LMFDB - Embedded newform 450.4.c.j.199.2

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 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -4.00000 q^{4} +4.00000i q^{7} -8.00000i q^{8} +O(q^{10})$	$q+2.00000i q^{2} -4.00000 q^{4} +4.00000i q^{7} -8.00000i q^{8} +48.0000 q^{11} +2.00000i q^{13} -8.00000 q^{14} +16.0000 q^{16} -114.000i q^{17} -140.000 q^{19} +96.0000i q^{22} -72.0000i q^{23} -4.00000 q^{26} -16.0000i q^{28} +210.000 q^{29} +272.000 q^{31} +32.0000i q^{32} +228.000 q^{34} +334.000i q^{37} -280.000i q^{38} +198.000 q^{41} -268.000i q^{43} -192.000 q^{44} +144.000 q^{46} +216.000i q^{47} +327.000 q^{49} -8.00000i q^{52} +78.0000i q^{53} +32.0000 q^{56} +420.000i q^{58} +240.000 q^{59} +302.000 q^{61} +544.000i q^{62} -64.0000 q^{64} -596.000i q^{67} +456.000i q^{68} +768.000 q^{71} -478.000i q^{73} -668.000 q^{74} +560.000 q^{76} +192.000i q^{77} +640.000 q^{79} +396.000i q^{82} +348.000i q^{83} +536.000 q^{86} -384.000i q^{88} +210.000 q^{89} -8.00000 q^{91} +288.000i q^{92} -432.000 q^{94} +1534.00i q^{97} +654.000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4}+O(q^{10}) $ 2 * q - 8 * q^4	$ 2 q - 8 q^{4} + 96 q^{11} - 16 q^{14} + 32 q^{16} - 280 q^{19} - 8 q^{26} + 420 q^{29} + 544 q^{31} + 456 q^{34} + 396 q^{41} - 384 q^{44} + 288 q^{46} + 654 q^{49} + 64 q^{56} + 480 q^{59} + 604 q^{61} - 128 q^{64} + 1536 q^{71} - 1336 q^{74} + 1120 q^{76} + 1280 q^{79} + 1072 q^{86} + 420 q^{89} - 16 q^{91} - 864 q^{94}+O(q^{100}) $ 2 * q - 8 * q^4 + 96 * q^11 - 16 * q^14 + 32 * q^16 - 280 * q^19 - 8 * q^26 + 420 * q^29 + 544 * q^31 + 456 * q^34 + 396 * q^41 - 384 * q^44 + 288 * q^46 + 654 * q^49 + 64 * q^56 + 480 * q^59 + 604 * q^61 - 128 * q^64 + 1536 * q^71 - 1336 * q^74 + 1120 * q^76 + 1280 * q^79 + 1072 * q^86 + 420 * q^89 - 16 * q^91 - 864 * 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	$0.0344414\pi$
$7$	4.00000i	0.215980i	−0.994152	+	0.107990i	$0.965559\pi$
$8$	− 8.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	48.0000	1.31569	0.657843	−	0.753155i	$-0.271469\pi$
$11$	48.0000	1.31569	0.657843	+	0.753155i	$0.271469\pi$
$12$	0	0
$13$	2.00000i	0.0426692i	0.999772	+	0.0213346i	$0.00679154\pi$
$13$	2.00000i	0.0426692i	−0.999772	+	0.0213346i	$0.993208\pi$
$14$	−8.00000	−0.152721
$15$	0	0
$16$	16.0000	0.250000
$17$	− 114.000i	− 1.62642i	−0.581974	−	0.813208i	$-0.697719\pi$
$17$	− 114.000i	− 1.62642i	0.581974	−	0.813208i	$-0.302281\pi$
$18$	0	0
$19$	−140.000	−1.69043	−0.845216	−	0.534425i	$-0.820528\pi$
$19$	−140.000	−1.69043	−0.845216	+	0.534425i	$0.820528\pi$
$20$	0	0
$21$	0	0
$22$	96.0000i	0.930330i
$23$	− 72.0000i	− 0.652741i	−0.945242	−	0.326370i	$-0.894174\pi$
$23$	− 72.0000i	− 0.652741i	0.945242	−	0.326370i	$-0.105826\pi$
$24$	0	0
$25$	0	0
$26$	−4.00000	−0.0301717
$27$	0	0
$28$	− 16.0000i	− 0.107990i
$29$	210.000	1.34469	0.672345	−	0.740238i	$-0.265287\pi$
$29$	210.000	1.34469	0.672345	+	0.740238i	$0.265287\pi$
$30$	0	0
$31$	272.000	1.57589	0.787946	−	0.615745i	$-0.211145\pi$
$31$	272.000	1.57589	0.787946	+	0.615745i	$0.211145\pi$
$32$	32.0000i	0.176777i
$33$	0	0
$34$	228.000	1.15005
$35$	0	0
$36$	0	0
$37$	334.000i	1.48403i	0.670381	+	0.742017i	$0.266131\pi$
$37$	334.000i	1.48403i	−0.670381	+	0.742017i	$0.733869\pi$
$38$	− 280.000i	− 1.19532i
$39$	0	0
$40$	0	0
$41$	198.000	0.754205	0.377102	−	0.926172i	$-0.376920\pi$
$41$	198.000	0.754205	0.377102	+	0.926172i	$0.376920\pi$
$42$	0	0
$43$	− 268.000i	− 0.950456i	−0.879863	−	0.475228i	$-0.842366\pi$
$43$	− 268.000i	− 0.950456i	0.879863	−	0.475228i	$-0.157634\pi$
$44$	−192.000	−0.657843
$45$	0	0
$46$	144.000	0.461557
$47$	216.000i	0.670358i	0.942154	+	0.335179i	$0.108797\pi$
$47$	216.000i	0.670358i	−0.942154	+	0.335179i	$0.891203\pi$
$48$	0	0
$49$	327.000	0.953353
$50$	0	0
$51$	0	0
$52$	− 8.00000i	− 0.0213346i
$53$	78.0000i	0.202153i	0.994879	+	0.101077i	$0.0322287\pi$
$53$	78.0000i	0.202153i	−0.994879	+	0.101077i	$0.967771\pi$
$54$	0	0
$55$	0	0
$56$	32.0000	0.0763604
$57$	0	0
$58$	420.000i	0.950840i
$59$	240.000	0.529582	0.264791	−	0.964306i	$-0.414697\pi$
$59$	240.000	0.529582	0.264791	+	0.964306i	$0.414697\pi$
$60$	0	0
$61$	302.000	0.633888	0.316944	−	0.948444i	$-0.397343\pi$
$61$	302.000	0.633888	0.316944	+	0.948444i	$0.397343\pi$
$62$	544.000i	1.11432i
$63$	0	0
$64$	−64.0000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 596.000i	− 1.08676i	−0.839487	−	0.543381i	$-0.817144\pi$
$67$	− 596.000i	− 1.08676i	0.839487	−	0.543381i	$-0.182856\pi$
$68$	456.000i	0.813208i
$69$	0	0
$70$	0	0
$71$	768.000	1.28373	0.641865	−	0.766818i	$-0.278161\pi$
$71$	768.000	1.28373	0.641865	+	0.766818i	$0.278161\pi$
$72$	0	0
$73$	− 478.000i	− 0.766379i	−0.923670	−	0.383190i	$-0.874826\pi$
$73$	− 478.000i	− 0.766379i	0.923670	−	0.383190i	$-0.125174\pi$
$74$	−668.000	−1.04937
$75$	0	0
$76$	560.000	0.845216
$77$	192.000i	0.284161i
$78$	0	0
$79$	640.000	0.911464	0.455732	−	0.890117i	$-0.349378\pi$
$79$	640.000	0.911464	0.455732	+	0.890117i	$0.349378\pi$
$80$	0	0
$81$	0	0
$82$	396.000i	0.533303i
$83$	348.000i	0.460216i	0.973165	+	0.230108i	$0.0739080\pi$
$83$	348.000i	0.460216i	−0.973165	+	0.230108i	$0.926092\pi$
$84$	0	0
$85$	0	0
$86$	536.000	0.672074
$87$	0	0
$88$	− 384.000i	− 0.465165i
$89$	210.000	0.250112	0.125056	−	0.992150i	$-0.460089\pi$
$89$	210.000	0.250112	0.125056	+	0.992150i	$0.460089\pi$
$90$	0	0
$91$	−8.00000	−0.00921569
$92$	288.000i	0.326370i
$93$	0	0
$94$	−432.000	−0.474015
$95$	0	0
$96$	0	0
$97$	1534.00i	1.60571i	0.596173	+	0.802856i	$0.296687\pi$
$97$	1534.00i	1.60571i	−0.596173	+	0.802856i	$0.703313\pi$
$98$	654.000i	0.674122i
$99$	0	0
$100$	0	0
$101$	−1722.00	−1.69649	−0.848245	−	0.529605i	$-0.822340\pi$
$101$	−1722.00	−1.69649	−0.848245	+	0.529605i	$0.822340\pi$
$102$	0	0
$103$	1052.00i	1.00638i	0.864177	+	0.503188i	$0.167840\pi$
$103$	1052.00i	1.00638i	−0.864177	+	0.503188i	$0.832160\pi$
$104$	16.0000	0.0150859
$105$	0	0
$106$	−156.000	−0.142944
$107$	− 564.000i	− 0.509570i	−0.966998	−	0.254785i	$-0.917995\pi$
$107$	− 564.000i	− 0.509570i	0.966998	−	0.254785i	$-0.0820046\pi$
$108$	0	0
$109$	610.000	0.536031	0.268016	−	0.963415i	$-0.413632\pi$
$109$	610.000	0.536031	0.268016	+	0.963415i	$0.413632\pi$
$110$	0	0
$111$	0	0
$112$	64.0000i	0.0539949i
$113$	− 1302.00i	− 1.08391i	−0.840407	−	0.541955i	$-0.817684\pi$
$113$	− 1302.00i	− 1.08391i	0.840407	−	0.541955i	$-0.182316\pi$
$114$	0	0
$115$	0	0
$116$	−840.000	−0.672345
$117$	0	0
$118$	480.000i	0.374471i
$119$	456.000	0.351273
$120$	0	0
$121$	973.000	0.731029
$122$	604.000i	0.448226i
$123$	0	0
$124$	−1088.00	−0.787946
$125$	0	0
$126$	0	0
$127$	124.000i	0.0866395i	0.999061	+	0.0433198i	$0.0137934\pi$
$127$	124.000i	0.0866395i	−0.999061	+	0.0433198i	$0.986207\pi$
$128$	− 128.000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−192.000	−0.128054	−0.0640272	−	0.997948i	$-0.520394\pi$
$131$	−192.000	−0.128054	−0.0640272	+	0.997948i	$0.520394\pi$
$132$	0	0
$133$	− 560.000i	− 0.365099i
$134$	1192.00	0.768456
$135$	0	0
$136$	−912.000	−0.575025
$137$	− 2514.00i	− 1.56778i	−0.620901	−	0.783889i	$-0.713233\pi$
$137$	− 2514.00i	− 1.56778i	0.620901	−	0.783889i	$-0.286767\pi$
$138$	0	0
$139$	−1340.00	−0.817679	−0.408839	−	0.912606i	$-0.634066\pi$
$139$	−1340.00	−0.817679	−0.408839	+	0.912606i	$0.634066\pi$
$140$	0	0
$141$	0	0
$142$	1536.00i	0.907734i
$143$	96.0000i	0.0561393i
$144$	0	0
$145$	0	0
$146$	956.000	0.541912
$147$	0	0
$148$	− 1336.00i	− 0.742017i
$149$	1410.00	0.775246	0.387623	−	0.921818i	$-0.373296\pi$
$149$	1410.00	0.775246	0.387623	+	0.921818i	$0.373296\pi$
$150$	0	0
$151$	−2128.00	−1.14685	−0.573424	−	0.819258i	$-0.694385\pi$
$151$	−2128.00	−1.14685	−0.573424	+	0.819258i	$0.694385\pi$
$152$	1120.00i	0.597658i
$153$	0	0
$154$	−384.000	−0.200932
$155$	0	0
$156$	0	0
$157$	− 3026.00i	− 1.53822i	−0.639114	−	0.769112i	$-0.720699\pi$
$157$	− 3026.00i	− 1.53822i	0.639114	−	0.769112i	$-0.279301\pi$
$158$	1280.00i	0.644502i
$159$	0	0
$160$	0	0
$161$	288.000	0.140979
$162$	0	0
$163$	2612.00i	1.25514i	0.778561	+	0.627569i	$0.215950\pi$
$163$	2612.00i	1.25514i	−0.778561	+	0.627569i	$0.784050\pi$
$164$	−792.000	−0.377102
$165$	0	0
$166$	−696.000	−0.325422
$167$	− 24.0000i	− 0.0111208i	−0.999985	−	0.00556041i	$-0.998230\pi$
$167$	− 24.0000i	− 0.0111208i	0.999985	−	0.00556041i	$-0.00176994\pi$
$168$	0	0
$169$	2193.00	0.998179
$170$	0	0
$171$	0	0
$172$	1072.00i	0.475228i
$173$	− 1962.00i	− 0.862243i	−0.902294	−	0.431122i	$-0.858118\pi$
$173$	− 1962.00i	− 0.862243i	0.902294	−	0.431122i	$-0.141882\pi$
$174$	0	0
$175$	0	0
$176$	768.000	0.328921
$177$	0	0
$178$	420.000i	0.176856i
$179$	−120.000	−0.0501074	−0.0250537	−	0.999686i	$-0.507976\pi$
$179$	−120.000	−0.0501074	−0.0250537	+	0.999686i	$0.507976\pi$
$180$	0	0
$181$	902.000	0.370415	0.185208	−	0.982699i	$-0.440704\pi$
$181$	902.000	0.370415	0.185208	+	0.982699i	$0.440704\pi$
$182$	− 16.0000i	− 0.00651648i
$183$	0	0
$184$	−576.000	−0.230779
$185$	0	0
$186$	0	0
$187$	− 5472.00i	− 2.13985i
$188$	− 864.000i	− 0.335179i
$189$	0	0
$190$	0	0
$191$	168.000	0.0636443	0.0318221	−	0.999494i	$-0.489869\pi$
$191$	168.000	0.0636443	0.0318221	+	0.999494i	$0.489869\pi$
$192$	0	0
$193$	− 1318.00i	− 0.491563i	−0.969325	−	0.245782i	$-0.920955\pi$
$193$	− 1318.00i	− 0.491563i	0.969325	−	0.245782i	$-0.0790446\pi$
$194$	−3068.00	−1.13541
$195$	0	0
$196$	−1308.00	−0.476676
$197$	− 4014.00i	− 1.45170i	−0.687851	−	0.725852i	$-0.741446\pi$
$197$	− 4014.00i	− 1.45170i	0.687851	−	0.725852i	$-0.258554\pi$
$198$	0	0
$199$	−2000.00	−0.712443	−0.356222	−	0.934401i	$-0.615935\pi$
$199$	−2000.00	−0.712443	−0.356222	+	0.934401i	$0.615935\pi$
$200$	0	0
$201$	0	0
$202$	− 3444.00i	− 1.19960i
$203$	840.000i	0.290426i
$204$	0	0
$205$	0	0
$206$	−2104.00	−0.711615
$207$	0	0
$208$	32.0000i	0.0106673i
$209$	−6720.00	−2.22408
$210$	0	0
$211$	−3868.00	−1.26201	−0.631005	−	0.775779i	$-0.717357\pi$
$211$	−3868.00	−1.26201	−0.631005	+	0.775779i	$0.717357\pi$
$212$	− 312.000i	− 0.101077i
$213$	0	0
$214$	1128.00	0.360320
$215$	0	0
$216$	0	0
$217$	1088.00i	0.340361i
$218$	1220.00i	0.379031i
$219$	0	0
$220$	0	0
$221$	228.000	0.0693979
$222$	0	0
$223$	− 3148.00i	− 0.945317i	−0.881246	−	0.472658i	$-0.843294\pi$
$223$	− 3148.00i	− 0.945317i	0.881246	−	0.472658i	$-0.156706\pi$
$224$	−128.000	−0.0381802
$225$	0	0
$226$	2604.00	0.766440
$227$	2556.00i	0.747347i	0.927560	+	0.373673i	$0.121902\pi$
$227$	2556.00i	0.747347i	−0.927560	+	0.373673i	$0.878098\pi$
$228$	0	0
$229$	610.000	0.176026	0.0880130	−	0.996119i	$-0.471948\pi$
$229$	610.000	0.176026	0.0880130	+	0.996119i	$0.471948\pi$
$230$	0	0
$231$	0	0
$232$	− 1680.00i	− 0.475420i
$233$	2058.00i	0.578644i	0.957232	+	0.289322i	$0.0934298\pi$
$233$	2058.00i	0.578644i	−0.957232	+	0.289322i	$0.906570\pi$
$234$	0	0
$235$	0	0
$236$	−960.000	−0.264791
$237$	0	0
$238$	912.000i	0.248387i
$239$	4920.00	1.33158	0.665792	−	0.746138i	$-0.268094\pi$
$239$	4920.00	1.33158	0.665792	+	0.746138i	$0.268094\pi$
$240$	0	0
$241$	−1438.00	−0.384356	−0.192178	−	0.981360i	$-0.561555\pi$
$241$	−1438.00	−0.384356	−0.192178	+	0.981360i	$0.561555\pi$
$242$	1946.00i	0.516916i
$243$	0	0
$244$	−1208.00	−0.316944
$245$	0	0
$246$	0	0
$247$	− 280.000i	− 0.0721294i
$248$	− 2176.00i	− 0.557162i
$249$	0	0
$250$	0	0
$251$	−792.000	−0.199166	−0.0995829	−	0.995029i	$-0.531751\pi$
$251$	−792.000	−0.199166	−0.0995829	+	0.995029i	$0.531751\pi$
$252$	0	0
$253$	− 3456.00i	− 0.858802i
$254$	−248.000	−0.0612634
$255$	0	0
$256$	256.000	0.0625000
$257$	2166.00i	0.525725i	0.964833	+	0.262863i	$0.0846666\pi$
$257$	2166.00i	0.525725i	−0.964833	+	0.262863i	$0.915333\pi$
$258$	0	0
$259$	−1336.00	−0.320521
$260$	0	0
$261$	0	0
$262$	− 384.000i	− 0.0905481i
$263$	− 3192.00i	− 0.748392i	−0.927350	−	0.374196i	$-0.877919\pi$
$263$	− 3192.00i	− 0.748392i	0.927350	−	0.374196i	$-0.122081\pi$
$264$	0	0
$265$	0	0
$266$	1120.00	0.258164
$267$	0	0
$268$	2384.00i	0.543381i
$269$	5490.00	1.24435	0.622177	−	0.782877i	$-0.286248\pi$
$269$	5490.00	1.24435	0.622177	+	0.782877i	$0.286248\pi$
$270$	0	0
$271$	−6328.00	−1.41845	−0.709223	−	0.704985i	$-0.750954\pi$
$271$	−6328.00	−1.41845	−0.709223	+	0.704985i	$0.750954\pi$
$272$	− 1824.00i	− 0.406604i
$273$	0	0
$274$	5028.00	1.10859
$275$	0	0
$276$	0	0
$277$	574.000i	0.124507i	0.998060	+	0.0622533i	$0.0198287\pi$
$277$	574.000i	0.124507i	−0.998060	+	0.0622533i	$0.980171\pi$
$278$	− 2680.00i	− 0.578186i
$279$	0	0
$280$	0	0
$281$	−4242.00	−0.900557	−0.450278	−	0.892888i	$-0.648675\pi$
$281$	−4242.00	−0.900557	−0.450278	+	0.892888i	$0.648675\pi$
$282$	0	0
$283$	− 628.000i	− 0.131911i	−0.997823	−	0.0659553i	$-0.978991\pi$
$283$	− 628.000i	− 0.131911i	0.997823	−	0.0659553i	$-0.0210095\pi$
$284$	−3072.00	−0.641865
$285$	0	0
$286$	−192.000	−0.0396965
$287$	792.000i	0.162893i
$288$	0	0
$289$	−8083.00	−1.64523
$290$	0	0
$291$	0	0
$292$	1912.00i	0.383190i
$293$	558.000i	0.111258i	0.998451	+	0.0556292i	$0.0177165\pi$
$293$	558.000i	0.111258i	−0.998451	+	0.0556292i	$0.982284\pi$
$294$	0	0
$295$	0	0
$296$	2672.00	0.524685
$297$	0	0
$298$	2820.00i	0.548182i
$299$	144.000	0.0278520
$300$	0	0
$301$	1072.00	0.205279
$302$	− 4256.00i	− 0.810945i
$303$	0	0
$304$	−2240.00	−0.422608
$305$	0	0
$306$	0	0
$307$	6964.00i	1.29465i	0.762216	+	0.647323i	$0.224112\pi$
$307$	6964.00i	1.29465i	−0.762216	+	0.647323i	$0.775888\pi$
$308$	− 768.000i	− 0.142081i
$309$	0	0
$310$	0	0
$311$	−2832.00	−0.516360	−0.258180	−	0.966097i	$-0.583123\pi$
$311$	−2832.00	−0.516360	−0.258180	+	0.966097i	$0.583123\pi$
$312$	0	0
$313$	8642.00i	1.56062i	0.625392	+	0.780311i	$0.284939\pi$
$313$	8642.00i	1.56062i	−0.625392	+	0.780311i	$0.715061\pi$
$314$	6052.00	1.08769
$315$	0	0
$316$	−2560.00	−0.455732
$317$	− 2214.00i	− 0.392273i	−0.980577	−	0.196137i	$-0.937160\pi$
$317$	− 2214.00i	− 0.392273i	0.980577	−	0.196137i	$-0.0628396\pi$
$318$	0	0
$319$	10080.0	1.76919
$320$	0	0
$321$	0	0
$322$	576.000i	0.0996870i
$323$	15960.0i	2.74934i
$324$	0	0
$325$	0	0
$326$	−5224.00	−0.887517
$327$	0	0
$328$	− 1584.00i	− 0.266652i
$329$	−864.000	−0.144784
$330$	0	0
$331$	10772.0	1.78877	0.894385	−	0.447299i	$-0.147614\pi$
$331$	10772.0	1.78877	0.894385	+	0.447299i	$0.147614\pi$
$332$	− 1392.00i	− 0.230108i
$333$	0	0
$334$	48.0000	0.00786360
$335$	0	0
$336$	0	0
$337$	1654.00i	0.267356i	0.991025	+	0.133678i	$0.0426789\pi$
$337$	1654.00i	0.267356i	−0.991025	+	0.133678i	$0.957321\pi$
$338$	4386.00i	0.705819i
$339$	0	0
$340$	0	0
$341$	13056.0	2.07338
$342$	0	0
$343$	2680.00i	0.421885i
$344$	−2144.00	−0.336037
$345$	0	0
$346$	3924.00	0.609698
$347$	2196.00i	0.339733i	0.985467	+	0.169867i	$0.0543337\pi$
$347$	2196.00i	0.339733i	−0.985467	+	0.169867i	$0.945666\pi$
$348$	0	0
$349$	−8270.00	−1.26843	−0.634216	−	0.773156i	$-0.718677\pi$
$349$	−8270.00	−1.26843	−0.634216	+	0.773156i	$0.718677\pi$
$350$	0	0
$351$	0	0
$352$	1536.00i	0.232583i
$353$	− 10302.0i	− 1.55331i	−0.629923	−	0.776657i	$-0.716914\pi$
$353$	− 10302.0i	− 1.55331i	0.629923	−	0.776657i	$-0.283086\pi$
$354$	0	0
$355$	0	0
$356$	−840.000	−0.125056
$357$	0	0
$358$	− 240.000i	− 0.0354313i
$359$	−2280.00	−0.335192	−0.167596	−	0.985856i	$-0.553600\pi$
$359$	−2280.00	−0.335192	−0.167596	+	0.985856i	$0.553600\pi$
$360$	0	0
$361$	12741.0	1.85756
$362$	1804.00i	0.261923i
$363$	0	0
$364$	32.0000	0.00460785
$365$	0	0
$366$	0	0
$367$	8764.00i	1.24653i	0.782010	+	0.623266i	$0.214195\pi$
$367$	8764.00i	1.24653i	−0.782010	+	0.623266i	$0.785805\pi$
$368$	− 1152.00i	− 0.163185i
$369$	0	0
$370$	0	0
$371$	−312.000	−0.0436610
$372$	0	0
$373$	− 1318.00i	− 0.182958i	−0.995807	−	0.0914792i	$-0.970841\pi$
$373$	− 1318.00i	− 0.182958i	0.995807	−	0.0914792i	$-0.0291595\pi$
$374$	10944.0	1.51310
$375$	0	0
$376$	1728.00	0.237007
$377$	420.000i	0.0573769i
$378$	0	0
$379$	−1100.00	−0.149085	−0.0745425	−	0.997218i	$-0.523750\pi$
$379$	−1100.00	−0.149085	−0.0745425	+	0.997218i	$0.523750\pi$
$380$	0	0
$381$	0	0
$382$	336.000i	0.0450033i
$383$	3528.00i	0.470685i	0.971912	+	0.235343i	$0.0756212\pi$
$383$	3528.00i	0.470685i	−0.971912	+	0.235343i	$0.924379\pi$
$384$	0	0
$385$	0	0
$386$	2636.00	0.347588
$387$	0	0
$388$	− 6136.00i	− 0.802856i
$389$	−9630.00	−1.25517	−0.627584	−	0.778549i	$-0.715956\pi$
$389$	−9630.00	−1.25517	−0.627584	+	0.778549i	$0.715956\pi$
$390$	0	0
$391$	−8208.00	−1.06163
$392$	− 2616.00i	− 0.337061i
$393$	0	0
$394$	8028.00	1.02651
$395$	0	0
$396$	0	0
$397$	3094.00i	0.391142i	0.980690	+	0.195571i	$0.0626560\pi$
$397$	3094.00i	0.391142i	−0.980690	+	0.195571i	$0.937344\pi$
$398$	− 4000.00i	− 0.503774i
$399$	0	0
$400$	0	0
$401$	1638.00	0.203985	0.101992	−	0.994785i	$-0.467478\pi$
$401$	1638.00	0.203985	0.101992	+	0.994785i	$0.467478\pi$
$402$	0	0
$403$	544.000i	0.0672421i
$404$	6888.00	0.848245
$405$	0	0
$406$	−1680.00	−0.205362
$407$	16032.0i	1.95252i
$408$	0	0
$409$	13750.0	1.66233	0.831166	−	0.556024i	$-0.187674\pi$
$409$	13750.0	1.66233	0.831166	+	0.556024i	$0.187674\pi$
$410$	0	0
$411$	0	0
$412$	− 4208.00i	− 0.503188i
$413$	960.000i	0.114379i
$414$	0	0
$415$	0	0
$416$	−64.0000	−0.00754293
$417$	0	0
$418$	− 13440.0i	− 1.57266i
$419$	−12480.0	−1.45510	−0.727551	−	0.686053i	$-0.759342\pi$
$419$	−12480.0	−1.45510	−0.727551	+	0.686053i	$0.759342\pi$
$420$	0	0
$421$	7262.00	0.840685	0.420342	−	0.907366i	$-0.361910\pi$
$421$	7262.00	0.840685	0.420342	+	0.907366i	$0.361910\pi$
$422$	− 7736.00i	− 0.892376i
$423$	0	0
$424$	624.000	0.0714720
$425$	0	0
$426$	0	0
$427$	1208.00i	0.136907i
$428$	2256.00i	0.254785i
$429$	0	0
$430$	0	0
$431$	−9792.00	−1.09435	−0.547174	−	0.837019i	$-0.684296\pi$
$431$	−9792.00	−1.09435	−0.547174	+	0.837019i	$0.684296\pi$
$432$	0	0
$433$	1802.00i	0.199997i	0.994988	+	0.0999984i	$0.0318838\pi$
$433$	1802.00i	0.199997i	−0.994988	+	0.0999984i	$0.968116\pi$
$434$	−2176.00	−0.240671
$435$	0	0
$436$	−2440.00	−0.268016
$437$	10080.0i	1.10341i
$438$	0	0
$439$	2320.00	0.252227	0.126113	−	0.992016i	$-0.459750\pi$
$439$	2320.00	0.252227	0.126113	+	0.992016i	$0.459750\pi$
$440$	0	0
$441$	0	0
$442$	456.000i	0.0490717i
$443$	− 11172.0i	− 1.19819i	−0.800678	−	0.599095i	$-0.795527\pi$
$443$	− 11172.0i	− 1.19819i	0.800678	−	0.599095i	$-0.204473\pi$
$444$	0	0
$445$	0	0
$446$	6296.00	0.668440
$447$	0	0
$448$	− 256.000i	− 0.0269975i
$449$	6810.00	0.715777	0.357888	−	0.933764i	$-0.383497\pi$
$449$	6810.00	0.715777	0.357888	+	0.933764i	$0.383497\pi$
$450$	0	0
$451$	9504.00	0.992297
$452$	5208.00i	0.541955i
$453$	0	0
$454$	−5112.00	−0.528454
$455$	0	0
$456$	0	0
$457$	− 17066.0i	− 1.74686i	−0.486952	−	0.873429i	$-0.661891\pi$
$457$	− 17066.0i	− 1.74686i	0.486952	−	0.873429i	$-0.338109\pi$
$458$	1220.00i	0.124469i
$459$	0	0
$460$	0	0
$461$	18918.0	1.91128	0.955639	−	0.294541i	$-0.0951667\pi$
$461$	18918.0	1.91128	0.955639	+	0.294541i	$0.0951667\pi$
$462$	0	0
$463$	1052.00i	0.105595i	0.998605	+	0.0527976i	$0.0168138\pi$
$463$	1052.00i	0.105595i	−0.998605	+	0.0527976i	$0.983186\pi$
$464$	3360.00	0.336173
$465$	0	0
$466$	−4116.00	−0.409163
$467$	11076.0i	1.09751i	0.835984	+	0.548754i	$0.184898\pi$
$467$	11076.0i	1.09751i	−0.835984	+	0.548754i	$0.815102\pi$
$468$	0	0
$469$	2384.00	0.234718
$470$	0	0
$471$	0	0
$472$	− 1920.00i	− 0.187236i
$473$	− 12864.0i	− 1.25050i
$474$	0	0
$475$	0	0
$476$	−1824.00	−0.175636
$477$	0	0
$478$	9840.00i	0.941571i
$479$	−9000.00	−0.858498	−0.429249	−	0.903186i	$-0.641222\pi$
$479$	−9000.00	−0.858498	−0.429249	+	0.903186i	$0.641222\pi$
$480$	0	0
$481$	−668.000	−0.0633226
$482$	− 2876.00i	− 0.271781i
$483$	0	0
$484$	−3892.00	−0.365515
$485$	0	0
$486$	0	0
$487$	8764.00i	0.815472i	0.913100	+	0.407736i	$0.133682\pi$
$487$	8764.00i	0.815472i	−0.913100	+	0.407736i	$0.866318\pi$
$488$	− 2416.00i	− 0.224113i
$489$	0	0
$490$	0	0
$491$	−5592.00	−0.513978	−0.256989	−	0.966414i	$-0.582730\pi$
$491$	−5592.00	−0.513978	−0.256989	+	0.966414i	$0.582730\pi$
$492$	0	0
$493$	− 23940.0i	− 2.18703i
$494$	560.000	0.0510032
$495$	0	0
$496$	4352.00	0.393973
$497$	3072.00i	0.277260i
$498$	0	0
$499$	−4700.00	−0.421645	−0.210823	−	0.977524i	$-0.567614\pi$
$499$	−4700.00	−0.421645	−0.210823	+	0.977524i	$0.567614\pi$
$500$	0	0
$501$	0	0
$502$	− 1584.00i	− 0.140831i
$503$	11808.0i	1.04671i	0.852116	+	0.523353i	$0.175319\pi$
$503$	11808.0i	1.04671i	−0.852116	+	0.523353i	$0.824681\pi$
$504$	0	0
$505$	0	0
$506$	6912.00	0.607265
$507$	0	0
$508$	− 496.000i	− 0.0433198i
$509$	1170.00	0.101885	0.0509424	−	0.998702i	$-0.483778\pi$
$509$	1170.00	0.101885	0.0509424	+	0.998702i	$0.483778\pi$
$510$	0	0
$511$	1912.00	0.165522
$512$	512.000i	0.0441942i
$513$	0	0
$514$	−4332.00	−0.371744
$515$	0	0
$516$	0	0
$517$	10368.0i	0.881981i
$518$	− 2672.00i	− 0.226643i
$519$	0	0
$520$	0	0
$521$	16638.0	1.39909	0.699543	−	0.714590i	$-0.253387\pi$
$521$	16638.0	1.39909	0.699543	+	0.714590i	$0.253387\pi$
$522$	0	0
$523$	15692.0i	1.31198i	0.754771	+	0.655988i	$0.227748\pi$
$523$	15692.0i	1.31198i	−0.754771	+	0.655988i	$0.772252\pi$
$524$	768.000	0.0640272
$525$	0	0
$526$	6384.00	0.529193
$527$	− 31008.0i	− 2.56305i
$528$	0	0
$529$	6983.00	0.573929
$530$	0	0
$531$	0	0
$532$	2240.00i	0.182549i
$533$	396.000i	0.0321814i
$534$	0	0
$535$	0	0
$536$	−4768.00	−0.384228
$537$	0	0
$538$	10980.0i	0.879891i
$539$	15696.0	1.25431
$540$	0	0
$541$	−22018.0	−1.74977	−0.874887	−	0.484327i	$-0.839064\pi$
$541$	−22018.0	−1.74977	−0.874887	+	0.484327i	$0.839064\pi$
$542$	− 12656.0i	− 1.00299i
$543$	0	0
$544$	3648.00	0.287512
$545$	0	0
$546$	0	0
$547$	4564.00i	0.356751i	0.983963	+	0.178375i	$0.0570841\pi$
$547$	4564.00i	0.356751i	−0.983963	+	0.178375i	$0.942916\pi$
$548$	10056.0i	0.783889i
$549$	0	0
$550$	0	0
$551$	−29400.0	−2.27311
$552$	0	0
$553$	2560.00i	0.196858i
$554$	−1148.00	−0.0880394
$555$	0	0
$556$	5360.00	0.408839
$557$	− 7734.00i	− 0.588331i	−0.955755	−	0.294165i	$-0.904958\pi$
$557$	− 7734.00i	− 0.588331i	0.955755	−	0.294165i	$-0.0950416\pi$
$558$	0	0
$559$	536.000	0.0405552
$560$	0	0
$561$	0	0
$562$	− 8484.00i	− 0.636790i
$563$	20148.0i	1.50824i	0.656739	+	0.754118i	$0.271935\pi$
$563$	20148.0i	1.50824i	−0.656739	+	0.754118i	$0.728065\pi$
$564$	0	0
$565$	0	0
$566$	1256.00	0.0932749
$567$	0	0
$568$	− 6144.00i	− 0.453867i
$569$	−24030.0	−1.77046	−0.885228	−	0.465156i	$-0.845998\pi$
$569$	−24030.0	−1.77046	−0.885228	+	0.465156i	$0.845998\pi$
$570$	0	0
$571$	2372.00	0.173844	0.0869222	−	0.996215i	$-0.472297\pi$
$571$	2372.00	0.173844	0.0869222	+	0.996215i	$0.472297\pi$
$572$	− 384.000i	− 0.0280697i
$573$	0	0
$574$	−1584.00	−0.115183
$575$	0	0
$576$	0	0
$577$	− 8546.00i	− 0.616594i	−0.951290	−	0.308297i	$-0.900241\pi$
$577$	− 8546.00i	− 0.616594i	0.951290	−	0.308297i	$-0.0997590\pi$
$578$	− 16166.0i	− 1.16335i
$579$	0	0
$580$	0	0
$581$	−1392.00	−0.0993974
$582$	0	0
$583$	3744.00i	0.265970i
$584$	−3824.00	−0.270956
$585$	0	0
$586$	−1116.00	−0.0786716
$587$	− 15444.0i	− 1.08593i	−0.839755	−	0.542966i	$-0.817301\pi$
$587$	− 15444.0i	− 1.08593i	0.839755	−	0.542966i	$-0.182699\pi$
$588$	0	0
$589$	−38080.0	−2.66394
$590$	0	0
$591$	0	0
$592$	5344.00i	0.371009i
$593$	− 18342.0i	− 1.27018i	−0.772439	−	0.635089i	$-0.780963\pi$
$593$	− 18342.0i	− 1.27018i	0.772439	−	0.635089i	$-0.219037\pi$
$594$	0	0
$595$	0	0
$596$	−5640.00	−0.387623
$597$	0	0
$598$	288.000i	0.0196943i
$599$	24600.0	1.67801	0.839006	−	0.544123i	$-0.183137\pi$
$599$	24600.0	1.67801	0.839006	+	0.544123i	$0.183137\pi$
$600$	0	0
$601$	−8998.00	−0.610709	−0.305354	−	0.952239i	$-0.598775\pi$
$601$	−8998.00	−0.610709	−0.305354	+	0.952239i	$0.598775\pi$
$602$	2144.00i	0.145154i
$603$	0	0
$604$	8512.00	0.573424
$605$	0	0
$606$	0	0
$607$	− 4076.00i	− 0.272553i	−0.990671	−	0.136277i	$-0.956486\pi$
$607$	− 4076.00i	− 0.272553i	0.990671	−	0.136277i	$-0.0435136\pi$
$608$	− 4480.00i	− 0.298829i
$609$	0	0
$610$	0	0
$611$	−432.000	−0.0286037
$612$	0	0
$613$	− 4078.00i	− 0.268693i	−0.990934	−	0.134347i	$-0.957106\pi$
$613$	− 4078.00i	− 0.268693i	0.990934	−	0.134347i	$-0.0428935\pi$
$614$	−13928.0	−0.915453
$615$	0	0
$616$	1536.00	0.100466
$617$	10086.0i	0.658099i	0.944313	+	0.329049i	$0.106728\pi$
$617$	10086.0i	0.658099i	−0.944313	+	0.329049i	$0.893272\pi$
$618$	0	0
$619$	−8780.00	−0.570110	−0.285055	−	0.958511i	$-0.592012\pi$
$619$	−8780.00	−0.570110	−0.285055	+	0.958511i	$0.592012\pi$
$620$	0	0
$621$	0	0
$622$	− 5664.00i	− 0.365122i
$623$	840.000i	0.0540191i
$624$	0	0
$625$	0	0
$626$	−17284.0	−1.10353
$627$	0	0
$628$	12104.0i	0.769112i
$629$	38076.0	2.41366
$630$	0	0
$631$	2792.00	0.176145	0.0880727	−	0.996114i	$-0.471929\pi$
$631$	2792.00	0.176145	0.0880727	+	0.996114i	$0.471929\pi$
$632$	− 5120.00i	− 0.322251i
$633$	0	0
$634$	4428.00	0.277379
$635$	0	0
$636$	0	0
$637$	654.000i	0.0406788i
$638$	20160.0i	1.25101i
$639$	0	0
$640$	0	0
$641$	−7602.00	−0.468426	−0.234213	−	0.972185i	$-0.575251\pi$
$641$	−7602.00	−0.468426	−0.234213	+	0.972185i	$0.575251\pi$
$642$	0	0
$643$	24212.0i	1.48496i	0.669869	+	0.742479i	$0.266350\pi$
$643$	24212.0i	1.48496i	−0.669869	+	0.742479i	$0.733650\pi$
$644$	−1152.00	−0.0704894
$645$	0	0
$646$	−31920.0	−1.94408
$647$	9456.00i	0.574581i	0.957844	+	0.287290i	$0.0927545\pi$
$647$	9456.00i	0.574581i	−0.957844	+	0.287290i	$0.907246\pi$
$648$	0	0
$649$	11520.0	0.696764
$650$	0	0
$651$	0	0
$652$	− 10448.0i	− 0.627569i
$653$	9558.00i	0.572792i	0.958111	+	0.286396i	$0.0924574\pi$
$653$	9558.00i	0.572792i	−0.958111	+	0.286396i	$0.907543\pi$
$654$	0	0
$655$	0	0
$656$	3168.00	0.188551
$657$	0	0
$658$	− 1728.00i	− 0.102378i
$659$	−29280.0	−1.73078	−0.865392	−	0.501095i	$-0.832931\pi$
$659$	−29280.0	−1.73078	−0.865392	+	0.501095i	$0.832931\pi$
$660$	0	0
$661$	−29098.0	−1.71223	−0.856113	−	0.516789i	$-0.827127\pi$
$661$	−29098.0	−1.71223	−0.856113	+	0.516789i	$0.827127\pi$
$662$	21544.0i	1.26485i
$663$	0	0
$664$	2784.00	0.162711
$665$	0	0
$666$	0	0
$667$	− 15120.0i	− 0.877734i
$668$	96.0000i	0.00556041i
$669$	0	0
$670$	0	0
$671$	14496.0	0.833997
$672$	0	0
$673$	− 11638.0i	− 0.666585i	−0.942823	−	0.333293i	$-0.891840\pi$
$673$	− 11638.0i	− 0.666585i	0.942823	−	0.333293i	$-0.108160\pi$
$674$	−3308.00	−0.189050
$675$	0	0
$676$	−8772.00	−0.499090
$677$	3426.00i	0.194493i	0.995260	+	0.0972466i	$0.0310035\pi$
$677$	3426.00i	0.194493i	−0.995260	+	0.0972466i	$0.968996\pi$
$678$	0	0
$679$	−6136.00	−0.346801
$680$	0	0
$681$	0	0
$682$	26112.0i	1.46610i
$683$	20148.0i	1.12876i	0.825516	+	0.564379i	$0.190884\pi$
$683$	20148.0i	1.12876i	−0.825516	+	0.564379i	$0.809116\pi$
$684$	0	0
$685$	0	0
$686$	−5360.00	−0.298317
$687$	0	0
$688$	− 4288.00i	− 0.237614i
$689$	−156.000	−0.00862573
$690$	0	0
$691$	−29428.0	−1.62011	−0.810053	−	0.586356i	$-0.800562\pi$
$691$	−29428.0	−1.62011	−0.810053	+	0.586356i	$0.800562\pi$
$692$	7848.00i	0.431122i
$693$	0	0
$694$	−4392.00	−0.240228
$695$	0	0
$696$	0	0
$697$	− 22572.0i	− 1.22665i
$698$	− 16540.0i	− 0.896917i
$699$	0	0
$700$	0	0
$701$	−16242.0	−0.875110	−0.437555	−	0.899192i	$-0.644155\pi$
$701$	−16242.0	−0.875110	−0.437555	+	0.899192i	$0.644155\pi$
$702$	0	0
$703$	− 46760.0i	− 2.50866i
$704$	−3072.00	−0.164461
$705$	0	0
$706$	20604.0	1.09836
$707$	− 6888.00i	− 0.366407i
$708$	0	0
$709$	−2030.00	−0.107529	−0.0537646	−	0.998554i	$-0.517122\pi$
$709$	−2030.00	−0.107529	−0.0537646	+	0.998554i	$0.517122\pi$
$710$	0	0
$711$	0	0
$712$	− 1680.00i	− 0.0884279i
$713$	− 19584.0i	− 1.02865i
$714$	0	0
$715$	0	0
$716$	480.000	0.0250537
$717$	0	0
$718$	− 4560.00i	− 0.237016i
$719$	6960.00	0.361007	0.180504	−	0.983574i	$-0.442227\pi$
$719$	6960.00	0.361007	0.180504	+	0.983574i	$0.442227\pi$
$720$	0	0
$721$	−4208.00	−0.217357
$722$	25482.0i	1.31349i
$723$	0	0
$724$	−3608.00	−0.185208
$725$	0	0
$726$	0	0
$727$	− 18596.0i	− 0.948676i	−0.880343	−	0.474338i	$-0.842687\pi$
$727$	− 18596.0i	− 0.948676i	0.880343	−	0.474338i	$-0.157313\pi$
$728$	64.0000i	0.00325824i
$729$	0	0
$730$	0	0
$731$	−30552.0	−1.54584
$732$	0	0
$733$	21242.0i	1.07038i	0.844731	+	0.535192i	$0.179761\pi$
$733$	21242.0i	1.07038i	−0.844731	+	0.535192i	$0.820239\pi$
$734$	−17528.0	−0.881431
$735$	0	0
$736$	2304.00	0.115389
$737$	− 28608.0i	− 1.42984i
$738$	0	0
$739$	340.000	0.0169244	0.00846218	−	0.999964i	$-0.497306\pi$
$739$	340.000	0.0169244	0.00846218	+	0.999964i	$0.497306\pi$
$740$	0	0
$741$	0	0
$742$	− 624.000i	− 0.0308730i
$743$	21888.0i	1.08074i	0.841426	+	0.540372i	$0.181716\pi$
$743$	21888.0i	1.08074i	−0.841426	+	0.540372i	$0.818284\pi$
$744$	0	0
$745$	0	0
$746$	2636.00	0.129371
$747$	0	0
$748$	21888.0i	1.06993i
$749$	2256.00	0.110057
$750$	0	0
$751$	17792.0	0.864500	0.432250	−	0.901754i	$-0.357720\pi$
$751$	17792.0	0.864500	0.432250	+	0.901754i	$0.357720\pi$
$752$	3456.00i	0.167590i
$753$	0	0
$754$	−840.000	−0.0405716
$755$	0	0
$756$	0	0
$757$	− 37346.0i	− 1.79308i	−0.442960	−	0.896541i	$-0.646072\pi$
$757$	− 37346.0i	− 1.79308i	0.442960	−	0.896541i	$-0.353928\pi$
$758$	− 2200.00i	− 0.105419i
$759$	0	0
$760$	0	0
$761$	11358.0	0.541034	0.270517	−	0.962715i	$-0.412805\pi$
$761$	11358.0	0.541034	0.270517	+	0.962715i	$0.412805\pi$
$762$	0	0
$763$	2440.00i	0.115772i
$764$	−672.000	−0.0318221
$765$	0	0
$766$	−7056.00	−0.332825
$767$	480.000i	0.0225969i
$768$	0	0
$769$	34270.0	1.60703	0.803516	−	0.595283i	$-0.202960\pi$
$769$	34270.0	1.60703	0.803516	+	0.595283i	$0.202960\pi$
$770$	0	0
$771$	0	0
$772$	5272.00i	0.245782i
$773$	13278.0i	0.617822i	0.951091	+	0.308911i	$0.0999645\pi$
$773$	13278.0i	0.617822i	−0.951091	+	0.308911i	$0.900035\pi$
$774$	0	0
$775$	0	0
$776$	12272.0	0.567705
$777$	0	0
$778$	− 19260.0i	− 0.887538i
$779$	−27720.0	−1.27493
$780$	0	0
$781$	36864.0	1.68899
$782$	− 16416.0i	− 0.750684i
$783$	0	0
$784$	5232.00	0.238338
$785$	0	0
$786$	0	0
$787$	11164.0i	0.505659i	0.967511	+	0.252829i	$0.0813612\pi$
$787$	11164.0i	0.505659i	−0.967511	+	0.252829i	$0.918639\pi$
$788$	16056.0i	0.725852i
$789$	0	0
$790$	0	0
$791$	5208.00	0.234103
$792$	0	0
$793$	604.000i	0.0270475i
$794$	−6188.00	−0.276579
$795$	0	0
$796$	8000.00	0.356222
$797$	− 5094.00i	− 0.226397i	−0.993572	−	0.113199i	$-0.963890\pi$
$797$	− 5094.00i	− 0.226397i	0.993572	−	0.113199i	$-0.0361097\pi$
$798$	0	0
$799$	24624.0	1.09028
$800$	0	0
$801$	0	0
$802$	3276.00i	0.144239i
$803$	− 22944.0i	− 1.00831i
$804$	0	0
$805$	0	0
$806$	−1088.00	−0.0475474
$807$	0	0
$808$	13776.0i	0.599799i
$809$	−8790.00	−0.382002	−0.191001	−	0.981590i	$-0.561173\pi$
$809$	−8790.00	−0.382002	−0.191001	+	0.981590i	$0.561173\pi$
$810$	0	0
$811$	5852.00	0.253380	0.126690	−	0.991942i	$-0.459565\pi$
$811$	5852.00	0.253380	0.126690	+	0.991942i	$0.459565\pi$
$812$	− 3360.00i	− 0.145213i
$813$	0	0
$814$	−32064.0	−1.38064
$815$	0	0
$816$	0	0
$817$	37520.0i	1.60668i
$818$	27500.0i	1.17545i
$819$	0	0
$820$	0	0
$821$	29478.0	1.25309	0.626546	−	0.779384i	$-0.284468\pi$
$821$	29478.0	1.25309	0.626546	+	0.779384i	$0.284468\pi$
$822$	0	0
$823$	39332.0i	1.66589i	0.553356	+	0.832945i	$0.313347\pi$
$823$	39332.0i	1.66589i	−0.553356	+	0.832945i	$0.686653\pi$
$824$	8416.00	0.355807
$825$	0	0
$826$	−1920.00	−0.0808781
$827$	6756.00i	0.284074i	0.989861	+	0.142037i	$0.0453652\pi$
$827$	6756.00i	0.284074i	−0.989861	+	0.142037i	$0.954635\pi$
$828$	0	0
$829$	−3950.00	−0.165488	−0.0827438	−	0.996571i	$-0.526368\pi$
$829$	−3950.00	−0.165488	−0.0827438	+	0.996571i	$0.526368\pi$
$830$	0	0
$831$	0	0
$832$	− 128.000i	− 0.00533366i
$833$	− 37278.0i	− 1.55055i
$834$	0	0
$835$	0	0
$836$	26880.0	1.11204
$837$	0	0
$838$	− 24960.0i	− 1.02891i
$839$	12360.0	0.508599	0.254300	−	0.967126i	$-0.418155\pi$
$839$	12360.0	0.508599	0.254300	+	0.967126i	$0.418155\pi$
$840$	0	0
$841$	19711.0	0.808192
$842$	14524.0i	0.594454i
$843$	0	0
$844$	15472.0	0.631005
$845$	0	0
$846$	0	0
$847$	3892.00i	0.157887i
$848$	1248.00i	0.0505383i
$849$	0	0
$850$	0	0
$851$	24048.0	0.968690
$852$	0	0
$853$	− 35998.0i	− 1.44496i	−0.691394	−	0.722478i	$-0.743003\pi$
$853$	− 35998.0i	− 1.44496i	0.691394	−	0.722478i	$-0.256997\pi$
$854$	−2416.00	−0.0968077
$855$	0	0
$856$	−4512.00	−0.180160
$857$	− 21594.0i	− 0.860720i	−0.902657	−	0.430360i	$-0.858387\pi$
$857$	− 21594.0i	− 0.860720i	0.902657	−	0.430360i	$-0.141613\pi$
$858$	0	0
$859$	−9260.00	−0.367808	−0.183904	−	0.982944i	$-0.558874\pi$
$859$	−9260.00	−0.367808	−0.183904	+	0.982944i	$0.558874\pi$
$860$	0	0
$861$	0	0
$862$	− 19584.0i	− 0.773821i
$863$	− 31632.0i	− 1.24770i	−0.781544	−	0.623850i	$-0.785567\pi$
$863$	− 31632.0i	− 1.24770i	0.781544	−	0.623850i	$-0.214433\pi$
$864$	0	0
$865$	0	0
$866$	−3604.00	−0.141419
$867$	0	0
$868$	− 4352.00i	− 0.170180i
$869$	30720.0	1.19920
$870$	0	0
$871$	1192.00	0.0463713
$872$	− 4880.00i	− 0.189516i
$873$	0	0
$874$	−20160.0	−0.780231
$875$	0	0
$876$	0	0
$877$	39694.0i	1.52836i	0.645003	+	0.764180i	$0.276856\pi$
$877$	39694.0i	1.52836i	−0.645003	+	0.764180i	$0.723144\pi$
$878$	4640.00i	0.178351i
$879$	0	0
$880$	0	0
$881$	−1242.00	−0.0474961	−0.0237480	−	0.999718i	$-0.507560\pi$
$881$	−1242.00	−0.0474961	−0.0237480	+	0.999718i	$0.507560\pi$
$882$	0	0
$883$	− 2668.00i	− 0.101682i	−0.998707	−	0.0508411i	$-0.983810\pi$
$883$	− 2668.00i	− 0.101682i	0.998707	−	0.0508411i	$-0.0161902\pi$
$884$	−912.000	−0.0346990
$885$	0	0
$886$	22344.0	0.847248
$887$	− 4344.00i	− 0.164439i	−0.996614	−	0.0822194i	$-0.973799\pi$
$887$	− 4344.00i	− 0.164439i	0.996614	−	0.0822194i	$-0.0262008\pi$
$888$	0	0
$889$	−496.000	−0.0187124
$890$	0	0
$891$	0	0
$892$	12592.0i	0.472658i
$893$	− 30240.0i	− 1.13319i
$894$	0	0
$895$	0	0
$896$	512.000	0.0190901
$897$	0	0
$898$	13620.0i	0.506131i
$899$	57120.0	2.11909
$900$	0	0
$901$	8892.00	0.328785
$902$	19008.0i	0.701660i
$903$	0	0
$904$	−10416.0	−0.383220
$905$	0	0
$906$	0	0
$907$	− 4436.00i	− 0.162398i	−0.996698	−	0.0811990i	$-0.974125\pi$
$907$	− 4436.00i	− 0.162398i	0.996698	−	0.0811990i	$-0.0258749\pi$
$908$	− 10224.0i	− 0.373673i
$909$	0	0
$910$	0	0
$911$	−22752.0	−0.827450	−0.413725	−	0.910402i	$-0.635773\pi$
$911$	−22752.0	−0.827450	−0.413725	+	0.910402i	$0.635773\pi$
$912$	0	0
$913$	16704.0i	0.605500i
$914$	34132.0	1.23521
$915$	0	0
$916$	−2440.00	−0.0880130
$917$	− 768.000i	− 0.0276571i
$918$	0	0
$919$	27160.0	0.974892	0.487446	−	0.873153i	$-0.337929\pi$
$919$	27160.0	0.974892	0.487446	+	0.873153i	$0.337929\pi$
$920$	0	0
$921$	0	0
$922$	37836.0i	1.35148i
$923$	1536.00i	0.0547758i
$924$	0	0
$925$	0	0
$926$	−2104.00	−0.0746671
$927$	0	0
$928$	6720.00i	0.237710i
$929$	−33030.0	−1.16650	−0.583250	−	0.812292i	$-0.698219\pi$
$929$	−33030.0	−1.16650	−0.583250	+	0.812292i	$0.698219\pi$
$930$	0	0
$931$	−45780.0	−1.61158
$932$	− 8232.00i	− 0.289322i
$933$	0	0
$934$	−22152.0	−0.776055
$935$	0	0
$936$	0	0
$937$	29974.0i	1.04505i	0.852625	+	0.522523i	$0.175009\pi$
$937$	29974.0i	1.04505i	−0.852625	+	0.522523i	$0.824991\pi$
$938$	4768.00i	0.165971i
$939$	0	0
$940$	0	0
$941$	−13962.0	−0.483686	−0.241843	−	0.970315i	$-0.577752\pi$
$941$	−13962.0	−0.483686	−0.241843	+	0.970315i	$0.577752\pi$
$942$	0	0
$943$	− 14256.0i	− 0.492300i
$944$	3840.00	0.132396
$945$	0	0
$946$	25728.0	0.884238
$947$	35196.0i	1.20773i	0.797088	+	0.603863i	$0.206373\pi$
$947$	35196.0i	1.20773i	−0.797088	+	0.603863i	$0.793627\pi$
$948$	0	0
$949$	956.000	0.0327008
$950$	0	0
$951$	0	0
$952$	− 3648.00i	− 0.124194i
$953$	28338.0i	0.963230i	0.876383	+	0.481615i	$0.159950\pi$
$953$	28338.0i	0.963230i	−0.876383	+	0.481615i	$0.840050\pi$
$954$	0	0
$955$	0	0
$956$	−19680.0	−0.665792
$957$	0	0
$958$	− 18000.0i	− 0.607050i
$959$	10056.0	0.338608
$960$	0	0
$961$	44193.0	1.48343
$962$	− 1336.00i	− 0.0447759i
$963$	0	0
$964$	5752.00	0.192178
$965$	0	0
$966$	0	0
$967$	17524.0i	0.582765i	0.956607	+	0.291383i	$0.0941153\pi$
$967$	17524.0i	0.582765i	−0.956607	+	0.291383i	$0.905885\pi$
$968$	− 7784.00i	− 0.258458i
$969$	0	0
$970$	0	0
$971$	26808.0	0.886004	0.443002	−	0.896521i	$-0.353913\pi$
$971$	26808.0	0.886004	0.443002	+	0.896521i	$0.353913\pi$
$972$	0	0
$973$	− 5360.00i	− 0.176602i
$974$	−17528.0	−0.576626
$975$	0	0
$976$	4832.00	0.158472
$977$	− 10914.0i	− 0.357390i	−0.983905	−	0.178695i	$-0.942813\pi$
$977$	− 10914.0i	− 0.357390i	0.983905	−	0.178695i	$-0.0571875\pi$
$978$	0	0
$979$	10080.0	0.329069
$980$	0	0
$981$	0	0
$982$	− 11184.0i	− 0.363438i
$983$	− 22272.0i	− 0.722652i	−0.932440	−	0.361326i	$-0.882324\pi$
$983$	− 22272.0i	− 0.722652i	0.932440	−	0.361326i	$-0.117676\pi$
$984$	0	0
$985$	0	0
$986$	47880.0	1.54646
$987$	0	0
$988$	1120.00i	0.0360647i
$989$	−19296.0	−0.620402
$990$	0	0
$991$	14072.0	0.451071	0.225536	−	0.974235i	$-0.427587\pi$
$991$	14072.0	0.451071	0.225536	+	0.974235i	$0.427587\pi$
$992$	8704.00i	0.278581i
$993$	0	0
$994$	−6144.00	−0.196052
$995$	0	0
$996$	0	0
$997$	− 4826.00i	− 0.153301i	−0.997058	−	0.0766504i	$-0.975577\pi$
$997$	− 4826.00i	− 0.153301i	0.997058	−	0.0766504i	$-0.0244225\pi$
$998$	− 9400.00i	− 0.298148i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.4.c.j.199.2		2
3.2	odd	2		150.4.c.c.49.1		2
5.2	odd	4		90.4.a.c.1.1		1
5.3	odd	4		450.4.a.r.1.1		1
5.4	even	2	inner	450.4.c.j.199.1		2
12.11	even	2		1200.4.f.r.49.1		2
15.2	even	4		30.4.a.b.1.1	✓	1
15.8	even	4		150.4.a.b.1.1		1
15.14	odd	2		150.4.c.c.49.2		2
20.7	even	4		720.4.a.y.1.1		1
45.2	even	12		810.4.e.i.271.1		2
45.7	odd	12		810.4.e.p.271.1		2
45.22	odd	12		810.4.e.p.541.1		2
45.32	even	12		810.4.e.i.541.1		2
60.23	odd	4		1200.4.a.ba.1.1		1
60.47	odd	4		240.4.a.b.1.1		1
60.59	even	2		1200.4.f.r.49.2		2
105.62	odd	4		1470.4.a.r.1.1		1
120.77	even	4		960.4.a.n.1.1		1
120.107	odd	4		960.4.a.bg.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.4.a.b.1.1	✓	1	15.2	even	4
90.4.a.c.1.1		1	5.2	odd	4
150.4.a.b.1.1		1	15.8	even	4
150.4.c.c.49.1		2	3.2	odd	2
150.4.c.c.49.2		2	15.14	odd	2
240.4.a.b.1.1		1	60.47	odd	4
450.4.a.r.1.1		1	5.3	odd	4
450.4.c.j.199.1		2	5.4	even	2	inner
450.4.c.j.199.2		2	1.1	even	1	trivial
720.4.a.y.1.1		1	20.7	even	4
810.4.e.i.271.1		2	45.2	even	12
810.4.e.i.541.1		2	45.32	even	12
810.4.e.p.271.1		2	45.7	odd	12
810.4.e.p.541.1		2	45.22	odd	12
960.4.a.n.1.1		1	120.77	even	4
960.4.a.bg.1.1		1	120.107	odd	4
1200.4.a.ba.1.1		1	60.23	odd	4
1200.4.f.r.49.1		2	12.11	even	2
1200.4.f.r.49.2		2	60.59	even	2
1470.4.a.r.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)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -4.00000 q^{4} +4.00000i q^{7} -8.00000i q^{8} +O(q^{10})\)	\(q+2.00000i q^{2} -4.00000 q^{4} +4.00000i q^{7} -8.00000i q^{8} +48.0000 q^{11} +2.00000i q^{13} -8.00000 q^{14} +16.0000 q^{16} -114.000i q^{17} -140.000 q^{19} +96.0000i q^{22} -72.0000i q^{23} -4.00000 q^{26} -16.0000i q^{28} +210.000 q^{29} +272.000 q^{31} +32.0000i q^{32} +228.000 q^{34} +334.000i q^{37} -280.000i q^{38} +198.000 q^{41} -268.000i q^{43} -192.000 q^{44} +144.000 q^{46} +216.000i q^{47} +327.000 q^{49} -8.00000i q^{52} +78.0000i q^{53} +32.0000 q^{56} +420.000i q^{58} +240.000 q^{59} +302.000 q^{61} +544.000i q^{62} -64.0000 q^{64} -596.000i q^{67} +456.000i q^{68} +768.000 q^{71} -478.000i q^{73} -668.000 q^{74} +560.000 q^{76} +192.000i q^{77} +640.000 q^{79} +396.000i q^{82} +348.000i q^{83} +536.000 q^{86} -384.000i q^{88} +210.000 q^{89} -8.00000 q^{91} +288.000i q^{92} -432.000 q^{94} +1534.00i q^{97} +654.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4}+O(q^{10}) \) 2 * q - 8 * q^4	\( 2 q - 8 q^{4} + 96 q^{11} - 16 q^{14} + 32 q^{16} - 280 q^{19} - 8 q^{26} + 420 q^{29} + 544 q^{31} + 456 q^{34} + 396 q^{41} - 384 q^{44} + 288 q^{46} + 654 q^{49} + 64 q^{56} + 480 q^{59} + 604 q^{61} - 128 q^{64} + 1536 q^{71} - 1336 q^{74} + 1120 q^{76} + 1280 q^{79} + 1072 q^{86} + 420 q^{89} - 16 q^{91} - 864 q^{94}+O(q^{100}) \) 2 * q - 8 * q^4 + 96 * q^11 - 16 * q^14 + 32 * q^16 - 280 * q^19 - 8 * q^26 + 420 * q^29 + 544 * q^31 + 456 * q^34 + 396 * q^41 - 384 * q^44 + 288 * q^46 + 654 * q^49 + 64 * q^56 + 480 * q^59 + 604 * q^61 - 128 * q^64 + 1536 * q^71 - 1336 * q^74 + 1120 * q^76 + 1280 * q^79 + 1072 * q^86 + 420 * q^89 - 16 * q^91 - 864 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more