LMFDB - Embedded newform 1815.4.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,4,Mod(1,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1815.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1815 = 3 \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1815.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$107.088466660$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	1815.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -8.00000 q^{4} -5.00000 q^{5} +26.8328 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -8.00000 q^{4} -5.00000 q^{5} +26.8328 q^{7} +9.00000 q^{9} -24.0000 q^{12} +26.8328 q^{13} -15.0000 q^{15} +64.0000 q^{16} -53.6656 q^{17} -80.4984 q^{19} +40.0000 q^{20} +80.4984 q^{21} -72.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -214.663 q^{28} -295.161 q^{29} -20.0000 q^{31} -134.164 q^{35} -72.0000 q^{36} +214.000 q^{37} +80.4984 q^{39} -26.8328 q^{41} +241.495 q^{43} -45.0000 q^{45} -264.000 q^{47} +192.000 q^{48} +377.000 q^{49} -160.997 q^{51} -214.663 q^{52} +78.0000 q^{53} -241.495 q^{57} +480.000 q^{59} +120.000 q^{60} +107.331 q^{61} +241.495 q^{63} -512.000 q^{64} -134.164 q^{65} -524.000 q^{67} +429.325 q^{68} -216.000 q^{69} +492.000 q^{71} +939.149 q^{73} +75.0000 q^{75} +643.988 q^{76} -295.161 q^{79} -320.000 q^{80} +81.0000 q^{81} -590.322 q^{83} -643.988 q^{84} +268.328 q^{85} -885.483 q^{87} -1206.00 q^{89} +720.000 q^{91} +576.000 q^{92} -60.0000 q^{93} +402.492 q^{95} -1186.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 16 q^{4} - 10 q^{5} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 - 16 * q^4 - 10 * q^5 + 18 * q^9	$ 2 q + 6 q^{3} - 16 q^{4} - 10 q^{5} + 18 q^{9} - 48 q^{12} - 30 q^{15} + 128 q^{16} + 80 q^{20} - 144 q^{23} + 50 q^{25} + 54 q^{27} - 40 q^{31} - 144 q^{36} + 428 q^{37} - 90 q^{45} - 528 q^{47} + 384 q^{48} + 754 q^{49} + 156 q^{53} + 960 q^{59} + 240 q^{60} - 1024 q^{64} - 1048 q^{67} - 432 q^{69} + 984 q^{71} + 150 q^{75} - 640 q^{80} + 162 q^{81} - 2412 q^{89} + 1440 q^{91} + 1152 q^{92} - 120 q^{93} - 2372 q^{97}+O(q^{100}) $ 2 * q + 6 * q^3 - 16 * q^4 - 10 * q^5 + 18 * q^9 - 48 * q^12 - 30 * q^15 + 128 * q^16 + 80 * q^20 - 144 * q^23 + 50 * q^25 + 54 * q^27 - 40 * q^31 - 144 * q^36 + 428 * q^37 - 90 * q^45 - 528 * q^47 + 384 * q^48 + 754 * q^49 + 156 * q^53 + 960 * q^59 + 240 * q^60 - 1024 * q^64 - 1048 * q^67 - 432 * q^69 + 984 * q^71 + 150 * q^75 - 640 * q^80 + 162 * q^81 - 2412 * q^89 + 1440 * q^91 + 1152 * q^92 - 120 * q^93 - 2372 * q^97

Display 100 coefficients 10 coefficients

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−8.00000	−1.00000
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	26.8328	1.44884	0.724418	−	0.689361i	$-0.242109\pi$
$7$	26.8328	1.44884	0.724418	+	0.689361i	$0.242109\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−24.0000	−0.577350
$13$	26.8328	0.572468	0.286234	−	0.958160i	$-0.407597\pi$
$13$	26.8328	0.572468	0.286234	+	0.958160i	$0.407597\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	64.0000	1.00000
$17$	−53.6656	−0.765637	−0.382818	−	0.923824i	$-0.625047\pi$
$17$	−53.6656	−0.765637	−0.382818	+	0.923824i	$0.625047\pi$
$18$	0	0
$19$	−80.4984	−0.971979	−0.485990	−	0.873965i	$-0.661541\pi$
$19$	−80.4984	−0.971979	−0.485990	+	0.873965i	$0.661541\pi$
$20$	40.0000	0.447214
$21$	80.4984	0.836486
$22$	0	0
$23$	−72.0000	−0.652741	−0.326370	−	0.945242i	$-0.605826\pi$
$23$	−72.0000	−0.652741	−0.326370	+	0.945242i	$0.605826\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	−214.663	−1.44884
$29$	−295.161	−1.89000	−0.945000	−	0.327069i	$-0.893939\pi$
$29$	−295.161	−1.89000	−0.945000	+	0.327069i	$0.893939\pi$
$30$	0	0
$31$	−20.0000	−0.115874	−0.0579372	−	0.998320i	$-0.518452\pi$
$31$	−20.0000	−0.115874	−0.0579372	+	0.998320i	$0.518452\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−134.164	−0.647939
$36$	−72.0000	−0.333333
$37$	214.000	0.950848	0.475424	−	0.879757i	$-0.342295\pi$
$37$	214.000	0.950848	0.475424	+	0.879757i	$0.342295\pi$
$38$	0	0
$39$	80.4984	0.330515
$40$	0	0
$41$	−26.8328	−0.102209	−0.0511047	−	0.998693i	$-0.516274\pi$
$41$	−26.8328	−0.102209	−0.0511047	+	0.998693i	$0.516274\pi$
$42$	0	0
$43$	241.495	0.856458	0.428229	−	0.903670i	$-0.359138\pi$
$43$	241.495	0.856458	0.428229	+	0.903670i	$0.359138\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	−264.000	−0.819327	−0.409663	−	0.912237i	$-0.634354\pi$
$47$	−264.000	−0.819327	−0.409663	+	0.912237i	$0.634354\pi$
$48$	192.000	0.577350
$49$	377.000	1.09913
$50$	0	0
$51$	−160.997	−0.442041
$52$	−214.663	−0.572468
$53$	78.0000	0.202153	0.101077	−	0.994879i	$-0.467771\pi$
$53$	78.0000	0.202153	0.101077	+	0.994879i	$0.467771\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−241.495	−0.561173
$58$	0	0
$59$	480.000	1.05916	0.529582	−	0.848259i	$-0.322349\pi$
$59$	480.000	1.05916	0.529582	+	0.848259i	$0.322349\pi$
$60$	120.000	0.258199
$61$	107.331	0.225285	0.112642	−	0.993636i	$-0.464069\pi$
$61$	107.331	0.225285	0.112642	+	0.993636i	$0.464069\pi$
$62$	0	0
$63$	241.495	0.482945
$64$	−512.000	−1.00000
$65$	−134.164	−0.256015
$66$	0	0
$67$	−524.000	−0.955474	−0.477737	−	0.878503i	$-0.658543\pi$
$67$	−524.000	−0.955474	−0.477737	+	0.878503i	$0.658543\pi$
$68$	429.325	0.765637
$69$	−216.000	−0.376860
$70$	0	0
$71$	492.000	0.822390	0.411195	−	0.911548i	$-0.365112\pi$
$71$	492.000	0.822390	0.411195	+	0.911548i	$0.365112\pi$
$72$	0	0
$73$	939.149	1.50574	0.752870	−	0.658169i	$-0.228669\pi$
$73$	939.149	1.50574	0.752870	+	0.658169i	$0.228669\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	643.988	0.971979
$77$	0	0
$78$	0	0
$79$	−295.161	−0.420357	−0.210179	−	0.977663i	$-0.567405\pi$
$79$	−295.161	−0.420357	−0.210179	+	0.977663i	$0.567405\pi$
$80$	−320.000	−0.447214
$81$	81.0000	0.111111
$82$	0	0
$83$	−590.322	−0.780678	−0.390339	−	0.920671i	$-0.627642\pi$
$83$	−590.322	−0.780678	−0.390339	+	0.920671i	$0.627642\pi$
$84$	−643.988	−0.836486
$85$	268.328	0.342403
$86$	0	0
$87$	−885.483	−1.09119
$88$	0	0
$89$	−1206.00	−1.43636	−0.718178	−	0.695859i	$-0.755024\pi$
$89$	−1206.00	−1.43636	−0.718178	+	0.695859i	$0.755024\pi$
$90$	0	0
$91$	720.000	0.829412
$92$	576.000	0.652741
$93$	−60.0000	−0.0669001
$94$	0	0
$95$	402.492	0.434682
$96$	0	0
$97$	−1186.00	−1.24144	−0.620722	−	0.784031i	$-0.713160\pi$
$97$	−1186.00	−1.24144	−0.620722	+	0.784031i	$0.713160\pi$
$98$	0	0
$99$	0	0
$100$	−200.000	−0.200000
$101$	−456.158	−0.449400	−0.224700	−	0.974428i	$-0.572140\pi$
$101$	−456.158	−0.449400	−0.224700	+	0.974428i	$0.572140\pi$
$102$	0	0
$103$	128.000	0.122449	0.0612243	−	0.998124i	$-0.480499\pi$
$103$	128.000	0.122449	0.0612243	+	0.998124i	$0.480499\pi$
$104$	0	0
$105$	−402.492	−0.374088
$106$	0	0
$107$	53.6656	0.0484865	0.0242432	−	0.999706i	$-0.492282\pi$
$107$	53.6656	0.0484865	0.0242432	+	0.999706i	$0.492282\pi$
$108$	−216.000	−0.192450
$109$	−321.994	−0.282949	−0.141474	−	0.989942i	$-0.545184\pi$
$109$	−321.994	−0.282949	−0.141474	+	0.989942i	$0.545184\pi$
$110$	0	0
$111$	642.000	0.548972
$112$	1717.30	1.44884
$113$	−762.000	−0.634362	−0.317181	−	0.948365i	$-0.602736\pi$
$113$	−762.000	−0.634362	−0.317181	+	0.948365i	$0.602736\pi$
$114$	0	0
$115$	360.000	0.291915
$116$	2361.29	1.89000
$117$	241.495	0.190823
$118$	0	0
$119$	−1440.00	−1.10928
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−80.4984	−0.0590106
$124$	160.000	0.115874
$125$	−125.000	−0.0894427
$126$	0	0
$127$	2602.78	1.81858	0.909290	−	0.416163i	$-0.136625\pi$
$127$	2602.78	1.81858	0.909290	+	0.416163i	$0.136625\pi$
$128$	0	0
$129$	724.486	0.494476
$130$	0	0
$131$	−2039.29	−1.36011	−0.680053	−	0.733163i	$-0.738043\pi$
$131$	−2039.29	−1.36011	−0.680053	+	0.733163i	$0.738043\pi$
$132$	0	0
$133$	−2160.00	−1.40824
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	−786.000	−0.490164	−0.245082	−	0.969502i	$-0.578815\pi$
$137$	−786.000	−0.490164	−0.245082	+	0.969502i	$0.578815\pi$
$138$	0	0
$139$	2173.46	1.32626	0.663131	−	0.748504i	$-0.269227\pi$
$139$	2173.46	1.32626	0.663131	+	0.748504i	$0.269227\pi$
$140$	1073.31	0.647939
$141$	−792.000	−0.473039
$142$	0	0
$143$	0	0
$144$	576.000	0.333333
$145$	1475.80	0.845234
$146$	0	0
$147$	1131.00	0.634580
$148$	−1712.00	−0.950848
$149$	−2119.79	−1.16550	−0.582752	−	0.812650i	$-0.698024\pi$
$149$	−2119.79	−1.16550	−0.582752	+	0.812650i	$0.698024\pi$
$150$	0	0
$151$	−3515.10	−1.89440	−0.947201	−	0.320641i	$-0.896102\pi$
$151$	−3515.10	−1.89440	−0.947201	+	0.320641i	$0.896102\pi$
$152$	0	0
$153$	−482.991	−0.255212
$154$	0	0
$155$	100.000	0.0518206
$156$	−643.988	−0.330515
$157$	−686.000	−0.348718	−0.174359	−	0.984682i	$-0.555785\pi$
$157$	−686.000	−0.348718	−0.174359	+	0.984682i	$0.555785\pi$
$158$	0	0
$159$	234.000	0.116713
$160$	0	0
$161$	−1931.96	−0.945714
$162$	0	0
$163$	−988.000	−0.474762	−0.237381	−	0.971417i	$-0.576289\pi$
$163$	−988.000	−0.474762	−0.237381	+	0.971417i	$0.576289\pi$
$164$	214.663	0.102209
$165$	0	0
$166$	0	0
$167$	−3327.27	−1.54175	−0.770874	−	0.636988i	$-0.780180\pi$
$167$	−3327.27	−1.54175	−0.770874	+	0.636988i	$0.780180\pi$
$168$	0	0
$169$	−1477.00	−0.672280
$170$	0	0
$171$	−724.486	−0.323993
$172$	−1931.96	−0.856458
$173$	3219.94	1.41507	0.707536	−	0.706678i	$-0.249807\pi$
$173$	3219.94	1.41507	0.707536	+	0.706678i	$0.249807\pi$
$174$	0	0
$175$	670.820	0.289767
$176$	0	0
$177$	1440.00	0.611509
$178$	0	0
$179$	−3336.00	−1.39299	−0.696493	−	0.717564i	$-0.745257\pi$
$179$	−3336.00	−1.39299	−0.696493	+	0.717564i	$0.745257\pi$
$180$	360.000	0.149071
$181$	−3490.00	−1.43320	−0.716601	−	0.697483i	$-0.754303\pi$
$181$	−3490.00	−1.43320	−0.716601	+	0.697483i	$0.754303\pi$
$182$	0	0
$183$	321.994	0.130068
$184$	0	0
$185$	−1070.00	−0.425232
$186$	0	0
$187$	0	0
$188$	2112.00	0.819327
$189$	724.486	0.278829
$190$	0	0
$191$	2820.00	1.06831	0.534157	−	0.845385i	$-0.320629\pi$
$191$	2820.00	1.06831	0.534157	+	0.845385i	$0.320629\pi$
$192$	−1536.00	−0.577350
$193$	−4373.75	−1.63124	−0.815620	−	0.578588i	$-0.803604\pi$
$193$	−4373.75	−1.63124	−0.815620	+	0.578588i	$0.803604\pi$
$194$	0	0
$195$	−402.492	−0.147811
$196$	−3016.00	−1.09913
$197$	3112.61	1.12571	0.562853	−	0.826557i	$-0.309704\pi$
$197$	3112.61	1.12571	0.562853	+	0.826557i	$0.309704\pi$
$198$	0	0
$199$	−5060.00	−1.80248	−0.901241	−	0.433318i	$-0.857343\pi$
$199$	−5060.00	−1.80248	−0.901241	+	0.433318i	$0.857343\pi$
$200$	0	0
$201$	−1572.00	−0.551643
$202$	0	0
$203$	−7920.00	−2.73830
$204$	1287.98	0.442041
$205$	134.164	0.0457094
$206$	0	0
$207$	−648.000	−0.217580
$208$	1717.30	0.572468
$209$	0	0
$210$	0	0
$211$	−3085.77	−1.00679	−0.503397	−	0.864055i	$-0.667917\pi$
$211$	−3085.77	−1.00679	−0.503397	+	0.864055i	$0.667917\pi$
$212$	−624.000	−0.202153
$213$	1476.00	0.474807
$214$	0	0
$215$	−1207.48	−0.383020
$216$	0	0
$217$	−536.656	−0.167883
$218$	0	0
$219$	2817.45	0.869339
$220$	0	0
$221$	−1440.00	−0.438303
$222$	0	0
$223$	5312.00	1.59515	0.797574	−	0.603222i	$-0.206117\pi$
$223$	5312.00	1.59515	0.797574	+	0.603222i	$0.206117\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	912.316	0.266751	0.133376	−	0.991066i	$-0.457418\pi$
$227$	912.316	0.266751	0.133376	+	0.991066i	$0.457418\pi$
$228$	1931.96	0.561173
$229$	−5330.00	−1.53806	−0.769031	−	0.639211i	$-0.779261\pi$
$229$	−5330.00	−1.53806	−0.769031	+	0.639211i	$0.779261\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6815.54	1.91631	0.958156	−	0.286248i	$-0.0924080\pi$
$233$	6815.54	1.91631	0.958156	+	0.286248i	$0.0924080\pi$
$234$	0	0
$235$	1320.00	0.366414
$236$	−3840.00	−1.05916
$237$	−885.483	−0.242693
$238$	0	0
$239$	−5956.89	−1.61221	−0.806106	−	0.591771i	$-0.798429\pi$
$239$	−5956.89	−1.61221	−0.806106	+	0.591771i	$0.798429\pi$
$240$	−960.000	−0.258199
$241$	−1180.64	−0.315568	−0.157784	−	0.987474i	$-0.550435\pi$
$241$	−1180.64	−0.315568	−0.157784	+	0.987474i	$0.550435\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	−858.650	−0.225285
$245$	−1885.00	−0.491544
$246$	0	0
$247$	−2160.00	−0.556427
$248$	0	0
$249$	−1770.97	−0.450724
$250$	0	0
$251$	−3312.00	−0.832875	−0.416437	−	0.909164i	$-0.636722\pi$
$251$	−3312.00	−0.832875	−0.416437	+	0.909164i	$0.636722\pi$
$252$	−1931.96	−0.482945
$253$	0	0
$254$	0	0
$255$	804.984	0.197687
$256$	4096.00	1.00000
$257$	4266.00	1.03543	0.517716	−	0.855553i	$-0.326783\pi$
$257$	4266.00	1.03543	0.517716	+	0.855553i	$0.326783\pi$
$258$	0	0
$259$	5742.22	1.37762
$260$	1073.31	0.256015
$261$	−2656.45	−0.630000
$262$	0	0
$263$	2253.96	0.528460	0.264230	−	0.964460i	$-0.414882\pi$
$263$	2253.96	0.528460	0.264230	+	0.964460i	$0.414882\pi$
$264$	0	0
$265$	−390.000	−0.0904057
$266$	0	0
$267$	−3618.00	−0.829281
$268$	4192.00	0.955474
$269$	−1770.00	−0.401185	−0.200593	−	0.979675i	$-0.564287\pi$
$269$	−1770.00	−0.401185	−0.200593	+	0.979675i	$0.564287\pi$
$270$	0	0
$271$	−1422.14	−0.318778	−0.159389	−	0.987216i	$-0.550952\pi$
$271$	−1422.14	−0.318778	−0.159389	+	0.987216i	$0.550952\pi$
$272$	−3434.60	−0.765637
$273$	2160.00	0.478861
$274$	0	0
$275$	0	0
$276$	1728.00	0.376860
$277$	−5017.74	−1.08840	−0.544200	−	0.838956i	$-0.683167\pi$
$277$	−5017.74	−1.08840	−0.544200	+	0.838956i	$0.683167\pi$
$278$	0	0
$279$	−180.000	−0.0386248
$280$	0	0
$281$	−1207.48	−0.256342	−0.128171	−	0.991752i	$-0.540911\pi$
$281$	−1207.48	−0.256342	−0.128171	+	0.991752i	$0.540911\pi$
$282$	0	0
$283$	6144.71	1.29069	0.645345	−	0.763891i	$-0.276714\pi$
$283$	6144.71	1.29069	0.645345	+	0.763891i	$0.276714\pi$
$284$	−3936.00	−0.822390
$285$	1207.48	0.250964
$286$	0	0
$287$	−720.000	−0.148085
$288$	0	0
$289$	−2033.00	−0.413800
$290$	0	0
$291$	−3558.00	−0.716748
$292$	−7513.19	−1.50574
$293$	−2253.96	−0.449411	−0.224706	−	0.974427i	$-0.572142\pi$
$293$	−2253.96	−0.449411	−0.224706	+	0.974427i	$0.572142\pi$
$294$	0	0
$295$	−2400.00	−0.473673
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1931.96	−0.373673
$300$	−600.000	−0.115470
$301$	6480.00	1.24087
$302$	0	0
$303$	−1368.47	−0.259461
$304$	−5151.90	−0.971979
$305$	−536.656	−0.100750
$306$	0	0
$307$	−3676.10	−0.683407	−0.341703	−	0.939808i	$-0.611004\pi$
$307$	−3676.10	−0.683407	−0.341703	+	0.939808i	$0.611004\pi$
$308$	0	0
$309$	384.000	0.0706958
$310$	0	0
$311$	1428.00	0.260368	0.130184	−	0.991490i	$-0.458443\pi$
$311$	1428.00	0.260368	0.130184	+	0.991490i	$0.458443\pi$
$312$	0	0
$313$	682.000	0.123159	0.0615797	−	0.998102i	$-0.480386\pi$
$313$	682.000	0.123159	0.0615797	+	0.998102i	$0.480386\pi$
$314$	0	0
$315$	−1207.48	−0.215980
$316$	2361.29	0.420357
$317$	7866.00	1.39369	0.696843	−	0.717224i	$-0.254587\pi$
$317$	7866.00	1.39369	0.696843	+	0.717224i	$0.254587\pi$
$318$	0	0
$319$	0	0
$320$	2560.00	0.447214
$321$	160.997	0.0279937
$322$	0	0
$323$	4320.00	0.744183
$324$	−648.000	−0.111111
$325$	670.820	0.114494
$326$	0	0
$327$	−965.981	−0.163361
$328$	0	0
$329$	−7083.86	−1.18707
$330$	0	0
$331$	−3040.00	−0.504814	−0.252407	−	0.967621i	$-0.581222\pi$
$331$	−3040.00	−0.504814	−0.252407	+	0.967621i	$0.581222\pi$
$332$	4722.58	0.780678
$333$	1926.00	0.316949
$334$	0	0
$335$	2620.00	0.427301
$336$	5151.90	0.836486
$337$	939.149	0.151806	0.0759031	−	0.997115i	$-0.475816\pi$
$337$	939.149	0.151806	0.0759031	+	0.997115i	$0.475816\pi$
$338$	0	0
$339$	−2286.00	−0.366249
$340$	−2146.63	−0.342403
$341$	0	0
$342$	0	0
$343$	912.316	0.143616
$344$	0	0
$345$	1080.00	0.168537
$346$	0	0
$347$	7352.19	1.13742	0.568712	−	0.822537i	$-0.307442\pi$
$347$	7352.19	1.13742	0.568712	+	0.822537i	$0.307442\pi$
$348$	7083.86	1.09119
$349$	8371.84	1.28405	0.642026	−	0.766683i	$-0.278094\pi$
$349$	8371.84	1.28405	0.642026	+	0.766683i	$0.278094\pi$
$350$	0	0
$351$	724.486	0.110172
$352$	0	0
$353$	−1062.00	−0.160126	−0.0800631	−	0.996790i	$-0.525512\pi$
$353$	−1062.00	−0.160126	−0.0800631	+	0.996790i	$0.525512\pi$
$354$	0	0
$355$	−2460.00	−0.367784
$356$	9648.00	1.43636
$357$	−4320.00	−0.640444
$358$	0	0
$359$	11055.1	1.62526	0.812628	−	0.582783i	$-0.198036\pi$
$359$	11055.1	1.62526	0.812628	+	0.582783i	$0.198036\pi$
$360$	0	0
$361$	−379.000	−0.0552559
$362$	0	0
$363$	0	0
$364$	−5760.00	−0.829412
$365$	−4695.74	−0.673387
$366$	0	0
$367$	12656.0	1.80010	0.900052	−	0.435783i	$-0.143529\pi$
$367$	12656.0	1.80010	0.900052	+	0.435783i	$0.143529\pi$
$368$	−4608.00	−0.652741
$369$	−241.495	−0.0340698
$370$	0	0
$371$	2092.96	0.292887
$372$	480.000	0.0669001
$373$	−4588.41	−0.636941	−0.318470	−	0.947933i	$-0.603169\pi$
$373$	−4588.41	−0.636941	−0.318470	+	0.947933i	$0.603169\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	−7920.00	−1.08196
$378$	0	0
$379$	8840.00	1.19810	0.599051	−	0.800711i	$-0.295545\pi$
$379$	8840.00	1.19810	0.599051	+	0.800711i	$0.295545\pi$
$380$	−3219.94	−0.434682
$381$	7808.35	1.04996
$382$	0	0
$383$	−10968.0	−1.46329	−0.731643	−	0.681688i	$-0.761246\pi$
$383$	−10968.0	−1.46329	−0.731643	+	0.681688i	$0.761246\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2173.46	0.285486
$388$	9488.00	1.24144
$389$	−4830.00	−0.629539	−0.314770	−	0.949168i	$-0.601927\pi$
$389$	−4830.00	−0.629539	−0.314770	+	0.949168i	$0.601927\pi$
$390$	0	0
$391$	3863.93	0.499762
$392$	0	0
$393$	−6117.88	−0.785258
$394$	0	0
$395$	1475.80	0.187989
$396$	0	0
$397$	−13966.0	−1.76558	−0.882788	−	0.469772i	$-0.844336\pi$
$397$	−13966.0	−1.76558	−0.882788	+	0.469772i	$0.844336\pi$
$398$	0	0
$399$	−6480.00	−0.813047
$400$	1600.00	0.200000
$401$	750.000	0.0933995	0.0466998	−	0.998909i	$-0.485130\pi$
$401$	750.000	0.0933995	0.0466998	+	0.998909i	$0.485130\pi$
$402$	0	0
$403$	−536.656	−0.0663344
$404$	3649.26	0.449400
$405$	−405.000	−0.0496904
$406$	0	0
$407$	0	0
$408$	0	0
$409$	13792.1	1.66742	0.833709	−	0.552204i	$-0.186213\pi$
$409$	13792.1	1.66742	0.833709	+	0.552204i	$0.186213\pi$
$410$	0	0
$411$	−2358.00	−0.282997
$412$	−1024.00	−0.122449
$413$	12879.8	1.53455
$414$	0	0
$415$	2951.61	0.349130
$416$	0	0
$417$	6520.37	0.765717
$418$	0	0
$419$	−12144.0	−1.41593	−0.707963	−	0.706249i	$-0.750386\pi$
$419$	−12144.0	−1.41593	−0.707963	+	0.706249i	$0.750386\pi$
$420$	3219.94	0.374088
$421$	3490.00	0.404019	0.202010	−	0.979384i	$-0.435253\pi$
$421$	3490.00	0.404019	0.202010	+	0.979384i	$0.435253\pi$
$422$	0	0
$423$	−2376.00	−0.273109
$424$	0	0
$425$	−1341.64	−0.153127
$426$	0	0
$427$	2880.00	0.326400
$428$	−429.325	−0.0484865
$429$	0	0
$430$	0	0
$431$	1985.63	0.221913	0.110956	−	0.993825i	$-0.464609\pi$
$431$	1985.63	0.221913	0.110956	+	0.993825i	$0.464609\pi$
$432$	1728.00	0.192450
$433$	7742.00	0.859254	0.429627	−	0.903007i	$-0.358645\pi$
$433$	7742.00	0.859254	0.429627	+	0.903007i	$0.358645\pi$
$434$	0	0
$435$	4427.41	0.487996
$436$	2575.95	0.282949
$437$	5795.89	0.634451
$438$	0	0
$439$	−2602.78	−0.282971	−0.141485	−	0.989940i	$-0.545188\pi$
$439$	−2602.78	−0.282971	−0.141485	+	0.989940i	$0.545188\pi$
$440$	0	0
$441$	3393.00	0.366375
$442$	0	0
$443$	6348.00	0.680818	0.340409	−	0.940277i	$-0.389434\pi$
$443$	6348.00	0.680818	0.340409	+	0.940277i	$0.389434\pi$
$444$	−5136.00	−0.548972
$445$	6030.00	0.642358
$446$	0	0
$447$	−6359.38	−0.672904
$448$	−13738.4	−1.44884
$449$	−17214.0	−1.80931	−0.904654	−	0.426148i	$-0.859870\pi$
$449$	−17214.0	−1.80931	−0.904654	+	0.426148i	$0.859870\pi$
$450$	0	0
$451$	0	0
$452$	6096.00	0.634362
$453$	−10545.3	−1.09373
$454$	0	0
$455$	−3600.00	−0.370924
$456$	0	0
$457$	−1583.14	−0.162048	−0.0810241	−	0.996712i	$-0.525819\pi$
$457$	−1583.14	−0.162048	−0.0810241	+	0.996712i	$0.525819\pi$
$458$	0	0
$459$	−1448.97	−0.147347
$460$	−2880.00	−0.291915
$461$	12047.9	1.21720	0.608599	−	0.793478i	$-0.291732\pi$
$461$	12047.9	1.21720	0.608599	+	0.793478i	$0.291732\pi$
$462$	0	0
$463$	−14488.0	−1.45424	−0.727121	−	0.686509i	$-0.759142\pi$
$463$	−14488.0	−1.45424	−0.727121	+	0.686509i	$0.759142\pi$
$464$	−18890.3	−1.89000
$465$	300.000	0.0299186
$466$	0	0
$467$	−1596.00	−0.158146	−0.0790729	−	0.996869i	$-0.525196\pi$
$467$	−1596.00	−0.158146	−0.0790729	+	0.996869i	$0.525196\pi$
$468$	−1931.96	−0.190823
$469$	−14060.4	−1.38433
$470$	0	0
$471$	−2058.00	−0.201333
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−2012.46	−0.194396
$476$	11520.0	1.10928
$477$	702.000	0.0673844
$478$	0	0
$479$	−19695.3	−1.87871	−0.939353	−	0.342951i	$-0.888574\pi$
$479$	−19695.3	−1.87871	−0.939353	+	0.342951i	$0.888574\pi$
$480$	0	0
$481$	5742.22	0.544330
$482$	0	0
$483$	−5795.89	−0.546008
$484$	0	0
$485$	5930.00	0.555191
$486$	0	0
$487$	−15464.0	−1.43889	−0.719446	−	0.694548i	$-0.755604\pi$
$487$	−15464.0	−1.43889	−0.719446	+	0.694548i	$0.755604\pi$
$488$	0	0
$489$	−2964.00	−0.274104
$490$	0	0
$491$	−14114.1	−1.29727	−0.648634	−	0.761100i	$-0.724659\pi$
$491$	−14114.1	−1.29727	−0.648634	+	0.761100i	$0.724659\pi$
$492$	643.988	0.0590106
$493$	15840.0	1.44705
$494$	0	0
$495$	0	0
$496$	−1280.00	−0.115874
$497$	13201.7	1.19151
$498$	0	0
$499$	−4880.00	−0.437793	−0.218897	−	0.975748i	$-0.570246\pi$
$499$	−4880.00	−0.437793	−0.218897	+	0.975748i	$0.570246\pi$
$500$	1000.00	0.0894427
$501$	−9981.81	−0.890128
$502$	0	0
$503$	−14275.1	−1.26539	−0.632697	−	0.774399i	$-0.718052\pi$
$503$	−14275.1	−1.26539	−0.632697	+	0.774399i	$0.718052\pi$
$504$	0	0
$505$	2280.79	0.200978
$506$	0	0
$507$	−4431.00	−0.388141
$508$	−20822.3	−1.81858
$509$	−11610.0	−1.01101	−0.505505	−	0.862824i	$-0.668694\pi$
$509$	−11610.0	−1.01101	−0.505505	+	0.862824i	$0.668694\pi$
$510$	0	0
$511$	25200.0	2.18157
$512$	0	0
$513$	−2173.46	−0.187058
$514$	0	0
$515$	−640.000	−0.0547607
$516$	−5795.89	−0.494476
$517$	0	0
$518$	0	0
$519$	9659.81	0.816992
$520$	0	0
$521$	−16170.0	−1.35973	−0.679866	−	0.733336i	$-0.737962\pi$
$521$	−16170.0	−1.35973	−0.679866	+	0.733336i	$0.737962\pi$
$522$	0	0
$523$	3568.76	0.298377	0.149189	−	0.988809i	$-0.452334\pi$
$523$	3568.76	0.298377	0.149189	+	0.988809i	$0.452334\pi$
$524$	16314.4	1.36011
$525$	2012.46	0.167297
$526$	0	0
$527$	1073.31	0.0887177
$528$	0	0
$529$	−6983.00	−0.573929
$530$	0	0
$531$	4320.00	0.353055
$532$	17280.0	1.40824
$533$	−720.000	−0.0585116
$534$	0	0
$535$	−268.328	−0.0216838
$536$	0	0
$537$	−10008.0	−0.804240
$538$	0	0
$539$	0	0
$540$	1080.00	0.0860663
$541$	10518.5	0.835904	0.417952	−	0.908469i	$-0.362748\pi$
$541$	10518.5	0.835904	0.417952	+	0.908469i	$0.362748\pi$
$542$	0	0
$543$	−10470.0	−0.827460
$544$	0	0
$545$	1609.97	0.126539
$546$	0	0
$547$	4856.74	0.379633	0.189816	−	0.981820i	$-0.439211\pi$
$547$	4856.74	0.379633	0.189816	+	0.981820i	$0.439211\pi$
$548$	6288.00	0.490164
$549$	965.981	0.0750949
$550$	0	0
$551$	23760.0	1.83704
$552$	0	0
$553$	−7920.00	−0.609028
$554$	0	0
$555$	−3210.00	−0.245508
$556$	−17387.7	−1.32626
$557$	2307.62	0.175542	0.0877712	−	0.996141i	$-0.472026\pi$
$557$	2307.62	0.175542	0.0877712	+	0.996141i	$0.472026\pi$
$558$	0	0
$559$	6480.00	0.490295
$560$	−8586.50	−0.647939
$561$	0	0
$562$	0	0
$563$	3434.60	0.257107	0.128553	−	0.991703i	$-0.458967\pi$
$563$	3434.60	0.257107	0.128553	+	0.991703i	$0.458967\pi$
$564$	6336.00	0.473039
$565$	3810.00	0.283695
$566$	0	0
$567$	2173.46	0.160982
$568$	0	0
$569$	−885.483	−0.0652397	−0.0326198	−	0.999468i	$-0.510385\pi$
$569$	−885.483	−0.0652397	−0.0326198	+	0.999468i	$0.510385\pi$
$570$	0	0
$571$	−670.820	−0.0491646	−0.0245823	−	0.999698i	$-0.507826\pi$
$571$	−670.820	−0.0491646	−0.0245823	+	0.999698i	$0.507826\pi$
$572$	0	0
$573$	8460.00	0.616792
$574$	0	0
$575$	−1800.00	−0.130548
$576$	−4608.00	−0.333333
$577$	8926.00	0.644011	0.322005	−	0.946738i	$-0.395643\pi$
$577$	8926.00	0.644011	0.322005	+	0.946738i	$0.395643\pi$
$578$	0	0
$579$	−13121.2	−0.941797
$580$	−11806.4	−0.845234
$581$	−15840.0	−1.13107
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−1207.48	−0.0853385
$586$	0	0
$587$	−28236.0	−1.98539	−0.992695	−	0.120647i	$-0.961503\pi$
$587$	−28236.0	−1.98539	−0.992695	+	0.120647i	$0.961503\pi$
$588$	−9048.00	−0.634580
$589$	1609.97	0.112628
$590$	0	0
$591$	9337.82	0.649927
$592$	13696.0	0.950848
$593$	12235.8	0.847323	0.423662	−	0.905821i	$-0.360745\pi$
$593$	12235.8	0.847323	0.423662	+	0.905821i	$0.360745\pi$
$594$	0	0
$595$	7200.00	0.496086
$596$	16958.3	1.16550
$597$	−15180.0	−1.04066
$598$	0	0
$599$	−16884.0	−1.15169	−0.575844	−	0.817559i	$-0.695327\pi$
$599$	−16884.0	−1.15169	−0.575844	+	0.817559i	$0.695327\pi$
$600$	0	0
$601$	19480.6	1.32218	0.661091	−	0.750306i	$-0.270094\pi$
$601$	19480.6	1.32218	0.661091	+	0.750306i	$0.270094\pi$
$602$	0	0
$603$	−4716.00	−0.318491
$604$	28120.8	1.89440
$605$	0	0
$606$	0	0
$607$	−11672.3	−0.780499	−0.390250	−	0.920709i	$-0.627611\pi$
$607$	−11672.3	−0.780499	−0.390250	+	0.920709i	$0.627611\pi$
$608$	0	0
$609$	−23760.0	−1.58096
$610$	0	0
$611$	−7083.86	−0.469038
$612$	3863.93	0.255212
$613$	−7701.02	−0.507408	−0.253704	−	0.967282i	$-0.581649\pi$
$613$	−7701.02	−0.507408	−0.253704	+	0.967282i	$0.581649\pi$
$614$	0	0
$615$	402.492	0.0263903
$616$	0	0
$617$	10686.0	0.697248	0.348624	−	0.937263i	$-0.386649\pi$
$617$	10686.0	0.697248	0.348624	+	0.937263i	$0.386649\pi$
$618$	0	0
$619$	−1744.00	−0.113243	−0.0566214	−	0.998396i	$-0.518033\pi$
$619$	−1744.00	−0.113243	−0.0566214	+	0.998396i	$0.518033\pi$
$620$	−800.000	−0.0518206
$621$	−1944.00	−0.125620
$622$	0	0
$623$	−32360.4	−2.08105
$624$	5151.90	0.330515
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	5488.00	0.348718
$629$	−11484.4	−0.728004
$630$	0	0
$631$	5092.00	0.321251	0.160625	−	0.987015i	$-0.448649\pi$
$631$	5092.00	0.321251	0.160625	+	0.987015i	$0.448649\pi$
$632$	0	0
$633$	−9257.32	−0.581273
$634$	0	0
$635$	−13013.9	−0.813294
$636$	−1872.00	−0.116713
$637$	10116.0	0.629214
$638$	0	0
$639$	4428.00	0.274130
$640$	0	0
$641$	11970.0	0.737577	0.368788	−	0.929513i	$-0.379773\pi$
$641$	11970.0	0.737577	0.368788	+	0.929513i	$0.379773\pi$
$642$	0	0
$643$	17228.0	1.05662	0.528309	−	0.849052i	$-0.322826\pi$
$643$	17228.0	1.05662	0.528309	+	0.849052i	$0.322826\pi$
$644$	15455.7	0.945714
$645$	−3622.43	−0.221137
$646$	0	0
$647$	20376.0	1.23812	0.619060	−	0.785344i	$-0.287514\pi$
$647$	20376.0	1.23812	0.619060	+	0.785344i	$0.287514\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−1609.97	−0.0969273
$652$	7904.00	0.474762
$653$	−21882.0	−1.31135	−0.655673	−	0.755045i	$-0.727615\pi$
$653$	−21882.0	−1.31135	−0.655673	+	0.755045i	$0.727615\pi$
$654$	0	0
$655$	10196.5	0.608258
$656$	−1717.30	−0.102209
$657$	8452.34	0.501913
$658$	0	0
$659$	−5795.89	−0.342604	−0.171302	−	0.985219i	$-0.554797\pi$
$659$	−5795.89	−0.342604	−0.171302	+	0.985219i	$0.554797\pi$
$660$	0	0
$661$	20338.0	1.19676	0.598379	−	0.801213i	$-0.295812\pi$
$661$	20338.0	1.19676	0.598379	+	0.801213i	$0.295812\pi$
$662$	0	0
$663$	−4320.00	−0.253054
$664$	0	0
$665$	10800.0	0.629784
$666$	0	0
$667$	21251.6	1.23368
$668$	26618.2	1.54175
$669$	15936.0	0.920959
$670$	0	0
$671$	0	0
$672$	0	0
$673$	8506.00	0.487195	0.243598	−	0.969876i	$-0.421672\pi$
$673$	8506.00	0.487195	0.243598	+	0.969876i	$0.421672\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	11816.0	0.672280
$677$	−21090.6	−1.19731	−0.598654	−	0.801008i	$-0.704297\pi$
$677$	−21090.6	−1.19731	−0.598654	+	0.801008i	$0.704297\pi$
$678$	0	0
$679$	−31823.7	−1.79865
$680$	0	0
$681$	2736.95	0.154009
$682$	0	0
$683$	−12108.0	−0.678331	−0.339165	−	0.940727i	$-0.610145\pi$
$683$	−12108.0	−0.678331	−0.339165	+	0.940727i	$0.610145\pi$
$684$	5795.89	0.323993
$685$	3930.00	0.219208
$686$	0	0
$687$	−15990.0	−0.888001
$688$	15455.7	0.856458
$689$	2092.96	0.115726
$690$	0	0
$691$	−13768.0	−0.757973	−0.378987	−	0.925402i	$-0.623727\pi$
$691$	−13768.0	−0.757973	−0.378987	+	0.925402i	$0.623727\pi$
$692$	−25759.5	−1.41507
$693$	0	0
$694$	0	0
$695$	−10867.3	−0.593122
$696$	0	0
$697$	1440.00	0.0782552
$698$	0	0
$699$	20446.6	1.10638
$700$	−5366.56	−0.289767
$701$	21224.8	1.14358	0.571789	−	0.820401i	$-0.306250\pi$
$701$	21224.8	1.14358	0.571789	+	0.820401i	$0.306250\pi$
$702$	0	0
$703$	−17226.7	−0.924205
$704$	0	0
$705$	3960.00	0.211549
$706$	0	0
$707$	−12240.0	−0.651107
$708$	−11520.0	−0.611509
$709$	31246.0	1.65510	0.827552	−	0.561390i	$-0.189733\pi$
$709$	31246.0	1.65510	0.827552	+	0.561390i	$0.189733\pi$
$710$	0	0
$711$	−2656.45	−0.140119
$712$	0	0
$713$	1440.00	0.0756359
$714$	0	0
$715$	0	0
$716$	26688.0	1.39299
$717$	−17870.7	−0.930812
$718$	0	0
$719$	−11316.0	−0.586948	−0.293474	−	0.955967i	$-0.594811\pi$
$719$	−11316.0	−0.586948	−0.293474	+	0.955967i	$0.594811\pi$
$720$	−2880.00	−0.149071
$721$	3434.60	0.177408
$722$	0	0
$723$	−3541.93	−0.182193
$724$	27920.0	1.43320
$725$	−7379.02	−0.378000
$726$	0	0
$727$	−9056.00	−0.461992	−0.230996	−	0.972955i	$-0.574198\pi$
$727$	−9056.00	−0.461992	−0.230996	+	0.972955i	$0.574198\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−12960.0	−0.655736
$732$	−2575.95	−0.130068
$733$	21546.8	1.08574	0.542870	−	0.839817i	$-0.317338\pi$
$733$	21546.8	1.08574	0.542870	+	0.839817i	$0.317338\pi$
$734$	0	0
$735$	−5655.00	−0.283793
$736$	0	0
$737$	0	0
$738$	0	0
$739$	7271.69	0.361967	0.180983	−	0.983486i	$-0.442072\pi$
$739$	7271.69	0.361967	0.180983	+	0.983486i	$0.442072\pi$
$740$	8560.00	0.425232
$741$	−6480.00	−0.321253
$742$	0	0
$743$	33970.3	1.67732	0.838662	−	0.544653i	$-0.183339\pi$
$743$	33970.3	1.67732	0.838662	+	0.544653i	$0.183339\pi$
$744$	0	0
$745$	10599.0	0.521229
$746$	0	0
$747$	−5312.90	−0.260226
$748$	0	0
$749$	1440.00	0.0702489
$750$	0	0
$751$	20932.0	1.01707	0.508535	−	0.861041i	$-0.330187\pi$
$751$	20932.0	1.01707	0.508535	+	0.861041i	$0.330187\pi$
$752$	−16896.0	−0.819327
$753$	−9936.00	−0.480861
$754$	0	0
$755$	17575.5	0.847202
$756$	−5795.89	−0.278829
$757$	28474.0	1.36711	0.683557	−	0.729897i	$-0.260432\pi$
$757$	28474.0	1.36711	0.683557	+	0.729897i	$0.260432\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−3032.11	−0.144433	−0.0722167	−	0.997389i	$-0.523007\pi$
$761$	−3032.11	−0.144433	−0.0722167	+	0.997389i	$0.523007\pi$
$762$	0	0
$763$	−8640.00	−0.409946
$764$	−22560.0	−1.06831
$765$	2414.95	0.114134
$766$	0	0
$767$	12879.8	0.606338
$768$	12288.0	0.577350
$769$	−14221.4	−0.666888	−0.333444	−	0.942770i	$-0.608211\pi$
$769$	−14221.4	−0.666888	−0.333444	+	0.942770i	$0.608211\pi$
$770$	0	0
$771$	12798.0	0.597806
$772$	34990.0	1.63124
$773$	−30258.0	−1.40790	−0.703949	−	0.710251i	$-0.748581\pi$
$773$	−30258.0	−1.40790	−0.703949	+	0.710251i	$0.748581\pi$
$774$	0	0
$775$	−500.000	−0.0231749
$776$	0	0
$777$	17226.7	0.795371
$778$	0	0
$779$	2160.00	0.0993454
$780$	3219.94	0.147811
$781$	0	0
$782$	0	0
$783$	−7969.35	−0.363731
$784$	24128.0	1.09913
$785$	3430.00	0.155952
$786$	0	0
$787$	25142.3	1.13879	0.569395	−	0.822064i	$-0.307177\pi$
$787$	25142.3	1.13879	0.569395	+	0.822064i	$0.307177\pi$
$788$	−24900.9	−1.12571
$789$	6761.87	0.305106
$790$	0	0
$791$	−20446.6	−0.919087
$792$	0	0
$793$	2880.00	0.128968
$794$	0	0
$795$	−1170.00	−0.0521958
$796$	40480.0	1.80248
$797$	6954.00	0.309063	0.154532	−	0.987988i	$-0.450613\pi$
$797$	6954.00	0.309063	0.154532	+	0.987988i	$0.450613\pi$
$798$	0	0
$799$	14167.7	0.627307
$800$	0	0
$801$	−10854.0	−0.478786
$802$	0	0
$803$	0	0
$804$	12576.0	0.551643
$805$	9659.81	0.422936
$806$	0	0
$807$	−5310.00	−0.231624
$808$	0	0
$809$	−25947.3	−1.12764	−0.563819	−	0.825898i	$-0.690669\pi$
$809$	−25947.3	−1.12764	−0.563819	+	0.825898i	$0.690669\pi$
$810$	0	0
$811$	−5017.74	−0.217258	−0.108629	−	0.994082i	$-0.534646\pi$
$811$	−5017.74	−0.217258	−0.108629	+	0.994082i	$0.534646\pi$
$812$	63360.0	2.73830
$813$	−4266.42	−0.184046
$814$	0	0
$815$	4940.00	0.212320
$816$	−10303.8	−0.442041
$817$	−19440.0	−0.832460
$818$	0	0
$819$	6480.00	0.276471
$820$	−1073.31	−0.0457094
$821$	35660.8	1.51592	0.757960	−	0.652301i	$-0.226196\pi$
$821$	35660.8	1.51592	0.757960	+	0.652301i	$0.226196\pi$
$822$	0	0
$823$	20288.0	0.859289	0.429645	−	0.902998i	$-0.358639\pi$
$823$	20288.0	0.859289	0.429645	+	0.902998i	$0.358639\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−17548.7	−0.737880	−0.368940	−	0.929453i	$-0.620279\pi$
$827$	−17548.7	−0.737880	−0.368940	+	0.929453i	$0.620279\pi$
$828$	5184.00	0.217580
$829$	9146.00	0.383177	0.191588	−	0.981475i	$-0.438636\pi$
$829$	9146.00	0.383177	0.191588	+	0.981475i	$0.438636\pi$
$830$	0	0
$831$	−15053.2	−0.628388
$832$	−13738.4	−0.572468
$833$	−20231.9	−0.841531
$834$	0	0
$835$	16636.3	0.689490
$836$	0	0
$837$	−540.000	−0.0223000
$838$	0	0
$839$	9900.00	0.407373	0.203687	−	0.979036i	$-0.434708\pi$
$839$	9900.00	0.407373	0.203687	+	0.979036i	$0.434708\pi$
$840$	0	0
$841$	62731.0	2.57210
$842$	0	0
$843$	−3622.43	−0.147999
$844$	24686.2	1.00679
$845$	7385.00	0.300653
$846$	0	0
$847$	0	0
$848$	4992.00	0.202153
$849$	18434.1	0.745180
$850$	0	0
$851$	−15408.0	−0.620657
$852$	−11808.0	−0.474807
$853$	−23264.1	−0.933817	−0.466909	−	0.884306i	$-0.654632\pi$
$853$	−23264.1	−0.933817	−0.466909	+	0.884306i	$0.654632\pi$
$854$	0	0
$855$	3622.43	0.144894
$856$	0	0
$857$	23988.5	0.956164	0.478082	−	0.878315i	$-0.341332\pi$
$857$	23988.5	0.956164	0.478082	+	0.878315i	$0.341332\pi$
$858$	0	0
$859$	−11104.0	−0.441052	−0.220526	−	0.975381i	$-0.570777\pi$
$859$	−11104.0	−0.441052	−0.220526	+	0.975381i	$0.570777\pi$
$860$	9659.81	0.383020
$861$	−2160.00	−0.0854966
$862$	0	0
$863$	42192.0	1.66423	0.832116	−	0.554601i	$-0.187129\pi$
$863$	42192.0	1.66423	0.832116	+	0.554601i	$0.187129\pi$
$864$	0	0
$865$	−16099.7	−0.632839
$866$	0	0
$867$	−6099.00	−0.238908
$868$	4293.25	0.167883
$869$	0	0
$870$	0	0
$871$	−14060.4	−0.546979
$872$	0	0
$873$	−10674.0	−0.413815
$874$	0	0
$875$	−3354.10	−0.129588
$876$	−22539.6	−0.869339
$877$	1744.13	0.0671553	0.0335776	−	0.999436i	$-0.489310\pi$
$877$	1744.13	0.0671553	0.0335776	+	0.999436i	$0.489310\pi$
$878$	0	0
$879$	−6761.87	−0.259468
$880$	0	0
$881$	32142.0	1.22916	0.614581	−	0.788854i	$-0.289325\pi$
$881$	32142.0	1.22916	0.614581	+	0.788854i	$0.289325\pi$
$882$	0	0
$883$	20572.0	0.784035	0.392018	−	0.919958i	$-0.371777\pi$
$883$	20572.0	0.784035	0.392018	+	0.919958i	$0.371777\pi$
$884$	11520.0	0.438303
$885$	−7200.00	−0.273475
$886$	0	0
$887$	−30535.7	−1.15591	−0.577954	−	0.816070i	$-0.696149\pi$
$887$	−30535.7	−1.15591	−0.577954	+	0.816070i	$0.696149\pi$
$888$	0	0
$889$	69840.0	2.63482
$890$	0	0
$891$	0	0
$892$	−42496.0	−1.59515
$893$	21251.6	0.796369
$894$	0	0
$895$	16680.0	0.622962
$896$	0	0
$897$	−5795.89	−0.215740
$898$	0	0
$899$	5903.22	0.219003
$900$	−1800.00	−0.0666667
$901$	−4185.92	−0.154776
$902$	0	0
$903$	19440.0	0.716415
$904$	0	0
$905$	17450.0	0.640948
$906$	0	0
$907$	32636.0	1.19477	0.597387	−	0.801953i	$-0.296206\pi$
$907$	32636.0	1.19477	0.597387	+	0.801953i	$0.296206\pi$
$908$	−7298.53	−0.266751
$909$	−4105.42	−0.149800
$910$	0	0
$911$	−37860.0	−1.37690	−0.688451	−	0.725283i	$-0.741709\pi$
$911$	−37860.0	−1.37690	−0.688451	+	0.725283i	$0.741709\pi$
$912$	−15455.7	−0.561173
$913$	0	0
$914$	0	0
$915$	−1609.97	−0.0581682
$916$	42640.0	1.53806
$917$	−54720.0	−1.97057
$918$	0	0
$919$	−26859.6	−0.964111	−0.482056	−	0.876141i	$-0.660110\pi$
$919$	−26859.6	−0.964111	−0.482056	+	0.876141i	$0.660110\pi$
$920$	0	0
$921$	−11028.3	−0.394565
$922$	0	0
$923$	13201.7	0.470792
$924$	0	0
$925$	5350.00	0.190170
$926$	0	0
$927$	1152.00	0.0408162
$928$	0	0
$929$	−38130.0	−1.34661	−0.673307	−	0.739363i	$-0.735127\pi$
$929$	−38130.0	−1.34661	−0.673307	+	0.739363i	$0.735127\pi$
$930$	0	0
$931$	−30347.9	−1.06833
$932$	−54524.3	−1.91631
$933$	4284.00	0.150324
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2227.12	0.0776488	0.0388244	−	0.999246i	$-0.487639\pi$
$937$	2227.12	0.0776488	0.0388244	+	0.999246i	$0.487639\pi$
$938$	0	0
$939$	2046.00	0.0711062
$940$	−10560.0	−0.366414
$941$	50204.2	1.73922	0.869612	−	0.493735i	$-0.164369\pi$
$941$	50204.2	1.73922	0.869612	+	0.493735i	$0.164369\pi$
$942$	0	0
$943$	1931.96	0.0667162
$944$	30720.0	1.05916
$945$	−3622.43	−0.124696
$946$	0	0
$947$	−276.000	−0.00947074	−0.00473537	−	0.999989i	$-0.501507\pi$
$947$	−276.000	−0.00947074	−0.00473537	+	0.999989i	$0.501507\pi$
$948$	7083.86	0.242693
$949$	25200.0	0.861988
$950$	0	0
$951$	23598.0	0.804645
$952$	0	0
$953$	42073.9	1.43012	0.715061	−	0.699062i	$-0.246399\pi$
$953$	42073.9	1.43012	0.715061	+	0.699062i	$0.246399\pi$
$954$	0	0
$955$	−14100.0	−0.477765
$956$	47655.1	1.61221
$957$	0	0
$958$	0	0
$959$	−21090.6	−0.710168
$960$	7680.00	0.258199
$961$	−29391.0	−0.986573
$962$	0	0
$963$	482.991	0.0161622
$964$	9445.15	0.315568
$965$	21868.7	0.729513
$966$	0	0
$967$	18809.8	0.625525	0.312762	−	0.949831i	$-0.398746\pi$
$967$	18809.8	0.625525	0.312762	+	0.949831i	$0.398746\pi$
$968$	0	0
$969$	12960.0	0.429654
$970$	0	0
$971$	5400.00	0.178470	0.0892349	−	0.996011i	$-0.471558\pi$
$971$	5400.00	0.178470	0.0892349	+	0.996011i	$0.471558\pi$
$972$	−1944.00	−0.0641500
$973$	58320.0	1.92153
$974$	0	0
$975$	2012.46	0.0661029
$976$	6869.20	0.225285
$977$	54486.0	1.78420	0.892099	−	0.451840i	$-0.149232\pi$
$977$	54486.0	1.78420	0.892099	+	0.451840i	$0.149232\pi$
$978$	0	0
$979$	0	0
$980$	15080.0	0.491544
$981$	−2897.94	−0.0943162
$982$	0	0
$983$	−4872.00	−0.158080	−0.0790400	−	0.996871i	$-0.525185\pi$
$983$	−4872.00	−0.158080	−0.0790400	+	0.996871i	$0.525185\pi$
$984$	0	0
$985$	−15563.0	−0.503431
$986$	0	0
$987$	−21251.6	−0.685355
$988$	17280.0	0.556427
$989$	−17387.7	−0.559045
$990$	0	0
$991$	5020.00	0.160914	0.0804569	−	0.996758i	$-0.474362\pi$
$991$	5020.00	0.160914	0.0804569	+	0.996758i	$0.474362\pi$
$992$	0	0
$993$	−9120.00	−0.291455
$994$	0	0
$995$	25300.0	0.806094
$996$	14167.7	0.450724
$997$	−26484.0	−0.841280	−0.420640	−	0.907228i	$-0.638194\pi$
$997$	−26484.0	−0.841280	−0.420640	+	0.907228i	$0.638194\pi$
$998$	0	0
$999$	5778.00	0.182991

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.4.a.j.1.2	yes	2
11.10	odd	2	inner	1815.4.a.j.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.4.a.j.1.1	✓	2	11.10	odd	2	inner
1815.4.a.j.1.2	yes	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1815.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -8.00000 q^{4} -5.00000 q^{5} +26.8328 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -8.00000 q^{4} -5.00000 q^{5} +26.8328 q^{7} +9.00000 q^{9} -24.0000 q^{12} +26.8328 q^{13} -15.0000 q^{15} +64.0000 q^{16} -53.6656 q^{17} -80.4984 q^{19} +40.0000 q^{20} +80.4984 q^{21} -72.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -214.663 q^{28} -295.161 q^{29} -20.0000 q^{31} -134.164 q^{35} -72.0000 q^{36} +214.000 q^{37} +80.4984 q^{39} -26.8328 q^{41} +241.495 q^{43} -45.0000 q^{45} -264.000 q^{47} +192.000 q^{48} +377.000 q^{49} -160.997 q^{51} -214.663 q^{52} +78.0000 q^{53} -241.495 q^{57} +480.000 q^{59} +120.000 q^{60} +107.331 q^{61} +241.495 q^{63} -512.000 q^{64} -134.164 q^{65} -524.000 q^{67} +429.325 q^{68} -216.000 q^{69} +492.000 q^{71} +939.149 q^{73} +75.0000 q^{75} +643.988 q^{76} -295.161 q^{79} -320.000 q^{80} +81.0000 q^{81} -590.322 q^{83} -643.988 q^{84} +268.328 q^{85} -885.483 q^{87} -1206.00 q^{89} +720.000 q^{91} +576.000 q^{92} -60.0000 q^{93} +402.492 q^{95} -1186.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 16 q^{4} - 10 q^{5} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 - 16 * q^4 - 10 * q^5 + 18 * q^9	\( 2 q + 6 q^{3} - 16 q^{4} - 10 q^{5} + 18 q^{9} - 48 q^{12} - 30 q^{15} + 128 q^{16} + 80 q^{20} - 144 q^{23} + 50 q^{25} + 54 q^{27} - 40 q^{31} - 144 q^{36} + 428 q^{37} - 90 q^{45} - 528 q^{47} + 384 q^{48} + 754 q^{49} + 156 q^{53} + 960 q^{59} + 240 q^{60} - 1024 q^{64} - 1048 q^{67} - 432 q^{69} + 984 q^{71} + 150 q^{75} - 640 q^{80} + 162 q^{81} - 2412 q^{89} + 1440 q^{91} + 1152 q^{92} - 120 q^{93} - 2372 q^{97}+O(q^{100}) \) 2 * q + 6 * q^3 - 16 * q^4 - 10 * q^5 + 18 * q^9 - 48 * q^12 - 30 * q^15 + 128 * q^16 + 80 * q^20 - 144 * q^23 + 50 * q^25 + 54 * q^27 - 40 * q^31 - 144 * q^36 + 428 * q^37 - 90 * q^45 - 528 * q^47 + 384 * q^48 + 754 * q^49 + 156 * q^53 + 960 * q^59 + 240 * q^60 - 1024 * q^64 - 1048 * q^67 - 432 * q^69 + 984 * q^71 + 150 * q^75 - 640 * q^80 + 162 * q^81 - 2412 * q^89 + 1440 * q^91 + 1152 * q^92 - 120 * q^93 - 2372 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more