LMFDB - Embedded newform 114.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [114,6,Mod(1,114)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(114, 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("114.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 114 = 2 \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	114.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:	$18.2837554587$
Analytic rank:	$1$
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	$\chi$	$=$	114.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +81.0000 q^{5} -36.0000 q^{6} -247.000 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q-4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +81.0000 q^{5} -36.0000 q^{6} -247.000 q^{7} -64.0000 q^{8} +81.0000 q^{9} -324.000 q^{10} -465.000 q^{11} +144.000 q^{12} -694.000 q^{13} +988.000 q^{14} +729.000 q^{15} +256.000 q^{16} +543.000 q^{17} -324.000 q^{18} +361.000 q^{19} +1296.00 q^{20} -2223.00 q^{21} +1860.00 q^{22} -2724.00 q^{23} -576.000 q^{24} +3436.00 q^{25} +2776.00 q^{26} +729.000 q^{27} -3952.00 q^{28} +342.000 q^{29} -2916.00 q^{30} -9442.00 q^{31} -1024.00 q^{32} -4185.00 q^{33} -2172.00 q^{34} -20007.0 q^{35} +1296.00 q^{36} +13088.0 q^{37} -1444.00 q^{38} -6246.00 q^{39} -5184.00 q^{40} -16272.0 q^{41} +8892.00 q^{42} -391.000 q^{43} -7440.00 q^{44} +6561.00 q^{45} +10896.0 q^{46} -8523.00 q^{47} +2304.00 q^{48} +44202.0 q^{49} -13744.0 q^{50} +4887.00 q^{51} -11104.0 q^{52} -10110.0 q^{53} -2916.00 q^{54} -37665.0 q^{55} +15808.0 q^{56} +3249.00 q^{57} -1368.00 q^{58} -27144.0 q^{59} +11664.0 q^{60} -48829.0 q^{61} +37768.0 q^{62} -20007.0 q^{63} +4096.00 q^{64} -56214.0 q^{65} +16740.0 q^{66} +55448.0 q^{67} +8688.00 q^{68} -24516.0 q^{69} +80028.0 q^{70} +43212.0 q^{71} -5184.00 q^{72} +37685.0 q^{73} -52352.0 q^{74} +30924.0 q^{75} +5776.00 q^{76} +114855. q^{77} +24984.0 q^{78} -78016.0 q^{79} +20736.0 q^{80} +6561.00 q^{81} +65088.0 q^{82} +83892.0 q^{83} -35568.0 q^{84} +43983.0 q^{85} +1564.00 q^{86} +3078.00 q^{87} +29760.0 q^{88} +25530.0 q^{89} -26244.0 q^{90} +171418. q^{91} -43584.0 q^{92} -84978.0 q^{93} +34092.0 q^{94} +29241.0 q^{95} -9216.00 q^{96} -76378.0 q^{97} -176808. q^{98} -37665.0 q^{99} +O(q^{100})\)

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
$5$	81.0000	1.44897	0.724486	−	0.689289i	$-0.242077\pi$
$5$	81.0000	1.44897	0.724486	+	0.689289i	$0.242077\pi$
$6$	−36.0000	−0.408248
$7$	−247.000	−1.90525	−0.952625	−	0.304148i	$-0.901628\pi$
$7$	−247.000	−1.90525	−0.952625	+	0.304148i	$0.901628\pi$
$8$	−64.0000	−0.353553
$9$	81.0000	0.333333
$10$	−324.000	−1.02458
$11$	−465.000	−1.15870	−0.579350	−	0.815079i	$-0.696694\pi$
$11$	−465.000	−1.15870	−0.579350	+	0.815079i	$0.696694\pi$
$12$	144.000	0.288675
$13$	−694.000	−1.13894	−0.569470	−	0.822012i	$-0.692852\pi$
$13$	−694.000	−1.13894	−0.569470	+	0.822012i	$0.692852\pi$
$14$	988.000	1.34721
$15$	729.000	0.836564
$16$	256.000	0.250000
$17$	543.000	0.455698	0.227849	−	0.973696i	$-0.426831\pi$
$17$	543.000	0.455698	0.227849	+	0.973696i	$0.426831\pi$
$18$	−324.000	−0.235702
$19$	361.000	0.229416
$19$	361.000	0.229416
$20$	1296.00	0.724486
$21$	−2223.00	−1.10000
$22$	1860.00	0.819325
$23$	−2724.00	−1.07371	−0.536856	−	0.843674i	$-0.680388\pi$
$23$	−2724.00	−1.07371	−0.536856	+	0.843674i	$0.680388\pi$
$24$	−576.000	−0.204124
$25$	3436.00	1.09952
$26$	2776.00	0.805353
$27$	729.000	0.192450
$28$	−3952.00	−0.952625
$29$	342.000	0.0755146	0.0377573	−	0.999287i	$-0.487979\pi$
$29$	342.000	0.0755146	0.0377573	+	0.999287i	$0.487979\pi$
$30$	−2916.00	−0.591540
$31$	−9442.00	−1.76465	−0.882327	−	0.470636i	$-0.844024\pi$
$31$	−9442.00	−1.76465	−0.882327	+	0.470636i	$0.844024\pi$
$32$	−1024.00	−0.176777
$33$	−4185.00	−0.668976
$34$	−2172.00	−0.322227
$35$	−20007.0	−2.76065
$36$	1296.00	0.166667
$37$	13088.0	1.57170	0.785849	−	0.618419i	$-0.212226\pi$
$37$	13088.0	1.57170	0.785849	+	0.618419i	$0.212226\pi$
$38$	−1444.00	−0.162221
$39$	−6246.00	−0.657568
$40$	−5184.00	−0.512289
$41$	−16272.0	−1.51175	−0.755877	−	0.654713i	$-0.772789\pi$
$41$	−16272.0	−1.51175	−0.755877	+	0.654713i	$0.772789\pi$
$42$	8892.00	0.777815
$43$	−391.000	−0.0322482	−0.0161241	−	0.999870i	$-0.505133\pi$
$43$	−391.000	−0.0322482	−0.0161241	+	0.999870i	$0.505133\pi$
$44$	−7440.00	−0.579350
$45$	6561.00	0.482991
$46$	10896.0	0.759229
$47$	−8523.00	−0.562792	−0.281396	−	0.959592i	$-0.590797\pi$
$47$	−8523.00	−0.562792	−0.281396	+	0.959592i	$0.590797\pi$
$48$	2304.00	0.144338
$49$	44202.0	2.62998
$50$	−13744.0	−0.777478
$51$	4887.00	0.263098
$52$	−11104.0	−0.569470
$53$	−10110.0	−0.494381	−0.247190	−	0.968967i	$-0.579507\pi$
$53$	−10110.0	−0.494381	−0.247190	+	0.968967i	$0.579507\pi$
$54$	−2916.00	−0.136083
$55$	−37665.0	−1.67892
$56$	15808.0	0.673607
$57$	3249.00	0.132453
$58$	−1368.00	−0.0533969
$59$	−27144.0	−1.01518	−0.507591	−	0.861598i	$-0.669464\pi$
$59$	−27144.0	−1.01518	−0.507591	+	0.861598i	$0.669464\pi$
$60$	11664.0	0.418282
$61$	−48829.0	−1.68017	−0.840085	−	0.542455i	$-0.817495\pi$
$61$	−48829.0	−1.68017	−0.840085	+	0.542455i	$0.817495\pi$
$62$	37768.0	1.24780
$63$	−20007.0	−0.635083
$64$	4096.00	0.125000
$65$	−56214.0	−1.65029
$66$	16740.0	0.473038
$67$	55448.0	1.50903	0.754517	−	0.656281i	$-0.227871\pi$
$67$	55448.0	1.50903	0.754517	+	0.656281i	$0.227871\pi$
$68$	8688.00	0.227849
$69$	−24516.0	−0.619907
$70$	80028.0	1.95208
$71$	43212.0	1.01732	0.508661	−	0.860967i	$-0.330141\pi$
$71$	43212.0	1.01732	0.508661	+	0.860967i	$0.330141\pi$
$72$	−5184.00	−0.117851
$73$	37685.0	0.827678	0.413839	−	0.910350i	$-0.364188\pi$
$73$	37685.0	0.827678	0.413839	+	0.910350i	$0.364188\pi$
$74$	−52352.0	−1.11136
$75$	30924.0	0.634808
$76$	5776.00	0.114708
$77$	114855.	2.20761
$78$	24984.0	0.464971
$79$	−78016.0	−1.40642	−0.703211	−	0.710981i	$-0.748251\pi$
$79$	−78016.0	−1.40642	−0.703211	+	0.710981i	$0.748251\pi$
$80$	20736.0	0.362243
$81$	6561.00	0.111111
$82$	65088.0	1.06897
$83$	83892.0	1.33667	0.668337	−	0.743859i	$-0.267007\pi$
$83$	83892.0	1.33667	0.668337	+	0.743859i	$0.267007\pi$
$84$	−35568.0	−0.549998
$85$	43983.0	0.660294
$86$	1564.00	0.0228029
$87$	3078.00	0.0435984
$88$	29760.0	0.409663
$89$	25530.0	0.341646	0.170823	−	0.985302i	$-0.445357\pi$
$89$	25530.0	0.341646	0.170823	+	0.985302i	$0.445357\pi$
$90$	−26244.0	−0.341526
$91$	171418.	2.16997
$92$	−43584.0	−0.536856
$93$	−84978.0	−1.01882
$94$	34092.0	0.397954
$95$	29241.0	0.332417
$96$	−9216.00	−0.102062
$97$	−76378.0	−0.824212	−0.412106	−	0.911136i	$-0.635207\pi$
$97$	−76378.0	−0.824212	−0.412106	+	0.911136i	$0.635207\pi$
$98$	−176808.	−1.85967
$99$	−37665.0	−0.386234
$100$	54976.0	0.549760
$101$	−132606.	−1.29348	−0.646740	−	0.762711i	$-0.723868\pi$
$101$	−132606.	−1.29348	−0.646740	+	0.762711i	$0.723868\pi$
$102$	−19548.0	−0.186038
$103$	67994.0	0.631506	0.315753	−	0.948841i	$-0.397743\pi$
$103$	67994.0	0.631506	0.315753	+	0.948841i	$0.397743\pi$
$104$	44416.0	0.402676
$105$	−180063.	−1.59386
$106$	40440.0	0.349580
$107$	−87606.0	−0.739732	−0.369866	−	0.929085i	$-0.620596\pi$
$107$	−87606.0	−0.739732	−0.369866	+	0.929085i	$0.620596\pi$
$108$	11664.0	0.0962250
$109$	75908.0	0.611958	0.305979	−	0.952038i	$-0.401016\pi$
$109$	75908.0	0.611958	0.305979	+	0.952038i	$0.401016\pi$
$110$	150660.	1.18718
$111$	117792.	0.907420
$112$	−63232.0	−0.476312
$113$	50946.0	0.375331	0.187665	−	0.982233i	$-0.439908\pi$
$113$	50946.0	0.375331	0.187665	+	0.982233i	$0.439908\pi$
$114$	−12996.0	−0.0936586
$115$	−220644.	−1.55578
$116$	5472.00	0.0377573
$117$	−56214.0	−0.379647
$118$	108576.	0.717842
$119$	−134121.	−0.868219
$120$	−46656.0	−0.295770
$121$	55174.0	0.342587
$122$	195316.	1.18806
$123$	−146448.	−0.872812
$124$	−151072.	−0.882327
$125$	25191.0	0.144202
$126$	80028.0	0.449072
$127$	116282.	0.639740	0.319870	−	0.947462i	$-0.396361\pi$
$127$	116282.	0.639740	0.319870	+	0.947462i	$0.396361\pi$
$128$	−16384.0	−0.0883883
$129$	−3519.00	−0.0186185
$130$	224856.	1.16693
$131$	−172215.	−0.876784	−0.438392	−	0.898784i	$-0.644452\pi$
$131$	−172215.	−0.876784	−0.438392	+	0.898784i	$0.644452\pi$
$132$	−66960.0	−0.334488
$133$	−89167.0	−0.437094
$134$	−221792.	−1.06705
$135$	59049.0	0.278855
$136$	−34752.0	−0.161114
$137$	−10593.0	−0.0482189	−0.0241095	−	0.999709i	$-0.507675\pi$
$137$	−10593.0	−0.0482189	−0.0241095	+	0.999709i	$0.507675\pi$
$138$	98064.0	0.438341
$139$	−240427.	−1.05547	−0.527735	−	0.849409i	$-0.676959\pi$
$139$	−240427.	−1.05547	−0.527735	+	0.849409i	$0.676959\pi$
$140$	−320112.	−1.38033
$141$	−76707.0	−0.324928
$142$	−172848.	−0.719355
$143$	322710.	1.31969
$144$	20736.0	0.0833333
$145$	27702.0	0.109419
$146$	−150740.	−0.585257
$147$	397818.	1.51842
$148$	209408.	0.785849
$149$	109935.	0.405668	0.202834	−	0.979213i	$-0.434985\pi$
$149$	109935.	0.405668	0.202834	+	0.979213i	$0.434985\pi$
$150$	−123696.	−0.448877
$151$	537008.	1.91663	0.958315	−	0.285713i	$-0.0922304\pi$
$151$	537008.	1.91663	0.958315	+	0.285713i	$0.0922304\pi$
$152$	−23104.0	−0.0811107
$153$	43983.0	0.151899
$154$	−459420.	−1.56102
$155$	−764802.	−2.55694
$156$	−99936.0	−0.328784
$157$	158606.	0.513536	0.256768	−	0.966473i	$-0.417342\pi$
$157$	158606.	0.513536	0.256768	+	0.966473i	$0.417342\pi$
$158$	312064.	0.994491
$159$	−90990.0	−0.285431
$160$	−82944.0	−0.256144
$161$	672828.	2.04569
$162$	−26244.0	−0.0785674
$163$	249968.	0.736912	0.368456	−	0.929645i	$-0.379887\pi$
$163$	249968.	0.736912	0.368456	+	0.929645i	$0.379887\pi$
$164$	−260352.	−0.755877
$165$	−338985.	−0.969328
$166$	−335568.	−0.945171
$167$	73038.0	0.202655	0.101328	−	0.994853i	$-0.467691\pi$
$167$	73038.0	0.202655	0.101328	+	0.994853i	$0.467691\pi$
$168$	142272.	0.388907
$169$	110343.	0.297186
$170$	−175932.	−0.466899
$171$	29241.0	0.0764719
$172$	−6256.00	−0.0161241
$173$	67182.0	0.170662	0.0853312	−	0.996353i	$-0.472805\pi$
$173$	67182.0	0.170662	0.0853312	+	0.996353i	$0.472805\pi$
$174$	−12312.0	−0.0308287
$175$	−848692.	−2.09486
$176$	−119040.	−0.289675
$177$	−244296.	−0.586115
$178$	−102120.	−0.241580
$179$	525330.	1.22546	0.612731	−	0.790292i	$-0.290071\pi$
$179$	525330.	1.22546	0.612731	+	0.790292i	$0.290071\pi$
$180$	104976.	0.241495
$181$	−74662.0	−0.169396	−0.0846980	−	0.996407i	$-0.526993\pi$
$181$	−74662.0	−0.169396	−0.0846980	+	0.996407i	$0.526993\pi$
$182$	−685672.	−1.53440
$183$	−439461.	−0.970047
$184$	174336.	0.379614
$185$	1.06013e6	2.27735
$186$	339912.	0.720417
$187$	−252495.	−0.528018
$188$	−136368.	−0.281396
$189$	−180063.	−0.366665
$190$	−116964.	−0.235054
$191$	404355.	0.802009	0.401005	−	0.916076i	$-0.368661\pi$
$191$	404355.	0.802009	0.401005	+	0.916076i	$0.368661\pi$
$192$	36864.0	0.0721688
$193$	835748.	1.61504	0.807518	−	0.589843i	$-0.200810\pi$
$193$	835748.	1.61504	0.807518	+	0.589843i	$0.200810\pi$
$194$	305512.	0.582806
$195$	−505926.	−0.952797
$196$	707232.	1.31499
$197$	−191682.	−0.351897	−0.175949	−	0.984399i	$-0.556299\pi$
$197$	−191682.	−0.351897	−0.175949	+	0.984399i	$0.556299\pi$
$198$	150660.	0.273108
$199$	−343231.	−0.614404	−0.307202	−	0.951644i	$-0.599393\pi$
$199$	−343231.	−0.614404	−0.307202	+	0.951644i	$0.599393\pi$
$200$	−219904.	−0.388739
$201$	499032.	0.871241
$202$	530424.	0.914629
$203$	−84474.0	−0.143874
$204$	78192.0	0.131549
$205$	−1.31803e6	−2.19049
$206$	−271976.	−0.446542
$207$	−220644.	−0.357904
$208$	−177664.	−0.284735
$209$	−167865.	−0.265824
$210$	720252.	1.12703
$211$	−353116.	−0.546023	−0.273012	−	0.962011i	$-0.588020\pi$
$211$	−353116.	−0.546023	−0.273012	+	0.962011i	$0.588020\pi$
$212$	−161760.	−0.247190
$213$	388908.	0.587351
$214$	350424.	0.523070
$215$	−31671.0	−0.0467268
$216$	−46656.0	−0.0680414
$217$	2.33217e6	3.36211
$218$	−303632.	−0.432719
$219$	339165.	0.477860
$220$	−602640.	−0.839462
$221$	−376842.	−0.519013
$222$	−471168.	−0.641643
$223$	−443884.	−0.597733	−0.298867	−	0.954295i	$-0.596609\pi$
$223$	−443884.	−0.597733	−0.298867	+	0.954295i	$0.596609\pi$
$224$	252928.	0.336804
$225$	278316.	0.366507
$226$	−203784.	−0.265399
$227$	−110130.	−0.141854	−0.0709269	−	0.997482i	$-0.522596\pi$
$227$	−110130.	−0.141854	−0.0709269	+	0.997482i	$0.522596\pi$
$228$	51984.0	0.0662266
$229$	−543979.	−0.685478	−0.342739	−	0.939431i	$-0.611355\pi$
$229$	−543979.	−0.685478	−0.342739	+	0.939431i	$0.611355\pi$
$230$	882576.	1.10010
$231$	1.03369e6	1.27457
$232$	−21888.0	−0.0266985
$233$	−842991.	−1.01726	−0.508631	−	0.860984i	$-0.669848\pi$
$233$	−842991.	−1.01726	−0.508631	+	0.860984i	$0.669848\pi$
$234$	224856.	0.268451
$235$	−690363.	−0.815470
$236$	−434304.	−0.507591
$237$	−702144.	−0.811999
$238$	536484.	0.613924
$239$	1.08729e6	1.23126	0.615630	−	0.788036i	$-0.288902\pi$
$239$	1.08729e6	1.23126	0.615630	+	0.788036i	$0.288902\pi$
$240$	186624.	0.209141
$241$	392036.	0.434794	0.217397	−	0.976083i	$-0.430243\pi$
$241$	392036.	0.434794	0.217397	+	0.976083i	$0.430243\pi$
$242$	−220696.	−0.242246
$243$	59049.0	0.0641500
$244$	−781264.	−0.840085
$245$	3.58036e6	3.81076
$246$	585792.	0.617171
$247$	−250534.	−0.261291
$248$	604288.	0.623900
$249$	755028.	0.771729
$250$	−100764.	−0.101966
$251$	−1.25925e6	−1.26161	−0.630807	−	0.775940i	$-0.717276\pi$
$251$	−1.25925e6	−1.26161	−0.630807	+	0.775940i	$0.717276\pi$
$252$	−320112.	−0.317542
$253$	1.26666e6	1.24411
$254$	−465128.	−0.452364
$255$	395847.	0.381221
$256$	65536.0	0.0625000
$257$	−22200.0	−0.0209662	−0.0104831	−	0.999945i	$-0.503337\pi$
$257$	−22200.0	−0.0209662	−0.0104831	+	0.999945i	$0.503337\pi$
$258$	14076.0	0.0131653
$259$	−3.23274e6	−2.99448
$260$	−899424.	−0.825147
$261$	27702.0	0.0251715
$262$	688860.	0.619980
$263$	−1.76991e6	−1.57783	−0.788917	−	0.614500i	$-0.789358\pi$
$263$	−1.76991e6	−1.57783	−0.788917	+	0.614500i	$0.789358\pi$
$264$	267840.	0.236519
$265$	−818910.	−0.716344
$266$	356668.	0.309072
$267$	229770.	0.197249
$268$	887168.	0.754517
$269$	333210.	0.280761	0.140381	−	0.990098i	$-0.455167\pi$
$269$	333210.	0.280761	0.140381	+	0.990098i	$0.455167\pi$
$270$	−236196.	−0.197180
$271$	1.21056e6	1.00129	0.500647	−	0.865652i	$-0.333095\pi$
$271$	1.21056e6	1.00129	0.500647	+	0.865652i	$0.333095\pi$
$272$	139008.	0.113925
$273$	1.54276e6	1.25283
$274$	42372.0	0.0340959
$275$	−1.59774e6	−1.27401
$276$	−392256.	−0.309954
$277$	51449.0	0.0402882	0.0201441	−	0.999797i	$-0.493588\pi$
$277$	51449.0	0.0402882	0.0201441	+	0.999797i	$0.493588\pi$
$278$	961708.	0.746331
$279$	−764802.	−0.588218
$280$	1.28045e6	0.976038
$281$	1.57761e6	1.19188	0.595942	−	0.803028i	$-0.296779\pi$
$281$	1.57761e6	1.19188	0.595942	+	0.803028i	$0.296779\pi$
$282$	306828.	0.229759
$283$	833525.	0.618661	0.309330	−	0.950955i	$-0.399895\pi$
$283$	833525.	0.618661	0.309330	+	0.950955i	$0.399895\pi$
$284$	691392.	0.508661
$285$	263169.	0.191921
$286$	−1.29084e6	−0.933163
$287$	4.01918e6	2.88027
$288$	−82944.0	−0.0589256
$289$	−1.12501e6	−0.792339
$290$	−110808.	−0.0773706
$291$	−687402.	−0.475859
$292$	602960.	0.413839
$293$	−2.86547e6	−1.94996	−0.974982	−	0.222285i	$-0.928649\pi$
$293$	−2.86547e6	−1.94996	−0.974982	+	0.222285i	$0.928649\pi$
$294$	−1.59127e6	−1.07368
$295$	−2.19866e6	−1.47097
$296$	−837632.	−0.555679
$297$	−338985.	−0.222992
$298$	−439740.	−0.286850
$299$	1.89046e6	1.22289
$300$	494784.	0.317404
$301$	96577.0	0.0614409
$302$	−2.14803e6	−1.35526
$303$	−1.19345e6	−0.746791
$304$	92416.0	0.0573539
$305$	−3.95515e6	−2.43452
$306$	−175932.	−0.107409
$307$	−418228.	−0.253260	−0.126630	−	0.991950i	$-0.540416\pi$
$307$	−418228.	−0.253260	−0.126630	+	0.991950i	$0.540416\pi$
$308$	1.83768e6	1.10381
$309$	611946.	0.364600
$310$	3.05921e6	1.80803
$311$	−1.76735e6	−1.03615	−0.518074	−	0.855336i	$-0.673351\pi$
$311$	−1.76735e6	−1.03615	−0.518074	+	0.855336i	$0.673351\pi$
$312$	399744.	0.232485
$313$	−1.64835e6	−0.951020	−0.475510	−	0.879710i	$-0.657736\pi$
$313$	−1.64835e6	−0.951020	−0.475510	+	0.879710i	$0.657736\pi$
$314$	−634424.	−0.363124
$315$	−1.62057e6	−0.920218
$316$	−1.24826e6	−0.703211
$317$	−1.94101e6	−1.08488	−0.542438	−	0.840096i	$-0.682499\pi$
$317$	−1.94101e6	−1.08488	−0.542438	+	0.840096i	$0.682499\pi$
$318$	363960.	0.201830
$319$	−159030.	−0.0874989
$320$	331776.	0.181122
$321$	−788454.	−0.427084
$322$	−2.69131e6	−1.44652
$323$	196023.	0.104544
$324$	104976.	0.0555556
$325$	−2.38458e6	−1.25229
$326$	−999872.	−0.521075
$327$	683172.	0.353314
$328$	1.04141e6	0.534486
$329$	2.10518e6	1.07226
$330$	1.35594e6	0.685418
$331$	−1.22566e6	−0.614894	−0.307447	−	0.951565i	$-0.599475\pi$
$331$	−1.22566e6	−0.614894	−0.307447	+	0.951565i	$0.599475\pi$
$332$	1.34227e6	0.668337
$333$	1.06013e6	0.523899
$334$	−292152.	−0.143299
$335$	4.49129e6	2.18655
$336$	−569088.	−0.274999
$337$	−1.14470e6	−0.549057	−0.274529	−	0.961579i	$-0.588522\pi$
$337$	−1.14470e6	−0.549057	−0.274529	+	0.961579i	$0.588522\pi$
$338$	−441372.	−0.210142
$339$	458514.	0.216697
$340$	703728.	0.330147
$341$	4.39053e6	2.04471
$342$	−116964.	−0.0540738
$343$	−6.76656e6	−3.10551
$344$	25024.0	0.0114015
$345$	−1.98580e6	−0.898229
$346$	−268728.	−0.120676
$347$	−3.51661e6	−1.56784	−0.783919	−	0.620863i	$-0.786782\pi$
$347$	−3.51661e6	−1.56784	−0.783919	+	0.620863i	$0.786782\pi$
$348$	49248.0	0.0217992
$349$	611789.	0.268867	0.134434	−	0.990923i	$-0.457078\pi$
$349$	611789.	0.268867	0.134434	+	0.990923i	$0.457078\pi$
$350$	3.39477e6	1.48129
$351$	−505926.	−0.219189
$352$	476160.	0.204831
$353$	1.49043e6	0.636612	0.318306	−	0.947988i	$-0.396886\pi$
$353$	1.49043e6	0.636612	0.318306	+	0.947988i	$0.396886\pi$
$354$	977184.	0.414446
$355$	3.50017e6	1.47407
$356$	408480.	0.170823
$357$	−1.20709e6	−0.501267
$358$	−2.10132e6	−0.866532
$359$	−1.79830e6	−0.736423	−0.368211	−	0.929742i	$-0.620030\pi$
$359$	−1.79830e6	−0.736423	−0.368211	+	0.929742i	$0.620030\pi$
$360$	−419904.	−0.170763
$361$	130321.	0.0526316
$362$	298648.	0.119781
$363$	496566.	0.197793
$364$	2.74269e6	1.08498
$365$	3.05248e6	1.19928
$366$	1.75784e6	0.685927
$367$	−453136.	−0.175616	−0.0878079	−	0.996137i	$-0.527986\pi$
$367$	−453136.	−0.175616	−0.0878079	+	0.996137i	$0.527986\pi$
$368$	−697344.	−0.268428
$369$	−1.31803e6	−0.503918
$370$	−4.24051e6	−1.61033
$371$	2.49717e6	0.941919
$372$	−1.35965e6	−0.509412
$373$	−2.01030e6	−0.748152	−0.374076	−	0.927398i	$-0.622040\pi$
$373$	−2.01030e6	−0.748152	−0.374076	+	0.927398i	$0.622040\pi$
$374$	1.00998e6	0.373365
$375$	226719.	0.0832549
$376$	545472.	0.198977
$377$	−237348.	−0.0860067
$378$	720252.	0.259272
$379$	2.02633e6	0.724624	0.362312	−	0.932057i	$-0.381987\pi$
$379$	2.02633e6	0.724624	0.362312	+	0.932057i	$0.381987\pi$
$380$	467856.	0.166208
$381$	1.04654e6	0.369354
$382$	−1.61742e6	−0.567106
$383$	−143910.	−0.0501296	−0.0250648	−	0.999686i	$-0.507979\pi$
$383$	−143910.	−0.0501296	−0.0250648	+	0.999686i	$0.507979\pi$
$384$	−147456.	−0.0510310
$385$	9.30326e6	3.19877
$386$	−3.34299e6	−1.14200
$387$	−31671.0	−0.0107494
$388$	−1.22205e6	−0.412106
$389$	−4.49337e6	−1.50556	−0.752780	−	0.658273i	$-0.771287\pi$
$389$	−4.49337e6	−1.50556	−0.752780	+	0.658273i	$0.771287\pi$
$390$	2.02370e6	0.673729
$391$	−1.47913e6	−0.489289
$392$	−2.82893e6	−0.929837
$393$	−1.54994e6	−0.506212
$394$	766728.	0.248829
$395$	−6.31930e6	−2.03787
$396$	−602640.	−0.193117
$397$	4.70645e6	1.49871	0.749355	−	0.662169i	$-0.230364\pi$
$397$	4.70645e6	1.49871	0.749355	+	0.662169i	$0.230364\pi$
$398$	1.37292e6	0.434449
$399$	−802503.	−0.252356
$400$	879616.	0.274880
$401$	−2.85534e6	−0.886741	−0.443371	−	0.896338i	$-0.646217\pi$
$401$	−2.85534e6	−0.886741	−0.443371	+	0.896338i	$0.646217\pi$
$402$	−1.99613e6	−0.616060
$403$	6.55275e6	2.00984
$404$	−2.12170e6	−0.646740
$405$	531441.	0.160997
$406$	337896.	0.101734
$407$	−6.08592e6	−1.82113
$408$	−312768.	−0.0930191
$409$	−3.50069e6	−1.03477	−0.517386	−	0.855752i	$-0.673095\pi$
$409$	−3.50069e6	−1.03477	−0.517386	+	0.855752i	$0.673095\pi$
$410$	5.27213e6	1.54891
$411$	−95337.0	−0.0278392
$412$	1.08790e6	0.315753
$413$	6.70457e6	1.93417
$414$	882576.	0.253076
$415$	6.79525e6	1.93680
$416$	710656.	0.201338
$417$	−2.16384e6	−0.609376
$418$	671460.	0.187966
$419$	−5.67707e6	−1.57975	−0.789876	−	0.613266i	$-0.789855\pi$
$419$	−5.67707e6	−1.57975	−0.789876	+	0.613266i	$0.789855\pi$
$420$	−2.88101e6	−0.796932
$421$	263714.	0.0725150	0.0362575	−	0.999342i	$-0.488456\pi$
$421$	263714.	0.0725150	0.0362575	+	0.999342i	$0.488456\pi$
$422$	1.41246e6	0.386097
$423$	−690363.	−0.187597
$424$	647040.	0.174790
$425$	1.86575e6	0.501050
$426$	−1.55563e6	−0.415320
$427$	1.20608e7	3.20114
$428$	−1.40170e6	−0.369866
$429$	2.90439e6	0.761924
$430$	126684.	0.0330408
$431$	−1.17097e6	−0.303634	−0.151817	−	0.988409i	$-0.548513\pi$
$431$	−1.17097e6	−0.303634	−0.151817	+	0.988409i	$0.548513\pi$
$432$	186624.	0.0481125
$433$	5.82657e6	1.49346	0.746729	−	0.665129i	$-0.231623\pi$
$433$	5.82657e6	1.49346	0.746729	+	0.665129i	$0.231623\pi$
$434$	−9.32870e6	−2.37737
$435$	249318.	0.0631729
$436$	1.21453e6	0.305979
$437$	−983364.	−0.246326
$438$	−1.35666e6	−0.337898
$439$	−892210.	−0.220956	−0.110478	−	0.993879i	$-0.535238\pi$
$439$	−892210.	−0.220956	−0.110478	+	0.993879i	$0.535238\pi$
$440$	2.41056e6	0.593590
$441$	3.58036e6	0.876659
$442$	1.50737e6	0.366998
$443$	2.33954e6	0.566398	0.283199	−	0.959061i	$-0.408604\pi$
$443$	2.33954e6	0.566398	0.283199	+	0.959061i	$0.408604\pi$
$444$	1.88467e6	0.453710
$445$	2.06793e6	0.495035
$446$	1.77554e6	0.422661
$447$	989415.	0.234212
$448$	−1.01171e6	−0.238156
$449$	2.82784e6	0.661970	0.330985	−	0.943636i	$-0.392619\pi$
$449$	2.82784e6	0.661970	0.330985	+	0.943636i	$0.392619\pi$
$450$	−1.11326e6	−0.259159
$451$	7.56648e6	1.75167
$452$	815136.	0.187665
$453$	4.83307e6	1.10657
$454$	440520.	0.100306
$455$	1.38849e7	3.14422
$456$	−207936.	−0.0468293
$457$	287195.	0.0643260	0.0321630	−	0.999483i	$-0.489760\pi$
$457$	287195.	0.0643260	0.0321630	+	0.999483i	$0.489760\pi$
$458$	2.17592e6	0.484706
$459$	395847.	0.0876992
$460$	−3.53030e6	−0.777889
$461$	5.47137e6	1.19907	0.599534	−	0.800350i	$-0.295353\pi$
$461$	5.47137e6	1.19907	0.599534	+	0.800350i	$0.295353\pi$
$462$	−4.13478e6	−0.901255
$463$	6.44627e6	1.39751	0.698756	−	0.715360i	$-0.253737\pi$
$463$	6.44627e6	1.39751	0.698756	+	0.715360i	$0.253737\pi$
$464$	87552.0	0.0188787
$465$	−6.88322e6	−1.47625
$466$	3.37196e6	0.719313
$467$	−7.21695e6	−1.53130	−0.765652	−	0.643255i	$-0.777583\pi$
$467$	−7.21695e6	−1.53130	−0.765652	+	0.643255i	$0.777583\pi$
$468$	−899424.	−0.189823
$469$	−1.36957e7	−2.87509
$470$	2.76145e6	0.576624
$471$	1.42745e6	0.296490
$472$	1.73722e6	0.358921
$473$	181815.	0.0373660
$474$	2.80858e6	0.574170
$475$	1.24040e6	0.252247
$476$	−2.14594e6	−0.434110
$477$	−818910.	−0.164794
$478$	−4.34915e6	−0.870632
$479$	−5.96484e6	−1.18785	−0.593923	−	0.804522i	$-0.702422\pi$
$479$	−5.96484e6	−1.18785	−0.593923	+	0.804522i	$0.702422\pi$
$480$	−746496.	−0.147885
$481$	−9.08307e6	−1.79007
$482$	−1.56814e6	−0.307446
$483$	6.05545e6	1.18108
$484$	882784.	0.171294
$485$	−6.18662e6	−1.19426
$486$	−236196.	−0.0453609
$487$	−5.30728e6	−1.01403	−0.507014	−	0.861938i	$-0.669251\pi$
$487$	−5.30728e6	−1.01403	−0.507014	+	0.861938i	$0.669251\pi$
$488$	3.12506e6	0.594030
$489$	2.24971e6	0.425456
$490$	−1.43214e7	−2.69462
$491$	−6.47410e6	−1.21192	−0.605962	−	0.795494i	$-0.707212\pi$
$491$	−6.47410e6	−1.21192	−0.605962	+	0.795494i	$0.707212\pi$
$492$	−2.34317e6	−0.436406
$493$	185706.	0.0344119
$494$	1.00214e6	0.184761
$495$	−3.05086e6	−0.559642
$496$	−2.41715e6	−0.441164
$497$	−1.06734e7	−1.93825
$498$	−3.02011e6	−0.545695
$499$	3.10316e6	0.557896	0.278948	−	0.960306i	$-0.410014\pi$
$499$	3.10316e6	0.557896	0.278948	+	0.960306i	$0.410014\pi$
$500$	403056.	0.0721008
$501$	657342.	0.117003
$502$	5.03699e6	0.892096
$503$	−9.26422e6	−1.63263	−0.816317	−	0.577604i	$-0.803988\pi$
$503$	−9.26422e6	−1.63263	−0.816317	+	0.577604i	$0.803988\pi$
$504$	1.28045e6	0.224536
$505$	−1.07411e7	−1.87422
$506$	−5.06664e6	−0.879719
$507$	993087.	0.171580
$508$	1.86051e6	0.319870
$509$	3.70270e6	0.633467	0.316734	−	0.948514i	$-0.397414\pi$
$509$	3.70270e6	0.633467	0.316734	+	0.948514i	$0.397414\pi$
$510$	−1.58339e6	−0.269564
$511$	−9.30820e6	−1.57693
$512$	−262144.	−0.0441942
$513$	263169.	0.0441511
$514$	88800.0	0.0148254
$515$	5.50751e6	0.915035
$516$	−56304.0	−0.00930926
$517$	3.96320e6	0.652107
$518$	1.29309e7	2.11741
$519$	604638.	0.0985319
$520$	3.59770e6	0.583467
$521$	8.31762e6	1.34247	0.671235	−	0.741244i	$-0.265764\pi$
$521$	8.31762e6	1.34247	0.671235	+	0.741244i	$0.265764\pi$
$522$	−110808.	−0.0177990
$523$	1.10321e7	1.76362	0.881810	−	0.471604i	$-0.156325\pi$
$523$	1.10321e7	1.76362	0.881810	+	0.471604i	$0.156325\pi$
$524$	−2.75544e6	−0.438392
$525$	−7.63823e6	−1.20947
$526$	7.07963e6	1.11570
$527$	−5.12701e6	−0.804150
$528$	−1.07136e6	−0.167244
$529$	983833.	0.152856
$530$	3.27564e6	0.506532
$531$	−2.19866e6	−0.338394
$532$	−1.42667e6	−0.218547
$533$	1.12928e7	1.72180
$534$	−919080.	−0.139476
$535$	−7.09609e6	−1.07185
$536$	−3.54867e6	−0.533524
$537$	4.72797e6	0.707520
$538$	−1.33284e6	−0.198528
$539$	−2.05539e7	−3.04735
$540$	944784.	0.139427
$541$	−1.09731e7	−1.61189	−0.805947	−	0.591987i	$-0.798344\pi$
$541$	−1.09731e7	−1.61189	−0.805947	+	0.591987i	$0.798344\pi$
$542$	−4.84222e6	−0.708022
$543$	−671958.	−0.0978008
$544$	−556032.	−0.0805569
$545$	6.14855e6	0.886709
$546$	−6.17105e6	−0.885885
$547$	1.71559e6	0.245158	0.122579	−	0.992459i	$-0.460884\pi$
$547$	1.71559e6	0.245158	0.122579	+	0.992459i	$0.460884\pi$
$548$	−169488.	−0.0241095
$549$	−3.95515e6	−0.560057
$550$	6.39096e6	0.900864
$551$	123462.	0.0173242
$552$	1.56902e6	0.219170
$553$	1.92700e7	2.67959
$554$	−205796.	−0.0284880
$555$	9.54115e6	1.31483
$556$	−3.84683e6	−0.527735
$557$	582603.	0.0795673	0.0397837	−	0.999208i	$-0.487333\pi$
$557$	582603.	0.0795673	0.0397837	+	0.999208i	$0.487333\pi$
$558$	3.05921e6	0.415933
$559$	271354.	0.0367288
$560$	−5.12179e6	−0.690163
$561$	−2.27246e6	−0.304851
$562$	−6.31044e6	−0.842789
$563$	8.98868e6	1.19516	0.597579	−	0.801810i	$-0.296130\pi$
$563$	8.98868e6	1.19516	0.597579	+	0.801810i	$0.296130\pi$
$564$	−1.22731e6	−0.162464
$565$	4.12663e6	0.543844
$566$	−3.33410e6	−0.437459
$567$	−1.62057e6	−0.211694
$568$	−2.76557e6	−0.359678
$569$	−539826.	−0.0698993	−0.0349497	−	0.999389i	$-0.511127\pi$
$569$	−539826.	−0.0698993	−0.0349497	+	0.999389i	$0.511127\pi$
$570$	−1.05268e6	−0.135709
$571$	−498448.	−0.0639778	−0.0319889	−	0.999488i	$-0.510184\pi$
$571$	−498448.	−0.0639778	−0.0319889	+	0.999488i	$0.510184\pi$
$572$	5.16336e6	0.659846
$573$	3.63920e6	0.463040
$574$	−1.60767e7	−2.03666
$575$	−9.35966e6	−1.18057
$576$	331776.	0.0416667
$577$	1.95388e6	0.244319	0.122160	−	0.992510i	$-0.461018\pi$
$577$	1.95388e6	0.244319	0.122160	+	0.992510i	$0.461018\pi$
$578$	4.50003e6	0.560268
$579$	7.52173e6	0.932441
$580$	443232.	0.0547093
$581$	−2.07213e7	−2.54670
$582$	2.74961e6	0.336483
$583$	4.70115e6	0.572839
$584$	−2.41184e6	−0.292628
$585$	−4.55333e6	−0.550098
$586$	1.14619e7	1.37883
$587$	2.34205e6	0.280544	0.140272	−	0.990113i	$-0.455202\pi$
$587$	2.34205e6	0.280544	0.140272	+	0.990113i	$0.455202\pi$
$588$	6.36509e6	0.759209
$589$	−3.40856e6	−0.404840
$590$	8.79466e6	1.04013
$591$	−1.72514e6	−0.203168
$592$	3.35053e6	0.392924
$593$	−1.01216e7	−1.18198	−0.590992	−	0.806678i	$-0.701263\pi$
$593$	−1.01216e7	−1.18198	−0.590992	+	0.806678i	$0.701263\pi$
$594$	1.35594e6	0.157679
$595$	−1.08638e7	−1.25803
$596$	1.75896e6	0.202834
$597$	−3.08908e6	−0.354726
$598$	−7.56182e6	−0.864716
$599$	3.41198e6	0.388544	0.194272	−	0.980948i	$-0.437766\pi$
$599$	3.41198e6	0.388544	0.194272	+	0.980948i	$0.437766\pi$
$600$	−1.97914e6	−0.224439
$601$	1.03264e7	1.16617	0.583087	−	0.812410i	$-0.301845\pi$
$601$	1.03264e7	1.16617	0.583087	+	0.812410i	$0.301845\pi$
$602$	−386308.	−0.0434453
$603$	4.49129e6	0.503011
$604$	8.59213e6	0.958315
$605$	4.46909e6	0.496399
$606$	4.77382e6	0.528061
$607$	1.59247e7	1.75429	0.877143	−	0.480229i	$-0.159446\pi$
$607$	1.59247e7	1.75429	0.877143	+	0.480229i	$0.159446\pi$
$608$	−369664.	−0.0405554
$609$	−760266.	−0.0830658
$610$	1.58206e7	1.72147
$611$	5.91496e6	0.640987
$612$	703728.	0.0759497
$613$	−1.17105e6	−0.125871	−0.0629353	−	0.998018i	$-0.520046\pi$
$613$	−1.17105e6	−0.125871	−0.0629353	+	0.998018i	$0.520046\pi$
$614$	1.67291e6	0.179082
$615$	−1.18623e7	−1.26468
$616$	−7.35072e6	−0.780509
$617$	1.63844e7	1.73268	0.866338	−	0.499458i	$-0.166467\pi$
$617$	1.63844e7	1.73268	0.866338	+	0.499458i	$0.166467\pi$
$618$	−2.44778e6	−0.257811
$619$	−1.63675e7	−1.71694	−0.858470	−	0.512864i	$-0.828585\pi$
$619$	−1.63675e7	−1.71694	−0.858470	+	0.512864i	$0.828585\pi$
$620$	−1.22368e7	−1.27847
$621$	−1.98580e6	−0.206636
$622$	7.06940e6	0.732667
$623$	−6.30591e6	−0.650920
$624$	−1.59898e6	−0.164392
$625$	−8.69703e6	−0.890576
$626$	6.59342e6	0.672473
$627$	−1.51078e6	−0.153474
$628$	2.53770e6	0.256768
$629$	7.10678e6	0.716220
$630$	6.48227e6	0.650692
$631$	−6.63506e6	−0.663394	−0.331697	−	0.943386i	$-0.607621\pi$
$631$	−6.63506e6	−0.663394	−0.331697	+	0.943386i	$0.607621\pi$
$632$	4.99302e6	0.497246
$633$	−3.17804e6	−0.315247
$634$	7.76405e6	0.767123
$635$	9.41884e6	0.926965
$636$	−1.45584e6	−0.142715
$637$	−3.06762e7	−2.99539
$638$	636120.	0.0618710
$639$	3.50017e6	0.339107
$640$	−1.32710e6	−0.128072
$641$	−1.33964e7	−1.28778	−0.643890	−	0.765118i	$-0.722680\pi$
$641$	−1.33964e7	−1.28778	−0.643890	+	0.765118i	$0.722680\pi$
$642$	3.15382e6	0.301994
$643$	−1.39471e7	−1.33032	−0.665161	−	0.746700i	$-0.731637\pi$
$643$	−1.39471e7	−1.33032	−0.665161	+	0.746700i	$0.731637\pi$
$644$	1.07652e7	1.02284
$645$	−285039.	−0.0269777
$646$	−784092.	−0.0739240
$647$	1.13734e6	0.106814	0.0534071	−	0.998573i	$-0.482992\pi$
$647$	1.13734e6	0.106814	0.0534071	+	0.998573i	$0.482992\pi$
$648$	−419904.	−0.0392837
$649$	1.26220e7	1.17629
$650$	9.53834e6	0.885501
$651$	2.09896e7	1.94111
$652$	3.99949e6	0.368456
$653$	9.31890e6	0.855228	0.427614	−	0.903961i	$-0.359354\pi$
$653$	9.31890e6	0.855228	0.427614	+	0.903961i	$0.359354\pi$
$654$	−2.73269e6	−0.249831
$655$	−1.39494e7	−1.27044
$656$	−4.16563e6	−0.377939
$657$	3.05248e6	0.275893
$658$	−8.42072e6	−0.758202
$659$	−1.39783e6	−0.125383	−0.0626916	−	0.998033i	$-0.519968\pi$
$659$	−1.39783e6	−0.125383	−0.0626916	+	0.998033i	$0.519968\pi$
$660$	−5.42376e6	−0.484664
$661$	−3.23088e6	−0.287619	−0.143810	−	0.989605i	$-0.545935\pi$
$661$	−3.23088e6	−0.287619	−0.143810	+	0.989605i	$0.545935\pi$
$662$	4.90264e6	0.434795
$663$	−3.39158e6	−0.299653
$664$	−5.36909e6	−0.472585
$665$	−7.22253e6	−0.633337
$666$	−4.24051e6	−0.370453
$667$	−931608.	−0.0810809
$668$	1.16861e6	0.101328
$669$	−3.99496e6	−0.345101
$670$	−1.79652e7	−1.54612
$671$	2.27055e7	1.94681
$672$	2.27635e6	0.194454
$673$	1.26083e7	1.07305	0.536524	−	0.843885i	$-0.319737\pi$
$673$	1.26083e7	1.07305	0.536524	+	0.843885i	$0.319737\pi$
$674$	4.57881e6	0.388242
$675$	2.50484e6	0.211603
$676$	1.76549e6	0.148593
$677$	−2.55485e6	−0.214237	−0.107118	−	0.994246i	$-0.534162\pi$
$677$	−2.55485e6	−0.214237	−0.107118	+	0.994246i	$0.534162\pi$
$678$	−1.83406e6	−0.153228
$679$	1.88654e7	1.57033
$680$	−2.81491e6	−0.233449
$681$	−991170.	−0.0818993
$682$	−1.75621e7	−1.44583
$683$	7.98629e6	0.655079	0.327539	−	0.944838i	$-0.393781\pi$
$683$	7.98629e6	0.655079	0.327539	+	0.944838i	$0.393781\pi$
$684$	467856.	0.0382360
$685$	−858033.	−0.0698679
$686$	2.70663e7	2.19593
$687$	−4.89581e6	−0.395761
$688$	−100096.	−0.00806205
$689$	7.01634e6	0.563070
$690$	7.94318e6	0.635144
$691$	−445657.	−0.0355063	−0.0177532	−	0.999842i	$-0.505651\pi$
$691$	−445657.	−0.0355063	−0.0177532	+	0.999842i	$0.505651\pi$
$692$	1.07491e6	0.0853312
$693$	9.30326e6	0.735871
$694$	1.40665e7	1.10863
$695$	−1.94746e7	−1.52935
$696$	−196992.	−0.0154144
$697$	−8.83570e6	−0.688904
$698$	−2.44716e6	−0.190118
$699$	−7.58692e6	−0.587317
$700$	−1.35791e7	−1.04743
$701$	−1.80137e7	−1.38455	−0.692273	−	0.721635i	$-0.743391\pi$
$701$	−1.80137e7	−1.38455	−0.692273	+	0.721635i	$0.743391\pi$
$702$	2.02370e6	0.154990
$703$	4.72477e6	0.360572
$704$	−1.90464e6	−0.144838
$705$	−6.21327e6	−0.470812
$706$	−5.96172e6	−0.450153
$707$	3.27537e7	2.46440
$708$	−3.90874e6	−0.293058
$709$	−1.46038e7	−1.09107	−0.545533	−	0.838090i	$-0.683673\pi$
$709$	−1.46038e7	−1.09107	−0.545533	+	0.838090i	$0.683673\pi$
$710$	−1.40007e7	−1.04233
$711$	−6.31930e6	−0.468808
$712$	−1.63392e6	−0.120790
$713$	2.57200e7	1.89473
$714$	4.82836e6	0.354449
$715$	2.61395e7	1.91220
$716$	8.40528e6	0.612731
$717$	9.78558e6	0.710868
$718$	7.19322e6	0.520730
$719$	−1.77317e7	−1.27917	−0.639583	−	0.768722i	$-0.720893\pi$
$719$	−1.77317e7	−1.27917	−0.639583	+	0.768722i	$0.720893\pi$
$720$	1.67962e6	0.120748
$721$	−1.67945e7	−1.20318
$722$	−521284.	−0.0372161
$723$	3.52832e6	0.251028
$724$	−1.19459e6	−0.0846980
$725$	1.17511e6	0.0830298
$726$	−1.98626e6	−0.139861
$727$	5.34167e6	0.374836	0.187418	−	0.982280i	$-0.439988\pi$
$727$	5.34167e6	0.374836	0.187418	+	0.982280i	$0.439988\pi$
$728$	−1.09708e7	−0.767199
$729$	531441.	0.0370370
$730$	−1.22099e7	−0.848021
$731$	−212313.	−0.0146955
$732$	−7.03138e6	−0.485023
$733$	−251182.	−0.0172675	−0.00863373	−	0.999963i	$-0.502748\pi$
$733$	−251182.	−0.0172675	−0.00863373	+	0.999963i	$0.502748\pi$
$734$	1.81254e6	0.124179
$735$	3.22233e7	2.20014
$736$	2.78938e6	0.189807
$737$	−2.57833e7	−1.74852
$738$	5.27213e6	0.356324
$739$	2.53900e7	1.71022	0.855108	−	0.518450i	$-0.173491\pi$
$739$	2.53900e7	1.71022	0.855108	+	0.518450i	$0.173491\pi$
$740$	1.69620e7	1.13867
$741$	−2.25481e6	−0.150856
$742$	−9.98868e6	−0.666037
$743$	−1.03864e7	−0.690226	−0.345113	−	0.938561i	$-0.612159\pi$
$743$	−1.03864e7	−0.690226	−0.345113	+	0.938561i	$0.612159\pi$
$744$	5.43859e6	0.360209
$745$	8.90474e6	0.587801
$746$	8.04122e6	0.529023
$747$	6.79525e6	0.445558
$748$	−4.03992e6	−0.264009
$749$	2.16387e7	1.40937
$750$	−906876.	−0.0588701
$751$	6.35473e6	0.411147	0.205573	−	0.978642i	$-0.434094\pi$
$751$	6.35473e6	0.411147	0.205573	+	0.978642i	$0.434094\pi$
$752$	−2.18189e6	−0.140698
$753$	−1.13332e7	−0.728393
$754$	949392.	0.0608159
$755$	4.34976e7	2.77714
$756$	−2.88101e6	−0.183333
$757$	−1.96396e7	−1.24564	−0.622822	−	0.782364i	$-0.714014\pi$
$757$	−1.96396e7	−1.24564	−0.622822	+	0.782364i	$0.714014\pi$
$758$	−8.10534e6	−0.512387
$759$	1.13999e7	0.718287
$760$	−1.87142e6	−0.117527
$761$	−2.38787e7	−1.49468	−0.747342	−	0.664440i	$-0.768670\pi$
$761$	−2.38787e7	−1.49468	−0.747342	+	0.664440i	$0.768670\pi$
$762$	−4.18615e6	−0.261173
$763$	−1.87493e7	−1.16593
$764$	6.46968e6	0.401005
$765$	3.56262e6	0.220098
$766$	575640.	0.0354470
$767$	1.88379e7	1.15623
$768$	589824.	0.0360844
$769$	−8.58553e6	−0.523542	−0.261771	−	0.965130i	$-0.584306\pi$
$769$	−8.58553e6	−0.523542	−0.261771	+	0.965130i	$0.584306\pi$
$770$	−3.72130e7	−2.26187
$771$	−199800.	−0.0121049
$772$	1.33720e7	0.807518
$773$	−2.09415e7	−1.26055	−0.630274	−	0.776373i	$-0.717057\pi$
$773$	−2.09415e7	−1.26055	−0.630274	+	0.776373i	$0.717057\pi$
$774$	126684.	0.00760098
$775$	−3.24427e7	−1.94027
$776$	4.88819e6	0.291403
$777$	−2.90946e7	−1.72886
$778$	1.79735e7	1.06459
$779$	−5.87419e6	−0.346820
$780$	−8.09482e6	−0.476399
$781$	−2.00936e7	−1.17877
$782$	5.91653e6	0.345979
$783$	249318.	0.0145328
$784$	1.13157e7	0.657494
$785$	1.28471e7	0.744099
$786$	6.19974e6	0.357946
$787$	−6.27868e6	−0.361353	−0.180676	−	0.983543i	$-0.557829\pi$
$787$	−6.27868e6	−0.361353	−0.180676	+	0.983543i	$0.557829\pi$
$788$	−3.06691e6	−0.175949
$789$	−1.59292e7	−0.910962
$790$	2.52772e7	1.44099
$791$	−1.25837e7	−0.715098
$792$	2.41056e6	0.136554
$793$	3.38873e7	1.91361
$794$	−1.88258e7	−1.05975
$795$	−7.37019e6	−0.413581
$796$	−5.49170e6	−0.307202
$797$	−2.07776e7	−1.15864	−0.579322	−	0.815099i	$-0.696683\pi$
$797$	−2.07776e7	−1.15864	−0.579322	+	0.815099i	$0.696683\pi$
$798$	3.21001e6	0.178443
$799$	−4.62799e6	−0.256463
$800$	−3.51846e6	−0.194370
$801$	2.06793e6	0.113882
$802$	1.14214e7	0.627021
$803$	−1.75235e7	−0.959031
$804$	7.98451e6	0.435620
$805$	5.44991e7	2.96414
$806$	−2.62110e7	−1.42117
$807$	2.99889e6	0.162098
$808$	8.48678e6	0.457314
$809$	−1.98777e6	−0.106781	−0.0533905	−	0.998574i	$-0.517003\pi$
$809$	−1.98777e6	−0.106781	−0.0533905	+	0.998574i	$0.517003\pi$
$810$	−2.12576e6	−0.113842
$811$	−7.95135e6	−0.424511	−0.212256	−	0.977214i	$-0.568081\pi$
$811$	−7.95135e6	−0.424511	−0.212256	+	0.977214i	$0.568081\pi$
$812$	−1.35158e6	−0.0719371
$813$	1.08950e7	0.578097
$814$	2.43437e7	1.28773
$815$	2.02474e7	1.06776
$816$	1.25107e6	0.0657744
$817$	−141151.	−0.00739825
$818$	1.40027e7	0.731695
$819$	1.38849e7	0.723322
$820$	−2.10885e7	−1.09525
$821$	−1.83609e7	−0.950682	−0.475341	−	0.879802i	$-0.657675\pi$
$821$	−1.83609e7	−0.950682	−0.475341	+	0.879802i	$0.657675\pi$
$822$	381348.	0.0196853
$823$	−2.41203e7	−1.24132	−0.620658	−	0.784081i	$-0.713135\pi$
$823$	−2.41203e7	−1.24132	−0.620658	+	0.784081i	$0.713135\pi$
$824$	−4.35162e6	−0.223271
$825$	−1.43797e7	−0.735553
$826$	−2.68183e7	−1.36767
$827$	2.72605e6	0.138602	0.0693011	−	0.997596i	$-0.477923\pi$
$827$	2.72605e6	0.138602	0.0693011	+	0.997596i	$0.477923\pi$
$828$	−3.53030e6	−0.178952
$829$	−9.37985e6	−0.474034	−0.237017	−	0.971505i	$-0.576170\pi$
$829$	−9.37985e6	−0.474034	−0.237017	+	0.971505i	$0.576170\pi$
$830$	−2.71810e7	−1.36953
$831$	463041.	0.0232604
$832$	−2.84262e6	−0.142368
$833$	2.40017e7	1.19848
$834$	8.65537e6	0.430894
$835$	5.91608e6	0.293642
$836$	−2.68584e6	−0.132912
$837$	−6.88322e6	−0.339608
$838$	2.27083e7	1.11705
$839$	9.77541e6	0.479435	0.239718	−	0.970843i	$-0.422945\pi$
$839$	9.77541e6	0.479435	0.239718	+	0.970843i	$0.422945\pi$
$840$	1.15240e7	0.563516
$841$	−2.03942e7	−0.994298
$842$	−1.05486e6	−0.0512759
$843$	1.41985e7	0.688134
$844$	−5.64986e6	−0.273012
$845$	8.93778e6	0.430614
$846$	2.76145e6	0.132651
$847$	−1.36280e7	−0.652714
$848$	−2.58816e6	−0.123595
$849$	7.50172e6	0.357184
$850$	−7.46299e6	−0.354296
$851$	−3.56517e7	−1.68755
$852$	6.22253e6	0.293676
$853$	6.31128e6	0.296992	0.148496	−	0.988913i	$-0.452557\pi$
$853$	6.31128e6	0.296992	0.148496	+	0.988913i	$0.452557\pi$
$854$	−4.82431e7	−2.26355
$855$	2.36852e6	0.110806
$856$	5.60678e6	0.261535
$857$	−5.14481e6	−0.239286	−0.119643	−	0.992817i	$-0.538175\pi$
$857$	−5.14481e6	−0.239286	−0.119643	+	0.992817i	$0.538175\pi$
$858$	−1.16176e7	−0.538762
$859$	2.00958e7	0.929226	0.464613	−	0.885514i	$-0.346193\pi$
$859$	2.00958e7	0.929226	0.464613	+	0.885514i	$0.346193\pi$
$860$	−506736.	−0.0233634
$861$	3.61727e7	1.66292
$862$	4.68386e6	0.214702
$863$	−2.76609e7	−1.26427	−0.632133	−	0.774860i	$-0.717820\pi$
$863$	−2.76609e7	−1.26427	−0.632133	+	0.774860i	$0.717820\pi$
$864$	−746496.	−0.0340207
$865$	5.44174e6	0.247285
$866$	−2.33063e7	−1.05603
$867$	−1.01251e7	−0.457457
$868$	3.73148e7	1.68105
$869$	3.62774e7	1.62962
$870$	−997272.	−0.0446700
$871$	−3.84809e7	−1.71870
$872$	−4.85811e6	−0.216360
$873$	−6.18662e6	−0.274737
$874$	3.93346e6	0.174179
$875$	−6.22218e6	−0.274740
$876$	5.42664e6	0.238930
$877$	−2.53036e7	−1.11092	−0.555460	−	0.831543i	$-0.687458\pi$
$877$	−2.53036e7	−1.11092	−0.555460	+	0.831543i	$0.687458\pi$
$878$	3.56884e6	0.156239
$879$	−2.57892e7	−1.12581
$880$	−9.64224e6	−0.419731
$881$	3.70933e7	1.61011	0.805056	−	0.593198i	$-0.202135\pi$
$881$	3.70933e7	1.61011	0.805056	+	0.593198i	$0.202135\pi$
$882$	−1.43214e7	−0.619891
$883$	4.35101e7	1.87797	0.938985	−	0.343958i	$-0.111768\pi$
$883$	4.35101e7	1.87797	0.938985	+	0.343958i	$0.111768\pi$
$884$	−6.02947e6	−0.259507
$885$	−1.97880e7	−0.849265
$886$	−9.35816e6	−0.400504
$887$	2.37449e7	1.01335	0.506677	−	0.862136i	$-0.330874\pi$
$887$	2.37449e7	1.01335	0.506677	+	0.862136i	$0.330874\pi$
$888$	−7.53869e6	−0.320821
$889$	−2.87217e7	−1.21886
$890$	−8.27172e6	−0.350043
$891$	−3.05086e6	−0.128745
$892$	−7.10214e6	−0.298867
$893$	−3.07680e6	−0.129113
$894$	−3.95766e6	−0.165613
$895$	4.25517e7	1.77566
$896$	4.04685e6	0.168402
$897$	1.70141e7	0.706038
$898$	−1.13113e7	−0.468084
$899$	−3.22916e6	−0.133257
$900$	4.45306e6	0.183253
$901$	−5.48973e6	−0.225288
$902$	−3.02659e7	−1.23862
$903$	869193.	0.0354729
$904$	−3.26054e6	−0.132699
$905$	−6.04762e6	−0.245450
$906$	−1.93323e7	−0.782461
$907$	−2.01882e7	−0.814855	−0.407428	−	0.913238i	$-0.633574\pi$
$907$	−2.01882e7	−0.814855	−0.407428	+	0.913238i	$0.633574\pi$
$908$	−1.76208e6	−0.0709269
$909$	−1.07411e7	−0.431160
$910$	−5.55394e7	−2.22330
$911$	8.39955e6	0.335320	0.167660	−	0.985845i	$-0.446379\pi$
$911$	8.39955e6	0.335320	0.167660	+	0.985845i	$0.446379\pi$
$912$	831744.	0.0331133
$913$	−3.90098e7	−1.54880
$914$	−1.14878e6	−0.0454853
$915$	−3.55963e7	−1.40557
$916$	−8.70366e6	−0.342739
$917$	4.25371e7	1.67049
$918$	−1.58339e6	−0.0620127
$919$	4.33552e7	1.69337	0.846686	−	0.532092i	$-0.178594\pi$
$919$	4.33552e7	1.69337	0.846686	+	0.532092i	$0.178594\pi$
$920$	1.41212e7	0.550050
$921$	−3.76405e6	−0.146220
$922$	−2.18855e7	−0.847869
$923$	−2.99891e7	−1.15867
$924$	1.65391e7	0.637283
$925$	4.49704e7	1.72811
$926$	−2.57851e7	−0.988191
$927$	5.50751e6	0.210502
$928$	−350208.	−0.0133492
$929$	2.04854e7	0.778762	0.389381	−	0.921077i	$-0.372689\pi$
$929$	2.04854e7	0.778762	0.389381	+	0.921077i	$0.372689\pi$
$930$	2.75329e7	1.04386
$931$	1.59569e7	0.603358
$932$	−1.34879e7	−0.508631
$933$	−1.59062e7	−0.598220
$934$	2.88678e7	1.08280
$935$	−2.04521e7	−0.765083
$936$	3.59770e6	0.134225
$937$	1.70989e7	0.636237	0.318118	−	0.948051i	$-0.396949\pi$
$937$	1.70989e7	0.636237	0.318118	+	0.948051i	$0.396949\pi$
$938$	5.47826e7	2.03299
$939$	−1.48352e7	−0.549072
$940$	−1.10458e7	−0.407735
$941$	2.46405e6	0.0907142	0.0453571	−	0.998971i	$-0.485557\pi$
$941$	2.46405e6	0.0907142	0.0453571	+	0.998971i	$0.485557\pi$
$942$	−5.70982e6	−0.209650
$943$	4.43249e7	1.62319
$944$	−6.94886e6	−0.253795
$945$	−1.45851e7	−0.531288
$946$	−727260.	−0.0264218
$947$	3.37152e7	1.22166	0.610831	−	0.791761i	$-0.290836\pi$
$947$	3.37152e7	1.22166	0.610831	+	0.791761i	$0.290836\pi$
$948$	−1.12343e7	−0.405999
$949$	−2.61534e7	−0.942676
$950$	−4.96158e6	−0.178366
$951$	−1.74691e7	−0.626353
$952$	8.58374e6	0.306962
$953$	−1.19024e7	−0.424525	−0.212262	−	0.977213i	$-0.568083\pi$
$953$	−1.19024e7	−0.424525	−0.212262	+	0.977213i	$0.568083\pi$
$954$	3.27564e6	0.116527
$955$	3.27528e7	1.16209
$956$	1.73966e7	0.615630
$957$	−1.43127e6	−0.0505175
$958$	2.38594e7	0.839934
$959$	2.61647e6	0.0918691
$960$	2.98598e6	0.104571
$961$	6.05222e7	2.11401
$962$	3.63323e7	1.26577
$963$	−7.09609e6	−0.246577
$964$	6.27258e6	0.217397
$965$	6.76956e7	2.34014
$966$	−2.42218e7	−0.835149
$967$	1.22453e7	0.421119	0.210559	−	0.977581i	$-0.432471\pi$
$967$	1.22453e7	0.421119	0.210559	+	0.977581i	$0.432471\pi$
$968$	−3.53114e6	−0.121123
$969$	1.76421e6	0.0603587
$970$	2.47465e7	0.844469
$971$	−1.19930e7	−0.408205	−0.204102	−	0.978950i	$-0.565428\pi$
$971$	−1.19930e7	−0.408205	−0.204102	+	0.978950i	$0.565428\pi$
$972$	944784.	0.0320750
$973$	5.93855e7	2.01094
$974$	2.12291e7	0.717025
$975$	−2.14613e7	−0.723009
$976$	−1.25002e7	−0.420043
$977$	5.17458e7	1.73436	0.867179	−	0.497996i	$-0.165931\pi$
$977$	5.17458e7	1.73436	0.867179	+	0.497996i	$0.165931\pi$
$978$	−8.99885e6	−0.300843
$979$	−1.18714e7	−0.395865
$980$	5.72858e7	1.90538
$981$	6.14855e6	0.203986
$982$	2.58964e7	0.856960
$983$	−5.48877e7	−1.81172	−0.905860	−	0.423577i	$-0.860774\pi$
$983$	−5.48877e7	−1.81172	−0.905860	+	0.423577i	$0.860774\pi$
$984$	9.37267e6	0.308586
$985$	−1.55262e7	−0.509889
$986$	−742824.	−0.0243329
$987$	1.89466e7	0.619069
$988$	−4.00854e6	−0.130645
$989$	1.06508e6	0.0346253
$990$	1.22035e7	0.395726
$991$	−9.62512e6	−0.311331	−0.155665	−	0.987810i	$-0.549752\pi$
$991$	−9.62512e6	−0.311331	−0.155665	+	0.987810i	$0.549752\pi$
$992$	9.66861e6	0.311950
$993$	−1.10309e7	−0.355009
$994$	4.26935e7	1.37055
$995$	−2.78017e7	−0.890254
$996$	1.20804e7	0.385864
$997$	−1.67975e7	−0.535187	−0.267593	−	0.963532i	$-0.586228\pi$
$997$	−1.67975e7	−0.535187	−0.267593	+	0.963532i	$0.586228\pi$
$998$	−1.24127e7	−0.394492
$999$	9.54115e6	0.302473

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.6.a.b.1.1	✓	1
3.2	odd	2		342.6.a.d.1.1		1
4.3	odd	2		912.6.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.6.a.b.1.1	✓	1	1.1	even	1	trivial
342.6.a.d.1.1		1	3.2	odd	2
912.6.a.e.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 114 = 2 \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	114.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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