LMFDB - Embedded newform 825.2.c.c.199.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("825.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	825.c (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:	$6.58765816676$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.4
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.2.c.c.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.73205 q^{6} -2.00000i q^{7} +1.73205i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.73205i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.73205 q^{6} -2.00000i q^{7} +1.73205i q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.00000i q^{12} +5.46410i q^{13} +3.46410 q^{14} -5.00000 q^{16} -1.73205i q^{18} -5.46410 q^{19} +2.00000 q^{21} -1.73205i q^{22} +6.92820i q^{23} -1.73205 q^{24} -9.46410 q^{26} -1.00000i q^{27} +2.00000i q^{28} +3.46410 q^{29} -10.9282 q^{31} -5.19615i q^{32} -1.00000i q^{33} +1.00000 q^{36} +4.92820i q^{37} -9.46410i q^{38} -5.46410 q^{39} +3.46410 q^{41} +3.46410i q^{42} -4.92820i q^{43} +1.00000 q^{44} -12.0000 q^{46} +6.92820i q^{47} -5.00000i q^{48} +3.00000 q^{49} -5.46410i q^{52} +0.928203i q^{53} +1.73205 q^{54} +3.46410 q^{56} -5.46410i q^{57} +6.00000i q^{58} +6.92820 q^{59} +2.00000 q^{61} -18.9282i q^{62} +2.00000i q^{63} -1.00000 q^{64} +1.73205 q^{66} -8.00000i q^{67} -6.92820 q^{69} +13.8564 q^{71} -1.73205i q^{72} -8.39230i q^{73} -8.53590 q^{74} +5.46410 q^{76} +2.00000i q^{77} -9.46410i q^{78} +6.53590 q^{79} +1.00000 q^{81} +6.00000i q^{82} +8.53590i q^{83} -2.00000 q^{84} +8.53590 q^{86} +3.46410i q^{87} -1.73205i q^{88} -0.928203 q^{89} +10.9282 q^{91} -6.92820i q^{92} -10.9282i q^{93} -12.0000 q^{94} +5.19615 q^{96} +10.0000i q^{97} +5.19615i q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 - 4 * q^9	$ 4 q - 4 q^{4} - 4 q^{9} - 4 q^{11} - 20 q^{16} - 8 q^{19} + 8 q^{21} - 24 q^{26} - 16 q^{31} + 4 q^{36} - 8 q^{39} + 4 q^{44} - 48 q^{46} + 12 q^{49} + 8 q^{61} - 4 q^{64} - 48 q^{74} + 8 q^{76} + 40 q^{79} + 4 q^{81} - 8 q^{84} + 48 q^{86} + 24 q^{89} + 16 q^{91} - 48 q^{94} + 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^9 - 4 * q^11 - 20 * q^16 - 8 * q^19 + 8 * q^21 - 24 * q^26 - 16 * q^31 + 4 * q^36 - 8 * q^39 + 4 * q^44 - 48 * q^46 + 12 * q^49 + 8 * q^61 - 4 * q^64 - 48 * q^74 + 8 * q^76 + 40 * q^79 + 4 * q^81 - 8 * q^84 + 48 * q^86 + 24 * q^89 + 16 * q^91 - 48 * q^94 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	1.73205i	1.22474i	0.790569	+	0.612372i	$0.209785\pi$
$2$	1.73205i	1.22474i	−0.790569	+	0.612372i	$0.790215\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.73205	−0.707107
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	1.73205i	0.612372i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	− 1.00000i	− 0.288675i
$13$	5.46410i	1.51547i	0.652563	+	0.757735i	$0.273694\pi$
$13$	5.46410i	1.51547i	−0.652563	+	0.757735i	$0.726306\pi$
$14$	3.46410	0.925820
$15$	0	0
$16$	−5.00000	−1.25000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	− 1.73205i	− 0.408248i
$19$	−5.46410	−1.25355	−0.626775	−	0.779200i	$-0.715626\pi$
$19$	−5.46410	−1.25355	−0.626775	+	0.779200i	$0.715626\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	− 1.73205i	− 0.369274i
$23$	6.92820i	1.44463i	0.691564	+	0.722315i	$0.256922\pi$
$23$	6.92820i	1.44463i	−0.691564	+	0.722315i	$0.743078\pi$
$24$	−1.73205	−0.353553
$25$	0	0
$26$	−9.46410	−1.85606
$27$	− 1.00000i	− 0.192450i
$28$	2.00000i	0.377964i
$29$	3.46410	0.643268	0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$30$	0	0
$31$	−10.9282	−1.96276	−0.981382	−	0.192068i	$-0.938481\pi$
$31$	−10.9282	−1.96276	−0.981382	+	0.192068i	$0.938481\pi$
$32$	− 5.19615i	− 0.918559i
$33$	− 1.00000i	− 0.174078i
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	4.92820i	0.810192i	0.914274	+	0.405096i	$0.132762\pi$
$37$	4.92820i	0.810192i	−0.914274	+	0.405096i	$0.867238\pi$
$38$	− 9.46410i	− 1.53528i
$39$	−5.46410	−0.874957
$40$	0	0
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	3.46410i	0.534522i
$43$	− 4.92820i	− 0.751544i	−0.926712	−	0.375772i	$-0.877378\pi$
$43$	− 4.92820i	− 0.751544i	0.926712	−	0.375772i	$-0.122622\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−12.0000	−1.76930
$47$	6.92820i	1.01058i	0.862949	+	0.505291i	$0.168615\pi$
$47$	6.92820i	1.01058i	−0.862949	+	0.505291i	$0.831385\pi$
$48$	− 5.00000i	− 0.721688i
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	− 5.46410i	− 0.757735i
$53$	0.928203i	0.127499i	0.997966	+	0.0637493i	$0.0203058\pi$
$53$	0.928203i	0.127499i	−0.997966	+	0.0637493i	$0.979694\pi$
$54$	1.73205	0.235702
$55$	0	0
$56$	3.46410	0.462910
$57$	− 5.46410i	− 0.723738i
$58$	6.00000i	0.787839i
$59$	6.92820	0.901975	0.450988	−	0.892530i	$-0.351072\pi$
$59$	6.92820	0.901975	0.450988	+	0.892530i	$0.351072\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	− 18.9282i	− 2.40388i
$63$	2.00000i	0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	1.73205	0.213201
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	0	0
$69$	−6.92820	−0.834058
$70$	0	0
$71$	13.8564	1.64445	0.822226	−	0.569160i	$-0.192732\pi$
$71$	13.8564	1.64445	0.822226	+	0.569160i	$0.192732\pi$
$72$	− 1.73205i	− 0.204124i
$73$	− 8.39230i	− 0.982245i	−0.871091	−	0.491122i	$-0.836587\pi$
$73$	− 8.39230i	− 0.982245i	0.871091	−	0.491122i	$-0.163413\pi$
$74$	−8.53590	−0.992278
$75$	0	0
$76$	5.46410	0.626775
$77$	2.00000i	0.227921i
$78$	− 9.46410i	− 1.07160i
$79$	6.53590	0.735346	0.367673	−	0.929955i	$-0.380155\pi$
$79$	6.53590	0.735346	0.367673	+	0.929955i	$0.380155\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000i	0.662589i
$83$	8.53590i	0.936937i	0.883480	+	0.468468i	$0.155194\pi$
$83$	8.53590i	0.936937i	−0.883480	+	0.468468i	$0.844806\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	8.53590	0.920450
$87$	3.46410i	0.371391i
$88$	− 1.73205i	− 0.184637i
$89$	−0.928203	−0.0983893	−0.0491947	−	0.998789i	$-0.515665\pi$
$89$	−0.928203	−0.0983893	−0.0491947	+	0.998789i	$0.515665\pi$
$90$	0	0
$91$	10.9282	1.14559
$92$	− 6.92820i	− 0.722315i
$93$	− 10.9282i	− 1.13320i
$94$	−12.0000	−1.23771
$95$	0	0
$96$	5.19615	0.530330
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	5.19615i	0.524891i
$99$	1.00000	0.100504
$100$	0	0
$101$	−10.3923	−1.03407	−0.517036	−	0.855963i	$-0.672965\pi$
$101$	−10.3923	−1.03407	−0.517036	+	0.855963i	$0.672965\pi$
$102$	0	0
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	−9.46410	−0.928032
$105$	0	0
$106$	−1.60770	−0.156153
$107$	− 8.53590i	− 0.825196i	−0.910913	−	0.412598i	$-0.864621\pi$
$107$	− 8.53590i	− 0.825196i	0.910913	−	0.412598i	$-0.135379\pi$
$108$	1.00000i	0.0962250i
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−4.92820	−0.467764
$112$	10.0000i	0.944911i
$113$	− 12.9282i	− 1.21618i	−0.793867	−	0.608092i	$-0.791935\pi$
$113$	− 12.9282i	− 1.21618i	0.793867	−	0.608092i	$-0.208065\pi$
$114$	9.46410	0.886394
$115$	0	0
$116$	−3.46410	−0.321634
$117$	− 5.46410i	− 0.505156i
$118$	12.0000i	1.10469i
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	3.46410i	0.313625i
$123$	3.46410i	0.312348i
$124$	10.9282	0.981382
$125$	0	0
$126$	−3.46410	−0.308607
$127$	− 8.92820i	− 0.792250i	−0.918197	−	0.396125i	$-0.870355\pi$
$127$	− 8.92820i	− 0.792250i	0.918197	−	0.396125i	$-0.129645\pi$
$128$	− 12.1244i	− 1.07165i
$129$	4.92820	0.433904
$130$	0	0
$131$	18.9282	1.65376	0.826882	−	0.562375i	$-0.190112\pi$
$131$	18.9282	1.65376	0.826882	+	0.562375i	$0.190112\pi$
$132$	1.00000i	0.0870388i
$133$	10.9282i	0.947595i
$134$	13.8564	1.19701
$135$	0	0
$136$	0	0
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	− 12.0000i	− 1.02151i
$139$	−12.3923	−1.05110	−0.525551	−	0.850762i	$-0.676141\pi$
$139$	−12.3923	−1.05110	−0.525551	+	0.850762i	$0.676141\pi$
$140$	0	0
$141$	−6.92820	−0.583460
$142$	24.0000i	2.01404i
$143$	− 5.46410i	− 0.456931i
$144$	5.00000	0.416667
$145$	0	0
$146$	14.5359	1.20300
$147$	3.00000i	0.247436i
$148$	− 4.92820i	− 0.405096i
$149$	15.4641	1.26687	0.633434	−	0.773796i	$-0.281645\pi$
$149$	15.4641	1.26687	0.633434	+	0.773796i	$0.281645\pi$
$150$	0	0
$151$	−20.3923	−1.65950	−0.829751	−	0.558134i	$-0.811518\pi$
$151$	−20.3923	−1.65950	−0.829751	+	0.558134i	$0.811518\pi$
$152$	− 9.46410i	− 0.767640i
$153$	0	0
$154$	−3.46410	−0.279145
$155$	0	0
$156$	5.46410	0.437478
$157$	3.07180i	0.245156i	0.992459	+	0.122578i	$0.0391162\pi$
$157$	3.07180i	0.245156i	−0.992459	+	0.122578i	$0.960884\pi$
$158$	11.3205i	0.900611i
$159$	−0.928203	−0.0736113
$160$	0	0
$161$	13.8564	1.09204
$162$	1.73205i	0.136083i
$163$	9.85641i	0.772013i	0.922496	+	0.386007i	$0.126146\pi$
$163$	9.85641i	0.772013i	−0.922496	+	0.386007i	$0.873854\pi$
$164$	−3.46410	−0.270501
$165$	0	0
$166$	−14.7846	−1.14751
$167$	10.3923i	0.804181i	0.915600	+	0.402090i	$0.131716\pi$
$167$	10.3923i	0.804181i	−0.915600	+	0.402090i	$0.868284\pi$
$168$	3.46410i	0.267261i
$169$	−16.8564	−1.29665
$170$	0	0
$171$	5.46410	0.417850
$172$	4.92820i	0.375772i
$173$	− 12.0000i	− 0.912343i	−0.889892	−	0.456172i	$-0.849220\pi$
$173$	− 12.0000i	− 0.912343i	0.889892	−	0.456172i	$-0.150780\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	5.00000	0.376889
$177$	6.92820i	0.520756i
$178$	− 1.60770i	− 0.120502i
$179$	−6.92820	−0.517838	−0.258919	−	0.965899i	$-0.583366\pi$
$179$	−6.92820	−0.517838	−0.258919	+	0.965899i	$0.583366\pi$
$180$	0	0
$181$	15.8564	1.17860	0.589299	−	0.807915i	$-0.299404\pi$
$181$	15.8564	1.17860	0.589299	+	0.807915i	$0.299404\pi$
$182$	18.9282i	1.40305i
$183$	2.00000i	0.147844i
$184$	−12.0000	−0.884652
$185$	0	0
$186$	18.9282	1.38788
$187$	0	0
$188$	− 6.92820i	− 0.505291i
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−18.9282	−1.36960	−0.684798	−	0.728733i	$-0.740110\pi$
$191$	−18.9282	−1.36960	−0.684798	+	0.728733i	$0.740110\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	24.3923i	1.75580i	0.478847	+	0.877898i	$0.341055\pi$
$193$	24.3923i	1.75580i	−0.478847	+	0.877898i	$0.658945\pi$
$194$	−17.3205	−1.24354
$195$	0	0
$196$	−3.00000	−0.214286
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	1.73205i	0.123091i
$199$	24.7846	1.75693	0.878467	−	0.477803i	$-0.158567\pi$
$199$	24.7846	1.75693	0.878467	+	0.477803i	$0.158567\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	− 18.0000i	− 1.26648i
$203$	− 6.92820i	− 0.486265i
$204$	0	0
$205$	0	0
$206$	−13.8564	−0.965422
$207$	− 6.92820i	− 0.481543i
$208$	− 27.3205i	− 1.89434i
$209$	5.46410	0.377960
$210$	0	0
$211$	−8.39230	−0.577750	−0.288875	−	0.957367i	$-0.593281\pi$
$211$	−8.39230	−0.577750	−0.288875	+	0.957367i	$0.593281\pi$
$212$	− 0.928203i	− 0.0637493i
$213$	13.8564i	0.949425i
$214$	14.7846	1.01066
$215$	0	0
$216$	1.73205	0.117851
$217$	21.8564i	1.48371i
$218$	17.3205i	1.17309i
$219$	8.39230	0.567099
$220$	0	0
$221$	0	0
$222$	− 8.53590i	− 0.572892i
$223$	9.85641i	0.660034i	0.943975	+	0.330017i	$0.107054\pi$
$223$	9.85641i	0.660034i	−0.943975	+	0.330017i	$0.892946\pi$
$224$	−10.3923	−0.694365
$225$	0	0
$226$	22.3923	1.48951
$227$	− 15.4641i	− 1.02639i	−0.858272	−	0.513194i	$-0.828462\pi$
$227$	− 15.4641i	− 1.02639i	0.858272	−	0.513194i	$-0.171538\pi$
$228$	5.46410i	0.361869i
$229$	23.8564	1.57648	0.788238	−	0.615371i	$-0.210994\pi$
$229$	23.8564	1.57648	0.788238	+	0.615371i	$0.210994\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	6.00000i	0.393919i
$233$	12.0000i	0.786146i	0.919507	+	0.393073i	$0.128588\pi$
$233$	12.0000i	0.786146i	−0.919507	+	0.393073i	$0.871412\pi$
$234$	9.46410	0.618688
$235$	0	0
$236$	−6.92820	−0.450988
$237$	6.53590i	0.424552i
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	0.143594	0.00924967	0.00462484	−	0.999989i	$-0.498528\pi$
$241$	0.143594	0.00924967	0.00462484	+	0.999989i	$0.498528\pi$
$242$	1.73205i	0.111340i
$243$	1.00000i	0.0641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	−6.00000	−0.382546
$247$	− 29.8564i	− 1.89972i
$248$	− 18.9282i	− 1.20194i
$249$	−8.53590	−0.540941
$250$	0	0
$251$	−1.85641	−0.117175	−0.0585877	−	0.998282i	$-0.518660\pi$
$251$	−1.85641	−0.117175	−0.0585877	+	0.998282i	$0.518660\pi$
$252$	− 2.00000i	− 0.125988i
$253$	− 6.92820i	− 0.435572i
$254$	15.4641	0.970304
$255$	0	0
$256$	19.0000	1.18750
$257$	19.8564i	1.23861i	0.785151	+	0.619304i	$0.212585\pi$
$257$	19.8564i	1.23861i	−0.785151	+	0.619304i	$0.787415\pi$
$258$	8.53590i	0.531422i
$259$	9.85641	0.612447
$260$	0	0
$261$	−3.46410	−0.214423
$262$	32.7846i	2.02544i
$263$	20.5359i	1.26630i	0.774030	+	0.633149i	$0.218238\pi$
$263$	20.5359i	1.26630i	−0.774030	+	0.633149i	$0.781762\pi$
$264$	1.73205	0.106600
$265$	0	0
$266$	−18.9282	−1.16056
$267$	− 0.928203i	− 0.0568051i
$268$	8.00000i	0.488678i
$269$	−19.8564	−1.21067	−0.605333	−	0.795972i	$-0.706960\pi$
$269$	−19.8564	−1.21067	−0.605333	+	0.795972i	$0.706960\pi$
$270$	0	0
$271$	−11.6077	−0.705117	−0.352559	−	0.935790i	$-0.614688\pi$
$271$	−11.6077	−0.705117	−0.352559	+	0.935790i	$0.614688\pi$
$272$	0	0
$273$	10.9282i	0.661405i
$274$	−31.1769	−1.88347
$275$	0	0
$276$	6.92820	0.417029
$277$	− 29.4641i	− 1.77033i	−0.465281	−	0.885163i	$-0.654047\pi$
$277$	− 29.4641i	− 1.77033i	0.465281	−	0.885163i	$-0.345953\pi$
$278$	− 21.4641i	− 1.28733i
$279$	10.9282	0.654254
$280$	0	0
$281$	3.46410	0.206651	0.103325	−	0.994648i	$-0.467052\pi$
$281$	3.46410	0.206651	0.103325	+	0.994648i	$0.467052\pi$
$282$	− 12.0000i	− 0.714590i
$283$	− 4.92820i	− 0.292951i	−0.989214	−	0.146476i	$-0.953207\pi$
$283$	− 4.92820i	− 0.292951i	0.989214	−	0.146476i	$-0.0467930\pi$
$284$	−13.8564	−0.822226
$285$	0	0
$286$	9.46410	0.559624
$287$	− 6.92820i	− 0.408959i
$288$	5.19615i	0.306186i
$289$	17.0000	1.00000
$290$	0	0
$291$	−10.0000	−0.586210
$292$	8.39230i	0.491122i
$293$	− 13.8564i	− 0.809500i	−0.914427	−	0.404750i	$-0.867359\pi$
$293$	− 13.8564i	− 0.809500i	0.914427	−	0.404750i	$-0.132641\pi$
$294$	−5.19615	−0.303046
$295$	0	0
$296$	−8.53590	−0.496139
$297$	1.00000i	0.0580259i
$298$	26.7846i	1.55159i
$299$	−37.8564	−2.18929
$300$	0	0
$301$	−9.85641	−0.568114
$302$	− 35.3205i	− 2.03247i
$303$	− 10.3923i	− 0.597022i
$304$	27.3205	1.56694
$305$	0	0
$306$	0	0
$307$	− 14.0000i	− 0.799022i	−0.916728	−	0.399511i	$-0.869180\pi$
$307$	− 14.0000i	− 0.799022i	0.916728	−	0.399511i	$-0.130820\pi$
$308$	− 2.00000i	− 0.113961i
$309$	−8.00000	−0.455104
$310$	0	0
$311$	5.07180	0.287595	0.143798	−	0.989607i	$-0.454069\pi$
$311$	5.07180	0.287595	0.143798	+	0.989607i	$0.454069\pi$
$312$	− 9.46410i	− 0.535799i
$313$	20.9282i	1.18293i	0.806330	+	0.591466i	$0.201451\pi$
$313$	20.9282i	1.18293i	−0.806330	+	0.591466i	$0.798549\pi$
$314$	−5.32051	−0.300254
$315$	0	0
$316$	−6.53590	−0.367673
$317$	24.9282i	1.40011i	0.714090	+	0.700054i	$0.246841\pi$
$317$	24.9282i	1.40011i	−0.714090	+	0.700054i	$0.753159\pi$
$318$	− 1.60770i	− 0.0901551i
$319$	−3.46410	−0.193952
$320$	0	0
$321$	8.53590	0.476427
$322$	24.0000i	1.33747i
$323$	0	0
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−17.0718	−0.945519
$327$	10.0000i	0.553001i
$328$	6.00000i	0.331295i
$329$	13.8564	0.763928
$330$	0	0
$331$	9.85641	0.541757	0.270879	−	0.962614i	$-0.412686\pi$
$331$	9.85641	0.541757	0.270879	+	0.962614i	$0.412686\pi$
$332$	− 8.53590i	− 0.468468i
$333$	− 4.92820i	− 0.270064i
$334$	−18.0000	−0.984916
$335$	0	0
$336$	−10.0000	−0.545545
$337$	− 33.1769i	− 1.80726i	−0.428312	−	0.903631i	$-0.640892\pi$
$337$	− 33.1769i	− 1.80726i	0.428312	−	0.903631i	$-0.359108\pi$
$338$	− 29.1962i	− 1.58806i
$339$	12.9282	0.702164
$340$	0	0
$341$	10.9282	0.591795
$342$	9.46410i	0.511760i
$343$	− 20.0000i	− 1.07990i
$344$	8.53590	0.460225
$345$	0	0
$346$	20.7846	1.11739
$347$	− 22.3923i	− 1.20208i	−0.799218	−	0.601041i	$-0.794753\pi$
$347$	− 22.3923i	− 1.20208i	0.799218	−	0.601041i	$-0.205247\pi$
$348$	− 3.46410i	− 0.185695i
$349$	8.14359	0.435917	0.217958	−	0.975958i	$-0.430060\pi$
$349$	8.14359	0.435917	0.217958	+	0.975958i	$0.430060\pi$
$350$	0	0
$351$	5.46410	0.291652
$352$	5.19615i	0.276956i
$353$	12.9282i	0.688099i	0.938952	+	0.344049i	$0.111799\pi$
$353$	12.9282i	0.688099i	−0.938952	+	0.344049i	$0.888201\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	0.928203	0.0491947
$357$	0	0
$358$	− 12.0000i	− 0.634220i
$359$	20.7846	1.09697	0.548485	−	0.836160i	$-0.315205\pi$
$359$	20.7846	1.09697	0.548485	+	0.836160i	$0.315205\pi$
$360$	0	0
$361$	10.8564	0.571390
$362$	27.4641i	1.44348i
$363$	1.00000i	0.0524864i
$364$	−10.9282	−0.572793
$365$	0	0
$366$	−3.46410	−0.181071
$367$	− 20.0000i	− 1.04399i	−0.852948	−	0.521996i	$-0.825188\pi$
$367$	− 20.0000i	− 1.04399i	0.852948	−	0.521996i	$-0.174812\pi$
$368$	− 34.6410i	− 1.80579i
$369$	−3.46410	−0.180334
$370$	0	0
$371$	1.85641	0.0963798
$372$	10.9282i	0.566601i
$373$	− 20.3923i	− 1.05587i	−0.849284	−	0.527937i	$-0.822966\pi$
$373$	− 20.3923i	− 1.05587i	0.849284	−	0.527937i	$-0.177034\pi$
$374$	0	0
$375$	0	0
$376$	−12.0000	−0.618853
$377$	18.9282i	0.974852i
$378$	− 3.46410i	− 0.178174i
$379$	17.8564	0.917222	0.458611	−	0.888637i	$-0.348347\pi$
$379$	17.8564	0.917222	0.458611	+	0.888637i	$0.348347\pi$
$380$	0	0
$381$	8.92820	0.457406
$382$	− 32.7846i	− 1.67741i
$383$	− 13.8564i	− 0.708029i	−0.935240	−	0.354015i	$-0.884816\pi$
$383$	− 13.8564i	− 0.708029i	0.935240	−	0.354015i	$-0.115184\pi$
$384$	12.1244	0.618718
$385$	0	0
$386$	−42.2487	−2.15040
$387$	4.92820i	0.250515i
$388$	− 10.0000i	− 0.507673i
$389$	−11.0718	−0.561362	−0.280681	−	0.959801i	$-0.590560\pi$
$389$	−11.0718	−0.561362	−0.280681	+	0.959801i	$0.590560\pi$
$390$	0	0
$391$	0	0
$392$	5.19615i	0.262445i
$393$	18.9282i	0.954802i
$394$	−20.7846	−1.04711
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	42.9282i	2.15180i
$399$	−10.9282	−0.547094
$400$	0	0
$401$	−7.85641	−0.392330	−0.196165	−	0.980571i	$-0.562849\pi$
$401$	−7.85641	−0.392330	−0.196165	+	0.980571i	$0.562849\pi$
$402$	13.8564i	0.691095i
$403$	− 59.7128i	− 2.97451i
$404$	10.3923	0.517036
$405$	0	0
$406$	12.0000	0.595550
$407$	− 4.92820i	− 0.244282i
$408$	0	0
$409$	6.78461	0.335477	0.167739	−	0.985831i	$-0.446354\pi$
$409$	6.78461	0.335477	0.167739	+	0.985831i	$0.446354\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	− 8.00000i	− 0.394132i
$413$	− 13.8564i	− 0.681829i
$414$	12.0000	0.589768
$415$	0	0
$416$	28.3923	1.39205
$417$	− 12.3923i	− 0.606854i
$418$	9.46410i	0.462904i
$419$	−30.9282	−1.51094	−0.755471	−	0.655182i	$-0.772592\pi$
$419$	−30.9282	−1.51094	−0.755471	+	0.655182i	$0.772592\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	− 14.5359i	− 0.707596i
$423$	− 6.92820i	− 0.336861i
$424$	−1.60770	−0.0780766
$425$	0	0
$426$	−24.0000	−1.16280
$427$	− 4.00000i	− 0.193574i
$428$	8.53590i	0.412598i
$429$	5.46410	0.263809
$430$	0	0
$431$	8.78461	0.423140	0.211570	−	0.977363i	$-0.432142\pi$
$431$	8.78461	0.423140	0.211570	+	0.977363i	$0.432142\pi$
$432$	5.00000i	0.240563i
$433$	0.143594i	0.00690067i	0.999994	+	0.00345033i	$0.00109828\pi$
$433$	0.143594i	0.00690067i	−0.999994	+	0.00345033i	$0.998902\pi$
$434$	−37.8564	−1.81717
$435$	0	0
$436$	−10.0000	−0.478913
$437$	− 37.8564i	− 1.81092i
$438$	14.5359i	0.694552i
$439$	−33.1769	−1.58345	−0.791724	−	0.610879i	$-0.790816\pi$
$439$	−33.1769	−1.58345	−0.791724	+	0.610879i	$0.790816\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	4.92820	0.233882
$445$	0	0
$446$	−17.0718	−0.808373
$447$	15.4641i	0.731427i
$448$	2.00000i	0.0944911i
$449$	26.7846	1.26404	0.632022	−	0.774950i	$-0.282225\pi$
$449$	26.7846	1.26404	0.632022	+	0.774950i	$0.282225\pi$
$450$	0	0
$451$	−3.46410	−0.163118
$452$	12.9282i	0.608092i
$453$	− 20.3923i	− 0.958114i
$454$	26.7846	1.25706
$455$	0	0
$456$	9.46410	0.443197
$457$	− 12.3923i	− 0.579688i	−0.957074	−	0.289844i	$-0.906397\pi$
$457$	− 12.3923i	− 0.579688i	0.957074	−	0.289844i	$-0.0936034\pi$
$458$	41.3205i	1.93078i
$459$	0	0
$460$	0	0
$461$	36.2487	1.68827	0.844135	−	0.536130i	$-0.180114\pi$
$461$	36.2487	1.68827	0.844135	+	0.536130i	$0.180114\pi$
$462$	− 3.46410i	− 0.161165i
$463$	− 28.0000i	− 1.30127i	−0.759390	−	0.650635i	$-0.774503\pi$
$463$	− 28.0000i	− 1.30127i	0.759390	−	0.650635i	$-0.225497\pi$
$464$	−17.3205	−0.804084
$465$	0	0
$466$	−20.7846	−0.962828
$467$	− 5.07180i	− 0.234695i	−0.993091	−	0.117347i	$-0.962561\pi$
$467$	− 5.07180i	− 0.234695i	0.993091	−	0.117347i	$-0.0374391\pi$
$468$	5.46410i	0.252578i
$469$	−16.0000	−0.738811
$470$	0	0
$471$	−3.07180	−0.141541
$472$	12.0000i	0.552345i
$473$	4.92820i	0.226599i
$474$	−11.3205	−0.519968
$475$	0	0
$476$	0	0
$477$	− 0.928203i	− 0.0424995i
$478$	20.7846i	0.950666i
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	−26.9282	−1.22782
$482$	0.248711i	0.0113285i
$483$	13.8564i	0.630488i
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	−1.73205	−0.0785674
$487$	31.7128i	1.43704i	0.695504	+	0.718522i	$0.255181\pi$
$487$	31.7128i	1.43704i	−0.695504	+	0.718522i	$0.744819\pi$
$488$	3.46410i	0.156813i
$489$	−9.85641	−0.445722
$490$	0	0
$491$	−30.9282	−1.39577	−0.697885	−	0.716210i	$-0.745875\pi$
$491$	−30.9282	−1.39577	−0.697885	+	0.716210i	$0.745875\pi$
$492$	− 3.46410i	− 0.156174i
$493$	0	0
$494$	51.7128	2.32667
$495$	0	0
$496$	54.6410	2.45345
$497$	− 27.7128i	− 1.24309i
$498$	− 14.7846i	− 0.662514i
$499$	−28.7846	−1.28858	−0.644288	−	0.764783i	$-0.722846\pi$
$499$	−28.7846	−1.28858	−0.644288	+	0.764783i	$0.722846\pi$
$500$	0	0
$501$	−10.3923	−0.464294
$502$	− 3.21539i	− 0.143510i
$503$	31.1769i	1.39011i	0.718957	+	0.695055i	$0.244620\pi$
$503$	31.1769i	1.39011i	−0.718957	+	0.695055i	$0.755380\pi$
$504$	−3.46410	−0.154303
$505$	0	0
$506$	12.0000	0.533465
$507$	− 16.8564i	− 0.748619i
$508$	8.92820i	0.396125i
$509$	−19.8564	−0.880120	−0.440060	−	0.897968i	$-0.645043\pi$
$509$	−19.8564	−0.880120	−0.440060	+	0.897968i	$0.645043\pi$
$510$	0	0
$511$	−16.7846	−0.742507
$512$	8.66025i	0.382733i
$513$	5.46410i	0.241246i
$514$	−34.3923	−1.51698
$515$	0	0
$516$	−4.92820	−0.216952
$517$	− 6.92820i	− 0.304702i
$518$	17.0718i	0.750092i
$519$	12.0000	0.526742
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	− 6.00000i	− 0.262613i
$523$	− 22.0000i	− 0.961993i	−0.876723	−	0.480996i	$-0.840275\pi$
$523$	− 22.0000i	− 0.961993i	0.876723	−	0.480996i	$-0.159725\pi$
$524$	−18.9282	−0.826882
$525$	0	0
$526$	−35.5692	−1.55089
$527$	0	0
$528$	5.00000i	0.217597i
$529$	−25.0000	−1.08696
$530$	0	0
$531$	−6.92820	−0.300658
$532$	− 10.9282i	− 0.473798i
$533$	18.9282i	0.819871i
$534$	1.60770	0.0695718
$535$	0	0
$536$	13.8564	0.598506
$537$	− 6.92820i	− 0.298974i
$538$	− 34.3923i	− 1.48276i
$539$	−3.00000	−0.129219
$540$	0	0
$541$	27.8564	1.19764	0.598820	−	0.800883i	$-0.295636\pi$
$541$	27.8564	1.19764	0.598820	+	0.800883i	$0.295636\pi$
$542$	− 20.1051i	− 0.863589i
$543$	15.8564i	0.680464i
$544$	0	0
$545$	0	0
$546$	−18.9282	−0.810052
$547$	− 2.00000i	− 0.0855138i	−0.999086	−	0.0427569i	$-0.986386\pi$
$547$	− 2.00000i	− 0.0855138i	0.999086	−	0.0427569i	$-0.0136141\pi$
$548$	− 18.0000i	− 0.768922i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−18.9282	−0.806369
$552$	− 12.0000i	− 0.510754i
$553$	− 13.0718i	− 0.555869i
$554$	51.0333	2.16820
$555$	0	0
$556$	12.3923	0.525551
$557$	− 3.21539i	− 0.136240i	−0.997677	−	0.0681202i	$-0.978300\pi$
$557$	− 3.21539i	− 0.136240i	0.997677	−	0.0681202i	$-0.0217001\pi$
$558$	18.9282i	0.801295i
$559$	26.9282	1.13894
$560$	0	0
$561$	0	0
$562$	6.00000i	0.253095i
$563$	10.3923i	0.437983i	0.975727	+	0.218992i	$0.0702768\pi$
$563$	10.3923i	0.437983i	−0.975727	+	0.218992i	$0.929723\pi$
$564$	6.92820	0.291730
$565$	0	0
$566$	8.53590	0.358791
$567$	− 2.00000i	− 0.0839921i
$568$	24.0000i	1.00702i
$569$	−5.32051	−0.223047	−0.111524	−	0.993762i	$-0.535573\pi$
$569$	−5.32051	−0.223047	−0.111524	+	0.993762i	$0.535573\pi$
$570$	0	0
$571$	3.60770	0.150977	0.0754887	−	0.997147i	$-0.475948\pi$
$571$	3.60770	0.150977	0.0754887	+	0.997147i	$0.475948\pi$
$572$	5.46410i	0.228466i
$573$	− 18.9282i	− 0.790737i
$574$	12.0000	0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	18.7846i	0.782014i	0.920388	+	0.391007i	$0.127873\pi$
$577$	18.7846i	0.782014i	−0.920388	+	0.391007i	$0.872127\pi$
$578$	29.4449i	1.22474i
$579$	−24.3923	−1.01371
$580$	0	0
$581$	17.0718	0.708257
$582$	− 17.3205i	− 0.717958i
$583$	− 0.928203i	− 0.0384422i
$584$	14.5359	0.601500
$585$	0	0
$586$	24.0000	0.991431
$587$	18.9282i	0.781251i	0.920550	+	0.390625i	$0.127741\pi$
$587$	18.9282i	0.781251i	−0.920550	+	0.390625i	$0.872259\pi$
$588$	− 3.00000i	− 0.123718i
$589$	59.7128	2.46042
$590$	0	0
$591$	−12.0000	−0.493614
$592$	− 24.6410i	− 1.01274i
$593$	8.78461i	0.360741i	0.983599	+	0.180370i	$0.0577296\pi$
$593$	8.78461i	0.360741i	−0.983599	+	0.180370i	$0.942270\pi$
$594$	−1.73205	−0.0710669
$595$	0	0
$596$	−15.4641	−0.633434
$597$	24.7846i	1.01437i
$598$	− 65.5692i	− 2.68132i
$599$	37.8564	1.54677	0.773385	−	0.633936i	$-0.218562\pi$
$599$	37.8564	1.54677	0.773385	+	0.633936i	$0.218562\pi$
$600$	0	0
$601$	−32.6410	−1.33145	−0.665727	−	0.746195i	$-0.731879\pi$
$601$	−32.6410	−1.33145	−0.665727	+	0.746195i	$0.731879\pi$
$602$	− 17.0718i	− 0.695794i
$603$	8.00000i	0.325785i
$604$	20.3923	0.829751
$605$	0	0
$606$	18.0000	0.731200
$607$	18.7846i	0.762444i	0.924484	+	0.381222i	$0.124497\pi$
$607$	18.7846i	0.762444i	−0.924484	+	0.381222i	$0.875503\pi$
$608$	28.3923i	1.15146i
$609$	6.92820	0.280745
$610$	0	0
$611$	−37.8564	−1.53151
$612$	0	0
$613$	− 20.3923i	− 0.823637i	−0.911266	−	0.411819i	$-0.864894\pi$
$613$	− 20.3923i	− 0.823637i	0.911266	−	0.411819i	$-0.135106\pi$
$614$	24.2487	0.978598
$615$	0	0
$616$	−3.46410	−0.139573
$617$	36.9282i	1.48667i	0.668917	+	0.743337i	$0.266758\pi$
$617$	36.9282i	1.48667i	−0.668917	+	0.743337i	$0.733242\pi$
$618$	− 13.8564i	− 0.557386i
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	6.92820	0.278019
$622$	8.78461i	0.352231i
$623$	1.85641i	0.0743754i
$624$	27.3205	1.09370
$625$	0	0
$626$	−36.2487	−1.44879
$627$	5.46410i	0.218215i
$628$	− 3.07180i	− 0.122578i
$629$	0	0
$630$	0	0
$631$	−21.0718	−0.838855	−0.419427	−	0.907789i	$-0.637769\pi$
$631$	−21.0718	−0.838855	−0.419427	+	0.907789i	$0.637769\pi$
$632$	11.3205i	0.450306i
$633$	− 8.39230i	− 0.333564i
$634$	−43.1769	−1.71477
$635$	0	0
$636$	0.928203	0.0368057
$637$	16.3923i	0.649487i
$638$	− 6.00000i	− 0.237542i
$639$	−13.8564	−0.548151
$640$	0	0
$641$	−12.9282	−0.510633	−0.255317	−	0.966857i	$-0.582180\pi$
$641$	−12.9282	−0.510633	−0.255317	+	0.966857i	$0.582180\pi$
$642$	14.7846i	0.583502i
$643$	37.5692i	1.48159i	0.671734	+	0.740793i	$0.265550\pi$
$643$	37.5692i	1.48159i	−0.671734	+	0.740793i	$0.734450\pi$
$644$	−13.8564	−0.546019
$645$	0	0
$646$	0	0
$647$	− 27.7128i	− 1.08950i	−0.838597	−	0.544752i	$-0.816624\pi$
$647$	− 27.7128i	− 1.08950i	0.838597	−	0.544752i	$-0.183376\pi$
$648$	1.73205i	0.0680414i
$649$	−6.92820	−0.271956
$650$	0	0
$651$	−21.8564	−0.856620
$652$	− 9.85641i	− 0.386007i
$653$	19.8564i	0.777041i	0.921440	+	0.388521i	$0.127014\pi$
$653$	19.8564i	0.777041i	−0.921440	+	0.388521i	$0.872986\pi$
$654$	−17.3205	−0.677285
$655$	0	0
$656$	−17.3205	−0.676252
$657$	8.39230i	0.327415i
$658$	24.0000i	0.935617i
$659$	15.7128	0.612084	0.306042	−	0.952018i	$-0.400995\pi$
$659$	15.7128	0.612084	0.306042	+	0.952018i	$0.400995\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	17.0718i	0.663514i
$663$	0	0
$664$	−14.7846	−0.573754
$665$	0	0
$666$	8.53590	0.330759
$667$	24.0000i	0.929284i
$668$	− 10.3923i	− 0.402090i
$669$	−9.85641	−0.381071
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	− 10.3923i	− 0.400892i
$673$	− 3.32051i	− 0.127996i	−0.997950	−	0.0639981i	$-0.979615\pi$
$673$	− 3.32051i	− 0.127996i	0.997950	−	0.0639981i	$-0.0203852\pi$
$674$	57.4641	2.21343
$675$	0	0
$676$	16.8564	0.648323
$677$	− 8.78461i	− 0.337620i	−0.985649	−	0.168810i	$-0.946008\pi$
$677$	− 8.78461i	− 0.337620i	0.985649	−	0.168810i	$-0.0539924\pi$
$678$	22.3923i	0.859971i
$679$	20.0000	0.767530
$680$	0	0
$681$	15.4641	0.592586
$682$	18.9282i	0.724798i
$683$	− 32.7846i	− 1.25447i	−0.778831	−	0.627234i	$-0.784187\pi$
$683$	− 32.7846i	− 1.25447i	0.778831	−	0.627234i	$-0.215813\pi$
$684$	−5.46410	−0.208925
$685$	0	0
$686$	34.6410	1.32260
$687$	23.8564i	0.910179i
$688$	24.6410i	0.939430i
$689$	−5.07180	−0.193220
$690$	0	0
$691$	47.7128	1.81508	0.907540	−	0.419965i	$-0.137958\pi$
$691$	47.7128	1.81508	0.907540	+	0.419965i	$0.137958\pi$
$692$	12.0000i	0.456172i
$693$	− 2.00000i	− 0.0759737i
$694$	38.7846	1.47224
$695$	0	0
$696$	−6.00000	−0.227429
$697$	0	0
$698$	14.1051i	0.533887i
$699$	−12.0000	−0.453882
$700$	0	0
$701$	39.4641	1.49054	0.745269	−	0.666764i	$-0.232321\pi$
$701$	39.4641	1.49054	0.745269	+	0.666764i	$0.232321\pi$
$702$	9.46410i	0.357199i
$703$	− 26.9282i	− 1.01562i
$704$	1.00000	0.0376889
$705$	0	0
$706$	−22.3923	−0.842746
$707$	20.7846i	0.781686i
$708$	− 6.92820i	− 0.260378i
$709$	11.8564	0.445277	0.222638	−	0.974901i	$-0.428533\pi$
$709$	11.8564	0.445277	0.222638	+	0.974901i	$0.428533\pi$
$710$	0	0
$711$	−6.53590	−0.245115
$712$	− 1.60770i	− 0.0602509i
$713$	− 75.7128i	− 2.83547i
$714$	0	0
$715$	0	0
$716$	6.92820	0.258919
$717$	12.0000i	0.448148i
$718$	36.0000i	1.34351i
$719$	5.07180	0.189146	0.0945731	−	0.995518i	$-0.469851\pi$
$719$	5.07180	0.189146	0.0945731	+	0.995518i	$0.469851\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	18.8038i	0.699807i
$723$	0.143594i	0.00534030i
$724$	−15.8564	−0.589299
$725$	0	0
$726$	−1.73205	−0.0642824
$727$	− 8.00000i	− 0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$	− 8.00000i	− 0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$	18.9282i	0.701526i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	− 2.00000i	− 0.0739221i
$733$	53.9615i	1.99311i	0.0829082	+	0.996557i	$0.473579\pi$
$733$	53.9615i	1.99311i	−0.0829082	+	0.996557i	$0.526421\pi$
$734$	34.6410	1.27862
$735$	0	0
$736$	36.0000	1.32698
$737$	8.00000i	0.294684i
$738$	− 6.00000i	− 0.220863i
$739$	−17.4641	−0.642427	−0.321214	−	0.947007i	$-0.604091\pi$
$739$	−17.4641	−0.642427	−0.321214	+	0.947007i	$0.604091\pi$
$740$	0	0
$741$	29.8564	1.09680
$742$	3.21539i	0.118041i
$743$	25.6077i	0.939455i	0.882811	+	0.469728i	$0.155648\pi$
$743$	25.6077i	0.939455i	−0.882811	+	0.469728i	$0.844352\pi$
$744$	18.9282	0.693942
$745$	0	0
$746$	35.3205	1.29318
$747$	− 8.53590i	− 0.312312i
$748$	0	0
$749$	−17.0718	−0.623790
$750$	0	0
$751$	26.9282	0.982624	0.491312	−	0.870984i	$-0.336517\pi$
$751$	26.9282	0.982624	0.491312	+	0.870984i	$0.336517\pi$
$752$	− 34.6410i	− 1.26323i
$753$	− 1.85641i	− 0.0676512i
$754$	−32.7846	−1.19395
$755$	0	0
$756$	2.00000	0.0727393
$757$	− 34.7846i	− 1.26427i	−0.774859	−	0.632134i	$-0.782179\pi$
$757$	− 34.7846i	− 1.26427i	0.774859	−	0.632134i	$-0.217821\pi$
$758$	30.9282i	1.12336i
$759$	6.92820	0.251478
$760$	0	0
$761$	−32.5359	−1.17943	−0.589713	−	0.807613i	$-0.700759\pi$
$761$	−32.5359	−1.17943	−0.589713	+	0.807613i	$0.700759\pi$
$762$	15.4641i	0.560205i
$763$	− 20.0000i	− 0.724049i
$764$	18.9282	0.684798
$765$	0	0
$766$	24.0000	0.867155
$767$	37.8564i	1.36692i
$768$	19.0000i	0.685603i
$769$	−50.4974	−1.82098	−0.910492	−	0.413527i	$-0.864297\pi$
$769$	−50.4974	−1.82098	−0.910492	+	0.413527i	$0.864297\pi$
$770$	0	0
$771$	−19.8564	−0.715111
$772$	− 24.3923i	− 0.877898i
$773$	− 4.14359i	− 0.149035i	−0.997220	−	0.0745174i	$-0.976258\pi$
$773$	− 4.14359i	− 0.149035i	0.997220	−	0.0745174i	$-0.0237416\pi$
$774$	−8.53590	−0.306817
$775$	0	0
$776$	−17.3205	−0.621770
$777$	9.85641i	0.353597i
$778$	− 19.1769i	− 0.687526i
$779$	−18.9282	−0.678173
$780$	0	0
$781$	−13.8564	−0.495821
$782$	0	0
$783$	− 3.46410i	− 0.123797i
$784$	−15.0000	−0.535714
$785$	0	0
$786$	−32.7846	−1.16939
$787$	− 22.7846i	− 0.812184i	−0.913832	−	0.406092i	$-0.866891\pi$
$787$	− 22.7846i	− 0.812184i	0.913832	−	0.406092i	$-0.133109\pi$
$788$	− 12.0000i	− 0.427482i
$789$	−20.5359	−0.731097
$790$	0	0
$791$	−25.8564	−0.919348
$792$	1.73205i	0.0615457i
$793$	10.9282i	0.388072i
$794$	3.46410	0.122936
$795$	0	0
$796$	−24.7846	−0.878467
$797$	52.6410i	1.86464i	0.361634	+	0.932320i	$0.382219\pi$
$797$	52.6410i	1.86464i	−0.361634	+	0.932320i	$0.617781\pi$
$798$	− 18.9282i	− 0.670051i
$799$	0	0
$800$	0	0
$801$	0.928203	0.0327964
$802$	− 13.6077i	− 0.480504i
$803$	8.39230i	0.296158i
$804$	−8.00000	−0.282138
$805$	0	0
$806$	103.426	3.64301
$807$	− 19.8564i	− 0.698979i
$808$	− 18.0000i	− 0.633238i
$809$	15.4641	0.543689	0.271844	−	0.962341i	$-0.412366\pi$
$809$	15.4641	0.543689	0.271844	+	0.962341i	$0.412366\pi$
$810$	0	0
$811$	12.3923	0.435153	0.217576	−	0.976043i	$-0.430185\pi$
$811$	12.3923	0.435153	0.217576	+	0.976043i	$0.430185\pi$
$812$	6.92820i	0.243132i
$813$	− 11.6077i	− 0.407100i
$814$	8.53590	0.299183
$815$	0	0
$816$	0	0
$817$	26.9282i	0.942099i
$818$	11.7513i	0.410874i
$819$	−10.9282	−0.381862
$820$	0	0
$821$	20.5359	0.716708	0.358354	−	0.933586i	$-0.383338\pi$
$821$	20.5359	0.716708	0.358354	+	0.933586i	$0.383338\pi$
$822$	− 31.1769i	− 1.08742i
$823$	− 33.5692i	− 1.17015i	−0.810979	−	0.585075i	$-0.801065\pi$
$823$	− 33.5692i	− 1.17015i	0.810979	−	0.585075i	$-0.198935\pi$
$824$	−13.8564	−0.482711
$825$	0	0
$826$	24.0000	0.835067
$827$	− 22.3923i	− 0.778657i	−0.921099	−	0.389328i	$-0.872707\pi$
$827$	− 22.3923i	− 0.778657i	0.921099	−	0.389328i	$-0.127293\pi$
$828$	6.92820i	0.240772i
$829$	−29.7128	−1.03197	−0.515984	−	0.856598i	$-0.672574\pi$
$829$	−29.7128	−1.03197	−0.515984	+	0.856598i	$0.672574\pi$
$830$	0	0
$831$	29.4641	1.02210
$832$	− 5.46410i	− 0.189434i
$833$	0	0
$834$	21.4641	0.743241
$835$	0	0
$836$	−5.46410	−0.188980
$837$	10.9282i	0.377734i
$838$	− 53.5692i	− 1.85052i
$839$	−56.7846	−1.96042	−0.980211	−	0.197954i	$-0.936570\pi$
$839$	−56.7846	−1.96042	−0.980211	+	0.197954i	$0.936570\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	3.46410i	0.119381i
$843$	3.46410i	0.119310i
$844$	8.39230	0.288875
$845$	0	0
$846$	12.0000	0.412568
$847$	− 2.00000i	− 0.0687208i
$848$	− 4.64102i	− 0.159373i
$849$	4.92820	0.169135
$850$	0	0
$851$	−34.1436	−1.17043
$852$	− 13.8564i	− 0.474713i
$853$	3.60770i	0.123525i	0.998091	+	0.0617626i	$0.0196722\pi$
$853$	3.60770i	0.123525i	−0.998091	+	0.0617626i	$0.980328\pi$
$854$	6.92820	0.237078
$855$	0	0
$856$	14.7846	0.505328
$857$	37.8564i	1.29315i	0.762850	+	0.646575i	$0.223799\pi$
$857$	37.8564i	1.29315i	−0.762850	+	0.646575i	$0.776201\pi$
$858$	9.46410i	0.323099i
$859$	7.71281	0.263158	0.131579	−	0.991306i	$-0.457995\pi$
$859$	7.71281	0.263158	0.131579	+	0.991306i	$0.457995\pi$
$860$	0	0
$861$	6.92820	0.236113
$862$	15.2154i	0.518238i
$863$	− 37.8564i	− 1.28865i	−0.764753	−	0.644324i	$-0.777139\pi$
$863$	− 37.8564i	− 1.28865i	0.764753	−	0.644324i	$-0.222861\pi$
$864$	−5.19615	−0.176777
$865$	0	0
$866$	−0.248711	−0.00845155
$867$	17.0000i	0.577350i
$868$	− 21.8564i	− 0.741855i
$869$	−6.53590	−0.221715
$870$	0	0
$871$	43.7128	1.48115
$872$	17.3205i	0.586546i
$873$	− 10.0000i	− 0.338449i
$874$	65.5692	2.21791
$875$	0	0
$876$	−8.39230	−0.283550
$877$	34.2487i	1.15650i	0.815861	+	0.578248i	$0.196264\pi$
$877$	34.2487i	1.15650i	−0.815861	+	0.578248i	$0.803736\pi$
$878$	− 57.4641i	− 1.93932i
$879$	13.8564	0.467365
$880$	0	0
$881$	−0.928203	−0.0312720	−0.0156360	−	0.999878i	$-0.504977\pi$
$881$	−0.928203	−0.0312720	−0.0156360	+	0.999878i	$0.504977\pi$
$882$	− 5.19615i	− 0.174964i
$883$	− 4.00000i	− 0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$	− 4.00000i	− 0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$	0	0
$885$	0	0
$886$	−20.7846	−0.698273
$887$	− 12.2487i	− 0.411271i	−0.978629	−	0.205636i	$-0.934074\pi$
$887$	− 12.2487i	− 0.411271i	0.978629	−	0.205636i	$-0.0659262\pi$
$888$	− 8.53590i	− 0.286446i
$889$	−17.8564	−0.598885
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	− 9.85641i	− 0.330017i
$893$	− 37.8564i	− 1.26682i
$894$	−26.7846	−0.895811
$895$	0	0
$896$	−24.2487	−0.810093
$897$	− 37.8564i	− 1.26399i
$898$	46.3923i	1.54813i
$899$	−37.8564	−1.26258
$900$	0	0
$901$	0	0
$902$	− 6.00000i	− 0.199778i
$903$	− 9.85641i	− 0.328001i
$904$	22.3923	0.744757
$905$	0	0
$906$	35.3205	1.17345
$907$	− 18.1436i	− 0.602448i	−0.953553	−	0.301224i	$-0.902605\pi$
$907$	− 18.1436i	− 0.602448i	0.953553	−	0.301224i	$-0.0973952\pi$
$908$	15.4641i	0.513194i
$909$	10.3923	0.344691
$910$	0	0
$911$	18.9282	0.627119	0.313560	−	0.949568i	$-0.398478\pi$
$911$	18.9282	0.627119	0.313560	+	0.949568i	$0.398478\pi$
$912$	27.3205i	0.904672i
$913$	− 8.53590i	− 0.282497i
$914$	21.4641	0.709969
$915$	0	0
$916$	−23.8564	−0.788238
$917$	− 37.8564i	− 1.25013i
$918$	0	0
$919$	32.3923	1.06852	0.534262	−	0.845319i	$-0.320590\pi$
$919$	32.3923	1.06852	0.534262	+	0.845319i	$0.320590\pi$
$920$	0	0
$921$	14.0000	0.461316
$922$	62.7846i	2.06770i
$923$	75.7128i	2.49212i
$924$	2.00000	0.0657952
$925$	0	0
$926$	48.4974	1.59372
$927$	− 8.00000i	− 0.262754i
$928$	− 18.0000i	− 0.590879i
$929$	−2.78461	−0.0913601	−0.0456800	−	0.998956i	$-0.514545\pi$
$929$	−2.78461	−0.0913601	−0.0456800	+	0.998956i	$0.514545\pi$
$930$	0	0
$931$	−16.3923	−0.537236
$932$	− 12.0000i	− 0.393073i
$933$	5.07180i	0.166043i
$934$	8.78461	0.287441
$935$	0	0
$936$	9.46410	0.309344
$937$	20.3923i	0.666188i	0.942894	+	0.333094i	$0.108093\pi$
$937$	20.3923i	0.666188i	−0.942894	+	0.333094i	$0.891907\pi$
$938$	− 27.7128i	− 0.904855i
$939$	−20.9282	−0.682966
$940$	0	0
$941$	27.4641	0.895304	0.447652	−	0.894208i	$-0.352260\pi$
$941$	27.4641	0.895304	0.447652	+	0.894208i	$0.352260\pi$
$942$	− 5.32051i	− 0.173352i
$943$	24.0000i	0.781548i
$944$	−34.6410	−1.12747
$945$	0	0
$946$	−8.53590	−0.277526
$947$	18.9282i	0.615084i	0.951535	+	0.307542i	$0.0995065\pi$
$947$	18.9282i	0.615084i	−0.951535	+	0.307542i	$0.900494\pi$
$948$	− 6.53590i	− 0.212276i
$949$	45.8564	1.48856
$950$	0	0
$951$	−24.9282	−0.808352
$952$	0	0
$953$	− 3.21539i	− 0.104157i	−0.998643	−	0.0520784i	$-0.983415\pi$
$953$	− 3.21539i	− 0.104157i	0.998643	−	0.0520784i	$-0.0165846\pi$
$954$	1.60770	0.0520511
$955$	0	0
$956$	−12.0000	−0.388108
$957$	− 3.46410i	− 0.111979i
$958$	− 20.7846i	− 0.671520i
$959$	36.0000	1.16250
$960$	0	0
$961$	88.4256	2.85244
$962$	− 46.6410i	− 1.50377i
$963$	8.53590i	0.275065i
$964$	−0.143594	−0.00462484
$965$	0	0
$966$	−24.0000	−0.772187
$967$	− 22.7846i	− 0.732704i	−0.930476	−	0.366352i	$-0.880607\pi$
$967$	− 22.7846i	− 0.732704i	0.930476	−	0.366352i	$-0.119393\pi$
$968$	1.73205i	0.0556702i
$969$	0	0
$970$	0	0
$971$	25.8564	0.829772	0.414886	−	0.909873i	$-0.363822\pi$
$971$	25.8564	0.829772	0.414886	+	0.909873i	$0.363822\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	24.7846i	0.794558i
$974$	−54.9282	−1.76001
$975$	0	0
$976$	−10.0000	−0.320092
$977$	− 47.5692i	− 1.52187i	−0.648826	−	0.760937i	$-0.724740\pi$
$977$	− 47.5692i	− 1.52187i	0.648826	−	0.760937i	$-0.275260\pi$
$978$	− 17.0718i	− 0.545896i
$979$	0.928203	0.0296655
$980$	0	0
$981$	−10.0000	−0.319275
$982$	− 53.5692i	− 1.70946i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	0	0
$987$	13.8564i	0.441054i
$988$	29.8564i	0.949859i
$989$	34.1436	1.08570
$990$	0	0
$991$	−7.21539	−0.229204	−0.114602	−	0.993411i	$-0.536559\pi$
$991$	−7.21539	−0.229204	−0.114602	+	0.993411i	$0.536559\pi$
$992$	56.7846i	1.80291i
$993$	9.85641i	0.312784i
$994$	48.0000	1.52247
$995$	0	0
$996$	8.53590	0.270470
$997$	− 27.6077i	− 0.874344i	−0.899378	−	0.437172i	$-0.855980\pi$
$997$	− 27.6077i	− 0.874344i	0.899378	−	0.437172i	$-0.144020\pi$
$998$	− 49.8564i	− 1.57818i
$999$	4.92820	0.155921

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.2.c.c.199.4		4
3.2	odd	2		2475.2.c.n.199.1		4
5.2	odd	4		165.2.a.b.1.1	✓	2
5.3	odd	4		825.2.a.e.1.2		2
5.4	even	2	inner	825.2.c.c.199.1		4
15.2	even	4		495.2.a.c.1.2		2
15.8	even	4		2475.2.a.r.1.1		2
15.14	odd	2		2475.2.c.n.199.4		4
20.7	even	4		2640.2.a.x.1.2		2
35.27	even	4		8085.2.a.bd.1.1		2
55.32	even	4		1815.2.a.i.1.2		2
55.43	even	4		9075.2.a.bh.1.1		2
60.47	odd	4		7920.2.a.bz.1.2		2
165.32	odd	4		5445.2.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.b.1.1	✓	2	5.2	odd	4
495.2.a.c.1.2		2	15.2	even	4
825.2.a.e.1.2		2	5.3	odd	4
825.2.c.c.199.1		4	5.4	even	2	inner
825.2.c.c.199.4		4	1.1	even	1	trivial
1815.2.a.i.1.2		2	55.32	even	4
2475.2.a.r.1.1		2	15.8	even	4
2475.2.c.n.199.1		4	3.2	odd	2
2475.2.c.n.199.4		4	15.14	odd	2
2640.2.a.x.1.2		2	20.7	even	4
5445.2.a.s.1.1		2	165.32	odd	4
7920.2.a.bz.1.2		2	60.47	odd	4
8085.2.a.bd.1.1		2	35.27	even	4
9075.2.a.bh.1.1		2	55.43	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.73205 q^{6} -2.00000i q^{7} +1.73205i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.73205 q^{6} -2.00000i q^{7} +1.73205i q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.00000i q^{12} +5.46410i q^{13} +3.46410 q^{14} -5.00000 q^{16} -1.73205i q^{18} -5.46410 q^{19} +2.00000 q^{21} -1.73205i q^{22} +6.92820i q^{23} -1.73205 q^{24} -9.46410 q^{26} -1.00000i q^{27} +2.00000i q^{28} +3.46410 q^{29} -10.9282 q^{31} -5.19615i q^{32} -1.00000i q^{33} +1.00000 q^{36} +4.92820i q^{37} -9.46410i q^{38} -5.46410 q^{39} +3.46410 q^{41} +3.46410i q^{42} -4.92820i q^{43} +1.00000 q^{44} -12.0000 q^{46} +6.92820i q^{47} -5.00000i q^{48} +3.00000 q^{49} -5.46410i q^{52} +0.928203i q^{53} +1.73205 q^{54} +3.46410 q^{56} -5.46410i q^{57} +6.00000i q^{58} +6.92820 q^{59} +2.00000 q^{61} -18.9282i q^{62} +2.00000i q^{63} -1.00000 q^{64} +1.73205 q^{66} -8.00000i q^{67} -6.92820 q^{69} +13.8564 q^{71} -1.73205i q^{72} -8.39230i q^{73} -8.53590 q^{74} +5.46410 q^{76} +2.00000i q^{77} -9.46410i q^{78} +6.53590 q^{79} +1.00000 q^{81} +6.00000i q^{82} +8.53590i q^{83} -2.00000 q^{84} +8.53590 q^{86} +3.46410i q^{87} -1.73205i q^{88} -0.928203 q^{89} +10.9282 q^{91} -6.92820i q^{92} -10.9282i q^{93} -12.0000 q^{94} +5.19615 q^{96} +10.0000i q^{97} +5.19615i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 - 4 * q^9	\( 4 q - 4 q^{4} - 4 q^{9} - 4 q^{11} - 20 q^{16} - 8 q^{19} + 8 q^{21} - 24 q^{26} - 16 q^{31} + 4 q^{36} - 8 q^{39} + 4 q^{44} - 48 q^{46} + 12 q^{49} + 8 q^{61} - 4 q^{64} - 48 q^{74} + 8 q^{76} + 40 q^{79} + 4 q^{81} - 8 q^{84} + 48 q^{86} + 24 q^{89} + 16 q^{91} - 48 q^{94} + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^9 - 4 * q^11 - 20 * q^16 - 8 * q^19 + 8 * q^21 - 24 * q^26 - 16 * q^31 + 4 * q^36 - 8 * q^39 + 4 * q^44 - 48 * q^46 + 12 * q^49 + 8 * q^61 - 4 * q^64 - 48 * q^74 + 8 * q^76 + 40 * q^79 + 4 * q^81 - 8 * q^84 + 48 * q^86 + 24 * q^89 + 16 * q^91 - 48 * q^94 + 4 * q^99

Introduction

L-functions

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Database

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