LMFDB - Embedded newform 2550.2.d.d.2449.1

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			2449.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2550.2449
Dual form			2550.2.d.d.2449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} +1.00000i q^{12} +4.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} +1.00000i q^{18} +5.00000 q^{19} +1.00000 q^{21} +3.00000i q^{22} -8.00000i q^{23} +1.00000 q^{24} +4.00000 q^{26} +1.00000i q^{27} -1.00000i q^{28} +4.00000 q^{29} -3.00000 q^{31} -1.00000i q^{32} +3.00000i q^{33} +1.00000 q^{34} +1.00000 q^{36} -7.00000i q^{37} -5.00000i q^{38} +4.00000 q^{39} +2.00000 q^{41} -1.00000i q^{42} +1.00000i q^{43} +3.00000 q^{44} -8.00000 q^{46} -7.00000i q^{47} -1.00000i q^{48} +6.00000 q^{49} +1.00000 q^{51} -4.00000i q^{52} -7.00000i q^{53} +1.00000 q^{54} -1.00000 q^{56} -5.00000i q^{57} -4.00000i q^{58} +8.00000 q^{59} -2.00000 q^{61} +3.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} -11.0000i q^{67} -1.00000i q^{68} -8.00000 q^{69} -6.00000 q^{71} -1.00000i q^{72} +2.00000i q^{73} -7.00000 q^{74} -5.00000 q^{76} -3.00000i q^{77} -4.00000i q^{78} +15.0000 q^{79} +1.00000 q^{81} -2.00000i q^{82} -16.0000i q^{83} -1.00000 q^{84} +1.00000 q^{86} -4.00000i q^{87} -3.00000i q^{88} -2.00000 q^{89} -4.00000 q^{91} +8.00000i q^{92} +3.00000i q^{93} -7.00000 q^{94} -1.00000 q^{96} +10.0000i q^{97} -6.00000i q^{98} +3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{14} + 2 q^{16} + 10 q^{19} + 2 q^{21} + 2 q^{24} + 8 q^{26} + 8 q^{29} - 6 q^{31} + 2 q^{34} + 2 q^{36} + 8 q^{39} + 4 q^{41} + 6 q^{44} - 16 q^{46} + 12 q^{49} + 2 q^{51} + 2 q^{54} - 2 q^{56} + 16 q^{59} - 4 q^{61} - 2 q^{64} + 6 q^{66} - 16 q^{69} - 12 q^{71} - 14 q^{74} - 10 q^{76} + 30 q^{79} + 2 q^{81} - 2 q^{84} + 2 q^{86} - 4 q^{89} - 8 q^{91} - 14 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 + 2 * q^14 + 2 * q^16 + 10 * q^19 + 2 * q^21 + 2 * q^24 + 8 * q^26 + 8 * q^29 - 6 * q^31 + 2 * q^34 + 2 * q^36 + 8 * q^39 + 4 * q^41 + 6 * q^44 - 16 * q^46 + 12 * q^49 + 2 * q^51 + 2 * q^54 - 2 * q^56 + 16 * q^59 - 4 * q^61 - 2 * q^64 + 6 * q^66 - 16 * q^69 - 12 * q^71 - 14 * q^74 - 10 * q^76 + 30 * q^79 + 2 * q^81 - 2 * q^84 + 2 * q^86 - 4 * q^89 - 8 * q^91 - 14 * 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
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	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	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	1.00000	0.267261
$15$	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
$21$	1.00000	0.218218
$22$	3.00000i	0.639602i
$23$	− 8.00000i	− 1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$	− 8.00000i	− 1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	4.00000	0.784465
$27$	1.00000i	0.192450i
$28$	− 1.00000i	− 0.188982i
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	− 1.00000i	− 0.176777i
$33$	3.00000i	0.522233i
$34$	1.00000	0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	− 7.00000i	− 1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$	− 7.00000i	− 1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$38$	− 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	$0.450084\pi$
$42$	− 1.00000i	− 0.154303i
$43$	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
$46$	−8.00000	−1.17954
$47$	− 7.00000i	− 1.02105i	−0.859861	−	0.510527i	$-0.829450\pi$
$47$	− 7.00000i	− 1.02105i	0.859861	−	0.510527i	$-0.170550\pi$
$48$	− 1.00000i	− 0.144338i
$49$	6.00000	0.857143
$50$	0	0
$51$	1.00000	0.140028
$52$	− 4.00000i	− 0.554700i
$53$	− 7.00000i	− 0.961524i	−0.876851	−	0.480762i	$-0.840360\pi$
$53$	− 7.00000i	− 0.961524i	0.876851	−	0.480762i	$-0.159640\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−1.00000	−0.133631
$57$	− 5.00000i	− 0.662266i
$58$	− 4.00000i	− 0.525226i
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	3.00000i	0.381000i
$63$	− 1.00000i	− 0.125988i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	3.00000	0.369274
$67$	− 11.0000i	− 1.34386i	−0.740613	−	0.671932i	$-0.765465\pi$
$67$	− 11.0000i	− 1.34386i	0.740613	−	0.671932i	$-0.234535\pi$
$68$	− 1.00000i	− 0.121268i
$69$	−8.00000	−0.963087
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	− 1.00000i	− 0.117851i
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	−7.00000	−0.813733
$75$	0	0
$76$	−5.00000	−0.573539
$77$	− 3.00000i	− 0.341882i
$78$	− 4.00000i	− 0.452911i
$79$	15.0000	1.68763	0.843816	−	0.536633i	$-0.180304\pi$
$79$	15.0000	1.68763	0.843816	+	0.536633i	$0.180304\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 2.00000i	− 0.220863i
$83$	− 16.0000i	− 1.75623i	−0.478451	−	0.878114i	$-0.658802\pi$
$83$	− 16.0000i	− 1.75623i	0.478451	−	0.878114i	$-0.341198\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	1.00000	0.107833
$87$	− 4.00000i	− 0.428845i
$88$	− 3.00000i	− 0.319801i
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	8.00000i	0.834058i
$93$	3.00000i	0.311086i
$94$	−7.00000	−0.721995
$95$	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	$0.830506\pi$
$98$	− 6.00000i	− 0.606092i
$99$	3.00000	0.301511
$100$	0	0
$101$	5.00000	0.497519	0.248759	−	0.968565i	$-0.419977\pi$
$101$	5.00000	0.497519	0.248759	+	0.968565i	$0.419977\pi$
$102$	− 1.00000i	− 0.0990148i
$103$	− 10.0000i	− 0.985329i	−0.870219	−	0.492665i	$-0.836023\pi$
$103$	− 10.0000i	− 0.985329i	0.870219	−	0.492665i	$-0.163977\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	−7.00000	−0.679900
$107$	15.0000i	1.45010i	0.688694	+	0.725052i	$0.258184\pi$
$107$	15.0000i	1.45010i	−0.688694	+	0.725052i	$0.741816\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	9.00000	0.862044	0.431022	−	0.902342i	$-0.358153\pi$
$109$	9.00000	0.862044	0.431022	+	0.902342i	$0.358153\pi$
$110$	0	0
$111$	−7.00000	−0.664411
$112$	1.00000i	0.0944911i
$113$	9.00000i	0.846649i	0.905978	+	0.423324i	$0.139137\pi$
$113$	9.00000i	0.846649i	−0.905978	+	0.423324i	$0.860863\pi$
$114$	−5.00000	−0.468293
$115$	0	0
$116$	−4.00000	−0.371391
$117$	− 4.00000i	− 0.369800i
$118$	− 8.00000i	− 0.736460i
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−2.00000	−0.181818
$122$	2.00000i	0.181071i
$123$	− 2.00000i	− 0.180334i
$124$	3.00000	0.269408
$125$	0	0
$126$	−1.00000	−0.0890871
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	1.00000i	0.0883883i
$129$	1.00000	0.0880451
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	− 3.00000i	− 0.261116i
$133$	5.00000i	0.433555i
$134$	−11.0000	−0.950255
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	8.00000i	0.681005i
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	−7.00000	−0.589506
$142$	6.00000i	0.503509i
$143$	− 12.0000i	− 1.00349i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	− 6.00000i	− 0.494872i
$148$	7.00000i	0.575396i
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	5.00000i	0.405554i
$153$	− 1.00000i	− 0.0808452i
$154$	−3.00000	−0.241747
$155$	0	0
$156$	−4.00000	−0.320256
$157$	− 4.00000i	− 0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$	− 4.00000i	− 0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$	− 15.0000i	− 1.19334i
$159$	−7.00000	−0.555136
$160$	0	0
$161$	8.00000	0.630488
$162$	− 1.00000i	− 0.0785674i
$163$	18.0000i	1.40987i	0.709273	+	0.704934i	$0.249024\pi$
$163$	18.0000i	1.40987i	−0.709273	+	0.704934i	$0.750976\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−16.0000	−1.24184
$167$	− 10.0000i	− 0.773823i	−0.922117	−	0.386912i	$-0.873542\pi$
$167$	− 10.0000i	− 0.773823i	0.922117	−	0.386912i	$-0.126458\pi$
$168$	1.00000i	0.0771517i
$169$	−3.00000	−0.230769
$170$	0	0
$171$	−5.00000	−0.382360
$172$	− 1.00000i	− 0.0762493i
$173$	− 2.00000i	− 0.152057i	−0.997106	−	0.0760286i	$-0.975776\pi$
$173$	− 2.00000i	− 0.152057i	0.997106	−	0.0760286i	$-0.0242240\pi$
$174$	−4.00000	−0.303239
$175$	0	0
$176$	−3.00000	−0.226134
$177$	− 8.00000i	− 0.601317i
$178$	2.00000i	0.149906i
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−19.0000	−1.41226	−0.706129	−	0.708083i	$-0.749560\pi$
$181$	−19.0000	−1.41226	−0.706129	+	0.708083i	$0.749560\pi$
$182$	4.00000i	0.296500i
$183$	2.00000i	0.147844i
$184$	8.00000	0.589768
$185$	0	0
$186$	3.00000	0.219971
$187$	− 3.00000i	− 0.219382i
$188$	7.00000i	0.510527i
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	1.00000i	0.0721688i
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−6.00000	−0.428571
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	− 3.00000i	− 0.213201i
$199$	−13.0000	−0.921546	−0.460773	−	0.887518i	$-0.652428\pi$
$199$	−13.0000	−0.921546	−0.460773	+	0.887518i	$0.652428\pi$
$200$	0	0
$201$	−11.0000	−0.775880
$202$	− 5.00000i	− 0.351799i
$203$	4.00000i	0.280745i
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	−10.0000	−0.696733
$207$	8.00000i	0.556038i
$208$	4.00000i	0.277350i
$209$	−15.0000	−1.03757
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	7.00000i	0.480762i
$213$	6.00000i	0.411113i
$214$	15.0000	1.02538
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	− 3.00000i	− 0.203653i
$218$	− 9.00000i	− 0.609557i
$219$	2.00000	0.135147
$220$	0	0
$221$	−4.00000	−0.269069
$222$	7.00000i	0.469809i
$223$	2.00000i	0.133930i	0.997755	+	0.0669650i	$0.0213316\pi$
$223$	2.00000i	0.133930i	−0.997755	+	0.0669650i	$0.978668\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	9.00000	0.598671
$227$	− 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
$229$	−28.0000	−1.85029	−0.925146	−	0.379611i	$-0.876058\pi$
$229$	−28.0000	−1.85029	−0.925146	+	0.379611i	$0.876058\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	4.00000i	0.262613i
$233$	− 22.0000i	− 1.44127i	−0.693316	−	0.720634i	$-0.743851\pi$
$233$	− 22.0000i	− 1.44127i	0.693316	−	0.720634i	$-0.256149\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	−8.00000	−0.520756
$237$	− 15.0000i	− 0.974355i
$238$	1.00000i	0.0648204i
$239$	3.00000	0.194054	0.0970269	−	0.995282i	$-0.469067\pi$
$239$	3.00000	0.194054	0.0970269	+	0.995282i	$0.469067\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	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
$248$	− 3.00000i	− 0.190500i
$249$	−16.0000	−1.01396
$250$	0	0
$251$	10.0000	0.631194	0.315597	−	0.948893i	$-0.397795\pi$
$251$	10.0000	0.631194	0.315597	+	0.948893i	$0.397795\pi$
$252$	1.00000i	0.0629941i
$253$	24.0000i	1.50887i
$254$	2.00000	0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	20.0000i	1.24757i	0.781598	+	0.623783i	$0.214405\pi$
$257$	20.0000i	1.24757i	−0.781598	+	0.623783i	$0.785595\pi$
$258$	− 1.00000i	− 0.0622573i
$259$	7.00000	0.434959
$260$	0	0
$261$	−4.00000	−0.247594
$262$	12.0000i	0.741362i
$263$	− 9.00000i	− 0.554964i	−0.960731	−	0.277482i	$-0.910500\pi$
$263$	− 9.00000i	− 0.554964i	0.960731	−	0.277482i	$-0.0894999\pi$
$264$	−3.00000	−0.184637
$265$	0	0
$266$	5.00000	0.306570
$267$	2.00000i	0.122398i
$268$	11.0000i	0.671932i
$269$	8.00000	0.487769	0.243884	−	0.969804i	$-0.421578\pi$
$269$	8.00000	0.487769	0.243884	+	0.969804i	$0.421578\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	1.00000i	0.0606339i
$273$	4.00000i	0.242091i
$274$	−18.0000	−1.08742
$275$	0	0
$276$	8.00000	0.481543
$277$	5.00000i	0.300421i	0.988654	+	0.150210i	$0.0479951\pi$
$277$	5.00000i	0.300421i	−0.988654	+	0.150210i	$0.952005\pi$
$278$	0	0
$279$	3.00000	0.179605
$280$	0	0
$281$	−14.0000	−0.835170	−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000	−0.835170	−0.417585	+	0.908638i	$0.637123\pi$
$282$	7.00000i	0.416844i
$283$	26.0000i	1.54554i	0.634686	+	0.772770i	$0.281129\pi$
$283$	26.0000i	1.54554i	−0.634686	+	0.772770i	$0.718871\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	−12.0000	−0.709575
$287$	2.00000i	0.118056i
$288$	1.00000i	0.0589256i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	10.0000	0.586210
$292$	− 2.00000i	− 0.117041i
$293$	10.0000i	0.584206i	0.956387	+	0.292103i	$0.0943550\pi$
$293$	10.0000i	0.584206i	−0.956387	+	0.292103i	$0.905645\pi$
$294$	−6.00000	−0.349927
$295$	0	0
$296$	7.00000	0.406867
$297$	− 3.00000i	− 0.174078i
$298$	− 18.0000i	− 1.04271i
$299$	32.0000	1.85061
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	− 20.0000i	− 1.15087i
$303$	− 5.00000i	− 0.287242i
$304$	5.00000	0.286770
$305$	0	0
$306$	−1.00000	−0.0571662
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	3.00000i	0.170941i
$309$	−10.0000	−0.568880
$310$	0	0
$311$	2.00000	0.113410	0.0567048	−	0.998391i	$-0.481941\pi$
$311$	2.00000	0.113410	0.0567048	+	0.998391i	$0.481941\pi$
$312$	4.00000i	0.226455i
$313$	8.00000i	0.452187i	0.974106	+	0.226093i	$0.0725954\pi$
$313$	8.00000i	0.452187i	−0.974106	+	0.226093i	$0.927405\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	−15.0000	−0.843816
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	7.00000i	0.392541i
$319$	−12.0000	−0.671871
$320$	0	0
$321$	15.0000	0.837218
$322$	− 8.00000i	− 0.445823i
$323$	5.00000i	0.278207i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	18.0000	0.996928
$327$	− 9.00000i	− 0.497701i
$328$	2.00000i	0.110432i
$329$	7.00000	0.385922
$330$	0	0
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	16.0000i	0.878114i
$333$	7.00000i	0.383598i
$334$	−10.0000	−0.547176
$335$	0	0
$336$	1.00000	0.0545545
$337$	− 24.0000i	− 1.30736i	−0.756770	−	0.653682i	$-0.773224\pi$
$337$	− 24.0000i	− 1.30736i	0.756770	−	0.653682i	$-0.226776\pi$
$338$	3.00000i	0.163178i
$339$	9.00000	0.488813
$340$	0	0
$341$	9.00000	0.487377
$342$	5.00000i	0.270369i
$343$	13.0000i	0.701934i
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	−2.00000	−0.107521
$347$	5.00000i	0.268414i	0.990953	+	0.134207i	$0.0428487\pi$
$347$	5.00000i	0.268414i	−0.990953	+	0.134207i	$0.957151\pi$
$348$	4.00000i	0.214423i
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	3.00000i	0.159901i
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	−8.00000	−0.425195
$355$	0	0
$356$	2.00000	0.106000
$357$	1.00000i	0.0529256i
$358$	0	0
$359$	25.0000	1.31945	0.659725	−	0.751507i	$-0.270673\pi$
$359$	25.0000	1.31945	0.659725	+	0.751507i	$0.270673\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	19.0000i	0.998618i
$363$	2.00000i	0.104973i
$364$	4.00000	0.209657
$365$	0	0
$366$	2.00000	0.104542
$367$	− 13.0000i	− 0.678594i	−0.940679	−	0.339297i	$-0.889811\pi$
$367$	− 13.0000i	− 0.678594i	0.940679	−	0.339297i	$-0.110189\pi$
$368$	− 8.00000i	− 0.417029i
$369$	−2.00000	−0.104116
$370$	0	0
$371$	7.00000	0.363422
$372$	− 3.00000i	− 0.155543i
$373$	38.0000i	1.96757i	0.179364	+	0.983783i	$0.442596\pi$
$373$	38.0000i	1.96757i	−0.179364	+	0.983783i	$0.557404\pi$
$374$	−3.00000	−0.155126
$375$	0	0
$376$	7.00000	0.360997
$377$	16.0000i	0.824042i
$378$	1.00000i	0.0514344i
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	− 15.0000i	− 0.767467i
$383$	− 20.0000i	− 1.02195i	−0.859595	−	0.510976i	$-0.829284\pi$
$383$	− 20.0000i	− 1.02195i	0.859595	−	0.510976i	$-0.170716\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	− 1.00000i	− 0.0508329i
$388$	− 10.0000i	− 0.507673i
$389$	−9.00000	−0.456318	−0.228159	−	0.973624i	$-0.573271\pi$
$389$	−9.00000	−0.456318	−0.228159	+	0.973624i	$0.573271\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	6.00000i	0.303046i
$393$	12.0000i	0.605320i
$394$	0	0
$395$	0	0
$396$	−3.00000	−0.150756
$397$	25.0000i	1.25471i	0.778732	+	0.627357i	$0.215863\pi$
$397$	25.0000i	1.25471i	−0.778732	+	0.627357i	$0.784137\pi$
$398$	13.0000i	0.651631i
$399$	5.00000	0.250313
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	11.0000i	0.548630i
$403$	− 12.0000i	− 0.597763i
$404$	−5.00000	−0.248759
$405$	0	0
$406$	4.00000	0.198517
$407$	21.0000i	1.04093i
$408$	1.00000i	0.0495074i
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	10.0000i	0.492665i
$413$	8.00000i	0.393654i
$414$	8.00000	0.393179
$415$	0	0
$416$	4.00000	0.196116
$417$	0	0
$418$	15.0000i	0.733674i
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	0	0
$423$	7.00000i	0.340352i
$424$	7.00000	0.339950
$425$	0	0
$426$	6.00000	0.290701
$427$	− 2.00000i	− 0.0967868i
$428$	− 15.0000i	− 0.725052i
$429$	−12.0000	−0.579365
$430$	0	0
$431$	26.0000	1.25238	0.626188	−	0.779672i	$-0.284614\pi$
$431$	26.0000	1.25238	0.626188	+	0.779672i	$0.284614\pi$
$432$	1.00000i	0.0481125i
$433$	5.00000i	0.240285i	0.992757	+	0.120142i	$0.0383351\pi$
$433$	5.00000i	0.240285i	−0.992757	+	0.120142i	$0.961665\pi$
$434$	−3.00000	−0.144005
$435$	0	0
$436$	−9.00000	−0.431022
$437$	− 40.0000i	− 1.91346i
$438$	− 2.00000i	− 0.0955637i
$439$	12.0000	0.572729	0.286364	−	0.958121i	$-0.407553\pi$
$439$	12.0000	0.572729	0.286364	+	0.958121i	$0.407553\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	4.00000i	0.190261i
$443$	− 26.0000i	− 1.23530i	−0.786454	−	0.617649i	$-0.788085\pi$
$443$	− 26.0000i	− 1.23530i	0.786454	−	0.617649i	$-0.211915\pi$
$444$	7.00000	0.332205
$445$	0	0
$446$	2.00000	0.0947027
$447$	− 18.0000i	− 0.851371i
$448$	− 1.00000i	− 0.0472456i
$449$	−13.0000	−0.613508	−0.306754	−	0.951789i	$-0.599243\pi$
$449$	−13.0000	−0.613508	−0.306754	+	0.951789i	$0.599243\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	− 9.00000i	− 0.423324i
$453$	− 20.0000i	− 0.939682i
$454$	−27.0000	−1.26717
$455$	0	0
$456$	5.00000	0.234146
$457$	− 11.0000i	− 0.514558i	−0.966337	−	0.257279i	$-0.917174\pi$
$457$	− 11.0000i	− 0.514558i	0.966337	−	0.257279i	$-0.0828260\pi$
$458$	28.0000i	1.30835i
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	−9.00000	−0.419172	−0.209586	−	0.977790i	$-0.567212\pi$
$461$	−9.00000	−0.419172	−0.209586	+	0.977790i	$0.567212\pi$
$462$	3.00000i	0.139573i
$463$	16.0000i	0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$	16.0000i	0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	4.00000	0.185695
$465$	0	0
$466$	−22.0000	−1.01913
$467$	− 18.0000i	− 0.832941i	−0.909149	−	0.416470i	$-0.863267\pi$
$467$	− 18.0000i	− 0.832941i	0.909149	−	0.416470i	$-0.136733\pi$
$468$	4.00000i	0.184900i
$469$	11.0000	0.507933
$470$	0	0
$471$	−4.00000	−0.184310
$472$	8.00000i	0.368230i
$473$	− 3.00000i	− 0.137940i
$474$	−15.0000	−0.688973
$475$	0	0
$476$	1.00000	0.0458349
$477$	7.00000i	0.320508i
$478$	− 3.00000i	− 0.137217i
$479$	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
$481$	28.0000	1.27669
$482$	0	0
$483$	− 8.00000i	− 0.364013i
$484$	2.00000	0.0909091
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	40.0000i	1.81257i	0.422664	+	0.906287i	$0.361095\pi$
$487$	40.0000i	1.81257i	−0.422664	+	0.906287i	$0.638905\pi$
$488$	− 2.00000i	− 0.0905357i
$489$	18.0000	0.813988
$490$	0	0
$491$	22.0000	0.992846	0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000	0.992846	0.496423	+	0.868081i	$0.334646\pi$
$492$	2.00000i	0.0901670i
$493$	4.00000i	0.180151i
$494$	20.0000	0.899843
$495$	0	0
$496$	−3.00000	−0.134704
$497$	− 6.00000i	− 0.269137i
$498$	16.0000i	0.716977i
$499$	6.00000	0.268597	0.134298	−	0.990941i	$-0.457122\pi$
$499$	6.00000	0.268597	0.134298	+	0.990941i	$0.457122\pi$
$500$	0	0
$501$	−10.0000	−0.446767
$502$	− 10.0000i	− 0.446322i
$503$	− 2.00000i	− 0.0891756i	−0.999005	−	0.0445878i	$-0.985803\pi$
$503$	− 2.00000i	− 0.0891756i	0.999005	−	0.0445878i	$-0.0141974\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	24.0000	1.06693
$507$	3.00000i	0.133235i
$508$	− 2.00000i	− 0.0887357i
$509$	29.0000	1.28540	0.642701	−	0.766117i	$-0.277814\pi$
$509$	29.0000	1.28540	0.642701	+	0.766117i	$0.277814\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	− 1.00000i	− 0.0441942i
$513$	5.00000i	0.220755i
$514$	20.0000	0.882162
$515$	0	0
$516$	−1.00000	−0.0440225
$517$	21.0000i	0.923579i
$518$	− 7.00000i	− 0.307562i
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	−11.0000	−0.481919	−0.240959	−	0.970535i	$-0.577462\pi$
$521$	−11.0000	−0.481919	−0.240959	+	0.970535i	$0.577462\pi$
$522$	4.00000i	0.175075i
$523$	− 36.0000i	− 1.57417i	−0.616844	−	0.787085i	$-0.711589\pi$
$523$	− 36.0000i	− 1.57417i	0.616844	−	0.787085i	$-0.288411\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−9.00000	−0.392419
$527$	− 3.00000i	− 0.130682i
$528$	3.00000i	0.130558i
$529$	−41.0000	−1.78261
$530$	0	0
$531$	−8.00000	−0.347170
$532$	− 5.00000i	− 0.216777i
$533$	8.00000i	0.346518i
$534$	2.00000	0.0865485
$535$	0	0
$536$	11.0000	0.475128
$537$	0	0
$538$	− 8.00000i	− 0.344904i
$539$	−18.0000	−0.775315
$540$	0	0
$541$	17.0000	0.730887	0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000	0.730887	0.365444	+	0.930834i	$0.380917\pi$
$542$	20.0000i	0.859074i
$543$	19.0000i	0.815368i
$544$	1.00000	0.0428746
$545$	0	0
$546$	4.00000	0.171184
$547$	− 42.0000i	− 1.79579i	−0.440209	−	0.897895i	$-0.645096\pi$
$547$	− 42.0000i	− 1.79579i	0.440209	−	0.897895i	$-0.354904\pi$
$548$	18.0000i	0.768922i
$549$	2.00000	0.0853579
$550$	0	0
$551$	20.0000	0.852029
$552$	− 8.00000i	− 0.340503i
$553$	15.0000i	0.637865i
$554$	5.00000	0.212430
$555$	0	0
$556$	0	0
$557$	− 11.0000i	− 0.466085i	−0.972467	−	0.233042i	$-0.925132\pi$
$557$	− 11.0000i	− 0.466085i	0.972467	−	0.233042i	$-0.0748681\pi$
$558$	− 3.00000i	− 0.127000i
$559$	−4.00000	−0.169182
$560$	0	0
$561$	−3.00000	−0.126660
$562$	14.0000i	0.590554i
$563$	− 20.0000i	− 0.842900i	−0.906852	−	0.421450i	$-0.861521\pi$
$563$	− 20.0000i	− 0.842900i	0.906852	−	0.421450i	$-0.138479\pi$
$564$	7.00000	0.294753
$565$	0	0
$566$	26.0000	1.09286
$567$	1.00000i	0.0419961i
$568$	− 6.00000i	− 0.251754i
$569$	−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
$571$	30.0000	1.25546	0.627730	−	0.778431i	$-0.283984\pi$
$571$	30.0000	1.25546	0.627730	+	0.778431i	$0.283984\pi$
$572$	12.0000i	0.501745i
$573$	− 15.0000i	− 0.626634i
$574$	2.00000	0.0834784
$575$	0	0
$576$	1.00000	0.0416667
$577$	11.0000i	0.457936i	0.973434	+	0.228968i	$0.0735351\pi$
$577$	11.0000i	0.457936i	−0.973434	+	0.228968i	$0.926465\pi$
$578$	1.00000i	0.0415945i
$579$	−14.0000	−0.581820
$580$	0	0
$581$	16.0000	0.663792
$582$	− 10.0000i	− 0.414513i
$583$	21.0000i	0.869731i
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	10.0000	0.413096
$587$	22.0000i	0.908037i	0.890992	+	0.454019i	$0.150010\pi$
$587$	22.0000i	0.908037i	−0.890992	+	0.454019i	$0.849990\pi$
$588$	6.00000i	0.247436i
$589$	−15.0000	−0.618064
$590$	0	0
$591$	0	0
$592$	− 7.00000i	− 0.287698i
$593$	12.0000i	0.492781i	0.969171	+	0.246390i	$0.0792446\pi$
$593$	12.0000i	0.492781i	−0.969171	+	0.246390i	$0.920755\pi$
$594$	−3.00000	−0.123091
$595$	0	0
$596$	−18.0000	−0.737309
$597$	13.0000i	0.532055i
$598$	− 32.0000i	− 1.30858i
$599$	−15.0000	−0.612883	−0.306442	−	0.951889i	$-0.599138\pi$
$599$	−15.0000	−0.612883	−0.306442	+	0.951889i	$0.599138\pi$
$600$	0	0
$601$	−16.0000	−0.652654	−0.326327	−	0.945257i	$-0.605811\pi$
$601$	−16.0000	−0.652654	−0.326327	+	0.945257i	$0.605811\pi$
$602$	1.00000i	0.0407570i
$603$	11.0000i	0.447955i
$604$	−20.0000	−0.813788
$605$	0	0
$606$	−5.00000	−0.203111
$607$	32.0000i	1.29884i	0.760430	+	0.649420i	$0.224988\pi$
$607$	32.0000i	1.29884i	−0.760430	+	0.649420i	$0.775012\pi$
$608$	− 5.00000i	− 0.202777i
$609$	4.00000	0.162088
$610$	0	0
$611$	28.0000	1.13276
$612$	1.00000i	0.0404226i
$613$	30.0000i	1.21169i	0.795583	+	0.605844i	$0.207165\pi$
$613$	30.0000i	1.21169i	−0.795583	+	0.605844i	$0.792835\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	3.00000	0.120873
$617$	33.0000i	1.32853i	0.747497	+	0.664265i	$0.231255\pi$
$617$	33.0000i	1.32853i	−0.747497	+	0.664265i	$0.768745\pi$
$618$	10.0000i	0.402259i
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	− 2.00000i	− 0.0801927i
$623$	− 2.00000i	− 0.0801283i
$624$	4.00000	0.160128
$625$	0	0
$626$	8.00000	0.319744
$627$	15.0000i	0.599042i
$628$	4.00000i	0.159617i
$629$	7.00000	0.279108
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	15.0000i	0.596668i
$633$	0	0
$634$	0	0
$635$	0	0
$636$	7.00000	0.277568
$637$	24.0000i	0.950915i
$638$	12.0000i	0.475085i
$639$	6.00000	0.237356
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	− 15.0000i	− 0.592003i
$643$	− 32.0000i	− 1.26196i	−0.775800	−	0.630978i	$-0.782654\pi$
$643$	− 32.0000i	− 1.26196i	0.775800	−	0.630978i	$-0.217346\pi$
$644$	−8.00000	−0.315244
$645$	0	0
$646$	5.00000	0.196722
$647$	32.0000i	1.25805i	0.777385	+	0.629025i	$0.216546\pi$
$647$	32.0000i	1.25805i	−0.777385	+	0.629025i	$0.783454\pi$
$648$	1.00000i	0.0392837i
$649$	−24.0000	−0.942082
$650$	0	0
$651$	−3.00000	−0.117579
$652$	− 18.0000i	− 0.704934i
$653$	− 2.00000i	− 0.0782660i	−0.999234	−	0.0391330i	$-0.987540\pi$
$653$	− 2.00000i	− 0.0782660i	0.999234	−	0.0391330i	$-0.0124596\pi$
$654$	−9.00000	−0.351928
$655$	0	0
$656$	2.00000	0.0780869
$657$	− 2.00000i	− 0.0780274i
$658$	− 7.00000i	− 0.272888i
$659$	48.0000	1.86981	0.934907	−	0.354892i	$-0.115482\pi$
$659$	48.0000	1.86981	0.934907	+	0.354892i	$0.115482\pi$
$660$	0	0
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	− 17.0000i	− 0.660724i
$663$	4.00000i	0.155347i
$664$	16.0000	0.620920
$665$	0	0
$666$	7.00000	0.271244
$667$	− 32.0000i	− 1.23904i
$668$	10.0000i	0.386912i
$669$	2.00000	0.0773245
$670$	0	0
$671$	6.00000	0.231627
$672$	− 1.00000i	− 0.0385758i
$673$	− 4.00000i	− 0.154189i	−0.997024	−	0.0770943i	$-0.975436\pi$
$673$	− 4.00000i	− 0.154189i	0.997024	−	0.0770943i	$-0.0245643\pi$
$674$	−24.0000	−0.924445
$675$	0	0
$676$	3.00000	0.115385
$677$	− 26.0000i	− 0.999261i	−0.866239	−	0.499631i	$-0.833469\pi$
$677$	− 26.0000i	− 0.999261i	0.866239	−	0.499631i	$-0.166531\pi$
$678$	− 9.00000i	− 0.345643i
$679$	−10.0000	−0.383765
$680$	0	0
$681$	−27.0000	−1.03464
$682$	− 9.00000i	− 0.344628i
$683$	− 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
$686$	13.0000	0.496342
$687$	28.0000i	1.06827i
$688$	1.00000i	0.0381246i
$689$	28.0000	1.06672
$690$	0	0
$691$	18.0000	0.684752	0.342376	−	0.939563i	$-0.388768\pi$
$691$	18.0000	0.684752	0.342376	+	0.939563i	$0.388768\pi$
$692$	2.00000i	0.0760286i
$693$	3.00000i	0.113961i
$694$	5.00000	0.189797
$695$	0	0
$696$	4.00000	0.151620
$697$	2.00000i	0.0757554i
$698$	− 22.0000i	− 0.832712i
$699$	−22.0000	−0.832116
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	4.00000i	0.150970i
$703$	− 35.0000i	− 1.32005i
$704$	3.00000	0.113067
$705$	0	0
$706$	0	0
$707$	5.00000i	0.188044i
$708$	8.00000i	0.300658i
$709$	1.00000	0.0375558	0.0187779	−	0.999824i	$-0.494022\pi$
$709$	1.00000	0.0375558	0.0187779	+	0.999824i	$0.494022\pi$
$710$	0	0
$711$	−15.0000	−0.562544
$712$	− 2.00000i	− 0.0749532i
$713$	24.0000i	0.898807i
$714$	1.00000	0.0374241
$715$	0	0
$716$	0	0
$717$	− 3.00000i	− 0.112037i
$718$	− 25.0000i	− 0.932992i
$719$	10.0000	0.372937	0.186469	−	0.982461i	$-0.440296\pi$
$719$	10.0000	0.372937	0.186469	+	0.982461i	$0.440296\pi$
$720$	0	0
$721$	10.0000	0.372419
$722$	− 6.00000i	− 0.223297i
$723$	0	0
$724$	19.0000	0.706129
$725$	0	0
$726$	2.00000	0.0742270
$727$	− 12.0000i	− 0.445055i	−0.974926	−	0.222528i	$-0.928569\pi$
$727$	− 12.0000i	− 0.445055i	0.974926	−	0.222528i	$-0.0714308\pi$
$728$	− 4.00000i	− 0.148250i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−1.00000	−0.0369863
$732$	− 2.00000i	− 0.0739221i
$733$	14.0000i	0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$	14.0000i	0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	−13.0000	−0.479839
$735$	0	0
$736$	−8.00000	−0.294884
$737$	33.0000i	1.21557i
$738$	2.00000i	0.0736210i
$739$	−11.0000	−0.404642	−0.202321	−	0.979319i	$-0.564848\pi$
$739$	−11.0000	−0.404642	−0.202321	+	0.979319i	$0.564848\pi$
$740$	0	0
$741$	20.0000	0.734718
$742$	− 7.00000i	− 0.256978i
$743$	− 6.00000i	− 0.220119i	−0.993925	−	0.110059i	$-0.964896\pi$
$743$	− 6.00000i	− 0.220119i	0.993925	−	0.110059i	$-0.0351041\pi$
$744$	−3.00000	−0.109985
$745$	0	0
$746$	38.0000	1.39128
$747$	16.0000i	0.585409i
$748$	3.00000i	0.109691i
$749$	−15.0000	−0.548088
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	− 7.00000i	− 0.255264i
$753$	− 10.0000i	− 0.364420i
$754$	16.0000	0.582686
$755$	0	0
$756$	1.00000	0.0363696
$757$	50.0000i	1.81728i	0.417579	+	0.908640i	$0.362879\pi$
$757$	50.0000i	1.81728i	−0.417579	+	0.908640i	$0.637121\pi$
$758$	16.0000i	0.581146i
$759$	24.0000	0.871145
$760$	0	0
$761$	−48.0000	−1.74000	−0.869999	−	0.493053i	$-0.835881\pi$
$761$	−48.0000	−1.74000	−0.869999	+	0.493053i	$0.835881\pi$
$762$	− 2.00000i	− 0.0724524i
$763$	9.00000i	0.325822i
$764$	−15.0000	−0.542681
$765$	0	0
$766$	−20.0000	−0.722629
$767$	32.0000i	1.15545i
$768$	− 1.00000i	− 0.0360844i
$769$	−11.0000	−0.396670	−0.198335	−	0.980134i	$-0.563553\pi$
$769$	−11.0000	−0.396670	−0.198335	+	0.980134i	$0.563553\pi$
$770$	0	0
$771$	20.0000	0.720282
$772$	14.0000i	0.503871i
$773$	18.0000i	0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$	18.0000i	0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$	−1.00000	−0.0359443
$775$	0	0
$776$	−10.0000	−0.358979
$777$	− 7.00000i	− 0.251124i
$778$	9.00000i	0.322666i
$779$	10.0000	0.358287
$780$	0	0
$781$	18.0000	0.644091
$782$	− 8.00000i	− 0.286079i
$783$	4.00000i	0.142948i
$784$	6.00000	0.214286
$785$	0	0
$786$	12.0000	0.428026
$787$	22.0000i	0.784215i	0.919919	+	0.392108i	$0.128254\pi$
$787$	22.0000i	0.784215i	−0.919919	+	0.392108i	$0.871746\pi$
$788$	0	0
$789$	−9.00000	−0.320408
$790$	0	0
$791$	−9.00000	−0.320003
$792$	3.00000i	0.106600i
$793$	− 8.00000i	− 0.284088i
$794$	25.0000	0.887217
$795$	0	0
$796$	13.0000	0.460773
$797$	− 51.0000i	− 1.80651i	−0.429101	−	0.903256i	$-0.641170\pi$
$797$	− 51.0000i	− 1.80651i	0.429101	−	0.903256i	$-0.358830\pi$
$798$	− 5.00000i	− 0.176998i
$799$	7.00000	0.247642
$800$	0	0
$801$	2.00000	0.0706665
$802$	− 30.0000i	− 1.05934i
$803$	− 6.00000i	− 0.211735i
$804$	11.0000	0.387940
$805$	0	0
$806$	−12.0000	−0.422682
$807$	− 8.00000i	− 0.281613i
$808$	5.00000i	0.175899i
$809$	−5.00000	−0.175791	−0.0878953	−	0.996130i	$-0.528014\pi$
$809$	−5.00000	−0.175791	−0.0878953	+	0.996130i	$0.528014\pi$
$810$	0	0
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	− 4.00000i	− 0.140372i
$813$	20.0000i	0.701431i
$814$	21.0000	0.736050
$815$	0	0
$816$	1.00000	0.0350070
$817$	5.00000i	0.174928i
$818$	− 2.00000i	− 0.0699284i
$819$	4.00000	0.139771
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	18.0000i	0.627822i
$823$	− 32.0000i	− 1.11545i	−0.830026	−	0.557725i	$-0.811674\pi$
$823$	− 32.0000i	− 1.11545i	0.830026	−	0.557725i	$-0.188326\pi$
$824$	10.0000	0.348367
$825$	0	0
$826$	8.00000	0.278356
$827$	47.0000i	1.63435i	0.576390	+	0.817175i	$0.304461\pi$
$827$	47.0000i	1.63435i	−0.576390	+	0.817175i	$0.695539\pi$
$828$	− 8.00000i	− 0.278019i
$829$	28.0000	0.972480	0.486240	−	0.873825i	$-0.338368\pi$
$829$	28.0000	0.972480	0.486240	+	0.873825i	$0.338368\pi$
$830$	0	0
$831$	5.00000	0.173448
$832$	− 4.00000i	− 0.138675i
$833$	6.00000i	0.207888i
$834$	0	0
$835$	0	0
$836$	15.0000	0.518786
$837$	− 3.00000i	− 0.103695i
$838$	20.0000i	0.690889i
$839$	34.0000	1.17381	0.586905	−	0.809656i	$-0.300346\pi$
$839$	34.0000	1.17381	0.586905	+	0.809656i	$0.300346\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	− 20.0000i	− 0.689246i
$843$	14.0000i	0.482186i
$844$	0	0
$845$	0	0
$846$	7.00000	0.240665
$847$	− 2.00000i	− 0.0687208i
$848$	− 7.00000i	− 0.240381i
$849$	26.0000	0.892318
$850$	0	0
$851$	−56.0000	−1.91966
$852$	− 6.00000i	− 0.205557i
$853$	9.00000i	0.308154i	0.988059	+	0.154077i	$0.0492404\pi$
$853$	9.00000i	0.308154i	−0.988059	+	0.154077i	$0.950760\pi$
$854$	−2.00000	−0.0684386
$855$	0	0
$856$	−15.0000	−0.512689
$857$	41.0000i	1.40053i	0.713881	+	0.700267i	$0.246936\pi$
$857$	41.0000i	1.40053i	−0.713881	+	0.700267i	$0.753064\pi$
$858$	12.0000i	0.409673i
$859$	45.0000	1.53538	0.767690	−	0.640821i	$-0.221406\pi$
$859$	45.0000	1.53538	0.767690	+	0.640821i	$0.221406\pi$
$860$	0	0
$861$	2.00000	0.0681598
$862$	− 26.0000i	− 0.885564i
$863$	− 1.00000i	− 0.0340404i	−0.999855	−	0.0170202i	$-0.994582\pi$
$863$	− 1.00000i	− 0.0340404i	0.999855	−	0.0170202i	$-0.00541796\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	5.00000	0.169907
$867$	1.00000i	0.0339618i
$868$	3.00000i	0.101827i
$869$	−45.0000	−1.52652
$870$	0	0
$871$	44.0000	1.49088
$872$	9.00000i	0.304778i
$873$	− 10.0000i	− 0.338449i
$874$	−40.0000	−1.35302
$875$	0	0
$876$	−2.00000	−0.0675737
$877$	− 46.0000i	− 1.55331i	−0.629926	−	0.776655i	$-0.716915\pi$
$877$	− 46.0000i	− 1.55331i	0.629926	−	0.776655i	$-0.283085\pi$
$878$	− 12.0000i	− 0.404980i
$879$	10.0000	0.337292
$880$	0	0
$881$	−57.0000	−1.92038	−0.960189	−	0.279350i	$-0.909881\pi$
$881$	−57.0000	−1.92038	−0.960189	+	0.279350i	$0.909881\pi$
$882$	6.00000i	0.202031i
$883$	− 56.0000i	− 1.88455i	−0.334840	−	0.942275i	$-0.608682\pi$
$883$	− 56.0000i	− 1.88455i	0.334840	−	0.942275i	$-0.391318\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	−26.0000	−0.873487
$887$	28.0000i	0.940148i	0.882627	+	0.470074i	$0.155773\pi$
$887$	28.0000i	0.940148i	−0.882627	+	0.470074i	$0.844227\pi$
$888$	− 7.00000i	− 0.234905i
$889$	−2.00000	−0.0670778
$890$	0	0
$891$	−3.00000	−0.100504
$892$	− 2.00000i	− 0.0669650i
$893$	− 35.0000i	− 1.17123i
$894$	−18.0000	−0.602010
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	− 32.0000i	− 1.06845i
$898$	13.0000i	0.433816i
$899$	−12.0000	−0.400222
$900$	0	0
$901$	7.00000	0.233204
$902$	6.00000i	0.199778i
$903$	1.00000i	0.0332779i
$904$	−9.00000	−0.299336
$905$	0	0
$906$	−20.0000	−0.664455
$907$	− 40.0000i	− 1.32818i	−0.747653	−	0.664089i	$-0.768820\pi$
$907$	− 40.0000i	− 1.32818i	0.747653	−	0.664089i	$-0.231180\pi$
$908$	27.0000i	0.896026i
$909$	−5.00000	−0.165840
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	− 5.00000i	− 0.165567i
$913$	48.0000i	1.58857i
$914$	−11.0000	−0.363848
$915$	0	0
$916$	28.0000	0.925146
$917$	− 12.0000i	− 0.396275i
$918$	1.00000i	0.0330049i
$919$	−26.0000	−0.857661	−0.428830	−	0.903385i	$-0.641074\pi$
$919$	−26.0000	−0.857661	−0.428830	+	0.903385i	$0.641074\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	9.00000i	0.296399i
$923$	− 24.0000i	− 0.789970i
$924$	3.00000	0.0986928
$925$	0	0
$926$	16.0000	0.525793
$927$	10.0000i	0.328443i
$928$	− 4.00000i	− 0.131306i
$929$	−19.0000	−0.623370	−0.311685	−	0.950186i	$-0.600893\pi$
$929$	−19.0000	−0.623370	−0.311685	+	0.950186i	$0.600893\pi$
$930$	0	0
$931$	30.0000	0.983210
$932$	22.0000i	0.720634i
$933$	− 2.00000i	− 0.0654771i
$934$	−18.0000	−0.588978
$935$	0	0
$936$	4.00000	0.130744
$937$	42.0000i	1.37208i	0.727564	+	0.686040i	$0.240653\pi$
$937$	42.0000i	1.37208i	−0.727564	+	0.686040i	$0.759347\pi$
$938$	− 11.0000i	− 0.359163i
$939$	8.00000	0.261070
$940$	0	0
$941$	−58.0000	−1.89075	−0.945373	−	0.325991i	$-0.894302\pi$
$941$	−58.0000	−1.89075	−0.945373	+	0.325991i	$0.894302\pi$
$942$	4.00000i	0.130327i
$943$	− 16.0000i	− 0.521032i
$944$	8.00000	0.260378
$945$	0	0
$946$	−3.00000	−0.0975384
$947$	13.0000i	0.422443i	0.977438	+	0.211222i	$0.0677442\pi$
$947$	13.0000i	0.422443i	−0.977438	+	0.211222i	$0.932256\pi$
$948$	15.0000i	0.487177i
$949$	−8.00000	−0.259691
$950$	0	0
$951$	0	0
$952$	− 1.00000i	− 0.0324102i
$953$	− 4.00000i	− 0.129573i	−0.997899	−	0.0647864i	$-0.979363\pi$
$953$	− 4.00000i	− 0.129573i	0.997899	−	0.0647864i	$-0.0206366\pi$
$954$	7.00000	0.226633
$955$	0	0
$956$	−3.00000	−0.0970269
$957$	12.0000i	0.387905i
$958$	− 10.0000i	− 0.323085i
$959$	18.0000	0.581250
$960$	0	0
$961$	−22.0000	−0.709677
$962$	− 28.0000i	− 0.902756i
$963$	− 15.0000i	− 0.483368i
$964$	0	0
$965$	0	0
$966$	−8.00000	−0.257396
$967$	28.0000i	0.900419i	0.892923	+	0.450210i	$0.148651\pi$
$967$	28.0000i	0.900419i	−0.892923	+	0.450210i	$0.851349\pi$
$968$	− 2.00000i	− 0.0642824i
$969$	5.00000	0.160623
$970$	0	0
$971$	6.00000	0.192549	0.0962746	−	0.995355i	$-0.469307\pi$
$971$	6.00000	0.192549	0.0962746	+	0.995355i	$0.469307\pi$
$972$	1.00000i	0.0320750i
$973$	0	0
$974$	40.0000	1.28168
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	− 12.0000i	− 0.383914i	−0.981403	−	0.191957i	$-0.938517\pi$
$977$	− 12.0000i	− 0.383914i	0.981403	−	0.191957i	$-0.0614834\pi$
$978$	− 18.0000i	− 0.575577i
$979$	6.00000	0.191761
$980$	0	0
$981$	−9.00000	−0.287348
$982$	− 22.0000i	− 0.702048i
$983$	32.0000i	1.02064i	0.859984	+	0.510321i	$0.170473\pi$
$983$	32.0000i	1.02064i	−0.859984	+	0.510321i	$0.829527\pi$
$984$	2.00000	0.0637577
$985$	0	0
$986$	4.00000	0.127386
$987$	− 7.00000i	− 0.222812i
$988$	− 20.0000i	− 0.636285i
$989$	8.00000	0.254385
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	3.00000i	0.0952501i
$993$	− 17.0000i	− 0.539479i
$994$	−6.00000	−0.190308
$995$	0	0
$996$	16.0000	0.506979
$997$	25.0000i	0.791758i	0.918303	+	0.395879i	$0.129560\pi$
$997$	25.0000i	0.791758i	−0.918303	+	0.395879i	$0.870440\pi$
$998$	− 6.00000i	− 0.189927i
$999$	7.00000	0.221470

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2550.2.a.m.1.1	✓	1	5.3	odd	4
2550.2.a.v.1.1	yes	1	5.2	odd	4
2550.2.d.d.2449.1		2	1.1	even	1	trivial
2550.2.d.d.2449.2		2	5.4	even	2	inner
7650.2.a.o.1.1		1	15.2	even	4
7650.2.a.cc.1.1		1	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} +1.00000i q^{12} +4.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} +1.00000i q^{18} +5.00000 q^{19} +1.00000 q^{21} +3.00000i q^{22} -8.00000i q^{23} +1.00000 q^{24} +4.00000 q^{26} +1.00000i q^{27} -1.00000i q^{28} +4.00000 q^{29} -3.00000 q^{31} -1.00000i q^{32} +3.00000i q^{33} +1.00000 q^{34} +1.00000 q^{36} -7.00000i q^{37} -5.00000i q^{38} +4.00000 q^{39} +2.00000 q^{41} -1.00000i q^{42} +1.00000i q^{43} +3.00000 q^{44} -8.00000 q^{46} -7.00000i q^{47} -1.00000i q^{48} +6.00000 q^{49} +1.00000 q^{51} -4.00000i q^{52} -7.00000i q^{53} +1.00000 q^{54} -1.00000 q^{56} -5.00000i q^{57} -4.00000i q^{58} +8.00000 q^{59} -2.00000 q^{61} +3.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} -11.0000i q^{67} -1.00000i q^{68} -8.00000 q^{69} -6.00000 q^{71} -1.00000i q^{72} +2.00000i q^{73} -7.00000 q^{74} -5.00000 q^{76} -3.00000i q^{77} -4.00000i q^{78} +15.0000 q^{79} +1.00000 q^{81} -2.00000i q^{82} -16.0000i q^{83} -1.00000 q^{84} +1.00000 q^{86} -4.00000i q^{87} -3.00000i q^{88} -2.00000 q^{89} -4.00000 q^{91} +8.00000i q^{92} +3.00000i q^{93} -7.00000 q^{94} -1.00000 q^{96} +10.0000i q^{97} -6.00000i q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 6 q^{11} + 2 q^{14} + 2 q^{16} + 10 q^{19} + 2 q^{21} + 2 q^{24} + 8 q^{26} + 8 q^{29} - 6 q^{31} + 2 q^{34} + 2 q^{36} + 8 q^{39} + 4 q^{41} + 6 q^{44} - 16 q^{46} + 12 q^{49} + 2 q^{51} + 2 q^{54} - 2 q^{56} + 16 q^{59} - 4 q^{61} - 2 q^{64} + 6 q^{66} - 16 q^{69} - 12 q^{71} - 14 q^{74} - 10 q^{76} + 30 q^{79} + 2 q^{81} - 2 q^{84} + 2 q^{86} - 4 q^{89} - 8 q^{91} - 14 q^{94} - 2 q^{96} + 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 6 * q^11 + 2 * q^14 + 2 * q^16 + 10 * q^19 + 2 * q^21 + 2 * q^24 + 8 * q^26 + 8 * q^29 - 6 * q^31 + 2 * q^34 + 2 * q^36 + 8 * q^39 + 4 * q^41 + 6 * q^44 - 16 * q^46 + 12 * q^49 + 2 * q^51 + 2 * q^54 - 2 * q^56 + 16 * q^59 - 4 * q^61 - 2 * q^64 + 6 * q^66 - 16 * q^69 - 12 * q^71 - 14 * q^74 - 10 * q^76 + 30 * q^79 + 2 * q^81 - 2 * q^84 + 2 * q^86 - 4 * q^89 - 8 * q^91 - 14 * q^94 - 2 * q^96 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more