LMFDB - Embedded newform 74.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [74,6,Mod(1,74)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(74, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("74.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 74 = 2 \cdot 37 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	74.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:	$11.8684026662$
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	$\chi$	$=$	74.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +7.00000 q^{3} +16.0000 q^{4} -84.0000 q^{5} +28.0000 q^{6} -139.000 q^{7} +64.0000 q^{8} -194.000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} +7.00000 q^{3} +16.0000 q^{4} -84.0000 q^{5} +28.0000 q^{6} -139.000 q^{7} +64.0000 q^{8} -194.000 q^{9} -336.000 q^{10} -147.000 q^{11} +112.000 q^{12} -280.000 q^{13} -556.000 q^{14} -588.000 q^{15} +256.000 q^{16} +978.000 q^{17} -776.000 q^{18} +1154.00 q^{19} -1344.00 q^{20} -973.000 q^{21} -588.000 q^{22} -1158.00 q^{23} +448.000 q^{24} +3931.00 q^{25} -1120.00 q^{26} -3059.00 q^{27} -2224.00 q^{28} -3198.00 q^{29} -2352.00 q^{30} -5932.00 q^{31} +1024.00 q^{32} -1029.00 q^{33} +3912.00 q^{34} +11676.0 q^{35} -3104.00 q^{36} +1369.00 q^{37} +4616.00 q^{38} -1960.00 q^{39} -5376.00 q^{40} +10023.0 q^{41} -3892.00 q^{42} -4036.00 q^{43} -2352.00 q^{44} +16296.0 q^{45} -4632.00 q^{46} -11631.0 q^{47} +1792.00 q^{48} +2514.00 q^{49} +15724.0 q^{50} +6846.00 q^{51} -4480.00 q^{52} +11193.0 q^{53} -12236.0 q^{54} +12348.0 q^{55} -8896.00 q^{56} +8078.00 q^{57} -12792.0 q^{58} +24660.0 q^{59} -9408.00 q^{60} -13360.0 q^{61} -23728.0 q^{62} +26966.0 q^{63} +4096.00 q^{64} +23520.0 q^{65} -4116.00 q^{66} -32860.0 q^{67} +15648.0 q^{68} -8106.00 q^{69} +46704.0 q^{70} +60123.0 q^{71} -12416.0 q^{72} +41915.0 q^{73} +5476.00 q^{74} +27517.0 q^{75} +18464.0 q^{76} +20433.0 q^{77} -7840.00 q^{78} -60898.0 q^{79} -21504.0 q^{80} +25729.0 q^{81} +40092.0 q^{82} -80169.0 q^{83} -15568.0 q^{84} -82152.0 q^{85} -16144.0 q^{86} -22386.0 q^{87} -9408.00 q^{88} -131358. q^{89} +65184.0 q^{90} +38920.0 q^{91} -18528.0 q^{92} -41524.0 q^{93} -46524.0 q^{94} -96936.0 q^{95} +7168.00 q^{96} -122872. q^{97} +10056.0 q^{98} +28518.0 q^{99} +O(q^{100})\)

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
$3$	7.00000	0.449050	0.224525	−	0.974468i	$-0.427917\pi$
$3$	7.00000	0.449050	0.224525	+	0.974468i	$0.427917\pi$
$4$	16.0000	0.500000
$5$	−84.0000	−1.50264	−0.751319	−	0.659939i	$-0.770582\pi$
$5$	−84.0000	−1.50264	−0.751319	+	0.659939i	$0.770582\pi$
$6$	28.0000	0.317526
$7$	−139.000	−1.07218	−0.536092	−	0.844159i	$-0.680100\pi$
$7$	−139.000	−1.07218	−0.536092	+	0.844159i	$0.680100\pi$
$8$	64.0000	0.353553
$9$	−194.000	−0.798354
$10$	−336.000	−1.06253
$11$	−147.000	−0.366299	−0.183149	−	0.983085i	$-0.558629\pi$
$11$	−147.000	−0.366299	−0.183149	+	0.983085i	$0.558629\pi$
$12$	112.000	0.224525
$13$	−280.000	−0.459515	−0.229757	−	0.973248i	$-0.573793\pi$
$13$	−280.000	−0.459515	−0.229757	+	0.973248i	$0.573793\pi$
$14$	−556.000	−0.758149
$15$	−588.000	−0.674760
$16$	256.000	0.250000
$17$	978.000	0.820761	0.410380	−	0.911914i	$-0.365396\pi$
$17$	978.000	0.820761	0.410380	+	0.911914i	$0.365396\pi$
$18$	−776.000	−0.564521
$19$	1154.00	0.733368	0.366684	−	0.930346i	$-0.380493\pi$
$19$	1154.00	0.733368	0.366684	+	0.930346i	$0.380493\pi$
$20$	−1344.00	−0.751319
$21$	−973.000	−0.481465
$22$	−588.000	−0.259012
$23$	−1158.00	−0.456446	−0.228223	−	0.973609i	$-0.573291\pi$
$23$	−1158.00	−0.456446	−0.228223	+	0.973609i	$0.573291\pi$
$24$	448.000	0.158763
$25$	3931.00	1.25792
$26$	−1120.00	−0.324926
$27$	−3059.00	−0.807551
$28$	−2224.00	−0.536092
$29$	−3198.00	−0.706128	−0.353064	−	0.935599i	$-0.614860\pi$
$29$	−3198.00	−0.706128	−0.353064	+	0.935599i	$0.614860\pi$
$30$	−2352.00	−0.477127
$31$	−5932.00	−1.10866	−0.554328	−	0.832298i	$-0.687025\pi$
$31$	−5932.00	−1.10866	−0.554328	+	0.832298i	$0.687025\pi$
$32$	1024.00	0.176777
$33$	−1029.00	−0.164487
$34$	3912.00	0.580365
$35$	11676.0	1.61111
$36$	−3104.00	−0.399177
$37$	1369.00	0.164399
$37$	1369.00	0.164399
$38$	4616.00	0.518569
$39$	−1960.00	−0.206345
$40$	−5376.00	−0.531263
$41$	10023.0	0.931190	0.465595	−	0.884998i	$-0.345840\pi$
$41$	10023.0	0.931190	0.465595	+	0.884998i	$0.345840\pi$
$42$	−3892.00	−0.340447
$43$	−4036.00	−0.332874	−0.166437	−	0.986052i	$-0.553226\pi$
$43$	−4036.00	−0.332874	−0.166437	+	0.986052i	$0.553226\pi$
$44$	−2352.00	−0.183149
$45$	16296.0	1.19964
$46$	−4632.00	−0.322756
$47$	−11631.0	−0.768020	−0.384010	−	0.923329i	$-0.625457\pi$
$47$	−11631.0	−0.768020	−0.384010	+	0.923329i	$0.625457\pi$
$48$	1792.00	0.112263
$49$	2514.00	0.149581
$50$	15724.0	0.889484
$51$	6846.00	0.368563
$52$	−4480.00	−0.229757
$53$	11193.0	0.547340	0.273670	−	0.961824i	$-0.411762\pi$
$53$	11193.0	0.547340	0.273670	+	0.961824i	$0.411762\pi$
$54$	−12236.0	−0.571025
$55$	12348.0	0.550415
$56$	−8896.00	−0.379075
$57$	8078.00	0.329319
$58$	−12792.0	−0.499308
$59$	24660.0	0.922281	0.461140	−	0.887327i	$-0.347440\pi$
$59$	24660.0	0.922281	0.461140	+	0.887327i	$0.347440\pi$
$60$	−9408.00	−0.337380
$61$	−13360.0	−0.459708	−0.229854	−	0.973225i	$-0.573825\pi$
$61$	−13360.0	−0.459708	−0.229854	+	0.973225i	$0.573825\pi$
$62$	−23728.0	−0.783938
$63$	26966.0	0.855983
$64$	4096.00	0.125000
$65$	23520.0	0.690484
$66$	−4116.00	−0.116310
$67$	−32860.0	−0.894294	−0.447147	−	0.894460i	$-0.647560\pi$
$67$	−32860.0	−0.894294	−0.447147	+	0.894460i	$0.647560\pi$
$68$	15648.0	0.410380
$69$	−8106.00	−0.204967
$70$	46704.0	1.13922
$71$	60123.0	1.41545	0.707725	−	0.706488i	$-0.249721\pi$
$71$	60123.0	1.41545	0.707725	+	0.706488i	$0.249721\pi$
$72$	−12416.0	−0.282261
$73$	41915.0	0.920582	0.460291	−	0.887768i	$-0.347745\pi$
$73$	41915.0	0.920582	0.460291	+	0.887768i	$0.347745\pi$
$74$	5476.00	0.116248
$75$	27517.0	0.564869
$76$	18464.0	0.366684
$77$	20433.0	0.392740
$78$	−7840.00	−0.145908
$79$	−60898.0	−1.09783	−0.548915	−	0.835878i	$-0.684959\pi$
$79$	−60898.0	−1.09783	−0.548915	+	0.835878i	$0.684959\pi$
$80$	−21504.0	−0.375659
$81$	25729.0	0.435723
$82$	40092.0	0.658450
$83$	−80169.0	−1.27735	−0.638677	−	0.769475i	$-0.720518\pi$
$83$	−80169.0	−1.27735	−0.638677	+	0.769475i	$0.720518\pi$
$84$	−15568.0	−0.240732
$85$	−82152.0	−1.23331
$86$	−16144.0	−0.235378
$87$	−22386.0	−0.317087
$88$	−9408.00	−0.129506
$89$	−131358.	−1.75785	−0.878924	−	0.476961i	$-0.841738\pi$
$89$	−131358.	−1.75785	−0.878924	+	0.476961i	$0.841738\pi$
$90$	65184.0	0.848271
$91$	38920.0	0.492685
$92$	−18528.0	−0.228223
$93$	−41524.0	−0.497842
$94$	−46524.0	−0.543072
$95$	−96936.0	−1.10199
$96$	7168.00	0.0793816
$97$	−122872.	−1.32594	−0.662970	−	0.748646i	$-0.730704\pi$
$97$	−122872.	−1.32594	−0.662970	+	0.748646i	$0.730704\pi$
$98$	10056.0	0.105769
$99$	28518.0	0.292436
$100$	62896.0	0.628960
$101$	−188265.	−1.83640	−0.918198	−	0.396123i	$-0.870356\pi$
$101$	−188265.	−1.83640	−0.918198	+	0.396123i	$0.870356\pi$
$102$	27384.0	0.260613
$103$	165248.	1.53477	0.767385	−	0.641187i	$-0.221558\pi$
$103$	165248.	1.53477	0.767385	+	0.641187i	$0.221558\pi$
$104$	−17920.0	−0.162463
$105$	81732.0	0.723467
$106$	44772.0	0.387028
$107$	100812.	0.851241	0.425621	−	0.904902i	$-0.360056\pi$
$107$	100812.	0.851241	0.425621	+	0.904902i	$0.360056\pi$
$108$	−48944.0	−0.403776
$109$	−66058.0	−0.532549	−0.266274	−	0.963897i	$-0.585793\pi$
$109$	−66058.0	−0.532549	−0.266274	+	0.963897i	$0.585793\pi$
$110$	49392.0	0.389202
$111$	9583.00	0.0738234
$112$	−35584.0	−0.268046
$113$	−122742.	−0.904268	−0.452134	−	0.891950i	$-0.649337\pi$
$113$	−122742.	−0.904268	−0.452134	+	0.891950i	$0.649337\pi$
$114$	32312.0	0.232864
$115$	97272.0	0.685872
$116$	−51168.0	−0.353064
$117$	54320.0	0.366856
$118$	98640.0	0.652151
$119$	−135942.	−0.880007
$120$	−37632.0	−0.238564
$121$	−139442.	−0.865825
$122$	−53440.0	−0.325063
$123$	70161.0	0.418151
$124$	−94912.0	−0.554328
$125$	−67704.0	−0.387560
$126$	107864.	0.605271
$127$	−159469.	−0.877338	−0.438669	−	0.898649i	$-0.644550\pi$
$127$	−159469.	−0.877338	−0.438669	+	0.898649i	$0.644550\pi$
$128$	16384.0	0.0883883
$129$	−28252.0	−0.149477
$130$	94080.0	0.488246
$131$	1818.00	0.00925584	0.00462792	−	0.999989i	$-0.498527\pi$
$131$	1818.00	0.00925584	0.00462792	+	0.999989i	$0.498527\pi$
$132$	−16464.0	−0.0822433
$133$	−160406.	−0.786306
$134$	−131440.	−0.632362
$135$	256956.	1.21346
$136$	62592.0	0.290183
$137$	368178.	1.67593	0.837966	−	0.545722i	$-0.183745\pi$
$137$	368178.	1.67593	0.837966	+	0.545722i	$0.183745\pi$
$138$	−32424.0	−0.144934
$139$	138788.	0.609277	0.304639	−	0.952468i	$-0.401464\pi$
$139$	138788.	0.609277	0.304639	+	0.952468i	$0.401464\pi$
$140$	186816.	0.805553
$141$	−81417.0	−0.344879
$142$	240492.	1.00087
$143$	41160.0	0.168320
$144$	−49664.0	−0.199588
$145$	268632.	1.06105
$146$	167660.	0.650950
$147$	17598.0	0.0671692
$148$	21904.0	0.0821995
$149$	495363.	1.82792	0.913961	−	0.405801i	$-0.133007\pi$
$149$	495363.	1.82792	0.913961	+	0.405801i	$0.133007\pi$
$150$	110068.	0.399423
$151$	−129784.	−0.463211	−0.231605	−	0.972810i	$-0.574398\pi$
$151$	−129784.	−0.463211	−0.231605	+	0.972810i	$0.574398\pi$
$152$	73856.0	0.259285
$153$	−189732.	−0.655258
$154$	81732.0	0.277709
$155$	498288.	1.66591
$156$	−31360.0	−0.103173
$157$	−277993.	−0.900087	−0.450044	−	0.893007i	$-0.648592\pi$
$157$	−277993.	−0.900087	−0.450044	+	0.893007i	$0.648592\pi$
$158$	−243592.	−0.776283
$159$	78351.0	0.245783
$160$	−86016.0	−0.265631
$161$	160962.	0.489394
$162$	102916.	0.308103
$163$	255620.	0.753574	0.376787	−	0.926300i	$-0.377029\pi$
$163$	255620.	0.753574	0.376787	+	0.926300i	$0.377029\pi$
$164$	160368.	0.465595
$165$	86436.0	0.247164
$166$	−320676.	−0.903226
$167$	−308478.	−0.855920	−0.427960	−	0.903798i	$-0.640768\pi$
$167$	−308478.	−0.855920	−0.427960	+	0.903798i	$0.640768\pi$
$168$	−62272.0	−0.170224
$169$	−292893.	−0.788846
$170$	−328608.	−0.872079
$171$	−223876.	−0.585487
$172$	−64576.0	−0.166437
$173$	−314691.	−0.799409	−0.399705	−	0.916644i	$-0.630887\pi$
$173$	−314691.	−0.799409	−0.399705	+	0.916644i	$0.630887\pi$
$174$	−89544.0	−0.224214
$175$	−546409.	−1.34872
$176$	−37632.0	−0.0915747
$177$	172620.	0.414150
$178$	−525432.	−1.24299
$179$	717894.	1.67466	0.837332	−	0.546695i	$-0.184114\pi$
$179$	717894.	1.67466	0.837332	+	0.546695i	$0.184114\pi$
$180$	260736.	0.599818
$181$	−493603.	−1.11991	−0.559953	−	0.828525i	$-0.689181\pi$
$181$	−493603.	−1.11991	−0.559953	+	0.828525i	$0.689181\pi$
$182$	155680.	0.348381
$183$	−93520.0	−0.206432
$184$	−74112.0	−0.161378
$185$	−114996.	−0.247032
$186$	−166096.	−0.352028
$187$	−143766.	−0.300644
$188$	−186096.	−0.384010
$189$	425201.	0.865844
$190$	−387744.	−0.779222
$191$	890376.	1.76600	0.882999	−	0.469376i	$-0.155521\pi$
$191$	890376.	1.76600	0.882999	+	0.469376i	$0.155521\pi$
$192$	28672.0	0.0561313
$193$	643760.	1.24403	0.622015	−	0.783005i	$-0.286314\pi$
$193$	643760.	1.24403	0.622015	+	0.783005i	$0.286314\pi$
$194$	−491488.	−0.937581
$195$	164640.	0.310062
$196$	40224.0	0.0747903
$197$	−767253.	−1.40855	−0.704276	−	0.709926i	$-0.748728\pi$
$197$	−767253.	−1.40855	−0.704276	+	0.709926i	$0.748728\pi$
$198$	114072.	0.206784
$199$	−652642.	−1.16827	−0.584134	−	0.811657i	$-0.698566\pi$
$199$	−652642.	−1.16827	−0.584134	+	0.811657i	$0.698566\pi$
$200$	251584.	0.444742
$201$	−230020.	−0.401583
$202$	−753060.	−1.29853
$203$	444522.	0.757100
$204$	109536.	0.184281
$205$	−841932.	−1.39924
$206$	660992.	1.08525
$207$	224652.	0.364405
$208$	−71680.0	−0.114879
$209$	−169638.	−0.268632
$210$	326928.	0.511569
$211$	−554923.	−0.858078	−0.429039	−	0.903286i	$-0.641148\pi$
$211$	−554923.	−0.858078	−0.429039	+	0.903286i	$0.641148\pi$
$212$	179088.	0.273670
$213$	420861.	0.635608
$214$	403248.	0.601919
$215$	339024.	0.500189
$216$	−195776.	−0.285512
$217$	824548.	1.18868
$218$	−264232.	−0.376569
$219$	293405.	0.413387
$220$	197568.	0.275207
$221$	−273840.	−0.377152
$222$	38332.0	0.0522010
$223$	−1.22263e6	−1.64639	−0.823193	−	0.567761i	$-0.807810\pi$
$223$	−1.22263e6	−1.64639	−0.823193	+	0.567761i	$0.807810\pi$
$224$	−142336.	−0.189537
$225$	−762614.	−1.00427
$226$	−490968.	−0.639414
$227$	385128.	0.496067	0.248034	−	0.968751i	$-0.420216\pi$
$227$	385128.	0.496067	0.248034	+	0.968751i	$0.420216\pi$
$228$	129248.	0.164659
$229$	834299.	1.05132	0.525658	−	0.850696i	$-0.323819\pi$
$229$	834299.	1.05132	0.525658	+	0.850696i	$0.323819\pi$
$230$	389088.	0.484985
$231$	143031.	0.176360
$232$	−204672.	−0.249654
$233$	−1.32475e6	−1.59861	−0.799306	−	0.600925i	$-0.794799\pi$
$233$	−1.32475e6	−1.59861	−0.799306	+	0.600925i	$0.794799\pi$
$234$	217280.	0.259406
$235$	977004.	1.15406
$236$	394560.	0.461140
$237$	−426286.	−0.492981
$238$	−543768.	−0.622259
$239$	1.64021e6	1.85740	0.928701	−	0.370830i	$-0.120927\pi$
$239$	1.64021e6	1.85740	0.928701	+	0.370830i	$0.120927\pi$
$240$	−150528.	−0.168690
$241$	−1.03175e6	−1.14427	−0.572137	−	0.820158i	$-0.693886\pi$
$241$	−1.03175e6	−1.14427	−0.572137	+	0.820158i	$0.693886\pi$
$242$	−557768.	−0.612231
$243$	923440.	1.00321
$244$	−213760.	−0.229854
$245$	−211176.	−0.224765
$246$	280644.	0.295677
$247$	−323120.	−0.336993
$248$	−379648.	−0.391969
$249$	−561183.	−0.573596
$250$	−270816.	−0.274047
$251$	1.50124e6	1.50406	0.752029	−	0.659130i	$-0.229075\pi$
$251$	1.50124e6	1.50406	0.752029	+	0.659130i	$0.229075\pi$
$252$	431456.	0.427992
$253$	170226.	0.167196
$254$	−637876.	−0.620372
$255$	−575064.	−0.553816
$256$	65536.0	0.0625000
$257$	126288.	0.119269	0.0596347	−	0.998220i	$-0.481006\pi$
$257$	126288.	0.119269	0.0596347	+	0.998220i	$0.481006\pi$
$258$	−113008.	−0.105696
$259$	−190291.	−0.176266
$260$	376320.	0.345242
$261$	620412.	0.563740
$262$	7272.00	0.00654486
$263$	1.06975e6	0.953658	0.476829	−	0.878996i	$-0.341786\pi$
$263$	1.06975e6	0.953658	0.476829	+	0.878996i	$0.341786\pi$
$264$	−65856.0	−0.0581548
$265$	−940212.	−0.822453
$266$	−641624.	−0.556002
$267$	−919506.	−0.789362
$268$	−525760.	−0.447147
$269$	−680130.	−0.573075	−0.286537	−	0.958069i	$-0.592504\pi$
$269$	−680130.	−0.573075	−0.286537	+	0.958069i	$0.592504\pi$
$270$	1.02782e6	0.858044
$271$	984575.	0.814377	0.407189	−	0.913344i	$-0.366509\pi$
$271$	984575.	0.814377	0.407189	+	0.913344i	$0.366509\pi$
$272$	250368.	0.205190
$273$	272440.	0.221240
$274$	1.47271e6	1.18506
$275$	−577857.	−0.460775
$276$	−129696.	−0.102483
$277$	1.76000e6	1.37820	0.689102	−	0.724665i	$-0.258005\pi$
$277$	1.76000e6	1.37820	0.689102	+	0.724665i	$0.258005\pi$
$278$	555152.	0.430824
$279$	1.15081e6	0.885100
$280$	747264.	0.569612
$281$	−102264.	−0.0772604	−0.0386302	−	0.999254i	$-0.512299\pi$
$281$	−102264.	−0.0772604	−0.0386302	+	0.999254i	$0.512299\pi$
$282$	−325668.	−0.243867
$283$	−188740.	−0.140087	−0.0700435	−	0.997544i	$-0.522314\pi$
$283$	−188740.	−0.140087	−0.0700435	+	0.997544i	$0.522314\pi$
$284$	961968.	0.707725
$285$	−678552.	−0.494847
$286$	164640.	0.119020
$287$	−1.39320e6	−0.998407
$288$	−198656.	−0.141130
$289$	−463373.	−0.326352
$290$	1.07453e6	0.750279
$291$	−860104.	−0.595413
$292$	670640.	0.460291
$293$	252798.	0.172030	0.0860151	−	0.996294i	$-0.472587\pi$
$293$	252798.	0.172030	0.0860151	+	0.996294i	$0.472587\pi$
$294$	70392.0	0.0474958
$295$	−2.07144e6	−1.38585
$296$	87616.0	0.0581238
$297$	449673.	0.295805
$298$	1.98145e6	1.29254
$299$	324240.	0.209744
$300$	440272.	0.282435
$301$	561004.	0.356903
$302$	−519136.	−0.327540
$303$	−1.31786e6	−0.824634
$304$	295424.	0.183342
$305$	1.12224e6	0.690774
$306$	−758928.	−0.463337
$307$	−1.72815e6	−1.04649	−0.523246	−	0.852182i	$-0.675279\pi$
$307$	−1.72815e6	−1.04649	−0.523246	+	0.852182i	$0.675279\pi$
$308$	326928.	0.196370
$309$	1.15674e6	0.689189
$310$	1.99315e6	1.17798
$311$	−1.09755e6	−0.643463	−0.321731	−	0.946831i	$-0.604265\pi$
$311$	−1.09755e6	−0.643463	−0.321731	+	0.946831i	$0.604265\pi$
$312$	−125440.	−0.0729541
$313$	1.45314e6	0.838389	0.419194	−	0.907897i	$-0.362313\pi$
$313$	1.45314e6	0.838389	0.419194	+	0.907897i	$0.362313\pi$
$314$	−1.11197e6	−0.636458
$315$	−2.26514e6	−1.28623
$316$	−974368.	−0.548915
$317$	42726.0	0.0238805	0.0119403	−	0.999929i	$-0.496199\pi$
$317$	42726.0	0.0238805	0.0119403	+	0.999929i	$0.496199\pi$
$318$	313404.	0.173795
$319$	470106.	0.258654
$320$	−344064.	−0.187830
$321$	705684.	0.382250
$322$	643848.	0.346054
$323$	1.12861e6	0.601919
$324$	411664.	0.217861
$325$	−1.10068e6	−0.578033
$326$	1.02248e6	0.532857
$327$	−462406.	−0.239141
$328$	641472.	0.329225
$329$	1.61671e6	0.823459
$330$	345744.	0.174771
$331$	−2.39863e6	−1.20335	−0.601677	−	0.798740i	$-0.705500\pi$
$331$	−2.39863e6	−1.20335	−0.601677	+	0.798740i	$0.705500\pi$
$332$	−1.28270e6	−0.638677
$333$	−265586.	−0.131249
$334$	−1.23391e6	−0.605227
$335$	2.76024e6	1.34380
$336$	−249088.	−0.120366
$337$	−787393.	−0.377674	−0.188837	−	0.982008i	$-0.560472\pi$
$337$	−787393.	−0.377674	−0.188837	+	0.982008i	$0.560472\pi$
$338$	−1.17157e6	−0.557798
$339$	−859194.	−0.406062
$340$	−1.31443e6	−0.616653
$341$	872004.	0.406100
$342$	−895504.	−0.414002
$343$	1.98673e6	0.911807
$344$	−258304.	−0.117689
$345$	680904.	0.307991
$346$	−1.25876e6	−0.565268
$347$	−275118.	−0.122658	−0.0613289	−	0.998118i	$-0.519534\pi$
$347$	−275118.	−0.122658	−0.0613289	+	0.998118i	$0.519534\pi$
$348$	−358176.	−0.158543
$349$	−952402.	−0.418559	−0.209280	−	0.977856i	$-0.567112\pi$
$349$	−952402.	−0.418559	−0.209280	+	0.977856i	$0.567112\pi$
$350$	−2.18564e6	−0.953691
$351$	856520.	0.371082
$352$	−150528.	−0.0647531
$353$	3.27239e6	1.39775	0.698874	−	0.715245i	$-0.253685\pi$
$353$	3.27239e6	1.39775	0.698874	+	0.715245i	$0.253685\pi$
$354$	690480.	0.292848
$355$	−5.05033e6	−2.12691
$356$	−2.10173e6	−0.878924
$357$	−951594.	−0.395167
$358$	2.87158e6	1.18417
$359$	747351.	0.306047	0.153024	−	0.988223i	$-0.451099\pi$
$359$	747351.	0.306047	0.153024	+	0.988223i	$0.451099\pi$
$360$	1.04294e6	0.424136
$361$	−1.14438e6	−0.462172
$362$	−1.97441e6	−0.791893
$363$	−976094.	−0.388799
$364$	622720.	0.246343
$365$	−3.52086e6	−1.38330
$366$	−374080.	−0.145969
$367$	−4.08359e6	−1.58262	−0.791311	−	0.611414i	$-0.790601\pi$
$367$	−4.08359e6	−1.58262	−0.791311	+	0.611414i	$0.790601\pi$
$368$	−296448.	−0.114111
$369$	−1.94446e6	−0.743419
$370$	−459984.	−0.174678
$371$	−1.55583e6	−0.586849
$372$	−664384.	−0.248921
$373$	1.57898e6	0.587630	0.293815	−	0.955862i	$-0.405075\pi$
$373$	1.57898e6	0.587630	0.293815	+	0.955862i	$0.405075\pi$
$374$	−575064.	−0.212587
$375$	−473928.	−0.174034
$376$	−744384.	−0.271536
$377$	895440.	0.324476
$378$	1.70080e6	0.612244
$379$	2.69866e6	0.965051	0.482526	−	0.875882i	$-0.339720\pi$
$379$	2.69866e6	0.965051	0.482526	+	0.875882i	$0.339720\pi$
$380$	−1.55098e6	−0.550993
$381$	−1.11628e6	−0.393969
$382$	3.56150e6	1.24875
$383$	2.39651e6	0.834799	0.417400	−	0.908723i	$-0.362942\pi$
$383$	2.39651e6	0.834799	0.417400	+	0.908723i	$0.362942\pi$
$384$	114688.	0.0396908
$385$	−1.71637e6	−0.590146
$386$	2.57504e6	0.879662
$387$	782984.	0.265751
$388$	−1.96595e6	−0.662970
$389$	2.49714e6	0.836698	0.418349	−	0.908286i	$-0.362609\pi$
$389$	2.49714e6	0.836698	0.418349	+	0.908286i	$0.362609\pi$
$390$	658560.	0.219247
$391$	−1.13252e6	−0.374633
$392$	160896.	0.0528847
$393$	12726.0	0.00415633
$394$	−3.06901e6	−0.995997
$395$	5.11543e6	1.64964
$396$	456288.	0.146218
$397$	4.78574e6	1.52396	0.761979	−	0.647602i	$-0.224228\pi$
$397$	4.78574e6	1.52396	0.761979	+	0.647602i	$0.224228\pi$
$398$	−2.61057e6	−0.826090
$399$	−1.12284e6	−0.353091
$400$	1.00634e6	0.314480
$401$	−1.46224e6	−0.454105	−0.227053	−	0.973882i	$-0.572909\pi$
$401$	−1.46224e6	−0.454105	−0.227053	+	0.973882i	$0.572909\pi$
$402$	−920080.	−0.283962
$403$	1.66096e6	0.509444
$404$	−3.01224e6	−0.918198
$405$	−2.16124e6	−0.654734
$406$	1.77809e6	0.535350
$407$	−201243.	−0.0602192
$408$	438144.	0.130307
$409$	−76780.0	−0.0226955	−0.0113478	−	0.999936i	$-0.503612\pi$
$409$	−76780.0	−0.0226955	−0.0113478	+	0.999936i	$0.503612\pi$
$410$	−3.36773e6	−0.989412
$411$	2.57725e6	0.752578
$412$	2.64397e6	0.767385
$413$	−3.42774e6	−0.988855
$414$	898608.	0.257673
$415$	6.73420e6	1.91940
$416$	−286720.	−0.0812315
$417$	971516.	0.273596
$418$	−678552.	−0.189951
$419$	286677.	0.0797733	0.0398867	−	0.999204i	$-0.487300\pi$
$419$	286677.	0.0797733	0.0398867	+	0.999204i	$0.487300\pi$
$420$	1.30771e6	0.361734
$421$	−3.45581e6	−0.950264	−0.475132	−	0.879915i	$-0.657600\pi$
$421$	−3.45581e6	−0.950264	−0.475132	+	0.879915i	$0.657600\pi$
$422$	−2.21969e6	−0.606753
$423$	2.25641e6	0.613152
$424$	716352.	0.193514
$425$	3.84452e6	1.03245
$426$	1.68344e6	0.449443
$427$	1.85704e6	0.492892
$428$	1.61299e6	0.425621
$429$	288120.	0.0755841
$430$	1.35610e6	0.353687
$431$	−5.92498e6	−1.53636	−0.768181	−	0.640233i	$-0.778838\pi$
$431$	−5.92498e6	−1.53636	−0.768181	+	0.640233i	$0.778838\pi$
$432$	−783104.	−0.201888
$433$	889733.	0.228055	0.114028	−	0.993478i	$-0.463625\pi$
$433$	889733.	0.228055	0.114028	+	0.993478i	$0.463625\pi$
$434$	3.29819e6	0.840527
$435$	1.88042e6	0.476467
$436$	−1.05693e6	−0.266274
$437$	−1.33633e6	−0.334742
$438$	1.17362e6	0.292309
$439$	−2.88771e6	−0.715141	−0.357571	−	0.933886i	$-0.616395\pi$
$439$	−2.88771e6	−0.715141	−0.357571	+	0.933886i	$0.616395\pi$
$440$	790272.	0.194601
$441$	−487716.	−0.119418
$442$	−1.09536e6	−0.266687
$443$	−1.20829e6	−0.292524	−0.146262	−	0.989246i	$-0.546724\pi$
$443$	−1.20829e6	−0.292524	−0.146262	+	0.989246i	$0.546724\pi$
$444$	153328.	0.0369117
$445$	1.10341e7	2.64141
$446$	−4.89051e6	−1.16417
$447$	3.46754e6	0.820829
$448$	−569344.	−0.134023
$449$	4.58427e6	1.07313	0.536567	−	0.843857i	$-0.319721\pi$
$449$	4.58427e6	1.07313	0.536567	+	0.843857i	$0.319721\pi$
$450$	−3.05046e6	−0.710123
$451$	−1.47338e6	−0.341094
$452$	−1.96387e6	−0.452134
$453$	−908488.	−0.208005
$454$	1.54051e6	0.350772
$455$	−3.26928e6	−0.740327
$456$	516992.	0.116432
$457$	2.16838e6	0.485675	0.242837	−	0.970067i	$-0.421922\pi$
$457$	2.16838e6	0.485675	0.242837	+	0.970067i	$0.421922\pi$
$458$	3.33720e6	0.743392
$459$	−2.99170e6	−0.662806
$460$	1.55635e6	0.342936
$461$	−2.19754e6	−0.481597	−0.240798	−	0.970575i	$-0.577409\pi$
$461$	−2.19754e6	−0.481597	−0.240798	+	0.970575i	$0.577409\pi$
$462$	572124.	0.124705
$463$	1.59900e6	0.346654	0.173327	−	0.984864i	$-0.444548\pi$
$463$	1.59900e6	0.346654	0.173327	+	0.984864i	$0.444548\pi$
$464$	−818688.	−0.176532
$465$	3.48802e6	0.748077
$466$	−5.29898e6	−1.13039
$467$	2.98676e6	0.633735	0.316868	−	0.948470i	$-0.397369\pi$
$467$	2.98676e6	0.633735	0.316868	+	0.948470i	$0.397369\pi$
$468$	869120.	0.183428
$469$	4.56754e6	0.958849
$470$	3.90802e6	0.816040
$471$	−1.94595e6	−0.404184
$472$	1.57824e6	0.326075
$473$	593292.	0.121931
$474$	−1.70514e6	−0.348590
$475$	4.53637e6	0.922518
$476$	−2.17507e6	−0.440004
$477$	−2.17144e6	−0.436971
$478$	6.56086e6	1.31338
$479$	−2.17790e6	−0.433709	−0.216855	−	0.976204i	$-0.569580\pi$
$479$	−2.17790e6	−0.433709	−0.216855	+	0.976204i	$0.569580\pi$
$480$	−602112.	−0.119282
$481$	−383320.	−0.0755438
$482$	−4.12698e6	−0.809124
$483$	1.12673e6	0.219762
$484$	−2.23107e6	−0.432913
$485$	1.03212e7	1.99241
$486$	3.69376e6	0.709378
$487$	−7.87061e6	−1.50379	−0.751893	−	0.659286i	$-0.770859\pi$
$487$	−7.87061e6	−1.50379	−0.751893	+	0.659286i	$0.770859\pi$
$488$	−855040.	−0.162531
$489$	1.78934e6	0.338392
$490$	−844704.	−0.158933
$491$	2.03394e6	0.380745	0.190373	−	0.981712i	$-0.439030\pi$
$491$	2.03394e6	0.380745	0.190373	+	0.981712i	$0.439030\pi$
$492$	1.12258e6	0.209075
$493$	−3.12764e6	−0.579562
$494$	−1.29248e6	−0.238290
$495$	−2.39551e6	−0.439426
$496$	−1.51859e6	−0.277164
$497$	−8.35710e6	−1.51762
$498$	−2.24473e6	−0.405594
$499$	2.50273e6	0.449948	0.224974	−	0.974365i	$-0.427770\pi$
$499$	2.50273e6	0.449948	0.224974	+	0.974365i	$0.427770\pi$
$500$	−1.08326e6	−0.193780
$501$	−2.15935e6	−0.384351
$502$	6.00494e6	1.06353
$503$	−3.53321e6	−0.622659	−0.311329	−	0.950302i	$-0.600774\pi$
$503$	−3.53321e6	−0.622659	−0.311329	+	0.950302i	$0.600774\pi$
$504$	1.72582e6	0.302636
$505$	1.58143e7	2.75944
$506$	680904.	0.118225
$507$	−2.05025e6	−0.354231
$508$	−2.55150e6	−0.438669
$509$	−8.39836e6	−1.43681	−0.718406	−	0.695624i	$-0.755128\pi$
$509$	−8.39836e6	−1.43681	−0.718406	+	0.695624i	$0.755128\pi$
$510$	−2.30026e6	−0.391607
$511$	−5.82618e6	−0.987034
$512$	262144.	0.0441942
$513$	−3.53009e6	−0.592232
$514$	505152.	0.0843362
$515$	−1.38808e7	−2.30620
$516$	−452032.	−0.0747386
$517$	1.70976e6	0.281325
$518$	−761164.	−0.124639
$519$	−2.20284e6	−0.358975
$520$	1.50528e6	0.244123
$521$	−1.00759e7	−1.62626	−0.813129	−	0.582084i	$-0.802237\pi$
$521$	−1.00759e7	−1.62626	−0.813129	+	0.582084i	$0.802237\pi$
$522$	2.48165e6	0.398624
$523$	6.17640e6	0.987372	0.493686	−	0.869640i	$-0.335649\pi$
$523$	6.17640e6	0.987372	0.493686	+	0.869640i	$0.335649\pi$
$524$	29088.0	0.00462792
$525$	−3.82486e6	−0.605644
$526$	4.27900e6	0.674338
$527$	−5.80150e6	−0.909941
$528$	−263424.	−0.0411217
$529$	−5.09538e6	−0.791657
$530$	−3.76085e6	−0.581562
$531$	−4.78404e6	−0.736306
$532$	−2.56650e6	−0.393153
$533$	−2.80644e6	−0.427896
$534$	−3.67802e6	−0.558163
$535$	−8.46821e6	−1.27911
$536$	−2.10304e6	−0.316181
$537$	5.02526e6	0.752008
$538$	−2.72052e6	−0.405225
$539$	−369558.	−0.0547912
$540$	4.11130e6	0.606728
$541$	−7.89089e6	−1.15913	−0.579566	−	0.814925i	$-0.696778\pi$
$541$	−7.89089e6	−1.15913	−0.579566	+	0.814925i	$0.696778\pi$
$542$	3.93830e6	0.575852
$543$	−3.45522e6	−0.502894
$544$	1.00147e6	0.145091
$545$	5.54887e6	0.800227
$546$	1.08976e6	0.156441
$547$	−30892.0	−0.00441446	−0.00220723	−	0.999998i	$-0.500703\pi$
$547$	−30892.0	−0.00441446	−0.00220723	+	0.999998i	$0.500703\pi$
$548$	5.89085e6	0.837966
$549$	2.59184e6	0.367010
$550$	−2.31143e6	−0.325817
$551$	−3.69049e6	−0.517852
$552$	−518784.	−0.0724668
$553$	8.46482e6	1.17708
$554$	7.04000e6	0.974537
$555$	−804972.	−0.110930
$556$	2.22061e6	0.304639
$557$	1.66821e6	0.227831	0.113915	−	0.993490i	$-0.463661\pi$
$557$	1.66821e6	0.227831	0.113915	+	0.993490i	$0.463661\pi$
$558$	4.60323e6	0.625860
$559$	1.13008e6	0.152961
$560$	2.98906e6	0.402776
$561$	−1.00636e6	−0.135004
$562$	−409056.	−0.0546314
$563$	3.53330e6	0.469797	0.234898	−	0.972020i	$-0.424524\pi$
$563$	3.53330e6	0.469797	0.234898	+	0.972020i	$0.424524\pi$
$564$	−1.30267e6	−0.172440
$565$	1.03103e7	1.35879
$566$	−754960.	−0.0990565
$567$	−3.57633e6	−0.467176
$568$	3.84787e6	0.500437
$569$	2.92326e6	0.378518	0.189259	−	0.981927i	$-0.439391\pi$
$569$	2.92326e6	0.378518	0.189259	+	0.981927i	$0.439391\pi$
$570$	−2.71421e6	−0.349910
$571$	3.16341e6	0.406036	0.203018	−	0.979175i	$-0.434925\pi$
$571$	3.16341e6	0.406036	0.203018	+	0.979175i	$0.434925\pi$
$572$	658560.	0.0841599
$573$	6.23263e6	0.793021
$574$	−5.57279e6	−0.705981
$575$	−4.55210e6	−0.574172
$576$	−794624.	−0.0997942
$577$	−8.92771e6	−1.11635	−0.558175	−	0.829723i	$-0.688498\pi$
$577$	−8.92771e6	−1.11635	−0.558175	+	0.829723i	$0.688498\pi$
$578$	−1.85349e6	−0.230766
$579$	4.50632e6	0.558632
$580$	4.29811e6	0.530527
$581$	1.11435e7	1.36956
$582$	−3.44042e6	−0.421021
$583$	−1.64537e6	−0.200490
$584$	2.68256e6	0.325475
$585$	−4.56288e6	−0.551251
$586$	1.01119e6	0.121644
$587$	4.59495e6	0.550409	0.275205	−	0.961386i	$-0.411254\pi$
$587$	4.59495e6	0.550409	0.275205	+	0.961386i	$0.411254\pi$
$588$	281568.	0.0335846
$589$	−6.84553e6	−0.813053
$590$	−8.28576e6	−0.979946
$591$	−5.37077e6	−0.632511
$592$	350464.	0.0410997
$593$	−1.41159e7	−1.64844	−0.824219	−	0.566271i	$-0.808386\pi$
$593$	−1.41159e7	−1.64844	−0.824219	+	0.566271i	$0.808386\pi$
$594$	1.79869e6	0.209166
$595$	1.14191e7	1.32233
$596$	7.92581e6	0.913961
$597$	−4.56849e6	−0.524611
$598$	1.29696e6	0.148311
$599$	2.44585e6	0.278525	0.139262	−	0.990256i	$-0.455527\pi$
$599$	2.44585e6	0.278525	0.139262	+	0.990256i	$0.455527\pi$
$600$	1.76109e6	0.199711
$601$	−1.43970e7	−1.62587	−0.812937	−	0.582352i	$-0.802133\pi$
$601$	−1.43970e7	−1.62587	−0.812937	+	0.582352i	$0.802133\pi$
$602$	2.24402e6	0.252368
$603$	6.37484e6	0.713963
$604$	−2.07654e6	−0.231605
$605$	1.17131e7	1.30102
$606$	−5.27142e6	−0.583104
$607$	−9.28038e6	−1.02234	−0.511168	−	0.859481i	$-0.670787\pi$
$607$	−9.28038e6	−1.02234	−0.511168	+	0.859481i	$0.670787\pi$
$608$	1.18170e6	0.129642
$609$	3.11165e6	0.339976
$610$	4.48896e6	0.488451
$611$	3.25668e6	0.352917
$612$	−3.03571e6	−0.327629
$613$	6.78601e6	0.729396	0.364698	−	0.931126i	$-0.381172\pi$
$613$	6.78601e6	0.729396	0.364698	+	0.931126i	$0.381172\pi$
$614$	−6.91260e6	−0.739981
$615$	−5.89352e6	−0.628329
$616$	1.30771e6	0.138855
$617$	1.31689e7	1.39263	0.696315	−	0.717736i	$-0.254822\pi$
$617$	1.31689e7	1.39263	0.696315	+	0.717736i	$0.254822\pi$
$618$	4.62694e6	0.487330
$619$	5.84201e6	0.612824	0.306412	−	0.951899i	$-0.400871\pi$
$619$	5.84201e6	0.612824	0.306412	+	0.951899i	$0.400871\pi$
$620$	7.97261e6	0.832954
$621$	3.54232e6	0.368603
$622$	−4.39020e6	−0.454997
$623$	1.82588e7	1.88474
$624$	−501760.	−0.0515863
$625$	−6.59724e6	−0.675557
$626$	5.81254e6	0.592830
$627$	−1.18747e6	−0.120629
$628$	−4.44789e6	−0.450044
$629$	1.33888e6	0.134932
$630$	−9.06058e6	−0.909504
$631$	−9.40619e6	−0.940460	−0.470230	−	0.882544i	$-0.655829\pi$
$631$	−9.40619e6	−0.940460	−0.470230	+	0.882544i	$0.655829\pi$
$632$	−3.89747e6	−0.388142
$633$	−3.88446e6	−0.385320
$634$	170904.	0.0168861
$635$	1.33954e7	1.31832
$636$	1.25362e6	0.122891
$637$	−703920.	−0.0687345
$638$	1.88042e6	0.182896
$639$	−1.16639e7	−1.13003
$640$	−1.37626e6	−0.132816
$641$	−1.36098e7	−1.30830	−0.654151	−	0.756364i	$-0.726974\pi$
$641$	−1.36098e7	−1.30830	−0.654151	+	0.756364i	$0.726974\pi$
$642$	2.82274e6	0.270292
$643$	2.03645e7	1.94243	0.971217	−	0.238195i	$-0.0765557\pi$
$643$	2.03645e7	1.94243	0.971217	+	0.238195i	$0.0765557\pi$
$644$	2.57539e6	0.244697
$645$	2.37317e6	0.224610
$646$	4.51445e6	0.425621
$647$	−8.69847e6	−0.816925	−0.408462	−	0.912775i	$-0.633935\pi$
$647$	−8.69847e6	−0.816925	−0.408462	+	0.912775i	$0.633935\pi$
$648$	1.64666e6	0.154051
$649$	−3.62502e6	−0.337830
$650$	−4.40272e6	−0.408731
$651$	5.77184e6	0.533779
$652$	4.08992e6	0.376787
$653$	1.08785e7	0.998361	0.499181	−	0.866498i	$-0.333634\pi$
$653$	1.08785e7	0.998361	0.499181	+	0.866498i	$0.333634\pi$
$654$	−1.84962e6	−0.169098
$655$	−152712.	−0.0139082
$656$	2.56589e6	0.232797
$657$	−8.13151e6	−0.734950
$658$	6.46684e6	0.582274
$659$	−1.04726e7	−0.939378	−0.469689	−	0.882832i	$-0.655634\pi$
$659$	−1.04726e7	−0.939378	−0.469689	+	0.882832i	$0.655634\pi$
$660$	1.38298e6	0.123582
$661$	1.43929e7	1.28128	0.640639	−	0.767842i	$-0.278669\pi$
$661$	1.43929e7	1.28128	0.640639	+	0.767842i	$0.278669\pi$
$662$	−9.59452e6	−0.850900
$663$	−1.91688e6	−0.169360
$664$	−5.13082e6	−0.451613
$665$	1.34741e7	1.18153
$666$	−1.06234e6	−0.0928068
$667$	3.70328e6	0.322309
$668$	−4.93565e6	−0.427960
$669$	−8.55839e6	−0.739310
$670$	1.10410e7	0.950211
$671$	1.96392e6	0.168390
$672$	−996352.	−0.0851118
$673$	2.28451e6	0.194427	0.0972133	−	0.995264i	$-0.469007\pi$
$673$	2.28451e6	0.194427	0.0972133	+	0.995264i	$0.469007\pi$
$674$	−3.14957e6	−0.267056
$675$	−1.20249e7	−1.01583
$676$	−4.68629e6	−0.394423
$677$	3.91453e6	0.328252	0.164126	−	0.986439i	$-0.447520\pi$
$677$	3.91453e6	0.328252	0.164126	+	0.986439i	$0.447520\pi$
$678$	−3.43678e6	−0.287129
$679$	1.70792e7	1.42165
$680$	−5.25773e6	−0.436040
$681$	2.69590e6	0.222759
$682$	3.48802e6	0.287156
$683$	2.22302e7	1.82344	0.911720	−	0.410811i	$-0.134754\pi$
$683$	2.22302e7	1.82344	0.911720	+	0.410811i	$0.134754\pi$
$684$	−3.58202e6	−0.292744
$685$	−3.09270e7	−2.51832
$686$	7.94691e6	0.644745
$687$	5.84009e6	0.472093
$688$	−1.03322e6	−0.0832185
$689$	−3.13404e6	−0.251511
$690$	2.72362e6	0.217783
$691$	8.01410e6	0.638498	0.319249	−	0.947671i	$-0.396569\pi$
$691$	8.01410e6	0.638498	0.319249	+	0.947671i	$0.396569\pi$
$692$	−5.03506e6	−0.399705
$693$	−3.96400e6	−0.313546
$694$	−1.10047e6	−0.0867322
$695$	−1.16582e7	−0.915523
$696$	−1.43270e6	−0.112107
$697$	9.80249e6	0.764284
$698$	−3.80961e6	−0.295966
$699$	−9.27322e6	−0.717857
$700$	−8.74254e6	−0.674361
$701$	−1.72312e7	−1.32441	−0.662203	−	0.749324i	$-0.730378\pi$
$701$	−1.72312e7	−1.32441	−0.662203	+	0.749324i	$0.730378\pi$
$702$	3.42608e6	0.262394
$703$	1.57983e6	0.120565
$704$	−602112.	−0.0457874
$705$	6.83903e6	0.518229
$706$	1.30896e7	0.988357
$707$	2.61688e7	1.96896
$708$	2.76192e6	0.207075
$709$	3.37761e6	0.252344	0.126172	−	0.992008i	$-0.459731\pi$
$709$	3.37761e6	0.252344	0.126172	+	0.992008i	$0.459731\pi$
$710$	−2.02013e7	−1.50395
$711$	1.18142e7	0.876457
$712$	−8.40691e6	−0.621493
$713$	6.86926e6	0.506041
$714$	−3.80638e6	−0.279426
$715$	−3.45744e6	−0.252924
$716$	1.14863e7	0.837332
$717$	1.14815e7	0.834066
$718$	2.98940e6	0.216408
$719$	−7.43292e6	−0.536213	−0.268107	−	0.963389i	$-0.586398\pi$
$719$	−7.43292e6	−0.536213	−0.268107	+	0.963389i	$0.586398\pi$
$720$	4.17178e6	0.299909
$721$	−2.29695e7	−1.64556
$722$	−4.57753e6	−0.326805
$723$	−7.22222e6	−0.513837
$724$	−7.89765e6	−0.559953
$725$	−1.25713e7	−0.888253
$726$	−3.90438e6	−0.274922
$727$	1.28188e7	0.899523	0.449761	−	0.893149i	$-0.351509\pi$
$727$	1.28188e7	0.899523	0.449761	+	0.893149i	$0.351509\pi$
$728$	2.49088e6	0.174190
$729$	211933.	0.0147700
$730$	−1.40834e7	−0.978141
$731$	−3.94721e6	−0.273210
$732$	−1.49632e6	−0.103216
$733$	8.75160e6	0.601627	0.300814	−	0.953683i	$-0.402742\pi$
$733$	8.75160e6	0.601627	0.300814	+	0.953683i	$0.402742\pi$
$734$	−1.63344e7	−1.11908
$735$	−1.47823e6	−0.100931
$736$	−1.18579e6	−0.0806889
$737$	4.83042e6	0.327579
$738$	−7.77785e6	−0.525676
$739$	1.15693e7	0.779283	0.389641	−	0.920967i	$-0.372599\pi$
$739$	1.15693e7	0.779283	0.389641	+	0.920967i	$0.372599\pi$
$740$	−1.83994e6	−0.123516
$741$	−2.26184e6	−0.151327
$742$	−6.22331e6	−0.414965
$743$	2.21079e7	1.46918	0.734591	−	0.678510i	$-0.237374\pi$
$743$	2.21079e7	1.46918	0.734591	+	0.678510i	$0.237374\pi$
$744$	−2.65754e6	−0.176014
$745$	−4.16105e7	−2.74671
$746$	6.31591e6	0.415517
$747$	1.55528e7	1.01978
$748$	−2.30026e6	−0.150322
$749$	−1.40129e7	−0.912688
$750$	−1.89571e6	−0.123061
$751$	−1.65053e7	−1.06788	−0.533942	−	0.845521i	$-0.679290\pi$
$751$	−1.65053e7	−1.06788	−0.533942	+	0.845521i	$0.679290\pi$
$752$	−2.97754e6	−0.192005
$753$	1.05087e7	0.675398
$754$	3.58176e6	0.229439
$755$	1.09019e7	0.696038
$756$	6.80322e6	0.432922
$757$	1.35366e7	0.858562	0.429281	−	0.903171i	$-0.358767\pi$
$757$	1.35366e7	0.858562	0.429281	+	0.903171i	$0.358767\pi$
$758$	1.07946e7	0.682394
$759$	1.19158e6	0.0750792
$760$	−6.20390e6	−0.389611
$761$	−7.40555e6	−0.463549	−0.231775	−	0.972770i	$-0.574453\pi$
$761$	−7.40555e6	−0.463549	−0.231775	+	0.972770i	$0.574453\pi$
$762$	−4.46513e6	−0.278578
$763$	9.18206e6	0.570990
$764$	1.42460e7	0.882999
$765$	1.59375e7	0.984615
$766$	9.58603e6	0.590292
$767$	−6.90480e6	−0.423802
$768$	458752.	0.0280656
$769$	−2.61625e7	−1.59538	−0.797690	−	0.603068i	$-0.793945\pi$
$769$	−2.61625e7	−1.59538	−0.797690	+	0.603068i	$0.793945\pi$
$770$	−6.86549e6	−0.417296
$771$	884016.	0.0535580
$772$	1.03002e7	0.622015
$773$	−5.90612e6	−0.355511	−0.177756	−	0.984075i	$-0.556884\pi$
$773$	−5.90612e6	−0.355511	−0.177756	+	0.984075i	$0.556884\pi$
$774$	3.13194e6	0.187915
$775$	−2.33187e7	−1.39460
$776$	−7.86381e6	−0.468790
$777$	−1.33204e6	−0.0791523
$778$	9.98856e6	0.591635
$779$	1.15665e7	0.682904
$780$	2.63424e6	0.155031
$781$	−8.83808e6	−0.518478
$782$	−4.53010e6	−0.264905
$783$	9.78268e6	0.570235
$784$	643584.	0.0373951
$785$	2.33514e7	1.35251
$786$	50904.0	0.00293897
$787$	−1.46812e7	−0.844939	−0.422469	−	0.906377i	$-0.638837\pi$
$787$	−1.46812e7	−0.844939	−0.422469	+	0.906377i	$0.638837\pi$
$788$	−1.22760e7	−0.704276
$789$	7.48824e6	0.428240
$790$	2.04617e7	1.16647
$791$	1.70611e7	0.969542
$792$	1.82515e6	0.103392
$793$	3.74080e6	0.211243
$794$	1.91430e7	1.07760
$795$	−6.58148e6	−0.369323
$796$	−1.04423e7	−0.584134
$797$	−2.81229e7	−1.56825	−0.784123	−	0.620606i	$-0.786887\pi$
$797$	−2.81229e7	−1.56825	−0.784123	+	0.620606i	$0.786887\pi$
$798$	−4.49137e6	−0.249673
$799$	−1.13751e7	−0.630360
$800$	4.02534e6	0.222371
$801$	2.54835e7	1.40339
$802$	−5.84894e6	−0.321101
$803$	−6.16151e6	−0.337208
$804$	−3.68032e6	−0.200792
$805$	−1.35208e7	−0.735382
$806$	6.64384e6	0.360231
$807$	−4.76091e6	−0.257339
$808$	−1.20490e7	−0.649264
$809$	−2.44219e7	−1.31192	−0.655960	−	0.754795i	$-0.727736\pi$
$809$	−2.44219e7	−1.31192	−0.655960	+	0.754795i	$0.727736\pi$
$810$	−8.64494e6	−0.462967
$811$	−5.68689e6	−0.303615	−0.151807	−	0.988410i	$-0.548509\pi$
$811$	−5.68689e6	−0.303615	−0.151807	+	0.988410i	$0.548509\pi$
$812$	7.11235e6	0.378550
$813$	6.89202e6	0.365696
$814$	−804972.	−0.0425814
$815$	−2.14721e7	−1.13235
$816$	1.75258e6	0.0921407
$817$	−4.65754e6	−0.244119
$818$	−307120.	−0.0160481
$819$	−7.55048e6	−0.393337
$820$	−1.34709e7	−0.699620
$821$	−2.83725e7	−1.46906	−0.734531	−	0.678575i	$-0.762598\pi$
$821$	−2.83725e7	−1.46906	−0.734531	+	0.678575i	$0.762598\pi$
$822$	1.03090e7	0.532153
$823$	−2.12488e7	−1.09354	−0.546770	−	0.837283i	$-0.684143\pi$
$823$	−2.12488e7	−1.09354	−0.546770	+	0.837283i	$0.684143\pi$
$824$	1.05759e7	0.542623
$825$	−4.04500e6	−0.206911
$826$	−1.37110e7	−0.699226
$827$	−3.27949e7	−1.66741	−0.833706	−	0.552209i	$-0.813785\pi$
$827$	−3.27949e7	−1.66741	−0.833706	+	0.552209i	$0.813785\pi$
$828$	3.59443e6	0.182203
$829$	−3.55433e7	−1.79627	−0.898134	−	0.439721i	$-0.855077\pi$
$829$	−3.55433e7	−1.79627	−0.898134	+	0.439721i	$0.855077\pi$
$830$	2.69368e7	1.35722
$831$	1.23200e7	0.618882
$832$	−1.14688e6	−0.0574394
$833$	2.45869e6	0.122770
$834$	3.88606e6	0.193462
$835$	2.59122e7	1.28614
$836$	−2.71421e6	−0.134316
$837$	1.81460e7	0.895297
$838$	1.14671e6	0.0564083
$839$	−4.54394e6	−0.222858	−0.111429	−	0.993772i	$-0.535543\pi$
$839$	−4.54394e6	−0.222858	−0.111429	+	0.993772i	$0.535543\pi$
$840$	5.23085e6	0.255784
$841$	−1.02839e7	−0.501383
$842$	−1.38232e7	−0.671938
$843$	−715848.	−0.0346938
$844$	−8.87877e6	−0.429039
$845$	2.46030e7	1.18535
$846$	9.02566e6	0.433564
$847$	1.93824e7	0.928325
$848$	2.86541e6	0.136835
$849$	−1.32118e6	−0.0629061
$850$	1.53781e7	0.730053
$851$	−1.58530e6	−0.0750392
$852$	6.73378e6	0.317804
$853$	1.82388e6	0.0858270	0.0429135	−	0.999079i	$-0.486336\pi$
$853$	1.82388e6	0.0858270	0.0429135	+	0.999079i	$0.486336\pi$
$854$	7.42816e6	0.348527
$855$	1.88056e7	0.879775
$856$	6.45197e6	0.300959
$857$	−3.78424e6	−0.176006	−0.0880029	−	0.996120i	$-0.528048\pi$
$857$	−3.78424e6	−0.176006	−0.0880029	+	0.996120i	$0.528048\pi$
$858$	1.15248e6	0.0534460
$859$	2.13312e7	0.986354	0.493177	−	0.869929i	$-0.335836\pi$
$859$	2.13312e7	0.986354	0.493177	+	0.869929i	$0.335836\pi$
$860$	5.42438e6	0.250095
$861$	−9.75238e6	−0.448335
$862$	−2.36999e7	−1.08637
$863$	4.09733e6	0.187272	0.0936362	−	0.995606i	$-0.470151\pi$
$863$	4.09733e6	0.187272	0.0936362	+	0.995606i	$0.470151\pi$
$864$	−3.13242e6	−0.142756
$865$	2.64340e7	1.20122
$866$	3.55893e6	0.161259
$867$	−3.24361e6	−0.146548
$868$	1.31928e7	0.594342
$869$	8.95201e6	0.402134
$870$	7.52170e6	0.336913
$871$	9.20080e6	0.410942
$872$	−4.22771e6	−0.188284
$873$	2.38372e7	1.05857
$874$	−5.34533e6	−0.236699
$875$	9.41086e6	0.415536
$876$	4.69448e6	0.206694
$877$	1.82256e7	0.800171	0.400086	−	0.916478i	$-0.368980\pi$
$877$	1.82256e7	0.800171	0.400086	+	0.916478i	$0.368980\pi$
$878$	−1.15508e7	−0.505681
$879$	1.76959e6	0.0772502
$880$	3.16109e6	0.137604
$881$	1.11601e7	0.484425	0.242213	−	0.970223i	$-0.422127\pi$
$881$	1.11601e7	0.484425	0.242213	+	0.970223i	$0.422127\pi$
$882$	−1.95086e6	−0.0844414
$883$	2.59857e7	1.12159	0.560793	−	0.827956i	$-0.310496\pi$
$883$	2.59857e7	1.12159	0.560793	+	0.827956i	$0.310496\pi$
$884$	−4.38144e6	−0.188576
$885$	−1.45001e7	−0.622318
$886$	−4.83316e6	−0.206846
$887$	8.83725e6	0.377145	0.188572	−	0.982059i	$-0.439614\pi$
$887$	8.83725e6	0.377145	0.188572	+	0.982059i	$0.439614\pi$
$888$	613312.	0.0261005
$889$	2.21662e7	0.940669
$890$	4.41363e7	1.86776
$891$	−3.78216e6	−0.159605
$892$	−1.95620e7	−0.823193
$893$	−1.34222e7	−0.563241
$894$	1.38702e7	0.580414
$895$	−6.03031e7	−2.51641
$896$	−2.27738e6	−0.0947687
$897$	2.26968e6	0.0941854
$898$	1.83371e7	0.758821
$899$	1.89705e7	0.782853
$900$	−1.22018e7	−0.502133
$901$	1.09468e7	0.449235
$902$	−5.89352e6	−0.241190
$903$	3.92703e6	0.160267
$904$	−7.85549e6	−0.319707
$905$	4.14627e7	1.68281
$906$	−3.63395e6	−0.147082
$907$	6.35281e6	0.256417	0.128209	−	0.991747i	$-0.459077\pi$
$907$	6.35281e6	0.256417	0.128209	+	0.991747i	$0.459077\pi$
$908$	6.16205e6	0.248034
$909$	3.65234e7	1.46609
$910$	−1.30771e7	−0.523490
$911$	1.75069e7	0.698896	0.349448	−	0.936956i	$-0.386369\pi$
$911$	1.75069e7	0.698896	0.349448	+	0.936956i	$0.386369\pi$
$912$	2.06797e6	0.0823297
$913$	1.17848e7	0.467893
$914$	8.67354e6	0.343424
$915$	7.85568e6	0.310192
$916$	1.33488e7	0.525658
$917$	−252702.	−0.00992397
$918$	−1.19668e7	−0.468675
$919$	3.34259e7	1.30555	0.652776	−	0.757551i	$-0.273604\pi$
$919$	3.34259e7	1.30555	0.652776	+	0.757551i	$0.273604\pi$
$920$	6.22541e6	0.242492
$921$	−1.20971e7	−0.469927
$922$	−8.79014e6	−0.340540
$923$	−1.68344e7	−0.650421
$924$	2.28850e6	0.0881800
$925$	5.38154e6	0.206801
$926$	6.39601e6	0.245122
$927$	−3.20581e7	−1.22529
$928$	−3.27475e6	−0.124827
$929$	−4.53366e7	−1.72349	−0.861747	−	0.507338i	$-0.830630\pi$
$929$	−4.53366e7	−1.72349	−0.861747	+	0.507338i	$0.830630\pi$
$930$	1.39521e7	0.528970
$931$	2.90116e6	0.109698
$932$	−2.11959e7	−0.799306
$933$	−7.68285e6	−0.288947
$934$	1.19470e7	0.448118
$935$	1.20763e7	0.451759
$936$	3.47648e6	0.129703
$937$	−3.12117e7	−1.16136	−0.580682	−	0.814130i	$-0.697214\pi$
$937$	−3.12117e7	−1.16136	−0.580682	+	0.814130i	$0.697214\pi$
$938$	1.82702e7	0.678009
$939$	1.01720e7	0.376479
$940$	1.56321e7	0.577028
$941$	−9.86035e6	−0.363010	−0.181505	−	0.983390i	$-0.558097\pi$
$941$	−9.86035e6	−0.363010	−0.181505	+	0.983390i	$0.558097\pi$
$942$	−7.78380e6	−0.285802
$943$	−1.16066e7	−0.425037
$944$	6.31296e6	0.230570
$945$	−3.57169e7	−1.30105
$946$	2.37317e6	0.0862185
$947$	−1.11578e7	−0.404302	−0.202151	−	0.979354i	$-0.564793\pi$
$947$	−1.11578e7	−0.404302	−0.202151	+	0.979354i	$0.564793\pi$
$948$	−6.82058e6	−0.246491
$949$	−1.17362e7	−0.423021
$950$	1.81455e7	0.652319
$951$	299082.	0.0107236
$952$	−8.70029e6	−0.311130
$953$	240981.	0.00859509	0.00429755	−	0.999991i	$-0.498632\pi$
$953$	240981.	0.00859509	0.00429755	+	0.999991i	$0.498632\pi$
$954$	−8.68577e6	−0.308985
$955$	−7.47916e7	−2.65365
$956$	2.62434e7	0.928701
$957$	3.29074e6	0.116149
$958$	−8.71159e6	−0.306679
$959$	−5.11767e7	−1.79691
$960$	−2.40845e6	−0.0843450
$961$	6.55947e6	0.229119
$962$	−1.53328e6	−0.0534175
$963$	−1.95575e7	−0.679592
$964$	−1.65079e7	−0.572137
$965$	−5.40758e7	−1.86933
$966$	4.50694e6	0.155396
$967$	4.32780e7	1.48834	0.744169	−	0.667992i	$-0.232846\pi$
$967$	4.32780e7	1.48834	0.744169	+	0.667992i	$0.232846\pi$
$968$	−8.92429e6	−0.306115
$969$	7.90028e6	0.270292
$970$	4.12850e7	1.40884
$971$	−4.51121e7	−1.53548	−0.767742	−	0.640759i	$-0.778620\pi$
$971$	−4.51121e7	−1.53548	−0.767742	+	0.640759i	$0.778620\pi$
$972$	1.47750e7	0.501606
$973$	−1.92915e7	−0.653258
$974$	−3.14824e7	−1.06334
$975$	−7.70476e6	−0.259566
$976$	−3.42016e6	−0.114927
$977$	−2.08737e7	−0.699621	−0.349811	−	0.936820i	$-0.613754\pi$
$977$	−2.08737e7	−0.699621	−0.349811	+	0.936820i	$0.613754\pi$
$978$	7.15736e6	0.239280
$979$	1.93096e7	0.643898
$980$	−3.37882e6	−0.112383
$981$	1.28153e7	0.425162
$982$	8.13576e6	0.269227
$983$	2.40935e7	0.795274	0.397637	−	0.917543i	$-0.369830\pi$
$983$	2.40935e7	0.795274	0.397637	+	0.917543i	$0.369830\pi$
$984$	4.49030e6	0.147839
$985$	6.44493e7	2.11654
$986$	−1.25106e7	−0.409812
$987$	1.13170e7	0.369775
$988$	−5.16992e6	−0.168497
$989$	4.67369e6	0.151939
$990$	−9.58205e6	−0.310721
$991$	−5.18446e7	−1.67695	−0.838473	−	0.544943i	$-0.816551\pi$
$991$	−5.18446e7	−1.67695	−0.838473	+	0.544943i	$0.816551\pi$
$992$	−6.07437e6	−0.195985
$993$	−1.67904e7	−0.540366
$994$	−3.34284e7	−1.07312
$995$	5.48219e7	1.75548
$996$	−8.97893e6	−0.286798
$997$	−3.10444e7	−0.989113	−0.494557	−	0.869145i	$-0.664670\pi$
$997$	−3.10444e7	−0.989113	−0.494557	+	0.869145i	$0.664670\pi$
$998$	1.00109e7	0.318161
$999$	−4.18777e6	−0.132761

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.6.a.b.1.1	✓	1
3.2	odd	2		666.6.a.b.1.1		1
4.3	odd	2		592.6.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.6.a.b.1.1	✓	1	1.1	even	1	trivial
592.6.a.a.1.1		1	4.3	odd	2
666.6.a.b.1.1		1	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	74.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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