LMFDB - Embedded newform 2550.2.d.o.2449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2550,2,Mod(2449,2550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, 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("2550.2449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2550.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:	$20.3618525154$
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			2449.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2550.2449
Dual form			2550.2.d.o.2449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} +1.00000i q^{12} -4.00000i q^{13} -3.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} +5.00000 q^{19} +3.00000 q^{21} -3.00000i q^{22} -4.00000i q^{23} -1.00000 q^{24} +4.00000 q^{26} +1.00000i q^{27} -3.00000i q^{28} +7.00000 q^{31} +1.00000i q^{32} +3.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} +3.00000i q^{37} +5.00000i q^{38} -4.00000 q^{39} +2.00000 q^{41} +3.00000i q^{42} +1.00000i q^{43} +3.00000 q^{44} +4.00000 q^{46} +3.00000i q^{47} -1.00000i q^{48} -2.00000 q^{49} +1.00000 q^{51} +4.00000i q^{52} +11.0000i q^{53} -1.00000 q^{54} +3.00000 q^{56} -5.00000i q^{57} +2.00000 q^{61} +7.00000i q^{62} -3.00000i q^{63} -1.00000 q^{64} -3.00000 q^{66} +13.0000i q^{67} -1.00000i q^{68} -4.00000 q^{69} +2.00000 q^{71} +1.00000i q^{72} +6.00000i q^{73} -3.00000 q^{74} -5.00000 q^{76} -9.00000i q^{77} -4.00000i q^{78} +5.00000 q^{79} +1.00000 q^{81} +2.00000i q^{82} +16.0000i q^{83} -3.00000 q^{84} -1.00000 q^{86} +3.00000i q^{88} +10.0000 q^{89} +12.0000 q^{91} +4.00000i q^{92} -7.00000i q^{93} -3.00000 q^{94} +1.00000 q^{96} -2.00000i q^{97} -2.00000i q^{98} +3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 6 q^{11} - 6 q^{14} + 2 q^{16} + 10 q^{19} + 6 q^{21} - 2 q^{24} + 8 q^{26} + 14 q^{31} - 2 q^{34} + 2 q^{36} - 8 q^{39} + 4 q^{41} + 6 q^{44} + 8 q^{46} - 4 q^{49} + 2 q^{51} - 2 q^{54} + 6 q^{56} + 4 q^{61} - 2 q^{64} - 6 q^{66} - 8 q^{69} + 4 q^{71} - 6 q^{74} - 10 q^{76} + 10 q^{79} + 2 q^{81} - 6 q^{84} - 2 q^{86} + 20 q^{89} + 24 q^{91} - 6 q^{94} + 2 q^{96} + 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 6 * q^11 - 6 * q^14 + 2 * q^16 + 10 * q^19 + 6 * q^21 - 2 * q^24 + 8 * q^26 + 14 * q^31 - 2 * q^34 + 2 * q^36 - 8 * q^39 + 4 * q^41 + 6 * q^44 + 8 * q^46 - 4 * q^49 + 2 * q^51 - 2 * q^54 + 6 * q^56 + 4 * q^61 - 2 * q^64 - 6 * q^66 - 8 * q^69 + 4 * q^71 - 6 * q^74 - 10 * q^76 + 10 * q^79 + 2 * q^81 - 6 * q^84 - 2 * q^86 + 20 * q^89 + 24 * q^91 - 6 * q^94 + 2 * q^96 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2550\mathbb{Z}\right)^\times$.

$n$	$751$	$851$	$1327$
$\chi(n)$	$1$	$1$	$-1$

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
$7$	3.00000i	1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$	3.00000i	1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	1.00000i	0.288675i
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	−3.00000	−0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000i	0.242536i
$17$	1.00000i	0.242536i
$18$	− 1.00000i	− 0.235702i
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	− 3.00000i	− 0.639602i
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	4.00000	0.784465
$27$	1.00000i	0.192450i
$28$	− 3.00000i	− 0.566947i
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	1.00000i	0.176777i
$33$	3.00000i	0.522233i
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	3.00000i	0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$	3.00000i	0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$	5.00000i	0.811107i
$39$	−4.00000	−0.640513
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	3.00000i	0.462910i
$43$	1.00000i	0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$	1.00000i	0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	4.00000	0.589768
$47$	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
$49$	−2.00000	−0.285714
$50$	0	0
$51$	1.00000	0.140028
$52$	4.00000i	0.554700i
$53$	11.0000i	1.51097i	0.655168	+	0.755483i	$0.272598\pi$
$53$	11.0000i	1.51097i	−0.655168	+	0.755483i	$0.727402\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	3.00000	0.400892
$57$	− 5.00000i	− 0.662266i
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	7.00000i	0.889001i
$63$	− 3.00000i	− 0.377964i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−3.00000	−0.369274
$67$	13.0000i	1.58820i	0.607785	+	0.794101i	$0.292058\pi$
$67$	13.0000i	1.58820i	−0.607785	+	0.794101i	$0.707942\pi$
$68$	− 1.00000i	− 0.121268i
$69$	−4.00000	−0.481543
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	1.00000i	0.117851i
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	−3.00000	−0.348743
$75$	0	0
$76$	−5.00000	−0.573539
$77$	− 9.00000i	− 1.02565i
$78$	− 4.00000i	− 0.452911i
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.00000i	0.220863i
$83$	16.0000i	1.75623i	0.478451	+	0.878114i	$0.341198\pi$
$83$	16.0000i	1.75623i	−0.478451	+	0.878114i	$0.658802\pi$
$84$	−3.00000	−0.327327
$85$	0	0
$86$	−1.00000	−0.107833
$87$	0	0
$88$	3.00000i	0.319801i
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	12.0000	1.25794
$92$	4.00000i	0.417029i
$93$	− 7.00000i	− 0.725866i
$94$	−3.00000	−0.309426
$95$	0	0
$96$	1.00000	0.102062
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	− 2.00000i	− 0.202031i
$99$	3.00000	0.301511
$100$	0	0
$101$	7.00000	0.696526	0.348263	−	0.937397i	$-0.386772\pi$
$101$	7.00000	0.696526	0.348263	+	0.937397i	$0.386772\pi$
$102$	1.00000i	0.0990148i
$103$	− 14.0000i	− 1.37946i	−0.724066	−	0.689730i	$-0.757729\pi$
$103$	− 14.0000i	− 1.37946i	0.724066	−	0.689730i	$-0.242271\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	−11.0000	−1.06841
$107$	− 17.0000i	− 1.64345i	−0.569883	−	0.821726i	$-0.693011\pi$
$107$	− 17.0000i	− 1.64345i	0.569883	−	0.821726i	$-0.306989\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	0	0
$111$	3.00000	0.284747
$112$	3.00000i	0.283473i
$113$	1.00000i	0.0940721i	0.998893	+	0.0470360i	$0.0149776\pi$
$113$	1.00000i	0.0940721i	−0.998893	+	0.0470360i	$0.985022\pi$
$114$	5.00000	0.468293
$115$	0	0
$116$	0	0
$117$	4.00000i	0.369800i
$118$	0	0
$119$	−3.00000	−0.275010
$120$	0	0
$121$	−2.00000	−0.181818
$122$	2.00000i	0.181071i
$123$	− 2.00000i	− 0.180334i
$124$	−7.00000	−0.628619
$125$	0	0
$126$	3.00000	0.267261
$127$	18.0000i	1.59724i	0.601834	+	0.798621i	$0.294437\pi$
$127$	18.0000i	1.59724i	−0.601834	+	0.798621i	$0.705563\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	1.00000	0.0880451
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	− 3.00000i	− 0.261116i
$133$	15.0000i	1.30066i
$134$	−13.0000	−1.12303
$135$	0	0
$136$	1.00000	0.0857493
$137$	− 2.00000i	− 0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$	− 2.00000i	− 0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$	− 4.00000i	− 0.340503i
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	2.00000i	0.167836i
$143$	12.0000i	1.00349i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−6.00000	−0.496564
$147$	2.00000i	0.164957i
$148$	− 3.00000i	− 0.246598i
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	− 5.00000i	− 0.405554i
$153$	− 1.00000i	− 0.0808452i
$154$	9.00000	0.725241
$155$	0	0
$156$	4.00000	0.320256
$157$	− 12.0000i	− 0.957704i	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	− 12.0000i	− 0.957704i	0.877896	−	0.478852i	$-0.158947\pi$
$158$	5.00000i	0.397779i
$159$	11.0000	0.872357
$160$	0	0
$161$	12.0000	0.945732
$162$	1.00000i	0.0785674i
$163$	6.00000i	0.469956i	0.972001	+	0.234978i	$0.0755019\pi$
$163$	6.00000i	0.469956i	−0.972001	+	0.234978i	$0.924498\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−16.0000	−1.24184
$167$	18.0000i	1.39288i	0.717614	+	0.696441i	$0.245234\pi$
$167$	18.0000i	1.39288i	−0.717614	+	0.696441i	$0.754766\pi$
$168$	− 3.00000i	− 0.231455i
$169$	−3.00000	−0.230769
$170$	0	0
$171$	−5.00000	−0.382360
$172$	− 1.00000i	− 0.0762493i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	−3.00000	−0.226134
$177$	0	0
$178$	10.0000i	0.749532i
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	12.0000i	0.889499i
$183$	− 2.00000i	− 0.147844i
$184$	−4.00000	−0.294884
$185$	0	0
$186$	7.00000	0.513265
$187$	− 3.00000i	− 0.219382i
$188$	− 3.00000i	− 0.218797i
$189$	−3.00000	−0.218218
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	1.00000i	0.0721688i
$193$	6.00000i	0.431889i	0.976406	+	0.215945i	$0.0692831\pi$
$193$	6.00000i	0.431889i	−0.976406	+	0.215945i	$0.930717\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	2.00000	0.142857
$197$	− 12.0000i	− 0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$	− 12.0000i	− 0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$	3.00000i	0.213201i
$199$	25.0000	1.77220	0.886102	−	0.463491i	$-0.153403\pi$
$199$	25.0000	1.77220	0.886102	+	0.463491i	$0.153403\pi$
$200$	0	0
$201$	13.0000	0.916949
$202$	7.00000i	0.492518i
$203$	0	0
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	14.0000	0.975426
$207$	4.00000i	0.278019i
$208$	− 4.00000i	− 0.277350i
$209$	−15.0000	−1.03757
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	− 11.0000i	− 0.755483i
$213$	− 2.00000i	− 0.137038i
$214$	17.0000	1.16210
$215$	0	0
$216$	1.00000	0.0680414
$217$	21.0000i	1.42557i
$218$	− 5.00000i	− 0.338643i
$219$	6.00000	0.405442
$220$	0	0
$221$	4.00000	0.269069
$222$	3.00000i	0.201347i
$223$	26.0000i	1.74109i	0.492090	+	0.870544i	$0.336233\pi$
$223$	26.0000i	1.74109i	−0.492090	+	0.870544i	$0.663767\pi$
$224$	−3.00000	−0.200446
$225$	0	0
$226$	−1.00000	−0.0665190
$227$	− 27.0000i	− 1.79205i	−0.444001	−	0.896026i	$-0.646441\pi$
$227$	− 27.0000i	− 1.79205i	0.444001	−	0.896026i	$-0.353559\pi$
$228$	5.00000i	0.331133i
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	−9.00000	−0.592157
$232$	0	0
$233$	26.0000i	1.70332i	0.524097	+	0.851658i	$0.324403\pi$
$233$	26.0000i	1.70332i	−0.524097	+	0.851658i	$0.675597\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	0	0
$237$	− 5.00000i	− 0.324785i
$238$	− 3.00000i	− 0.194461i
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	0	0
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	− 2.00000i	− 0.128565i
$243$	− 1.00000i	− 0.0641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	2.00000	0.127515
$247$	− 20.0000i	− 1.27257i
$248$	− 7.00000i	− 0.444500i
$249$	16.0000	1.01396
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	3.00000i	0.188982i
$253$	12.0000i	0.754434i
$254$	−18.0000	−1.12942
$255$	0	0
$256$	1.00000	0.0625000
$257$	8.00000i	0.499026i	0.968371	+	0.249513i	$0.0802706\pi$
$257$	8.00000i	0.499026i	−0.968371	+	0.249513i	$0.919729\pi$
$258$	1.00000i	0.0622573i
$259$	−9.00000	−0.559233
$260$	0	0
$261$	0	0
$262$	12.0000i	0.741362i
$263$	− 19.0000i	− 1.17159i	−0.810459	−	0.585795i	$-0.800782\pi$
$263$	− 19.0000i	− 1.17159i	0.810459	−	0.585795i	$-0.199218\pi$
$264$	3.00000	0.184637
$265$	0	0
$266$	−15.0000	−0.919709
$267$	− 10.0000i	− 0.611990i
$268$	− 13.0000i	− 0.794101i
$269$	−20.0000	−1.21942	−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000	−1.21942	−0.609711	+	0.792624i	$0.708714\pi$
$270$	0	0
$271$	32.0000	1.94386	0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000	1.94386	0.971931	+	0.235267i	$0.0755965\pi$
$272$	1.00000i	0.0606339i
$273$	− 12.0000i	− 0.726273i
$274$	2.00000	0.120824
$275$	0	0
$276$	4.00000	0.240772
$277$	23.0000i	1.38194i	0.722885	+	0.690968i	$0.242815\pi$
$277$	23.0000i	1.38194i	−0.722885	+	0.690968i	$0.757185\pi$
$278$	20.0000i	1.19952i
$279$	−7.00000	−0.419079
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	3.00000i	0.178647i
$283$	− 14.0000i	− 0.832214i	−0.909316	−	0.416107i	$-0.863394\pi$
$283$	− 14.0000i	− 0.832214i	0.909316	−	0.416107i	$-0.136606\pi$
$284$	−2.00000	−0.118678
$285$	0	0
$286$	−12.0000	−0.709575
$287$	6.00000i	0.354169i
$288$	− 1.00000i	− 0.0589256i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−2.00000	−0.117242
$292$	− 6.00000i	− 0.351123i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	−2.00000	−0.116642
$295$	0	0
$296$	3.00000	0.174371
$297$	− 3.00000i	− 0.174078i
$298$	− 10.0000i	− 0.579284i
$299$	−16.0000	−0.925304
$300$	0	0
$301$	−3.00000	−0.172917
$302$	12.0000i	0.690522i
$303$	− 7.00000i	− 0.402139i
$304$	5.00000	0.286770
$305$	0	0
$306$	1.00000	0.0571662
$307$	− 32.0000i	− 1.82634i	−0.407583	−	0.913168i	$-0.633628\pi$
$307$	− 32.0000i	− 1.82634i	0.407583	−	0.913168i	$-0.366372\pi$
$308$	9.00000i	0.512823i
$309$	−14.0000	−0.796432
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	4.00000i	0.226455i
$313$	− 4.00000i	− 0.226093i	−0.993590	−	0.113047i	$-0.963939\pi$
$313$	− 4.00000i	− 0.226093i	0.993590	−	0.113047i	$-0.0360610\pi$
$314$	12.0000	0.677199
$315$	0	0
$316$	−5.00000	−0.281272
$317$	− 32.0000i	− 1.79730i	−0.438667	−	0.898650i	$-0.644549\pi$
$317$	− 32.0000i	− 1.79730i	0.438667	−	0.898650i	$-0.355451\pi$
$318$	11.0000i	0.616849i
$319$	0	0
$320$	0	0
$321$	−17.0000	−0.948847
$322$	12.0000i	0.668734i
$323$	5.00000i	0.278207i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−6.00000	−0.332309
$327$	5.00000i	0.276501i
$328$	− 2.00000i	− 0.110432i
$329$	−9.00000	−0.496186
$330$	0	0
$331$	−23.0000	−1.26419	−0.632097	−	0.774889i	$-0.717806\pi$
$331$	−23.0000	−1.26419	−0.632097	+	0.774889i	$0.717806\pi$
$332$	− 16.0000i	− 0.878114i
$333$	− 3.00000i	− 0.164399i
$334$	−18.0000	−0.984916
$335$	0	0
$336$	3.00000	0.163663
$337$	− 12.0000i	− 0.653682i	−0.945079	−	0.326841i	$-0.894016\pi$
$337$	− 12.0000i	− 0.653682i	0.945079	−	0.326841i	$-0.105984\pi$
$338$	− 3.00000i	− 0.163178i
$339$	1.00000	0.0543125
$340$	0	0
$341$	−21.0000	−1.13721
$342$	− 5.00000i	− 0.270369i
$343$	15.0000i	0.809924i
$344$	1.00000	0.0539164
$345$	0	0
$346$	−6.00000	−0.322562
$347$	− 27.0000i	− 1.44944i	−0.689046	−	0.724718i	$-0.741970\pi$
$347$	− 27.0000i	− 1.44944i	0.689046	−	0.724718i	$-0.258030\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	− 3.00000i	− 0.159901i
$353$	16.0000i	0.851594i	0.904819	+	0.425797i	$0.140006\pi$
$353$	16.0000i	0.851594i	−0.904819	+	0.425797i	$0.859994\pi$
$354$	0	0
$355$	0	0
$356$	−10.0000	−0.529999
$357$	3.00000i	0.158777i
$358$	0	0
$359$	−5.00000	−0.263890	−0.131945	−	0.991257i	$-0.542122\pi$
$359$	−5.00000	−0.263890	−0.131945	+	0.991257i	$0.542122\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	7.00000i	0.367912i
$363$	2.00000i	0.104973i
$364$	−12.0000	−0.628971
$365$	0	0
$366$	2.00000	0.104542
$367$	− 7.00000i	− 0.365397i	−0.983169	−	0.182699i	$-0.941517\pi$
$367$	− 7.00000i	− 0.365397i	0.983169	−	0.182699i	$-0.0584832\pi$
$368$	− 4.00000i	− 0.208514i
$369$	−2.00000	−0.104116
$370$	0	0
$371$	−33.0000	−1.71327
$372$	7.00000i	0.362933i
$373$	− 14.0000i	− 0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	− 14.0000i	− 0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$	3.00000	0.155126
$375$	0	0
$376$	3.00000	0.154713
$377$	0	0
$378$	− 3.00000i	− 0.154303i
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	18.0000	0.922168
$382$	− 3.00000i	− 0.153493i
$383$	− 4.00000i	− 0.204390i	−0.994764	−	0.102195i	$-0.967413\pi$
$383$	− 4.00000i	− 0.204390i	0.994764	−	0.102195i	$-0.0325866\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−6.00000	−0.305392
$387$	− 1.00000i	− 0.0508329i
$388$	2.00000i	0.101535i
$389$	5.00000	0.253510	0.126755	−	0.991934i	$-0.459544\pi$
$389$	5.00000	0.253510	0.126755	+	0.991934i	$0.459544\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	2.00000i	0.101015i
$393$	− 12.0000i	− 0.605320i
$394$	12.0000	0.604551
$395$	0	0
$396$	−3.00000	−0.150756
$397$	3.00000i	0.150566i	0.997162	+	0.0752828i	$0.0239860\pi$
$397$	3.00000i	0.150566i	−0.997162	+	0.0752828i	$0.976014\pi$
$398$	25.0000i	1.25314i
$399$	15.0000	0.750939
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	13.0000i	0.648381i
$403$	− 28.0000i	− 1.39478i
$404$	−7.00000	−0.348263
$405$	0	0
$406$	0	0
$407$	− 9.00000i	− 0.446113i
$408$	− 1.00000i	− 0.0495074i
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	14.0000i	0.689730i
$413$	0	0
$414$	−4.00000	−0.196589
$415$	0	0
$416$	4.00000	0.196116
$417$	− 20.0000i	− 0.979404i
$418$	− 15.0000i	− 0.733674i
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	0	0
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	12.0000i	0.584151i
$423$	− 3.00000i	− 0.145865i
$424$	11.0000	0.534207
$425$	0	0
$426$	2.00000	0.0969003
$427$	6.00000i	0.290360i
$428$	17.0000i	0.821726i
$429$	12.0000	0.579365
$430$	0	0
$431$	2.00000	0.0963366	0.0481683	−	0.998839i	$-0.484662\pi$
$431$	2.00000	0.0963366	0.0481683	+	0.998839i	$0.484662\pi$
$432$	1.00000i	0.0481125i
$433$	21.0000i	1.00920i	0.863355	+	0.504598i	$0.168359\pi$
$433$	21.0000i	1.00920i	−0.863355	+	0.504598i	$0.831641\pi$
$434$	−21.0000	−1.00803
$435$	0	0
$436$	5.00000	0.239457
$437$	− 20.0000i	− 0.956730i
$438$	6.00000i	0.286691i
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	4.00000i	0.190261i
$443$	− 14.0000i	− 0.665160i	−0.943075	−	0.332580i	$-0.892081\pi$
$443$	− 14.0000i	− 0.665160i	0.943075	−	0.332580i	$-0.107919\pi$
$444$	−3.00000	−0.142374
$445$	0	0
$446$	−26.0000	−1.23114
$447$	10.0000i	0.472984i
$448$	− 3.00000i	− 0.141737i
$449$	−5.00000	−0.235965	−0.117982	−	0.993016i	$-0.537643\pi$
$449$	−5.00000	−0.235965	−0.117982	+	0.993016i	$0.537643\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	− 1.00000i	− 0.0470360i
$453$	− 12.0000i	− 0.563809i
$454$	27.0000	1.26717
$455$	0	0
$456$	−5.00000	−0.234146
$457$	13.0000i	0.608114i	0.952654	+	0.304057i	$0.0983414\pi$
$457$	13.0000i	0.608114i	−0.952654	+	0.304057i	$0.901659\pi$
$458$	0	0
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	37.0000	1.72326	0.861631	−	0.507535i	$-0.169443\pi$
$461$	37.0000	1.72326	0.861631	+	0.507535i	$0.169443\pi$
$462$	− 9.00000i	− 0.418718i
$463$	− 24.0000i	− 1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$	− 24.0000i	− 1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$	0	0
$465$	0	0
$466$	−26.0000	−1.20443
$467$	18.0000i	0.832941i	0.909149	+	0.416470i	$0.136733\pi$
$467$	18.0000i	0.832941i	−0.909149	+	0.416470i	$0.863267\pi$
$468$	− 4.00000i	− 0.184900i
$469$	−39.0000	−1.80085
$470$	0	0
$471$	−12.0000	−0.552931
$472$	0	0
$473$	− 3.00000i	− 0.137940i
$474$	5.00000	0.229658
$475$	0	0
$476$	3.00000	0.137505
$477$	− 11.0000i	− 0.503655i
$478$	− 15.0000i	− 0.686084i
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	12.0000i	0.546585i
$483$	− 12.0000i	− 0.546019i
$484$	2.00000	0.0909091
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 32.0000i	− 1.45006i	−0.688718	−	0.725029i	$-0.741826\pi$
$487$	− 32.0000i	− 1.45006i	0.688718	−	0.725029i	$-0.258174\pi$
$488$	− 2.00000i	− 0.0905357i
$489$	6.00000	0.271329
$490$	0	0
$491$	2.00000	0.0902587	0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000	0.0902587	0.0451294	+	0.998981i	$0.485630\pi$
$492$	2.00000i	0.0901670i
$493$	0	0
$494$	20.0000	0.899843
$495$	0	0
$496$	7.00000	0.314309
$497$	6.00000i	0.269137i
$498$	16.0000i	0.716977i
$499$	−10.0000	−0.447661	−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000	−0.447661	−0.223831	+	0.974628i	$0.571856\pi$
$500$	0	0
$501$	18.0000	0.804181
$502$	− 18.0000i	− 0.803379i
$503$	− 14.0000i	− 0.624229i	−0.950044	−	0.312115i	$-0.898963\pi$
$503$	− 14.0000i	− 0.624229i	0.950044	−	0.312115i	$-0.101037\pi$
$504$	−3.00000	−0.133631
$505$	0	0
$506$	−12.0000	−0.533465
$507$	3.00000i	0.133235i
$508$	− 18.0000i	− 0.798621i
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	0	0
$511$	−18.0000	−0.796273
$512$	1.00000i	0.0441942i
$513$	5.00000i	0.220755i
$514$	−8.00000	−0.352865
$515$	0	0
$516$	−1.00000	−0.0440225
$517$	− 9.00000i	− 0.395820i
$518$	− 9.00000i	− 0.395437i
$519$	6.00000	0.263371
$520$	0	0
$521$	−3.00000	−0.131432	−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000	−0.131432	−0.0657162	+	0.997838i	$0.520933\pi$
$522$	0	0
$523$	− 44.0000i	− 1.92399i	−0.273075	−	0.961993i	$-0.588041\pi$
$523$	− 44.0000i	− 1.92399i	0.273075	−	0.961993i	$-0.411959\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	19.0000	0.828439
$527$	7.00000i	0.304925i
$528$	3.00000i	0.130558i
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	− 15.0000i	− 0.650332i
$533$	− 8.00000i	− 0.346518i
$534$	10.0000	0.432742
$535$	0	0
$536$	13.0000	0.561514
$537$	0	0
$538$	− 20.0000i	− 0.862261i
$539$	6.00000	0.258438
$540$	0	0
$541$	−13.0000	−0.558914	−0.279457	−	0.960158i	$-0.590154\pi$
$541$	−13.0000	−0.558914	−0.279457	+	0.960158i	$0.590154\pi$
$542$	32.0000i	1.37452i
$543$	− 7.00000i	− 0.300399i
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	12.0000	0.513553
$547$	− 2.00000i	− 0.0855138i	−0.999086	−	0.0427569i	$-0.986386\pi$
$547$	− 2.00000i	− 0.0855138i	0.999086	−	0.0427569i	$-0.0136141\pi$
$548$	2.00000i	0.0854358i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	0	0
$552$	4.00000i	0.170251i
$553$	15.0000i	0.637865i
$554$	−23.0000	−0.977176
$555$	0	0
$556$	−20.0000	−0.848189
$557$	23.0000i	0.974541i	0.873251	+	0.487271i	$0.162007\pi$
$557$	23.0000i	0.974541i	−0.873251	+	0.487271i	$0.837993\pi$
$558$	− 7.00000i	− 0.296334i
$559$	4.00000	0.169182
$560$	0	0
$561$	−3.00000	−0.126660
$562$	− 18.0000i	− 0.759284i
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	−3.00000	−0.126323
$565$	0	0
$566$	14.0000	0.588464
$567$	3.00000i	0.125988i
$568$	− 2.00000i	− 0.0839181i
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	2.00000	0.0836974	0.0418487	−	0.999124i	$-0.486675\pi$
$571$	2.00000	0.0836974	0.0418487	+	0.999124i	$0.486675\pi$
$572$	− 12.0000i	− 0.501745i
$573$	3.00000i	0.125327i
$574$	−6.00000	−0.250435
$575$	0	0
$576$	1.00000	0.0416667
$577$	43.0000i	1.79011i	0.445952	+	0.895057i	$0.352865\pi$
$577$	43.0000i	1.79011i	−0.445952	+	0.895057i	$0.647135\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	6.00000	0.249351
$580$	0	0
$581$	−48.0000	−1.99138
$582$	− 2.00000i	− 0.0829027i
$583$	− 33.0000i	− 1.36672i
$584$	6.00000	0.248282
$585$	0	0
$586$	−6.00000	−0.247858
$587$	18.0000i	0.742940i	0.928445	+	0.371470i	$0.121146\pi$
$587$	18.0000i	0.742940i	−0.928445	+	0.371470i	$0.878854\pi$
$588$	− 2.00000i	− 0.0824786i
$589$	35.0000	1.44215
$590$	0	0
$591$	−12.0000	−0.493614
$592$	3.00000i	0.123299i
$593$	36.0000i	1.47834i	0.673517	+	0.739171i	$0.264783\pi$
$593$	36.0000i	1.47834i	−0.673517	+	0.739171i	$0.735217\pi$
$594$	3.00000	0.123091
$595$	0	0
$596$	10.0000	0.409616
$597$	− 25.0000i	− 1.02318i
$598$	− 16.0000i	− 0.654289i
$599$	35.0000	1.43006	0.715031	−	0.699093i	$-0.246413\pi$
$599$	35.0000	1.43006	0.715031	+	0.699093i	$0.246413\pi$
$600$	0	0
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	− 3.00000i	− 0.122271i
$603$	− 13.0000i	− 0.529401i
$604$	−12.0000	−0.488273
$605$	0	0
$606$	7.00000	0.284356
$607$	8.00000i	0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$	8.00000i	0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$	5.00000i	0.202777i
$609$	0	0
$610$	0	0
$611$	12.0000	0.485468
$612$	1.00000i	0.0404226i
$613$	46.0000i	1.85792i	0.370177	+	0.928961i	$0.379297\pi$
$613$	46.0000i	1.85792i	−0.370177	+	0.928961i	$0.620703\pi$
$614$	32.0000	1.29141
$615$	0	0
$616$	−9.00000	−0.362620
$617$	33.0000i	1.32853i	0.747497	+	0.664265i	$0.231255\pi$
$617$	33.0000i	1.32853i	−0.747497	+	0.664265i	$0.768745\pi$
$618$	− 14.0000i	− 0.563163i
$619$	−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
$621$	4.00000	0.160514
$622$	− 18.0000i	− 0.721734i
$623$	30.0000i	1.20192i
$624$	−4.00000	−0.160128
$625$	0	0
$626$	4.00000	0.159872
$627$	15.0000i	0.599042i
$628$	12.0000i	0.478852i
$629$	−3.00000	−0.119618
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	− 5.00000i	− 0.198889i
$633$	− 12.0000i	− 0.476957i
$634$	32.0000	1.27088
$635$	0	0
$636$	−11.0000	−0.436178
$637$	8.00000i	0.316972i
$638$	0	0
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	− 17.0000i	− 0.670936i
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	−5.00000	−0.196722
$647$	8.00000i	0.314512i	0.987558	+	0.157256i	$0.0502649\pi$
$647$	8.00000i	0.314512i	−0.987558	+	0.157256i	$0.949735\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	0	0
$650$	0	0
$651$	21.0000	0.823055
$652$	− 6.00000i	− 0.234978i
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	−5.00000	−0.195515
$655$	0	0
$656$	2.00000	0.0780869
$657$	− 6.00000i	− 0.234082i
$658$	− 9.00000i	− 0.350857i
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	− 23.0000i	− 0.893920i
$663$	− 4.00000i	− 0.155347i
$664$	16.0000	0.620920
$665$	0	0
$666$	3.00000	0.116248
$667$	0	0
$668$	− 18.0000i	− 0.696441i
$669$	26.0000	1.00522
$670$	0	0
$671$	−6.00000	−0.231627
$672$	3.00000i	0.115728i
$673$	− 44.0000i	− 1.69608i	−0.529936	−	0.848038i	$-0.677784\pi$
$673$	− 44.0000i	− 1.69608i	0.529936	−	0.848038i	$-0.322216\pi$
$674$	12.0000	0.462223
$675$	0	0
$676$	3.00000	0.115385
$677$	− 42.0000i	− 1.61419i	−0.590421	−	0.807096i	$-0.701038\pi$
$677$	− 42.0000i	− 1.61419i	0.590421	−	0.807096i	$-0.298962\pi$
$678$	1.00000i	0.0384048i
$679$	6.00000	0.230259
$680$	0	0
$681$	−27.0000	−1.03464
$682$	− 21.0000i	− 0.804132i
$683$	− 24.0000i	− 0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	− 24.0000i	− 0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$	5.00000	0.191180
$685$	0	0
$686$	−15.0000	−0.572703
$687$	0	0
$688$	1.00000i	0.0381246i
$689$	44.0000	1.67627
$690$	0	0
$691$	42.0000	1.59776	0.798878	−	0.601494i	$-0.205427\pi$
$691$	42.0000	1.59776	0.798878	+	0.601494i	$0.205427\pi$
$692$	− 6.00000i	− 0.228086i
$693$	9.00000i	0.341882i
$694$	27.0000	1.02491
$695$	0	0
$696$	0	0
$697$	2.00000i	0.0757554i
$698$	10.0000i	0.378506i
$699$	26.0000	0.983410
$700$	0	0
$701$	−38.0000	−1.43524	−0.717620	−	0.696435i	$-0.754769\pi$
$701$	−38.0000	−1.43524	−0.717620	+	0.696435i	$0.754769\pi$
$702$	4.00000i	0.150970i
$703$	15.0000i	0.565736i
$704$	3.00000	0.113067
$705$	0	0
$706$	−16.0000	−0.602168
$707$	21.0000i	0.789786i
$708$	0	0
$709$	−5.00000	−0.187779	−0.0938895	−	0.995583i	$-0.529930\pi$
$709$	−5.00000	−0.187779	−0.0938895	+	0.995583i	$0.529930\pi$
$710$	0	0
$711$	−5.00000	−0.187515
$712$	− 10.0000i	− 0.374766i
$713$	− 28.0000i	− 1.04861i
$714$	−3.00000	−0.112272
$715$	0	0
$716$	0	0
$717$	15.0000i	0.560185i
$718$	− 5.00000i	− 0.186598i
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	42.0000	1.56416
$722$	6.00000i	0.223297i
$723$	− 12.0000i	− 0.446285i
$724$	−7.00000	−0.260153
$725$	0	0
$726$	−2.00000	−0.0742270
$727$	28.0000i	1.03846i	0.854634	+	0.519231i	$0.173782\pi$
$727$	28.0000i	1.03846i	−0.854634	+	0.519231i	$0.826218\pi$
$728$	− 12.0000i	− 0.444750i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−1.00000	−0.0369863
$732$	2.00000i	0.0739221i
$733$	26.0000i	0.960332i	0.877178	+	0.480166i	$0.159424\pi$
$733$	26.0000i	0.960332i	−0.877178	+	0.480166i	$0.840576\pi$
$734$	7.00000	0.258375
$735$	0	0
$736$	4.00000	0.147442
$737$	− 39.0000i	− 1.43658i
$738$	− 2.00000i	− 0.0736210i
$739$	−35.0000	−1.28750	−0.643748	−	0.765238i	$-0.722621\pi$
$739$	−35.0000	−1.28750	−0.643748	+	0.765238i	$0.722621\pi$
$740$	0	0
$741$	−20.0000	−0.734718
$742$	− 33.0000i	− 1.21147i
$743$	26.0000i	0.953847i	0.878945	+	0.476924i	$0.158248\pi$
$743$	26.0000i	0.953847i	−0.878945	+	0.476924i	$0.841752\pi$
$744$	−7.00000	−0.256632
$745$	0	0
$746$	14.0000	0.512576
$747$	− 16.0000i	− 0.585409i
$748$	3.00000i	0.109691i
$749$	51.0000	1.86350
$750$	0	0
$751$	12.0000	0.437886	0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000	0.437886	0.218943	+	0.975738i	$0.429739\pi$
$752$	3.00000i	0.109399i
$753$	18.0000i	0.655956i
$754$	0	0
$755$	0	0
$756$	3.00000	0.109109
$757$	− 22.0000i	− 0.799604i	−0.916602	−	0.399802i	$-0.869079\pi$
$757$	− 22.0000i	− 0.799604i	0.916602	−	0.399802i	$-0.130921\pi$
$758$	20.0000i	0.726433i
$759$	12.0000	0.435572
$760$	0	0
$761$	−48.0000	−1.74000	−0.869999	−	0.493053i	$-0.835881\pi$
$761$	−48.0000	−1.74000	−0.869999	+	0.493053i	$0.835881\pi$
$762$	18.0000i	0.652071i
$763$	− 15.0000i	− 0.543036i
$764$	3.00000	0.108536
$765$	0	0
$766$	4.00000	0.144526
$767$	0	0
$768$	− 1.00000i	− 0.0360844i
$769$	−35.0000	−1.26213	−0.631066	−	0.775729i	$-0.717382\pi$
$769$	−35.0000	−1.26213	−0.631066	+	0.775729i	$0.717382\pi$
$770$	0	0
$771$	8.00000	0.288113
$772$	− 6.00000i	− 0.215945i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	1.00000	0.0359443
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	9.00000i	0.322873i
$778$	5.00000i	0.179259i
$779$	10.0000	0.358287
$780$	0	0
$781$	−6.00000	−0.214697
$782$	4.00000i	0.143040i
$783$	0	0
$784$	−2.00000	−0.0714286
$785$	0	0
$786$	12.0000	0.428026
$787$	− 42.0000i	− 1.49714i	−0.663057	−	0.748569i	$-0.730741\pi$
$787$	− 42.0000i	− 1.49714i	0.663057	−	0.748569i	$-0.269259\pi$
$788$	12.0000i	0.427482i
$789$	−19.0000	−0.676418
$790$	0	0
$791$	−3.00000	−0.106668
$792$	− 3.00000i	− 0.106600i
$793$	− 8.00000i	− 0.284088i
$794$	−3.00000	−0.106466
$795$	0	0
$796$	−25.0000	−0.886102
$797$	− 17.0000i	− 0.602171i	−0.953597	−	0.301085i	$-0.902651\pi$
$797$	− 17.0000i	− 0.602171i	0.953597	−	0.301085i	$-0.0973489\pi$
$798$	15.0000i	0.530994i
$799$	−3.00000	−0.106132
$800$	0	0
$801$	−10.0000	−0.353333
$802$	− 18.0000i	− 0.635602i
$803$	− 18.0000i	− 0.635206i
$804$	−13.0000	−0.458475
$805$	0	0
$806$	28.0000	0.986258
$807$	20.0000i	0.704033i
$808$	− 7.00000i	− 0.246259i
$809$	−45.0000	−1.58212	−0.791058	−	0.611741i	$-0.790469\pi$
$809$	−45.0000	−1.58212	−0.791058	+	0.611741i	$0.790469\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	− 32.0000i	− 1.12229i
$814$	9.00000	0.315450
$815$	0	0
$816$	1.00000	0.0350070
$817$	5.00000i	0.174928i
$818$	10.0000i	0.349642i
$819$	−12.0000	−0.419314
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	− 2.00000i	− 0.0697580i
$823$	16.0000i	0.557725i	0.960331	+	0.278862i	$0.0899574\pi$
$823$	16.0000i	0.557725i	−0.960331	+	0.278862i	$0.910043\pi$
$824$	−14.0000	−0.487713
$825$	0	0
$826$	0	0
$827$	− 17.0000i	− 0.591148i	−0.955320	−	0.295574i	$-0.904489\pi$
$827$	− 17.0000i	− 0.591148i	0.955320	−	0.295574i	$-0.0955109\pi$
$828$	− 4.00000i	− 0.139010i
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	0	0
$831$	23.0000	0.797861
$832$	4.00000i	0.138675i
$833$	− 2.00000i	− 0.0692959i
$834$	20.0000	0.692543
$835$	0	0
$836$	15.0000	0.518786
$837$	7.00000i	0.241955i
$838$	20.0000i	0.690889i
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	− 28.0000i	− 0.964944i
$843$	18.0000i	0.619953i
$844$	−12.0000	−0.413057
$845$	0	0
$846$	3.00000	0.103142
$847$	− 6.00000i	− 0.206162i
$848$	11.0000i	0.377742i
$849$	−14.0000	−0.480479
$850$	0	0
$851$	12.0000	0.411355
$852$	2.00000i	0.0685189i
$853$	51.0000i	1.74621i	0.487535	+	0.873103i	$0.337896\pi$
$853$	51.0000i	1.74621i	−0.487535	+	0.873103i	$0.662104\pi$
$854$	−6.00000	−0.205316
$855$	0	0
$856$	−17.0000	−0.581048
$857$	− 47.0000i	− 1.60549i	−0.596323	−	0.802745i	$-0.703372\pi$
$857$	− 47.0000i	− 1.60549i	0.596323	−	0.802745i	$-0.296628\pi$
$858$	12.0000i	0.409673i
$859$	−35.0000	−1.19418	−0.597092	−	0.802173i	$-0.703677\pi$
$859$	−35.0000	−1.19418	−0.597092	+	0.802173i	$0.703677\pi$
$860$	0	0
$861$	6.00000	0.204479
$862$	2.00000i	0.0681203i
$863$	− 19.0000i	− 0.646768i	−0.946268	−	0.323384i	$-0.895180\pi$
$863$	− 19.0000i	− 0.646768i	0.946268	−	0.323384i	$-0.104820\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−21.0000	−0.713609
$867$	1.00000i	0.0339618i
$868$	− 21.0000i	− 0.712786i
$869$	−15.0000	−0.508840
$870$	0	0
$871$	52.0000	1.76195
$872$	5.00000i	0.169321i
$873$	2.00000i	0.0676897i
$874$	20.0000	0.676510
$875$	0	0
$876$	−6.00000	−0.202721
$877$	38.0000i	1.28317i	0.767052	+	0.641584i	$0.221723\pi$
$877$	38.0000i	1.28317i	−0.767052	+	0.641584i	$0.778277\pi$
$878$	− 20.0000i	− 0.674967i
$879$	6.00000	0.202375
$880$	0	0
$881$	7.00000	0.235836	0.117918	−	0.993023i	$-0.462378\pi$
$881$	7.00000	0.235836	0.117918	+	0.993023i	$0.462378\pi$
$882$	2.00000i	0.0673435i
$883$	− 24.0000i	− 0.807664i	−0.914833	−	0.403832i	$-0.867678\pi$
$883$	− 24.0000i	− 0.807664i	0.914833	−	0.403832i	$-0.132322\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	14.0000	0.470339
$887$	− 12.0000i	− 0.402921i	−0.979497	−	0.201460i	$-0.935431\pi$
$887$	− 12.0000i	− 0.402921i	0.979497	−	0.201460i	$-0.0645687\pi$
$888$	− 3.00000i	− 0.100673i
$889$	−54.0000	−1.81110
$890$	0	0
$891$	−3.00000	−0.100504
$892$	− 26.0000i	− 0.870544i
$893$	15.0000i	0.501956i
$894$	−10.0000	−0.334450
$895$	0	0
$896$	3.00000	0.100223
$897$	16.0000i	0.534224i
$898$	− 5.00000i	− 0.166852i
$899$	0	0
$900$	0	0
$901$	−11.0000	−0.366463
$902$	− 6.00000i	− 0.199778i
$903$	3.00000i	0.0998337i
$904$	1.00000	0.0332595
$905$	0	0
$906$	12.0000	0.398673
$907$	28.0000i	0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$	28.0000i	0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$	27.0000i	0.896026i
$909$	−7.00000	−0.232175
$910$	0	0
$911$	−8.00000	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$912$	− 5.00000i	− 0.165567i
$913$	− 48.0000i	− 1.58857i
$914$	−13.0000	−0.430002
$915$	0	0
$916$	0	0
$917$	36.0000i	1.18882i
$918$	− 1.00000i	− 0.0330049i
$919$	−10.0000	−0.329870	−0.164935	−	0.986304i	$-0.552741\pi$
$919$	−10.0000	−0.329870	−0.164935	+	0.986304i	$0.552741\pi$
$920$	0	0
$921$	−32.0000	−1.05444
$922$	37.0000i	1.21853i
$923$	− 8.00000i	− 0.263323i
$924$	9.00000	0.296078
$925$	0	0
$926$	24.0000	0.788689
$927$	14.0000i	0.459820i
$928$	0	0
$929$	−35.0000	−1.14831	−0.574156	−	0.818746i	$-0.694670\pi$
$929$	−35.0000	−1.14831	−0.574156	+	0.818746i	$0.694670\pi$
$930$	0	0
$931$	−10.0000	−0.327737
$932$	− 26.0000i	− 0.851658i
$933$	18.0000i	0.589294i
$934$	−18.0000	−0.588978
$935$	0	0
$936$	4.00000	0.130744
$937$	− 22.0000i	− 0.718709i	−0.933201	−	0.359354i	$-0.882997\pi$
$937$	− 22.0000i	− 0.718709i	0.933201	−	0.359354i	$-0.117003\pi$
$938$	− 39.0000i	− 1.27340i
$939$	−4.00000	−0.130535
$940$	0	0
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	− 12.0000i	− 0.390981i
$943$	− 8.00000i	− 0.260516i
$944$	0	0
$945$	0	0
$946$	3.00000	0.0975384
$947$	− 27.0000i	− 0.877382i	−0.898638	−	0.438691i	$-0.855442\pi$
$947$	− 27.0000i	− 0.877382i	0.898638	−	0.438691i	$-0.144558\pi$
$948$	5.00000i	0.162392i
$949$	24.0000	0.779073
$950$	0	0
$951$	−32.0000	−1.03767
$952$	3.00000i	0.0972306i
$953$	16.0000i	0.518291i	0.965838	+	0.259145i	$0.0834409\pi$
$953$	16.0000i	0.518291i	−0.965838	+	0.259145i	$0.916559\pi$
$954$	11.0000	0.356138
$955$	0	0
$956$	15.0000	0.485135
$957$	0	0
$958$	10.0000i	0.323085i
$959$	6.00000	0.193750
$960$	0	0
$961$	18.0000	0.580645
$962$	12.0000i	0.386896i
$963$	17.0000i	0.547817i
$964$	−12.0000	−0.386494
$965$	0	0
$966$	12.0000	0.386094
$967$	48.0000i	1.54358i	0.635880	+	0.771788i	$0.280637\pi$
$967$	48.0000i	1.54358i	−0.635880	+	0.771788i	$0.719363\pi$
$968$	2.00000i	0.0642824i
$969$	5.00000	0.160623
$970$	0	0
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	1.00000i	0.0320750i
$973$	60.0000i	1.92351i
$974$	32.0000	1.02535
$975$	0	0
$976$	2.00000	0.0640184
$977$	28.0000i	0.895799i	0.894084	+	0.447900i	$0.147828\pi$
$977$	28.0000i	0.895799i	−0.894084	+	0.447900i	$0.852172\pi$
$978$	6.00000i	0.191859i
$979$	−30.0000	−0.958804
$980$	0	0
$981$	5.00000	0.159638
$982$	2.00000i	0.0638226i
$983$	− 44.0000i	− 1.40338i	−0.712481	−	0.701691i	$-0.752429\pi$
$983$	− 44.0000i	− 1.40338i	0.712481	−	0.701691i	$-0.247571\pi$
$984$	−2.00000	−0.0637577
$985$	0	0
$986$	0	0
$987$	9.00000i	0.286473i
$988$	20.0000i	0.636285i
$989$	4.00000	0.127193
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	7.00000i	0.222250i
$993$	23.0000i	0.729883i
$994$	−6.00000	−0.190308
$995$	0	0
$996$	−16.0000	−0.506979
$997$	− 37.0000i	− 1.17180i	−0.810383	−	0.585901i	$-0.800741\pi$
$997$	− 37.0000i	− 1.17180i	0.810383	−	0.585901i	$-0.199259\pi$
$998$	− 10.0000i	− 0.316544i
$999$	−3.00000	−0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.d.o.2449.2		2
5.2	odd	4		2550.2.a.a.1.1	✓	1
5.3	odd	4		2550.2.a.bf.1.1	yes	1
5.4	even	2	inner	2550.2.d.o.2449.1		2
15.2	even	4		7650.2.a.bk.1.1		1
15.8	even	4		7650.2.a.bd.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2550.2.a.a.1.1	✓	1	5.2	odd	4
2550.2.a.bf.1.1	yes	1	5.3	odd	4
2550.2.d.o.2449.1		2	5.4	even	2	inner
2550.2.d.o.2449.2		2	1.1	even	1	trivial
7650.2.a.bd.1.1		1	15.8	even	4
7650.2.a.bk.1.1		1	15.2	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:	\( N \)	\(=\)	\( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2550.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} +1.00000i q^{12} -4.00000i q^{13} -3.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} -1.00000i q^{18} +5.00000 q^{19} +3.00000 q^{21} -3.00000i q^{22} -4.00000i q^{23} -1.00000 q^{24} +4.00000 q^{26} +1.00000i q^{27} -3.00000i q^{28} +7.00000 q^{31} +1.00000i q^{32} +3.00000i q^{33} -1.00000 q^{34} +1.00000 q^{36} +3.00000i q^{37} +5.00000i q^{38} -4.00000 q^{39} +2.00000 q^{41} +3.00000i q^{42} +1.00000i q^{43} +3.00000 q^{44} +4.00000 q^{46} +3.00000i q^{47} -1.00000i q^{48} -2.00000 q^{49} +1.00000 q^{51} +4.00000i q^{52} +11.0000i q^{53} -1.00000 q^{54} +3.00000 q^{56} -5.00000i q^{57} +2.00000 q^{61} +7.00000i q^{62} -3.00000i q^{63} -1.00000 q^{64} -3.00000 q^{66} +13.0000i q^{67} -1.00000i q^{68} -4.00000 q^{69} +2.00000 q^{71} +1.00000i q^{72} +6.00000i q^{73} -3.00000 q^{74} -5.00000 q^{76} -9.00000i q^{77} -4.00000i q^{78} +5.00000 q^{79} +1.00000 q^{81} +2.00000i q^{82} +16.0000i q^{83} -3.00000 q^{84} -1.00000 q^{86} +3.00000i q^{88} +10.0000 q^{89} +12.0000 q^{91} +4.00000i q^{92} -7.00000i q^{93} -3.00000 q^{94} +1.00000 q^{96} -2.00000i q^{97} -2.00000i q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 6 q^{11} - 6 q^{14} + 2 q^{16} + 10 q^{19} + 6 q^{21} - 2 q^{24} + 8 q^{26} + 14 q^{31} - 2 q^{34} + 2 q^{36} - 8 q^{39} + 4 q^{41} + 6 q^{44} + 8 q^{46} - 4 q^{49} + 2 q^{51} - 2 q^{54} + 6 q^{56} + 4 q^{61} - 2 q^{64} - 6 q^{66} - 8 q^{69} + 4 q^{71} - 6 q^{74} - 10 q^{76} + 10 q^{79} + 2 q^{81} - 6 q^{84} - 2 q^{86} + 20 q^{89} + 24 q^{91} - 6 q^{94} + 2 q^{96} + 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 6 * q^11 - 6 * q^14 + 2 * q^16 + 10 * q^19 + 6 * q^21 - 2 * q^24 + 8 * q^26 + 14 * q^31 - 2 * q^34 + 2 * q^36 - 8 * q^39 + 4 * q^41 + 6 * q^44 + 8 * q^46 - 4 * q^49 + 2 * q^51 - 2 * q^54 + 6 * q^56 + 4 * q^61 - 2 * q^64 - 6 * q^66 - 8 * q^69 + 4 * q^71 - 6 * q^74 - 10 * q^76 + 10 * q^79 + 2 * q^81 - 6 * q^84 - 2 * q^86 + 20 * q^89 + 24 * q^91 - 6 * q^94 + 2 * q^96 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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