LMFDB - Embedded newform 945.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [945,2,Mod(1,945)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(945, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("945.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 945 = 3^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	945.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:	$7.54586299101$
Analytic rank:	$0$
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:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	945.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.58579 q^{8} +O(q^{10})$	\(q-0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.58579 q^{8} -0.414214 q^{10} -2.41421 q^{11} -0.414214 q^{13} +0.414214 q^{14} +3.00000 q^{16} -0.828427 q^{17} +4.41421 q^{19} -1.82843 q^{20} +1.00000 q^{22} +4.82843 q^{23} +1.00000 q^{25} +0.171573 q^{26} +1.82843 q^{28} -4.00000 q^{29} +6.00000 q^{31} -4.41421 q^{32} +0.343146 q^{34} -1.00000 q^{35} +8.48528 q^{37} -1.82843 q^{38} +1.58579 q^{40} -2.17157 q^{41} +4.65685 q^{43} +4.41421 q^{44} -2.00000 q^{46} -9.48528 q^{47} +1.00000 q^{49} -0.414214 q^{50} +0.757359 q^{52} +8.89949 q^{53} -2.41421 q^{55} -1.58579 q^{56} +1.65685 q^{58} +10.4853 q^{59} +8.48528 q^{61} -2.48528 q^{62} -4.17157 q^{64} -0.414214 q^{65} +2.17157 q^{67} +1.51472 q^{68} +0.414214 q^{70} +2.00000 q^{71} +12.0711 q^{73} -3.51472 q^{74} -8.07107 q^{76} +2.41421 q^{77} +9.17157 q^{79} +3.00000 q^{80} +0.899495 q^{82} +9.48528 q^{83} -0.828427 q^{85} -1.92893 q^{86} -3.82843 q^{88} +2.65685 q^{89} +0.414214 q^{91} -8.82843 q^{92} +3.92893 q^{94} +4.41421 q^{95} -14.1421 q^{97} -0.414214 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8} + 2 q^{10} - 2 q^{11} + 2 q^{13} - 2 q^{14} + 6 q^{16} + 4 q^{17} + 6 q^{19} + 2 q^{20} + 2 q^{22} + 4 q^{23} + 2 q^{25} + 6 q^{26} - 2 q^{28} - 8 q^{29} + 12 q^{31} - 6 q^{32} + 12 q^{34} - 2 q^{35} + 2 q^{38} + 6 q^{40} - 10 q^{41} - 2 q^{43} + 6 q^{44} - 4 q^{46} - 2 q^{47} + 2 q^{49} + 2 q^{50} + 10 q^{52} - 2 q^{53} - 2 q^{55} - 6 q^{56} - 8 q^{58} + 4 q^{59} + 12 q^{62} - 14 q^{64} + 2 q^{65} + 10 q^{67} + 20 q^{68} - 2 q^{70} + 4 q^{71} + 10 q^{73} - 24 q^{74} - 2 q^{76} + 2 q^{77} + 24 q^{79} + 6 q^{80} - 18 q^{82} + 2 q^{83} + 4 q^{85} - 18 q^{86} - 2 q^{88} - 6 q^{89} - 2 q^{91} - 12 q^{92} + 22 q^{94} + 6 q^{95} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8 + 2 * q^10 - 2 * q^11 + 2 * q^13 - 2 * q^14 + 6 * q^16 + 4 * q^17 + 6 * q^19 + 2 * q^20 + 2 * q^22 + 4 * q^23 + 2 * q^25 + 6 * q^26 - 2 * q^28 - 8 * q^29 + 12 * q^31 - 6 * q^32 + 12 * q^34 - 2 * q^35 + 2 * q^38 + 6 * q^40 - 10 * q^41 - 2 * q^43 + 6 * q^44 - 4 * q^46 - 2 * q^47 + 2 * q^49 + 2 * q^50 + 10 * q^52 - 2 * q^53 - 2 * q^55 - 6 * q^56 - 8 * q^58 + 4 * q^59 + 12 * q^62 - 14 * q^64 + 2 * q^65 + 10 * q^67 + 20 * q^68 - 2 * q^70 + 4 * q^71 + 10 * q^73 - 24 * q^74 - 2 * q^76 + 2 * q^77 + 24 * q^79 + 6 * q^80 - 18 * q^82 + 2 * q^83 + 4 * q^85 - 18 * q^86 - 2 * q^88 - 6 * q^89 - 2 * q^91 - 12 * q^92 + 22 * q^94 + 6 * q^95 + 2 * q^98

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 $

$2$	−0.414214	−0.292893	−0.146447	−	0.989219i	$-0.546784\pi$
$2$	−0.414214	−0.292893	−0.146447	+	0.989219i	$0.546784\pi$
$3$	0	0
$3$	0	0
$4$	−1.82843	−0.914214
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.58579	0.560660
$9$	0	0
$10$	−0.414214	−0.130986
$11$	−2.41421	−0.727913	−0.363956	−	0.931416i	$-0.618574\pi$
$11$	−2.41421	−0.727913	−0.363956	+	0.931416i	$0.618574\pi$
$12$	0	0
$13$	−0.414214	−0.114882	−0.0574411	−	0.998349i	$-0.518294\pi$
$13$	−0.414214	−0.114882	−0.0574411	+	0.998349i	$0.518294\pi$
$14$	0.414214	0.110703
$15$	0	0
$16$	3.00000	0.750000
$17$	−0.828427	−0.200923	−0.100462	−	0.994941i	$-0.532032\pi$
$17$	−0.828427	−0.200923	−0.100462	+	0.994941i	$0.532032\pi$
$18$	0	0
$19$	4.41421	1.01269	0.506345	−	0.862331i	$-0.330996\pi$
$19$	4.41421	1.01269	0.506345	+	0.862331i	$0.330996\pi$
$20$	−1.82843	−0.408849
$21$	0	0
$22$	1.00000	0.213201
$23$	4.82843	1.00680	0.503398	−	0.864054i	$-0.332083\pi$
$23$	4.82843	1.00680	0.503398	+	0.864054i	$0.332083\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.171573	0.0336482
$27$	0	0
$28$	1.82843	0.345540
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−4.41421	−0.780330
$33$	0	0
$34$	0.343146	0.0588490
$35$	−1.00000	−0.169031
$36$	0	0
$37$	8.48528	1.39497	0.697486	−	0.716599i	$-0.254302\pi$
$37$	8.48528	1.39497	0.697486	+	0.716599i	$0.254302\pi$
$38$	−1.82843	−0.296610
$39$	0	0
$40$	1.58579	0.250735
$41$	−2.17157	−0.339143	−0.169571	−	0.985518i	$-0.554238\pi$
$41$	−2.17157	−0.339143	−0.169571	+	0.985518i	$0.554238\pi$
$42$	0	0
$43$	4.65685	0.710164	0.355082	−	0.934835i	$-0.384453\pi$
$43$	4.65685	0.710164	0.355082	+	0.934835i	$0.384453\pi$
$44$	4.41421	0.665468
$45$	0	0
$46$	−2.00000	−0.294884
$47$	−9.48528	−1.38357	−0.691785	−	0.722103i	$-0.743176\pi$
$47$	−9.48528	−1.38357	−0.691785	+	0.722103i	$0.743176\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−0.414214	−0.0585786
$51$	0	0
$52$	0.757359	0.105027
$53$	8.89949	1.22244	0.611220	−	0.791461i	$-0.290679\pi$
$53$	8.89949	1.22244	0.611220	+	0.791461i	$0.290679\pi$
$54$	0	0
$55$	−2.41421	−0.325532
$56$	−1.58579	−0.211910
$57$	0	0
$58$	1.65685	0.217556
$59$	10.4853	1.36507	0.682534	−	0.730854i	$-0.260878\pi$
$59$	10.4853	1.36507	0.682534	+	0.730854i	$0.260878\pi$
$60$	0	0
$61$	8.48528	1.08643	0.543214	−	0.839594i	$-0.317207\pi$
$61$	8.48528	1.08643	0.543214	+	0.839594i	$0.317207\pi$
$62$	−2.48528	−0.315631
$63$	0	0
$64$	−4.17157	−0.521447
$65$	−0.414214	−0.0513769
$66$	0	0
$67$	2.17157	0.265300	0.132650	−	0.991163i	$-0.457651\pi$
$67$	2.17157	0.265300	0.132650	+	0.991163i	$0.457651\pi$
$68$	1.51472	0.183687
$69$	0	0
$70$	0.414214	0.0495080
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	12.0711	1.41281	0.706406	−	0.707807i	$-0.250315\pi$
$73$	12.0711	1.41281	0.706406	+	0.707807i	$0.250315\pi$
$74$	−3.51472	−0.408578
$75$	0	0
$76$	−8.07107	−0.925815
$77$	2.41421	0.275125
$78$	0	0
$79$	9.17157	1.03188	0.515941	−	0.856624i	$-0.327442\pi$
$79$	9.17157	1.03188	0.515941	+	0.856624i	$0.327442\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	0.899495	0.0993326
$83$	9.48528	1.04114	0.520572	−	0.853818i	$-0.325719\pi$
$83$	9.48528	1.04114	0.520572	+	0.853818i	$0.325719\pi$
$84$	0	0
$85$	−0.828427	−0.0898555
$86$	−1.92893	−0.208002
$87$	0	0
$88$	−3.82843	−0.408112
$89$	2.65685	0.281626	0.140813	−	0.990036i	$-0.455028\pi$
$89$	2.65685	0.281626	0.140813	+	0.990036i	$0.455028\pi$
$90$	0	0
$91$	0.414214	0.0434214
$92$	−8.82843	−0.920427
$93$	0	0
$94$	3.92893	0.405238
$95$	4.41421	0.452889
$96$	0	0
$97$	−14.1421	−1.43592	−0.717958	−	0.696086i	$-0.754923\pi$
$97$	−14.1421	−1.43592	−0.717958	+	0.696086i	$0.754923\pi$
$98$	−0.414214	−0.0418419
$99$	0	0
$100$	−1.82843	−0.182843
$101$	−14.1716	−1.41012	−0.705062	−	0.709146i	$-0.749081\pi$
$101$	−14.1716	−1.41012	−0.705062	+	0.709146i	$0.749081\pi$
$102$	0	0
$103$	5.17157	0.509570	0.254785	−	0.966998i	$-0.417995\pi$
$103$	5.17157	0.509570	0.254785	+	0.966998i	$0.417995\pi$
$104$	−0.656854	−0.0644099
$105$	0	0
$106$	−3.68629	−0.358044
$107$	6.48528	0.626956	0.313478	−	0.949595i	$-0.398506\pi$
$107$	6.48528	0.626956	0.313478	+	0.949595i	$0.398506\pi$
$108$	0	0
$109$	−13.4853	−1.29166	−0.645828	−	0.763483i	$-0.723488\pi$
$109$	−13.4853	−1.29166	−0.645828	+	0.763483i	$0.723488\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	−3.00000	−0.283473
$113$	11.2426	1.05762	0.528809	−	0.848741i	$-0.322639\pi$
$113$	11.2426	1.05762	0.528809	+	0.848741i	$0.322639\pi$
$114$	0	0
$115$	4.82843	0.450253
$116$	7.31371	0.679061
$117$	0	0
$118$	−4.34315	−0.399819
$119$	0.828427	0.0759418
$120$	0	0
$121$	−5.17157	−0.470143
$122$	−3.51472	−0.318208
$123$	0	0
$124$	−10.9706	−0.985186
$125$	1.00000	0.0894427
$126$	0	0
$127$	5.34315	0.474128	0.237064	−	0.971494i	$-0.423815\pi$
$127$	5.34315	0.474128	0.237064	+	0.971494i	$0.423815\pi$
$128$	10.5563	0.933058
$129$	0	0
$130$	0.171573	0.0150479
$131$	−16.1421	−1.41034	−0.705172	−	0.709036i	$-0.749130\pi$
$131$	−16.1421	−1.41034	−0.705172	+	0.709036i	$0.749130\pi$
$132$	0	0
$133$	−4.41421	−0.382761
$134$	−0.899495	−0.0777045
$135$	0	0
$136$	−1.31371	−0.112650
$137$	−5.58579	−0.477226	−0.238613	−	0.971115i	$-0.576693\pi$
$137$	−5.58579	−0.477226	−0.238613	+	0.971115i	$0.576693\pi$
$138$	0	0
$139$	−1.31371	−0.111427	−0.0557137	−	0.998447i	$-0.517743\pi$
$139$	−1.31371	−0.111427	−0.0557137	+	0.998447i	$0.517743\pi$
$140$	1.82843	0.154530
$141$	0	0
$142$	−0.828427	−0.0695201
$143$	1.00000	0.0836242
$144$	0	0
$145$	−4.00000	−0.332182
$146$	−5.00000	−0.413803
$147$	0	0
$148$	−15.5147	−1.27530
$149$	−21.7990	−1.78584	−0.892921	−	0.450213i	$-0.851348\pi$
$149$	−21.7990	−1.78584	−0.892921	+	0.450213i	$0.851348\pi$
$150$	0	0
$151$	−0.686292	−0.0558496	−0.0279248	−	0.999610i	$-0.508890\pi$
$151$	−0.686292	−0.0558496	−0.0279248	+	0.999610i	$0.508890\pi$
$152$	7.00000	0.567775
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	6.00000	0.481932
$156$	0	0
$157$	9.17157	0.731971	0.365986	−	0.930621i	$-0.380732\pi$
$157$	9.17157	0.731971	0.365986	+	0.930621i	$0.380732\pi$
$158$	−3.79899	−0.302231
$159$	0	0
$160$	−4.41421	−0.348974
$161$	−4.82843	−0.380533
$162$	0	0
$163$	17.6569	1.38299	0.691496	−	0.722380i	$-0.256952\pi$
$163$	17.6569	1.38299	0.691496	+	0.722380i	$0.256952\pi$
$164$	3.97056	0.310049
$165$	0	0
$166$	−3.92893	−0.304944
$167$	11.3137	0.875481	0.437741	−	0.899101i	$-0.355779\pi$
$167$	11.3137	0.875481	0.437741	+	0.899101i	$0.355779\pi$
$168$	0	0
$169$	−12.8284	−0.986802
$170$	0.343146	0.0263181
$171$	0	0
$172$	−8.51472	−0.649241
$173$	−7.31371	−0.556051	−0.278025	−	0.960574i	$-0.589680\pi$
$173$	−7.31371	−0.556051	−0.278025	+	0.960574i	$0.589680\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−7.24264	−0.545935
$177$	0	0
$178$	−1.10051	−0.0824863
$179$	11.7279	0.876586	0.438293	−	0.898832i	$-0.355583\pi$
$179$	11.7279	0.876586	0.438293	+	0.898832i	$0.355583\pi$
$180$	0	0
$181$	−5.65685	−0.420471	−0.210235	−	0.977651i	$-0.567423\pi$
$181$	−5.65685	−0.420471	−0.210235	+	0.977651i	$0.567423\pi$
$182$	−0.171573	−0.0127178
$183$	0	0
$184$	7.65685	0.564471
$185$	8.48528	0.623850
$186$	0	0
$187$	2.00000	0.146254
$188$	17.3431	1.26488
$189$	0	0
$190$	−1.82843	−0.132648
$191$	−5.92893	−0.429002	−0.214501	−	0.976724i	$-0.568813\pi$
$191$	−5.92893	−0.429002	−0.214501	+	0.976724i	$0.568813\pi$
$192$	0	0
$193$	−22.4853	−1.61853	−0.809263	−	0.587447i	$-0.800133\pi$
$193$	−22.4853	−1.61853	−0.809263	+	0.587447i	$0.800133\pi$
$194$	5.85786	0.420570
$195$	0	0
$196$	−1.82843	−0.130602
$197$	−6.89949	−0.491569	−0.245784	−	0.969325i	$-0.579045\pi$
$197$	−6.89949	−0.491569	−0.245784	+	0.969325i	$0.579045\pi$
$198$	0	0
$199$	0.414214	0.0293628	0.0146814	−	0.999892i	$-0.495327\pi$
$199$	0.414214	0.0293628	0.0146814	+	0.999892i	$0.495327\pi$
$200$	1.58579	0.112132
$201$	0	0
$202$	5.87006	0.413016
$203$	4.00000	0.280745
$204$	0	0
$205$	−2.17157	−0.151669
$206$	−2.14214	−0.149250
$207$	0	0
$208$	−1.24264	−0.0861616
$209$	−10.6569	−0.737150
$210$	0	0
$211$	−14.4853	−0.997208	−0.498604	−	0.866830i	$-0.666154\pi$
$211$	−14.4853	−0.997208	−0.498604	+	0.866830i	$0.666154\pi$
$212$	−16.2721	−1.11757
$213$	0	0
$214$	−2.68629	−0.183631
$215$	4.65685	0.317595
$216$	0	0
$217$	−6.00000	−0.407307
$218$	5.58579	0.378317
$219$	0	0
$220$	4.41421	0.297606
$221$	0.343146	0.0230825
$222$	0	0
$223$	5.65685	0.378811	0.189405	−	0.981899i	$-0.439344\pi$
$223$	5.65685	0.378811	0.189405	+	0.981899i	$0.439344\pi$
$224$	4.41421	0.294937
$225$	0	0
$226$	−4.65685	−0.309769
$227$	−7.48528	−0.496816	−0.248408	−	0.968656i	$-0.579907\pi$
$227$	−7.48528	−0.496816	−0.248408	+	0.968656i	$0.579907\pi$
$228$	0	0
$229$	17.6569	1.16680	0.583399	−	0.812186i	$-0.301722\pi$
$229$	17.6569	1.16680	0.583399	+	0.812186i	$0.301722\pi$
$230$	−2.00000	−0.131876
$231$	0	0
$232$	−6.34315	−0.416448
$233$	16.5563	1.08464	0.542321	−	0.840171i	$-0.317546\pi$
$233$	16.5563	1.08464	0.542321	+	0.840171i	$0.317546\pi$
$234$	0	0
$235$	−9.48528	−0.618752
$236$	−19.1716	−1.24796
$237$	0	0
$238$	−0.343146	−0.0222428
$239$	−20.6274	−1.33428	−0.667138	−	0.744934i	$-0.732481\pi$
$239$	−20.6274	−1.33428	−0.667138	+	0.744934i	$0.732481\pi$
$240$	0	0
$241$	23.7990	1.53303	0.766514	−	0.642228i	$-0.221990\pi$
$241$	23.7990	1.53303	0.766514	+	0.642228i	$0.221990\pi$
$242$	2.14214	0.137702
$243$	0	0
$244$	−15.5147	−0.993228
$245$	1.00000	0.0638877
$246$	0	0
$247$	−1.82843	−0.116340
$248$	9.51472	0.604185
$249$	0	0
$250$	−0.414214	−0.0261972
$251$	−30.1421	−1.90255	−0.951277	−	0.308336i	$-0.900228\pi$
$251$	−30.1421	−1.90255	−0.951277	+	0.308336i	$0.900228\pi$
$252$	0	0
$253$	−11.6569	−0.732860
$254$	−2.21320	−0.138869
$255$	0	0
$256$	3.97056	0.248160
$257$	17.6569	1.10140	0.550702	−	0.834702i	$-0.314360\pi$
$257$	17.6569	1.10140	0.550702	+	0.834702i	$0.314360\pi$
$258$	0	0
$259$	−8.48528	−0.527250
$260$	0.757359	0.0469694
$261$	0	0
$262$	6.68629	0.413080
$263$	−7.65685	−0.472142	−0.236071	−	0.971736i	$-0.575860\pi$
$263$	−7.65685	−0.472142	−0.236071	+	0.971736i	$0.575860\pi$
$264$	0	0
$265$	8.89949	0.546692
$266$	1.82843	0.112108
$267$	0	0
$268$	−3.97056	−0.242541
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	21.2426	1.29040	0.645199	−	0.764014i	$-0.276774\pi$
$271$	21.2426	1.29040	0.645199	+	0.764014i	$0.276774\pi$
$272$	−2.48528	−0.150692
$273$	0	0
$274$	2.31371	0.139776
$275$	−2.41421	−0.145583
$276$	0	0
$277$	−15.6569	−0.940729	−0.470365	−	0.882472i	$-0.655878\pi$
$277$	−15.6569	−0.940729	−0.470365	+	0.882472i	$0.655878\pi$
$278$	0.544156	0.0326363
$279$	0	0
$280$	−1.58579	−0.0947689
$281$	16.3431	0.974950	0.487475	−	0.873137i	$-0.337918\pi$
$281$	16.3431	0.974950	0.487475	+	0.873137i	$0.337918\pi$
$282$	0	0
$283$	4.48528	0.266622	0.133311	−	0.991074i	$-0.457439\pi$
$283$	4.48528	0.266622	0.133311	+	0.991074i	$0.457439\pi$
$284$	−3.65685	−0.216994
$285$	0	0
$286$	−0.414214	−0.0244930
$287$	2.17157	0.128184
$288$	0	0
$289$	−16.3137	−0.959630
$290$	1.65685	0.0972938
$291$	0	0
$292$	−22.0711	−1.29161
$293$	−1.51472	−0.0884908	−0.0442454	−	0.999021i	$-0.514088\pi$
$293$	−1.51472	−0.0884908	−0.0442454	+	0.999021i	$0.514088\pi$
$294$	0	0
$295$	10.4853	0.610477
$296$	13.4558	0.782105
$297$	0	0
$298$	9.02944	0.523061
$299$	−2.00000	−0.115663
$300$	0	0
$301$	−4.65685	−0.268417
$302$	0.284271	0.0163580
$303$	0	0
$304$	13.2426	0.759518
$305$	8.48528	0.485866
$306$	0	0
$307$	2.48528	0.141843	0.0709213	−	0.997482i	$-0.477406\pi$
$307$	2.48528	0.141843	0.0709213	+	0.997482i	$0.477406\pi$
$308$	−4.41421	−0.251523
$309$	0	0
$310$	−2.48528	−0.141154
$311$	25.6569	1.45487	0.727433	−	0.686178i	$-0.240713\pi$
$311$	25.6569	1.45487	0.727433	+	0.686178i	$0.240713\pi$
$312$	0	0
$313$	−9.38478	−0.530459	−0.265229	−	0.964185i	$-0.585448\pi$
$313$	−9.38478	−0.530459	−0.265229	+	0.964185i	$0.585448\pi$
$314$	−3.79899	−0.214389
$315$	0	0
$316$	−16.7696	−0.943361
$317$	25.5858	1.43704	0.718520	−	0.695506i	$-0.244820\pi$
$317$	25.5858	1.43704	0.718520	+	0.695506i	$0.244820\pi$
$318$	0	0
$319$	9.65685	0.540680
$320$	−4.17157	−0.233198
$321$	0	0
$322$	2.00000	0.111456
$323$	−3.65685	−0.203473
$324$	0	0
$325$	−0.414214	−0.0229764
$326$	−7.31371	−0.405069
$327$	0	0
$328$	−3.44365	−0.190144
$329$	9.48528	0.522940
$330$	0	0
$331$	6.00000	0.329790	0.164895	−	0.986311i	$-0.447272\pi$
$331$	6.00000	0.329790	0.164895	+	0.986311i	$0.447272\pi$
$332$	−17.3431	−0.951829
$333$	0	0
$334$	−4.68629	−0.256422
$335$	2.17157	0.118646
$336$	0	0
$337$	−16.6274	−0.905753	−0.452877	−	0.891573i	$-0.649602\pi$
$337$	−16.6274	−0.905753	−0.452877	+	0.891573i	$0.649602\pi$
$338$	5.31371	0.289028
$339$	0	0
$340$	1.51472	0.0821472
$341$	−14.4853	−0.784422
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	7.38478	0.398160
$345$	0	0
$346$	3.02944	0.162864
$347$	8.82843	0.473935	0.236967	−	0.971518i	$-0.423847\pi$
$347$	8.82843	0.473935	0.236967	+	0.971518i	$0.423847\pi$
$348$	0	0
$349$	13.7990	0.738643	0.369321	−	0.929302i	$-0.379590\pi$
$349$	13.7990	0.738643	0.369321	+	0.929302i	$0.379590\pi$
$350$	0.414214	0.0221406
$351$	0	0
$352$	10.6569	0.568012
$353$	−2.14214	−0.114014	−0.0570072	−	0.998374i	$-0.518156\pi$
$353$	−2.14214	−0.114014	−0.0570072	+	0.998374i	$0.518156\pi$
$354$	0	0
$355$	2.00000	0.106149
$356$	−4.85786	−0.257466
$357$	0	0
$358$	−4.85786	−0.256746
$359$	−30.6985	−1.62020	−0.810102	−	0.586289i	$-0.800588\pi$
$359$	−30.6985	−1.62020	−0.810102	+	0.586289i	$0.800588\pi$
$360$	0	0
$361$	0.485281	0.0255411
$362$	2.34315	0.123153
$363$	0	0
$364$	−0.757359	−0.0396964
$365$	12.0711	0.631829
$366$	0	0
$367$	26.0000	1.35719	0.678594	−	0.734513i	$-0.262589\pi$
$367$	26.0000	1.35719	0.678594	+	0.734513i	$0.262589\pi$
$368$	14.4853	0.755097
$369$	0	0
$370$	−3.51472	−0.182722
$371$	−8.89949	−0.462039
$372$	0	0
$373$	−14.9706	−0.775146	−0.387573	−	0.921839i	$-0.626687\pi$
$373$	−14.9706	−0.775146	−0.387573	+	0.921839i	$0.626687\pi$
$374$	−0.828427	−0.0428369
$375$	0	0
$376$	−15.0416	−0.775713
$377$	1.65685	0.0853323
$378$	0	0
$379$	−34.2843	−1.76106	−0.880532	−	0.473986i	$-0.842815\pi$
$379$	−34.2843	−1.76106	−0.880532	+	0.473986i	$0.842815\pi$
$380$	−8.07107	−0.414037
$381$	0	0
$382$	2.45584	0.125652
$383$	21.6569	1.10661	0.553307	−	0.832978i	$-0.313366\pi$
$383$	21.6569	1.10661	0.553307	+	0.832978i	$0.313366\pi$
$384$	0	0
$385$	2.41421	0.123040
$386$	9.31371	0.474055
$387$	0	0
$388$	25.8579	1.31273
$389$	5.31371	0.269416	0.134708	−	0.990885i	$-0.456990\pi$
$389$	5.31371	0.269416	0.134708	+	0.990885i	$0.456990\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	1.58579	0.0800943
$393$	0	0
$394$	2.85786	0.143977
$395$	9.17157	0.461472
$396$	0	0
$397$	9.17157	0.460308	0.230154	−	0.973154i	$-0.426077\pi$
$397$	9.17157	0.460308	0.230154	+	0.973154i	$0.426077\pi$
$398$	−0.171573	−0.00860017
$399$	0	0
$400$	3.00000	0.150000
$401$	−33.7990	−1.68784	−0.843921	−	0.536468i	$-0.819758\pi$
$401$	−33.7990	−1.68784	−0.843921	+	0.536468i	$0.819758\pi$
$402$	0	0
$403$	−2.48528	−0.123801
$404$	25.9117	1.28915
$405$	0	0
$406$	−1.65685	−0.0822283
$407$	−20.4853	−1.01542
$408$	0	0
$409$	19.3137	0.955001	0.477501	−	0.878631i	$-0.341543\pi$
$409$	19.3137	0.955001	0.477501	+	0.878631i	$0.341543\pi$
$410$	0.899495	0.0444229
$411$	0	0
$412$	−9.45584	−0.465856
$413$	−10.4853	−0.515947
$414$	0	0
$415$	9.48528	0.465614
$416$	1.82843	0.0896460
$417$	0	0
$418$	4.41421	0.215906
$419$	−8.82843	−0.431297	−0.215648	−	0.976471i	$-0.569187\pi$
$419$	−8.82843	−0.431297	−0.215648	+	0.976471i	$0.569187\pi$
$420$	0	0
$421$	−7.00000	−0.341159	−0.170580	−	0.985344i	$-0.554564\pi$
$421$	−7.00000	−0.341159	−0.170580	+	0.985344i	$0.554564\pi$
$422$	6.00000	0.292075
$423$	0	0
$424$	14.1127	0.685373
$425$	−0.828427	−0.0401846
$426$	0	0
$427$	−8.48528	−0.410632
$428$	−11.8579	−0.573172
$429$	0	0
$430$	−1.92893	−0.0930214
$431$	−33.5858	−1.61777	−0.808885	−	0.587967i	$-0.799929\pi$
$431$	−33.5858	−1.61777	−0.808885	+	0.587967i	$0.799929\pi$
$432$	0	0
$433$	23.8701	1.14712	0.573561	−	0.819163i	$-0.305562\pi$
$433$	23.8701	1.14712	0.573561	+	0.819163i	$0.305562\pi$
$434$	2.48528	0.119297
$435$	0	0
$436$	24.6569	1.18085
$437$	21.3137	1.01957
$438$	0	0
$439$	19.2426	0.918401	0.459201	−	0.888333i	$-0.348136\pi$
$439$	19.2426	0.918401	0.459201	+	0.888333i	$0.348136\pi$
$440$	−3.82843	−0.182513
$441$	0	0
$442$	−0.142136	−0.00676070
$443$	40.9706	1.94657	0.973285	−	0.229600i	$-0.0737418\pi$
$443$	40.9706	1.94657	0.973285	+	0.229600i	$0.0737418\pi$
$444$	0	0
$445$	2.65685	0.125947
$446$	−2.34315	−0.110951
$447$	0	0
$448$	4.17157	0.197088
$449$	−16.8284	−0.794183	−0.397091	−	0.917779i	$-0.629980\pi$
$449$	−16.8284	−0.794183	−0.397091	+	0.917779i	$0.629980\pi$
$450$	0	0
$451$	5.24264	0.246866
$452$	−20.5563	−0.966889
$453$	0	0
$454$	3.10051	0.145514
$455$	0.414214	0.0194186
$456$	0	0
$457$	−1.85786	−0.0869072	−0.0434536	−	0.999055i	$-0.513836\pi$
$457$	−1.85786	−0.0869072	−0.0434536	+	0.999055i	$0.513836\pi$
$458$	−7.31371	−0.341747
$459$	0	0
$460$	−8.82843	−0.411628
$461$	31.6274	1.47304	0.736518	−	0.676418i	$-0.236469\pi$
$461$	31.6274	1.47304	0.736518	+	0.676418i	$0.236469\pi$
$462$	0	0
$463$	11.3431	0.527161	0.263580	−	0.964637i	$-0.415097\pi$
$463$	11.3431	0.527161	0.263580	+	0.964637i	$0.415097\pi$
$464$	−12.0000	−0.557086
$465$	0	0
$466$	−6.85786	−0.317684
$467$	−9.65685	−0.446866	−0.223433	−	0.974719i	$-0.571726\pi$
$467$	−9.65685	−0.446866	−0.223433	+	0.974719i	$0.571726\pi$
$468$	0	0
$469$	−2.17157	−0.100274
$470$	3.92893	0.181228
$471$	0	0
$472$	16.6274	0.765339
$473$	−11.2426	−0.516937
$474$	0	0
$475$	4.41421	0.202538
$476$	−1.51472	−0.0694270
$477$	0	0
$478$	8.54416	0.390801
$479$	2.14214	0.0978767	0.0489383	−	0.998802i	$-0.484416\pi$
$479$	2.14214	0.0978767	0.0489383	+	0.998802i	$0.484416\pi$
$480$	0	0
$481$	−3.51472	−0.160257
$482$	−9.85786	−0.449013
$483$	0	0
$484$	9.45584	0.429811
$485$	−14.1421	−0.642161
$486$	0	0
$487$	7.68629	0.348299	0.174150	−	0.984719i	$-0.444282\pi$
$487$	7.68629	0.348299	0.174150	+	0.984719i	$0.444282\pi$
$488$	13.4558	0.609117
$489$	0	0
$490$	−0.414214	−0.0187123
$491$	32.6274	1.47245	0.736227	−	0.676734i	$-0.236605\pi$
$491$	32.6274	1.47245	0.736227	+	0.676734i	$0.236605\pi$
$492$	0	0
$493$	3.31371	0.149242
$494$	0.757359	0.0340752
$495$	0	0
$496$	18.0000	0.808224
$497$	−2.00000	−0.0897123
$498$	0	0
$499$	39.7990	1.78165	0.890824	−	0.454349i	$-0.150128\pi$
$499$	39.7990	1.78165	0.890824	+	0.454349i	$0.150128\pi$
$500$	−1.82843	−0.0817697
$501$	0	0
$502$	12.4853	0.557245
$503$	9.62742	0.429265	0.214633	−	0.976695i	$-0.431145\pi$
$503$	9.62742	0.429265	0.214633	+	0.976695i	$0.431145\pi$
$504$	0	0
$505$	−14.1716	−0.630627
$506$	4.82843	0.214650
$507$	0	0
$508$	−9.76955	−0.433454
$509$	15.6569	0.693978	0.346989	−	0.937869i	$-0.387204\pi$
$509$	15.6569	0.693978	0.346989	+	0.937869i	$0.387204\pi$
$510$	0	0
$511$	−12.0711	−0.533993
$512$	−22.7574	−1.00574
$513$	0	0
$514$	−7.31371	−0.322594
$515$	5.17157	0.227887
$516$	0	0
$517$	22.8995	1.00712
$518$	3.51472	0.154428
$519$	0	0
$520$	−0.656854	−0.0288050
$521$	10.5147	0.460658	0.230329	−	0.973113i	$-0.426020\pi$
$521$	10.5147	0.460658	0.230329	+	0.973113i	$0.426020\pi$
$522$	0	0
$523$	26.8284	1.17313	0.586563	−	0.809904i	$-0.300481\pi$
$523$	26.8284	1.17313	0.586563	+	0.809904i	$0.300481\pi$
$524$	29.5147	1.28936
$525$	0	0
$526$	3.17157	0.138287
$527$	−4.97056	−0.216521
$528$	0	0
$529$	0.313708	0.0136395
$530$	−3.68629	−0.160122
$531$	0	0
$532$	8.07107	0.349925
$533$	0.899495	0.0389615
$534$	0	0
$535$	6.48528	0.280383
$536$	3.44365	0.148743
$537$	0	0
$538$	5.79899	0.250012
$539$	−2.41421	−0.103988
$540$	0	0
$541$	17.6274	0.757862	0.378931	−	0.925425i	$-0.376292\pi$
$541$	17.6274	0.757862	0.378931	+	0.925425i	$0.376292\pi$
$542$	−8.79899	−0.377949
$543$	0	0
$544$	3.65685	0.156786
$545$	−13.4853	−0.577646
$546$	0	0
$547$	16.2843	0.696265	0.348133	−	0.937445i	$-0.386816\pi$
$547$	16.2843	0.696265	0.348133	+	0.937445i	$0.386816\pi$
$548$	10.2132	0.436286
$549$	0	0
$550$	1.00000	0.0426401
$551$	−17.6569	−0.752207
$552$	0	0
$553$	−9.17157	−0.390015
$554$	6.48528	0.275533
$555$	0	0
$556$	2.40202	0.101868
$557$	4.20101	0.178003	0.0890013	−	0.996032i	$-0.471632\pi$
$557$	4.20101	0.178003	0.0890013	+	0.996032i	$0.471632\pi$
$558$	0	0
$559$	−1.92893	−0.0815851
$560$	−3.00000	−0.126773
$561$	0	0
$562$	−6.76955	−0.285556
$563$	−5.97056	−0.251629	−0.125815	−	0.992054i	$-0.540154\pi$
$563$	−5.97056	−0.251629	−0.125815	+	0.992054i	$0.540154\pi$
$564$	0	0
$565$	11.2426	0.472981
$566$	−1.85786	−0.0780919
$567$	0	0
$568$	3.17157	0.133076
$569$	31.3137	1.31274	0.656369	−	0.754440i	$-0.272091\pi$
$569$	31.3137	1.31274	0.656369	+	0.754440i	$0.272091\pi$
$570$	0	0
$571$	−4.20101	−0.175807	−0.0879034	−	0.996129i	$-0.528017\pi$
$571$	−4.20101	−0.175807	−0.0879034	+	0.996129i	$0.528017\pi$
$572$	−1.82843	−0.0764504
$573$	0	0
$574$	−0.899495	−0.0375442
$575$	4.82843	0.201359
$576$	0	0
$577$	−19.7279	−0.821284	−0.410642	−	0.911797i	$-0.634695\pi$
$577$	−19.7279	−0.821284	−0.410642	+	0.911797i	$0.634695\pi$
$578$	6.75736	0.281069
$579$	0	0
$580$	7.31371	0.303685
$581$	−9.48528	−0.393516
$582$	0	0
$583$	−21.4853	−0.889829
$584$	19.1421	0.792107
$585$	0	0
$586$	0.627417	0.0259184
$587$	35.3137	1.45755	0.728776	−	0.684752i	$-0.240089\pi$
$587$	35.3137	1.45755	0.728776	+	0.684752i	$0.240089\pi$
$588$	0	0
$589$	26.4853	1.09131
$590$	−4.34315	−0.178804
$591$	0	0
$592$	25.4558	1.04623
$593$	−38.1421	−1.56631	−0.783155	−	0.621827i	$-0.786391\pi$
$593$	−38.1421	−1.56631	−0.783155	+	0.621827i	$0.786391\pi$
$594$	0	0
$595$	0.828427	0.0339622
$596$	39.8579	1.63264
$597$	0	0
$598$	0.828427	0.0338769
$599$	24.8995	1.01737	0.508683	−	0.860954i	$-0.330133\pi$
$599$	24.8995	1.01737	0.508683	+	0.860954i	$0.330133\pi$
$600$	0	0
$601$	−20.6274	−0.841410	−0.420705	−	0.907198i	$-0.638217\pi$
$601$	−20.6274	−0.841410	−0.420705	+	0.907198i	$0.638217\pi$
$602$	1.92893	0.0786174
$603$	0	0
$604$	1.25483	0.0510585
$605$	−5.17157	−0.210254
$606$	0	0
$607$	23.5147	0.954433	0.477216	−	0.878786i	$-0.341646\pi$
$607$	23.5147	0.954433	0.477216	+	0.878786i	$0.341646\pi$
$608$	−19.4853	−0.790233
$609$	0	0
$610$	−3.51472	−0.142307
$611$	3.92893	0.158948
$612$	0	0
$613$	−29.9411	−1.20931	−0.604655	−	0.796487i	$-0.706689\pi$
$613$	−29.9411	−1.20931	−0.604655	+	0.796487i	$0.706689\pi$
$614$	−1.02944	−0.0415447
$615$	0	0
$616$	3.82843	0.154252
$617$	−48.7696	−1.96339	−0.981694	−	0.190464i	$-0.939001\pi$
$617$	−48.7696	−1.96339	−0.981694	+	0.190464i	$0.939001\pi$
$618$	0	0
$619$	16.2132	0.651664	0.325832	−	0.945428i	$-0.394356\pi$
$619$	16.2132	0.651664	0.325832	+	0.945428i	$0.394356\pi$
$620$	−10.9706	−0.440588
$621$	0	0
$622$	−10.6274	−0.426121
$623$	−2.65685	−0.106445
$624$	0	0
$625$	1.00000	0.0400000
$626$	3.88730	0.155368
$627$	0	0
$628$	−16.7696	−0.669178
$629$	−7.02944	−0.280282
$630$	0	0
$631$	46.6274	1.85621	0.928104	−	0.372321i	$-0.121438\pi$
$631$	46.6274	1.85621	0.928104	+	0.372321i	$0.121438\pi$
$632$	14.5442	0.578535
$633$	0	0
$634$	−10.5980	−0.420900
$635$	5.34315	0.212036
$636$	0	0
$637$	−0.414214	−0.0164117
$638$	−4.00000	−0.158362
$639$	0	0
$640$	10.5563	0.417276
$641$	16.4853	0.651129	0.325565	−	0.945520i	$-0.394446\pi$
$641$	16.4853	0.651129	0.325565	+	0.945520i	$0.394446\pi$
$642$	0	0
$643$	−44.7696	−1.76554	−0.882769	−	0.469807i	$-0.844324\pi$
$643$	−44.7696	−1.76554	−0.882769	+	0.469807i	$0.844324\pi$
$644$	8.82843	0.347889
$645$	0	0
$646$	1.51472	0.0595958
$647$	−36.6569	−1.44113	−0.720565	−	0.693388i	$-0.756117\pi$
$647$	−36.6569	−1.44113	−0.720565	+	0.693388i	$0.756117\pi$
$648$	0	0
$649$	−25.3137	−0.993650
$650$	0.171573	0.00672964
$651$	0	0
$652$	−32.2843	−1.26435
$653$	−47.7990	−1.87052	−0.935260	−	0.353963i	$-0.884834\pi$
$653$	−47.7990	−1.87052	−0.935260	+	0.353963i	$0.884834\pi$
$654$	0	0
$655$	−16.1421	−0.630725
$656$	−6.51472	−0.254357
$657$	0	0
$658$	−3.92893	−0.153166
$659$	−23.6569	−0.921540	−0.460770	−	0.887520i	$-0.652427\pi$
$659$	−23.6569	−0.921540	−0.460770	+	0.887520i	$0.652427\pi$
$660$	0	0
$661$	−47.7990	−1.85917	−0.929583	−	0.368614i	$-0.879832\pi$
$661$	−47.7990	−1.85917	−0.929583	+	0.368614i	$0.879832\pi$
$662$	−2.48528	−0.0965932
$663$	0	0
$664$	15.0416	0.583728
$665$	−4.41421	−0.171176
$666$	0	0
$667$	−19.3137	−0.747830
$668$	−20.6863	−0.800377
$669$	0	0
$670$	−0.899495	−0.0347505
$671$	−20.4853	−0.790826
$672$	0	0
$673$	8.82843	0.340311	0.170155	−	0.985417i	$-0.445573\pi$
$673$	8.82843	0.340311	0.170155	+	0.985417i	$0.445573\pi$
$674$	6.88730	0.265289
$675$	0	0
$676$	23.4558	0.902148
$677$	42.1421	1.61965	0.809827	−	0.586669i	$-0.199561\pi$
$677$	42.1421	1.61965	0.809827	+	0.586669i	$0.199561\pi$
$678$	0	0
$679$	14.1421	0.542725
$680$	−1.31371	−0.0503784
$681$	0	0
$682$	6.00000	0.229752
$683$	−32.4853	−1.24301	−0.621507	−	0.783408i	$-0.713479\pi$
$683$	−32.4853	−1.24301	−0.621507	+	0.783408i	$0.713479\pi$
$684$	0	0
$685$	−5.58579	−0.213422
$686$	0.414214	0.0158147
$687$	0	0
$688$	13.9706	0.532623
$689$	−3.68629	−0.140437
$690$	0	0
$691$	−2.55635	−0.0972481	−0.0486241	−	0.998817i	$-0.515484\pi$
$691$	−2.55635	−0.0972481	−0.0486241	+	0.998817i	$0.515484\pi$
$692$	13.3726	0.508349
$693$	0	0
$694$	−3.65685	−0.138812
$695$	−1.31371	−0.0498318
$696$	0	0
$697$	1.79899	0.0681416
$698$	−5.71573	−0.216344
$699$	0	0
$700$	1.82843	0.0691080
$701$	39.6569	1.49782	0.748909	−	0.662672i	$-0.230578\pi$
$701$	39.6569	1.49782	0.748909	+	0.662672i	$0.230578\pi$
$702$	0	0
$703$	37.4558	1.41267
$704$	10.0711	0.379568
$705$	0	0
$706$	0.887302	0.0333940
$707$	14.1716	0.532977
$708$	0	0
$709$	−43.6274	−1.63846	−0.819231	−	0.573464i	$-0.805599\pi$
$709$	−43.6274	−1.63846	−0.819231	+	0.573464i	$0.805599\pi$
$710$	−0.828427	−0.0310903
$711$	0	0
$712$	4.21320	0.157896
$713$	28.9706	1.08496
$714$	0	0
$715$	1.00000	0.0373979
$716$	−21.4437	−0.801387
$717$	0	0
$718$	12.7157	0.474547
$719$	3.79899	0.141678	0.0708392	−	0.997488i	$-0.477432\pi$
$719$	3.79899	0.141678	0.0708392	+	0.997488i	$0.477432\pi$
$720$	0	0
$721$	−5.17157	−0.192599
$722$	−0.201010	−0.00748082
$723$	0	0
$724$	10.3431	0.384400
$725$	−4.00000	−0.148556
$726$	0	0
$727$	−48.0833	−1.78331	−0.891655	−	0.452716i	$-0.850455\pi$
$727$	−48.0833	−1.78331	−0.891655	+	0.452716i	$0.850455\pi$
$728$	0.656854	0.0243446
$729$	0	0
$730$	−5.00000	−0.185058
$731$	−3.85786	−0.142688
$732$	0	0
$733$	−32.7574	−1.20992	−0.604960	−	0.796256i	$-0.706811\pi$
$733$	−32.7574	−1.20992	−0.604960	+	0.796256i	$0.706811\pi$
$734$	−10.7696	−0.397511
$735$	0	0
$736$	−21.3137	−0.785634
$737$	−5.24264	−0.193115
$738$	0	0
$739$	−40.8284	−1.50190	−0.750949	−	0.660360i	$-0.770404\pi$
$739$	−40.8284	−1.50190	−0.750949	+	0.660360i	$0.770404\pi$
$740$	−15.5147	−0.570332
$741$	0	0
$742$	3.68629	0.135328
$743$	0.343146	0.0125888	0.00629440	−	0.999980i	$-0.497996\pi$
$743$	0.343146	0.0125888	0.00629440	+	0.999980i	$0.497996\pi$
$744$	0	0
$745$	−21.7990	−0.798653
$746$	6.20101	0.227035
$747$	0	0
$748$	−3.65685	−0.133708
$749$	−6.48528	−0.236967
$750$	0	0
$751$	−15.1716	−0.553619	−0.276809	−	0.960925i	$-0.589277\pi$
$751$	−15.1716	−0.553619	−0.276809	+	0.960925i	$0.589277\pi$
$752$	−28.4558	−1.03768
$753$	0	0
$754$	−0.686292	−0.0249933
$755$	−0.686292	−0.0249767
$756$	0	0
$757$	−21.5147	−0.781966	−0.390983	−	0.920398i	$-0.627865\pi$
$757$	−21.5147	−0.781966	−0.390983	+	0.920398i	$0.627865\pi$
$758$	14.2010	0.515804
$759$	0	0
$760$	7.00000	0.253917
$761$	20.6274	0.747743	0.373872	−	0.927480i	$-0.378030\pi$
$761$	20.6274	0.747743	0.373872	+	0.927480i	$0.378030\pi$
$762$	0	0
$763$	13.4853	0.488200
$764$	10.8406	0.392200
$765$	0	0
$766$	−8.97056	−0.324120
$767$	−4.34315	−0.156822
$768$	0	0
$769$	−26.7696	−0.965335	−0.482667	−	0.875804i	$-0.660332\pi$
$769$	−26.7696	−0.965335	−0.482667	+	0.875804i	$0.660332\pi$
$770$	−1.00000	−0.0360375
$771$	0	0
$772$	41.1127	1.47968
$773$	−15.8579	−0.570368	−0.285184	−	0.958473i	$-0.592055\pi$
$773$	−15.8579	−0.570368	−0.285184	+	0.958473i	$0.592055\pi$
$774$	0	0
$775$	6.00000	0.215526
$776$	−22.4264	−0.805061
$777$	0	0
$778$	−2.20101	−0.0789100
$779$	−9.58579	−0.343446
$780$	0	0
$781$	−4.82843	−0.172775
$782$	1.65685	0.0592490
$783$	0	0
$784$	3.00000	0.107143
$785$	9.17157	0.327347
$786$	0	0
$787$	−35.7990	−1.27610	−0.638048	−	0.769997i	$-0.720258\pi$
$787$	−35.7990	−1.27610	−0.638048	+	0.769997i	$0.720258\pi$
$788$	12.6152	0.449399
$789$	0	0
$790$	−3.79899	−0.135162
$791$	−11.2426	−0.399742
$792$	0	0
$793$	−3.51472	−0.124811
$794$	−3.79899	−0.134821
$795$	0	0
$796$	−0.757359	−0.0268439
$797$	16.0000	0.566749	0.283375	−	0.959009i	$-0.408546\pi$
$797$	16.0000	0.566749	0.283375	+	0.959009i	$0.408546\pi$
$798$	0	0
$799$	7.85786	0.277991
$800$	−4.41421	−0.156066
$801$	0	0
$802$	14.0000	0.494357
$803$	−29.1421	−1.02840
$804$	0	0
$805$	−4.82843	−0.170180
$806$	1.02944	0.0362604
$807$	0	0
$808$	−22.4731	−0.790600
$809$	−44.0833	−1.54988	−0.774942	−	0.632032i	$-0.782221\pi$
$809$	−44.0833	−1.54988	−0.774942	+	0.632032i	$0.782221\pi$
$810$	0	0
$811$	17.3848	0.610462	0.305231	−	0.952278i	$-0.401266\pi$
$811$	17.3848	0.610462	0.305231	+	0.952278i	$0.401266\pi$
$812$	−7.31371	−0.256661
$813$	0	0
$814$	8.48528	0.297409
$815$	17.6569	0.618493
$816$	0	0
$817$	20.5563	0.719176
$818$	−8.00000	−0.279713
$819$	0	0
$820$	3.97056	0.138658
$821$	42.9706	1.49968	0.749841	−	0.661618i	$-0.230130\pi$
$821$	42.9706	1.49968	0.749841	+	0.661618i	$0.230130\pi$
$822$	0	0
$823$	−27.2843	−0.951070	−0.475535	−	0.879697i	$-0.657745\pi$
$823$	−27.2843	−0.951070	−0.475535	+	0.879697i	$0.657745\pi$
$824$	8.20101	0.285696
$825$	0	0
$826$	4.34315	0.151117
$827$	19.5147	0.678593	0.339297	−	0.940679i	$-0.389811\pi$
$827$	19.5147	0.678593	0.339297	+	0.940679i	$0.389811\pi$
$828$	0	0
$829$	−26.8284	−0.931790	−0.465895	−	0.884840i	$-0.654268\pi$
$829$	−26.8284	−0.931790	−0.465895	+	0.884840i	$0.654268\pi$
$830$	−3.92893	−0.136375
$831$	0	0
$832$	1.72792	0.0599049
$833$	−0.828427	−0.0287033
$834$	0	0
$835$	11.3137	0.391527
$836$	19.4853	0.673913
$837$	0	0
$838$	3.65685	0.126324
$839$	13.7990	0.476394	0.238197	−	0.971217i	$-0.423444\pi$
$839$	13.7990	0.476394	0.238197	+	0.971217i	$0.423444\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	2.89949	0.0999232
$843$	0	0
$844$	26.4853	0.911661
$845$	−12.8284	−0.441311
$846$	0	0
$847$	5.17157	0.177697
$848$	26.6985	0.916830
$849$	0	0
$850$	0.343146	0.0117698
$851$	40.9706	1.40445
$852$	0	0
$853$	12.4853	0.427488	0.213744	−	0.976890i	$-0.431434\pi$
$853$	12.4853	0.427488	0.213744	+	0.976890i	$0.431434\pi$
$854$	3.51472	0.120271
$855$	0	0
$856$	10.2843	0.351509
$857$	−5.31371	−0.181513	−0.0907564	−	0.995873i	$-0.528928\pi$
$857$	−5.31371	−0.181513	−0.0907564	+	0.995873i	$0.528928\pi$
$858$	0	0
$859$	32.3553	1.10395	0.551975	−	0.833861i	$-0.313874\pi$
$859$	32.3553	1.10395	0.551975	+	0.833861i	$0.313874\pi$
$860$	−8.51472	−0.290349
$861$	0	0
$862$	13.9117	0.473834
$863$	−28.3431	−0.964812	−0.482406	−	0.875948i	$-0.660237\pi$
$863$	−28.3431	−0.964812	−0.482406	+	0.875948i	$0.660237\pi$
$864$	0	0
$865$	−7.31371	−0.248674
$866$	−9.88730	−0.335984
$867$	0	0
$868$	10.9706	0.372365
$869$	−22.1421	−0.751121
$870$	0	0
$871$	−0.899495	−0.0304782
$872$	−21.3848	−0.724180
$873$	0	0
$874$	−8.82843	−0.298626
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−29.1716	−0.985054	−0.492527	−	0.870297i	$-0.663927\pi$
$877$	−29.1716	−0.985054	−0.492527	+	0.870297i	$0.663927\pi$
$878$	−7.97056	−0.268993
$879$	0	0
$880$	−7.24264	−0.244149
$881$	−4.34315	−0.146324	−0.0731621	−	0.997320i	$-0.523309\pi$
$881$	−4.34315	−0.146324	−0.0731621	+	0.997320i	$0.523309\pi$
$882$	0	0
$883$	13.6569	0.459590	0.229795	−	0.973239i	$-0.426194\pi$
$883$	13.6569	0.459590	0.229795	+	0.973239i	$0.426194\pi$
$884$	−0.627417	−0.0211023
$885$	0	0
$886$	−16.9706	−0.570137
$887$	55.7696	1.87256	0.936279	−	0.351257i	$-0.114246\pi$
$887$	55.7696	1.87256	0.936279	+	0.351257i	$0.114246\pi$
$888$	0	0
$889$	−5.34315	−0.179203
$890$	−1.10051	−0.0368890
$891$	0	0
$892$	−10.3431	−0.346314
$893$	−41.8701	−1.40113
$894$	0	0
$895$	11.7279	0.392021
$896$	−10.5563	−0.352663
$897$	0	0
$898$	6.97056	0.232611
$899$	−24.0000	−0.800445
$900$	0	0
$901$	−7.37258	−0.245616
$902$	−2.17157	−0.0723055
$903$	0	0
$904$	17.8284	0.592965
$905$	−5.65685	−0.188040
$906$	0	0
$907$	−43.4853	−1.44391	−0.721953	−	0.691943i	$-0.756755\pi$
$907$	−43.4853	−1.44391	−0.721953	+	0.691943i	$0.756755\pi$
$908$	13.6863	0.454196
$909$	0	0
$910$	−0.171573	−0.00568759
$911$	46.8406	1.55190	0.775949	−	0.630795i	$-0.217271\pi$
$911$	46.8406	1.55190	0.775949	+	0.630795i	$0.217271\pi$
$912$	0	0
$913$	−22.8995	−0.757863
$914$	0.769553	0.0254545
$915$	0	0
$916$	−32.2843	−1.06670
$917$	16.1421	0.533060
$918$	0	0
$919$	−36.7696	−1.21292	−0.606458	−	0.795116i	$-0.707410\pi$
$919$	−36.7696	−1.21292	−0.606458	+	0.795116i	$0.707410\pi$
$920$	7.65685	0.252439
$921$	0	0
$922$	−13.1005	−0.431442
$923$	−0.828427	−0.0272680
$924$	0	0
$925$	8.48528	0.278994
$926$	−4.69848	−0.154402
$927$	0	0
$928$	17.6569	0.579615
$929$	43.9706	1.44263	0.721314	−	0.692609i	$-0.243539\pi$
$929$	43.9706	1.44263	0.721314	+	0.692609i	$0.243539\pi$
$930$	0	0
$931$	4.41421	0.144670
$932$	−30.2721	−0.991595
$933$	0	0
$934$	4.00000	0.130884
$935$	2.00000	0.0654070
$936$	0	0
$937$	−0.899495	−0.0293852	−0.0146926	−	0.999892i	$-0.504677\pi$
$937$	−0.899495	−0.0293852	−0.0146926	+	0.999892i	$0.504677\pi$
$938$	0.899495	0.0293696
$939$	0	0
$940$	17.3431	0.565671
$941$	26.3137	0.857802	0.428901	−	0.903351i	$-0.358901\pi$
$941$	26.3137	0.857802	0.428901	+	0.903351i	$0.358901\pi$
$942$	0	0
$943$	−10.4853	−0.341448
$944$	31.4558	1.02380
$945$	0	0
$946$	4.65685	0.151407
$947$	41.4558	1.34713	0.673567	−	0.739126i	$-0.264761\pi$
$947$	41.4558	1.34713	0.673567	+	0.739126i	$0.264761\pi$
$948$	0	0
$949$	−5.00000	−0.162307
$950$	−1.82843	−0.0593220
$951$	0	0
$952$	1.31371	0.0425775
$953$	−17.8579	−0.578473	−0.289236	−	0.957258i	$-0.593401\pi$
$953$	−17.8579	−0.578473	−0.289236	+	0.957258i	$0.593401\pi$
$954$	0	0
$955$	−5.92893	−0.191856
$956$	37.7157	1.21981
$957$	0	0
$958$	−0.887302	−0.0286674
$959$	5.58579	0.180374
$960$	0	0
$961$	5.00000	0.161290
$962$	1.45584	0.0469383
$963$	0	0
$964$	−43.5147	−1.40151
$965$	−22.4853	−0.723827
$966$	0	0
$967$	8.17157	0.262780	0.131390	−	0.991331i	$-0.458056\pi$
$967$	8.17157	0.262780	0.131390	+	0.991331i	$0.458056\pi$
$968$	−8.20101	−0.263590
$969$	0	0
$970$	5.85786	0.188085
$971$	−37.4558	−1.20202	−0.601008	−	0.799243i	$-0.705234\pi$
$971$	−37.4558	−1.20202	−0.601008	+	0.799243i	$0.705234\pi$
$972$	0	0
$973$	1.31371	0.0421156
$974$	−3.18377	−0.102014
$975$	0	0
$976$	25.4558	0.814822
$977$	21.5269	0.688707	0.344353	−	0.938840i	$-0.388098\pi$
$977$	21.5269	0.688707	0.344353	+	0.938840i	$0.388098\pi$
$978$	0	0
$979$	−6.41421	−0.204999
$980$	−1.82843	−0.0584070
$981$	0	0
$982$	−13.5147	−0.431272
$983$	−36.9706	−1.17918	−0.589589	−	0.807703i	$-0.700710\pi$
$983$	−36.9706	−1.17918	−0.589589	+	0.807703i	$0.700710\pi$
$984$	0	0
$985$	−6.89949	−0.219836
$986$	−1.37258	−0.0437119
$987$	0	0
$988$	3.34315	0.106360
$989$	22.4853	0.714990
$990$	0	0
$991$	28.8284	0.915765	0.457883	−	0.889013i	$-0.348608\pi$
$991$	28.8284	0.915765	0.457883	+	0.889013i	$0.348608\pi$
$992$	−26.4853	−0.840909
$993$	0	0
$994$	0.828427	0.0262761
$995$	0.414214	0.0131315
$996$	0	0
$997$	−48.4853	−1.53554	−0.767772	−	0.640723i	$-0.778635\pi$
$997$	−48.4853	−1.53554	−0.767772	+	0.640723i	$0.778635\pi$
$998$	−16.4853	−0.521832
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	945.2.a.k.1.1	yes	2
3.2	odd	2		945.2.a.b.1.2	✓	2
5.4	even	2		4725.2.a.v.1.2		2
7.6	odd	2		6615.2.a.w.1.1		2
15.14	odd	2		4725.2.a.bg.1.1		2
21.20	even	2		6615.2.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.a.b.1.2	✓	2	3.2	odd	2
945.2.a.k.1.1	yes	2	1.1	even	1	trivial
4725.2.a.v.1.2		2	5.4	even	2
4725.2.a.bg.1.1		2	15.14	odd	2
6615.2.a.l.1.2		2	21.20	even	2
6615.2.a.w.1.1		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	945.a (trivial)

\(f(q)\)	\(=\)	\(q-0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.58579 q^{8} +O(q^{10})\)	\(q-0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.58579 q^{8} -0.414214 q^{10} -2.41421 q^{11} -0.414214 q^{13} +0.414214 q^{14} +3.00000 q^{16} -0.828427 q^{17} +4.41421 q^{19} -1.82843 q^{20} +1.00000 q^{22} +4.82843 q^{23} +1.00000 q^{25} +0.171573 q^{26} +1.82843 q^{28} -4.00000 q^{29} +6.00000 q^{31} -4.41421 q^{32} +0.343146 q^{34} -1.00000 q^{35} +8.48528 q^{37} -1.82843 q^{38} +1.58579 q^{40} -2.17157 q^{41} +4.65685 q^{43} +4.41421 q^{44} -2.00000 q^{46} -9.48528 q^{47} +1.00000 q^{49} -0.414214 q^{50} +0.757359 q^{52} +8.89949 q^{53} -2.41421 q^{55} -1.58579 q^{56} +1.65685 q^{58} +10.4853 q^{59} +8.48528 q^{61} -2.48528 q^{62} -4.17157 q^{64} -0.414214 q^{65} +2.17157 q^{67} +1.51472 q^{68} +0.414214 q^{70} +2.00000 q^{71} +12.0711 q^{73} -3.51472 q^{74} -8.07107 q^{76} +2.41421 q^{77} +9.17157 q^{79} +3.00000 q^{80} +0.899495 q^{82} +9.48528 q^{83} -0.828427 q^{85} -1.92893 q^{86} -3.82843 q^{88} +2.65685 q^{89} +0.414214 q^{91} -8.82843 q^{92} +3.92893 q^{94} +4.41421 q^{95} -14.1421 q^{97} -0.414214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8} + 2 q^{10} - 2 q^{11} + 2 q^{13} - 2 q^{14} + 6 q^{16} + 4 q^{17} + 6 q^{19} + 2 q^{20} + 2 q^{22} + 4 q^{23} + 2 q^{25} + 6 q^{26} - 2 q^{28} - 8 q^{29} + 12 q^{31} - 6 q^{32} + 12 q^{34} - 2 q^{35} + 2 q^{38} + 6 q^{40} - 10 q^{41} - 2 q^{43} + 6 q^{44} - 4 q^{46} - 2 q^{47} + 2 q^{49} + 2 q^{50} + 10 q^{52} - 2 q^{53} - 2 q^{55} - 6 q^{56} - 8 q^{58} + 4 q^{59} + 12 q^{62} - 14 q^{64} + 2 q^{65} + 10 q^{67} + 20 q^{68} - 2 q^{70} + 4 q^{71} + 10 q^{73} - 24 q^{74} - 2 q^{76} + 2 q^{77} + 24 q^{79} + 6 q^{80} - 18 q^{82} + 2 q^{83} + 4 q^{85} - 18 q^{86} - 2 q^{88} - 6 q^{89} - 2 q^{91} - 12 q^{92} + 22 q^{94} + 6 q^{95} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8 + 2 * q^10 - 2 * q^11 + 2 * q^13 - 2 * q^14 + 6 * q^16 + 4 * q^17 + 6 * q^19 + 2 * q^20 + 2 * q^22 + 4 * q^23 + 2 * q^25 + 6 * q^26 - 2 * q^28 - 8 * q^29 + 12 * q^31 - 6 * q^32 + 12 * q^34 - 2 * q^35 + 2 * q^38 + 6 * q^40 - 10 * q^41 - 2 * q^43 + 6 * q^44 - 4 * q^46 - 2 * q^47 + 2 * q^49 + 2 * q^50 + 10 * q^52 - 2 * q^53 - 2 * q^55 - 6 * q^56 - 8 * q^58 + 4 * q^59 + 12 * q^62 - 14 * q^64 + 2 * q^65 + 10 * q^67 + 20 * q^68 - 2 * q^70 + 4 * q^71 + 10 * q^73 - 24 * q^74 - 2 * q^76 + 2 * q^77 + 24 * q^79 + 6 * q^80 - 18 * q^82 + 2 * q^83 + 4 * q^85 - 18 * q^86 - 2 * q^88 - 6 * q^89 - 2 * q^91 - 12 * q^92 + 22 * q^94 + 6 * q^95 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more