LMFDB - Embedded newform 224.3.c.b.97.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,3,Mod(97,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("224.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	224.c (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:	$6.10355792167$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 20x^{6} + 116x^{4} + 180x^{2} + 81 $ x^8 + 20x^6 + 116x^4 + 180*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.6
Root			$2.71359i$ of defining polynomial
Character	$\chi$	$=$	224.97
Dual form			224.3.c.b.97.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.27411i q^{3} +5.40107i q^{5} +(4.34256 + 5.49019i) q^{7} +3.82843 q^{9} +O(q^{10})$	$q+2.27411i q^{3} +5.40107i q^{5} +(4.34256 + 5.49019i) q^{7} +3.82843 q^{9} -20.9678 q^{11} -20.6776i q^{13} -12.2826 q^{15} +15.2765i q^{17} +30.6670i q^{19} +(-12.4853 + 9.87547i) q^{21} -3.59750 q^{23} -4.17157 q^{25} +29.1732i q^{27} -14.9706 q^{29} -23.0643i q^{31} -47.6830i q^{33} +(-29.6529 + 23.4545i) q^{35} +33.5980 q^{37} +47.0231 q^{39} +1.85335i q^{41} +3.59750 q^{43} +20.6776i q^{45} +1.10358i q^{47} +(-11.2843 + 47.6830i) q^{49} -34.7405 q^{51} +56.6274 q^{53} -113.248i q^{55} -69.7401 q^{57} -74.5885i q^{59} +59.4118i q^{61} +(16.6252 + 21.0188i) q^{63} +111.681 q^{65} +52.7280 q^{67} -8.18110i q^{69} +92.5561 q^{71} -78.2360i q^{73} -9.48661i q^{75} +(-91.0538 - 115.117i) q^{77} +82.3809 q^{79} -31.8873 q^{81} +34.8920i q^{83} -82.5097 q^{85} -34.0447i q^{87} +65.5805i q^{89} +(113.524 - 89.7938i) q^{91} +52.4508 q^{93} -165.635 q^{95} +85.3315i q^{97} -80.2735 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^9	$ 8 q + 8 q^{9} - 32 q^{21} - 56 q^{25} + 16 q^{29} - 48 q^{37} + 136 q^{49} + 272 q^{53} - 128 q^{57} + 192 q^{65} - 208 q^{77} - 504 q^{81} + 64 q^{85} - 576 q^{93}+O(q^{100}) $ 8 * q + 8 * q^9 - 32 * q^21 - 56 * q^25 + 16 * q^29 - 48 * q^37 + 136 * q^49 + 272 * q^53 - 128 * q^57 + 192 * q^65 - 208 * q^77 - 504 * q^81 + 64 * q^85 - 576 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$.

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

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
$3$			2.27411i			0.758036i	0.925389	+	0.379018i	$0.123738\pi$
$3$			2.27411i			0.758036i	−0.925389	+	0.379018i	$0.876262\pi$
$4$		0			0
$5$			5.40107i			1.08021i	0.841596	+	0.540107i	$0.181616\pi$
$5$			5.40107i			1.08021i	−0.841596	+	0.540107i	$0.818384\pi$
$6$		0			0
$7$	4.34256	+	5.49019i	0.620366	+	0.784312i
$7$	4.34256	+	5.49019i	0.620366	+	0.784312i
$8$		0			0
$9$	3.82843			0.425381
$10$		0			0
$11$	−20.9678			−1.90616			−0.953080	−	0.302719i	$-0.902106\pi$
$11$	−20.9678			−1.90616			−0.953080	+	0.302719i	$0.902106\pi$
$12$		0			0
$13$		−	20.6776i		−	1.59059i	−0.606226	−	0.795293i	$-0.707317\pi$
$13$		−	20.6776i		−	1.59059i	0.606226	−	0.795293i	$-0.292683\pi$
$14$		0			0
$15$	−12.2826			−0.818842
$16$		0			0
$17$			15.2765i			0.898620i	0.893376	+	0.449310i	$0.148330\pi$
$17$			15.2765i			0.898620i	−0.893376	+	0.449310i	$0.851670\pi$
$18$		0			0
$19$			30.6670i			1.61405i	0.590515	+	0.807026i	$0.298925\pi$
$19$			30.6670i			1.61405i	−0.590515	+	0.807026i	$0.701075\pi$
$20$		0			0
$21$	−12.4853	+	9.87547i	−0.594537	+	0.470260i
$22$		0			0
$23$	−3.59750			−0.156413			−0.0782065	−	0.996937i	$-0.524919\pi$
$23$	−3.59750			−0.156413			−0.0782065	+	0.996937i	$0.524919\pi$
$24$		0			0
$25$	−4.17157			−0.166863
$26$		0			0
$27$			29.1732i			1.08049i
$28$		0			0
$29$	−14.9706			−0.516226			−0.258113	−	0.966115i	$-0.583101\pi$
$29$	−14.9706			−0.516226			−0.258113	+	0.966115i	$0.583101\pi$
$30$		0			0
$31$		−	23.0643i		−	0.744010i	−0.928231	−	0.372005i	$-0.878670\pi$
$31$		−	23.0643i		−	0.744010i	0.928231	−	0.372005i	$-0.121330\pi$
$32$		0			0
$33$		−	47.6830i		−	1.44494i
$34$		0			0
$35$	−29.6529	+	23.4545i	−0.847225	+	0.670129i
$36$		0			0
$37$	33.5980			0.908054			0.454027	−	0.890988i	$-0.349987\pi$
$37$	33.5980			0.908054			0.454027	+	0.890988i	$0.349987\pi$
$38$		0			0
$39$	47.0231			1.20572
$40$		0			0
$41$			1.85335i			0.0452038i	0.999745	+	0.0226019i	$0.00719502\pi$
$41$			1.85335i			0.0452038i	−0.999745	+	0.0226019i	$0.992805\pi$
$42$		0			0
$43$	3.59750			0.0836627			0.0418314	−	0.999125i	$-0.486681\pi$
$43$	3.59750			0.0836627			0.0418314	+	0.999125i	$0.486681\pi$
$44$		0			0
$45$			20.6776i			0.459502i
$46$		0			0
$47$			1.10358i			0.0234805i	0.999931	+	0.0117402i	$0.00373712\pi$
$47$			1.10358i			0.0234805i	−0.999931	+	0.0117402i	$0.996263\pi$
$48$		0			0
$49$	−11.2843	+	47.6830i	−0.230291	+	0.973122i
$50$		0			0
$51$	−34.7405			−0.681187
$52$		0			0
$53$	56.6274			1.06844			0.534221	−	0.845345i	$-0.320605\pi$
$53$	56.6274			1.06844			0.534221	+	0.845345i	$0.320605\pi$
$54$		0			0
$55$		−	113.248i		−	2.05906i
$56$		0			0
$57$	−69.7401			−1.22351
$58$		0			0
$59$		−	74.5885i		−	1.26421i	−0.774882	−	0.632106i	$-0.782191\pi$
$59$		−	74.5885i		−	1.26421i	0.774882	−	0.632106i	$-0.217809\pi$
$60$		0			0
$61$			59.4118i			0.973964i	0.873412	+	0.486982i	$0.161902\pi$
$61$			59.4118i			0.973964i	−0.873412	+	0.486982i	$0.838098\pi$
$62$		0			0
$63$	16.6252	+	21.0188i	0.263892	+	0.333631i
$64$		0			0
$65$	111.681			1.71817
$66$		0			0
$67$	52.7280			0.786985			0.393493	−	0.919328i	$-0.371267\pi$
$67$	52.7280			0.786985			0.393493	+	0.919328i	$0.371267\pi$
$68$		0			0
$69$		−	8.18110i		−	0.118567i
$70$		0			0
$71$	92.5561			1.30361			0.651804	−	0.758388i	$-0.274013\pi$
$71$	92.5561			1.30361			0.651804	+	0.758388i	$0.274013\pi$
$72$		0			0
$73$		−	78.2360i		−	1.07173i	−0.844305	−	0.535863i	$-0.819986\pi$
$73$		−	78.2360i		−	1.07173i	0.844305	−	0.535863i	$-0.180014\pi$
$74$		0			0
$75$		−	9.48661i		−	0.126488i
$76$		0			0
$77$	−91.0538	−	115.117i	−1.18252	−	1.49502i
$78$		0			0
$79$	82.3809			1.04280			0.521398	−	0.853314i	$-0.325411\pi$
$79$	82.3809			1.04280			0.521398	+	0.853314i	$0.325411\pi$
$80$		0			0
$81$	−31.8873			−0.393670
$82$		0			0
$83$			34.8920i			0.420385i	0.977660	+	0.210193i	$0.0674091\pi$
$83$			34.8920i			0.420385i	−0.977660	+	0.210193i	$0.932591\pi$
$84$		0			0
$85$	−82.5097			−0.970702
$86$		0			0
$87$		−	34.0447i		−	0.391318i
$88$		0			0
$89$			65.5805i			0.736860i	0.929655	+	0.368430i	$0.120105\pi$
$89$			65.5805i			0.736860i	−0.929655	+	0.368430i	$0.879895\pi$
$90$		0			0
$91$	113.524	−	89.7938i	1.24752	−	0.986746i
$92$		0			0
$93$	52.4508			0.563987
$94$		0			0
$95$	−165.635			−1.74352
$96$		0			0
$97$			85.3315i			0.879706i	0.898070	+	0.439853i	$0.144969\pi$
$97$			85.3315i			0.879706i	−0.898070	+	0.439853i	$0.855031\pi$
$98$		0			0
$99$	−80.2735			−0.810844
$100$		0			0
$101$			2.78003i			0.0275251i	0.999905	+	0.0137625i	$0.00438089\pi$
$101$			2.78003i			0.0275251i	−0.999905	+	0.0137625i	$0.995619\pi$
$102$		0			0
$103$		−	61.3340i		−	0.595476i	−0.954648	−	0.297738i	$-0.903768\pi$
$103$		−	61.3340i		−	0.595476i	0.954648	−	0.297738i	$-0.0962321\pi$
$104$		0			0
$105$	−53.3381	−	67.4339i	−0.507982	−	0.642228i
$106$		0			0
$107$	107.819			1.00765			0.503827	−	0.863804i	$-0.331925\pi$
$107$	107.819			1.00765			0.503827	+	0.863804i	$0.331925\pi$
$108$		0			0
$109$	93.3137			0.856089			0.428045	−	0.903758i	$-0.359203\pi$
$109$	93.3137			0.856089			0.428045	+	0.903758i	$0.359203\pi$
$110$		0			0
$111$			76.4055i			0.688338i
$112$		0			0
$113$	−102.652			−0.908423			−0.454212	−	0.890894i	$-0.650079\pi$
$113$	−102.652			−0.908423			−0.454212	+	0.890894i	$0.650079\pi$
$114$		0			0
$115$		−	19.4303i		−	0.168960i
$116$		0			0
$117$		−	79.1627i		−	0.676604i
$118$		0			0
$119$	−83.8710	+	66.3393i	−0.704798	+	0.557473i
$120$		0			0
$121$	318.647			2.63344
$122$		0			0
$123$	−4.21473			−0.0342661
$124$		0			0
$125$			112.496i			0.899967i
$126$		0			0
$127$	−107.819			−0.848969			−0.424484	−	0.905435i	$-0.639545\pi$
$127$	−107.819			−0.848969			−0.424484	+	0.905435i	$0.639545\pi$
$128$		0			0
$129$			8.18110i			0.0634194i
$130$		0			0
$131$			36.4527i			0.278265i	0.990274	+	0.139132i	$0.0444314\pi$
$131$			36.4527i			0.278265i	−0.990274	+	0.139132i	$0.955569\pi$
$132$		0			0
$133$	−168.368	+	133.173i	−1.26592	+	1.00130i
$134$		0			0
$135$	−157.567			−1.16716
$136$		0			0
$137$	−66.1177			−0.482611			−0.241306	−	0.970449i	$-0.577576\pi$
$137$	−66.1177			−0.482611			−0.241306	+	0.970449i	$0.577576\pi$
$138$		0			0
$139$		−	41.9706i		−	0.301947i	−0.988538	−	0.150973i	$-0.951759\pi$
$139$		−	41.9706i		−	0.301947i	0.988538	−	0.150973i	$-0.0482408\pi$
$140$		0			0
$141$	−2.50967			−0.0177991
$142$		0			0
$143$			433.563i			3.03191i
$144$		0			0
$145$		−	80.8571i		−	0.557635i
$146$		0			0
$147$	−108.436	−	25.6617i	−0.737662	−	0.174569i
$148$		0			0
$149$	88.3919			0.593234			0.296617	−	0.954996i	$-0.404141\pi$
$149$	88.3919			0.593234			0.296617	+	0.954996i	$0.404141\pi$
$150$		0			0
$151$	164.145			1.08705			0.543525	−	0.839393i	$-0.317089\pi$
$151$	164.145			1.08705			0.543525	+	0.839393i	$0.317089\pi$
$152$		0			0
$153$			58.4851i			0.382256i
$154$		0			0
$155$	124.572			0.803691
$156$		0			0
$157$		−	171.908i		−	1.09495i	−0.836821	−	0.547476i	$-0.815589\pi$
$157$		−	171.908i		−	1.09495i	0.836821	−	0.547476i	$-0.184411\pi$
$158$		0			0
$159$			128.777i			0.809918i
$160$		0			0
$161$	−15.6224	−	19.7509i	−0.0970333	−	0.122677i
$162$		0			0
$163$	−139.579			−0.856315			−0.428157	−	0.903704i	$-0.640837\pi$
$163$	−139.579			−0.856315			−0.428157	+	0.903704i	$0.640837\pi$
$164$		0			0
$165$	257.539			1.56084
$166$		0			0
$167$			83.2947i			0.498771i	0.968404	+	0.249385i	$0.0802286\pi$
$167$			83.2947i			0.498771i	−0.968404	+	0.249385i	$0.919771\pi$
$168$		0			0
$169$	−258.563			−1.52996
$170$		0			0
$171$			117.406i			0.686587i
$172$		0			0
$173$			45.6706i			0.263992i	0.991250	+	0.131996i	$0.0421386\pi$
$173$			45.6706i			0.263992i	−0.991250	+	0.131996i	$0.957861\pi$
$174$		0			0
$175$	−18.1153	−	22.9027i	−0.103516	−	0.130873i
$176$		0			0
$177$	169.622			0.958318
$178$		0			0
$179$	−35.3577			−0.197529			−0.0987647	−	0.995111i	$-0.531489\pi$
$179$	−35.3577			−0.197529			−0.0987647	+	0.995111i	$0.531489\pi$
$180$		0			0
$181$		−	151.071i		−	0.834646i	−0.908758	−	0.417323i	$-0.862968\pi$
$181$		−	151.071i		−	0.834646i	0.908758	−	0.417323i	$-0.137032\pi$
$182$		0			0
$183$	−135.109			−0.738300
$184$		0			0
$185$			181.465i			0.980892i
$186$		0			0
$187$		−	320.315i		−	1.71291i
$188$		0			0
$189$	−160.167	+	126.687i	−0.847442	+	0.670300i
$190$		0			0
$191$	−238.713			−1.24981			−0.624904	−	0.780702i	$-0.714862\pi$
$191$	−238.713			−1.24981			−0.624904	+	0.780702i	$0.714862\pi$
$192$		0			0
$193$	−262.083			−1.35794			−0.678972	−	0.734164i	$-0.737574\pi$
$193$	−262.083			−1.35794			−0.678972	+	0.734164i	$0.737574\pi$
$194$		0			0
$195$			253.975i			1.30244i
$196$		0			0
$197$	−184.108			−0.934557			−0.467278	−	0.884110i	$-0.654765\pi$
$197$	−184.108			−0.934557			−0.467278	+	0.884110i	$0.654765\pi$
$198$		0			0
$199$		−	166.077i		−	0.834557i	−0.908779	−	0.417279i	$-0.862984\pi$
$199$		−	166.077i		−	0.834557i	0.908779	−	0.417279i	$-0.137016\pi$
$200$		0			0
$201$			119.909i			0.596563i
$202$		0			0
$203$	−65.0106	−	82.1912i	−0.320249	−	0.404883i
$204$		0			0
$205$	−10.0101			−0.0488298
$206$		0			0
$207$	−13.7728			−0.0665351
$208$		0			0
$209$		−	643.018i		−	3.07664i
$210$		0			0
$211$	200.120			0.948434			0.474217	−	0.880408i	$-0.342731\pi$
$211$	200.120			0.948434			0.474217	+	0.880408i	$0.342731\pi$
$212$		0			0
$213$			210.483i			0.988182i
$214$		0			0
$215$			19.4303i			0.0903737i
$216$		0			0
$217$	126.627	−	100.158i	0.583536	−	0.461559i
$218$		0			0
$219$	177.917			0.812408
$220$		0			0
$221$	315.882			1.42933
$222$		0			0
$223$		−	66.5287i		−	0.298335i	−0.988812	−	0.149167i	$-0.952341\pi$
$223$		−	66.5287i		−	0.298335i	0.988812	−	0.149167i	$-0.0476593\pi$
$224$		0			0
$225$	−15.9706			−0.0709803
$226$		0			0
$227$		−	12.6080i		−	0.0555419i	−0.999614	−	0.0277710i	$-0.991159\pi$
$227$		−	12.6080i		−	0.0555419i	0.999614	−	0.0277710i	$-0.00884091\pi$
$228$		0			0
$229$		−	187.952i		−	0.820750i	−0.911917	−	0.410375i	$-0.865398\pi$
$229$		−	187.952i		−	0.820750i	0.911917	−	0.410375i	$-0.134602\pi$
$230$		0			0
$231$	261.788	−	207.066i	1.13328	−	0.896391i
$232$		0			0
$233$	288.627			1.23874			0.619372	−	0.785098i	$-0.287387\pi$
$233$	288.627			1.23874			0.619372	+	0.785098i	$0.287387\pi$
$234$		0			0
$235$	−5.96053			−0.0253640
$236$		0			0
$237$			187.343i			0.790477i
$238$		0			0
$239$	−58.6885			−0.245559			−0.122779	−	0.992434i	$-0.539181\pi$
$239$	−58.6885			−0.245559			−0.122779	+	0.992434i	$0.539181\pi$
$240$		0			0
$241$			231.319i			0.959832i	0.877315	+	0.479916i	$0.159333\pi$
$241$			231.319i			0.959832i	−0.877315	+	0.479916i	$0.840667\pi$
$242$		0			0
$243$			190.044i			0.782074i
$244$		0			0
$245$	−257.539	−	60.9472i	−1.05118	−	0.248764i
$246$		0			0
$247$	634.120			2.56729
$248$		0			0
$249$	−79.3482			−0.318667
$250$		0			0
$251$		−	417.321i		−	1.66263i	−0.555799	−	0.831317i	$-0.687588\pi$
$251$		−	417.321i		−	1.66263i	0.555799	−	0.831317i	$-0.312412\pi$
$252$		0			0
$253$	75.4315			0.298148
$254$		0			0
$255$		−	187.636i		−	0.735827i
$256$		0			0
$257$			361.395i			1.40621i	0.711088	+	0.703103i	$0.248203\pi$
$257$			361.395i			1.40621i	−0.711088	+	0.703103i	$0.751797\pi$
$258$		0			0
$259$	145.901	+	184.459i	0.563326	+	0.712197i
$260$		0			0
$261$	−57.3137			−0.219593
$262$		0			0
$263$	−124.316			−0.472686			−0.236343	−	0.971670i	$-0.575949\pi$
$263$	−124.316			−0.472686			−0.236343	+	0.971670i	$0.575949\pi$
$264$		0			0
$265$			305.849i			1.15415i
$266$		0			0
$267$	−149.137			−0.558567
$268$		0			0
$269$		−	2.78003i		−	0.0103347i	−0.999987	−	0.00516735i	$-0.998355\pi$
$269$		−	2.78003i		−	0.0103347i	0.999987	−	0.00516735i	$-0.00164482\pi$
$270$		0			0
$271$		−	478.076i		−	1.76412i	−0.471141	−	0.882058i	$-0.656158\pi$
$271$		−	478.076i		−	1.76412i	0.471141	−	0.882058i	$-0.343842\pi$
$272$		0			0
$273$	204.201	+	258.166i	0.747989	+	0.945662i
$274$		0			0
$275$	87.4685			0.318067
$276$		0			0
$277$	89.2649			0.322256			0.161128	−	0.986934i	$-0.448487\pi$
$277$	89.2649			0.322256			0.161128	+	0.986934i	$0.448487\pi$
$278$		0			0
$279$		−	88.3001i		−	0.316488i
$280$		0			0
$281$	35.1371			0.125043			0.0625215	−	0.998044i	$-0.480086\pi$
$281$	35.1371			0.125043			0.0625215	+	0.998044i	$0.480086\pi$
$282$		0			0
$283$			173.680i			0.613709i	0.951756	+	0.306854i	$0.0992765\pi$
$283$			173.680i			0.613709i	−0.951756	+	0.306854i	$0.900724\pi$
$284$		0			0
$285$		−	376.671i		−	1.32165i
$286$		0			0
$287$	−10.1753	+	8.04831i	−0.0354539	+	0.0280429i
$288$		0			0
$289$	55.6274			0.192482
$290$		0			0
$291$	−194.053			−0.666849
$292$		0			0
$293$		−	170.054i		−	0.580390i	−0.956968	−	0.290195i	$-0.906280\pi$
$293$		−	170.054i		−	0.580390i	0.956968	−	0.290195i	$-0.0937202\pi$
$294$		0			0
$295$	402.858			1.36562
$296$		0			0
$297$		−	611.697i		−	2.05959i
$298$		0			0
$299$			74.3877i			0.248788i
$300$		0			0
$301$	15.6224	+	19.7509i	0.0519015	+	0.0656177i
$302$		0			0
$303$	−6.32210			−0.0208650
$304$		0			0
$305$	−320.887			−1.05209
$306$		0			0
$307$		−	307.651i		−	1.00212i	−0.865412	−	0.501061i	$-0.832943\pi$
$307$		−	307.651i		−	1.00212i	0.865412	−	0.501061i	$-0.167057\pi$
$308$		0			0
$309$	139.480			0.451392
$310$		0			0
$311$		−	505.688i		−	1.62601i	−0.582259	−	0.813003i	$-0.697831\pi$
$311$		−	505.688i		−	1.62601i	0.582259	−	0.813003i	$-0.302169\pi$
$312$		0			0
$313$		−	413.870i		−	1.32227i	−0.750267	−	0.661134i	$-0.770075\pi$
$313$		−	413.870i		−	1.32227i	0.750267	−	0.661134i	$-0.229925\pi$
$314$		0			0
$315$	−113.524	+	89.7938i	−0.360393	+	0.285060i
$316$		0			0
$317$	−601.245			−1.89667			−0.948336	−	0.317269i	$-0.897234\pi$
$317$	−601.245			−1.89667			−0.948336	+	0.317269i	$0.897234\pi$
$318$		0			0
$319$	313.899			0.984010
$320$		0			0
$321$			245.192i			0.763839i
$322$		0			0
$323$	−468.486			−1.45042
$324$		0			0
$325$			86.2582i			0.265410i
$326$		0			0
$327$			212.206i			0.648947i
$328$		0			0
$329$	−6.05887	+	4.79238i	−0.0184160	+	0.0145665i
$330$		0			0
$331$	−570.855			−1.72464			−0.862319	−	0.506365i	$-0.830989\pi$
$331$	−570.855			−1.72464			−0.862319	+	0.506365i	$0.830989\pi$
$332$		0			0
$333$	128.627			0.386269
$334$		0			0
$335$			284.788i			0.850113i
$336$		0			0
$337$	65.1127			0.193213			0.0966064	−	0.995323i	$-0.469201\pi$
$337$	65.1127			0.193213			0.0966064	+	0.995323i	$0.469201\pi$
$338$		0			0
$339$		−	233.441i		−	0.688618i
$340$		0			0
$341$			483.607i			1.41820i
$342$		0			0
$343$	−310.791	+	145.114i	−0.906096	+	0.423072i
$344$		0			0
$345$	44.1867			0.128077
$346$		0			0
$347$	617.006			1.77811			0.889057	−	0.457796i	$-0.151361\pi$
$347$	617.006			1.77811			0.889057	+	0.457796i	$0.151361\pi$
$348$		0			0
$349$			309.078i			0.885612i	0.896618	+	0.442806i	$0.146017\pi$
$349$			309.078i			0.885612i	−0.896618	+	0.442806i	$0.853983\pi$
$350$		0			0
$351$	603.233			1.71861
$352$		0			0
$353$		−	183.318i		−	0.519316i	−0.965701	−	0.259658i	$-0.916390\pi$
$353$		−	183.318i		−	0.519316i	0.965701	−	0.259658i	$-0.0836098\pi$
$354$		0			0
$355$			499.902i			1.40818i
$356$		0			0
$357$	−150.863	−	190.732i	−0.422585	−	0.534263i
$358$		0			0
$359$	−408.563			−1.13806			−0.569029	−	0.822318i	$-0.692681\pi$
$359$	−408.563			−1.13806			−0.569029	+	0.822318i	$0.692681\pi$
$360$		0			0
$361$	−579.465			−1.60517
$362$		0			0
$363$			724.638i			1.99625i
$364$		0			0
$365$	422.558			1.15769
$366$		0			0
$367$		−	139.947i		−	0.381326i	−0.981656	−	0.190663i	$-0.938936\pi$
$367$		−	139.947i		−	0.381326i	0.981656	−	0.190663i	$-0.0610638\pi$
$368$		0			0
$369$			7.09543i			0.0192288i
$370$		0			0
$371$	245.908	+	310.895i	0.662825	+	0.837992i
$372$		0			0
$373$	−20.7939			−0.0557478			−0.0278739	−	0.999611i	$-0.508874\pi$
$373$	−20.7939			−0.0557478			−0.0278739	+	0.999611i	$0.508874\pi$
$374$		0			0
$375$	−255.828			−0.682207
$376$		0			0
$377$			309.555i			0.821102i
$378$		0			0
$379$	−84.4883			−0.222924			−0.111462	−	0.993769i	$-0.535553\pi$
$379$	−84.4883			−0.222924			−0.111462	+	0.993769i	$0.535553\pi$
$380$		0			0
$381$		−	245.192i		−	0.643549i
$382$		0			0
$383$			459.158i			1.19885i	0.800433	+	0.599423i	$0.204603\pi$
$383$			459.158i			1.19885i	−0.800433	+	0.599423i	$0.795397\pi$
$384$		0			0
$385$	621.754	−	491.788i	1.61495	−	1.27737i
$386$		0			0
$387$	13.7728			0.0355885
$388$		0			0
$389$	551.421			1.41754			0.708768	−	0.705442i	$-0.249251\pi$
$389$	551.421			1.41754			0.708768	+	0.705442i	$0.249251\pi$
$390$		0			0
$391$		−	54.9573i		−	0.140556i
$392$		0			0
$393$	−82.8974			−0.210935
$394$		0			0
$395$			444.945i			1.12644i
$396$		0			0
$397$			469.893i			1.18361i	0.806081	+	0.591805i	$0.201584\pi$
$397$			469.893i			1.18361i	−0.806081	+	0.591805i	$0.798416\pi$
$398$		0			0
$399$	−302.851	−	382.886i	−0.759025	−	0.959614i
$400$		0			0
$401$	−468.808			−1.16910			−0.584549	−	0.811358i	$-0.698728\pi$
$401$	−468.808			−1.16910			−0.584549	+	0.811358i	$0.698728\pi$
$402$		0			0
$403$	−476.915			−1.18341
$404$		0			0
$405$		−	172.226i		−	0.425248i
$406$		0			0
$407$	−704.474			−1.73089
$408$		0			0
$409$			498.752i			1.21944i	0.792616	+	0.609721i	$0.208719\pi$
$409$			498.752i			1.21944i	−0.792616	+	0.609721i	$0.791281\pi$
$410$		0			0
$411$		−	150.359i		−	0.365837i
$412$		0			0
$413$	409.505	−	323.905i	0.991537	−	0.784274i
$414$		0			0
$415$	−188.454			−0.454106
$416$		0			0
$417$	95.4457			0.228887
$418$		0			0
$419$		−	555.841i		−	1.32659i	−0.748358	−	0.663295i	$-0.769158\pi$
$419$		−	555.841i		−	1.32659i	0.748358	−	0.663295i	$-0.230842\pi$
$420$		0			0
$421$	271.352			0.644542			0.322271	−	0.946647i	$-0.395554\pi$
$421$	271.352			0.644542			0.322271	+	0.946647i	$0.395554\pi$
$422$		0			0
$423$			4.22499i			0.00998815i
$424$		0			0
$425$		−	63.7272i		−	0.149946i
$426$		0			0
$427$	−326.182	+	257.999i	−0.763892	+	0.604214i
$428$		0			0
$429$	−985.970			−2.29830
$430$		0			0
$431$	227.665			0.528225			0.264113	−	0.964492i	$-0.414921\pi$
$431$	227.665			0.528225			0.264113	+	0.964492i	$0.414921\pi$
$432$		0			0
$433$			53.2430i			0.122963i	0.998108	+	0.0614816i	$0.0195825\pi$
$433$			53.2430i			0.122963i	−0.998108	+	0.0614816i	$0.980417\pi$
$434$		0			0
$435$	183.878			0.422708
$436$		0			0
$437$		−	110.324i		−	0.252459i
$438$		0			0
$439$			537.392i			1.22413i	0.790809	+	0.612063i	$0.209660\pi$
$439$			537.392i			1.22413i	−0.790809	+	0.612063i	$0.790340\pi$
$440$		0			0
$441$	−43.2010	+	182.551i	−0.0979615	+	0.413947i
$442$		0			0
$443$	213.786			0.482588			0.241294	−	0.970452i	$-0.422428\pi$
$443$	213.786			0.482588			0.241294	+	0.970452i	$0.422428\pi$
$444$		0			0
$445$	−354.205			−0.795967
$446$		0			0
$447$			201.013i			0.449693i
$448$		0			0
$449$	503.588			1.12158			0.560788	−	0.827959i	$-0.310498\pi$
$449$	503.588			1.12158			0.560788	+	0.827959i	$0.310498\pi$
$450$		0			0
$451$		−	38.8607i		−	0.0861656i
$452$		0			0
$453$			373.283i			0.824023i
$454$		0			0
$455$	484.983	+	613.151i	1.06590	+	1.34758i
$456$		0			0
$457$	193.750			0.423961			0.211981	−	0.977274i	$-0.432009\pi$
$457$	193.750			0.423961			0.211981	+	0.977274i	$0.432009\pi$
$458$		0			0
$459$	−445.666			−0.970950
$460$		0			0
$461$		−	305.690i		−	0.663101i	−0.943437	−	0.331551i	$-0.892428\pi$
$461$		−	305.690i		−	0.663101i	0.943437	−	0.331551i	$-0.107572\pi$
$462$		0			0
$463$	−513.146			−1.10831			−0.554153	−	0.832415i	$-0.686958\pi$
$463$	−513.146			−1.10831			−0.554153	+	0.832415i	$0.686958\pi$
$464$		0			0
$465$			283.290i			0.609227i
$466$		0			0
$467$		−	773.163i		−	1.65559i	−0.561027	−	0.827797i	$-0.689594\pi$
$467$		−	773.163i		−	1.65559i	0.561027	−	0.827797i	$-0.310406\pi$
$468$		0			0
$469$	228.975	+	289.487i	0.488219	+	0.617242i
$470$		0			0
$471$	390.937			0.830014
$472$		0			0
$473$	−75.4315			−0.159475
$474$		0			0
$475$		−	127.930i		−	0.269326i
$476$		0			0
$477$	216.794			0.454495
$478$		0			0
$479$			397.958i			0.830809i	0.909637	+	0.415405i	$0.136360\pi$
$479$			397.958i			0.830809i	−0.909637	+	0.415405i	$0.863640\pi$
$480$		0			0
$481$		−	694.726i		−	1.44434i
$482$		0			0
$483$	44.9158	−	35.5270i	0.0929933	−	0.0735548i
$484$		0			0
$485$	−460.881			−0.950271
$486$		0			0
$487$	−686.487			−1.40962			−0.704812	−	0.709394i	$-0.748969\pi$
$487$	−686.487			−1.40962			−0.704812	+	0.709394i	$0.748969\pi$
$488$		0			0
$489$		−	317.419i		−	0.649118i
$490$		0			0
$491$	445.772			0.907886			0.453943	−	0.891031i	$-0.350017\pi$
$491$	445.772			0.907886			0.453943	+	0.891031i	$0.350017\pi$
$492$		0			0
$493$		−	228.698i		−	0.463891i
$494$		0			0
$495$		−	433.563i		−	0.875885i
$496$		0			0
$497$	401.931	+	508.150i	0.808714	+	1.02244i
$498$		0			0
$499$	780.533			1.56419			0.782097	−	0.623156i	$-0.214150\pi$
$499$	780.533			1.56419			0.782097	+	0.623156i	$0.214150\pi$
$500$		0			0
$501$	−189.421			−0.378087
$502$		0			0
$503$			263.206i			0.523272i	0.965167	+	0.261636i	$0.0842619\pi$
$503$			263.206i			0.523272i	−0.965167	+	0.261636i	$0.915738\pi$
$504$		0			0
$505$	−15.0152			−0.0297330
$506$		0			0
$507$		−	588.002i		−	1.15977i
$508$		0			0
$509$		−	266.188i		−	0.522962i	−0.965209	−	0.261481i	$-0.915789\pi$
$509$		−	266.188i		−	0.522962i	0.965209	−	0.261481i	$-0.0842109\pi$
$510$		0			0
$511$	429.530	−	339.745i	0.840568	−	0.664863i
$512$		0			0
$513$	−894.656			−1.74397
$514$		0			0
$515$	331.269			0.643241
$516$		0			0
$517$		−	23.1397i		−	0.0447576i
$518$		0			0
$519$	−103.860			−0.200116
$520$		0			0
$521$		−	498.752i		−	0.957297i	−0.878007	−	0.478649i	$-0.841127\pi$
$521$		−	498.752i		−	0.957297i	0.878007	−	0.478649i	$-0.158873\pi$
$522$		0			0
$523$			586.709i			1.12181i	0.827879	+	0.560907i	$0.189547\pi$
$523$			586.709i			1.12181i	−0.827879	+	0.560907i	$0.810453\pi$
$524$		0			0
$525$	52.0833	−	41.1962i	0.0992062	−	0.0784690i
$526$		0			0
$527$	352.343			0.668583
$528$		0			0
$529$	−516.058			−0.975535
$530$		0			0
$531$		−	285.557i		−	0.537771i
$532$		0			0
$533$	38.3229			0.0719005
$534$		0			0
$535$			582.338i			1.08848i
$536$		0			0
$537$		−	80.4074i		−	0.149734i
$538$		0			0
$539$	236.606	−	999.805i	0.438972	−	1.85493i
$540$		0			0
$541$	−385.314			−0.712225			−0.356112	−	0.934443i	$-0.615898\pi$
$541$	−385.314			−0.712225			−0.356112	+	0.934443i	$0.615898\pi$
$542$		0			0
$543$	343.552			0.632692
$544$		0			0
$545$			503.994i			0.924760i
$546$		0			0
$547$	641.571			1.17289			0.586445	−	0.809989i	$-0.300527\pi$
$547$	641.571			1.17289			0.586445	+	0.809989i	$0.300527\pi$
$548$		0			0
$549$			227.454i			0.414305i
$550$		0			0
$551$		−	459.102i		−	0.833217i
$552$		0			0
$553$	357.744	+	452.286i	0.646916	+	0.817878i
$554$		0			0
$555$	−412.671			−0.743552
$556$		0			0
$557$	218.098			0.391558			0.195779	−	0.980648i	$-0.437277\pi$
$557$	218.098			0.391558			0.195779	+	0.980648i	$0.437277\pi$
$558$		0			0
$559$		−	74.3877i		−	0.133073i
$560$		0			0
$561$	728.431			1.29845
$562$		0			0
$563$		−	606.084i		−	1.07653i	−0.842777	−	0.538263i	$-0.819081\pi$
$563$		−	606.084i		−	1.07653i	0.842777	−	0.538263i	$-0.180919\pi$
$564$		0			0
$565$		−	554.430i		−	0.981292i
$566$		0			0
$567$	−138.473	−	175.067i	−0.244220	−	0.308760i
$568$		0			0
$569$	−217.426			−0.382119			−0.191059	−	0.981578i	$-0.561192\pi$
$569$	−217.426			−0.382119			−0.191059	+	0.981578i	$0.561192\pi$
$570$		0			0
$571$	150.989			0.264429			0.132215	−	0.991221i	$-0.457791\pi$
$571$	150.989			0.264429			0.132215	+	0.991221i	$0.457791\pi$
$572$		0			0
$573$		−	542.860i		−	0.947399i
$574$		0			0
$575$	15.0072			0.0260995
$576$		0			0
$577$			131.797i			0.228418i	0.993457	+	0.114209i	$0.0364333\pi$
$577$			131.797i			0.228418i	−0.993457	+	0.114209i	$0.963567\pi$
$578$		0			0
$579$		−	596.006i		−	1.02937i
$580$		0			0
$581$	−191.563	+	151.521i	−0.329713	+	0.260793i
$582$		0			0
$583$	−1187.35			−2.03662
$584$		0			0
$585$	427.563			0.730878
$586$		0			0
$587$		−	14.0348i		−	0.0239094i	−0.999929	−	0.0119547i	$-0.996195\pi$
$587$		−	14.0348i		−	0.0239094i	0.999929	−	0.0119547i	$-0.00380539\pi$
$588$		0			0
$589$	707.314			1.20087
$590$		0			0
$591$		−	418.681i		−	0.708428i
$592$		0			0
$593$			234.576i			0.395576i	0.980245	+	0.197788i	$0.0633757\pi$
$593$			234.576i			0.395576i	−0.980245	+	0.197788i	$0.936624\pi$
$594$		0			0
$595$	−358.304	−	452.993i	−0.602191	−	0.761333i
$596$		0			0
$597$	377.677			0.632625
$598$		0			0
$599$	528.770			0.882755			0.441377	−	0.897322i	$-0.354490\pi$
$599$	528.770			0.882755			0.441377	+	0.897322i	$0.354490\pi$
$600$		0			0
$601$		−	1026.02i		−	1.70718i	−0.520943	−	0.853592i	$-0.674419\pi$
$601$		−	1026.02i		−	1.70718i	0.520943	−	0.853592i	$-0.325581\pi$
$602$		0			0
$603$	201.865			0.334768
$604$		0			0
$605$			1721.03i			2.84468i
$606$		0			0
$607$		−	690.258i		−	1.13716i	−0.822627	−	0.568582i	$-0.807492\pi$
$607$		−	690.258i		−	1.13716i	0.822627	−	0.568582i	$-0.192508\pi$
$608$		0			0
$609$	186.912	−	147.841i	0.306916	−	0.242761i
$610$		0			0
$611$	22.8195			0.0373477
$612$		0			0
$613$	−681.744			−1.11214			−0.556072	−	0.831134i	$-0.687692\pi$
$613$	−681.744			−1.11214			−0.556072	+	0.831134i	$0.687692\pi$
$614$		0			0
$615$		−	22.7641i		−	0.0370147i
$616$		0			0
$617$	410.416			0.665180			0.332590	−	0.943071i	$-0.392077\pi$
$617$	410.416			0.665180			0.332590	+	0.943071i	$0.392077\pi$
$618$		0			0
$619$			1215.96i			1.96439i	0.187870	+	0.982194i	$0.439842\pi$
$619$			1215.96i			1.96439i	−0.187870	+	0.982194i	$0.560158\pi$
$620$		0			0
$621$		−	104.951i		−	0.169003i
$622$		0			0
$623$	−360.049	+	284.788i	−0.577928	+	0.457123i
$624$		0			0
$625$	−711.887			−1.13902
$626$		0			0
$627$	1462.29			2.33221
$628$		0			0
$629$			513.261i			0.815995i
$630$		0			0
$631$	295.039			0.467573			0.233787	−	0.972288i	$-0.424888\pi$
$631$	295.039			0.467573			0.233787	+	0.972288i	$0.424888\pi$
$632$		0			0
$633$			455.094i			0.718947i
$634$		0			0
$635$		−	582.338i		−	0.917068i
$636$		0			0
$637$	985.970	+	233.332i	1.54783	+	0.366298i
$638$		0			0
$639$	354.344			0.554530
$640$		0			0
$641$	−178.201			−0.278005			−0.139002	−	0.990292i	$-0.544390\pi$
$641$	−178.201			−0.278005			−0.139002	+	0.990292i	$0.544390\pi$
$642$		0			0
$643$			455.078i			0.707742i	0.935294	+	0.353871i	$0.115135\pi$
$643$			455.078i			0.707742i	−0.935294	+	0.353871i	$0.884865\pi$
$644$		0			0
$645$	−44.1867			−0.0685066
$646$		0			0
$647$			397.367i			0.614168i	0.951682	+	0.307084i	$0.0993532\pi$
$647$			397.367i			0.614168i	−0.951682	+	0.307084i	$0.900647\pi$
$648$		0			0
$649$			1563.95i			2.40979i
$650$		0			0
$651$	227.771	+	287.965i	0.349879	+	0.442342i
$652$		0			0
$653$	−554.215			−0.848722			−0.424361	−	0.905493i	$-0.639501\pi$
$653$	−554.215			−0.848722			−0.424361	+	0.905493i	$0.639501\pi$
$654$		0			0
$655$	−196.884			−0.300586
$656$		0			0
$657$		−	299.521i		−	0.455892i
$658$		0			0
$659$	79.0391			0.119938			0.0599689	−	0.998200i	$-0.480900\pi$
$659$	79.0391			0.119938			0.0599689	+	0.998200i	$0.480900\pi$
$660$		0			0
$661$		−	870.340i		−	1.31670i	−0.752711	−	0.658351i	$-0.771254\pi$
$661$		−	870.340i		−	1.31670i	0.752711	−	0.658351i	$-0.228746\pi$
$662$		0			0
$663$			718.351i			1.08349i
$664$		0			0
$665$	−719.279	−	909.365i	−1.08162	−	1.36747i
$666$		0			0
$667$	53.8566			0.0807445
$668$		0			0
$669$	151.294			0.226149
$670$		0			0
$671$		−	1245.73i		−	1.85653i
$672$		0			0
$673$	−66.2153			−0.0983883			−0.0491941	−	0.998789i	$-0.515665\pi$
$673$	−66.2153			−0.0983883			−0.0491941	+	0.998789i	$0.515665\pi$
$674$		0			0
$675$		−	121.698i		−	0.180294i
$676$		0			0
$677$		−	780.666i		−	1.15313i	−0.817053	−	0.576563i	$-0.804394\pi$
$677$		−	780.666i		−	1.15313i	0.817053	−	0.576563i	$-0.195606\pi$
$678$		0			0
$679$	−468.486	+	370.557i	−0.689964	+	0.545740i
$680$		0			0
$681$	28.6720			0.0421028
$682$		0			0
$683$	−1127.43			−1.65070			−0.825349	−	0.564623i	$-0.809022\pi$
$683$	−1127.43			−1.65070			−0.825349	+	0.564623i	$0.809022\pi$
$684$		0			0
$685$		−	357.107i		−	0.521324i
$686$		0			0
$687$	427.423			0.622159
$688$		0			0
$689$		−	1170.92i		−	1.69945i
$690$		0			0
$691$			391.214i			0.566156i	0.959097	+	0.283078i	$0.0913555\pi$
$691$			391.214i			0.566156i	−0.959097	+	0.283078i	$0.908644\pi$
$692$		0			0
$693$	−348.593	−	440.717i	−0.503020	−	0.635955i
$694$		0			0
$695$	226.686			0.326167
$696$		0			0
$697$	−28.3128			−0.0406210
$698$		0			0
$699$			656.370i			0.939013i
$700$		0			0
$701$	−71.2750			−0.101676			−0.0508381	−	0.998707i	$-0.516189\pi$
$701$	−71.2750			−0.101676			−0.0508381	+	0.998707i	$0.516189\pi$
$702$		0			0
$703$			1030.35i			1.46565i
$704$		0			0
$705$		−	13.5549i		−	0.0192268i
$706$		0			0
$707$	−15.2629	+	12.0725i	−0.0215882	+	0.0170756i
$708$		0			0
$709$	1039.92			1.46674			0.733372	−	0.679828i	$-0.237946\pi$
$709$	1039.92			1.46674			0.733372	+	0.679828i	$0.237946\pi$
$710$		0			0
$711$	315.389			0.443585
$712$		0			0
$713$			82.9739i			0.116373i
$714$		0			0
$715$	−2341.70			−3.27511
$716$		0			0
$717$		−	133.464i		−	0.186142i
$718$		0			0
$719$		−	148.096i		−	0.205975i	−0.994683	−	0.102988i	$-0.967160\pi$
$719$		−	148.096i		−	0.205975i	0.994683	−	0.102988i	$-0.0328402\pi$
$720$		0			0
$721$	336.735	−	266.347i	0.467039	−	0.369413i
$722$		0			0
$723$	−526.046			−0.727587
$724$		0			0
$725$	62.4508			0.0861390
$726$		0			0
$727$			530.692i			0.729975i	0.931012	+	0.364987i	$0.118927\pi$
$727$			530.692i			0.729975i	−0.931012	+	0.364987i	$0.881073\pi$
$728$		0			0
$729$	−719.167			−0.986511
$730$		0			0
$731$			54.9573i			0.0751810i
$732$		0			0
$733$			1110.87i			1.51551i	0.652537	+	0.757757i	$0.273705\pi$
$733$			1110.87i			1.51551i	−0.652537	+	0.757757i	$0.726295\pi$
$734$		0			0
$735$	138.600	−	585.672i	0.188572	−	0.796833i
$736$		0			0
$737$	−1105.59			−1.50012
$738$		0			0
$739$	−483.493			−0.654253			−0.327126	−	0.944981i	$-0.606080\pi$
$739$	−483.493			−0.654253			−0.327126	+	0.944981i	$0.606080\pi$
$740$		0			0
$741$			1442.06i			1.94610i
$742$		0			0
$743$	581.031			0.782006			0.391003	−	0.920389i	$-0.372128\pi$
$743$	581.031			0.782006			0.391003	+	0.920389i	$0.372128\pi$
$744$		0			0
$745$			477.411i			0.640820i
$746$		0			0
$747$			133.581i			0.178824i
$748$		0			0
$749$	468.211	+	591.947i	0.625115	+	0.790316i
$750$		0			0
$751$	509.081			0.677871			0.338935	−	0.940810i	$-0.389933\pi$
$751$	509.081			0.677871			0.338935	+	0.940810i	$0.389933\pi$
$752$		0			0
$753$	949.034			1.26034
$754$		0			0
$755$			886.556i			1.17425i
$756$		0			0
$757$	−1215.90			−1.60621			−0.803105	−	0.595838i	$-0.796820\pi$
$757$	−1215.90			−1.60621			−0.803105	+	0.595838i	$0.796820\pi$
$758$		0			0
$759$			171.539i			0.226007i
$760$		0			0
$761$		−	800.576i		−	1.05201i	−0.850483	−	0.526003i	$-0.823690\pi$
$761$		−	800.576i		−	1.05201i	0.850483	−	0.526003i	$-0.176310\pi$
$762$		0			0
$763$	405.221	+	512.310i	0.531089	+	0.671441i
$764$		0			0
$765$	−315.882			−0.412918
$766$		0			0
$767$	−1542.31			−2.01084
$768$		0			0
$769$			591.179i			0.768763i	0.923174	+	0.384382i	$0.125585\pi$
$769$			591.179i			0.768763i	−0.923174	+	0.384382i	$0.874415\pi$
$770$		0			0
$771$	−821.851			−1.06595
$772$		0			0
$773$		−	1170.44i		−	1.51416i	−0.653325	−	0.757078i	$-0.726626\pi$
$773$		−	1170.44i		−	1.51416i	0.653325	−	0.757078i	$-0.273374\pi$
$774$		0			0
$775$			96.2145i			0.124148i
$776$		0			0
$777$	−419.480	+	331.796i	−0.539872	+	0.427021i
$778$		0			0
$779$	−56.8368			−0.0729613
$780$		0			0
$781$	−1940.69			−2.48488
$782$		0			0
$783$		−	436.740i		−	0.557778i
$784$		0			0
$785$	928.485			1.18278
$786$		0			0
$787$		−	414.255i		−	0.526372i	−0.964745	−	0.263186i	$-0.915227\pi$
$787$		−	414.255i		−	0.526372i	0.964745	−	0.263186i	$-0.0847734\pi$
$788$		0			0
$789$		−	282.709i		−	0.358313i
$790$		0			0
$791$	−445.772	−	563.577i	−0.563555	−	0.712487i
$792$		0			0
$793$	1228.49			1.54917
$794$		0			0
$795$	−695.533			−0.874885
$796$		0			0
$797$			103.070i			0.129323i	0.997907	+	0.0646613i	$0.0205967\pi$
$797$			103.070i			0.129323i	−0.997907	+	0.0646613i	$0.979403\pi$
$798$		0			0
$799$	−16.8589			−0.0211000
$800$		0			0
$801$			251.070i			0.313446i
$802$		0			0
$803$			1640.43i			2.04288i
$804$		0			0
$805$	106.676	−	84.3775i	0.132517	−	0.104817i
$806$		0			0
$807$	6.32210			0.00783407
$808$		0			0
$809$	1545.24			1.91006			0.955030	−	0.296509i	$-0.0958224\pi$
$809$	1545.24			1.91006			0.955030	+	0.296509i	$0.0958224\pi$
$810$		0			0
$811$			1021.70i			1.25980i	0.776676	+	0.629900i	$0.216904\pi$
$811$			1021.70i			1.25980i	−0.776676	+	0.629900i	$0.783096\pi$
$812$		0			0
$813$	1087.20			1.33726
$814$		0			0
$815$		−	753.878i		−	0.925003i
$816$		0			0
$817$			110.324i			0.135036i
$818$		0			0
$819$	434.618	−	343.769i	0.530669	−	0.419743i
$820$		0			0
$821$	868.912			1.05836			0.529179	−	0.848510i	$-0.322500\pi$
$821$	868.912			1.05836			0.529179	+	0.848510i	$0.322500\pi$
$822$		0			0
$823$	55.3467			0.0672499			0.0336250	−	0.999435i	$-0.489295\pi$
$823$	55.3467			0.0672499			0.0336250	+	0.999435i	$0.489295\pi$
$824$		0			0
$825$			198.913i			0.241107i
$826$		0			0
$827$	1234.63			1.49290			0.746450	−	0.665441i	$-0.231757\pi$
$827$	1234.63			1.49290			0.746450	+	0.665441i	$0.231757\pi$
$828$		0			0
$829$		−	1551.59i		−	1.87164i	−0.352480	−	0.935819i	$-0.614662\pi$
$829$		−	1551.59i		−	1.87164i	0.352480	−	0.935819i	$-0.385338\pi$
$830$		0			0
$831$			202.998i			0.244282i
$832$		0			0
$833$	−728.431	−	172.385i	−0.874466	−	0.206944i
$834$		0			0
$835$	−449.881			−0.538780
$836$		0			0
$837$	672.861			0.803896
$838$		0			0
$839$		−	312.377i		−	0.372321i	−0.982519	−	0.186160i	$-0.940396\pi$
$839$		−	312.377i		−	0.372321i	0.982519	−	0.186160i	$-0.0596044\pi$
$840$		0			0
$841$	−616.882			−0.733510
$842$		0			0
$843$			79.9056i			0.0947872i
$844$		0			0
$845$		−	1396.52i		−	1.65269i
$846$		0			0
$847$	1383.74	+	1749.43i	1.63370	+	2.06544i
$848$		0			0
$849$	−394.966			−0.465214
$850$		0			0
$851$	−120.869			−0.142031
$852$		0			0
$853$			764.172i			0.895864i	0.894068	+	0.447932i	$0.147839\pi$
$853$			764.172i			0.895864i	−0.894068	+	0.447932i	$0.852161\pi$
$854$		0			0
$855$	−634.120			−0.741661
$856$		0			0
$857$			1368.17i			1.59646i	0.602353	+	0.798230i	$0.294230\pi$
$857$			1368.17i			1.59646i	−0.602353	+	0.798230i	$0.705770\pi$
$858$		0			0
$859$			374.559i			0.436040i	0.975944	+	0.218020i	$0.0699598\pi$
$859$			374.559i			0.436040i	−0.975944	+	0.218020i	$0.930040\pi$
$860$		0			0
$861$	−18.3027	−	23.1397i	−0.0212575	−	0.0268753i
$862$		0			0
$863$	472.956			0.548037			0.274019	−	0.961724i	$-0.411647\pi$
$863$	472.956			0.548037			0.274019	+	0.961724i	$0.411647\pi$
$864$		0			0
$865$	−246.670			−0.285168
$866$		0			0
$867$			126.503i			0.145909i
$868$		0			0
$869$	−1727.34			−1.98774
$870$		0			0
$871$		−	1090.29i		−	1.25177i
$872$		0			0
$873$			326.685i			0.374210i
$874$		0			0
$875$	−617.623	+	488.520i	−0.705855	+	0.558309i
$876$		0			0
$877$	996.067			1.13577			0.567883	−	0.823109i	$-0.307762\pi$
$877$	996.067			1.13577			0.567883	+	0.823109i	$0.307762\pi$
$878$		0			0
$879$	386.722			0.439957
$880$		0			0
$881$			1093.13i			1.24079i	0.784291	+	0.620393i	$0.213027\pi$
$881$			1093.13i			1.24079i	−0.784291	+	0.620393i	$0.786973\pi$
$882$		0			0
$883$	316.262			0.358168			0.179084	−	0.983834i	$-0.442687\pi$
$883$	316.262			0.358168			0.179084	+	0.983834i	$0.442687\pi$
$884$		0			0
$885$			916.143i			1.03519i
$886$		0			0
$887$		−	1315.30i		−	1.48287i	−0.671026	−	0.741434i	$-0.734146\pi$
$887$		−	1315.30i		−	1.48287i	0.671026	−	0.741434i	$-0.265854\pi$
$888$		0			0
$889$	−468.211	−	591.947i	−0.526672	−	0.665857i
$890$		0			0
$891$	668.605			0.750399
$892$		0			0
$893$	−33.8436			−0.0378987
$894$		0			0
$895$		−	190.970i		−	0.213374i
$896$		0			0
$897$	−169.166			−0.188590
$898$		0			0
$899$			345.286i			0.384078i
$900$		0			0
$901$			865.071i			0.960123i
$902$		0			0
$903$	−44.9158	+	35.5270i	−0.0497406	+	0.0393433i
$904$		0			0
$905$	815.945			0.901597
$906$		0			0
$907$	860.912			0.949187			0.474593	−	0.880205i	$-0.342595\pi$
$907$	860.912			0.949187			0.474593	+	0.880205i	$0.342595\pi$
$908$		0			0
$909$			10.6432i			0.0117086i
$910$		0			0
$911$	248.016			0.272245			0.136123	−	0.990692i	$-0.456536\pi$
$911$	248.016			0.272245			0.136123	+	0.990692i	$0.456536\pi$
$912$		0			0
$913$		−	731.607i		−	0.801322i
$914$		0			0
$915$		−	729.733i		−	0.797522i
$916$		0			0
$917$	−200.132	+	158.298i	−0.218246	+	0.172626i
$918$		0			0
$919$	−1134.26			−1.23423			−0.617117	−	0.786872i	$-0.711699\pi$
$919$	−1134.26			−1.23423			−0.617117	+	0.786872i	$0.711699\pi$
$920$		0			0
$921$	699.632			0.759644
$922$		0			0
$923$		−	1913.84i		−	2.07350i
$924$		0			0
$925$	−140.156			−0.151520
$926$		0			0
$927$		−	234.813i		−	0.253304i
$928$		0			0
$929$			220.199i			0.237028i	0.992952	+	0.118514i	$0.0378131\pi$
$929$			220.199i			0.237028i	−0.992952	+	0.118514i	$0.962187\pi$
$930$		0			0
$931$	−1462.29	−	346.055i	−1.57067	−	0.371702i
$932$		0			0
$933$	1149.99			1.23257
$934$		0			0
$935$	1730.04			1.85031
$936$		0			0
$937$		−	1443.91i		−	1.54099i	−0.637443	−	0.770497i	$-0.720008\pi$
$937$		−	1443.91i		−	1.54099i	0.637443	−	0.770497i	$-0.279992\pi$
$938$		0			0
$939$	941.186			1.00233
$940$		0			0
$941$			594.727i			0.632015i	0.948757	+	0.316008i	$0.102343\pi$
$941$			594.727i			0.632015i	−0.948757	+	0.316008i	$0.897657\pi$
$942$		0			0
$943$		−	6.66744i		−	0.00707046i
$944$		0			0
$945$	−684.244	−	865.071i	−0.724068	−	0.915419i
$946$		0			0
$947$	−1057.95			−1.11715			−0.558577	−	0.829452i	$-0.688653\pi$
$947$	−1057.95			−1.11715			−0.558577	+	0.829452i	$0.688653\pi$
$948$		0			0
$949$	−1617.73			−1.70467
$950$		0			0
$951$		−	1367.30i		−	1.43775i
$952$		0			0
$953$	−656.469			−0.688845			−0.344422	−	0.938815i	$-0.611925\pi$
$953$	−656.469			−0.688845			−0.344422	+	0.938815i	$0.611925\pi$
$954$		0			0
$955$		−	1289.31i		−	1.35006i
$956$		0			0
$957$			713.841i			0.745915i
$958$		0			0
$959$	−287.121	−	362.999i	−0.299396	−	0.378518i
$960$		0			0
$961$	429.037			0.446448
$962$		0			0
$963$	412.777			0.428637
$964$		0			0
$965$		−	1415.53i		−	1.46687i
$966$		0			0
$967$	−293.443			−0.303457			−0.151728	−	0.988422i	$-0.548484\pi$
$967$	−293.443			−0.303457			−0.151728	+	0.988422i	$0.548484\pi$
$968$		0			0
$969$		−	1065.39i		−	1.09947i
$970$		0			0
$971$			1088.42i			1.12092i	0.828180	+	0.560462i	$0.189376\pi$
$971$			1088.42i			1.12092i	−0.828180	+	0.560462i	$0.810624\pi$
$972$		0			0
$973$	230.426	−	182.260i	0.236821	−	0.187318i
$974$		0			0
$975$	−196.160			−0.201190
$976$		0			0
$977$	810.861			0.829950			0.414975	−	0.909833i	$-0.363790\pi$
$977$	810.861			0.829950			0.414975	+	0.909833i	$0.363790\pi$
$978$		0			0
$979$		−	1375.08i		−	1.40457i
$980$		0			0
$981$	357.245			0.364164
$982$		0			0
$983$		−	443.874i		−	0.451550i	−0.974179	−	0.225775i	$-0.927509\pi$
$983$		−	443.874i		−	0.451550i	0.974179	−	0.225775i	$-0.0724915\pi$
$984$		0			0
$985$		−	994.379i		−	1.00952i
$986$		0			0
$987$	−10.8984	−	13.7785i	−0.0110419	−	0.0139600i
$988$		0			0
$989$	−12.9420			−0.0130859
$990$		0			0
$991$	1137.96			1.14830			0.574149	−	0.818751i	$-0.305333\pi$
$991$	1137.96			1.14830			0.574149	+	0.818751i	$0.305333\pi$
$992$		0			0
$993$		−	1298.19i		−	1.30734i
$994$		0			0
$995$	896.993			0.901501
$996$		0			0
$997$			1433.98i			1.43830i	0.694856	+	0.719149i	$0.255468\pi$
$997$			1433.98i			1.43830i	−0.694856	+	0.719149i	$0.744532\pi$
$998$		0			0
$999$			980.162i			0.981143i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.c.b.97.6	yes	8
3.2	odd	2		2016.3.f.b.1441.4		8
4.3	odd	2	inner	224.3.c.b.97.4	yes	8
7.6	odd	2	inner	224.3.c.b.97.3	✓	8
8.3	odd	2		448.3.c.h.321.5		8
8.5	even	2		448.3.c.h.321.3		8
12.11	even	2		2016.3.f.b.1441.3		8
21.20	even	2		2016.3.f.b.1441.6		8
28.27	even	2	inner	224.3.c.b.97.5	yes	8
56.13	odd	2		448.3.c.h.321.6		8
56.27	even	2		448.3.c.h.321.4		8
84.83	odd	2		2016.3.f.b.1441.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.c.b.97.3	✓	8	7.6	odd	2	inner
224.3.c.b.97.4	yes	8	4.3	odd	2	inner
224.3.c.b.97.5	yes	8	28.27	even	2	inner
224.3.c.b.97.6	yes	8	1.1	even	1	trivial
448.3.c.h.321.3		8	8.5	even	2
448.3.c.h.321.4		8	56.27	even	2
448.3.c.h.321.5		8	8.3	odd	2
448.3.c.h.321.6		8	56.13	odd	2
2016.3.f.b.1441.3		8	12.11	even	2
2016.3.f.b.1441.4		8	3.2	odd	2
2016.3.f.b.1441.5		8	84.83	odd	2
2016.3.f.b.1441.6		8	21.20	even	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:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	224.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.27411i q^{3} +5.40107i q^{5} +(4.34256 + 5.49019i) q^{7} +3.82843 q^{9} +O(q^{10})\)	\(q+2.27411i q^{3} +5.40107i q^{5} +(4.34256 + 5.49019i) q^{7} +3.82843 q^{9} -20.9678 q^{11} -20.6776i q^{13} -12.2826 q^{15} +15.2765i q^{17} +30.6670i q^{19} +(-12.4853 + 9.87547i) q^{21} -3.59750 q^{23} -4.17157 q^{25} +29.1732i q^{27} -14.9706 q^{29} -23.0643i q^{31} -47.6830i q^{33} +(-29.6529 + 23.4545i) q^{35} +33.5980 q^{37} +47.0231 q^{39} +1.85335i q^{41} +3.59750 q^{43} +20.6776i q^{45} +1.10358i q^{47} +(-11.2843 + 47.6830i) q^{49} -34.7405 q^{51} +56.6274 q^{53} -113.248i q^{55} -69.7401 q^{57} -74.5885i q^{59} +59.4118i q^{61} +(16.6252 + 21.0188i) q^{63} +111.681 q^{65} +52.7280 q^{67} -8.18110i q^{69} +92.5561 q^{71} -78.2360i q^{73} -9.48661i q^{75} +(-91.0538 - 115.117i) q^{77} +82.3809 q^{79} -31.8873 q^{81} +34.8920i q^{83} -82.5097 q^{85} -34.0447i q^{87} +65.5805i q^{89} +(113.524 - 89.7938i) q^{91} +52.4508 q^{93} -165.635 q^{95} +85.3315i q^{97} -80.2735 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^9	\( 8 q + 8 q^{9} - 32 q^{21} - 56 q^{25} + 16 q^{29} - 48 q^{37} + 136 q^{49} + 272 q^{53} - 128 q^{57} + 192 q^{65} - 208 q^{77} - 504 q^{81} + 64 q^{85} - 576 q^{93}+O(q^{100}) \) 8 * q + 8 * q^9 - 32 * q^21 - 56 * q^25 + 16 * q^29 - 48 * q^37 + 136 * q^49 + 272 * q^53 - 128 * q^57 + 192 * q^65 - 208 * q^77 - 504 * q^81 + 64 * q^85 - 576 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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