LMFDB - Embedded newform 448.2.b.d.225.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,2,Mod(225,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("448.225");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	448.b (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:	$3.57729801055$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			225.3
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	448.225
Dual form			448.2.b.d.225.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051i q^{3} -0.732051i q^{5} +1.00000 q^{7} +2.46410 q^{9} +O(q^{10})$	$q+0.732051i q^{3} -0.732051i q^{5} +1.00000 q^{7} +2.46410 q^{9} +1.46410i q^{11} -3.26795i q^{13} +0.535898 q^{15} +2.00000 q^{17} +4.73205i q^{19} +0.732051i q^{21} +3.46410 q^{23} +4.46410 q^{25} +4.00000i q^{27} -5.46410i q^{29} +4.00000 q^{31} -1.07180 q^{33} -0.732051i q^{35} +5.46410i q^{37} +2.39230 q^{39} -2.00000 q^{41} +1.46410i q^{43} -1.80385i q^{45} -10.9282 q^{47} +1.00000 q^{49} +1.46410i q^{51} +12.0000i q^{53} +1.07180 q^{55} -3.46410 q^{57} -7.66025i q^{59} -13.1244i q^{61} +2.46410 q^{63} -2.39230 q^{65} -8.00000i q^{67} +2.53590i q^{69} -10.9282 q^{71} -0.928203 q^{73} +3.26795i q^{75} +1.46410i q^{77} +2.92820 q^{79} +4.46410 q^{81} -11.6603i q^{83} -1.46410i q^{85} +4.00000 q^{87} -15.8564 q^{89} -3.26795i q^{91} +2.92820i q^{93} +3.46410 q^{95} -4.92820 q^{97} +3.60770i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^7 - 4 * q^9	$ 4 q + 4 q^{7} - 4 q^{9} + 16 q^{15} + 8 q^{17} + 4 q^{25} + 16 q^{31} - 32 q^{33} - 32 q^{39} - 8 q^{41} - 16 q^{47} + 4 q^{49} + 32 q^{55} - 4 q^{63} + 32 q^{65} - 16 q^{71} + 24 q^{73} - 16 q^{79} + 4 q^{81} + 16 q^{87} - 8 q^{89} + 8 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 - 4 * q^9 + 16 * q^15 + 8 * q^17 + 4 * q^25 + 16 * q^31 - 32 * q^33 - 32 * q^39 - 8 * q^41 - 16 * q^47 + 4 * q^49 + 32 * q^55 - 4 * q^63 + 32 * q^65 - 16 * q^71 + 24 * q^73 - 16 * q^79 + 4 * q^81 + 16 * q^87 - 8 * q^89 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	0.732051i	0.422650i	0.977416	+	0.211325i	$0.0677778\pi$
$3$	0.732051i	0.422650i	−0.977416	+	0.211325i	$0.932222\pi$
$4$	0	0
$5$	− 0.732051i	− 0.327383i	−0.986512	−	0.163692i	$-0.947660\pi$
$5$	− 0.732051i	− 0.327383i	0.986512	−	0.163692i	$-0.0523402\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	2.46410	0.821367
$10$	0	0
$11$	1.46410i	0.441443i	0.975337	+	0.220722i	$0.0708412\pi$
$11$	1.46410i	0.441443i	−0.975337	+	0.220722i	$0.929159\pi$
$12$	0	0
$13$	− 3.26795i	− 0.906366i	−0.891417	−	0.453183i	$-0.850288\pi$
$13$	− 3.26795i	− 0.906366i	0.891417	−	0.453183i	$-0.149712\pi$
$14$	0	0
$15$	0.535898	0.138368
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	4.73205i	1.08561i	0.839860	+	0.542803i	$0.182637\pi$
$19$	4.73205i	1.08561i	−0.839860	+	0.542803i	$0.817363\pi$
$20$	0	0
$21$	0.732051i	0.159747i
$22$	0	0
$23$	3.46410	0.722315	0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	0	0
$25$	4.46410	0.892820
$26$	0	0
$27$	4.00000i	0.769800i
$28$	0	0
$29$	− 5.46410i	− 1.01466i	−0.861752	−	0.507329i	$-0.830633\pi$
$29$	− 5.46410i	− 1.01466i	0.861752	−	0.507329i	$-0.169367\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	−1.07180	−0.186576
$34$	0	0
$35$	− 0.732051i	− 0.123739i
$36$	0	0
$37$	5.46410i	0.898293i	0.893458	+	0.449146i	$0.148272\pi$
$37$	5.46410i	0.898293i	−0.893458	+	0.449146i	$0.851728\pi$
$38$	0	0
$39$	2.39230	0.383075
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	1.46410i	0.223273i	0.993749	+	0.111637i	$0.0356093\pi$
$43$	1.46410i	0.223273i	−0.993749	+	0.111637i	$0.964391\pi$
$44$	0	0
$45$	− 1.80385i	− 0.268902i
$46$	0	0
$47$	−10.9282	−1.59404	−0.797021	−	0.603951i	$-0.793592\pi$
$47$	−10.9282	−1.59404	−0.797021	+	0.603951i	$0.793592\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.46410i	0.205015i
$52$	0	0
$53$	12.0000i	1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$	12.0000i	1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$	0	0
$55$	1.07180	0.144521
$56$	0	0
$57$	−3.46410	−0.458831
$58$	0	0
$59$	− 7.66025i	− 0.997280i	−0.866809	−	0.498640i	$-0.833833\pi$
$59$	− 7.66025i	− 0.997280i	0.866809	−	0.498640i	$-0.166167\pi$
$60$	0	0
$61$	− 13.1244i	− 1.68040i	−0.542275	−	0.840201i	$-0.682437\pi$
$61$	− 13.1244i	− 1.68040i	0.542275	−	0.840201i	$-0.317563\pi$
$62$	0	0
$63$	2.46410	0.310448
$64$	0	0
$65$	−2.39230	−0.296729
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	0	0
$69$	2.53590i	0.305286i
$70$	0	0
$71$	−10.9282	−1.29694	−0.648470	−	0.761241i	$-0.724591\pi$
$71$	−10.9282	−1.29694	−0.648470	+	0.761241i	$0.724591\pi$
$72$	0	0
$73$	−0.928203	−0.108638	−0.0543190	−	0.998524i	$-0.517299\pi$
$73$	−0.928203	−0.108638	−0.0543190	+	0.998524i	$0.517299\pi$
$74$	0	0
$75$	3.26795i	0.377350i
$76$	0	0
$77$	1.46410i	0.166850i
$78$	0	0
$79$	2.92820	0.329449	0.164724	−	0.986340i	$-0.447327\pi$
$79$	2.92820	0.329449	0.164724	+	0.986340i	$0.447327\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	− 11.6603i	− 1.27988i	−0.768425	−	0.639940i	$-0.778959\pi$
$83$	− 11.6603i	− 1.27988i	0.768425	−	0.639940i	$-0.221041\pi$
$84$	0	0
$85$	− 1.46410i	− 0.158804i
$86$	0	0
$87$	4.00000	0.428845
$88$	0	0
$89$	−15.8564	−1.68078	−0.840388	−	0.541985i	$-0.817673\pi$
$89$	−15.8564	−1.68078	−0.840388	+	0.541985i	$0.817673\pi$
$90$	0	0
$91$	− 3.26795i	− 0.342574i
$92$	0	0
$93$	2.92820i	0.303641i
$94$	0	0
$95$	3.46410	0.355409
$96$	0	0
$97$	−4.92820	−0.500383	−0.250192	−	0.968196i	$-0.580494\pi$
$97$	−4.92820	−0.500383	−0.250192	+	0.968196i	$0.580494\pi$
$98$	0	0
$99$	3.60770i	0.362587i
$100$	0	0
$101$	15.6603i	1.55825i	0.626866	+	0.779127i	$0.284337\pi$
$101$	15.6603i	1.55825i	−0.626866	+	0.779127i	$0.715663\pi$
$102$	0	0
$103$	−17.8564	−1.75944	−0.879722	−	0.475488i	$-0.842271\pi$
$103$	−17.8564	−1.75944	−0.879722	+	0.475488i	$0.842271\pi$
$104$	0	0
$105$	0.535898	0.0522983
$106$	0	0
$107$	− 18.9282i	− 1.82986i	−0.403614	−	0.914929i	$-0.632246\pi$
$107$	− 18.9282i	− 1.82986i	0.403614	−	0.914929i	$-0.367754\pi$
$108$	0	0
$109$	8.39230i	0.803837i	0.915675	+	0.401919i	$0.131656\pi$
$109$	8.39230i	0.803837i	−0.915675	+	0.401919i	$0.868344\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	−13.4641	−1.26660	−0.633298	−	0.773908i	$-0.718299\pi$
$113$	−13.4641	−1.26660	−0.633298	+	0.773908i	$0.718299\pi$
$114$	0	0
$115$	− 2.53590i	− 0.236474i
$116$	0	0
$117$	− 8.05256i	− 0.744459i
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	8.85641	0.805128
$122$	0	0
$123$	− 1.46410i	− 0.132014i
$124$	0	0
$125$	− 6.92820i	− 0.619677i
$126$	0	0
$127$	−4.53590	−0.402496	−0.201248	−	0.979540i	$-0.564500\pi$
$127$	−4.53590	−0.402496	−0.201248	+	0.979540i	$0.564500\pi$
$128$	0	0
$129$	−1.07180	−0.0943664
$130$	0	0
$131$	7.26795i	0.635004i	0.948258	+	0.317502i	$0.102844\pi$
$131$	7.26795i	0.635004i	−0.948258	+	0.317502i	$0.897156\pi$
$132$	0	0
$133$	4.73205i	0.410321i
$134$	0	0
$135$	2.92820	0.252020
$136$	0	0
$137$	−7.85641	−0.671218	−0.335609	−	0.942001i	$-0.608942\pi$
$137$	−7.85641	−0.671218	−0.335609	+	0.942001i	$0.608942\pi$
$138$	0	0
$139$	12.7321i	1.07992i	0.841691	+	0.539959i	$0.181560\pi$
$139$	12.7321i	1.07992i	−0.841691	+	0.539959i	$0.818440\pi$
$140$	0	0
$141$	− 8.00000i	− 0.673722i
$142$	0	0
$143$	4.78461	0.400109
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	0.732051i	0.0603785i
$148$	0	0
$149$	14.9282i	1.22297i	0.791258	+	0.611483i	$0.209427\pi$
$149$	14.9282i	1.22297i	−0.791258	+	0.611483i	$0.790573\pi$
$150$	0	0
$151$	18.3923	1.49674	0.748372	−	0.663279i	$-0.230836\pi$
$151$	18.3923	1.49674	0.748372	+	0.663279i	$0.230836\pi$
$152$	0	0
$153$	4.92820	0.398422
$154$	0	0
$155$	− 2.92820i	− 0.235199i
$156$	0	0
$157$	1.80385i	0.143963i	0.997406	+	0.0719814i	$0.0229322\pi$
$157$	1.80385i	0.143963i	−0.997406	+	0.0719814i	$0.977068\pi$
$158$	0	0
$159$	−8.78461	−0.696665
$160$	0	0
$161$	3.46410	0.273009
$162$	0	0
$163$	6.53590i	0.511931i	0.966686	+	0.255966i	$0.0823934\pi$
$163$	6.53590i	0.511931i	−0.966686	+	0.255966i	$0.917607\pi$
$164$	0	0
$165$	0.784610i	0.0610818i
$166$	0	0
$167$	−18.9282	−1.46471	−0.732354	−	0.680924i	$-0.761578\pi$
$167$	−18.9282	−1.46471	−0.732354	+	0.680924i	$0.761578\pi$
$168$	0	0
$169$	2.32051	0.178501
$170$	0	0
$171$	11.6603i	0.891682i
$172$	0	0
$173$	− 14.1962i	− 1.07931i	−0.841885	−	0.539657i	$-0.818554\pi$
$173$	− 14.1962i	− 1.07931i	0.841885	−	0.539657i	$-0.181446\pi$
$174$	0	0
$175$	4.46410	0.337454
$176$	0	0
$177$	5.60770	0.421500
$178$	0	0
$179$	− 10.9282i	− 0.816812i	−0.912800	−	0.408406i	$-0.866085\pi$
$179$	− 10.9282i	− 0.816812i	0.912800	−	0.408406i	$-0.133915\pi$
$180$	0	0
$181$	− 7.26795i	− 0.540222i	−0.962829	−	0.270111i	$-0.912940\pi$
$181$	− 7.26795i	− 0.540222i	0.962829	−	0.270111i	$-0.0870605\pi$
$182$	0	0
$183$	9.60770	0.710221
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	2.92820i	0.214131i
$188$	0	0
$189$	4.00000i	0.290957i
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	0	0
$193$	13.4641	0.969167	0.484584	−	0.874745i	$-0.338971\pi$
$193$	13.4641	0.969167	0.484584	+	0.874745i	$0.338971\pi$
$194$	0	0
$195$	− 1.75129i	− 0.125412i
$196$	0	0
$197$	− 1.85641i	− 0.132263i	−0.997811	−	0.0661317i	$-0.978934\pi$
$197$	− 1.85641i	− 0.132263i	0.997811	−	0.0661317i	$-0.0210658\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	5.85641	0.413079
$202$	0	0
$203$	− 5.46410i	− 0.383505i
$204$	0	0
$205$	1.46410i	0.102257i
$206$	0	0
$207$	8.53590	0.593286
$208$	0	0
$209$	−6.92820	−0.479234
$210$	0	0
$211$	24.7846i	1.70624i	0.521712	+	0.853121i	$0.325293\pi$
$211$	24.7846i	1.70624i	−0.521712	+	0.853121i	$0.674707\pi$
$212$	0	0
$213$	− 8.00000i	− 0.548151i
$214$	0	0
$215$	1.07180	0.0730959
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	− 0.679492i	− 0.0459158i
$220$	0	0
$221$	− 6.53590i	− 0.439652i
$222$	0	0
$223$	6.92820	0.463947	0.231973	−	0.972722i	$-0.425482\pi$
$223$	6.92820	0.463947	0.231973	+	0.972722i	$0.425482\pi$
$224$	0	0
$225$	11.0000	0.733333
$226$	0	0
$227$	− 14.1962i	− 0.942232i	−0.882071	−	0.471116i	$-0.843851\pi$
$227$	− 14.1962i	− 0.942232i	0.882071	−	0.471116i	$-0.156149\pi$
$228$	0	0
$229$	− 20.7321i	− 1.37001i	−0.728537	−	0.685006i	$-0.759799\pi$
$229$	− 20.7321i	− 1.37001i	0.728537	−	0.685006i	$-0.240201\pi$
$230$	0	0
$231$	−1.07180	−0.0705191
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	8.00000i	0.521862i
$236$	0	0
$237$	2.14359i	0.139241i
$238$	0	0
$239$	−19.4641	−1.25903	−0.629514	−	0.776989i	$-0.716746\pi$
$239$	−19.4641	−1.25903	−0.629514	+	0.776989i	$0.716746\pi$
$240$	0	0
$241$	22.7846	1.46769	0.733843	−	0.679319i	$-0.237725\pi$
$241$	22.7846	1.46769	0.733843	+	0.679319i	$0.237725\pi$
$242$	0	0
$243$	15.2679i	0.979439i
$244$	0	0
$245$	− 0.732051i	− 0.0467690i
$246$	0	0
$247$	15.4641	0.983957
$248$	0	0
$249$	8.53590	0.540941
$250$	0	0
$251$	26.1962i	1.65349i	0.562579	+	0.826743i	$0.309809\pi$
$251$	26.1962i	1.65349i	−0.562579	+	0.826743i	$0.690191\pi$
$252$	0	0
$253$	5.07180i	0.318861i
$254$	0	0
$255$	1.07180	0.0671185
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	5.46410i	0.339523i
$260$	0	0
$261$	− 13.4641i	− 0.833407i
$262$	0	0
$263$	−2.92820	−0.180561	−0.0902804	−	0.995916i	$-0.528776\pi$
$263$	−2.92820	−0.180561	−0.0902804	+	0.995916i	$0.528776\pi$
$264$	0	0
$265$	8.78461	0.539634
$266$	0	0
$267$	− 11.6077i	− 0.710379i
$268$	0	0
$269$	18.5885i	1.13336i	0.823939	+	0.566679i	$0.191772\pi$
$269$	18.5885i	1.13336i	−0.823939	+	0.566679i	$0.808228\pi$
$270$	0	0
$271$	1.07180	0.0651070	0.0325535	−	0.999470i	$-0.489636\pi$
$271$	1.07180	0.0651070	0.0325535	+	0.999470i	$0.489636\pi$
$272$	0	0
$273$	2.39230	0.144789
$274$	0	0
$275$	6.53590i	0.394130i
$276$	0	0
$277$	− 4.00000i	− 0.240337i	−0.992754	−	0.120168i	$-0.961657\pi$
$277$	− 4.00000i	− 0.240337i	0.992754	−	0.120168i	$-0.0383434\pi$
$278$	0	0
$279$	9.85641	0.590088
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	14.5885i	0.867194i	0.901107	+	0.433597i	$0.142756\pi$
$283$	14.5885i	0.867194i	−0.901107	+	0.433597i	$0.857244\pi$
$284$	0	0
$285$	2.53590i	0.150214i
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	− 3.60770i	− 0.211487i
$292$	0	0
$293$	1.80385i	0.105382i	0.998611	+	0.0526910i	$0.0167798\pi$
$293$	1.80385i	0.105382i	−0.998611	+	0.0526910i	$0.983220\pi$
$294$	0	0
$295$	−5.60770	−0.326493
$296$	0	0
$297$	−5.85641	−0.339823
$298$	0	0
$299$	− 11.3205i	− 0.654682i
$300$	0	0
$301$	1.46410i	0.0843894i
$302$	0	0
$303$	−11.4641	−0.658595
$304$	0	0
$305$	−9.60770	−0.550135
$306$	0	0
$307$	− 22.5885i	− 1.28919i	−0.764524	−	0.644596i	$-0.777026\pi$
$307$	− 22.5885i	− 1.28919i	0.764524	−	0.644596i	$-0.222974\pi$
$308$	0	0
$309$	− 13.0718i	− 0.743629i
$310$	0	0
$311$	21.8564	1.23936	0.619682	−	0.784853i	$-0.287262\pi$
$311$	21.8564	1.23936	0.619682	+	0.784853i	$0.287262\pi$
$312$	0	0
$313$	−7.85641	−0.444070	−0.222035	−	0.975039i	$-0.571270\pi$
$313$	−7.85641	−0.444070	−0.222035	+	0.975039i	$0.571270\pi$
$314$	0	0
$315$	− 1.80385i	− 0.101635i
$316$	0	0
$317$	12.7846i	0.718055i	0.933327	+	0.359028i	$0.116892\pi$
$317$	12.7846i	0.718055i	−0.933327	+	0.359028i	$0.883108\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	0	0
$321$	13.8564	0.773389
$322$	0	0
$323$	9.46410i	0.526597i
$324$	0	0
$325$	− 14.5885i	− 0.809222i
$326$	0	0
$327$	−6.14359	−0.339741
$328$	0	0
$329$	−10.9282	−0.602491
$330$	0	0
$331$	− 4.39230i	− 0.241423i	−0.992688	−	0.120711i	$-0.961482\pi$
$331$	− 4.39230i	− 0.241423i	0.992688	−	0.120711i	$-0.0385176\pi$
$332$	0	0
$333$	13.4641i	0.737828i
$334$	0	0
$335$	−5.85641	−0.319970
$336$	0	0
$337$	−4.39230	−0.239264	−0.119632	−	0.992818i	$-0.538171\pi$
$337$	−4.39230	−0.239264	−0.119632	+	0.992818i	$0.538171\pi$
$338$	0	0
$339$	− 9.85641i	− 0.535327i
$340$	0	0
$341$	5.85641i	0.317142i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	1.85641	0.0999456
$346$	0	0
$347$	0.679492i	0.0364770i	0.999834	+	0.0182385i	$0.00580582\pi$
$347$	0.679492i	0.0364770i	−0.999834	+	0.0182385i	$0.994194\pi$
$348$	0	0
$349$	24.0526i	1.28750i	0.765234	+	0.643752i	$0.222623\pi$
$349$	24.0526i	1.28750i	−0.765234	+	0.643752i	$0.777377\pi$
$350$	0	0
$351$	13.0718	0.697721
$352$	0	0
$353$	−12.9282	−0.688099	−0.344049	−	0.938952i	$-0.611799\pi$
$353$	−12.9282	−0.688099	−0.344049	+	0.938952i	$0.611799\pi$
$354$	0	0
$355$	8.00000i	0.424596i
$356$	0	0
$357$	1.46410i	0.0774885i
$358$	0	0
$359$	−20.5359	−1.08384	−0.541922	−	0.840429i	$-0.682303\pi$
$359$	−20.5359	−1.08384	−0.541922	+	0.840429i	$0.682303\pi$
$360$	0	0
$361$	−3.39230	−0.178542
$362$	0	0
$363$	6.48334i	0.340287i
$364$	0	0
$365$	0.679492i	0.0355662i
$366$	0	0
$367$	−20.7846	−1.08495	−0.542474	−	0.840073i	$-0.682512\pi$
$367$	−20.7846	−1.08495	−0.542474	+	0.840073i	$0.682512\pi$
$368$	0	0
$369$	−4.92820	−0.256552
$370$	0	0
$371$	12.0000i	0.623009i
$372$	0	0
$373$	− 17.0718i	− 0.883944i	−0.897029	−	0.441972i	$-0.854279\pi$
$373$	− 17.0718i	− 0.883944i	0.897029	−	0.441972i	$-0.145721\pi$
$374$	0	0
$375$	5.07180	0.261906
$376$	0	0
$377$	−17.8564	−0.919652
$378$	0	0
$379$	− 34.2487i	− 1.75924i	−0.475679	−	0.879619i	$-0.657798\pi$
$379$	− 34.2487i	− 1.75924i	0.475679	−	0.879619i	$-0.342202\pi$
$380$	0	0
$381$	− 3.32051i	− 0.170115i
$382$	0	0
$383$	8.78461	0.448873	0.224436	−	0.974489i	$-0.427946\pi$
$383$	8.78461	0.448873	0.224436	+	0.974489i	$0.427946\pi$
$384$	0	0
$385$	1.07180	0.0546238
$386$	0	0
$387$	3.60770i	0.183389i
$388$	0	0
$389$	− 11.3205i	− 0.573973i	−0.957935	−	0.286986i	$-0.907347\pi$
$389$	− 11.3205i	− 0.573973i	0.957935	−	0.286986i	$-0.0926534\pi$
$390$	0	0
$391$	6.92820	0.350374
$392$	0	0
$393$	−5.32051	−0.268384
$394$	0	0
$395$	− 2.14359i	− 0.107856i
$396$	0	0
$397$	29.1244i	1.46171i	0.682533	+	0.730855i	$0.260878\pi$
$397$	29.1244i	1.46171i	−0.682533	+	0.730855i	$0.739122\pi$
$398$	0	0
$399$	−3.46410	−0.173422
$400$	0	0
$401$	6.53590	0.326387	0.163194	−	0.986594i	$-0.447820\pi$
$401$	6.53590	0.326387	0.163194	+	0.986594i	$0.447820\pi$
$402$	0	0
$403$	− 13.0718i	− 0.651153i
$404$	0	0
$405$	− 3.26795i	− 0.162386i
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	−16.9282	−0.837046	−0.418523	−	0.908206i	$-0.637452\pi$
$409$	−16.9282	−0.837046	−0.418523	+	0.908206i	$0.637452\pi$
$410$	0	0
$411$	− 5.75129i	− 0.283690i
$412$	0	0
$413$	− 7.66025i	− 0.376936i
$414$	0	0
$415$	−8.53590	−0.419011
$416$	0	0
$417$	−9.32051	−0.456427
$418$	0	0
$419$	− 34.1962i	− 1.67059i	−0.549801	−	0.835296i	$-0.685296\pi$
$419$	− 34.1962i	− 1.67059i	0.549801	−	0.835296i	$-0.314704\pi$
$420$	0	0
$421$	1.85641i	0.0904757i	0.998976	+	0.0452379i	$0.0144046\pi$
$421$	1.85641i	0.0904757i	−0.998976	+	0.0452379i	$0.985595\pi$
$422$	0	0
$423$	−26.9282	−1.30929
$424$	0	0
$425$	8.92820	0.433081
$426$	0	0
$427$	− 13.1244i	− 0.635132i
$428$	0	0
$429$	3.50258i	0.169106i
$430$	0	0
$431$	−28.5359	−1.37453	−0.687263	−	0.726409i	$-0.741188\pi$
$431$	−28.5359	−1.37453	−0.687263	+	0.726409i	$0.741188\pi$
$432$	0	0
$433$	−11.8564	−0.569783	−0.284891	−	0.958560i	$-0.591957\pi$
$433$	−11.8564	−0.569783	−0.284891	+	0.958560i	$0.591957\pi$
$434$	0	0
$435$	− 2.92820i	− 0.140397i
$436$	0	0
$437$	16.3923i	0.784150i
$438$	0	0
$439$	17.0718	0.814792	0.407396	−	0.913252i	$-0.366437\pi$
$439$	17.0718	0.814792	0.407396	+	0.913252i	$0.366437\pi$
$440$	0	0
$441$	2.46410	0.117338
$442$	0	0
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	0	0
$445$	11.6077i	0.550258i
$446$	0	0
$447$	−10.9282	−0.516886
$448$	0	0
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	− 2.92820i	− 0.137884i
$452$	0	0
$453$	13.4641i	0.632599i
$454$	0	0
$455$	−2.39230	−0.112153
$456$	0	0
$457$	−19.3205	−0.903775	−0.451888	−	0.892075i	$-0.649249\pi$
$457$	−19.3205	−0.903775	−0.451888	+	0.892075i	$0.649249\pi$
$458$	0	0
$459$	8.00000i	0.373408i
$460$	0	0
$461$	− 13.8038i	− 0.642909i	−0.946925	−	0.321455i	$-0.895828\pi$
$461$	− 13.8038i	− 0.642909i	0.946925	−	0.321455i	$-0.104172\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	0	0
$465$	2.14359	0.0994068
$466$	0	0
$467$	− 25.1244i	− 1.16262i	−0.813683	−	0.581308i	$-0.802541\pi$
$467$	− 25.1244i	− 1.16262i	0.813683	−	0.581308i	$-0.197459\pi$
$468$	0	0
$469$	− 8.00000i	− 0.369406i
$470$	0	0
$471$	−1.32051	−0.0608458
$472$	0	0
$473$	−2.14359	−0.0985625
$474$	0	0
$475$	21.1244i	0.969252i
$476$	0	0
$477$	29.5692i	1.35388i
$478$	0	0
$479$	17.8564	0.815880	0.407940	−	0.913009i	$-0.366247\pi$
$479$	17.8564	0.815880	0.407940	+	0.913009i	$0.366247\pi$
$480$	0	0
$481$	17.8564	0.814182
$482$	0	0
$483$	2.53590i	0.115387i
$484$	0	0
$485$	3.60770i	0.163817i
$486$	0	0
$487$	−42.3923	−1.92098	−0.960489	−	0.278317i	$-0.910224\pi$
$487$	−42.3923	−1.92098	−0.960489	+	0.278317i	$0.910224\pi$
$488$	0	0
$489$	−4.78461	−0.216368
$490$	0	0
$491$	8.78461i	0.396444i	0.980157	+	0.198222i	$0.0635167\pi$
$491$	8.78461i	0.396444i	−0.980157	+	0.198222i	$0.936483\pi$
$492$	0	0
$493$	− 10.9282i	− 0.492182i
$494$	0	0
$495$	2.64102	0.118705
$496$	0	0
$497$	−10.9282	−0.490197
$498$	0	0
$499$	26.9282i	1.20547i	0.797941	+	0.602736i	$0.205923\pi$
$499$	26.9282i	1.20547i	−0.797941	+	0.602736i	$0.794077\pi$
$500$	0	0
$501$	− 13.8564i	− 0.619059i
$502$	0	0
$503$	17.0718	0.761194	0.380597	−	0.924741i	$-0.375719\pi$
$503$	17.0718	0.761194	0.380597	+	0.924741i	$0.375719\pi$
$504$	0	0
$505$	11.4641	0.510146
$506$	0	0
$507$	1.69873i	0.0754432i
$508$	0	0
$509$	21.1244i	0.936321i	0.883644	+	0.468160i	$0.155083\pi$
$509$	21.1244i	0.936321i	−0.883644	+	0.468160i	$0.844917\pi$
$510$	0	0
$511$	−0.928203	−0.0410613
$512$	0	0
$513$	−18.9282	−0.835701
$514$	0	0
$515$	13.0718i	0.576012i
$516$	0	0
$517$	− 16.0000i	− 0.703679i
$518$	0	0
$519$	10.3923	0.456172
$520$	0	0
$521$	26.7846	1.17346	0.586728	−	0.809784i	$-0.300416\pi$
$521$	26.7846	1.17346	0.586728	+	0.809784i	$0.300416\pi$
$522$	0	0
$523$	− 34.5885i	− 1.51245i	−0.654313	−	0.756224i	$-0.727042\pi$
$523$	− 34.5885i	− 1.51245i	0.654313	−	0.756224i	$-0.272958\pi$
$524$	0	0
$525$	3.26795i	0.142625i
$526$	0	0
$527$	8.00000	0.348485
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	− 18.8756i	− 0.819133i
$532$	0	0
$533$	6.53590i	0.283101i
$534$	0	0
$535$	−13.8564	−0.599065
$536$	0	0
$537$	8.00000	0.345225
$538$	0	0
$539$	1.46410i	0.0630633i
$540$	0	0
$541$	− 4.00000i	− 0.171973i	−0.996296	−	0.0859867i	$-0.972596\pi$
$541$	− 4.00000i	− 0.171973i	0.996296	−	0.0859867i	$-0.0274043\pi$
$542$	0	0
$543$	5.32051	0.228325
$544$	0	0
$545$	6.14359	0.263163
$546$	0	0
$547$	− 2.24871i	− 0.0961480i	−0.998844	−	0.0480740i	$-0.984692\pi$
$547$	− 2.24871i	− 0.0961480i	0.998844	−	0.0480740i	$-0.0153083\pi$
$548$	0	0
$549$	− 32.3397i	− 1.38023i
$550$	0	0
$551$	25.8564	1.10152
$552$	0	0
$553$	2.92820	0.124520
$554$	0	0
$555$	2.92820i	0.124295i
$556$	0	0
$557$	− 31.7128i	− 1.34372i	−0.740680	−	0.671858i	$-0.765497\pi$
$557$	− 31.7128i	− 1.34372i	0.740680	−	0.671858i	$-0.234503\pi$
$558$	0	0
$559$	4.78461	0.202367
$560$	0	0
$561$	−2.14359	−0.0905026
$562$	0	0
$563$	4.05256i	0.170795i	0.996347	+	0.0853975i	$0.0272160\pi$
$563$	4.05256i	0.170795i	−0.996347	+	0.0853975i	$0.972784\pi$
$564$	0	0
$565$	9.85641i	0.414662i
$566$	0	0
$567$	4.46410	0.187475
$568$	0	0
$569$	1.46410	0.0613783	0.0306892	−	0.999529i	$-0.490230\pi$
$569$	1.46410	0.0613783	0.0306892	+	0.999529i	$0.490230\pi$
$570$	0	0
$571$	37.1769i	1.55581i	0.628385	+	0.777903i	$0.283716\pi$
$571$	37.1769i	1.55581i	−0.628385	+	0.777903i	$0.716284\pi$
$572$	0	0
$573$	11.7128i	0.489310i
$574$	0	0
$575$	15.4641	0.644898
$576$	0	0
$577$	−44.9282	−1.87039	−0.935193	−	0.354139i	$-0.884774\pi$
$577$	−44.9282	−1.87039	−0.935193	+	0.354139i	$0.884774\pi$
$578$	0	0
$579$	9.85641i	0.409618i
$580$	0	0
$581$	− 11.6603i	− 0.483749i
$582$	0	0
$583$	−17.5692	−0.727643
$584$	0	0
$585$	−5.89488	−0.243723
$586$	0	0
$587$	36.7321i	1.51609i	0.652200	+	0.758047i	$0.273846\pi$
$587$	36.7321i	1.51609i	−0.652200	+	0.758047i	$0.726154\pi$
$588$	0	0
$589$	18.9282i	0.779923i
$590$	0	0
$591$	1.35898	0.0559011
$592$	0	0
$593$	39.8564	1.63671	0.818353	−	0.574716i	$-0.194887\pi$
$593$	39.8564	1.63671	0.818353	+	0.574716i	$0.194887\pi$
$594$	0	0
$595$	− 1.46410i	− 0.0600223i
$596$	0	0
$597$	− 2.92820i	− 0.119843i
$598$	0	0
$599$	−27.7128	−1.13231	−0.566157	−	0.824297i	$-0.691571\pi$
$599$	−27.7128	−1.13231	−0.566157	+	0.824297i	$0.691571\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	0	0
$603$	− 19.7128i	− 0.802768i
$604$	0	0
$605$	− 6.48334i	− 0.263585i
$606$	0	0
$607$	27.7128	1.12483	0.562414	−	0.826856i	$-0.309873\pi$
$607$	27.7128	1.12483	0.562414	+	0.826856i	$0.309873\pi$
$608$	0	0
$609$	4.00000	0.162088
$610$	0	0
$611$	35.7128i	1.44479i
$612$	0	0
$613$	4.67949i	0.189003i	0.995525	+	0.0945014i	$0.0301257\pi$
$613$	4.67949i	0.189003i	−0.995525	+	0.0945014i	$0.969874\pi$
$614$	0	0
$615$	−1.07180	−0.0432190
$616$	0	0
$617$	−3.60770	−0.145240	−0.0726202	−	0.997360i	$-0.523136\pi$
$617$	−3.60770	−0.145240	−0.0726202	+	0.997360i	$0.523136\pi$
$618$	0	0
$619$	13.8038i	0.554823i	0.960751	+	0.277412i	$0.0894766\pi$
$619$	13.8038i	0.554823i	−0.960751	+	0.277412i	$0.910523\pi$
$620$	0	0
$621$	13.8564i	0.556038i
$622$	0	0
$623$	−15.8564	−0.635274
$624$	0	0
$625$	17.2487	0.689948
$626$	0	0
$627$	− 5.07180i	− 0.202548i
$628$	0	0
$629$	10.9282i	0.435736i
$630$	0	0
$631$	33.5692	1.33637	0.668185	−	0.743995i	$-0.267072\pi$
$631$	33.5692	1.33637	0.668185	+	0.743995i	$0.267072\pi$
$632$	0	0
$633$	−18.1436	−0.721143
$634$	0	0
$635$	3.32051i	0.131770i
$636$	0	0
$637$	− 3.26795i	− 0.129481i
$638$	0	0
$639$	−26.9282	−1.06526
$640$	0	0
$641$	26.2487	1.03676	0.518381	−	0.855150i	$-0.326535\pi$
$641$	26.2487	1.03676	0.518381	+	0.855150i	$0.326535\pi$
$642$	0	0
$643$	− 31.6603i	− 1.24856i	−0.781201	−	0.624279i	$-0.785393\pi$
$643$	− 31.6603i	− 1.24856i	0.781201	−	0.624279i	$-0.214607\pi$
$644$	0	0
$645$	0.784610i	0.0308940i
$646$	0	0
$647$	25.8564	1.01652	0.508260	−	0.861204i	$-0.330289\pi$
$647$	25.8564	1.01652	0.508260	+	0.861204i	$0.330289\pi$
$648$	0	0
$649$	11.2154	0.440243
$650$	0	0
$651$	2.92820i	0.114765i
$652$	0	0
$653$	− 45.4641i	− 1.77915i	−0.456791	−	0.889574i	$-0.651001\pi$
$653$	− 45.4641i	− 1.77915i	0.456791	−	0.889574i	$-0.348999\pi$
$654$	0	0
$655$	5.32051	0.207889
$656$	0	0
$657$	−2.28719	−0.0892317
$658$	0	0
$659$	31.3205i	1.22007i	0.792373	+	0.610037i	$0.208845\pi$
$659$	31.3205i	1.22007i	−0.792373	+	0.610037i	$0.791155\pi$
$660$	0	0
$661$	− 22.9808i	− 0.893848i	−0.894572	−	0.446924i	$-0.852519\pi$
$661$	− 22.9808i	− 0.893848i	0.894572	−	0.446924i	$-0.147481\pi$
$662$	0	0
$663$	4.78461	0.185819
$664$	0	0
$665$	3.46410	0.134332
$666$	0	0
$667$	− 18.9282i	− 0.732903i
$668$	0	0
$669$	5.07180i	0.196087i
$670$	0	0
$671$	19.2154	0.741802
$672$	0	0
$673$	31.8564	1.22797	0.613987	−	0.789316i	$-0.289565\pi$
$673$	31.8564	1.22797	0.613987	+	0.789316i	$0.289565\pi$
$674$	0	0
$675$	17.8564i	0.687293i
$676$	0	0
$677$	44.4449i	1.70815i	0.520146	+	0.854077i	$0.325878\pi$
$677$	44.4449i	1.70815i	−0.520146	+	0.854077i	$0.674122\pi$
$678$	0	0
$679$	−4.92820	−0.189127
$680$	0	0
$681$	10.3923	0.398234
$682$	0	0
$683$	38.6410i	1.47856i	0.673400	+	0.739279i	$0.264833\pi$
$683$	38.6410i	1.47856i	−0.673400	+	0.739279i	$0.735167\pi$
$684$	0	0
$685$	5.75129i	0.219745i
$686$	0	0
$687$	15.1769	0.579035
$688$	0	0
$689$	39.2154	1.49399
$690$	0	0
$691$	13.1244i	0.499274i	0.968339	+	0.249637i	$0.0803113\pi$
$691$	13.1244i	0.499274i	−0.968339	+	0.249637i	$0.919689\pi$
$692$	0	0
$693$	3.60770i	0.137045i
$694$	0	0
$695$	9.32051	0.353547
$696$	0	0
$697$	−4.00000	−0.151511
$698$	0	0
$699$	4.39230i	0.166132i
$700$	0	0
$701$	− 40.3923i	− 1.52560i	−0.646637	−	0.762798i	$-0.723825\pi$
$701$	− 40.3923i	− 1.52560i	0.646637	−	0.762798i	$-0.276175\pi$
$702$	0	0
$703$	−25.8564	−0.975193
$704$	0	0
$705$	−5.85641	−0.220565
$706$	0	0
$707$	15.6603i	0.588964i
$708$	0	0
$709$	− 28.1051i	− 1.05551i	−0.849397	−	0.527755i	$-0.823034\pi$
$709$	− 28.1051i	− 1.05551i	0.849397	−	0.527755i	$-0.176966\pi$
$710$	0	0
$711$	7.21539	0.270598
$712$	0	0
$713$	13.8564	0.518927
$714$	0	0
$715$	− 3.50258i	− 0.130989i
$716$	0	0
$717$	− 14.2487i	− 0.532128i
$718$	0	0
$719$	34.9282	1.30260	0.651301	−	0.758819i	$-0.274223\pi$
$719$	34.9282	1.30260	0.651301	+	0.758819i	$0.274223\pi$
$720$	0	0
$721$	−17.8564	−0.665007
$722$	0	0
$723$	16.6795i	0.620317i
$724$	0	0
$725$	− 24.3923i	− 0.905907i
$726$	0	0
$727$	−2.92820	−0.108601	−0.0543005	−	0.998525i	$-0.517293\pi$
$727$	−2.92820	−0.108601	−0.0543005	+	0.998525i	$0.517293\pi$
$728$	0	0
$729$	2.21539	0.0820515
$730$	0	0
$731$	2.92820i	0.108304i
$732$	0	0
$733$	3.66025i	0.135195i	0.997713	+	0.0675973i	$0.0215333\pi$
$733$	3.66025i	0.135195i	−0.997713	+	0.0675973i	$0.978467\pi$
$734$	0	0
$735$	0.535898	0.0197669
$736$	0	0
$737$	11.7128	0.431447
$738$	0	0
$739$	− 31.3205i	− 1.15214i	−0.817399	−	0.576072i	$-0.804585\pi$
$739$	− 31.3205i	− 1.15214i	0.817399	−	0.576072i	$-0.195415\pi$
$740$	0	0
$741$	11.3205i	0.415869i
$742$	0	0
$743$	−16.2487	−0.596107	−0.298054	−	0.954549i	$-0.596337\pi$
$743$	−16.2487	−0.596107	−0.298054	+	0.954549i	$0.596337\pi$
$744$	0	0
$745$	10.9282	0.400378
$746$	0	0
$747$	− 28.7321i	− 1.05125i
$748$	0	0
$749$	− 18.9282i	− 0.691621i
$750$	0	0
$751$	−24.2487	−0.884848	−0.442424	−	0.896806i	$-0.645881\pi$
$751$	−24.2487	−0.884848	−0.442424	+	0.896806i	$0.645881\pi$
$752$	0	0
$753$	−19.1769	−0.698846
$754$	0	0
$755$	− 13.4641i	− 0.490009i
$756$	0	0
$757$	38.2487i	1.39017i	0.718926	+	0.695087i	$0.244634\pi$
$757$	38.2487i	1.39017i	−0.718926	+	0.695087i	$0.755366\pi$
$758$	0	0
$759$	−3.71281	−0.134767
$760$	0	0
$761$	33.7128	1.22209	0.611044	−	0.791596i	$-0.290750\pi$
$761$	33.7128	1.22209	0.611044	+	0.791596i	$0.290750\pi$
$762$	0	0
$763$	8.39230i	0.303822i
$764$	0	0
$765$	− 3.60770i	− 0.130436i
$766$	0	0
$767$	−25.0333	−0.903901
$768$	0	0
$769$	30.7846	1.11012	0.555061	−	0.831810i	$-0.312695\pi$
$769$	30.7846	1.11012	0.555061	+	0.831810i	$0.312695\pi$
$770$	0	0
$771$	− 4.39230i	− 0.158185i
$772$	0	0
$773$	45.1244i	1.62301i	0.584345	+	0.811505i	$0.301351\pi$
$773$	45.1244i	1.62301i	−0.584345	+	0.811505i	$0.698649\pi$
$774$	0	0
$775$	17.8564	0.641421
$776$	0	0
$777$	−4.00000	−0.143499
$778$	0	0
$779$	− 9.46410i	− 0.339087i
$780$	0	0
$781$	− 16.0000i	− 0.572525i
$782$	0	0
$783$	21.8564	0.781084
$784$	0	0
$785$	1.32051	0.0471310
$786$	0	0
$787$	10.5885i	0.377438i	0.982031	+	0.188719i	$0.0604335\pi$
$787$	10.5885i	0.377438i	−0.982031	+	0.188719i	$0.939567\pi$
$788$	0	0
$789$	− 2.14359i	− 0.0763140i
$790$	0	0
$791$	−13.4641	−0.478728
$792$	0	0
$793$	−42.8897	−1.52306
$794$	0	0
$795$	6.43078i	0.228076i
$796$	0	0
$797$	− 38.9808i	− 1.38077i	−0.723442	−	0.690385i	$-0.757441\pi$
$797$	− 38.9808i	− 1.38077i	0.723442	−	0.690385i	$-0.242559\pi$
$798$	0	0
$799$	−21.8564	−0.773224
$800$	0	0
$801$	−39.0718	−1.38053
$802$	0	0
$803$	− 1.35898i	− 0.0479575i
$804$	0	0
$805$	− 2.53590i	− 0.0893787i
$806$	0	0
$807$	−13.6077	−0.479014
$808$	0	0
$809$	−11.3205	−0.398008	−0.199004	−	0.979999i	$-0.563771\pi$
$809$	−11.3205	−0.398008	−0.199004	+	0.979999i	$0.563771\pi$
$810$	0	0
$811$	2.87564i	0.100978i	0.998725	+	0.0504888i	$0.0160779\pi$
$811$	2.87564i	0.100978i	−0.998725	+	0.0504888i	$0.983922\pi$
$812$	0	0
$813$	0.784610i	0.0275175i
$814$	0	0
$815$	4.78461	0.167598
$816$	0	0
$817$	−6.92820	−0.242387
$818$	0	0
$819$	− 8.05256i	− 0.281379i
$820$	0	0
$821$	6.92820i	0.241796i	0.992665	+	0.120898i	$0.0385774\pi$
$821$	6.92820i	0.241796i	−0.992665	+	0.120898i	$0.961423\pi$
$822$	0	0
$823$	10.9282	0.380933	0.190467	−	0.981694i	$-0.439000\pi$
$823$	10.9282	0.380933	0.190467	+	0.981694i	$0.439000\pi$
$824$	0	0
$825$	−4.78461	−0.166579
$826$	0	0
$827$	− 13.8564i	− 0.481834i	−0.970546	−	0.240917i	$-0.922552\pi$
$827$	− 13.8564i	− 0.481834i	0.970546	−	0.240917i	$-0.0774482\pi$
$828$	0	0
$829$	− 10.9808i	− 0.381378i	−0.981651	−	0.190689i	$-0.938928\pi$
$829$	− 10.9808i	− 0.381378i	0.981651	−	0.190689i	$-0.0610721\pi$
$830$	0	0
$831$	2.92820	0.101578
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	13.8564i	0.479521i
$836$	0	0
$837$	16.0000i	0.553041i
$838$	0	0
$839$	15.7128	0.542467	0.271233	−	0.962514i	$-0.412569\pi$
$839$	15.7128	0.542467	0.271233	+	0.962514i	$0.412569\pi$
$840$	0	0
$841$	−0.856406	−0.0295313
$842$	0	0
$843$	1.46410i	0.0504263i
$844$	0	0
$845$	− 1.69873i	− 0.0584381i
$846$	0	0
$847$	8.85641	0.304310
$848$	0	0
$849$	−10.6795	−0.366519
$850$	0	0
$851$	18.9282i	0.648850i
$852$	0	0
$853$	46.9808i	1.60859i	0.594230	+	0.804295i	$0.297457\pi$
$853$	46.9808i	1.60859i	−0.594230	+	0.804295i	$0.702543\pi$
$854$	0	0
$855$	8.53590	0.291922
$856$	0	0
$857$	10.7846	0.368395	0.184198	−	0.982889i	$-0.441031\pi$
$857$	10.7846	0.368395	0.184198	+	0.982889i	$0.441031\pi$
$858$	0	0
$859$	− 28.4449i	− 0.970526i	−0.874368	−	0.485263i	$-0.838724\pi$
$859$	− 28.4449i	− 0.970526i	0.874368	−	0.485263i	$-0.161276\pi$
$860$	0	0
$861$	− 1.46410i	− 0.0498964i
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	−10.3923	−0.353349
$866$	0	0
$867$	− 9.51666i	− 0.323203i
$868$	0	0
$869$	4.28719i	0.145433i
$870$	0	0
$871$	−26.1436	−0.885842
$872$	0	0
$873$	−12.1436	−0.410998
$874$	0	0
$875$	− 6.92820i	− 0.234216i
$876$	0	0
$877$	− 0.392305i	− 0.0132472i	−0.999978	−	0.00662360i	$-0.997892\pi$
$877$	− 0.392305i	− 0.0132472i	0.999978	−	0.00662360i	$-0.00210837\pi$
$878$	0	0
$879$	−1.32051	−0.0445396
$880$	0	0
$881$	−9.21539	−0.310474	−0.155237	−	0.987877i	$-0.549614\pi$
$881$	−9.21539	−0.310474	−0.155237	+	0.987877i	$0.549614\pi$
$882$	0	0
$883$	− 18.9282i	− 0.636985i	−0.947925	−	0.318492i	$-0.896823\pi$
$883$	− 18.9282i	− 0.636985i	0.947925	−	0.318492i	$-0.103177\pi$
$884$	0	0
$885$	− 4.10512i	− 0.137992i
$886$	0	0
$887$	22.6410	0.760211	0.380105	−	0.924943i	$-0.375888\pi$
$887$	22.6410	0.760211	0.380105	+	0.924943i	$0.375888\pi$
$888$	0	0
$889$	−4.53590	−0.152129
$890$	0	0
$891$	6.53590i	0.218961i
$892$	0	0
$893$	− 51.7128i	− 1.73050i
$894$	0	0
$895$	−8.00000	−0.267411
$896$	0	0
$897$	8.28719	0.276701
$898$	0	0
$899$	− 21.8564i	− 0.728952i
$900$	0	0
$901$	24.0000i	0.799556i
$902$	0	0
$903$	−1.07180	−0.0356672
$904$	0	0
$905$	−5.32051	−0.176860
$906$	0	0
$907$	− 11.7128i	− 0.388918i	−0.980911	−	0.194459i	$-0.937705\pi$
$907$	− 11.7128i	− 0.388918i	0.980911	−	0.194459i	$-0.0622951\pi$
$908$	0	0
$909$	38.5885i	1.27990i
$910$	0	0
$911$	−11.4641	−0.379823	−0.189911	−	0.981801i	$-0.560820\pi$
$911$	−11.4641	−0.379823	−0.189911	+	0.981801i	$0.560820\pi$
$912$	0	0
$913$	17.0718	0.564994
$914$	0	0
$915$	− 7.03332i	− 0.232514i
$916$	0	0
$917$	7.26795i	0.240009i
$918$	0	0
$919$	−48.7846	−1.60926	−0.804628	−	0.593779i	$-0.797635\pi$
$919$	−48.7846	−1.60926	−0.804628	+	0.593779i	$0.797635\pi$
$920$	0	0
$921$	16.5359	0.544876
$922$	0	0
$923$	35.7128i	1.17550i
$924$	0	0
$925$	24.3923i	0.802014i
$926$	0	0
$927$	−44.0000	−1.44515
$928$	0	0
$929$	−17.7128	−0.581139	−0.290569	−	0.956854i	$-0.593845\pi$
$929$	−17.7128	−0.581139	−0.290569	+	0.956854i	$0.593845\pi$
$930$	0	0
$931$	4.73205i	0.155087i
$932$	0	0
$933$	16.0000i	0.523816i
$934$	0	0
$935$	2.14359	0.0701030
$936$	0	0
$937$	−60.6410	−1.98106	−0.990528	−	0.137312i	$-0.956154\pi$
$937$	−60.6410	−1.98106	−0.990528	+	0.137312i	$0.956154\pi$
$938$	0	0
$939$	− 5.75129i	− 0.187686i
$940$	0	0
$941$	6.98076i	0.227566i	0.993506	+	0.113783i	$0.0362969\pi$
$941$	6.98076i	0.227566i	−0.993506	+	0.113783i	$0.963703\pi$
$942$	0	0
$943$	−6.92820	−0.225613
$944$	0	0
$945$	2.92820	0.0952545
$946$	0	0
$947$	3.60770i	0.117234i	0.998281	+	0.0586172i	$0.0186691\pi$
$947$	3.60770i	0.117234i	−0.998281	+	0.0586172i	$0.981331\pi$
$948$	0	0
$949$	3.03332i	0.0984658i
$950$	0	0
$951$	−9.35898	−0.303486
$952$	0	0
$953$	7.85641	0.254494	0.127247	−	0.991871i	$-0.459386\pi$
$953$	7.85641	0.254494	0.127247	+	0.991871i	$0.459386\pi$
$954$	0	0
$955$	− 11.7128i	− 0.379018i
$956$	0	0
$957$	5.85641i	0.189311i
$958$	0	0
$959$	−7.85641	−0.253697
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	− 46.6410i	− 1.50299i
$964$	0	0
$965$	− 9.85641i	− 0.317289i
$966$	0	0
$967$	47.1769	1.51711	0.758554	−	0.651611i	$-0.225906\pi$
$967$	47.1769	1.51711	0.758554	+	0.651611i	$0.225906\pi$
$968$	0	0
$969$	−6.92820	−0.222566
$970$	0	0
$971$	10.4833i	0.336426i	0.985751	+	0.168213i	$0.0537997\pi$
$971$	10.4833i	0.336426i	−0.985751	+	0.168213i	$0.946200\pi$
$972$	0	0
$973$	12.7321i	0.408171i
$974$	0	0
$975$	10.6795	0.342017
$976$	0	0
$977$	−55.8564	−1.78700	−0.893502	−	0.449058i	$-0.851759\pi$
$977$	−55.8564	−1.78700	−0.893502	+	0.449058i	$0.851759\pi$
$978$	0	0
$979$	− 23.2154i	− 0.741967i
$980$	0	0
$981$	20.6795i	0.660245i
$982$	0	0
$983$	7.71281	0.246001	0.123000	−	0.992407i	$-0.460748\pi$
$983$	7.71281	0.246001	0.123000	+	0.992407i	$0.460748\pi$
$984$	0	0
$985$	−1.35898	−0.0433008
$986$	0	0
$987$	− 8.00000i	− 0.254643i
$988$	0	0
$989$	5.07180i	0.161274i
$990$	0	0
$991$	−16.7846	−0.533181	−0.266590	−	0.963810i	$-0.585897\pi$
$991$	−16.7846	−0.533181	−0.266590	+	0.963810i	$0.585897\pi$
$992$	0	0
$993$	3.21539	0.102037
$994$	0	0
$995$	2.92820i	0.0928303i
$996$	0	0
$997$	− 8.73205i	− 0.276547i	−0.990394	−	0.138273i	$-0.955845\pi$
$997$	− 8.73205i	− 0.276547i	0.990394	−	0.138273i	$-0.0441553\pi$
$998$	0	0
$999$	−21.8564	−0.691506

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.b.d.225.3	yes	4
3.2	odd	2		4032.2.c.n.2017.3		4
4.3	odd	2		448.2.b.c.225.2	✓	4
7.6	odd	2		3136.2.b.h.1569.2		4
8.3	odd	2		448.2.b.c.225.3	yes	4
8.5	even	2	inner	448.2.b.d.225.2	yes	4
12.11	even	2		4032.2.c.k.2017.3		4
16.3	odd	4		1792.2.a.k.1.2		2
16.5	even	4		1792.2.a.i.1.2		2
16.11	odd	4		1792.2.a.q.1.1		2
16.13	even	4		1792.2.a.s.1.1		2
24.5	odd	2		4032.2.c.n.2017.2		4
24.11	even	2		4032.2.c.k.2017.2		4
28.27	even	2		3136.2.b.g.1569.3		4
56.13	odd	2		3136.2.b.h.1569.3		4
56.27	even	2		3136.2.b.g.1569.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.b.c.225.2	✓	4	4.3	odd	2
448.2.b.c.225.3	yes	4	8.3	odd	2
448.2.b.d.225.2	yes	4	8.5	even	2	inner
448.2.b.d.225.3	yes	4	1.1	even	1	trivial
1792.2.a.i.1.2		2	16.5	even	4
1792.2.a.k.1.2		2	16.3	odd	4
1792.2.a.q.1.1		2	16.11	odd	4
1792.2.a.s.1.1		2	16.13	even	4
3136.2.b.g.1569.2		4	56.27	even	2
3136.2.b.g.1569.3		4	28.27	even	2
3136.2.b.h.1569.2		4	7.6	odd	2
3136.2.b.h.1569.3		4	56.13	odd	2
4032.2.c.k.2017.2		4	24.11	even	2
4032.2.c.k.2017.3		4	12.11	even	2
4032.2.c.n.2017.2		4	24.5	odd	2
4032.2.c.n.2017.3		4	3.2	odd	2

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	448.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.732051i q^{3} -0.732051i q^{5} +1.00000 q^{7} +2.46410 q^{9} +O(q^{10})\)	\(q+0.732051i q^{3} -0.732051i q^{5} +1.00000 q^{7} +2.46410 q^{9} +1.46410i q^{11} -3.26795i q^{13} +0.535898 q^{15} +2.00000 q^{17} +4.73205i q^{19} +0.732051i q^{21} +3.46410 q^{23} +4.46410 q^{25} +4.00000i q^{27} -5.46410i q^{29} +4.00000 q^{31} -1.07180 q^{33} -0.732051i q^{35} +5.46410i q^{37} +2.39230 q^{39} -2.00000 q^{41} +1.46410i q^{43} -1.80385i q^{45} -10.9282 q^{47} +1.00000 q^{49} +1.46410i q^{51} +12.0000i q^{53} +1.07180 q^{55} -3.46410 q^{57} -7.66025i q^{59} -13.1244i q^{61} +2.46410 q^{63} -2.39230 q^{65} -8.00000i q^{67} +2.53590i q^{69} -10.9282 q^{71} -0.928203 q^{73} +3.26795i q^{75} +1.46410i q^{77} +2.92820 q^{79} +4.46410 q^{81} -11.6603i q^{83} -1.46410i q^{85} +4.00000 q^{87} -15.8564 q^{89} -3.26795i q^{91} +2.92820i q^{93} +3.46410 q^{95} -4.92820 q^{97} +3.60770i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^7 - 4 * q^9	\( 4 q + 4 q^{7} - 4 q^{9} + 16 q^{15} + 8 q^{17} + 4 q^{25} + 16 q^{31} - 32 q^{33} - 32 q^{39} - 8 q^{41} - 16 q^{47} + 4 q^{49} + 32 q^{55} - 4 q^{63} + 32 q^{65} - 16 q^{71} + 24 q^{73} - 16 q^{79} + 4 q^{81} + 16 q^{87} - 8 q^{89} + 8 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 - 4 * q^9 + 16 * q^15 + 8 * q^17 + 4 * q^25 + 16 * q^31 - 32 * q^33 - 32 * q^39 - 8 * q^41 - 16 * q^47 + 4 * q^49 + 32 * q^55 - 4 * q^63 + 32 * q^65 - 16 * q^71 + 24 * q^73 - 16 * q^79 + 4 * q^81 + 16 * q^87 - 8 * q^89 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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