LMFDB - Embedded newform 150.8.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,8,Mod(1,150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("150.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$N$	$=$	$150 = 2 \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	150.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:	$46.8577538226$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 30)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	150.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +216.000 q^{6} +1126.00 q^{7} -512.000 q^{8} +729.000 q^{9} -5518.00 q^{11} -1728.00 q^{12} -12798.0 q^{13} -9008.00 q^{14} +4096.00 q^{16} +32206.0 q^{17} -5832.00 q^{18} -4440.00 q^{19} -30402.0 q^{21} +44144.0 q^{22} +95452.0 q^{23} +13824.0 q^{24} +102384. q^{26} -19683.0 q^{27} +72064.0 q^{28} +19440.0 q^{29} -240248. q^{31} -32768.0 q^{32} +148986. q^{33} -257648. q^{34} +46656.0 q^{36} -77834.0 q^{37} +35520.0 q^{38} +345546. q^{39} +299522. q^{41} +243216. q^{42} +416212. q^{43} -353152. q^{44} -763616. q^{46} +322976. q^{47} -110592. q^{48} +444333. q^{49} -869562. q^{51} -819072. q^{52} -880878. q^{53} +157464. q^{54} -576512. q^{56} +119880. q^{57} -155520. q^{58} -1.84511e6 q^{59} -861718. q^{61} +1.92198e6 q^{62} +820854. q^{63} +262144. q^{64} -1.19189e6 q^{66} -673864. q^{67} +2.06118e6 q^{68} -2.57720e6 q^{69} -3.42695e6 q^{71} -373248. q^{72} -4.67875e6 q^{73} +622672. q^{74} -284160. q^{76} -6.21327e6 q^{77} -2.76437e6 q^{78} -3.13776e6 q^{79} +531441. q^{81} -2.39618e6 q^{82} +484132. q^{83} -1.94573e6 q^{84} -3.32970e6 q^{86} -524880. q^{87} +2.82522e6 q^{88} +6.25871e6 q^{89} -1.44105e7 q^{91} +6.10893e6 q^{92} +6.48670e6 q^{93} -2.58381e6 q^{94} +884736. q^{96} +8.65758e6 q^{97} -3.55466e6 q^{98} -4.02262e6 q^{99} +O(q^{100})

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

)

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

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	64.0000	0.500000
$5$	0	0
$5$	0	0
$6$	216.000	0.408248
$7$	1126.00	1.24078	0.620391	−	0.784293i	$-0.286974\pi$
$7$	1126.00	1.24078	0.620391	+	0.784293i	$0.286974\pi$
$8$	−512.000	−0.353553
$9$	729.000	0.333333
$10$	0	0
$11$	−5518.00	−1.24999	−0.624996	−	0.780628i	$-0.714899\pi$
$11$	−5518.00	−1.24999	−0.624996	+	0.780628i	$0.714899\pi$
$12$	−1728.00	−0.288675
$13$	−12798.0	−1.61562	−0.807812	−	0.589440i	$-0.799348\pi$
$13$	−12798.0	−1.61562	−0.807812	+	0.589440i	$0.799348\pi$
$14$	−9008.00	−0.877365
$15$	0	0
$16$	4096.00	0.250000
$17$	32206.0	1.58988	0.794942	−	0.606685i	$-0.207501\pi$
$17$	32206.0	1.58988	0.794942	+	0.606685i	$0.207501\pi$
$18$	−5832.00	−0.235702
$19$	−4440.00	−0.148506	−0.0742532	−	0.997239i	$-0.523657\pi$
$19$	−4440.00	−0.148506	−0.0742532	+	0.997239i	$0.523657\pi$
$20$	0	0
$21$	−30402.0	−0.716365
$22$	44144.0	0.883878
$23$	95452.0	1.63583	0.817914	−	0.575341i	$-0.195130\pi$
$23$	95452.0	1.63583	0.817914	+	0.575341i	$0.195130\pi$
$24$	13824.0	0.204124
$25$	0	0
$26$	102384.	1.14242
$27$	−19683.0	−0.192450
$28$	72064.0	0.620391
$29$	19440.0	0.148014	0.0740071	−	0.997258i	$-0.476421\pi$
$29$	19440.0	0.148014	0.0740071	+	0.997258i	$0.476421\pi$
$30$	0	0
$31$	−240248.	−1.44842	−0.724209	−	0.689581i	$-0.757795\pi$
$31$	−240248.	−1.44842	−0.724209	+	0.689581i	$0.757795\pi$
$32$	−32768.0	−0.176777
$33$	148986.	0.721683
$34$	−257648.	−1.12422
$35$	0	0
$36$	46656.0	0.166667
$37$	−77834.0	−0.252617	−0.126309	−	0.991991i	$-0.540313\pi$
$37$	−77834.0	−0.252617	−0.126309	+	0.991991i	$0.540313\pi$
$38$	35520.0	0.105010
$39$	345546.	0.932781
$40$	0	0
$41$	299522.	0.678712	0.339356	−	0.940658i	$-0.389791\pi$
$41$	299522.	0.678712	0.339356	+	0.940658i	$0.389791\pi$
$42$	243216.	0.506547
$43$	416212.	0.798316	0.399158	−	0.916882i	$-0.369302\pi$
$43$	416212.	0.798316	0.399158	+	0.916882i	$0.369302\pi$
$44$	−353152.	−0.624996
$45$	0	0
$46$	−763616.	−1.15670
$47$	322976.	0.453762	0.226881	−	0.973923i	$-0.427147\pi$
$47$	322976.	0.453762	0.226881	+	0.973923i	$0.427147\pi$
$48$	−110592.	−0.144338
$49$	444333.	0.539538
$50$	0	0
$51$	−869562.	−0.917920
$52$	−819072.	−0.807812
$53$	−880878.	−0.812737	−0.406369	−	0.913709i	$-0.633205\pi$
$53$	−880878.	−0.812737	−0.406369	+	0.913709i	$0.633205\pi$
$54$	157464.	0.136083
$55$	0	0
$56$	−576512.	−0.438682
$57$	119880.	0.0857402
$58$	−155520.	−0.104662
$59$	−1.84511e6	−1.16961	−0.584804	−	0.811175i	$-0.698829\pi$
$59$	−1.84511e6	−1.16961	−0.584804	+	0.811175i	$0.698829\pi$
$60$	0	0
$61$	−861718.	−0.486083	−0.243042	−	0.970016i	$-0.578145\pi$
$61$	−861718.	−0.486083	−0.243042	+	0.970016i	$0.578145\pi$
$62$	1.92198e6	1.02419
$63$	820854.	0.413594
$64$	262144.	0.125000
$65$	0	0
$66$	−1.19189e6	−0.510307
$67$	−673864.	−0.273722	−0.136861	−	0.990590i	$-0.543701\pi$
$67$	−673864.	−0.273722	−0.136861	+	0.990590i	$0.543701\pi$
$68$	2.06118e6	0.794942
$69$	−2.57720e6	−0.944446
$70$	0	0
$71$	−3.42695e6	−1.13633	−0.568163	−	0.822916i	$-0.692346\pi$
$71$	−3.42695e6	−1.13633	−0.568163	+	0.822916i	$0.692346\pi$
$72$	−373248.	−0.117851
$73$	−4.67875e6	−1.40767	−0.703833	−	0.710365i	$-0.748530\pi$
$73$	−4.67875e6	−1.40767	−0.703833	+	0.710365i	$0.748530\pi$
$74$	622672.	0.178627
$75$	0	0
$76$	−284160.	−0.0742532
$77$	−6.21327e6	−1.55097
$78$	−2.76437e6	−0.659576
$79$	−3.13776e6	−0.716020	−0.358010	−	0.933718i	$-0.616545\pi$
$79$	−3.13776e6	−0.716020	−0.358010	+	0.933718i	$0.616545\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	−2.39618e6	−0.479922
$83$	484132.	0.0929374	0.0464687	−	0.998920i	$-0.485203\pi$
$83$	484132.	0.0929374	0.0464687	+	0.998920i	$0.485203\pi$
$84$	−1.94573e6	−0.358183
$85$	0	0
$86$	−3.32970e6	−0.564495
$87$	−524880.	−0.0854560
$88$	2.82522e6	0.441939
$89$	6.25871e6	0.941065	0.470533	−	0.882383i	$-0.344062\pi$
$89$	6.25871e6	0.941065	0.470533	+	0.882383i	$0.344062\pi$
$90$	0	0
$91$	−1.44105e7	−2.00464
$92$	6.10893e6	0.817914
$93$	6.48670e6	0.836244
$94$	−2.58381e6	−0.320858
$95$	0	0
$96$	884736.	0.102062
$97$	8.65758e6	0.963153	0.481576	−	0.876404i	$-0.340064\pi$
$97$	8.65758e6	0.963153	0.481576	+	0.876404i	$0.340064\pi$
$98$	−3.55466e6	−0.381511
$99$	−4.02262e6	−0.416664
$100$	0	0
$101$	−7.52329e6	−0.726579	−0.363290	−	0.931676i	$-0.618346\pi$
$101$	−7.52329e6	−0.726579	−0.363290	+	0.931676i	$0.618346\pi$
$102$	6.95650e6	0.649068
$103$	−1.40261e7	−1.26476	−0.632380	−	0.774659i	$-0.717922\pi$
$103$	−1.40261e7	−1.26476	−0.632380	+	0.774659i	$0.717922\pi$
$104$	6.55258e6	0.571209
$105$	0	0
$106$	7.04702e6	0.574692
$107$	−1.59380e6	−0.125774	−0.0628871	−	0.998021i	$-0.520031\pi$
$107$	−1.59380e6	−0.125774	−0.0628871	+	0.998021i	$0.520031\pi$
$108$	−1.25971e6	−0.0962250
$109$	−1.66129e7	−1.22872	−0.614360	−	0.789026i	$-0.710586\pi$
$109$	−1.66129e7	−1.22872	−0.614360	+	0.789026i	$0.710586\pi$
$110$	0	0
$111$	2.10152e6	0.145849
$112$	4.61210e6	0.310195
$113$	−536598.	−0.0349844	−0.0174922	−	0.999847i	$-0.505568\pi$
$113$	−536598.	−0.0349844	−0.0174922	+	0.999847i	$0.505568\pi$
$114$	−959040.	−0.0606275
$115$	0	0
$116$	1.24416e6	0.0740071
$117$	−9.32974e6	−0.538541
$118$	1.47609e7	0.827038
$119$	3.62640e7	1.97270
$120$	0	0
$121$	1.09612e7	0.562480
$122$	6.89374e6	0.343713
$123$	−8.08709e6	−0.391854
$124$	−1.53759e7	−0.724209
$125$	0	0
$126$	−6.56683e6	−0.292455
$127$	−3.90459e7	−1.69146	−0.845732	−	0.533608i	$-0.820836\pi$
$127$	−3.90459e7	−1.69146	−0.845732	+	0.533608i	$0.820836\pi$
$128$	−2.09715e6	−0.0883883
$129$	−1.12377e7	−0.460908
$130$	0	0
$131$	4.08046e6	0.158584	0.0792921	−	0.996851i	$-0.474734\pi$
$131$	4.08046e6	0.158584	0.0792921	+	0.996851i	$0.474734\pi$
$132$	9.53510e6	0.360842
$133$	−4.99944e6	−0.184264
$134$	5.39091e6	0.193551
$135$	0	0
$136$	−1.64895e7	−0.562109
$137$	−2.34170e7	−0.778052	−0.389026	−	0.921227i	$-0.627188\pi$
$137$	−2.34170e7	−0.778052	−0.389026	+	0.921227i	$0.627188\pi$
$138$	2.06176e7	0.667824
$139$	−1.35950e6	−0.0429366	−0.0214683	−	0.999770i	$-0.506834\pi$
$139$	−1.35950e6	−0.0429366	−0.0214683	+	0.999770i	$0.506834\pi$
$140$	0	0
$141$	−8.72035e6	−0.261979
$142$	2.74156e7	0.803504
$143$	7.06194e7	2.01952
$144$	2.98598e6	0.0833333
$145$	0	0
$146$	3.74300e7	0.995370
$147$	−1.19970e7	−0.311503
$148$	−4.98138e6	−0.126309
$149$	−6.79776e7	−1.68350	−0.841752	−	0.539865i	$-0.818475\pi$
$149$	−6.79776e7	−1.68350	−0.841752	+	0.539865i	$0.818475\pi$
$150$	0	0
$151$	3.00991e7	0.711434	0.355717	−	0.934594i	$-0.384237\pi$
$151$	3.00991e7	0.711434	0.355717	+	0.934594i	$0.384237\pi$
$152$	2.27328e6	0.0525050
$153$	2.34782e7	0.529961
$154$	4.97061e7	1.09670
$155$	0	0
$156$	2.21149e7	0.466391
$157$	1.09929e7	0.226706	0.113353	−	0.993555i	$-0.463841\pi$
$157$	1.09929e7	0.226706	0.113353	+	0.993555i	$0.463841\pi$
$158$	2.51021e7	0.506302
$159$	2.37837e7	0.469234
$160$	0	0
$161$	1.07479e8	2.02970
$162$	−4.25153e6	−0.0785674
$163$	−8.99208e7	−1.62631	−0.813155	−	0.582047i	$-0.802252\pi$
$163$	−8.99208e7	−1.62631	−0.813155	+	0.582047i	$0.802252\pi$
$164$	1.91694e7	0.339356
$165$	0	0
$166$	−3.87306e6	−0.0657167
$167$	4.57557e7	0.760217	0.380108	−	0.924942i	$-0.375887\pi$
$167$	4.57557e7	0.760217	0.380108	+	0.924942i	$0.375887\pi$
$168$	1.55658e7	0.253273
$169$	1.01040e8	1.61024
$170$	0	0
$171$	−3.23676e6	−0.0495022
$172$	2.66376e7	0.399158
$173$	−9.47012e7	−1.39057	−0.695287	−	0.718732i	$-0.744723\pi$
$173$	−9.47012e7	−1.39057	−0.695287	+	0.718732i	$0.744723\pi$
$174$	4.19904e6	0.0604265
$175$	0	0
$176$	−2.26017e7	−0.312498
$177$	4.98180e7	0.675273
$178$	−5.00697e7	−0.665434
$179$	7.12070e7	0.927977	0.463989	−	0.885841i	$-0.346418\pi$
$179$	7.12070e7	0.927977	0.463989	+	0.885841i	$0.346418\pi$
$180$	0	0
$181$	6.10292e7	0.765003	0.382501	−	0.923955i	$-0.375063\pi$
$181$	6.10292e7	0.765003	0.382501	+	0.923955i	$0.375063\pi$
$182$	1.15284e8	1.41749
$183$	2.32664e7	0.280640
$184$	−4.88714e7	−0.578352
$185$	0	0
$186$	−5.18936e7	−0.591314
$187$	−1.77713e8	−1.98734
$188$	2.06705e7	0.226881
$189$	−2.21631e7	−0.238788
$190$	0	0
$191$	5.63172e7	0.584823	0.292411	−	0.956293i	$-0.405542\pi$
$191$	5.63172e7	0.584823	0.292411	+	0.956293i	$0.405542\pi$
$192$	−7.07789e6	−0.0721688
$193$	−9.97730e7	−0.998993	−0.499496	−	0.866316i	$-0.666482\pi$
$193$	−9.97730e7	−0.998993	−0.499496	+	0.866316i	$0.666482\pi$
$194$	−6.92606e7	−0.681052
$195$	0	0
$196$	2.84373e7	0.269769
$197$	−1.25457e8	−1.16913	−0.584566	−	0.811346i	$-0.698735\pi$
$197$	−1.25457e8	−1.16913	−0.584566	+	0.811346i	$0.698735\pi$
$198$	3.21810e7	0.294626
$199$	−2.32320e6	−0.0208978	−0.0104489	−	0.999945i	$-0.503326\pi$
$199$	−2.32320e6	−0.0208978	−0.0104489	+	0.999945i	$0.503326\pi$
$200$	0	0
$201$	1.81943e7	0.158034
$202$	6.01863e7	0.513769
$203$	2.18894e7	0.183653
$204$	−5.56520e7	−0.458960
$205$	0	0
$206$	1.12209e8	0.894320
$207$	6.95845e7	0.545276
$208$	−5.24206e7	−0.403906
$209$	2.44999e7	0.185632
$210$	0	0
$211$	8.85076e7	0.648622	0.324311	−	0.945950i	$-0.394868\pi$
$211$	8.85076e7	0.648622	0.324311	+	0.945950i	$0.394868\pi$
$212$	−5.63762e7	−0.406369
$213$	9.25276e7	0.656059
$214$	1.27504e7	0.0889358
$215$	0	0
$216$	1.00777e7	0.0680414
$217$	−2.70519e8	−1.79717
$218$	1.32903e8	0.868836
$219$	1.26326e8	0.812716
$220$	0	0
$221$	−4.12172e8	−2.56866
$222$	−1.68121e7	−0.103131
$223$	−629858.	−0.00380343	−0.00190172	−	0.999998i	$-0.500605\pi$
$223$	−629858.	−0.00380343	−0.00190172	+	0.999998i	$0.500605\pi$
$224$	−3.68968e7	−0.219341
$225$	0	0
$226$	4.29278e6	0.0247377
$227$	1.74337e8	0.989236	0.494618	−	0.869111i	$-0.335308\pi$
$227$	1.74337e8	0.989236	0.494618	+	0.869111i	$0.335308\pi$
$228$	7.67232e6	0.0428701
$229$	−2.00644e8	−1.10408	−0.552042	−	0.833816i	$-0.686151\pi$
$229$	−2.00644e8	−1.10408	−0.552042	+	0.833816i	$0.686151\pi$
$230$	0	0
$231$	1.67758e8	0.895451
$232$	−9.95328e6	−0.0523309
$233$	1.66522e8	0.862434	0.431217	−	0.902248i	$-0.358084\pi$
$233$	1.66522e8	0.862434	0.431217	+	0.902248i	$0.358084\pi$
$234$	7.46379e7	0.380806
$235$	0	0
$236$	−1.18087e8	−0.584804
$237$	8.47195e7	0.413394
$238$	−2.90112e8	−1.39491
$239$	−1.17669e8	−0.557531	−0.278765	−	0.960359i	$-0.589925\pi$
$239$	−1.17669e8	−0.557531	−0.278765	+	0.960359i	$0.589925\pi$
$240$	0	0
$241$	−4.04961e8	−1.86360	−0.931802	−	0.362967i	$-0.881764\pi$
$241$	−4.04961e8	−1.86360	−0.931802	+	0.362967i	$0.881764\pi$
$242$	−8.76892e7	−0.397734
$243$	−1.43489e7	−0.0641500
$244$	−5.51500e7	−0.243042
$245$	0	0
$246$	6.46968e7	0.277083
$247$	5.68231e7	0.239931
$248$	1.23007e8	0.512093
$249$	−1.30716e7	−0.0536574
$250$	0	0
$251$	−5.93170e7	−0.236767	−0.118383	−	0.992968i	$-0.537771\pi$
$251$	−5.93170e7	−0.236767	−0.118383	+	0.992968i	$0.537771\pi$
$252$	5.25347e7	0.206797
$253$	−5.26704e8	−2.04477
$254$	3.12367e8	1.19605
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	2.48636e7	0.0913688	0.0456844	−	0.998956i	$-0.485453\pi$
$257$	2.48636e7	0.0913688	0.0456844	+	0.998956i	$0.485453\pi$
$258$	8.99018e7	0.325911
$259$	−8.76411e7	−0.313443
$260$	0	0
$261$	1.41718e7	0.0493381
$262$	−3.26437e7	−0.112136
$263$	−5.36503e6	−0.0181856	−0.00909278	−	0.999959i	$-0.502894\pi$
$263$	−5.36503e6	−0.0181856	−0.00909278	+	0.999959i	$0.502894\pi$
$264$	−7.62808e7	−0.255154
$265$	0	0
$266$	3.99955e7	0.130294
$267$	−1.68985e8	−0.543324
$268$	−4.31273e7	−0.136861
$269$	−1.83159e8	−0.573713	−0.286856	−	0.957974i	$-0.592610\pi$
$269$	−1.83159e8	−0.573713	−0.286856	+	0.957974i	$0.592610\pi$
$270$	0	0
$271$	2.54405e8	0.776486	0.388243	−	0.921557i	$-0.373082\pi$
$271$	2.54405e8	0.776486	0.388243	+	0.921557i	$0.373082\pi$
$272$	1.31916e8	0.397471
$273$	3.89085e8	1.15738
$274$	1.87336e8	0.550166
$275$	0	0
$276$	−1.64941e8	−0.472223
$277$	5.92863e8	1.67601	0.838003	−	0.545666i	$-0.183723\pi$
$277$	5.92863e8	1.67601	0.838003	+	0.545666i	$0.183723\pi$
$278$	1.08760e7	0.0303607
$279$	−1.75141e8	−0.482806
$280$	0	0
$281$	−6.22030e8	−1.67240	−0.836199	−	0.548427i	$-0.815227\pi$
$281$	−6.22030e8	−1.67240	−0.836199	+	0.548427i	$0.815227\pi$
$282$	6.97628e7	0.185247
$283$	3.94755e8	1.03532	0.517661	−	0.855586i	$-0.326803\pi$
$283$	3.94755e8	1.03532	0.517661	+	0.855586i	$0.326803\pi$
$284$	−2.19325e8	−0.568163
$285$	0	0
$286$	−5.64955e8	−1.42801
$287$	3.37262e8	0.842133
$288$	−2.38879e7	−0.0589256
$289$	6.26888e8	1.52773
$290$	0	0
$291$	−2.33755e8	−0.556077
$292$	−2.99440e8	−0.703833
$293$	6.89404e8	1.60117	0.800585	−	0.599220i	$-0.204522\pi$
$293$	6.89404e8	1.60117	0.800585	+	0.599220i	$0.204522\pi$
$294$	9.59759e7	0.220266
$295$	0	0
$296$	3.98510e7	0.0893137
$297$	1.08611e8	0.240561
$298$	5.43821e8	1.19042
$299$	−1.22159e9	−2.64288
$300$	0	0
$301$	4.68655e8	0.990536
$302$	−2.40793e8	−0.503060
$303$	2.03129e8	0.419491
$304$	−1.81862e7	−0.0371266
$305$	0	0
$306$	−1.87825e8	−0.374739
$307$	−2.67769e8	−0.528174	−0.264087	−	0.964499i	$-0.585071\pi$
$307$	−2.67769e8	−0.528174	−0.264087	+	0.964499i	$0.585071\pi$
$308$	−3.97649e8	−0.775483
$309$	3.78706e8	0.730209
$310$	0	0
$311$	−1.94551e8	−0.366752	−0.183376	−	0.983043i	$-0.558703\pi$
$311$	−1.94551e8	−0.366752	−0.183376	+	0.983043i	$0.558703\pi$
$312$	−1.76920e8	−0.329788
$313$	−8.81802e8	−1.62542	−0.812711	−	0.582667i	$-0.802009\pi$
$313$	−8.81802e8	−1.62542	−0.812711	+	0.582667i	$0.802009\pi$
$314$	−8.79429e7	−0.160305
$315$	0	0
$316$	−2.00817e8	−0.358010
$317$	−4.88296e8	−0.860946	−0.430473	−	0.902603i	$-0.641653\pi$
$317$	−4.88296e8	−0.860946	−0.430473	+	0.902603i	$0.641653\pi$
$318$	−1.90270e8	−0.331799
$319$	−1.07270e8	−0.185017
$320$	0	0
$321$	4.30327e7	0.0726158
$322$	−8.59832e8	−1.43522
$323$	−1.42995e8	−0.236108
$324$	3.40122e7	0.0555556
$325$	0	0
$326$	7.19366e8	1.14997
$327$	4.48549e8	0.709402
$328$	−1.53355e8	−0.239961
$329$	3.63671e8	0.563019
$330$	0	0
$331$	−7.08199e8	−1.07339	−0.536695	−	0.843777i	$-0.680327\pi$
$331$	−7.08199e8	−1.07339	−0.536695	+	0.843777i	$0.680327\pi$
$332$	3.09844e7	0.0464687
$333$	−5.67410e7	−0.0842058
$334$	−3.66045e8	−0.537554
$335$	0	0
$336$	−1.24527e8	−0.179091
$337$	6.69707e8	0.953192	0.476596	−	0.879122i	$-0.341870\pi$
$337$	6.69707e8	0.953192	0.476596	+	0.879122i	$0.341870\pi$
$338$	−8.08322e8	−1.13861
$339$	1.44881e7	0.0201983
$340$	0	0
$341$	1.32569e9	1.81051
$342$	2.58941e7	0.0350033
$343$	−4.26990e8	−0.571332
$344$	−2.13101e8	−0.282247
$345$	0	0
$346$	7.57609e8	0.983284
$347$	1.01172e9	1.29989	0.649945	−	0.759982i	$-0.274792\pi$
$347$	1.01172e9	1.29989	0.649945	+	0.759982i	$0.274792\pi$
$348$	−3.35923e7	−0.0427280
$349$	3.46636e8	0.436501	0.218250	−	0.975893i	$-0.429965\pi$
$349$	3.46636e8	0.436501	0.218250	+	0.975893i	$0.429965\pi$
$350$	0	0
$351$	2.51903e8	0.310927
$352$	1.80814e8	0.220969
$353$	2.93766e6	0.00355460	0.00177730	−	0.999998i	$-0.499434\pi$
$353$	2.93766e6	0.00355460	0.00177730	+	0.999998i	$0.499434\pi$
$354$	−3.98544e8	−0.477490
$355$	0	0
$356$	4.00557e8	0.470533
$357$	−9.79127e8	−1.13894
$358$	−5.69656e8	−0.656179
$359$	1.64977e9	1.88188	0.940942	−	0.338568i	$-0.109942\pi$
$359$	1.64977e9	1.88188	0.940942	+	0.338568i	$0.109942\pi$
$360$	0	0
$361$	−8.74158e8	−0.977946
$362$	−4.88234e8	−0.540939
$363$	−2.95951e8	−0.324748
$364$	−9.22275e8	−1.00232
$365$	0	0
$366$	−1.86131e8	−0.198443
$367$	−4.33054e8	−0.457311	−0.228655	−	0.973507i	$-0.573433\pi$
$367$	−4.33054e8	−0.457311	−0.228655	+	0.973507i	$0.573433\pi$
$368$	3.90971e8	0.408957
$369$	2.18352e8	0.226237
$370$	0	0
$371$	−9.91869e8	−1.00843
$372$	4.15149e8	0.418122
$373$	−7.13249e8	−0.711640	−0.355820	−	0.934555i	$-0.615798\pi$
$373$	−7.13249e8	−0.711640	−0.355820	+	0.934555i	$0.615798\pi$
$374$	1.42170e9	1.40526
$375$	0	0
$376$	−1.65364e8	−0.160429
$377$	−2.48793e8	−0.239135
$378$	1.77304e8	0.168849
$379$	1.92795e8	0.181911	0.0909555	−	0.995855i	$-0.471008\pi$
$379$	1.92795e8	0.181911	0.0909555	+	0.995855i	$0.471008\pi$
$380$	0	0
$381$	1.05424e9	0.976567
$382$	−4.50538e8	−0.413532
$383$	1.30544e9	1.18730	0.593649	−	0.804724i	$-0.297687\pi$
$383$	1.30544e9	1.18730	0.593649	+	0.804724i	$0.297687\pi$
$384$	5.66231e7	0.0510310
$385$	0	0
$386$	7.98184e8	0.706395
$387$	3.03419e8	0.266105
$388$	5.54085e8	0.481576
$389$	2.53075e7	0.0217985	0.0108992	−	0.999941i	$-0.496531\pi$
$389$	2.53075e7	0.0217985	0.0108992	+	0.999941i	$0.496531\pi$
$390$	0	0
$391$	3.07413e9	2.60078
$392$	−2.27498e8	−0.190756
$393$	−1.10172e8	−0.0915586
$394$	1.00366e9	0.826702
$395$	0	0
$396$	−2.57448e8	−0.208332
$397$	1.34997e9	1.08282	0.541410	−	0.840759i	$-0.317891\pi$
$397$	1.34997e9	1.08282	0.541410	+	0.840759i	$0.317891\pi$
$398$	1.85856e7	0.0147770
$399$	1.34985e8	0.106385
$400$	0	0
$401$	−9.20762e7	−0.0713086	−0.0356543	−	0.999364i	$-0.511352\pi$
$401$	−9.20762e7	−0.0713086	−0.0356543	+	0.999364i	$0.511352\pi$
$402$	−1.45555e8	−0.111747
$403$	3.07469e9	2.34010
$404$	−4.81490e8	−0.363290
$405$	0	0
$406$	−1.75116e8	−0.129862
$407$	4.29488e8	0.315770
$408$	4.45216e8	0.324534
$409$	−2.68396e9	−1.93975	−0.969874	−	0.243609i	$-0.921669\pi$
$409$	−2.68396e9	−1.93975	−0.969874	+	0.243609i	$0.921669\pi$
$410$	0	0
$411$	6.32258e8	0.449209
$412$	−8.97673e8	−0.632380
$413$	−2.07759e9	−1.45123
$414$	−5.56676e8	−0.385568
$415$	0	0
$416$	4.19365e8	0.285605
$417$	3.67065e7	0.0247894
$418$	−1.95999e8	−0.131262
$419$	8.45201e8	0.561320	0.280660	−	0.959807i	$-0.409447\pi$
$419$	8.45201e8	0.561320	0.280660	+	0.959807i	$0.409447\pi$
$420$	0	0
$421$	2.66227e9	1.73886	0.869430	−	0.494056i	$-0.164486\pi$
$421$	2.66227e9	1.73886	0.869430	+	0.494056i	$0.164486\pi$
$422$	−7.08061e8	−0.458645
$423$	2.35450e8	0.151254
$424$	4.51010e8	0.287346
$425$	0	0
$426$	−7.40221e8	−0.463904
$427$	−9.70294e8	−0.603123
$428$	−1.02003e8	−0.0628871
$429$	−1.90672e9	−1.16597
$430$	0	0
$431$	−9.83169e8	−0.591504	−0.295752	−	0.955265i	$-0.595570\pi$
$431$	−9.83169e8	−0.591504	−0.295752	+	0.955265i	$0.595570\pi$
$432$	−8.06216e7	−0.0481125
$433$	8.30277e8	0.491491	0.245745	−	0.969334i	$-0.420967\pi$
$433$	8.30277e8	0.491491	0.245745	+	0.969334i	$0.420967\pi$
$434$	2.16415e9	1.27079
$435$	0	0
$436$	−1.06323e9	−0.614360
$437$	−4.23807e8	−0.242931
$438$	−1.01061e9	−0.574677
$439$	2.79662e9	1.57764	0.788820	−	0.614625i	$-0.210692\pi$
$439$	2.79662e9	1.57764	0.788820	+	0.614625i	$0.210692\pi$
$440$	0	0
$441$	3.23919e8	0.179846
$442$	3.29738e9	1.81631
$443$	3.97081e8	0.217003	0.108502	−	0.994096i	$-0.465395\pi$
$443$	3.97081e8	0.217003	0.108502	+	0.994096i	$0.465395\pi$
$444$	1.34497e8	0.0729244
$445$	0	0
$446$	5.03886e6	0.00268943
$447$	1.83540e9	0.971971
$448$	2.95174e8	0.155098
$449$	1.74461e8	0.0909569	0.0454785	−	0.998965i	$-0.485519\pi$
$449$	1.74461e8	0.0909569	0.0454785	+	0.998965i	$0.485519\pi$
$450$	0	0
$451$	−1.65276e9	−0.848384
$452$	−3.43423e7	−0.0174922
$453$	−8.12676e8	−0.410746
$454$	−1.39470e9	−0.699495
$455$	0	0
$456$	−6.13786e7	−0.0303138
$457$	3.60027e8	0.176453	0.0882264	−	0.996100i	$-0.471880\pi$
$457$	3.60027e8	0.176453	0.0882264	+	0.996100i	$0.471880\pi$
$458$	1.60515e9	0.780705
$459$	−6.33911e8	−0.305973
$460$	0	0
$461$	−1.33487e8	−0.0634579	−0.0317290	−	0.999497i	$-0.510101\pi$
$461$	−1.33487e8	−0.0634579	−0.0317290	+	0.999497i	$0.510101\pi$
$462$	−1.34207e9	−0.633180
$463$	2.93605e9	1.37477	0.687384	−	0.726294i	$-0.258759\pi$
$463$	2.93605e9	1.37477	0.687384	+	0.726294i	$0.258759\pi$
$464$	7.96262e7	0.0370035
$465$	0	0
$466$	−1.33218e9	−0.609833
$467$	1.72777e9	0.785011	0.392506	−	0.919750i	$-0.371608\pi$
$467$	1.72777e9	0.785011	0.392506	+	0.919750i	$0.371608\pi$
$468$	−5.97103e8	−0.269271
$469$	−7.58771e8	−0.339630
$470$	0	0
$471$	−2.96807e8	−0.130889
$472$	9.44696e8	0.413519
$473$	−2.29666e9	−0.997889
$474$	−6.77756e8	−0.292314
$475$	0	0
$476$	2.32089e9	0.986349
$477$	−6.42160e8	−0.270912
$478$	9.41351e8	0.394234
$479$	−1.69230e9	−0.703562	−0.351781	−	0.936082i	$-0.614424\pi$
$479$	−1.69230e9	−0.703562	−0.351781	+	0.936082i	$0.614424\pi$
$480$	0	0
$481$	9.96120e8	0.408135
$482$	3.23969e9	1.31777
$483$	−2.90193e9	−1.17185
$484$	7.01514e8	0.281240
$485$	0	0
$486$	1.14791e8	0.0453609
$487$	−1.46319e9	−0.574048	−0.287024	−	0.957923i	$-0.592666\pi$
$487$	−1.46319e9	−0.574048	−0.287024	+	0.957923i	$0.592666\pi$
$488$	4.41200e8	0.171856
$489$	2.42786e9	0.938951
$490$	0	0
$491$	−2.86378e9	−1.09183	−0.545915	−	0.837840i	$-0.683818\pi$
$491$	−2.86378e9	−1.09183	−0.545915	+	0.837840i	$0.683818\pi$
$492$	−5.17574e8	−0.195927
$493$	6.26085e8	0.235325
$494$	−4.54585e8	−0.169657
$495$	0	0
$496$	−9.84056e8	−0.362104
$497$	−3.85874e9	−1.40993
$498$	1.04573e8	0.0379415
$499$	−1.17178e9	−0.422176	−0.211088	−	0.977467i	$-0.567701\pi$
$499$	−1.17178e9	−0.422176	−0.211088	+	0.977467i	$0.567701\pi$
$500$	0	0
$501$	−1.23540e9	−0.438911
$502$	4.74536e8	0.167419
$503$	−3.67913e9	−1.28901	−0.644506	−	0.764599i	$-0.722937\pi$
$503$	−3.67913e9	−1.28901	−0.644506	+	0.764599i	$0.722937\pi$
$504$	−4.20277e8	−0.146227
$505$	0	0
$506$	4.21363e9	1.44587
$507$	−2.72809e9	−0.929674
$508$	−2.49894e9	−0.845732
$509$	−4.56520e9	−1.53443	−0.767216	−	0.641389i	$-0.778359\pi$
$509$	−4.56520e9	−1.53443	−0.767216	+	0.641389i	$0.778359\pi$
$510$	0	0
$511$	−5.26827e9	−1.74661
$512$	−1.34218e8	−0.0441942
$513$	8.73925e7	0.0285801
$514$	−1.98909e8	−0.0646075
$515$	0	0
$516$	−7.19214e8	−0.230454
$517$	−1.78218e9	−0.567198
$518$	7.01129e8	0.221638
$519$	2.55693e9	0.802848
$520$	0	0
$521$	−5.34662e9	−1.65633	−0.828166	−	0.560484i	$-0.810615\pi$
$521$	−5.34662e9	−1.65633	−0.828166	+	0.560484i	$0.810615\pi$
$522$	−1.13374e8	−0.0348873
$523$	5.14610e9	1.57298	0.786488	−	0.617605i	$-0.211897\pi$
$523$	5.14610e9	1.57298	0.786488	+	0.617605i	$0.211897\pi$
$524$	2.61150e8	0.0792921
$525$	0	0
$526$	4.29202e7	0.0128591
$527$	−7.73743e9	−2.30282
$528$	6.10247e8	0.180421
$529$	5.70626e9	1.67593
$530$	0	0
$531$	−1.34509e9	−0.389869
$532$	−3.19964e8	−0.0921320
$533$	−3.83328e9	−1.09654
$534$	1.35188e9	0.384188
$535$	0	0
$536$	3.45018e8	0.0967755
$537$	−1.92259e9	−0.535768
$538$	1.46527e9	0.405676
$539$	−2.45183e9	−0.674419
$540$	0	0
$541$	−2.32073e8	−0.0630137	−0.0315069	−	0.999504i	$-0.510031\pi$
$541$	−2.32073e8	−0.0630137	−0.0315069	+	0.999504i	$0.510031\pi$
$542$	−2.03524e9	−0.549058
$543$	−1.64779e9	−0.441675
$544$	−1.05533e9	−0.281055
$545$	0	0
$546$	−3.11268e9	−0.818389
$547$	7.09990e9	1.85480	0.927399	−	0.374075i	$-0.122040\pi$
$547$	7.09990e9	1.85480	0.927399	+	0.374075i	$0.122040\pi$
$548$	−1.49869e9	−0.389026
$549$	−6.28192e8	−0.162028
$550$	0	0
$551$	−8.63136e7	−0.0219811
$552$	1.31953e9	0.333912
$553$	−3.53312e9	−0.888424
$554$	−4.74291e9	−1.18511
$555$	0	0
$556$	−8.70080e7	−0.0214683
$557$	2.92398e9	0.716937	0.358469	−	0.933542i	$-0.383299\pi$
$557$	2.92398e9	0.716937	0.358469	+	0.933542i	$0.383299\pi$
$558$	1.40113e9	0.341395
$559$	−5.32668e9	−1.28978
$560$	0	0
$561$	4.79824e9	1.14739
$562$	4.97624e9	1.18256
$563$	3.17221e9	0.749174	0.374587	−	0.927192i	$-0.377785\pi$
$563$	3.17221e9	0.749174	0.374587	+	0.927192i	$0.377785\pi$
$564$	−5.58103e8	−0.130990
$565$	0	0
$566$	−3.15804e9	−0.732083
$567$	5.98403e8	0.137865
$568$	1.75460e9	0.401752
$569$	2.75561e9	0.627082	0.313541	−	0.949575i	$-0.398485\pi$
$569$	2.75561e9	0.627082	0.313541	+	0.949575i	$0.398485\pi$
$570$	0	0
$571$	4.36061e9	0.980213	0.490107	−	0.871662i	$-0.336958\pi$
$571$	4.36061e9	0.980213	0.490107	+	0.871662i	$0.336958\pi$
$572$	4.51964e9	1.00976
$573$	−1.52056e9	−0.337648
$574$	−2.69809e9	−0.595478
$575$	0	0
$576$	1.91103e8	0.0416667
$577$	−6.07328e9	−1.31616	−0.658079	−	0.752949i	$-0.728631\pi$
$577$	−6.07328e9	−1.31616	−0.658079	+	0.752949i	$0.728631\pi$
$578$	−5.01510e9	−1.08027
$579$	2.69387e9	0.576769
$580$	0	0
$581$	5.45133e8	0.115315
$582$	1.87004e9	0.393206
$583$	4.86068e9	1.01592
$584$	2.39552e9	0.497685
$585$	0	0
$586$	−5.51523e9	−1.13220
$587$	4.35319e9	0.888329	0.444165	−	0.895945i	$-0.353501\pi$
$587$	4.35319e9	0.888329	0.444165	+	0.895945i	$0.353501\pi$
$588$	−7.67807e8	−0.155751
$589$	1.06670e9	0.215099
$590$	0	0
$591$	3.38734e9	0.674999
$592$	−3.18808e8	−0.0631544
$593$	−2.22483e9	−0.438133	−0.219067	−	0.975710i	$-0.570301\pi$
$593$	−2.22483e9	−0.438133	−0.219067	+	0.975710i	$0.570301\pi$
$594$	−8.68886e8	−0.170102
$595$	0	0
$596$	−4.35057e9	−0.841752
$597$	6.27264e7	0.0120654
$598$	9.77276e9	1.86880
$599$	−3.63376e9	−0.690815	−0.345408	−	0.938453i	$-0.612259\pi$
$599$	−3.63376e9	−0.690815	−0.345408	+	0.938453i	$0.612259\pi$
$600$	0	0
$601$	−6.00206e9	−1.12782	−0.563909	−	0.825837i	$-0.690703\pi$
$601$	−6.00206e9	−1.12782	−0.563909	+	0.825837i	$0.690703\pi$
$602$	−3.74924e9	−0.700415
$603$	−4.91247e8	−0.0912408
$604$	1.92634e9	0.355717
$605$	0	0
$606$	−1.62503e9	−0.296625
$607$	2.80807e9	0.509621	0.254811	−	0.966991i	$-0.417987\pi$
$607$	2.80807e9	0.509621	0.254811	+	0.966991i	$0.417987\pi$
$608$	1.45490e8	0.0262525
$609$	−5.91015e8	−0.106032
$610$	0	0
$611$	−4.13345e9	−0.733108
$612$	1.50260e9	0.264981
$613$	−8.34916e9	−1.46397	−0.731983	−	0.681323i	$-0.761405\pi$
$613$	−8.34916e9	−1.46397	−0.731983	+	0.681323i	$0.761405\pi$
$614$	2.14216e9	0.373475
$615$	0	0
$616$	3.18119e9	0.548350
$617$	2.16207e9	0.370571	0.185286	−	0.982685i	$-0.440679\pi$
$617$	2.16207e9	0.370571	0.185286	+	0.982685i	$0.440679\pi$
$618$	−3.02965e9	−0.516336
$619$	−2.82281e9	−0.478370	−0.239185	−	0.970974i	$-0.576880\pi$
$619$	−2.82281e9	−0.478370	−0.239185	+	0.970974i	$0.576880\pi$
$620$	0	0
$621$	−1.87878e9	−0.314815
$622$	1.55641e9	0.259333
$623$	7.04731e9	1.16766
$624$	1.41536e9	0.233195
$625$	0	0
$626$	7.05442e9	1.14935
$627$	−6.61498e8	−0.107175
$628$	7.03543e8	0.113353
$629$	−2.50672e9	−0.401633
$630$	0	0
$631$	7.87109e9	1.24719	0.623595	−	0.781748i	$-0.285672\pi$
$631$	7.87109e9	1.24719	0.623595	+	0.781748i	$0.285672\pi$
$632$	1.60653e9	0.253151
$633$	−2.38970e9	−0.374482
$634$	3.90637e9	0.608781
$635$	0	0
$636$	1.52216e9	0.234617
$637$	−5.68657e9	−0.871691
$638$	8.58159e8	0.130826
$639$	−2.49825e9	−0.378776
$640$	0	0
$641$	−3.77527e9	−0.566168	−0.283084	−	0.959095i	$-0.591357\pi$
$641$	−3.77527e9	−0.566168	−0.283084	+	0.959095i	$0.591357\pi$
$642$	−3.44262e8	−0.0513471
$643$	−2.51923e9	−0.373705	−0.186852	−	0.982388i	$-0.559829\pi$
$643$	−2.51923e9	−0.373705	−0.186852	+	0.982388i	$0.559829\pi$
$644$	6.87865e9	1.01485
$645$	0	0
$646$	1.14396e9	0.166954
$647$	−3.83911e9	−0.557269	−0.278635	−	0.960397i	$-0.589882\pi$
$647$	−3.83911e9	−0.557269	−0.278635	+	0.960397i	$0.589882\pi$
$648$	−2.72098e8	−0.0392837
$649$	1.01813e10	1.46200
$650$	0	0
$651$	7.30402e9	1.03760
$652$	−5.75493e9	−0.813155
$653$	1.00785e10	1.41644	0.708222	−	0.705989i	$-0.249497\pi$
$653$	1.00785e10	1.41644	0.708222	+	0.705989i	$0.249497\pi$
$654$	−3.58839e9	−0.501623
$655$	0	0
$656$	1.22684e9	0.169678
$657$	−3.41081e9	−0.469222
$658$	−2.90937e9	−0.398115
$659$	−8.94469e8	−0.121749	−0.0608746	−	0.998145i	$-0.519389\pi$
$659$	−8.94469e8	−0.121749	−0.0608746	+	0.998145i	$0.519389\pi$
$660$	0	0
$661$	1.19420e10	1.60832	0.804159	−	0.594414i	$-0.202616\pi$
$661$	1.19420e10	1.60832	0.804159	+	0.594414i	$0.202616\pi$
$662$	5.66559e9	0.759001
$663$	1.11287e10	1.48301
$664$	−2.47876e8	−0.0328583
$665$	0	0
$666$	4.53928e8	0.0595425
$667$	1.85559e9	0.242126
$668$	2.92836e9	0.380108
$669$	1.70062e7	0.00219591
$670$	0	0
$671$	4.75496e9	0.607600
$672$	9.96213e8	0.126637
$673$	1.66611e9	0.210693	0.105347	−	0.994436i	$-0.466405\pi$
$673$	1.66611e9	0.210693	0.105347	+	0.994436i	$0.466405\pi$
$674$	−5.35766e9	−0.674009
$675$	0	0
$676$	6.46658e9	0.805121
$677$	−3.07840e9	−0.381298	−0.190649	−	0.981658i	$-0.561059\pi$
$677$	−3.07840e9	−0.381298	−0.190649	+	0.981658i	$0.561059\pi$
$678$	−1.15905e8	−0.0142823
$679$	9.74843e9	1.19506
$680$	0	0
$681$	−4.70711e9	−0.571135
$682$	−1.06055e10	−1.28022
$683$	3.66304e9	0.439915	0.219958	−	0.975509i	$-0.429408\pi$
$683$	3.66304e9	0.439915	0.219958	+	0.975509i	$0.429408\pi$
$684$	−2.07153e8	−0.0247511
$685$	0	0
$686$	3.41592e9	0.403993
$687$	5.41739e9	0.637443
$688$	1.70480e9	0.199579
$689$	1.12735e10	1.31308
$690$	0	0
$691$	1.03135e10	1.18914	0.594569	−	0.804045i	$-0.297323\pi$
$691$	1.03135e10	1.18914	0.594569	+	0.804045i	$0.297323\pi$
$692$	−6.06088e9	−0.695287
$693$	−4.52947e9	−0.516989
$694$	−8.09374e9	−0.919161
$695$	0	0
$696$	2.68739e8	0.0302133
$697$	9.64641e9	1.07907
$698$	−2.77309e9	−0.308653
$699$	−4.49609e9	−0.497926
$700$	0	0
$701$	−5.82173e9	−0.638321	−0.319161	−	0.947701i	$-0.603401\pi$
$701$	−5.82173e9	−0.638321	−0.319161	+	0.947701i	$0.603401\pi$
$702$	−2.01522e9	−0.219859
$703$	3.45583e8	0.0375153
$704$	−1.44651e9	−0.156249
$705$	0	0
$706$	−2.35013e7	−0.00251348
$707$	−8.47122e9	−0.901526
$708$	3.18835e9	0.337637
$709$	−8.52806e9	−0.898645	−0.449323	−	0.893370i	$-0.648335\pi$
$709$	−8.52806e9	−0.898645	−0.449323	+	0.893370i	$0.648335\pi$
$710$	0	0
$711$	−2.28743e9	−0.238673
$712$	−3.20446e9	−0.332717
$713$	−2.29322e10	−2.36936
$714$	7.83301e9	0.805351
$715$	0	0
$716$	4.55725e9	0.463989
$717$	3.17706e9	0.321891
$718$	−1.31982e10	−1.33069
$719$	1.73546e9	0.174126	0.0870631	−	0.996203i	$-0.472252\pi$
$719$	1.73546e9	0.174126	0.0870631	+	0.996203i	$0.472252\pi$
$720$	0	0
$721$	−1.57934e10	−1.56929
$722$	6.99327e9	0.691512
$723$	1.09339e10	1.07595
$724$	3.90587e9	0.382501
$725$	0	0
$726$	2.36761e9	0.229632
$727$	1.16762e10	1.12702	0.563509	−	0.826110i	$-0.309451\pi$
$727$	1.16762e10	1.12702	0.563509	+	0.826110i	$0.309451\pi$
$728$	7.37820e9	0.708746
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	1.34045e10	1.26923
$732$	1.48905e9	0.140320
$733$	−4.92260e9	−0.461669	−0.230834	−	0.972993i	$-0.574146\pi$
$733$	−4.92260e9	−0.461669	−0.230834	+	0.972993i	$0.574146\pi$
$734$	3.46444e9	0.323367
$735$	0	0
$736$	−3.12777e9	−0.289176
$737$	3.71838e9	0.342151
$738$	−1.74681e9	−0.159974
$739$	−4.46577e9	−0.407043	−0.203522	−	0.979070i	$-0.565239\pi$
$739$	−4.46577e9	−0.407043	−0.203522	+	0.979070i	$0.565239\pi$
$740$	0	0
$741$	−1.53422e9	−0.138524
$742$	7.93495e9	0.713067
$743$	−8.36112e8	−0.0747831	−0.0373916	−	0.999301i	$-0.511905\pi$
$743$	−8.36112e8	−0.0747831	−0.0373916	+	0.999301i	$0.511905\pi$
$744$	−3.32119e9	−0.295657
$745$	0	0
$746$	5.70599e9	0.503205
$747$	3.52932e8	0.0309791
$748$	−1.13736e10	−0.993672
$749$	−1.79462e9	−0.156058
$750$	0	0
$751$	−1.37161e10	−1.18166	−0.590829	−	0.806797i	$-0.701199\pi$
$751$	−1.37161e10	−1.18166	−0.590829	+	0.806797i	$0.701199\pi$
$752$	1.32291e9	0.113440
$753$	1.60156e9	0.136697
$754$	1.99034e9	0.169094
$755$	0	0
$756$	−1.41844e9	−0.119394
$757$	−1.41449e10	−1.18513	−0.592563	−	0.805524i	$-0.701884\pi$
$757$	−1.41449e10	−1.18513	−0.592563	+	0.805524i	$0.701884\pi$
$758$	−1.54236e9	−0.128631
$759$	1.42210e10	1.18055
$760$	0	0
$761$	2.16828e10	1.78348	0.891741	−	0.452547i	$-0.149484\pi$
$761$	2.16828e10	1.78348	0.891741	+	0.452547i	$0.149484\pi$
$762$	−8.43392e9	−0.690537
$763$	−1.87061e10	−1.52457
$764$	3.60430e9	0.292411
$765$	0	0
$766$	−1.04435e10	−0.839547
$767$	2.36137e10	1.88965
$768$	−4.52985e8	−0.0360844
$769$	−1.45804e10	−1.15618	−0.578092	−	0.815972i	$-0.696202\pi$
$769$	−1.45804e10	−1.15618	−0.578092	+	0.815972i	$0.696202\pi$
$770$	0	0
$771$	−6.71317e8	−0.0527518
$772$	−6.38547e9	−0.499496
$773$	1.83373e9	0.142793	0.0713966	−	0.997448i	$-0.477254\pi$
$773$	1.83373e9	0.142793	0.0713966	+	0.997448i	$0.477254\pi$
$774$	−2.42735e9	−0.188165
$775$	0	0
$776$	−4.43268e9	−0.340526
$777$	2.36631e9	0.180966
$778$	−2.02460e8	−0.0154139
$779$	−1.32988e9	−0.100793
$780$	0	0
$781$	1.89099e10	1.42040
$782$	−2.45930e10	−1.83903
$783$	−3.82638e8	−0.0284853
$784$	1.81999e9	0.134885
$785$	0	0
$786$	8.81380e8	0.0647417
$787$	−1.48611e10	−1.08678	−0.543388	−	0.839482i	$-0.682859\pi$
$787$	−1.48611e10	−1.08678	−0.543388	+	0.839482i	$0.682859\pi$
$788$	−8.02926e9	−0.584566
$789$	1.44856e8	0.0104994
$790$	0	0
$791$	−6.04209e8	−0.0434080
$792$	2.05958e9	0.147313
$793$	1.10283e10	0.785328
$794$	−1.07997e10	−0.765669
$795$	0	0
$796$	−1.48685e8	−0.0104489
$797$	1.46595e10	1.02569	0.512843	−	0.858483i	$-0.328592\pi$
$797$	1.46595e10	1.02569	0.512843	+	0.858483i	$0.328592\pi$
$798$	−1.07988e9	−0.0752255
$799$	1.04018e10	0.721429
$800$	0	0
$801$	4.56260e9	0.313688
$802$	7.36610e8	0.0504228
$803$	2.58173e10	1.75957
$804$	1.16444e9	0.0790169
$805$	0	0
$806$	−2.45976e10	−1.65470
$807$	4.94528e9	0.331233
$808$	3.85192e9	0.256884
$809$	−9.95193e9	−0.660827	−0.330413	−	0.943836i	$-0.607188\pi$
$809$	−9.95193e9	−0.660827	−0.330413	+	0.943836i	$0.607188\pi$
$810$	0	0
$811$	−1.78463e10	−1.17483	−0.587415	−	0.809286i	$-0.699854\pi$
$811$	−1.78463e10	−1.17483	−0.587415	+	0.809286i	$0.699854\pi$
$812$	1.40092e9	0.0918266
$813$	−6.86894e9	−0.448304
$814$	−3.43590e9	−0.223283
$815$	0	0
$816$	−3.56173e9	−0.229480
$817$	−1.84798e9	−0.118555
$818$	2.14717e10	1.37161
$819$	−1.05053e10	−0.668212
$820$	0	0
$821$	−3.93673e9	−0.248276	−0.124138	−	0.992265i	$-0.539617\pi$
$821$	−3.93673e9	−0.248276	−0.124138	+	0.992265i	$0.539617\pi$
$822$	−5.05807e9	−0.317639
$823$	8.31828e9	0.520156	0.260078	−	0.965588i	$-0.416252\pi$
$823$	8.31828e9	0.520156	0.260078	+	0.965588i	$0.416252\pi$
$824$	7.18138e9	0.447160
$825$	0	0
$826$	1.66208e10	1.02617
$827$	1.30781e10	0.804033	0.402017	−	0.915632i	$-0.368309\pi$
$827$	1.30781e10	0.804033	0.402017	+	0.915632i	$0.368309\pi$
$828$	4.45341e9	0.272638
$829$	−1.78307e10	−1.08700	−0.543499	−	0.839410i	$-0.682901\pi$
$829$	−1.78307e10	−1.08700	−0.543499	+	0.839410i	$0.682901\pi$
$830$	0	0
$831$	−1.60073e10	−0.967642
$832$	−3.35492e9	−0.201953
$833$	1.43102e10	0.857804
$834$	−2.93652e8	−0.0175288
$835$	0	0
$836$	1.56799e9	0.0928160
$837$	4.72880e9	0.278748
$838$	−6.76160e9	−0.396913
$839$	2.05093e10	1.19890	0.599451	−	0.800411i	$-0.295386\pi$
$839$	2.05093e10	1.19890	0.599451	+	0.800411i	$0.295386\pi$
$840$	0	0
$841$	−1.68720e10	−0.978092
$842$	−2.12981e10	−1.22956
$843$	1.67948e10	0.965559
$844$	5.66448e9	0.324311
$845$	0	0
$846$	−1.88360e9	−0.106953
$847$	1.23423e10	0.697915
$848$	−3.60808e9	−0.203184
$849$	−1.06584e10	−0.597744
$850$	0	0
$851$	−7.42941e9	−0.413239
$852$	5.92177e9	0.328029
$853$	−2.30130e10	−1.26955	−0.634777	−	0.772695i	$-0.718908\pi$
$853$	−2.30130e10	−1.26955	−0.634777	+	0.772695i	$0.718908\pi$
$854$	7.76236e9	0.426472
$855$	0	0
$856$	8.16028e8	0.0444679
$857$	−1.45901e10	−0.791819	−0.395910	−	0.918289i	$-0.629571\pi$
$857$	−1.45901e10	−0.791819	−0.395910	+	0.918289i	$0.629571\pi$
$858$	1.52538e10	0.824465
$859$	3.04784e10	1.64065	0.820324	−	0.571898i	$-0.193793\pi$
$859$	3.04784e10	1.64065	0.820324	+	0.571898i	$0.193793\pi$
$860$	0	0
$861$	−9.10607e9	−0.486205
$862$	7.86535e9	0.418257
$863$	−1.15324e10	−0.610775	−0.305387	−	0.952228i	$-0.598786\pi$
$863$	−1.15324e10	−0.610775	−0.305387	+	0.952228i	$0.598786\pi$
$864$	6.44973e8	0.0340207
$865$	0	0
$866$	−6.64222e9	−0.347536
$867$	−1.69260e10	−0.882037
$868$	−1.73132e10	−0.898585
$869$	1.73142e10	0.895019
$870$	0	0
$871$	8.62411e9	0.442233
$872$	8.50581e9	0.434418
$873$	6.31137e9	0.321051
$874$	3.39046e9	0.171778
$875$	0	0
$876$	8.08488e9	0.406358
$877$	−1.15209e10	−0.576748	−0.288374	−	0.957518i	$-0.593115\pi$
$877$	−1.15209e10	−0.576748	−0.288374	+	0.957518i	$0.593115\pi$
$878$	−2.23730e10	−1.11556
$879$	−1.86139e10	−0.924436
$880$	0	0
$881$	−7.26258e8	−0.0357829	−0.0178914	−	0.999840i	$-0.505695\pi$
$881$	−7.26258e8	−0.0357829	−0.0178914	+	0.999840i	$0.505695\pi$
$882$	−2.59135e9	−0.127170
$883$	−1.81507e10	−0.887219	−0.443609	−	0.896220i	$-0.646302\pi$
$883$	−1.81507e10	−0.887219	−0.443609	+	0.896220i	$0.646302\pi$
$884$	−2.63790e10	−1.28433
$885$	0	0
$886$	−3.17665e9	−0.153444
$887$	−8.91198e9	−0.428787	−0.214393	−	0.976747i	$-0.568777\pi$
$887$	−8.91198e9	−0.428787	−0.214393	+	0.976747i	$0.568777\pi$
$888$	−1.07598e9	−0.0515653
$889$	−4.39657e10	−2.09874
$890$	0	0
$891$	−2.93249e9	−0.138888
$892$	−4.03109e7	−0.00190172
$893$	−1.43401e9	−0.0673865
$894$	−1.46832e10	−0.687287
$895$	0	0
$896$	−2.36139e9	−0.109671
$897$	3.29831e10	1.52587
$898$	−1.39569e9	−0.0643163
$899$	−4.67042e9	−0.214386
$900$	0	0
$901$	−2.83696e10	−1.29216
$902$	1.32221e10	0.599898
$903$	−1.26537e10	−0.571886
$904$	2.74738e8	0.0123689
$905$	0	0
$906$	6.50141e9	0.290442
$907$	1.76727e10	0.786462	0.393231	−	0.919440i	$-0.371357\pi$
$907$	1.76727e10	0.786462	0.393231	+	0.919440i	$0.371357\pi$
$908$	1.11576e10	0.494618
$909$	−5.48448e9	−0.242193
$910$	0	0
$911$	−2.84979e10	−1.24882	−0.624409	−	0.781098i	$-0.714660\pi$
$911$	−2.84979e10	−1.24882	−0.624409	+	0.781098i	$0.714660\pi$
$912$	4.91028e8	0.0214351
$913$	−2.67144e9	−0.116171
$914$	−2.88022e9	−0.124771
$915$	0	0
$916$	−1.28412e10	−0.552042
$917$	4.59460e9	0.196768
$918$	5.07129e9	0.216356
$919$	−9.46354e9	−0.402207	−0.201103	−	0.979570i	$-0.564453\pi$
$919$	−9.46354e9	−0.402207	−0.201103	+	0.979570i	$0.564453\pi$
$920$	0	0
$921$	7.22977e9	0.304941
$922$	1.06790e9	0.0448715
$923$	4.38581e10	1.83588
$924$	1.07365e10	0.447726
$925$	0	0
$926$	−2.34884e10	−0.972108
$927$	−1.02251e10	−0.421586
$928$	−6.37010e8	−0.0261655
$929$	1.56995e10	0.642438	0.321219	−	0.947005i	$-0.395907\pi$
$929$	1.56995e10	0.642438	0.321219	+	0.947005i	$0.395907\pi$
$930$	0	0
$931$	−1.97284e9	−0.0801249
$932$	1.06574e10	0.431217
$933$	5.25289e9	0.211745
$934$	−1.38221e10	−0.555087
$935$	0	0
$936$	4.77683e9	0.190403
$937$	−8.78711e9	−0.348946	−0.174473	−	0.984662i	$-0.555822\pi$
$937$	−8.78711e9	−0.348946	−0.174473	+	0.984662i	$0.555822\pi$
$938$	6.07017e9	0.240154
$939$	2.38087e10	0.938438
$940$	0	0
$941$	−1.58721e10	−0.620971	−0.310485	−	0.950578i	$-0.600492\pi$
$941$	−1.58721e10	−0.620971	−0.310485	+	0.950578i	$0.600492\pi$
$942$	2.37446e9	0.0925522
$943$	2.85900e10	1.11026
$944$	−7.55757e9	−0.292402
$945$	0	0
$946$	1.83733e10	0.705614
$947$	5.08496e9	0.194564	0.0972820	−	0.995257i	$-0.468985\pi$
$947$	5.08496e9	0.194564	0.0972820	+	0.995257i	$0.468985\pi$
$948$	5.42205e9	0.206697
$949$	5.98786e10	2.27426
$950$	0	0
$951$	1.31840e10	0.497068
$952$	−1.85671e10	−0.697454
$953$	−4.59917e10	−1.72129	−0.860645	−	0.509205i	$-0.829940\pi$
$953$	−4.59917e10	−1.72129	−0.860645	+	0.509205i	$0.829940\pi$
$954$	5.13728e9	0.191564
$955$	0	0
$956$	−7.53081e9	−0.278765
$957$	2.89629e9	0.106819
$958$	1.35384e10	0.497493
$959$	−2.63675e10	−0.965393
$960$	0	0
$961$	3.02065e10	1.09791
$962$	−7.96896e9	−0.288595
$963$	−1.16188e9	−0.0419247
$964$	−2.59175e10	−0.931802
$965$	0	0
$966$	2.32155e10	0.828623
$967$	−3.82819e10	−1.36145	−0.680723	−	0.732541i	$-0.738334\pi$
$967$	−3.82819e10	−1.36145	−0.680723	+	0.732541i	$0.738334\pi$
$968$	−5.61211e9	−0.198867
$969$	3.86086e9	0.136317
$970$	0	0
$971$	2.85329e10	1.00018	0.500090	−	0.865974i	$-0.333300\pi$
$971$	2.85329e10	1.00018	0.500090	+	0.865974i	$0.333300\pi$
$972$	−9.18330e8	−0.0320750
$973$	−1.53080e9	−0.0532749
$974$	1.17055e10	0.405914
$975$	0	0
$976$	−3.52960e9	−0.121521
$977$	−8.40644e9	−0.288391	−0.144195	−	0.989549i	$-0.546059\pi$
$977$	−8.40644e9	−0.288391	−0.144195	+	0.989549i	$0.546059\pi$
$978$	−1.94229e10	−0.663938
$979$	−3.45356e10	−1.17632
$980$	0	0
$981$	−1.21108e10	−0.409573
$982$	2.29103e10	0.772041
$983$	5.00100e10	1.67927	0.839633	−	0.543154i	$-0.182770\pi$
$983$	5.00100e10	1.67927	0.839633	+	0.543154i	$0.182770\pi$
$984$	4.14059e9	0.138541
$985$	0	0
$986$	−5.00868e9	−0.166400
$987$	−9.81912e9	−0.325059
$988$	3.63668e9	0.119965
$989$	3.97283e10	1.30591
$990$	0	0
$991$	−1.17819e10	−0.384555	−0.192277	−	0.981341i	$-0.561587\pi$
$991$	−1.17819e10	−0.384555	−0.192277	+	0.981341i	$0.561587\pi$
$992$	7.87245e9	0.256047
$993$	1.91214e10	0.619722
$994$	3.08699e10	0.996973
$995$	0	0
$996$	−8.36580e8	−0.0268287
$997$	−2.36572e10	−0.756014	−0.378007	−	0.925803i	$-0.623391\pi$
$997$	−2.36572e10	−0.756014	−0.378007	+	0.925803i	$0.623391\pi$
$998$	9.37423e9	0.298524
$999$	1.53201e9	0.0486162

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.8.a.d.1.1		1
3.2	odd	2		450.8.a.y.1.1		1
5.2	odd	4		30.8.c.a.19.1	✓	2
5.3	odd	4		30.8.c.a.19.2	yes	2
5.4	even	2		150.8.a.m.1.1		1
15.2	even	4		90.8.c.a.19.2		2
15.8	even	4		90.8.c.a.19.1		2
15.14	odd	2		450.8.a.b.1.1		1
20.3	even	4		240.8.f.a.49.2		2
20.7	even	4		240.8.f.a.49.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.8.c.a.19.1	✓	2	5.2	odd	4
30.8.c.a.19.2	yes	2	5.3	odd	4
90.8.c.a.19.1		2	15.8	even	4
90.8.c.a.19.2		2	15.2	even	4
150.8.a.d.1.1		1	1.1	even	1	trivial
150.8.a.m.1.1		1	5.4	even	2
240.8.f.a.49.1		2	20.7	even	4
240.8.f.a.49.2		2	20.3	even	4
450.8.a.b.1.1		1	15.14	odd	2
450.8.a.y.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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	150.8.a.d.1.1

Level	$150$
Weight	$8$
Character	150.1
Self dual	yes
Analytic conductor	$46.858$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$