LMFDB - Embedded newform 1840.2.i.a.1471.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(1471,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1840.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.i (of order $2$, degree $1$, 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:	$14.6924739719$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 32x^{14} + 400x^{12} - 2398x^{10} + 7128x^{8} - 9200x^{6} + 4705x^{4} + 2696x^{2} + 784 $ x^16 - 32x^14 + 400x^12 - 2398x^10 + 7128x^8 - 9200x^6 + 4705x^4 + 2696*x^2 + 784
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.10
Root			$0.208533 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1840.1471
Dual form			1840.2.i.a.1471.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.657492i q^{3} +1.00000i q^{5} -1.07456 q^{7} +2.56770 q^{9} +O(q^{10})$	$q+0.657492i q^{3} +1.00000i q^{5} -1.07456 q^{7} +2.56770 q^{9} -5.98282 q^{11} -3.56770 q^{13} -0.657492 q^{15} -4.56770i q^{17} +4.31821 q^{19} -0.706514i q^{21} +(-3.59328 - 3.17621i) q^{23} -1.00000 q^{25} +3.66072i q^{27} +9.42889 q^{29} -9.28822i q^{31} -3.93366i q^{33} -1.07456i q^{35} -8.93026i q^{37} -2.34574i q^{39} +0.656040 q^{41} +2.14912 q^{43} +2.56770i q^{45} -1.86123i q^{47} -5.84532 q^{49} +3.00323 q^{51} +1.79485i q^{53} -5.98282i q^{55} +2.83919i q^{57} +3.68455i q^{59} +4.08493i q^{61} -2.75915 q^{63} -3.56770i q^{65} +3.50201 q^{67} +(2.08834 - 2.36255i) q^{69} -3.90115i q^{71} +12.6368 q^{73} -0.657492i q^{75} +6.42889 q^{77} -7.92577 q^{79} +5.29621 q^{81} -3.50201 q^{83} +4.56770 q^{85} +6.19943i q^{87} +5.13541i q^{89} +3.83371 q^{91} +6.10693 q^{93} +4.31821i q^{95} -15.7756i q^{97} -15.3621 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{9}+O(q^{10}) $ 16 * q - 32 * q^9	$ 16 q - 32 q^{9} + 16 q^{13} - 16 q^{25} + 72 q^{29} - 4 q^{41} - 32 q^{49} + 92 q^{69} - 20 q^{73} + 24 q^{77} + 60 q^{93}+O(q^{100}) $ 16 * q - 32 * q^9 + 16 * q^13 - 16 * q^25 + 72 * q^29 - 4 * q^41 - 32 * q^49 + 92 * q^69 - 20 * q^73 + 24 * q^77 + 60 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$			0.657492i			0.379603i	0.981822	+	0.189802i	$0.0607845\pi$
$3$			0.657492i			0.379603i	−0.981822	+	0.189802i	$0.939215\pi$
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	−1.07456			−0.406145			−0.203072	−	0.979164i	$-0.565093\pi$
$7$	−1.07456			−0.406145			−0.203072	+	0.979164i	$0.565093\pi$
$8$		0			0
$9$	2.56770			0.855901
$10$		0			0
$11$	−5.98282			−1.80389			−0.901945	−	0.431852i	$-0.857860\pi$
$11$	−5.98282			−1.80389			−0.901945	+	0.431852i	$0.857860\pi$
$12$		0			0
$13$	−3.56770			−0.989503			−0.494752	−	0.869034i	$-0.664741\pi$
$13$	−3.56770			−0.989503			−0.494752	+	0.869034i	$0.664741\pi$
$14$		0			0
$15$	−0.657492			−0.169764
$16$		0			0
$17$		−	4.56770i		−	1.10783i	−0.832573	−	0.553915i	$-0.813133\pi$
$17$		−	4.56770i		−	1.10783i	0.832573	−	0.553915i	$-0.186867\pi$
$18$		0			0
$19$	4.31821			0.990666			0.495333	−	0.868703i	$-0.335046\pi$
$19$	4.31821			0.990666			0.495333	+	0.868703i	$0.335046\pi$
$20$		0			0
$21$		−	0.706514i		−	0.154174i
$22$		0			0
$23$	−3.59328	−	3.17621i	−0.749251	−	0.662286i
$23$	−3.59328	−	3.17621i	−0.749251	−	0.662286i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$			3.66072i			0.704506i
$28$		0			0
$29$	9.42889			1.75090			0.875451	−	0.483307i	$-0.160565\pi$
$29$	9.42889			1.75090			0.875451	+	0.483307i	$0.160565\pi$
$30$		0			0
$31$		−	9.28822i		−	1.66821i	−0.551604	−	0.834106i	$-0.685984\pi$
$31$		−	9.28822i		−	1.66821i	0.551604	−	0.834106i	$-0.314016\pi$
$32$		0			0
$33$		−	3.93366i		−	0.684762i
$34$		0			0
$35$		−	1.07456i		−	0.181634i
$36$		0			0
$37$		−	8.93026i		−	1.46813i	−0.679082	−	0.734063i	$-0.737622\pi$
$37$		−	8.93026i		−	1.46813i	0.679082	−	0.734063i	$-0.262378\pi$
$38$		0			0
$39$		−	2.34574i		−	0.375619i
$40$		0			0
$41$	0.656040			0.102456			0.0512281	−	0.998687i	$-0.483686\pi$
$41$	0.656040			0.102456			0.0512281	+	0.998687i	$0.483686\pi$
$42$		0			0
$43$	2.14912			0.327737			0.163869	−	0.986482i	$-0.447603\pi$
$43$	2.14912			0.327737			0.163869	+	0.986482i	$0.447603\pi$
$44$		0			0
$45$			2.56770i			0.382771i
$46$		0			0
$47$		−	1.86123i		−	0.271488i	−0.990744	−	0.135744i	$-0.956657\pi$
$47$		−	1.86123i		−	0.271488i	0.990744	−	0.135744i	$-0.0433425\pi$
$48$		0			0
$49$	−5.84532			−0.835046
$50$		0			0
$51$	3.00323			0.420536
$52$		0			0
$53$			1.79485i			0.246542i	0.992373	+	0.123271i	$0.0393384\pi$
$53$			1.79485i			0.246542i	−0.992373	+	0.123271i	$0.960662\pi$
$54$		0			0
$55$		−	5.98282i		−	0.806724i
$56$		0			0
$57$			2.83919i			0.376060i
$58$		0			0
$59$			3.68455i			0.479687i	0.970812	+	0.239844i	$0.0770962\pi$
$59$			3.68455i			0.479687i	−0.970812	+	0.239844i	$0.922904\pi$
$60$		0			0
$61$			4.08493i			0.523022i	0.965200	+	0.261511i	$0.0842208\pi$
$61$			4.08493i			0.523022i	−0.965200	+	0.261511i	$0.915779\pi$
$62$		0			0
$63$	−2.75915			−0.347620
$64$		0			0
$65$		−	3.56770i		−	0.442519i
$66$		0			0
$67$	3.50201			0.427839			0.213920	−	0.976851i	$-0.431377\pi$
$67$	3.50201			0.427839			0.213920	+	0.976851i	$0.431377\pi$
$68$		0			0
$69$	2.08834	−	2.36255i	0.251406	−	0.284418i
$70$		0			0
$71$		−	3.90115i		−	0.462981i	−0.972837	−	0.231491i	$-0.925640\pi$
$71$		−	3.90115i		−	0.462981i	0.972837	−	0.231491i	$-0.0743602\pi$
$72$		0			0
$73$	12.6368			1.47902			0.739511	−	0.673144i	$-0.235057\pi$
$73$	12.6368			1.47902			0.739511	+	0.673144i	$0.235057\pi$
$74$		0			0
$75$		−	0.657492i		−	0.0759207i
$76$		0			0
$77$	6.42889			0.732640
$78$		0			0
$79$	−7.92577			−0.891719			−0.445859	−	0.895103i	$-0.647102\pi$
$79$	−7.92577			−0.891719			−0.445859	+	0.895103i	$0.647102\pi$
$80$		0			0
$81$	5.29621			0.588468
$82$		0			0
$83$	−3.50201			−0.384396			−0.192198	−	0.981356i	$-0.561562\pi$
$83$	−3.50201			−0.384396			−0.192198	+	0.981356i	$0.561562\pi$
$84$		0			0
$85$	4.56770			0.495437
$86$		0			0
$87$			6.19943i			0.664648i
$88$		0			0
$89$			5.13541i			0.544352i	0.962247	+	0.272176i	$0.0877433\pi$
$89$			5.13541i			0.544352i	−0.962247	+	0.272176i	$0.912257\pi$
$90$		0			0
$91$	3.83371			0.401882
$92$		0			0
$93$	6.10693			0.633259
$94$		0			0
$95$			4.31821i			0.443039i
$96$		0			0
$97$		−	15.7756i		−	1.60177i	−0.598820	−	0.800884i	$-0.704363\pi$
$97$		−	15.7756i		−	1.60177i	0.598820	−	0.800884i	$-0.295637\pi$
$98$		0			0
$99$	−15.3621			−1.54395
$100$		0			0
$101$	0.482770			0.0480374			0.0240187	−	0.999712i	$-0.492354\pi$
$101$	0.482770			0.0480374			0.0240187	+	0.999712i	$0.492354\pi$
$102$		0			0
$103$	−14.6193			−1.44048			−0.720239	−	0.693726i	$-0.755968\pi$
$103$	−14.6193			−1.44048			−0.720239	+	0.693726i	$0.755968\pi$
$104$		0			0
$105$	0.706514			0.0689487
$106$		0			0
$107$	−1.92867			−0.186452			−0.0932259	−	0.995645i	$-0.529718\pi$
$107$	−1.92867			−0.186452			−0.0932259	+	0.995645i	$0.529718\pi$
$108$		0			0
$109$		−	11.0691i		−	1.06022i	−0.847927	−	0.530112i	$-0.822150\pi$
$109$		−	11.0691i		−	1.06022i	0.847927	−	0.530112i	$-0.177850\pi$
$110$		0			0
$111$	5.87158			0.557305
$112$		0			0
$113$		−	8.65264i		−	0.813972i	−0.913434	−	0.406986i	$-0.866580\pi$
$113$		−	8.65264i		−	0.813972i	0.913434	−	0.406986i	$-0.133420\pi$
$114$		0			0
$115$	3.17621	−	3.59328i	0.296183	−	0.335075i
$116$		0			0
$117$	−9.16081			−0.846917
$118$		0			0
$119$			4.90826i			0.449940i
$120$		0			0
$121$	24.7942			2.25402
$122$		0			0
$123$			0.431341i			0.0388927i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	10.9385i		−	0.970639i	−0.874337	−	0.485320i	$-0.838703\pi$
$127$		−	10.9385i		−	0.970639i	0.874337	−	0.485320i	$-0.161297\pi$
$128$		0			0
$129$			1.41303i			0.124410i
$130$		0			0
$131$			3.17621i			0.277507i	0.990327	+	0.138754i	$0.0443096\pi$
$131$			3.17621i			0.277507i	−0.990327	+	0.138754i	$0.955690\pi$
$132$		0			0
$133$	−4.64017			−0.402354
$134$		0			0
$135$	−3.66072			−0.315065
$136$		0			0
$137$			2.62158i			0.223977i	0.993710	+	0.111988i	$0.0357219\pi$
$137$			2.62158i			0.223977i	−0.993710	+	0.111988i	$0.964278\pi$
$138$		0			0
$139$		−	15.4972i		−	1.31445i	−0.753692	−	0.657227i	$-0.771729\pi$
$139$		−	15.4972i		−	1.31445i	0.753692	−	0.657227i	$-0.228271\pi$
$140$		0			0
$141$	1.22374			0.103058
$142$		0			0
$143$	21.3449			1.78495
$144$		0			0
$145$			9.42889i			0.783027i
$146$		0			0
$147$		−	3.84326i		−	0.316986i
$148$		0			0
$149$		−	7.75699i		−	0.635477i	−0.948178	−	0.317739i	$-0.897077\pi$
$149$		−	7.75699i		−	0.635477i	0.948178	−	0.317739i	$-0.102923\pi$
$150$		0			0
$151$			13.5922i			1.10611i	0.833143	+	0.553057i	$0.186539\pi$
$151$			13.5922i			1.10611i	−0.833143	+	0.553057i	$0.813461\pi$
$152$		0			0
$153$		−	11.7285i		−	0.948194i
$154$		0			0
$155$	9.28822			0.746047
$156$		0			0
$157$		−	2.65264i		−	0.211704i	−0.994382	−	0.105852i	$-0.966243\pi$
$157$		−	2.65264i		−	0.211704i	0.994382	−	0.105852i	$-0.0337569\pi$
$158$		0			0
$159$	−1.18010			−0.0935880
$160$		0			0
$161$	3.86119	+	3.41303i	0.304304	+	0.268984i
$162$		0			0
$163$		−	1.40036i		−	0.109685i	−0.998495	−	0.0548423i	$-0.982534\pi$
$163$		−	1.40036i		−	0.109685i	0.998495	−	0.0548423i	$-0.0174656\pi$
$164$		0			0
$165$	3.93366			0.306235
$166$		0			0
$167$		−	24.3245i		−	1.88229i	−0.338004	−	0.941145i	$-0.609752\pi$
$167$		−	24.3245i		−	1.88229i	0.338004	−	0.941145i	$-0.390248\pi$
$168$		0			0
$169$	−0.271489			−0.0208838
$170$		0			0
$171$	11.0879			0.847913
$172$		0			0
$173$	−15.2203			−1.15718			−0.578591	−	0.815618i	$-0.696397\pi$
$173$	−15.2203			−1.15718			−0.578591	+	0.815618i	$0.696397\pi$
$174$		0			0
$175$	1.07456			0.0812290
$176$		0			0
$177$	−2.42256			−0.182091
$178$		0			0
$179$			7.47445i			0.558666i	0.960194	+	0.279333i	$0.0901134\pi$
$179$			7.47445i			0.558666i	−0.960194	+	0.279333i	$0.909887\pi$
$180$		0			0
$181$		−	0.211281i		−	0.0157044i	−0.999969	−	0.00785218i	$-0.997501\pi$
$181$		−	0.211281i		−	0.0157044i	0.999969	−	0.00785218i	$-0.00249945\pi$
$182$		0			0
$183$	−2.68581			−0.198541
$184$		0			0
$185$	8.93026			0.656566
$186$		0			0
$187$			27.3278i			1.99840i
$188$		0			0
$189$		−	3.93366i		−	0.286132i
$190$		0			0
$191$	−2.05419			−0.148636			−0.0743181	−	0.997235i	$-0.523678\pi$
$191$	−2.05419			−0.148636			−0.0743181	+	0.997235i	$0.523678\pi$
$192$		0			0
$193$	12.3592			0.889631			0.444816	−	0.895622i	$-0.353269\pi$
$193$	12.3592			0.889631			0.444816	+	0.895622i	$0.353269\pi$
$194$		0			0
$195$	2.34574			0.167982
$196$		0			0
$197$	−25.3087			−1.80317			−0.901584	−	0.432603i	$-0.857595\pi$
$197$	−25.3087			−1.80317			−0.901584	+	0.432603i	$0.857595\pi$
$198$		0			0
$199$	0.575775			0.0408156			0.0204078	−	0.999792i	$-0.493504\pi$
$199$	0.575775			0.0408156			0.0204078	+	0.999792i	$0.493504\pi$
$200$		0			0
$201$			2.30255i			0.162409i
$202$		0			0
$203$	−10.1319			−0.711120
$204$		0			0
$205$			0.656040i			0.0458198i
$206$		0			0
$207$	−9.22648	−	8.15558i	−0.641285	−	0.566852i
$208$		0			0
$209$	−25.8351			−1.78705
$210$		0			0
$211$		−	1.09454i		−	0.0753512i	−0.999290	−	0.0376756i	$-0.988005\pi$
$211$		−	1.09454i		−	0.0753512i	0.999290	−	0.0376756i	$-0.0119954\pi$
$212$		0			0
$213$	2.56498			0.175749
$214$		0			0
$215$			2.14912i			0.146569i
$216$		0			0
$217$			9.98073i			0.677536i
$218$		0			0
$219$			8.30858i			0.561442i
$220$		0			0
$221$			16.2962i			1.09620i
$222$		0			0
$223$		−	3.72246i		−	0.249274i	−0.992202	−	0.124637i	$-0.960223\pi$
$223$		−	3.72246i		−	0.249274i	0.992202	−	0.124637i	$-0.0397767\pi$
$224$		0			0
$225$	−2.56770			−0.171180
$226$		0			0
$227$	−15.4297			−1.02411			−0.512054	−	0.858953i	$-0.671115\pi$
$227$	−15.4297			−1.02411			−0.512054	+	0.858953i	$0.671115\pi$
$228$		0			0
$229$		−	8.85779i		−	0.585339i	−0.956214	−	0.292670i	$-0.905456\pi$
$229$		−	8.85779i		−	0.585339i	0.956214	−	0.292670i	$-0.0945436\pi$
$230$		0			0
$231$			4.22695i			0.278113i
$232$		0			0
$233$	−22.2265			−1.45610			−0.728052	−	0.685522i	$-0.759574\pi$
$233$	−22.2265			−1.45610			−0.728052	+	0.685522i	$0.759574\pi$
$234$		0			0
$235$	1.86123			0.121413
$236$		0			0
$237$		−	5.21113i		−	0.338499i
$238$		0			0
$239$			23.9987i			1.55235i	0.630518	+	0.776175i	$0.282842\pi$
$239$			23.9987i			1.55235i	−0.630518	+	0.776175i	$0.717158\pi$
$240$		0			0
$241$			27.1726i			1.75034i	0.483815	+	0.875170i	$0.339251\pi$
$241$			27.1726i			1.75034i	−0.483815	+	0.875170i	$0.660749\pi$
$242$		0			0
$243$			14.4644i			0.927891i
$244$		0			0
$245$		−	5.84532i		−	0.373444i
$246$		0			0
$247$	−15.4061			−0.980267
$248$		0			0
$249$		−	2.30255i		−	0.145918i
$250$		0			0
$251$	24.6029			1.55292			0.776460	−	0.630167i	$-0.217013\pi$
$251$	24.6029			1.55292			0.776460	+	0.630167i	$0.217013\pi$
$252$		0			0
$253$	21.4980	+	19.0027i	1.35157	+	1.19469i
$254$		0			0
$255$			3.00323i			0.188070i
$256$		0			0
$257$	−6.35915			−0.396673			−0.198336	−	0.980134i	$-0.563554\pi$
$257$	−6.35915			−0.396673			−0.198336	+	0.980134i	$0.563554\pi$
$258$		0			0
$259$			9.59608i			0.596272i
$260$		0			0
$261$	24.2106			1.49860
$262$		0			0
$263$	−8.48727			−0.523347			−0.261674	−	0.965156i	$-0.584274\pi$
$263$	−8.48727			−0.523347			−0.261674	+	0.965156i	$0.584274\pi$
$264$		0			0
$265$	−1.79485			−0.110257
$266$		0			0
$267$	−3.37649			−0.206638
$268$		0			0
$269$	−7.29621			−0.444858			−0.222429	−	0.974949i	$-0.571399\pi$
$269$	−7.29621			−0.444858			−0.222429	+	0.974949i	$0.571399\pi$
$270$		0			0
$271$		−	24.3425i		−	1.47870i	−0.673322	−	0.739350i	$-0.735133\pi$
$271$		−	24.3425i		−	1.47870i	0.673322	−	0.739350i	$-0.264867\pi$
$272$		0			0
$273$			2.52063i			0.152556i
$274$		0			0
$275$	5.98282			0.360778
$276$		0			0
$277$	−9.94885			−0.597769			−0.298884	−	0.954289i	$-0.596614\pi$
$277$	−9.94885			−0.597769			−0.298884	+	0.954289i	$0.596614\pi$
$278$		0			0
$279$		−	23.8494i		−	1.42783i
$280$		0			0
$281$		−	19.5457i		−	1.16600i	−0.812472	−	0.583000i	$-0.801879\pi$
$281$		−	19.5457i		−	1.16600i	0.812472	−	0.583000i	$-0.198121\pi$
$282$		0			0
$283$	24.5849			1.46142			0.730712	−	0.682686i	$-0.239188\pi$
$283$	24.5849			1.46142			0.730712	+	0.682686i	$0.239188\pi$
$284$		0			0
$285$	−2.83919			−0.168179
$286$		0			0
$287$	−0.704954			−0.0416121
$288$		0			0
$289$	−3.86392			−0.227289
$290$		0			0
$291$	10.3723			0.608036
$292$		0			0
$293$			21.9647i			1.28319i	0.767043	+	0.641596i	$0.221728\pi$
$293$			21.9647i			1.28319i	−0.767043	+	0.641596i	$0.778272\pi$
$294$		0			0
$295$	−3.68455			−0.214523
$296$		0			0
$297$		−	21.9015i		−	1.27085i
$298$		0			0
$299$	12.8198	+	11.3318i	0.741386	+	0.655334i
$300$		0			0
$301$	−2.30935			−0.133109
$302$		0			0
$303$			0.317417i			0.0182352i
$304$		0			0
$305$	−4.08493			−0.233903
$306$		0			0
$307$			16.9509i			0.967439i	0.875223	+	0.483720i	$0.160715\pi$
$307$			16.9509i			0.967439i	−0.875223	+	0.483720i	$0.839285\pi$
$308$		0			0
$309$		−	9.61205i		−	0.546810i
$310$		0			0
$311$		−	23.2025i		−	1.31569i	−0.753152	−	0.657847i	$-0.771467\pi$
$311$		−	23.2025i		−	1.31569i	0.753152	−	0.657847i	$-0.228533\pi$
$312$		0			0
$313$			23.8227i			1.34654i	0.739399	+	0.673268i	$0.235110\pi$
$313$			23.8227i			1.34654i	−0.739399	+	0.673268i	$0.764890\pi$
$314$		0			0
$315$		−	2.75915i		−	0.155460i
$316$		0			0
$317$	13.4794			0.757077			0.378538	−	0.925586i	$-0.376427\pi$
$317$	13.4794			0.757077			0.378538	+	0.925586i	$0.376427\pi$
$318$		0			0
$319$	−56.4114			−3.15843
$320$		0			0
$321$		−	1.26809i		−	0.0707777i
$322$		0			0
$323$		−	19.7243i		−	1.09749i
$324$		0			0
$325$	3.56770			0.197901
$326$		0			0
$327$	7.27783			0.402465
$328$		0			0
$329$			2.00000i			0.110264i
$330$		0			0
$331$		−	20.0937i		−	1.10445i	−0.833694	−	0.552226i	$-0.813779\pi$
$331$		−	20.0937i		−	1.10445i	0.833694	−	0.552226i	$-0.186221\pi$
$332$		0			0
$333$		−	22.9303i		−	1.25657i
$334$		0			0
$335$			3.50201i			0.191335i
$336$		0			0
$337$		−	24.8924i		−	1.35598i	−0.735073	−	0.677988i	$-0.762852\pi$
$337$		−	24.8924i		−	1.35598i	0.735073	−	0.677988i	$-0.237148\pi$
$338$		0			0
$339$	5.68904			0.308986
$340$		0			0
$341$			55.5698i			3.00927i
$342$		0			0
$343$	13.8031			0.745295
$344$		0			0
$345$	2.36255	+	2.08834i	0.127196	+	0.112432i
$346$		0			0
$347$			6.98039i			0.374727i	0.982291	+	0.187364i	$0.0599942\pi$
$347$			6.98039i			0.374727i	−0.982291	+	0.187364i	$0.940006\pi$
$348$		0			0
$349$	−5.88319			−0.314920			−0.157460	−	0.987525i	$-0.550330\pi$
$349$	−5.88319			−0.314920			−0.157460	+	0.987525i	$0.550330\pi$
$350$		0			0
$351$		−	13.0604i		−	0.697111i
$352$		0			0
$353$	−25.2609			−1.34450			−0.672252	−	0.740322i	$-0.734673\pi$
$353$	−25.2609			−1.34450			−0.672252	+	0.740322i	$0.734673\pi$
$354$		0			0
$355$	3.90115			0.207051
$356$		0			0
$357$	−3.22715			−0.170799
$358$		0			0
$359$	23.7129			1.25152			0.625759	−	0.780016i	$-0.284789\pi$
$359$	23.7129			1.25152			0.625759	+	0.780016i	$0.284789\pi$
$360$		0			0
$361$	−0.353021			−0.0185800
$362$		0			0
$363$			16.3020i			0.855632i
$364$		0			0
$365$			12.6368i			0.661439i
$366$		0			0
$367$	19.2891			1.00688			0.503442	−	0.864029i	$-0.332067\pi$
$367$	19.2891			1.00688			0.503442	+	0.864029i	$0.332067\pi$
$368$		0			0
$369$	1.68452			0.0876925
$370$		0			0
$371$		−	1.92867i		−	0.100132i
$372$		0			0
$373$			18.1699i			0.940800i	0.882453	+	0.470400i	$0.155890\pi$
$373$			18.1699i			0.940800i	−0.882453	+	0.470400i	$0.844110\pi$
$374$		0			0
$375$	0.657492			0.0339528
$376$		0			0
$377$	−33.6395			−1.73252
$378$		0			0
$379$	4.44534			0.228342			0.114171	−	0.993461i	$-0.463579\pi$
$379$	4.44534			0.228342			0.114171	+	0.993461i	$0.463579\pi$
$380$		0			0
$381$	7.19201			0.368458
$382$		0			0
$383$	−5.73874			−0.293236			−0.146618	−	0.989193i	$-0.546839\pi$
$383$	−5.73874			−0.293236			−0.146618	+	0.989193i	$0.546839\pi$
$384$		0			0
$385$			6.42889i			0.327647i
$386$		0			0
$387$	5.51830			0.280511
$388$		0			0
$389$			31.3213i			1.58805i	0.607884	+	0.794026i	$0.292019\pi$
$389$			31.3213i			1.58805i	−0.607884	+	0.794026i	$0.707981\pi$
$390$		0			0
$391$	−14.5080	+	16.4130i	−0.733701	+	0.830043i
$392$		0			0
$393$	−2.08834			−0.105343
$394$		0			0
$395$		−	7.92577i		−	0.398789i
$396$		0			0
$397$	−30.6460			−1.53808			−0.769039	−	0.639202i	$-0.779265\pi$
$397$	−30.6460			−1.53808			−0.769039	+	0.639202i	$0.779265\pi$
$398$		0			0
$399$		−	3.05088i		−	0.152735i
$400$		0			0
$401$			37.4434i			1.86983i	0.354865	+	0.934917i	$0.384527\pi$
$401$			37.4434i			1.86983i	−0.354865	+	0.934917i	$0.615473\pi$
$402$		0			0
$403$			33.1376i			1.65070i
$404$		0			0
$405$			5.29621i			0.263171i
$406$		0			0
$407$			53.4282i			2.64834i
$408$		0			0
$409$	−15.9110			−0.786748			−0.393374	−	0.919379i	$-0.628692\pi$
$409$	−15.9110			−0.786748			−0.393374	+	0.919379i	$0.628692\pi$
$410$		0			0
$411$	−1.72367			−0.0850223
$412$		0			0
$413$		−	3.95926i		−	0.194823i
$414$		0			0
$415$		−	3.50201i		−	0.171907i
$416$		0			0
$417$	10.1893			0.498971
$418$		0			0
$419$	27.5303			1.34494			0.672471	−	0.740123i	$-0.265233\pi$
$419$	27.5303			1.34494			0.672471	+	0.740123i	$0.265233\pi$
$420$		0			0
$421$		−	16.0663i		−	0.783025i	−0.920173	−	0.391513i	$-0.871952\pi$
$421$		−	16.0663i		−	0.783025i	0.920173	−	0.391513i	$-0.128048\pi$
$422$		0			0
$423$		−	4.77909i		−	0.232367i
$424$		0			0
$425$			4.56770i			0.221566i
$426$		0			0
$427$		−	4.38950i		−	0.212423i
$428$		0			0
$429$			14.0341i			0.677574i
$430$		0			0
$431$	−6.22896			−0.300038			−0.150019	−	0.988683i	$-0.547934\pi$
$431$	−6.22896			−0.300038			−0.150019	+	0.988683i	$0.547934\pi$
$432$		0			0
$433$		−	29.8598i		−	1.43497i	−0.696573	−	0.717486i	$-0.745293\pi$
$433$		−	29.8598i		−	1.43497i	0.696573	−	0.717486i	$-0.254707\pi$
$434$		0			0
$435$	−6.19943			−0.297240
$436$		0			0
$437$	−15.5166	−	13.7156i	−0.742258	−	0.656105i
$438$		0			0
$439$			3.63709i			0.173589i	0.996226	+	0.0867943i	$0.0276623\pi$
$439$			3.63709i			0.173589i	−0.996226	+	0.0867943i	$0.972338\pi$
$440$		0			0
$441$	−15.0091			−0.714717
$442$		0			0
$443$			33.8732i			1.60936i	0.593707	+	0.804681i	$0.297664\pi$
$443$			33.8732i			1.60936i	−0.593707	+	0.804681i	$0.702336\pi$
$444$		0			0
$445$	−5.13541			−0.243442
$446$		0			0
$447$	5.10016			0.241229
$448$		0			0
$449$	22.2330			1.04924			0.524619	−	0.851337i	$-0.324208\pi$
$449$	22.2330			1.04924			0.524619	+	0.851337i	$0.324208\pi$
$450$		0			0
$451$	−3.92497			−0.184820
$452$		0			0
$453$	−8.93674			−0.419885
$454$		0			0
$455$			3.83371i			0.179727i
$456$		0			0
$457$		−	4.79758i		−	0.224421i	−0.993684	−	0.112211i	$-0.964207\pi$
$457$		−	4.79758i		−	0.224421i	0.993684	−	0.112211i	$-0.0357931\pi$
$458$		0			0
$459$	16.7211			0.780474
$460$		0			0
$461$	−12.0883			−0.563010			−0.281505	−	0.959560i	$-0.590834\pi$
$461$	−12.0883			−0.563010			−0.281505	+	0.959560i	$0.590834\pi$
$462$		0			0
$463$		−	35.2926i		−	1.64019i	−0.572230	−	0.820093i	$-0.693921\pi$
$463$		−	35.2926i		−	1.64019i	0.572230	−	0.820093i	$-0.306079\pi$
$464$		0			0
$465$			6.10693i			0.283202i
$466$		0			0
$467$	11.2285			0.519592			0.259796	−	0.965663i	$-0.416345\pi$
$467$	11.2285			0.519592			0.259796	+	0.965663i	$0.416345\pi$
$468$		0			0
$469$	−3.76312			−0.173765
$470$		0			0
$471$	1.74409			0.0803634
$472$		0			0
$473$	−12.8578			−0.591202
$474$		0			0
$475$	−4.31821			−0.198133
$476$		0			0
$477$			4.60864i			0.211015i
$478$		0			0
$479$	−22.8461			−1.04387			−0.521933	−	0.852987i	$-0.674789\pi$
$479$	−22.8461			−1.04387			−0.521933	+	0.852987i	$0.674789\pi$
$480$		0			0
$481$			31.8605i			1.45271i
$482$		0			0
$483$	−2.24404	+	2.53870i	−0.102107	+	0.115515i
$484$		0			0
$485$	15.7756			0.716332
$486$		0			0
$487$			29.9559i			1.35743i	0.734402	+	0.678715i	$0.237463\pi$
$487$			29.9559i			1.35743i	−0.734402	+	0.678715i	$0.762537\pi$
$488$		0			0
$489$	0.920725			0.0416366
$490$		0			0
$491$			14.8815i			0.671591i	0.941935	+	0.335795i	$0.109005\pi$
$491$			14.8815i			0.671591i	−0.941935	+	0.335795i	$0.890995\pi$
$492$		0			0
$493$		−	43.0684i		−	1.93970i
$494$		0			0
$495$		−	15.3621i		−	0.690476i
$496$		0			0
$497$			4.19201i			0.188037i
$498$		0			0
$499$		−	10.2373i		−	0.458282i	−0.973393	−	0.229141i	$-0.926408\pi$
$499$		−	10.2373i		−	0.458282i	0.973393	−	0.229141i	$-0.0735917\pi$
$500$		0			0
$501$	15.9932			0.714523
$502$		0			0
$503$	19.1701			0.854754			0.427377	−	0.904073i	$-0.359438\pi$
$503$	19.1701			0.854754			0.427377	+	0.904073i	$0.359438\pi$
$504$		0			0
$505$			0.482770i			0.0214830i
$506$		0			0
$507$		−	0.178502i		−	0.00792755i
$508$		0			0
$509$	−12.2582			−0.543335			−0.271668	−	0.962391i	$-0.587575\pi$
$509$	−12.2582			−0.543335			−0.271668	+	0.962391i	$0.587575\pi$
$510$		0			0
$511$	−13.5790			−0.600697
$512$		0			0
$513$			15.8078i			0.697931i
$514$		0			0
$515$		−	14.6193i		−	0.644201i
$516$		0			0
$517$			11.1354i			0.489735i
$518$		0			0
$519$		−	10.0073i		−	0.439270i
$520$		0			0
$521$			34.0304i			1.49090i	0.666563	+	0.745449i	$0.267765\pi$
$521$			34.0304i			1.49090i	−0.666563	+	0.745449i	$0.732235\pi$
$522$		0			0
$523$	−7.80230			−0.341171			−0.170585	−	0.985343i	$-0.554566\pi$
$523$	−7.80230			−0.341171			−0.170585	+	0.985343i	$0.554566\pi$
$524$		0			0
$525$			0.706514i			0.0308348i
$526$		0			0
$527$	−42.4258			−1.84810
$528$		0			0
$529$	2.82333	+	22.8261i	0.122753	+	0.992437i
$530$		0			0
$531$			9.46083i			0.410565i
$532$		0			0
$533$	−2.34056			−0.101381
$534$		0			0
$535$		−	1.92867i		−	0.0833837i
$536$		0			0
$537$	−4.91439			−0.212072
$538$		0			0
$539$	34.9715			1.50633
$540$		0			0
$541$	−7.73499			−0.332553			−0.166277	−	0.986079i	$-0.553174\pi$
$541$	−7.73499			−0.332553			−0.166277	+	0.986079i	$0.553174\pi$
$542$		0			0
$543$	0.138915			0.00596143
$544$		0			0
$545$	11.0691			0.474147
$546$		0			0
$547$			2.93374i			0.125438i	0.998031	+	0.0627188i	$0.0199771\pi$
$547$			2.93374i			0.125438i	−0.998031	+	0.0627188i	$0.980023\pi$
$548$		0			0
$549$			10.4889i			0.447655i
$550$		0			0
$551$	40.7160			1.73456
$552$		0			0
$553$	8.51670			0.362167
$554$		0			0
$555$			5.87158i			0.249235i
$556$		0			0
$557$		−	35.3406i		−	1.49743i	−0.662893	−	0.748714i	$-0.730672\pi$
$557$		−	35.3406i		−	1.49743i	0.662893	−	0.748714i	$-0.269328\pi$
$558$		0			0
$559$	−7.66741			−0.324297
$560$		0			0
$561$	−17.9678			−0.758601
$562$		0			0
$563$	−41.8655			−1.76442			−0.882210	−	0.470856i	$-0.843945\pi$
$563$	−41.8655			−1.76442			−0.882210	+	0.470856i	$0.843945\pi$
$564$		0			0
$565$	8.65264			0.364019
$566$		0			0
$567$	−5.69109			−0.239003
$568$		0			0
$569$		−	32.4035i		−	1.35843i	−0.733942	−	0.679213i	$-0.762321\pi$
$569$		−	32.4035i		−	1.35843i	0.733942	−	0.679213i	$-0.237679\pi$
$570$		0			0
$571$	17.4318			0.729496			0.364748	−	0.931106i	$-0.381155\pi$
$571$	17.4318			0.729496			0.364748	+	0.931106i	$0.381155\pi$
$572$		0			0
$573$		−	1.35062i		−	0.0564228i
$574$		0			0
$575$	3.59328	+	3.17621i	0.149850	+	0.132457i
$576$		0			0
$577$	−2.74452			−0.114256			−0.0571280	−	0.998367i	$-0.518194\pi$
$577$	−2.74452			−0.114256			−0.0571280	+	0.998367i	$0.518194\pi$
$578$		0			0
$579$			8.12605i			0.337707i
$580$		0			0
$581$	3.76312			0.156120
$582$		0			0
$583$		−	10.7383i		−	0.444734i
$584$		0			0
$585$		−	9.16081i		−	0.378753i
$586$		0			0
$587$			32.0337i			1.32217i	0.750310	+	0.661086i	$0.229904\pi$
$587$			32.0337i			1.32217i	−0.750310	+	0.661086i	$0.770096\pi$
$588$		0			0
$589$		−	40.1085i		−	1.65264i
$590$		0			0
$591$		−	16.6403i		−	0.684489i
$592$		0			0
$593$	28.9587			1.18919			0.594596	−	0.804024i	$-0.297312\pi$
$593$	28.9587			1.18919			0.594596	+	0.804024i	$0.297312\pi$
$594$		0			0
$595$	−4.90826			−0.201219
$596$		0			0
$597$			0.378568i			0.0154937i
$598$		0			0
$599$			34.5865i			1.41317i	0.707630	+	0.706583i	$0.249764\pi$
$599$			34.5865i			1.41317i	−0.707630	+	0.706583i	$0.750236\pi$
$600$		0			0
$601$	23.5324			0.959908			0.479954	−	0.877294i	$-0.340653\pi$
$601$	23.5324			0.959908			0.479954	+	0.877294i	$0.340653\pi$
$602$		0			0
$603$	8.99213			0.366188
$604$		0			0
$605$			24.7942i			1.00803i
$606$		0			0
$607$			3.68660i			0.149634i	0.997197	+	0.0748172i	$0.0238373\pi$
$607$			3.68660i			0.149634i	−0.997197	+	0.0748172i	$0.976163\pi$
$608$		0			0
$609$		−	6.66164i		−	0.269943i
$610$		0			0
$611$			6.64032i			0.268638i
$612$		0			0
$613$		−	39.4130i		−	1.59188i	−0.605377	−	0.795939i	$-0.706978\pi$
$613$		−	39.4130i		−	1.59188i	0.605377	−	0.795939i	$-0.293022\pi$
$614$		0			0
$615$	−0.431341			−0.0173934
$616$		0			0
$617$			33.9862i			1.36823i	0.729373	+	0.684116i	$0.239812\pi$
$617$			33.9862i			1.36823i	−0.729373	+	0.684116i	$0.760188\pi$
$618$		0			0
$619$	−11.2464			−0.452032			−0.226016	−	0.974124i	$-0.572570\pi$
$619$	−11.2464			−0.452032			−0.226016	+	0.974124i	$0.572570\pi$
$620$		0			0
$621$	11.6272	−	13.1540i	0.466585	−	0.527852i
$622$		0			0
$623$		−	5.51830i		−	0.221086i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	16.9864i		−	0.678371i
$628$		0			0
$629$	−40.7908			−1.62643
$630$		0			0
$631$	45.4624			1.80983			0.904915	−	0.425592i	$-0.139934\pi$
$631$	45.4624			1.80983			0.904915	+	0.425592i	$0.139934\pi$
$632$		0			0
$633$	0.719651			0.0286036
$634$		0			0
$635$	10.9385			0.434083
$636$		0			0
$637$	20.8544			0.826281
$638$		0			0
$639$		−	10.0170i		−	0.396266i
$640$		0			0
$641$		−	2.51125i		−	0.0991883i	−0.998769	−	0.0495941i	$-0.984207\pi$
$641$		−	2.51125i		−	0.0991883i	0.998769	−	0.0495941i	$-0.0157928\pi$
$642$		0			0
$643$	27.1350			1.07010			0.535050	−	0.844821i	$-0.320293\pi$
$643$	27.1350			1.07010			0.535050	+	0.844821i	$0.320293\pi$
$644$		0			0
$645$	−1.41303			−0.0556379
$646$		0			0
$647$		−	11.9003i		−	0.467847i	−0.972255	−	0.233924i	$-0.924843\pi$
$647$		−	11.9003i		−	0.467847i	0.972255	−	0.233924i	$-0.0751566\pi$
$648$		0			0
$649$		−	22.0440i		−	0.865302i
$650$		0			0
$651$	−6.56225			−0.257195
$652$		0			0
$653$	−15.5677			−0.609211			−0.304606	−	0.952479i	$-0.598525\pi$
$653$	−15.5677			−0.609211			−0.304606	+	0.952479i	$0.598525\pi$
$654$		0			0
$655$	−3.17621			−0.124105
$656$		0			0
$657$	32.4475			1.26590
$658$		0			0
$659$	0.123473			0.00480983			0.00240492	−	0.999997i	$-0.499234\pi$
$659$	0.123473			0.00480983			0.00240492	+	0.999997i	$0.499234\pi$
$660$		0			0
$661$		−	37.7370i		−	1.46780i	−0.679257	−	0.733900i	$-0.737698\pi$
$661$		−	37.7370i		−	1.46780i	0.679257	−	0.733900i	$-0.262302\pi$
$662$		0			0
$663$	−10.7146			−0.416122
$664$		0			0
$665$		−	4.64017i		−	0.179938i
$666$		0			0
$667$	−33.8807	−	29.9482i	−1.31186	−	1.15960i
$668$		0			0
$669$	2.44749			0.0946254
$670$		0			0
$671$		−	24.4394i		−	0.943474i
$672$		0			0
$673$	−43.4946			−1.67659			−0.838296	−	0.545215i	$-0.816448\pi$
$673$	−43.4946			−1.67659			−0.838296	+	0.545215i	$0.816448\pi$
$674$		0			0
$675$		−	3.66072i		−	0.140901i
$676$		0			0
$677$		−	7.37775i		−	0.283550i	−0.989899	−	0.141775i	$-0.954719\pi$
$677$		−	7.37775i		−	0.283550i	0.989899	−	0.141775i	$-0.0452809\pi$
$678$		0			0
$679$			16.9518i			0.650550i
$680$		0			0
$681$		−	10.1449i		−	0.388755i
$682$		0			0
$683$			3.45455i			0.132185i	0.997814	+	0.0660924i	$0.0210532\pi$
$683$			3.45455i			0.132185i	−0.997814	+	0.0660924i	$0.978947\pi$
$684$		0			0
$685$	−2.62158			−0.100165
$686$		0			0
$687$	5.82393			0.222197
$688$		0			0
$689$		−	6.40349i		−	0.243954i
$690$		0			0
$691$		−	3.46410i		−	0.131781i	−0.997827	−	0.0658903i	$-0.979011\pi$
$691$		−	3.46410i		−	0.131781i	0.997827	−	0.0658903i	$-0.0209887\pi$
$692$		0			0
$693$	16.5075			0.627068
$694$		0			0
$695$	15.4972			0.587842
$696$		0			0
$697$		−	2.99660i		−	0.113504i
$698$		0			0
$699$		−	14.6137i		−	0.552742i
$700$		0			0
$701$			1.24028i			0.0468449i	0.999726	+	0.0234224i	$0.00745627\pi$
$701$			1.24028i			0.0468449i	−0.999726	+	0.0234224i	$0.992544\pi$
$702$		0			0
$703$		−	38.5628i		−	1.45442i
$704$		0			0
$705$			1.22374i			0.0460889i
$706$		0			0
$707$	−0.518764			−0.0195101
$708$		0			0
$709$			19.6999i			0.739844i	0.929063	+	0.369922i	$0.120616\pi$
$709$			19.6999i			0.739844i	−0.929063	+	0.369922i	$0.879384\pi$
$710$		0			0
$711$	−20.3510			−0.763223
$712$		0			0
$713$	−29.5014	+	33.3752i	−1.10483	+	1.24991i
$714$		0			0
$715$			21.3449i			0.798256i
$716$		0			0
$717$	−15.7790			−0.589277
$718$		0			0
$719$			3.04114i			0.113415i	0.998391	+	0.0567077i	$0.0180603\pi$
$719$			3.04114i			0.113415i	−0.998391	+	0.0567077i	$0.981940\pi$
$720$		0			0
$721$	15.7092			0.585043
$722$		0			0
$723$	−17.8658			−0.664435
$724$		0			0
$725$	−9.42889			−0.350180
$726$		0			0
$727$	−3.83166			−0.142108			−0.0710541	−	0.997472i	$-0.522636\pi$
$727$	−3.83166			−0.142108			−0.0710541	+	0.997472i	$0.522636\pi$
$728$		0			0
$729$	6.37842			0.236238
$730$		0			0
$731$		−	9.81653i		−	0.363077i
$732$		0			0
$733$			42.0906i			1.55465i	0.629098	+	0.777326i	$0.283424\pi$
$733$			42.0906i			1.55465i	−0.629098	+	0.777326i	$0.716576\pi$
$734$		0			0
$735$	3.84326			0.141761
$736$		0			0
$737$	−20.9519			−0.771774
$738$		0			0
$739$		−	31.3487i		−	1.15318i	−0.817033	−	0.576591i	$-0.804383\pi$
$739$		−	31.3487i		−	1.15318i	0.817033	−	0.576591i	$-0.195617\pi$
$740$		0			0
$741$		−	10.1294i		−	0.372113i
$742$		0			0
$743$	34.1496			1.25283			0.626414	−	0.779490i	$-0.284522\pi$
$743$	34.1496			1.25283			0.626414	+	0.779490i	$0.284522\pi$
$744$		0			0
$745$	7.75699			0.284194
$746$		0			0
$747$	−8.99213			−0.329005
$748$		0			0
$749$	2.07247			0.0757264
$750$		0			0
$751$	51.0565			1.86308			0.931540	−	0.363639i	$-0.118466\pi$
$751$	51.0565			1.86308			0.931540	+	0.363639i	$0.118466\pi$
$752$		0			0
$753$			16.1762i			0.589494i
$754$		0			0
$755$	−13.5922			−0.494669
$756$		0			0
$757$			11.2396i			0.408511i	0.978918	+	0.204255i	$0.0654773\pi$
$757$			11.2396i			0.408511i	−0.978918	+	0.204255i	$0.934523\pi$
$758$		0			0
$759$	−12.4941	+	14.1347i	−0.453509	+	0.513059i
$760$		0			0
$761$	50.9137			1.84562			0.922810	−	0.385254i	$-0.125886\pi$
$761$	50.9137			1.84562			0.922810	+	0.385254i	$0.125886\pi$
$762$		0			0
$763$			11.8944i			0.430605i
$764$		0			0
$765$	11.7285			0.424045
$766$		0			0
$767$		−	13.1454i		−	0.474652i
$768$		0			0
$769$		−	43.4502i		−	1.56685i	−0.621483	−	0.783427i	$-0.713470\pi$
$769$		−	43.4502i		−	1.56685i	0.621483	−	0.783427i	$-0.286530\pi$
$770$		0			0
$771$		−	4.18109i		−	0.150578i
$772$		0			0
$773$			3.41983i			0.123003i	0.998107	+	0.0615014i	$0.0195889\pi$
$773$			3.41983i			0.123003i	−0.998107	+	0.0615014i	$0.980411\pi$
$774$		0			0
$775$			9.28822i			0.333643i
$776$		0			0
$777$	−6.30935			−0.226347
$778$		0			0
$779$	2.83292			0.101500
$780$		0			0
$781$			23.3399i			0.835167i
$782$		0			0
$783$			34.5166i			1.23352i
$784$		0			0
$785$	2.65264			0.0946767
$786$		0			0
$787$	−21.1567			−0.754155			−0.377078	−	0.926182i	$-0.623071\pi$
$787$	−21.1567			−0.754155			−0.377078	+	0.926182i	$0.623071\pi$
$788$		0			0
$789$		−	5.58031i		−	0.198664i
$790$		0			0
$791$			9.29777i			0.330590i
$792$		0			0
$793$		−	14.5738i		−	0.517532i
$794$		0			0
$795$		−	1.18010i		−	0.0418538i
$796$		0			0
$797$		−	3.72593i		−	0.131979i	−0.997820	−	0.0659896i	$-0.978980\pi$
$797$		−	3.72593i		−	0.131979i	0.997820	−	0.0659896i	$-0.0210204\pi$
$798$		0			0
$799$	−8.50155			−0.300763
$800$		0			0
$801$			13.1862i			0.465912i
$802$		0			0
$803$	−75.6036			−2.66799
$804$		0			0
$805$	−3.41303	+	3.86119i	−0.120293	+	0.136089i
$806$		0			0
$807$		−	4.79721i		−	0.168870i
$808$		0			0
$809$	11.8798			0.417671			0.208835	−	0.977951i	$-0.433033\pi$
$809$	11.8798			0.417671			0.208835	+	0.977951i	$0.433033\pi$
$810$		0			0
$811$			34.5618i			1.21363i	0.794844	+	0.606814i	$0.207553\pi$
$811$			34.5618i			1.21363i	−0.794844	+	0.606814i	$0.792447\pi$
$812$		0			0
$813$	16.0050			0.561319
$814$		0			0
$815$	1.40036			0.0490524
$816$		0			0
$817$	9.28035			0.324678
$818$		0			0
$819$	9.84382			0.343971
$820$		0			0
$821$	24.5235			0.855876			0.427938	−	0.903808i	$-0.359240\pi$
$821$	24.5235			0.855876			0.427938	+	0.903808i	$0.359240\pi$
$822$		0			0
$823$		−	3.99894i		−	0.139394i	−0.997568	−	0.0696971i	$-0.977797\pi$
$823$		−	3.99894i		−	0.139394i	0.997568	−	0.0696971i	$-0.0222033\pi$
$824$		0			0
$825$			3.93366i			0.136952i
$826$		0			0
$827$	11.4583			0.398444			0.199222	−	0.979954i	$-0.436159\pi$
$827$	11.4583			0.398444			0.199222	+	0.979954i	$0.436159\pi$
$828$		0			0
$829$	−10.3818			−0.360576			−0.180288	−	0.983614i	$-0.557703\pi$
$829$	−10.3818			−0.360576			−0.180288	+	0.983614i	$0.557703\pi$
$830$		0			0
$831$		−	6.54129i		−	0.226915i
$832$		0			0
$833$			26.6997i			0.925090i
$834$		0			0
$835$	24.3245			0.841785
$836$		0			0
$837$	34.0016			1.17527
$838$		0			0
$839$	−31.4353			−1.08527			−0.542633	−	0.839970i	$-0.682573\pi$
$839$	−31.4353			−1.08527			−0.542633	+	0.839970i	$0.682573\pi$
$840$		0			0
$841$	59.9040			2.06566
$842$		0			0
$843$	12.8512			0.442617
$844$		0			0
$845$		−	0.271489i		−	0.00933951i
$846$		0			0
$847$	−26.6428			−0.915457
$848$		0			0
$849$			16.1644i			0.554761i
$850$		0			0
$851$	−28.3644	+	32.0889i	−0.972320	+	1.09999i
$852$		0			0
$853$	30.2430			1.03550			0.517751	−	0.855532i	$-0.326770\pi$
$853$	30.2430			1.03550			0.517751	+	0.855532i	$0.326770\pi$
$854$		0			0
$855$			11.0879i			0.379198i
$856$		0			0
$857$	−28.9198			−0.987883			−0.493942	−	0.869495i	$-0.664444\pi$
$857$	−28.9198			−0.987883			−0.493942	+	0.869495i	$0.664444\pi$
$858$		0			0
$859$			28.1144i			0.959252i	0.877473	+	0.479626i	$0.159228\pi$
$859$			28.1144i			0.959252i	−0.877473	+	0.479626i	$0.840772\pi$
$860$		0			0
$861$		−	0.463502i		−	0.0157961i
$862$		0			0
$863$		−	28.9107i		−	0.984130i	−0.870558	−	0.492065i	$-0.836242\pi$
$863$		−	28.9107i		−	0.984130i	0.870558	−	0.492065i	$-0.163758\pi$
$864$		0			0
$865$		−	15.2203i		−	0.517507i
$866$		0			0
$867$		−	2.54050i		−	0.0862798i
$868$		0			0
$869$	47.4185			1.60856
$870$		0			0
$871$	−12.4941			−0.423348
$872$		0			0
$873$		−	40.5070i		−	1.37096i
$874$		0			0
$875$			1.07456i			0.0363267i
$876$		0			0
$877$	27.2203			0.919166			0.459583	−	0.888135i	$-0.347999\pi$
$877$	27.2203			0.919166			0.459583	+	0.888135i	$0.347999\pi$
$878$		0			0
$879$	−14.4416			−0.487104
$880$		0			0
$881$		−	43.0648i		−	1.45089i	−0.688279	−	0.725446i	$-0.741634\pi$
$881$		−	43.0648i		−	1.45089i	0.688279	−	0.725446i	$-0.258366\pi$
$882$		0			0
$883$		−	47.2383i		−	1.58969i	−0.606809	−	0.794847i	$-0.707551\pi$
$883$		−	47.2383i		−	1.58969i	0.606809	−	0.794847i	$-0.292449\pi$
$884$		0			0
$885$		−	2.42256i		−	0.0814335i
$886$		0			0
$887$			10.8795i			0.365298i	0.983178	+	0.182649i	$0.0584671\pi$
$887$			10.8795i			0.365298i	−0.983178	+	0.182649i	$0.941533\pi$
$888$		0			0
$889$			11.7541i			0.394220i
$890$		0			0
$891$	−31.6863			−1.06153
$892$		0			0
$893$		−	8.03719i		−	0.268954i
$894$		0			0
$895$	−7.47445			−0.249843
$896$		0			0
$897$	−7.45057	+	8.42889i	−0.248767	+	0.281433i
$898$		0			0
$899$		−	87.5776i		−	2.92088i
$900$		0			0
$901$	8.19834			0.273126
$902$		0			0
$903$		−	1.51838i		−	0.0505286i
$904$		0			0
$905$	0.211281			0.00702321
$906$		0			0
$907$	42.8272			1.42205			0.711027	−	0.703165i	$-0.248230\pi$
$907$	42.8272			1.42205			0.711027	+	0.703165i	$0.248230\pi$
$908$		0			0
$909$	1.23961			0.0411153
$910$		0			0
$911$	14.1189			0.467779			0.233889	−	0.972263i	$-0.424855\pi$
$911$	14.1189			0.467779			0.233889	+	0.972263i	$0.424855\pi$
$912$		0			0
$913$	20.9519			0.693408
$914$		0			0
$915$		−	2.68581i		−	0.0887902i
$916$		0			0
$917$		−	3.41303i		−	0.112708i
$918$		0			0
$919$	−20.3205			−0.670312			−0.335156	−	0.942163i	$-0.608789\pi$
$919$	−20.3205			−0.670312			−0.335156	+	0.942163i	$0.608789\pi$
$920$		0			0
$921$	−11.1451			−0.367243
$922$		0			0
$923$			13.9181i			0.458121i
$924$		0			0
$925$			8.93026i			0.293625i
$926$		0			0
$927$	−37.5379			−1.23291
$928$		0			0
$929$	−8.11681			−0.266304			−0.133152	−	0.991096i	$-0.542510\pi$
$929$	−8.11681			−0.266304			−0.133152	+	0.991096i	$0.542510\pi$
$930$		0			0
$931$	−25.2414			−0.827252
$932$		0			0
$933$	15.2555			0.499442
$934$		0			0
$935$	−27.3278			−0.893714
$936$		0			0
$937$		−	58.0333i		−	1.89586i	−0.318473	−	0.947932i	$-0.603170\pi$
$937$		−	58.0333i		−	1.89586i	0.318473	−	0.947932i	$-0.396830\pi$
$938$		0			0
$939$	−15.6632			−0.511150
$940$		0			0
$941$			3.64290i			0.118755i	0.998236	+	0.0593776i	$0.0189116\pi$
$941$			3.64290i			0.118755i	−0.998236	+	0.0593776i	$0.981088\pi$
$942$		0			0
$943$	−2.35734	−	2.08372i	−0.0767654	−	0.0678554i
$944$		0			0
$945$	3.93366			0.127962
$946$		0			0
$947$		−	53.1562i		−	1.72734i	−0.504055	−	0.863671i	$-0.668159\pi$
$947$		−	53.1562i		−	1.72734i	0.504055	−	0.863671i	$-0.331841\pi$
$948$		0			0
$949$	−45.0843			−1.46350
$950$		0			0
$951$			8.86258i			0.287389i
$952$		0			0
$953$		−	35.0437i		−	1.13518i	−0.823313	−	0.567588i	$-0.807877\pi$
$953$		−	35.0437i		−	1.13518i	0.823313	−	0.567588i	$-0.192123\pi$
$954$		0			0
$955$		−	2.05419i		−	0.0664722i
$956$		0			0
$957$		−	37.0901i		−	1.19895i
$958$		0			0
$959$		−	2.81704i		−	0.0909670i
$960$		0			0
$961$	−55.2710			−1.78293
$962$		0			0
$963$	−4.95226			−0.159584
$964$		0			0
$965$			12.3592i			0.397855i
$966$		0			0
$967$		−	46.9513i		−	1.50985i	−0.655810	−	0.754926i	$-0.727673\pi$
$967$		−	46.9513i		−	1.50985i	0.655810	−	0.754926i	$-0.272327\pi$
$968$		0			0
$969$	12.9686			0.416611
$970$		0			0
$971$	32.3807			1.03914			0.519572	−	0.854426i	$-0.326091\pi$
$971$	32.3807			1.03914			0.519572	+	0.854426i	$0.326091\pi$
$972$		0			0
$973$			16.6526i			0.533859i
$974$		0			0
$975$			2.34574i			0.0751237i
$976$		0			0
$977$			43.2357i			1.38323i	0.722265	+	0.691616i	$0.243101\pi$
$977$			43.2357i			1.38323i	−0.722265	+	0.691616i	$0.756899\pi$
$978$		0			0
$979$		−	30.7242i		−	0.981951i
$980$		0			0
$981$		−	28.4221i		−	0.907447i
$982$		0			0
$983$	36.7453			1.17199			0.585997	−	0.810313i	$-0.300703\pi$
$983$	36.7453			1.17199			0.585997	+	0.810313i	$0.300703\pi$
$984$		0			0
$985$		−	25.3087i		−	0.806402i
$986$		0			0
$987$	−1.31498			−0.0418564
$988$		0			0
$989$	−7.72238	−	6.82606i	−0.245557	−	0.217056i
$990$		0			0
$991$		−	36.4508i		−	1.15790i	−0.815364	−	0.578948i	$-0.803463\pi$
$991$		−	36.4508i		−	1.15790i	0.815364	−	0.578948i	$-0.196537\pi$
$992$		0			0
$993$	13.2115			0.419254
$994$		0			0
$995$			0.575775i			0.0182533i
$996$		0			0
$997$	−30.1313			−0.954269			−0.477134	−	0.878830i	$-0.658325\pi$
$997$	−30.1313			−0.954269			−0.477134	+	0.878830i	$0.658325\pi$
$998$		0			0
$999$	32.6912			1.03430

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.i.a.1471.10	yes	16
4.3	odd	2	inner	1840.2.i.a.1471.8	yes	16
23.22	odd	2	inner	1840.2.i.a.1471.9	yes	16
92.91	even	2	inner	1840.2.i.a.1471.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1840.2.i.a.1471.7	✓	16	92.91	even	2	inner
1840.2.i.a.1471.8	yes	16	4.3	odd	2	inner
1840.2.i.a.1471.9	yes	16	23.22	odd	2	inner
1840.2.i.a.1471.10	yes	16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.657492i q^{3} +1.00000i q^{5} -1.07456 q^{7} +2.56770 q^{9} +O(q^{10})\)	\(q+0.657492i q^{3} +1.00000i q^{5} -1.07456 q^{7} +2.56770 q^{9} -5.98282 q^{11} -3.56770 q^{13} -0.657492 q^{15} -4.56770i q^{17} +4.31821 q^{19} -0.706514i q^{21} +(-3.59328 - 3.17621i) q^{23} -1.00000 q^{25} +3.66072i q^{27} +9.42889 q^{29} -9.28822i q^{31} -3.93366i q^{33} -1.07456i q^{35} -8.93026i q^{37} -2.34574i q^{39} +0.656040 q^{41} +2.14912 q^{43} +2.56770i q^{45} -1.86123i q^{47} -5.84532 q^{49} +3.00323 q^{51} +1.79485i q^{53} -5.98282i q^{55} +2.83919i q^{57} +3.68455i q^{59} +4.08493i q^{61} -2.75915 q^{63} -3.56770i q^{65} +3.50201 q^{67} +(2.08834 - 2.36255i) q^{69} -3.90115i q^{71} +12.6368 q^{73} -0.657492i q^{75} +6.42889 q^{77} -7.92577 q^{79} +5.29621 q^{81} -3.50201 q^{83} +4.56770 q^{85} +6.19943i q^{87} +5.13541i q^{89} +3.83371 q^{91} +6.10693 q^{93} +4.31821i q^{95} -15.7756i q^{97} -15.3621 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{9}+O(q^{10}) \) 16 * q - 32 * q^9	\( 16 q - 32 q^{9} + 16 q^{13} - 16 q^{25} + 72 q^{29} - 4 q^{41} - 32 q^{49} + 92 q^{69} - 20 q^{73} + 24 q^{77} + 60 q^{93}+O(q^{100}) \) 16 * q - 32 * q^9 + 16 * q^13 - 16 * q^25 + 72 * q^29 - 4 * q^41 - 32 * q^49 + 92 * q^69 - 20 * q^73 + 24 * q^77 + 60 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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