LMFDB - Embedded newform 3450.2.d.h.2899.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3450,2,Mod(2899,3450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3450, 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("3450.2899");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3450.d (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:	$27.5483886973$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2899.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3450.2899
Dual form			3450.2.d.h.2899.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -5.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -5.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} +2.00000i q^{13} +5.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} -1.00000i q^{18} -2.00000 q^{19} +5.00000 q^{21} +1.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +5.00000i q^{28} -3.00000 q^{29} +2.00000 q^{31} +1.00000i q^{32} +3.00000 q^{34} +1.00000 q^{36} +7.00000i q^{37} -2.00000i q^{38} -2.00000 q^{39} +5.00000i q^{42} +2.00000i q^{43} -1.00000 q^{46} +3.00000i q^{47} +1.00000i q^{48} -18.0000 q^{49} +3.00000 q^{51} -2.00000i q^{52} -12.0000i q^{53} +1.00000 q^{54} -5.00000 q^{56} -2.00000i q^{57} -3.00000i q^{58} -6.00000 q^{59} +2.00000 q^{61} +2.00000i q^{62} +5.00000i q^{63} -1.00000 q^{64} -2.00000i q^{67} +3.00000i q^{68} -1.00000 q^{69} -15.0000 q^{71} +1.00000i q^{72} +11.0000i q^{73} -7.00000 q^{74} +2.00000 q^{76} -2.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} -9.00000i q^{83} -5.00000 q^{84} -2.00000 q^{86} -3.00000i q^{87} -3.00000 q^{89} +10.0000 q^{91} -1.00000i q^{92} +2.00000i q^{93} -3.00000 q^{94} -1.00000 q^{96} +10.0000i q^{97} -18.0000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 10 q^{14} + 2 q^{16} - 4 q^{19} + 10 q^{21} + 2 q^{24} - 4 q^{26} - 6 q^{29} + 4 q^{31} + 6 q^{34} + 2 q^{36} - 4 q^{39} - 2 q^{46} - 36 q^{49} + 6 q^{51} + 2 q^{54} - 10 q^{56} - 12 q^{59} + 4 q^{61} - 2 q^{64} - 2 q^{69} - 30 q^{71} - 14 q^{74} + 4 q^{76} - 16 q^{79} + 2 q^{81} - 10 q^{84} - 4 q^{86} - 6 q^{89} + 20 q^{91} - 6 q^{94} - 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 10 * q^14 + 2 * q^16 - 4 * q^19 + 10 * q^21 + 2 * q^24 - 4 * q^26 - 6 * q^29 + 4 * q^31 + 6 * q^34 + 2 * q^36 - 4 * q^39 - 2 * q^46 - 36 * q^49 + 6 * q^51 + 2 * q^54 - 10 * q^56 - 12 * q^59 + 4 * q^61 - 2 * q^64 - 2 * q^69 - 30 * q^71 - 14 * q^74 + 4 * q^76 - 16 * q^79 + 2 * q^81 - 10 * q^84 - 4 * q^86 - 6 * q^89 + 20 * q^91 - 6 * q^94 - 2 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	− 5.00000i	− 1.88982i	−0.327327	−	0.944911i	$-0.606148\pi$
$7$	− 5.00000i	− 1.88982i	0.327327	−	0.944911i	$-0.393852\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	− 1.00000i	− 0.288675i
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	5.00000	1.33631
$15$	0	0
$16$	1.00000	0.250000
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	− 1.00000i	− 0.235702i
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	5.00000	1.09109
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	1.00000	0.204124
$25$	0	0
$26$	−2.00000	−0.392232
$27$	− 1.00000i	− 0.192450i
$28$	5.00000i	0.944911i
$29$	−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
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	3.00000	0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	7.00000i	1.15079i	0.817875	+	0.575396i	$0.195152\pi$
$37$	7.00000i	1.15079i	−0.817875	+	0.575396i	$0.804848\pi$
$38$	− 2.00000i	− 0.324443i
$39$	−2.00000	−0.320256
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	5.00000i	0.771517i
$43$	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
$46$	−1.00000	−0.147442
$47$	3.00000i	0.437595i	0.975770	+	0.218797i	$0.0702134\pi$
$47$	3.00000i	0.437595i	−0.975770	+	0.218797i	$0.929787\pi$
$48$	1.00000i	0.144338i
$49$	−18.0000	−2.57143
$50$	0	0
$51$	3.00000	0.420084
$52$	− 2.00000i	− 0.277350i
$53$	− 12.0000i	− 1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$	− 12.0000i	− 1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−5.00000	−0.668153
$57$	− 2.00000i	− 0.264906i
$58$	− 3.00000i	− 0.393919i
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	2.00000i	0.254000i
$63$	5.00000i	0.629941i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 2.00000i	− 0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$	− 2.00000i	− 0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	3.00000i	0.363803i
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−15.0000	−1.78017	−0.890086	−	0.455792i	$-0.849356\pi$
$71$	−15.0000	−1.78017	−0.890086	+	0.455792i	$0.849356\pi$
$72$	1.00000i	0.117851i
$73$	11.0000i	1.28745i	0.765256	+	0.643726i	$0.222612\pi$
$73$	11.0000i	1.28745i	−0.765256	+	0.643726i	$0.777388\pi$
$74$	−7.00000	−0.813733
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	− 2.00000i	− 0.226455i
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 9.00000i	− 0.987878i	−0.869496	−	0.493939i	$-0.835557\pi$
$83$	− 9.00000i	− 0.987878i	0.869496	−	0.493939i	$-0.164443\pi$
$84$	−5.00000	−0.545545
$85$	0	0
$86$	−2.00000	−0.215666
$87$	− 3.00000i	− 0.321634i
$88$	0	0
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	10.0000	1.04828
$92$	− 1.00000i	− 0.104257i
$93$	2.00000i	0.207390i
$94$	−3.00000	−0.309426
$95$	0	0
$96$	−1.00000	−0.102062
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	− 18.0000i	− 1.81827i
$99$	0	0
$100$	0	0
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	3.00000i	0.297044i
$103$	− 13.0000i	− 1.28093i	−0.767988	−	0.640464i	$-0.778742\pi$
$103$	− 13.0000i	− 1.28093i	0.767988	−	0.640464i	$-0.221258\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	12.0000	1.16554
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	1.00000i	0.0962250i
$109$	−17.0000	−1.62830	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	−17.0000	−1.62830	−0.814152	+	0.580651i	$0.802798\pi$
$110$	0	0
$111$	−7.00000	−0.664411
$112$	− 5.00000i	− 0.472456i
$113$	− 9.00000i	− 0.846649i	−0.905978	−	0.423324i	$-0.860863\pi$
$113$	− 9.00000i	− 0.846649i	0.905978	−	0.423324i	$-0.139137\pi$
$114$	2.00000	0.187317
$115$	0	0
$116$	3.00000	0.278543
$117$	− 2.00000i	− 0.184900i
$118$	− 6.00000i	− 0.552345i
$119$	−15.0000	−1.37505
$120$	0	0
$121$	−11.0000	−1.00000
$122$	2.00000i	0.181071i
$123$	0	0
$124$	−2.00000	−0.179605
$125$	0	0
$126$	−5.00000	−0.445435
$127$	16.0000i	1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$	16.0000i	1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−2.00000	−0.176090
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	10.0000i	0.867110i
$134$	2.00000	0.172774
$135$	0	0
$136$	−3.00000	−0.257248
$137$	15.0000i	1.28154i	0.767734	+	0.640768i	$0.221384\pi$
$137$	15.0000i	1.28154i	−0.767734	+	0.640768i	$0.778616\pi$
$138$	− 1.00000i	− 0.0851257i
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	− 15.0000i	− 1.25877i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−11.0000	−0.910366
$147$	− 18.0000i	− 1.48461i
$148$	− 7.00000i	− 0.575396i
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	2.00000i	0.162221i
$153$	3.00000i	0.242536i
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	− 8.00000i	− 0.636446i
$159$	12.0000	0.951662
$160$	0	0
$161$	5.00000	0.394055
$162$	1.00000i	0.0785674i
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	0	0
$165$	0	0
$166$	9.00000	0.698535
$167$	− 15.0000i	− 1.16073i	−0.814355	−	0.580367i	$-0.802909\pi$
$167$	− 15.0000i	− 1.16073i	0.814355	−	0.580367i	$-0.197091\pi$
$168$	− 5.00000i	− 0.385758i
$169$	9.00000	0.692308
$170$	0	0
$171$	2.00000	0.152944
$172$	− 2.00000i	− 0.152499i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	3.00000	0.227429
$175$	0	0
$176$	0	0
$177$	− 6.00000i	− 0.450988i
$178$	− 3.00000i	− 0.224860i
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	10.0000i	0.741249i
$183$	2.00000i	0.147844i
$184$	1.00000	0.0737210
$185$	0	0
$186$	−2.00000	−0.146647
$187$	0	0
$188$	− 3.00000i	− 0.218797i
$189$	−5.00000	−0.363696
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 1.00000i	− 0.0719816i	−0.999352	−	0.0359908i	$-0.988541\pi$
$193$	− 1.00000i	− 0.0719816i	0.999352	−	0.0359908i	$-0.0114587\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	18.0000	1.28571
$197$	− 3.00000i	− 0.213741i	−0.994273	−	0.106871i	$-0.965917\pi$
$197$	− 3.00000i	− 0.213741i	0.994273	−	0.106871i	$-0.0340831\pi$
$198$	0	0
$199$	19.0000	1.34687	0.673437	−	0.739244i	$-0.264817\pi$
$199$	19.0000	1.34687	0.673437	+	0.739244i	$0.264817\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	− 3.00000i	− 0.211079i
$203$	15.0000i	1.05279i
$204$	−3.00000	−0.210042
$205$	0	0
$206$	13.0000	0.905753
$207$	− 1.00000i	− 0.0695048i
$208$	2.00000i	0.138675i
$209$	0	0
$210$	0	0
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	12.0000i	0.824163i
$213$	− 15.0000i	− 1.02778i
$214$	12.0000	0.820303
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	− 10.0000i	− 0.678844i
$218$	− 17.0000i	− 1.15139i
$219$	−11.0000	−0.743311
$220$	0	0
$221$	6.00000	0.403604
$222$	− 7.00000i	− 0.469809i
$223$	26.0000i	1.74109i	0.492090	+	0.870544i	$0.336233\pi$
$223$	26.0000i	1.74109i	−0.492090	+	0.870544i	$0.663767\pi$
$224$	5.00000	0.334077
$225$	0	0
$226$	9.00000	0.598671
$227$	− 3.00000i	− 0.199117i	−0.995032	−	0.0995585i	$-0.968257\pi$
$227$	− 3.00000i	− 0.199117i	0.995032	−	0.0995585i	$-0.0317430\pi$
$228$	2.00000i	0.132453i
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	3.00000i	0.196960i
$233$	12.0000i	0.786146i	0.919507	+	0.393073i	$0.128588\pi$
$233$	12.0000i	0.786146i	−0.919507	+	0.393073i	$0.871412\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	6.00000	0.390567
$237$	− 8.00000i	− 0.519656i
$238$	− 15.0000i	− 0.972306i
$239$	15.0000	0.970269	0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000	0.970269	0.485135	+	0.874439i	$0.338771\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	− 11.0000i	− 0.707107i
$243$	1.00000i	0.0641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	0	0
$247$	− 4.00000i	− 0.254514i
$248$	− 2.00000i	− 0.127000i
$249$	9.00000	0.570352
$250$	0	0
$251$	3.00000	0.189358	0.0946792	−	0.995508i	$-0.469817\pi$
$251$	3.00000	0.189358	0.0946792	+	0.995508i	$0.469817\pi$
$252$	− 5.00000i	− 0.314970i
$253$	0	0
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	− 2.00000i	− 0.124515i
$259$	35.0000	2.17479
$260$	0	0
$261$	3.00000	0.185695
$262$	− 12.0000i	− 0.741362i
$263$	− 18.0000i	− 1.10993i	−0.831875	−	0.554964i	$-0.812732\pi$
$263$	− 18.0000i	− 1.10993i	0.831875	−	0.554964i	$-0.187268\pi$
$264$	0	0
$265$	0	0
$266$	−10.0000	−0.613139
$267$	− 3.00000i	− 0.183597i
$268$	2.00000i	0.122169i
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	− 3.00000i	− 0.181902i
$273$	10.0000i	0.605228i
$274$	−15.0000	−0.906183
$275$	0	0
$276$	1.00000	0.0601929
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	− 5.00000i	− 0.299880i
$279$	−2.00000	−0.119737
$280$	0	0
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	− 3.00000i	− 0.178647i
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	15.0000	0.890086
$285$	0	0
$286$	0	0
$287$	0	0
$288$	− 1.00000i	− 0.0589256i
$289$	8.00000	0.470588
$290$	0	0
$291$	−10.0000	−0.586210
$292$	− 11.0000i	− 0.643726i
$293$	− 30.0000i	− 1.75262i	−0.481749	−	0.876309i	$-0.659998\pi$
$293$	− 30.0000i	− 1.75262i	0.481749	−	0.876309i	$-0.340002\pi$
$294$	18.0000	1.04978
$295$	0	0
$296$	7.00000	0.406867
$297$	0	0
$298$	− 12.0000i	− 0.695141i
$299$	−2.00000	−0.115663
$300$	0	0
$301$	10.0000	0.576390
$302$	− 4.00000i	− 0.230174i
$303$	− 3.00000i	− 0.172345i
$304$	−2.00000	−0.114708
$305$	0	0
$306$	−3.00000	−0.171499
$307$	− 29.0000i	− 1.65512i	−0.561379	−	0.827559i	$-0.689729\pi$
$307$	− 29.0000i	− 1.65512i	0.561379	−	0.827559i	$-0.310271\pi$
$308$	0	0
$309$	13.0000	0.739544
$310$	0	0
$311$	27.0000	1.53103	0.765515	−	0.643418i	$-0.222484\pi$
$311$	27.0000	1.53103	0.765515	+	0.643418i	$0.222484\pi$
$312$	2.00000i	0.113228i
$313$	8.00000i	0.452187i	0.974106	+	0.226093i	$0.0725954\pi$
$313$	8.00000i	0.452187i	−0.974106	+	0.226093i	$0.927405\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	8.00000	0.450035
$317$	9.00000i	0.505490i	0.967533	+	0.252745i	$0.0813334\pi$
$317$	9.00000i	0.505490i	−0.967533	+	0.252745i	$0.918667\pi$
$318$	12.0000i	0.672927i
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	5.00000i	0.278639i
$323$	6.00000i	0.333849i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	16.0000	0.886158
$327$	− 17.0000i	− 0.940102i
$328$	0	0
$329$	15.0000	0.826977
$330$	0	0
$331$	−19.0000	−1.04433	−0.522167	−	0.852843i	$-0.674876\pi$
$331$	−19.0000	−1.04433	−0.522167	+	0.852843i	$0.674876\pi$
$332$	9.00000i	0.493939i
$333$	− 7.00000i	− 0.383598i
$334$	15.0000	0.820763
$335$	0	0
$336$	5.00000	0.272772
$337$	− 20.0000i	− 1.08947i	−0.838608	−	0.544735i	$-0.816630\pi$
$337$	− 20.0000i	− 1.08947i	0.838608	−	0.544735i	$-0.183370\pi$
$338$	9.00000i	0.489535i
$339$	9.00000	0.488813
$340$	0	0
$341$	0	0
$342$	2.00000i	0.108148i
$343$	55.0000i	2.96972i
$344$	2.00000	0.107833
$345$	0	0
$346$	−6.00000	−0.322562
$347$	18.0000i	0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$	18.0000i	0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$	3.00000i	0.160817i
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	6.00000	0.318896
$355$	0	0
$356$	3.00000	0.159000
$357$	− 15.0000i	− 0.793884i
$358$	− 12.0000i	− 0.634220i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	5.00000i	0.262794i
$363$	− 11.0000i	− 0.577350i
$364$	−10.0000	−0.524142
$365$	0	0
$366$	−2.00000	−0.104542
$367$	16.0000i	0.835193i	0.908633	+	0.417597i	$0.137127\pi$
$367$	16.0000i	0.835193i	−0.908633	+	0.417597i	$0.862873\pi$
$368$	1.00000i	0.0521286i
$369$	0	0
$370$	0	0
$371$	−60.0000	−3.11504
$372$	− 2.00000i	− 0.103695i
$373$	17.0000i	0.880227i	0.897942	+	0.440113i	$0.145062\pi$
$373$	17.0000i	0.880227i	−0.897942	+	0.440113i	$0.854938\pi$
$374$	0	0
$375$	0	0
$376$	3.00000	0.154713
$377$	− 6.00000i	− 0.309016i
$378$	− 5.00000i	− 0.257172i
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	− 6.00000i	− 0.306987i
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	1.00000	0.0508987
$387$	− 2.00000i	− 0.101666i
$388$	− 10.0000i	− 0.507673i
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	18.0000i	0.909137i
$393$	− 12.0000i	− 0.605320i
$394$	3.00000	0.151138
$395$	0	0
$396$	0	0
$397$	− 20.0000i	− 1.00377i	−0.864934	−	0.501886i	$-0.832640\pi$
$397$	− 20.0000i	− 1.00377i	0.864934	−	0.501886i	$-0.167360\pi$
$398$	19.0000i	0.952384i
$399$	−10.0000	−0.500626
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	2.00000i	0.0997509i
$403$	4.00000i	0.199254i
$404$	3.00000	0.149256
$405$	0	0
$406$	−15.0000	−0.744438
$407$	0	0
$408$	− 3.00000i	− 0.148522i
$409$	7.00000	0.346128	0.173064	−	0.984911i	$-0.444633\pi$
$409$	7.00000	0.346128	0.173064	+	0.984911i	$0.444633\pi$
$410$	0	0
$411$	−15.0000	−0.739895
$412$	13.0000i	0.640464i
$413$	30.0000i	1.47620i
$414$	1.00000	0.0491473
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	− 5.00000i	− 0.244851i
$418$	0	0
$419$	−15.0000	−0.732798	−0.366399	−	0.930458i	$-0.619409\pi$
$419$	−15.0000	−0.732798	−0.366399	+	0.930458i	$0.619409\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	23.0000i	1.11962i
$423$	− 3.00000i	− 0.145865i
$424$	−12.0000	−0.582772
$425$	0	0
$426$	15.0000	0.726752
$427$	− 10.0000i	− 0.483934i
$428$	12.0000i	0.580042i
$429$	0	0
$430$	0	0
$431$	6.00000	0.289010	0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000	0.289010	0.144505	+	0.989504i	$0.453841\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	2.00000i	0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$	2.00000i	0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$	10.0000	0.480015
$435$	0	0
$436$	17.0000	0.814152
$437$	− 2.00000i	− 0.0956730i
$438$	− 11.0000i	− 0.525600i
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	18.0000	0.857143
$442$	6.00000i	0.285391i
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	7.00000	0.332205
$445$	0	0
$446$	−26.0000	−1.23114
$447$	− 12.0000i	− 0.567581i
$448$	5.00000i	0.236228i
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	0	0
$452$	9.00000i	0.423324i
$453$	− 4.00000i	− 0.187936i
$454$	3.00000	0.140797
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	28.0000i	1.30978i	0.755722	+	0.654892i	$0.227286\pi$
$457$	28.0000i	1.30978i	−0.755722	+	0.654892i	$0.772714\pi$
$458$	− 14.0000i	− 0.654177i
$459$	−3.00000	−0.140028
$460$	0	0
$461$	−15.0000	−0.698620	−0.349310	−	0.937007i	$-0.613584\pi$
$461$	−15.0000	−0.698620	−0.349310	+	0.937007i	$0.613584\pi$
$462$	0	0
$463$	− 40.0000i	− 1.85896i	−0.368875	−	0.929479i	$-0.620257\pi$
$463$	− 40.0000i	− 1.85896i	0.368875	−	0.929479i	$-0.379743\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−12.0000	−0.555889
$467$	− 15.0000i	− 0.694117i	−0.937843	−	0.347059i	$-0.887180\pi$
$467$	− 15.0000i	− 0.694117i	0.937843	−	0.347059i	$-0.112820\pi$
$468$	2.00000i	0.0924500i
$469$	−10.0000	−0.461757
$470$	0	0
$471$	2.00000	0.0921551
$472$	6.00000i	0.276172i
$473$	0	0
$474$	8.00000	0.367452
$475$	0	0
$476$	15.0000	0.687524
$477$	12.0000i	0.549442i
$478$	15.0000i	0.686084i
$479$	−30.0000	−1.37073	−0.685367	−	0.728197i	$-0.740358\pi$
$479$	−30.0000	−1.37073	−0.685367	+	0.728197i	$0.740358\pi$
$480$	0	0
$481$	−14.0000	−0.638345
$482$	8.00000i	0.364390i
$483$	5.00000i	0.227508i
$484$	11.0000	0.500000
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	− 2.00000i	− 0.0905357i
$489$	16.0000	0.723545
$490$	0	0
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	0	0
$493$	9.00000i	0.405340i
$494$	4.00000	0.179969
$495$	0	0
$496$	2.00000	0.0898027
$497$	75.0000i	3.36421i
$498$	9.00000i	0.403300i
$499$	−5.00000	−0.223831	−0.111915	−	0.993718i	$-0.535699\pi$
$499$	−5.00000	−0.223831	−0.111915	+	0.993718i	$0.535699\pi$
$500$	0	0
$501$	15.0000	0.670151
$502$	3.00000i	0.133897i
$503$	12.0000i	0.535054i	0.963550	+	0.267527i	$0.0862064\pi$
$503$	12.0000i	0.535054i	−0.963550	+	0.267527i	$0.913794\pi$
$504$	5.00000	0.222718
$505$	0	0
$506$	0	0
$507$	9.00000i	0.399704i
$508$	− 16.0000i	− 0.709885i
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	0	0
$511$	55.0000	2.43306
$512$	1.00000i	0.0441942i
$513$	2.00000i	0.0883022i
$514$	−18.0000	−0.793946
$515$	0	0
$516$	2.00000	0.0880451
$517$	0	0
$518$	35.0000i	1.53781i
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−39.0000	−1.70862	−0.854311	−	0.519763i	$-0.826020\pi$
$521$	−39.0000	−1.70862	−0.854311	+	0.519763i	$0.826020\pi$
$522$	3.00000i	0.131306i
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	18.0000	0.784837
$527$	− 6.00000i	− 0.261364i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	6.00000	0.260378
$532$	− 10.0000i	− 0.433555i
$533$	0	0
$534$	3.00000	0.129823
$535$	0	0
$536$	−2.00000	−0.0863868
$537$	− 12.0000i	− 0.517838i
$538$	− 18.0000i	− 0.776035i
$539$	0	0
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	− 16.0000i	− 0.687259i
$543$	5.00000i	0.214571i
$544$	3.00000	0.128624
$545$	0	0
$546$	−10.0000	−0.427960
$547$	1.00000i	0.0427569i	0.999771	+	0.0213785i	$0.00680549\pi$
$547$	1.00000i	0.0427569i	−0.999771	+	0.0213785i	$0.993195\pi$
$548$	− 15.0000i	− 0.640768i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	6.00000	0.255609
$552$	1.00000i	0.0425628i
$553$	40.0000i	1.70097i
$554$	2.00000	0.0849719
$555$	0	0
$556$	5.00000	0.212047
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	− 2.00000i	− 0.0846668i
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	27.0000i	1.13893i
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	3.00000	0.126323
$565$	0	0
$566$	4.00000	0.168133
$567$	− 5.00000i	− 0.209980i
$568$	15.0000i	0.629386i
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	26.0000	1.08807	0.544033	−	0.839064i	$-0.316897\pi$
$571$	26.0000	1.08807	0.544033	+	0.839064i	$0.316897\pi$
$572$	0	0
$573$	− 6.00000i	− 0.250654i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 2.00000i	− 0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$	− 2.00000i	− 0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$	8.00000i	0.332756i
$579$	1.00000	0.0415586
$580$	0	0
$581$	−45.0000	−1.86691
$582$	− 10.0000i	− 0.414513i
$583$	0	0
$584$	11.0000	0.455183
$585$	0	0
$586$	30.0000	1.23929
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	18.0000i	0.742307i
$589$	−4.00000	−0.164817
$590$	0	0
$591$	3.00000	0.123404
$592$	7.00000i	0.287698i
$593$	− 24.0000i	− 0.985562i	−0.870153	−	0.492781i	$-0.835980\pi$
$593$	− 24.0000i	− 0.985562i	0.870153	−	0.492781i	$-0.164020\pi$
$594$	0	0
$595$	0	0
$596$	12.0000	0.491539
$597$	19.0000i	0.777618i
$598$	− 2.00000i	− 0.0817861i
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−43.0000	−1.75401	−0.877003	−	0.480484i	$-0.840461\pi$
$601$	−43.0000	−1.75401	−0.877003	+	0.480484i	$0.840461\pi$
$602$	10.0000i	0.407570i
$603$	2.00000i	0.0814463i
$604$	4.00000	0.162758
$605$	0	0
$606$	3.00000	0.121867
$607$	− 14.0000i	− 0.568242i	−0.958788	−	0.284121i	$-0.908298\pi$
$607$	− 14.0000i	− 0.568242i	0.958788	−	0.284121i	$-0.0917018\pi$
$608$	− 2.00000i	− 0.0811107i
$609$	−15.0000	−0.607831
$610$	0	0
$611$	−6.00000	−0.242734
$612$	− 3.00000i	− 0.121268i
$613$	− 25.0000i	− 1.00974i	−0.863195	−	0.504870i	$-0.831540\pi$
$613$	− 25.0000i	− 1.00974i	0.863195	−	0.504870i	$-0.168460\pi$
$614$	29.0000	1.17034
$615$	0	0
$616$	0	0
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	13.0000i	0.522937i
$619$	40.0000	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	27.0000i	1.08260i
$623$	15.0000i	0.600962i
$624$	−2.00000	−0.0800641
$625$	0	0
$626$	−8.00000	−0.319744
$627$	0	0
$628$	2.00000i	0.0798087i
$629$	21.0000	0.837325
$630$	0	0
$631$	−1.00000	−0.0398094	−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	−1.00000	−0.0398094	−0.0199047	+	0.999802i	$0.506336\pi$
$632$	8.00000i	0.318223i
$633$	23.0000i	0.914168i
$634$	−9.00000	−0.357436
$635$	0	0
$636$	−12.0000	−0.475831
$637$	− 36.0000i	− 1.42637i
$638$	0	0
$639$	15.0000	0.593391
$640$	0	0
$641$	3.00000	0.118493	0.0592464	−	0.998243i	$-0.481130\pi$
$641$	3.00000	0.118493	0.0592464	+	0.998243i	$0.481130\pi$
$642$	12.0000i	0.473602i
$643$	14.0000i	0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$	14.0000i	0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$	−5.00000	−0.197028
$645$	0	0
$646$	−6.00000	−0.236067
$647$	9.00000i	0.353827i	0.984226	+	0.176913i	$0.0566112\pi$
$647$	9.00000i	0.353827i	−0.984226	+	0.176913i	$0.943389\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	0	0
$650$	0	0
$651$	10.0000	0.391931
$652$	16.0000i	0.626608i
$653$	− 45.0000i	− 1.76099i	−0.474059	−	0.880493i	$-0.657212\pi$
$653$	− 45.0000i	− 1.76099i	0.474059	−	0.880493i	$-0.342788\pi$
$654$	17.0000	0.664753
$655$	0	0
$656$	0	0
$657$	− 11.0000i	− 0.429151i
$658$	15.0000i	0.584761i
$659$	33.0000	1.28550	0.642749	−	0.766077i	$-0.277794\pi$
$659$	33.0000	1.28550	0.642749	+	0.766077i	$0.277794\pi$
$660$	0	0
$661$	−13.0000	−0.505641	−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000	−0.505641	−0.252821	+	0.967513i	$0.581358\pi$
$662$	− 19.0000i	− 0.738456i
$663$	6.00000i	0.233021i
$664$	−9.00000	−0.349268
$665$	0	0
$666$	7.00000	0.271244
$667$	− 3.00000i	− 0.116160i
$668$	15.0000i	0.580367i
$669$	−26.0000	−1.00522
$670$	0	0
$671$	0	0
$672$	5.00000i	0.192879i
$673$	17.0000i	0.655302i	0.944799	+	0.327651i	$0.106257\pi$
$673$	17.0000i	0.655302i	−0.944799	+	0.327651i	$0.893743\pi$
$674$	20.0000	0.770371
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 48.0000i	− 1.84479i	−0.386248	−	0.922395i	$-0.626229\pi$
$677$	− 48.0000i	− 1.84479i	0.386248	−	0.922395i	$-0.373771\pi$
$678$	9.00000i	0.345643i
$679$	50.0000	1.91882
$680$	0	0
$681$	3.00000	0.114960
$682$	0	0
$683$	24.0000i	0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$	24.0000i	0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$	−2.00000	−0.0764719
$685$	0	0
$686$	−55.0000	−2.09991
$687$	− 14.0000i	− 0.534133i
$688$	2.00000i	0.0762493i
$689$	24.0000	0.914327
$690$	0	0
$691$	35.0000	1.33146	0.665731	−	0.746191i	$-0.268120\pi$
$691$	35.0000	1.33146	0.665731	+	0.746191i	$0.268120\pi$
$692$	− 6.00000i	− 0.228086i
$693$	0	0
$694$	−18.0000	−0.683271
$695$	0	0
$696$	−3.00000	−0.113715
$697$	0	0
$698$	− 26.0000i	− 0.984115i
$699$	−12.0000	−0.453882
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	2.00000i	0.0754851i
$703$	− 14.0000i	− 0.528020i
$704$	0	0
$705$	0	0
$706$	6.00000	0.225813
$707$	15.0000i	0.564133i
$708$	6.00000i	0.225494i
$709$	49.0000	1.84023	0.920117	−	0.391644i	$-0.128094\pi$
$709$	49.0000	1.84023	0.920117	+	0.391644i	$0.128094\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	3.00000i	0.112430i
$713$	2.00000i	0.0749006i
$714$	15.0000	0.561361
$715$	0	0
$716$	12.0000	0.448461
$717$	15.0000i	0.560185i
$718$	0	0
$719$	−3.00000	−0.111881	−0.0559406	−	0.998434i	$-0.517816\pi$
$719$	−3.00000	−0.111881	−0.0559406	+	0.998434i	$0.517816\pi$
$720$	0	0
$721$	−65.0000	−2.42073
$722$	− 15.0000i	− 0.558242i
$723$	8.00000i	0.297523i
$724$	−5.00000	−0.185824
$725$	0	0
$726$	11.0000	0.408248
$727$	40.0000i	1.48352i	0.670667	+	0.741759i	$0.266008\pi$
$727$	40.0000i	1.48352i	−0.670667	+	0.741759i	$0.733992\pi$
$728$	− 10.0000i	− 0.370625i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	6.00000	0.221918
$732$	− 2.00000i	− 0.0739221i
$733$	− 49.0000i	− 1.80986i	−0.425564	−	0.904928i	$-0.639924\pi$
$733$	− 49.0000i	− 1.80986i	0.425564	−	0.904928i	$-0.360076\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	0	0
$738$	0	0
$739$	25.0000	0.919640	0.459820	−	0.888012i	$-0.347914\pi$
$739$	25.0000	0.919640	0.459820	+	0.888012i	$0.347914\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	− 60.0000i	− 2.20267i
$743$	− 6.00000i	− 0.220119i	−0.993925	−	0.110059i	$-0.964896\pi$
$743$	− 6.00000i	− 0.220119i	0.993925	−	0.110059i	$-0.0351041\pi$
$744$	2.00000	0.0733236
$745$	0	0
$746$	−17.0000	−0.622414
$747$	9.00000i	0.329293i
$748$	0	0
$749$	−60.0000	−2.19235
$750$	0	0
$751$	41.0000	1.49611	0.748056	−	0.663636i	$-0.230988\pi$
$751$	41.0000	1.49611	0.748056	+	0.663636i	$0.230988\pi$
$752$	3.00000i	0.109399i
$753$	3.00000i	0.109326i
$754$	6.00000	0.218507
$755$	0	0
$756$	5.00000	0.181848
$757$	7.00000i	0.254419i	0.991876	+	0.127210i	$0.0406021\pi$
$757$	7.00000i	0.254419i	−0.991876	+	0.127210i	$0.959398\pi$
$758$	16.0000i	0.581146i
$759$	0	0
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	− 16.0000i	− 0.579619i
$763$	85.0000i	3.07721i
$764$	6.00000	0.217072
$765$	0	0
$766$	−24.0000	−0.867155
$767$	− 12.0000i	− 0.433295i
$768$	1.00000i	0.0360844i
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	1.00000i	0.0359908i
$773$	− 42.0000i	− 1.51064i	−0.655359	−	0.755318i	$-0.727483\pi$
$773$	− 42.0000i	− 1.51064i	0.655359	−	0.755318i	$-0.272517\pi$
$774$	2.00000	0.0718885
$775$	0	0
$776$	10.0000	0.358979
$777$	35.0000i	1.25562i
$778$	− 18.0000i	− 0.645331i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	3.00000i	0.107280i
$783$	3.00000i	0.107211i
$784$	−18.0000	−0.642857
$785$	0	0
$786$	12.0000	0.428026
$787$	− 32.0000i	− 1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$	− 32.0000i	− 1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$	3.00000i	0.106871i
$789$	18.0000	0.640817
$790$	0	0
$791$	−45.0000	−1.60002
$792$	0	0
$793$	4.00000i	0.142044i
$794$	20.0000	0.709773
$795$	0	0
$796$	−19.0000	−0.673437
$797$	42.0000i	1.48772i	0.668338	+	0.743858i	$0.267006\pi$
$797$	42.0000i	1.48772i	−0.668338	+	0.743858i	$0.732994\pi$
$798$	− 10.0000i	− 0.353996i
$799$	9.00000	0.318397
$800$	0	0
$801$	3.00000	0.106000
$802$	6.00000i	0.211867i
$803$	0	0
$804$	−2.00000	−0.0705346
$805$	0	0
$806$	−4.00000	−0.140894
$807$	− 18.0000i	− 0.633630i
$808$	3.00000i	0.105540i
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	− 15.0000i	− 0.526397i
$813$	− 16.0000i	− 0.561144i
$814$	0	0
$815$	0	0
$816$	3.00000	0.105021
$817$	− 4.00000i	− 0.139942i
$818$	7.00000i	0.244749i
$819$	−10.0000	−0.349428
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	− 15.0000i	− 0.523185i
$823$	− 28.0000i	− 0.976019i	−0.872838	−	0.488009i	$-0.837723\pi$
$823$	− 28.0000i	− 0.976019i	0.872838	−	0.488009i	$-0.162277\pi$
$824$	−13.0000	−0.452876
$825$	0	0
$826$	−30.0000	−1.04383
$827$	9.00000i	0.312961i	0.987681	+	0.156480i	$0.0500148\pi$
$827$	9.00000i	0.312961i	−0.987681	+	0.156480i	$0.949985\pi$
$828$	1.00000i	0.0347524i
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	− 2.00000i	− 0.0693375i
$833$	54.0000i	1.87099i
$834$	5.00000	0.173136
$835$	0	0
$836$	0	0
$837$	− 2.00000i	− 0.0691301i
$838$	− 15.0000i	− 0.518166i
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	− 10.0000i	− 0.344623i
$843$	27.0000i	0.929929i
$844$	−23.0000	−0.791693
$845$	0	0
$846$	3.00000	0.103142
$847$	55.0000i	1.88982i
$848$	− 12.0000i	− 0.412082i
$849$	4.00000	0.137280
$850$	0	0
$851$	−7.00000	−0.239957
$852$	15.0000i	0.513892i
$853$	− 4.00000i	− 0.136957i	−0.997653	−	0.0684787i	$-0.978185\pi$
$853$	− 4.00000i	− 0.136957i	0.997653	−	0.0684787i	$-0.0218145\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	−12.0000	−0.410152
$857$	− 54.0000i	− 1.84460i	−0.386469	−	0.922302i	$-0.626305\pi$
$857$	− 54.0000i	− 1.84460i	0.386469	−	0.922302i	$-0.373695\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	0	0
$862$	6.00000i	0.204361i
$863$	33.0000i	1.12333i	0.827364	+	0.561667i	$0.189840\pi$
$863$	33.0000i	1.12333i	−0.827364	+	0.561667i	$0.810160\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	8.00000i	0.271694i
$868$	10.0000i	0.339422i
$869$	0	0
$870$	0	0
$871$	4.00000	0.135535
$872$	17.0000i	0.575693i
$873$	− 10.0000i	− 0.338449i
$874$	2.00000	0.0676510
$875$	0	0
$876$	11.0000	0.371656
$877$	22.0000i	0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$	22.0000i	0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$	28.0000i	0.944954i
$879$	30.0000	1.01187
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	18.0000i	0.606092i
$883$	11.0000i	0.370179i	0.982722	+	0.185090i	$0.0592576\pi$
$883$	11.0000i	0.370179i	−0.982722	+	0.185090i	$0.940742\pi$
$884$	−6.00000	−0.201802
$885$	0	0
$886$	24.0000	0.806296
$887$	27.0000i	0.906571i	0.891365	+	0.453286i	$0.149748\pi$
$887$	27.0000i	0.906571i	−0.891365	+	0.453286i	$0.850252\pi$
$888$	7.00000i	0.234905i
$889$	80.0000	2.68311
$890$	0	0
$891$	0	0
$892$	− 26.0000i	− 0.870544i
$893$	− 6.00000i	− 0.200782i
$894$	12.0000	0.401340
$895$	0	0
$896$	−5.00000	−0.167038
$897$	− 2.00000i	− 0.0667781i
$898$	− 30.0000i	− 1.00111i
$899$	−6.00000	−0.200111
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	10.0000i	0.332779i
$904$	−9.00000	−0.299336
$905$	0	0
$906$	4.00000	0.132891
$907$	− 26.0000i	− 0.863316i	−0.902037	−	0.431658i	$-0.857929\pi$
$907$	− 26.0000i	− 0.863316i	0.902037	−	0.431658i	$-0.142071\pi$
$908$	3.00000i	0.0995585i
$909$	3.00000	0.0995037
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	− 2.00000i	− 0.0662266i
$913$	0	0
$914$	−28.0000	−0.926158
$915$	0	0
$916$	14.0000	0.462573
$917$	60.0000i	1.98137i
$918$	− 3.00000i	− 0.0990148i
$919$	−11.0000	−0.362857	−0.181428	−	0.983404i	$-0.558072\pi$
$919$	−11.0000	−0.362857	−0.181428	+	0.983404i	$0.558072\pi$
$920$	0	0
$921$	29.0000	0.955582
$922$	− 15.0000i	− 0.493999i
$923$	− 30.0000i	− 0.987462i
$924$	0	0
$925$	0	0
$926$	40.0000	1.31448
$927$	13.0000i	0.426976i
$928$	− 3.00000i	− 0.0984798i
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	− 12.0000i	− 0.393073i
$933$	27.0000i	0.883940i
$934$	15.0000	0.490815
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	58.0000i	1.89478i	0.320085	+	0.947389i	$0.396288\pi$
$937$	58.0000i	1.89478i	−0.320085	+	0.947389i	$0.603712\pi$
$938$	− 10.0000i	− 0.326512i
$939$	−8.00000	−0.261070
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	2.00000i	0.0651635i
$943$	0	0
$944$	−6.00000	−0.195283
$945$	0	0
$946$	0	0
$947$	24.0000i	0.779895i	0.920837	+	0.389948i	$0.127507\pi$
$947$	24.0000i	0.779895i	−0.920837	+	0.389948i	$0.872493\pi$
$948$	8.00000i	0.259828i
$949$	−22.0000	−0.714150
$950$	0	0
$951$	−9.00000	−0.291845
$952$	15.0000i	0.486153i
$953$	− 33.0000i	− 1.06897i	−0.845176	−	0.534487i	$-0.820505\pi$
$953$	− 33.0000i	− 1.06897i	0.845176	−	0.534487i	$-0.179495\pi$
$954$	−12.0000	−0.388514
$955$	0	0
$956$	−15.0000	−0.485135
$957$	0	0
$958$	− 30.0000i	− 0.969256i
$959$	75.0000	2.42188
$960$	0	0
$961$	−27.0000	−0.870968
$962$	− 14.0000i	− 0.451378i
$963$	12.0000i	0.386695i
$964$	−8.00000	−0.257663
$965$	0	0
$966$	−5.00000	−0.160872
$967$	58.0000i	1.86515i	0.360971	+	0.932577i	$0.382445\pi$
$967$	58.0000i	1.86515i	−0.360971	+	0.932577i	$0.617555\pi$
$968$	11.0000i	0.353553i
$969$	−6.00000	−0.192748
$970$	0	0
$971$	27.0000	0.866471	0.433236	−	0.901281i	$-0.357372\pi$
$971$	27.0000	0.866471	0.433236	+	0.901281i	$0.357372\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	25.0000i	0.801463i
$974$	8.00000	0.256337
$975$	0	0
$976$	2.00000	0.0640184
$977$	51.0000i	1.63163i	0.578310	+	0.815817i	$0.303713\pi$
$977$	51.0000i	1.63163i	−0.578310	+	0.815817i	$0.696287\pi$
$978$	16.0000i	0.511624i
$979$	0	0
$980$	0	0
$981$	17.0000	0.542768
$982$	− 6.00000i	− 0.191468i
$983$	− 48.0000i	− 1.53096i	−0.643458	−	0.765481i	$-0.722501\pi$
$983$	− 48.0000i	− 1.53096i	0.643458	−	0.765481i	$-0.277499\pi$
$984$	0	0
$985$	0	0
$986$	−9.00000	−0.286618
$987$	15.0000i	0.477455i
$988$	4.00000i	0.127257i
$989$	−2.00000	−0.0635963
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	2.00000i	0.0635001i
$993$	− 19.0000i	− 0.602947i
$994$	−75.0000	−2.37886
$995$	0	0
$996$	−9.00000	−0.285176
$997$	− 26.0000i	− 0.823428i	−0.911313	−	0.411714i	$-0.864930\pi$
$997$	− 26.0000i	− 0.823428i	0.911313	−	0.411714i	$-0.135070\pi$
$998$	− 5.00000i	− 0.158272i
$999$	7.00000	0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.d.h.2899.2		2
5.2	odd	4		3450.2.a.m.1.1	✓	1
5.3	odd	4		3450.2.a.n.1.1	yes	1
5.4	even	2	inner	3450.2.d.h.2899.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3450.2.a.m.1.1	✓	1	5.2	odd	4
3450.2.a.n.1.1	yes	1	5.3	odd	4
3450.2.d.h.2899.1		2	5.4	even	2	inner
3450.2.d.h.2899.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -5.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -5.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} +2.00000i q^{13} +5.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} -1.00000i q^{18} -2.00000 q^{19} +5.00000 q^{21} +1.00000i q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +5.00000i q^{28} -3.00000 q^{29} +2.00000 q^{31} +1.00000i q^{32} +3.00000 q^{34} +1.00000 q^{36} +7.00000i q^{37} -2.00000i q^{38} -2.00000 q^{39} +5.00000i q^{42} +2.00000i q^{43} -1.00000 q^{46} +3.00000i q^{47} +1.00000i q^{48} -18.0000 q^{49} +3.00000 q^{51} -2.00000i q^{52} -12.0000i q^{53} +1.00000 q^{54} -5.00000 q^{56} -2.00000i q^{57} -3.00000i q^{58} -6.00000 q^{59} +2.00000 q^{61} +2.00000i q^{62} +5.00000i q^{63} -1.00000 q^{64} -2.00000i q^{67} +3.00000i q^{68} -1.00000 q^{69} -15.0000 q^{71} +1.00000i q^{72} +11.0000i q^{73} -7.00000 q^{74} +2.00000 q^{76} -2.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} -9.00000i q^{83} -5.00000 q^{84} -2.00000 q^{86} -3.00000i q^{87} -3.00000 q^{89} +10.0000 q^{91} -1.00000i q^{92} +2.00000i q^{93} -3.00000 q^{94} -1.00000 q^{96} +10.0000i q^{97} -18.0000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 10 q^{14} + 2 q^{16} - 4 q^{19} + 10 q^{21} + 2 q^{24} - 4 q^{26} - 6 q^{29} + 4 q^{31} + 6 q^{34} + 2 q^{36} - 4 q^{39} - 2 q^{46} - 36 q^{49} + 6 q^{51} + 2 q^{54} - 10 q^{56} - 12 q^{59} + 4 q^{61} - 2 q^{64} - 2 q^{69} - 30 q^{71} - 14 q^{74} + 4 q^{76} - 16 q^{79} + 2 q^{81} - 10 q^{84} - 4 q^{86} - 6 q^{89} + 20 q^{91} - 6 q^{94} - 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 10 * q^14 + 2 * q^16 - 4 * q^19 + 10 * q^21 + 2 * q^24 - 4 * q^26 - 6 * q^29 + 4 * q^31 + 6 * q^34 + 2 * q^36 - 4 * q^39 - 2 * q^46 - 36 * q^49 + 6 * q^51 + 2 * q^54 - 10 * q^56 - 12 * q^59 + 4 * q^61 - 2 * q^64 - 2 * q^69 - 30 * q^71 - 14 * q^74 + 4 * q^76 - 16 * q^79 + 2 * q^81 - 10 * q^84 - 4 * q^86 - 6 * q^89 + 20 * q^91 - 6 * q^94 - 2 * q^96

Introduction

L-functions

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Varieties

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