LMFDB - Embedded newform 1425.2.c.p.799.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1425,2,Mod(799,1425)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1425.799"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1425 = 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1425.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,-12,0,0,0,0,-6,0,-6,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	no
Analytic conductor:	$11.3786822880$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.24681024.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 12x^{4} + 36x^{2} + 9 $ x^6 + 12x^4 + 36x^2 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.5
Root			$2.14510i$ of defining polynomial
Character	$\chi$	$=$	1425.799
Dual form			1425.2.c.p.799.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.14510i q^{2} -1.00000i q^{3} -2.60147 q^{4} +2.14510 q^{6} +2.74657i q^{7} -1.29021i q^{8} -1.00000 q^{9} -3.74657 q^{11} +2.60147i q^{12} -6.29021i q^{13} -5.89167 q^{14} -2.43531 q^{16} -2.14510i q^{18} -1.00000 q^{19} +2.74657 q^{21} -8.03677i q^{22} -0.543637i q^{23} -1.29021 q^{24} +13.4931 q^{26} +1.00000i q^{27} -7.14510i q^{28} -3.00000 q^{29} +1.45636 q^{31} -7.80440i q^{32} +3.74657i q^{33} +2.60147 q^{36} +5.20293i q^{37} -2.14510i q^{38} -6.29021 q^{39} -12.5299 q^{41} +5.89167i q^{42} -8.00000i q^{43} +9.74657 q^{44} +1.16616 q^{46} -11.7833i q^{47} +2.43531i q^{48} -0.543637 q^{49} +16.3638i q^{52} +5.58041i q^{53} -2.14510 q^{54} +3.54364 q^{56} +1.00000i q^{57} -6.43531i q^{58} -5.25343 q^{59} -8.49314 q^{61} +3.12405i q^{62} -2.74657i q^{63} +11.8706 q^{64} -8.03677 q^{66} -2.83384i q^{67} -0.543637 q^{69} +7.83384 q^{71} +1.29021i q^{72} -7.58041i q^{73} -11.1608 q^{74} +2.60147 q^{76} -10.2902i q^{77} -13.4931i q^{78} +7.52991 q^{79} +1.00000 q^{81} -26.8779i q^{82} +2.25343i q^{83} -7.14510 q^{84} +17.1608 q^{86} +3.00000i q^{87} +4.83384i q^{88} -4.49314 q^{89} +17.2765 q^{91} +1.41425i q^{92} -1.45636i q^{93} +25.2765 q^{94} -7.80440 q^{96} -16.8706i q^{97} -1.16616i q^{98} +3.74657 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{4} - 6 q^{9} - 6 q^{11} - 6 q^{14} + 24 q^{16} - 6 q^{19} + 18 q^{24} + 48 q^{26} - 18 q^{29} + 18 q^{31} + 12 q^{36} - 12 q^{39} + 42 q^{44} + 42 q^{46} + 6 q^{49} + 12 q^{56} - 48 q^{59} - 18 q^{61}+ \cdots + 6 q^{99}+O(q^{100}) $ 6 * q - 12 * q^4 - 6 * q^9 - 6 * q^11 - 6 * q^14 + 24 * q^16 - 6 * q^19 + 18 * q^24 + 48 * q^26 - 18 * q^29 + 18 * q^31 + 12 * q^36 - 12 * q^39 + 42 * q^44 + 42 * q^46 + 6 * q^49 + 12 * q^56 - 48 * q^59 - 18 * q^61 - 6 * q^64 - 6 * q^66 + 6 * q^69 + 12 * q^71 + 36 * q^74 + 12 * q^76 - 30 * q^79 + 6 * q^81 - 30 * q^84 + 6 * q^89 + 12 * q^91 + 60 * q^94 - 36 * q^96 + 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times$.

$n$	$476$	$1027$	$1351$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	2.14510i	1.51682i	0.651780	+	0.758408i	$0.274023\pi$
$2$	2.14510i	1.51682i	−0.651780	+	0.758408i	$0.725977\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−2.60147	−1.30073
$5$	0	0
$5$	0	0
$6$	2.14510	0.875735
$7$	2.74657i	1.03811i	0.854742	+	0.519053i	$0.173715\pi$
$7$	2.74657i	1.03811i	−0.854742	+	0.519053i	$0.826285\pi$
$8$	− 1.29021i	− 0.456156i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−3.74657	−1.12963	−0.564816	−	0.825217i	$-0.691053\pi$
$11$	−3.74657	−1.12963	−0.564816	+	0.825217i	$0.691053\pi$
$12$	2.60147i	0.750978i
$13$	− 6.29021i	− 1.74459i	−0.488981	−	0.872295i	$-0.662631\pi$
$13$	− 6.29021i	− 1.74459i	0.488981	−	0.872295i	$-0.337369\pi$
$14$	−5.89167	−1.57462
$15$	0	0
$16$	−2.43531	−0.608827
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	− 2.14510i	− 0.505606i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.74657	0.599350
$22$	− 8.03677i	− 1.71345i
$23$	− 0.543637i	− 0.113356i	−0.998393	−	0.0566781i	$-0.981949\pi$
$23$	− 0.543637i	− 0.113356i	0.998393	−	0.0566781i	$-0.0180509\pi$
$24$	−1.29021	−0.263362
$25$	0	0
$26$	13.4931	2.64622
$27$	1.00000i	0.192450i
$28$	− 7.14510i	− 1.35030i
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	1.45636	0.261570	0.130785	−	0.991411i	$-0.458250\pi$
$31$	1.45636	0.261570	0.130785	+	0.991411i	$0.458250\pi$
$32$	− 7.80440i	− 1.37964i
$33$	3.74657i	0.652194i
$34$	0	0
$35$	0	0
$36$	2.60147	0.433578
$37$	5.20293i	0.855357i	0.903931	+	0.427678i	$0.140668\pi$
$37$	5.20293i	0.855357i	−0.903931	+	0.427678i	$0.859332\pi$
$38$	− 2.14510i	− 0.347982i
$39$	−6.29021	−1.00724
$40$	0	0
$41$	−12.5299	−1.95684	−0.978422	−	0.206618i	$-0.933754\pi$
$41$	−12.5299	−1.95684	−0.978422	+	0.206618i	$0.933754\pi$
$42$	5.89167i	0.909105i
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	9.74657	1.46935
$45$	0	0
$46$	1.16616	0.171941
$47$	− 11.7833i	− 1.71878i	−0.511323	−	0.859389i	$-0.670844\pi$
$47$	− 11.7833i	− 1.71878i	0.511323	−	0.859389i	$-0.329156\pi$
$48$	2.43531i	0.351506i
$49$	−0.543637	−0.0776624
$50$	0	0
$51$	0	0
$52$	16.3638i	2.26924i
$53$	5.58041i	0.766528i	0.923639	+	0.383264i	$0.125200\pi$
$53$	5.58041i	0.766528i	−0.923639	+	0.383264i	$0.874800\pi$
$54$	−2.14510	−0.291912
$55$	0	0
$56$	3.54364	0.473538
$57$	1.00000i	0.132453i
$58$	− 6.43531i	− 0.844997i
$59$	−5.25343	−0.683939	−0.341969	−	0.939711i	$-0.611094\pi$
$59$	−5.25343	−0.683939	−0.341969	+	0.939711i	$0.611094\pi$
$60$	0	0
$61$	−8.49314	−1.08743	−0.543717	−	0.839268i	$-0.682984\pi$
$61$	−8.49314	−1.08743	−0.543717	+	0.839268i	$0.682984\pi$
$62$	3.12405i	0.396754i
$63$	− 2.74657i	− 0.346035i
$64$	11.8706	1.48383
$65$	0	0
$66$	−8.03677	−0.989258
$67$	− 2.83384i	− 0.346209i	−0.984903	−	0.173104i	$-0.944620\pi$
$67$	− 2.83384i	− 0.346209i	0.984903	−	0.173104i	$-0.0553798\pi$
$68$	0	0
$69$	−0.543637	−0.0654462
$70$	0	0
$71$	7.83384	0.929706	0.464853	−	0.885388i	$-0.346107\pi$
$71$	7.83384	0.929706	0.464853	+	0.885388i	$0.346107\pi$
$72$	1.29021i	0.152052i
$73$	− 7.58041i	− 0.887220i	−0.896220	−	0.443610i	$-0.853698\pi$
$73$	− 7.58041i	− 0.887220i	0.896220	−	0.443610i	$-0.146302\pi$
$74$	−11.1608	−1.29742
$75$	0	0
$76$	2.60147	0.298409
$77$	− 10.2902i	− 1.17268i
$78$	− 13.4931i	− 1.52780i
$79$	7.52991	0.847181	0.423591	−	0.905854i	$-0.360770\pi$
$79$	7.52991	0.847181	0.423591	+	0.905854i	$0.360770\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 26.8779i	− 2.96817i
$83$	2.25343i	0.247346i	0.992323	+	0.123673i	$0.0394674\pi$
$83$	2.25343i	0.247346i	−0.992323	+	0.123673i	$0.960533\pi$
$84$	−7.14510	−0.779595
$85$	0	0
$86$	17.1608	1.85050
$87$	3.00000i	0.321634i
$88$	4.83384i	0.515289i
$89$	−4.49314	−0.476272	−0.238136	−	0.971232i	$-0.576536\pi$
$89$	−4.49314	−0.476272	−0.238136	+	0.971232i	$0.576536\pi$
$90$	0	0
$91$	17.2765	1.81107
$92$	1.41425i	0.147446i
$93$	− 1.45636i	− 0.151018i
$94$	25.2765	2.60707
$95$	0	0
$96$	−7.80440	−0.796533
$97$	− 16.8706i	− 1.71295i	−0.516187	−	0.856476i	$-0.672649\pi$
$97$	− 16.8706i	− 1.71295i	0.516187	−	0.856476i	$-0.327351\pi$
$98$	− 1.16616i	− 0.117800i
$99$	3.74657	0.376544
$100$	0	0
$101$	−13.2765	−1.32106	−0.660529	−	0.750800i	$-0.729668\pi$
$101$	−13.2765	−1.32106	−0.660529	+	0.750800i	$0.729668\pi$
$102$	0	0
$103$	0.253432i	0.0249714i	0.999922	+	0.0124857i	$0.00397442\pi$
$103$	0.253432i	0.0249714i	−0.999922	+	0.0124857i	$0.996026\pi$
$104$	−8.11566	−0.795806
$105$	0	0
$106$	−11.9706	−1.16268
$107$	2.23970i	0.216520i	0.994123	+	0.108260i	$0.0345280\pi$
$107$	2.23970i	0.216520i	−0.994123	+	0.108260i	$0.965472\pi$
$108$	− 2.60147i	− 0.250326i
$109$	1.20293	0.115220	0.0576100	−	0.998339i	$-0.481652\pi$
$109$	1.20293	0.115220	0.0576100	+	0.998339i	$0.481652\pi$
$110$	0	0
$111$	5.20293	0.493840
$112$	− 6.68874i	− 0.632027i
$113$	12.5436i	1.18001i	0.807401	+	0.590003i	$0.200873\pi$
$113$	12.5436i	1.18001i	−0.807401	+	0.590003i	$0.799127\pi$
$114$	−2.14510	−0.200907
$115$	0	0
$116$	7.80440	0.724620
$117$	6.29021i	0.581530i
$118$	− 11.2692i	− 1.03741i
$119$	0	0
$120$	0	0
$121$	3.03677	0.276070
$122$	− 18.2186i	− 1.64944i
$123$	12.5299i	1.12978i
$124$	−3.78868	−0.340233
$125$	0	0
$126$	5.89167	0.524872
$127$	− 12.4426i	− 1.10411i	−0.833809	−	0.552053i	$-0.813845\pi$
$127$	− 12.4426i	− 1.10411i	0.833809	−	0.552053i	$-0.186155\pi$
$128$	9.85490i	0.871058i
$129$	−8.00000	−0.704361
$130$	0	0
$131$	18.9495	1.65563	0.827813	−	0.561005i	$-0.189585\pi$
$131$	18.9495	1.65563	0.827813	+	0.561005i	$0.189585\pi$
$132$	− 9.74657i	− 0.848330i
$133$	− 2.74657i	− 0.238158i
$134$	6.07888	0.525136
$135$	0	0
$136$	0	0
$137$	21.3921i	1.82765i	0.406104	+	0.913827i	$0.366887\pi$
$137$	21.3921i	1.82765i	−0.406104	+	0.913827i	$0.633113\pi$
$138$	− 1.16616i	− 0.0992699i
$139$	−20.3133	−1.72295	−0.861474	−	0.507802i	$-0.830458\pi$
$139$	−20.3133	−1.72295	−0.861474	+	0.507802i	$0.830458\pi$
$140$	0	0
$141$	−11.7833	−0.992336
$142$	16.8044i	1.41019i
$143$	23.5667i	1.97075i
$144$	2.43531	0.202942
$145$	0	0
$146$	16.2608	1.34575
$147$	0.543637i	0.0448384i
$148$	− 13.5352i	− 1.11259i
$149$	−4.69607	−0.384717	−0.192358	−	0.981325i	$-0.561614\pi$
$149$	−4.69607	−0.384717	−0.192358	+	0.981325i	$0.561614\pi$
$150$	0	0
$151$	1.59414	0.129729	0.0648645	−	0.997894i	$-0.479338\pi$
$151$	1.59414	0.129729	0.0648645	+	0.997894i	$0.479338\pi$
$152$	1.29021i	0.104649i
$153$	0	0
$154$	22.0735	1.77874
$155$	0	0
$156$	16.3638	1.31015
$157$	− 13.1103i	− 1.04632i	−0.852235	−	0.523159i	$-0.824753\pi$
$157$	− 13.1103i	− 1.04632i	0.852235	−	0.523159i	$-0.175247\pi$
$158$	16.1524i	1.28502i
$159$	5.58041	0.442555
$160$	0	0
$161$	1.49314	0.117676
$162$	2.14510i	0.168535i
$163$	7.73284i	0.605683i	0.953041	+	0.302841i	$0.0979353\pi$
$163$	7.73284i	0.605683i	−0.953041	+	0.302841i	$0.902065\pi$
$164$	32.5961	2.54533
$165$	0	0
$166$	−4.83384	−0.375179
$167$	− 9.32698i	− 0.721743i	−0.932615	−	0.360872i	$-0.882479\pi$
$167$	− 9.32698i	− 0.721743i	0.932615	−	0.360872i	$-0.117521\pi$
$168$	− 3.54364i	− 0.273398i
$169$	−26.5667	−2.04359
$170$	0	0
$171$	1.00000	0.0764719
$172$	20.8117i	1.58688i
$173$	5.17455i	0.393414i	0.980462	+	0.196707i	$0.0630247\pi$
$173$	5.17455i	0.393414i	−0.980462	+	0.196707i	$0.936975\pi$
$174$	−6.43531	−0.487859
$175$	0	0
$176$	9.12405	0.687751
$177$	5.25343i	0.394872i
$178$	− 9.63824i	− 0.722417i
$179$	−15.3270	−1.14559	−0.572796	−	0.819698i	$-0.694141\pi$
$179$	−15.3270	−1.14559	−0.572796	+	0.819698i	$0.694141\pi$
$180$	0	0
$181$	−10.4059	−0.773462	−0.386731	−	0.922193i	$-0.626396\pi$
$181$	−10.4059	−0.773462	−0.386731	+	0.922193i	$0.626396\pi$
$182$	37.0598i	2.74706i
$183$	8.49314i	0.627831i
$184$	−0.701404	−0.0517082
$185$	0	0
$186$	3.12405	0.229066
$187$	0	0
$188$	30.6540i	2.23567i
$189$	−2.74657	−0.199783
$190$	0	0
$191$	−6.54364	−0.473481	−0.236740	−	0.971573i	$-0.576079\pi$
$191$	−6.54364	−0.473481	−0.236740	+	0.971573i	$0.576079\pi$
$192$	− 11.8706i	− 0.856688i
$193$	24.9863i	1.79855i	0.437382	+	0.899276i	$0.355906\pi$
$193$	24.9863i	1.79855i	−0.437382	+	0.899276i	$0.644094\pi$
$194$	36.1892	2.59823
$195$	0	0
$196$	1.41425	0.101018
$197$	8.79707i	0.626765i	0.949627	+	0.313383i	$0.101462\pi$
$197$	8.79707i	0.626765i	−0.949627	+	0.313383i	$0.898538\pi$
$198$	8.03677i	0.571149i
$199$	−8.74657	−0.620028	−0.310014	−	0.950732i	$-0.600334\pi$
$199$	−8.74657	−0.620028	−0.310014	+	0.950732i	$0.600334\pi$
$200$	0	0
$201$	−2.83384	−0.199884
$202$	− 28.4794i	− 2.00380i
$203$	− 8.23970i	− 0.578314i
$204$	0	0
$205$	0	0
$206$	−0.543637	−0.0378770
$207$	0.543637i	0.0377854i
$208$	15.3186i	1.06215i
$209$	3.74657	0.259156
$210$	0	0
$211$	−12.7182	−0.875556	−0.437778	−	0.899083i	$-0.644234\pi$
$211$	−12.7182	−0.875556	−0.437778	+	0.899083i	$0.644234\pi$
$212$	− 14.5172i	− 0.997049i
$213$	− 7.83384i	− 0.536766i
$214$	−4.80440	−0.328422
$215$	0	0
$216$	1.29021	0.0877874
$217$	4.00000i	0.271538i
$218$	2.58041i	0.174767i
$219$	−7.58041	−0.512237
$220$	0	0
$221$	0	0
$222$	11.1608i	0.749065i
$223$	10.5436i	0.706054i	0.935613	+	0.353027i	$0.114848\pi$
$223$	10.5436i	0.706054i	−0.935613	+	0.353027i	$0.885152\pi$
$224$	21.4353	1.43221
$225$	0	0
$226$	−26.9074	−1.78985
$227$	7.83384i	0.519950i	0.965615	+	0.259975i	$0.0837144\pi$
$227$	7.83384i	0.519950i	−0.965615	+	0.259975i	$0.916286\pi$
$228$	− 2.60147i	− 0.172286i
$229$	23.6035	1.55976	0.779880	−	0.625929i	$-0.215280\pi$
$229$	23.6035	1.55976	0.779880	+	0.625929i	$0.215280\pi$
$230$	0	0
$231$	−10.2902	−0.677046
$232$	3.87062i	0.254118i
$233$	23.7833i	1.55810i	0.626963	+	0.779049i	$0.284298\pi$
$233$	23.7833i	1.55810i	−0.626963	+	0.779049i	$0.715702\pi$
$234$	−13.4931	−0.882074
$235$	0	0
$236$	13.6666	0.889621
$237$	− 7.52991i	− 0.489120i
$238$	0	0
$239$	1.27648	0.0825685	0.0412843	−	0.999147i	$-0.486855\pi$
$239$	1.27648	0.0825685	0.0412843	+	0.999147i	$0.486855\pi$
$240$	0	0
$241$	21.4931	1.38449	0.692247	−	0.721660i	$-0.256621\pi$
$241$	21.4931	1.38449	0.692247	+	0.721660i	$0.256621\pi$
$242$	6.51419i	0.418748i
$243$	− 1.00000i	− 0.0641500i
$244$	22.0946	1.41446
$245$	0	0
$246$	−26.8779	−1.71368
$247$	6.29021i	0.400236i
$248$	− 1.87901i	− 0.119317i
$249$	2.25343	0.142805
$250$	0	0
$251$	−29.9725	−1.89185	−0.945925	−	0.324385i	$-0.894843\pi$
$251$	−29.9725	−1.89185	−0.945925	+	0.324385i	$0.894843\pi$
$252$	7.14510i	0.450099i
$253$	2.03677i	0.128051i
$254$	26.6907	1.67473
$255$	0	0
$256$	2.60147	0.162592
$257$	19.0735i	1.18978i	0.803809	+	0.594888i	$0.202803\pi$
$257$	19.0735i	1.18978i	−0.803809	+	0.594888i	$0.797197\pi$
$258$	− 17.1608i	− 1.06839i
$259$	−14.2902	−0.887950
$260$	0	0
$261$	3.00000	0.185695
$262$	40.6486i	2.51128i
$263$	− 11.6456i	− 0.718096i	−0.933319	−	0.359048i	$-0.883101\pi$
$263$	− 11.6456i	− 0.718096i	0.933319	−	0.359048i	$-0.116899\pi$
$264$	4.83384	0.297502
$265$	0	0
$266$	5.89167	0.361242
$267$	4.49314i	0.274975i
$268$	7.37214i	0.450325i
$269$	−26.1471	−1.59422	−0.797108	−	0.603836i	$-0.793638\pi$
$269$	−26.1471	−1.59422	−0.797108	+	0.603836i	$0.793638\pi$
$270$	0	0
$271$	15.8338	0.961837	0.480919	−	0.876765i	$-0.340303\pi$
$271$	15.8338	0.961837	0.480919	+	0.876765i	$0.340303\pi$
$272$	0	0
$273$	− 17.2765i	− 1.04562i
$274$	−45.8883	−2.77222
$275$	0	0
$276$	1.41425	0.0851280
$277$	20.6677i	1.24180i	0.783889	+	0.620900i	$0.213233\pi$
$277$	20.6677i	1.24180i	−0.783889	+	0.620900i	$0.786767\pi$
$278$	− 43.5740i	− 2.61340i
$279$	−1.45636	−0.0871902
$280$	0	0
$281$	1.35536	0.0808541	0.0404270	−	0.999182i	$-0.487128\pi$
$281$	1.35536	0.0808541	0.0404270	+	0.999182i	$0.487128\pi$
$282$	− 25.2765i	− 1.50519i
$283$	− 28.1471i	− 1.67317i	−0.547836	−	0.836586i	$-0.684548\pi$
$283$	− 28.1471i	− 1.67317i	0.547836	−	0.836586i	$-0.315452\pi$
$284$	−20.3795	−1.20930
$285$	0	0
$286$	−50.5530	−2.98926
$287$	− 34.4143i	− 2.03141i
$288$	7.80440i	0.459878i
$289$	17.0000	1.00000
$290$	0	0
$291$	−16.8706	−0.988973
$292$	19.7202i	1.15404i
$293$	8.47009i	0.494828i	0.968910	+	0.247414i	$0.0795808\pi$
$293$	8.47009i	0.494828i	−0.968910	+	0.247414i	$0.920419\pi$
$294$	−1.16616	−0.0680117
$295$	0	0
$296$	6.71285	0.390176
$297$	− 3.74657i	− 0.217398i
$298$	− 10.0735i	− 0.583545i
$299$	−3.41959	−0.197760
$300$	0	0
$301$	21.9725	1.26648
$302$	3.41959i	0.196775i
$303$	13.2765i	0.762714i
$304$	2.43531	0.139674
$305$	0	0
$306$	0	0
$307$	− 19.3133i	− 1.10227i	−0.834418	−	0.551133i	$-0.814196\pi$
$307$	− 19.3133i	− 1.10227i	0.834418	−	0.551133i	$-0.185804\pi$
$308$	26.7696i	1.52534i
$309$	0.253432	0.0144172
$310$	0	0
$311$	20.5804	1.16701	0.583504	−	0.812110i	$-0.301681\pi$
$311$	20.5804	1.16701	0.583504	+	0.812110i	$0.301681\pi$
$312$	8.11566i	0.459459i
$313$	− 22.4426i	− 1.26853i	−0.773115	−	0.634266i	$-0.781302\pi$
$313$	− 22.4426i	− 1.26853i	0.773115	−	0.634266i	$-0.218698\pi$
$314$	28.1230	1.58707
$315$	0	0
$316$	−19.5888	−1.10196
$317$	11.1745i	0.627625i	0.949485	+	0.313813i	$0.101606\pi$
$317$	11.1745i	0.627625i	−0.949485	+	0.313813i	$0.898394\pi$
$318$	11.9706i	0.671275i
$319$	11.2397	0.629303
$320$	0	0
$321$	2.23970	0.125008
$322$	3.20293i	0.178492i
$323$	0	0
$324$	−2.60147	−0.144526
$325$	0	0
$326$	−16.5877	−0.918710
$327$	− 1.20293i	− 0.0665222i
$328$	16.1662i	0.892627i
$329$	32.3638	1.78427
$330$	0	0
$331$	−25.9358	−1.42556	−0.712779	−	0.701388i	$-0.752564\pi$
$331$	−25.9358	−1.42556	−0.712779	+	0.701388i	$0.752564\pi$
$332$	− 5.86223i	− 0.321731i
$333$	− 5.20293i	− 0.285119i
$334$	20.0073	1.09475
$335$	0	0
$336$	−6.68874	−0.364901
$337$	15.0873i	0.821856i	0.911668	+	0.410928i	$0.134795\pi$
$337$	15.0873i	0.821856i	−0.911668	+	0.410928i	$0.865205\pi$
$338$	− 56.9883i	− 3.09975i
$339$	12.5436	0.681277
$340$	0	0
$341$	−5.45636	−0.295479
$342$	2.14510i	0.115994i
$343$	17.7328i	0.957483i
$344$	−10.3216	−0.556506
$345$	0	0
$346$	−11.0999	−0.596736
$347$	− 6.62252i	− 0.355516i	−0.984074	−	0.177758i	$-0.943116\pi$
$347$	− 6.62252i	− 0.355516i	0.984074	−	0.177758i	$-0.0568843\pi$
$348$	− 7.80440i	− 0.418360i
$349$	−9.07355	−0.485696	−0.242848	−	0.970064i	$-0.578082\pi$
$349$	−9.07355	−0.485696	−0.242848	+	0.970064i	$0.578082\pi$
$350$	0	0
$351$	6.29021	0.335746
$352$	29.2397i	1.55848i
$353$	− 20.1471i	− 1.07232i	−0.844116	−	0.536161i	$-0.819874\pi$
$353$	− 20.1471i	− 1.07232i	0.844116	−	0.536161i	$-0.180126\pi$
$354$	−11.2692	−0.598949
$355$	0	0
$356$	11.6887	0.619502
$357$	0	0
$358$	− 32.8779i	− 1.73765i
$359$	−15.8843	−0.838344	−0.419172	−	0.907907i	$-0.637679\pi$
$359$	−15.8843	−0.838344	−0.419172	+	0.907907i	$0.637679\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 22.3216i	− 1.17320i
$363$	− 3.03677i	− 0.159389i
$364$	−44.9442	−2.35571
$365$	0	0
$366$	−18.2186	−0.952304
$367$	24.8853i	1.29900i	0.760361	+	0.649500i	$0.225022\pi$
$367$	24.8853i	1.29900i	−0.760361	+	0.649500i	$0.774978\pi$
$368$	1.32392i	0.0690143i
$369$	12.5299	0.652281
$370$	0	0
$371$	−15.3270	−0.795737
$372$	3.78868i	0.196434i
$373$	− 11.6677i	− 0.604130i	−0.953287	−	0.302065i	$-0.902324\pi$
$373$	− 11.6677i	− 0.604130i	0.953287	−	0.302065i	$-0.0976759\pi$
$374$	0	0
$375$	0	0
$376$	−15.2029	−0.784031
$377$	18.8706i	0.971886i
$378$	− 5.89167i	− 0.303035i
$379$	−6.29021	−0.323106	−0.161553	−	0.986864i	$-0.551650\pi$
$379$	−6.29021	−0.323106	−0.161553	+	0.986864i	$0.551650\pi$
$380$	0	0
$381$	−12.4426	−0.637456
$382$	− 14.0368i	− 0.718184i
$383$	− 12.0652i	− 0.616501i	−0.951305	−	0.308250i	$-0.900257\pi$
$383$	− 12.0652i	− 0.616501i	0.951305	−	0.308250i	$-0.0997434\pi$
$384$	9.85490	0.502906
$385$	0	0
$386$	−53.5981	−2.72807
$387$	8.00000i	0.406663i
$388$	43.8883i	2.22809i
$389$	14.3638	0.728271	0.364136	−	0.931346i	$-0.381364\pi$
$389$	14.3638	0.728271	0.364136	+	0.931346i	$0.381364\pi$
$390$	0	0
$391$	0	0
$392$	0.701404i	0.0354262i
$393$	− 18.9495i	− 0.955876i
$394$	−18.8706	−0.950688
$395$	0	0
$396$	−9.74657	−0.489783
$397$	− 31.0966i	− 1.56069i	−0.625347	−	0.780347i	$-0.715043\pi$
$397$	− 31.0966i	− 1.56069i	0.625347	−	0.780347i	$-0.284957\pi$
$398$	− 18.7623i	− 0.940468i
$399$	−2.74657	−0.137500
$400$	0	0
$401$	−0.543637	−0.0271479	−0.0135740	−	0.999908i	$-0.504321\pi$
$401$	−0.543637	−0.0271479	−0.0135740	+	0.999908i	$0.504321\pi$
$402$	− 6.07888i	− 0.303187i
$403$	− 9.16082i	− 0.456333i
$404$	34.5383	1.71834
$405$	0	0
$406$	17.6750	0.877196
$407$	− 19.4931i	− 0.966239i
$408$	0	0
$409$	−26.4648	−1.30860	−0.654299	−	0.756236i	$-0.727036\pi$
$409$	−26.4648	−1.30860	−0.654299	+	0.756236i	$0.727036\pi$
$410$	0	0
$411$	21.3921	1.05520
$412$	− 0.659294i	− 0.0324811i
$413$	− 14.4289i	− 0.710000i
$414$	−1.16616	−0.0573135
$415$	0	0
$416$	−49.0913	−2.40690
$417$	20.3133i	0.994744i
$418$	8.03677i	0.393091i
$419$	6.18920	0.302362	0.151181	−	0.988506i	$-0.451692\pi$
$419$	6.18920	0.302362	0.151181	+	0.988506i	$0.451692\pi$
$420$	0	0
$421$	15.2765	0.744530	0.372265	−	0.928126i	$-0.378581\pi$
$421$	15.2765	0.744530	0.372265	+	0.928126i	$0.378581\pi$
$422$	− 27.2818i	− 1.32806i
$423$	11.7833i	0.572926i
$424$	7.19988	0.349657
$425$	0	0
$426$	16.8044	0.814176
$427$	− 23.3270i	− 1.12887i
$428$	− 5.82651i	− 0.281635i
$429$	23.5667	1.13781
$430$	0	0
$431$	13.5583	0.653080	0.326540	−	0.945183i	$-0.394117\pi$
$431$	13.5583	0.653080	0.326540	+	0.945183i	$0.394117\pi$
$432$	− 2.43531i	− 0.117169i
$433$	29.6823i	1.42644i	0.700939	+	0.713221i	$0.252764\pi$
$433$	29.6823i	1.42644i	−0.700939	+	0.713221i	$0.747236\pi$
$434$	−8.58041	−0.411873
$435$	0	0
$436$	−3.12938	−0.149870
$437$	0.543637i	0.0260057i
$438$	− 16.2608i	− 0.776969i
$439$	−34.0093	−1.62318	−0.811588	−	0.584230i	$-0.801397\pi$
$439$	−34.0093	−1.62318	−0.811588	+	0.584230i	$0.801397\pi$
$440$	0	0
$441$	0.543637	0.0258875
$442$	0	0
$443$	16.3416i	0.776414i	0.921572	+	0.388207i	$0.126906\pi$
$443$	16.3416i	0.776414i	−0.921572	+	0.388207i	$0.873094\pi$
$444$	−13.5352	−0.642354
$445$	0	0
$446$	−22.6172	−1.07095
$447$	4.69607i	0.222116i
$448$	32.6035i	1.54037i
$449$	−16.8990	−0.797513	−0.398757	−	0.917057i	$-0.630558\pi$
$449$	−16.8990	−0.797513	−0.398757	+	0.917057i	$0.630558\pi$
$450$	0	0
$451$	46.9442	2.21051
$452$	− 32.6318i	− 1.53487i
$453$	− 1.59414i	− 0.0748991i
$454$	−16.8044	−0.788669
$455$	0	0
$456$	1.29021	0.0604194
$457$	16.8622i	0.788782i	0.918943	+	0.394391i	$0.129044\pi$
$457$	16.8622i	0.788782i	−0.918943	+	0.394391i	$0.870956\pi$
$458$	50.6318i	2.36587i
$459$	0	0
$460$	0	0
$461$	−24.4648	−1.13944	−0.569719	−	0.821840i	$-0.692948\pi$
$461$	−24.4648	−1.13944	−0.569719	+	0.821840i	$0.692948\pi$
$462$	− 22.0735i	− 1.02695i
$463$	14.7550i	0.685721i	0.939386	+	0.342861i	$0.111396\pi$
$463$	14.7550i	0.685721i	−0.939386	+	0.342861i	$0.888604\pi$
$464$	7.30592	0.339169
$465$	0	0
$466$	−51.0177	−2.36335
$467$	12.3270i	0.570425i	0.958464	+	0.285212i	$0.0920641\pi$
$467$	12.3270i	0.570425i	−0.958464	+	0.285212i	$0.907936\pi$
$468$	− 16.3638i	− 0.756415i
$469$	7.78334	0.359401
$470$	0	0
$471$	−13.1103	−0.604092
$472$	6.77801i	0.311983i
$473$	29.9725i	1.37814i
$474$	16.1524	0.741906
$475$	0	0
$476$	0	0
$477$	− 5.58041i	− 0.255509i
$478$	2.73818i	0.125241i
$479$	19.0084	0.868516	0.434258	−	0.900789i	$-0.357011\pi$
$479$	19.0084	0.868516	0.434258	+	0.900789i	$0.357011\pi$
$480$	0	0
$481$	32.7275	1.49225
$482$	46.1050i	2.10002i
$483$	− 1.49314i	− 0.0679401i
$484$	−7.90006	−0.359094
$485$	0	0
$486$	2.14510	0.0973038
$487$	33.2765i	1.50790i	0.656931	+	0.753951i	$0.271854\pi$
$487$	33.2765i	1.50790i	−0.656931	+	0.753951i	$0.728146\pi$
$488$	10.9579i	0.496040i
$489$	7.73284	0.349691
$490$	0	0
$491$	28.7275	1.29645	0.648227	−	0.761447i	$-0.275511\pi$
$491$	28.7275	1.29645	0.648227	+	0.761447i	$0.275511\pi$
$492$	− 32.5961i	− 1.46955i
$493$	0	0
$494$	−13.4931	−0.607085
$495$	0	0
$496$	−3.54669	−0.159251
$497$	21.5162i	0.965133i
$498$	4.83384i	0.216610i
$499$	24.4878	1.09622	0.548112	−	0.836405i	$-0.315347\pi$
$499$	24.4878	1.09622	0.548112	+	0.836405i	$0.315347\pi$
$500$	0	0
$501$	−9.32698	−0.416699
$502$	− 64.2942i	− 2.86959i
$503$	7.30393i	0.325666i	0.986654	+	0.162833i	$0.0520632\pi$
$503$	7.30393i	0.325666i	−0.986654	+	0.162833i	$0.947937\pi$
$504$	−3.54364	−0.157846
$505$	0	0
$506$	−4.36909	−0.194230
$507$	26.5667i	1.17987i
$508$	32.3691i	1.43615i
$509$	−33.3784	−1.47947	−0.739736	−	0.672897i	$-0.765050\pi$
$509$	−33.3784	−1.47947	−0.739736	+	0.672897i	$0.765050\pi$
$510$	0	0
$511$	20.8201	0.921028
$512$	25.2902i	1.11768i
$513$	− 1.00000i	− 0.0441511i
$514$	−40.9147	−1.80467
$515$	0	0
$516$	20.8117	0.916185
$517$	44.1471i	1.94159i
$518$	− 30.6540i	− 1.34686i
$519$	5.17455	0.227137
$520$	0	0
$521$	−4.08727	−0.179067	−0.0895334	−	0.995984i	$-0.528538\pi$
$521$	−4.08727	−0.179067	−0.0895334	+	0.995984i	$0.528538\pi$
$522$	6.43531i	0.281666i
$523$	− 27.9304i	− 1.22131i	−0.791896	−	0.610656i	$-0.790906\pi$
$523$	− 27.9304i	− 1.22131i	0.791896	−	0.610656i	$-0.209094\pi$
$524$	−49.2965	−2.15353
$525$	0	0
$526$	24.9809	1.08922
$527$	0	0
$528$	− 9.12405i	− 0.397073i
$529$	22.7045	0.987150
$530$	0	0
$531$	5.25343	0.227980
$532$	7.14510i	0.309779i
$533$	78.8157i	3.41389i
$534$	−9.63824	−0.417087
$535$	0	0
$536$	−3.65624	−0.157925
$537$	15.3270i	0.661408i
$538$	− 56.0882i	− 2.41813i
$539$	2.03677	0.0877301
$540$	0	0
$541$	26.7917	1.15187	0.575933	−	0.817497i	$-0.304639\pi$
$541$	26.7917	1.15187	0.575933	+	0.817497i	$0.304639\pi$
$542$	33.9652i	1.45893i
$543$	10.4059i	0.446558i
$544$	0	0
$545$	0	0
$546$	37.0598	1.58601
$547$	− 9.26716i	− 0.396235i	−0.980178	−	0.198118i	$-0.936517\pi$
$547$	− 9.26716i	− 0.396235i	0.980178	−	0.198118i	$-0.0634828\pi$
$548$	− 55.6509i	− 2.37729i
$549$	8.49314	0.362478
$550$	0	0
$551$	3.00000	0.127804
$552$	0.701404i	0.0298537i
$553$	20.6814i	0.879463i
$554$	−44.3343	−1.88358
$555$	0	0
$556$	52.8442	2.24109
$557$	− 32.5804i	− 1.38048i	−0.723582	−	0.690238i	$-0.757506\pi$
$557$	− 32.5804i	− 1.38048i	0.723582	−	0.690238i	$-0.242494\pi$
$558$	− 3.12405i	− 0.132251i
$559$	−50.3216	−2.12838
$560$	0	0
$561$	0	0
$562$	2.90739i	0.122641i
$563$	− 33.3270i	− 1.40456i	−0.711899	−	0.702282i	$-0.752164\pi$
$563$	− 33.3270i	− 1.40456i	0.711899	−	0.702282i	$-0.247836\pi$
$564$	30.6540	1.29076
$565$	0	0
$566$	60.3784	2.53789
$567$	2.74657i	0.115345i
$568$	− 10.1073i	− 0.424091i
$569$	27.3859	1.14808	0.574038	−	0.818829i	$-0.305376\pi$
$569$	27.3859	1.14808	0.574038	+	0.818829i	$0.305376\pi$
$570$	0	0
$571$	−43.2995	−1.81203	−0.906014	−	0.423247i	$-0.860890\pi$
$571$	−43.2995	−1.81203	−0.906014	+	0.423247i	$0.860890\pi$
$572$	− 61.3079i	− 2.56341i
$573$	6.54364i	0.273364i
$574$	73.8221	3.08128
$575$	0	0
$576$	−11.8706	−0.494609
$577$	47.3721i	1.97213i	0.166366	+	0.986064i	$0.446797\pi$
$577$	47.3721i	1.97213i	−0.166366	+	0.986064i	$0.553203\pi$
$578$	36.4667i	1.51682i
$579$	24.9863	1.03839
$580$	0	0
$581$	−6.18920	−0.256771
$582$	− 36.1892i	− 1.50009i
$583$	− 20.9074i	− 0.865896i
$584$	−9.78029	−0.404711
$585$	0	0
$586$	−18.1692	−0.750563
$587$	− 29.0230i	− 1.19791i	−0.800783	−	0.598955i	$-0.795583\pi$
$587$	− 29.0230i	− 1.19791i	0.800783	−	0.598955i	$-0.204417\pi$
$588$	− 1.41425i	− 0.0583228i
$589$	−1.45636	−0.0600084
$590$	0	0
$591$	8.79707	0.361863
$592$	− 12.6707i	− 0.520764i
$593$	− 30.8706i	− 1.26770i	−0.773454	−	0.633852i	$-0.781473\pi$
$593$	− 30.8706i	− 1.26770i	0.773454	−	0.633852i	$-0.218527\pi$
$594$	8.03677	0.329753
$595$	0	0
$596$	12.2167	0.500414
$597$	8.74657i	0.357973i
$598$	− 7.33537i	− 0.299966i
$599$	−17.1608	−0.701172	−0.350586	−	0.936531i	$-0.614018\pi$
$599$	−17.1608	−0.701172	−0.350586	+	0.936531i	$0.614018\pi$
$600$	0	0
$601$	−13.2029	−0.538559	−0.269279	−	0.963062i	$-0.586786\pi$
$601$	−13.2029	−0.538559	−0.269279	+	0.963062i	$0.586786\pi$
$602$	47.1334i	1.92101i
$603$	2.83384i	0.115403i
$604$	−4.14709	−0.168743
$605$	0	0
$606$	−28.4794	−1.15690
$607$	− 16.7603i	− 0.680279i	−0.940375	−	0.340140i	$-0.889526\pi$
$607$	− 16.7603i	− 0.680279i	0.940375	−	0.340140i	$-0.110474\pi$
$608$	7.80440i	0.316510i
$609$	−8.23970	−0.333890
$610$	0	0
$611$	−74.1196	−2.99856
$612$	0	0
$613$	− 32.7780i	− 1.32389i	−0.749552	−	0.661946i	$-0.769731\pi$
$613$	− 32.7780i	− 1.32389i	0.749552	−	0.661946i	$-0.230269\pi$
$614$	41.4289	1.67193
$615$	0	0
$616$	−13.2765	−0.534925
$617$	− 33.4510i	− 1.34669i	−0.739329	−	0.673344i	$-0.764857\pi$
$617$	− 33.4510i	− 1.34669i	0.739329	−	0.673344i	$-0.235143\pi$
$618$	0.543637i	0.0218683i
$619$	−39.3731	−1.58254	−0.791269	−	0.611469i	$-0.790579\pi$
$619$	−39.3731	−1.58254	−0.791269	+	0.611469i	$0.790579\pi$
$620$	0	0
$621$	0.543637	0.0218154
$622$	44.1471i	1.77014i
$623$	− 12.3407i	− 0.494420i
$624$	15.3186	0.613234
$625$	0	0
$626$	48.1418	1.92413
$627$	− 3.74657i	− 0.149624i
$628$	34.1060i	1.36098i
$629$	0	0
$630$	0	0
$631$	40.5804	1.61548	0.807740	−	0.589538i	$-0.200690\pi$
$631$	40.5804	1.61548	0.807740	+	0.589538i	$0.200690\pi$
$632$	− 9.71513i	− 0.386447i
$633$	12.7182i	0.505502i
$634$	−23.9706	−0.951992
$635$	0	0
$636$	−14.5172	−0.575646
$637$	3.41959i	0.135489i
$638$	24.1103i	0.954537i
$639$	−7.83384	−0.309902
$640$	0	0
$641$	−6.81172	−0.269047	−0.134523	−	0.990910i	$-0.542950\pi$
$641$	−6.81172	−0.269047	−0.134523	+	0.990910i	$0.542950\pi$
$642$	4.80440i	0.189614i
$643$	14.3868i	0.567360i	0.958919	+	0.283680i	$0.0915553\pi$
$643$	14.3868i	0.567360i	−0.958919	+	0.283680i	$0.908445\pi$
$644$	−3.88434	−0.153065
$645$	0	0
$646$	0	0
$647$	0.137775i	0.00541649i	0.999996	+	0.00270825i	$0.000862062\pi$
$647$	0.137775i	0.00541649i	−0.999996	+	0.00270825i	$0.999138\pi$
$648$	− 1.29021i	− 0.0506841i
$649$	19.6823	0.772599
$650$	0	0
$651$	4.00000	0.156772
$652$	− 20.1167i	− 0.787832i
$653$	11.1019i	0.434452i	0.976121	+	0.217226i	$0.0697009\pi$
$653$	11.1019i	0.434452i	−0.976121	+	0.217226i	$0.930299\pi$
$654$	2.58041	0.100902
$655$	0	0
$656$	30.5142	1.19138
$657$	7.58041i	0.295740i
$658$	69.4236i	2.70641i
$659$	29.9725	1.16756	0.583782	−	0.811910i	$-0.301572\pi$
$659$	29.9725	1.16756	0.583782	+	0.811910i	$0.301572\pi$
$660$	0	0
$661$	−42.5530	−1.65512	−0.827559	−	0.561379i	$-0.810271\pi$
$661$	−42.5530	−1.65512	−0.827559	+	0.561379i	$0.810271\pi$
$662$	− 55.6349i	− 2.16231i
$663$	0	0
$664$	2.90739	0.112829
$665$	0	0
$666$	11.1608	0.432473
$667$	1.63091i	0.0631491i
$668$	24.2638i	0.938795i
$669$	10.5436	0.407641
$670$	0	0
$671$	31.8201	1.22840
$672$	− 21.4353i	− 0.826885i
$673$	− 12.6961i	− 0.489397i	−0.969599	−	0.244699i	$-0.921311\pi$
$673$	− 12.6961i	− 0.489397i	0.969599	−	0.244699i	$-0.0786891\pi$
$674$	−32.3638	−1.24661
$675$	0	0
$676$	69.1123	2.65817
$677$	− 21.3784i	− 0.821639i	−0.911717	−	0.410819i	$-0.865243\pi$
$677$	− 21.3784i	− 0.821639i	0.911717	−	0.410819i	$-0.134757\pi$
$678$	26.9074i	1.03337i
$679$	46.3363	1.77822
$680$	0	0
$681$	7.83384	0.300193
$682$	− 11.7045i	− 0.448187i
$683$	− 24.0652i	− 0.920828i	−0.887704	−	0.460414i	$-0.847701\pi$
$683$	− 24.0652i	− 0.920828i	0.887704	−	0.460414i	$-0.152299\pi$
$684$	−2.60147	−0.0994695
$685$	0	0
$686$	−38.0388	−1.45233
$687$	− 23.6035i	− 0.900528i
$688$	19.4825i	0.742762i
$689$	35.1019	1.33728
$690$	0	0
$691$	29.6677	1.12861	0.564306	−	0.825566i	$-0.309144\pi$
$691$	29.6677	1.12861	0.564306	+	0.825566i	$0.309144\pi$
$692$	− 13.4614i	− 0.511726i
$693$	10.2902i	0.390893i
$694$	14.2060	0.539252
$695$	0	0
$696$	3.87062	0.146715
$697$	0	0
$698$	− 19.4637i	− 0.736712i
$699$	23.7833	0.899569
$700$	0	0
$701$	6.65396	0.251317	0.125658	−	0.992074i	$-0.459896\pi$
$701$	6.65396	0.251317	0.125658	+	0.992074i	$0.459896\pi$
$702$	13.4931i	0.509266i
$703$	− 5.20293i	− 0.196232i
$704$	−44.4741	−1.67618
$705$	0	0
$706$	43.2176	1.62652
$707$	− 36.4648i	− 1.37140i
$708$	− 13.6666i	− 0.513623i
$709$	29.0735	1.09188	0.545940	−	0.837824i	$-0.316173\pi$
$709$	29.0735	1.09188	0.545940	+	0.837824i	$0.316173\pi$
$710$	0	0
$711$	−7.52991	−0.282394
$712$	5.79707i	0.217254i
$713$	− 0.791733i	− 0.0296506i
$714$	0	0
$715$	0	0
$716$	39.8726	1.49011
$717$	− 1.27648i	− 0.0476710i
$718$	− 34.0735i	− 1.27161i
$719$	−12.6739	−0.472659	−0.236329	−	0.971673i	$-0.575944\pi$
$719$	−12.6739	−0.472659	−0.236329	+	0.971673i	$0.575944\pi$
$720$	0	0
$721$	−0.696068	−0.0259229
$722$	2.14510i	0.0798325i
$723$	− 21.4931i	− 0.799338i
$724$	27.0705	1.00607
$725$	0	0
$726$	6.51419	0.241764
$727$	13.2260i	0.490524i	0.969457	+	0.245262i	$0.0788740\pi$
$727$	13.2260i	0.490524i	−0.969457	+	0.245262i	$0.921126\pi$
$728$	− 22.2902i	− 0.826130i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	− 22.0946i	− 0.816640i
$733$	− 33.0735i	− 1.22160i	−0.791785	−	0.610800i	$-0.790848\pi$
$733$	− 33.0735i	− 1.22160i	0.791785	−	0.610800i	$-0.209152\pi$
$734$	−53.3815	−1.97035
$735$	0	0
$736$	−4.24276	−0.156390
$737$	10.6172i	0.391089i
$738$	26.8779i	0.989391i
$739$	27.2260	1.00152	0.500762	−	0.865585i	$-0.333053\pi$
$739$	27.2260	1.00152	0.500762	+	0.865585i	$0.333053\pi$
$740$	0	0
$741$	6.29021	0.231076
$742$	− 32.8779i	− 1.20699i
$743$	− 51.5751i	− 1.89211i	−0.324012	−	0.946053i	$-0.605032\pi$
$743$	− 51.5751i	− 1.89211i	0.324012	−	0.946053i	$-0.394968\pi$
$744$	−1.87901	−0.0688877
$745$	0	0
$746$	25.0284	0.916354
$747$	− 2.25343i	− 0.0824488i
$748$	0	0
$749$	−6.15150	−0.224771
$750$	0	0
$751$	16.7971	0.612934	0.306467	−	0.951881i	$-0.400853\pi$
$751$	16.7971	0.612934	0.306467	+	0.951881i	$0.400853\pi$
$752$	28.6961i	1.04644i
$753$	29.9725i	1.09226i
$754$	−40.4794	−1.47417
$755$	0	0
$756$	7.14510	0.259865
$757$	− 34.6402i	− 1.25902i	−0.776992	−	0.629510i	$-0.783256\pi$
$757$	− 34.6402i	− 1.25902i	0.776992	−	0.629510i	$-0.216744\pi$
$758$	− 13.4931i	− 0.490093i
$759$	2.03677	0.0739302
$760$	0	0
$761$	33.8294	1.22632	0.613158	−	0.789960i	$-0.289899\pi$
$761$	33.8294	1.22632	0.613158	+	0.789960i	$0.289899\pi$
$762$	− 26.6907i	− 0.966903i
$763$	3.30393i	0.119610i
$764$	17.0230	0.615872
$765$	0	0
$766$	25.8810	0.935119
$767$	33.0452i	1.19319i
$768$	− 2.60147i	− 0.0938723i
$769$	−21.7550	−0.784504	−0.392252	−	0.919858i	$-0.628304\pi$
$769$	−21.7550	−0.784504	−0.392252	+	0.919858i	$0.628304\pi$
$770$	0	0
$771$	19.0735	0.686917
$772$	− 65.0009i	− 2.33943i
$773$	1.77495i	0.0638405i	0.999490	+	0.0319203i	$0.0101623\pi$
$773$	1.77495i	0.0638405i	−0.999490	+	0.0319203i	$0.989838\pi$
$774$	−17.1608	−0.616833
$775$	0	0
$776$	−21.7666	−0.781374
$777$	14.2902i	0.512658i
$778$	30.8117i	1.10465i
$779$	12.5299	0.448931
$780$	0	0
$781$	−29.3500	−1.05023
$782$	0	0
$783$	− 3.00000i	− 0.107211i
$784$	1.32392	0.0472830
$785$	0	0
$786$	40.6486	1.44989
$787$	19.8937i	0.709132i	0.935031	+	0.354566i	$0.115371\pi$
$787$	19.8937i	0.709132i	−0.935031	+	0.354566i	$0.884629\pi$
$788$	− 22.8853i	− 0.815254i
$789$	−11.6456	−0.414593
$790$	0	0
$791$	−34.4520	−1.22497
$792$	− 4.83384i	− 0.171763i
$793$	53.4236i	1.89713i
$794$	66.7054	2.36729
$795$	0	0
$796$	22.7539	0.806490
$797$	− 7.36909i	− 0.261027i	−0.991447	−	0.130513i	$-0.958337\pi$
$797$	− 7.36909i	− 0.261027i	0.991447	−	0.130513i	$-0.0416625\pi$
$798$	− 5.89167i	− 0.208563i
$799$	0	0
$800$	0	0
$801$	4.49314	0.158757
$802$	− 1.16616i	− 0.0411785i
$803$	28.4005i	1.00223i
$804$	7.37214	0.259995
$805$	0	0
$806$	19.6509	0.692174
$807$	26.1471i	0.920421i
$808$	17.1294i	0.602610i
$809$	10.4205	0.366366	0.183183	−	0.983079i	$-0.441360\pi$
$809$	10.4205	0.366366	0.183183	+	0.983079i	$0.441360\pi$
$810$	0	0
$811$	46.9074	1.64714	0.823571	−	0.567214i	$-0.191979\pi$
$811$	46.9074	1.64714	0.823571	+	0.567214i	$0.191979\pi$
$812$	21.4353i	0.752232i
$813$	− 15.8338i	− 0.555317i
$814$	41.8148	1.46561
$815$	0	0
$816$	0	0
$817$	8.00000i	0.279885i
$818$	− 56.7696i	− 1.98490i
$819$	−17.2765	−0.603689
$820$	0	0
$821$	35.5392	1.24033	0.620164	−	0.784472i	$-0.287066\pi$
$821$	35.5392	1.24033	0.620164	+	0.784472i	$0.287066\pi$
$822$	45.8883i	1.60054i
$823$	− 17.7603i	− 0.619085i	−0.950886	−	0.309542i	$-0.899824\pi$
$823$	− 17.7603i	− 0.619085i	0.950886	−	0.309542i	$-0.100176\pi$
$824$	0.326979	0.0113909
$825$	0	0
$826$	30.9515	1.07694
$827$	− 30.7843i	− 1.07047i	−0.844702	−	0.535237i	$-0.820222\pi$
$827$	− 30.7843i	− 1.07047i	0.844702	−	0.535237i	$-0.179778\pi$
$828$	− 1.41425i	− 0.0491487i
$829$	23.7098	0.823475	0.411738	−	0.911302i	$-0.364922\pi$
$829$	23.7098	0.823475	0.411738	+	0.911302i	$0.364922\pi$
$830$	0	0
$831$	20.6677	0.716954
$832$	− 74.6686i	− 2.58867i
$833$	0	0
$834$	−43.5740	−1.50884
$835$	0	0
$836$	−9.74657	−0.337092
$837$	1.45636i	0.0503393i
$838$	13.2765i	0.458628i
$839$	−34.1662	−1.17955	−0.589773	−	0.807569i	$-0.700783\pi$
$839$	−34.1662	−1.17955	−0.589773	+	0.807569i	$0.700783\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	32.7696i	1.12932i
$843$	− 1.35536i	− 0.0466811i
$844$	33.0859	1.13886
$845$	0	0
$846$	−25.2765	−0.869023
$847$	8.34071i	0.286590i
$848$	− 13.5900i	− 0.466683i
$849$	−28.1471	−0.966006
$850$	0	0
$851$	2.82851	0.0969600
$852$	20.3795i	0.698189i
$853$	28.7780i	0.985340i	0.870216	+	0.492670i	$0.163979\pi$
$853$	28.7780i	0.985340i	−0.870216	+	0.492670i	$0.836021\pi$
$854$	50.0388	1.71229
$855$	0	0
$856$	2.88968	0.0987672
$857$	− 47.2849i	− 1.61522i	−0.589717	−	0.807610i	$-0.700761\pi$
$857$	− 47.2849i	− 1.61522i	0.589717	−	0.807610i	$-0.299239\pi$
$858$	50.5530i	1.72585i
$859$	23.0221	0.785505	0.392752	−	0.919644i	$-0.371523\pi$
$859$	23.0221	0.785505	0.392752	+	0.919644i	$0.371523\pi$
$860$	0	0
$861$	−34.4143	−1.17283
$862$	29.0839i	0.990603i
$863$	− 56.0545i	− 1.90812i	−0.299621	−	0.954058i	$-0.596860\pi$
$863$	− 56.0545i	− 1.90812i	0.299621	−	0.954058i	$-0.403140\pi$
$864$	7.80440	0.265511
$865$	0	0
$866$	−63.6717	−2.16365
$867$	− 17.0000i	− 0.577350i
$868$	− 10.4059i	− 0.353198i
$869$	−28.2113	−0.957004
$870$	0	0
$871$	−17.8255	−0.603992
$872$	− 1.55203i	− 0.0525583i
$873$	16.8706i	0.570984i
$874$	−1.16616	−0.0394459
$875$	0	0
$876$	19.7202	0.666283
$877$	− 3.37748i	− 0.114049i	−0.998373	−	0.0570247i	$-0.981839\pi$
$877$	− 3.37748i	− 0.114049i	0.998373	−	0.0570247i	$-0.0181614\pi$
$878$	− 72.9535i	− 2.46206i
$879$	8.47009	0.285689
$880$	0	0
$881$	36.0314	1.21393	0.606965	−	0.794729i	$-0.292387\pi$
$881$	36.0314	1.21393	0.606965	+	0.794729i	$0.292387\pi$
$882$	1.16616i	0.0392666i
$883$	6.23970i	0.209983i	0.994473	+	0.104991i	$0.0334815\pi$
$883$	6.23970i	0.209983i	−0.994473	+	0.104991i	$0.966518\pi$
$884$	0	0
$885$	0	0
$886$	−35.0545	−1.17768
$887$	40.2942i	1.35295i	0.736467	+	0.676473i	$0.236493\pi$
$887$	40.2942i	1.35295i	−0.736467	+	0.676473i	$0.763507\pi$
$888$	− 6.71285i	− 0.225268i
$889$	34.1745	1.14618
$890$	0	0
$891$	−3.74657	−0.125515
$892$	− 27.4289i	− 0.918388i
$893$	11.7833i	0.394315i
$894$	−10.0735	−0.336910
$895$	0	0
$896$	−27.0671	−0.904250
$897$	3.41959i	0.114177i
$898$	− 36.2501i	− 1.20968i
$899$	−4.36909	−0.145717
$900$	0	0
$901$	0	0
$902$	100.700i	3.35294i
$903$	− 21.9725i	− 0.731201i
$904$	16.1839	0.538267
$905$	0	0
$906$	3.41959	0.113608
$907$	1.59414i	0.0529325i	0.999650	+	0.0264662i	$0.00842545\pi$
$907$	1.59414i	0.0529325i	−0.999650	+	0.0264662i	$0.991575\pi$
$908$	− 20.3795i	− 0.676317i
$909$	13.2765	0.440353
$910$	0	0
$911$	−6.58880	−0.218297	−0.109148	−	0.994025i	$-0.534812\pi$
$911$	−6.58880	−0.218297	−0.109148	+	0.994025i	$0.534812\pi$
$912$	− 2.43531i	− 0.0806411i
$913$	− 8.44264i	− 0.279410i
$914$	−36.1712	−1.19644
$915$	0	0
$916$	−61.4036	−2.02883
$917$	52.0461i	1.71871i
$918$	0	0
$919$	13.3270	0.439616	0.219808	−	0.975543i	$-0.429457\pi$
$919$	13.3270	0.439616	0.219808	+	0.975543i	$0.429457\pi$
$920$	0	0
$921$	−19.3133	−0.636393
$922$	− 52.4794i	− 1.72832i
$923$	− 49.2765i	− 1.62196i
$924$	26.7696	0.880656
$925$	0	0
$926$	−31.6509	−1.04011
$927$	− 0.253432i	− 0.00832379i
$928$	23.4132i	0.768576i
$929$	22.0735	0.724210	0.362105	−	0.932137i	$-0.382058\pi$
$929$	22.0735	0.724210	0.362105	+	0.932137i	$0.382058\pi$
$930$	0	0
$931$	0.543637	0.0178170
$932$	− 61.8715i	− 2.02667i
$933$	− 20.5804i	− 0.673772i
$934$	−26.4426	−0.865229
$935$	0	0
$936$	8.11566	0.265269
$937$	− 13.5436i	− 0.442451i	−0.975223	−	0.221226i	$-0.928994\pi$
$937$	− 13.5436i	− 0.442451i	0.975223	−	0.221226i	$-0.0710057\pi$
$938$	16.6961i	0.545146i
$939$	−22.4426	−0.732388
$940$	0	0
$941$	3.96323	0.129197	0.0645987	−	0.997911i	$-0.479423\pi$
$941$	3.96323	0.129197	0.0645987	+	0.997911i	$0.479423\pi$
$942$	− 28.1230i	− 0.916296i
$943$	6.81172i	0.221820i
$944$	12.7937	0.416400
$945$	0	0
$946$	−64.2942	−2.09038
$947$	24.3784i	0.792192i	0.918209	+	0.396096i	$0.129635\pi$
$947$	24.3784i	0.792192i	−0.918209	+	0.396096i	$0.870365\pi$
$948$	19.5888i	0.636215i
$949$	−47.6823	−1.54783
$950$	0	0
$951$	11.1745	0.362360
$952$	0	0
$953$	41.4226i	1.34181i	0.741543	+	0.670906i	$0.234094\pi$
$953$	41.4226i	1.34181i	−0.741543	+	0.670906i	$0.765906\pi$
$954$	11.9706	0.387561
$955$	0	0
$956$	−3.32071	−0.107400
$957$	− 11.2397i	− 0.363328i
$958$	40.7750i	1.31738i
$959$	−58.7550	−1.89730
$960$	0	0
$961$	−28.8790	−0.931581
$962$	70.2039i	2.26346i
$963$	− 2.23970i	− 0.0721735i
$964$	−55.9137	−1.80086
$965$	0	0
$966$	3.20293	0.103053
$967$	− 34.5888i	− 1.11230i	−0.831082	−	0.556150i	$-0.812278\pi$
$967$	− 34.5888i	− 1.11230i	0.831082	−	0.556150i	$-0.187722\pi$
$968$	− 3.91806i	− 0.125931i
$969$	0	0
$970$	0	0
$971$	9.48475	0.304380	0.152190	−	0.988351i	$-0.451367\pi$
$971$	9.48475	0.304380	0.152190	+	0.988351i	$0.451367\pi$
$972$	2.60147i	0.0834420i
$973$	− 55.7917i	− 1.78860i
$974$	−71.3815	−2.28721
$975$	0	0
$976$	20.6834	0.662060
$977$	− 46.7275i	− 1.49495i	−0.664293	−	0.747473i	$-0.731267\pi$
$977$	− 46.7275i	− 1.49495i	0.664293	−	0.747473i	$-0.268733\pi$
$978$	16.5877i	0.530417i
$979$	16.8338	0.538012
$980$	0	0
$981$	−1.20293	−0.0384066
$982$	61.6234i	1.96648i
$983$	21.2344i	0.677271i	0.940918	+	0.338636i	$0.109965\pi$
$983$	21.2344i	0.677271i	−0.940918	+	0.338636i	$0.890035\pi$
$984$	16.1662	0.515358
$985$	0	0
$986$	0	0
$987$	− 32.3638i	− 1.03015i
$988$	− 16.3638i	− 0.520600i
$989$	−4.34910	−0.138293
$990$	0	0
$991$	29.7466	0.944931	0.472465	−	0.881349i	$-0.343364\pi$
$991$	29.7466	0.944931	0.472465	+	0.881349i	$0.343364\pi$
$992$	− 11.3660i	− 0.360872i
$993$	25.9358i	0.823047i
$994$	−46.1544	−1.46393
$995$	0	0
$996$	−5.86223	−0.185752
$997$	4.84850i	0.153553i	0.997048	+	0.0767767i	$0.0244629\pi$
$997$	4.84850i	0.153553i	−0.997048	+	0.0767767i	$0.975537\pi$
$998$	52.5288i	1.66277i
$999$	−5.20293	−0.164613

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1425.2.c.p.799.5		6
5.2	odd	4		1425.2.a.u.1.1	✓	3
5.3	odd	4		1425.2.a.v.1.3	yes	3
5.4	even	2	inner	1425.2.c.p.799.2		6
15.2	even	4		4275.2.a.be.1.3		3
15.8	even	4		4275.2.a.bh.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1425.2.a.u.1.1	✓	3	5.2	odd	4
1425.2.a.v.1.3	yes	3	5.3	odd	4
1425.2.c.p.799.2		6	5.4	even	2	inner
1425.2.c.p.799.5		6	1.1	even	1	trivial
4275.2.a.be.1.3		3	15.2	even	4
4275.2.a.bh.1.1		3	15.8	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1425.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.14510i q^{2} -1.00000i q^{3} -2.60147 q^{4} +2.14510 q^{6} +2.74657i q^{7} -1.29021i q^{8} -1.00000 q^{9} -3.74657 q^{11} +2.60147i q^{12} -6.29021i q^{13} -5.89167 q^{14} -2.43531 q^{16} -2.14510i q^{18} -1.00000 q^{19} +2.74657 q^{21} -8.03677i q^{22} -0.543637i q^{23} -1.29021 q^{24} +13.4931 q^{26} +1.00000i q^{27} -7.14510i q^{28} -3.00000 q^{29} +1.45636 q^{31} -7.80440i q^{32} +3.74657i q^{33} +2.60147 q^{36} +5.20293i q^{37} -2.14510i q^{38} -6.29021 q^{39} -12.5299 q^{41} +5.89167i q^{42} -8.00000i q^{43} +9.74657 q^{44} +1.16616 q^{46} -11.7833i q^{47} +2.43531i q^{48} -0.543637 q^{49} +16.3638i q^{52} +5.58041i q^{53} -2.14510 q^{54} +3.54364 q^{56} +1.00000i q^{57} -6.43531i q^{58} -5.25343 q^{59} -8.49314 q^{61} +3.12405i q^{62} -2.74657i q^{63} +11.8706 q^{64} -8.03677 q^{66} -2.83384i q^{67} -0.543637 q^{69} +7.83384 q^{71} +1.29021i q^{72} -7.58041i q^{73} -11.1608 q^{74} +2.60147 q^{76} -10.2902i q^{77} -13.4931i q^{78} +7.52991 q^{79} +1.00000 q^{81} -26.8779i q^{82} +2.25343i q^{83} -7.14510 q^{84} +17.1608 q^{86} +3.00000i q^{87} +4.83384i q^{88} -4.49314 q^{89} +17.2765 q^{91} +1.41425i q^{92} -1.45636i q^{93} +25.2765 q^{94} -7.80440 q^{96} -16.8706i q^{97} -1.16616i q^{98} +3.74657 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{4} - 6 q^{9} - 6 q^{11} - 6 q^{14} + 24 q^{16} - 6 q^{19} + 18 q^{24} + 48 q^{26} - 18 q^{29} + 18 q^{31} + 12 q^{36} - 12 q^{39} + 42 q^{44} + 42 q^{46} + 6 q^{49} + 12 q^{56} - 48 q^{59} - 18 q^{61}+ \cdots + 6 q^{99}+O(q^{100}) \) 6 * q - 12 * q^4 - 6 * q^9 - 6 * q^11 - 6 * q^14 + 24 * q^16 - 6 * q^19 + 18 * q^24 + 48 * q^26 - 18 * q^29 + 18 * q^31 + 12 * q^36 - 12 * q^39 + 42 * q^44 + 42 * q^46 + 6 * q^49 + 12 * q^56 - 48 * q^59 - 18 * q^61 - 6 * q^64 - 6 * q^66 + 6 * q^69 + 12 * q^71 + 36 * q^74 + 12 * q^76 - 30 * q^79 + 6 * q^81 - 30 * q^84 + 6 * q^89 + 12 * q^91 + 60 * q^94 - 36 * q^96 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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