LMFDB - Embedded newform 1260.2.d.a.881.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1260,2,Mod(881,1260)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1260, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1260.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1260.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$10.0611506547$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 6x^{2} + 4 $ x^4 + 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.1
Root			$-2.28825i$ of defining polynomial
Character	$\chi$	$=$	1260.881
Dual form			1260.2.d.a.881.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +(-2.23607 - 1.41421i) q^{7} +O(q^{10})$	$q-1.00000 q^{5} +(-2.23607 - 1.41421i) q^{7} -1.41421i q^{11} +4.57649i q^{13} +4.47214 q^{17} +4.57649i q^{19} +3.16228i q^{23} +1.00000 q^{25} -10.5672i q^{29} +1.74806i q^{31} +(2.23607 + 1.41421i) q^{35} +10.4721 q^{37} +8.47214 q^{41} +8.47214 q^{43} +6.47214 q^{47} +(3.00000 + 6.32456i) q^{49} -2.49458i q^{53} +1.41421i q^{55} -10.4721 q^{59} +9.15298i q^{61} -4.57649i q^{65} +4.91034i q^{71} +2.41577i q^{73} +(-2.00000 + 3.16228i) q^{77} -8.94427 q^{79} +14.4721 q^{83} -4.47214 q^{85} +6.94427 q^{89} +(6.47214 - 10.2333i) q^{91} -4.57649i q^{95} +19.3863i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5}+O(q^{10}) $ 4 * q - 4 * q^5	$ 4 q - 4 q^{5} + 4 q^{25} + 24 q^{37} + 16 q^{41} + 16 q^{43} + 8 q^{47} + 12 q^{49} - 24 q^{59} - 8 q^{77} + 40 q^{83} - 8 q^{89} + 8 q^{91}+O(q^{100}) $ 4 * q - 4 * q^5 + 4 * q^25 + 24 * q^37 + 16 * q^41 + 16 * q^43 + 8 * q^47 + 12 * q^49 - 24 * q^59 - 8 * q^77 + 40 * q^83 - 8 * q^89 + 8 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times$.

$n$	$281$	$631$	$757$	$1081$
$\chi(n)$	$-1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	−2.23607	−	1.41421i	−0.845154	−	0.534522i
$7$	−2.23607	−	1.41421i	−0.845154	−	0.534522i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.41421i		−	0.426401i	−0.977008	−	0.213201i	$-0.931611\pi$
$11$		−	1.41421i		−	0.426401i	0.977008	−	0.213201i	$-0.0683888\pi$
$12$		0			0
$13$			4.57649i			1.26929i	0.772804	+	0.634645i	$0.218854\pi$
$13$			4.57649i			1.26929i	−0.772804	+	0.634645i	$0.781146\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.47214			1.08465			0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214			1.08465			0.542326	+	0.840168i	$0.317544\pi$
$18$		0			0
$19$			4.57649i			1.04992i	0.851127	+	0.524960i	$0.175920\pi$
$19$			4.57649i			1.04992i	−0.851127	+	0.524960i	$0.824080\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			3.16228i			0.659380i	0.944089	+	0.329690i	$0.106944\pi$
$23$			3.16228i			0.659380i	−0.944089	+	0.329690i	$0.893056\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	10.5672i		−	1.96228i	−0.193301	−	0.981140i	$-0.561919\pi$
$29$		−	10.5672i		−	1.96228i	0.193301	−	0.981140i	$-0.438081\pi$
$30$		0			0
$31$			1.74806i			0.313962i	0.987602	+	0.156981i	$0.0501761\pi$
$31$			1.74806i			0.313962i	−0.987602	+	0.156981i	$0.949824\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.23607	+	1.41421i	0.377964	+	0.239046i
$36$		0			0
$37$	10.4721			1.72161			0.860804	−	0.508936i	$-0.169961\pi$
$37$	10.4721			1.72161			0.860804	+	0.508936i	$0.169961\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.47214			1.32313			0.661563	−	0.749890i	$-0.269894\pi$
$41$	8.47214			1.32313			0.661563	+	0.749890i	$0.269894\pi$
$42$		0			0
$43$	8.47214			1.29199			0.645994	−	0.763342i	$-0.276443\pi$
$43$	8.47214			1.29199			0.645994	+	0.763342i	$0.276443\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.47214			0.944058			0.472029	−	0.881583i	$-0.343522\pi$
$47$	6.47214			0.944058			0.472029	+	0.881583i	$0.343522\pi$
$48$		0			0
$49$	3.00000	+	6.32456i	0.428571	+	0.903508i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	2.49458i		−	0.342656i	−0.985214	−	0.171328i	$-0.945194\pi$
$53$		−	2.49458i		−	0.342656i	0.985214	−	0.171328i	$-0.0548059\pi$
$54$		0			0
$55$			1.41421i			0.190693i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−10.4721			−1.36336			−0.681678	−	0.731652i	$-0.738749\pi$
$59$	−10.4721			−1.36336			−0.681678	+	0.731652i	$0.738749\pi$
$60$		0			0
$61$			9.15298i			1.17192i	0.810340	+	0.585960i	$0.199282\pi$
$61$			9.15298i			1.17192i	−0.810340	+	0.585960i	$0.800718\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	4.57649i		−	0.567644i
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			4.91034i			0.582750i	0.956609	+	0.291375i	$0.0941128\pi$
$71$			4.91034i			0.582750i	−0.956609	+	0.291375i	$0.905887\pi$
$72$		0			0
$73$			2.41577i			0.282744i	0.989957	+	0.141372i	$0.0451514\pi$
$73$			2.41577i			0.282744i	−0.989957	+	0.141372i	$0.954849\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.00000	+	3.16228i	−0.227921	+	0.360375i
$78$		0			0
$79$	−8.94427			−1.00631			−0.503155	−	0.864196i	$-0.667827\pi$
$79$	−8.94427			−1.00631			−0.503155	+	0.864196i	$0.667827\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	14.4721			1.58852			0.794262	−	0.607576i	$-0.207858\pi$
$83$	14.4721			1.58852			0.794262	+	0.607576i	$0.207858\pi$
$84$		0			0
$85$	−4.47214			−0.485071
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.94427			0.736091			0.368046	−	0.929808i	$-0.380027\pi$
$89$	6.94427			0.736091			0.368046	+	0.929808i	$0.380027\pi$
$90$		0			0
$91$	6.47214	−	10.2333i	0.678464	−	1.07275i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	4.57649i		−	0.469538i
$96$		0			0
$97$			19.3863i			1.96838i	0.177107	+	0.984192i	$0.443326\pi$
$97$			19.3863i			1.96838i	−0.177107	+	0.984192i	$0.556674\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	0.472136			0.0469793			0.0234896	−	0.999724i	$-0.492522\pi$
$101$	0.472136			0.0469793			0.0234896	+	0.999724i	$0.492522\pi$
$102$		0			0
$103$		−	9.15298i		−	0.901870i	−0.892557	−	0.450935i	$-0.851091\pi$
$103$		−	9.15298i		−	0.901870i	0.892557	−	0.450935i	$-0.148909\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	3.16228i		−	0.305709i	−0.988249	−	0.152854i	$-0.951153\pi$
$107$		−	3.16228i		−	0.305709i	0.988249	−	0.152854i	$-0.0488466\pi$
$108$		0			0
$109$	−8.94427			−0.856706			−0.428353	−	0.903612i	$-0.640906\pi$
$109$	−8.94427			−0.856706			−0.428353	+	0.903612i	$0.640906\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			12.3153i			1.15852i	0.815142	+	0.579261i	$0.196659\pi$
$113$			12.3153i			1.15852i	−0.815142	+	0.579261i	$0.803341\pi$
$114$		0			0
$115$		−	3.16228i		−	0.294884i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−10.0000	−	6.32456i	−0.916698	−	0.579771i
$120$		0			0
$121$	9.00000			0.818182
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−4.00000			−0.354943			−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000			−0.354943			−0.177471	+	0.984126i	$0.556792\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−11.4164			−0.997456			−0.498728	−	0.866758i	$-0.666199\pi$
$131$	−11.4164			−0.997456			−0.498728	+	0.866758i	$0.666199\pi$
$132$		0			0
$133$	6.47214	−	10.2333i	0.561205	−	0.887344i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			0.333851i			0.0285228i	0.999898	+	0.0142614i	$0.00453969\pi$
$137$			0.333851i			0.0285228i	−0.999898	+	0.0142614i	$0.995460\pi$
$138$		0			0
$139$			15.2225i			1.29116i	0.763695	+	0.645578i	$0.223383\pi$
$139$			15.2225i			1.29116i	−0.763695	+	0.645578i	$0.776617\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.47214			0.541227
$144$		0			0
$145$			10.5672i			0.877558i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	8.40647i		−	0.688685i	−0.938844	−	0.344342i	$-0.888102\pi$
$149$		−	8.40647i		−	0.688685i	0.938844	−	0.344342i	$-0.111898\pi$
$150$		0			0
$151$	2.94427			0.239601			0.119801	−	0.992798i	$-0.461774\pi$
$151$	2.94427			0.239601			0.119801	+	0.992798i	$0.461774\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	1.74806i		−	0.140408i
$156$		0			0
$157$		−	18.7186i		−	1.49391i	−0.664875	−	0.746955i	$-0.731515\pi$
$157$		−	18.7186i		−	1.49391i	0.664875	−	0.746955i	$-0.268485\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	4.47214	−	7.07107i	0.352454	−	0.557278i
$162$		0			0
$163$	−3.05573			−0.239343			−0.119672	−	0.992814i	$-0.538184\pi$
$163$	−3.05573			−0.239343			−0.119672	+	0.992814i	$0.538184\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−13.8885			−1.07473			−0.537364	−	0.843350i	$-0.680580\pi$
$167$	−13.8885			−1.07473			−0.537364	+	0.843350i	$0.680580\pi$
$168$		0			0
$169$	−7.94427			−0.611098
$170$		0			0
$171$		0			0
$172$		0			0
$173$	14.0000			1.06440			0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000			1.06440			0.532200	+	0.846619i	$0.321365\pi$
$174$		0			0
$175$	−2.23607	−	1.41421i	−0.169031	−	0.106904i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	20.3879i		−	1.52386i	−0.647658	−	0.761931i	$-0.724252\pi$
$179$		−	20.3879i		−	1.52386i	0.647658	−	0.761931i	$-0.275748\pi$
$180$		0			0
$181$			4.98915i			0.370841i	0.982659	+	0.185420i	$0.0593647\pi$
$181$			4.98915i			0.370841i	−0.982659	+	0.185420i	$0.940635\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−10.4721			−0.769927
$186$		0			0
$187$		−	6.32456i		−	0.462497i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			22.5486i			1.63156i	0.578362	+	0.815780i	$0.303692\pi$
$191$			22.5486i			1.63156i	−0.578362	+	0.815780i	$0.696308\pi$
$192$		0			0
$193$	19.4164			1.39762			0.698812	−	0.715306i	$-0.253712\pi$
$193$	19.4164			1.39762			0.698812	+	0.715306i	$0.253712\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			0.333851i			0.0237859i	0.999929	+	0.0118929i	$0.00378573\pi$
$197$			0.333851i			0.0237859i	−0.999929	+	0.0118929i	$0.996214\pi$
$198$		0			0
$199$		−	17.2256i		−	1.22109i	−0.791981	−	0.610545i	$-0.790950\pi$
$199$		−	17.2256i		−	1.22109i	0.791981	−	0.610545i	$-0.209050\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−14.9443	+	23.6290i	−1.04888	+	1.65843i
$204$		0			0
$205$	−8.47214			−0.591720
$206$		0			0
$207$		0			0
$208$		0			0
$209$	6.47214			0.447687
$210$		0			0
$211$	−14.0000			−0.963800			−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000			−0.963800			−0.481900	+	0.876226i	$0.660053\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−8.47214			−0.577795
$216$		0			0
$217$	2.47214	−	3.90879i	0.167820	−	0.265346i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			20.4667i			1.37674i
$222$		0			0
$223$		−	4.16383i		−	0.278831i	−0.990234	−	0.139415i	$-0.955478\pi$
$223$		−	4.16383i		−	0.278831i	0.990234	−	0.139415i	$-0.0445223\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.4721			−0.695060			−0.347530	−	0.937669i	$-0.612980\pi$
$227$	−10.4721			−0.695060			−0.347530	+	0.937669i	$0.612980\pi$
$228$		0			0
$229$			4.98915i			0.329693i	0.986319	+	0.164846i	$0.0527128\pi$
$229$			4.98915i			0.329693i	−0.986319	+	0.164846i	$0.947287\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			4.49768i			0.294653i	0.989088	+	0.147326i	$0.0470668\pi$
$233$			4.49768i			0.294653i	−0.989088	+	0.147326i	$0.952933\pi$
$234$		0			0
$235$	−6.47214			−0.422196
$236$		0			0
$237$		0			0
$238$		0			0
$239$			2.90724i			0.188054i	0.995570	+	0.0940268i	$0.0299739\pi$
$239$			2.90724i			0.188054i	−0.995570	+	0.0940268i	$0.970026\pi$
$240$		0			0
$241$		−	19.7990i		−	1.27537i	−0.770299	−	0.637683i	$-0.779893\pi$
$241$		−	19.7990i		−	1.27537i	0.770299	−	0.637683i	$-0.220107\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.00000	−	6.32456i	−0.191663	−	0.404061i
$246$		0			0
$247$	−20.9443			−1.33265
$248$		0			0
$249$		0			0
$250$		0			0
$251$	30.4721			1.92338			0.961692	−	0.274132i	$-0.0883905\pi$
$251$	30.4721			1.92338			0.961692	+	0.274132i	$0.0883905\pi$
$252$		0			0
$253$	4.47214			0.281161
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.52786			0.469575			0.234788	−	0.972047i	$-0.424561\pi$
$257$	7.52786			0.469575			0.234788	+	0.972047i	$0.424561\pi$
$258$		0			0
$259$	−23.4164	−	14.8098i	−1.45502	−	0.920238i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			24.9644i			1.53937i	0.638424	+	0.769685i	$0.279586\pi$
$263$			24.9644i			1.53937i	−0.638424	+	0.769685i	$0.720414\pi$
$264$		0			0
$265$			2.49458i			0.153241i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−28.4721			−1.73598			−0.867988	−	0.496584i	$-0.834587\pi$
$269$	−28.4721			−1.73598			−0.867988	+	0.496584i	$0.834587\pi$
$270$		0			0
$271$		−	20.0540i		−	1.21820i	−0.793095	−	0.609098i	$-0.791532\pi$
$271$		−	20.0540i		−	1.21820i	0.793095	−	0.609098i	$-0.208468\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	1.41421i		−	0.0852803i
$276$		0			0
$277$	−6.00000			−0.360505			−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000			−0.360505			−0.180253	+	0.983620i	$0.557691\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	19.0525i		−	1.13658i	−0.822830	−	0.568288i	$-0.807606\pi$
$281$		−	19.0525i		−	1.13658i	0.822830	−	0.568288i	$-0.192394\pi$
$282$		0			0
$283$		−	26.7912i		−	1.59257i	−0.604919	−	0.796287i	$-0.706794\pi$
$283$		−	26.7912i		−	1.59257i	0.604919	−	0.796287i	$-0.293206\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−18.9443	−	11.9814i	−1.11825	−	0.707240i
$288$		0			0
$289$	3.00000			0.176471
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.58359			0.150935			0.0754675	−	0.997148i	$-0.475955\pi$
$293$	2.58359			0.150935			0.0754675	+	0.997148i	$0.475955\pi$
$294$		0			0
$295$	10.4721			0.609711
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−14.4721			−0.836945
$300$		0			0
$301$	−18.9443	−	11.9814i	−1.09193	−	0.690597i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	9.15298i		−	0.524098i
$306$		0			0
$307$			7.14988i			0.408065i	0.978964	+	0.204033i	$0.0654049\pi$
$307$			7.14988i			0.408065i	−0.978964	+	0.204033i	$0.934595\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−20.3607			−1.15455			−0.577274	−	0.816550i	$-0.695884\pi$
$311$	−20.3607			−1.15455			−0.577274	+	0.816550i	$0.695884\pi$
$312$		0			0
$313$			7.40492i			0.418551i	0.977857	+	0.209275i	$0.0671105\pi$
$313$			7.40492i			0.418551i	−0.977857	+	0.209275i	$0.932889\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			27.1251i			1.52350i	0.647873	+	0.761749i	$0.275659\pi$
$317$			27.1251i			1.52350i	−0.647873	+	0.761749i	$0.724341\pi$
$318$		0			0
$319$	−14.9443			−0.836719
$320$		0			0
$321$		0			0
$322$		0			0
$323$			20.4667i			1.13880i
$324$		0			0
$325$			4.57649i			0.253858i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−14.4721	−	9.15298i	−0.797875	−	0.504620i
$330$		0			0
$331$	23.8885			1.31303			0.656517	−	0.754312i	$-0.272029\pi$
$331$	23.8885			1.31303			0.656517	+	0.754312i	$0.272029\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	15.4164			0.839785			0.419893	−	0.907574i	$-0.362068\pi$
$337$	15.4164			0.839785			0.419893	+	0.907574i	$0.362068\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.47214			0.133874
$342$		0			0
$343$	2.23607	−	18.3848i	0.120736	−	0.992685i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			16.6367i			0.893106i	0.894757	+	0.446553i	$0.147348\pi$
$347$			16.6367i			0.893106i	−0.894757	+	0.446553i	$0.852652\pi$
$348$		0			0
$349$			22.6274i			1.21122i	0.795762	+	0.605609i	$0.207070\pi$
$349$			22.6274i			1.21122i	−0.795762	+	0.605609i	$0.792930\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.00000			−0.319348			−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000			−0.319348			−0.159674	+	0.987170i	$0.551044\pi$
$354$		0			0
$355$		−	4.91034i		−	0.260614i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	32.3693i		−	1.70839i	−0.519956	−	0.854193i	$-0.674052\pi$
$359$		−	32.3693i		−	1.70839i	0.519956	−	0.854193i	$-0.325948\pi$
$360$		0			0
$361$	−1.94427			−0.102330
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	2.41577i		−	0.126447i
$366$		0			0
$367$			2.82843i			0.147643i	0.997271	+	0.0738213i	$0.0235195\pi$
$367$			2.82843i			0.147643i	−0.997271	+	0.0738213i	$0.976481\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.52786	+	5.57804i	−0.183158	+	0.289598i
$372$		0			0
$373$	−9.05573			−0.468888			−0.234444	−	0.972130i	$-0.575327\pi$
$373$	−9.05573			−0.468888			−0.234444	+	0.972130i	$0.575327\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	48.3607			2.49070
$378$		0			0
$379$	−23.8885			−1.22707			−0.613536	−	0.789667i	$-0.710253\pi$
$379$	−23.8885			−1.22707			−0.613536	+	0.789667i	$0.710253\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.944272			−0.0482500			−0.0241250	−	0.999709i	$-0.507680\pi$
$383$	−0.944272			−0.0482500			−0.0241250	+	0.999709i	$0.507680\pi$
$384$		0			0
$385$	2.00000	−	3.16228i	0.101929	−	0.161165i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	5.57804i		−	0.282818i	−0.989951	−	0.141409i	$-0.954837\pi$
$389$		−	5.57804i		−	0.282818i	0.989951	−	0.141409i	$-0.0451633\pi$
$390$		0			0
$391$			14.1421i			0.715199i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8.94427			0.450035
$396$		0			0
$397$		−	5.24419i		−	0.263198i	−0.991303	−	0.131599i	$-0.957989\pi$
$397$		−	5.24419i		−	0.263198i	0.991303	−	0.131599i	$-0.0420112\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			2.90724i			0.145181i	0.997362	+	0.0725903i	$0.0231265\pi$
$401$			2.90724i			0.145181i	−0.997362	+	0.0725903i	$0.976873\pi$
$402$		0			0
$403$	−8.00000			−0.398508
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	14.8098i		−	0.734096i
$408$		0			0
$409$			2.00310i			0.0990471i	0.998773	+	0.0495235i	$0.0157703\pi$
$409$			2.00310i			0.0990471i	−0.998773	+	0.0495235i	$0.984230\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	23.4164	+	14.8098i	1.15225	+	0.728744i
$414$		0			0
$415$	−14.4721			−0.710409
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−30.8328			−1.50628			−0.753141	−	0.657859i	$-0.771462\pi$
$419$	−30.8328			−1.50628			−0.753141	+	0.657859i	$0.771462\pi$
$420$		0			0
$421$	15.8885			0.774360			0.387180	−	0.922004i	$-0.373449\pi$
$421$	15.8885			0.774360			0.387180	+	0.922004i	$0.373449\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.47214			0.216930
$426$		0			0
$427$	12.9443	−	20.4667i	0.626417	−	0.990453i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			20.3879i			0.982050i	0.871146	+	0.491025i	$0.163378\pi$
$431$			20.3879i			0.982050i	−0.871146	+	0.491025i	$0.836622\pi$
$432$		0			0
$433$			31.3677i			1.50744i	0.657197	+	0.753719i	$0.271742\pi$
$433$			31.3677i			1.50744i	−0.657197	+	0.753719i	$0.728258\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−14.4721			−0.692296
$438$		0			0
$439$			6.73722i			0.321550i	0.986991	+	0.160775i	$0.0513993\pi$
$439$			6.73722i			0.321550i	−0.986991	+	0.160775i	$0.948601\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	39.9318i		−	1.89722i	−0.316450	−	0.948609i	$-0.602491\pi$
$443$		−	39.9318i		−	1.89722i	0.316450	−	0.948609i	$-0.397509\pi$
$444$		0			0
$445$	−6.94427			−0.329190
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	15.5563i		−	0.734150i	−0.930191	−	0.367075i	$-0.880359\pi$
$449$		−	15.5563i		−	0.734150i	0.930191	−	0.367075i	$-0.119641\pi$
$450$		0			0
$451$		−	11.9814i		−	0.564183i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−6.47214	+	10.2333i	−0.303418	+	0.479747i
$456$		0			0
$457$	13.5279			0.632807			0.316403	−	0.948625i	$-0.397525\pi$
$457$	13.5279			0.632807			0.316403	+	0.948625i	$0.397525\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−19.8885			−0.926302			−0.463151	−	0.886279i	$-0.653281\pi$
$461$	−19.8885			−0.926302			−0.463151	+	0.886279i	$0.653281\pi$
$462$		0			0
$463$	24.9443			1.15926			0.579629	−	0.814880i	$-0.303197\pi$
$463$	24.9443			1.15926			0.579629	+	0.814880i	$0.303197\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4.58359			−0.212103			−0.106052	−	0.994361i	$-0.533821\pi$
$467$	−4.58359			−0.212103			−0.106052	+	0.994361i	$0.533821\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	11.9814i		−	0.550906i
$474$		0			0
$475$			4.57649i			0.209984i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	20.3607			0.930303			0.465152	−	0.885231i	$-0.346000\pi$
$479$	20.3607			0.930303			0.465152	+	0.885231i	$0.346000\pi$
$480$		0			0
$481$			47.9256i			2.18522i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	19.3863i		−	0.880288i
$486$		0			0
$487$	12.4721			0.565166			0.282583	−	0.959243i	$-0.408809\pi$
$487$	12.4721			0.565166			0.282583	+	0.959243i	$0.408809\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			14.7310i			0.664802i	0.943138	+	0.332401i	$0.107859\pi$
$491$			14.7310i			0.664802i	−0.943138	+	0.332401i	$0.892141\pi$
$492$		0			0
$493$		−	47.2579i		−	2.12839i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	6.94427	−	10.9799i	0.311493	−	0.492514i
$498$		0			0
$499$	−2.94427			−0.131804			−0.0659019	−	0.997826i	$-0.520992\pi$
$499$	−2.94427			−0.131804			−0.0659019	+	0.997826i	$0.520992\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	32.9443			1.46891			0.734456	−	0.678656i	$-0.237437\pi$
$503$	32.9443			1.46891			0.734456	+	0.678656i	$0.237437\pi$
$504$		0			0
$505$	−0.472136			−0.0210098
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−18.3607			−0.813823			−0.406911	−	0.913468i	$-0.633394\pi$
$509$	−18.3607			−0.813823			−0.406911	+	0.913468i	$0.633394\pi$
$510$		0			0
$511$	3.41641	−	5.40182i	0.151133	−	0.238962i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			9.15298i			0.403329i
$516$		0			0
$517$		−	9.15298i		−	0.402548i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	2.36068			0.103423			0.0517116	−	0.998662i	$-0.483532\pi$
$521$	2.36068			0.103423			0.0517116	+	0.998662i	$0.483532\pi$
$522$		0			0
$523$			10.4884i			0.458625i	0.973353	+	0.229313i	$0.0736478\pi$
$523$			10.4884i			0.458625i	−0.973353	+	0.229313i	$0.926352\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			7.81758i			0.340539i
$528$		0			0
$529$	13.0000			0.565217
$530$		0			0
$531$		0			0
$532$		0			0
$533$			38.7727i			1.67943i
$534$		0			0
$535$			3.16228i			0.136717i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	8.94427	−	4.24264i	0.385257	−	0.182743i
$540$		0			0
$541$	−7.88854			−0.339155			−0.169577	−	0.985517i	$-0.554240\pi$
$541$	−7.88854			−0.339155			−0.169577	+	0.985517i	$0.554240\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	8.94427			0.383131
$546$		0			0
$547$	−30.8328			−1.31832			−0.659158	−	0.752004i	$-0.729087\pi$
$547$	−30.8328			−1.31832			−0.659158	+	0.752004i	$0.729087\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	48.3607			2.06023
$552$		0			0
$553$	20.0000	+	12.6491i	0.850487	+	0.537895i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			3.82998i			0.162281i	0.996703	+	0.0811407i	$0.0258563\pi$
$557$			3.82998i			0.162281i	−0.996703	+	0.0811407i	$0.974144\pi$
$558$		0			0
$559$			38.7727i			1.63991i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−13.8885			−0.585332			−0.292666	−	0.956215i	$-0.594542\pi$
$563$	−13.8885			−0.585332			−0.292666	+	0.956215i	$0.594542\pi$
$564$		0			0
$565$		−	12.3153i		−	0.518107i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			14.0633i			0.589565i	0.955564	+	0.294783i	$0.0952472\pi$
$569$			14.0633i			0.589565i	−0.955564	+	0.294783i	$0.904753\pi$
$570$		0			0
$571$	16.0000			0.669579			0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000			0.669579			0.334790	+	0.942293i	$0.391335\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			3.16228i			0.131876i
$576$		0			0
$577$		−	17.3832i		−	0.723673i	−0.932242	−	0.361837i	$-0.882150\pi$
$577$		−	17.3832i		−	0.723673i	0.932242	−	0.361837i	$-0.117850\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−32.3607	−	20.4667i	−1.34255	−	0.849101i
$582$		0			0
$583$	−3.52786			−0.146109
$584$		0			0
$585$		0			0
$586$		0			0
$587$	34.8328			1.43770			0.718852	−	0.695163i	$-0.244668\pi$
$587$	34.8328			1.43770			0.718852	+	0.695163i	$0.244668\pi$
$588$		0			0
$589$	−8.00000			−0.329634
$590$		0			0
$591$		0			0
$592$		0			0
$593$	11.8885			0.488204			0.244102	−	0.969750i	$-0.421507\pi$
$593$	11.8885			0.488204			0.244102	+	0.969750i	$0.421507\pi$
$594$		0			0
$595$	10.0000	+	6.32456i	0.409960	+	0.259281i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	1.41421i		−	0.0577832i	−0.999583	−	0.0288916i	$-0.990802\pi$
$599$		−	1.41421i		−	0.0577832i	0.999583	−	0.0288916i	$-0.00919776\pi$
$600$		0			0
$601$			23.9628i			0.977464i	0.872434	+	0.488732i	$0.162540\pi$
$601$			23.9628i			0.977464i	−0.872434	+	0.488732i	$0.837460\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−9.00000			−0.365902
$606$		0			0
$607$			37.2796i			1.51313i	0.653916	+	0.756567i	$0.273125\pi$
$607$			37.2796i			1.51313i	−0.653916	+	0.756567i	$0.726875\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			29.6197i			1.19828i
$612$		0			0
$613$	2.94427			0.118918			0.0594590	−	0.998231i	$-0.481062\pi$
$613$	2.94427			0.118918			0.0594590	+	0.998231i	$0.481062\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			15.8114i			0.636543i	0.948000	+	0.318271i	$0.103102\pi$
$617$			15.8114i			0.636543i	−0.948000	+	0.318271i	$0.896898\pi$
$618$		0			0
$619$			47.5130i			1.90971i	0.297076	+	0.954854i	$0.403989\pi$
$619$			47.5130i			1.90971i	−0.297076	+	0.954854i	$0.596011\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−15.5279	−	9.82068i	−0.622111	−	0.393457i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	46.8328			1.86735
$630$		0			0
$631$	8.94427			0.356066			0.178033	−	0.984025i	$-0.443027\pi$
$631$	8.94427			0.356066			0.178033	+	0.984025i	$0.443027\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.00000			0.158735
$636$		0			0
$637$	−28.9443	+	13.7295i	−1.14681	+	0.543982i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	44.3507i		−	1.75175i	−0.482541	−	0.875874i	$-0.660286\pi$
$641$		−	44.3507i		−	1.75175i	0.482541	−	0.875874i	$-0.339714\pi$
$642$		0			0
$643$			23.2951i			0.918670i	0.888263	+	0.459335i	$0.151912\pi$
$643$			23.2951i			0.918670i	−0.888263	+	0.459335i	$0.848088\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.00000			−0.314512			−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000			−0.314512			−0.157256	+	0.987558i	$0.550265\pi$
$648$		0			0
$649$			14.8098i			0.581337i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	13.1406i		−	0.514231i	−0.966381	−	0.257115i	$-0.917228\pi$
$653$		−	13.1406i		−	0.514231i	0.966381	−	0.257115i	$-0.0827720\pi$
$654$		0			0
$655$	11.4164			0.446076
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	37.3584i		−	1.45528i	−0.685960	−	0.727639i	$-0.740618\pi$
$659$		−	37.3584i		−	1.45528i	0.685960	−	0.727639i	$-0.259382\pi$
$660$		0			0
$661$		−	14.8098i		−	0.576036i	−0.957625	−	0.288018i	$-0.907004\pi$
$661$		−	14.8098i		−	0.576036i	0.957625	−	0.288018i	$-0.0929963\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.47214	+	10.2333i	−0.250979	+	0.396832i
$666$		0			0
$667$	33.4164			1.29389
$668$		0			0
$669$		0			0
$670$		0			0
$671$	12.9443			0.499708
$672$		0			0
$673$	25.5279			0.984027			0.492013	−	0.870588i	$-0.336261\pi$
$673$	25.5279			0.984027			0.492013	+	0.870588i	$0.336261\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	10.5836			0.406760			0.203380	−	0.979100i	$-0.434807\pi$
$677$	10.5836			0.406760			0.203380	+	0.979100i	$0.434807\pi$
$678$		0			0
$679$	27.4164	−	43.3491i	1.05215	−	1.66359i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	34.9427i		−	1.33704i	−0.743692	−	0.668522i	$-0.766927\pi$
$683$		−	34.9427i		−	1.33704i	0.743692	−	0.668522i	$-0.233073\pi$
$684$		0			0
$685$		−	0.333851i		−	0.0127558i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	11.4164			0.434931
$690$		0			0
$691$		−	44.8422i		−	1.70588i	−0.522012	−	0.852938i	$-0.674818\pi$
$691$		−	44.8422i		−	1.70588i	0.522012	−	0.852938i	$-0.325182\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	15.2225i		−	0.577422i
$696$		0			0
$697$	37.8885			1.43513
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	45.8437i		−	1.73149i	−0.500482	−	0.865747i	$-0.666844\pi$
$701$		−	45.8437i		−	1.73149i	0.500482	−	0.865747i	$-0.333156\pi$
$702$		0			0
$703$			47.9256i			1.80755i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.05573	−	0.667701i	−0.0397047	−	0.0251115i
$708$		0			0
$709$	−16.9443			−0.636355			−0.318178	−	0.948031i	$-0.603071\pi$
$709$	−16.9443			−0.636355			−0.318178	+	0.948031i	$0.603071\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−5.52786			−0.207020
$714$		0			0
$715$	−6.47214			−0.242044
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−13.5279			−0.504504			−0.252252	−	0.967662i	$-0.581171\pi$
$719$	−13.5279			−0.504504			−0.252252	+	0.967662i	$0.581171\pi$
$720$		0			0
$721$	−12.9443	+	20.4667i	−0.482070	+	0.762219i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	10.5672i		−	0.392456i
$726$		0			0
$727$			16.9706i			0.629403i	0.949191	+	0.314702i	$0.101904\pi$
$727$			16.9706i			0.629403i	−0.949191	+	0.314702i	$0.898096\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	37.8885			1.40136
$732$		0			0
$733$		−	2.41577i		−	0.0892283i	−0.999004	−	0.0446142i	$-0.985794\pi$
$733$		−	2.41577i		−	0.0892283i	0.999004	−	0.0446142i	$-0.0142058\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	20.9443			0.770447			0.385224	−	0.922823i	$-0.374124\pi$
$739$	20.9443			0.770447			0.385224	+	0.922823i	$0.374124\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			27.2827i			1.00091i	0.865764	+	0.500453i	$0.166833\pi$
$743$			27.2827i			1.00091i	−0.865764	+	0.500453i	$0.833167\pi$
$744$		0			0
$745$			8.40647i			0.307989i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4.47214	+	7.07107i	−0.163408	+	0.258371i
$750$		0			0
$751$	34.0000			1.24068			0.620339	−	0.784334i	$-0.286995\pi$
$751$	34.0000			1.24068			0.620339	+	0.784334i	$0.286995\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.94427			−0.107153
$756$		0			0
$757$	−9.52786			−0.346296			−0.173148	−	0.984896i	$-0.555394\pi$
$757$	−9.52786			−0.346296			−0.173148	+	0.984896i	$0.555394\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	23.8885			0.865959			0.432980	−	0.901404i	$-0.357462\pi$
$761$	23.8885			0.865959			0.432980	+	0.901404i	$0.357462\pi$
$762$		0			0
$763$	20.0000	+	12.6491i	0.724049	+	0.457929i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	47.9256i		−	1.73049i
$768$		0			0
$769$		−	16.9706i		−	0.611974i	−0.952036	−	0.305987i	$-0.901014\pi$
$769$		−	16.9706i		−	0.611974i	0.952036	−	0.305987i	$-0.0989864\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−18.0000			−0.647415			−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000			−0.647415			−0.323708	+	0.946157i	$0.604929\pi$
$774$		0			0
$775$			1.74806i			0.0627923i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			38.7727i			1.38917i
$780$		0			0
$781$	6.94427			0.248486
$782$		0			0
$783$		0			0
$784$		0			0
$785$			18.7186i			0.668096i
$786$		0			0
$787$			29.6197i			1.05583i	0.849298	+	0.527914i	$0.177026\pi$
$787$			29.6197i			1.05583i	−0.849298	+	0.527914i	$0.822974\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	17.4164	−	27.5378i	0.619256	−	0.979130i
$792$		0			0
$793$	−41.8885			−1.48751
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−28.8328			−1.02131			−0.510655	−	0.859785i	$-0.670597\pi$
$797$	−28.8328			−1.02131			−0.510655	+	0.859785i	$0.670597\pi$
$798$		0			0
$799$	28.9443			1.02397
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.41641			0.120562
$804$		0			0
$805$	−4.47214	+	7.07107i	−0.157622	+	0.249222i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			33.0370i			1.16152i	0.814075	+	0.580759i	$0.197244\pi$
$809$			33.0370i			1.16152i	−0.814075	+	0.580759i	$0.802756\pi$
$810$		0			0
$811$			34.8639i			1.22424i	0.790766	+	0.612118i	$0.209682\pi$
$811$			34.8639i			1.22424i	−0.790766	+	0.612118i	$0.790318\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.05573			0.107037
$816$		0			0
$817$			38.7727i			1.35648i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			40.3445i			1.40803i	0.710184	+	0.704016i	$0.248612\pi$
$821$			40.3445i			1.40803i	−0.710184	+	0.704016i	$0.751388\pi$
$822$		0			0
$823$	−37.8885			−1.32071			−0.660356	−	0.750953i	$-0.729595\pi$
$823$	−37.8885			−1.32071			−0.660356	+	0.750953i	$0.729595\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	43.4280i		−	1.51014i	−0.655645	−	0.755069i	$-0.727603\pi$
$827$		−	43.4280i		−	1.51014i	0.655645	−	0.755069i	$-0.272397\pi$
$828$		0			0
$829$		−	56.4109i		−	1.95923i	−0.200879	−	0.979616i	$-0.564380\pi$
$829$		−	56.4109i		−	1.95923i	0.200879	−	0.979616i	$-0.435620\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	13.4164	+	28.2843i	0.464851	+	0.979992i
$834$		0			0
$835$	13.8885			0.480633
$836$		0			0
$837$		0			0
$838$		0			0
$839$	13.8885			0.479486			0.239743	−	0.970836i	$-0.422937\pi$
$839$	13.8885			0.479486			0.239743	+	0.970836i	$0.422937\pi$
$840$		0			0
$841$	−82.6656			−2.85054
$842$		0			0
$843$		0			0
$844$		0			0
$845$	7.94427			0.273291
$846$		0			0
$847$	−20.1246	−	12.7279i	−0.691490	−	0.437337i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			33.1158i			1.13520i
$852$		0			0
$853$		−	20.0540i		−	0.686637i	−0.939219	−	0.343318i	$-0.888449\pi$
$853$		−	20.0540i		−	0.686637i	0.939219	−	0.343318i	$-0.111551\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−14.9443			−0.510487			−0.255243	−	0.966877i	$-0.582156\pi$
$857$	−14.9443			−0.510487			−0.255243	+	0.966877i	$0.582156\pi$
$858$		0			0
$859$		−	23.7078i		−	0.808899i	−0.914560	−	0.404450i	$-0.867463\pi$
$859$		−	23.7078i		−	0.808899i	0.914560	−	0.404450i	$-0.132537\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	36.4357i		−	1.24029i	−0.784489	−	0.620143i	$-0.787075\pi$
$863$		−	36.4357i		−	1.24029i	0.784489	−	0.620143i	$-0.212925\pi$
$864$		0			0
$865$	−14.0000			−0.476014
$866$		0			0
$867$		0			0
$868$		0			0
$869$			12.6491i			0.429092i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.23607	+	1.41421i	0.0755929	+	0.0478091i
$876$		0			0
$877$	37.7771			1.27564			0.637821	−	0.770185i	$-0.279836\pi$
$877$	37.7771			1.27564			0.637821	+	0.770185i	$0.279836\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−40.8328			−1.37569			−0.687846	−	0.725856i	$-0.741444\pi$
$881$	−40.8328			−1.37569			−0.687846	+	0.725856i	$0.741444\pi$
$882$		0			0
$883$	11.5279			0.387944			0.193972	−	0.981007i	$-0.437863\pi$
$883$	11.5279			0.387944			0.193972	+	0.981007i	$0.437863\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−45.3050			−1.52119			−0.760596	−	0.649226i	$-0.775093\pi$
$887$	−45.3050			−1.52119			−0.760596	+	0.649226i	$0.775093\pi$
$888$		0			0
$889$	8.94427	+	5.65685i	0.299981	+	0.189725i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			29.6197i			0.991185i
$894$		0			0
$895$			20.3879i			0.681492i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	18.4721			0.616080
$900$		0			0
$901$		−	11.1561i		−	0.371663i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	4.98915i		−	0.165845i
$906$		0			0
$907$	−53.1935			−1.76626			−0.883131	−	0.469127i	$-0.844569\pi$
$907$	−53.1935			−1.76626			−0.883131	+	0.469127i	$0.844569\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	19.8778i		−	0.658581i	−0.944229	−	0.329290i	$-0.893190\pi$
$911$		−	19.8778i		−	0.658581i	0.944229	−	0.329290i	$-0.106810\pi$
$912$		0			0
$913$		−	20.4667i		−	0.677349i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	25.5279	+	16.1452i	0.843004	+	0.533163i
$918$		0			0
$919$	14.9443			0.492966			0.246483	−	0.969147i	$-0.420725\pi$
$919$	14.9443			0.492966			0.246483	+	0.969147i	$0.420725\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−22.4721			−0.739679
$924$		0			0
$925$	10.4721			0.344322
$926$		0			0
$927$		0			0
$928$		0			0
$929$	54.9443			1.80266			0.901332	−	0.433130i	$-0.142591\pi$
$929$	54.9443			1.80266			0.901332	+	0.433130i	$0.142591\pi$
$930$		0			0
$931$	−28.9443	+	13.7295i	−0.948610	+	0.449965i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			6.32456i			0.206835i
$936$		0			0
$937$		−	15.2225i		−	0.497297i	−0.968594	−	0.248649i	$-0.920014\pi$
$937$		−	15.2225i		−	0.497297i	0.968594	−	0.248649i	$-0.0799865\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−48.8328			−1.59190			−0.795952	−	0.605360i	$-0.793029\pi$
$941$	−48.8328			−1.59190			−0.795952	+	0.605360i	$0.793029\pi$
$942$		0			0
$943$			26.7912i			0.872443i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	20.6429i		−	0.670805i	−0.942075	−	0.335402i	$-0.891128\pi$
$947$		−	20.6429i		−	0.670805i	0.942075	−	0.335402i	$-0.108872\pi$
$948$		0			0
$949$	−11.0557			−0.358884
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	38.9489i		−	1.26168i	−0.775914	−	0.630839i	$-0.782711\pi$
$953$		−	38.9489i		−	1.26168i	0.775914	−	0.630839i	$-0.217289\pi$
$954$		0			0
$955$		−	22.5486i		−	0.729656i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	0.472136	−	0.746512i	0.0152461	−	0.0241061i
$960$		0			0
$961$	27.9443			0.901428
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−19.4164			−0.625036
$966$		0			0
$967$	5.88854			0.189363			0.0946814	−	0.995508i	$-0.469817\pi$
$967$	5.88854			0.189363			0.0946814	+	0.995508i	$0.469817\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−32.0000			−1.02693			−0.513464	−	0.858111i	$-0.671638\pi$
$971$	−32.0000			−1.02693			−0.513464	+	0.858111i	$0.671638\pi$
$972$		0			0
$973$	21.5279	−	34.0385i	0.690152	−	1.09123i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			28.4605i			0.910532i	0.890355	+	0.455266i	$0.150456\pi$
$977$			28.4605i			0.910532i	−0.890355	+	0.455266i	$0.849544\pi$
$978$		0			0
$979$		−	9.82068i		−	0.313870i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	27.4164			0.874448			0.437224	−	0.899353i	$-0.355962\pi$
$983$	27.4164			0.874448			0.437224	+	0.899353i	$0.355962\pi$
$984$		0			0
$985$		−	0.333851i		−	0.0106374i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			26.7912i			0.851912i
$990$		0			0
$991$	18.9443			0.601785			0.300892	−	0.953658i	$-0.402716\pi$
$991$	18.9443			0.601785			0.300892	+	0.953658i	$0.402716\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			17.2256i			0.546088i
$996$		0			0
$997$		−	23.5502i		−	0.745841i	−0.927863	−	0.372920i	$-0.878356\pi$
$997$		−	23.5502i		−	0.745841i	0.927863	−	0.372920i	$-0.121644\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1260.2.d.a.881.1	✓	4
3.2	odd	2		1260.2.d.b.881.1	yes	4
4.3	odd	2		5040.2.f.b.881.4		4
5.2	odd	4		6300.2.f.c.3149.5		8
5.3	odd	4		6300.2.f.c.3149.3		8
5.4	even	2		6300.2.d.b.3401.4		4
7.6	odd	2		1260.2.d.b.881.2	yes	4
12.11	even	2		5040.2.f.d.881.4		4
15.2	even	4		6300.2.f.a.3149.6		8
15.8	even	4		6300.2.f.a.3149.4		8
15.14	odd	2		6300.2.d.a.3401.4		4
21.20	even	2	inner	1260.2.d.a.881.2	yes	4
28.27	even	2		5040.2.f.d.881.3		4
35.13	even	4		6300.2.f.a.3149.7		8
35.27	even	4		6300.2.f.a.3149.1		8
35.34	odd	2		6300.2.d.a.3401.3		4
84.83	odd	2		5040.2.f.b.881.3		4
105.62	odd	4		6300.2.f.c.3149.2		8
105.83	odd	4		6300.2.f.c.3149.8		8
105.104	even	2		6300.2.d.b.3401.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1260.2.d.a.881.1	✓	4	1.1	even	1	trivial
1260.2.d.a.881.2	yes	4	21.20	even	2	inner
1260.2.d.b.881.1	yes	4	3.2	odd	2
1260.2.d.b.881.2	yes	4	7.6	odd	2
5040.2.f.b.881.3		4	84.83	odd	2
5040.2.f.b.881.4		4	4.3	odd	2
5040.2.f.d.881.3		4	28.27	even	2
5040.2.f.d.881.4		4	12.11	even	2
6300.2.d.a.3401.3		4	35.34	odd	2
6300.2.d.a.3401.4		4	15.14	odd	2
6300.2.d.b.3401.3		4	105.104	even	2
6300.2.d.b.3401.4		4	5.4	even	2
6300.2.f.a.3149.1		8	35.27	even	4
6300.2.f.a.3149.4		8	15.8	even	4
6300.2.f.a.3149.6		8	15.2	even	4
6300.2.f.a.3149.7		8	35.13	even	4
6300.2.f.c.3149.2		8	105.62	odd	4
6300.2.f.c.3149.3		8	5.3	odd	4
6300.2.f.c.3149.5		8	5.2	odd	4
6300.2.f.c.3149.8		8	105.83	odd	4

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:	\( N \)	\(=\)	\( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1260.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +(-2.23607 - 1.41421i) q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +(-2.23607 - 1.41421i) q^{7} -1.41421i q^{11} +4.57649i q^{13} +4.47214 q^{17} +4.57649i q^{19} +3.16228i q^{23} +1.00000 q^{25} -10.5672i q^{29} +1.74806i q^{31} +(2.23607 + 1.41421i) q^{35} +10.4721 q^{37} +8.47214 q^{41} +8.47214 q^{43} +6.47214 q^{47} +(3.00000 + 6.32456i) q^{49} -2.49458i q^{53} +1.41421i q^{55} -10.4721 q^{59} +9.15298i q^{61} -4.57649i q^{65} +4.91034i q^{71} +2.41577i q^{73} +(-2.00000 + 3.16228i) q^{77} -8.94427 q^{79} +14.4721 q^{83} -4.47214 q^{85} +6.94427 q^{89} +(6.47214 - 10.2333i) q^{91} -4.57649i q^{95} +19.3863i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5}+O(q^{10}) \) 4 * q - 4 * q^5	\( 4 q - 4 q^{5} + 4 q^{25} + 24 q^{37} + 16 q^{41} + 16 q^{43} + 8 q^{47} + 12 q^{49} - 24 q^{59} - 8 q^{77} + 40 q^{83} - 8 q^{89} + 8 q^{91}+O(q^{100}) \) 4 * q - 4 * q^5 + 4 * q^25 + 24 * q^37 + 16 * q^41 + 16 * q^43 + 8 * q^47 + 12 * q^49 - 24 * q^59 - 8 * q^77 + 40 * q^83 - 8 * q^89 + 8 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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