LMFDB - Embedded newform 1725.4.a.m.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1725,4,Mod(1,1725)] 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("1725.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1725, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1725 = 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1725.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,6,2,0,6,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$101.778294760$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1725.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.82843 q^{2} +3.00000 q^{3} +6.65685 q^{4} +11.4853 q^{6} +16.8284 q^{7} -5.14214 q^{8} +9.00000 q^{9} -55.7990 q^{11} +19.9706 q^{12} +46.9706 q^{13} +64.4264 q^{14} -72.9411 q^{16} -62.1421 q^{17} +34.4558 q^{18} -141.882 q^{19} +50.4853 q^{21} -213.622 q^{22} -23.0000 q^{23} -15.4264 q^{24} +179.823 q^{26} +27.0000 q^{27} +112.024 q^{28} -288.676 q^{29} +68.7351 q^{31} -238.113 q^{32} -167.397 q^{33} -237.907 q^{34} +59.9117 q^{36} -179.622 q^{37} -543.186 q^{38} +140.912 q^{39} -71.5391 q^{41} +193.279 q^{42} +159.088 q^{43} -371.446 q^{44} -88.0538 q^{46} +272.902 q^{47} -218.823 q^{48} -59.8040 q^{49} -186.426 q^{51} +312.676 q^{52} +12.2843 q^{53} +103.368 q^{54} -86.5341 q^{56} -425.647 q^{57} -1105.18 q^{58} +426.891 q^{59} -243.407 q^{61} +263.147 q^{62} +151.456 q^{63} -328.068 q^{64} -640.867 q^{66} +81.0496 q^{67} -413.671 q^{68} -69.0000 q^{69} +696.930 q^{71} -46.2792 q^{72} -568.392 q^{73} -687.671 q^{74} -944.489 q^{76} -939.009 q^{77} +539.470 q^{78} +719.123 q^{79} +81.0000 q^{81} -273.882 q^{82} -1336.80 q^{83} +336.073 q^{84} +609.058 q^{86} -866.029 q^{87} +286.926 q^{88} -337.935 q^{89} +790.441 q^{91} -153.108 q^{92} +206.205 q^{93} +1044.78 q^{94} -714.338 q^{96} +1419.76 q^{97} -228.955 q^{98} -502.191 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 6 q^{3} + 2 q^{4} + 6 q^{6} + 28 q^{7} + 18 q^{8} + 18 q^{9} - 72 q^{11} + 6 q^{12} + 60 q^{13} + 44 q^{14} - 78 q^{16} - 96 q^{17} + 18 q^{18} - 148 q^{19} + 84 q^{21} - 184 q^{22} - 46 q^{23}+ \cdots - 648 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 6 * q^3 + 2 * q^4 + 6 * q^6 + 28 * q^7 + 18 * q^8 + 18 * q^9 - 72 * q^11 + 6 * q^12 + 60 * q^13 + 44 * q^14 - 78 * q^16 - 96 * q^17 + 18 * q^18 - 148 * q^19 + 84 * q^21 - 184 * q^22 - 46 * q^23 + 54 * q^24 + 156 * q^26 + 54 * q^27 + 60 * q^28 - 204 * q^29 - 168 * q^31 - 414 * q^32 - 216 * q^33 - 176 * q^34 + 18 * q^36 - 116 * q^37 - 532 * q^38 + 180 * q^39 + 4 * q^41 + 132 * q^42 + 420 * q^43 - 296 * q^44 - 46 * q^46 + 48 * q^47 - 234 * q^48 - 278 * q^49 - 288 * q^51 + 252 * q^52 - 32 * q^53 + 54 * q^54 + 172 * q^56 - 444 * q^57 - 1260 * q^58 - 40 * q^59 - 764 * q^61 + 696 * q^62 + 252 * q^63 + 34 * q^64 - 552 * q^66 + 988 * q^67 - 256 * q^68 - 138 * q^69 - 224 * q^71 + 162 * q^72 - 820 * q^73 - 804 * q^74 - 916 * q^76 - 1120 * q^77 + 468 * q^78 + 1772 * q^79 + 162 * q^81 - 412 * q^82 - 1480 * q^83 + 180 * q^84 + 132 * q^86 - 612 * q^87 - 88 * q^88 + 744 * q^89 + 936 * q^91 - 46 * q^92 - 504 * q^93 + 1456 * q^94 - 1242 * q^96 + 260 * q^97 + 170 * q^98 - 648 * 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 $

$2$	3.82843	1.35355	0.676777	−	0.736188i	$-0.263376\pi$
$2$	3.82843	1.35355	0.676777	+	0.736188i	$0.263376\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	6.65685	0.832107
$5$	0	0
$5$	0	0
$6$	11.4853	0.781474
$7$	16.8284	0.908650	0.454325	−	0.890836i	$-0.349881\pi$
$7$	16.8284	0.908650	0.454325	+	0.890836i	$0.349881\pi$
$8$	−5.14214	−0.227252
$9$	9.00000	0.333333
$10$	0	0
$11$	−55.7990	−1.52946	−0.764729	−	0.644353i	$-0.777127\pi$
$11$	−55.7990	−1.52946	−0.764729	+	0.644353i	$0.777127\pi$
$12$	19.9706	0.480417
$13$	46.9706	1.00210	0.501050	−	0.865419i	$-0.332947\pi$
$13$	46.9706	1.00210	0.501050	+	0.865419i	$0.332947\pi$
$14$	64.4264	1.22991
$15$	0	0
$16$	−72.9411	−1.13971
$17$	−62.1421	−0.886570	−0.443285	−	0.896381i	$-0.646187\pi$
$17$	−62.1421	−0.886570	−0.443285	+	0.896381i	$0.646187\pi$
$18$	34.4558	0.451184
$19$	−141.882	−1.71316	−0.856579	−	0.516015i	$-0.827415\pi$
$19$	−141.882	−1.71316	−0.856579	+	0.516015i	$0.827415\pi$
$20$	0	0
$21$	50.4853	0.524609
$22$	−213.622	−2.07020
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−15.4264	−0.131204
$25$	0	0
$26$	179.823	1.35639
$27$	27.0000	0.192450
$28$	112.024	0.756094
$29$	−288.676	−1.84848	−0.924238	−	0.381816i	$-0.875299\pi$
$29$	−288.676	−1.84848	−0.924238	+	0.381816i	$0.875299\pi$
$30$	0	0
$31$	68.7351	0.398232	0.199116	−	0.979976i	$-0.436193\pi$
$31$	68.7351	0.398232	0.199116	+	0.979976i	$0.436193\pi$
$32$	−238.113	−1.31540
$33$	−167.397	−0.883032
$34$	−237.907	−1.20002
$35$	0	0
$36$	59.9117	0.277369
$37$	−179.622	−0.798101	−0.399050	−	0.916929i	$-0.630660\pi$
$37$	−179.622	−0.798101	−0.399050	+	0.916929i	$0.630660\pi$
$38$	−543.186	−2.31885
$39$	140.912	0.578562
$40$	0	0
$41$	−71.5391	−0.272501	−0.136250	−	0.990674i	$-0.543505\pi$
$41$	−71.5391	−0.272501	−0.136250	+	0.990674i	$0.543505\pi$
$42$	193.279	0.710086
$43$	159.088	0.564203	0.282102	−	0.959385i	$-0.408968\pi$
$43$	159.088	0.564203	0.282102	+	0.959385i	$0.408968\pi$
$44$	−371.446	−1.27267
$45$	0	0
$46$	−88.0538	−0.282235
$47$	272.902	0.846953	0.423476	−	0.905907i	$-0.360810\pi$
$47$	272.902	0.846953	0.423476	+	0.905907i	$0.360810\pi$
$48$	−218.823	−0.658009
$49$	−59.8040	−0.174356
$50$	0	0
$51$	−186.426	−0.511861
$52$	312.676	0.833854
$53$	12.2843	0.0318373	0.0159186	−	0.999873i	$-0.494933\pi$
$53$	12.2843	0.0318373	0.0159186	+	0.999873i	$0.494933\pi$
$54$	103.368	0.260491
$55$	0	0
$56$	−86.5341	−0.206493
$57$	−425.647	−0.989093
$58$	−1105.18	−2.50201
$59$	426.891	0.941975	0.470988	−	0.882140i	$-0.343898\pi$
$59$	426.891	0.941975	0.470988	+	0.882140i	$0.343898\pi$
$60$	0	0
$61$	−243.407	−0.510903	−0.255451	−	0.966822i	$-0.582224\pi$
$61$	−243.407	−0.510903	−0.255451	+	0.966822i	$0.582224\pi$
$62$	263.147	0.539028
$63$	151.456	0.302883
$64$	−328.068	−0.640758
$65$	0	0
$66$	−640.867	−1.19523
$67$	81.0496	0.147788	0.0738940	−	0.997266i	$-0.476457\pi$
$67$	81.0496	0.147788	0.0738940	+	0.997266i	$0.476457\pi$
$68$	−413.671	−0.737721
$69$	−69.0000	−0.120386
$70$	0	0
$71$	696.930	1.16494	0.582468	−	0.812854i	$-0.302087\pi$
$71$	696.930	1.16494	0.582468	+	0.812854i	$0.302087\pi$
$72$	−46.2792	−0.0757508
$73$	−568.392	−0.911305	−0.455652	−	0.890158i	$-0.650594\pi$
$73$	−568.392	−0.911305	−0.455652	+	0.890158i	$0.650594\pi$
$74$	−687.671	−1.08027
$75$	0	0
$76$	−944.489	−1.42553
$77$	−939.009	−1.38974
$78$	539.470	0.783115
$79$	719.123	1.02415	0.512074	−	0.858942i	$-0.328877\pi$
$79$	719.123	1.02415	0.512074	+	0.858942i	$0.328877\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−273.882	−0.368844
$83$	−1336.80	−1.76786	−0.883932	−	0.467616i	$-0.845113\pi$
$83$	−1336.80	−1.76786	−0.883932	+	0.467616i	$0.845113\pi$
$84$	336.073	0.436531
$85$	0	0
$86$	609.058	0.763679
$87$	−866.029	−1.06722
$88$	286.926	0.347573
$89$	−337.935	−0.402484	−0.201242	−	0.979542i	$-0.564498\pi$
$89$	−337.935	−0.402484	−0.201242	+	0.979542i	$0.564498\pi$
$90$	0	0
$91$	790.441	0.910557
$92$	−153.108	−0.173506
$93$	206.205	0.229919
$94$	1044.78	1.14640
$95$	0	0
$96$	−714.338	−0.759446
$97$	1419.76	1.48613	0.743067	−	0.669217i	$-0.233370\pi$
$97$	1419.76	1.48613	0.743067	+	0.669217i	$0.233370\pi$
$98$	−228.955	−0.236000
$99$	−502.191	−0.509819
$100$	0	0
$101$	471.275	0.464293	0.232147	−	0.972681i	$-0.425425\pi$
$101$	471.275	0.464293	0.232147	+	0.972681i	$0.425425\pi$
$102$	−713.720	−0.692831
$103$	1129.00	1.08004	0.540019	−	0.841653i	$-0.318417\pi$
$103$	1129.00	1.08004	0.540019	+	0.841653i	$0.318417\pi$
$104$	−241.529	−0.227729
$105$	0	0
$106$	47.0294	0.0430934
$107$	−203.543	−0.183900	−0.0919499	−	0.995764i	$-0.529310\pi$
$107$	−203.543	−0.183900	−0.0919499	+	0.995764i	$0.529310\pi$
$108$	179.735	0.160139
$109$	−1938.34	−1.70329	−0.851646	−	0.524117i	$-0.824395\pi$
$109$	−1938.34	−1.70329	−0.851646	+	0.524117i	$0.824395\pi$
$110$	0	0
$111$	−538.867	−0.460784
$112$	−1227.48	−1.03559
$113$	−1663.79	−1.38510	−0.692548	−	0.721371i	$-0.743512\pi$
$113$	−1663.79	−1.38510	−0.692548	+	0.721371i	$0.743512\pi$
$114$	−1629.56	−1.33879
$115$	0	0
$116$	−1921.68	−1.53813
$117$	422.735	0.334033
$118$	1634.32	1.27501
$119$	−1045.75	−0.805581
$120$	0	0
$121$	1782.53	1.33924
$122$	−931.866	−0.691534
$123$	−214.617	−0.157328
$124$	457.559	0.331371
$125$	0	0
$126$	579.838	0.409969
$127$	734.676	0.513323	0.256661	−	0.966501i	$-0.417378\pi$
$127$	734.676	0.513323	0.256661	+	0.966501i	$0.417378\pi$
$128$	648.917	0.448099
$129$	477.265	0.325743
$130$	0	0
$131$	−2855.44	−1.90443	−0.952217	−	0.305421i	$-0.901203\pi$
$131$	−2855.44	−1.90443	−0.952217	+	0.305421i	$0.901203\pi$
$132$	−1114.34	−0.734777
$133$	−2387.66	−1.55666
$134$	310.293	0.200039
$135$	0	0
$136$	319.543	0.201475
$137$	−1430.56	−0.892127	−0.446063	−	0.895001i	$-0.647174\pi$
$137$	−1430.56	−0.892127	−0.446063	+	0.895001i	$0.647174\pi$
$138$	−264.161	−0.162949
$139$	288.410	0.175990	0.0879951	−	0.996121i	$-0.471954\pi$
$139$	288.410	0.175990	0.0879951	+	0.996121i	$0.471954\pi$
$140$	0	0
$141$	818.705	0.488989
$142$	2668.15	1.57680
$143$	−2620.91	−1.53267
$144$	−656.470	−0.379902
$145$	0	0
$146$	−2176.05	−1.23350
$147$	−179.412	−0.100664
$148$	−1195.72	−0.664105
$149$	−1937.08	−1.06504	−0.532522	−	0.846416i	$-0.678756\pi$
$149$	−1937.08	−1.06504	−0.532522	+	0.846416i	$0.678756\pi$
$150$	0	0
$151$	1847.98	0.995935	0.497967	−	0.867196i	$-0.334080\pi$
$151$	1847.98	0.995935	0.497967	+	0.867196i	$0.334080\pi$
$152$	729.578	0.389320
$153$	−559.279	−0.295523
$154$	−3594.93	−1.88109
$155$	0	0
$156$	938.029	0.481426
$157$	−51.2489	−0.0260516	−0.0130258	−	0.999915i	$-0.504146\pi$
$157$	−51.2489	−0.0260516	−0.0130258	+	0.999915i	$0.504146\pi$
$158$	2753.11	1.38624
$159$	36.8528	0.0183812
$160$	0	0
$161$	−387.054	−0.189467
$162$	310.103	0.150395
$163$	607.960	0.292142	0.146071	−	0.989274i	$-0.453337\pi$
$163$	607.960	0.292142	0.146071	+	0.989274i	$0.453337\pi$
$164$	−476.225	−0.226750
$165$	0	0
$166$	−5117.83	−2.39290
$167$	3253.06	1.50736	0.753682	−	0.657240i	$-0.228276\pi$
$167$	3253.06	1.50736	0.753682	+	0.657240i	$0.228276\pi$
$168$	−259.602	−0.119219
$169$	9.23376	0.00420290
$170$	0	0
$171$	−1276.94	−0.571053
$172$	1059.03	0.469477
$173$	−1975.75	−0.868288	−0.434144	−	0.900843i	$-0.642949\pi$
$173$	−1975.75	−0.868288	−0.434144	+	0.900843i	$0.642949\pi$
$174$	−3315.53	−1.44454
$175$	0	0
$176$	4070.04	1.74313
$177$	1280.67	0.543850
$178$	−1293.76	−0.544783
$179$	4261.58	1.77947	0.889737	−	0.456474i	$-0.150888\pi$
$179$	4261.58	1.77947	0.889737	+	0.456474i	$0.150888\pi$
$180$	0	0
$181$	2043.29	0.839095	0.419548	−	0.907733i	$-0.362189\pi$
$181$	2043.29	0.839095	0.419548	+	0.907733i	$0.362189\pi$
$182$	3026.14	1.23249
$183$	−730.221	−0.294970
$184$	118.269	0.0473854
$185$	0	0
$186$	789.442	0.311208
$187$	3467.47	1.35597
$188$	1816.67	0.704755
$189$	454.368	0.174870
$190$	0	0
$191$	−73.1001	−0.0276929	−0.0138464	−	0.999904i	$-0.504408\pi$
$191$	−73.1001	−0.0276929	−0.0138464	+	0.999904i	$0.504408\pi$
$192$	−984.204	−0.369942
$193$	227.901	0.0849982	0.0424991	−	0.999097i	$-0.486468\pi$
$193$	227.901	0.0849982	0.0424991	+	0.999097i	$0.486468\pi$
$194$	5435.46	2.01156
$195$	0	0
$196$	−398.107	−0.145083
$197$	−577.781	−0.208960	−0.104480	−	0.994527i	$-0.533318\pi$
$197$	−577.781	−0.208960	−0.104480	+	0.994527i	$0.533318\pi$
$198$	−1922.60	−0.690067
$199$	−4569.01	−1.62758	−0.813790	−	0.581159i	$-0.802599\pi$
$199$	−4569.01	−1.62758	−0.813790	+	0.581159i	$0.802599\pi$
$200$	0	0
$201$	243.149	0.0853254
$202$	1804.24	0.628446
$203$	−4857.97	−1.67962
$204$	−1241.01	−0.425923
$205$	0	0
$206$	4322.31	1.46189
$207$	−207.000	−0.0695048
$208$	−3426.09	−1.14210
$209$	7916.89	2.62020
$210$	0	0
$211$	−1816.71	−0.592736	−0.296368	−	0.955074i	$-0.595775\pi$
$211$	−1816.71	−0.592736	−0.296368	+	0.955074i	$0.595775\pi$
$212$	81.7746	0.0264920
$213$	2090.79	0.672576
$214$	−779.251	−0.248918
$215$	0	0
$216$	−138.838	−0.0437348
$217$	1156.70	0.361853
$218$	−7420.78	−2.30550
$219$	−1705.18	−0.526142
$220$	0	0
$221$	−2918.85	−0.888431
$222$	−2063.01	−0.623695
$223$	−3494.47	−1.04936	−0.524679	−	0.851300i	$-0.675815\pi$
$223$	−3494.47	−1.04936	−0.524679	+	0.851300i	$0.675815\pi$
$224$	−4007.06	−1.19524
$225$	0	0
$226$	−6369.69	−1.87480
$227$	−1044.51	−0.305402	−0.152701	−	0.988272i	$-0.548797\pi$
$227$	−1044.51	−0.305402	−0.152701	+	0.988272i	$0.548797\pi$
$228$	−2833.47	−0.823031
$229$	2141.54	0.617979	0.308989	−	0.951065i	$-0.400009\pi$
$229$	2141.54	0.617979	0.308989	+	0.951065i	$0.400009\pi$
$230$	0	0
$231$	−2817.03	−0.802367
$232$	1484.41	0.420071
$233$	−716.802	−0.201542	−0.100771	−	0.994910i	$-0.532131\pi$
$233$	−716.802	−0.201542	−0.100771	+	0.994910i	$0.532131\pi$
$234$	1618.41	0.452132
$235$	0	0
$236$	2841.75	0.783824
$237$	2157.37	0.591292
$238$	−4003.59	−1.09040
$239$	−3515.96	−0.951585	−0.475792	−	0.879558i	$-0.657839\pi$
$239$	−3515.96	−0.951585	−0.475792	+	0.879558i	$0.657839\pi$
$240$	0	0
$241$	5684.64	1.51942	0.759710	−	0.650262i	$-0.225341\pi$
$241$	5684.64	1.51942	0.759710	+	0.650262i	$0.225341\pi$
$242$	6824.28	1.81273
$243$	243.000	0.0641500
$244$	−1620.33	−0.425126
$245$	0	0
$246$	−821.647	−0.212952
$247$	−6664.29	−1.71676
$248$	−353.445	−0.0904991
$249$	−4010.39	−1.02068
$250$	0	0
$251$	−6946.90	−1.74695	−0.873475	−	0.486870i	$-0.838139\pi$
$251$	−6946.90	−1.74695	−0.873475	+	0.486870i	$0.838139\pi$
$252$	1008.22	0.252031
$253$	1283.38	0.318914
$254$	2812.65	0.694810
$255$	0	0
$256$	5108.88	1.24728
$257$	1776.30	0.431139	0.215569	−	0.976489i	$-0.430839\pi$
$257$	1776.30	0.431139	0.215569	+	0.976489i	$0.430839\pi$
$258$	1827.17	0.440910
$259$	−3022.76	−0.725194
$260$	0	0
$261$	−2598.09	−0.616159
$262$	−10931.8	−2.57775
$263$	−112.405	−0.0263544	−0.0131772	−	0.999913i	$-0.504195\pi$
$263$	−112.405	−0.0263544	−0.0131772	+	0.999913i	$0.504195\pi$
$264$	860.778	0.200671
$265$	0	0
$266$	−9140.96	−2.10702
$267$	−1013.81	−0.232374
$268$	539.536	0.122975
$269$	−5916.43	−1.34101	−0.670504	−	0.741906i	$-0.733922\pi$
$269$	−5916.43	−1.34101	−0.670504	+	0.741906i	$0.733922\pi$
$270$	0	0
$271$	−1011.96	−0.226836	−0.113418	−	0.993547i	$-0.536180\pi$
$271$	−1011.96	−0.226836	−0.113418	+	0.993547i	$0.536180\pi$
$272$	4532.72	1.01043
$273$	2371.32	0.525710
$274$	−5476.81	−1.20754
$275$	0	0
$276$	−459.323	−0.100174
$277$	3632.42	0.787910	0.393955	−	0.919130i	$-0.371107\pi$
$277$	3632.42	0.787910	0.393955	+	0.919130i	$0.371107\pi$
$278$	1104.16	0.238212
$279$	618.616	0.132744
$280$	0	0
$281$	4382.47	0.930378	0.465189	−	0.885211i	$-0.345986\pi$
$281$	4382.47	0.930378	0.465189	+	0.885211i	$0.345986\pi$
$282$	3134.35	0.661872
$283$	2692.29	0.565513	0.282757	−	0.959192i	$-0.408751\pi$
$283$	2692.29	0.565513	0.282757	+	0.959192i	$0.408751\pi$
$284$	4639.36	0.969350
$285$	0	0
$286$	−10034.0	−2.07455
$287$	−1203.89	−0.247608
$288$	−2143.01	−0.438466
$289$	−1051.35	−0.213995
$290$	0	0
$291$	4259.29	0.858020
$292$	−3783.70	−0.758303
$293$	−4023.60	−0.802256	−0.401128	−	0.916022i	$-0.631382\pi$
$293$	−4023.60	−0.802256	−0.401128	+	0.916022i	$0.631382\pi$
$294$	−686.866	−0.136255
$295$	0	0
$296$	923.643	0.181370
$297$	−1506.57	−0.294344
$298$	−7415.96	−1.44159
$299$	−1080.32	−0.208952
$300$	0	0
$301$	2677.21	0.512663
$302$	7074.84	1.34805
$303$	1413.83	0.268060
$304$	10349.1	1.95250
$305$	0	0
$306$	−2141.16	−0.400006
$307$	−5195.21	−0.965818	−0.482909	−	0.875670i	$-0.660420\pi$
$307$	−5195.21	−0.965818	−0.482909	+	0.875670i	$0.660420\pi$
$308$	−6250.85	−1.15641
$309$	3387.01	0.623561
$310$	0	0
$311$	5440.59	0.991986	0.495993	−	0.868327i	$-0.334804\pi$
$311$	5440.59	0.991986	0.495993	+	0.868327i	$0.334804\pi$
$312$	−724.587	−0.131480
$313$	7390.51	1.33462	0.667310	−	0.744780i	$-0.267445\pi$
$313$	7390.51	1.33462	0.667310	+	0.744780i	$0.267445\pi$
$314$	−196.203	−0.0352623
$315$	0	0
$316$	4787.10	0.852200
$317$	8785.02	1.55652	0.778258	−	0.627944i	$-0.216103\pi$
$317$	8785.02	1.55652	0.778258	+	0.627944i	$0.216103\pi$
$318$	141.088	0.0248800
$319$	16107.8	2.82717
$320$	0	0
$321$	−610.630	−0.106175
$322$	−1481.81	−0.256453
$323$	8816.87	1.51883
$324$	539.205	0.0924563
$325$	0	0
$326$	2327.53	0.395429
$327$	−5815.01	−0.983396
$328$	367.864	0.0619265
$329$	4592.50	0.769583
$330$	0	0
$331$	−7998.03	−1.32813	−0.664066	−	0.747674i	$-0.731170\pi$
$331$	−7998.03	−1.32813	−0.664066	+	0.747674i	$0.731170\pi$
$332$	−8898.87	−1.47105
$333$	−1616.60	−0.266034
$334$	12454.1	2.04030
$335$	0	0
$336$	−3682.45	−0.597900
$337$	5572.96	0.900827	0.450413	−	0.892820i	$-0.351277\pi$
$337$	5572.96	0.900827	0.450413	+	0.892820i	$0.351277\pi$
$338$	35.3508	0.00568885
$339$	−4991.36	−0.799686
$340$	0	0
$341$	−3835.35	−0.609078
$342$	−4888.67	−0.772951
$343$	−6778.56	−1.06708
$344$	−818.054	−0.128217
$345$	0	0
$346$	−7564.03	−1.17527
$347$	8070.32	1.24852	0.624262	−	0.781215i	$-0.285400\pi$
$347$	8070.32	1.24852	0.624262	+	0.781215i	$0.285400\pi$
$348$	−5765.03	−0.888040
$349$	−10343.3	−1.58644	−0.793218	−	0.608937i	$-0.791596\pi$
$349$	−10343.3	−1.58644	−0.793218	+	0.608937i	$0.791596\pi$
$350$	0	0
$351$	1268.21	0.192854
$352$	13286.4	2.01185
$353$	2326.06	0.350718	0.175359	−	0.984505i	$-0.443891\pi$
$353$	2326.06	0.350718	0.175359	+	0.984505i	$0.443891\pi$
$354$	4902.97	0.736130
$355$	0	0
$356$	−2249.59	−0.334910
$357$	−3137.26	−0.465102
$358$	16315.2	2.40861
$359$	−3333.39	−0.490054	−0.245027	−	0.969516i	$-0.578797\pi$
$359$	−3333.39	−0.490054	−0.245027	+	0.969516i	$0.578797\pi$
$360$	0	0
$361$	13271.6	1.93491
$362$	7822.57	1.13576
$363$	5347.58	0.773210
$364$	5261.85	0.757681
$365$	0	0
$366$	−2795.60	−0.399258
$367$	−6803.73	−0.967716	−0.483858	−	0.875147i	$-0.660765\pi$
$367$	−6803.73	−0.967716	−0.483858	+	0.875147i	$0.660765\pi$
$368$	1677.65	0.237645
$369$	−643.852	−0.0908336
$370$	0	0
$371$	206.725	0.0289289
$372$	1372.68	0.191317
$373$	−628.598	−0.0872589	−0.0436294	−	0.999048i	$-0.513892\pi$
$373$	−628.598	−0.0872589	−0.0436294	+	0.999048i	$0.513892\pi$
$374$	13275.0	1.83538
$375$	0	0
$376$	−1403.30	−0.192472
$377$	−13559.3	−1.85236
$378$	1739.51	0.236695
$379$	3289.15	0.445785	0.222892	−	0.974843i	$-0.428450\pi$
$379$	3289.15	0.445785	0.222892	+	0.974843i	$0.428450\pi$
$380$	0	0
$381$	2204.03	0.296367
$382$	−279.859	−0.0374838
$383$	1889.81	0.252128	0.126064	−	0.992022i	$-0.459766\pi$
$383$	1889.81	0.252128	0.126064	+	0.992022i	$0.459766\pi$
$384$	1946.75	0.258710
$385$	0	0
$386$	872.501	0.115050
$387$	1431.79	0.188068
$388$	9451.15	1.23662
$389$	−569.046	−0.0741692	−0.0370846	−	0.999312i	$-0.511807\pi$
$389$	−569.046	−0.0741692	−0.0370846	+	0.999312i	$0.511807\pi$
$390$	0	0
$391$	1429.27	0.184863
$392$	307.520	0.0396228
$393$	−8566.32	−1.09953
$394$	−2211.99	−0.282839
$395$	0	0
$396$	−3343.01	−0.424224
$397$	−4092.11	−0.517323	−0.258661	−	0.965968i	$-0.583281\pi$
$397$	−4092.11	−0.517323	−0.258661	+	0.965968i	$0.583281\pi$
$398$	−17492.1	−2.20302
$399$	−7162.97	−0.898739
$400$	0	0
$401$	3867.00	0.481568	0.240784	−	0.970579i	$-0.422595\pi$
$401$	3867.00	0.481568	0.240784	+	0.970579i	$0.422595\pi$
$402$	930.878	0.115492
$403$	3228.52	0.399068
$404$	3137.21	0.386342
$405$	0	0
$406$	−18598.4	−2.27345
$407$	10022.7	1.22066
$408$	958.630	0.116322
$409$	−15631.6	−1.88981	−0.944906	−	0.327343i	$-0.893847\pi$
$409$	−15631.6	−1.88981	−0.944906	+	0.327343i	$0.893847\pi$
$410$	0	0
$411$	−4291.69	−0.515070
$412$	7515.61	0.898708
$413$	7183.91	0.855925
$414$	−792.484	−0.0940785
$415$	0	0
$416$	−11184.3	−1.31816
$417$	865.231	0.101608
$418$	30309.2	3.54658
$419$	−3562.17	−0.415330	−0.207665	−	0.978200i	$-0.566586\pi$
$419$	−3562.17	−0.415330	−0.207665	+	0.978200i	$0.566586\pi$
$420$	0	0
$421$	10576.0	1.22433	0.612166	−	0.790729i	$-0.290298\pi$
$421$	10576.0	1.22433	0.612166	+	0.790729i	$0.290298\pi$
$422$	−6955.13	−0.802299
$423$	2456.11	0.282318
$424$	−63.1674	−0.00723509
$425$	0	0
$426$	8004.44	0.910367
$427$	−4096.16	−0.464232
$428$	−1354.96	−0.153024
$429$	−7862.73	−0.884886
$430$	0	0
$431$	6965.77	0.778490	0.389245	−	0.921134i	$-0.372736\pi$
$431$	6965.77	0.778490	0.389245	+	0.921134i	$0.372736\pi$
$432$	−1969.41	−0.219336
$433$	16073.5	1.78393	0.891967	−	0.452100i	$-0.149325\pi$
$433$	16073.5	1.78393	0.891967	+	0.452100i	$0.149325\pi$
$434$	4428.35	0.489787
$435$	0	0
$436$	−12903.2	−1.41732
$437$	3263.29	0.357218
$438$	−6528.14	−0.712161
$439$	−613.988	−0.0667518	−0.0333759	−	0.999443i	$-0.510626\pi$
$439$	−613.988	−0.0667518	−0.0333759	+	0.999443i	$0.510626\pi$
$440$	0	0
$441$	−538.236	−0.0581186
$442$	−11174.6	−1.20254
$443$	263.475	0.0282575	0.0141288	−	0.999900i	$-0.495503\pi$
$443$	263.475	0.0282575	0.0141288	+	0.999900i	$0.495503\pi$
$444$	−3587.16	−0.383421
$445$	0	0
$446$	−13378.3	−1.42036
$447$	−5811.23	−0.614903
$448$	−5520.87	−0.582225
$449$	1644.47	0.172845	0.0864226	−	0.996259i	$-0.472456\pi$
$449$	1644.47	0.172845	0.0864226	+	0.996259i	$0.472456\pi$
$450$	0	0
$451$	3991.81	0.416778
$452$	−11075.6	−1.15255
$453$	5543.93	0.575003
$454$	−3998.81	−0.413378
$455$	0	0
$456$	2188.73	0.224774
$457$	16190.9	1.65728	0.828641	−	0.559781i	$-0.189115\pi$
$457$	16190.9	1.65728	0.828641	+	0.559781i	$0.189115\pi$
$458$	8198.74	0.836467
$459$	−1677.84	−0.170620
$460$	0	0
$461$	5783.08	0.584262	0.292131	−	0.956378i	$-0.405636\pi$
$461$	5783.08	0.584262	0.292131	+	0.956378i	$0.405636\pi$
$462$	−10784.8	−1.08605
$463$	3753.34	0.376744	0.188372	−	0.982098i	$-0.439679\pi$
$463$	3753.34	0.376744	0.188372	+	0.982098i	$0.439679\pi$
$464$	21056.4	2.10672
$465$	0	0
$466$	−2744.23	−0.272798
$467$	8441.21	0.836430	0.418215	−	0.908348i	$-0.362656\pi$
$467$	8441.21	0.836430	0.418215	+	0.908348i	$0.362656\pi$
$468$	2814.09	0.277951
$469$	1363.94	0.134287
$470$	0	0
$471$	−153.747	−0.0150409
$472$	−2195.13	−0.214066
$473$	−8876.97	−0.862925
$474$	8259.33	0.800345
$475$	0	0
$476$	−6961.43	−0.670329
$477$	110.558	0.0106124
$478$	−13460.6	−1.28802
$479$	−1001.54	−0.0955359	−0.0477680	−	0.998858i	$-0.515211\pi$
$479$	−1001.54	−0.0955359	−0.0477680	+	0.998858i	$0.515211\pi$
$480$	0	0
$481$	−8436.96	−0.799776
$482$	21763.2	2.05662
$483$	−1161.16	−0.109389
$484$	11866.0	1.11439
$485$	0	0
$486$	930.308	0.0868305
$487$	−13740.3	−1.27851	−0.639253	−	0.768997i	$-0.720756\pi$
$487$	−13740.3	−1.27851	−0.639253	+	0.768997i	$0.720756\pi$
$488$	1251.63	0.116104
$489$	1823.88	0.168668
$490$	0	0
$491$	−10708.1	−0.984217	−0.492109	−	0.870534i	$-0.663774\pi$
$491$	−10708.1	−0.984217	−0.492109	+	0.870534i	$0.663774\pi$
$492$	−1428.68	−0.130914
$493$	17939.0	1.63880
$494$	−25513.7	−2.32372
$495$	0	0
$496$	−5013.61	−0.453867
$497$	11728.2	1.05852
$498$	−15353.5	−1.38154
$499$	9630.22	0.863944	0.431972	−	0.901887i	$-0.357818\pi$
$499$	9630.22	0.863944	0.431972	+	0.901887i	$0.357818\pi$
$500$	0	0
$501$	9759.19	0.870277
$502$	−26595.7	−2.36459
$503$	−18170.1	−1.61067	−0.805334	−	0.592822i	$-0.798014\pi$
$503$	−18170.1	−1.61067	−0.805334	+	0.592822i	$0.798014\pi$
$504$	−778.806	−0.0688309
$505$	0	0
$506$	4913.31	0.431667
$507$	27.7013	0.00242654
$508$	4890.63	0.427139
$509$	12525.5	1.09073	0.545365	−	0.838199i	$-0.316391\pi$
$509$	12525.5	1.09073	0.545365	+	0.838199i	$0.316391\pi$
$510$	0	0
$511$	−9565.14	−0.828057
$512$	14367.6	1.24017
$513$	−3830.82	−0.329698
$514$	6800.45	0.583570
$515$	0	0
$516$	3177.08	0.271053
$517$	−15227.6	−1.29538
$518$	−11572.4	−0.981589
$519$	−5927.26	−0.501306
$520$	0	0
$521$	3631.35	0.305360	0.152680	−	0.988276i	$-0.451210\pi$
$521$	3631.35	0.305360	0.152680	+	0.988276i	$0.451210\pi$
$522$	−9946.58	−0.834004
$523$	−12118.1	−1.01317	−0.506584	−	0.862191i	$-0.669092\pi$
$523$	−12118.1	−1.01317	−0.506584	+	0.862191i	$0.669092\pi$
$524$	−19008.2	−1.58469
$525$	0	0
$526$	−430.336	−0.0356721
$527$	−4271.34	−0.353060
$528$	12210.1	1.00640
$529$	529.000	0.0434783
$530$	0	0
$531$	3842.02	0.313992
$532$	−15894.3	−1.29531
$533$	−3360.23	−0.273073
$534$	−3881.28	−0.314531
$535$	0	0
$536$	−416.768	−0.0335852
$537$	12784.8	1.02738
$538$	−22650.6	−1.81513
$539$	3337.01	0.266670
$540$	0	0
$541$	−24466.6	−1.94437	−0.972184	−	0.234220i	$-0.924746\pi$
$541$	−24466.6	−1.94437	−0.972184	+	0.234220i	$0.924746\pi$
$542$	−3874.23	−0.307034
$543$	6129.86	0.484452
$544$	14796.8	1.16619
$545$	0	0
$546$	9078.43	0.711577
$547$	9781.99	0.764621	0.382311	−	0.924034i	$-0.375128\pi$
$547$	9781.99	0.764621	0.382311	+	0.924034i	$0.375128\pi$
$548$	−9523.06	−0.742345
$549$	−2190.66	−0.170301
$550$	0	0
$551$	40958.0	3.16673
$552$	354.807	0.0273580
$553$	12101.7	0.930591
$554$	13906.5	1.06648
$555$	0	0
$556$	1919.91	0.146443
$557$	2198.74	0.167260	0.0836300	−	0.996497i	$-0.473349\pi$
$557$	2198.74	0.167260	0.0836300	+	0.996497i	$0.473349\pi$
$558$	2368.32	0.179676
$559$	7472.47	0.565388
$560$	0	0
$561$	10402.4	0.782870
$562$	16778.0	1.25932
$563$	−7927.38	−0.593427	−0.296714	−	0.954967i	$-0.595891\pi$
$563$	−7927.38	−0.593427	−0.296714	+	0.954967i	$0.595891\pi$
$564$	5450.00	0.406891
$565$	0	0
$566$	10307.3	0.765453
$567$	1363.10	0.100961
$568$	−3583.71	−0.264734
$569$	−18860.7	−1.38960	−0.694799	−	0.719204i	$-0.744507\pi$
$569$	−18860.7	−1.38960	−0.694799	+	0.719204i	$0.744507\pi$
$570$	0	0
$571$	−1044.79	−0.0765726	−0.0382863	−	0.999267i	$-0.512190\pi$
$571$	−1044.79	−0.0765726	−0.0382863	+	0.999267i	$0.512190\pi$
$572$	−17447.0	−1.27534
$573$	−219.300	−0.0159885
$574$	−4609.01	−0.335150
$575$	0	0
$576$	−2952.61	−0.213586
$577$	3366.04	0.242860	0.121430	−	0.992600i	$-0.461252\pi$
$577$	3366.04	0.242860	0.121430	+	0.992600i	$0.461252\pi$
$578$	−4025.04	−0.289653
$579$	683.702	0.0490737
$580$	0	0
$581$	−22496.2	−1.60637
$582$	16306.4	1.16138
$583$	−685.450	−0.0486937
$584$	2922.75	0.207096
$585$	0	0
$586$	−15404.1	−1.08590
$587$	17910.9	1.25939	0.629696	−	0.776842i	$-0.283180\pi$
$587$	17910.9	1.25939	0.629696	+	0.776842i	$0.283180\pi$
$588$	−1194.32	−0.0837635
$589$	−9752.29	−0.682234
$590$	0	0
$591$	−1733.34	−0.120643
$592$	13101.9	0.909600
$593$	−3396.99	−0.235240	−0.117620	−	0.993059i	$-0.537527\pi$
$593$	−3396.99	−0.235240	−0.117620	+	0.993059i	$0.537527\pi$
$594$	−5767.80	−0.398411
$595$	0	0
$596$	−12894.8	−0.886230
$597$	−13707.0	−0.939684
$598$	−4135.94	−0.282828
$599$	−139.199	−0.00949505	−0.00474752	−	0.999989i	$-0.501511\pi$
$599$	−139.199	−0.00949505	−0.00474752	+	0.999989i	$0.501511\pi$
$600$	0	0
$601$	1384.91	0.0939959	0.0469979	−	0.998895i	$-0.485035\pi$
$601$	1384.91	0.0939959	0.0469979	+	0.998895i	$0.485035\pi$
$602$	10249.5	0.693917
$603$	729.447	0.0492626
$604$	12301.7	0.828724
$605$	0	0
$606$	5412.73	0.362833
$607$	−1928.79	−0.128974	−0.0644871	−	0.997919i	$-0.520541\pi$
$607$	−1928.79	−0.128974	−0.0644871	+	0.997919i	$0.520541\pi$
$608$	33784.0	2.25349
$609$	−14573.9	−0.969728
$610$	0	0
$611$	12818.3	0.848731
$612$	−3723.04	−0.245907
$613$	−14251.4	−0.939001	−0.469500	−	0.882932i	$-0.655566\pi$
$613$	−14251.4	−0.939001	−0.469500	+	0.882932i	$0.655566\pi$
$614$	−19889.5	−1.30729
$615$	0	0
$616$	4828.51	0.315822
$617$	6803.77	0.443938	0.221969	−	0.975054i	$-0.428752\pi$
$617$	6803.77	0.443938	0.221969	+	0.975054i	$0.428752\pi$
$618$	12966.9	0.844023
$619$	11856.9	0.769904	0.384952	−	0.922937i	$-0.374218\pi$
$619$	11856.9	0.769904	0.384952	+	0.922937i	$0.374218\pi$
$620$	0	0
$621$	−621.000	−0.0401286
$622$	20828.9	1.34271
$623$	−5686.92	−0.365717
$624$	−10278.3	−0.659390
$625$	0	0
$626$	28294.0	1.80648
$627$	23750.7	1.51278
$628$	−341.157	−0.0216778
$629$	11162.1	0.707572
$630$	0	0
$631$	−15491.7	−0.977362	−0.488681	−	0.872463i	$-0.662522\pi$
$631$	−15491.7	−0.977362	−0.488681	+	0.872463i	$0.662522\pi$
$632$	−3697.83	−0.232740
$633$	−5450.12	−0.342216
$634$	33632.8	2.10683
$635$	0	0
$636$	245.324	0.0152952
$637$	−2809.03	−0.174722
$638$	61667.7	3.82672
$639$	6272.37	0.388312
$640$	0	0
$641$	−18427.5	−1.13548	−0.567740	−	0.823208i	$-0.692182\pi$
$641$	−18427.5	−1.13548	−0.567740	+	0.823208i	$0.692182\pi$
$642$	−2337.75	−0.143713
$643$	−4164.58	−0.255420	−0.127710	−	0.991812i	$-0.540763\pi$
$643$	−4164.58	−0.255420	−0.127710	+	0.991812i	$0.540763\pi$
$644$	−2576.56	−0.157656
$645$	0	0
$646$	33754.7	2.05582
$647$	−24965.0	−1.51697	−0.758483	−	0.651693i	$-0.774059\pi$
$647$	−24965.0	−1.51697	−0.758483	+	0.651693i	$0.774059\pi$
$648$	−416.513	−0.0252503
$649$	−23820.1	−1.44071
$650$	0	0
$651$	3470.11	0.208916
$652$	4047.10	0.243093
$653$	19702.6	1.18074	0.590370	−	0.807133i	$-0.298982\pi$
$653$	19702.6	1.18074	0.590370	+	0.807133i	$0.298982\pi$
$654$	−22262.3	−1.33108
$655$	0	0
$656$	5218.14	0.310571
$657$	−5115.53	−0.303768
$658$	17582.1	1.04167
$659$	−19243.1	−1.13749	−0.568744	−	0.822515i	$-0.692571\pi$
$659$	−19243.1	−1.13749	−0.568744	+	0.822515i	$0.692571\pi$
$660$	0	0
$661$	11725.0	0.689939	0.344969	−	0.938614i	$-0.387889\pi$
$661$	11725.0	0.689939	0.344969	+	0.938614i	$0.387889\pi$
$662$	−30619.9	−1.79770
$663$	−8756.55	−0.512936
$664$	6874.00	0.401751
$665$	0	0
$666$	−6189.04	−0.360091
$667$	6639.55	0.385434
$668$	21655.2	1.25429
$669$	−10483.4	−0.605847
$670$	0	0
$671$	13581.9	0.781404
$672$	−12021.2	−0.690070
$673$	27664.1	1.58450	0.792252	−	0.610194i	$-0.208909\pi$
$673$	27664.1	1.58450	0.792252	+	0.610194i	$0.208909\pi$
$674$	21335.7	1.21932
$675$	0	0
$676$	61.4678	0.00349726
$677$	−16852.0	−0.956683	−0.478342	−	0.878174i	$-0.658762\pi$
$677$	−16852.0	−0.956683	−0.478342	+	0.878174i	$0.658762\pi$
$678$	−19109.1	−1.08242
$679$	23892.4	1.35038
$680$	0	0
$681$	−3133.52	−0.176324
$682$	−14683.3	−0.824420
$683$	−24478.8	−1.37138	−0.685692	−	0.727891i	$-0.740500\pi$
$683$	−24478.8	−1.37138	−0.685692	+	0.727891i	$0.740500\pi$
$684$	−8500.41	−0.475177
$685$	0	0
$686$	−25951.2	−1.44435
$687$	6424.63	0.356790
$688$	−11604.1	−0.643025
$689$	576.999	0.0319041
$690$	0	0
$691$	22681.5	1.24869	0.624346	−	0.781148i	$-0.285365\pi$
$691$	22681.5	1.24869	0.624346	+	0.781148i	$0.285365\pi$
$692$	−13152.3	−0.722508
$693$	−8451.08	−0.463247
$694$	30896.6	1.68994
$695$	0	0
$696$	4453.24	0.242528
$697$	4445.59	0.241591
$698$	−39598.7	−2.14733
$699$	−2150.41	−0.116360
$700$	0	0
$701$	6898.78	0.371703	0.185851	−	0.982578i	$-0.440496\pi$
$701$	6898.78	0.371703	0.185851	+	0.982578i	$0.440496\pi$
$702$	4855.23	0.261038
$703$	25485.2	1.36727
$704$	18305.9	0.980012
$705$	0	0
$706$	8905.15	0.474716
$707$	7930.82	0.421880
$708$	8525.26	0.452541
$709$	−1725.21	−0.0913845	−0.0456922	−	0.998956i	$-0.514549\pi$
$709$	−1725.21	−0.0913845	−0.0456922	+	0.998956i	$0.514549\pi$
$710$	0	0
$711$	6472.11	0.341382
$712$	1737.71	0.0914654
$713$	−1580.91	−0.0830370
$714$	−12010.8	−0.629541
$715$	0	0
$716$	28368.7	1.48071
$717$	−10547.9	−0.549398
$718$	−12761.6	−0.663315
$719$	−9213.52	−0.477895	−0.238947	−	0.971033i	$-0.576802\pi$
$719$	−9213.52	−0.477895	−0.238947	+	0.971033i	$0.576802\pi$
$720$	0	0
$721$	18999.4	0.981377
$722$	50809.3	2.61901
$723$	17053.9	0.877237
$724$	13601.9	0.698217
$725$	0	0
$726$	20472.8	1.04658
$727$	−23768.6	−1.21256	−0.606278	−	0.795253i	$-0.707338\pi$
$727$	−23768.6	−1.21256	−0.606278	+	0.795253i	$0.707338\pi$
$728$	−4064.55	−0.206926
$729$	729.000	0.0370370
$730$	0	0
$731$	−9886.09	−0.500205
$732$	−4860.98	−0.245447
$733$	−10711.8	−0.539767	−0.269884	−	0.962893i	$-0.586985\pi$
$733$	−10711.8	−0.539767	−0.269884	+	0.962893i	$0.586985\pi$
$734$	−26047.6	−1.30986
$735$	0	0
$736$	5476.59	0.274280
$737$	−4522.49	−0.226035
$738$	−2464.94	−0.122948
$739$	−25572.2	−1.27292	−0.636461	−	0.771309i	$-0.719602\pi$
$739$	−25572.2	−1.27292	−0.636461	+	0.771309i	$0.719602\pi$
$740$	0	0
$741$	−19992.9	−0.991169
$742$	791.431	0.0391568
$743$	3613.01	0.178397	0.0891983	−	0.996014i	$-0.471570\pi$
$743$	3613.01	0.178397	0.0891983	+	0.996014i	$0.471570\pi$
$744$	−1060.34	−0.0522497
$745$	0	0
$746$	−2406.54	−0.118110
$747$	−12031.2	−0.589288
$748$	23082.4	1.12831
$749$	−3425.31	−0.167100
$750$	0	0
$751$	26247.1	1.27532	0.637662	−	0.770316i	$-0.279902\pi$
$751$	26247.1	1.27532	0.637662	+	0.770316i	$0.279902\pi$
$752$	−19905.7	−0.965277
$753$	−20840.7	−1.00860
$754$	−51910.7	−2.50726
$755$	0	0
$756$	3024.66	0.145510
$757$	12700.0	0.609761	0.304880	−	0.952391i	$-0.401383\pi$
$757$	12700.0	0.609761	0.304880	+	0.952391i	$0.401383\pi$
$758$	12592.3	0.603393
$759$	3850.13	0.184125
$760$	0	0
$761$	22485.4	1.07108	0.535542	−	0.844508i	$-0.320107\pi$
$761$	22485.4	1.07108	0.535542	+	0.844508i	$0.320107\pi$
$762$	8437.96	0.401148
$763$	−32619.1	−1.54770
$764$	−486.617	−0.0230434
$765$	0	0
$766$	7235.01	0.341268
$767$	20051.3	0.943953
$768$	15326.6	0.720120
$769$	−7141.34	−0.334881	−0.167440	−	0.985882i	$-0.553550\pi$
$769$	−7141.34	−0.334881	−0.167440	+	0.985882i	$0.553550\pi$
$770$	0	0
$771$	5328.91	0.248918
$772$	1517.10	0.0707276
$773$	−15819.1	−0.736060	−0.368030	−	0.929814i	$-0.619968\pi$
$773$	−15819.1	−0.736060	−0.368030	+	0.929814i	$0.619968\pi$
$774$	5481.52	0.254560
$775$	0	0
$776$	−7300.61	−0.337728
$777$	−9068.29	−0.418691
$778$	−2178.55	−0.100392
$779$	10150.1	0.466837
$780$	0	0
$781$	−38888.0	−1.78172
$782$	5471.85	0.250221
$783$	−7794.26	−0.355739
$784$	4362.17	0.198714
$785$	0	0
$786$	−32795.5	−1.48827
$787$	2959.52	0.134048	0.0670239	−	0.997751i	$-0.478650\pi$
$787$	2959.52	0.134048	0.0670239	+	0.997751i	$0.478650\pi$
$788$	−3846.21	−0.173877
$789$	−337.216	−0.0152157
$790$	0	0
$791$	−27998.9	−1.25857
$792$	2582.33	0.115858
$793$	−11433.0	−0.511975
$794$	−15666.3	−0.700224
$795$	0	0
$796$	−30415.2	−1.35432
$797$	−19932.4	−0.885876	−0.442938	−	0.896552i	$-0.646064\pi$
$797$	−19932.4	−0.885876	−0.442938	+	0.896552i	$0.646064\pi$
$798$	−27422.9	−1.21649
$799$	−16958.7	−0.750883
$800$	0	0
$801$	−3041.42	−0.134161
$802$	14804.5	0.651828
$803$	31715.7	1.39380
$804$	1618.61	0.0709998
$805$	0	0
$806$	12360.2	0.540159
$807$	−17749.3	−0.774232
$808$	−2423.36	−0.105512
$809$	−7334.79	−0.318761	−0.159380	−	0.987217i	$-0.550950\pi$
$809$	−7334.79	−0.318761	−0.159380	+	0.987217i	$0.550950\pi$
$810$	0	0
$811$	−15908.5	−0.688807	−0.344404	−	0.938822i	$-0.611919\pi$
$811$	−15908.5	−0.688807	−0.344404	+	0.938822i	$0.611919\pi$
$812$	−32338.8	−1.39762
$813$	−3035.89	−0.130964
$814$	38371.4	1.65223
$815$	0	0
$816$	13598.2	0.583371
$817$	−22571.8	−0.966570
$818$	−59844.4	−2.55796
$819$	7113.97	0.303519
$820$	0	0
$821$	−6167.56	−0.262180	−0.131090	−	0.991371i	$-0.541848\pi$
$821$	−6167.56	−0.262180	−0.131090	+	0.991371i	$0.541848\pi$
$822$	−16430.4	−0.697174
$823$	45695.7	1.93542	0.967710	−	0.252066i	$-0.0811101\pi$
$823$	45695.7	1.93542	0.967710	+	0.252066i	$0.0811101\pi$
$824$	−5805.49	−0.245441
$825$	0	0
$826$	27503.1	1.15854
$827$	−10905.6	−0.458554	−0.229277	−	0.973361i	$-0.573636\pi$
$827$	−10905.6	−0.458554	−0.229277	+	0.973361i	$0.573636\pi$
$828$	−1377.97	−0.0578354
$829$	−4256.81	−0.178341	−0.0891707	−	0.996016i	$-0.528422\pi$
$829$	−4256.81	−0.178341	−0.0891707	+	0.996016i	$0.528422\pi$
$830$	0	0
$831$	10897.3	0.454900
$832$	−15409.5	−0.642103
$833$	3716.35	0.154579
$834$	3312.47	0.137532
$835$	0	0
$836$	52701.6	2.18029
$837$	1855.85	0.0766397
$838$	−13637.5	−0.562171
$839$	−15166.7	−0.624092	−0.312046	−	0.950067i	$-0.601014\pi$
$839$	−15166.7	−0.624092	−0.312046	+	0.950067i	$0.601014\pi$
$840$	0	0
$841$	58944.9	2.41687
$842$	40489.5	1.65720
$843$	13147.4	0.537154
$844$	−12093.6	−0.493219
$845$	0	0
$846$	9403.05	0.382132
$847$	29997.1	1.21690
$848$	−896.029	−0.0362851
$849$	8076.88	0.326499
$850$	0	0
$851$	4131.31	0.166416
$852$	13918.1	0.559655
$853$	24050.4	0.965379	0.482690	−	0.875791i	$-0.339660\pi$
$853$	24050.4	0.965379	0.482690	+	0.875791i	$0.339660\pi$
$854$	−15681.8	−0.628363
$855$	0	0
$856$	1046.65	0.0417917
$857$	−39207.3	−1.56277	−0.781387	−	0.624047i	$-0.785487\pi$
$857$	−39207.3	−1.56277	−0.781387	+	0.624047i	$0.785487\pi$
$858$	−30101.9	−1.19774
$859$	−10527.2	−0.418142	−0.209071	−	0.977900i	$-0.567044\pi$
$859$	−10527.2	−0.418142	−0.209071	+	0.977900i	$0.567044\pi$
$860$	0	0
$861$	−3611.67	−0.142956
$862$	26667.9	1.05373
$863$	11856.0	0.467651	0.233825	−	0.972279i	$-0.424876\pi$
$863$	11856.0	0.467651	0.233825	+	0.972279i	$0.424876\pi$
$864$	−6429.04	−0.253149
$865$	0	0
$866$	61536.3	2.41465
$867$	−3154.06	−0.123550
$868$	7700.00	0.301100
$869$	−40126.3	−1.56639
$870$	0	0
$871$	3806.95	0.148098
$872$	9967.18	0.387077
$873$	12777.9	0.495378
$874$	12493.3	0.483514
$875$	0	0
$876$	−11351.1	−0.437806
$877$	−51435.9	−1.98046	−0.990232	−	0.139432i	$-0.955472\pi$
$877$	−51435.9	−1.98046	−0.990232	+	0.139432i	$0.955472\pi$
$878$	−2350.61	−0.0903521
$879$	−12070.8	−0.463183
$880$	0	0
$881$	8069.52	0.308592	0.154296	−	0.988025i	$-0.450689\pi$
$881$	8069.52	0.308592	0.154296	+	0.988025i	$0.450689\pi$
$882$	−2060.60	−0.0786666
$883$	43901.7	1.67317	0.836586	−	0.547835i	$-0.184548\pi$
$883$	43901.7	1.67317	0.836586	+	0.547835i	$0.184548\pi$
$884$	−19430.4	−0.739269
$885$	0	0
$886$	1008.70	0.0382481
$887$	5877.02	0.222470	0.111235	−	0.993794i	$-0.464519\pi$
$887$	5877.02	0.222470	0.111235	+	0.993794i	$0.464519\pi$
$888$	2770.93	0.104714
$889$	12363.4	0.466430
$890$	0	0
$891$	−4519.72	−0.169940
$892$	−23262.1	−0.873178
$893$	−38719.9	−1.45097
$894$	−22247.9	−0.832304
$895$	0	0
$896$	10920.2	0.407165
$897$	−3240.97	−0.120639
$898$	6295.75	0.233955
$899$	−19842.2	−0.736122
$900$	0	0
$901$	−763.371	−0.0282259
$902$	15282.4	0.564132
$903$	8031.62	0.295986
$904$	8555.42	0.314767
$905$	0	0
$906$	21224.5	0.778298
$907$	−1967.03	−0.0720110	−0.0360055	−	0.999352i	$-0.511463\pi$
$907$	−1967.03	−0.0720110	−0.0360055	+	0.999352i	$0.511463\pi$
$908$	−6953.12	−0.254127
$909$	4241.48	0.154764
$910$	0	0
$911$	21073.2	0.766396	0.383198	−	0.923666i	$-0.374823\pi$
$911$	21073.2	0.766396	0.383198	+	0.923666i	$0.374823\pi$
$912$	31047.2	1.12727
$913$	74592.0	2.70387
$914$	61985.6	2.24322
$915$	0	0
$916$	14255.9	0.514224
$917$	−48052.6	−1.73046
$918$	−6423.48	−0.230944
$919$	−7748.69	−0.278135	−0.139067	−	0.990283i	$-0.544410\pi$
$919$	−7748.69	−0.278135	−0.139067	+	0.990283i	$0.544410\pi$
$920$	0	0
$921$	−15585.6	−0.557616
$922$	22140.1	0.790830
$923$	32735.2	1.16738
$924$	−18752.5	−0.667655
$925$	0	0
$926$	14369.4	0.509943
$927$	10161.0	0.360013
$928$	68737.5	2.43148
$929$	35683.4	1.26021	0.630105	−	0.776510i	$-0.283012\pi$
$929$	35683.4	1.26021	0.630105	+	0.776510i	$0.283012\pi$
$930$	0	0
$931$	8485.13	0.298699
$932$	−4771.65	−0.167704
$933$	16321.8	0.572723
$934$	32316.6	1.13215
$935$	0	0
$936$	−2173.76	−0.0759098
$937$	−46259.2	−1.61283	−0.806414	−	0.591351i	$-0.798595\pi$
$937$	−46259.2	−1.61283	−0.806414	+	0.591351i	$0.798595\pi$
$938$	5221.74	0.181765
$939$	22171.5	0.770544
$940$	0	0
$941$	23519.2	0.814777	0.407388	−	0.913255i	$-0.366440\pi$
$941$	23519.2	0.814777	0.407388	+	0.913255i	$0.366440\pi$
$942$	−588.608	−0.0203587
$943$	1645.40	0.0568203
$944$	−31137.9	−1.07357
$945$	0	0
$946$	−33984.8	−1.16801
$947$	52061.1	1.78644	0.893220	−	0.449621i	$-0.148441\pi$
$947$	52061.1	1.78644	0.893220	+	0.449621i	$0.148441\pi$
$948$	14361.3	0.492018
$949$	−26697.7	−0.913218
$950$	0	0
$951$	26355.1	0.898655
$952$	5377.41	0.183070
$953$	15306.5	0.520278	0.260139	−	0.965571i	$-0.416232\pi$
$953$	15306.5	0.520278	0.260139	+	0.965571i	$0.416232\pi$
$954$	423.265	0.0143645
$955$	0	0
$956$	−23405.3	−0.791820
$957$	48323.5	1.63226
$958$	−3834.34	−0.129313
$959$	−24074.1	−0.810631
$960$	0	0
$961$	−25066.5	−0.841412
$962$	−32300.3	−1.08254
$963$	−1831.89	−0.0612999
$964$	37841.8	1.26432
$965$	0	0
$966$	−4445.42	−0.148063
$967$	−19036.6	−0.633068	−0.316534	−	0.948581i	$-0.602519\pi$
$967$	−19036.6	−0.633068	−0.316534	+	0.948581i	$0.602519\pi$
$968$	−9166.00	−0.304345
$969$	26450.6	0.876900
$970$	0	0
$971$	9160.05	0.302740	0.151370	−	0.988477i	$-0.451632\pi$
$971$	9160.05	0.302740	0.151370	+	0.988477i	$0.451632\pi$
$972$	1617.62	0.0533797
$973$	4853.49	0.159914
$974$	−52603.7	−1.73053
$975$	0	0
$976$	17754.4	0.582279
$977$	−27640.7	−0.905122	−0.452561	−	0.891733i	$-0.649490\pi$
$977$	−27640.7	−0.905122	−0.452561	+	0.891733i	$0.649490\pi$
$978$	6982.59	0.228301
$979$	18856.4	0.615582
$980$	0	0
$981$	−17445.0	−0.567764
$982$	−40995.2	−1.33219
$983$	−40382.6	−1.31028	−0.655140	−	0.755508i	$-0.727390\pi$
$983$	−40382.6	−1.31028	−0.655140	+	0.755508i	$0.727390\pi$
$984$	1103.59	0.0357533
$985$	0	0
$986$	68678.0	2.21821
$987$	13777.5	0.444319
$988$	−44363.2	−1.42852
$989$	−3659.03	−0.117645
$990$	0	0
$991$	17541.4	0.562283	0.281141	−	0.959666i	$-0.409287\pi$
$991$	17541.4	0.562283	0.281141	+	0.959666i	$0.409287\pi$
$992$	−16366.7	−0.523834
$993$	−23994.1	−0.766797
$994$	44900.7	1.43276
$995$	0	0
$996$	−26696.6	−0.849312
$997$	42798.5	1.35952	0.679760	−	0.733435i	$-0.262084\pi$
$997$	42798.5	1.35952	0.679760	+	0.733435i	$0.262084\pi$
$998$	36868.6	1.16939
$999$	−4849.80	−0.153595

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1725.4.a.m.1.2		2
5.4	even	2		69.4.a.b.1.1	✓	2
15.14	odd	2		207.4.a.b.1.2		2
20.19	odd	2		1104.4.a.q.1.2		2
115.114	odd	2		1587.4.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.a.b.1.1	✓	2	5.4	even	2
207.4.a.b.1.2		2	15.14	odd	2
1104.4.a.q.1.2		2	20.19	odd	2
1587.4.a.c.1.1		2	115.114	odd	2
1725.4.a.m.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1725 = 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1725.a (trivial)

\(f(q)\)	\(=\)	\(q+3.82843 q^{2} +3.00000 q^{3} +6.65685 q^{4} +11.4853 q^{6} +16.8284 q^{7} -5.14214 q^{8} +9.00000 q^{9} -55.7990 q^{11} +19.9706 q^{12} +46.9706 q^{13} +64.4264 q^{14} -72.9411 q^{16} -62.1421 q^{17} +34.4558 q^{18} -141.882 q^{19} +50.4853 q^{21} -213.622 q^{22} -23.0000 q^{23} -15.4264 q^{24} +179.823 q^{26} +27.0000 q^{27} +112.024 q^{28} -288.676 q^{29} +68.7351 q^{31} -238.113 q^{32} -167.397 q^{33} -237.907 q^{34} +59.9117 q^{36} -179.622 q^{37} -543.186 q^{38} +140.912 q^{39} -71.5391 q^{41} +193.279 q^{42} +159.088 q^{43} -371.446 q^{44} -88.0538 q^{46} +272.902 q^{47} -218.823 q^{48} -59.8040 q^{49} -186.426 q^{51} +312.676 q^{52} +12.2843 q^{53} +103.368 q^{54} -86.5341 q^{56} -425.647 q^{57} -1105.18 q^{58} +426.891 q^{59} -243.407 q^{61} +263.147 q^{62} +151.456 q^{63} -328.068 q^{64} -640.867 q^{66} +81.0496 q^{67} -413.671 q^{68} -69.0000 q^{69} +696.930 q^{71} -46.2792 q^{72} -568.392 q^{73} -687.671 q^{74} -944.489 q^{76} -939.009 q^{77} +539.470 q^{78} +719.123 q^{79} +81.0000 q^{81} -273.882 q^{82} -1336.80 q^{83} +336.073 q^{84} +609.058 q^{86} -866.029 q^{87} +286.926 q^{88} -337.935 q^{89} +790.441 q^{91} -153.108 q^{92} +206.205 q^{93} +1044.78 q^{94} -714.338 q^{96} +1419.76 q^{97} -228.955 q^{98} -502.191 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 6 q^{3} + 2 q^{4} + 6 q^{6} + 28 q^{7} + 18 q^{8} + 18 q^{9} - 72 q^{11} + 6 q^{12} + 60 q^{13} + 44 q^{14} - 78 q^{16} - 96 q^{17} + 18 q^{18} - 148 q^{19} + 84 q^{21} - 184 q^{22} - 46 q^{23}+ \cdots - 648 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 6 * q^3 + 2 * q^4 + 6 * q^6 + 28 * q^7 + 18 * q^8 + 18 * q^9 - 72 * q^11 + 6 * q^12 + 60 * q^13 + 44 * q^14 - 78 * q^16 - 96 * q^17 + 18 * q^18 - 148 * q^19 + 84 * q^21 - 184 * q^22 - 46 * q^23 + 54 * q^24 + 156 * q^26 + 54 * q^27 + 60 * q^28 - 204 * q^29 - 168 * q^31 - 414 * q^32 - 216 * q^33 - 176 * q^34 + 18 * q^36 - 116 * q^37 - 532 * q^38 + 180 * q^39 + 4 * q^41 + 132 * q^42 + 420 * q^43 - 296 * q^44 - 46 * q^46 + 48 * q^47 - 234 * q^48 - 278 * q^49 - 288 * q^51 + 252 * q^52 - 32 * q^53 + 54 * q^54 + 172 * q^56 - 444 * q^57 - 1260 * q^58 - 40 * q^59 - 764 * q^61 + 696 * q^62 + 252 * q^63 + 34 * q^64 - 552 * q^66 + 988 * q^67 - 256 * q^68 - 138 * q^69 - 224 * q^71 + 162 * q^72 - 820 * q^73 - 804 * q^74 - 916 * q^76 - 1120 * q^77 + 468 * q^78 + 1772 * q^79 + 162 * q^81 - 412 * q^82 - 1480 * q^83 + 180 * q^84 + 132 * q^86 - 612 * q^87 - 88 * q^88 + 744 * q^89 + 936 * q^91 - 46 * q^92 - 504 * q^93 + 1456 * q^94 - 1242 * q^96 + 260 * q^97 + 170 * q^98 - 648 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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