LMFDB - Embedded newform 2205.2.b.d.881.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2205,2,Mod(881,2205)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2205, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2205.881"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2205.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,-16,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$17.6070136457$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 790 x^{12} - 2192 x^{11} + 5036 x^{10} - 9472 x^{9} + \cdots + 17 $ x^16 - 8x^15 + 52x^14 - 224x^13 + 790x^12 - 2192x^11 + 5036x^10 - 9472x^9 + 14669x^8 - 18520x^7 + 18744x^6 - 14872x^5 + 8956x^4 - 3912x^3 + 1160x^2 - 208*x + 17
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.12
Root			$0.500000 + 2.12518i$ of defining polynomial
Character	$\chi$	$=$	2205.881
Dual form			2205.2.b.d.881.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.69637i q^{2} -0.877672 q^{4} +1.00000 q^{5} +1.90388i q^{8} +1.69637i q^{10} -0.421924i q^{11} +2.17300i q^{13} -4.98504 q^{16} +5.88935 q^{17} -1.60310i q^{19} -0.877672 q^{20} +0.715740 q^{22} -0.156191i q^{23} +1.00000 q^{25} -3.68621 q^{26} +6.40873i q^{29} +8.33203i q^{31} -4.64870i q^{32} +9.99051i q^{34} -5.87346 q^{37} +2.71946 q^{38} +1.90388i q^{40} +0.916349 q^{41} +1.68638 q^{43} +0.370311i q^{44} +0.264958 q^{46} +6.12938 q^{47} +1.69637i q^{50} -1.90718i q^{52} -13.1297i q^{53} -0.421924i q^{55} -10.8716 q^{58} +8.44691 q^{59} +6.71530i q^{61} -14.1342 q^{62} -2.08416 q^{64} +2.17300i q^{65} -10.2055 q^{67} -5.16892 q^{68} +16.5279i q^{71} -0.804332i q^{73} -9.96356i q^{74} +1.40700i q^{76} +2.06547 q^{79} -4.98504 q^{80} +1.55447i q^{82} +4.20554 q^{83} +5.88935 q^{85} +2.86073i q^{86} +0.803295 q^{88} +0.722456 q^{89} +0.137085i q^{92} +10.3977i q^{94} -1.60310i q^{95} +14.4435i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4} + 16 q^{5} - 16 q^{20} - 16 q^{22} + 16 q^{25} + 32 q^{26} - 16 q^{38} - 32 q^{41} + 32 q^{43} - 16 q^{46} + 32 q^{47} + 48 q^{58} - 32 q^{59} - 32 q^{62} + 16 q^{64} - 16 q^{68} + 32 q^{79}+ \cdots + 64 q^{89}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^5 - 16 * q^20 - 16 * q^22 + 16 * q^25 + 32 * q^26 - 16 * q^38 - 32 * q^41 + 32 * q^43 - 16 * q^46 + 32 * q^47 + 48 * q^58 - 32 * q^59 - 32 * q^62 + 16 * q^64 - 16 * q^68 + 32 * q^79 + 16 * q^88 + 64 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2205\mathbb{Z}\right)^\times$.

$n$	$442$	$1081$	$1226$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.69637i	1.19951i	0.800182	+	0.599757i	$0.204736\pi$
$2$	1.69637i	1.19951i	−0.800182	+	0.599757i	$0.795264\pi$
$3$	0	0
$3$	0	0
$4$	−0.877672	−0.438836
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.90388i	0.673124i
$9$	0	0
$10$	1.69637i	0.536439i
$11$	− 0.421924i	− 0.127215i	−0.997975	−	0.0636075i	$-0.979739\pi$
$11$	− 0.421924i	− 0.127215i	0.997975	−	0.0636075i	$-0.0202606\pi$
$12$	0	0
$13$	2.17300i	0.602681i	0.953517	+	0.301341i	$0.0974341\pi$
$13$	2.17300i	0.602681i	−0.953517	+	0.301341i	$0.902566\pi$
$14$	0	0
$15$	0	0
$16$	−4.98504	−1.24626
$17$	5.88935	1.42838	0.714188	−	0.699954i	$-0.246796\pi$
$17$	5.88935	1.42838	0.714188	+	0.699954i	$0.246796\pi$
$18$	0	0
$19$	− 1.60310i	− 0.367777i	−0.982947	−	0.183889i	$-0.941131\pi$
$19$	− 1.60310i	− 0.367777i	0.982947	−	0.183889i	$-0.0588686\pi$
$20$	−0.877672	−0.196253
$21$	0	0
$22$	0.715740	0.152596
$23$	− 0.156191i	− 0.0325681i	−0.999867	−	0.0162841i	$-0.994816\pi$
$23$	− 0.156191i	− 0.0325681i	0.999867	−	0.0162841i	$-0.00518360\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−3.68621	−0.722925
$27$	0	0
$28$	0	0
$29$	6.40873i	1.19007i	0.803699	+	0.595036i	$0.202862\pi$
$29$	6.40873i	1.19007i	−0.803699	+	0.595036i	$0.797138\pi$
$30$	0	0
$31$	8.33203i	1.49648i	0.663430	+	0.748238i	$0.269100\pi$
$31$	8.33203i	1.49648i	−0.663430	+	0.748238i	$0.730900\pi$
$32$	− 4.64870i	− 0.821782i
$33$	0	0
$34$	9.99051i	1.71336i
$35$	0	0
$36$	0	0
$37$	−5.87346	−0.965591	−0.482795	−	0.875733i	$-0.660379\pi$
$37$	−5.87346	−0.965591	−0.482795	+	0.875733i	$0.660379\pi$
$38$	2.71946	0.441154
$39$	0	0
$40$	1.90388i	0.301030i
$41$	0.916349	0.143110	0.0715548	−	0.997437i	$-0.477204\pi$
$41$	0.916349	0.143110	0.0715548	+	0.997437i	$0.477204\pi$
$42$	0	0
$43$	1.68638	0.257171	0.128585	−	0.991698i	$-0.458956\pi$
$43$	1.68638	0.257171	0.128585	+	0.991698i	$0.458956\pi$
$44$	0.370311i	0.0558265i
$45$	0	0
$46$	0.264958	0.0390659
$47$	6.12938	0.894062	0.447031	−	0.894519i	$-0.352481\pi$
$47$	6.12938	0.894062	0.447031	+	0.894519i	$0.352481\pi$
$48$	0	0
$49$	0	0
$50$	1.69637i	0.239903i
$51$	0	0
$52$	− 1.90718i	− 0.264478i
$53$	− 13.1297i	− 1.80351i	−0.432252	−	0.901753i	$-0.642281\pi$
$53$	− 13.1297i	− 1.80351i	0.432252	−	0.901753i	$-0.357719\pi$
$54$	0	0
$55$	− 0.421924i	− 0.0568923i
$56$	0	0
$57$	0	0
$58$	−10.8716	−1.42751
$59$	8.44691	1.09969	0.549847	−	0.835265i	$-0.314686\pi$
$59$	8.44691	1.09969	0.549847	+	0.835265i	$0.314686\pi$
$60$	0	0
$61$	6.71530i	0.859806i	0.902875	+	0.429903i	$0.141452\pi$
$61$	6.71530i	0.859806i	−0.902875	+	0.429903i	$0.858548\pi$
$62$	−14.1342	−1.79505
$63$	0	0
$64$	−2.08416	−0.260519
$65$	2.17300i	0.269527i
$66$	0	0
$67$	−10.2055	−1.24680	−0.623402	−	0.781901i	$-0.714250\pi$
$67$	−10.2055	−1.24680	−0.623402	+	0.781901i	$0.714250\pi$
$68$	−5.16892	−0.626823
$69$	0	0
$70$	0	0
$71$	16.5279i	1.96150i	0.195274	+	0.980749i	$0.437440\pi$
$71$	16.5279i	1.96150i	−0.195274	+	0.980749i	$0.562560\pi$
$72$	0	0
$73$	− 0.804332i	− 0.0941399i	−0.998892	−	0.0470700i	$-0.985012\pi$
$73$	− 0.804332i	− 0.0941399i	0.998892	−	0.0470700i	$-0.0149884\pi$
$74$	− 9.96356i	− 1.15824i
$75$	0	0
$76$	1.40700i	0.161394i
$77$	0	0
$78$	0	0
$79$	2.06547	0.232383	0.116192	−	0.993227i	$-0.462931\pi$
$79$	2.06547	0.232383	0.116192	+	0.993227i	$0.462931\pi$
$80$	−4.98504	−0.557344
$81$	0	0
$82$	1.55447i	0.171662i
$83$	4.20554	0.461617	0.230809	−	0.972999i	$-0.425863\pi$
$83$	4.20554	0.461617	0.230809	+	0.972999i	$0.425863\pi$
$84$	0	0
$85$	5.88935	0.638789
$86$	2.86073i	0.308480i
$87$	0	0
$88$	0.803295	0.0856315
$89$	0.722456	0.0765802	0.0382901	−	0.999267i	$-0.487809\pi$
$89$	0.722456	0.0765802	0.0382901	+	0.999267i	$0.487809\pi$
$90$	0	0
$91$	0	0
$92$	0.137085i	0.0142921i
$93$	0	0
$94$	10.3977i	1.07244i
$95$	− 1.60310i	− 0.164475i
$96$	0	0
$97$	14.4435i	1.46652i	0.679951	+	0.733258i	$0.262001\pi$
$97$	14.4435i	1.46652i	−0.679951	+	0.733258i	$0.737999\pi$
$98$	0	0
$99$	0	0
$100$	−0.877672	−0.0877672
$101$	0.0839075	0.00834911	0.00417455	−	0.999991i	$-0.498671\pi$
$101$	0.0839075	0.00834911	0.00417455	+	0.999991i	$0.498671\pi$
$102$	0	0
$103$	9.56809i	0.942772i	0.881927	+	0.471386i	$0.156246\pi$
$103$	9.56809i	0.942772i	−0.881927	+	0.471386i	$0.843754\pi$
$104$	−4.13713	−0.405679
$105$	0	0
$106$	22.2729	2.16333
$107$	5.06681i	0.489827i	0.969545	+	0.244913i	$0.0787595\pi$
$107$	5.06681i	0.489827i	−0.969545	+	0.244913i	$0.921240\pi$
$108$	0	0
$109$	−0.705584	−0.0675827	−0.0337913	−	0.999429i	$-0.510758\pi$
$109$	−0.705584	−0.0675827	−0.0337913	+	0.999429i	$0.510758\pi$
$110$	0.715740	0.0682431
$111$	0	0
$112$	0	0
$113$	− 3.26734i	− 0.307366i	−0.988120	−	0.153683i	$-0.950887\pi$
$113$	− 3.26734i	− 0.307366i	0.988120	−	0.153683i	$-0.0491134\pi$
$114$	0	0
$115$	− 0.156191i	− 0.0145649i
$116$	− 5.62476i	− 0.522246i
$117$	0	0
$118$	14.3291i	1.31910i
$119$	0	0
$120$	0	0
$121$	10.8220	0.983816
$122$	−11.3916	−1.03135
$123$	0	0
$124$	− 7.31279i	− 0.656708i
$125$	1.00000	0.0894427
$126$	0	0
$127$	−20.4480	−1.81446	−0.907231	−	0.420632i	$-0.861808\pi$
$127$	−20.4480	−1.81446	−0.907231	+	0.420632i	$0.861808\pi$
$128$	− 12.8329i	− 1.13428i
$129$	0	0
$130$	−3.68621	−0.323302
$131$	−13.5344	−1.18250	−0.591252	−	0.806487i	$-0.701366\pi$
$131$	−13.5344	−1.18250	−0.591252	+	0.806487i	$0.701366\pi$
$132$	0	0
$133$	0	0
$134$	− 17.3124i	− 1.49556i
$135$	0	0
$136$	11.2126i	0.961475i
$137$	4.86423i	0.415579i	0.978174	+	0.207790i	$0.0666270\pi$
$137$	4.86423i	0.415579i	−0.978174	+	0.207790i	$0.933373\pi$
$138$	0	0
$139$	17.7105i	1.50218i	0.660198	+	0.751092i	$0.270472\pi$
$139$	17.7105i	1.50218i	−0.660198	+	0.751092i	$0.729528\pi$
$140$	0	0
$141$	0	0
$142$	−28.0374	−2.35285
$143$	0.916841	0.0766701
$144$	0	0
$145$	6.40873i	0.532216i
$146$	1.36444	0.112922
$147$	0	0
$148$	5.15497	0.423736
$149$	− 20.6400i	− 1.69089i	−0.534060	−	0.845447i	$-0.679334\pi$
$149$	− 20.6400i	− 1.69089i	0.534060	−	0.845447i	$-0.320666\pi$
$150$	0	0
$151$	−19.1135	−1.55543	−0.777716	−	0.628616i	$-0.783622\pi$
$151$	−19.1135	−1.55543	−0.777716	+	0.628616i	$0.783622\pi$
$152$	3.05212	0.247560
$153$	0	0
$154$	0	0
$155$	8.33203i	0.669245i
$156$	0	0
$157$	− 21.1807i	− 1.69040i	−0.534450	−	0.845200i	$-0.679481\pi$
$157$	− 21.1807i	− 1.69040i	0.534450	−	0.845200i	$-0.320519\pi$
$158$	3.50380i	0.278747i
$159$	0	0
$160$	− 4.64870i	− 0.367512i
$161$	0	0
$162$	0	0
$163$	18.7515	1.46873	0.734367	−	0.678753i	$-0.237479\pi$
$163$	18.7515	1.46873	0.734367	+	0.678753i	$0.237479\pi$
$164$	−0.804254	−0.0628017
$165$	0	0
$166$	7.13415i	0.553717i
$167$	−18.6004	−1.43934	−0.719672	−	0.694314i	$-0.755708\pi$
$167$	−18.6004	−1.43934	−0.719672	+	0.694314i	$0.755708\pi$
$168$	0	0
$169$	8.27808	0.636775
$170$	9.99051i	0.766237i
$171$	0	0
$172$	−1.48009	−0.112856
$173$	18.2051	1.38411	0.692054	−	0.721845i	$-0.256706\pi$
$173$	18.2051	1.38411	0.692054	+	0.721845i	$0.256706\pi$
$174$	0	0
$175$	0	0
$176$	2.10331i	0.158543i
$177$	0	0
$178$	1.22555i	0.0918591i
$179$	5.05092i	0.377524i	0.982023	+	0.188762i	$0.0604474\pi$
$179$	5.05092i	0.377524i	−0.982023	+	0.188762i	$0.939553\pi$
$180$	0	0
$181$	− 25.2795i	− 1.87901i	−0.342534	−	0.939505i	$-0.611285\pi$
$181$	− 25.2795i	− 1.87901i	0.342534	−	0.939505i	$-0.388715\pi$
$182$	0	0
$183$	0	0
$184$	0.297370	0.0219224
$185$	−5.87346	−0.431825
$186$	0	0
$187$	− 2.48486i	− 0.181711i
$188$	−5.37959	−0.392347
$189$	0	0
$190$	2.71946	0.197290
$191$	− 3.60523i	− 0.260865i	−0.991457	−	0.130433i	$-0.958363\pi$
$191$	− 3.60523i	− 0.260865i	0.991457	−	0.130433i	$-0.0416366\pi$
$192$	0	0
$193$	−16.3036	−1.17356	−0.586781	−	0.809746i	$-0.699605\pi$
$193$	−16.3036	−1.17356	−0.586781	+	0.809746i	$0.699605\pi$
$194$	−24.5015	−1.75911
$195$	0	0
$196$	0	0
$197$	− 1.96777i	− 0.140198i	−0.997540	−	0.0700989i	$-0.977668\pi$
$197$	− 1.96777i	− 0.140198i	0.997540	−	0.0700989i	$-0.0223315\pi$
$198$	0	0
$199$	− 19.7989i	− 1.40351i	−0.712420	−	0.701753i	$-0.752401\pi$
$199$	− 19.7989i	− 1.40351i	0.712420	−	0.701753i	$-0.247599\pi$
$200$	1.90388i	0.134625i
$201$	0	0
$202$	0.142338i	0.0100149i
$203$	0	0
$204$	0	0
$205$	0.916349	0.0640006
$206$	−16.2310	−1.13087
$207$	0	0
$208$	− 10.8325i	− 0.751097i
$209$	−0.676389	−0.0467868
$210$	0	0
$211$	22.9079	1.57705	0.788524	−	0.615004i	$-0.210846\pi$
$211$	22.9079	1.57705	0.788524	+	0.615004i	$0.210846\pi$
$212$	11.5236i	0.791444i
$213$	0	0
$214$	−8.59518	−0.587554
$215$	1.68638	0.115010
$216$	0	0
$217$	0	0
$218$	− 1.19693i	− 0.0810664i
$219$	0	0
$220$	0.370311i	0.0249664i
$221$	12.7975i	0.860855i
$222$	0	0
$223$	4.50216i	0.301487i	0.988573	+	0.150744i	$0.0481668\pi$
$223$	4.50216i	0.301487i	−0.988573	+	0.150744i	$0.951833\pi$
$224$	0	0
$225$	0	0
$226$	5.54262	0.368690
$227$	1.28684	0.0854107	0.0427053	−	0.999088i	$-0.486402\pi$
$227$	1.28684	0.0854107	0.0427053	+	0.999088i	$0.486402\pi$
$228$	0	0
$229$	− 6.87038i	− 0.454007i	−0.973894	−	0.227004i	$-0.927107\pi$
$229$	− 6.87038i	− 0.454007i	0.973894	−	0.227004i	$-0.0728929\pi$
$230$	0.264958	0.0174708
$231$	0	0
$232$	−12.2015	−0.801066
$233$	− 17.1479i	− 1.12340i	−0.827342	−	0.561698i	$-0.810148\pi$
$233$	− 17.1479i	− 1.12340i	0.827342	−	0.561698i	$-0.189852\pi$
$234$	0	0
$235$	6.12938	0.399837
$236$	−7.41362	−0.482585
$237$	0	0
$238$	0	0
$239$	− 10.7501i	− 0.695367i	−0.937612	−	0.347684i	$-0.886968\pi$
$239$	− 10.7501i	− 0.695367i	0.937612	−	0.347684i	$-0.113032\pi$
$240$	0	0
$241$	7.36348i	0.474323i	0.971470	+	0.237162i	$0.0762171\pi$
$241$	7.36348i	0.474323i	−0.971470	+	0.237162i	$0.923783\pi$
$242$	18.3581i	1.18010i
$243$	0	0
$244$	− 5.89383i	− 0.377314i
$245$	0	0
$246$	0	0
$247$	3.48354	0.221652
$248$	−15.8632	−1.00731
$249$	0	0
$250$	1.69637i	0.107288i
$251$	16.1171	1.01730	0.508651	−	0.860973i	$-0.330144\pi$
$251$	16.1171	1.01730	0.508651	+	0.860973i	$0.330144\pi$
$252$	0	0
$253$	−0.0659009	−0.00414315
$254$	− 34.6873i	− 2.17648i
$255$	0	0
$256$	17.6010	1.10006
$257$	25.0530	1.56277	0.781383	−	0.624052i	$-0.214515\pi$
$257$	25.0530	1.56277	0.781383	+	0.624052i	$0.214515\pi$
$258$	0	0
$259$	0	0
$260$	− 1.90718i	− 0.118278i
$261$	0	0
$262$	− 22.9593i	− 1.41843i
$263$	3.19594i	0.197070i	0.995134	+	0.0985352i	$0.0314157\pi$
$263$	3.19594i	0.197070i	−0.995134	+	0.0985352i	$0.968584\pi$
$264$	0	0
$265$	− 13.1297i	− 0.806552i
$266$	0	0
$267$	0	0
$268$	8.95711	0.547143
$269$	18.6820	1.13906	0.569532	−	0.821969i	$-0.307125\pi$
$269$	18.6820	1.13906	0.569532	+	0.821969i	$0.307125\pi$
$270$	0	0
$271$	1.71027i	0.103892i	0.998650	+	0.0519458i	$0.0165423\pi$
$271$	1.71027i	0.103892i	−0.998650	+	0.0519458i	$0.983458\pi$
$272$	−29.3586	−1.78013
$273$	0	0
$274$	−8.25154	−0.498494
$275$	− 0.421924i	− 0.0254430i
$276$	0	0
$277$	−19.4613	−1.16932	−0.584658	−	0.811280i	$-0.698771\pi$
$277$	−19.4613	−1.16932	−0.584658	+	0.811280i	$0.698771\pi$
$278$	−30.0435	−1.80189
$279$	0	0
$280$	0	0
$281$	26.5135i	1.58166i	0.612036	+	0.790830i	$0.290351\pi$
$281$	26.5135i	1.58166i	−0.612036	+	0.790830i	$0.709649\pi$
$282$	0	0
$283$	− 2.82100i	− 0.167691i	−0.996479	−	0.0838456i	$-0.973280\pi$
$283$	− 2.82100i	− 0.167691i	0.996479	−	0.0838456i	$-0.0267202\pi$
$284$	− 14.5061i	− 0.860776i
$285$	0	0
$286$	1.55530i	0.0919669i
$287$	0	0
$288$	0	0
$289$	17.6844	1.04026
$290$	−10.8716	−0.638401
$291$	0	0
$292$	0.705940i	0.0413120i
$293$	0.356494	0.0208266	0.0104133	−	0.999946i	$-0.496685\pi$
$293$	0.356494	0.0208266	0.0104133	+	0.999946i	$0.496685\pi$
$294$	0	0
$295$	8.44691	0.491798
$296$	− 11.1824i	− 0.649963i
$297$	0	0
$298$	35.0131	2.02825
$299$	0.339403	0.0196282
$300$	0	0
$301$	0	0
$302$	− 32.4235i	− 1.86576i
$303$	0	0
$304$	7.99153i	0.458346i
$305$	6.71530i	0.384517i
$306$	0	0
$307$	− 26.4987i	− 1.51236i	−0.654365	−	0.756179i	$-0.727064\pi$
$307$	− 26.4987i	− 1.51236i	0.654365	−	0.756179i	$-0.272936\pi$
$308$	0	0
$309$	0	0
$310$	−14.1342	−0.802769
$311$	−28.1716	−1.59747	−0.798733	−	0.601686i	$-0.794496\pi$
$311$	−28.1716	−1.59747	−0.798733	+	0.601686i	$0.794496\pi$
$312$	0	0
$313$	− 22.2111i	− 1.25544i	−0.778438	−	0.627722i	$-0.783988\pi$
$313$	− 22.2111i	− 1.25544i	0.778438	−	0.627722i	$-0.216012\pi$
$314$	35.9302	2.02766
$315$	0	0
$316$	−1.81280	−0.101978
$317$	7.45636i	0.418791i	0.977831	+	0.209396i	$0.0671496\pi$
$317$	7.45636i	0.418791i	−0.977831	+	0.209396i	$0.932850\pi$
$318$	0	0
$319$	2.70400	0.151395
$320$	−2.08416	−0.116508
$321$	0	0
$322$	0	0
$323$	− 9.44123i	− 0.525324i
$324$	0	0
$325$	2.17300i	0.120536i
$326$	31.8095i	1.76177i
$327$	0	0
$328$	1.74462i	0.0963306i
$329$	0	0
$330$	0	0
$331$	16.0494	0.882157	0.441079	−	0.897468i	$-0.354596\pi$
$331$	16.0494	0.882157	0.441079	+	0.897468i	$0.354596\pi$
$332$	−3.69108	−0.202574
$333$	0	0
$334$	− 31.5532i	− 1.72651i
$335$	−10.2055	−0.557588
$336$	0	0
$337$	3.87759	0.211226	0.105613	−	0.994407i	$-0.466320\pi$
$337$	3.87759	0.211226	0.105613	+	0.994407i	$0.466320\pi$
$338$	14.0427i	0.763822i
$339$	0	0
$340$	−5.16892	−0.280324
$341$	3.51549	0.190374
$342$	0	0
$343$	0	0
$344$	3.21068i	0.173108i
$345$	0	0
$346$	30.8826i	1.66026i
$347$	− 0.312664i	− 0.0167847i	−0.999965	−	0.00839234i	$-0.997329\pi$
$347$	− 0.312664i	− 0.0167847i	0.999965	−	0.00839234i	$-0.00267139\pi$
$348$	0	0
$349$	10.7227i	0.573973i	0.957935	+	0.286987i	$0.0926535\pi$
$349$	10.7227i	0.573973i	−0.957935	+	0.286987i	$0.907346\pi$
$350$	0	0
$351$	0	0
$352$	−1.96140	−0.104543
$353$	20.0165	1.06537	0.532684	−	0.846314i	$-0.321183\pi$
$353$	20.0165	1.06537	0.532684	+	0.846314i	$0.321183\pi$
$354$	0	0
$355$	16.5279i	0.877208i
$356$	−0.634080	−0.0336062
$357$	0	0
$358$	−8.56824	−0.452845
$359$	11.1677i	0.589411i	0.955588	+	0.294706i	$0.0952216\pi$
$359$	11.1677i	0.589411i	−0.955588	+	0.294706i	$0.904778\pi$
$360$	0	0
$361$	16.4301	0.864740
$362$	42.8834	2.25390
$363$	0	0
$364$	0	0
$365$	− 0.804332i	− 0.0421007i
$366$	0	0
$367$	− 32.4067i	− 1.69162i	−0.533488	−	0.845808i	$-0.679119\pi$
$367$	− 32.4067i	− 1.69162i	0.533488	−	0.845808i	$-0.320881\pi$
$368$	0.778619i	0.0405883i
$369$	0	0
$370$	− 9.96356i	− 0.517981i
$371$	0	0
$372$	0	0
$373$	3.62970	0.187939	0.0939695	−	0.995575i	$-0.470044\pi$
$373$	3.62970	0.187939	0.0939695	+	0.995575i	$0.470044\pi$
$374$	4.21524	0.217965
$375$	0	0
$376$	11.6696i	0.601815i
$377$	−13.9262	−0.717233
$378$	0	0
$379$	−27.9146	−1.43387	−0.716937	−	0.697138i	$-0.754457\pi$
$379$	−27.9146	−1.43387	−0.716937	+	0.697138i	$0.754457\pi$
$380$	1.40700i	0.0721775i
$381$	0	0
$382$	6.11581	0.312912
$383$	10.5449	0.538820	0.269410	−	0.963026i	$-0.413171\pi$
$383$	10.5449	0.538820	0.269410	+	0.963026i	$0.413171\pi$
$384$	0	0
$385$	0	0
$386$	− 27.6570i	− 1.40770i
$387$	0	0
$388$	− 12.6767i	− 0.643560i
$389$	11.2605i	0.570929i	0.958389	+	0.285465i	$0.0921479\pi$
$389$	11.2605i	0.570929i	−0.958389	+	0.285465i	$0.907852\pi$
$390$	0	0
$391$	− 0.919864i	− 0.0465195i
$392$	0	0
$393$	0	0
$394$	3.33807	0.168169
$395$	2.06547	0.103925
$396$	0	0
$397$	− 1.15486i	− 0.0579608i	−0.999580	−	0.0289804i	$-0.990774\pi$
$397$	− 1.15486i	− 0.0579608i	0.999580	−	0.0289804i	$-0.00922604\pi$
$398$	33.5863	1.68353
$399$	0	0
$400$	−4.98504	−0.249252
$401$	24.7946i	1.23818i	0.785319	+	0.619091i	$0.212499\pi$
$401$	24.7946i	1.23818i	−0.785319	+	0.619091i	$0.787501\pi$
$402$	0	0
$403$	−18.1055	−0.901898
$404$	−0.0736433	−0.00366389
$405$	0	0
$406$	0	0
$407$	2.47816i	0.122838i
$408$	0	0
$409$	− 27.9661i	− 1.38283i	−0.722456	−	0.691417i	$-0.756987\pi$
$409$	− 27.9661i	− 1.38283i	0.722456	−	0.691417i	$-0.243013\pi$
$410$	1.55447i	0.0767697i
$411$	0	0
$412$	− 8.39765i	− 0.413722i
$413$	0	0
$414$	0	0
$415$	4.20554	0.206442
$416$	10.1016	0.495272
$417$	0	0
$418$	− 1.14741i	− 0.0561214i
$419$	33.9771	1.65989	0.829945	−	0.557845i	$-0.188371\pi$
$419$	33.9771	1.65989	0.829945	+	0.557845i	$0.188371\pi$
$420$	0	0
$421$	8.14359	0.396895	0.198447	−	0.980112i	$-0.436410\pi$
$421$	8.14359	0.396895	0.198447	+	0.980112i	$0.436410\pi$
$422$	38.8603i	1.89169i
$423$	0	0
$424$	24.9975	1.21398
$425$	5.88935	0.285675
$426$	0	0
$427$	0	0
$428$	− 4.44700i	− 0.214954i
$429$	0	0
$430$	2.86073i	0.137957i
$431$	− 33.1596i	− 1.59724i	−0.601834	−	0.798621i	$-0.705563\pi$
$431$	− 33.1596i	− 1.59724i	0.601834	−	0.798621i	$-0.294437\pi$
$432$	0	0
$433$	− 26.5417i	− 1.27551i	−0.770239	−	0.637756i	$-0.779863\pi$
$433$	− 26.5417i	− 1.27551i	0.770239	−	0.637756i	$-0.220137\pi$
$434$	0	0
$435$	0	0
$436$	0.619271	0.0296577
$437$	−0.250391	−0.0119778
$438$	0	0
$439$	24.3693i	1.16308i	0.813517	+	0.581541i	$0.197550\pi$
$439$	24.3693i	1.16308i	−0.813517	+	0.581541i	$0.802450\pi$
$440$	0.803295	0.0382956
$441$	0	0
$442$	−21.7094	−1.03261
$443$	− 3.49881i	− 0.166233i	−0.996540	−	0.0831167i	$-0.973513\pi$
$443$	− 3.49881i	− 0.166233i	0.996540	−	0.0831167i	$-0.0264874\pi$
$444$	0	0
$445$	0.722456	0.0342477
$446$	−7.63733	−0.361638
$447$	0	0
$448$	0	0
$449$	4.58599i	0.216426i	0.994128	+	0.108213i	$0.0345129\pi$
$449$	4.58599i	0.216426i	−0.994128	+	0.108213i	$0.965487\pi$
$450$	0	0
$451$	− 0.386630i	− 0.0182057i
$452$	2.86766i	0.134883i
$453$	0	0
$454$	2.18296i	0.102451i
$455$	0	0
$456$	0	0
$457$	−21.3080	−0.996746	−0.498373	−	0.866963i	$-0.666069\pi$
$457$	−21.3080	−0.996746	−0.498373	+	0.866963i	$0.666069\pi$
$458$	11.6547	0.544589
$459$	0	0
$460$	0.137085i	0.00639160i
$461$	39.4276	1.83633	0.918164	−	0.396201i	$-0.129672\pi$
$461$	39.4276	1.83633	0.918164	+	0.396201i	$0.129672\pi$
$462$	0	0
$463$	−9.87407	−0.458887	−0.229443	−	0.973322i	$-0.573691\pi$
$463$	−9.87407	−0.458887	−0.229443	+	0.973322i	$0.573691\pi$
$464$	− 31.9477i	− 1.48314i
$465$	0	0
$466$	29.0892	1.34753
$467$	2.71056	0.125430	0.0627148	−	0.998031i	$-0.480024\pi$
$467$	2.71056	0.125430	0.0627148	+	0.998031i	$0.480024\pi$
$468$	0	0
$469$	0	0
$470$	10.3977i	0.479610i
$471$	0	0
$472$	16.0819i	0.740231i
$473$	− 0.711526i	− 0.0327160i
$474$	0	0
$475$	− 1.60310i	− 0.0735554i
$476$	0	0
$477$	0	0
$478$	18.2362	0.834103
$479$	29.7608	1.35980	0.679902	−	0.733303i	$-0.262022\pi$
$479$	29.7608	1.35980	0.679902	+	0.733303i	$0.262022\pi$
$480$	0	0
$481$	− 12.7630i	− 0.581943i
$482$	−12.4912	−0.568958
$483$	0	0
$484$	−9.49815	−0.431734
$485$	14.4435i	0.655846i
$486$	0	0
$487$	12.0256	0.544931	0.272465	−	0.962166i	$-0.412161\pi$
$487$	12.0256	0.544931	0.272465	+	0.962166i	$0.412161\pi$
$488$	−12.7851	−0.578756
$489$	0	0
$490$	0	0
$491$	− 0.923603i	− 0.0416816i	−0.999783	−	0.0208408i	$-0.993366\pi$
$491$	− 0.923603i	− 0.0416816i	0.999783	−	0.0208408i	$-0.00663431\pi$
$492$	0	0
$493$	37.7432i	1.69987i
$494$	5.90937i	0.265875i
$495$	0	0
$496$	− 41.5355i	− 1.86500i
$497$	0	0
$498$	0	0
$499$	30.7766	1.37775	0.688876	−	0.724880i	$-0.258105\pi$
$499$	30.7766	1.37775	0.688876	+	0.724880i	$0.258105\pi$
$500$	−0.877672	−0.0392507
$501$	0	0
$502$	27.3406i	1.22027i
$503$	6.28875	0.280401	0.140201	−	0.990123i	$-0.455225\pi$
$503$	6.28875	0.280401	0.140201	+	0.990123i	$0.455225\pi$
$504$	0	0
$505$	0.0839075	0.00373384
$506$	− 0.111792i	− 0.00496977i
$507$	0	0
$508$	17.9466	0.796252
$509$	−36.5692	−1.62090	−0.810451	−	0.585806i	$-0.800778\pi$
$509$	−36.5692	−1.62090	−0.810451	+	0.585806i	$0.800778\pi$
$510$	0	0
$511$	0	0
$512$	4.19208i	0.185266i
$513$	0	0
$514$	42.4992i	1.87456i
$515$	9.56809i	0.421621i
$516$	0	0
$517$	− 2.58613i	− 0.113738i
$518$	0	0
$519$	0	0
$520$	−4.13713	−0.181425
$521$	4.94611	0.216693	0.108347	−	0.994113i	$-0.465444\pi$
$521$	4.94611	0.216693	0.108347	+	0.994113i	$0.465444\pi$
$522$	0	0
$523$	− 12.2773i	− 0.536850i	−0.963301	−	0.268425i	$-0.913497\pi$
$523$	− 12.2773i	− 0.536850i	0.963301	−	0.268425i	$-0.0865032\pi$
$524$	11.8788	0.518926
$525$	0	0
$526$	−5.42150	−0.236389
$527$	49.0702i	2.13753i
$528$	0	0
$529$	22.9756	0.998939
$530$	22.2729	0.967472
$531$	0	0
$532$	0	0
$533$	1.99122i	0.0862495i
$534$	0	0
$535$	5.06681i	0.219057i
$536$	− 19.4301i	− 0.839254i
$537$	0	0
$538$	31.6916i	1.36632i
$539$	0	0
$540$	0	0
$541$	0.589732	0.0253546	0.0126773	−	0.999920i	$-0.495965\pi$
$541$	0.589732	0.0253546	0.0126773	+	0.999920i	$0.495965\pi$
$542$	−2.90126	−0.124620
$543$	0	0
$544$	− 27.3778i	− 1.17381i
$545$	−0.705584	−0.0302239
$546$	0	0
$547$	−35.3759	−1.51257	−0.756283	−	0.654245i	$-0.772987\pi$
$547$	−35.3759	−1.51257	−0.756283	+	0.654245i	$0.772987\pi$
$548$	− 4.26920i	− 0.182371i
$549$	0	0
$550$	0.715740	0.0305193
$551$	10.2739	0.437681
$552$	0	0
$553$	0	0
$554$	− 33.0135i	− 1.40261i
$555$	0	0
$556$	− 15.5440i	− 0.659212i
$557$	− 10.7404i	− 0.455086i	−0.973768	−	0.227543i	$-0.926931\pi$
$557$	− 10.7404i	− 0.455086i	0.973768	−	0.227543i	$-0.0730692\pi$
$558$	0	0
$559$	3.66451i	0.154992i
$560$	0	0
$561$	0	0
$562$	−44.9766	−1.89722
$563$	20.7643	0.875110	0.437555	−	0.899192i	$-0.355845\pi$
$563$	20.7643	0.875110	0.437555	+	0.899192i	$0.355845\pi$
$564$	0	0
$565$	− 3.26734i	− 0.137458i
$566$	4.78546	0.201148
$567$	0	0
$568$	−31.4671	−1.32033
$569$	− 0.616823i	− 0.0258586i	−0.999916	−	0.0129293i	$-0.995884\pi$
$569$	− 0.616823i	− 0.0258586i	0.999916	−	0.0129293i	$-0.00411563\pi$
$570$	0	0
$571$	−12.2339	−0.511971	−0.255986	−	0.966681i	$-0.582400\pi$
$571$	−12.2339	−0.511971	−0.255986	+	0.966681i	$0.582400\pi$
$572$	−0.804686	−0.0336456
$573$	0	0
$574$	0	0
$575$	− 0.156191i	− 0.00651362i
$576$	0	0
$577$	− 2.02832i	− 0.0844402i	−0.999108	−	0.0422201i	$-0.986557\pi$
$577$	− 2.02832i	− 0.0844402i	0.999108	−	0.0422201i	$-0.0134431\pi$
$578$	29.9993i	1.24781i
$579$	0	0
$580$	− 5.62476i	− 0.233556i
$581$	0	0
$582$	0	0
$583$	−5.53975	−0.229433
$584$	1.53135	0.0633679
$585$	0	0
$586$	0.604746i	0.0249818i
$587$	22.6594	0.935253	0.467627	−	0.883926i	$-0.345109\pi$
$587$	22.6594	0.935253	0.467627	+	0.883926i	$0.345109\pi$
$588$	0	0
$589$	13.3571	0.550370
$590$	14.3291i	0.589919i
$591$	0	0
$592$	29.2794	1.20338
$593$	−10.1329	−0.416110	−0.208055	−	0.978117i	$-0.566713\pi$
$593$	−10.1329	−0.416110	−0.208055	+	0.978117i	$0.566713\pi$
$594$	0	0
$595$	0	0
$596$	18.1151i	0.742025i
$597$	0	0
$598$	0.575753i	0.0235443i
$599$	− 33.0325i	− 1.34967i	−0.737969	−	0.674835i	$-0.764215\pi$
$599$	− 33.0325i	− 1.34967i	0.737969	−	0.674835i	$-0.235785\pi$
$600$	0	0
$601$	1.82019i	0.0742472i	0.999311	+	0.0371236i	$0.0118195\pi$
$601$	1.82019i	0.0742472i	−0.999311	+	0.0371236i	$0.988180\pi$
$602$	0	0
$603$	0	0
$604$	16.7754	0.682580
$605$	10.8220	0.439976
$606$	0	0
$607$	6.79608i	0.275844i	0.990443	+	0.137922i	$0.0440424\pi$
$607$	6.79608i	0.275844i	−0.990443	+	0.137922i	$0.955958\pi$
$608$	−7.45235	−0.302233
$609$	0	0
$610$	−11.3916	−0.461234
$611$	13.3191i	0.538834i
$612$	0	0
$613$	23.0064	0.929219	0.464609	−	0.885516i	$-0.346195\pi$
$613$	23.0064	0.929219	0.464609	+	0.885516i	$0.346195\pi$
$614$	44.9515	1.81410
$615$	0	0
$616$	0	0
$617$	15.5535i	0.626162i	0.949726	+	0.313081i	$0.101361\pi$
$617$	15.5535i	0.626162i	−0.949726	+	0.313081i	$0.898639\pi$
$618$	0	0
$619$	26.4304i	1.06233i	0.847269	+	0.531164i	$0.178245\pi$
$619$	26.4304i	1.06233i	−0.847269	+	0.531164i	$0.821755\pi$
$620$	− 7.31279i	− 0.293689i
$621$	0	0
$622$	− 47.7895i	− 1.91618i
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	37.6782	1.50592
$627$	0	0
$628$	18.5897i	0.741809i
$629$	−34.5908	−1.37923
$630$	0	0
$631$	−29.6661	−1.18099	−0.590494	−	0.807042i	$-0.701067\pi$
$631$	−29.6661	−1.18099	−0.590494	+	0.807042i	$0.701067\pi$
$632$	3.93241i	0.156423i
$633$	0	0
$634$	−12.6488	−0.502346
$635$	−20.4480	−0.811452
$636$	0	0
$637$	0	0
$638$	4.58698i	0.181600i
$639$	0	0
$640$	− 12.8329i	− 0.507265i
$641$	− 3.08663i	− 0.121915i	−0.998140	−	0.0609573i	$-0.980585\pi$
$641$	− 3.08663i	− 0.121915i	0.998140	−	0.0609573i	$-0.0194154\pi$
$642$	0	0
$643$	− 31.9782i	− 1.26110i	−0.776150	−	0.630549i	$-0.782830\pi$
$643$	− 31.9782i	− 1.26110i	0.776150	−	0.630549i	$-0.217170\pi$
$644$	0	0
$645$	0	0
$646$	16.0158	0.630134
$647$	−22.9939	−0.903982	−0.451991	−	0.892022i	$-0.649286\pi$
$647$	−22.9939	−0.903982	−0.451991	+	0.892022i	$0.649286\pi$
$648$	0	0
$649$	− 3.56396i	− 0.139898i
$650$	−3.68621	−0.144585
$651$	0	0
$652$	−16.4577	−0.644533
$653$	30.9691i	1.21191i	0.795497	+	0.605957i	$0.207210\pi$
$653$	30.9691i	1.21191i	−0.795497	+	0.605957i	$0.792790\pi$
$654$	0	0
$655$	−13.5344	−0.528832
$656$	−4.56803	−0.178352
$657$	0	0
$658$	0	0
$659$	11.6761i	0.454835i	0.973797	+	0.227417i	$0.0730281\pi$
$659$	11.6761i	0.454835i	−0.973797	+	0.227417i	$0.926972\pi$
$660$	0	0
$661$	− 32.8962i	− 1.27951i	−0.768578	−	0.639757i	$-0.779035\pi$
$661$	− 32.8962i	− 1.27951i	0.768578	−	0.639757i	$-0.220965\pi$
$662$	27.2258i	1.05816i
$663$	0	0
$664$	8.00685i	0.310726i
$665$	0	0
$666$	0	0
$667$	1.00099	0.0387584
$668$	16.3251	0.631636
$669$	0	0
$670$	− 17.3124i	− 0.668835i
$671$	2.83335	0.109380
$672$	0	0
$673$	40.9627	1.57900	0.789498	−	0.613753i	$-0.210341\pi$
$673$	40.9627	1.57900	0.789498	+	0.613753i	$0.210341\pi$
$674$	6.57783i	0.253368i
$675$	0	0
$676$	−7.26544	−0.279440
$677$	34.2063	1.31466	0.657328	−	0.753604i	$-0.271686\pi$
$677$	34.2063	1.31466	0.657328	+	0.753604i	$0.271686\pi$
$678$	0	0
$679$	0	0
$680$	11.2126i	0.429985i
$681$	0	0
$682$	5.96357i	0.228357i
$683$	− 29.6923i	− 1.13615i	−0.822978	−	0.568073i	$-0.807689\pi$
$683$	− 29.6923i	− 1.13615i	0.822978	−	0.568073i	$-0.192311\pi$
$684$	0	0
$685$	4.86423i	0.185853i
$686$	0	0
$687$	0	0
$688$	−8.40668	−0.320502
$689$	28.5309	1.08694
$690$	0	0
$691$	9.07297i	0.345152i	0.984996	+	0.172576i	$0.0552090\pi$
$691$	9.07297i	0.345152i	−0.984996	+	0.172576i	$0.944791\pi$
$692$	−15.9781	−0.607397
$693$	0	0
$694$	0.530394	0.0201335
$695$	17.7105i	0.671797i
$696$	0	0
$697$	5.39670	0.204414
$698$	−18.1897	−0.688489
$699$	0	0
$700$	0	0
$701$	3.16585i	0.119573i	0.998211	+	0.0597863i	$0.0190419\pi$
$701$	3.16585i	0.119573i	−0.998211	+	0.0597863i	$0.980958\pi$
$702$	0	0
$703$	9.41576i	0.355122i
$704$	0.879356i	0.0331420i
$705$	0	0
$706$	33.9553i	1.27793i
$707$	0	0
$708$	0	0
$709$	2.44533	0.0918363	0.0459181	−	0.998945i	$-0.485379\pi$
$709$	2.44533	0.0918363	0.0459181	+	0.998945i	$0.485379\pi$
$710$	−28.0374	−1.05222
$711$	0	0
$712$	1.37547i	0.0515480i
$713$	1.30139	0.0487374
$714$	0	0
$715$	0.916841	0.0342879
$716$	− 4.43306i	− 0.165671i
$717$	0	0
$718$	−18.9446	−0.707008
$719$	−36.8543	−1.37444	−0.687218	−	0.726452i	$-0.741168\pi$
$719$	−36.8543	−1.37444	−0.687218	+	0.726452i	$0.741168\pi$
$720$	0	0
$721$	0	0
$722$	27.8715i	1.03727i
$723$	0	0
$724$	22.1871i	0.824578i
$725$	6.40873i	0.238014i
$726$	0	0
$727$	6.77506i	0.251273i	0.992076	+	0.125637i	$0.0400973\pi$
$727$	6.77506i	0.251273i	−0.992076	+	0.125637i	$0.959903\pi$
$728$	0	0
$729$	0	0
$730$	1.36444	0.0505004
$731$	9.93169	0.367337
$732$	0	0
$733$	− 46.3529i	− 1.71208i	−0.516906	−	0.856042i	$-0.672916\pi$
$733$	− 46.3529i	− 1.71208i	0.516906	−	0.856042i	$-0.327084\pi$
$734$	54.9737	2.02912
$735$	0	0
$736$	−0.726086	−0.0267639
$737$	4.30596i	0.158612i
$738$	0	0
$739$	43.0028	1.58188	0.790942	−	0.611892i	$-0.209591\pi$
$739$	43.0028	1.58188	0.790942	+	0.611892i	$0.209591\pi$
$740$	5.15497	0.189501
$741$	0	0
$742$	0	0
$743$	− 34.0373i	− 1.24871i	−0.781142	−	0.624353i	$-0.785363\pi$
$743$	− 34.0373i	− 1.24871i	0.781142	−	0.624353i	$-0.214637\pi$
$744$	0	0
$745$	− 20.6400i	− 0.756191i
$746$	6.15732i	0.225436i
$747$	0	0
$748$	2.18089i	0.0797413i
$749$	0	0
$750$	0	0
$751$	−16.3270	−0.595781	−0.297891	−	0.954600i	$-0.596283\pi$
$751$	−16.3270	−0.595781	−0.297891	+	0.954600i	$0.596283\pi$
$752$	−30.5552	−1.11423
$753$	0	0
$754$	− 23.6239i	− 0.860332i
$755$	−19.1135	−0.695610
$756$	0	0
$757$	27.4559	0.997900	0.498950	−	0.866631i	$-0.333719\pi$
$757$	27.4559	0.997900	0.498950	+	0.866631i	$0.333719\pi$
$758$	− 47.3534i	− 1.71995i
$759$	0	0
$760$	3.05212	0.110712
$761$	21.9331	0.795075	0.397537	−	0.917586i	$-0.369865\pi$
$761$	21.9331	0.795075	0.397537	+	0.917586i	$0.369865\pi$
$762$	0	0
$763$	0	0
$764$	3.16421i	0.114477i
$765$	0	0
$766$	17.8881i	0.646323i
$767$	18.3551i	0.662765i
$768$	0	0
$769$	19.7466i	0.712082i	0.934470	+	0.356041i	$0.115874\pi$
$769$	19.7466i	0.712082i	−0.934470	+	0.356041i	$0.884126\pi$
$770$	0	0
$771$	0	0
$772$	14.3092	0.515001
$773$	35.9983	1.29477	0.647385	−	0.762163i	$-0.275863\pi$
$773$	35.9983	1.29477	0.647385	+	0.762163i	$0.275863\pi$
$774$	0	0
$775$	8.33203i	0.299295i
$776$	−27.4987	−0.987148
$777$	0	0
$778$	−19.1020	−0.684838
$779$	− 1.46900i	− 0.0526325i
$780$	0	0
$781$	6.97351	0.249532
$782$	1.56043	0.0558009
$783$	0	0
$784$	0	0
$785$	− 21.1807i	− 0.755970i
$786$	0	0
$787$	1.13004i	0.0402817i	0.999797	+	0.0201409i	$0.00641147\pi$
$787$	1.13004i	0.0402817i	−0.999797	+	0.0201409i	$0.993589\pi$
$788$	1.72706i	0.0615239i
$789$	0	0
$790$	3.50380i	0.124659i
$791$	0	0
$792$	0	0
$793$	−14.5923	−0.518189
$794$	1.95907	0.0695248
$795$	0	0
$796$	17.3769i	0.615909i
$797$	−4.06109	−0.143851	−0.0719257	−	0.997410i	$-0.522914\pi$
$797$	−4.06109	−0.143851	−0.0719257	+	0.997410i	$0.522914\pi$
$798$	0	0
$799$	36.0980	1.27706
$800$	− 4.64870i	− 0.164356i
$801$	0	0
$802$	−42.0608	−1.48522
$803$	−0.339367	−0.0119760
$804$	0	0
$805$	0	0
$806$	− 30.7136i	− 1.08184i
$807$	0	0
$808$	0.159750i	0.00561999i
$809$	18.9378i	0.665817i	0.942959	+	0.332908i	$0.108030\pi$
$809$	18.9378i	0.665817i	−0.942959	+	0.332908i	$0.891970\pi$
$810$	0	0
$811$	29.5835i	1.03882i	0.854526	+	0.519408i	$0.173848\pi$
$811$	29.5835i	1.03882i	−0.854526	+	0.519408i	$0.826152\pi$
$812$	0	0
$813$	0	0
$814$	−4.20387	−0.147346
$815$	18.7515	0.656837
$816$	0	0
$817$	− 2.70345i	− 0.0945816i
$818$	47.4408	1.65873
$819$	0	0
$820$	−0.804254	−0.0280858
$821$	− 37.9510i	− 1.32450i	−0.749284	−	0.662249i	$-0.769602\pi$
$821$	− 37.9510i	− 1.32450i	0.749284	−	0.662249i	$-0.230398\pi$
$822$	0	0
$823$	−38.3882	−1.33813	−0.669065	−	0.743204i	$-0.733305\pi$
$823$	−38.3882	−1.33813	−0.669065	+	0.743204i	$0.733305\pi$
$824$	−18.2165	−0.634603
$825$	0	0
$826$	0	0
$827$	− 7.85718i	− 0.273221i	−0.990625	−	0.136610i	$-0.956379\pi$
$827$	− 7.85718i	− 0.273221i	0.990625	−	0.136610i	$-0.0436208\pi$
$828$	0	0
$829$	− 9.22844i	− 0.320517i	−0.987075	−	0.160259i	$-0.948767\pi$
$829$	− 9.22844i	− 0.320517i	0.987075	−	0.160259i	$-0.0512328\pi$
$830$	7.13415i	0.247630i
$831$	0	0
$832$	− 4.52886i	− 0.157010i
$833$	0	0
$834$	0	0
$835$	−18.6004	−0.643694
$836$	0.593647	0.0205317
$837$	0	0
$838$	57.6378i	1.99106i
$839$	1.21933	0.0420959	0.0210479	−	0.999778i	$-0.493300\pi$
$839$	1.21933	0.0420959	0.0210479	+	0.999778i	$0.493300\pi$
$840$	0	0
$841$	−12.0718	−0.416269
$842$	13.8146i	0.476081i
$843$	0	0
$844$	−20.1057	−0.692065
$845$	8.27808	0.284775
$846$	0	0
$847$	0	0
$848$	65.4521i	2.24764i
$849$	0	0
$850$	9.99051i	0.342672i
$851$	0.917383i	0.0314475i
$852$	0	0
$853$	− 29.4604i	− 1.00871i	−0.863498	−	0.504353i	$-0.831731\pi$
$853$	− 29.4604i	− 1.00871i	0.863498	−	0.504353i	$-0.168269\pi$
$854$	0	0
$855$	0	0
$856$	−9.64661	−0.329714
$857$	−39.1712	−1.33806	−0.669032	−	0.743234i	$-0.733291\pi$
$857$	−39.1712	−1.33806	−0.669032	+	0.743234i	$0.733291\pi$
$858$	0	0
$859$	10.9319i	0.372991i	0.982456	+	0.186495i	$0.0597129\pi$
$859$	10.9319i	0.372991i	−0.982456	+	0.186495i	$0.940287\pi$
$860$	−1.48009	−0.0504707
$861$	0	0
$862$	56.2510	1.91592
$863$	− 41.5258i	− 1.41356i	−0.707436	−	0.706778i	$-0.750148\pi$
$863$	− 41.5258i	− 1.41356i	0.707436	−	0.706778i	$-0.249852\pi$
$864$	0	0
$865$	18.2051	0.618992
$866$	45.0245	1.53000
$867$	0	0
$868$	0	0
$869$	− 0.871471i	− 0.0295626i
$870$	0	0
$871$	− 22.1766i	− 0.751425i
$872$	− 1.34335i	− 0.0454916i
$873$	0	0
$874$	− 0.424755i	− 0.0143676i
$875$	0	0
$876$	0	0
$877$	−6.55222	−0.221253	−0.110626	−	0.993862i	$-0.535286\pi$
$877$	−6.55222	−0.221253	−0.110626	+	0.993862i	$0.535286\pi$
$878$	−41.3393	−1.39513
$879$	0	0
$880$	2.10331i	0.0709025i
$881$	15.4985	0.522156	0.261078	−	0.965318i	$-0.415922\pi$
$881$	15.4985	0.522156	0.261078	+	0.965318i	$0.415922\pi$
$882$	0	0
$883$	12.8809	0.433477	0.216738	−	0.976230i	$-0.430458\pi$
$883$	12.8809	0.433477	0.216738	+	0.976230i	$0.430458\pi$
$884$	− 11.2320i	− 0.377774i
$885$	0	0
$886$	5.93528	0.199400
$887$	−42.8287	−1.43805	−0.719024	−	0.694985i	$-0.755411\pi$
$887$	−42.8287	−1.43805	−0.719024	+	0.694985i	$0.755411\pi$
$888$	0	0
$889$	0	0
$890$	1.22555i	0.0410806i
$891$	0	0
$892$	− 3.95142i	− 0.132303i
$893$	− 9.82603i	− 0.328816i
$894$	0	0
$895$	5.05092i	0.168834i
$896$	0	0
$897$	0	0
$898$	−7.77954	−0.259607
$899$	−53.3977	−1.78091
$900$	0	0
$901$	− 77.3255i	− 2.57609i
$902$	0.655868	0.0218380
$903$	0	0
$904$	6.22064	0.206895
$905$	− 25.2795i	− 0.840319i
$906$	0	0
$907$	−27.1932	−0.902936	−0.451468	−	0.892287i	$-0.649099\pi$
$907$	−27.1932	−0.902936	−0.451468	+	0.892287i	$0.649099\pi$
$908$	−1.12943	−0.0374813
$909$	0	0
$910$	0	0
$911$	− 30.4716i	− 1.00957i	−0.863245	−	0.504785i	$-0.831572\pi$
$911$	− 30.4716i	− 1.00957i	0.863245	−	0.504785i	$-0.168428\pi$
$912$	0	0
$913$	− 1.77442i	− 0.0587247i
$914$	− 36.1462i	− 1.19561i
$915$	0	0
$916$	6.02994i	0.199235i
$917$	0	0
$918$	0	0
$919$	21.0277	0.693640	0.346820	−	0.937932i	$-0.387261\pi$
$919$	21.0277	0.693640	0.346820	+	0.937932i	$0.387261\pi$
$920$	0.297370	0.00980399
$921$	0	0
$922$	66.8838i	2.20270i
$923$	−35.9150	−1.18216
$924$	0	0
$925$	−5.87346	−0.193118
$926$	− 16.7501i	− 0.550442i
$927$	0	0
$928$	29.7923	0.977979
$929$	−4.85466	−0.159276	−0.0796382	−	0.996824i	$-0.525376\pi$
$929$	−4.85466	−0.159276	−0.0796382	+	0.996824i	$0.525376\pi$
$930$	0	0
$931$	0	0
$932$	15.0502i	0.492987i
$933$	0	0
$934$	4.59811i	0.150455i
$935$	− 2.48486i	− 0.0812636i
$936$	0	0
$937$	− 17.1825i	− 0.561329i	−0.959806	−	0.280664i	$-0.909445\pi$
$937$	− 17.1825i	− 0.561329i	0.959806	−	0.280664i	$-0.0905548\pi$
$938$	0	0
$939$	0	0
$940$	−5.37959	−0.175463
$941$	−36.7030	−1.19648	−0.598242	−	0.801316i	$-0.704134\pi$
$941$	−36.7030	−1.19648	−0.598242	+	0.801316i	$0.704134\pi$
$942$	0	0
$943$	− 0.143126i	− 0.00466081i
$944$	−42.1081	−1.37050
$945$	0	0
$946$	1.20701	0.0392433
$947$	− 3.79258i	− 0.123242i	−0.998100	−	0.0616211i	$-0.980373\pi$
$947$	− 3.79258i	− 0.123242i	0.998100	−	0.0616211i	$-0.0196271\pi$
$948$	0	0
$949$	1.74781	0.0567364
$950$	2.71946	0.0882308
$951$	0	0
$952$	0	0
$953$	− 20.4628i	− 0.662856i	−0.943481	−	0.331428i	$-0.892470\pi$
$953$	− 20.4628i	− 0.662856i	0.943481	−	0.331428i	$-0.107530\pi$
$954$	0	0
$955$	− 3.60523i	− 0.116663i
$956$	9.43508i	0.305152i
$957$	0	0
$958$	50.4853i	1.63111i
$959$	0	0
$960$	0	0
$961$	−38.4227	−1.23944
$962$	21.6508	0.698050
$963$	0	0
$964$	− 6.46272i	− 0.208150i
$965$	−16.3036	−0.524833
$966$	0	0
$967$	56.6843	1.82284	0.911422	−	0.411473i	$-0.134986\pi$
$967$	56.6843	1.82284	0.911422	+	0.411473i	$0.134986\pi$
$968$	20.6038i	0.662231i
$969$	0	0
$970$	−24.5015	−0.786697
$971$	−43.4291	−1.39370	−0.696852	−	0.717215i	$-0.745417\pi$
$971$	−43.4291	−1.39370	−0.696852	+	0.717215i	$0.745417\pi$
$972$	0	0
$973$	0	0
$974$	20.3998i	0.653653i
$975$	0	0
$976$	− 33.4760i	− 1.07154i
$977$	− 41.8099i	− 1.33762i	−0.743435	−	0.668809i	$-0.766805\pi$
$977$	− 41.8099i	− 1.33762i	0.743435	−	0.668809i	$-0.233195\pi$
$978$	0	0
$979$	− 0.304822i	− 0.00974215i
$980$	0	0
$981$	0	0
$982$	1.56677	0.0499977
$983$	−32.2607	−1.02896	−0.514478	−	0.857504i	$-0.672014\pi$
$983$	−32.2607	−1.02896	−0.514478	+	0.857504i	$0.672014\pi$
$984$	0	0
$985$	− 1.96777i	− 0.0626984i
$986$	−64.0265	−2.03902
$987$	0	0
$988$	−3.05741	−0.0972691
$989$	− 0.263398i	− 0.00837557i
$990$	0	0
$991$	−41.2806	−1.31132	−0.655661	−	0.755056i	$-0.727610\pi$
$991$	−41.2806	−1.31132	−0.655661	+	0.755056i	$0.727610\pi$
$992$	38.7331	1.22978
$993$	0	0
$994$	0	0
$995$	− 19.7989i	− 0.627667i
$996$	0	0
$997$	− 9.14133i	− 0.289509i	−0.989468	−	0.144754i	$-0.953761\pi$
$997$	− 9.14133i	− 0.289509i	0.989468	−	0.144754i	$-0.0462392\pi$
$998$	52.2086i	1.65263i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.2.b.d.881.12	yes	16
3.2	odd	2		2205.2.b.c.881.5	✓	16
7.6	odd	2		2205.2.b.c.881.12	yes	16
21.20	even	2	inner	2205.2.b.d.881.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2205.2.b.c.881.5	✓	16	3.2	odd	2
2205.2.b.c.881.12	yes	16	7.6	odd	2
2205.2.b.d.881.5	yes	16	21.20	even	2	inner
2205.2.b.d.881.12	yes	16	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2205.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.69637i q^{2} -0.877672 q^{4} +1.00000 q^{5} +1.90388i q^{8} +1.69637i q^{10} -0.421924i q^{11} +2.17300i q^{13} -4.98504 q^{16} +5.88935 q^{17} -1.60310i q^{19} -0.877672 q^{20} +0.715740 q^{22} -0.156191i q^{23} +1.00000 q^{25} -3.68621 q^{26} +6.40873i q^{29} +8.33203i q^{31} -4.64870i q^{32} +9.99051i q^{34} -5.87346 q^{37} +2.71946 q^{38} +1.90388i q^{40} +0.916349 q^{41} +1.68638 q^{43} +0.370311i q^{44} +0.264958 q^{46} +6.12938 q^{47} +1.69637i q^{50} -1.90718i q^{52} -13.1297i q^{53} -0.421924i q^{55} -10.8716 q^{58} +8.44691 q^{59} +6.71530i q^{61} -14.1342 q^{62} -2.08416 q^{64} +2.17300i q^{65} -10.2055 q^{67} -5.16892 q^{68} +16.5279i q^{71} -0.804332i q^{73} -9.96356i q^{74} +1.40700i q^{76} +2.06547 q^{79} -4.98504 q^{80} +1.55447i q^{82} +4.20554 q^{83} +5.88935 q^{85} +2.86073i q^{86} +0.803295 q^{88} +0.722456 q^{89} +0.137085i q^{92} +10.3977i q^{94} -1.60310i q^{95} +14.4435i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} + 16 q^{5} - 16 q^{20} - 16 q^{22} + 16 q^{25} + 32 q^{26} - 16 q^{38} - 32 q^{41} + 32 q^{43} - 16 q^{46} + 32 q^{47} + 48 q^{58} - 32 q^{59} - 32 q^{62} + 16 q^{64} - 16 q^{68} + 32 q^{79}+ \cdots + 64 q^{89}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^5 - 16 * q^20 - 16 * q^22 + 16 * q^25 + 32 * q^26 - 16 * q^38 - 32 * q^41 + 32 * q^43 - 16 * q^46 + 32 * q^47 + 48 * q^58 - 32 * q^59 - 32 * q^62 + 16 * q^64 - 16 * q^68 + 32 * q^79 + 16 * q^88 + 64 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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