LMFDB - Embedded newform 825.4.c.g.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, 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("825.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	825.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:	$48.6765757547$
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 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.4.c.g.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +8.00000 q^{4} -2.00000i q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} +8.00000 q^{4} -2.00000i q^{7} -9.00000 q^{9} -11.0000 q^{11} -24.0000i q^{12} -22.0000i q^{13} +64.0000 q^{16} -72.0000i q^{17} -122.000 q^{19} -6.00000 q^{21} +72.0000i q^{23} +27.0000i q^{27} -16.0000i q^{28} -96.0000 q^{29} -112.000 q^{31} +33.0000i q^{33} -72.0000 q^{36} -266.000i q^{37} -66.0000 q^{39} -96.0000 q^{41} -382.000i q^{43} -88.0000 q^{44} -360.000i q^{47} -192.000i q^{48} +339.000 q^{49} -216.000 q^{51} -176.000i q^{52} +318.000i q^{53} +366.000i q^{57} -660.000 q^{59} -430.000 q^{61} +18.0000i q^{63} +512.000 q^{64} -380.000i q^{67} -576.000i q^{68} +216.000 q^{69} +168.000 q^{71} +218.000i q^{73} -976.000 q^{76} +22.0000i q^{77} +706.000 q^{79} +81.0000 q^{81} +1068.00i q^{83} -48.0000 q^{84} +288.000i q^{87} +6.00000 q^{89} -44.0000 q^{91} +576.000i q^{92} +336.000i q^{93} -686.000i q^{97} +99.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 16 q^{4} - 18 q^{9}+O(q^{10}) $ 2 * q + 16 * q^4 - 18 * q^9	$ 2 q + 16 q^{4} - 18 q^{9} - 22 q^{11} + 128 q^{16} - 244 q^{19} - 12 q^{21} - 192 q^{29} - 224 q^{31} - 144 q^{36} - 132 q^{39} - 192 q^{41} - 176 q^{44} + 678 q^{49} - 432 q^{51} - 1320 q^{59} - 860 q^{61} + 1024 q^{64} + 432 q^{69} + 336 q^{71} - 1952 q^{76} + 1412 q^{79} + 162 q^{81} - 96 q^{84} + 12 q^{89} - 88 q^{91} + 198 q^{99}+O(q^{100}) $ 2 * q + 16 * q^4 - 18 * q^9 - 22 * q^11 + 128 * q^16 - 244 * q^19 - 12 * q^21 - 192 * q^29 - 224 * q^31 - 144 * q^36 - 132 * q^39 - 192 * q^41 - 176 * q^44 + 678 * q^49 - 432 * q^51 - 1320 * q^59 - 860 * q^61 + 1024 * q^64 + 432 * q^69 + 336 * q^71 - 1952 * q^76 + 1412 * q^79 + 162 * q^81 - 96 * q^84 + 12 * q^89 - 88 * q^91 + 198 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$1$	$-1$

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	1.00000			$0$
$2$	0	0	−1.00000			$\pi$
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	8.00000	1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.107990i	−0.998541	−	0.0539949i	$-0.982805\pi$
$7$	− 2.00000i	− 0.107990i	0.998541	−	0.0539949i	$-0.0171955\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	− 24.0000i	− 0.577350i
$13$	− 22.0000i	− 0.469362i	−0.972072	−	0.234681i	$-0.924595\pi$
$13$	− 22.0000i	− 0.469362i	0.972072	−	0.234681i	$-0.0754045\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	1.00000
$17$	− 72.0000i	− 1.02721i	−0.858027	−	0.513605i	$-0.828310\pi$
$17$	− 72.0000i	− 1.02721i	0.858027	−	0.513605i	$-0.171690\pi$
$18$	0	0
$19$	−122.000	−1.47309	−0.736545	−	0.676388i	$-0.763544\pi$
$19$	−122.000	−1.47309	−0.736545	+	0.676388i	$0.763544\pi$
$20$	0	0
$21$	−6.00000	−0.0623480
$22$	0	0
$23$	72.0000i	0.652741i	0.945242	+	0.326370i	$0.105826\pi$
$23$	72.0000i	0.652741i	−0.945242	+	0.326370i	$0.894174\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	0.192450i
$28$	− 16.0000i	− 0.107990i
$29$	−96.0000	−0.614716	−0.307358	−	0.951594i	$-0.599445\pi$
$29$	−96.0000	−0.614716	−0.307358	+	0.951594i	$0.599445\pi$
$30$	0	0
$31$	−112.000	−0.648897	−0.324448	−	0.945903i	$-0.605179\pi$
$31$	−112.000	−0.648897	−0.324448	+	0.945903i	$0.605179\pi$
$32$	0	0
$33$	33.0000i	0.174078i
$34$	0	0
$35$	0	0
$36$	−72.0000	−0.333333
$37$	− 266.000i	− 1.18190i	−0.806710	−	0.590948i	$-0.798754\pi$
$37$	− 266.000i	− 1.18190i	0.806710	−	0.590948i	$-0.201246\pi$
$38$	0	0
$39$	−66.0000	−0.270986
$40$	0	0
$41$	−96.0000	−0.365675	−0.182838	−	0.983143i	$-0.558528\pi$
$41$	−96.0000	−0.365675	−0.182838	+	0.983143i	$0.558528\pi$
$42$	0	0
$43$	− 382.000i	− 1.35475i	−0.735636	−	0.677377i	$-0.763116\pi$
$43$	− 382.000i	− 1.35475i	0.735636	−	0.677377i	$-0.236884\pi$
$44$	−88.0000	−0.301511
$45$	0	0
$46$	0	0
$47$	− 360.000i	− 1.11726i	−0.829416	−	0.558632i	$-0.811326\pi$
$47$	− 360.000i	− 1.11726i	0.829416	−	0.558632i	$-0.188674\pi$
$48$	− 192.000i	− 0.577350i
$49$	339.000	0.988338
$50$	0	0
$51$	−216.000	−0.593060
$52$	− 176.000i	− 0.469362i
$53$	318.000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	318.000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	366.000i	0.850489i
$58$	0	0
$59$	−660.000	−1.45635	−0.728175	−	0.685391i	$-0.759631\pi$
$59$	−660.000	−1.45635	−0.728175	+	0.685391i	$0.759631\pi$
$60$	0	0
$61$	−430.000	−0.902555	−0.451278	−	0.892384i	$-0.649032\pi$
$61$	−430.000	−0.902555	−0.451278	+	0.892384i	$0.649032\pi$
$62$	0	0
$63$	18.0000i	0.0359966i
$64$	512.000	1.00000
$65$	0	0
$66$	0	0
$67$	− 380.000i	− 0.692901i	−0.938068	−	0.346451i	$-0.887387\pi$
$67$	− 380.000i	− 0.692901i	0.938068	−	0.346451i	$-0.112613\pi$
$68$	− 576.000i	− 1.02721i
$69$	216.000	0.376860
$70$	0	0
$71$	168.000	0.280816	0.140408	−	0.990094i	$-0.455159\pi$
$71$	168.000	0.280816	0.140408	+	0.990094i	$0.455159\pi$
$72$	0	0
$73$	218.000i	0.349520i	0.984611	+	0.174760i	$0.0559150\pi$
$73$	218.000i	0.349520i	−0.984611	+	0.174760i	$0.944085\pi$
$74$	0	0
$75$	0	0
$76$	−976.000	−1.47309
$77$	22.0000i	0.0325602i
$78$	0	0
$79$	706.000	1.00546	0.502729	−	0.864444i	$-0.332329\pi$
$79$	706.000	1.00546	0.502729	+	0.864444i	$0.332329\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1068.00i	1.41239i	0.708018	+	0.706194i	$0.249589\pi$
$83$	1068.00i	1.41239i	−0.708018	+	0.706194i	$0.750411\pi$
$84$	−48.0000	−0.0623480
$85$	0	0
$86$	0	0
$87$	288.000i	0.354906i
$88$	0	0
$89$	6.00000	0.00714605	0.00357303	−	0.999994i	$-0.498863\pi$
$89$	6.00000	0.00714605	0.00357303	+	0.999994i	$0.498863\pi$
$90$	0	0
$91$	−44.0000	−0.0506863
$92$	576.000i	0.652741i
$93$	336.000i	0.374641i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 686.000i	− 0.718070i	−0.933324	−	0.359035i	$-0.883106\pi$
$97$	− 686.000i	− 0.718070i	0.933324	−	0.359035i	$-0.116894\pi$
$98$	0	0
$99$	99.0000	0.100504
$100$	0	0
$101$	−960.000	−0.945778	−0.472889	−	0.881122i	$-0.656789\pi$
$101$	−960.000	−0.945778	−0.472889	+	0.881122i	$0.656789\pi$
$102$	0	0
$103$	− 844.000i	− 0.807396i	−0.914892	−	0.403698i	$-0.867725\pi$
$103$	− 844.000i	− 0.807396i	0.914892	−	0.403698i	$-0.132275\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2172.00i	− 1.96238i	−0.193033	−	0.981192i	$-0.561832\pi$
$107$	− 2172.00i	− 1.96238i	0.193033	−	0.981192i	$-0.438168\pi$
$108$	216.000i	0.192450i
$109$	−614.000	−0.539546	−0.269773	−	0.962924i	$-0.586949\pi$
$109$	−614.000	−0.539546	−0.269773	+	0.962924i	$0.586949\pi$
$110$	0	0
$111$	−798.000	−0.682368
$112$	− 128.000i	− 0.107990i
$113$	1254.00i	1.04395i	0.852961	+	0.521975i	$0.174805\pi$
$113$	1254.00i	1.04395i	−0.852961	+	0.521975i	$0.825195\pi$
$114$	0	0
$115$	0	0
$116$	−768.000	−0.614716
$117$	198.000i	0.156454i
$118$	0	0
$119$	−144.000	−0.110928
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	288.000i	0.211123i
$124$	−896.000	−0.648897
$125$	0	0
$126$	0	0
$127$	− 1394.00i	− 0.973996i	−0.873403	−	0.486998i	$-0.838092\pi$
$127$	− 1394.00i	− 0.973996i	0.873403	−	0.486998i	$-0.161908\pi$
$128$	0	0
$129$	−1146.00	−0.782168
$130$	0	0
$131$	−252.000	−0.168071	−0.0840357	−	0.996463i	$-0.526781\pi$
$131$	−252.000	−0.168071	−0.0840357	+	0.996463i	$0.526781\pi$
$132$	264.000i	0.174078i
$133$	244.000i	0.159079i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1050.00i	0.654800i	0.944886	+	0.327400i	$0.106172\pi$
$137$	1050.00i	0.654800i	−0.944886	+	0.327400i	$0.893828\pi$
$138$	0	0
$139$	−1874.00	−1.14353	−0.571765	−	0.820418i	$-0.693741\pi$
$139$	−1874.00	−1.14353	−0.571765	+	0.820418i	$0.693741\pi$
$140$	0	0
$141$	−1080.00	−0.645053
$142$	0	0
$143$	242.000i	0.141518i
$144$	−576.000	−0.333333
$145$	0	0
$146$	0	0
$147$	− 1017.00i	− 0.570617i
$148$	− 2128.00i	− 1.18190i
$149$	−1476.00	−0.811534	−0.405767	−	0.913976i	$-0.632996\pi$
$149$	−1476.00	−0.811534	−0.405767	+	0.913976i	$0.632996\pi$
$150$	0	0
$151$	1478.00	0.796543	0.398271	−	0.917268i	$-0.369610\pi$
$151$	1478.00	0.796543	0.398271	+	0.917268i	$0.369610\pi$
$152$	0	0
$153$	648.000i	0.342403i
$154$	0	0
$155$	0	0
$156$	−528.000	−0.270986
$157$	− 854.000i	− 0.434119i	−0.976158	−	0.217059i	$-0.930354\pi$
$157$	− 854.000i	− 0.434119i	0.976158	−	0.217059i	$-0.0696465\pi$
$158$	0	0
$159$	954.000	0.475831
$160$	0	0
$161$	144.000	0.0704894
$162$	0	0
$163$	1544.00i	0.741935i	0.928646	+	0.370968i	$0.120974\pi$
$163$	1544.00i	0.741935i	−0.928646	+	0.370968i	$0.879026\pi$
$164$	−768.000	−0.365675
$165$	0	0
$166$	0	0
$167$	− 240.000i	− 0.111208i	−0.998453	−	0.0556041i	$-0.982292\pi$
$167$	− 240.000i	− 0.111208i	0.998453	−	0.0556041i	$-0.0177085\pi$
$168$	0	0
$169$	1713.00	0.779700
$170$	0	0
$171$	1098.00	0.491030
$172$	− 3056.00i	− 1.35475i
$173$	2532.00i	1.11274i	0.830934	+	0.556371i	$0.187807\pi$
$173$	2532.00i	1.11274i	−0.830934	+	0.556371i	$0.812193\pi$
$174$	0	0
$175$	0	0
$176$	−704.000	−0.301511
$177$	1980.00i	0.840824i
$178$	0	0
$179$	−1092.00	−0.455977	−0.227989	−	0.973664i	$-0.573215\pi$
$179$	−1092.00	−0.455977	−0.227989	+	0.973664i	$0.573215\pi$
$180$	0	0
$181$	−2290.00	−0.940411	−0.470205	−	0.882557i	$-0.655820\pi$
$181$	−2290.00	−0.940411	−0.470205	+	0.882557i	$0.655820\pi$
$182$	0	0
$183$	1290.00i	0.521090i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	792.000i	0.309715i
$188$	− 2880.00i	− 1.11726i
$189$	54.0000	0.0207827
$190$	0	0
$191$	−4392.00	−1.66384	−0.831921	−	0.554894i	$-0.812759\pi$
$191$	−4392.00	−1.66384	−0.831921	+	0.554894i	$0.812759\pi$
$192$	− 1536.00i	− 0.577350i
$193$	− 5074.00i	− 1.89241i	−0.323572	−	0.946203i	$-0.604884\pi$
$193$	− 5074.00i	− 1.89241i	0.323572	−	0.946203i	$-0.395116\pi$
$194$	0	0
$195$	0	0
$196$	2712.00	0.988338
$197$	− 1692.00i	− 0.611929i	−0.952043	−	0.305964i	$-0.901021\pi$
$197$	− 1692.00i	− 0.611929i	0.952043	−	0.305964i	$-0.0989789\pi$
$198$	0	0
$199$	−4664.00	−1.66142	−0.830709	−	0.556707i	$-0.812065\pi$
$199$	−4664.00	−1.66142	−0.830709	+	0.556707i	$0.812065\pi$
$200$	0	0
$201$	−1140.00	−0.400047
$202$	0	0
$203$	192.000i	0.0663830i
$204$	−1728.00	−0.593060
$205$	0	0
$206$	0	0
$207$	− 648.000i	− 0.217580i
$208$	− 1408.00i	− 0.469362i
$209$	1342.00	0.444153
$210$	0	0
$211$	−1870.00	−0.610124	−0.305062	−	0.952333i	$-0.598677\pi$
$211$	−1870.00	−0.610124	−0.305062	+	0.952333i	$0.598677\pi$
$212$	2544.00i	0.824163i
$213$	− 504.000i	− 0.162129i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	224.000i	0.0700742i
$218$	0	0
$219$	654.000	0.201796
$220$	0	0
$221$	−1584.00	−0.482133
$222$	0	0
$223$	2300.00i	0.690670i	0.938479	+	0.345335i	$0.112235\pi$
$223$	2300.00i	0.690670i	−0.938479	+	0.345335i	$0.887765\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1332.00i	0.389462i	0.980857	+	0.194731i	$0.0623834\pi$
$227$	1332.00i	0.389462i	−0.980857	+	0.194731i	$0.937617\pi$
$228$	2928.00i	0.850489i
$229$	6022.00	1.73775	0.868875	−	0.495031i	$-0.164843\pi$
$229$	6022.00	1.73775	0.868875	+	0.495031i	$0.164843\pi$
$230$	0	0
$231$	66.0000	0.0187986
$232$	0	0
$233$	− 4716.00i	− 1.32599i	−0.748624	−	0.662994i	$-0.769285\pi$
$233$	− 4716.00i	− 1.32599i	0.748624	−	0.662994i	$-0.230715\pi$
$234$	0	0
$235$	0	0
$236$	−5280.00	−1.45635
$237$	− 2118.00i	− 0.580502i
$238$	0	0
$239$	6420.00	1.73755	0.868777	−	0.495204i	$-0.164907\pi$
$239$	6420.00	1.73755	0.868777	+	0.495204i	$0.164907\pi$
$240$	0	0
$241$	3302.00	0.882575	0.441287	−	0.897366i	$-0.354522\pi$
$241$	3302.00	0.882575	0.441287	+	0.897366i	$0.354522\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	−3440.00	−0.902555
$245$	0	0
$246$	0	0
$247$	2684.00i	0.691412i
$248$	0	0
$249$	3204.00	0.815443
$250$	0	0
$251$	732.000	0.184077	0.0920387	−	0.995755i	$-0.470662\pi$
$251$	732.000	0.184077	0.0920387	+	0.995755i	$0.470662\pi$
$252$	144.000i	0.0359966i
$253$	− 792.000i	− 0.196809i
$254$	0	0
$255$	0	0
$256$	4096.00	1.00000
$257$	3438.00i	0.834461i	0.908801	+	0.417231i	$0.136999\pi$
$257$	3438.00i	0.834461i	−0.908801	+	0.417231i	$0.863001\pi$
$258$	0	0
$259$	−532.000	−0.127633
$260$	0	0
$261$	864.000	0.204905
$262$	0	0
$263$	− 696.000i	− 0.163183i	−0.996666	−	0.0815916i	$-0.974000\pi$
$263$	− 696.000i	− 0.163183i	0.996666	−	0.0815916i	$-0.0260003\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 18.0000i	− 0.00412578i
$268$	− 3040.00i	− 0.692901i
$269$	7338.00	1.66322	0.831609	−	0.555361i	$-0.187420\pi$
$269$	7338.00	1.66322	0.831609	+	0.555361i	$0.187420\pi$
$270$	0	0
$271$	5114.00	1.14632	0.573161	−	0.819443i	$-0.305717\pi$
$271$	5114.00	1.14632	0.573161	+	0.819443i	$0.305717\pi$
$272$	− 4608.00i	− 1.02721i
$273$	132.000i	0.0292637i
$274$	0	0
$275$	0	0
$276$	1728.00	0.376860
$277$	− 986.000i	− 0.213874i	−0.994266	−	0.106937i	$-0.965896\pi$
$277$	− 986.000i	− 0.213874i	0.994266	−	0.106937i	$-0.0341043\pi$
$278$	0	0
$279$	1008.00	0.216299
$280$	0	0
$281$	3312.00	0.703122	0.351561	−	0.936165i	$-0.385651\pi$
$281$	3312.00	0.703122	0.351561	+	0.936165i	$0.385651\pi$
$282$	0	0
$283$	4298.00i	0.902790i	0.892324	+	0.451395i	$0.149073\pi$
$283$	4298.00i	0.902790i	−0.892324	+	0.451395i	$0.850927\pi$
$284$	1344.00	0.280816
$285$	0	0
$286$	0	0
$287$	192.000i	0.0394892i
$288$	0	0
$289$	−271.000	−0.0551598
$290$	0	0
$291$	−2058.00	−0.414578
$292$	1744.00i	0.349520i
$293$	2736.00i	0.545525i	0.962081	+	0.272763i	$0.0879373\pi$
$293$	2736.00i	0.545525i	−0.962081	+	0.272763i	$0.912063\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 297.000i	− 0.0580259i
$298$	0	0
$299$	1584.00	0.306372
$300$	0	0
$301$	−764.000	−0.146300
$302$	0	0
$303$	2880.00i	0.546045i
$304$	−7808.00	−1.47309
$305$	0	0
$306$	0	0
$307$	250.000i	0.0464764i	0.999730	+	0.0232382i	$0.00739761\pi$
$307$	250.000i	0.0464764i	−0.999730	+	0.0232382i	$0.992602\pi$
$308$	176.000i	0.0325602i
$309$	−2532.00	−0.466150
$310$	0	0
$311$	7248.00	1.32153	0.660766	−	0.750592i	$-0.270232\pi$
$311$	7248.00	1.32153	0.660766	+	0.750592i	$0.270232\pi$
$312$	0	0
$313$	− 7786.00i	− 1.40604i	−0.711170	−	0.703020i	$-0.751834\pi$
$313$	− 7786.00i	− 1.40604i	0.711170	−	0.703020i	$-0.248166\pi$
$314$	0	0
$315$	0	0
$316$	5648.00	1.00546
$317$	4230.00i	0.749465i	0.927133	+	0.374733i	$0.122265\pi$
$317$	4230.00i	0.749465i	−0.927133	+	0.374733i	$0.877735\pi$
$318$	0	0
$319$	1056.00	0.185344
$320$	0	0
$321$	−6516.00	−1.13298
$322$	0	0
$323$	8784.00i	1.51317i
$324$	648.000	0.111111
$325$	0	0
$326$	0	0
$327$	1842.00i	0.311507i
$328$	0	0
$329$	−720.000	−0.120653
$330$	0	0
$331$	7736.00	1.28462	0.642310	−	0.766445i	$-0.277976\pi$
$331$	7736.00	1.28462	0.642310	+	0.766445i	$0.277976\pi$
$332$	8544.00i	1.41239i
$333$	2394.00i	0.393965i
$334$	0	0
$335$	0	0
$336$	−384.000	−0.0623480
$337$	2014.00i	0.325548i	0.986663	+	0.162774i	$0.0520441\pi$
$337$	2014.00i	0.325548i	−0.986663	+	0.162774i	$0.947956\pi$
$338$	0	0
$339$	3762.00	0.602725
$340$	0	0
$341$	1232.00	0.195650
$342$	0	0
$343$	− 1364.00i	− 0.214720i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 7692.00i	− 1.18999i	−0.803728	−	0.594997i	$-0.797153\pi$
$347$	− 7692.00i	− 1.18999i	0.803728	−	0.594997i	$-0.202847\pi$
$348$	2304.00i	0.354906i
$349$	1750.00	0.268411	0.134205	−	0.990954i	$-0.457152\pi$
$349$	1750.00	0.268411	0.134205	+	0.990954i	$0.457152\pi$
$350$	0	0
$351$	594.000	0.0903287
$352$	0	0
$353$	8034.00i	1.21135i	0.795712	+	0.605675i	$0.207097\pi$
$353$	8034.00i	1.21135i	−0.795712	+	0.605675i	$0.792903\pi$
$354$	0	0
$355$	0	0
$356$	48.0000	0.00714605
$357$	432.000i	0.0640444i
$358$	0	0
$359$	−2304.00	−0.338720	−0.169360	−	0.985554i	$-0.554170\pi$
$359$	−2304.00	−0.338720	−0.169360	+	0.985554i	$0.554170\pi$
$360$	0	0
$361$	8025.00	1.17000
$362$	0	0
$363$	− 363.000i	− 0.0524864i
$364$	−352.000	−0.0506863
$365$	0	0
$366$	0	0
$367$	2356.00i	0.335101i	0.985863	+	0.167551i	$0.0535858\pi$
$367$	2356.00i	0.335101i	−0.985863	+	0.167551i	$0.946414\pi$
$368$	4608.00i	0.652741i
$369$	864.000	0.121892
$370$	0	0
$371$	636.000	0.0890013
$372$	2688.00i	0.374641i
$373$	− 8602.00i	− 1.19409i	−0.802209	−	0.597044i	$-0.796342\pi$
$373$	− 8602.00i	− 1.19409i	0.802209	−	0.597044i	$-0.203658\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2112.00i	0.288524i
$378$	0	0
$379$	12016.0	1.62855	0.814275	−	0.580479i	$-0.197135\pi$
$379$	12016.0	1.62855	0.814275	+	0.580479i	$0.197135\pi$
$380$	0	0
$381$	−4182.00	−0.562337
$382$	0	0
$383$	1728.00i	0.230540i	0.993334	+	0.115270i	$0.0367733\pi$
$383$	1728.00i	0.230540i	−0.993334	+	0.115270i	$0.963227\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3438.00i	0.451585i
$388$	− 5488.00i	− 0.718070i
$389$	−8010.00	−1.04402	−0.522009	−	0.852940i	$-0.674817\pi$
$389$	−8010.00	−1.04402	−0.522009	+	0.852940i	$0.674817\pi$
$390$	0	0
$391$	5184.00	0.670502
$392$	0	0
$393$	756.000i	0.0970360i
$394$	0	0
$395$	0	0
$396$	792.000	0.100504
$397$	10150.0i	1.28316i	0.767057	+	0.641579i	$0.221720\pi$
$397$	10150.0i	1.28316i	−0.767057	+	0.641579i	$0.778280\pi$
$398$	0	0
$399$	732.000	0.0918442
$400$	0	0
$401$	11862.0	1.47721	0.738604	−	0.674140i	$-0.235486\pi$
$401$	11862.0	1.47721	0.738604	+	0.674140i	$0.235486\pi$
$402$	0	0
$403$	2464.00i	0.304567i
$404$	−7680.00	−0.945778
$405$	0	0
$406$	0	0
$407$	2926.00i	0.356355i
$408$	0	0
$409$	682.000	0.0824517	0.0412258	−	0.999150i	$-0.486874\pi$
$409$	682.000	0.0824517	0.0412258	+	0.999150i	$0.486874\pi$
$410$	0	0
$411$	3150.00	0.378049
$412$	− 6752.00i	− 0.807396i
$413$	1320.00i	0.157271i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5622.00i	0.660217i
$418$	0	0
$419$	−10836.0	−1.26342	−0.631710	−	0.775205i	$-0.717647\pi$
$419$	−10836.0	−1.26342	−0.631710	+	0.775205i	$0.717647\pi$
$420$	0	0
$421$	12350.0	1.42970	0.714848	−	0.699280i	$-0.246496\pi$
$421$	12350.0	1.42970	0.714848	+	0.699280i	$0.246496\pi$
$422$	0	0
$423$	3240.00i	0.372421i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	860.000i	0.0974668i
$428$	− 17376.0i	− 1.96238i
$429$	726.000	0.0817054
$430$	0	0
$431$	−5940.00	−0.663851	−0.331925	−	0.943306i	$-0.607698\pi$
$431$	−5940.00	−0.663851	−0.331925	+	0.943306i	$0.607698\pi$
$432$	1728.00i	0.192450i
$433$	− 12898.0i	− 1.43150i	−0.698358	−	0.715749i	$-0.746086\pi$
$433$	− 12898.0i	− 1.43150i	0.698358	−	0.715749i	$-0.253914\pi$
$434$	0	0
$435$	0	0
$436$	−4912.00	−0.539546
$437$	− 8784.00i	− 0.961546i
$438$	0	0
$439$	−11450.0	−1.24483	−0.622413	−	0.782689i	$-0.713848\pi$
$439$	−11450.0	−1.24483	−0.622413	+	0.782689i	$0.713848\pi$
$440$	0	0
$441$	−3051.00	−0.329446
$442$	0	0
$443$	2100.00i	0.225224i	0.993639	+	0.112612i	$0.0359216\pi$
$443$	2100.00i	0.225224i	−0.993639	+	0.112612i	$0.964078\pi$
$444$	−6384.00	−0.682368
$445$	0	0
$446$	0	0
$447$	4428.00i	0.468540i
$448$	− 1024.00i	− 0.107990i
$449$	11934.0	1.25434	0.627172	−	0.778881i	$-0.284212\pi$
$449$	11934.0	1.25434	0.627172	+	0.778881i	$0.284212\pi$
$450$	0	0
$451$	1056.00	0.110255
$452$	10032.0i	1.04395i
$453$	− 4434.00i	− 0.459884i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 578.000i	− 0.0591635i	−0.999562	−	0.0295817i	$-0.990582\pi$
$457$	− 578.000i	− 0.0591635i	0.999562	−	0.0295817i	$-0.00941753\pi$
$458$	0	0
$459$	1944.00	0.197687
$460$	0	0
$461$	−324.000	−0.0327336	−0.0163668	−	0.999866i	$-0.505210\pi$
$461$	−324.000	−0.0327336	−0.0163668	+	0.999866i	$0.505210\pi$
$462$	0	0
$463$	− 11788.0i	− 1.18323i	−0.806221	−	0.591614i	$-0.798491\pi$
$463$	− 11788.0i	− 1.18323i	0.806221	−	0.591614i	$-0.201509\pi$
$464$	−6144.00	−0.614716
$465$	0	0
$466$	0	0
$467$	14484.0i	1.43520i	0.696454	+	0.717601i	$0.254760\pi$
$467$	14484.0i	1.43520i	−0.696454	+	0.717601i	$0.745240\pi$
$468$	1584.00i	0.156454i
$469$	−760.000	−0.0748263
$470$	0	0
$471$	−2562.00	−0.250638
$472$	0	0
$473$	4202.00i	0.408474i
$474$	0	0
$475$	0	0
$476$	−1152.00	−0.110928
$477$	− 2862.00i	− 0.274721i
$478$	0	0
$479$	3084.00	0.294179	0.147089	−	0.989123i	$-0.453010\pi$
$479$	3084.00	0.294179	0.147089	+	0.989123i	$0.453010\pi$
$480$	0	0
$481$	−5852.00	−0.554736
$482$	0	0
$483$	− 432.000i	− 0.0406971i
$484$	968.000	0.0909091
$485$	0	0
$486$	0	0
$487$	5584.00i	0.519579i	0.965665	+	0.259790i	$0.0836533\pi$
$487$	5584.00i	0.519579i	−0.965665	+	0.259790i	$0.916347\pi$
$488$	0	0
$489$	4632.00	0.428356
$490$	0	0
$491$	−10752.0	−0.988250	−0.494125	−	0.869391i	$-0.664512\pi$
$491$	−10752.0	−0.988250	−0.494125	+	0.869391i	$0.664512\pi$
$492$	2304.00i	0.211123i
$493$	6912.00i	0.631442i
$494$	0	0
$495$	0	0
$496$	−7168.00	−0.648897
$497$	− 336.000i	− 0.0303253i
$498$	0	0
$499$	13372.0	1.19963	0.599813	−	0.800141i	$-0.295242\pi$
$499$	13372.0	1.19963	0.599813	+	0.800141i	$0.295242\pi$
$500$	0	0
$501$	−720.000	−0.0642060
$502$	0	0
$503$	9072.00i	0.804176i	0.915601	+	0.402088i	$0.131715\pi$
$503$	9072.00i	0.804176i	−0.915601	+	0.402088i	$0.868285\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 5139.00i	− 0.450160i
$508$	− 11152.0i	− 0.973996i
$509$	14586.0	1.27016	0.635082	−	0.772445i	$-0.280966\pi$
$509$	14586.0	1.27016	0.635082	+	0.772445i	$0.280966\pi$
$510$	0	0
$511$	436.000	0.0377446
$512$	0	0
$513$	− 3294.00i	− 0.283496i
$514$	0	0
$515$	0	0
$516$	−9168.00	−0.782168
$517$	3960.00i	0.336868i
$518$	0	0
$519$	7596.00	0.642442
$520$	0	0
$521$	2718.00	0.228556	0.114278	−	0.993449i	$-0.463545\pi$
$521$	2718.00	0.228556	0.114278	+	0.993449i	$0.463545\pi$
$522$	0	0
$523$	− 2086.00i	− 0.174406i	−0.996191	−	0.0872031i	$-0.972207\pi$
$523$	− 2086.00i	− 0.174406i	0.996191	−	0.0872031i	$-0.0277929\pi$
$524$	−2016.00	−0.168071
$525$	0	0
$526$	0	0
$527$	8064.00i	0.666553i
$528$	2112.00i	0.174078i
$529$	6983.00	0.573929
$530$	0	0
$531$	5940.00	0.485450
$532$	1952.00i	0.159079i
$533$	2112.00i	0.171634i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3276.00i	0.263259i
$538$	0	0
$539$	−3729.00	−0.297995
$540$	0	0
$541$	13838.0	1.09971	0.549854	−	0.835261i	$-0.314683\pi$
$541$	13838.0	1.09971	0.549854	+	0.835261i	$0.314683\pi$
$542$	0	0
$543$	6870.00i	0.542946i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 23546.0i	− 1.84050i	−0.391329	−	0.920251i	$-0.627985\pi$
$547$	− 23546.0i	− 1.84050i	0.391329	−	0.920251i	$-0.372015\pi$
$548$	8400.00i	0.654800i
$549$	3870.00	0.300852
$550$	0	0
$551$	11712.0	0.905532
$552$	0	0
$553$	− 1412.00i	− 0.108579i
$554$	0	0
$555$	0	0
$556$	−14992.0	−1.14353
$557$	− 15624.0i	− 1.18853i	−0.804270	−	0.594264i	$-0.797443\pi$
$557$	− 15624.0i	− 1.18853i	0.804270	−	0.594264i	$-0.202557\pi$
$558$	0	0
$559$	−8404.00	−0.635870
$560$	0	0
$561$	2376.00	0.178814
$562$	0	0
$563$	2400.00i	0.179659i	0.995957	+	0.0898294i	$0.0286322\pi$
$563$	2400.00i	0.179659i	−0.995957	+	0.0898294i	$0.971368\pi$
$564$	−8640.00	−0.645053
$565$	0	0
$566$	0	0
$567$	− 162.000i	− 0.0119989i
$568$	0	0
$569$	18300.0	1.34829	0.674144	−	0.738600i	$-0.264513\pi$
$569$	18300.0	1.34829	0.674144	+	0.738600i	$0.264513\pi$
$570$	0	0
$571$	25454.0	1.86553	0.932764	−	0.360487i	$-0.117389\pi$
$571$	25454.0	1.86553	0.932764	+	0.360487i	$0.117389\pi$
$572$	1936.00i	0.141518i
$573$	13176.0i	0.960620i
$574$	0	0
$575$	0	0
$576$	−4608.00	−0.333333
$577$	− 19802.0i	− 1.42871i	−0.699781	−	0.714357i	$-0.746719\pi$
$577$	− 19802.0i	− 1.42871i	0.699781	−	0.714357i	$-0.253281\pi$
$578$	0	0
$579$	−15222.0	−1.09258
$580$	0	0
$581$	2136.00	0.152524
$582$	0	0
$583$	− 3498.00i	− 0.248495i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 18396.0i	− 1.29350i	−0.762702	−	0.646750i	$-0.776128\pi$
$587$	− 18396.0i	− 1.29350i	0.762702	−	0.646750i	$-0.223872\pi$
$588$	− 8136.00i	− 0.570617i
$589$	13664.0	0.955883
$590$	0	0
$591$	−5076.00	−0.353297
$592$	− 17024.0i	− 1.18190i
$593$	15012.0i	1.03958i	0.854295	+	0.519788i	$0.173989\pi$
$593$	15012.0i	1.03958i	−0.854295	+	0.519788i	$0.826011\pi$
$594$	0	0
$595$	0	0
$596$	−11808.0	−0.811534
$597$	13992.0i	0.959220i
$598$	0	0
$599$	−15408.0	−1.05101	−0.525504	−	0.850791i	$-0.676123\pi$
$599$	−15408.0	−1.05101	−0.525504	+	0.850791i	$0.676123\pi$
$600$	0	0
$601$	−1558.00	−0.105744	−0.0528720	−	0.998601i	$-0.516838\pi$
$601$	−1558.00	−0.105744	−0.0528720	+	0.998601i	$0.516838\pi$
$602$	0	0
$603$	3420.00i	0.230967i
$604$	11824.0	0.796543
$605$	0	0
$606$	0	0
$607$	− 22970.0i	− 1.53595i	−0.640478	−	0.767977i	$-0.721264\pi$
$607$	− 22970.0i	− 1.53595i	0.640478	−	0.767977i	$-0.278736\pi$
$608$	0	0
$609$	576.000	0.0383263
$610$	0	0
$611$	−7920.00	−0.524401
$612$	5184.00i	0.342403i
$613$	− 11482.0i	− 0.756531i	−0.925697	−	0.378266i	$-0.876521\pi$
$613$	− 11482.0i	− 0.756531i	0.925697	−	0.378266i	$-0.123479\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 246.000i	− 0.0160512i	−0.999968	−	0.00802560i	$-0.997445\pi$
$617$	− 246.000i	− 0.0160512i	0.999968	−	0.00802560i	$-0.00255465\pi$
$618$	0	0
$619$	−11648.0	−0.756337	−0.378169	−	0.925737i	$-0.623446\pi$
$619$	−11648.0	−0.756337	−0.378169	+	0.925737i	$0.623446\pi$
$620$	0	0
$621$	−1944.00	−0.125620
$622$	0	0
$623$	− 12.0000i	0 0.000771701i
$624$	−4224.00	−0.270986
$625$	0	0
$626$	0	0
$627$	− 4026.00i	− 0.256432i
$628$	− 6832.00i	− 0.434119i
$629$	−19152.0	−1.21405
$630$	0	0
$631$	−22024.0	−1.38948	−0.694740	−	0.719261i	$-0.744480\pi$
$631$	−22024.0	−1.38948	−0.694740	+	0.719261i	$0.744480\pi$
$632$	0	0
$633$	5610.00i	0.352255i
$634$	0	0
$635$	0	0
$636$	7632.00	0.475831
$637$	− 7458.00i	− 0.463888i
$638$	0	0
$639$	−1512.00	−0.0936053
$640$	0	0
$641$	2322.00	0.143079	0.0715394	−	0.997438i	$-0.477209\pi$
$641$	2322.00	0.143079	0.0715394	+	0.997438i	$0.477209\pi$
$642$	0	0
$643$	14024.0i	0.860113i	0.902802	+	0.430056i	$0.141506\pi$
$643$	14024.0i	0.860113i	−0.902802	+	0.430056i	$0.858494\pi$
$644$	1152.00	0.0704894
$645$	0	0
$646$	0	0
$647$	7152.00i	0.434581i	0.976107	+	0.217291i	$0.0697219\pi$
$647$	7152.00i	0.434581i	−0.976107	+	0.217291i	$0.930278\pi$
$648$	0	0
$649$	7260.00	0.439106
$650$	0	0
$651$	672.000	0.0404574
$652$	12352.0i	0.741935i
$653$	− 3138.00i	− 0.188054i	−0.995570	−	0.0940271i	$-0.970026\pi$
$653$	− 3138.00i	− 0.188054i	0.995570	−	0.0940271i	$-0.0299740\pi$
$654$	0	0
$655$	0	0
$656$	−6144.00	−0.365675
$657$	− 1962.00i	− 0.116507i
$658$	0	0
$659$	15876.0	0.938454	0.469227	−	0.883078i	$-0.344533\pi$
$659$	15876.0	0.938454	0.469227	+	0.883078i	$0.344533\pi$
$660$	0	0
$661$	−20554.0	−1.20947	−0.604734	−	0.796428i	$-0.706720\pi$
$661$	−20554.0	−1.20947	−0.604734	+	0.796428i	$0.706720\pi$
$662$	0	0
$663$	4752.00i	0.278360i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 6912.00i	− 0.401250i
$668$	− 1920.00i	− 0.111208i
$669$	6900.00	0.398758
$670$	0	0
$671$	4730.00	0.272131
$672$	0	0
$673$	27806.0i	1.59263i	0.604880	+	0.796317i	$0.293221\pi$
$673$	27806.0i	1.59263i	−0.604880	+	0.796317i	$0.706779\pi$
$674$	0	0
$675$	0	0
$676$	13704.0	0.779700
$677$	− 20820.0i	− 1.18195i	−0.806691	−	0.590973i	$-0.798744\pi$
$677$	− 20820.0i	− 1.18195i	0.806691	−	0.590973i	$-0.201256\pi$
$678$	0	0
$679$	−1372.00	−0.0775442
$680$	0	0
$681$	3996.00	0.224856
$682$	0	0
$683$	− 7020.00i	− 0.393284i	−0.980475	−	0.196642i	$-0.936996\pi$
$683$	− 7020.00i	− 0.393284i	0.980475	−	0.196642i	$-0.0630036\pi$
$684$	8784.00	0.491030
$685$	0	0
$686$	0	0
$687$	− 18066.0i	− 1.00329i
$688$	− 24448.0i	− 1.35475i
$689$	6996.00	0.386831
$690$	0	0
$691$	12536.0	0.690147	0.345074	−	0.938576i	$-0.387854\pi$
$691$	12536.0	0.690147	0.345074	+	0.938576i	$0.387854\pi$
$692$	20256.0i	1.11274i
$693$	− 198.000i	− 0.0108534i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6912.00i	0.375625i
$698$	0	0
$699$	−14148.0	−0.765560
$700$	0	0
$701$	−33276.0	−1.79289	−0.896446	−	0.443153i	$-0.853860\pi$
$701$	−33276.0	−1.79289	−0.896446	+	0.443153i	$0.853860\pi$
$702$	0	0
$703$	32452.0i	1.74104i
$704$	−5632.00	−0.301511
$705$	0	0
$706$	0	0
$707$	1920.00i	0.102134i
$708$	15840.0i	0.840824i
$709$	−9818.00	−0.520060	−0.260030	−	0.965601i	$-0.583733\pi$
$709$	−9818.00	−0.520060	−0.260030	+	0.965601i	$0.583733\pi$
$710$	0	0
$711$	−6354.00	−0.335153
$712$	0	0
$713$	− 8064.00i	− 0.423561i
$714$	0	0
$715$	0	0
$716$	−8736.00	−0.455977
$717$	− 19260.0i	− 1.00318i
$718$	0	0
$719$	−3216.00	−0.166810	−0.0834051	−	0.996516i	$-0.526580\pi$
$719$	−3216.00	−0.166810	−0.0834051	+	0.996516i	$0.526580\pi$
$720$	0	0
$721$	−1688.00	−0.0871906
$722$	0	0
$723$	− 9906.00i	− 0.509555i
$724$	−18320.0	−0.940411
$725$	0	0
$726$	0	0
$727$	10960.0i	0.559125i	0.960127	+	0.279563i	$0.0901894\pi$
$727$	10960.0i	0.559125i	−0.960127	+	0.279563i	$0.909811\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−27504.0	−1.39162
$732$	10320.0i	0.521090i
$733$	14618.0i	0.736600i	0.929707	+	0.368300i	$0.120060\pi$
$733$	14618.0i	0.736600i	−0.929707	+	0.368300i	$0.879940\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4180.00i	0.208918i
$738$	0	0
$739$	−36518.0	−1.81778	−0.908888	−	0.417040i	$-0.863067\pi$
$739$	−36518.0	−1.81778	−0.908888	+	0.417040i	$0.863067\pi$
$740$	0	0
$741$	8052.00	0.399187
$742$	0	0
$743$	− 37452.0i	− 1.84923i	−0.380899	−	0.924617i	$-0.624385\pi$
$743$	− 37452.0i	− 1.84923i	0.380899	−	0.924617i	$-0.375615\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 9612.00i	− 0.470796i
$748$	6336.00i	0.309715i
$749$	−4344.00	−0.211918
$750$	0	0
$751$	−10648.0	−0.517378	−0.258689	−	0.965961i	$-0.583291\pi$
$751$	−10648.0	−0.517378	−0.258689	+	0.965961i	$0.583291\pi$
$752$	− 23040.0i	− 1.11726i
$753$	− 2196.00i	− 0.106277i
$754$	0	0
$755$	0	0
$756$	432.000	0.0207827
$757$	1258.00i	0.0604000i	0.999544	+	0.0302000i	$0.00961442\pi$
$757$	1258.00i	0.0604000i	−0.999544	+	0.0302000i	$0.990386\pi$
$758$	0	0
$759$	−2376.00	−0.113628
$760$	0	0
$761$	1740.00	0.0828843	0.0414421	−	0.999141i	$-0.486805\pi$
$761$	1740.00	0.0828843	0.0414421	+	0.999141i	$0.486805\pi$
$762$	0	0
$763$	1228.00i	0.0582655i
$764$	−35136.0	−1.66384
$765$	0	0
$766$	0	0
$767$	14520.0i	0.683555i
$768$	− 12288.0i	− 0.577350i
$769$	10774.0	0.505228	0.252614	−	0.967567i	$-0.418710\pi$
$769$	10774.0	0.505228	0.252614	+	0.967567i	$0.418710\pi$
$770$	0	0
$771$	10314.0	0.481776
$772$	− 40592.0i	− 1.89241i
$773$	− 19146.0i	− 0.890859i	−0.895317	−	0.445429i	$-0.853051\pi$
$773$	− 19146.0i	− 0.890859i	0.895317	−	0.445429i	$-0.146949\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1596.00i	0.0736888i
$778$	0	0
$779$	11712.0	0.538673
$780$	0	0
$781$	−1848.00	−0.0846692
$782$	0	0
$783$	− 2592.00i	− 0.118302i
$784$	21696.0	0.988338
$785$	0	0
$786$	0	0
$787$	30670.0i	1.38916i	0.719416	+	0.694579i	$0.244409\pi$
$787$	30670.0i	1.38916i	−0.719416	+	0.694579i	$0.755591\pi$
$788$	− 13536.0i	− 0.611929i
$789$	−2088.00	−0.0942139
$790$	0	0
$791$	2508.00	0.112736
$792$	0	0
$793$	9460.00i	0.423625i
$794$	0	0
$795$	0	0
$796$	−37312.0	−1.66142
$797$	− 11970.0i	− 0.531994i	−0.963974	−	0.265997i	$-0.914299\pi$
$797$	− 11970.0i	− 0.531994i	0.963974	−	0.265997i	$-0.0857011\pi$
$798$	0	0
$799$	−25920.0	−1.14766
$800$	0	0
$801$	−54.0000	−0.00238202
$802$	0	0
$803$	− 2398.00i	− 0.105384i
$804$	−9120.00	−0.400047
$805$	0	0
$806$	0	0
$807$	− 22014.0i	− 0.960260i
$808$	0	0
$809$	19932.0	0.866220	0.433110	−	0.901341i	$-0.357416\pi$
$809$	19932.0	0.866220	0.433110	+	0.901341i	$0.357416\pi$
$810$	0	0
$811$	−31462.0	−1.36224	−0.681122	−	0.732170i	$-0.738508\pi$
$811$	−31462.0	−1.36224	−0.681122	+	0.732170i	$0.738508\pi$
$812$	1536.00i	0.0663830i
$813$	− 15342.0i	− 0.661830i
$814$	0	0
$815$	0	0
$816$	−13824.0	−0.593060
$817$	46604.0i	1.99568i
$818$	0	0
$819$	396.000	0.0168954
$820$	0	0
$821$	−39720.0	−1.68847	−0.844237	−	0.535970i	$-0.819946\pi$
$821$	−39720.0	−1.68847	−0.844237	+	0.535970i	$0.819946\pi$
$822$	0	0
$823$	− 28492.0i	− 1.20677i	−0.797451	−	0.603383i	$-0.793819\pi$
$823$	− 28492.0i	− 1.20677i	0.797451	−	0.603383i	$-0.206181\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	18324.0i	0.770481i	0.922816	+	0.385241i	$0.125882\pi$
$827$	18324.0i	0.770481i	−0.922816	+	0.385241i	$0.874118\pi$
$828$	− 5184.00i	− 0.217580i
$829$	−21626.0	−0.906034	−0.453017	−	0.891502i	$-0.649652\pi$
$829$	−21626.0	−0.906034	−0.453017	+	0.891502i	$0.649652\pi$
$830$	0	0
$831$	−2958.00	−0.123480
$832$	− 11264.0i	− 0.469362i
$833$	− 24408.0i	− 1.01523i
$834$	0	0
$835$	0	0
$836$	10736.0	0.444153
$837$	− 3024.00i	− 0.124880i
$838$	0	0
$839$	36960.0	1.52086	0.760430	−	0.649420i	$-0.224988\pi$
$839$	36960.0	1.52086	0.760430	+	0.649420i	$0.224988\pi$
$840$	0	0
$841$	−15173.0	−0.622125
$842$	0	0
$843$	− 9936.00i	− 0.405948i
$844$	−14960.0	−0.610124
$845$	0	0
$846$	0	0
$847$	− 242.000i	− 0.00981726i
$848$	20352.0i	0.824163i
$849$	12894.0	0.521226
$850$	0	0
$851$	19152.0	0.771471
$852$	− 4032.00i	− 0.162129i
$853$	31502.0i	1.26449i	0.774769	+	0.632244i	$0.217866\pi$
$853$	31502.0i	1.26449i	−0.774769	+	0.632244i	$0.782134\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 5640.00i	− 0.224806i	−0.993663	−	0.112403i	$-0.964145\pi$
$857$	− 5640.00i	− 0.224806i	0.993663	−	0.112403i	$-0.0358548\pi$
$858$	0	0
$859$	2056.00	0.0816645	0.0408323	−	0.999166i	$-0.486999\pi$
$859$	2056.00	0.0816645	0.0408323	+	0.999166i	$0.486999\pi$
$860$	0	0
$861$	576.000	0.0227991
$862$	0	0
$863$	18336.0i	0.723250i	0.932324	+	0.361625i	$0.117778\pi$
$863$	18336.0i	0.723250i	−0.932324	+	0.361625i	$0.882222\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	813.000i	0.0318465i
$868$	1792.00i	0.0700742i
$869$	−7766.00	−0.303157
$870$	0	0
$871$	−8360.00	−0.325221
$872$	0	0
$873$	6174.00i	0.239357i
$874$	0	0
$875$	0	0
$876$	5232.00	0.201796
$877$	51346.0i	1.97700i	0.151213	+	0.988501i	$0.451682\pi$
$877$	51346.0i	1.97700i	−0.151213	+	0.988501i	$0.548318\pi$
$878$	0	0
$879$	8208.00	0.314959
$880$	0	0
$881$	32910.0	1.25853	0.629266	−	0.777190i	$-0.283356\pi$
$881$	32910.0	1.25853	0.629266	+	0.777190i	$0.283356\pi$
$882$	0	0
$883$	15356.0i	0.585244i	0.956228	+	0.292622i	$0.0945278\pi$
$883$	15356.0i	0.585244i	−0.956228	+	0.292622i	$0.905472\pi$
$884$	−12672.0	−0.482133
$885$	0	0
$886$	0	0
$887$	− 18372.0i	− 0.695458i	−0.937595	−	0.347729i	$-0.886953\pi$
$887$	− 18372.0i	− 0.695458i	0.937595	−	0.347729i	$-0.113047\pi$
$888$	0	0
$889$	−2788.00	−0.105182
$890$	0	0
$891$	−891.000	−0.0335013
$892$	18400.0i	0.690670i
$893$	43920.0i	1.64583i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 4752.00i	− 0.176884i
$898$	0	0
$899$	10752.0	0.398887
$900$	0	0
$901$	22896.0	0.846589
$902$	0	0
$903$	2292.00i	0.0844662i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 7640.00i	− 0.279694i	−0.990173	−	0.139847i	$-0.955339\pi$
$907$	− 7640.00i	− 0.279694i	0.990173	−	0.139847i	$-0.0446610\pi$
$908$	10656.0i	0.389462i
$909$	8640.00	0.315259
$910$	0	0
$911$	−53040.0	−1.92897	−0.964486	−	0.264134i	$-0.914914\pi$
$911$	−53040.0	−1.92897	−0.964486	+	0.264134i	$0.914914\pi$
$912$	23424.0i	0.850489i
$913$	− 11748.0i	− 0.425851i
$914$	0	0
$915$	0	0
$916$	48176.0	1.73775
$917$	504.000i	0.0181500i
$918$	0	0
$919$	11302.0	0.405679	0.202839	−	0.979212i	$-0.434983\pi$
$919$	11302.0	0.405679	0.202839	+	0.979212i	$0.434983\pi$
$920$	0	0
$921$	750.000	0.0268332
$922$	0	0
$923$	− 3696.00i	− 0.131804i
$924$	528.000	0.0187986
$925$	0	0
$926$	0	0
$927$	7596.00i	0.269132i
$928$	0	0
$929$	−19254.0	−0.679982	−0.339991	−	0.940429i	$-0.610424\pi$
$929$	−19254.0	−0.679982	−0.339991	+	0.940429i	$0.610424\pi$
$930$	0	0
$931$	−41358.0	−1.45591
$932$	− 37728.0i	− 1.32599i
$933$	− 21744.0i	− 0.762987i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 22214.0i	− 0.774493i	−0.921976	−	0.387246i	$-0.873426\pi$
$937$	− 22214.0i	− 0.774493i	0.921976	−	0.387246i	$-0.126574\pi$
$938$	0	0
$939$	−23358.0	−0.811778
$940$	0	0
$941$	41736.0	1.44586	0.722930	−	0.690921i	$-0.242795\pi$
$941$	41736.0	1.44586	0.722930	+	0.690921i	$0.242795\pi$
$942$	0	0
$943$	− 6912.00i	− 0.238691i
$944$	−42240.0	−1.45635
$945$	0	0
$946$	0	0
$947$	− 42732.0i	− 1.46632i	−0.680057	−	0.733159i	$-0.738045\pi$
$947$	− 42732.0i	− 1.46632i	0.680057	−	0.733159i	$-0.261955\pi$
$948$	− 16944.0i	− 0.580502i
$949$	4796.00	0.164051
$950$	0	0
$951$	12690.0	0.432704
$952$	0	0
$953$	− 25056.0i	− 0.851672i	−0.904800	−	0.425836i	$-0.859980\pi$
$953$	− 25056.0i	− 0.851672i	0.904800	−	0.425836i	$-0.140020\pi$
$954$	0	0
$955$	0	0
$956$	51360.0	1.73755
$957$	− 3168.00i	− 0.107008i
$958$	0	0
$959$	2100.00	0.0707117
$960$	0	0
$961$	−17247.0	−0.578933
$962$	0	0
$963$	19548.0i	0.654128i
$964$	26416.0	0.882575
$965$	0	0
$966$	0	0
$967$	14326.0i	0.476415i	0.971214	+	0.238207i	$0.0765598\pi$
$967$	14326.0i	0.476415i	−0.971214	+	0.238207i	$0.923440\pi$
$968$	0	0
$969$	26352.0	0.873631
$970$	0	0
$971$	45924.0	1.51779	0.758894	−	0.651215i	$-0.225740\pi$
$971$	45924.0	1.51779	0.758894	+	0.651215i	$0.225740\pi$
$972$	− 1944.00i	− 0.0641500i
$973$	3748.00i	0.123490i
$974$	0	0
$975$	0	0
$976$	−27520.0	−0.902555
$977$	38946.0i	1.27533i	0.770316	+	0.637663i	$0.220099\pi$
$977$	38946.0i	1.27533i	−0.770316	+	0.637663i	$0.779901\pi$
$978$	0	0
$979$	−66.0000	−0.00215462
$980$	0	0
$981$	5526.00	0.179849
$982$	0	0
$983$	21000.0i	0.681379i	0.940176	+	0.340690i	$0.110661\pi$
$983$	21000.0i	0.681379i	−0.940176	+	0.340690i	$0.889339\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2160.00i	0.0696591i
$988$	21472.0i	0.691412i
$989$	27504.0	0.884304
$990$	0	0
$991$	7760.00	0.248743	0.124372	−	0.992236i	$-0.460309\pi$
$991$	7760.00	0.248743	0.124372	+	0.992236i	$0.460309\pi$
$992$	0	0
$993$	− 23208.0i	− 0.741675i
$994$	0	0
$995$	0	0
$996$	25632.0	0.815443
$997$	− 21350.0i	− 0.678196i	−0.940751	−	0.339098i	$-0.889878\pi$
$997$	− 21350.0i	− 0.678196i	0.940751	−	0.339098i	$-0.110122\pi$
$998$	0	0
$999$	7182.00	0.227456

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.c.g.199.1		2
5.2	odd	4		165.4.a.a.1.1	✓	1
5.3	odd	4		825.4.a.e.1.1		1
5.4	even	2	inner	825.4.c.g.199.2		2
15.2	even	4		495.4.a.c.1.1		1
15.8	even	4		2475.4.a.f.1.1		1
55.32	even	4		1815.4.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.a.1.1	✓	1	5.2	odd	4
495.4.a.c.1.1		1	15.2	even	4
825.4.a.e.1.1		1	5.3	odd	4
825.4.c.g.199.1		2	1.1	even	1	trivial
825.4.c.g.199.2		2	5.4	even	2	inner
1815.4.a.f.1.1		1	55.32	even	4
2475.4.a.f.1.1		1	15.8	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +8.00000 q^{4} -2.00000i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} +8.00000 q^{4} -2.00000i q^{7} -9.00000 q^{9} -11.0000 q^{11} -24.0000i q^{12} -22.0000i q^{13} +64.0000 q^{16} -72.0000i q^{17} -122.000 q^{19} -6.00000 q^{21} +72.0000i q^{23} +27.0000i q^{27} -16.0000i q^{28} -96.0000 q^{29} -112.000 q^{31} +33.0000i q^{33} -72.0000 q^{36} -266.000i q^{37} -66.0000 q^{39} -96.0000 q^{41} -382.000i q^{43} -88.0000 q^{44} -360.000i q^{47} -192.000i q^{48} +339.000 q^{49} -216.000 q^{51} -176.000i q^{52} +318.000i q^{53} +366.000i q^{57} -660.000 q^{59} -430.000 q^{61} +18.0000i q^{63} +512.000 q^{64} -380.000i q^{67} -576.000i q^{68} +216.000 q^{69} +168.000 q^{71} +218.000i q^{73} -976.000 q^{76} +22.0000i q^{77} +706.000 q^{79} +81.0000 q^{81} +1068.00i q^{83} -48.0000 q^{84} +288.000i q^{87} +6.00000 q^{89} -44.0000 q^{91} +576.000i q^{92} +336.000i q^{93} -686.000i q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 16 q^{4} - 18 q^{9}+O(q^{10}) \) 2 * q + 16 * q^4 - 18 * q^9	\( 2 q + 16 q^{4} - 18 q^{9} - 22 q^{11} + 128 q^{16} - 244 q^{19} - 12 q^{21} - 192 q^{29} - 224 q^{31} - 144 q^{36} - 132 q^{39} - 192 q^{41} - 176 q^{44} + 678 q^{49} - 432 q^{51} - 1320 q^{59} - 860 q^{61} + 1024 q^{64} + 432 q^{69} + 336 q^{71} - 1952 q^{76} + 1412 q^{79} + 162 q^{81} - 96 q^{84} + 12 q^{89} - 88 q^{91} + 198 q^{99}+O(q^{100}) \) 2 * q + 16 * q^4 - 18 * q^9 - 22 * q^11 + 128 * q^16 - 244 * q^19 - 12 * q^21 - 192 * q^29 - 224 * q^31 - 144 * q^36 - 132 * q^39 - 192 * q^41 - 176 * q^44 + 678 * q^49 - 432 * q^51 - 1320 * q^59 - 860 * q^61 + 1024 * q^64 + 432 * q^69 + 336 * q^71 - 1952 * q^76 + 1412 * q^79 + 162 * q^81 - 96 * q^84 + 12 * q^89 - 88 * q^91 + 198 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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