LMFDB - Embedded newform 560.4.e.c.559.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,4,Mod(559,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("560.559");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	560.e (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:	$33.0410696032$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.121550625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 $ x^8 - x^7 - 4x^6 - 9x^5 + 23x^4 + 18x^3 - 16x^2 + 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			559.1
Root			$-1.44918 - 1.77086i$ of defining polynomial
Character	$\chi$	$=$	560.559
Dual form			560.4.e.c.559.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.3917i q^{3} -11.1803 q^{5} +18.5203i q^{7} -80.9878 q^{9} +O(q^{10})$	$q-10.3917i q^{3} -11.1803 q^{5} +18.5203i q^{7} -80.9878 q^{9} -68.2699i q^{11} -53.1966 q^{13} +116.183i q^{15} -31.0568 q^{17} +192.457 q^{21} +125.000 q^{25} +561.026i q^{27} +239.421 q^{29} -709.442 q^{33} -207.063i q^{35} +552.804i q^{39} +905.471 q^{45} +154.634i q^{47} -343.000 q^{49} +322.733i q^{51} +763.281i q^{55} -1499.92i q^{63} +594.756 q^{65} -863.748i q^{71} +523.240 q^{73} -1298.96i q^{75} +1264.38 q^{77} +896.563i q^{79} +3643.35 q^{81} +47.6235i q^{83} +347.226 q^{85} -2487.99i q^{87} -985.215i q^{91} -1659.14 q^{97} +5529.03i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 320 q^{9}+O(q^{10}) $ 8 * q - 320 * q^9	$ 8 q - 320 q^{9} + 392 q^{21} + 1000 q^{25} - 216 q^{29} - 2744 q^{49} - 1800 q^{65} + 11768 q^{81} - 4600 q^{85}+O(q^{100}) $ 8 * q - 320 * q^9 + 392 * q^21 + 1000 * q^25 - 216 * q^29 - 2744 * q^49 - 1800 * q^65 + 11768 * q^81 - 4600 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/560\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$351$	$421$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$3$	− 10.3917i	− 1.99989i	−0.0106271	−	0.999944i	$-0.503383\pi$
$3$	− 10.3917i	− 1.99989i	0.0106271	−	0.999944i	$-0.496617\pi$
$4$	0	0
$5$	−11.1803	−1.00000
$5$	−11.1803	−1.00000
$6$	0	0
$7$	18.5203i	1.00000i
$7$	18.5203i	1.00000i
$8$	0	0
$9$	−80.9878	−2.99955
$10$	0	0
$11$	− 68.2699i	− 1.87129i	−0.352947	−	0.935643i	$-0.614820\pi$
$11$	− 68.2699i	− 1.87129i	0.352947	−	0.935643i	$-0.385180\pi$
$12$	0	0
$13$	−53.1966	−1.13493	−0.567465	−	0.823398i	$-0.692076\pi$
$13$	−53.1966	−1.13493	−0.567465	+	0.823398i	$0.692076\pi$
$14$	0	0
$15$	116.183i	1.99989i
$16$	0	0
$17$	−31.0568	−0.443081	−0.221541	−	0.975151i	$-0.571109\pi$
$17$	−31.0568	−0.443081	−0.221541	+	0.975151i	$0.571109\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	192.457	1.99989
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	125.000	1.00000
$26$	0	0
$27$	561.026i	3.99887i
$28$	0	0
$29$	239.421	1.53308	0.766540	−	0.642197i	$-0.221977\pi$
$29$	239.421	1.53308	0.766540	+	0.642197i	$0.221977\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−709.442	−3.74236
$34$	0	0
$35$	− 207.063i	− 1.00000i
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	552.804i	2.26973i
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	905.471	2.99955
$46$	0	0
$47$	154.634i	0.479907i	0.970784	+	0.239953i	$0.0771322\pi$
$47$	154.634i	0.479907i	−0.970784	+	0.239953i	$0.922868\pi$
$48$	0	0
$49$	−343.000	−1.00000
$50$	0	0
$51$	322.733i	0.886112i
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	763.281i	1.87129i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	− 1499.92i	− 2.99955i
$64$	0	0
$65$	594.756	1.13493
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 863.748i	− 1.44377i	−0.692011	−	0.721887i	$-0.743275\pi$
$71$	− 863.748i	− 1.44377i	0.692011	−	0.721887i	$-0.256725\pi$
$72$	0	0
$73$	523.240	0.838912	0.419456	−	0.907776i	$-0.362221\pi$
$73$	523.240	0.838912	0.419456	+	0.907776i	$0.362221\pi$
$74$	0	0
$75$	− 1298.96i	− 1.99989i
$76$	0	0
$77$	1264.38	1.87129
$78$	0	0
$79$	896.563i	1.27685i	0.769683	+	0.638426i	$0.220414\pi$
$79$	896.563i	1.27685i	−0.769683	+	0.638426i	$0.779586\pi$
$80$	0	0
$81$	3643.35	4.99774
$82$	0	0
$83$	47.6235i	0.0629803i	0.999504	+	0.0314901i	$0.0100253\pi$
$83$	47.6235i	0.0629803i	−0.999504	+	0.0314901i	$0.989975\pi$
$84$	0	0
$85$	347.226	0.443081
$86$	0	0
$87$	− 2487.99i	− 3.06599i
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	− 985.215i	− 1.13493i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1659.14	−1.73670	−0.868350	−	0.495952i	$-0.834819\pi$
$97$	−1659.14	−1.73670	−0.868350	+	0.495952i	$0.834819\pi$
$98$	0	0
$99$	5529.03i	5.61301i
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	1273.62i	1.21838i	0.793023	+	0.609192i	$0.208506\pi$
$103$	1273.62i	1.21838i	−0.793023	+	0.609192i	$0.791494\pi$
$104$	0	0
$105$	−2151.74	−1.99989
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	948.555	0.833533	0.416766	−	0.909014i	$-0.363163\pi$
$109$	948.555	0.833533	0.416766	+	0.909014i	$0.363163\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4308.28	3.40428
$118$	0	0
$119$	− 575.180i	− 0.443081i
$120$	0	0
$121$	−3329.78	−2.50171
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1397.54	−1.00000
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 6272.46i	− 3.99887i
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	1606.91	0.959760
$142$	0	0
$143$	3631.73i	2.12378i
$144$	0	0
$145$	−2676.81	−1.53308
$146$	0	0
$147$	3564.36i	1.99989i
$148$	0	0
$149$	−2466.00	−1.35586	−0.677928	−	0.735128i	$-0.737122\pi$
$149$	−2466.00	−1.35586	−0.677928	+	0.735128i	$0.737122\pi$
$150$	0	0
$151$	1189.85i	0.641249i	0.947206	+	0.320625i	$0.103893\pi$
$151$	1189.85i	0.641249i	−0.947206	+	0.320625i	$0.896107\pi$
$152$	0	0
$153$	2515.22	1.32904
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3904.17	−1.98463	−0.992315	−	0.123734i	$-0.960513\pi$
$157$	−3904.17	−1.98463	−0.992315	+	0.123734i	$0.960513\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	7931.80	3.74236
$166$	0	0
$167$	− 477.088i	− 0.221067i	−0.993872	−	0.110533i	$-0.964744\pi$
$167$	− 477.088i	− 0.221067i	0.993872	−	0.110533i	$-0.0352559\pi$
$168$	0	0
$169$	632.878	0.288065
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3192.36	1.40295	0.701477	−	0.712693i	$-0.252525\pi$
$173$	3192.36	1.40295	0.701477	+	0.712693i	$0.252525\pi$
$174$	0	0
$175$	2315.03i	1.00000i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4697.37i	1.96144i	0.195419	+	0.980720i	$0.437393\pi$
$179$	4697.37i	1.96144i	−0.195419	+	0.980720i	$0.562607\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2120.24i	0.829132i
$188$	0	0
$189$	−10390.3	−3.99887
$190$	0	0
$191$	5239.12i	1.98476i	0.123203	+	0.992382i	$0.460683\pi$
$191$	5239.12i	1.98476i	−0.123203	+	0.992382i	$0.539317\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	− 6180.54i	− 2.26973i
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4434.13i	1.53308i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 4893.50i	− 1.59660i	−0.602260	−	0.798300i	$-0.705733\pi$
$211$	− 4893.50i	− 1.59660i	0.602260	−	0.798300i	$-0.294267\pi$
$212$	0	0
$213$	−8975.82	−2.88739
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 5437.36i	− 1.67773i
$220$	0	0
$221$	1652.12	0.502866
$222$	0	0
$223$	1034.92i	0.310778i	0.987853	+	0.155389i	$0.0496631\pi$
$223$	1034.92i	0.310778i	−0.987853	+	0.155389i	$0.950337\pi$
$224$	0	0
$225$	−10123.5	−2.99955
$226$	0	0
$227$	6581.21i	1.92427i	0.272565	+	0.962137i	$0.412128\pi$
$227$	6581.21i	1.92427i	−0.272565	+	0.962137i	$0.587872\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	− 13139.0i	− 3.74236i
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	− 1728.86i	− 0.479907i
$236$	0	0
$237$	9316.83	2.55356
$238$	0	0
$239$	2620.28i	0.709170i	0.935024	+	0.354585i	$0.115378\pi$
$239$	2620.28i	0.709170i	−0.935024	+	0.354585i	$0.884622\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	− 22713.0i	− 5.99605i
$244$	0	0
$245$	3834.86	1.00000
$246$	0	0
$247$	0	0
$248$	0	0
$249$	494.890	0.125953
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	− 3608.27i	− 0.886112i
$256$	0	0
$257$	−3358.57	−0.815183	−0.407592	−	0.913164i	$-0.633631\pi$
$257$	−3358.57	−0.815183	−0.407592	+	0.913164i	$0.633631\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−19390.2	−4.59855
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	−10238.1	−2.26973
$274$	0	0
$275$	− 8533.74i	− 1.87129i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8198.77	1.74056	0.870280	−	0.492557i	$-0.163938\pi$
$281$	8198.77	1.74056	0.870280	+	0.492557i	$0.163938\pi$
$282$	0	0
$283$	− 9130.94i	− 1.91794i	−0.283503	−	0.958971i	$-0.591496\pi$
$283$	− 9130.94i	− 1.91794i	0.283503	−	0.958971i	$-0.408504\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−3948.48	−0.803679
$290$	0	0
$291$	17241.3i	3.47320i
$292$	0	0
$293$	−9803.69	−1.95474	−0.977368	−	0.211546i	$-0.932150\pi$
$293$	−9803.69	−1.95474	−0.977368	+	0.211546i	$0.932150\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	38301.2	7.48303
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	9594.81i	1.78373i	0.452302	+	0.891865i	$0.350603\pi$
$307$	9594.81i	1.78373i	−0.452302	+	0.891865i	$0.649397\pi$
$308$	0	0
$309$	13235.1	2.43663
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	7692.05	1.38908	0.694538	−	0.719456i	$-0.255609\pi$
$313$	7692.05	1.38908	0.694538	+	0.719456i	$0.255609\pi$
$314$	0	0
$315$	16769.6i	2.99955i
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	− 16345.2i	− 2.86883i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−6649.57	−1.13493
$326$	0	0
$327$	− 9857.11i	− 1.66697i
$328$	0	0
$329$	−2863.85	−0.479907
$330$	0	0
$331$	9477.56i	1.57382i	0.617069	+	0.786909i	$0.288320\pi$
$331$	9477.56i	1.57382i	−0.617069	+	0.786909i	$0.711680\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 6352.45i	− 1.00000i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	− 29844.7i	− 4.53844i
$352$	0	0
$353$	−7035.45	−1.06079	−0.530396	−	0.847750i	$-0.677957\pi$
$353$	−7035.45	−1.06079	−0.530396	+	0.847750i	$0.677957\pi$
$354$	0	0
$355$	9656.99i	1.44377i
$356$	0	0
$357$	−5977.11	−0.886112
$358$	0	0
$359$	− 11086.7i	− 1.62990i	−0.579529	−	0.814952i	$-0.696763\pi$
$359$	− 11086.7i	− 1.62990i	0.579529	−	0.814952i	$-0.303237\pi$
$360$	0	0
$361$	−6859.00	−1.00000
$362$	0	0
$363$	34602.1i	5.00314i
$364$	0	0
$365$	−5850.00	−0.838912
$366$	0	0
$367$	6322.04i	0.899204i	0.893229	+	0.449602i	$0.148434\pi$
$367$	6322.04i	0.899204i	−0.893229	+	0.449602i	$0.851566\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	14522.9i	1.99989i
$376$	0	0
$377$	−12736.4	−1.73994
$378$	0	0
$379$	4366.07i	0.591741i	0.955228	+	0.295870i	$0.0956097\pi$
$379$	4366.07i	0.591741i	−0.955228	+	0.295870i	$0.904390\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	14239.4i	1.89974i	0.312643	+	0.949871i	$0.398786\pi$
$383$	14239.4i	1.89974i	−0.312643	+	0.949871i	$0.601214\pi$
$384$	0	0
$385$	−14136.2	−1.87129
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1103.23	−0.143794	−0.0718969	−	0.997412i	$-0.522905\pi$
$389$	−1103.23	−0.143794	−0.0718969	+	0.997412i	$0.522905\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 10023.9i	− 1.27685i
$396$	0	0
$397$	7670.68	0.969724	0.484862	−	0.874591i	$-0.338870\pi$
$397$	7670.68	0.969724	0.484862	+	0.874591i	$0.338870\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14801.5	1.84327	0.921635	−	0.388057i	$-0.126854\pi$
$401$	14801.5	1.84327	0.921635	+	0.388057i	$0.126854\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−40733.9	−4.99774
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 532.447i	− 0.0629803i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−12610.2	−1.45981	−0.729907	−	0.683546i	$-0.760437\pi$
$421$	−12610.2	−1.45981	−0.729907	+	0.683546i	$0.760437\pi$
$422$	0	0
$423$	− 12523.4i	− 1.43950i
$424$	0	0
$425$	−3882.10	−0.443081
$426$	0	0
$427$	0	0
$428$	0	0
$429$	37739.9	4.24732
$430$	0	0
$431$	5402.12i	0.603738i	0.953349	+	0.301869i	$0.0976105\pi$
$431$	5402.12i	0.603738i	−0.953349	+	0.301869i	$0.902390\pi$
$432$	0	0
$433$	12920.0	1.43394	0.716970	−	0.697105i	$-0.245529\pi$
$433$	12920.0	1.43394	0.716970	+	0.697105i	$0.245529\pi$
$434$	0	0
$435$	27816.6i	3.06599i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	27778.8	2.99955
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	25626.0i	2.71156i
$448$	0	0
$449$	7501.07	0.788413	0.394207	−	0.919022i	$-0.371019\pi$
$449$	7501.07	0.788413	0.394207	+	0.919022i	$0.371019\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	12364.6	1.28243
$454$	0	0
$455$	11015.0i	1.13493i
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	− 17423.7i	− 1.77182i
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 9554.49i	− 0.946743i	−0.880863	−	0.473371i	$-0.843037\pi$
$467$	− 9554.49i	− 0.946743i	0.880863	−	0.473371i	$-0.156963\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	40571.1i	3.96904i
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	18549.7	1.73670
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	19818.8i	1.82161i	0.412837	+	0.910805i	$0.364538\pi$
$491$	19818.8i	1.82161i	−0.412837	+	0.910805i	$0.635462\pi$
$492$	0	0
$493$	−7435.64	−0.679279
$494$	0	0
$495$	− 61816.4i	− 5.61301i
$496$	0	0
$497$	15996.8	1.44377
$498$	0	0
$499$	− 17022.5i	− 1.52712i	−0.645738	−	0.763559i	$-0.723450\pi$
$499$	− 17022.5i	− 1.52712i	0.645738	−	0.763559i	$-0.276550\pi$
$500$	0	0
$501$	−4957.76	−0.442109
$502$	0	0
$503$	− 13322.2i	− 1.18093i	−0.807062	−	0.590467i	$-0.798944\pi$
$503$	− 13322.2i	− 1.18093i	0.807062	−	0.590467i	$-0.201056\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 6576.69i	− 0.576097i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	9690.54i	0.838912i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 14239.5i	− 1.21838i
$516$	0	0
$517$	10556.8	0.898043
$518$	0	0
$519$	− 33174.1i	− 2.80575i
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	− 7296.98i	− 0.610086i	−0.952339	−	0.305043i	$-0.901329\pi$
$523$	− 7296.98i	− 0.610086i	0.952339	−	0.305043i	$-0.0986708\pi$
$524$	0	0
$525$	24057.2	1.99989
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−12167.0	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	48813.7	3.92266
$538$	0	0
$539$	23416.6i	1.87129i
$540$	0	0
$541$	−24963.5	−1.98385	−0.991927	−	0.126807i	$-0.959527\pi$
$541$	−24963.5	−1.98385	−0.991927	+	0.126807i	$0.959527\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−10605.2	−0.833533
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−16604.6	−1.27685
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	22033.0	1.65817
$562$	0	0
$563$	− 22811.7i	− 1.70763i	−0.520574	−	0.853816i	$-0.674282\pi$
$563$	− 22811.7i	− 1.70763i	0.520574	−	0.853816i	$-0.325718\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	67475.9i	4.99774i
$568$	0	0
$569$	10026.0	0.738685	0.369343	−	0.929293i	$-0.379583\pi$
$569$	10026.0	0.738685	0.369343	+	0.929293i	$0.379583\pi$
$570$	0	0
$571$	− 16079.9i	− 1.17850i	−0.807951	−	0.589250i	$-0.799423\pi$
$571$	− 16079.9i	− 1.17850i	0.807951	−	0.589250i	$-0.200577\pi$
$572$	0	0
$573$	54443.5	3.96930
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−26486.1	−1.91097	−0.955485	−	0.295041i	$-0.904667\pi$
$577$	−26486.1	−1.91097	−0.955485	+	0.295041i	$0.904667\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−882.000	−0.0629803
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−48168.0	−3.40428
$586$	0	0
$587$	− 24145.1i	− 1.69774i	−0.528598	−	0.848872i	$-0.677282\pi$
$587$	− 24145.1i	− 1.69774i	0.528598	−	0.848872i	$-0.322718\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−24441.1	−1.69254	−0.846268	−	0.532757i	$-0.821156\pi$
$593$	−24441.1	−1.69254	−0.846268	+	0.532757i	$0.821156\pi$
$594$	0	0
$595$	6430.71i	0.443081i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4314.67i	0.294311i	0.989113	+	0.147156i	$0.0470118\pi$
$599$	4314.67i	0.294311i	−0.989113	+	0.147156i	$0.952988\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	37228.1	2.50171
$606$	0	0
$607$	− 20135.9i	− 1.34644i	−0.739441	−	0.673221i	$-0.764910\pi$
$607$	− 20135.9i	− 1.34644i	0.739441	−	0.673221i	$-0.235090\pi$
$608$	0	0
$609$	46078.3	3.06599
$610$	0	0
$611$	− 8225.98i	− 0.544660i
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	15625.0	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 1141.15i	− 0.0719945i	−0.999352	−	0.0359972i	$-0.988539\pi$
$631$	− 1141.15i	− 0.0719945i	0.999352	−	0.0359972i	$-0.0114608\pi$
$632$	0	0
$633$	−50851.9	−3.19302
$634$	0	0
$635$	0	0
$636$	0	0
$637$	18246.4	1.13493
$638$	0	0
$639$	69953.0i	4.33067i
$640$	0	0
$641$	−28782.0	−1.77351	−0.886756	−	0.462239i	$-0.847046\pi$
$641$	−28782.0	−1.77351	−0.886756	+	0.462239i	$0.847046\pi$
$642$	0	0
$643$	− 4085.46i	− 0.250568i	−0.992121	−	0.125284i	$-0.960016\pi$
$643$	− 4085.46i	− 0.250568i	0.992121	−	0.125284i	$-0.0399841\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15572.9i	0.946265i	0.880991	+	0.473133i	$0.156877\pi$
$647$	15572.9i	0.946265i	−0.880991	+	0.473133i	$0.843123\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−42376.1	−2.51636
$658$	0	0
$659$	33304.9i	1.96870i	0.176218	+	0.984351i	$0.443614\pi$
$659$	33304.9i	1.96870i	−0.176218	+	0.984351i	$0.556386\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	− 17168.3i	− 1.00567i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	10754.6	0.621521
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	70128.3i	3.99887i
$676$	0	0
$677$	29316.6	1.66430	0.832149	−	0.554552i	$-0.187110\pi$
$677$	29316.6	1.66430	0.832149	+	0.554552i	$0.187110\pi$
$678$	0	0
$679$	− 30727.7i	− 1.73670i
$680$	0	0
$681$	68390.1	3.84833
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	−102399.	−5.61301
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−30646.4	−1.65121	−0.825604	−	0.564250i	$-0.809166\pi$
$701$	−30646.4	−1.65121	−0.825604	+	0.564250i	$0.809166\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−17965.8	−0.959760
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−33709.8	−1.78561	−0.892805	−	0.450443i	$-0.851266\pi$
$709$	−33709.8	−1.78561	−0.892805	+	0.450443i	$0.851266\pi$
$710$	0	0
$711$	− 72610.7i	− 3.82998i
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 40603.9i	− 2.12378i
$716$	0	0
$717$	27229.2	1.41826
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−23587.8	−1.21838
$722$	0	0
$723$	0	0
$724$	0	0
$725$	29927.6	1.53308
$726$	0	0
$727$	− 11392.6i	− 0.581194i	−0.956845	−	0.290597i	$-0.906146\pi$
$727$	− 11392.6i	− 0.581194i	0.956845	−	0.290597i	$-0.0938540\pi$
$728$	0	0
$729$	−137657.	−6.99368
$730$	0	0
$731$	0	0
$732$	0	0
$733$	3772.47	0.190095	0.0950473	−	0.995473i	$-0.469700\pi$
$733$	3772.47	0.190095	0.0950473	+	0.995473i	$0.469700\pi$
$734$	0	0
$735$	− 39850.7i	− 1.99989i
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 36419.7i	− 1.81288i	−0.422332	−	0.906441i	$-0.638788\pi$
$739$	− 36419.7i	− 1.81288i	0.422332	−	0.906441i	$-0.361212\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	27570.7	1.35586
$746$	0	0
$747$	− 3856.92i	− 0.188912i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	7968.15i	0.387166i	0.981084	+	0.193583i	$0.0620109\pi$
$751$	7968.15i	0.387166i	−0.981084	+	0.193583i	$0.937989\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 13302.9i	− 0.641249i
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	17567.5i	0.833533i
$764$	0	0
$765$	−28121.0	−1.32904
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	34901.4i	1.63027i
$772$	0	0
$773$	39541.0	1.83983	0.919917	−	0.392113i	$-0.128256\pi$
$773$	39541.0	1.83983	0.919917	+	0.392113i	$0.128256\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−58968.0	−2.70172
$782$	0	0
$783$	134321.i	6.13059i
$784$	0	0
$785$	43650.0	1.98463
$786$	0	0
$787$	28301.3i	1.28187i	0.767595	+	0.640936i	$0.221453\pi$
$787$	28301.3i	1.28187i	−0.767595	+	0.640936i	$0.778547\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−4513.57	−0.200601	−0.100300	−	0.994957i	$-0.531980\pi$
$797$	−4513.57	−0.200601	−0.100300	+	0.994957i	$0.531980\pi$
$798$	0	0
$799$	− 4802.42i	− 0.212638i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 35721.5i	− 1.56985i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−7868.40	−0.341951	−0.170975	−	0.985275i	$-0.554692\pi$
$809$	−7868.40	−0.341951	−0.170975	+	0.985275i	$0.554692\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	79790.4i	3.40428i
$820$	0	0
$821$	34840.8	1.48106	0.740531	−	0.672022i	$-0.234574\pi$
$821$	34840.8	1.48106	0.740531	+	0.672022i	$0.234574\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	−88680.2	−3.74236
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	10652.5	0.443081
$834$	0	0
$835$	5334.00i	0.221067i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	32933.3	1.35033
$842$	0	0
$843$	− 85199.3i	− 3.48092i
$844$	0	0
$845$	−7075.79	−0.288065
$846$	0	0
$847$	− 61668.4i	− 2.50171i
$848$	0	0
$849$	−94886.1	−3.83567
$850$	0	0
$851$	0	0
$852$	0	0
$853$	37794.0	1.51705	0.758524	−	0.651645i	$-0.225921\pi$
$853$	37794.0	1.51705	0.758524	+	0.651645i	$0.225921\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7428.22	0.296083	0.148041	−	0.988981i	$-0.452703\pi$
$857$	7428.22	0.296083	0.148041	+	0.988981i	$0.452703\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	−35691.7	−1.40295
$866$	0	0
$867$	41031.4i	1.60727i
$868$	0	0
$869$	61208.3	2.38935
$870$	0	0
$871$	0	0
$872$	0	0
$873$	134370.	5.20932
$874$	0	0
$875$	− 25882.8i	− 1.00000i
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	101877.i	3.90925i
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 18716.0i	− 0.708482i	−0.935154	−	0.354241i	$-0.884739\pi$
$887$	− 18716.0i	− 0.708482i	0.935154	−	0.354241i	$-0.115261\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 248731.i	− 9.35221i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	− 52518.2i	− 1.96144i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	45376.3i	1.65026i	0.564944	+	0.825129i	$0.308898\pi$
$911$	45376.3i	1.65026i	−0.564944	+	0.825129i	$0.691102\pi$
$912$	0	0
$913$	3251.25	0.117854
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	46043.3i	1.65270i	0.563159	+	0.826348i	$0.309586\pi$
$919$	46043.3i	1.65270i	−0.563159	+	0.826348i	$0.690414\pi$
$920$	0	0
$921$	99706.6	3.56726
$922$	0	0
$923$	45948.4i	1.63858i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 103148.i	− 3.65460i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 23705.1i	− 0.829132i
$936$	0	0
$937$	−56285.5	−1.96240	−0.981199	−	0.192998i	$-0.938179\pi$
$937$	−56285.5	−1.96240	−0.981199	+	0.192998i	$0.938179\pi$
$938$	0	0
$939$	− 79933.7i	− 2.77799i
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	116168.	3.99887
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	−27834.6	−0.952106
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	− 58575.2i	− 1.98476i
$956$	0	0
$957$	−169855.	−5.73734
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29791.0	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	69100.5i	2.26973i
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−76821.4	−2.50022
$982$	0	0
$983$	− 49133.2i	− 1.59421i	−0.603842	−	0.797104i	$-0.706364\pi$
$983$	− 49133.2i	− 1.59421i	0.603842	−	0.797104i	$-0.293636\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	29760.4i	0.959760i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	60805.5i	1.94909i	0.224189	+	0.974546i	$0.428027\pi$
$991$	60805.5i	1.94909i	−0.224189	+	0.974546i	$0.571973\pi$
$992$	0	0
$993$	98488.1	3.14746
$994$	0	0
$995$	0	0
$996$	0	0
$997$	62213.6	1.97625	0.988126	−	0.153646i	$-0.0491016\pi$
$997$	62213.6	1.97625	0.988126	+	0.153646i	$0.0491016\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.4.e.c.559.1	✓	8
4.3	odd	2	inner	560.4.e.c.559.7	yes	8
5.4	even	2	inner	560.4.e.c.559.8	yes	8
7.6	odd	2	inner	560.4.e.c.559.8	yes	8
20.19	odd	2	inner	560.4.e.c.559.2	yes	8
28.27	even	2	inner	560.4.e.c.559.2	yes	8
35.34	odd	2	CM	560.4.e.c.559.1	✓	8
140.139	even	2	inner	560.4.e.c.559.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.4.e.c.559.1	✓	8	1.1	even	1	trivial
560.4.e.c.559.1	✓	8	35.34	odd	2	CM
560.4.e.c.559.2	yes	8	20.19	odd	2	inner
560.4.e.c.559.2	yes	8	28.27	even	2	inner
560.4.e.c.559.7	yes	8	4.3	odd	2	inner
560.4.e.c.559.7	yes	8	140.139	even	2	inner
560.4.e.c.559.8	yes	8	5.4	even	2	inner
560.4.e.c.559.8	yes	8	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	560.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-10.3917i q^{3} -11.1803 q^{5} +18.5203i q^{7} -80.9878 q^{9} +O(q^{10})\)	\(q-10.3917i q^{3} -11.1803 q^{5} +18.5203i q^{7} -80.9878 q^{9} -68.2699i q^{11} -53.1966 q^{13} +116.183i q^{15} -31.0568 q^{17} +192.457 q^{21} +125.000 q^{25} +561.026i q^{27} +239.421 q^{29} -709.442 q^{33} -207.063i q^{35} +552.804i q^{39} +905.471 q^{45} +154.634i q^{47} -343.000 q^{49} +322.733i q^{51} +763.281i q^{55} -1499.92i q^{63} +594.756 q^{65} -863.748i q^{71} +523.240 q^{73} -1298.96i q^{75} +1264.38 q^{77} +896.563i q^{79} +3643.35 q^{81} +47.6235i q^{83} +347.226 q^{85} -2487.99i q^{87} -985.215i q^{91} -1659.14 q^{97} +5529.03i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 320 q^{9}+O(q^{10}) \) 8 * q - 320 * q^9	\( 8 q - 320 q^{9} + 392 q^{21} + 1000 q^{25} - 216 q^{29} - 2744 q^{49} - 1800 q^{65} + 11768 q^{81} - 4600 q^{85}+O(q^{100}) \) 8 * q - 320 * q^9 + 392 * q^21 + 1000 * q^25 - 216 * q^29 - 2744 * q^49 - 1800 * q^65 + 11768 * q^81 - 4600 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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