LMFDB - Embedded newform 1617.4.a.o.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1617,4,Mod(1,1617)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1617, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1617.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1617 = 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1617.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$95.4060884793$
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			1.5
Root			$-5.02173$ of defining polynomial
Character	$\chi$	$=$	1617.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.02173 q^{2} +3.00000 q^{3} +17.2178 q^{4} -20.4371 q^{5} +15.0652 q^{6} +46.2894 q^{8} +9.00000 q^{9} -102.630 q^{10} -11.0000 q^{11} +51.6534 q^{12} +0.0115563 q^{13} -61.3113 q^{15} +94.7106 q^{16} +9.52800 q^{17} +45.1956 q^{18} +93.4260 q^{19} -351.882 q^{20} -55.2391 q^{22} +99.9552 q^{23} +138.868 q^{24} +292.676 q^{25} +0.0580324 q^{26} +27.0000 q^{27} +276.183 q^{29} -307.889 q^{30} +181.425 q^{31} +105.296 q^{32} -33.0000 q^{33} +47.8471 q^{34} +154.960 q^{36} -404.794 q^{37} +469.160 q^{38} +0.0346688 q^{39} -946.022 q^{40} +27.8263 q^{41} +76.9875 q^{43} -189.396 q^{44} -183.934 q^{45} +501.948 q^{46} +136.979 q^{47} +284.132 q^{48} +1469.74 q^{50} +28.5840 q^{51} +0.198973 q^{52} +170.391 q^{53} +135.587 q^{54} +224.808 q^{55} +280.278 q^{57} +1386.92 q^{58} +585.593 q^{59} -1055.65 q^{60} +530.953 q^{61} +911.070 q^{62} -228.916 q^{64} -0.236177 q^{65} -165.717 q^{66} -354.952 q^{67} +164.051 q^{68} +299.866 q^{69} +1119.80 q^{71} +416.605 q^{72} +785.389 q^{73} -2032.77 q^{74} +878.027 q^{75} +1608.59 q^{76} +0.174097 q^{78} -937.289 q^{79} -1935.61 q^{80} +81.0000 q^{81} +139.736 q^{82} -471.454 q^{83} -194.725 q^{85} +386.611 q^{86} +828.548 q^{87} -509.183 q^{88} -563.040 q^{89} -923.668 q^{90} +1721.01 q^{92} +544.276 q^{93} +687.874 q^{94} -1909.36 q^{95} +315.888 q^{96} -895.067 q^{97} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} + 15 q^{3} + 21 q^{4} - 7 q^{5} - 3 q^{6} - 12 q^{8} + 45 q^{9} - 113 q^{10} - 55 q^{11} + 63 q^{12} - 23 q^{13} - 21 q^{15} + 281 q^{16} + 102 q^{17} - 9 q^{18} + 155 q^{19} - 291 q^{20} + 11 q^{22}+ \cdots - 495 q^{99}+O(q^{100}) $ 5 * q - q^2 + 15 * q^3 + 21 * q^4 - 7 * q^5 - 3 * q^6 - 12 * q^8 + 45 * q^9 - 113 * q^10 - 55 * q^11 + 63 * q^12 - 23 * q^13 - 21 * q^15 + 281 * q^16 + 102 * q^17 - 9 * q^18 + 155 * q^19 - 291 * q^20 + 11 * q^22 + 192 * q^23 - 36 * q^24 + 394 * q^25 - 41 * q^26 + 135 * q^27 + 591 * q^29 - 339 * q^30 - 366 * q^31 - 87 * q^32 - 165 * q^33 - 594 * q^34 + 189 * q^36 + 259 * q^37 - 175 * q^38 - 69 * q^39 - 1890 * q^40 + 104 * q^41 + 224 * q^43 - 231 * q^44 - 63 * q^45 + 1416 * q^46 + 453 * q^47 + 843 * q^48 + 1350 * q^50 + 306 * q^51 - 815 * q^52 + 1032 * q^53 - 27 * q^54 + 77 * q^55 + 465 * q^57 + 293 * q^58 + 517 * q^59 - 873 * q^60 - 958 * q^61 + 3030 * q^62 + 516 * q^64 - 197 * q^65 + 33 * q^66 + 361 * q^67 + 1680 * q^68 + 576 * q^69 + 548 * q^71 - 108 * q^72 + 1035 * q^73 - 2555 * q^74 + 1182 * q^75 + 5201 * q^76 - 123 * q^78 + 776 * q^79 + 1273 * q^80 + 405 * q^81 - 376 * q^82 + 1974 * q^83 + 1838 * q^85 + 1224 * q^86 + 1773 * q^87 + 132 * q^88 - 2710 * q^89 - 1017 * q^90 - 2834 * q^92 - 1098 * q^93 + 563 * q^94 - 355 * q^95 - 261 * q^96 - 1988 * q^97 - 495 * q^99

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	$-0.152284\pi$
$2$	5.02173	1.77545	0.887726	+	0.460373i	$0.152284\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	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	$0.867004\pi$
$6$	15.0652	1.02506
$7$	0	0
$7$	0	0
$8$	46.2894	2.04572
$9$	9.00000	0.333333
$10$	−102.630	−3.24544
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	51.6534	1.24259
$13$	0.0115563	0.000246548 0	0.000123274	−	1.00000i	$-0.499961\pi$
$13$	0.0115563	0.000246548 0	0.000123274		1.00000i	$0.499961\pi$
$14$	0	0
$15$	−61.3113	−1.05537
$16$	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	$0.478349\pi$
$18$	45.1956	0.591817
$19$	93.4260	1.12807	0.564037	−	0.825750i	$-0.309248\pi$
$19$	93.4260	1.12807	0.564037	+	0.825750i	$0.309248\pi$
$20$	−351.882	−3.93416
$21$	0	0
$22$	−55.2391	−0.535319
$23$	99.9552	0.906178	0.453089	−	0.891465i	$-0.350322\pi$
$23$	99.9552	0.906178	0.453089	+	0.891465i	$0.350322\pi$
$24$	138.868	1.18110
$25$	292.676	2.34140
$26$	0.0580324	0.000437735 0
$27$	27.0000	0.192450
$28$	0	0
$29$	276.183	1.76848	0.884239	−	0.467035i	$-0.154678\pi$
$29$	276.183	1.76848	0.884239	+	0.467035i	$0.154678\pi$
$30$	−307.889	−1.87375
$31$	181.425	1.05113	0.525564	−	0.850754i	$-0.323854\pi$
$31$	181.425	1.05113	0.525564	+	0.850754i	$0.323854\pi$
$32$	105.296	0.581684
$33$	−33.0000	−0.174078
$34$	47.8471	0.241344
$35$	0	0
$36$	154.960	0.717409
$37$	−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
$39$	0.0346688	0.000142345 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
$45$	−183.934	−0.609317
$46$	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	$0.431820\pi$
$48$	284.132	0.854393
$49$	0	0
$50$	1469.74	4.15705
$51$	28.5840	0.0784816
$52$	0.198973	0.000530628 0
$53$	170.391	0.441604	0.220802	−	0.975319i	$-0.429133\pi$
$53$	170.391	0.441604	0.220802	+	0.975319i	$0.429133\pi$
$54$	135.587	0.341686
$55$	224.808	0.551148
$56$	0	0
$57$	280.278	0.651293
$58$	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	$0.276406\pi$
$60$	−1055.65	−2.27139
$61$	530.953	1.11445	0.557226	−	0.830361i	$-0.311866\pi$
$61$	530.953	1.11445	0.557226	+	0.830361i	$0.311866\pi$
$62$	911.070	1.86623
$63$	0	0
$64$	−228.916	−0.447101
$65$	−0.236177	−0.000450678 0
$66$	−165.717	−0.309066
$67$	−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
$69$	299.866	0.523182
$70$	0	0
$71$	1119.80	1.87177	0.935887	−	0.352301i	$-0.114601\pi$
$71$	1119.80	1.87177	0.935887	+	0.352301i	$0.114601\pi$
$72$	416.605	0.681907
$73$	785.389	1.25922	0.629608	−	0.776913i	$-0.283215\pi$
$73$	785.389	1.25922	0.629608	+	0.776913i	$0.283215\pi$
$74$	−2032.77	−3.19331
$75$	878.027	1.35181
$76$	1608.59	2.42787
$77$	0	0
$78$	0.174097	0.000252726 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
$81$	81.0000	0.111111
$82$	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
$87$	828.548	1.02103
$88$	−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	$0.608835\pi$
$90$	−923.668	−1.08181
$91$	0	0
$92$	1721.01	1.95030
$93$	544.276	0.606869
$94$	687.874	0.754775
$95$	−1909.36	−2.06206
$96$	315.888	0.335835
$97$	−895.067	−0.936910	−0.468455	−	0.883487i	$-0.655189\pi$
$97$	−895.067	−0.936910	−0.468455	+	0.883487i	$0.655189\pi$
$98$	0	0
$99$	−99.0000	−0.100504
$100$	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	$0.435414\pi$
$102$	143.541	0.139340
$103$	−207.199	−0.198213	−0.0991065	−	0.995077i	$-0.531598\pi$
$103$	−207.199	−0.198213	−0.0991065	+	0.995077i	$0.531598\pi$
$104$	0.534932	0.000504369 0
$105$	0	0
$106$	855.658	0.784046
$107$	−1475.53	−1.33313	−0.666566	−	0.745446i	$-0.732236\pi$
$107$	−1475.53	−1.33313	−0.666566	+	0.745446i	$0.732236\pi$
$108$	464.881	0.414196
$109$	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
$111$	−1214.38	−1.03842
$112$	0	0
$113$	−1307.55	−1.08853	−0.544265	−	0.838913i	$-0.683192\pi$
$113$	−1307.55	−1.08853	−0.544265	+	0.838913i	$0.683192\pi$
$114$	1407.48	1.15634
$115$	−2042.80	−1.65645
$116$	4755.26	3.80616
$117$	0.104006	8.21828e−5 0
$118$	2940.69	2.29417
$119$	0	0
$120$	−2838.07	−2.15899
$121$	121.000	0.0909091
$122$	2666.31	1.97866
$123$	83.4788	0.0611954
$124$	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
$129$	230.962	0.157636
$130$	−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	$0.316984\pi$
$132$	−568.188	−0.374655
$133$	0	0
$134$	−1782.47	−1.14912
$135$	−551.802	−0.351789
$136$	441.045	0.278083
$137$	2939.97	1.83342	0.916711	−	0.399552i	$-0.130834\pi$
$137$	2939.97	1.83342	0.916711	+	0.399552i	$0.130834\pi$
$138$	1505.85	0.928884
$139$	−1519.07	−0.926948	−0.463474	−	0.886111i	$-0.653397\pi$
$139$	−1519.07	−0.926948	−0.463474	+	0.886111i	$0.653397\pi$
$140$	0	0
$141$	410.938	0.245441
$142$	5623.34	3.32324
$143$	−0.127119	−7.43371e−5 0
$144$	852.395	0.493284
$145$	−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	$-0.519599\pi$
$149$	−223.829	−0.123066	−0.0615329	+	0.998105i	$0.519599\pi$
$150$	4409.22	2.40007
$151$	−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
$153$	85.7520	0.0453114
$154$	0	0
$155$	−3707.81	−1.92141
$156$	0.596920	0.000306358 0
$157$	−2358.25	−1.19878	−0.599392	−	0.800456i	$-0.704591\pi$
$157$	−2358.25	−1.19878	−0.599392	+	0.800456i	$0.704591\pi$
$158$	−4706.81	−2.36996
$159$	511.173	0.254960
$160$	−2151.95	−1.06329
$161$	0	0
$162$	406.760	0.197272
$163$	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
$165$	674.425	0.318205
$166$	−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
$171$	840.834	0.376024
$172$	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	$0.359549\pi$
$174$	4160.75	1.81279
$175$	0	0
$176$	−1041.82	−0.446192
$177$	1756.78	0.746031
$178$	−2827.44	−1.19059
$179$	3511.90	1.46643	0.733217	−	0.679995i	$-0.238018\pi$
$179$	3511.90	1.46643	0.733217	+	0.679995i	$0.238018\pi$
$180$	−3166.94	−1.31139
$181$	−3058.76	−1.25611	−0.628055	−	0.778169i	$-0.716149\pi$
$181$	−3058.76	−1.25611	−0.628055	+	0.778169i	$0.716149\pi$
$182$	0	0
$183$	1592.86	0.643430
$184$	4626.87	1.85379
$185$	8272.83	3.28773
$186$	2733.21	1.07747
$187$	−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	$-0.164929\pi$
$191$	4586.39	1.73749	0.868743	+	0.495264i	$0.164929\pi$
$192$	−686.747	−0.258134
$193$	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
$195$	−0.708530	−0.000260199 0
$196$	0	0
$197$	−239.271	−0.0865346	−0.0432673	−	0.999064i	$-0.513777\pi$
$197$	−239.271	−0.0865346	−0.0432673	+	0.999064i	$0.513777\pi$
$198$	−497.152	−0.178440
$199$	677.963	0.241505	0.120753	−	0.992683i	$-0.461469\pi$
$199$	677.963	0.241505	0.120753	+	0.992683i	$0.461469\pi$
$200$	13547.8	4.78986
$201$	−1064.86	−0.373677
$202$	2054.33	0.715555
$203$	0	0
$204$	492.154	0.168910
$205$	−568.688	−0.193751
$206$	−1040.50	−0.351917
$207$	899.597	0.302059
$208$	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
$213$	3359.40	1.08067
$214$	−7409.73	−2.36691
$215$	−1573.40	−0.499093
$216$	1249.81	0.393699
$217$	0	0
$218$	7320.76	2.27442
$219$	2356.17	0.727009
$220$	3870.71	1.18620
$221$	0.110108	3.35143e−5 0
$222$	−6098.31	−1.84366
$223$	1229.51	0.369210	0.184605	−	0.982813i	$-0.440899\pi$
$223$	1229.51	0.369210	0.184605	+	0.982813i	$0.440899\pi$
$224$	0	0
$225$	2634.08	0.780468
$226$	−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	$0.324548\pi$
$228$	4825.77	1.40173
$229$	−4732.43	−1.36562	−0.682812	−	0.730594i	$-0.739243\pi$
$229$	−4732.43	−1.36562	−0.682812	+	0.730594i	$0.739243\pi$
$230$	−10258.4	−2.94094
$231$	0	0
$232$	12784.3	3.61781
$233$	−3018.63	−0.848744	−0.424372	−	0.905488i	$-0.639505\pi$
$233$	−3018.63	−0.848744	−0.424372	+	0.905488i	$0.639505\pi$
$234$	0.522292	0.000145912 0
$235$	−2799.46	−0.777093
$236$	10082.6	2.78103
$237$	−2811.87	−0.770677
$238$	0	0
$239$	−2915.73	−0.789133	−0.394567	−	0.918867i	$-0.629105\pi$
$239$	−2915.73	−0.789133	−0.394567	+	0.918867i	$0.629105\pi$
$240$	−5806.83	−1.56179
$241$	−5714.30	−1.52735	−0.763674	−	0.645603i	$-0.776606\pi$
$241$	−5714.30	−1.52735	−0.763674	+	0.645603i	$0.776606\pi$
$242$	607.630	0.161405
$243$	243.000	0.0641500
$244$	9141.86	2.39855
$245$	0	0
$246$	419.208	0.108649
$247$	1.07965	0.000278125 0
$248$	8398.07	2.15031
$249$	−1414.36	−0.359966
$250$	−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
$255$	−584.174	−0.143460
$256$	−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	$0.389433\pi$
$258$	1159.83	0.279876
$259$	0	0
$260$	−4.06644	−0.000969962 0
$261$	2485.64	0.589492
$262$	8189.03	1.93099
$263$	2178.21	0.510699	0.255350	−	0.966849i	$-0.417809\pi$
$263$	2178.21	0.510699	0.255350	+	0.966849i	$0.417809\pi$
$264$	−1527.55	−0.356114
$265$	−3482.30	−0.807230
$266$	0	0
$267$	−1689.12	−0.387163
$268$	−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	$0.651252\pi$
$270$	−2771.00	−0.624585
$271$	2166.33	0.485591	0.242796	−	0.970077i	$-0.421936\pi$
$271$	2166.33	0.485591	0.242796	+	0.970077i	$0.421936\pi$
$272$	902.402	0.201162
$273$	0	0
$274$	14763.8	3.25515
$275$	−3219.43	−0.705960
$276$	5163.03	1.12601
$277$	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
$279$	1632.83	0.350376
$280$	0	0
$281$	−1200.61	−0.254883	−0.127442	−	0.991846i	$-0.540677\pi$
$281$	−1200.61	−0.254883	−0.127442	+	0.991846i	$0.540677\pi$
$282$	2063.62	0.435769
$283$	−6393.02	−1.34285	−0.671423	−	0.741074i	$-0.734317\pi$
$283$	−6393.02	−1.34285	−0.671423	+	0.741074i	$0.734317\pi$
$284$	19280.5	4.02848
$285$	−5728.07	−1.19053
$286$	−0.638357	−0.000131982 0
$287$	0	0
$288$	947.664	0.193895
$289$	−4822.22	−0.981522
$290$	−28344.6	−5.73948
$291$	−2685.20	−0.540925
$292$	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
$297$	−297.000	−0.0580259
$298$	−1124.01	−0.218497
$299$	1.15511	0.000223417 0
$300$	15117.7	2.90940
$301$	0	0
$302$	−15923.8	−3.03415
$303$	1227.26	0.232688
$304$	8848.43	1.66938
$305$	−10851.2	−2.03717
$306$	430.624	0.0804481
$307$	2792.13	0.519073	0.259536	−	0.965733i	$-0.416430\pi$
$307$	2792.13	0.519073	0.259536	+	0.965733i	$0.416430\pi$
$308$	0	0
$309$	−621.597	−0.114438
$310$	−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	$0.662522\pi$
$312$	1.60480	0.000291198 0
$313$	9998.72	1.80563	0.902813	−	0.430033i	$-0.141498\pi$
$313$	9998.72	1.80563	0.902813	+	0.430033i	$0.141498\pi$
$314$	−11842.5	−2.12838
$315$	0	0
$316$	−16138.1	−2.87290
$317$	1043.98	0.184971	0.0924855	−	0.995714i	$-0.470519\pi$
$317$	1043.98	0.184971	0.0924855	+	0.995714i	$0.470519\pi$
$318$	2566.97	0.452669
$319$	−3038.01	−0.533216
$320$	4678.38	0.817279
$321$	−4426.60	−0.769684
$322$	0	0
$323$	890.163	0.153344
$324$	1394.64	0.239136
$325$	3.38223	0.000577269 0
$326$	11473.7	1.94929
$327$	4373.45	0.739609
$328$	1288.06	0.216833
$329$	0	0
$330$	3386.78	0.564958
$331$	−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
$333$	−3643.15	−0.599530
$334$	−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
$339$	−3922.65	−0.628464
$340$	−3352.74	−0.534787
$341$	−1995.68	−0.316927
$342$	4222.44	0.667613
$343$	0	0
$344$	3563.70	0.558552
$345$	−6128.39	−0.956351
$346$	9759.87	1.51646
$347$	4459.48	0.689906	0.344953	−	0.938620i	$-0.387895\pi$
$347$	4459.48	0.689906	0.344953	+	0.938620i	$0.387895\pi$
$348$	14265.8	2.19749
$349$	−7902.74	−1.21210	−0.606051	−	0.795426i	$-0.707247\pi$
$349$	−7902.74	−1.21210	−0.606051	+	0.795426i	$0.707247\pi$
$350$	0	0
$351$	0.312019	4.74483e−5 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	$0.472146\pi$
$354$	8822.07	1.32454
$355$	−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	$-0.425284\pi$
$359$	3164.04	0.465158	0.232579	+	0.972577i	$0.425284\pi$
$360$	−8514.20	−1.24649
$361$	1869.42	0.272549
$362$	−15360.3	−2.23016
$363$	363.000	0.0524864
$364$	0	0
$365$	−16051.1	−2.30179
$366$	7998.92	1.14238
$367$	−6586.11	−0.936763	−0.468382	−	0.883526i	$-0.655163\pi$
$367$	−6586.11	−0.936763	−0.468382	+	0.883526i	$0.655163\pi$
$368$	9466.81	1.34101
$369$	250.436	0.0353312
$370$	41543.9	5.83721
$371$	0	0
$372$	9371.24	1.30612
$373$	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
$375$	−10280.4	−1.41568
$376$	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
$381$	6468.51	0.869795
$382$	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	$0.420507\pi$
$384$	−5975.77	−0.794140
$385$	0	0
$386$	3481.74	0.459109
$387$	692.887	0.0910115
$388$	−15411.1	−2.01644
$389$	3365.36	0.438638	0.219319	−	0.975653i	$-0.429616\pi$
$389$	3365.36	0.438638	0.219319	+	0.975653i	$0.429616\pi$
$390$	−3.55805	−0.000461971 0
$391$	952.373	0.123180
$392$	0	0
$393$	4892.15	0.627930
$394$	−1201.55	−0.153638
$395$	19155.5	2.44004
$396$	−1704.56	−0.216307
$397$	10914.1	1.37976	0.689879	−	0.723925i	$-0.257664\pi$
$397$	10914.1	1.37976	0.689879	+	0.723925i	$0.257664\pi$
$398$	3404.55	0.428781
$399$	0	0
$400$	27719.5	3.46493
$401$	4980.45	0.620228	0.310114	−	0.950699i	$-0.399633\pi$
$401$	4980.45	0.620228	0.310114	+	0.950699i	$0.399633\pi$
$402$	−5347.42	−0.663446
$403$	2.09660	0.000259154 0
$404$	7043.59	0.867406
$405$	−1655.41	−0.203106
$406$	0	0
$407$	4452.74	0.542295
$408$	1323.14	0.160551
$409$	9813.28	1.18639	0.593197	−	0.805057i	$-0.297865\pi$
$409$	9813.28	1.18639	0.593197	+	0.805057i	$0.297865\pi$
$410$	−2855.80	−0.343995
$411$	8819.92	1.05853
$412$	−3567.51	−0.426599
$413$	0	0
$414$	4517.54	0.536292
$415$	9635.15	1.13969
$416$	1.21683	0.000143413 0
$417$	−4557.21	−0.535174
$418$	−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
$423$	1232.81	0.141706
$424$	7887.29	0.903398
$425$	2788.61	0.318277
$426$	16870.0	1.91867
$427$	0	0
$428$	−25405.4	−2.86920
$429$	−0.381356	−4.29186e−5 0
$430$	−7901.20	−0.886116
$431$	1181.89	0.132087	0.0660437	−	0.997817i	$-0.478962\pi$
$431$	1181.89	0.132087	0.0660437	+	0.997817i	$0.478962\pi$
$432$	2557.19	0.284798
$433$	7028.94	0.780113	0.390057	−	0.920791i	$-0.372455\pi$
$433$	7028.94	0.780113	0.390057	+	0.920791i	$0.372455\pi$
$434$	0	0
$435$	−16933.1	−1.86639
$436$	25100.4	2.75709
$437$	9338.41	1.02224
$438$	11832.0	1.29077
$439$	−5758.63	−0.626069	−0.313034	−	0.949742i	$-0.601346\pi$
$439$	−5758.63	−0.626069	−0.313034	+	0.949742i	$0.601346\pi$
$440$	10406.2	1.12750
$441$	0	0
$442$	0.552933	5.95030e−5 0
$443$	−13270.7	−1.42327	−0.711636	−	0.702548i	$-0.752045\pi$
$443$	−13270.7	−1.42327	−0.711636	+	0.702548i	$0.752045\pi$
$444$	−20909.0	−2.23491
$445$	11506.9	1.22580
$446$	6174.25	0.655514
$447$	−671.487	−0.0710520
$448$	0	0
$449$	8609.07	0.904871	0.452435	−	0.891797i	$-0.350555\pi$
$449$	8609.07	0.904871	0.452435	+	0.891797i	$0.350555\pi$
$450$	13227.6	1.38568
$451$	−306.089	−0.0319582
$452$	−22513.2	−2.34276
$453$	−9512.95	−0.986661
$454$	17989.3	1.85964
$455$	0	0
$456$	12973.9	1.33237
$457$	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
$459$	257.256	0.0261605
$460$	−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
$465$	−11123.4	−1.10933
$466$	−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	$0.585723\pi$
$468$	1.79076	0.000176876 0
$469$	0	0
$470$	−14058.2	−1.37969
$471$	−7074.76	−0.692118
$472$	27106.7	2.64341
$473$	−846.862	−0.0823230
$474$	−14120.4	−1.36830
$475$	27343.5	2.64128
$476$	0	0
$477$	1533.52	0.147201
$478$	−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	$0.445080\pi$
$480$	−6455.84	−0.613891
$481$	−4.67791	−0.000443439 0
$482$	−28695.7	−2.71173
$483$	0	0
$484$	2083.36	0.195657
$485$	18292.6	1.71263
$486$	1220.28	0.113895
$487$	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
$489$	6854.41	0.633880
$490$	0	0
$491$	−20613.6	−1.89466	−0.947330	−	0.320259i	$-0.896230\pi$
$491$	−20613.6	−1.89466	−0.947330	+	0.320259i	$0.896230\pi$
$492$	1437.32	0.131706
$493$	2631.47	0.240396
$494$	5.42174	0.000493797 0
$495$	2023.27	0.183716
$496$	17182.9	1.55551
$497$	0	0
$498$	−7102.54	−0.639102
$499$	−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
$501$	−6991.44	−0.623462
$502$	−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
$507$	−6591.00	−0.577350
$508$	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	$0.733370\pi$
$510$	−2933.57	−0.254707
$511$	0	0
$512$	−25100.1	−2.16656
$513$	2522.50	0.217098
$514$	14086.0	1.20877
$515$	4234.55	0.362324
$516$	3976.67	0.339269
$517$	−1506.77	−0.128178
$518$	0	0
$519$	5830.58	0.493129
$520$	−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	$0.214056\pi$
$522$	12482.2	1.04662
$523$	1484.42	0.124109	0.0620545	−	0.998073i	$-0.480235\pi$
$523$	1484.42	0.124109	0.0620545	+	0.998073i	$0.480235\pi$
$524$	28077.4	2.34078
$525$	0	0
$526$	10938.4	0.906721
$527$	1728.62	0.142884
$528$	−3125.45	−0.257609
$529$	−2175.96	−0.178841
$530$	−17487.2	−1.43320
$531$	5270.33	0.430721
$532$	0	0
$533$	0.321567	2.61325e−5 0
$534$	−8482.31	−0.687389
$535$	30155.6	2.43690
$536$	−16430.5	−1.32405
$537$	10535.7	0.846646
$538$	−20272.0	−1.62451
$539$	0	0
$540$	−9500.82	−0.757130
$541$	−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
$543$	−9176.29	−0.725216
$544$	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
$549$	4778.58	0.371484
$550$	−16167.1	−1.25340
$551$	25802.6	1.99497
$552$	13880.6	1.07029
$553$	0	0
$554$	11732.8	0.899783
$555$	24818.5	1.89817
$556$	−26155.0	−1.99500
$557$	17492.7	1.33068	0.665340	−	0.746541i	$-0.268287\pi$
$557$	17492.7	1.33068	0.665340	+	0.746541i	$0.268287\pi$
$558$	8199.63	0.622075
$559$	0.889687	6.73162e−5 0
$560$	0	0
$561$	−314.424	−0.0236631
$562$	−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	$0.359977\pi$
$564$	7075.46	0.528246
$565$	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	$-0.340661\pi$
$569$	13028.1	0.959868	0.479934	+	0.877305i	$0.340661\pi$
$570$	−28764.9	−2.11373
$571$	−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
$573$	13759.2	1.00314
$574$	0	0
$575$	29254.4	2.12173
$576$	−2060.24	−0.149034
$577$	24253.5	1.74989	0.874944	−	0.484224i	$-0.160898\pi$
$577$	24253.5	1.74989	0.874944	+	0.484224i	$0.160898\pi$
$578$	−24215.9	−1.74264
$579$	2080.00	0.149295
$580$	−97183.8	−6.95748
$581$	0	0
$582$	−13484.4	−0.960387
$583$	−1874.30	−0.133149
$584$	36355.2	2.57601
$585$	−2.12559	−0.000150226 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
$591$	−717.812	−0.0499608
$592$	−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	$0.480072\pi$
$594$	−1491.45	−0.103022
$595$	0	0
$596$	−3853.85	−0.264865
$597$	2033.89	0.139433
$598$	5.80064	0.000396665 0
$599$	−16213.0	−1.10592	−0.552958	−	0.833209i	$-0.686501\pi$
$599$	−16213.0	−1.10592	−0.552958	+	0.833209i	$0.686501\pi$
$600$	40643.3	2.76543
$601$	−28306.2	−1.92119	−0.960594	−	0.277957i	$-0.910343\pi$
$601$	−28306.2	−1.92119	−0.960594	+	0.277957i	$0.910343\pi$
$602$	0	0
$603$	−3194.57	−0.215743
$604$	−54597.4	−3.67804
$605$	−2472.89	−0.166177
$606$	6162.99	0.413126
$607$	−13513.0	−0.903587	−0.451794	−	0.892122i	$-0.649216\pi$
$607$	−13513.0	−0.903587	−0.451794	+	0.892122i	$0.649216\pi$
$608$	9837.39	0.656182
$609$	0	0
$610$	−54491.6	−3.61689
$611$	1.58297	0.000104812 0
$612$	1476.46	0.0975203
$613$	−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
$615$	−1706.07	−0.111862
$616$	0	0
$617$	−6054.73	−0.395063	−0.197532	−	0.980297i	$-0.563293\pi$
$617$	−6054.73	−0.395063	−0.197532	+	0.980297i	$0.563293\pi$
$618$	−3121.50	−0.203180
$619$	−2193.31	−0.142418	−0.0712089	−	0.997461i	$-0.522686\pi$
$619$	−2193.31	−0.142418	−0.0712089	+	0.997461i	$0.522686\pi$
$620$	−63840.4	−4.13531
$621$	2698.79	0.174394
$622$	−26918.5	−1.73526
$623$	0	0
$624$	3.28350	0.000210649 0
$625$	33449.5	2.14077
$626$	50210.9	3.20580
$627$	−3083.06	−0.196372
$628$	−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
$633$	2209.07	0.138709
$634$	5242.59	0.328407
$635$	−44065.9	−2.75386
$636$	8801.28	0.548732
$637$	0	0
$638$	−15256.1	−0.946699
$639$	10078.2	0.623925
$640$	40709.1	2.51433
$641$	−1276.44	−0.0786524	−0.0393262	−	0.999226i	$-0.512521\pi$
$641$	−1276.44	−0.0786524	−0.0393262	+	0.999226i	$0.512521\pi$
$642$	−22229.2	−1.36654
$643$	−19610.9	−1.20276	−0.601382	−	0.798961i	$-0.705383\pi$
$643$	−19610.9	−1.20276	−0.601382	+	0.798961i	$0.705383\pi$
$644$	0	0
$645$	−4720.20	−0.288152
$646$	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	$0.486926\pi$
$648$	3749.44	0.227302
$649$	−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	$-0.525757\pi$
$653$	−2697.61	−0.161662	−0.0808312	+	0.996728i	$0.525757\pi$
$654$	21962.3	1.31314
$655$	−33327.2	−1.98809
$656$	2635.44	0.156855
$657$	7068.50	0.419739
$658$	0	0
$659$	−22130.4	−1.30816	−0.654080	−	0.756426i	$-0.726944\pi$
$659$	−22130.4	−1.30816	−0.654080	+	0.756426i	$0.726944\pi$
$660$	11612.1	0.684850
$661$	12681.1	0.746199	0.373100	−	0.927791i	$-0.378295\pi$
$661$	12681.1	0.746199	0.373100	+	0.927791i	$0.378295\pi$
$662$	−4149.68	−0.243628
$663$	0.330324	1.93495e−5 0
$664$	−21823.3	−1.27546
$665$	0	0
$666$	−18294.9	−1.06444
$667$	27605.9	1.60256
$668$	−40125.8	−2.32412
$669$	3688.52	0.213164
$670$	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
$675$	7902.24	0.450604
$676$	−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	$0.554515\pi$
$678$	−19698.5	−1.11581
$679$	0	0
$680$	−9013.69	−0.508323
$681$	10746.8	0.604728
$682$	−10021.8	−0.562688
$683$	15747.3	0.882217	0.441109	−	0.897454i	$-0.354585\pi$
$683$	15747.3	0.882217	0.441109	+	0.897454i	$0.354585\pi$
$684$	14477.3	0.809290
$685$	−60084.5	−3.35140
$686$	0	0
$687$	−14197.3	−0.788443
$688$	7291.53	0.404051
$689$	1.96908	0.000108877 0
$690$	−30775.1	−1.69795
$691$	206.020	0.0113421	0.00567105	−	0.999984i	$-0.498195\pi$
$691$	206.020	0.0113421	0.00567105	+	0.999984i	$0.498195\pi$
$692$	33463.3	1.83827
$693$	0	0
$694$	22394.3	1.22489
$695$	31045.4	1.69442
$696$	38353.0	2.08875
$697$	265.129	0.0144081
$698$	−39685.4	−2.15203
$699$	−9055.90	−0.490022
$700$	0	0
$701$	−2413.37	−0.130031	−0.0650154	−	0.997884i	$-0.520710\pi$
$701$	−2413.37	−0.130031	−0.0650154	+	0.997884i	$0.520710\pi$
$702$	1.56688	8.42421e−5 0
$703$	−37818.3	−2.02894
$704$	2518.07	0.134806
$705$	−8398.39	−0.448655
$706$	5821.37	0.310326
$707$	0	0
$708$	30247.9	1.60563
$709$	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
$711$	−8435.60	−0.444950
$712$	−26062.8	−1.37183
$713$	18134.4	0.952509
$714$	0	0
$715$	2.59794	0.000135885 0
$716$	60467.2	3.15610
$717$	−8747.19	−0.455606
$718$	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	$0.638401\pi$
$720$	−17420.5	−0.901699
$721$	0	0
$722$	9387.71	0.483898
$723$	−17142.9	−0.881814
$724$	−52665.2	−2.70344
$725$	80831.9	4.14072
$726$	1822.89	0.0931870
$727$	247.150	0.0126084	0.00630418	−	0.999980i	$-0.497993\pi$
$727$	247.150	0.0126084	0.00630418	+	0.999980i	$0.497993\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−80604.3	−4.08671
$731$	733.537	0.0371147
$732$	27425.6	1.38481
$733$	−27692.4	−1.39542	−0.697711	−	0.716380i	$-0.745798\pi$
$733$	−27692.4	−1.39542	−0.697711	+	0.716380i	$0.745798\pi$
$734$	−33073.7	−1.66318
$735$	0	0
$736$	10524.9	0.527109
$737$	3904.47	0.195147
$738$	1257.62	0.0627287
$739$	−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
$741$	3.23896	0.000160575 0
$742$	0	0
$743$	−10527.8	−0.519822	−0.259911	−	0.965633i	$-0.583693\pi$
$743$	−10527.8	−0.519822	−0.259911	+	0.965633i	$0.583693\pi$
$744$	25194.2	1.24148
$745$	4574.42	0.224958
$746$	6078.49	0.298324
$747$	−4243.08	−0.207826
$748$	−1804.56	−0.0882104
$749$	0	0
$750$	−51625.5	−2.51346
$751$	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
$753$	−3143.14	−0.152115
$754$	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
$759$	−3298.52	−0.157745
$760$	−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	$0.488011\pi$
$762$	32483.2	1.54428
$763$	0	0
$764$	78967.6	3.73946
$765$	−1752.52	−0.0828269
$766$	18605.3	0.877596
$767$	6.76726	0.000318581 0
$768$	−24514.7	−1.15182
$769$	24477.4	1.14783	0.573913	−	0.818916i	$-0.305425\pi$
$769$	24477.4	1.14783	0.573913	+	0.818916i	$0.305425\pi$
$770$	0	0
$771$	8415.04	0.393074
$772$	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	$0.116334\pi$
$774$	3479.50	0.161586
$775$	53098.8	2.46111
$776$	−41432.1	−1.91666
$777$	0	0
$778$	16899.9	0.778781
$779$	2599.70	0.119568
$780$	−12.1993	−0.000560008 0
$781$	−12317.8	−0.564361
$782$	4782.56	0.218701
$783$	7456.93	0.340344
$784$	0	0
$785$	48195.9	2.19132
$786$	24567.1	1.11486
$787$	38095.3	1.72548	0.862739	−	0.505650i	$-0.168747\pi$
$787$	38095.3	1.72548	0.862739	+	0.505650i	$0.168747\pi$
$788$	−4119.72	−0.186242
$789$	6534.62	0.294852
$790$	96193.7	4.33217
$791$	0	0
$792$	−4582.65	−0.205603
$793$	6.13583	0.000274767 0
$794$	54807.8	2.44969
$795$	−10446.9	−0.466054
$796$	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
$801$	−5067.36	−0.223529
$802$	25010.5	1.10119
$803$	−8639.28	−0.379668
$804$	−18334.5	−0.804238
$805$	0	0
$806$	10.5286	0.000460115 0
$807$	−12110.5	−0.528267
$808$	18936.4	0.824481
$809$	−39595.8	−1.72078	−0.860392	−	0.509633i	$-0.829781\pi$
$809$	−39595.8	−1.72078	−0.860392	+	0.509633i	$0.829781\pi$
$810$	−8313.01	−0.360604
$811$	−7072.51	−0.306226	−0.153113	−	0.988209i	$-0.548930\pi$
$811$	−7072.51	−0.306226	−0.153113	+	0.988209i	$0.548930\pi$
$812$	0	0
$813$	6498.99	0.280356
$814$	22360.5	0.962818
$815$	−46694.8	−2.00693
$816$	2707.21	0.116141
$817$	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	$-0.474461\pi$
$821$	3770.85	0.160297	0.0801484	+	0.996783i	$0.474461\pi$
$822$	44291.3	1.87936
$823$	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
$825$	−9658.29	−0.407586
$826$	0	0
$827$	13104.1	0.550995	0.275497	−	0.961302i	$-0.411157\pi$
$827$	13104.1	0.550995	0.275497	+	0.961302i	$0.411157\pi$
$828$	15489.1	0.650100
$829$	−25672.5	−1.07556	−0.537782	−	0.843084i	$-0.680738\pi$
$829$	−25672.5	−1.07556	−0.537782	+	0.843084i	$0.680738\pi$
$830$	48385.2	2.02346
$831$	7009.23	0.292596
$832$	−2.64541	−0.000110232 0
$833$	0	0
$834$	−22885.1	−0.950174
$835$	47628.3	1.97395
$836$	−17694.5	−0.732030
$837$	4898.48	0.202290
$838$	−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
$843$	−3601.82	−0.147157
$844$	12678.4	0.517073
$845$	44900.3	1.82795
$846$	6190.87	0.251592
$847$	0	0
$848$	16137.8	0.653508
$849$	−19179.1	−0.775293
$850$	14003.7	0.565085
$851$	−40461.3	−1.62984
$852$	57841.6	2.32584
$853$	38411.5	1.54183	0.770917	−	0.636936i	$-0.219798\pi$
$853$	38411.5	1.54183	0.770917	+	0.636936i	$0.219798\pi$
$854$	0	0
$855$	−17184.2	−0.687354
$856$	−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	$0.702736\pi$
$858$	−1.91507	−7.61998e−5 0
$859$	6603.57	0.262294	0.131147	−	0.991363i	$-0.458134\pi$
$859$	6603.57	0.262294	0.131147	+	0.991363i	$0.458134\pi$
$860$	−27090.5	−1.07416
$861$	0	0
$862$	5935.14	0.234515
$863$	−35177.7	−1.38756	−0.693780	−	0.720187i	$-0.744056\pi$
$863$	−35177.7	−1.38756	−0.693780	+	0.720187i	$0.744056\pi$
$864$	2842.99	0.111945
$865$	−39720.1	−1.56130
$866$	35297.4	1.38505
$867$	−14466.7	−0.566682
$868$	0	0
$869$	10310.2	0.402473
$870$	−85033.7	−3.31369
$871$	−4.10191	−0.000159573 0
$872$	67481.4	2.62065
$873$	−8055.60	−0.312303
$874$	46895.0	1.81493
$875$	0	0
$876$	40568.0	1.56469
$877$	−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
$879$	18113.5	0.695056
$880$	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
$885$	−35903.5	−1.36371
$886$	−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	$0.286596\pi$
$888$	−56213.1	−2.12431
$889$	0	0
$890$	57784.7	2.17634
$891$	−891.000	−0.0335013
$892$	21169.4	0.794624
$893$	12797.4	0.479563
$894$	−3372.03	−0.126149
$895$	−71773.1	−2.68057
$896$	0	0
$897$	3.46532	0.000128990 0
$898$	43232.4	1.60655
$899$	50106.5	1.85890
$900$	45353.1	1.67974
$901$	1623.48	0.0600290
$902$	−1537.10	−0.0567403
$903$	0	0
$904$	−60525.7	−2.22683
$905$	62512.3	2.29611
$906$	−47771.5	−1.75177
$907$	−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
$909$	3681.79	0.134342
$910$	0	0
$911$	11001.3	0.400098	0.200049	−	0.979786i	$-0.435890\pi$
$911$	11001.3	0.400098	0.200049	+	0.979786i	$0.435890\pi$
$912$	26545.3	0.963818
$913$	5185.99	0.187986
$914$	5586.65	0.202177
$915$	−32553.5	−1.17616
$916$	−81482.1	−2.93913
$917$	0	0
$918$	1291.87	0.0464467
$919$	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
$921$	8376.39	0.299687
$922$	−1879.64	−0.0671396
$923$	12.9407	0.000461483 0
$924$	0	0
$925$	−118473.	−4.21122
$926$	13716.7	0.486782
$927$	−1864.79	−0.0660710
$928$	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	$0.764305\pi$
$930$	−55858.9	−1.96955
$931$	0	0
$932$	−51974.3	−1.82669
$933$	−16081.2	−0.564282
$934$	−26967.6	−0.944761
$935$	2141.97	0.0749198
$936$	4.81439	0.000168123 0
$937$	−30060.9	−1.04808	−0.524038	−	0.851695i	$-0.675575\pi$
$937$	−30060.9	−1.04808	−0.524038	+	0.851695i	$0.675575\pi$
$938$	0	0
$939$	29996.2	1.04248
$940$	−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	$0.286938\pi$
$942$	−35527.5	−1.22882
$943$	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	$-0.537877\pi$
$947$	−6919.22	−0.237428	−0.118714	+	0.992928i	$0.537877\pi$
$948$	−48414.2	−1.65867
$949$	9.07616	0.000310458 0
$950$	137312.	4.68946
$951$	3131.94	0.106793
$952$	0	0
$953$	−29103.4	−0.989248	−0.494624	−	0.869107i	$-0.664694\pi$
$953$	−29103.4	−0.989248	−0.494624	+	0.869107i	$0.664694\pi$
$954$	7700.92	0.261349
$955$	−93732.6	−3.17604
$956$	−50202.5	−1.69839
$957$	−9114.03	−0.307852
$958$	18076.3	0.609624
$959$	0	0
$960$	14035.1	0.471856
$961$	3124.15	0.104869
$962$	−23.4912	−0.000787304 0
$963$	−13279.8	−0.444377
$964$	−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
$969$	2670.49	0.0885330
$970$	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	$0.629576\pi$
$972$	4183.93	0.138065
$973$	0	0
$974$	66459.8	2.18636
$975$	10.1467	0.000333287 0
$976$	50286.9	1.64923
$977$	3709.07	0.121457	0.0607286	−	0.998154i	$-0.480658\pi$
$977$	3709.07	0.121457	0.0607286	+	0.998154i	$0.480658\pi$
$978$	34421.0	1.12542
$979$	6193.44	0.202189
$980$	0	0
$981$	13120.3	0.427013
$982$	−103516.	−3.36388
$983$	29542.7	0.958561	0.479281	−	0.877662i	$-0.340898\pi$
$983$	29542.7	0.958561	0.479281	+	0.877662i	$0.340898\pi$
$984$	3864.18	0.125189
$985$	4890.00	0.158181
$986$	13214.5	0.426812
$987$	0	0
$988$	18.5893	0.000598587 0
$989$	7695.30	0.247418
$990$	10160.3	0.326179
$991$	−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
$993$	−2479.03	−0.0792242
$994$	0	0
$995$	−13855.6	−0.441460
$996$	−24352.2	−0.774728
$997$	−24918.4	−0.791547	−0.395774	−	0.918348i	$-0.629524\pi$
$997$	−24918.4	−0.791547	−0.395774	+	0.918348i	$0.629524\pi$
$998$	−58331.6	−1.85016
$999$	−10929.4	−0.346139

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.j.1.5	✓	5	7.6	odd	2
693.4.a.q.1.1		5	21.20	even	2
1617.4.a.o.1.5		5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1617.a (trivial)

\(f(q)\)	\(=\)	\(q+5.02173 q^{2} +3.00000 q^{3} +17.2178 q^{4} -20.4371 q^{5} +15.0652 q^{6} +46.2894 q^{8} +9.00000 q^{9} -102.630 q^{10} -11.0000 q^{11} +51.6534 q^{12} +0.0115563 q^{13} -61.3113 q^{15} +94.7106 q^{16} +9.52800 q^{17} +45.1956 q^{18} +93.4260 q^{19} -351.882 q^{20} -55.2391 q^{22} +99.9552 q^{23} +138.868 q^{24} +292.676 q^{25} +0.0580324 q^{26} +27.0000 q^{27} +276.183 q^{29} -307.889 q^{30} +181.425 q^{31} +105.296 q^{32} -33.0000 q^{33} +47.8471 q^{34} +154.960 q^{36} -404.794 q^{37} +469.160 q^{38} +0.0346688 q^{39} -946.022 q^{40} +27.8263 q^{41} +76.9875 q^{43} -189.396 q^{44} -183.934 q^{45} +501.948 q^{46} +136.979 q^{47} +284.132 q^{48} +1469.74 q^{50} +28.5840 q^{51} +0.198973 q^{52} +170.391 q^{53} +135.587 q^{54} +224.808 q^{55} +280.278 q^{57} +1386.92 q^{58} +585.593 q^{59} -1055.65 q^{60} +530.953 q^{61} +911.070 q^{62} -228.916 q^{64} -0.236177 q^{65} -165.717 q^{66} -354.952 q^{67} +164.051 q^{68} +299.866 q^{69} +1119.80 q^{71} +416.605 q^{72} +785.389 q^{73} -2032.77 q^{74} +878.027 q^{75} +1608.59 q^{76} +0.174097 q^{78} -937.289 q^{79} -1935.61 q^{80} +81.0000 q^{81} +139.736 q^{82} -471.454 q^{83} -194.725 q^{85} +386.611 q^{86} +828.548 q^{87} -509.183 q^{88} -563.040 q^{89} -923.668 q^{90} +1721.01 q^{92} +544.276 q^{93} +687.874 q^{94} -1909.36 q^{95} +315.888 q^{96} -895.067 q^{97} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + 15 q^{3} + 21 q^{4} - 7 q^{5} - 3 q^{6} - 12 q^{8} + 45 q^{9} - 113 q^{10} - 55 q^{11} + 63 q^{12} - 23 q^{13} - 21 q^{15} + 281 q^{16} + 102 q^{17} - 9 q^{18} + 155 q^{19} - 291 q^{20} + 11 q^{22}+ \cdots - 495 q^{99}+O(q^{100}) \) 5 * q - q^2 + 15 * q^3 + 21 * q^4 - 7 * q^5 - 3 * q^6 - 12 * q^8 + 45 * q^9 - 113 * q^10 - 55 * q^11 + 63 * q^12 - 23 * q^13 - 21 * q^15 + 281 * q^16 + 102 * q^17 - 9 * q^18 + 155 * q^19 - 291 * q^20 + 11 * q^22 + 192 * q^23 - 36 * q^24 + 394 * q^25 - 41 * q^26 + 135 * q^27 + 591 * q^29 - 339 * q^30 - 366 * q^31 - 87 * q^32 - 165 * q^33 - 594 * q^34 + 189 * q^36 + 259 * q^37 - 175 * q^38 - 69 * q^39 - 1890 * q^40 + 104 * q^41 + 224 * q^43 - 231 * q^44 - 63 * q^45 + 1416 * q^46 + 453 * q^47 + 843 * q^48 + 1350 * q^50 + 306 * q^51 - 815 * q^52 + 1032 * q^53 - 27 * q^54 + 77 * q^55 + 465 * q^57 + 293 * q^58 + 517 * q^59 - 873 * q^60 - 958 * q^61 + 3030 * q^62 + 516 * q^64 - 197 * q^65 + 33 * q^66 + 361 * q^67 + 1680 * q^68 + 576 * q^69 + 548 * q^71 - 108 * q^72 + 1035 * q^73 - 2555 * q^74 + 1182 * q^75 + 5201 * q^76 - 123 * q^78 + 776 * q^79 + 1273 * q^80 + 405 * q^81 - 376 * q^82 + 1974 * q^83 + 1838 * q^85 + 1224 * q^86 + 1773 * q^87 + 132 * q^88 - 2710 * q^89 - 1017 * q^90 - 2834 * q^92 - 1098 * q^93 + 563 * q^94 - 355 * q^95 - 261 * q^96 - 1988 * q^97 - 495 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more