LMFDB - Embedded newform 1680.2.d.d.1231.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,2,Mod(1231,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("1680.1231");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.d (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:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + 35 x^{10} - 120 x^{9} + 328 x^{8} - 658 x^{7} + 1045 x^{6} - 1270 x^{5} + 1183 x^{4} + \cdots + 13 $ x^12 - 6x^11 + 35x^10 - 120x^9 + 328x^8 - 658x^7 + 1045x^6 - 1270x^5 + 1183x^4 - 810x^3 + 376x^2 - 104*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1231.8
Root			$0.500000 + 1.60175i$ of defining polynomial
Character	$\chi$	$=$	1680.1231
Dual form			1680.2.d.d.1231.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000i q^{5} +(-2.46778 - 0.953976i) q^{7} +1.00000 q^{9} -4.70840i q^{11} +2.93556i q^{13} +1.00000i q^{15} +3.31169i q^{17} +4.77579 q^{19} +(-2.46778 - 0.953976i) q^{21} +6.24724i q^{23} -1.00000 q^{25} +1.00000 q^{27} +10.0805 q^{29} +10.0201 q^{31} -4.70840i q^{33} +(0.953976 - 2.46778i) q^{35} -4.80045 q^{37} +2.93556i q^{39} -9.92804i q^{41} -9.55191i q^{43} +1.00000i q^{45} +7.14490 q^{47} +(5.17986 + 4.70840i) q^{49} +3.31169i q^{51} +5.21964 q^{53} +4.70840 q^{55} +4.77579 q^{57} -4.31496 q^{59} +7.53182i q^{61} +(-2.46778 - 0.953976i) q^{63} -2.93556 q^{65} +6.48827i q^{67} +6.24724i q^{69} +1.65555i q^{71} -2.06738i q^{73} -1.00000 q^{75} +(-4.49171 + 11.6193i) q^{77} -12.8809i q^{79} +1.00000 q^{81} +16.0402 q^{83} -3.31169 q^{85} +10.0805 q^{87} +13.0403i q^{89} +(2.80045 - 7.24430i) q^{91} +10.0201 q^{93} +4.77579i q^{95} +1.74852i q^{97} -4.70840i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{3} - 4 q^{7} + 12 q^{9} - 16 q^{19} - 4 q^{21} - 12 q^{25} + 12 q^{27} + 8 q^{29} + 32 q^{31} + 4 q^{35} - 16 q^{37} + 24 q^{47} - 4 q^{49} + 16 q^{53} - 16 q^{57} + 24 q^{59} - 4 q^{63} + 16 q^{65}+ \cdots + 32 q^{93}+O(q^{100}) $ 12 * q + 12 * q^3 - 4 * q^7 + 12 * q^9 - 16 * q^19 - 4 * q^21 - 12 * q^25 + 12 * q^27 + 8 * q^29 + 32 * q^31 + 4 * q^35 - 16 * q^37 + 24 * q^47 - 4 * q^49 + 16 * q^53 - 16 * q^57 + 24 * q^59 - 4 * q^63 + 16 * q^65 - 12 * q^75 + 24 * q^77 + 12 * q^81 + 16 * q^83 - 8 * q^85 + 8 * q^87 - 8 * q^91 + 32 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$-1$	$1$	$1$	$1$	$-1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	−2.46778	−	0.953976i	−0.932732	−	0.360569i
$7$	−2.46778	−	0.953976i	−0.932732	−	0.360569i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	4.70840i		−	1.41964i	−0.704385	−	0.709819i	$-0.748777\pi$
$11$		−	4.70840i		−	1.41964i	0.704385	−	0.709819i	$-0.251223\pi$
$12$		0			0
$13$			2.93556i			0.814177i	0.913389	+	0.407088i	$0.133456\pi$
$13$			2.93556i			0.814177i	−0.913389	+	0.407088i	$0.866544\pi$
$14$		0			0
$15$			1.00000i			0.258199i
$16$		0			0
$17$			3.31169i			0.803202i	0.915815	+	0.401601i	$0.131546\pi$
$17$			3.31169i			0.803202i	−0.915815	+	0.401601i	$0.868454\pi$
$18$		0			0
$19$	4.77579			1.09564			0.547820	−	0.836596i	$-0.315458\pi$
$19$	4.77579			1.09564			0.547820	+	0.836596i	$0.315458\pi$
$20$		0			0
$21$	−2.46778	−	0.953976i	−0.538513	−	0.208175i
$22$		0			0
$23$			6.24724i			1.30264i	0.758803	+	0.651320i	$0.225784\pi$
$23$			6.24724i			1.30264i	−0.758803	+	0.651320i	$0.774216\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	10.0805			1.87189			0.935947	−	0.352141i	$-0.114546\pi$
$29$	10.0805			1.87189			0.935947	+	0.352141i	$0.114546\pi$
$30$		0			0
$31$	10.0201			1.79966			0.899831	−	0.436240i	$-0.143690\pi$
$31$	10.0201			1.79966			0.899831	+	0.436240i	$0.143690\pi$
$32$		0			0
$33$		−	4.70840i		−	0.819628i
$34$		0			0
$35$	0.953976	−	2.46778i	0.161251	−	0.417131i
$36$		0			0
$37$	−4.80045			−0.789189			−0.394595	−	0.918855i	$-0.629115\pi$
$37$	−4.80045			−0.789189			−0.394595	+	0.918855i	$0.629115\pi$
$38$		0			0
$39$			2.93556i			0.470065i
$40$		0			0
$41$		−	9.92804i		−	1.55050i	−0.631655	−	0.775250i	$-0.717624\pi$
$41$		−	9.92804i		−	1.55050i	0.631655	−	0.775250i	$-0.282376\pi$
$42$		0			0
$43$		−	9.55191i		−	1.45665i	−0.685230	−	0.728327i	$-0.740298\pi$
$43$		−	9.55191i		−	1.45665i	0.685230	−	0.728327i	$-0.259702\pi$
$44$		0			0
$45$			1.00000i			0.149071i
$46$		0			0
$47$	7.14490			1.04219			0.521095	−	0.853498i	$-0.325524\pi$
$47$	7.14490			1.04219			0.521095	+	0.853498i	$0.325524\pi$
$48$		0			0
$49$	5.17986	+	4.70840i	0.739980	+	0.672629i
$50$		0			0
$51$			3.31169i			0.463729i
$52$		0			0
$53$	5.21964			0.716972			0.358486	−	0.933535i	$-0.383293\pi$
$53$	5.21964			0.716972			0.358486	+	0.933535i	$0.383293\pi$
$54$		0			0
$55$	4.70840			0.634881
$56$		0			0
$57$	4.77579			0.632568
$58$		0			0
$59$	−4.31496			−0.561760			−0.280880	−	0.959743i	$-0.590626\pi$
$59$	−4.31496			−0.561760			−0.280880	+	0.959743i	$0.590626\pi$
$60$		0			0
$61$			7.53182i			0.964351i	0.876075	+	0.482176i	$0.160153\pi$
$61$			7.53182i			0.964351i	−0.876075	+	0.482176i	$0.839847\pi$
$62$		0			0
$63$	−2.46778	−	0.953976i	−0.310911	−	0.120190i
$64$		0			0
$65$	−2.93556			−0.364111
$66$		0			0
$67$			6.48827i			0.792668i	0.918106	+	0.396334i	$0.129718\pi$
$67$			6.48827i			0.792668i	−0.918106	+	0.396334i	$0.870282\pi$
$68$		0			0
$69$			6.24724i			0.752080i
$70$		0			0
$71$			1.65555i			0.196478i	0.995163	+	0.0982388i	$0.0313209\pi$
$71$			1.65555i			0.196478i	−0.995163	+	0.0982388i	$0.968679\pi$
$72$		0			0
$73$		−	2.06738i		−	0.241969i	−0.992654	−	0.120984i	$-0.961395\pi$
$73$		−	2.06738i		−	0.241969i	0.992654	−	0.120984i	$-0.0386051\pi$
$74$		0			0
$75$	−1.00000			−0.115470
$76$		0			0
$77$	−4.49171	+	11.6193i	−0.511877	+	1.32414i
$78$		0			0
$79$		−	12.8809i		−	1.44922i	−0.689162	−	0.724608i	$-0.742021\pi$
$79$		−	12.8809i		−	1.44922i	0.689162	−	0.724608i	$-0.257979\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	16.0402			1.76064			0.880319	−	0.474382i	$-0.157328\pi$
$83$	16.0402			1.76064			0.880319	+	0.474382i	$0.157328\pi$
$84$		0			0
$85$	−3.31169			−0.359203
$86$		0			0
$87$	10.0805			1.08074
$88$		0			0
$89$			13.0403i			1.38227i	0.722724	+	0.691137i	$0.242890\pi$
$89$			13.0403i			1.38227i	−0.722724	+	0.691137i	$0.757110\pi$
$90$		0			0
$91$	2.80045	−	7.24430i	0.293567	−	0.759409i
$92$		0			0
$93$	10.0201			1.03903
$94$		0			0
$95$			4.77579i			0.489985i
$96$		0			0
$97$			1.74852i			0.177535i	0.996052	+	0.0887677i	$0.0282929\pi$
$97$			1.74852i			0.177535i	−0.996052	+	0.0887677i	$0.971707\pi$
$98$		0			0
$99$		−	4.70840i		−	0.473212i
$100$		0			0
$101$			4.94291i			0.491838i	0.969290	+	0.245919i	$0.0790897\pi$
$101$			4.94291i			0.491838i	−0.969290	+	0.245919i	$0.920910\pi$
$102$		0			0
$103$	2.17674			0.214481			0.107240	−	0.994233i	$-0.465799\pi$
$103$	2.17674			0.214481			0.107240	+	0.994233i	$0.465799\pi$
$104$		0			0
$105$	0.953976	−	2.46778i	0.0930986	−	0.240830i
$106$		0			0
$107$			7.30433i			0.706136i	0.935598	+	0.353068i	$0.114862\pi$
$107$			7.30433i			0.706136i	−0.935598	+	0.353068i	$0.885138\pi$
$108$		0			0
$109$	−17.1525			−1.64291			−0.821455	−	0.570274i	$-0.806837\pi$
$109$	−17.1525			−1.64291			−0.821455	+	0.570274i	$0.806837\pi$
$110$		0			0
$111$	−4.80045			−0.455639
$112$		0			0
$113$	−7.70824			−0.725130			−0.362565	−	0.931958i	$-0.618099\pi$
$113$	−7.70824			−0.725130			−0.362565	+	0.931958i	$0.618099\pi$
$114$		0			0
$115$	−6.24724			−0.582558
$116$		0			0
$117$			2.93556i			0.271392i
$118$		0			0
$119$	3.15927	−	8.17251i	0.289610	−	0.749172i
$120$		0			0
$121$	−11.1691			−1.01537
$122$		0			0
$123$		−	9.92804i		−	0.895181i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			9.58841i			0.850834i	0.904997	+	0.425417i	$0.139873\pi$
$127$			9.58841i			0.850834i	−0.904997	+	0.425417i	$0.860127\pi$
$128$		0			0
$129$		−	9.55191i		−	0.840999i
$130$		0			0
$131$	1.55615			0.135961			0.0679807	−	0.997687i	$-0.478344\pi$
$131$	1.55615			0.135961			0.0679807	+	0.997687i	$0.478344\pi$
$132$		0			0
$133$	−11.7856	−	4.55599i	−1.02194	−	0.395054i
$134$		0			0
$135$			1.00000i			0.0860663i
$136$		0			0
$137$	−2.73103			−0.233328			−0.116664	−	0.993171i	$-0.537220\pi$
$137$	−2.73103			−0.233328			−0.116664	+	0.993171i	$0.537220\pi$
$138$		0			0
$139$	−17.2562			−1.46366			−0.731828	−	0.681490i	$-0.761332\pi$
$139$	−17.2562			−1.46366			−0.731828	+	0.681490i	$0.761332\pi$
$140$		0			0
$141$	7.14490			0.601709
$142$		0			0
$143$	13.8218			1.15584
$144$		0			0
$145$			10.0805i			0.837136i
$146$		0			0
$147$	5.17986	+	4.70840i	0.427228	+	0.388343i
$148$		0			0
$149$	15.9516			1.30680			0.653402	−	0.757011i	$-0.273341\pi$
$149$	15.9516			1.30680			0.653402	+	0.757011i	$0.273341\pi$
$150$		0			0
$151$		−	21.3352i		−	1.73624i	−0.496359	−	0.868118i	$-0.665330\pi$
$151$		−	21.3352i		−	1.73624i	0.496359	−	0.868118i	$-0.334670\pi$
$152$		0			0
$153$			3.31169i			0.267734i
$154$		0			0
$155$			10.0201i			0.804833i
$156$		0			0
$157$			0.128090i			0.0102227i	0.999987	+	0.00511137i	$0.00162701\pi$
$157$			0.128090i			0.0102227i	−0.999987	+	0.00511137i	$0.998373\pi$
$158$		0			0
$159$	5.21964			0.413944
$160$		0			0
$161$	5.95972	−	15.4168i	0.469692	−	1.21501i
$162$		0			0
$163$		−	17.2324i		−	1.34974i	−0.737935	−	0.674872i	$-0.764199\pi$
$163$		−	17.2324i		−	1.34974i	0.737935	−	0.674872i	$-0.235801\pi$
$164$		0			0
$165$	4.70840			0.366549
$166$		0			0
$167$	−6.40667			−0.495763			−0.247882	−	0.968790i	$-0.579734\pi$
$167$	−6.40667			−0.495763			−0.247882	+	0.968790i	$0.579734\pi$
$168$		0			0
$169$	4.38251			0.337116
$170$		0			0
$171$	4.77579			0.365214
$172$		0			0
$173$			1.12700i			0.0856846i	0.999082	+	0.0428423i	$0.0136413\pi$
$173$			1.12700i			0.0856846i	−0.999082	+	0.0428423i	$0.986359\pi$
$174$		0			0
$175$	2.46778	+	0.953976i	0.186546	+	0.0721138i
$176$		0			0
$177$	−4.31496			−0.324333
$178$		0			0
$179$		−	12.3889i		−	0.925987i	−0.886362	−	0.462994i	$-0.846775\pi$
$179$		−	12.3889i		−	0.925987i	0.886362	−	0.462994i	$-0.153225\pi$
$180$		0			0
$181$			12.6291i			0.938713i	0.883009	+	0.469357i	$0.155514\pi$
$181$			12.6291i			0.938713i	−0.883009	+	0.469357i	$0.844486\pi$
$182$		0			0
$183$			7.53182i			0.556768i
$184$		0			0
$185$		−	4.80045i		−	0.352936i
$186$		0			0
$187$	15.5928			1.14025
$188$		0			0
$189$	−2.46778	−	0.953976i	−0.179504	−	0.0693916i
$190$		0			0
$191$		−	3.91073i		−	0.282971i	−0.989940	−	0.141485i	$-0.954812\pi$
$191$		−	3.91073i		−	0.282971i	0.989940	−	0.141485i	$-0.0451878\pi$
$192$		0			0
$193$	−19.6723			−1.41604			−0.708022	−	0.706190i	$-0.750412\pi$
$193$	−19.6723			−1.41604			−0.708022	+	0.706190i	$0.750412\pi$
$194$		0			0
$195$	−2.93556			−0.210220
$196$		0			0
$197$	22.6217			1.61173			0.805866	−	0.592098i	$-0.201700\pi$
$197$	22.6217			1.61173			0.805866	+	0.592098i	$0.201700\pi$
$198$		0			0
$199$	−24.5801			−1.74244			−0.871219	−	0.490895i	$-0.836670\pi$
$199$	−24.5801			−1.74244			−0.871219	+	0.490895i	$0.836670\pi$
$200$		0			0
$201$			6.48827i			0.457647i
$202$		0			0
$203$	−24.8763	−	9.61652i	−1.74598	−	0.674947i
$204$		0			0
$205$	9.92804			0.693405
$206$		0			0
$207$			6.24724i			0.434213i
$208$		0			0
$209$		−	22.4863i		−	1.55541i
$210$		0			0
$211$			27.6397i			1.90280i	0.307962	+	0.951399i	$0.400353\pi$
$211$			27.6397i			1.90280i	−0.307962	+	0.951399i	$0.599647\pi$
$212$		0			0
$213$			1.65555i			0.113436i
$214$		0			0
$215$	9.55191			0.651435
$216$		0			0
$217$	−24.7274	−	9.55893i	−1.67860	−	0.648902i
$218$		0			0
$219$		−	2.06738i		−	0.139701i
$220$		0			0
$221$	−9.72164			−0.653948
$222$		0			0
$223$	−19.2254			−1.28743			−0.643713	−	0.765267i	$-0.722607\pi$
$223$	−19.2254			−1.28743			−0.643713	+	0.765267i	$0.722607\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$	6.60934			0.438677			0.219339	−	0.975649i	$-0.429610\pi$
$227$	6.60934			0.438677			0.219339	+	0.975649i	$0.429610\pi$
$228$		0			0
$229$			14.1898i			0.937689i	0.883281	+	0.468845i	$0.155330\pi$
$229$			14.1898i			0.937689i	−0.883281	+	0.468845i	$0.844670\pi$
$230$		0			0
$231$	−4.49171	+	11.6193i	−0.295533	+	0.764494i
$232$		0			0
$233$	18.5010			1.21204			0.606020	−	0.795449i	$-0.292765\pi$
$233$	18.5010			1.21204			0.606020	+	0.795449i	$0.292765\pi$
$234$		0			0
$235$			7.14490i			0.466082i
$236$		0			0
$237$		−	12.8809i		−	0.836705i
$238$		0			0
$239$		−	0.881706i		−	0.0570328i	−0.999593	−	0.0285164i	$-0.990922\pi$
$239$		−	0.881706i		−	0.0570328i	0.999593	−	0.0285164i	$-0.00907828\pi$
$240$		0			0
$241$		−	8.45431i		−	0.544590i	−0.962214	−	0.272295i	$-0.912217\pi$
$241$		−	8.45431i		−	0.544590i	0.962214	−	0.272295i	$-0.0877826\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	−4.70840	+	5.17986i	−0.300809	+	0.330929i
$246$		0			0
$247$			14.0196i			0.892045i
$248$		0			0
$249$	16.0402			1.01651
$250$		0			0
$251$	−1.55615			−0.0982233			−0.0491116	−	0.998793i	$-0.515639\pi$
$251$	−1.55615			−0.0982233			−0.0491116	+	0.998793i	$0.515639\pi$
$252$		0			0
$253$	29.4145			1.84928
$254$		0			0
$255$	−3.31169			−0.207386
$256$		0			0
$257$		−	20.6548i		−	1.28841i	−0.764852	−	0.644206i	$-0.777188\pi$
$257$		−	20.6548i		−	1.28841i	0.764852	−	0.644206i	$-0.222812\pi$
$258$		0			0
$259$	11.8464	+	4.57952i	0.736103	+	0.284557i
$260$		0			0
$261$	10.0805			0.623965
$262$		0			0
$263$		−	5.44793i		−	0.335934i	−0.985793	−	0.167967i	$-0.946280\pi$
$263$		−	5.44793i		−	0.335934i	0.985793	−	0.167967i	$-0.0537201\pi$
$264$		0			0
$265$			5.21964i			0.320640i
$266$		0			0
$267$			13.0403i			0.798056i
$268$		0			0
$269$			12.1900i			0.743236i	0.928386	+	0.371618i	$0.121197\pi$
$269$			12.1900i			0.743236i	−0.928386	+	0.371618i	$0.878803\pi$
$270$		0			0
$271$	−14.3798			−0.873511			−0.436756	−	0.899580i	$-0.643873\pi$
$271$	−14.3798			−0.873511			−0.436756	+	0.899580i	$0.643873\pi$
$272$		0			0
$273$	2.80045	−	7.24430i	0.169491	−	0.438445i
$274$		0			0
$275$			4.70840i			0.283927i
$276$		0			0
$277$	−22.4728			−1.35026			−0.675129	−	0.737700i	$-0.735912\pi$
$277$	−22.4728			−1.35026			−0.675129	+	0.737700i	$0.735912\pi$
$278$		0			0
$279$	10.0201			0.599887
$280$		0			0
$281$	22.6639			1.35201			0.676007	−	0.736896i	$-0.263709\pi$
$281$	22.6639			1.35201			0.676007	+	0.736896i	$0.263709\pi$
$282$		0			0
$283$	2.63808			0.156817			0.0784087	−	0.996921i	$-0.475016\pi$
$283$	2.63808			0.156817			0.0784087	+	0.996921i	$0.475016\pi$
$284$		0			0
$285$			4.77579i			0.282893i
$286$		0			0
$287$	−9.47112	+	24.5002i	−0.559062	+	1.44620i
$288$		0			0
$289$	6.03274			0.354867
$290$		0			0
$291$			1.74852i			0.102500i
$292$		0			0
$293$		−	2.88770i		−	0.168701i	−0.996436	−	0.0843507i	$-0.973118\pi$
$293$		−	2.88770i		−	0.168701i	0.996436	−	0.0843507i	$-0.0268816\pi$
$294$		0			0
$295$		−	4.31496i		−	0.251227i
$296$		0			0
$297$		−	4.70840i		−	0.273209i
$298$		0			0
$299$	−18.3391			−1.06058
$300$		0			0
$301$	−9.11230	+	23.5720i	−0.525224	+	1.35867i
$302$		0			0
$303$			4.94291i			0.283963i
$304$		0			0
$305$	−7.53182			−0.431271
$306$		0			0
$307$	13.3682			0.762961			0.381480	−	0.924377i	$-0.375414\pi$
$307$	13.3682			0.762961			0.381480	+	0.924377i	$0.375414\pi$
$308$		0			0
$309$	2.17674			0.123831
$310$		0			0
$311$	7.70205			0.436743			0.218372	−	0.975866i	$-0.429926\pi$
$311$	7.70205			0.436743			0.218372	+	0.975866i	$0.429926\pi$
$312$		0			0
$313$		−	20.4676i		−	1.15690i	−0.815719	−	0.578449i	$-0.803658\pi$
$313$		−	20.4676i		−	1.15690i	0.815719	−	0.578449i	$-0.196342\pi$
$314$		0			0
$315$	0.953976	−	2.46778i	0.0537505	−	0.139044i
$316$		0			0
$317$	−24.8411			−1.39522			−0.697608	−	0.716479i	$-0.745752\pi$
$317$	−24.8411			−1.39522			−0.697608	+	0.716479i	$0.745752\pi$
$318$		0			0
$319$		−	47.4629i		−	2.65741i
$320$		0			0
$321$			7.30433i			0.407688i
$322$		0			0
$323$			15.8159i			0.880021i
$324$		0			0
$325$		−	2.93556i		−	0.162835i
$326$		0			0
$327$	−17.1525			−0.948534
$328$		0			0
$329$	−17.6320	−	6.81607i	−0.972085	−	0.375782i
$330$		0			0
$331$			1.40889i			0.0774398i	0.999250	+	0.0387199i	$0.0123280\pi$
$331$			1.40889i			0.0774398i	−0.999250	+	0.0387199i	$0.987672\pi$
$332$		0			0
$333$	−4.80045			−0.263063
$334$		0			0
$335$	−6.48827			−0.354492
$336$		0			0
$337$	−31.5578			−1.71906			−0.859532	−	0.511082i	$-0.829245\pi$
$337$	−31.5578			−1.71906			−0.859532	+	0.511082i	$0.829245\pi$
$338$		0			0
$339$	−7.70824			−0.418654
$340$		0			0
$341$		−	47.1786i		−	2.55487i
$342$		0			0
$343$	−8.29104	−	16.5608i	−0.447674	−	0.894197i
$344$		0			0
$345$	−6.24724			−0.336340
$346$		0			0
$347$			24.0827i			1.29283i	0.762986	+	0.646415i	$0.223732\pi$
$347$			24.0827i			1.29283i	−0.762986	+	0.646415i	$0.776268\pi$
$348$		0			0
$349$		−	22.8927i		−	1.22542i	−0.790308	−	0.612710i	$-0.790079\pi$
$349$		−	22.8927i		−	1.22542i	0.790308	−	0.612710i	$-0.209921\pi$
$350$		0			0
$351$			2.93556i			0.156688i
$352$		0			0
$353$		−	9.68643i		−	0.515557i	−0.966204	−	0.257778i	$-0.917010\pi$
$353$		−	9.68643i		−	0.515557i	0.966204	−	0.257778i	$-0.0829904\pi$
$354$		0			0
$355$	−1.65555			−0.0878675
$356$		0			0
$357$	3.15927	−	8.17251i	0.167206	−	0.432535i
$358$		0			0
$359$			15.4633i			0.816122i	0.912955	+	0.408061i	$0.133795\pi$
$359$			15.4633i			0.816122i	−0.912955	+	0.408061i	$0.866205\pi$
$360$		0			0
$361$	3.80814			0.200429
$362$		0			0
$363$	−11.1691			−0.586224
$364$		0			0
$365$	2.06738			0.108212
$366$		0			0
$367$	−10.1274			−0.528647			−0.264323	−	0.964434i	$-0.585149\pi$
$367$	−10.1274			−0.528647			−0.264323	+	0.964434i	$0.585149\pi$
$368$		0			0
$369$		−	9.92804i		−	0.516833i
$370$		0			0
$371$	−12.8809	−	4.97941i	−0.668743	−	0.258518i
$372$		0			0
$373$	0.760273			0.0393654			0.0196827	−	0.999806i	$-0.493734\pi$
$373$	0.760273			0.0393654			0.0196827	+	0.999806i	$0.493734\pi$
$374$		0			0
$375$		−	1.00000i		−	0.0516398i
$376$		0			0
$377$			29.5918i			1.52405i
$378$		0			0
$379$		−	11.6390i		−	0.597858i	−0.954275	−	0.298929i	$-0.903371\pi$
$379$		−	11.6390i		−	0.597858i	0.954275	−	0.298929i	$-0.0966293\pi$
$380$		0			0
$381$			9.58841i			0.491229i
$382$		0			0
$383$	−14.5676			−0.744369			−0.372185	−	0.928159i	$-0.621391\pi$
$383$	−14.5676			−0.744369			−0.372185	+	0.928159i	$0.621391\pi$
$384$		0			0
$385$	−11.6193	−	4.49171i	−0.592174	−	0.228919i
$386$		0			0
$387$		−	9.55191i		−	0.485551i
$388$		0			0
$389$	12.7679			0.647356			0.323678	−	0.946167i	$-0.395080\pi$
$389$	12.7679			0.647356			0.323678	+	0.946167i	$0.395080\pi$
$390$		0			0
$391$	−20.6889			−1.04628
$392$		0			0
$393$	1.55615			0.0784973
$394$		0			0
$395$	12.8809			0.648109
$396$		0			0
$397$			37.5505i			1.88460i	0.334765	+	0.942302i	$0.391343\pi$
$397$			37.5505i			1.88460i	−0.334765	+	0.942302i	$0.608657\pi$
$398$		0			0
$399$	−11.7856	−	4.55599i	−0.590017	−	0.228085i
$400$		0			0
$401$	−3.62593			−0.181070			−0.0905352	−	0.995893i	$-0.528858\pi$
$401$	−3.62593			−0.181070			−0.0905352	+	0.995893i	$0.528858\pi$
$402$		0			0
$403$			29.4145i			1.46524i
$404$		0			0
$405$			1.00000i			0.0496904i
$406$		0			0
$407$			22.6025i			1.12036i
$408$		0			0
$409$		−	26.9831i		−	1.33423i	−0.744956	−	0.667114i	$-0.767529\pi$
$409$		−	26.9831i		−	1.33423i	0.744956	−	0.667114i	$-0.232471\pi$
$410$		0			0
$411$	−2.73103			−0.134712
$412$		0			0
$413$	10.6484	+	4.11637i	0.523972	+	0.202553i
$414$		0			0
$415$			16.0402i			0.787381i
$416$		0			0
$417$	−17.2562			−0.845042
$418$		0			0
$419$	−10.9027			−0.532633			−0.266316	−	0.963886i	$-0.585807\pi$
$419$	−10.9027			−0.532633			−0.266316	+	0.963886i	$0.585807\pi$
$420$		0			0
$421$	5.23304			0.255043			0.127521	−	0.991836i	$-0.459298\pi$
$421$	5.23304			0.255043			0.127521	+	0.991836i	$0.459298\pi$
$422$		0			0
$423$	7.14490			0.347397
$424$		0			0
$425$		−	3.31169i		−	0.160640i
$426$		0			0
$427$	7.18518	−	18.5869i	0.347715	−	0.899482i
$428$		0			0
$429$	13.8218			0.667322
$430$		0			0
$431$		−	1.98473i		−	0.0956012i	−0.998857	−	0.0478006i	$-0.984779\pi$
$431$		−	1.98473i		−	0.0956012i	0.998857	−	0.0478006i	$-0.0152212\pi$
$432$		0			0
$433$		−	6.57657i		−	0.316050i	−0.987435	−	0.158025i	$-0.949487\pi$
$433$		−	6.57657i		−	0.316050i	0.987435	−	0.158025i	$-0.0505127\pi$
$434$		0			0
$435$			10.0805i			0.483321i
$436$		0			0
$437$			29.8355i			1.42723i
$438$		0			0
$439$	14.2653			0.680844			0.340422	−	0.940273i	$-0.389430\pi$
$439$	14.2653			0.680844			0.340422	+	0.940273i	$0.389430\pi$
$440$		0			0
$441$	5.17986	+	4.70840i	0.246660	+	0.224210i
$442$		0			0
$443$		−	10.1118i		−	0.480426i	−0.970720	−	0.240213i	$-0.922783\pi$
$443$		−	10.1118i		−	0.480426i	0.970720	−	0.240213i	$-0.0772173\pi$
$444$		0			0
$445$	−13.0403			−0.618171
$446$		0			0
$447$	15.9516			0.754483
$448$		0			0
$449$	32.3855			1.52837			0.764183	−	0.644999i	$-0.223142\pi$
$449$	32.3855			1.52837			0.764183	+	0.644999i	$0.223142\pi$
$450$		0			0
$451$	−46.7452			−2.20115
$452$		0			0
$453$		−	21.3352i		−	1.00242i
$454$		0			0
$455$	7.24430	+	2.80045i	0.339618	+	0.131287i
$456$		0			0
$457$	17.5722			0.821991			0.410995	−	0.911637i	$-0.365181\pi$
$457$	17.5722			0.821991			0.410995	+	0.911637i	$0.365181\pi$
$458$		0			0
$459$			3.31169i			0.154576i
$460$		0			0
$461$			20.8059i			0.969026i	0.874784	+	0.484513i	$0.161003\pi$
$461$			20.8059i			0.969026i	−0.874784	+	0.484513i	$0.838997\pi$
$462$		0			0
$463$			0.885484i			0.0411519i	0.999788	+	0.0205760i	$0.00654999\pi$
$463$			0.885484i			0.0411519i	−0.999788	+	0.0205760i	$0.993450\pi$
$464$		0			0
$465$			10.0201i			0.464671i
$466$		0			0
$467$	−22.5350			−1.04279			−0.521397	−	0.853314i	$-0.674589\pi$
$467$	−22.5350			−1.04279			−0.521397	+	0.853314i	$0.674589\pi$
$468$		0			0
$469$	6.18965	−	16.0116i	0.285812	−	0.739347i
$470$		0			0
$471$			0.128090i			0.00590210i
$472$		0			0
$473$	−44.9743			−2.06792
$474$		0			0
$475$	−4.77579			−0.219128
$476$		0			0
$477$	5.21964			0.238991
$478$		0			0
$479$	−30.3516			−1.38680			−0.693399	−	0.720554i	$-0.743888\pi$
$479$	−30.3516			−1.38680			−0.693399	+	0.720554i	$0.743888\pi$
$480$		0			0
$481$		−	14.0920i		−	0.642540i
$482$		0			0
$483$	5.95972	−	15.4168i	0.271177	−	0.701489i
$484$		0			0
$485$	−1.74852			−0.0793963
$486$		0			0
$487$		−	18.5464i		−	0.840415i	−0.907428	−	0.420208i	$-0.861957\pi$
$487$		−	18.5464i		−	0.840415i	0.907428	−	0.420208i	$-0.138043\pi$
$488$		0			0
$489$		−	17.2324i		−	0.779275i
$490$		0			0
$491$		−	26.6197i		−	1.20133i	−0.799501	−	0.600665i	$-0.794903\pi$
$491$		−	26.6197i		−	1.20133i	0.799501	−	0.600665i	$-0.205097\pi$
$492$		0			0
$493$			33.3833i			1.50351i
$494$		0			0
$495$	4.70840			0.211627
$496$		0			0
$497$	1.57936	−	4.08553i	0.0708438	−	0.183261i
$498$		0			0
$499$		−	1.13868i		−	0.0509745i	−0.999675	−	0.0254872i	$-0.991886\pi$
$499$		−	1.13868i		−	0.0509745i	0.999675	−	0.0254872i	$-0.00811372\pi$
$500$		0			0
$501$	−6.40667			−0.286229
$502$		0			0
$503$	−22.9365			−1.02269			−0.511343	−	0.859377i	$-0.670852\pi$
$503$	−22.9365			−1.02269			−0.511343	+	0.859377i	$0.670852\pi$
$504$		0			0
$505$	−4.94291			−0.219957
$506$		0			0
$507$	4.38251			0.194634
$508$		0			0
$509$			24.2014i			1.07271i	0.843993	+	0.536354i	$0.180199\pi$
$509$			24.2014i			1.07271i	−0.843993	+	0.536354i	$0.819801\pi$
$510$		0			0
$511$	−1.97223	+	5.10184i	−0.0872465	+	0.225692i
$512$		0			0
$513$	4.77579			0.210856
$514$		0			0
$515$			2.17674i			0.0959187i
$516$		0			0
$517$		−	33.6411i		−	1.47953i
$518$		0			0
$519$			1.12700i			0.0494700i
$520$		0			0
$521$		−	26.0396i		−	1.14082i	−0.821361	−	0.570408i	$-0.806785\pi$
$521$		−	26.0396i		−	1.14082i	0.821361	−	0.570408i	$-0.193215\pi$
$522$		0			0
$523$	11.3825			0.497722			0.248861	−	0.968539i	$-0.419944\pi$
$523$	11.3825			0.497722			0.248861	+	0.968539i	$0.419944\pi$
$524$		0			0
$525$	2.46778	+	0.953976i	0.107703	+	0.0416349i
$526$		0			0
$527$			33.1834i			1.44549i
$528$		0			0
$529$	−16.0280			−0.696871
$530$		0			0
$531$	−4.31496			−0.187253
$532$		0			0
$533$	29.1443			1.26238
$534$		0			0
$535$	−7.30433			−0.315794
$536$		0			0
$537$		−	12.3889i		−	0.534619i
$538$		0			0
$539$	22.1691	−	24.3889i	0.954889	−	1.05050i
$540$		0			0
$541$	−7.13288			−0.306667			−0.153333	−	0.988175i	$-0.549001\pi$
$541$	−7.13288			−0.306667			−0.153333	+	0.988175i	$0.549001\pi$
$542$		0			0
$543$			12.6291i			0.541966i
$544$		0			0
$545$		−	17.1525i		−	0.734731i
$546$		0			0
$547$			32.0960i			1.37233i	0.727447	+	0.686164i	$0.240707\pi$
$547$			32.0960i			1.37233i	−0.727447	+	0.686164i	$0.759293\pi$
$548$		0			0
$549$			7.53182i			0.321450i
$550$		0			0
$551$	48.1421			2.05092
$552$		0			0
$553$	−12.2881	+	31.7872i	−0.522542	+	1.35173i
$554$		0			0
$555$		−	4.80045i		−	0.203768i
$556$		0			0
$557$	−23.3853			−0.990865			−0.495433	−	0.868646i	$-0.664990\pi$
$557$	−23.3853			−0.990865			−0.495433	+	0.868646i	$0.664990\pi$
$558$		0			0
$559$	28.0402			1.18597
$560$		0			0
$561$	15.5928			0.658327
$562$		0			0
$563$	−5.77540			−0.243404			−0.121702	−	0.992567i	$-0.538835\pi$
$563$	−5.77540			−0.243404			−0.121702	+	0.992567i	$0.538835\pi$
$564$		0			0
$565$		−	7.70824i		−	0.324288i
$566$		0			0
$567$	−2.46778	−	0.953976i	−0.103637	−	0.0400632i
$568$		0			0
$569$	−17.2018			−0.721137			−0.360569	−	0.932733i	$-0.617417\pi$
$569$	−17.2018			−0.721137			−0.360569	+	0.932733i	$0.617417\pi$
$570$		0			0
$571$		−	2.99509i		−	0.125341i	−0.998034	−	0.0626703i	$-0.980038\pi$
$571$		−	2.99509i		−	0.125341i	0.998034	−	0.0626703i	$-0.0199617\pi$
$572$		0			0
$573$		−	3.91073i		−	0.163373i
$574$		0			0
$575$		−	6.24724i		−	0.260528i
$576$		0			0
$577$		−	11.4933i		−	0.478474i	−0.970961	−	0.239237i	$-0.923103\pi$
$577$		−	11.4933i		−	0.478474i	0.970961	−	0.239237i	$-0.0768973\pi$
$578$		0			0
$579$	−19.6723			−0.817553
$580$		0			0
$581$	−39.5836	−	15.3019i	−1.64220	−	0.634832i
$582$		0			0
$583$		−	24.5762i		−	1.01784i
$584$		0			0
$585$	−2.93556			−0.121370
$586$		0			0
$587$	5.40210			0.222969			0.111484	−	0.993766i	$-0.464440\pi$
$587$	5.40210			0.222969			0.111484	+	0.993766i	$0.464440\pi$
$588$		0			0
$589$	47.8538			1.97178
$590$		0			0
$591$	22.6217			0.930534
$592$		0			0
$593$		−	1.43275i		−	0.0588358i	−0.999567	−	0.0294179i	$-0.990635\pi$
$593$		−	1.43275i		−	0.0588358i	0.999567	−	0.0294179i	$-0.00936536\pi$
$594$		0			0
$595$	8.17251	+	3.15927i	0.335040	+	0.129517i
$596$		0			0
$597$	−24.5801			−1.00600
$598$		0			0
$599$		−	20.8886i		−	0.853484i	−0.904373	−	0.426742i	$-0.859661\pi$
$599$		−	20.8886i		−	0.853484i	0.904373	−	0.426742i	$-0.140339\pi$
$600$		0			0
$601$			13.3760i			0.545616i	0.962068	+	0.272808i	$0.0879525\pi$
$601$			13.3760i			0.545616i	−0.962068	+	0.272808i	$0.912048\pi$
$602$		0			0
$603$			6.48827i			0.264223i
$604$		0			0
$605$		−	11.1691i		−	0.454087i
$606$		0			0
$607$	28.2581			1.14696			0.573480	−	0.819219i	$-0.305593\pi$
$607$	28.2581			1.14696			0.573480	+	0.819219i	$0.305593\pi$
$608$		0			0
$609$	−24.8763	−	9.61652i	−1.00804	−	0.389681i
$610$		0			0
$611$			20.9743i			0.848528i
$612$		0			0
$613$	20.5829			0.831334			0.415667	−	0.909517i	$-0.363548\pi$
$613$	20.5829			0.831334			0.415667	+	0.909517i	$0.363548\pi$
$614$		0			0
$615$	9.92804			0.400337
$616$		0			0
$617$	39.6919			1.59794			0.798969	−	0.601373i	$-0.205379\pi$
$617$	39.6919			1.59794			0.798969	+	0.601373i	$0.205379\pi$
$618$		0			0
$619$	−11.7386			−0.471815			−0.235907	−	0.971776i	$-0.575806\pi$
$619$	−11.7386			−0.471815			−0.235907	+	0.971776i	$0.575806\pi$
$620$		0			0
$621$			6.24724i			0.250693i
$622$		0			0
$623$	12.4402	−	32.1807i	0.498405	−	1.28929i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	22.4863i		−	0.898018i
$628$		0			0
$629$		−	15.8976i		−	0.633878i
$630$		0			0
$631$		−	23.2958i		−	0.927393i	−0.885994	−	0.463696i	$-0.846523\pi$
$631$		−	23.2958i		−	0.927393i	0.885994	−	0.463696i	$-0.153477\pi$
$632$		0			0
$633$			27.6397i			1.09858i
$634$		0			0
$635$	−9.58841			−0.380505
$636$		0			0
$637$	−13.8218	+	15.2058i	−0.547639	+	0.602474i
$638$		0			0
$639$			1.65555i			0.0654925i
$640$		0			0
$641$	−16.4507			−0.649764			−0.324882	−	0.945755i	$-0.605325\pi$
$641$	−16.4507			−0.649764			−0.324882	+	0.945755i	$0.605325\pi$
$642$		0			0
$643$	−9.48017			−0.373861			−0.186931	−	0.982373i	$-0.559854\pi$
$643$	−9.48017			−0.373861			−0.186931	+	0.982373i	$0.559854\pi$
$644$		0			0
$645$	9.55191			0.376106
$646$		0			0
$647$	−14.9652			−0.588342			−0.294171	−	0.955753i	$-0.595044\pi$
$647$	−14.9652			−0.588342			−0.294171	+	0.955753i	$0.595044\pi$
$648$		0			0
$649$			20.3166i			0.797496i
$650$		0			0
$651$	−24.7274	−	9.55893i	−0.969142	−	0.374644i
$652$		0			0
$653$	28.1610			1.10202			0.551012	−	0.834498i	$-0.314242\pi$
$653$	28.1610			1.10202			0.551012	+	0.834498i	$0.314242\pi$
$654$		0			0
$655$			1.55615i			0.0608038i
$656$		0			0
$657$		−	2.06738i		−	0.0806563i
$658$		0			0
$659$			1.46913i			0.0572294i	0.999591	+	0.0286147i	$0.00910958\pi$
$659$			1.46913i			0.0572294i	−0.999591	+	0.0286147i	$0.990890\pi$
$660$		0			0
$661$		−	9.01199i		−	0.350526i	−0.984522	−	0.175263i	$-0.943922\pi$
$661$		−	9.01199i		−	0.350526i	0.984522	−	0.175263i	$-0.0560775\pi$
$662$		0			0
$663$	−9.72164			−0.377557
$664$		0			0
$665$	4.55599	−	11.7856i	0.176674	−	0.457025i
$666$		0			0
$667$			62.9751i			2.43840i
$668$		0			0
$669$	−19.2254			−0.743295
$670$		0			0
$671$	35.4629			1.36903
$672$		0			0
$673$	9.53099			0.367393			0.183696	−	0.982983i	$-0.441194\pi$
$673$	9.53099			0.367393			0.183696	+	0.982983i	$0.441194\pi$
$674$		0			0
$675$	−1.00000			−0.0384900
$676$		0			0
$677$			43.6581i			1.67792i	0.544196	+	0.838958i	$0.316835\pi$
$677$			43.6581i			1.67792i	−0.544196	+	0.838958i	$0.683165\pi$
$678$		0			0
$679$	1.66805	−	4.31496i	0.0640138	−	0.165593i
$680$		0			0
$681$	6.60934			0.253271
$682$		0			0
$683$			3.26415i			0.124899i	0.998048	+	0.0624497i	$0.0198913\pi$
$683$			3.26415i			0.124899i	−0.998048	+	0.0624497i	$0.980109\pi$
$684$		0			0
$685$		−	2.73103i		−	0.104347i
$686$		0			0
$687$			14.1898i			0.541375i
$688$		0			0
$689$			15.3225i			0.583742i
$690$		0			0
$691$	5.72246			0.217693			0.108846	−	0.994059i	$-0.465284\pi$
$691$	5.72246			0.217693			0.108846	+	0.994059i	$0.465284\pi$
$692$		0			0
$693$	−4.49171	+	11.6193i	−0.170626	+	0.441381i
$694$		0			0
$695$		−	17.2562i		−	0.654567i
$696$		0			0
$697$	32.8786			1.24536
$698$		0			0
$699$	18.5010			0.699772
$700$		0			0
$701$	−11.6967			−0.441780			−0.220890	−	0.975299i	$-0.570896\pi$
$701$	−11.6967			−0.441780			−0.220890	+	0.975299i	$0.570896\pi$
$702$		0			0
$703$	−22.9259			−0.864668
$704$		0			0
$705$			7.14490i			0.269093i
$706$		0			0
$707$	4.71542	−	12.1980i	0.177342	−	0.458753i
$708$		0			0
$709$	−41.0217			−1.54060			−0.770301	−	0.637681i	$-0.779894\pi$
$709$	−41.0217			−1.54060			−0.770301	+	0.637681i	$0.779894\pi$
$710$		0			0
$711$		−	12.8809i		−	0.483072i
$712$		0			0
$713$			62.5979i			2.34431i
$714$		0			0
$715$			13.8218i			0.516905i
$716$		0			0
$717$		−	0.881706i		−	0.0329279i
$718$		0			0
$719$	−46.2772			−1.72585			−0.862924	−	0.505333i	$-0.831370\pi$
$719$	−46.2772			−1.72585			−0.862924	+	0.505333i	$0.831370\pi$
$720$		0			0
$721$	−5.37172	−	2.07656i	−0.200053	−	0.0773351i
$722$		0			0
$723$		−	8.45431i		−	0.314419i
$724$		0			0
$725$	−10.0805			−0.374379
$726$		0			0
$727$	−9.56548			−0.354764			−0.177382	−	0.984142i	$-0.556763\pi$
$727$	−9.56548			−0.354764			−0.177382	+	0.984142i	$0.556763\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	31.6329			1.16999
$732$		0			0
$733$		−	26.7136i		−	0.986688i	−0.869834	−	0.493344i	$-0.835774\pi$
$733$		−	26.7136i		−	0.986688i	0.869834	−	0.493344i	$-0.164226\pi$
$734$		0			0
$735$	−4.70840	+	5.17986i	−0.173672	+	0.191062i
$736$		0			0
$737$	30.5494			1.12530
$738$		0			0
$739$			10.4279i			0.383597i	0.981434	+	0.191799i	$0.0614320\pi$
$739$			10.4279i			0.383597i	−0.981434	+	0.191799i	$0.938568\pi$
$740$		0			0
$741$			14.0196i			0.515023i
$742$		0			0
$743$		−	4.84970i		−	0.177918i	−0.996035	−	0.0889592i	$-0.971646\pi$
$743$		−	4.84970i		−	0.177918i	0.996035	−	0.0889592i	$-0.0283541\pi$
$744$		0			0
$745$			15.9516i			0.584420i
$746$		0			0
$747$	16.0402			0.586879
$748$		0			0
$749$	6.96816	−	18.0255i	0.254611	−	0.658636i
$750$		0			0
$751$			18.8466i			0.687723i	0.939020	+	0.343861i	$0.111735\pi$
$751$			18.8466i			0.687723i	−0.939020	+	0.343861i	$0.888265\pi$
$752$		0			0
$753$	−1.55615			−0.0567092
$754$		0			0
$755$	21.3352			0.776468
$756$		0			0
$757$	−6.31808			−0.229635			−0.114817	−	0.993387i	$-0.536628\pi$
$757$	−6.31808			−0.229635			−0.114817	+	0.993387i	$0.536628\pi$
$758$		0			0
$759$	29.4145			1.06768
$760$		0			0
$761$			8.79795i			0.318925i	0.987204	+	0.159463i	$0.0509762\pi$
$761$			8.79795i			0.318925i	−0.987204	+	0.159463i	$0.949024\pi$
$762$		0			0
$763$	42.3285	+	16.3631i	1.53239	+	0.592382i
$764$		0			0
$765$	−3.31169			−0.119734
$766$		0			0
$767$		−	12.6668i		−	0.457372i
$768$		0			0
$769$			6.51734i			0.235021i	0.993072	+	0.117511i	$0.0374914\pi$
$769$			6.51734i			0.235021i	−0.993072	+	0.117511i	$0.962509\pi$
$770$		0			0
$771$		−	20.6548i		−	0.743865i
$772$		0			0
$773$		−	32.2077i		−	1.15843i	−0.815175	−	0.579214i	$-0.803359\pi$
$773$		−	32.2077i		−	1.15843i	0.815175	−	0.579214i	$-0.196641\pi$
$774$		0			0
$775$	−10.0201			−0.359932
$776$		0			0
$777$	11.8464	+	4.57952i	0.424989	+	0.164289i
$778$		0			0
$779$		−	47.4142i		−	1.69879i
$780$		0			0
$781$	7.79500			0.278927
$782$		0			0
$783$	10.0805			0.360246
$784$		0			0
$785$	−0.128090			−0.00457175
$786$		0			0
$787$	10.1351			0.361278			0.180639	−	0.983549i	$-0.442183\pi$
$787$	10.1351			0.361278			0.180639	+	0.983549i	$0.442183\pi$
$788$		0			0
$789$		−	5.44793i		−	0.193951i
$790$		0			0
$791$	19.0222	+	7.35348i	0.676353	+	0.261460i
$792$		0			0
$793$	−22.1101			−0.785152
$794$		0			0
$795$			5.21964i			0.185121i
$796$		0			0
$797$			43.9618i			1.55721i	0.627516	+	0.778603i	$0.284072\pi$
$797$			43.9618i			1.55721i	−0.627516	+	0.778603i	$0.715928\pi$
$798$		0			0
$799$			23.6617i			0.837090i
$800$		0			0
$801$			13.0403i			0.460758i
$802$		0			0
$803$	−9.73408			−0.343508
$804$		0			0
$805$	15.4168	+	5.95972i	0.543371	+	0.210053i
$806$		0			0
$807$			12.1900i			0.429107i
$808$		0			0
$809$	24.6002			0.864897			0.432448	−	0.901659i	$-0.357650\pi$
$809$	24.6002			0.864897			0.432448	+	0.901659i	$0.357650\pi$
$810$		0			0
$811$	25.5542			0.897329			0.448665	−	0.893700i	$-0.351900\pi$
$811$	25.5542			0.897329			0.448665	+	0.893700i	$0.351900\pi$
$812$		0			0
$813$	−14.3798			−0.504322
$814$		0			0
$815$	17.2324			0.603624
$816$		0			0
$817$		−	45.6179i		−	1.59597i
$818$		0			0
$819$	2.80045	−	7.24430i	0.0978557	−	0.253136i
$820$		0			0
$821$	−10.5701			−0.368898			−0.184449	−	0.982842i	$-0.559050\pi$
$821$	−10.5701			−0.368898			−0.184449	+	0.982842i	$0.559050\pi$
$822$		0			0
$823$		−	30.1329i		−	1.05037i	−0.850989	−	0.525183i	$-0.823997\pi$
$823$		−	30.1329i		−	1.05037i	0.850989	−	0.525183i	$-0.176003\pi$
$824$		0			0
$825$			4.70840i			0.163926i
$826$		0			0
$827$			29.0609i			1.01055i	0.862959	+	0.505274i	$0.168609\pi$
$827$			29.0609i			1.01055i	−0.862959	+	0.505274i	$0.831391\pi$
$828$		0			0
$829$			7.01787i			0.243741i	0.992546	+	0.121870i	$0.0388892\pi$
$829$			7.01787i			0.243741i	−0.992546	+	0.121870i	$0.961111\pi$
$830$		0			0
$831$	−22.4728			−0.779572
$832$		0			0
$833$	−15.5928	+	17.1541i	−0.540257	+	0.594353i
$834$		0			0
$835$		−	6.40667i		−	0.221712i
$836$		0			0
$837$	10.0201			0.346345
$838$		0			0
$839$	−13.3369			−0.460442			−0.230221	−	0.973138i	$-0.573945\pi$
$839$	−13.3369			−0.460442			−0.230221	+	0.973138i	$0.573945\pi$
$840$		0			0
$841$	72.6156			2.50399
$842$		0			0
$843$	22.6639			0.780585
$844$		0			0
$845$			4.38251i			0.150763i
$846$		0			0
$847$	27.5628	+	10.6550i	0.947068	+	0.366111i
$848$		0			0
$849$	2.63808			0.0905386
$850$		0			0
$851$		−	29.9896i		−	1.02803i
$852$		0			0
$853$			27.5042i			0.941725i	0.882207	+	0.470862i	$0.156057\pi$
$853$			27.5042i			0.941725i	−0.882207	+	0.470862i	$0.843943\pi$
$854$		0			0
$855$			4.77579i			0.163328i
$856$		0			0
$857$			30.1662i			1.03046i	0.857053	+	0.515229i	$0.172293\pi$
$857$			30.1662i			1.03046i	−0.857053	+	0.515229i	$0.827707\pi$
$858$		0			0
$859$	−14.6355			−0.499355			−0.249678	−	0.968329i	$-0.580325\pi$
$859$	−14.6355			−0.499355			−0.249678	+	0.968329i	$0.580325\pi$
$860$		0			0
$861$	−9.47112	+	24.5002i	−0.322775	+	0.834965i
$862$		0			0
$863$			8.75903i			0.298161i	0.988825	+	0.149080i	$0.0476313\pi$
$863$			8.75903i			0.298161i	−0.988825	+	0.149080i	$0.952369\pi$
$864$		0			0
$865$	−1.12700			−0.0383193
$866$		0			0
$867$	6.03274			0.204883
$868$		0			0
$869$	−60.6485			−2.05736
$870$		0			0
$871$	−19.0467			−0.645372
$872$		0			0
$873$			1.74852i			0.0591785i
$874$		0			0
$875$	−0.953976	+	2.46778i	−0.0322503	+	0.0834261i
$876$		0			0
$877$	51.4317			1.73672			0.868362	−	0.495931i	$-0.165173\pi$
$877$	51.4317			1.73672			0.868362	+	0.495931i	$0.165173\pi$
$878$		0			0
$879$		−	2.88770i		−	0.0973998i
$880$		0			0
$881$		−	27.7635i		−	0.935377i	−0.883893	−	0.467689i	$-0.845087\pi$
$881$		−	27.7635i		−	0.935377i	0.883893	−	0.467689i	$-0.154913\pi$
$882$		0			0
$883$			39.7073i			1.33626i	0.744046	+	0.668128i	$0.232904\pi$
$883$			39.7073i			1.33626i	−0.744046	+	0.668128i	$0.767096\pi$
$884$		0			0
$885$		−	4.31496i		−	0.145046i
$886$		0			0
$887$	−4.40767			−0.147995			−0.0739976	−	0.997258i	$-0.523576\pi$
$887$	−4.40767			−0.147995			−0.0739976	+	0.997258i	$0.523576\pi$
$888$		0			0
$889$	9.14712	−	23.6621i	0.306785	−	0.793601i
$890$		0			0
$891$		−	4.70840i		−	0.157737i
$892$		0			0
$893$	34.1225			1.14187
$894$		0			0
$895$	12.3889			0.414114
$896$		0			0
$897$	−18.3391			−0.612326
$898$		0			0
$899$	101.007			3.36877
$900$		0			0
$901$			17.2858i			0.575873i
$902$		0			0
$903$	−9.11230	+	23.5720i	−0.303238	+	0.784427i
$904$		0			0
$905$	−12.6291			−0.419805
$906$		0			0
$907$		−	20.0167i		−	0.664642i	−0.943166	−	0.332321i	$-0.892168\pi$
$907$		−	20.0167i		−	0.664642i	0.943166	−	0.332321i	$-0.107832\pi$
$908$		0			0
$909$			4.94291i			0.163946i
$910$		0			0
$911$			35.2875i			1.16913i	0.811348	+	0.584563i	$0.198734\pi$
$911$			35.2875i			1.16913i	−0.811348	+	0.584563i	$0.801266\pi$
$912$		0			0
$913$		−	75.5236i		−	2.49947i
$914$		0			0
$915$	−7.53182			−0.248994
$916$		0			0
$917$	−3.84023	−	1.48453i	−0.126816	−	0.0490235i
$918$		0			0
$919$		−	27.6306i		−	0.911449i	−0.890121	−	0.455724i	$-0.849380\pi$
$919$		−	27.6306i		−	0.911449i	0.890121	−	0.455724i	$-0.150620\pi$
$920$		0			0
$921$	13.3682			0.440496
$922$		0			0
$923$	−4.85996			−0.159968
$924$		0			0
$925$	4.80045			0.157838
$926$		0			0
$927$	2.17674			0.0714936
$928$		0			0
$929$			3.55394i			0.116601i	0.998299	+	0.0583005i	$0.0185682\pi$
$929$			3.55394i			0.116601i	−0.998299	+	0.0583005i	$0.981432\pi$
$930$		0			0
$931$	24.7379	+	22.4863i	0.810752	+	0.736960i
$932$		0			0
$933$	7.70205			0.252154
$934$		0			0
$935$			15.5928i			0.509938i
$936$		0			0
$937$			5.91674i			0.193291i	0.995319	+	0.0966457i	$0.0308114\pi$
$937$			5.91674i			0.193291i	−0.995319	+	0.0966457i	$0.969189\pi$
$938$		0			0
$939$		−	20.4676i		−	0.667935i
$940$		0			0
$941$		−	5.27481i		−	0.171954i	−0.996297	−	0.0859769i	$-0.972599\pi$
$941$		−	5.27481i		−	0.171954i	0.996297	−	0.0859769i	$-0.0274011\pi$
$942$		0			0
$943$	62.0229			2.01974
$944$		0			0
$945$	0.953976	−	2.46778i	0.0310329	−	0.0802768i
$946$		0			0
$947$		−	52.4634i		−	1.70483i	−0.522866	−	0.852415i	$-0.675137\pi$
$947$		−	52.4634i		−	1.70483i	0.522866	−	0.852415i	$-0.324863\pi$
$948$		0			0
$949$	6.06892			0.197005
$950$		0			0
$951$	−24.8411			−0.805529
$952$		0			0
$953$	18.4167			0.596577			0.298288	−	0.954476i	$-0.403584\pi$
$953$	18.4167			0.596577			0.298288	+	0.954476i	$0.403584\pi$
$954$		0			0
$955$	3.91073			0.126548
$956$		0			0
$957$		−	47.4629i		−	1.53426i
$958$		0			0
$959$	6.73959	+	2.60534i	0.217633	+	0.0841309i
$960$		0			0
$961$	69.4022			2.23878
$962$		0			0
$963$			7.30433i			0.235379i
$964$		0			0
$965$		−	19.6723i		−	0.633274i
$966$		0			0
$967$		−	39.0931i		−	1.25715i	−0.777749	−	0.628575i	$-0.783638\pi$
$967$		−	39.0931i		−	1.25715i	0.777749	−	0.628575i	$-0.216362\pi$
$968$		0			0
$969$			15.8159i			0.508080i
$970$		0			0
$971$	−13.5676			−0.435405			−0.217702	−	0.976015i	$-0.569856\pi$
$971$	−13.5676			−0.435405			−0.217702	+	0.976015i	$0.569856\pi$
$972$		0			0
$973$	42.5846	+	16.4620i	1.36520	+	0.527749i
$974$		0			0
$975$		−	2.93556i		−	0.0940130i
$976$		0			0
$977$	−35.1649			−1.12502			−0.562512	−	0.826789i	$-0.690165\pi$
$977$	−35.1649			−1.12502			−0.562512	+	0.826789i	$0.690165\pi$
$978$		0			0
$979$	61.3992			1.96233
$980$		0			0
$981$	−17.1525			−0.547636
$982$		0			0
$983$	10.2663			0.327446			0.163723	−	0.986506i	$-0.447650\pi$
$983$	10.2663			0.327446			0.163723	+	0.986506i	$0.447650\pi$
$984$		0			0
$985$			22.6217i			0.720789i
$986$		0			0
$987$	−17.6320	−	6.81607i	−0.561234	−	0.216958i
$988$		0			0
$989$	59.6731			1.89749
$990$		0			0
$991$		−	3.28848i		−	0.104462i	−0.998635	−	0.0522310i	$-0.983367\pi$
$991$		−	3.28848i		−	0.104462i	0.998635	−	0.0522310i	$-0.0166332\pi$
$992$		0			0
$993$			1.40889i			0.0447099i
$994$		0			0
$995$		−	24.5801i		−	0.779242i
$996$		0			0
$997$		−	55.2574i		−	1.75002i	−0.484106	−	0.875010i	$-0.660855\pi$
$997$		−	55.2574i		−	1.75002i	0.484106	−	0.875010i	$-0.339145\pi$
$998$		0			0
$999$	−4.80045			−0.151880

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.d.d.1231.8	yes	12
3.2	odd	2		5040.2.d.f.4591.2		12
4.3	odd	2		1680.2.d.c.1231.11	yes	12
7.6	odd	2		1680.2.d.c.1231.5	✓	12
12.11	even	2		5040.2.d.g.4591.5		12
21.20	even	2		5040.2.d.g.4591.11		12
28.27	even	2	inner	1680.2.d.d.1231.2	yes	12
84.83	odd	2		5040.2.d.f.4591.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1680.2.d.c.1231.5	✓	12	7.6	odd	2
1680.2.d.c.1231.11	yes	12	4.3	odd	2
1680.2.d.d.1231.2	yes	12	28.27	even	2	inner
1680.2.d.d.1231.8	yes	12	1.1	even	1	trivial
5040.2.d.f.4591.2		12	3.2	odd	2
5040.2.d.f.4591.8		12	84.83	odd	2
5040.2.d.g.4591.5		12	12.11	even	2
5040.2.d.g.4591.11		12	21.20	even	2

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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000i q^{5} +(-2.46778 - 0.953976i) q^{7} +1.00000 q^{9} -4.70840i q^{11} +2.93556i q^{13} +1.00000i q^{15} +3.31169i q^{17} +4.77579 q^{19} +(-2.46778 - 0.953976i) q^{21} +6.24724i q^{23} -1.00000 q^{25} +1.00000 q^{27} +10.0805 q^{29} +10.0201 q^{31} -4.70840i q^{33} +(0.953976 - 2.46778i) q^{35} -4.80045 q^{37} +2.93556i q^{39} -9.92804i q^{41} -9.55191i q^{43} +1.00000i q^{45} +7.14490 q^{47} +(5.17986 + 4.70840i) q^{49} +3.31169i q^{51} +5.21964 q^{53} +4.70840 q^{55} +4.77579 q^{57} -4.31496 q^{59} +7.53182i q^{61} +(-2.46778 - 0.953976i) q^{63} -2.93556 q^{65} +6.48827i q^{67} +6.24724i q^{69} +1.65555i q^{71} -2.06738i q^{73} -1.00000 q^{75} +(-4.49171 + 11.6193i) q^{77} -12.8809i q^{79} +1.00000 q^{81} +16.0402 q^{83} -3.31169 q^{85} +10.0805 q^{87} +13.0403i q^{89} +(2.80045 - 7.24430i) q^{91} +10.0201 q^{93} +4.77579i q^{95} +1.74852i q^{97} -4.70840i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} - 4 q^{7} + 12 q^{9} - 16 q^{19} - 4 q^{21} - 12 q^{25} + 12 q^{27} + 8 q^{29} + 32 q^{31} + 4 q^{35} - 16 q^{37} + 24 q^{47} - 4 q^{49} + 16 q^{53} - 16 q^{57} + 24 q^{59} - 4 q^{63} + 16 q^{65}+ \cdots + 32 q^{93}+O(q^{100}) \) 12 * q + 12 * q^3 - 4 * q^7 + 12 * q^9 - 16 * q^19 - 4 * q^21 - 12 * q^25 + 12 * q^27 + 8 * q^29 + 32 * q^31 + 4 * q^35 - 16 * q^37 + 24 * q^47 - 4 * q^49 + 16 * q^53 - 16 * q^57 + 24 * q^59 - 4 * q^63 + 16 * q^65 - 12 * q^75 + 24 * q^77 + 12 * q^81 + 16 * q^83 - 8 * q^85 + 8 * q^87 - 8 * q^91 + 32 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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