LMFDB - Embedded newform 75.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,2,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, 2, names="a")

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

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
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:	$0.598878015160$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -2.00000 q^{6} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -2.00000 q^{6} +3.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} +2.00000 q^{12} -1.00000 q^{13} -6.00000 q^{14} -4.00000 q^{16} -2.00000 q^{17} -2.00000 q^{18} -5.00000 q^{19} +3.00000 q^{21} -4.00000 q^{22} -6.00000 q^{23} +2.00000 q^{26} +1.00000 q^{27} +6.00000 q^{28} +10.0000 q^{29} -3.00000 q^{31} +8.00000 q^{32} +2.00000 q^{33} +4.00000 q^{34} +2.00000 q^{36} -2.00000 q^{37} +10.0000 q^{38} -1.00000 q^{39} -8.00000 q^{41} -6.00000 q^{42} -1.00000 q^{43} +4.00000 q^{44} +12.0000 q^{46} -2.00000 q^{47} -4.00000 q^{48} +2.00000 q^{49} -2.00000 q^{51} -2.00000 q^{52} +4.00000 q^{53} -2.00000 q^{54} -5.00000 q^{57} -20.0000 q^{58} -10.0000 q^{59} +7.00000 q^{61} +6.00000 q^{62} +3.00000 q^{63} -8.00000 q^{64} -4.00000 q^{66} +3.00000 q^{67} -4.00000 q^{68} -6.00000 q^{69} -8.00000 q^{71} +14.0000 q^{73} +4.00000 q^{74} -10.0000 q^{76} +6.00000 q^{77} +2.00000 q^{78} +1.00000 q^{81} +16.0000 q^{82} -6.00000 q^{83} +6.00000 q^{84} +2.00000 q^{86} +10.0000 q^{87} -3.00000 q^{91} -12.0000 q^{92} -3.00000 q^{93} +4.00000 q^{94} +8.00000 q^{96} -17.0000 q^{97} -4.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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 $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.00000	1.00000
$5$	0	0
$5$	0	0
$6$	−2.00000	−0.816497
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	2.00000	0.577350
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	−6.00000	−1.60357
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−2.00000	−0.471405
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	−4.00000	−0.852803
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	0	0
$26$	2.00000	0.392232
$27$	1.00000	0.192450
$28$	6.00000	1.13389
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	8.00000	1.41421
$33$	2.00000	0.348155
$34$	4.00000	0.685994
$35$	0	0
$36$	2.00000	0.333333
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	10.0000	1.62221
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	−6.00000	−0.925820
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	12.0000	1.76930
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	−4.00000	−0.577350
$49$	2.00000	0.285714
$50$	0	0
$51$	−2.00000	−0.280056
$52$	−2.00000	−0.277350
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	−2.00000	−0.272166
$55$	0	0
$56$	0	0
$57$	−5.00000	−0.662266
$58$	−20.0000	−2.62613
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	6.00000	0.762001
$63$	3.00000	0.377964
$64$	−8.00000	−1.00000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	3.00000	0.366508	0.183254	−	0.983066i	$-0.441337\pi$
$67$	3.00000	0.366508	0.183254	+	0.983066i	$0.441337\pi$
$68$	−4.00000	−0.485071
$69$	−6.00000	−0.722315
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	−10.0000	−1.14708
$77$	6.00000	0.683763
$78$	2.00000	0.226455
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	16.0000	1.76690
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	6.00000	0.654654
$85$	0	0
$86$	2.00000	0.215666
$87$	10.0000	1.07211
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	−12.0000	−1.25109
$93$	−3.00000	−0.311086
$94$	4.00000	0.412568
$95$	0	0
$96$	8.00000	0.816497
$97$	−17.0000	−1.72609	−0.863044	−	0.505128i	$-0.831445\pi$
$97$	−17.0000	−1.72609	−0.863044	+	0.505128i	$0.831445\pi$
$98$	−4.00000	−0.404061
$99$	2.00000	0.201008
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	4.00000	0.396059
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0	0
$106$	−8.00000	−0.777029
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	2.00000	0.192450
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	−12.0000	−1.13389
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	10.0000	0.936586
$115$	0	0
$116$	20.0000	1.85695
$117$	−1.00000	−0.0924500
$118$	20.0000	1.84115
$119$	−6.00000	−0.550019
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−14.0000	−1.26750
$123$	−8.00000	−0.721336
$124$	−6.00000	−0.538816
$125$	0	0
$126$	−6.00000	−0.534522
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	4.00000	0.348155
$133$	−15.0000	−1.30066
$134$	−6.00000	−0.518321
$135$	0	0
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	12.0000	1.02151
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	16.0000	1.34269
$143$	−2.00000	−0.167248
$144$	−4.00000	−0.333333
$145$	0	0
$146$	−28.0000	−2.31730
$147$	2.00000	0.164957
$148$	−4.00000	−0.328798
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	7.00000	0.569652	0.284826	−	0.958579i	$-0.408064\pi$
$151$	7.00000	0.569652	0.284826	+	0.958579i	$0.408064\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	−12.0000	−0.966988
$155$	0	0
$156$	−2.00000	−0.160128
$157$	13.0000	1.03751	0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000	1.03751	0.518756	+	0.854922i	$0.326395\pi$
$158$	0	0
$159$	4.00000	0.317221
$160$	0	0
$161$	−18.0000	−1.41860
$162$	−2.00000	−0.157135
$163$	−11.0000	−0.861586	−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000	−0.861586	−0.430793	+	0.902451i	$0.641766\pi$
$164$	−16.0000	−1.24939
$165$	0	0
$166$	12.0000	0.931381
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−5.00000	−0.382360
$172$	−2.00000	−0.152499
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−20.0000	−1.51620
$175$	0	0
$176$	−8.00000	−0.603023
$177$	−10.0000	−0.751646
$178$	0	0
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	0	0
$181$	17.0000	1.26360	0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000	1.26360	0.631800	+	0.775131i	$0.282316\pi$
$182$	6.00000	0.444750
$183$	7.00000	0.517455
$184$	0	0
$185$	0	0
$186$	6.00000	0.439941
$187$	−4.00000	−0.292509
$188$	−4.00000	−0.291730
$189$	3.00000	0.218218
$190$	0	0
$191$	22.0000	1.59186	0.795932	−	0.605386i	$-0.206981\pi$
$191$	22.0000	1.59186	0.795932	+	0.605386i	$0.206981\pi$
$192$	−8.00000	−0.577350
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	34.0000	2.44106
$195$	0	0
$196$	4.00000	0.285714
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	−4.00000	−0.284268
$199$	−5.00000	−0.354441	−0.177220	−	0.984171i	$-0.556711\pi$
$199$	−5.00000	−0.354441	−0.177220	+	0.984171i	$0.556711\pi$
$200$	0	0
$201$	3.00000	0.211604
$202$	−24.0000	−1.68863
$203$	30.0000	2.10559
$204$	−4.00000	−0.280056
$205$	0	0
$206$	−8.00000	−0.557386
$207$	−6.00000	−0.417029
$208$	4.00000	0.277350
$209$	−10.0000	−0.691714
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	8.00000	0.549442
$213$	−8.00000	−0.548151
$214$	24.0000	1.64061
$215$	0	0
$216$	0	0
$217$	−9.00000	−0.610960
$218$	−10.0000	−0.677285
$219$	14.0000	0.946032
$220$	0	0
$221$	2.00000	0.134535
$222$	4.00000	0.268462
$223$	19.0000	1.27233	0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000	1.27233	0.636167	+	0.771551i	$0.280519\pi$
$224$	24.0000	1.60357
$225$	0	0
$226$	−8.00000	−0.532152
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	−10.0000	−0.662266
$229$	−15.0000	−0.991228	−0.495614	−	0.868543i	$-0.665057\pi$
$229$	−15.0000	−0.991228	−0.495614	+	0.868543i	$0.665057\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−20.0000	−1.30189
$237$	0	0
$238$	12.0000	0.777844
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	−23.0000	−1.48156	−0.740780	−	0.671748i	$-0.765544\pi$
$241$	−23.0000	−1.48156	−0.740780	+	0.671748i	$0.765544\pi$
$242$	14.0000	0.899954
$243$	1.00000	0.0641500
$244$	14.0000	0.896258
$245$	0	0
$246$	16.0000	1.02012
$247$	5.00000	0.318142
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	6.00000	0.377964
$253$	−12.0000	−0.754434
$254$	−16.0000	−1.00393
$255$	0	0
$256$	16.0000	1.00000
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	2.00000	0.124515
$259$	−6.00000	−0.372822
$260$	0	0
$261$	10.0000	0.618984
$262$	−24.0000	−1.48272
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	0	0
$266$	30.0000	1.83942
$267$	0	0
$268$	6.00000	0.366508
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	8.00000	0.485071
$273$	−3.00000	−0.181568
$274$	−36.0000	−2.17484
$275$	0	0
$276$	−12.0000	−0.722315
$277$	3.00000	0.180253	0.0901263	−	0.995930i	$-0.471273\pi$
$277$	3.00000	0.180253	0.0901263	+	0.995930i	$0.471273\pi$
$278$	−40.0000	−2.39904
$279$	−3.00000	−0.179605
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	4.00000	0.238197
$283$	9.00000	0.534994	0.267497	−	0.963559i	$-0.413803\pi$
$283$	9.00000	0.534994	0.267497	+	0.963559i	$0.413803\pi$
$284$	−16.0000	−0.949425
$285$	0	0
$286$	4.00000	0.236525
$287$	−24.0000	−1.41668
$288$	8.00000	0.471405
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−17.0000	−0.996558
$292$	28.0000	1.63858
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	−4.00000	−0.233285
$295$	0	0
$296$	0	0
$297$	2.00000	0.116052
$298$	−20.0000	−1.15857
$299$	6.00000	0.346989
$300$	0	0
$301$	−3.00000	−0.172917
$302$	−14.0000	−0.805609
$303$	12.0000	0.689382
$304$	20.0000	1.14708
$305$	0	0
$306$	4.00000	0.228665
$307$	−7.00000	−0.399511	−0.199756	−	0.979846i	$-0.564015\pi$
$307$	−7.00000	−0.399511	−0.199756	+	0.979846i	$0.564015\pi$
$308$	12.0000	0.683763
$309$	4.00000	0.227552
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	−11.0000	−0.621757	−0.310878	−	0.950450i	$-0.600623\pi$
$313$	−11.0000	−0.621757	−0.310878	+	0.950450i	$0.600623\pi$
$314$	−26.0000	−1.46726
$315$	0	0
$316$	0	0
$317$	8.00000	0.449325	0.224662	−	0.974437i	$-0.427872\pi$
$317$	8.00000	0.449325	0.224662	+	0.974437i	$0.427872\pi$
$318$	−8.00000	−0.448618
$319$	20.0000	1.11979
$320$	0	0
$321$	−12.0000	−0.669775
$322$	36.0000	2.00620
$323$	10.0000	0.556415
$324$	2.00000	0.111111
$325$	0	0
$326$	22.0000	1.21847
$327$	5.00000	0.276501
$328$	0	0
$329$	−6.00000	−0.330791
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	−12.0000	−0.658586
$333$	−2.00000	−0.109599
$334$	24.0000	1.31322
$335$	0	0
$336$	−12.0000	−0.654654
$337$	23.0000	1.25289	0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000	1.25289	0.626445	+	0.779466i	$0.284509\pi$
$338$	24.0000	1.30543
$339$	4.00000	0.217250
$340$	0	0
$341$	−6.00000	−0.324918
$342$	10.0000	0.540738
$343$	−15.0000	−0.809924
$344$	0	0
$345$	0	0
$346$	12.0000	0.645124
$347$	−2.00000	−0.107366	−0.0536828	−	0.998558i	$-0.517096\pi$
$347$	−2.00000	−0.107366	−0.0536828	+	0.998558i	$0.517096\pi$
$348$	20.0000	1.07211
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	16.0000	0.852803
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	20.0000	1.06299
$355$	0	0
$356$	0	0
$357$	−6.00000	−0.317554
$358$	20.0000	1.05703
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	−34.0000	−1.78700
$363$	−7.00000	−0.367405
$364$	−6.00000	−0.314485
$365$	0	0
$366$	−14.0000	−0.731792
$367$	−27.0000	−1.40939	−0.704694	−	0.709511i	$-0.748916\pi$
$367$	−27.0000	−1.40939	−0.704694	+	0.709511i	$0.748916\pi$
$368$	24.0000	1.25109
$369$	−8.00000	−0.416463
$370$	0	0
$371$	12.0000	0.623009
$372$	−6.00000	−0.311086
$373$	29.0000	1.50156	0.750782	−	0.660551i	$-0.229677\pi$
$373$	29.0000	1.50156	0.750782	+	0.660551i	$0.229677\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	0	0
$377$	−10.0000	−0.515026
$378$	−6.00000	−0.308607
$379$	25.0000	1.28416	0.642082	−	0.766636i	$-0.278071\pi$
$379$	25.0000	1.28416	0.642082	+	0.766636i	$0.278071\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	−44.0000	−2.25124
$383$	−36.0000	−1.83951	−0.919757	−	0.392488i	$-0.871614\pi$
$383$	−36.0000	−1.83951	−0.919757	+	0.392488i	$0.871614\pi$
$384$	0	0
$385$	0	0
$386$	22.0000	1.11977
$387$	−1.00000	−0.0508329
$388$	−34.0000	−1.72609
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	12.0000	0.605320
$394$	−36.0000	−1.81365
$395$	0	0
$396$	4.00000	0.201008
$397$	−7.00000	−0.351320	−0.175660	−	0.984451i	$-0.556206\pi$
$397$	−7.00000	−0.351320	−0.175660	+	0.984451i	$0.556206\pi$
$398$	10.0000	0.501255
$399$	−15.0000	−0.750939
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	−6.00000	−0.299253
$403$	3.00000	0.149441
$404$	24.0000	1.19404
$405$	0	0
$406$	−60.0000	−2.97775
$407$	−4.00000	−0.198273
$408$	0	0
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	8.00000	0.394132
$413$	−30.0000	−1.47620
$414$	12.0000	0.589768
$415$	0	0
$416$	−8.00000	−0.392232
$417$	20.0000	0.979404
$418$	20.0000	0.978232
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	26.0000	1.26566
$423$	−2.00000	−0.0972433
$424$	0	0
$425$	0	0
$426$	16.0000	0.775203
$427$	21.0000	1.01626
$428$	−24.0000	−1.16008
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	−4.00000	−0.192450
$433$	29.0000	1.39365	0.696826	−	0.717241i	$-0.254595\pi$
$433$	29.0000	1.39365	0.696826	+	0.717241i	$0.254595\pi$
$434$	18.0000	0.864028
$435$	0	0
$436$	10.0000	0.478913
$437$	30.0000	1.43509
$438$	−28.0000	−1.33789
$439$	−35.0000	−1.67046	−0.835229	−	0.549902i	$-0.814665\pi$
$439$	−35.0000	−1.67046	−0.835229	+	0.549902i	$0.814665\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	−4.00000	−0.190261
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	−38.0000	−1.79935
$447$	10.0000	0.472984
$448$	−24.0000	−1.13389
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	−16.0000	−0.753411
$452$	8.00000	0.376288
$453$	7.00000	0.328889
$454$	−16.0000	−0.750917
$455$	0	0
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	30.0000	1.40181
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	−12.0000	−0.558291
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	−40.0000	−1.85695
$465$	0	0
$466$	−48.0000	−2.22356
$467$	38.0000	1.75843	0.879215	−	0.476425i	$-0.158068\pi$
$467$	38.0000	1.75843	0.879215	+	0.476425i	$0.158068\pi$
$468$	−2.00000	−0.0924500
$469$	9.00000	0.415581
$470$	0	0
$471$	13.0000	0.599008
$472$	0	0
$473$	−2.00000	−0.0919601
$474$	0	0
$475$	0	0
$476$	−12.0000	−0.550019
$477$	4.00000	0.183147
$478$	−40.0000	−1.82956
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	46.0000	2.09524
$483$	−18.0000	−0.819028
$484$	−14.0000	−0.636364
$485$	0	0
$486$	−2.00000	−0.0907218
$487$	13.0000	0.589086	0.294543	−	0.955638i	$-0.404833\pi$
$487$	13.0000	0.589086	0.294543	+	0.955638i	$0.404833\pi$
$488$	0	0
$489$	−11.0000	−0.497437
$490$	0	0
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	−16.0000	−0.721336
$493$	−20.0000	−0.900755
$494$	−10.0000	−0.449921
$495$	0	0
$496$	12.0000	0.538816
$497$	−24.0000	−1.07655
$498$	12.0000	0.537733
$499$	−5.00000	−0.223831	−0.111915	−	0.993718i	$-0.535699\pi$
$499$	−5.00000	−0.223831	−0.111915	+	0.993718i	$0.535699\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	−24.0000	−1.07117
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	0	0
$506$	24.0000	1.06693
$507$	−12.0000	−0.532939
$508$	16.0000	0.709885
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	0	0
$511$	42.0000	1.85797
$512$	−32.0000	−1.41421
$513$	−5.00000	−0.220755
$514$	24.0000	1.05859
$515$	0	0
$516$	−2.00000	−0.0880451
$517$	−4.00000	−0.175920
$518$	12.0000	0.527250
$519$	−6.00000	−0.263371
$520$	0	0
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	−20.0000	−0.875376
$523$	−31.0000	−1.35554	−0.677768	−	0.735276i	$-0.737052\pi$
$523$	−31.0000	−1.35554	−0.677768	+	0.735276i	$0.737052\pi$
$524$	24.0000	1.04844
$525$	0	0
$526$	32.0000	1.39527
$527$	6.00000	0.261364
$528$	−8.00000	−0.348155
$529$	13.0000	0.565217
$530$	0	0
$531$	−10.0000	−0.433963
$532$	−30.0000	−1.30066
$533$	8.00000	0.346518
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−10.0000	−0.431532
$538$	20.0000	0.862261
$539$	4.00000	0.172292
$540$	0	0
$541$	−3.00000	−0.128980	−0.0644900	−	0.997918i	$-0.520542\pi$
$541$	−3.00000	−0.128980	−0.0644900	+	0.997918i	$0.520542\pi$
$542$	16.0000	0.687259
$543$	17.0000	0.729540
$544$	−16.0000	−0.685994
$545$	0	0
$546$	6.00000	0.256776
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	36.0000	1.53784
$549$	7.00000	0.298753
$550$	0	0
$551$	−50.0000	−2.13007
$552$	0	0
$553$	0	0
$554$	−6.00000	−0.254916
$555$	0	0
$556$	40.0000	1.69638
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	6.00000	0.254000
$559$	1.00000	0.0422955
$560$	0	0
$561$	−4.00000	−0.168880
$562$	36.0000	1.51857
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	−4.00000	−0.168430
$565$	0	0
$566$	−18.0000	−0.756596
$567$	3.00000	0.125988
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	−13.0000	−0.544033	−0.272017	−	0.962293i	$-0.587691\pi$
$571$	−13.0000	−0.544033	−0.272017	+	0.962293i	$0.587691\pi$
$572$	−4.00000	−0.167248
$573$	22.0000	0.919063
$574$	48.0000	2.00348
$575$	0	0
$576$	−8.00000	−0.333333
$577$	13.0000	0.541197	0.270599	−	0.962692i	$-0.412778\pi$
$577$	13.0000	0.541197	0.270599	+	0.962692i	$0.412778\pi$
$578$	26.0000	1.08146
$579$	−11.0000	−0.457144
$580$	0	0
$581$	−18.0000	−0.746766
$582$	34.0000	1.40935
$583$	8.00000	0.331326
$584$	0	0
$585$	0	0
$586$	12.0000	0.495715
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	4.00000	0.164957
$589$	15.0000	0.618064
$590$	0	0
$591$	18.0000	0.740421
$592$	8.00000	0.328798
$593$	−16.0000	−0.657041	−0.328521	−	0.944497i	$-0.606550\pi$
$593$	−16.0000	−0.657041	−0.328521	+	0.944497i	$0.606550\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	20.0000	0.819232
$597$	−5.00000	−0.204636
$598$	−12.0000	−0.490716
$599$	−20.0000	−0.817178	−0.408589	−	0.912719i	$-0.633979\pi$
$599$	−20.0000	−0.817178	−0.408589	+	0.912719i	$0.633979\pi$
$600$	0	0
$601$	−13.0000	−0.530281	−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000	−0.530281	−0.265141	+	0.964210i	$0.585418\pi$
$602$	6.00000	0.244542
$603$	3.00000	0.122169
$604$	14.0000	0.569652
$605$	0	0
$606$	−24.0000	−0.974933
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	−40.0000	−1.62221
$609$	30.0000	1.21566
$610$	0	0
$611$	2.00000	0.0809113
$612$	−4.00000	−0.161690
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	14.0000	0.564994
$615$	0	0
$616$	0	0
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	−8.00000	−0.321807
$619$	−25.0000	−1.00483	−0.502417	−	0.864625i	$-0.667556\pi$
$619$	−25.0000	−1.00483	−0.502417	+	0.864625i	$0.667556\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	36.0000	1.44347
$623$	0	0
$624$	4.00000	0.160128
$625$	0	0
$626$	22.0000	0.879297
$627$	−10.0000	−0.399362
$628$	26.0000	1.03751
$629$	4.00000	0.159490
$630$	0	0
$631$	−23.0000	−0.915616	−0.457808	−	0.889051i	$-0.651365\pi$
$631$	−23.0000	−0.915616	−0.457808	+	0.889051i	$0.651365\pi$
$632$	0	0
$633$	−13.0000	−0.516704
$634$	−16.0000	−0.635441
$635$	0	0
$636$	8.00000	0.317221
$637$	−2.00000	−0.0792429
$638$	−40.0000	−1.58362
$639$	−8.00000	−0.316475
$640$	0	0
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	24.0000	0.947204
$643$	−36.0000	−1.41970	−0.709851	−	0.704352i	$-0.751238\pi$
$643$	−36.0000	−1.41970	−0.709851	+	0.704352i	$0.751238\pi$
$644$	−36.0000	−1.41860
$645$	0	0
$646$	−20.0000	−0.786889
$647$	28.0000	1.10079	0.550397	−	0.834903i	$-0.314476\pi$
$647$	28.0000	1.10079	0.550397	+	0.834903i	$0.314476\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	0	0
$651$	−9.00000	−0.352738
$652$	−22.0000	−0.861586
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	−10.0000	−0.391031
$655$	0	0
$656$	32.0000	1.24939
$657$	14.0000	0.546192
$658$	12.0000	0.467809
$659$	−40.0000	−1.55818	−0.779089	−	0.626913i	$-0.784318\pi$
$659$	−40.0000	−1.55818	−0.779089	+	0.626913i	$0.784318\pi$
$660$	0	0
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	−24.0000	−0.932786
$663$	2.00000	0.0776736
$664$	0	0
$665$	0	0
$666$	4.00000	0.154997
$667$	−60.0000	−2.32321
$668$	−24.0000	−0.928588
$669$	19.0000	0.734582
$670$	0	0
$671$	14.0000	0.540464
$672$	24.0000	0.925820
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	−46.0000	−1.77185
$675$	0	0
$676$	−24.0000	−0.923077
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	−8.00000	−0.307238
$679$	−51.0000	−1.95720
$680$	0	0
$681$	8.00000	0.306561
$682$	12.0000	0.459504
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	−10.0000	−0.382360
$685$	0	0
$686$	30.0000	1.14541
$687$	−15.0000	−0.572286
$688$	4.00000	0.152499
$689$	−4.00000	−0.152388
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	−12.0000	−0.456172
$693$	6.00000	0.227921
$694$	4.00000	0.151838
$695$	0	0
$696$	0	0
$697$	16.0000	0.606043
$698$	−20.0000	−0.757011
$699$	24.0000	0.907763
$700$	0	0
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	2.00000	0.0754851
$703$	10.0000	0.377157
$704$	−16.0000	−0.603023
$705$	0	0
$706$	12.0000	0.451626
$707$	36.0000	1.35392
$708$	−20.0000	−0.751646
$709$	25.0000	0.938895	0.469447	−	0.882960i	$-0.344453\pi$
$709$	25.0000	0.938895	0.469447	+	0.882960i	$0.344453\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	18.0000	0.674105
$714$	12.0000	0.449089
$715$	0	0
$716$	−20.0000	−0.747435
$717$	20.0000	0.746914
$718$	0	0
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	−12.0000	−0.446594
$723$	−23.0000	−0.855379
$724$	34.0000	1.26360
$725$	0	0
$726$	14.0000	0.519589
$727$	43.0000	1.59478	0.797391	−	0.603463i	$-0.206213\pi$
$727$	43.0000	1.59478	0.797391	+	0.603463i	$0.206213\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.00000	0.0739727
$732$	14.0000	0.517455
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	54.0000	1.99318
$735$	0	0
$736$	−48.0000	−1.76930
$737$	6.00000	0.221013
$738$	16.0000	0.588968
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	5.00000	0.183680
$742$	−24.0000	−0.881068
$743$	4.00000	0.146746	0.0733729	−	0.997305i	$-0.476624\pi$
$743$	4.00000	0.146746	0.0733729	+	0.997305i	$0.476624\pi$
$744$	0	0
$745$	0	0
$746$	−58.0000	−2.12353
$747$	−6.00000	−0.219529
$748$	−8.00000	−0.292509
$749$	−36.0000	−1.31541
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	8.00000	0.291730
$753$	12.0000	0.437304
$754$	20.0000	0.728357
$755$	0	0
$756$	6.00000	0.218218
$757$	23.0000	0.835949	0.417975	−	0.908459i	$-0.362740\pi$
$757$	23.0000	0.835949	0.417975	+	0.908459i	$0.362740\pi$
$758$	−50.0000	−1.81608
$759$	−12.0000	−0.435572
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	−16.0000	−0.579619
$763$	15.0000	0.543036
$764$	44.0000	1.59186
$765$	0	0
$766$	72.0000	2.60147
$767$	10.0000	0.361079
$768$	16.0000	0.577350
$769$	35.0000	1.26213	0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000	1.26213	0.631066	+	0.775729i	$0.282618\pi$
$770$	0	0
$771$	−12.0000	−0.432169
$772$	−22.0000	−0.791797
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	2.00000	0.0718885
$775$	0	0
$776$	0	0
$777$	−6.00000	−0.215249
$778$	0	0
$779$	40.0000	1.43315
$780$	0	0
$781$	−16.0000	−0.572525
$782$	−24.0000	−0.858238
$783$	10.0000	0.357371
$784$	−8.00000	−0.285714
$785$	0	0
$786$	−24.0000	−0.856052
$787$	−7.00000	−0.249523	−0.124762	−	0.992187i	$-0.539817\pi$
$787$	−7.00000	−0.249523	−0.124762	+	0.992187i	$0.539817\pi$
$788$	36.0000	1.28245
$789$	−16.0000	−0.569615
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	−7.00000	−0.248577
$794$	14.0000	0.496841
$795$	0	0
$796$	−10.0000	−0.354441
$797$	−52.0000	−1.84193	−0.920967	−	0.389640i	$-0.872599\pi$
$797$	−52.0000	−1.84193	−0.920967	+	0.389640i	$0.872599\pi$
$798$	30.0000	1.06199
$799$	4.00000	0.141510
$800$	0	0
$801$	0	0
$802$	−24.0000	−0.847469
$803$	28.0000	0.988099
$804$	6.00000	0.211604
$805$	0	0
$806$	−6.00000	−0.211341
$807$	−10.0000	−0.352017
$808$	0	0
$809$	−20.0000	−0.703163	−0.351581	−	0.936157i	$-0.614356\pi$
$809$	−20.0000	−0.703163	−0.351581	+	0.936157i	$0.614356\pi$
$810$	0	0
$811$	27.0000	0.948098	0.474049	−	0.880498i	$-0.342792\pi$
$811$	27.0000	0.948098	0.474049	+	0.880498i	$0.342792\pi$
$812$	60.0000	2.10559
$813$	−8.00000	−0.280572
$814$	8.00000	0.280400
$815$	0	0
$816$	8.00000	0.280056
$817$	5.00000	0.174928
$818$	−10.0000	−0.349642
$819$	−3.00000	−0.104828
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	−36.0000	−1.25564
$823$	−41.0000	−1.42917	−0.714585	−	0.699549i	$-0.753384\pi$
$823$	−41.0000	−1.42917	−0.714585	+	0.699549i	$0.753384\pi$
$824$	0	0
$825$	0	0
$826$	60.0000	2.08767
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	−12.0000	−0.417029
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	0	0
$831$	3.00000	0.104069
$832$	8.00000	0.277350
$833$	−4.00000	−0.138592
$834$	−40.0000	−1.38509
$835$	0	0
$836$	−20.0000	−0.691714
$837$	−3.00000	−0.103695
$838$	40.0000	1.38178
$839$	10.0000	0.345238	0.172619	−	0.984989i	$-0.444777\pi$
$839$	10.0000	0.345238	0.172619	+	0.984989i	$0.444777\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	−44.0000	−1.51634
$843$	−18.0000	−0.619953
$844$	−26.0000	−0.894957
$845$	0	0
$846$	4.00000	0.137523
$847$	−21.0000	−0.721569
$848$	−16.0000	−0.549442
$849$	9.00000	0.308879
$850$	0	0
$851$	12.0000	0.411355
$852$	−16.0000	−0.548151
$853$	−51.0000	−1.74621	−0.873103	−	0.487535i	$-0.837896\pi$
$853$	−51.0000	−1.74621	−0.873103	+	0.487535i	$0.837896\pi$
$854$	−42.0000	−1.43721
$855$	0	0
$856$	0	0
$857$	28.0000	0.956462	0.478231	−	0.878234i	$-0.341278\pi$
$857$	28.0000	0.956462	0.478231	+	0.878234i	$0.341278\pi$
$858$	4.00000	0.136558
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	0	0
$861$	−24.0000	−0.817918
$862$	36.0000	1.22616
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	8.00000	0.272166
$865$	0	0
$866$	−58.0000	−1.97092
$867$	−13.0000	−0.441503
$868$	−18.0000	−0.610960
$869$	0	0
$870$	0	0
$871$	−3.00000	−0.101651
$872$	0	0
$873$	−17.0000	−0.575363
$874$	−60.0000	−2.02953
$875$	0	0
$876$	28.0000	0.946032
$877$	−27.0000	−0.911725	−0.455863	−	0.890050i	$-0.650669\pi$
$877$	−27.0000	−0.911725	−0.455863	+	0.890050i	$0.650669\pi$
$878$	70.0000	2.36239
$879$	−6.00000	−0.202375
$880$	0	0
$881$	32.0000	1.07811	0.539054	−	0.842271i	$-0.318782\pi$
$881$	32.0000	1.07811	0.539054	+	0.842271i	$0.318782\pi$
$882$	−4.00000	−0.134687
$883$	−41.0000	−1.37976	−0.689880	−	0.723924i	$-0.742337\pi$
$883$	−41.0000	−1.37976	−0.689880	+	0.723924i	$0.742337\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	−48.0000	−1.61259
$887$	18.0000	0.604381	0.302190	−	0.953248i	$-0.402282\pi$
$887$	18.0000	0.604381	0.302190	+	0.953248i	$0.402282\pi$
$888$	0	0
$889$	24.0000	0.804934
$890$	0	0
$891$	2.00000	0.0670025
$892$	38.0000	1.27233
$893$	10.0000	0.334637
$894$	−20.0000	−0.668900
$895$	0	0
$896$	0	0
$897$	6.00000	0.200334
$898$	−40.0000	−1.33482
$899$	−30.0000	−1.00056
$900$	0	0
$901$	−8.00000	−0.266519
$902$	32.0000	1.06548
$903$	−3.00000	−0.0998337
$904$	0	0
$905$	0	0
$906$	−14.0000	−0.465119
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	16.0000	0.530979
$909$	12.0000	0.398015
$910$	0	0
$911$	−58.0000	−1.92163	−0.960813	−	0.277198i	$-0.910594\pi$
$911$	−58.0000	−1.92163	−0.960813	+	0.277198i	$0.910594\pi$
$912$	20.0000	0.662266
$913$	−12.0000	−0.397142
$914$	44.0000	1.45539
$915$	0	0
$916$	−30.0000	−0.991228
$917$	36.0000	1.18882
$918$	4.00000	0.132020
$919$	55.0000	1.81428	0.907141	−	0.420826i	$-0.138260\pi$
$919$	55.0000	1.81428	0.907141	+	0.420826i	$0.138260\pi$
$920$	0	0
$921$	−7.00000	−0.230658
$922$	−24.0000	−0.790398
$923$	8.00000	0.263323
$924$	12.0000	0.394771
$925$	0	0
$926$	−48.0000	−1.57738
$927$	4.00000	0.131377
$928$	80.0000	2.62613
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	0	0
$931$	−10.0000	−0.327737
$932$	48.0000	1.57229
$933$	−18.0000	−0.589294
$934$	−76.0000	−2.48680
$935$	0	0
$936$	0	0
$937$	33.0000	1.07806	0.539032	−	0.842286i	$-0.318790\pi$
$937$	33.0000	1.07806	0.539032	+	0.842286i	$0.318790\pi$
$938$	−18.0000	−0.587721
$939$	−11.0000	−0.358971
$940$	0	0
$941$	22.0000	0.717180	0.358590	−	0.933495i	$-0.383258\pi$
$941$	22.0000	0.717180	0.358590	+	0.933495i	$0.383258\pi$
$942$	−26.0000	−0.847126
$943$	48.0000	1.56310
$944$	40.0000	1.30189
$945$	0	0
$946$	4.00000	0.130051
$947$	18.0000	0.584921	0.292461	−	0.956278i	$-0.405526\pi$
$947$	18.0000	0.584921	0.292461	+	0.956278i	$0.405526\pi$
$948$	0	0
$949$	−14.0000	−0.454459
$950$	0	0
$951$	8.00000	0.259418
$952$	0	0
$953$	−56.0000	−1.81402	−0.907009	−	0.421111i	$-0.861640\pi$
$953$	−56.0000	−1.81402	−0.907009	+	0.421111i	$0.861640\pi$
$954$	−8.00000	−0.259010
$955$	0	0
$956$	40.0000	1.29369
$957$	20.0000	0.646508
$958$	−60.0000	−1.93851
$959$	54.0000	1.74375
$960$	0	0
$961$	−22.0000	−0.709677
$962$	−4.00000	−0.128965
$963$	−12.0000	−0.386695
$964$	−46.0000	−1.48156
$965$	0	0
$966$	36.0000	1.15828
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	0	0
$969$	10.0000	0.321246
$970$	0	0
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	2.00000	0.0641500
$973$	60.0000	1.92351
$974$	−26.0000	−0.833094
$975$	0	0
$976$	−28.0000	−0.896258
$977$	−2.00000	−0.0639857	−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000	−0.0639857	−0.0319928	+	0.999488i	$0.510185\pi$
$978$	22.0000	0.703482
$979$	0	0
$980$	0	0
$981$	5.00000	0.159638
$982$	16.0000	0.510581
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	0	0
$985$	0	0
$986$	40.0000	1.27386
$987$	−6.00000	−0.190982
$988$	10.0000	0.318142
$989$	6.00000	0.190789
$990$	0	0
$991$	17.0000	0.540023	0.270011	−	0.962857i	$-0.412973\pi$
$991$	17.0000	0.540023	0.270011	+	0.962857i	$0.412973\pi$
$992$	−24.0000	−0.762001
$993$	12.0000	0.380808
$994$	48.0000	1.52247
$995$	0	0
$996$	−12.0000	−0.380235
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	10.0000	0.316544
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.2.a.a.1.1	✓	1
3.2	odd	2		225.2.a.e.1.1		1
4.3	odd	2		1200.2.a.c.1.1		1
5.2	odd	4		75.2.b.a.49.1		2
5.3	odd	4		75.2.b.a.49.2		2
5.4	even	2		75.2.a.c.1.1	yes	1
7.6	odd	2		3675.2.a.b.1.1		1
8.3	odd	2		4800.2.a.br.1.1		1
8.5	even	2		4800.2.a.bb.1.1		1
11.10	odd	2		9075.2.a.s.1.1		1
12.11	even	2		3600.2.a.j.1.1		1
15.2	even	4		225.2.b.a.199.2		2
15.8	even	4		225.2.b.a.199.1		2
15.14	odd	2		225.2.a.a.1.1		1
20.3	even	4		1200.2.f.d.49.1		2
20.7	even	4		1200.2.f.d.49.2		2
20.19	odd	2		1200.2.a.p.1.1		1
35.34	odd	2		3675.2.a.q.1.1		1
40.3	even	4		4800.2.f.y.3649.2		2
40.13	odd	4		4800.2.f.l.3649.1		2
40.19	odd	2		4800.2.a.be.1.1		1
40.27	even	4		4800.2.f.y.3649.1		2
40.29	even	2		4800.2.a.bq.1.1		1
40.37	odd	4		4800.2.f.l.3649.2		2
55.54	odd	2		9075.2.a.a.1.1		1
60.23	odd	4		3600.2.f.p.2449.2		2
60.47	odd	4		3600.2.f.p.2449.1		2
60.59	even	2		3600.2.a.bk.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.a.a.1.1	✓	1	1.1	even	1	trivial
75.2.a.c.1.1	yes	1	5.4	even	2
75.2.b.a.49.1		2	5.2	odd	4
75.2.b.a.49.2		2	5.3	odd	4
225.2.a.a.1.1		1	15.14	odd	2
225.2.a.e.1.1		1	3.2	odd	2
225.2.b.a.199.1		2	15.8	even	4
225.2.b.a.199.2		2	15.2	even	4
1200.2.a.c.1.1		1	4.3	odd	2
1200.2.a.p.1.1		1	20.19	odd	2
1200.2.f.d.49.1		2	20.3	even	4
1200.2.f.d.49.2		2	20.7	even	4
3600.2.a.j.1.1		1	12.11	even	2
3600.2.a.bk.1.1		1	60.59	even	2
3600.2.f.p.2449.1		2	60.47	odd	4
3600.2.f.p.2449.2		2	60.23	odd	4
3675.2.a.b.1.1		1	7.6	odd	2
3675.2.a.q.1.1		1	35.34	odd	2
4800.2.a.bb.1.1		1	8.5	even	2
4800.2.a.be.1.1		1	40.19	odd	2
4800.2.a.bq.1.1		1	40.29	even	2
4800.2.a.br.1.1		1	8.3	odd	2
4800.2.f.l.3649.1		2	40.13	odd	4
4800.2.f.l.3649.2		2	40.37	odd	4
4800.2.f.y.3649.1		2	40.27	even	4
4800.2.f.y.3649.2		2	40.3	even	4
9075.2.a.a.1.1		1	55.54	odd	2
9075.2.a.s.1.1		1	11.10	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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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