LMFDB - Embedded newform 225.4.b.f.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,Mod(199,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, 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("225.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	225.b (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:	$13.2754297513$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 25)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.4.b.f.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +7.00000 q^{4} +6.00000i q^{7} -15.0000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} +7.00000 q^{4} +6.00000i q^{7} -15.0000i q^{8} +43.0000 q^{11} +28.0000i q^{13} +6.00000 q^{14} +41.0000 q^{16} -91.0000i q^{17} +35.0000 q^{19} -43.0000i q^{22} +162.000i q^{23} +28.0000 q^{26} +42.0000i q^{28} +160.000 q^{29} +42.0000 q^{31} -161.000i q^{32} -91.0000 q^{34} -314.000i q^{37} -35.0000i q^{38} +203.000 q^{41} -92.0000i q^{43} +301.000 q^{44} +162.000 q^{46} -196.000i q^{47} +307.000 q^{49} +196.000i q^{52} +82.0000i q^{53} +90.0000 q^{56} -160.000i q^{58} -280.000 q^{59} -518.000 q^{61} -42.0000i q^{62} +167.000 q^{64} +141.000i q^{67} -637.000i q^{68} -412.000 q^{71} +763.000i q^{73} -314.000 q^{74} +245.000 q^{76} +258.000i q^{77} -510.000 q^{79} -203.000i q^{82} +777.000i q^{83} -92.0000 q^{86} -645.000i q^{88} -945.000 q^{89} -168.000 q^{91} +1134.00i q^{92} -196.000 q^{94} +1246.00i q^{97} -307.000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{4}+O(q^{10}) $ 2 * q + 14 * q^4	$ 2 q + 14 q^{4} + 86 q^{11} + 12 q^{14} + 82 q^{16} + 70 q^{19} + 56 q^{26} + 320 q^{29} + 84 q^{31} - 182 q^{34} + 406 q^{41} + 602 q^{44} + 324 q^{46} + 614 q^{49} + 180 q^{56} - 560 q^{59} - 1036 q^{61} + 334 q^{64} - 824 q^{71} - 628 q^{74} + 490 q^{76} - 1020 q^{79} - 184 q^{86} - 1890 q^{89} - 336 q^{91} - 392 q^{94}+O(q^{100}) $ 2 * q + 14 * q^4 + 86 * q^11 + 12 * q^14 + 82 * q^16 + 70 * q^19 + 56 * q^26 + 320 * q^29 + 84 * q^31 - 182 * q^34 + 406 * q^41 + 602 * q^44 + 324 * q^46 + 614 * q^49 + 180 * q^56 - 560 * q^59 - 1036 * q^61 + 334 * q^64 - 824 * q^71 - 628 * q^74 + 490 * q^76 - 1020 * q^79 - 184 * q^86 - 1890 * q^89 - 336 * q^91 - 392 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

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

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 $

$2$	− 1.00000i	− 0.353553i	−0.984251	−	0.176777i	$-0.943433\pi$
$2$	− 1.00000i	− 0.353553i	0.984251	−	0.176777i	$-0.0565670\pi$
$3$	0	0
$3$	0	0
$4$	7.00000	0.875000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	6.00000i	0.323970i	0.986793	+	0.161985i	$0.0517895\pi$
$7$	6.00000i	0.323970i	−0.986793	+	0.161985i	$0.948210\pi$
$8$	− 15.0000i	− 0.662913i
$9$	0	0
$10$	0	0
$11$	43.0000	1.17864	0.589318	−	0.807901i	$-0.299397\pi$
$11$	43.0000	1.17864	0.589318	+	0.807901i	$0.299397\pi$
$12$	0	0
$13$	28.0000i	0.597369i	0.954352	+	0.298685i	$0.0965479\pi$
$13$	28.0000i	0.597369i	−0.954352	+	0.298685i	$0.903452\pi$
$14$	6.00000	0.114541
$15$	0	0
$16$	41.0000	0.640625
$17$	− 91.0000i	− 1.29828i	−0.760669	−	0.649139i	$-0.775129\pi$
$17$	− 91.0000i	− 1.29828i	0.760669	−	0.649139i	$-0.224871\pi$
$18$	0	0
$19$	35.0000	0.422608	0.211304	−	0.977420i	$-0.432229\pi$
$19$	35.0000	0.422608	0.211304	+	0.977420i	$0.432229\pi$
$20$	0	0
$21$	0	0
$22$	− 43.0000i	− 0.416710i
$23$	162.000i	1.46867i	0.678789	+	0.734333i	$0.262505\pi$
$23$	162.000i	1.46867i	−0.678789	+	0.734333i	$0.737495\pi$
$24$	0	0
$25$	0	0
$26$	28.0000	0.211202
$27$	0	0
$28$	42.0000i	0.283473i
$29$	160.000	1.02453	0.512263	−	0.858829i	$-0.328807\pi$
$29$	160.000	1.02453	0.512263	+	0.858829i	$0.328807\pi$
$30$	0	0
$31$	42.0000	0.243336	0.121668	−	0.992571i	$-0.461176\pi$
$31$	42.0000	0.243336	0.121668	+	0.992571i	$0.461176\pi$
$32$	− 161.000i	− 0.889408i
$33$	0	0
$34$	−91.0000	−0.459011
$35$	0	0
$36$	0	0
$37$	− 314.000i	− 1.39517i	−0.716502	−	0.697585i	$-0.754258\pi$
$37$	− 314.000i	− 1.39517i	0.716502	−	0.697585i	$-0.245742\pi$
$38$	− 35.0000i	− 0.149414i
$39$	0	0
$40$	0	0
$41$	203.000	0.773251	0.386625	−	0.922237i	$-0.373641\pi$
$41$	203.000	0.773251	0.386625	+	0.922237i	$0.373641\pi$
$42$	0	0
$43$	− 92.0000i	− 0.326276i	−0.986603	−	0.163138i	$-0.947838\pi$
$43$	− 92.0000i	− 0.326276i	0.986603	−	0.163138i	$-0.0521616\pi$
$44$	301.000	1.03131
$45$	0	0
$46$	162.000	0.519252
$47$	− 196.000i	− 0.608288i	−0.952626	−	0.304144i	$-0.901630\pi$
$47$	− 196.000i	− 0.608288i	0.952626	−	0.304144i	$-0.0983704\pi$
$48$	0	0
$49$	307.000	0.895044
$50$	0	0
$51$	0	0
$52$	196.000i	0.522698i
$53$	82.0000i	0.212520i	0.994338	+	0.106260i	$0.0338876\pi$
$53$	82.0000i	0.212520i	−0.994338	+	0.106260i	$0.966112\pi$
$54$	0	0
$55$	0	0
$56$	90.0000	0.214763
$57$	0	0
$58$	− 160.000i	− 0.362225i
$59$	−280.000	−0.617846	−0.308923	−	0.951087i	$-0.599968\pi$
$59$	−280.000	−0.617846	−0.308923	+	0.951087i	$0.599968\pi$
$60$	0	0
$61$	−518.000	−1.08726	−0.543632	−	0.839324i	$-0.682951\pi$
$61$	−518.000	−1.08726	−0.543632	+	0.839324i	$0.682951\pi$
$62$	− 42.0000i	− 0.0860323i
$63$	0	0
$64$	167.000	0.326172
$65$	0	0
$66$	0	0
$67$	141.000i	0.257103i	0.991703	+	0.128551i	$0.0410327\pi$
$67$	141.000i	0.257103i	−0.991703	+	0.128551i	$0.958967\pi$
$68$	− 637.000i	− 1.13599i
$69$	0	0
$70$	0	0
$71$	−412.000	−0.688668	−0.344334	−	0.938847i	$-0.611895\pi$
$71$	−412.000	−0.688668	−0.344334	+	0.938847i	$0.611895\pi$
$72$	0	0
$73$	763.000i	1.22332i	0.791121	+	0.611660i	$0.209498\pi$
$73$	763.000i	1.22332i	−0.791121	+	0.611660i	$0.790502\pi$
$74$	−314.000	−0.493267
$75$	0	0
$76$	245.000	0.369782
$77$	258.000i	0.381842i
$78$	0	0
$79$	−510.000	−0.726323	−0.363161	−	0.931726i	$-0.618303\pi$
$79$	−510.000	−0.726323	−0.363161	+	0.931726i	$0.618303\pi$
$80$	0	0
$81$	0	0
$82$	− 203.000i	− 0.273385i
$83$	777.000i	1.02755i	0.857924	+	0.513776i	$0.171754\pi$
$83$	777.000i	1.02755i	−0.857924	+	0.513776i	$0.828246\pi$
$84$	0	0
$85$	0	0
$86$	−92.0000	−0.115356
$87$	0	0
$88$	− 645.000i	− 0.781332i
$89$	−945.000	−1.12550	−0.562752	−	0.826626i	$-0.690257\pi$
$89$	−945.000	−1.12550	−0.562752	+	0.826626i	$0.690257\pi$
$90$	0	0
$91$	−168.000	−0.193530
$92$	1134.00i	1.28508i
$93$	0	0
$94$	−196.000	−0.215062
$95$	0	0
$96$	0	0
$97$	1246.00i	1.30425i	0.758112	+	0.652124i	$0.226122\pi$
$97$	1246.00i	1.30425i	−0.758112	+	0.652124i	$0.773878\pi$
$98$	− 307.000i	− 0.316446i
$99$	0	0
$100$	0	0
$101$	−1302.00	−1.28271	−0.641356	−	0.767244i	$-0.721628\pi$
$101$	−1302.00	−1.28271	−0.641356	+	0.767244i	$0.721628\pi$
$102$	0	0
$103$	− 532.000i	− 0.508927i	−0.967082	−	0.254464i	$-0.918101\pi$
$103$	− 532.000i	− 0.508927i	0.967082	−	0.254464i	$-0.0818989\pi$
$104$	420.000	0.396004
$105$	0	0
$106$	82.0000	0.0751372
$107$	1269.00i	1.14653i	0.819370	+	0.573266i	$0.194324\pi$
$107$	1269.00i	1.14653i	−0.819370	+	0.573266i	$0.805676\pi$
$108$	0	0
$109$	−1070.00	−0.940251	−0.470126	−	0.882599i	$-0.655791\pi$
$109$	−1070.00	−0.940251	−0.470126	+	0.882599i	$0.655791\pi$
$110$	0	0
$111$	0	0
$112$	246.000i	0.207543i
$113$	− 503.000i	− 0.418746i	−0.977836	−	0.209373i	$-0.932858\pi$
$113$	− 503.000i	− 0.418746i	0.977836	−	0.209373i	$-0.0671422\pi$
$114$	0	0
$115$	0	0
$116$	1120.00	0.896460
$117$	0	0
$118$	280.000i	0.218441i
$119$	546.000	0.420603
$120$	0	0
$121$	518.000	0.389181
$122$	518.000i	0.384406i
$123$	0	0
$124$	294.000	0.212919
$125$	0	0
$126$	0	0
$127$	− 874.000i	− 0.610669i	−0.952245	−	0.305334i	$-0.901232\pi$
$127$	− 874.000i	− 0.610669i	0.952245	−	0.305334i	$-0.0987683\pi$
$128$	− 1455.00i	− 1.00473i
$129$	0	0
$130$	0	0
$131$	−1092.00	−0.728309	−0.364155	−	0.931339i	$-0.618642\pi$
$131$	−1092.00	−0.728309	−0.364155	+	0.931339i	$0.618642\pi$
$132$	0	0
$133$	210.000i	0.136912i
$134$	141.000	0.0908996
$135$	0	0
$136$	−1365.00	−0.860645
$137$	− 411.000i	− 0.256307i	−0.991754	−	0.128154i	$-0.959095\pi$
$137$	− 411.000i	− 0.256307i	0.991754	−	0.128154i	$-0.0409051\pi$
$138$	0	0
$139$	595.000	0.363074	0.181537	−	0.983384i	$-0.441893\pi$
$139$	595.000	0.363074	0.181537	+	0.983384i	$0.441893\pi$
$140$	0	0
$141$	0	0
$142$	412.000i	0.243481i
$143$	1204.00i	0.704081i
$144$	0	0
$145$	0	0
$146$	763.000	0.432509
$147$	0	0
$148$	− 2198.00i	− 1.22077i
$149$	−3200.00	−1.75942	−0.879712	−	0.475507i	$-0.842265\pi$
$149$	−3200.00	−1.75942	−0.879712	+	0.475507i	$0.842265\pi$
$150$	0	0
$151$	202.000	0.108864	0.0544322	−	0.998517i	$-0.482665\pi$
$151$	202.000	0.108864	0.0544322	+	0.998517i	$0.482665\pi$
$152$	− 525.000i	− 0.280152i
$153$	0	0
$154$	258.000	0.135002
$155$	0	0
$156$	0	0
$157$	406.000i	0.206384i	0.994661	+	0.103192i	$0.0329057\pi$
$157$	406.000i	0.206384i	−0.994661	+	0.103192i	$0.967094\pi$
$158$	510.000i	0.256794i
$159$	0	0
$160$	0	0
$161$	−972.000	−0.475803
$162$	0	0
$163$	3803.00i	1.82745i	0.406336	+	0.913724i	$0.366806\pi$
$163$	3803.00i	1.82745i	−0.406336	+	0.913724i	$0.633194\pi$
$164$	1421.00	0.676594
$165$	0	0
$166$	777.000	0.363295
$167$	− 4116.00i	− 1.90722i	−0.301046	−	0.953610i	$-0.597336\pi$
$167$	− 4116.00i	− 1.90722i	0.301046	−	0.953610i	$-0.402664\pi$
$168$	0	0
$169$	1413.00	0.643150
$170$	0	0
$171$	0	0
$172$	− 644.000i	− 0.285492i
$173$	1512.00i	0.664481i	0.943195	+	0.332241i	$0.107805\pi$
$173$	1512.00i	0.664481i	−0.943195	+	0.332241i	$0.892195\pi$
$174$	0	0
$175$	0	0
$176$	1763.00	0.755063
$177$	0	0
$178$	945.000i	0.397926i
$179$	2585.00	1.07940	0.539698	−	0.841859i	$-0.318538\pi$
$179$	2585.00	1.07940	0.539698	+	0.841859i	$0.318538\pi$
$180$	0	0
$181$	−2758.00	−1.13260	−0.566300	−	0.824199i	$-0.691626\pi$
$181$	−2758.00	−1.13260	−0.566300	+	0.824199i	$0.691626\pi$
$182$	168.000i	0.0684230i
$183$	0	0
$184$	2430.00	0.973598
$185$	0	0
$186$	0	0
$187$	− 3913.00i	− 1.53020i
$188$	− 1372.00i	− 0.532252i
$189$	0	0
$190$	0	0
$191$	2378.00	0.900869	0.450435	−	0.892809i	$-0.351269\pi$
$191$	2378.00	0.900869	0.450435	+	0.892809i	$0.351269\pi$
$192$	0	0
$193$	− 3067.00i	− 1.14387i	−0.820298	−	0.571937i	$-0.806192\pi$
$193$	− 3067.00i	− 1.14387i	0.820298	−	0.571937i	$-0.193808\pi$
$194$	1246.00	0.461122
$195$	0	0
$196$	2149.00	0.783163
$197$	− 2346.00i	− 0.848455i	−0.905556	−	0.424227i	$-0.860546\pi$
$197$	− 2346.00i	− 0.848455i	0.905556	−	0.424227i	$-0.139454\pi$
$198$	0	0
$199$	−4900.00	−1.74549	−0.872743	−	0.488180i	$-0.837661\pi$
$199$	−4900.00	−1.74549	−0.872743	+	0.488180i	$0.837661\pi$
$200$	0	0
$201$	0	0
$202$	1302.00i	0.453507i
$203$	960.000i	0.331915i
$204$	0	0
$205$	0	0
$206$	−532.000	−0.179933
$207$	0	0
$208$	1148.00i	0.382690i
$209$	1505.00	0.498101
$210$	0	0
$211$	4307.00	1.40524	0.702621	−	0.711564i	$-0.252013\pi$
$211$	4307.00	1.40524	0.702621	+	0.711564i	$0.252013\pi$
$212$	574.000i	0.185955i
$213$	0	0
$214$	1269.00	0.405360
$215$	0	0
$216$	0	0
$217$	252.000i	0.0788335i
$218$	1070.00i	0.332429i
$219$	0	0
$220$	0	0
$221$	2548.00	0.775552
$222$	0	0
$223$	− 2212.00i	− 0.664244i	−0.943236	−	0.332122i	$-0.892235\pi$
$223$	− 2212.00i	− 0.664244i	0.943236	−	0.332122i	$-0.107765\pi$
$224$	966.000	0.288141
$225$	0	0
$226$	−503.000	−0.148049
$227$	− 476.000i	− 0.139177i	−0.997576	−	0.0695886i	$-0.977831\pi$
$227$	− 476.000i	− 0.139177i	0.997576	−	0.0695886i	$-0.0221687\pi$
$228$	0	0
$229$	2940.00	0.848387	0.424194	−	0.905572i	$-0.360558\pi$
$229$	2940.00	0.848387	0.424194	+	0.905572i	$0.360558\pi$
$230$	0	0
$231$	0	0
$232$	− 2400.00i	− 0.679171i
$233$	1002.00i	0.281730i	0.990029	+	0.140865i	$0.0449884\pi$
$233$	1002.00i	0.281730i	−0.990029	+	0.140865i	$0.955012\pi$
$234$	0	0
$235$	0	0
$236$	−1960.00	−0.540615
$237$	0	0
$238$	− 546.000i	− 0.148706i
$239$	2480.00	0.671204	0.335602	−	0.942004i	$-0.391060\pi$
$239$	2480.00	0.671204	0.335602	+	0.942004i	$0.391060\pi$
$240$	0	0
$241$	1897.00	0.507039	0.253520	−	0.967330i	$-0.418412\pi$
$241$	1897.00	0.507039	0.253520	+	0.967330i	$0.418412\pi$
$242$	− 518.000i	− 0.137596i
$243$	0	0
$244$	−3626.00	−0.951356
$245$	0	0
$246$	0	0
$247$	980.000i	0.252453i
$248$	− 630.000i	− 0.161311i
$249$	0	0
$250$	0	0
$251$	2373.00	0.596743	0.298371	−	0.954450i	$-0.403557\pi$
$251$	2373.00	0.596743	0.298371	+	0.954450i	$0.403557\pi$
$252$	0	0
$253$	6966.00i	1.73102i
$254$	−874.000	−0.215904
$255$	0	0
$256$	−119.000	−0.0290527
$257$	4494.00i	1.09077i	0.838185	+	0.545385i	$0.183617\pi$
$257$	4494.00i	1.09077i	−0.838185	+	0.545385i	$0.816383\pi$
$258$	0	0
$259$	1884.00	0.451993
$260$	0	0
$261$	0	0
$262$	1092.00i	0.257496i
$263$	722.000i	0.169279i	0.996412	+	0.0846396i	$0.0269739\pi$
$263$	722.000i	0.169279i	−0.996412	+	0.0846396i	$0.973026\pi$
$264$	0	0
$265$	0	0
$266$	210.000	0.0484057
$267$	0	0
$268$	987.000i	0.224965i
$269$	−6160.00	−1.39621	−0.698107	−	0.715993i	$-0.745974\pi$
$269$	−6160.00	−1.39621	−0.698107	+	0.715993i	$0.745974\pi$
$270$	0	0
$271$	−7238.00	−1.62243	−0.811213	−	0.584751i	$-0.801192\pi$
$271$	−7238.00	−1.62243	−0.811213	+	0.584751i	$0.801192\pi$
$272$	− 3731.00i	− 0.831710i
$273$	0	0
$274$	−411.000	−0.0906183
$275$	0	0
$276$	0	0
$277$	1776.00i	0.385233i	0.981274	+	0.192616i	$0.0616973\pi$
$277$	1776.00i	0.385233i	−0.981274	+	0.192616i	$0.938303\pi$
$278$	− 595.000i	− 0.128366i
$279$	0	0
$280$	0	0
$281$	−4542.00	−0.964246	−0.482123	−	0.876104i	$-0.660134\pi$
$281$	−4542.00	−0.964246	−0.482123	+	0.876104i	$0.660134\pi$
$282$	0	0
$283$	− 7077.00i	− 1.48652i	−0.669005	−	0.743258i	$-0.733280\pi$
$283$	− 7077.00i	− 1.48652i	0.669005	−	0.743258i	$-0.266720\pi$
$284$	−2884.00	−0.602584
$285$	0	0
$286$	1204.00	0.248930
$287$	1218.00i	0.250510i
$288$	0	0
$289$	−3368.00	−0.685528
$290$	0	0
$291$	0	0
$292$	5341.00i	1.07041i
$293$	− 4158.00i	− 0.829054i	−0.910037	−	0.414527i	$-0.863947\pi$
$293$	− 4158.00i	− 0.829054i	0.910037	−	0.414527i	$-0.136053\pi$
$294$	0	0
$295$	0	0
$296$	−4710.00	−0.924876
$297$	0	0
$298$	3200.00i	0.622050i
$299$	−4536.00	−0.877337
$300$	0	0
$301$	552.000	0.105703
$302$	− 202.000i	− 0.0384894i
$303$	0	0
$304$	1435.00	0.270733
$305$	0	0
$306$	0	0
$307$	− 2569.00i	− 0.477591i	−0.971070	−	0.238796i	$-0.923247\pi$
$307$	− 2569.00i	− 0.477591i	0.971070	−	0.238796i	$-0.0767526\pi$
$308$	1806.00i	0.334112i
$309$	0	0
$310$	0	0
$311$	−2982.00	−0.543710	−0.271855	−	0.962338i	$-0.587637\pi$
$311$	−2982.00	−0.543710	−0.271855	+	0.962338i	$0.587637\pi$
$312$	0	0
$313$	− 2422.00i	− 0.437379i	−0.975795	−	0.218689i	$-0.929822\pi$
$313$	− 2422.00i	− 0.437379i	0.975795	−	0.218689i	$-0.0701781\pi$
$314$	406.000	0.0729679
$315$	0	0
$316$	−3570.00	−0.635532
$317$	9484.00i	1.68036i	0.542307	+	0.840181i	$0.317551\pi$
$317$	9484.00i	1.68036i	−0.542307	+	0.840181i	$0.682449\pi$
$318$	0	0
$319$	6880.00	1.20754
$320$	0	0
$321$	0	0
$322$	972.000i	0.168222i
$323$	− 3185.00i	− 0.548663i
$324$	0	0
$325$	0	0
$326$	3803.00	0.646100
$327$	0	0
$328$	− 3045.00i	− 0.512598i
$329$	1176.00	0.197067
$330$	0	0
$331$	−183.000	−0.0303885	−0.0151942	−	0.999885i	$-0.504837\pi$
$331$	−183.000	−0.0303885	−0.0151942	+	0.999885i	$0.504837\pi$
$332$	5439.00i	0.899108i
$333$	0	0
$334$	−4116.00	−0.674304
$335$	0	0
$336$	0	0
$337$	2861.00i	0.462459i	0.972899	+	0.231229i	$0.0742748\pi$
$337$	2861.00i	0.462459i	−0.972899	+	0.231229i	$0.925725\pi$
$338$	− 1413.00i	− 0.227388i
$339$	0	0
$340$	0	0
$341$	1806.00	0.286805
$342$	0	0
$343$	3900.00i	0.613936i
$344$	−1380.00	−0.216292
$345$	0	0
$346$	1512.00	0.234930
$347$	629.000i	0.0973098i	0.998816	+	0.0486549i	$0.0154934\pi$
$347$	629.000i	0.0973098i	−0.998816	+	0.0486549i	$0.984507\pi$
$348$	0	0
$349$	−5950.00	−0.912597	−0.456298	−	0.889827i	$-0.650825\pi$
$349$	−5950.00	−0.912597	−0.456298	+	0.889827i	$0.650825\pi$
$350$	0	0
$351$	0	0
$352$	− 6923.00i	− 1.04829i
$353$	− 11718.0i	− 1.76682i	−0.468604	−	0.883408i	$-0.655243\pi$
$353$	− 11718.0i	− 1.76682i	0.468604	−	0.883408i	$-0.344757\pi$
$354$	0	0
$355$	0	0
$356$	−6615.00	−0.984815
$357$	0	0
$358$	− 2585.00i	− 0.381624i
$359$	8070.00	1.18640	0.593201	−	0.805054i	$-0.297864\pi$
$359$	8070.00	1.18640	0.593201	+	0.805054i	$0.297864\pi$
$360$	0	0
$361$	−5634.00	−0.821403
$362$	2758.00i	0.400434i
$363$	0	0
$364$	−1176.00	−0.169338
$365$	0	0
$366$	0	0
$367$	8316.00i	1.18281i	0.806374	+	0.591406i	$0.201427\pi$
$367$	8316.00i	1.18281i	−0.806374	+	0.591406i	$0.798573\pi$
$368$	6642.00i	0.940865i
$369$	0	0
$370$	0	0
$371$	−492.000	−0.0688500
$372$	0	0
$373$	− 12062.0i	− 1.67439i	−0.546906	−	0.837194i	$-0.684195\pi$
$373$	− 12062.0i	− 1.67439i	0.546906	−	0.837194i	$-0.315805\pi$
$374$	−3913.00	−0.541006
$375$	0	0
$376$	−2940.00	−0.403242
$377$	4480.00i	0.612021i
$378$	0	0
$379$	−1735.00	−0.235148	−0.117574	−	0.993064i	$-0.537512\pi$
$379$	−1735.00	−0.235148	−0.117574	+	0.993064i	$0.537512\pi$
$380$	0	0
$381$	0	0
$382$	− 2378.00i	− 0.318505i
$383$	7602.00i	1.01421i	0.861883	+	0.507107i	$0.169285\pi$
$383$	7602.00i	1.01421i	−0.861883	+	0.507107i	$0.830715\pi$
$384$	0	0
$385$	0	0
$386$	−3067.00	−0.404420
$387$	0	0
$388$	8722.00i	1.14122i
$389$	3030.00	0.394928	0.197464	−	0.980310i	$-0.436729\pi$
$389$	3030.00	0.394928	0.197464	+	0.980310i	$0.436729\pi$
$390$	0	0
$391$	14742.0	1.90674
$392$	− 4605.00i	− 0.593336i
$393$	0	0
$394$	−2346.00	−0.299974
$395$	0	0
$396$	0	0
$397$	− 1204.00i	− 0.152209i	−0.997100	−	0.0761046i	$-0.975752\pi$
$397$	− 1204.00i	− 0.152209i	0.997100	−	0.0761046i	$-0.0242483\pi$
$398$	4900.00i	0.617123i
$399$	0	0
$400$	0	0
$401$	−1077.00	−0.134122	−0.0670609	−	0.997749i	$-0.521362\pi$
$401$	−1077.00	−0.134122	−0.0670609	+	0.997749i	$0.521362\pi$
$402$	0	0
$403$	1176.00i	0.145362i
$404$	−9114.00	−1.12237
$405$	0	0
$406$	960.000	0.117350
$407$	− 13502.0i	− 1.64440i
$408$	0	0
$409$	3955.00	0.478147	0.239074	−	0.971001i	$-0.423156\pi$
$409$	3955.00	0.478147	0.239074	+	0.971001i	$0.423156\pi$
$410$	0	0
$411$	0	0
$412$	− 3724.00i	− 0.445311i
$413$	− 1680.00i	− 0.200163i
$414$	0	0
$415$	0	0
$416$	4508.00	0.531305
$417$	0	0
$418$	− 1505.00i	− 0.176105i
$419$	6265.00	0.730466	0.365233	−	0.930916i	$-0.380989\pi$
$419$	6265.00	0.730466	0.365233	+	0.930916i	$0.380989\pi$
$420$	0	0
$421$	−3788.00	−0.438517	−0.219259	−	0.975667i	$-0.570364\pi$
$421$	−3788.00	−0.438517	−0.219259	+	0.975667i	$0.570364\pi$
$422$	− 4307.00i	− 0.496828i
$423$	0	0
$424$	1230.00	0.140882
$425$	0	0
$426$	0	0
$427$	− 3108.00i	− 0.352240i
$428$	8883.00i	1.00321i
$429$	0	0
$430$	0	0
$431$	15258.0	1.70523	0.852613	−	0.522544i	$-0.175017\pi$
$431$	15258.0	1.70523	0.852613	+	0.522544i	$0.175017\pi$
$432$	0	0
$433$	13573.0i	1.50641i	0.657784	+	0.753206i	$0.271494\pi$
$433$	13573.0i	1.50641i	−0.657784	+	0.753206i	$0.728506\pi$
$434$	252.000	0.0278719
$435$	0	0
$436$	−7490.00	−0.822720
$437$	5670.00i	0.620670i
$438$	0	0
$439$	8120.00	0.882794	0.441397	−	0.897312i	$-0.354483\pi$
$439$	8120.00	0.882794	0.441397	+	0.897312i	$0.354483\pi$
$440$	0	0
$441$	0	0
$442$	− 2548.00i	− 0.274199i
$443$	− 6183.00i	− 0.663122i	−0.943434	−	0.331561i	$-0.892425\pi$
$443$	− 6183.00i	− 0.663122i	0.943434	−	0.331561i	$-0.107575\pi$
$444$	0	0
$445$	0	0
$446$	−2212.00	−0.234846
$447$	0	0
$448$	1002.00i	0.105670i
$449$	−1975.00	−0.207586	−0.103793	−	0.994599i	$-0.533098\pi$
$449$	−1975.00	−0.207586	−0.103793	+	0.994599i	$0.533098\pi$
$450$	0	0
$451$	8729.00	0.911380
$452$	− 3521.00i	− 0.366402i
$453$	0	0
$454$	−476.000	−0.0492066
$455$	0	0
$456$	0	0
$457$	11831.0i	1.21101i	0.795842	+	0.605504i	$0.207029\pi$
$457$	11831.0i	1.21101i	−0.795842	+	0.605504i	$0.792971\pi$
$458$	− 2940.00i	− 0.299950i
$459$	0	0
$460$	0	0
$461$	−1932.00	−0.195189	−0.0975946	−	0.995226i	$-0.531115\pi$
$461$	−1932.00	−0.195189	−0.0975946	+	0.995226i	$0.531115\pi$
$462$	0	0
$463$	9228.00i	0.926267i	0.886289	+	0.463133i	$0.153275\pi$
$463$	9228.00i	0.926267i	−0.886289	+	0.463133i	$0.846725\pi$
$464$	6560.00	0.656337
$465$	0	0
$466$	1002.00	0.0996068
$467$	− 13916.0i	− 1.37892i	−0.724324	−	0.689460i	$-0.757848\pi$
$467$	− 13916.0i	− 1.37892i	0.724324	−	0.689460i	$-0.242152\pi$
$468$	0	0
$469$	−846.000	−0.0832935
$470$	0	0
$471$	0	0
$472$	4200.00i	0.409578i
$473$	− 3956.00i	− 0.384560i
$474$	0	0
$475$	0	0
$476$	3822.00	0.368027
$477$	0	0
$478$	− 2480.00i	− 0.237307i
$479$	2310.00	0.220348	0.110174	−	0.993912i	$-0.464859\pi$
$479$	2310.00	0.220348	0.110174	+	0.993912i	$0.464859\pi$
$480$	0	0
$481$	8792.00	0.833432
$482$	− 1897.00i	− 0.179266i
$483$	0	0
$484$	3626.00	0.340533
$485$	0	0
$486$	0	0
$487$	− 17114.0i	− 1.59242i	−0.605019	−	0.796211i	$-0.706835\pi$
$487$	− 17114.0i	− 1.59242i	0.605019	−	0.796211i	$-0.293165\pi$
$488$	7770.00i	0.720761i
$489$	0	0
$490$	0	0
$491$	17228.0	1.58348	0.791740	−	0.610858i	$-0.209175\pi$
$491$	17228.0	1.58348	0.791740	+	0.610858i	$0.209175\pi$
$492$	0	0
$493$	− 14560.0i	− 1.33012i
$494$	980.000	0.0892556
$495$	0	0
$496$	1722.00	0.155887
$497$	− 2472.00i	− 0.223107i
$498$	0	0
$499$	12500.0	1.12140	0.560698	−	0.828020i	$-0.310533\pi$
$499$	12500.0	1.12140	0.560698	+	0.828020i	$0.310533\pi$
$500$	0	0
$501$	0	0
$502$	− 2373.00i	− 0.210980i
$503$	− 868.000i	− 0.0769428i	−0.999260	−	0.0384714i	$-0.987751\pi$
$503$	− 868.000i	− 0.0769428i	0.999260	−	0.0384714i	$-0.0122488\pi$
$504$	0	0
$505$	0	0
$506$	6966.00	0.612009
$507$	0	0
$508$	− 6118.00i	− 0.534335i
$509$	13370.0	1.16427	0.582136	−	0.813091i	$-0.302217\pi$
$509$	13370.0	1.16427	0.582136	+	0.813091i	$0.302217\pi$
$510$	0	0
$511$	−4578.00	−0.396319
$512$	− 11521.0i	− 0.994455i
$513$	0	0
$514$	4494.00	0.385646
$515$	0	0
$516$	0	0
$517$	− 8428.00i	− 0.716950i
$518$	− 1884.00i	− 0.159803i
$519$	0	0
$520$	0	0
$521$	−21637.0	−1.81945	−0.909726	−	0.415210i	$-0.863708\pi$
$521$	−21637.0	−1.81945	−0.909726	+	0.415210i	$0.863708\pi$
$522$	0	0
$523$	− 287.000i	− 0.0239955i	−0.999928	−	0.0119977i	$-0.996181\pi$
$523$	− 287.000i	− 0.0239955i	0.999928	−	0.0119977i	$-0.00381909\pi$
$524$	−7644.00	−0.637270
$525$	0	0
$526$	722.000	0.0598492
$527$	− 3822.00i	− 0.315918i
$528$	0	0
$529$	−14077.0	−1.15698
$530$	0	0
$531$	0	0
$532$	1470.00i	0.119798i
$533$	5684.00i	0.461916i
$534$	0	0
$535$	0	0
$536$	2115.00	0.170437
$537$	0	0
$538$	6160.00i	0.493637i
$539$	13201.0	1.05493
$540$	0	0
$541$	−5328.00	−0.423417	−0.211709	−	0.977333i	$-0.567903\pi$
$541$	−5328.00	−0.423417	−0.211709	+	0.977333i	$0.567903\pi$
$542$	7238.00i	0.573614i
$543$	0	0
$544$	−14651.0	−1.15470
$545$	0	0
$546$	0	0
$547$	71.0000i	0.00554980i	0.999996	+	0.00277490i	$0.000883279\pi$
$547$	71.0000i	0.00554980i	−0.999996	+	0.00277490i	$0.999117\pi$
$548$	− 2877.00i	− 0.224269i
$549$	0	0
$550$	0	0
$551$	5600.00	0.432973
$552$	0	0
$553$	− 3060.00i	− 0.235306i
$554$	1776.00	0.136200
$555$	0	0
$556$	4165.00	0.317689
$557$	18444.0i	1.40305i	0.712646	+	0.701524i	$0.247497\pi$
$557$	18444.0i	1.40305i	−0.712646	+	0.701524i	$0.752503\pi$
$558$	0	0
$559$	2576.00	0.194907
$560$	0	0
$561$	0	0
$562$	4542.00i	0.340912i
$563$	672.000i	0.0503045i	0.999684	+	0.0251522i	$0.00800705\pi$
$563$	672.000i	0.0503045i	−0.999684	+	0.0251522i	$0.991993\pi$
$564$	0	0
$565$	0	0
$566$	−7077.00	−0.525563
$567$	0	0
$568$	6180.00i	0.456526i
$569$	−10935.0	−0.805657	−0.402829	−	0.915275i	$-0.631973\pi$
$569$	−10935.0	−0.805657	−0.402829	+	0.915275i	$0.631973\pi$
$570$	0	0
$571$	−13588.0	−0.995867	−0.497934	−	0.867215i	$-0.665908\pi$
$571$	−13588.0	−0.995867	−0.497934	+	0.867215i	$0.665908\pi$
$572$	8428.00i	0.616071i
$573$	0	0
$574$	1218.00	0.0885685
$575$	0	0
$576$	0	0
$577$	8701.00i	0.627777i	0.949460	+	0.313889i	$0.101632\pi$
$577$	8701.00i	0.627777i	−0.949460	+	0.313889i	$0.898368\pi$
$578$	3368.00i	0.242371i
$579$	0	0
$580$	0	0
$581$	−4662.00	−0.332896
$582$	0	0
$583$	3526.00i	0.250484i
$584$	11445.0	0.810955
$585$	0	0
$586$	−4158.00	−0.293115
$587$	− 11361.0i	− 0.798839i	−0.916768	−	0.399420i	$-0.869212\pi$
$587$	− 11361.0i	− 0.798839i	0.916768	−	0.399420i	$-0.130788\pi$
$588$	0	0
$589$	1470.00	0.102836
$590$	0	0
$591$	0	0
$592$	− 12874.0i	− 0.893781i
$593$	11417.0i	0.790624i	0.918547	+	0.395312i	$0.129364\pi$
$593$	11417.0i	0.790624i	−0.918547	+	0.395312i	$0.870636\pi$
$594$	0	0
$595$	0	0
$596$	−22400.0	−1.53950
$597$	0	0
$598$	4536.00i	0.310185i
$599$	−21050.0	−1.43586	−0.717930	−	0.696116i	$-0.754910\pi$
$599$	−21050.0	−1.43586	−0.717930	+	0.696116i	$0.754910\pi$
$600$	0	0
$601$	7427.00	0.504083	0.252041	−	0.967716i	$-0.418898\pi$
$601$	7427.00	0.504083	0.252041	+	0.967716i	$0.418898\pi$
$602$	− 552.000i	− 0.0373718i
$603$	0	0
$604$	1414.00	0.0952564
$605$	0	0
$606$	0	0
$607$	− 4144.00i	− 0.277100i	−0.990355	−	0.138550i	$-0.955756\pi$
$607$	− 4144.00i	− 0.277100i	0.990355	−	0.138550i	$-0.0442442\pi$
$608$	− 5635.00i	− 0.375871i
$609$	0	0
$610$	0	0
$611$	5488.00	0.363373
$612$	0	0
$613$	− 30122.0i	− 1.98469i	−0.123489	−	0.992346i	$-0.539408\pi$
$613$	− 30122.0i	− 1.98469i	0.123489	−	0.992346i	$-0.460592\pi$
$614$	−2569.00	−0.168854
$615$	0	0
$616$	3870.00	0.253128
$617$	11934.0i	0.778679i	0.921094	+	0.389339i	$0.127297\pi$
$617$	11934.0i	0.778679i	−0.921094	+	0.389339i	$0.872703\pi$
$618$	0	0
$619$	−8540.00	−0.554526	−0.277263	−	0.960794i	$-0.589427\pi$
$619$	−8540.00	−0.554526	−0.277263	+	0.960794i	$0.589427\pi$
$620$	0	0
$621$	0	0
$622$	2982.00i	0.192230i
$623$	− 5670.00i	− 0.364629i
$624$	0	0
$625$	0	0
$626$	−2422.00	−0.154637
$627$	0	0
$628$	2842.00i	0.180586i
$629$	−28574.0	−1.81132
$630$	0	0
$631$	−3158.00	−0.199236	−0.0996181	−	0.995026i	$-0.531762\pi$
$631$	−3158.00	−0.199236	−0.0996181	+	0.995026i	$0.531762\pi$
$632$	7650.00i	0.481488i
$633$	0	0
$634$	9484.00	0.594097
$635$	0	0
$636$	0	0
$637$	8596.00i	0.534672i
$638$	− 6880.00i	− 0.426931i
$639$	0	0
$640$	0	0
$641$	4278.00	0.263605	0.131803	−	0.991276i	$-0.457924\pi$
$641$	4278.00	0.263605	0.131803	+	0.991276i	$0.457924\pi$
$642$	0	0
$643$	11508.0i	0.705803i	0.935661	+	0.352901i	$0.114805\pi$
$643$	11508.0i	0.705803i	−0.935661	+	0.352901i	$0.885195\pi$
$644$	−6804.00	−0.416328
$645$	0	0
$646$	−3185.00	−0.193982
$647$	8204.00i	0.498505i	0.968439	+	0.249252i	$0.0801849\pi$
$647$	8204.00i	0.498505i	−0.968439	+	0.249252i	$0.919815\pi$
$648$	0	0
$649$	−12040.0	−0.728215
$650$	0	0
$651$	0	0
$652$	26621.0i	1.59902i
$653$	− 5518.00i	− 0.330683i	−0.986236	−	0.165342i	$-0.947127\pi$
$653$	− 5518.00i	− 0.330683i	0.986236	−	0.165342i	$-0.0528726\pi$
$654$	0	0
$655$	0	0
$656$	8323.00	0.495364
$657$	0	0
$658$	− 1176.00i	− 0.0696736i
$659$	13295.0	0.785887	0.392944	−	0.919563i	$-0.371457\pi$
$659$	13295.0	0.785887	0.392944	+	0.919563i	$0.371457\pi$
$660$	0	0
$661$	−9968.00	−0.586551	−0.293276	−	0.956028i	$-0.594745\pi$
$661$	−9968.00	−0.586551	−0.293276	+	0.956028i	$0.594745\pi$
$662$	183.000i	0.0107440i
$663$	0	0
$664$	11655.0	0.681177
$665$	0	0
$666$	0	0
$667$	25920.0i	1.50469i
$668$	− 28812.0i	− 1.66882i
$669$	0	0
$670$	0	0
$671$	−22274.0	−1.28149
$672$	0	0
$673$	15738.0i	0.901419i	0.892671	+	0.450710i	$0.148829\pi$
$673$	15738.0i	0.901419i	−0.892671	+	0.450710i	$0.851171\pi$
$674$	2861.00	0.163504
$675$	0	0
$676$	9891.00	0.562756
$677$	19824.0i	1.12540i	0.826660	+	0.562702i	$0.190238\pi$
$677$	19824.0i	1.12540i	−0.826660	+	0.562702i	$0.809762\pi$
$678$	0	0
$679$	−7476.00	−0.422537
$680$	0	0
$681$	0	0
$682$	− 1806.00i	− 0.101401i
$683$	− 11073.0i	− 0.620346i	−0.950680	−	0.310173i	$-0.899613\pi$
$683$	− 11073.0i	− 0.620346i	0.950680	−	0.310173i	$-0.100387\pi$
$684$	0	0
$685$	0	0
$686$	3900.00	0.217059
$687$	0	0
$688$	− 3772.00i	− 0.209021i
$689$	−2296.00	−0.126953
$690$	0	0
$691$	−6503.00	−0.358011	−0.179006	−	0.983848i	$-0.557288\pi$
$691$	−6503.00	−0.358011	−0.179006	+	0.983848i	$0.557288\pi$
$692$	10584.0i	0.581421i
$693$	0	0
$694$	629.000	0.0344042
$695$	0	0
$696$	0	0
$697$	− 18473.0i	− 1.00389i
$698$	5950.00i	0.322652i
$699$	0	0
$700$	0	0
$701$	10148.0	0.546768	0.273384	−	0.961905i	$-0.411857\pi$
$701$	10148.0	0.546768	0.273384	+	0.961905i	$0.411857\pi$
$702$	0	0
$703$	− 10990.0i	− 0.589610i
$704$	7181.00	0.384438
$705$	0	0
$706$	−11718.0	−0.624664
$707$	− 7812.00i	− 0.415559i
$708$	0	0
$709$	9980.00	0.528641	0.264321	−	0.964435i	$-0.414852\pi$
$709$	9980.00	0.528641	0.264321	+	0.964435i	$0.414852\pi$
$710$	0	0
$711$	0	0
$712$	14175.0i	0.746110i
$713$	6804.00i	0.357380i
$714$	0	0
$715$	0	0
$716$	18095.0	0.944472
$717$	0	0
$718$	− 8070.00i	− 0.419456i
$719$	−27510.0	−1.42691	−0.713456	−	0.700700i	$-0.752871\pi$
$719$	−27510.0	−1.42691	−0.713456	+	0.700700i	$0.752871\pi$
$720$	0	0
$721$	3192.00	0.164877
$722$	5634.00i	0.290410i
$723$	0	0
$724$	−19306.0	−0.991025
$725$	0	0
$726$	0	0
$727$	− 17024.0i	− 0.868480i	−0.900797	−	0.434240i	$-0.857017\pi$
$727$	− 17024.0i	− 0.868480i	0.900797	−	0.434240i	$-0.142983\pi$
$728$	2520.00i	0.128293i
$729$	0	0
$730$	0	0
$731$	−8372.00	−0.423597
$732$	0	0
$733$	34748.0i	1.75095i	0.483263	+	0.875475i	$0.339451\pi$
$733$	34748.0i	1.75095i	−0.483263	+	0.875475i	$0.660549\pi$
$734$	8316.00	0.418187
$735$	0	0
$736$	26082.0	1.30624
$737$	6063.00i	0.303030i
$738$	0	0
$739$	12020.0	0.598326	0.299163	−	0.954202i	$-0.403293\pi$
$739$	12020.0	0.598326	0.299163	+	0.954202i	$0.403293\pi$
$740$	0	0
$741$	0	0
$742$	492.000i	0.0243422i
$743$	28642.0i	1.41423i	0.707098	+	0.707115i	$0.250004\pi$
$743$	28642.0i	1.41423i	−0.707098	+	0.707115i	$0.749996\pi$
$744$	0	0
$745$	0	0
$746$	−12062.0	−0.591986
$747$	0	0
$748$	− 27391.0i	− 1.33892i
$749$	−7614.00	−0.371441
$750$	0	0
$751$	8752.00	0.425253	0.212627	−	0.977134i	$-0.431798\pi$
$751$	8752.00	0.425253	0.212627	+	0.977134i	$0.431798\pi$
$752$	− 8036.00i	− 0.389685i
$753$	0	0
$754$	4480.00	0.216382
$755$	0	0
$756$	0	0
$757$	10256.0i	0.492418i	0.969217	+	0.246209i	$0.0791850\pi$
$757$	10256.0i	0.492418i	−0.969217	+	0.246209i	$0.920815\pi$
$758$	1735.00i	0.0831373i
$759$	0	0
$760$	0	0
$761$	−33957.0	−1.61753	−0.808765	−	0.588132i	$-0.799864\pi$
$761$	−33957.0	−1.61753	−0.808765	+	0.588132i	$0.799864\pi$
$762$	0	0
$763$	− 6420.00i	− 0.304613i
$764$	16646.0	0.788261
$765$	0	0
$766$	7602.00	0.358579
$767$	− 7840.00i	− 0.369082i
$768$	0	0
$769$	−27965.0	−1.31137	−0.655685	−	0.755034i	$-0.727620\pi$
$769$	−27965.0	−1.31137	−0.655685	+	0.755034i	$0.727620\pi$
$770$	0	0
$771$	0	0
$772$	− 21469.0i	− 1.00089i
$773$	9912.00i	0.461203i	0.973048	+	0.230601i	$0.0740694\pi$
$773$	9912.00i	0.461203i	−0.973048	+	0.230601i	$0.925931\pi$
$774$	0	0
$775$	0	0
$776$	18690.0	0.864603
$777$	0	0
$778$	− 3030.00i	− 0.139628i
$779$	7105.00	0.326782
$780$	0	0
$781$	−17716.0	−0.811688
$782$	− 14742.0i	− 0.674134i
$783$	0	0
$784$	12587.0	0.573387
$785$	0	0
$786$	0	0
$787$	− 25564.0i	− 1.15789i	−0.815367	−	0.578944i	$-0.803465\pi$
$787$	− 25564.0i	− 1.15789i	0.815367	−	0.578944i	$-0.196535\pi$
$788$	− 16422.0i	− 0.742398i
$789$	0	0
$790$	0	0
$791$	3018.00	0.135661
$792$	0	0
$793$	− 14504.0i	− 0.649498i
$794$	−1204.00	−0.0538141
$795$	0	0
$796$	−34300.0	−1.52730
$797$	− 12446.0i	− 0.553149i	−0.960992	−	0.276575i	$-0.910801\pi$
$797$	− 12446.0i	− 0.553149i	0.960992	−	0.276575i	$-0.0891993\pi$
$798$	0	0
$799$	−17836.0	−0.789728
$800$	0	0
$801$	0	0
$802$	1077.00i	0.0474192i
$803$	32809.0i	1.44185i
$804$	0	0
$805$	0	0
$806$	1176.00	0.0513931
$807$	0	0
$808$	19530.0i	0.850325i
$809$	33970.0	1.47629	0.738147	−	0.674640i	$-0.235701\pi$
$809$	33970.0	1.47629	0.738147	+	0.674640i	$0.235701\pi$
$810$	0	0
$811$	18732.0	0.811060	0.405530	−	0.914082i	$-0.367087\pi$
$811$	18732.0	0.811060	0.405530	+	0.914082i	$0.367087\pi$
$812$	6720.00i	0.290426i
$813$	0	0
$814$	−13502.0	−0.581382
$815$	0	0
$816$	0	0
$817$	− 3220.00i	− 0.137887i
$818$	− 3955.00i	− 0.169051i
$819$	0	0
$820$	0	0
$821$	−6162.00	−0.261943	−0.130972	−	0.991386i	$-0.541810\pi$
$821$	−6162.00	−0.261943	−0.130972	+	0.991386i	$0.541810\pi$
$822$	0	0
$823$	25388.0i	1.07530i	0.843169	+	0.537649i	$0.180687\pi$
$823$	25388.0i	1.07530i	−0.843169	+	0.537649i	$0.819313\pi$
$824$	−7980.00	−0.337374
$825$	0	0
$826$	−1680.00	−0.0707684
$827$	− 25201.0i	− 1.05964i	−0.848109	−	0.529821i	$-0.822259\pi$
$827$	− 25201.0i	− 1.05964i	0.848109	−	0.529821i	$-0.177741\pi$
$828$	0	0
$829$	19740.0	0.827019	0.413509	−	0.910500i	$-0.364303\pi$
$829$	19740.0	0.827019	0.413509	+	0.910500i	$0.364303\pi$
$830$	0	0
$831$	0	0
$832$	4676.00i	0.194845i
$833$	− 27937.0i	− 1.16202i
$834$	0	0
$835$	0	0
$836$	10535.0	0.435838
$837$	0	0
$838$	− 6265.00i	− 0.258259i
$839$	29680.0	1.22130	0.610648	−	0.791902i	$-0.290909\pi$
$839$	29680.0	1.22130	0.610648	+	0.791902i	$0.290909\pi$
$840$	0	0
$841$	1211.00	0.0496535
$842$	3788.00i	0.155039i
$843$	0	0
$844$	30149.0	1.22959
$845$	0	0
$846$	0	0
$847$	3108.00i	0.126083i
$848$	3362.00i	0.136146i
$849$	0	0
$850$	0	0
$851$	50868.0	2.04904
$852$	0	0
$853$	1218.00i	0.0488904i	0.999701	+	0.0244452i	$0.00778193\pi$
$853$	1218.00i	0.0488904i	−0.999701	+	0.0244452i	$0.992218\pi$
$854$	−3108.00	−0.124536
$855$	0	0
$856$	19035.0	0.760050
$857$	− 38731.0i	− 1.54379i	−0.635752	−	0.771894i	$-0.719310\pi$
$857$	− 38731.0i	− 1.54379i	0.635752	−	0.771894i	$-0.280690\pi$
$858$	0	0
$859$	23555.0	0.935607	0.467803	−	0.883833i	$-0.345046\pi$
$859$	23555.0	0.935607	0.467803	+	0.883833i	$0.345046\pi$
$860$	0	0
$861$	0	0
$862$	− 15258.0i	− 0.602888i
$863$	24872.0i	0.981058i	0.871425	+	0.490529i	$0.163196\pi$
$863$	24872.0i	0.981058i	−0.871425	+	0.490529i	$0.836804\pi$
$864$	0	0
$865$	0	0
$866$	13573.0	0.532597
$867$	0	0
$868$	1764.00i	0.0689793i
$869$	−21930.0	−0.856069
$870$	0	0
$871$	−3948.00	−0.153585
$872$	16050.0i	0.623305i
$873$	0	0
$874$	5670.00	0.219440
$875$	0	0
$876$	0	0
$877$	− 17124.0i	− 0.659335i	−0.944097	−	0.329667i	$-0.893063\pi$
$877$	− 17124.0i	− 0.659335i	0.944097	−	0.329667i	$-0.106937\pi$
$878$	− 8120.00i	− 0.312115i
$879$	0	0
$880$	0	0
$881$	658.000	0.0251630	0.0125815	−	0.999921i	$-0.495995\pi$
$881$	658.000	0.0251630	0.0125815	+	0.999921i	$0.495995\pi$
$882$	0	0
$883$	− 33727.0i	− 1.28540i	−0.766120	−	0.642698i	$-0.777815\pi$
$883$	− 33727.0i	− 1.28540i	0.766120	−	0.642698i	$-0.222185\pi$
$884$	17836.0	0.678608
$885$	0	0
$886$	−6183.00	−0.234449
$887$	− 36036.0i	− 1.36412i	−0.731298	−	0.682058i	$-0.761085\pi$
$887$	− 36036.0i	− 1.36412i	0.731298	−	0.682058i	$-0.238915\pi$
$888$	0	0
$889$	5244.00	0.197838
$890$	0	0
$891$	0	0
$892$	− 15484.0i	− 0.581214i
$893$	− 6860.00i	− 0.257067i
$894$	0	0
$895$	0	0
$896$	8730.00	0.325501
$897$	0	0
$898$	1975.00i	0.0733927i
$899$	6720.00	0.249304
$900$	0	0
$901$	7462.00	0.275910
$902$	− 8729.00i	− 0.322222i
$903$	0	0
$904$	−7545.00	−0.277592
$905$	0	0
$906$	0	0
$907$	39156.0i	1.43347i	0.697348	+	0.716733i	$0.254363\pi$
$907$	39156.0i	1.43347i	−0.697348	+	0.716733i	$0.745637\pi$
$908$	− 3332.00i	− 0.121780i
$909$	0	0
$910$	0	0
$911$	−43532.0	−1.58318	−0.791591	−	0.611051i	$-0.790747\pi$
$911$	−43532.0	−1.58318	−0.791591	+	0.611051i	$0.790747\pi$
$912$	0	0
$913$	33411.0i	1.21111i
$914$	11831.0	0.428156
$915$	0	0
$916$	20580.0	0.742339
$917$	− 6552.00i	− 0.235950i
$918$	0	0
$919$	28610.0	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$919$	28610.0	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$920$	0	0
$921$	0	0
$922$	1932.00i	0.0690098i
$923$	− 11536.0i	− 0.411389i
$924$	0	0
$925$	0	0
$926$	9228.00	0.327485
$927$	0	0
$928$	− 25760.0i	− 0.911221i
$929$	−24290.0	−0.857835	−0.428918	−	0.903344i	$-0.641105\pi$
$929$	−24290.0	−0.857835	−0.428918	+	0.903344i	$0.641105\pi$
$930$	0	0
$931$	10745.0	0.378253
$932$	7014.00i	0.246514i
$933$	0	0
$934$	−13916.0	−0.487522
$935$	0	0
$936$	0	0
$937$	34461.0i	1.20149i	0.799442	+	0.600743i	$0.205128\pi$
$937$	34461.0i	1.20149i	−0.799442	+	0.600743i	$0.794872\pi$
$938$	846.000i	0.0294487i
$939$	0	0
$940$	0	0
$941$	40628.0	1.40748	0.703738	−	0.710460i	$-0.251513\pi$
$941$	40628.0	1.40748	0.703738	+	0.710460i	$0.251513\pi$
$942$	0	0
$943$	32886.0i	1.13565i
$944$	−11480.0	−0.395807
$945$	0	0
$946$	−3956.00	−0.135963
$947$	20904.0i	0.717306i	0.933471	+	0.358653i	$0.116764\pi$
$947$	20904.0i	0.717306i	−0.933471	+	0.358653i	$0.883236\pi$
$948$	0	0
$949$	−21364.0	−0.730774
$950$	0	0
$951$	0	0
$952$	− 8190.00i	− 0.278823i
$953$	1807.00i	0.0614213i	0.999528	+	0.0307106i	$0.00977704\pi$
$953$	1807.00i	0.0614213i	−0.999528	+	0.0307106i	$0.990223\pi$
$954$	0	0
$955$	0	0
$956$	17360.0	0.587304
$957$	0	0
$958$	− 2310.00i	− 0.0779047i
$959$	2466.00	0.0830358
$960$	0	0
$961$	−28027.0	−0.940787
$962$	− 8792.00i	− 0.294663i
$963$	0	0
$964$	13279.0	0.443660
$965$	0	0
$966$	0	0
$967$	− 57584.0i	− 1.91497i	−0.288482	−	0.957485i	$-0.593151\pi$
$967$	− 57584.0i	− 1.91497i	0.288482	−	0.957485i	$-0.406849\pi$
$968$	− 7770.00i	− 0.257993i
$969$	0	0
$970$	0	0
$971$	−27237.0	−0.900182	−0.450091	−	0.892983i	$-0.648608\pi$
$971$	−27237.0	−0.900182	−0.450091	+	0.892983i	$0.648608\pi$
$972$	0	0
$973$	3570.00i	0.117625i
$974$	−17114.0	−0.563006
$975$	0	0
$976$	−21238.0	−0.696528
$977$	13649.0i	0.446950i	0.974710	+	0.223475i	$0.0717401\pi$
$977$	13649.0i	0.446950i	−0.974710	+	0.223475i	$0.928260\pi$
$978$	0	0
$979$	−40635.0	−1.32656
$980$	0	0
$981$	0	0
$982$	− 17228.0i	− 0.559845i
$983$	16002.0i	0.519211i	0.965715	+	0.259606i	$0.0835925\pi$
$983$	16002.0i	0.519211i	−0.965715	+	0.259606i	$0.916407\pi$
$984$	0	0
$985$	0	0
$986$	−14560.0	−0.470269
$987$	0	0
$988$	6860.00i	0.220896i
$989$	14904.0	0.479191
$990$	0	0
$991$	37022.0	1.18672	0.593362	−	0.804936i	$-0.297800\pi$
$991$	37022.0	1.18672	0.593362	+	0.804936i	$0.297800\pi$
$992$	− 6762.00i	− 0.216425i
$993$	0	0
$994$	−2472.00	−0.0788804
$995$	0	0
$996$	0	0
$997$	18396.0i	0.584360i	0.956363	+	0.292180i	$0.0943807\pi$
$997$	18396.0i	0.584360i	−0.956363	+	0.292180i	$0.905619\pi$
$998$	− 12500.0i	− 0.396474i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.b.f.199.1		2
3.2	odd	2		25.4.b.b.24.2		2
5.2	odd	4		225.4.a.e.1.1		1
5.3	odd	4		225.4.a.c.1.1		1
5.4	even	2	inner	225.4.b.f.199.2		2
12.11	even	2		400.4.c.e.49.2		2
15.2	even	4		25.4.a.a.1.1	✓	1
15.8	even	4		25.4.a.b.1.1	yes	1
15.14	odd	2		25.4.b.b.24.1		2
60.23	odd	4		400.4.a.c.1.1		1
60.47	odd	4		400.4.a.s.1.1		1
60.59	even	2		400.4.c.e.49.1		2
105.62	odd	4		1225.4.a.h.1.1		1
105.83	odd	4		1225.4.a.i.1.1		1
120.53	even	4		1600.4.a.i.1.1		1
120.77	even	4		1600.4.a.bt.1.1		1
120.83	odd	4		1600.4.a.bs.1.1		1
120.107	odd	4		1600.4.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.4.a.a.1.1	✓	1	15.2	even	4
25.4.a.b.1.1	yes	1	15.8	even	4
25.4.b.b.24.1		2	15.14	odd	2
25.4.b.b.24.2		2	3.2	odd	2
225.4.a.c.1.1		1	5.3	odd	4
225.4.a.e.1.1		1	5.2	odd	4
225.4.b.f.199.1		2	1.1	even	1	trivial
225.4.b.f.199.2		2	5.4	even	2	inner
400.4.a.c.1.1		1	60.23	odd	4
400.4.a.s.1.1		1	60.47	odd	4
400.4.c.e.49.1		2	60.59	even	2
400.4.c.e.49.2		2	12.11	even	2
1225.4.a.h.1.1		1	105.62	odd	4
1225.4.a.i.1.1		1	105.83	odd	4
1600.4.a.h.1.1		1	120.107	odd	4
1600.4.a.i.1.1		1	120.53	even	4
1600.4.a.bs.1.1		1	120.83	odd	4
1600.4.a.bt.1.1		1	120.77	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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +7.00000 q^{4} +6.00000i q^{7} -15.0000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} +7.00000 q^{4} +6.00000i q^{7} -15.0000i q^{8} +43.0000 q^{11} +28.0000i q^{13} +6.00000 q^{14} +41.0000 q^{16} -91.0000i q^{17} +35.0000 q^{19} -43.0000i q^{22} +162.000i q^{23} +28.0000 q^{26} +42.0000i q^{28} +160.000 q^{29} +42.0000 q^{31} -161.000i q^{32} -91.0000 q^{34} -314.000i q^{37} -35.0000i q^{38} +203.000 q^{41} -92.0000i q^{43} +301.000 q^{44} +162.000 q^{46} -196.000i q^{47} +307.000 q^{49} +196.000i q^{52} +82.0000i q^{53} +90.0000 q^{56} -160.000i q^{58} -280.000 q^{59} -518.000 q^{61} -42.0000i q^{62} +167.000 q^{64} +141.000i q^{67} -637.000i q^{68} -412.000 q^{71} +763.000i q^{73} -314.000 q^{74} +245.000 q^{76} +258.000i q^{77} -510.000 q^{79} -203.000i q^{82} +777.000i q^{83} -92.0000 q^{86} -645.000i q^{88} -945.000 q^{89} -168.000 q^{91} +1134.00i q^{92} -196.000 q^{94} +1246.00i q^{97} -307.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{4}+O(q^{10}) \) 2 * q + 14 * q^4	\( 2 q + 14 q^{4} + 86 q^{11} + 12 q^{14} + 82 q^{16} + 70 q^{19} + 56 q^{26} + 320 q^{29} + 84 q^{31} - 182 q^{34} + 406 q^{41} + 602 q^{44} + 324 q^{46} + 614 q^{49} + 180 q^{56} - 560 q^{59} - 1036 q^{61} + 334 q^{64} - 824 q^{71} - 628 q^{74} + 490 q^{76} - 1020 q^{79} - 184 q^{86} - 1890 q^{89} - 336 q^{91} - 392 q^{94}+O(q^{100}) \) 2 * q + 14 * q^4 + 86 * q^11 + 12 * q^14 + 82 * q^16 + 70 * q^19 + 56 * q^26 + 320 * q^29 + 84 * q^31 - 182 * q^34 + 406 * q^41 + 602 * q^44 + 324 * q^46 + 614 * q^49 + 180 * q^56 - 560 * q^59 - 1036 * q^61 + 334 * q^64 - 824 * q^71 - 628 * q^74 + 490 * q^76 - 1020 * q^79 - 184 * q^86 - 1890 * q^89 - 336 * q^91 - 392 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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