LMFDB - Embedded newform 531.6.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,6,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, 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("531.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	531.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:	$85.1638083207$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 283 x^{10} + 1045 x^{9} + 27968 x^{8} - 94393 x^{7} - 1130486 x^{6} + \cdots - 50564480 $ x^12 - 4x^11 - 283x^10 + 1045x^9 + 27968x^8 - 94393x^7 - 1130486x^6 + 3566264x^5 + 15496192x^4 - 53008480x^3 - 16576192x^2 + 120303168*x - 50564480
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-7.32600$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.32600 q^{2} +21.6702 q^{4} -46.7845 q^{5} -85.8026 q^{7} +75.6759 q^{8} +O(q^{10})$	\(q-7.32600 q^{2} +21.6702 q^{4} -46.7845 q^{5} -85.8026 q^{7} +75.6759 q^{8} +342.743 q^{10} +408.472 q^{11} +27.9269 q^{13} +628.589 q^{14} -1247.85 q^{16} -1587.94 q^{17} -2204.29 q^{19} -1013.83 q^{20} -2992.47 q^{22} +3968.63 q^{23} -936.208 q^{25} -204.593 q^{26} -1859.36 q^{28} +2905.16 q^{29} +4857.43 q^{31} +6720.11 q^{32} +11633.3 q^{34} +4014.23 q^{35} +2836.94 q^{37} +16148.6 q^{38} -3540.46 q^{40} -3698.67 q^{41} -8798.36 q^{43} +8851.69 q^{44} -29074.2 q^{46} +21002.5 q^{47} -9444.92 q^{49} +6858.66 q^{50} +605.183 q^{52} +11032.7 q^{53} -19110.2 q^{55} -6493.18 q^{56} -21283.2 q^{58} +3481.00 q^{59} +45308.5 q^{61} -35585.5 q^{62} -9300.32 q^{64} -1306.55 q^{65} -40826.6 q^{67} -34411.1 q^{68} -29408.2 q^{70} +27052.4 q^{71} +70459.2 q^{73} -20783.4 q^{74} -47767.4 q^{76} -35048.0 q^{77} +71673.4 q^{79} +58380.0 q^{80} +27096.4 q^{82} -93280.1 q^{83} +74291.2 q^{85} +64456.8 q^{86} +30911.5 q^{88} +116570. q^{89} -2396.20 q^{91} +86001.2 q^{92} -153864. q^{94} +103127. q^{95} +91107.3 q^{97} +69193.5 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} + 198 q^{4} - 36 q^{5} - 411 q^{7} + 69 q^{8}+O(q^{10}) $ 12 * q + 4 * q^2 + 198 * q^4 - 36 * q^5 - 411 * q^7 + 69 * q^8	$ 12 q + 4 q^{2} + 198 q^{4} - 36 q^{5} - 411 q^{7} + 69 q^{8} - 863 q^{10} - 492 q^{11} - 974 q^{13} + 967 q^{14} + 6370 q^{16} + 1463 q^{17} - 3189 q^{19} + 835 q^{20} - 2726 q^{22} + 2617 q^{23} + 8642 q^{25} - 2414 q^{26} - 20458 q^{28} + 1963 q^{29} - 11929 q^{31} + 14382 q^{32} - 20744 q^{34} - 1829 q^{35} - 28105 q^{37} + 23475 q^{38} - 100576 q^{40} + 7585 q^{41} - 33146 q^{43} - 26014 q^{44} - 142851 q^{46} + 79215 q^{47} - 32569 q^{49} + 136019 q^{50} - 248218 q^{52} + 12220 q^{53} - 117770 q^{55} + 186728 q^{56} - 188072 q^{58} + 41772 q^{59} - 54195 q^{61} - 36230 q^{62} + 45197 q^{64} - 42368 q^{65} + 24224 q^{67} + 209639 q^{68} - 35684 q^{70} - 60254 q^{71} - 15385 q^{73} - 214638 q^{74} - 167504 q^{76} + 17169 q^{77} - 27054 q^{79} - 216899 q^{80} + 37917 q^{82} + 117595 q^{83} - 121585 q^{85} - 306756 q^{86} - 105799 q^{88} + 36033 q^{89} - 32217 q^{91} + 30906 q^{92} + 128392 q^{94} + 50721 q^{95} - 196914 q^{97} - 574100 q^{98}+O(q^{100}) $ 12 * q + 4 * q^2 + 198 * q^4 - 36 * q^5 - 411 * q^7 + 69 * q^8 - 863 * q^10 - 492 * q^11 - 974 * q^13 + 967 * q^14 + 6370 * q^16 + 1463 * q^17 - 3189 * q^19 + 835 * q^20 - 2726 * q^22 + 2617 * q^23 + 8642 * q^25 - 2414 * q^26 - 20458 * q^28 + 1963 * q^29 - 11929 * q^31 + 14382 * q^32 - 20744 * q^34 - 1829 * q^35 - 28105 * q^37 + 23475 * q^38 - 100576 * q^40 + 7585 * q^41 - 33146 * q^43 - 26014 * q^44 - 142851 * q^46 + 79215 * q^47 - 32569 * q^49 + 136019 * q^50 - 248218 * q^52 + 12220 * q^53 - 117770 * q^55 + 186728 * q^56 - 188072 * q^58 + 41772 * q^59 - 54195 * q^61 - 36230 * q^62 + 45197 * q^64 - 42368 * q^65 + 24224 * q^67 + 209639 * q^68 - 35684 * q^70 - 60254 * q^71 - 15385 * q^73 - 214638 * q^74 - 167504 * q^76 + 17169 * q^77 - 27054 * q^79 - 216899 * q^80 + 37917 * q^82 + 117595 * q^83 - 121585 * q^85 - 306756 * q^86 - 105799 * q^88 + 36033 * q^89 - 32217 * q^91 + 30906 * q^92 + 128392 * q^94 + 50721 * q^95 - 196914 * q^97 - 574100 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−7.32600	−1.29507	−0.647533	−	0.762038i	$-0.724199\pi$
$2$	−7.32600	−1.29507	−0.647533	+	0.762038i	$0.724199\pi$
$3$	0	0
$3$	0	0
$4$	21.6702	0.677195
$5$	−46.7845	−0.836907	−0.418453	−	0.908238i	$-0.637428\pi$
$5$	−46.7845	−0.836907	−0.418453	+	0.908238i	$0.637428\pi$
$6$	0	0
$7$	−85.8026	−0.661843	−0.330922	−	0.943658i	$-0.607360\pi$
$7$	−85.8026	−0.661843	−0.330922	+	0.943658i	$0.607360\pi$
$8$	75.6759	0.418054
$9$	0	0
$10$	342.743	1.08385
$11$	408.472	1.01784	0.508921	−	0.860813i	$-0.330044\pi$
$11$	408.472	1.01784	0.508921	+	0.860813i	$0.330044\pi$
$12$	0	0
$13$	27.9269	0.0458316	0.0229158	−	0.999737i	$-0.492705\pi$
$13$	27.9269	0.0458316	0.0229158	+	0.999737i	$0.492705\pi$
$14$	628.589	0.857130
$15$	0	0
$16$	−1247.85	−1.21860
$17$	−1587.94	−1.33264	−0.666320	−	0.745666i	$-0.732131\pi$
$17$	−1587.94	−1.33264	−0.666320	+	0.745666i	$0.732131\pi$
$18$	0	0
$19$	−2204.29	−1.40083	−0.700413	−	0.713738i	$-0.747001\pi$
$19$	−2204.29	−1.40083	−0.700413	+	0.713738i	$0.747001\pi$
$20$	−1013.83	−0.566749
$21$	0	0
$22$	−2992.47	−1.31817
$23$	3968.63	1.56430	0.782152	−	0.623087i	$-0.214122\pi$
$23$	3968.63	1.56430	0.782152	+	0.623087i	$0.214122\pi$
$24$	0	0
$25$	−936.208	−0.299587
$26$	−204.593	−0.0593549
$27$	0	0
$28$	−1859.36	−0.448197
$29$	2905.16	0.641468	0.320734	−	0.947169i	$-0.396070\pi$
$29$	2905.16	0.641468	0.320734	+	0.947169i	$0.396070\pi$
$30$	0	0
$31$	4857.43	0.907826	0.453913	−	0.891046i	$-0.350028\pi$
$31$	4857.43	0.907826	0.453913	+	0.891046i	$0.350028\pi$
$32$	6720.11	1.16012
$33$	0	0
$34$	11633.3	1.72586
$35$	4014.23	0.553901
$36$	0	0
$37$	2836.94	0.340679	0.170340	−	0.985385i	$-0.445514\pi$
$37$	2836.94	0.340679	0.170340	+	0.985385i	$0.445514\pi$
$38$	16148.6	1.81416
$39$	0	0
$40$	−3540.46	−0.349872
$41$	−3698.67	−0.343625	−0.171813	−	0.985130i	$-0.554962\pi$
$41$	−3698.67	−0.343625	−0.171813	+	0.985130i	$0.554962\pi$
$42$	0	0
$43$	−8798.36	−0.725656	−0.362828	−	0.931856i	$-0.618189\pi$
$43$	−8798.36	−0.725656	−0.362828	+	0.931856i	$0.618189\pi$
$44$	8851.69	0.689278
$45$	0	0
$46$	−29074.2	−2.02588
$47$	21002.5	1.38684	0.693421	−	0.720533i	$-0.256103\pi$
$47$	21002.5	1.38684	0.693421	+	0.720533i	$0.256103\pi$
$48$	0	0
$49$	−9444.92	−0.561963
$50$	6858.66	0.387984
$51$	0	0
$52$	605.183	0.0310369
$53$	11032.7	0.539499	0.269750	−	0.962930i	$-0.413059\pi$
$53$	11032.7	0.539499	0.269750	+	0.962930i	$0.413059\pi$
$54$	0	0
$55$	−19110.2	−0.851840
$56$	−6493.18	−0.276686
$57$	0	0
$58$	−21283.2	−0.830743
$59$	3481.00	0.130189
$59$	3481.00	0.130189
$60$	0	0
$61$	45308.5	1.55903	0.779516	−	0.626382i	$-0.215465\pi$
$61$	45308.5	1.55903	0.779516	+	0.626382i	$0.215465\pi$
$62$	−35585.5	−1.17569
$63$	0	0
$64$	−9300.32	−0.283823
$65$	−1306.55	−0.0383568
$66$	0	0
$67$	−40826.6	−1.11111	−0.555553	−	0.831481i	$-0.687494\pi$
$67$	−40826.6	−1.11111	−0.555553	+	0.831481i	$0.687494\pi$
$68$	−34411.1	−0.902456
$69$	0	0
$70$	−29408.2	−0.717338
$71$	27052.4	0.636884	0.318442	−	0.947942i	$-0.396840\pi$
$71$	27052.4	0.636884	0.318442	+	0.947942i	$0.396840\pi$
$72$	0	0
$73$	70459.2	1.54750	0.773749	−	0.633492i	$-0.218379\pi$
$73$	70459.2	1.54750	0.773749	+	0.633492i	$0.218379\pi$
$74$	−20783.4	−0.441202
$75$	0	0
$76$	−47767.4	−0.948632
$77$	−35048.0	−0.673653
$78$	0	0
$79$	71673.4	1.29208	0.646042	−	0.763302i	$-0.276423\pi$
$79$	71673.4	1.29208	0.646042	+	0.763302i	$0.276423\pi$
$80$	58380.0	1.01986
$81$	0	0
$82$	27096.4	0.445018
$83$	−93280.1	−1.48626	−0.743128	−	0.669149i	$-0.766659\pi$
$83$	−93280.1	−1.48626	−0.743128	+	0.669149i	$0.766659\pi$
$84$	0	0
$85$	74291.2	1.11530
$86$	64456.8	0.939772
$87$	0	0
$88$	30911.5	0.425513
$89$	116570.	1.55996	0.779978	−	0.625806i	$-0.215230\pi$
$89$	116570.	1.55996	0.779978	+	0.625806i	$0.215230\pi$
$90$	0	0
$91$	−2396.20	−0.0303333
$92$	86001.2	1.05934
$93$	0	0
$94$	−153864.	−1.79605
$95$	103127.	1.17236
$96$	0	0
$97$	91107.3	0.983159	0.491579	−	0.870833i	$-0.336420\pi$
$97$	91107.3	0.983159	0.491579	+	0.870833i	$0.336420\pi$
$98$	69193.5	0.727780
$99$	0	0
$100$	−20287.9	−0.202879
$101$	−121110.	−1.18135	−0.590674	−	0.806910i	$-0.701138\pi$
$101$	−121110.	−1.18135	−0.590674	+	0.806910i	$0.701138\pi$
$102$	0	0
$103$	99956.2	0.928360	0.464180	−	0.885741i	$-0.346349\pi$
$103$	99956.2	0.928360	0.464180	+	0.885741i	$0.346349\pi$
$104$	2113.39	0.0191601
$105$	0	0
$106$	−80825.3	−0.698687
$107$	−191304.	−1.61534	−0.807669	−	0.589636i	$-0.799271\pi$
$107$	−191304.	−1.61534	−0.807669	+	0.589636i	$0.799271\pi$
$108$	0	0
$109$	−142907.	−1.15209	−0.576046	−	0.817417i	$-0.695405\pi$
$109$	−142907.	−1.15209	−0.576046	+	0.817417i	$0.695405\pi$
$110$	140001.	1.10319
$111$	0	0
$112$	107069.	0.806524
$113$	5623.98	0.0414332	0.0207166	−	0.999785i	$-0.493405\pi$
$113$	5623.98	0.0414332	0.0207166	+	0.999785i	$0.493405\pi$
$114$	0	0
$115$	−185671.	−1.30918
$116$	62955.5	0.434399
$117$	0	0
$118$	−25501.8	−0.168603
$119$	136250.	0.881999
$120$	0	0
$121$	5798.53	0.0360043
$122$	−331930.	−2.01905
$123$	0	0
$124$	105262.	0.614775
$125$	190002.	1.08763
$126$	0	0
$127$	−128360.	−0.706190	−0.353095	−	0.935587i	$-0.614871\pi$
$127$	−128360.	−0.706190	−0.353095	+	0.935587i	$0.614871\pi$
$128$	−146909.	−0.792546
$129$	0	0
$130$	9571.77	0.0496745
$131$	−175357.	−0.892783	−0.446391	−	0.894838i	$-0.647291\pi$
$131$	−175357.	−0.892783	−0.446391	+	0.894838i	$0.647291\pi$
$132$	0	0
$133$	189134.	0.927128
$134$	299095.	1.43896
$135$	0	0
$136$	−120169.	−0.557115
$137$	204387.	0.930360	0.465180	−	0.885216i	$-0.345990\pi$
$137$	204387.	0.930360	0.465180	+	0.885216i	$0.345990\pi$
$138$	0	0
$139$	2129.04	0.00934644	0.00467322	−	0.999989i	$-0.498512\pi$
$139$	2129.04	0.00934644	0.00467322	+	0.999989i	$0.498512\pi$
$140$	86989.3	0.375099
$141$	0	0
$142$	−198186.	−0.824807
$143$	11407.4	0.0466493
$144$	0	0
$145$	−135916.	−0.536849
$146$	−516184.	−2.00411
$147$	0	0
$148$	61477.1	0.230706
$149$	1847.03	0.00681567	0.00340784	−	0.999994i	$-0.498915\pi$
$149$	1847.03	0.00681567	0.00340784	+	0.999994i	$0.498915\pi$
$150$	0	0
$151$	−67707.1	−0.241653	−0.120826	−	0.992674i	$-0.538554\pi$
$151$	−67707.1	−0.241653	−0.120826	+	0.992674i	$0.538554\pi$
$152$	−166811.	−0.585621
$153$	0	0
$154$	256761.	0.872424
$155$	−227253.	−0.759766
$156$	0	0
$157$	−208769.	−0.675953	−0.337976	−	0.941155i	$-0.609742\pi$
$157$	−208769.	−0.675953	−0.337976	+	0.941155i	$0.609742\pi$
$158$	−525079.	−1.67333
$159$	0	0
$160$	−314397.	−0.970909
$161$	−340519.	−1.03532
$162$	0	0
$163$	19124.2	0.0563787	0.0281894	−	0.999603i	$-0.491026\pi$
$163$	19124.2	0.0563787	0.0281894	+	0.999603i	$0.491026\pi$
$164$	−80150.9	−0.232701
$165$	0	0
$166$	683370.	1.92480
$167$	313422.	0.869638	0.434819	−	0.900518i	$-0.356812\pi$
$167$	313422.	0.869638	0.434819	+	0.900518i	$0.356812\pi$
$168$	0	0
$169$	−370513.	−0.997899
$170$	−544257.	−1.44438
$171$	0	0
$172$	−190662.	−0.491410
$173$	−620202.	−1.57550	−0.787750	−	0.615996i	$-0.788754\pi$
$173$	−620202.	−1.57550	−0.787750	+	0.615996i	$0.788754\pi$
$174$	0	0
$175$	80329.1	0.198279
$176$	−509711.	−1.24035
$177$	0	0
$178$	−853993.	−2.02025
$179$	402042.	0.937861	0.468931	−	0.883235i	$-0.344639\pi$
$179$	402042.	0.937861	0.468931	+	0.883235i	$0.344639\pi$
$180$	0	0
$181$	326602.	0.741007	0.370503	−	0.928831i	$-0.379185\pi$
$181$	326602.	0.741007	0.370503	+	0.928831i	$0.379185\pi$
$182$	17554.6	0.0392836
$183$	0	0
$184$	300330.	0.653964
$185$	−132725.	−0.285117
$186$	0	0
$187$	−648631.	−1.35642
$188$	455130.	0.939162
$189$	0	0
$190$	−755505.	−1.51828
$191$	−322656.	−0.639964	−0.319982	−	0.947424i	$-0.603677\pi$
$191$	−322656.	−0.639964	−0.319982	+	0.947424i	$0.603677\pi$
$192$	0	0
$193$	−234977.	−0.454080	−0.227040	−	0.973885i	$-0.572905\pi$
$193$	−234977.	−0.454080	−0.227040	+	0.973885i	$0.572905\pi$
$194$	−667451.	−1.27325
$195$	0	0
$196$	−204674.	−0.380559
$197$	658997.	1.20981	0.604906	−	0.796297i	$-0.293211\pi$
$197$	658997.	1.20981	0.604906	+	0.796297i	$0.293211\pi$
$198$	0	0
$199$	−339500.	−0.607725	−0.303863	−	0.952716i	$-0.598276\pi$
$199$	−339500.	−0.607725	−0.303863	+	0.952716i	$0.598276\pi$
$200$	−70848.4	−0.125243
$201$	0	0
$202$	887254.	1.52992
$203$	−249270.	−0.424551
$204$	0	0
$205$	173040.	0.287583
$206$	−732279.	−1.20229
$207$	0	0
$208$	−34848.6	−0.0558505
$209$	−900390.	−1.42582
$210$	0	0
$211$	−862627.	−1.33388	−0.666940	−	0.745111i	$-0.732396\pi$
$211$	−862627.	−1.33388	−0.666940	+	0.745111i	$0.732396\pi$
$212$	239080.	0.365346
$213$	0	0
$214$	1.40149e6	2.09197
$215$	411627.	0.607306
$216$	0	0
$217$	−416780.	−0.600839
$218$	1.04694e6	1.49203
$219$	0	0
$220$	−414122.	−0.576861
$221$	−44346.4	−0.0610770
$222$	0	0
$223$	−1.00055e6	−1.34734	−0.673671	−	0.739032i	$-0.735283\pi$
$223$	−1.00055e6	−1.34734	−0.673671	+	0.739032i	$0.735283\pi$
$224$	−576602.	−0.767815
$225$	0	0
$226$	−41201.3	−0.0536587
$227$	39203.6	0.0504965	0.0252482	−	0.999681i	$-0.491962\pi$
$227$	39203.6	0.0504965	0.0252482	+	0.999681i	$0.491962\pi$
$228$	0	0
$229$	−1.53017e6	−1.92819	−0.964096	−	0.265555i	$-0.914445\pi$
$229$	−1.53017e6	−1.92819	−0.964096	+	0.265555i	$0.914445\pi$
$230$	1.36022e6	1.69547
$231$	0	0
$232$	219850.	0.268168
$233$	3398.03	0.00410050	0.00205025	−	0.999998i	$-0.499347\pi$
$233$	3398.03	0.00410050	0.00205025	+	0.999998i	$0.499347\pi$
$234$	0	0
$235$	−982593.	−1.16066
$236$	75434.1	0.0881632
$237$	0	0
$238$	−998164.	−1.14225
$239$	−514521.	−0.582651	−0.291326	−	0.956624i	$-0.594096\pi$
$239$	−514521.	−0.582651	−0.291326	+	0.956624i	$0.594096\pi$
$240$	0	0
$241$	−1.54065e6	−1.70868	−0.854342	−	0.519711i	$-0.826040\pi$
$241$	−1.54065e6	−1.70868	−0.854342	+	0.519711i	$0.826040\pi$
$242$	−42480.0	−0.0466279
$243$	0	0
$244$	981845.	1.05577
$245$	441876.	0.470311
$246$	0	0
$247$	−61559.0	−0.0642021
$248$	367591.	0.379520
$249$	0	0
$250$	−1.39195e6	−1.40856
$251$	−1.00599e6	−1.00789	−0.503943	−	0.863737i	$-0.668118\pi$
$251$	−1.00599e6	−1.00789	−0.503943	+	0.863737i	$0.668118\pi$
$252$	0	0
$253$	1.62108e6	1.59222
$254$	940368.	0.914563
$255$	0	0
$256$	1.37387e6	1.31022
$257$	821444.	0.775792	0.387896	−	0.921703i	$-0.373202\pi$
$257$	821444.	0.775792	0.387896	+	0.921703i	$0.373202\pi$
$258$	0	0
$259$	−243417.	−0.225476
$260$	−28313.2	−0.0259750
$261$	0	0
$262$	1.28467e6	1.15621
$263$	−336622.	−0.300091	−0.150045	−	0.988679i	$-0.547942\pi$
$263$	−336622.	−0.300091	−0.150045	+	0.988679i	$0.547942\pi$
$264$	0	0
$265$	−516158.	−0.451511
$266$	−1.38559e6	−1.20069
$267$	0	0
$268$	−884721.	−0.752436
$269$	2.12624e6	1.79156	0.895781	−	0.444496i	$-0.146617\pi$
$269$	2.12624e6	1.79156	0.895781	+	0.444496i	$0.146617\pi$
$270$	0	0
$271$	−781161.	−0.646126	−0.323063	−	0.946377i	$-0.604713\pi$
$271$	−781161.	−0.646126	−0.323063	+	0.946377i	$0.604713\pi$
$272$	1.98151e6	1.62396
$273$	0	0
$274$	−1.49734e6	−1.20488
$275$	−382415.	−0.304932
$276$	0	0
$277$	−22946.2	−0.0179685	−0.00898426	−	0.999960i	$-0.502860\pi$
$277$	−22946.2	−0.0179685	−0.00898426	+	0.999960i	$0.502860\pi$
$278$	−15597.3	−0.0121043
$279$	0	0
$280$	303781.	0.231561
$281$	−1.46132e6	−1.10403	−0.552015	−	0.833834i	$-0.686141\pi$
$281$	−1.46132e6	−1.10403	−0.552015	+	0.833834i	$0.686141\pi$
$282$	0	0
$283$	1.25766e6	0.933466	0.466733	−	0.884398i	$-0.345431\pi$
$283$	1.25766e6	0.933466	0.466733	+	0.884398i	$0.345431\pi$
$284$	586233.	0.431295
$285$	0	0
$286$	−83570.4	−0.0604140
$287$	317355.	0.227426
$288$	0	0
$289$	1.10171e6	0.775928
$290$	995724.	0.695255
$291$	0	0
$292$	1.52687e6	1.04796
$293$	−1.57391e6	−1.07105	−0.535525	−	0.844519i	$-0.679886\pi$
$293$	−1.57391e6	−1.07105	−0.535525	+	0.844519i	$0.679886\pi$
$294$	0	0
$295$	−162857.	−0.108956
$296$	214688.	0.142422
$297$	0	0
$298$	−13531.4	−0.00882675
$299$	110832.	0.0716945
$300$	0	0
$301$	754922.	0.480270
$302$	496022.	0.312956
$303$	0	0
$304$	2.75062e6	1.70705
$305$	−2.11974e6	−1.30476
$306$	0	0
$307$	1.48288e6	0.897969	0.448985	−	0.893539i	$-0.351786\pi$
$307$	1.48288e6	0.897969	0.448985	+	0.893539i	$0.351786\pi$
$308$	−759497.	−0.456194
$309$	0	0
$310$	1.66485e6	0.983947
$311$	−762503.	−0.447034	−0.223517	−	0.974700i	$-0.571754\pi$
$311$	−762503.	−0.447034	−0.223517	+	0.974700i	$0.571754\pi$
$312$	0	0
$313$	−1.35503e6	−0.781786	−0.390893	−	0.920436i	$-0.627834\pi$
$313$	−1.35503e6	−0.781786	−0.390893	+	0.920436i	$0.627834\pi$
$314$	1.52944e6	0.875403
$315$	0	0
$316$	1.55318e6	0.874992
$317$	897102.	0.501411	0.250705	−	0.968063i	$-0.419337\pi$
$317$	897102.	0.501411	0.250705	+	0.968063i	$0.419337\pi$
$318$	0	0
$319$	1.18668e6	0.652914
$320$	435111.	0.237534
$321$	0	0
$322$	2.49464e6	1.34081
$323$	3.50028e6	1.86680
$324$	0	0
$325$	−26145.4	−0.0137305
$326$	−140104.	−0.0730141
$327$	0	0
$328$	−279900.	−0.143654
$329$	−1.80207e6	−0.917872
$330$	0	0
$331$	−1.45033e6	−0.727606	−0.363803	−	0.931476i	$-0.618522\pi$
$331$	−1.45033e6	−0.727606	−0.363803	+	0.931476i	$0.618522\pi$
$332$	−2.02140e6	−1.00648
$333$	0	0
$334$	−2.29613e6	−1.12624
$335$	1.91005e6	0.929893
$336$	0	0
$337$	−2.03832e6	−0.977680	−0.488840	−	0.872373i	$-0.662580\pi$
$337$	−2.03832e6	−0.977680	−0.488840	+	0.872373i	$0.662580\pi$
$338$	2.71438e6	1.29235
$339$	0	0
$340$	1.60991e6	0.755272
$341$	1.98413e6	0.924024
$342$	0	0
$343$	2.25248e6	1.03378
$344$	−665824.	−0.303363
$345$	0	0
$346$	4.54360e6	2.04037
$347$	−2.94852e6	−1.31456	−0.657280	−	0.753646i	$-0.728293\pi$
$347$	−2.94852e6	−1.31456	−0.657280	+	0.753646i	$0.728293\pi$
$348$	0	0
$349$	4.25738e6	1.87102	0.935511	−	0.353299i	$-0.114940\pi$
$349$	4.25738e6	1.87102	0.935511	+	0.353299i	$0.114940\pi$
$350$	−588491.	−0.256785
$351$	0	0
$352$	2.74498e6	1.18082
$353$	2.89165e6	1.23512	0.617560	−	0.786524i	$-0.288121\pi$
$353$	2.89165e6	1.23512	0.617560	+	0.786524i	$0.288121\pi$
$354$	0	0
$355$	−1.26564e6	−0.533013
$356$	2.52610e6	1.05639
$357$	0	0
$358$	−2.94536e6	−1.21459
$359$	−566386.	−0.231940	−0.115970	−	0.993253i	$-0.536998\pi$
$359$	−566386.	−0.231940	−0.115970	+	0.993253i	$0.536998\pi$
$360$	0	0
$361$	2.38279e6	0.962315
$362$	−2.39268e6	−0.959652
$363$	0	0
$364$	−51926.2	−0.0205416
$365$	−3.29640e6	−1.29511
$366$	0	0
$367$	314932.	0.122054	0.0610270	−	0.998136i	$-0.480562\pi$
$367$	314932.	0.122054	0.0610270	+	0.998136i	$0.480562\pi$
$368$	−4.95225e6	−1.90626
$369$	0	0
$370$	972341.	0.369245
$371$	−946631.	−0.357064
$372$	0	0
$373$	−1.61192e6	−0.599891	−0.299946	−	0.953956i	$-0.596969\pi$
$373$	−1.61192e6	−0.599891	−0.299946	+	0.953956i	$0.596969\pi$
$374$	4.75187e6	1.75665
$375$	0	0
$376$	1.58938e6	0.579775
$377$	81132.2	0.0293995
$378$	0	0
$379$	−3.46978e6	−1.24081	−0.620404	−	0.784283i	$-0.713031\pi$
$379$	−3.46978e6	−1.24081	−0.620404	+	0.784283i	$0.713031\pi$
$380$	2.23478e6	0.793917
$381$	0	0
$382$	2.36377e6	0.828796
$383$	632278.	0.220248	0.110124	−	0.993918i	$-0.464875\pi$
$383$	632278.	0.220248	0.110124	+	0.993918i	$0.464875\pi$
$384$	0	0
$385$	1.63970e6	0.563784
$386$	1.72144e6	0.588063
$387$	0	0
$388$	1.97432e6	0.665790
$389$	1.99683e6	0.669063	0.334531	−	0.942385i	$-0.391422\pi$
$389$	1.99683e6	0.669063	0.334531	+	0.942385i	$0.391422\pi$
$390$	0	0
$391$	−6.30196e6	−2.08465
$392$	−714753.	−0.234931
$393$	0	0
$394$	−4.82781e6	−1.56679
$395$	−3.35321e6	−1.08135
$396$	0	0
$397$	−3.53468e6	−1.12557	−0.562787	−	0.826602i	$-0.690271\pi$
$397$	−3.53468e6	−1.12557	−0.562787	+	0.826602i	$0.690271\pi$
$398$	2.48718e6	0.787044
$399$	0	0
$400$	1.16825e6	0.365077
$401$	−3.76927e6	−1.17057	−0.585283	−	0.810829i	$-0.699017\pi$
$401$	−3.76927e6	−1.17057	−0.585283	+	0.810829i	$0.699017\pi$
$402$	0	0
$403$	135653.	0.0416071
$404$	−2.62449e6	−0.800002
$405$	0	0
$406$	1.82615e6	0.549822
$407$	1.15881e6	0.346758
$408$	0	0
$409$	−108345.	−0.0320259	−0.0160130	−	0.999872i	$-0.505097\pi$
$409$	−108345.	−0.0320259	−0.0160130	+	0.999872i	$0.505097\pi$
$410$	−1.26769e6	−0.372438
$411$	0	0
$412$	2.16607e6	0.628681
$413$	−298679.	−0.0861647
$414$	0	0
$415$	4.36407e6	1.24386
$416$	187672.	0.0531699
$417$	0	0
$418$	6.59626e6	1.84653
$419$	4.59866e6	1.27966	0.639832	−	0.768515i	$-0.279004\pi$
$419$	4.59866e6	1.27966	0.639832	+	0.768515i	$0.279004\pi$
$420$	0	0
$421$	5.27760e6	1.45121	0.725606	−	0.688110i	$-0.241560\pi$
$421$	5.27760e6	1.45121	0.725606	+	0.688110i	$0.241560\pi$
$422$	6.31960e6	1.72746
$423$	0	0
$424$	834907.	0.225540
$425$	1.48665e6	0.399241
$426$	0	0
$427$	−3.88758e6	−1.03183
$428$	−4.14559e6	−1.09390
$429$	0	0
$430$	−3.01558e6	−0.786502
$431$	3.25854e6	0.844948	0.422474	−	0.906375i	$-0.361162\pi$
$431$	3.25854e6	0.844948	0.422474	+	0.906375i	$0.361162\pi$
$432$	0	0
$433$	2.53904e6	0.650803	0.325402	−	0.945576i	$-0.394501\pi$
$433$	2.53904e6	0.650803	0.325402	+	0.945576i	$0.394501\pi$
$434$	3.05333e6	0.778125
$435$	0	0
$436$	−3.09683e6	−0.780190
$437$	−8.74801e6	−2.19132
$438$	0	0
$439$	5.11072e6	1.26567	0.632835	−	0.774286i	$-0.281891\pi$
$439$	5.11072e6	1.26567	0.632835	+	0.774286i	$0.281891\pi$
$440$	−1.44618e6	−0.356115
$441$	0	0
$442$	324881.	0.0790987
$443$	2.08543e6	0.504877	0.252438	−	0.967613i	$-0.418767\pi$
$443$	2.08543e6	0.504877	0.252438	+	0.967613i	$0.418767\pi$
$444$	0	0
$445$	−5.45368e6	−1.30554
$446$	7.33004e6	1.74489
$447$	0	0
$448$	797992.	0.187847
$449$	2.92355e6	0.684376	0.342188	−	0.939632i	$-0.388832\pi$
$449$	2.92355e6	0.684376	0.342188	+	0.939632i	$0.388832\pi$
$450$	0	0
$451$	−1.51080e6	−0.349757
$452$	121873.	0.0280583
$453$	0	0
$454$	−287205.	−0.0653962
$455$	112105.	0.0253862
$456$	0	0
$457$	−8.29298e6	−1.85746	−0.928731	−	0.370754i	$-0.879099\pi$
$457$	−8.29298e6	−1.85746	−0.928731	+	0.370754i	$0.879099\pi$
$458$	1.12100e7	2.49713
$459$	0	0
$460$	−4.02352e6	−0.886568
$461$	2.53751e6	0.556102	0.278051	−	0.960566i	$-0.410312\pi$
$461$	2.53751e6	0.556102	0.278051	+	0.960566i	$0.410312\pi$
$462$	0	0
$463$	3.55974e6	0.771731	0.385865	−	0.922555i	$-0.373903\pi$
$463$	3.55974e6	0.771731	0.385865	+	0.922555i	$0.373903\pi$
$464$	−3.62520e6	−0.781694
$465$	0	0
$466$	−24893.9	−0.00531042
$467$	−6.73742e6	−1.42956	−0.714779	−	0.699351i	$-0.753473\pi$
$467$	−6.73742e6	−1.42956	−0.714779	+	0.699351i	$0.753473\pi$
$468$	0	0
$469$	3.50302e6	0.735378
$470$	7.19848e6	1.50313
$471$	0	0
$472$	263428.	0.0544260
$473$	−3.59389e6	−0.738604
$474$	0	0
$475$	2.06367e6	0.419669
$476$	2.95256e6	0.597285
$477$	0	0
$478$	3.76938e6	0.754571
$479$	−7.29265e6	−1.45227	−0.726134	−	0.687554i	$-0.758685\pi$
$479$	−7.29265e6	−1.45227	−0.726134	+	0.687554i	$0.758685\pi$
$480$	0	0
$481$	79227.0	0.0156139
$482$	1.12868e7	2.21286
$483$	0	0
$484$	125655.	0.0243819
$485$	−4.26241e6	−0.822812
$486$	0	0
$487$	1.72557e6	0.329694	0.164847	−	0.986319i	$-0.447287\pi$
$487$	1.72557e6	0.329694	0.164847	+	0.986319i	$0.447287\pi$
$488$	3.42876e6	0.651760
$489$	0	0
$490$	−3.23718e6	−0.609084
$491$	−8.06758e6	−1.51022	−0.755109	−	0.655599i	$-0.772416\pi$
$491$	−8.06758e6	−1.51022	−0.755109	+	0.655599i	$0.772416\pi$
$492$	0	0
$493$	−4.61323e6	−0.854846
$494$	450981.	0.0831459
$495$	0	0
$496$	−6.06134e6	−1.10628
$497$	−2.32117e6	−0.421518
$498$	0	0
$499$	−7.26673e6	−1.30644	−0.653218	−	0.757170i	$-0.726581\pi$
$499$	−7.26673e6	−1.30644	−0.653218	+	0.757170i	$0.726581\pi$
$500$	4.11738e6	0.736539
$501$	0	0
$502$	7.36991e6	1.30528
$503$	6.90011e6	1.21601	0.608004	−	0.793934i	$-0.291971\pi$
$503$	6.90011e6	1.21601	0.608004	+	0.793934i	$0.291971\pi$
$504$	0	0
$505$	5.66609e6	0.988678
$506$	−1.18760e7	−2.06202
$507$	0	0
$508$	−2.78160e6	−0.478228
$509$	−2.10129e6	−0.359495	−0.179747	−	0.983713i	$-0.557528\pi$
$509$	−2.10129e6	−0.359495	−0.179747	+	0.983713i	$0.557528\pi$
$510$	0	0
$511$	−6.04558e6	−1.02420
$512$	−5.36385e6	−0.904277
$513$	0	0
$514$	−6.01790e6	−1.00470
$515$	−4.67640e6	−0.776951
$516$	0	0
$517$	8.57895e6	1.41159
$518$	1.78327e6	0.292006
$519$	0	0
$520$	−98874.2	−0.0160352
$521$	−5.71866e6	−0.922996	−0.461498	−	0.887141i	$-0.652688\pi$
$521$	−5.71866e6	−0.922996	−0.461498	+	0.887141i	$0.652688\pi$
$522$	0	0
$523$	−548224.	−0.0876403	−0.0438202	−	0.999039i	$-0.513953\pi$
$523$	−548224.	−0.0876403	−0.0438202	+	0.999039i	$0.513953\pi$
$524$	−3.80003e6	−0.604588
$525$	0	0
$526$	2.46609e6	0.388637
$527$	−7.71333e6	−1.20980
$528$	0	0
$529$	9.31370e6	1.44705
$530$	3.78137e6	0.584736
$531$	0	0
$532$	4.09857e6	0.627846
$533$	−103292.	−0.0157489
$534$	0	0
$535$	8.95005e6	1.35189
$536$	−3.08959e6	−0.464503
$537$	0	0
$538$	−1.55768e7	−2.32019
$539$	−3.85799e6	−0.571991
$540$	0	0
$541$	−1.44297e6	−0.211965	−0.105982	−	0.994368i	$-0.533799\pi$
$541$	−1.44297e6	−0.211965	−0.105982	+	0.994368i	$0.533799\pi$
$542$	5.72278e6	0.836775
$543$	0	0
$544$	−1.06711e7	−1.54602
$545$	6.68583e6	0.964193
$546$	0	0
$547$	5.77418e6	0.825130	0.412565	−	0.910928i	$-0.364633\pi$
$547$	5.77418e6	0.825130	0.412565	+	0.910928i	$0.364633\pi$
$548$	4.42910e6	0.630035
$549$	0	0
$550$	2.80157e6	0.394907
$551$	−6.40381e6	−0.898585
$552$	0	0
$553$	−6.14976e6	−0.855156
$554$	168104.	0.0232704
$555$	0	0
$556$	46136.7	0.00632936
$557$	3.90827e6	0.533761	0.266880	−	0.963730i	$-0.414007\pi$
$557$	3.90827e6	0.533761	0.266880	+	0.963730i	$0.414007\pi$
$558$	0	0
$559$	−245711.	−0.0332579
$560$	−5.00915e6	−0.674985
$561$	0	0
$562$	1.07057e7	1.42979
$563$	8.26195e6	1.09853	0.549264	−	0.835649i	$-0.314908\pi$
$563$	8.26195e6	1.09853	0.549264	+	0.835649i	$0.314908\pi$
$564$	0	0
$565$	−263115.	−0.0346757
$566$	−9.21364e6	−1.20890
$567$	0	0
$568$	2.04722e6	0.266252
$569$	−9.52546e6	−1.23340	−0.616702	−	0.787197i	$-0.711532\pi$
$569$	−9.52546e6	−1.23340	−0.616702	+	0.787197i	$0.711532\pi$
$570$	0	0
$571$	−5.73979e6	−0.736725	−0.368363	−	0.929682i	$-0.620081\pi$
$571$	−5.73979e6	−0.736725	−0.368363	+	0.929682i	$0.620081\pi$
$572$	247200.	0.0315907
$573$	0	0
$574$	−2.32494e6	−0.294532
$575$	−3.71547e6	−0.468645
$576$	0	0
$577$	7.36689e6	0.921180	0.460590	−	0.887613i	$-0.347638\pi$
$577$	7.36689e6	0.921180	0.460590	+	0.887613i	$0.347638\pi$
$578$	−8.07111e6	−1.00488
$579$	0	0
$580$	−2.94534e6	−0.363551
$581$	8.00367e6	0.983669
$582$	0	0
$583$	4.50654e6	0.549126
$584$	5.33206e6	0.646938
$585$	0	0
$586$	1.15304e7	1.38708
$587$	1.28494e7	1.53918	0.769590	−	0.638539i	$-0.220461\pi$
$587$	1.28494e7	1.53918	0.769590	+	0.638539i	$0.220461\pi$
$588$	0	0
$589$	−1.07072e7	−1.27171
$590$	1.19309e6	0.141105
$591$	0	0
$592$	−3.54007e6	−0.415152
$593$	1.31628e7	1.53714	0.768568	−	0.639768i	$-0.220969\pi$
$593$	1.31628e7	1.53714	0.768568	+	0.639768i	$0.220969\pi$
$594$	0	0
$595$	−6.37437e6	−0.738151
$596$	40025.6	0.00461554
$597$	0	0
$598$	−811953.	−0.0928491
$599$	9.54195e6	1.08660	0.543300	−	0.839538i	$-0.317175\pi$
$599$	9.54195e6	1.08660	0.543300	+	0.839538i	$0.317175\pi$
$600$	0	0
$601$	−6.64300e6	−0.750201	−0.375101	−	0.926984i	$-0.622392\pi$
$601$	−6.64300e6	−0.750201	−0.375101	+	0.926984i	$0.622392\pi$
$602$	−5.53056e6	−0.621982
$603$	0	0
$604$	−1.46723e6	−0.163646
$605$	−271281.	−0.0301323
$606$	0	0
$607$	−1.26387e7	−1.39229	−0.696147	−	0.717899i	$-0.745104\pi$
$607$	−1.26387e7	−1.39229	−0.696147	+	0.717899i	$0.745104\pi$
$608$	−1.48130e7	−1.62512
$609$	0	0
$610$	1.55292e7	1.68976
$611$	586536.	0.0635611
$612$	0	0
$613$	−1.20014e7	−1.28997	−0.644985	−	0.764195i	$-0.723136\pi$
$613$	−1.20014e7	−1.28997	−0.644985	+	0.764195i	$0.723136\pi$
$614$	−1.08636e7	−1.16293
$615$	0	0
$616$	−2.65229e6	−0.281623
$617$	−8.14931e6	−0.861803	−0.430901	−	0.902399i	$-0.641804\pi$
$617$	−8.14931e6	−0.861803	−0.430901	+	0.902399i	$0.641804\pi$
$618$	0	0
$619$	1.19794e6	0.125663	0.0628314	−	0.998024i	$-0.479987\pi$
$619$	1.19794e6	0.125663	0.0628314	+	0.998024i	$0.479987\pi$
$620$	−4.92462e6	−0.514509
$621$	0	0
$622$	5.58609e6	0.578938
$623$	−1.00020e7	−1.03245
$624$	0	0
$625$	−5.96349e6	−0.610661
$626$	9.92694e6	1.01246
$627$	0	0
$628$	−4.52407e6	−0.457752
$629$	−4.50490e6	−0.454003
$630$	0	0
$631$	2.94186e6	0.294136	0.147068	−	0.989126i	$-0.453016\pi$
$631$	2.94186e6	0.294136	0.147068	+	0.989126i	$0.453016\pi$
$632$	5.42395e6	0.540161
$633$	0	0
$634$	−6.57217e6	−0.649360
$635$	6.00528e6	0.591016
$636$	0	0
$637$	−263768.	−0.0257557
$638$	−8.69359e6	−0.845566
$639$	0	0
$640$	6.87308e6	0.663287
$641$	−1.94544e7	−1.87014	−0.935068	−	0.354469i	$-0.884661\pi$
$641$	−1.94544e7	−1.87014	−0.935068	+	0.354469i	$0.884661\pi$
$642$	0	0
$643$	−1.76373e7	−1.68231	−0.841153	−	0.540797i	$-0.818123\pi$
$643$	−1.76373e7	−1.68231	−0.841153	+	0.540797i	$0.818123\pi$
$644$	−7.37912e6	−0.701116
$645$	0	0
$646$	−2.56431e7	−2.41762
$647$	−1.72256e7	−1.61776	−0.808878	−	0.587976i	$-0.799925\pi$
$647$	−1.72256e7	−1.61776	−0.808878	+	0.587976i	$0.799925\pi$
$648$	0	0
$649$	1.42189e6	0.132512
$650$	191541.	0.0177819
$651$	0	0
$652$	414427.	0.0381794
$653$	4.60913e6	0.422996	0.211498	−	0.977378i	$-0.432166\pi$
$653$	4.60913e6	0.422996	0.211498	+	0.977378i	$0.432166\pi$
$654$	0	0
$655$	8.20401e6	0.747176
$656$	4.61537e6	0.418743
$657$	0	0
$658$	1.32020e7	1.18870
$659$	−858283.	−0.0769869	−0.0384935	−	0.999259i	$-0.512256\pi$
$659$	−858283.	−0.0769869	−0.0384935	+	0.999259i	$0.512256\pi$
$660$	0	0
$661$	1.26702e7	1.12793	0.563964	−	0.825799i	$-0.309276\pi$
$661$	1.26702e7	1.12793	0.563964	+	0.825799i	$0.309276\pi$
$662$	1.06251e7	0.942297
$663$	0	0
$664$	−7.05905e6	−0.621336
$665$	−8.84852e6	−0.775920
$666$	0	0
$667$	1.15295e7	1.00345
$668$	6.79193e6	0.588914
$669$	0	0
$670$	−1.39930e7	−1.20427
$671$	1.85073e7	1.58685
$672$	0	0
$673$	−2.56765e6	−0.218524	−0.109262	−	0.994013i	$-0.534849\pi$
$673$	−2.56765e6	−0.218524	−0.109262	+	0.994013i	$0.534849\pi$
$674$	1.49327e7	1.26616
$675$	0	0
$676$	−8.02910e6	−0.675772
$677$	−1.74857e7	−1.46626	−0.733131	−	0.680087i	$-0.761942\pi$
$677$	−1.74857e7	−1.46626	−0.733131	+	0.680087i	$0.761942\pi$
$678$	0	0
$679$	−7.81724e6	−0.650697
$680$	5.62205e6	0.466254
$681$	0	0
$682$	−1.45357e7	−1.19667
$683$	−6.06090e6	−0.497147	−0.248574	−	0.968613i	$-0.579962\pi$
$683$	−6.06090e6	−0.497147	−0.248574	+	0.968613i	$0.579962\pi$
$684$	0	0
$685$	−9.56213e6	−0.778625
$686$	−1.65017e7	−1.33881
$687$	0	0
$688$	1.09790e7	0.884286
$689$	308109.	0.0247261
$690$	0	0
$691$	−7.50002e6	−0.597541	−0.298770	−	0.954325i	$-0.596576\pi$
$691$	−7.50002e6	−0.597541	−0.298770	+	0.954325i	$0.596576\pi$
$692$	−1.34399e7	−1.06692
$693$	0	0
$694$	2.16009e7	1.70244
$695$	−99606.0	−0.00782210
$696$	0	0
$697$	5.87327e6	0.457929
$698$	−3.11895e7	−2.42310
$699$	0	0
$700$	1.74075e6	0.134274
$701$	1.97391e7	1.51717	0.758583	−	0.651576i	$-0.225892\pi$
$701$	1.97391e7	1.51717	0.758583	+	0.651576i	$0.225892\pi$
$702$	0	0
$703$	−6.25343e6	−0.477232
$704$	−3.79892e6	−0.288888
$705$	0	0
$706$	−2.11842e7	−1.59956
$707$	1.03916e7	0.781867
$708$	0	0
$709$	−2.68377e6	−0.200507	−0.100254	−	0.994962i	$-0.531965\pi$
$709$	−2.68377e6	−0.200507	−0.100254	+	0.994962i	$0.531965\pi$
$710$	9.27204e6	0.690287
$711$	0	0
$712$	8.82155e6	0.652146
$713$	1.92774e7	1.42012
$714$	0	0
$715$	−533689.	−0.0390412
$716$	8.71234e6	0.635115
$717$	0	0
$718$	4.14934e6	0.300378
$719$	1.26395e7	0.911816	0.455908	−	0.890027i	$-0.349315\pi$
$719$	1.26395e7	0.911816	0.455908	+	0.890027i	$0.349315\pi$
$720$	0	0
$721$	−8.57650e6	−0.614429
$722$	−1.74563e7	−1.24626
$723$	0	0
$724$	7.07753e6	0.501806
$725$	−2.71983e6	−0.192175
$726$	0	0
$727$	1.30124e7	0.913105	0.456552	−	0.889697i	$-0.349084\pi$
$727$	1.30124e7	0.913105	0.456552	+	0.889697i	$0.349084\pi$
$728$	−181335.	−0.0126810
$729$	0	0
$730$	2.41494e7	1.67726
$731$	1.39713e7	0.967038
$732$	0	0
$733$	−5.49665e6	−0.377866	−0.188933	−	0.981990i	$-0.560503\pi$
$733$	−5.49665e6	−0.377866	−0.188933	+	0.981990i	$0.560503\pi$
$734$	−2.30719e6	−0.158068
$735$	0	0
$736$	2.66696e7	1.81477
$737$	−1.66765e7	−1.13093
$738$	0	0
$739$	−2.85074e6	−0.192020	−0.0960101	−	0.995380i	$-0.530608\pi$
$739$	−2.85074e6	−0.192020	−0.0960101	+	0.995380i	$0.530608\pi$
$740$	−2.87618e6	−0.193080
$741$	0	0
$742$	6.93502e6	0.462421
$743$	8.85765e6	0.588635	0.294318	−	0.955708i	$-0.404908\pi$
$743$	8.85765e6	0.588635	0.294318	+	0.955708i	$0.404908\pi$
$744$	0	0
$745$	−86412.5	−0.00570409
$746$	1.18090e7	0.776899
$747$	0	0
$748$	−1.40560e7	−0.918559
$749$	1.64143e7	1.06910
$750$	0	0
$751$	−1.95283e7	−1.26347	−0.631733	−	0.775186i	$-0.717656\pi$
$751$	−1.95283e7	−1.26347	−0.631733	+	0.775186i	$0.717656\pi$
$752$	−2.62080e7	−1.69001
$753$	0	0
$754$	−594374.	−0.0380743
$755$	3.16764e6	0.202241
$756$	0	0
$757$	2.35067e7	1.49091	0.745457	−	0.666554i	$-0.232231\pi$
$757$	2.35067e7	1.49091	0.745457	+	0.666554i	$0.232231\pi$
$758$	2.54196e7	1.60693
$759$	0	0
$760$	7.80419e6	0.490111
$761$	−1.76666e7	−1.10584	−0.552920	−	0.833234i	$-0.686487\pi$
$761$	−1.76666e7	−1.10584	−0.552920	+	0.833234i	$0.686487\pi$
$762$	0	0
$763$	1.22618e7	0.762504
$764$	−6.99202e6	−0.433380
$765$	0	0
$766$	−4.63207e6	−0.285235
$767$	97213.6	0.00596676
$768$	0	0
$769$	−6.78452e6	−0.413717	−0.206858	−	0.978371i	$-0.566324\pi$
$769$	−6.78452e6	−0.413717	−0.206858	+	0.978371i	$0.566324\pi$
$770$	−1.20125e7	−0.730138
$771$	0	0
$772$	−5.09201e6	−0.307501
$773$	1.32057e7	0.794901	0.397451	−	0.917624i	$-0.369895\pi$
$773$	1.32057e7	0.794901	0.397451	+	0.917624i	$0.369895\pi$
$774$	0	0
$775$	−4.54757e6	−0.271973
$776$	6.89462e6	0.411014
$777$	0	0
$778$	−1.46288e7	−0.866480
$779$	8.15292e6	0.481360
$780$	0	0
$781$	1.10502e7	0.648248
$782$	4.61682e7	2.69976
$783$	0	0
$784$	1.17858e7	0.684810
$785$	9.76714e6	0.565710
$786$	0	0
$787$	−5.65588e6	−0.325509	−0.162755	−	0.986667i	$-0.552038\pi$
$787$	−5.65588e6	−0.325509	−0.162755	+	0.986667i	$0.552038\pi$
$788$	1.42806e7	0.819279
$789$	0	0
$790$	2.45656e7	1.40042
$791$	−482552.	−0.0274223
$792$	0	0
$793$	1.26533e6	0.0714529
$794$	2.58951e7	1.45769
$795$	0	0
$796$	−7.35705e6	−0.411548
$797$	−2.41632e6	−0.134744	−0.0673718	−	0.997728i	$-0.521461\pi$
$797$	−2.41632e6	−0.134744	−0.0673718	+	0.997728i	$0.521461\pi$
$798$	0	0
$799$	−3.33508e7	−1.84816
$800$	−6.29142e6	−0.347555
$801$	0	0
$802$	2.76136e7	1.51596
$803$	2.87806e7	1.57511
$804$	0	0
$805$	1.59310e7	0.866470
$806$	−993795.	−0.0538839
$807$	0	0
$808$	−9.16513e6	−0.493867
$809$	−1.19990e7	−0.644576	−0.322288	−	0.946642i	$-0.604452\pi$
$809$	−1.19990e7	−0.644576	−0.322288	+	0.946642i	$0.604452\pi$
$810$	0	0
$811$	−1.49699e7	−0.799220	−0.399610	−	0.916685i	$-0.630854\pi$
$811$	−1.49699e7	−0.799220	−0.399610	+	0.916685i	$0.630854\pi$
$812$	−5.40174e6	−0.287504
$813$	0	0
$814$	−8.48944e6	−0.449074
$815$	−894719.	−0.0471837
$816$	0	0
$817$	1.93941e7	1.01652
$818$	793738.	0.0414757
$819$	0	0
$820$	3.74982e6	0.194749
$821$	−2.52512e7	−1.30745	−0.653724	−	0.756733i	$-0.726794\pi$
$821$	−2.52512e7	−1.30745	−0.653724	+	0.756733i	$0.726794\pi$
$822$	0	0
$823$	1.71441e6	0.0882299	0.0441150	−	0.999026i	$-0.485953\pi$
$823$	1.71441e6	0.0882299	0.0441150	+	0.999026i	$0.485953\pi$
$824$	7.56427e6	0.388105
$825$	0	0
$826$	2.18812e6	0.111589
$827$	1.24001e7	0.630464	0.315232	−	0.949015i	$-0.397918\pi$
$827$	1.24001e7	0.630464	0.315232	+	0.949015i	$0.397918\pi$
$828$	0	0
$829$	3.26664e7	1.65088	0.825439	−	0.564491i	$-0.190928\pi$
$829$	3.26664e7	1.65088	0.825439	+	0.564491i	$0.190928\pi$
$830$	−3.19711e7	−1.61088
$831$	0	0
$832$	−259729.	−0.0130081
$833$	1.49980e7	0.748895
$834$	0	0
$835$	−1.46633e7	−0.727806
$836$	−1.95117e7	−0.965559
$837$	0	0
$838$	−3.36897e7	−1.65725
$839$	−3.43057e7	−1.68252	−0.841261	−	0.540629i	$-0.818186\pi$
$839$	−3.43057e7	−1.68252	−0.841261	+	0.540629i	$0.818186\pi$
$840$	0	0
$841$	−1.20712e7	−0.588519
$842$	−3.86636e7	−1.87941
$843$	0	0
$844$	−1.86933e7	−0.903297
$845$	1.73343e7	0.835149
$846$	0	0
$847$	−497529.	−0.0238292
$848$	−1.37671e7	−0.657435
$849$	0	0
$850$	−1.08912e7	−0.517043
$851$	1.12588e7	0.532926
$852$	0	0
$853$	1.96117e6	0.0922874	0.0461437	−	0.998935i	$-0.485307\pi$
$853$	1.96117e6	0.0922874	0.0461437	+	0.998935i	$0.485307\pi$
$854$	2.84804e7	1.33629
$855$	0	0
$856$	−1.44771e7	−0.675299
$857$	1.73109e7	0.805134	0.402567	−	0.915391i	$-0.368118\pi$
$857$	1.73109e7	0.805134	0.402567	+	0.915391i	$0.368118\pi$
$858$	0	0
$859$	1.08498e7	0.501692	0.250846	−	0.968027i	$-0.419291\pi$
$859$	1.08498e7	0.501692	0.250846	+	0.968027i	$0.419291\pi$
$860$	8.92005e6	0.411265
$861$	0	0
$862$	−2.38721e7	−1.09426
$863$	3.39433e6	0.155141	0.0775705	−	0.996987i	$-0.475284\pi$
$863$	3.39433e6	0.155141	0.0775705	+	0.996987i	$0.475284\pi$
$864$	0	0
$865$	2.90159e7	1.31855
$866$	−1.86010e7	−0.842833
$867$	0	0
$868$	−9.03172e6	−0.406885
$869$	2.92766e7	1.31514
$870$	0	0
$871$	−1.14016e6	−0.0509238
$872$	−1.08146e7	−0.481637
$873$	0	0
$874$	6.40879e7	2.83790
$875$	−1.63026e7	−0.719843
$876$	0	0
$877$	−2.12843e7	−0.934460	−0.467230	−	0.884136i	$-0.654748\pi$
$877$	−2.12843e7	−0.934460	−0.467230	+	0.884136i	$0.654748\pi$
$878$	−3.74411e7	−1.63913
$879$	0	0
$880$	2.38466e7	1.03805
$881$	4.15740e7	1.80461	0.902303	−	0.431103i	$-0.141876\pi$
$881$	4.15740e7	1.80461	0.902303	+	0.431103i	$0.141876\pi$
$882$	0	0
$883$	−3.74730e6	−0.161740	−0.0808699	−	0.996725i	$-0.525770\pi$
$883$	−3.74730e6	−0.161740	−0.0808699	+	0.996725i	$0.525770\pi$
$884$	−960996.	−0.0413610
$885$	0	0
$886$	−1.52778e7	−0.653849
$887$	6.86645e6	0.293038	0.146519	−	0.989208i	$-0.453193\pi$
$887$	6.86645e6	0.293038	0.146519	+	0.989208i	$0.453193\pi$
$888$	0	0
$889$	1.10137e7	0.467387
$890$	3.99537e7	1.69076
$891$	0	0
$892$	−2.16822e7	−0.912412
$893$	−4.62956e7	−1.94272
$894$	0	0
$895$	−1.88093e7	−0.784903
$896$	1.26052e7	0.524541
$897$	0	0
$898$	−2.14179e7	−0.886312
$899$	1.41116e7	0.582341
$900$	0	0
$901$	−1.75193e7	−0.718958
$902$	1.10681e7	0.452958
$903$	0	0
$904$	425600.	0.0173213
$905$	−1.52799e7	−0.620154
$906$	0	0
$907$	−4.68280e7	−1.89011	−0.945056	−	0.326910i	$-0.893993\pi$
$907$	−4.68280e7	−1.89011	−0.945056	+	0.326910i	$0.893993\pi$
$908$	849550.	0.0341959
$909$	0	0
$910$	−821282.	−0.0328767
$911$	−7.20062e6	−0.287458	−0.143729	−	0.989617i	$-0.545909\pi$
$911$	−7.20062e6	−0.287458	−0.143729	+	0.989617i	$0.545909\pi$
$912$	0	0
$913$	−3.81023e7	−1.51278
$914$	6.07543e7	2.40554
$915$	0	0
$916$	−3.31591e7	−1.30576
$917$	1.50461e7	0.590882
$918$	0	0
$919$	−3.30525e7	−1.29097	−0.645485	−	0.763773i	$-0.723345\pi$
$919$	−3.30525e7	−1.29097	−0.645485	+	0.763773i	$0.723345\pi$
$920$	−1.40508e7	−0.547307
$921$	0	0
$922$	−1.85898e7	−0.720189
$923$	755492.	0.0291894
$924$	0	0
$925$	−2.65597e6	−0.102063
$926$	−2.60786e7	−0.999442
$927$	0	0
$928$	1.95230e7	0.744177
$929$	−2.05879e7	−0.782658	−0.391329	−	0.920251i	$-0.627985\pi$
$929$	−2.05879e7	−0.782658	−0.391329	+	0.920251i	$0.627985\pi$
$930$	0	0
$931$	2.08193e7	0.787213
$932$	73636.0	0.00277684
$933$	0	0
$934$	4.93583e7	1.85137
$935$	3.03459e7	1.13520
$936$	0	0
$937$	−2.98879e7	−1.11211	−0.556054	−	0.831146i	$-0.687685\pi$
$937$	−2.98879e7	−1.11211	−0.556054	+	0.831146i	$0.687685\pi$
$938$	−2.56631e7	−0.952363
$939$	0	0
$940$	−2.12930e7	−0.785991
$941$	−3.84112e7	−1.41411	−0.707056	−	0.707158i	$-0.749977\pi$
$941$	−3.84112e7	−1.41411	−0.707056	+	0.707158i	$0.749977\pi$
$942$	0	0
$943$	−1.46786e7	−0.537535
$944$	−4.34376e6	−0.158648
$945$	0	0
$946$	2.63288e7	0.956540
$947$	827143.	0.0299713	0.0149857	−	0.999888i	$-0.495230\pi$
$947$	827143.	0.0299713	0.0149857	+	0.999888i	$0.495230\pi$
$948$	0	0
$949$	1.96771e6	0.0709243
$950$	−1.51185e7	−0.543499
$951$	0	0
$952$	1.03108e7	0.368723
$953$	1.38653e7	0.494535	0.247267	−	0.968947i	$-0.420467\pi$
$953$	1.38653e7	0.494535	0.247267	+	0.968947i	$0.420467\pi$
$954$	0	0
$955$	1.50953e7	0.535591
$956$	−1.11498e7	−0.394568
$957$	0	0
$958$	5.34259e7	1.88078
$959$	−1.75369e7	−0.615753
$960$	0	0
$961$	−5.03449e6	−0.175852
$962$	−580417.	−0.0202210
$963$	0	0
$964$	−3.33863e7	−1.15711
$965$	1.09933e7	0.380023
$966$	0	0
$967$	−1.09394e7	−0.376208	−0.188104	−	0.982149i	$-0.560234\pi$
$967$	−1.09394e7	−0.376208	−0.188104	+	0.982149i	$0.560234\pi$
$968$	438809.	0.0150517
$969$	0	0
$970$	3.12264e7	1.06560
$971$	−4.27950e7	−1.45661	−0.728307	−	0.685251i	$-0.759693\pi$
$971$	−4.27950e7	−1.45661	−0.728307	+	0.685251i	$0.759693\pi$
$972$	0	0
$973$	−182677.	−0.00618588
$974$	−1.26415e7	−0.426975
$975$	0	0
$976$	−5.65381e7	−1.89984
$977$	−1.09971e7	−0.368590	−0.184295	−	0.982871i	$-0.559000\pi$
$977$	−1.09971e7	−0.368590	−0.184295	+	0.982871i	$0.559000\pi$
$978$	0	0
$979$	4.76157e7	1.58779
$980$	9.57556e6	0.318492
$981$	0	0
$982$	5.91031e7	1.95583
$983$	−2.08290e7	−0.687518	−0.343759	−	0.939058i	$-0.611700\pi$
$983$	−2.08290e7	−0.687518	−0.343759	+	0.939058i	$0.611700\pi$
$984$	0	0
$985$	−3.08309e7	−1.01250
$986$	3.37965e7	1.10708
$987$	0	0
$988$	−1.33400e6	−0.0434773
$989$	−3.49175e7	−1.13515
$990$	0	0
$991$	1.73858e7	0.562355	0.281177	−	0.959656i	$-0.409275\pi$
$991$	1.73858e7	0.562355	0.281177	+	0.959656i	$0.409275\pi$
$992$	3.26425e7	1.05318
$993$	0	0
$994$	1.70049e7	0.545893
$995$	1.58834e7	0.508610
$996$	0	0
$997$	−1.37910e7	−0.439397	−0.219699	−	0.975568i	$-0.570507\pi$
$997$	−1.37910e7	−0.439397	−0.219699	+	0.975568i	$0.570507\pi$
$998$	5.32361e7	1.69192
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.6.a.d.1.3		12
3.2	odd	2		177.6.a.b.1.10	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.a.b.1.10	✓	12	3.2	odd	2
531.6.a.d.1.3		12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q-7.32600 q^{2} +21.6702 q^{4} -46.7845 q^{5} -85.8026 q^{7} +75.6759 q^{8} +O(q^{10})\)	\(q-7.32600 q^{2} +21.6702 q^{4} -46.7845 q^{5} -85.8026 q^{7} +75.6759 q^{8} +342.743 q^{10} +408.472 q^{11} +27.9269 q^{13} +628.589 q^{14} -1247.85 q^{16} -1587.94 q^{17} -2204.29 q^{19} -1013.83 q^{20} -2992.47 q^{22} +3968.63 q^{23} -936.208 q^{25} -204.593 q^{26} -1859.36 q^{28} +2905.16 q^{29} +4857.43 q^{31} +6720.11 q^{32} +11633.3 q^{34} +4014.23 q^{35} +2836.94 q^{37} +16148.6 q^{38} -3540.46 q^{40} -3698.67 q^{41} -8798.36 q^{43} +8851.69 q^{44} -29074.2 q^{46} +21002.5 q^{47} -9444.92 q^{49} +6858.66 q^{50} +605.183 q^{52} +11032.7 q^{53} -19110.2 q^{55} -6493.18 q^{56} -21283.2 q^{58} +3481.00 q^{59} +45308.5 q^{61} -35585.5 q^{62} -9300.32 q^{64} -1306.55 q^{65} -40826.6 q^{67} -34411.1 q^{68} -29408.2 q^{70} +27052.4 q^{71} +70459.2 q^{73} -20783.4 q^{74} -47767.4 q^{76} -35048.0 q^{77} +71673.4 q^{79} +58380.0 q^{80} +27096.4 q^{82} -93280.1 q^{83} +74291.2 q^{85} +64456.8 q^{86} +30911.5 q^{88} +116570. q^{89} -2396.20 q^{91} +86001.2 q^{92} -153864. q^{94} +103127. q^{95} +91107.3 q^{97} +69193.5 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} + 198 q^{4} - 36 q^{5} - 411 q^{7} + 69 q^{8}+O(q^{10}) \) 12 * q + 4 * q^2 + 198 * q^4 - 36 * q^5 - 411 * q^7 + 69 * q^8	\( 12 q + 4 q^{2} + 198 q^{4} - 36 q^{5} - 411 q^{7} + 69 q^{8} - 863 q^{10} - 492 q^{11} - 974 q^{13} + 967 q^{14} + 6370 q^{16} + 1463 q^{17} - 3189 q^{19} + 835 q^{20} - 2726 q^{22} + 2617 q^{23} + 8642 q^{25} - 2414 q^{26} - 20458 q^{28} + 1963 q^{29} - 11929 q^{31} + 14382 q^{32} - 20744 q^{34} - 1829 q^{35} - 28105 q^{37} + 23475 q^{38} - 100576 q^{40} + 7585 q^{41} - 33146 q^{43} - 26014 q^{44} - 142851 q^{46} + 79215 q^{47} - 32569 q^{49} + 136019 q^{50} - 248218 q^{52} + 12220 q^{53} - 117770 q^{55} + 186728 q^{56} - 188072 q^{58} + 41772 q^{59} - 54195 q^{61} - 36230 q^{62} + 45197 q^{64} - 42368 q^{65} + 24224 q^{67} + 209639 q^{68} - 35684 q^{70} - 60254 q^{71} - 15385 q^{73} - 214638 q^{74} - 167504 q^{76} + 17169 q^{77} - 27054 q^{79} - 216899 q^{80} + 37917 q^{82} + 117595 q^{83} - 121585 q^{85} - 306756 q^{86} - 105799 q^{88} + 36033 q^{89} - 32217 q^{91} + 30906 q^{92} + 128392 q^{94} + 50721 q^{95} - 196914 q^{97} - 574100 q^{98}+O(q^{100}) \) 12 * q + 4 * q^2 + 198 * q^4 - 36 * q^5 - 411 * q^7 + 69 * q^8 - 863 * q^10 - 492 * q^11 - 974 * q^13 + 967 * q^14 + 6370 * q^16 + 1463 * q^17 - 3189 * q^19 + 835 * q^20 - 2726 * q^22 + 2617 * q^23 + 8642 * q^25 - 2414 * q^26 - 20458 * q^28 + 1963 * q^29 - 11929 * q^31 + 14382 * q^32 - 20744 * q^34 - 1829 * q^35 - 28105 * q^37 + 23475 * q^38 - 100576 * q^40 + 7585 * q^41 - 33146 * q^43 - 26014 * q^44 - 142851 * q^46 + 79215 * q^47 - 32569 * q^49 + 136019 * q^50 - 248218 * q^52 + 12220 * q^53 - 117770 * q^55 + 186728 * q^56 - 188072 * q^58 + 41772 * q^59 - 54195 * q^61 - 36230 * q^62 + 45197 * q^64 - 42368 * q^65 + 24224 * q^67 + 209639 * q^68 - 35684 * q^70 - 60254 * q^71 - 15385 * q^73 - 214638 * q^74 - 167504 * q^76 + 17169 * q^77 - 27054 * q^79 - 216899 * q^80 + 37917 * q^82 + 117595 * q^83 - 121585 * q^85 - 306756 * q^86 - 105799 * q^88 + 36033 * q^89 - 32217 * q^91 + 30906 * q^92 + 128392 * q^94 + 50721 * q^95 - 196914 * q^97 - 574100 * q^98

Introduction

L-functions

Modular forms

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Database

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