0.382918	−	0.923782i	$-0.374919\pi$
$797$	1.37335e7	0.765837	0.382918	+	0.923782i	$0.374919\pi$
$798$	−1.09502e7	−0.608715
$799$	−275390.	−0.0152610
$800$	0	0
$801$	−4.74908e7	−2.61534
$802$	906680.	0.0497758
$803$	1.44284e7	0.789638
$804$	−1.04048e7	−0.567664
$805$	0	0
$806$	3.48495e6	0.188955
$807$	−894009.	−0.0483235
$808$	−2.60283e6	−0.140255
$809$	−1.72731e6	−0.0927895	−0.0463947	−	0.998923i	$-0.514773\pi$
$809$	−1.72731e6	−0.0927895	−0.0463947	+	0.998923i	$0.514773\pi$
$810$	0	0
$811$	2.78088e7	1.48467	0.742335	−	0.670029i	$-0.233718\pi$
$811$	2.78088e7	1.48467	0.742335	+	0.670029i	$0.233718\pi$
$812$	−3.79523e6	−0.201998
$813$	8.52412e6	0.452297
$814$	−864515.	−0.0457311
$815$	0	0
$816$	−167609.	−0.00881197
$817$	1.52172e7	0.797588
$818$	1.22798e7	0.641663
$819$	5.53733e6	0.288464
$820$	0	0
$821$	−1.34763e7	−0.697771	−0.348886	−	0.937165i	$-0.613440\pi$
$821$	−1.34763e7	−0.697771	−0.348886	+	0.937165i	$0.613440\pi$
$822$	2.65079e7	1.36835
$823$	2.98539e7	1.53639	0.768196	−	0.640215i	$-0.221155\pi$
$823$	2.98539e7	1.53639	0.768196	+	0.640215i	$0.221155\pi$
$824$	6.72782e6	0.345188
$825$	0	0
$826$	2.07232e6	0.105683
$827$	−1.86881e7	−0.950170	−0.475085	−	0.879940i	$-0.657583\pi$
$827$	−1.86881e7	−0.950170	−0.475085	+	0.879940i	$0.657583\pi$
$828$	5.14178e7	2.60638
$829$	2.92764e7	1.47955	0.739777	−	0.672852i	$-0.234931\pi$
$829$	2.92764e7	1.47955	0.739777	+	0.672852i	$0.234931\pi$
$830$	0	0
$831$	6.28795e7	3.15869
$832$	−692224.	−0.0346688
$833$	−312297.	−0.0155939
$834$	−4.01143e7	−1.99703
$835$	0	0
$836$	−5.45301e6	−0.269849
$837$	−6.71791e7	−3.31452
$838$	1.10063e7	0.541417
$839$	5.09345e6	0.249808	0.124904	−	0.992169i	$-0.460138\pi$
$839$	5.09345e6	0.249808	0.124904	+	0.992169i	$0.460138\pi$
$840$	0	0
$841$	3.25869e6	0.158874
$842$	−1.16566e7	−0.566619
$843$	1.01220e6	0.0490567
$844$	1.48965e6	0.0719826
$845$	0	0
$846$	3.43017e7	1.64774
$847$	6.19970e6	0.296936
$848$	−1.40169e6	−0.0669362
$849$	−6.82769e7	−3.25091
$850$	0	0
$851$	−5.62446e6	−0.266230
$852$	3.52249e7	1.66246
$853$	−2.71182e7	−1.27611	−0.638056	−	0.769990i	$-0.720261\pi$
$853$	−2.71182e7	−1.27611	−0.638056	+	0.769990i	$0.720261\pi$
$854$	−5.44199e6	−0.255337
$855$	0	0
$856$	4.38611e6	0.204595
$857$	−2.90233e7	−1.34988	−0.674939	−	0.737873i	$-0.735830\pi$
$857$	−2.90233e7	−1.34988	−0.674939	+	0.737873i	$0.735830\pi$
$858$	3.75248e6	0.174020
$859$	9.21734e6	0.426209	0.213105	−	0.977029i	$-0.431643\pi$
$859$	9.21734e6	0.426209	0.213105	+	0.977029i	$0.431643\pi$
$860$	0	0
$861$	1.40337e6	0.0645156
$862$	5.14230e6	0.235716
$863$	5.18993e6	0.237211	0.118605	−	0.992941i	$-0.462158\pi$
$863$	5.18993e6	0.237211	0.118605	+	0.992941i	$0.462158\pi$
$864$	1.33440e7	0.608135
$865$	0	0
$866$	−1.92902e7	−0.874064
$867$	4.29692e7	1.94138
$868$	−4.01305e6	−0.180790
$869$	−1.61500e7	−0.725478
$870$	0	0
$871$	−3.63030e6	−0.162143
$872$	−1.02119e7	−0.454793
$873$	−7.58389e6	−0.336788
$874$	−3.54768e7	−1.57096
$875$	0	0
$876$	3.81133e7	1.67810
$877$	−1.62757e6	−0.0714563	−0.0357281	−	0.999362i	$-0.511375\pi$
$877$	−1.62757e6	−0.0714563	−0.0357281	+	0.999362i	$0.511375\pi$
$878$	−2.74736e6	−0.120276
$879$	−4.53420e7	−1.97938
$880$	0	0
$881$	−1.51075e6	−0.0655771	−0.0327885	−	0.999462i	$-0.510439\pi$
$881$	−1.51075e6	−0.0655771	−0.0327885	+	0.999462i	$0.510439\pi$
$882$	3.88986e7	1.68369
$883$	4.23022e7	1.82583	0.912917	−	0.408144i	$-0.133824\pi$
$883$	4.23022e7	1.82583	0.912917	+	0.408144i	$0.133824\pi$
$884$	−58480.2	−0.00251697
$885$	0	0
$886$	−1.43741e7	−0.615171
$887$	−4.11822e7	−1.75752	−0.878759	−	0.477265i	$-0.841628\pi$
$887$	−4.11822e7	−1.75752	−0.878759	+	0.477265i	$0.841628\pi$
$888$	−2.28366e6	−0.0971852
$889$	−1.26434e7	−0.536549
$890$	0	0
$891$	−4.23287e7	−1.78624
$892$	8.64419e6	0.363758
$893$	−2.36672e7	−0.993157
$894$	−6.88402e6	−0.288070
$895$	0	0
$896$	797122.	0.0331707
$897$	2.44133e7	1.01308
$898$	1.51070e7	0.625155
$899$	2.51341e7	1.03720
$900$	0	0
$901$	−118417.	−0.00485961
$902$	698857.	0.0286004
$903$	1.20585e7	0.492124
$904$	6.40744e6	0.260774
$905$	0	0
$906$	8.18485e6	0.331276
$907$	8.04757e6	0.324823	0.162411	−	0.986723i	$-0.448073\pi$
$907$	8.04757e6	0.324823	0.162411	+	0.986723i	$0.448073\pi$
$908$	−4.76477e6	−0.191791
$909$	2.73890e7	1.09943
$910$	0	0
$911$	4.68266e7	1.86938	0.934688	−	0.355468i	$-0.115679\pi$
$911$	4.68266e7	1.86938	0.934688	+	0.355468i	$0.115679\pi$
$912$	−1.44044e7	−0.573468
$913$	−2.17428e7	−0.863254
$914$	−1.94506e7	−0.770134
$915$	0	0
$916$	1.33664e7	0.526352
$917$	8.35123e6	0.327965
$918$	1.12732e6	0.0441510
$919$	4.47821e7	1.74910	0.874552	−	0.484931i	$-0.161155\pi$
$919$	4.47821e7	1.74910	0.874552	+	0.484931i	$0.161155\pi$
$920$	0	0
$921$	−2.27831e7	−0.885040
$922$	9.29885e6	0.360248
$923$	1.22902e7	0.474850
$924$	−4.32112e6	−0.166501
$925$	0	0
$926$	−1.79387e7	−0.687487
$927$	−7.07951e7	−2.70585
$928$	−4.99244e6	−0.190302
$929$	2.24234e7	0.852439	0.426219	−	0.904620i	$-0.359845\pi$
$929$	2.24234e7	0.852439	0.426219	+	0.904620i	$0.359845\pi$
$930$	0	0
$931$	−2.68390e7	−1.01483
$932$	−1.15287e6	−0.0434751
$933$	−6.79315e7	−2.55486
$934$	1.49340e6	0.0560155
$935$	0	0
$936$	7.28410e6	0.271760
$937$	−9.70701e6	−0.361191	−0.180595	−	0.983557i	$-0.557802\pi$
$937$	−9.70701e6	−0.361191	−0.180595	+	0.983557i	$0.557802\pi$
$938$	4.18043e6	0.155136
$939$	6.66456e7	2.46665
$940$	0	0
$941$	3.83951e7	1.41352	0.706760	−	0.707453i	$-0.250156\pi$
$941$	3.83951e7	1.41352	0.706760	+	0.707453i	$0.250156\pi$
$942$	−2.70005e7	−0.991391
$943$	4.54670e6	0.166501
$944$	2.72604e6	0.0995639
$945$	0	0
$946$	6.00495e6	0.218163
$947$	−2.96988e7	−1.07613	−0.538064	−	0.842904i	$-0.680844\pi$
$947$	−2.96988e7	−1.07613	−0.538064	+	0.842904i	$0.680844\pi$
$948$	−4.26612e7	−1.54174
$949$	1.32980e7	0.479316
$950$	0	0
$951$	−6.89226e7	−2.47122
$952$	67342.2	0.00240821
$953$	−3.29694e7	−1.17592	−0.587961	−	0.808889i	$-0.700069\pi$
$953$	−3.29694e7	−1.17592	−0.587961	+	0.808889i	$0.700069\pi$
$954$	1.47496e7	0.524697
$955$	0	0
$956$	−1.80907e7	−0.640191
$957$	2.70636e7	0.955224
$958$	−2.03316e7	−0.715744
$959$	−1.06504e7	−0.373954
$960$	0	0
$961$	−2.05255e6	−0.0716943
$962$	−796789.	−0.0277591
$963$	−4.61540e7	−1.60377
$964$	−1.74780e7	−0.605758
$965$	0	0
$966$	−2.81128e7	−0.969308
$967$	3.22700e6	0.110977	0.0554885	−	0.998459i	$-0.482328\pi$
$967$	3.22700e6	0.110977	0.0554885	+	0.998459i	$0.482328\pi$
$968$	8.15541e6	0.279742
$969$	−1.21691e6	−0.0416341
$970$	0	0
$971$	−3.78306e7	−1.28764	−0.643822	−	0.765176i	$-0.722652\pi$
$971$	−3.78306e7	−1.28764	−0.643822	+	0.765176i	$0.722652\pi$
$972$	−6.11483e7	−2.07596
$973$	1.61172e7	0.545766
$974$	−2.02348e7	−0.683441
$975$	0	0
$976$	−7.15869e6	−0.240552
$977$	−2.42424e7	−0.812529	−0.406265	−	0.913755i	$-0.633169\pi$
$977$	−2.42424e7	−0.812529	−0.406265	+	0.913755i	$0.633169\pi$
$978$	7.34556e7	2.45571
$979$	1.29305e7	0.431181
$980$	0	0
$981$	1.07457e8	3.56502
$982$	1.17551e7	0.389000
$983$	−4.25830e6	−0.140557	−0.0702785	−	0.997527i	$-0.522389\pi$
$983$	−4.25830e6	−0.140557	−0.0702785	+	0.997527i	$0.522389\pi$
$984$	1.84607e6	0.0607799
$985$	0	0
$986$	−421770.	−0.0138160
$987$	−1.87546e7	−0.612793
$988$	−5.02582e6	−0.163800
$989$	3.90677e7	1.27007
$990$	0	0
$991$	1.84537e6	0.0596897	0.0298448	−	0.999555i	$-0.490499\pi$
$991$	1.84537e6	0.0596897	0.0298448	+	0.999555i	$0.490499\pi$
$992$	−5.27898e6	−0.170322
$993$	1.29174e7	0.415720
$994$	−1.41527e7	−0.454331
$995$	0	0
$996$	−5.74348e7	−1.83454
$997$	−9.84877e6	−0.313794	−0.156897	−	0.987615i	$-0.550149\pi$
$997$	−9.84877e6	−0.313794	−0.156897	+	0.987615i	$0.550149\pi$
$998$	−3.77930e6	−0.120112
$999$	1.53596e7	0.486931

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.6.a.v.1.1		9
5.2	odd	4		130.6.b.b.79.9	✓	18
5.3	odd	4		130.6.b.b.79.10	yes	18
5.4	even	2		650.6.a.w.1.9		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.6.b.b.79.9	✓	18	5.2	odd	4
130.6.b.b.79.10	yes	18	5.3	odd	4
650.6.a.v.1.1		9	1.1	even	1	trivial
650.6.a.w.1.9		9	5.4	even	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:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	650.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} -30.2730 q^{3} +16.0000 q^{4} +121.092 q^{6} -48.6525 q^{7} -64.0000 q^{8} +673.456 q^{9} -183.365 q^{11} -484.368 q^{12} -169.000 q^{13} +194.610 q^{14} +256.000 q^{16} +21.6273 q^{17} -2693.82 q^{18} +1858.66 q^{19} +1472.86 q^{21} +733.459 q^{22} +4771.82 q^{23} +1937.47 q^{24} +676.000 q^{26} -13031.2 q^{27} -778.439 q^{28} +4875.43 q^{29} +5155.25 q^{31} -1024.00 q^{32} +5551.01 q^{33} -86.5092 q^{34} +10775.3 q^{36} -1178.68 q^{37} -7434.65 q^{38} +5116.14 q^{39} +952.823 q^{41} -5891.43 q^{42} +8187.16 q^{43} -2933.84 q^{44} -19087.3 q^{46} -12733.5 q^{47} -7749.89 q^{48} -14439.9 q^{49} -654.724 q^{51} -2704.00 q^{52} -5475.33 q^{53} +52124.8 q^{54} +3113.76 q^{56} -56267.3 q^{57} -19501.7 q^{58} +10648.6 q^{59} -27963.6 q^{61} -20621.0 q^{62} -32765.3 q^{63} +4096.00 q^{64} -22204.0 q^{66} +21481.1 q^{67} +346.037 q^{68} -144457. q^{69} -72723.3 q^{71} -43101.2 q^{72} -78686.6 q^{73} +4714.73 q^{74} +29738.6 q^{76} +8921.15 q^{77} -20464.6 q^{78} +88076.0 q^{79} +230844. q^{81} -3811.29 q^{82} +118577. q^{83} +23565.7 q^{84} -32748.6 q^{86} -147594. q^{87} +11735.4 q^{88} -70518.1 q^{89} +8222.27 q^{91} +76349.1 q^{92} -156065. q^{93} +50933.8 q^{94} +30999.6 q^{96} -11261.1 q^{97} +57759.8 q^{98} -123488. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 36 q^{2} - 14 q^{3} + 144 q^{4} + 56 q^{6} - 164 q^{7} - 576 q^{8} + 1019 q^{9} + 326 q^{11} - 224 q^{12} - 1521 q^{13} + 656 q^{14} + 2304 q^{16} - 2144 q^{17} - 4076 q^{18} + 5558 q^{19} - 236 q^{21}+ \cdots + 231774 q^{99}+O(q^{100}) \) 9 * q - 36 * q^2 - 14 * q^3 + 144 * q^4 + 56 * q^6 - 164 * q^7 - 576 * q^8 + 1019 * q^9 + 326 * q^11 - 224 * q^12 - 1521 * q^13 + 656 * q^14 + 2304 * q^16 - 2144 * q^17 - 4076 * q^18 + 5558 * q^19 - 236 * q^21 - 1304 * q^22 - 660 * q^23 + 896 * q^24 + 6084 * q^26 - 1796 * q^27 - 2624 * q^28 + 6742 * q^29 + 3100 * q^31 - 9216 * q^32 + 5672 * q^33 + 8576 * q^34 + 16304 * q^36 - 4236 * q^37 - 22232 * q^38 + 2366 * q^39 + 15494 * q^41 + 944 * q^42 - 22630 * q^43 + 5216 * q^44 + 2640 * q^46 - 11620 * q^47 - 3584 * q^48 + 35575 * q^49 + 40698 * q^51 - 24336 * q^52 - 80830 * q^53 + 7184 * q^54 + 10496 * q^56 + 33508 * q^57 - 26968 * q^58 + 57474 * q^59 + 69486 * q^61 - 12400 * q^62 - 72522 * q^63 + 36864 * q^64 - 22688 * q^66 + 16450 * q^67 - 34304 * q^68 - 32980 * q^69 - 7228 * q^71 - 65216 * q^72 - 115278 * q^73 + 16944 * q^74 + 88928 * q^76 - 5568 * q^77 - 9464 * q^78 + 81836 * q^79 + 254885 * q^81 - 61976 * q^82 - 108670 * q^83 - 3776 * q^84 + 90520 * q^86 - 333048 * q^87 - 20864 * q^88 + 164994 * q^89 + 27716 * q^91 - 10560 * q^92 - 316844 * q^93 + 46480 * q^94 + 14336 * q^96 - 155190 * q^97 - 142300 * q^98 + 231774 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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