LMFDB - Embedded newform 98.6.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [98,6,Mod(1,98)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("98.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$N$	$=$	$98 = 2 \cdot 7^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	98.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-8,-14,32,-42,56,0,-128,652] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$15.7176143417$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{130})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 130$ x^2 - 130
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-11.4018$ of defining polynomial
Character	$\chi$	$=$	98.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000 q^{2} -29.8035 q^{3} +16.0000 q^{4} -21.0000 q^{5} +119.214 q^{6} -64.0000 q^{8} +645.249 q^{9} +84.0000 q^{10} +331.874 q^{11} -476.856 q^{12} +66.8211 q^{13} +625.874 q^{15} +256.000 q^{16} +240.537 q^{17} -2581.00 q^{18} +441.979 q^{19} -336.000 q^{20} -1327.49 q^{22} -1071.37 q^{23} +1907.42 q^{24} -2684.00 q^{25} -267.284 q^{26} -11988.4 q^{27} +1791.75 q^{29} -2503.49 q^{30} -5688.74 q^{31} -1024.00 q^{32} -9891.00 q^{33} -962.147 q^{34} +10324.0 q^{36} +11210.7 q^{37} -1767.92 q^{38} -1991.50 q^{39} +1344.00 q^{40} +12077.9 q^{41} -9921.98 q^{43} +5309.98 q^{44} -13550.2 q^{45} +4285.47 q^{46} -16869.2 q^{47} -7629.70 q^{48} +10736.0 q^{50} -7168.84 q^{51} +1069.14 q^{52} +5298.77 q^{53} +47953.7 q^{54} -6969.35 q^{55} -13172.5 q^{57} -7166.99 q^{58} +41384.8 q^{59} +10014.0 q^{60} -21523.3 q^{61} +22754.9 q^{62} +4096.00 q^{64} -1403.24 q^{65} +39564.0 q^{66} -26618.8 q^{67} +3848.59 q^{68} +31930.5 q^{69} -58096.5 q^{71} -41295.9 q^{72} -39987.5 q^{73} -44842.9 q^{74} +79992.6 q^{75} +7071.66 q^{76} +7966.01 q^{78} -43949.8 q^{79} -5376.00 q^{80} +200502. q^{81} -48311.4 q^{82} +22421.4 q^{83} -5051.27 q^{85} +39687.9 q^{86} -53400.4 q^{87} -21239.9 q^{88} +24062.0 q^{89} +54200.9 q^{90} -17141.9 q^{92} +169544. q^{93} +67477.0 q^{94} -9281.56 q^{95} +30518.8 q^{96} +71896.4 q^{97} +214141. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 8 q^{2} - 14 q^{3} + 32 q^{4} - 42 q^{5} + 56 q^{6} - 128 q^{8} + 652 q^{9} + 168 q^{10} - 294 q^{11} - 224 q^{12} - 140 q^{13} + 294 q^{15} + 512 q^{16} + 1302 q^{17} - 2608 q^{18} - 1442 q^{19}+ \cdots + 209916 q^{99}+O(q^{100})$ 2 * q - 8 * q^2 - 14 * q^3 + 32 * q^4 - 42 * q^5 + 56 * q^6 - 128 * q^8 + 652 * q^9 + 168 * q^10 - 294 * q^11 - 224 * q^12 - 140 * q^13 + 294 * q^15 + 512 * q^16 + 1302 * q^17 - 2608 * q^18 - 1442 * q^19 - 672 * q^20 + 1176 * q^22 + 2646 * q^23 + 896 * q^24 - 5368 * q^25 + 560 * q^26 - 15722 * q^27 + 1668 * q^29 - 1176 * q^30 - 14798 * q^31 - 2048 * q^32 - 19782 * q^33 - 5208 * q^34 + 10432 * q^36 + 5182 * q^37 + 5768 * q^38 - 5260 * q^39 + 2688 * q^40 - 5124 * q^41 - 4520 * q^43 - 4704 * q^44 - 13692 * q^45 - 10584 * q^46 - 14994 * q^47 - 3584 * q^48 + 21472 * q^50 + 9606 * q^51 - 2240 * q^52 + 24006 * q^53 + 62888 * q^54 + 6174 * q^55 - 42946 * q^57 - 6672 * q^58 + 38850 * q^59 + 4704 * q^60 - 23618 * q^61 + 59192 * q^62 + 8192 * q^64 + 2940 * q^65 + 79128 * q^66 + 32002 * q^67 + 20832 * q^68 + 90678 * q^69 - 89376 * q^71 - 41728 * q^72 - 47138 * q^73 - 20728 * q^74 + 37576 * q^75 - 23072 * q^76 + 21040 * q^78 - 40970 * q^79 - 10752 * q^80 + 139858 * q^81 + 20496 * q^82 + 68376 * q^83 - 27342 * q^85 + 18080 * q^86 - 55356 * q^87 + 18816 * q^88 + 123102 * q^89 + 54768 * q^90 + 42336 * q^92 + 25586 * q^93 + 59976 * q^94 + 30282 * q^95 + 14336 * q^96 - 43652 * q^97 + 209916 * 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
$3$	−29.8035	−1.91190	−0.955948	−	0.293536i	$-0.905168\pi$
$3$	−29.8035	−1.91190	−0.955948	+	0.293536i	$0.905168\pi$
$4$	16.0000	0.500000
$5$	−21.0000	−0.375659	−0.187830	−	0.982202i	$-0.560145\pi$
$5$	−21.0000	−0.375659	−0.187830	+	0.982202i	$0.560145\pi$
$6$	119.214	1.35191
$7$	0	0
$7$	0	0
$8$	−64.0000	−0.353553
$9$	645.249	2.65535
$10$	84.0000	0.265631
$11$	331.874	0.826973	0.413486	−	0.910510i	$-0.364311\pi$
$11$	331.874	0.826973	0.413486	+	0.910510i	$0.364311\pi$
$12$	−476.856	−0.955948
$13$	66.8211	0.109662	0.0548308	−	0.998496i	$-0.482538\pi$
$13$	66.8211	0.109662	0.0548308	+	0.998496i	$0.482538\pi$
$14$	0	0
$15$	625.874	0.718222
$16$	256.000	0.250000
$17$	240.537	0.201864	0.100932	−	0.994893i	$-0.467818\pi$
$17$	240.537	0.201864	0.100932	+	0.994893i	$0.467818\pi$
$18$	−2581.00	−1.87761
$19$	441.979	0.280878	0.140439	−	0.990089i	$-0.455149\pi$
$19$	441.979	0.280878	0.140439	+	0.990089i	$0.455149\pi$
$20$	−336.000	−0.187830
$21$	0	0
$22$	−1327.49	−0.584758
$23$	−1071.37	−0.422298	−0.211149	−	0.977454i	$-0.567721\pi$
$23$	−1071.37	−0.422298	−0.211149	+	0.977454i	$0.567721\pi$
$24$	1907.42	0.675957
$25$	−2684.00	−0.858880
$26$	−267.284	−0.0775425
$27$	−11988.4	−3.16485
$28$	0	0
$29$	1791.75	0.395623	0.197812	−	0.980240i	$-0.436617\pi$
$29$	1791.75	0.395623	0.197812	+	0.980240i	$0.436617\pi$
$30$	−2503.49	−0.507859
$31$	−5688.74	−1.06319	−0.531596	−	0.846998i	$-0.678407\pi$
$31$	−5688.74	−1.06319	−0.531596	+	0.846998i	$0.678407\pi$
$32$	−1024.00	−0.176777
$33$	−9891.00	−1.58109
$34$	−962.147	−0.142740
$35$	0	0
$36$	10324.0	1.32767
$37$	11210.7	1.34626	0.673131	−	0.739523i	$-0.264949\pi$
$37$	11210.7	1.34626	0.673131	+	0.739523i	$0.264949\pi$
$38$	−1767.92	−0.198611
$39$	−1991.50	−0.209662
$40$	1344.00	0.132816
$41$	12077.9	1.12210	0.561048	−	0.827783i	$-0.310398\pi$
$41$	12077.9	1.12210	0.561048	+	0.827783i	$0.310398\pi$
$42$	0	0
$43$	−9921.98	−0.818328	−0.409164	−	0.912461i	$-0.634180\pi$
$43$	−9921.98	−0.818328	−0.409164	+	0.912461i	$0.634180\pi$
$44$	5309.98	0.413486
$45$	−13550.2	−0.997506
$46$	4285.47	0.298610
$47$	−16869.2	−1.11391	−0.556956	−	0.830542i	$-0.688031\pi$
$47$	−16869.2	−1.11391	−0.556956	+	0.830542i	$0.688031\pi$
$48$	−7629.70	−0.477974
$49$	0	0
$50$	10736.0	0.607320
$51$	−7168.84	−0.385943
$52$	1069.14	0.0548308
$53$	5298.77	0.259111	0.129555	−	0.991572i	$-0.458645\pi$
$53$	5298.77	0.259111	0.129555	+	0.991572i	$0.458645\pi$
$54$	47953.7	2.23789
$55$	−6969.35	−0.310660
$56$	0	0
$57$	−13172.5	−0.537009
$58$	−7166.99	−0.279748
$59$	41384.8	1.54778	0.773892	−	0.633317i	$-0.218307\pi$
$59$	41384.8	1.54778	0.773892	+	0.633317i	$0.218307\pi$
$60$	10014.0	0.359111
$61$	−21523.3	−0.740601	−0.370300	−	0.928912i	$-0.620745\pi$
$61$	−21523.3	−0.740601	−0.370300	+	0.928912i	$0.620745\pi$
$62$	22754.9	0.751790
$63$	0	0
$64$	4096.00	0.125000
$65$	−1403.24	−0.0411954
$66$	39564.0	1.11800
$67$	−26618.8	−0.724437	−0.362219	−	0.932093i	$-0.617981\pi$
$67$	−26618.8	−0.724437	−0.362219	+	0.932093i	$0.617981\pi$
$68$	3848.59	0.100932
$69$	31930.5	0.807390
$70$	0	0
$71$	−58096.5	−1.36774	−0.683870	−	0.729603i	$-0.739705\pi$
$71$	−58096.5	−1.36774	−0.683870	+	0.729603i	$0.739705\pi$
$72$	−41295.9	−0.938807
$73$	−39987.5	−0.878248	−0.439124	−	0.898426i	$-0.644711\pi$
$73$	−39987.5	−0.878248	−0.439124	+	0.898426i	$0.644711\pi$
$74$	−44842.9	−0.951951
$75$	79992.6	1.64209
$76$	7071.66	0.140439
$77$	0	0
$78$	7966.01	0.148253
$79$	−43949.8	−0.792299	−0.396150	−	0.918186i	$-0.629654\pi$
$79$	−43949.8	−0.792299	−0.396150	+	0.918186i	$0.629654\pi$
$80$	−5376.00	−0.0939149
$81$	200502.	3.39552
$82$	−48311.4	−0.793442
$83$	22421.4	0.357246	0.178623	−	0.983918i	$-0.442836\pi$
$83$	22421.4	0.357246	0.178623	+	0.983918i	$0.442836\pi$
$84$	0	0
$85$	−5051.27	−0.0758322
$86$	39687.9	0.578645
$87$	−53400.4	−0.756390
$88$	−21239.9	−0.292379
$89$	24062.0	0.322001	0.161001	−	0.986954i	$-0.448528\pi$
$89$	24062.0	0.322001	0.161001	+	0.986954i	$0.448528\pi$
$90$	54200.9	0.705343
$91$	0	0
$92$	−17141.9	−0.211149
$93$	169544.	2.03271
$94$	67477.0	0.787655
$95$	−9281.56	−0.105514
$96$	30518.8	0.337979
$97$	71896.4	0.775850	0.387925	−	0.921691i	$-0.373192\pi$
$97$	71896.4	0.775850	0.387925	+	0.921691i	$0.373192\pi$
$98$	0	0
$99$	214141.	2.19590
$100$	−42944.0	−0.429440
$101$	−15865.5	−0.154757	−0.0773783	−	0.997002i	$-0.524655\pi$
$101$	−15865.5	−0.154757	−0.0773783	+	0.997002i	$0.524655\pi$
$102$	28675.4	0.272903
$103$	−160574.	−1.49135	−0.745677	−	0.666307i	$-0.767874\pi$
$103$	−160574.	−1.49135	−0.745677	+	0.666307i	$0.767874\pi$
$104$	−4276.55	−0.0387713
$105$	0	0
$106$	−21195.1	−0.183219
$107$	−205047.	−1.73139	−0.865693	−	0.500575i	$-0.833122\pi$
$107$	−205047.	−1.73139	−0.865693	+	0.500575i	$0.833122\pi$
$108$	−191815.	−1.58242
$109$	−112544.	−0.907313	−0.453657	−	0.891177i	$-0.649881\pi$
$109$	−112544.	−0.907313	−0.453657	+	0.891177i	$0.649881\pi$
$110$	27877.4	0.219670
$111$	−334119.	−2.57391
$112$	0	0
$113$	−118332.	−0.871782	−0.435891	−	0.900000i	$-0.643567\pi$
$113$	−118332.	−0.871782	−0.435891	+	0.900000i	$0.643567\pi$
$114$	52690.1	0.379723
$115$	22498.7	0.158640
$116$	28668.0	0.197812
$117$	43116.2	0.291190
$118$	−165539.	−1.09445
$119$	0	0
$120$	−40055.9	−0.253930
$121$	−50910.9	−0.316116
$122$	86093.2	0.523684
$123$	−359962.	−2.14533
$124$	−91019.8	−0.531596
$125$	121989.	0.698306
$126$	0	0
$127$	−245195.	−1.34897	−0.674485	−	0.738289i	$-0.735634\pi$
$127$	−245195.	−1.34897	−0.674485	+	0.738289i	$0.735634\pi$
$128$	−16384.0	−0.0883883
$129$	295710.	1.56456
$130$	5612.97	0.0291296
$131$	69993.7	0.356353	0.178177	−	0.983999i	$-0.442980\pi$
$131$	69993.7	0.356353	0.178177	+	0.983999i	$0.442980\pi$
$132$	−158256.	−0.790543
$133$	0	0
$134$	106475.	0.512254
$135$	251757.	1.18891
$136$	−15394.4	−0.0713698
$137$	−187664.	−0.854239	−0.427119	−	0.904195i	$-0.640472\pi$
$137$	−187664.	−0.854239	−0.427119	+	0.904195i	$0.640472\pi$
$138$	−127722.	−0.570911
$139$	−78272.7	−0.343616	−0.171808	−	0.985130i	$-0.554961\pi$
$139$	−78272.7	−0.343616	−0.171808	+	0.985130i	$0.554961\pi$
$140$	0	0
$141$	502763.	2.12968
$142$	232386.	0.967139
$143$	22176.1	0.0906872
$144$	165184.	0.663837
$145$	−37626.7	−0.148620
$146$	159950.	0.621015
$147$	0	0
$148$	179372.	0.673131
$149$	−92166.4	−0.340100	−0.170050	−	0.985435i	$-0.554393\pi$
$149$	−92166.4	−0.340100	−0.170050	+	0.985435i	$0.554393\pi$
$150$	−319970.	−1.16113
$151$	53416.0	0.190646	0.0953232	−	0.995446i	$-0.469612\pi$
$151$	53416.0	0.190646	0.0953232	+	0.995446i	$0.469612\pi$
$152$	−28286.7	−0.0993053
$153$	155206.	0.536019
$154$	0	0
$155$	119463.	0.399398
$156$	−31864.0	−0.104831
$157$	−280501.	−0.908206	−0.454103	−	0.890949i	$-0.650040\pi$
$157$	−280501.	−0.908206	−0.454103	+	0.890949i	$0.650040\pi$
$158$	175799.	0.560240
$159$	−157922.	−0.495393
$160$	21504.0	0.0664078
$161$	0	0
$162$	−802008.	−2.40099
$163$	626810.	1.84785	0.923925	−	0.382574i	$-0.124962\pi$
$163$	626810.	1.84785	0.923925	+	0.382574i	$0.124962\pi$
$164$	193246.	0.561048
$165$	207711.	0.593950
$166$	−89685.6	−0.252611
$167$	293997.	0.815741	0.407870	−	0.913040i	$-0.366272\pi$
$167$	293997.	0.815741	0.407870	+	0.913040i	$0.366272\pi$
$168$	0	0
$169$	−366828.	−0.987974
$170$	20205.1	0.0536215
$171$	285187.	0.745828
$172$	−158752.	−0.409164
$173$	714187.	1.81425	0.907124	−	0.420864i	$-0.138273\pi$
$173$	714187.	1.81425	0.907124	+	0.420864i	$0.138273\pi$
$174$	213601.	0.534849
$175$	0	0
$176$	84959.7	0.206743
$177$	−1.23341e6	−2.95920
$178$	−96248.1	−0.227689
$179$	580050.	1.35311	0.676554	−	0.736393i	$-0.263472\pi$
$179$	580050.	1.35311	0.676554	+	0.736393i	$0.263472\pi$
$180$	−216804.	−0.498753
$181$	−308046.	−0.698907	−0.349454	−	0.936954i	$-0.613633\pi$
$181$	−308046.	−0.698907	−0.349454	+	0.936954i	$0.613633\pi$
$182$	0	0
$183$	641470.	1.41595
$184$	68567.6	0.149305
$185$	−235425.	−0.505736
$186$	−678177.	−1.43734
$187$	79827.8	0.166936
$188$	−269908.	−0.556956
$189$	0	0
$190$	37126.2	0.0746100
$191$	−202789.	−0.402218	−0.201109	−	0.979569i	$-0.564455\pi$
$191$	−202789.	−0.402218	−0.201109	+	0.979569i	$0.564455\pi$
$192$	−122075.	−0.238987
$193$	605988.	1.17104	0.585519	−	0.810659i	$-0.300891\pi$
$193$	605988.	1.17104	0.585519	+	0.810659i	$0.300891\pi$
$194$	−287586.	−0.548609
$195$	41821.5	0.0787614
$196$	0	0
$197$	−310.819	−0.000570614 0	−0.000285307	−	1.00000i	$-0.500091\pi$
$197$	−310.819	−0.000570614 0	−0.000285307		1.00000i	$0.500091\pi$
$198$	−856565.	−1.55273
$199$	−409666.	−0.733327	−0.366664	−	0.930354i	$-0.619500\pi$
$199$	−409666.	−0.733327	−0.366664	+	0.930354i	$0.619500\pi$
$200$	171776.	0.303660
$201$	793332.	1.38505
$202$	63461.9	0.109429
$203$	0	0
$204$	−114701.	−0.192972
$205$	−253635.	−0.421526
$206$	642294.	1.05455
$207$	−691300.	−1.12135
$208$	17106.2	0.0274154
$209$	146681.	0.232278
$210$	0	0
$211$	−441339.	−0.682442	−0.341221	−	0.939983i	$-0.610840\pi$
$211$	−441339.	−0.682442	−0.341221	+	0.939983i	$0.610840\pi$
$212$	84780.3	0.129555
$213$	1.73148e6	2.61498
$214$	820188.	1.22427
$215$	208362.	0.307412
$216$	767260.	1.11894
$217$	0	0
$218$	450177.	0.641567
$219$	1.19177e6	1.67912
$220$	−111510.	−0.155330
$221$	16072.9	0.0221368
$222$	1.33648e6	1.82003
$223$	265133.	0.357028	0.178514	−	0.983937i	$-0.442871\pi$
$223$	265133.	0.357028	0.178514	+	0.983937i	$0.442871\pi$
$224$	0	0
$225$	−1.73185e6	−2.28062
$226$	473330.	0.616443
$227$	−1.43044e6	−1.84249	−0.921244	−	0.388985i	$-0.872826\pi$
$227$	−1.43044e6	−1.84249	−0.921244	+	0.388985i	$0.872826\pi$
$228$	−210760.	−0.268505
$229$	−1.25805e6	−1.58529	−0.792646	−	0.609682i	$-0.791297\pi$
$229$	−1.25805e6	−1.58529	−0.792646	+	0.609682i	$0.791297\pi$
$230$	−89994.9	−0.112176
$231$	0	0
$232$	−114672.	−0.139874
$233$	224723.	0.271180	0.135590	−	0.990765i	$-0.456707\pi$
$233$	224723.	0.271180	0.135590	+	0.990765i	$0.456707\pi$
$234$	−172465.	−0.205902
$235$	354254.	0.418452
$236$	662156.	0.773892
$237$	1.30986e6	1.51479
$238$	0	0
$239$	1.28193e6	1.45167	0.725837	−	0.687867i	$-0.241453\pi$
$239$	1.28193e6	1.45167	0.725837	+	0.687867i	$0.241453\pi$
$240$	160224.	0.179555
$241$	−576125.	−0.638961	−0.319480	−	0.947593i	$-0.603508\pi$
$241$	−576125.	−0.638961	−0.319480	+	0.947593i	$0.603508\pi$
$242$	203643.	0.223528
$243$	−3.06247e6	−3.32703
$244$	−344373.	−0.370300
$245$	0	0
$246$	1.43985e6	1.51698
$247$	29533.5	0.0308015
$248$	364079.	0.375895
$249$	−668236.	−0.683017
$250$	−487956.	−0.493777
$251$	−609040.	−0.610185	−0.305092	−	0.952323i	$-0.598687\pi$
$251$	−609040.	−0.610185	−0.305092	+	0.952323i	$0.598687\pi$
$252$	0	0
$253$	−355559.	−0.349229
$254$	980780.	0.953865
$255$	150546.	0.144983
$256$	65536.0	0.0625000
$257$	1.03067e6	0.973389	0.486695	−	0.873572i	$-0.338202\pi$
$257$	1.03067e6	0.973389	0.486695	+	0.873572i	$0.338202\pi$
$258$	−1.18284e6	−1.10631
$259$	0	0
$260$	−22451.9	−0.0205977
$261$	1.15612e6	1.05052
$262$	−279975.	−0.251980
$263$	−910585.	−0.811767	−0.405883	−	0.913925i	$-0.633036\pi$
$263$	−910585.	−0.811767	−0.405883	+	0.913925i	$0.633036\pi$
$264$	633024.	0.558998
$265$	−111274.	−0.0973374
$266$	0	0
$267$	−717133.	−0.615633
$268$	−425900.	−0.362219
$269$	1.42100e6	1.19733	0.598663	−	0.801001i	$-0.295699\pi$
$269$	1.42100e6	1.19733	0.598663	+	0.801001i	$0.295699\pi$
$270$	−1.00703e6	−0.840683
$271$	−297530.	−0.246097	−0.123049	−	0.992401i	$-0.539267\pi$
$271$	−297530.	−0.246097	−0.123049	+	0.992401i	$0.539267\pi$
$272$	61577.4	0.0504661
$273$	0	0
$274$	750655.	0.604038
$275$	−890749.	−0.710270
$276$	510889.	0.403695
$277$	849612.	0.665305	0.332653	−	0.943049i	$-0.392056\pi$
$277$	849612.	0.665305	0.332653	+	0.943049i	$0.392056\pi$
$278$	313091.	0.242973
$279$	−3.67065e6	−2.82314
$280$	0	0
$281$	−7680.49	−0.00580261	−0.00290130	−	0.999996i	$-0.500924\pi$
$281$	−7680.49	−0.00580261	−0.00290130	+	0.999996i	$0.500924\pi$
$282$	−2.01105e6	−1.50591
$283$	−985368.	−0.731362	−0.365681	−	0.930740i	$-0.619164\pi$
$283$	−985368.	−0.731362	−0.365681	+	0.930740i	$0.619164\pi$
$284$	−929543.	−0.683870
$285$	276623.	0.201733
$286$	−88704.6	−0.0641255
$287$	0	0
$288$	−660735.	−0.469403
$289$	−1.36200e6	−0.959251
$290$	150507.	0.105090
$291$	−2.14277e6	−1.48335
$292$	−639800.	−0.439124
$293$	−2.16934e6	−1.47625	−0.738124	−	0.674665i	$-0.764288\pi$
$293$	−2.16934e6	−1.47625	−0.738124	+	0.674665i	$0.764288\pi$
$294$	0	0
$295$	−869080.	−0.581440
$296$	−717486.	−0.475975
$297$	−3.97865e6	−2.61724
$298$	368666.	0.240487
$299$	−71590.0	−0.0463099
$300$	1.27988e6	0.821045
$301$	0	0
$302$	−213664.	−0.134807
$303$	472846.	0.295879
$304$	113147.	0.0702195
$305$	451989.	0.278214
$306$	−620825.	−0.379023
$307$	−1.39093e6	−0.842287	−0.421143	−	0.906994i	$-0.638371\pi$
$307$	−1.39093e6	−0.842287	−0.421143	+	0.906994i	$0.638371\pi$
$308$	0	0
$309$	4.78566e6	2.85132
$310$	−477854.	−0.282417
$311$	2.03794e6	1.19479	0.597394	−	0.801948i	$-0.296203\pi$
$311$	2.03794e6	1.19479	0.597394	+	0.801948i	$0.296203\pi$
$312$	127456.	0.0741266
$313$	−1.16958e6	−0.674789	−0.337395	−	0.941363i	$-0.609546\pi$
$313$	−1.16958e6	−0.674789	−0.337395	+	0.941363i	$0.609546\pi$
$314$	1.12200e6	0.642199
$315$	0	0
$316$	−703197.	−0.396150
$317$	801137.	0.447774	0.223887	−	0.974615i	$-0.428125\pi$
$317$	801137.	0.447774	0.223887	+	0.974615i	$0.428125\pi$
$318$	631688.	0.350295
$319$	594634.	0.327170
$320$	−86016.0	−0.0469574
$321$	6.11112e6	3.31023
$322$	0	0
$323$	106312.	0.0566992
$324$	3.20803e6	1.69776
$325$	−179348.	−0.0941862
$326$	−2.50724e6	−1.30663
$327$	3.35421e6	1.73469
$328$	−772983.	−0.396721
$329$	0	0
$330$	−830844.	−0.419986
$331$	2.64951e6	1.32922	0.664608	−	0.747192i	$-0.268598\pi$
$331$	2.64951e6	1.32922	0.664608	+	0.747192i	$0.268598\pi$
$332$	358742.	0.178623
$333$	7.23371e6	3.57479
$334$	−1.17599e6	−0.576816
$335$	558994.	0.272142
$336$	0	0
$337$	−3.40056e6	−1.63108	−0.815541	−	0.578699i	$-0.803560\pi$
$337$	−3.40056e6	−1.63108	−0.815541	+	0.578699i	$0.803560\pi$
$338$	1.46731e6	0.698603
$339$	3.52672e6	1.66676
$340$	−80820.4	−0.0379161
$341$	−1.88794e6	−0.879230
$342$	−1.14075e6	−0.527380
$343$	0	0
$344$	635007.	0.289322
$345$	−670541.	−0.303304
$346$	−2.85675e6	−1.28287
$347$	−617709.	−0.275398	−0.137699	−	0.990474i	$-0.543971\pi$
$347$	−617709.	−0.275398	−0.137699	+	0.990474i	$0.543971\pi$
$348$	−854406.	−0.378195
$349$	2.70539e6	1.18896	0.594479	−	0.804111i	$-0.297358\pi$
$349$	2.70539e6	1.18896	0.594479	+	0.804111i	$0.297358\pi$
$350$	0	0
$351$	−801080.	−0.347063
$352$	−339839.	−0.146189
$353$	−3.18064e6	−1.35856	−0.679279	−	0.733880i	$-0.737707\pi$
$353$	−3.18064e6	−1.35856	−0.679279	+	0.733880i	$0.737707\pi$
$354$	4.93365e6	2.09247
$355$	1.22003e6	0.513805
$356$	384993.	0.161001
$357$	0	0
$358$	−2.32020e6	−0.956792
$359$	−4.45970e6	−1.82629	−0.913145	−	0.407635i	$-0.866354\pi$
$359$	−4.45970e6	−1.82629	−0.913145	+	0.407635i	$0.866354\pi$
$360$	867215.	0.352672
$361$	−2.28075e6	−0.921108
$362$	1.23219e6	0.494202
$363$	1.51732e6	0.604382
$364$	0	0
$365$	839738.	0.329922
$366$	−2.56588e6	−1.00123
$367$	461247.	0.178759	0.0893795	−	0.995998i	$-0.471512\pi$
$367$	461247.	0.178759	0.0893795	+	0.995998i	$0.471512\pi$
$368$	−274270.	−0.105575
$369$	7.79322e6	2.97955
$370$	941701.	0.357609
$371$	0	0
$372$	2.71271e6	1.01636
$373$	3.41954e6	1.27261	0.636305	−	0.771437i	$-0.280462\pi$
$373$	3.41954e6	1.27261	0.636305	+	0.771437i	$0.280462\pi$
$374$	−319311.	−0.118042
$375$	−3.63570e6	−1.33509
$376$	1.07963e6	0.393827
$377$	119726.	0.0433847
$378$	0	0
$379$	16355.6	0.00584882	0.00292441	−	0.999996i	$-0.499069\pi$
$379$	16355.6	0.00584882	0.00292441	+	0.999996i	$0.499069\pi$
$380$	−148505.	−0.0527572
$381$	7.30767e6	2.57909
$382$	811158.	0.284411
$383$	−3.43643e6	−1.19705	−0.598523	−	0.801105i	$-0.704246\pi$
$383$	−3.43643e6	−1.19705	−0.598523	+	0.801105i	$0.704246\pi$
$384$	488301.	0.168989
$385$	0	0
$386$	−2.42395e6	−0.828048
$387$	−6.40215e6	−2.17294
$388$	1.15034e6	0.387925
$389$	−4.81052e6	−1.61183	−0.805914	−	0.592033i	$-0.798325\pi$
$389$	−4.81052e6	−1.61183	−0.805914	+	0.592033i	$0.798325\pi$
$390$	−167286.	−0.0556927
$391$	−257704.	−0.0852469
$392$	0	0
$393$	−2.08606e6	−0.681310
$394$	1243.28	0.000403485 0
$395$	922946.	0.297635
$396$	3.42626e6	1.09795
$397$	−3.29848e6	−1.05036	−0.525180	−	0.850991i	$-0.676002\pi$
$397$	−3.29848e6	−1.05036	−0.525180	+	0.850991i	$0.676002\pi$
$398$	1.63867e6	0.518540
$399$	0	0
$400$	−687104.	−0.214720
$401$	136164.	0.0422865	0.0211432	−	0.999776i	$-0.493269\pi$
$401$	136164.	0.0422865	0.0211432	+	0.999776i	$0.493269\pi$
$402$	−3.17333e6	−0.979377
$403$	−380127.	−0.116591
$404$	−253847.	−0.0773783
$405$	−4.21054e6	−1.27556
$406$	0	0
$407$	3.72054e6	1.11332
$408$	458806.	0.136452
$409$	2.74869e6	0.812490	0.406245	−	0.913764i	$-0.366838\pi$
$409$	2.74869e6	0.812490	0.406245	+	0.913764i	$0.366838\pi$
$410$	1.01454e6	0.298064
$411$	5.59304e6	1.63322
$412$	−2.56918e6	−0.745677
$413$	0	0
$414$	2.76520e6	0.792913
$415$	−470849.	−0.134203
$416$	−68424.8	−0.0193856
$417$	2.33280e6	0.656958
$418$	−586725.	−0.164246
$419$	−1.87219e6	−0.520973	−0.260486	−	0.965478i	$-0.583883\pi$
$419$	−1.87219e6	−0.520973	−0.260486	+	0.965478i	$0.583883\pi$
$420$	0	0
$421$	655225.	0.180171	0.0900856	−	0.995934i	$-0.471286\pi$
$421$	655225.	0.180171	0.0900856	+	0.995934i	$0.471286\pi$
$422$	1.76535e6	0.482559
$423$	−1.08849e7	−2.95782
$424$	−339121.	−0.0916095
$425$	−645601.	−0.173377
$426$	−6.92591e6	−1.84907
$427$	0	0
$428$	−3.28075e6	−0.865693
$429$	−660927.	−0.173385
$430$	−833446.	−0.217373
$431$	6.37690e6	1.65355	0.826773	−	0.562535i	$-0.190174\pi$
$431$	6.37690e6	1.65355	0.826773	+	0.562535i	$0.190174\pi$
$432$	−3.06904e6	−0.791212
$433$	4.80003e6	1.23034	0.615168	−	0.788396i	$-0.289088\pi$
$433$	4.80003e6	1.23034	0.615168	+	0.788396i	$0.289088\pi$
$434$	0	0
$435$	1.12141e6	0.284145
$436$	−1.80071e6	−0.453657
$437$	−473522.	−0.118614
$438$	−4.76707e6	−1.18732
$439$	−5.59508e6	−1.38562	−0.692811	−	0.721119i	$-0.743628\pi$
$439$	−5.59508e6	−1.38562	−0.692811	+	0.721119i	$0.743628\pi$
$440$	446038.	0.109835
$441$	0	0
$442$	−64291.7	−0.0156531
$443$	4.14010e6	1.00231	0.501154	−	0.865358i	$-0.332909\pi$
$443$	4.14010e6	1.00231	0.501154	+	0.865358i	$0.332909\pi$
$444$	−5.34590e6	−1.28696
$445$	−505303.	−0.120963
$446$	−1.06053e6	−0.252457
$447$	2.74688e6	0.650237
$448$	0	0
$449$	−305966.	−0.0716239	−0.0358119	−	0.999359i	$-0.511402\pi$
$449$	−305966.	−0.0716239	−0.0358119	+	0.999359i	$0.511402\pi$
$450$	6.92739e6	1.61264
$451$	4.00832e6	0.927943
$452$	−1.89332e6	−0.435891
$453$	−1.59198e6	−0.364496
$454$	5.72176e6	1.30284
$455$	0	0
$456$	843041.	0.189861
$457$	892608.	0.199926	0.0999632	−	0.994991i	$-0.468127\pi$
$457$	892608.	0.199926	0.0999632	+	0.994991i	$0.468127\pi$
$458$	5.03220e6	1.12097
$459$	−2.88366e6	−0.638870
$460$	359980.	0.0793202
$461$	−3.01465e6	−0.660670	−0.330335	−	0.943864i	$-0.607162\pi$
$461$	−3.01465e6	−0.660670	−0.330335	+	0.943864i	$0.607162\pi$
$462$	0	0
$463$	3.45497e6	0.749017	0.374508	−	0.927224i	$-0.377812\pi$
$463$	3.45497e6	0.749017	0.374508	+	0.927224i	$0.377812\pi$
$464$	458687.	0.0989058
$465$	−3.56043e6	−0.763607
$466$	−898893.	−0.191753
$467$	2.83776e6	0.602120	0.301060	−	0.953605i	$-0.402660\pi$
$467$	2.83776e6	0.602120	0.301060	+	0.953605i	$0.402660\pi$
$468$	689860.	0.145595
$469$	0	0
$470$	−1.41702e6	−0.295890
$471$	8.35990e6	1.73640
$472$	−2.64863e6	−0.547225
$473$	−3.29284e6	−0.676734
$474$	−5.23943e6	−1.07112
$475$	−1.18627e6	−0.241240
$476$	0	0
$477$	3.41903e6	0.688028
$478$	−5.12772e6	−1.02649
$479$	−4.70442e6	−0.936845	−0.468422	−	0.883505i	$-0.655177\pi$
$479$	−4.70442e6	−0.936845	−0.468422	+	0.883505i	$0.655177\pi$
$480$	−640895.	−0.126965
$481$	749113.	0.147633
$482$	2.30450e6	0.451813
$483$	0	0
$484$	−814574.	−0.158058
$485$	−1.50982e6	−0.291455
$486$	1.22499e7	2.35256
$487$	4.16634e6	0.796036	0.398018	−	0.917378i	$-0.369698\pi$
$487$	4.16634e6	0.796036	0.398018	+	0.917378i	$0.369698\pi$
$488$	1.37749e6	0.261842
$489$	−1.86811e7	−3.53290
$490$	0	0
$491$	1.57876e6	0.295537	0.147768	−	0.989022i	$-0.452791\pi$
$491$	1.57876e6	0.295537	0.147768	+	0.989022i	$0.452791\pi$
$492$	−5.75940e6	−1.07267
$493$	430981.	0.0798622
$494$	−118134.	−0.0217800
$495$	−4.49697e6	−0.824910
$496$	−1.45632e6	−0.265798
$497$	0	0
$498$	2.67294e6	0.482966
$499$	1.55712e6	0.279943	0.139972	−	0.990156i	$-0.455299\pi$
$499$	1.55712e6	0.279943	0.139972	+	0.990156i	$0.455299\pi$
$500$	1.95182e6	0.349153
$501$	−8.76215e6	−1.55961
$502$	2.43616e6	0.431466
$503$	−1.19459e6	−0.210523	−0.105261	−	0.994445i	$-0.533568\pi$
$503$	−1.19459e6	−0.210523	−0.105261	+	0.994445i	$0.533568\pi$
$504$	0	0
$505$	333175.	0.0581358
$506$	1.42224e6	0.246942
$507$	1.09328e7	1.88890
$508$	−3.92312e6	−0.674485
$509$	3.16857e6	0.542087	0.271043	−	0.962567i	$-0.412631\pi$
$509$	3.16857e6	0.542087	0.271043	+	0.962567i	$0.412631\pi$
$510$	−602183.	−0.102519
$511$	0	0
$512$	−262144.	−0.0441942
$513$	−5.29864e6	−0.888936
$514$	−4.12268e6	−0.688290
$515$	3.37204e6	0.560241
$516$	4.73136e6	0.782279
$517$	−5.59846e6	−0.921175
$518$	0	0
$519$	−2.12853e7	−3.46865
$520$	89807.5	0.0145648
$521$	−1.00658e6	−0.162462	−0.0812312	−	0.996695i	$-0.525885\pi$
$521$	−1.00658e6	−0.162462	−0.0812312	+	0.996695i	$0.525885\pi$
$522$	−4.62449e6	−0.742827
$523$	8.78985e6	1.40517	0.702583	−	0.711602i	$-0.252030\pi$
$523$	8.78985e6	1.40517	0.702583	+	0.711602i	$0.252030\pi$
$524$	1.11990e6	0.178177
$525$	0	0
$526$	3.64234e6	0.574006
$527$	−1.36835e6	−0.214620
$528$	−2.53210e6	−0.395271
$529$	−5.28851e6	−0.821664
$530$	445097.	0.0688279
$531$	2.67035e7	4.10990
$532$	0	0
$533$	807055.	0.123051
$534$	2.86853e6	0.435318
$535$	4.30599e6	0.650412
$536$	1.70360e6	0.256127
$537$	−1.72875e7	−2.58700
$538$	−5.68399e6	−0.846638
$539$	0	0
$540$	4.02811e6	0.594453
$541$	−3.54864e6	−0.521277	−0.260639	−	0.965436i	$-0.583933\pi$
$541$	−3.54864e6	−0.521277	−0.260639	+	0.965436i	$0.583933\pi$
$542$	1.19012e6	0.174017
$543$	9.18086e6	1.33624
$544$	−246310.	−0.0356849
$545$	2.36343e6	0.340841
$546$	0	0
$547$	4.68179e6	0.669027	0.334513	−	0.942391i	$-0.391428\pi$
$547$	4.68179e6	0.669027	0.334513	+	0.942391i	$0.391428\pi$
$548$	−3.00262e6	−0.427119
$549$	−1.38879e7	−1.96655
$550$	3.56300e6	0.502237
$551$	791915.	0.111122
$552$	−2.04355e6	−0.285456
$553$	0	0
$554$	−3.39845e6	−0.470442
$555$	7.01650e6	0.966914
$556$	−1.25236e6	−0.171808
$557$	5.79507e6	0.791445	0.395723	−	0.918370i	$-0.370494\pi$
$557$	5.79507e6	0.791445	0.395723	+	0.918370i	$0.370494\pi$
$558$	1.46826e7	1.99626
$559$	−662997.	−0.0897392
$560$	0	0
$561$	−2.37915e6	−0.319165
$562$	30722.0	0.00410306
$563$	−7.59703e6	−1.01012	−0.505060	−	0.863084i	$-0.668530\pi$
$563$	−7.59703e6	−1.01012	−0.505060	+	0.863084i	$0.668530\pi$
$564$	8.04420e6	1.06484
$565$	2.48498e6	0.327493
$566$	3.94147e6	0.517151
$567$	0	0
$568$	3.71817e6	0.483569
$569$	−9.81699e6	−1.27115	−0.635576	−	0.772038i	$-0.719237\pi$
$569$	−9.81699e6	−1.27115	−0.635576	+	0.772038i	$0.719237\pi$
$570$	−1.10649e6	−0.142646
$571$	5.25888e6	0.674999	0.337499	−	0.941326i	$-0.390419\pi$
$571$	5.25888e6	0.674999	0.337499	+	0.941326i	$0.390419\pi$
$572$	354818.	0.0453436
$573$	6.04384e6	0.769000
$574$	0	0
$575$	2.87555e6	0.362703
$576$	2.64294e6	0.331918
$577$	8.63468e6	1.07971	0.539855	−	0.841758i	$-0.318479\pi$
$577$	8.63468e6	1.07971	0.539855	+	0.841758i	$0.318479\pi$
$578$	5.44800e6	0.678293
$579$	−1.80606e7	−2.23890
$580$	−602027.	−0.0743098
$581$	0	0
$582$	8.57106e6	1.04888
$583$	1.75852e6	0.214277
$584$	2.55920e6	0.310508
$585$	−905441.	−0.109388
$586$	8.67737e6	1.04386
$587$	−3.32014e6	−0.397705	−0.198852	−	0.980029i	$-0.563721\pi$
$587$	−3.32014e6	−0.397705	−0.198852	+	0.980029i	$0.563721\pi$
$588$	0	0
$589$	−2.51430e6	−0.298627
$590$	3.47632e6	0.411140
$591$	9263.50	0.00109095
$592$	2.86995e6	0.336565
$593$	−6.40001e6	−0.747384	−0.373692	−	0.927553i	$-0.621908\pi$
$593$	−6.40001e6	−0.747384	−0.373692	+	0.927553i	$0.621908\pi$
$594$	1.59146e7	1.85067
$595$	0	0
$596$	−1.47466e6	−0.170050
$597$	1.22095e7	1.40204
$598$	286360.	0.0327461
$599$	7.49862e6	0.853915	0.426957	−	0.904272i	$-0.359585\pi$
$599$	7.49862e6	0.853915	0.426957	+	0.904272i	$0.359585\pi$
$600$	−5.11953e6	−0.580566
$601$	−227052.	−0.0256412	−0.0128206	−	0.999918i	$-0.504081\pi$
$601$	−227052.	−0.0256412	−0.0128206	+	0.999918i	$0.504081\pi$
$602$	0	0
$603$	−1.71757e7	−1.92363
$604$	854656.	0.0953232
$605$	1.06913e6	0.118752
$606$	−1.89139e6	−0.209218
$607$	1.63037e7	1.79603	0.898015	−	0.439965i	$-0.145009\pi$
$607$	1.63037e7	1.79603	0.898015	+	0.439965i	$0.145009\pi$
$608$	−452586.	−0.0496527
$609$	0	0
$610$	−1.80796e6	−0.196727
$611$	−1.12722e6	−0.122153
$612$	2.48330e6	0.268010
$613$	−1.04062e7	−1.11852	−0.559259	−	0.828993i	$-0.688914\pi$
$613$	−1.04062e7	−1.11852	−0.559259	+	0.828993i	$0.688914\pi$
$614$	5.56373e6	0.595587
$615$	7.55921e6	0.805914
$616$	0	0
$617$	4.74140e6	0.501411	0.250705	−	0.968063i	$-0.419337\pi$
$617$	4.74140e6	0.501411	0.250705	+	0.968063i	$0.419337\pi$
$618$	−1.91426e7	−2.01618
$619$	−1.08534e7	−1.13851	−0.569256	−	0.822160i	$-0.692769\pi$
$619$	−1.08534e7	−1.13851	−0.569256	+	0.822160i	$0.692769\pi$
$620$	1.91142e6	0.199699
$621$	1.28440e7	1.33651
$622$	−8.15177e6	−0.844843
$623$	0	0
$624$	−509824.	−0.0524154
$625$	5.82573e6	0.596555
$626$	4.67831e6	0.477148
$627$	−4.37161e6	−0.444092
$628$	−4.48801e6	−0.454103
$629$	2.69659e6	0.271762
$630$	0	0
$631$	1.58978e7	1.58951	0.794757	−	0.606928i	$-0.207598\pi$
$631$	1.58978e7	1.58951	0.794757	+	0.606928i	$0.207598\pi$
$632$	2.81279e6	0.280120
$633$	1.31534e7	1.30476
$634$	−3.20455e6	−0.316624
$635$	5.14909e6	0.506753
$636$	−2.52675e6	−0.247696
$637$	0	0
$638$	−2.37854e6	−0.231344
$639$	−3.74867e7	−3.63183
$640$	344064.	0.0332039
$641$	1.11828e7	1.07499	0.537496	−	0.843267i	$-0.319371\pi$
$641$	1.11828e7	1.07499	0.537496	+	0.843267i	$0.319371\pi$
$642$	−2.44445e7	−2.34069
$643$	1.55983e6	0.148782	0.0743909	−	0.997229i	$-0.476299\pi$
$643$	1.55983e6	0.148782	0.0743909	+	0.997229i	$0.476299\pi$
$644$	0	0
$645$	−6.20991e6	−0.587741
$646$	−425249.	−0.0400924
$647$	−1.54824e7	−1.45404	−0.727020	−	0.686616i	$-0.759095\pi$
$647$	−1.54824e7	−1.45404	−0.727020	+	0.686616i	$0.759095\pi$
$648$	−1.28321e7	−1.20050
$649$	1.37345e7	1.27998
$650$	717391.	0.0665997
$651$	0	0
$652$	1.00290e7	0.923925
$653$	614752.	0.0564179	0.0282090	−	0.999602i	$-0.491020\pi$
$653$	614752.	0.0564179	0.0282090	+	0.999602i	$0.491020\pi$
$654$	−1.34169e7	−1.22661
$655$	−1.46987e6	−0.133867
$656$	3.09193e6	0.280524
$657$	−2.58019e7	−2.33205
$658$	0	0
$659$	−1.32697e7	−1.19028	−0.595139	−	0.803623i	$-0.702903\pi$
$659$	−1.32697e7	−1.19028	−0.595139	+	0.803623i	$0.702903\pi$
$660$	3.32338e6	0.296975
$661$	5.03584e6	0.448299	0.224150	−	0.974555i	$-0.428040\pi$
$661$	5.03584e6	0.448299	0.224150	+	0.974555i	$0.428040\pi$
$662$	−1.05980e7	−0.939898
$663$	−479030.	−0.0423232
$664$	−1.43497e6	−0.126306
$665$	0	0
$666$	−2.89348e7	−2.52776
$667$	−1.91962e6	−0.167071
$668$	4.70396e6	0.407870
$669$	−7.90190e6	−0.682600
$670$	−2.23598e6	−0.192433
$671$	−7.14301e6	−0.612457
$672$	0	0
$673$	8.28068e6	0.704739	0.352369	−	0.935861i	$-0.385376\pi$
$673$	8.28068e6	0.704739	0.352369	+	0.935861i	$0.385376\pi$
$674$	1.36022e7	1.15335
$675$	3.21770e7	2.71823
$676$	−5.86925e6	−0.493987
$677$	−1.52513e7	−1.27889	−0.639446	−	0.768836i	$-0.720836\pi$
$677$	−1.52513e7	−1.27889	−0.639446	+	0.768836i	$0.720836\pi$
$678$	−1.41069e7	−1.17857
$679$	0	0
$680$	323282.	0.0268107
$681$	4.26321e7	3.52265
$682$	7.55177e6	0.621710
$683$	7.88673e6	0.646912	0.323456	−	0.946243i	$-0.395155\pi$
$683$	7.88673e6	0.646912	0.323456	+	0.946243i	$0.395155\pi$
$684$	4.56298e6	0.372914
$685$	3.94094e6	0.320903
$686$	0	0
$687$	3.74943e7	3.03091
$688$	−2.54003e6	−0.204582
$689$	354069.	0.0284145
$690$	2.68217e6	0.214468
$691$	−1.66190e6	−0.132407	−0.0662033	−	0.997806i	$-0.521089\pi$
$691$	−1.66190e6	−0.132407	−0.0662033	+	0.997806i	$0.521089\pi$
$692$	1.14270e7	0.907124
$693$	0	0
$694$	2.47084e6	0.194736
$695$	1.64373e6	0.129083
$696$	3.41762e6	0.267424
$697$	2.90517e6	0.226511
$698$	−1.08216e7	−0.840721
$699$	−6.69754e6	−0.518469
$700$	0	0
$701$	−1.39364e7	−1.07117	−0.535583	−	0.844483i	$-0.679908\pi$
$701$	−1.39364e7	−1.07117	−0.535583	+	0.844483i	$0.679908\pi$
$702$	3.20432e6	0.245410
$703$	4.95490e6	0.378135
$704$	1.35935e6	0.103372
$705$	−1.05580e7	−0.800036
$706$	1.27226e7	0.960646
$707$	0	0
$708$	−1.97346e7	−1.47960
$709$	−1.01874e7	−0.761108	−0.380554	−	0.924759i	$-0.624267\pi$
$709$	−1.01874e7	−0.761108	−0.380554	+	0.924759i	$0.624267\pi$
$710$	−4.88010e6	−0.363315
$711$	−2.83586e7	−2.10383
$712$	−1.53997e6	−0.113845
$713$	6.09473e6	0.448984
$714$	0	0
$715$	−465699.	−0.0340675
$716$	9.28079e6	0.676554
$717$	−3.82060e7	−2.77545
$718$	1.78388e7	1.29138
$719$	1.39169e7	1.00397	0.501983	−	0.864877i	$-0.332604\pi$
$719$	1.39169e7	1.00397	0.501983	+	0.864877i	$0.332604\pi$
$720$	−3.46886e6	−0.249376
$721$	0	0
$722$	9.12301e6	0.651321
$723$	1.71705e7	1.22163
$724$	−4.92874e6	−0.349454
$725$	−4.80905e6	−0.339793
$726$	−6.06929e6	−0.427362
$727$	3.42063e6	0.240033	0.120016	−	0.992772i	$-0.461705\pi$
$727$	3.42063e6	0.240033	0.120016	+	0.992772i	$0.461705\pi$
$728$	0	0
$729$	4.25504e7	2.96541
$730$	−3.35895e6	−0.233290
$731$	−2.38660e6	−0.165191
$732$	1.02635e7	0.707976
$733$	−1.03162e7	−0.709186	−0.354593	−	0.935021i	$-0.615381\pi$
$733$	−1.03162e7	−0.709186	−0.354593	+	0.935021i	$0.615381\pi$
$734$	−1.84499e6	−0.126402
$735$	0	0
$736$	1.09708e6	0.0746525
$737$	−8.83406e6	−0.599090
$738$	−3.11729e7	−2.10686
$739$	−1.03890e7	−0.699779	−0.349889	−	0.936791i	$-0.613781\pi$
$739$	−1.03890e7	−0.699779	−0.349889	+	0.936791i	$0.613781\pi$
$740$	−3.76680e6	−0.252868
$741$	−880202.	−0.0588893
$742$	0	0
$743$	−1.04738e7	−0.696035	−0.348018	−	0.937488i	$-0.613145\pi$
$743$	−1.04738e7	−0.696035	−0.348018	+	0.937488i	$0.613145\pi$
$744$	−1.08508e7	−0.718672
$745$	1.93550e6	0.127762
$746$	−1.36782e7	−0.899872
$747$	1.44674e7	0.948612
$748$	1.27725e6	0.0834681
$749$	0	0
$750$	1.45428e7	0.944050
$751$	−9.40445e6	−0.608462	−0.304231	−	0.952598i	$-0.598400\pi$
$751$	−9.40445e6	−0.608462	−0.304231	+	0.952598i	$0.598400\pi$
$752$	−4.31853e6	−0.278478
$753$	1.81515e7	1.16661
$754$	−478906.	−0.0306776
$755$	−1.12174e6	−0.0716181
$756$	0	0
$757$	−1.33677e7	−0.847848	−0.423924	−	0.905698i	$-0.639348\pi$
$757$	−1.33677e7	−0.847848	−0.423924	+	0.905698i	$0.639348\pi$
$758$	−65422.4	−0.00413574
$759$	1.05969e7	0.667690
$760$	594020.	0.0373050
$761$	−2.22623e7	−1.39350	−0.696752	−	0.717312i	$-0.745372\pi$
$761$	−2.22623e7	−1.39350	−0.696752	+	0.717312i	$0.745372\pi$
$762$	−2.92307e7	−1.82369
$763$	0	0
$764$	−3.24463e6	−0.201109
$765$	−3.25933e6	−0.201361
$766$	1.37457e7	0.846440
$767$	2.76537e6	0.169733
$768$	−1.95320e6	−0.119493
$769$	9.65833e6	0.588961	0.294480	−	0.955658i	$-0.404853\pi$
$769$	9.65833e6	0.588961	0.294480	+	0.955658i	$0.404853\pi$
$770$	0	0
$771$	−3.07176e7	−1.86102
$772$	9.69581e6	0.585519
$773$	−5.10951e6	−0.307561	−0.153780	−	0.988105i	$-0.549145\pi$
$773$	−5.10951e6	−0.307561	−0.153780	+	0.988105i	$0.549145\pi$
$774$	2.56086e7	1.53650
$775$	1.52686e7	0.913154
$776$	−4.60137e6	−0.274305
$777$	0	0
$778$	1.92421e7	1.13973
$779$	5.33816e6	0.315172
$780$	669145.	0.0393807
$781$	−1.92807e7	−1.13108
$782$	1.03081e6	0.0602787
$783$	−2.14802e7	−1.25209
$784$	0	0
$785$	5.89051e6	0.341176
$786$	8.34423e6	0.481759
$787$	2.51592e7	1.44797	0.723984	−	0.689816i	$-0.242309\pi$
$787$	2.51592e7	1.44797	0.723984	+	0.689816i	$0.242309\pi$
$788$	−4973.11	−0.000285307 0
$789$	2.71386e7	1.55201
$790$	−3.69178e6	−0.210460
$791$	0	0
$792$	−1.37050e7	−0.776367
$793$	−1.43821e6	−0.0812155
$794$	1.31939e7	0.742716
$795$	3.31636e6	0.186099
$796$	−6.55466e6	−0.366664
$797$	−3.35976e6	−0.187354	−0.0936768	−	0.995603i	$-0.529862\pi$
$797$	−3.35976e6	−0.187354	−0.0936768	+	0.995603i	$0.529862\pi$
$798$	0	0
$799$	−4.05767e6	−0.224859
$800$	2.74842e6	0.151830
$801$	1.55260e7	0.855024
$802$	−544656.	−0.0299010
$803$	−1.32708e7	−0.726287
$804$	1.26933e7	0.692524
$805$	0	0
$806$	1.52051e6	0.0824426
$807$	−4.23507e7	−2.28916
$808$	1.01539e6	0.0547147
$809$	−2.97050e7	−1.59572	−0.797862	−	0.602840i	$-0.794036\pi$
$809$	−2.97050e7	−1.59572	−0.797862	+	0.602840i	$0.794036\pi$
$810$	1.68422e7	0.901956
$811$	1.26386e7	0.674757	0.337379	−	0.941369i	$-0.390460\pi$
$811$	1.26386e7	0.674757	0.337379	+	0.941369i	$0.390460\pi$
$812$	0	0
$813$	8.86743e6	0.470513
$814$	−1.48822e7	−0.787237
$815$	−1.31630e7	−0.694162
$816$	−1.83522e6	−0.0964858
$817$	−4.38531e6	−0.229850
$818$	−1.09948e7	−0.574517
$819$	0	0
$820$	−4.05816e6	−0.210763
$821$	−2.61760e7	−1.35533	−0.677666	−	0.735370i	$-0.737008\pi$
$821$	−2.61760e7	−1.35533	−0.677666	+	0.735370i	$0.737008\pi$
$822$	−2.23722e7	−1.15486
$823$	−7.29843e6	−0.375604	−0.187802	−	0.982207i	$-0.560136\pi$
$823$	−7.29843e6	−0.375604	−0.187802	+	0.982207i	$0.560136\pi$
$824$	1.02767e7	0.527274
$825$	2.65474e7	1.35796
$826$	0	0
$827$	1.18681e7	0.603418	0.301709	−	0.953400i	$-0.402443\pi$
$827$	1.18681e7	0.603418	0.301709	+	0.953400i	$0.402443\pi$
$828$	−1.10608e7	−0.560674
$829$	−1.04338e7	−0.527298	−0.263649	−	0.964619i	$-0.584926\pi$
$829$	−1.04338e7	−0.527298	−0.263649	+	0.964619i	$0.584926\pi$
$830$	1.88340e6	0.0948957
$831$	−2.53214e7	−1.27199
$832$	273699.	0.0137077
$833$	0	0
$834$	−9.33120e6	−0.464539
$835$	−6.17394e6	−0.306441
$836$	2.34690e6	0.116139
$837$	6.81991e7	3.36484
$838$	7.48876e6	0.368383
$839$	−2.91444e7	−1.42939	−0.714695	−	0.699436i	$-0.753435\pi$
$839$	−2.91444e7	−1.42939	−0.714695	+	0.699436i	$0.753435\pi$
$840$	0	0
$841$	−1.73008e7	−0.843482
$842$	−2.62090e6	−0.127400
$843$	228906.	0.0110940
$844$	−7.06142e6	−0.341221
$845$	7.70339e6	0.371142
$846$	4.35395e7	2.09150
$847$	0	0
$848$	1.35648e6	0.0647777
$849$	2.93674e7	1.39829
$850$	2.58240e6	0.122596
$851$	−1.20108e7	−0.568524
$852$	2.77037e7	1.30749
$853$	2.63032e7	1.23776	0.618879	−	0.785487i	$-0.287587\pi$
$853$	2.63032e7	1.23776	0.618879	+	0.785487i	$0.287587\pi$
$854$	0	0
$855$	−5.98892e6	−0.280177
$856$	1.31230e7	0.612137
$857$	3.43598e6	0.159808	0.0799040	−	0.996803i	$-0.474539\pi$
$857$	3.43598e6	0.159808	0.0799040	+	0.996803i	$0.474539\pi$
$858$	2.64371e6	0.122601
$859$	1.09139e7	0.504659	0.252329	−	0.967641i	$-0.418803\pi$
$859$	1.09139e7	0.504659	0.252329	+	0.967641i	$0.418803\pi$
$860$	3.33378e6	0.153706
$861$	0	0
$862$	−2.55076e7	−1.16923
$863$	2.19058e7	1.00123	0.500614	−	0.865671i	$-0.333108\pi$
$863$	2.19058e7	1.00123	0.500614	+	0.865671i	$0.333108\pi$
$864$	1.22762e7	0.559472
$865$	−1.49979e7	−0.681539
$866$	−1.92001e7	−0.869979
$867$	4.05923e7	1.83399
$868$	0	0
$869$	−1.45858e7	−0.655210
$870$	−4.48563e6	−0.200921
$871$	−1.77869e6	−0.0794430
$872$	7.20283e6	0.320784
$873$	4.63911e7	2.06015
$874$	1.89409e6	0.0838729
$875$	0	0
$876$	1.90683e7	0.839560
$877$	6.46927e6	0.284025	0.142012	−	0.989865i	$-0.454643\pi$
$877$	6.46927e6	0.284025	0.142012	+	0.989865i	$0.454643\pi$
$878$	2.23803e7	0.979782
$879$	6.46540e7	2.82243
$880$	−1.78415e6	−0.0776650
$881$	2.03983e7	0.885430	0.442715	−	0.896662i	$-0.354015\pi$
$881$	2.03983e7	0.885430	0.442715	+	0.896662i	$0.354015\pi$
$882$	0	0
$883$	1.62381e7	0.700862	0.350431	−	0.936589i	$-0.386035\pi$
$883$	1.62381e7	0.700862	0.350431	+	0.936589i	$0.386035\pi$
$884$	257167.	0.0110684
$885$	2.59016e7	1.11165
$886$	−1.65604e7	−0.708739
$887$	−6.79027e6	−0.289786	−0.144893	−	0.989447i	$-0.546284\pi$
$887$	−6.79027e6	−0.289786	−0.144893	+	0.989447i	$0.546284\pi$
$888$	2.13836e7	0.910015
$889$	0	0
$890$	2.02121e6	0.0855336
$891$	6.65413e7	2.80800
$892$	4.24213e6	0.178514
$893$	−7.45585e6	−0.312873
$894$	−1.09875e7	−0.459787
$895$	−1.21810e7	−0.508308
$896$	0	0
$897$	2.13363e6	0.0885398
$898$	1.22387e6	0.0506457
$899$	−1.01928e7	−0.420623
$900$	−2.77096e7	−1.14031
$901$	1.27455e6	0.0523052
$902$	−1.60333e7	−0.656155
$903$	0	0
$904$	7.57328e6	0.308221
$905$	6.46897e6	0.262551
$906$	6.36793e6	0.257738
$907$	3.42142e7	1.38098	0.690492	−	0.723340i	$-0.257394\pi$
$907$	3.42142e7	1.38098	0.690492	+	0.723340i	$0.257394\pi$
$908$	−2.28870e7	−0.921244
$909$	−1.02372e7	−0.410932
$910$	0	0
$911$	−4.47390e6	−0.178604	−0.0893019	−	0.996005i	$-0.528464\pi$
$911$	−4.47390e6	−0.178604	−0.0893019	+	0.996005i	$0.528464\pi$
$912$	−3.37217e6	−0.134252
$913$	7.44107e6	0.295433
$914$	−3.57043e6	−0.141369
$915$	−1.34709e7	−0.531916
$916$	−2.01288e7	−0.792646
$917$	0	0
$918$	1.15346e7	0.451749
$919$	−537900.	−0.0210094	−0.0105047	−	0.999945i	$-0.503344\pi$
$919$	−537900.	−0.0210094	−0.0105047	+	0.999945i	$0.503344\pi$
$920$	−1.43992e6	−0.0560878
$921$	4.14547e7	1.61036
$922$	1.20586e7	0.467164
$923$	−3.88207e6	−0.149989
$924$	0	0
$925$	−3.00896e7	−1.15628
$926$	−1.38199e7	−0.529635
$927$	−1.03610e8	−3.96006
$928$	−1.83475e6	−0.0699370
$929$	−1.77241e7	−0.673792	−0.336896	−	0.941542i	$-0.609377\pi$
$929$	−1.77241e7	−0.673792	−0.336896	+	0.941542i	$0.609377\pi$
$930$	1.42417e7	0.539952
$931$	0	0
$932$	3.59557e6	0.135590
$933$	−6.07379e7	−2.28431
$934$	−1.13510e7	−0.425763
$935$	−1.67638e6	−0.0627111
$936$	−2.75944e6	−0.102951
$937$	−3.32444e7	−1.23700	−0.618499	−	0.785786i	$-0.712259\pi$
$937$	−3.32444e7	−1.23700	−0.618499	+	0.785786i	$0.712259\pi$
$938$	0	0
$939$	3.48575e7	1.29013
$940$	5.66807e6	0.209226
$941$	2.78480e7	1.02523	0.512613	−	0.858620i	$-0.328678\pi$
$941$	2.78480e7	1.02523	0.512613	+	0.858620i	$0.328678\pi$
$942$	−3.34396e7	−1.22782
$943$	−1.29398e7	−0.473859
$944$	1.05945e7	0.386946
$945$	0	0
$946$	1.31714e7	0.478523
$947$	−2.30569e7	−0.835459	−0.417729	−	0.908571i	$-0.637174\pi$
$947$	−2.30569e7	−0.835459	−0.417729	+	0.908571i	$0.637174\pi$
$948$	2.09577e7	0.757397
$949$	−2.67201e6	−0.0963102
$950$	4.74509e6	0.170583
$951$	−2.38767e7	−0.856097
$952$	0	0
$953$	2.85536e7	1.01843	0.509213	−	0.860641i	$-0.329937\pi$
$953$	2.85536e7	1.01843	0.509213	+	0.860641i	$0.329937\pi$
$954$	−1.36761e7	−0.486510
$955$	4.25858e6	0.151097
$956$	2.05109e7	0.725837
$957$	−1.77222e7	−0.625514
$958$	1.88177e7	0.662449
$959$	0	0
$960$	2.56358e6	0.0897777
$961$	3.73258e6	0.130377
$962$	−2.99645e6	−0.104393
$963$	−1.32306e8	−4.59743
$964$	−9.21800e6	−0.319480
$965$	−1.27257e7	−0.439911
$966$	0	0
$967$	−3.45310e7	−1.18753	−0.593763	−	0.804640i	$-0.702358\pi$
$967$	−3.45310e7	−1.18753	−0.593763	+	0.804640i	$0.702358\pi$
$968$	3.25830e6	0.111764
$969$	−3.16848e6	−0.108403
$970$	6.03930e6	0.206090
$971$	1.24933e7	0.425236	0.212618	−	0.977135i	$-0.431801\pi$
$971$	1.24933e7	0.425236	0.212618	+	0.977135i	$0.431801\pi$
$972$	−4.89995e7	−1.66351
$973$	0	0
$974$	−1.66654e7	−0.562883
$975$	5.34519e6	0.180074
$976$	−5.50996e6	−0.185150
$977$	−3.89440e7	−1.30528	−0.652641	−	0.757667i	$-0.726339\pi$
$977$	−3.89440e7	−1.30528	−0.652641	+	0.757667i	$0.726339\pi$
$978$	7.47245e7	2.49813
$979$	7.98556e6	0.266286
$980$	0	0
$981$	−7.26191e7	−2.40923
$982$	−6.31503e6	−0.208976
$983$	279901.	0.00923889	0.00461945	−	0.999989i	$-0.498530\pi$
$983$	279901.	0.00923889	0.00461945	+	0.999989i	$0.498530\pi$
$984$	2.30376e7	0.758489
$985$	6527.20	0.000214356 0
$986$	−1.72393e6	−0.0564711
$987$	0	0
$988$	472536.	0.0154008
$989$	1.06301e7	0.345578
$990$	1.79879e7	0.583299
$991$	1.96343e7	0.635084	0.317542	−	0.948244i	$-0.397142\pi$
$991$	1.96343e7	0.635084	0.317542	+	0.948244i	$0.397142\pi$
$992$	5.82527e6	0.187948
$993$	−7.89647e7	−2.54132
$994$	0	0
$995$	8.60299e6	0.275481
$996$	−1.06918e7	−0.341509
$997$	−3.73374e7	−1.18961	−0.594807	−	0.803868i	$-0.702772\pi$
$997$	−3.73374e7	−1.18961	−0.594807	+	0.803868i	$0.702772\pi$
$998$	−6.22847e6	−0.197950
$999$	−1.34399e8	−4.26072

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	98.6.a.c.1.1		2
3.2	odd	2		882.6.a.bt.1.1		2
4.3	odd	2		784.6.a.bc.1.2		2
7.2	even	3		14.6.c.b.11.2	yes	4
7.3	odd	6		98.6.c.f.79.1		4
7.4	even	3		14.6.c.b.9.2	✓	4
7.5	odd	6		98.6.c.f.67.1		4
7.6	odd	2		98.6.a.f.1.2		2
21.2	odd	6		126.6.g.e.109.1		4
21.11	odd	6		126.6.g.e.37.1		4
21.20	even	2		882.6.a.bl.1.1		2
28.11	odd	6		112.6.i.b.65.1		4
28.23	odd	6		112.6.i.b.81.1		4
28.27	even	2		784.6.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.6.c.b.9.2	✓	4	7.4	even	3
14.6.c.b.11.2	yes	4	7.2	even	3
98.6.a.c.1.1		2	1.1	even	1	trivial
98.6.a.f.1.2		2	7.6	odd	2
98.6.c.f.67.1		4	7.5	odd	6
98.6.c.f.79.1		4	7.3	odd	6
112.6.i.b.65.1		4	28.11	odd	6
112.6.i.b.81.1		4	28.23	odd	6
126.6.g.e.37.1		4	21.11	odd	6
126.6.g.e.109.1		4	21.2	odd	6
784.6.a.r.1.1		2	28.27	even	2
784.6.a.bc.1.2		2	4.3	odd	2
882.6.a.bl.1.1		2	21.20	even	2
882.6.a.bt.1.1		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	98.6.a.c.1.1

Level	$98$
Weight	$6$
Character	98.1
Self dual	yes
Analytic conductor	$15.718$
Analytic rank	$1$
Dimension	$2$
CM	no
Inner twists	$1$