LMFDB - Embedded newform 3447.2.a.j.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3447,2,Mod(1,3447)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3447, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3447.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3447 = 3^{2} \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3447.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:	$27.5244335767$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 383)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	3447.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.30644 q^{2} -0.293212 q^{4} +0.910566 q^{5} +2.08417 q^{7} +2.99595 q^{8} -1.18960 q^{10} -1.45014 q^{11} -5.62225 q^{13} -2.72284 q^{14} -3.32760 q^{16} -2.35320 q^{17} -1.05653 q^{19} -0.266989 q^{20} +1.89453 q^{22} +5.63172 q^{23} -4.17087 q^{25} +7.34513 q^{26} -0.611103 q^{28} -4.34646 q^{29} +8.73676 q^{31} -1.64457 q^{32} +3.07432 q^{34} +1.89777 q^{35} -4.46866 q^{37} +1.38029 q^{38} +2.72801 q^{40} +10.5553 q^{41} +4.24845 q^{43} +0.425199 q^{44} -7.35751 q^{46} +8.72689 q^{47} -2.65624 q^{49} +5.44899 q^{50} +1.64851 q^{52} -1.87073 q^{53} -1.32045 q^{55} +6.24405 q^{56} +5.67839 q^{58} -4.30187 q^{59} -6.48215 q^{61} -11.4141 q^{62} +8.80375 q^{64} -5.11943 q^{65} +14.6140 q^{67} +0.689986 q^{68} -2.47933 q^{70} +5.12074 q^{71} +7.67008 q^{73} +5.83804 q^{74} +0.309786 q^{76} -3.02234 q^{77} +11.7685 q^{79} -3.03000 q^{80} -13.7899 q^{82} -7.99857 q^{83} -2.14274 q^{85} -5.55035 q^{86} -4.34455 q^{88} +2.63315 q^{89} -11.7177 q^{91} -1.65129 q^{92} -11.4012 q^{94} -0.962038 q^{95} +11.6053 q^{97} +3.47023 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 5 q^{2} + 29 q^{4} - 3 q^{5} + 17 q^{7} - 15 q^{8} + q^{10} + 28 q^{13} + 8 q^{14} + 35 q^{16} - 16 q^{17} + 13 q^{19} + 4 q^{20} + 12 q^{22} - 7 q^{23} + 67 q^{25} + 14 q^{26} + 39 q^{28} + 2 q^{29}+ \cdots + 29 q^{98}+O(q^{100}) $ 24 * q - 5 * q^2 + 29 * q^4 - 3 * q^5 + 17 * q^7 - 15 * q^8 + q^10 + 28 * q^13 + 8 * q^14 + 35 * q^16 - 16 * q^17 + 13 * q^19 + 4 * q^20 + 12 * q^22 - 7 * q^23 + 67 * q^25 + 14 * q^26 + 39 * q^28 + 2 * q^29 - 20 * q^32 - 18 * q^34 + 10 * q^35 + 35 * q^37 + 5 * q^38 - 36 * q^40 + 34 * q^43 + 36 * q^44 - 17 * q^46 + 3 * q^47 + 49 * q^49 + 20 * q^50 + 29 * q^52 + 9 * q^53 + 10 * q^55 + 47 * q^56 + 12 * q^58 + 6 * q^59 + 3 * q^61 + 58 * q^62 + 15 * q^64 - 8 * q^65 + 42 * q^67 + 23 * q^68 - 59 * q^70 + 13 * q^71 + 83 * q^73 + 45 * q^74 + 4 * q^76 + 21 * q^77 - 3 * q^79 + 93 * q^80 - 13 * q^82 + 23 * q^83 + 52 * q^85 + 43 * q^86 - 25 * q^88 + 38 * q^89 - 17 * q^91 + 61 * q^92 - 71 * q^94 + 66 * q^95 + 63 * q^97 + 29 * q^98

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 $

$2$	−1.30644	−0.923793	−0.461897	−	0.886934i	$-0.652831\pi$
$2$	−1.30644	−0.923793	−0.461897	+	0.886934i	$0.652831\pi$
$3$	0	0
$3$	0	0
$4$	−0.293212	−0.146606
$5$	0.910566	0.407218	0.203609	−	0.979052i	$-0.434733\pi$
$5$	0.910566	0.407218	0.203609	+	0.979052i	$0.434733\pi$
$6$	0	0
$7$	2.08417	0.787741	0.393871	−	0.919166i	$-0.371136\pi$
$7$	2.08417	0.787741	0.393871	+	0.919166i	$0.371136\pi$
$8$	2.99595	1.05923
$9$	0	0
$10$	−1.18960	−0.376185
$11$	−1.45014	−0.437234	−0.218617	−	0.975811i	$-0.570155\pi$
$11$	−1.45014	−0.437234	−0.218617	+	0.975811i	$0.570155\pi$
$12$	0	0
$13$	−5.62225	−1.55933	−0.779665	−	0.626197i	$-0.784611\pi$
$13$	−5.62225	−1.55933	−0.779665	+	0.626197i	$0.784611\pi$
$14$	−2.72284	−0.727710
$15$	0	0
$16$	−3.32760	−0.831901
$17$	−2.35320	−0.570735	−0.285367	−	0.958418i	$-0.592116\pi$
$17$	−2.35320	−0.570735	−0.285367	+	0.958418i	$0.592116\pi$
$18$	0	0
$19$	−1.05653	−0.242384	−0.121192	−	0.992629i	$-0.538672\pi$
$19$	−1.05653	−0.242384	−0.121192	+	0.992629i	$0.538672\pi$
$20$	−0.266989	−0.0597005
$21$	0	0
$22$	1.89453	0.403914
$23$	5.63172	1.17430	0.587148	−	0.809480i	$-0.300251\pi$
$23$	5.63172	1.17430	0.587148	+	0.809480i	$0.300251\pi$
$24$	0	0
$25$	−4.17087	−0.834174
$26$	7.34513	1.44050
$27$	0	0
$28$	−0.611103	−0.115488
$29$	−4.34646	−0.807117	−0.403559	−	0.914954i	$-0.632227\pi$
$29$	−4.34646	−0.807117	−0.403559	+	0.914954i	$0.632227\pi$
$30$	0	0
$31$	8.73676	1.56917	0.784584	−	0.620022i	$-0.212876\pi$
$31$	8.73676	1.56917	0.784584	+	0.620022i	$0.212876\pi$
$32$	−1.64457	−0.290722
$33$	0	0
$34$	3.07432	0.527241
$35$	1.89777	0.320782
$36$	0	0
$37$	−4.46866	−0.734643	−0.367321	−	0.930094i	$-0.619725\pi$
$37$	−4.46866	−0.734643	−0.367321	+	0.930094i	$0.619725\pi$
$38$	1.38029	0.223913
$39$	0	0
$40$	2.72801	0.431336
$41$	10.5553	1.64847	0.824233	−	0.566251i	$-0.191607\pi$
$41$	10.5553	1.64847	0.824233	+	0.566251i	$0.191607\pi$
$42$	0	0
$43$	4.24845	0.647883	0.323942	−	0.946077i	$-0.394992\pi$
$43$	4.24845	0.647883	0.323942	+	0.946077i	$0.394992\pi$
$44$	0.425199	0.0641011
$45$	0	0
$46$	−7.35751	−1.08481
$47$	8.72689	1.27295	0.636474	−	0.771298i	$-0.280392\pi$
$47$	8.72689	1.27295	0.636474	+	0.771298i	$0.280392\pi$
$48$	0	0
$49$	−2.65624	−0.379463
$50$	5.44899	0.770604
$51$	0	0
$52$	1.64851	0.228607
$53$	−1.87073	−0.256965	−0.128483	−	0.991712i	$-0.541011\pi$
$53$	−1.87073	−0.256965	−0.128483	+	0.991712i	$0.541011\pi$
$54$	0	0
$55$	−1.32045	−0.178050
$56$	6.24405	0.834397
$57$	0	0
$58$	5.67839	0.745610
$59$	−4.30187	−0.560056	−0.280028	−	0.959992i	$-0.590344\pi$
$59$	−4.30187	−0.560056	−0.280028	+	0.959992i	$0.590344\pi$
$60$	0	0
$61$	−6.48215	−0.829954	−0.414977	−	0.909832i	$-0.636210\pi$
$61$	−6.48215	−0.829954	−0.414977	+	0.909832i	$0.636210\pi$
$62$	−11.4141	−1.44959
$63$	0	0
$64$	8.80375	1.10047
$65$	−5.11943	−0.634987
$66$	0	0
$67$	14.6140	1.78539	0.892693	−	0.450665i	$-0.148813\pi$
$67$	14.6140	1.78539	0.892693	+	0.450665i	$0.148813\pi$
$68$	0.689986	0.0836731
$69$	0	0
$70$	−2.47933	−0.296336
$71$	5.12074	0.607720	0.303860	−	0.952717i	$-0.401724\pi$
$71$	5.12074	0.607720	0.303860	+	0.952717i	$0.401724\pi$
$72$	0	0
$73$	7.67008	0.897715	0.448858	−	0.893603i	$-0.351831\pi$
$73$	7.67008	0.897715	0.448858	+	0.893603i	$0.351831\pi$
$74$	5.83804	0.678658
$75$	0	0
$76$	0.309786	0.0355349
$77$	−3.02234	−0.344428
$78$	0	0
$79$	11.7685	1.32406	0.662029	−	0.749479i	$-0.269696\pi$
$79$	11.7685	1.32406	0.662029	+	0.749479i	$0.269696\pi$
$80$	−3.03000	−0.338765
$81$	0	0
$82$	−13.7899	−1.52284
$83$	−7.99857	−0.877957	−0.438979	−	0.898497i	$-0.644660\pi$
$83$	−7.99857	−0.877957	−0.438979	+	0.898497i	$0.644660\pi$
$84$	0	0
$85$	−2.14274	−0.232413
$86$	−5.55035	−0.598510
$87$	0	0
$88$	−4.34455	−0.463130
$89$	2.63315	0.279113	0.139557	−	0.990214i	$-0.455432\pi$
$89$	2.63315	0.279113	0.139557	+	0.990214i	$0.455432\pi$
$90$	0	0
$91$	−11.7177	−1.22835
$92$	−1.65129	−0.172159
$93$	0	0
$94$	−11.4012	−1.17594
$95$	−0.962038	−0.0987030
$96$	0	0
$97$	11.6053	1.17834	0.589169	−	0.808010i	$-0.299455\pi$
$97$	11.6053	1.17834	0.589169	+	0.808010i	$0.299455\pi$
$98$	3.47023	0.350546
$99$	0	0
$100$	1.22295	0.122295
$101$	7.93506	0.789568	0.394784	−	0.918774i	$-0.370819\pi$
$101$	7.93506	0.789568	0.394784	+	0.918774i	$0.370819\pi$
$102$	0	0
$103$	−6.37732	−0.628376	−0.314188	−	0.949361i	$-0.601732\pi$
$103$	−6.37732	−0.628376	−0.314188	+	0.949361i	$0.601732\pi$
$104$	−16.8439	−1.65168
$105$	0	0
$106$	2.44400	0.237383
$107$	0.919710	0.0889117	0.0444559	−	0.999011i	$-0.485845\pi$
$107$	0.919710	0.0889117	0.0444559	+	0.999011i	$0.485845\pi$
$108$	0	0
$109$	−16.3419	−1.56527	−0.782634	−	0.622482i	$-0.786124\pi$
$109$	−16.3419	−1.56527	−0.782634	+	0.622482i	$0.786124\pi$
$110$	1.72509	0.164481
$111$	0	0
$112$	−6.93528	−0.655323
$113$	−10.2230	−0.961698	−0.480849	−	0.876803i	$-0.659671\pi$
$113$	−10.2230	−0.961698	−0.480849	+	0.876803i	$0.659671\pi$
$114$	0	0
$115$	5.12806	0.478194
$116$	1.27443	0.118328
$117$	0	0
$118$	5.62014	0.517376
$119$	−4.90446	−0.449591
$120$	0	0
$121$	−8.89709	−0.808826
$122$	8.46855	0.766706
$123$	0	0
$124$	−2.56172	−0.230049
$125$	−8.35068	−0.746908
$126$	0	0
$127$	6.20139	0.550284	0.275142	−	0.961404i	$-0.411275\pi$
$127$	6.20139	0.550284	0.275142	+	0.961404i	$0.411275\pi$
$128$	−8.21243	−0.725883
$129$	0	0
$130$	6.68823	0.586597
$131$	3.80228	0.332207	0.166104	−	0.986108i	$-0.446881\pi$
$131$	3.80228	0.332207	0.166104	+	0.986108i	$0.446881\pi$
$132$	0	0
$133$	−2.20198	−0.190936
$134$	−19.0924	−1.64933
$135$	0	0
$136$	−7.05006	−0.604538
$137$	7.18556	0.613904	0.306952	−	0.951725i	$-0.400691\pi$
$137$	7.18556	0.613904	0.306952	+	0.951725i	$0.400691\pi$
$138$	0	0
$139$	22.5012	1.90853	0.954264	−	0.298964i	$-0.0966412\pi$
$139$	22.5012	1.90853	0.954264	+	0.298964i	$0.0966412\pi$
$140$	−0.556449	−0.0470286
$141$	0	0
$142$	−6.68994	−0.561408
$143$	8.15305	0.681793
$144$	0	0
$145$	−3.95774	−0.328672
$146$	−10.0205	−0.829303
$147$	0	0
$148$	1.31026	0.107703
$149$	−10.1145	−0.828608	−0.414304	−	0.910139i	$-0.635975\pi$
$149$	−10.1145	−0.828608	−0.414304	+	0.910139i	$0.635975\pi$
$150$	0	0
$151$	19.4558	1.58329	0.791647	−	0.610979i	$-0.209224\pi$
$151$	19.4558	1.58329	0.791647	+	0.610979i	$0.209224\pi$
$152$	−3.16530	−0.256740
$153$	0	0
$154$	3.94851	0.318180
$155$	7.95540	0.638993
$156$	0	0
$157$	4.94110	0.394343	0.197171	−	0.980369i	$-0.436824\pi$
$157$	4.94110	0.394343	0.197171	+	0.980369i	$0.436824\pi$
$158$	−15.3748	−1.22316
$159$	0	0
$160$	−1.49749	−0.118387
$161$	11.7375	0.925041
$162$	0	0
$163$	−14.3804	−1.12636	−0.563182	−	0.826333i	$-0.690423\pi$
$163$	−14.3804	−1.12636	−0.563182	+	0.826333i	$0.690423\pi$
$164$	−3.09495	−0.241675
$165$	0	0
$166$	10.4497	0.811051
$167$	−22.3066	−1.72613	−0.863067	−	0.505090i	$-0.831459\pi$
$167$	−22.3066	−1.72613	−0.863067	+	0.505090i	$0.831459\pi$
$168$	0	0
$169$	18.6096	1.43151
$170$	2.79937	0.214702
$171$	0	0
$172$	−1.24570	−0.0949835
$173$	13.2300	1.00586	0.502930	−	0.864327i	$-0.332255\pi$
$173$	13.2300	1.00586	0.502930	+	0.864327i	$0.332255\pi$
$174$	0	0
$175$	−8.69279	−0.657113
$176$	4.82550	0.363736
$177$	0	0
$178$	−3.44005	−0.257843
$179$	−6.68349	−0.499548	−0.249774	−	0.968304i	$-0.580356\pi$
$179$	−6.68349	−0.499548	−0.249774	+	0.968304i	$0.580356\pi$
$180$	0	0
$181$	−9.78911	−0.727619	−0.363809	−	0.931473i	$-0.618524\pi$
$181$	−9.78911	−0.727619	−0.363809	+	0.931473i	$0.618524\pi$
$182$	15.3085	1.13474
$183$	0	0
$184$	16.8723	1.24385
$185$	−4.06901	−0.299159
$186$	0	0
$187$	3.41247	0.249545
$188$	−2.55883	−0.186622
$189$	0	0
$190$	1.25685	0.0911812
$191$	21.6986	1.57005	0.785027	−	0.619462i	$-0.212649\pi$
$191$	21.6986	1.57005	0.785027	+	0.619462i	$0.212649\pi$
$192$	0	0
$193$	11.3873	0.819674	0.409837	−	0.912159i	$-0.365586\pi$
$193$	11.3873	0.819674	0.409837	+	0.912159i	$0.365586\pi$
$194$	−15.1616	−1.08854
$195$	0	0
$196$	0.778842	0.0556316
$197$	5.15003	0.366924	0.183462	−	0.983027i	$-0.441270\pi$
$197$	5.15003	0.366924	0.183462	+	0.983027i	$0.441270\pi$
$198$	0	0
$199$	15.9165	1.12829	0.564145	−	0.825676i	$-0.309206\pi$
$199$	15.9165	1.12829	0.564145	+	0.825676i	$0.309206\pi$
$200$	−12.4957	−0.883579
$201$	0	0
$202$	−10.3667	−0.729398
$203$	−9.05875	−0.635800
$204$	0	0
$205$	9.61133	0.671284
$206$	8.33159	0.580489
$207$	0	0
$208$	18.7086	1.29721
$209$	1.53211	0.105979
$210$	0	0
$211$	−4.51738	−0.310989	−0.155494	−	0.987837i	$-0.549697\pi$
$211$	−4.51738	−0.310989	−0.155494	+	0.987837i	$0.549697\pi$
$212$	0.548521	0.0376726
$213$	0	0
$214$	−1.20155	−0.0821360
$215$	3.86850	0.263829
$216$	0	0
$217$	18.2089	1.23610
$218$	21.3497	1.44598
$219$	0	0
$220$	0.387172	0.0261031
$221$	13.2303	0.889964
$222$	0	0
$223$	13.5001	0.904036	0.452018	−	0.892009i	$-0.350704\pi$
$223$	13.5001	0.904036	0.452018	+	0.892009i	$0.350704\pi$
$224$	−3.42757	−0.229014
$225$	0	0
$226$	13.3557	0.888410
$227$	18.5652	1.23222	0.616109	−	0.787661i	$-0.288708\pi$
$227$	18.5652	1.23222	0.616109	+	0.787661i	$0.288708\pi$
$228$	0	0
$229$	3.86141	0.255169	0.127584	−	0.991828i	$-0.459278\pi$
$229$	3.86141	0.255169	0.127584	+	0.991828i	$0.459278\pi$
$230$	−6.69950	−0.441752
$231$	0	0
$232$	−13.0218	−0.854921
$233$	−7.30219	−0.478382	−0.239191	−	0.970973i	$-0.576882\pi$
$233$	−7.30219	−0.478382	−0.239191	+	0.970973i	$0.576882\pi$
$234$	0	0
$235$	7.94641	0.518367
$236$	1.26136	0.0821075
$237$	0	0
$238$	6.40739	0.415330
$239$	7.87705	0.509524	0.254762	−	0.967004i	$-0.418003\pi$
$239$	7.87705	0.509524	0.254762	+	0.967004i	$0.418003\pi$
$240$	0	0
$241$	29.1082	1.87502	0.937511	−	0.347955i	$-0.113124\pi$
$241$	29.1082	1.87502	0.937511	+	0.347955i	$0.113124\pi$
$242$	11.6235	0.747188
$243$	0	0
$244$	1.90064	0.121676
$245$	−2.41869	−0.154524
$246$	0	0
$247$	5.94006	0.377957
$248$	26.1749	1.66211
$249$	0	0
$250$	10.9097	0.689989
$251$	16.9063	1.06712	0.533559	−	0.845763i	$-0.320854\pi$
$251$	16.9063	1.06712	0.533559	+	0.845763i	$0.320854\pi$
$252$	0	0
$253$	−8.16680	−0.513442
$254$	−8.10175	−0.508349
$255$	0	0
$256$	−6.87844	−0.429903
$257$	11.7786	0.734726	0.367363	−	0.930078i	$-0.380261\pi$
$257$	11.7786	0.734726	0.367363	+	0.930078i	$0.380261\pi$
$258$	0	0
$259$	−9.31343	−0.578709
$260$	1.50108	0.0930928
$261$	0	0
$262$	−4.96746	−0.306891
$263$	17.2051	1.06091	0.530456	−	0.847713i	$-0.322021\pi$
$263$	17.2051	1.06091	0.530456	+	0.847713i	$0.322021\pi$
$264$	0	0
$265$	−1.70343	−0.104641
$266$	2.87676	0.176385
$267$	0	0
$268$	−4.28500	−0.261748
$269$	12.1597	0.741390	0.370695	−	0.928755i	$-0.379120\pi$
$269$	12.1597	0.741390	0.370695	+	0.928755i	$0.379120\pi$
$270$	0	0
$271$	0.144543	0.00878034	0.00439017	−	0.999990i	$-0.498603\pi$
$271$	0.144543	0.00878034	0.00439017	+	0.999990i	$0.498603\pi$
$272$	7.83052	0.474795
$273$	0	0
$274$	−9.38751	−0.567120
$275$	6.04835	0.364729
$276$	0	0
$277$	4.61563	0.277327	0.138663	−	0.990340i	$-0.455719\pi$
$277$	4.61563	0.277327	0.138663	+	0.990340i	$0.455719\pi$
$278$	−29.3965	−1.76309
$279$	0	0
$280$	5.68563	0.339781
$281$	−2.37703	−0.141801	−0.0709007	−	0.997483i	$-0.522587\pi$
$281$	−2.37703	−0.141801	−0.0709007	+	0.997483i	$0.522587\pi$
$282$	0	0
$283$	13.4835	0.801513	0.400756	−	0.916185i	$-0.368747\pi$
$283$	13.4835	0.801513	0.400756	+	0.916185i	$0.368747\pi$
$284$	−1.50146	−0.0890953
$285$	0	0
$286$	−10.6515	−0.629836
$287$	21.9991	1.29856
$288$	0	0
$289$	−11.4624	−0.674262
$290$	5.17055	0.303625
$291$	0	0
$292$	−2.24896	−0.131610
$293$	29.2780	1.71044	0.855219	−	0.518268i	$-0.173423\pi$
$293$	29.2780	1.71044	0.855219	+	0.518268i	$0.173423\pi$
$294$	0	0
$295$	−3.91714	−0.228065
$296$	−13.3879	−0.778153
$297$	0	0
$298$	13.2139	0.765463
$299$	−31.6629	−1.83111
$300$	0	0
$301$	8.85449	0.510364
$302$	−25.4179	−1.46264
$303$	0	0
$304$	3.51570	0.201639
$305$	−5.90243	−0.337972
$306$	0	0
$307$	−27.2949	−1.55780	−0.778901	−	0.627147i	$-0.784222\pi$
$307$	−27.2949	−1.55780	−0.778901	+	0.627147i	$0.784222\pi$
$308$	0.886186	0.0504951
$309$	0	0
$310$	−10.3933	−0.590297
$311$	−14.7646	−0.837223	−0.418612	−	0.908165i	$-0.637483\pi$
$311$	−14.7646	−0.837223	−0.418612	+	0.908165i	$0.637483\pi$
$312$	0	0
$313$	4.02085	0.227272	0.113636	−	0.993522i	$-0.463750\pi$
$313$	4.02085	0.227272	0.113636	+	0.993522i	$0.463750\pi$
$314$	−6.45526	−0.364291
$315$	0	0
$316$	−3.45066	−0.194115
$317$	−30.2041	−1.69643	−0.848214	−	0.529653i	$-0.822322\pi$
$317$	−30.2041	−1.69643	−0.848214	+	0.529653i	$0.822322\pi$
$318$	0	0
$319$	6.30299	0.352899
$320$	8.01639	0.448130
$321$	0	0
$322$	−15.3343	−0.854547
$323$	2.48622	0.138337
$324$	0	0
$325$	23.4496	1.30075
$326$	18.7872	1.04053
$327$	0	0
$328$	31.6232	1.74610
$329$	18.1883	1.00275
$330$	0	0
$331$	−18.7445	−1.03029	−0.515145	−	0.857103i	$-0.672262\pi$
$331$	−18.7445	−1.03029	−0.515145	+	0.857103i	$0.672262\pi$
$332$	2.34528	0.128714
$333$	0	0
$334$	29.1422	1.59459
$335$	13.3070	0.727041
$336$	0	0
$337$	23.4653	1.27824	0.639119	−	0.769108i	$-0.279299\pi$
$337$	23.4653	1.27824	0.639119	+	0.769108i	$0.279299\pi$
$338$	−24.3124	−1.32242
$339$	0	0
$340$	0.628278	0.0340732
$341$	−12.6695	−0.686094
$342$	0	0
$343$	−20.1252	−1.08666
$344$	12.7281	0.686255
$345$	0	0
$346$	−17.2843	−0.929208
$347$	−8.14300	−0.437139	−0.218570	−	0.975821i	$-0.570139\pi$
$347$	−8.14300	−0.437139	−0.218570	+	0.975821i	$0.570139\pi$
$348$	0	0
$349$	−25.3501	−1.35696	−0.678480	−	0.734619i	$-0.737361\pi$
$349$	−25.3501	−1.35696	−0.678480	+	0.734619i	$0.737361\pi$
$350$	11.3566	0.607037
$351$	0	0
$352$	2.38487	0.127114
$353$	−24.1384	−1.28476	−0.642378	−	0.766388i	$-0.722052\pi$
$353$	−24.1384	−1.28476	−0.642378	+	0.766388i	$0.722052\pi$
$354$	0	0
$355$	4.66277	0.247474
$356$	−0.772070	−0.0409196
$357$	0	0
$358$	8.73159	0.461479
$359$	36.0517	1.90274	0.951368	−	0.308055i	$-0.0996782\pi$
$359$	36.0517	1.90274	0.951368	+	0.308055i	$0.0996782\pi$
$360$	0	0
$361$	−17.8837	−0.941250
$362$	12.7889	0.672169
$363$	0	0
$364$	3.43577	0.180083
$365$	6.98412	0.365565
$366$	0	0
$367$	−14.3193	−0.747461	−0.373730	−	0.927537i	$-0.621921\pi$
$367$	−14.3193	−0.747461	−0.373730	+	0.927537i	$0.621921\pi$
$368$	−18.7401	−0.976897
$369$	0	0
$370$	5.31592	0.276362
$371$	−3.89892	−0.202422
$372$	0	0
$373$	29.7969	1.54283	0.771413	−	0.636335i	$-0.219550\pi$
$373$	29.7969	1.54283	0.771413	+	0.636335i	$0.219550\pi$
$374$	−4.45820	−0.230528
$375$	0	0
$376$	26.1453	1.34834
$377$	24.4369	1.25856
$378$	0	0
$379$	−18.1874	−0.934226	−0.467113	−	0.884198i	$-0.654706\pi$
$379$	−18.1874	−0.934226	−0.467113	+	0.884198i	$0.654706\pi$
$380$	0.282081	0.0144704
$381$	0	0
$382$	−28.3479	−1.45041
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	0	0
$385$	−2.75204	−0.140257
$386$	−14.8768	−0.757210
$387$	0	0
$388$	−3.40281	−0.172751
$389$	12.4224	0.629840	0.314920	−	0.949118i	$-0.398022\pi$
$389$	12.4224	0.629840	0.314920	+	0.949118i	$0.398022\pi$
$390$	0	0
$391$	−13.2526	−0.670211
$392$	−7.95796	−0.401938
$393$	0	0
$394$	−6.72821	−0.338962
$395$	10.7160	0.539179
$396$	0	0
$397$	−16.8357	−0.844958	−0.422479	−	0.906373i	$-0.638840\pi$
$397$	−16.8357	−0.844958	−0.422479	+	0.906373i	$0.638840\pi$
$398$	−20.7940	−1.04231
$399$	0	0
$400$	13.8790	0.693950
$401$	5.37256	0.268293	0.134146	−	0.990962i	$-0.457171\pi$
$401$	5.37256	0.268293	0.134146	+	0.990962i	$0.457171\pi$
$402$	0	0
$403$	−49.1202	−2.44685
$404$	−2.32665	−0.115755
$405$	0	0
$406$	11.8347	0.587348
$407$	6.48019	0.321211
$408$	0	0
$409$	23.8792	1.18075	0.590375	−	0.807129i	$-0.298980\pi$
$409$	23.8792	1.18075	0.590375	+	0.807129i	$0.298980\pi$
$410$	−12.5566	−0.620128
$411$	0	0
$412$	1.86990	0.0921236
$413$	−8.96582	−0.441179
$414$	0	0
$415$	−7.28323	−0.357520
$416$	9.24620	0.453332
$417$	0	0
$418$	−2.00162	−0.0979023
$419$	8.42586	0.411630	0.205815	−	0.978591i	$-0.434015\pi$
$419$	8.42586	0.411630	0.205815	+	0.978591i	$0.434015\pi$
$420$	0	0
$421$	23.9189	1.16573	0.582867	−	0.812567i	$-0.301931\pi$
$421$	23.9189	1.16573	0.582867	+	0.812567i	$0.301931\pi$
$422$	5.90168	0.287290
$423$	0	0
$424$	−5.60462	−0.272184
$425$	9.81489	0.476092
$426$	0	0
$427$	−13.5099	−0.653789
$428$	−0.269670	−0.0130350
$429$	0	0
$430$	−5.05397	−0.243724
$431$	−24.6046	−1.18516	−0.592580	−	0.805511i	$-0.701891\pi$
$431$	−24.6046	−1.18516	−0.592580	+	0.805511i	$0.701891\pi$
$432$	0	0
$433$	−21.8659	−1.05081	−0.525404	−	0.850853i	$-0.676086\pi$
$433$	−21.8659	−1.05081	−0.525404	+	0.850853i	$0.676086\pi$
$434$	−23.7888	−1.14190
$435$	0	0
$436$	4.79163	0.229478
$437$	−5.95007	−0.284630
$438$	0	0
$439$	−2.76635	−0.132031	−0.0660154	−	0.997819i	$-0.521029\pi$
$439$	−2.76635	−0.132031	−0.0660154	+	0.997819i	$0.521029\pi$
$440$	−3.95600	−0.188595
$441$	0	0
$442$	−17.2846	−0.822143
$443$	15.7990	0.750633	0.375316	−	0.926897i	$-0.377534\pi$
$443$	15.7990	0.750633	0.375316	+	0.926897i	$0.377534\pi$
$444$	0	0
$445$	2.39766	0.113660
$446$	−17.6371	−0.835142
$447$	0	0
$448$	18.3485	0.866884
$449$	−8.09725	−0.382133	−0.191066	−	0.981577i	$-0.561195\pi$
$449$	−8.09725	−0.382133	−0.191066	+	0.981577i	$0.561195\pi$
$450$	0	0
$451$	−15.3067	−0.720766
$452$	2.99750	0.140991
$453$	0	0
$454$	−24.2544	−1.13832
$455$	−10.6697	−0.500205
$456$	0	0
$457$	16.4155	0.767885	0.383943	−	0.923357i	$-0.374566\pi$
$457$	16.4155	0.767885	0.383943	+	0.923357i	$0.374566\pi$
$458$	−5.04470	−0.235723
$459$	0	0
$460$	−1.50361	−0.0701060
$461$	11.4672	0.534082	0.267041	−	0.963685i	$-0.413954\pi$
$461$	11.4672	0.534082	0.267041	+	0.963685i	$0.413954\pi$
$462$	0	0
$463$	11.8489	0.550666	0.275333	−	0.961349i	$-0.411212\pi$
$463$	11.8489	0.550666	0.275333	+	0.961349i	$0.411212\pi$
$464$	14.4633	0.671442
$465$	0	0
$466$	9.53988	0.441926
$467$	21.6852	1.00347	0.501735	−	0.865021i	$-0.332695\pi$
$467$	21.6852	1.00347	0.501735	+	0.865021i	$0.332695\pi$
$468$	0	0
$469$	30.4581	1.40642
$470$	−10.3815	−0.478864
$471$	0	0
$472$	−12.8882	−0.593226
$473$	−6.16086	−0.283277
$474$	0	0
$475$	4.40664	0.202190
$476$	1.43805	0.0659128
$477$	0	0
$478$	−10.2909	−0.470695
$479$	38.7480	1.77044	0.885220	−	0.465173i	$-0.154008\pi$
$479$	38.7480	1.77044	0.885220	+	0.465173i	$0.154008\pi$
$480$	0	0
$481$	25.1239	1.14555
$482$	−38.0281	−1.73213
$483$	0	0
$484$	2.60873	0.118579
$485$	10.5674	0.479840
$486$	0	0
$487$	12.6258	0.572129	0.286065	−	0.958210i	$-0.407653\pi$
$487$	12.6258	0.572129	0.286065	+	0.958210i	$0.407653\pi$
$488$	−19.4202	−0.879110
$489$	0	0
$490$	3.15987	0.142748
$491$	17.3101	0.781193	0.390597	−	0.920562i	$-0.372269\pi$
$491$	17.3101	0.781193	0.390597	+	0.920562i	$0.372269\pi$
$492$	0	0
$493$	10.2281	0.460650
$494$	−7.76033	−0.349154
$495$	0	0
$496$	−29.0725	−1.30539
$497$	10.6725	0.478726
$498$	0	0
$499$	−27.8854	−1.24832	−0.624160	−	0.781296i	$-0.714559\pi$
$499$	−27.8854	−1.24832	−0.624160	+	0.781296i	$0.714559\pi$
$500$	2.44852	0.109501
$501$	0	0
$502$	−22.0871	−0.985796
$503$	12.9536	0.577572	0.288786	−	0.957394i	$-0.406748\pi$
$503$	12.9536	0.577572	0.288786	+	0.957394i	$0.406748\pi$
$504$	0	0
$505$	7.22540	0.321526
$506$	10.6694	0.474314
$507$	0	0
$508$	−1.81832	−0.0806749
$509$	42.3593	1.87754	0.938772	−	0.344540i	$-0.111965\pi$
$509$	42.3593	1.87754	0.938772	+	0.344540i	$0.111965\pi$
$510$	0	0
$511$	15.9857	0.707167
$512$	25.4111	1.12302
$513$	0	0
$514$	−15.3880	−0.678735
$515$	−5.80697	−0.255886
$516$	0	0
$517$	−12.6552	−0.556577
$518$	12.1675	0.534607
$519$	0	0
$520$	−15.3375	−0.672595
$521$	24.9719	1.09404	0.547019	−	0.837120i	$-0.315762\pi$
$521$	24.9719	1.09404	0.547019	+	0.837120i	$0.315762\pi$
$522$	0	0
$523$	−39.9277	−1.74592	−0.872958	−	0.487795i	$-0.837801\pi$
$523$	−39.9277	−1.74592	−0.872958	+	0.487795i	$0.837801\pi$
$524$	−1.11487	−0.0487035
$525$	0	0
$526$	−22.4774	−0.980063
$527$	−20.5593	−0.895579
$528$	0	0
$529$	8.71630	0.378970
$530$	2.22543	0.0966664
$531$	0	0
$532$	0.645647	0.0279923
$533$	−59.3446	−2.57050
$534$	0	0
$535$	0.837457	0.0362064
$536$	43.7828	1.89113
$537$	0	0
$538$	−15.8859	−0.684891
$539$	3.85193	0.165914
$540$	0	0
$541$	16.6446	0.715607	0.357804	−	0.933797i	$-0.383526\pi$
$541$	16.6446	0.715607	0.357804	+	0.933797i	$0.383526\pi$
$542$	−0.188837	−0.00811122
$543$	0	0
$544$	3.87001	0.165925
$545$	−14.8804	−0.637405
$546$	0	0
$547$	38.3956	1.64168	0.820839	−	0.571159i	$-0.193506\pi$
$547$	38.3956	1.64168	0.820839	+	0.571159i	$0.193506\pi$
$548$	−2.10689	−0.0900019
$549$	0	0
$550$	−7.90182	−0.336935
$551$	4.59216	0.195632
$552$	0	0
$553$	24.5275	1.04301
$554$	−6.03005	−0.256192
$555$	0	0
$556$	−6.59762	−0.279802
$557$	−30.1962	−1.27945	−0.639726	−	0.768603i	$-0.720952\pi$
$557$	−30.1962	−1.27945	−0.639726	+	0.768603i	$0.720952\pi$
$558$	0	0
$559$	−23.8859	−1.01026
$560$	−6.31504	−0.266859
$561$	0	0
$562$	3.10544	0.130995
$563$	36.1995	1.52563	0.762814	−	0.646618i	$-0.223817\pi$
$563$	36.1995	1.52563	0.762814	+	0.646618i	$0.223817\pi$
$564$	0	0
$565$	−9.30871	−0.391620
$566$	−17.6154	−0.740432
$567$	0	0
$568$	15.3415	0.643713
$569$	−31.1371	−1.30533	−0.652667	−	0.757645i	$-0.726350\pi$
$569$	−31.1371	−1.30533	−0.652667	+	0.757645i	$0.726350\pi$
$570$	0	0
$571$	−9.41933	−0.394187	−0.197093	−	0.980385i	$-0.563150\pi$
$571$	−9.41933	−0.394187	−0.197093	+	0.980385i	$0.563150\pi$
$572$	−2.39057	−0.0999548
$573$	0	0
$574$	−28.7405	−1.19961
$575$	−23.4892	−0.979566
$576$	0	0
$577$	−27.0205	−1.12488	−0.562438	−	0.826839i	$-0.690137\pi$
$577$	−27.0205	−1.12488	−0.562438	+	0.826839i	$0.690137\pi$
$578$	14.9750	0.622878
$579$	0	0
$580$	1.16046	0.0481853
$581$	−16.6704	−0.691603
$582$	0	0
$583$	2.71283	0.112354
$584$	22.9791	0.950884
$585$	0	0
$586$	−38.2499	−1.58009
$587$	−32.0201	−1.32161	−0.660806	−	0.750557i	$-0.729785\pi$
$587$	−32.0201	−1.32161	−0.660806	+	0.750557i	$0.729785\pi$
$588$	0	0
$589$	−9.23063	−0.380341
$590$	5.11751	0.210685
$591$	0	0
$592$	14.8699	0.611150
$593$	−20.8329	−0.855503	−0.427751	−	0.903896i	$-0.640694\pi$
$593$	−20.8329	−0.855503	−0.427751	+	0.903896i	$0.640694\pi$
$594$	0	0
$595$	−4.46584	−0.183082
$596$	2.96568	0.121479
$597$	0	0
$598$	41.3657	1.69157
$599$	−22.7908	−0.931208	−0.465604	−	0.884993i	$-0.654163\pi$
$599$	−22.7908	−0.931208	−0.465604	+	0.884993i	$0.654163\pi$
$600$	0	0
$601$	−17.4469	−0.711672	−0.355836	−	0.934548i	$-0.615804\pi$
$601$	−17.4469	−0.711672	−0.355836	+	0.934548i	$0.615804\pi$
$602$	−11.5679	−0.471471
$603$	0	0
$604$	−5.70468	−0.232120
$605$	−8.10139	−0.329368
$606$	0	0
$607$	36.8843	1.49709	0.748543	−	0.663086i	$-0.230754\pi$
$607$	36.8843	1.49709	0.748543	+	0.663086i	$0.230754\pi$
$608$	1.73754	0.0704665
$609$	0	0
$610$	7.71117	0.312216
$611$	−49.0647	−1.98495
$612$	0	0
$613$	9.45500	0.381884	0.190942	−	0.981601i	$-0.438846\pi$
$613$	9.45500	0.381884	0.190942	+	0.981601i	$0.438846\pi$
$614$	35.6592	1.43909
$615$	0	0
$616$	−9.05477	−0.364827
$617$	31.2901	1.25969	0.629846	−	0.776720i	$-0.283118\pi$
$617$	31.2901	1.25969	0.629846	+	0.776720i	$0.283118\pi$
$618$	0	0
$619$	27.7405	1.11498	0.557492	−	0.830182i	$-0.311764\pi$
$619$	27.7405	1.11498	0.557492	+	0.830182i	$0.311764\pi$
$620$	−2.33262	−0.0936801
$621$	0	0
$622$	19.2891	0.773421
$623$	5.48792	0.219869
$624$	0	0
$625$	13.2505	0.530020
$626$	−5.25300	−0.209952
$627$	0	0
$628$	−1.44879	−0.0578130
$629$	10.5156	0.419286
$630$	0	0
$631$	−5.80111	−0.230938	−0.115469	−	0.993311i	$-0.536837\pi$
$631$	−5.80111	−0.230938	−0.115469	+	0.993311i	$0.536837\pi$
$632$	35.2577	1.40248
$633$	0	0
$634$	39.4598	1.56715
$635$	5.64678	0.224086
$636$	0	0
$637$	14.9341	0.591709
$638$	−8.23448	−0.326006
$639$	0	0
$640$	−7.47796	−0.295592
$641$	12.6431	0.499374	0.249687	−	0.968327i	$-0.419672\pi$
$641$	12.6431	0.499374	0.249687	+	0.968327i	$0.419672\pi$
$642$	0	0
$643$	0.739741	0.0291725	0.0145863	−	0.999894i	$-0.495357\pi$
$643$	0.739741	0.0291725	0.0145863	+	0.999894i	$0.495357\pi$
$644$	−3.44156	−0.135616
$645$	0	0
$646$	−3.24810	−0.127795
$647$	5.76141	0.226505	0.113252	−	0.993566i	$-0.463873\pi$
$647$	5.76141	0.226505	0.113252	+	0.993566i	$0.463873\pi$
$648$	0	0
$649$	6.23833	0.244876
$650$	−30.6356	−1.20163
$651$	0	0
$652$	4.21652	0.165132
$653$	23.8932	0.935015	0.467507	−	0.883989i	$-0.345152\pi$
$653$	23.8932	0.935015	0.467507	+	0.883989i	$0.345152\pi$
$654$	0	0
$655$	3.46223	0.135281
$656$	−35.1239	−1.37136
$657$	0	0
$658$	−23.7620	−0.926337
$659$	−0.450480	−0.0175482	−0.00877410	−	0.999962i	$-0.502793\pi$
$659$	−0.450480	−0.0175482	−0.00877410	+	0.999962i	$0.502793\pi$
$660$	0	0
$661$	−27.4188	−1.06647	−0.533234	−	0.845968i	$-0.679023\pi$
$661$	−27.4188	−1.06647	−0.533234	+	0.845968i	$0.679023\pi$
$662$	24.4886	0.951775
$663$	0	0
$664$	−23.9633	−0.929956
$665$	−2.00505	−0.0777525
$666$	0	0
$667$	−24.4781	−0.947794
$668$	6.54055	0.253061
$669$	0	0
$670$	−17.3849	−0.671636
$671$	9.40004	0.362884
$672$	0	0
$673$	−39.9618	−1.54041	−0.770207	−	0.637794i	$-0.779847\pi$
$673$	−39.9618	−1.54041	−0.770207	+	0.637794i	$0.779847\pi$
$674$	−30.6561	−1.18083
$675$	0	0
$676$	−5.45657	−0.209868
$677$	−22.5667	−0.867308	−0.433654	−	0.901079i	$-0.642776\pi$
$677$	−22.5667	−0.867308	−0.433654	+	0.901079i	$0.642776\pi$
$678$	0	0
$679$	24.1874	0.928226
$680$	−6.41955	−0.246178
$681$	0	0
$682$	16.5520	0.633809
$683$	−38.1485	−1.45971	−0.729855	−	0.683602i	$-0.760412\pi$
$683$	−38.1485	−1.45971	−0.729855	+	0.683602i	$0.760412\pi$
$684$	0	0
$685$	6.54293	0.249992
$686$	26.2924	1.00385
$687$	0	0
$688$	−14.1372	−0.538975
$689$	10.5177	0.400693
$690$	0	0
$691$	−6.53892	−0.248752	−0.124376	−	0.992235i	$-0.539693\pi$
$691$	−6.53892	−0.248752	−0.124376	+	0.992235i	$0.539693\pi$
$692$	−3.87920	−0.147465
$693$	0	0
$694$	10.6383	0.403826
$695$	20.4889	0.777186
$696$	0	0
$697$	−24.8388	−0.940837
$698$	33.1184	1.25355
$699$	0	0
$700$	2.54883	0.0963367
$701$	−10.4051	−0.392996	−0.196498	−	0.980504i	$-0.562957\pi$
$701$	−10.4051	−0.392996	−0.196498	+	0.980504i	$0.562957\pi$
$702$	0	0
$703$	4.72126	0.178066
$704$	−12.7667	−0.481162
$705$	0	0
$706$	31.5354	1.18685
$707$	16.5380	0.621975
$708$	0	0
$709$	12.9924	0.487938	0.243969	−	0.969783i	$-0.421550\pi$
$709$	12.9924	0.487938	0.243969	+	0.969783i	$0.421550\pi$
$710$	−6.09164	−0.228615
$711$	0	0
$712$	7.88877	0.295644
$713$	49.2030	1.84267
$714$	0	0
$715$	7.42390	0.277638
$716$	1.95968	0.0732366
$717$	0	0
$718$	−47.0994	−1.75774
$719$	−36.1495	−1.34815	−0.674074	−	0.738663i	$-0.735457\pi$
$719$	−36.1495	−1.34815	−0.674074	+	0.738663i	$0.735457\pi$
$720$	0	0
$721$	−13.2914	−0.494998
$722$	23.3641	0.869520
$723$	0	0
$724$	2.87028	0.106673
$725$	18.1285	0.673276
$726$	0	0
$727$	−18.2522	−0.676935	−0.338468	−	0.940978i	$-0.609909\pi$
$727$	−18.2522	−0.676935	−0.338468	+	0.940978i	$0.609909\pi$
$728$	−35.1056	−1.30110
$729$	0	0
$730$	−9.12434	−0.337707
$731$	−9.99746	−0.369769
$732$	0	0
$733$	−33.7251	−1.24567	−0.622833	−	0.782355i	$-0.714018\pi$
$733$	−33.7251	−1.24567	−0.622833	+	0.782355i	$0.714018\pi$
$734$	18.7073	0.690499
$735$	0	0
$736$	−9.26179	−0.341394
$737$	−21.1924	−0.780632
$738$	0	0
$739$	31.2341	1.14897	0.574483	−	0.818516i	$-0.305203\pi$
$739$	31.2341	1.14897	0.574483	+	0.818516i	$0.305203\pi$
$740$	1.19308	0.0438585
$741$	0	0
$742$	5.09371	0.186996
$743$	−21.4211	−0.785865	−0.392932	−	0.919567i	$-0.628539\pi$
$743$	−21.4211	−0.785865	−0.392932	+	0.919567i	$0.628539\pi$
$744$	0	0
$745$	−9.20988	−0.337424
$746$	−38.9279	−1.42525
$747$	0	0
$748$	−1.00058	−0.0365847
$749$	1.91683	0.0700394
$750$	0	0
$751$	−15.0581	−0.549479	−0.274740	−	0.961519i	$-0.588592\pi$
$751$	−15.0581	−0.549479	−0.274740	+	0.961519i	$0.588592\pi$
$752$	−29.0396	−1.05897
$753$	0	0
$754$	−31.9253	−1.16265
$755$	17.7158	0.644745
$756$	0	0
$757$	5.16450	0.187707	0.0938534	−	0.995586i	$-0.470082\pi$
$757$	5.16450	0.187707	0.0938534	+	0.995586i	$0.470082\pi$
$758$	23.7608	0.863032
$759$	0	0
$760$	−2.88221	−0.104549
$761$	−29.8494	−1.08204	−0.541021	−	0.841009i	$-0.681962\pi$
$761$	−29.8494	−1.08204	−0.541021	+	0.841009i	$0.681962\pi$
$762$	0	0
$763$	−34.0592	−1.23303
$764$	−6.36228	−0.230179
$765$	0	0
$766$	1.30644	0.0472036
$767$	24.1862	0.873312
$768$	0	0
$769$	15.6136	0.563042	0.281521	−	0.959555i	$-0.409161\pi$
$769$	15.6136	0.563042	0.281521	+	0.959555i	$0.409161\pi$
$770$	3.59538	0.129568
$771$	0	0
$772$	−3.33888	−0.120169
$773$	−33.2962	−1.19758	−0.598790	−	0.800906i	$-0.704352\pi$
$773$	−33.2962	−1.19758	−0.598790	+	0.800906i	$0.704352\pi$
$774$	0	0
$775$	−36.4399	−1.30896
$776$	34.7688	1.24813
$777$	0	0
$778$	−16.2291	−0.581842
$779$	−11.1520	−0.399562
$780$	0	0
$781$	−7.42580	−0.265716
$782$	17.3137	0.619137
$783$	0	0
$784$	8.83893	0.315676
$785$	4.49920	0.160583
$786$	0	0
$787$	25.0491	0.892905	0.446453	−	0.894807i	$-0.352687\pi$
$787$	25.0491	0.892905	0.446453	+	0.894807i	$0.352687\pi$
$788$	−1.51005	−0.0537933
$789$	0	0
$790$	−13.9998	−0.498090
$791$	−21.3064	−0.757569
$792$	0	0
$793$	36.4442	1.29417
$794$	21.9948	0.780566
$795$	0	0
$796$	−4.66690	−0.165414
$797$	51.0298	1.80757	0.903785	−	0.427987i	$-0.140777\pi$
$797$	51.0298	1.80757	0.903785	+	0.427987i	$0.140777\pi$
$798$	0	0
$799$	−20.5361	−0.726516
$800$	6.85930	0.242513
$801$	0	0
$802$	−7.01893	−0.247847
$803$	−11.1227	−0.392512
$804$	0	0
$805$	10.6877	0.376693
$806$	64.1727	2.26039
$807$	0	0
$808$	23.7730	0.836332
$809$	−25.1705	−0.884947	−0.442473	−	0.896782i	$-0.645899\pi$
$809$	−25.1705	−0.884947	−0.442473	+	0.896782i	$0.645899\pi$
$810$	0	0
$811$	−5.64533	−0.198235	−0.0991173	−	0.995076i	$-0.531602\pi$
$811$	−5.64533	−0.198235	−0.0991173	+	0.995076i	$0.531602\pi$
$812$	2.65613	0.0932120
$813$	0	0
$814$	−8.46598	−0.296733
$815$	−13.0944	−0.458675
$816$	0	0
$817$	−4.48861	−0.157037
$818$	−31.1967	−1.09077
$819$	0	0
$820$	−2.81815	−0.0984142
$821$	−10.6903	−0.373096	−0.186548	−	0.982446i	$-0.559730\pi$
$821$	−10.6903	−0.373096	−0.186548	+	0.982446i	$0.559730\pi$
$822$	0	0
$823$	12.4143	0.432734	0.216367	−	0.976312i	$-0.430579\pi$
$823$	12.4143	0.432734	0.216367	+	0.976312i	$0.430579\pi$
$824$	−19.1061	−0.665593
$825$	0	0
$826$	11.7133	0.407558
$827$	44.5662	1.54972	0.774859	−	0.632134i	$-0.217821\pi$
$827$	44.5662	1.54972	0.774859	+	0.632134i	$0.217821\pi$
$828$	0	0
$829$	−18.5663	−0.644834	−0.322417	−	0.946598i	$-0.604495\pi$
$829$	−18.5663	−0.644834	−0.322417	+	0.946598i	$0.604495\pi$
$830$	9.51511	0.330274
$831$	0	0
$832$	−49.4968	−1.71599
$833$	6.25067	0.216573
$834$	0	0
$835$	−20.3116	−0.702912
$836$	−0.449234	−0.0155371
$837$	0	0
$838$	−11.0079	−0.380261
$839$	−43.3676	−1.49722	−0.748608	−	0.663012i	$-0.769278\pi$
$839$	−43.3676	−1.49722	−0.748608	+	0.663012i	$0.769278\pi$
$840$	0	0
$841$	−10.1083	−0.348561
$842$	−31.2486	−1.07690
$843$	0	0
$844$	1.32455	0.0455928
$845$	16.9453	0.582937
$846$	0	0
$847$	−18.5430	−0.637146
$848$	6.22506	0.213769
$849$	0	0
$850$	−12.8226	−0.439811
$851$	−25.1662	−0.862688
$852$	0	0
$853$	19.3360	0.662050	0.331025	−	0.943622i	$-0.392605\pi$
$853$	19.3360	0.662050	0.331025	+	0.943622i	$0.392605\pi$
$854$	17.6499	0.603966
$855$	0	0
$856$	2.75540	0.0941777
$857$	33.8462	1.15616	0.578082	−	0.815979i	$-0.303801\pi$
$857$	33.8462	1.15616	0.578082	+	0.815979i	$0.303801\pi$
$858$	0	0
$859$	−12.4242	−0.423908	−0.211954	−	0.977280i	$-0.567983\pi$
$859$	−12.4242	−0.423908	−0.211954	+	0.977280i	$0.567983\pi$
$860$	−1.13429	−0.0386790
$861$	0	0
$862$	32.1444	1.09484
$863$	46.9463	1.59807	0.799035	−	0.601284i	$-0.205344\pi$
$863$	46.9463	1.59807	0.799035	+	0.601284i	$0.205344\pi$
$864$	0	0
$865$	12.0468	0.409604
$866$	28.5665	0.970729
$867$	0	0
$868$	−5.33906	−0.181219
$869$	−17.0660	−0.578923
$870$	0	0
$871$	−82.1636	−2.78401
$872$	−48.9594	−1.65797
$873$	0	0
$874$	7.77342	0.262940
$875$	−17.4042	−0.588370
$876$	0	0
$877$	32.6960	1.10407	0.552033	−	0.833822i	$-0.313852\pi$
$877$	32.6960	1.10407	0.552033	+	0.833822i	$0.313852\pi$
$878$	3.61408	0.121969
$879$	0	0
$880$	4.39394	0.148120
$881$	12.2877	0.413982	0.206991	−	0.978343i	$-0.433633\pi$
$881$	12.2877	0.413982	0.206991	+	0.978343i	$0.433633\pi$
$882$	0	0
$883$	−2.03500	−0.0684831	−0.0342415	−	0.999414i	$-0.510902\pi$
$883$	−2.03500	−0.0684831	−0.0342415	+	0.999414i	$0.510902\pi$
$884$	−3.87927	−0.130474
$885$	0	0
$886$	−20.6405	−0.693430
$887$	−47.0813	−1.58083	−0.790417	−	0.612569i	$-0.790136\pi$
$887$	−47.0813	−1.58083	−0.790417	+	0.612569i	$0.790136\pi$
$888$	0	0
$889$	12.9247	0.433482
$890$	−3.13240	−0.104998
$891$	0	0
$892$	−3.95840	−0.132537
$893$	−9.22020	−0.308542
$894$	0	0
$895$	−6.08576	−0.203425
$896$	−17.1161	−0.571808
$897$	0	0
$898$	10.5786	0.353012
$899$	−37.9740	−1.26650
$900$	0	0
$901$	4.40221	0.146659
$902$	19.9973	0.665838
$903$	0	0
$904$	−30.6275	−1.01866
$905$	−8.91363	−0.296299
$906$	0	0
$907$	−21.3028	−0.707349	−0.353674	−	0.935369i	$-0.615068\pi$
$907$	−21.3028	−0.707349	−0.353674	+	0.935369i	$0.615068\pi$
$908$	−5.44355	−0.180651
$909$	0	0
$910$	13.9394	0.462086
$911$	14.2825	0.473199	0.236600	−	0.971607i	$-0.423967\pi$
$911$	14.2825	0.473199	0.236600	+	0.971607i	$0.423967\pi$
$912$	0	0
$913$	11.5991	0.383873
$914$	−21.4459	−0.709367
$915$	0	0
$916$	−1.13221	−0.0374093
$917$	7.92460	0.261693
$918$	0	0
$919$	10.2972	0.339673	0.169837	−	0.985472i	$-0.445676\pi$
$919$	10.2972	0.339673	0.169837	+	0.985472i	$0.445676\pi$
$920$	15.3634	0.506516
$921$	0	0
$922$	−14.9813	−0.493381
$923$	−28.7901	−0.947636
$924$	0	0
$925$	18.6382	0.612820
$926$	−15.4799	−0.508702
$927$	0	0
$928$	7.14808	0.234647
$929$	−3.77218	−0.123761	−0.0618806	−	0.998084i	$-0.519710\pi$
$929$	−3.77218	−0.123761	−0.0618806	+	0.998084i	$0.519710\pi$
$930$	0	0
$931$	2.80640	0.0919759
$932$	2.14109	0.0701337
$933$	0	0
$934$	−28.3304	−0.926999
$935$	3.10728	0.101619
$936$	0	0
$937$	55.6015	1.81642	0.908211	−	0.418513i	$-0.137449\pi$
$937$	55.6015	1.81642	0.908211	+	0.418513i	$0.137449\pi$
$938$	−39.7917	−1.29924
$939$	0	0
$940$	−2.32998	−0.0759956
$941$	−1.44384	−0.0470680	−0.0235340	−	0.999723i	$-0.507492\pi$
$941$	−1.44384	−0.0470680	−0.0235340	+	0.999723i	$0.507492\pi$
$942$	0	0
$943$	59.4447	1.93579
$944$	14.3149	0.465911
$945$	0	0
$946$	8.04880	0.261689
$947$	−41.0784	−1.33487	−0.667434	−	0.744669i	$-0.732607\pi$
$947$	−41.0784	−1.33487	−0.667434	+	0.744669i	$0.732607\pi$
$948$	0	0
$949$	−43.1231	−1.39983
$950$	−5.75701	−0.186782
$951$	0	0
$952$	−14.6935	−0.476219
$953$	−26.5179	−0.859000	−0.429500	−	0.903067i	$-0.641310\pi$
$953$	−26.5179	−0.859000	−0.429500	+	0.903067i	$0.641310\pi$
$954$	0	0
$955$	19.7580	0.639354
$956$	−2.30964	−0.0746992
$957$	0	0
$958$	−50.6219	−1.63552
$959$	14.9759	0.483597
$960$	0	0
$961$	45.3310	1.46229
$962$	−32.8229	−1.05825
$963$	0	0
$964$	−8.53486	−0.274889
$965$	10.3689	0.333786
$966$	0	0
$967$	30.1124	0.968349	0.484175	−	0.874971i	$-0.339120\pi$
$967$	30.1124	0.968349	0.484175	+	0.874971i	$0.339120\pi$
$968$	−26.6552	−0.856730
$969$	0	0
$970$	−13.8057	−0.443273
$971$	−3.49524	−0.112168	−0.0560838	−	0.998426i	$-0.517861\pi$
$971$	−3.49524	−0.112168	−0.0560838	+	0.998426i	$0.517861\pi$
$972$	0	0
$973$	46.8963	1.50343
$974$	−16.4949	−0.528529
$975$	0	0
$976$	21.5700	0.690439
$977$	−55.8789	−1.78772	−0.893862	−	0.448341i	$-0.852015\pi$
$977$	−55.8789	−1.78772	−0.893862	+	0.448341i	$0.852015\pi$
$978$	0	0
$979$	−3.81844	−0.122038
$980$	0.709187	0.0226542
$981$	0	0
$982$	−22.6146	−0.721661
$983$	−52.8623	−1.68604	−0.843022	−	0.537879i	$-0.819226\pi$
$983$	−52.8623	−1.68604	−0.843022	+	0.537879i	$0.819226\pi$
$984$	0	0
$985$	4.68944	0.149418
$986$	−13.3624	−0.425545
$987$	0	0
$988$	−1.74169	−0.0554107
$989$	23.9261	0.760806
$990$	0	0
$991$	8.06158	0.256084	0.128042	−	0.991769i	$-0.459131\pi$
$991$	8.06158	0.256084	0.128042	+	0.991769i	$0.459131\pi$
$992$	−14.3683	−0.456192
$993$	0	0
$994$	−13.9430	−0.442244
$995$	14.4930	0.459460
$996$	0	0
$997$	−18.7766	−0.594662	−0.297331	−	0.954775i	$-0.596096\pi$
$997$	−18.7766	−0.594662	−0.297331	+	0.954775i	$0.596096\pi$
$998$	36.4306	1.15319
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3447.2.a.j.1.9		24
3.2	odd	2		383.2.a.c.1.16	✓	24
12.11	even	2		6128.2.a.p.1.1		24
15.14	odd	2		9575.2.a.e.1.9		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
383.2.a.c.1.16	✓	24	3.2	odd	2
3447.2.a.j.1.9		24	1.1	even	1	trivial
6128.2.a.p.1.1		24	12.11	even	2
9575.2.a.e.1.9		24	15.14	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:	\( N \)	\(=\)	\( 3447 = 3^{2} \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3447.a (trivial)

\(f(q)\)	\(=\)	\(q-1.30644 q^{2} -0.293212 q^{4} +0.910566 q^{5} +2.08417 q^{7} +2.99595 q^{8} -1.18960 q^{10} -1.45014 q^{11} -5.62225 q^{13} -2.72284 q^{14} -3.32760 q^{16} -2.35320 q^{17} -1.05653 q^{19} -0.266989 q^{20} +1.89453 q^{22} +5.63172 q^{23} -4.17087 q^{25} +7.34513 q^{26} -0.611103 q^{28} -4.34646 q^{29} +8.73676 q^{31} -1.64457 q^{32} +3.07432 q^{34} +1.89777 q^{35} -4.46866 q^{37} +1.38029 q^{38} +2.72801 q^{40} +10.5553 q^{41} +4.24845 q^{43} +0.425199 q^{44} -7.35751 q^{46} +8.72689 q^{47} -2.65624 q^{49} +5.44899 q^{50} +1.64851 q^{52} -1.87073 q^{53} -1.32045 q^{55} +6.24405 q^{56} +5.67839 q^{58} -4.30187 q^{59} -6.48215 q^{61} -11.4141 q^{62} +8.80375 q^{64} -5.11943 q^{65} +14.6140 q^{67} +0.689986 q^{68} -2.47933 q^{70} +5.12074 q^{71} +7.67008 q^{73} +5.83804 q^{74} +0.309786 q^{76} -3.02234 q^{77} +11.7685 q^{79} -3.03000 q^{80} -13.7899 q^{82} -7.99857 q^{83} -2.14274 q^{85} -5.55035 q^{86} -4.34455 q^{88} +2.63315 q^{89} -11.7177 q^{91} -1.65129 q^{92} -11.4012 q^{94} -0.962038 q^{95} +11.6053 q^{97} +3.47023 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 5 q^{2} + 29 q^{4} - 3 q^{5} + 17 q^{7} - 15 q^{8} + q^{10} + 28 q^{13} + 8 q^{14} + 35 q^{16} - 16 q^{17} + 13 q^{19} + 4 q^{20} + 12 q^{22} - 7 q^{23} + 67 q^{25} + 14 q^{26} + 39 q^{28} + 2 q^{29}+ \cdots + 29 q^{98}+O(q^{100}) \) 24 * q - 5 * q^2 + 29 * q^4 - 3 * q^5 + 17 * q^7 - 15 * q^8 + q^10 + 28 * q^13 + 8 * q^14 + 35 * q^16 - 16 * q^17 + 13 * q^19 + 4 * q^20 + 12 * q^22 - 7 * q^23 + 67 * q^25 + 14 * q^26 + 39 * q^28 + 2 * q^29 - 20 * q^32 - 18 * q^34 + 10 * q^35 + 35 * q^37 + 5 * q^38 - 36 * q^40 + 34 * q^43 + 36 * q^44 - 17 * q^46 + 3 * q^47 + 49 * q^49 + 20 * q^50 + 29 * q^52 + 9 * q^53 + 10 * q^55 + 47 * q^56 + 12 * q^58 + 6 * q^59 + 3 * q^61 + 58 * q^62 + 15 * q^64 - 8 * q^65 + 42 * q^67 + 23 * q^68 - 59 * q^70 + 13 * q^71 + 83 * q^73 + 45 * q^74 + 4 * q^76 + 21 * q^77 - 3 * q^79 + 93 * q^80 - 13 * q^82 + 23 * q^83 + 52 * q^85 + 43 * q^86 - 25 * q^88 + 38 * q^89 - 17 * q^91 + 61 * q^92 - 71 * q^94 + 66 * q^95 + 63 * q^97 + 29 * q^98

Introduction

L-functions

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