LMFDB - Embedded newform 1575.2.d.l.1324.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(1324,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1575.1324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$12.5764383184$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.30599805184.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 13x^{4} + 27x^{2} + 81 $ x^8 + 3x^6 + 13x^4 + 27*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1324.1
Root			$1.34095 - 1.09629i$ of defining polynomial
Character	$\chi$	$=$	1575.1324
Dual form			1575.2.d.l.1324.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.68190i q^{2} -5.19258 q^{4} -1.00000i q^{7} +8.56218i q^{8} +O(q^{10})$	$q-2.68190i q^{2} -5.19258 q^{4} -1.00000i q^{7} +8.56218i q^{8} +3.19839 q^{11} -6.38516i q^{13} -2.68190 q^{14} +12.5777 q^{16} -5.36380i q^{17} +2.38516 q^{19} -8.57775i q^{22} -3.19839i q^{23} -17.1244 q^{26} +5.19258i q^{28} -2.16541 q^{29} +6.00000 q^{31} -16.6079i q^{32} -14.3852 q^{34} +3.00000i q^{37} -6.39677i q^{38} -10.7276 q^{41} +9.38516i q^{43} -16.6079 q^{44} -8.57775 q^{46} +11.7606i q^{47} -1.00000 q^{49} +33.1555i q^{52} -10.7276i q^{53} +8.56218 q^{56} +5.80742i q^{58} -11.7606 q^{59} -16.0914i q^{62} -19.3852 q^{64} -9.38516i q^{67} +27.8520i q^{68} -13.9260 q^{71} -4.38516i q^{73} +8.04570 q^{74} -12.3852 q^{76} -3.19839i q^{77} +5.38516 q^{79} +28.7703i q^{82} -4.33082i q^{83} +25.1701 q^{86} +27.3852i q^{88} +1.03297 q^{89} -6.38516 q^{91} +16.6079i q^{92} +31.5407 q^{94} -10.7703i q^{97} +2.68190i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 20 q^{4}+O(q^{10}) $ 8 * q - 20 * q^4	$ 8 q - 20 q^{4} + 36 q^{16} - 24 q^{19} + 48 q^{31} - 72 q^{34} - 4 q^{46} - 8 q^{49} - 112 q^{64} - 56 q^{76} - 8 q^{91} + 80 q^{94}+O(q^{100}) $ 8 * q - 20 * q^4 + 36 * q^16 - 24 * q^19 + 48 * q^31 - 72 * q^34 - 4 * q^46 - 8 * q^49 - 112 * q^64 - 56 * q^76 - 8 * q^91 + 80 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$1226$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	− 2.68190i	− 1.89639i	−0.317690	−	0.948194i	$-0.602907\pi$
$2$	− 2.68190i	− 1.89639i	0.317690	−	0.948194i	$-0.397093\pi$
$3$	0	0
$3$	0	0
$4$	−5.19258	−2.59629
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	8.56218i	3.02719i
$9$	0	0
$10$	0	0
$11$	3.19839	0.964350	0.482175	−	0.876075i	$-0.339847\pi$
$11$	3.19839	0.964350	0.482175	+	0.876075i	$0.339847\pi$
$12$	0	0
$13$	− 6.38516i	− 1.77093i	−0.464710	−	0.885463i	$-0.653841\pi$
$13$	− 6.38516i	− 1.77093i	0.464710	−	0.885463i	$-0.346159\pi$
$14$	−2.68190	−0.716768
$15$	0	0
$16$	12.5777	3.14444
$17$	− 5.36380i	− 1.30091i	−0.759544	−	0.650456i	$-0.774578\pi$
$17$	− 5.36380i	− 1.30091i	0.759544	−	0.650456i	$-0.225422\pi$
$18$	0	0
$19$	2.38516	0.547194	0.273597	−	0.961844i	$-0.411786\pi$
$19$	2.38516	0.547194	0.273597	+	0.961844i	$0.411786\pi$
$20$	0	0
$21$	0	0
$22$	− 8.57775i	− 1.82878i
$23$	− 3.19839i	− 0.666909i	−0.942766	−	0.333455i	$-0.891786\pi$
$23$	− 3.19839i	− 0.666909i	0.942766	−	0.333455i	$-0.108214\pi$
$24$	0	0
$25$	0	0
$26$	−17.1244	−3.35836
$27$	0	0
$28$	5.19258i	0.981306i
$29$	−2.16541	−0.402107	−0.201054	−	0.979580i	$-0.564437\pi$
$29$	−2.16541	−0.402107	−0.201054	+	0.979580i	$0.564437\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	− 16.6079i	− 2.93589i
$33$	0	0
$34$	−14.3852	−2.46704
$35$	0	0
$36$	0	0
$37$	3.00000i	0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$	3.00000i	0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$	− 6.39677i	− 1.03769i
$39$	0	0
$40$	0	0
$41$	−10.7276	−1.67537	−0.837685	−	0.546154i	$-0.816091\pi$
$41$	−10.7276	−1.67537	−0.837685	+	0.546154i	$0.816091\pi$
$42$	0	0
$43$	9.38516i	1.43122i	0.698498	+	0.715612i	$0.253852\pi$
$43$	9.38516i	1.43122i	−0.698498	+	0.715612i	$0.746148\pi$
$44$	−16.6079	−2.50373
$45$	0	0
$46$	−8.57775	−1.26472
$47$	11.7606i	1.71546i	0.514104	+	0.857728i	$0.328124\pi$
$47$	11.7606i	1.71546i	−0.514104	+	0.857728i	$0.671876\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	33.1555i	4.59784i
$53$	− 10.7276i	− 1.47355i	−0.676139	−	0.736774i	$-0.736348\pi$
$53$	− 10.7276i	− 1.47355i	0.676139	−	0.736774i	$-0.263652\pi$
$54$	0	0
$55$	0	0
$56$	8.56218	1.14417
$57$	0	0
$58$	5.80742i	0.762551i
$59$	−11.7606	−1.53110	−0.765548	−	0.643379i	$-0.777532\pi$
$59$	−11.7606	−1.53110	−0.765548	+	0.643379i	$0.777532\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	− 16.0914i	− 2.04361i
$63$	0	0
$64$	−19.3852	−2.42315
$65$	0	0
$66$	0	0
$67$	− 9.38516i	− 1.14658i	−0.819352	−	0.573290i	$-0.805667\pi$
$67$	− 9.38516i	− 1.14658i	0.819352	−	0.573290i	$-0.194333\pi$
$68$	27.8520i	3.37755i
$69$	0	0
$70$	0	0
$71$	−13.9260	−1.65271	−0.826355	−	0.563150i	$-0.809589\pi$
$71$	−13.9260	−1.65271	−0.826355	+	0.563150i	$0.809589\pi$
$72$	0	0
$73$	− 4.38516i	− 0.513245i	−0.966512	−	0.256622i	$-0.917390\pi$
$73$	− 4.38516i	− 0.513245i	0.966512	−	0.256622i	$-0.0826097\pi$
$74$	8.04570	0.935293
$75$	0	0
$76$	−12.3852	−1.42068
$77$	− 3.19839i	− 0.364490i
$78$	0	0
$79$	5.38516	0.605878	0.302939	−	0.953010i	$-0.402032\pi$
$79$	5.38516	0.605878	0.302939	+	0.953010i	$0.402032\pi$
$80$	0	0
$81$	0	0
$82$	28.7703i	3.17715i
$83$	− 4.33082i	− 0.475370i	−0.971342	−	0.237685i	$-0.923611\pi$
$83$	− 4.33082i	− 0.475370i	0.971342	−	0.237685i	$-0.0763886\pi$
$84$	0	0
$85$	0	0
$86$	25.1701	2.71416
$87$	0	0
$88$	27.3852i	2.91927i
$89$	1.03297	0.109495	0.0547475	−	0.998500i	$-0.482565\pi$
$89$	1.03297	0.109495	0.0547475	+	0.998500i	$0.482565\pi$
$90$	0	0
$91$	−6.38516	−0.669347
$92$	16.6079i	1.73149i
$93$	0	0
$94$	31.5407	3.25317
$95$	0	0
$96$	0	0
$97$	− 10.7703i	− 1.09356i	−0.837276	−	0.546781i	$-0.815853\pi$
$97$	− 10.7703i	− 1.09356i	0.837276	−	0.546781i	$-0.184147\pi$
$98$	2.68190i	0.270913i
$99$	0	0
$100$	0	0
$101$	5.36380	0.533718	0.266859	−	0.963736i	$-0.414014\pi$
$101$	5.36380	0.533718	0.266859	+	0.963736i	$0.414014\pi$
$102$	0	0
$103$	− 5.61484i	− 0.553246i	−0.960978	−	0.276623i	$-0.910785\pi$
$103$	− 5.61484i	− 0.553246i	0.960978	−	0.276623i	$-0.0892153\pi$
$104$	54.6710	5.36093
$105$	0	0
$106$	−28.7703	−2.79442
$107$	4.33082i	0.418677i	0.977843	+	0.209338i	$0.0671310\pi$
$107$	4.33082i	0.418677i	−0.977843	+	0.209338i	$0.932869\pi$
$108$	0	0
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	0	0
$111$	0	0
$112$	− 12.5777i	− 1.18849i
$113$	2.16541i	0.203705i	0.994800	+	0.101852i	$0.0324770\pi$
$113$	2.16541i	0.203705i	−0.994800	+	0.101852i	$0.967523\pi$
$114$	0	0
$115$	0	0
$116$	11.2441	1.04399
$117$	0	0
$118$	31.5407i	2.90355i
$119$	−5.36380	−0.491699
$120$	0	0
$121$	−0.770330	−0.0700300
$122$	0	0
$123$	0	0
$124$	−31.1555	−2.79785
$125$	0	0
$126$	0	0
$127$	− 4.61484i	− 0.409500i	−0.978814	−	0.204750i	$-0.934362\pi$
$127$	− 4.61484i	− 0.409500i	0.978814	−	0.204750i	$-0.0656382\pi$
$128$	18.7733i	1.65934i
$129$	0	0
$130$	0	0
$131$	5.36380	0.468637	0.234319	−	0.972160i	$-0.424714\pi$
$131$	5.36380	0.468637	0.234319	+	0.972160i	$0.424714\pi$
$132$	0	0
$133$	− 2.38516i	− 0.206820i
$134$	−25.1701	−2.17436
$135$	0	0
$136$	45.9258	3.93811
$137$	10.7276i	0.916520i	0.888818	+	0.458260i	$0.151527\pi$
$137$	10.7276i	0.916520i	−0.888818	+	0.458260i	$0.848473\pi$
$138$	0	0
$139$	4.38516	0.371945	0.185972	−	0.982555i	$-0.440456\pi$
$139$	4.38516	0.371945	0.185972	+	0.982555i	$0.440456\pi$
$140$	0	0
$141$	0	0
$142$	37.3481i	3.13418i
$143$	− 20.4222i	− 1.70779i
$144$	0	0
$145$	0	0
$146$	−11.7606	−0.973312
$147$	0	0
$148$	− 15.5777i	− 1.28048i
$149$	−12.8930	−1.05624	−0.528118	−	0.849171i	$-0.677102\pi$
$149$	−12.8930	−1.05624	−0.528118	+	0.849171i	$0.677102\pi$
$150$	0	0
$151$	11.3852	0.926512	0.463256	−	0.886225i	$-0.346681\pi$
$151$	11.3852	0.926512	0.463256	+	0.886225i	$0.346681\pi$
$152$	20.4222i	1.65646i
$153$	0	0
$154$	−8.57775	−0.691215
$155$	0	0
$156$	0	0
$157$	16.7703i	1.33842i	0.743074	+	0.669209i	$0.233367\pi$
$157$	16.7703i	1.33842i	−0.743074	+	0.669209i	$0.766633\pi$
$158$	− 14.4425i	− 1.14898i
$159$	0	0
$160$	0	0
$161$	−3.19839	−0.252068
$162$	0	0
$163$	− 16.7703i	− 1.31355i	−0.754085	−	0.656777i	$-0.771919\pi$
$163$	− 16.7703i	− 1.31355i	0.754085	−	0.656777i	$-0.228081\pi$
$164$	55.7039	4.34975
$165$	0	0
$166$	−11.6148	−0.901486
$167$	1.03297i	0.0799339i	0.999201	+	0.0399669i	$0.0127253\pi$
$167$	1.03297i	0.0799339i	−0.999201	+	0.0399669i	$0.987275\pi$
$168$	0	0
$169$	−27.7703	−2.13618
$170$	0	0
$171$	0	0
$172$	− 48.7332i	− 3.71587i
$173$	− 22.4882i	− 1.70974i	−0.518839	−	0.854872i	$-0.673636\pi$
$173$	− 22.4882i	− 1.70974i	0.518839	−	0.854872i	$-0.326364\pi$
$174$	0	0
$175$	0	0
$176$	40.2285	3.03234
$177$	0	0
$178$	− 2.77033i	− 0.207645i
$179$	4.33082	0.323701	0.161851	−	0.986815i	$-0.448254\pi$
$179$	4.33082	0.323701	0.161851	+	0.986815i	$0.448254\pi$
$180$	0	0
$181$	20.3852	1.51522	0.757609	−	0.652709i	$-0.226368\pi$
$181$	20.3852	1.51522	0.757609	+	0.652709i	$0.226368\pi$
$182$	17.1244i	1.26934i
$183$	0	0
$184$	27.3852	2.01886
$185$	0	0
$186$	0	0
$187$	− 17.1555i	− 1.25453i
$188$	− 61.0677i	− 4.45382i
$189$	0	0
$190$	0	0
$191$	−6.39677	−0.462854	−0.231427	−	0.972852i	$-0.574339\pi$
$191$	−6.39677	−0.462854	−0.231427	+	0.972852i	$0.574339\pi$
$192$	0	0
$193$	− 1.00000i	− 0.0719816i	−0.999352	−	0.0359908i	$-0.988541\pi$
$193$	− 1.00000i	− 0.0719816i	0.999352	−	0.0359908i	$-0.0114587\pi$
$194$	−28.8849	−2.07382
$195$	0	0
$196$	5.19258	0.370899
$197$	23.6206i	1.68290i	0.540336	+	0.841449i	$0.318297\pi$
$197$	23.6206i	1.68290i	−0.540336	+	0.841449i	$0.681703\pi$
$198$	0	0
$199$	−0.770330	−0.0546072	−0.0273036	−	0.999627i	$-0.508692\pi$
$199$	−0.770330	−0.0546072	−0.0273036	+	0.999627i	$0.508692\pi$
$200$	0	0
$201$	0	0
$202$	− 14.3852i	− 1.01214i
$203$	2.16541i	0.151982i
$204$	0	0
$205$	0	0
$206$	−15.0584	−1.04917
$207$	0	0
$208$	− 80.3110i	− 5.56857i
$209$	7.62868	0.527687
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	55.7039i	3.82576i
$213$	0	0
$214$	11.6148	0.793974
$215$	0	0
$216$	0	0
$217$	− 6.00000i	− 0.407307i
$218$	18.7733i	1.27149i
$219$	0	0
$220$	0	0
$221$	−34.2487	−2.30382
$222$	0	0
$223$	− 3.61484i	− 0.242067i	−0.992648	−	0.121034i	$-0.961379\pi$
$223$	− 3.61484i	− 0.242067i	0.992648	−	0.121034i	$-0.0386209\pi$
$224$	−16.6079	−1.10966
$225$	0	0
$226$	5.80742	0.386304
$227$	− 11.7606i	− 0.780576i	−0.920693	−	0.390288i	$-0.872375\pi$
$227$	− 11.7606i	− 0.780576i	0.920693	−	0.390288i	$-0.127625\pi$
$228$	0	0
$229$	15.5407	1.02696	0.513478	−	0.858103i	$-0.328357\pi$
$229$	15.5407	1.02696	0.513478	+	0.858103i	$0.328357\pi$
$230$	0	0
$231$	0	0
$232$	− 18.5407i	− 1.21725i
$233$	4.23136i	0.277206i	0.990348	+	0.138603i	$0.0442611\pi$
$233$	4.23136i	0.277206i	−0.990348	+	0.138603i	$0.955739\pi$
$234$	0	0
$235$	0	0
$236$	61.0677	3.97517
$237$	0	0
$238$	14.3852i	0.932452i
$239$	6.39677	0.413773	0.206886	−	0.978365i	$-0.433667\pi$
$239$	6.39677	0.413773	0.206886	+	0.978365i	$0.433667\pi$
$240$	0	0
$241$	18.3852	1.18429	0.592146	−	0.805830i	$-0.298281\pi$
$241$	18.3852	1.18429	0.592146	+	0.805830i	$0.298281\pi$
$242$	2.06595i	0.132804i
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 15.2297i	− 0.969041i
$248$	51.3731i	3.26220i
$249$	0	0
$250$	0	0
$251$	−3.29785	−0.208159	−0.104079	−	0.994569i	$-0.533190\pi$
$251$	−3.29785	−0.208159	−0.104079	+	0.994569i	$0.533190\pi$
$252$	0	0
$253$	− 10.2297i	− 0.643134i
$254$	−12.3765	−0.776572
$255$	0	0
$256$	11.5777	0.723609
$257$	− 17.1244i	− 1.06819i	−0.845425	−	0.534094i	$-0.820653\pi$
$257$	− 17.1244i	− 1.06819i	0.845425	−	0.534094i	$-0.179347\pi$
$258$	0	0
$259$	3.00000	0.186411
$260$	0	0
$261$	0	0
$262$	− 14.3852i	− 0.888718i
$263$	20.3228i	1.25315i	0.779359	+	0.626577i	$0.215545\pi$
$263$	20.3228i	1.25315i	−0.779359	+	0.626577i	$0.784455\pi$
$264$	0	0
$265$	0	0
$266$	−6.39677	−0.392211
$267$	0	0
$268$	48.7332i	2.97686i
$269$	−4.33082	−0.264055	−0.132028	−	0.991246i	$-0.542149\pi$
$269$	−4.33082	−0.264055	−0.132028	+	0.991246i	$0.542149\pi$
$270$	0	0
$271$	15.1555	0.920631	0.460315	−	0.887755i	$-0.347736\pi$
$271$	15.1555	0.920631	0.460315	+	0.887755i	$0.347736\pi$
$272$	− 67.4645i	− 4.09064i
$273$	0	0
$274$	28.7703	1.73808
$275$	0	0
$276$	0	0
$277$	23.5407i	1.41442i	0.707003	+	0.707211i	$0.250047\pi$
$277$	23.5407i	1.41442i	−0.707003	+	0.707211i	$0.749953\pi$
$278$	− 11.7606i	− 0.705352i
$279$	0	0
$280$	0	0
$281$	14.9590	0.892376	0.446188	−	0.894939i	$-0.352781\pi$
$281$	14.9590	0.892376	0.446188	+	0.894939i	$0.352781\pi$
$282$	0	0
$283$	− 14.0000i	− 0.832214i	−0.909316	−	0.416107i	$-0.863394\pi$
$283$	− 14.0000i	− 0.832214i	0.909316	−	0.416107i	$-0.136606\pi$
$284$	72.3118	4.29092
$285$	0	0
$286$	−54.7703	−3.23864
$287$	10.7276i	0.633230i
$288$	0	0
$289$	−11.7703	−0.692372
$290$	0	0
$291$	0	0
$292$	22.7703i	1.33253i
$293$	− 11.7606i	− 0.687060i	−0.939142	−	0.343530i	$-0.888377\pi$
$293$	− 11.7606i	− 0.687060i	0.939142	−	0.343530i	$-0.111623\pi$
$294$	0	0
$295$	0	0
$296$	−25.6866	−1.49300
$297$	0	0
$298$	34.5777i	2.00304i
$299$	−20.4222	−1.18105
$300$	0	0
$301$	9.38516	0.540952
$302$	− 30.5339i	− 1.75703i
$303$	0	0
$304$	30.0000	1.72062
$305$	0	0
$306$	0	0
$307$	13.1555i	0.750824i	0.926858	+	0.375412i	$0.122499\pi$
$307$	13.1555i	0.750824i	−0.926858	+	0.375412i	$0.877501\pi$
$308$	16.6079i	0.946322i
$309$	0	0
$310$	0	0
$311$	26.8190	1.52077	0.760383	−	0.649475i	$-0.225011\pi$
$311$	26.8190	1.52077	0.760383	+	0.649475i	$0.225011\pi$
$312$	0	0
$313$	23.9258i	1.35237i	0.736733	+	0.676184i	$0.236367\pi$
$313$	23.9258i	1.35237i	−0.736733	+	0.676184i	$0.763633\pi$
$314$	44.9763	2.53816
$315$	0	0
$316$	−27.9629	−1.57304
$317$	− 8.56218i	− 0.480900i	−0.970662	−	0.240450i	$-0.922705\pi$
$317$	− 8.56218i	− 0.480900i	0.970662	−	0.240450i	$-0.0772950\pi$
$318$	0	0
$319$	−6.92582	−0.387772
$320$	0	0
$321$	0	0
$322$	8.57775i	0.478019i
$323$	− 12.7935i	− 0.711852i
$324$	0	0
$325$	0	0
$326$	−44.9763	−2.49101
$327$	0	0
$328$	− 91.8516i	− 5.07166i
$329$	11.7606	0.648381
$330$	0	0
$331$	10.1555	0.558196	0.279098	−	0.960263i	$-0.409964\pi$
$331$	10.1555	0.558196	0.279098	+	0.960263i	$0.409964\pi$
$332$	22.4882i	1.23420i
$333$	0	0
$334$	2.77033	0.151586
$335$	0	0
$336$	0	0
$337$	− 10.7703i	− 0.586697i	−0.956006	−	0.293349i	$-0.905230\pi$
$337$	− 10.7703i	− 0.586697i	0.956006	−	0.293349i	$-0.0947697\pi$
$338$	74.4772i	4.05103i
$339$	0	0
$340$	0	0
$341$	19.1903	1.03921
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	−80.3575	−4.33259
$345$	0	0
$346$	−60.3110	−3.24234
$347$	− 24.6536i	− 1.32347i	−0.749736	−	0.661737i	$-0.769820\pi$
$347$	− 24.6536i	− 1.32347i	0.749736	−	0.661737i	$-0.230180\pi$
$348$	0	0
$349$	−2.38516	−0.127675	−0.0638375	−	0.997960i	$-0.520334\pi$
$349$	−2.38516	−0.127675	−0.0638375	+	0.997960i	$0.520334\pi$
$350$	0	0
$351$	0	0
$352$	− 53.1184i	− 2.83122i
$353$	− 17.1244i	− 0.911438i	−0.890124	−	0.455719i	$-0.849382\pi$
$353$	− 17.1244i	− 0.911438i	0.890124	−	0.455719i	$-0.150618\pi$
$354$	0	0
$355$	0	0
$356$	−5.36380	−0.284281
$357$	0	0
$358$	− 11.6148i	− 0.613863i
$359$	−18.2568	−0.963557	−0.481779	−	0.876293i	$-0.660009\pi$
$359$	−18.2568	−0.963557	−0.481779	+	0.876293i	$0.660009\pi$
$360$	0	0
$361$	−13.3110	−0.700578
$362$	− 54.6710i	− 2.87344i
$363$	0	0
$364$	33.1555	1.73782
$365$	0	0
$366$	0	0
$367$	− 4.00000i	− 0.208798i	−0.994535	−	0.104399i	$-0.966708\pi$
$367$	− 4.00000i	− 0.208798i	0.994535	−	0.104399i	$-0.0332919\pi$
$368$	− 40.2285i	− 2.09705i
$369$	0	0
$370$	0	0
$371$	−10.7276	−0.556949
$372$	0	0
$373$	25.0000i	1.29445i	0.762299	+	0.647225i	$0.224071\pi$
$373$	25.0000i	1.29445i	−0.762299	+	0.647225i	$0.775929\pi$
$374$	−46.0093	−2.37908
$375$	0	0
$376$	−100.696	−5.19301
$377$	13.8265i	0.712102i
$378$	0	0
$379$	11.3852	0.584817	0.292408	−	0.956294i	$-0.405543\pi$
$379$	11.3852	0.584817	0.292408	+	0.956294i	$0.405543\pi$
$380$	0	0
$381$	0	0
$382$	17.1555i	0.877751i
$383$	9.69462i	0.495372i	0.968840	+	0.247686i	$0.0796701\pi$
$383$	9.69462i	0.495372i	−0.968840	+	0.247686i	$0.920330\pi$
$384$	0	0
$385$	0	0
$386$	−2.68190	−0.136505
$387$	0	0
$388$	55.9258i	2.83920i
$389$	−10.6281	−0.538868	−0.269434	−	0.963019i	$-0.586837\pi$
$389$	−10.6281	−0.538868	−0.269434	+	0.963019i	$0.586837\pi$
$390$	0	0
$391$	−17.1555	−0.867591
$392$	− 8.56218i	− 0.432456i
$393$	0	0
$394$	63.3481	3.19143
$395$	0	0
$396$	0	0
$397$	− 1.22967i	− 0.0617154i	−0.999524	−	0.0308577i	$-0.990176\pi$
$397$	− 1.22967i	− 0.0617154i	0.999524	−	0.0308577i	$-0.00982387\pi$
$398$	2.06595i	0.103557i
$399$	0	0
$400$	0	0
$401$	12.8930	0.643846	0.321923	−	0.946766i	$-0.395671\pi$
$401$	12.8930	0.643846	0.321923	+	0.946766i	$0.395671\pi$
$402$	0	0
$403$	− 38.3110i	− 1.90841i
$404$	−27.8520	−1.38569
$405$	0	0
$406$	5.80742	0.288217
$407$	9.59516i	0.475614i
$408$	0	0
$409$	−22.3852	−1.10688	−0.553438	−	0.832891i	$-0.686684\pi$
$409$	−22.3852	−1.10688	−0.553438	+	0.832891i	$0.686684\pi$
$410$	0	0
$411$	0	0
$412$	29.1555i	1.43639i
$413$	11.7606i	0.578700i
$414$	0	0
$415$	0	0
$416$	−106.044	−5.19924
$417$	0	0
$418$	− 20.4593i	− 1.00070i
$419$	17.1244	0.836580	0.418290	−	0.908314i	$-0.362630\pi$
$419$	17.1244	0.836580	0.418290	+	0.908314i	$0.362630\pi$
$420$	0	0
$421$	25.3110	1.23358	0.616791	−	0.787127i	$-0.288432\pi$
$421$	25.3110	1.23358	0.616791	+	0.787127i	$0.288432\pi$
$422$	0	0
$423$	0	0
$424$	91.8516	4.46071
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 22.4882i	− 1.08701i
$429$	0	0
$430$	0	0
$431$	−25.7860	−1.24207	−0.621034	−	0.783783i	$-0.713287\pi$
$431$	−25.7860	−1.24207	−0.621034	+	0.783783i	$0.713287\pi$
$432$	0	0
$433$	9.54066i	0.458495i	0.973368	+	0.229247i	$0.0736265\pi$
$433$	9.54066i	0.458495i	−0.973368	+	0.229247i	$0.926374\pi$
$434$	−16.0914	−0.772412
$435$	0	0
$436$	36.3481	1.74076
$437$	− 7.62868i	− 0.364929i
$438$	0	0
$439$	−19.1555	−0.914242	−0.457121	−	0.889405i	$-0.651119\pi$
$439$	−19.1555	−0.914242	−0.457121	+	0.889405i	$0.651119\pi$
$440$	0	0
$441$	0	0
$442$	91.8516i	4.36894i
$443$	− 19.1903i	− 0.911759i	−0.890041	−	0.455880i	$-0.849325\pi$
$443$	− 19.1903i	− 0.911759i	0.890041	−	0.455880i	$-0.150675\pi$
$444$	0	0
$445$	0	0
$446$	−9.69462	−0.459054
$447$	0	0
$448$	19.3852i	0.915863i
$449$	−12.8930	−0.608459	−0.304229	−	0.952599i	$-0.598399\pi$
$449$	−12.8930	−0.608459	−0.304229	+	0.952599i	$0.598399\pi$
$450$	0	0
$451$	−34.3110	−1.61564
$452$	− 11.2441i	− 0.528877i
$453$	0	0
$454$	−31.5407	−1.48028
$455$	0	0
$456$	0	0
$457$	− 28.5407i	− 1.33508i	−0.744576	−	0.667538i	$-0.767348\pi$
$457$	− 28.5407i	− 1.33508i	0.744576	−	0.667538i	$-0.232652\pi$
$458$	− 41.6785i	− 1.94751i
$459$	0	0
$460$	0	0
$461$	−1.03297	−0.0481104	−0.0240552	−	0.999711i	$-0.507658\pi$
$461$	−1.03297	−0.0481104	−0.0240552	+	0.999711i	$0.507658\pi$
$462$	0	0
$463$	8.77033i	0.407592i	0.979013	+	0.203796i	$0.0653279\pi$
$463$	8.77033i	0.407592i	−0.979013	+	0.203796i	$0.934672\pi$
$464$	−27.2360	−1.26440
$465$	0	0
$466$	11.3481	0.525690
$467$	− 6.39677i	− 0.296007i	−0.988987	−	0.148004i	$-0.952715\pi$
$467$	− 6.39677i	− 0.296007i	0.988987	−	0.148004i	$-0.0472847\pi$
$468$	0	0
$469$	−9.38516	−0.433367
$470$	0	0
$471$	0	0
$472$	− 100.696i	− 4.63492i
$473$	30.0174i	1.38020i
$474$	0	0
$475$	0	0
$476$	27.8520	1.27659
$477$	0	0
$478$	− 17.1555i	− 0.784674i
$479$	33.2158	1.51767	0.758833	−	0.651285i	$-0.225770\pi$
$479$	33.2158	1.51767	0.758833	+	0.651285i	$0.225770\pi$
$480$	0	0
$481$	19.1555	0.873415
$482$	− 49.3072i	− 2.24588i
$483$	0	0
$484$	4.00000	0.181818
$485$	0	0
$486$	0	0
$487$	− 13.3852i	− 0.606540i	−0.952905	−	0.303270i	$-0.901922\pi$
$487$	− 13.3852i	− 0.606540i	0.952905	−	0.303270i	$-0.0980784\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−35.3812	−1.59673	−0.798365	−	0.602174i	$-0.794301\pi$
$491$	−35.3812	−1.59673	−0.798365	+	0.602174i	$0.794301\pi$
$492$	0	0
$493$	11.6148i	0.523106i
$494$	−40.8444	−1.83768
$495$	0	0
$496$	75.4665	3.38855
$497$	13.9260i	0.624666i
$498$	0	0
$499$	−29.5407	−1.32242	−0.661211	−	0.750200i	$-0.729957\pi$
$499$	−29.5407	−1.32242	−0.661211	+	0.750200i	$0.729957\pi$
$500$	0	0
$501$	0	0
$502$	8.84451i	0.394750i
$503$	40.6455i	1.81229i	0.422963	+	0.906147i	$0.360990\pi$
$503$	40.6455i	1.81229i	−0.422963	+	0.906147i	$0.639010\pi$
$504$	0	0
$505$	0	0
$506$	−27.4349	−1.21963
$507$	0	0
$508$	23.9629i	1.06318i
$509$	9.69462	0.429707	0.214853	−	0.976646i	$-0.431073\pi$
$509$	9.69462	0.429707	0.214853	+	0.976646i	$0.431073\pi$
$510$	0	0
$511$	−4.38516	−0.193988
$512$	6.49624i	0.287096i
$513$	0	0
$514$	−45.9258	−2.02570
$515$	0	0
$516$	0	0
$517$	37.6148i	1.65430i
$518$	− 8.04570i	− 0.353508i
$519$	0	0
$520$	0	0
$521$	36.5136	1.59969	0.799845	−	0.600206i	$-0.204915\pi$
$521$	36.5136	1.59969	0.799845	+	0.600206i	$0.204915\pi$
$522$	0	0
$523$	− 19.2297i	− 0.840855i	−0.907326	−	0.420427i	$-0.861880\pi$
$523$	− 19.2297i	− 0.840855i	0.907326	−	0.420427i	$-0.138120\pi$
$524$	−27.8520	−1.21672
$525$	0	0
$526$	54.5036	2.37647
$527$	− 32.1828i	− 1.40190i
$528$	0	0
$529$	12.7703	0.555232
$530$	0	0
$531$	0	0
$532$	12.3852i	0.536965i
$533$	68.4975i	2.96695i
$534$	0	0
$535$	0	0
$536$	80.3575	3.47092
$537$	0	0
$538$	11.6148i	0.500751i
$539$	−3.19839	−0.137764
$540$	0	0
$541$	−14.5407	−0.625152	−0.312576	−	0.949893i	$-0.601192\pi$
$541$	−14.5407	−0.625152	−0.312576	+	0.949893i	$0.601192\pi$
$542$	− 40.6455i	− 1.74587i
$543$	0	0
$544$	−89.0813	−3.81933
$545$	0	0
$546$	0	0
$547$	− 12.6148i	− 0.539371i	−0.962948	−	0.269686i	$-0.913080\pi$
$547$	− 12.6148i	− 0.539371i	0.962948	−	0.269686i	$-0.0869198\pi$
$548$	− 55.7039i	− 2.37955i
$549$	0	0
$550$	0	0
$551$	−5.16487	−0.220031
$552$	0	0
$553$	− 5.38516i	− 0.229001i
$554$	63.1337	2.68229
$555$	0	0
$556$	−22.7703	−0.965677
$557$	8.56218i	0.362791i	0.983410	+	0.181396i	$0.0580615\pi$
$557$	8.56218i	0.362791i	−0.983410	+	0.181396i	$0.941939\pi$
$558$	0	0
$559$	59.9258	2.53459
$560$	0	0
$561$	0	0
$562$	− 40.1184i	− 1.69229i
$563$	4.33082i	0.182523i	0.995827	+	0.0912613i	$0.0290898\pi$
$563$	4.33082i	0.182523i	−0.995827	+	0.0912613i	$0.970910\pi$
$564$	0	0
$565$	0	0
$566$	−37.5466	−1.57820
$567$	0	0
$568$	− 119.237i	− 5.00307i
$569$	32.0833	1.34500	0.672501	−	0.740096i	$-0.265220\pi$
$569$	32.0833	1.34500	0.672501	+	0.740096i	$0.265220\pi$
$570$	0	0
$571$	35.3852	1.48082	0.740412	−	0.672154i	$-0.234631\pi$
$571$	35.3852	1.48082	0.740412	+	0.672154i	$0.234631\pi$
$572$	106.044i	4.43392i
$573$	0	0
$574$	28.7703	1.20085
$575$	0	0
$576$	0	0
$577$	− 47.1555i	− 1.96311i	−0.191183	−	0.981554i	$-0.561232\pi$
$577$	− 47.1555i	− 1.96311i	0.191183	−	0.981554i	$-0.438768\pi$
$578$	31.5668i	1.31301i
$579$	0	0
$580$	0	0
$581$	−4.33082	−0.179673
$582$	0	0
$583$	− 34.3110i	− 1.42102i
$584$	37.5466	1.55369
$585$	0	0
$586$	−31.5407	−1.30293
$587$	14.0254i	0.578892i	0.957194	+	0.289446i	$0.0934711\pi$
$587$	14.0254i	0.578892i	−0.957194	+	0.289446i	$0.906529\pi$
$588$	0	0
$589$	14.3110	0.589674
$590$	0	0
$591$	0	0
$592$	37.7332i	1.55083i
$593$	34.2487i	1.40643i	0.710979	+	0.703213i	$0.248252\pi$
$593$	34.2487i	1.40643i	−0.710979	+	0.703213i	$0.751748\pi$
$594$	0	0
$595$	0	0
$596$	66.9480	2.74230
$597$	0	0
$598$	54.7703i	2.23973i
$599$	26.7195	1.09173	0.545865	−	0.837873i	$-0.316201\pi$
$599$	26.7195	1.09173	0.545865	+	0.837873i	$0.316201\pi$
$600$	0	0
$601$	41.5407	1.69448	0.847239	−	0.531211i	$-0.178263\pi$
$601$	41.5407	1.69448	0.847239	+	0.531211i	$0.178263\pi$
$602$	− 25.1701i	− 1.02586i
$603$	0	0
$604$	−59.1184	−2.40549
$605$	0	0
$606$	0	0
$607$	− 23.1555i	− 0.939853i	−0.882706	−	0.469926i	$-0.844280\pi$
$607$	− 23.1555i	− 0.939853i	0.882706	−	0.469926i	$-0.155720\pi$
$608$	− 39.6125i	− 1.60650i
$609$	0	0
$610$	0	0
$611$	75.0932	3.03794
$612$	0	0
$613$	23.0000i	0.928961i	0.885583	+	0.464481i	$0.153759\pi$
$613$	23.0000i	0.928961i	−0.885583	+	0.464481i	$0.846241\pi$
$614$	35.2817	1.42385
$615$	0	0
$616$	27.3852	1.10338
$617$	− 10.6281i	− 0.427872i	−0.976848	−	0.213936i	$-0.931372\pi$
$617$	− 10.6281i	− 0.427872i	0.976848	−	0.213936i	$-0.0686285\pi$
$618$	0	0
$619$	27.1555	1.09147	0.545736	−	0.837957i	$-0.316250\pi$
$619$	27.1555	1.09147	0.545736	+	0.837957i	$0.316250\pi$
$620$	0	0
$621$	0	0
$622$	− 71.9258i	− 2.88396i
$623$	− 1.03297i	− 0.0413852i
$624$	0	0
$625$	0	0
$626$	64.1666	2.56461
$627$	0	0
$628$	− 87.0813i	− 3.47492i
$629$	16.0914	0.641606
$630$	0	0
$631$	−5.38516	−0.214380	−0.107190	−	0.994239i	$-0.534185\pi$
$631$	−5.38516	−0.214380	−0.107190	+	0.994239i	$0.534185\pi$
$632$	46.1088i	1.83411i
$633$	0	0
$634$	−22.9629	−0.911974
$635$	0	0
$636$	0	0
$637$	6.38516i	0.252989i
$638$	18.5744i	0.735366i
$639$	0	0
$640$	0	0
$641$	21.3557	0.843500	0.421750	−	0.906712i	$-0.361416\pi$
$641$	21.3557	0.843500	0.421750	+	0.906712i	$0.361416\pi$
$642$	0	0
$643$	− 6.45934i	− 0.254732i	−0.991856	−	0.127366i	$-0.959348\pi$
$643$	− 6.45934i	− 0.254732i	0.991856	−	0.127366i	$-0.0406522\pi$
$644$	16.6079	0.654442
$645$	0	0
$646$	−34.3110	−1.34995
$647$	− 38.5796i	− 1.51672i	−0.651837	−	0.758359i	$-0.726001\pi$
$647$	− 38.5796i	− 1.51672i	0.651837	−	0.758359i	$-0.273999\pi$
$648$	0	0
$649$	−37.6148	−1.47651
$650$	0	0
$651$	0	0
$652$	87.0813i	3.41037i
$653$	− 10.7276i	− 0.419803i	−0.977723	−	0.209902i	$-0.932686\pi$
$653$	− 10.7276i	− 0.419803i	0.977723	−	0.209902i	$-0.0673144\pi$
$654$	0	0
$655$	0	0
$656$	−134.929	−5.26809
$657$	0	0
$658$	− 31.5407i	− 1.22958i
$659$	47.2412	1.84026	0.920128	−	0.391617i	$-0.128084\pi$
$659$	47.2412	1.84026	0.920128	+	0.391617i	$0.128084\pi$
$660$	0	0
$661$	31.9258	1.24177	0.620885	−	0.783901i	$-0.286773\pi$
$661$	31.9258	1.24177	0.620885	+	0.783901i	$0.286773\pi$
$662$	− 27.2360i	− 1.05856i
$663$	0	0
$664$	37.0813	1.43903
$665$	0	0
$666$	0	0
$667$	6.92582i	0.268169i
$668$	− 5.36380i	− 0.207532i
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 3.54066i	− 0.136482i	−0.997669	−	0.0682412i	$-0.978261\pi$
$673$	− 3.54066i	− 0.136482i	0.997669	−	0.0682412i	$-0.0217387\pi$
$674$	−28.8849	−1.11261
$675$	0	0
$676$	144.200	5.54614
$677$	6.39677i	0.245848i	0.992416	+	0.122924i	$0.0392271\pi$
$677$	6.39677i	0.245848i	−0.992416	+	0.122924i	$0.960773\pi$
$678$	0	0
$679$	−10.7703	−0.413327
$680$	0	0
$681$	0	0
$682$	− 51.4665i	− 1.97075i
$683$	− 9.59516i	− 0.367148i	−0.983006	−	0.183574i	$-0.941233\pi$
$683$	− 9.59516i	− 0.367148i	0.983006	−	0.183574i	$-0.0587668\pi$
$684$	0	0
$685$	0	0
$686$	2.68190	0.102395
$687$	0	0
$688$	118.044i	4.50039i
$689$	−68.4975	−2.60955
$690$	0	0
$691$	−15.2297	−0.579364	−0.289682	−	0.957123i	$-0.593550\pi$
$691$	−15.2297	−0.579364	−0.289682	+	0.957123i	$0.593550\pi$
$692$	116.772i	4.43899i
$693$	0	0
$694$	−66.1184	−2.50982
$695$	0	0
$696$	0	0
$697$	57.5407i	2.17951i
$698$	6.39677i	0.242121i
$699$	0	0
$700$	0	0
$701$	−44.9763	−1.69873	−0.849366	−	0.527804i	$-0.823016\pi$
$701$	−44.9763	−1.69873	−0.849366	+	0.527804i	$0.823016\pi$
$702$	0	0
$703$	7.15549i	0.269875i
$704$	−62.0012	−2.33676
$705$	0	0
$706$	−45.9258	−1.72844
$707$	− 5.36380i	− 0.201726i
$708$	0	0
$709$	−39.5407	−1.48498	−0.742490	−	0.669857i	$-0.766355\pi$
$709$	−39.5407	−1.48498	−0.742490	+	0.669857i	$0.766355\pi$
$710$	0	0
$711$	0	0
$712$	8.84451i	0.331462i
$713$	− 19.1903i	− 0.718683i
$714$	0	0
$715$	0	0
$716$	−22.4882	−0.840422
$717$	0	0
$718$	48.9629i	1.82728i
$719$	19.3892	0.723097	0.361548	−	0.932353i	$-0.382248\pi$
$719$	19.3892	0.723097	0.361548	+	0.932353i	$0.382248\pi$
$720$	0	0
$721$	−5.61484	−0.209107
$722$	35.6987i	1.32857i
$723$	0	0
$724$	−105.852	−3.93395
$725$	0	0
$726$	0	0
$727$	32.3852i	1.20110i	0.799587	+	0.600550i	$0.205052\pi$
$727$	32.3852i	1.20110i	−0.799587	+	0.600550i	$0.794948\pi$
$728$	− 54.6710i	− 2.02624i
$729$	0	0
$730$	0	0
$731$	50.3401	1.86190
$732$	0	0
$733$	28.0000i	1.03420i	0.855924	+	0.517102i	$0.172989\pi$
$733$	28.0000i	1.03420i	−0.855924	+	0.517102i	$0.827011\pi$
$734$	−10.7276	−0.395963
$735$	0	0
$736$	−53.1184	−1.95797
$737$	− 30.0174i	− 1.10570i
$738$	0	0
$739$	43.3852	1.59595	0.797975	−	0.602691i	$-0.205905\pi$
$739$	43.3852	1.59595	0.797975	+	0.602691i	$0.205905\pi$
$740$	0	0
$741$	0	0
$742$	28.7703i	1.05619i
$743$	15.0584i	0.552440i	0.961094	+	0.276220i	$0.0890818\pi$
$743$	15.0584i	0.552440i	−0.961094	+	0.276220i	$0.910918\pi$
$744$	0	0
$745$	0	0
$746$	67.0475	2.45478
$747$	0	0
$748$	89.0813i	3.25714i
$749$	4.33082	0.158245
$750$	0	0
$751$	−28.7703	−1.04984	−0.524922	−	0.851150i	$-0.675906\pi$
$751$	−28.7703	−1.04984	−0.524922	+	0.851150i	$0.675906\pi$
$752$	147.921i	5.39414i
$753$	0	0
$754$	37.0813	1.35042
$755$	0	0
$756$	0	0
$757$	− 11.3110i	− 0.411105i	−0.978646	−	0.205552i	$-0.934101\pi$
$757$	− 11.3110i	− 0.411105i	0.978646	−	0.205552i	$-0.0658991\pi$
$758$	− 30.5339i	− 1.10904i
$759$	0	0
$760$	0	0
$761$	−12.7935	−0.463766	−0.231883	−	0.972744i	$-0.574489\pi$
$761$	−12.7935	−0.463766	−0.231883	+	0.972744i	$0.574489\pi$
$762$	0	0
$763$	7.00000i	0.253417i
$764$	33.2158	1.20170
$765$	0	0
$766$	26.0000	0.939418
$767$	75.0932i	2.71146i
$768$	0	0
$769$	−44.7703	−1.61446	−0.807230	−	0.590237i	$-0.799034\pi$
$769$	−44.7703	−1.61446	−0.807230	+	0.590237i	$0.799034\pi$
$770$	0	0
$771$	0	0
$772$	5.19258i	0.186885i
$773$	− 15.0584i	− 0.541614i	−0.962634	−	0.270807i	$-0.912710\pi$
$773$	− 15.0584i	− 0.541614i	0.962634	−	0.270807i	$-0.0872905\pi$
$774$	0	0
$775$	0	0
$776$	92.2175	3.31042
$777$	0	0
$778$	28.5036i	1.02190i
$779$	−25.5871	−0.916752
$780$	0	0
$781$	−44.5407	−1.59379
$782$	46.0093i	1.64529i
$783$	0	0
$784$	−12.5777	−0.449205
$785$	0	0
$786$	0	0
$787$	− 0.385165i	− 0.0137296i	−0.999976	−	0.00686482i	$-0.997815\pi$
$787$	− 0.385165i	− 0.0137296i	0.999976	−	0.00686482i	$-0.00218516\pi$
$788$	− 122.652i	− 4.36929i
$789$	0	0
$790$	0	0
$791$	2.16541	0.0769932
$792$	0	0
$793$	0	0
$794$	−3.29785	−0.117036
$795$	0	0
$796$	4.00000	0.141776
$797$	− 30.9509i	− 1.09634i	−0.836368	−	0.548168i	$-0.815325\pi$
$797$	− 30.9509i	− 1.09634i	0.836368	−	0.548168i	$-0.184675\pi$
$798$	0	0
$799$	63.0813	2.23166
$800$	0	0
$801$	0	0
$802$	− 34.5777i	− 1.22098i
$803$	− 14.0254i	− 0.494947i
$804$	0	0
$805$	0	0
$806$	−102.746	−3.61908
$807$	0	0
$808$	45.9258i	1.61566i
$809$	36.4141	1.28025	0.640127	−	0.768269i	$-0.278882\pi$
$809$	36.4141	1.28025	0.640127	+	0.768269i	$0.278882\pi$
$810$	0	0
$811$	30.7703	1.08049	0.540246	−	0.841507i	$-0.318331\pi$
$811$	30.7703	1.08049	0.540246	+	0.841507i	$0.318331\pi$
$812$	− 11.2441i	− 0.394590i
$813$	0	0
$814$	25.7332	0.901950
$815$	0	0
$816$	0	0
$817$	22.3852i	0.783158i
$818$	60.0348i	2.09907i
$819$	0	0
$820$	0	0
$821$	47.0423	1.64179	0.820893	−	0.571081i	$-0.193476\pi$
$821$	47.0423	1.64179	0.820893	+	0.571081i	$0.193476\pi$
$822$	0	0
$823$	− 24.9258i	− 0.868860i	−0.900706	−	0.434430i	$-0.856950\pi$
$823$	− 24.9258i	− 0.868860i	0.900706	−	0.434430i	$-0.143050\pi$
$824$	48.0753	1.67478
$825$	0	0
$826$	31.5407	1.09744
$827$	13.9260i	0.484254i	0.970245	+	0.242127i	$0.0778450\pi$
$827$	13.9260i	0.484254i	−0.970245	+	0.242127i	$0.922155\pi$
$828$	0	0
$829$	8.38516	0.291229	0.145614	−	0.989341i	$-0.453484\pi$
$829$	8.38516	0.291229	0.145614	+	0.989341i	$0.453484\pi$
$830$	0	0
$831$	0	0
$832$	123.777i	4.29121i
$833$	5.36380i	0.185845i
$834$	0	0
$835$	0	0
$836$	−39.6125	−1.37003
$837$	0	0
$838$	− 45.9258i	− 1.58648i
$839$	−14.0254	−0.484212	−0.242106	−	0.970250i	$-0.577838\pi$
$839$	−14.0254	−0.484212	−0.242106	+	0.970250i	$0.577838\pi$
$840$	0	0
$841$	−24.3110	−0.838310
$842$	− 67.8815i	− 2.33935i
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.770330i	0.0264688i
$848$	− 134.929i	− 4.63348i
$849$	0	0
$850$	0	0
$851$	9.59516	0.328918
$852$	0	0
$853$	− 30.3110i	− 1.03783i	−0.854826	−	0.518914i	$-0.826336\pi$
$853$	− 30.3110i	− 1.03783i	0.854826	−	0.518914i	$-0.173664\pi$
$854$	0	0
$855$	0	0
$856$	−37.0813	−1.26741
$857$	− 36.3147i	− 1.24049i	−0.784410	−	0.620243i	$-0.787034\pi$
$857$	− 36.3147i	− 1.24049i	0.784410	−	0.620243i	$-0.212966\pi$
$858$	0	0
$859$	27.5407	0.939675	0.469838	−	0.882753i	$-0.344312\pi$
$859$	27.5407	0.939675	0.469838	+	0.882753i	$0.344312\pi$
$860$	0	0
$861$	0	0
$862$	69.1555i	2.35545i
$863$	− 1.13244i	− 0.0385487i	−0.999814	−	0.0192743i	$-0.993864\pi$
$863$	− 1.13244i	− 0.0385487i	0.999814	−	0.0192743i	$-0.00613559\pi$
$864$	0	0
$865$	0	0
$866$	25.5871	0.869485
$867$	0	0
$868$	31.1555i	1.05749i
$869$	17.2238	0.584279
$870$	0	0
$871$	−59.9258	−2.03051
$872$	− 59.9353i	− 2.02966i
$873$	0	0
$874$	−20.4593	−0.692048
$875$	0	0
$876$	0	0
$877$	19.5407i	0.659841i	0.944009	+	0.329921i	$0.107022\pi$
$877$	19.5407i	0.659841i	−0.944009	+	0.329921i	$0.892978\pi$
$878$	51.3731i	1.73376i
$879$	0	0
$880$	0	0
$881$	−40.6455	−1.36938	−0.684691	−	0.728834i	$-0.740063\pi$
$881$	−40.6455	−1.36938	−0.684691	+	0.728834i	$0.740063\pi$
$882$	0	0
$883$	− 8.61484i	− 0.289912i	−0.989438	−	0.144956i	$-0.953696\pi$
$883$	− 8.61484i	− 0.289912i	0.989438	−	0.144956i	$-0.0463041\pi$
$884$	177.839	5.98139
$885$	0	0
$886$	−51.4665	−1.72905
$887$	− 10.7276i	− 0.360197i	−0.983649	−	0.180099i	$-0.942358\pi$
$887$	− 10.7276i	− 0.360197i	0.983649	−	0.180099i	$-0.0576417\pi$
$888$	0	0
$889$	−4.61484	−0.154777
$890$	0	0
$891$	0	0
$892$	18.7703i	0.628477i
$893$	28.0509i	0.938687i
$894$	0	0
$895$	0	0
$896$	18.7733	0.627172
$897$	0	0
$898$	34.5777i	1.15387i
$899$	−12.9925	−0.433323
$900$	0	0
$901$	−57.5407	−1.91696
$902$	92.0186i	3.06388i
$903$	0	0
$904$	−18.5407	−0.616653
$905$	0	0
$906$	0	0
$907$	− 22.3110i	− 0.740824i	−0.928868	−	0.370412i	$-0.879216\pi$
$907$	− 22.3110i	− 0.740824i	0.928868	−	0.370412i	$-0.120784\pi$
$908$	61.0677i	2.02660i
$909$	0	0
$910$	0	0
$911$	5.46326	0.181006	0.0905030	−	0.995896i	$-0.471153\pi$
$911$	5.46326	0.181006	0.0905030	+	0.995896i	$0.471153\pi$
$912$	0	0
$913$	− 13.8516i	− 0.458423i
$914$	−76.5432	−2.53182
$915$	0	0
$916$	−80.6962	−2.66628
$917$	− 5.36380i	− 0.177128i
$918$	0	0
$919$	49.6962	1.63932	0.819662	−	0.572847i	$-0.194161\pi$
$919$	49.6962	1.63932	0.819662	+	0.572847i	$0.194161\pi$
$920$	0	0
$921$	0	0
$922$	2.77033i	0.0912359i
$923$	88.9197i	2.92683i
$924$	0	0
$925$	0	0
$926$	23.5211	0.772953
$927$	0	0
$928$	35.9629i	1.18054i
$929$	16.0914	0.527942	0.263971	−	0.964531i	$-0.414968\pi$
$929$	16.0914	0.527942	0.263971	+	0.964531i	$0.414968\pi$
$930$	0	0
$931$	−2.38516	−0.0781706
$932$	− 21.9717i	− 0.719706i
$933$	0	0
$934$	−17.1555	−0.561345
$935$	0	0
$936$	0	0
$937$	19.2297i	0.628206i	0.949389	+	0.314103i	$0.101704\pi$
$937$	19.2297i	0.628206i	−0.949389	+	0.314103i	$0.898296\pi$
$938$	25.1701i	0.821832i
$939$	0	0
$940$	0	0
$941$	49.3072	1.60737	0.803684	−	0.595057i	$-0.202870\pi$
$941$	49.3072	1.60737	0.803684	+	0.595057i	$0.202870\pi$
$942$	0	0
$943$	34.3110i	1.11732i
$944$	−147.921	−4.81443
$945$	0	0
$946$	80.5036	2.61740
$947$	42.7115i	1.38794i	0.720006	+	0.693968i	$0.244139\pi$
$947$	42.7115i	1.38794i	−0.720006	+	0.693968i	$0.755861\pi$
$948$	0	0
$949$	−28.0000	−0.908918
$950$	0	0
$951$	0	0
$952$	− 45.9258i	− 1.48846i
$953$	0.0994662i	0.00322203i	0.999999	+	0.00161101i	$0.000512802\pi$
$953$	0.0994662i	0.00322203i	−0.999999	+	0.00161101i	$0.999487\pi$
$954$	0	0
$955$	0	0
$956$	−33.2158	−1.07427
$957$	0	0
$958$	− 89.0813i	− 2.87809i
$959$	10.7276	0.346412
$960$	0	0
$961$	5.00000	0.161290
$962$	− 51.3731i	− 1.65634i
$963$	0	0
$964$	−95.4665	−3.07477
$965$	0	0
$966$	0	0
$967$	− 37.5407i	− 1.20723i	−0.797277	−	0.603613i	$-0.793727\pi$
$967$	− 37.5407i	− 1.20723i	0.797277	−	0.603613i	$-0.206273\pi$
$968$	− 6.59570i	− 0.211994i
$969$	0	0
$970$	0	0
$971$	3.09892	0.0994491	0.0497245	−	0.998763i	$-0.484166\pi$
$971$	3.09892	0.0994491	0.0497245	+	0.998763i	$0.484166\pi$
$972$	0	0
$973$	− 4.38516i	− 0.140582i
$974$	−35.8977	−1.15024
$975$	0	0
$976$	0	0
$977$	34.1493i	1.09253i	0.837612	+	0.546266i	$0.183951\pi$
$977$	34.1493i	1.09253i	−0.837612	+	0.546266i	$0.816049\pi$
$978$	0	0
$979$	3.30385	0.105591
$980$	0	0
$981$	0	0
$982$	94.8887i	3.02802i
$983$	− 36.5136i	− 1.16460i	−0.812973	−	0.582302i	$-0.802152\pi$
$983$	− 36.5136i	− 1.16460i	0.812973	−	0.582302i	$-0.197848\pi$
$984$	0	0
$985$	0	0
$986$	31.1498	0.992012
$987$	0	0
$988$	79.0813i	2.51591i
$989$	30.0174	0.954497
$990$	0	0
$991$	39.6962	1.26099	0.630495	−	0.776193i	$-0.282852\pi$
$991$	39.6962	1.26099	0.630495	+	0.776193i	$0.282852\pi$
$992$	− 99.6473i	− 3.16380i
$993$	0	0
$994$	37.3481	1.18461
$995$	0	0
$996$	0	0
$997$	24.8445i	0.786833i	0.919360	+	0.393417i	$0.128707\pi$
$997$	24.8445i	0.786833i	−0.919360	+	0.393417i	$0.871293\pi$
$998$	79.2251i	2.50783i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.d.l.1324.1		8
3.2	odd	2	inner	1575.2.d.l.1324.7		8
5.2	odd	4		1575.2.a.z.1.4	yes	4
5.3	odd	4		1575.2.a.y.1.1	✓	4
5.4	even	2	inner	1575.2.d.l.1324.8		8
15.2	even	4		1575.2.a.z.1.1	yes	4
15.8	even	4		1575.2.a.y.1.4	yes	4
15.14	odd	2	inner	1575.2.d.l.1324.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1575.2.a.y.1.1	✓	4	5.3	odd	4
1575.2.a.y.1.4	yes	4	15.8	even	4
1575.2.a.z.1.1	yes	4	15.2	even	4
1575.2.a.z.1.4	yes	4	5.2	odd	4
1575.2.d.l.1324.1		8	1.1	even	1	trivial
1575.2.d.l.1324.2		8	15.14	odd	2	inner
1575.2.d.l.1324.7		8	3.2	odd	2	inner
1575.2.d.l.1324.8		8	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.68190i q^{2} -5.19258 q^{4} -1.00000i q^{7} +8.56218i q^{8} +O(q^{10})\)	\(q-2.68190i q^{2} -5.19258 q^{4} -1.00000i q^{7} +8.56218i q^{8} +3.19839 q^{11} -6.38516i q^{13} -2.68190 q^{14} +12.5777 q^{16} -5.36380i q^{17} +2.38516 q^{19} -8.57775i q^{22} -3.19839i q^{23} -17.1244 q^{26} +5.19258i q^{28} -2.16541 q^{29} +6.00000 q^{31} -16.6079i q^{32} -14.3852 q^{34} +3.00000i q^{37} -6.39677i q^{38} -10.7276 q^{41} +9.38516i q^{43} -16.6079 q^{44} -8.57775 q^{46} +11.7606i q^{47} -1.00000 q^{49} +33.1555i q^{52} -10.7276i q^{53} +8.56218 q^{56} +5.80742i q^{58} -11.7606 q^{59} -16.0914i q^{62} -19.3852 q^{64} -9.38516i q^{67} +27.8520i q^{68} -13.9260 q^{71} -4.38516i q^{73} +8.04570 q^{74} -12.3852 q^{76} -3.19839i q^{77} +5.38516 q^{79} +28.7703i q^{82} -4.33082i q^{83} +25.1701 q^{86} +27.3852i q^{88} +1.03297 q^{89} -6.38516 q^{91} +16.6079i q^{92} +31.5407 q^{94} -10.7703i q^{97} +2.68190i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 20 q^{4}+O(q^{10}) \) 8 * q - 20 * q^4	\( 8 q - 20 q^{4} + 36 q^{16} - 24 q^{19} + 48 q^{31} - 72 q^{34} - 4 q^{46} - 8 q^{49} - 112 q^{64} - 56 q^{76} - 8 q^{91} + 80 q^{94}+O(q^{100}) \) 8 * q - 20 * q^4 + 36 * q^16 - 24 * q^19 + 48 * q^31 - 72 * q^34 - 4 * q^46 - 8 * q^49 - 112 * q^64 - 56 * q^76 - 8 * q^91 + 80 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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