LMFDB - Embedded newform 1280.4.d.b.641.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,4,Mod(641,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1280.641");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1280.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$75.5224448073$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			641.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1280.641
Dual form			1280.4.d.b.641.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.0000i q^{3} -5.00000i q^{5} -18.0000 q^{7} -73.0000 q^{9} +O(q^{10})$	$q+10.0000i q^{3} -5.00000i q^{5} -18.0000 q^{7} -73.0000 q^{9} +16.0000i q^{11} +6.00000i q^{13} +50.0000 q^{15} -6.00000 q^{17} -124.000i q^{19} -180.000i q^{21} +42.0000 q^{23} -25.0000 q^{25} -460.000i q^{27} -142.000i q^{29} +188.000 q^{31} -160.000 q^{33} +90.0000i q^{35} +202.000i q^{37} -60.0000 q^{39} -54.0000 q^{41} -66.0000i q^{43} +365.000i q^{45} -38.0000 q^{47} -19.0000 q^{49} -60.0000i q^{51} +738.000i q^{53} +80.0000 q^{55} +1240.00 q^{57} -564.000i q^{59} +262.000i q^{61} +1314.00 q^{63} +30.0000 q^{65} -554.000i q^{67} +420.000i q^{69} +140.000 q^{71} -882.000 q^{73} -250.000i q^{75} -288.000i q^{77} +1160.00 q^{79} +2629.00 q^{81} +642.000i q^{83} +30.0000i q^{85} +1420.00 q^{87} +854.000 q^{89} -108.000i q^{91} +1880.00i q^{93} -620.000 q^{95} -478.000 q^{97} -1168.00i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 36 q^{7} - 146 q^{9}+O(q^{10}) $ 2 * q - 36 * q^7 - 146 * q^9	$ 2 q - 36 q^{7} - 146 q^{9} + 100 q^{15} - 12 q^{17} + 84 q^{23} - 50 q^{25} + 376 q^{31} - 320 q^{33} - 120 q^{39} - 108 q^{41} - 76 q^{47} - 38 q^{49} + 160 q^{55} + 2480 q^{57} + 2628 q^{63} + 60 q^{65} + 280 q^{71} - 1764 q^{73} + 2320 q^{79} + 5258 q^{81} + 2840 q^{87} + 1708 q^{89} - 1240 q^{95} - 956 q^{97}+O(q^{100}) $ 2 * q - 36 * q^7 - 146 * q^9 + 100 * q^15 - 12 * q^17 + 84 * q^23 - 50 * q^25 + 376 * q^31 - 320 * q^33 - 120 * q^39 - 108 * q^41 - 76 * q^47 - 38 * q^49 + 160 * q^55 + 2480 * q^57 + 2628 * q^63 + 60 * q^65 + 280 * q^71 - 1764 * q^73 + 2320 * q^79 + 5258 * q^81 + 2840 * q^87 + 1708 * q^89 - 1240 * q^95 - 956 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	10.0000i	1.92450i	0.272166	+	0.962250i	$0.412260\pi$
$3$	10.0000i	1.92450i	−0.272166	+	0.962250i	$0.587740\pi$
$4$	0	0
$5$	− 5.00000i	− 0.447214i
$5$	− 5.00000i	− 0.447214i
$6$	0	0
$7$	−18.0000	−0.971909	−0.485954	−	0.873984i	$-0.661528\pi$
$7$	−18.0000	−0.971909	−0.485954	+	0.873984i	$0.661528\pi$
$8$	0	0
$9$	−73.0000	−2.70370
$10$	0	0
$11$	16.0000i	0.438562i	0.975662	+	0.219281i	$0.0703711\pi$
$11$	16.0000i	0.438562i	−0.975662	+	0.219281i	$0.929629\pi$
$12$	0	0
$13$	6.00000i	0.128008i	0.997950	+	0.0640039i	$0.0203870\pi$
$13$	6.00000i	0.128008i	−0.997950	+	0.0640039i	$0.979613\pi$
$14$	0	0
$15$	50.0000	0.860663
$16$	0	0
$17$	−6.00000	−0.0856008	−0.0428004	−	0.999084i	$-0.513628\pi$
$17$	−6.00000	−0.0856008	−0.0428004	+	0.999084i	$0.513628\pi$
$18$	0	0
$19$	− 124.000i	− 1.49724i	−0.663000	−	0.748620i	$-0.730717\pi$
$19$	− 124.000i	− 1.49724i	0.663000	−	0.748620i	$-0.269283\pi$
$20$	0	0
$21$	− 180.000i	− 1.87044i
$22$	0	0
$23$	42.0000	0.380765	0.190383	−	0.981710i	$-0.439027\pi$
$23$	42.0000	0.380765	0.190383	+	0.981710i	$0.439027\pi$
$24$	0	0
$25$	−25.0000	−0.200000
$26$	0	0
$27$	− 460.000i	− 3.27878i
$28$	0	0
$29$	− 142.000i	− 0.909267i	−0.890679	−	0.454633i	$-0.849770\pi$
$29$	− 142.000i	− 0.909267i	0.890679	−	0.454633i	$-0.150230\pi$
$30$	0	0
$31$	188.000	1.08922	0.544610	−	0.838690i	$-0.316678\pi$
$31$	188.000	1.08922	0.544610	+	0.838690i	$0.316678\pi$
$32$	0	0
$33$	−160.000	−0.844013
$34$	0	0
$35$	90.0000i	0.434651i
$36$	0	0
$37$	202.000i	0.897530i	0.893650	+	0.448765i	$0.148136\pi$
$37$	202.000i	0.897530i	−0.893650	+	0.448765i	$0.851864\pi$
$38$	0	0
$39$	−60.0000	−0.246351
$40$	0	0
$41$	−54.0000	−0.205692	−0.102846	−	0.994697i	$-0.532795\pi$
$41$	−54.0000	−0.205692	−0.102846	+	0.994697i	$0.532795\pi$
$42$	0	0
$43$	− 66.0000i	− 0.234068i	−0.993128	−	0.117034i	$-0.962661\pi$
$43$	− 66.0000i	− 0.234068i	0.993128	−	0.117034i	$-0.0373386\pi$
$44$	0	0
$45$	365.000i	1.20913i
$46$	0	0
$47$	−38.0000	−0.117933	−0.0589667	−	0.998260i	$-0.518781\pi$
$47$	−38.0000	−0.117933	−0.0589667	+	0.998260i	$0.518781\pi$
$48$	0	0
$49$	−19.0000	−0.0553936
$50$	0	0
$51$	− 60.0000i	− 0.164739i
$52$	0	0
$53$	738.000i	1.91268i	0.292255	+	0.956341i	$0.405595\pi$
$53$	738.000i	1.91268i	−0.292255	+	0.956341i	$0.594405\pi$
$54$	0	0
$55$	80.0000	0.196131
$56$	0	0
$57$	1240.00	2.88144
$58$	0	0
$59$	− 564.000i	− 1.24452i	−0.782812	−	0.622259i	$-0.786215\pi$
$59$	− 564.000i	− 1.24452i	0.782812	−	0.622259i	$-0.213785\pi$
$60$	0	0
$61$	262.000i	0.549929i	0.961454	+	0.274964i	$0.0886661\pi$
$61$	262.000i	0.549929i	−0.961454	+	0.274964i	$0.911334\pi$
$62$	0	0
$63$	1314.00	2.62775
$64$	0	0
$65$	30.0000	0.0572468
$66$	0	0
$67$	− 554.000i	− 1.01018i	−0.863067	−	0.505089i	$-0.831460\pi$
$67$	− 554.000i	− 1.01018i	0.863067	−	0.505089i	$-0.168540\pi$
$68$	0	0
$69$	420.000i	0.732783i
$70$	0	0
$71$	140.000	0.234013	0.117007	−	0.993131i	$-0.462670\pi$
$71$	140.000	0.234013	0.117007	+	0.993131i	$0.462670\pi$
$72$	0	0
$73$	−882.000	−1.41411	−0.707057	−	0.707157i	$-0.749977\pi$
$73$	−882.000	−1.41411	−0.707057	+	0.707157i	$0.749977\pi$
$74$	0	0
$75$	− 250.000i	− 0.384900i
$76$	0	0
$77$	− 288.000i	− 0.426242i
$78$	0	0
$79$	1160.00	1.65203	0.826014	−	0.563650i	$-0.190603\pi$
$79$	1160.00	1.65203	0.826014	+	0.563650i	$0.190603\pi$
$80$	0	0
$81$	2629.00	3.60631
$82$	0	0
$83$	642.000i	0.849020i	0.905423	+	0.424510i	$0.139554\pi$
$83$	642.000i	0.849020i	−0.905423	+	0.424510i	$0.860446\pi$
$84$	0	0
$85$	30.0000i	0.0382818i
$86$	0	0
$87$	1420.00	1.74988
$88$	0	0
$89$	854.000	1.01712	0.508561	−	0.861026i	$-0.330178\pi$
$89$	854.000	1.01712	0.508561	+	0.861026i	$0.330178\pi$
$90$	0	0
$91$	− 108.000i	− 0.124412i
$92$	0	0
$93$	1880.00i	2.09620i
$94$	0	0
$95$	−620.000	−0.669586
$96$	0	0
$97$	−478.000	−0.500346	−0.250173	−	0.968201i	$-0.580487\pi$
$97$	−478.000	−0.500346	−0.250173	+	0.968201i	$0.580487\pi$
$98$	0	0
$99$	− 1168.00i	− 1.18574i
$100$	0	0
$101$	− 1794.00i	− 1.76742i	−0.468033	−	0.883711i	$-0.655037\pi$
$101$	− 1794.00i	− 1.76742i	0.468033	−	0.883711i	$-0.344963\pi$
$102$	0	0
$103$	642.000	0.614157	0.307078	−	0.951684i	$-0.400649\pi$
$103$	642.000	0.614157	0.307078	+	0.951684i	$0.400649\pi$
$104$	0	0
$105$	−900.000	−0.836486
$106$	0	0
$107$	850.000i	0.767968i	0.923340	+	0.383984i	$0.125448\pi$
$107$	850.000i	0.767968i	−0.923340	+	0.383984i	$0.874552\pi$
$108$	0	0
$109$	− 666.000i	− 0.585241i	−0.956229	−	0.292620i	$-0.905473\pi$
$109$	− 666.000i	− 0.585241i	0.956229	−	0.292620i	$-0.0945272\pi$
$110$	0	0
$111$	−2020.00	−1.72730
$112$	0	0
$113$	−1446.00	−1.20379	−0.601895	−	0.798575i	$-0.705587\pi$
$113$	−1446.00	−1.20379	−0.601895	+	0.798575i	$0.705587\pi$
$114$	0	0
$115$	− 210.000i	− 0.170283i
$116$	0	0
$117$	− 438.000i	− 0.346095i
$118$	0	0
$119$	108.000	0.0831962
$120$	0	0
$121$	1075.00	0.807663
$122$	0	0
$123$	− 540.000i	− 0.395855i
$124$	0	0
$125$	125.000i	0.0894427i
$126$	0	0
$127$	1154.00	0.806307	0.403153	−	0.915132i	$-0.367914\pi$
$127$	1154.00	0.806307	0.403153	+	0.915132i	$0.367914\pi$
$128$	0	0
$129$	660.000	0.450463
$130$	0	0
$131$	− 368.000i	− 0.245437i	−0.992441	−	0.122719i	$-0.960839\pi$
$131$	− 368.000i	− 0.245437i	0.992441	−	0.122719i	$-0.0391613\pi$
$132$	0	0
$133$	2232.00i	1.45518i
$134$	0	0
$135$	−2300.00	−1.46631
$136$	0	0
$137$	670.000	0.417825	0.208912	−	0.977934i	$-0.433008\pi$
$137$	670.000	0.417825	0.208912	+	0.977934i	$0.433008\pi$
$138$	0	0
$139$	572.000i	0.349039i	0.984654	+	0.174519i	$0.0558372\pi$
$139$	572.000i	0.349039i	−0.984654	+	0.174519i	$0.944163\pi$
$140$	0	0
$141$	− 380.000i	− 0.226963i
$142$	0	0
$143$	−96.0000	−0.0561393
$144$	0	0
$145$	−710.000	−0.406636
$146$	0	0
$147$	− 190.000i	− 0.106605i
$148$	0	0
$149$	1730.00i	0.951189i	0.879665	+	0.475594i	$0.157767\pi$
$149$	1730.00i	0.951189i	−0.879665	+	0.475594i	$0.842233\pi$
$150$	0	0
$151$	1324.00	0.713547	0.356773	−	0.934191i	$-0.383877\pi$
$151$	1324.00	0.713547	0.356773	+	0.934191i	$0.383877\pi$
$152$	0	0
$153$	438.000	0.231439
$154$	0	0
$155$	− 940.000i	− 0.487114i
$156$	0	0
$157$	− 2946.00i	− 1.49756i	−0.662820	−	0.748778i	$-0.730641\pi$
$157$	− 2946.00i	− 1.49756i	0.662820	−	0.748778i	$-0.269359\pi$
$158$	0	0
$159$	−7380.00	−3.68096
$160$	0	0
$161$	−756.000	−0.370069
$162$	0	0
$163$	2098.00i	1.00815i	0.863661	+	0.504074i	$0.168166\pi$
$163$	2098.00i	1.00815i	−0.863661	+	0.504074i	$0.831834\pi$
$164$	0	0
$165$	800.000i	0.377454i
$166$	0	0
$167$	−866.000	−0.401276	−0.200638	−	0.979665i	$-0.564301\pi$
$167$	−866.000	−0.401276	−0.200638	+	0.979665i	$0.564301\pi$
$168$	0	0
$169$	2161.00	0.983614
$170$	0	0
$171$	9052.00i	4.04809i
$172$	0	0
$173$	1678.00i	0.737433i	0.929542	+	0.368717i	$0.120203\pi$
$173$	1678.00i	0.737433i	−0.929542	+	0.368717i	$0.879797\pi$
$174$	0	0
$175$	450.000	0.194382
$176$	0	0
$177$	5640.00	2.39508
$178$	0	0
$179$	− 1620.00i	− 0.676450i	−0.941065	−	0.338225i	$-0.890174\pi$
$179$	− 1620.00i	− 0.676450i	0.941065	−	0.338225i	$-0.109826\pi$
$180$	0	0
$181$	2510.00i	1.03076i	0.856963	+	0.515378i	$0.172348\pi$
$181$	2510.00i	1.03076i	−0.856963	+	0.515378i	$0.827652\pi$
$182$	0	0
$183$	−2620.00	−1.05834
$184$	0	0
$185$	1010.00	0.401387
$186$	0	0
$187$	− 96.0000i	− 0.0375413i
$188$	0	0
$189$	8280.00i	3.18667i
$190$	0	0
$191$	372.000	0.140927	0.0704633	−	0.997514i	$-0.477552\pi$
$191$	372.000	0.140927	0.0704633	+	0.997514i	$0.477552\pi$
$192$	0	0
$193$	2938.00	1.09576	0.547880	−	0.836557i	$-0.315435\pi$
$193$	2938.00	1.09576	0.547880	+	0.836557i	$0.315435\pi$
$194$	0	0
$195$	300.000i	0.110172i
$196$	0	0
$197$	2234.00i	0.807949i	0.914770	+	0.403974i	$0.132372\pi$
$197$	2234.00i	0.807949i	−0.914770	+	0.403974i	$0.867628\pi$
$198$	0	0
$199$	−3048.00	−1.08576	−0.542882	−	0.839809i	$-0.682667\pi$
$199$	−3048.00	−1.08576	−0.542882	+	0.839809i	$0.682667\pi$
$200$	0	0
$201$	5540.00	1.94409
$202$	0	0
$203$	2556.00i	0.883724i
$204$	0	0
$205$	270.000i	0.0919884i
$206$	0	0
$207$	−3066.00	−1.02948
$208$	0	0
$209$	1984.00	0.656632
$210$	0	0
$211$	4896.00i	1.59741i	0.601720	+	0.798707i	$0.294482\pi$
$211$	4896.00i	1.59741i	−0.601720	+	0.798707i	$0.705518\pi$
$212$	0	0
$213$	1400.00i	0.450359i
$214$	0	0
$215$	−330.000	−0.104678
$216$	0	0
$217$	−3384.00	−1.05862
$218$	0	0
$219$	− 8820.00i	− 2.72146i
$220$	0	0
$221$	− 36.0000i	− 0.0109576i
$222$	0	0
$223$	5302.00	1.59214	0.796072	−	0.605202i	$-0.206908\pi$
$223$	5302.00	1.59214	0.796072	+	0.605202i	$0.206908\pi$
$224$	0	0
$225$	1825.00	0.540741
$226$	0	0
$227$	− 3778.00i	− 1.10465i	−0.833630	−	0.552323i	$-0.813741\pi$
$227$	− 3778.00i	− 1.10465i	0.833630	−	0.552323i	$-0.186259\pi$
$228$	0	0
$229$	− 3034.00i	− 0.875513i	−0.899094	−	0.437756i	$-0.855773\pi$
$229$	− 3034.00i	− 0.875513i	0.899094	−	0.437756i	$-0.144227\pi$
$230$	0	0
$231$	2880.00	0.820303
$232$	0	0
$233$	3478.00	0.977903	0.488951	−	0.872311i	$-0.337380\pi$
$233$	3478.00	0.977903	0.488951	+	0.872311i	$0.337380\pi$
$234$	0	0
$235$	190.000i	0.0527414i
$236$	0	0
$237$	11600.0i	3.17933i
$238$	0	0
$239$	−1560.00	−0.422209	−0.211105	−	0.977463i	$-0.567706\pi$
$239$	−1560.00	−0.422209	−0.211105	+	0.977463i	$0.567706\pi$
$240$	0	0
$241$	−3218.00	−0.860123	−0.430061	−	0.902800i	$-0.641508\pi$
$241$	−3218.00	−0.860123	−0.430061	+	0.902800i	$0.641508\pi$
$242$	0	0
$243$	13870.0i	3.66157i
$244$	0	0
$245$	95.0000i	0.0247728i
$246$	0	0
$247$	744.000	0.191658
$248$	0	0
$249$	−6420.00	−1.63394
$250$	0	0
$251$	− 688.000i	− 0.173013i	−0.996251	−	0.0865063i	$-0.972430\pi$
$251$	− 688.000i	− 0.173013i	0.996251	−	0.0865063i	$-0.0275703\pi$
$252$	0	0
$253$	672.000i	0.166989i
$254$	0	0
$255$	−300.000	−0.0736734
$256$	0	0
$257$	2170.00	0.526696	0.263348	−	0.964701i	$-0.415173\pi$
$257$	2170.00	0.526696	0.263348	+	0.964701i	$0.415173\pi$
$258$	0	0
$259$	− 3636.00i	− 0.872317i
$260$	0	0
$261$	10366.0i	2.45839i
$262$	0	0
$263$	2274.00	0.533159	0.266580	−	0.963813i	$-0.414106\pi$
$263$	2274.00	0.533159	0.266580	+	0.963813i	$0.414106\pi$
$264$	0	0
$265$	3690.00	0.855377
$266$	0	0
$267$	8540.00i	1.95745i
$268$	0	0
$269$	− 7146.00i	− 1.61970i	−0.586637	−	0.809850i	$-0.699548\pi$
$269$	− 7146.00i	− 1.61970i	0.586637	−	0.809850i	$-0.300452\pi$
$270$	0	0
$271$	−2604.00	−0.583696	−0.291848	−	0.956465i	$-0.594270\pi$
$271$	−2604.00	−0.583696	−0.291848	+	0.956465i	$0.594270\pi$
$272$	0	0
$273$	1080.00	0.239431
$274$	0	0
$275$	− 400.000i	− 0.0877124i
$276$	0	0
$277$	− 5150.00i	− 1.11709i	−0.829475	−	0.558544i	$-0.811360\pi$
$277$	− 5150.00i	− 1.11709i	0.829475	−	0.558544i	$-0.188640\pi$
$278$	0	0
$279$	−13724.0	−2.94493
$280$	0	0
$281$	−5270.00	−1.11880	−0.559398	−	0.828899i	$-0.688968\pi$
$281$	−5270.00	−1.11880	−0.559398	+	0.828899i	$0.688968\pi$
$282$	0	0
$283$	− 3434.00i	− 0.721308i	−0.932700	−	0.360654i	$-0.882554\pi$
$283$	− 3434.00i	− 0.721308i	0.932700	−	0.360654i	$-0.117446\pi$
$284$	0	0
$285$	− 6200.00i	− 1.28862i
$286$	0	0
$287$	972.000	0.199914
$288$	0	0
$289$	−4877.00	−0.992673
$290$	0	0
$291$	− 4780.00i	− 0.962916i
$292$	0	0
$293$	− 9878.00i	− 1.96955i	−0.173826	−	0.984776i	$-0.555613\pi$
$293$	− 9878.00i	− 1.96955i	0.173826	−	0.984776i	$-0.444387\pi$
$294$	0	0
$295$	−2820.00	−0.556565
$296$	0	0
$297$	7360.00	1.43795
$298$	0	0
$299$	252.000i	0.0487409i
$300$	0	0
$301$	1188.00i	0.227492i
$302$	0	0
$303$	17940.0	3.40141
$304$	0	0
$305$	1310.00	0.245936
$306$	0	0
$307$	8054.00i	1.49728i	0.662975	+	0.748642i	$0.269294\pi$
$307$	8054.00i	1.49728i	−0.662975	+	0.748642i	$0.730706\pi$
$308$	0	0
$309$	6420.00i	1.18195i
$310$	0	0
$311$	5492.00	1.00136	0.500680	−	0.865633i	$-0.333083\pi$
$311$	5492.00	1.00136	0.500680	+	0.865633i	$0.333083\pi$
$312$	0	0
$313$	422.000	0.0762072	0.0381036	−	0.999274i	$-0.487868\pi$
$313$	422.000	0.0762072	0.0381036	+	0.999274i	$0.487868\pi$
$314$	0	0
$315$	− 6570.00i	− 1.17517i
$316$	0	0
$317$	− 6194.00i	− 1.09744i	−0.836005	−	0.548722i	$-0.815115\pi$
$317$	− 6194.00i	− 1.09744i	0.836005	−	0.548722i	$-0.184885\pi$
$318$	0	0
$319$	2272.00	0.398770
$320$	0	0
$321$	−8500.00	−1.47796
$322$	0	0
$323$	744.000i	0.128165i
$324$	0	0
$325$	− 150.000i	− 0.0256015i
$326$	0	0
$327$	6660.00	1.12630
$328$	0	0
$329$	684.000	0.114620
$330$	0	0
$331$	− 7688.00i	− 1.27665i	−0.769768	−	0.638324i	$-0.779628\pi$
$331$	− 7688.00i	− 1.27665i	0.769768	−	0.638324i	$-0.220372\pi$
$332$	0	0
$333$	− 14746.0i	− 2.42665i
$334$	0	0
$335$	−2770.00	−0.451765
$336$	0	0
$337$	−1438.00	−0.232442	−0.116221	−	0.993223i	$-0.537078\pi$
$337$	−1438.00	−0.232442	−0.116221	+	0.993223i	$0.537078\pi$
$338$	0	0
$339$	− 14460.0i	− 2.31669i
$340$	0	0
$341$	3008.00i	0.477690i
$342$	0	0
$343$	6516.00	1.02575
$344$	0	0
$345$	2100.00	0.327711
$346$	0	0
$347$	− 8838.00i	− 1.36729i	−0.729816	−	0.683644i	$-0.760394\pi$
$347$	− 8838.00i	− 1.36729i	0.729816	−	0.683644i	$-0.239606\pi$
$348$	0	0
$349$	7810.00i	1.19788i	0.800794	+	0.598939i	$0.204411\pi$
$349$	7810.00i	1.19788i	−0.800794	+	0.598939i	$0.795589\pi$
$350$	0	0
$351$	2760.00	0.419709
$352$	0	0
$353$	5906.00	0.890495	0.445247	−	0.895408i	$-0.353116\pi$
$353$	5906.00	0.890495	0.445247	+	0.895408i	$0.353116\pi$
$354$	0	0
$355$	− 700.000i	− 0.104654i
$356$	0	0
$357$	1080.00i	0.160111i
$358$	0	0
$359$	8904.00	1.30901	0.654506	−	0.756057i	$-0.272877\pi$
$359$	8904.00	1.30901	0.654506	+	0.756057i	$0.272877\pi$
$360$	0	0
$361$	−8517.00	−1.24173
$362$	0	0
$363$	10750.0i	1.55435i
$364$	0	0
$365$	4410.00i	0.632411i
$366$	0	0
$367$	7370.00	1.04826	0.524129	−	0.851639i	$-0.324391\pi$
$367$	7370.00	1.04826	0.524129	+	0.851639i	$0.324391\pi$
$368$	0	0
$369$	3942.00	0.556131
$370$	0	0
$371$	− 13284.0i	− 1.85895i
$372$	0	0
$373$	− 734.000i	− 0.101890i	−0.998701	−	0.0509451i	$-0.983777\pi$
$373$	− 734.000i	− 0.101890i	0.998701	−	0.0509451i	$-0.0162234\pi$
$374$	0	0
$375$	−1250.00	−0.172133
$376$	0	0
$377$	852.000	0.116393
$378$	0	0
$379$	− 10300.0i	− 1.39598i	−0.716109	−	0.697989i	$-0.754079\pi$
$379$	− 10300.0i	− 1.39598i	0.716109	−	0.697989i	$-0.245921\pi$
$380$	0	0
$381$	11540.0i	1.55174i
$382$	0	0
$383$	−2682.00	−0.357817	−0.178908	−	0.983866i	$-0.557257\pi$
$383$	−2682.00	−0.357817	−0.178908	+	0.983866i	$0.557257\pi$
$384$	0	0
$385$	−1440.00	−0.190621
$386$	0	0
$387$	4818.00i	0.632849i
$388$	0	0
$389$	6114.00i	0.796895i	0.917191	+	0.398447i	$0.130451\pi$
$389$	6114.00i	0.796895i	−0.917191	+	0.398447i	$0.869549\pi$
$390$	0	0
$391$	−252.000	−0.0325938
$392$	0	0
$393$	3680.00	0.472345
$394$	0	0
$395$	− 5800.00i	− 0.738809i
$396$	0	0
$397$	7174.00i	0.906934i	0.891273	+	0.453467i	$0.149813\pi$
$397$	7174.00i	0.906934i	−0.891273	+	0.453467i	$0.850187\pi$
$398$	0	0
$399$	−22320.0	−2.80050
$400$	0	0
$401$	10498.0	1.30734	0.653672	−	0.756778i	$-0.273228\pi$
$401$	10498.0	1.30734	0.653672	+	0.756778i	$0.273228\pi$
$402$	0	0
$403$	1128.00i	0.139428i
$404$	0	0
$405$	− 13145.0i	− 1.61279i
$406$	0	0
$407$	−3232.00	−0.393622
$408$	0	0
$409$	1810.00	0.218823	0.109412	−	0.993997i	$-0.465103\pi$
$409$	1810.00	0.218823	0.109412	+	0.993997i	$0.465103\pi$
$410$	0	0
$411$	6700.00i	0.804104i
$412$	0	0
$413$	10152.0i	1.20956i
$414$	0	0
$415$	3210.00	0.379693
$416$	0	0
$417$	−5720.00	−0.671726
$418$	0	0
$419$	− 3396.00i	− 0.395956i	−0.980206	−	0.197978i	$-0.936563\pi$
$419$	− 3396.00i	− 0.395956i	0.980206	−	0.197978i	$-0.0634374\pi$
$420$	0	0
$421$	− 14974.0i	− 1.73346i	−0.498775	−	0.866732i	$-0.666216\pi$
$421$	− 14974.0i	− 1.73346i	0.498775	−	0.866732i	$-0.333784\pi$
$422$	0	0
$423$	2774.00	0.318857
$424$	0	0
$425$	150.000	0.0171202
$426$	0	0
$427$	− 4716.00i	− 0.534481i
$428$	0	0
$429$	− 960.000i	− 0.108040i
$430$	0	0
$431$	13540.0	1.51322	0.756611	−	0.653865i	$-0.226854\pi$
$431$	13540.0	1.51322	0.756611	+	0.653865i	$0.226854\pi$
$432$	0	0
$433$	15426.0	1.71207	0.856035	−	0.516918i	$-0.172921\pi$
$433$	15426.0	1.71207	0.856035	+	0.516918i	$0.172921\pi$
$434$	0	0
$435$	− 7100.00i	− 0.782572i
$436$	0	0
$437$	− 5208.00i	− 0.570097i
$438$	0	0
$439$	−10472.0	−1.13850	−0.569250	−	0.822165i	$-0.692766\pi$
$439$	−10472.0	−1.13850	−0.569250	+	0.822165i	$0.692766\pi$
$440$	0	0
$441$	1387.00	0.149768
$442$	0	0
$443$	− 722.000i	− 0.0774340i	−0.999250	−	0.0387170i	$-0.987673\pi$
$443$	− 722.000i	− 0.0774340i	0.999250	−	0.0387170i	$-0.0123271\pi$
$444$	0	0
$445$	− 4270.00i	− 0.454871i
$446$	0	0
$447$	−17300.0	−1.83056
$448$	0	0
$449$	−11898.0	−1.25056	−0.625280	−	0.780401i	$-0.715015\pi$
$449$	−11898.0	−1.25056	−0.625280	+	0.780401i	$0.715015\pi$
$450$	0	0
$451$	− 864.000i	− 0.0902088i
$452$	0	0
$453$	13240.0i	1.37322i
$454$	0	0
$455$	−540.000	−0.0556387
$456$	0	0
$457$	790.000	0.0808635	0.0404318	−	0.999182i	$-0.487127\pi$
$457$	790.000	0.0808635	0.0404318	+	0.999182i	$0.487127\pi$
$458$	0	0
$459$	2760.00i	0.280666i
$460$	0	0
$461$	3418.00i	0.345319i	0.984982	+	0.172660i	$0.0552360\pi$
$461$	3418.00i	0.345319i	−0.984982	+	0.172660i	$0.944764\pi$
$462$	0	0
$463$	7534.00	0.756230	0.378115	−	0.925759i	$-0.376572\pi$
$463$	7534.00	0.756230	0.378115	+	0.925759i	$0.376572\pi$
$464$	0	0
$465$	9400.00	0.937451
$466$	0	0
$467$	− 14314.0i	− 1.41836i	−0.705029	−	0.709179i	$-0.749066\pi$
$467$	− 14314.0i	− 1.41836i	0.705029	−	0.709179i	$-0.250934\pi$
$468$	0	0
$469$	9972.00i	0.981800i
$470$	0	0
$471$	29460.0	2.88205
$472$	0	0
$473$	1056.00	0.102653
$474$	0	0
$475$	3100.00i	0.299448i
$476$	0	0
$477$	− 53874.0i	− 5.17132i
$478$	0	0
$479$	7016.00	0.669247	0.334623	−	0.942352i	$-0.391391\pi$
$479$	7016.00	0.669247	0.334623	+	0.942352i	$0.391391\pi$
$480$	0	0
$481$	−1212.00	−0.114891
$482$	0	0
$483$	− 7560.00i	− 0.712199i
$484$	0	0
$485$	2390.00i	0.223761i
$486$	0	0
$487$	15190.0	1.41340	0.706699	−	0.707515i	$-0.250184\pi$
$487$	15190.0	1.41340	0.706699	+	0.707515i	$0.250184\pi$
$488$	0	0
$489$	−20980.0	−1.94018
$490$	0	0
$491$	12624.0i	1.16031i	0.814505	+	0.580156i	$0.197008\pi$
$491$	12624.0i	1.16031i	−0.814505	+	0.580156i	$0.802992\pi$
$492$	0	0
$493$	852.000i	0.0778340i
$494$	0	0
$495$	−5840.00	−0.530280
$496$	0	0
$497$	−2520.00	−0.227440
$498$	0	0
$499$	− 2492.00i	− 0.223562i	−0.993733	−	0.111781i	$-0.964345\pi$
$499$	− 2492.00i	− 0.223562i	0.993733	−	0.111781i	$-0.0356555\pi$
$500$	0	0
$501$	− 8660.00i	− 0.772256i
$502$	0	0
$503$	11714.0	1.03837	0.519186	−	0.854661i	$-0.326235\pi$
$503$	11714.0	1.03837	0.519186	+	0.854661i	$0.326235\pi$
$504$	0	0
$505$	−8970.00	−0.790415
$506$	0	0
$507$	21610.0i	1.89297i
$508$	0	0
$509$	5618.00i	0.489221i	0.969621	+	0.244610i	$0.0786601\pi$
$509$	5618.00i	0.489221i	−0.969621	+	0.244610i	$0.921340\pi$
$510$	0	0
$511$	15876.0	1.37439
$512$	0	0
$513$	−57040.0	−4.90912
$514$	0	0
$515$	− 3210.00i	− 0.274659i
$516$	0	0
$517$	− 608.000i	− 0.0517211i
$518$	0	0
$519$	−16780.0	−1.41919
$520$	0	0
$521$	−13770.0	−1.15792	−0.578958	−	0.815357i	$-0.696541\pi$
$521$	−13770.0	−1.15792	−0.578958	+	0.815357i	$0.696541\pi$
$522$	0	0
$523$	− 6986.00i	− 0.584085i	−0.956405	−	0.292042i	$-0.905665\pi$
$523$	− 6986.00i	− 0.584085i	0.956405	−	0.292042i	$-0.0943349\pi$
$524$	0	0
$525$	4500.00i	0.374088i
$526$	0	0
$527$	−1128.00	−0.0932380
$528$	0	0
$529$	−10403.0	−0.855018
$530$	0	0
$531$	41172.0i	3.36481i
$532$	0	0
$533$	− 324.000i	− 0.0263302i
$534$	0	0
$535$	4250.00	0.343446
$536$	0	0
$537$	16200.0	1.30183
$538$	0	0
$539$	− 304.000i	− 0.0242935i
$540$	0	0
$541$	− 11958.0i	− 0.950304i	−0.879904	−	0.475152i	$-0.842393\pi$
$541$	− 11958.0i	− 0.950304i	0.879904	−	0.475152i	$-0.157607\pi$
$542$	0	0
$543$	−25100.0	−1.98369
$544$	0	0
$545$	−3330.00	−0.261728
$546$	0	0
$547$	− 4194.00i	− 0.327829i	−0.986475	−	0.163915i	$-0.947588\pi$
$547$	− 4194.00i	− 0.327829i	0.986475	−	0.163915i	$-0.0524121\pi$
$548$	0	0
$549$	− 19126.0i	− 1.48684i
$550$	0	0
$551$	−17608.0	−1.36139
$552$	0	0
$553$	−20880.0	−1.60562
$554$	0	0
$555$	10100.0i	0.772470i
$556$	0	0
$557$	5382.00i	0.409412i	0.978823	+	0.204706i	$0.0656239\pi$
$557$	5382.00i	0.409412i	−0.978823	+	0.204706i	$0.934376\pi$
$558$	0	0
$559$	396.000	0.0299625
$560$	0	0
$561$	960.000	0.0722482
$562$	0	0
$563$	15418.0i	1.15416i	0.816688	+	0.577079i	$0.195808\pi$
$563$	15418.0i	1.15416i	−0.816688	+	0.577079i	$0.804192\pi$
$564$	0	0
$565$	7230.00i	0.538351i
$566$	0	0
$567$	−47322.0	−3.50500
$568$	0	0
$569$	5778.00	0.425705	0.212853	−	0.977084i	$-0.431725\pi$
$569$	5778.00	0.425705	0.212853	+	0.977084i	$0.431725\pi$
$570$	0	0
$571$	− 6024.00i	− 0.441500i	−0.975330	−	0.220750i	$-0.929149\pi$
$571$	− 6024.00i	− 0.441500i	0.975330	−	0.220750i	$-0.0708505\pi$
$572$	0	0
$573$	3720.00i	0.271213i
$574$	0	0
$575$	−1050.00	−0.0761531
$576$	0	0
$577$	554.000	0.0399711	0.0199855	−	0.999800i	$-0.493638\pi$
$577$	554.000	0.0399711	0.0199855	+	0.999800i	$0.493638\pi$
$578$	0	0
$579$	29380.0i	2.10879i
$580$	0	0
$581$	− 11556.0i	− 0.825170i
$582$	0	0
$583$	−11808.0	−0.838829
$584$	0	0
$585$	−2190.00	−0.154778
$586$	0	0
$587$	2386.00i	0.167770i	0.996475	+	0.0838848i	$0.0267328\pi$
$587$	2386.00i	0.167770i	−0.996475	+	0.0838848i	$0.973267\pi$
$588$	0	0
$589$	− 23312.0i	− 1.63082i
$590$	0	0
$591$	−22340.0	−1.55490
$592$	0	0
$593$	−846.000	−0.0585853	−0.0292926	−	0.999571i	$-0.509325\pi$
$593$	−846.000	−0.0585853	−0.0292926	+	0.999571i	$0.509325\pi$
$594$	0	0
$595$	− 540.000i	− 0.0372065i
$596$	0	0
$597$	− 30480.0i	− 2.08955i
$598$	0	0
$599$	22304.0	1.52140	0.760698	−	0.649105i	$-0.224857\pi$
$599$	22304.0	1.52140	0.760698	+	0.649105i	$0.224857\pi$
$600$	0	0
$601$	−5510.00	−0.373973	−0.186986	−	0.982363i	$-0.559872\pi$
$601$	−5510.00	−0.373973	−0.186986	+	0.982363i	$0.559872\pi$
$602$	0	0
$603$	40442.0i	2.73122i
$604$	0	0
$605$	− 5375.00i	− 0.361198i
$606$	0	0
$607$	8234.00	0.550589	0.275295	−	0.961360i	$-0.411225\pi$
$607$	8234.00	0.550589	0.275295	+	0.961360i	$0.411225\pi$
$608$	0	0
$609$	−25560.0	−1.70073
$610$	0	0
$611$	− 228.000i	− 0.0150964i
$612$	0	0
$613$	− 1046.00i	− 0.0689193i	−0.999406	−	0.0344597i	$-0.989029\pi$
$613$	− 1046.00i	− 0.0689193i	0.999406	−	0.0344597i	$-0.0109710\pi$
$614$	0	0
$615$	−2700.00	−0.177032
$616$	0	0
$617$	3862.00	0.251991	0.125995	−	0.992031i	$-0.459788\pi$
$617$	3862.00	0.251991	0.125995	+	0.992031i	$0.459788\pi$
$618$	0	0
$619$	− 13964.0i	− 0.906721i	−0.891327	−	0.453361i	$-0.850225\pi$
$619$	− 13964.0i	− 0.906721i	0.891327	−	0.453361i	$-0.149775\pi$
$620$	0	0
$621$	− 19320.0i	− 1.24845i
$622$	0	0
$623$	−15372.0	−0.988549
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	19840.0i	1.26369i
$628$	0	0
$629$	− 1212.00i	− 0.0768293i
$630$	0	0
$631$	−14884.0	−0.939022	−0.469511	−	0.882927i	$-0.655570\pi$
$631$	−14884.0	−0.939022	−0.469511	+	0.882927i	$0.655570\pi$
$632$	0	0
$633$	−48960.0	−3.07423
$634$	0	0
$635$	− 5770.00i	− 0.360591i
$636$	0	0
$637$	− 114.000i	− 0.00709081i
$638$	0	0
$639$	−10220.0	−0.632703
$640$	0	0
$641$	17838.0	1.09916	0.549578	−	0.835443i	$-0.314789\pi$
$641$	17838.0	1.09916	0.549578	+	0.835443i	$0.314789\pi$
$642$	0	0
$643$	− 7814.00i	− 0.479244i	−0.970866	−	0.239622i	$-0.922976\pi$
$643$	− 7814.00i	− 0.479244i	0.970866	−	0.239622i	$-0.0770236\pi$
$644$	0	0
$645$	− 3300.00i	− 0.201453i
$646$	0	0
$647$	774.000	0.0470310	0.0235155	−	0.999723i	$-0.492514\pi$
$647$	774.000	0.0470310	0.0235155	+	0.999723i	$0.492514\pi$
$648$	0	0
$649$	9024.00	0.545798
$650$	0	0
$651$	− 33840.0i	− 2.03732i
$652$	0	0
$653$	23422.0i	1.40364i	0.712357	+	0.701818i	$0.247628\pi$
$653$	23422.0i	1.40364i	−0.712357	+	0.701818i	$0.752372\pi$
$654$	0	0
$655$	−1840.00	−0.109763
$656$	0	0
$657$	64386.0	3.82334
$658$	0	0
$659$	− 13508.0i	− 0.798478i	−0.916847	−	0.399239i	$-0.869274\pi$
$659$	− 13508.0i	− 0.798478i	0.916847	−	0.399239i	$-0.130726\pi$
$660$	0	0
$661$	− 6222.00i	− 0.366124i	−0.983101	−	0.183062i	$-0.941399\pi$
$661$	− 6222.00i	− 0.366124i	0.983101	−	0.183062i	$-0.0586009\pi$
$662$	0	0
$663$	360.000	0.0210878
$664$	0	0
$665$	11160.0	0.650776
$666$	0	0
$667$	− 5964.00i	− 0.346217i
$668$	0	0
$669$	53020.0i	3.06408i
$670$	0	0
$671$	−4192.00	−0.241178
$672$	0	0
$673$	−15566.0	−0.891568	−0.445784	−	0.895141i	$-0.647075\pi$
$673$	−15566.0	−0.891568	−0.445784	+	0.895141i	$0.647075\pi$
$674$	0	0
$675$	11500.0i	0.655756i
$676$	0	0
$677$	2234.00i	0.126824i	0.997987	+	0.0634118i	$0.0201982\pi$
$677$	2234.00i	0.126824i	−0.997987	+	0.0634118i	$0.979802\pi$
$678$	0	0
$679$	8604.00	0.486290
$680$	0	0
$681$	37780.0	2.12589
$682$	0	0
$683$	− 13282.0i	− 0.744102i	−0.928212	−	0.372051i	$-0.878655\pi$
$683$	− 13282.0i	− 0.744102i	0.928212	−	0.372051i	$-0.121345\pi$
$684$	0	0
$685$	− 3350.00i	− 0.186857i
$686$	0	0
$687$	30340.0	1.68492
$688$	0	0
$689$	−4428.00	−0.244838
$690$	0	0
$691$	− 27416.0i	− 1.50934i	−0.656105	−	0.754670i	$-0.727797\pi$
$691$	− 27416.0i	− 1.50934i	0.656105	−	0.754670i	$-0.272203\pi$
$692$	0	0
$693$	21024.0i	1.15243i
$694$	0	0
$695$	2860.00	0.156095
$696$	0	0
$697$	324.000	0.0176074
$698$	0	0
$699$	34780.0i	1.88197i
$700$	0	0
$701$	− 25626.0i	− 1.38071i	−0.723469	−	0.690357i	$-0.757453\pi$
$701$	− 25626.0i	− 1.38071i	0.723469	−	0.690357i	$-0.242547\pi$
$702$	0	0
$703$	25048.0	1.34382
$704$	0	0
$705$	−1900.00	−0.101501
$706$	0	0
$707$	32292.0i	1.71777i
$708$	0	0
$709$	11702.0i	0.619856i	0.950760	+	0.309928i	$0.100305\pi$
$709$	11702.0i	0.619856i	−0.950760	+	0.309928i	$0.899695\pi$
$710$	0	0
$711$	−84680.0	−4.46659
$712$	0	0
$713$	7896.00	0.414737
$714$	0	0
$715$	480.000i	0.0251063i
$716$	0	0
$717$	− 15600.0i	− 0.812542i
$718$	0	0
$719$	28008.0	1.45274	0.726371	−	0.687302i	$-0.241205\pi$
$719$	28008.0	1.45274	0.726371	+	0.687302i	$0.241205\pi$
$720$	0	0
$721$	−11556.0	−0.596904
$722$	0	0
$723$	− 32180.0i	− 1.65531i
$724$	0	0
$725$	3550.00i	0.181853i
$726$	0	0
$727$	−7682.00	−0.391898	−0.195949	−	0.980614i	$-0.562779\pi$
$727$	−7682.00	−0.391898	−0.195949	+	0.980614i	$0.562779\pi$
$728$	0	0
$729$	−67717.0	−3.44038
$730$	0	0
$731$	396.000i	0.0200364i
$732$	0	0
$733$	14270.0i	0.719065i	0.933133	+	0.359532i	$0.117064\pi$
$733$	14270.0i	0.719065i	−0.933133	+	0.359532i	$0.882936\pi$
$734$	0	0
$735$	−950.000	−0.0476752
$736$	0	0
$737$	8864.00	0.443025
$738$	0	0
$739$	29324.0i	1.45968i	0.683620	+	0.729838i	$0.260405\pi$
$739$	29324.0i	1.45968i	−0.683620	+	0.729838i	$0.739595\pi$
$740$	0	0
$741$	7440.00i	0.368846i
$742$	0	0
$743$	29258.0	1.44465	0.722323	−	0.691556i	$-0.243074\pi$
$743$	29258.0	1.44465	0.722323	+	0.691556i	$0.243074\pi$
$744$	0	0
$745$	8650.00	0.425385
$746$	0	0
$747$	− 46866.0i	− 2.29550i
$748$	0	0
$749$	− 15300.0i	− 0.746395i
$750$	0	0
$751$	−19076.0	−0.926888	−0.463444	−	0.886126i	$-0.653387\pi$
$751$	−19076.0	−0.926888	−0.463444	+	0.886126i	$0.653387\pi$
$752$	0	0
$753$	6880.00	0.332963
$754$	0	0
$755$	− 6620.00i	− 0.319108i
$756$	0	0
$757$	− 22670.0i	− 1.08845i	−0.838940	−	0.544224i	$-0.816824\pi$
$757$	− 22670.0i	− 1.08845i	0.838940	−	0.544224i	$-0.183176\pi$
$758$	0	0
$759$	−6720.00	−0.321371
$760$	0	0
$761$	23206.0	1.10541	0.552705	−	0.833377i	$-0.313596\pi$
$761$	23206.0	1.10541	0.552705	+	0.833377i	$0.313596\pi$
$762$	0	0
$763$	11988.0i	0.568800i
$764$	0	0
$765$	− 2190.00i	− 0.103503i
$766$	0	0
$767$	3384.00	0.159308
$768$	0	0
$769$	−1854.00	−0.0869401	−0.0434701	−	0.999055i	$-0.513841\pi$
$769$	−1854.00	−0.0869401	−0.0434701	+	0.999055i	$0.513841\pi$
$770$	0	0
$771$	21700.0i	1.01363i
$772$	0	0
$773$	6474.00i	0.301234i	0.988592	+	0.150617i	$0.0481260\pi$
$773$	6474.00i	0.301234i	−0.988592	+	0.150617i	$0.951874\pi$
$774$	0	0
$775$	−4700.00	−0.217844
$776$	0	0
$777$	36360.0	1.67877
$778$	0	0
$779$	6696.00i	0.307971i
$780$	0	0
$781$	2240.00i	0.102629i
$782$	0	0
$783$	−65320.0	−2.98129
$784$	0	0
$785$	−14730.0	−0.669728
$786$	0	0
$787$	− 20354.0i	− 0.921908i	−0.887424	−	0.460954i	$-0.847507\pi$
$787$	− 20354.0i	− 0.921908i	0.887424	−	0.460954i	$-0.152493\pi$
$788$	0	0
$789$	22740.0i	1.02607i
$790$	0	0
$791$	26028.0	1.16997
$792$	0	0
$793$	−1572.00	−0.0703952
$794$	0	0
$795$	36900.0i	1.64617i
$796$	0	0
$797$	1886.00i	0.0838213i	0.999121	+	0.0419106i	$0.0133445\pi$
$797$	1886.00i	0.0838213i	−0.999121	+	0.0419106i	$0.986656\pi$
$798$	0	0
$799$	228.000	0.0100952
$800$	0	0
$801$	−62342.0	−2.75000
$802$	0	0
$803$	− 14112.0i	− 0.620176i
$804$	0	0
$805$	3780.00i	0.165500i
$806$	0	0
$807$	71460.0	3.11711
$808$	0	0
$809$	9462.00	0.411207	0.205603	−	0.978635i	$-0.434084\pi$
$809$	9462.00	0.411207	0.205603	+	0.978635i	$0.434084\pi$
$810$	0	0
$811$	− 24512.0i	− 1.06132i	−0.847584	−	0.530661i	$-0.821944\pi$
$811$	− 24512.0i	− 1.06132i	0.847584	−	0.530661i	$-0.178056\pi$
$812$	0	0
$813$	− 26040.0i	− 1.12332i
$814$	0	0
$815$	10490.0	0.450857
$816$	0	0
$817$	−8184.00	−0.350455
$818$	0	0
$819$	7884.00i	0.336373i
$820$	0	0
$821$	36242.0i	1.54063i	0.637666	+	0.770313i	$0.279900\pi$
$821$	36242.0i	1.54063i	−0.637666	+	0.770313i	$0.720100\pi$
$822$	0	0
$823$	−17718.0	−0.750438	−0.375219	−	0.926936i	$-0.622433\pi$
$823$	−17718.0	−0.750438	−0.375219	+	0.926936i	$0.622433\pi$
$824$	0	0
$825$	4000.00	0.168803
$826$	0	0
$827$	− 6726.00i	− 0.282812i	−0.989952	−	0.141406i	$-0.954838\pi$
$827$	− 6726.00i	− 0.282812i	0.989952	−	0.141406i	$-0.0451624\pi$
$828$	0	0
$829$	− 41722.0i	− 1.74797i	−0.485955	−	0.873984i	$-0.661528\pi$
$829$	− 41722.0i	− 1.74797i	0.485955	−	0.873984i	$-0.338472\pi$
$830$	0	0
$831$	51500.0	2.14984
$832$	0	0
$833$	114.000	0.00474174
$834$	0	0
$835$	4330.00i	0.179456i
$836$	0	0
$837$	− 86480.0i	− 3.57131i
$838$	0	0
$839$	16720.0	0.688008	0.344004	−	0.938968i	$-0.388217\pi$
$839$	16720.0	0.688008	0.344004	+	0.938968i	$0.388217\pi$
$840$	0	0
$841$	4225.00	0.173234
$842$	0	0
$843$	− 52700.0i	− 2.15313i
$844$	0	0
$845$	− 10805.0i	− 0.439886i
$846$	0	0
$847$	−19350.0	−0.784975
$848$	0	0
$849$	34340.0	1.38816
$850$	0	0
$851$	8484.00i	0.341748i
$852$	0	0
$853$	− 33286.0i	− 1.33610i	−0.744118	−	0.668049i	$-0.767130\pi$
$853$	− 33286.0i	− 1.33610i	0.744118	−	0.668049i	$-0.232870\pi$
$854$	0	0
$855$	45260.0	1.81036
$856$	0	0
$857$	−38978.0	−1.55363	−0.776816	−	0.629727i	$-0.783167\pi$
$857$	−38978.0	−1.55363	−0.776816	+	0.629727i	$0.783167\pi$
$858$	0	0
$859$	1916.00i	0.0761037i	0.999276	+	0.0380518i	$0.0121152\pi$
$859$	1916.00i	0.0761037i	−0.999276	+	0.0380518i	$0.987885\pi$
$860$	0	0
$861$	9720.00i	0.384735i
$862$	0	0
$863$	2374.00	0.0936407	0.0468203	−	0.998903i	$-0.485091\pi$
$863$	2374.00	0.0936407	0.0468203	+	0.998903i	$0.485091\pi$
$864$	0	0
$865$	8390.00	0.329790
$866$	0	0
$867$	− 48770.0i	− 1.91040i
$868$	0	0
$869$	18560.0i	0.724517i
$870$	0	0
$871$	3324.00	0.129310
$872$	0	0
$873$	34894.0	1.35279
$874$	0	0
$875$	− 2250.00i	− 0.0869302i
$876$	0	0
$877$	− 32722.0i	− 1.25991i	−0.776631	−	0.629956i	$-0.783073\pi$
$877$	− 32722.0i	− 1.25991i	0.776631	−	0.629956i	$-0.216927\pi$
$878$	0	0
$879$	98780.0	3.79041
$880$	0	0
$881$	5390.00	0.206122	0.103061	−	0.994675i	$-0.467136\pi$
$881$	5390.00	0.206122	0.103061	+	0.994675i	$0.467136\pi$
$882$	0	0
$883$	− 43238.0i	− 1.64788i	−0.566680	−	0.823938i	$-0.691772\pi$
$883$	− 43238.0i	− 1.64788i	0.566680	−	0.823938i	$-0.308228\pi$
$884$	0	0
$885$	− 28200.0i	− 1.07111i
$886$	0	0
$887$	−11010.0	−0.416775	−0.208388	−	0.978046i	$-0.566822\pi$
$887$	−11010.0	−0.416775	−0.208388	+	0.978046i	$0.566822\pi$
$888$	0	0
$889$	−20772.0	−0.783656
$890$	0	0
$891$	42064.0i	1.58159i
$892$	0	0
$893$	4712.00i	0.176575i
$894$	0	0
$895$	−8100.00	−0.302517
$896$	0	0
$897$	−2520.00	−0.0938020
$898$	0	0
$899$	− 26696.0i	− 0.990391i
$900$	0	0
$901$	− 4428.00i	− 0.163727i
$902$	0	0
$903$	−11880.0	−0.437809
$904$	0	0
$905$	12550.0	0.460968
$906$	0	0
$907$	74.0000i	0.00270907i	0.999999	+	0.00135454i	$0.000431163\pi$
$907$	74.0000i	0.00270907i	−0.999999	+	0.00135454i	$0.999569\pi$
$908$	0	0
$909$	130962.i	4.77859i
$910$	0	0
$911$	−17460.0	−0.634990	−0.317495	−	0.948260i	$-0.602842\pi$
$911$	−17460.0	−0.634990	−0.317495	+	0.948260i	$0.602842\pi$
$912$	0	0
$913$	−10272.0	−0.372348
$914$	0	0
$915$	13100.0i	0.473303i
$916$	0	0
$917$	6624.00i	0.238543i
$918$	0	0
$919$	17072.0	0.612789	0.306395	−	0.951905i	$-0.400877\pi$
$919$	17072.0	0.612789	0.306395	+	0.951905i	$0.400877\pi$
$920$	0	0
$921$	−80540.0	−2.88152
$922$	0	0
$923$	840.000i	0.0299555i
$924$	0	0
$925$	− 5050.00i	− 0.179506i
$926$	0	0
$927$	−46866.0	−1.66050
$928$	0	0
$929$	−14826.0	−0.523601	−0.261800	−	0.965122i	$-0.584316\pi$
$929$	−14826.0	−0.523601	−0.261800	+	0.965122i	$0.584316\pi$
$930$	0	0
$931$	2356.00i	0.0829375i
$932$	0	0
$933$	54920.0i	1.92712i
$934$	0	0
$935$	−480.000	−0.0167890
$936$	0	0
$937$	−3354.00	−0.116937	−0.0584687	−	0.998289i	$-0.518622\pi$
$937$	−3354.00	−0.116937	−0.0584687	+	0.998289i	$0.518622\pi$
$938$	0	0
$939$	4220.00i	0.146661i
$940$	0	0
$941$	15434.0i	0.534680i	0.963602	+	0.267340i	$0.0861447\pi$
$941$	15434.0i	0.534680i	−0.963602	+	0.267340i	$0.913855\pi$
$942$	0	0
$943$	−2268.00	−0.0783205
$944$	0	0
$945$	41400.0	1.42512
$946$	0	0
$947$	− 9306.00i	− 0.319329i	−0.987171	−	0.159664i	$-0.948959\pi$
$947$	− 9306.00i	− 0.319329i	0.987171	−	0.159664i	$-0.0510412\pi$
$948$	0	0
$949$	− 5292.00i	− 0.181017i
$950$	0	0
$951$	61940.0	2.11203
$952$	0	0
$953$	−12202.0	−0.414755	−0.207378	−	0.978261i	$-0.566493\pi$
$953$	−12202.0	−0.414755	−0.207378	+	0.978261i	$0.566493\pi$
$954$	0	0
$955$	− 1860.00i	− 0.0630243i
$956$	0	0
$957$	22720.0i	0.767433i
$958$	0	0
$959$	−12060.0	−0.406087
$960$	0	0
$961$	5553.00	0.186399
$962$	0	0
$963$	− 62050.0i	− 2.07636i
$964$	0	0
$965$	− 14690.0i	− 0.490039i
$966$	0	0
$967$	17478.0	0.581235	0.290618	−	0.956839i	$-0.406139\pi$
$967$	17478.0	0.581235	0.290618	+	0.956839i	$0.406139\pi$
$968$	0	0
$969$	−7440.00	−0.246653
$970$	0	0
$971$	− 10920.0i	− 0.360906i	−0.983584	−	0.180453i	$-0.942244\pi$
$971$	− 10920.0i	− 0.360906i	0.983584	−	0.180453i	$-0.0577563\pi$
$972$	0	0
$973$	− 10296.0i	− 0.339234i
$974$	0	0
$975$	1500.00	0.0492702
$976$	0	0
$977$	10834.0	0.354770	0.177385	−	0.984142i	$-0.443236\pi$
$977$	10834.0	0.354770	0.177385	+	0.984142i	$0.443236\pi$
$978$	0	0
$979$	13664.0i	0.446071i
$980$	0	0
$981$	48618.0i	1.58232i
$982$	0	0
$983$	−36862.0	−1.19605	−0.598024	−	0.801478i	$-0.704047\pi$
$983$	−36862.0	−1.19605	−0.598024	+	0.801478i	$0.704047\pi$
$984$	0	0
$985$	11170.0	0.361326
$986$	0	0
$987$	6840.00i	0.220587i
$988$	0	0
$989$	− 2772.00i	− 0.0891248i
$990$	0	0
$991$	−5380.00	−0.172453	−0.0862267	−	0.996276i	$-0.527481\pi$
$991$	−5380.00	−0.172453	−0.0862267	+	0.996276i	$0.527481\pi$
$992$	0	0
$993$	76880.0	2.45691
$994$	0	0
$995$	15240.0i	0.485568i
$996$	0	0
$997$	31266.0i	0.993184i	0.867984	+	0.496592i	$0.165415\pi$
$997$	31266.0i	0.993184i	−0.867984	+	0.496592i	$0.834585\pi$
$998$	0	0
$999$	92920.0	2.94280

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.4.d.b.641.2		2
4.3	odd	2		1280.4.d.o.641.1		2
8.3	odd	2		1280.4.d.o.641.2		2
8.5	even	2	inner	1280.4.d.b.641.1		2
16.3	odd	4		40.4.a.c.1.1	✓	1
16.5	even	4		320.4.a.n.1.1		1
16.11	odd	4		320.4.a.a.1.1		1
16.13	even	4		80.4.a.a.1.1		1
48.29	odd	4		720.4.a.ba.1.1		1
48.35	even	4		360.4.a.i.1.1		1
80.3	even	4		200.4.c.a.49.2		2
80.13	odd	4		400.4.c.a.49.1		2
80.19	odd	4		200.4.a.a.1.1		1
80.29	even	4		400.4.a.u.1.1		1
80.59	odd	4		1600.4.a.ca.1.1		1
80.67	even	4		200.4.c.a.49.1		2
80.69	even	4		1600.4.a.a.1.1		1
80.77	odd	4		400.4.c.a.49.2		2
112.83	even	4		1960.4.a.a.1.1		1
240.83	odd	4		1800.4.f.n.649.2		2
240.179	even	4		1800.4.a.bd.1.1		1
240.227	odd	4		1800.4.f.n.649.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.a.c.1.1	✓	1	16.3	odd	4
80.4.a.a.1.1		1	16.13	even	4
200.4.a.a.1.1		1	80.19	odd	4
200.4.c.a.49.1		2	80.67	even	4
200.4.c.a.49.2		2	80.3	even	4
320.4.a.a.1.1		1	16.11	odd	4
320.4.a.n.1.1		1	16.5	even	4
360.4.a.i.1.1		1	48.35	even	4
400.4.a.u.1.1		1	80.29	even	4
400.4.c.a.49.1		2	80.13	odd	4
400.4.c.a.49.2		2	80.77	odd	4
720.4.a.ba.1.1		1	48.29	odd	4
1280.4.d.b.641.1		2	8.5	even	2	inner
1280.4.d.b.641.2		2	1.1	even	1	trivial
1280.4.d.o.641.1		2	4.3	odd	2
1280.4.d.o.641.2		2	8.3	odd	2
1600.4.a.a.1.1		1	80.69	even	4
1600.4.a.ca.1.1		1	80.59	odd	4
1800.4.a.bd.1.1		1	240.179	even	4
1800.4.f.n.649.1		2	240.227	odd	4
1800.4.f.n.649.2		2	240.83	odd	4
1960.4.a.a.1.1		1	112.83	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1280.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+10.0000i q^{3} -5.00000i q^{5} -18.0000 q^{7} -73.0000 q^{9} +O(q^{10})\)	\(q+10.0000i q^{3} -5.00000i q^{5} -18.0000 q^{7} -73.0000 q^{9} +16.0000i q^{11} +6.00000i q^{13} +50.0000 q^{15} -6.00000 q^{17} -124.000i q^{19} -180.000i q^{21} +42.0000 q^{23} -25.0000 q^{25} -460.000i q^{27} -142.000i q^{29} +188.000 q^{31} -160.000 q^{33} +90.0000i q^{35} +202.000i q^{37} -60.0000 q^{39} -54.0000 q^{41} -66.0000i q^{43} +365.000i q^{45} -38.0000 q^{47} -19.0000 q^{49} -60.0000i q^{51} +738.000i q^{53} +80.0000 q^{55} +1240.00 q^{57} -564.000i q^{59} +262.000i q^{61} +1314.00 q^{63} +30.0000 q^{65} -554.000i q^{67} +420.000i q^{69} +140.000 q^{71} -882.000 q^{73} -250.000i q^{75} -288.000i q^{77} +1160.00 q^{79} +2629.00 q^{81} +642.000i q^{83} +30.0000i q^{85} +1420.00 q^{87} +854.000 q^{89} -108.000i q^{91} +1880.00i q^{93} -620.000 q^{95} -478.000 q^{97} -1168.00i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 36 q^{7} - 146 q^{9}+O(q^{10}) \) 2 * q - 36 * q^7 - 146 * q^9	\( 2 q - 36 q^{7} - 146 q^{9} + 100 q^{15} - 12 q^{17} + 84 q^{23} - 50 q^{25} + 376 q^{31} - 320 q^{33} - 120 q^{39} - 108 q^{41} - 76 q^{47} - 38 q^{49} + 160 q^{55} + 2480 q^{57} + 2628 q^{63} + 60 q^{65} + 280 q^{71} - 1764 q^{73} + 2320 q^{79} + 5258 q^{81} + 2840 q^{87} + 1708 q^{89} - 1240 q^{95} - 956 q^{97}+O(q^{100}) \) 2 * q - 36 * q^7 - 146 * q^9 + 100 * q^15 - 12 * q^17 + 84 * q^23 - 50 * q^25 + 376 * q^31 - 320 * q^33 - 120 * q^39 - 108 * q^41 - 76 * q^47 - 38 * q^49 + 160 * q^55 + 2480 * q^57 + 2628 * q^63 + 60 * q^65 + 280 * q^71 - 1764 * q^73 + 2320 * q^79 + 5258 * q^81 + 2840 * q^87 + 1708 * q^89 - 1240 * q^95 - 956 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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