LMFDB - Embedded newform 1040.2.d.c.209.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,2,Mod(209,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1040.209");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1040.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:	$8.30444181021$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.4
Root			$0.403032 - 0.403032i$ of defining polynomial
Character	$\chi$	$=$	1040.209
Dual form			1040.2.d.c.209.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.481194i q^{3} +(1.67513 + 1.48119i) q^{5} +0.806063i q^{7} +2.76845 q^{9} +3.67513 q^{11} -1.00000i q^{13} +(-0.712742 + 0.806063i) q^{15} +1.35026i q^{17} -1.67513 q^{19} -0.387873 q^{21} -6.48119i q^{23} +(0.612127 + 4.96239i) q^{25} +2.77575i q^{27} -2.41819 q^{29} +5.28726 q^{31} +1.76845i q^{33} +(-1.19394 + 1.35026i) q^{35} -3.76845i q^{37} +0.481194 q^{39} -8.31265 q^{41} +6.79384i q^{43} +(4.63752 + 4.10062i) q^{45} +3.19394i q^{47} +6.35026 q^{49} -0.649738 q^{51} +5.73813i q^{53} +(6.15633 + 5.44358i) q^{55} -0.806063i q^{57} +5.98778 q^{59} -1.76845 q^{61} +2.23155i q^{63} +(1.48119 - 1.67513i) q^{65} +9.89446i q^{67} +3.11871 q^{69} -8.56230 q^{71} -11.7685i q^{73} +(-2.38787 + 0.294552i) q^{75} +2.96239i q^{77} -2.26187 q^{79} +6.96968 q^{81} -3.84367i q^{83} +(-2.00000 + 2.26187i) q^{85} -1.16362i q^{87} -2.77575 q^{89} +0.806063 q^{91} +2.54420i q^{93} +(-2.80606 - 2.48119i) q^{95} +1.87399i q^{97} +10.1744 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9} + 12 q^{11} - 16 q^{15} - 4 q^{21} + 2 q^{25} - 12 q^{29} + 20 q^{31} - 8 q^{35} - 8 q^{39} - 8 q^{41} - 4 q^{45} + 18 q^{49} - 24 q^{51} + 16 q^{55} - 16 q^{59} + 12 q^{61} - 2 q^{65} - 24 q^{69}+ \cdots - 16 q^{99}+O(q^{100}) $ 6 * q - 6 * q^9 + 12 * q^11 - 16 * q^15 - 4 * q^21 + 2 * q^25 - 12 * q^29 + 20 * q^31 - 8 * q^35 - 8 * q^39 - 8 * q^41 - 4 * q^45 + 18 * q^49 - 24 * q^51 + 16 * q^55 - 16 * q^59 + 12 * q^61 - 2 * q^65 - 24 * q^69 + 24 * q^71 - 16 * q^75 - 32 * q^79 + 46 * q^81 - 12 * q^85 - 20 * q^89 + 4 * q^91 - 16 * q^95 - 16 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times$.

$n$	$261$	$417$	$561$	$911$
$\chi(n)$	$1$	$-1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$			0.481194i			0.277818i	0.990305	+	0.138909i	$0.0443595\pi$
$3$			0.481194i			0.277818i	−0.990305	+	0.138909i	$0.955641\pi$
$4$		0			0
$5$	1.67513	+	1.48119i	0.749141	+	0.662410i
$5$	1.67513	+	1.48119i	0.749141	+	0.662410i
$6$		0			0
$7$			0.806063i			0.304663i	0.988329	+	0.152332i	$0.0486782\pi$
$7$			0.806063i			0.304663i	−0.988329	+	0.152332i	$0.951322\pi$
$8$		0			0
$9$	2.76845			0.922817
$10$		0			0
$11$	3.67513			1.10809			0.554047	−	0.832486i	$-0.313083\pi$
$11$	3.67513			1.10809			0.554047	+	0.832486i	$0.313083\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$		0			0
$15$	−0.712742	+	0.806063i	−0.184029	+	0.208125i
$16$		0			0
$17$			1.35026i			0.327487i	0.986503	+	0.163743i	$0.0523569\pi$
$17$			1.35026i			0.327487i	−0.986503	+	0.163743i	$0.947643\pi$
$18$		0			0
$19$	−1.67513			−0.384301			−0.192151	−	0.981365i	$-0.561546\pi$
$19$	−1.67513			−0.384301			−0.192151	+	0.981365i	$0.561546\pi$
$20$		0			0
$21$	−0.387873			−0.0846409
$22$		0			0
$23$		−	6.48119i		−	1.35142i	−0.737166	−	0.675711i	$-0.763837\pi$
$23$		−	6.48119i		−	1.35142i	0.737166	−	0.675711i	$-0.236163\pi$
$24$		0			0
$25$	0.612127	+	4.96239i	0.122425	+	0.992478i
$26$		0			0
$27$			2.77575i			0.534193i
$28$		0			0
$29$	−2.41819			−0.449047			−0.224523	−	0.974469i	$-0.572083\pi$
$29$	−2.41819			−0.449047			−0.224523	+	0.974469i	$0.572083\pi$
$30$		0			0
$31$	5.28726			0.949620			0.474810	−	0.880088i	$-0.342517\pi$
$31$	5.28726			0.949620			0.474810	+	0.880088i	$0.342517\pi$
$32$		0			0
$33$			1.76845i			0.307848i
$34$		0			0
$35$	−1.19394	+	1.35026i	−0.201812	+	0.228236i
$36$		0			0
$37$		−	3.76845i		−	0.619530i	−0.950813	−	0.309765i	$-0.899750\pi$
$37$		−	3.76845i		−	0.619530i	0.950813	−	0.309765i	$-0.100250\pi$
$38$		0			0
$39$	0.481194			0.0770528
$40$		0			0
$41$	−8.31265			−1.29822			−0.649109	−	0.760695i	$-0.724858\pi$
$41$	−8.31265			−1.29822			−0.649109	+	0.760695i	$0.724858\pi$
$42$		0			0
$43$			6.79384i			1.03605i	0.855365	+	0.518026i	$0.173333\pi$
$43$			6.79384i			1.03605i	−0.855365	+	0.518026i	$0.826667\pi$
$44$		0			0
$45$	4.63752	+	4.10062i	0.691321	+	0.611284i
$46$		0			0
$47$			3.19394i			0.465884i	0.972491	+	0.232942i	$0.0748352\pi$
$47$			3.19394i			0.465884i	−0.972491	+	0.232942i	$0.925165\pi$
$48$		0			0
$49$	6.35026			0.907180
$50$		0			0
$51$	−0.649738			−0.0909816
$52$		0			0
$53$			5.73813i			0.788193i	0.919069	+	0.394097i	$0.128943\pi$
$53$			5.73813i			0.788193i	−0.919069	+	0.394097i	$0.871057\pi$
$54$		0			0
$55$	6.15633	+	5.44358i	0.830119	+	0.734013i
$56$		0			0
$57$		−	0.806063i		−	0.106766i
$58$		0			0
$59$	5.98778			0.779543			0.389771	−	0.920912i	$-0.372554\pi$
$59$	5.98778			0.779543			0.389771	+	0.920912i	$0.372554\pi$
$60$		0			0
$61$	−1.76845			−0.226427			−0.113214	−	0.993571i	$-0.536114\pi$
$61$	−1.76845			−0.226427			−0.113214	+	0.993571i	$0.536114\pi$
$62$		0			0
$63$			2.23155i			0.281149i
$64$		0			0
$65$	1.48119	−	1.67513i	0.183720	−	0.207774i
$66$		0			0
$67$			9.89446i			1.20880i	0.796681	+	0.604400i	$0.206587\pi$
$67$			9.89446i			1.20880i	−0.796681	+	0.604400i	$0.793413\pi$
$68$		0			0
$69$	3.11871			0.375449
$70$		0			0
$71$	−8.56230			−1.01616			−0.508079	−	0.861311i	$-0.669644\pi$
$71$	−8.56230			−1.01616			−0.508079	+	0.861311i	$0.669644\pi$
$72$		0			0
$73$		−	11.7685i		−	1.37739i	−0.725050	−	0.688697i	$-0.758183\pi$
$73$		−	11.7685i		−	1.37739i	0.725050	−	0.688697i	$-0.241817\pi$
$74$		0			0
$75$	−2.38787	+	0.294552i	−0.275728	+	0.0340119i
$76$		0			0
$77$			2.96239i			0.337596i
$78$		0			0
$79$	−2.26187			−0.254480			−0.127240	−	0.991872i	$-0.540612\pi$
$79$	−2.26187			−0.254480			−0.127240	+	0.991872i	$0.540612\pi$
$80$		0			0
$81$	6.96968			0.774409
$82$		0			0
$83$		−	3.84367i		−	0.421898i	−0.977497	−	0.210949i	$-0.932345\pi$
$83$		−	3.84367i		−	0.421898i	0.977497	−	0.210949i	$-0.0676554\pi$
$84$		0			0
$85$	−2.00000	+	2.26187i	−0.216930	+	0.245334i
$86$		0			0
$87$		−	1.16362i		−	0.124753i
$88$		0			0
$89$	−2.77575			−0.294229			−0.147114	−	0.989120i	$-0.546999\pi$
$89$	−2.77575			−0.294229			−0.147114	+	0.989120i	$0.546999\pi$
$90$		0			0
$91$	0.806063			0.0844984
$92$		0			0
$93$			2.54420i			0.263821i
$94$		0			0
$95$	−2.80606	−	2.48119i	−0.287896	−	0.254565i
$96$		0			0
$97$			1.87399i			0.190275i	0.995464	+	0.0951375i	$0.0303291\pi$
$97$			1.87399i			0.190275i	−0.995464	+	0.0951375i	$0.969671\pi$
$98$		0			0
$99$	10.1744			1.02257
$100$		0			0
$101$	10.4993			1.04472			0.522359	−	0.852725i	$-0.325052\pi$
$101$	10.4993			1.04472			0.522359	+	0.852725i	$0.325052\pi$
$102$		0			0
$103$		−	15.3684i		−	1.51429i	−0.653247	−	0.757145i	$-0.726594\pi$
$103$		−	15.3684i		−	1.51429i	0.653247	−	0.757145i	$-0.273406\pi$
$104$		0			0
$105$	−0.649738	−	0.574515i	−0.0634080	−	0.0560670i
$106$		0			0
$107$			11.1309i			1.07607i	0.842923	+	0.538034i	$0.180833\pi$
$107$			11.1309i			1.07607i	−0.842923	+	0.538034i	$0.819167\pi$
$108$		0			0
$109$	−9.58769			−0.918334			−0.459167	−	0.888350i	$-0.651852\pi$
$109$	−9.58769			−0.918334			−0.459167	+	0.888350i	$0.651852\pi$
$110$		0			0
$111$	1.81336			0.172116
$112$		0			0
$113$			0.574515i			0.0540459i	0.999635	+	0.0270229i	$0.00860271\pi$
$113$			0.574515i			0.0540459i	−0.999635	+	0.0270229i	$0.991397\pi$
$114$		0			0
$115$	9.59991	−	10.8568i	0.895196	−	1.01241i
$116$		0			0
$117$		−	2.76845i		−	0.255943i
$118$		0			0
$119$	−1.08840			−0.0997732
$120$		0			0
$121$	2.50659			0.227872
$122$		0			0
$123$		−	4.00000i		−	0.360668i
$124$		0			0
$125$	−6.32487	+	9.21933i	−0.565713	+	0.824602i
$126$		0			0
$127$		−	4.29455i		−	0.381080i	−0.981679	−	0.190540i	$-0.938976\pi$
$127$		−	4.29455i		−	0.381080i	0.981679	−	0.190540i	$-0.0610239\pi$
$128$		0			0
$129$	−3.26916			−0.287833
$130$		0			0
$131$	0.836381			0.0730749			0.0365375	−	0.999332i	$-0.488367\pi$
$131$	0.836381			0.0730749			0.0365375	+	0.999332i	$0.488367\pi$
$132$		0			0
$133$		−	1.35026i		−	0.117083i
$134$		0			0
$135$	−4.11142	+	4.64974i	−0.353855	+	0.400186i
$136$		0			0
$137$			14.9380i			1.27624i	0.769939	+	0.638118i	$0.220287\pi$
$137$			14.9380i			1.27624i	−0.769939	+	0.638118i	$0.779713\pi$
$138$		0			0
$139$	−8.43866			−0.715758			−0.357879	−	0.933768i	$-0.616500\pi$
$139$	−8.43866			−0.715758			−0.357879	+	0.933768i	$0.616500\pi$
$140$		0			0
$141$	−1.53690			−0.129431
$142$		0			0
$143$		−	3.67513i		−	0.307330i
$144$		0			0
$145$	−4.05079	−	3.58181i	−0.336399	−	0.297453i
$146$		0			0
$147$			3.05571i			0.252031i
$148$		0			0
$149$	11.3503			0.929850			0.464925	−	0.885350i	$-0.346081\pi$
$149$	11.3503			0.929850			0.464925	+	0.885350i	$0.346081\pi$
$150$		0			0
$151$	−13.9878			−1.13831			−0.569155	−	0.822230i	$-0.692729\pi$
$151$	−13.9878			−1.13831			−0.569155	+	0.822230i	$0.692729\pi$
$152$		0			0
$153$			3.73813i			0.302210i
$154$		0			0
$155$	8.85685	+	7.83146i	0.711399	+	0.629038i
$156$		0			0
$157$		−	2.77575i		−	0.221529i	−0.993847	−	0.110764i	$-0.964670\pi$
$157$		−	2.77575i		−	0.221529i	0.993847	−	0.110764i	$-0.0353299\pi$
$158$		0			0
$159$	−2.76116			−0.218974
$160$		0			0
$161$	5.22425			0.411729
$162$		0			0
$163$		−	2.23155i		−	0.174788i	−0.996174	−	0.0873942i	$-0.972146\pi$
$163$		−	2.23155i		−	0.174788i	0.996174	−	0.0873942i	$-0.0278540\pi$
$164$		0			0
$165$	−2.61942	+	2.96239i	−0.203922	+	0.230622i
$166$		0			0
$167$		−	15.6932i		−	1.21438i	−0.794557	−	0.607189i	$-0.792297\pi$
$167$		−	15.6932i		−	1.21438i	0.794557	−	0.607189i	$-0.207703\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$	−4.63752			−0.354640
$172$		0			0
$173$		−	25.5877i		−	1.94540i	−0.232075	−	0.972698i	$-0.574551\pi$
$173$		−	25.5877i		−	1.94540i	0.232075	−	0.972698i	$-0.425449\pi$
$174$		0			0
$175$	−4.00000	+	0.493413i	−0.302372	+	0.0372985i
$176$		0			0
$177$			2.88129i			0.216571i
$178$		0			0
$179$	12.1260			0.906340			0.453170	−	0.891424i	$-0.350293\pi$
$179$	12.1260			0.906340			0.453170	+	0.891424i	$0.350293\pi$
$180$		0			0
$181$	−2.73084			−0.202982			−0.101491	−	0.994836i	$-0.532361\pi$
$181$	−2.73084			−0.202982			−0.101491	+	0.994836i	$0.532361\pi$
$182$		0			0
$183$		−	0.850969i		−	0.0629054i
$184$		0			0
$185$	5.58181	−	6.31265i	0.410383	−	0.464115i
$186$		0			0
$187$			4.96239i			0.362886i
$188$		0			0
$189$	−2.23743			−0.162749
$190$		0			0
$191$	−20.6253			−1.49239			−0.746197	−	0.665725i	$-0.768122\pi$
$191$	−20.6253			−1.49239			−0.746197	+	0.665725i	$0.768122\pi$
$192$		0			0
$193$		−	21.7889i		−	1.56840i	−0.620508	−	0.784200i	$-0.713073\pi$
$193$		−	21.7889i		−	1.56840i	0.620508	−	0.784200i	$-0.286927\pi$
$194$		0			0
$195$	0.806063	+	0.712742i	0.0577234	+	0.0510405i
$196$		0			0
$197$		−	2.00000i		−	0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$		−	2.00000i		−	0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$		0			0
$199$	−16.7513			−1.18747			−0.593734	−	0.804661i	$-0.702347\pi$
$199$	−16.7513			−1.18747			−0.593734	+	0.804661i	$0.702347\pi$
$200$		0			0
$201$	−4.76116			−0.335826
$202$		0			0
$203$		−	1.94921i		−	0.136808i
$204$		0			0
$205$	−13.9248	−	12.3127i	−0.972549	−	0.859953i
$206$		0			0
$207$		−	17.9429i		−	1.24712i
$208$		0			0
$209$	−6.15633			−0.425842
$210$		0			0
$211$	4.90175			0.337451			0.168725	−	0.985663i	$-0.446035\pi$
$211$	4.90175			0.337451			0.168725	+	0.985663i	$0.446035\pi$
$212$		0			0
$213$		−	4.12013i		−	0.282307i
$214$		0			0
$215$	−10.0630	+	11.3806i	−0.686291	+	0.776149i
$216$		0			0
$217$			4.26187i			0.289314i
$218$		0			0
$219$	5.66291			0.382664
$220$		0			0
$221$	1.35026			0.0908284
$222$		0			0
$223$		−	24.9076i		−	1.66794i	−0.551811	−	0.833969i	$-0.686063\pi$
$223$		−	24.9076i		−	1.66794i	0.551811	−	0.833969i	$-0.313937\pi$
$224$		0			0
$225$	1.69464	+	13.7381i	0.112976	+	0.915876i
$226$		0			0
$227$			9.95509i			0.660743i	0.943851	+	0.330371i	$0.107174\pi$
$227$			9.95509i			0.660743i	−0.943851	+	0.330371i	$0.892826\pi$
$228$		0			0
$229$	−5.35026			−0.353555			−0.176778	−	0.984251i	$-0.556567\pi$
$229$	−5.35026			−0.353555			−0.176778	+	0.984251i	$0.556567\pi$
$230$		0			0
$231$	−1.42548			−0.0937900
$232$		0			0
$233$		−	10.7612i		−	0.704987i	−0.935814	−	0.352493i	$-0.885334\pi$
$233$		−	10.7612i		−	0.704987i	0.935814	−	0.352493i	$-0.114666\pi$
$234$		0			0
$235$	−4.73084	+	5.35026i	−0.308606	+	0.349013i
$236$		0			0
$237$		−	1.08840i		−	0.0706990i
$238$		0			0
$239$	−11.8618			−0.767274			−0.383637	−	0.923484i	$-0.625329\pi$
$239$	−11.8618			−0.767274			−0.383637	+	0.923484i	$0.625329\pi$
$240$		0			0
$241$	−28.6253			−1.84392			−0.921959	−	0.387288i	$-0.873412\pi$
$241$	−28.6253			−1.84392			−0.921959	+	0.387288i	$0.873412\pi$
$242$		0			0
$243$			11.6810i			0.749337i
$244$		0			0
$245$	10.6375	+	9.40597i	0.679606	+	0.600925i
$246$		0			0
$247$			1.67513i			0.106586i
$248$		0			0
$249$	1.84955			0.117211
$250$		0			0
$251$	19.3865			1.22366			0.611831	−	0.790988i	$-0.290433\pi$
$251$	19.3865			1.22366			0.611831	+	0.790988i	$0.290433\pi$
$252$		0			0
$253$		−	23.8192i		−	1.49750i
$254$		0			0
$255$	−1.08840	−	0.962389i	−0.0681580	−	0.0602671i
$256$		0			0
$257$		−	22.8627i		−	1.42614i	−0.701094	−	0.713069i	$-0.747305\pi$
$257$		−	22.8627i		−	1.42614i	0.701094	−	0.713069i	$-0.252695\pi$
$258$		0			0
$259$	3.03761			0.188748
$260$		0			0
$261$	−6.69464			−0.414388
$262$		0			0
$263$		−	21.8822i		−	1.34932i	−0.738130	−	0.674658i	$-0.764291\pi$
$263$		−	21.8822i		−	1.34932i	0.738130	−	0.674658i	$-0.235709\pi$
$264$		0			0
$265$	−8.49929	+	9.61213i	−0.522107	+	0.590468i
$266$		0			0
$267$		−	1.33567i		−	0.0817419i
$268$		0			0
$269$	−22.7513			−1.38717			−0.693586	−	0.720374i	$-0.743970\pi$
$269$	−22.7513			−1.38717			−0.693586	+	0.720374i	$0.743970\pi$
$270$		0			0
$271$	−0.123638			−0.00751049			−0.00375525	−	0.999993i	$-0.501195\pi$
$271$	−0.123638			−0.00751049			−0.00375525	+	0.999993i	$0.501195\pi$
$272$		0			0
$273$			0.387873i			0.0234751i
$274$		0			0
$275$	2.24965	+	18.2374i	0.135659	+	1.09976i
$276$		0			0
$277$			15.3503i			0.922308i	0.887320	+	0.461154i	$0.152564\pi$
$277$			15.3503i			0.922308i	−0.887320	+	0.461154i	$0.847436\pi$
$278$		0			0
$279$	14.6375			0.876325
$280$		0			0
$281$	13.9248			0.830683			0.415341	−	0.909666i	$-0.363662\pi$
$281$	13.9248			0.830683			0.415341	+	0.909666i	$0.363662\pi$
$282$		0			0
$283$		−	20.3815i		−	1.21156i	−0.795634	−	0.605778i	$-0.792862\pi$
$283$		−	20.3815i		−	1.21156i	0.795634	−	0.605778i	$-0.207138\pi$
$284$		0			0
$285$	1.19394	−	1.35026i	0.0707227	−	0.0799826i
$286$		0			0
$287$		−	6.70052i		−	0.395519i
$288$		0			0
$289$	15.1768			0.892753
$290$		0			0
$291$	−0.901754			−0.0528618
$292$		0			0
$293$			5.38058i			0.314337i	0.987572	+	0.157168i	$0.0502365\pi$
$293$			5.38058i			0.314337i	−0.987572	+	0.157168i	$0.949763\pi$
$294$		0			0
$295$	10.0303	+	8.86907i	0.583988	+	0.516377i
$296$		0			0
$297$			10.2012i			0.591935i
$298$		0			0
$299$	−6.48119			−0.374817
$300$		0			0
$301$	−5.47627			−0.315647
$302$		0			0
$303$			5.05220i			0.290241i
$304$		0			0
$305$	−2.96239	−	2.61942i	−0.169626	−	0.149988i
$306$		0			0
$307$		−	19.1695i		−	1.09406i	−0.837113	−	0.547031i	$-0.815758\pi$
$307$		−	19.1695i		−	1.09406i	0.837113	−	0.547031i	$-0.184242\pi$
$308$		0			0
$309$	7.39517			0.420696
$310$		0			0
$311$	25.2506			1.43183			0.715915	−	0.698187i	$-0.246010\pi$
$311$	25.2506			1.43183			0.715915	+	0.698187i	$0.246010\pi$
$312$		0			0
$313$			2.81194i			0.158940i	0.996837	+	0.0794702i	$0.0253229\pi$
$313$			2.81194i			0.158940i	−0.996837	+	0.0794702i	$0.974677\pi$
$314$		0			0
$315$	−3.30536	+	3.73813i	−0.186236	+	0.210620i
$316$		0			0
$317$			23.7685i			1.33497i	0.744624	+	0.667485i	$0.232629\pi$
$317$			23.7685i			1.33497i	−0.744624	+	0.667485i	$0.767371\pi$
$318$		0			0
$319$	−8.88717			−0.497586
$320$		0			0
$321$	−5.35614			−0.298951
$322$		0			0
$323$		−	2.26187i		−	0.125854i
$324$		0			0
$325$	4.96239	−	0.612127i	0.275264	−	0.0339547i
$326$		0			0
$327$		−	4.61354i		−	0.255129i
$328$		0			0
$329$	−2.57452			−0.141938
$330$		0			0
$331$	11.8011			0.648649			0.324325	−	0.945946i	$-0.394863\pi$
$331$	11.8011			0.648649			0.324325	+	0.945946i	$0.394863\pi$
$332$		0			0
$333$		−	10.4328i		−	0.571713i
$334$		0			0
$335$	−14.6556	+	16.5745i	−0.800722	+	0.905563i
$336$		0			0
$337$		−	16.1114i		−	0.877645i	−0.898574	−	0.438822i	$-0.855396\pi$
$337$		−	16.1114i		−	0.877645i	0.898574	−	0.438822i	$-0.144604\pi$
$338$		0			0
$339$	−0.276454			−0.0150149
$340$		0			0
$341$	19.4314			1.05227
$342$		0			0
$343$			10.7612i			0.581048i
$344$		0			0
$345$	5.22425	+	4.61942i	0.281264	+	0.248701i
$346$		0			0
$347$			27.4944i			1.47598i	0.674814	+	0.737988i	$0.264224\pi$
$347$			27.4944i			1.47598i	−0.674814	+	0.737988i	$0.735776\pi$
$348$		0			0
$349$	17.6023			0.942228			0.471114	−	0.882072i	$-0.343852\pi$
$349$	17.6023			0.942228			0.471114	+	0.882072i	$0.343852\pi$
$350$		0			0
$351$	2.77575			0.148158
$352$		0			0
$353$			15.7685i			0.839270i	0.907693	+	0.419635i	$0.137842\pi$
$353$			15.7685i			0.839270i	−0.907693	+	0.419635i	$0.862158\pi$
$354$		0			0
$355$	−14.3430	−	12.6824i	−0.761246	−	0.673113i
$356$		0			0
$357$		−	0.523730i		−	0.0277187i
$358$		0			0
$359$	−14.8242			−0.782389			−0.391195	−	0.920308i	$-0.627938\pi$
$359$	−14.8242			−0.782389			−0.391195	+	0.920308i	$0.627938\pi$
$360$		0			0
$361$	−16.1939			−0.852312
$362$		0			0
$363$			1.20616i			0.0633067i
$364$		0			0
$365$	17.4314	−	19.7137i	0.912399	−	1.03186i
$366$		0			0
$367$		−	27.0313i		−	1.41102i	−0.708700	−	0.705510i	$-0.750718\pi$
$367$		−	27.0313i		−	1.41102i	0.708700	−	0.705510i	$-0.249282\pi$
$368$		0			0
$369$	−23.0132			−1.19802
$370$		0			0
$371$	−4.62530			−0.240134
$372$		0			0
$373$			12.9525i			0.670657i	0.942101	+	0.335329i	$0.108847\pi$
$373$			12.9525i			0.670657i	−0.942101	+	0.335329i	$0.891153\pi$
$374$		0			0
$375$	−4.43629	−	3.04349i	−0.229089	−	0.157165i
$376$		0			0
$377$			2.41819i			0.124543i
$378$		0			0
$379$	−30.2858			−1.55568			−0.777840	−	0.628463i	$-0.783684\pi$
$379$	−30.2858			−1.55568			−0.777840	+	0.628463i	$0.783684\pi$
$380$		0			0
$381$	2.06651			0.105871
$382$		0			0
$383$			21.0943i			1.07787i	0.842348	+	0.538934i	$0.181173\pi$
$383$			21.0943i			1.07787i	−0.842348	+	0.538934i	$0.818827\pi$
$384$		0			0
$385$	−4.38787	+	4.96239i	−0.223627	+	0.252907i
$386$		0			0
$387$			18.8084i			0.956086i
$388$		0			0
$389$	6.77575			0.343544			0.171772	−	0.985137i	$-0.445051\pi$
$389$	6.77575			0.343544			0.171772	+	0.985137i	$0.445051\pi$
$390$		0			0
$391$	8.75131			0.442573
$392$		0			0
$393$			0.402462i			0.0203015i
$394$		0			0
$395$	−3.78892	−	3.35026i	−0.190641	−	0.168570i
$396$		0			0
$397$		−	10.4690i		−	0.525423i	−0.964874	−	0.262711i	$-0.915383\pi$
$397$		−	10.4690i		−	0.525423i	0.964874	−	0.262711i	$-0.0846167\pi$
$398$		0			0
$399$	0.649738			0.0325276
$400$		0			0
$401$	5.01317			0.250346			0.125173	−	0.992135i	$-0.460051\pi$
$401$	5.01317			0.250346			0.125173	+	0.992135i	$0.460051\pi$
$402$		0			0
$403$		−	5.28726i		−	0.263377i
$404$		0			0
$405$	11.6751	+	10.3235i	0.580142	+	0.512977i
$406$		0			0
$407$		−	13.8496i		−	0.686497i
$408$		0			0
$409$	14.3879			0.711435			0.355717	−	0.934594i	$-0.384237\pi$
$409$	14.3879			0.711435			0.355717	+	0.934594i	$0.384237\pi$
$410$		0			0
$411$	−7.18806			−0.354561
$412$		0			0
$413$			4.82653i			0.237498i
$414$		0			0
$415$	5.69323	−	6.43866i	0.279470	−	0.316061i
$416$		0			0
$417$		−	4.06063i		−	0.198850i
$418$		0			0
$419$	17.4617			0.853059			0.426529	−	0.904474i	$-0.359736\pi$
$419$	17.4617			0.853059			0.426529	+	0.904474i	$0.359736\pi$
$420$		0			0
$421$	2.88717			0.140712			0.0703559	−	0.997522i	$-0.477586\pi$
$421$	2.88717			0.140712			0.0703559	+	0.997522i	$0.477586\pi$
$422$		0			0
$423$			8.84226i			0.429925i
$424$		0			0
$425$	−6.70052	+	0.826531i	−0.325023	+	0.0400927i
$426$		0			0
$427$		−	1.42548i		−	0.0689840i
$428$		0			0
$429$	1.76845			0.0853817
$430$		0			0
$431$	−0.889535			−0.0428474			−0.0214237	−	0.999770i	$-0.506820\pi$
$431$	−0.889535			−0.0428474			−0.0214237	+	0.999770i	$0.506820\pi$
$432$		0			0
$433$		−	25.2506i		−	1.21347i	−0.794906	−	0.606733i	$-0.792480\pi$
$433$		−	25.2506i		−	1.21347i	0.794906	−	0.606733i	$-0.207520\pi$
$434$		0			0
$435$	1.72355	−	1.94921i	0.0826377	−	0.0934577i
$436$		0			0
$437$			10.8568i			0.519354i
$438$		0			0
$439$	28.8119			1.37512			0.687560	−	0.726128i	$-0.258682\pi$
$439$	28.8119			1.37512			0.687560	+	0.726128i	$0.258682\pi$
$440$		0			0
$441$	17.5804			0.837162
$442$		0			0
$443$		−	36.9805i		−	1.75700i	−0.477746	−	0.878498i	$-0.658546\pi$
$443$		−	36.9805i		−	1.75700i	0.477746	−	0.878498i	$-0.341454\pi$
$444$		0			0
$445$	−4.64974	−	4.11142i	−0.220419	−	0.194900i
$446$		0			0
$447$			5.46168i			0.258329i
$448$		0			0
$449$	12.6859			0.598686			0.299343	−	0.954146i	$-0.403232\pi$
$449$	12.6859			0.598686			0.299343	+	0.954146i	$0.403232\pi$
$450$		0			0
$451$	−30.5501			−1.43855
$452$		0			0
$453$		−	6.73084i		−	0.316242i
$454$		0			0
$455$	1.35026	+	1.19394i	0.0633012	+	0.0559726i
$456$		0			0
$457$			25.0494i			1.17176i	0.810398	+	0.585880i	$0.199251\pi$
$457$			25.0494i			1.17176i	−0.810398	+	0.585880i	$0.800749\pi$
$458$		0			0
$459$	−3.74798			−0.174941
$460$		0			0
$461$	−36.8872			−1.71801			−0.859003	−	0.511970i	$-0.828916\pi$
$461$	−36.8872			−1.71801			−0.859003	+	0.511970i	$0.828916\pi$
$462$		0			0
$463$			39.0191i			1.81337i	0.421809	+	0.906685i	$0.361395\pi$
$463$			39.0191i			1.81337i	−0.421809	+	0.906685i	$0.638605\pi$
$464$		0			0
$465$	−3.76845	+	4.26187i	−0.174758	+	0.197639i
$466$		0			0
$467$			32.7694i			1.51639i	0.652029	+	0.758194i	$0.273918\pi$
$467$			32.7694i			1.51639i	−0.652029	+	0.758194i	$0.726082\pi$
$468$		0			0
$469$	−7.97556			−0.368277
$470$		0			0
$471$	1.33567			0.0615446
$472$		0			0
$473$			24.9683i			1.14804i
$474$		0			0
$475$	−1.02539	−	8.31265i	−0.0470482	−	0.381411i
$476$		0			0
$477$			15.8858i			0.727359i
$478$		0			0
$479$	16.8749			0.771036			0.385518	−	0.922700i	$-0.374023\pi$
$479$	16.8749			0.771036			0.385518	+	0.922700i	$0.374023\pi$
$480$		0			0
$481$	−3.76845			−0.171827
$482$		0			0
$483$			2.51388i			0.114386i
$484$		0			0
$485$	−2.77575	+	3.13918i	−0.126040	+	0.142543i
$486$		0			0
$487$		−	9.24472i		−	0.418918i	−0.977817	−	0.209459i	$-0.932830\pi$
$487$		−	9.24472i		−	0.418918i	0.977817	−	0.209459i	$-0.0671703\pi$
$488$		0			0
$489$	1.07381			0.0485593
$490$		0			0
$491$	25.7499			1.16208			0.581038	−	0.813876i	$-0.302647\pi$
$491$	25.7499			1.16208			0.581038	+	0.813876i	$0.302647\pi$
$492$		0			0
$493$		−	3.26519i		−	0.147057i
$494$		0			0
$495$	17.0435	+	15.0703i	0.766048	+	0.677360i
$496$		0			0
$497$		−	6.90175i		−	0.309586i
$498$		0			0
$499$	−27.7015			−1.24009			−0.620044	−	0.784567i	$-0.712885\pi$
$499$	−27.7015			−1.24009			−0.620044	+	0.784567i	$0.712885\pi$
$500$		0			0
$501$	7.55149			0.337376
$502$		0			0
$503$		−	2.35519i		−	0.105013i	−0.998621	−	0.0525063i	$-0.983279\pi$
$503$		−	2.35519i		−	0.105013i	0.998621	−	0.0525063i	$-0.0167210\pi$
$504$		0			0
$505$	17.5877	+	15.5515i	0.782642	+	0.692032i
$506$		0			0
$507$		−	0.481194i		−	0.0213706i
$508$		0			0
$509$	−21.5125			−0.953523			−0.476762	−	0.879033i	$-0.658190\pi$
$509$	−21.5125			−0.953523			−0.476762	+	0.879033i	$0.658190\pi$
$510$		0			0
$511$	9.48612			0.419641
$512$		0			0
$513$		−	4.64974i		−	0.205291i
$514$		0			0
$515$	22.7635	−	25.7440i	1.00308	−	1.13442i
$516$		0			0
$517$			11.7381i			0.516243i
$518$		0			0
$519$	12.3127			0.540465
$520$		0			0
$521$	37.7440			1.65360			0.826798	−	0.562499i	$-0.190160\pi$
$521$	37.7440			1.65360			0.826798	+	0.562499i	$0.190160\pi$
$522$		0			0
$523$		−	23.7416i		−	1.03815i	−0.854729	−	0.519075i	$-0.826277\pi$
$523$		−	23.7416i		−	1.03815i	0.854729	−	0.519075i	$-0.173723\pi$
$524$		0			0
$525$	−0.237428	−	1.92478i	−0.0103622	−	0.0840042i
$526$		0			0
$527$			7.13918i			0.310988i
$528$		0			0
$529$	−19.0059			−0.826343
$530$		0			0
$531$	16.5769			0.719376
$532$		0			0
$533$			8.31265i			0.360061i
$534$		0			0
$535$	−16.4871	+	18.6458i	−0.712798	+	0.806127i
$536$		0			0
$537$			5.83497i			0.251797i
$538$		0			0
$539$	23.3380			1.00524
$540$		0			0
$541$	13.0376			0.560531			0.280265	−	0.959923i	$-0.409578\pi$
$541$	13.0376			0.560531			0.280265	+	0.959923i	$0.409578\pi$
$542$		0			0
$543$		−	1.31406i		−	0.0563919i
$544$		0			0
$545$	−16.0606	−	14.2012i	−0.687962	−	0.608314i
$546$		0			0
$547$			8.43041i			0.360458i	0.983625	+	0.180229i	$0.0576839\pi$
$547$			8.43041i			0.360458i	−0.983625	+	0.180229i	$0.942316\pi$
$548$		0			0
$549$	−4.89587			−0.208951
$550$		0			0
$551$	4.05079			0.172569
$552$		0			0
$553$		−	1.82321i		−	0.0775306i
$554$		0			0
$555$	3.03761	+	2.68594i	0.128939	+	0.114012i
$556$		0			0
$557$		−	13.6932i		−	0.580201i	−0.956996	−	0.290100i	$-0.906311\pi$
$557$		−	13.6932i		−	0.580201i	0.956996	−	0.290100i	$-0.0936887\pi$
$558$		0			0
$559$	6.79384			0.287349
$560$		0			0
$561$	−2.38787			−0.100816
$562$		0			0
$563$			8.86907i			0.373787i	0.982380	+	0.186893i	$0.0598419\pi$
$563$			8.86907i			0.373787i	−0.982380	+	0.186893i	$0.940158\pi$
$564$		0			0
$565$	−0.850969	+	0.962389i	−0.0358005	+	0.0404880i
$566$		0			0
$567$			5.61801i			0.235934i
$568$		0			0
$569$	32.7816			1.37428			0.687139	−	0.726526i	$-0.258866\pi$
$569$	32.7816			1.37428			0.687139	+	0.726526i	$0.258866\pi$
$570$		0			0
$571$	−40.2882			−1.68601			−0.843005	−	0.537906i	$-0.819215\pi$
$571$	−40.2882			−1.68601			−0.843005	+	0.537906i	$0.819215\pi$
$572$		0			0
$573$		−	9.92478i		−	0.414614i
$574$		0			0
$575$	32.1622	−	3.96731i	1.34126	−	0.165448i
$576$		0			0
$577$		−	28.8568i		−	1.20133i	−0.799502	−	0.600663i	$-0.794903\pi$
$577$		−	28.8568i		−	1.20133i	0.799502	−	0.600663i	$-0.205097\pi$
$578$		0			0
$579$	10.4847			0.435729
$580$		0			0
$581$	3.09825			0.128537
$582$		0			0
$583$			21.0884i			0.873392i
$584$		0			0
$585$	4.10062	−	4.63752i	0.169540	−	0.191738i
$586$		0			0
$587$			41.6786i			1.72026i	0.510074	+	0.860131i	$0.329618\pi$
$587$			41.6786i			1.72026i	−0.510074	+	0.860131i	$0.670382\pi$
$588$		0			0
$589$	−8.85685			−0.364940
$590$		0			0
$591$	0.962389			0.0395874
$592$		0			0
$593$			22.4993i			0.923935i	0.886897	+	0.461968i	$0.152856\pi$
$593$			22.4993i			0.923935i	−0.886897	+	0.461968i	$0.847144\pi$
$594$		0			0
$595$	−1.82321	−	1.61213i	−0.0747442	−	0.0660908i
$596$		0			0
$597$		−	8.06063i		−	0.329900i
$598$		0			0
$599$	4.15045			0.169583			0.0847913	−	0.996399i	$-0.472978\pi$
$599$	4.15045			0.169583			0.0847913	+	0.996399i	$0.472978\pi$
$600$		0			0
$601$	27.9248			1.13908			0.569538	−	0.821965i	$-0.307122\pi$
$601$	27.9248			1.13908			0.569538	+	0.821965i	$0.307122\pi$
$602$		0			0
$603$			27.3923i			1.11550i
$604$		0			0
$605$	4.19886	+	3.71274i	0.170708	+	0.150944i
$606$		0			0
$607$			8.19489i			0.332620i	0.986073	+	0.166310i	$0.0531853\pi$
$607$			8.19489i			0.332620i	−0.986073	+	0.166310i	$0.946815\pi$
$608$		0			0
$609$	0.937951			0.0380077
$610$		0			0
$611$	3.19394			0.129213
$612$		0			0
$613$			33.1392i			1.33848i	0.743047	+	0.669239i	$0.233380\pi$
$613$			33.1392i			1.33848i	−0.743047	+	0.669239i	$0.766620\pi$
$614$		0			0
$615$	5.92478	−	6.70052i	0.238910	−	0.270191i
$616$		0			0
$617$			29.0132i			1.16803i	0.811744	+	0.584013i	$0.198518\pi$
$617$			29.0132i			1.16803i	−0.811744	+	0.584013i	$0.801482\pi$
$618$		0			0
$619$	12.2134			0.490900			0.245450	−	0.969409i	$-0.421064\pi$
$619$	12.2134			0.490900			0.245450	+	0.969409i	$0.421064\pi$
$620$		0			0
$621$	17.9902			0.721920
$622$		0			0
$623$		−	2.23743i		−	0.0896406i
$624$		0			0
$625$	−24.2506	+	6.07522i	−0.970024	+	0.243009i
$626$		0			0
$627$		−	2.96239i		−	0.118306i
$628$		0			0
$629$	5.08840			0.202888
$630$		0			0
$631$	1.22188			0.0486424			0.0243212	−	0.999704i	$-0.492258\pi$
$631$	1.22188			0.0486424			0.0243212	+	0.999704i	$0.492258\pi$
$632$		0			0
$633$			2.35870i			0.0937498i
$634$		0			0
$635$	6.36107	−	7.19394i	0.252431	−	0.285483i
$636$		0			0
$637$		−	6.35026i		−	0.251607i
$638$		0			0
$639$	−23.7043			−0.937728
$640$		0			0
$641$	−22.1016			−0.872960			−0.436480	−	0.899714i	$-0.643775\pi$
$641$	−22.1016			−0.872960			−0.436480	+	0.899714i	$0.643775\pi$
$642$		0			0
$643$		−	11.6688i		−	0.460172i	−0.973170	−	0.230086i	$-0.926099\pi$
$643$		−	11.6688i		−	0.460172i	0.973170	−	0.230086i	$-0.0739008\pi$
$644$		0			0
$645$	−5.47627	−	4.84226i	−0.215628	−	0.190664i
$646$		0			0
$647$			11.9575i			0.470096i	0.971984	+	0.235048i	$0.0755248\pi$
$647$			11.9575i			0.470096i	−0.971984	+	0.235048i	$0.924475\pi$
$648$		0			0
$649$	22.0059			0.863806
$650$		0			0
$651$	−2.05079			−0.0803766
$652$		0			0
$653$		−	10.9986i		−	0.430408i	−0.976569	−	0.215204i	$-0.930958\pi$
$653$		−	10.9986i		−	0.430408i	0.976569	−	0.215204i	$-0.0690416\pi$
$654$		0			0
$655$	1.40105	+	1.23884i	0.0547434	+	0.0484056i
$656$		0			0
$657$		−	32.5804i		−	1.27108i
$658$		0			0
$659$	−2.63989			−0.102835			−0.0514177	−	0.998677i	$-0.516374\pi$
$659$	−2.63989			−0.102835			−0.0514177	+	0.998677i	$0.516374\pi$
$660$		0			0
$661$	18.3028			0.711896			0.355948	−	0.934506i	$-0.384158\pi$
$661$	18.3028			0.711896			0.355948	+	0.934506i	$0.384158\pi$
$662$		0			0
$663$			0.649738i			0.0252337i
$664$		0			0
$665$	2.00000	−	2.26187i	0.0775567	−	0.0877114i
$666$		0			0
$667$			15.6728i			0.606852i
$668$		0			0
$669$	11.9854			0.463383
$670$		0			0
$671$	−6.49929			−0.250902
$672$		0			0
$673$			6.71037i			0.258666i	0.991601	+	0.129333i	$0.0412836\pi$
$673$			6.71037i			0.258666i	−0.991601	+	0.129333i	$0.958716\pi$
$674$		0			0
$675$	−13.7743	+	1.69911i	−0.530174	+	0.0653987i
$676$		0			0
$677$			1.57593i			0.0605679i	0.999541	+	0.0302840i	$0.00964116\pi$
$677$			1.57593i			0.0605679i	−0.999541	+	0.0302840i	$0.990359\pi$
$678$		0			0
$679$	−1.51056			−0.0579698
$680$		0			0
$681$	−4.79033			−0.183566
$682$		0			0
$683$		−	15.1939i		−	0.581380i	−0.956817	−	0.290690i	$-0.906115\pi$
$683$		−	15.1939i		−	0.581380i	0.956817	−	0.290690i	$-0.0938848\pi$
$684$		0			0
$685$	−22.1260	+	25.0230i	−0.845391	+	0.956081i
$686$		0			0
$687$		−	2.57452i		−	0.0982239i
$688$		0			0
$689$	5.73813			0.218606
$690$		0			0
$691$	18.7127			0.711866			0.355933	−	0.934511i	$-0.384163\pi$
$691$	18.7127			0.711866			0.355933	+	0.934511i	$0.384163\pi$
$692$		0			0
$693$			8.20123i			0.311539i
$694$		0			0
$695$	−14.1359	−	12.4993i	−0.536204	−	0.474125i
$696$		0			0
$697$		−	11.2243i		−	0.425149i
$698$		0			0
$699$	5.17821			0.195858
$700$		0			0
$701$	24.3028			0.917904			0.458952	−	0.888461i	$-0.348225\pi$
$701$	24.3028			0.917904			0.458952	+	0.888461i	$0.348225\pi$
$702$		0			0
$703$			6.31265i			0.238086i
$704$		0			0
$705$	−2.57452	−	2.27645i	−0.0969619	−	0.0857362i
$706$		0			0
$707$			8.46310i			0.318287i
$708$		0			0
$709$	9.66291			0.362898			0.181449	−	0.983400i	$-0.441921\pi$
$709$	9.66291			0.362898			0.181449	+	0.983400i	$0.441921\pi$
$710$		0			0
$711$	−6.26187			−0.234838
$712$		0			0
$713$		−	34.2677i		−	1.28334i
$714$		0			0
$715$	5.44358	−	6.15633i	0.203578	−	0.230234i
$716$		0			0
$717$		−	5.70782i		−	0.213162i
$718$		0			0
$719$	28.4142			1.05967			0.529836	−	0.848100i	$-0.322254\pi$
$719$	28.4142			1.05967			0.529836	+	0.848100i	$0.322254\pi$
$720$		0			0
$721$	12.3879			0.461349
$722$		0			0
$723$		−	13.7743i		−	0.512273i
$724$		0			0
$725$	−1.48024	−	12.0000i	−0.0549747	−	0.445669i
$726$		0			0
$727$		−	34.8545i		−	1.29268i	−0.763049	−	0.646341i	$-0.776299\pi$
$727$		−	34.8545i		−	1.29268i	0.763049	−	0.646341i	$-0.223701\pi$
$728$		0			0
$729$	15.2882			0.566230
$730$		0			0
$731$	−9.17347			−0.339293
$732$		0			0
$733$			6.25202i			0.230923i	0.993312	+	0.115462i	$0.0368348\pi$
$733$			6.25202i			0.230923i	−0.993312	+	0.115462i	$0.963165\pi$
$734$		0			0
$735$	−4.52610	+	5.11871i	−0.166948	+	0.188807i
$736$		0			0
$737$			36.3634i			1.33946i
$738$		0			0
$739$	−32.0846			−1.18025			−0.590126	−	0.807311i	$-0.700922\pi$
$739$	−32.0846			−1.18025			−0.590126	+	0.807311i	$0.700922\pi$
$740$		0			0
$741$	−0.806063			−0.0296115
$742$		0			0
$743$			30.5442i			1.12056i	0.828304	+	0.560279i	$0.189306\pi$
$743$			30.5442i			1.12056i	−0.828304	+	0.560279i	$0.810694\pi$
$744$		0			0
$745$	19.0132	+	16.8119i	0.696589	+	0.615942i
$746$		0			0
$747$		−	10.6410i		−	0.389335i
$748$		0			0
$749$	−8.97224			−0.327838
$750$		0			0
$751$	−28.1622			−1.02765			−0.513827	−	0.857894i	$-0.671773\pi$
$751$	−28.1622			−1.02765			−0.513827	+	0.857894i	$0.671773\pi$
$752$		0			0
$753$			9.32865i			0.339955i
$754$		0			0
$755$	−23.4314	−	20.7186i	−0.852755	−	0.754028i
$756$		0			0
$757$		−	35.4109i		−	1.28703i	−0.765433	−	0.643515i	$-0.777475\pi$
$757$		−	35.4109i		−	1.28703i	0.765433	−	0.643515i	$-0.222525\pi$
$758$		0			0
$759$	11.4617			0.416033
$760$		0			0
$761$	−19.2388			−0.697407			−0.348704	−	0.937233i	$-0.613378\pi$
$761$	−19.2388			−0.697407			−0.348704	+	0.937233i	$0.613378\pi$
$762$		0			0
$763$		−	7.72829i		−	0.279783i
$764$		0			0
$765$	−5.53690	+	6.26187i	−0.200187	+	0.226398i
$766$		0			0
$767$		−	5.98778i		−	0.216206i
$768$		0			0
$769$	−48.9643			−1.76570			−0.882849	−	0.469657i	$-0.844378\pi$
$769$	−48.9643			−1.76570			−0.882849	+	0.469657i	$0.844378\pi$
$770$		0			0
$771$	11.0014			0.396206
$772$		0			0
$773$			46.1681i			1.66055i	0.557354	+	0.830275i	$0.311817\pi$
$773$			46.1681i			1.66055i	−0.557354	+	0.830275i	$0.688183\pi$
$774$		0			0
$775$	3.23647	+	26.2374i	0.116258	+	0.942476i
$776$		0			0
$777$			1.46168i			0.0524375i
$778$		0			0
$779$	13.9248			0.498907
$780$		0			0
$781$	−31.4676			−1.12600
$782$		0			0
$783$		−	6.71228i		−	0.239877i
$784$		0			0
$785$	4.11142	−	4.64974i	0.146743	−	0.165956i
$786$		0			0
$787$		−	22.6458i		−	0.807234i	−0.914928	−	0.403617i	$-0.867753\pi$
$787$		−	22.6458i		−	0.807234i	0.914928	−	0.403617i	$-0.132247\pi$
$788$		0			0
$789$	10.5296			0.374864
$790$		0			0
$791$	−0.463096			−0.0164658
$792$		0			0
$793$			1.76845i			0.0627996i
$794$		0			0
$795$	−4.62530	−	4.08981i	−0.164043	−	0.145051i
$796$		0			0
$797$		−	8.23743i		−	0.291785i	−0.989300	−	0.145892i	$-0.953395\pi$
$797$		−	8.23743i		−	0.291785i	0.989300	−	0.145892i	$-0.0466053\pi$
$798$		0			0
$799$	−4.31265			−0.152571
$800$		0			0
$801$	−7.68452			−0.271519
$802$		0			0
$803$		−	43.2506i		−	1.52628i
$804$		0			0
$805$	8.75131	+	7.73813i	0.308443	+	0.272733i
$806$		0			0
$807$		−	10.9478i		−	0.385381i
$808$		0			0
$809$	−44.1319			−1.55159			−0.775797	−	0.630982i	$-0.782652\pi$
$809$	−44.1319			−1.55159			−0.775797	+	0.630982i	$0.782652\pi$
$810$		0			0
$811$	−22.6883			−0.796694			−0.398347	−	0.917235i	$-0.630416\pi$
$811$	−22.6883			−0.796694			−0.398347	+	0.917235i	$0.630416\pi$
$812$		0			0
$813$		−	0.0594941i		−	0.00208655i
$814$		0			0
$815$	3.30536	−	3.73813i	0.115782	−	0.130941i
$816$		0			0
$817$		−	11.3806i		−	0.398156i
$818$		0			0
$819$	2.23155			0.0779766
$820$		0			0
$821$	−50.2736			−1.75456			−0.877281	−	0.479978i	$-0.840645\pi$
$821$	−50.2736			−1.75456			−0.877281	+	0.479978i	$0.840645\pi$
$822$		0			0
$823$		−	5.13093i		−	0.178853i	−0.995993	−	0.0894265i	$-0.971497\pi$
$823$		−	5.13093i		−	0.178853i	0.995993	−	0.0894265i	$-0.0285034\pi$
$824$		0			0
$825$	−8.77575	+	1.08252i	−0.305532	+	0.0376884i
$826$		0			0
$827$			18.6946i			0.650076i	0.945701	+	0.325038i	$0.105377\pi$
$827$			18.6946i			0.650076i	−0.945701	+	0.325038i	$0.894623\pi$
$828$		0			0
$829$	3.44121			0.119518			0.0597591	−	0.998213i	$-0.480967\pi$
$829$	3.44121			0.119518			0.0597591	+	0.998213i	$0.480967\pi$
$830$		0			0
$831$	−7.38646			−0.256233
$832$		0			0
$833$			8.57452i			0.297089i
$834$		0			0
$835$	23.2447	−	26.2882i	0.804417	−	0.909741i
$836$		0			0
$837$			14.6761i			0.507280i
$838$		0			0
$839$	−52.6248			−1.81681			−0.908406	−	0.418090i	$-0.862700\pi$
$839$	−52.6248			−1.81681			−0.908406	+	0.418090i	$0.862700\pi$
$840$		0			0
$841$	−23.1524			−0.798357
$842$		0			0
$843$			6.70052i			0.230778i
$844$		0			0
$845$	−1.67513	−	1.48119i	−0.0576263	−	0.0509546i
$846$		0			0
$847$			2.02047i			0.0694241i
$848$		0			0
$849$	9.80748			0.336592
$850$		0			0
$851$	−24.4241			−0.837246
$852$		0			0
$853$		−	6.31853i		−	0.216342i	−0.994132	−	0.108171i	$-0.965501\pi$
$853$		−	6.31853i		−	0.216342i	0.994132	−	0.108171i	$-0.0344995\pi$
$854$		0			0
$855$	−7.76845	−	6.86907i	−0.265675	−	0.234917i
$856$		0			0
$857$			0.775746i			0.0264990i	0.999912	+	0.0132495i	$0.00421757\pi$
$857$			0.775746i			0.0264990i	−0.999912	+	0.0132495i	$0.995782\pi$
$858$		0			0
$859$	3.24869			0.110844			0.0554220	−	0.998463i	$-0.482350\pi$
$859$	3.24869			0.110844			0.0554220	+	0.998463i	$0.482350\pi$
$860$		0			0
$861$	3.22425			0.109882
$862$		0			0
$863$			19.9208i			0.678112i	0.940766	+	0.339056i	$0.110108\pi$
$863$			19.9208i			0.678112i	−0.940766	+	0.339056i	$0.889892\pi$
$864$		0			0
$865$	37.9003	−	42.8627i	1.28865	−	1.45738i
$866$		0			0
$867$			7.30299i			0.248022i
$868$		0			0
$869$	−8.31265			−0.281987
$870$		0			0
$871$	9.89446			0.335261
$872$		0			0
$873$			5.18806i			0.175589i
$874$		0			0
$875$	−7.43136	−	5.09825i	−0.251226	−	0.172352i
$876$		0			0
$877$		−	22.1378i		−	0.747539i	−0.927522	−	0.373770i	$-0.878065\pi$
$877$		−	22.1378i		−	0.747539i	0.927522	−	0.373770i	$-0.121935\pi$
$878$		0			0
$879$	−2.58910			−0.0873283
$880$		0			0
$881$	2.23155			0.0751828			0.0375914	−	0.999293i	$-0.488031\pi$
$881$	2.23155			0.0751828			0.0375914	+	0.999293i	$0.488031\pi$
$882$		0			0
$883$			4.30440i			0.144855i	0.997374	+	0.0724273i	$0.0230745\pi$
$883$			4.30440i			0.144855i	−0.997374	+	0.0724273i	$0.976925\pi$
$884$		0			0
$885$	−4.26774	+	4.82653i	−0.143459	+	0.162242i
$886$		0			0
$887$			15.9330i			0.534979i	0.963561	+	0.267489i	$0.0861940\pi$
$887$			15.9330i			0.534979i	−0.963561	+	0.267489i	$0.913806\pi$
$888$		0			0
$889$	3.46168			0.116101
$890$		0			0
$891$	25.6145			0.858118
$892$		0			0
$893$		−	5.35026i		−	0.179040i
$894$		0			0
$895$	20.3127	+	17.9610i	0.678977	+	0.600369i
$896$		0			0
$897$		−	3.11871i		−	0.104131i
$898$		0			0
$899$	−12.7856			−0.426423
$900$		0			0
$901$	−7.74798			−0.258123
$902$		0			0
$903$		−	2.63515i		−	0.0876923i
$904$		0			0
$905$	−4.57452	−	4.04491i	−0.152062	−	0.134457i
$906$		0			0
$907$		−	51.9086i		−	1.72360i	−0.507251	−	0.861798i	$-0.669338\pi$
$907$		−	51.9086i		−	1.72360i	0.507251	−	0.861798i	$-0.330662\pi$
$908$		0			0
$909$	29.0668			0.964085
$910$		0			0
$911$	9.67750			0.320630			0.160315	−	0.987066i	$-0.448749\pi$
$911$	9.67750			0.320630			0.160315	+	0.987066i	$0.448749\pi$
$912$		0			0
$913$		−	14.1260i		−	0.467503i
$914$		0			0
$915$	1.26045	−	1.42548i	0.0416692	−	0.0471251i
$916$		0			0
$917$			0.674176i			0.0222632i
$918$		0			0
$919$	13.5515			0.447022			0.223511	−	0.974701i	$-0.428248\pi$
$919$	13.5515			0.447022			0.223511	+	0.974701i	$0.428248\pi$
$920$		0			0
$921$	9.22425			0.303949
$922$		0			0
$923$			8.56230i			0.281831i
$924$		0			0
$925$	18.7005	−	2.30677i	0.614869	−	0.0758462i
$926$		0			0
$927$		−	42.5466i		−	1.39741i
$928$		0			0
$929$	9.44992			0.310042			0.155021	−	0.987911i	$-0.450455\pi$
$929$	9.44992			0.310042			0.155021	+	0.987911i	$0.450455\pi$
$930$		0			0
$931$	−10.6375			−0.348631
$932$		0			0
$933$			12.1504i			0.397788i
$934$		0			0
$935$	−7.35026	+	8.31265i	−0.240379	+	0.271853i
$936$		0			0
$937$		−	16.0409i		−	0.524035i	−0.965063	−	0.262017i	$-0.915612\pi$
$937$		−	16.0409i		−	0.524035i	0.965063	−	0.262017i	$-0.0843877\pi$
$938$		0			0
$939$	−1.35309			−0.0441565
$940$		0			0
$941$	−21.6747			−0.706574			−0.353287	−	0.935515i	$-0.614936\pi$
$941$	−21.6747			−0.706574			−0.353287	+	0.935515i	$0.614936\pi$
$942$		0			0
$943$			53.8759i			1.75444i
$944$		0			0
$945$	−3.74798	−	3.31406i	−0.121922	−	0.107807i
$946$		0			0
$947$		−	4.63118i		−	0.150493i	−0.997165	−	0.0752466i	$-0.976026\pi$
$947$		−	4.63118i		−	0.150493i	0.997165	−	0.0752466i	$-0.0239744\pi$
$948$		0			0
$949$	−11.7685			−0.382020
$950$		0			0
$951$	−11.4372			−0.370878
$952$		0			0
$953$			26.2981i			0.851878i	0.904752	+	0.425939i	$0.140056\pi$
$953$			26.2981i			0.851878i	−0.904752	+	0.425939i	$0.859944\pi$
$954$		0			0
$955$	−34.5501	−	30.5501i	−1.11801	−	0.988577i
$956$		0			0
$957$		−	4.27645i		−	0.138238i
$958$		0			0
$959$	−12.0409			−0.388822
$960$		0			0
$961$	−3.04491			−0.0982228
$962$		0			0
$963$			30.8155i			0.993014i
$964$		0			0
$965$	32.2736	−	36.4993i	1.03892	−	1.17495i
$966$		0			0
$967$		−	11.9405i		−	0.383981i	−0.981397	−	0.191990i	$-0.938506\pi$
$967$		−	11.9405i		−	0.383981i	0.981397	−	0.191990i	$-0.0614942\pi$
$968$		0			0
$969$	1.08840			0.0349643
$970$		0			0
$971$	−30.1524			−0.967635			−0.483818	−	0.875169i	$-0.660750\pi$
$971$	−30.1524			−0.967635			−0.483818	+	0.875169i	$0.660750\pi$
$972$		0			0
$973$		−	6.80209i		−	0.218065i
$974$		0			0
$975$	0.294552	+	2.38787i	0.00943321	+	0.0764731i
$976$		0			0
$977$		−	26.9321i		−	0.861633i	−0.902439	−	0.430817i	$-0.858226\pi$
$977$		−	26.9321i		−	0.861633i	0.902439	−	0.430817i	$-0.141774\pi$
$978$		0			0
$979$	−10.2012			−0.326033
$980$		0			0
$981$	−26.5431			−0.847455
$982$		0			0
$983$			20.5902i			0.656727i	0.944551	+	0.328363i	$0.106497\pi$
$983$			20.5902i			0.656727i	−0.944551	+	0.328363i	$0.893503\pi$
$984$		0			0
$985$	2.96239	−	3.35026i	0.0943895	−	0.106748i
$986$		0			0
$987$		−	1.23884i		−	0.0394328i
$988$		0			0
$989$	44.0322			1.40014
$990$		0			0
$991$	48.1378			1.52915			0.764573	−	0.644537i	$-0.222950\pi$
$991$	48.1378			1.52915			0.764573	+	0.644537i	$0.222950\pi$
$992$		0			0
$993$			5.67864i			0.180206i
$994$		0			0
$995$	−28.0606	−	24.8119i	−0.889582	−	0.786591i
$996$		0			0
$997$		−	33.4255i		−	1.05860i	−0.848436	−	0.529298i	$-0.822455\pi$
$997$		−	33.4255i		−	1.05860i	0.848436	−	0.529298i	$-0.177545\pi$
$998$		0			0
$999$	10.4603			0.330948

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.2.d.c.209.4		6
4.3	odd	2		65.2.b.a.14.1	✓	6
5.2	odd	4		5200.2.a.cb.1.3		3
5.3	odd	4		5200.2.a.cj.1.1		3
5.4	even	2	inner	1040.2.d.c.209.3		6
12.11	even	2		585.2.c.b.469.6		6
20.3	even	4		325.2.a.j.1.1		3
20.7	even	4		325.2.a.k.1.3		3
20.19	odd	2		65.2.b.a.14.6	yes	6
52.3	odd	6		845.2.n.f.529.6		12
52.7	even	12		845.2.l.d.699.1		12
52.11	even	12		845.2.l.d.654.2		12
52.15	even	12		845.2.l.e.654.6		12
52.19	even	12		845.2.l.e.699.5		12
52.23	odd	6		845.2.n.g.529.1		12
52.31	even	4		845.2.d.a.844.1		6
52.35	odd	6		845.2.n.f.484.1		12
52.43	odd	6		845.2.n.g.484.6		12
52.47	even	4		845.2.d.b.844.5		6
52.51	odd	2		845.2.b.c.339.6		6
60.23	odd	4		2925.2.a.bj.1.3		3
60.47	odd	4		2925.2.a.bf.1.1		3
60.59	even	2		585.2.c.b.469.1		6
260.19	even	12		845.2.l.d.699.2		12
260.59	even	12		845.2.l.e.699.6		12
260.99	even	4		845.2.d.a.844.2		6
260.103	even	4		4225.2.a.bh.1.3		3
260.119	even	12		845.2.l.d.654.1		12
260.139	odd	6		845.2.n.f.484.6		12
260.159	odd	6		845.2.n.f.529.1		12
260.179	odd	6		845.2.n.g.529.6		12
260.199	odd	6		845.2.n.g.484.1		12
260.207	even	4		4225.2.a.ba.1.1		3
260.219	even	12		845.2.l.e.654.5		12
260.239	even	4		845.2.d.b.844.6		6
260.259	odd	2		845.2.b.c.339.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.b.a.14.1	✓	6	4.3	odd	2
65.2.b.a.14.6	yes	6	20.19	odd	2
325.2.a.j.1.1		3	20.3	even	4
325.2.a.k.1.3		3	20.7	even	4
585.2.c.b.469.1		6	60.59	even	2
585.2.c.b.469.6		6	12.11	even	2
845.2.b.c.339.1		6	260.259	odd	2
845.2.b.c.339.6		6	52.51	odd	2
845.2.d.a.844.1		6	52.31	even	4
845.2.d.a.844.2		6	260.99	even	4
845.2.d.b.844.5		6	52.47	even	4
845.2.d.b.844.6		6	260.239	even	4
845.2.l.d.654.1		12	260.119	even	12
845.2.l.d.654.2		12	52.11	even	12
845.2.l.d.699.1		12	52.7	even	12
845.2.l.d.699.2		12	260.19	even	12
845.2.l.e.654.5		12	260.219	even	12
845.2.l.e.654.6		12	52.15	even	12
845.2.l.e.699.5		12	52.19	even	12
845.2.l.e.699.6		12	260.59	even	12
845.2.n.f.484.1		12	52.35	odd	6
845.2.n.f.484.6		12	260.139	odd	6
845.2.n.f.529.1		12	260.159	odd	6
845.2.n.f.529.6		12	52.3	odd	6
845.2.n.g.484.1		12	260.199	odd	6
845.2.n.g.484.6		12	52.43	odd	6
845.2.n.g.529.1		12	52.23	odd	6
845.2.n.g.529.6		12	260.179	odd	6
1040.2.d.c.209.3		6	5.4	even	2	inner
1040.2.d.c.209.4		6	1.1	even	1	trivial
2925.2.a.bf.1.1		3	60.47	odd	4
2925.2.a.bj.1.3		3	60.23	odd	4
4225.2.a.ba.1.1		3	260.207	even	4
4225.2.a.bh.1.3		3	260.103	even	4
5200.2.a.cb.1.3		3	5.2	odd	4
5200.2.a.cj.1.1		3	5.3	odd	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:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1040.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.481194i q^{3} +(1.67513 + 1.48119i) q^{5} +0.806063i q^{7} +2.76845 q^{9} +3.67513 q^{11} -1.00000i q^{13} +(-0.712742 + 0.806063i) q^{15} +1.35026i q^{17} -1.67513 q^{19} -0.387873 q^{21} -6.48119i q^{23} +(0.612127 + 4.96239i) q^{25} +2.77575i q^{27} -2.41819 q^{29} +5.28726 q^{31} +1.76845i q^{33} +(-1.19394 + 1.35026i) q^{35} -3.76845i q^{37} +0.481194 q^{39} -8.31265 q^{41} +6.79384i q^{43} +(4.63752 + 4.10062i) q^{45} +3.19394i q^{47} +6.35026 q^{49} -0.649738 q^{51} +5.73813i q^{53} +(6.15633 + 5.44358i) q^{55} -0.806063i q^{57} +5.98778 q^{59} -1.76845 q^{61} +2.23155i q^{63} +(1.48119 - 1.67513i) q^{65} +9.89446i q^{67} +3.11871 q^{69} -8.56230 q^{71} -11.7685i q^{73} +(-2.38787 + 0.294552i) q^{75} +2.96239i q^{77} -2.26187 q^{79} +6.96968 q^{81} -3.84367i q^{83} +(-2.00000 + 2.26187i) q^{85} -1.16362i q^{87} -2.77575 q^{89} +0.806063 q^{91} +2.54420i q^{93} +(-2.80606 - 2.48119i) q^{95} +1.87399i q^{97} +10.1744 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9} + 12 q^{11} - 16 q^{15} - 4 q^{21} + 2 q^{25} - 12 q^{29} + 20 q^{31} - 8 q^{35} - 8 q^{39} - 8 q^{41} - 4 q^{45} + 18 q^{49} - 24 q^{51} + 16 q^{55} - 16 q^{59} + 12 q^{61} - 2 q^{65} - 24 q^{69}+ \cdots - 16 q^{99}+O(q^{100}) \) 6 * q - 6 * q^9 + 12 * q^11 - 16 * q^15 - 4 * q^21 + 2 * q^25 - 12 * q^29 + 20 * q^31 - 8 * q^35 - 8 * q^39 - 8 * q^41 - 4 * q^45 + 18 * q^49 - 24 * q^51 + 16 * q^55 - 16 * q^59 + 12 * q^61 - 2 * q^65 - 24 * q^69 + 24 * q^71 - 16 * q^75 - 32 * q^79 + 46 * q^81 - 12 * q^85 - 20 * q^89 + 4 * q^91 - 16 * q^95 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more