LMFDB - Embedded newform 1225.2.b.j.99.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,2,Mod(99,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1225.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1225.b (of order $2$, degree $1$, not 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:	$9.78167424761$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 245)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.2
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1225.99
Dual form			1225.2.b.j.99.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} +2.41421i q^{3} +3.41421 q^{6} -2.82843i q^{8} -2.82843 q^{9} +O(q^{10})$	$q-1.41421i q^{2} +2.41421i q^{3} +3.41421 q^{6} -2.82843i q^{8} -2.82843 q^{9} -5.82843 q^{11} +1.58579i q^{13} -4.00000 q^{16} +5.24264i q^{17} +4.00000i q^{18} -6.00000 q^{19} +8.24264i q^{22} +4.58579i q^{23} +6.82843 q^{24} +2.24264 q^{26} +0.414214i q^{27} -2.65685 q^{29} +1.75736 q^{31} -14.0711i q^{33} +7.41421 q^{34} +6.24264i q^{37} +8.48528i q^{38} -3.82843 q^{39} +2.24264 q^{41} +2.00000i q^{43} +6.48528 q^{46} -1.24264i q^{47} -9.65685i q^{48} -12.6569 q^{51} -4.24264i q^{53} +0.585786 q^{54} -14.4853i q^{57} +3.75736i q^{58} -6.24264 q^{59} +2.82843 q^{61} -2.48528i q^{62} -8.00000 q^{64} -19.8995 q^{66} -0.242641i q^{67} -11.0711 q^{69} -8.82843 q^{71} +8.00000i q^{72} +8.48528i q^{73} +8.82843 q^{74} +5.41421i q^{78} +15.4853 q^{79} -9.48528 q^{81} -3.17157i q^{82} +2.82843 q^{86} -6.41421i q^{87} +16.4853i q^{88} +8.00000 q^{89} +4.24264i q^{93} -1.75736 q^{94} -4.75736i q^{97} +16.4853 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{6}+O(q^{10}) $ 4 * q + 8 * q^6	$ 4 q + 8 q^{6} - 12 q^{11} - 16 q^{16} - 24 q^{19} + 16 q^{24} - 8 q^{26} + 12 q^{29} + 24 q^{31} + 24 q^{34} - 4 q^{39} - 8 q^{41} - 8 q^{46} - 28 q^{51} + 8 q^{54} - 8 q^{59} - 32 q^{64} - 40 q^{66} - 16 q^{69} - 24 q^{71} + 24 q^{74} + 28 q^{79} - 4 q^{81} + 32 q^{89} - 24 q^{94} + 32 q^{99}+O(q^{100}) $ 4 * q + 8 * q^6 - 12 * q^11 - 16 * q^16 - 24 * q^19 + 16 * q^24 - 8 * q^26 + 12 * q^29 + 24 * q^31 + 24 * q^34 - 4 * q^39 - 8 * q^41 - 8 * q^46 - 28 * q^51 + 8 * q^54 - 8 * q^59 - 32 * q^64 - 40 * q^66 - 16 * q^69 - 24 * q^71 + 24 * q^74 + 28 * q^79 - 4 * q^81 + 32 * q^89 - 24 * q^94 + 32 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.41421i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$2$	− 1.41421i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$3$	2.41421i	1.39385i	0.717146	+	0.696923i	$0.245448\pi$
$3$	2.41421i	1.39385i	−0.717146	+	0.696923i	$0.754552\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	3.41421	1.39385
$7$	0	0
$7$	0	0
$8$	− 2.82843i	− 1.00000i
$9$	−2.82843	−0.942809
$10$	0	0
$11$	−5.82843	−1.75734	−0.878668	−	0.477432i	$-0.841568\pi$
$11$	−5.82843	−1.75734	−0.878668	+	0.477432i	$0.841568\pi$
$12$	0	0
$13$	1.58579i	0.439818i	0.975520	+	0.219909i	$0.0705760\pi$
$13$	1.58579i	0.439818i	−0.975520	+	0.219909i	$0.929424\pi$
$14$	0	0
$15$	0	0
$16$	−4.00000	−1.00000
$17$	5.24264i	1.27153i	0.771884	+	0.635764i	$0.219315\pi$
$17$	5.24264i	1.27153i	−0.771884	+	0.635764i	$0.780685\pi$
$18$	4.00000i	0.942809i
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	0	0
$22$	8.24264i	1.75734i
$23$	4.58579i	0.956203i	0.878305	+	0.478101i	$0.158675\pi$
$23$	4.58579i	0.956203i	−0.878305	+	0.478101i	$0.841325\pi$
$24$	6.82843	1.39385
$25$	0	0
$26$	2.24264	0.439818
$27$	0.414214i	0.0797154i
$28$	0	0
$29$	−2.65685	−0.493365	−0.246683	−	0.969096i	$-0.579341\pi$
$29$	−2.65685	−0.493365	−0.246683	+	0.969096i	$0.579341\pi$
$30$	0	0
$31$	1.75736	0.315631	0.157816	−	0.987469i	$-0.449555\pi$
$31$	1.75736	0.315631	0.157816	+	0.987469i	$0.449555\pi$
$32$	0	0
$33$	− 14.0711i	− 2.44946i
$34$	7.41421	1.27153
$35$	0	0
$36$	0	0
$37$	6.24264i	1.02628i	0.858304	+	0.513142i	$0.171519\pi$
$37$	6.24264i	1.02628i	−0.858304	+	0.513142i	$0.828481\pi$
$38$	8.48528i	1.37649i
$39$	−3.82843	−0.613039
$40$	0	0
$41$	2.24264	0.350242	0.175121	−	0.984547i	$-0.443968\pi$
$41$	2.24264	0.350242	0.175121	+	0.984547i	$0.443968\pi$
$42$	0	0
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	0	0
$45$	0	0
$46$	6.48528	0.956203
$47$	− 1.24264i	− 0.181258i	−0.995885	−	0.0906289i	$-0.971112\pi$
$47$	− 1.24264i	− 0.181258i	0.995885	−	0.0906289i	$-0.0288877\pi$
$48$	− 9.65685i	− 1.39385i
$49$	0	0
$50$	0	0
$51$	−12.6569	−1.77231
$52$	0	0
$53$	− 4.24264i	− 0.582772i	−0.956606	−	0.291386i	$-0.905884\pi$
$53$	− 4.24264i	− 0.582772i	0.956606	−	0.291386i	$-0.0941163\pi$
$54$	0.585786	0.0797154
$55$	0	0
$56$	0	0
$57$	− 14.4853i	− 1.91862i
$58$	3.75736i	0.493365i
$59$	−6.24264	−0.812723	−0.406361	−	0.913712i	$-0.633203\pi$
$59$	−6.24264	−0.812723	−0.406361	+	0.913712i	$0.633203\pi$
$60$	0	0
$61$	2.82843	0.362143	0.181071	−	0.983470i	$-0.442043\pi$
$61$	2.82843	0.362143	0.181071	+	0.983470i	$0.442043\pi$
$62$	− 2.48528i	− 0.315631i
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	−19.8995	−2.44946
$67$	− 0.242641i	− 0.0296433i	−0.999890	−	0.0148216i	$-0.995282\pi$
$67$	− 0.242641i	− 0.0296433i	0.999890	−	0.0148216i	$-0.00471805\pi$
$68$	0	0
$69$	−11.0711	−1.33280
$70$	0	0
$71$	−8.82843	−1.04774	−0.523871	−	0.851798i	$-0.675513\pi$
$71$	−8.82843	−1.04774	−0.523871	+	0.851798i	$0.675513\pi$
$72$	8.00000i	0.942809i
$73$	8.48528i	0.993127i	0.868000	+	0.496564i	$0.165405\pi$
$73$	8.48528i	0.993127i	−0.868000	+	0.496564i	$0.834595\pi$
$74$	8.82843	1.02628
$75$	0	0
$76$	0	0
$77$	0	0
$78$	5.41421i	0.613039i
$79$	15.4853	1.74223	0.871115	−	0.491079i	$-0.163397\pi$
$79$	15.4853	1.74223	0.871115	+	0.491079i	$0.163397\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	− 3.17157i	− 0.350242i
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	2.82843	0.304997
$87$	− 6.41421i	− 0.687676i
$88$	16.4853i	1.75734i
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.24264i	0.439941i
$94$	−1.75736	−0.181258
$95$	0	0
$96$	0	0
$97$	− 4.75736i	− 0.483037i	−0.970396	−	0.241518i	$-0.922355\pi$
$97$	− 4.75736i	− 0.483037i	0.970396	−	0.241518i	$-0.0776454\pi$
$98$	0	0
$99$	16.4853	1.65683
$100$	0	0
$101$	−14.4853	−1.44134	−0.720670	−	0.693279i	$-0.756166\pi$
$101$	−14.4853	−1.44134	−0.720670	+	0.693279i	$0.756166\pi$
$102$	17.8995i	1.77231i
$103$	10.7574i	1.05995i	0.848012	+	0.529977i	$0.177799\pi$
$103$	10.7574i	1.05995i	−0.848012	+	0.529977i	$0.822201\pi$
$104$	4.48528	0.439818
$105$	0	0
$106$	−6.00000	−0.582772
$107$	− 14.4853i	− 1.40035i	−0.713974	−	0.700173i	$-0.753106\pi$
$107$	− 14.4853i	− 1.40035i	0.713974	−	0.700173i	$-0.246894\pi$
$108$	0	0
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	0	0
$111$	−15.0711	−1.43048
$112$	0	0
$113$	1.07107i	0.100758i	0.998730	+	0.0503788i	$0.0160429\pi$
$113$	1.07107i	0.100758i	−0.998730	+	0.0503788i	$0.983957\pi$
$114$	−20.4853	−1.91862
$115$	0	0
$116$	0	0
$117$	− 4.48528i	− 0.414664i
$118$	8.82843i	0.812723i
$119$	0	0
$120$	0	0
$121$	22.9706	2.08823
$122$	− 4.00000i	− 0.362143i
$123$	5.41421i	0.488183i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0.242641i	0.0215309i	0.999942	+	0.0107654i	$0.00342681\pi$
$127$	0.242641i	0.0215309i	−0.999942	+	0.0107654i	$0.996573\pi$
$128$	11.3137i	1.00000i
$129$	−4.82843	−0.425119
$130$	0	0
$131$	3.75736	0.328282	0.164141	−	0.986437i	$-0.447515\pi$
$131$	3.75736	0.328282	0.164141	+	0.986437i	$0.447515\pi$
$132$	0	0
$133$	0	0
$134$	−0.343146	−0.0296433
$135$	0	0
$136$	14.8284	1.27153
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	15.6569i	1.33280i
$139$	−16.2426	−1.37768	−0.688841	−	0.724912i	$-0.741880\pi$
$139$	−16.2426	−1.37768	−0.688841	+	0.724912i	$0.741880\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	12.4853i	1.04774i
$143$	− 9.24264i	− 0.772908i
$144$	11.3137	0.942809
$145$	0	0
$146$	12.0000	0.993127
$147$	0	0
$148$	0	0
$149$	14.8284	1.21479	0.607396	−	0.794399i	$-0.292214\pi$
$149$	14.8284	1.21479	0.607396	+	0.794399i	$0.292214\pi$
$150$	0	0
$151$	9.48528	0.771901	0.385951	−	0.922519i	$-0.373874\pi$
$151$	9.48528	0.771901	0.385951	+	0.922519i	$0.373874\pi$
$152$	16.9706i	1.37649i
$153$	− 14.8284i	− 1.19881i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	20.8284i	1.66229i	0.556056	+	0.831145i	$0.312314\pi$
$157$	20.8284i	1.66229i	−0.556056	+	0.831145i	$0.687686\pi$
$158$	− 21.8995i	− 1.74223i
$159$	10.2426	0.812294
$160$	0	0
$161$	0	0
$162$	13.4142i	1.05392i
$163$	− 1.75736i	− 0.137647i	−0.997629	−	0.0688235i	$-0.978075\pi$
$163$	− 1.75736i	− 0.137647i	0.997629	−	0.0688235i	$-0.0219245\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 9.24264i	− 0.715217i	−0.933872	−	0.357609i	$-0.883592\pi$
$167$	− 9.24264i	− 0.715217i	0.933872	−	0.357609i	$-0.116408\pi$
$168$	0	0
$169$	10.4853	0.806560
$170$	0	0
$171$	16.9706	1.29777
$172$	0	0
$173$	1.24264i	0.0944762i	0.998884	+	0.0472381i	$0.0150420\pi$
$173$	1.24264i	0.0944762i	−0.998884	+	0.0472381i	$0.984958\pi$
$174$	−9.07107	−0.687676
$175$	0	0
$176$	23.3137	1.75734
$177$	− 15.0711i	− 1.13281i
$178$	− 11.3137i	− 0.847998i
$179$	−2.48528	−0.185759	−0.0928793	−	0.995677i	$-0.529607\pi$
$179$	−2.48528	−0.185759	−0.0928793	+	0.995677i	$0.529607\pi$
$180$	0	0
$181$	6.72792	0.500083	0.250041	−	0.968235i	$-0.419556\pi$
$181$	6.72792	0.500083	0.250041	+	0.968235i	$0.419556\pi$
$182$	0	0
$183$	6.82843i	0.504772i
$184$	12.9706	0.956203
$185$	0	0
$186$	6.00000	0.439941
$187$	− 30.5563i	− 2.23450i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.9706	−1.44502	−0.722510	−	0.691361i	$-0.757011\pi$
$191$	−19.9706	−1.44502	−0.722510	+	0.691361i	$0.757011\pi$
$192$	− 19.3137i	− 1.39385i
$193$	− 16.0000i	− 1.15171i	−0.817554	−	0.575853i	$-0.804670\pi$
$193$	− 16.0000i	− 1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$	−6.72792	−0.483037
$195$	0	0
$196$	0	0
$197$	− 10.5858i	− 0.754206i	−0.926171	−	0.377103i	$-0.876920\pi$
$197$	− 10.5858i	− 0.754206i	0.926171	−	0.377103i	$-0.123080\pi$
$198$	− 23.3137i	− 1.65683i
$199$	4.58579	0.325078	0.162539	−	0.986702i	$-0.448032\pi$
$199$	4.58579	0.325078	0.162539	+	0.986702i	$0.448032\pi$
$200$	0	0
$201$	0.585786	0.0413182
$202$	20.4853i	1.44134i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	15.2132	1.05995
$207$	− 12.9706i	− 0.901516i
$208$	− 6.34315i	− 0.439818i
$209$	34.9706	2.41896
$210$	0	0
$211$	9.00000	0.619586	0.309793	−	0.950804i	$-0.399740\pi$
$211$	9.00000	0.619586	0.309793	+	0.950804i	$0.399740\pi$
$212$	0	0
$213$	− 21.3137i	− 1.46039i
$214$	−20.4853	−1.40035
$215$	0	0
$216$	1.17157	0.0797154
$217$	0	0
$218$	7.07107i	0.478913i
$219$	−20.4853	−1.38427
$220$	0	0
$221$	−8.31371	−0.559241
$222$	21.3137i	1.43048i
$223$	18.2132i	1.21965i	0.792538	+	0.609823i	$0.208760\pi$
$223$	18.2132i	1.21965i	−0.792538	+	0.609823i	$0.791240\pi$
$224$	0	0
$225$	0	0
$226$	1.51472	0.100758
$227$	9.72792i	0.645665i	0.946456	+	0.322832i	$0.104635\pi$
$227$	9.72792i	0.645665i	−0.946456	+	0.322832i	$0.895365\pi$
$228$	0	0
$229$	30.0416	1.98521	0.992603	−	0.121402i	$-0.0387390\pi$
$229$	30.0416	1.98521	0.992603	+	0.121402i	$0.0387390\pi$
$230$	0	0
$231$	0	0
$232$	7.51472i	0.493365i
$233$	− 14.8284i	− 0.971443i	−0.874114	−	0.485721i	$-0.838557\pi$
$233$	− 14.8284i	− 0.971443i	0.874114	−	0.485721i	$-0.161443\pi$
$234$	−6.34315	−0.414664
$235$	0	0
$236$	0	0
$237$	37.3848i	2.42840i
$238$	0	0
$239$	0.514719	0.0332944	0.0166472	−	0.999861i	$-0.494701\pi$
$239$	0.514719	0.0332944	0.0166472	+	0.999861i	$0.494701\pi$
$240$	0	0
$241$	24.7279	1.59287	0.796433	−	0.604727i	$-0.206718\pi$
$241$	24.7279	1.59287	0.796433	+	0.604727i	$0.206718\pi$
$242$	− 32.4853i	− 2.08823i
$243$	− 21.6569i	− 1.38929i
$244$	0	0
$245$	0	0
$246$	7.65685	0.488183
$247$	− 9.51472i	− 0.605407i
$248$	− 4.97056i	− 0.315631i
$249$	0	0
$250$	0	0
$251$	−25.2132	−1.59144	−0.795722	−	0.605663i	$-0.792908\pi$
$251$	−25.2132	−1.59144	−0.795722	+	0.605663i	$0.792908\pi$
$252$	0	0
$253$	− 26.7279i	− 1.68037i
$254$	0.343146	0.0215309
$255$	0	0
$256$	0	0
$257$	26.4853i	1.65211i	0.563592	+	0.826053i	$0.309419\pi$
$257$	26.4853i	1.65211i	−0.563592	+	0.826053i	$0.690581\pi$
$258$	6.82843i	0.425119i
$259$	0	0
$260$	0	0
$261$	7.51472	0.465149
$262$	− 5.31371i	− 0.328282i
$263$	28.6274i	1.76524i	0.470085	+	0.882621i	$0.344223\pi$
$263$	28.6274i	1.76524i	−0.470085	+	0.882621i	$0.655777\pi$
$264$	−39.7990	−2.44946
$265$	0	0
$266$	0	0
$267$	19.3137i	1.18198i
$268$	0	0
$269$	7.75736	0.472975	0.236487	−	0.971635i	$-0.424004\pi$
$269$	7.75736	0.472975	0.236487	+	0.971635i	$0.424004\pi$
$270$	0	0
$271$	−23.3137	−1.41621	−0.708103	−	0.706109i	$-0.750449\pi$
$271$	−23.3137	−1.41621	−0.708103	+	0.706109i	$0.750449\pi$
$272$	− 20.9706i	− 1.27153i
$273$	0	0
$274$	−16.9706	−1.02523
$275$	0	0
$276$	0	0
$277$	27.2132i	1.63508i	0.575870	+	0.817541i	$0.304664\pi$
$277$	27.2132i	1.63508i	−0.575870	+	0.817541i	$0.695336\pi$
$278$	22.9706i	1.37768i
$279$	−4.97056	−0.297580
$280$	0	0
$281$	−20.3137	−1.21181	−0.605907	−	0.795535i	$-0.707190\pi$
$281$	−20.3137	−1.21181	−0.605907	+	0.795535i	$0.707190\pi$
$282$	− 4.24264i	− 0.252646i
$283$	− 6.55635i	− 0.389735i	−0.980830	−	0.194867i	$-0.937572\pi$
$283$	− 6.55635i	− 0.389735i	0.980830	−	0.194867i	$-0.0624276\pi$
$284$	0	0
$285$	0	0
$286$	−13.0711	−0.772908
$287$	0	0
$288$	0	0
$289$	−10.4853	−0.616781
$290$	0	0
$291$	11.4853	0.673279
$292$	0	0
$293$	0.272078i	0.0158950i	0.999968	+	0.00794748i	$0.00252979\pi$
$293$	0.272078i	0.0158950i	−0.999968	+	0.00794748i	$0.997470\pi$
$294$	0	0
$295$	0	0
$296$	17.6569	1.02628
$297$	− 2.41421i	− 0.140087i
$298$	− 20.9706i	− 1.21479i
$299$	−7.27208	−0.420555
$300$	0	0
$301$	0	0
$302$	− 13.4142i	− 0.771901i
$303$	− 34.9706i	− 2.00901i
$304$	24.0000	1.37649
$305$	0	0
$306$	−20.9706	−1.19881
$307$	− 11.1005i	− 0.633539i	−0.948503	−	0.316770i	$-0.897402\pi$
$307$	− 11.1005i	− 0.633539i	0.948503	−	0.316770i	$-0.102598\pi$
$308$	0	0
$309$	−25.9706	−1.47741
$310$	0	0
$311$	−10.0000	−0.567048	−0.283524	−	0.958965i	$-0.591504\pi$
$311$	−10.0000	−0.567048	−0.283524	+	0.958965i	$0.591504\pi$
$312$	10.8284i	0.613039i
$313$	24.2132i	1.36861i	0.729195	+	0.684306i	$0.239895\pi$
$313$	24.2132i	1.36861i	−0.729195	+	0.684306i	$0.760105\pi$
$314$	29.4558	1.66229
$315$	0	0
$316$	0	0
$317$	11.6569i	0.654714i	0.944901	+	0.327357i	$0.106158\pi$
$317$	11.6569i	0.654714i	−0.944901	+	0.327357i	$0.893842\pi$
$318$	− 14.4853i	− 0.812294i
$319$	15.4853	0.867009
$320$	0	0
$321$	34.9706	1.95187
$322$	0	0
$323$	− 31.4558i	− 1.75025i
$324$	0	0
$325$	0	0
$326$	−2.48528	−0.137647
$327$	− 12.0711i	− 0.667532i
$328$	− 6.34315i	− 0.350242i
$329$	0	0
$330$	0	0
$331$	23.4558	1.28925	0.644625	−	0.764499i	$-0.277014\pi$
$331$	23.4558	1.28925	0.644625	+	0.764499i	$0.277014\pi$
$332$	0	0
$333$	− 17.6569i	− 0.967590i
$334$	−13.0711	−0.715217
$335$	0	0
$336$	0	0
$337$	− 13.7574i	− 0.749411i	−0.927144	−	0.374706i	$-0.877744\pi$
$337$	− 13.7574i	− 0.749411i	0.927144	−	0.374706i	$-0.122256\pi$
$338$	− 14.8284i	− 0.806560i
$339$	−2.58579	−0.140441
$340$	0	0
$341$	−10.2426	−0.554670
$342$	− 24.0000i	− 1.29777i
$343$	0	0
$344$	5.65685	0.304997
$345$	0	0
$346$	1.75736	0.0944762
$347$	1.07107i	0.0574979i	0.999587	+	0.0287490i	$0.00915234\pi$
$347$	1.07107i	0.0574979i	−0.999587	+	0.0287490i	$0.990848\pi$
$348$	0	0
$349$	−22.9706	−1.22959	−0.614793	−	0.788688i	$-0.710760\pi$
$349$	−22.9706	−1.22959	−0.614793	+	0.788688i	$0.710760\pi$
$350$	0	0
$351$	−0.656854	−0.0350603
$352$	0	0
$353$	36.2132i	1.92743i	0.266925	+	0.963717i	$0.413992\pi$
$353$	36.2132i	1.92743i	−0.266925	+	0.963717i	$0.586008\pi$
$354$	−21.3137	−1.13281
$355$	0	0
$356$	0	0
$357$	0	0
$358$	3.51472i	0.185759i
$359$	−11.3137	−0.597115	−0.298557	−	0.954392i	$-0.596505\pi$
$359$	−11.3137	−0.597115	−0.298557	+	0.954392i	$0.596505\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	− 9.51472i	− 0.500083i
$363$	55.4558i	2.91068i
$364$	0	0
$365$	0	0
$366$	9.65685	0.504772
$367$	− 29.8701i	− 1.55920i	−0.626275	−	0.779602i	$-0.715421\pi$
$367$	− 29.8701i	− 1.55920i	0.626275	−	0.779602i	$-0.284579\pi$
$368$	− 18.3431i	− 0.956203i
$369$	−6.34315	−0.330211
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0.485281i	0.0251269i	0.999921	+	0.0125635i	$0.00399918\pi$
$373$	0.485281i	0.0251269i	−0.999921	+	0.0125635i	$0.996001\pi$
$374$	−43.2132	−2.23450
$375$	0	0
$376$	−3.51472	−0.181258
$377$	− 4.21320i	− 0.216991i
$378$	0	0
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	0	0
$381$	−0.585786	−0.0300107
$382$	28.2426i	1.44502i
$383$	− 12.4853i	− 0.637968i	−0.947760	−	0.318984i	$-0.896658\pi$
$383$	− 12.4853i	− 0.637968i	0.947760	−	0.318984i	$-0.103342\pi$
$384$	−27.3137	−1.39385
$385$	0	0
$386$	−22.6274	−1.15171
$387$	− 5.65685i	− 0.287554i
$388$	0	0
$389$	35.1421	1.78178	0.890889	−	0.454222i	$-0.150083\pi$
$389$	35.1421	1.78178	0.890889	+	0.454222i	$0.150083\pi$
$390$	0	0
$391$	−24.0416	−1.21584
$392$	0	0
$393$	9.07107i	0.457575i
$394$	−14.9706	−0.754206
$395$	0	0
$396$	0	0
$397$	− 4.41421i	− 0.221543i	−0.993846	−	0.110772i	$-0.964668\pi$
$397$	− 4.41421i	− 0.221543i	0.993846	−	0.110772i	$-0.0353322\pi$
$398$	− 6.48528i	− 0.325078i
$399$	0	0
$400$	0	0
$401$	6.17157	0.308194	0.154097	−	0.988056i	$-0.450753\pi$
$401$	6.17157	0.308194	0.154097	+	0.988056i	$0.450753\pi$
$402$	− 0.828427i	− 0.0413182i
$403$	2.78680i	0.138820i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 36.3848i	− 1.80353i
$408$	35.7990i	1.77231i
$409$	14.4853	0.716251	0.358126	−	0.933673i	$-0.383416\pi$
$409$	14.4853	0.716251	0.358126	+	0.933673i	$0.383416\pi$
$410$	0	0
$411$	28.9706	1.42901
$412$	0	0
$413$	0	0
$414$	−18.3431	−0.901516
$415$	0	0
$416$	0	0
$417$	− 39.2132i	− 1.92028i
$418$	− 49.4558i	− 2.41896i
$419$	6.72792	0.328681	0.164340	−	0.986404i	$-0.447450\pi$
$419$	6.72792	0.328681	0.164340	+	0.986404i	$0.447450\pi$
$420$	0	0
$421$	−19.0000	−0.926003	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000	−0.926003	−0.463002	+	0.886357i	$0.653228\pi$
$422$	− 12.7279i	− 0.619586i
$423$	3.51472i	0.170891i
$424$	−12.0000	−0.582772
$425$	0	0
$426$	−30.1421	−1.46039
$427$	0	0
$428$	0	0
$429$	22.3137	1.07732
$430$	0	0
$431$	10.7990	0.520169	0.260085	−	0.965586i	$-0.416250\pi$
$431$	10.7990	0.520169	0.260085	+	0.965586i	$0.416250\pi$
$432$	− 1.65685i	− 0.0797154i
$433$	− 22.9706i	− 1.10389i	−0.833879	−	0.551947i	$-0.813885\pi$
$433$	− 22.9706i	− 1.10389i	0.833879	−	0.551947i	$-0.186115\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 27.5147i	− 1.31621i
$438$	28.9706i	1.38427i
$439$	−6.38478	−0.304729	−0.152364	−	0.988324i	$-0.548689\pi$
$439$	−6.38478	−0.304729	−0.152364	+	0.988324i	$0.548689\pi$
$440$	0	0
$441$	0	0
$442$	11.7574i	0.559241i
$443$	20.8284i	0.989588i	0.869010	+	0.494794i	$0.164757\pi$
$443$	20.8284i	0.989588i	−0.869010	+	0.494794i	$0.835243\pi$
$444$	0	0
$445$	0	0
$446$	25.7574	1.21965
$447$	35.7990i	1.69323i
$448$	0	0
$449$	29.8284	1.40769	0.703845	−	0.710353i	$-0.251465\pi$
$449$	29.8284	1.40769	0.703845	+	0.710353i	$0.251465\pi$
$450$	0	0
$451$	−13.0711	−0.615493
$452$	0	0
$453$	22.8995i	1.07591i
$454$	13.7574	0.645665
$455$	0	0
$456$	−40.9706	−1.91862
$457$	20.2426i	0.946911i	0.880818	+	0.473455i	$0.156993\pi$
$457$	20.2426i	0.946911i	−0.880818	+	0.473455i	$0.843007\pi$
$458$	− 42.4853i	− 1.98521i
$459$	−2.17157	−0.101360
$460$	0	0
$461$	36.9706	1.72189	0.860945	−	0.508697i	$-0.169873\pi$
$461$	36.9706	1.72189	0.860945	+	0.508697i	$0.169873\pi$
$462$	0	0
$463$	29.4558i	1.36893i	0.729046	+	0.684465i	$0.239964\pi$
$463$	29.4558i	1.36893i	−0.729046	+	0.684465i	$0.760036\pi$
$464$	10.6274	0.493365
$465$	0	0
$466$	−20.9706	−0.971443
$467$	− 19.7279i	− 0.912899i	−0.889749	−	0.456450i	$-0.849121\pi$
$467$	− 19.7279i	− 0.912899i	0.889749	−	0.456450i	$-0.150879\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−50.2843	−2.31698
$472$	17.6569i	0.812723i
$473$	− 11.6569i	− 0.535983i
$474$	52.8701	2.42840
$475$	0	0
$476$	0	0
$477$	12.0000i	0.549442i
$478$	− 0.727922i	− 0.0332944i
$479$	−20.2426	−0.924910	−0.462455	−	0.886643i	$-0.653031\pi$
$479$	−20.2426	−0.924910	−0.462455	+	0.886643i	$0.653031\pi$
$480$	0	0
$481$	−9.89949	−0.451378
$482$	− 34.9706i	− 1.59287i
$483$	0	0
$484$	0	0
$485$	0	0
$486$	−30.6274	−1.38929
$487$	31.6985i	1.43640i	0.695839	+	0.718198i	$0.255033\pi$
$487$	31.6985i	1.43640i	−0.695839	+	0.718198i	$0.744967\pi$
$488$	− 8.00000i	− 0.362143i
$489$	4.24264	0.191859
$490$	0	0
$491$	19.2843	0.870287	0.435143	−	0.900361i	$-0.356698\pi$
$491$	19.2843	0.870287	0.435143	+	0.900361i	$0.356698\pi$
$492$	0	0
$493$	− 13.9289i	− 0.627328i
$494$	−13.4558	−0.605407
$495$	0	0
$496$	−7.02944	−0.315631
$497$	0	0
$498$	0	0
$499$	−3.00000	−0.134298	−0.0671492	−	0.997743i	$-0.521390\pi$
$499$	−3.00000	−0.134298	−0.0671492	+	0.997743i	$0.521390\pi$
$500$	0	0
$501$	22.3137	0.996903
$502$	35.6569i	1.59144i
$503$	− 32.7574i	− 1.46058i	−0.683138	−	0.730289i	$-0.739385\pi$
$503$	− 32.7574i	− 1.46058i	0.683138	−	0.730289i	$-0.260615\pi$
$504$	0	0
$505$	0	0
$506$	−37.7990	−1.68037
$507$	25.3137i	1.12422i
$508$	0	0
$509$	−17.2132	−0.762962	−0.381481	−	0.924377i	$-0.624586\pi$
$509$	−17.2132	−0.762962	−0.381481	+	0.924377i	$0.624586\pi$
$510$	0	0
$511$	0	0
$512$	22.6274i	1.00000i
$513$	− 2.48528i	− 0.109728i
$514$	37.4558	1.65211
$515$	0	0
$516$	0	0
$517$	7.24264i	0.318531i
$518$	0	0
$519$	−3.00000	−0.131685
$520$	0	0
$521$	−18.9706	−0.831115	−0.415558	−	0.909567i	$-0.636414\pi$
$521$	−18.9706	−0.831115	−0.415558	+	0.909567i	$0.636414\pi$
$522$	− 10.6274i	− 0.465149i
$523$	15.5147i	0.678411i	0.940712	+	0.339206i	$0.110158\pi$
$523$	15.5147i	0.678411i	−0.940712	+	0.339206i	$0.889842\pi$
$524$	0	0
$525$	0	0
$526$	40.4853	1.76524
$527$	9.21320i	0.401333i
$528$	56.2843i	2.44946i
$529$	1.97056	0.0856766
$530$	0	0
$531$	17.6569	0.766242
$532$	0	0
$533$	3.55635i	0.154043i
$534$	27.3137	1.18198
$535$	0	0
$536$	−0.686292	−0.0296433
$537$	− 6.00000i	− 0.258919i
$538$	− 10.9706i	− 0.472975i
$539$	0	0
$540$	0	0
$541$	11.9706	0.514655	0.257327	−	0.966324i	$-0.417158\pi$
$541$	11.9706	0.514655	0.257327	+	0.966324i	$0.417158\pi$
$542$	32.9706i	1.41621i
$543$	16.2426i	0.697038i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	7.51472i	0.321306i	0.987011	+	0.160653i	$0.0513600\pi$
$547$	7.51472i	0.321306i	−0.987011	+	0.160653i	$0.948640\pi$
$548$	0	0
$549$	−8.00000	−0.341432
$550$	0	0
$551$	15.9411	0.679115
$552$	31.3137i	1.33280i
$553$	0	0
$554$	38.4853	1.63508
$555$	0	0
$556$	0	0
$557$	− 31.7990i	− 1.34737i	−0.739020	−	0.673683i	$-0.764711\pi$
$557$	− 31.7990i	− 1.34737i	0.739020	−	0.673683i	$-0.235289\pi$
$558$	7.02944i	0.297580i
$559$	−3.17157	−0.134143
$560$	0	0
$561$	73.7696	3.11455
$562$	28.7279i	1.21181i
$563$	35.9411i	1.51474i	0.652987	+	0.757369i	$0.273516\pi$
$563$	35.9411i	1.51474i	−0.652987	+	0.757369i	$0.726484\pi$
$564$	0	0
$565$	0	0
$566$	−9.27208	−0.389735
$567$	0	0
$568$	24.9706i	1.04774i
$569$	2.14214	0.0898030	0.0449015	−	0.998991i	$-0.485703\pi$
$569$	2.14214	0.0898030	0.0449015	+	0.998991i	$0.485703\pi$
$570$	0	0
$571$	−34.4853	−1.44316	−0.721582	−	0.692329i	$-0.756585\pi$
$571$	−34.4853	−1.44316	−0.721582	+	0.692329i	$0.756585\pi$
$572$	0	0
$573$	− 48.2132i	− 2.01414i
$574$	0	0
$575$	0	0
$576$	22.6274	0.942809
$577$	9.72792i	0.404979i	0.979284	+	0.202489i	$0.0649032\pi$
$577$	9.72792i	0.404979i	−0.979284	+	0.202489i	$0.935097\pi$
$578$	14.8284i	0.616781i
$579$	38.6274	1.60530
$580$	0	0
$581$	0	0
$582$	− 16.2426i	− 0.673279i
$583$	24.7279i	1.02413i
$584$	24.0000	0.993127
$585$	0	0
$586$	0.384776	0.0158950
$587$	− 13.4558i	− 0.555382i	−0.960670	−	0.277691i	$-0.910431\pi$
$587$	− 13.4558i	− 0.555382i	0.960670	−	0.277691i	$-0.0895692\pi$
$588$	0	0
$589$	−10.5442	−0.434464
$590$	0	0
$591$	25.5563	1.05125
$592$	− 24.9706i	− 1.02628i
$593$	10.7574i	0.441752i	0.975302	+	0.220876i	$0.0708916\pi$
$593$	10.7574i	0.441752i	−0.975302	+	0.220876i	$0.929108\pi$
$594$	−3.41421	−0.140087
$595$	0	0
$596$	0	0
$597$	11.0711i	0.453109i
$598$	10.2843i	0.420555i
$599$	12.1716	0.497317	0.248658	−	0.968591i	$-0.420010\pi$
$599$	12.1716	0.497317	0.248658	+	0.968591i	$0.420010\pi$
$600$	0	0
$601$	22.9706	0.936989	0.468494	−	0.883466i	$-0.344797\pi$
$601$	22.9706	0.936989	0.468494	+	0.883466i	$0.344797\pi$
$602$	0	0
$603$	0.686292i	0.0279480i
$604$	0	0
$605$	0	0
$606$	−49.4558	−2.00901
$607$	24.8995i	1.01064i	0.862932	+	0.505320i	$0.168625\pi$
$607$	24.8995i	1.01064i	−0.862932	+	0.505320i	$0.831375\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.97056	0.0797204
$612$	0	0
$613$	43.9411i	1.77477i	0.461034	+	0.887383i	$0.347479\pi$
$613$	43.9411i	1.77477i	−0.461034	+	0.887383i	$0.652521\pi$
$614$	−15.6985	−0.633539
$615$	0	0
$616$	0	0
$617$	− 7.41421i	− 0.298485i	−0.988801	−	0.149242i	$-0.952316\pi$
$617$	− 7.41421i	− 0.298485i	0.988801	−	0.149242i	$-0.0476835\pi$
$618$	36.7279i	1.47741i
$619$	−31.0711	−1.24885	−0.624426	−	0.781084i	$-0.714667\pi$
$619$	−31.0711	−1.24885	−0.624426	+	0.781084i	$0.714667\pi$
$620$	0	0
$621$	−1.89949	−0.0762241
$622$	14.1421i	0.567048i
$623$	0	0
$624$	15.3137	0.613039
$625$	0	0
$626$	34.2426	1.36861
$627$	84.4264i	3.37167i
$628$	0	0
$629$	−32.7279	−1.30495
$630$	0	0
$631$	8.45584	0.336622	0.168311	−	0.985734i	$-0.446169\pi$
$631$	8.45584	0.336622	0.168311	+	0.985734i	$0.446169\pi$
$632$	− 43.7990i	− 1.74223i
$633$	21.7279i	0.863607i
$634$	16.4853	0.654714
$635$	0	0
$636$	0	0
$637$	0	0
$638$	− 21.8995i	− 0.867009i
$639$	24.9706	0.987820
$640$	0	0
$641$	−0.686292	−0.0271069	−0.0135534	−	0.999908i	$-0.504314\pi$
$641$	−0.686292	−0.0271069	−0.0135534	+	0.999908i	$0.504314\pi$
$642$	− 49.4558i	− 1.95187i
$643$	27.7279i	1.09348i	0.837302	+	0.546741i	$0.184132\pi$
$643$	27.7279i	1.09348i	−0.837302	+	0.546741i	$0.815868\pi$
$644$	0	0
$645$	0	0
$646$	−44.4853	−1.75025
$647$	28.4853i	1.11987i	0.828536	+	0.559936i	$0.189174\pi$
$647$	28.4853i	1.11987i	−0.828536	+	0.559936i	$0.810826\pi$
$648$	26.8284i	1.05392i
$649$	36.3848	1.42823
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.02944i	0.0402850i	0.999797	+	0.0201425i	$0.00641199\pi$
$653$	1.02944i	0.0402850i	−0.999797	+	0.0201425i	$0.993588\pi$
$654$	−17.0711	−0.667532
$655$	0	0
$656$	−8.97056	−0.350242
$657$	− 24.0000i	− 0.936329i
$658$	0	0
$659$	13.9706	0.544216	0.272108	−	0.962267i	$-0.412279\pi$
$659$	13.9706	0.544216	0.272108	+	0.962267i	$0.412279\pi$
$660$	0	0
$661$	−37.4558	−1.45686	−0.728432	−	0.685118i	$-0.759750\pi$
$661$	−37.4558	−1.45686	−0.728432	+	0.685118i	$0.759750\pi$
$662$	− 33.1716i	− 1.28925i
$663$	− 20.0711i	− 0.779496i
$664$	0	0
$665$	0	0
$666$	−24.9706	−0.967590
$667$	− 12.1838i	− 0.471757i
$668$	0	0
$669$	−43.9706	−1.70000
$670$	0	0
$671$	−16.4853	−0.636407
$672$	0	0
$673$	20.4853i	0.789650i	0.918756	+	0.394825i	$0.129195\pi$
$673$	20.4853i	0.789650i	−0.918756	+	0.394825i	$0.870805\pi$
$674$	−19.4558	−0.749411
$675$	0	0
$676$	0	0
$677$	− 1.78680i	− 0.0686722i	−0.999410	−	0.0343361i	$-0.989068\pi$
$677$	− 1.78680i	− 0.0686722i	0.999410	−	0.0343361i	$-0.0109317\pi$
$678$	3.65685i	0.140441i
$679$	0	0
$680$	0	0
$681$	−23.4853	−0.899958
$682$	14.4853i	0.554670i
$683$	− 7.79899i	− 0.298420i	−0.988806	−	0.149210i	$-0.952327\pi$
$683$	− 7.79899i	− 0.298420i	0.988806	−	0.149210i	$-0.0476731\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	72.5269i	2.76707i
$688$	− 8.00000i	− 0.304997i
$689$	6.72792	0.256313
$690$	0	0
$691$	9.17157	0.348903	0.174452	−	0.984666i	$-0.444185\pi$
$691$	9.17157	0.348903	0.174452	+	0.984666i	$0.444185\pi$
$692$	0	0
$693$	0	0
$694$	1.51472	0.0574979
$695$	0	0
$696$	−18.1421	−0.687676
$697$	11.7574i	0.445342i
$698$	32.4853i	1.22959i
$699$	35.7990	1.35404
$700$	0	0
$701$	−4.45584	−0.168295	−0.0841475	−	0.996453i	$-0.526817\pi$
$701$	−4.45584	−0.168295	−0.0841475	+	0.996453i	$0.526817\pi$
$702$	0.928932i	0.0350603i
$703$	− 37.4558i	− 1.41267i
$704$	46.6274	1.75734
$705$	0	0
$706$	51.2132	1.92743
$707$	0	0
$708$	0	0
$709$	−17.0000	−0.638448	−0.319224	−	0.947679i	$-0.603422\pi$
$709$	−17.0000	−0.638448	−0.319224	+	0.947679i	$0.603422\pi$
$710$	0	0
$711$	−43.7990	−1.64259
$712$	− 22.6274i	− 0.847998i
$713$	8.05887i	0.301807i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.24264i	0.0464073i
$718$	16.0000i	0.597115i
$719$	−17.2132	−0.641944	−0.320972	−	0.947089i	$-0.604010\pi$
$719$	−17.2132	−0.641944	−0.320972	+	0.947089i	$0.604010\pi$
$720$	0	0
$721$	0	0
$722$	− 24.0416i	− 0.894737i
$723$	59.6985i	2.22021i
$724$	0	0
$725$	0	0
$726$	78.4264	2.91068
$727$	29.3137i	1.08719i	0.839349	+	0.543593i	$0.182936\pi$
$727$	29.3137i	1.08719i	−0.839349	+	0.543593i	$0.817064\pi$
$728$	0	0
$729$	23.8284	0.882534
$730$	0	0
$731$	−10.4853	−0.387812
$732$	0	0
$733$	− 44.6985i	− 1.65098i	−0.564420	−	0.825488i	$-0.690900\pi$
$733$	− 44.6985i	− 1.65098i	0.564420	−	0.825488i	$-0.309100\pi$
$734$	−42.2426	−1.55920
$735$	0	0
$736$	0	0
$737$	1.41421i	0.0520932i
$738$	8.97056i	0.330211i
$739$	3.97056	0.146060	0.0730298	−	0.997330i	$-0.476733\pi$
$739$	3.97056	0.146060	0.0730298	+	0.997330i	$0.476733\pi$
$740$	0	0
$741$	22.9706	0.843845
$742$	0	0
$743$	36.7279i	1.34742i	0.738997	+	0.673708i	$0.235300\pi$
$743$	36.7279i	1.34742i	−0.738997	+	0.673708i	$0.764700\pi$
$744$	12.0000	0.439941
$745$	0	0
$746$	0.686292	0.0251269
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	12.5147	0.456669	0.228334	−	0.973583i	$-0.426672\pi$
$751$	12.5147	0.456669	0.228334	+	0.973583i	$0.426672\pi$
$752$	4.97056i	0.181258i
$753$	− 60.8701i	− 2.21823i
$754$	−5.95837	−0.216991
$755$	0	0
$756$	0	0
$757$	16.4853i	0.599168i	0.954070	+	0.299584i	$0.0968478\pi$
$757$	16.4853i	0.599168i	−0.954070	+	0.299584i	$0.903152\pi$
$758$	2.82843i	0.102733i
$759$	64.5269	2.34218
$760$	0	0
$761$	12.7279	0.461387	0.230693	−	0.973026i	$-0.425901\pi$
$761$	12.7279	0.461387	0.230693	+	0.973026i	$0.425901\pi$
$762$	0.828427i	0.0300107i
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−17.6569	−0.637968
$767$	− 9.89949i	− 0.357450i
$768$	0	0
$769$	3.17157	0.114370	0.0571849	−	0.998364i	$-0.481788\pi$
$769$	3.17157	0.114370	0.0571849	+	0.998364i	$0.481788\pi$
$770$	0	0
$771$	−63.9411	−2.30278
$772$	0	0
$773$	36.2132i	1.30250i	0.758864	+	0.651249i	$0.225755\pi$
$773$	36.2132i	1.30250i	−0.758864	+	0.651249i	$0.774245\pi$
$774$	−8.00000	−0.287554
$775$	0	0
$776$	−13.4558	−0.483037
$777$	0	0
$778$	− 49.6985i	− 1.78178i
$779$	−13.4558	−0.482106
$780$	0	0
$781$	51.4558	1.84123
$782$	34.0000i	1.21584i
$783$	− 1.10051i	− 0.0393288i
$784$	0	0
$785$	0	0
$786$	12.8284	0.457575
$787$	− 45.7279i	− 1.63002i	−0.579444	−	0.815012i	$-0.696730\pi$
$787$	− 45.7279i	− 1.63002i	0.579444	−	0.815012i	$-0.303270\pi$
$788$	0	0
$789$	−69.1127	−2.46048
$790$	0	0
$791$	0	0
$792$	− 46.6274i	− 1.65683i
$793$	4.48528i	0.159277i
$794$	−6.24264	−0.221543
$795$	0	0
$796$	0	0
$797$	− 55.1838i	− 1.95471i	−0.211608	−	0.977355i	$-0.567870\pi$
$797$	− 55.1838i	− 1.95471i	0.211608	−	0.977355i	$-0.432130\pi$
$798$	0	0
$799$	6.51472	0.230474
$800$	0	0
$801$	−22.6274	−0.799500
$802$	− 8.72792i	− 0.308194i
$803$	− 49.4558i	− 1.74526i
$804$	0	0
$805$	0	0
$806$	3.94113	0.138820
$807$	18.7279i	0.659254i
$808$	40.9706i	1.44134i
$809$	−30.5980	−1.07577	−0.537884	−	0.843019i	$-0.680776\pi$
$809$	−30.5980	−1.07577	−0.537884	+	0.843019i	$0.680776\pi$
$810$	0	0
$811$	−23.3553	−0.820117	−0.410058	−	0.912059i	$-0.634492\pi$
$811$	−23.3553	−0.820117	−0.410058	+	0.912059i	$0.634492\pi$
$812$	0	0
$813$	− 56.2843i	− 1.97398i
$814$	−51.4558	−1.80353
$815$	0	0
$816$	50.6274	1.77231
$817$	− 12.0000i	− 0.419827i
$818$	− 20.4853i	− 0.716251i
$819$	0	0
$820$	0	0
$821$	−6.51472	−0.227365	−0.113683	−	0.993517i	$-0.536265\pi$
$821$	−6.51472	−0.227365	−0.113683	+	0.993517i	$0.536265\pi$
$822$	− 40.9706i	− 1.42901i
$823$	− 8.72792i	− 0.304236i	−0.988362	−	0.152118i	$-0.951391\pi$
$823$	− 8.72792i	− 0.304236i	0.988362	−	0.152118i	$-0.0486094\pi$
$824$	30.4264	1.05995
$825$	0	0
$826$	0	0
$827$	6.04163i	0.210088i	0.994468	+	0.105044i	$0.0334984\pi$
$827$	6.04163i	0.210088i	−0.994468	+	0.105044i	$0.966502\pi$
$828$	0	0
$829$	−54.0416	−1.87694	−0.938472	−	0.345356i	$-0.887758\pi$
$829$	−54.0416	−1.87694	−0.938472	+	0.345356i	$0.887758\pi$
$830$	0	0
$831$	−65.6985	−2.27906
$832$	− 12.6863i	− 0.439818i
$833$	0	0
$834$	−55.4558	−1.92028
$835$	0	0
$836$	0	0
$837$	0.727922i	0.0251607i
$838$	− 9.51472i	− 0.328681i
$839$	48.7279	1.68227	0.841137	−	0.540822i	$-0.181887\pi$
$839$	48.7279	1.68227	0.841137	+	0.540822i	$0.181887\pi$
$840$	0	0
$841$	−21.9411	−0.756591
$842$	26.8701i	0.926003i
$843$	− 49.0416i	− 1.68908i
$844$	0	0
$845$	0	0
$846$	4.97056	0.170891
$847$	0	0
$848$	16.9706i	0.582772i
$849$	15.8284	0.543230
$850$	0	0
$851$	−28.6274	−0.981335
$852$	0	0
$853$	− 22.9706i	− 0.786497i	−0.919432	−	0.393249i	$-0.871351\pi$
$853$	− 22.9706i	− 0.786497i	0.919432	−	0.393249i	$-0.128649\pi$
$854$	0	0
$855$	0	0
$856$	−40.9706	−1.40035
$857$	5.51472i	0.188379i	0.995554	+	0.0941896i	$0.0300260\pi$
$857$	5.51472i	0.188379i	−0.995554	+	0.0941896i	$0.969974\pi$
$858$	− 31.5563i	− 1.07732i
$859$	−48.7696	−1.66400	−0.831998	−	0.554779i	$-0.812803\pi$
$859$	−48.7696	−1.66400	−0.831998	+	0.554779i	$0.812803\pi$
$860$	0	0
$861$	0	0
$862$	− 15.2721i	− 0.520169i
$863$	− 18.3848i	− 0.625825i	−0.949782	−	0.312913i	$-0.898695\pi$
$863$	− 18.3848i	− 0.625825i	0.949782	−	0.312913i	$-0.101305\pi$
$864$	0	0
$865$	0	0
$866$	−32.4853	−1.10389
$867$	− 25.3137i	− 0.859699i
$868$	0	0
$869$	−90.2548	−3.06169
$870$	0	0
$871$	0.384776	0.0130376
$872$	14.1421i	0.478913i
$873$	13.4558i	0.455411i
$874$	−38.9117	−1.31621
$875$	0	0
$876$	0	0
$877$	− 46.9706i	− 1.58608i	−0.609167	−	0.793042i	$-0.708496\pi$
$877$	− 46.9706i	− 1.58608i	0.609167	−	0.793042i	$-0.291504\pi$
$878$	9.02944i	0.304729i
$879$	−0.656854	−0.0221551
$880$	0	0
$881$	−19.0294	−0.641118	−0.320559	−	0.947229i	$-0.603871\pi$
$881$	−19.0294	−0.641118	−0.320559	+	0.947229i	$0.603871\pi$
$882$	0	0
$883$	48.4853i	1.63166i	0.578292	+	0.815830i	$0.303719\pi$
$883$	48.4853i	1.63166i	−0.578292	+	0.815830i	$0.696281\pi$
$884$	0	0
$885$	0	0
$886$	29.4558	0.989588
$887$	30.9706i	1.03989i	0.854200	+	0.519945i	$0.174048\pi$
$887$	30.9706i	1.03989i	−0.854200	+	0.519945i	$0.825952\pi$
$888$	42.6274i	1.43048i
$889$	0	0
$890$	0	0
$891$	55.2843	1.85209
$892$	0	0
$893$	7.45584i	0.249500i
$894$	50.6274	1.69323
$895$	0	0
$896$	0	0
$897$	− 17.5563i	− 0.586189i
$898$	− 42.1838i	− 1.40769i
$899$	−4.66905	−0.155721
$900$	0	0
$901$	22.2426	0.741010
$902$	18.4853i	0.615493i
$903$	0	0
$904$	3.02944	0.100758
$905$	0	0
$906$	32.3848	1.07591
$907$	− 32.1838i	− 1.06864i	−0.845281	−	0.534322i	$-0.820567\pi$
$907$	− 32.1838i	− 1.06864i	0.845281	−	0.534322i	$-0.179433\pi$
$908$	0	0
$909$	40.9706	1.35891
$910$	0	0
$911$	−5.65685	−0.187420	−0.0937100	−	0.995600i	$-0.529873\pi$
$911$	−5.65685	−0.187420	−0.0937100	+	0.995600i	$0.529873\pi$
$912$	57.9411i	1.91862i
$913$	0	0
$914$	28.6274	0.946911
$915$	0	0
$916$	0	0
$917$	0	0
$918$	3.07107i	0.101360i
$919$	38.4558	1.26854	0.634271	−	0.773111i	$-0.281301\pi$
$919$	38.4558	1.26854	0.634271	+	0.773111i	$0.281301\pi$
$920$	0	0
$921$	26.7990	0.883057
$922$	− 52.2843i	− 1.72189i
$923$	− 14.0000i	− 0.460816i
$924$	0	0
$925$	0	0
$926$	41.6569	1.36893
$927$	− 30.4264i	− 0.999334i
$928$	0	0
$929$	−54.7279	−1.79556	−0.897782	−	0.440439i	$-0.854823\pi$
$929$	−54.7279	−1.79556	−0.897782	+	0.440439i	$0.854823\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 24.1421i	− 0.790378i
$934$	−27.8995	−0.912899
$935$	0	0
$936$	−12.6863	−0.414664
$937$	17.4437i	0.569859i	0.958548	+	0.284930i	$0.0919702\pi$
$937$	17.4437i	0.569859i	−0.958548	+	0.284930i	$0.908030\pi$
$938$	0	0
$939$	−58.4558	−1.90763
$940$	0	0
$941$	−4.97056	−0.162036	−0.0810179	−	0.996713i	$-0.525817\pi$
$941$	−4.97056	−0.162036	−0.0810179	+	0.996713i	$0.525817\pi$
$942$	71.1127i	2.31698i
$943$	10.2843i	0.334902i
$944$	24.9706	0.812723
$945$	0	0
$946$	−16.4853	−0.535983
$947$	43.7574i	1.42192i	0.703231	+	0.710962i	$0.251740\pi$
$947$	43.7574i	1.42192i	−0.703231	+	0.710962i	$0.748260\pi$
$948$	0	0
$949$	−13.4558	−0.436795
$950$	0	0
$951$	−28.1421	−0.912571
$952$	0	0
$953$	− 29.0122i	− 0.939797i	−0.882720	−	0.469899i	$-0.844290\pi$
$953$	− 29.0122i	− 0.939797i	0.882720	−	0.469899i	$-0.155710\pi$
$954$	16.9706	0.549442
$955$	0	0
$956$	0	0
$957$	37.3848i	1.20848i
$958$	28.6274i	0.924910i
$959$	0	0
$960$	0	0
$961$	−27.9117	−0.900377
$962$	14.0000i	0.451378i
$963$	40.9706i	1.32026i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 24.4264i	− 0.785500i	−0.919645	−	0.392750i	$-0.871524\pi$
$967$	− 24.4264i	− 0.785500i	0.919645	−	0.392750i	$-0.128476\pi$
$968$	− 64.9706i	− 2.08823i
$969$	75.9411	2.43958
$970$	0	0
$971$	−42.7279	−1.37120	−0.685602	−	0.727976i	$-0.740461\pi$
$971$	−42.7279	−1.37120	−0.685602	+	0.727976i	$0.740461\pi$
$972$	0	0
$973$	0	0
$974$	44.8284	1.43640
$975$	0	0
$976$	−11.3137	−0.362143
$977$	30.7696i	0.984405i	0.870481	+	0.492203i	$0.163808\pi$
$977$	30.7696i	0.984405i	−0.870481	+	0.492203i	$0.836192\pi$
$978$	− 6.00000i	− 0.191859i
$979$	−46.6274	−1.49022
$980$	0	0
$981$	14.1421	0.451524
$982$	− 27.2721i	− 0.870287i
$983$	− 42.2132i	− 1.34639i	−0.739464	−	0.673196i	$-0.764921\pi$
$983$	− 42.2132i	− 1.34639i	0.739464	−	0.673196i	$-0.235079\pi$
$984$	15.3137	0.488183
$985$	0	0
$986$	−19.6985	−0.627328
$987$	0	0
$988$	0	0
$989$	−9.17157	−0.291639
$990$	0	0
$991$	−47.9411	−1.52290	−0.761450	−	0.648224i	$-0.775512\pi$
$991$	−47.9411	−1.52290	−0.761450	+	0.648224i	$0.775512\pi$
$992$	0	0
$993$	56.6274i	1.79702i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 3.72792i	− 0.118064i	−0.998256	−	0.0590322i	$-0.981199\pi$
$997$	− 3.72792i	− 0.118064i	0.998256	−	0.0590322i	$-0.0188015\pi$
$998$	4.24264i	0.134298i
$999$	−2.58579	−0.0818107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.2.b.j.99.2		4
5.2	odd	4		245.2.a.f.1.2	yes	2
5.3	odd	4		1225.2.a.p.1.1		2
5.4	even	2	inner	1225.2.b.j.99.3		4
7.6	odd	2		1225.2.b.i.99.1		4
15.2	even	4		2205.2.a.t.1.1		2
20.7	even	4		3920.2.a.br.1.1		2
35.2	odd	12		245.2.e.f.116.1		4
35.12	even	12		245.2.e.g.116.1		4
35.13	even	4		1225.2.a.r.1.1		2
35.17	even	12		245.2.e.g.226.1		4
35.27	even	4		245.2.a.e.1.2	✓	2
35.32	odd	12		245.2.e.f.226.1		4
35.34	odd	2		1225.2.b.i.99.4		4
105.62	odd	4		2205.2.a.v.1.1		2
140.27	odd	4		3920.2.a.bw.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.a.e.1.2	✓	2	35.27	even	4
245.2.a.f.1.2	yes	2	5.2	odd	4
245.2.e.f.116.1		4	35.2	odd	12
245.2.e.f.226.1		4	35.32	odd	12
245.2.e.g.116.1		4	35.12	even	12
245.2.e.g.226.1		4	35.17	even	12
1225.2.a.p.1.1		2	5.3	odd	4
1225.2.a.r.1.1		2	35.13	even	4
1225.2.b.i.99.1		4	7.6	odd	2
1225.2.b.i.99.4		4	35.34	odd	2
1225.2.b.j.99.2		4	1.1	even	1	trivial
1225.2.b.j.99.3		4	5.4	even	2	inner
2205.2.a.t.1.1		2	15.2	even	4
2205.2.a.v.1.1		2	105.62	odd	4
3920.2.a.br.1.1		2	20.7	even	4
3920.2.a.bw.1.2		2	140.27	odd	4

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:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +2.41421i q^{3} +3.41421 q^{6} -2.82843i q^{8} -2.82843 q^{9} +O(q^{10})\)	\(q-1.41421i q^{2} +2.41421i q^{3} +3.41421 q^{6} -2.82843i q^{8} -2.82843 q^{9} -5.82843 q^{11} +1.58579i q^{13} -4.00000 q^{16} +5.24264i q^{17} +4.00000i q^{18} -6.00000 q^{19} +8.24264i q^{22} +4.58579i q^{23} +6.82843 q^{24} +2.24264 q^{26} +0.414214i q^{27} -2.65685 q^{29} +1.75736 q^{31} -14.0711i q^{33} +7.41421 q^{34} +6.24264i q^{37} +8.48528i q^{38} -3.82843 q^{39} +2.24264 q^{41} +2.00000i q^{43} +6.48528 q^{46} -1.24264i q^{47} -9.65685i q^{48} -12.6569 q^{51} -4.24264i q^{53} +0.585786 q^{54} -14.4853i q^{57} +3.75736i q^{58} -6.24264 q^{59} +2.82843 q^{61} -2.48528i q^{62} -8.00000 q^{64} -19.8995 q^{66} -0.242641i q^{67} -11.0711 q^{69} -8.82843 q^{71} +8.00000i q^{72} +8.48528i q^{73} +8.82843 q^{74} +5.41421i q^{78} +15.4853 q^{79} -9.48528 q^{81} -3.17157i q^{82} +2.82843 q^{86} -6.41421i q^{87} +16.4853i q^{88} +8.00000 q^{89} +4.24264i q^{93} -1.75736 q^{94} -4.75736i q^{97} +16.4853 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{6}+O(q^{10}) \) 4 * q + 8 * q^6	\( 4 q + 8 q^{6} - 12 q^{11} - 16 q^{16} - 24 q^{19} + 16 q^{24} - 8 q^{26} + 12 q^{29} + 24 q^{31} + 24 q^{34} - 4 q^{39} - 8 q^{41} - 8 q^{46} - 28 q^{51} + 8 q^{54} - 8 q^{59} - 32 q^{64} - 40 q^{66} - 16 q^{69} - 24 q^{71} + 24 q^{74} + 28 q^{79} - 4 q^{81} + 32 q^{89} - 24 q^{94} + 32 q^{99}+O(q^{100}) \) 4 * q + 8 * q^6 - 12 * q^11 - 16 * q^16 - 24 * q^19 + 16 * q^24 - 8 * q^26 + 12 * q^29 + 24 * q^31 + 24 * q^34 - 4 * q^39 - 8 * q^41 - 8 * q^46 - 28 * q^51 + 8 * q^54 - 8 * q^59 - 32 * q^64 - 40 * q^66 - 16 * q^69 - 24 * q^71 + 24 * q^74 + 28 * q^79 - 4 * q^81 + 32 * q^89 - 24 * q^94 + 32 * q^99

Introduction

L-functions

Modular forms

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Database

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