LMFDB - Embedded newform 450.4.c.d.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,4,Mod(199,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("450.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	450.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:	$26.5508595026$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	450.199
Dual form			450.4.c.d.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -4.00000 q^{4} -4.00000i q^{7} +8.00000i q^{8} +O(q^{10})$	$q-2.00000i q^{2} -4.00000 q^{4} -4.00000i q^{7} +8.00000i q^{8} -12.0000 q^{11} +58.0000i q^{13} -8.00000 q^{14} +16.0000 q^{16} -66.0000i q^{17} +100.000 q^{19} +24.0000i q^{22} +132.000i q^{23} +116.000 q^{26} +16.0000i q^{28} -90.0000 q^{29} +152.000 q^{31} -32.0000i q^{32} -132.000 q^{34} -34.0000i q^{37} -200.000i q^{38} +438.000 q^{41} -32.0000i q^{43} +48.0000 q^{44} +264.000 q^{46} +204.000i q^{47} +327.000 q^{49} -232.000i q^{52} +222.000i q^{53} +32.0000 q^{56} +180.000i q^{58} +420.000 q^{59} +902.000 q^{61} -304.000i q^{62} -64.0000 q^{64} -1024.00i q^{67} +264.000i q^{68} -432.000 q^{71} -362.000i q^{73} -68.0000 q^{74} -400.000 q^{76} +48.0000i q^{77} +160.000 q^{79} -876.000i q^{82} +72.0000i q^{83} -64.0000 q^{86} -96.0000i q^{88} +810.000 q^{89} +232.000 q^{91} -528.000i q^{92} +408.000 q^{94} +1106.00i q^{97} -654.000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4}+O(q^{10}) $ 2 * q - 8 * q^4	$ 2 q - 8 q^{4} - 24 q^{11} - 16 q^{14} + 32 q^{16} + 200 q^{19} + 232 q^{26} - 180 q^{29} + 304 q^{31} - 264 q^{34} + 876 q^{41} + 96 q^{44} + 528 q^{46} + 654 q^{49} + 64 q^{56} + 840 q^{59} + 1804 q^{61} - 128 q^{64} - 864 q^{71} - 136 q^{74} - 800 q^{76} + 320 q^{79} - 128 q^{86} + 1620 q^{89} + 464 q^{91} + 816 q^{94}+O(q^{100}) $ 2 * q - 8 * q^4 - 24 * q^11 - 16 * q^14 + 32 * q^16 + 200 * q^19 + 232 * q^26 - 180 * q^29 + 304 * q^31 - 264 * q^34 + 876 * q^41 + 96 * q^44 + 528 * q^46 + 654 * q^49 + 64 * q^56 + 840 * q^59 + 1804 * q^61 - 128 * q^64 - 864 * q^71 - 136 * q^74 - 800 * q^76 + 320 * q^79 - 128 * q^86 + 1620 * q^89 + 464 * q^91 + 816 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.00000i	− 0.707107i
$2$	− 2.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.00000i	− 0.215980i	−0.994152	−	0.107990i	$-0.965559\pi$
$7$	− 4.00000i	− 0.215980i	0.994152	−	0.107990i	$-0.0344414\pi$
$8$	8.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	−12.0000	−0.328921	−0.164461	−	0.986384i	$-0.552588\pi$
$11$	−12.0000	−0.328921	−0.164461	+	0.986384i	$0.552588\pi$
$12$	0	0
$13$	58.0000i	1.23741i	0.785624	+	0.618704i	$0.212342\pi$
$13$	58.0000i	1.23741i	−0.785624	+	0.618704i	$0.787658\pi$
$14$	−8.00000	−0.152721
$15$	0	0
$16$	16.0000	0.250000
$17$	− 66.0000i	− 0.941609i	−0.882238	−	0.470804i	$-0.843964\pi$
$17$	− 66.0000i	− 0.941609i	0.882238	−	0.470804i	$-0.156036\pi$
$18$	0	0
$19$	100.000	1.20745	0.603726	−	0.797192i	$-0.293682\pi$
$19$	100.000	1.20745	0.603726	+	0.797192i	$0.293682\pi$
$20$	0	0
$21$	0	0
$22$	24.0000i	0.232583i
$23$	132.000i	1.19669i	0.801238	+	0.598346i	$0.204175\pi$
$23$	132.000i	1.19669i	−0.801238	+	0.598346i	$0.795825\pi$
$24$	0	0
$25$	0	0
$26$	116.000	0.874980
$27$	0	0
$28$	16.0000i	0.107990i
$29$	−90.0000	−0.576296	−0.288148	−	0.957586i	$-0.593039\pi$
$29$	−90.0000	−0.576296	−0.288148	+	0.957586i	$0.593039\pi$
$30$	0	0
$31$	152.000	0.880645	0.440323	−	0.897840i	$-0.354864\pi$
$31$	152.000	0.880645	0.440323	+	0.897840i	$0.354864\pi$
$32$	− 32.0000i	− 0.176777i
$33$	0	0
$34$	−132.000	−0.665818
$35$	0	0
$36$	0	0
$37$	− 34.0000i	− 0.151069i	−0.997143	−	0.0755347i	$-0.975934\pi$
$37$	− 34.0000i	− 0.151069i	0.997143	−	0.0755347i	$-0.0240664\pi$
$38$	− 200.000i	− 0.853797i
$39$	0	0
$40$	0	0
$41$	438.000	1.66839	0.834196	−	0.551467i	$-0.185932\pi$
$41$	438.000	1.66839	0.834196	+	0.551467i	$0.185932\pi$
$42$	0	0
$43$	− 32.0000i	− 0.113487i	−0.998389	−	0.0567437i	$-0.981928\pi$
$43$	− 32.0000i	− 0.113487i	0.998389	−	0.0567437i	$-0.0180718\pi$
$44$	48.0000	0.164461
$45$	0	0
$46$	264.000	0.846189
$47$	204.000i	0.633116i	0.948573	+	0.316558i	$0.102527\pi$
$47$	204.000i	0.633116i	−0.948573	+	0.316558i	$0.897473\pi$
$48$	0	0
$49$	327.000	0.953353
$50$	0	0
$51$	0	0
$52$	− 232.000i	− 0.618704i
$53$	222.000i	0.575359i	0.957727	+	0.287680i	$0.0928838\pi$
$53$	222.000i	0.575359i	−0.957727	+	0.287680i	$0.907116\pi$
$54$	0	0
$55$	0	0
$56$	32.0000	0.0763604
$57$	0	0
$58$	180.000i	0.407503i
$59$	420.000	0.926769	0.463384	−	0.886157i	$-0.346635\pi$
$59$	420.000	0.926769	0.463384	+	0.886157i	$0.346635\pi$
$60$	0	0
$61$	902.000	1.89327	0.946633	−	0.322312i	$-0.104460\pi$
$61$	902.000	1.89327	0.946633	+	0.322312i	$0.104460\pi$
$62$	− 304.000i	− 0.622710i
$63$	0	0
$64$	−64.0000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 1024.00i	− 1.86719i	−0.358334	−	0.933593i	$-0.616655\pi$
$67$	− 1024.00i	− 1.86719i	0.358334	−	0.933593i	$-0.383345\pi$
$68$	264.000i	0.470804i
$69$	0	0
$70$	0	0
$71$	−432.000	−0.722098	−0.361049	−	0.932547i	$-0.617581\pi$
$71$	−432.000	−0.722098	−0.361049	+	0.932547i	$0.617581\pi$
$72$	0	0
$73$	− 362.000i	− 0.580396i	−0.956967	−	0.290198i	$-0.906279\pi$
$73$	− 362.000i	− 0.580396i	0.956967	−	0.290198i	$-0.0937211\pi$
$74$	−68.0000	−0.106822
$75$	0	0
$76$	−400.000	−0.603726
$77$	48.0000i	0.0710404i
$78$	0	0
$79$	160.000	0.227866	0.113933	−	0.993488i	$-0.463655\pi$
$79$	160.000	0.227866	0.113933	+	0.993488i	$0.463655\pi$
$80$	0	0
$81$	0	0
$82$	− 876.000i	− 1.17973i
$83$	72.0000i	0.0952172i	0.998866	+	0.0476086i	$0.0151600\pi$
$83$	72.0000i	0.0952172i	−0.998866	+	0.0476086i	$0.984840\pi$
$84$	0	0
$85$	0	0
$86$	−64.0000	−0.0802476
$87$	0	0
$88$	− 96.0000i	− 0.116291i
$89$	810.000	0.964717	0.482359	−	0.875974i	$-0.339780\pi$
$89$	810.000	0.964717	0.482359	+	0.875974i	$0.339780\pi$
$90$	0	0
$91$	232.000	0.267255
$92$	− 528.000i	− 0.598346i
$93$	0	0
$94$	408.000	0.447681
$95$	0	0
$96$	0	0
$97$	1106.00i	1.15770i	0.815433	+	0.578852i	$0.196499\pi$
$97$	1106.00i	1.15770i	−0.815433	+	0.578852i	$0.803501\pi$
$98$	− 654.000i	− 0.674122i
$99$	0	0
$100$	0	0
$101$	258.000	0.254178	0.127089	−	0.991891i	$-0.459437\pi$
$101$	258.000	0.254178	0.127089	+	0.991891i	$0.459437\pi$
$102$	0	0
$103$	988.000i	0.945151i	0.881290	+	0.472575i	$0.156676\pi$
$103$	988.000i	0.945151i	−0.881290	+	0.472575i	$0.843324\pi$
$104$	−464.000	−0.437490
$105$	0	0
$106$	444.000	0.406840
$107$	24.0000i	0.0216838i	0.999941	+	0.0108419i	$0.00345115\pi$
$107$	24.0000i	0.0216838i	−0.999941	+	0.0108419i	$0.996549\pi$
$108$	0	0
$109$	−950.000	−0.834803	−0.417401	−	0.908722i	$-0.637059\pi$
$109$	−950.000	−0.834803	−0.417401	+	0.908722i	$0.637059\pi$
$110$	0	0
$111$	0	0
$112$	− 64.0000i	− 0.0539949i
$113$	− 1038.00i	− 0.864131i	−0.901842	−	0.432066i	$-0.857785\pi$
$113$	− 1038.00i	− 0.864131i	0.901842	−	0.432066i	$-0.142215\pi$
$114$	0	0
$115$	0	0
$116$	360.000	0.288148
$117$	0	0
$118$	− 840.000i	− 0.655324i
$119$	−264.000	−0.203368
$120$	0	0
$121$	−1187.00	−0.891811
$122$	− 1804.00i	− 1.33874i
$123$	0	0
$124$	−608.000	−0.440323
$125$	0	0
$126$	0	0
$127$	− 124.000i	− 0.0866395i	−0.999061	−	0.0433198i	$-0.986207\pi$
$127$	− 124.000i	− 0.0866395i	0.999061	−	0.0433198i	$-0.0137934\pi$
$128$	128.000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−132.000	−0.0880374	−0.0440187	−	0.999031i	$-0.514016\pi$
$131$	−132.000	−0.0880374	−0.0440187	+	0.999031i	$0.514016\pi$
$132$	0	0
$133$	− 400.000i	− 0.260785i
$134$	−2048.00	−1.32030
$135$	0	0
$136$	528.000	0.332909
$137$	1254.00i	0.782018i	0.920387	+	0.391009i	$0.127874\pi$
$137$	1254.00i	0.782018i	−0.920387	+	0.391009i	$0.872126\pi$
$138$	0	0
$139$	2860.00	1.74519	0.872597	−	0.488440i	$-0.162434\pi$
$139$	2860.00	1.74519	0.872597	+	0.488440i	$0.162434\pi$
$140$	0	0
$141$	0	0
$142$	864.000i	0.510600i
$143$	− 696.000i	− 0.407010i
$144$	0	0
$145$	0	0
$146$	−724.000	−0.410402
$147$	0	0
$148$	136.000i	0.0755347i
$149$	750.000	0.412365	0.206183	−	0.978514i	$-0.433896\pi$
$149$	750.000	0.412365	0.206183	+	0.978514i	$0.433896\pi$
$150$	0	0
$151$	−448.000	−0.241442	−0.120721	−	0.992686i	$-0.538521\pi$
$151$	−448.000	−0.241442	−0.120721	+	0.992686i	$0.538521\pi$
$152$	800.000i	0.426898i
$153$	0	0
$154$	96.0000	0.0502331
$155$	0	0
$156$	0	0
$157$	2246.00i	1.14172i	0.821047	+	0.570861i	$0.193390\pi$
$157$	2246.00i	1.14172i	−0.821047	+	0.570861i	$0.806610\pi$
$158$	− 320.000i	− 0.161126i
$159$	0	0
$160$	0	0
$161$	528.000	0.258461
$162$	0	0
$163$	568.000i	0.272940i	0.990644	+	0.136470i	$0.0435757\pi$
$163$	568.000i	0.272940i	−0.990644	+	0.136470i	$0.956424\pi$
$164$	−1752.00	−0.834196
$165$	0	0
$166$	144.000	0.0673287
$167$	1524.00i	0.706172i	0.935591	+	0.353086i	$0.114868\pi$
$167$	1524.00i	0.706172i	−0.935591	+	0.353086i	$0.885132\pi$
$168$	0	0
$169$	−1167.00	−0.531179
$170$	0	0
$171$	0	0
$172$	128.000i	0.0567437i
$173$	3702.00i	1.62692i	0.581618	+	0.813462i	$0.302420\pi$
$173$	3702.00i	1.62692i	−0.581618	+	0.813462i	$0.697580\pi$
$174$	0	0
$175$	0	0
$176$	−192.000	−0.0822304
$177$	0	0
$178$	− 1620.00i	− 0.682158i
$179$	3180.00	1.32785	0.663923	−	0.747801i	$-0.268890\pi$
$179$	3180.00	1.32785	0.663923	+	0.747801i	$0.268890\pi$
$180$	0	0
$181$	−2098.00	−0.861564	−0.430782	−	0.902456i	$-0.641762\pi$
$181$	−2098.00	−0.861564	−0.430782	+	0.902456i	$0.641762\pi$
$182$	− 464.000i	− 0.188978i
$183$	0	0
$184$	−1056.00	−0.423094
$185$	0	0
$186$	0	0
$187$	792.000i	0.309715i
$188$	− 816.000i	− 0.316558i
$189$	0	0
$190$	0	0
$191$	−4392.00	−1.66384	−0.831921	−	0.554894i	$-0.812759\pi$
$191$	−4392.00	−1.66384	−0.831921	+	0.554894i	$0.812759\pi$
$192$	0	0
$193$	2158.00i	0.804851i	0.915453	+	0.402425i	$0.131833\pi$
$193$	2158.00i	0.804851i	−0.915453	+	0.402425i	$0.868167\pi$
$194$	2212.00	0.818620
$195$	0	0
$196$	−1308.00	−0.476676
$197$	1074.00i	0.388423i	0.980960	+	0.194212i	$0.0622148\pi$
$197$	1074.00i	0.388423i	−0.980960	+	0.194212i	$0.937785\pi$
$198$	0	0
$199$	−2840.00	−1.01167	−0.505835	−	0.862630i	$-0.668815\pi$
$199$	−2840.00	−1.01167	−0.505835	+	0.862630i	$0.668815\pi$
$200$	0	0
$201$	0	0
$202$	− 516.000i	− 0.179731i
$203$	360.000i	0.124468i
$204$	0	0
$205$	0	0
$206$	1976.00	0.668323
$207$	0	0
$208$	928.000i	0.309352i
$209$	−1200.00	−0.397157
$210$	0	0
$211$	−2668.00	−0.870487	−0.435243	−	0.900313i	$-0.643338\pi$
$211$	−2668.00	−0.870487	−0.435243	+	0.900313i	$0.643338\pi$
$212$	− 888.000i	− 0.287680i
$213$	0	0
$214$	48.0000	0.0153328
$215$	0	0
$216$	0	0
$217$	− 608.000i	− 0.190202i
$218$	1900.00i	0.590295i
$219$	0	0
$220$	0	0
$221$	3828.00	1.16515
$222$	0	0
$223$	− 1772.00i	− 0.532116i	−0.963957	−	0.266058i	$-0.914279\pi$
$223$	− 1772.00i	− 0.532116i	0.963957	−	0.266058i	$-0.0857213\pi$
$224$	−128.000	−0.0381802
$225$	0	0
$226$	−2076.00	−0.611033
$227$	2784.00i	0.814011i	0.913426	+	0.407006i	$0.133427\pi$
$227$	2784.00i	0.814011i	−0.913426	+	0.407006i	$0.866573\pi$
$228$	0	0
$229$	−350.000	−0.100998	−0.0504992	−	0.998724i	$-0.516081\pi$
$229$	−350.000	−0.100998	−0.0504992	+	0.998724i	$0.516081\pi$
$230$	0	0
$231$	0	0
$232$	− 720.000i	− 0.203751i
$233$	1962.00i	0.551652i	0.961208	+	0.275826i	$0.0889513\pi$
$233$	1962.00i	0.551652i	−0.961208	+	0.275826i	$0.911049\pi$
$234$	0	0
$235$	0	0
$236$	−1680.00	−0.463384
$237$	0	0
$238$	528.000i	0.143803i
$239$	−4320.00	−1.16919	−0.584597	−	0.811324i	$-0.698748\pi$
$239$	−4320.00	−1.16919	−0.584597	+	0.811324i	$0.698748\pi$
$240$	0	0
$241$	−478.000	−0.127762	−0.0638811	−	0.997958i	$-0.520348\pi$
$241$	−478.000	−0.127762	−0.0638811	+	0.997958i	$0.520348\pi$
$242$	2374.00i	0.630605i
$243$	0	0
$244$	−3608.00	−0.946633
$245$	0	0
$246$	0	0
$247$	5800.00i	1.49411i
$248$	1216.00i	0.311355i
$249$	0	0
$250$	0	0
$251$	−2652.00	−0.666903	−0.333452	−	0.942767i	$-0.608213\pi$
$251$	−2652.00	−0.666903	−0.333452	+	0.942767i	$0.608213\pi$
$252$	0	0
$253$	− 1584.00i	− 0.393617i
$254$	−248.000	−0.0612634
$255$	0	0
$256$	256.000	0.0625000
$257$	2334.00i	0.566502i	0.959046	+	0.283251i	$0.0914129\pi$
$257$	2334.00i	0.566502i	−0.959046	+	0.283251i	$0.908587\pi$
$258$	0	0
$259$	−136.000	−0.0326279
$260$	0	0
$261$	0	0
$262$	264.000i	0.0622518i
$263$	− 3948.00i	− 0.925643i	−0.886451	−	0.462822i	$-0.846837\pi$
$263$	− 3948.00i	− 0.925643i	0.886451	−	0.462822i	$-0.153163\pi$
$264$	0	0
$265$	0	0
$266$	−800.000	−0.184403
$267$	0	0
$268$	4096.00i	0.933593i
$269$	1590.00	0.360387	0.180193	−	0.983631i	$-0.442328\pi$
$269$	1590.00	0.360387	0.180193	+	0.983631i	$0.442328\pi$
$270$	0	0
$271$	4952.00	1.11001	0.555005	−	0.831847i	$-0.312716\pi$
$271$	4952.00	1.11001	0.555005	+	0.831847i	$0.312716\pi$
$272$	− 1056.00i	− 0.235402i
$273$	0	0
$274$	2508.00	0.552970
$275$	0	0
$276$	0	0
$277$	1646.00i	0.357034i	0.983937	+	0.178517i	$0.0571300\pi$
$277$	1646.00i	0.357034i	−0.983937	+	0.178517i	$0.942870\pi$
$278$	− 5720.00i	− 1.23404i
$279$	0	0
$280$	0	0
$281$	1158.00	0.245838	0.122919	−	0.992417i	$-0.460774\pi$
$281$	1158.00	0.245838	0.122919	+	0.992417i	$0.460774\pi$
$282$	0	0
$283$	− 6992.00i	− 1.46866i	−0.678792	−	0.734331i	$-0.737496\pi$
$283$	− 6992.00i	− 1.46866i	0.678792	−	0.734331i	$-0.262504\pi$
$284$	1728.00	0.361049
$285$	0	0
$286$	−1392.00	−0.287800
$287$	− 1752.00i	− 0.360339i
$288$	0	0
$289$	557.000	0.113373
$290$	0	0
$291$	0	0
$292$	1448.00i	0.290198i
$293$	− 258.000i	− 0.0514421i	−0.999669	−	0.0257210i	$-0.991812\pi$
$293$	− 258.000i	− 0.0514421i	0.999669	−	0.0257210i	$-0.00818816\pi$
$294$	0	0
$295$	0	0
$296$	272.000	0.0534111
$297$	0	0
$298$	− 1500.00i	− 0.291586i
$299$	−7656.00	−1.48080
$300$	0	0
$301$	−128.000	−0.0245110
$302$	896.000i	0.170725i
$303$	0	0
$304$	1600.00	0.301863
$305$	0	0
$306$	0	0
$307$	− 8944.00i	− 1.66274i	−0.555720	−	0.831370i	$-0.687557\pi$
$307$	− 8944.00i	− 1.66274i	0.555720	−	0.831370i	$-0.312443\pi$
$308$	− 192.000i	− 0.0355202i
$309$	0	0
$310$	0	0
$311$	−1392.00	−0.253804	−0.126902	−	0.991915i	$-0.540503\pi$
$311$	−1392.00	−0.253804	−0.126902	+	0.991915i	$0.540503\pi$
$312$	0	0
$313$	5878.00i	1.06148i	0.847534	+	0.530742i	$0.178087\pi$
$313$	5878.00i	1.06148i	−0.847534	+	0.530742i	$0.821913\pi$
$314$	4492.00	0.807319
$315$	0	0
$316$	−640.000	−0.113933
$317$	− 10326.0i	− 1.82955i	−0.403969	−	0.914773i	$-0.632370\pi$
$317$	− 10326.0i	− 1.82955i	0.403969	−	0.914773i	$-0.367630\pi$
$318$	0	0
$319$	1080.00	0.189556
$320$	0	0
$321$	0	0
$322$	− 1056.00i	− 0.182760i
$323$	− 6600.00i	− 1.13695i
$324$	0	0
$325$	0	0
$326$	1136.00	0.192998
$327$	0	0
$328$	3504.00i	0.589866i
$329$	816.000	0.136740
$330$	0	0
$331$	−4228.00	−0.702090	−0.351045	−	0.936359i	$-0.614174\pi$
$331$	−4228.00	−0.702090	−0.351045	+	0.936359i	$0.614174\pi$
$332$	− 288.000i	− 0.0476086i
$333$	0	0
$334$	3048.00	0.499339
$335$	0	0
$336$	0	0
$337$	1106.00i	0.178776i	0.995997	+	0.0893882i	$0.0284912\pi$
$337$	1106.00i	0.178776i	−0.995997	+	0.0893882i	$0.971509\pi$
$338$	2334.00i	0.375600i
$339$	0	0
$340$	0	0
$341$	−1824.00	−0.289663
$342$	0	0
$343$	− 2680.00i	− 0.421885i
$344$	256.000	0.0401238
$345$	0	0
$346$	7404.00	1.15041
$347$	− 9336.00i	− 1.44433i	−0.691720	−	0.722165i	$-0.743147\pi$
$347$	− 9336.00i	− 1.44433i	0.691720	−	0.722165i	$-0.256853\pi$
$348$	0	0
$349$	11770.0	1.80525	0.902627	−	0.430424i	$-0.141636\pi$
$349$	11770.0	1.80525	0.902627	+	0.430424i	$0.141636\pi$
$350$	0	0
$351$	0	0
$352$	384.000i	0.0581456i
$353$	8322.00i	1.25477i	0.778707	+	0.627387i	$0.215876\pi$
$353$	8322.00i	1.25477i	−0.778707	+	0.627387i	$0.784124\pi$
$354$	0	0
$355$	0	0
$356$	−3240.00	−0.482359
$357$	0	0
$358$	− 6360.00i	− 0.938929i
$359$	10680.0	1.57011	0.785054	−	0.619427i	$-0.212635\pi$
$359$	10680.0	1.57011	0.785054	+	0.619427i	$0.212635\pi$
$360$	0	0
$361$	3141.00	0.457938
$362$	4196.00i	0.609218i
$363$	0	0
$364$	−928.000	−0.133628
$365$	0	0
$366$	0	0
$367$	− 5884.00i	− 0.836900i	−0.908240	−	0.418450i	$-0.862574\pi$
$367$	− 5884.00i	− 0.836900i	0.908240	−	0.418450i	$-0.137426\pi$
$368$	2112.00i	0.299173i
$369$	0	0
$370$	0	0
$371$	888.000	0.124266
$372$	0	0
$373$	2098.00i	0.291234i	0.989341	+	0.145617i	$0.0465167\pi$
$373$	2098.00i	0.291234i	−0.989341	+	0.145617i	$0.953483\pi$
$374$	1584.00	0.219002
$375$	0	0
$376$	−1632.00	−0.223840
$377$	− 5220.00i	− 0.713113i
$378$	0	0
$379$	−3860.00	−0.523153	−0.261576	−	0.965183i	$-0.584242\pi$
$379$	−3860.00	−0.523153	−0.261576	+	0.965183i	$0.584242\pi$
$380$	0	0
$381$	0	0
$382$	8784.00i	1.17651i
$383$	− 9588.00i	− 1.27917i	−0.768718	−	0.639587i	$-0.779105\pi$
$383$	− 9588.00i	− 1.27917i	0.768718	−	0.639587i	$-0.220895\pi$
$384$	0	0
$385$	0	0
$386$	4316.00	0.569116
$387$	0	0
$388$	− 4424.00i	− 0.578852i
$389$	−13410.0	−1.74785	−0.873925	−	0.486060i	$-0.838434\pi$
$389$	−13410.0	−1.74785	−0.873925	+	0.486060i	$0.838434\pi$
$390$	0	0
$391$	8712.00	1.12682
$392$	2616.00i	0.337061i
$393$	0	0
$394$	2148.00	0.274657
$395$	0	0
$396$	0	0
$397$	− 13114.0i	− 1.65787i	−0.559348	−	0.828933i	$-0.688948\pi$
$397$	− 13114.0i	− 1.65787i	0.559348	−	0.828933i	$-0.311052\pi$
$398$	5680.00i	0.715358i
$399$	0	0
$400$	0	0
$401$	5838.00	0.727022	0.363511	−	0.931590i	$-0.381578\pi$
$401$	5838.00	0.727022	0.363511	+	0.931590i	$0.381578\pi$
$402$	0	0
$403$	8816.00i	1.08972i
$404$	−1032.00	−0.127089
$405$	0	0
$406$	720.000	0.0880123
$407$	408.000i	0.0496899i
$408$	0	0
$409$	−9530.00	−1.15215	−0.576074	−	0.817398i	$-0.695416\pi$
$409$	−9530.00	−1.15215	−0.576074	+	0.817398i	$0.695416\pi$
$410$	0	0
$411$	0	0
$412$	− 3952.00i	− 0.472575i
$413$	− 1680.00i	− 0.200163i
$414$	0	0
$415$	0	0
$416$	1856.00	0.218745
$417$	0	0
$418$	2400.00i	0.280832i
$419$	7260.00	0.846478	0.423239	−	0.906018i	$-0.360893\pi$
$419$	7260.00	0.846478	0.423239	+	0.906018i	$0.360893\pi$
$420$	0	0
$421$	12062.0	1.39636	0.698178	−	0.715924i	$-0.253994\pi$
$421$	12062.0	1.39636	0.698178	+	0.715924i	$0.253994\pi$
$422$	5336.00i	0.615527i
$423$	0	0
$424$	−1776.00	−0.203420
$425$	0	0
$426$	0	0
$427$	− 3608.00i	− 0.408907i
$428$	− 96.0000i	− 0.0108419i
$429$	0	0
$430$	0	0
$431$	13608.0	1.52082	0.760411	−	0.649442i	$-0.224998\pi$
$431$	13608.0	1.52082	0.760411	+	0.649442i	$0.224998\pi$
$432$	0	0
$433$	3838.00i	0.425964i	0.977056	+	0.212982i	$0.0683176\pi$
$433$	3838.00i	0.425964i	−0.977056	+	0.212982i	$0.931682\pi$
$434$	−1216.00	−0.134493
$435$	0	0
$436$	3800.00	0.417401
$437$	13200.0i	1.44495i
$438$	0	0
$439$	−7400.00	−0.804516	−0.402258	−	0.915526i	$-0.631775\pi$
$439$	−7400.00	−0.804516	−0.402258	+	0.915526i	$0.631775\pi$
$440$	0	0
$441$	0	0
$442$	− 7656.00i	− 0.823889i
$443$	8352.00i	0.895746i	0.894097	+	0.447873i	$0.147818\pi$
$443$	8352.00i	0.895746i	−0.894097	+	0.447873i	$0.852182\pi$
$444$	0	0
$445$	0	0
$446$	−3544.00	−0.376263
$447$	0	0
$448$	256.000i	0.0269975i
$449$	10770.0	1.13200	0.566000	−	0.824405i	$-0.308490\pi$
$449$	10770.0	1.13200	0.566000	+	0.824405i	$0.308490\pi$
$450$	0	0
$451$	−5256.00	−0.548770
$452$	4152.00i	0.432066i
$453$	0	0
$454$	5568.00	0.575593
$455$	0	0
$456$	0	0
$457$	− 6694.00i	− 0.685191i	−0.939483	−	0.342595i	$-0.888694\pi$
$457$	− 6694.00i	− 0.685191i	0.939483	−	0.342595i	$-0.111306\pi$
$458$	700.000i	0.0714167i
$459$	0	0
$460$	0	0
$461$	3018.00	0.304907	0.152454	−	0.988311i	$-0.451283\pi$
$461$	3018.00	0.304907	0.152454	+	0.988311i	$0.451283\pi$
$462$	0	0
$463$	− 14492.0i	− 1.45464i	−0.686296	−	0.727322i	$-0.740765\pi$
$463$	− 14492.0i	− 1.45464i	0.686296	−	0.727322i	$-0.259235\pi$
$464$	−1440.00	−0.144074
$465$	0	0
$466$	3924.00	0.390077
$467$	− 7776.00i	− 0.770515i	−0.922809	−	0.385257i	$-0.874113\pi$
$467$	− 7776.00i	− 0.770515i	0.922809	−	0.385257i	$-0.125887\pi$
$468$	0	0
$469$	−4096.00	−0.403274
$470$	0	0
$471$	0	0
$472$	3360.00i	0.327662i
$473$	384.000i	0.0373284i
$474$	0	0
$475$	0	0
$476$	1056.00	0.101684
$477$	0	0
$478$	8640.00i	0.826746i
$479$	−13680.0	−1.30492	−0.652458	−	0.757825i	$-0.726262\pi$
$479$	−13680.0	−1.30492	−0.652458	+	0.757825i	$0.726262\pi$
$480$	0	0
$481$	1972.00	0.186934
$482$	956.000i	0.0903415i
$483$	0	0
$484$	4748.00	0.445905
$485$	0	0
$486$	0	0
$487$	7916.00i	0.736567i	0.929714	+	0.368284i	$0.120054\pi$
$487$	7916.00i	0.736567i	−0.929714	+	0.368284i	$0.879946\pi$
$488$	7216.00i	0.669371i
$489$	0	0
$490$	0	0
$491$	−13932.0	−1.28053	−0.640267	−	0.768152i	$-0.721176\pi$
$491$	−13932.0	−1.28053	−0.640267	+	0.768152i	$0.721176\pi$
$492$	0	0
$493$	5940.00i	0.542645i
$494$	11600.0	1.05650
$495$	0	0
$496$	2432.00	0.220161
$497$	1728.00i	0.155959i
$498$	0	0
$499$	8260.00	0.741019	0.370509	−	0.928829i	$-0.379183\pi$
$499$	8260.00	0.741019	0.370509	+	0.928829i	$0.379183\pi$
$500$	0	0
$501$	0	0
$502$	5304.00i	0.471572i
$503$	− 11148.0i	− 0.988200i	−0.869405	−	0.494100i	$-0.835498\pi$
$503$	− 11148.0i	− 0.988200i	0.869405	−	0.494100i	$-0.164502\pi$
$504$	0	0
$505$	0	0
$506$	−3168.00	−0.278330
$507$	0	0
$508$	496.000i	0.0433198i
$509$	−9690.00	−0.843815	−0.421907	−	0.906639i	$-0.638639\pi$
$509$	−9690.00	−0.843815	−0.421907	+	0.906639i	$0.638639\pi$
$510$	0	0
$511$	−1448.00	−0.125354
$512$	− 512.000i	− 0.0441942i
$513$	0	0
$514$	4668.00	0.400577
$515$	0	0
$516$	0	0
$517$	− 2448.00i	− 0.208245i
$518$	272.000i	0.0230714i
$519$	0	0
$520$	0	0
$521$	16038.0	1.34863	0.674316	−	0.738443i	$-0.264438\pi$
$521$	16038.0	1.34863	0.674316	+	0.738443i	$0.264438\pi$
$522$	0	0
$523$	− 992.000i	− 0.0829391i	−0.999140	−	0.0414695i	$-0.986796\pi$
$523$	− 992.000i	− 0.0829391i	0.999140	−	0.0414695i	$-0.0132039\pi$
$524$	528.000	0.0440187
$525$	0	0
$526$	−7896.00	−0.654528
$527$	− 10032.0i	− 0.829223i
$528$	0	0
$529$	−5257.00	−0.432070
$530$	0	0
$531$	0	0
$532$	1600.00i	0.130392i
$533$	25404.0i	2.06448i
$534$	0	0
$535$	0	0
$536$	8192.00	0.660150
$537$	0	0
$538$	− 3180.00i	− 0.254832i
$539$	−3924.00	−0.313578
$540$	0	0
$541$	7142.00	0.567576	0.283788	−	0.958887i	$-0.408409\pi$
$541$	7142.00	0.567576	0.283788	+	0.958887i	$0.408409\pi$
$542$	− 9904.00i	− 0.784895i
$543$	0	0
$544$	−2112.00	−0.166455
$545$	0	0
$546$	0	0
$547$	7616.00i	0.595314i	0.954673	+	0.297657i	$0.0962051\pi$
$547$	7616.00i	0.595314i	−0.954673	+	0.297657i	$0.903795\pi$
$548$	− 5016.00i	− 0.391009i
$549$	0	0
$550$	0	0
$551$	−9000.00	−0.695849
$552$	0	0
$553$	− 640.000i	− 0.0492144i
$554$	3292.00	0.252462
$555$	0	0
$556$	−11440.0	−0.872597
$557$	10314.0i	0.784593i	0.919839	+	0.392296i	$0.128319\pi$
$557$	10314.0i	0.784593i	−0.919839	+	0.392296i	$0.871681\pi$
$558$	0	0
$559$	1856.00	0.140430
$560$	0	0
$561$	0	0
$562$	− 2316.00i	− 0.173834i
$563$	− 7128.00i	− 0.533587i	−0.963754	−	0.266793i	$-0.914036\pi$
$563$	− 7128.00i	− 0.533587i	0.963754	−	0.266793i	$-0.0859641\pi$
$564$	0	0
$565$	0	0
$566$	−13984.0	−1.03850
$567$	0	0
$568$	− 3456.00i	− 0.255300i
$569$	2010.00	0.148091	0.0740453	−	0.997255i	$-0.476409\pi$
$569$	2010.00	0.148091	0.0740453	+	0.997255i	$0.476409\pi$
$570$	0	0
$571$	−23188.0	−1.69945	−0.849726	−	0.527224i	$-0.823233\pi$
$571$	−23188.0	−1.69945	−0.849726	+	0.527224i	$0.823233\pi$
$572$	2784.00i	0.203505i
$573$	0	0
$574$	−3504.00	−0.254798
$575$	0	0
$576$	0	0
$577$	22466.0i	1.62092i	0.585793	+	0.810461i	$0.300783\pi$
$577$	22466.0i	1.62092i	−0.585793	+	0.810461i	$0.699217\pi$
$578$	− 1114.00i	− 0.0801666i
$579$	0	0
$580$	0	0
$581$	288.000	0.0205650
$582$	0	0
$583$	− 2664.00i	− 0.189248i
$584$	2896.00	0.205201
$585$	0	0
$586$	−516.000	−0.0363750
$587$	− 22776.0i	− 1.60148i	−0.599015	−	0.800738i	$-0.704441\pi$
$587$	− 22776.0i	− 1.60148i	0.599015	−	0.800738i	$-0.295559\pi$
$588$	0	0
$589$	15200.0	1.06334
$590$	0	0
$591$	0	0
$592$	− 544.000i	− 0.0377673i
$593$	− 21198.0i	− 1.46796i	−0.679174	−	0.733978i	$-0.737662\pi$
$593$	− 21198.0i	− 1.46796i	0.679174	−	0.733978i	$-0.262338\pi$
$594$	0	0
$595$	0	0
$596$	−3000.00	−0.206183
$597$	0	0
$598$	15312.0i	1.04708i
$599$	15960.0	1.08866	0.544330	−	0.838871i	$-0.316784\pi$
$599$	15960.0	1.08866	0.544330	+	0.838871i	$0.316784\pi$
$600$	0	0
$601$	5882.00	0.399221	0.199610	−	0.979875i	$-0.436032\pi$
$601$	5882.00	0.399221	0.199610	+	0.979875i	$0.436032\pi$
$602$	256.000i	0.0173319i
$603$	0	0
$604$	1792.00	0.120721
$605$	0	0
$606$	0	0
$607$	8516.00i	0.569446i	0.958610	+	0.284723i	$0.0919016\pi$
$607$	8516.00i	0.569446i	−0.958610	+	0.284723i	$0.908098\pi$
$608$	− 3200.00i	− 0.213449i
$609$	0	0
$610$	0	0
$611$	−11832.0	−0.783423
$612$	0	0
$613$	− 8462.00i	− 0.557548i	−0.960357	−	0.278774i	$-0.910072\pi$
$613$	− 8462.00i	− 0.557548i	0.960357	−	0.278774i	$-0.0899280\pi$
$614$	−17888.0	−1.17573
$615$	0	0
$616$	−384.000	−0.0251166
$617$	11094.0i	0.723870i	0.932203	+	0.361935i	$0.117884\pi$
$617$	11094.0i	0.723870i	−0.932203	+	0.361935i	$0.882116\pi$
$618$	0	0
$619$	−2180.00	−0.141553	−0.0707767	−	0.997492i	$-0.522548\pi$
$619$	−2180.00	−0.141553	−0.0707767	+	0.997492i	$0.522548\pi$
$620$	0	0
$621$	0	0
$622$	2784.00i	0.179467i
$623$	− 3240.00i	− 0.208359i
$624$	0	0
$625$	0	0
$626$	11756.0	0.750582
$627$	0	0
$628$	− 8984.00i	− 0.570861i
$629$	−2244.00	−0.142248
$630$	0	0
$631$	−26848.0	−1.69382	−0.846911	−	0.531734i	$-0.821541\pi$
$631$	−26848.0	−1.69382	−0.846911	+	0.531734i	$0.821541\pi$
$632$	1280.00i	0.0805628i
$633$	0	0
$634$	−20652.0	−1.29368
$635$	0	0
$636$	0	0
$637$	18966.0i	1.17969i
$638$	− 2160.00i	− 0.134036i
$639$	0	0
$640$	0	0
$641$	−26322.0	−1.62193	−0.810965	−	0.585095i	$-0.801057\pi$
$641$	−26322.0	−1.62193	−0.810965	+	0.585095i	$0.801057\pi$
$642$	0	0
$643$	10168.0i	0.623619i	0.950145	+	0.311809i	$0.100935\pi$
$643$	10168.0i	0.623619i	−0.950145	+	0.311809i	$0.899065\pi$
$644$	−2112.00	−0.129231
$645$	0	0
$646$	−13200.0	−0.803943
$647$	23604.0i	1.43426i	0.696937	+	0.717132i	$0.254546\pi$
$647$	23604.0i	1.43426i	−0.696937	+	0.717132i	$0.745454\pi$
$648$	0	0
$649$	−5040.00	−0.304834
$650$	0	0
$651$	0	0
$652$	− 2272.00i	− 0.136470i
$653$	16422.0i	0.984139i	0.870556	+	0.492069i	$0.163759\pi$
$653$	16422.0i	0.984139i	−0.870556	+	0.492069i	$0.836241\pi$
$654$	0	0
$655$	0	0
$656$	7008.00	0.417098
$657$	0	0
$658$	− 1632.00i	− 0.0966899i
$659$	−26100.0	−1.54281	−0.771405	−	0.636345i	$-0.780446\pi$
$659$	−26100.0	−1.54281	−0.771405	+	0.636345i	$0.780446\pi$
$660$	0	0
$661$	−3058.00	−0.179943	−0.0899716	−	0.995944i	$-0.528678\pi$
$661$	−3058.00	−0.179943	−0.0899716	+	0.995944i	$0.528678\pi$
$662$	8456.00i	0.496453i
$663$	0	0
$664$	−576.000	−0.0336644
$665$	0	0
$666$	0	0
$667$	− 11880.0i	− 0.689648i
$668$	− 6096.00i	− 0.353086i
$669$	0	0
$670$	0	0
$671$	−10824.0	−0.622736
$672$	0	0
$673$	− 10802.0i	− 0.618702i	−0.950948	−	0.309351i	$-0.899888\pi$
$673$	− 10802.0i	− 0.618702i	0.950948	−	0.309351i	$-0.100112\pi$
$674$	2212.00	0.126414
$675$	0	0
$676$	4668.00	0.265589
$677$	10674.0i	0.605960i	0.952997	+	0.302980i	$0.0979816\pi$
$677$	10674.0i	0.605960i	−0.952997	+	0.302980i	$0.902018\pi$
$678$	0	0
$679$	4424.00	0.250041
$680$	0	0
$681$	0	0
$682$	3648.00i	0.204823i
$683$	− 28608.0i	− 1.60272i	−0.598185	−	0.801358i	$-0.704111\pi$
$683$	− 28608.0i	− 1.60272i	0.598185	−	0.801358i	$-0.295889\pi$
$684$	0	0
$685$	0	0
$686$	−5360.00	−0.298317
$687$	0	0
$688$	− 512.000i	− 0.0283718i
$689$	−12876.0	−0.711954
$690$	0	0
$691$	−2428.00	−0.133669	−0.0668346	−	0.997764i	$-0.521290\pi$
$691$	−2428.00	−0.133669	−0.0668346	+	0.997764i	$0.521290\pi$
$692$	− 14808.0i	− 0.813462i
$693$	0	0
$694$	−18672.0	−1.02130
$695$	0	0
$696$	0	0
$697$	− 28908.0i	− 1.57097i
$698$	− 23540.0i	− 1.27651i
$699$	0	0
$700$	0	0
$701$	6618.00	0.356574	0.178287	−	0.983979i	$-0.442944\pi$
$701$	6618.00	0.356574	0.178287	+	0.983979i	$0.442944\pi$
$702$	0	0
$703$	− 3400.00i	− 0.182409i
$704$	768.000	0.0411152
$705$	0	0
$706$	16644.0	0.887259
$707$	− 1032.00i	− 0.0548972i
$708$	0	0
$709$	−20510.0	−1.08642	−0.543208	−	0.839598i	$-0.682791\pi$
$709$	−20510.0	−1.08642	−0.543208	+	0.839598i	$0.682791\pi$
$710$	0	0
$711$	0	0
$712$	6480.00i	0.341079i
$713$	20064.0i	1.05386i
$714$	0	0
$715$	0	0
$716$	−12720.0	−0.663923
$717$	0	0
$718$	− 21360.0i	− 1.11023i
$719$	31680.0	1.64321	0.821603	−	0.570061i	$-0.193080\pi$
$719$	31680.0	1.64321	0.821603	+	0.570061i	$0.193080\pi$
$720$	0	0
$721$	3952.00	0.204133
$722$	− 6282.00i	− 0.323811i
$723$	0	0
$724$	8392.00	0.430782
$725$	0	0
$726$	0	0
$727$	13196.0i	0.673195i	0.941649	+	0.336597i	$0.109276\pi$
$727$	13196.0i	0.673195i	−0.941649	+	0.336597i	$0.890724\pi$
$728$	1856.00i	0.0944889i
$729$	0	0
$730$	0	0
$731$	−2112.00	−0.106861
$732$	0	0
$733$	− 8102.00i	− 0.408259i	−0.978944	−	0.204130i	$-0.934564\pi$
$733$	− 8102.00i	− 0.408259i	0.978944	−	0.204130i	$-0.0654364\pi$
$734$	−11768.0	−0.591778
$735$	0	0
$736$	4224.00	0.211547
$737$	12288.0i	0.614158i
$738$	0	0
$739$	12580.0	0.626201	0.313101	−	0.949720i	$-0.398632\pi$
$739$	12580.0	0.626201	0.313101	+	0.949720i	$0.398632\pi$
$740$	0	0
$741$	0	0
$742$	− 1776.00i	− 0.0878693i
$743$	29892.0i	1.47595i	0.674828	+	0.737975i	$0.264218\pi$
$743$	29892.0i	1.47595i	−0.674828	+	0.737975i	$0.735782\pi$
$744$	0	0
$745$	0	0
$746$	4196.00	0.205934
$747$	0	0
$748$	− 3168.00i	− 0.154858i
$749$	96.0000	0.00468326
$750$	0	0
$751$	−40408.0	−1.96339	−0.981697	−	0.190450i	$-0.939005\pi$
$751$	−40408.0	−1.96339	−0.981697	+	0.190450i	$0.939005\pi$
$752$	3264.00i	0.158279i
$753$	0	0
$754$	−10440.0	−0.504247
$755$	0	0
$756$	0	0
$757$	32366.0i	1.55398i	0.629513	+	0.776990i	$0.283254\pi$
$757$	32366.0i	1.55398i	−0.629513	+	0.776990i	$0.716746\pi$
$758$	7720.00i	0.369925i
$759$	0	0
$760$	0	0
$761$	17238.0	0.821126	0.410563	−	0.911832i	$-0.365332\pi$
$761$	17238.0	0.821126	0.410563	+	0.911832i	$0.365332\pi$
$762$	0	0
$763$	3800.00i	0.180300i
$764$	17568.0	0.831921
$765$	0	0
$766$	−19176.0	−0.904513
$767$	24360.0i	1.14679i
$768$	0	0
$769$	−10850.0	−0.508792	−0.254396	−	0.967100i	$-0.581877\pi$
$769$	−10850.0	−0.508792	−0.254396	+	0.967100i	$0.581877\pi$
$770$	0	0
$771$	0	0
$772$	− 8632.00i	− 0.402425i
$773$	9102.00i	0.423514i	0.977322	+	0.211757i	$0.0679185\pi$
$773$	9102.00i	0.423514i	−0.977322	+	0.211757i	$0.932081\pi$
$774$	0	0
$775$	0	0
$776$	−8848.00	−0.409310
$777$	0	0
$778$	26820.0i	1.23592i
$779$	43800.0	2.01450
$780$	0	0
$781$	5184.00	0.237514
$782$	− 17424.0i	− 0.796779i
$783$	0	0
$784$	5232.00	0.238338
$785$	0	0
$786$	0	0
$787$	− 25504.0i	− 1.15517i	−0.816330	−	0.577585i	$-0.803995\pi$
$787$	− 25504.0i	− 1.15517i	0.816330	−	0.577585i	$-0.196005\pi$
$788$	− 4296.00i	− 0.194212i
$789$	0	0
$790$	0	0
$791$	−4152.00	−0.186635
$792$	0	0
$793$	52316.0i	2.34274i
$794$	−26228.0	−1.17229
$795$	0	0
$796$	11360.0	0.505835
$797$	− 14166.0i	− 0.629593i	−0.949159	−	0.314796i	$-0.898064\pi$
$797$	− 14166.0i	− 0.629593i	0.949159	−	0.314796i	$-0.101936\pi$
$798$	0	0
$799$	13464.0	0.596148
$800$	0	0
$801$	0	0
$802$	− 11676.0i	− 0.514082i
$803$	4344.00i	0.190905i
$804$	0	0
$805$	0	0
$806$	17632.0	0.770547
$807$	0	0
$808$	2064.00i	0.0898654i
$809$	33210.0	1.44327	0.721633	−	0.692276i	$-0.243392\pi$
$809$	33210.0	1.44327	0.721633	+	0.692276i	$0.243392\pi$
$810$	0	0
$811$	39212.0	1.69780	0.848902	−	0.528550i	$-0.177264\pi$
$811$	39212.0	1.69780	0.848902	+	0.528550i	$0.177264\pi$
$812$	− 1440.00i	− 0.0622341i
$813$	0	0
$814$	816.000	0.0351361
$815$	0	0
$816$	0	0
$817$	− 3200.00i	− 0.137030i
$818$	19060.0i	0.814691i
$819$	0	0
$820$	0	0
$821$	−6222.00	−0.264494	−0.132247	−	0.991217i	$-0.542219\pi$
$821$	−6222.00	−0.264494	−0.132247	+	0.991217i	$0.542219\pi$
$822$	0	0
$823$	− 31172.0i	− 1.32028i	−0.751144	−	0.660138i	$-0.770498\pi$
$823$	− 31172.0i	− 1.32028i	0.751144	−	0.660138i	$-0.229502\pi$
$824$	−7904.00	−0.334161
$825$	0	0
$826$	−3360.00	−0.141537
$827$	264.000i	0.0111006i	0.999985	+	0.00555029i	$0.00176672\pi$
$827$	264.000i	0.0111006i	−0.999985	+	0.00555029i	$0.998233\pi$
$828$	0	0
$829$	29050.0	1.21707	0.608533	−	0.793528i	$-0.291758\pi$
$829$	29050.0	1.21707	0.608533	+	0.793528i	$0.291758\pi$
$830$	0	0
$831$	0	0
$832$	− 3712.00i	− 0.154676i
$833$	− 21582.0i	− 0.897685i
$834$	0	0
$835$	0	0
$836$	4800.00	0.198578
$837$	0	0
$838$	− 14520.0i	− 0.598550i
$839$	−21720.0	−0.893752	−0.446876	−	0.894596i	$-0.647463\pi$
$839$	−21720.0	−0.893752	−0.446876	+	0.894596i	$0.647463\pi$
$840$	0	0
$841$	−16289.0	−0.667883
$842$	− 24124.0i	− 0.987373i
$843$	0	0
$844$	10672.0	0.435243
$845$	0	0
$846$	0	0
$847$	4748.00i	0.192613i
$848$	3552.00i	0.143840i
$849$	0	0
$850$	0	0
$851$	4488.00	0.180783
$852$	0	0
$853$	6658.00i	0.267252i	0.991032	+	0.133626i	$0.0426620\pi$
$853$	6658.00i	0.267252i	−0.991032	+	0.133626i	$0.957338\pi$
$854$	−7216.00	−0.289141
$855$	0	0
$856$	−192.000	−0.00766638
$857$	13974.0i	0.556993i	0.960437	+	0.278496i	$0.0898360\pi$
$857$	13974.0i	0.556993i	−0.960437	+	0.278496i	$0.910164\pi$
$858$	0	0
$859$	−23780.0	−0.944544	−0.472272	−	0.881453i	$-0.656566\pi$
$859$	−23780.0	−0.944544	−0.472272	+	0.881453i	$0.656566\pi$
$860$	0	0
$861$	0	0
$862$	− 27216.0i	− 1.07538i
$863$	− 12228.0i	− 0.482324i	−0.970485	−	0.241162i	$-0.922471\pi$
$863$	− 12228.0i	− 0.482324i	0.970485	−	0.241162i	$-0.0775286\pi$
$864$	0	0
$865$	0	0
$866$	7676.00	0.301202
$867$	0	0
$868$	2432.00i	0.0951008i
$869$	−1920.00	−0.0749500
$870$	0	0
$871$	59392.0	2.31047
$872$	− 7600.00i	− 0.295147i
$873$	0	0
$874$	26400.0	1.02173
$875$	0	0
$876$	0	0
$877$	11606.0i	0.446872i	0.974719	+	0.223436i	$0.0717274\pi$
$877$	11606.0i	0.446872i	−0.974719	+	0.223436i	$0.928273\pi$
$878$	14800.0i	0.568879i
$879$	0	0
$880$	0	0
$881$	32958.0	1.26037	0.630183	−	0.776446i	$-0.282980\pi$
$881$	32958.0	1.26037	0.630183	+	0.776446i	$0.282980\pi$
$882$	0	0
$883$	− 8072.00i	− 0.307638i	−0.988099	−	0.153819i	$-0.950843\pi$
$883$	− 8072.00i	− 0.307638i	0.988099	−	0.153819i	$-0.0491573\pi$
$884$	−15312.0	−0.582577
$885$	0	0
$886$	16704.0	0.633388
$887$	− 15756.0i	− 0.596431i	−0.954498	−	0.298216i	$-0.903609\pi$
$887$	− 15756.0i	− 0.596431i	0.954498	−	0.298216i	$-0.0963915\pi$
$888$	0	0
$889$	−496.000	−0.0187124
$890$	0	0
$891$	0	0
$892$	7088.00i	0.266058i
$893$	20400.0i	0.764457i
$894$	0	0
$895$	0	0
$896$	512.000	0.0190901
$897$	0	0
$898$	− 21540.0i	− 0.800444i
$899$	−13680.0	−0.507512
$900$	0	0
$901$	14652.0	0.541763
$902$	10512.0i	0.388039i
$903$	0	0
$904$	8304.00	0.305517
$905$	0	0
$906$	0	0
$907$	18776.0i	0.687372i	0.939085	+	0.343686i	$0.111676\pi$
$907$	18776.0i	0.687372i	−0.939085	+	0.343686i	$0.888324\pi$
$908$	− 11136.0i	− 0.407006i
$909$	0	0
$910$	0	0
$911$	20568.0	0.748022	0.374011	−	0.927424i	$-0.377982\pi$
$911$	20568.0	0.748022	0.374011	+	0.927424i	$0.377982\pi$
$912$	0	0
$913$	− 864.000i	− 0.0313190i
$914$	−13388.0	−0.484503
$915$	0	0
$916$	1400.00	0.0504992
$917$	528.000i	0.0190143i
$918$	0	0
$919$	6280.00	0.225417	0.112708	−	0.993628i	$-0.464047\pi$
$919$	6280.00	0.225417	0.112708	+	0.993628i	$0.464047\pi$
$920$	0	0
$921$	0	0
$922$	− 6036.00i	− 0.215602i
$923$	− 25056.0i	− 0.893530i
$924$	0	0
$925$	0	0
$926$	−28984.0	−1.02859
$927$	0	0
$928$	2880.00i	0.101876i
$929$	−20430.0	−0.721514	−0.360757	−	0.932660i	$-0.617482\pi$
$929$	−20430.0	−0.721514	−0.360757	+	0.932660i	$0.617482\pi$
$930$	0	0
$931$	32700.0	1.15113
$932$	− 7848.00i	− 0.275826i
$933$	0	0
$934$	−15552.0	−0.544836
$935$	0	0
$936$	0	0
$937$	8906.00i	0.310508i	0.987875	+	0.155254i	$0.0496197\pi$
$937$	8906.00i	0.310508i	−0.987875	+	0.155254i	$0.950380\pi$
$938$	8192.00i	0.285158i
$939$	0	0
$940$	0	0
$941$	17418.0	0.603412	0.301706	−	0.953401i	$-0.402444\pi$
$941$	17418.0	0.603412	0.301706	+	0.953401i	$0.402444\pi$
$942$	0	0
$943$	57816.0i	1.99655i
$944$	6720.00	0.231692
$945$	0	0
$946$	768.000	0.0263952
$947$	2544.00i	0.0872956i	0.999047	+	0.0436478i	$0.0138979\pi$
$947$	2544.00i	0.0872956i	−0.999047	+	0.0436478i	$0.986102\pi$
$948$	0	0
$949$	20996.0	0.718187
$950$	0	0
$951$	0	0
$952$	− 2112.00i	− 0.0719016i
$953$	15402.0i	0.523525i	0.965132	+	0.261763i	$0.0843038\pi$
$953$	15402.0i	0.523525i	−0.965132	+	0.261763i	$0.915696\pi$
$954$	0	0
$955$	0	0
$956$	17280.0	0.584597
$957$	0	0
$958$	27360.0i	0.922716i
$959$	5016.00	0.168900
$960$	0	0
$961$	−6687.00	−0.224464
$962$	− 3944.00i	− 0.132183i
$963$	0	0
$964$	1912.00	0.0638811
$965$	0	0
$966$	0	0
$967$	− 49444.0i	− 1.64427i	−0.569291	−	0.822136i	$-0.692782\pi$
$967$	− 49444.0i	− 1.64427i	0.569291	−	0.822136i	$-0.307218\pi$
$968$	− 9496.00i	− 0.315303i
$969$	0	0
$970$	0	0
$971$	25188.0	0.832463	0.416231	−	0.909259i	$-0.363351\pi$
$971$	25188.0	0.832463	0.416231	+	0.909259i	$0.363351\pi$
$972$	0	0
$973$	− 11440.0i	− 0.376927i
$974$	15832.0	0.520832
$975$	0	0
$976$	14432.0	0.473317
$977$	− 2946.00i	− 0.0964697i	−0.998836	−	0.0482348i	$-0.984640\pi$
$977$	− 2946.00i	− 0.0964697i	0.998836	−	0.0482348i	$-0.0153596\pi$
$978$	0	0
$979$	−9720.00	−0.317316
$980$	0	0
$981$	0	0
$982$	27864.0i	0.905475i
$983$	15012.0i	0.487089i	0.969890	+	0.243544i	$0.0783102\pi$
$983$	15012.0i	0.487089i	−0.969890	+	0.243544i	$0.921690\pi$
$984$	0	0
$985$	0	0
$986$	11880.0	0.383708
$987$	0	0
$988$	− 23200.0i	− 0.747055i
$989$	4224.00	0.135809
$990$	0	0
$991$	−5128.00	−0.164376	−0.0821878	−	0.996617i	$-0.526191\pi$
$991$	−5128.00	−0.164376	−0.0821878	+	0.996617i	$0.526191\pi$
$992$	− 4864.00i	− 0.155678i
$993$	0	0
$994$	3456.00	0.110279
$995$	0	0
$996$	0	0
$997$	− 49714.0i	− 1.57920i	−0.613625	−	0.789598i	$-0.710289\pi$
$997$	− 49714.0i	− 1.57920i	0.613625	−	0.789598i	$-0.289711\pi$
$998$	− 16520.0i	− 0.523979i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.4.c.d.199.1		2
3.2	odd	2		50.4.b.a.49.2		2
5.2	odd	4		450.4.a.q.1.1		1
5.3	odd	4		90.4.a.a.1.1		1
5.4	even	2	inner	450.4.c.d.199.2		2
12.11	even	2		400.4.c.c.49.1		2
15.2	even	4		50.4.a.c.1.1		1
15.8	even	4		10.4.a.a.1.1	✓	1
15.14	odd	2		50.4.b.a.49.1		2
20.3	even	4		720.4.a.j.1.1		1
45.13	odd	12		810.4.e.w.541.1		2
45.23	even	12		810.4.e.c.541.1		2
45.38	even	12		810.4.e.c.271.1		2
45.43	odd	12		810.4.e.w.271.1		2
60.23	odd	4		80.4.a.f.1.1		1
60.47	odd	4		400.4.a.b.1.1		1
60.59	even	2		400.4.c.c.49.2		2
105.23	even	12		490.4.e.i.361.1		2
105.38	odd	12		490.4.e.a.471.1		2
105.53	even	12		490.4.e.i.471.1		2
105.62	odd	4		2450.4.a.b.1.1		1
105.68	odd	12		490.4.e.a.361.1		2
105.83	odd	4		490.4.a.o.1.1		1
120.53	even	4		320.4.a.m.1.1		1
120.77	even	4		1600.4.a.d.1.1		1
120.83	odd	4		320.4.a.b.1.1		1
120.107	odd	4		1600.4.a.bx.1.1		1
165.98	odd	4		1210.4.a.b.1.1		1
195.38	even	4		1690.4.a.a.1.1		1
240.53	even	4		1280.4.d.j.641.2		2
240.83	odd	4		1280.4.d.g.641.2		2
240.173	even	4		1280.4.d.j.641.1		2
240.203	odd	4		1280.4.d.g.641.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.4.a.a.1.1	✓	1	15.8	even	4
50.4.a.c.1.1		1	15.2	even	4
50.4.b.a.49.1		2	15.14	odd	2
50.4.b.a.49.2		2	3.2	odd	2
80.4.a.f.1.1		1	60.23	odd	4
90.4.a.a.1.1		1	5.3	odd	4
320.4.a.b.1.1		1	120.83	odd	4
320.4.a.m.1.1		1	120.53	even	4
400.4.a.b.1.1		1	60.47	odd	4
400.4.c.c.49.1		2	12.11	even	2
400.4.c.c.49.2		2	60.59	even	2
450.4.a.q.1.1		1	5.2	odd	4
450.4.c.d.199.1		2	1.1	even	1	trivial
450.4.c.d.199.2		2	5.4	even	2	inner
490.4.a.o.1.1		1	105.83	odd	4
490.4.e.a.361.1		2	105.68	odd	12
490.4.e.a.471.1		2	105.38	odd	12
490.4.e.i.361.1		2	105.23	even	12
490.4.e.i.471.1		2	105.53	even	12
720.4.a.j.1.1		1	20.3	even	4
810.4.e.c.271.1		2	45.38	even	12
810.4.e.c.541.1		2	45.23	even	12
810.4.e.w.271.1		2	45.43	odd	12
810.4.e.w.541.1		2	45.13	odd	12
1210.4.a.b.1.1		1	165.98	odd	4
1280.4.d.g.641.1		2	240.203	odd	4
1280.4.d.g.641.2		2	240.83	odd	4
1280.4.d.j.641.1		2	240.173	even	4
1280.4.d.j.641.2		2	240.53	even	4
1600.4.a.d.1.1		1	120.77	even	4
1600.4.a.bx.1.1		1	120.107	odd	4
1690.4.a.a.1.1		1	195.38	even	4
2450.4.a.b.1.1		1	105.62	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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	450.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -4.00000 q^{4} -4.00000i q^{7} +8.00000i q^{8} +O(q^{10})\)	\(q-2.00000i q^{2} -4.00000 q^{4} -4.00000i q^{7} +8.00000i q^{8} -12.0000 q^{11} +58.0000i q^{13} -8.00000 q^{14} +16.0000 q^{16} -66.0000i q^{17} +100.000 q^{19} +24.0000i q^{22} +132.000i q^{23} +116.000 q^{26} +16.0000i q^{28} -90.0000 q^{29} +152.000 q^{31} -32.0000i q^{32} -132.000 q^{34} -34.0000i q^{37} -200.000i q^{38} +438.000 q^{41} -32.0000i q^{43} +48.0000 q^{44} +264.000 q^{46} +204.000i q^{47} +327.000 q^{49} -232.000i q^{52} +222.000i q^{53} +32.0000 q^{56} +180.000i q^{58} +420.000 q^{59} +902.000 q^{61} -304.000i q^{62} -64.0000 q^{64} -1024.00i q^{67} +264.000i q^{68} -432.000 q^{71} -362.000i q^{73} -68.0000 q^{74} -400.000 q^{76} +48.0000i q^{77} +160.000 q^{79} -876.000i q^{82} +72.0000i q^{83} -64.0000 q^{86} -96.0000i q^{88} +810.000 q^{89} +232.000 q^{91} -528.000i q^{92} +408.000 q^{94} +1106.00i q^{97} -654.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4}+O(q^{10}) \) 2 * q - 8 * q^4	\( 2 q - 8 q^{4} - 24 q^{11} - 16 q^{14} + 32 q^{16} + 200 q^{19} + 232 q^{26} - 180 q^{29} + 304 q^{31} - 264 q^{34} + 876 q^{41} + 96 q^{44} + 528 q^{46} + 654 q^{49} + 64 q^{56} + 840 q^{59} + 1804 q^{61} - 128 q^{64} - 864 q^{71} - 136 q^{74} - 800 q^{76} + 320 q^{79} - 128 q^{86} + 1620 q^{89} + 464 q^{91} + 816 q^{94}+O(q^{100}) \) 2 * q - 8 * q^4 - 24 * q^11 - 16 * q^14 + 32 * q^16 + 200 * q^19 + 232 * q^26 - 180 * q^29 + 304 * q^31 - 264 * q^34 + 876 * q^41 + 96 * q^44 + 528 * q^46 + 654 * q^49 + 64 * q^56 + 840 * q^59 + 1804 * q^61 - 128 * q^64 - 864 * q^71 - 136 * q^74 - 800 * q^76 + 320 * q^79 - 128 * q^86 + 1620 * q^89 + 464 * q^91 + 816 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more