LMFDB - Embedded newform 756.4.t.c.593.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,4,Mod(269,756)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(756, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 1]))

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

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

chi := DirichletCharacter("756.269");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	756.t (of order $6$, degree $2$, minimal)

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:	no
Analytic conductor:	$44.6054439643$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 43x^{10} + 1500x^{8} - 13455x^{6} + 88433x^{4} - 270824x^{2} + 602176 $ x^12 - 43x^10 + 1500x^8 - 13455x^6 + 88433x^4 - 270824*x^2 + 602176
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{9}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			593.6
Root			$-2.07339 - 1.19707i$ of defining polynomial
Character	$\chi$	$=$	756.593
Dual form			756.4.t.c.269.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(7.05863 - 12.2259i) q^{5} +(-2.66851 - 18.3270i) q^{7} +O(q^{10})$	$q+(7.05863 - 12.2259i) q^{5} +(-2.66851 - 18.3270i) q^{7} +(-14.4837 + 8.36214i) q^{11} -21.2956i q^{13} +(-38.7712 - 67.1538i) q^{17} +(-1.43186 - 0.826683i) q^{19} +(-35.6595 - 20.5880i) q^{23} +(-37.1485 - 64.3430i) q^{25} -100.859i q^{29} +(-45.9022 + 26.5017i) q^{31} +(-242.900 - 96.7386i) q^{35} +(-101.803 + 176.328i) q^{37} +347.031 q^{41} -113.500 q^{43} +(-45.7274 + 79.2022i) q^{47} +(-328.758 + 97.8116i) q^{49} +(-22.2751 + 12.8605i) q^{53} +236.101i q^{55} +(450.976 + 781.113i) q^{59} +(-402.078 - 232.140i) q^{61} +(-260.358 - 150.318i) q^{65} +(-173.573 - 300.638i) q^{67} +620.693i q^{71} +(-564.784 + 326.078i) q^{73} +(191.903 + 243.127i) q^{77} +(68.3109 - 118.318i) q^{79} +298.505 q^{83} -1094.69 q^{85} +(422.682 - 732.108i) q^{89} +(-390.285 + 56.8276i) q^{91} +(-20.2139 + 11.6705i) q^{95} -494.533i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 42 q^{7}+O(q^{10}) $ 12 * q - 42 * q^7	$ 12 q - 42 q^{7} + 186 q^{19} - 246 q^{25} + 738 q^{31} - 336 q^{37} + 756 q^{43} - 1218 q^{49} - 426 q^{61} - 1440 q^{67} + 2730 q^{73} + 1992 q^{79} + 3288 q^{85} + 1764 q^{91}+O(q^{100}) $ 12 * q - 42 * q^7 + 186 * q^19 - 246 * q^25 + 738 * q^31 - 336 * q^37 + 756 * q^43 - 1218 * q^49 - 426 * q^61 - 1440 * q^67 + 2730 * q^73 + 1992 * q^79 + 3288 * q^85 + 1764 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	7.05863	−	12.2259i	0.631343	−	1.09352i	−0.355935	−	0.934511i	$-0.615837\pi$
$5$	7.05863	−	12.2259i	0.631343	−	1.09352i	0.987277	−	0.159007i	$-0.0508293\pi$
$6$		0			0
$7$	−2.66851	−	18.3270i	−0.144086	−	0.989565i
$7$	−2.66851	−	18.3270i	−0.144086	−	0.989565i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−14.4837	+	8.36214i	−0.396999	+	0.229207i	−0.685188	−	0.728366i	$-0.740280\pi$
$11$	−14.4837	+	8.36214i	−0.396999	+	0.229207i	0.288189	+	0.957573i	$0.406947\pi$
$12$		0			0
$13$		−	21.2956i		−	0.454334i	−0.973856	−	0.227167i	$-0.927054\pi$
$13$		−	21.2956i		−	0.454334i	0.973856	−	0.227167i	$-0.0729464\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−38.7712	−	67.1538i	−0.553142	−	0.958069i	−0.998046	−	0.0624910i	$-0.980096\pi$
$17$	−38.7712	−	67.1538i	−0.553142	−	0.958069i	0.444904	−	0.895578i	$-0.353238\pi$
$18$		0			0
$19$	−1.43186	−	0.826683i	−0.0172890	−	0.00998179i	0.491331	−	0.870973i	$-0.336511\pi$
$19$	−1.43186	−	0.826683i	−0.0172890	−	0.00998179i	−0.508620	+	0.860991i	$0.669844\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−35.6595	−	20.5880i	−0.323284	−	0.186648i	0.329572	−	0.944131i	$-0.393096\pi$
$23$	−35.6595	−	20.5880i	−0.323284	−	0.186648i	−0.652855	+	0.757483i	$0.726429\pi$
$24$		0			0
$25$	−37.1485	−	64.3430i	−0.297188	−	0.514744i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	100.859i		−	0.645829i	−0.946428	−	0.322914i	$-0.895337\pi$
$29$		−	100.859i		−	0.645829i	0.946428	−	0.322914i	$-0.104663\pi$
$30$		0			0
$31$	−45.9022	+	26.5017i	−0.265945	+	0.153543i	−0.627043	−	0.778984i	$-0.715735\pi$
$31$	−45.9022	+	26.5017i	−0.265945	+	0.153543i	0.361099	+	0.932528i	$0.382402\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−242.900	−	96.7386i	−1.17308	−	0.467194i
$36$		0			0
$37$	−101.803	+	176.328i	−0.452334	+	0.783465i	−0.998531	−	0.0541919i	$-0.982742\pi$
$37$	−101.803	+	176.328i	−0.452334	+	0.783465i	0.546197	+	0.837657i	$0.316075\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	347.031			1.32188			0.660941	−	0.750438i	$-0.270157\pi$
$41$	347.031			1.32188			0.660941	+	0.750438i	$0.270157\pi$
$42$		0			0
$43$	−113.500			−0.402525			−0.201262	−	0.979537i	$-0.564504\pi$
$43$	−113.500			−0.402525			−0.201262	+	0.979537i	$0.564504\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−45.7274	+	79.2022i	−0.141916	+	0.245805i	−0.928218	−	0.372037i	$-0.878659\pi$
$47$	−45.7274	+	79.2022i	−0.141916	+	0.245805i	0.786302	+	0.617842i	$0.211993\pi$
$48$		0			0
$49$	−328.758	+	97.8116i	−0.958478	+	0.285165i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−22.2751	+	12.8605i	−0.0577305	+	0.0333307i	−0.528587	−	0.848879i	$-0.677278\pi$
$53$	−22.2751	+	12.8605i	−0.0577305	+	0.0333307i	0.470857	+	0.882210i	$0.343945\pi$
$54$		0			0
$55$			236.101i			0.578834i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	450.976	+	781.113i	0.995120	+	1.72360i	0.583012	+	0.812463i	$0.301874\pi$
$59$	450.976	+	781.113i	0.995120	+	1.72360i	0.412108	+	0.911135i	$0.364793\pi$
$60$		0			0
$61$	−402.078	−	232.140i	−0.843947	−	0.487253i	0.0146569	−	0.999893i	$-0.495334\pi$
$61$	−402.078	−	232.140i	−0.843947	−	0.487253i	−0.858604	+	0.512640i	$0.828668\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−260.358	−	150.318i	−0.496823	−	0.286841i
$66$		0			0
$67$	−173.573	−	300.638i	−0.316498	−	0.548191i	0.663257	−	0.748392i	$-0.269174\pi$
$67$	−173.573	−	300.638i	−0.316498	−	0.548191i	−0.979755	+	0.200201i	$0.935840\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			620.693i			1.03750i	0.854925	+	0.518751i	$0.173603\pi$
$71$			620.693i			1.03750i	−0.854925	+	0.518751i	$0.826397\pi$
$72$		0			0
$73$	−564.784	+	326.078i	−0.905521	+	0.522803i	−0.878987	−	0.476845i	$-0.841780\pi$
$73$	−564.784	+	326.078i	−0.905521	+	0.522803i	−0.0265336	+	0.999648i	$0.508447\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	191.903	+	243.127i	0.284018	+	0.359830i
$78$		0			0
$79$	68.3109	−	118.318i	0.0972858	−	0.168504i	−0.813274	−	0.581880i	$-0.802317\pi$
$79$	68.3109	−	118.318i	0.0972858	−	0.168504i	0.910560	+	0.413376i	$0.135651\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	298.505			0.394761			0.197380	−	0.980327i	$-0.436757\pi$
$83$	298.505			0.394761			0.197380	+	0.980327i	$0.436757\pi$
$84$		0			0
$85$	−1094.69			−1.39689
$86$		0			0
$87$		0			0
$88$		0			0
$89$	422.682	−	732.108i	0.503419	−	0.871947i	−0.496574	−	0.867995i	$-0.665409\pi$
$89$	422.682	−	732.108i	0.503419	−	0.871947i	0.999992	−	0.00395192i	$-0.00125794\pi$
$90$		0			0
$91$	−390.285	+	56.8276i	−0.449593	+	0.0654632i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−20.2139	+	11.6705i	−0.0218305	+	0.0126039i
$96$		0			0
$97$		−	494.533i		−	0.517652i	−0.965924	−	0.258826i	$-0.916664\pi$
$97$		−	494.533i		−	0.517652i	0.965924	−	0.258826i	$-0.0833356\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−181.996	−	315.227i	−0.179300	−	0.310557i	0.762341	−	0.647176i	$-0.224050\pi$
$101$	−181.996	−	315.227i	−0.179300	−	0.310557i	−0.941641	+	0.336619i	$0.890717\pi$
$102$		0			0
$103$	−700.114	−	404.211i	−0.669750	−	0.386681i	0.126232	−	0.992001i	$-0.459712\pi$
$103$	−700.114	−	404.211i	−0.669750	−	0.386681i	−0.795982	+	0.605320i	$0.793045\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1481.56	−	855.379i	−1.33858	−	0.772828i	−0.351981	−	0.936007i	$-0.614492\pi$
$107$	−1481.56	−	855.379i	−1.33858	−	0.772828i	−0.986597	+	0.163179i	$0.947825\pi$
$108$		0			0
$109$	−784.949	−	1359.57i	−0.689766	−	1.19471i	−0.971913	−	0.235339i	$-0.924380\pi$
$109$	−784.949	−	1359.57i	−0.689766	−	1.19471i	0.282148	−	0.959371i	$-0.408953\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	83.7798i		−	0.0697464i	−0.999392	−	0.0348732i	$-0.988897\pi$
$113$		−	83.7798i		−	0.0697464i	0.999392	−	0.0348732i	$-0.0111027\pi$
$114$		0			0
$115$	−503.415	+	290.647i	−0.408206	+	0.235678i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1127.27	+	889.761i	−0.868372	+	0.685414i
$120$		0			0
$121$	−525.649	+	910.451i	−0.394928	+	0.684035i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	715.788			0.512176
$126$		0			0
$127$	−1611.24			−1.12578			−0.562892	−	0.826530i	$-0.690311\pi$
$127$	−1611.24			−1.12578			−0.562892	+	0.826530i	$0.690311\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	945.525	−	1637.70i	0.630617	−	1.09226i	−0.356808	−	0.934178i	$-0.616135\pi$
$131$	945.525	−	1637.70i	0.630617	−	1.09226i	0.987426	−	0.158084i	$-0.0505315\pi$
$132$		0			0
$133$	−11.3297	+	28.4476i	−0.00738653	+	0.0185468i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	414.207	−	239.142i	0.258307	−	0.149134i	−0.365255	−	0.930908i	$-0.619018\pi$
$137$	414.207	−	239.142i	0.258307	−	0.149134i	0.623562	+	0.781774i	$0.285685\pi$
$138$		0			0
$139$			1024.01i			0.624856i	0.949941	+	0.312428i	$0.101142\pi$
$139$			1024.01i			0.624856i	−0.949941	+	0.312428i	$0.898858\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	178.077	+	308.439i	0.104137	+	0.180370i
$144$		0			0
$145$	−1233.09	−	711.926i	−0.706225	−	0.407739i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2161.22	−	1247.78i	−1.18828	−	0.686054i	−0.230365	−	0.973104i	$-0.573992\pi$
$149$	−2161.22	−	1247.78i	−1.18828	−	0.686054i	−0.957915	+	0.287050i	$0.907325\pi$
$150$		0			0
$151$	207.605	+	359.583i	0.111885	+	0.193791i	0.916530	−	0.399965i	$-0.130978\pi$
$151$	207.605	+	359.583i	0.111885	+	0.193791i	−0.804645	+	0.593756i	$0.797644\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			748.261i			0.387754i
$156$		0			0
$157$	−2957.44	+	1707.48i	−1.50337	+	0.867972i	−0.503379	+	0.864066i	$0.667910\pi$
$157$	−2957.44	+	1707.48i	−1.50337	+	0.867972i	−0.999992	+	0.00390617i	$0.998757\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−282.159	+	708.472i	−0.138120	+	0.346804i
$162$		0			0
$163$	−789.449	+	1367.37i	−0.379352	+	0.657057i	−0.990968	−	0.134098i	$-0.957186\pi$
$163$	−789.449	+	1367.37i	−0.379352	+	0.657057i	0.611616	+	0.791155i	$0.290520\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1986.05			−0.920272			−0.460136	−	0.887848i	$-0.652199\pi$
$167$	−1986.05			−0.920272			−0.460136	+	0.887848i	$0.652199\pi$
$168$		0			0
$169$	1743.50			0.793580
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1789.64	−	3099.75i	0.786496	−	1.36225i	−0.141606	−	0.989923i	$-0.545227\pi$
$173$	1789.64	−	3099.75i	0.786496	−	1.36225i	0.928101	−	0.372327i	$-0.121440\pi$
$174$		0			0
$175$	−1080.08	+	852.520i	−0.466552	+	0.368254i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2211.83	−	1277.00i	0.923574	−	0.533226i	0.0388005	−	0.999247i	$-0.487646\pi$
$179$	2211.83	−	1277.00i	0.923574	−	0.533226i	0.884774	+	0.466021i	$0.154313\pi$
$180$		0			0
$181$		−	1958.55i		−	0.804300i	−0.915574	−	0.402150i	$-0.868263\pi$
$181$		−	1958.55i		−	0.804300i	0.915574	−	0.402150i	$-0.131737\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1437.18	+	2489.27i	0.571155	+	0.989270i
$186$		0			0
$187$	1123.10	+	648.421i	0.439193	+	0.253568i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1047.80	+	604.948i	0.396944	+	0.229176i	0.685164	−	0.728388i	$-0.259730\pi$
$191$	1047.80	+	604.948i	0.396944	+	0.229176i	−0.288221	+	0.957564i	$0.593064\pi$
$192$		0			0
$193$	−533.166	−	923.471i	−0.198850	−	0.344419i	0.749305	−	0.662224i	$-0.230387\pi$
$193$	−533.166	−	923.471i	−0.198850	−	0.344419i	−0.948156	+	0.317805i	$0.897054\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4190.09i			1.51539i	0.652610	+	0.757694i	$0.273674\pi$
$197$			4190.09i			1.51539i	−0.652610	+	0.757694i	$0.726326\pi$
$198$		0			0
$199$	−2224.02	+	1284.04i	−0.792245	+	0.457403i	−0.840752	−	0.541420i	$-0.817887\pi$
$199$	−2224.02	+	1284.04i	−0.792245	+	0.457403i	0.0485073	+	0.998823i	$0.484554\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1848.44	+	269.143i	−0.639090	+	0.0930549i
$204$		0			0
$205$	2449.56	−	4242.77i	0.834561	−	1.44550i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	27.6513			0.00915160
$210$		0			0
$211$	668.025			0.217956			0.108978	−	0.994044i	$-0.465242\pi$
$211$	668.025			0.217956			0.108978	+	0.994044i	$0.465242\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−801.154	+	1387.64i	−0.254131	+	0.440168i
$216$		0			0
$217$	608.187	+	770.530i	0.190260	+	0.241046i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1430.08	+	825.658i	−0.435284	+	0.251311i
$222$		0			0
$223$		−	1802.53i		−	0.541283i	−0.962680	−	0.270642i	$-0.912764\pi$
$223$		−	1802.53i		−	0.541283i	0.962680	−	0.270642i	$-0.0872359\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1848.54	+	3201.77i	0.540493	+	0.936162i	0.998876	+	0.0474067i	$0.0150957\pi$
$227$	1848.54	+	3201.77i	0.540493	+	0.936162i	−0.458382	+	0.888755i	$0.651571\pi$
$228$		0			0
$229$	−1852.26	−	1069.40i	−0.534500	−	0.308594i	0.208347	−	0.978055i	$-0.433192\pi$
$229$	−1852.26	−	1069.40i	−0.534500	−	0.308594i	−0.742847	+	0.669461i	$0.766525\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−794.607	−	458.766i	−0.223418	−	0.128990i	0.384114	−	0.923286i	$-0.374507\pi$
$233$	−794.607	−	458.766i	−0.223418	−	0.128990i	−0.607532	+	0.794295i	$0.707840\pi$
$234$		0			0
$235$	645.546	+	1118.12i	0.179195	+	0.310374i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	4966.27i		−	1.34411i	−0.740503	−	0.672053i	$-0.765413\pi$
$239$		−	4966.27i		−	1.34411i	0.740503	−	0.672053i	$-0.234587\pi$
$240$		0			0
$241$	5588.13	−	3226.31i	1.49362	−	0.862344i	0.493650	−	0.869660i	$-0.335662\pi$
$241$	5588.13	−	3226.31i	1.49362	−	0.862344i	0.999973	+	0.00731642i	$0.00232891\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1124.75	+	4709.78i	−0.293295	+	1.22815i
$246$		0			0
$247$	−17.6047	+	30.4923i	−0.00453507	+	0.00785497i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7731.71			1.94431			0.972153	−	0.234345i	$-0.0752946\pi$
$251$	7731.71			1.94431			0.972153	+	0.234345i	$0.0752946\pi$
$252$		0			0
$253$	688.640			0.171124
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−44.6006	+	77.2505i	−0.0108253	+	0.0187500i	−0.871387	−	0.490596i	$-0.836779\pi$
$257$	−44.6006	+	77.2505i	−0.0108253	+	0.0187500i	0.860562	+	0.509346i	$0.170113\pi$
$258$		0			0
$259$	3503.23	+	1395.21i	0.840465	+	0.334727i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−713.061	+	411.686i	−0.167183	+	0.0965233i	−0.581257	−	0.813720i	$-0.697439\pi$
$263$	−713.061	+	411.686i	−0.167183	+	0.0965233i	0.414074	+	0.910243i	$0.364106\pi$
$264$		0			0
$265$			363.110i			0.0841724i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−22.6151	−	39.1705i	−0.00512590	−	0.00887832i	0.863451	−	0.504433i	$-0.168298\pi$
$269$	−22.6151	−	39.1705i	−0.00512590	−	0.00887832i	−0.868577	+	0.495554i	$0.834965\pi$
$270$		0			0
$271$	5476.64	+	3161.94i	1.22761	+	0.708761i	0.966529	−	0.256556i	$-0.0825879\pi$
$271$	5476.64	+	3161.94i	1.22761	+	0.708761i	0.261080	+	0.965317i	$0.415921\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1076.09	+	621.281i	0.235966	+	0.136235i
$276$		0			0
$277$	−2615.18	−	4529.63i	−0.567260	−	0.982523i	−0.996835	−	0.0794922i	$-0.974670\pi$
$277$	−2615.18	−	4529.63i	−0.567260	−	0.982523i	0.429575	−	0.903031i	$-0.358663\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	6587.84i		−	1.39857i	−0.714844	−	0.699284i	$-0.753502\pi$
$281$		−	6587.84i		−	1.39857i	0.714844	−	0.699284i	$-0.246498\pi$
$282$		0			0
$283$	−4031.60	+	2327.64i	−0.846832	+	0.488919i	−0.859581	−	0.511000i	$-0.829275\pi$
$283$	−4031.60	+	2327.64i	−0.846832	+	0.488919i	0.0127484	+	0.999919i	$0.495942\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−926.056	−	6360.04i	−0.190465	−	1.30809i
$288$		0			0
$289$	−549.918	+	952.485i	−0.111931	+	0.193870i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2987.28			−0.595627			−0.297813	−	0.954624i	$-0.596257\pi$
$293$	−2987.28			−0.595627			−0.297813	+	0.954624i	$0.596257\pi$
$294$		0			0
$295$	12733.1			2.51305
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−438.435	+	759.392i	−0.0848006	+	0.146879i
$300$		0			0
$301$	302.876	+	2080.11i	0.0579982	+	0.398325i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5676.23	+	3277.17i	−1.06564	+	0.615248i
$306$		0			0
$307$			6351.43i			1.18077i	0.807123	+	0.590383i	$0.201023\pi$
$307$			6351.43i			1.18077i	−0.807123	+	0.590383i	$0.798977\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2152.75	−	3728.67i	−0.392512	−	0.679850i	0.600268	−	0.799799i	$-0.295060\pi$
$311$	−2152.75	−	3728.67i	−0.392512	−	0.679850i	−0.992780	+	0.119948i	$0.961727\pi$
$312$		0			0
$313$	6376.15	+	3681.27i	1.15144	+	0.664786i	0.949238	−	0.314558i	$-0.101856\pi$
$313$	6376.15	+	3681.27i	1.15144	+	0.664786i	0.202204	+	0.979343i	$0.435189\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−647.462	−	373.812i	−0.114716	−	0.0662315i	0.441544	−	0.897240i	$-0.354431\pi$
$317$	−647.462	−	373.812i	−0.114716	−	0.0662315i	−0.556260	+	0.831008i	$0.687764\pi$
$318$		0			0
$319$	843.397	+	1460.81i	0.148029	+	0.256393i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			128.206i			0.0220854i
$324$		0			0
$325$	−1370.23	+	791.100i	−0.233866	+	0.135023i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1573.56	+	626.695i	0.263688	+	0.105018i
$330$		0			0
$331$	3970.37	−	6876.89i	0.659309	−	1.14196i	−0.321486	−	0.946914i	$-0.604182\pi$
$331$	3970.37	−	6876.89i	0.659309	−	1.14196i	0.980795	−	0.195043i	$-0.0624845\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4900.76			−0.799275
$336$		0			0
$337$	−1880.67			−0.303996			−0.151998	−	0.988381i	$-0.548571\pi$
$337$	−1880.67			−0.303996			−0.151998	+	0.988381i	$0.548571\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	443.221	−	767.682i	0.0703864	−	0.121913i
$342$		0			0
$343$	2669.89	+	5764.14i	0.420293	+	0.907389i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	9038.98	−	5218.66i	1.39838	−	0.807355i	0.404157	−	0.914690i	$-0.367565\pi$
$347$	9038.98	−	5218.66i	1.39838	−	0.807355i	0.994223	+	0.107335i	$0.0342317\pi$
$348$		0			0
$349$		−	5570.29i		−	0.854357i	−0.904167	−	0.427179i	$-0.859508\pi$
$349$		−	5570.29i		−	0.854357i	0.904167	−	0.427179i	$-0.140492\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2947.40	−	5105.05i	−0.444403	−	0.769729i	0.553607	−	0.832778i	$-0.313251\pi$
$353$	−2947.40	−	5105.05i	−0.444403	−	0.769729i	−0.998010	+	0.0630488i	$0.979918\pi$
$354$		0			0
$355$	7588.53	+	4381.24i	1.13453	+	0.655020i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6504.85	−	3755.58i	−0.956303	−	0.552122i	−0.0612698	−	0.998121i	$-0.519515\pi$
$359$	−6504.85	−	3755.58i	−0.956303	−	0.552122i	−0.895033	+	0.445999i	$0.852848\pi$
$360$		0			0
$361$	−3428.13	−	5937.70i	−0.499801	−	0.865680i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			9206.67i			1.32027i
$366$		0			0
$367$	7104.72	−	4101.91i	1.01053	−	0.583428i	0.0991809	−	0.995069i	$-0.468378\pi$
$367$	7104.72	−	4101.91i	1.01053	−	0.583428i	0.911346	+	0.411642i	$0.135044\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	295.136	+	373.917i	0.0413011	+	0.0523256i
$372$		0			0
$373$	1702.24	−	2948.36i	0.236296	−	0.409277i	−0.723353	−	0.690479i	$-0.757400\pi$
$373$	1702.24	−	2948.36i	0.236296	−	0.409277i	0.959649	+	0.281202i	$0.0907332\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2147.86			−0.293422
$378$		0			0
$379$	11782.0			1.59684			0.798419	−	0.602103i	$-0.205670\pi$
$379$	11782.0			1.59684			0.798419	+	0.602103i	$0.205670\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4515.01	+	7820.23i	−0.602367	+	1.04333i	0.390095	+	0.920775i	$0.372442\pi$
$383$	−4515.01	+	7820.23i	−0.602367	+	1.04333i	−0.992462	+	0.122555i	$0.960891\pi$
$384$		0			0
$385$	4327.02	−	630.038i	0.572794	−	0.0834018i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	4977.97	−	2874.03i	0.648826	−	0.374600i	−0.139180	−	0.990267i	$-0.544447\pi$
$389$	4977.97	−	2874.03i	0.648826	−	0.374600i	0.788006	+	0.615667i	$0.211113\pi$
$390$		0			0
$391$			3192.90i			0.412971i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−964.363	−	1670.33i	−0.122841	−	0.212768i
$396$		0			0
$397$	−7579.38	−	4375.96i	−0.958182	−	0.553206i	−0.0625687	−	0.998041i	$-0.519929\pi$
$397$	−7579.38	−	4375.96i	−0.958182	−	0.553206i	−0.895613	+	0.444834i	$0.853263\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	13106.9	+	7567.27i	1.63224	+	0.942373i	0.983401	+	0.181445i	$0.0580773\pi$
$401$	13106.9	+	7567.27i	1.63224	+	0.942373i	0.648836	+	0.760928i	$0.275256\pi$
$402$		0			0
$403$	564.370	+	977.517i	0.0697599	+	0.120828i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	3405.17i		−	0.414713i
$408$		0			0
$409$	−6470.35	+	3735.66i	−0.782245	+	0.451630i	−0.837225	−	0.546858i	$-0.815824\pi$
$409$	−6470.35	+	3735.66i	−0.782245	+	0.451630i	0.0549801	+	0.998487i	$0.482490\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	13112.0	−	10349.4i	1.56223	−	1.23308i
$414$		0			0
$415$	2107.03	−	3649.49i	0.249229	−	0.431678i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7134.02			0.831789			0.415894	−	0.909413i	$-0.363469\pi$
$419$	7134.02			0.831789			0.415894	+	0.909413i	$0.363469\pi$
$420$		0			0
$421$	16443.9			1.90363			0.951813	−	0.306679i	$-0.0992177\pi$
$421$	16443.9			1.90363			0.951813	+	0.306679i	$0.0992177\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2880.58	+	4989.32i	−0.328774	+	0.569453i
$426$		0			0
$427$	−3181.48	+	7988.35i	−0.360568	+	0.905347i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−14085.5	+	8132.26i	−1.57419	+	0.908856i	−0.578538	+	0.815656i	$0.696376\pi$
$431$	−14085.5	+	8132.26i	−1.57419	+	0.908856i	−0.995647	+	0.0932007i	$0.970290\pi$
$432$		0			0
$433$		−	9052.67i		−	1.00472i	−0.864659	−	0.502360i	$-0.832465\pi$
$433$		−	9052.67i		−	1.00472i	0.864659	−	0.502360i	$-0.167535\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	34.0396	+	58.9582i	0.00372616	+	0.00645390i
$438$		0			0
$439$	7170.58	+	4139.94i	0.779575	+	0.450088i	0.836279	−	0.548303i	$-0.184726\pi$
$439$	7170.58	+	4139.94i	0.779575	+	0.450088i	−0.0567050	+	0.998391i	$0.518059\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	15166.5	+	8756.39i	1.62660	+	0.939117i	0.985098	+	0.171993i	$0.0550207\pi$
$443$	15166.5	+	8756.39i	1.62660	+	0.939117i	0.641500	+	0.767123i	$0.278313\pi$
$444$		0			0
$445$	−5967.12	−	10335.3i	−0.635659	−	1.10099i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			11731.8i			1.23309i	0.787318	+	0.616547i	$0.211469\pi$
$449$			11731.8i			1.23309i	−0.787318	+	0.616547i	$0.788531\pi$
$450$		0			0
$451$	−5026.28	+	2901.92i	−0.524785	+	0.302985i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2060.11	+	5172.71i	−0.212262	+	0.532968i
$456$		0			0
$457$	3010.39	−	5214.16i	0.308141	−	0.533715i	−0.669815	−	0.742528i	$-0.733627\pi$
$457$	3010.39	−	5214.16i	0.308141	−	0.533715i	0.977956	+	0.208813i	$0.0669599\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−15644.8			−1.58058			−0.790292	−	0.612730i	$-0.790071\pi$
$461$	−15644.8			−1.58058			−0.790292	+	0.612730i	$0.790071\pi$
$462$		0			0
$463$	4525.62			0.454262			0.227131	−	0.973864i	$-0.427065\pi$
$463$	4525.62			0.454262			0.227131	+	0.973864i	$0.427065\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4693.66	+	8129.66i	−0.465090	+	0.805559i	−0.999206	−	0.0398525i	$-0.987311\pi$
$467$	−4693.66	+	8129.66i	−0.465090	+	0.805559i	0.534116	+	0.845411i	$0.320645\pi$
$468$		0			0
$469$	−5046.61	+	3983.34i	−0.496867	+	0.392182i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1643.89	−	949.102i	0.159802	−	0.0922616i
$474$		0			0
$475$			122.840i			0.0118659i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−469.441	−	813.096i	−0.0447794	−	0.0775601i	0.842767	−	0.538279i	$-0.180925\pi$
$479$	−469.441	−	813.096i	−0.0447794	−	0.0775601i	−0.887546	+	0.460718i	$0.847592\pi$
$480$		0			0
$481$	3755.02	+	2167.96i	0.355955	+	0.205511i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−6046.12	−	3490.73i	−0.566062	−	0.326816i
$486$		0			0
$487$	−1581.32	−	2738.92i	−0.147138	−	0.254851i	0.783030	−	0.621983i	$-0.213673\pi$
$487$	−1581.32	−	2738.92i	−0.147138	−	0.254851i	−0.930169	+	0.367133i	$0.880340\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	5092.84i		−	0.468099i	−0.972225	−	0.234049i	$-0.924802\pi$
$491$		−	5092.84i		−	0.468099i	0.972225	−	0.234049i	$-0.0751978\pi$
$492$		0			0
$493$	−6773.06	+	3910.43i	−0.618749	+	0.357235i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	11375.4	−	1656.33i	1.02668	−	0.149490i
$498$		0			0
$499$	−3434.22	+	5948.25i	−0.308090	+	0.533628i	−0.977945	−	0.208865i	$-0.933023\pi$
$499$	−3434.22	+	5948.25i	−0.308090	+	0.533628i	0.669854	+	0.742493i	$0.266357\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	14365.3			1.27340			0.636698	−	0.771113i	$-0.280300\pi$
$503$	14365.3			1.27340			0.636698	+	0.771113i	$0.280300\pi$
$504$		0			0
$505$	−5138.58			−0.452800
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5989.45	−	10374.0i	0.521567	−	0.903381i	−0.478118	−	0.878296i	$-0.658681\pi$
$509$	5989.45	−	10374.0i	0.521567	−	0.903381i	0.999685	−	0.0250853i	$-0.00798573\pi$
$510$		0			0
$511$	7483.17	+	9480.66i	0.647820	+	0.820743i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−9883.69	+	5706.35i	−0.845684	+	0.488256i
$516$		0			0
$517$		−	1529.52i		−	0.130112i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9351.25	−	16196.8i	−0.786345	−	1.36199i	−0.928193	−	0.372100i	$-0.878638\pi$
$521$	−9351.25	−	16196.8i	−0.786345	−	1.36199i	0.141848	−	0.989888i	$-0.454696\pi$
$522$		0			0
$523$	2002.32	+	1156.04i	0.167410	+	0.0966543i	0.581364	−	0.813644i	$-0.302519\pi$
$523$	2002.32	+	1156.04i	0.167410	+	0.0966543i	−0.413954	+	0.910298i	$0.635853\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3559.37	+	2055.00i	0.294210	+	0.169862i
$528$		0			0
$529$	−5235.76	−	9068.61i	−0.430325	−	0.745345i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	7390.25i		−	0.600576i
$534$		0			0
$535$	−20915.6	+	12075.6i	−1.69020	+	0.975839i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3943.70	−	4165.79i	0.315153	−	0.332900i
$540$		0			0
$541$	342.908	−	593.933i	0.0272509	−	0.0472000i	−0.852078	−	0.523414i	$-0.824658\pi$
$541$	342.908	−	593.933i	0.0272509	−	0.0472000i	0.879329	+	0.476214i	$0.157991\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−22162.7			−1.74192
$546$		0			0
$547$	−6325.53			−0.494443			−0.247221	−	0.968959i	$-0.579518\pi$
$547$	−6325.53			−0.494443			−0.247221	+	0.968959i	$0.579518\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−83.3783	+	144.416i	−0.00644653	+	0.0111657i
$552$		0			0
$553$	−2350.70	−	936.202i	−0.180763	−	0.0719916i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	12570.4	−	7257.52i	0.956239	−	0.552085i	0.0612254	−	0.998124i	$-0.480499\pi$
$557$	12570.4	−	7257.52i	0.956239	−	0.552085i	0.895013	+	0.446039i	$0.147166\pi$
$558$		0			0
$559$			2417.05i			0.182881i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6640.99	−	11502.5i	−0.497130	−	0.861055i	0.502864	−	0.864365i	$-0.332280\pi$
$563$	−6640.99	−	11502.5i	−0.497130	−	0.861055i	−0.999995	+	0.00331036i	$0.998946\pi$
$564$		0			0
$565$	−1024.28	−	591.371i	−0.0762689	−	0.0440339i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	2749.92	+	1587.67i	0.202606	+	0.116974i	0.597870	−	0.801593i	$-0.296014\pi$
$569$	2749.92	+	1587.67i	0.202606	+	0.116974i	−0.395265	+	0.918567i	$0.629347\pi$
$570$		0			0
$571$	−3099.49	−	5368.48i	−0.227162	−	0.393457i	0.729804	−	0.683657i	$-0.239611\pi$
$571$	−3099.49	−	5368.48i	−0.227162	−	0.393457i	−0.956966	+	0.290200i	$0.906278\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			3059.26i			0.221878i
$576$		0			0
$577$	−18335.4	+	10585.9i	−1.32290	+	0.763776i	−0.984190	−	0.177116i	$-0.943323\pi$
$577$	−18335.4	+	10585.9i	−1.32290	+	0.763776i	−0.338708	+	0.940891i	$0.609990\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−796.563	−	5470.69i	−0.0568795	−	0.390641i
$582$		0			0
$583$	215.083	−	372.535i	0.0152793	−	0.0264645i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6456.95			−0.454015			−0.227008	−	0.973893i	$-0.572894\pi$
$587$	−6456.95			−0.454015			−0.227008	+	0.973893i	$0.572894\pi$
$588$		0			0
$589$	87.6339			0.00613054
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−4817.40	+	8343.98i	−0.333604	+	0.577818i	−0.983216	−	0.182448i	$-0.941598\pi$
$593$	−4817.40	+	8343.98i	−0.333604	+	0.577818i	0.649612	+	0.760266i	$0.274931\pi$
$594$		0			0
$595$	2921.18	+	20062.3i	0.201272	+	1.38231i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12078.3	+	6973.39i	−0.823881	+	0.475668i	−0.851753	−	0.523944i	$-0.824460\pi$
$599$	−12078.3	+	6973.39i	−0.823881	+	0.475668i	0.0278719	+	0.999612i	$0.491127\pi$
$600$		0			0
$601$		−	14889.6i		−	1.01058i	−0.862950	−	0.505289i	$-0.831386\pi$
$601$		−	14889.6i		−	1.01058i	0.862950	−	0.505289i	$-0.168614\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7420.72	+	12853.1i	0.498670	+	0.863722i
$606$		0			0
$607$	−9421.45	−	5439.48i	−0.629992	−	0.363726i	0.150757	−	0.988571i	$-0.451829\pi$
$607$	−9421.45	−	5439.48i	−0.629992	−	0.363726i	−0.780749	+	0.624845i	$0.785162\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1686.66	+	973.795i	0.111678	+	0.0644771i
$612$		0			0
$613$	−12365.0	−	21416.7i	−0.814708	−	1.41112i	−0.909537	−	0.415623i	$-0.863564\pi$
$613$	−12365.0	−	21416.7i	−0.814708	−	1.41112i	0.0948289	−	0.995494i	$-0.469770\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	21456.1i		−	1.39999i	−0.714149	−	0.699993i	$-0.753186\pi$
$617$		−	21456.1i		−	1.39999i	0.714149	−	0.699993i	$-0.246814\pi$
$618$		0			0
$619$	−21326.4	+	12312.8i	−1.38478	+	0.799506i	−0.992722	−	0.120432i	$-0.961572\pi$
$619$	−21326.4	+	12312.8i	−1.38478	+	0.799506i	−0.392063	+	0.919938i	$0.628239\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−14545.3	−	5792.87i	−0.935383	−	0.372530i
$624$		0			0
$625$	9696.04	−	16794.0i	0.620547	−	1.07482i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	15788.1			1.00082
$630$		0			0
$631$	11824.3			0.745988			0.372994	−	0.927834i	$-0.378331\pi$
$631$	11824.3			0.745988			0.372994	+	0.927834i	$0.378331\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−11373.2	+	19698.9i	−0.710756	+	1.23107i
$636$		0			0
$637$	2082.96	+	7001.11i	0.129560	+	0.435470i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	17132.3	−	9891.33i	1.05567	−	0.609492i	0.131439	−	0.991324i	$-0.458040\pi$
$641$	17132.3	−	9891.33i	1.05567	−	0.609492i	0.924232	+	0.381833i	$0.124707\pi$
$642$		0			0
$643$			9218.81i			0.565403i	0.959208	+	0.282702i	$0.0912306\pi$
$643$			9218.81i			0.565403i	−0.959208	+	0.282702i	$0.908769\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	6396.37	+	11078.8i	0.388666	+	0.673190i	0.992270	−	0.124094i	$-0.0396026\pi$
$647$	6396.37	+	11078.8i	0.388666	+	0.673190i	−0.603604	+	0.797284i	$0.706269\pi$
$648$		0			0
$649$	−13063.6	−	7542.25i	−0.790123	−	0.456178i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−7100.22	−	4099.32i	−0.425503	−	0.245664i	0.271926	−	0.962318i	$-0.412339\pi$
$653$	−7100.22	−	4099.32i	−0.425503	−	0.245664i	−0.697429	+	0.716654i	$0.745673\pi$
$654$		0			0
$655$	−13348.2	−	23119.8i	−0.796271	−	1.37918i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	27391.7i		−	1.61917i	−0.587006	−	0.809583i	$-0.699693\pi$
$659$		−	27391.7i		−	1.61917i	0.587006	−	0.809583i	$-0.300307\pi$
$660$		0			0
$661$	−16024.0	+	9251.48i	−0.942909	+	0.544389i	−0.890871	−	0.454256i	$-0.849905\pi$
$661$	−16024.0	+	9251.48i	−0.942909	+	0.544389i	−0.0520381	+	0.998645i	$0.516572\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	267.826	+	339.317i	0.0156178	+	0.0197867i
$666$		0			0
$667$	−2076.49	+	3596.58i	−0.120543	+	0.208786i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	7764.74			0.446728
$672$		0			0
$673$	−3129.18			−0.179229			−0.0896145	−	0.995977i	$-0.528563\pi$
$673$	−3129.18			−0.179229			−0.0896145	+	0.995977i	$0.528563\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3088.55	+	5349.53i	−0.175336	+	0.303692i	−0.940278	−	0.340408i	$-0.889435\pi$
$677$	−3088.55	+	5349.53i	−0.175336	+	0.303692i	0.764941	+	0.644100i	$0.222768\pi$
$678$		0			0
$679$	−9063.31	+	1319.67i	−0.512250	+	0.0745864i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−2664.39	+	1538.29i	−0.149268	+	0.0861799i	−0.572774	−	0.819713i	$-0.694133\pi$
$683$	−2664.39	+	1538.29i	−0.149268	+	0.0861799i	0.423506	+	0.905893i	$0.360799\pi$
$684$		0			0
$685$		−	6752.07i		−	0.376618i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	273.873	+	474.362i	0.0151433	+	0.0262289i
$690$		0			0
$691$	−17103.0	−	9874.41i	−0.941575	−	0.543619i	−0.0511213	−	0.998692i	$-0.516280\pi$
$691$	−17103.0	−	9874.41i	−0.941575	−	0.543619i	−0.890454	+	0.455074i	$0.849613\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	12519.4	+	7228.07i	0.683291	+	0.394499i
$696$		0			0
$697$	−13454.8	−	23304.4i	−0.731188	−	1.26645i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			26015.3i			1.40169i	0.713314	+	0.700845i	$0.247194\pi$
$701$			26015.3i			1.40169i	−0.713314	+	0.700845i	$0.752806\pi$
$702$		0			0
$703$	291.535	−	168.318i	0.0156408	−	0.00903020i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5291.51	+	4176.63i	−0.281482	+	0.222176i
$708$		0			0
$709$	−2763.16	+	4785.93i	−0.146365	+	0.253511i	−0.929881	−	0.367860i	$-0.880091\pi$
$709$	−2763.16	+	4785.93i	−0.146365	+	0.253511i	0.783517	+	0.621371i	$0.213424\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2182.47			0.114634
$714$		0			0
$715$	5027.92			0.262984
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4116.00	−	7129.12i	0.213492	−	0.369779i	−0.739313	−	0.673362i	$-0.764850\pi$
$719$	4116.00	−	7129.12i	0.213492	−	0.369779i	0.952805	+	0.303583i	$0.0981829\pi$
$720$		0			0
$721$	−5539.72	+	13909.6i	−0.286144	+	0.718477i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−6489.57	+	3746.76i	−0.332437	+	0.191932i
$726$		0			0
$727$		−	5504.31i		−	0.280802i	−0.990095	−	0.140401i	$-0.955161\pi$
$727$		−	5504.31i		−	0.280802i	0.990095	−	0.140401i	$-0.0448392\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4400.53	+	7621.94i	0.222653	+	0.385647i
$732$		0			0
$733$	20792.6	+	12004.6i	1.04774	+	0.604912i	0.922015	−	0.387154i	$-0.126542\pi$
$733$	20792.6	+	12004.6i	1.04774	+	0.604912i	0.125723	+	0.992065i	$0.459875\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5027.95	+	2902.89i	0.251299	+	0.145087i
$738$		0			0
$739$	5290.32	+	9163.10i	0.263339	+	0.456117i	0.967127	−	0.254293i	$-0.0818429\pi$
$739$	5290.32	+	9163.10i	0.263339	+	0.456117i	−0.703788	+	0.710410i	$0.748510\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	3359.87i		−	0.165897i	−0.996554	−	0.0829487i	$-0.973566\pi$
$743$		−	3359.87i		−	0.165897i	0.996554	−	0.0829487i	$-0.0264338\pi$
$744$		0			0
$745$	−30510.5	+	17615.2i	−1.50042	+	0.866271i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−11723.0	+	29435.1i	−0.571894	+	1.43596i
$750$		0			0
$751$	9949.87	−	17233.7i	0.483457	−	0.837372i	−0.516363	−	0.856370i	$-0.672714\pi$
$751$	9949.87	−	17233.7i	0.483457	−	0.837372i	0.999820	+	0.0189983i	$0.00604772\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	5861.63			0.282552
$756$		0			0
$757$	1321.99			0.0634725			0.0317363	−	0.999496i	$-0.489896\pi$
$757$	1321.99			0.0634725			0.0317363	+	0.999496i	$0.489896\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	11213.7	−	19422.7i	0.534161	−	0.925194i	−0.465042	−	0.885288i	$-0.653961\pi$
$761$	11213.7	−	19422.7i	0.534161	−	0.925194i	0.999203	−	0.0399057i	$-0.0127058\pi$
$762$		0			0
$763$	−22822.2	+	18013.8i	−1.08286	+	0.854709i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	16634.3	−	9603.82i	0.783090	−	0.452117i
$768$		0			0
$769$		−	4120.19i		−	0.193209i	−0.995323	−	0.0966045i	$-0.969202\pi$
$769$		−	4120.19i		−	0.193209i	0.995323	−	0.0966045i	$-0.0307982\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−3390.45	−	5872.43i	−0.157757	−	0.273243i	0.776303	−	0.630360i	$-0.217093\pi$
$773$	−3390.45	−	5872.43i	−0.157757	−	0.273243i	−0.934059	+	0.357118i	$0.883760\pi$
$774$		0			0
$775$	3410.39	+	1968.99i	0.158071	+	0.0912623i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−496.899	−	286.885i	−0.0228540	−	0.0131947i
$780$		0			0
$781$	−5190.32	−	8989.90i	−0.237803	−	0.411887i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			48209.8i			2.19195i
$786$		0			0
$787$	−22055.2	+	12733.6i	−0.998961	+	0.576751i	−0.907941	−	0.419099i	$-0.862346\pi$
$787$	−22055.2	+	12733.6i	−0.998961	+	0.576751i	−0.0910204	+	0.995849i	$0.529013\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1535.43	+	223.567i	−0.0690186	+	0.0100495i
$792$		0			0
$793$	−4943.56	+	8562.50i	−0.221376	+	0.383434i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−5174.72			−0.229985			−0.114992	−	0.993366i	$-0.536684\pi$
$797$	−5174.72			−0.229985			−0.114992	+	0.993366i	$0.536684\pi$
$798$		0			0
$799$	7091.64			0.313998
$800$		0			0
$801$		0			0
$802$		0			0
$803$	5453.43	−	9445.61i	0.239660	−	0.415104i
$804$		0			0
$805$	6670.05	+	8450.49i	0.292035	+	0.369988i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−18774.6	+	10839.5i	−0.815921	+	0.471072i	−0.849008	−	0.528380i	$-0.822800\pi$
$809$	−18774.6	+	10839.5i	−0.815921	+	0.471072i	0.0330866	+	0.999452i	$0.489466\pi$
$810$		0			0
$811$			929.429i			0.0402425i	0.999798	+	0.0201213i	$0.00640523\pi$
$811$			929.429i			0.0402425i	−0.999798	+	0.0201213i	$0.993595\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	11144.8	+	19303.4i	0.479003	+	0.829657i
$816$		0			0
$817$	162.516	+	93.8284i	0.00695924	+	0.00401792i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−21375.6	−	12341.2i	−0.908667	−	0.524619i	−0.0286647	−	0.999589i	$-0.509126\pi$
$821$	−21375.6	−	12341.2i	−0.908667	−	0.524619i	−0.880002	+	0.474970i	$0.842459\pi$
$822$		0			0
$823$	−6454.73	−	11179.9i	−0.273387	−	0.473521i	0.696340	−	0.717712i	$-0.254811\pi$
$823$	−6454.73	−	11179.9i	−0.273387	−	0.473521i	−0.969727	+	0.244192i	$0.921477\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	15358.0i		−	0.645768i	−0.946439	−	0.322884i	$-0.895348\pi$
$827$		−	15358.0i		−	0.645768i	0.946439	−	0.322884i	$-0.104652\pi$
$828$		0			0
$829$	−9480.96	+	5473.83i	−0.397210	+	0.229329i	−0.685280	−	0.728280i	$-0.740320\pi$
$829$	−9480.96	+	5473.83i	−0.397210	+	0.229329i	0.288069	+	0.957610i	$0.406987\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	19314.8	+	18285.1i	0.803382	+	0.760552i
$834$		0			0
$835$	−14018.8	+	24281.3i	−0.581007	+	1.00633i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	8223.91			0.338404			0.169202	−	0.985581i	$-0.445881\pi$
$839$	8223.91			0.338404			0.169202	+	0.985581i	$0.445881\pi$
$840$		0			0
$841$	14216.5			0.582905
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12306.7	−	21315.8i	0.501021	−	0.867794i
$846$		0			0
$847$	18088.5	+	7204.03i	0.733801	+	0.292247i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	7260.51	−	4191.86i	0.292464	−	0.168854i
$852$		0			0
$853$			28381.4i			1.13923i	0.821913	+	0.569613i	$0.192907\pi$
$853$			28381.4i			1.13923i	−0.821913	+	0.569613i	$0.807093\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−5195.47	−	8998.82i	−0.207088	−	0.358686i	0.743708	−	0.668504i	$-0.233065\pi$
$857$	−5195.47	−	8998.82i	−0.207088	−	0.358686i	−0.950796	+	0.309818i	$0.899732\pi$
$858$		0			0
$859$	30647.1	+	17694.1i	1.21731	+	0.702812i	0.964341	−	0.264663i	$-0.0852608\pi$
$859$	30647.1	+	17694.1i	1.21731	+	0.702812i	0.252965	+	0.967475i	$0.418594\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−21702.3	−	12529.8i	−0.856031	−	0.494230i	0.00665032	−	0.999978i	$-0.497883\pi$
$863$	−21702.3	−	12529.8i	−0.856031	−	0.494230i	−0.862681	+	0.505748i	$0.831216\pi$
$864$		0			0
$865$	−25264.8	−	43759.9i	−0.993097	−	1.72009i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			2284.90i			0.0891945i
$870$		0			0
$871$	−6402.28	+	3696.36i	−0.249062	+	0.143796i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1910.09	−	13118.3i	−0.0737975	−	0.506832i
$876$		0			0
$877$	17309.3	−	29980.6i	0.666470	−	1.15436i	−0.312415	−	0.949946i	$-0.601138\pi$
$877$	17309.3	−	29980.6i	0.666470	−	1.15436i	0.978885	−	0.204414i	$-0.0655287\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18543.7			−0.709140			−0.354570	−	0.935029i	$-0.615373\pi$
$881$	−18543.7			−0.709140			−0.354570	+	0.935029i	$0.615373\pi$
$882$		0			0
$883$	42520.3			1.62052			0.810261	−	0.586069i	$-0.199325\pi$
$883$	42520.3			1.62052			0.810261	+	0.586069i	$0.199325\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	12927.0	−	22390.2i	0.489341	−	0.847563i	−0.510584	−	0.859828i	$-0.670571\pi$
$887$	12927.0	−	22390.2i	0.489341	−	0.847563i	0.999925	+	0.0122646i	$0.00390406\pi$
$888$		0			0
$889$	4299.62	+	29529.3i	0.162210	+	1.11404i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	130.950	−	75.6041i	0.00490715	−	0.00283314i
$894$		0			0
$895$		−	36055.4i		−	1.34659i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2672.93	+	4629.65i	0.0991626	+	0.171755i
$900$		0			0
$901$	1727.26	+	997.236i	0.0638663	+	0.0368732i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−23945.1	−	13824.7i	−0.879516	−	0.507789i
$906$		0			0
$907$	−15588.6	−	27000.2i	−0.570684	−	0.988454i	−0.996496	−	0.0836425i	$-0.973345\pi$
$907$	−15588.6	−	27000.2i	−0.570684	−	0.988454i	0.425811	−	0.904812i	$-0.359989\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	24793.5i		−	0.901695i	−0.892601	−	0.450847i	$-0.851122\pi$
$911$		−	24793.5i		−	0.901695i	0.892601	−	0.450847i	$-0.148878\pi$
$912$		0			0
$913$	−4323.44	+	2496.14i	−0.156719	+	0.0904820i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−32537.2	−	12958.4i	−1.17173	−	0.466657i
$918$		0			0
$919$	15319.2	−	26533.7i	0.549875	−	0.952411i	−0.448408	−	0.893829i	$-0.648009\pi$
$919$	15319.2	−	26533.7i	0.549875	−	0.952411i	0.998283	−	0.0585822i	$-0.0186580\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	13218.1			0.471373
$924$		0			0
$925$	15127.3			0.537712
$926$		0			0
$927$		0			0
$928$		0			0
$929$	12767.1	−	22113.2i	0.450887	−	0.780959i	−0.547554	−	0.836770i	$-0.684441\pi$
$929$	12767.1	−	22113.2i	0.450887	−	0.780959i	0.998441	+	0.0558110i	$0.0177744\pi$
$930$		0			0
$931$	551.594	+	131.726i	0.0194176	+	0.00463712i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	15855.1	−	9153.93i	0.554563	−	0.320177i
$936$		0			0
$937$		−	38358.6i		−	1.33738i	−0.743544	−	0.668688i	$-0.766856\pi$
$937$		−	38358.6i		−	1.33738i	0.743544	−	0.668688i	$-0.233144\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	19423.6	+	33642.7i	0.672892	+	1.16548i	0.977080	+	0.212871i	$0.0682814\pi$
$941$	19423.6	+	33642.7i	0.672892	+	1.16548i	−0.304189	+	0.952612i	$0.598385\pi$
$942$		0			0
$943$	−12375.0	−	7144.69i	−0.427343	−	0.246727i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−47466.8	−	27404.9i	−1.62879	−	0.940381i	−0.984455	−	0.175636i	$-0.943802\pi$
$947$	−47466.8	−	27404.9i	−1.62879	−	0.940381i	−0.644333	−	0.764745i	$-0.722865\pi$
$948$		0			0
$949$	6944.05	+	12027.4i	0.237527	+	0.411409i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			11947.0i			0.406087i	0.979170	+	0.203044i	$0.0650833\pi$
$953$			11947.0i			0.406087i	−0.979170	+	0.203044i	$0.934917\pi$
$954$		0			0
$955$	14792.1	−	8540.21i	0.501215	−	0.289377i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−5488.08	−	6953.01i	−0.184796	−	0.234124i
$960$		0			0
$961$	−13490.8	+	23366.8i	−0.452849	+	0.784357i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−15053.7			−0.502171
$966$		0			0
$967$	38713.1			1.28741			0.643707	−	0.765272i	$-0.277396\pi$
$967$	38713.1			1.28741			0.643707	+	0.765272i	$0.277396\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1978.83	−	3427.43i	0.0654001	−	0.113276i	−0.831471	−	0.555568i	$-0.812501\pi$
$971$	1978.83	−	3427.43i	0.0654001	−	0.113276i	0.896871	+	0.442291i	$0.145834\pi$
$972$		0			0
$973$	18767.0	−	2732.57i	0.618336	−	0.0900331i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−6009.57	+	3469.63i	−0.196789	+	0.113616i	−0.595157	−	0.803609i	$-0.702910\pi$
$977$	−6009.57	+	3469.63i	−0.196789	+	0.113616i	0.398368	+	0.917226i	$0.369577\pi$
$978$		0			0
$979$			14138.1i			0.461549i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1900.65	+	3292.02i	0.0616696	+	0.106815i	0.895212	−	0.445641i	$-0.147024\pi$
$983$	1900.65	+	3292.02i	0.0616696	+	0.106815i	−0.833542	+	0.552456i	$0.813691\pi$
$984$		0			0
$985$	51227.6	+	29576.3i	1.65710	+	0.956729i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	4047.35	+	2336.74i	0.130130	+	0.0751305i
$990$		0			0
$991$	−4425.63	−	7665.42i	−0.141862	−	0.245712i	0.786336	−	0.617799i	$-0.211975\pi$
$991$	−4425.63	−	7665.42i	−0.141862	−	0.245712i	−0.928198	+	0.372087i	$0.878642\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			36254.2i			1.15511i
$996$		0			0
$997$	−7868.04	+	4542.61i	−0.249933	+	0.144299i	−0.619734	−	0.784812i	$-0.712759\pi$
$997$	−7868.04	+	4542.61i	−0.249933	+	0.144299i	0.369801	+	0.929111i	$0.379426\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.4.t.c.593.6	yes	12
3.2	odd	2	inner	756.4.t.c.593.1	yes	12
7.3	odd	6	inner	756.4.t.c.269.1	✓	12
21.17	even	6	inner	756.4.t.c.269.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.4.t.c.269.1	✓	12	7.3	odd	6	inner
756.4.t.c.269.6	yes	12	21.17	even	6	inner
756.4.t.c.593.1	yes	12	3.2	odd	2	inner
756.4.t.c.593.6	yes	12	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:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	756.t (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(7.05863 - 12.2259i) q^{5} +(-2.66851 - 18.3270i) q^{7} +O(q^{10})\)	\(q+(7.05863 - 12.2259i) q^{5} +(-2.66851 - 18.3270i) q^{7} +(-14.4837 + 8.36214i) q^{11} -21.2956i q^{13} +(-38.7712 - 67.1538i) q^{17} +(-1.43186 - 0.826683i) q^{19} +(-35.6595 - 20.5880i) q^{23} +(-37.1485 - 64.3430i) q^{25} -100.859i q^{29} +(-45.9022 + 26.5017i) q^{31} +(-242.900 - 96.7386i) q^{35} +(-101.803 + 176.328i) q^{37} +347.031 q^{41} -113.500 q^{43} +(-45.7274 + 79.2022i) q^{47} +(-328.758 + 97.8116i) q^{49} +(-22.2751 + 12.8605i) q^{53} +236.101i q^{55} +(450.976 + 781.113i) q^{59} +(-402.078 - 232.140i) q^{61} +(-260.358 - 150.318i) q^{65} +(-173.573 - 300.638i) q^{67} +620.693i q^{71} +(-564.784 + 326.078i) q^{73} +(191.903 + 243.127i) q^{77} +(68.3109 - 118.318i) q^{79} +298.505 q^{83} -1094.69 q^{85} +(422.682 - 732.108i) q^{89} +(-390.285 + 56.8276i) q^{91} +(-20.2139 + 11.6705i) q^{95} -494.533i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 42 q^{7}+O(q^{10}) \) 12 * q - 42 * q^7	\( 12 q - 42 q^{7} + 186 q^{19} - 246 q^{25} + 738 q^{31} - 336 q^{37} + 756 q^{43} - 1218 q^{49} - 426 q^{61} - 1440 q^{67} + 2730 q^{73} + 1992 q^{79} + 3288 q^{85} + 1764 q^{91}+O(q^{100}) \) 12 * q - 42 * q^7 + 186 * q^19 - 246 * q^25 + 738 * q^31 - 336 * q^37 + 756 * q^43 - 1218 * q^49 - 426 * q^61 - 1440 * q^67 + 2730 * q^73 + 1992 * q^79 + 3288 * q^85 + 1764 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more