LMFDB - Embedded newform 75.6.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,6,Mod(1,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, 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("75.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	75.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:	$12.0287864860$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	75.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+6.00000 q^{2} -9.00000 q^{3} +4.00000 q^{4} -54.0000 q^{6} +40.0000 q^{7} -168.000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+6.00000 q^{2} -9.00000 q^{3} +4.00000 q^{4} -54.0000 q^{6} +40.0000 q^{7} -168.000 q^{8} +81.0000 q^{9} -564.000 q^{11} -36.0000 q^{12} -638.000 q^{13} +240.000 q^{14} -1136.00 q^{16} -882.000 q^{17} +486.000 q^{18} -556.000 q^{19} -360.000 q^{21} -3384.00 q^{22} +840.000 q^{23} +1512.00 q^{24} -3828.00 q^{26} -729.000 q^{27} +160.000 q^{28} +4638.00 q^{29} +4400.00 q^{31} -1440.00 q^{32} +5076.00 q^{33} -5292.00 q^{34} +324.000 q^{36} +2410.00 q^{37} -3336.00 q^{38} +5742.00 q^{39} -6870.00 q^{41} -2160.00 q^{42} -9644.00 q^{43} -2256.00 q^{44} +5040.00 q^{46} +18672.0 q^{47} +10224.0 q^{48} -15207.0 q^{49} +7938.00 q^{51} -2552.00 q^{52} -33750.0 q^{53} -4374.00 q^{54} -6720.00 q^{56} +5004.00 q^{57} +27828.0 q^{58} -18084.0 q^{59} +39758.0 q^{61} +26400.0 q^{62} +3240.00 q^{63} +27712.0 q^{64} +30456.0 q^{66} +23068.0 q^{67} -3528.00 q^{68} -7560.00 q^{69} -4248.00 q^{71} -13608.0 q^{72} +41110.0 q^{73} +14460.0 q^{74} -2224.00 q^{76} -22560.0 q^{77} +34452.0 q^{78} +21920.0 q^{79} +6561.00 q^{81} -41220.0 q^{82} -82452.0 q^{83} -1440.00 q^{84} -57864.0 q^{86} -41742.0 q^{87} +94752.0 q^{88} -94086.0 q^{89} -25520.0 q^{91} +3360.00 q^{92} -39600.0 q^{93} +112032. q^{94} +12960.0 q^{96} -49442.0 q^{97} -91242.0 q^{98} -45684.0 q^{99} +O(q^{100})\)

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 $

$2$	6.00000	1.06066	0.530330	−	0.847791i	$-0.322068\pi$
$2$	6.00000	1.06066	0.530330	+	0.847791i	$0.322068\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	4.00000	0.125000
$5$	0	0
$5$	0	0
$6$	−54.0000	−0.612372
$7$	40.0000	0.308542	0.154271	−	0.988029i	$-0.450697\pi$
$7$	40.0000	0.308542	0.154271	+	0.988029i	$0.450697\pi$
$8$	−168.000	−0.928078
$9$	81.0000	0.333333
$10$	0	0
$11$	−564.000	−1.40539	−0.702696	−	0.711490i	$-0.748021\pi$
$11$	−564.000	−1.40539	−0.702696	+	0.711490i	$0.748021\pi$
$12$	−36.0000	−0.0721688
$13$	−638.000	−1.04704	−0.523519	−	0.852014i	$-0.675381\pi$
$13$	−638.000	−1.04704	−0.523519	+	0.852014i	$0.675381\pi$
$14$	240.000	0.327259
$15$	0	0
$16$	−1136.00	−1.10938
$17$	−882.000	−0.740195	−0.370098	−	0.928993i	$-0.620676\pi$
$17$	−882.000	−0.740195	−0.370098	+	0.928993i	$0.620676\pi$
$18$	486.000	0.353553
$19$	−556.000	−0.353338	−0.176669	−	0.984270i	$-0.556532\pi$
$19$	−556.000	−0.353338	−0.176669	+	0.984270i	$0.556532\pi$
$20$	0	0
$21$	−360.000	−0.178137
$22$	−3384.00	−1.49064
$23$	840.000	0.331100	0.165550	−	0.986201i	$-0.447060\pi$
$23$	840.000	0.331100	0.165550	+	0.986201i	$0.447060\pi$
$24$	1512.00	0.535826
$25$	0	0
$26$	−3828.00	−1.11055
$27$	−729.000	−0.192450
$28$	160.000	0.0385678
$29$	4638.00	1.02408	0.512042	−	0.858960i	$-0.328889\pi$
$29$	4638.00	1.02408	0.512042	+	0.858960i	$0.328889\pi$
$30$	0	0
$31$	4400.00	0.822334	0.411167	−	0.911560i	$-0.365121\pi$
$31$	4400.00	0.822334	0.411167	+	0.911560i	$0.365121\pi$
$32$	−1440.00	−0.248592
$33$	5076.00	0.811403
$34$	−5292.00	−0.785096
$35$	0	0
$36$	324.000	0.0416667
$37$	2410.00	0.289409	0.144705	−	0.989475i	$-0.453777\pi$
$37$	2410.00	0.289409	0.144705	+	0.989475i	$0.453777\pi$
$38$	−3336.00	−0.374772
$39$	5742.00	0.604507
$40$	0	0
$41$	−6870.00	−0.638259	−0.319130	−	0.947711i	$-0.603391\pi$
$41$	−6870.00	−0.638259	−0.319130	+	0.947711i	$0.603391\pi$
$42$	−2160.00	−0.188943
$43$	−9644.00	−0.795401	−0.397700	−	0.917515i	$-0.630192\pi$
$43$	−9644.00	−0.795401	−0.397700	+	0.917515i	$0.630192\pi$
$44$	−2256.00	−0.175674
$45$	0	0
$46$	5040.00	0.351185
$47$	18672.0	1.23295	0.616476	−	0.787374i	$-0.288560\pi$
$47$	18672.0	1.23295	0.616476	+	0.787374i	$0.288560\pi$
$48$	10224.0	0.640498
$49$	−15207.0	−0.904802
$50$	0	0
$51$	7938.00	0.427352
$52$	−2552.00	−0.130880
$53$	−33750.0	−1.65038	−0.825190	−	0.564855i	$-0.808932\pi$
$53$	−33750.0	−1.65038	−0.825190	+	0.564855i	$0.808932\pi$
$54$	−4374.00	−0.204124
$55$	0	0
$56$	−6720.00	−0.286351
$57$	5004.00	0.204000
$58$	27828.0	1.08621
$59$	−18084.0	−0.676339	−0.338170	−	0.941085i	$-0.609808\pi$
$59$	−18084.0	−0.676339	−0.338170	+	0.941085i	$0.609808\pi$
$60$	0	0
$61$	39758.0	1.36804	0.684022	−	0.729462i	$-0.260229\pi$
$61$	39758.0	1.36804	0.684022	+	0.729462i	$0.260229\pi$
$62$	26400.0	0.872217
$63$	3240.00	0.102847
$64$	27712.0	0.845703
$65$	0	0
$66$	30456.0	0.860623
$67$	23068.0	0.627802	0.313901	−	0.949456i	$-0.398364\pi$
$67$	23068.0	0.627802	0.313901	+	0.949456i	$0.398364\pi$
$68$	−3528.00	−0.0925244
$69$	−7560.00	−0.191161
$70$	0	0
$71$	−4248.00	−0.100009	−0.0500044	−	0.998749i	$-0.515924\pi$
$71$	−4248.00	−0.100009	−0.0500044	+	0.998749i	$0.515924\pi$
$72$	−13608.0	−0.309359
$73$	41110.0	0.902901	0.451451	−	0.892296i	$-0.350907\pi$
$73$	41110.0	0.902901	0.451451	+	0.892296i	$0.350907\pi$
$74$	14460.0	0.306965
$75$	0	0
$76$	−2224.00	−0.0441673
$77$	−22560.0	−0.433623
$78$	34452.0	0.641177
$79$	21920.0	0.395160	0.197580	−	0.980287i	$-0.436692\pi$
$79$	21920.0	0.395160	0.197580	+	0.980287i	$0.436692\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−41220.0	−0.676976
$83$	−82452.0	−1.31373	−0.656865	−	0.754008i	$-0.728118\pi$
$83$	−82452.0	−1.31373	−0.656865	+	0.754008i	$0.728118\pi$
$84$	−1440.00	−0.0222671
$85$	0	0
$86$	−57864.0	−0.843650
$87$	−41742.0	−0.591255
$88$	94752.0	1.30431
$89$	−94086.0	−1.25907	−0.629535	−	0.776972i	$-0.716755\pi$
$89$	−94086.0	−1.25907	−0.629535	+	0.776972i	$0.716755\pi$
$90$	0	0
$91$	−25520.0	−0.323056
$92$	3360.00	0.0413875
$93$	−39600.0	−0.474775
$94$	112032.	1.30774
$95$	0	0
$96$	12960.0	0.143525
$97$	−49442.0	−0.533540	−0.266770	−	0.963760i	$-0.585956\pi$
$97$	−49442.0	−0.533540	−0.266770	+	0.963760i	$0.585956\pi$
$98$	−91242.0	−0.959687
$99$	−45684.0	−0.468464
$100$	0	0
$101$	−143034.	−1.39520	−0.697599	−	0.716488i	$-0.745748\pi$
$101$	−143034.	−1.39520	−0.697599	+	0.716488i	$0.745748\pi$
$102$	47628.0	0.453275
$103$	−53144.0	−0.493584	−0.246792	−	0.969068i	$-0.579376\pi$
$103$	−53144.0	−0.493584	−0.246792	+	0.969068i	$0.579376\pi$
$104$	107184.	0.971732
$105$	0	0
$106$	−202500.	−1.75049
$107$	−90828.0	−0.766938	−0.383469	−	0.923554i	$-0.625271\pi$
$107$	−90828.0	−0.766938	−0.383469	+	0.923554i	$0.625271\pi$
$108$	−2916.00	−0.0240563
$109$	−61666.0	−0.497141	−0.248570	−	0.968614i	$-0.579961\pi$
$109$	−61666.0	−0.497141	−0.248570	+	0.968614i	$0.579961\pi$
$110$	0	0
$111$	−21690.0	−0.167091
$112$	−45440.0	−0.342289
$113$	−10482.0	−0.0772232	−0.0386116	−	0.999254i	$-0.512294\pi$
$113$	−10482.0	−0.0772232	−0.0386116	+	0.999254i	$0.512294\pi$
$114$	30024.0	0.216375
$115$	0	0
$116$	18552.0	0.128011
$117$	−51678.0	−0.349013
$118$	−108504.	−0.717366
$119$	−35280.0	−0.228382
$120$	0	0
$121$	157045.	0.975126
$122$	238548.	1.45103
$123$	61830.0	0.368499
$124$	17600.0	0.102792
$125$	0	0
$126$	19440.0	0.109086
$127$	171088.	0.941261	0.470631	−	0.882330i	$-0.344026\pi$
$127$	171088.	0.941261	0.470631	+	0.882330i	$0.344026\pi$
$128$	212352.	1.14560
$129$	86796.0	0.459225
$130$	0	0
$131$	258468.	1.31592	0.657959	−	0.753054i	$-0.271420\pi$
$131$	258468.	1.31592	0.657959	+	0.753054i	$0.271420\pi$
$132$	20304.0	0.101425
$133$	−22240.0	−0.109020
$134$	138408.	0.665885
$135$	0	0
$136$	148176.	0.686959
$137$	−300234.	−1.36665	−0.683327	−	0.730113i	$-0.739468\pi$
$137$	−300234.	−1.36665	−0.683327	+	0.730113i	$0.739468\pi$
$138$	−45360.0	−0.202757
$139$	−350164.	−1.53721	−0.768607	−	0.639721i	$-0.779050\pi$
$139$	−350164.	−1.53721	−0.768607	+	0.639721i	$0.779050\pi$
$140$	0	0
$141$	−168048.	−0.711845
$142$	−25488.0	−0.106075
$143$	359832.	1.47150
$144$	−92016.0	−0.369792
$145$	0	0
$146$	246660.	0.957672
$147$	136863.	0.522387
$148$	9640.00	0.0361762
$149$	−105258.	−0.388409	−0.194205	−	0.980961i	$-0.562213\pi$
$149$	−105258.	−0.388409	−0.194205	+	0.980961i	$0.562213\pi$
$150$	0	0
$151$	396392.	1.41476	0.707380	−	0.706834i	$-0.249877\pi$
$151$	396392.	1.41476	0.707380	+	0.706834i	$0.249877\pi$
$152$	93408.0	0.327925
$153$	−71442.0	−0.246732
$154$	−135360.	−0.459927
$155$	0	0
$156$	22968.0	0.0755634
$157$	137746.	0.445995	0.222997	−	0.974819i	$-0.428416\pi$
$157$	137746.	0.445995	0.222997	+	0.974819i	$0.428416\pi$
$158$	131520.	0.419130
$159$	303750.	0.952848
$160$	0	0
$161$	33600.0	0.102159
$162$	39366.0	0.117851
$163$	−352676.	−1.03970	−0.519849	−	0.854258i	$-0.674012\pi$
$163$	−352676.	−1.03970	−0.519849	+	0.854258i	$0.674012\pi$
$164$	−27480.0	−0.0797824
$165$	0	0
$166$	−494712.	−1.39342
$167$	217560.	0.603654	0.301827	−	0.953363i	$-0.402404\pi$
$167$	217560.	0.603654	0.301827	+	0.953363i	$0.402404\pi$
$168$	60480.0	0.165325
$169$	35751.0	0.0962878
$170$	0	0
$171$	−45036.0	−0.117779
$172$	−38576.0	−0.0994251
$173$	163698.	0.415842	0.207921	−	0.978146i	$-0.433330\pi$
$173$	163698.	0.415842	0.207921	+	0.978146i	$0.433330\pi$
$174$	−250452.	−0.627121
$175$	0	0
$176$	640704.	1.55911
$177$	162756.	0.390485
$178$	−564516.	−1.33545
$179$	358740.	0.836849	0.418425	−	0.908252i	$-0.362582\pi$
$179$	358740.	0.836849	0.418425	+	0.908252i	$0.362582\pi$
$180$	0	0
$181$	−507130.	−1.15060	−0.575298	−	0.817944i	$-0.695114\pi$
$181$	−507130.	−1.15060	−0.575298	+	0.817944i	$0.695114\pi$
$182$	−153120.	−0.342652
$183$	−357822.	−0.789840
$184$	−141120.	−0.307287
$185$	0	0
$186$	−237600.	−0.503575
$187$	497448.	1.04026
$188$	74688.0	0.154119
$189$	−29160.0	−0.0593790
$190$	0	0
$191$	−648384.	−1.28602	−0.643012	−	0.765856i	$-0.722315\pi$
$191$	−648384.	−1.28602	−0.643012	+	0.765856i	$0.722315\pi$
$192$	−249408.	−0.488267
$193$	27838.0	0.0537954	0.0268977	−	0.999638i	$-0.491437\pi$
$193$	27838.0	0.0537954	0.0268977	+	0.999638i	$0.491437\pi$
$194$	−296652.	−0.565904
$195$	0	0
$196$	−60828.0	−0.113100
$197$	−611046.	−1.12178	−0.560891	−	0.827890i	$-0.689541\pi$
$197$	−611046.	−1.12178	−0.560891	+	0.827890i	$0.689541\pi$
$198$	−274104.	−0.496881
$199$	879032.	1.57352	0.786760	−	0.617260i	$-0.211757\pi$
$199$	879032.	1.57352	0.786760	+	0.617260i	$0.211757\pi$
$200$	0	0
$201$	−207612.	−0.362462
$202$	−858204.	−1.47983
$203$	185520.	0.315973
$204$	31752.0	0.0534190
$205$	0	0
$206$	−318864.	−0.523525
$207$	68040.0	0.110367
$208$	724768.	1.16156
$209$	313584.	0.496579
$210$	0	0
$211$	48500.0	0.0749956	0.0374978	−	0.999297i	$-0.488061\pi$
$211$	48500.0	0.0749956	0.0374978	+	0.999297i	$0.488061\pi$
$212$	−135000.	−0.206298
$213$	38232.0	0.0577402
$214$	−544968.	−0.813461
$215$	0	0
$216$	122472.	0.178609
$217$	176000.	0.253725
$218$	−369996.	−0.527298
$219$	−369990.	−0.521290
$220$	0	0
$221$	562716.	0.775012
$222$	−130140.	−0.177226
$223$	999472.	1.34589	0.672943	−	0.739694i	$-0.265030\pi$
$223$	999472.	1.34589	0.672943	+	0.739694i	$0.265030\pi$
$224$	−57600.0	−0.0767012
$225$	0	0
$226$	−62892.0	−0.0819076
$227$	−606180.	−0.780795	−0.390397	−	0.920646i	$-0.627662\pi$
$227$	−606180.	−0.780795	−0.390397	+	0.920646i	$0.627662\pi$
$228$	20016.0	0.0255000
$229$	1.35993e6	1.71367	0.856834	−	0.515593i	$-0.172428\pi$
$229$	1.35993e6	1.71367	0.856834	+	0.515593i	$0.172428\pi$
$230$	0	0
$231$	203040.	0.250352
$232$	−779184.	−0.950430
$233$	392886.	0.474107	0.237054	−	0.971497i	$-0.423818\pi$
$233$	392886.	0.474107	0.237054	+	0.971497i	$0.423818\pi$
$234$	−310068.	−0.370184
$235$	0	0
$236$	−72336.0	−0.0845424
$237$	−197280.	−0.228146
$238$	−211680.	−0.242235
$239$	−1.32514e6	−1.50060	−0.750301	−	0.661096i	$-0.770092\pi$
$239$	−1.32514e6	−1.50060	−0.750301	+	0.661096i	$0.770092\pi$
$240$	0	0
$241$	−990094.	−1.09808	−0.549040	−	0.835796i	$-0.685006\pi$
$241$	−990094.	−1.09808	−0.549040	+	0.835796i	$0.685006\pi$
$242$	942270.	1.03428
$243$	−59049.0	−0.0641500
$244$	159032.	0.171005
$245$	0	0
$246$	370980.	0.390852
$247$	354728.	0.369959
$248$	−739200.	−0.763190
$249$	742068.	0.758482
$250$	0	0
$251$	147132.	0.147409	0.0737043	−	0.997280i	$-0.476518\pi$
$251$	147132.	0.147409	0.0737043	+	0.997280i	$0.476518\pi$
$252$	12960.0	0.0128559
$253$	−473760.	−0.465326
$254$	1.02653e6	0.998358
$255$	0	0
$256$	387328.	0.369385
$257$	483582.	0.456707	0.228353	−	0.973578i	$-0.426666\pi$
$257$	483582.	0.456707	0.228353	+	0.973578i	$0.426666\pi$
$258$	520776.	0.487082
$259$	96400.0	0.0892951
$260$	0	0
$261$	375678.	0.341361
$262$	1.55081e6	1.39574
$263$	−813576.	−0.725285	−0.362643	−	0.931928i	$-0.618125\pi$
$263$	−813576.	−0.725285	−0.362643	+	0.931928i	$0.618125\pi$
$264$	−852768.	−0.753045
$265$	0	0
$266$	−133440.	−0.115633
$267$	846774.	0.726925
$268$	92272.0	0.0784753
$269$	−461106.	−0.388526	−0.194263	−	0.980949i	$-0.562232\pi$
$269$	−461106.	−0.388526	−0.194263	+	0.980949i	$0.562232\pi$
$270$	0	0
$271$	1.67514e6	1.38556	0.692782	−	0.721147i	$-0.256385\pi$
$271$	1.67514e6	1.38556	0.692782	+	0.721147i	$0.256385\pi$
$272$	1.00195e6	0.821154
$273$	229680.	0.186516
$274$	−1.80140e6	−1.44956
$275$	0	0
$276$	−30240.0	−0.0238951
$277$	−401126.	−0.314110	−0.157055	−	0.987590i	$-0.550200\pi$
$277$	−401126.	−0.314110	−0.157055	+	0.987590i	$0.550200\pi$
$278$	−2.10098e6	−1.63046
$279$	356400.	0.274111
$280$	0	0
$281$	−2.30977e6	−1.74503	−0.872514	−	0.488590i	$-0.837511\pi$
$281$	−2.30977e6	−1.74503	−0.872514	+	0.488590i	$0.837511\pi$
$282$	−1.00829e6	−0.755026
$283$	1.12877e6	0.837800	0.418900	−	0.908032i	$-0.362416\pi$
$283$	1.12877e6	0.837800	0.418900	+	0.908032i	$0.362416\pi$
$284$	−16992.0	−0.0125011
$285$	0	0
$286$	2.15899e6	1.56076
$287$	−274800.	−0.196930
$288$	−116640.	−0.0828641
$289$	−641933.	−0.452111
$290$	0	0
$291$	444978.	0.308039
$292$	164440.	0.112863
$293$	938874.	0.638908	0.319454	−	0.947602i	$-0.396501\pi$
$293$	938874.	0.638908	0.319454	+	0.947602i	$0.396501\pi$
$294$	821178.	0.554076
$295$	0	0
$296$	−404880.	−0.268594
$297$	411156.	0.270468
$298$	−631548.	−0.411970
$299$	−535920.	−0.346675
$300$	0	0
$301$	−385760.	−0.245415
$302$	2.37835e6	1.50058
$303$	1.28731e6	0.805518
$304$	631616.	0.391985
$305$	0	0
$306$	−428652.	−0.261699
$307$	−692948.	−0.419619	−0.209809	−	0.977742i	$-0.567284\pi$
$307$	−692948.	−0.419619	−0.209809	+	0.977742i	$0.567284\pi$
$308$	−90240.0	−0.0542029
$309$	478296.	0.284971
$310$	0	0
$311$	2.94310e6	1.72545	0.862727	−	0.505670i	$-0.168755\pi$
$311$	2.94310e6	1.72545	0.862727	+	0.505670i	$0.168755\pi$
$312$	−964656.	−0.561030
$313$	−885146.	−0.510686	−0.255343	−	0.966851i	$-0.582188\pi$
$313$	−885146.	−0.510686	−0.255343	+	0.966851i	$0.582188\pi$
$314$	826476.	0.473049
$315$	0	0
$316$	87680.0	0.0493950
$317$	−2.50880e6	−1.40222	−0.701112	−	0.713051i	$-0.747313\pi$
$317$	−2.50880e6	−1.40222	−0.701112	+	0.713051i	$0.747313\pi$
$318$	1.82250e6	1.01065
$319$	−2.61583e6	−1.43924
$320$	0	0
$321$	817452.	0.442792
$322$	201600.	0.108355
$323$	490392.	0.261539
$324$	26244.0	0.0138889
$325$	0	0
$326$	−2.11606e6	−1.10277
$327$	554994.	0.287024
$328$	1.15416e6	0.592354
$329$	746880.	0.380418
$330$	0	0
$331$	−216148.	−0.108438	−0.0542190	−	0.998529i	$-0.517267\pi$
$331$	−216148.	−0.108438	−0.0542190	+	0.998529i	$0.517267\pi$
$332$	−329808.	−0.164216
$333$	195210.	0.0964698
$334$	1.30536e6	0.640271
$335$	0	0
$336$	408960.	0.197621
$337$	−3.25263e6	−1.56012	−0.780062	−	0.625702i	$-0.784813\pi$
$337$	−3.25263e6	−1.56012	−0.780062	+	0.625702i	$0.784813\pi$
$338$	214506.	0.102129
$339$	94338.0	0.0445849
$340$	0	0
$341$	−2.48160e6	−1.15570
$342$	−270216.	−0.124924
$343$	−1.28056e6	−0.587712
$344$	1.62019e6	0.738194
$345$	0	0
$346$	982188.	0.441067
$347$	2.93207e6	1.30723	0.653613	−	0.756829i	$-0.273253\pi$
$347$	2.93207e6	1.30723	0.653613	+	0.756829i	$0.273253\pi$
$348$	−166968.	−0.0739069
$349$	905198.	0.397814	0.198907	−	0.980018i	$-0.436261\pi$
$349$	905198.	0.397814	0.198907	+	0.980018i	$0.436261\pi$
$350$	0	0
$351$	465102.	0.201502
$352$	812160.	0.349369
$353$	−1.91786e6	−0.819181	−0.409590	−	0.912270i	$-0.634328\pi$
$353$	−1.91786e6	−0.819181	−0.409590	+	0.912270i	$0.634328\pi$
$354$	976536.	0.414171
$355$	0	0
$356$	−376344.	−0.157384
$357$	317520.	0.131856
$358$	2.15244e6	0.887613
$359$	−2.43698e6	−0.997968	−0.498984	−	0.866611i	$-0.666293\pi$
$359$	−2.43698e6	−0.997968	−0.498984	+	0.866611i	$0.666293\pi$
$360$	0	0
$361$	−2.16696e6	−0.875152
$362$	−3.04278e6	−1.22039
$363$	−1.41340e6	−0.562989
$364$	−102080.	−0.0403819
$365$	0	0
$366$	−2.14693e6	−0.837752
$367$	984064.	0.381380	0.190690	−	0.981650i	$-0.438927\pi$
$367$	984064.	0.381380	0.190690	+	0.981650i	$0.438927\pi$
$368$	−954240.	−0.367314
$369$	−556470.	−0.212753
$370$	0	0
$371$	−1.35000e6	−0.509212
$372$	−158400.	−0.0593469
$373$	−1.70365e6	−0.634029	−0.317015	−	0.948421i	$-0.602680\pi$
$373$	−1.70365e6	−0.634029	−0.317015	+	0.948421i	$0.602680\pi$
$374$	2.98469e6	1.10337
$375$	0	0
$376$	−3.13690e6	−1.14428
$377$	−2.95904e6	−1.07225
$378$	−174960.	−0.0629810
$379$	2.75654e6	0.985749	0.492874	−	0.870100i	$-0.335946\pi$
$379$	2.75654e6	0.985749	0.492874	+	0.870100i	$0.335946\pi$
$380$	0	0
$381$	−1.53979e6	−0.543438
$382$	−3.89030e6	−1.36403
$383$	−456576.	−0.159044	−0.0795218	−	0.996833i	$-0.525339\pi$
$383$	−456576.	−0.159044	−0.0795218	+	0.996833i	$0.525339\pi$
$384$	−1.91117e6	−0.661410
$385$	0	0
$386$	167028.	0.0570586
$387$	−781164.	−0.265134
$388$	−197768.	−0.0666925
$389$	−2.00639e6	−0.672268	−0.336134	−	0.941814i	$-0.609119\pi$
$389$	−2.00639e6	−0.672268	−0.336134	+	0.941814i	$0.609119\pi$
$390$	0	0
$391$	−740880.	−0.245079
$392$	2.55478e6	0.839726
$393$	−2.32621e6	−0.759745
$394$	−3.66628e6	−1.18983
$395$	0	0
$396$	−182736.	−0.0585580
$397$	5.77040e6	1.83751	0.918755	−	0.394828i	$-0.129196\pi$
$397$	5.77040e6	1.83751	0.918755	+	0.394828i	$0.129196\pi$
$398$	5.27419e6	1.66897
$399$	200160.	0.0629427
$400$	0	0
$401$	3.00626e6	0.933610	0.466805	−	0.884360i	$-0.345405\pi$
$401$	3.00626e6	0.933610	0.466805	+	0.884360i	$0.345405\pi$
$402$	−1.24567e6	−0.384449
$403$	−2.80720e6	−0.861015
$404$	−572136.	−0.174400
$405$	0	0
$406$	1.11312e6	0.335140
$407$	−1.35924e6	−0.406734
$408$	−1.33358e6	−0.396616
$409$	1.53363e6	0.453327	0.226663	−	0.973973i	$-0.427218\pi$
$409$	1.53363e6	0.453327	0.226663	+	0.973973i	$0.427218\pi$
$410$	0	0
$411$	2.70211e6	0.789038
$412$	−212576.	−0.0616980
$413$	−723360.	−0.208679
$414$	408240.	0.117062
$415$	0	0
$416$	918720.	0.260285
$417$	3.15148e6	0.887511
$418$	1.88150e6	0.526701
$419$	−3.87376e6	−1.07795	−0.538973	−	0.842323i	$-0.681188\pi$
$419$	−3.87376e6	−1.07795	−0.538973	+	0.842323i	$0.681188\pi$
$420$	0	0
$421$	−1.33307e6	−0.366561	−0.183281	−	0.983061i	$-0.558672\pi$
$421$	−1.33307e6	−0.366561	−0.183281	+	0.983061i	$0.558672\pi$
$422$	291000.	0.0795448
$423$	1.51243e6	0.410984
$424$	5.67000e6	1.53168
$425$	0	0
$426$	229392.	0.0612427
$427$	1.59032e6	0.422099
$428$	−363312.	−0.0958673
$429$	−3.23849e6	−0.849570
$430$	0	0
$431$	6.45192e6	1.67300	0.836500	−	0.547967i	$-0.184598\pi$
$431$	6.45192e6	1.67300	0.836500	+	0.547967i	$0.184598\pi$
$432$	828144.	0.213499
$433$	4.16577e6	1.06777	0.533883	−	0.845558i	$-0.320732\pi$
$433$	4.16577e6	1.06777	0.533883	+	0.845558i	$0.320732\pi$
$434$	1.05600e6	0.269116
$435$	0	0
$436$	−246664.	−0.0621426
$437$	−467040.	−0.116990
$438$	−2.21994e6	−0.552912
$439$	792680.	0.196307	0.0981537	−	0.995171i	$-0.468706\pi$
$439$	792680.	0.196307	0.0981537	+	0.995171i	$0.468706\pi$
$440$	0	0
$441$	−1.23177e6	−0.301601
$442$	3.37630e6	0.822025
$443$	1.39981e6	0.338891	0.169446	−	0.985540i	$-0.445802\pi$
$443$	1.39981e6	0.338891	0.169446	+	0.985540i	$0.445802\pi$
$444$	−86760.0	−0.0208863
$445$	0	0
$446$	5.99683e6	1.42753
$447$	947322.	0.224248
$448$	1.10848e6	0.260935
$449$	2.99248e6	0.700512	0.350256	−	0.936654i	$-0.386095\pi$
$449$	2.99248e6	0.700512	0.350256	+	0.936654i	$0.386095\pi$
$450$	0	0
$451$	3.87468e6	0.897004
$452$	−41928.0	−0.00965291
$453$	−3.56753e6	−0.816812
$454$	−3.63708e6	−0.828158
$455$	0	0
$456$	−840672.	−0.189328
$457$	−6.29969e6	−1.41101	−0.705503	−	0.708707i	$-0.749279\pi$
$457$	−6.29969e6	−1.41101	−0.705503	+	0.708707i	$0.749279\pi$
$458$	8.15956e6	1.81762
$459$	642978.	0.142451
$460$	0	0
$461$	3.40318e6	0.745818	0.372909	−	0.927868i	$-0.378360\pi$
$461$	3.40318e6	0.745818	0.372909	+	0.927868i	$0.378360\pi$
$462$	1.21824e6	0.265539
$463$	2.23034e6	0.483524	0.241762	−	0.970336i	$-0.422275\pi$
$463$	2.23034e6	0.483524	0.241762	+	0.970336i	$0.422275\pi$
$464$	−5.26877e6	−1.13609
$465$	0	0
$466$	2.35732e6	0.502867
$467$	6.51409e6	1.38217	0.691085	−	0.722773i	$-0.257133\pi$
$467$	6.51409e6	1.38217	0.691085	+	0.722773i	$0.257133\pi$
$468$	−206712.	−0.0436266
$469$	922720.	0.193704
$470$	0	0
$471$	−1.23971e6	−0.257495
$472$	3.03811e6	0.627695
$473$	5.43922e6	1.11785
$474$	−1.18368e6	−0.241985
$475$	0	0
$476$	−141120.	−0.0285477
$477$	−2.73375e6	−0.550127
$478$	−7.95082e6	−1.59163
$479$	2.39232e6	0.476410	0.238205	−	0.971215i	$-0.423441\pi$
$479$	2.39232e6	0.476410	0.238205	+	0.971215i	$0.423441\pi$
$480$	0	0
$481$	−1.53758e6	−0.303023
$482$	−5.94056e6	−1.16469
$483$	−302400.	−0.0589812
$484$	628180.	0.121891
$485$	0	0
$486$	−354294.	−0.0680414
$487$	6.13089e6	1.17139	0.585694	−	0.810532i	$-0.300822\pi$
$487$	6.13089e6	1.17139	0.585694	+	0.810532i	$0.300822\pi$
$488$	−6.67934e6	−1.26965
$489$	3.17408e6	0.600269
$490$	0	0
$491$	−1.23589e6	−0.231354	−0.115677	−	0.993287i	$-0.536904\pi$
$491$	−1.23589e6	−0.231354	−0.115677	+	0.993287i	$0.536904\pi$
$492$	247320.	0.0460624
$493$	−4.09072e6	−0.758022
$494$	2.12837e6	0.392400
$495$	0	0
$496$	−4.99840e6	−0.912277
$497$	−169920.	−0.0308570
$498$	4.45241e6	0.804492
$499$	−9.85496e6	−1.77175	−0.885877	−	0.463921i	$-0.846442\pi$
$499$	−9.85496e6	−1.77175	−0.885877	+	0.463921i	$0.846442\pi$
$500$	0	0
$501$	−1.95804e6	−0.348520
$502$	882792.	0.156350
$503$	−1.16777e6	−0.205796	−0.102898	−	0.994692i	$-0.532812\pi$
$503$	−1.16777e6	−0.205796	−0.102898	+	0.994692i	$0.532812\pi$
$504$	−544320.	−0.0954504
$505$	0	0
$506$	−2.84256e6	−0.493552
$507$	−321759.	−0.0555918
$508$	684352.	0.117658
$509$	1.04941e6	0.179535	0.0897675	−	0.995963i	$-0.471388\pi$
$509$	1.04941e6	0.179535	0.0897675	+	0.995963i	$0.471388\pi$
$510$	0	0
$511$	1.64440e6	0.278583
$512$	−4.47130e6	−0.753804
$513$	405324.	0.0680000
$514$	2.90149e6	0.484411
$515$	0	0
$516$	347184.	0.0574031
$517$	−1.05310e7	−1.73278
$518$	578400.	0.0947118
$519$	−1.47328e6	−0.240086
$520$	0	0
$521$	−9.61407e6	−1.55172	−0.775859	−	0.630906i	$-0.782683\pi$
$521$	−9.61407e6	−1.55172	−0.775859	+	0.630906i	$0.782683\pi$
$522$	2.25407e6	0.362069
$523$	−6.96148e6	−1.11288	−0.556439	−	0.830888i	$-0.687833\pi$
$523$	−6.96148e6	−1.11288	−0.556439	+	0.830888i	$0.687833\pi$
$524$	1.03387e6	0.164490
$525$	0	0
$526$	−4.88146e6	−0.769281
$527$	−3.88080e6	−0.608688
$528$	−5.76634e6	−0.900151
$529$	−5.73074e6	−0.890373
$530$	0	0
$531$	−1.46480e6	−0.225446
$532$	−88960.0	−0.0136275
$533$	4.38306e6	0.668281
$534$	5.08064e6	0.771020
$535$	0	0
$536$	−3.87542e6	−0.582649
$537$	−3.22866e6	−0.483155
$538$	−2.76664e6	−0.412094
$539$	8.57675e6	1.27160
$540$	0	0
$541$	−712690.	−0.104691	−0.0523453	−	0.998629i	$-0.516670\pi$
$541$	−712690.	−0.104691	−0.0523453	+	0.998629i	$0.516670\pi$
$542$	1.00508e7	1.46961
$543$	4.56417e6	0.664297
$544$	1.27008e6	0.184007
$545$	0	0
$546$	1.37808e6	0.197830
$547$	3.62614e6	0.518175	0.259087	−	0.965854i	$-0.416578\pi$
$547$	3.62614e6	0.518175	0.259087	+	0.965854i	$0.416578\pi$
$548$	−1.20094e6	−0.170832
$549$	3.22040e6	0.456015
$550$	0	0
$551$	−2.57873e6	−0.361848
$552$	1.27008e6	0.177412
$553$	876800.	0.121924
$554$	−2.40676e6	−0.333164
$555$	0	0
$556$	−1.40066e6	−0.192152
$557$	−4.84846e6	−0.662165	−0.331082	−	0.943602i	$-0.607414\pi$
$557$	−4.84846e6	−0.662165	−0.331082	+	0.943602i	$0.607414\pi$
$558$	2.13840e6	0.290739
$559$	6.15287e6	0.832815
$560$	0	0
$561$	−4.47703e6	−0.600597
$562$	−1.38586e7	−1.85088
$563$	−8.50405e6	−1.13072	−0.565360	−	0.824844i	$-0.691263\pi$
$563$	−8.50405e6	−1.13072	−0.565360	+	0.824844i	$0.691263\pi$
$564$	−672192.	−0.0889807
$565$	0	0
$566$	6.77263e6	0.888621
$567$	262440.	0.0342825
$568$	713664.	0.0928160
$569$	362874.	0.0469867	0.0234934	−	0.999724i	$-0.492521\pi$
$569$	362874.	0.0469867	0.0234934	+	0.999724i	$0.492521\pi$
$570$	0	0
$571$	4.11024e6	0.527566	0.263783	−	0.964582i	$-0.415030\pi$
$571$	4.11024e6	0.527566	0.263783	+	0.964582i	$0.415030\pi$
$572$	1.43933e6	0.183937
$573$	5.83546e6	0.742486
$574$	−1.64880e6	−0.208876
$575$	0	0
$576$	2.24467e6	0.281901
$577$	7.87680e6	0.984941	0.492470	−	0.870329i	$-0.336094\pi$
$577$	7.87680e6	0.984941	0.492470	+	0.870329i	$0.336094\pi$
$578$	−3.85160e6	−0.479536
$579$	−250542.	−0.0310588
$580$	0	0
$581$	−3.29808e6	−0.405341
$582$	2.66987e6	0.326725
$583$	1.90350e7	2.31943
$584$	−6.90648e6	−0.837963
$585$	0	0
$586$	5.63324e6	0.677664
$587$	−603948.	−0.0723443	−0.0361721	−	0.999346i	$-0.511516\pi$
$587$	−603948.	−0.0723443	−0.0361721	+	0.999346i	$0.511516\pi$
$588$	547452.	0.0652984
$589$	−2.44640e6	−0.290562
$590$	0	0
$591$	5.49941e6	0.647661
$592$	−2.73776e6	−0.321064
$593$	5.39077e6	0.629526	0.314763	−	0.949170i	$-0.398075\pi$
$593$	5.39077e6	0.629526	0.314763	+	0.949170i	$0.398075\pi$
$594$	2.46694e6	0.286874
$595$	0	0
$596$	−421032.	−0.0485511
$597$	−7.91129e6	−0.908472
$598$	−3.21552e6	−0.367704
$599$	4.27999e6	0.487389	0.243695	−	0.969852i	$-0.421641\pi$
$599$	4.27999e6	0.487389	0.243695	+	0.969852i	$0.421641\pi$
$600$	0	0
$601$	1.02483e6	0.115735	0.0578674	−	0.998324i	$-0.481570\pi$
$601$	1.02483e6	0.115735	0.0578674	+	0.998324i	$0.481570\pi$
$602$	−2.31456e6	−0.260302
$603$	1.86851e6	0.209267
$604$	1.58557e6	0.176845
$605$	0	0
$606$	7.72384e6	0.854381
$607$	−1.24342e7	−1.36976	−0.684882	−	0.728654i	$-0.740146\pi$
$607$	−1.24342e7	−1.36976	−0.684882	+	0.728654i	$0.740146\pi$
$608$	800640.	0.0878372
$609$	−1.66968e6	−0.182427
$610$	0	0
$611$	−1.19127e7	−1.29095
$612$	−285768.	−0.0308415
$613$	−4.21506e6	−0.453057	−0.226528	−	0.974005i	$-0.572738\pi$
$613$	−4.21506e6	−0.453057	−0.226528	+	0.974005i	$0.572738\pi$
$614$	−4.15769e6	−0.445073
$615$	0	0
$616$	3.79008e6	0.402436
$617$	4.40665e6	0.466010	0.233005	−	0.972476i	$-0.425144\pi$
$617$	4.40665e6	0.466010	0.233005	+	0.972476i	$0.425144\pi$
$618$	2.86978e6	0.302257
$619$	4.80168e6	0.503693	0.251847	−	0.967767i	$-0.418962\pi$
$619$	4.80168e6	0.503693	0.251847	+	0.967767i	$0.418962\pi$
$620$	0	0
$621$	−612360.	−0.0637203
$622$	1.76586e7	1.83012
$623$	−3.76344e6	−0.388477
$624$	−6.52291e6	−0.670625
$625$	0	0
$626$	−5.31088e6	−0.541664
$627$	−2.82226e6	−0.286700
$628$	550984.	0.0557494
$629$	−2.12562e6	−0.214220
$630$	0	0
$631$	8.30727e6	0.830587	0.415293	−	0.909688i	$-0.363679\pi$
$631$	8.30727e6	0.830587	0.415293	+	0.909688i	$0.363679\pi$
$632$	−3.68256e6	−0.366739
$633$	−436500.	−0.0432987
$634$	−1.50528e7	−1.48728
$635$	0	0
$636$	1.21500e6	0.119106
$637$	9.70207e6	0.947361
$638$	−1.56950e7	−1.52654
$639$	−344088.	−0.0333363
$640$	0	0
$641$	1.76956e7	1.70107	0.850534	−	0.525921i	$-0.176279\pi$
$641$	1.76956e7	1.70107	0.850534	+	0.525921i	$0.176279\pi$
$642$	4.90471e6	0.469652
$643$	1.28394e7	1.22466	0.612330	−	0.790602i	$-0.290232\pi$
$643$	1.28394e7	1.22466	0.612330	+	0.790602i	$0.290232\pi$
$644$	134400.	0.0127698
$645$	0	0
$646$	2.94235e6	0.277404
$647$	2.08468e7	1.95785	0.978924	−	0.204226i	$-0.0654678\pi$
$647$	2.08468e7	1.95785	0.978924	+	0.204226i	$0.0654678\pi$
$648$	−1.10225e6	−0.103120
$649$	1.01994e7	0.950521
$650$	0	0
$651$	−1.58400e6	−0.146488
$652$	−1.41070e6	−0.129962
$653$	−1.29632e7	−1.18968	−0.594841	−	0.803843i	$-0.702785\pi$
$653$	−1.29632e7	−1.18968	−0.594841	+	0.803843i	$0.702785\pi$
$654$	3.32996e6	0.304435
$655$	0	0
$656$	7.80432e6	0.708069
$657$	3.32991e6	0.300967
$658$	4.48128e6	0.403494
$659$	−5.66862e6	−0.508468	−0.254234	−	0.967143i	$-0.581823\pi$
$659$	−5.66862e6	−0.508468	−0.254234	+	0.967143i	$0.581823\pi$
$660$	0	0
$661$	−3.11430e6	−0.277240	−0.138620	−	0.990346i	$-0.544267\pi$
$661$	−3.11430e6	−0.277240	−0.138620	+	0.990346i	$0.544267\pi$
$662$	−1.29689e6	−0.115016
$663$	−5.06444e6	−0.447454
$664$	1.38519e7	1.21924
$665$	0	0
$666$	1.17126e6	0.102322
$667$	3.89592e6	0.339075
$668$	870240.	0.0754567
$669$	−8.99525e6	−0.777048
$670$	0	0
$671$	−2.24235e7	−1.92264
$672$	518400.	0.0442835
$673$	−105890.	−0.00901192	−0.00450596	−	0.999990i	$-0.501434\pi$
$673$	−105890.	−0.00901192	−0.00450596	+	0.999990i	$0.501434\pi$
$674$	−1.95158e7	−1.65476
$675$	0	0
$676$	143004.	0.0120360
$677$	1.60910e7	1.34931	0.674656	−	0.738132i	$-0.264292\pi$
$677$	1.60910e7	1.34931	0.674656	+	0.738132i	$0.264292\pi$
$678$	566028.	0.0472894
$679$	−1.97768e6	−0.164620
$680$	0	0
$681$	5.45562e6	0.450792
$682$	−1.48896e7	−1.22581
$683$	−1.60780e7	−1.31880	−0.659402	−	0.751791i	$-0.729190\pi$
$683$	−1.60780e7	−1.31880	−0.659402	+	0.751791i	$0.729190\pi$
$684$	−180144.	−0.0147224
$685$	0	0
$686$	−7.68336e6	−0.623363
$687$	−1.22393e7	−0.989386
$688$	1.09556e7	0.882398
$689$	2.15325e7	1.72801
$690$	0	0
$691$	−165964.	−0.0132227	−0.00661133	−	0.999978i	$-0.502104\pi$
$691$	−165964.	−0.0132227	−0.00661133	+	0.999978i	$0.502104\pi$
$692$	654792.	0.0519802
$693$	−1.82736e6	−0.144541
$694$	1.75924e7	1.38652
$695$	0	0
$696$	7.01266e6	0.548731
$697$	6.05934e6	0.472436
$698$	5.43119e6	0.421945
$699$	−3.53597e6	−0.273726
$700$	0	0
$701$	1.77248e7	1.36234	0.681171	−	0.732124i	$-0.261471\pi$
$701$	1.77248e7	1.36234	0.681171	+	0.732124i	$0.261471\pi$
$702$	2.79061e6	0.213726
$703$	−1.33996e6	−0.102259
$704$	−1.56296e7	−1.18854
$705$	0	0
$706$	−1.15071e7	−0.868872
$707$	−5.72136e6	−0.430478
$708$	651024.	0.0488106
$709$	−1.06023e7	−0.792112	−0.396056	−	0.918226i	$-0.629621\pi$
$709$	−1.06023e7	−0.792112	−0.396056	+	0.918226i	$0.629621\pi$
$710$	0	0
$711$	1.77552e6	0.131720
$712$	1.58064e7	1.16852
$713$	3.69600e6	0.272275
$714$	1.90512e6	0.139855
$715$	0	0
$716$	1.43496e6	0.104606
$717$	1.19262e7	0.866373
$718$	−1.46219e7	−1.05850
$719$	9.03211e6	0.651579	0.325790	−	0.945442i	$-0.394370\pi$
$719$	9.03211e6	0.651579	0.325790	+	0.945442i	$0.394370\pi$
$720$	0	0
$721$	−2.12576e6	−0.152292
$722$	−1.30018e7	−0.928239
$723$	8.91085e6	0.633977
$724$	−2.02852e6	−0.143825
$725$	0	0
$726$	−8.48043e6	−0.597140
$727$	−1.87575e7	−1.31625	−0.658127	−	0.752907i	$-0.728651\pi$
$727$	−1.87575e7	−1.31625	−0.658127	+	0.752907i	$0.728651\pi$
$728$	4.28736e6	0.299821
$729$	531441.	0.0370370
$730$	0	0
$731$	8.50601e6	0.588752
$732$	−1.43129e6	−0.0987300
$733$	1.17773e7	0.809626	0.404813	−	0.914399i	$-0.367337\pi$
$733$	1.17773e7	0.809626	0.404813	+	0.914399i	$0.367337\pi$
$734$	5.90438e6	0.404515
$735$	0	0
$736$	−1.20960e6	−0.0823090
$737$	−1.30104e7	−0.882308
$738$	−3.33882e6	−0.225659
$739$	5.88948e6	0.396703	0.198352	−	0.980131i	$-0.436441\pi$
$739$	5.88948e6	0.396703	0.198352	+	0.980131i	$0.436441\pi$
$740$	0	0
$741$	−3.19255e6	−0.213596
$742$	−8.10000e6	−0.540101
$743$	1.00476e7	0.667712	0.333856	−	0.942624i	$-0.391650\pi$
$743$	1.00476e7	0.667712	0.333856	+	0.942624i	$0.391650\pi$
$744$	6.65280e6	0.440628
$745$	0	0
$746$	−1.02219e7	−0.672490
$747$	−6.67861e6	−0.437910
$748$	1.98979e6	0.130033
$749$	−3.63312e6	−0.236633
$750$	0	0
$751$	4.81530e6	0.311547	0.155773	−	0.987793i	$-0.450213\pi$
$751$	4.81530e6	0.311547	0.155773	+	0.987793i	$0.450213\pi$
$752$	−2.12114e7	−1.36781
$753$	−1.32419e6	−0.0851064
$754$	−1.77543e7	−1.13730
$755$	0	0
$756$	−116640.	−0.00742238
$757$	−3.12973e6	−0.198503	−0.0992516	−	0.995062i	$-0.531645\pi$
$757$	−3.12973e6	−0.198503	−0.0992516	+	0.995062i	$0.531645\pi$
$758$	1.65392e7	1.04554
$759$	4.26384e6	0.268656
$760$	0	0
$761$	−1.17773e7	−0.737197	−0.368599	−	0.929589i	$-0.620162\pi$
$761$	−1.17773e7	−0.737197	−0.368599	+	0.929589i	$0.620162\pi$
$762$	−9.23875e6	−0.576403
$763$	−2.46664e6	−0.153389
$764$	−2.59354e6	−0.160753
$765$	0	0
$766$	−2.73946e6	−0.168691
$767$	1.15376e7	0.708152
$768$	−3.48595e6	−0.213264
$769$	−1.49376e6	−0.0910887	−0.0455443	−	0.998962i	$-0.514502\pi$
$769$	−1.49376e6	−0.0910887	−0.0455443	+	0.998962i	$0.514502\pi$
$770$	0	0
$771$	−4.35224e6	−0.263680
$772$	111352.	0.00672442
$773$	2.25125e7	1.35511	0.677555	−	0.735472i	$-0.263040\pi$
$773$	2.25125e7	1.35511	0.677555	+	0.735472i	$0.263040\pi$
$774$	−4.68698e6	−0.281217
$775$	0	0
$776$	8.30626e6	0.495166
$777$	−867600.	−0.0515545
$778$	−1.20384e7	−0.713048
$779$	3.81972e6	0.225521
$780$	0	0
$781$	2.39587e6	0.140552
$782$	−4.44528e6	−0.259945
$783$	−3.38110e6	−0.197085
$784$	1.72752e7	1.00376
$785$	0	0
$786$	−1.39573e7	−0.805831
$787$	−1.19547e7	−0.688022	−0.344011	−	0.938966i	$-0.611786\pi$
$787$	−1.19547e7	−0.688022	−0.344011	+	0.938966i	$0.611786\pi$
$788$	−2.44418e6	−0.140223
$789$	7.32218e6	0.418744
$790$	0	0
$791$	−419280.	−0.0238266
$792$	7.67491e6	0.434771
$793$	−2.53656e7	−1.43239
$794$	3.46224e7	1.94897
$795$	0	0
$796$	3.51613e6	0.196690
$797$	−540798.	−0.0301571	−0.0150785	−	0.999886i	$-0.504800\pi$
$797$	−540798.	−0.0301571	−0.0150785	+	0.999886i	$0.504800\pi$
$798$	1.20096e6	0.0667608
$799$	−1.64687e7	−0.912625
$800$	0	0
$801$	−7.62097e6	−0.419690
$802$	1.80375e7	0.990243
$803$	−2.31860e7	−1.26893
$804$	−830448.	−0.0453077
$805$	0	0
$806$	−1.68432e7	−0.913244
$807$	4.14995e6	0.224316
$808$	2.40297e7	1.29485
$809$	−6.14223e6	−0.329955	−0.164978	−	0.986297i	$-0.552755\pi$
$809$	−6.14223e6	−0.329955	−0.164978	+	0.986297i	$0.552755\pi$
$810$	0	0
$811$	−3.16734e7	−1.69100	−0.845499	−	0.533977i	$-0.820697\pi$
$811$	−3.16734e7	−1.69100	−0.845499	+	0.533977i	$0.820697\pi$
$812$	742080.	0.0394967
$813$	−1.50762e7	−0.799956
$814$	−8.15544e6	−0.431406
$815$	0	0
$816$	−9.01757e6	−0.474094
$817$	5.36206e6	0.281046
$818$	9.20176e6	0.480825
$819$	−2.06712e6	−0.107685
$820$	0	0
$821$	2.66175e7	1.37819	0.689095	−	0.724671i	$-0.258008\pi$
$821$	2.66175e7	1.37819	0.689095	+	0.724671i	$0.258008\pi$
$822$	1.62126e7	0.836901
$823$	−3.62817e7	−1.86719	−0.933593	−	0.358335i	$-0.883345\pi$
$823$	−3.62817e7	−1.86719	−0.933593	+	0.358335i	$0.883345\pi$
$824$	8.92819e6	0.458084
$825$	0	0
$826$	−4.34016e6	−0.221338
$827$	−1.09033e6	−0.0554364	−0.0277182	−	0.999616i	$-0.508824\pi$
$827$	−1.09033e6	−0.0554364	−0.0277182	+	0.999616i	$0.508824\pi$
$828$	272160.	0.0137958
$829$	−1.03016e7	−0.520620	−0.260310	−	0.965525i	$-0.583825\pi$
$829$	−1.03016e7	−0.520620	−0.260310	+	0.965525i	$0.583825\pi$
$830$	0	0
$831$	3.61013e6	0.181351
$832$	−1.76803e7	−0.885483
$833$	1.34126e7	0.669730
$834$	1.89089e7	0.941348
$835$	0	0
$836$	1.25434e6	0.0620724
$837$	−3.20760e6	−0.158258
$838$	−2.32425e7	−1.14333
$839$	−1.96134e7	−0.961940	−0.480970	−	0.876737i	$-0.659715\pi$
$839$	−1.96134e7	−0.961940	−0.480970	+	0.876737i	$0.659715\pi$
$840$	0	0
$841$	999895.	0.0487489
$842$	−7.99840e6	−0.388797
$843$	2.07879e7	1.00749
$844$	194000.	0.00937445
$845$	0	0
$846$	9.07459e6	0.435914
$847$	6.28180e6	0.300868
$848$	3.83400e7	1.83089
$849$	−1.01589e7	−0.483704
$850$	0	0
$851$	2.02440e6	0.0958236
$852$	152928.	0.00721752
$853$	−3.27565e7	−1.54143	−0.770717	−	0.637178i	$-0.780102\pi$
$853$	−3.27565e7	−1.54143	−0.770717	+	0.637178i	$0.780102\pi$
$854$	9.54192e6	0.447704
$855$	0	0
$856$	1.52591e7	0.711778
$857$	2.57953e7	1.19974	0.599872	−	0.800096i	$-0.295218\pi$
$857$	2.57953e7	1.19974	0.599872	+	0.800096i	$0.295218\pi$
$858$	−1.94309e7	−0.901105
$859$	−1.98548e7	−0.918085	−0.459043	−	0.888414i	$-0.651807\pi$
$859$	−1.98548e7	−0.918085	−0.459043	+	0.888414i	$0.651807\pi$
$860$	0	0
$861$	2.47320e6	0.113698
$862$	3.87115e7	1.77448
$863$	673056.	0.0307627	0.0153813	−	0.999882i	$-0.495104\pi$
$863$	673056.	0.0307627	0.0153813	+	0.999882i	$0.495104\pi$
$864$	1.04976e6	0.0478416
$865$	0	0
$866$	2.49946e7	1.13254
$867$	5.77740e6	0.261026
$868$	704000.	0.0317156
$869$	−1.23629e7	−0.555354
$870$	0	0
$871$	−1.47174e7	−0.657333
$872$	1.03599e7	0.461385
$873$	−4.00480e6	−0.177847
$874$	−2.80224e6	−0.124087
$875$	0	0
$876$	−1.47996e6	−0.0651613
$877$	−5.32115e6	−0.233618	−0.116809	−	0.993154i	$-0.537267\pi$
$877$	−5.32115e6	−0.233618	−0.116809	+	0.993154i	$0.537267\pi$
$878$	4.75608e6	0.208215
$879$	−8.44987e6	−0.368874
$880$	0	0
$881$	2.78891e7	1.21058	0.605291	−	0.796004i	$-0.293057\pi$
$881$	2.78891e7	1.21058	0.605291	+	0.796004i	$0.293057\pi$
$882$	−7.39060e6	−0.319896
$883$	2.83786e7	1.22487	0.612435	−	0.790521i	$-0.290190\pi$
$883$	2.83786e7	1.22487	0.612435	+	0.790521i	$0.290190\pi$
$884$	2.25086e6	0.0968765
$885$	0	0
$886$	8.39887e6	0.359448
$887$	−4.22678e7	−1.80385	−0.901925	−	0.431893i	$-0.857846\pi$
$887$	−4.22678e7	−1.80385	−0.901925	+	0.431893i	$0.857846\pi$
$888$	3.64392e6	0.155073
$889$	6.84352e6	0.290419
$890$	0	0
$891$	−3.70040e6	−0.156155
$892$	3.99789e6	0.168236
$893$	−1.03816e7	−0.435649
$894$	5.68393e6	0.237851
$895$	0	0
$896$	8.49408e6	0.353465
$897$	4.82328e6	0.200153
$898$	1.79549e7	0.743005
$899$	2.04072e7	0.842140
$900$	0	0
$901$	2.97675e7	1.22160
$902$	2.32481e7	0.951417
$903$	3.47184e6	0.141690
$904$	1.76098e6	0.0716692
$905$	0	0
$906$	−2.14052e7	−0.866360
$907$	−3.19526e7	−1.28970	−0.644849	−	0.764310i	$-0.723080\pi$
$907$	−3.19526e7	−1.28970	−0.644849	+	0.764310i	$0.723080\pi$
$908$	−2.42472e6	−0.0975994
$909$	−1.15858e7	−0.465066
$910$	0	0
$911$	−1.16429e7	−0.464800	−0.232400	−	0.972620i	$-0.574658\pi$
$911$	−1.16429e7	−0.464800	−0.232400	+	0.972620i	$0.574658\pi$
$912$	−5.68454e6	−0.226312
$913$	4.65029e7	1.84630
$914$	−3.77981e7	−1.49660
$915$	0	0
$916$	5.43970e6	0.214208
$917$	1.03387e7	0.406016
$918$	3.85787e6	0.151092
$919$	1.39844e6	0.0546204	0.0273102	−	0.999627i	$-0.491306\pi$
$919$	1.39844e6	0.0546204	0.0273102	+	0.999627i	$0.491306\pi$
$920$	0	0
$921$	6.23653e6	0.242267
$922$	2.04191e7	0.791059
$923$	2.71022e6	0.104713
$924$	812160.	0.0312940
$925$	0	0
$926$	1.33820e7	0.512854
$927$	−4.30466e6	−0.164528
$928$	−6.67872e6	−0.254579
$929$	−1.66792e7	−0.634067	−0.317033	−	0.948414i	$-0.602687\pi$
$929$	−1.66792e7	−0.634067	−0.317033	+	0.948414i	$0.602687\pi$
$930$	0	0
$931$	8.45509e6	0.319701
$932$	1.57154e6	0.0592634
$933$	−2.64879e7	−0.996191
$934$	3.90846e7	1.46601
$935$	0	0
$936$	8.68190e6	0.323911
$937$	2.47956e7	0.922625	0.461312	−	0.887238i	$-0.347379\pi$
$937$	2.47956e7	0.922625	0.461312	+	0.887238i	$0.347379\pi$
$938$	5.53632e6	0.205454
$939$	7.96631e6	0.294845
$940$	0	0
$941$	2.79574e7	1.02925	0.514627	−	0.857414i	$-0.327930\pi$
$941$	2.79574e7	1.02925	0.514627	+	0.857414i	$0.327930\pi$
$942$	−7.43828e6	−0.273115
$943$	−5.77080e6	−0.211328
$944$	2.05434e7	0.750314
$945$	0	0
$946$	3.26353e7	1.18566
$947$	−7.64936e6	−0.277173	−0.138586	−	0.990350i	$-0.544256\pi$
$947$	−7.64936e6	−0.277173	−0.138586	+	0.990350i	$0.544256\pi$
$948$	−789120.	−0.0285182
$949$	−2.62282e7	−0.945372
$950$	0	0
$951$	2.25792e7	0.809575
$952$	5.92704e6	0.211956
$953$	4.62179e7	1.64846	0.824228	−	0.566257i	$-0.191609\pi$
$953$	4.62179e7	1.64846	0.824228	+	0.566257i	$0.191609\pi$
$954$	−1.64025e7	−0.583498
$955$	0	0
$956$	−5.30054e6	−0.187575
$957$	2.35425e7	0.830945
$958$	1.43539e7	0.505309
$959$	−1.20094e7	−0.421671
$960$	0	0
$961$	−9.26915e6	−0.323766
$962$	−9.22548e6	−0.321404
$963$	−7.35707e6	−0.255646
$964$	−3.96038e6	−0.137260
$965$	0	0
$966$	−1.81440e6	−0.0625591
$967$	−2.08557e7	−0.717229	−0.358615	−	0.933486i	$-0.616751\pi$
$967$	−2.08557e7	−0.717229	−0.358615	+	0.933486i	$0.616751\pi$
$968$	−2.63836e7	−0.904993
$969$	−4.41353e6	−0.151000
$970$	0	0
$971$	−4.58152e7	−1.55941	−0.779707	−	0.626144i	$-0.784632\pi$
$971$	−4.58152e7	−1.55941	−0.779707	+	0.626144i	$0.784632\pi$
$972$	−236196.	−0.00801875
$973$	−1.40066e7	−0.474296
$974$	3.67853e7	1.24245
$975$	0	0
$976$	−4.51651e7	−1.51767
$977$	1.09544e6	0.0367157	0.0183578	−	0.999831i	$-0.494156\pi$
$977$	1.09544e6	0.0367157	0.0183578	+	0.999831i	$0.494156\pi$
$978$	1.90445e7	0.636682
$979$	5.30645e7	1.76949
$980$	0	0
$981$	−4.99495e6	−0.165714
$982$	−7.41535e6	−0.245388
$983$	−5.25817e7	−1.73561	−0.867803	−	0.496909i	$-0.834468\pi$
$983$	−5.25817e7	−1.73561	−0.867803	+	0.496909i	$0.834468\pi$
$984$	−1.03874e7	−0.341996
$985$	0	0
$986$	−2.45443e7	−0.804004
$987$	−6.72192e6	−0.219634
$988$	1.41891e6	0.0462448
$989$	−8.10096e6	−0.263358
$990$	0	0
$991$	−4.90389e7	−1.58620	−0.793098	−	0.609094i	$-0.791533\pi$
$991$	−4.90389e7	−1.58620	−0.793098	+	0.609094i	$0.791533\pi$
$992$	−6.33600e6	−0.204426
$993$	1.94533e6	0.0626067
$994$	−1.01952e6	−0.0327288
$995$	0	0
$996$	2.96827e6	0.0948103
$997$	−3.05461e6	−0.0973237	−0.0486618	−	0.998815i	$-0.515496\pi$
$997$	−3.05461e6	−0.0973237	−0.0486618	+	0.998815i	$0.515496\pi$
$998$	−5.91297e7	−1.87923
$999$	−1.75689e6	−0.0556969

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.6.a.e.1.1		1
3.2	odd	2		225.6.a.a.1.1		1
5.2	odd	4		75.6.b.b.49.2		2
5.3	odd	4		75.6.b.b.49.1		2
5.4	even	2		3.6.a.a.1.1	✓	1
15.2	even	4		225.6.b.b.199.1		2
15.8	even	4		225.6.b.b.199.2		2
15.14	odd	2		9.6.a.a.1.1		1
20.19	odd	2		48.6.a.a.1.1		1
35.4	even	6		147.6.e.h.79.1		2
35.9	even	6		147.6.e.h.67.1		2
35.19	odd	6		147.6.e.k.67.1		2
35.24	odd	6		147.6.e.k.79.1		2
35.34	odd	2		147.6.a.a.1.1		1
40.19	odd	2		192.6.a.l.1.1		1
40.29	even	2		192.6.a.d.1.1		1
45.4	even	6		81.6.c.c.55.1		2
45.14	odd	6		81.6.c.a.55.1		2
45.29	odd	6		81.6.c.a.28.1		2
45.34	even	6		81.6.c.c.28.1		2
55.54	odd	2		363.6.a.d.1.1		1
60.59	even	2		144.6.a.f.1.1		1
65.64	even	2		507.6.a.b.1.1		1
80.19	odd	4		768.6.d.h.385.1		2
80.29	even	4		768.6.d.k.385.2		2
80.59	odd	4		768.6.d.h.385.2		2
80.69	even	4		768.6.d.k.385.1		2
85.84	even	2		867.6.a.a.1.1		1
95.94	odd	2		1083.6.a.c.1.1		1
105.104	even	2		441.6.a.i.1.1		1
120.29	odd	2		576.6.a.s.1.1		1
120.59	even	2		576.6.a.t.1.1		1
165.164	even	2		1089.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.6.a.a.1.1	✓	1	5.4	even	2
9.6.a.a.1.1		1	15.14	odd	2
48.6.a.a.1.1		1	20.19	odd	2
75.6.a.e.1.1		1	1.1	even	1	trivial
75.6.b.b.49.1		2	5.3	odd	4
75.6.b.b.49.2		2	5.2	odd	4
81.6.c.a.28.1		2	45.29	odd	6
81.6.c.a.55.1		2	45.14	odd	6
81.6.c.c.28.1		2	45.34	even	6
81.6.c.c.55.1		2	45.4	even	6
144.6.a.f.1.1		1	60.59	even	2
147.6.a.a.1.1		1	35.34	odd	2
147.6.e.h.67.1		2	35.9	even	6
147.6.e.h.79.1		2	35.4	even	6
147.6.e.k.67.1		2	35.19	odd	6
147.6.e.k.79.1		2	35.24	odd	6
192.6.a.d.1.1		1	40.29	even	2
192.6.a.l.1.1		1	40.19	odd	2
225.6.a.a.1.1		1	3.2	odd	2
225.6.b.b.199.1		2	15.2	even	4
225.6.b.b.199.2		2	15.8	even	4
363.6.a.d.1.1		1	55.54	odd	2
441.6.a.i.1.1		1	105.104	even	2
507.6.a.b.1.1		1	65.64	even	2
576.6.a.s.1.1		1	120.29	odd	2
576.6.a.t.1.1		1	120.59	even	2
768.6.d.h.385.1		2	80.19	odd	4
768.6.d.h.385.2		2	80.59	odd	4
768.6.d.k.385.1		2	80.69	even	4
768.6.d.k.385.2		2	80.29	even	4
867.6.a.a.1.1		1	85.84	even	2
1083.6.a.c.1.1		1	95.94	odd	2
1089.6.a.b.1.1		1	165.164	even	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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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