LMFDB - Embedded newform 675.4.a.ba.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,4,Mod(1,675)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("675.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,6,0,-6,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$39.8262892539$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.467024.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 26x^{2} + 101 $ x^4 - 26*x^2 + 101
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.18032$ of defining polynomial
Character	$\chi$	$=$	675.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.561553 q^{2} -7.68466 q^{4} +29.8369 q^{7} +8.80776 q^{8} -56.0007 q^{11} -20.4281 q^{13} -16.7550 q^{14} +56.5312 q^{16} +18.5464 q^{17} -70.7926 q^{19} +31.4474 q^{22} +111.885 q^{23} +11.4715 q^{26} -229.286 q^{28} -72.7557 q^{29} +106.170 q^{31} -102.207 q^{32} -10.4148 q^{34} -100.530 q^{37} +39.7538 q^{38} +307.326 q^{41} +479.001 q^{43} +430.347 q^{44} -62.8296 q^{46} -472.786 q^{47} +547.241 q^{49} +156.983 q^{52} -583.816 q^{53} +262.796 q^{56} +40.8562 q^{58} -429.894 q^{59} -443.315 q^{61} -59.6203 q^{62} -394.855 q^{64} -465.015 q^{67} -142.523 q^{68} -1056.92 q^{71} +234.118 q^{73} +56.4529 q^{74} +544.017 q^{76} -1670.89 q^{77} +275.639 q^{79} -172.580 q^{82} -1188.62 q^{83} -268.984 q^{86} -493.241 q^{88} -309.134 q^{89} -609.511 q^{91} -859.801 q^{92} +265.494 q^{94} -637.341 q^{97} -307.305 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{2} - 6 q^{4} - 6 q^{8} - 46 q^{16} - 66 q^{17} - 110 q^{19} - 6 q^{23} - 268 q^{31} - 582 q^{32} - 388 q^{34} + 192 q^{38} - 944 q^{46} - 1116 q^{47} + 482 q^{49} - 1824 q^{53} - 1798 q^{61}+ \cdots - 2796 q^{98}+O(q^{100}) $ 4 * q + 6 * q^2 - 6 * q^4 - 6 * q^8 - 46 * q^16 - 66 * q^17 - 110 * q^19 - 6 * q^23 - 268 * q^31 - 582 * q^32 - 388 * q^34 + 192 * q^38 - 944 * q^46 - 1116 * q^47 + 482 * q^49 - 1824 * q^53 - 1798 * q^61 - 1830 * q^62 - 862 * q^64 - 768 * q^68 + 1236 * q^76 - 4086 * q^77 + 682 * q^79 - 384 * q^83 - 360 * q^91 - 2796 * q^92 - 76 * q^94 - 2796 * 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 $

$2$	−0.561553	−0.198539	−0.0992695	−	0.995061i	$-0.531651\pi$
$2$	−0.561553	−0.198539	−0.0992695	+	0.995061i	$0.531651\pi$
$3$	0	0
$3$	0	0
$4$	−7.68466	−0.960582
$5$	0	0
$5$	0	0
$6$	0	0
$7$	29.8369	1.61104	0.805521	−	0.592567i	$-0.201886\pi$
$7$	29.8369	1.61104	0.805521	+	0.592567i	$0.201886\pi$
$8$	8.80776	0.389252
$9$	0	0
$10$	0	0
$11$	−56.0007	−1.53499	−0.767494	−	0.641057i	$-0.778496\pi$
$11$	−56.0007	−1.53499	−0.767494	+	0.641057i	$0.778496\pi$
$12$	0	0
$13$	−20.4281	−0.435826	−0.217913	−	0.975968i	$-0.569925\pi$
$13$	−20.4281	−0.435826	−0.217913	+	0.975968i	$0.569925\pi$
$14$	−16.7550	−0.319854
$15$	0	0
$16$	56.5312	0.883301
$17$	18.5464	0.264598	0.132299	−	0.991210i	$-0.457764\pi$
$17$	18.5464	0.264598	0.132299	+	0.991210i	$0.457764\pi$
$18$	0	0
$19$	−70.7926	−0.854786	−0.427393	−	0.904066i	$-0.640568\pi$
$19$	−70.7926	−0.854786	−0.427393	+	0.904066i	$0.640568\pi$
$20$	0	0
$21$	0	0
$22$	31.4474	0.304755
$23$	111.885	1.01434	0.507168	−	0.861847i	$-0.330692\pi$
$23$	111.885	1.01434	0.507168	+	0.861847i	$0.330692\pi$
$24$	0	0
$25$	0	0
$26$	11.4715	0.0865284
$27$	0	0
$28$	−229.286	−1.54754
$29$	−72.7557	−0.465876	−0.232938	−	0.972492i	$-0.574834\pi$
$29$	−72.7557	−0.465876	−0.232938	+	0.972492i	$0.574834\pi$
$30$	0	0
$31$	106.170	0.615122	0.307561	−	0.951528i	$-0.400487\pi$
$31$	106.170	0.615122	0.307561	+	0.951528i	$0.400487\pi$
$32$	−102.207	−0.564621
$33$	0	0
$34$	−10.4148	−0.0525330
$35$	0	0
$36$	0	0
$37$	−100.530	−0.446677	−0.223338	−	0.974741i	$-0.571695\pi$
$37$	−100.530	−0.446677	−0.223338	+	0.974741i	$0.571695\pi$
$38$	39.7538	0.169708
$39$	0	0
$40$	0	0
$41$	307.326	1.17064	0.585320	−	0.810803i	$-0.300969\pi$
$41$	307.326	1.17064	0.585320	+	0.810803i	$0.300969\pi$
$42$	0	0
$43$	479.001	1.69877	0.849383	−	0.527776i	$-0.176974\pi$
$43$	479.001	1.69877	0.849383	+	0.527776i	$0.176974\pi$
$44$	430.347	1.47448
$45$	0	0
$46$	−62.8296	−0.201385
$47$	−472.786	−1.46730	−0.733648	−	0.679530i	$-0.762184\pi$
$47$	−472.786	−1.46730	−0.733648	+	0.679530i	$0.762184\pi$
$48$	0	0
$49$	547.241	1.59546
$50$	0	0
$51$	0	0
$52$	156.983	0.418646
$53$	−583.816	−1.51308	−0.756541	−	0.653946i	$-0.773112\pi$
$53$	−583.816	−1.51308	−0.756541	+	0.653946i	$0.773112\pi$
$54$	0	0
$55$	0	0
$56$	262.796	0.627101
$57$	0	0
$58$	40.8562	0.0924945
$59$	−429.894	−0.948601	−0.474301	−	0.880363i	$-0.657299\pi$
$59$	−429.894	−0.948601	−0.474301	+	0.880363i	$0.657299\pi$
$60$	0	0
$61$	−443.315	−0.930503	−0.465252	−	0.885178i	$-0.654036\pi$
$61$	−443.315	−0.930503	−0.465252	+	0.885178i	$0.654036\pi$
$62$	−59.6203	−0.122126
$63$	0	0
$64$	−394.855	−0.771201
$65$	0	0
$66$	0	0
$67$	−465.015	−0.847919	−0.423960	−	0.905681i	$-0.639360\pi$
$67$	−465.015	−0.847919	−0.423960	+	0.905681i	$0.639360\pi$
$68$	−142.523	−0.254168
$69$	0	0
$70$	0	0
$71$	−1056.92	−1.76667	−0.883335	−	0.468743i	$-0.844707\pi$
$71$	−1056.92	−1.76667	−0.883335	+	0.468743i	$0.844707\pi$
$72$	0	0
$73$	234.118	0.375362	0.187681	−	0.982230i	$-0.439903\pi$
$73$	234.118	0.375362	0.187681	+	0.982230i	$0.439903\pi$
$74$	56.4529	0.0886827
$75$	0	0
$76$	544.017	0.821093
$77$	−1670.89	−2.47293
$78$	0	0
$79$	275.639	0.392555	0.196277	−	0.980548i	$-0.437115\pi$
$79$	275.639	0.392555	0.196277	+	0.980548i	$0.437115\pi$
$80$	0	0
$81$	0	0
$82$	−172.580	−0.232417
$83$	−1188.62	−1.57191	−0.785954	−	0.618285i	$-0.787828\pi$
$83$	−1188.62	−1.57191	−0.785954	+	0.618285i	$0.787828\pi$
$84$	0	0
$85$	0	0
$86$	−268.984	−0.337271
$87$	0	0
$88$	−493.241	−0.597497
$89$	−309.134	−0.368182	−0.184091	−	0.982909i	$-0.558934\pi$
$89$	−309.134	−0.368182	−0.184091	+	0.982909i	$0.558934\pi$
$90$	0	0
$91$	−609.511	−0.702133
$92$	−859.801	−0.974353
$93$	0	0
$94$	265.494	0.291315
$95$	0	0
$96$	0	0
$97$	−637.341	−0.667135	−0.333568	−	0.942726i	$-0.608253\pi$
$97$	−637.341	−0.667135	−0.333568	+	0.942726i	$0.608253\pi$
$98$	−307.305	−0.316760
$99$	0	0
$100$	0	0
$101$	174.642	0.172055	0.0860275	−	0.996293i	$-0.472583\pi$
$101$	174.642	0.172055	0.0860275	+	0.996293i	$0.472583\pi$
$102$	0	0
$103$	−295.148	−0.282348	−0.141174	−	0.989985i	$-0.545088\pi$
$103$	−295.148	−0.282348	−0.141174	+	0.989985i	$0.545088\pi$
$104$	−179.926	−0.169646
$105$	0	0
$106$	327.844	0.300406
$107$	−1356.65	−1.22572	−0.612862	−	0.790190i	$-0.709982\pi$
$107$	−1356.65	−1.22572	−0.612862	+	0.790190i	$0.709982\pi$
$108$	0	0
$109$	−1178.35	−1.03547	−0.517734	−	0.855542i	$-0.673224\pi$
$109$	−1178.35	−1.03547	−0.517734	+	0.855542i	$0.673224\pi$
$110$	0	0
$111$	0	0
$112$	1686.72	1.42303
$113$	−394.674	−0.328565	−0.164282	−	0.986413i	$-0.552531\pi$
$113$	−394.674	−0.328565	−0.164282	+	0.986413i	$0.552531\pi$
$114$	0	0
$115$	0	0
$116$	559.103	0.447512
$117$	0	0
$118$	241.408	0.188334
$119$	553.367	0.426278
$120$	0	0
$121$	1805.08	1.35619
$122$	248.945	0.184741
$123$	0	0
$124$	−815.884	−0.590875
$125$	0	0
$126$	0	0
$127$	1987.80	1.38889	0.694444	−	0.719547i	$-0.255650\pi$
$127$	1987.80	1.38889	0.694444	+	0.719547i	$0.255650\pi$
$128$	1039.39	0.717735
$129$	0	0
$130$	0	0
$131$	911.862	0.608166	0.304083	−	0.952645i	$-0.401650\pi$
$131$	911.862	0.608166	0.304083	+	0.952645i	$0.401650\pi$
$132$	0	0
$133$	−2112.23	−1.37710
$134$	261.130	0.168345
$135$	0	0
$136$	163.352	0.102995
$137$	−739.670	−0.461272	−0.230636	−	0.973040i	$-0.574081\pi$
$137$	−739.670	−0.461272	−0.230636	+	0.973040i	$0.574081\pi$
$138$	0	0
$139$	641.841	0.391656	0.195828	−	0.980638i	$-0.437261\pi$
$139$	641.841	0.391656	0.195828	+	0.980638i	$0.437261\pi$
$140$	0	0
$141$	0	0
$142$	593.517	0.350753
$143$	1143.99	0.668987
$144$	0	0
$145$	0	0
$146$	−131.470	−0.0745240
$147$	0	0
$148$	772.539	0.429070
$149$	2070.22	1.13825	0.569124	−	0.822252i	$-0.307282\pi$
$149$	2070.22	1.13825	0.569124	+	0.822252i	$0.307282\pi$
$150$	0	0
$151$	−2975.42	−1.60355	−0.801776	−	0.597624i	$-0.796112\pi$
$151$	−2975.42	−1.60355	−0.801776	+	0.597624i	$0.796112\pi$
$152$	−623.525	−0.332727
$153$	0	0
$154$	938.292	0.490972
$155$	0	0
$156$	0	0
$157$	−2030.01	−1.03193	−0.515964	−	0.856610i	$-0.672566\pi$
$157$	−2030.01	−1.03193	−0.515964	+	0.856610i	$0.672566\pi$
$158$	−154.786	−0.0779374
$159$	0	0
$160$	0	0
$161$	3338.32	1.63414
$162$	0	0
$163$	2000.94	0.961506	0.480753	−	0.876856i	$-0.340363\pi$
$163$	2000.94	0.961506	0.480753	+	0.876856i	$0.340363\pi$
$164$	−2361.69	−1.12450
$165$	0	0
$166$	667.475	0.312085
$167$	−432.425	−0.200372	−0.100186	−	0.994969i	$-0.531944\pi$
$167$	−432.425	−0.200372	−0.100186	+	0.994969i	$0.531944\pi$
$168$	0	0
$169$	−1779.69	−0.810056
$170$	0	0
$171$	0	0
$172$	−3680.96	−1.63181
$173$	−1961.11	−0.861850	−0.430925	−	0.902388i	$-0.641813\pi$
$173$	−1961.11	−0.861850	−0.430925	+	0.902388i	$0.641813\pi$
$174$	0	0
$175$	0	0
$176$	−3165.79	−1.35586
$177$	0	0
$178$	173.595	0.0730984
$179$	−1727.57	−0.721368	−0.360684	−	0.932688i	$-0.617457\pi$
$179$	−1727.57	−0.721368	−0.360684	+	0.932688i	$0.617457\pi$
$180$	0	0
$181$	−778.122	−0.319543	−0.159772	−	0.987154i	$-0.551076\pi$
$181$	−778.122	−0.319543	−0.159772	+	0.987154i	$0.551076\pi$
$182$	342.273	0.139401
$183$	0	0
$184$	985.460	0.394832
$185$	0	0
$186$	0	0
$187$	−1038.61	−0.406154
$188$	3633.20	1.40946
$189$	0	0
$190$	0	0
$191$	−4093.65	−1.55082	−0.775408	−	0.631460i	$-0.782456\pi$
$191$	−4093.65	−1.55082	−0.775408	+	0.631460i	$0.782456\pi$
$192$	0	0
$193$	390.593	0.145676	0.0728381	−	0.997344i	$-0.476794\pi$
$193$	390.593	0.145676	0.0728381	+	0.997344i	$0.476794\pi$
$194$	357.900	0.132452
$195$	0	0
$196$	−4205.36	−1.53257
$197$	−736.133	−0.266230	−0.133115	−	0.991101i	$-0.542498\pi$
$197$	−736.133	−0.266230	−0.133115	+	0.991101i	$0.542498\pi$
$198$	0	0
$199$	2412.02	0.859214	0.429607	−	0.903016i	$-0.358652\pi$
$199$	2412.02	0.859214	0.429607	+	0.903016i	$0.358652\pi$
$200$	0	0
$201$	0	0
$202$	−98.0709	−0.0341596
$203$	−2170.81	−0.750546
$204$	0	0
$205$	0	0
$206$	165.741	0.0560570
$207$	0	0
$208$	−1154.83	−0.384965
$209$	3964.44	1.31209
$210$	0	0
$211$	3547.01	1.15728	0.578640	−	0.815583i	$-0.303584\pi$
$211$	3547.01	1.15728	0.578640	+	0.815583i	$0.303584\pi$
$212$	4486.43	1.45344
$213$	0	0
$214$	761.832	0.243354
$215$	0	0
$216$	0	0
$217$	3167.80	0.990987
$218$	661.709	0.205581
$219$	0	0
$220$	0	0
$221$	−378.868	−0.115319
$222$	0	0
$223$	4851.89	1.45698	0.728490	−	0.685056i	$-0.240222\pi$
$223$	4851.89	1.45698	0.728490	+	0.685056i	$0.240222\pi$
$224$	−3049.55	−0.909629
$225$	0	0
$226$	221.630	0.0652329
$227$	1458.12	0.426339	0.213169	−	0.977015i	$-0.431621\pi$
$227$	1458.12	0.426339	0.213169	+	0.977015i	$0.431621\pi$
$228$	0	0
$229$	−2260.72	−0.652371	−0.326185	−	0.945306i	$-0.605763\pi$
$229$	−2260.72	−0.652371	−0.326185	+	0.945306i	$0.605763\pi$
$230$	0	0
$231$	0	0
$232$	−640.815	−0.181343
$233$	−2131.54	−0.599321	−0.299660	−	0.954046i	$-0.596873\pi$
$233$	−2131.54	−0.599321	−0.299660	+	0.954046i	$0.596873\pi$
$234$	0	0
$235$	0	0
$236$	3303.59	0.911210
$237$	0	0
$238$	−310.745	−0.0846328
$239$	−2859.82	−0.774002	−0.387001	−	0.922079i	$-0.626489\pi$
$239$	−2859.82	−0.774002	−0.387001	+	0.922079i	$0.626489\pi$
$240$	0	0
$241$	−1438.84	−0.384580	−0.192290	−	0.981338i	$-0.561591\pi$
$241$	−1438.84	−0.384580	−0.192290	+	0.981338i	$0.561591\pi$
$242$	−1013.65	−0.269256
$243$	0	0
$244$	3406.73	0.893825
$245$	0	0
$246$	0	0
$247$	1446.16	0.372538
$248$	935.124	0.239437
$249$	0	0
$250$	0	0
$251$	−2911.13	−0.732068	−0.366034	−	0.930601i	$-0.619285\pi$
$251$	−2911.13	−0.732068	−0.366034	+	0.930601i	$0.619285\pi$
$252$	0	0
$253$	−6265.66	−1.55699
$254$	−1116.25	−0.275748
$255$	0	0
$256$	2575.17	0.628703
$257$	2092.28	0.507832	0.253916	−	0.967226i	$-0.418281\pi$
$257$	2092.28	0.507832	0.253916	+	0.967226i	$0.418281\pi$
$258$	0	0
$259$	−2999.51	−0.719615
$260$	0	0
$261$	0	0
$262$	−512.059	−0.120745
$263$	−5033.58	−1.18017	−0.590083	−	0.807343i	$-0.700905\pi$
$263$	−5033.58	−1.18017	−0.590083	+	0.807343i	$0.700905\pi$
$264$	0	0
$265$	0	0
$266$	1186.13	0.273407
$267$	0	0
$268$	3573.48	0.814496
$269$	3875.07	0.878317	0.439158	−	0.898410i	$-0.355277\pi$
$269$	3875.07	0.878317	0.439158	+	0.898410i	$0.355277\pi$
$270$	0	0
$271$	1414.37	0.317037	0.158519	−	0.987356i	$-0.449328\pi$
$271$	1414.37	0.317037	0.158519	+	0.987356i	$0.449328\pi$
$272$	1048.45	0.233719
$273$	0	0
$274$	415.363	0.0915804
$275$	0	0
$276$	0	0
$277$	7330.64	1.59009	0.795046	−	0.606549i	$-0.207447\pi$
$277$	7330.64	1.59009	0.795046	+	0.606549i	$0.207447\pi$
$278$	−360.428	−0.0777590
$279$	0	0
$280$	0	0
$281$	7704.56	1.63564	0.817822	−	0.575472i	$-0.195182\pi$
$281$	7704.56	1.63564	0.817822	+	0.575472i	$0.195182\pi$
$282$	0	0
$283$	−5207.81	−1.09390	−0.546948	−	0.837167i	$-0.684210\pi$
$283$	−5207.81	−1.09390	−0.546948	+	0.837167i	$0.684210\pi$
$284$	8122.08	1.69703
$285$	0	0
$286$	−642.410	−0.132820
$287$	9169.65	1.88595
$288$	0	0
$289$	−4569.03	−0.929988
$290$	0	0
$291$	0	0
$292$	−1799.12	−0.360566
$293$	5182.24	1.03328	0.516638	−	0.856204i	$-0.327183\pi$
$293$	5182.24	1.03328	0.516638	+	0.856204i	$0.327183\pi$
$294$	0	0
$295$	0	0
$296$	−885.445	−0.173870
$297$	0	0
$298$	−1162.54	−0.225986
$299$	−2285.61	−0.442074
$300$	0	0
$301$	14291.9	2.73678
$302$	1670.86	0.318368
$303$	0	0
$304$	−4001.99	−0.755033
$305$	0	0
$306$	0	0
$307$	−5655.62	−1.05141	−0.525706	−	0.850666i	$-0.676199\pi$
$307$	−5655.62	−1.05141	−0.525706	+	0.850666i	$0.676199\pi$
$308$	12840.2	2.37545
$309$	0	0
$310$	0	0
$311$	3372.73	0.614952	0.307476	−	0.951556i	$-0.400516\pi$
$311$	3372.73	0.614952	0.307476	+	0.951556i	$0.400516\pi$
$312$	0	0
$313$	−6300.42	−1.13777	−0.568883	−	0.822418i	$-0.692624\pi$
$313$	−6300.42	−1.13777	−0.568883	+	0.822418i	$0.692624\pi$
$314$	1139.96	0.204878
$315$	0	0
$316$	−2118.19	−0.377081
$317$	7278.61	1.28961	0.644806	−	0.764346i	$-0.276938\pi$
$317$	7278.61	1.28961	0.644806	+	0.764346i	$0.276938\pi$
$318$	0	0
$319$	4074.37	0.715113
$320$	0	0
$321$	0	0
$322$	−1874.64	−0.324440
$323$	−1312.95	−0.226175
$324$	0	0
$325$	0	0
$326$	−1123.63	−0.190896
$327$	0	0
$328$	2706.85	0.455674
$329$	−14106.5	−2.36388
$330$	0	0
$331$	−11103.1	−1.84374	−0.921872	−	0.387494i	$-0.873341\pi$
$331$	−11103.1	−1.84374	−0.921872	+	0.387494i	$0.873341\pi$
$332$	9134.16	1.50995
$333$	0	0
$334$	242.830	0.0397816
$335$	0	0
$336$	0	0
$337$	3993.32	0.645489	0.322744	−	0.946486i	$-0.395395\pi$
$337$	3993.32	0.645489	0.322744	+	0.946486i	$0.395395\pi$
$338$	999.392	0.160828
$339$	0	0
$340$	0	0
$341$	−5945.62	−0.944204
$342$	0	0
$343$	6093.93	0.959305
$344$	4218.93	0.661248
$345$	0	0
$346$	1101.26	0.171111
$347$	−4130.36	−0.638989	−0.319494	−	0.947588i	$-0.603513\pi$
$347$	−4130.36	−0.638989	−0.319494	+	0.947588i	$0.603513\pi$
$348$	0	0
$349$	−245.389	−0.0376372	−0.0188186	−	0.999823i	$-0.505991\pi$
$349$	−245.389	−0.0376372	−0.0188186	+	0.999823i	$0.505991\pi$
$350$	0	0
$351$	0	0
$352$	5723.69	0.866687
$353$	1404.54	0.211774	0.105887	−	0.994378i	$-0.466232\pi$
$353$	1404.54	0.211774	0.105887	+	0.994378i	$0.466232\pi$
$354$	0	0
$355$	0	0
$356$	2375.59	0.353669
$357$	0	0
$358$	970.124	0.143220
$359$	5456.09	0.802120	0.401060	−	0.916052i	$-0.368642\pi$
$359$	5456.09	0.802120	0.401060	+	0.916052i	$0.368642\pi$
$360$	0	0
$361$	−1847.41	−0.269341
$362$	436.957	0.0634418
$363$	0	0
$364$	4683.89	0.674457
$365$	0	0
$366$	0	0
$367$	−10023.2	−1.42563	−0.712813	−	0.701354i	$-0.752579\pi$
$367$	−10023.2	−1.42563	−0.712813	+	0.701354i	$0.752579\pi$
$368$	6325.02	0.895963
$369$	0	0
$370$	0	0
$371$	−17419.3	−2.43764
$372$	0	0
$373$	6378.15	0.885384	0.442692	−	0.896674i	$-0.354024\pi$
$373$	6378.15	0.885384	0.442692	+	0.896674i	$0.354024\pi$
$374$	583.235	0.0806374
$375$	0	0
$376$	−4164.19	−0.571148
$377$	1486.26	0.203041
$378$	0	0
$379$	271.207	0.0367572	0.0183786	−	0.999831i	$-0.494150\pi$
$379$	271.207	0.0367572	0.0183786	+	0.999831i	$0.494150\pi$
$380$	0	0
$381$	0	0
$382$	2298.80	0.307897
$383$	4609.55	0.614979	0.307490	−	0.951551i	$-0.400511\pi$
$383$	4609.55	0.614979	0.307490	+	0.951551i	$0.400511\pi$
$384$	0	0
$385$	0	0
$386$	−219.339	−0.0289224
$387$	0	0
$388$	4897.74	0.640838
$389$	−8190.17	−1.06750	−0.533751	−	0.845642i	$-0.679218\pi$
$389$	−8190.17	−1.06750	−0.533751	+	0.845642i	$0.679218\pi$
$390$	0	0
$391$	2075.07	0.268391
$392$	4819.97	0.621034
$393$	0	0
$394$	413.377	0.0528570
$395$	0	0
$396$	0	0
$397$	983.856	0.124379	0.0621893	−	0.998064i	$-0.480192\pi$
$397$	983.856	0.124379	0.0621893	+	0.998064i	$0.480192\pi$
$398$	−1354.48	−0.170587
$399$	0	0
$400$	0	0
$401$	938.899	0.116924	0.0584618	−	0.998290i	$-0.481380\pi$
$401$	938.899	0.116924	0.0584618	+	0.998290i	$0.481380\pi$
$402$	0	0
$403$	−2168.86	−0.268086
$404$	−1342.07	−0.165273
$405$	0	0
$406$	1219.02	0.149013
$407$	5629.75	0.685643
$408$	0	0
$409$	11342.4	1.37127	0.685633	−	0.727948i	$-0.259526\pi$
$409$	11342.4	1.37127	0.685633	+	0.727948i	$0.259526\pi$
$410$	0	0
$411$	0	0
$412$	2268.11	0.271218
$413$	−12826.7	−1.52824
$414$	0	0
$415$	0	0
$416$	2087.90	0.246077
$417$	0	0
$418$	−2226.24	−0.260500
$419$	3547.23	0.413588	0.206794	−	0.978384i	$-0.433697\pi$
$419$	3547.23	0.413588	0.206794	+	0.978384i	$0.433697\pi$
$420$	0	0
$421$	−5067.83	−0.586677	−0.293338	−	0.956009i	$-0.594766\pi$
$421$	−5067.83	−0.586677	−0.293338	+	0.956009i	$0.594766\pi$
$422$	−1991.83	−0.229765
$423$	0	0
$424$	−5142.12	−0.588970
$425$	0	0
$426$	0	0
$427$	−13227.2	−1.49908
$428$	10425.4	1.17741
$429$	0	0
$430$	0	0
$431$	11730.9	1.31104	0.655520	−	0.755178i	$-0.272450\pi$
$431$	11730.9	1.31104	0.655520	+	0.755178i	$0.272450\pi$
$432$	0	0
$433$	5764.46	0.639774	0.319887	−	0.947456i	$-0.396355\pi$
$433$	5764.46	0.639774	0.319887	+	0.947456i	$0.396355\pi$
$434$	−1778.89	−0.196749
$435$	0	0
$436$	9055.26	0.994652
$437$	−7920.66	−0.867040
$438$	0	0
$439$	7745.05	0.842030	0.421015	−	0.907054i	$-0.361674\pi$
$439$	7745.05	0.842030	0.421015	+	0.907054i	$0.361674\pi$
$440$	0	0
$441$	0	0
$442$	212.754	0.0228952
$443$	729.748	0.0782650	0.0391325	−	0.999234i	$-0.487541\pi$
$443$	729.748	0.0782650	0.0391325	+	0.999234i	$0.487541\pi$
$444$	0	0
$445$	0	0
$446$	−2724.59	−0.289267
$447$	0	0
$448$	−11781.3	−1.24244
$449$	7915.00	0.831920	0.415960	−	0.909383i	$-0.363446\pi$
$449$	7915.00	0.831920	0.415960	+	0.909383i	$0.363446\pi$
$450$	0	0
$451$	−17210.5	−1.79692
$452$	3032.94	0.315614
$453$	0	0
$454$	−818.812	−0.0846448
$455$	0	0
$456$	0	0
$457$	−1031.32	−0.105565	−0.0527825	−	0.998606i	$-0.516809\pi$
$457$	−1031.32	−0.105565	−0.0527825	+	0.998606i	$0.516809\pi$
$458$	1269.52	0.129521
$459$	0	0
$460$	0	0
$461$	14251.2	1.43980	0.719898	−	0.694080i	$-0.244189\pi$
$461$	14251.2	1.43980	0.719898	+	0.694080i	$0.244189\pi$
$462$	0	0
$463$	5253.76	0.527349	0.263675	−	0.964612i	$-0.415065\pi$
$463$	5253.76	0.527349	0.263675	+	0.964612i	$0.415065\pi$
$464$	−4112.97	−0.411509
$465$	0	0
$466$	1196.97	0.118988
$467$	−7785.91	−0.771497	−0.385748	−	0.922604i	$-0.626057\pi$
$467$	−7785.91	−0.771497	−0.385748	+	0.922604i	$0.626057\pi$
$468$	0	0
$469$	−13874.6	−1.36603
$470$	0	0
$471$	0	0
$472$	−3786.41	−0.369245
$473$	−26824.4	−2.60759
$474$	0	0
$475$	0	0
$476$	−4252.44	−0.409475
$477$	0	0
$478$	1605.94	0.153670
$479$	12558.1	1.19790	0.598950	−	0.800787i	$-0.295585\pi$
$479$	12558.1	1.19790	0.598950	+	0.800787i	$0.295585\pi$
$480$	0	0
$481$	2053.64	0.194673
$482$	807.984	0.0763540
$483$	0	0
$484$	−13871.4	−1.30273
$485$	0	0
$486$	0	0
$487$	−1913.38	−0.178036	−0.0890179	−	0.996030i	$-0.528373\pi$
$487$	−1913.38	−0.178036	−0.0890179	+	0.996030i	$0.528373\pi$
$488$	−3904.62	−0.362200
$489$	0	0
$490$	0	0
$491$	7333.07	0.674006	0.337003	−	0.941504i	$-0.390587\pi$
$491$	7333.07	0.674006	0.337003	+	0.941504i	$0.390587\pi$
$492$	0	0
$493$	−1349.36	−0.123270
$494$	−812.094	−0.0739632
$495$	0	0
$496$	6001.95	0.543337
$497$	−31535.3	−2.84618
$498$	0	0
$499$	4940.20	0.443193	0.221597	−	0.975138i	$-0.428873\pi$
$499$	4940.20	0.443193	0.221597	+	0.975138i	$0.428873\pi$
$500$	0	0
$501$	0	0
$502$	1634.76	0.145344
$503$	−7988.42	−0.708123	−0.354062	−	0.935222i	$-0.615200\pi$
$503$	−7988.42	−0.708123	−0.354062	+	0.935222i	$0.615200\pi$
$504$	0	0
$505$	0	0
$506$	3518.50	0.309123
$507$	0	0
$508$	−15275.6	−1.33414
$509$	10056.8	0.875753	0.437876	−	0.899035i	$-0.355731\pi$
$509$	10056.8	0.875753	0.437876	+	0.899035i	$0.355731\pi$
$510$	0	0
$511$	6985.35	0.604724
$512$	−9761.22	−0.842557
$513$	0	0
$514$	−1174.93	−0.100824
$515$	0	0
$516$	0	0
$517$	26476.4	2.25228
$518$	1684.38	0.142872
$519$	0	0
$520$	0	0
$521$	−8552.26	−0.719158	−0.359579	−	0.933115i	$-0.617080\pi$
$521$	−8552.26	−0.719158	−0.359579	+	0.933115i	$0.617080\pi$
$522$	0	0
$523$	−9523.15	−0.796211	−0.398105	−	0.917340i	$-0.630332\pi$
$523$	−9523.15	−0.796211	−0.398105	+	0.917340i	$0.630332\pi$
$524$	−7007.35	−0.584194
$525$	0	0
$526$	2826.62	0.234309
$527$	1969.08	0.162760
$528$	0	0
$529$	351.344	0.0288768
$530$	0	0
$531$	0	0
$532$	16231.8	1.32281
$533$	−6278.08	−0.510195
$534$	0	0
$535$	0	0
$536$	−4095.74	−0.330054
$537$	0	0
$538$	−2176.06	−0.174380
$539$	−30645.9	−2.44900
$540$	0	0
$541$	5247.11	0.416988	0.208494	−	0.978024i	$-0.433144\pi$
$541$	5247.11	0.416988	0.208494	+	0.978024i	$0.433144\pi$
$542$	−794.246	−0.0629443
$543$	0	0
$544$	−1895.58	−0.149398
$545$	0	0
$546$	0	0
$547$	14173.3	1.10787	0.553936	−	0.832559i	$-0.313125\pi$
$547$	14173.3	1.10787	0.553936	+	0.832559i	$0.313125\pi$
$548$	5684.11	0.443089
$549$	0	0
$550$	0	0
$551$	5150.57	0.398224
$552$	0	0
$553$	8224.22	0.632422
$554$	−4116.54	−0.315695
$555$	0	0
$556$	−4932.33	−0.376218
$557$	−14105.2	−1.07299	−0.536496	−	0.843903i	$-0.680252\pi$
$557$	−14105.2	−1.07299	−0.536496	+	0.843903i	$0.680252\pi$
$558$	0	0
$559$	−9785.08	−0.740366
$560$	0	0
$561$	0	0
$562$	−4326.52	−0.324739
$563$	−12040.2	−0.901307	−0.450653	−	0.892699i	$-0.648809\pi$
$563$	−12040.2	−0.901307	−0.450653	+	0.892699i	$0.648809\pi$
$564$	0	0
$565$	0	0
$566$	2924.46	0.217181
$567$	0	0
$568$	−9309.12	−0.687679
$569$	6433.56	0.474005	0.237003	−	0.971509i	$-0.423835\pi$
$569$	6433.56	0.474005	0.237003	+	0.971509i	$0.423835\pi$
$570$	0	0
$571$	3127.79	0.229236	0.114618	−	0.993410i	$-0.463436\pi$
$571$	3127.79	0.229236	0.114618	+	0.993410i	$0.463436\pi$
$572$	−8791.16	−0.642617
$573$	0	0
$574$	−5149.24	−0.374434
$575$	0	0
$576$	0	0
$577$	−22052.3	−1.59107	−0.795536	−	0.605907i	$-0.792810\pi$
$577$	−22052.3	−1.59107	−0.795536	+	0.605907i	$0.792810\pi$
$578$	2565.75	0.184639
$579$	0	0
$580$	0	0
$581$	−35464.8	−2.53241
$582$	0	0
$583$	32694.1	2.32256
$584$	2062.05	0.146110
$585$	0	0
$586$	−2910.10	−0.205145
$587$	9281.07	0.652591	0.326295	−	0.945268i	$-0.394200\pi$
$587$	9281.07	0.652591	0.326295	+	0.945268i	$0.394200\pi$
$588$	0	0
$589$	−7516.08	−0.525798
$590$	0	0
$591$	0	0
$592$	−5683.09	−0.394550
$593$	11156.9	0.772613	0.386306	−	0.922370i	$-0.373751\pi$
$593$	11156.9	0.772613	0.386306	+	0.922370i	$0.373751\pi$
$594$	0	0
$595$	0	0
$596$	−15908.9	−1.09338
$597$	0	0
$598$	1283.49	0.0877688
$599$	26468.6	1.80547	0.902736	−	0.430195i	$-0.141555\pi$
$599$	26468.6	1.80547	0.902736	+	0.430195i	$0.141555\pi$
$600$	0	0
$601$	−2633.36	−0.178730	−0.0893651	−	0.995999i	$-0.528484\pi$
$601$	−2633.36	−0.178730	−0.0893651	+	0.995999i	$0.528484\pi$
$602$	−8025.66	−0.543358
$603$	0	0
$604$	22865.1	1.54034
$605$	0	0
$606$	0	0
$607$	−9050.50	−0.605187	−0.302594	−	0.953120i	$-0.597852\pi$
$607$	−9050.50	−0.605187	−0.302594	+	0.953120i	$0.597852\pi$
$608$	7235.53	0.482631
$609$	0	0
$610$	0	0
$611$	9658.12	0.639485
$612$	0	0
$613$	−15873.7	−1.04590	−0.522948	−	0.852364i	$-0.675168\pi$
$613$	−15873.7	−1.04590	−0.522948	+	0.852364i	$0.675168\pi$
$614$	3175.93	0.208746
$615$	0	0
$616$	−14716.8	−0.962592
$617$	−6178.82	−0.403161	−0.201580	−	0.979472i	$-0.564608\pi$
$617$	−6178.82	−0.403161	−0.201580	+	0.979472i	$0.564608\pi$
$618$	0	0
$619$	24531.2	1.59288	0.796440	−	0.604717i	$-0.206714\pi$
$619$	24531.2	1.59288	0.796440	+	0.604717i	$0.206714\pi$
$620$	0	0
$621$	0	0
$622$	−1893.97	−0.122092
$623$	−9223.62	−0.593157
$624$	0	0
$625$	0	0
$626$	3538.02	0.225891
$627$	0	0
$628$	15600.0	0.991252
$629$	−1864.47	−0.118190
$630$	0	0
$631$	6495.69	0.409809	0.204904	−	0.978782i	$-0.434312\pi$
$631$	6495.69	0.409809	0.204904	+	0.978782i	$0.434312\pi$
$632$	2427.76	0.152803
$633$	0	0
$634$	−4087.32	−0.256038
$635$	0	0
$636$	0	0
$637$	−11179.1	−0.695341
$638$	−2287.98	−0.141978
$639$	0	0
$640$	0	0
$641$	−7193.13	−0.443232	−0.221616	−	0.975134i	$-0.571133\pi$
$641$	−7193.13	−0.443232	−0.221616	+	0.975134i	$0.571133\pi$
$642$	0	0
$643$	−21896.7	−1.34295	−0.671477	−	0.741025i	$-0.734340\pi$
$643$	−21896.7	−1.34295	−0.671477	+	0.741025i	$0.734340\pi$
$644$	−25653.8	−1.56972
$645$	0	0
$646$	737.290	0.0449044
$647$	−32602.6	−1.98105	−0.990525	−	0.137334i	$-0.956147\pi$
$647$	−32602.6	−1.98105	−0.990525	+	0.137334i	$0.956147\pi$
$648$	0	0
$649$	24074.4	1.45609
$650$	0	0
$651$	0	0
$652$	−15376.5	−0.923606
$653$	−5153.57	−0.308843	−0.154422	−	0.988005i	$-0.549351\pi$
$653$	−5153.57	−0.308843	−0.154422	+	0.988005i	$0.549351\pi$
$654$	0	0
$655$	0	0
$656$	17373.5	1.03403
$657$	0	0
$658$	7921.53	0.469321
$659$	−10838.0	−0.640652	−0.320326	−	0.947307i	$-0.603793\pi$
$659$	−10838.0	−0.640652	−0.320326	+	0.947307i	$0.603793\pi$
$660$	0	0
$661$	−25123.0	−1.47832	−0.739160	−	0.673529i	$-0.764777\pi$
$661$	−25123.0	−1.47832	−0.739160	+	0.673529i	$0.764777\pi$
$662$	6234.96	0.366055
$663$	0	0
$664$	−10469.1	−0.611868
$665$	0	0
$666$	0	0
$667$	−8140.31	−0.472555
$668$	3323.04	0.192473
$669$	0	0
$670$	0	0
$671$	24826.0	1.42831
$672$	0	0
$673$	−5134.50	−0.294087	−0.147043	−	0.989130i	$-0.546976\pi$
$673$	−5134.50	−0.294087	−0.147043	+	0.989130i	$0.546976\pi$
$674$	−2242.46	−0.128155
$675$	0	0
$676$	13676.3	0.778125
$677$	−25534.3	−1.44957	−0.724787	−	0.688973i	$-0.758062\pi$
$677$	−25534.3	−1.44957	−0.724787	+	0.688973i	$0.758062\pi$
$678$	0	0
$679$	−19016.3	−1.07478
$680$	0	0
$681$	0	0
$682$	3338.78	0.187461
$683$	15701.4	0.879644	0.439822	−	0.898085i	$-0.355042\pi$
$683$	15701.4	0.879644	0.439822	+	0.898085i	$0.355042\pi$
$684$	0	0
$685$	0	0
$686$	−3422.07	−0.190459
$687$	0	0
$688$	27078.5	1.50052
$689$	11926.3	0.659440
$690$	0	0
$691$	25785.3	1.41957	0.709783	−	0.704420i	$-0.248793\pi$
$691$	25785.3	1.41957	0.709783	+	0.704420i	$0.248793\pi$
$692$	15070.4	0.827878
$693$	0	0
$694$	2319.41	0.126864
$695$	0	0
$696$	0	0
$697$	5699.79	0.309749
$698$	137.799	0.00747245
$699$	0	0
$700$	0	0
$701$	7310.13	0.393866	0.196933	−	0.980417i	$-0.436902\pi$
$701$	7310.13	0.393866	0.196933	+	0.980417i	$0.436902\pi$
$702$	0	0
$703$	7116.78	0.381813
$704$	22112.2	1.18378
$705$	0	0
$706$	−788.724	−0.0420453
$707$	5210.79	0.277188
$708$	0	0
$709$	13616.1	0.721246	0.360623	−	0.932712i	$-0.382564\pi$
$709$	13616.1	0.721246	0.360623	+	0.932712i	$0.382564\pi$
$710$	0	0
$711$	0	0
$712$	−2722.78	−0.143315
$713$	11878.9	0.623940
$714$	0	0
$715$	0	0
$716$	13275.8	0.692934
$717$	0	0
$718$	−3063.88	−0.159252
$719$	16018.4	0.830854	0.415427	−	0.909627i	$-0.363632\pi$
$719$	16018.4	0.830854	0.415427	+	0.909627i	$0.363632\pi$
$720$	0	0
$721$	−8806.31	−0.454874
$722$	1037.42	0.0534746
$723$	0	0
$724$	5979.60	0.306948
$725$	0	0
$726$	0	0
$727$	20662.3	1.05409	0.527045	−	0.849838i	$-0.323300\pi$
$727$	20662.3	1.05409	0.527045	+	0.849838i	$0.323300\pi$
$728$	−5368.43	−0.273307
$729$	0	0
$730$	0	0
$731$	8883.74	0.449490
$732$	0	0
$733$	3992.89	0.201202	0.100601	−	0.994927i	$-0.467923\pi$
$733$	3992.89	0.201202	0.100601	+	0.994927i	$0.467923\pi$
$734$	5628.54	0.283042
$735$	0	0
$736$	−11435.5	−0.572716
$737$	26041.2	1.30155
$738$	0	0
$739$	−12595.1	−0.626954	−0.313477	−	0.949596i	$-0.601494\pi$
$739$	−12595.1	−0.626954	−0.313477	+	0.949596i	$0.601494\pi$
$740$	0	0
$741$	0	0
$742$	9781.84	0.483966
$743$	2179.02	0.107591	0.0537957	−	0.998552i	$-0.482868\pi$
$743$	2179.02	0.107591	0.0537957	+	0.998552i	$0.482868\pi$
$744$	0	0
$745$	0	0
$746$	−3581.67	−0.175783
$747$	0	0
$748$	7981.38	0.390144
$749$	−40478.3	−1.97469
$750$	0	0
$751$	−23412.8	−1.13761	−0.568806	−	0.822472i	$-0.692594\pi$
$751$	−23412.8	−1.13761	−0.568806	+	0.822472i	$0.692594\pi$
$752$	−26727.2	−1.29606
$753$	0	0
$754$	−834.614	−0.0403115
$755$	0	0
$756$	0	0
$757$	19971.1	0.958865	0.479432	−	0.877579i	$-0.340843\pi$
$757$	19971.1	0.958865	0.479432	+	0.877579i	$0.340843\pi$
$758$	−152.297	−0.00729774
$759$	0	0
$760$	0	0
$761$	−11501.1	−0.547853	−0.273926	−	0.961751i	$-0.588322\pi$
$761$	−11501.1	−0.547853	−0.273926	+	0.961751i	$0.588322\pi$
$762$	0	0
$763$	−35158.5	−1.66818
$764$	31458.3	1.48969
$765$	0	0
$766$	−2588.51	−0.122097
$767$	8781.92	0.413425
$768$	0	0
$769$	−31214.0	−1.46372	−0.731862	−	0.681453i	$-0.761348\pi$
$769$	−31214.0	−1.46372	−0.731862	+	0.681453i	$0.761348\pi$
$770$	0	0
$771$	0	0
$772$	−3001.57	−0.139934
$773$	−39290.8	−1.82819	−0.914095	−	0.405501i	$-0.867097\pi$
$773$	−39290.8	−1.82819	−0.914095	+	0.405501i	$0.867097\pi$
$774$	0	0
$775$	0	0
$776$	−5613.55	−0.259684
$777$	0	0
$778$	4599.22	0.211941
$779$	−21756.4	−1.00065
$780$	0	0
$781$	59188.4	2.71181
$782$	−1165.26	−0.0532860
$783$	0	0
$784$	30936.2	1.40927
$785$	0	0
$786$	0	0
$787$	4599.54	0.208330	0.104165	−	0.994560i	$-0.466783\pi$
$787$	4599.54	0.208330	0.104165	+	0.994560i	$0.466783\pi$
$788$	5656.93	0.255736
$789$	0	0
$790$	0	0
$791$	−11775.9	−0.529332
$792$	0	0
$793$	9056.09	0.405537
$794$	−552.487	−0.0246940
$795$	0	0
$796$	−18535.5	−0.825346
$797$	−15673.8	−0.696604	−0.348302	−	0.937382i	$-0.613242\pi$
$797$	−15673.8	−0.696604	−0.348302	+	0.937382i	$0.613242\pi$
$798$	0	0
$799$	−8768.48	−0.388243
$800$	0	0
$801$	0	0
$802$	−527.242	−0.0232139
$803$	−13110.8	−0.576176
$804$	0	0
$805$	0	0
$806$	1217.93	0.0532255
$807$	0	0
$808$	1538.21	0.0669727
$809$	−16311.7	−0.708887	−0.354444	−	0.935077i	$-0.615330\pi$
$809$	−16311.7	−0.708887	−0.354444	+	0.935077i	$0.615330\pi$
$810$	0	0
$811$	44438.8	1.92412	0.962058	−	0.272844i	$-0.0879644\pi$
$811$	44438.8	1.92412	0.962058	+	0.272844i	$0.0879644\pi$
$812$	16681.9	0.720961
$813$	0	0
$814$	−3161.40	−0.136127
$815$	0	0
$816$	0	0
$817$	−33909.7	−1.45208
$818$	−6369.38	−0.272250
$819$	0	0
$820$	0	0
$821$	12887.8	0.547852	0.273926	−	0.961751i	$-0.411678\pi$
$821$	12887.8	0.547852	0.273926	+	0.961751i	$0.411678\pi$
$822$	0	0
$823$	−29778.1	−1.26124	−0.630619	−	0.776093i	$-0.717199\pi$
$823$	−29778.1	−1.26124	−0.630619	+	0.776093i	$0.717199\pi$
$824$	−2599.60	−0.109904
$825$	0	0
$826$	7202.88	0.303414
$827$	39185.8	1.64767	0.823835	−	0.566830i	$-0.191830\pi$
$827$	39185.8	1.64767	0.823835	+	0.566830i	$0.191830\pi$
$828$	0	0
$829$	−16187.1	−0.678167	−0.339083	−	0.940756i	$-0.610117\pi$
$829$	−16187.1	−0.678167	−0.339083	+	0.940756i	$0.610117\pi$
$830$	0	0
$831$	0	0
$832$	8066.14	0.336109
$833$	10149.4	0.422154
$834$	0	0
$835$	0	0
$836$	−30465.4	−1.26037
$837$	0	0
$838$	−1991.96	−0.0821134
$839$	24729.1	1.01757	0.508786	−	0.860893i	$-0.330094\pi$
$839$	24729.1	1.01757	0.508786	+	0.860893i	$0.330094\pi$
$840$	0	0
$841$	−19095.6	−0.782960
$842$	2845.86	0.116478
$843$	0	0
$844$	−27257.5	−1.11166
$845$	0	0
$846$	0	0
$847$	53858.1	2.18487
$848$	−33003.9	−1.33651
$849$	0	0
$850$	0	0
$851$	−11247.8	−0.453080
$852$	0	0
$853$	−35957.2	−1.44332	−0.721660	−	0.692248i	$-0.756620\pi$
$853$	−35957.2	−1.44332	−0.721660	+	0.692248i	$0.756620\pi$
$854$	7427.75	0.297626
$855$	0	0
$856$	−11949.1	−0.477116
$857$	−15454.4	−0.615999	−0.307999	−	0.951387i	$-0.599659\pi$
$857$	−15454.4	−0.615999	−0.307999	+	0.951387i	$0.599659\pi$
$858$	0	0
$859$	−10352.1	−0.411188	−0.205594	−	0.978637i	$-0.565913\pi$
$859$	−10352.1	−0.411188	−0.205594	+	0.978637i	$0.565913\pi$
$860$	0	0
$861$	0	0
$862$	−6587.52	−0.260292
$863$	31401.6	1.23861	0.619307	−	0.785149i	$-0.287414\pi$
$863$	31401.6	1.23861	0.619307	+	0.785149i	$0.287414\pi$
$864$	0	0
$865$	0	0
$866$	−3237.05	−0.127020
$867$	0	0
$868$	−24343.4	−0.951924
$869$	−15436.0	−0.602567
$870$	0	0
$871$	9499.37	0.369545
$872$	−10378.7	−0.403058
$873$	0	0
$874$	4447.87	0.172141
$875$	0	0
$876$	0	0
$877$	−26798.0	−1.03182	−0.515910	−	0.856643i	$-0.672546\pi$
$877$	−26798.0	−1.03182	−0.515910	+	0.856643i	$0.672546\pi$
$878$	−4349.25	−0.167176
$879$	0	0
$880$	0	0
$881$	−46926.9	−1.79456	−0.897281	−	0.441461i	$-0.854460\pi$
$881$	−46926.9	−1.79456	−0.897281	+	0.441461i	$0.854460\pi$
$882$	0	0
$883$	5665.21	0.215911	0.107956	−	0.994156i	$-0.465570\pi$
$883$	5665.21	0.215911	0.107956	+	0.994156i	$0.465570\pi$
$884$	2911.47	0.110773
$885$	0	0
$886$	−409.792	−0.0155386
$887$	38617.1	1.46182	0.730910	−	0.682474i	$-0.239096\pi$
$887$	38617.1	1.46182	0.730910	+	0.682474i	$0.239096\pi$
$888$	0	0
$889$	59309.8	2.23756
$890$	0	0
$891$	0	0
$892$	−37285.1	−1.39955
$893$	33469.8	1.25422
$894$	0	0
$895$	0	0
$896$	31012.2	1.15630
$897$	0	0
$898$	−4444.69	−0.165168
$899$	−7724.51	−0.286570
$900$	0	0
$901$	−10827.7	−0.400358
$902$	9664.59	0.356758
$903$	0	0
$904$	−3476.20	−0.127894
$905$	0	0
$906$	0	0
$907$	30205.0	1.10578	0.552888	−	0.833256i	$-0.313526\pi$
$907$	30205.0	1.10578	0.552888	+	0.833256i	$0.313526\pi$
$908$	−11205.2	−0.409533
$909$	0	0
$910$	0	0
$911$	−3271.34	−0.118973	−0.0594865	−	0.998229i	$-0.518946\pi$
$911$	−3271.34	−0.118973	−0.0594865	+	0.998229i	$0.518946\pi$
$912$	0	0
$913$	66563.8	2.41286
$914$	579.141	0.0209587
$915$	0	0
$916$	17372.9	0.626656
$917$	27207.2	0.979781
$918$	0	0
$919$	−13978.1	−0.501734	−0.250867	−	0.968022i	$-0.580716\pi$
$919$	−13978.1	−0.501734	−0.250867	+	0.968022i	$0.580716\pi$
$920$	0	0
$921$	0	0
$922$	−8002.81	−0.285855
$923$	21590.9	0.769960
$924$	0	0
$925$	0	0
$926$	−2950.26	−0.104699
$927$	0	0
$928$	7436.17	0.263044
$929$	230.879	0.00815383	0.00407691	−	0.999992i	$-0.498702\pi$
$929$	230.879	0.00815383	0.00407691	+	0.999992i	$0.498702\pi$
$930$	0	0
$931$	−38740.6	−1.36377
$932$	16380.1	0.575697
$933$	0	0
$934$	4372.20	0.153172
$935$	0	0
$936$	0	0
$937$	−23309.8	−0.812699	−0.406349	−	0.913718i	$-0.633198\pi$
$937$	−23309.8	−0.812699	−0.406349	+	0.913718i	$0.633198\pi$
$938$	7791.32	0.271211
$939$	0	0
$940$	0	0
$941$	14023.8	0.485828	0.242914	−	0.970048i	$-0.421897\pi$
$941$	14023.8	0.485828	0.242914	+	0.970048i	$0.421897\pi$
$942$	0	0
$943$	34385.3	1.18742
$944$	−24302.5	−0.837900
$945$	0	0
$946$	15063.3	0.517707
$947$	52264.5	1.79342	0.896710	−	0.442618i	$-0.145950\pi$
$947$	52264.5	1.79342	0.896710	+	0.442618i	$0.145950\pi$
$948$	0	0
$949$	−4782.58	−0.163592
$950$	0	0
$951$	0	0
$952$	4873.93	0.165930
$953$	5585.19	0.189845	0.0949224	−	0.995485i	$-0.469740\pi$
$953$	5585.19	0.189845	0.0949224	+	0.995485i	$0.469740\pi$
$954$	0	0
$955$	0	0
$956$	21976.8	0.743493
$957$	0	0
$958$	−7052.03	−0.237830
$959$	−22069.5	−0.743128
$960$	0	0
$961$	−18518.8	−0.621625
$962$	−1153.23	−0.0386502
$963$	0	0
$964$	11057.0	0.369421
$965$	0	0
$966$	0	0
$967$	36555.7	1.21567	0.607835	−	0.794063i	$-0.292038\pi$
$967$	36555.7	1.21567	0.607835	+	0.794063i	$0.292038\pi$
$968$	15898.7	0.527898
$969$	0	0
$970$	0	0
$971$	38068.1	1.25815	0.629075	−	0.777345i	$-0.283434\pi$
$971$	38068.1	1.25815	0.629075	+	0.777345i	$0.283434\pi$
$972$	0	0
$973$	19150.6	0.630975
$974$	1074.46	0.0353470
$975$	0	0
$976$	−25061.2	−0.821914
$977$	22970.9	0.752204	0.376102	−	0.926578i	$-0.377264\pi$
$977$	22970.9	0.752204	0.376102	+	0.926578i	$0.377264\pi$
$978$	0	0
$979$	17311.8	0.565154
$980$	0	0
$981$	0	0
$982$	−4117.91	−0.133816
$983$	21172.1	0.686962	0.343481	−	0.939160i	$-0.388394\pi$
$983$	21172.1	0.686962	0.343481	+	0.939160i	$0.388394\pi$
$984$	0	0
$985$	0	0
$986$	757.735	0.0244738
$987$	0	0
$988$	−11113.2	−0.357853
$989$	53593.2	1.72312
$990$	0	0
$991$	−28688.6	−0.919601	−0.459800	−	0.888022i	$-0.652079\pi$
$991$	−28688.6	−0.919601	−0.459800	+	0.888022i	$0.652079\pi$
$992$	−10851.4	−0.347311
$993$	0	0
$994$	17708.7	0.565077
$995$	0	0
$996$	0	0
$997$	−24557.0	−0.780069	−0.390035	−	0.920800i	$-0.627537\pi$
$997$	−24557.0	−0.780069	−0.390035	+	0.920800i	$0.627537\pi$
$998$	−2774.18	−0.0879911
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.a.ba.1.2		4
3.2	odd	2		675.4.a.t.1.4		4
5.2	odd	4		135.4.b.b.109.4	yes	8
5.3	odd	4		135.4.b.b.109.6	yes	8
5.4	even	2		675.4.a.t.1.3		4
15.2	even	4		135.4.b.b.109.5	yes	8
15.8	even	4		135.4.b.b.109.3	✓	8
15.14	odd	2	inner	675.4.a.ba.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.b.b.109.3	✓	8	15.8	even	4
135.4.b.b.109.4	yes	8	5.2	odd	4
135.4.b.b.109.5	yes	8	15.2	even	4
135.4.b.b.109.6	yes	8	5.3	odd	4
675.4.a.t.1.3		4	5.4	even	2
675.4.a.t.1.4		4	3.2	odd	2
675.4.a.ba.1.1		4	15.14	odd	2	inner
675.4.a.ba.1.2		4	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

\(f(q)\)	\(=\)	\(q-0.561553 q^{2} -7.68466 q^{4} +29.8369 q^{7} +8.80776 q^{8} -56.0007 q^{11} -20.4281 q^{13} -16.7550 q^{14} +56.5312 q^{16} +18.5464 q^{17} -70.7926 q^{19} +31.4474 q^{22} +111.885 q^{23} +11.4715 q^{26} -229.286 q^{28} -72.7557 q^{29} +106.170 q^{31} -102.207 q^{32} -10.4148 q^{34} -100.530 q^{37} +39.7538 q^{38} +307.326 q^{41} +479.001 q^{43} +430.347 q^{44} -62.8296 q^{46} -472.786 q^{47} +547.241 q^{49} +156.983 q^{52} -583.816 q^{53} +262.796 q^{56} +40.8562 q^{58} -429.894 q^{59} -443.315 q^{61} -59.6203 q^{62} -394.855 q^{64} -465.015 q^{67} -142.523 q^{68} -1056.92 q^{71} +234.118 q^{73} +56.4529 q^{74} +544.017 q^{76} -1670.89 q^{77} +275.639 q^{79} -172.580 q^{82} -1188.62 q^{83} -268.984 q^{86} -493.241 q^{88} -309.134 q^{89} -609.511 q^{91} -859.801 q^{92} +265.494 q^{94} -637.341 q^{97} -307.305 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{2} - 6 q^{4} - 6 q^{8} - 46 q^{16} - 66 q^{17} - 110 q^{19} - 6 q^{23} - 268 q^{31} - 582 q^{32} - 388 q^{34} + 192 q^{38} - 944 q^{46} - 1116 q^{47} + 482 q^{49} - 1824 q^{53} - 1798 q^{61}+ \cdots - 2796 q^{98}+O(q^{100}) \) 4 * q + 6 * q^2 - 6 * q^4 - 6 * q^8 - 46 * q^16 - 66 * q^17 - 110 * q^19 - 6 * q^23 - 268 * q^31 - 582 * q^32 - 388 * q^34 + 192 * q^38 - 944 * q^46 - 1116 * q^47 + 482 * q^49 - 1824 * q^53 - 1798 * q^61 - 1830 * q^62 - 862 * q^64 - 768 * q^68 + 1236 * q^76 - 4086 * q^77 + 682 * q^79 - 384 * q^83 - 360 * q^91 - 2796 * q^92 - 76 * q^94 - 2796 * q^98

Introduction

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Database

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