LMFDB - Embedded newform 448.4.j.b.111.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,4,Mod(111,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("448.111");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	448.j (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$26.4328556826$
Analytic rank:	$0$
Dimension:	$88$
Relative dimension:	$44$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			111.19
Character	$\chi$	$=$	448.111
Dual form			448.4.j.b.335.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.83086 - 1.83086i) q^{3} +(-11.2147 - 11.2147i) q^{5} +(-18.0262 - 4.24923i) q^{7} -20.2959i q^{9} +(3.79046 + 3.79046i) q^{11} +(-12.4135 + 12.4135i) q^{13} +41.0653i q^{15} -92.7849i q^{17} +(-112.714 - 112.714i) q^{19} +(25.2238 + 40.7833i) q^{21} -65.3480 q^{23} +126.541i q^{25} +(-86.5923 + 86.5923i) q^{27} +(179.260 + 179.260i) q^{29} +287.429 q^{31} -13.8796i q^{33} +(154.505 + 249.813i) q^{35} +(-194.602 + 194.602i) q^{37} +45.4548 q^{39} +291.553 q^{41} +(-186.279 - 186.279i) q^{43} +(-227.613 + 227.613i) q^{45} -203.179 q^{47} +(306.888 + 153.195i) q^{49} +(-169.876 + 169.876i) q^{51} +(-205.330 + 205.330i) q^{53} -85.0182i q^{55} +412.727i q^{57} +(-176.660 + 176.660i) q^{59} +(410.414 - 410.414i) q^{61} +(-86.2419 + 365.858i) q^{63} +278.428 q^{65} +(13.4406 - 13.4406i) q^{67} +(119.643 + 119.643i) q^{69} +416.602 q^{71} +268.844 q^{73} +(231.679 - 231.679i) q^{75} +(-52.2211 - 84.4342i) q^{77} +705.142i q^{79} -230.911 q^{81} +(296.671 + 296.671i) q^{83} +(-1040.56 + 1040.56i) q^{85} -656.402i q^{87} -1109.14 q^{89} +(276.516 - 171.020i) q^{91} +(-526.243 - 526.243i) q^{93} +2528.11i q^{95} +383.187i q^{97} +(76.9308 - 76.9308i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q + 4 q^{7} - 112 q^{11} + 52 q^{21} - 160 q^{23} + 528 q^{29} - 476 q^{35} - 896 q^{37} + 8 q^{39} - 40 q^{43} - 1376 q^{49} - 1504 q^{51} - 1560 q^{53} - 8 q^{65} - 648 q^{67} + 456 q^{71} + 292 q^{77}+ \cdots - 7592 q^{99}+O(q^{100}) $ 88 * q + 4 * q^7 - 112 * q^11 + 52 * q^21 - 160 * q^23 + 528 * q^29 - 476 * q^35 - 896 * q^37 + 8 * q^39 - 40 * q^43 - 1376 * q^49 - 1504 * q^51 - 1560 * q^53 - 8 * q^65 - 648 * q^67 + 456 * q^71 + 292 * q^77 - 1952 * q^81 + 496 * q^85 + 1964 * q^91 - 112 * q^93 - 7592 * q^99

Character values

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

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$

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
$3$	−1.83086	−	1.83086i	−0.352350	−	0.352350i	0.508633	−	0.860983i	$-0.330151\pi$
$3$	−1.83086	−	1.83086i	−0.352350	−	0.352350i	−0.860983	+	0.508633i	$0.830151\pi$
$4$		0			0
$5$	−11.2147	−	11.2147i	−1.00308	−	1.00308i	−0.999995	−	0.00308220i	$-0.999019\pi$
$5$	−11.2147	−	11.2147i	−1.00308	−	1.00308i	−0.00308220	−	0.999995i	$-0.500981\pi$
$6$		0			0
$7$	−18.0262	−	4.24923i	−0.973324	−	0.229437i
$7$	−18.0262	−	4.24923i	−0.973324	−	0.229437i
$8$		0			0
$9$		−	20.2959i		−	0.751699i
$10$		0			0
$11$	3.79046	+	3.79046i	0.103897	+	0.103897i	0.757144	−	0.653247i	$-0.226594\pi$
$11$	3.79046	+	3.79046i	0.103897	+	0.103897i	−0.653247	+	0.757144i	$0.726594\pi$
$12$		0			0
$13$	−12.4135	+	12.4135i	−0.264837	+	0.264837i	−0.827016	−	0.562179i	$-0.809963\pi$
$13$	−12.4135	+	12.4135i	−0.264837	+	0.264837i	0.562179	+	0.827016i	$0.309963\pi$
$14$		0			0
$15$			41.0653i			0.706868i
$16$		0			0
$17$		−	92.7849i		−	1.32374i	−0.749617	−	0.661872i	$-0.769762\pi$
$17$		−	92.7849i		−	1.32374i	0.749617	−	0.661872i	$-0.230238\pi$
$18$		0			0
$19$	−112.714	−	112.714i	−1.36096	−	1.36096i	−0.872693	−	0.488270i	$-0.837628\pi$
$19$	−112.714	−	112.714i	−1.36096	−	1.36096i	−0.488270	−	0.872693i	$-0.662372\pi$
$20$		0			0
$21$	25.2238	+	40.7833i	0.262108	+	0.423792i
$22$		0			0
$23$	−65.3480			−0.592435			−0.296218	−	0.955121i	$-0.595725\pi$
$23$	−65.3480			−0.592435			−0.296218	+	0.955121i	$0.595725\pi$
$24$		0			0
$25$			126.541i			1.01233i
$26$		0			0
$27$	−86.5923	+	86.5923i	−0.617211	+	0.617211i
$28$		0			0
$29$	179.260	+	179.260i	1.14786	+	1.14786i	0.986974	+	0.160882i	$0.0514337\pi$
$29$	179.260	+	179.260i	1.14786	+	1.14786i	0.160882	+	0.986974i	$0.448566\pi$
$30$		0			0
$31$	287.429			1.66528			0.832641	−	0.553813i	$-0.186828\pi$
$31$	287.429			1.66528			0.832641	+	0.553813i	$0.186828\pi$
$32$		0			0
$33$		−	13.8796i		−	0.0732162i
$34$		0			0
$35$	154.505	+	249.813i	0.746176	+	1.20646i
$36$		0			0
$37$	−194.602	+	194.602i	−0.864660	+	0.864660i	−0.991875	−	0.127215i	$-0.959396\pi$
$37$	−194.602	+	194.602i	−0.864660	+	0.864660i	0.127215	+	0.991875i	$0.459396\pi$
$38$		0			0
$39$	45.4548			0.186631
$40$		0			0
$41$	291.553			1.11056			0.555280	−	0.831663i	$-0.312611\pi$
$41$	291.553			1.11056			0.555280	+	0.831663i	$0.312611\pi$
$42$		0			0
$43$	−186.279	−	186.279i	−0.660635	−	0.660635i	0.294895	−	0.955530i	$-0.404715\pi$
$43$	−186.279	−	186.279i	−0.660635	−	0.660635i	−0.955530	+	0.294895i	$0.904715\pi$
$44$		0			0
$45$	−227.613	+	227.613i	−0.754012	+	0.754012i
$46$		0			0
$47$	−203.179			−0.630569			−0.315285	−	0.948997i	$-0.602100\pi$
$47$	−203.179			−0.630569			−0.315285	+	0.948997i	$0.602100\pi$
$48$		0			0
$49$	306.888	+	153.195i	0.894717	+	0.446633i
$50$		0			0
$51$	−169.876	+	169.876i	−0.466421	+	0.466421i
$52$		0			0
$53$	−205.330	+	205.330i	−0.532155	+	0.532155i	−0.921213	−	0.389058i	$-0.872801\pi$
$53$	−205.330	+	205.330i	−0.532155	+	0.532155i	0.389058	+	0.921213i	$0.372801\pi$
$54$		0			0
$55$		−	85.0182i		−	0.208434i
$56$		0			0
$57$			412.727i			0.959070i
$58$		0			0
$59$	−176.660	+	176.660i	−0.389816	+	0.389816i	−0.874622	−	0.484806i	$-0.838890\pi$
$59$	−176.660	+	176.660i	−0.389816	+	0.389816i	0.484806	+	0.874622i	$0.338890\pi$
$60$		0			0
$61$	410.414	−	410.414i	0.861446	−	0.861446i	−0.130060	−	0.991506i	$-0.541517\pi$
$61$	410.414	−	410.414i	0.861446	−	0.861446i	0.991506	+	0.130060i	$0.0415171\pi$
$62$		0			0
$63$	−86.2419	+	365.858i	−0.172468	+	0.731646i
$64$		0			0
$65$	278.428			0.531304
$66$		0			0
$67$	13.4406	−	13.4406i	0.0245079	−	0.0245079i	−0.694747	−	0.719255i	$-0.744484\pi$
$67$	13.4406	−	13.4406i	0.0245079	−	0.0245079i	0.719255	+	0.694747i	$0.244484\pi$
$68$		0			0
$69$	119.643	+	119.643i	0.208744	+	0.208744i
$70$		0			0
$71$	416.602			0.696360			0.348180	−	0.937428i	$-0.386800\pi$
$71$	416.602			0.696360			0.348180	+	0.937428i	$0.386800\pi$
$72$		0			0
$73$	268.844			0.431039			0.215519	−	0.976500i	$-0.430856\pi$
$73$	268.844			0.431039			0.215519	+	0.976500i	$0.430856\pi$
$74$		0			0
$75$	231.679	−	231.679i	0.356694	−	0.356694i
$76$		0			0
$77$	−52.2211	−	84.4342i	−0.0772876	−	0.124963i
$78$		0			0
$79$			705.142i			1.00424i	0.864799	+	0.502118i	$0.167446\pi$
$79$			705.142i			1.00424i	−0.864799	+	0.502118i	$0.832554\pi$
$80$		0			0
$81$	−230.911			−0.316751
$82$		0			0
$83$	296.671	+	296.671i	0.392335	+	0.392335i	0.875519	−	0.483184i	$-0.160520\pi$
$83$	296.671	+	296.671i	0.392335	+	0.392335i	−0.483184	+	0.875519i	$0.660520\pi$
$84$		0			0
$85$	−1040.56	+	1040.56i	−1.32782	+	1.32782i
$86$		0			0
$87$		−	656.402i		−	0.808893i
$88$		0			0
$89$	−1109.14			−1.32100			−0.660500	−	0.750826i	$-0.729656\pi$
$89$	−1109.14			−1.32100			−0.660500	+	0.750826i	$0.729656\pi$
$90$		0			0
$91$	276.516	−	171.020i	0.318536	−	0.197009i
$92$		0			0
$93$	−526.243	−	526.243i	−0.586762	−	0.586762i
$94$		0			0
$95$			2528.11i			2.73030i
$96$		0			0
$97$			383.187i			0.401101i	0.979683	+	0.200550i	$0.0642730\pi$
$97$			383.187i			0.401101i	−0.979683	+	0.200550i	$0.935727\pi$
$98$		0			0
$99$	76.9308	−	76.9308i	0.0780993	−	0.0780993i
$100$		0			0
$101$	−565.573	−	565.573i	−0.557194	−	0.557194i	0.371314	−	0.928508i	$-0.378907\pi$
$101$	−565.573	−	565.573i	−0.557194	−	0.557194i	−0.928508	+	0.371314i	$0.878907\pi$
$102$		0			0
$103$		−	152.956i		−	0.146323i	−0.997320	−	0.0731613i	$-0.976691\pi$
$103$		−	152.956i		−	0.146323i	0.997320	−	0.0731613i	$-0.0233088\pi$
$104$		0			0
$105$	174.496	−	740.252i	0.162182	−	0.688012i
$106$		0			0
$107$	147.332	+	147.332i	0.133113	+	0.133113i	0.770524	−	0.637411i	$-0.219995\pi$
$107$	147.332	+	147.332i	0.133113	+	0.133113i	−0.637411	+	0.770524i	$0.719995\pi$
$108$		0			0
$109$	−287.137	−	287.137i	−0.252319	−	0.252319i	0.569602	−	0.821921i	$-0.307097\pi$
$109$	−287.137	−	287.137i	−0.252319	−	0.252319i	−0.821921	+	0.569602i	$0.807097\pi$
$110$		0			0
$111$	712.580			0.609326
$112$		0			0
$113$	1290.14			1.07404			0.537020	−	0.843570i	$-0.319550\pi$
$113$	1290.14			1.07404			0.537020	+	0.843570i	$0.319550\pi$
$114$		0			0
$115$	732.862	+	732.862i	0.594258	+	0.594258i
$116$		0			0
$117$	251.943	+	251.943i	0.199078	+	0.199078i
$118$		0			0
$119$	−394.264	+	1672.56i	−0.303716	+	1.28843i
$120$		0			0
$121$		−	1302.26i		−	0.978411i
$122$		0			0
$123$	−533.794	−	533.794i	−0.391306	−	0.391306i
$124$		0			0
$125$	17.2830	−	17.2830i	0.0123667	−	0.0123667i
$126$		0			0
$127$		−	402.112i		−	0.280958i	−0.990084	−	0.140479i	$-0.955136\pi$
$127$		−	402.112i		−	0.280958i	0.990084	−	0.140479i	$-0.0448643\pi$
$128$		0			0
$129$			682.103i			0.465549i
$130$		0			0
$131$	−1908.64	−	1908.64i	−1.27297	−	1.27297i	−0.944524	−	0.328442i	$-0.893476\pi$
$131$	−1908.64	−	1908.64i	−1.27297	−	1.27297i	−0.328442	−	0.944524i	$-0.606524\pi$
$132$		0			0
$133$	1552.85	+	2510.75i	1.01240	+	1.63691i
$134$		0			0
$135$	1942.22			1.23822
$136$		0			0
$137$		−	1156.17i		−	0.721006i	−0.932758	−	0.360503i	$-0.882605\pi$
$137$		−	1156.17i		−	0.721006i	0.932758	−	0.360503i	$-0.117395\pi$
$138$		0			0
$139$	−41.7681	+	41.7681i	−0.0254872	+	0.0254872i	−0.719736	−	0.694248i	$-0.755737\pi$
$139$	−41.7681	+	41.7681i	−0.0254872	+	0.0254872i	0.694248	+	0.719736i	$0.255737\pi$
$140$		0			0
$141$	371.994	+	371.994i	0.222181	+	0.222181i
$142$		0			0
$143$	−94.1058			−0.0550316
$144$		0			0
$145$		−	4020.72i		−	2.30278i
$146$		0			0
$147$	−281.391	−	842.349i	−0.157883	−	0.472625i
$148$		0			0
$149$	−697.420	+	697.420i	−0.383456	+	0.383456i	−0.872346	−	0.488890i	$-0.837402\pi$
$149$	−697.420	+	697.420i	−0.383456	+	0.383456i	0.488890	+	0.872346i	$0.337402\pi$
$150$		0			0
$151$	1440.69			0.776438			0.388219	−	0.921567i	$-0.373090\pi$
$151$	1440.69			0.776438			0.388219	+	0.921567i	$0.373090\pi$
$152$		0			0
$153$	−1883.15			−0.995057
$154$		0			0
$155$	−3223.44	−	3223.44i	−1.67041	−	1.67041i
$156$		0			0
$157$	−257.853	+	257.853i	−0.131076	+	0.131076i	−0.769601	−	0.638525i	$-0.779545\pi$
$157$	−257.853	+	257.853i	−0.131076	+	0.131076i	0.638525	+	0.769601i	$0.279545\pi$
$158$		0			0
$159$	751.862			0.375010
$160$		0			0
$161$	1177.98	+	277.679i	0.576631	+	0.135926i
$162$		0			0
$163$	1931.95	−	1931.95i	0.928354	−	0.928354i	−0.0692461	−	0.997600i	$-0.522059\pi$
$163$	1931.95	−	1931.95i	0.928354	−	0.928354i	0.997600	+	0.0692461i	$0.0220594\pi$
$164$		0			0
$165$	−155.657	+	155.657i	−0.0734415	+	0.0734415i
$166$		0			0
$167$		−	694.423i		−	0.321773i	−0.986973	−	0.160886i	$-0.948565\pi$
$167$		−	694.423i		−	0.321773i	0.986973	−	0.160886i	$-0.0514353\pi$
$168$		0			0
$169$			1888.81i			0.859723i
$170$		0			0
$171$	−2287.62	+	2287.62i	−1.02303	+	1.02303i
$172$		0			0
$173$	−2284.55	+	2284.55i	−1.00400	+	1.00400i	−0.00400438	+	0.999992i	$0.501275\pi$
$173$	−2284.55	+	2284.55i	−1.00400	+	1.00400i	−0.999992	+	0.00400438i	$0.998725\pi$
$174$		0			0
$175$	537.702	−	2281.06i	0.232266	−	0.985323i
$176$		0			0
$177$	646.880			0.274703
$178$		0			0
$179$	−1164.37	+	1164.37i	−0.486195	+	0.486195i	−0.907103	−	0.420908i	$-0.861712\pi$
$179$	−1164.37	+	1164.37i	−0.486195	+	0.486195i	0.420908	+	0.907103i	$0.361712\pi$
$180$		0			0
$181$	−1563.58	−	1563.58i	−0.642099	−	0.642099i	0.308972	−	0.951071i	$-0.400015\pi$
$181$	−1563.58	−	1563.58i	−0.642099	−	0.642099i	−0.951071	+	0.308972i	$0.900015\pi$
$182$		0			0
$183$	−1502.83			−0.607061
$184$		0			0
$185$	4364.83			1.73464
$186$		0			0
$187$	351.698	−	351.698i	0.137533	−	0.137533i
$188$		0			0
$189$	1928.88	−	1192.98i	0.742357	−	0.459135i
$190$		0			0
$191$		−	887.128i		−	0.336075i	−0.985781	−	0.168037i	$-0.946257\pi$
$191$		−	887.128i		−	0.336075i	0.985781	−	0.168037i	$-0.0537429\pi$
$192$		0			0
$193$	2230.55			0.831908			0.415954	−	0.909386i	$-0.363448\pi$
$193$	2230.55			0.831908			0.415954	+	0.909386i	$0.363448\pi$
$194$		0			0
$195$	−509.764	−	509.764i	−0.187205	−	0.187205i
$196$		0			0
$197$	788.206	−	788.206i	0.285063	−	0.285063i	−0.550061	−	0.835124i	$-0.685396\pi$
$197$	788.206	−	788.206i	0.285063	−	0.285063i	0.835124	+	0.550061i	$0.185396\pi$
$198$		0			0
$199$		−	3530.96i		−	1.25780i	−0.777485	−	0.628902i	$-0.783505\pi$
$199$		−	3530.96i		−	1.25780i	0.777485	−	0.628902i	$-0.216495\pi$
$200$		0			0
$201$	−49.2157			−0.0172707
$202$		0			0
$203$	−2469.66	−	3993.10i	−0.853874	−	1.38059i
$204$		0			0
$205$	−3269.70	−	3269.70i	−1.11398	−	1.11398i
$206$		0			0
$207$			1326.30i			0.445333i
$208$		0			0
$209$		−	854.474i		−	0.282800i
$210$		0			0
$211$	1565.47	−	1565.47i	0.510763	−	0.510763i	−0.403997	−	0.914760i	$-0.632379\pi$
$211$	1565.47	−	1565.47i	0.510763	−	0.510763i	0.914760	+	0.403997i	$0.132379\pi$
$212$		0			0
$213$	−762.741	−	762.741i	−0.245362	−	0.245362i
$214$		0			0
$215$			4178.15i			1.32534i
$216$		0			0
$217$	−5181.25	−	1221.35i	−1.62086	−	0.382077i
$218$		0			0
$219$	−492.217	−	492.217i	−0.151876	−	0.151876i
$220$		0			0
$221$	1151.78	+	1151.78i	0.350576	+	0.350576i
$222$		0			0
$223$	−4375.56			−1.31394			−0.656972	−	0.753915i	$-0.728163\pi$
$223$	−4375.56			−1.31394			−0.656972	+	0.753915i	$0.728163\pi$
$224$		0			0
$225$	2568.26			0.760967
$226$		0			0
$227$	810.029	+	810.029i	0.236844	+	0.236844i	0.815542	−	0.578698i	$-0.196439\pi$
$227$	810.029	+	810.029i	0.236844	+	0.236844i	−0.578698	+	0.815542i	$0.696439\pi$
$228$		0			0
$229$	3364.38	+	3364.38i	0.970849	+	0.970849i	0.999587	−	0.0287379i	$-0.00914881\pi$
$229$	3364.38	+	3364.38i	0.970849	+	0.970849i	−0.0287379	+	0.999587i	$0.509149\pi$
$230$		0			0
$231$	−58.9778	+	250.197i	−0.0167985	+	0.0712631i
$232$		0			0
$233$		−	5545.33i		−	1.55917i	−0.626296	−	0.779585i	$-0.715430\pi$
$233$		−	5545.33i		−	1.55917i	0.626296	−	0.779585i	$-0.284570\pi$
$234$		0			0
$235$	2278.60	+	2278.60i	0.632510	+	0.632510i
$236$		0			0
$237$	1291.02	−	1291.02i	0.353842	−	0.353842i
$238$		0			0
$239$		−	1675.51i		−	0.453471i	−0.973956	−	0.226735i	$-0.927195\pi$
$239$		−	1675.51i		−	0.453471i	0.973956	−	0.226735i	$-0.0728053\pi$
$240$		0			0
$241$			1991.63i			0.532333i	0.963927	+	0.266167i	$0.0857571\pi$
$241$			1991.63i			0.532333i	−0.963927	+	0.266167i	$0.914243\pi$
$242$		0			0
$243$	2760.76	+	2760.76i	0.728818	+	0.728818i
$244$		0			0
$245$	−1723.63	−	5159.72i	−0.449464	−	1.34548i
$246$		0			0
$247$	2798.34			0.720867
$248$		0			0
$249$		−	1086.33i		−	0.276479i
$250$		0			0
$251$	238.611	−	238.611i	0.0600040	−	0.0600040i	−0.676468	−	0.736472i	$-0.736490\pi$
$251$	238.611	−	238.611i	0.0600040	−	0.0600040i	0.736472	+	0.676468i	$0.236490\pi$
$252$		0			0
$253$	−247.699	−	247.699i	−0.0615523	−	0.0615523i
$254$		0			0
$255$	3810.24			0.935712
$256$		0			0
$257$			6431.44i			1.56102i	0.625143	+	0.780510i	$0.285041\pi$
$257$			6431.44i			1.56102i	−0.625143	+	0.780510i	$0.714959\pi$
$258$		0			0
$259$	4334.85	−	2681.03i	1.03998	−	0.643209i
$260$		0			0
$261$	3638.25	−	3638.25i	0.862842	−	0.862842i
$262$		0			0
$263$	−5721.88			−1.34154			−0.670772	−	0.741663i	$-0.734037\pi$
$263$	−5721.88			−1.34154			−0.670772	+	0.741663i	$0.734037\pi$
$264$		0			0
$265$	4605.44			1.06759
$266$		0			0
$267$	2030.69	+	2030.69i	0.465454	+	0.465454i
$268$		0			0
$269$	−2618.36	+	2618.36i	−0.593474	+	0.593474i	−0.938568	−	0.345094i	$-0.887847\pi$
$269$	−2618.36	+	2618.36i	−0.593474	+	0.593474i	0.345094	+	0.938568i	$0.387847\pi$
$270$		0			0
$271$	910.567			0.204107			0.102054	−	0.994779i	$-0.467459\pi$
$271$	910.567			0.204107			0.102054	+	0.994779i	$0.467459\pi$
$272$		0			0
$273$	−819.378	−	193.148i	−0.181652	−	0.0428200i
$274$		0			0
$275$	−479.649	+	479.649i	−0.105178	+	0.105178i
$276$		0			0
$277$	−4584.43	+	4584.43i	−0.994409	+	0.994409i	−0.999984	−	0.00557497i	$-0.998225\pi$
$277$	−4584.43	+	4584.43i	−0.994409	+	0.994409i	0.00557497	+	0.999984i	$0.498225\pi$
$278$		0			0
$279$		−	5833.62i		−	1.25179i
$280$		0			0
$281$			2784.34i			0.591103i	0.955327	+	0.295552i	$0.0955034\pi$
$281$			2784.34i			0.591103i	−0.955327	+	0.295552i	$0.904497\pi$
$282$		0			0
$283$	−2601.85	+	2601.85i	−0.546516	+	0.546516i	−0.925431	−	0.378915i	$-0.876297\pi$
$283$	−2601.85	+	2601.85i	−0.546516	+	0.546516i	0.378915	+	0.925431i	$0.376297\pi$
$284$		0			0
$285$	4628.62	−	4628.62i	0.962021	−	0.962021i
$286$		0			0
$287$	−5255.60	−	1238.88i	−1.08093	−	0.254804i
$288$		0			0
$289$	−3696.03			−0.752296
$290$		0			0
$291$	701.563	−	701.563i	0.141328	−	0.141328i
$292$		0			0
$293$	−1789.88	−	1789.88i	−0.356880	−	0.356880i	0.505782	−	0.862662i	$-0.331204\pi$
$293$	−1789.88	−	1789.88i	−0.356880	−	0.356880i	−0.862662	+	0.505782i	$0.831204\pi$
$294$		0			0
$295$	3962.39			0.782032
$296$		0			0
$297$	−656.450			−0.128253
$298$		0			0
$299$	811.197	−	811.197i	0.156899	−	0.156899i
$300$		0			0
$301$	2566.36	+	4149.45i	0.491437	+	0.794585i
$302$		0			0
$303$			2070.97i			0.392654i
$304$		0			0
$305$	−9205.39			−1.72819
$306$		0			0
$307$	−437.170	−	437.170i	−0.0812724	−	0.0812724i	0.665302	−	0.746574i	$-0.268303\pi$
$307$	−437.170	−	437.170i	−0.0812724	−	0.0812724i	−0.746574	+	0.665302i	$0.768303\pi$
$308$		0			0
$309$	−280.042	+	280.042i	−0.0515567	+	0.0515567i
$310$		0			0
$311$			3087.74i			0.562989i	0.959563	+	0.281495i	$0.0908301\pi$
$311$			3087.74i			0.562989i	−0.959563	+	0.281495i	$0.909170\pi$
$312$		0			0
$313$	−2848.48			−0.514395			−0.257197	−	0.966359i	$-0.582799\pi$
$313$	−2848.48			−0.514395			−0.257197	+	0.966359i	$0.582799\pi$
$314$		0			0
$315$	5070.18	−	3135.82i	0.906896	−	0.560900i
$316$		0			0
$317$	−3347.44	−	3347.44i	−0.593095	−	0.593095i	0.345371	−	0.938466i	$-0.387753\pi$
$317$	−3347.44	−	3347.44i	−0.593095	−	0.593095i	−0.938466	+	0.345371i	$0.887753\pi$
$318$		0			0
$319$			1358.96i			0.238518i
$320$		0			0
$321$		−	539.490i		−	0.0938050i
$322$		0			0
$323$	−10458.1	+	10458.1i	−1.80156	+	1.80156i
$324$		0			0
$325$	−1570.82	−	1570.82i	−0.268102	−	0.268102i
$326$		0			0
$327$			1051.42i			0.177809i
$328$		0			0
$329$	3662.55	+	863.356i	0.613748	+	0.144676i
$330$		0			0
$331$	3102.79	+	3102.79i	0.515241	+	0.515241i	0.916128	−	0.400887i	$-0.131298\pi$
$331$	3102.79	+	3102.79i	0.515241	+	0.515241i	−0.400887	+	0.916128i	$0.631298\pi$
$332$		0			0
$333$	3949.62	+	3949.62i	0.649964	+	0.649964i
$334$		0			0
$335$	−301.465			−0.0491666
$336$		0			0
$337$	−188.583			−0.0304830			−0.0152415	−	0.999884i	$-0.504852\pi$
$337$	−188.583			−0.0304830			−0.0152415	+	0.999884i	$0.504852\pi$
$338$		0			0
$339$	−2362.08	−	2362.08i	−0.378438	−	0.378438i
$340$		0			0
$341$	1089.49	+	1089.49i	0.173018	+	0.173018i
$342$		0			0
$343$	−4881.07	−	4065.56i	−0.768375	−	0.639999i
$344$		0			0
$345$		−	2683.54i		−	0.418774i
$346$		0			0
$347$	6366.71	+	6366.71i	0.984965	+	0.984965i	0.999889	−	0.0149239i	$-0.00475059\pi$
$347$	6366.71	+	6366.71i	0.984965	+	0.984965i	−0.0149239	+	0.999889i	$0.504751\pi$
$348$		0			0
$349$	−1932.62	+	1932.62i	−0.296421	+	0.296421i	−0.839610	−	0.543189i	$-0.817217\pi$
$349$	−1932.62	+	1932.62i	−0.296421	+	0.296421i	0.543189	+	0.839610i	$0.317217\pi$
$350$		0			0
$351$		−	2149.83i		−	0.326921i
$352$		0			0
$353$		−	7298.63i		−	1.10047i	−0.835009	−	0.550236i	$-0.814538\pi$
$353$		−	7298.63i		−	1.10047i	0.835009	−	0.550236i	$-0.185462\pi$
$354$		0			0
$355$	−4672.08	−	4672.08i	−0.698503	−	0.698503i
$356$		0			0
$357$	3784.07	−	2340.38i	0.560992	−	0.346964i
$358$		0			0
$359$	−357.053			−0.0524918			−0.0262459	−	0.999656i	$-0.508355\pi$
$359$	−357.053			−0.0524918			−0.0262459	+	0.999656i	$0.508355\pi$
$360$		0			0
$361$			18549.7i			2.70444i
$362$		0			0
$363$	−2384.27	+	2384.27i	−0.344743	+	0.344743i
$364$		0			0
$365$	−3015.02	−	3015.02i	−0.432365	−	0.432365i
$366$		0			0
$367$	1135.13			0.161453			0.0807266	−	0.996736i	$-0.474276\pi$
$367$	1135.13			0.161453			0.0807266	+	0.996736i	$0.474276\pi$
$368$		0			0
$369$		−	5917.33i		−	0.834807i
$370$		0			0
$371$	4573.81	−	2828.82i	0.640055	−	0.395863i
$372$		0			0
$373$	7327.70	−	7327.70i	1.01720	−	1.01720i	0.0173464	−	0.999850i	$-0.494478\pi$
$373$	7327.70	−	7327.70i	1.01720	−	1.01720i	0.999850	−	0.0173464i	$-0.00552181\pi$
$374$		0			0
$375$	−63.2855			−0.00871479
$376$		0			0
$377$	−4450.49			−0.607989
$378$		0			0
$379$	6349.98	+	6349.98i	0.860625	+	0.860625i	0.991411	−	0.130786i	$-0.0417501\pi$
$379$	6349.98	+	6349.98i	0.860625	+	0.860625i	−0.130786	+	0.991411i	$0.541750\pi$
$380$		0			0
$381$	−736.212	+	736.212i	−0.0989955	+	0.0989955i
$382$		0			0
$383$	7528.58			1.00442			0.502209	−	0.864746i	$-0.332521\pi$
$383$	7528.58			1.00442			0.502209	+	0.864746i	$0.332521\pi$
$384$		0			0
$385$	−361.262	+	1532.55i	−0.0478224	+	0.202873i
$386$		0			0
$387$	−3780.70	+	3780.70i	−0.496599	+	0.496599i
$388$		0			0
$389$	−6789.02	+	6789.02i	−0.884877	+	0.884877i	−0.994025	−	0.109149i	$-0.965187\pi$
$389$	−6789.02	+	6789.02i	−0.884877	+	0.884877i	0.109149	+	0.994025i	$0.465187\pi$
$390$		0			0
$391$			6063.31i			0.784232i
$392$		0			0
$393$			6988.92i			0.897059i
$394$		0			0
$395$	7907.99	−	7907.99i	1.00733	−	1.00733i
$396$		0			0
$397$	−7961.75	+	7961.75i	−1.00652	+	1.00652i	−0.00654246	+	0.999979i	$0.502083\pi$
$397$	−7961.75	+	7961.75i	−1.00652	+	1.00652i	−0.999979	+	0.00654246i	$0.997917\pi$
$398$		0			0
$399$	1753.77	−	7439.89i	0.220046	−	0.933485i
$400$		0			0
$401$	−10277.5			−1.27988			−0.639940	−	0.768425i	$-0.721041\pi$
$401$	−10277.5			−1.27988			−0.639940	+	0.768425i	$0.721041\pi$
$402$		0			0
$403$	−3567.99	+	3567.99i	−0.441028	+	0.441028i
$404$		0			0
$405$	2589.61	+	2589.61i	0.317726	+	0.317726i
$406$		0			0
$407$	−1475.27			−0.179671
$408$		0			0
$409$	684.166			0.0827135			0.0413567	−	0.999144i	$-0.486832\pi$
$409$	684.166			0.0827135			0.0413567	+	0.999144i	$0.486832\pi$
$410$		0			0
$411$	−2116.78	+	2116.78i	−0.254046	+	0.254046i
$412$		0			0
$413$	3935.18	−	2433.84i	0.468855	−	0.289979i
$414$		0			0
$415$		−	6654.17i		−	0.787086i
$416$		0			0
$417$	152.943			0.0179608
$418$		0			0
$419$	5968.84	+	5968.84i	0.695936	+	0.695936i	0.963531	−	0.267596i	$-0.0862292\pi$
$419$	5968.84	+	5968.84i	0.695936	+	0.695936i	−0.267596	+	0.963531i	$0.586229\pi$
$420$		0			0
$421$	6775.82	−	6775.82i	0.784402	−	0.784402i	−0.196168	−	0.980570i	$-0.562850\pi$
$421$	6775.82	−	6775.82i	0.784402	−	0.784402i	0.980570	+	0.196168i	$0.0628498\pi$
$422$		0			0
$423$			4123.70i			0.473998i
$424$		0			0
$425$	11741.1			1.34006
$426$		0			0
$427$	−9142.16	+	5654.27i	−1.03611	+	0.640818i
$428$		0			0
$429$	172.295	+	172.295i	0.0193904	+	0.0193904i
$430$		0			0
$431$			8756.78i			0.978652i	0.872101	+	0.489326i	$0.162757\pi$
$431$			8756.78i			0.978652i	−0.872101	+	0.489326i	$0.837243\pi$
$432$		0			0
$433$			10685.4i			1.18593i	0.805229	+	0.592964i	$0.202043\pi$
$433$			10685.4i			1.18593i	−0.805229	+	0.592964i	$0.797957\pi$
$434$		0			0
$435$	−7361.39	+	7361.39i	−0.811383	+	0.811383i
$436$		0			0
$437$	7365.62	+	7365.62i	0.806282	+	0.806282i
$438$		0			0
$439$			7688.54i			0.835886i	0.908473	+	0.417943i	$0.137249\pi$
$439$			7688.54i			0.835886i	−0.908473	+	0.417943i	$0.862751\pi$
$440$		0			0
$441$	3109.23	−	6228.56i	0.335733	−	0.672558i
$442$		0			0
$443$	852.302	+	852.302i	0.0914088	+	0.0914088i	0.751333	−	0.659924i	$-0.229411\pi$
$443$	852.302	+	852.302i	0.0914088	+	0.0914088i	−0.659924	+	0.751333i	$0.729411\pi$
$444$		0			0
$445$	12438.8	+	12438.8i	1.32506	+	1.32506i
$446$		0			0
$447$	2553.76			0.270221
$448$		0			0
$449$	−4449.83			−0.467707			−0.233853	−	0.972272i	$-0.575134\pi$
$449$	−4449.83			−0.467707			−0.233853	+	0.972272i	$0.575134\pi$
$450$		0			0
$451$	1105.12	+	1105.12i	0.115384	+	0.115384i
$452$		0			0
$453$	−2637.72	−	2637.72i	−0.273578	−	0.273578i
$454$		0			0
$455$	−5019.01	−	1183.11i	−0.517131	−	0.121901i
$456$		0			0
$457$			7852.46i			0.803769i	0.915690	+	0.401885i	$0.131645\pi$
$457$			7852.46i			0.803769i	−0.915690	+	0.401885i	$0.868355\pi$
$458$		0			0
$459$	8034.45	+	8034.45i	0.817029	+	0.817029i
$460$		0			0
$461$	7341.95	−	7341.95i	0.741755	−	0.741755i	−0.231161	−	0.972916i	$-0.574252\pi$
$461$	7341.95	−	7341.95i	0.741755	−	0.741755i	0.972916	+	0.231161i	$0.0742524\pi$
$462$		0			0
$463$			3439.86i			0.345278i	0.984985	+	0.172639i	$0.0552294\pi$
$463$			3439.86i			0.345278i	−0.984985	+	0.172639i	$0.944771\pi$
$464$		0			0
$465$			11803.4i			1.17714i
$466$		0			0
$467$	2011.27	+	2011.27i	0.199295	+	0.199295i	0.799698	−	0.600403i	$-0.204993\pi$
$467$	2011.27	+	2011.27i	0.199295	+	0.199295i	−0.600403	+	0.799698i	$0.704993\pi$
$468$		0			0
$469$	−299.395	+	185.170i	−0.0294771	+	0.0182311i
$470$		0			0
$471$	944.187			0.0923690
$472$		0			0
$473$		−	1412.17i		−	0.137276i
$474$		0			0
$475$	14262.9	−	14262.9i	1.37774	−	1.37774i
$476$		0			0
$477$	4167.35	+	4167.35i	0.400021	+	0.400021i
$478$		0			0
$479$	−17755.0			−1.69363			−0.846814	−	0.531890i	$-0.821482\pi$
$479$	−17755.0			−1.69363			−0.846814	+	0.531890i	$0.821482\pi$
$480$		0			0
$481$		−	4831.39i		−	0.457988i
$482$		0			0
$483$	−1648.32	−	2665.11i	−0.155282	−	0.251070i
$484$		0			0
$485$	4297.35	−	4297.35i	0.402335	−	0.402335i
$486$		0			0
$487$	−18871.9			−1.75599			−0.877997	−	0.478666i	$-0.841120\pi$
$487$	−18871.9			−1.75599			−0.877997	+	0.478666i	$0.841120\pi$
$488$		0			0
$489$	−7074.26			−0.654210
$490$		0			0
$491$	−5489.27	−	5489.27i	−0.504537	−	0.504537i	0.408308	−	0.912844i	$-0.366119\pi$
$491$	−5489.27	−	5489.27i	−0.504537	−	0.504537i	−0.912844	+	0.408308i	$0.866119\pi$
$492$		0			0
$493$	16632.6	−	16632.6i	1.51947	−	1.51947i
$494$		0			0
$495$	−1725.52			−0.156679
$496$		0			0
$497$	−7509.75	−	1770.24i	−0.677783	−	0.159771i
$498$		0			0
$499$	9410.32	−	9410.32i	0.844216	−	0.844216i	−0.145188	−	0.989404i	$-0.546379\pi$
$499$	9410.32	−	9410.32i	0.844216	−	0.844216i	0.989404	+	0.145188i	$0.0463788\pi$
$500$		0			0
$501$	−1271.39	+	1271.39i	−0.113377	+	0.113377i
$502$		0			0
$503$			10971.4i			0.972548i	0.873806	+	0.486274i	$0.161644\pi$
$503$			10971.4i			0.972548i	−0.873806	+	0.486274i	$0.838356\pi$
$504$		0			0
$505$			12685.5i			1.11782i
$506$		0			0
$507$	3458.15	−	3458.15i	0.302923	−	0.302923i
$508$		0			0
$509$	−3007.58	+	3007.58i	−0.261903	+	0.261903i	−0.825827	−	0.563924i	$-0.809291\pi$
$509$	−3007.58	+	3007.58i	−0.261903	+	0.261903i	0.563924	+	0.825827i	$0.309291\pi$
$510$		0			0
$511$	−4846.24	−	1142.38i	−0.419540	−	0.0988962i
$512$		0			0
$513$	19520.3			1.68000
$514$		0			0
$515$	−1715.37	+	1715.37i	−0.146773	+	0.146773i
$516$		0			0
$517$	−770.144	−	770.144i	−0.0655143	−	0.0655143i
$518$		0			0
$519$	8365.41			0.707516
$520$		0			0
$521$	−23290.9			−1.95853			−0.979265	−	0.202585i	$-0.935066\pi$
$521$	−23290.9			−1.95853			−0.979265	+	0.202585i	$0.935066\pi$
$522$		0			0
$523$	−6591.18	+	6591.18i	−0.551075	+	0.551075i	−0.926751	−	0.375676i	$-0.877411\pi$
$523$	−6591.18	+	6591.18i	−0.551075	+	0.551075i	0.375676	+	0.926751i	$0.377411\pi$
$524$		0			0
$525$	−5160.76	+	3191.84i	−0.429017	+	0.265340i
$526$		0			0
$527$		−	26669.0i		−	2.20441i
$528$		0			0
$529$	−7896.63			−0.649021
$530$		0			0
$531$	3585.47	+	3585.47i	0.293024	+	0.293024i
$532$		0			0
$533$	−3619.19	+	3619.19i	−0.294118	+	0.294118i
$534$		0			0
$535$		−	3304.59i		−	0.267046i
$536$		0			0
$537$	4263.60			0.342622
$538$		0			0
$539$	582.568	+	1743.93i	0.0465547	+	0.139362i
$540$		0			0
$541$	3915.48	+	3915.48i	0.311164	+	0.311164i	0.845360	−	0.534196i	$-0.179386\pi$
$541$	3915.48	+	3915.48i	0.311164	+	0.311164i	−0.534196	+	0.845360i	$0.679386\pi$
$542$		0			0
$543$			5725.40i			0.452487i
$544$		0			0
$545$			6440.33i			0.506190i
$546$		0			0
$547$	285.556	−	285.556i	0.0223208	−	0.0223208i	−0.695858	−	0.718179i	$-0.744976\pi$
$547$	285.556	−	285.556i	0.0223208	−	0.0223208i	0.718179	+	0.695858i	$0.244976\pi$
$548$		0			0
$549$	−8329.72	−	8329.72i	−0.647548	−	0.647548i
$550$		0			0
$551$		−	40410.2i		−	3.12438i
$552$		0			0
$553$	2996.31	−	12711.0i	0.230409	−	0.977447i
$554$		0			0
$555$	−7991.41	−	7991.41i	−0.611201	−	0.611201i
$556$		0			0
$557$	5713.13	+	5713.13i	0.434601	+	0.434601i	0.890190	−	0.455589i	$-0.150571\pi$
$557$	5713.13	+	5713.13i	0.434601	+	0.434601i	−0.455589	+	0.890190i	$0.650571\pi$
$558$		0			0
$559$	4624.75			0.349921
$560$		0			0
$561$	−1287.82			−0.0969195
$562$		0			0
$563$	−4321.65	−	4321.65i	−0.323510	−	0.323510i	0.526602	−	0.850112i	$-0.323466\pi$
$563$	−4321.65	−	4321.65i	−0.323510	−	0.323510i	−0.850112	+	0.526602i	$0.823466\pi$
$564$		0			0
$565$	−14468.6	−	14468.6i	−1.07734	−	1.07734i
$566$		0			0
$567$	4162.45	+	981.196i	0.308301	+	0.0726743i
$568$		0			0
$569$		−	7330.40i		−	0.540082i	−0.962849	−	0.270041i	$-0.912963\pi$
$569$		−	7330.40i		−	0.540082i	0.962849	−	0.270041i	$-0.0870372\pi$
$570$		0			0
$571$	−2589.85	−	2589.85i	−0.189811	−	0.189811i	0.605803	−	0.795614i	$-0.292852\pi$
$571$	−2589.85	−	2589.85i	−0.189811	−	0.189811i	−0.795614	+	0.605803i	$0.792852\pi$
$572$		0			0
$573$	−1624.21	+	1624.21i	−0.118416	+	0.118416i
$574$		0			0
$575$		−	8269.21i		−	0.599739i
$576$		0			0
$577$		−	21935.0i		−	1.58261i	−0.611423	−	0.791304i	$-0.709403\pi$
$577$		−	21935.0i		−	1.58261i	0.611423	−	0.791304i	$-0.290597\pi$
$578$		0			0
$579$	−4083.83	−	4083.83i	−0.293123	−	0.293123i
$580$		0			0
$581$	−4087.22	−	6608.47i	−0.291853	−	0.471886i
$582$		0			0
$583$	−1556.59			−0.110579
$584$		0			0
$585$		−	5650.95i		−	0.399381i
$586$		0			0
$587$	−12339.1	+	12339.1i	−0.867610	+	0.867610i	−0.992207	−	0.124597i	$-0.960236\pi$
$587$	−12339.1	+	12339.1i	−0.867610	+	0.867610i	0.124597	+	0.992207i	$0.460236\pi$
$588$		0			0
$589$	−32397.2	−	32397.2i	−2.26639	−	2.26639i
$590$		0			0
$591$	−2886.19			−0.200884
$592$		0			0
$593$			5060.82i			0.350460i	0.984528	+	0.175230i	$0.0560669\pi$
$593$			5060.82i			0.350460i	−0.984528	+	0.175230i	$0.943933\pi$
$594$		0			0
$595$	23178.9	−	14335.7i	1.59705	−	0.987745i
$596$		0			0
$597$	−6464.70	+	6464.70i	−0.443187	+	0.443187i
$598$		0			0
$599$	−4676.57			−0.318997			−0.159499	−	0.987198i	$-0.550988\pi$
$599$	−4676.57			−0.318997			−0.159499	+	0.987198i	$0.550988\pi$
$600$		0			0
$601$	21898.3			1.48627			0.743137	−	0.669140i	$-0.233337\pi$
$601$	21898.3			1.48627			0.743137	+	0.669140i	$0.233337\pi$
$602$		0			0
$603$	−272.788	−	272.788i	−0.0184225	−	0.0184225i
$604$		0			0
$605$	−14604.6	+	14604.6i	−0.981422	+	0.981422i
$606$		0			0
$607$	−19840.3			−1.32667			−0.663337	−	0.748321i	$-0.730860\pi$
$607$	−19840.3			−1.32667			−0.663337	+	0.748321i	$0.730860\pi$
$608$		0			0
$609$	−2789.21	+	11832.4i	−0.185590	+	0.787315i
$610$		0			0
$611$	2522.16	−	2522.16i	0.166998	−	0.166998i
$612$		0			0
$613$	11909.5	−	11909.5i	0.784696	−	0.784696i	−0.195923	−	0.980619i	$-0.562770\pi$
$613$	11909.5	−	11909.5i	0.784696	−	0.784696i	0.980619	+	0.195923i	$0.0627703\pi$
$614$		0			0
$615$			11972.7i			0.785020i
$616$		0			0
$617$			13162.6i			0.858842i	0.903104	+	0.429421i	$0.141282\pi$
$617$			13162.6i			0.858842i	−0.903104	+	0.429421i	$0.858718\pi$
$618$		0			0
$619$	11398.3	−	11398.3i	0.740125	−	0.740125i	−0.232477	−	0.972602i	$-0.574683\pi$
$619$	11398.3	−	11398.3i	0.740125	−	0.740125i	0.972602	+	0.232477i	$0.0746830\pi$
$620$		0			0
$621$	5658.64	−	5658.64i	0.365657	−	0.365657i
$622$		0			0
$623$	19993.6	+	4713.01i	1.28576	+	0.303086i
$624$		0			0
$625$	15430.0			0.987519
$626$		0			0
$627$	−1564.43	+	1564.43i	−0.0996445	+	0.0996445i
$628$		0			0
$629$	18056.1	+	18056.1i	1.14459	+	1.14459i
$630$		0			0
$631$	29952.0			1.88965			0.944826	−	0.327574i	$-0.106231\pi$
$631$	29952.0			1.88965			0.944826	+	0.327574i	$0.106231\pi$
$632$		0			0
$633$	−5732.31			−0.359935
$634$		0			0
$635$	−4509.58	+	4509.58i	−0.281823	+	0.281823i
$636$		0			0
$637$	−5711.24	+	1907.87i	−0.355239	+	0.118669i
$638$		0			0
$639$		−	8455.30i		−	0.523453i
$640$		0			0
$641$	−12639.3			−0.778817			−0.389409	−	0.921065i	$-0.627321\pi$
$641$	−12639.3			−0.778817			−0.389409	+	0.921065i	$0.627321\pi$
$642$		0			0
$643$	−3582.49	−	3582.49i	−0.219720	−	0.219720i	0.588661	−	0.808380i	$-0.299655\pi$
$643$	−3582.49	−	3582.49i	−0.219720	−	0.219720i	−0.808380	+	0.588661i	$0.799655\pi$
$644$		0			0
$645$	7649.62	−	7649.62i	0.466982	−	0.466982i
$646$		0			0
$647$			21300.9i			1.29432i	0.762354	+	0.647160i	$0.224044\pi$
$647$			21300.9i			1.29432i	−0.762354	+	0.647160i	$0.775956\pi$
$648$		0			0
$649$	−1339.25			−0.0810015
$650$		0			0
$651$	7250.03	+	11722.3i	0.436484	+	0.705734i
$652$		0			0
$653$	−17283.7	−	17283.7i	−1.03578	−	1.03578i	−0.999336	−	0.0364411i	$-0.988398\pi$
$653$	−17283.7	−	17283.7i	−1.03578	−	1.03578i	−0.0364411	−	0.999336i	$-0.511602\pi$
$654$		0			0
$655$			42809.8i			2.55377i
$656$		0			0
$657$		−	5456.43i		−	0.324011i
$658$		0			0
$659$	−16431.8	+	16431.8i	−0.971309	+	0.971309i	−0.999600	−	0.0282905i	$-0.990994\pi$
$659$	−16431.8	+	16431.8i	−0.971309	+	0.971309i	0.0282905	+	0.999600i	$0.490994\pi$
$660$		0			0
$661$	13671.4	+	13671.4i	0.804471	+	0.804471i	0.983791	−	0.179320i	$-0.0573898\pi$
$661$	13671.4	+	13671.4i	0.804471	+	0.804471i	−0.179320	+	0.983791i	$0.557390\pi$
$662$		0			0
$663$		−	4217.52i		−	0.247051i
$664$		0			0
$665$	10742.5	−	45572.2i	0.626432	−	2.65747i
$666$		0			0
$667$	−11714.3	−	11714.3i	−0.680030	−	0.680030i
$668$		0			0
$669$	8011.06	+	8011.06i	0.462968	+	0.462968i
$670$		0			0
$671$	3111.32			0.179003
$672$		0			0
$673$	28888.0			1.65460			0.827302	−	0.561757i	$-0.189874\pi$
$673$	28888.0			1.65460			0.827302	+	0.561757i	$0.189874\pi$
$674$		0			0
$675$	−10957.5	−	10957.5i	−0.624820	−	0.624820i
$676$		0			0
$677$	−5592.63	−	5592.63i	−0.317492	−	0.317492i	0.530311	−	0.847803i	$-0.322075\pi$
$677$	−5592.63	−	5592.63i	−0.317492	−	0.317492i	−0.847803	+	0.530311i	$0.822075\pi$
$678$		0			0
$679$	1628.25	−	6907.41i	0.0920273	−	0.390401i
$680$		0			0
$681$		−	2966.10i		−	0.166904i
$682$		0			0
$683$	−3564.66	−	3564.66i	−0.199704	−	0.199704i	0.600169	−	0.799873i	$-0.295100\pi$
$683$	−3564.66	−	3564.66i	−0.199704	−	0.199704i	−0.799873	+	0.600169i	$0.795100\pi$
$684$		0			0
$685$	−12966.1	+	12966.1i	−0.723225	+	0.723225i
$686$		0			0
$687$		−	12319.4i		−	0.684157i
$688$		0			0
$689$		−	5097.72i		−	0.281869i
$690$		0			0
$691$	10009.9	+	10009.9i	0.551080	+	0.551080i	0.926752	−	0.375673i	$-0.122588\pi$
$691$	10009.9	+	10009.9i	0.551080	+	0.551080i	−0.375673	+	0.926752i	$0.622588\pi$
$692$		0			0
$693$	−1713.67	+	1059.87i	−0.0939348	+	0.0580970i
$694$		0			0
$695$	936.838			0.0511313
$696$		0			0
$697$		−	27051.7i		−	1.47010i
$698$		0			0
$699$	−10152.7	+	10152.7i	−0.549374	+	0.549374i
$700$		0			0
$701$	−5491.44	−	5491.44i	−0.295875	−	0.295875i	0.543520	−	0.839396i	$-0.317091\pi$
$701$	−5491.44	−	5491.44i	−0.295875	−	0.295875i	−0.839396	+	0.543520i	$0.817091\pi$
$702$		0			0
$703$	43868.7			2.35354
$704$		0			0
$705$		−	8343.63i		−	0.445729i
$706$		0			0
$707$	7791.88	+	12598.4i	0.414489	+	0.670171i
$708$		0			0
$709$	7585.98	−	7585.98i	0.401830	−	0.401830i	−0.477047	−	0.878878i	$-0.658293\pi$
$709$	7585.98	−	7585.98i	0.401830	−	0.401830i	0.878878	+	0.477047i	$0.158293\pi$
$710$		0			0
$711$	14311.5			0.754883
$712$		0			0
$713$	−18782.9			−0.986571
$714$		0			0
$715$	1055.37	+	1055.37i	0.0552010	+	0.0552010i
$716$		0			0
$717$	−3067.62	+	3067.62i	−0.159780	+	0.159780i
$718$		0			0
$719$	16056.7			0.832842			0.416421	−	0.909172i	$-0.363284\pi$
$719$	16056.7			0.832842			0.416421	+	0.909172i	$0.363284\pi$
$720$		0			0
$721$	−649.946	+	2757.22i	−0.0335718	+	0.142419i
$722$		0			0
$723$	3646.41	−	3646.41i	0.187567	−	0.187567i
$724$		0			0
$725$	−22683.8	+	22683.8i	−1.16201	+	1.16201i
$726$		0			0
$727$		−	28984.2i		−	1.47863i	−0.673360	−	0.739314i	$-0.735150\pi$
$727$		−	28984.2i		−	1.47863i	0.673360	−	0.739314i	$-0.264850\pi$
$728$		0			0
$729$		−	3874.54i		−	0.196847i
$730$		0			0
$731$	−17283.9	+	17283.9i	−0.874511	+	0.874511i
$732$		0			0
$733$	−6059.21	+	6059.21i	−0.305323	+	0.305323i	−0.843092	−	0.537769i	$-0.819267\pi$
$733$	−6059.21	+	6059.21i	−0.305323	+	0.305323i	0.537769	+	0.843092i	$0.319267\pi$
$734$		0			0
$735$	−6291.01	+	12602.5i	−0.315711	+	0.632447i
$736$		0			0
$737$	101.892			0.00509259
$738$		0			0
$739$	19859.9	−	19859.9i	0.988578	−	0.988578i	−0.0113576	−	0.999936i	$-0.503615\pi$
$739$	19859.9	−	19859.9i	0.988578	−	0.988578i	0.999936	+	0.0113576i	$0.00361530\pi$
$740$		0			0
$741$	−5123.38	−	5123.38i	−0.253997	−	0.253997i
$742$		0			0
$743$	8710.40			0.430086			0.215043	−	0.976605i	$-0.431011\pi$
$743$	8710.40			0.430086			0.215043	+	0.976605i	$0.431011\pi$
$744$		0			0
$745$	15642.8			0.769272
$746$		0			0
$747$	6021.19	−	6021.19i	0.294918	−	0.294918i
$748$		0			0
$749$	−2029.79	−	3281.89i	−0.0990213	−	0.160104i
$750$		0			0
$751$		−	12623.6i		−	0.613373i	−0.951811	−	0.306686i	$-0.900780\pi$
$751$		−	12623.6i		−	0.613373i	0.951811	−	0.306686i	$-0.0992202\pi$
$752$		0			0
$753$	−873.729			−0.0422848
$754$		0			0
$755$	−16157.0	−	16157.0i	−0.778827	−	0.778827i
$756$		0			0
$757$	−14538.8	+	14538.8i	−0.698046	+	0.698046i	−0.963989	−	0.265943i	$-0.914317\pi$
$757$	−14538.8	+	14538.8i	−0.698046	+	0.698046i	0.265943	+	0.963989i	$0.414317\pi$
$758$		0			0
$759$			907.007i			0.0433759i
$760$		0			0
$761$	−13774.9			−0.656164			−0.328082	−	0.944649i	$-0.606402\pi$
$761$	−13774.9			−0.656164			−0.328082	+	0.944649i	$0.606402\pi$
$762$		0			0
$763$	3955.88	+	6396.10i	0.187696	+	0.303479i
$764$		0			0
$765$	21119.1	+	21119.1i	0.998119	+	0.998119i
$766$		0			0
$767$		−	4385.93i		−	0.206476i
$768$		0			0
$769$		−	39212.0i		−	1.83878i	−0.393351	−	0.919388i	$-0.628684\pi$
$769$		−	39212.0i		−	1.83878i	0.393351	−	0.919388i	$-0.371316\pi$
$770$		0			0
$771$	11775.1	−	11775.1i	0.550025	−	0.550025i
$772$		0			0
$773$	12077.5	+	12077.5i	0.561961	+	0.561961i	0.929864	−	0.367903i	$-0.119924\pi$
$773$	12077.5	+	12077.5i	0.561961	+	0.561961i	−0.367903	+	0.929864i	$0.619924\pi$
$774$		0			0
$775$			36371.6i			1.68581i
$776$		0			0
$777$	−12845.1	−	3027.92i	−0.593071	−	0.139802i
$778$		0			0
$779$	−32862.0	−	32862.0i	−1.51143	−	1.51143i
$780$		0			0
$781$	1579.11	+	1579.11i	0.0723497	+	0.0723497i
$782$		0			0
$783$	−31045.1			−1.41694
$784$		0			0
$785$	5783.51			0.262958
$786$		0			0
$787$	−12974.2	−	12974.2i	−0.587649	−	0.587649i	0.349345	−	0.936994i	$-0.386404\pi$
$787$	−12974.2	−	12974.2i	−0.587649	−	0.587649i	−0.936994	+	0.349345i	$0.886404\pi$
$788$		0			0
$789$	10476.0	+	10476.0i	0.472693	+	0.472693i
$790$		0			0
$791$	−23256.4	−	5482.11i	−1.04539	−	0.246424i
$792$		0			0
$793$			10189.4i			0.456286i
$794$		0			0
$795$	−8431.94	−	8431.94i	−0.376164	−	0.376164i
$796$		0			0
$797$	10200.6	−	10200.6i	0.453353	−	0.453353i	−0.443113	−	0.896466i	$-0.646126\pi$
$797$	10200.6	−	10200.6i	0.453353	−	0.453353i	0.896466	+	0.443113i	$0.146126\pi$
$798$		0			0
$799$			18852.0i			0.834711i
$800$		0			0
$801$			22511.0i			0.992994i
$802$		0			0
$803$	1019.04	+	1019.04i	0.0447836	+	0.0447836i
$804$		0			0
$805$	−10096.6	−	16324.8i	−0.442061	−	0.714750i
$806$		0			0
$807$	9587.73			0.418221
$808$		0			0
$809$			5095.00i			0.221422i	0.993853	+	0.110711i	$0.0353128\pi$
$809$			5095.00i			0.221422i	−0.993853	+	0.110711i	$0.964687\pi$
$810$		0			0
$811$	16365.5	−	16365.5i	0.708595	−	0.708595i	−0.257644	−	0.966240i	$-0.582946\pi$
$811$	16365.5	−	16365.5i	0.708595	−	0.708595i	0.966240	+	0.257644i	$0.0829463\pi$
$812$		0			0
$813$	−1667.12	−	1667.12i	−0.0719171	−	0.0719171i
$814$		0			0
$815$	−43332.6			−1.86242
$816$		0			0
$817$			41992.4i			1.79820i
$818$		0			0
$819$	−3471.01	−	5612.13i	−0.148091	−	0.239443i
$820$		0			0
$821$	−17759.4	+	17759.4i	−0.754943	+	0.754943i	−0.975397	−	0.220455i	$-0.929246\pi$
$821$	−17759.4	+	17759.4i	−0.754943	+	0.754943i	0.220455	+	0.975397i	$0.429246\pi$
$822$		0			0
$823$	−40905.7			−1.73254			−0.866272	−	0.499572i	$-0.833490\pi$
$823$	−40905.7			−1.73254			−0.866272	+	0.499572i	$0.833490\pi$
$824$		0			0
$825$	1756.35			0.0741189
$826$		0			0
$827$	−28790.3	−	28790.3i	−1.21056	−	1.21056i	−0.970842	−	0.239722i	$-0.922944\pi$
$827$	−28790.3	−	28790.3i	−1.21056	−	1.21056i	−0.239722	−	0.970842i	$-0.577056\pi$
$828$		0			0
$829$	23367.1	−	23367.1i	0.978977	−	0.978977i	−0.0208065	−	0.999784i	$-0.506623\pi$
$829$	23367.1	−	23367.1i	0.978977	−	0.978977i	0.999784	+	0.0208065i	$0.00662340\pi$
$830$		0			0
$831$	16786.9			0.700760
$832$		0			0
$833$	14214.2	−	28474.6i	0.591227	−	1.18438i
$834$		0			0
$835$	−7787.77	+	7787.77i	−0.322763	+	0.322763i
$836$		0			0
$837$	−24889.1	+	24889.1i	−1.02783	+	1.02783i
$838$		0			0
$839$		−	11206.9i		−	0.461152i	−0.973054	−	0.230576i	$-0.925939\pi$
$839$		−	11206.9i		−	0.461152i	0.973054	−	0.230576i	$-0.0740610\pi$
$840$		0			0
$841$			39879.5i			1.63514i
$842$		0			0
$843$	5097.75	−	5097.75i	0.208275	−	0.208275i
$844$		0			0
$845$	21182.5	−	21182.5i	0.862368	−	0.862368i
$846$		0			0
$847$	−5533.62	+	23474.9i	−0.224484	+	0.952310i
$848$		0			0
$849$	9527.28			0.385130
$850$		0			0
$851$	12716.9	−	12716.9i	0.512255	−	0.512255i
$852$		0			0
$853$	−13523.7	−	13523.7i	−0.542839	−	0.542839i	0.381521	−	0.924360i	$-0.375400\pi$
$853$	−13523.7	−	13523.7i	−0.542839	−	0.542839i	−0.924360	+	0.381521i	$0.875400\pi$
$854$		0			0
$855$	51310.2			2.05237
$856$		0			0
$857$	−10498.9			−0.418477			−0.209238	−	0.977865i	$-0.567098\pi$
$857$	−10498.9			−0.418477			−0.209238	+	0.977865i	$0.567098\pi$
$858$		0			0
$859$	13133.0	−	13133.0i	0.521643	−	0.521643i	−0.396425	−	0.918067i	$-0.629749\pi$
$859$	13133.0	−	13133.0i	0.521643	−	0.521643i	0.918067	+	0.396425i	$0.129749\pi$
$860$		0			0
$861$	7354.07	+	11890.5i	0.291087	+	0.470647i
$862$		0			0
$863$			29386.8i			1.15914i	0.814922	+	0.579571i	$0.196780\pi$
$863$			29386.8i			1.15914i	−0.814922	+	0.579571i	$0.803220\pi$
$864$		0			0
$865$	51241.4			2.01417
$866$		0			0
$867$	6766.93	+	6766.93i	0.265071	+	0.265071i
$868$		0			0
$869$	−2672.81	+	2672.81i	−0.104337	+	0.104337i
$870$		0			0
$871$			333.689i			0.0129812i
$872$		0			0
$873$	7777.12			0.301507
$874$		0			0
$875$	−384.985	+	238.107i	−0.0148741	+	0.00919940i
$876$		0			0
$877$	−9197.32	−	9197.32i	−0.354129	−	0.354129i	0.507514	−	0.861643i	$-0.330564\pi$
$877$	−9197.32	−	9197.32i	−0.354129	−	0.354129i	−0.861643	+	0.507514i	$0.830564\pi$
$878$		0			0
$879$			6554.05i			0.251493i
$880$		0			0
$881$			13824.6i			0.528676i	0.964430	+	0.264338i	$0.0851534\pi$
$881$			13824.6i			0.528676i	−0.964430	+	0.264338i	$0.914847\pi$
$882$		0			0
$883$	−32329.5	+	32329.5i	−1.23213	+	1.23213i	−0.268993	+	0.963142i	$0.586691\pi$
$883$	−32329.5	+	32329.5i	−1.23213	+	1.23213i	−0.963142	+	0.268993i	$0.913309\pi$
$884$		0			0
$885$	−7254.60	−	7254.60i	−0.275549	−	0.275549i
$886$		0			0
$887$			23638.4i			0.894815i	0.894330	+	0.447408i	$0.147653\pi$
$887$			23638.4i			0.894815i	−0.894330	+	0.447408i	$0.852347\pi$
$888$		0			0
$889$	−1708.67	+	7248.55i	−0.0644621	+	0.273463i
$890$		0			0
$891$	−875.261	−	875.261i	−0.0329095	−	0.0329095i
$892$		0			0
$893$	22901.1	+	22901.1i	0.858181	+	0.858181i
$894$		0			0
$895$	26116.2			0.975383
$896$		0			0
$897$	−2970.38			−0.110567
$898$		0			0
$899$	51524.6	+	51524.6i	1.91150	+	1.91150i
$900$		0			0
$901$	19051.5	+	19051.5i	0.704437	+	0.704437i
$902$		0			0
$903$	2898.41	−	12295.7i	0.106814	−	0.453130i
$904$		0			0
$905$			35070.3i			1.28815i
$906$		0			0
$907$	17491.9	+	17491.9i	0.640365	+	0.640365i	0.950645	−	0.310281i	$-0.100423\pi$
$907$	17491.9	+	17491.9i	0.640365	+	0.640365i	−0.310281	+	0.950645i	$0.600423\pi$
$908$		0			0
$909$	−11478.8	+	11478.8i	−0.418842	+	0.418842i
$910$		0			0
$911$		−	8516.20i		−	0.309719i	−0.987936	−	0.154860i	$-0.950507\pi$
$911$		−	8516.20i		−	0.309719i	0.987936	−	0.154860i	$-0.0494926\pi$
$912$		0			0
$913$			2249.04i			0.0815250i
$914$		0			0
$915$	16853.8	+	16853.8i	0.608929	+	0.608929i
$916$		0			0
$917$	26295.3	+	42515.8i	0.946942	+	1.53107i
$918$		0			0
$919$	−26011.3			−0.933661			−0.466830	−	0.884347i	$-0.654604\pi$
$919$	−26011.3			−0.933661			−0.466830	+	0.884347i	$0.654604\pi$
$920$		0			0
$921$			1600.80i			0.0572726i
$922$		0			0
$923$	−5171.48	+	5171.48i	−0.184422	+	0.184422i
$924$		0			0
$925$	−24625.2	−	24625.2i	−0.875320	−	0.875320i
$926$		0			0
$927$	−3104.38			−0.109991
$928$		0			0
$929$		−	4227.95i		−	0.149316i	−0.997209	−	0.0746579i	$-0.976214\pi$
$929$		−	4227.95i		−	0.149316i	0.997209	−	0.0746579i	$-0.0237865\pi$
$930$		0			0
$931$	−17323.3	−	51857.6i	−0.609826	−	1.82553i
$932$		0			0
$933$	5653.23	−	5653.23i	0.198369	−	0.198369i
$934$		0			0
$935$	−7888.40			−0.275913
$936$		0			0
$937$	−3414.62			−0.119051			−0.0595255	−	0.998227i	$-0.518959\pi$
$937$	−3414.62			−0.119051			−0.0595255	+	0.998227i	$0.518959\pi$
$938$		0			0
$939$	5215.17	+	5215.17i	0.181247	+	0.181247i
$940$		0			0
$941$	−10219.7	+	10219.7i	−0.354042	+	0.354042i	−0.861611	−	0.507569i	$-0.830544\pi$
$941$	−10219.7	+	10219.7i	−0.354042	+	0.354042i	0.507569	+	0.861611i	$0.330544\pi$
$942$		0			0
$943$	−19052.4			−0.657935
$944$		0			0
$945$	−35010.9	−	8252.95i	−1.20519	−	0.284094i
$946$		0			0
$947$	−21349.8	+	21349.8i	−0.732603	+	0.732603i	−0.971135	−	0.238532i	$-0.923334\pi$
$947$	−21349.8	+	21349.8i	−0.732603	+	0.732603i	0.238532	+	0.971135i	$0.423334\pi$
$948$		0			0
$949$	−3337.29	+	3337.29i	−0.114155	+	0.114155i
$950$		0			0
$951$			12257.4i			0.417954i
$952$		0			0
$953$			664.145i			0.0225748i	0.999936	+	0.0112874i	$0.00359296\pi$
$953$			664.145i			0.0225748i	−0.999936	+	0.0112874i	$0.996407\pi$
$954$		0			0
$955$	−9948.91	+	9948.91i	−0.337109	+	0.337109i
$956$		0			0
$957$	2488.07	−	2488.07i	0.0840416	−	0.0840416i
$958$		0			0
$959$	−4912.81	+	20841.3i	−0.165425	+	0.701772i
$960$		0			0
$961$	52824.3			1.77316
$962$		0			0
$963$	2990.24	−	2990.24i	0.100061	−	0.100061i
$964$		0			0
$965$	−25015.0	−	25015.0i	−0.834468	−	0.834468i
$966$		0			0
$967$	9585.77			0.318777			0.159389	−	0.987216i	$-0.449048\pi$
$967$	9585.77			0.318777			0.159389	+	0.987216i	$0.449048\pi$
$968$		0			0
$969$	38294.8			1.26956
$970$		0			0
$971$	−15043.6	+	15043.6i	−0.497190	+	0.497190i	−0.910562	−	0.413372i	$-0.864351\pi$
$971$	−15043.6	+	15043.6i	−0.497190	+	0.497190i	0.413372	+	0.910562i	$0.364351\pi$
$972$		0			0
$973$	930.403	−	575.438i	0.0306550	−	0.0189596i
$974$		0			0
$975$			5751.90i			0.188932i
$976$		0			0
$977$	−45426.9			−1.48755			−0.743774	−	0.668431i	$-0.766966\pi$
$977$	−45426.9			−1.48755			−0.743774	+	0.668431i	$0.766966\pi$
$978$		0			0
$979$	−4204.17	−	4204.17i	−0.137248	−	0.137248i
$980$		0			0
$981$	−5827.69	+	5827.69i	−0.189668	+	0.189668i
$982$		0			0
$983$		−	52579.6i		−	1.70603i	−0.521885	−	0.853016i	$-0.674771\pi$
$983$		−	52579.6i		−	1.70603i	0.521885	−	0.853016i	$-0.325229\pi$
$984$		0			0
$985$	−17679.1			−0.571880
$986$		0			0
$987$	−5124.95	−	8286.32i	−0.165277	−	0.267230i
$988$		0			0
$989$	12173.0	+	12173.0i	0.391383	+	0.391383i
$990$		0			0
$991$			7892.45i			0.252989i	0.991967	+	0.126494i	$0.0403726\pi$
$991$			7892.45i			0.252989i	−0.991967	+	0.126494i	$0.959627\pi$
$992$		0			0
$993$		−	11361.6i		−	0.363090i
$994$		0			0
$995$	−39598.8	+	39598.8i	−1.26167	+	1.26167i
$996$		0			0
$997$	−10004.4	−	10004.4i	−0.317796	−	0.317796i	0.530124	−	0.847920i	$-0.322145\pi$
$997$	−10004.4	−	10004.4i	−0.317796	−	0.317796i	−0.847920	+	0.530124i	$0.822145\pi$
$998$		0			0
$999$		−	33702.1i		−	1.06736i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.j.b.111.19		88
4.3	odd	2		112.4.j.b.83.34	yes	88
7.6	odd	2	inner	448.4.j.b.111.26		88
16.5	even	4		112.4.j.b.27.33	✓	88
16.11	odd	4	inner	448.4.j.b.335.26		88
28.27	even	2		112.4.j.b.83.33	yes	88
112.27	even	4	inner	448.4.j.b.335.19		88
112.69	odd	4		112.4.j.b.27.34	yes	88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.4.j.b.27.33	✓	88	16.5	even	4
112.4.j.b.27.34	yes	88	112.69	odd	4
112.4.j.b.83.33	yes	88	28.27	even	2
112.4.j.b.83.34	yes	88	4.3	odd	2
448.4.j.b.111.19		88	1.1	even	1	trivial
448.4.j.b.111.26		88	7.6	odd	2	inner
448.4.j.b.335.19		88	112.27	even	4	inner
448.4.j.b.335.26		88	16.11	odd	4	inner

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.83086 - 1.83086i) q^{3} +(-11.2147 - 11.2147i) q^{5} +(-18.0262 - 4.24923i) q^{7} -20.2959i q^{9} +(3.79046 + 3.79046i) q^{11} +(-12.4135 + 12.4135i) q^{13} +41.0653i q^{15} -92.7849i q^{17} +(-112.714 - 112.714i) q^{19} +(25.2238 + 40.7833i) q^{21} -65.3480 q^{23} +126.541i q^{25} +(-86.5923 + 86.5923i) q^{27} +(179.260 + 179.260i) q^{29} +287.429 q^{31} -13.8796i q^{33} +(154.505 + 249.813i) q^{35} +(-194.602 + 194.602i) q^{37} +45.4548 q^{39} +291.553 q^{41} +(-186.279 - 186.279i) q^{43} +(-227.613 + 227.613i) q^{45} -203.179 q^{47} +(306.888 + 153.195i) q^{49} +(-169.876 + 169.876i) q^{51} +(-205.330 + 205.330i) q^{53} -85.0182i q^{55} +412.727i q^{57} +(-176.660 + 176.660i) q^{59} +(410.414 - 410.414i) q^{61} +(-86.2419 + 365.858i) q^{63} +278.428 q^{65} +(13.4406 - 13.4406i) q^{67} +(119.643 + 119.643i) q^{69} +416.602 q^{71} +268.844 q^{73} +(231.679 - 231.679i) q^{75} +(-52.2211 - 84.4342i) q^{77} +705.142i q^{79} -230.911 q^{81} +(296.671 + 296.671i) q^{83} +(-1040.56 + 1040.56i) q^{85} -656.402i q^{87} -1109.14 q^{89} +(276.516 - 171.020i) q^{91} +(-526.243 - 526.243i) q^{93} +2528.11i q^{95} +383.187i q^{97} +(76.9308 - 76.9308i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 4 q^{7} - 112 q^{11} + 52 q^{21} - 160 q^{23} + 528 q^{29} - 476 q^{35} - 896 q^{37} + 8 q^{39} - 40 q^{43} - 1376 q^{49} - 1504 q^{51} - 1560 q^{53} - 8 q^{65} - 648 q^{67} + 456 q^{71} + 292 q^{77}+ \cdots - 7592 q^{99}+O(q^{100}) \) 88 * q + 4 * q^7 - 112 * q^11 + 52 * q^21 - 160 * q^23 + 528 * q^29 - 476 * q^35 - 896 * q^37 + 8 * q^39 - 40 * q^43 - 1376 * q^49 - 1504 * q^51 - 1560 * q^53 - 8 * q^65 - 648 * q^67 + 456 * q^71 + 292 * q^77 - 1952 * q^81 + 496 * q^85 + 1964 * q^91 - 112 * q^93 - 7592 * q^99

Introduction

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