LMFDB - Embedded newform 600.4.f.g.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,4,Mod(49,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("600.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	600.f (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:	$35.4011460034$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	600.49
Dual form			600.4.f.g.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} -8.00000i q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} -8.00000i q^{7} -9.00000 q^{9} +20.0000 q^{11} +22.0000i q^{13} +14.0000i q^{17} -76.0000 q^{19} +24.0000 q^{21} +56.0000i q^{23} -27.0000i q^{27} +154.000 q^{29} +160.000 q^{31} +60.0000i q^{33} +162.000i q^{37} -66.0000 q^{39} -390.000 q^{41} +388.000i q^{43} +544.000i q^{47} +279.000 q^{49} -42.0000 q^{51} -210.000i q^{53} -228.000i q^{57} +380.000 q^{59} -794.000 q^{61} +72.0000i q^{63} +148.000i q^{67} -168.000 q^{69} -840.000 q^{71} +858.000i q^{73} -160.000i q^{77} -144.000 q^{79} +81.0000 q^{81} +316.000i q^{83} +462.000i q^{87} -1098.00 q^{89} +176.000 q^{91} +480.000i q^{93} -994.000i q^{97} -180.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9}+O(q^{10}) $ 2 * q - 18 * q^9	$ 2 q - 18 q^{9} + 40 q^{11} - 152 q^{19} + 48 q^{21} + 308 q^{29} + 320 q^{31} - 132 q^{39} - 780 q^{41} + 558 q^{49} - 84 q^{51} + 760 q^{59} - 1588 q^{61} - 336 q^{69} - 1680 q^{71} - 288 q^{79} + 162 q^{81} - 2196 q^{89} + 352 q^{91} - 360 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 + 40 * q^11 - 152 * q^19 + 48 * q^21 + 308 * q^29 + 320 * q^31 - 132 * q^39 - 780 * q^41 + 558 * q^49 - 84 * q^51 + 760 * q^59 - 1588 * q^61 - 336 * q^69 - 1680 * q^71 - 288 * q^79 + 162 * q^81 - 2196 * q^89 + 352 * q^91 - 360 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 8.00000i	− 0.431959i	−0.976398	−	0.215980i	$-0.930705\pi$
$7$	− 8.00000i	− 0.431959i	0.976398	−	0.215980i	$-0.0692945\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	20.0000	0.548202	0.274101	−	0.961701i	$-0.411620\pi$
$11$	20.0000	0.548202	0.274101	+	0.961701i	$0.411620\pi$
$12$	0	0
$13$	22.0000i	0.469362i	0.972072	+	0.234681i	$0.0754045\pi$
$13$	22.0000i	0.469362i	−0.972072	+	0.234681i	$0.924595\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	14.0000i	0.199735i	0.995001	+	0.0998676i	$0.0318419\pi$
$17$	14.0000i	0.199735i	−0.995001	+	0.0998676i	$0.968158\pi$
$18$	0	0
$19$	−76.0000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−76.0000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	24.0000	0.249392
$22$	0	0
$23$	56.0000i	0.507687i	0.967245	+	0.253844i	$0.0816949\pi$
$23$	56.0000i	0.507687i	−0.967245	+	0.253844i	$0.918305\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	154.000	0.986106	0.493053	−	0.869999i	$-0.335881\pi$
$29$	154.000	0.986106	0.493053	+	0.869999i	$0.335881\pi$
$30$	0	0
$31$	160.000	0.926995	0.463498	−	0.886098i	$-0.346594\pi$
$31$	160.000	0.926995	0.463498	+	0.886098i	$0.346594\pi$
$32$	0	0
$33$	60.0000i	0.316505i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	162.000i	0.719801i	0.932991	+	0.359900i	$0.117189\pi$
$37$	162.000i	0.719801i	−0.932991	+	0.359900i	$0.882811\pi$
$38$	0	0
$39$	−66.0000	−0.270986
$40$	0	0
$41$	−390.000	−1.48556	−0.742778	−	0.669538i	$-0.766492\pi$
$41$	−390.000	−1.48556	−0.742778	+	0.669538i	$0.766492\pi$
$42$	0	0
$43$	388.000i	1.37603i	0.725695	+	0.688017i	$0.241518\pi$
$43$	388.000i	1.37603i	−0.725695	+	0.688017i	$0.758482\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	544.000i	1.68831i	0.536099	+	0.844155i	$0.319897\pi$
$47$	544.000i	1.68831i	−0.536099	+	0.844155i	$0.680103\pi$
$48$	0	0
$49$	279.000	0.813411
$50$	0	0
$51$	−42.0000	−0.115317
$52$	0	0
$53$	− 210.000i	− 0.544259i	−0.962261	−	0.272129i	$-0.912272\pi$
$53$	− 210.000i	− 0.544259i	0.962261	−	0.272129i	$-0.0877279\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 228.000i	− 0.529813i
$58$	0	0
$59$	380.000	0.838505	0.419252	−	0.907870i	$-0.362292\pi$
$59$	380.000	0.838505	0.419252	+	0.907870i	$0.362292\pi$
$60$	0	0
$61$	−794.000	−1.66658	−0.833289	−	0.552837i	$-0.813545\pi$
$61$	−794.000	−1.66658	−0.833289	+	0.552837i	$0.813545\pi$
$62$	0	0
$63$	72.0000i	0.143986i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	148.000i	0.269867i	0.990855	+	0.134933i	$0.0430821\pi$
$67$	148.000i	0.269867i	−0.990855	+	0.134933i	$0.956918\pi$
$68$	0	0
$69$	−168.000	−0.293113
$70$	0	0
$71$	−840.000	−1.40408	−0.702040	−	0.712138i	$-0.747727\pi$
$71$	−840.000	−1.40408	−0.702040	+	0.712138i	$0.747727\pi$
$72$	0	0
$73$	858.000i	1.37563i	0.725884	+	0.687817i	$0.241431\pi$
$73$	858.000i	1.37563i	−0.725884	+	0.687817i	$0.758569\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 160.000i	− 0.236801i
$78$	0	0
$79$	−144.000	−0.205079	−0.102540	−	0.994729i	$-0.532697\pi$
$79$	−144.000	−0.205079	−0.102540	+	0.994729i	$0.532697\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	316.000i	0.417898i	0.977927	+	0.208949i	$0.0670042\pi$
$83$	316.000i	0.417898i	−0.977927	+	0.208949i	$0.932996\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	462.000i	0.569329i
$88$	0	0
$89$	−1098.00	−1.30773	−0.653864	−	0.756612i	$-0.726853\pi$
$89$	−1098.00	−1.30773	−0.653864	+	0.756612i	$0.726853\pi$
$90$	0	0
$91$	176.000	0.202745
$92$	0	0
$93$	480.000i	0.535201i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 994.000i	− 1.04047i	−0.854024	−	0.520234i	$-0.825845\pi$
$97$	− 994.000i	− 1.04047i	0.854024	−	0.520234i	$-0.174155\pi$
$98$	0	0
$99$	−180.000	−0.182734
$100$	0	0
$101$	−834.000	−0.821645	−0.410822	−	0.911715i	$-0.634758\pi$
$101$	−834.000	−0.821645	−0.410822	+	0.911715i	$0.634758\pi$
$102$	0	0
$103$	1672.00i	1.59949i	0.600343	+	0.799743i	$0.295031\pi$
$103$	1672.00i	1.59949i	−0.600343	+	0.799743i	$0.704969\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	732.000i	0.661356i	0.943744	+	0.330678i	$0.107277\pi$
$107$	732.000i	0.661356i	−0.943744	+	0.330678i	$0.892723\pi$
$108$	0	0
$109$	970.000	0.852378	0.426189	−	0.904634i	$-0.359856\pi$
$109$	970.000	0.852378	0.426189	+	0.904634i	$0.359856\pi$
$110$	0	0
$111$	−486.000	−0.415577
$112$	0	0
$113$	1938.00i	1.61338i	0.590976	+	0.806689i	$0.298743\pi$
$113$	1938.00i	1.61338i	−0.590976	+	0.806689i	$0.701257\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 198.000i	− 0.156454i
$118$	0	0
$119$	112.000	0.0862775
$120$	0	0
$121$	−931.000	−0.699474
$122$	0	0
$123$	− 1170.00i	− 0.857686i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	528.000i	0.368917i	0.982840	+	0.184458i	$0.0590531\pi$
$127$	528.000i	0.368917i	−0.982840	+	0.184458i	$0.940947\pi$
$128$	0	0
$129$	−1164.00	−0.794453
$130$	0	0
$131$	636.000	0.424180	0.212090	−	0.977250i	$-0.431973\pi$
$131$	636.000	0.424180	0.212090	+	0.977250i	$0.431973\pi$
$132$	0	0
$133$	608.000i	0.396393i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1754.00i	− 1.09383i	−0.837189	−	0.546914i	$-0.815803\pi$
$137$	− 1754.00i	− 1.09383i	0.837189	−	0.546914i	$-0.184197\pi$
$138$	0	0
$139$	2508.00	1.53040	0.765201	−	0.643792i	$-0.222640\pi$
$139$	2508.00	1.53040	0.765201	+	0.643792i	$0.222640\pi$
$140$	0	0
$141$	−1632.00	−0.974746
$142$	0	0
$143$	440.000i	0.257305i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	837.000i	0.469623i
$148$	0	0
$149$	−1486.00	−0.817033	−0.408516	−	0.912751i	$-0.633954\pi$
$149$	−1486.00	−0.817033	−0.408516	+	0.912751i	$0.633954\pi$
$150$	0	0
$151$	2120.00	1.14254	0.571269	−	0.820763i	$-0.306451\pi$
$151$	2120.00	1.14254	0.571269	+	0.820763i	$0.306451\pi$
$152$	0	0
$153$	− 126.000i	− 0.0665784i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1850.00i	0.940421i	0.882554	+	0.470210i	$0.155822\pi$
$157$	1850.00i	0.940421i	−0.882554	+	0.470210i	$0.844178\pi$
$158$	0	0
$159$	630.000	0.314228
$160$	0	0
$161$	448.000	0.219300
$162$	0	0
$163$	− 1172.00i	− 0.563179i	−0.959535	−	0.281589i	$-0.909138\pi$
$163$	− 1172.00i	− 0.563179i	0.959535	−	0.281589i	$-0.0908616\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1656.00i	0.767336i	0.923471	+	0.383668i	$0.125339\pi$
$167$	1656.00i	0.767336i	−0.923471	+	0.383668i	$0.874661\pi$
$168$	0	0
$169$	1713.00	0.779700
$170$	0	0
$171$	684.000	0.305888
$172$	0	0
$173$	− 2666.00i	− 1.17163i	−0.810444	−	0.585816i	$-0.800774\pi$
$173$	− 2666.00i	− 1.17163i	0.810444	−	0.585816i	$-0.199226\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1140.00i	0.484111i
$178$	0	0
$179$	−1132.00	−0.472680	−0.236340	−	0.971670i	$-0.575948\pi$
$179$	−1132.00	−0.472680	−0.236340	+	0.971670i	$0.575948\pi$
$180$	0	0
$181$	−2866.00	−1.17695	−0.588475	−	0.808515i	$-0.700272\pi$
$181$	−2866.00	−1.17695	−0.588475	+	0.808515i	$0.700272\pi$
$182$	0	0
$183$	− 2382.00i	− 0.962199i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	280.000i	0.109495i
$188$	0	0
$189$	−216.000	−0.0831306
$190$	0	0
$191$	1888.00	0.715240	0.357620	−	0.933867i	$-0.383588\pi$
$191$	1888.00	0.715240	0.357620	+	0.933867i	$0.383588\pi$
$192$	0	0
$193$	1282.00i	0.478137i	0.971003	+	0.239068i	$0.0768420\pi$
$193$	1282.00i	0.478137i	−0.971003	+	0.239068i	$0.923158\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 350.000i	− 0.126581i	−0.997995	−	0.0632905i	$-0.979841\pi$
$197$	− 350.000i	− 0.126581i	0.997995	−	0.0632905i	$-0.0201595\pi$
$198$	0	0
$199$	3400.00	1.21115	0.605577	−	0.795787i	$-0.292942\pi$
$199$	3400.00	1.21115	0.605577	+	0.795787i	$0.292942\pi$
$200$	0	0
$201$	−444.000	−0.155808
$202$	0	0
$203$	− 1232.00i	− 0.425958i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 504.000i	− 0.169229i
$208$	0	0
$209$	−1520.00	−0.503065
$210$	0	0
$211$	4652.00	1.51781	0.758903	−	0.651204i	$-0.225736\pi$
$211$	4652.00	1.51781	0.758903	+	0.651204i	$0.225736\pi$
$212$	0	0
$213$	− 2520.00i	− 0.810646i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1280.00i	− 0.400424i
$218$	0	0
$219$	−2574.00	−0.794223
$220$	0	0
$221$	−308.000	−0.0937481
$222$	0	0
$223$	4016.00i	1.20597i	0.797753	+	0.602985i	$0.206022\pi$
$223$	4016.00i	1.20597i	−0.797753	+	0.602985i	$0.793978\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2316.00i	− 0.677173i	−0.940935	−	0.338587i	$-0.890051\pi$
$227$	− 2316.00i	− 0.677173i	0.940935	−	0.338587i	$-0.109949\pi$
$228$	0	0
$229$	−94.0000	−0.0271253	−0.0135627	−	0.999908i	$-0.504317\pi$
$229$	−94.0000	−0.0271253	−0.0135627	+	0.999908i	$0.504317\pi$
$230$	0	0
$231$	480.000	0.136717
$232$	0	0
$233$	− 4230.00i	− 1.18934i	−0.803969	−	0.594671i	$-0.797282\pi$
$233$	− 4230.00i	− 1.18934i	0.803969	−	0.594671i	$-0.202718\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 432.000i	− 0.118403i
$238$	0	0
$239$	−2064.00	−0.558615	−0.279308	−	0.960202i	$-0.590105\pi$
$239$	−2064.00	−0.558615	−0.279308	+	0.960202i	$0.590105\pi$
$240$	0	0
$241$	4562.00	1.21935	0.609677	−	0.792650i	$-0.291299\pi$
$241$	4562.00	1.21935	0.609677	+	0.792650i	$0.291299\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1672.00i	− 0.430716i
$248$	0	0
$249$	−948.000	−0.241273
$250$	0	0
$251$	2532.00	0.636727	0.318363	−	0.947969i	$-0.396867\pi$
$251$	2532.00	0.636727	0.318363	+	0.947969i	$0.396867\pi$
$252$	0	0
$253$	1120.00i	0.278315i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3522.00i	− 0.854850i	−0.904051	−	0.427425i	$-0.859421\pi$
$257$	− 3522.00i	− 0.854850i	0.904051	−	0.427425i	$-0.140579\pi$
$258$	0	0
$259$	1296.00	0.310925
$260$	0	0
$261$	−1386.00	−0.328702
$262$	0	0
$263$	− 2232.00i	− 0.523312i	−0.965161	−	0.261656i	$-0.915731\pi$
$263$	− 2232.00i	− 0.523312i	0.965161	−	0.261656i	$-0.0842686\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 3294.00i	− 0.755017i
$268$	0	0
$269$	−2806.00	−0.636003	−0.318002	−	0.948090i	$-0.603012\pi$
$269$	−2806.00	−0.636003	−0.318002	+	0.948090i	$0.603012\pi$
$270$	0	0
$271$	4848.00	1.08670	0.543349	−	0.839507i	$-0.317156\pi$
$271$	4848.00	1.08670	0.543349	+	0.839507i	$0.317156\pi$
$272$	0	0
$273$	528.000i	0.117055i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 7790.00i	− 1.68973i	−0.534978	−	0.844866i	$-0.679680\pi$
$277$	− 7790.00i	− 1.68973i	0.534978	−	0.844866i	$-0.320320\pi$
$278$	0	0
$279$	−1440.00	−0.308998
$280$	0	0
$281$	−118.000	−0.0250509	−0.0125254	−	0.999922i	$-0.503987\pi$
$281$	−118.000	−0.0250509	−0.0125254	+	0.999922i	$0.503987\pi$
$282$	0	0
$283$	− 6508.00i	− 1.36700i	−0.729951	−	0.683499i	$-0.760457\pi$
$283$	− 6508.00i	− 1.36700i	0.729951	−	0.683499i	$-0.239543\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3120.00i	0.641700i
$288$	0	0
$289$	4717.00	0.960106
$290$	0	0
$291$	2982.00	0.600715
$292$	0	0
$293$	− 8770.00i	− 1.74863i	−0.485358	−	0.874315i	$-0.661311\pi$
$293$	− 8770.00i	− 1.74863i	0.485358	−	0.874315i	$-0.338689\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 540.000i	− 0.105502i
$298$	0	0
$299$	−1232.00	−0.238289
$300$	0	0
$301$	3104.00	0.594391
$302$	0	0
$303$	− 2502.00i	− 0.474377i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4292.00i	0.797907i	0.916971	+	0.398953i	$0.130626\pi$
$307$	4292.00i	0.797907i	−0.916971	+	0.398953i	$0.869374\pi$
$308$	0	0
$309$	−5016.00	−0.923464
$310$	0	0
$311$	−9464.00	−1.72558	−0.862788	−	0.505566i	$-0.831284\pi$
$311$	−9464.00	−1.72558	−0.862788	+	0.505566i	$0.831284\pi$
$312$	0	0
$313$	9578.00i	1.72965i	0.502073	+	0.864825i	$0.332571\pi$
$313$	9578.00i	1.72965i	−0.502073	+	0.864825i	$0.667429\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	186.000i	0.0329552i	0.999864	+	0.0164776i	$0.00524522\pi$
$317$	186.000i	0.0329552i	−0.999864	+	0.0164776i	$0.994755\pi$
$318$	0	0
$319$	3080.00	0.540586
$320$	0	0
$321$	−2196.00	−0.381834
$322$	0	0
$323$	− 1064.00i	− 0.183290i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2910.00i	0.492120i
$328$	0	0
$329$	4352.00	0.729281
$330$	0	0
$331$	−492.000	−0.0817002	−0.0408501	−	0.999165i	$-0.513007\pi$
$331$	−492.000	−0.0817002	−0.0408501	+	0.999165i	$0.513007\pi$
$332$	0	0
$333$	− 1458.00i	− 0.239934i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 2290.00i	− 0.370161i	−0.982723	−	0.185080i	$-0.940745\pi$
$337$	− 2290.00i	− 0.370161i	0.982723	−	0.185080i	$-0.0592546\pi$
$338$	0	0
$339$	−5814.00	−0.931484
$340$	0	0
$341$	3200.00	0.508181
$342$	0	0
$343$	− 4976.00i	− 0.783320i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6092.00i	0.942466i	0.882009	+	0.471233i	$0.156191\pi$
$347$	6092.00i	0.942466i	−0.882009	+	0.471233i	$0.843809\pi$
$348$	0	0
$349$	−5766.00	−0.884375	−0.442188	−	0.896923i	$-0.645797\pi$
$349$	−5766.00	−0.884375	−0.442188	+	0.896923i	$0.645797\pi$
$350$	0	0
$351$	594.000	0.0903287
$352$	0	0
$353$	− 9374.00i	− 1.41339i	−0.707517	−	0.706696i	$-0.750185\pi$
$353$	− 9374.00i	− 1.41339i	0.707517	−	0.706696i	$-0.249815\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	336.000i	0.0498123i
$358$	0	0
$359$	3528.00	0.518665	0.259332	−	0.965788i	$-0.416497\pi$
$359$	3528.00	0.518665	0.259332	+	0.965788i	$0.416497\pi$
$360$	0	0
$361$	−1083.00	−0.157895
$362$	0	0
$363$	− 2793.00i	− 0.403842i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 7616.00i	− 1.08325i	−0.840621	−	0.541624i	$-0.817810\pi$
$367$	− 7616.00i	− 1.08325i	0.840621	−	0.541624i	$-0.182190\pi$
$368$	0	0
$369$	3510.00	0.495185
$370$	0	0
$371$	−1680.00	−0.235098
$372$	0	0
$373$	3406.00i	0.472804i	0.971655	+	0.236402i	$0.0759683\pi$
$373$	3406.00i	0.472804i	−0.971655	+	0.236402i	$0.924032\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3388.00i	0.462841i
$378$	0	0
$379$	12284.0	1.66487	0.832436	−	0.554121i	$-0.186945\pi$
$379$	12284.0	1.66487	0.832436	+	0.554121i	$0.186945\pi$
$380$	0	0
$381$	−1584.00	−0.212994
$382$	0	0
$383$	5424.00i	0.723638i	0.932248	+	0.361819i	$0.117844\pi$
$383$	5424.00i	0.723638i	−0.932248	+	0.361819i	$0.882156\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 3492.00i	− 0.458678i
$388$	0	0
$389$	−3486.00	−0.454363	−0.227182	−	0.973852i	$-0.572951\pi$
$389$	−3486.00	−0.454363	−0.227182	+	0.973852i	$0.572951\pi$
$390$	0	0
$391$	−784.000	−0.101403
$392$	0	0
$393$	1908.00i	0.244900i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3626.00i	0.458397i	0.973380	+	0.229199i	$0.0736105\pi$
$397$	3626.00i	0.458397i	−0.973380	+	0.229199i	$0.926389\pi$
$398$	0	0
$399$	−1824.00	−0.228858
$400$	0	0
$401$	5874.00	0.731505	0.365753	−	0.930712i	$-0.380812\pi$
$401$	5874.00	0.731505	0.365753	+	0.930712i	$0.380812\pi$
$402$	0	0
$403$	3520.00i	0.435096i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3240.00i	0.394597i
$408$	0	0
$409$	12662.0	1.53080	0.765398	−	0.643557i	$-0.222542\pi$
$409$	12662.0	1.53080	0.765398	+	0.643557i	$0.222542\pi$
$410$	0	0
$411$	5262.00	0.631521
$412$	0	0
$413$	− 3040.00i	− 0.362200i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7524.00i	0.883578i
$418$	0	0
$419$	−6396.00	−0.745740	−0.372870	−	0.927884i	$-0.621626\pi$
$419$	−6396.00	−0.745740	−0.372870	+	0.927884i	$0.621626\pi$
$420$	0	0
$421$	8286.00	0.959228	0.479614	−	0.877480i	$-0.340777\pi$
$421$	8286.00	0.959228	0.479614	+	0.877480i	$0.340777\pi$
$422$	0	0
$423$	− 4896.00i	− 0.562770i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6352.00i	0.719894i
$428$	0	0
$429$	−1320.00	−0.148555
$430$	0	0
$431$	−4112.00	−0.459555	−0.229777	−	0.973243i	$-0.573800\pi$
$431$	−4112.00	−0.459555	−0.229777	+	0.973243i	$0.573800\pi$
$432$	0	0
$433$	5330.00i	0.591555i	0.955257	+	0.295778i	$0.0955788\pi$
$433$	5330.00i	0.591555i	−0.955257	+	0.295778i	$0.904421\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4256.00i	− 0.465886i
$438$	0	0
$439$	−11272.0	−1.22547	−0.612737	−	0.790287i	$-0.709932\pi$
$439$	−11272.0	−1.22547	−0.612737	+	0.790287i	$0.709932\pi$
$440$	0	0
$441$	−2511.00	−0.271137
$442$	0	0
$443$	14196.0i	1.52251i	0.648452	+	0.761255i	$0.275417\pi$
$443$	14196.0i	1.52251i	−0.648452	+	0.761255i	$0.724583\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 4458.00i	− 0.471714i
$448$	0	0
$449$	5886.00	0.618658	0.309329	−	0.950955i	$-0.399896\pi$
$449$	5886.00	0.618658	0.309329	+	0.950955i	$0.399896\pi$
$450$	0	0
$451$	−7800.00	−0.814385
$452$	0	0
$453$	6360.00i	0.659644i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7526.00i	0.770353i	0.922843	+	0.385177i	$0.125859\pi$
$457$	7526.00i	0.770353i	−0.922843	+	0.385177i	$0.874141\pi$
$458$	0	0
$459$	378.000	0.0384391
$460$	0	0
$461$	8502.00	0.858954	0.429477	−	0.903078i	$-0.358698\pi$
$461$	8502.00	0.858954	0.429477	+	0.903078i	$0.358698\pi$
$462$	0	0
$463$	12672.0i	1.27196i	0.771705	+	0.635980i	$0.219404\pi$
$463$	12672.0i	1.27196i	−0.771705	+	0.635980i	$0.780596\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 16540.0i	− 1.63893i	−0.573130	−	0.819465i	$-0.694271\pi$
$467$	− 16540.0i	− 1.63893i	0.573130	−	0.819465i	$-0.305729\pi$
$468$	0	0
$469$	1184.00	0.116572
$470$	0	0
$471$	−5550.00	−0.542952
$472$	0	0
$473$	7760.00i	0.754345i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1890.00i	0.181420i
$478$	0	0
$479$	−8864.00	−0.845525	−0.422763	−	0.906241i	$-0.638940\pi$
$479$	−8864.00	−0.845525	−0.422763	+	0.906241i	$0.638940\pi$
$480$	0	0
$481$	−3564.00	−0.337847
$482$	0	0
$483$	1344.00i	0.126613i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 3688.00i	− 0.343161i	−0.985170	−	0.171580i	$-0.945113\pi$
$487$	− 3688.00i	− 0.343161i	0.985170	−	0.171580i	$-0.0548873\pi$
$488$	0	0
$489$	3516.00	0.325151
$490$	0	0
$491$	−16140.0	−1.48348	−0.741739	−	0.670688i	$-0.765999\pi$
$491$	−16140.0	−1.48348	−0.741739	+	0.670688i	$0.765999\pi$
$492$	0	0
$493$	2156.00i	0.196960i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6720.00i	0.606505i
$498$	0	0
$499$	−1580.00	−0.141745	−0.0708723	−	0.997485i	$-0.522578\pi$
$499$	−1580.00	−0.141745	−0.0708723	+	0.997485i	$0.522578\pi$
$500$	0	0
$501$	−4968.00	−0.443022
$502$	0	0
$503$	15000.0i	1.32966i	0.746996	+	0.664828i	$0.231495\pi$
$503$	15000.0i	1.32966i	−0.746996	+	0.664828i	$0.768505\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	5139.00i	0.450160i
$508$	0	0
$509$	−20486.0	−1.78394	−0.891971	−	0.452094i	$-0.850677\pi$
$509$	−20486.0	−1.78394	−0.891971	+	0.452094i	$0.850677\pi$
$510$	0	0
$511$	6864.00	0.594218
$512$	0	0
$513$	2052.00i	0.176604i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	10880.0i	0.925535i
$518$	0	0
$519$	7998.00	0.676442
$520$	0	0
$521$	7706.00	0.647996	0.323998	−	0.946058i	$-0.394973\pi$
$521$	7706.00	0.647996	0.323998	+	0.946058i	$0.394973\pi$
$522$	0	0
$523$	− 3932.00i	− 0.328746i	−0.986398	−	0.164373i	$-0.947440\pi$
$523$	− 3932.00i	− 0.328746i	0.986398	−	0.164373i	$-0.0525601\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2240.00i	0.185154i
$528$	0	0
$529$	9031.00	0.742254
$530$	0	0
$531$	−3420.00	−0.279502
$532$	0	0
$533$	− 8580.00i	− 0.697263i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 3396.00i	− 0.272902i
$538$	0	0
$539$	5580.00	0.445914
$540$	0	0
$541$	−23930.0	−1.90172	−0.950860	−	0.309620i	$-0.899798\pi$
$541$	−23930.0	−1.90172	−0.950860	+	0.309620i	$0.899798\pi$
$542$	0	0
$543$	− 8598.00i	− 0.679513i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 11468.0i	− 0.896410i	−0.893931	−	0.448205i	$-0.852063\pi$
$547$	− 11468.0i	− 0.896410i	0.893931	−	0.448205i	$-0.147937\pi$
$548$	0	0
$549$	7146.00	0.555526
$550$	0	0
$551$	−11704.0	−0.904913
$552$	0	0
$553$	1152.00i	0.0885859i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11498.0i	0.874660i	0.899301	+	0.437330i	$0.144076\pi$
$557$	11498.0i	0.874660i	−0.899301	+	0.437330i	$0.855924\pi$
$558$	0	0
$559$	−8536.00	−0.645857
$560$	0	0
$561$	−840.000	−0.0632172
$562$	0	0
$563$	16988.0i	1.27169i	0.771819	+	0.635843i	$0.219347\pi$
$563$	16988.0i	1.27169i	−0.771819	+	0.635843i	$0.780653\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 648.000i	− 0.0479955i
$568$	0	0
$569$	17366.0	1.27947	0.639737	−	0.768594i	$-0.279043\pi$
$569$	17366.0	1.27947	0.639737	+	0.768594i	$0.279043\pi$
$570$	0	0
$571$	−24860.0	−1.82199	−0.910997	−	0.412413i	$-0.864686\pi$
$571$	−24860.0	−1.82199	−0.910997	+	0.412413i	$0.864686\pi$
$572$	0	0
$573$	5664.00i	0.412944i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	26302.0i	1.89769i	0.315744	+	0.948845i	$0.397746\pi$
$577$	26302.0i	1.89769i	−0.315744	+	0.948845i	$0.602254\pi$
$578$	0	0
$579$	−3846.00	−0.276052
$580$	0	0
$581$	2528.00	0.180515
$582$	0	0
$583$	− 4200.00i	− 0.298364i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 7812.00i	− 0.549294i	−0.961545	−	0.274647i	$-0.911439\pi$
$587$	− 7812.00i	− 0.549294i	0.961545	−	0.274647i	$-0.0885610\pi$
$588$	0	0
$589$	−12160.0	−0.850669
$590$	0	0
$591$	1050.00	0.0730816
$592$	0	0
$593$	7986.00i	0.553028i	0.961010	+	0.276514i	$0.0891792\pi$
$593$	7986.00i	0.553028i	−0.961010	+	0.276514i	$0.910821\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10200.0i	0.699260i
$598$	0	0
$599$	21048.0	1.43572	0.717861	−	0.696186i	$-0.245121\pi$
$599$	21048.0	1.43572	0.717861	+	0.696186i	$0.245121\pi$
$600$	0	0
$601$	1738.00	0.117961	0.0589804	−	0.998259i	$-0.481215\pi$
$601$	1738.00	0.117961	0.0589804	+	0.998259i	$0.481215\pi$
$602$	0	0
$603$	− 1332.00i	− 0.0899556i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 18576.0i	− 1.24214i	−0.783757	−	0.621068i	$-0.786699\pi$
$607$	− 18576.0i	− 1.24214i	0.783757	−	0.621068i	$-0.213301\pi$
$608$	0	0
$609$	3696.00	0.245927
$610$	0	0
$611$	−11968.0	−0.792428
$612$	0	0
$613$	− 13602.0i	− 0.896215i	−0.893980	−	0.448107i	$-0.852098\pi$
$613$	− 13602.0i	− 0.896215i	0.893980	−	0.448107i	$-0.147902\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 19578.0i	− 1.27744i	−0.769439	−	0.638720i	$-0.779464\pi$
$617$	− 19578.0i	− 1.27744i	0.769439	−	0.638720i	$-0.220536\pi$
$618$	0	0
$619$	−12308.0	−0.799193	−0.399596	−	0.916691i	$-0.630850\pi$
$619$	−12308.0	−0.799193	−0.399596	+	0.916691i	$0.630850\pi$
$620$	0	0
$621$	1512.00	0.0977045
$622$	0	0
$623$	8784.00i	0.564885i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 4560.00i	− 0.290445i
$628$	0	0
$629$	−2268.00	−0.143770
$630$	0	0
$631$	−8600.00	−0.542568	−0.271284	−	0.962499i	$-0.587448\pi$
$631$	−8600.00	−0.542568	−0.271284	+	0.962499i	$0.587448\pi$
$632$	0	0
$633$	13956.0i	0.876305i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6138.00i	0.381784i
$638$	0	0
$639$	7560.00	0.468027
$640$	0	0
$641$	6978.00	0.429976	0.214988	−	0.976617i	$-0.431029\pi$
$641$	6978.00	0.429976	0.214988	+	0.976617i	$0.431029\pi$
$642$	0	0
$643$	− 7668.00i	− 0.470290i	−0.971960	−	0.235145i	$-0.924444\pi$
$643$	− 7668.00i	− 0.470290i	0.971960	−	0.235145i	$-0.0755565\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15384.0i	0.934787i	0.884049	+	0.467394i	$0.154807\pi$
$647$	15384.0i	0.934787i	−0.884049	+	0.467394i	$0.845193\pi$
$648$	0	0
$649$	7600.00	0.459670
$650$	0	0
$651$	3840.00	0.231185
$652$	0	0
$653$	− 2186.00i	− 0.131003i	−0.997852	−	0.0655014i	$-0.979135\pi$
$653$	− 2186.00i	− 0.131003i	0.997852	−	0.0655014i	$-0.0208647\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 7722.00i	− 0.458545i
$658$	0	0
$659$	1524.00	0.0900859	0.0450430	−	0.998985i	$-0.485658\pi$
$659$	1524.00	0.0900859	0.0450430	+	0.998985i	$0.485658\pi$
$660$	0	0
$661$	−4242.00	−0.249614	−0.124807	−	0.992181i	$-0.539831\pi$
$661$	−4242.00	−0.249614	−0.124807	+	0.992181i	$0.539831\pi$
$662$	0	0
$663$	− 924.000i	− 0.0541255i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8624.00i	0.500634i
$668$	0	0
$669$	−12048.0	−0.696267
$670$	0	0
$671$	−15880.0	−0.913622
$672$	0	0
$673$	24354.0i	1.39491i	0.716626	+	0.697457i	$0.245685\pi$
$673$	24354.0i	1.39491i	−0.716626	+	0.697457i	$0.754315\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	322.000i	0.0182799i	0.999958	+	0.00913993i	$0.00290937\pi$
$677$	322.000i	0.0182799i	−0.999958	+	0.00913993i	$0.997091\pi$
$678$	0	0
$679$	−7952.00	−0.449440
$680$	0	0
$681$	6948.00	0.390966
$682$	0	0
$683$	− 7932.00i	− 0.444377i	−0.975004	−	0.222189i	$-0.928680\pi$
$683$	− 7932.00i	− 0.444377i	0.975004	−	0.222189i	$-0.0713201\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 282.000i	− 0.0156608i
$688$	0	0
$689$	4620.00	0.255454
$690$	0	0
$691$	20684.0	1.13872	0.569361	−	0.822088i	$-0.307191\pi$
$691$	20684.0	1.13872	0.569361	+	0.822088i	$0.307191\pi$
$692$	0	0
$693$	1440.00i	0.0789337i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 5460.00i	− 0.296718i
$698$	0	0
$699$	12690.0	0.686666
$700$	0	0
$701$	25222.0	1.35895	0.679473	−	0.733700i	$-0.262208\pi$
$701$	25222.0	1.35895	0.679473	+	0.733700i	$0.262208\pi$
$702$	0	0
$703$	− 12312.0i	− 0.660535i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6672.00i	0.354917i
$708$	0	0
$709$	−23678.0	−1.25423	−0.627113	−	0.778928i	$-0.715764\pi$
$709$	−23678.0	−1.25423	−0.627113	+	0.778928i	$0.715764\pi$
$710$	0	0
$711$	1296.00	0.0683598
$712$	0	0
$713$	8960.00i	0.470624i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 6192.00i	− 0.322517i
$718$	0	0
$719$	−8432.00	−0.437358	−0.218679	−	0.975797i	$-0.570175\pi$
$719$	−8432.00	−0.437358	−0.218679	+	0.975797i	$0.570175\pi$
$720$	0	0
$721$	13376.0	0.690913
$722$	0	0
$723$	13686.0i	0.703994i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 8312.00i	− 0.424037i	−0.977266	−	0.212019i	$-0.931996\pi$
$727$	− 8312.00i	− 0.424037i	0.977266	−	0.212019i	$-0.0680037\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−5432.00	−0.274842
$732$	0	0
$733$	− 26298.0i	− 1.32516i	−0.748993	−	0.662578i	$-0.769462\pi$
$733$	− 26298.0i	− 1.32516i	0.748993	−	0.662578i	$-0.230538\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2960.00i	0.147942i
$738$	0	0
$739$	−16956.0	−0.844028	−0.422014	−	0.906589i	$-0.638677\pi$
$739$	−16956.0	−0.844028	−0.422014	+	0.906589i	$0.638677\pi$
$740$	0	0
$741$	5016.00	0.248674
$742$	0	0
$743$	− 17880.0i	− 0.882845i	−0.897299	−	0.441422i	$-0.854474\pi$
$743$	− 17880.0i	− 0.882845i	0.897299	−	0.441422i	$-0.145526\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 2844.00i	− 0.139299i
$748$	0	0
$749$	5856.00	0.285679
$750$	0	0
$751$	22032.0	1.07052	0.535259	−	0.844688i	$-0.320214\pi$
$751$	22032.0	1.07052	0.535259	+	0.844688i	$0.320214\pi$
$752$	0	0
$753$	7596.00i	0.367614i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 11534.0i	− 0.553779i	−0.960902	−	0.276889i	$-0.910696\pi$
$757$	− 11534.0i	− 0.553779i	0.960902	−	0.276889i	$-0.0893035\pi$
$758$	0	0
$759$	−3360.00	−0.160685
$760$	0	0
$761$	38250.0	1.82203	0.911013	−	0.412378i	$-0.135302\pi$
$761$	38250.0	1.82203	0.911013	+	0.412378i	$0.135302\pi$
$762$	0	0
$763$	− 7760.00i	− 0.368192i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8360.00i	0.393562i
$768$	0	0
$769$	−19330.0	−0.906447	−0.453223	−	0.891397i	$-0.649726\pi$
$769$	−19330.0	−0.906447	−0.453223	+	0.891397i	$0.649726\pi$
$770$	0	0
$771$	10566.0	0.493548
$772$	0	0
$773$	− 40674.0i	− 1.89255i	−0.323361	−	0.946276i	$-0.604813\pi$
$773$	− 40674.0i	− 1.89255i	0.323361	−	0.946276i	$-0.395187\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3888.00i	0.179513i
$778$	0	0
$779$	29640.0	1.36324
$780$	0	0
$781$	−16800.0	−0.769720
$782$	0	0
$783$	− 4158.00i	− 0.189776i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 7004.00i	− 0.317237i	−0.987340	−	0.158619i	$-0.949296\pi$
$787$	− 7004.00i	− 0.317237i	0.987340	−	0.158619i	$-0.0507040\pi$
$788$	0	0
$789$	6696.00	0.302134
$790$	0	0
$791$	15504.0	0.696914
$792$	0	0
$793$	− 17468.0i	− 0.782228i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 12198.0i	− 0.542127i	−0.962561	−	0.271064i	$-0.912625\pi$
$797$	− 12198.0i	− 0.542127i	0.962561	−	0.271064i	$-0.0873754\pi$
$798$	0	0
$799$	−7616.00	−0.337215
$800$	0	0
$801$	9882.00	0.435909
$802$	0	0
$803$	17160.0i	0.754126i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 8418.00i	− 0.367197i
$808$	0	0
$809$	25734.0	1.11837	0.559184	−	0.829044i	$-0.311115\pi$
$809$	25734.0	1.11837	0.559184	+	0.829044i	$0.311115\pi$
$810$	0	0
$811$	15668.0	0.678394	0.339197	−	0.940715i	$-0.389845\pi$
$811$	15668.0	0.678394	0.339197	+	0.940715i	$0.389845\pi$
$812$	0	0
$813$	14544.0i	0.627405i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 29488.0i	− 1.26274i
$818$	0	0
$819$	−1584.00	−0.0675817
$820$	0	0
$821$	−34450.0	−1.46445	−0.732225	−	0.681063i	$-0.761518\pi$
$821$	−34450.0	−1.46445	−0.732225	+	0.681063i	$0.761518\pi$
$822$	0	0
$823$	− 38792.0i	− 1.64302i	−0.570195	−	0.821509i	$-0.693132\pi$
$823$	− 38792.0i	− 1.64302i	0.570195	−	0.821509i	$-0.306868\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20460.0i	0.860295i	0.902759	+	0.430147i	$0.141538\pi$
$827$	20460.0i	0.860295i	−0.902759	+	0.430147i	$0.858462\pi$
$828$	0	0
$829$	−5542.00	−0.232185	−0.116093	−	0.993238i	$-0.537037\pi$
$829$	−5542.00	−0.232185	−0.116093	+	0.993238i	$0.537037\pi$
$830$	0	0
$831$	23370.0	0.975567
$832$	0	0
$833$	3906.00i	0.162467i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 4320.00i	− 0.178400i
$838$	0	0
$839$	−25240.0	−1.03860	−0.519298	−	0.854593i	$-0.673806\pi$
$839$	−25240.0	−1.03860	−0.519298	+	0.854593i	$0.673806\pi$
$840$	0	0
$841$	−673.000	−0.0275944
$842$	0	0
$843$	− 354.000i	− 0.0144631i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	7448.00i	0.302144i
$848$	0	0
$849$	19524.0	0.789237
$850$	0	0
$851$	−9072.00	−0.365434
$852$	0	0
$853$	− 37330.0i	− 1.49842i	−0.662331	−	0.749212i	$-0.730433\pi$
$853$	− 37330.0i	− 1.49842i	0.662331	−	0.749212i	$-0.269567\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3894.00i	0.155212i	0.996984	+	0.0776059i	$0.0247276\pi$
$857$	3894.00i	0.155212i	−0.996984	+	0.0776059i	$0.975272\pi$
$858$	0	0
$859$	−20324.0	−0.807271	−0.403636	−	0.914920i	$-0.632254\pi$
$859$	−20324.0	−0.807271	−0.403636	+	0.914920i	$0.632254\pi$
$860$	0	0
$861$	−9360.00	−0.370485
$862$	0	0
$863$	6288.00i	0.248026i	0.992281	+	0.124013i	$0.0395764\pi$
$863$	6288.00i	0.248026i	−0.992281	+	0.124013i	$0.960424\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	14151.0i	0.554317i
$868$	0	0
$869$	−2880.00	−0.112425
$870$	0	0
$871$	−3256.00	−0.126665
$872$	0	0
$873$	8946.00i	0.346823i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	24650.0i	0.949112i	0.880225	+	0.474556i	$0.157391\pi$
$877$	24650.0i	0.949112i	−0.880225	+	0.474556i	$0.842609\pi$
$878$	0	0
$879$	26310.0	1.00957
$880$	0	0
$881$	9426.00	0.360465	0.180233	−	0.983624i	$-0.442315\pi$
$881$	9426.00	0.360465	0.180233	+	0.983624i	$0.442315\pi$
$882$	0	0
$883$	− 9316.00i	− 0.355049i	−0.984116	−	0.177525i	$-0.943191\pi$
$883$	− 9316.00i	− 0.355049i	0.984116	−	0.177525i	$-0.0568089\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 6968.00i	− 0.263768i	−0.991265	−	0.131884i	$-0.957897\pi$
$887$	− 6968.00i	− 0.263768i	0.991265	−	0.131884i	$-0.0421027\pi$
$888$	0	0
$889$	4224.00	0.159357
$890$	0	0
$891$	1620.00	0.0609114
$892$	0	0
$893$	− 41344.0i	− 1.54930i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 3696.00i	− 0.137576i
$898$	0	0
$899$	24640.0	0.914116
$900$	0	0
$901$	2940.00	0.108708
$902$	0	0
$903$	9312.00i	0.343172i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	35964.0i	1.31661i	0.752751	+	0.658305i	$0.228726\pi$
$907$	35964.0i	1.31661i	−0.752751	+	0.658305i	$0.771274\pi$
$908$	0	0
$909$	7506.00	0.273882
$910$	0	0
$911$	47888.0	1.74160	0.870801	−	0.491635i	$-0.163601\pi$
$911$	47888.0	1.74160	0.870801	+	0.491635i	$0.163601\pi$
$912$	0	0
$913$	6320.00i	0.229093i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 5088.00i	− 0.183229i
$918$	0	0
$919$	12760.0	0.458013	0.229006	−	0.973425i	$-0.426452\pi$
$919$	12760.0	0.458013	0.229006	+	0.973425i	$0.426452\pi$
$920$	0	0
$921$	−12876.0	−0.460672
$922$	0	0
$923$	− 18480.0i	− 0.659021i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 15048.0i	− 0.533162i
$928$	0	0
$929$	25054.0	0.884817	0.442409	−	0.896814i	$-0.354124\pi$
$929$	25054.0	0.884817	0.442409	+	0.896814i	$0.354124\pi$
$930$	0	0
$931$	−21204.0	−0.746437
$932$	0	0
$933$	− 28392.0i	− 0.996262i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 28282.0i	− 0.986054i	−0.870014	−	0.493027i	$-0.835890\pi$
$937$	− 28282.0i	− 0.986054i	0.870014	−	0.493027i	$-0.164110\pi$
$938$	0	0
$939$	−28734.0	−0.998614
$940$	0	0
$941$	−30634.0	−1.06125	−0.530627	−	0.847605i	$-0.678043\pi$
$941$	−30634.0	−1.06125	−0.530627	+	0.847605i	$0.678043\pi$
$942$	0	0
$943$	− 21840.0i	− 0.754198i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 48572.0i	− 1.66671i	−0.552735	−	0.833357i	$-0.686416\pi$
$947$	− 48572.0i	− 1.66671i	0.552735	−	0.833357i	$-0.313584\pi$
$948$	0	0
$949$	−18876.0	−0.645670
$950$	0	0
$951$	−558.000	−0.0190267
$952$	0	0
$953$	12906.0i	0.438685i	0.975648	+	0.219342i	$0.0703911\pi$
$953$	12906.0i	0.438685i	−0.975648	+	0.219342i	$0.929609\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9240.00i	0.312107i
$958$	0	0
$959$	−14032.0	−0.472489
$960$	0	0
$961$	−4191.00	−0.140680
$962$	0	0
$963$	− 6588.00i	− 0.220452i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	5880.00i	0.195541i	0.995209	+	0.0977705i	$0.0311711\pi$
$967$	5880.00i	0.195541i	−0.995209	+	0.0977705i	$0.968829\pi$
$968$	0	0
$969$	3192.00	0.105822
$970$	0	0
$971$	55444.0	1.83242	0.916211	−	0.400695i	$-0.131231\pi$
$971$	55444.0	1.83242	0.916211	+	0.400695i	$0.131231\pi$
$972$	0	0
$973$	− 20064.0i	− 0.661071i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 32050.0i	− 1.04951i	−0.851254	−	0.524755i	$-0.824157\pi$
$977$	− 32050.0i	− 1.04951i	0.851254	−	0.524755i	$-0.175843\pi$
$978$	0	0
$979$	−21960.0	−0.716900
$980$	0	0
$981$	−8730.00	−0.284126
$982$	0	0
$983$	29880.0i	0.969506i	0.874651	+	0.484753i	$0.161090\pi$
$983$	29880.0i	0.969506i	−0.874651	+	0.484753i	$0.838910\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	13056.0i	0.421051i
$988$	0	0
$989$	−21728.0	−0.698595
$990$	0	0
$991$	5216.00	0.167196	0.0835982	−	0.996500i	$-0.473359\pi$
$991$	5216.00	0.167196	0.0835982	+	0.996500i	$0.473359\pi$
$992$	0	0
$993$	− 1476.00i	− 0.0471696i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 6750.00i	− 0.214418i	−0.994237	−	0.107209i	$-0.965809\pi$
$997$	− 6750.00i	− 0.214418i	0.994237	−	0.107209i	$-0.0341914\pi$
$998$	0	0
$999$	4374.00	0.138526

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.4.f.g.49.2		2
3.2	odd	2		1800.4.f.h.649.1		2
4.3	odd	2		1200.4.f.g.49.1		2
5.2	odd	4		120.4.a.f.1.1	✓	1
5.3	odd	4		600.4.a.d.1.1		1
5.4	even	2	inner	600.4.f.g.49.1		2
15.2	even	4		360.4.a.e.1.1		1
15.8	even	4		1800.4.a.k.1.1		1
15.14	odd	2		1800.4.f.h.649.2		2
20.3	even	4		1200.4.a.bf.1.1		1
20.7	even	4		240.4.a.d.1.1		1
20.19	odd	2		1200.4.f.g.49.2		2
40.27	even	4		960.4.a.v.1.1		1
40.37	odd	4		960.4.a.g.1.1		1
60.47	odd	4		720.4.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.a.f.1.1	✓	1	5.2	odd	4
240.4.a.d.1.1		1	20.7	even	4
360.4.a.e.1.1		1	15.2	even	4
600.4.a.d.1.1		1	5.3	odd	4
600.4.f.g.49.1		2	5.4	even	2	inner
600.4.f.g.49.2		2	1.1	even	1	trivial
720.4.a.f.1.1		1	60.47	odd	4
960.4.a.g.1.1		1	40.37	odd	4
960.4.a.v.1.1		1	40.27	even	4
1200.4.a.bf.1.1		1	20.3	even	4
1200.4.f.g.49.1		2	4.3	odd	2
1200.4.f.g.49.2		2	20.19	odd	2
1800.4.a.k.1.1		1	15.8	even	4
1800.4.f.h.649.1		2	3.2	odd	2
1800.4.f.h.649.2		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	600.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} -8.00000i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} -8.00000i q^{7} -9.00000 q^{9} +20.0000 q^{11} +22.0000i q^{13} +14.0000i q^{17} -76.0000 q^{19} +24.0000 q^{21} +56.0000i q^{23} -27.0000i q^{27} +154.000 q^{29} +160.000 q^{31} +60.0000i q^{33} +162.000i q^{37} -66.0000 q^{39} -390.000 q^{41} +388.000i q^{43} +544.000i q^{47} +279.000 q^{49} -42.0000 q^{51} -210.000i q^{53} -228.000i q^{57} +380.000 q^{59} -794.000 q^{61} +72.0000i q^{63} +148.000i q^{67} -168.000 q^{69} -840.000 q^{71} +858.000i q^{73} -160.000i q^{77} -144.000 q^{79} +81.0000 q^{81} +316.000i q^{83} +462.000i q^{87} -1098.00 q^{89} +176.000 q^{91} +480.000i q^{93} -994.000i q^{97} -180.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9}+O(q^{10}) \) 2 * q - 18 * q^9	\( 2 q - 18 q^{9} + 40 q^{11} - 152 q^{19} + 48 q^{21} + 308 q^{29} + 320 q^{31} - 132 q^{39} - 780 q^{41} + 558 q^{49} - 84 q^{51} + 760 q^{59} - 1588 q^{61} - 336 q^{69} - 1680 q^{71} - 288 q^{79} + 162 q^{81} - 2196 q^{89} + 352 q^{91} - 360 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 + 40 * q^11 - 152 * q^19 + 48 * q^21 + 308 * q^29 + 320 * q^31 - 132 * q^39 - 780 * q^41 + 558 * q^49 - 84 * q^51 + 760 * q^59 - 1588 * q^61 - 336 * q^69 - 1680 * q^71 - 288 * q^79 + 162 * q^81 - 2196 * q^89 + 352 * q^91 - 360 * q^99

Introduction

L-functions

Modular forms

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Database

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