LMFDB - Embedded newform 135.4.a.e.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [135,4,Mod(1,135)] 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("135.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(135, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	135.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.96525785077$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1772.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 12x + 8 $ x^3 - x^2 - 12*x + 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.32803$ of defining polynomial
Character	$\chi$	$=$	135.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.12612 q^{2} -3.47962 q^{4} -5.00000 q^{5} +30.7000 q^{7} +24.4070 q^{8} +10.6306 q^{10} -50.1548 q^{11} -15.9592 q^{13} -65.2720 q^{14} -24.0553 q^{16} -105.668 q^{17} -21.3040 q^{19} +17.3981 q^{20} +106.635 q^{22} -136.137 q^{23} +25.0000 q^{25} +33.9312 q^{26} -106.824 q^{28} +224.323 q^{29} -225.982 q^{31} -144.112 q^{32} +224.663 q^{34} -153.500 q^{35} -416.386 q^{37} +45.2948 q^{38} -122.035 q^{40} -76.1411 q^{41} +31.7372 q^{43} +174.519 q^{44} +289.443 q^{46} -60.8026 q^{47} +599.493 q^{49} -53.1530 q^{50} +55.5320 q^{52} +466.532 q^{53} +250.774 q^{55} +749.297 q^{56} -476.938 q^{58} -95.4239 q^{59} -357.174 q^{61} +480.464 q^{62} +498.842 q^{64} +79.7962 q^{65} +87.8344 q^{67} +367.685 q^{68} +326.360 q^{70} -412.693 q^{71} -331.133 q^{73} +885.286 q^{74} +74.1296 q^{76} -1539.75 q^{77} -248.123 q^{79} +120.277 q^{80} +161.885 q^{82} +552.505 q^{83} +528.341 q^{85} -67.4771 q^{86} -1224.13 q^{88} +291.478 q^{89} -489.949 q^{91} +473.704 q^{92} +129.274 q^{94} +106.520 q^{95} +198.606 q^{97} -1274.59 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 5 q^{2} + 17 q^{4} - 15 q^{5} - 4 q^{7} - 75 q^{8} + 25 q^{10} - 5 q^{11} + 7 q^{13} - 60 q^{14} + 161 q^{16} - 155 q^{17} - 50 q^{19} - 85 q^{20} - 229 q^{22} - 285 q^{23} + 75 q^{25} - 185 q^{26}+ \cdots + 305 q^{98}+O(q^{100}) $ 3 * q - 5 * q^2 + 17 * q^4 - 15 * q^5 - 4 * q^7 - 75 * q^8 + 25 * q^10 - 5 * q^11 + 7 * q^13 - 60 * q^14 + 161 * q^16 - 155 * q^17 - 50 * q^19 - 85 * q^20 - 229 * q^22 - 285 * q^23 + 75 * q^25 - 185 * q^26 - 334 * q^28 - 115 * q^29 - 115 * q^31 - 775 * q^32 + 413 * q^34 + 20 * q^35 - 384 * q^37 + 1150 * q^38 + 375 * q^40 - 580 * q^41 - 797 * q^43 + 1415 * q^44 - 285 * q^46 + 145 * q^47 + 577 * q^49 - 125 * q^50 + 825 * q^52 + 400 * q^53 + 25 * q^55 + 2190 * q^56 - 59 * q^58 - 380 * q^59 - 152 * q^61 + 1005 * q^62 + 2937 * q^64 - 35 * q^65 + 2 * q^67 - 475 * q^68 + 300 * q^70 - 40 * q^71 - 980 * q^73 + 2720 * q^74 - 3276 * q^76 - 1950 * q^77 + 1013 * q^79 - 805 * q^80 + 4 * q^82 - 270 * q^83 + 775 * q^85 + 1555 * q^86 - 5193 * q^88 - 1020 * q^89 - 632 * q^91 + 1215 * q^92 + 3833 * q^94 + 250 * q^95 + 720 * q^97 + 305 * 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 $

$2$	−2.12612	−0.751697	−0.375848	−	0.926681i	$-0.622649\pi$
$2$	−2.12612	−0.751697	−0.375848	+	0.926681i	$0.622649\pi$
$3$	0	0
$3$	0	0
$4$	−3.47962	−0.434952
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	30.7000	1.65765	0.828823	−	0.559511i	$-0.189011\pi$
$7$	30.7000	1.65765	0.828823	+	0.559511i	$0.189011\pi$
$8$	24.4070	1.07865
$9$	0	0
$10$	10.6306	0.336169
$11$	−50.1548	−1.37475	−0.687375	−	0.726303i	$-0.741237\pi$
$11$	−50.1548	−1.37475	−0.687375	+	0.726303i	$0.741237\pi$
$12$	0	0
$13$	−15.9592	−0.340484	−0.170242	−	0.985402i	$-0.554455\pi$
$13$	−15.9592	−0.340484	−0.170242	+	0.985402i	$0.554455\pi$
$14$	−65.2720	−1.24605
$15$	0	0
$16$	−24.0553	−0.375865
$17$	−105.668	−1.50755	−0.753774	−	0.657134i	$-0.771769\pi$
$17$	−105.668	−1.50755	−0.753774	+	0.657134i	$0.771769\pi$
$18$	0	0
$19$	−21.3040	−0.257235	−0.128618	−	0.991694i	$-0.541054\pi$
$19$	−21.3040	−0.257235	−0.128618	+	0.991694i	$0.541054\pi$
$20$	17.3981	0.194516
$21$	0	0
$22$	106.635	1.03339
$23$	−136.137	−1.23420	−0.617098	−	0.786886i	$-0.711692\pi$
$23$	−136.137	−1.23420	−0.617098	+	0.786886i	$0.711692\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	33.9312	0.255941
$27$	0	0
$28$	−106.824	−0.720997
$29$	224.323	1.43641	0.718203	−	0.695834i	$-0.244965\pi$
$29$	224.323	1.43641	0.718203	+	0.695834i	$0.244965\pi$
$30$	0	0
$31$	−225.982	−1.30928	−0.654638	−	0.755943i	$-0.727179\pi$
$31$	−225.982	−1.30928	−0.654638	+	0.755943i	$0.727179\pi$
$32$	−144.112	−0.796112
$33$	0	0
$34$	224.663	1.13322
$35$	−153.500	−0.741322
$36$	0	0
$37$	−416.386	−1.85009	−0.925046	−	0.379854i	$-0.875974\pi$
$37$	−416.386	−1.85009	−0.925046	+	0.379854i	$0.875974\pi$
$38$	45.2948	0.193363
$39$	0	0
$40$	−122.035	−0.482386
$41$	−76.1411	−0.290030	−0.145015	−	0.989429i	$-0.546323\pi$
$41$	−76.1411	−0.290030	−0.145015	+	0.989429i	$0.546323\pi$
$42$	0	0
$43$	31.7372	0.112555	0.0562777	−	0.998415i	$-0.482077\pi$
$43$	31.7372	0.112555	0.0562777	+	0.998415i	$0.482077\pi$
$44$	174.519	0.597950
$45$	0	0
$46$	289.443	0.927741
$47$	−60.8026	−0.188701	−0.0943507	−	0.995539i	$-0.530078\pi$
$47$	−60.8026	−0.188701	−0.0943507	+	0.995539i	$0.530078\pi$
$48$	0	0
$49$	599.493	1.74779
$50$	−53.1530	−0.150339
$51$	0	0
$52$	55.5320	0.148094
$53$	466.532	1.20911	0.604557	−	0.796562i	$-0.293350\pi$
$53$	466.532	1.20911	0.604557	+	0.796562i	$0.293350\pi$
$54$	0	0
$55$	250.774	0.614806
$56$	749.297	1.78802
$57$	0	0
$58$	−476.938	−1.07974
$59$	−95.4239	−0.210562	−0.105281	−	0.994443i	$-0.533574\pi$
$59$	−95.4239	−0.210562	−0.105281	+	0.994443i	$0.533574\pi$
$60$	0	0
$61$	−357.174	−0.749696	−0.374848	−	0.927086i	$-0.622305\pi$
$61$	−357.174	−0.749696	−0.374848	+	0.927086i	$0.622305\pi$
$62$	480.464	0.984178
$63$	0	0
$64$	498.842	0.974300
$65$	79.7962	0.152269
$66$	0	0
$67$	87.8344	0.160159	0.0800797	−	0.996788i	$-0.474483\pi$
$67$	87.8344	0.160159	0.0800797	+	0.996788i	$0.474483\pi$
$68$	367.685	0.655711
$69$	0	0
$70$	326.360	0.557249
$71$	−412.693	−0.689826	−0.344913	−	0.938635i	$-0.612092\pi$
$71$	−412.693	−0.689826	−0.344913	+	0.938635i	$0.612092\pi$
$72$	0	0
$73$	−331.133	−0.530906	−0.265453	−	0.964124i	$-0.585522\pi$
$73$	−331.133	−0.530906	−0.265453	+	0.964124i	$0.585522\pi$
$74$	885.286	1.39071
$75$	0	0
$76$	74.1296	0.111885
$77$	−1539.75	−2.27885
$78$	0	0
$79$	−248.123	−0.353368	−0.176684	−	0.984268i	$-0.556537\pi$
$79$	−248.123	−0.353368	−0.176684	+	0.984268i	$0.556537\pi$
$80$	120.277	0.168092
$81$	0	0
$82$	161.885	0.218015
$83$	552.505	0.730667	0.365333	−	0.930877i	$-0.380955\pi$
$83$	552.505	0.730667	0.365333	+	0.930877i	$0.380955\pi$
$84$	0	0
$85$	528.341	0.674196
$86$	−67.4771	−0.0846075
$87$	0	0
$88$	−1224.13	−1.48287
$89$	291.478	0.347153	0.173577	−	0.984820i	$-0.444468\pi$
$89$	291.478	0.347153	0.173577	+	0.984820i	$0.444468\pi$
$90$	0	0
$91$	−489.949	−0.564402
$92$	473.704	0.536816
$93$	0	0
$94$	129.274	0.141846
$95$	106.520	0.115039
$96$	0	0
$97$	198.606	0.207891	0.103946	−	0.994583i	$-0.466853\pi$
$97$	198.606	0.207891	0.103946	+	0.994583i	$0.466853\pi$
$98$	−1274.59	−1.31381
$99$	0	0
$100$	−86.9904	−0.0869904
$101$	−816.235	−0.804143	−0.402071	−	0.915608i	$-0.631710\pi$
$101$	−816.235	−0.804143	−0.402071	+	0.915608i	$0.631710\pi$
$102$	0	0
$103$	−1402.37	−1.34155	−0.670776	−	0.741660i	$-0.734039\pi$
$103$	−1402.37	−1.34155	−0.670776	+	0.741660i	$0.734039\pi$
$104$	−389.518	−0.367263
$105$	0	0
$106$	−991.902	−0.908887
$107$	978.996	0.884515	0.442258	−	0.896888i	$-0.354178\pi$
$107$	978.996	0.884515	0.442258	+	0.896888i	$0.354178\pi$
$108$	0	0
$109$	2122.96	1.86553	0.932766	−	0.360484i	$-0.117388\pi$
$109$	2122.96	1.86553	0.932766	+	0.360484i	$0.117388\pi$
$110$	−533.175	−0.462148
$111$	0	0
$112$	−738.500	−0.623051
$113$	−1794.80	−1.49416	−0.747082	−	0.664732i	$-0.768546\pi$
$113$	−1794.80	−1.49416	−0.747082	+	0.664732i	$0.768546\pi$
$114$	0	0
$115$	680.684	0.551949
$116$	−780.559	−0.624768
$117$	0	0
$118$	202.883	0.158278
$119$	−3244.02	−2.49898
$120$	0	0
$121$	1184.50	0.889935
$122$	759.395	0.563544
$123$	0	0
$124$	786.330	0.569472
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−748.894	−0.523257	−0.261628	−	0.965169i	$-0.584259\pi$
$127$	−748.894	−0.523257	−0.261628	+	0.965169i	$0.584259\pi$
$128$	92.2971	0.0637343
$129$	0	0
$130$	−169.656	−0.114460
$131$	2396.11	1.59808	0.799042	−	0.601275i	$-0.205340\pi$
$131$	2396.11	1.59808	0.799042	+	0.601275i	$0.205340\pi$
$132$	0	0
$133$	−654.033	−0.426405
$134$	−186.746	−0.120391
$135$	0	0
$136$	−2579.05	−1.62611
$137$	−1004.48	−0.626409	−0.313205	−	0.949686i	$-0.601403\pi$
$137$	−1004.48	−0.626409	−0.313205	+	0.949686i	$0.601403\pi$
$138$	0	0
$139$	2403.02	1.46634	0.733172	−	0.680043i	$-0.238039\pi$
$139$	2403.02	1.46634	0.733172	+	0.680043i	$0.238039\pi$
$140$	534.122	0.322439
$141$	0	0
$142$	877.435	0.518540
$143$	800.432	0.468080
$144$	0	0
$145$	−1121.62	−0.642380
$146$	704.028	0.399081
$147$	0	0
$148$	1448.86	0.804701
$149$	−509.648	−0.280215	−0.140107	−	0.990136i	$-0.544745\pi$
$149$	−509.648	−0.280215	−0.140107	+	0.990136i	$0.544745\pi$
$150$	0	0
$151$	1443.30	0.777840	0.388920	−	0.921272i	$-0.372848\pi$
$151$	1443.30	0.777840	0.388920	+	0.921272i	$0.372848\pi$
$152$	−519.967	−0.277466
$153$	0	0
$154$	3273.70	1.71300
$155$	1129.91	0.585526
$156$	0	0
$157$	−2155.64	−1.09579	−0.547895	−	0.836547i	$-0.684571\pi$
$157$	−2155.64	−1.09579	−0.547895	+	0.836547i	$0.684571\pi$
$158$	527.539	0.265625
$159$	0	0
$160$	720.559	0.356032
$161$	−4179.41	−2.04586
$162$	0	0
$163$	−529.909	−0.254636	−0.127318	−	0.991862i	$-0.540637\pi$
$163$	−529.909	−0.254636	−0.127318	+	0.991862i	$0.540637\pi$
$164$	264.942	0.126149
$165$	0	0
$166$	−1174.69	−0.549240
$167$	2979.28	1.38050	0.690250	−	0.723571i	$-0.257501\pi$
$167$	2979.28	1.38050	0.690250	+	0.723571i	$0.257501\pi$
$168$	0	0
$169$	−1942.30	−0.884071
$170$	−1123.32	−0.506791
$171$	0	0
$172$	−110.433	−0.0489562
$173$	−1779.88	−0.782205	−0.391103	−	0.920347i	$-0.627906\pi$
$173$	−1779.88	−0.782205	−0.391103	+	0.920347i	$0.627906\pi$
$174$	0	0
$175$	767.501	0.331529
$176$	1206.49	0.516720
$177$	0	0
$178$	−619.718	−0.260954
$179$	2836.26	1.18431	0.592157	−	0.805823i	$-0.298276\pi$
$179$	2836.26	1.18431	0.592157	+	0.805823i	$0.298276\pi$
$180$	0	0
$181$	811.890	0.333410	0.166705	−	0.986007i	$-0.446687\pi$
$181$	811.890	0.333410	0.166705	+	0.986007i	$0.446687\pi$
$182$	1041.69	0.424259
$183$	0	0
$184$	−3322.70	−1.33126
$185$	2081.93	0.827387
$186$	0	0
$187$	5299.77	2.07250
$188$	211.570	0.0820761
$189$	0	0
$190$	−226.474	−0.0864745
$191$	−1148.46	−0.435077	−0.217539	−	0.976052i	$-0.569803\pi$
$191$	−1148.46	−0.435077	−0.217539	+	0.976052i	$0.569803\pi$
$192$	0	0
$193$	−2150.93	−0.802215	−0.401107	−	0.916031i	$-0.631375\pi$
$193$	−2150.93	−0.802215	−0.401107	+	0.916031i	$0.631375\pi$
$194$	−422.261	−0.156271
$195$	0	0
$196$	−2086.00	−0.760206
$197$	5057.51	1.82910	0.914551	−	0.404471i	$-0.132544\pi$
$197$	5057.51	1.82910	0.914551	+	0.404471i	$0.132544\pi$
$198$	0	0
$199$	−3554.12	−1.26605	−0.633027	−	0.774130i	$-0.718188\pi$
$199$	−3554.12	−1.26605	−0.633027	+	0.774130i	$0.718188\pi$
$200$	610.176	0.215730
$201$	0	0
$202$	1735.41	0.604471
$203$	6886.73	2.38105
$204$	0	0
$205$	380.706	0.129706
$206$	2981.61	1.00844
$207$	0	0
$208$	383.905	0.127976
$209$	1068.50	0.353634
$210$	0	0
$211$	−107.909	−0.0352075	−0.0176038	−	0.999845i	$-0.505604\pi$
$211$	−107.909	−0.0352075	−0.0176038	+	0.999845i	$0.505604\pi$
$212$	−1623.35	−0.525907
$213$	0	0
$214$	−2081.46	−0.664887
$215$	−158.686	−0.0503363
$216$	0	0
$217$	−6937.65	−2.17032
$218$	−4513.67	−1.40231
$219$	0	0
$220$	−872.597	−0.267411
$221$	1686.38	0.513296
$222$	0	0
$223$	−1942.03	−0.583174	−0.291587	−	0.956544i	$-0.594183\pi$
$223$	−1942.03	−0.583174	−0.291587	+	0.956544i	$0.594183\pi$
$224$	−4424.24	−1.31967
$225$	0	0
$226$	3815.96	1.12316
$227$	−3443.99	−1.00698	−0.503492	−	0.864000i	$-0.667952\pi$
$227$	−3443.99	−1.00698	−0.503492	+	0.864000i	$0.667952\pi$
$228$	0	0
$229$	2069.39	0.597159	0.298579	−	0.954385i	$-0.403487\pi$
$229$	2069.39	0.597159	0.298579	+	0.954385i	$0.403487\pi$
$230$	−1447.22	−0.414898
$231$	0	0
$232$	5475.07	1.54938
$233$	2769.24	0.778622	0.389311	−	0.921106i	$-0.372713\pi$
$233$	2769.24	0.778622	0.389311	+	0.921106i	$0.372713\pi$
$234$	0	0
$235$	304.013	0.0843898
$236$	332.038	0.0915842
$237$	0	0
$238$	6897.17	1.87848
$239$	2943.84	0.796742	0.398371	−	0.917224i	$-0.369576\pi$
$239$	2943.84	0.796742	0.398371	+	0.917224i	$0.369576\pi$
$240$	0	0
$241$	4474.54	1.19598	0.597988	−	0.801505i	$-0.295967\pi$
$241$	4474.54	1.19598	0.597988	+	0.801505i	$0.295967\pi$
$242$	−2518.40	−0.668961
$243$	0	0
$244$	1242.83	0.326082
$245$	−2997.46	−0.781636
$246$	0	0
$247$	339.995	0.0875845
$248$	−5515.55	−1.41225
$249$	0	0
$250$	265.765	0.0672338
$251$	2121.41	0.533474	0.266737	−	0.963769i	$-0.414055\pi$
$251$	2121.41	0.533474	0.266737	+	0.963769i	$0.414055\pi$
$252$	0	0
$253$	6827.92	1.69671
$254$	1592.24	0.393330
$255$	0	0
$256$	−4186.97	−1.02221
$257$	−4548.24	−1.10393	−0.551967	−	0.833866i	$-0.686123\pi$
$257$	−4548.24	−1.10393	−0.551967	+	0.833866i	$0.686123\pi$
$258$	0	0
$259$	−12783.1	−3.06680
$260$	−277.660	−0.0662298
$261$	0	0
$262$	−5094.42	−1.20127
$263$	3499.84	0.820567	0.410284	−	0.911958i	$-0.365430\pi$
$263$	3499.84	0.820567	0.410284	+	0.911958i	$0.365430\pi$
$264$	0	0
$265$	−2332.66	−0.540732
$266$	1390.55	0.320527
$267$	0	0
$268$	−305.630	−0.0696616
$269$	−2594.25	−0.588008	−0.294004	−	0.955804i	$-0.594988\pi$
$269$	−2594.25	−0.588008	−0.294004	+	0.955804i	$0.594988\pi$
$270$	0	0
$271$	8518.20	1.90939	0.954693	−	0.297593i	$-0.0961838\pi$
$271$	8518.20	1.90939	0.954693	+	0.297593i	$0.0961838\pi$
$272$	2541.89	0.566634
$273$	0	0
$274$	2135.63	0.470870
$275$	−1253.87	−0.274950
$276$	0	0
$277$	−1890.50	−0.410068	−0.205034	−	0.978755i	$-0.565731\pi$
$277$	−1890.50	−0.410068	−0.205034	+	0.978755i	$0.565731\pi$
$278$	−5109.12	−1.10225
$279$	0	0
$280$	−3746.49	−0.799626
$281$	−3138.68	−0.666328	−0.333164	−	0.942869i	$-0.608116\pi$
$281$	−3138.68	−0.666328	−0.333164	+	0.942869i	$0.608116\pi$
$282$	0	0
$283$	4069.09	0.854708	0.427354	−	0.904084i	$-0.359446\pi$
$283$	4069.09	0.854708	0.427354	+	0.904084i	$0.359446\pi$
$284$	1436.01	0.300041
$285$	0	0
$286$	−1701.81	−0.351854
$287$	−2337.54	−0.480768
$288$	0	0
$289$	6252.77	1.27270
$290$	2384.69	0.482875
$291$	0	0
$292$	1152.22	0.230919
$293$	−3637.95	−0.725362	−0.362681	−	0.931913i	$-0.618139\pi$
$293$	−3637.95	−0.725362	−0.362681	+	0.931913i	$0.618139\pi$
$294$	0	0
$295$	477.119	0.0941660
$296$	−10162.7	−1.99560
$297$	0	0
$298$	1083.57	0.210637
$299$	2172.64	0.420224
$300$	0	0
$301$	974.334	0.186577
$302$	−3068.62	−0.584700
$303$	0	0
$304$	512.474	0.0966856
$305$	1785.87	0.335274
$306$	0	0
$307$	−6829.07	−1.26956	−0.634781	−	0.772692i	$-0.718910\pi$
$307$	−6829.07	−1.26956	−0.634781	+	0.772692i	$0.718910\pi$
$308$	5357.75	0.991189
$309$	0	0
$310$	−2402.32	−0.440138
$311$	−6601.03	−1.20357	−0.601785	−	0.798659i	$-0.705543\pi$
$311$	−6601.03	−1.20357	−0.601785	+	0.798659i	$0.705543\pi$
$312$	0	0
$313$	2766.59	0.499607	0.249803	−	0.968297i	$-0.419634\pi$
$313$	2766.59	0.499607	0.249803	+	0.968297i	$0.419634\pi$
$314$	4583.15	0.823701
$315$	0	0
$316$	863.373	0.153698
$317$	−4564.41	−0.808716	−0.404358	−	0.914601i	$-0.632505\pi$
$317$	−4564.41	−0.808716	−0.404358	+	0.914601i	$0.632505\pi$
$318$	0	0
$319$	−11250.9	−1.97470
$320$	−2494.21	−0.435720
$321$	0	0
$322$	8885.92	1.53787
$323$	2251.15	0.387794
$324$	0	0
$325$	−398.981	−0.0680968
$326$	1126.65	0.191409
$327$	0	0
$328$	−1858.38	−0.312841
$329$	−1866.64	−0.312800
$330$	0	0
$331$	−4665.62	−0.774760	−0.387380	−	0.921920i	$-0.626620\pi$
$331$	−4665.62	−0.774760	−0.387380	+	0.921920i	$0.626620\pi$
$332$	−1922.51	−0.317805
$333$	0	0
$334$	−6334.30	−1.03772
$335$	−439.172	−0.0716254
$336$	0	0
$337$	−3807.06	−0.615382	−0.307691	−	0.951486i	$-0.599556\pi$
$337$	−3807.06	−0.615382	−0.307691	+	0.951486i	$0.599556\pi$
$338$	4129.57	0.664553
$339$	0	0
$340$	−1838.42	−0.293243
$341$	11334.1	1.79993
$342$	0	0
$343$	7874.33	1.23957
$344$	774.612	0.121408
$345$	0	0
$346$	3784.23	0.587981
$347$	−2311.26	−0.357565	−0.178783	−	0.983889i	$-0.557216\pi$
$347$	−2311.26	−0.357565	−0.178783	+	0.983889i	$0.557216\pi$
$348$	0	0
$349$	−10873.2	−1.66771	−0.833854	−	0.551984i	$-0.813871\pi$
$349$	−10873.2	−1.66771	−0.833854	+	0.551984i	$0.813871\pi$
$350$	−1631.80	−0.249209
$351$	0	0
$352$	7227.89	1.09445
$353$	2893.74	0.436312	0.218156	−	0.975914i	$-0.429996\pi$
$353$	2893.74	0.436312	0.218156	+	0.975914i	$0.429996\pi$
$354$	0	0
$355$	2063.46	0.308500
$356$	−1014.23	−0.150995
$357$	0	0
$358$	−6030.23	−0.890245
$359$	9875.47	1.45183	0.725915	−	0.687784i	$-0.241416\pi$
$359$	9875.47	1.45183	0.725915	+	0.687784i	$0.241416\pi$
$360$	0	0
$361$	−6405.14	−0.933830
$362$	−1726.17	−0.250624
$363$	0	0
$364$	1704.83	0.245488
$365$	1655.66	0.237429
$366$	0	0
$367$	10722.2	1.52505	0.762523	−	0.646961i	$-0.223960\pi$
$367$	10722.2	1.52505	0.762523	+	0.646961i	$0.223960\pi$
$368$	3274.82	0.463891
$369$	0	0
$370$	−4426.43	−0.621944
$371$	14322.5	2.00428
$372$	0	0
$373$	4175.90	0.579678	0.289839	−	0.957075i	$-0.406398\pi$
$373$	4175.90	0.579678	0.289839	+	0.957075i	$0.406398\pi$
$374$	−11267.9	−1.55789
$375$	0	0
$376$	−1484.01	−0.203543
$377$	−3580.03	−0.489074
$378$	0	0
$379$	1715.14	0.232457	0.116228	−	0.993223i	$-0.462920\pi$
$379$	1715.14	0.232457	0.116228	+	0.993223i	$0.462920\pi$
$380$	−370.648	−0.0500364
$381$	0	0
$382$	2441.77	0.327046
$383$	−5139.06	−0.685624	−0.342812	−	0.939404i	$-0.611379\pi$
$383$	−5139.06	−0.685624	−0.342812	+	0.939404i	$0.611379\pi$
$384$	0	0
$385$	7698.77	1.01913
$386$	4573.14	0.603022
$387$	0	0
$388$	−691.074	−0.0904226
$389$	−8996.53	−1.17260	−0.586301	−	0.810093i	$-0.699416\pi$
$389$	−8996.53	−1.17260	−0.586301	+	0.810093i	$0.699416\pi$
$390$	0	0
$391$	14385.3	1.86061
$392$	14631.8	1.88525
$393$	0	0
$394$	−10752.9	−1.37493
$395$	1240.62	0.158031
$396$	0	0
$397$	−105.823	−0.0133781	−0.00668905	−	0.999978i	$-0.502129\pi$
$397$	−105.823	−0.0133781	−0.00668905	+	0.999978i	$0.502129\pi$
$398$	7556.48	0.951688
$399$	0	0
$400$	−601.384	−0.0751730
$401$	−14481.8	−1.80346	−0.901730	−	0.432300i	$-0.857702\pi$
$401$	−14481.8	−1.80346	−0.901730	+	0.432300i	$0.857702\pi$
$402$	0	0
$403$	3606.50	0.445788
$404$	2840.18	0.349763
$405$	0	0
$406$	−14642.0	−1.78983
$407$	20883.8	2.54341
$408$	0	0
$409$	−861.065	−0.104100	−0.0520500	−	0.998644i	$-0.516576\pi$
$409$	−861.065	−0.104100	−0.0520500	+	0.998644i	$0.516576\pi$
$410$	−809.426	−0.0974993
$411$	0	0
$412$	4879.72	0.583511
$413$	−2929.52	−0.349037
$414$	0	0
$415$	−2762.53	−0.326764
$416$	2299.91	0.271064
$417$	0	0
$418$	−2271.75	−0.265825
$419$	13017.7	1.51780	0.758900	−	0.651207i	$-0.225737\pi$
$419$	13017.7	1.51780	0.758900	+	0.651207i	$0.225737\pi$
$420$	0	0
$421$	14546.6	1.68398	0.841991	−	0.539492i	$-0.181384\pi$
$421$	14546.6	1.68398	0.841991	+	0.539492i	$0.181384\pi$
$422$	229.428	0.0264654
$423$	0	0
$424$	11386.7	1.30421
$425$	−2641.71	−0.301510
$426$	0	0
$427$	−10965.3	−1.24273
$428$	−3406.53	−0.384722
$429$	0	0
$430$	337.386	0.0378376
$431$	3539.94	0.395622	0.197811	−	0.980240i	$-0.436617\pi$
$431$	3539.94	0.395622	0.197811	+	0.980240i	$0.436617\pi$
$432$	0	0
$433$	669.471	0.0743019	0.0371509	−	0.999310i	$-0.488172\pi$
$433$	669.471	0.0743019	0.0371509	+	0.999310i	$0.488172\pi$
$434$	14750.3	1.63142
$435$	0	0
$436$	−7387.09	−0.811416
$437$	2900.26	0.317478
$438$	0	0
$439$	12568.5	1.36643	0.683216	−	0.730216i	$-0.260581\pi$
$439$	12568.5	1.36643	0.683216	+	0.730216i	$0.260581\pi$
$440$	6120.65	0.663160
$441$	0	0
$442$	−3585.45	−0.385843
$443$	−11060.3	−1.18621	−0.593106	−	0.805124i	$-0.702099\pi$
$443$	−11060.3	−1.18621	−0.593106	+	0.805124i	$0.702099\pi$
$444$	0	0
$445$	−1457.39	−0.155252
$446$	4128.98	0.438370
$447$	0	0
$448$	15314.5	1.61504
$449$	−18553.9	−1.95014	−0.975072	−	0.221891i	$-0.928777\pi$
$449$	−18553.9	−1.95014	−0.975072	+	0.221891i	$0.928777\pi$
$450$	0	0
$451$	3818.84	0.398719
$452$	6245.21	0.649889
$453$	0	0
$454$	7322.33	0.756947
$455$	2449.75	0.252408
$456$	0	0
$457$	−6802.26	−0.696272	−0.348136	−	0.937444i	$-0.613185\pi$
$457$	−6802.26	−0.696272	−0.348136	+	0.937444i	$0.613185\pi$
$458$	−4399.78	−0.448882
$459$	0	0
$460$	−2368.52	−0.240071
$461$	−14894.4	−1.50477	−0.752386	−	0.658722i	$-0.771097\pi$
$461$	−14894.4	−1.50477	−0.752386	+	0.658722i	$0.771097\pi$
$462$	0	0
$463$	−14288.7	−1.43423	−0.717117	−	0.696952i	$-0.754539\pi$
$463$	−14288.7	−1.43423	−0.717117	+	0.696952i	$0.754539\pi$
$464$	−5396.17	−0.539895
$465$	0	0
$466$	−5887.74	−0.585288
$467$	−13115.1	−1.29956	−0.649780	−	0.760122i	$-0.725139\pi$
$467$	−13115.1	−1.29956	−0.649780	+	0.760122i	$0.725139\pi$
$468$	0	0
$469$	2696.52	0.265488
$470$	−646.368	−0.0634356
$471$	0	0
$472$	−2329.01	−0.227122
$473$	−1591.77	−0.154735
$474$	0	0
$475$	−532.599	−0.0514470
$476$	11287.9	1.08694
$477$	0	0
$478$	−6258.96	−0.598908
$479$	−7919.79	−0.755458	−0.377729	−	0.925916i	$-0.623295\pi$
$479$	−7919.79	−0.755458	−0.377729	+	0.925916i	$0.623295\pi$
$480$	0	0
$481$	6645.20	0.629927
$482$	−9513.40	−0.899011
$483$	0	0
$484$	−4121.62	−0.387079
$485$	−993.032	−0.0929717
$486$	0	0
$487$	−17003.1	−1.58210	−0.791051	−	0.611751i	$-0.790466\pi$
$487$	−17003.1	−1.58210	−0.791051	+	0.611751i	$0.790466\pi$
$488$	−8717.56	−0.808659
$489$	0	0
$490$	6372.96	0.587553
$491$	−6391.74	−0.587485	−0.293743	−	0.955885i	$-0.594901\pi$
$491$	−6391.74	−0.587485	−0.293743	+	0.955885i	$0.594901\pi$
$492$	0	0
$493$	−23703.8	−2.16545
$494$	−722.870	−0.0658370
$495$	0	0
$496$	5436.07	0.492111
$497$	−12669.7	−1.14349
$498$	0	0
$499$	−2674.12	−0.239900	−0.119950	−	0.992780i	$-0.538273\pi$
$499$	−2674.12	−0.239900	−0.119950	+	0.992780i	$0.538273\pi$
$500$	434.952	0.0389033
$501$	0	0
$502$	−4510.36	−0.401011
$503$	−1263.13	−0.111969	−0.0559843	−	0.998432i	$-0.517830\pi$
$503$	−1263.13	−0.111969	−0.0559843	+	0.998432i	$0.517830\pi$
$504$	0	0
$505$	4081.17	0.359624
$506$	−14517.0	−1.27541
$507$	0	0
$508$	2605.86	0.227592
$509$	−1380.84	−0.120245	−0.0601226	−	0.998191i	$-0.519149\pi$
$509$	−1380.84	−0.120245	−0.0601226	+	0.998191i	$0.519149\pi$
$510$	0	0
$511$	−10165.8	−0.880055
$512$	8163.62	0.704657
$513$	0	0
$514$	9670.09	0.829824
$515$	7011.86	0.599960
$516$	0	0
$517$	3049.54	0.259417
$518$	27178.3	2.30530
$519$	0	0
$520$	1947.59	0.164245
$521$	−2689.80	−0.226185	−0.113092	−	0.993584i	$-0.536076\pi$
$521$	−2689.80	−0.226185	−0.113092	+	0.993584i	$0.536076\pi$
$522$	0	0
$523$	7144.18	0.597310	0.298655	−	0.954361i	$-0.403462\pi$
$523$	7144.18	0.597310	0.298655	+	0.954361i	$0.403462\pi$
$524$	−8337.54	−0.695090
$525$	0	0
$526$	−7441.07	−0.616818
$527$	23879.1	1.97379
$528$	0	0
$529$	6366.25	0.523239
$530$	4959.51	0.406467
$531$	0	0
$532$	2275.78	0.185466
$533$	1215.15	0.0987508
$534$	0	0
$535$	−4894.98	−0.395567
$536$	2143.78	0.172756
$537$	0	0
$538$	5515.68	0.442004
$539$	−30067.4	−2.40278
$540$	0	0
$541$	−3310.57	−0.263091	−0.131546	−	0.991310i	$-0.541994\pi$
$541$	−3310.57	−0.263091	−0.131546	+	0.991310i	$0.541994\pi$
$542$	−18110.7	−1.43528
$543$	0	0
$544$	15228.0	1.20018
$545$	−10614.8	−0.834291
$546$	0	0
$547$	2286.52	0.178729	0.0893643	−	0.995999i	$-0.471516\pi$
$547$	2286.52	0.178729	0.0893643	+	0.995999i	$0.471516\pi$
$548$	3495.19	0.272458
$549$	0	0
$550$	2665.88	0.206679
$551$	−4778.98	−0.369494
$552$	0	0
$553$	−7617.39	−0.585758
$554$	4019.42	0.308247
$555$	0	0
$556$	−8361.60	−0.637789
$557$	13846.6	1.05332	0.526658	−	0.850077i	$-0.323445\pi$
$557$	13846.6	1.05332	0.526658	+	0.850077i	$0.323445\pi$
$558$	0	0
$559$	−506.502	−0.0383233
$560$	3692.50	0.278637
$561$	0	0
$562$	6673.21	0.500876
$563$	−16164.4	−1.21003	−0.605017	−	0.796213i	$-0.706834\pi$
$563$	−16164.4	−1.21003	−0.605017	+	0.796213i	$0.706834\pi$
$564$	0	0
$565$	8973.99	0.668210
$566$	−8651.37	−0.642481
$567$	0	0
$568$	−10072.6	−0.744080
$569$	496.334	0.0365684	0.0182842	−	0.999833i	$-0.494180\pi$
$569$	496.334	0.0365684	0.0182842	+	0.999833i	$0.494180\pi$
$570$	0	0
$571$	5971.18	0.437629	0.218815	−	0.975766i	$-0.429781\pi$
$571$	5971.18	0.437629	0.218815	+	0.975766i	$0.429781\pi$
$572$	−2785.20	−0.203592
$573$	0	0
$574$	4969.88	0.361392
$575$	−3403.42	−0.246839
$576$	0	0
$577$	9571.82	0.690607	0.345304	−	0.938491i	$-0.387776\pi$
$577$	9571.82	0.690607	0.345304	+	0.938491i	$0.387776\pi$
$578$	−13294.1	−0.956684
$579$	0	0
$580$	3902.79	0.279405
$581$	16961.9	1.21119
$582$	0	0
$583$	−23398.8	−1.66223
$584$	−8081.97	−0.572662
$585$	0	0
$586$	7734.71	0.545253
$587$	5322.18	0.374225	0.187112	−	0.982339i	$-0.440087\pi$
$587$	5322.18	0.374225	0.187112	+	0.982339i	$0.440087\pi$
$588$	0	0
$589$	4814.31	0.336791
$590$	−1014.41	−0.0707843
$591$	0	0
$592$	10016.3	0.695385
$593$	−11066.5	−0.766354	−0.383177	−	0.923675i	$-0.625170\pi$
$593$	−11066.5	−0.766354	−0.383177	+	0.923675i	$0.625170\pi$
$594$	0	0
$595$	16220.1	1.11758
$596$	1773.38	0.121880
$597$	0	0
$598$	−4619.29	−0.315881
$599$	18385.0	1.25407	0.627037	−	0.778989i	$-0.284267\pi$
$599$	18385.0	1.25407	0.627037	+	0.778989i	$0.284267\pi$
$600$	0	0
$601$	−617.830	−0.0419331	−0.0209666	−	0.999780i	$-0.506674\pi$
$601$	−617.830	−0.0419331	−0.0209666	+	0.999780i	$0.506674\pi$
$602$	−2071.55	−0.140249
$603$	0	0
$604$	−5022.12	−0.338323
$605$	−5922.52	−0.397991
$606$	0	0
$607$	13839.8	0.925436	0.462718	−	0.886506i	$-0.346874\pi$
$607$	13839.8	0.925436	0.462718	+	0.886506i	$0.346874\pi$
$608$	3070.15	0.204788
$609$	0	0
$610$	−3796.97	−0.252025
$611$	970.362	0.0642498
$612$	0	0
$613$	−25655.4	−1.69039	−0.845196	−	0.534457i	$-0.820516\pi$
$613$	−25655.4	−1.69039	−0.845196	+	0.534457i	$0.820516\pi$
$614$	14519.4	0.954326
$615$	0	0
$616$	−37580.8	−2.45808
$617$	4218.26	0.275236	0.137618	−	0.990485i	$-0.456055\pi$
$617$	4218.26	0.275236	0.137618	+	0.990485i	$0.456055\pi$
$618$	0	0
$619$	12182.6	0.791047	0.395524	−	0.918456i	$-0.370563\pi$
$619$	12182.6	0.791047	0.395524	+	0.918456i	$0.370563\pi$
$620$	−3931.65	−0.254676
$621$	0	0
$622$	14034.6	0.904719
$623$	8948.40	0.575458
$624$	0	0
$625$	625.000	0.0400000
$626$	−5882.10	−0.375553
$627$	0	0
$628$	7500.80	0.476616
$629$	43998.8	2.78910
$630$	0	0
$631$	16673.2	1.05190	0.525951	−	0.850515i	$-0.323709\pi$
$631$	16673.2	1.05190	0.525951	+	0.850515i	$0.323709\pi$
$632$	−6055.95	−0.381159
$633$	0	0
$634$	9704.48	0.607909
$635$	3744.47	0.234007
$636$	0	0
$637$	−9567.44	−0.595096
$638$	23920.7	1.48437
$639$	0	0
$640$	−461.486	−0.0285028
$641$	26270.8	1.61878	0.809388	−	0.587275i	$-0.199799\pi$
$641$	26270.8	1.61878	0.809388	+	0.587275i	$0.199799\pi$
$642$	0	0
$643$	11556.3	0.708767	0.354384	−	0.935100i	$-0.384691\pi$
$643$	11556.3	0.708767	0.354384	+	0.935100i	$0.384691\pi$
$644$	14542.7	0.889851
$645$	0	0
$646$	−4786.22	−0.291504
$647$	−25395.2	−1.54310	−0.771552	−	0.636166i	$-0.780519\pi$
$647$	−25395.2	−1.54310	−0.771552	+	0.636166i	$0.780519\pi$
$648$	0	0
$649$	4785.96	0.289469
$650$	848.281	0.0511882
$651$	0	0
$652$	1843.88	0.110754
$653$	−6182.84	−0.370526	−0.185263	−	0.982689i	$-0.559314\pi$
$653$	−6182.84	−0.370526	−0.185263	+	0.982689i	$0.559314\pi$
$654$	0	0
$655$	−11980.5	−0.714685
$656$	1831.60	0.109012
$657$	0	0
$658$	3968.70	0.235131
$659$	−8885.32	−0.525224	−0.262612	−	0.964901i	$-0.584584\pi$
$659$	−8885.32	−0.525224	−0.262612	+	0.964901i	$0.584584\pi$
$660$	0	0
$661$	−7171.60	−0.422001	−0.211001	−	0.977486i	$-0.567672\pi$
$661$	−7171.60	−0.422001	−0.211001	+	0.977486i	$0.567672\pi$
$662$	9919.66	0.582385
$663$	0	0
$664$	13485.0	0.788133
$665$	3270.16	0.190694
$666$	0	0
$667$	−30538.7	−1.77281
$668$	−10366.7	−0.600451
$669$	0	0
$670$	933.732	0.0538406
$671$	17914.0	1.03064
$672$	0	0
$673$	−4250.52	−0.243455	−0.121728	−	0.992564i	$-0.538843\pi$
$673$	−4250.52	−0.243455	−0.121728	+	0.992564i	$0.538843\pi$
$674$	8094.27	0.462581
$675$	0	0
$676$	6758.47	0.384528
$677$	2233.65	0.126804	0.0634019	−	0.997988i	$-0.479805\pi$
$677$	2233.65	0.126804	0.0634019	+	0.997988i	$0.479805\pi$
$678$	0	0
$679$	6097.23	0.344610
$680$	12895.2	0.727220
$681$	0	0
$682$	−24097.6	−1.35300
$683$	−6071.62	−0.340153	−0.170076	−	0.985431i	$-0.554401\pi$
$683$	−6071.62	−0.340153	−0.170076	+	0.985431i	$0.554401\pi$
$684$	0	0
$685$	5022.38	0.280139
$686$	−16741.8	−0.931784
$687$	0	0
$688$	−763.450	−0.0423056
$689$	−7445.49	−0.411684
$690$	0	0
$691$	−765.784	−0.0421589	−0.0210794	−	0.999778i	$-0.506710\pi$
$691$	−765.784	−0.0421589	−0.0210794	+	0.999778i	$0.506710\pi$
$692$	6193.29	0.340222
$693$	0	0
$694$	4914.02	0.268780
$695$	−12015.1	−0.655769
$696$	0	0
$697$	8045.70	0.437235
$698$	23117.8	1.25361
$699$	0	0
$700$	−2670.61	−0.144199
$701$	23564.3	1.26963	0.634814	−	0.772665i	$-0.281077\pi$
$701$	23564.3	1.26963	0.634814	+	0.772665i	$0.281077\pi$
$702$	0	0
$703$	8870.67	0.475909
$704$	−25019.3	−1.33942
$705$	0	0
$706$	−6152.43	−0.327974
$707$	−25058.4	−1.33298
$708$	0	0
$709$	24172.3	1.28041	0.640204	−	0.768205i	$-0.278850\pi$
$709$	24172.3	1.28041	0.640204	+	0.768205i	$0.278850\pi$
$710$	−4387.17	−0.231898
$711$	0	0
$712$	7114.13	0.374457
$713$	30764.5	1.61590
$714$	0	0
$715$	−4002.16	−0.209332
$716$	−9869.11	−0.515120
$717$	0	0
$718$	−20996.4	−1.09134
$719$	12829.0	0.665424	0.332712	−	0.943028i	$-0.392036\pi$
$719$	12829.0	0.665424	0.332712	+	0.943028i	$0.392036\pi$
$720$	0	0
$721$	−43052.9	−2.22382
$722$	13618.1	0.701957
$723$	0	0
$724$	−2825.06	−0.145018
$725$	5608.08	0.287281
$726$	0	0
$727$	−24724.7	−1.26133	−0.630666	−	0.776054i	$-0.717218\pi$
$727$	−24724.7	−1.26133	−0.630666	+	0.776054i	$0.717218\pi$
$728$	−11958.2	−0.608792
$729$	0	0
$730$	−3520.14	−0.178474
$731$	−3353.62	−0.169683
$732$	0	0
$733$	−20172.2	−1.01648	−0.508239	−	0.861216i	$-0.669703\pi$
$733$	−20172.2	−1.01648	−0.508239	+	0.861216i	$0.669703\pi$
$734$	−22796.6	−1.14637
$735$	0	0
$736$	19618.9	0.982559
$737$	−4405.32	−0.220179
$738$	0	0
$739$	−16452.8	−0.818979	−0.409489	−	0.912315i	$-0.634293\pi$
$739$	−16452.8	−0.818979	−0.409489	+	0.912315i	$0.634293\pi$
$740$	−7244.32	−0.359873
$741$	0	0
$742$	−30451.4	−1.50661
$743$	20864.6	1.03021	0.515107	−	0.857126i	$-0.327752\pi$
$743$	20864.6	1.03021	0.515107	+	0.857126i	$0.327752\pi$
$744$	0	0
$745$	2548.24	0.125316
$746$	−8878.47	−0.435742
$747$	0	0
$748$	−18441.2	−0.901438
$749$	30055.2	1.46621
$750$	0	0
$751$	15525.7	0.754381	0.377191	−	0.926136i	$-0.376890\pi$
$751$	15525.7	0.754381	0.377191	+	0.926136i	$0.376890\pi$
$752$	1462.63	0.0709262
$753$	0	0
$754$	7611.56	0.367635
$755$	−7216.48	−0.347861
$756$	0	0
$757$	30105.7	1.44546	0.722729	−	0.691132i	$-0.242888\pi$
$757$	30105.7	1.44546	0.722729	+	0.691132i	$0.242888\pi$
$758$	−3646.60	−0.174737
$759$	0	0
$760$	2599.83	0.124087
$761$	−21739.9	−1.03557	−0.517786	−	0.855510i	$-0.673244\pi$
$761$	−21739.9	−1.03557	−0.517786	+	0.855510i	$0.673244\pi$
$762$	0	0
$763$	65175.0	3.09239
$764$	3996.21	0.189238
$765$	0	0
$766$	10926.3	0.515381
$767$	1522.89	0.0716929
$768$	0	0
$769$	1942.22	0.0910772	0.0455386	−	0.998963i	$-0.485500\pi$
$769$	1942.22	0.0910772	0.0455386	+	0.998963i	$0.485500\pi$
$770$	−16368.5	−0.766078
$771$	0	0
$772$	7484.42	0.348925
$773$	7921.02	0.368563	0.184282	−	0.982873i	$-0.441004\pi$
$773$	7921.02	0.368563	0.184282	+	0.982873i	$0.441004\pi$
$774$	0	0
$775$	−5649.55	−0.261855
$776$	4847.40	0.224241
$777$	0	0
$778$	19127.7	0.881441
$779$	1622.11	0.0746060
$780$	0	0
$781$	20698.5	0.948338
$782$	−30585.0	−1.39861
$783$	0	0
$784$	−14421.0	−0.656933
$785$	10778.2	0.490052
$786$	0	0
$787$	−1361.40	−0.0616630	−0.0308315	−	0.999525i	$-0.509816\pi$
$787$	−1361.40	−0.0616630	−0.0308315	+	0.999525i	$0.509816\pi$
$788$	−17598.2	−0.795571
$789$	0	0
$790$	−2637.70	−0.118791
$791$	−55100.4	−2.47679
$792$	0	0
$793$	5700.22	0.255260
$794$	224.992	0.0100563
$795$	0	0
$796$	12367.0	0.550672
$797$	−27917.5	−1.24076	−0.620381	−	0.784300i	$-0.713022\pi$
$797$	−27917.5	−1.24076	−0.620381	+	0.784300i	$0.713022\pi$
$798$	0	0
$799$	6424.90	0.284476
$800$	−3602.79	−0.159222
$801$	0	0
$802$	30790.1	1.35565
$803$	16607.9	0.729863
$804$	0	0
$805$	20897.0	0.914937
$806$	−7667.84	−0.335097
$807$	0	0
$808$	−19921.9	−0.867388
$809$	−5275.87	−0.229283	−0.114641	−	0.993407i	$-0.536572\pi$
$809$	−5275.87	−0.229283	−0.114641	+	0.993407i	$0.536572\pi$
$810$	0	0
$811$	−23769.1	−1.02916	−0.514579	−	0.857443i	$-0.672052\pi$
$811$	−23769.1	−1.02916	−0.514579	+	0.857443i	$0.672052\pi$
$812$	−23963.2	−1.03564
$813$	0	0
$814$	−44401.4	−1.91188
$815$	2649.54	0.113877
$816$	0	0
$817$	−676.129	−0.0289532
$818$	1830.73	0.0782516
$819$	0	0
$820$	−1324.71	−0.0564157
$821$	34209.3	1.45422	0.727108	−	0.686523i	$-0.240864\pi$
$821$	34209.3	1.45422	0.727108	+	0.686523i	$0.240864\pi$
$822$	0	0
$823$	1240.13	0.0525252	0.0262626	−	0.999655i	$-0.491639\pi$
$823$	1240.13	0.0525252	0.0262626	+	0.999655i	$0.491639\pi$
$824$	−34227.8	−1.44706
$825$	0	0
$826$	6228.50	0.262370
$827$	26971.0	1.13407	0.567034	−	0.823694i	$-0.308091\pi$
$827$	26971.0	1.13407	0.567034	+	0.823694i	$0.308091\pi$
$828$	0	0
$829$	−26743.1	−1.12042	−0.560208	−	0.828352i	$-0.689279\pi$
$829$	−26743.1	−1.12042	−0.560208	+	0.828352i	$0.689279\pi$
$830$	5873.46	0.245628
$831$	0	0
$832$	−7961.13	−0.331734
$833$	−63347.3	−2.63488
$834$	0	0
$835$	−14896.4	−0.617378
$836$	−3717.96	−0.153814
$837$	0	0
$838$	−27677.3	−1.14093
$839$	5273.14	0.216983	0.108492	−	0.994097i	$-0.465398\pi$
$839$	5273.14	0.216983	0.108492	+	0.994097i	$0.465398\pi$
$840$	0	0
$841$	25931.9	1.06326
$842$	−30927.7	−1.26584
$843$	0	0
$844$	375.483	0.0153136
$845$	9711.51	0.395368
$846$	0	0
$847$	36364.3	1.47520
$848$	−11222.6	−0.454464
$849$	0	0
$850$	5616.58	0.226644
$851$	56685.5	2.28338
$852$	0	0
$853$	5043.46	0.202444	0.101222	−	0.994864i	$-0.467725\pi$
$853$	5043.46	0.202444	0.101222	+	0.994864i	$0.467725\pi$
$854$	23313.5	0.934157
$855$	0	0
$856$	23894.4	0.954081
$857$	14478.5	0.577103	0.288552	−	0.957464i	$-0.406826\pi$
$857$	14478.5	0.577103	0.288552	+	0.957464i	$0.406826\pi$
$858$	0	0
$859$	−7873.88	−0.312751	−0.156376	−	0.987698i	$-0.549981\pi$
$859$	−7873.88	−0.312751	−0.156376	+	0.987698i	$0.549981\pi$
$860$	552.167	0.0218939
$861$	0	0
$862$	−7526.35	−0.297388
$863$	−24166.0	−0.953209	−0.476604	−	0.879118i	$-0.658133\pi$
$863$	−24166.0	−0.953209	−0.476604	+	0.879118i	$0.658133\pi$
$864$	0	0
$865$	8899.39	0.349813
$866$	−1423.38	−0.0558525
$867$	0	0
$868$	24140.4	0.943983
$869$	12444.6	0.485792
$870$	0	0
$871$	−1401.77	−0.0545317
$872$	51815.2	2.01225
$873$	0	0
$874$	−6166.29	−0.238647
$875$	−3837.51	−0.148264
$876$	0	0
$877$	6411.82	0.246878	0.123439	−	0.992352i	$-0.460608\pi$
$877$	6411.82	0.246878	0.123439	+	0.992352i	$0.460608\pi$
$878$	−26722.2	−1.02714
$879$	0	0
$880$	−6032.46	−0.231084
$881$	−22908.5	−0.876059	−0.438029	−	0.898961i	$-0.644323\pi$
$881$	−22908.5	−0.876059	−0.438029	+	0.898961i	$0.644323\pi$
$882$	0	0
$883$	39198.4	1.49392	0.746960	−	0.664869i	$-0.231513\pi$
$883$	39198.4	1.49392	0.746960	+	0.664869i	$0.231513\pi$
$884$	−5867.97	−0.223259
$885$	0	0
$886$	23515.6	0.891672
$887$	−5422.19	−0.205253	−0.102626	−	0.994720i	$-0.532725\pi$
$887$	−5422.19	−0.205253	−0.102626	+	0.994720i	$0.532725\pi$
$888$	0	0
$889$	−22991.1	−0.867375
$890$	3098.59	0.116702
$891$	0	0
$892$	6757.51	0.253653
$893$	1295.34	0.0485406
$894$	0	0
$895$	−14181.3	−0.529641
$896$	2833.53	0.105649
$897$	0	0
$898$	39447.9	1.46592
$899$	−50693.0	−1.88065
$900$	0	0
$901$	−49297.6	−1.82280
$902$	−8119.32	−0.299716
$903$	0	0
$904$	−43805.7	−1.61168
$905$	−4059.45	−0.149106
$906$	0	0
$907$	−30727.0	−1.12489	−0.562444	−	0.826836i	$-0.690139\pi$
$907$	−30727.0	−1.12489	−0.562444	+	0.826836i	$0.690139\pi$
$908$	11983.8	0.437990
$909$	0	0
$910$	−5208.45	−0.189735
$911$	−4101.34	−0.149159	−0.0745793	−	0.997215i	$-0.523761\pi$
$911$	−4101.34	−0.149159	−0.0745793	+	0.997215i	$0.523761\pi$
$912$	0	0
$913$	−27710.8	−1.00448
$914$	14462.4	0.523385
$915$	0	0
$916$	−7200.69	−0.259735
$917$	73560.7	2.64906
$918$	0	0
$919$	−50926.3	−1.82797	−0.913984	−	0.405750i	$-0.867010\pi$
$919$	−50926.3	−1.82797	−0.913984	+	0.405750i	$0.867010\pi$
$920$	16613.5	0.595359
$921$	0	0
$922$	31667.2	1.13113
$923$	6586.26	0.234875
$924$	0	0
$925$	−10409.6	−0.370019
$926$	30379.4	1.07811
$927$	0	0
$928$	−32327.6	−1.14354
$929$	−22040.8	−0.778402	−0.389201	−	0.921153i	$-0.627249\pi$
$929$	−22040.8	−0.778402	−0.389201	+	0.921153i	$0.627249\pi$
$930$	0	0
$931$	−12771.6	−0.449593
$932$	−9635.89	−0.338663
$933$	0	0
$934$	27884.3	0.976875
$935$	−26498.8	−0.926850
$936$	0	0
$937$	28111.2	0.980097	0.490049	−	0.871695i	$-0.336979\pi$
$937$	28111.2	0.980097	0.490049	+	0.871695i	$0.336979\pi$
$938$	−5733.12	−0.199566
$939$	0	0
$940$	−1057.85	−0.0367055
$941$	−52811.3	−1.82954	−0.914770	−	0.403974i	$-0.867629\pi$
$941$	−52811.3	−1.82954	−0.914770	+	0.403974i	$0.867629\pi$
$942$	0	0
$943$	10365.6	0.357954
$944$	2295.45	0.0791427
$945$	0	0
$946$	3384.30	0.116314
$947$	1826.23	0.0626657	0.0313329	−	0.999509i	$-0.490025\pi$
$947$	1826.23	0.0626657	0.0313329	+	0.999509i	$0.490025\pi$
$948$	0	0
$949$	5284.63	0.180765
$950$	1132.37	0.0386725
$951$	0	0
$952$	−79176.9	−2.69552
$953$	41164.2	1.39920	0.699601	−	0.714534i	$-0.253361\pi$
$953$	41164.2	1.39920	0.699601	+	0.714534i	$0.253361\pi$
$954$	0	0
$955$	5742.31	0.194573
$956$	−10243.4	−0.346545
$957$	0	0
$958$	16838.4	0.567875
$959$	−30837.4	−1.03837
$960$	0	0
$961$	21276.8	0.714202
$962$	−14128.5	−0.473514
$963$	0	0
$964$	−15569.7	−0.520192
$965$	10754.7	0.358761
$966$	0	0
$967$	−7088.78	−0.235739	−0.117870	−	0.993029i	$-0.537606\pi$
$967$	−7088.78	−0.235739	−0.117870	+	0.993029i	$0.537606\pi$
$968$	28910.2	0.959927
$969$	0	0
$970$	2111.31	0.0698865
$971$	−2355.90	−0.0778625	−0.0389312	−	0.999242i	$-0.512395\pi$
$971$	−2355.90	−0.0778625	−0.0389312	+	0.999242i	$0.512395\pi$
$972$	0	0
$973$	73773.0	2.43068
$974$	36150.6	1.18926
$975$	0	0
$976$	8591.95	0.281784
$977$	16795.1	0.549972	0.274986	−	0.961448i	$-0.411327\pi$
$977$	16795.1	0.549972	0.274986	+	0.961448i	$0.411327\pi$
$978$	0	0
$979$	−14619.0	−0.477249
$980$	10430.0	0.339974
$981$	0	0
$982$	13589.6	0.441611
$983$	14733.8	0.478063	0.239032	−	0.971012i	$-0.423170\pi$
$983$	14733.8	0.478063	0.239032	+	0.971012i	$0.423170\pi$
$984$	0	0
$985$	−25287.6	−0.817999
$986$	50397.2	1.62776
$987$	0	0
$988$	−1183.05	−0.0380950
$989$	−4320.61	−0.138915
$990$	0	0
$991$	36809.2	1.17990	0.589950	−	0.807440i	$-0.299147\pi$
$991$	36809.2	1.17990	0.589950	+	0.807440i	$0.299147\pi$
$992$	32566.6	1.04233
$993$	0	0
$994$	26937.3	0.859556
$995$	17770.6	0.566196
$996$	0	0
$997$	−21210.0	−0.673749	−0.336875	−	0.941550i	$-0.609370\pi$
$997$	−21210.0	−0.673749	−0.336875	+	0.941550i	$0.609370\pi$
$998$	5685.49	0.180332
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.a.e.1.2	✓	3
3.2	odd	2		135.4.a.h.1.2	yes	3
4.3	odd	2		2160.4.a.bi.1.1		3
5.2	odd	4		675.4.b.m.649.3		6
5.3	odd	4		675.4.b.m.649.4		6
5.4	even	2		675.4.a.s.1.2		3
9.2	odd	6		405.4.e.q.271.2		6
9.4	even	3		405.4.e.v.136.2		6
9.5	odd	6		405.4.e.q.136.2		6
9.7	even	3		405.4.e.v.271.2		6
12.11	even	2		2160.4.a.bq.1.1		3
15.2	even	4		675.4.b.n.649.4		6
15.8	even	4		675.4.b.n.649.3		6
15.14	odd	2		675.4.a.p.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.e.1.2	✓	3	1.1	even	1	trivial
135.4.a.h.1.2	yes	3	3.2	odd	2
405.4.e.q.136.2		6	9.5	odd	6
405.4.e.q.271.2		6	9.2	odd	6
405.4.e.v.136.2		6	9.4	even	3
405.4.e.v.271.2		6	9.7	even	3
675.4.a.p.1.2		3	15.14	odd	2
675.4.a.s.1.2		3	5.4	even	2
675.4.b.m.649.3		6	5.2	odd	4
675.4.b.m.649.4		6	5.3	odd	4
675.4.b.n.649.3		6	15.8	even	4
675.4.b.n.649.4		6	15.2	even	4
2160.4.a.bi.1.1		3	4.3	odd	2
2160.4.a.bq.1.1		3	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	135.a (trivial)

\(f(q)\)	\(=\)	\(q-2.12612 q^{2} -3.47962 q^{4} -5.00000 q^{5} +30.7000 q^{7} +24.4070 q^{8} +10.6306 q^{10} -50.1548 q^{11} -15.9592 q^{13} -65.2720 q^{14} -24.0553 q^{16} -105.668 q^{17} -21.3040 q^{19} +17.3981 q^{20} +106.635 q^{22} -136.137 q^{23} +25.0000 q^{25} +33.9312 q^{26} -106.824 q^{28} +224.323 q^{29} -225.982 q^{31} -144.112 q^{32} +224.663 q^{34} -153.500 q^{35} -416.386 q^{37} +45.2948 q^{38} -122.035 q^{40} -76.1411 q^{41} +31.7372 q^{43} +174.519 q^{44} +289.443 q^{46} -60.8026 q^{47} +599.493 q^{49} -53.1530 q^{50} +55.5320 q^{52} +466.532 q^{53} +250.774 q^{55} +749.297 q^{56} -476.938 q^{58} -95.4239 q^{59} -357.174 q^{61} +480.464 q^{62} +498.842 q^{64} +79.7962 q^{65} +87.8344 q^{67} +367.685 q^{68} +326.360 q^{70} -412.693 q^{71} -331.133 q^{73} +885.286 q^{74} +74.1296 q^{76} -1539.75 q^{77} -248.123 q^{79} +120.277 q^{80} +161.885 q^{82} +552.505 q^{83} +528.341 q^{85} -67.4771 q^{86} -1224.13 q^{88} +291.478 q^{89} -489.949 q^{91} +473.704 q^{92} +129.274 q^{94} +106.520 q^{95} +198.606 q^{97} -1274.59 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 5 q^{2} + 17 q^{4} - 15 q^{5} - 4 q^{7} - 75 q^{8} + 25 q^{10} - 5 q^{11} + 7 q^{13} - 60 q^{14} + 161 q^{16} - 155 q^{17} - 50 q^{19} - 85 q^{20} - 229 q^{22} - 285 q^{23} + 75 q^{25} - 185 q^{26}+ \cdots + 305 q^{98}+O(q^{100}) \) 3 * q - 5 * q^2 + 17 * q^4 - 15 * q^5 - 4 * q^7 - 75 * q^8 + 25 * q^10 - 5 * q^11 + 7 * q^13 - 60 * q^14 + 161 * q^16 - 155 * q^17 - 50 * q^19 - 85 * q^20 - 229 * q^22 - 285 * q^23 + 75 * q^25 - 185 * q^26 - 334 * q^28 - 115 * q^29 - 115 * q^31 - 775 * q^32 + 413 * q^34 + 20 * q^35 - 384 * q^37 + 1150 * q^38 + 375 * q^40 - 580 * q^41 - 797 * q^43 + 1415 * q^44 - 285 * q^46 + 145 * q^47 + 577 * q^49 - 125 * q^50 + 825 * q^52 + 400 * q^53 + 25 * q^55 + 2190 * q^56 - 59 * q^58 - 380 * q^59 - 152 * q^61 + 1005 * q^62 + 2937 * q^64 - 35 * q^65 + 2 * q^67 - 475 * q^68 + 300 * q^70 - 40 * q^71 - 980 * q^73 + 2720 * q^74 - 3276 * q^76 - 1950 * q^77 + 1013 * q^79 - 805 * q^80 + 4 * q^82 - 270 * q^83 + 775 * q^85 + 1555 * q^86 - 5193 * q^88 - 1020 * q^89 - 632 * q^91 + 1215 * q^92 + 3833 * q^94 + 250 * q^95 + 720 * q^97 + 305 * q^98

Introduction

L-functions

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