LMFDB - Embedded newform 400.4.c.a.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(49,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	400.c (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:	$23.6007640023$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.4.c.a.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.0000i q^{3} -18.0000i q^{7} -73.0000 q^{9} +O(q^{10})$	$q-10.0000i q^{3} -18.0000i q^{7} -73.0000 q^{9} +16.0000 q^{11} -6.00000i q^{13} +6.00000i q^{17} -124.000 q^{19} -180.000 q^{21} -42.0000i q^{23} +460.000i q^{27} -142.000 q^{29} +188.000 q^{31} -160.000i q^{33} -202.000i q^{37} -60.0000 q^{39} +54.0000 q^{41} -66.0000i q^{43} +38.0000i q^{47} +19.0000 q^{49} +60.0000 q^{51} +738.000i q^{53} +1240.00i q^{57} +564.000 q^{59} -262.000 q^{61} +1314.00i q^{63} -554.000i q^{67} -420.000 q^{69} -140.000 q^{71} +882.000i q^{73} -288.000i q^{77} -1160.00 q^{79} +2629.00 q^{81} -642.000i q^{83} +1420.00i q^{87} +854.000 q^{89} -108.000 q^{91} -1880.00i q^{93} +478.000i q^{97} -1168.00 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 146 q^{9}+O(q^{10}) $ 2 * q - 146 * q^9	$ 2 q - 146 q^{9} + 32 q^{11} - 248 q^{19} - 360 q^{21} - 284 q^{29} + 376 q^{31} - 120 q^{39} + 108 q^{41} + 38 q^{49} + 120 q^{51} + 1128 q^{59} - 524 q^{61} - 840 q^{69} - 280 q^{71} - 2320 q^{79} + 5258 q^{81} + 1708 q^{89} - 216 q^{91} - 2336 q^{99}+O(q^{100}) $ 2 * q - 146 * q^9 + 32 * q^11 - 248 * q^19 - 360 * q^21 - 284 * q^29 + 376 * q^31 - 120 * q^39 + 108 * q^41 + 38 * q^49 + 120 * q^51 + 1128 * q^59 - 524 * q^61 - 840 * q^69 - 280 * q^71 - 2320 * q^79 + 5258 * q^81 + 1708 * q^89 - 216 * q^91 - 2336 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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	$-0.587740\pi$
$3$	− 10.0000i	− 1.92450i	0.272166	−	0.962250i	$-0.412260\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 18.0000i	− 0.971909i	−0.873984	−	0.485954i	$-0.838472\pi$
$7$	− 18.0000i	− 0.971909i	0.873984	−	0.485954i	$-0.161528\pi$
$8$	0	0
$9$	−73.0000	−2.70370
$10$	0	0
$11$	16.0000	0.438562	0.219281	−	0.975662i	$-0.429629\pi$
$11$	16.0000	0.438562	0.219281	+	0.975662i	$0.429629\pi$
$12$	0	0
$13$	− 6.00000i	− 0.128008i	−0.997950	−	0.0640039i	$-0.979613\pi$
$13$	− 6.00000i	− 0.128008i	0.997950	−	0.0640039i	$-0.0203870\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000i	0.0856008i	0.999084	+	0.0428004i	$0.0136280\pi$
$17$	6.00000i	0.0856008i	−0.999084	+	0.0428004i	$0.986372\pi$
$18$	0	0
$19$	−124.000	−1.49724	−0.748620	−	0.663000i	$-0.769283\pi$
$19$	−124.000	−1.49724	−0.748620	+	0.663000i	$0.769283\pi$
$20$	0	0
$21$	−180.000	−1.87044
$22$	0	0
$23$	− 42.0000i	− 0.380765i	−0.981710	−	0.190383i	$-0.939027\pi$
$23$	− 42.0000i	− 0.380765i	0.981710	−	0.190383i	$-0.0609729\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	460.000i	3.27878i
$28$	0	0
$29$	−142.000	−0.909267	−0.454633	−	0.890679i	$-0.650230\pi$
$29$	−142.000	−0.909267	−0.454633	+	0.890679i	$0.650230\pi$
$30$	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
$33$	− 160.000i	− 0.844013i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 202.000i	− 0.897530i	−0.893650	−	0.448765i	$-0.851864\pi$
$37$	− 202.000i	− 0.897530i	0.893650	−	0.448765i	$-0.148136\pi$
$38$	0	0
$39$	−60.0000	−0.246351
$40$	0	0
$41$	54.0000	0.205692	0.102846	−	0.994697i	$-0.467205\pi$
$41$	54.0000	0.205692	0.102846	+	0.994697i	$0.467205\pi$
$42$	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
$45$	0	0
$46$	0	0
$47$	38.0000i	0.117933i	0.998260	+	0.0589667i	$0.0187806\pi$
$47$	38.0000i	0.117933i	−0.998260	+	0.0589667i	$0.981219\pi$
$48$	0	0
$49$	19.0000	0.0553936
$50$	0	0
$51$	60.0000	0.164739
$52$	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
$55$	0	0
$56$	0	0
$57$	1240.00i	2.88144i
$58$	0	0
$59$	564.000	1.24452	0.622259	−	0.782812i	$-0.286215\pi$
$59$	564.000	1.24452	0.622259	+	0.782812i	$0.286215\pi$
$60$	0	0
$61$	−262.000	−0.549929	−0.274964	−	0.961454i	$-0.588666\pi$
$61$	−262.000	−0.549929	−0.274964	+	0.961454i	$0.588666\pi$
$62$	0	0
$63$	1314.00i	2.62775i
$64$	0	0
$65$	0	0
$66$	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
$69$	−420.000	−0.732783
$70$	0	0
$71$	−140.000	−0.234013	−0.117007	−	0.993131i	$-0.537330\pi$
$71$	−140.000	−0.234013	−0.117007	+	0.993131i	$0.537330\pi$
$72$	0	0
$73$	882.000i	1.41411i	0.707157	+	0.707057i	$0.249977\pi$
$73$	882.000i	1.41411i	−0.707157	+	0.707057i	$0.750023\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 288.000i	− 0.426242i
$78$	0	0
$79$	−1160.00	−1.65203	−0.826014	−	0.563650i	$-0.809397\pi$
$79$	−1160.00	−1.65203	−0.826014	+	0.563650i	$0.809397\pi$
$80$	0	0
$81$	2629.00	3.60631
$82$	0	0
$83$	− 642.000i	− 0.849020i	−0.905423	−	0.424510i	$-0.860446\pi$
$83$	− 642.000i	− 0.849020i	0.905423	−	0.424510i	$-0.139554\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1420.00i	1.74988i
$88$	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
$91$	−108.000	−0.124412
$92$	0	0
$93$	− 1880.00i	− 2.09620i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	478.000i	0.500346i	0.968201	+	0.250173i	$0.0804875\pi$
$97$	478.000i	0.500346i	−0.968201	+	0.250173i	$0.919513\pi$
$98$	0	0
$99$	−1168.00	−1.18574
$100$	0	0
$101$	−1794.00	−1.76742	−0.883711	−	0.468033i	$-0.844963\pi$
$101$	−1794.00	−1.76742	−0.883711	+	0.468033i	$0.844963\pi$
$102$	0	0
$103$	− 642.000i	− 0.614157i	−0.951684	−	0.307078i	$-0.900649\pi$
$103$	− 642.000i	− 0.614157i	0.951684	−	0.307078i	$-0.0993514\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 850.000i	− 0.767968i	−0.923340	−	0.383984i	$-0.874552\pi$
$107$	− 850.000i	− 0.767968i	0.923340	−	0.383984i	$-0.125448\pi$
$108$	0	0
$109$	−666.000	−0.585241	−0.292620	−	0.956229i	$-0.594527\pi$
$109$	−666.000	−0.585241	−0.292620	+	0.956229i	$0.594527\pi$
$110$	0	0
$111$	−2020.00	−1.72730
$112$	0	0
$113$	− 1446.00i	− 1.20379i	−0.798575	−	0.601895i	$-0.794413\pi$
$113$	− 1446.00i	− 1.20379i	0.798575	−	0.601895i	$-0.205587\pi$
$114$	0	0
$115$	0	0
$116$	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
$125$	0	0
$126$	0	0
$127$	− 1154.00i	− 0.806307i	−0.915132	−	0.403153i	$-0.867914\pi$
$127$	− 1154.00i	− 0.806307i	0.915132	−	0.403153i	$-0.132086\pi$
$128$	0	0
$129$	−660.000	−0.450463
$130$	0	0
$131$	368.000	0.245437	0.122719	−	0.992441i	$-0.460839\pi$
$131$	368.000	0.245437	0.122719	+	0.992441i	$0.460839\pi$
$132$	0	0
$133$	2232.00i	1.45518i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	670.000i	0.417825i	0.977934	+	0.208912i	$0.0669923\pi$
$137$	670.000i	0.417825i	−0.977934	+	0.208912i	$0.933008\pi$
$138$	0	0
$139$	−572.000	−0.349039	−0.174519	−	0.984654i	$-0.555837\pi$
$139$	−572.000	−0.349039	−0.174519	+	0.984654i	$0.555837\pi$
$140$	0	0
$141$	380.000	0.226963
$142$	0	0
$143$	− 96.0000i	− 0.0561393i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 190.000i	− 0.106605i
$148$	0	0
$149$	−1730.00	−0.951189	−0.475594	−	0.879665i	$-0.657767\pi$
$149$	−1730.00	−0.951189	−0.475594	+	0.879665i	$0.657767\pi$
$150$	0	0
$151$	−1324.00	−0.713547	−0.356773	−	0.934191i	$-0.616123\pi$
$151$	−1324.00	−0.713547	−0.356773	+	0.934191i	$0.616123\pi$
$152$	0	0
$153$	− 438.000i	− 0.231439i
$154$	0	0
$155$	0	0
$156$	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	$-0.831834\pi$
$163$	− 2098.00i	− 1.00815i	0.863661	−	0.504074i	$-0.168166\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 866.000i	− 0.401276i	−0.979665	−	0.200638i	$-0.935699\pi$
$167$	− 866.000i	− 0.401276i	0.979665	−	0.200638i	$-0.0643015\pi$
$168$	0	0
$169$	2161.00	0.983614
$170$	0	0
$171$	9052.00	4.04809
$172$	0	0
$173$	− 1678.00i	− 0.737433i	−0.929542	−	0.368717i	$-0.879797\pi$
$173$	− 1678.00i	− 0.737433i	0.929542	−	0.368717i	$-0.120203\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 5640.00i	− 2.39508i
$178$	0	0
$179$	−1620.00	−0.676450	−0.338225	−	0.941065i	$-0.609826\pi$
$179$	−1620.00	−0.676450	−0.338225	+	0.941065i	$0.609826\pi$
$180$	0	0
$181$	2510.00	1.03076	0.515378	−	0.856963i	$-0.327652\pi$
$181$	2510.00	1.03076	0.515378	+	0.856963i	$0.327652\pi$
$182$	0	0
$183$	2620.00i	1.05834i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	96.0000i	0.0375413i
$188$	0	0
$189$	8280.00	3.18667
$190$	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
$193$	2938.00i	1.09576i	0.836557	+	0.547880i	$0.184565\pi$
$193$	2938.00i	1.09576i	−0.836557	+	0.547880i	$0.815435\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2234.00i	− 0.807949i	−0.914770	−	0.403974i	$-0.867628\pi$
$197$	− 2234.00i	− 0.807949i	0.914770	−	0.403974i	$-0.132372\pi$
$198$	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
$205$	0	0
$206$	0	0
$207$	3066.00i	1.02948i
$208$	0	0
$209$	−1984.00	−0.656632
$210$	0	0
$211$	−4896.00	−1.59741	−0.798707	−	0.601720i	$-0.794482\pi$
$211$	−4896.00	−1.59741	−0.798707	+	0.601720i	$0.794482\pi$
$212$	0	0
$213$	1400.00i	0.450359i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 3384.00i	− 1.05862i
$218$	0	0
$219$	8820.00	2.72146
$220$	0	0
$221$	36.0000	0.0109576
$222$	0	0
$223$	5302.00i	1.59214i	0.605202	+	0.796072i	$0.293092\pi$
$223$	5302.00i	1.59214i	−0.605202	+	0.796072i	$0.706908\pi$
$224$	0	0
$225$	0	0
$226$	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
$229$	3034.00	0.875513	0.437756	−	0.899094i	$-0.355773\pi$
$229$	3034.00	0.875513	0.437756	+	0.899094i	$0.355773\pi$
$230$	0	0
$231$	−2880.00	−0.820303
$232$	0	0
$233$	− 3478.00i	− 0.977903i	−0.872311	−	0.488951i	$-0.837380\pi$
$233$	− 3478.00i	− 0.977903i	0.872311	−	0.488951i	$-0.162620\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	11600.0i	3.17933i
$238$	0	0
$239$	1560.00	0.422209	0.211105	−	0.977463i	$-0.432294\pi$
$239$	1560.00	0.422209	0.211105	+	0.977463i	$0.432294\pi$
$240$	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
$245$	0	0
$246$	0	0
$247$	744.000i	0.191658i
$248$	0	0
$249$	−6420.00	−1.63394
$250$	0	0
$251$	−688.000	−0.173013	−0.0865063	−	0.996251i	$-0.527570\pi$
$251$	−688.000	−0.173013	−0.0865063	+	0.996251i	$0.527570\pi$
$252$	0	0
$253$	− 672.000i	− 0.166989i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2170.00i	− 0.526696i	−0.964701	−	0.263348i	$-0.915173\pi$
$257$	− 2170.00i	− 0.526696i	0.964701	−	0.263348i	$-0.0848267\pi$
$258$	0	0
$259$	−3636.00	−0.872317
$260$	0	0
$261$	10366.0	2.45839
$262$	0	0
$263$	− 2274.00i	− 0.533159i	−0.963813	−	0.266580i	$-0.914106\pi$
$263$	− 2274.00i	− 0.533159i	0.963813	−	0.266580i	$-0.0858935\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 8540.00i	− 1.95745i
$268$	0	0
$269$	−7146.00	−1.61970	−0.809850	−	0.586637i	$-0.800452\pi$
$269$	−7146.00	−1.61970	−0.809850	+	0.586637i	$0.800452\pi$
$270$	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
$273$	1080.00i	0.239431i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5150.00i	1.11709i	0.829475	+	0.558544i	$0.188640\pi$
$277$	5150.00i	1.11709i	−0.829475	+	0.558544i	$0.811360\pi$
$278$	0	0
$279$	−13724.0	−2.94493
$280$	0	0
$281$	5270.00	1.11880	0.559398	−	0.828899i	$-0.311032\pi$
$281$	5270.00	1.11880	0.559398	+	0.828899i	$0.311032\pi$
$282$	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
$285$	0	0
$286$	0	0
$287$	− 972.000i	− 0.199914i
$288$	0	0
$289$	4877.00	0.992673
$290$	0	0
$291$	4780.00	0.962916
$292$	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
$295$	0	0
$296$	0	0
$297$	7360.00i	1.43795i
$298$	0	0
$299$	−252.000	−0.0487409
$300$	0	0
$301$	−1188.00	−0.227492
$302$	0	0
$303$	17940.0i	3.40141i
$304$	0	0
$305$	0	0
$306$	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
$309$	−6420.00	−1.18195
$310$	0	0
$311$	−5492.00	−1.00136	−0.500680	−	0.865633i	$-0.666917\pi$
$311$	−5492.00	−1.00136	−0.500680	+	0.865633i	$0.666917\pi$
$312$	0	0
$313$	− 422.000i	− 0.0762072i	−0.999274	−	0.0381036i	$-0.987868\pi$
$313$	− 422.000i	− 0.0762072i	0.999274	−	0.0381036i	$-0.0121317\pi$
$314$	0	0
$315$	0	0
$316$	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
$325$	0	0
$326$	0	0
$327$	6660.00i	1.12630i
$328$	0	0
$329$	684.000	0.114620
$330$	0	0
$331$	−7688.00	−1.27665	−0.638324	−	0.769768i	$-0.720372\pi$
$331$	−7688.00	−1.27665	−0.638324	+	0.769768i	$0.720372\pi$
$332$	0	0
$333$	14746.0i	2.42665i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1438.00i	0.232442i	0.993223	+	0.116221i	$0.0370780\pi$
$337$	1438.00i	0.232442i	−0.993223	+	0.116221i	$0.962922\pi$
$338$	0	0
$339$	−14460.0	−2.31669
$340$	0	0
$341$	3008.00	0.477690
$342$	0	0
$343$	− 6516.00i	− 1.02575i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8838.00i	1.36729i	0.729816	+	0.683644i	$0.239606\pi$
$347$	8838.00i	1.36729i	−0.729816	+	0.683644i	$0.760394\pi$
$348$	0	0
$349$	7810.00	1.19788	0.598939	−	0.800794i	$-0.295589\pi$
$349$	7810.00	1.19788	0.598939	+	0.800794i	$0.295589\pi$
$350$	0	0
$351$	2760.00	0.419709
$352$	0	0
$353$	5906.00i	0.890495i	0.895408	+	0.445247i	$0.146884\pi$
$353$	5906.00i	0.890495i	−0.895408	+	0.445247i	$0.853116\pi$
$354$	0	0
$355$	0	0
$356$	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
$365$	0	0
$366$	0	0
$367$	− 7370.00i	− 1.04826i	−0.851639	−	0.524129i	$-0.824391\pi$
$367$	− 7370.00i	− 1.04826i	0.851639	−	0.524129i	$-0.175609\pi$
$368$	0	0
$369$	−3942.00	−0.556131
$370$	0	0
$371$	13284.0	1.85895
$372$	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
$375$	0	0
$376$	0	0
$377$	852.000i	0.116393i
$378$	0	0
$379$	10300.0	1.39598	0.697989	−	0.716109i	$-0.254079\pi$
$379$	10300.0	1.39598	0.697989	+	0.716109i	$0.254079\pi$
$380$	0	0
$381$	−11540.0	−1.55174
$382$	0	0
$383$	− 2682.00i	− 0.357817i	−0.983866	−	0.178908i	$-0.942743\pi$
$383$	− 2682.00i	− 0.357817i	0.983866	−	0.178908i	$-0.0572566\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4818.00i	0.632849i
$388$	0	0
$389$	−6114.00	−0.796895	−0.398447	−	0.917191i	$-0.630451\pi$
$389$	−6114.00	−0.796895	−0.398447	+	0.917191i	$0.630451\pi$
$390$	0	0
$391$	252.000	0.0325938
$392$	0	0
$393$	− 3680.00i	− 0.472345i
$394$	0	0
$395$	0	0
$396$	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
$405$	0	0
$406$	0	0
$407$	− 3232.00i	− 0.393622i
$408$	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
$411$	6700.00	0.804104
$412$	0	0
$413$	− 10152.0i	− 1.20956i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5720.00i	0.671726i
$418$	0	0
$419$	−3396.00	−0.395956	−0.197978	−	0.980206i	$-0.563437\pi$
$419$	−3396.00	−0.395956	−0.197978	+	0.980206i	$0.563437\pi$
$420$	0	0
$421$	−14974.0	−1.73346	−0.866732	−	0.498775i	$-0.833784\pi$
$421$	−14974.0	−1.73346	−0.866732	+	0.498775i	$0.833784\pi$
$422$	0	0
$423$	− 2774.00i	− 0.318857i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4716.00i	0.534481i
$428$	0	0
$429$	−960.000	−0.108040
$430$	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
$433$	15426.0i	1.71207i	0.516918	+	0.856035i	$0.327079\pi$
$433$	15426.0i	1.71207i	−0.516918	+	0.856035i	$0.672921\pi$
$434$	0	0
$435$	0	0
$436$	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
$445$	0	0
$446$	0	0
$447$	17300.0i	1.83056i
$448$	0	0
$449$	11898.0	1.25056	0.625280	−	0.780401i	$-0.284985\pi$
$449$	11898.0	1.25056	0.625280	+	0.780401i	$0.284985\pi$
$450$	0	0
$451$	864.000	0.0902088
$452$	0	0
$453$	13240.0i	1.37322i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	790.000i	0.0808635i	0.999182	+	0.0404318i	$0.0128733\pi$
$457$	790.000i	0.0808635i	−0.999182	+	0.0404318i	$0.987127\pi$
$458$	0	0
$459$	−2760.00	−0.280666
$460$	0	0
$461$	−3418.00	−0.345319	−0.172660	−	0.984982i	$-0.555236\pi$
$461$	−3418.00	−0.345319	−0.172660	+	0.984982i	$0.555236\pi$
$462$	0	0
$463$	7534.00i	0.756230i	0.925759	+	0.378115i	$0.123428\pi$
$463$	7534.00i	0.756230i	−0.925759	+	0.378115i	$0.876572\pi$
$464$	0	0
$465$	0	0
$466$	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
$469$	−9972.00	−0.981800
$470$	0	0
$471$	−29460.0	−2.88205
$472$	0	0
$473$	− 1056.00i	− 0.102653i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 53874.0i	− 5.17132i
$478$	0	0
$479$	−7016.00	−0.669247	−0.334623	−	0.942352i	$-0.608609\pi$
$479$	−7016.00	−0.669247	−0.334623	+	0.942352i	$0.608609\pi$
$480$	0	0
$481$	−1212.00	−0.114891
$482$	0	0
$483$	7560.00i	0.712199i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	15190.0i	1.41340i	0.707515	+	0.706699i	$0.249816\pi$
$487$	15190.0i	1.41340i	−0.707515	+	0.706699i	$0.750184\pi$
$488$	0	0
$489$	−20980.0	−1.94018
$490$	0	0
$491$	12624.0	1.16031	0.580156	−	0.814505i	$-0.302992\pi$
$491$	12624.0	1.16031	0.580156	+	0.814505i	$0.302992\pi$
$492$	0	0
$493$	− 852.000i	− 0.0778340i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2520.00i	0.227440i
$498$	0	0
$499$	−2492.00	−0.223562	−0.111781	−	0.993733i	$-0.535655\pi$
$499$	−2492.00	−0.223562	−0.111781	+	0.993733i	$0.535655\pi$
$500$	0	0
$501$	−8660.00	−0.772256
$502$	0	0
$503$	− 11714.0i	− 1.03837i	−0.854661	−	0.519186i	$-0.826235\pi$
$503$	− 11714.0i	− 1.03837i	0.854661	−	0.519186i	$-0.173765\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 21610.0i	− 1.89297i
$508$	0	0
$509$	5618.00	0.489221	0.244610	−	0.969621i	$-0.421340\pi$
$509$	5618.00	0.489221	0.244610	+	0.969621i	$0.421340\pi$
$510$	0	0
$511$	15876.0	1.37439
$512$	0	0
$513$	− 57040.0i	− 4.90912i
$514$	0	0
$515$	0	0
$516$	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	$-0.303459\pi$
$521$	13770.0	1.15792	0.578958	+	0.815357i	$0.303459\pi$
$522$	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
$525$	0	0
$526$	0	0
$527$	1128.00i	0.0932380i
$528$	0	0
$529$	10403.0	0.855018
$530$	0	0
$531$	−41172.0	−3.36481
$532$	0	0
$533$	− 324.000i	− 0.0263302i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	16200.0i	1.30183i
$538$	0	0
$539$	304.000	0.0242935
$540$	0	0
$541$	11958.0	0.950304	0.475152	−	0.879904i	$-0.342393\pi$
$541$	11958.0	0.950304	0.475152	+	0.879904i	$0.342393\pi$
$542$	0	0
$543$	− 25100.0i	− 1.98369i
$544$	0	0
$545$	0	0
$546$	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
$549$	19126.0	1.48684
$550$	0	0
$551$	17608.0	1.36139
$552$	0	0
$553$	20880.0i	1.60562i
$554$	0	0
$555$	0	0
$556$	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	$-0.804192\pi$
$563$	− 15418.0i	− 1.15416i	0.816688	−	0.577079i	$-0.195808\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 47322.0i	− 3.50500i
$568$	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
$571$	−6024.00	−0.441500	−0.220750	−	0.975330i	$-0.570851\pi$
$571$	−6024.00	−0.441500	−0.220750	+	0.975330i	$0.570851\pi$
$572$	0	0
$573$	− 3720.00i	− 0.271213i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 554.000i	− 0.0399711i	−0.999800	−	0.0199855i	$-0.993638\pi$
$577$	− 554.000i	− 0.0399711i	0.999800	−	0.0199855i	$-0.00636202\pi$
$578$	0	0
$579$	29380.0	2.10879
$580$	0	0
$581$	−11556.0	−0.825170
$582$	0	0
$583$	11808.0i	0.838829i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2386.00i	− 0.167770i	−0.996475	−	0.0838848i	$-0.973267\pi$
$587$	− 2386.00i	− 0.167770i	0.996475	−	0.0838848i	$-0.0267328\pi$
$588$	0	0
$589$	−23312.0	−1.63082
$590$	0	0
$591$	−22340.0	−1.55490
$592$	0	0
$593$	− 846.000i	− 0.0585853i	−0.999571	−	0.0292926i	$-0.990675\pi$
$593$	− 846.000i	− 0.0585853i	0.999571	−	0.0292926i	$-0.00932547\pi$
$594$	0	0
$595$	0	0
$596$	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	$-0.440128\pi$
$601$	5510.00	0.373973	0.186986	+	0.982363i	$0.440128\pi$
$602$	0	0
$603$	40442.0i	2.73122i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 8234.00i	− 0.550589i	−0.961360	−	0.275295i	$-0.911225\pi$
$607$	− 8234.00i	− 0.550589i	0.961360	−	0.275295i	$-0.0887754\pi$
$608$	0	0
$609$	25560.0	1.70073
$610$	0	0
$611$	228.000	0.0150964
$612$	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
$615$	0	0
$616$	0	0
$617$	3862.00i	0.251991i	0.992031	+	0.125995i	$0.0402124\pi$
$617$	3862.00i	0.251991i	−0.992031	+	0.125995i	$0.959788\pi$
$618$	0	0
$619$	13964.0	0.906721	0.453361	−	0.891327i	$-0.350225\pi$
$619$	13964.0	0.906721	0.453361	+	0.891327i	$0.350225\pi$
$620$	0	0
$621$	19320.0	1.24845
$622$	0	0
$623$	− 15372.0i	− 0.988549i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	19840.0i	1.26369i
$628$	0	0
$629$	1212.00	0.0768293
$630$	0	0
$631$	14884.0	0.939022	0.469511	−	0.882927i	$-0.344430\pi$
$631$	14884.0	0.939022	0.469511	+	0.882927i	$0.344430\pi$
$632$	0	0
$633$	48960.0i	3.07423i
$634$	0	0
$635$	0	0
$636$	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	$0.0770236\pi$
$643$	7814.00i	0.479244i	−0.970866	+	0.239622i	$0.922976\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	774.000i	0.0470310i	0.999723	+	0.0235155i	$0.00748591\pi$
$647$	774.000i	0.0470310i	−0.999723	+	0.0235155i	$0.992514\pi$
$648$	0	0
$649$	9024.00	0.545798
$650$	0	0
$651$	−33840.0	−2.03732
$652$	0	0
$653$	− 23422.0i	− 1.40364i	−0.712357	−	0.701818i	$-0.752372\pi$
$653$	− 23422.0i	− 1.40364i	0.712357	−	0.701818i	$-0.247628\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 64386.0i	− 3.82334i
$658$	0	0
$659$	−13508.0	−0.798478	−0.399239	−	0.916847i	$-0.630726\pi$
$659$	−13508.0	−0.798478	−0.399239	+	0.916847i	$0.630726\pi$
$660$	0	0
$661$	−6222.00	−0.366124	−0.183062	−	0.983101i	$-0.558601\pi$
$661$	−6222.00	−0.366124	−0.183062	+	0.983101i	$0.558601\pi$
$662$	0	0
$663$	− 360.000i	− 0.0210878i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5964.00i	0.346217i
$668$	0	0
$669$	53020.0	3.06408
$670$	0	0
$671$	−4192.00	−0.241178
$672$	0	0
$673$	− 15566.0i	− 0.891568i	−0.895141	−	0.445784i	$-0.852925\pi$
$673$	− 15566.0i	− 0.891568i	0.895141	−	0.445784i	$-0.147075\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 2234.00i	− 0.126824i	−0.997987	−	0.0634118i	$-0.979802\pi$
$677$	− 2234.00i	− 0.126824i	0.997987	−	0.0634118i	$-0.0201982\pi$
$678$	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
$685$	0	0
$686$	0	0
$687$	− 30340.0i	− 1.68492i
$688$	0	0
$689$	4428.00	0.244838
$690$	0	0
$691$	27416.0	1.50934	0.754670	−	0.656105i	$-0.227797\pi$
$691$	27416.0	1.50934	0.754670	+	0.656105i	$0.227797\pi$
$692$	0	0
$693$	21024.0i	1.15243i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	324.000i	0.0176074i
$698$	0	0
$699$	−34780.0	−1.88197
$700$	0	0
$701$	25626.0	1.38071	0.690357	−	0.723469i	$-0.257453\pi$
$701$	25626.0	1.38071	0.690357	+	0.723469i	$0.257453\pi$
$702$	0	0
$703$	25048.0i	1.34382i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	32292.0i	1.71777i
$708$	0	0
$709$	−11702.0	−0.619856	−0.309928	−	0.950760i	$-0.600305\pi$
$709$	−11702.0	−0.619856	−0.309928	+	0.950760i	$0.600305\pi$
$710$	0	0
$711$	84680.0	4.46659
$712$	0	0
$713$	− 7896.00i	− 0.414737i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 15600.0i	− 0.812542i
$718$	0	0
$719$	−28008.0	−1.45274	−0.726371	−	0.687302i	$-0.758795\pi$
$719$	−28008.0	−1.45274	−0.726371	+	0.687302i	$0.758795\pi$
$720$	0	0
$721$	−11556.0	−0.596904
$722$	0	0
$723$	32180.0i	1.65531i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 7682.00i	− 0.391898i	−0.980614	−	0.195949i	$-0.937221\pi$
$727$	− 7682.00i	− 0.391898i	0.980614	−	0.195949i	$-0.0627787\pi$
$728$	0	0
$729$	−67717.0	−3.44038
$730$	0	0
$731$	396.000	0.0200364
$732$	0	0
$733$	− 14270.0i	− 0.719065i	−0.933133	−	0.359532i	$-0.882936\pi$
$733$	− 14270.0i	− 0.719065i	0.933133	−	0.359532i	$-0.117064\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8864.00i	− 0.443025i
$738$	0	0
$739$	29324.0	1.45968	0.729838	−	0.683620i	$-0.239595\pi$
$739$	29324.0	1.45968	0.729838	+	0.683620i	$0.239595\pi$
$740$	0	0
$741$	7440.00	0.368846
$742$	0	0
$743$	− 29258.0i	− 1.44465i	−0.691556	−	0.722323i	$-0.743074\pi$
$743$	− 29258.0i	− 1.44465i	0.691556	−	0.722323i	$-0.256926\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	46866.0i	2.29550i
$748$	0	0
$749$	−15300.0	−0.746395
$750$	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
$753$	6880.00i	0.332963i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	22670.0i	1.08845i	0.838940	+	0.544224i	$0.183176\pi$
$757$	22670.0i	1.08845i	−0.838940	+	0.544224i	$0.816824\pi$
$758$	0	0
$759$	−6720.00	−0.321371
$760$	0	0
$761$	−23206.0	−1.10541	−0.552705	−	0.833377i	$-0.686404\pi$
$761$	−23206.0	−1.10541	−0.552705	+	0.833377i	$0.686404\pi$
$762$	0	0
$763$	11988.0i	0.568800i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 3384.00i	− 0.159308i
$768$	0	0
$769$	1854.00	0.0869401	0.0434701	−	0.999055i	$-0.486159\pi$
$769$	1854.00	0.0869401	0.0434701	+	0.999055i	$0.486159\pi$
$770$	0	0
$771$	−21700.0	−1.01363
$772$	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
$775$	0	0
$776$	0	0
$777$	36360.0i	1.67877i
$778$	0	0
$779$	−6696.00	−0.307971
$780$	0	0
$781$	−2240.00	−0.102629
$782$	0	0
$783$	− 65320.0i	− 2.98129i
$784$	0	0
$785$	0	0
$786$	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
$789$	−22740.0	−1.02607
$790$	0	0
$791$	−26028.0	−1.16997
$792$	0	0
$793$	1572.00i	0.0703952i
$794$	0	0
$795$	0	0
$796$	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
$805$	0	0
$806$	0	0
$807$	71460.0i	3.11711i
$808$	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
$811$	−24512.0	−1.06132	−0.530661	−	0.847584i	$-0.678056\pi$
$811$	−24512.0	−1.06132	−0.530661	+	0.847584i	$0.678056\pi$
$812$	0	0
$813$	26040.0i	1.12332i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8184.00i	0.350455i
$818$	0	0
$819$	7884.00	0.336373
$820$	0	0
$821$	36242.0	1.54063	0.770313	−	0.637666i	$-0.220100\pi$
$821$	36242.0	1.54063	0.770313	+	0.637666i	$0.220100\pi$
$822$	0	0
$823$	17718.0i	0.750438i	0.926936	+	0.375219i	$0.122433\pi$
$823$	17718.0i	0.750438i	−0.926936	+	0.375219i	$0.877567\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6726.00i	0.282812i	0.989952	+	0.141406i	$0.0451624\pi$
$827$	6726.00i	0.282812i	−0.989952	+	0.141406i	$0.954838\pi$
$828$	0	0
$829$	−41722.0	−1.74797	−0.873984	−	0.485955i	$-0.838472\pi$
$829$	−41722.0	−1.74797	−0.873984	+	0.485955i	$0.838472\pi$
$830$	0	0
$831$	51500.0	2.14984
$832$	0	0
$833$	114.000i	0.00474174i
$834$	0	0
$835$	0	0
$836$	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
$845$	0	0
$846$	0	0
$847$	19350.0i	0.784975i
$848$	0	0
$849$	−34340.0	−1.38816
$850$	0	0
$851$	−8484.00	−0.341748
$852$	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
$855$	0	0
$856$	0	0
$857$	− 38978.0i	− 1.55363i	−0.629727	−	0.776816i	$-0.716833\pi$
$857$	− 38978.0i	− 1.55363i	0.629727	−	0.776816i	$-0.283167\pi$
$858$	0	0
$859$	−1916.00	−0.0761037	−0.0380518	−	0.999276i	$-0.512115\pi$
$859$	−1916.00	−0.0761037	−0.0380518	+	0.999276i	$0.512115\pi$
$860$	0	0
$861$	−9720.00	−0.384735
$862$	0	0
$863$	2374.00i	0.0936407i	0.998903	+	0.0468203i	$0.0149088\pi$
$863$	2374.00i	0.0936407i	−0.998903	+	0.0468203i	$0.985091\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 48770.0i	− 1.91040i
$868$	0	0
$869$	−18560.0	−0.724517
$870$	0	0
$871$	−3324.00	−0.129310
$872$	0	0
$873$	− 34894.0i	− 1.35279i
$874$	0	0
$875$	0	0
$876$	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	$0.308228\pi$
$883$	43238.0i	1.64788i	−0.566680	+	0.823938i	$0.691772\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 11010.0i	− 0.416775i	−0.978046	−	0.208388i	$-0.933178\pi$
$887$	− 11010.0i	− 0.416775i	0.978046	−	0.208388i	$-0.0668215\pi$
$888$	0	0
$889$	−20772.0	−0.783656
$890$	0	0
$891$	42064.0	1.58159
$892$	0	0
$893$	− 4712.00i	− 0.176575i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2520.00i	0.0938020i
$898$	0	0
$899$	−26696.0	−0.990391
$900$	0	0
$901$	−4428.00	−0.163727
$902$	0	0
$903$	11880.0i	0.437809i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 74.0000i	− 0.00270907i	−0.999999	−	0.00135454i	$-0.999569\pi$
$907$	− 74.0000i	− 0.00270907i	0.999999	−	0.00135454i	$-0.000431163\pi$
$908$	0	0
$909$	130962.	4.77859
$910$	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
$913$	− 10272.0i	− 0.372348i
$914$	0	0
$915$	0	0
$916$	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
$925$	0	0
$926$	0	0
$927$	46866.0i	1.66050i
$928$	0	0
$929$	14826.0	0.523601	0.261800	−	0.965122i	$-0.415684\pi$
$929$	14826.0	0.523601	0.261800	+	0.965122i	$0.415684\pi$
$930$	0	0
$931$	−2356.00	−0.0829375
$932$	0	0
$933$	54920.0i	1.92712i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 3354.00i	− 0.116937i	−0.998289	−	0.0584687i	$-0.981378\pi$
$937$	− 3354.00i	− 0.116937i	0.998289	−	0.0584687i	$-0.0186218\pi$
$938$	0	0
$939$	−4220.00	−0.146661
$940$	0	0
$941$	−15434.0	−0.534680	−0.267340	−	0.963602i	$-0.586145\pi$
$941$	−15434.0	−0.534680	−0.267340	+	0.963602i	$0.586145\pi$
$942$	0	0
$943$	− 2268.00i	− 0.0783205i
$944$	0	0
$945$	0	0
$946$	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
$949$	5292.00	0.181017
$950$	0	0
$951$	−61940.0	−2.11203
$952$	0	0
$953$	12202.0i	0.414755i	0.978261	+	0.207378i	$0.0664928\pi$
$953$	12202.0i	0.414755i	−0.978261	+	0.207378i	$0.933507\pi$
$954$	0	0
$955$	0	0
$956$	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
$965$	0	0
$966$	0	0
$967$	17478.0i	0.581235i	0.956839	+	0.290618i	$0.0938608\pi$
$967$	17478.0i	0.581235i	−0.956839	+	0.290618i	$0.906139\pi$
$968$	0	0
$969$	−7440.00	−0.246653
$970$	0	0
$971$	−10920.0	−0.360906	−0.180453	−	0.983584i	$-0.557756\pi$
$971$	−10920.0	−0.360906	−0.180453	+	0.983584i	$0.557756\pi$
$972$	0	0
$973$	10296.0i	0.339234i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 10834.0i	− 0.354770i	−0.984142	−	0.177385i	$-0.943236\pi$
$977$	− 10834.0i	− 0.354770i	0.984142	−	0.177385i	$-0.0567638\pi$
$978$	0	0
$979$	13664.0	0.446071
$980$	0	0
$981$	48618.0	1.58232
$982$	0	0
$983$	36862.0i	1.19605i	0.801478	+	0.598024i	$0.204047\pi$
$983$	36862.0i	1.19605i	−0.801478	+	0.598024i	$0.795953\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 6840.00i	− 0.220587i
$988$	0	0
$989$	−2772.00	−0.0891248
$990$	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
$993$	76880.0i	2.45691i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 31266.0i	− 0.993184i	−0.867984	−	0.496592i	$-0.834585\pi$
$997$	− 31266.0i	− 0.993184i	0.867984	−	0.496592i	$-0.165415\pi$
$998$	0	0
$999$	92920.0	2.94280

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.4.c.a.49.1		2
4.3	odd	2		200.4.c.a.49.2		2
5.2	odd	4		80.4.a.a.1.1		1
5.3	odd	4		400.4.a.u.1.1		1
5.4	even	2	inner	400.4.c.a.49.2		2
12.11	even	2		1800.4.f.n.649.2		2
15.2	even	4		720.4.a.ba.1.1		1
20.3	even	4		200.4.a.a.1.1		1
20.7	even	4		40.4.a.c.1.1	✓	1
20.19	odd	2		200.4.c.a.49.1		2
40.3	even	4		1600.4.a.ca.1.1		1
40.13	odd	4		1600.4.a.a.1.1		1
40.27	even	4		320.4.a.a.1.1		1
40.37	odd	4		320.4.a.n.1.1		1
60.23	odd	4		1800.4.a.bd.1.1		1
60.47	odd	4		360.4.a.i.1.1		1
60.59	even	2		1800.4.f.n.649.1		2
80.27	even	4		1280.4.d.o.641.1		2
80.37	odd	4		1280.4.d.b.641.2		2
80.67	even	4		1280.4.d.o.641.2		2
80.77	odd	4		1280.4.d.b.641.1		2
140.27	odd	4		1960.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.a.c.1.1	✓	1	20.7	even	4
80.4.a.a.1.1		1	5.2	odd	4
200.4.a.a.1.1		1	20.3	even	4
200.4.c.a.49.1		2	20.19	odd	2
200.4.c.a.49.2		2	4.3	odd	2
320.4.a.a.1.1		1	40.27	even	4
320.4.a.n.1.1		1	40.37	odd	4
360.4.a.i.1.1		1	60.47	odd	4
400.4.a.u.1.1		1	5.3	odd	4
400.4.c.a.49.1		2	1.1	even	1	trivial
400.4.c.a.49.2		2	5.4	even	2	inner
720.4.a.ba.1.1		1	15.2	even	4
1280.4.d.b.641.1		2	80.77	odd	4
1280.4.d.b.641.2		2	80.37	odd	4
1280.4.d.o.641.1		2	80.27	even	4
1280.4.d.o.641.2		2	80.67	even	4
1600.4.a.a.1.1		1	40.13	odd	4
1600.4.a.ca.1.1		1	40.3	even	4
1800.4.a.bd.1.1		1	60.23	odd	4
1800.4.f.n.649.1		2	60.59	even	2
1800.4.f.n.649.2		2	12.11	even	2
1960.4.a.a.1.1		1	140.27	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-10.0000i q^{3} -18.0000i q^{7} -73.0000 q^{9} +O(q^{10})\)	\(q-10.0000i q^{3} -18.0000i q^{7} -73.0000 q^{9} +16.0000 q^{11} -6.00000i q^{13} +6.00000i q^{17} -124.000 q^{19} -180.000 q^{21} -42.0000i q^{23} +460.000i q^{27} -142.000 q^{29} +188.000 q^{31} -160.000i q^{33} -202.000i q^{37} -60.0000 q^{39} +54.0000 q^{41} -66.0000i q^{43} +38.0000i q^{47} +19.0000 q^{49} +60.0000 q^{51} +738.000i q^{53} +1240.00i q^{57} +564.000 q^{59} -262.000 q^{61} +1314.00i q^{63} -554.000i q^{67} -420.000 q^{69} -140.000 q^{71} +882.000i q^{73} -288.000i q^{77} -1160.00 q^{79} +2629.00 q^{81} -642.000i q^{83} +1420.00i q^{87} +854.000 q^{89} -108.000 q^{91} -1880.00i q^{93} +478.000i q^{97} -1168.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 146 q^{9}+O(q^{10}) \) 2 * q - 146 * q^9	\( 2 q - 146 q^{9} + 32 q^{11} - 248 q^{19} - 360 q^{21} - 284 q^{29} + 376 q^{31} - 120 q^{39} + 108 q^{41} + 38 q^{49} + 120 q^{51} + 1128 q^{59} - 524 q^{61} - 840 q^{69} - 280 q^{71} - 2320 q^{79} + 5258 q^{81} + 1708 q^{89} - 216 q^{91} - 2336 q^{99}+O(q^{100}) \) 2 * q - 146 * q^9 + 32 * q^11 - 248 * q^19 - 360 * q^21 - 284 * q^29 + 376 * q^31 - 120 * q^39 + 108 * q^41 + 38 * q^49 + 120 * q^51 + 1128 * q^59 - 524 * q^61 - 840 * q^69 - 280 * q^71 - 2320 * q^79 + 5258 * q^81 + 1708 * q^89 - 216 * q^91 - 2336 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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