LMFDB - Embedded newform 1035.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1035,2,Mod(1,1035)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1035, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1035.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1035 = 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1035.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:	$8.26451660920$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.15317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 115)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.69353$ of defining polynomial
Character	$\chi$	$=$	1035.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69353 q^{2} +5.25508 q^{4} -1.00000 q^{5} -2.74252 q^{7} -8.76763 q^{8} +O(q^{10})$	\(q-2.69353 q^{2} +5.25508 q^{4} -1.00000 q^{5} -2.74252 q^{7} -8.76763 q^{8} +2.69353 q^{10} +3.38705 q^{11} -2.46356 q^{13} +7.38705 q^{14} +13.1057 q^{16} +2.64453 q^{17} +3.38705 q^{19} -5.25508 q^{20} -9.12311 q^{22} +1.00000 q^{23} +1.00000 q^{25} +6.63566 q^{26} -14.4122 q^{28} -9.20608 q^{29} -5.10809 q^{31} -17.7652 q^{32} -7.12311 q^{34} +2.74252 q^{35} +5.50369 q^{37} -9.12311 q^{38} +8.76763 q^{40} +1.20608 q^{41} +3.02511 q^{43} +17.7992 q^{44} -2.69353 q^{46} -8.21255 q^{47} +0.521423 q^{49} -2.69353 q^{50} -12.9462 q^{52} +10.5417 q^{53} -3.38705 q^{55} +24.0454 q^{56} +24.7968 q^{58} -8.28259 q^{59} +0.263945 q^{61} +13.7588 q^{62} +21.6397 q^{64} +2.46356 q^{65} -7.66964 q^{67} +13.8972 q^{68} -7.38705 q^{70} +0.0150161 q^{71} -5.53644 q^{73} -14.8243 q^{74} +17.7992 q^{76} -9.28906 q^{77} -8.67611 q^{79} -13.1057 q^{80} -3.24861 q^{82} +3.52142 q^{83} -2.64453 q^{85} -8.14822 q^{86} -29.6964 q^{88} +4.66318 q^{89} +6.75637 q^{91} +5.25508 q^{92} +22.1207 q^{94} -3.38705 q^{95} -4.09799 q^{97} -1.40447 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 4 q^{4} - 4 q^{5} - 3 q^{7} - 9 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 - 3 * q^7 - 9 * q^8	$ 4 q - 2 q^{2} + 4 q^{4} - 4 q^{5} - 3 q^{7} - 9 q^{8} + 2 q^{10} - 4 q^{11} + 12 q^{14} + 8 q^{16} + q^{17} - 4 q^{19} - 4 q^{20} - 20 q^{22} + 4 q^{23} + 4 q^{25} + q^{26} - 22 q^{28} - 19 q^{29} - q^{31} - 20 q^{32} - 12 q^{34} + 3 q^{35} - 3 q^{37} - 20 q^{38} + 9 q^{40} - 13 q^{41} - 6 q^{43} + 18 q^{44} - 2 q^{46} - 6 q^{47} + 9 q^{49} - 2 q^{50} - q^{52} - 19 q^{53} + 4 q^{55} + 10 q^{56} + 21 q^{58} - 23 q^{59} + 13 q^{62} + 27 q^{64} - 3 q^{67} + 4 q^{68} - 12 q^{70} + 3 q^{71} - 32 q^{73} + 12 q^{74} + 18 q^{76} - 18 q^{77} + 2 q^{79} - 8 q^{80} - 5 q^{82} + 21 q^{83} - q^{85} + 2 q^{86} - 14 q^{88} - 40 q^{91} + 4 q^{92} + 47 q^{94} + 4 q^{95} - 18 q^{97} - 16 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 - 3 * q^7 - 9 * q^8 + 2 * q^10 - 4 * q^11 + 12 * q^14 + 8 * q^16 + q^17 - 4 * q^19 - 4 * q^20 - 20 * q^22 + 4 * q^23 + 4 * q^25 + q^26 - 22 * q^28 - 19 * q^29 - q^31 - 20 * q^32 - 12 * q^34 + 3 * q^35 - 3 * q^37 - 20 * q^38 + 9 * q^40 - 13 * q^41 - 6 * q^43 + 18 * q^44 - 2 * q^46 - 6 * q^47 + 9 * q^49 - 2 * q^50 - q^52 - 19 * q^53 + 4 * q^55 + 10 * q^56 + 21 * q^58 - 23 * q^59 + 13 * q^62 + 27 * q^64 - 3 * q^67 + 4 * q^68 - 12 * q^70 + 3 * q^71 - 32 * q^73 + 12 * q^74 + 18 * q^76 - 18 * q^77 + 2 * q^79 - 8 * q^80 - 5 * q^82 + 21 * q^83 - q^85 + 2 * q^86 - 14 * q^88 - 40 * q^91 + 4 * q^92 + 47 * q^94 + 4 * q^95 - 18 * q^97 - 16 * q^98

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 $

$2$	−2.69353	−1.90461	−0.952305	−	0.305148i	$-0.901294\pi$
$2$	−2.69353	−1.90461	−0.952305	+	0.305148i	$0.901294\pi$
$3$	0	0
$3$	0	0
$4$	5.25508	2.62754
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.74252	−1.03658	−0.518288	−	0.855206i	$-0.673430\pi$
$7$	−2.74252	−1.03658	−0.518288	+	0.855206i	$0.673430\pi$
$8$	−8.76763	−3.09983
$9$	0	0
$10$	2.69353	0.851767
$11$	3.38705	1.02123	0.510617	−	0.859808i	$-0.329417\pi$
$11$	3.38705	1.02123	0.510617	+	0.859808i	$0.329417\pi$
$12$	0	0
$13$	−2.46356	−0.683269	−0.341634	−	0.939833i	$-0.610980\pi$
$13$	−2.46356	−0.683269	−0.341634	+	0.939833i	$0.610980\pi$
$14$	7.38705	1.97427
$15$	0	0
$16$	13.1057	3.27642
$17$	2.64453	0.641393	0.320696	−	0.947182i	$-0.396083\pi$
$17$	2.64453	0.641393	0.320696	+	0.947182i	$0.396083\pi$
$18$	0	0
$19$	3.38705	0.777043	0.388521	−	0.921440i	$-0.372986\pi$
$19$	3.38705	0.777043	0.388521	+	0.921440i	$0.372986\pi$
$20$	−5.25508	−1.17507
$21$	0	0
$22$	−9.12311	−1.94505
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	6.63566	1.30136
$27$	0	0
$28$	−14.4122	−2.72364
$29$	−9.20608	−1.70953	−0.854763	−	0.519018i	$-0.826298\pi$
$29$	−9.20608	−1.70953	−0.854763	+	0.519018i	$0.826298\pi$
$30$	0	0
$31$	−5.10809	−0.917440	−0.458720	−	0.888581i	$-0.651692\pi$
$31$	−5.10809	−0.917440	−0.458720	+	0.888581i	$0.651692\pi$
$32$	−17.7652	−3.14048
$33$	0	0
$34$	−7.12311	−1.22160
$35$	2.74252	0.463571
$36$	0	0
$37$	5.50369	0.904801	0.452401	−	0.891815i	$-0.350568\pi$
$37$	5.50369	0.904801	0.452401	+	0.891815i	$0.350568\pi$
$38$	−9.12311	−1.47996
$39$	0	0
$40$	8.76763	1.38628
$41$	1.20608	0.188358	0.0941792	−	0.995555i	$-0.469977\pi$
$41$	1.20608	0.188358	0.0941792	+	0.995555i	$0.469977\pi$
$42$	0	0
$43$	3.02511	0.461325	0.230663	−	0.973034i	$-0.425911\pi$
$43$	3.02511	0.461325	0.230663	+	0.973034i	$0.425911\pi$
$44$	17.7992	2.68333
$45$	0	0
$46$	−2.69353	−0.397139
$47$	−8.21255	−1.19792	−0.598962	−	0.800778i	$-0.704420\pi$
$47$	−8.21255	−1.19792	−0.598962	+	0.800778i	$0.704420\pi$
$48$	0	0
$49$	0.521423	0.0744891
$50$	−2.69353	−0.380922
$51$	0	0
$52$	−12.9462	−1.79532
$53$	10.5417	1.44802	0.724009	−	0.689790i	$-0.242297\pi$
$53$	10.5417	1.44802	0.724009	+	0.689790i	$0.242297\pi$
$54$	0	0
$55$	−3.38705	−0.456710
$56$	24.0454	3.21321
$57$	0	0
$58$	24.7968	3.25598
$59$	−8.28259	−1.07830	−0.539151	−	0.842209i	$-0.681255\pi$
$59$	−8.28259	−1.07830	−0.539151	+	0.842209i	$0.681255\pi$
$60$	0	0
$61$	0.263945	0.0337947	0.0168973	−	0.999857i	$-0.494621\pi$
$61$	0.263945	0.0337947	0.0168973	+	0.999857i	$0.494621\pi$
$62$	13.7588	1.74737
$63$	0	0
$64$	21.6397	2.70497
$65$	2.46356	0.305567
$66$	0	0
$67$	−7.66964	−0.936996	−0.468498	−	0.883465i	$-0.655205\pi$
$67$	−7.66964	−0.936996	−0.468498	+	0.883465i	$0.655205\pi$
$68$	13.8972	1.68528
$69$	0	0
$70$	−7.38705	−0.882921
$71$	0.0150161	0.00178208	0.000891041	−	1.00000i	$-0.499716\pi$
$71$	0.0150161	0.00178208	0.000891041		1.00000i	$0.499716\pi$
$72$	0	0
$73$	−5.53644	−0.647991	−0.323996	−	0.946059i	$-0.605026\pi$
$73$	−5.53644	−0.647991	−0.323996	+	0.946059i	$0.605026\pi$
$74$	−14.8243	−1.72329
$75$	0	0
$76$	17.7992	2.04171
$77$	−9.28906	−1.05859
$78$	0	0
$79$	−8.67611	−0.976138	−0.488069	−	0.872805i	$-0.662299\pi$
$79$	−8.67611	−0.976138	−0.488069	+	0.872805i	$0.662299\pi$
$80$	−13.1057	−1.46526
$81$	0	0
$82$	−3.24861	−0.358749
$83$	3.52142	0.386526	0.193263	−	0.981147i	$-0.438093\pi$
$83$	3.52142	0.386526	0.193263	+	0.981147i	$0.438093\pi$
$84$	0	0
$85$	−2.64453	−0.286839
$86$	−8.14822	−0.878645
$87$	0	0
$88$	−29.6964	−3.16565
$89$	4.66318	0.494296	0.247148	−	0.968978i	$-0.420507\pi$
$89$	4.66318	0.494296	0.247148	+	0.968978i	$0.420507\pi$
$90$	0	0
$91$	6.75637	0.708260
$92$	5.25508	0.547880
$93$	0	0
$94$	22.1207	2.28158
$95$	−3.38705	−0.347504
$96$	0	0
$97$	−4.09799	−0.416088	−0.208044	−	0.978119i	$-0.566710\pi$
$97$	−4.09799	−0.416088	−0.208044	+	0.978119i	$0.566710\pi$
$98$	−1.40447	−0.141873
$99$	0	0
$100$	5.25508	0.525508
$101$	−12.2955	−1.22345	−0.611725	−	0.791070i	$-0.709524\pi$
$101$	−12.2955	−1.22345	−0.611725	+	0.791070i	$0.709524\pi$
$102$	0	0
$103$	6.05023	0.596147	0.298073	−	0.954543i	$-0.403656\pi$
$103$	6.05023	0.596147	0.298073	+	0.954543i	$0.403656\pi$
$104$	21.5996	2.11801
$105$	0	0
$106$	−28.3944	−2.75791
$107$	−13.1547	−1.27171	−0.635856	−	0.771808i	$-0.719353\pi$
$107$	−13.1547	−1.27171	−0.635856	+	0.771808i	$0.719353\pi$
$108$	0	0
$109$	−17.1183	−1.63964	−0.819818	−	0.572624i	$-0.805925\pi$
$109$	−17.1183	−1.63964	−0.819818	+	0.572624i	$0.805925\pi$
$110$	9.12311	0.869854
$111$	0	0
$112$	−35.9426	−3.39626
$113$	−4.38058	−0.412091	−0.206045	−	0.978542i	$-0.566059\pi$
$113$	−4.38058	−0.412091	−0.206045	+	0.978542i	$0.566059\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	−48.3787	−4.49185
$117$	0	0
$118$	22.3094	2.05374
$119$	−7.25268	−0.664852
$120$	0	0
$121$	0.472110	0.0429191
$122$	−0.710942	−0.0643657
$123$	0	0
$124$	−26.8434	−2.41061
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−18.4588	−1.63795	−0.818975	−	0.573829i	$-0.805457\pi$
$127$	−18.4588	−1.63795	−0.818975	+	0.573829i	$0.805457\pi$
$128$	−22.7567	−2.01143
$129$	0	0
$130$	−6.63566	−0.581986
$131$	3.86834	0.337979	0.168989	−	0.985618i	$-0.445950\pi$
$131$	3.86834	0.337979	0.168989	+	0.985618i	$0.445950\pi$
$132$	0	0
$133$	−9.28906	−0.805463
$134$	20.6584	1.78461
$135$	0	0
$136$	−23.1863	−1.98821
$137$	−20.6713	−1.76607	−0.883034	−	0.469308i	$-0.844503\pi$
$137$	−20.6713	−1.76607	−0.883034	+	0.469308i	$0.844503\pi$
$138$	0	0
$139$	12.5802	1.06704	0.533519	−	0.845788i	$-0.320869\pi$
$139$	12.5802	1.06704	0.533519	+	0.845788i	$0.320869\pi$
$140$	14.4122	1.21805
$141$	0	0
$142$	−0.0404462	−0.00339417
$143$	−8.34420	−0.697777
$144$	0	0
$145$	9.20608	0.764523
$146$	14.9125	1.23417
$147$	0	0
$148$	28.9223	2.37740
$149$	−11.4850	−0.940891	−0.470446	−	0.882429i	$-0.655907\pi$
$149$	−11.4850	−0.940891	−0.470446	+	0.882429i	$0.655907\pi$
$150$	0	0
$151$	−5.58667	−0.454636	−0.227318	−	0.973821i	$-0.572996\pi$
$151$	−5.58667	−0.454636	−0.227318	+	0.973821i	$0.572996\pi$
$152$	−29.6964	−2.40870
$153$	0	0
$154$	25.0203	2.01619
$155$	5.10809	0.410292
$156$	0	0
$157$	5.86563	0.468128	0.234064	−	0.972221i	$-0.424797\pi$
$157$	5.86563	0.468128	0.234064	+	0.972221i	$0.424797\pi$
$158$	23.3693	1.85916
$159$	0	0
$160$	17.7652	1.40447
$161$	−2.74252	−0.216141
$162$	0	0
$163$	0.0465955	0.00364964	0.00182482	−	0.999998i	$-0.499419\pi$
$163$	0.0465955	0.00364964	0.00182482	+	0.999998i	$0.499419\pi$
$164$	6.33805	0.494919
$165$	0	0
$166$	−9.48504	−0.736182
$167$	2.24621	0.173817	0.0869085	−	0.996216i	$-0.472301\pi$
$167$	2.24621	0.173817	0.0869085	+	0.996216i	$0.472301\pi$
$168$	0	0
$169$	−6.93087	−0.533144
$170$	7.12311	0.546317
$171$	0	0
$172$	15.8972	1.21215
$173$	−8.01293	−0.609212	−0.304606	−	0.952478i	$-0.598525\pi$
$173$	−8.01293	−0.609212	−0.304606	+	0.952478i	$0.598525\pi$
$174$	0	0
$175$	−2.74252	−0.207315
$176$	44.3896	3.34599
$177$	0	0
$178$	−12.5604	−0.941440
$179$	11.9615	0.894047	0.447024	−	0.894522i	$-0.352484\pi$
$179$	11.9615	0.894047	0.447024	+	0.894522i	$0.352484\pi$
$180$	0	0
$181$	−17.4802	−1.29930	−0.649648	−	0.760235i	$-0.725084\pi$
$181$	−17.4802	−1.29930	−0.649648	+	0.760235i	$0.725084\pi$
$182$	−18.1984	−1.34896
$183$	0	0
$184$	−8.76763	−0.646359
$185$	−5.50369	−0.404639
$186$	0	0
$187$	8.95715	0.655012
$188$	−43.1576	−3.14759
$189$	0	0
$190$	9.12311	0.661860
$191$	13.1231	0.949555	0.474777	−	0.880106i	$-0.342529\pi$
$191$	13.1231	0.949555	0.474777	+	0.880106i	$0.342529\pi$
$192$	0	0
$193$	−12.5438	−0.902924	−0.451462	−	0.892290i	$-0.649097\pi$
$193$	−12.5438	−0.902924	−0.451462	+	0.892290i	$0.649097\pi$
$194$	11.0380	0.792485
$195$	0	0
$196$	2.74012	0.195723
$197$	−3.88544	−0.276826	−0.138413	−	0.990375i	$-0.544200\pi$
$197$	−3.88544	−0.276826	−0.138413	+	0.990375i	$0.544200\pi$
$198$	0	0
$199$	22.1612	1.57096	0.785481	−	0.618886i	$-0.212416\pi$
$199$	22.1612	1.57096	0.785481	+	0.618886i	$0.212416\pi$
$200$	−8.76763	−0.619965
$201$	0	0
$202$	33.1183	2.33020
$203$	25.2479	1.77205
$204$	0	0
$205$	−1.20608	−0.0842364
$206$	−16.2964	−1.13543
$207$	0	0
$208$	−32.2867	−2.23868
$209$	11.4721	0.793542
$210$	0	0
$211$	4.81048	0.331167	0.165584	−	0.986196i	$-0.447049\pi$
$211$	4.81048	0.331167	0.165584	+	0.986196i	$0.447049\pi$
$212$	55.3976	3.80473
$213$	0	0
$214$	35.4325	2.42211
$215$	−3.02511	−0.206311
$216$	0	0
$217$	14.0090	0.950996
$218$	46.1086	3.12287
$219$	0	0
$220$	−17.7992	−1.20002
$221$	−6.51496	−0.438243
$222$	0	0
$223$	13.7815	0.922876	0.461438	−	0.887172i	$-0.347334\pi$
$223$	13.7815	0.922876	0.461438	+	0.887172i	$0.347334\pi$
$224$	48.7215	3.25534
$225$	0	0
$226$	11.7992	0.784872
$227$	−29.1012	−1.93151	−0.965757	−	0.259447i	$-0.916460\pi$
$227$	−29.1012	−1.93151	−0.965757	+	0.259447i	$0.916460\pi$
$228$	0	0
$229$	−9.38705	−0.620314	−0.310157	−	0.950685i	$-0.600382\pi$
$229$	−9.38705	−0.620314	−0.310157	+	0.950685i	$0.600382\pi$
$230$	2.69353	0.177606
$231$	0	0
$232$	80.7156	5.29924
$233$	12.6725	0.830202	0.415101	−	0.909775i	$-0.363746\pi$
$233$	12.6725	0.830202	0.415101	+	0.909775i	$0.363746\pi$
$234$	0	0
$235$	8.21255	0.535728
$236$	−43.5257	−2.83328
$237$	0	0
$238$	19.5353	1.26628
$239$	−19.1883	−1.24119	−0.620596	−	0.784131i	$-0.713109\pi$
$239$	−19.1883	−1.24119	−0.620596	+	0.784131i	$0.713109\pi$
$240$	0	0
$241$	16.7741	1.08051	0.540257	−	0.841500i	$-0.318327\pi$
$241$	16.7741	1.08051	0.540257	+	0.841500i	$0.318327\pi$
$242$	−1.27164	−0.0817442
$243$	0	0
$244$	1.38705	0.0887968
$245$	−0.521423	−0.0333125
$246$	0	0
$247$	−8.34420	−0.530929
$248$	44.7859	2.84391
$249$	0	0
$250$	2.69353	0.170353
$251$	22.8114	1.43984	0.719921	−	0.694056i	$-0.244178\pi$
$251$	22.8114	1.43984	0.719921	+	0.694056i	$0.244178\pi$
$252$	0	0
$253$	3.38705	0.212942
$254$	49.7191	3.11966
$255$	0	0
$256$	18.0162	1.12602
$257$	23.7526	1.48165	0.740824	−	0.671699i	$-0.234435\pi$
$257$	23.7526	1.48165	0.740824	+	0.671699i	$0.234435\pi$
$258$	0	0
$259$	−15.0940	−0.937895
$260$	12.9462	0.802889
$261$	0	0
$262$	−10.4195	−0.643718
$263$	−16.5895	−1.02295	−0.511476	−	0.859297i	$-0.670901\pi$
$263$	−16.5895	−1.02295	−0.511476	+	0.859297i	$0.670901\pi$
$264$	0	0
$265$	−10.5417	−0.647574
$266$	25.0203	1.53409
$267$	0	0
$268$	−40.3046	−2.46199
$269$	8.47495	0.516727	0.258363	−	0.966048i	$-0.416817\pi$
$269$	8.47495	0.516727	0.258363	+	0.966048i	$0.416817\pi$
$270$	0	0
$271$	−21.3029	−1.29406	−0.647030	−	0.762465i	$-0.723989\pi$
$271$	−21.3029	−1.29406	−0.647030	+	0.762465i	$0.723989\pi$
$272$	34.6584	2.10147
$273$	0	0
$274$	55.6787	3.36367
$275$	3.38705	0.204247
$276$	0	0
$277$	21.9615	1.31954	0.659770	−	0.751467i	$-0.270654\pi$
$277$	21.9615	1.31954	0.659770	+	0.751467i	$0.270654\pi$
$278$	−33.8851	−2.03229
$279$	0	0
$280$	−24.0454	−1.43699
$281$	−19.9020	−1.18725	−0.593627	−	0.804740i	$-0.702305\pi$
$281$	−19.9020	−1.18725	−0.593627	+	0.804740i	$0.702305\pi$
$282$	0	0
$283$	−8.87781	−0.527731	−0.263865	−	0.964559i	$-0.584997\pi$
$283$	−8.87781	−0.527731	−0.263865	+	0.964559i	$0.584997\pi$
$284$	0.0789107	0.00468249
$285$	0	0
$286$	22.4753	1.32899
$287$	−3.30771	−0.195248
$288$	0	0
$289$	−10.0065	−0.588616
$290$	−24.7968	−1.45612
$291$	0	0
$292$	−29.0944	−1.70262
$293$	−23.9709	−1.40039	−0.700197	−	0.713950i	$-0.746904\pi$
$293$	−23.9709	−1.40039	−0.700197	+	0.713950i	$0.746904\pi$
$294$	0	0
$295$	8.28259	0.482231
$296$	−48.2543	−2.80473
$297$	0	0
$298$	30.9353	1.79203
$299$	−2.46356	−0.142471
$300$	0	0
$301$	−8.29644	−0.478199
$302$	15.0478	0.865905
$303$	0	0
$304$	44.3896	2.54592
$305$	−0.263945	−0.0151134
$306$	0	0
$307$	12.0632	0.688481	0.344240	−	0.938882i	$-0.388136\pi$
$307$	12.0632	0.688481	0.344240	+	0.938882i	$0.388136\pi$
$308$	−48.8147	−2.78148
$309$	0	0
$310$	−13.7588	−0.781446
$311$	6.81257	0.386305	0.193153	−	0.981169i	$-0.438129\pi$
$311$	6.81257	0.386305	0.193153	+	0.981169i	$0.438129\pi$
$312$	0	0
$313$	−6.38539	−0.360923	−0.180462	−	0.983582i	$-0.557759\pi$
$313$	−6.38539	−0.360923	−0.180462	+	0.983582i	$0.557759\pi$
$314$	−15.7992	−0.891601
$315$	0	0
$316$	−45.5936	−2.56484
$317$	−16.8114	−0.944222	−0.472111	−	0.881539i	$-0.656508\pi$
$317$	−16.8114	−0.944222	−0.472111	+	0.881539i	$0.656508\pi$
$318$	0	0
$319$	−31.1815	−1.74583
$320$	−21.6397	−1.20970
$321$	0	0
$322$	7.38705	0.411664
$323$	8.95715	0.498389
$324$	0	0
$325$	−2.46356	−0.136654
$326$	−0.125506	−0.00695115
$327$	0	0
$328$	−10.5745	−0.583878
$329$	22.5231	1.24174
$330$	0	0
$331$	3.29114	0.180898	0.0904488	−	0.995901i	$-0.471170\pi$
$331$	3.29114	0.180898	0.0904488	+	0.995901i	$0.471170\pi$
$332$	18.5054	1.01561
$333$	0	0
$334$	−6.05023	−0.331054
$335$	7.66964	0.419037
$336$	0	0
$337$	−7.90201	−0.430450	−0.215225	−	0.976565i	$-0.569048\pi$
$337$	−7.90201	−0.430450	−0.215225	+	0.976565i	$0.569048\pi$
$338$	18.6685	1.01543
$339$	0	0
$340$	−13.8972	−0.753682
$341$	−17.3014	−0.936921
$342$	0	0
$343$	17.7676	0.959362
$344$	−26.5231	−1.43003
$345$	0	0
$346$	21.5830	1.16031
$347$	19.4024	1.04158	0.520789	−	0.853686i	$-0.325638\pi$
$347$	19.4024	1.04158	0.520789	+	0.853686i	$0.325638\pi$
$348$	0	0
$349$	−33.1664	−1.77536	−0.887680	−	0.460462i	$-0.847684\pi$
$349$	−33.1664	−1.77536	−0.887680	+	0.460462i	$0.847684\pi$
$350$	7.38705	0.394854
$351$	0	0
$352$	−60.1717	−3.20716
$353$	29.8960	1.59121	0.795603	−	0.605819i	$-0.207154\pi$
$353$	29.8960	1.59121	0.795603	+	0.605819i	$0.207154\pi$
$354$	0	0
$355$	−0.0150161	−0.000796971 0
$356$	24.5054	1.29878
$357$	0	0
$358$	−32.2187	−1.70281
$359$	24.9725	1.31800	0.659000	−	0.752143i	$-0.270980\pi$
$359$	24.9725	1.31800	0.659000	+	0.752143i	$0.270980\pi$
$360$	0	0
$361$	−7.52789	−0.396205
$362$	47.0835	2.47465
$363$	0	0
$364$	35.5052	1.86098
$365$	5.53644	0.289790
$366$	0	0
$367$	15.8179	0.825686	0.412843	−	0.910802i	$-0.364536\pi$
$367$	15.8179	0.825686	0.412843	+	0.910802i	$0.364536\pi$
$368$	13.1057	0.683181
$369$	0	0
$370$	14.8243	0.770680
$371$	−28.9109	−1.50098
$372$	0	0
$373$	22.6283	1.17165	0.585826	−	0.810437i	$-0.300770\pi$
$373$	22.6283	1.17165	0.585826	+	0.810437i	$0.300770\pi$
$374$	−24.1263	−1.24754
$375$	0	0
$376$	72.0046	3.71335
$377$	22.6797	1.16807
$378$	0	0
$379$	10.2721	0.527641	0.263821	−	0.964572i	$-0.415017\pi$
$379$	10.2721	0.527641	0.263821	+	0.964572i	$0.415017\pi$
$380$	−17.7992	−0.913080
$381$	0	0
$382$	−35.3474	−1.80853
$383$	−6.21463	−0.317553	−0.158776	−	0.987315i	$-0.550755\pi$
$383$	−6.21463	−0.317553	−0.158776	+	0.987315i	$0.550755\pi$
$384$	0	0
$385$	9.28906	0.473414
$386$	33.7871	1.71972
$387$	0	0
$388$	−21.5353	−1.09329
$389$	−27.6414	−1.40147	−0.700737	−	0.713420i	$-0.747145\pi$
$389$	−27.6414	−1.40147	−0.700737	+	0.713420i	$0.747145\pi$
$390$	0	0
$391$	2.64453	0.133740
$392$	−4.57165	−0.230903
$393$	0	0
$394$	10.4655	0.527246
$395$	8.67611	0.436542
$396$	0	0
$397$	1.07171	0.0537875	0.0268938	−	0.999638i	$-0.491438\pi$
$397$	1.07171	0.0537875	0.0268938	+	0.999638i	$0.491438\pi$
$398$	−59.6916	−2.99207
$399$	0	0
$400$	13.1057	0.655284
$401$	−1.49798	−0.0748053	−0.0374027	−	0.999300i	$-0.511908\pi$
$401$	−1.49798	−0.0748053	−0.0374027	+	0.999300i	$0.511908\pi$
$402$	0	0
$403$	12.5841	0.626858
$404$	−64.6139	−3.21466
$405$	0	0
$406$	−68.0058	−3.37507
$407$	18.6413	0.924014
$408$	0	0
$409$	20.9373	1.03528	0.517642	−	0.855597i	$-0.326810\pi$
$409$	20.9373	1.03528	0.517642	+	0.855597i	$0.326810\pi$
$410$	3.24861	0.160438
$411$	0	0
$412$	31.7944	1.56640
$413$	22.7152	1.11774
$414$	0	0
$415$	−3.52142	−0.172860
$416$	43.7657	2.14579
$417$	0	0
$418$	−30.9004	−1.51139
$419$	18.7564	0.916309	0.458154	−	0.888873i	$-0.348511\pi$
$419$	18.7564	0.916309	0.458154	+	0.888873i	$0.348511\pi$
$420$	0	0
$421$	20.6163	1.00478	0.502388	−	0.864642i	$-0.332455\pi$
$421$	20.6163	1.00478	0.502388	+	0.864642i	$0.332455\pi$
$422$	−12.9572	−0.630744
$423$	0	0
$424$	−92.4261	−4.48861
$425$	2.64453	0.128279
$426$	0	0
$427$	−0.723874	−0.0350307
$428$	−69.1289	−3.34147
$429$	0	0
$430$	8.14822	0.392942
$431$	−27.0155	−1.30129	−0.650646	−	0.759381i	$-0.725502\pi$
$431$	−27.0155	−1.30129	−0.650646	+	0.759381i	$0.725502\pi$
$432$	0	0
$433$	−25.2900	−1.21536	−0.607679	−	0.794183i	$-0.707899\pi$
$433$	−25.2900	−1.21536	−0.607679	+	0.794183i	$0.707899\pi$
$434$	−37.7337	−1.81128
$435$	0	0
$436$	−89.9580	−4.30821
$437$	3.38705	0.162025
$438$	0	0
$439$	34.4110	1.64235	0.821174	−	0.570679i	$-0.193320\pi$
$439$	34.4110	1.64235	0.821174	+	0.570679i	$0.193320\pi$
$440$	29.6964	1.41572
$441$	0	0
$442$	17.5482	0.834683
$443$	29.8281	1.41717	0.708587	−	0.705623i	$-0.249333\pi$
$443$	29.8281	1.41717	0.708587	+	0.705623i	$0.249333\pi$
$444$	0	0
$445$	−4.66318	−0.221056
$446$	−37.1208	−1.75772
$447$	0	0
$448$	−59.3474	−2.80390
$449$	2.67456	0.126220	0.0631102	−	0.998007i	$-0.479898\pi$
$449$	2.67456	0.126220	0.0631102	+	0.998007i	$0.479898\pi$
$450$	0	0
$451$	4.08506	0.192358
$452$	−23.0203	−1.08278
$453$	0	0
$454$	78.3848	3.67878
$455$	−6.75637	−0.316743
$456$	0	0
$457$	−30.9037	−1.44561	−0.722806	−	0.691051i	$-0.757148\pi$
$457$	−30.9037	−1.44561	−0.722806	+	0.691051i	$0.757148\pi$
$458$	25.2843	1.18146
$459$	0	0
$460$	−5.25508	−0.245019
$461$	26.6022	1.23899	0.619493	−	0.785002i	$-0.287338\pi$
$461$	26.6022	1.23899	0.619493	+	0.785002i	$0.287338\pi$
$462$	0	0
$463$	26.7013	1.24092	0.620458	−	0.784239i	$-0.286947\pi$
$463$	26.7013	1.24092	0.620458	+	0.784239i	$0.286947\pi$
$464$	−120.652	−5.60113
$465$	0	0
$466$	−34.1336	−1.58121
$467$	−35.2503	−1.63119	−0.815596	−	0.578622i	$-0.803590\pi$
$467$	−35.2503	−1.63119	−0.815596	+	0.578622i	$0.803590\pi$
$468$	0	0
$469$	21.0342	0.971267
$470$	−22.1207	−1.02035
$471$	0	0
$472$	72.6187	3.34255
$473$	10.2462	0.471121
$474$	0	0
$475$	3.38705	0.155409
$476$	−38.1134	−1.74692
$477$	0	0
$478$	51.6843	2.36398
$479$	39.0705	1.78518	0.892589	−	0.450871i	$-0.148886\pi$
$479$	39.0705	1.78518	0.892589	+	0.450871i	$0.148886\pi$
$480$	0	0
$481$	−13.5587	−0.618222
$482$	−45.1815	−2.05796
$483$	0	0
$484$	2.48098	0.112772
$485$	4.09799	0.186080
$486$	0	0
$487$	24.0839	1.09135	0.545673	−	0.837998i	$-0.316274\pi$
$487$	24.0839	1.09135	0.545673	+	0.837998i	$0.316274\pi$
$488$	−2.31417	−0.104758
$489$	0	0
$490$	1.40447	0.0634474
$491$	−5.60051	−0.252748	−0.126374	−	0.991983i	$-0.540334\pi$
$491$	−5.60051	−0.252748	−0.126374	+	0.991983i	$0.540334\pi$
$492$	0	0
$493$	−24.3458	−1.09648
$494$	22.4753	1.01121
$495$	0	0
$496$	−66.9450	−3.00592
$497$	−0.0411819	−0.00184726
$498$	0	0
$499$	9.53319	0.426764	0.213382	−	0.976969i	$-0.431552\pi$
$499$	9.53319	0.426764	0.213382	+	0.976969i	$0.431552\pi$
$500$	−5.25508	−0.235014
$501$	0	0
$502$	−61.4431	−2.74234
$503$	−14.0568	−0.626762	−0.313381	−	0.949627i	$-0.601462\pi$
$503$	−14.0568	−0.626762	−0.313381	+	0.949627i	$0.601462\pi$
$504$	0	0
$505$	12.2955	0.547144
$506$	−9.12311	−0.405572
$507$	0	0
$508$	−97.0022	−4.30378
$509$	−21.7105	−0.962302	−0.481151	−	0.876638i	$-0.659781\pi$
$509$	−21.7105	−0.962302	−0.481151	+	0.876638i	$0.659781\pi$
$510$	0	0
$511$	15.1838	0.671692
$512$	−3.01385	−0.133194
$513$	0	0
$514$	−63.9783	−2.82196
$515$	−6.05023	−0.266605
$516$	0	0
$517$	−27.8163	−1.22336
$518$	40.6560	1.78632
$519$	0	0
$520$	−21.5996	−0.947205
$521$	8.87689	0.388904	0.194452	−	0.980912i	$-0.437707\pi$
$521$	8.87689	0.388904	0.194452	+	0.980912i	$0.437707\pi$
$522$	0	0
$523$	−0.0550279	−0.00240620	−0.00120310	−	0.999999i	$-0.500383\pi$
$523$	−0.0550279	−0.00240620	−0.00120310	+	0.999999i	$0.500383\pi$
$524$	20.3285	0.888053
$525$	0	0
$526$	44.6842	1.94833
$527$	−13.5085	−0.588439
$528$	0	0
$529$	1.00000	0.0434783
$530$	28.3944	1.23338
$531$	0	0
$532$	−48.8147	−2.11639
$533$	−2.97126	−0.128699
$534$	0	0
$535$	13.1547	0.568727
$536$	67.2446	2.90453
$537$	0	0
$538$	−22.8275	−0.984163
$539$	1.76609	0.0760708
$540$	0	0
$541$	5.99070	0.257560	0.128780	−	0.991673i	$-0.458894\pi$
$541$	5.99070	0.257560	0.128780	+	0.991673i	$0.458894\pi$
$542$	57.3799	2.46468
$543$	0	0
$544$	−46.9807	−2.01428
$545$	17.1183	0.733268
$546$	0	0
$547$	−0.408533	−0.0174676	−0.00873380	−	0.999962i	$-0.502780\pi$
$547$	−0.408533	−0.0174676	−0.00873380	+	0.999962i	$0.502780\pi$
$548$	−108.629	−4.64042
$549$	0	0
$550$	−9.12311	−0.389011
$551$	−31.1815	−1.32837
$552$	0	0
$553$	23.7944	1.01184
$554$	−59.1539	−2.51321
$555$	0	0
$556$	66.1099	2.80369
$557$	5.46406	0.231519	0.115760	−	0.993277i	$-0.463070\pi$
$557$	5.46406	0.231519	0.115760	+	0.993277i	$0.463070\pi$
$558$	0	0
$559$	−7.45255	−0.315209
$560$	35.9426	1.51885
$561$	0	0
$562$	53.6066	2.26126
$563$	41.8212	1.76255	0.881277	−	0.472601i	$-0.156685\pi$
$563$	41.8212	1.76255	0.881277	+	0.472601i	$0.156685\pi$
$564$	0	0
$565$	4.38058	0.184293
$566$	23.9126	1.00512
$567$	0	0
$568$	−0.131656	−0.00552415
$569$	−39.5127	−1.65646	−0.828230	−	0.560388i	$-0.810652\pi$
$569$	−39.5127	−1.65646	−0.828230	+	0.560388i	$0.810652\pi$
$570$	0	0
$571$	−24.4924	−1.02498	−0.512488	−	0.858694i	$-0.671276\pi$
$571$	−24.4924	−1.02498	−0.512488	+	0.858694i	$0.671276\pi$
$572$	−43.8494	−1.83344
$573$	0	0
$574$	8.90939	0.371871
$575$	1.00000	0.0417029
$576$	0	0
$577$	−36.3382	−1.51278	−0.756390	−	0.654121i	$-0.773039\pi$
$577$	−36.3382	−1.51278	−0.756390	+	0.654121i	$0.773039\pi$
$578$	26.9527	1.12108
$579$	0	0
$580$	48.3787	2.00882
$581$	−9.65758	−0.400664
$582$	0	0
$583$	35.7054	1.47877
$584$	48.5415	2.00866
$585$	0	0
$586$	64.5662	2.66720
$587$	−5.89358	−0.243254	−0.121627	−	0.992576i	$-0.538811\pi$
$587$	−5.89358	−0.243254	−0.121627	+	0.992576i	$0.538811\pi$
$588$	0	0
$589$	−17.3014	−0.712890
$590$	−22.3094	−0.918462
$591$	0	0
$592$	72.1296	2.96451
$593$	11.0931	0.455538	0.227769	−	0.973715i	$-0.426857\pi$
$593$	11.0931	0.455538	0.227769	+	0.973715i	$0.426857\pi$
$594$	0	0
$595$	7.25268	0.297331
$596$	−60.3548	−2.47223
$597$	0	0
$598$	6.63566	0.271352
$599$	−40.1864	−1.64197	−0.820986	−	0.570949i	$-0.806575\pi$
$599$	−40.1864	−1.64197	−0.820986	+	0.570949i	$0.806575\pi$
$600$	0	0
$601$	41.1965	1.68044	0.840220	−	0.542246i	$-0.182426\pi$
$601$	41.1965	1.68044	0.840220	+	0.542246i	$0.182426\pi$
$602$	22.3467	0.910782
$603$	0	0
$604$	−29.3584	−1.19457
$605$	−0.472110	−0.0191940
$606$	0	0
$607$	−16.6283	−0.674924	−0.337462	−	0.941339i	$-0.609568\pi$
$607$	−16.6283	−0.674924	−0.337462	+	0.941339i	$0.609568\pi$
$608$	−60.1717	−2.44029
$609$	0	0
$610$	0.710942	0.0287852
$611$	20.2321	0.818504
$612$	0	0
$613$	25.5052	1.03015	0.515073	−	0.857146i	$-0.327765\pi$
$613$	25.5052	1.03015	0.515073	+	0.857146i	$0.327765\pi$
$614$	−32.4924	−1.31129
$615$	0	0
$616$	81.4431	3.28143
$617$	3.14742	0.126710	0.0633552	−	0.997991i	$-0.479820\pi$
$617$	3.14742	0.126710	0.0633552	+	0.997991i	$0.479820\pi$
$618$	0	0
$619$	−39.2665	−1.57825	−0.789127	−	0.614230i	$-0.789467\pi$
$619$	−39.2665	−1.57825	−0.789127	+	0.614230i	$0.789467\pi$
$620$	26.8434	1.07806
$621$	0	0
$622$	−18.3498	−0.735761
$623$	−12.7889	−0.512375
$624$	0	0
$625$	1.00000	0.0400000
$626$	17.1992	0.687418
$627$	0	0
$628$	30.8243	1.23002
$629$	14.5547	0.580333
$630$	0	0
$631$	18.2916	0.728179	0.364089	−	0.931364i	$-0.381380\pi$
$631$	18.2916	0.728179	0.364089	+	0.931364i	$0.381380\pi$
$632$	76.0689	3.02586
$633$	0	0
$634$	45.2819	1.79837
$635$	18.4588	0.732514
$636$	0	0
$637$	−1.28456	−0.0508960
$638$	83.9881	3.32512
$639$	0	0
$640$	22.7567	0.899537
$641$	−28.5304	−1.12688	−0.563441	−	0.826157i	$-0.690523\pi$
$641$	−28.5304	−1.12688	−0.563441	+	0.826157i	$0.690523\pi$
$642$	0	0
$643$	−12.2430	−0.482815	−0.241408	−	0.970424i	$-0.577609\pi$
$643$	−12.2430	−0.482815	−0.241408	+	0.970424i	$0.577609\pi$
$644$	−14.4122	−0.567919
$645$	0	0
$646$	−24.1263	−0.949237
$647$	−3.11947	−0.122639	−0.0613196	−	0.998118i	$-0.519531\pi$
$647$	−3.11947	−0.122639	−0.0613196	+	0.998118i	$0.519531\pi$
$648$	0	0
$649$	−28.0536	−1.10120
$650$	6.63566	0.260272
$651$	0	0
$652$	0.244863	0.00958958
$653$	21.7301	0.850364	0.425182	−	0.905108i	$-0.360210\pi$
$653$	21.7301	0.850364	0.425182	+	0.905108i	$0.360210\pi$
$654$	0	0
$655$	−3.86834	−0.151149
$656$	15.8065	0.617141
$657$	0	0
$658$	−60.6665	−2.36503
$659$	12.2333	0.476541	0.238270	−	0.971199i	$-0.423420\pi$
$659$	12.2333	0.476541	0.238270	+	0.971199i	$0.423420\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	−8.86477	−0.344539
$663$	0	0
$664$	−30.8746	−1.19817
$665$	9.28906	0.360214
$666$	0	0
$667$	−9.20608	−0.356461
$668$	11.8040	0.456711
$669$	0	0
$670$	−20.6584	−0.798103
$671$	0.893994	0.0345123
$672$	0	0
$673$	10.7900	0.415925	0.207963	−	0.978137i	$-0.433317\pi$
$673$	10.7900	0.415925	0.207963	+	0.978137i	$0.433317\pi$
$674$	21.2843	0.819839
$675$	0	0
$676$	−36.4223	−1.40086
$677$	−23.6721	−0.909793	−0.454896	−	0.890544i	$-0.650324\pi$
$677$	−23.6721	−0.909793	−0.454896	+	0.890544i	$0.650324\pi$
$678$	0	0
$679$	11.2388	0.431307
$680$	23.1863	0.889153
$681$	0	0
$682$	46.6016	1.78447
$683$	10.3583	0.396350	0.198175	−	0.980167i	$-0.436499\pi$
$683$	10.3583	0.396350	0.198175	+	0.980167i	$0.436499\pi$
$684$	0	0
$685$	20.6713	0.789810
$686$	−47.8576	−1.82721
$687$	0	0
$688$	39.6462	1.51150
$689$	−25.9702	−0.989386
$690$	0	0
$691$	13.5578	0.515763	0.257882	−	0.966177i	$-0.416976\pi$
$691$	13.5578	0.515763	0.257882	+	0.966177i	$0.416976\pi$
$692$	−42.1086	−1.60073
$693$	0	0
$694$	−52.2610	−1.98380
$695$	−12.5802	−0.477194
$696$	0	0
$697$	3.18952	0.120812
$698$	89.3347	3.38137
$699$	0	0
$700$	−14.4122	−0.544729
$701$	21.3774	0.807415	0.403708	−	0.914888i	$-0.367721\pi$
$701$	21.3774	0.807415	0.403708	+	0.914888i	$0.367721\pi$
$702$	0	0
$703$	18.6413	0.703069
$704$	73.2948	2.76240
$705$	0	0
$706$	−80.5257	−3.03063
$707$	33.7207	1.26820
$708$	0	0
$709$	30.3719	1.14064	0.570320	−	0.821422i	$-0.306819\pi$
$709$	30.3719	1.14064	0.570320	+	0.821422i	$0.306819\pi$
$710$	0.0404462	0.00151792
$711$	0	0
$712$	−40.8850	−1.53223
$713$	−5.10809	−0.191299
$714$	0	0
$715$	8.34420	0.312056
$716$	62.8588	2.34914
$717$	0	0
$718$	−67.2642	−2.51028
$719$	−8.41851	−0.313958	−0.156979	−	0.987602i	$-0.550175\pi$
$719$	−8.41851	−0.313958	−0.156979	+	0.987602i	$0.550175\pi$
$720$	0	0
$721$	−16.5929	−0.617951
$722$	20.2766	0.754615
$723$	0	0
$724$	−91.8600	−3.41395
$725$	−9.20608	−0.341905
$726$	0	0
$727$	19.7505	0.732507	0.366253	−	0.930515i	$-0.380640\pi$
$727$	19.7505	0.732507	0.366253	+	0.930515i	$0.380640\pi$
$728$	−59.2374	−2.19548
$729$	0	0
$730$	−14.9125	−0.551938
$731$	8.00000	0.295891
$732$	0	0
$733$	19.7280	0.728670	0.364335	−	0.931268i	$-0.381296\pi$
$733$	19.7280	0.728670	0.364335	+	0.931268i	$0.381296\pi$
$734$	−42.6058	−1.57261
$735$	0	0
$736$	−17.7652	−0.654835
$737$	−25.9775	−0.956892
$738$	0	0
$739$	−0.180969	−0.00665704	−0.00332852	−	0.999994i	$-0.501060\pi$
$739$	−0.180969	−0.00665704	−0.00332852	+	0.999994i	$0.501060\pi$
$740$	−28.9223	−1.06321
$741$	0	0
$742$	77.8723	2.85878
$743$	−6.05023	−0.221961	−0.110981	−	0.993823i	$-0.535399\pi$
$743$	−6.05023	−0.221961	−0.110981	+	0.993823i	$0.535399\pi$
$744$	0	0
$745$	11.4850	0.420779
$746$	−60.9500	−2.23154
$747$	0	0
$748$	47.0705	1.72107
$749$	36.0770	1.31823
$750$	0	0
$751$	−21.5659	−0.786952	−0.393476	−	0.919335i	$-0.628728\pi$
$751$	−21.5659	−0.786952	−0.393476	+	0.919335i	$0.628728\pi$
$752$	−107.631	−3.92490
$753$	0	0
$754$	−61.0885	−2.22471
$755$	5.58667	0.203320
$756$	0	0
$757$	−24.7798	−0.900638	−0.450319	−	0.892868i	$-0.648690\pi$
$757$	−24.7798	−0.900638	−0.450319	+	0.892868i	$0.648690\pi$
$758$	−27.6681	−1.00495
$759$	0	0
$760$	29.6964	1.07720
$761$	−2.29916	−0.0833443	−0.0416722	−	0.999131i	$-0.513269\pi$
$761$	−2.29916	−0.0833443	−0.0416722	+	0.999131i	$0.513269\pi$
$762$	0	0
$763$	46.9473	1.69961
$764$	68.9629	2.49499
$765$	0	0
$766$	16.7393	0.604814
$767$	20.4047	0.736770
$768$	0	0
$769$	−10.3692	−0.373923	−0.186961	−	0.982367i	$-0.559864\pi$
$769$	−10.3692	−0.373923	−0.186961	+	0.982367i	$0.559864\pi$
$770$	−25.0203	−0.901669
$771$	0	0
$772$	−65.9187	−2.37247
$773$	−12.7864	−0.459895	−0.229947	−	0.973203i	$-0.573855\pi$
$773$	−12.7864	−0.459895	−0.229947	+	0.973203i	$0.573855\pi$
$774$	0	0
$775$	−5.10809	−0.183488
$776$	35.9297	1.28980
$777$	0	0
$778$	74.4528	2.66926
$779$	4.08506	0.146362
$780$	0	0
$781$	0.0508602	0.00181992
$782$	−7.12311	−0.254722
$783$	0	0
$784$	6.83361	0.244058
$785$	−5.86563	−0.209353
$786$	0	0
$787$	−34.9239	−1.24490	−0.622451	−	0.782659i	$-0.713863\pi$
$787$	−34.9239	−1.24490	−0.622451	+	0.782659i	$0.713863\pi$
$788$	−20.4183	−0.727372
$789$	0	0
$790$	−23.3693	−0.831443
$791$	12.0138	0.427163
$792$	0	0
$793$	−0.650244	−0.0230908
$794$	−2.88667	−0.102444
$795$	0	0
$796$	116.459	4.12776
$797$	−4.87844	−0.172803	−0.0864016	−	0.996260i	$-0.527537\pi$
$797$	−4.87844	−0.172803	−0.0864016	+	0.996260i	$0.527537\pi$
$798$	0	0
$799$	−21.7183	−0.768339
$800$	−17.7652	−0.628096
$801$	0	0
$802$	4.03483	0.142475
$803$	−18.7522	−0.661751
$804$	0	0
$805$	2.74252	0.0966612
$806$	−33.8956	−1.19392
$807$	0	0
$808$	107.803	3.79248
$809$	−18.1498	−0.638112	−0.319056	−	0.947736i	$-0.603366\pi$
$809$	−18.1498	−0.638112	−0.319056	+	0.947736i	$0.603366\pi$
$810$	0	0
$811$	17.8019	0.625110	0.312555	−	0.949900i	$-0.398815\pi$
$811$	17.8019	0.625110	0.312555	+	0.949900i	$0.398815\pi$
$812$	132.680	4.65614
$813$	0	0
$814$	−50.2107	−1.75989
$815$	−0.0465955	−0.00163217
$816$	0	0
$817$	10.2462	0.358470
$818$	−56.3952	−1.97181
$819$	0	0
$820$	−6.33805	−0.221334
$821$	−32.9701	−1.15066	−0.575332	−	0.817920i	$-0.695127\pi$
$821$	−32.9701	−1.15066	−0.575332	+	0.817920i	$0.695127\pi$
$822$	0	0
$823$	−39.5819	−1.37974	−0.689869	−	0.723935i	$-0.742332\pi$
$823$	−39.5819	−1.37974	−0.689869	+	0.723935i	$0.742332\pi$
$824$	−53.0462	−1.84795
$825$	0	0
$826$	−61.1839	−2.12886
$827$	1.50849	0.0524554	0.0262277	−	0.999656i	$-0.491651\pi$
$827$	1.50849	0.0524554	0.0262277	+	0.999656i	$0.491651\pi$
$828$	0	0
$829$	−7.66718	−0.266292	−0.133146	−	0.991096i	$-0.542508\pi$
$829$	−7.66718	−0.266292	−0.133146	+	0.991096i	$0.542508\pi$
$830$	9.48504	0.329231
$831$	0	0
$832$	−53.3108	−1.84822
$833$	1.37892	0.0477767
$834$	0	0
$835$	−2.24621	−0.0777333
$836$	60.2868	2.08506
$837$	0	0
$838$	−50.5207	−1.74521
$839$	8.53602	0.294696	0.147348	−	0.989085i	$-0.452926\pi$
$839$	8.53602	0.294696	0.147348	+	0.989085i	$0.452926\pi$
$840$	0	0
$841$	55.7519	1.92248
$842$	−55.5305	−1.91371
$843$	0	0
$844$	25.2795	0.870155
$845$	6.93087	0.238429
$846$	0	0
$847$	−1.29477	−0.0444889
$848$	138.157	4.74432
$849$	0	0
$850$	−7.12311	−0.244321
$851$	5.50369	0.188664
$852$	0	0
$853$	−48.0665	−1.64577	−0.822883	−	0.568211i	$-0.807636\pi$
$853$	−48.0665	−1.64577	−0.822883	+	0.568211i	$0.807636\pi$
$854$	1.94977	0.0667199
$855$	0	0
$856$	115.335	3.94209
$857$	−29.4313	−1.00535	−0.502677	−	0.864474i	$-0.667652\pi$
$857$	−29.4313	−1.00535	−0.502677	+	0.864474i	$0.667652\pi$
$858$	0	0
$859$	−33.5308	−1.14406	−0.572029	−	0.820234i	$-0.693843\pi$
$859$	−33.5308	−1.14406	−0.572029	+	0.820234i	$0.693843\pi$
$860$	−15.8972	−0.542090
$861$	0	0
$862$	72.7670	2.47845
$863$	−6.79483	−0.231299	−0.115649	−	0.993290i	$-0.536895\pi$
$863$	−6.79483	−0.231299	−0.115649	+	0.993290i	$0.536895\pi$
$864$	0	0
$865$	8.01293	0.272448
$866$	68.1192	2.31478
$867$	0	0
$868$	73.6186	2.49878
$869$	−29.3864	−0.996866
$870$	0	0
$871$	18.8946	0.640220
$872$	150.087	5.08259
$873$	0	0
$874$	−9.12311	−0.308594
$875$	2.74252	0.0927141
$876$	0	0
$877$	−21.9904	−0.742563	−0.371281	−	0.928520i	$-0.621082\pi$
$877$	−21.9904	−0.742563	−0.371281	+	0.928520i	$0.621082\pi$
$878$	−92.6869	−3.12803
$879$	0	0
$880$	−44.3896	−1.49637
$881$	−3.66242	−0.123390	−0.0616951	−	0.998095i	$-0.519651\pi$
$881$	−3.66242	−0.123390	−0.0616951	+	0.998095i	$0.519651\pi$
$882$	0	0
$883$	10.6640	0.358874	0.179437	−	0.983769i	$-0.442572\pi$
$883$	10.6640	0.358874	0.179437	+	0.983769i	$0.442572\pi$
$884$	−34.2366	−1.15150
$885$	0	0
$886$	−80.3427	−2.69916
$887$	−0.481294	−0.0161603	−0.00808014	−	0.999967i	$-0.502572\pi$
$887$	−0.481294	−0.0161603	−0.00808014	+	0.999967i	$0.502572\pi$
$888$	0	0
$889$	50.6235	1.69786
$890$	12.5604	0.421025
$891$	0	0
$892$	72.4228	2.42489
$893$	−27.8163	−0.930837
$894$	0	0
$895$	−11.9615	−0.399830
$896$	62.4107	2.08499
$897$	0	0
$898$	−7.20400	−0.240401
$899$	47.0255	1.56839
$900$	0	0
$901$	27.8779	0.928748
$902$	−11.0032	−0.366367
$903$	0	0
$904$	38.4074	1.27741
$905$	17.4802	0.581063
$906$	0	0
$907$	−50.3078	−1.67044	−0.835222	−	0.549913i	$-0.814661\pi$
$907$	−50.3078	−1.67044	−0.835222	+	0.549913i	$0.814661\pi$
$908$	−152.929	−5.07513
$909$	0	0
$910$	18.1984	0.603273
$911$	39.5103	1.30903	0.654517	−	0.756047i	$-0.272872\pi$
$911$	39.5103	1.30903	0.654517	+	0.756047i	$0.272872\pi$
$912$	0	0
$913$	11.9272	0.394734
$914$	83.2398	2.75333
$915$	0	0
$916$	−49.3297	−1.62990
$917$	−10.6090	−0.350341
$918$	0	0
$919$	−10.0307	−0.330881	−0.165441	−	0.986220i	$-0.552905\pi$
$919$	−10.0307	−0.330881	−0.165441	+	0.986220i	$0.552905\pi$
$920$	8.76763	0.289060
$921$	0	0
$922$	−71.6536	−2.35979
$923$	−0.0369930	−0.00121764
$924$	0	0
$925$	5.50369	0.180960
$926$	−71.9207	−2.36346
$927$	0	0
$928$	163.548	5.36873
$929$	−5.83676	−0.191498	−0.0957490	−	0.995406i	$-0.530525\pi$
$929$	−5.83676	−0.191498	−0.0957490	+	0.995406i	$0.530525\pi$
$930$	0	0
$931$	1.76609	0.0578812
$932$	66.5949	2.18139
$933$	0	0
$934$	94.9477	3.10678
$935$	−8.95715	−0.292930
$936$	0	0
$937$	37.5175	1.22564	0.612822	−	0.790221i	$-0.290034\pi$
$937$	37.5175	1.22564	0.612822	+	0.790221i	$0.290034\pi$
$938$	−56.6560	−1.84989
$939$	0	0
$940$	43.1576	1.40765
$941$	−45.6189	−1.48713	−0.743566	−	0.668662i	$-0.766867\pi$
$941$	−45.6189	−1.48713	−0.743566	+	0.668662i	$0.766867\pi$
$942$	0	0
$943$	1.20608	0.0392754
$944$	−108.549	−3.53297
$945$	0	0
$946$	−27.5984	−0.897302
$947$	17.3357	0.563333	0.281667	−	0.959512i	$-0.409113\pi$
$947$	17.3357	0.563333	0.281667	+	0.959512i	$0.409113\pi$
$948$	0	0
$949$	13.6394	0.442752
$950$	−9.12311	−0.295993
$951$	0	0
$952$	63.5888	2.06093
$953$	−14.2706	−0.462269	−0.231135	−	0.972922i	$-0.574244\pi$
$953$	−14.2706	−0.462269	−0.231135	+	0.972922i	$0.574244\pi$
$954$	0	0
$955$	−13.1231	−0.424654
$956$	−100.836	−3.26128
$957$	0	0
$958$	−105.237	−3.40007
$959$	56.6915	1.83066
$960$	0	0
$961$	−4.90742	−0.158304
$962$	36.5206	1.17747
$963$	0	0
$964$	88.1492	2.83909
$965$	12.5438	0.403800
$966$	0	0
$967$	−4.07663	−0.131096	−0.0655478	−	0.997849i	$-0.520879\pi$
$967$	−4.07663	−0.131096	−0.0655478	+	0.997849i	$0.520879\pi$
$968$	−4.13929	−0.133042
$969$	0	0
$970$	−11.0380	−0.354410
$971$	20.2988	0.651419	0.325709	−	0.945470i	$-0.394397\pi$
$971$	20.2988	0.651419	0.325709	+	0.945470i	$0.394397\pi$
$972$	0	0
$973$	−34.5015	−1.10607
$974$	−64.8706	−2.07859
$975$	0	0
$976$	3.45918	0.110726
$977$	40.3782	1.29181	0.645907	−	0.763416i	$-0.276479\pi$
$977$	40.3782	1.29181	0.645907	+	0.763416i	$0.276479\pi$
$978$	0	0
$979$	15.7944	0.504792
$980$	−2.74012	−0.0875299
$981$	0	0
$982$	15.0851	0.481386
$983$	53.1628	1.69563	0.847815	−	0.530292i	$-0.177918\pi$
$983$	53.1628	1.69563	0.847815	+	0.530292i	$0.177918\pi$
$984$	0	0
$985$	3.88544	0.123801
$986$	65.5759	2.08836
$987$	0	0
$988$	−43.8494	−1.39504
$989$	3.02511	0.0961930
$990$	0	0
$991$	36.6746	1.16501	0.582503	−	0.812829i	$-0.302073\pi$
$991$	36.6746	1.16501	0.582503	+	0.812829i	$0.302073\pi$
$992$	90.7464	2.88120
$993$	0	0
$994$	0.110925	0.00351831
$995$	−22.1612	−0.702556
$996$	0	0
$997$	−11.1189	−0.352140	−0.176070	−	0.984378i	$-0.556339\pi$
$997$	−11.1189	−0.352140	−0.176070	+	0.984378i	$0.556339\pi$
$998$	−25.6779	−0.812819
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1035.2.a.o.1.1		4
3.2	odd	2		115.2.a.c.1.4	✓	4
5.4	even	2		5175.2.a.bx.1.4		4
12.11	even	2		1840.2.a.u.1.4		4
15.2	even	4		575.2.b.e.24.8		8
15.8	even	4		575.2.b.e.24.1		8
15.14	odd	2		575.2.a.h.1.1		4
21.20	even	2		5635.2.a.v.1.4		4
24.5	odd	2		7360.2.a.cj.1.3		4
24.11	even	2		7360.2.a.cg.1.2		4
60.59	even	2		9200.2.a.cl.1.1		4
69.68	even	2		2645.2.a.m.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.a.c.1.4	✓	4	3.2	odd	2
575.2.a.h.1.1		4	15.14	odd	2
575.2.b.e.24.1		8	15.8	even	4
575.2.b.e.24.8		8	15.2	even	4
1035.2.a.o.1.1		4	1.1	even	1	trivial
1840.2.a.u.1.4		4	12.11	even	2
2645.2.a.m.1.4		4	69.68	even	2
5175.2.a.bx.1.4		4	5.4	even	2
5635.2.a.v.1.4		4	21.20	even	2
7360.2.a.cg.1.2		4	24.11	even	2
7360.2.a.cj.1.3		4	24.5	odd	2
9200.2.a.cl.1.1		4	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1035 = 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1035.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69353 q^{2} +5.25508 q^{4} -1.00000 q^{5} -2.74252 q^{7} -8.76763 q^{8} +O(q^{10})\)	\(q-2.69353 q^{2} +5.25508 q^{4} -1.00000 q^{5} -2.74252 q^{7} -8.76763 q^{8} +2.69353 q^{10} +3.38705 q^{11} -2.46356 q^{13} +7.38705 q^{14} +13.1057 q^{16} +2.64453 q^{17} +3.38705 q^{19} -5.25508 q^{20} -9.12311 q^{22} +1.00000 q^{23} +1.00000 q^{25} +6.63566 q^{26} -14.4122 q^{28} -9.20608 q^{29} -5.10809 q^{31} -17.7652 q^{32} -7.12311 q^{34} +2.74252 q^{35} +5.50369 q^{37} -9.12311 q^{38} +8.76763 q^{40} +1.20608 q^{41} +3.02511 q^{43} +17.7992 q^{44} -2.69353 q^{46} -8.21255 q^{47} +0.521423 q^{49} -2.69353 q^{50} -12.9462 q^{52} +10.5417 q^{53} -3.38705 q^{55} +24.0454 q^{56} +24.7968 q^{58} -8.28259 q^{59} +0.263945 q^{61} +13.7588 q^{62} +21.6397 q^{64} +2.46356 q^{65} -7.66964 q^{67} +13.8972 q^{68} -7.38705 q^{70} +0.0150161 q^{71} -5.53644 q^{73} -14.8243 q^{74} +17.7992 q^{76} -9.28906 q^{77} -8.67611 q^{79} -13.1057 q^{80} -3.24861 q^{82} +3.52142 q^{83} -2.64453 q^{85} -8.14822 q^{86} -29.6964 q^{88} +4.66318 q^{89} +6.75637 q^{91} +5.25508 q^{92} +22.1207 q^{94} -3.38705 q^{95} -4.09799 q^{97} -1.40447 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 4 q^{4} - 4 q^{5} - 3 q^{7} - 9 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 - 3 * q^7 - 9 * q^8	\( 4 q - 2 q^{2} + 4 q^{4} - 4 q^{5} - 3 q^{7} - 9 q^{8} + 2 q^{10} - 4 q^{11} + 12 q^{14} + 8 q^{16} + q^{17} - 4 q^{19} - 4 q^{20} - 20 q^{22} + 4 q^{23} + 4 q^{25} + q^{26} - 22 q^{28} - 19 q^{29} - q^{31} - 20 q^{32} - 12 q^{34} + 3 q^{35} - 3 q^{37} - 20 q^{38} + 9 q^{40} - 13 q^{41} - 6 q^{43} + 18 q^{44} - 2 q^{46} - 6 q^{47} + 9 q^{49} - 2 q^{50} - q^{52} - 19 q^{53} + 4 q^{55} + 10 q^{56} + 21 q^{58} - 23 q^{59} + 13 q^{62} + 27 q^{64} - 3 q^{67} + 4 q^{68} - 12 q^{70} + 3 q^{71} - 32 q^{73} + 12 q^{74} + 18 q^{76} - 18 q^{77} + 2 q^{79} - 8 q^{80} - 5 q^{82} + 21 q^{83} - q^{85} + 2 q^{86} - 14 q^{88} - 40 q^{91} + 4 q^{92} + 47 q^{94} + 4 q^{95} - 18 q^{97} - 16 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 4 * q^4 - 4 * q^5 - 3 * q^7 - 9 * q^8 + 2 * q^10 - 4 * q^11 + 12 * q^14 + 8 * q^16 + q^17 - 4 * q^19 - 4 * q^20 - 20 * q^22 + 4 * q^23 + 4 * q^25 + q^26 - 22 * q^28 - 19 * q^29 - q^31 - 20 * q^32 - 12 * q^34 + 3 * q^35 - 3 * q^37 - 20 * q^38 + 9 * q^40 - 13 * q^41 - 6 * q^43 + 18 * q^44 - 2 * q^46 - 6 * q^47 + 9 * q^49 - 2 * q^50 - q^52 - 19 * q^53 + 4 * q^55 + 10 * q^56 + 21 * q^58 - 23 * q^59 + 13 * q^62 + 27 * q^64 - 3 * q^67 + 4 * q^68 - 12 * q^70 + 3 * q^71 - 32 * q^73 + 12 * q^74 + 18 * q^76 - 18 * q^77 + 2 * q^79 - 8 * q^80 - 5 * q^82 + 21 * q^83 - q^85 + 2 * q^86 - 14 * q^88 - 40 * q^91 + 4 * q^92 + 47 * q^94 + 4 * q^95 - 18 * q^97 - 16 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more