LMFDB - Embedded newform 350.6.a.w.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,6,Mod(1,350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("350.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	350.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,12,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$56.1343369345$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1378776.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 336x + 840 $ x^3 - x^2 - 336*x + 840
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$17.4760$ of defining polynomial
Character	$\chi$	$=$	350.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.00000 q^{2} +17.4760 q^{3} +16.0000 q^{4} +69.9039 q^{6} -49.0000 q^{7} +64.0000 q^{8} +62.4100 q^{9} -282.586 q^{11} +279.616 q^{12} -823.190 q^{13} -196.000 q^{14} +256.000 q^{16} -2181.07 q^{17} +249.640 q^{18} -2832.64 q^{19} -856.323 q^{21} -1130.34 q^{22} +2700.53 q^{23} +1118.46 q^{24} -3292.76 q^{26} -3155.99 q^{27} -784.000 q^{28} +3641.17 q^{29} +378.139 q^{31} +1024.00 q^{32} -4938.46 q^{33} -8724.27 q^{34} +998.560 q^{36} +6385.29 q^{37} -11330.5 q^{38} -14386.1 q^{39} -19008.9 q^{41} -3425.29 q^{42} +13402.5 q^{43} -4521.37 q^{44} +10802.1 q^{46} +16712.0 q^{47} +4473.85 q^{48} +2401.00 q^{49} -38116.3 q^{51} -13171.0 q^{52} -5764.23 q^{53} -12624.0 q^{54} -3136.00 q^{56} -49503.1 q^{57} +14564.7 q^{58} +21798.6 q^{59} +25712.7 q^{61} +1512.55 q^{62} -3058.09 q^{63} +4096.00 q^{64} -19753.8 q^{66} +2244.75 q^{67} -34897.1 q^{68} +47194.4 q^{69} -49986.5 q^{71} +3994.24 q^{72} -39482.6 q^{73} +25541.1 q^{74} -45322.2 q^{76} +13846.7 q^{77} -57544.2 q^{78} +77115.4 q^{79} -70319.6 q^{81} -76035.7 q^{82} +15123.4 q^{83} -13701.2 q^{84} +53610.1 q^{86} +63633.0 q^{87} -18085.5 q^{88} +96816.4 q^{89} +40336.3 q^{91} +43208.5 q^{92} +6608.34 q^{93} +66848.1 q^{94} +17895.4 q^{96} -70141.8 q^{97} +9604.00 q^{98} -17636.2 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 12 q^{2} + q^{3} + 48 q^{4} + 4 q^{6} - 147 q^{7} + 192 q^{8} - 56 q^{9} - 11 q^{11} + 16 q^{12} - 482 q^{13} - 588 q^{14} + 768 q^{16} - 1185 q^{17} - 224 q^{18} - 1811 q^{19} - 49 q^{21} - 44 q^{22}+ \cdots + 109448 q^{99}+O(q^{100}) $ 3 * q + 12 * q^2 + q^3 + 48 * q^4 + 4 * q^6 - 147 * q^7 + 192 * q^8 - 56 * q^9 - 11 * q^11 + 16 * q^12 - 482 * q^13 - 588 * q^14 + 768 * q^16 - 1185 * q^17 - 224 * q^18 - 1811 * q^19 - 49 * q^21 - 44 * q^22 - 3350 * q^23 + 64 * q^24 - 1928 * q^26 - 1997 * q^27 - 2352 * q^28 + 1924 * q^29 - 6560 * q^31 + 3072 * q^32 - 15865 * q^33 - 4740 * q^34 - 896 * q^36 + 5194 * q^37 - 7244 * q^38 - 1574 * q^39 - 26981 * q^41 - 196 * q^42 - 18938 * q^43 - 176 * q^44 - 13400 * q^46 + 2878 * q^47 + 256 * q^48 + 7203 * q^49 - 55729 * q^51 - 7712 * q^52 - 6416 * q^53 - 7988 * q^54 - 9408 * q^56 - 33983 * q^57 + 7696 * q^58 + 17480 * q^59 - 40538 * q^61 - 26240 * q^62 + 2744 * q^63 + 12288 * q^64 - 63460 * q^66 - 43761 * q^67 - 18960 * q^68 + 70000 * q^69 - 65476 * q^71 - 3584 * q^72 - 59143 * q^73 + 20776 * q^74 - 28976 * q^76 + 539 * q^77 - 6296 * q^78 + 133128 * q^79 - 89693 * q^81 - 107924 * q^82 - 97205 * q^83 - 784 * q^84 - 75752 * q^86 - 63566 * q^87 - 704 * q^88 + 61353 * q^89 + 23618 * q^91 - 53600 * q^92 - 53964 * q^93 + 11512 * q^94 + 1024 * q^96 - 288942 * q^97 + 28812 * q^98 + 109448 * 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
$3$	17.4760	1.12108	0.560542	−	0.828126i	$-0.310593\pi$
$3$	17.4760	1.12108	0.560542	+	0.828126i	$0.310593\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	69.9039	0.792727
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	64.0000	0.353553
$9$	62.4100	0.256831
$10$	0	0
$11$	−282.586	−0.704155	−0.352078	−	0.935971i	$-0.614525\pi$
$11$	−282.586	−0.704155	−0.352078	+	0.935971i	$0.614525\pi$
$12$	279.616	0.560542
$13$	−823.190	−1.35096	−0.675479	−	0.737380i	$-0.736063\pi$
$13$	−823.190	−1.35096	−0.675479	+	0.737380i	$0.736063\pi$
$14$	−196.000	−0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	−2181.07	−1.83040	−0.915202	−	0.402995i	$-0.867969\pi$
$17$	−2181.07	−1.83040	−0.915202	+	0.402995i	$0.867969\pi$
$18$	249.640	0.181607
$19$	−2832.64	−1.80014	−0.900071	−	0.435743i	$-0.856486\pi$
$19$	−2832.64	−1.80014	−0.900071	+	0.435743i	$0.856486\pi$
$20$	0	0
$21$	−856.323	−0.423730
$22$	−1130.34	−0.497913
$23$	2700.53	1.06446	0.532230	−	0.846600i	$-0.321354\pi$
$23$	2700.53	1.06446	0.532230	+	0.846600i	$0.321354\pi$
$24$	1118.46	0.396363
$25$	0	0
$26$	−3292.76	−0.955271
$27$	−3155.99	−0.833155
$28$	−784.000	−0.188982
$29$	3641.17	0.803982	0.401991	−	0.915644i	$-0.368318\pi$
$29$	3641.17	0.803982	0.401991	+	0.915644i	$0.368318\pi$
$30$	0	0
$31$	378.139	0.0706719	0.0353360	−	0.999375i	$-0.488750\pi$
$31$	378.139	0.0706719	0.0353360	+	0.999375i	$0.488750\pi$
$32$	1024.00	0.176777
$33$	−4938.46	−0.789418
$34$	−8724.27	−1.29429
$35$	0	0
$36$	998.560	0.128416
$37$	6385.29	0.766789	0.383395	−	0.923585i	$-0.374755\pi$
$37$	6385.29	0.766789	0.383395	+	0.923585i	$0.374755\pi$
$38$	−11330.5	−1.27289
$39$	−14386.1	−1.51454
$40$	0	0
$41$	−19008.9	−1.76603	−0.883015	−	0.469345i	$-0.844490\pi$
$41$	−19008.9	−1.76603	−0.883015	+	0.469345i	$0.844490\pi$
$42$	−3425.29	−0.299623
$43$	13402.5	1.10539	0.552695	−	0.833383i	$-0.313599\pi$
$43$	13402.5	1.10539	0.552695	+	0.833383i	$0.313599\pi$
$44$	−4521.37	−0.352078
$45$	0	0
$46$	10802.1	0.752687
$47$	16712.0	1.10353	0.551765	−	0.833999i	$-0.313954\pi$
$47$	16712.0	1.10353	0.551765	+	0.833999i	$0.313954\pi$
$48$	4473.85	0.280271
$49$	2401.00	0.142857
$50$	0	0
$51$	−38116.3	−2.05204
$52$	−13171.0	−0.675479
$53$	−5764.23	−0.281872	−0.140936	−	0.990019i	$-0.545011\pi$
$53$	−5764.23	−0.281872	−0.140936	+	0.990019i	$0.545011\pi$
$54$	−12624.0	−0.589130
$55$	0	0
$56$	−3136.00	−0.133631
$57$	−49503.1	−2.01811
$58$	14564.7	0.568501
$59$	21798.6	0.815266	0.407633	−	0.913146i	$-0.366354\pi$
$59$	21798.6	0.815266	0.407633	+	0.913146i	$0.366354\pi$
$60$	0	0
$61$	25712.7	0.884757	0.442378	−	0.896829i	$-0.354135\pi$
$61$	25712.7	0.884757	0.442378	+	0.896829i	$0.354135\pi$
$62$	1512.55	0.0499726
$63$	−3058.09	−0.0970731
$64$	4096.00	0.125000
$65$	0	0
$66$	−19753.8	−0.558203
$67$	2244.75	0.0610915	0.0305457	−	0.999533i	$-0.490275\pi$
$67$	2244.75	0.0610915	0.0305457	+	0.999533i	$0.490275\pi$
$68$	−34897.1	−0.915202
$69$	47194.4	1.19335
$70$	0	0
$71$	−49986.5	−1.17681	−0.588405	−	0.808566i	$-0.700244\pi$
$71$	−49986.5	−1.17681	−0.588405	+	0.808566i	$0.700244\pi$
$72$	3994.24	0.0908036
$73$	−39482.6	−0.867159	−0.433580	−	0.901115i	$-0.642750\pi$
$73$	−39482.6	−0.867159	−0.433580	+	0.901115i	$0.642750\pi$
$74$	25541.1	0.542202
$75$	0	0
$76$	−45322.2	−0.900071
$77$	13846.7	0.266146
$78$	−57544.2	−1.07094
$79$	77115.4	1.39019	0.695094	−	0.718919i	$-0.255363\pi$
$79$	77115.4	1.39019	0.695094	+	0.718919i	$0.255363\pi$
$80$	0	0
$81$	−70319.6	−1.19087
$82$	−76035.7	−1.24877
$83$	15123.4	0.240965	0.120483	−	0.992715i	$-0.461556\pi$
$83$	15123.4	0.240965	0.120483	+	0.992715i	$0.461556\pi$
$84$	−13701.2	−0.211865
$85$	0	0
$86$	53610.1	0.781629
$87$	63633.0	0.901332
$88$	−18085.5	−0.248956
$89$	96816.4	1.29561	0.647804	−	0.761807i	$-0.275687\pi$
$89$	96816.4	1.29561	0.647804	+	0.761807i	$0.275687\pi$
$90$	0	0
$91$	40336.3	0.510614
$92$	43208.5	0.532230
$93$	6608.34	0.0792292
$94$	66848.1	0.780314
$95$	0	0
$96$	17895.4	0.198182
$97$	−70141.8	−0.756915	−0.378458	−	0.925619i	$-0.623545\pi$
$97$	−70141.8	−0.756915	−0.378458	+	0.925619i	$0.623545\pi$
$98$	9604.00	0.101015
$99$	−17636.2	−0.180849
$100$	0	0
$101$	−57879.8	−0.564577	−0.282289	−	0.959330i	$-0.591094\pi$
$101$	−57879.8	−0.564577	−0.282289	+	0.959330i	$0.591094\pi$
$102$	−152465.	−1.45101
$103$	−136738.	−1.26998	−0.634988	−	0.772522i	$-0.718995\pi$
$103$	−136738.	−1.26998	−0.634988	+	0.772522i	$0.718995\pi$
$104$	−52684.2	−0.477636
$105$	0	0
$106$	−23056.9	−0.199314
$107$	−189123.	−1.59693	−0.798463	−	0.602044i	$-0.794353\pi$
$107$	−189123.	−1.59693	−0.798463	+	0.602044i	$0.794353\pi$
$108$	−50495.8	−0.416578
$109$	142624.	1.14981	0.574905	−	0.818220i	$-0.305039\pi$
$109$	142624.	1.14981	0.574905	+	0.818220i	$0.305039\pi$
$110$	0	0
$111$	111589.	0.859636
$112$	−12544.0	−0.0944911
$113$	98940.3	0.728915	0.364458	−	0.931220i	$-0.381254\pi$
$113$	98940.3	0.728915	0.364458	+	0.931220i	$0.381254\pi$
$114$	−198012.	−1.42702
$115$	0	0
$116$	58258.7	0.401991
$117$	−51375.3	−0.346968
$118$	87194.5	0.576480
$119$	106872.	0.691828
$120$	0	0
$121$	−81196.4	−0.504165
$122$	102851.	0.625617
$123$	−332200.	−1.97987
$124$	6050.22	0.0353360
$125$	0	0
$126$	−12232.4	−0.0686411
$127$	−71821.4	−0.395134	−0.197567	−	0.980289i	$-0.563304\pi$
$127$	−71821.4	−0.395134	−0.197567	+	0.980289i	$0.563304\pi$
$128$	16384.0	0.0883883
$129$	234222.	1.23924
$130$	0	0
$131$	−114126.	−0.581043	−0.290521	−	0.956868i	$-0.593829\pi$
$131$	−114126.	−0.581043	−0.290521	+	0.956868i	$0.593829\pi$
$132$	−79015.4	−0.394709
$133$	138799.	0.680390
$134$	8978.99	0.0431982
$135$	0	0
$136$	−139588.	−0.647146
$137$	−233482.	−1.06280	−0.531401	−	0.847120i	$-0.678334\pi$
$137$	−233482.	−1.06280	−0.531401	+	0.847120i	$0.678334\pi$
$138$	188778.	0.843826
$139$	42919.9	0.188418	0.0942088	−	0.995552i	$-0.469968\pi$
$139$	42919.9	0.188418	0.0942088	+	0.995552i	$0.469968\pi$
$140$	0	0
$141$	292059.	1.23715
$142$	−199946.	−0.832130
$143$	232622.	0.951284
$144$	15977.0	0.0642078
$145$	0	0
$146$	−157930.	−0.613174
$147$	41959.8	0.160155
$148$	102165.	0.383395
$149$	−334911.	−1.23585	−0.617923	−	0.786239i	$-0.712026\pi$
$149$	−334911.	−1.23585	−0.617923	+	0.786239i	$0.712026\pi$
$150$	0	0
$151$	296718.	1.05901	0.529507	−	0.848305i	$-0.322377\pi$
$151$	296718.	1.05901	0.529507	+	0.848305i	$0.322377\pi$
$152$	−181289.	−0.636447
$153$	−136121.	−0.470105
$154$	55386.8	0.188193
$155$	0	0
$156$	−230177.	−0.757269
$157$	−295780.	−0.957677	−0.478839	−	0.877903i	$-0.658942\pi$
$157$	−295780.	−0.957677	−0.478839	+	0.877903i	$0.658942\pi$
$158$	308462.	0.983012
$159$	−100736.	−0.316002
$160$	0	0
$161$	−132326.	−0.402328
$162$	−281278.	−0.842072
$163$	−307013.	−0.905083	−0.452541	−	0.891743i	$-0.649483\pi$
$163$	−307013.	−0.905083	−0.452541	+	0.891743i	$0.649483\pi$
$164$	−304143.	−0.883015
$165$	0	0
$166$	60493.7	0.170388
$167$	−465474.	−1.29153	−0.645765	−	0.763536i	$-0.723461\pi$
$167$	−465474.	−1.29153	−0.645765	+	0.763536i	$0.723461\pi$
$168$	−54804.7	−0.149811
$169$	306349.	0.825086
$170$	0	0
$171$	−176785.	−0.462333
$172$	214441.	0.552695
$173$	−255450.	−0.648918	−0.324459	−	0.945900i	$-0.605182\pi$
$173$	−255450.	−0.648918	−0.324459	+	0.945900i	$0.605182\pi$
$174$	254532.	0.637338
$175$	0	0
$176$	−72341.9	−0.176039
$177$	380953.	0.913982
$178$	387266.	0.916134
$179$	−187030.	−0.436294	−0.218147	−	0.975916i	$-0.570001\pi$
$179$	−187030.	−0.436294	−0.218147	+	0.975916i	$0.570001\pi$
$180$	0	0
$181$	−597517.	−1.35567	−0.677835	−	0.735214i	$-0.737082\pi$
$181$	−597517.	−1.35567	−0.677835	+	0.735214i	$0.737082\pi$
$182$	161345.	0.361059
$183$	449356.	0.991887
$184$	172834.	0.376344
$185$	0	0
$186$	26433.4	0.0560235
$187$	616339.	1.28889
$188$	267392.	0.551765
$189$	154643.	0.314903
$190$	0	0
$191$	336777.	0.667974	0.333987	−	0.942578i	$-0.391606\pi$
$191$	336777.	0.667974	0.333987	+	0.942578i	$0.391606\pi$
$192$	71581.6	0.140136
$193$	−686902.	−1.32740	−0.663699	−	0.748000i	$-0.731014\pi$
$193$	−686902.	−1.32740	−0.663699	+	0.748000i	$0.731014\pi$
$194$	−280567.	−0.535220
$195$	0	0
$196$	38416.0	0.0714286
$197$	547971.	1.00599	0.502993	−	0.864290i	$-0.332232\pi$
$197$	547971.	1.00599	0.502993	+	0.864290i	$0.332232\pi$
$198$	−70544.7	−0.127880
$199$	456643.	0.817418	0.408709	−	0.912665i	$-0.365979\pi$
$199$	456643.	0.817418	0.408709	+	0.912665i	$0.365979\pi$
$200$	0	0
$201$	39229.2	0.0684887
$202$	−231519.	−0.399216
$203$	−178417.	−0.303876
$204$	−609861.	−1.02602
$205$	0	0
$206$	−546951.	−0.898008
$207$	168540.	0.273387
$208$	−210737.	−0.337739
$209$	800463.	1.26758
$210$	0	0
$211$	319210.	0.493594	0.246797	−	0.969067i	$-0.420622\pi$
$211$	319210.	0.493594	0.246797	+	0.969067i	$0.420622\pi$
$212$	−92227.7	−0.140936
$213$	−873562.	−1.31930
$214$	−756491.	−1.12920
$215$	0	0
$216$	−201983.	−0.294565
$217$	−18528.8	−0.0267115
$218$	570496.	0.813039
$219$	−689998.	−0.972159
$220$	0	0
$221$	1.79543e6	2.47280
$222$	446357.	0.607854
$223$	871349.	1.17336	0.586678	−	0.809820i	$-0.300435\pi$
$223$	871349.	1.17336	0.586678	+	0.809820i	$0.300435\pi$
$224$	−50176.0	−0.0668153
$225$	0	0
$226$	395761.	0.515421
$227$	1.44405e6	1.86002	0.930010	−	0.367535i	$-0.119798\pi$
$227$	1.44405e6	1.86002	0.930010	+	0.367535i	$0.119798\pi$
$228$	−792050.	−1.00906
$229$	−1.02961e6	−1.29744	−0.648718	−	0.761029i	$-0.724695\pi$
$229$	−1.02961e6	−1.29744	−0.648718	+	0.761029i	$0.724695\pi$
$230$	0	0
$231$	241985.	0.298372
$232$	233035.	0.284250
$233$	94836.5	0.114442	0.0572210	−	0.998362i	$-0.481776\pi$
$233$	94836.5	0.114442	0.0572210	+	0.998362i	$0.481776\pi$
$234$	−205501.	−0.245344
$235$	0	0
$236$	348778.	0.407633
$237$	1.34767e6	1.55852
$238$	427489.	0.489196
$239$	1.07387e6	1.21607	0.608035	−	0.793910i	$-0.291958\pi$
$239$	1.07387e6	1.21607	0.608035	+	0.793910i	$0.291958\pi$
$240$	0	0
$241$	−663240.	−0.735577	−0.367789	−	0.929909i	$-0.619885\pi$
$241$	−663240.	−0.735577	−0.367789	+	0.929909i	$0.619885\pi$
$242$	−324785.	−0.356499
$243$	−462000.	−0.501910
$244$	411404.	0.442378
$245$	0	0
$246$	−1.32880e6	−1.39998
$247$	2.33180e6	2.43192
$248$	24200.9	0.0249863
$249$	264297.	0.270143
$250$	0	0
$251$	57498.6	0.0576067	0.0288034	−	0.999585i	$-0.490830\pi$
$251$	57498.6	0.0576067	0.0288034	+	0.999585i	$0.490830\pi$
$252$	−48929.5	−0.0485366
$253$	−763131.	−0.749545
$254$	−287286.	−0.279402
$255$	0	0
$256$	65536.0	0.0625000
$257$	−1.17039e6	−1.10535	−0.552673	−	0.833398i	$-0.686392\pi$
$257$	−1.17039e6	−1.10535	−0.552673	+	0.833398i	$0.686392\pi$
$258$	936890.	0.876273
$259$	−312879.	−0.289819
$260$	0	0
$261$	227246.	0.206488
$262$	−456506.	−0.410859
$263$	−1.87583e6	−1.67226	−0.836131	−	0.548530i	$-0.815187\pi$
$263$	−1.87583e6	−1.67226	−0.836131	+	0.548530i	$0.815187\pi$
$264$	−316062.	−0.279101
$265$	0	0
$266$	555197.	0.481108
$267$	1.69196e6	1.45249
$268$	35916.0	0.0305457
$269$	235810.	0.198693	0.0993463	−	0.995053i	$-0.468325\pi$
$269$	235810.	0.198693	0.0993463	+	0.995053i	$0.468325\pi$
$270$	0	0
$271$	1.11599e6	0.923071	0.461536	−	0.887122i	$-0.347299\pi$
$271$	1.11599e6	0.923071	0.461536	+	0.887122i	$0.347299\pi$
$272$	−558354.	−0.457601
$273$	704917.	0.572442
$274$	−933929.	−0.751515
$275$	0	0
$276$	755111.	0.596675
$277$	−2.23954e6	−1.75372	−0.876858	−	0.480750i	$-0.840364\pi$
$277$	−2.23954e6	−1.75372	−0.876858	+	0.480750i	$0.840364\pi$
$278$	171680.	0.133231
$279$	23599.6	0.0181508
$280$	0	0
$281$	246922.	0.186550	0.0932749	−	0.995640i	$-0.470266\pi$
$281$	246922.	0.186550	0.0932749	+	0.995640i	$0.470266\pi$
$282$	1.16824e6	0.874798
$283$	1.06876e6	0.793261	0.396630	−	0.917978i	$-0.370180\pi$
$283$	1.06876e6	0.793261	0.396630	+	0.917978i	$0.370180\pi$
$284$	−799783.	−0.588405
$285$	0	0
$286$	930487.	0.672659
$287$	931438.	0.667497
$288$	63907.9	0.0454018
$289$	3.33720e6	2.35038
$290$	0	0
$291$	−1.22580e6	−0.848566
$292$	−631722.	−0.433580
$293$	445723.	0.303317	0.151658	−	0.988433i	$-0.451539\pi$
$293$	445723.	0.303317	0.151658	+	0.988433i	$0.451539\pi$
$294$	167839.	0.113247
$295$	0	0
$296$	408658.	0.271101
$297$	891837.	0.586671
$298$	−1.33965e6	−0.873875
$299$	−2.22305e6	−1.43804
$300$	0	0
$301$	−656724.	−0.417798
$302$	1.18687e6	0.748836
$303$	−1.01151e6	−0.632939
$304$	−725155.	−0.450036
$305$	0	0
$306$	−544482.	−0.332415
$307$	−425921.	−0.257919	−0.128959	−	0.991650i	$-0.541164\pi$
$307$	−425921.	−0.257919	−0.128959	+	0.991650i	$0.541164\pi$
$308$	221547.	0.133073
$309$	−2.38963e6	−1.42375
$310$	0	0
$311$	473646.	0.277685	0.138843	−	0.990314i	$-0.455662\pi$
$311$	473646.	0.277685	0.138843	+	0.990314i	$0.455662\pi$
$312$	−920707.	−0.535470
$313$	2.93119e6	1.69115	0.845577	−	0.533853i	$-0.179257\pi$
$313$	2.93119e6	1.69115	0.845577	+	0.533853i	$0.179257\pi$
$314$	−1.18312e6	−0.677180
$315$	0	0
$316$	1.23385e6	0.695094
$317$	−112972.	−0.0631427	−0.0315713	−	0.999502i	$-0.510051\pi$
$317$	−112972.	−0.0631427	−0.0315713	+	0.999502i	$0.510051\pi$
$318$	−402943.	−0.223447
$319$	−1.02894e6	−0.566128
$320$	0	0
$321$	−3.30511e6	−1.79029
$322$	−529304.	−0.284489
$323$	6.17818e6	3.29499
$324$	−1.12511e6	−0.595435
$325$	0	0
$326$	−1.22805e6	−0.639990
$327$	2.49249e6	1.28904
$328$	−1.21657e6	−0.624386
$329$	−818889.	−0.417095
$330$	0	0
$331$	2.30582e6	1.15679	0.578397	−	0.815755i	$-0.303678\pi$
$331$	2.30582e6	1.15679	0.578397	+	0.815755i	$0.303678\pi$
$332$	241975.	0.120483
$333$	398506.	0.196936
$334$	−1.86190e6	−0.913249
$335$	0	0
$336$	−219219.	−0.105933
$337$	−2.92219e6	−1.40163	−0.700815	−	0.713343i	$-0.747180\pi$
$337$	−2.92219e6	−1.40163	−0.700815	+	0.713343i	$0.747180\pi$
$338$	1.22539e6	0.583424
$339$	1.72908e6	0.817176
$340$	0	0
$341$	−106857.	−0.0497640
$342$	−707140.	−0.326919
$343$	−117649.	−0.0539949
$344$	857762.	0.390815
$345$	0	0
$346$	−1.02180e6	−0.458854
$347$	787976.	0.351309	0.175654	−	0.984452i	$-0.443796\pi$
$347$	787976.	0.351309	0.175654	+	0.984452i	$0.443796\pi$
$348$	1.01813e6	0.450666
$349$	4.04864e6	1.77929	0.889643	−	0.456657i	$-0.150953\pi$
$349$	4.04864e6	1.77929	0.889643	+	0.456657i	$0.150953\pi$
$350$	0	0
$351$	2.59798e6	1.12556
$352$	−289368.	−0.124478
$353$	4.15543e6	1.77492	0.887460	−	0.460884i	$-0.152468\pi$
$353$	4.15543e6	1.77492	0.887460	+	0.460884i	$0.152468\pi$
$354$	1.52381e6	0.646283
$355$	0	0
$356$	1.54906e6	0.647804
$357$	1.86770e6	0.775598
$358$	−748120.	−0.308506
$359$	−3.14726e6	−1.28883	−0.644416	−	0.764675i	$-0.722900\pi$
$359$	−3.14726e6	−1.28883	−0.644416	+	0.764675i	$0.722900\pi$
$360$	0	0
$361$	5.54773e6	2.24051
$362$	−2.39007e6	−0.958604
$363$	−1.41899e6	−0.565212
$364$	645381.	0.255307
$365$	0	0
$366$	1.79742e6	0.701370
$367$	−1.84691e6	−0.715784	−0.357892	−	0.933763i	$-0.616504\pi$
$367$	−1.84691e6	−0.715784	−0.357892	+	0.933763i	$0.616504\pi$
$368$	691336.	0.266115
$369$	−1.18635e6	−0.453572
$370$	0	0
$371$	282447.	0.106538
$372$	105734.	0.0396146
$373$	−4.08390e6	−1.51986	−0.759929	−	0.650006i	$-0.774766\pi$
$373$	−4.08390e6	−1.51986	−0.759929	+	0.650006i	$0.774766\pi$
$374$	2.46535e6	0.911382
$375$	0	0
$376$	1.06957e6	0.390157
$377$	−2.99737e6	−1.08614
$378$	618574.	0.222670
$379$	−538962.	−0.192735	−0.0963674	−	0.995346i	$-0.530722\pi$
$379$	−538962.	−0.192735	−0.0963674	+	0.995346i	$0.530722\pi$
$380$	0	0
$381$	−1.25515e6	−0.442979
$382$	1.34711e6	0.472329
$383$	−4.73328e6	−1.64879	−0.824395	−	0.566015i	$-0.808484\pi$
$383$	−4.73328e6	−1.64879	−0.824395	+	0.566015i	$0.808484\pi$
$384$	286327.	0.0990908
$385$	0	0
$386$	−2.74761e6	−0.938612
$387$	836452.	0.283899
$388$	−1.12227e6	−0.378458
$389$	2.49369e6	0.835542	0.417771	−	0.908552i	$-0.362811\pi$
$389$	2.49369e6	0.835542	0.417771	+	0.908552i	$0.362811\pi$
$390$	0	0
$391$	−5.89004e6	−1.94839
$392$	153664.	0.0505076
$393$	−1.99447e6	−0.651398
$394$	2.19189e6	0.711340
$395$	0	0
$396$	−282179.	−0.0904246
$397$	−2.06171e6	−0.656524	−0.328262	−	0.944587i	$-0.606463\pi$
$397$	−2.06171e6	−0.656524	−0.328262	+	0.944587i	$0.606463\pi$
$398$	1.82657e6	0.578002
$399$	2.42565e6	0.762775
$400$	0	0
$401$	1.81036e6	0.562217	0.281109	−	0.959676i	$-0.409298\pi$
$401$	1.81036e6	0.562217	0.281109	+	0.959676i	$0.409298\pi$
$402$	156917.	0.0484288
$403$	−311280.	−0.0954747
$404$	−926076.	−0.282289
$405$	0	0
$406$	−713669.	−0.214873
$407$	−1.80439e6	−0.539939
$408$	−2.43944e6	−0.725505
$409$	962792.	0.284593	0.142296	−	0.989824i	$-0.454551\pi$
$409$	962792.	0.284593	0.142296	+	0.989824i	$0.454551\pi$
$410$	0	0
$411$	−4.08033e6	−1.19149
$412$	−2.18780e6	−0.634988
$413$	−1.06813e6	−0.308142
$414$	674161.	0.193314
$415$	0	0
$416$	−842946.	−0.238818
$417$	750067.	0.211232
$418$	3.20185e6	0.896314
$419$	−711918.	−0.198105	−0.0990524	−	0.995082i	$-0.531581\pi$
$419$	−711918.	−0.198105	−0.0990524	+	0.995082i	$0.531581\pi$
$420$	0	0
$421$	2.92216e6	0.803522	0.401761	−	0.915744i	$-0.368398\pi$
$421$	2.92216e6	0.803522	0.401761	+	0.915744i	$0.368398\pi$
$422$	1.27684e6	0.349024
$423$	1.04300e6	0.283421
$424$	−368911.	−0.0996568
$425$	0	0
$426$	−3.49425e6	−0.932889
$427$	−1.25992e6	−0.334407
$428$	−3.02597e6	−0.798463
$429$	4.06529e6	1.06647
$430$	0	0
$431$	−5.89406e6	−1.52835	−0.764173	−	0.645012i	$-0.776852\pi$
$431$	−5.89406e6	−1.52835	−0.764173	+	0.645012i	$0.776852\pi$
$432$	−807933.	−0.208289
$433$	−6.31234e6	−1.61797	−0.808985	−	0.587829i	$-0.799983\pi$
$433$	−6.31234e6	−1.61797	−0.808985	+	0.587829i	$0.799983\pi$
$434$	−74115.2	−0.0188879
$435$	0	0
$436$	2.28198e6	0.574905
$437$	−7.64962e6	−1.91618
$438$	−2.75999e6	−0.687420
$439$	−5.07241e6	−1.25618	−0.628092	−	0.778139i	$-0.716164\pi$
$439$	−5.07241e6	−1.25618	−0.628092	+	0.778139i	$0.716164\pi$
$440$	0	0
$441$	149846.	0.0366902
$442$	7.18173e6	1.74853
$443$	2.08233e6	0.504127	0.252064	−	0.967711i	$-0.418891\pi$
$443$	2.08233e6	0.504127	0.252064	+	0.967711i	$0.418891\pi$
$444$	1.78543e6	0.429818
$445$	0	0
$446$	3.48540e6	0.829688
$447$	−5.85291e6	−1.38549
$448$	−200704.	−0.0472456
$449$	−3.64122e6	−0.852375	−0.426188	−	0.904635i	$-0.640144\pi$
$449$	−3.64122e6	−0.852375	−0.426188	+	0.904635i	$0.640144\pi$
$450$	0	0
$451$	5.37165e6	1.24356
$452$	1.58305e6	0.364458
$453$	5.18544e6	1.18725
$454$	5.77620e6	1.31523
$455$	0	0
$456$	−3.16820e6	−0.713511
$457$	4.59232e6	1.02859	0.514294	−	0.857614i	$-0.328054\pi$
$457$	4.59232e6	1.02859	0.514294	+	0.857614i	$0.328054\pi$
$458$	−4.11846e6	−0.917426
$459$	6.88343e6	1.52501
$460$	0	0
$461$	1.21001e6	0.265177	0.132588	−	0.991171i	$-0.457671\pi$
$461$	1.21001e6	0.265177	0.132588	+	0.991171i	$0.457671\pi$
$462$	967939.	0.210981
$463$	−5.15318e6	−1.11718	−0.558589	−	0.829444i	$-0.688657\pi$
$463$	−5.15318e6	−1.11718	−0.558589	+	0.829444i	$0.688657\pi$
$464$	932140.	0.200995
$465$	0	0
$466$	379346.	0.0809228
$467$	−5.64439e6	−1.19764	−0.598818	−	0.800885i	$-0.704363\pi$
$467$	−5.64439e6	−1.19764	−0.598818	+	0.800885i	$0.704363\pi$
$468$	−822005.	−0.173484
$469$	−109993.	−0.0230904
$470$	0	0
$471$	−5.16904e6	−1.07364
$472$	1.39511e6	0.288240
$473$	−3.78736e6	−0.778366
$474$	5.39067e6	1.10204
$475$	0	0
$476$	1.70996e6	0.345914
$477$	−359746.	−0.0723936
$478$	4.29550e6	0.859892
$479$	4.64735e6	0.925478	0.462739	−	0.886494i	$-0.346867\pi$
$479$	4.64735e6	0.925478	0.462739	+	0.886494i	$0.346867\pi$
$480$	0	0
$481$	−5.25630e6	−1.03590
$482$	−2.65296e6	−0.520132
$483$	−2.31253e6	−0.451044
$484$	−1.29914e6	−0.252083
$485$	0	0
$486$	−1.84800e6	−0.354904
$487$	2.24265e6	0.428489	0.214244	−	0.976780i	$-0.431271\pi$
$487$	2.24265e6	0.428489	0.214244	+	0.976780i	$0.431271\pi$
$488$	1.64562e6	0.312809
$489$	−5.36536e6	−1.01467
$490$	0	0
$491$	−7.21876e6	−1.35132	−0.675661	−	0.737213i	$-0.736142\pi$
$491$	−7.21876e6	−1.35132	−0.675661	+	0.737213i	$0.736142\pi$
$492$	−5.31520e6	−0.989935
$493$	−7.94164e6	−1.47161
$494$	9.32719e6	1.71962
$495$	0	0
$496$	96803.5	0.0176680
$497$	2.44934e6	0.444792
$498$	1.05719e6	0.191020
$499$	−4.78946e6	−0.861064	−0.430532	−	0.902575i	$-0.641674\pi$
$499$	−4.78946e6	−0.861064	−0.430532	+	0.902575i	$0.641674\pi$
$500$	0	0
$501$	−8.13462e6	−1.44791
$502$	229994.	0.0407341
$503$	1.45950e6	0.257208	0.128604	−	0.991696i	$-0.458950\pi$
$503$	1.45950e6	0.257208	0.128604	+	0.991696i	$0.458950\pi$
$504$	−195718.	−0.0343205
$505$	0	0
$506$	−3.05252e6	−0.530008
$507$	5.35374e6	0.924991
$508$	−1.14914e6	−0.197567
$509$	−6.36689e6	−1.08926	−0.544632	−	0.838675i	$-0.683330\pi$
$509$	−6.36689e6	−1.08926	−0.544632	+	0.838675i	$0.683330\pi$
$510$	0	0
$511$	1.93465e6	0.327755
$512$	262144.	0.0441942
$513$	8.93977e6	1.49980
$514$	−4.68157e6	−0.781598
$515$	0	0
$516$	3.74756e6	0.619618
$517$	−4.72258e6	−0.777057
$518$	−1.25152e6	−0.204933
$519$	−4.46423e6	−0.727492
$520$	0	0
$521$	2.07512e6	0.334926	0.167463	−	0.985878i	$-0.446443\pi$
$521$	2.07512e6	0.334926	0.167463	+	0.985878i	$0.446443\pi$
$522$	908982.	0.146009
$523$	3.28342e6	0.524895	0.262447	−	0.964946i	$-0.415470\pi$
$523$	3.28342e6	0.524895	0.262447	+	0.964946i	$0.415470\pi$
$524$	−1.82602e6	−0.290521
$525$	0	0
$526$	−7.50332e6	−1.18247
$527$	−824746.	−0.129358
$528$	−1.26425e6	−0.197354
$529$	856519.	0.133075
$530$	0	0
$531$	1.36045e6	0.209386
$532$	2.22079e6	0.340195
$533$	1.56480e7	2.38583
$534$	6.76785e6	1.02706
$535$	0	0
$536$	143664.	0.0215991
$537$	−3.26854e6	−0.489122
$538$	943240.	0.140497
$539$	−678488.	−0.100594
$540$	0	0
$541$	618352.	0.0908328	0.0454164	−	0.998968i	$-0.485539\pi$
$541$	618352.	0.0908328	0.0454164	+	0.998968i	$0.485539\pi$
$542$	4.46394e6	0.652710
$543$	−1.04422e7	−1.51982
$544$	−2.23341e6	−0.323573
$545$	0	0
$546$	2.81967e6	0.404777
$547$	−9.34342e6	−1.33517	−0.667587	−	0.744532i	$-0.732673\pi$
$547$	−9.34342e6	−1.33517	−0.667587	+	0.744532i	$0.732673\pi$
$548$	−3.73572e6	−0.531401
$549$	1.60473e6	0.227233
$550$	0	0
$551$	−1.03141e7	−1.44728
$552$	3.02044e6	0.421913
$553$	−3.77866e6	−0.525442
$554$	−8.95815e6	−1.24006
$555$	0	0
$556$	686718.	0.0942088
$557$	−1.01532e7	−1.38665	−0.693324	−	0.720626i	$-0.743854\pi$
$557$	−1.01532e7	−1.38665	−0.693324	+	0.720626i	$0.743854\pi$
$558$	94398.6	0.0128345
$559$	−1.10328e7	−1.49334
$560$	0	0
$561$	1.07711e7	1.44495
$562$	987690.	0.131911
$563$	−1.51019e6	−0.200799	−0.100399	−	0.994947i	$-0.532012\pi$
$563$	−1.51019e6	−0.200799	−0.100399	+	0.994947i	$0.532012\pi$
$564$	4.67295e6	0.618576
$565$	0	0
$566$	4.27506e6	0.560920
$567$	3.44566e6	0.450106
$568$	−3.19913e6	−0.416065
$569$	4.38494e6	0.567783	0.283892	−	0.958856i	$-0.408374\pi$
$569$	4.38494e6	0.567783	0.283892	+	0.958856i	$0.408374\pi$
$570$	0	0
$571$	−738836.	−0.0948326	−0.0474163	−	0.998875i	$-0.515099\pi$
$571$	−738836.	−0.0948326	−0.0474163	+	0.998875i	$0.515099\pi$
$572$	3.72195e6	0.475642
$573$	5.88552e6	0.748856
$574$	3.72575e6	0.471991
$575$	0	0
$576$	255631.	0.0321039
$577$	−9.51498e6	−1.18978	−0.594892	−	0.803805i	$-0.702805\pi$
$577$	−9.51498e6	−1.18978	−0.594892	+	0.803805i	$0.702805\pi$
$578$	1.33488e7	1.66197
$579$	−1.20043e7	−1.48813
$580$	0	0
$581$	−741048.	−0.0910764
$582$	−4.90319e6	−0.600027
$583$	1.62889e6	0.198482
$584$	−2.52689e6	−0.306587
$585$	0	0
$586$	1.78289e6	0.214477
$587$	−82033.2	−0.00982640	−0.00491320	−	0.999988i	$-0.501564\pi$
$587$	−82033.2	−0.00982640	−0.00491320	+	0.999988i	$0.501564\pi$
$588$	671357.	0.0800775
$589$	−1.07113e6	−0.127220
$590$	0	0
$591$	9.57634e6	1.12780
$592$	1.63463e6	0.191697
$593$	1.07166e7	1.25147	0.625736	−	0.780035i	$-0.284799\pi$
$593$	1.07166e7	1.25147	0.625736	+	0.780035i	$0.284799\pi$
$594$	3.56735e6	0.414839
$595$	0	0
$596$	−5.35858e6	−0.617923
$597$	7.98029e6	0.916395
$598$	−8.89220e6	−1.01685
$599$	1.14355e7	1.30223	0.651116	−	0.758978i	$-0.274301\pi$
$599$	1.14355e7	1.30223	0.651116	+	0.758978i	$0.274301\pi$
$600$	0	0
$601$	−5.01913e6	−0.566816	−0.283408	−	0.958999i	$-0.591465\pi$
$601$	−5.01913e6	−0.566816	−0.283408	+	0.958999i	$0.591465\pi$
$602$	−2.62690e6	−0.295428
$603$	140095.	0.0156902
$604$	4.74749e6	0.529507
$605$	0	0
$606$	−4.04602e6	−0.447555
$607$	1.92272e6	0.211809	0.105905	−	0.994376i	$-0.466226\pi$
$607$	1.92272e6	0.211809	0.105905	+	0.994376i	$0.466226\pi$
$608$	−2.90062e6	−0.318223
$609$	−3.11802e6	−0.340671
$610$	0	0
$611$	−1.37572e7	−1.49082
$612$	−2.17793e6	−0.235053
$613$	1.53621e7	1.65120	0.825601	−	0.564254i	$-0.190836\pi$
$613$	1.53621e7	1.65120	0.825601	+	0.564254i	$0.190836\pi$
$614$	−1.70368e6	−0.182376
$615$	0	0
$616$	886189.	0.0940967
$617$	−352050.	−0.0372299	−0.0186149	−	0.999827i	$-0.505926\pi$
$617$	−352050.	−0.0372299	−0.0186149	+	0.999827i	$0.505926\pi$
$618$	−9.55850e6	−1.00674
$619$	1.10144e7	1.15541	0.577705	−	0.816246i	$-0.303949\pi$
$619$	1.10144e7	1.15541	0.577705	+	0.816246i	$0.303949\pi$
$620$	0	0
$621$	−8.52284e6	−0.886860
$622$	1.89458e6	0.196353
$623$	−4.74400e6	−0.489694
$624$	−3.68283e6	−0.378634
$625$	0	0
$626$	1.17248e7	1.19583
$627$	1.39889e7	1.42106
$628$	−4.73247e6	−0.478839
$629$	−1.39267e7	−1.40353
$630$	0	0
$631$	1.20676e7	1.20655	0.603277	−	0.797532i	$-0.293862\pi$
$631$	1.20676e7	1.20655	0.603277	+	0.797532i	$0.293862\pi$
$632$	4.93539e6	0.491506
$633$	5.57850e6	0.553361
$634$	−451888.	−0.0446486
$635$	0	0
$636$	−1.61177e6	−0.158001
$637$	−1.97648e6	−0.192994
$638$	−4.11577e6	−0.400313
$639$	−3.11966e6	−0.302242
$640$	0	0
$641$	1.68056e7	1.61551	0.807755	−	0.589518i	$-0.200682\pi$
$641$	1.68056e7	1.61551	0.807755	+	0.589518i	$0.200682\pi$
$642$	−1.32204e7	−1.26593
$643$	−1.53856e7	−1.46753	−0.733763	−	0.679405i	$-0.762238\pi$
$643$	−1.53856e7	−1.46753	−0.733763	+	0.679405i	$0.762238\pi$
$644$	−2.11722e6	−0.201164
$645$	0	0
$646$	2.47127e7	2.32991
$647$	1.03532e7	0.972335	0.486167	−	0.873866i	$-0.338395\pi$
$647$	1.03532e7	0.972335	0.486167	+	0.873866i	$0.338395\pi$
$648$	−4.50046e6	−0.421036
$649$	−6.15998e6	−0.574074
$650$	0	0
$651$	−323809.	−0.0299458
$652$	−4.91221e6	−0.452541
$653$	−7.50862e6	−0.689092	−0.344546	−	0.938769i	$-0.611967\pi$
$653$	−7.50862e6	−0.689092	−0.344546	+	0.938769i	$0.611967\pi$
$654$	9.96998e6	0.911486
$655$	0	0
$656$	−4.86629e6	−0.441508
$657$	−2.46411e6	−0.222714
$658$	−3.27556e6	−0.294931
$659$	−1.53907e7	−1.38052	−0.690262	−	0.723560i	$-0.742505\pi$
$659$	−1.53907e7	−1.38052	−0.690262	+	0.723560i	$0.742505\pi$
$660$	0	0
$661$	1.03263e7	0.919263	0.459632	−	0.888110i	$-0.347981\pi$
$661$	1.03263e7	0.919263	0.459632	+	0.888110i	$0.347981\pi$
$662$	9.22330e6	0.817977
$663$	3.13770e7	2.77222
$664$	967899.	0.0851942
$665$	0	0
$666$	1.59402e6	0.139254
$667$	9.83309e6	0.855806
$668$	−7.44758e6	−0.645765
$669$	1.52277e7	1.31543
$670$	0	0
$671$	−7.26605e6	−0.623006
$672$	−876875.	−0.0749056
$673$	4.46576e6	0.380065	0.190033	−	0.981778i	$-0.439141\pi$
$673$	4.46576e6	0.380065	0.190033	+	0.981778i	$0.439141\pi$
$674$	−1.16888e7	−0.991102
$675$	0	0
$676$	4.90158e6	0.412543
$677$	2.93723e6	0.246301	0.123151	−	0.992388i	$-0.460700\pi$
$677$	2.93723e6	0.246301	0.123151	+	0.992388i	$0.460700\pi$
$678$	6.91632e6	0.577831
$679$	3.43695e6	0.286087
$680$	0	0
$681$	2.52362e7	2.08524
$682$	−427426.	−0.0351885
$683$	1.90246e7	1.56050	0.780252	−	0.625465i	$-0.215091\pi$
$683$	1.90246e7	1.56050	0.780252	+	0.625465i	$0.215091\pi$
$684$	−2.82856e6	−0.231167
$685$	0	0
$686$	−470596.	−0.0381802
$687$	−1.79935e7	−1.45454
$688$	3.43105e6	0.276348
$689$	4.74506e6	0.380797
$690$	0	0
$691$	−1.50379e6	−0.119810	−0.0599049	−	0.998204i	$-0.519080\pi$
$691$	−1.50379e6	−0.119810	−0.0599049	+	0.998204i	$0.519080\pi$
$692$	−4.08719e6	−0.324459
$693$	864173.	0.0683545
$694$	3.15190e6	0.248413
$695$	0	0
$696$	4.07251e6	0.318669
$697$	4.14598e7	3.23255
$698$	1.61946e7	1.25815
$699$	1.65736e6	0.128299
$700$	0	0
$701$	6.85393e6	0.526799	0.263399	−	0.964687i	$-0.415156\pi$
$701$	6.85393e6	0.526799	0.263399	+	0.964687i	$0.415156\pi$
$702$	1.03919e7	0.795889
$703$	−1.80872e7	−1.38033
$704$	−1.15747e6	−0.0880194
$705$	0	0
$706$	1.66217e7	1.25506
$707$	2.83611e6	0.213390
$708$	6.09524e6	0.456991
$709$	1.78998e7	1.33731	0.668655	−	0.743573i	$-0.266870\pi$
$709$	1.78998e7	1.33731	0.668655	+	0.743573i	$0.266870\pi$
$710$	0	0
$711$	4.81278e6	0.357044
$712$	6.19625e6	0.458067
$713$	1.02117e6	0.0752274
$714$	7.47080e6	0.548430
$715$	0	0
$716$	−2.99248e6	−0.218147
$717$	1.87670e7	1.36332
$718$	−1.25890e7	−0.911342
$719$	−7.26497e6	−0.524097	−0.262049	−	0.965055i	$-0.584398\pi$
$719$	−7.26497e6	−0.524097	−0.262049	+	0.965055i	$0.584398\pi$
$720$	0	0
$721$	6.70015e6	0.480005
$722$	2.21909e7	1.58428
$723$	−1.15908e7	−0.824645
$724$	−9.56028e6	−0.677835
$725$	0	0
$726$	−5.67594e6	−0.399665
$727$	−2.96379e6	−0.207975	−0.103988	−	0.994579i	$-0.533160\pi$
$727$	−2.96379e6	−0.207975	−0.103988	+	0.994579i	$0.533160\pi$
$728$	2.58152e6	0.180529
$729$	9.01377e6	0.628185
$730$	0	0
$731$	−2.92318e7	−2.02331
$732$	7.18969e6	0.495944
$733$	2.65625e6	0.182604	0.0913019	−	0.995823i	$-0.470897\pi$
$733$	2.65625e6	0.182604	0.0913019	+	0.995823i	$0.470897\pi$
$734$	−7.38766e6	−0.506135
$735$	0	0
$736$	2.76534e6	0.188172
$737$	−634334.	−0.0430179
$738$	−4.74539e6	−0.320724
$739$	2.37780e6	0.160164	0.0800818	−	0.996788i	$-0.474482\pi$
$739$	2.37780e6	0.160164	0.0800818	+	0.996788i	$0.474482\pi$
$740$	0	0
$741$	4.07505e7	2.72638
$742$	1.12979e6	0.0753335
$743$	−1.37118e7	−0.911216	−0.455608	−	0.890180i	$-0.650578\pi$
$743$	−1.37118e7	−0.911216	−0.455608	+	0.890180i	$0.650578\pi$
$744$	422934.	0.0280118
$745$	0	0
$746$	−1.63356e7	−1.07470
$747$	943853.	0.0618875
$748$	9.86142e6	0.644444
$749$	9.26702e6	0.603581
$750$	0	0
$751$	−1.20276e7	−0.778180	−0.389090	−	0.921200i	$-0.627211\pi$
$751$	−1.20276e7	−0.778180	−0.389090	+	0.921200i	$0.627211\pi$
$752$	4.27828e6	0.275883
$753$	1.00485e6	0.0645820
$754$	−1.19895e7	−0.768020
$755$	0	0
$756$	2.47429e6	0.157452
$757$	−119922.	−0.00760606	−0.00380303	−	0.999993i	$-0.501211\pi$
$757$	−119922.	−0.00760606	−0.00380303	+	0.999993i	$0.501211\pi$
$758$	−2.15585e6	−0.136284
$759$	−1.33365e7	−0.840304
$760$	0	0
$761$	3.37460e6	0.211232	0.105616	−	0.994407i	$-0.466319\pi$
$761$	3.37460e6	0.211232	0.105616	+	0.994407i	$0.466319\pi$
$762$	−5.02060e6	−0.313233
$763$	−6.98858e6	−0.434588
$764$	5.38844e6	0.333987
$765$	0	0
$766$	−1.89331e7	−1.16587
$767$	−1.79444e7	−1.10139
$768$	1.14531e6	0.0700678
$769$	−2.78531e7	−1.69847	−0.849235	−	0.528014i	$-0.822937\pi$
$769$	−2.78531e7	−1.69847	−0.849235	+	0.528014i	$0.822937\pi$
$770$	0	0
$771$	−2.04538e7	−1.23919
$772$	−1.09904e7	−0.663699
$773$	2.72350e7	1.63938	0.819689	−	0.572809i	$-0.194146\pi$
$773$	2.72350e7	1.63938	0.819689	+	0.572809i	$0.194146\pi$
$774$	3.34581e6	0.200747
$775$	0	0
$776$	−4.48907e6	−0.267610
$777$	−5.46787e6	−0.324912
$778$	9.97475e6	0.590817
$779$	5.38454e7	3.17911
$780$	0	0
$781$	1.41255e7	0.828657
$782$	−2.35602e7	−1.37772
$783$	−1.14915e7	−0.669841
$784$	614656.	0.0357143
$785$	0	0
$786$	−7.97789e6	−0.460608
$787$	2.41156e7	1.38791	0.693955	−	0.720019i	$-0.255867\pi$
$787$	2.41156e7	1.38791	0.693955	+	0.720019i	$0.255867\pi$
$788$	8.76754e6	0.502993
$789$	−3.27820e7	−1.87475
$790$	0	0
$791$	−4.84808e6	−0.275504
$792$	−1.12872e6	−0.0639398
$793$	−2.11665e7	−1.19527
$794$	−8.24683e6	−0.464233
$795$	0	0
$796$	7.30629e6	0.408709
$797$	1.93741e7	1.08038	0.540190	−	0.841543i	$-0.318352\pi$
$797$	1.93741e7	1.08038	0.540190	+	0.841543i	$0.318352\pi$
$798$	9.70261e6	0.539363
$799$	−3.64501e7	−2.01991
$800$	0	0
$801$	6.04231e6	0.332753
$802$	7.24144e6	0.397547
$803$	1.11572e7	0.610615
$804$	627667.	0.0342444
$805$	0	0
$806$	−1.24512e6	−0.0675108
$807$	4.12101e6	0.222751
$808$	−3.70430e6	−0.199608
$809$	2.75238e6	0.147855	0.0739276	−	0.997264i	$-0.476447\pi$
$809$	2.75238e6	0.147855	0.0739276	+	0.997264i	$0.476447\pi$
$810$	0	0
$811$	−2.46882e7	−1.31807	−0.659034	−	0.752113i	$-0.729035\pi$
$811$	−2.46882e7	−1.31807	−0.659034	+	0.752113i	$0.729035\pi$
$812$	−2.85468e6	−0.151938
$813$	1.95029e7	1.03484
$814$	−7.21756e6	−0.381794
$815$	0	0
$816$	−9.75778e6	−0.513010
$817$	−3.79645e7	−1.98986
$818$	3.85117e6	0.201238
$819$	2.51739e6	0.131142
$820$	0	0
$821$	−1.00209e7	−0.518859	−0.259429	−	0.965762i	$-0.583534\pi$
$821$	−1.00209e7	−0.518859	−0.259429	+	0.965762i	$0.583534\pi$
$822$	−1.63213e7	−0.842512
$823$	−2.76244e7	−1.42165	−0.710827	−	0.703367i	$-0.751679\pi$
$823$	−2.76244e7	−1.42165	−0.710827	+	0.703367i	$0.751679\pi$
$824$	−8.75121e6	−0.449004
$825$	0	0
$826$	−4.27253e6	−0.217889
$827$	4.28505e6	0.217867	0.108934	−	0.994049i	$-0.465256\pi$
$827$	4.28505e6	0.217867	0.108934	+	0.994049i	$0.465256\pi$
$828$	2.69664e6	0.136693
$829$	−1.28054e7	−0.647154	−0.323577	−	0.946202i	$-0.604885\pi$
$829$	−1.28054e7	−0.647154	−0.323577	+	0.946202i	$0.604885\pi$
$830$	0	0
$831$	−3.91381e7	−1.96606
$832$	−3.37179e6	−0.168870
$833$	−5.23675e6	−0.261486
$834$	3.00027e6	0.149364
$835$	0	0
$836$	1.28074e7	0.633790
$837$	−1.19340e6	−0.0588807
$838$	−2.84767e6	−0.140081
$839$	3.16996e7	1.55471	0.777355	−	0.629062i	$-0.216561\pi$
$839$	3.16996e7	1.55471	0.777355	+	0.629062i	$0.216561\pi$
$840$	0	0
$841$	−7.25302e6	−0.353614
$842$	1.16886e7	0.568176
$843$	4.31521e6	0.209138
$844$	5.10735e6	0.246797
$845$	0	0
$846$	4.17199e6	0.200409
$847$	3.97862e6	0.190557
$848$	−1.47564e6	−0.0704680
$849$	1.86777e7	0.889312
$850$	0	0
$851$	1.72437e7	0.816217
$852$	−1.39770e7	−0.659652
$853$	−1.25810e6	−0.0592029	−0.0296014	−	0.999562i	$-0.509424\pi$
$853$	−1.25810e6	−0.0592029	−0.0296014	+	0.999562i	$0.509424\pi$
$854$	−5.03970e6	−0.236461
$855$	0	0
$856$	−1.21039e7	−0.564598
$857$	−6.46012e6	−0.300461	−0.150231	−	0.988651i	$-0.548002\pi$
$857$	−6.46012e6	−0.300461	−0.150231	+	0.988651i	$0.548002\pi$
$858$	1.62612e7	0.754108
$859$	−1.59546e7	−0.737740	−0.368870	−	0.929481i	$-0.620255\pi$
$859$	−1.59546e7	−0.737740	−0.368870	+	0.929481i	$0.620255\pi$
$860$	0	0
$861$	1.62778e7	0.748320
$862$	−2.35762e7	−1.08070
$863$	4.03274e6	0.184321	0.0921603	−	0.995744i	$-0.470623\pi$
$863$	4.03274e6	0.184321	0.0921603	+	0.995744i	$0.470623\pi$
$864$	−3.23173e6	−0.147282
$865$	0	0
$866$	−2.52494e7	−1.14408
$867$	5.83209e7	2.63498
$868$	−296461.	−0.0133557
$869$	−2.17917e7	−0.978908
$870$	0	0
$871$	−1.84785e6	−0.0825320
$872$	9.12794e6	0.406519
$873$	−4.37755e6	−0.194400
$874$	−3.05985e7	−1.35494
$875$	0	0
$876$	−1.10400e7	−0.486079
$877$	502061.	0.0220424	0.0110212	−	0.999939i	$-0.496492\pi$
$877$	502061.	0.0220424	0.0110212	+	0.999939i	$0.496492\pi$
$878$	−2.02896e7	−0.888256
$879$	7.78945e6	0.340044
$880$	0	0
$881$	1.26072e7	0.547243	0.273621	−	0.961837i	$-0.411778\pi$
$881$	1.26072e7	0.547243	0.273621	+	0.961837i	$0.411778\pi$
$882$	599386.	0.0259439
$883$	−3.24012e7	−1.39849	−0.699245	−	0.714882i	$-0.746480\pi$
$883$	−3.24012e7	−1.39849	−0.699245	+	0.714882i	$0.746480\pi$
$884$	2.87269e7	1.23640
$885$	0	0
$886$	8.32931e6	0.356472
$887$	−2.22191e7	−0.948238	−0.474119	−	0.880461i	$-0.657233\pi$
$887$	−2.22191e7	−0.948238	−0.474119	+	0.880461i	$0.657233\pi$
$888$	7.14171e6	0.303927
$889$	3.51925e6	0.149347
$890$	0	0
$891$	1.98713e7	0.838557
$892$	1.39416e7	0.586678
$893$	−4.73391e7	−1.98651
$894$	−2.34116e7	−0.979688
$895$	0	0
$896$	−802816.	−0.0334077
$897$	−3.88500e7	−1.61217
$898$	−1.45649e7	−0.602720
$899$	1.37687e6	0.0568189
$900$	0	0
$901$	1.25722e7	0.515940
$902$	2.14866e7	0.879329
$903$	−1.14769e7	−0.468387
$904$	6.33218e6	0.257711
$905$	0	0
$906$	2.07418e7	0.839509
$907$	−4.05126e7	−1.63520	−0.817602	−	0.575784i	$-0.804697\pi$
$907$	−4.05126e7	−1.63520	−0.817602	+	0.575784i	$0.804697\pi$
$908$	2.31048e7	0.930010
$909$	−3.61228e6	−0.145001
$910$	0	0
$911$	2.24420e7	0.895913	0.447956	−	0.894055i	$-0.352152\pi$
$911$	2.24420e7	0.895913	0.447956	+	0.894055i	$0.352152\pi$
$912$	−1.26728e7	−0.504528
$913$	−4.27366e6	−0.169677
$914$	1.83693e7	0.727321
$915$	0	0
$916$	−1.64738e7	−0.648718
$917$	5.59220e6	0.219614
$918$	2.75337e7	1.07835
$919$	−2.89932e7	−1.13242	−0.566211	−	0.824261i	$-0.691591\pi$
$919$	−2.89932e7	−1.13242	−0.566211	+	0.824261i	$0.691591\pi$
$920$	0	0
$921$	−7.44339e6	−0.289149
$922$	4.84003e6	0.187508
$923$	4.11483e7	1.58982
$924$	3.87175e6	0.149186
$925$	0	0
$926$	−2.06127e7	−0.789965
$927$	−8.53380e6	−0.326169
$928$	3.72856e6	0.142125
$929$	−1.19021e7	−0.452466	−0.226233	−	0.974073i	$-0.572641\pi$
$929$	−1.19021e7	−0.452466	−0.226233	+	0.974073i	$0.572641\pi$
$930$	0	0
$931$	−6.80116e6	−0.257163
$932$	1.51738e6	0.0572210
$933$	8.27743e6	0.311309
$934$	−2.25776e7	−0.846857
$935$	0	0
$936$	−3.28802e6	−0.122672
$937$	−4.06074e7	−1.51097	−0.755486	−	0.655165i	$-0.772599\pi$
$937$	−4.06074e7	−1.51097	−0.755486	+	0.655165i	$0.772599\pi$
$938$	−439971.	−0.0163274
$939$	5.12254e7	1.89593
$940$	0	0
$941$	−2.22687e7	−0.819825	−0.409912	−	0.912125i	$-0.634441\pi$
$941$	−2.22687e7	−0.819825	−0.409912	+	0.912125i	$0.634441\pi$
$942$	−2.06762e7	−0.759176
$943$	−5.13342e7	−1.87987
$944$	5.58045e6	0.203816
$945$	0	0
$946$	−1.51495e7	−0.550388
$947$	2.57504e7	0.933060	0.466530	−	0.884505i	$-0.345504\pi$
$947$	2.57504e7	0.933060	0.466530	+	0.884505i	$0.345504\pi$
$948$	2.15627e7	0.779260
$949$	3.25017e7	1.17149
$950$	0	0
$951$	−1.97430e6	−0.0707883
$952$	6.83983e6	0.244598
$953$	−1.10332e6	−0.0393523	−0.0196761	−	0.999806i	$-0.506264\pi$
$953$	−1.10332e6	−0.0393523	−0.0196761	+	0.999806i	$0.506264\pi$
$954$	−1.43898e6	−0.0511900
$955$	0	0
$956$	1.71820e7	0.608035
$957$	−1.79818e7	−0.634677
$958$	1.85894e7	0.654412
$959$	1.14406e7	0.401701
$960$	0	0
$961$	−2.84862e7	−0.995005
$962$	−2.10252e7	−0.732492
$963$	−1.18032e7	−0.410141
$964$	−1.06118e7	−0.367789
$965$	0	0
$966$	−9.25011e6	−0.318936
$967$	−8.41097e6	−0.289254	−0.144627	−	0.989486i	$-0.546198\pi$
$967$	−8.41097e6	−0.289254	−0.144627	+	0.989486i	$0.546198\pi$
$968$	−5.19657e6	−0.178249
$969$	1.07970e8	3.69396
$970$	0	0
$971$	−1.73671e7	−0.591125	−0.295563	−	0.955323i	$-0.595507\pi$
$971$	−1.73671e7	−0.591125	−0.295563	+	0.955323i	$0.595507\pi$
$972$	−7.39199e6	−0.250955
$973$	−2.10307e6	−0.0712152
$974$	8.97061e6	0.302987
$975$	0	0
$976$	6.58246e6	0.221189
$977$	4.89702e7	1.64133	0.820665	−	0.571410i	$-0.193603\pi$
$977$	4.89702e7	1.64133	0.820665	+	0.571410i	$0.193603\pi$
$978$	−2.14614e7	−0.717483
$979$	−2.73589e7	−0.912310
$980$	0	0
$981$	8.90117e6	0.295307
$982$	−2.88750e7	−0.955529
$983$	−2.70646e7	−0.893342	−0.446671	−	0.894698i	$-0.647391\pi$
$983$	−2.70646e7	−0.893342	−0.446671	+	0.894698i	$0.647391\pi$
$984$	−2.12608e7	−0.699990
$985$	0	0
$986$	−3.17666e7	−1.04059
$987$	−1.43109e7	−0.467599
$988$	3.73088e7	1.21596
$989$	3.61939e7	1.17664
$990$	0	0
$991$	3.29586e7	1.06607	0.533034	−	0.846094i	$-0.321052\pi$
$991$	3.29586e7	1.06607	0.533034	+	0.846094i	$0.321052\pi$
$992$	387214.	0.0124931
$993$	4.02966e7	1.29687
$994$	9.79734e6	0.314516
$995$	0	0
$996$	4.22875e6	0.135071
$997$	3.07051e7	0.978301	0.489150	−	0.872200i	$-0.337307\pi$
$997$	3.07051e7	0.978301	0.489150	+	0.872200i	$0.337307\pi$
$998$	−1.91579e7	−0.608864
$999$	−2.01519e7	−0.638855

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.a.w.1.3	yes	3
5.2	odd	4		350.6.c.m.99.4		6
5.3	odd	4		350.6.c.m.99.3		6
5.4	even	2		350.6.a.v.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.6.a.v.1.1	✓	3	5.4	even	2
350.6.a.w.1.3	yes	3	1.1	even	1	trivial
350.6.c.m.99.3		6	5.3	odd	4
350.6.c.m.99.4		6	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +17.4760 q^{3} +16.0000 q^{4} +69.9039 q^{6} -49.0000 q^{7} +64.0000 q^{8} +62.4100 q^{9} -282.586 q^{11} +279.616 q^{12} -823.190 q^{13} -196.000 q^{14} +256.000 q^{16} -2181.07 q^{17} +249.640 q^{18} -2832.64 q^{19} -856.323 q^{21} -1130.34 q^{22} +2700.53 q^{23} +1118.46 q^{24} -3292.76 q^{26} -3155.99 q^{27} -784.000 q^{28} +3641.17 q^{29} +378.139 q^{31} +1024.00 q^{32} -4938.46 q^{33} -8724.27 q^{34} +998.560 q^{36} +6385.29 q^{37} -11330.5 q^{38} -14386.1 q^{39} -19008.9 q^{41} -3425.29 q^{42} +13402.5 q^{43} -4521.37 q^{44} +10802.1 q^{46} +16712.0 q^{47} +4473.85 q^{48} +2401.00 q^{49} -38116.3 q^{51} -13171.0 q^{52} -5764.23 q^{53} -12624.0 q^{54} -3136.00 q^{56} -49503.1 q^{57} +14564.7 q^{58} +21798.6 q^{59} +25712.7 q^{61} +1512.55 q^{62} -3058.09 q^{63} +4096.00 q^{64} -19753.8 q^{66} +2244.75 q^{67} -34897.1 q^{68} +47194.4 q^{69} -49986.5 q^{71} +3994.24 q^{72} -39482.6 q^{73} +25541.1 q^{74} -45322.2 q^{76} +13846.7 q^{77} -57544.2 q^{78} +77115.4 q^{79} -70319.6 q^{81} -76035.7 q^{82} +15123.4 q^{83} -13701.2 q^{84} +53610.1 q^{86} +63633.0 q^{87} -18085.5 q^{88} +96816.4 q^{89} +40336.3 q^{91} +43208.5 q^{92} +6608.34 q^{93} +66848.1 q^{94} +17895.4 q^{96} -70141.8 q^{97} +9604.00 q^{98} -17636.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 12 q^{2} + q^{3} + 48 q^{4} + 4 q^{6} - 147 q^{7} + 192 q^{8} - 56 q^{9} - 11 q^{11} + 16 q^{12} - 482 q^{13} - 588 q^{14} + 768 q^{16} - 1185 q^{17} - 224 q^{18} - 1811 q^{19} - 49 q^{21} - 44 q^{22}+ \cdots + 109448 q^{99}+O(q^{100}) \) 3 * q + 12 * q^2 + q^3 + 48 * q^4 + 4 * q^6 - 147 * q^7 + 192 * q^8 - 56 * q^9 - 11 * q^11 + 16 * q^12 - 482 * q^13 - 588 * q^14 + 768 * q^16 - 1185 * q^17 - 224 * q^18 - 1811 * q^19 - 49 * q^21 - 44 * q^22 - 3350 * q^23 + 64 * q^24 - 1928 * q^26 - 1997 * q^27 - 2352 * q^28 + 1924 * q^29 - 6560 * q^31 + 3072 * q^32 - 15865 * q^33 - 4740 * q^34 - 896 * q^36 + 5194 * q^37 - 7244 * q^38 - 1574 * q^39 - 26981 * q^41 - 196 * q^42 - 18938 * q^43 - 176 * q^44 - 13400 * q^46 + 2878 * q^47 + 256 * q^48 + 7203 * q^49 - 55729 * q^51 - 7712 * q^52 - 6416 * q^53 - 7988 * q^54 - 9408 * q^56 - 33983 * q^57 + 7696 * q^58 + 17480 * q^59 - 40538 * q^61 - 26240 * q^62 + 2744 * q^63 + 12288 * q^64 - 63460 * q^66 - 43761 * q^67 - 18960 * q^68 + 70000 * q^69 - 65476 * q^71 - 3584 * q^72 - 59143 * q^73 + 20776 * q^74 - 28976 * q^76 + 539 * q^77 - 6296 * q^78 + 133128 * q^79 - 89693 * q^81 - 107924 * q^82 - 97205 * q^83 - 784 * q^84 - 75752 * q^86 - 63566 * q^87 - 704 * q^88 + 61353 * q^89 + 23618 * q^91 - 53600 * q^92 - 53964 * q^93 + 11512 * q^94 + 1024 * q^96 - 288942 * q^97 + 28812 * q^98 + 109448 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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