LMFDB - Embedded newform 1148.2.d.a.1065.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1148,2,Mod(1065,1148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1148, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1148.1065");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1148 = 2^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1148.d (of order $2$, degree $1$, 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:	$9.16682615204$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 39 x^{18} + 531 x^{16} + 3488 x^{14} + 12661 x^{12} + 27027 x^{10} + 34540 x^{8} + 25909 x^{6} + \cdots + 144 $ x^20 + 39x^18 + 531x^16 + 3488x^14 + 12661x^12 + 27027x^10 + 34540x^8 + 25909x^6 + 10677x^4 + 2092*x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1065.15
Root			$4.38287i$ of defining polynomial
Character	$\chi$	$=$	1148.1065
Dual form			1148.2.d.a.1065.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.40464i q^{3} -1.35828 q^{5} -1.00000i q^{7} +1.02697 q^{9} +O(q^{10})$	$q+1.40464i q^{3} -1.35828 q^{5} -1.00000i q^{7} +1.02697 q^{9} -1.15173i q^{11} +1.66501i q^{13} -1.90790i q^{15} +2.28853i q^{17} +5.03870i q^{19} +1.40464 q^{21} +4.00693 q^{23} -3.15507 q^{25} +5.65647i q^{27} +2.40457i q^{29} +5.53514 q^{31} +1.61777 q^{33} +1.35828i q^{35} -0.838223 q^{37} -2.33875 q^{39} +(6.21611 - 1.53621i) q^{41} -10.5361 q^{43} -1.39492 q^{45} +13.4336i q^{47} -1.00000 q^{49} -3.21457 q^{51} +6.80004i q^{53} +1.56437i q^{55} -7.07758 q^{57} -7.72782 q^{59} +3.76583 q^{61} -1.02697i q^{63} -2.26155i q^{65} -6.05601i q^{67} +5.62831i q^{69} +13.1629i q^{71} +8.45968 q^{73} -4.43176i q^{75} -1.15173 q^{77} +1.01755i q^{79} -4.86441 q^{81} -17.7169 q^{83} -3.10846i q^{85} -3.37756 q^{87} +7.88443i q^{89} +1.66501 q^{91} +7.77491i q^{93} -6.84396i q^{95} -9.62923i q^{97} -1.18280i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 4 q^{5} - 20 q^{9}+O(q^{10}) $ 20 * q + 4 * q^5 - 20 * q^9	$ 20 q + 4 q^{5} - 20 q^{9} + 4 q^{21} + 8 q^{31} + 20 q^{37} + 4 q^{39} - 16 q^{41} + 20 q^{43} - 4 q^{45} - 20 q^{49} + 52 q^{51} - 36 q^{57} + 20 q^{59} - 4 q^{61} - 12 q^{73} + 8 q^{77} + 20 q^{81} - 48 q^{83} + 44 q^{87} - 4 q^{91}+O(q^{100}) $ 20 * q + 4 * q^5 - 20 * q^9 + 4 * q^21 + 8 * q^31 + 20 * q^37 + 4 * q^39 - 16 * q^41 + 20 * q^43 - 4 * q^45 - 20 * q^49 + 52 * q^51 - 36 * q^57 + 20 * q^59 - 4 * q^61 - 12 * q^73 + 8 * q^77 + 20 * q^81 - 48 * q^83 + 44 * q^87 - 4 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times$.

$n$	$493$	$575$	$785$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			1.40464i			0.810972i	0.914101	+	0.405486i	$0.132898\pi$
$3$			1.40464i			0.810972i	−0.914101	+	0.405486i	$0.867102\pi$
$4$		0			0
$5$	−1.35828			−0.607442			−0.303721	−	0.952761i	$-0.598229\pi$
$5$	−1.35828			−0.607442			−0.303721	+	0.952761i	$0.598229\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.02697			0.342324
$10$		0			0
$11$		−	1.15173i		−	0.347260i	−0.984811	−	0.173630i	$-0.944450\pi$
$11$		−	1.15173i		−	0.347260i	0.984811	−	0.173630i	$-0.0555496\pi$
$12$		0			0
$13$			1.66501i			0.461791i	0.972979	+	0.230896i	$0.0741656\pi$
$13$			1.66501i			0.461791i	−0.972979	+	0.230896i	$0.925834\pi$
$14$		0			0
$15$		−	1.90790i		−	0.492618i
$16$		0			0
$17$			2.28853i			0.555049i	0.960719	+	0.277525i	$0.0895140\pi$
$17$			2.28853i			0.555049i	−0.960719	+	0.277525i	$0.910486\pi$
$18$		0			0
$19$			5.03870i			1.15596i	0.816052	+	0.577978i	$0.196158\pi$
$19$			5.03870i			1.15596i	−0.816052	+	0.577978i	$0.803842\pi$
$20$		0			0
$21$	1.40464			0.306519
$22$		0			0
$23$	4.00693			0.835502			0.417751	−	0.908562i	$-0.362818\pi$
$23$	4.00693			0.835502			0.417751	+	0.908562i	$0.362818\pi$
$24$		0			0
$25$	−3.15507			−0.631015
$26$		0			0
$27$			5.65647i			1.08859i
$28$		0			0
$29$			2.40457i			0.446517i	0.974759	+	0.223259i	$0.0716694\pi$
$29$			2.40457i			0.446517i	−0.974759	+	0.223259i	$0.928331\pi$
$30$		0			0
$31$	5.53514			0.994141			0.497070	−	0.867710i	$-0.334409\pi$
$31$	5.53514			0.994141			0.497070	+	0.867710i	$0.334409\pi$
$32$		0			0
$33$	1.61777			0.281618
$34$		0			0
$35$			1.35828i			0.229591i
$36$		0			0
$37$	−0.838223			−0.137803			−0.0689015	−	0.997623i	$-0.521949\pi$
$37$	−0.838223			−0.137803			−0.0689015	+	0.997623i	$0.521949\pi$
$38$		0			0
$39$	−2.33875			−0.374500
$40$		0			0
$41$	6.21611	−	1.53621i	0.970794	−	0.239916i
$41$	6.21611	−	1.53621i	0.970794	−	0.239916i
$42$		0			0
$43$	−10.5361			−1.60674			−0.803371	−	0.595479i	$-0.796962\pi$
$43$	−10.5361			−1.60674			−0.803371	+	0.595479i	$0.796962\pi$
$44$		0			0
$45$	−1.39492			−0.207942
$46$		0			0
$47$			13.4336i			1.95950i	0.200234	+	0.979748i	$0.435830\pi$
$47$			13.4336i			1.95950i	−0.200234	+	0.979748i	$0.564170\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−3.21457			−0.450129
$52$		0			0
$53$			6.80004i			0.934057i	0.884242	+	0.467028i	$0.154675\pi$
$53$			6.80004i			0.934057i	−0.884242	+	0.467028i	$0.845325\pi$
$54$		0			0
$55$			1.56437i			0.210940i
$56$		0			0
$57$	−7.07758			−0.937448
$58$		0			0
$59$	−7.72782			−1.00608			−0.503039	−	0.864264i	$-0.667785\pi$
$59$	−7.72782			−1.00608			−0.503039	+	0.864264i	$0.667785\pi$
$60$		0			0
$61$	3.76583			0.482165			0.241083	−	0.970505i	$-0.422497\pi$
$61$	3.76583			0.482165			0.241083	+	0.970505i	$0.422497\pi$
$62$		0			0
$63$		−	1.02697i		−	0.129386i
$64$		0			0
$65$		−	2.26155i		−	0.280511i
$66$		0			0
$67$		−	6.05601i		−	0.739859i	−0.929060	−	0.369930i	$-0.879382\pi$
$67$		−	6.05601i		−	0.739859i	0.929060	−	0.369930i	$-0.120618\pi$
$68$		0			0
$69$			5.62831i			0.677569i
$70$		0			0
$71$			13.1629i			1.56215i	0.624436	+	0.781076i	$0.285329\pi$
$71$			13.1629i			1.56215i	−0.624436	+	0.781076i	$0.714671\pi$
$72$		0			0
$73$	8.45968			0.990130			0.495065	−	0.868856i	$-0.335144\pi$
$73$	8.45968			0.990130			0.495065	+	0.868856i	$0.335144\pi$
$74$		0			0
$75$		−	4.43176i		−	0.511735i
$76$		0			0
$77$	−1.15173			−0.131252
$78$		0			0
$79$			1.01755i			0.114484i	0.998360	+	0.0572418i	$0.0182306\pi$
$79$			1.01755i			0.114484i	−0.998360	+	0.0572418i	$0.981769\pi$
$80$		0			0
$81$	−4.86441			−0.540490
$82$		0			0
$83$	−17.7169			−1.94468			−0.972340	−	0.233571i	$-0.924959\pi$
$83$	−17.7169			−1.94468			−0.972340	+	0.233571i	$0.924959\pi$
$84$		0			0
$85$		−	3.10846i		−	0.337160i
$86$		0			0
$87$	−3.37756			−0.362113
$88$		0			0
$89$			7.88443i			0.835748i	0.908505	+	0.417874i	$0.137225\pi$
$89$			7.88443i			0.835748i	−0.908505	+	0.417874i	$0.862775\pi$
$90$		0			0
$91$	1.66501			0.174541
$92$		0			0
$93$			7.77491i			0.806220i
$94$		0			0
$95$		−	6.84396i		−	0.702176i
$96$		0			0
$97$		−	9.62923i		−	0.977700i	−0.872368	−	0.488850i	$-0.837417\pi$
$97$		−	9.62923i		−	0.977700i	0.872368	−	0.488850i	$-0.162583\pi$
$98$		0			0
$99$		−	1.18280i		−	0.118875i
$100$		0			0
$101$		−	3.15207i		−	0.313643i	−0.987627	−	0.156821i	$-0.949875\pi$
$101$		−	3.15207i		−	0.313643i	0.987627	−	0.156821i	$-0.0501247\pi$
$102$		0			0
$103$	6.42106			0.632686			0.316343	−	0.948645i	$-0.397545\pi$
$103$	6.42106			0.632686			0.316343	+	0.948645i	$0.397545\pi$
$104$		0			0
$105$	−1.90790			−0.186192
$106$		0			0
$107$	2.68916			0.259971			0.129985	−	0.991516i	$-0.458507\pi$
$107$	2.68916			0.259971			0.129985	+	0.991516i	$0.458507\pi$
$108$		0			0
$109$		−	0.843693i		−	0.0808111i	−0.999183	−	0.0404056i	$-0.987135\pi$
$109$		−	0.843693i		−	0.0808111i	0.999183	−	0.0404056i	$-0.0128650\pi$
$110$		0			0
$111$		−	1.17741i		−	0.111754i
$112$		0			0
$113$	2.46309			0.231708			0.115854	−	0.993266i	$-0.463039\pi$
$113$	2.46309			0.231708			0.115854	+	0.993266i	$0.463039\pi$
$114$		0			0
$115$	−5.44253			−0.507519
$116$		0			0
$117$			1.70992i			0.158082i
$118$		0			0
$119$	2.28853			0.209789
$120$		0			0
$121$	9.67352			0.879411
$122$		0			0
$123$	2.15783	+	8.73143i	0.194565	+	0.787286i
$124$		0			0
$125$	11.0769			0.990746
$126$		0			0
$127$	−18.6916			−1.65861			−0.829305	−	0.558796i	$-0.811264\pi$
$127$	−18.6916			−1.65861			−0.829305	+	0.558796i	$0.811264\pi$
$128$		0			0
$129$		−	14.7995i		−	1.30302i
$130$		0			0
$131$	−3.67323			−0.320931			−0.160466	−	0.987041i	$-0.551300\pi$
$131$	−3.67323			−0.320931			−0.160466	+	0.987041i	$0.551300\pi$
$132$		0			0
$133$	5.03870			0.436910
$134$		0			0
$135$		−	7.68307i		−	0.661253i
$136$		0			0
$137$		−	13.6977i		−	1.17027i	−0.810935	−	0.585136i	$-0.801041\pi$
$137$		−	13.6977i		−	1.17027i	0.810935	−	0.585136i	$-0.198959\pi$
$138$		0			0
$139$	17.9939			1.52623			0.763113	−	0.646265i	$-0.223670\pi$
$139$	17.9939			1.52623			0.763113	+	0.646265i	$0.223670\pi$
$140$		0			0
$141$	−18.8695			−1.58910
$142$		0			0
$143$	1.91764			0.160361
$144$		0			0
$145$		−	3.26608i		−	0.271233i
$146$		0			0
$147$		−	1.40464i		−	0.115853i
$148$		0			0
$149$		−	7.97880i		−	0.653649i	−0.945085	−	0.326824i	$-0.894021\pi$
$149$		−	7.97880i		−	0.653649i	0.945085	−	0.326824i	$-0.105979\pi$
$150$		0			0
$151$			4.53026i			0.368667i	0.982864	+	0.184333i	$0.0590126\pi$
$151$			4.53026i			0.368667i	−0.982864	+	0.184333i	$0.940987\pi$
$152$		0			0
$153$			2.35025i			0.190007i
$154$		0			0
$155$	−7.51828			−0.603882
$156$		0			0
$157$			2.54513i			0.203124i	0.994829	+	0.101562i	$0.0323840\pi$
$157$			2.54513i			0.203124i	−0.994829	+	0.101562i	$0.967616\pi$
$158$		0			0
$159$	−9.55164			−0.757494
$160$		0			0
$161$		−	4.00693i		−	0.315790i
$162$		0			0
$163$	22.3553			1.75100			0.875502	−	0.483215i	$-0.160531\pi$
$163$	22.3553			1.75100			0.875502	+	0.483215i	$0.160531\pi$
$164$		0			0
$165$	−2.19739			−0.171066
$166$		0			0
$167$			22.9254i			1.77402i	0.461748	+	0.887011i	$0.347223\pi$
$167$			22.9254i			1.77402i	−0.461748	+	0.887011i	$0.652777\pi$
$168$		0			0
$169$	10.2277			0.786749
$170$		0			0
$171$			5.17461i			0.395712i
$172$		0			0
$173$	18.7451			1.42516			0.712581	−	0.701590i	$-0.247526\pi$
$173$	18.7451			1.42516			0.712581	+	0.701590i	$0.247526\pi$
$174$		0			0
$175$			3.15507i			0.238501i
$176$		0			0
$177$		−	10.8548i		−	0.815900i
$178$		0			0
$179$			13.3583i			0.998444i	0.866474	+	0.499222i	$0.166381\pi$
$179$			13.3583i			0.998444i	−0.866474	+	0.499222i	$0.833619\pi$
$180$		0			0
$181$		−	13.1556i		−	0.977846i	−0.872327	−	0.488923i	$-0.837390\pi$
$181$		−	13.1556i		−	0.977846i	0.872327	−	0.488923i	$-0.162610\pi$
$182$		0			0
$183$			5.28966i			0.391023i
$184$		0			0
$185$	1.13854			0.0837073
$186$		0			0
$187$	2.63576			0.192746
$188$		0			0
$189$	5.65647			0.411447
$190$		0			0
$191$			3.95650i			0.286283i	0.989702	+	0.143141i	$0.0457203\pi$
$191$			3.95650i			0.286283i	−0.989702	+	0.143141i	$0.954280\pi$
$192$		0			0
$193$		−	25.2978i		−	1.82098i	−0.413534	−	0.910488i	$-0.635706\pi$
$193$		−	25.2978i		−	1.82098i	0.413534	−	0.910488i	$-0.364294\pi$
$194$		0			0
$195$	3.17668			0.227487
$196$		0			0
$197$	−5.25097			−0.374116			−0.187058	−	0.982349i	$-0.559895\pi$
$197$	−5.25097			−0.374116			−0.187058	+	0.982349i	$0.559895\pi$
$198$		0			0
$199$		−	10.7283i		−	0.760511i	−0.924882	−	0.380255i	$-0.875836\pi$
$199$		−	10.7283i		−	0.760511i	0.924882	−	0.380255i	$-0.124164\pi$
$200$		0			0
$201$	8.50654			0.600005
$202$		0			0
$203$	2.40457			0.168768
$204$		0			0
$205$	−8.44323	+	2.08661i	−0.589700	+	0.145735i
$206$		0			0
$207$	4.11501			0.286013
$208$		0			0
$209$	5.80322			0.401417
$210$		0			0
$211$		−	14.2255i		−	0.979326i	−0.871912	−	0.489663i	$-0.837120\pi$
$211$		−	14.2255i		−	0.979326i	0.871912	−	0.489663i	$-0.162880\pi$
$212$		0			0
$213$	−18.4892			−1.26686
$214$		0			0
$215$	14.3110			0.976002
$216$		0			0
$217$		−	5.53514i		−	0.375750i
$218$		0			0
$219$			11.8828i			0.802968i
$220$		0			0
$221$	−3.81042			−0.256317
$222$		0			0
$223$	18.8314			1.26105			0.630523	−	0.776171i	$-0.282841\pi$
$223$	18.8314			1.26105			0.630523	+	0.776171i	$0.282841\pi$
$224$		0			0
$225$	−3.24018			−0.216012
$226$		0			0
$227$		−	1.40671i		−	0.0933667i	−0.998910	−	0.0466833i	$-0.985135\pi$
$227$		−	1.40671i		−	0.0933667i	0.998910	−	0.0466833i	$-0.0148652\pi$
$228$		0			0
$229$			15.8047i			1.04440i	0.852822	+	0.522202i	$0.174889\pi$
$229$			15.8047i			1.04440i	−0.852822	+	0.522202i	$0.825111\pi$
$230$		0			0
$231$		−	1.61777i		−	0.106442i
$232$		0			0
$233$		−	1.10843i		−	0.0726159i	−0.999341	−	0.0363080i	$-0.988440\pi$
$233$		−	1.10843i		−	0.0726159i	0.999341	−	0.0363080i	$-0.0115597\pi$
$234$		0			0
$235$		−	18.2466i		−	1.19028i
$236$		0			0
$237$	−1.42930			−0.0928430
$238$		0			0
$239$		−	17.6798i		−	1.14361i	−0.820389	−	0.571806i	$-0.806243\pi$
$239$		−	17.6798i		−	1.14361i	0.820389	−	0.571806i	$-0.193757\pi$
$240$		0			0
$241$	−8.94266			−0.576047			−0.288024	−	0.957623i	$-0.592998\pi$
$241$	−8.94266			−0.576047			−0.288024	+	0.957623i	$0.592998\pi$
$242$		0			0
$243$			10.1366i			0.650266i
$244$		0			0
$245$	1.35828			0.0867774
$246$		0			0
$247$	−8.38948			−0.533810
$248$		0			0
$249$		−	24.8859i		−	1.57708i
$250$		0			0
$251$	−7.20705			−0.454905			−0.227453	−	0.973789i	$-0.573040\pi$
$251$	−7.20705			−0.454905			−0.227453	+	0.973789i	$0.573040\pi$
$252$		0			0
$253$		−	4.61490i		−	0.290136i
$254$		0			0
$255$	4.36628			0.273427
$256$		0			0
$257$			5.59872i			0.349239i	0.984636	+	0.174619i	$0.0558695\pi$
$257$			5.59872i			0.349239i	−0.984636	+	0.174619i	$0.944131\pi$
$258$		0			0
$259$			0.838223i			0.0520847i
$260$		0			0
$261$			2.46943i			0.152854i
$262$		0			0
$263$			14.4777i			0.892730i	0.894851	+	0.446365i	$0.147282\pi$
$263$			14.4777i			0.892730i	−0.894851	+	0.446365i	$0.852718\pi$
$264$		0			0
$265$		−	9.23636i		−	0.567385i
$266$		0			0
$267$	−11.0748			−0.677768
$268$		0			0
$269$	0.913652			0.0557063			0.0278532	−	0.999612i	$-0.491133\pi$
$269$	0.913652			0.0557063			0.0278532	+	0.999612i	$0.491133\pi$
$270$		0			0
$271$	0.314962			0.0191326			0.00956630	−	0.999954i	$-0.496955\pi$
$271$	0.314962			0.0191326			0.00956630	+	0.999954i	$0.496955\pi$
$272$		0			0
$273$			2.33875i			0.141548i
$274$		0			0
$275$			3.63379i			0.219126i
$276$		0			0
$277$	5.19368			0.312058			0.156029	−	0.987752i	$-0.450131\pi$
$277$	5.19368			0.312058			0.156029	+	0.987752i	$0.450131\pi$
$278$		0			0
$279$	5.68444			0.340319
$280$		0			0
$281$		−	21.4131i		−	1.27740i	−0.769456	−	0.638699i	$-0.779473\pi$
$281$		−	21.4131i		−	1.27740i	0.769456	−	0.638699i	$-0.220527\pi$
$282$		0			0
$283$	−21.3292			−1.26789			−0.633945	−	0.773379i	$-0.718565\pi$
$283$	−21.3292			−1.26789			−0.633945	+	0.773379i	$0.718565\pi$
$284$		0			0
$285$	9.61334			0.569445
$286$		0			0
$287$	−1.53621	−	6.21611i	−0.0906798	−	0.366925i
$288$		0			0
$289$	11.7627			0.691921
$290$		0			0
$291$	13.5256			0.792887
$292$		0			0
$293$		−	21.2087i		−	1.23903i	−0.784986	−	0.619513i	$-0.787330\pi$
$293$		−	21.2087i		−	1.23903i	0.784986	−	0.619513i	$-0.212670\pi$
$294$		0			0
$295$	10.4966			0.611133
$296$		0			0
$297$	6.51472			0.378022
$298$		0			0
$299$			6.67158i			0.385827i
$300$		0			0
$301$			10.5361i			0.607291i
$302$		0			0
$303$	4.42754			0.254355
$304$		0			0
$305$	−5.11506			−0.292887
$306$		0			0
$307$	−6.94680			−0.396475			−0.198237	−	0.980154i	$-0.563522\pi$
$307$	−6.94680			−0.396475			−0.198237	+	0.980154i	$0.563522\pi$
$308$		0			0
$309$			9.01931i			0.513090i
$310$		0			0
$311$		−	18.5322i		−	1.05087i	−0.850835	−	0.525433i	$-0.823903\pi$
$311$		−	18.5322i		−	1.05087i	0.850835	−	0.525433i	$-0.176097\pi$
$312$		0			0
$313$		−	26.7851i		−	1.51398i	−0.653426	−	0.756991i	$-0.726669\pi$
$313$		−	26.7851i		−	1.51398i	0.653426	−	0.756991i	$-0.273331\pi$
$314$		0			0
$315$			1.39492i			0.0785947i
$316$		0			0
$317$		−	4.59346i		−	0.257995i	−0.991645	−	0.128997i	$-0.958824\pi$
$317$		−	4.59346i		−	0.257995i	0.991645	−	0.128997i	$-0.0411758\pi$
$318$		0			0
$319$	2.76941			0.155057
$320$		0			0
$321$			3.77731i			0.210829i
$322$		0			0
$323$	−11.5312			−0.641612
$324$		0			0
$325$		−	5.25323i		−	0.291397i
$326$		0			0
$327$	1.18509			0.0655355
$328$		0			0
$329$	13.4336			0.740620
$330$		0			0
$331$		−	29.1959i		−	1.60475i	−0.596820	−	0.802375i	$-0.703569\pi$
$331$		−	29.1959i		−	1.60475i	0.596820	−	0.802375i	$-0.296431\pi$
$332$		0			0
$333$	−0.860833			−0.0471734
$334$		0			0
$335$			8.22576i			0.449421i
$336$		0			0
$337$	−25.3198			−1.37925			−0.689627	−	0.724164i	$-0.742226\pi$
$337$	−25.3198			−1.37925			−0.689627	+	0.724164i	$0.742226\pi$
$338$		0			0
$339$			3.45977i			0.187909i
$340$		0			0
$341$		−	6.37499i		−	0.345225i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		−	7.64482i		−	0.411583i
$346$		0			0
$347$			14.5714i			0.782233i	0.920341	+	0.391116i	$0.127911\pi$
$347$			14.5714i			0.782233i	−0.920341	+	0.391116i	$0.872089\pi$
$348$		0			0
$349$	24.3554			1.30371			0.651857	−	0.758342i	$-0.273990\pi$
$349$	24.3554			1.30371			0.651857	+	0.758342i	$0.273990\pi$
$350$		0			0
$351$	−9.41808			−0.502700
$352$		0			0
$353$	−10.7267			−0.570924			−0.285462	−	0.958390i	$-0.592147\pi$
$353$	−10.7267			−0.570924			−0.285462	+	0.958390i	$0.592147\pi$
$354$		0			0
$355$		−	17.8790i		−	0.948917i
$356$		0			0
$357$			3.21457i			0.170133i
$358$		0			0
$359$	30.6428			1.61727			0.808634	−	0.588313i	$-0.200208\pi$
$359$	30.6428			1.61727			0.808634	+	0.588313i	$0.200208\pi$
$360$		0			0
$361$	−6.38845			−0.336234
$362$		0			0
$363$			13.5879i			0.713178i
$364$		0			0
$365$	−11.4906			−0.601446
$366$		0			0
$367$	−35.9390			−1.87600			−0.938002	−	0.346631i	$-0.887326\pi$
$367$	−35.9390			−1.87600			−0.938002	+	0.346631i	$0.887326\pi$
$368$		0			0
$369$	6.38378	−	1.57765i	0.332326	−	0.0821292i
$370$		0			0
$371$	6.80004			0.353040
$372$		0			0
$373$	−17.7915			−0.921207			−0.460603	−	0.887606i	$-0.652367\pi$
$373$	−17.7915			−0.921207			−0.460603	+	0.887606i	$0.652367\pi$
$374$		0			0
$375$			15.5591i			0.803467i
$376$		0			0
$377$	−4.00363			−0.206198
$378$		0			0
$379$	−9.86593			−0.506779			−0.253389	−	0.967364i	$-0.581545\pi$
$379$	−9.86593			−0.506779			−0.253389	+	0.967364i	$0.581545\pi$
$380$		0			0
$381$		−	26.2551i		−	1.34509i
$382$		0			0
$383$		−	3.98164i		−	0.203452i	−0.994812	−	0.101726i	$-0.967563\pi$
$383$		−	3.98164i		−	0.203452i	0.994812	−	0.101726i	$-0.0324365\pi$
$384$		0			0
$385$	1.56437			0.0797278
$386$		0			0
$387$	−10.8203			−0.550027
$388$		0			0
$389$	−34.3705			−1.74265			−0.871326	−	0.490705i	$-0.836739\pi$
$389$	−34.3705			−1.74265			−0.871326	+	0.490705i	$0.836739\pi$
$390$		0			0
$391$			9.16995i			0.463744i
$392$		0			0
$393$		−	5.15958i		−	0.260266i
$394$		0			0
$395$		−	1.38212i		−	0.0695421i
$396$		0			0
$397$			18.4252i			0.924733i	0.886689	+	0.462367i	$0.153000\pi$
$397$			18.4252i			0.924733i	−0.886689	+	0.462367i	$0.847000\pi$
$398$		0			0
$399$			7.07758i			0.354322i
$400$		0			0
$401$	−20.2007			−1.00877			−0.504387	−	0.863478i	$-0.668281\pi$
$401$	−20.2007			−1.00877			−0.504387	+	0.863478i	$0.668281\pi$
$402$		0			0
$403$			9.21607i			0.459085i
$404$		0			0
$405$	6.60723			0.328316
$406$		0			0
$407$			0.965407i			0.0478534i
$408$		0			0
$409$	−1.11502			−0.0551344			−0.0275672	−	0.999620i	$-0.508776\pi$
$409$	−1.11502			−0.0551344			−0.0275672	+	0.999620i	$0.508776\pi$
$410$		0			0
$411$	19.2404			0.949057
$412$		0			0
$413$			7.72782i			0.380261i
$414$		0			0
$415$	24.0645			1.18128
$416$		0			0
$417$			25.2751i			1.23773i
$418$		0			0
$419$	−22.4070			−1.09465			−0.547327	−	0.836919i	$-0.684354\pi$
$419$	−22.4070			−1.09465			−0.547327	+	0.836919i	$0.684354\pi$
$420$		0			0
$421$			7.94066i			0.387004i	0.981100	+	0.193502i	$0.0619846\pi$
$421$			7.94066i			0.387004i	−0.981100	+	0.193502i	$0.938015\pi$
$422$		0			0
$423$			13.7960i			0.670783i
$424$		0			0
$425$		−	7.22047i		−	0.350244i
$426$		0			0
$427$		−	3.76583i		−	0.182241i
$428$		0			0
$429$			2.69361i			0.130049i
$430$		0			0
$431$	37.7371			1.81773			0.908866	−	0.417089i	$-0.136950\pi$
$431$	37.7371			1.81773			0.908866	+	0.417089i	$0.136950\pi$
$432$		0			0
$433$	26.4996			1.27349			0.636744	−	0.771075i	$-0.280281\pi$
$433$	26.4996			1.27349			0.636744	+	0.771075i	$0.280281\pi$
$434$		0			0
$435$	4.58768			0.219962
$436$		0			0
$437$			20.1897i			0.965803i
$438$		0			0
$439$		−	27.4781i		−	1.31146i	−0.754996	−	0.655729i	$-0.772361\pi$
$439$		−	27.4781i		−	1.31146i	0.754996	−	0.655729i	$-0.227639\pi$
$440$		0			0
$441$	−1.02697			−0.0489035
$442$		0			0
$443$	6.04591			0.287250			0.143625	−	0.989632i	$-0.454124\pi$
$443$	6.04591			0.287250			0.143625	+	0.989632i	$0.454124\pi$
$444$		0			0
$445$		−	10.7093i		−	0.507668i
$446$		0			0
$447$	11.2074			0.530091
$448$		0			0
$449$	−5.55217			−0.262023			−0.131012	−	0.991381i	$-0.541822\pi$
$449$	−5.55217			−0.262023			−0.131012	+	0.991381i	$0.541822\pi$
$450$		0			0
$451$	−1.76930	−	7.15928i	−0.0833132	−	0.337117i
$452$		0			0
$453$	−6.36340			−0.298979
$454$		0			0
$455$	−2.26155			−0.106023
$456$		0			0
$457$			23.5007i			1.09932i	0.835389	+	0.549659i	$0.185242\pi$
$457$			23.5007i			1.09932i	−0.835389	+	0.549659i	$0.814758\pi$
$458$		0			0
$459$	−12.9450			−0.604219
$460$		0			0
$461$	25.8753			1.20513			0.602567	−	0.798069i	$-0.294145\pi$
$461$	25.8753			1.20513			0.602567	+	0.798069i	$0.294145\pi$
$462$		0			0
$463$		−	38.5903i		−	1.79344i	−0.442594	−	0.896722i	$-0.645942\pi$
$463$		−	38.5903i		−	1.79344i	0.442594	−	0.896722i	$-0.354058\pi$
$464$		0			0
$465$		−	10.5605i		−	0.489732i
$466$		0			0
$467$	11.2032			0.518424			0.259212	−	0.965820i	$-0.416537\pi$
$467$	11.2032			0.518424			0.259212	+	0.965820i	$0.416537\pi$
$468$		0			0
$469$	−6.05601			−0.279640
$470$		0			0
$471$	−3.57500			−0.164728
$472$		0			0
$473$			12.1348i			0.557956i
$474$		0			0
$475$		−	15.8975i		−	0.729425i
$476$		0			0
$477$			6.98346i			0.319750i
$478$		0			0
$479$			31.4877i			1.43871i	0.694643	+	0.719355i	$0.255562\pi$
$479$			31.4877i			1.43871i	−0.694643	+	0.719355i	$0.744438\pi$
$480$		0			0
$481$		−	1.39565i		−	0.0636362i
$482$		0			0
$483$	5.62831			0.256097
$484$		0			0
$485$			13.0792i			0.593896i
$486$		0			0
$487$	−19.9431			−0.903708			−0.451854	−	0.892092i	$-0.649237\pi$
$487$	−19.9431			−0.903708			−0.451854	+	0.892092i	$0.649237\pi$
$488$		0			0
$489$			31.4013i			1.42001i
$490$		0			0
$491$	−21.7545			−0.981768			−0.490884	−	0.871225i	$-0.663326\pi$
$491$	−21.7545			−0.981768			−0.490884	+	0.871225i	$0.663326\pi$
$492$		0			0
$493$	−5.50292			−0.247839
$494$		0			0
$495$			1.60657i			0.0722099i
$496$		0			0
$497$	13.1629			0.590438
$498$		0			0
$499$			25.9159i			1.16015i	0.814562	+	0.580077i	$0.196978\pi$
$499$			25.9159i			1.16015i	−0.814562	+	0.580077i	$0.803022\pi$
$500$		0			0
$501$	−32.2021			−1.43868
$502$		0			0
$503$			18.1880i			0.810961i	0.914104	+	0.405480i	$0.132896\pi$
$503$			18.1880i			0.810961i	−0.914104	+	0.405480i	$0.867104\pi$
$504$		0			0
$505$			4.28139i			0.190520i
$506$		0			0
$507$			14.3663i			0.638031i
$508$		0			0
$509$		−	4.58230i		−	0.203107i	−0.994830	−	0.101553i	$-0.967619\pi$
$509$		−	4.58230i		−	0.203107i	0.994830	−	0.101553i	$-0.0323813\pi$
$510$		0			0
$511$		−	8.45968i		−	0.374234i
$512$		0			0
$513$	−28.5012			−1.25836
$514$		0			0
$515$	−8.72160			−0.384320
$516$		0			0
$517$	15.4719			0.680454
$518$		0			0
$519$			26.3302i			1.15577i
$520$		0			0
$521$		−	4.35877i		−	0.190961i	−0.995431	−	0.0954806i	$-0.969561\pi$
$521$		−	4.35877i		−	0.190961i	0.995431	−	0.0954806i	$-0.0304388\pi$
$522$		0			0
$523$	16.3327			0.714179			0.357090	−	0.934070i	$-0.383769\pi$
$523$	16.3327			0.714179			0.357090	+	0.934070i	$0.383769\pi$
$524$		0			0
$525$	−4.43176			−0.193418
$526$		0			0
$527$			12.6673i			0.551797i
$528$		0			0
$529$	−6.94455			−0.301937
$530$		0			0
$531$	−7.93627			−0.344405
$532$		0			0
$533$	2.55781	+	10.3499i	0.110791	+	0.448304i
$534$		0			0
$535$	−3.65263			−0.157917
$536$		0			0
$537$	−18.7636			−0.809710
$538$		0			0
$539$			1.15173i			0.0496085i
$540$		0			0
$541$	28.9309			1.24384			0.621919	−	0.783082i	$-0.286353\pi$
$541$	28.9309			1.24384			0.621919	+	0.783082i	$0.286353\pi$
$542$		0			0
$543$	18.4789			0.793006
$544$		0			0
$545$			1.14597i			0.0490880i
$546$		0			0
$547$		−	23.9217i		−	1.02282i	−0.859337	−	0.511409i	$-0.829124\pi$
$547$		−	23.9217i		−	1.02282i	0.859337	−	0.511409i	$-0.170876\pi$
$548$		0			0
$549$	3.86741			0.165057
$550$		0			0
$551$	−12.1159			−0.516154
$552$		0			0
$553$	1.01755			0.0432707
$554$		0			0
$555$			1.59925i			0.0678843i
$556$		0			0
$557$			2.18023i			0.0923795i	0.998933	+	0.0461897i	$0.0147079\pi$
$557$			2.18023i			0.0923795i	−0.998933	+	0.0461897i	$0.985292\pi$
$558$		0			0
$559$		−	17.5427i		−	0.741979i
$560$		0			0
$561$			3.70231i			0.156312i
$562$		0			0
$563$			38.7732i			1.63409i	0.576571	+	0.817047i	$0.304391\pi$
$563$			38.7732i			1.63409i	−0.576571	+	0.817047i	$0.695609\pi$
$564$		0			0
$565$	−3.34557			−0.140749
$566$		0			0
$567$			4.86441i			0.204286i
$568$		0			0
$569$	11.0270			0.462278			0.231139	−	0.972921i	$-0.425755\pi$
$569$	11.0270			0.462278			0.231139	+	0.972921i	$0.425755\pi$
$570$		0			0
$571$			20.3493i			0.851592i	0.904819	+	0.425796i	$0.140006\pi$
$571$			20.3493i			0.851592i	−0.904819	+	0.425796i	$0.859994\pi$
$572$		0			0
$573$	−5.55748			−0.232167
$574$		0			0
$575$	−12.6421			−0.527214
$576$		0			0
$577$			20.9769i			0.873279i	0.899637	+	0.436640i	$0.143831\pi$
$577$			20.9769i			0.873279i	−0.899637	+	0.436640i	$0.856169\pi$
$578$		0			0
$579$	35.5344			1.47676
$580$		0			0
$581$			17.7169i			0.735020i
$582$		0			0
$583$	7.83181			0.324360
$584$		0			0
$585$		−	2.32255i		−	0.0960258i
$586$		0			0
$587$		−	37.6543i		−	1.55416i	−0.629402	−	0.777080i	$-0.716700\pi$
$587$		−	37.6543i		−	1.55416i	0.629402	−	0.777080i	$-0.283300\pi$
$588$		0			0
$589$			27.8899i			1.14918i
$590$		0			0
$591$		−	7.37574i		−	0.303398i
$592$		0			0
$593$			11.0466i			0.453629i	0.973938	+	0.226814i	$0.0728311\pi$
$593$			11.0466i			0.453629i	−0.973938	+	0.226814i	$0.927169\pi$
$594$		0			0
$595$	−3.10846			−0.127434
$596$		0			0
$597$	15.0695			0.616753
$598$		0			0
$599$	−10.6972			−0.437075			−0.218537	−	0.975829i	$-0.570129\pi$
$599$	−10.6972			−0.437075			−0.218537	+	0.975829i	$0.570129\pi$
$600$		0			0
$601$		−	14.4952i		−	0.591272i	−0.955301	−	0.295636i	$-0.904468\pi$
$601$		−	14.4952i		−	0.591272i	0.955301	−	0.295636i	$-0.0955316\pi$
$602$		0			0
$603$		−	6.21936i		−	0.253272i
$604$		0			0
$605$	−13.1394			−0.534191
$606$		0			0
$607$	2.74552			0.111437			0.0557186	−	0.998447i	$-0.482255\pi$
$607$	2.74552			0.111437			0.0557186	+	0.998447i	$0.482255\pi$
$608$		0			0
$609$			3.37756i			0.136866i
$610$		0			0
$611$	−22.3671			−0.904878
$612$		0			0
$613$	1.29092			0.0521396			0.0260698	−	0.999660i	$-0.491701\pi$
$613$	1.29092			0.0521396			0.0260698	+	0.999660i	$0.491701\pi$
$614$		0			0
$615$	−2.93094	−	11.8597i	−0.118187	−	0.478231i
$616$		0			0
$617$	26.6193			1.07165			0.535826	−	0.844328i	$-0.320000\pi$
$617$	26.6193			1.07165			0.535826	+	0.844328i	$0.320000\pi$
$618$		0			0
$619$	−6.50185			−0.261331			−0.130666	−	0.991426i	$-0.541711\pi$
$619$	−6.50185			−0.261331			−0.130666	+	0.991426i	$0.541711\pi$
$620$		0			0
$621$			22.6650i			0.909517i
$622$		0			0
$623$	7.88443			0.315883
$624$		0			0
$625$	0.729856			0.0291942
$626$		0			0
$627$			8.15146i			0.325538i
$628$		0			0
$629$		−	1.91830i		−	0.0764875i
$630$		0			0
$631$	14.4938			0.576988			0.288494	−	0.957482i	$-0.406845\pi$
$631$	14.4938			0.576988			0.288494	+	0.957482i	$0.406845\pi$
$632$		0			0
$633$	19.9818			0.794206
$634$		0			0
$635$	25.3884			1.00751
$636$		0			0
$637$		−	1.66501i		−	0.0659701i
$638$		0			0
$639$			13.5180i			0.534763i
$640$		0			0
$641$		−	42.0599i		−	1.66127i	−0.556820	−	0.830633i	$-0.687979\pi$
$641$		−	42.0599i		−	1.66127i	0.556820	−	0.830633i	$-0.312021\pi$
$642$		0			0
$643$			36.1067i			1.42391i	0.702225	+	0.711955i	$0.252190\pi$
$643$			36.1067i			1.42391i	−0.702225	+	0.711955i	$0.747810\pi$
$644$		0			0
$645$			20.1019i			0.791510i
$646$		0			0
$647$	−5.49156			−0.215895			−0.107948	−	0.994157i	$-0.534428\pi$
$647$	−5.49156			−0.215895			−0.107948	+	0.994157i	$0.534428\pi$
$648$		0			0
$649$			8.90037i			0.349370i
$650$		0			0
$651$	7.77491			0.304723
$652$		0			0
$653$			41.9683i			1.64235i	0.570680	+	0.821173i	$0.306680\pi$
$653$			41.9683i			1.64235i	−0.570680	+	0.821173i	$0.693320\pi$
$654$		0			0
$655$	4.98927			0.194947
$656$		0			0
$657$	8.68786			0.338946
$658$		0			0
$659$		−	14.4600i		−	0.563283i	−0.959520	−	0.281641i	$-0.909121\pi$
$659$		−	14.4600i		−	0.563283i	0.959520	−	0.281641i	$-0.0908788\pi$
$660$		0			0
$661$	11.5392			0.448824			0.224412	−	0.974494i	$-0.427954\pi$
$661$	11.5392			0.448824			0.224412	+	0.974494i	$0.427954\pi$
$662$		0			0
$663$		−	5.35229i		−	0.207866i
$664$		0			0
$665$	−6.84396			−0.265398
$666$		0			0
$667$			9.63493i			0.373066i
$668$		0			0
$669$			26.4515i			1.02267i
$670$		0			0
$671$		−	4.33722i		−	0.167437i
$672$		0			0
$673$		−	32.0633i		−	1.23595i	−0.786198	−	0.617975i	$-0.787953\pi$
$673$		−	32.0633i		−	1.23595i	0.786198	−	0.617975i	$-0.212047\pi$
$674$		0			0
$675$		−	17.8466i		−	0.686915i
$676$		0			0
$677$	9.06923			0.348559			0.174279	−	0.984696i	$-0.444240\pi$
$677$	9.06923			0.348559			0.174279	+	0.984696i	$0.444240\pi$
$678$		0			0
$679$	−9.62923			−0.369536
$680$		0			0
$681$	1.97593			0.0757178
$682$		0			0
$683$		−	2.83942i		−	0.108647i	−0.998523	−	0.0543237i	$-0.982700\pi$
$683$		−	2.83942i		−	0.108647i	0.998523	−	0.0543237i	$-0.0173003\pi$
$684$		0			0
$685$			18.6053i			0.710872i
$686$		0			0
$687$	−22.2000			−0.846982
$688$		0			0
$689$	−11.3221			−0.431339
$690$		0			0
$691$		−	10.8854i		−	0.414098i	−0.978331	−	0.207049i	$-0.933614\pi$
$691$		−	10.8854i		−	0.414098i	0.978331	−	0.207049i	$-0.0663860\pi$
$692$		0			0
$693$	−1.18280			−0.0449307
$694$		0			0
$695$	−24.4408			−0.927093
$696$		0			0
$697$	3.51566	+	14.2257i	0.133165	+	0.538838i
$698$		0			0
$699$	1.55696			0.0588895
$700$		0			0
$701$	−15.5887			−0.588775			−0.294388	−	0.955686i	$-0.595116\pi$
$701$	−15.5887			−0.588775			−0.294388	+	0.955686i	$0.595116\pi$
$702$		0			0
$703$		−	4.22355i		−	0.159294i
$704$		0			0
$705$	25.6300			0.965283
$706$		0			0
$707$	−3.15207			−0.118546
$708$		0			0
$709$			21.5808i			0.810485i	0.914209	+	0.405242i	$0.132813\pi$
$709$			21.5808i			0.810485i	−0.914209	+	0.405242i	$0.867187\pi$
$710$		0			0
$711$			1.04500i			0.0391905i
$712$		0			0
$713$	22.1789			0.830606
$714$		0			0
$715$	−2.60470			−0.0974102
$716$		0			0
$717$	24.8338			0.927437
$718$		0			0
$719$			6.91985i			0.258067i	0.991640	+	0.129033i	$0.0411874\pi$
$719$			6.91985i			0.258067i	−0.991640	+	0.129033i	$0.958813\pi$
$720$		0			0
$721$		−	6.42106i		−	0.239133i
$722$		0			0
$723$		−	12.5613i		−	0.467158i
$724$		0			0
$725$		−	7.58659i		−	0.281759i
$726$		0			0
$727$			25.5733i			0.948462i	0.880401	+	0.474231i	$0.157274\pi$
$727$			25.5733i			0.948462i	−0.880401	+	0.474231i	$0.842726\pi$
$728$		0			0
$729$	−28.8316			−1.06784
$730$		0			0
$731$		−	24.1122i		−	0.891820i
$732$		0			0
$733$	−31.6366			−1.16852			−0.584262	−	0.811565i	$-0.698616\pi$
$733$	−31.6366			−1.16852			−0.584262	+	0.811565i	$0.698616\pi$
$734$		0			0
$735$			1.90790i			0.0703740i
$736$		0			0
$737$	−6.97488			−0.256923
$738$		0			0
$739$	15.6332			0.575077			0.287539	−	0.957769i	$-0.407163\pi$
$739$	15.6332			0.575077			0.287539	+	0.957769i	$0.407163\pi$
$740$		0			0
$741$		−	11.7842i		−	0.432905i
$742$		0			0
$743$	−33.9097			−1.24403			−0.622013	−	0.783007i	$-0.713685\pi$
$743$	−33.9097			−1.24403			−0.622013	+	0.783007i	$0.713685\pi$
$744$		0			0
$745$			10.8374i			0.397053i
$746$		0			0
$747$	−18.1948			−0.665711
$748$		0			0
$749$		−	2.68916i		−	0.0982596i
$750$		0			0
$751$		−	27.7263i		−	1.01175i	−0.862607	−	0.505874i	$-0.831170\pi$
$751$		−	27.7263i		−	1.01175i	0.862607	−	0.505874i	$-0.168830\pi$
$752$		0			0
$753$		−	10.1233i		−	0.368915i
$754$		0			0
$755$		−	6.15336i		−	0.223944i
$756$		0			0
$757$		−	17.9832i		−	0.653611i	−0.945092	−	0.326806i	$-0.894028\pi$
$757$		−	17.9832i		−	0.653611i	0.945092	−	0.326806i	$-0.105972\pi$
$758$		0			0
$759$	6.48229			0.235292
$760$		0			0
$761$	34.5712			1.25320			0.626602	−	0.779340i	$-0.284445\pi$
$761$	34.5712			1.25320			0.626602	+	0.779340i	$0.284445\pi$
$762$		0			0
$763$	−0.843693			−0.0305437
$764$		0			0
$765$		−	3.19231i		−	0.115418i
$766$		0			0
$767$		−	12.8669i		−	0.464597i
$768$		0			0
$769$	−17.4330			−0.628649			−0.314324	−	0.949316i	$-0.601778\pi$
$769$	−17.4330			−0.628649			−0.314324	+	0.949316i	$0.601778\pi$
$770$		0			0
$771$	−7.86422			−0.283223
$772$		0			0
$773$		−	1.09666i		−	0.0394441i	−0.999806	−	0.0197220i	$-0.993722\pi$
$773$		−	1.09666i		−	0.0394441i	0.999806	−	0.0197220i	$-0.00627813\pi$
$774$		0			0
$775$	−17.4638			−0.627317
$776$		0			0
$777$	−1.17741			−0.0422392
$778$		0			0
$779$	7.74051	+	31.3211i	0.277333	+	1.12219i
$780$		0			0
$781$	15.1601			0.542472
$782$		0			0
$783$	−13.6014			−0.486073
$784$		0			0
$785$		−	3.45700i		−	0.123386i
$786$		0			0
$787$	−32.3899			−1.15458			−0.577288	−	0.816541i	$-0.695889\pi$
$787$	−32.3899			−1.15458			−0.577288	+	0.816541i	$0.695889\pi$
$788$		0			0
$789$	−20.3360			−0.723979
$790$		0			0
$791$		−	2.46309i		−	0.0875775i
$792$		0			0
$793$			6.27015i			0.222660i
$794$		0			0
$795$	12.9738			0.460133
$796$		0			0
$797$	−35.3364			−1.25168			−0.625840	−	0.779951i	$-0.715244\pi$
$797$	−35.3364			−1.25168			−0.625840	+	0.779951i	$0.715244\pi$
$798$		0			0
$799$	−30.7432			−1.08762
$800$		0			0
$801$			8.09710i			0.286097i
$802$		0			0
$803$		−	9.74326i		−	0.343832i
$804$		0			0
$805$			5.44253i			0.191824i
$806$		0			0
$807$			1.28336i			0.0451763i
$808$		0			0
$809$			6.16729i			0.216830i	0.994106	+	0.108415i	$0.0345776\pi$
$809$			6.16729i			0.216830i	−0.994106	+	0.108415i	$0.965422\pi$
$810$		0			0
$811$	−16.3704			−0.574842			−0.287421	−	0.957804i	$-0.592798\pi$
$811$	−16.3704			−0.574842			−0.287421	+	0.957804i	$0.592798\pi$
$812$		0			0
$813$			0.442410i			0.0155160i
$814$		0			0
$815$	−30.3648			−1.06363
$816$		0			0
$817$		−	53.0883i		−	1.85732i
$818$		0			0
$819$	1.70992			0.0597495
$820$		0			0
$821$	33.0231			1.15251			0.576257	−	0.817268i	$-0.304513\pi$
$821$	33.0231			1.15251			0.576257	+	0.817268i	$0.304513\pi$
$822$		0			0
$823$			10.6841i			0.372423i	0.982510	+	0.186212i	$0.0596210\pi$
$823$			10.6841i			0.372423i	−0.982510	+	0.186212i	$0.940379\pi$
$824$		0			0
$825$	−5.10419			−0.177705
$826$		0			0
$827$			20.7578i			0.721819i	0.932601	+	0.360910i	$0.117534\pi$
$827$			20.7578i			0.721819i	−0.932601	+	0.360910i	$0.882466\pi$
$828$		0			0
$829$	16.7353			0.581242			0.290621	−	0.956838i	$-0.406138\pi$
$829$	16.7353			0.581242			0.290621	+	0.956838i	$0.406138\pi$
$830$		0			0
$831$			7.29528i			0.253070i
$832$		0			0
$833$		−	2.28853i		−	0.0792927i
$834$		0			0
$835$		−	31.1392i		−	1.07762i
$836$		0			0
$837$			31.3093i			1.08221i
$838$		0			0
$839$		−	3.64975i		−	0.126003i	−0.998013	−	0.0630016i	$-0.979933\pi$
$839$		−	3.64975i		−	0.126003i	0.998013	−	0.0630016i	$-0.0200673\pi$
$840$		0			0
$841$	23.2181			0.800622
$842$		0			0
$843$	30.0778			1.03593
$844$		0			0
$845$	−13.8921			−0.477904
$846$		0			0
$847$		−	9.67352i		−	0.332386i
$848$		0			0
$849$		−	29.9599i		−	1.02822i
$850$		0			0
$851$	−3.35870			−0.115135
$852$		0			0
$853$	−9.80145			−0.335595			−0.167798	−	0.985821i	$-0.553666\pi$
$853$	−9.80145			−0.335595			−0.167798	+	0.985821i	$0.553666\pi$
$854$		0			0
$855$		−	7.02857i		−	0.240372i
$856$		0			0
$857$	−1.30429			−0.0445538			−0.0222769	−	0.999752i	$-0.507092\pi$
$857$	−1.30429			−0.0445538			−0.0222769	+	0.999752i	$0.507092\pi$
$858$		0			0
$859$	15.8869			0.542054			0.271027	−	0.962572i	$-0.412637\pi$
$859$	15.8869			0.542054			0.271027	+	0.962572i	$0.412637\pi$
$860$		0			0
$861$	8.73143	−	2.15783i	0.297566	−	0.0735388i
$862$		0			0
$863$	−2.41503			−0.0822084			−0.0411042	−	0.999155i	$-0.513088\pi$
$863$	−2.41503			−0.0822084			−0.0411042	+	0.999155i	$0.513088\pi$
$864$		0			0
$865$	−25.4611			−0.865703
$866$		0			0
$867$			16.5223i			0.561128i
$868$		0			0
$869$	1.17195			0.0397555
$870$		0			0
$871$	10.0833			0.341660
$872$		0			0
$873$		−	9.88896i		−	0.334691i
$874$		0			0
$875$		−	11.0769i		−	0.374467i
$876$		0			0
$877$	−3.76190			−0.127030			−0.0635152	−	0.997981i	$-0.520231\pi$
$877$	−3.76190			−0.127030			−0.0635152	+	0.997981i	$0.520231\pi$
$878$		0			0
$879$	29.7907			1.00482
$880$		0			0
$881$	2.32140			0.0782099			0.0391050	−	0.999235i	$-0.487549\pi$
$881$	2.32140			0.0782099			0.0391050	+	0.999235i	$0.487549\pi$
$882$		0			0
$883$		−	44.8647i		−	1.50982i	−0.655829	−	0.754909i	$-0.727681\pi$
$883$		−	44.8647i		−	1.50982i	0.655829	−	0.754909i	$-0.272319\pi$
$884$		0			0
$885$			14.7439i			0.495612i
$886$		0			0
$887$			15.4389i			0.518389i	0.965825	+	0.259194i	$0.0834571\pi$
$887$			15.4389i			0.518389i	−0.965825	+	0.259194i	$0.916543\pi$
$888$		0			0
$889$			18.6916i			0.626896i
$890$		0			0
$891$			5.60248i			0.187690i
$892$		0			0
$893$	−67.6880			−2.26509
$894$		0			0
$895$		−	18.1443i		−	0.606497i
$896$		0			0
$897$	−9.37119			−0.312895
$898$		0			0
$899$			13.3096i			0.443901i
$900$		0			0
$901$	−15.5621			−0.518447
$902$		0			0
$903$	−14.7995			−0.492496
$904$		0			0
$905$			17.8690i			0.593985i
$906$		0			0
$907$	10.2137			0.339140			0.169570	−	0.985518i	$-0.445762\pi$
$907$	10.2137			0.339140			0.169570	+	0.985518i	$0.445762\pi$
$908$		0			0
$909$		−	3.23709i		−	0.107367i
$910$		0			0
$911$	26.6314			0.882337			0.441169	−	0.897424i	$-0.354564\pi$
$911$	26.6314			0.882337			0.441169	+	0.897424i	$0.354564\pi$
$912$		0			0
$913$			20.4051i			0.675309i
$914$		0			0
$915$		−	7.18484i		−	0.237523i
$916$		0			0
$917$			3.67323i			0.121301i
$918$		0			0
$919$		−	47.2508i		−	1.55866i	−0.626614	−	0.779330i	$-0.715560\pi$
$919$		−	47.2508i		−	1.55866i	0.626614	−	0.779330i	$-0.284440\pi$
$920$		0			0
$921$		−	9.75779i		−	0.321530i
$922$		0			0
$923$	−21.9164			−0.721388
$924$		0			0
$925$	2.64466			0.0869558
$926$		0			0
$927$	6.59425			0.216584
$928$		0			0
$929$			24.2164i			0.794515i	0.917707	+	0.397257i	$0.130038\pi$
$929$			24.2164i			0.794515i	−0.917707	+	0.397257i	$0.869962\pi$
$930$		0			0
$931$		−	5.03870i		−	0.165137i
$932$		0			0
$933$	26.0312			0.852223
$934$		0			0
$935$	−3.58011			−0.117082
$936$		0			0
$937$		−	3.38219i		−	0.110491i	−0.998473	−	0.0552456i	$-0.982406\pi$
$937$		−	3.38219i		−	0.110491i	0.998473	−	0.0552456i	$-0.0175942\pi$
$938$		0			0
$939$	37.6235			1.22780
$940$		0			0
$941$	30.6330			0.998608			0.499304	−	0.866427i	$-0.333589\pi$
$941$	30.6330			0.998608			0.499304	+	0.866427i	$0.333589\pi$
$942$		0			0
$943$	24.9075	−	6.15549i	0.811100	−	0.200450i
$944$		0			0
$945$	−7.68307			−0.249930
$946$		0			0
$947$	1.53839			0.0499910			0.0249955	−	0.999688i	$-0.492043\pi$
$947$	1.53839			0.0499910			0.0249955	+	0.999688i	$0.492043\pi$
$948$		0			0
$949$			14.0855i			0.457233i
$950$		0			0
$951$	6.45218			0.209226
$952$		0			0
$953$	33.4184			1.08253			0.541264	−	0.840853i	$-0.317946\pi$
$953$	33.4184			1.08253			0.541264	+	0.840853i	$0.317946\pi$
$954$		0			0
$955$		−	5.37404i		−	0.173900i
$956$		0			0
$957$			3.89004i			0.125747i
$958$		0			0
$959$	−13.6977			−0.442321
$960$		0			0
$961$	−0.362214			−0.0116843
$962$		0			0
$963$	2.76169			0.0889943
$964$		0			0
$965$			34.3615i			1.10614i
$966$		0			0
$967$			23.7245i			0.762927i	0.924384	+	0.381464i	$0.124580\pi$
$967$			23.7245i			0.762927i	−0.924384	+	0.381464i	$0.875420\pi$
$968$		0			0
$969$		−	16.1972i		−	0.520330i
$970$		0			0
$971$			21.8464i			0.701085i	0.936547	+	0.350543i	$0.114003\pi$
$971$			21.8464i			0.701085i	−0.936547	+	0.350543i	$0.885997\pi$
$972$		0			0
$973$		−	17.9939i		−	0.576859i
$974$		0			0
$975$	7.37893			0.236315
$976$		0			0
$977$			30.4314i			0.973585i	0.873518	+	0.486793i	$0.161833\pi$
$977$			30.4314i			0.973585i	−0.873518	+	0.486793i	$0.838167\pi$
$978$		0			0
$979$	9.08073			0.290221
$980$		0			0
$981$		−	0.866450i		−	0.0276636i
$982$		0			0
$983$	10.4012			0.331748			0.165874	−	0.986147i	$-0.446956\pi$
$983$	10.4012			0.331748			0.165874	+	0.986147i	$0.446956\pi$
$984$		0			0
$985$	7.13229			0.227254
$986$		0			0
$987$			18.8695i			0.600622i
$988$		0			0
$989$	−42.2174			−1.34244
$990$		0			0
$991$		−	53.6551i		−	1.70441i	−0.523207	−	0.852206i	$-0.675264\pi$
$991$		−	53.6551i		−	1.70441i	0.523207	−	0.852206i	$-0.324736\pi$
$992$		0			0
$993$	41.0098			1.30141
$994$		0			0
$995$			14.5721i			0.461966i
$996$		0			0
$997$		−	39.6109i		−	1.25449i	−0.778822	−	0.627245i	$-0.784183\pi$
$997$		−	39.6109i		−	1.25449i	0.778822	−	0.627245i	$-0.215817\pi$
$998$		0			0
$999$		−	4.74138i		−	0.150011i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.2.d.a.1065.15	yes	20
41.40	even	2	inner	1148.2.d.a.1065.6	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.d.a.1065.6	✓	20	41.40	even	2	inner
1148.2.d.a.1065.15	yes	20	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1148.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.40464i q^{3} -1.35828 q^{5} -1.00000i q^{7} +1.02697 q^{9} +O(q^{10})\)	\(q+1.40464i q^{3} -1.35828 q^{5} -1.00000i q^{7} +1.02697 q^{9} -1.15173i q^{11} +1.66501i q^{13} -1.90790i q^{15} +2.28853i q^{17} +5.03870i q^{19} +1.40464 q^{21} +4.00693 q^{23} -3.15507 q^{25} +5.65647i q^{27} +2.40457i q^{29} +5.53514 q^{31} +1.61777 q^{33} +1.35828i q^{35} -0.838223 q^{37} -2.33875 q^{39} +(6.21611 - 1.53621i) q^{41} -10.5361 q^{43} -1.39492 q^{45} +13.4336i q^{47} -1.00000 q^{49} -3.21457 q^{51} +6.80004i q^{53} +1.56437i q^{55} -7.07758 q^{57} -7.72782 q^{59} +3.76583 q^{61} -1.02697i q^{63} -2.26155i q^{65} -6.05601i q^{67} +5.62831i q^{69} +13.1629i q^{71} +8.45968 q^{73} -4.43176i q^{75} -1.15173 q^{77} +1.01755i q^{79} -4.86441 q^{81} -17.7169 q^{83} -3.10846i q^{85} -3.37756 q^{87} +7.88443i q^{89} +1.66501 q^{91} +7.77491i q^{93} -6.84396i q^{95} -9.62923i q^{97} -1.18280i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{5} - 20 q^{9}+O(q^{10}) \) 20 * q + 4 * q^5 - 20 * q^9	\( 20 q + 4 q^{5} - 20 q^{9} + 4 q^{21} + 8 q^{31} + 20 q^{37} + 4 q^{39} - 16 q^{41} + 20 q^{43} - 4 q^{45} - 20 q^{49} + 52 q^{51} - 36 q^{57} + 20 q^{59} - 4 q^{61} - 12 q^{73} + 8 q^{77} + 20 q^{81} - 48 q^{83} + 44 q^{87} - 4 q^{91}+O(q^{100}) \) 20 * q + 4 * q^5 - 20 * q^9 + 4 * q^21 + 8 * q^31 + 20 * q^37 + 4 * q^39 - 16 * q^41 + 20 * q^43 - 4 * q^45 - 20 * q^49 + 52 * q^51 - 36 * q^57 + 20 * q^59 - 4 * q^61 - 12 * q^73 + 8 * q^77 + 20 * q^81 - 48 * q^83 + 44 * q^87 - 4 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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