LMFDB - Embedded newform 7448.2.a.bo.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7448,2,Mod(1,7448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7448, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7448 = 2^{3} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7448.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:	$59.4725794254$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 14x^{5} + 13x^{4} + 50x^{3} - 53x^{2} - 25x + 28 $ x^7 - x^6 - 14x^5 + 13x^4 + 50x^3 - 53x^2 - 25*x + 28
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.62810$ of defining polynomial
Character	$\chi$	$=$	7448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.62810 q^{3} -2.40717 q^{5} +3.90693 q^{9} +O(q^{10})$	$q+2.62810 q^{3} -2.40717 q^{5} +3.90693 q^{9} +0.814908 q^{11} -6.10535 q^{13} -6.32628 q^{15} -1.76022 q^{17} -1.00000 q^{19} +7.28837 q^{23} +0.794449 q^{25} +2.38351 q^{27} +8.27273 q^{29} +6.60511 q^{31} +2.14166 q^{33} +3.48082 q^{37} -16.0455 q^{39} -12.4553 q^{41} -6.29205 q^{43} -9.40463 q^{45} -7.68073 q^{47} -4.62604 q^{51} +0.143606 q^{53} -1.96162 q^{55} -2.62810 q^{57} -9.50725 q^{59} +3.68073 q^{61} +14.6966 q^{65} -2.28240 q^{67} +19.1546 q^{69} +2.05582 q^{71} +2.01756 q^{73} +2.08789 q^{75} -5.85018 q^{79} -5.45669 q^{81} -1.76088 q^{83} +4.23714 q^{85} +21.7416 q^{87} -9.23724 q^{89} +17.3589 q^{93} +2.40717 q^{95} +13.2291 q^{97} +3.18379 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{3} - q^{5} + 8 q^{9}+O(q^{10}) $ 7 * q - q^3 - q^5 + 8 * q^9	$ 7 q - q^{3} - q^{5} + 8 q^{9} + 3 q^{11} - 6 q^{13} - 4 q^{15} - 10 q^{17} - 7 q^{19} - 2 q^{23} + 8 q^{25} + 2 q^{27} + 3 q^{29} + 6 q^{31} - 21 q^{33} - 7 q^{37} - 2 q^{39} - 9 q^{41} + q^{43} - 24 q^{45} - 15 q^{47} + 6 q^{51} + 5 q^{53} + 6 q^{55} + q^{57} + 9 q^{59} - 13 q^{61} - 6 q^{65} - 2 q^{67} + 20 q^{69} - q^{71} - 42 q^{73} - 40 q^{75} - 3 q^{79} - q^{81} + 12 q^{83} - 16 q^{85} + 2 q^{87} - 41 q^{89} - 2 q^{93} + q^{95} - 5 q^{97} + 9 q^{99}+O(q^{100}) $ 7 * q - q^3 - q^5 + 8 * q^9 + 3 * q^11 - 6 * q^13 - 4 * q^15 - 10 * q^17 - 7 * q^19 - 2 * q^23 + 8 * q^25 + 2 * q^27 + 3 * q^29 + 6 * q^31 - 21 * q^33 - 7 * q^37 - 2 * q^39 - 9 * q^41 + q^43 - 24 * q^45 - 15 * q^47 + 6 * q^51 + 5 * q^53 + 6 * q^55 + q^57 + 9 * q^59 - 13 * q^61 - 6 * q^65 - 2 * q^67 + 20 * q^69 - q^71 - 42 * q^73 - 40 * q^75 - 3 * q^79 - q^81 + 12 * q^83 - 16 * q^85 + 2 * q^87 - 41 * q^89 - 2 * q^93 + q^95 - 5 * q^97 + 9 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	2.62810	1.51734	0.758668	−	0.651477i	$-0.225850\pi$
$3$	2.62810	1.51734	0.758668	+	0.651477i	$0.225850\pi$
$4$	0	0
$5$	−2.40717	−1.07652	−0.538259	−	0.842780i	$-0.680918\pi$
$5$	−2.40717	−1.07652	−0.538259	+	0.842780i	$0.680918\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	3.90693	1.30231
$10$	0	0
$11$	0.814908	0.245704	0.122852	−	0.992425i	$-0.460796\pi$
$11$	0.814908	0.245704	0.122852	+	0.992425i	$0.460796\pi$
$12$	0	0
$13$	−6.10535	−1.69332	−0.846659	−	0.532136i	$-0.821390\pi$
$13$	−6.10535	−1.69332	−0.846659	+	0.532136i	$0.821390\pi$
$14$	0	0
$15$	−6.32628	−1.63344
$16$	0	0
$17$	−1.76022	−0.426916	−0.213458	−	0.976952i	$-0.568473\pi$
$17$	−1.76022	−0.426916	−0.213458	+	0.976952i	$0.568473\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.28837	1.51973	0.759865	−	0.650080i	$-0.225265\pi$
$23$	7.28837	1.51973	0.759865	+	0.650080i	$0.225265\pi$
$24$	0	0
$25$	0.794449	0.158890
$26$	0	0
$27$	2.38351	0.458706
$28$	0	0
$29$	8.27273	1.53621	0.768104	−	0.640325i	$-0.221200\pi$
$29$	8.27273	1.53621	0.768104	+	0.640325i	$0.221200\pi$
$30$	0	0
$31$	6.60511	1.18631	0.593156	−	0.805087i	$-0.297882\pi$
$31$	6.60511	1.18631	0.593156	+	0.805087i	$0.297882\pi$
$32$	0	0
$33$	2.14166	0.372816
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.48082	0.572243	0.286121	−	0.958193i	$-0.407634\pi$
$37$	3.48082	0.572243	0.286121	+	0.958193i	$0.407634\pi$
$38$	0	0
$39$	−16.0455	−2.56933
$40$	0	0
$41$	−12.4553	−1.94519	−0.972595	−	0.232508i	$-0.925307\pi$
$41$	−12.4553	−1.94519	−0.972595	+	0.232508i	$0.925307\pi$
$42$	0	0
$43$	−6.29205	−0.959529	−0.479765	−	0.877397i	$-0.659278\pi$
$43$	−6.29205	−0.959529	−0.479765	+	0.877397i	$0.659278\pi$
$44$	0	0
$45$	−9.40463	−1.40196
$46$	0	0
$47$	−7.68073	−1.12035	−0.560175	−	0.828375i	$-0.689266\pi$
$47$	−7.68073	−1.12035	−0.560175	+	0.828375i	$0.689266\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−4.62604	−0.647775
$52$	0	0
$53$	0.143606	0.0197258	0.00986290	−	0.999951i	$-0.496860\pi$
$53$	0.143606	0.0197258	0.00986290	+	0.999951i	$0.496860\pi$
$54$	0	0
$55$	−1.96162	−0.264505
$56$	0	0
$57$	−2.62810	−0.348101
$58$	0	0
$59$	−9.50725	−1.23774	−0.618869	−	0.785494i	$-0.712409\pi$
$59$	−9.50725	−1.23774	−0.618869	+	0.785494i	$0.712409\pi$
$60$	0	0
$61$	3.68073	0.471269	0.235634	−	0.971842i	$-0.424283\pi$
$61$	3.68073	0.471269	0.235634	+	0.971842i	$0.424283\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	14.6966	1.82289
$66$	0	0
$67$	−2.28240	−0.278840	−0.139420	−	0.990233i	$-0.544524\pi$
$67$	−2.28240	−0.278840	−0.139420	+	0.990233i	$0.544524\pi$
$68$	0	0
$69$	19.1546	2.30594
$70$	0	0
$71$	2.05582	0.243981	0.121991	−	0.992531i	$-0.461072\pi$
$71$	2.05582	0.243981	0.121991	+	0.992531i	$0.461072\pi$
$72$	0	0
$73$	2.01756	0.236138	0.118069	−	0.993005i	$-0.462330\pi$
$73$	2.01756	0.236138	0.118069	+	0.993005i	$0.462330\pi$
$74$	0	0
$75$	2.08789	0.241089
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−5.85018	−0.658196	−0.329098	−	0.944296i	$-0.606745\pi$
$79$	−5.85018	−0.658196	−0.329098	+	0.944296i	$0.606745\pi$
$80$	0	0
$81$	−5.45669	−0.606299
$82$	0	0
$83$	−1.76088	−0.193282	−0.0966411	−	0.995319i	$-0.530810\pi$
$83$	−1.76088	−0.193282	−0.0966411	+	0.995319i	$0.530810\pi$
$84$	0	0
$85$	4.23714	0.459582
$86$	0	0
$87$	21.7416	2.33094
$88$	0	0
$89$	−9.23724	−0.979146	−0.489573	−	0.871962i	$-0.662847\pi$
$89$	−9.23724	−0.979146	−0.489573	+	0.871962i	$0.662847\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	17.3589	1.80004
$94$	0	0
$95$	2.40717	0.246970
$96$	0	0
$97$	13.2291	1.34321	0.671605	−	0.740909i	$-0.265605\pi$
$97$	13.2291	1.34321	0.671605	+	0.740909i	$0.265605\pi$
$98$	0	0
$99$	3.18379	0.319983
$100$	0	0
$101$	−18.0239	−1.79344	−0.896722	−	0.442595i	$-0.854058\pi$
$101$	−18.0239	−1.79344	−0.896722	+	0.442595i	$0.854058\pi$
$102$	0	0
$103$	−8.62626	−0.849970	−0.424985	−	0.905200i	$-0.639721\pi$
$103$	−8.62626	−0.849970	−0.424985	+	0.905200i	$0.639721\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.74965	0.169145	0.0845725	−	0.996417i	$-0.473048\pi$
$107$	1.74965	0.169145	0.0845725	+	0.996417i	$0.473048\pi$
$108$	0	0
$109$	−15.0101	−1.43771	−0.718854	−	0.695161i	$-0.755333\pi$
$109$	−15.0101	−1.43771	−0.718854	+	0.695161i	$0.755333\pi$
$110$	0	0
$111$	9.14795	0.868285
$112$	0	0
$113$	−9.97230	−0.938115	−0.469058	−	0.883168i	$-0.655406\pi$
$113$	−9.97230	−0.938115	−0.469058	+	0.883168i	$0.655406\pi$
$114$	0	0
$115$	−17.5443	−1.63602
$116$	0	0
$117$	−23.8532	−2.20523
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.3359	−0.939630
$122$	0	0
$123$	−32.7338	−2.95151
$124$	0	0
$125$	10.1235	0.905470
$126$	0	0
$127$	−10.0139	−0.888589	−0.444295	−	0.895881i	$-0.646546\pi$
$127$	−10.0139	−0.888589	−0.444295	+	0.895881i	$0.646546\pi$
$128$	0	0
$129$	−16.5362	−1.45593
$130$	0	0
$131$	−3.82228	−0.333954	−0.166977	−	0.985961i	$-0.553401\pi$
$131$	−3.82228	−0.333954	−0.166977	+	0.985961i	$0.553401\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−5.73750	−0.493805
$136$	0	0
$137$	−21.9818	−1.87803	−0.939017	−	0.343870i	$-0.888262\pi$
$137$	−21.9818	−1.87803	−0.939017	+	0.343870i	$0.888262\pi$
$138$	0	0
$139$	1.04618	0.0887360	0.0443680	−	0.999015i	$-0.485873\pi$
$139$	1.04618	0.0887360	0.0443680	+	0.999015i	$0.485873\pi$
$140$	0	0
$141$	−20.1858	−1.69995
$142$	0	0
$143$	−4.97529	−0.416055
$144$	0	0
$145$	−19.9138	−1.65375
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.4416	1.18310	0.591551	−	0.806267i	$-0.298516\pi$
$149$	14.4416	1.18310	0.591551	+	0.806267i	$0.298516\pi$
$150$	0	0
$151$	4.52553	0.368282	0.184141	−	0.982900i	$-0.441050\pi$
$151$	4.52553	0.368282	0.184141	+	0.982900i	$0.441050\pi$
$152$	0	0
$153$	−6.87705	−0.555977
$154$	0	0
$155$	−15.8996	−1.27709
$156$	0	0
$157$	−0.677397	−0.0540622	−0.0270311	−	0.999635i	$-0.508605\pi$
$157$	−0.677397	−0.0540622	−0.0270311	+	0.999635i	$0.508605\pi$
$158$	0	0
$159$	0.377411	0.0299307
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−12.9998	−1.01823	−0.509113	−	0.860700i	$-0.670026\pi$
$163$	−12.9998	−1.01823	−0.509113	+	0.860700i	$0.670026\pi$
$164$	0	0
$165$	−5.15534	−0.401342
$166$	0	0
$167$	−5.50170	−0.425735	−0.212867	−	0.977081i	$-0.568280\pi$
$167$	−5.50170	−0.425735	−0.212867	+	0.977081i	$0.568280\pi$
$168$	0	0
$169$	24.2752	1.86733
$170$	0	0
$171$	−3.90693	−0.298770
$172$	0	0
$173$	9.99791	0.760127	0.380063	−	0.924960i	$-0.375902\pi$
$173$	9.99791	0.760127	0.380063	+	0.924960i	$0.375902\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−24.9860	−1.87807
$178$	0	0
$179$	−17.5643	−1.31282	−0.656409	−	0.754405i	$-0.727925\pi$
$179$	−17.5643	−1.31282	−0.656409	+	0.754405i	$0.727925\pi$
$180$	0	0
$181$	−7.23540	−0.537803	−0.268901	−	0.963168i	$-0.586661\pi$
$181$	−7.23540	−0.537803	−0.268901	+	0.963168i	$0.586661\pi$
$182$	0	0
$183$	9.67334	0.715074
$184$	0	0
$185$	−8.37891	−0.616029
$186$	0	0
$187$	−1.43442	−0.104895
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−24.8902	−1.80099	−0.900494	−	0.434868i	$-0.856795\pi$
$191$	−24.8902	−1.80099	−0.900494	+	0.434868i	$0.856795\pi$
$192$	0	0
$193$	12.8667	0.926165	0.463083	−	0.886315i	$-0.346743\pi$
$193$	12.8667	0.926165	0.463083	+	0.886315i	$0.346743\pi$
$194$	0	0
$195$	38.6241	2.76593
$196$	0	0
$197$	15.5595	1.10857	0.554285	−	0.832327i	$-0.312992\pi$
$197$	15.5595	1.10857	0.554285	+	0.832327i	$0.312992\pi$
$198$	0	0
$199$	−16.7272	−1.18576	−0.592881	−	0.805290i	$-0.702010\pi$
$199$	−16.7272	−1.18576	−0.592881	+	0.805290i	$0.702010\pi$
$200$	0	0
$201$	−5.99839	−0.423094
$202$	0	0
$203$	0	0
$204$	0	0
$205$	29.9819	2.09403
$206$	0	0
$207$	28.4752	1.97916
$208$	0	0
$209$	−0.814908	−0.0563684
$210$	0	0
$211$	−12.6024	−0.867582	−0.433791	−	0.901014i	$-0.642824\pi$
$211$	−12.6024	−0.867582	−0.433791	+	0.901014i	$0.642824\pi$
$212$	0	0
$213$	5.40292	0.370202
$214$	0	0
$215$	15.1460	1.03295
$216$	0	0
$217$	0	0
$218$	0	0
$219$	5.30237	0.358301
$220$	0	0
$221$	10.7467	0.722904
$222$	0	0
$223$	−13.6077	−0.911236	−0.455618	−	0.890175i	$-0.650582\pi$
$223$	−13.6077	−0.911236	−0.455618	+	0.890175i	$0.650582\pi$
$224$	0	0
$225$	3.10386	0.206924
$226$	0	0
$227$	5.86226	0.389092	0.194546	−	0.980893i	$-0.437677\pi$
$227$	5.86226	0.389092	0.194546	+	0.980893i	$0.437677\pi$
$228$	0	0
$229$	20.8826	1.37996	0.689979	−	0.723829i	$-0.257620\pi$
$229$	20.8826	1.37996	0.689979	+	0.723829i	$0.257620\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	17.2717	1.13151	0.565753	−	0.824575i	$-0.308585\pi$
$233$	17.2717	1.13151	0.565753	+	0.824575i	$0.308585\pi$
$234$	0	0
$235$	18.4888	1.20608
$236$	0	0
$237$	−15.3749	−0.998705
$238$	0	0
$239$	−19.3446	−1.25130	−0.625650	−	0.780104i	$-0.715166\pi$
$239$	−19.3446	−1.25130	−0.625650	+	0.780104i	$0.715166\pi$
$240$	0	0
$241$	−5.56140	−0.358241	−0.179121	−	0.983827i	$-0.557325\pi$
$241$	−5.56140	−0.358241	−0.179121	+	0.983827i	$0.557325\pi$
$242$	0	0
$243$	−21.4913	−1.37867
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.10535	0.388474
$248$	0	0
$249$	−4.62779	−0.293274
$250$	0	0
$251$	11.5337	0.727999	0.364000	−	0.931399i	$-0.381411\pi$
$251$	11.5337	0.727999	0.364000	+	0.931399i	$0.381411\pi$
$252$	0	0
$253$	5.93935	0.373404
$254$	0	0
$255$	11.1356	0.697341
$256$	0	0
$257$	8.21831	0.512644	0.256322	−	0.966591i	$-0.417489\pi$
$257$	8.21831	0.512644	0.256322	+	0.966591i	$0.417489\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	32.3210	2.00062
$262$	0	0
$263$	5.09513	0.314179	0.157090	−	0.987584i	$-0.449789\pi$
$263$	5.09513	0.314179	0.157090	+	0.987584i	$0.449789\pi$
$264$	0	0
$265$	−0.345683	−0.0212352
$266$	0	0
$267$	−24.2764	−1.48569
$268$	0	0
$269$	2.49882	0.152356	0.0761779	−	0.997094i	$-0.475728\pi$
$269$	2.49882	0.152356	0.0761779	+	0.997094i	$0.475728\pi$
$270$	0	0
$271$	23.4838	1.42654	0.713268	−	0.700891i	$-0.247214\pi$
$271$	23.4838	1.42654	0.713268	+	0.700891i	$0.247214\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.647403	0.0390398
$276$	0	0
$277$	2.39777	0.144068	0.0720341	−	0.997402i	$-0.477051\pi$
$277$	2.39777	0.144068	0.0720341	+	0.997402i	$0.477051\pi$
$278$	0	0
$279$	25.8057	1.54495
$280$	0	0
$281$	−17.2156	−1.02700	−0.513498	−	0.858091i	$-0.671651\pi$
$281$	−17.2156	−1.02700	−0.513498	+	0.858091i	$0.671651\pi$
$282$	0	0
$283$	7.64072	0.454194	0.227097	−	0.973872i	$-0.427077\pi$
$283$	7.64072	0.454194	0.227097	+	0.973872i	$0.427077\pi$
$284$	0	0
$285$	6.32628	0.374737
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.9016	−0.817743
$290$	0	0
$291$	34.7674	2.03810
$292$	0	0
$293$	22.9755	1.34224	0.671122	−	0.741347i	$-0.265813\pi$
$293$	22.9755	1.34224	0.671122	+	0.741347i	$0.265813\pi$
$294$	0	0
$295$	22.8855	1.33245
$296$	0	0
$297$	1.94234	0.112706
$298$	0	0
$299$	−44.4980	−2.57339
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−47.3686	−2.72126
$304$	0	0
$305$	−8.86012	−0.507329
$306$	0	0
$307$	12.7122	0.725521	0.362761	−	0.931882i	$-0.381834\pi$
$307$	12.7122	0.725521	0.362761	+	0.931882i	$0.381834\pi$
$308$	0	0
$309$	−22.6707	−1.28969
$310$	0	0
$311$	10.8389	0.614618	0.307309	−	0.951610i	$-0.400571\pi$
$311$	10.8389	0.614618	0.307309	+	0.951610i	$0.400571\pi$
$312$	0	0
$313$	6.53882	0.369596	0.184798	−	0.982777i	$-0.440837\pi$
$313$	6.53882	0.369596	0.184798	+	0.982777i	$0.440837\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	28.7168	1.61289	0.806447	−	0.591307i	$-0.201388\pi$
$317$	28.7168	1.61289	0.806447	+	0.591307i	$0.201388\pi$
$318$	0	0
$319$	6.74151	0.377452
$320$	0	0
$321$	4.59826	0.256650
$322$	0	0
$323$	1.76022	0.0979412
$324$	0	0
$325$	−4.85038	−0.269051
$326$	0	0
$327$	−39.4481	−2.18149
$328$	0	0
$329$	0	0
$330$	0	0
$331$	15.8147	0.869252	0.434626	−	0.900611i	$-0.356880\pi$
$331$	15.8147	0.869252	0.434626	+	0.900611i	$0.356880\pi$
$332$	0	0
$333$	13.5993	0.745238
$334$	0	0
$335$	5.49412	0.300176
$336$	0	0
$337$	−13.6714	−0.744729	−0.372364	−	0.928087i	$-0.621453\pi$
$337$	−13.6714	−0.744729	−0.372364	+	0.928087i	$0.621453\pi$
$338$	0	0
$339$	−26.2082	−1.42344
$340$	0	0
$341$	5.38256	0.291482
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−46.1083	−2.48239
$346$	0	0
$347$	0.948768	0.0509325	0.0254663	−	0.999676i	$-0.491893\pi$
$347$	0.948768	0.0509325	0.0254663	+	0.999676i	$0.491893\pi$
$348$	0	0
$349$	−18.0346	−0.965369	−0.482684	−	0.875794i	$-0.660338\pi$
$349$	−18.0346	−0.965369	−0.482684	+	0.875794i	$0.660338\pi$
$350$	0	0
$351$	−14.5521	−0.776735
$352$	0	0
$353$	−6.42200	−0.341809	−0.170904	−	0.985288i	$-0.554669\pi$
$353$	−6.42200	−0.341809	−0.170904	+	0.985288i	$0.554669\pi$
$354$	0	0
$355$	−4.94871	−0.262650
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7.53859	0.397871	0.198936	−	0.980013i	$-0.436252\pi$
$359$	7.53859	0.397871	0.198936	+	0.980013i	$0.436252\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−27.1639	−1.42573
$364$	0	0
$365$	−4.85661	−0.254207
$366$	0	0
$367$	−36.2517	−1.89232	−0.946161	−	0.323696i	$-0.895075\pi$
$367$	−36.2517	−1.89232	−0.946161	+	0.323696i	$0.895075\pi$
$368$	0	0
$369$	−48.6619	−2.53324
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−27.4071	−1.41908	−0.709542	−	0.704663i	$-0.751098\pi$
$373$	−27.4071	−1.41908	−0.709542	+	0.704663i	$0.751098\pi$
$374$	0	0
$375$	26.6055	1.37390
$376$	0	0
$377$	−50.5079	−2.60129
$378$	0	0
$379$	26.6336	1.36808	0.684038	−	0.729446i	$-0.260222\pi$
$379$	26.6336	1.36808	0.684038	+	0.729446i	$0.260222\pi$
$380$	0	0
$381$	−26.3175	−1.34829
$382$	0	0
$383$	−14.8553	−0.759073	−0.379536	−	0.925177i	$-0.623916\pi$
$383$	−14.8553	−0.759073	−0.379536	+	0.925177i	$0.623916\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−24.5826	−1.24960
$388$	0	0
$389$	−12.5223	−0.634907	−0.317453	−	0.948274i	$-0.602828\pi$
$389$	−12.5223	−0.634907	−0.317453	+	0.948274i	$0.602828\pi$
$390$	0	0
$391$	−12.8291	−0.648797
$392$	0	0
$393$	−10.0453	−0.506721
$394$	0	0
$395$	14.0824	0.708560
$396$	0	0
$397$	21.9824	1.10326	0.551632	−	0.834087i	$-0.314005\pi$
$397$	21.9824	1.10326	0.551632	+	0.834087i	$0.314005\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11.4276	0.570666	0.285333	−	0.958428i	$-0.407896\pi$
$401$	11.4276	0.570666	0.285333	+	0.958428i	$0.407896\pi$
$402$	0	0
$403$	−40.3265	−2.00880
$404$	0	0
$405$	13.1352	0.652691
$406$	0	0
$407$	2.83655	0.140602
$408$	0	0
$409$	−12.2566	−0.606049	−0.303025	−	0.952983i	$-0.597996\pi$
$409$	−12.2566	−0.606049	−0.303025	+	0.952983i	$0.597996\pi$
$410$	0	0
$411$	−57.7705	−2.84961
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4.23874	0.208072
$416$	0	0
$417$	2.74948	0.134642
$418$	0	0
$419$	32.8794	1.60626	0.803131	−	0.595802i	$-0.203166\pi$
$419$	32.8794	1.60626	0.803131	+	0.595802i	$0.203166\pi$
$420$	0	0
$421$	34.9512	1.70342	0.851708	−	0.524017i	$-0.175567\pi$
$421$	34.9512	1.70342	0.851708	+	0.524017i	$0.175567\pi$
$422$	0	0
$423$	−30.0081	−1.45904
$424$	0	0
$425$	−1.39840	−0.0678326
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−13.0756	−0.631295
$430$	0	0
$431$	31.8404	1.53370	0.766848	−	0.641829i	$-0.221824\pi$
$431$	31.8404	1.53370	0.766848	+	0.641829i	$0.221824\pi$
$432$	0	0
$433$	6.42912	0.308964	0.154482	−	0.987996i	$-0.450629\pi$
$433$	6.42912	0.308964	0.154482	+	0.987996i	$0.450629\pi$
$434$	0	0
$435$	−52.3356	−2.50930
$436$	0	0
$437$	−7.28837	−0.348650
$438$	0	0
$439$	−16.8378	−0.803623	−0.401811	−	0.915723i	$-0.631619\pi$
$439$	−16.8378	−0.803623	−0.401811	+	0.915723i	$0.631619\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	24.0806	1.14411	0.572053	−	0.820217i	$-0.306147\pi$
$443$	24.0806	1.14411	0.572053	+	0.820217i	$0.306147\pi$
$444$	0	0
$445$	22.2356	1.05407
$446$	0	0
$447$	37.9540	1.79516
$448$	0	0
$449$	39.9004	1.88302	0.941508	−	0.336992i	$-0.109409\pi$
$449$	39.9004	1.88302	0.941508	+	0.336992i	$0.109409\pi$
$450$	0	0
$451$	−10.1499	−0.477941
$452$	0	0
$453$	11.8936	0.558808
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−2.82981	−0.132373	−0.0661864	−	0.997807i	$-0.521083\pi$
$457$	−2.82981	−0.132373	−0.0661864	+	0.997807i	$0.521083\pi$
$458$	0	0
$459$	−4.19550	−0.195829
$460$	0	0
$461$	−0.0268602	−0.00125101	−0.000625503	−	1.00000i	$-0.500199\pi$
$461$	−0.0268602	−0.00125101	−0.000625503		1.00000i	$0.500199\pi$
$462$	0	0
$463$	30.6762	1.42565	0.712823	−	0.701344i	$-0.247416\pi$
$463$	30.6762	1.42565	0.712823	+	0.701344i	$0.247416\pi$
$464$	0	0
$465$	−41.7858	−1.93777
$466$	0	0
$467$	2.94767	0.136402	0.0682010	−	0.997672i	$-0.478274\pi$
$467$	2.94767	0.136402	0.0682010	+	0.997672i	$0.478274\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1.78027	−0.0820305
$472$	0	0
$473$	−5.12744	−0.235760
$474$	0	0
$475$	−0.794449	−0.0364518
$476$	0	0
$477$	0.561059	0.0256891
$478$	0	0
$479$	34.6722	1.58421	0.792106	−	0.610384i	$-0.208985\pi$
$479$	34.6722	1.58421	0.792106	+	0.610384i	$0.208985\pi$
$480$	0	0
$481$	−21.2516	−0.968989
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−31.8446	−1.44599
$486$	0	0
$487$	25.7621	1.16739	0.583695	−	0.811973i	$-0.301606\pi$
$487$	25.7621	1.16739	0.583695	+	0.811973i	$0.301606\pi$
$488$	0	0
$489$	−34.1649	−1.54499
$490$	0	0
$491$	−16.9942	−0.766940	−0.383470	−	0.923553i	$-0.625271\pi$
$491$	−16.9942	−0.766940	−0.383470	+	0.923553i	$0.625271\pi$
$492$	0	0
$493$	−14.5618	−0.655831
$494$	0	0
$495$	−7.66391	−0.344467
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−18.1156	−0.810964	−0.405482	−	0.914103i	$-0.632896\pi$
$499$	−18.1156	−0.810964	−0.405482	+	0.914103i	$0.632896\pi$
$500$	0	0
$501$	−14.4590	−0.645983
$502$	0	0
$503$	−23.0997	−1.02996	−0.514981	−	0.857201i	$-0.672201\pi$
$503$	−23.0997	−1.02996	−0.514981	+	0.857201i	$0.672201\pi$
$504$	0	0
$505$	43.3865	1.93067
$506$	0	0
$507$	63.7978	2.83336
$508$	0	0
$509$	−7.15666	−0.317213	−0.158607	−	0.987342i	$-0.550700\pi$
$509$	−7.15666	−0.317213	−0.158607	+	0.987342i	$0.550700\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2.38351	−0.105234
$514$	0	0
$515$	20.7648	0.915008
$516$	0	0
$517$	−6.25909	−0.275274
$518$	0	0
$519$	26.2755	1.15337
$520$	0	0
$521$	−0.234066	−0.0102546	−0.00512731	−	0.999987i	$-0.501632\pi$
$521$	−0.234066	−0.0102546	−0.00512731	+	0.999987i	$0.501632\pi$
$522$	0	0
$523$	−10.0201	−0.438148	−0.219074	−	0.975708i	$-0.570304\pi$
$523$	−10.0201	−0.438148	−0.219074	+	0.975708i	$0.570304\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11.6264	−0.506456
$528$	0	0
$529$	30.1204	1.30958
$530$	0	0
$531$	−37.1442	−1.61192
$532$	0	0
$533$	76.0438	3.29382
$534$	0	0
$535$	−4.21170	−0.182088
$536$	0	0
$537$	−46.1608	−1.99199
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−6.68612	−0.287459	−0.143729	−	0.989617i	$-0.545910\pi$
$541$	−6.68612	−0.287459	−0.143729	+	0.989617i	$0.545910\pi$
$542$	0	0
$543$	−19.0154	−0.816028
$544$	0	0
$545$	36.1318	1.54772
$546$	0	0
$547$	5.12668	0.219201	0.109601	−	0.993976i	$-0.465043\pi$
$547$	5.12668	0.219201	0.109601	+	0.993976i	$0.465043\pi$
$548$	0	0
$549$	14.3803	0.613738
$550$	0	0
$551$	−8.27273	−0.352430
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−22.0206	−0.934724
$556$	0	0
$557$	27.2254	1.15358	0.576789	−	0.816893i	$-0.304305\pi$
$557$	27.2254	1.15358	0.576789	+	0.816893i	$0.304305\pi$
$558$	0	0
$559$	38.4152	1.62479
$560$	0	0
$561$	−3.76980	−0.159161
$562$	0	0
$563$	−9.28672	−0.391389	−0.195694	−	0.980665i	$-0.562696\pi$
$563$	−9.28672	−0.391389	−0.195694	+	0.980665i	$0.562696\pi$
$564$	0	0
$565$	24.0050	1.00990
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.12489	0.382535	0.191268	−	0.981538i	$-0.438740\pi$
$569$	9.12489	0.382535	0.191268	+	0.981538i	$0.438740\pi$
$570$	0	0
$571$	−41.4824	−1.73598	−0.867991	−	0.496579i	$-0.834589\pi$
$571$	−41.4824	−1.73598	−0.867991	+	0.496579i	$0.834589\pi$
$572$	0	0
$573$	−65.4139	−2.73271
$574$	0	0
$575$	5.79024	0.241470
$576$	0	0
$577$	−9.27999	−0.386331	−0.193166	−	0.981166i	$-0.561875\pi$
$577$	−9.27999	−0.386331	−0.193166	+	0.981166i	$0.561875\pi$
$578$	0	0
$579$	33.8150	1.40530
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0.117026	0.00484671
$584$	0	0
$585$	57.4185	2.37396
$586$	0	0
$587$	42.9919	1.77447	0.887233	−	0.461322i	$-0.152625\pi$
$587$	42.9919	1.77447	0.887233	+	0.461322i	$0.152625\pi$
$588$	0	0
$589$	−6.60511	−0.272159
$590$	0	0
$591$	40.8920	1.68207
$592$	0	0
$593$	−34.5692	−1.41959	−0.709793	−	0.704411i	$-0.751211\pi$
$593$	−34.5692	−1.41959	−0.709793	+	0.704411i	$0.751211\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−43.9609	−1.79920
$598$	0	0
$599$	33.1208	1.35328	0.676640	−	0.736314i	$-0.263435\pi$
$599$	33.1208	1.35328	0.676640	+	0.736314i	$0.263435\pi$
$600$	0	0
$601$	−24.9596	−1.01812	−0.509062	−	0.860730i	$-0.670007\pi$
$601$	−24.9596	−1.01812	−0.509062	+	0.860730i	$0.670007\pi$
$602$	0	0
$603$	−8.91719	−0.363136
$604$	0	0
$605$	24.8803	1.01153
$606$	0	0
$607$	−44.2177	−1.79474	−0.897371	−	0.441278i	$-0.854525\pi$
$607$	−44.2177	−1.79474	−0.897371	+	0.441278i	$0.854525\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	46.8935	1.89711
$612$	0	0
$613$	−23.5069	−0.949434	−0.474717	−	0.880139i	$-0.657450\pi$
$613$	−23.5069	−0.949434	−0.474717	+	0.880139i	$0.657450\pi$
$614$	0	0
$615$	78.7957	3.17735
$616$	0	0
$617$	−47.0280	−1.89327	−0.946637	−	0.322301i	$-0.895544\pi$
$617$	−47.0280	−1.89327	−0.946637	+	0.322301i	$0.895544\pi$
$618$	0	0
$619$	27.9989	1.12537	0.562686	−	0.826671i	$-0.309768\pi$
$619$	27.9989	1.12537	0.562686	+	0.826671i	$0.309768\pi$
$620$	0	0
$621$	17.3719	0.697110
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−28.3411	−1.13364
$626$	0	0
$627$	−2.14166	−0.0855298
$628$	0	0
$629$	−6.12700	−0.244300
$630$	0	0
$631$	10.5657	0.420614	0.210307	−	0.977635i	$-0.432554\pi$
$631$	10.5657	0.420614	0.210307	+	0.977635i	$0.432554\pi$
$632$	0	0
$633$	−33.1203	−1.31641
$634$	0	0
$635$	24.1051	0.956582
$636$	0	0
$637$	0	0
$638$	0	0
$639$	8.03196	0.317740
$640$	0	0
$641$	−21.2222	−0.838226	−0.419113	−	0.907934i	$-0.637659\pi$
$641$	−21.2222	−0.838226	−0.419113	+	0.907934i	$0.637659\pi$
$642$	0	0
$643$	−49.4634	−1.95065	−0.975323	−	0.220781i	$-0.929139\pi$
$643$	−49.4634	−1.95065	−0.975323	+	0.220781i	$0.929139\pi$
$644$	0	0
$645$	39.8053	1.56733
$646$	0	0
$647$	−16.6963	−0.656398	−0.328199	−	0.944609i	$-0.606442\pi$
$647$	−16.6963	−0.656398	−0.328199	+	0.944609i	$0.606442\pi$
$648$	0	0
$649$	−7.74753	−0.304117
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.0678	0.393982	0.196991	−	0.980405i	$-0.436883\pi$
$653$	10.0678	0.393982	0.196991	+	0.980405i	$0.436883\pi$
$654$	0	0
$655$	9.20086	0.359507
$656$	0	0
$657$	7.88248	0.307525
$658$	0	0
$659$	−1.48756	−0.0579472	−0.0289736	−	0.999580i	$-0.509224\pi$
$659$	−1.48756	−0.0579472	−0.0289736	+	0.999580i	$0.509224\pi$
$660$	0	0
$661$	−34.4726	−1.34083	−0.670415	−	0.741987i	$-0.733884\pi$
$661$	−34.4726	−1.34083	−0.670415	+	0.741987i	$0.733884\pi$
$662$	0	0
$663$	28.2436	1.09689
$664$	0	0
$665$	0	0
$666$	0	0
$667$	60.2947	2.33462
$668$	0	0
$669$	−35.7623	−1.38265
$670$	0	0
$671$	2.99945	0.115793
$672$	0	0
$673$	−8.50701	−0.327921	−0.163961	−	0.986467i	$-0.552427\pi$
$673$	−8.50701	−0.327921	−0.163961	+	0.986467i	$0.552427\pi$
$674$	0	0
$675$	1.89357	0.0728837
$676$	0	0
$677$	23.7273	0.911912	0.455956	−	0.890002i	$-0.349297\pi$
$677$	23.7273	0.911912	0.455956	+	0.890002i	$0.349297\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	15.4066	0.590383
$682$	0	0
$683$	13.6607	0.522712	0.261356	−	0.965242i	$-0.415830\pi$
$683$	13.6607	0.522712	0.261356	+	0.965242i	$0.415830\pi$
$684$	0	0
$685$	52.9139	2.02174
$686$	0	0
$687$	54.8815	2.09386
$688$	0	0
$689$	−0.876764	−0.0334020
$690$	0	0
$691$	49.3775	1.87841	0.939204	−	0.343361i	$-0.111565\pi$
$691$	49.3775	1.87841	0.939204	+	0.343361i	$0.111565\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.51833	−0.0955259
$696$	0	0
$697$	21.9240	0.830432
$698$	0	0
$699$	45.3918	1.71688
$700$	0	0
$701$	21.7523	0.821571	0.410786	−	0.911732i	$-0.365254\pi$
$701$	21.7523	0.821571	0.410786	+	0.911732i	$0.365254\pi$
$702$	0	0
$703$	−3.48082	−0.131282
$704$	0	0
$705$	48.5905	1.83002
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−25.8487	−0.970770	−0.485385	−	0.874301i	$-0.661321\pi$
$709$	−25.8487	−0.970770	−0.485385	+	0.874301i	$0.661321\pi$
$710$	0	0
$711$	−22.8562	−0.857176
$712$	0	0
$713$	48.1405	1.80288
$714$	0	0
$715$	11.9764	0.447890
$716$	0	0
$717$	−50.8397	−1.89864
$718$	0	0
$719$	11.5004	0.428894	0.214447	−	0.976736i	$-0.431205\pi$
$719$	11.5004	0.428894	0.214447	+	0.976736i	$0.431205\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−14.6159	−0.543572
$724$	0	0
$725$	6.57226	0.244088
$726$	0	0
$727$	−46.0959	−1.70960	−0.854801	−	0.518956i	$-0.826321\pi$
$727$	−46.0959	−1.70960	−0.854801	+	0.518956i	$0.826321\pi$
$728$	0	0
$729$	−40.1112	−1.48560
$730$	0	0
$731$	11.0754	0.409638
$732$	0	0
$733$	−1.38992	−0.0513378	−0.0256689	−	0.999670i	$-0.508172\pi$
$733$	−1.38992	−0.0513378	−0.0256689	+	0.999670i	$0.508172\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.85995	−0.0685121
$738$	0	0
$739$	−42.7670	−1.57321	−0.786605	−	0.617456i	$-0.788163\pi$
$739$	−42.7670	−1.57321	−0.786605	+	0.617456i	$0.788163\pi$
$740$	0	0
$741$	16.0455	0.589445
$742$	0	0
$743$	15.3881	0.564534	0.282267	−	0.959336i	$-0.408914\pi$
$743$	15.3881	0.564534	0.282267	+	0.959336i	$0.408914\pi$
$744$	0	0
$745$	−34.7633	−1.27363
$746$	0	0
$747$	−6.87965	−0.251713
$748$	0	0
$749$	0	0
$750$	0	0
$751$	13.4964	0.492492	0.246246	−	0.969207i	$-0.420803\pi$
$751$	13.4964	0.492492	0.246246	+	0.969207i	$0.420803\pi$
$752$	0	0
$753$	30.3117	1.10462
$754$	0	0
$755$	−10.8937	−0.396462
$756$	0	0
$757$	−27.7187	−1.00745	−0.503727	−	0.863863i	$-0.668038\pi$
$757$	−27.7187	−1.00745	−0.503727	+	0.863863i	$0.668038\pi$
$758$	0	0
$759$	15.6092	0.566579
$760$	0	0
$761$	11.7399	0.425571	0.212786	−	0.977099i	$-0.431746\pi$
$761$	11.7399	0.425571	0.212786	+	0.977099i	$0.431746\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	16.5542	0.598519
$766$	0	0
$767$	58.0450	2.09588
$768$	0	0
$769$	1.38162	0.0498224	0.0249112	−	0.999690i	$-0.492070\pi$
$769$	1.38162	0.0498224	0.0249112	+	0.999690i	$0.492070\pi$
$770$	0	0
$771$	21.5986	0.777854
$772$	0	0
$773$	−6.07902	−0.218647	−0.109324	−	0.994006i	$-0.534868\pi$
$773$	−6.07902	−0.218647	−0.109324	+	0.994006i	$0.534868\pi$
$774$	0	0
$775$	5.24742	0.188493
$776$	0	0
$777$	0	0
$778$	0	0
$779$	12.4553	0.446257
$780$	0	0
$781$	1.67531	0.0599472
$782$	0	0
$783$	19.7181	0.704668
$784$	0	0
$785$	1.63061	0.0581989
$786$	0	0
$787$	6.41204	0.228565	0.114282	−	0.993448i	$-0.463543\pi$
$787$	6.41204	0.228565	0.114282	+	0.993448i	$0.463543\pi$
$788$	0	0
$789$	13.3905	0.476715
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−22.4721	−0.798008
$794$	0	0
$795$	−0.908492	−0.0322209
$796$	0	0
$797$	−16.8287	−0.596105	−0.298052	−	0.954550i	$-0.596337\pi$
$797$	−16.8287	−0.596105	−0.298052	+	0.954550i	$0.596337\pi$
$798$	0	0
$799$	13.5198	0.478295
$800$	0	0
$801$	−36.0893	−1.27515
$802$	0	0
$803$	1.64413	0.0580200
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.56716	0.231175
$808$	0	0
$809$	−48.5636	−1.70740	−0.853702	−	0.520762i	$-0.825648\pi$
$809$	−48.5636	−1.70740	−0.853702	+	0.520762i	$0.825648\pi$
$810$	0	0
$811$	−0.286186	−0.0100493	−0.00502467	−	0.999987i	$-0.501599\pi$
$811$	−0.286186	−0.0100493	−0.00502467	+	0.999987i	$0.501599\pi$
$812$	0	0
$813$	61.7178	2.16454
$814$	0	0
$815$	31.2928	1.09614
$816$	0	0
$817$	6.29205	0.220131
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−13.5843	−0.474097	−0.237048	−	0.971498i	$-0.576180\pi$
$821$	−13.5843	−0.474097	−0.237048	+	0.971498i	$0.576180\pi$
$822$	0	0
$823$	−46.7226	−1.62865	−0.814323	−	0.580412i	$-0.802892\pi$
$823$	−46.7226	−1.62865	−0.814323	+	0.580412i	$0.802892\pi$
$824$	0	0
$825$	1.70144	0.0592366
$826$	0	0
$827$	−26.6207	−0.925692	−0.462846	−	0.886439i	$-0.653172\pi$
$827$	−26.6207	−0.925692	−0.462846	+	0.886439i	$0.653172\pi$
$828$	0	0
$829$	21.2059	0.736511	0.368256	−	0.929725i	$-0.379955\pi$
$829$	21.2059	0.736511	0.368256	+	0.929725i	$0.379955\pi$
$830$	0	0
$831$	6.30160	0.218600
$832$	0	0
$833$	0	0
$834$	0	0
$835$	13.2435	0.458311
$836$	0	0
$837$	15.7433	0.544169
$838$	0	0
$839$	−33.9019	−1.17042	−0.585212	−	0.810881i	$-0.698989\pi$
$839$	−33.9019	−1.17042	−0.585212	+	0.810881i	$0.698989\pi$
$840$	0	0
$841$	39.4381	1.35993
$842$	0	0
$843$	−45.2444	−1.55830
$844$	0	0
$845$	−58.4345	−2.01021
$846$	0	0
$847$	0	0
$848$	0	0
$849$	20.0806	0.689165
$850$	0	0
$851$	25.3695	0.869655
$852$	0	0
$853$	4.18645	0.143341	0.0716706	−	0.997428i	$-0.477167\pi$
$853$	4.18645	0.143341	0.0716706	+	0.997428i	$0.477167\pi$
$854$	0	0
$855$	9.40463	0.321632
$856$	0	0
$857$	11.1522	0.380953	0.190476	−	0.981692i	$-0.438997\pi$
$857$	11.1522	0.380953	0.190476	+	0.981692i	$0.438997\pi$
$858$	0	0
$859$	4.44105	0.151527	0.0757633	−	0.997126i	$-0.475861\pi$
$859$	4.44105	0.151527	0.0757633	+	0.997126i	$0.475861\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	44.1816	1.50396	0.751979	−	0.659187i	$-0.229099\pi$
$863$	44.1816	1.50396	0.751979	+	0.659187i	$0.229099\pi$
$864$	0	0
$865$	−24.0666	−0.818290
$866$	0	0
$867$	−36.5349	−1.24079
$868$	0	0
$869$	−4.76736	−0.161721
$870$	0	0
$871$	13.9349	0.472165
$872$	0	0
$873$	51.6851	1.74928
$874$	0	0
$875$	0	0
$876$	0	0
$877$	32.8751	1.11011	0.555056	−	0.831813i	$-0.312697\pi$
$877$	32.8751	1.11011	0.555056	+	0.831813i	$0.312697\pi$
$878$	0	0
$879$	60.3820	2.03664
$880$	0	0
$881$	−16.4442	−0.554019	−0.277010	−	0.960867i	$-0.589343\pi$
$881$	−16.4442	−0.554019	−0.277010	+	0.960867i	$0.589343\pi$
$882$	0	0
$883$	−26.6042	−0.895303	−0.447652	−	0.894208i	$-0.647740\pi$
$883$	−26.6042	−0.895303	−0.447652	+	0.894208i	$0.647740\pi$
$884$	0	0
$885$	60.1455	2.02177
$886$	0	0
$887$	8.24896	0.276973	0.138487	−	0.990364i	$-0.455776\pi$
$887$	8.24896	0.276973	0.138487	+	0.990364i	$0.455776\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.44670	−0.148970
$892$	0	0
$893$	7.68073	0.257026
$894$	0	0
$895$	42.2802	1.41327
$896$	0	0
$897$	−116.945	−3.90470
$898$	0	0
$899$	54.6423	1.82242
$900$	0	0
$901$	−0.252778	−0.00842126
$902$	0	0
$903$	0	0
$904$	0	0
$905$	17.4168	0.578954
$906$	0	0
$907$	−25.4813	−0.846093	−0.423047	−	0.906108i	$-0.639039\pi$
$907$	−25.4813	−0.846093	−0.423047	+	0.906108i	$0.639039\pi$
$908$	0	0
$909$	−70.4181	−2.33562
$910$	0	0
$911$	59.6649	1.97679	0.988393	−	0.151916i	$-0.0485444\pi$
$911$	59.6649	1.97679	0.988393	+	0.151916i	$0.0485444\pi$
$912$	0	0
$913$	−1.43496	−0.0474902
$914$	0	0
$915$	−23.2853	−0.769789
$916$	0	0
$917$	0	0
$918$	0	0
$919$	51.4372	1.69676	0.848378	−	0.529391i	$-0.177579\pi$
$919$	51.4372	1.69676	0.848378	+	0.529391i	$0.177579\pi$
$920$	0	0
$921$	33.4089	1.10086
$922$	0	0
$923$	−12.5515	−0.413138
$924$	0	0
$925$	2.76533	0.0909236
$926$	0	0
$927$	−33.7022	−1.10692
$928$	0	0
$929$	57.0874	1.87298	0.936488	−	0.350701i	$-0.114057\pi$
$929$	57.0874	1.87298	0.936488	+	0.350701i	$0.114057\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	28.4858	0.932583
$934$	0	0
$935$	3.45288	0.112921
$936$	0	0
$937$	36.9589	1.20740	0.603698	−	0.797213i	$-0.293693\pi$
$937$	36.9589	1.20740	0.603698	+	0.797213i	$0.293693\pi$
$938$	0	0
$939$	17.1847	0.560801
$940$	0	0
$941$	45.3279	1.47765	0.738824	−	0.673899i	$-0.235382\pi$
$941$	45.3279	1.47765	0.738824	+	0.673899i	$0.235382\pi$
$942$	0	0
$943$	−90.7788	−2.95616
$944$	0	0
$945$	0	0
$946$	0	0
$947$	33.5867	1.09142	0.545711	−	0.837973i	$-0.316260\pi$
$947$	33.5867	1.09142	0.545711	+	0.837973i	$0.316260\pi$
$948$	0	0
$949$	−12.3179	−0.399857
$950$	0	0
$951$	75.4706	2.44730
$952$	0	0
$953$	−33.5786	−1.08772	−0.543858	−	0.839177i	$-0.683037\pi$
$953$	−33.5786	−1.08772	−0.543858	+	0.839177i	$0.683037\pi$
$954$	0	0
$955$	59.9147	1.93880
$956$	0	0
$957$	17.7174	0.572722
$958$	0	0
$959$	0	0
$960$	0	0
$961$	12.6275	0.407338
$962$	0	0
$963$	6.83576	0.220279
$964$	0	0
$965$	−30.9723	−0.997033
$966$	0	0
$967$	−59.8122	−1.92343	−0.961715	−	0.274051i	$-0.911636\pi$
$967$	−59.8122	−1.92343	−0.961715	+	0.274051i	$0.911636\pi$
$968$	0	0
$969$	4.62604	0.148610
$970$	0	0
$971$	41.2260	1.32301	0.661503	−	0.749943i	$-0.269919\pi$
$971$	41.2260	1.32301	0.661503	+	0.749943i	$0.269919\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−12.7473	−0.408241
$976$	0	0
$977$	48.4848	1.55116	0.775582	−	0.631246i	$-0.217456\pi$
$977$	48.4848	1.55116	0.775582	+	0.631246i	$0.217456\pi$
$978$	0	0
$979$	−7.52750	−0.240580
$980$	0	0
$981$	−58.6435	−1.87234
$982$	0	0
$983$	−17.8001	−0.567736	−0.283868	−	0.958863i	$-0.591618\pi$
$983$	−17.8001	−0.567736	−0.283868	+	0.958863i	$0.591618\pi$
$984$	0	0
$985$	−37.4543	−1.19339
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−45.8588	−1.45823
$990$	0	0
$991$	−26.8128	−0.851735	−0.425868	−	0.904785i	$-0.640031\pi$
$991$	−26.8128	−0.851735	−0.425868	+	0.904785i	$0.640031\pi$
$992$	0	0
$993$	41.5626	1.31895
$994$	0	0
$995$	40.2652	1.27649
$996$	0	0
$997$	−30.7115	−0.972644	−0.486322	−	0.873780i	$-0.661662\pi$
$997$	−30.7115	−0.972644	−0.486322	+	0.873780i	$0.661662\pi$
$998$	0	0
$999$	8.29655	0.262491

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bo.1.7	✓	7
7.6	odd	2		7448.2.a.bp.1.1	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7448.2.a.bo.1.7	✓	7	1.1	even	1	trivial
7448.2.a.bp.1.1	yes	7	7.6	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:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q+2.62810 q^{3} -2.40717 q^{5} +3.90693 q^{9} +O(q^{10})\)	\(q+2.62810 q^{3} -2.40717 q^{5} +3.90693 q^{9} +0.814908 q^{11} -6.10535 q^{13} -6.32628 q^{15} -1.76022 q^{17} -1.00000 q^{19} +7.28837 q^{23} +0.794449 q^{25} +2.38351 q^{27} +8.27273 q^{29} +6.60511 q^{31} +2.14166 q^{33} +3.48082 q^{37} -16.0455 q^{39} -12.4553 q^{41} -6.29205 q^{43} -9.40463 q^{45} -7.68073 q^{47} -4.62604 q^{51} +0.143606 q^{53} -1.96162 q^{55} -2.62810 q^{57} -9.50725 q^{59} +3.68073 q^{61} +14.6966 q^{65} -2.28240 q^{67} +19.1546 q^{69} +2.05582 q^{71} +2.01756 q^{73} +2.08789 q^{75} -5.85018 q^{79} -5.45669 q^{81} -1.76088 q^{83} +4.23714 q^{85} +21.7416 q^{87} -9.23724 q^{89} +17.3589 q^{93} +2.40717 q^{95} +13.2291 q^{97} +3.18379 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{3} - q^{5} + 8 q^{9}+O(q^{10}) \) 7 * q - q^3 - q^5 + 8 * q^9	\( 7 q - q^{3} - q^{5} + 8 q^{9} + 3 q^{11} - 6 q^{13} - 4 q^{15} - 10 q^{17} - 7 q^{19} - 2 q^{23} + 8 q^{25} + 2 q^{27} + 3 q^{29} + 6 q^{31} - 21 q^{33} - 7 q^{37} - 2 q^{39} - 9 q^{41} + q^{43} - 24 q^{45} - 15 q^{47} + 6 q^{51} + 5 q^{53} + 6 q^{55} + q^{57} + 9 q^{59} - 13 q^{61} - 6 q^{65} - 2 q^{67} + 20 q^{69} - q^{71} - 42 q^{73} - 40 q^{75} - 3 q^{79} - q^{81} + 12 q^{83} - 16 q^{85} + 2 q^{87} - 41 q^{89} - 2 q^{93} + q^{95} - 5 q^{97} + 9 q^{99}+O(q^{100}) \) 7 * q - q^3 - q^5 + 8 * q^9 + 3 * q^11 - 6 * q^13 - 4 * q^15 - 10 * q^17 - 7 * q^19 - 2 * q^23 + 8 * q^25 + 2 * q^27 + 3 * q^29 + 6 * q^31 - 21 * q^33 - 7 * q^37 - 2 * q^39 - 9 * q^41 + q^43 - 24 * q^45 - 15 * q^47 + 6 * q^51 + 5 * q^53 + 6 * q^55 + q^57 + 9 * q^59 - 13 * q^61 - 6 * q^65 - 2 * q^67 + 20 * q^69 - q^71 - 42 * q^73 - 40 * q^75 - 3 * q^79 - q^81 + 12 * q^83 - 16 * q^85 + 2 * q^87 - 41 * q^89 - 2 * q^93 + q^95 - 5 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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