LMFDB - Embedded newform 546.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [546,6,Mod(1,546)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(546, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("546.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 546 = 2 \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	546.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:	$87.5695656179$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	546.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +99.0000 q^{5} -36.0000 q^{6} +49.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +99.0000 q^{5} -36.0000 q^{6} +49.0000 q^{7} -64.0000 q^{8} +81.0000 q^{9} -396.000 q^{10} +783.000 q^{11} +144.000 q^{12} +169.000 q^{13} -196.000 q^{14} +891.000 q^{15} +256.000 q^{16} +33.0000 q^{17} -324.000 q^{18} +1433.00 q^{19} +1584.00 q^{20} +441.000 q^{21} -3132.00 q^{22} -3087.00 q^{23} -576.000 q^{24} +6676.00 q^{25} -676.000 q^{26} +729.000 q^{27} +784.000 q^{28} +1515.00 q^{29} -3564.00 q^{30} +2780.00 q^{31} -1024.00 q^{32} +7047.00 q^{33} -132.000 q^{34} +4851.00 q^{35} +1296.00 q^{36} -6595.00 q^{37} -5732.00 q^{38} +1521.00 q^{39} -6336.00 q^{40} +4380.00 q^{41} -1764.00 q^{42} +16043.0 q^{43} +12528.0 q^{44} +8019.00 q^{45} +12348.0 q^{46} +3480.00 q^{47} +2304.00 q^{48} +2401.00 q^{49} -26704.0 q^{50} +297.000 q^{51} +2704.00 q^{52} +618.000 q^{53} -2916.00 q^{54} +77517.0 q^{55} -3136.00 q^{56} +12897.0 q^{57} -6060.00 q^{58} -52116.0 q^{59} +14256.0 q^{60} +887.000 q^{61} -11120.0 q^{62} +3969.00 q^{63} +4096.00 q^{64} +16731.0 q^{65} -28188.0 q^{66} +854.000 q^{67} +528.000 q^{68} -27783.0 q^{69} -19404.0 q^{70} -47340.0 q^{71} -5184.00 q^{72} -64915.0 q^{73} +26380.0 q^{74} +60084.0 q^{75} +22928.0 q^{76} +38367.0 q^{77} -6084.00 q^{78} -73282.0 q^{79} +25344.0 q^{80} +6561.00 q^{81} -17520.0 q^{82} -97062.0 q^{83} +7056.00 q^{84} +3267.00 q^{85} -64172.0 q^{86} +13635.0 q^{87} -50112.0 q^{88} +118218. q^{89} -32076.0 q^{90} +8281.00 q^{91} -49392.0 q^{92} +25020.0 q^{93} -13920.0 q^{94} +141867. q^{95} -9216.00 q^{96} -184522. q^{97} -9604.00 q^{98} +63423.0 q^{99} +O(q^{100})\)

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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	16.0000	0.500000
$5$	99.0000	1.77097	0.885483	−	0.464672i	$-0.153828\pi$
$5$	99.0000	1.77097	0.885483	+	0.464672i	$0.153828\pi$
$6$	−36.0000	−0.408248
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	−64.0000	−0.353553
$9$	81.0000	0.333333
$10$	−396.000	−1.25226
$11$	783.000	1.95110	0.975551	−	0.219772i	$-0.0705315\pi$
$11$	783.000	1.95110	0.975551	+	0.219772i	$0.0705315\pi$
$12$	144.000	0.288675
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	−196.000	−0.267261
$15$	891.000	1.02247
$16$	256.000	0.250000
$17$	33.0000	0.0276944	0.0138472	−	0.999904i	$-0.495592\pi$
$17$	33.0000	0.0276944	0.0138472	+	0.999904i	$0.495592\pi$
$18$	−324.000	−0.235702
$19$	1433.00	0.910672	0.455336	−	0.890320i	$-0.349519\pi$
$19$	1433.00	0.910672	0.455336	+	0.890320i	$0.349519\pi$
$20$	1584.00	0.885483
$21$	441.000	0.218218
$22$	−3132.00	−1.37964
$23$	−3087.00	−1.21679	−0.608397	−	0.793633i	$-0.708187\pi$
$23$	−3087.00	−1.21679	−0.608397	+	0.793633i	$0.708187\pi$
$24$	−576.000	−0.204124
$25$	6676.00	2.13632
$26$	−676.000	−0.196116
$27$	729.000	0.192450
$28$	784.000	0.188982
$29$	1515.00	0.334517	0.167258	−	0.985913i	$-0.446509\pi$
$29$	1515.00	0.334517	0.167258	+	0.985913i	$0.446509\pi$
$30$	−3564.00	−0.722994
$31$	2780.00	0.519566	0.259783	−	0.965667i	$-0.416349\pi$
$31$	2780.00	0.519566	0.259783	+	0.965667i	$0.416349\pi$
$32$	−1024.00	−0.176777
$33$	7047.00	1.12647
$34$	−132.000	−0.0195829
$35$	4851.00	0.669362
$36$	1296.00	0.166667
$37$	−6595.00	−0.791973	−0.395987	−	0.918256i	$-0.629597\pi$
$37$	−6595.00	−0.791973	−0.395987	+	0.918256i	$0.629597\pi$
$38$	−5732.00	−0.643943
$39$	1521.00	0.160128
$40$	−6336.00	−0.626131
$41$	4380.00	0.406925	0.203463	−	0.979083i	$-0.434780\pi$
$41$	4380.00	0.406925	0.203463	+	0.979083i	$0.434780\pi$
$42$	−1764.00	−0.154303
$43$	16043.0	1.32317	0.661583	−	0.749872i	$-0.269885\pi$
$43$	16043.0	1.32317	0.661583	+	0.749872i	$0.269885\pi$
$44$	12528.0	0.975551
$45$	8019.00	0.590322
$46$	12348.0	0.860403
$47$	3480.00	0.229792	0.114896	−	0.993378i	$-0.463347\pi$
$47$	3480.00	0.229792	0.114896	+	0.993378i	$0.463347\pi$
$48$	2304.00	0.144338
$49$	2401.00	0.142857
$50$	−26704.0	−1.51061
$51$	297.000	0.0159894
$52$	2704.00	0.138675
$53$	618.000	0.0302203	0.0151102	−	0.999886i	$-0.495190\pi$
$53$	618.000	0.0302203	0.0151102	+	0.999886i	$0.495190\pi$
$54$	−2916.00	−0.136083
$55$	77517.0	3.45534
$56$	−3136.00	−0.133631
$57$	12897.0	0.525777
$58$	−6060.00	−0.236539
$59$	−52116.0	−1.94913	−0.974566	−	0.224103i	$-0.928055\pi$
$59$	−52116.0	−1.94913	−0.974566	+	0.224103i	$0.928055\pi$
$60$	14256.0	0.511234
$61$	887.000	0.0305210	0.0152605	−	0.999884i	$-0.495142\pi$
$61$	887.000	0.0305210	0.0152605	+	0.999884i	$0.495142\pi$
$62$	−11120.0	−0.367389
$63$	3969.00	0.125988
$64$	4096.00	0.125000
$65$	16731.0	0.491178
$66$	−28188.0	−0.796534
$67$	854.000	0.0232419	0.0116209	−	0.999932i	$-0.496301\pi$
$67$	854.000	0.0232419	0.0116209	+	0.999932i	$0.496301\pi$
$68$	528.000	0.0138472
$69$	−27783.0	−0.702516
$70$	−19404.0	−0.473311
$71$	−47340.0	−1.11451	−0.557253	−	0.830343i	$-0.688145\pi$
$71$	−47340.0	−1.11451	−0.557253	+	0.830343i	$0.688145\pi$
$72$	−5184.00	−0.117851
$73$	−64915.0	−1.42573	−0.712866	−	0.701300i	$-0.752603\pi$
$73$	−64915.0	−1.42573	−0.712866	+	0.701300i	$0.752603\pi$
$74$	26380.0	0.560010
$75$	60084.0	1.23340
$76$	22928.0	0.455336
$77$	38367.0	0.737447
$78$	−6084.00	−0.113228
$79$	−73282.0	−1.32108	−0.660541	−	0.750790i	$-0.729673\pi$
$79$	−73282.0	−1.32108	−0.660541	+	0.750790i	$0.729673\pi$
$80$	25344.0	0.442741
$81$	6561.00	0.111111
$82$	−17520.0	−0.287739
$83$	−97062.0	−1.54651	−0.773257	−	0.634092i	$-0.781374\pi$
$83$	−97062.0	−1.54651	−0.773257	+	0.634092i	$0.781374\pi$
$84$	7056.00	0.109109
$85$	3267.00	0.0490458
$86$	−64172.0	−0.935620
$87$	13635.0	0.193133
$88$	−50112.0	−0.689819
$89$	118218.	1.58201	0.791004	−	0.611811i	$-0.209559\pi$
$89$	118218.	1.58201	0.791004	+	0.611811i	$0.209559\pi$
$90$	−32076.0	−0.417421
$91$	8281.00	0.104828
$92$	−49392.0	−0.608397
$93$	25020.0	0.299971
$94$	−13920.0	−0.162487
$95$	141867.	1.61277
$96$	−9216.00	−0.102062
$97$	−184522.	−1.99122	−0.995609	−	0.0936092i	$-0.970160\pi$
$97$	−184522.	−1.99122	−0.995609	+	0.0936092i	$0.970160\pi$
$98$	−9604.00	−0.101015
$99$	63423.0	0.650367
$100$	106816.	1.06816
$101$	−88452.0	−0.862788	−0.431394	−	0.902164i	$-0.641978\pi$
$101$	−88452.0	−0.862788	−0.431394	+	0.902164i	$0.641978\pi$
$102$	−1188.00	−0.0113062
$103$	3311.00	0.0307515	0.0153757	−	0.999882i	$-0.495106\pi$
$103$	3311.00	0.0307515	0.0153757	+	0.999882i	$0.495106\pi$
$104$	−10816.0	−0.0980581
$105$	43659.0	0.386456
$106$	−2472.00	−0.0213690
$107$	−67902.0	−0.573354	−0.286677	−	0.958027i	$-0.592551\pi$
$107$	−67902.0	−0.573354	−0.286677	+	0.958027i	$0.592551\pi$
$108$	11664.0	0.0962250
$109$	−4993.00	−0.0402527	−0.0201264	−	0.999797i	$-0.506407\pi$
$109$	−4993.00	−0.0402527	−0.0201264	+	0.999797i	$0.506407\pi$
$110$	−310068.	−2.44329
$111$	−59355.0	−0.457246
$112$	12544.0	0.0944911
$113$	152358.	1.12246	0.561228	−	0.827661i	$-0.310329\pi$
$113$	152358.	1.12246	0.561228	+	0.827661i	$0.310329\pi$
$114$	−51588.0	−0.371780
$115$	−305613.	−2.15490
$116$	24240.0	0.167258
$117$	13689.0	0.0924500
$118$	208464.	1.37824
$119$	1617.00	0.0104675
$120$	−57024.0	−0.361497
$121$	452038.	2.80680
$122$	−3548.00	−0.0215816
$123$	39420.0	0.234938
$124$	44480.0	0.259783
$125$	351549.	2.01238
$126$	−15876.0	−0.0890871
$127$	−203902.	−1.12179	−0.560896	−	0.827886i	$-0.689543\pi$
$127$	−203902.	−1.12179	−0.560896	+	0.827886i	$0.689543\pi$
$128$	−16384.0	−0.0883883
$129$	144387.	0.763930
$130$	−66924.0	−0.347315
$131$	214857.	1.09388	0.546942	−	0.837171i	$-0.315792\pi$
$131$	214857.	1.09388	0.546942	+	0.837171i	$0.315792\pi$
$132$	112752.	0.563235
$133$	70217.0	0.344202
$134$	−3416.00	−0.0164345
$135$	72171.0	0.340823
$136$	−2112.00	−0.00979144
$137$	−207927.	−0.946476	−0.473238	−	0.880935i	$-0.656915\pi$
$137$	−207927.	−0.946476	−0.473238	+	0.880935i	$0.656915\pi$
$138$	111132.	0.496754
$139$	−231364.	−1.01568	−0.507842	−	0.861450i	$-0.669557\pi$
$139$	−231364.	−1.01568	−0.507842	+	0.861450i	$0.669557\pi$
$140$	77616.0	0.334681
$141$	31320.0	0.132670
$142$	189360.	0.788075
$143$	132327.	0.541138
$144$	20736.0	0.0833333
$145$	149985.	0.592417
$146$	259660.	1.00814
$147$	21609.0	0.0824786
$148$	−105520.	−0.395987
$149$	242172.	0.893631	0.446816	−	0.894626i	$-0.352558\pi$
$149$	242172.	0.893631	0.446816	+	0.894626i	$0.352558\pi$
$150$	−240336.	−0.872149
$151$	268241.	0.957377	0.478688	−	0.877985i	$-0.341112\pi$
$151$	268241.	0.957377	0.478688	+	0.877985i	$0.341112\pi$
$152$	−91712.0	−0.321971
$153$	2673.00	0.00923146
$154$	−153468.	−0.521454
$155$	275220.	0.920133
$156$	24336.0	0.0800641
$157$	−340885.	−1.10372	−0.551860	−	0.833937i	$-0.686082\pi$
$157$	−340885.	−1.10372	−0.551860	+	0.833937i	$0.686082\pi$
$158$	293128.	0.934146
$159$	5562.00	0.0174477
$160$	−101376.	−0.313065
$161$	−151263.	−0.459905
$162$	−26244.0	−0.0785674
$163$	−10222.0	−0.0301347	−0.0150673	−	0.999886i	$-0.504796\pi$
$163$	−10222.0	−0.0301347	−0.0150673	+	0.999886i	$0.504796\pi$
$164$	70080.0	0.203463
$165$	697653.	1.99494
$166$	388248.	1.09355
$167$	397815.	1.10380	0.551899	−	0.833911i	$-0.313903\pi$
$167$	397815.	1.10380	0.551899	+	0.833911i	$0.313903\pi$
$168$	−28224.0	−0.0771517
$169$	28561.0	0.0769231
$170$	−13068.0	−0.0346806
$171$	116073.	0.303557
$172$	256688.	0.661583
$173$	−368886.	−0.937081	−0.468540	−	0.883442i	$-0.655220\pi$
$173$	−368886.	−0.937081	−0.468540	+	0.883442i	$0.655220\pi$
$174$	−54540.0	−0.136566
$175$	327124.	0.807453
$176$	200448.	0.487776
$177$	−469044.	−1.12533
$178$	−472872.	−1.11865
$179$	−201864.	−0.470897	−0.235449	−	0.971887i	$-0.575656\pi$
$179$	−201864.	−0.470897	−0.235449	+	0.971887i	$0.575656\pi$
$180$	128304.	0.295161
$181$	548210.	1.24380	0.621900	−	0.783097i	$-0.286361\pi$
$181$	548210.	1.24380	0.621900	+	0.783097i	$0.286361\pi$
$182$	−33124.0	−0.0741249
$183$	7983.00	0.0176213
$184$	197568.	0.430202
$185$	−652905.	−1.40256
$186$	−100080.	−0.212112
$187$	25839.0	0.0540346
$188$	55680.0	0.114896
$189$	35721.0	0.0727393
$190$	−567468.	−1.14040
$191$	291201.	0.577576	0.288788	−	0.957393i	$-0.406748\pi$
$191$	291201.	0.577576	0.288788	+	0.957393i	$0.406748\pi$
$192$	36864.0	0.0721688
$193$	−793324.	−1.53305	−0.766527	−	0.642212i	$-0.778017\pi$
$193$	−793324.	−1.53305	−0.766527	+	0.642212i	$0.778017\pi$
$194$	738088.	1.40800
$195$	150579.	0.283581
$196$	38416.0	0.0714286
$197$	396180.	0.727322	0.363661	−	0.931531i	$-0.381527\pi$
$197$	396180.	0.727322	0.363661	+	0.931531i	$0.381527\pi$
$198$	−253692.	−0.459879
$199$	624341.	1.11761	0.558804	−	0.829300i	$-0.311261\pi$
$199$	624341.	1.11761	0.558804	+	0.829300i	$0.311261\pi$
$200$	−427264.	−0.755303
$201$	7686.00	0.0134187
$202$	353808.	0.610083
$203$	74235.0	0.126435
$204$	4752.00	0.00799468
$205$	433620.	0.720650
$206$	−13244.0	−0.0217446
$207$	−250047.	−0.405598
$208$	43264.0	0.0693375
$209$	1.12204e6	1.77682
$210$	−174636.	−0.273266
$211$	−212095.	−0.327963	−0.163981	−	0.986463i	$-0.552434\pi$
$211$	−212095.	−0.327963	−0.163981	+	0.986463i	$0.552434\pi$
$212$	9888.00	0.0151102
$213$	−426060.	−0.643460
$214$	271608.	0.405423
$215$	1.58826e6	2.34328
$216$	−46656.0	−0.0680414
$217$	136220.	0.196377
$218$	19972.0	0.0284630
$219$	−584235.	−0.823147
$220$	1.24027e6	1.72767
$221$	5577.00	0.00768104
$222$	237420.	0.323322
$223$	945554.	1.27328	0.636640	−	0.771161i	$-0.280324\pi$
$223$	945554.	1.27328	0.636640	+	0.771161i	$0.280324\pi$
$224$	−50176.0	−0.0668153
$225$	540756.	0.712107
$226$	−609432.	−0.793696
$227$	462726.	0.596018	0.298009	−	0.954563i	$-0.403677\pi$
$227$	462726.	0.596018	0.298009	+	0.954563i	$0.403677\pi$
$228$	206352.	0.262888
$229$	752606.	0.948373	0.474186	−	0.880425i	$-0.342742\pi$
$229$	752606.	0.948373	0.474186	+	0.880425i	$0.342742\pi$
$230$	1.22245e6	1.52374
$231$	345303.	0.425765
$232$	−96960.0	−0.118269
$233$	−739032.	−0.891812	−0.445906	−	0.895080i	$-0.647118\pi$
$233$	−739032.	−0.891812	−0.445906	+	0.895080i	$0.647118\pi$
$234$	−54756.0	−0.0653720
$235$	344520.	0.406953
$236$	−833856.	−0.974566
$237$	−659538.	−0.762727
$238$	−6468.00	−0.00740163
$239$	−97752.0	−0.110696	−0.0553479	−	0.998467i	$-0.517627\pi$
$239$	−97752.0	−0.110696	−0.0553479	+	0.998467i	$0.517627\pi$
$240$	228096.	0.255617
$241$	−1.23313e6	−1.36762	−0.683811	−	0.729659i	$-0.739679\pi$
$241$	−1.23313e6	−1.36762	−0.683811	+	0.729659i	$0.739679\pi$
$242$	−1.80815e6	−1.98471
$243$	59049.0	0.0641500
$244$	14192.0	0.0152605
$245$	237699.	0.252995
$246$	−157680.	−0.166126
$247$	242177.	0.252575
$248$	−177920.	−0.183694
$249$	−873558.	−0.892881
$250$	−1.40620e6	−1.42297
$251$	198117.	0.198489	0.0992447	−	0.995063i	$-0.468357\pi$
$251$	198117.	0.198489	0.0992447	+	0.995063i	$0.468357\pi$
$252$	63504.0	0.0629941
$253$	−2.41712e6	−2.37409
$254$	815608.	0.793226
$255$	29403.0	0.0283166
$256$	65536.0	0.0625000
$257$	−277362.	−0.261947	−0.130974	−	0.991386i	$-0.541810\pi$
$257$	−277362.	−0.261947	−0.130974	+	0.991386i	$0.541810\pi$
$258$	−577548.	−0.540180
$259$	−323155.	−0.299338
$260$	267696.	0.245589
$261$	122715.	0.111506
$262$	−859428.	−0.773493
$263$	1.25996e6	1.12323	0.561615	−	0.827399i	$-0.310180\pi$
$263$	1.25996e6	1.12323	0.561615	+	0.827399i	$0.310180\pi$
$264$	−451008.	−0.398267
$265$	61182.0	0.0535191
$266$	−280868.	−0.243387
$267$	1.06396e6	0.913373
$268$	13664.0	0.0116209
$269$	1.77437e6	1.49508	0.747540	−	0.664217i	$-0.231235\pi$
$269$	1.77437e6	1.49508	0.747540	+	0.664217i	$0.231235\pi$
$270$	−288684.	−0.240998
$271$	509330.	0.421285	0.210643	−	0.977563i	$-0.432444\pi$
$271$	509330.	0.421285	0.210643	+	0.977563i	$0.432444\pi$
$272$	8448.00	0.00692359
$273$	74529.0	0.0605228
$274$	831708.	0.669259
$275$	5.22731e6	4.16818
$276$	−444528.	−0.351258
$277$	−418198.	−0.327478	−0.163739	−	0.986504i	$-0.552356\pi$
$277$	−418198.	−0.327478	−0.163739	+	0.986504i	$0.552356\pi$
$278$	925456.	0.718197
$279$	225180.	0.173189
$280$	−310464.	−0.236655
$281$	−1.09383e6	−0.826388	−0.413194	−	0.910643i	$-0.635587\pi$
$281$	−1.09383e6	−0.826388	−0.413194	+	0.910643i	$0.635587\pi$
$282$	−125280.	−0.0938121
$283$	2.48145e6	1.84179	0.920894	−	0.389812i	$-0.127460\pi$
$283$	2.48145e6	1.84179	0.920894	+	0.389812i	$0.127460\pi$
$284$	−757440.	−0.557253
$285$	1.27680e6	0.931133
$286$	−529308.	−0.382643
$287$	214620.	0.153803
$288$	−82944.0	−0.0589256
$289$	−1.41877e6	−0.999233
$290$	−599940.	−0.418902
$291$	−1.66070e6	−1.14963
$292$	−1.03864e6	−0.712866
$293$	2.36297e6	1.60801	0.804007	−	0.594620i	$-0.202697\pi$
$293$	2.36297e6	1.60801	0.804007	+	0.594620i	$0.202697\pi$
$294$	−86436.0	−0.0583212
$295$	−5.15948e6	−3.45184
$296$	422080.	0.280005
$297$	570807.	0.375490
$298$	−968688.	−0.631893
$299$	−521703.	−0.337478
$300$	961344.	0.616702
$301$	786107.	0.500110
$302$	−1.07296e6	−0.676967
$303$	−796068.	−0.498131
$304$	366848.	0.227668
$305$	87813.0	0.0540517
$306$	−10692.0	−0.00652763
$307$	−720880.	−0.436533	−0.218266	−	0.975889i	$-0.570040\pi$
$307$	−720880.	−0.436533	−0.218266	+	0.975889i	$0.570040\pi$
$308$	613872.	0.368724
$309$	29799.0	0.0177544
$310$	−1.10088e6	−0.650632
$311$	691194.	0.405228	0.202614	−	0.979259i	$-0.435056\pi$
$311$	691194.	0.405228	0.202614	+	0.979259i	$0.435056\pi$
$312$	−97344.0	−0.0566139
$313$	−2.26912e6	−1.30917	−0.654586	−	0.755988i	$-0.727157\pi$
$313$	−2.26912e6	−1.30917	−0.654586	+	0.755988i	$0.727157\pi$
$314$	1.36354e6	0.780448
$315$	392931.	0.223121
$316$	−1.17251e6	−0.660541
$317$	2.29798e6	1.28439	0.642196	−	0.766541i	$-0.278024\pi$
$317$	2.29798e6	1.28439	0.642196	+	0.766541i	$0.278024\pi$
$318$	−22248.0	−0.0123374
$319$	1.18624e6	0.652676
$320$	405504.	0.221371
$321$	−611118.	−0.331026
$322$	605052.	0.325202
$323$	47289.0	0.0252205
$324$	104976.	0.0555556
$325$	1.12824e6	0.592509
$326$	40888.0	0.0213085
$327$	−44937.0	−0.0232399
$328$	−280320.	−0.143870
$329$	170520.	0.0868532
$330$	−2.79061e6	−1.41063
$331$	−2.47614e6	−1.24224	−0.621119	−	0.783716i	$-0.713322\pi$
$331$	−2.47614e6	−1.24224	−0.621119	+	0.783716i	$0.713322\pi$
$332$	−1.55299e6	−0.773257
$333$	−534195.	−0.263991
$334$	−1.59126e6	−0.780504
$335$	84546.0	0.0411605
$336$	112896.	0.0545545
$337$	2.35926e6	1.13162	0.565810	−	0.824535i	$-0.308563\pi$
$337$	2.35926e6	1.13162	0.565810	+	0.824535i	$0.308563\pi$
$338$	−114244.	−0.0543928
$339$	1.37122e6	0.648050
$340$	52272.0	0.0245229
$341$	2.17674e6	1.01373
$342$	−464292.	−0.214648
$343$	117649.	0.0539949
$344$	−1.02675e6	−0.467810
$345$	−2.75052e6	−1.24413
$346$	1.47554e6	0.662616
$347$	−2.54144e6	−1.13307	−0.566535	−	0.824038i	$-0.691716\pi$
$347$	−2.54144e6	−1.13307	−0.566535	+	0.824038i	$0.691716\pi$
$348$	218160.	0.0965666
$349$	−2.86368e6	−1.25852	−0.629261	−	0.777194i	$-0.716642\pi$
$349$	−2.86368e6	−1.25852	−0.629261	+	0.777194i	$0.716642\pi$
$350$	−1.30850e6	−0.570956
$351$	123201.	0.0533761
$352$	−801792.	−0.344909
$353$	3.01156e6	1.28634	0.643169	−	0.765724i	$-0.277619\pi$
$353$	3.01156e6	1.28634	0.643169	+	0.765724i	$0.277619\pi$
$354$	1.87618e6	0.795729
$355$	−4.68666e6	−1.97375
$356$	1.89149e6	0.791004
$357$	14553.0	0.00604341
$358$	807456.	0.332975
$359$	986448.	0.403960	0.201980	−	0.979390i	$-0.435262\pi$
$359$	986448.	0.403960	0.201980	+	0.979390i	$0.435262\pi$
$360$	−513216.	−0.208710
$361$	−422610.	−0.170676
$362$	−2.19284e6	−0.879499
$363$	4.06834e6	1.62051
$364$	132496.	0.0524142
$365$	−6.42658e6	−2.52492
$366$	−31932.0	−0.0124602
$367$	225392.	0.0873521	0.0436760	−	0.999046i	$-0.486093\pi$
$367$	225392.	0.0873521	0.0436760	+	0.999046i	$0.486093\pi$
$368$	−790272.	−0.304198
$369$	354780.	0.135642
$370$	2.61162e6	0.991758
$371$	30282.0	0.0114222
$372$	400320.	0.149986
$373$	−286876.	−0.106763	−0.0533817	−	0.998574i	$-0.517000\pi$
$373$	−286876.	−0.106763	−0.0533817	+	0.998574i	$0.517000\pi$
$374$	−103356.	−0.0382082
$375$	3.16394e6	1.16185
$376$	−222720.	−0.0812437
$377$	256035.	0.0927782
$378$	−142884.	−0.0514344
$379$	4.67343e6	1.67124	0.835618	−	0.549311i	$-0.185110\pi$
$379$	4.67343e6	1.67124	0.835618	+	0.549311i	$0.185110\pi$
$380$	2.26987e6	0.806385
$381$	−1.83512e6	−0.647667
$382$	−1.16480e6	−0.408408
$383$	−3.52635e6	−1.22837	−0.614185	−	0.789162i	$-0.710515\pi$
$383$	−3.52635e6	−1.22837	−0.614185	+	0.789162i	$0.710515\pi$
$384$	−147456.	−0.0510310
$385$	3.79833e6	1.30599
$386$	3.17330e6	1.08403
$387$	1.29948e6	0.441055
$388$	−2.95235e6	−0.995609
$389$	−2.84427e6	−0.953009	−0.476504	−	0.879172i	$-0.658096\pi$
$389$	−2.84427e6	−0.953009	−0.476504	+	0.879172i	$0.658096\pi$
$390$	−602316.	−0.200522
$391$	−101871.	−0.0336984
$392$	−153664.	−0.0505076
$393$	1.93371e6	0.631554
$394$	−1.58472e6	−0.514295
$395$	−7.25492e6	−2.33959
$396$	1.01477e6	0.325184
$397$	−2.74727e6	−0.874832	−0.437416	−	0.899259i	$-0.644106\pi$
$397$	−2.74727e6	−0.874832	−0.437416	+	0.899259i	$0.644106\pi$
$398$	−2.49736e6	−0.790268
$399$	631953.	0.198725
$400$	1.70906e6	0.534080
$401$	−4.63621e6	−1.43980	−0.719899	−	0.694078i	$-0.755812\pi$
$401$	−4.63621e6	−1.43980	−0.719899	+	0.694078i	$0.755812\pi$
$402$	−30744.0	−0.00948845
$403$	469820.	0.144102
$404$	−1.41523e6	−0.431394
$405$	649539.	0.196774
$406$	−296940.	−0.0894033
$407$	−5.16388e6	−1.54522
$408$	−19008.0	−0.00565309
$409$	−4.55465e6	−1.34631	−0.673157	−	0.739499i	$-0.735062\pi$
$409$	−4.55465e6	−1.34631	−0.673157	+	0.739499i	$0.735062\pi$
$410$	−1.73448e6	−0.509577
$411$	−1.87134e6	−0.546448
$412$	52976.0	0.0153757
$413$	−2.55368e6	−0.736702
$414$	1.00019e6	0.286801
$415$	−9.60914e6	−2.73882
$416$	−173056.	−0.0490290
$417$	−2.08228e6	−0.586406
$418$	−4.48816e6	−1.25640
$419$	−2.01681e6	−0.561217	−0.280608	−	0.959822i	$-0.590536\pi$
$419$	−2.01681e6	−0.561217	−0.280608	+	0.959822i	$0.590536\pi$
$420$	698544.	0.193228
$421$	2.01148e6	0.553108	0.276554	−	0.960998i	$-0.410808\pi$
$421$	2.01148e6	0.553108	0.276554	+	0.960998i	$0.410808\pi$
$422$	848380.	0.231905
$423$	281880.	0.0765973
$424$	−39552.0	−0.0106845
$425$	220308.	0.0591641
$426$	1.70424e6	0.454995
$427$	43463.0	0.0115359
$428$	−1.08643e6	−0.286677
$429$	1.19094e6	0.312426
$430$	−6.35303e6	−1.65695
$431$	−3.13520e6	−0.812965	−0.406482	−	0.913659i	$-0.633245\pi$
$431$	−3.13520e6	−0.812965	−0.406482	+	0.913659i	$0.633245\pi$
$432$	186624.	0.0481125
$433$	−3.90933e6	−1.00203	−0.501017	−	0.865438i	$-0.667041\pi$
$433$	−3.90933e6	−1.00203	−0.501017	+	0.865438i	$0.667041\pi$
$434$	−544880.	−0.138860
$435$	1.34986e6	0.342032
$436$	−79888.0	−0.0201264
$437$	−4.42367e6	−1.10810
$438$	2.33694e6	0.582053
$439$	−3.44543e6	−0.853261	−0.426631	−	0.904426i	$-0.640300\pi$
$439$	−3.44543e6	−0.853261	−0.426631	+	0.904426i	$0.640300\pi$
$440$	−4.96109e6	−1.22165
$441$	194481.	0.0476190
$442$	−22308.0	−0.00543131
$443$	−535350.	−0.129607	−0.0648035	−	0.997898i	$-0.520642\pi$
$443$	−535350.	−0.129607	−0.0648035	+	0.997898i	$0.520642\pi$
$444$	−949680.	−0.228623
$445$	1.17036e7	2.80168
$446$	−3.78222e6	−0.900345
$447$	2.17955e6	0.515938
$448$	200704.	0.0472456
$449$	5.00800e6	1.17233	0.586164	−	0.810193i	$-0.300638\pi$
$449$	5.00800e6	1.17233	0.586164	+	0.810193i	$0.300638\pi$
$450$	−2.16302e6	−0.503535
$451$	3.42954e6	0.793952
$452$	2.43773e6	0.561228
$453$	2.41417e6	0.552742
$454$	−1.85090e6	−0.421448
$455$	819819.	0.185648
$456$	−825408.	−0.185890
$457$	938516.	0.210209	0.105104	−	0.994461i	$-0.466482\pi$
$457$	938516.	0.210209	0.105104	+	0.994461i	$0.466482\pi$
$458$	−3.01042e6	−0.670601
$459$	24057.0	0.00532979
$460$	−4.88981e6	−1.07745
$461$	−2.07095e6	−0.453855	−0.226928	−	0.973912i	$-0.572868\pi$
$461$	−2.07095e6	−0.453855	−0.226928	+	0.973912i	$0.572868\pi$
$462$	−1.38121e6	−0.301062
$463$	−4.38025e6	−0.949612	−0.474806	−	0.880091i	$-0.657482\pi$
$463$	−4.38025e6	−0.949612	−0.474806	+	0.880091i	$0.657482\pi$
$464$	387840.	0.0836291
$465$	2.47698e6	0.531239
$466$	2.95613e6	0.630607
$467$	−1.66822e6	−0.353965	−0.176983	−	0.984214i	$-0.556634\pi$
$467$	−1.66822e6	−0.353965	−0.176983	+	0.984214i	$0.556634\pi$
$468$	219024.	0.0462250
$469$	41846.0	0.00878460
$470$	−1.37808e6	−0.287760
$471$	−3.06796e6	−0.637233
$472$	3.33542e6	0.689122
$473$	1.25617e7	2.58163
$474$	2.63815e6	0.539329
$475$	9.56671e6	1.94549
$476$	25872.0	0.00523375
$477$	50058.0	0.0100734
$478$	391008.	0.0782737
$479$	8.40532e6	1.67385	0.836923	−	0.547320i	$-0.184352\pi$
$479$	8.40532e6	1.67385	0.836923	+	0.547320i	$0.184352\pi$
$480$	−912384.	−0.180748
$481$	−1.11456e6	−0.219654
$482$	4.93252e6	0.967055
$483$	−1.36137e6	−0.265526
$484$	7.23261e6	1.40340
$485$	−1.82677e7	−3.52638
$486$	−236196.	−0.0453609
$487$	5.45499e6	1.04225	0.521125	−	0.853481i	$-0.325513\pi$
$487$	5.45499e6	1.04225	0.521125	+	0.853481i	$0.325513\pi$
$488$	−56768.0	−0.0107908
$489$	−91998.0	−0.0173983
$490$	−950796.	−0.178895
$491$	9.07107e6	1.69807	0.849033	−	0.528339i	$-0.177185\pi$
$491$	9.07107e6	1.69807	0.849033	+	0.528339i	$0.177185\pi$
$492$	630720.	0.117469
$493$	49995.0	0.00926423
$494$	−968708.	−0.178598
$495$	6.27888e6	1.15178
$496$	711680.	0.129891
$497$	−2.31966e6	−0.421244
$498$	3.49423e6	0.631362
$499$	−4.62902e6	−0.832218	−0.416109	−	0.909315i	$-0.636607\pi$
$499$	−4.62902e6	−0.832218	−0.416109	+	0.909315i	$0.636607\pi$
$500$	5.62478e6	1.00619
$501$	3.58034e6	0.637279
$502$	−792468.	−0.140353
$503$	5.00935e6	0.882797	0.441399	−	0.897311i	$-0.354482\pi$
$503$	5.00935e6	0.882797	0.441399	+	0.897311i	$0.354482\pi$
$504$	−254016.	−0.0445435
$505$	−8.75675e6	−1.52797
$506$	9.66848e6	1.67873
$507$	257049.	0.0444116
$508$	−3.26243e6	−0.560896
$509$	7.66606e6	1.31153	0.655765	−	0.754965i	$-0.272346\pi$
$509$	7.66606e6	1.31153	0.655765	+	0.754965i	$0.272346\pi$
$510$	−117612.	−0.0200229
$511$	−3.18084e6	−0.538876
$512$	−262144.	−0.0441942
$513$	1.04466e6	0.175259
$514$	1.10945e6	0.185225
$515$	327789.	0.0544598
$516$	2.31019e6	0.381965
$517$	2.72484e6	0.448347
$518$	1.29262e6	0.211664
$519$	−3.31997e6	−0.541024
$520$	−1.07078e6	−0.173657
$521$	5.43684e6	0.877511	0.438755	−	0.898607i	$-0.355419\pi$
$521$	5.43684e6	0.877511	0.438755	+	0.898607i	$0.355419\pi$
$522$	−490860.	−0.0788463
$523$	7.35759e6	1.17620	0.588100	−	0.808788i	$-0.299876\pi$
$523$	7.35759e6	1.17620	0.588100	+	0.808788i	$0.299876\pi$
$524$	3.43771e6	0.546942
$525$	2.94412e6	0.466183
$526$	−5.03986e6	−0.794244
$527$	91740.0	0.0143891
$528$	1.80403e6	0.281617
$529$	3.09323e6	0.480588
$530$	−244728.	−0.0378437
$531$	−4.22140e6	−0.649710
$532$	1.12347e6	0.172101
$533$	740220.	0.112861
$534$	−4.25585e6	−0.645852
$535$	−6.72230e6	−1.01539
$536$	−54656.0	−0.00821724
$537$	−1.81678e6	−0.271873
$538$	−7.09750e6	−1.05718
$539$	1.87998e6	0.278729
$540$	1.15474e6	0.170411
$541$	−4.05946e6	−0.596314	−0.298157	−	0.954517i	$-0.596372\pi$
$541$	−4.05946e6	−0.596314	−0.298157	+	0.954517i	$0.596372\pi$
$542$	−2.03732e6	−0.297894
$543$	4.93389e6	0.718108
$544$	−33792.0	−0.00489572
$545$	−494307.	−0.0712862
$546$	−298116.	−0.0427960
$547$	8.30563e6	1.18687	0.593436	−	0.804881i	$-0.297771\pi$
$547$	8.30563e6	1.18687	0.593436	+	0.804881i	$0.297771\pi$
$548$	−3.32683e6	−0.473238
$549$	71847.0	0.0101737
$550$	−2.09092e7	−2.94735
$551$	2.17100e6	0.304635
$552$	1.77811e6	0.248377
$553$	−3.59082e6	−0.499322
$554$	1.67279e6	0.231562
$555$	−5.87614e6	−0.809767
$556$	−3.70182e6	−0.507842
$557$	−7.21366e6	−0.985184	−0.492592	−	0.870260i	$-0.663951\pi$
$557$	−7.21366e6	−0.985184	−0.492592	+	0.870260i	$0.663951\pi$
$558$	−900720.	−0.122463
$559$	2.71127e6	0.366980
$560$	1.24186e6	0.167341
$561$	232551.	0.0311969
$562$	4.37532e6	0.584345
$563$	1.11052e7	1.47657	0.738287	−	0.674487i	$-0.235635\pi$
$563$	1.11052e7	1.47657	0.738287	+	0.674487i	$0.235635\pi$
$564$	501120.	0.0663352
$565$	1.50834e7	1.98783
$566$	−9.92581e6	−1.30234
$567$	321489.	0.0419961
$568$	3.02976e6	0.394037
$569$	5.82358e6	0.754066	0.377033	−	0.926200i	$-0.376944\pi$
$569$	5.82358e6	0.754066	0.377033	+	0.926200i	$0.376944\pi$
$570$	−5.10721e6	−0.658410
$571$	7.76146e6	0.996216	0.498108	−	0.867115i	$-0.334028\pi$
$571$	7.76146e6	0.996216	0.498108	+	0.867115i	$0.334028\pi$
$572$	2.11723e6	0.270569
$573$	2.62081e6	0.333464
$574$	−858480.	−0.108755
$575$	−2.06088e7	−2.59946
$576$	331776.	0.0416667
$577$	−5.02239e6	−0.628017	−0.314008	−	0.949420i	$-0.601672\pi$
$577$	−5.02239e6	−0.628017	−0.314008	+	0.949420i	$0.601672\pi$
$578$	5.67507e6	0.706564
$579$	−7.13992e6	−0.885109
$580$	2.39976e6	0.296209
$581$	−4.75604e6	−0.584528
$582$	6.64279e6	0.812911
$583$	483894.	0.0589629
$584$	4.15456e6	0.504072
$585$	1.35521e6	0.163726
$586$	−9.45190e6	−1.13704
$587$	−1.80400e6	−0.216094	−0.108047	−	0.994146i	$-0.534460\pi$
$587$	−1.80400e6	−0.216094	−0.108047	+	0.994146i	$0.534460\pi$
$588$	345744.	0.0412393
$589$	3.98374e6	0.473154
$590$	2.06379e7	2.44082
$591$	3.56562e6	0.419920
$592$	−1.68832e6	−0.197993
$593$	167646.	0.0195775	0.00978873	−	0.999952i	$-0.496884\pi$
$593$	167646.	0.0195775	0.00978873	+	0.999952i	$0.496884\pi$
$594$	−2.28323e6	−0.265511
$595$	160083.	0.0185376
$596$	3.87475e6	0.446816
$597$	5.61907e6	0.645251
$598$	2.08681e6	0.238633
$599$	−1.46631e7	−1.66978	−0.834890	−	0.550417i	$-0.814469\pi$
$599$	−1.46631e7	−1.66978	−0.834890	+	0.550417i	$0.814469\pi$
$600$	−3.84538e6	−0.436074
$601$	3.50773e6	0.396132	0.198066	−	0.980189i	$-0.436534\pi$
$601$	3.50773e6	0.396132	0.198066	+	0.980189i	$0.436534\pi$
$602$	−3.14443e6	−0.353631
$603$	69174.0	0.00774729
$604$	4.29186e6	0.478688
$605$	4.47518e7	4.97075
$606$	3.18427e6	0.352232
$607$	−1.63805e7	−1.80449	−0.902245	−	0.431223i	$-0.858082\pi$
$607$	−1.63805e7	−1.80449	−0.902245	+	0.431223i	$0.858082\pi$
$608$	−1.46739e6	−0.160986
$609$	668115.	0.0729975
$610$	−351252.	−0.0382203
$611$	588120.	0.0637328
$612$	42768.0	0.00461573
$613$	−1.75848e6	−0.189011	−0.0945054	−	0.995524i	$-0.530127\pi$
$613$	−1.75848e6	−0.189011	−0.0945054	+	0.995524i	$0.530127\pi$
$614$	2.88352e6	0.308675
$615$	3.90258e6	0.416068
$616$	−2.45549e6	−0.260727
$617$	9.99581e6	1.05707	0.528537	−	0.848910i	$-0.322741\pi$
$617$	9.99581e6	1.05707	0.528537	+	0.848910i	$0.322741\pi$
$618$	−119196.	−0.0125542
$619$	−9.36392e6	−0.982270	−0.491135	−	0.871083i	$-0.663418\pi$
$619$	−9.36392e6	−0.982270	−0.491135	+	0.871083i	$0.663418\pi$
$620$	4.40352e6	0.460067
$621$	−2.25042e6	−0.234172
$622$	−2.76478e6	−0.286539
$623$	5.79268e6	0.597943
$624$	389376.	0.0400320
$625$	1.39409e7	1.42754
$626$	9.07648e6	0.925724
$627$	1.00984e7	1.02584
$628$	−5.45416e6	−0.551860
$629$	−217635.	−0.0219332
$630$	−1.57172e6	−0.157770
$631$	−1.26052e7	−1.26031	−0.630155	−	0.776469i	$-0.717009\pi$
$631$	−1.26052e7	−1.26031	−0.630155	+	0.776469i	$0.717009\pi$
$632$	4.69005e6	0.467073
$633$	−1.90886e6	−0.189349
$634$	−9.19190e6	−0.908202
$635$	−2.01863e7	−1.98665
$636$	88992.0	0.00872385
$637$	405769.	0.0396214
$638$	−4.74498e6	−0.461512
$639$	−3.83454e6	−0.371502
$640$	−1.62202e6	−0.156533
$641$	−1.56405e7	−1.50351	−0.751756	−	0.659442i	$-0.770793\pi$
$641$	−1.56405e7	−1.50351	−0.751756	+	0.659442i	$0.770793\pi$
$642$	2.44447e6	0.234071
$643$	−1.45253e6	−0.138547	−0.0692735	−	0.997598i	$-0.522068\pi$
$643$	−1.45253e6	−0.138547	−0.0692735	+	0.997598i	$0.522068\pi$
$644$	−2.42021e6	−0.229952
$645$	1.42943e7	1.35289
$646$	−189156.	−0.0178336
$647$	1.28264e7	1.20460	0.602301	−	0.798269i	$-0.294251\pi$
$647$	1.28264e7	1.20460	0.602301	+	0.798269i	$0.294251\pi$
$648$	−419904.	−0.0392837
$649$	−4.08068e7	−3.80295
$650$	−4.51298e6	−0.418967
$651$	1.22598e6	0.113379
$652$	−163552.	−0.0150673
$653$	−3.88644e6	−0.356672	−0.178336	−	0.983970i	$-0.557071\pi$
$653$	−3.88644e6	−0.356672	−0.178336	+	0.983970i	$0.557071\pi$
$654$	179748.	0.0164331
$655$	2.12708e7	1.93723
$656$	1.12128e6	0.101731
$657$	−5.25812e6	−0.475244
$658$	−682080.	−0.0614145
$659$	−6.80296e6	−0.610217	−0.305109	−	0.952318i	$-0.598693\pi$
$659$	−6.80296e6	−0.610217	−0.305109	+	0.952318i	$0.598693\pi$
$660$	1.11624e7	0.997469
$661$	1.36404e7	1.21429	0.607146	−	0.794590i	$-0.292314\pi$
$661$	1.36404e7	1.21429	0.607146	+	0.794590i	$0.292314\pi$
$662$	9.90455e6	0.878395
$663$	50193.0	0.00443465
$664$	6.21197e6	0.546775
$665$	6.95148e6	0.609570
$666$	2.13678e6	0.186670
$667$	−4.67680e6	−0.407038
$668$	6.36504e6	0.551899
$669$	8.50999e6	0.735129
$670$	−338184.	−0.0291049
$671$	694521.	0.0595496
$672$	−451584.	−0.0385758
$673$	−1.86150e7	−1.58426	−0.792128	−	0.610355i	$-0.791027\pi$
$673$	−1.86150e7	−1.58426	−0.792128	+	0.610355i	$0.791027\pi$
$674$	−9.43704e6	−0.800177
$675$	4.86680e6	0.411135
$676$	456976.	0.0384615
$677$	−7.31919e6	−0.613750	−0.306875	−	0.951750i	$-0.599283\pi$
$677$	−7.31919e6	−0.613750	−0.306875	+	0.951750i	$0.599283\pi$
$678$	−5.48489e6	−0.458241
$679$	−9.04158e6	−0.752610
$680$	−209088.	−0.0173403
$681$	4.16453e6	0.344111
$682$	−8.70696e6	−0.716813
$683$	4.87742e6	0.400072	0.200036	−	0.979789i	$-0.435894\pi$
$683$	4.87742e6	0.400072	0.200036	+	0.979789i	$0.435894\pi$
$684$	1.85717e6	0.151779
$685$	−2.05848e7	−1.67618
$686$	−470596.	−0.0381802
$687$	6.77345e6	0.547543
$688$	4.10701e6	0.330792
$689$	104442.	0.00838160
$690$	1.10021e7	0.879734
$691$	9.38815e6	0.747971	0.373986	−	0.927435i	$-0.377991\pi$
$691$	9.38815e6	0.747971	0.373986	+	0.927435i	$0.377991\pi$
$692$	−5.90218e6	−0.468540
$693$	3.10773e6	0.245816
$694$	1.01658e7	0.801202
$695$	−2.29050e7	−1.79874
$696$	−872640.	−0.0682829
$697$	144540.	0.0112695
$698$	1.14547e7	0.889909
$699$	−6.65129e6	−0.514888
$700$	5.23398e6	0.403727
$701$	−3.88816e6	−0.298847	−0.149424	−	0.988773i	$-0.547742\pi$
$701$	−3.88816e6	−0.298847	−0.149424	+	0.988773i	$0.547742\pi$
$702$	−492804.	−0.0377426
$703$	−9.45064e6	−0.721228
$704$	3.20717e6	0.243888
$705$	3.10068e6	0.234955
$706$	−1.20462e7	−0.909578
$707$	−4.33415e6	−0.326103
$708$	−7.50470e6	−0.562666
$709$	2.00356e6	0.149688	0.0748439	−	0.997195i	$-0.476154\pi$
$709$	2.00356e6	0.149688	0.0748439	+	0.997195i	$0.476154\pi$
$710$	1.87466e7	1.39565
$711$	−5.93584e6	−0.440360
$712$	−7.56595e6	−0.559324
$713$	−8.58186e6	−0.632205
$714$	−58212.0	−0.00427334
$715$	1.31004e7	0.958338
$716$	−3.22982e6	−0.235449
$717$	−879768.	−0.0639102
$718$	−3.94579e6	−0.285643
$719$	1.27107e7	0.916950	0.458475	−	0.888707i	$-0.348396\pi$
$719$	1.27107e7	0.916950	0.458475	+	0.888707i	$0.348396\pi$
$720$	2.05286e6	0.147580
$721$	162239.	0.0116230
$722$	1.69044e6	0.120686
$723$	−1.10982e7	−0.789597
$724$	8.77136e6	0.621900
$725$	1.01141e7	0.714634
$726$	−1.62734e7	−1.14587
$727$	1.38863e7	0.974432	0.487216	−	0.873281i	$-0.338012\pi$
$727$	1.38863e7	0.974432	0.487216	+	0.873281i	$0.338012\pi$
$728$	−529984.	−0.0370625
$729$	531441.	0.0370370
$730$	2.57063e7	1.78539
$731$	529419.	0.0366443
$732$	127728.	0.00881066
$733$	−2.87911e7	−1.97924	−0.989621	−	0.143703i	$-0.954099\pi$
$733$	−2.87911e7	−1.97924	−0.989621	+	0.143703i	$0.954099\pi$
$734$	−901568.	−0.0617673
$735$	2.13929e6	0.146067
$736$	3.16109e6	0.215101
$737$	668682.	0.0453472
$738$	−1.41912e6	−0.0959132
$739$	2.79137e7	1.88021	0.940106	−	0.340884i	$-0.110726\pi$
$739$	2.79137e7	1.88021	0.940106	+	0.340884i	$0.110726\pi$
$740$	−1.04465e7	−0.701279
$741$	2.17959e6	0.145824
$742$	−121128.	−0.00807672
$743$	−1.77697e7	−1.18089	−0.590444	−	0.807078i	$-0.701047\pi$
$743$	−1.77697e7	−1.18089	−0.590444	+	0.807078i	$0.701047\pi$
$744$	−1.60128e6	−0.106056
$745$	2.39750e7	1.58259
$746$	1.14750e6	0.0754931
$747$	−7.86202e6	−0.515505
$748$	413424.	0.0270173
$749$	−3.32720e6	−0.216708
$750$	−1.26558e7	−0.821552
$751$	1.74520e7	1.12913	0.564566	−	0.825388i	$-0.309043\pi$
$751$	1.74520e7	1.12913	0.564566	+	0.825388i	$0.309043\pi$
$752$	890880.	0.0574480
$753$	1.78305e6	0.114598
$754$	−1.02414e6	−0.0656041
$755$	2.65559e7	1.69548
$756$	571536.	0.0363696
$757$	1.27070e7	0.805938	0.402969	−	0.915214i	$-0.367978\pi$
$757$	1.27070e7	0.805938	0.402969	+	0.915214i	$0.367978\pi$
$758$	−1.86937e7	−1.18174
$759$	−2.17541e7	−1.37068
$760$	−9.07949e6	−0.570200
$761$	9.75769e6	0.610781	0.305390	−	0.952227i	$-0.401213\pi$
$761$	9.75769e6	0.610781	0.305390	+	0.952227i	$0.401213\pi$
$762$	7.34047e6	0.457970
$763$	−244657.	−0.0152141
$764$	4.65922e6	0.288788
$765$	264627.	0.0163486
$766$	1.41054e7	0.868588
$767$	−8.80760e6	−0.540592
$768$	589824.	0.0360844
$769$	2.53703e7	1.54707	0.773534	−	0.633755i	$-0.218487\pi$
$769$	2.53703e7	1.54707	0.773534	+	0.633755i	$0.218487\pi$
$770$	−1.51933e7	−0.923477
$771$	−2.49626e6	−0.151235
$772$	−1.26932e7	−0.766527
$773$	4.54316e6	0.273470	0.136735	−	0.990608i	$-0.456339\pi$
$773$	4.54316e6	0.273470	0.136735	+	0.990608i	$0.456339\pi$
$774$	−5.19793e6	−0.311873
$775$	1.85593e7	1.10996
$776$	1.18094e7	0.704002
$777$	−2.90840e6	−0.172823
$778$	1.13771e7	0.673879
$779$	6.27654e6	0.370575
$780$	2.40926e6	0.141791
$781$	−3.70672e7	−2.17452
$782$	407484.	0.0238283
$783$	1.10444e6	0.0643777
$784$	614656.	0.0357143
$785$	−3.37476e7	−1.95465
$786$	−7.73485e6	−0.446576
$787$	−2.17187e7	−1.24996	−0.624982	−	0.780639i	$-0.714894\pi$
$787$	−2.17187e7	−1.24996	−0.624982	+	0.780639i	$0.714894\pi$
$788$	6.33888e6	0.363661
$789$	1.13397e7	0.648497
$790$	2.90197e7	1.65434
$791$	7.46554e6	0.424248
$792$	−4.05907e6	−0.229940
$793$	149903.	0.00846501
$794$	1.09891e7	0.618599
$795$	550638.	0.0308993
$796$	9.98946e6	0.558804
$797$	−1.12329e7	−0.626395	−0.313197	−	0.949688i	$-0.601400\pi$
$797$	−1.12329e7	−0.626395	−0.313197	+	0.949688i	$0.601400\pi$
$798$	−2.52781e6	−0.140520
$799$	114840.	0.00636394
$800$	−6.83622e6	−0.377652
$801$	9.57566e6	0.527336
$802$	1.85448e7	1.01809
$803$	−5.08284e7	−2.78175
$804$	122976.	0.00670935
$805$	−1.49750e7	−0.814476
$806$	−1.87928e6	−0.101895
$807$	1.59694e7	0.863185
$808$	5.66093e6	0.305042
$809$	−3.51542e7	−1.88845	−0.944226	−	0.329298i	$-0.893188\pi$
$809$	−3.51542e7	−1.88845	−0.944226	+	0.329298i	$0.893188\pi$
$810$	−2.59816e6	−0.139140
$811$	−3.15188e7	−1.68274	−0.841372	−	0.540457i	$-0.818252\pi$
$811$	−3.15188e7	−1.68274	−0.841372	+	0.540457i	$0.818252\pi$
$812$	1.18776e6	0.0632177
$813$	4.58397e6	0.243229
$814$	2.06555e7	1.09264
$815$	−1.01198e6	−0.0533675
$816$	76032.0	0.00399734
$817$	2.29896e7	1.20497
$818$	1.82186e7	0.951988
$819$	670761.	0.0349428
$820$	6.93792e6	0.360325
$821$	1.58199e7	0.819115	0.409557	−	0.912284i	$-0.365683\pi$
$821$	1.58199e7	0.819115	0.409557	+	0.912284i	$0.365683\pi$
$822$	7.48537e6	0.386397
$823$	1.51130e7	0.777771	0.388886	−	0.921286i	$-0.372860\pi$
$823$	1.51130e7	0.777771	0.388886	+	0.921286i	$0.372860\pi$
$824$	−211904.	−0.0108723
$825$	4.70458e7	2.40650
$826$	1.02147e7	0.520927
$827$	4.27161e6	0.217184	0.108592	−	0.994086i	$-0.465366\pi$
$827$	4.27161e6	0.217184	0.108592	+	0.994086i	$0.465366\pi$
$828$	−4.00075e6	−0.202799
$829$	−2.64597e7	−1.33721	−0.668603	−	0.743620i	$-0.733107\pi$
$829$	−2.64597e7	−1.33721	−0.668603	+	0.743620i	$0.733107\pi$
$830$	3.84366e7	1.93664
$831$	−3.76378e6	−0.189070
$832$	692224.	0.0346688
$833$	79233.0	0.00395634
$834$	8.32910e6	0.414651
$835$	3.93837e7	1.95479
$836$	1.79526e7	0.888408
$837$	2.02662e6	0.0999905
$838$	8.06725e6	0.396840
$839$	−2.49933e7	−1.22580	−0.612900	−	0.790161i	$-0.709997\pi$
$839$	−2.49933e7	−1.22580	−0.612900	+	0.790161i	$0.709997\pi$
$840$	−2.79418e6	−0.136633
$841$	−1.82159e7	−0.888099
$842$	−8.04591e6	−0.391107
$843$	−9.84447e6	−0.477115
$844$	−3.39352e6	−0.163981
$845$	2.82754e6	0.136228
$846$	−1.12752e6	−0.0541625
$847$	2.21499e7	1.06087
$848$	158208.	0.00755508
$849$	2.23331e7	1.06336
$850$	−881232.	−0.0418353
$851$	2.03588e7	0.963668
$852$	−6.81696e6	−0.321730
$853$	1.10142e7	0.518299	0.259149	−	0.965837i	$-0.416558\pi$
$853$	1.10142e7	0.518299	0.259149	+	0.965837i	$0.416558\pi$
$854$	−173852.	−0.00815709
$855$	1.14912e7	0.537590
$856$	4.34573e6	0.202711
$857$	1.63028e7	0.758244	0.379122	−	0.925347i	$-0.376226\pi$
$857$	1.63028e7	0.758244	0.379122	+	0.925347i	$0.376226\pi$
$858$	−4.76377e6	−0.220919
$859$	934172.	0.0431960	0.0215980	−	0.999767i	$-0.493125\pi$
$859$	934172.	0.0431960	0.0215980	+	0.999767i	$0.493125\pi$
$860$	2.54121e7	1.17164
$861$	1.93158e6	0.0887983
$862$	1.25408e7	0.574853
$863$	3.36625e6	0.153858	0.0769288	−	0.997037i	$-0.475489\pi$
$863$	3.36625e6	0.153858	0.0769288	+	0.997037i	$0.475489\pi$
$864$	−746496.	−0.0340207
$865$	−3.65197e7	−1.65954
$866$	1.56373e7	0.708545
$867$	−1.27689e7	−0.576907
$868$	2.17952e6	0.0981887
$869$	−5.73798e7	−2.57757
$870$	−5.39946e6	−0.241853
$871$	144326.	0.00644613
$872$	319552.	0.0142315
$873$	−1.49463e7	−0.663739
$874$	1.76947e7	0.783546
$875$	1.72259e7	0.760610
$876$	−9.34776e6	−0.411573
$877$	−2.78648e7	−1.22337	−0.611684	−	0.791102i	$-0.709508\pi$
$877$	−2.78648e7	−1.22337	−0.611684	+	0.791102i	$0.709508\pi$
$878$	1.37817e7	0.603347
$879$	2.12668e7	0.928387
$880$	1.98444e7	0.863834
$881$	1.76751e7	0.767223	0.383611	−	0.923495i	$-0.374680\pi$
$881$	1.76751e7	0.767223	0.383611	+	0.923495i	$0.374680\pi$
$882$	−777924.	−0.0336718
$883$	1.80758e7	0.780183	0.390091	−	0.920776i	$-0.372443\pi$
$883$	1.80758e7	0.780183	0.390091	+	0.920776i	$0.372443\pi$
$884$	89232.0	0.00384052
$885$	−4.64354e7	−1.99292
$886$	2.14140e6	0.0916460
$887$	−2.04628e7	−0.873285	−0.436642	−	0.899635i	$-0.643833\pi$
$887$	−2.04628e7	−0.873285	−0.436642	+	0.899635i	$0.643833\pi$
$888$	3.79872e6	0.161661
$889$	−9.99120e6	−0.423997
$890$	−4.68143e7	−1.98109
$891$	5.13726e6	0.216789
$892$	1.51289e7	0.636640
$893$	4.98684e6	0.209265
$894$	−8.71819e6	−0.364823
$895$	−1.99845e7	−0.833943
$896$	−802816.	−0.0334077
$897$	−4.69533e6	−0.194843
$898$	−2.00320e7	−0.828961
$899$	4.21170e6	0.173803
$900$	8.65210e6	0.356053
$901$	20394.0	0.000836933 0
$902$	−1.37182e7	−0.561409
$903$	7.07496e6	0.288739
$904$	−9.75091e6	−0.396848
$905$	5.42728e7	2.20273
$906$	−9.65668e6	−0.390847
$907$	3.60854e7	1.45651	0.728254	−	0.685307i	$-0.240332\pi$
$907$	3.60854e7	1.45651	0.728254	+	0.685307i	$0.240332\pi$
$908$	7.40362e6	0.298009
$909$	−7.16461e6	−0.287596
$910$	−3.27928e6	−0.131273
$911$	−155841.	−0.00622137	−0.00311068	−	0.999995i	$-0.500990\pi$
$911$	−155841.	−0.00622137	−0.00311068	+	0.999995i	$0.500990\pi$
$912$	3.30163e6	0.131444
$913$	−7.59995e7	−3.01741
$914$	−3.75406e6	−0.148640
$915$	790317.	0.0312068
$916$	1.20417e7	0.474186
$917$	1.05280e7	0.413449
$918$	−96228.0	−0.00376873
$919$	−4.09049e7	−1.59767	−0.798834	−	0.601552i	$-0.794549\pi$
$919$	−4.09049e7	−1.59767	−0.798834	+	0.601552i	$0.794549\pi$
$920$	1.95592e7	0.761872
$921$	−6.48792e6	−0.252032
$922$	8.28380e6	0.320924
$923$	−8.00046e6	−0.309108
$924$	5.52485e6	0.212883
$925$	−4.40282e7	−1.69191
$926$	1.75210e7	0.671477
$927$	268191.	0.0102505
$928$	−1.55136e6	−0.0591347
$929$	1.22573e7	0.465969	0.232984	−	0.972480i	$-0.425151\pi$
$929$	1.22573e7	0.465969	0.232984	+	0.972480i	$0.425151\pi$
$930$	−9.90792e6	−0.375643
$931$	3.44063e6	0.130096
$932$	−1.18245e7	−0.445906
$933$	6.22075e6	0.233958
$934$	6.67288e6	0.250291
$935$	2.55806e6	0.0956934
$936$	−876096.	−0.0326860
$937$	−3.08207e7	−1.14682	−0.573408	−	0.819270i	$-0.694379\pi$
$937$	−3.08207e7	−1.14682	−0.573408	+	0.819270i	$0.694379\pi$
$938$	−167384.	−0.00621165
$939$	−2.04221e7	−0.755851
$940$	5.51232e6	0.203477
$941$	2.11450e7	0.778456	0.389228	−	0.921141i	$-0.372742\pi$
$941$	2.11450e7	0.778456	0.389228	+	0.921141i	$0.372742\pi$
$942$	1.22719e7	0.450592
$943$	−1.35211e7	−0.495144
$944$	−1.33417e7	−0.487283
$945$	3.53638e6	0.128819
$946$	−5.02467e7	−1.82549
$947$	1.97293e7	0.714888	0.357444	−	0.933935i	$-0.383648\pi$
$947$	1.97293e7	0.714888	0.357444	+	0.933935i	$0.383648\pi$
$948$	−1.05526e7	−0.381363
$949$	−1.09706e7	−0.395427
$950$	−3.82668e7	−1.37567
$951$	2.06818e7	0.741544
$952$	−103488.	−0.00370082
$953$	−2.94280e7	−1.04961	−0.524805	−	0.851222i	$-0.675862\pi$
$953$	−2.94280e7	−1.04961	−0.524805	+	0.851222i	$0.675862\pi$
$954$	−200232.	−0.00712299
$955$	2.88289e7	1.02287
$956$	−1.56403e6	−0.0553479
$957$	1.06762e7	0.376823
$958$	−3.36213e7	−1.18359
$959$	−1.01884e7	−0.357734
$960$	3.64954e6	0.127808
$961$	−2.09008e7	−0.730051
$962$	4.45822e6	0.155319
$963$	−5.50006e6	−0.191118
$964$	−1.97301e7	−0.683811
$965$	−7.85391e7	−2.71499
$966$	5.44547e6	0.187755
$967$	7.79389e6	0.268033	0.134016	−	0.990979i	$-0.457212\pi$
$967$	7.79389e6	0.268033	0.134016	+	0.990979i	$0.457212\pi$
$968$	−2.89304e7	−0.992354
$969$	425601.	0.0145611
$970$	7.30707e7	2.49353
$971$	−2.24472e7	−0.764038	−0.382019	−	0.924154i	$-0.624771\pi$
$971$	−2.24472e7	−0.764038	−0.382019	+	0.924154i	$0.624771\pi$
$972$	944784.	0.0320750
$973$	−1.13368e7	−0.383893
$974$	−2.18200e7	−0.736982
$975$	1.01542e7	0.342085
$976$	227072.	0.00763025
$977$	−5.22839e6	−0.175239	−0.0876196	−	0.996154i	$-0.527926\pi$
$977$	−5.22839e6	−0.175239	−0.0876196	+	0.996154i	$0.527926\pi$
$978$	367992.	0.0123024
$979$	9.25647e7	3.08666
$980$	3.80318e6	0.126498
$981$	−404433.	−0.0134176
$982$	−3.62843e7	−1.20071
$983$	−4.36835e7	−1.44190	−0.720948	−	0.692990i	$-0.756293\pi$
$983$	−4.36835e7	−1.44190	−0.720948	+	0.692990i	$0.756293\pi$
$984$	−2.52288e6	−0.0830632
$985$	3.92218e7	1.28806
$986$	−199980.	−0.00655080
$987$	1.53468e6	0.0501447
$988$	3.87483e6	0.126288
$989$	−4.95247e7	−1.61002
$990$	−2.51155e7	−0.814430
$991$	1.79892e7	0.581874	0.290937	−	0.956742i	$-0.406033\pi$
$991$	1.79892e7	0.581874	0.290937	+	0.956742i	$0.406033\pi$
$992$	−2.84672e6	−0.0918471
$993$	−2.22852e7	−0.717207
$994$	9.27864e6	0.297864
$995$	6.18098e7	1.97924
$996$	−1.39769e7	−0.446440
$997$	1.70256e7	0.542456	0.271228	−	0.962515i	$-0.412570\pi$
$997$	1.70256e7	0.542456	0.271228	+	0.962515i	$0.412570\pi$
$998$	1.85161e7	0.588467
$999$	−4.80776e6	−0.152415

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.6.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.6.a.b.1.1	✓	1	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:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	546.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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