LMFDB - Embedded newform 693.4.a.q.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [693,4,Mod(1,693)] 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("693.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(693, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	693.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,1,0,21,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$40.8883236340$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 30x^{3} + 21x^{2} + 103x + 18 $ x^5 - x^4 - 30x^3 + 21x^2 + 103*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-5.02173$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.02173 q^{2} +17.2178 q^{4} -20.4371 q^{5} -7.00000 q^{7} -46.2894 q^{8} +102.630 q^{10} +11.0000 q^{11} -0.0115563 q^{13} +35.1521 q^{14} +94.7106 q^{16} +9.52800 q^{17} -93.4260 q^{19} -351.882 q^{20} -55.2391 q^{22} -99.9552 q^{23} +292.676 q^{25} +0.0580324 q^{26} -120.525 q^{28} -276.183 q^{29} -181.425 q^{31} -105.296 q^{32} -47.8471 q^{34} +143.060 q^{35} -404.794 q^{37} +469.160 q^{38} +946.022 q^{40} +27.8263 q^{41} +76.9875 q^{43} +189.396 q^{44} +501.948 q^{46} +136.979 q^{47} +49.0000 q^{49} -1469.74 q^{50} -0.198973 q^{52} -170.391 q^{53} -224.808 q^{55} +324.026 q^{56} +1386.92 q^{58} +585.593 q^{59} -530.953 q^{61} +911.070 q^{62} -228.916 q^{64} +0.236177 q^{65} -354.952 q^{67} +164.051 q^{68} -718.408 q^{70} -1119.80 q^{71} -785.389 q^{73} +2032.77 q^{74} -1608.59 q^{76} -77.0000 q^{77} -937.289 q^{79} -1935.61 q^{80} -139.736 q^{82} -471.454 q^{83} -194.725 q^{85} -386.611 q^{86} -509.183 q^{88} -563.040 q^{89} +0.0808938 q^{91} -1721.01 q^{92} -687.874 q^{94} +1909.36 q^{95} +895.067 q^{97} -246.065 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 21 q^{4} - 7 q^{5} - 35 q^{7} + 12 q^{8} + 113 q^{10} + 55 q^{11} + 23 q^{13} - 7 q^{14} + 281 q^{16} + 102 q^{17} - 155 q^{19} - 291 q^{20} + 11 q^{22} - 192 q^{23} + 394 q^{25} - 41 q^{26}+ \cdots + 49 q^{98}+O(q^{100}) $ 5 * q + q^2 + 21 * q^4 - 7 * q^5 - 35 * q^7 + 12 * q^8 + 113 * q^10 + 55 * q^11 + 23 * q^13 - 7 * q^14 + 281 * q^16 + 102 * q^17 - 155 * q^19 - 291 * q^20 + 11 * q^22 - 192 * q^23 + 394 * q^25 - 41 * q^26 - 147 * q^28 - 591 * q^29 + 366 * q^31 + 87 * q^32 + 594 * q^34 + 49 * q^35 + 259 * q^37 - 175 * q^38 + 1890 * q^40 + 104 * q^41 + 224 * q^43 + 231 * q^44 + 1416 * q^46 + 453 * q^47 + 245 * q^49 - 1350 * q^50 + 815 * q^52 - 1032 * q^53 - 77 * q^55 - 84 * q^56 + 293 * q^58 + 517 * q^59 + 958 * q^61 + 3030 * q^62 + 516 * q^64 + 197 * q^65 + 361 * q^67 + 1680 * q^68 - 791 * q^70 - 548 * q^71 - 1035 * q^73 + 2555 * q^74 - 5201 * q^76 - 385 * q^77 + 776 * q^79 + 1273 * q^80 + 376 * q^82 + 1974 * q^83 + 1838 * q^85 - 1224 * q^86 + 132 * q^88 - 2710 * q^89 - 161 * q^91 + 2834 * q^92 - 563 * q^94 + 355 * q^95 + 1988 * q^97 + 49 * 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 $

$2$	−5.02173	−1.77545	−0.887726	−	0.460373i	$-0.847716\pi$
$2$	−5.02173	−1.77545	−0.887726	+	0.460373i	$0.847716\pi$
$3$	0	0
$3$	0	0
$4$	17.2178	2.15223
$5$	−20.4371	−1.82795	−0.913975	−	0.405769i	$-0.867004\pi$
$5$	−20.4371	−1.82795	−0.913975	+	0.405769i	$0.867004\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−46.2894	−2.04572
$9$	0	0
$10$	102.630	3.24544
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−0.0115563	−0.000246548 0	−0.000123274	−	1.00000i	$-0.500039\pi$
$13$	−0.0115563	−0.000246548 0	−0.000123274		1.00000i	$0.500039\pi$
$14$	35.1521	0.671057
$15$	0	0
$16$	94.7106	1.47985
$17$	9.52800	0.135934	0.0679670	−	0.997688i	$-0.478349\pi$
$17$	9.52800	0.135934	0.0679670	+	0.997688i	$0.478349\pi$
$18$	0	0
$19$	−93.4260	−1.12807	−0.564037	−	0.825750i	$-0.690752\pi$
$19$	−93.4260	−1.12807	−0.564037	+	0.825750i	$0.690752\pi$
$20$	−351.882	−3.93416
$21$	0	0
$22$	−55.2391	−0.535319
$23$	−99.9552	−0.906178	−0.453089	−	0.891465i	$-0.649678\pi$
$23$	−99.9552	−0.906178	−0.453089	+	0.891465i	$0.649678\pi$
$24$	0	0
$25$	292.676	2.34140
$26$	0.0580324	0.000437735 0
$27$	0	0
$28$	−120.525	−0.813465
$29$	−276.183	−1.76848	−0.884239	−	0.467035i	$-0.845322\pi$
$29$	−276.183	−1.76848	−0.884239	+	0.467035i	$0.845322\pi$
$30$	0	0
$31$	−181.425	−1.05113	−0.525564	−	0.850754i	$-0.676146\pi$
$31$	−181.425	−1.05113	−0.525564	+	0.850754i	$0.676146\pi$
$32$	−105.296	−0.581684
$33$	0	0
$34$	−47.8471	−0.241344
$35$	143.060	0.690901
$36$	0	0
$37$	−404.794	−1.79859	−0.899294	−	0.437344i	$-0.855919\pi$
$37$	−404.794	−1.79859	−0.899294	+	0.437344i	$0.855919\pi$
$38$	469.160	2.00284
$39$	0	0
$40$	946.022	3.73948
$41$	27.8263	0.105993	0.0529967	−	0.998595i	$-0.483123\pi$
$41$	27.8263	0.105993	0.0529967	+	0.998595i	$0.483123\pi$
$42$	0	0
$43$	76.9875	0.273034	0.136517	−	0.990638i	$-0.456409\pi$
$43$	76.9875	0.273034	0.136517	+	0.990638i	$0.456409\pi$
$44$	189.396	0.648921
$45$	0	0
$46$	501.948	1.60887
$47$	136.979	0.425117	0.212559	−	0.977148i	$-0.431820\pi$
$47$	136.979	0.425117	0.212559	+	0.977148i	$0.431820\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−1469.74	−4.15705
$51$	0	0
$52$	−0.198973	−0.000530628 0
$53$	−170.391	−0.441604	−0.220802	−	0.975319i	$-0.570867\pi$
$53$	−170.391	−0.441604	−0.220802	+	0.975319i	$0.570867\pi$
$54$	0	0
$55$	−224.808	−0.551148
$56$	324.026	0.773210
$57$	0	0
$58$	1386.92	3.13985
$59$	585.593	1.29216	0.646082	−	0.763268i	$-0.276406\pi$
$59$	585.593	1.29216	0.646082	+	0.763268i	$0.276406\pi$
$60$	0	0
$61$	−530.953	−1.11445	−0.557226	−	0.830361i	$-0.688134\pi$
$61$	−530.953	−1.11445	−0.557226	+	0.830361i	$0.688134\pi$
$62$	911.070	1.86623
$63$	0	0
$64$	−228.916	−0.447101
$65$	0.236177	0.000450678 0
$66$	0	0
$67$	−354.952	−0.647228	−0.323614	−	0.946189i	$-0.604898\pi$
$67$	−354.952	−0.647228	−0.323614	+	0.946189i	$0.604898\pi$
$68$	164.051	0.292561
$69$	0	0
$70$	−718.408	−1.22666
$71$	−1119.80	−1.87177	−0.935887	−	0.352301i	$-0.885399\pi$
$71$	−1119.80	−1.87177	−0.935887	+	0.352301i	$0.885399\pi$
$72$	0	0
$73$	−785.389	−1.25922	−0.629608	−	0.776913i	$-0.716785\pi$
$73$	−785.389	−1.25922	−0.629608	+	0.776913i	$0.716785\pi$
$74$	2032.77	3.19331
$75$	0	0
$76$	−1608.59	−2.42787
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−937.289	−1.33485	−0.667425	−	0.744677i	$-0.732604\pi$
$79$	−937.289	−1.33485	−0.667425	+	0.744677i	$0.732604\pi$
$80$	−1935.61	−2.70510
$81$	0	0
$82$	−139.736	−0.188186
$83$	−471.454	−0.623479	−0.311740	−	0.950168i	$-0.600912\pi$
$83$	−471.454	−0.623479	−0.311740	+	0.950168i	$0.600912\pi$
$84$	0	0
$85$	−194.725	−0.248481
$86$	−386.611	−0.484759
$87$	0	0
$88$	−509.183	−0.616808
$89$	−563.040	−0.670586	−0.335293	−	0.942114i	$-0.608835\pi$
$89$	−563.040	−0.670586	−0.335293	+	0.942114i	$0.608835\pi$
$90$	0	0
$91$	0.0808938	9.31865e−5 0
$92$	−1721.01	−1.95030
$93$	0	0
$94$	−687.874	−0.754775
$95$	1909.36	2.06206
$96$	0	0
$97$	895.067	0.936910	0.468455	−	0.883487i	$-0.344811\pi$
$97$	895.067	0.936910	0.468455	+	0.883487i	$0.344811\pi$
$98$	−246.065	−0.253636
$99$	0	0
$100$	5039.23	5.03923
$101$	409.088	0.403027	0.201514	−	0.979486i	$-0.435414\pi$
$101$	409.088	0.403027	0.201514	+	0.979486i	$0.435414\pi$
$102$	0	0
$103$	207.199	0.198213	0.0991065	−	0.995077i	$-0.468402\pi$
$103$	207.199	0.198213	0.0991065	+	0.995077i	$0.468402\pi$
$104$	0.534932	0.000504369 0
$105$	0	0
$106$	855.658	0.784046
$107$	1475.53	1.33313	0.666566	−	0.745446i	$-0.267764\pi$
$107$	1475.53	1.33313	0.666566	+	0.745446i	$0.267764\pi$
$108$	0	0
$109$	1457.82	1.28104	0.640520	−	0.767941i	$-0.278719\pi$
$109$	1457.82	1.28104	0.640520	+	0.767941i	$0.278719\pi$
$110$	1128.93	0.978536
$111$	0	0
$112$	−662.974	−0.559332
$113$	1307.55	1.08853	0.544265	−	0.838913i	$-0.316808\pi$
$113$	1307.55	1.08853	0.544265	+	0.838913i	$0.316808\pi$
$114$	0	0
$115$	2042.80	1.65645
$116$	−4755.26	−3.80616
$117$	0	0
$118$	−2940.69	−2.29417
$119$	−66.6960	−0.0513783
$120$	0	0
$121$	121.000	0.0909091
$122$	2666.31	1.97866
$123$	0	0
$124$	−3123.75	−2.26226
$125$	−3426.80	−2.45202
$126$	0	0
$127$	2156.17	1.50653	0.753265	−	0.657717i	$-0.228478\pi$
$127$	2156.17	1.50653	0.753265	+	0.657717i	$0.228478\pi$
$128$	1991.92	1.37549
$129$	0	0
$130$	−1.18602	−0.000800157 0
$131$	1630.72	1.08761	0.543803	−	0.839213i	$-0.316984\pi$
$131$	1630.72	1.08761	0.543803	+	0.839213i	$0.316984\pi$
$132$	0	0
$133$	653.982	0.426372
$134$	1782.47	1.14912
$135$	0	0
$136$	−441.045	−0.278083
$137$	−2939.97	−1.83342	−0.916711	−	0.399552i	$-0.869166\pi$
$137$	−2939.97	−1.83342	−0.916711	+	0.399552i	$0.869166\pi$
$138$	0	0
$139$	1519.07	0.926948	0.463474	−	0.886111i	$-0.346603\pi$
$139$	1519.07	0.926948	0.463474	+	0.886111i	$0.346603\pi$
$140$	2463.18	1.48697
$141$	0	0
$142$	5623.34	3.32324
$143$	−0.127119	−7.43371e−5 0
$144$	0	0
$145$	5644.38	3.23269
$146$	3944.01	2.23568
$147$	0	0
$148$	−6969.67	−3.87097
$149$	223.829	0.123066	0.0615329	−	0.998105i	$-0.480401\pi$
$149$	223.829	0.123066	0.0615329	+	0.998105i	$0.480401\pi$
$150$	0	0
$151$	−3170.98	−1.70895	−0.854474	−	0.519495i	$-0.826120\pi$
$151$	−3170.98	−1.70895	−0.854474	+	0.519495i	$0.826120\pi$
$152$	4324.63	2.30772
$153$	0	0
$154$	386.674	0.202331
$155$	3707.81	1.92141
$156$	0	0
$157$	2358.25	1.19878	0.599392	−	0.800456i	$-0.295409\pi$
$157$	2358.25	1.19878	0.599392	+	0.800456i	$0.295409\pi$
$158$	4706.81	2.36996
$159$	0	0
$160$	2151.95	1.06329
$161$	699.686	0.342503
$162$	0	0
$163$	2284.80	1.09791	0.548956	−	0.835851i	$-0.315025\pi$
$163$	2284.80	1.09791	0.548956	+	0.835851i	$0.315025\pi$
$164$	479.107	0.228122
$165$	0	0
$166$	2367.51	1.10696
$167$	−2330.48	−1.07987	−0.539934	−	0.841707i	$-0.681551\pi$
$167$	−2330.48	−1.07987	−0.539934	+	0.841707i	$0.681551\pi$
$168$	0	0
$169$	−2197.00	−1.00000
$170$	977.856	0.441166
$171$	0	0
$172$	1325.56	0.587632
$173$	1943.53	0.854125	0.427062	−	0.904222i	$-0.359549\pi$
$173$	1943.53	0.854125	0.427062	+	0.904222i	$0.359549\pi$
$174$	0	0
$175$	−2048.73	−0.884968
$176$	1041.82	0.446192
$177$	0	0
$178$	2827.44	1.19059
$179$	−3511.90	−1.46643	−0.733217	−	0.679995i	$-0.761982\pi$
$179$	−3511.90	−1.46643	−0.733217	+	0.679995i	$0.761982\pi$
$180$	0	0
$181$	3058.76	1.25611	0.628055	−	0.778169i	$-0.283851\pi$
$181$	3058.76	1.25611	0.628055	+	0.778169i	$0.283851\pi$
$182$	−0.406227	−0.000165448 0
$183$	0	0
$184$	4626.87	1.85379
$185$	8272.83	3.28773
$186$	0	0
$187$	104.808	0.0409857
$188$	2358.49	0.914948
$189$	0	0
$190$	−9588.29	−3.66109
$191$	−4586.39	−1.73749	−0.868743	−	0.495264i	$-0.835071\pi$
$191$	−4586.39	−1.73749	−0.868743	+	0.495264i	$0.835071\pi$
$192$	0	0
$193$	693.335	0.258587	0.129294	−	0.991606i	$-0.458729\pi$
$193$	693.335	0.258587	0.129294	+	0.991606i	$0.458729\pi$
$194$	−4494.79	−1.66344
$195$	0	0
$196$	843.673	0.307461
$197$	239.271	0.0865346	0.0432673	−	0.999064i	$-0.486223\pi$
$197$	239.271	0.0865346	0.0432673	+	0.999064i	$0.486223\pi$
$198$	0	0
$199$	−677.963	−0.241505	−0.120753	−	0.992683i	$-0.538531\pi$
$199$	−677.963	−0.241505	−0.120753	+	0.992683i	$0.538531\pi$
$200$	−13547.8	−4.78986
$201$	0	0
$202$	−2054.33	−0.715555
$203$	1933.28	0.668422
$204$	0	0
$205$	−568.688	−0.193751
$206$	−1040.50	−0.351917
$207$	0	0
$208$	−1.09450	−0.000364855 0
$209$	−1027.69	−0.340127
$210$	0	0
$211$	736.355	0.240250	0.120125	−	0.992759i	$-0.461670\pi$
$211$	736.355	0.240250	0.120125	+	0.992759i	$0.461670\pi$
$212$	−2933.76	−0.950431
$213$	0	0
$214$	−7409.73	−2.36691
$215$	−1573.40	−0.499093
$216$	0	0
$217$	1269.98	0.397289
$218$	−7320.76	−2.27442
$219$	0	0
$220$	−3870.71	−1.18620
$221$	−0.110108	−3.35143e−5 0
$222$	0	0
$223$	−1229.51	−0.369210	−0.184605	−	0.982813i	$-0.559101\pi$
$223$	−1229.51	−0.369210	−0.184605	+	0.982813i	$0.559101\pi$
$224$	737.072	0.219856
$225$	0	0
$226$	−6566.17	−1.93263
$227$	3582.28	1.04742	0.523710	−	0.851897i	$-0.324548\pi$
$227$	3582.28	1.04742	0.523710	+	0.851897i	$0.324548\pi$
$228$	0	0
$229$	4732.43	1.36562	0.682812	−	0.730594i	$-0.260757\pi$
$229$	4732.43	1.36562	0.682812	+	0.730594i	$0.260757\pi$
$230$	−10258.4	−2.94094
$231$	0	0
$232$	12784.3	3.61781
$233$	3018.63	0.848744	0.424372	−	0.905488i	$-0.360495\pi$
$233$	3018.63	0.848744	0.424372	+	0.905488i	$0.360495\pi$
$234$	0	0
$235$	−2799.46	−0.777093
$236$	10082.6	2.78103
$237$	0	0
$238$	334.930	0.0912196
$239$	2915.73	0.789133	0.394567	−	0.918867i	$-0.370895\pi$
$239$	2915.73	0.789133	0.394567	+	0.918867i	$0.370895\pi$
$240$	0	0
$241$	5714.30	1.52735	0.763674	−	0.645603i	$-0.223394\pi$
$241$	5714.30	1.52735	0.763674	+	0.645603i	$0.223394\pi$
$242$	−607.630	−0.161405
$243$	0	0
$244$	−9141.86	−2.39855
$245$	−1001.42	−0.261136
$246$	0	0
$247$	1.07965	0.000278125 0
$248$	8398.07	2.15031
$249$	0	0
$250$	17208.5	4.35344
$251$	−1047.71	−0.263471	−0.131735	−	0.991285i	$-0.542055\pi$
$251$	−1047.71	−0.263471	−0.131735	+	0.991285i	$0.542055\pi$
$252$	0	0
$253$	−1099.51	−0.273223
$254$	−10827.7	−2.67477
$255$	0	0
$256$	−8171.58	−1.99501
$257$	2805.01	0.680825	0.340412	−	0.940276i	$-0.389433\pi$
$257$	2805.01	0.680825	0.340412	+	0.940276i	$0.389433\pi$
$258$	0	0
$259$	2833.56	0.679803
$260$	4.06644	0.000969962 0
$261$	0	0
$262$	−8189.03	−1.93099
$263$	−2178.21	−0.510699	−0.255350	−	0.966849i	$-0.582191\pi$
$263$	−2178.21	−0.510699	−0.255350	+	0.966849i	$0.582191\pi$
$264$	0	0
$265$	3482.30	0.807230
$266$	−3284.12	−0.757002
$267$	0	0
$268$	−6111.49	−1.39298
$269$	−4036.85	−0.914985	−0.457493	−	0.889213i	$-0.651252\pi$
$269$	−4036.85	−0.914985	−0.457493	+	0.889213i	$0.651252\pi$
$270$	0	0
$271$	−2166.33	−0.485591	−0.242796	−	0.970077i	$-0.578064\pi$
$271$	−2166.33	−0.485591	−0.242796	+	0.970077i	$0.578064\pi$
$272$	902.402	0.201162
$273$	0	0
$274$	14763.8	3.25515
$275$	3219.43	0.705960
$276$	0	0
$277$	2336.41	0.506791	0.253396	−	0.967363i	$-0.418453\pi$
$277$	2336.41	0.506791	0.253396	+	0.967363i	$0.418453\pi$
$278$	−7628.36	−1.64575
$279$	0	0
$280$	−6622.15	−1.41339
$281$	1200.61	0.254883	0.127442	−	0.991846i	$-0.459323\pi$
$281$	1200.61	0.254883	0.127442	+	0.991846i	$0.459323\pi$
$282$	0	0
$283$	6393.02	1.34285	0.671423	−	0.741074i	$-0.265683\pi$
$283$	6393.02	1.34285	0.671423	+	0.741074i	$0.265683\pi$
$284$	−19280.5	−4.02848
$285$	0	0
$286$	0.638357	0.000131982 0
$287$	−194.784	−0.0400618
$288$	0	0
$289$	−4822.22	−0.981522
$290$	−28344.6	−5.73948
$291$	0	0
$292$	−13522.7	−2.71012
$293$	6037.85	1.20387	0.601936	−	0.798544i	$-0.294396\pi$
$293$	6037.85	1.20387	0.601936	+	0.798544i	$0.294396\pi$
$294$	0	0
$295$	−11967.8	−2.36201
$296$	18737.7	3.67941
$297$	0	0
$298$	−1124.01	−0.218497
$299$	1.15511	0.000223417 0
$300$	0	0
$301$	−538.912	−0.103197
$302$	15923.8	3.03415
$303$	0	0
$304$	−8848.43	−1.66938
$305$	10851.2	2.03717
$306$	0	0
$307$	−2792.13	−0.519073	−0.259536	−	0.965733i	$-0.583570\pi$
$307$	−2792.13	−0.519073	−0.259536	+	0.965733i	$0.583570\pi$
$308$	−1325.77	−0.245269
$309$	0	0
$310$	−18619.6	−3.41137
$311$	−5360.40	−0.977364	−0.488682	−	0.872462i	$-0.662522\pi$
$311$	−5360.40	−0.977364	−0.488682	+	0.872462i	$0.662522\pi$
$312$	0	0
$313$	−9998.72	−1.80563	−0.902813	−	0.430033i	$-0.858502\pi$
$313$	−9998.72	−1.80563	−0.902813	+	0.430033i	$0.858502\pi$
$314$	−11842.5	−2.12838
$315$	0	0
$316$	−16138.1	−2.87290
$317$	−1043.98	−0.184971	−0.0924855	−	0.995714i	$-0.529481\pi$
$317$	−1043.98	−0.184971	−0.0924855	+	0.995714i	$0.529481\pi$
$318$	0	0
$319$	−3038.01	−0.533216
$320$	4678.38	0.817279
$321$	0	0
$322$	−3513.64	−0.608098
$323$	−890.163	−0.153344
$324$	0	0
$325$	−3.38223	−0.000577269 0
$326$	−11473.7	−1.94929
$327$	0	0
$328$	−1288.06	−0.216833
$329$	−958.856	−0.160679
$330$	0	0
$331$	−826.343	−0.137220	−0.0686102	−	0.997644i	$-0.521856\pi$
$331$	−826.343	−0.137220	−0.0686102	+	0.997644i	$0.521856\pi$
$332$	−8117.40	−1.34187
$333$	0	0
$334$	11703.0	1.91725
$335$	7254.19	1.18310
$336$	0	0
$337$	460.594	0.0744515	0.0372258	−	0.999307i	$-0.488148\pi$
$337$	460.594	0.0744515	0.0372258	+	0.999307i	$0.488148\pi$
$338$	11032.7	1.77545
$339$	0	0
$340$	−3352.74	−0.534787
$341$	−1995.68	−0.316927
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−3563.70	−0.558552
$345$	0	0
$346$	−9759.87	−1.51646
$347$	−4459.48	−0.689906	−0.344953	−	0.938620i	$-0.612105\pi$
$347$	−4459.48	−0.689906	−0.344953	+	0.938620i	$0.612105\pi$
$348$	0	0
$349$	7902.74	1.21210	0.606051	−	0.795426i	$-0.292753\pi$
$349$	7902.74	1.21210	0.606051	+	0.795426i	$0.292753\pi$
$350$	10288.2	1.57122
$351$	0	0
$352$	−1158.26	−0.175384
$353$	1159.24	0.174787	0.0873936	−	0.996174i	$-0.472146\pi$
$353$	1159.24	0.174787	0.0873936	+	0.996174i	$0.472146\pi$
$354$	0	0
$355$	22885.5	3.42151
$356$	−9694.32	−1.44325
$357$	0	0
$358$	17635.8	2.60358
$359$	−3164.04	−0.465158	−0.232579	−	0.972577i	$-0.574716\pi$
$359$	−3164.04	−0.465158	−0.232579	+	0.972577i	$0.574716\pi$
$360$	0	0
$361$	1869.42	0.272549
$362$	−15360.3	−2.23016
$363$	0	0
$364$	1.39281	0.000200559 0
$365$	16051.1	2.30179
$366$	0	0
$367$	6586.11	0.936763	0.468382	−	0.883526i	$-0.344837\pi$
$367$	6586.11	0.936763	0.468382	+	0.883526i	$0.344837\pi$
$368$	−9466.81	−1.34101
$369$	0	0
$370$	−41543.9	−5.83721
$371$	1192.74	0.166911
$372$	0	0
$373$	1210.44	0.168027	0.0840135	−	0.996465i	$-0.473226\pi$
$373$	1210.44	0.168027	0.0840135	+	0.996465i	$0.473226\pi$
$374$	−526.318	−0.0727680
$375$	0	0
$376$	−6340.70	−0.869671
$377$	3.19164	0.000436015 0
$378$	0	0
$379$	−5802.28	−0.786393	−0.393197	−	0.919454i	$-0.628631\pi$
$379$	−5802.28	−0.786393	−0.393197	+	0.919454i	$0.628631\pi$
$380$	32875.0	4.43803
$381$	0	0
$382$	23031.6	3.08482
$383$	3704.97	0.494295	0.247147	−	0.968978i	$-0.420507\pi$
$383$	3704.97	0.494295	0.247147	+	0.968978i	$0.420507\pi$
$384$	0	0
$385$	1573.66	0.208314
$386$	−3481.74	−0.459109
$387$	0	0
$388$	15411.1	2.01644
$389$	−3365.36	−0.438638	−0.219319	−	0.975653i	$-0.570384\pi$
$389$	−3365.36	−0.438638	−0.219319	+	0.975653i	$0.570384\pi$
$390$	0	0
$391$	−952.373	−0.123180
$392$	−2268.18	−0.292246
$393$	0	0
$394$	−1201.55	−0.153638
$395$	19155.5	2.44004
$396$	0	0
$397$	−10914.1	−1.37976	−0.689879	−	0.723925i	$-0.742336\pi$
$397$	−10914.1	−1.37976	−0.689879	+	0.723925i	$0.742336\pi$
$398$	3404.55	0.428781
$399$	0	0
$400$	27719.5	3.46493
$401$	−4980.45	−0.620228	−0.310114	−	0.950699i	$-0.600367\pi$
$401$	−4980.45	−0.620228	−0.310114	+	0.950699i	$0.600367\pi$
$402$	0	0
$403$	2.09660	0.000259154 0
$404$	7043.59	0.867406
$405$	0	0
$406$	−9708.41	−1.18675
$407$	−4452.74	−0.542295
$408$	0	0
$409$	−9813.28	−1.18639	−0.593197	−	0.805057i	$-0.702135\pi$
$409$	−9813.28	−1.18639	−0.593197	+	0.805057i	$0.702135\pi$
$410$	2855.80	0.343995
$411$	0	0
$412$	3567.51	0.426599
$413$	−4099.15	−0.488392
$414$	0	0
$415$	9635.15	1.13969
$416$	1.21683	0.000143413 0
$417$	0	0
$418$	5160.77	0.603879
$419$	−14050.0	−1.63815	−0.819077	−	0.573683i	$-0.805514\pi$
$419$	−14050.0	−1.63815	−0.819077	+	0.573683i	$0.805514\pi$
$420$	0	0
$421$	10403.9	1.20441	0.602203	−	0.798343i	$-0.294290\pi$
$421$	10403.9	1.20441	0.602203	+	0.798343i	$0.294290\pi$
$422$	−3697.78	−0.426553
$423$	0	0
$424$	7887.29	0.903398
$425$	2788.61	0.318277
$426$	0	0
$427$	3716.67	0.421224
$428$	25405.4	2.86920
$429$	0	0
$430$	7901.20	0.886116
$431$	−1181.89	−0.132087	−0.0660437	−	0.997817i	$-0.521038\pi$
$431$	−1181.89	−0.132087	−0.0660437	+	0.997817i	$0.521038\pi$
$432$	0	0
$433$	−7028.94	−0.780113	−0.390057	−	0.920791i	$-0.627545\pi$
$433$	−7028.94	−0.780113	−0.390057	+	0.920791i	$0.627545\pi$
$434$	−6377.49	−0.705367
$435$	0	0
$436$	25100.4	2.75709
$437$	9338.41	1.02224
$438$	0	0
$439$	5758.63	0.626069	0.313034	−	0.949742i	$-0.398654\pi$
$439$	5758.63	0.626069	0.313034	+	0.949742i	$0.398654\pi$
$440$	10406.2	1.12750
$441$	0	0
$442$	0.552933	5.95030e−5 0
$443$	13270.7	1.42327	0.711636	−	0.702548i	$-0.247955\pi$
$443$	13270.7	1.42327	0.711636	+	0.702548i	$0.247955\pi$
$444$	0	0
$445$	11506.9	1.22580
$446$	6174.25	0.655514
$447$	0	0
$448$	1602.41	0.168988
$449$	−8609.07	−0.904871	−0.452435	−	0.891797i	$-0.649445\pi$
$449$	−8609.07	−0.904871	−0.452435	+	0.891797i	$0.649445\pi$
$450$	0	0
$451$	306.089	0.0319582
$452$	22513.2	2.34276
$453$	0	0
$454$	−17989.3	−1.85964
$455$	−1.65324	−0.000170340 0
$456$	0	0
$457$	1112.50	0.113874	0.0569369	−	0.998378i	$-0.481867\pi$
$457$	1112.50	0.113874	0.0569369	+	0.998378i	$0.481867\pi$
$458$	−23765.0	−2.42460
$459$	0	0
$460$	35172.5	3.56505
$461$	−374.302	−0.0378155	−0.0189078	−	0.999821i	$-0.506019\pi$
$461$	−374.302	−0.0378155	−0.0189078	+	0.999821i	$0.506019\pi$
$462$	0	0
$463$	2731.47	0.274174	0.137087	−	0.990559i	$-0.456226\pi$
$463$	2731.47	0.274174	0.137087	+	0.990559i	$0.456226\pi$
$464$	−26157.4	−2.61709
$465$	0	0
$466$	−15158.8	−1.50690
$467$	−5370.18	−0.532125	−0.266062	−	0.963956i	$-0.585723\pi$
$467$	−5370.18	−0.532125	−0.266062	+	0.963956i	$0.585723\pi$
$468$	0	0
$469$	2484.66	0.244629
$470$	14058.2	1.37969
$471$	0	0
$472$	−27106.7	−2.64341
$473$	846.862	0.0823230
$474$	0	0
$475$	−27343.5	−2.64128
$476$	−1148.36	−0.110578
$477$	0	0
$478$	−14642.0	−1.40107
$479$	3599.62	0.343363	0.171681	−	0.985153i	$-0.445080\pi$
$479$	3599.62	0.343363	0.171681	+	0.985153i	$0.445080\pi$
$480$	0	0
$481$	4.67791	0.000443439 0
$482$	−28695.7	−2.71173
$483$	0	0
$484$	2083.36	0.195657
$485$	−18292.6	−1.71263
$486$	0	0
$487$	13234.4	1.23144	0.615718	−	0.787966i	$-0.288866\pi$
$487$	13234.4	1.23144	0.615718	+	0.787966i	$0.288866\pi$
$488$	24577.5	2.27986
$489$	0	0
$490$	5028.86	0.463634
$491$	20613.6	1.89466	0.947330	−	0.320259i	$-0.103770\pi$
$491$	20613.6	1.89466	0.947330	+	0.320259i	$0.103770\pi$
$492$	0	0
$493$	−2631.47	−0.240396
$494$	−5.42174	−0.000493797 0
$495$	0	0
$496$	−17182.9	−1.55551
$497$	7838.61	0.707464
$498$	0	0
$499$	−11615.8	−1.04208	−0.521038	−	0.853533i	$-0.674455\pi$
$499$	−11615.8	−1.04208	−0.521038	+	0.853533i	$0.674455\pi$
$500$	−59002.1	−5.27731
$501$	0	0
$502$	5261.34	0.467779
$503$	−5414.81	−0.479989	−0.239994	−	0.970774i	$-0.577146\pi$
$503$	−5414.81	−0.479989	−0.239994	+	0.970774i	$0.577146\pi$
$504$	0	0
$505$	−8360.57	−0.736714
$506$	5521.43	0.485094
$507$	0	0
$508$	37124.5	3.24239
$509$	−15370.0	−1.33843	−0.669216	−	0.743068i	$-0.733370\pi$
$509$	−15370.0	−1.33843	−0.669216	+	0.743068i	$0.733370\pi$
$510$	0	0
$511$	5497.72	0.475939
$512$	25100.1	2.16656
$513$	0	0
$514$	−14086.0	−1.20877
$515$	−4234.55	−0.362324
$516$	0	0
$517$	1506.77	0.128178
$518$	−14229.4	−1.20696
$519$	0	0
$520$	−10.9325	−0.000921962 0
$521$	18605.8	1.56456	0.782281	−	0.622926i	$-0.214056\pi$
$521$	18605.8	1.56456	0.782281	+	0.622926i	$0.214056\pi$
$522$	0	0
$523$	−1484.42	−0.124109	−0.0620545	−	0.998073i	$-0.519765\pi$
$523$	−1484.42	−0.124109	−0.0620545	+	0.998073i	$0.519765\pi$
$524$	28077.4	2.34078
$525$	0	0
$526$	10938.4	0.906721
$527$	−1728.62	−0.142884
$528$	0	0
$529$	−2175.96	−0.178841
$530$	−17487.2	−1.43320
$531$	0	0
$532$	11260.1	0.917648
$533$	−0.321567	−2.61325e−5 0
$534$	0	0
$535$	−30155.6	−2.43690
$536$	16430.5	1.32405
$537$	0	0
$538$	20272.0	1.62451
$539$	539.000	0.0430730
$540$	0	0
$541$	−3695.09	−0.293649	−0.146825	−	0.989163i	$-0.546905\pi$
$541$	−3695.09	−0.293649	−0.146825	+	0.989163i	$0.546905\pi$
$542$	10878.7	0.862144
$543$	0	0
$544$	−1003.26	−0.0790707
$545$	−29793.5	−2.34168
$546$	0	0
$547$	−24243.6	−1.89503	−0.947516	−	0.319709i	$-0.896415\pi$
$547$	−24243.6	−1.89503	−0.947516	+	0.319709i	$0.896415\pi$
$548$	−50619.9	−3.94594
$549$	0	0
$550$	−16167.1	−1.25340
$551$	25802.6	1.99497
$552$	0	0
$553$	6561.02	0.504526
$554$	−11732.8	−0.899783
$555$	0	0
$556$	26155.0	1.99500
$557$	−17492.7	−1.33068	−0.665340	−	0.746541i	$-0.731713\pi$
$557$	−17492.7	−1.33068	−0.665340	+	0.746541i	$0.731713\pi$
$558$	0	0
$559$	−0.889687	−6.73162e−5 0
$560$	13549.3	1.02243
$561$	0	0
$562$	−6029.13	−0.452533
$563$	11377.4	0.851689	0.425845	−	0.904796i	$-0.359977\pi$
$563$	11377.4	0.851689	0.425845	+	0.904796i	$0.359977\pi$
$564$	0	0
$565$	−26722.6	−1.98978
$566$	−32104.0	−2.38416
$567$	0	0
$568$	51834.9	3.82913
$569$	−13028.1	−0.959868	−0.479934	−	0.877305i	$-0.659339\pi$
$569$	−13028.1	−0.959868	−0.479934	+	0.877305i	$0.659339\pi$
$570$	0	0
$571$	−15448.5	−1.13223	−0.566113	−	0.824327i	$-0.691554\pi$
$571$	−15448.5	−1.13223	−0.566113	+	0.824327i	$0.691554\pi$
$572$	−2.18871	−0.000159990 0
$573$	0	0
$574$	978.153	0.0711277
$575$	−29254.4	−2.12173
$576$	0	0
$577$	−24253.5	−1.74989	−0.874944	−	0.484224i	$-0.839102\pi$
$577$	−24253.5	−1.74989	−0.874944	+	0.484224i	$0.839102\pi$
$578$	24215.9	1.74264
$579$	0	0
$580$	97183.8	6.95748
$581$	3300.18	0.235653
$582$	0	0
$583$	−1874.30	−0.133149
$584$	36355.2	2.57601
$585$	0	0
$586$	−30320.5	−2.13742
$587$	−3081.95	−0.216705	−0.108352	−	0.994113i	$-0.534557\pi$
$587$	−3081.95	−0.216705	−0.108352	+	0.994113i	$0.534557\pi$
$588$	0	0
$589$	16949.8	1.18575
$590$	60099.2	4.19364
$591$	0	0
$592$	−38338.3	−2.66165
$593$	1806.94	0.125130	0.0625649	−	0.998041i	$-0.480072\pi$
$593$	1806.94	0.125130	0.0625649	+	0.998041i	$0.480072\pi$
$594$	0	0
$595$	1363.07	0.0939169
$596$	3853.85	0.264865
$597$	0	0
$598$	−5.80064	−0.000396665 0
$599$	16213.0	1.10592	0.552958	−	0.833209i	$-0.313499\pi$
$599$	16213.0	1.10592	0.552958	+	0.833209i	$0.313499\pi$
$600$	0	0
$601$	28306.2	1.92119	0.960594	−	0.277957i	$-0.0896572\pi$
$601$	28306.2	1.92119	0.960594	+	0.277957i	$0.0896572\pi$
$602$	2706.27	0.183222
$603$	0	0
$604$	−54597.4	−3.67804
$605$	−2472.89	−0.166177
$606$	0	0
$607$	13513.0	0.903587	0.451794	−	0.892122i	$-0.350784\pi$
$607$	13513.0	0.903587	0.451794	+	0.892122i	$0.350784\pi$
$608$	9837.39	0.656182
$609$	0	0
$610$	−54491.6	−3.61689
$611$	−1.58297	−0.000104812 0
$612$	0	0
$613$	−6645.66	−0.437873	−0.218936	−	0.975739i	$-0.570259\pi$
$613$	−6645.66	−0.437873	−0.218936	+	0.975739i	$0.570259\pi$
$614$	14021.3	0.921588
$615$	0	0
$616$	3564.28	0.233132
$617$	6054.73	0.395063	0.197532	−	0.980297i	$-0.436707\pi$
$617$	6054.73	0.395063	0.197532	+	0.980297i	$0.436707\pi$
$618$	0	0
$619$	2193.31	0.142418	0.0712089	−	0.997461i	$-0.477314\pi$
$619$	2193.31	0.142418	0.0712089	+	0.997461i	$0.477314\pi$
$620$	63840.4	4.13531
$621$	0	0
$622$	26918.5	1.73526
$623$	3941.28	0.253458
$624$	0	0
$625$	33449.5	2.14077
$626$	50210.9	3.20580
$627$	0	0
$628$	40603.9	2.58005
$629$	−3856.88	−0.244490
$630$	0	0
$631$	22314.2	1.40778	0.703892	−	0.710307i	$-0.251444\pi$
$631$	22314.2	1.40778	0.703892	+	0.710307i	$0.251444\pi$
$632$	43386.5	2.73073
$633$	0	0
$634$	5242.59	0.328407
$635$	−44065.9	−2.75386
$636$	0	0
$637$	−0.566257	−3.52212e−5 0
$638$	15256.1	0.946699
$639$	0	0
$640$	−40709.1	−2.51433
$641$	1276.44	0.0786524	0.0393262	−	0.999226i	$-0.487479\pi$
$641$	1276.44	0.0786524	0.0393262	+	0.999226i	$0.487479\pi$
$642$	0	0
$643$	19610.9	1.20276	0.601382	−	0.798961i	$-0.294617\pi$
$643$	19610.9	1.20276	0.601382	+	0.798961i	$0.294617\pi$
$644$	12047.1	0.737144
$645$	0	0
$646$	4470.16	0.272254
$647$	1351.52	0.0821232	0.0410616	−	0.999157i	$-0.486926\pi$
$647$	1351.52	0.0821232	0.0410616	+	0.999157i	$0.486926\pi$
$648$	0	0
$649$	6441.52	0.389602
$650$	16.9847	0.00102491
$651$	0	0
$652$	39339.3	2.36296
$653$	2697.61	0.161662	0.0808312	−	0.996728i	$-0.474243\pi$
$653$	2697.61	0.161662	0.0808312	+	0.996728i	$0.474243\pi$
$654$	0	0
$655$	−33327.2	−1.98809
$656$	2635.44	0.156855
$657$	0	0
$658$	4815.12	0.285278
$659$	22130.4	1.30816	0.654080	−	0.756426i	$-0.273056\pi$
$659$	22130.4	1.30816	0.654080	+	0.756426i	$0.273056\pi$
$660$	0	0
$661$	−12681.1	−0.746199	−0.373100	−	0.927791i	$-0.621705\pi$
$661$	−12681.1	−0.746199	−0.373100	+	0.927791i	$0.621705\pi$
$662$	4149.68	0.243628
$663$	0	0
$664$	21823.3	1.27546
$665$	−13365.5	−0.779386
$666$	0	0
$667$	27605.9	1.60256
$668$	−40125.8	−2.32412
$669$	0	0
$670$	−36428.6	−2.10054
$671$	−5840.49	−0.336020
$672$	0	0
$673$	5000.12	0.286390	0.143195	−	0.989694i	$-0.454262\pi$
$673$	5000.12	0.286390	0.143195	+	0.989694i	$0.454262\pi$
$674$	−2312.98	−0.132185
$675$	0	0
$676$	−37827.5	−2.15223
$677$	−6004.13	−0.340853	−0.170427	−	0.985370i	$-0.554515\pi$
$677$	−6004.13	−0.340853	−0.170427	+	0.985370i	$0.554515\pi$
$678$	0	0
$679$	−6265.47	−0.354119
$680$	9013.69	0.508323
$681$	0	0
$682$	10021.8	0.562688
$683$	−15747.3	−0.882217	−0.441109	−	0.897454i	$-0.645415\pi$
$683$	−15747.3	−0.882217	−0.441109	+	0.897454i	$0.645415\pi$
$684$	0	0
$685$	60084.5	3.35140
$686$	1722.45	0.0958653
$687$	0	0
$688$	7291.53	0.404051
$689$	1.96908	0.000108877 0
$690$	0	0
$691$	−206.020	−0.0113421	−0.00567105	−	0.999984i	$-0.501805\pi$
$691$	−206.020	−0.0113421	−0.00567105	+	0.999984i	$0.501805\pi$
$692$	33463.3	1.83827
$693$	0	0
$694$	22394.3	1.22489
$695$	−31045.4	−1.69442
$696$	0	0
$697$	265.129	0.0144081
$698$	−39685.4	−2.15203
$699$	0	0
$700$	−35274.6	−1.90465
$701$	2413.37	0.130031	0.0650154	−	0.997884i	$-0.479290\pi$
$701$	2413.37	0.130031	0.0650154	+	0.997884i	$0.479290\pi$
$702$	0	0
$703$	37818.3	2.02894
$704$	−2518.07	−0.134806
$705$	0	0
$706$	−5821.37	−0.310326
$707$	−2863.61	−0.152330
$708$	0	0
$709$	13211.5	0.699817	0.349908	−	0.936784i	$-0.386213\pi$
$709$	13211.5	0.699817	0.349908	+	0.936784i	$0.386213\pi$
$710$	−114925.	−6.07472
$711$	0	0
$712$	26062.8	1.37183
$713$	18134.4	0.952509
$714$	0	0
$715$	2.59794	0.000135885 0
$716$	−60467.2	−3.15610
$717$	0	0
$718$	15889.0	0.825866
$719$	−16242.1	−0.842458	−0.421229	−	0.906954i	$-0.638401\pi$
$719$	−16242.1	−0.842458	−0.421229	+	0.906954i	$0.638401\pi$
$720$	0	0
$721$	−1450.39	−0.0749174
$722$	−9387.71	−0.483898
$723$	0	0
$724$	52665.2	2.70344
$725$	−80831.9	−4.14072
$726$	0	0
$727$	−247.150	−0.0126084	−0.00630418	−	0.999980i	$-0.502007\pi$
$727$	−247.150	−0.0126084	−0.00630418	+	0.999980i	$0.502007\pi$
$728$	−3.74453	−0.000190634 0
$729$	0	0
$730$	−80604.3	−4.08671
$731$	733.537	0.0371147
$732$	0	0
$733$	27692.4	1.39542	0.697711	−	0.716380i	$-0.254202\pi$
$733$	27692.4	1.39542	0.697711	+	0.716380i	$0.254202\pi$
$734$	−33073.7	−1.66318
$735$	0	0
$736$	10524.9	0.527109
$737$	−3904.47	−0.195147
$738$	0	0
$739$	−30513.8	−1.51890	−0.759451	−	0.650564i	$-0.774532\pi$
$739$	−30513.8	−1.51890	−0.759451	+	0.650564i	$0.774532\pi$
$740$	142440.	7.07594
$741$	0	0
$742$	−5989.61	−0.296341
$743$	10527.8	0.519822	0.259911	−	0.965633i	$-0.416307\pi$
$743$	10527.8	0.519822	0.259911	+	0.965633i	$0.416307\pi$
$744$	0	0
$745$	−4574.42	−0.224958
$746$	−6078.49	−0.298324
$747$	0	0
$748$	1804.56	0.0882104
$749$	−10328.7	−0.503876
$750$	0	0
$751$	9111.92	0.442741	0.221371	−	0.975190i	$-0.428947\pi$
$751$	9111.92	0.442741	0.221371	+	0.975190i	$0.428947\pi$
$752$	12973.4	0.629111
$753$	0	0
$754$	−16.0276	−0.000774124 0
$755$	64805.8	3.12387
$756$	0	0
$757$	−2875.38	−0.138055	−0.0690273	−	0.997615i	$-0.521990\pi$
$757$	−2875.38	−0.138055	−0.0690273	+	0.997615i	$0.521990\pi$
$758$	29137.5	1.39620
$759$	0	0
$760$	−88383.0	−4.21841
$761$	1580.96	0.0753084	0.0376542	−	0.999291i	$-0.488011\pi$
$761$	1580.96	0.0753084	0.0376542	+	0.999291i	$0.488011\pi$
$762$	0	0
$763$	−10204.7	−0.484188
$764$	−78967.6	−3.73946
$765$	0	0
$766$	−18605.3	−0.877596
$767$	−6.76726	−0.000318581 0
$768$	0	0
$769$	−24477.4	−1.14783	−0.573913	−	0.818916i	$-0.694575\pi$
$769$	−24477.4	−1.14783	−0.573913	+	0.818916i	$0.694575\pi$
$770$	−7902.49	−0.369852
$771$	0	0
$772$	11937.7	0.556538
$773$	40144.4	1.86791	0.933954	−	0.357393i	$-0.116334\pi$
$773$	40144.4	1.86791	0.933954	+	0.357393i	$0.116334\pi$
$774$	0	0
$775$	−53098.8	−2.46111
$776$	−41432.1	−1.91666
$777$	0	0
$778$	16899.9	0.778781
$779$	−2599.70	−0.119568
$780$	0	0
$781$	−12317.8	−0.564361
$782$	4782.56	0.218701
$783$	0	0
$784$	4640.82	0.211408
$785$	−48195.9	−2.19132
$786$	0	0
$787$	−38095.3	−1.72548	−0.862739	−	0.505650i	$-0.831253\pi$
$787$	−38095.3	−1.72548	−0.862739	+	0.505650i	$0.831253\pi$
$788$	4119.72	0.186242
$789$	0	0
$790$	−96193.7	−4.33217
$791$	−9152.85	−0.411426
$792$	0	0
$793$	6.13583	0.000274767 0
$794$	54807.8	2.44969
$795$	0	0
$796$	−11673.0	−0.519774
$797$	13196.7	0.586511	0.293256	−	0.956034i	$-0.405261\pi$
$797$	13196.7	0.586511	0.293256	+	0.956034i	$0.405261\pi$
$798$	0	0
$799$	1305.14	0.0577879
$800$	−30817.6	−1.36196
$801$	0	0
$802$	25010.5	1.10119
$803$	−8639.28	−0.379668
$804$	0	0
$805$	−14299.6	−0.626079
$806$	−10.5286	−0.000460115 0
$807$	0	0
$808$	−18936.4	−0.824481
$809$	39595.8	1.72078	0.860392	−	0.509633i	$-0.170219\pi$
$809$	39595.8	1.72078	0.860392	+	0.509633i	$0.170219\pi$
$810$	0	0
$811$	7072.51	0.306226	0.153113	−	0.988209i	$-0.451070\pi$
$811$	7072.51	0.306226	0.153113	+	0.988209i	$0.451070\pi$
$812$	33286.8	1.43859
$813$	0	0
$814$	22360.5	0.962818
$815$	−46694.8	−2.00693
$816$	0	0
$817$	−7192.63	−0.308003
$818$	49279.7	2.10639
$819$	0	0
$820$	−9791.57	−0.416996
$821$	−3770.85	−0.160297	−0.0801484	−	0.996783i	$-0.525539\pi$
$821$	−3770.85	−0.160297	−0.0801484	+	0.996783i	$0.525539\pi$
$822$	0	0
$823$	34875.3	1.47713	0.738565	−	0.674182i	$-0.235504\pi$
$823$	34875.3	1.47713	0.738565	+	0.674182i	$0.235504\pi$
$824$	−9591.12	−0.405489
$825$	0	0
$826$	20584.8	0.867116
$827$	−13104.1	−0.550995	−0.275497	−	0.961302i	$-0.588843\pi$
$827$	−13104.1	−0.550995	−0.275497	+	0.961302i	$0.588843\pi$
$828$	0	0
$829$	25672.5	1.07556	0.537782	−	0.843084i	$-0.319262\pi$
$829$	25672.5	1.07556	0.537782	+	0.843084i	$0.319262\pi$
$830$	−48385.2	−2.02346
$831$	0	0
$832$	2.64541	0.000110232 0
$833$	466.872	0.0194192
$834$	0	0
$835$	47628.3	1.97395
$836$	−17694.5	−0.732030
$837$	0	0
$838$	70555.3	2.90846
$839$	12972.5	0.533804	0.266902	−	0.963724i	$-0.414000\pi$
$839$	12972.5	0.533804	0.266902	+	0.963724i	$0.414000\pi$
$840$	0	0
$841$	51887.9	2.12751
$842$	−52245.6	−2.13836
$843$	0	0
$844$	12678.4	0.517073
$845$	44900.3	1.82795
$846$	0	0
$847$	−847.000	−0.0343604
$848$	−16137.8	−0.653508
$849$	0	0
$850$	−14003.7	−0.565085
$851$	40461.3	1.62984
$852$	0	0
$853$	−38411.5	−1.54183	−0.770917	−	0.636936i	$-0.780202\pi$
$853$	−38411.5	−1.54183	−0.770917	+	0.636936i	$0.780202\pi$
$854$	−18664.1	−0.747862
$855$	0	0
$856$	−68301.5	−2.72722
$857$	−29840.9	−1.18943	−0.594717	−	0.803935i	$-0.702736\pi$
$857$	−29840.9	−1.18943	−0.594717	+	0.803935i	$0.702736\pi$
$858$	0	0
$859$	−6603.57	−0.262294	−0.131147	−	0.991363i	$-0.541866\pi$
$859$	−6603.57	−0.262294	−0.131147	+	0.991363i	$0.541866\pi$
$860$	−27090.5	−1.07416
$861$	0	0
$862$	5935.14	0.234515
$863$	35177.7	1.38756	0.693780	−	0.720187i	$-0.255944\pi$
$863$	35177.7	1.38756	0.693780	+	0.720187i	$0.255944\pi$
$864$	0	0
$865$	−39720.1	−1.56130
$866$	35297.4	1.38505
$867$	0	0
$868$	21866.2	0.855056
$869$	−10310.2	−0.402473
$870$	0	0
$871$	4.10191	0.000159573 0
$872$	−67481.4	−2.62065
$873$	0	0
$874$	−46895.0	−1.81493
$875$	23987.6	0.926777
$876$	0	0
$877$	−22669.6	−0.872860	−0.436430	−	0.899738i	$-0.643757\pi$
$877$	−22669.6	−0.872860	−0.436430	+	0.899738i	$0.643757\pi$
$878$	−28918.3	−1.11155
$879$	0	0
$880$	−21291.7	−0.815618
$881$	−18705.2	−0.715317	−0.357658	−	0.933852i	$-0.616425\pi$
$881$	−18705.2	−0.715317	−0.357658	+	0.933852i	$0.616425\pi$
$882$	0	0
$883$	−768.137	−0.0292750	−0.0146375	−	0.999893i	$-0.504659\pi$
$883$	−768.137	−0.0292750	−0.0146375	+	0.999893i	$0.504659\pi$
$884$	−1.89582	−7.21304e−5 0
$885$	0	0
$886$	−66641.9	−2.52695
$887$	32827.0	1.24264	0.621322	−	0.783556i	$-0.286596\pi$
$887$	32827.0	1.24264	0.621322	+	0.783556i	$0.286596\pi$
$888$	0	0
$889$	−15093.2	−0.569415
$890$	−57784.7	−2.17634
$891$	0	0
$892$	−21169.4	−0.794624
$893$	−12797.4	−0.479563
$894$	0	0
$895$	71773.1	2.68057
$896$	−13943.5	−0.519886
$897$	0	0
$898$	43232.4	1.60655
$899$	50106.5	1.85890
$900$	0	0
$901$	−1623.48	−0.0600290
$902$	−1537.10	−0.0567403
$903$	0	0
$904$	−60525.7	−2.22683
$905$	−62512.3	−2.29611
$906$	0	0
$907$	−12345.2	−0.451945	−0.225972	−	0.974134i	$-0.572556\pi$
$907$	−12345.2	−0.451945	−0.225972	+	0.974134i	$0.572556\pi$
$908$	61679.0	2.25428
$909$	0	0
$910$	8.30211	0.000302431 0
$911$	−11001.3	−0.400098	−0.200049	−	0.979786i	$-0.564110\pi$
$911$	−11001.3	−0.400098	−0.200049	+	0.979786i	$0.564110\pi$
$912$	0	0
$913$	−5185.99	−0.187986
$914$	−5586.65	−0.202177
$915$	0	0
$916$	81482.1	2.93913
$917$	−11415.0	−0.411077
$918$	0	0
$919$	22345.4	0.802075	0.401037	−	0.916062i	$-0.368650\pi$
$919$	22345.4	0.802075	0.401037	+	0.916062i	$0.368650\pi$
$920$	−94559.8	−3.38863
$921$	0	0
$922$	1879.64	0.0671396
$923$	12.9407	0.000461483 0
$924$	0	0
$925$	−118473.	−4.21122
$926$	−13716.7	−0.486782
$927$	0	0
$928$	29080.9	1.02869
$929$	−41802.6	−1.47632	−0.738160	−	0.674626i	$-0.764305\pi$
$929$	−41802.6	−1.47632	−0.738160	+	0.674626i	$0.764305\pi$
$930$	0	0
$931$	−4577.87	−0.161153
$932$	51974.3	1.82669
$933$	0	0
$934$	26967.6	0.944761
$935$	−2141.97	−0.0749198
$936$	0	0
$937$	30060.9	1.04808	0.524038	−	0.851695i	$-0.324425\pi$
$937$	30060.9	1.04808	0.524038	+	0.851695i	$0.324425\pi$
$938$	−12477.3	−0.434327
$939$	0	0
$940$	−48200.6	−1.67248
$941$	35821.3	1.24096	0.620479	−	0.784223i	$-0.286938\pi$
$941$	35821.3	1.24096	0.620479	+	0.784223i	$0.286938\pi$
$942$	0	0
$943$	−2781.38	−0.0960490
$944$	55461.8	1.91221
$945$	0	0
$946$	−4252.72	−0.146160
$947$	6919.22	0.237428	0.118714	−	0.992928i	$-0.462123\pi$
$947$	6919.22	0.237428	0.118714	+	0.992928i	$0.462123\pi$
$948$	0	0
$949$	9.07616	0.000310458 0
$950$	137312.	4.68946
$951$	0	0
$952$	3087.32	0.105106
$953$	29103.4	0.989248	0.494624	−	0.869107i	$-0.335306\pi$
$953$	29103.4	0.989248	0.494624	+	0.869107i	$0.335306\pi$
$954$	0	0
$955$	93732.6	3.17604
$956$	50202.5	1.69839
$957$	0	0
$958$	−18076.3	−0.609624
$959$	20579.8	0.692968
$960$	0	0
$961$	3124.15	0.104869
$962$	−23.4912	−0.000787304 0
$963$	0	0
$964$	98387.8	3.28720
$965$	−14169.8	−0.472685
$966$	0	0
$967$	−8723.55	−0.290104	−0.145052	−	0.989424i	$-0.546335\pi$
$967$	−8723.55	−0.290104	−0.145052	+	0.989424i	$0.546335\pi$
$968$	−5601.02	−0.185975
$969$	0	0
$970$	91860.5	3.04068
$971$	−23959.2	−0.791850	−0.395925	−	0.918283i	$-0.629576\pi$
$971$	−23959.2	−0.791850	−0.395925	+	0.918283i	$0.629576\pi$
$972$	0	0
$973$	−10633.5	−0.350353
$974$	−66459.8	−2.18636
$975$	0	0
$976$	−50286.9	−1.64923
$977$	−3709.07	−0.121457	−0.0607286	−	0.998154i	$-0.519342\pi$
$977$	−3709.07	−0.121457	−0.0607286	+	0.998154i	$0.519342\pi$
$978$	0	0
$979$	−6193.44	−0.202189
$980$	−17242.2	−0.562023
$981$	0	0
$982$	−103516.	−3.36388
$983$	29542.7	0.958561	0.479281	−	0.877662i	$-0.340898\pi$
$983$	29542.7	0.958561	0.479281	+	0.877662i	$0.340898\pi$
$984$	0	0
$985$	−4890.00	−0.158181
$986$	13214.5	0.426812
$987$	0	0
$988$	18.5893	0.000598587 0
$989$	−7695.30	−0.247418
$990$	0	0
$991$	−37600.9	−1.20528	−0.602640	−	0.798013i	$-0.705885\pi$
$991$	−37600.9	−1.20528	−0.602640	+	0.798013i	$0.705885\pi$
$992$	19103.4	0.611424
$993$	0	0
$994$	−39363.4	−1.25607
$995$	13855.6	0.441460
$996$	0	0
$997$	24918.4	0.791547	0.395774	−	0.918348i	$-0.370476\pi$
$997$	24918.4	0.791547	0.395774	+	0.918348i	$0.370476\pi$
$998$	58331.6	1.85016
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.q.1.1		5
3.2	odd	2		231.4.a.j.1.5	✓	5
21.20	even	2		1617.4.a.o.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.j.1.5	✓	5	3.2	odd	2
693.4.a.q.1.1		5	1.1	even	1	trivial
1617.4.a.o.1.5		5	21.20	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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q-5.02173 q^{2} +17.2178 q^{4} -20.4371 q^{5} -7.00000 q^{7} -46.2894 q^{8} +102.630 q^{10} +11.0000 q^{11} -0.0115563 q^{13} +35.1521 q^{14} +94.7106 q^{16} +9.52800 q^{17} -93.4260 q^{19} -351.882 q^{20} -55.2391 q^{22} -99.9552 q^{23} +292.676 q^{25} +0.0580324 q^{26} -120.525 q^{28} -276.183 q^{29} -181.425 q^{31} -105.296 q^{32} -47.8471 q^{34} +143.060 q^{35} -404.794 q^{37} +469.160 q^{38} +946.022 q^{40} +27.8263 q^{41} +76.9875 q^{43} +189.396 q^{44} +501.948 q^{46} +136.979 q^{47} +49.0000 q^{49} -1469.74 q^{50} -0.198973 q^{52} -170.391 q^{53} -224.808 q^{55} +324.026 q^{56} +1386.92 q^{58} +585.593 q^{59} -530.953 q^{61} +911.070 q^{62} -228.916 q^{64} +0.236177 q^{65} -354.952 q^{67} +164.051 q^{68} -718.408 q^{70} -1119.80 q^{71} -785.389 q^{73} +2032.77 q^{74} -1608.59 q^{76} -77.0000 q^{77} -937.289 q^{79} -1935.61 q^{80} -139.736 q^{82} -471.454 q^{83} -194.725 q^{85} -386.611 q^{86} -509.183 q^{88} -563.040 q^{89} +0.0808938 q^{91} -1721.01 q^{92} -687.874 q^{94} +1909.36 q^{95} +895.067 q^{97} -246.065 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 21 q^{4} - 7 q^{5} - 35 q^{7} + 12 q^{8} + 113 q^{10} + 55 q^{11} + 23 q^{13} - 7 q^{14} + 281 q^{16} + 102 q^{17} - 155 q^{19} - 291 q^{20} + 11 q^{22} - 192 q^{23} + 394 q^{25} - 41 q^{26}+ \cdots + 49 q^{98}+O(q^{100}) \) 5 * q + q^2 + 21 * q^4 - 7 * q^5 - 35 * q^7 + 12 * q^8 + 113 * q^10 + 55 * q^11 + 23 * q^13 - 7 * q^14 + 281 * q^16 + 102 * q^17 - 155 * q^19 - 291 * q^20 + 11 * q^22 - 192 * q^23 + 394 * q^25 - 41 * q^26 - 147 * q^28 - 591 * q^29 + 366 * q^31 + 87 * q^32 + 594 * q^34 + 49 * q^35 + 259 * q^37 - 175 * q^38 + 1890 * q^40 + 104 * q^41 + 224 * q^43 + 231 * q^44 + 1416 * q^46 + 453 * q^47 + 245 * q^49 - 1350 * q^50 + 815 * q^52 - 1032 * q^53 - 77 * q^55 - 84 * q^56 + 293 * q^58 + 517 * q^59 + 958 * q^61 + 3030 * q^62 + 516 * q^64 + 197 * q^65 + 361 * q^67 + 1680 * q^68 - 791 * q^70 - 548 * q^71 - 1035 * q^73 + 2555 * q^74 - 5201 * q^76 - 385 * q^77 + 776 * q^79 + 1273 * q^80 + 376 * q^82 + 1974 * q^83 + 1838 * q^85 - 1224 * q^86 + 132 * q^88 - 2710 * q^89 - 161 * q^91 + 2834 * q^92 - 563 * q^94 + 355 * q^95 + 1988 * q^97 + 49 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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