LMFDB - Embedded newform 1456.2.k.d.337.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1456,2,Mod(337,1456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1456, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1456.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1456.k (of order $2$, degree $1$, 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:	$11.6262185343$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.41589892096.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 16x^{6} + 80x^{4} + 132x^{2} + 64 $ x^8 + 16x^6 + 80x^4 + 132*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 364)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.5
Root			$-1.29363i$ of defining polynomial
Character	$\chi$	$=$	1456.337
Dual form			1456.2.k.d.337.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.29363 q^{3} -2.79846i q^{5} -1.00000i q^{7} -1.32653 q^{9} +O(q^{10})$	$q+1.29363 q^{3} -2.79846i q^{5} -1.00000i q^{7} -1.32653 q^{9} -1.03291i q^{11} +(-1.79846 - 3.12499i) q^{13} -3.62016i q^{15} -6.95635 q^{17} +2.53774i q^{19} -1.29363i q^{21} +3.83136 q^{23} -2.83136 q^{25} -5.59691 q^{27} -5.10175 q^{29} +3.78880i q^{31} -1.33619i q^{33} -2.79846 q^{35} -6.20741i q^{37} +(-2.32653 - 4.04257i) q^{39} +0.990338i q^{41} -0.560979 q^{43} +3.71224i q^{45} +7.19080i q^{47} -1.00000 q^{49} -8.99892 q^{51} +1.17830 q^{53} -2.89054 q^{55} +3.28288i q^{57} -11.9234i q^{59} +13.8469 q^{61} +1.32653i q^{63} +(-8.74515 + 5.03291i) q^{65} -13.7811i q^{67} +4.95635 q^{69} -3.80432i q^{71} +3.19080i q^{73} -3.66272 q^{75} -1.03291 q^{77} -7.33240 q^{79} -3.26072 q^{81} +16.6260i q^{83} +19.4670i q^{85} -6.59975 q^{87} -11.8964i q^{89} +(-3.12499 + 1.79846i) q^{91} +4.90128i q^{93} +7.10175 q^{95} -8.60355i q^{97} +1.37018i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^9	$ 8 q + 8 q^{9} + 2 q^{13} + 4 q^{17} + 6 q^{23} + 2 q^{25} - 12 q^{27} - 2 q^{29} - 6 q^{35} + 6 q^{43} - 8 q^{49} + 8 q^{51} + 22 q^{53} + 20 q^{55} + 8 q^{61} - 6 q^{65} - 20 q^{69} + 20 q^{75} + 26 q^{79} - 24 q^{81} + 32 q^{87} + 10 q^{91} + 18 q^{95}+O(q^{100}) $ 8 * q + 8 * q^9 + 2 * q^13 + 4 * q^17 + 6 * q^23 + 2 * q^25 - 12 * q^27 - 2 * q^29 - 6 * q^35 + 6 * q^43 - 8 * q^49 + 8 * q^51 + 22 * q^53 + 20 * q^55 + 8 * q^61 - 6 * q^65 - 20 * q^69 + 20 * q^75 + 26 * q^79 - 24 * q^81 + 32 * q^87 + 10 * q^91 + 18 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times$.

$n$	$561$	$911$	$1093$	$1249$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	1.29363			0.746875			0.373438	−	0.927655i	$-0.378179\pi$
$3$	1.29363			0.746875			0.373438	+	0.927655i	$0.378179\pi$
$4$		0			0
$5$		−	2.79846i		−	1.25151i	−0.780020	−	0.625754i	$-0.784791\pi$
$5$		−	2.79846i		−	1.25151i	0.780020	−	0.625754i	$-0.215209\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−1.32653			−0.442177
$10$		0			0
$11$		−	1.03291i		−	0.311433i	−0.987802	−	0.155716i	$-0.950231\pi$
$11$		−	1.03291i		−	0.311433i	0.987802	−	0.155716i	$-0.0497686\pi$
$12$		0			0
$13$	−1.79846	−	3.12499i	−0.498802	−	0.866716i
$13$	−1.79846	−	3.12499i	−0.498802	−	0.866716i
$14$		0			0
$15$		−	3.62016i		−	0.934721i
$16$		0			0
$17$	−6.95635			−1.68716			−0.843581	−	0.537001i	$-0.819557\pi$
$17$	−6.95635			−1.68716			−0.843581	+	0.537001i	$0.819557\pi$
$18$		0			0
$19$			2.53774i			0.582197i	0.956693	+	0.291098i	$0.0940207\pi$
$19$			2.53774i			0.582197i	−0.956693	+	0.291098i	$0.905979\pi$
$20$		0			0
$21$		−	1.29363i		−	0.282292i
$22$		0			0
$23$	3.83136			0.798894			0.399447	−	0.916756i	$-0.369202\pi$
$23$	3.83136			0.798894			0.399447	+	0.916756i	$0.369202\pi$
$24$		0			0
$25$	−2.83136			−0.566272
$26$		0			0
$27$	−5.59691			−1.07713
$28$		0			0
$29$	−5.10175			−0.947370			−0.473685	−	0.880694i	$-0.657077\pi$
$29$	−5.10175			−0.947370			−0.473685	+	0.880694i	$0.657077\pi$
$30$		0			0
$31$			3.78880i			0.680488i	0.940337	+	0.340244i	$0.110510\pi$
$31$			3.78880i			0.680488i	−0.940337	+	0.340244i	$0.889490\pi$
$32$		0			0
$33$		−	1.33619i		−	0.232601i
$34$		0			0
$35$	−2.79846			−0.473026
$36$		0			0
$37$		−	6.20741i		−	1.02049i	−0.860029	−	0.510246i	$-0.829554\pi$
$37$		−	6.20741i		−	1.02049i	0.860029	−	0.510246i	$-0.170446\pi$
$38$		0			0
$39$	−2.32653	−	4.04257i	−0.372543	−	0.647329i
$40$		0			0
$41$			0.990338i			0.154665i	0.997005	+	0.0773324i	$0.0246403\pi$
$41$			0.990338i			0.154665i	−0.997005	+	0.0773324i	$0.975360\pi$
$42$		0			0
$43$	−0.560979			−0.0855485			−0.0427743	−	0.999085i	$-0.513620\pi$
$43$	−0.560979			−0.0855485			−0.0427743	+	0.999085i	$0.513620\pi$
$44$		0			0
$45$			3.71224i			0.553388i
$46$		0			0
$47$			7.19080i			1.04889i	0.851446	+	0.524443i	$0.175726\pi$
$47$			7.19080i			1.04889i	−0.851446	+	0.524443i	$0.824274\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−8.99892			−1.26010
$52$		0			0
$53$	1.17830			0.161852			0.0809260	−	0.996720i	$-0.474212\pi$
$53$	1.17830			0.161852			0.0809260	+	0.996720i	$0.474212\pi$
$54$		0			0
$55$	−2.89054			−0.389760
$56$		0			0
$57$			3.28288i			0.434828i
$58$		0			0
$59$		−	11.9234i		−	1.55230i	−0.630548	−	0.776150i	$-0.717170\pi$
$59$		−	11.9234i		−	1.55230i	0.630548	−	0.776150i	$-0.282830\pi$
$60$		0			0
$61$	13.8469			1.77291			0.886456	−	0.462812i	$-0.153160\pi$
$61$	13.8469			1.77291			0.886456	+	0.462812i	$0.153160\pi$
$62$		0			0
$63$			1.32653i			0.167127i
$64$		0			0
$65$	−8.74515	+	5.03291i	−1.08470	+	0.624255i
$66$		0			0
$67$		−	13.7811i		−	1.68363i	−0.539769	−	0.841813i	$-0.681488\pi$
$67$		−	13.7811i		−	1.68363i	0.539769	−	0.841813i	$-0.318512\pi$
$68$		0			0
$69$	4.95635			0.596674
$70$		0			0
$71$		−	3.80432i		−	0.451490i	−0.974186	−	0.225745i	$-0.927518\pi$
$71$		−	3.80432i		−	0.451490i	0.974186	−	0.225745i	$-0.0724816\pi$
$72$		0			0
$73$			3.19080i			0.373455i	0.982412	+	0.186727i	$0.0597881\pi$
$73$			3.19080i			0.373455i	−0.982412	+	0.186727i	$0.940212\pi$
$74$		0			0
$75$	−3.66272			−0.422935
$76$		0			0
$77$	−1.03291			−0.117710
$78$		0			0
$79$	−7.33240			−0.824959			−0.412480	−	0.910967i	$-0.635337\pi$
$79$	−7.33240			−0.824959			−0.412480	+	0.910967i	$0.635337\pi$
$80$		0			0
$81$	−3.26072			−0.362302
$82$		0			0
$83$			16.6260i			1.82494i	0.409140	+	0.912472i	$0.365829\pi$
$83$			16.6260i			1.82494i	−0.409140	+	0.912472i	$0.634171\pi$
$84$		0			0
$85$			19.4670i			2.11150i
$86$		0			0
$87$	−6.59975			−0.707568
$88$		0			0
$89$		−	11.8964i		−	1.26102i	−0.776182	−	0.630508i	$-0.782846\pi$
$89$		−	11.8964i		−	1.26102i	0.776182	−	0.630508i	$-0.217154\pi$
$90$		0			0
$91$	−3.12499	+	1.79846i	−0.327588	+	0.188530i
$92$		0			0
$93$			4.90128i			0.508240i
$94$		0			0
$95$	7.10175			0.728624
$96$		0			0
$97$		−	8.60355i		−	0.873558i	−0.899569	−	0.436779i	$-0.856119\pi$
$97$		−	8.60355i		−	0.873558i	0.899569	−	0.436779i	$-0.143881\pi$
$98$		0			0
$99$			1.37018i			0.137708i
$100$		0			0
$101$	8.36910			0.832756			0.416378	−	0.909192i	$-0.363299\pi$
$101$	8.36910			0.832756			0.416378	+	0.909192i	$0.363299\pi$
$102$		0			0
$103$	−0.521442			−0.0513792			−0.0256896	−	0.999670i	$-0.508178\pi$
$103$	−0.521442			−0.0513792			−0.0256896	+	0.999670i	$0.508178\pi$
$104$		0			0
$105$	−3.62016			−0.353291
$106$		0			0
$107$	−9.43523			−0.912138			−0.456069	−	0.889944i	$-0.650743\pi$
$107$	−9.43523			−0.912138			−0.456069	+	0.889944i	$0.650743\pi$
$108$		0			0
$109$		−	13.6860i		−	1.31088i	−0.755248	−	0.655439i	$-0.772484\pi$
$109$		−	13.6860i		−	1.31088i	0.755248	−	0.655439i	$-0.227516\pi$
$110$		0			0
$111$		−	8.03007i		−	0.762180i
$112$		0			0
$113$	−12.2655			−1.15384			−0.576921	−	0.816800i	$-0.695746\pi$
$113$	−12.2655			−1.15384			−0.576921	+	0.816800i	$0.695746\pi$
$114$		0			0
$115$		−	10.7219i		−	0.999823i
$116$		0			0
$117$	2.38571	+	4.14539i	0.220559	+	0.383242i
$118$		0			0
$119$			6.95635i			0.637688i
$120$		0			0
$121$	9.93311			0.903010
$122$		0			0
$123$			1.28113i			0.115515i
$124$		0			0
$125$		−	6.06884i		−	0.542814i
$126$		0			0
$127$	10.9331			0.970156			0.485078	−	0.874471i	$-0.338791\pi$
$127$	10.9331			0.970156			0.485078	+	0.874471i	$0.338791\pi$
$128$		0			0
$129$	−0.725697			−0.0638941
$130$		0			0
$131$	6.58725			0.575531			0.287765	−	0.957701i	$-0.407088\pi$
$131$	6.58725			0.575531			0.287765	+	0.957701i	$0.407088\pi$
$132$		0			0
$133$	2.53774			0.220050
$134$		0			0
$135$			15.6627i			1.34803i
$136$		0			0
$137$		−	12.8333i		−	1.09642i	−0.836340	−	0.548212i	$-0.815309\pi$
$137$		−	12.8333i		−	1.09642i	0.836340	−	0.548212i	$-0.184691\pi$
$138$		0			0
$139$	18.5998			1.57761			0.788805	−	0.614643i	$-0.210700\pi$
$139$	18.5998			1.57761			0.788805	+	0.614643i	$0.210700\pi$
$140$		0			0
$141$			9.30221i			0.783387i
$142$		0			0
$143$	−3.22782	+	1.85764i	−0.269924	+	0.155343i
$144$		0			0
$145$			14.2770i			1.18564i
$146$		0			0
$147$	−1.29363			−0.106696
$148$		0			0
$149$		−	9.97676i		−	0.817328i	−0.912685	−	0.408664i	$-0.865995\pi$
$149$		−	9.97676i		−	0.817328i	0.912685	−	0.408664i	$-0.134005\pi$
$150$		0			0
$151$		−	14.0543i		−	1.14372i	−0.820350	−	0.571861i	$-0.806222\pi$
$151$		−	14.0543i		−	1.14372i	0.820350	−	0.571861i	$-0.193778\pi$
$152$		0			0
$153$	9.22782			0.746025
$154$		0			0
$155$	10.6028			0.851636
$156$		0			0
$157$	−7.76858			−0.620000			−0.310000	−	0.950737i	$-0.600329\pi$
$157$	−7.76858			−0.620000			−0.310000	+	0.950737i	$0.600329\pi$
$158$		0			0
$159$	1.52428			0.120883
$160$		0			0
$161$		−	3.83136i		−	0.301954i
$162$		0			0
$163$		−	7.39342i		−	0.579098i	−0.957163	−	0.289549i	$-0.906495\pi$
$163$		−	7.39342i		−	0.579098i	0.957163	−	0.289549i	$-0.0935053\pi$
$164$		0			0
$165$	−3.73928			−0.291102
$166$		0			0
$167$		−	15.7122i		−	1.21585i	−0.793995	−	0.607925i	$-0.792002\pi$
$167$		−	15.7122i		−	1.21585i	0.793995	−	0.607925i	$-0.207998\pi$
$168$		0			0
$169$	−6.53110	+	11.2403i	−0.502393	+	0.864640i
$170$		0			0
$171$		−	3.36639i		−	0.257434i
$172$		0			0
$173$	−0.415585			−0.0315963			−0.0157982	−	0.999875i	$-0.505029\pi$
$173$	−0.415585			−0.0315963			−0.0157982	+	0.999875i	$0.505029\pi$
$174$		0			0
$175$			2.83136i			0.214031i
$176$		0			0
$177$		−	15.4245i		−	1.15938i
$178$		0			0
$179$	−3.90792			−0.292091			−0.146046	−	0.989278i	$-0.546655\pi$
$179$	−3.90792			−0.292091			−0.146046	+	0.989278i	$0.546655\pi$
$180$		0			0
$181$	4.79151			0.356150			0.178075	−	0.984017i	$-0.443013\pi$
$181$	4.79151			0.356150			0.178075	+	0.984017i	$0.443013\pi$
$182$		0			0
$183$	17.9127			1.32414
$184$		0			0
$185$	−17.3712			−1.27715
$186$		0			0
$187$			7.18525i			0.525437i
$188$		0			0
$189$			5.59691i			0.407116i
$190$		0			0
$191$	8.62905			0.624376			0.312188	−	0.950020i	$-0.398938\pi$
$191$	8.62905			0.624376			0.312188	+	0.950020i	$0.398938\pi$
$192$		0			0
$193$		−	24.4535i		−	1.76020i	−0.474789	−	0.880100i	$-0.657476\pi$
$193$		−	24.4535i		−	1.76020i	0.474789	−	0.880100i	$-0.342524\pi$
$194$		0			0
$195$	−11.3129	+	6.51070i	−0.810137	+	0.466241i
$196$		0			0
$197$			7.14128i			0.508795i	0.967100	+	0.254398i	$0.0818772\pi$
$197$			7.14128i			0.508795i	−0.967100	+	0.254398i	$0.918123\pi$
$198$		0			0
$199$	1.00284			0.0710892			0.0355446	−	0.999368i	$-0.488683\pi$
$199$	1.00284			0.0710892			0.0355446	+	0.999368i	$0.488683\pi$
$200$		0			0
$201$		−	17.8276i		−	1.25746i
$202$		0			0
$203$			5.10175i			0.358072i
$204$		0			0
$205$	2.77142			0.193564
$206$		0			0
$207$	−5.08242			−0.353253
$208$		0			0
$209$	2.62124			0.181315
$210$		0			0
$211$	−26.2655			−1.80819			−0.904096	−	0.427329i	$-0.859454\pi$
$211$	−26.2655			−1.80819			−0.904096	+	0.427329i	$0.859454\pi$
$212$		0			0
$213$		−	4.92137i		−	0.337207i
$214$		0			0
$215$			1.56988i			0.107065i
$216$		0			0
$217$	3.78880			0.257200
$218$		0			0
$219$			4.12770i			0.278924i
$220$		0			0
$221$	12.5107	+	21.7385i	0.841561	+	1.46229i
$222$		0			0
$223$			17.7540i			1.18890i	0.804133	+	0.594449i	$0.202630\pi$
$223$			17.7540i			1.18890i	−0.804133	+	0.594449i	$0.797370\pi$
$224$		0			0
$225$	3.75589			0.250393
$226$		0			0
$227$			17.8276i			1.18326i	0.806211	+	0.591629i	$0.201515\pi$
$227$			17.8276i			1.18326i	−0.806211	+	0.591629i	$0.798485\pi$
$228$		0			0
$229$			4.21423i			0.278484i	0.990258	+	0.139242i	$0.0444667\pi$
$229$			4.21423i			0.278484i	−0.990258	+	0.139242i	$0.955533\pi$
$230$		0			0
$231$	−1.33619			−0.0879150
$232$		0			0
$233$	−28.2880			−1.85321			−0.926604	−	0.376039i	$-0.877286\pi$
$233$	−28.2880			−1.85321			−0.926604	+	0.376039i	$0.877286\pi$
$234$		0			0
$235$	20.1231			1.31269
$236$		0			0
$237$	−9.48538			−0.616142
$238$		0			0
$239$			8.94385i			0.578530i	0.957249	+	0.289265i	$0.0934108\pi$
$239$			8.94385i			0.578530i	−0.957249	+	0.289265i	$0.906589\pi$
$240$		0			0
$241$			17.0019i			1.09519i	0.836743	+	0.547596i	$0.184457\pi$
$241$			17.0019i			1.09519i	−0.836743	+	0.547596i	$0.815543\pi$
$242$		0			0
$243$	12.5726			0.806532
$244$		0			0
$245$			2.79846i			0.178787i
$246$		0			0
$247$	7.93040	−	4.56401i	0.504599	−	0.290401i
$248$		0			0
$249$			21.5079i			1.36301i
$250$		0			0
$251$	6.70637			0.423303			0.211651	−	0.977345i	$-0.432116\pi$
$251$	6.70637			0.423303			0.211651	+	0.977345i	$0.432116\pi$
$252$		0			0
$253$		−	3.95743i		−	0.248802i
$254$		0			0
$255$			25.1831i			1.57703i
$256$		0			0
$257$	−14.6656			−0.914813			−0.457406	−	0.889258i	$-0.651221\pi$
$257$	−14.6656			−0.914813			−0.457406	+	0.889258i	$0.651221\pi$
$258$		0			0
$259$	−6.20741			−0.385710
$260$		0			0
$261$	6.76762			0.418905
$262$		0			0
$263$	6.02627			0.371596			0.185798	−	0.982588i	$-0.440513\pi$
$263$	6.02627			0.371596			0.185798	+	0.982588i	$0.440513\pi$
$264$		0			0
$265$		−	3.29742i		−	0.202559i
$266$		0			0
$267$		−	15.3895i		−	0.941822i
$268$		0			0
$269$	−0.825495			−0.0503313			−0.0251657	−	0.999683i	$-0.508011\pi$
$269$	−0.825495			−0.0503313			−0.0251657	+	0.999683i	$0.508011\pi$
$270$		0			0
$271$		−	9.82757i		−	0.596982i	−0.954412	−	0.298491i	$-0.903517\pi$
$271$		−	9.82757i		−	0.596982i	0.954412	−	0.298491i	$-0.0964833\pi$
$272$		0			0
$273$	−4.04257	+	2.32653i	−0.244667	+	0.140808i
$274$		0			0
$275$			2.92453i			0.176356i
$276$		0			0
$277$	27.8517			1.67344			0.836722	−	0.547627i	$-0.184469\pi$
$277$	27.8517			1.67344			0.836722	+	0.547627i	$0.184469\pi$
$278$		0			0
$279$		−	5.02596i		−	0.300896i
$280$		0			0
$281$		−	4.04257i		−	0.241159i	−0.992704	−	0.120580i	$-0.961525\pi$
$281$		−	4.04257i		−	0.241159i	0.992704	−	0.120580i	$-0.0384753\pi$
$282$		0			0
$283$	29.7596			1.76902			0.884512	−	0.466517i	$-0.154491\pi$
$283$	29.7596			1.76902			0.884512	+	0.466517i	$0.154491\pi$
$284$		0			0
$285$	9.18700			0.544191
$286$		0			0
$287$	0.990338			0.0584578
$288$		0			0
$289$	31.3908			1.84652
$290$		0			0
$291$		−	11.1298i		−	0.652439i
$292$		0			0
$293$			29.8091i			1.74147i	0.491755	+	0.870733i	$0.336355\pi$
$293$			29.8091i			1.74147i	−0.491755	+	0.870733i	$0.663645\pi$
$294$		0			0
$295$	−33.3673			−1.94272
$296$		0			0
$297$			5.78108i			0.335452i
$298$		0			0
$299$	−6.89054	−	11.9730i	−0.398490	−	0.692414i
$300$		0			0
$301$			0.560979i			0.0323343i
$302$		0			0
$303$	10.8265			0.621965
$304$		0			0
$305$		−	38.7499i		−	2.21881i
$306$		0			0
$307$			18.5928i			1.06115i	0.847639	+	0.530574i	$0.178023\pi$
$307$			18.5928i			1.06115i	−0.847639	+	0.530574i	$0.821977\pi$
$308$		0			0
$309$	−0.674552			−0.0383739
$310$		0			0
$311$	−15.9119			−0.902283			−0.451142	−	0.892452i	$-0.648983\pi$
$311$	−15.9119			−0.902283			−0.451142	+	0.892452i	$0.648983\pi$
$312$		0			0
$313$	−1.79575			−0.101502			−0.0507508	−	0.998711i	$-0.516161\pi$
$313$	−1.79575			−0.101502			−0.0507508	+	0.998711i	$0.516161\pi$
$314$		0			0
$315$	3.71224			0.209161
$316$		0			0
$317$			16.0311i			0.900394i	0.892929	+	0.450197i	$0.148646\pi$
$317$			16.0311i			0.900394i	−0.892929	+	0.450197i	$0.851354\pi$
$318$		0			0
$319$			5.26962i			0.295042i
$320$		0			0
$321$	−12.2057			−0.681253
$322$		0			0
$323$		−	17.6534i		−	0.982260i
$324$		0			0
$325$	5.09208	+	8.84797i	0.282458	+	0.490797i
$326$		0			0
$327$		−	17.7045i		−	0.979063i
$328$		0			0
$329$	7.19080			0.396442
$330$		0			0
$331$			6.81791i			0.374746i	0.982289	+	0.187373i	$0.0599973\pi$
$331$			6.81791i			0.374746i	−0.982289	+	0.187373i	$0.940003\pi$
$332$		0			0
$333$			8.23432i			0.451238i
$334$		0			0
$335$	−38.5658			−2.10707
$336$		0			0
$337$	0.418615			0.0228034			0.0114017	−	0.999935i	$-0.496371\pi$
$337$	0.418615			0.0228034			0.0114017	+	0.999935i	$0.496371\pi$
$338$		0			0
$339$	−15.8670			−0.861776
$340$		0			0
$341$	3.91347			0.211926
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		−	13.8701i		−	0.746743i
$346$		0			0
$347$	10.0658			0.540361			0.270180	−	0.962810i	$-0.412917\pi$
$347$	10.0658			0.540361			0.270180	+	0.962810i	$0.412917\pi$
$348$		0			0
$349$			8.27233i			0.442808i	0.975182	+	0.221404i	$0.0710639\pi$
$349$			8.27233i			0.442808i	−0.975182	+	0.221404i	$0.928936\pi$
$350$		0			0
$351$	10.0658	+	17.4903i	0.537273	+	0.933563i
$352$		0			0
$353$		−	16.3908i		−	0.872395i	−0.899851	−	0.436197i	$-0.856325\pi$
$353$		−	16.3908i		−	0.872395i	0.899851	−	0.436197i	$-0.143675\pi$
$354$		0			0
$355$	−10.6462			−0.565044
$356$		0			0
$357$			8.99892i			0.476273i
$358$		0			0
$359$		−	32.1512i		−	1.69687i	−0.529297	−	0.848437i	$-0.677544\pi$
$359$		−	32.1512i		−	1.69687i	0.529297	−	0.848437i	$-0.322456\pi$
$360$		0			0
$361$	12.5599			0.661047
$362$		0			0
$363$	12.8497			0.674436
$364$		0			0
$365$	8.92931			0.467382
$366$		0			0
$367$	7.79575			0.406935			0.203467	−	0.979082i	$-0.434779\pi$
$367$	7.79575			0.406935			0.203467	+	0.979082i	$0.434779\pi$
$368$		0			0
$369$		−	1.31371i		−	0.0683892i
$370$		0			0
$371$		−	1.17830i		−	0.0611743i
$372$		0			0
$373$	31.7371			1.64329			0.821643	−	0.570003i	$-0.193058\pi$
$373$	31.7371			1.64329			0.821643	+	0.570003i	$0.193058\pi$
$374$		0			0
$375$		−	7.85081i		−	0.405414i
$376$		0			0
$377$	9.17527	+	15.9429i	0.472550	+	0.821101i
$378$		0			0
$379$			30.3236i			1.55762i	0.627261	+	0.778809i	$0.284176\pi$
$379$			30.3236i			1.55762i	−0.627261	+	0.778809i	$0.715824\pi$
$380$		0			0
$381$	14.1434			0.724586
$382$		0			0
$383$			14.0193i			0.716354i	0.933654	+	0.358177i	$0.116602\pi$
$383$			14.0193i			0.716354i	−0.933654	+	0.358177i	$0.883398\pi$
$384$		0			0
$385$			2.89054i			0.147316i
$386$		0			0
$387$	0.744156			0.0378276
$388$		0			0
$389$	19.8887			1.00840			0.504198	−	0.863588i	$-0.331788\pi$
$389$	19.8887			1.00840			0.504198	+	0.863588i	$0.331788\pi$
$390$		0			0
$391$	−26.6523			−1.34786
$392$		0			0
$393$	8.52144			0.429850
$394$		0			0
$395$			20.5194i			1.03244i
$396$		0			0
$397$		−	14.7219i		−	0.738871i	−0.929256	−	0.369436i	$-0.879551\pi$
$397$		−	14.7219i		−	0.738871i	0.929256	−	0.369436i	$-0.120449\pi$
$398$		0			0
$399$	3.28288			0.164350
$400$		0			0
$401$			19.6412i			0.980836i	0.871487	+	0.490418i	$0.163156\pi$
$401$			19.6412i			0.980836i	−0.871487	+	0.490418i	$0.836844\pi$
$402$		0			0
$403$	11.8399	−	6.81399i	0.589789	−	0.339429i
$404$		0			0
$405$			9.12499i			0.453424i
$406$		0			0
$407$	−6.41166			−0.317814
$408$		0			0
$409$			24.1132i			1.19232i	0.802866	+	0.596160i	$0.203307\pi$
$409$			24.1132i			1.19232i	−0.802866	+	0.596160i	$0.796693\pi$
$410$		0			0
$411$		−	16.6015i		−	0.818892i
$412$		0			0
$413$	−11.9234			−0.586714
$414$		0			0
$415$	46.5272			2.28393
$416$		0			0
$417$	24.0611			1.17828
$418$		0			0
$419$	19.7818			0.966406			0.483203	−	0.875508i	$-0.339473\pi$
$419$	19.7818			0.966406			0.483203	+	0.875508i	$0.339473\pi$
$420$		0			0
$421$		−	19.1237i		−	0.932033i	−0.884776	−	0.466016i	$-0.845689\pi$
$421$		−	19.1237i		−	0.932033i	0.884776	−	0.466016i	$-0.154311\pi$
$422$		0			0
$423$		−	9.53882i		−	0.463793i
$424$		0			0
$425$	19.6959			0.955394
$426$		0			0
$427$		−	13.8469i		−	0.670098i
$428$		0			0
$429$	−4.17559	+	2.40309i	−0.201599	+	0.116022i
$430$		0			0
$431$		−	8.87588i		−	0.427536i	−0.976884	−	0.213768i	$-0.931426\pi$
$431$		−	8.87588i		−	0.427536i	0.976884	−	0.213768i	$-0.0685736\pi$
$432$		0			0
$433$	−36.9341			−1.77494			−0.887470	−	0.460866i	$-0.847539\pi$
$433$	−36.9341			−1.77494			−0.887470	+	0.460866i	$0.847539\pi$
$434$		0			0
$435$			18.4691i			0.885527i
$436$		0			0
$437$			9.72299i			0.465113i
$438$		0			0
$439$	20.6591			0.986006			0.493003	−	0.870028i	$-0.335899\pi$
$439$	20.6591			0.986006			0.493003	+	0.870028i	$0.335899\pi$
$440$		0			0
$441$	1.32653			0.0631682
$442$		0			0
$443$	−2.85492			−0.135641			−0.0678207	−	0.997698i	$-0.521605\pi$
$443$	−2.85492			−0.135641			−0.0678207	+	0.997698i	$0.521605\pi$
$444$		0			0
$445$	−33.2916			−1.57817
$446$		0			0
$447$		−	12.9062i		−	0.610442i
$448$		0			0
$449$		−	30.0443i		−	1.41788i	−0.705269	−	0.708940i	$-0.749174\pi$
$449$		−	30.0443i		−	1.41788i	0.705269	−	0.708940i	$-0.250826\pi$
$450$		0			0
$451$	1.02293			0.0481677
$452$		0			0
$453$		−	18.1810i		−	0.854218i
$454$		0			0
$455$	5.03291	+	8.74515i	0.235946	+	0.409979i
$456$		0			0
$457$			28.6491i			1.34015i	0.742293	+	0.670075i	$0.233738\pi$
$457$			28.6491i			1.34015i	−0.742293	+	0.670075i	$0.766262\pi$
$458$		0			0
$459$	38.9341			1.81729
$460$		0			0
$461$			0.922363i			0.0429587i	0.999769	+	0.0214794i	$0.00683762\pi$
$461$			0.922363i			0.0429587i	−0.999769	+	0.0214794i	$0.993162\pi$
$462$		0			0
$463$			26.0929i			1.21264i	0.795220	+	0.606321i	$0.207355\pi$
$463$			26.0929i			1.21264i	−0.795220	+	0.606321i	$0.792645\pi$
$464$		0			0
$465$	13.7160			0.636066
$466$		0			0
$467$	−29.5553			−1.36766			−0.683829	−	0.729642i	$-0.739687\pi$
$467$	−29.5553			−1.36766			−0.683829	+	0.729642i	$0.739687\pi$
$468$		0			0
$469$	−13.7811			−0.636351
$470$		0			0
$471$	−10.0496			−0.463063
$472$		0			0
$473$			0.579438i			0.0266426i
$474$		0			0
$475$		−	7.18525i		−	0.329682i
$476$		0			0
$477$	−1.56305			−0.0715672
$478$		0			0
$479$		−	15.8954i		−	0.726280i	−0.931735	−	0.363140i	$-0.881705\pi$
$479$		−	15.8954i		−	0.726280i	0.931735	−	0.363140i	$-0.118295\pi$
$480$		0			0
$481$	−19.3981	+	11.1638i	−0.884476	+	0.509024i
$482$		0			0
$483$		−	4.95635i		−	0.225522i
$484$		0			0
$485$	−24.0767			−1.09326
$486$		0			0
$487$			0.0851336i			0.00385777i	0.999998	+	0.00192889i	$0.000613984\pi$
$487$			0.0851336i			0.00385777i	−0.999998	+	0.00192889i	$0.999386\pi$
$488$		0			0
$489$		−	9.56433i		−	0.432514i
$490$		0			0
$491$	−2.56324			−0.115678			−0.0578388	−	0.998326i	$-0.518421\pi$
$491$	−2.56324			−0.115678			−0.0578388	+	0.998326i	$0.518421\pi$
$492$		0			0
$493$	35.4895			1.59837
$494$		0			0
$495$	3.83439			0.172343
$496$		0			0
$497$	−3.80432			−0.170647
$498$		0			0
$499$			28.6941i			1.28452i	0.766485	+	0.642262i	$0.222004\pi$
$499$			28.6941i			1.28452i	−0.766485	+	0.642262i	$0.777996\pi$
$500$		0			0
$501$		−	20.3258i		−	0.908088i
$502$		0			0
$503$	32.7056			1.45827			0.729136	−	0.684369i	$-0.239922\pi$
$503$	32.7056			1.45827			0.729136	+	0.684369i	$0.239922\pi$
$504$		0			0
$505$		−	23.4206i		−	1.04220i
$506$		0			0
$507$	−8.44881	+	14.5408i	−0.375225	+	0.645778i
$508$		0			0
$509$		−	26.0938i		−	1.15659i	−0.815828	−	0.578295i	$-0.803718\pi$
$509$		−	26.0938i		−	1.15659i	0.815828	−	0.578295i	$-0.196282\pi$
$510$		0			0
$511$	3.19080			0.141153
$512$		0			0
$513$		−	14.2035i		−	0.627099i
$514$		0			0
$515$			1.45923i			0.0643015i
$516$		0			0
$517$	7.42741			0.326657
$518$		0			0
$519$	−0.537611			−0.0235985
$520$		0			0
$521$	−1.75570			−0.0769185			−0.0384592	−	0.999260i	$-0.512245\pi$
$521$	−1.75570			−0.0769185			−0.0384592	+	0.999260i	$0.512245\pi$
$522$		0			0
$523$	32.3779			1.41579			0.707893	−	0.706319i	$-0.249646\pi$
$523$	32.3779			1.41579			0.707893	+	0.706319i	$0.249646\pi$
$524$		0			0
$525$			3.66272i			0.159854i
$526$		0			0
$527$		−	26.3562i		−	1.14809i
$528$		0			0
$529$	−8.32066			−0.361768
$530$		0			0
$531$			15.8168i			0.686392i
$532$		0			0
$533$	3.09480	−	1.78108i	0.134050	−	0.0771472i
$534$		0			0
$535$			26.4041i			1.14155i
$536$		0			0
$537$	−5.05538			−0.218156
$538$		0			0
$539$			1.03291i			0.0444904i
$540$		0			0
$541$			8.52536i			0.366534i	0.983063	+	0.183267i	$0.0586673\pi$
$541$			8.52536i			0.366534i	−0.983063	+	0.183267i	$0.941333\pi$
$542$		0			0
$543$	6.19842			0.266000
$544$		0			0
$545$	−38.2996			−1.64057
$546$		0			0
$547$	−1.14355			−0.0488945			−0.0244473	−	0.999701i	$-0.507783\pi$
$547$	−1.14355			−0.0488945			−0.0244473	+	0.999701i	$0.507783\pi$
$548$		0			0
$549$	−18.3683			−0.783941
$550$		0			0
$551$		−	12.9469i		−	0.551556i
$552$		0			0
$553$			7.33240i			0.311805i
$554$		0			0
$555$	−22.4718			−0.953875
$556$		0			0
$557$		−	42.8275i		−	1.81466i	−0.420421	−	0.907329i	$-0.638118\pi$
$557$		−	42.8275i		−	1.81466i	0.420421	−	0.907329i	$-0.361882\pi$
$558$		0			0
$559$	1.00890	+	1.75305i	0.0426718	+	0.0741463i
$560$		0			0
$561$			9.29503i			0.392436i
$562$		0			0
$563$	−13.5754			−0.572136			−0.286068	−	0.958209i	$-0.592348\pi$
$563$	−13.5754			−0.572136			−0.286068	+	0.958209i	$0.592348\pi$
$564$		0			0
$565$			34.3245i			1.44404i
$566$		0			0
$567$			3.26072i			0.136937i
$568$		0			0
$569$	−8.87109			−0.371896			−0.185948	−	0.982560i	$-0.559536\pi$
$569$	−8.87109			−0.371896			−0.185948	+	0.982560i	$0.559536\pi$
$570$		0			0
$571$	−6.28093			−0.262849			−0.131424	−	0.991326i	$-0.541955\pi$
$571$	−6.28093			−0.262849			−0.131424	+	0.991326i	$0.541955\pi$
$572$		0			0
$573$	11.1628			0.466331
$574$		0			0
$575$	−10.8480			−0.452392
$576$		0			0
$577$			25.1005i			1.04495i	0.852656	+	0.522473i	$0.174990\pi$
$577$			25.1005i			1.04495i	−0.852656	+	0.522473i	$0.825010\pi$
$578$		0			0
$579$		−	31.6336i		−	1.31465i
$580$		0			0
$581$	16.6260			0.689764
$582$		0			0
$583$		−	1.21707i		−	0.0504060i
$584$		0			0
$585$	11.6007	−	6.67631i	0.479630	−	0.276031i
$586$		0			0
$587$			28.2963i			1.16791i	0.811784	+	0.583957i	$0.198497\pi$
$587$			28.2963i			1.16791i	−0.811784	+	0.583957i	$0.801503\pi$
$588$		0			0
$589$	−9.61496			−0.396178
$590$		0			0
$591$			9.23815i			0.380007i
$592$		0			0
$593$			12.8643i			0.528272i	0.964485	+	0.264136i	$0.0850868\pi$
$593$			12.8643i			0.528272i	−0.964485	+	0.264136i	$0.914913\pi$
$594$		0			0
$595$	19.4670			0.798071
$596$		0			0
$597$	1.29730			0.0530948
$598$		0			0
$599$	−25.5685			−1.04470			−0.522350	−	0.852731i	$-0.674944\pi$
$599$	−25.5685			−1.04470			−0.522350	+	0.852731i	$0.674944\pi$
$600$		0			0
$601$	6.63763			0.270755			0.135377	−	0.990794i	$-0.456775\pi$
$601$	6.63763			0.270755			0.135377	+	0.990794i	$0.456775\pi$
$602$		0			0
$603$			18.2810i			0.744461i
$604$		0			0
$605$		−	27.7974i		−	1.13012i
$606$		0			0
$607$	−22.3226			−0.906047			−0.453023	−	0.891499i	$-0.649655\pi$
$607$	−22.3226			−0.906047			−0.453023	+	0.891499i	$0.649655\pi$
$608$		0			0
$609$			6.59975i			0.267435i
$610$		0			0
$611$	22.4712	−	12.9323i	0.909086	−	0.523187i
$612$		0			0
$613$		−	4.23671i		−	0.171119i	−0.996333	−	0.0855596i	$-0.972732\pi$
$613$		−	4.23671i		−	0.171119i	0.996333	−	0.0855596i	$-0.0272678\pi$
$614$		0			0
$615$	3.58518			0.144568
$616$		0			0
$617$		−	14.0193i		−	0.564397i	−0.959356	−	0.282198i	$-0.908936\pi$
$617$		−	14.0193i		−	0.564397i	0.959356	−	0.282198i	$-0.0910636\pi$
$618$		0			0
$619$			23.8469i			0.958487i	0.877682	+	0.479244i	$0.159089\pi$
$619$			23.8469i			0.958487i	−0.877682	+	0.479244i	$0.840911\pi$
$620$		0			0
$621$	−21.4438			−0.860510
$622$		0			0
$623$	−11.8964			−0.476619
$624$		0			0
$625$	−31.1402			−1.24561
$626$		0			0
$627$	3.39090			0.135420
$628$		0			0
$629$			43.1809i			1.72174i
$630$		0			0
$631$			7.18624i			0.286080i	0.989717	+	0.143040i	$0.0456877\pi$
$631$			7.18624i			0.286080i	−0.989717	+	0.143040i	$0.954312\pi$
$632$		0			0
$633$	−33.9777			−1.35049
$634$		0			0
$635$		−	30.5958i		−	1.21416i
$636$		0			0
$637$	1.79846	+	3.12499i	0.0712575	+	0.123817i
$638$		0			0
$639$			5.04655i			0.199639i
$640$		0			0
$641$	−8.05886			−0.318306			−0.159153	−	0.987254i	$-0.550876\pi$
$641$	−8.05886			−0.318306			−0.159153	+	0.987254i	$0.550876\pi$
$642$		0			0
$643$		−	28.4148i		−	1.12057i	−0.828300	−	0.560286i	$-0.810691\pi$
$643$		−	28.4148i		−	1.12057i	0.828300	−	0.560286i	$-0.189309\pi$
$644$		0			0
$645$			2.03083i			0.0799640i
$646$		0			0
$647$	−37.6791			−1.48132			−0.740659	−	0.671881i	$-0.765487\pi$
$647$	−37.6791			−1.48132			−0.740659	+	0.671881i	$0.765487\pi$
$648$		0			0
$649$	−12.3158			−0.483437
$650$		0			0
$651$	4.90128			0.192097
$652$		0			0
$653$	37.0486			1.44982			0.724911	−	0.688842i	$-0.241881\pi$
$653$	37.0486			1.44982			0.724911	+	0.688842i	$0.241881\pi$
$654$		0			0
$655$		−	18.4341i		−	0.720282i
$656$		0			0
$657$		−	4.23269i		−	0.165133i
$658$		0			0
$659$	−44.4553			−1.73173			−0.865867	−	0.500274i	$-0.833233\pi$
$659$	−44.4553			−1.73173			−0.865867	+	0.500274i	$0.833233\pi$
$660$		0			0
$661$		−	40.3413i		−	1.56909i	−0.620069	−	0.784547i	$-0.712895\pi$
$661$		−	40.3413i		−	1.56909i	0.620069	−	0.784547i	$-0.287105\pi$
$662$		0			0
$663$	16.1842	+	28.1215i	0.628541	+	1.09215i
$664$		0			0
$665$		−	7.10175i		−	0.275394i
$666$		0			0
$667$	−19.5466			−0.756849
$668$		0			0
$669$			22.9671i			0.887959i
$670$		0			0
$671$		−	14.3025i		−	0.552143i
$672$		0			0
$673$	−16.8024			−0.647684			−0.323842	−	0.946111i	$-0.604975\pi$
$673$	−16.8024			−0.647684			−0.323842	+	0.946111i	$0.604975\pi$
$674$		0			0
$675$	15.8469			0.609947
$676$		0			0
$677$	−46.5650			−1.78964			−0.894819	−	0.446429i	$-0.852696\pi$
$677$	−46.5650			−1.78964			−0.894819	+	0.446429i	$0.852696\pi$
$678$		0			0
$679$	−8.60355			−0.330174
$680$		0			0
$681$			23.0622i			0.883746i
$682$		0			0
$683$		−	5.79498i		−	0.221739i	−0.993835	−	0.110869i	$-0.964636\pi$
$683$		−	5.79498i		−	0.221739i	0.993835	−	0.110869i	$-0.0353635\pi$
$684$		0			0
$685$	−35.9135			−1.37218
$686$		0			0
$687$			5.45164i			0.207993i
$688$		0			0
$689$	−2.11912	−	3.68217i	−0.0807321	−	0.140280i
$690$		0			0
$691$		−	27.1213i		−	1.03174i	−0.856666	−	0.515871i	$-0.827468\pi$
$691$		−	27.1213i		−	1.03174i	0.856666	−	0.515871i	$-0.172532\pi$
$692$		0			0
$693$	1.37018			0.0520489
$694$		0			0
$695$		−	52.0506i		−	1.97439i
$696$		0			0
$697$		−	6.88914i		−	0.260945i
$698$		0			0
$699$	−36.5941			−1.38412
$700$		0			0
$701$	11.0283			0.416535			0.208267	−	0.978072i	$-0.433218\pi$
$701$	11.0283			0.416535			0.208267	+	0.978072i	$0.433218\pi$
$702$		0			0
$703$	15.7528			0.594127
$704$		0			0
$705$	26.0318			0.980415
$706$		0			0
$707$		−	8.36910i		−	0.314752i
$708$		0			0
$709$		−	49.7496i		−	1.86839i	−0.356768	−	0.934193i	$-0.616121\pi$
$709$		−	49.7496i		−	1.86839i	0.356768	−	0.934193i	$-0.383879\pi$
$710$		0			0
$711$	9.72666			0.364778
$712$		0			0
$713$			14.5162i			0.543638i
$714$		0			0
$715$	5.19851	+	9.03291i	0.194413	+	0.337811i
$716$		0			0
$717$			11.5700i			0.432090i
$718$		0			0
$719$	−28.5650			−1.06529			−0.532647	−	0.846337i	$-0.678803\pi$
$719$	−28.5650			−1.06529			−0.532647	+	0.846337i	$0.678803\pi$
$720$		0			0
$721$			0.521442i			0.0194195i
$722$		0			0
$723$			21.9942i			0.817972i
$724$		0			0
$725$	14.4449			0.536470
$726$		0			0
$727$	−36.5250			−1.35464			−0.677318	−	0.735691i	$-0.736858\pi$
$727$	−36.5250			−1.35464			−0.677318	+	0.735691i	$0.736858\pi$
$728$		0			0
$729$	26.0464			0.964681
$730$		0			0
$731$	3.90237			0.144334
$732$		0			0
$733$		−	18.7444i		−	0.692340i	−0.938172	−	0.346170i	$-0.887482\pi$
$733$		−	18.7444i		−	0.692340i	0.938172	−	0.346170i	$-0.112518\pi$
$734$		0			0
$735$			3.62016i			0.133532i
$736$		0			0
$737$	−14.2345			−0.524336
$738$		0			0
$739$		−	11.1838i		−	0.411405i	−0.978615	−	0.205702i	$-0.934052\pi$
$739$		−	11.1838i		−	0.411405i	0.978615	−	0.205702i	$-0.0659478\pi$
$740$		0			0
$741$	10.2590	−	5.90412i	0.376873	−	0.216893i
$742$		0			0
$743$			32.7986i			1.20326i	0.798774	+	0.601632i	$0.205482\pi$
$743$			32.7986i			1.20326i	−0.798774	+	0.601632i	$0.794518\pi$
$744$		0			0
$745$	−27.9195			−1.02289
$746$		0			0
$747$		−	22.0549i		−	0.806948i
$748$		0			0
$749$			9.43523i			0.344756i
$750$		0			0
$751$	28.7451			1.04893			0.524463	−	0.851433i	$-0.324266\pi$
$751$	28.7451			1.04893			0.524463	+	0.851433i	$0.324266\pi$
$752$		0			0
$753$	8.67554			0.316154
$754$		0			0
$755$	−39.3304			−1.43138
$756$		0			0
$757$	12.4952			0.454145			0.227072	−	0.973878i	$-0.427085\pi$
$757$	12.4952			0.454145			0.227072	+	0.973878i	$0.427085\pi$
$758$		0			0
$759$		−	5.11944i		−	0.185824i
$760$		0			0
$761$		−	29.6628i		−	1.07528i	−0.843175	−	0.537639i	$-0.819316\pi$
$761$		−	29.6628i		−	1.07528i	0.843175	−	0.537639i	$-0.180684\pi$
$762$		0			0
$763$	−13.6860			−0.495465
$764$		0			0
$765$		−	25.8236i		−	0.933656i
$766$		0			0
$767$	−37.2606	+	21.4438i	−1.34540	+	0.774291i
$768$		0			0
$769$			40.8488i			1.47305i	0.676412	+	0.736523i	$0.263534\pi$
$769$			40.8488i			1.47305i	−0.676412	+	0.736523i	$0.736466\pi$
$770$		0			0
$771$	−18.9718			−0.683251
$772$		0			0
$773$		−	19.4045i		−	0.697932i	−0.937135	−	0.348966i	$-0.886533\pi$
$773$		−	19.4045i		−	0.697932i	0.937135	−	0.348966i	$-0.113467\pi$
$774$		0			0
$775$		−	10.7275i		−	0.385341i
$776$		0			0
$777$	−8.03007			−0.288077
$778$		0			0
$779$	−2.51322			−0.0900453
$780$		0			0
$781$	−3.92951			−0.140609
$782$		0			0
$783$	28.5540			1.02044
$784$		0			0
$785$			21.7400i			0.775935i
$786$		0			0
$787$		−	10.0913i		−	0.359717i	−0.983693	−	0.179858i	$-0.942436\pi$
$787$		−	10.0913i		−	0.359717i	0.983693	−	0.179858i	$-0.0575639\pi$
$788$		0			0
$789$	7.79575			0.277536
$790$		0			0
$791$			12.2655i			0.436111i
$792$		0			0
$793$	−24.9030	−	43.2714i	−0.884333	−	1.53661i
$794$		0			0
$795$		−	4.26563i		−	0.151286i
$796$		0			0
$797$	−11.2735			−0.399329			−0.199665	−	0.979864i	$-0.563985\pi$
$797$	−11.2735			−0.399329			−0.199665	+	0.979864i	$0.563985\pi$
$798$		0			0
$799$		−	50.0217i		−	1.76964i
$800$		0			0
$801$			15.7810i			0.557593i
$802$		0			0
$803$	3.29579			0.116306
$804$		0			0
$805$	−10.7219			−0.377897
$806$		0			0
$807$	−1.06788			−0.0375912
$808$		0			0
$809$	17.4236			0.612581			0.306290	−	0.951938i	$-0.400912\pi$
$809$	17.4236			0.612581			0.306290	+	0.951938i	$0.400912\pi$
$810$		0			0
$811$			35.9555i			1.26257i	0.775552	+	0.631284i	$0.217472\pi$
$811$			35.9555i			1.26257i	−0.775552	+	0.631284i	$0.782528\pi$
$812$		0			0
$813$		−	12.7132i		−	0.445871i
$814$		0			0
$815$	−20.6902			−0.724745
$816$		0			0
$817$		−	1.42362i		−	0.0498061i
$818$		0			0
$819$	4.14539	−	2.38571i	0.144852	−	0.0833634i
$820$		0			0
$821$			15.1183i			0.527631i	0.964573	+	0.263816i	$0.0849810\pi$
$821$			15.1183i			0.527631i	−0.964573	+	0.263816i	$0.915019\pi$
$822$		0			0
$823$	−43.2335			−1.50702			−0.753512	−	0.657434i	$-0.771642\pi$
$823$	−43.2335			−1.50702			−0.753512	+	0.657434i	$0.771642\pi$
$824$		0			0
$825$			3.78325i			0.131716i
$826$		0			0
$827$		−	50.4513i		−	1.75436i	−0.480158	−	0.877182i	$-0.659421\pi$
$827$		−	50.4513i		−	1.75436i	0.480158	−	0.877182i	$-0.340579\pi$
$828$		0			0
$829$	1.79727			0.0624219			0.0312110	−	0.999513i	$-0.490064\pi$
$829$	1.79727			0.0624219			0.0312110	+	0.999513i	$0.490064\pi$
$830$		0			0
$831$	36.0297			1.24985
$832$		0			0
$833$	6.95635			0.241023
$834$		0			0
$835$	−43.9700			−1.52165
$836$		0			0
$837$		−	21.2056i		−	0.732971i
$838$		0			0
$839$			4.72104i			0.162988i	0.996674	+	0.0814942i	$0.0259692\pi$
$839$			4.72104i			0.162988i	−0.996674	+	0.0814942i	$0.974031\pi$
$840$		0			0
$841$	−2.97220			−0.102490
$842$		0			0
$843$		−	5.22957i		−	0.180116i
$844$		0			0
$845$	31.4555	+	18.2770i	1.08210	+	0.628748i
$846$		0			0
$847$		−	9.93311i		−	0.341306i
$848$		0			0
$849$	38.4978			1.32124
$850$		0			0
$851$		−	23.7828i		−	0.815265i
$852$		0			0
$853$			48.8428i			1.67234i	0.548467	+	0.836172i	$0.315212\pi$
$853$			48.8428i			1.67234i	−0.548467	+	0.836172i	$0.684788\pi$
$854$		0			0
$855$	−9.42069			−0.322181
$856$		0			0
$857$	14.9964			0.512267			0.256134	−	0.966641i	$-0.417551\pi$
$857$	14.9964			0.512267			0.256134	+	0.966641i	$0.417551\pi$
$858$		0			0
$859$	33.0464			1.12753			0.563764	−	0.825936i	$-0.309353\pi$
$859$	33.0464			1.12753			0.563764	+	0.825936i	$0.309353\pi$
$860$		0			0
$861$	1.28113			0.0436607
$862$		0			0
$863$		−	47.1122i		−	1.60372i	−0.597513	−	0.801859i	$-0.703844\pi$
$863$		−	47.1122i		−	1.60372i	0.597513	−	0.801859i	$-0.296156\pi$
$864$		0			0
$865$			1.16300i			0.0395430i
$866$		0			0
$867$	40.6080			1.37912
$868$		0			0
$869$			7.57367i			0.256919i
$870$		0			0
$871$	−43.0657	+	24.7847i	−1.45923	+	0.839797i
$872$		0			0
$873$			11.4129i			0.386267i
$874$		0			0
$875$	−6.06884			−0.205164
$876$		0			0
$877$		−	6.74601i		−	0.227797i	−0.993492	−	0.113898i	$-0.963666\pi$
$877$		−	6.74601i		−	0.227797i	0.993492	−	0.113898i	$-0.0363338\pi$
$878$		0			0
$879$			38.5618i			1.30066i
$880$		0			0
$881$	22.5679			0.760333			0.380166	−	0.924918i	$-0.375867\pi$
$881$	22.5679			0.760333			0.380166	+	0.924918i	$0.375867\pi$
$882$		0			0
$883$	19.2628			0.648245			0.324122	−	0.946015i	$-0.394931\pi$
$883$	19.2628			0.648245			0.324122	+	0.946015i	$0.394931\pi$
$884$		0			0
$885$	−43.1647			−1.45097
$886$		0			0
$887$	0.852658			0.0286295			0.0143147	−	0.999898i	$-0.495443\pi$
$887$	0.852658			0.0286295			0.0143147	+	0.999898i	$0.495443\pi$
$888$		0			0
$889$		−	10.9331i		−	0.366685i
$890$		0			0
$891$			3.36802i			0.112833i
$892$		0			0
$893$	−18.2483			−0.610658
$894$		0			0
$895$			10.9361i			0.365555i
$896$		0			0
$897$	−8.91378	−	15.4885i	−0.297623	−	0.517147i
$898$		0			0
$899$		−	19.3295i		−	0.644674i
$900$		0			0
$901$	−8.19667			−0.273071
$902$		0			0
$903$			0.725697i			0.0241497i
$904$		0			0
$905$		−	13.4088i		−	0.445725i
$906$		0			0
$907$	−46.7594			−1.55262			−0.776310	−	0.630351i	$-0.782911\pi$
$907$	−46.7594			−1.55262			−0.776310	+	0.630351i	$0.782911\pi$
$908$		0			0
$909$	−11.1019			−0.368226
$910$		0			0
$911$	33.7415			1.11791			0.558954	−	0.829199i	$-0.311203\pi$
$911$	33.7415			1.11791			0.558954	+	0.829199i	$0.311203\pi$
$912$		0			0
$913$	17.1731			0.568347
$914$		0			0
$915$		−	50.1279i		−	1.65718i
$916$		0			0
$917$		−	6.58725i		−	0.217530i
$918$		0			0
$919$	23.4340			0.773018			0.386509	−	0.922286i	$-0.373681\pi$
$919$	23.4340			0.773018			0.386509	+	0.922286i	$0.373681\pi$
$920$		0			0
$921$			24.0521i			0.792545i
$922$		0			0
$923$	−11.8885	+	6.84191i	−0.391314	+	0.225204i
$924$		0			0
$925$			17.5754i			0.577876i
$926$		0			0
$927$	0.691710			0.0227187
$928$		0			0
$929$			10.9826i			0.360328i	0.983637	+	0.180164i	$0.0576629\pi$
$929$			10.9826i			0.360328i	−0.983637	+	0.180164i	$0.942337\pi$
$930$		0			0
$931$		−	2.53774i		−	0.0831709i
$932$		0			0
$933$	−20.5841			−0.673893
$934$		0			0
$935$	20.1076			0.657589
$936$		0			0
$937$	47.5028			1.55185			0.775924	−	0.630826i	$-0.217284\pi$
$937$	47.5028			1.55185			0.775924	+	0.630826i	$0.217284\pi$
$938$		0			0
$939$	−2.32302			−0.0758090
$940$		0			0
$941$		−	8.08915i		−	0.263699i	−0.991270	−	0.131849i	$-0.957908\pi$
$941$		−	8.08915i		−	0.263699i	0.991270	−	0.131849i	$-0.0420915\pi$
$942$		0			0
$943$			3.79434i			0.123561i
$944$		0			0
$945$	15.6627			0.509508
$946$		0			0
$947$		−	18.7265i		−	0.608528i	−0.952588	−	0.304264i	$-0.901589\pi$
$947$		−	18.7265i		−	0.608528i	0.952588	−	0.304264i	$-0.0984105\pi$
$948$		0			0
$949$	9.97121	−	5.73851i	0.323679	−	0.186280i
$950$		0			0
$951$			20.7382i			0.672482i
$952$		0			0
$953$	47.2451			1.53042			0.765209	−	0.643781i	$-0.222635\pi$
$953$	47.2451			1.53042			0.765209	+	0.643781i	$0.222635\pi$
$954$		0			0
$955$		−	24.1480i		−	0.781412i
$956$		0			0
$957$			6.81692i			0.220360i
$958$		0			0
$959$	−12.8333			−0.414409
$960$		0			0
$961$	16.6450			0.536936
$962$		0			0
$963$	12.5161			0.403326
$964$		0			0
$965$	−68.4320			−2.20290
$966$		0			0
$967$		−	39.7889i		−	1.27952i	−0.768573	−	0.639762i	$-0.779033\pi$
$967$		−	39.7889i		−	1.27952i	0.768573	−	0.639762i	$-0.220967\pi$
$968$		0			0
$969$		−	22.8369i		−	0.733626i
$970$		0			0
$971$	−10.0133			−0.321341			−0.160670	−	0.987008i	$-0.551366\pi$
$971$	−10.0133			−0.321341			−0.160670	+	0.987008i	$0.551366\pi$
$972$		0			0
$973$		−	18.5998i		−	0.596281i
$974$		0			0
$975$	6.58725	+	11.4460i	0.210961	+	0.366564i
$976$		0			0
$977$		−	58.9087i		−	1.88466i	−0.334691	−	0.942328i	$-0.608632\pi$
$977$		−	58.9087i		−	1.88466i	0.334691	−	0.942328i	$-0.391368\pi$
$978$		0			0
$979$	−12.2879			−0.392722
$980$		0			0
$981$			18.1549i			0.579640i
$982$		0			0
$983$			5.15605i			0.164452i	0.996614	+	0.0822262i	$0.0262030\pi$
$983$			5.15605i			0.164452i	−0.996614	+	0.0822262i	$0.973797\pi$
$984$		0			0
$985$	19.9846			0.636761
$986$		0			0
$987$	9.30221			0.296092
$988$		0			0
$989$	−2.14931			−0.0683442
$990$		0			0
$991$	62.6905			1.99143			0.995715	−	0.0924728i	$-0.0294771\pi$
$991$	62.6905			1.99143			0.995715	+	0.0924728i	$0.0294771\pi$
$992$		0			0
$993$			8.81982i			0.279889i
$994$		0			0
$995$		−	2.80640i		−	0.0889687i
$996$		0			0
$997$	−14.1377			−0.447745			−0.223872	−	0.974618i	$-0.571870\pi$
$997$	−14.1377			−0.447745			−0.223872	+	0.974618i	$0.571870\pi$
$998$		0			0
$999$			34.7423i			1.09920i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.2.k.d.337.5		8
4.3	odd	2		364.2.g.a.337.3	✓	8
12.11	even	2		3276.2.e.f.2521.8		8
13.12	even	2	inner	1456.2.k.d.337.6		8
28.3	even	6		2548.2.y.e.961.3		16
28.11	odd	6		2548.2.y.f.961.6		16
28.19	even	6		2548.2.y.e.753.4		16
28.23	odd	6		2548.2.y.f.753.5		16
28.27	even	2		2548.2.g.g.2157.6		8
52.31	even	4		4732.2.a.o.1.2		4
52.47	even	4		4732.2.a.p.1.2		4
52.51	odd	2		364.2.g.a.337.4	yes	8
156.155	even	2		3276.2.e.f.2521.1		8
364.51	odd	6		2548.2.y.f.753.6		16
364.103	even	6		2548.2.y.e.753.3		16
364.207	odd	6		2548.2.y.f.961.5		16
364.311	even	6		2548.2.y.e.961.4		16
364.363	even	2		2548.2.g.g.2157.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.g.a.337.3	✓	8	4.3	odd	2
364.2.g.a.337.4	yes	8	52.51	odd	2
1456.2.k.d.337.5		8	1.1	even	1	trivial
1456.2.k.d.337.6		8	13.12	even	2	inner
2548.2.g.g.2157.5		8	364.363	even	2
2548.2.g.g.2157.6		8	28.27	even	2
2548.2.y.e.753.3		16	364.103	even	6
2548.2.y.e.753.4		16	28.19	even	6
2548.2.y.e.961.3		16	28.3	even	6
2548.2.y.e.961.4		16	364.311	even	6
2548.2.y.f.753.5		16	28.23	odd	6
2548.2.y.f.753.6		16	364.51	odd	6
2548.2.y.f.961.5		16	364.207	odd	6
2548.2.y.f.961.6		16	28.11	odd	6
3276.2.e.f.2521.1		8	156.155	even	2
3276.2.e.f.2521.8		8	12.11	even	2
4732.2.a.o.1.2		4	52.31	even	4
4732.2.a.p.1.2		4	52.47	even	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:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1456.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.29363 q^{3} -2.79846i q^{5} -1.00000i q^{7} -1.32653 q^{9} +O(q^{10})\)	\(q+1.29363 q^{3} -2.79846i q^{5} -1.00000i q^{7} -1.32653 q^{9} -1.03291i q^{11} +(-1.79846 - 3.12499i) q^{13} -3.62016i q^{15} -6.95635 q^{17} +2.53774i q^{19} -1.29363i q^{21} +3.83136 q^{23} -2.83136 q^{25} -5.59691 q^{27} -5.10175 q^{29} +3.78880i q^{31} -1.33619i q^{33} -2.79846 q^{35} -6.20741i q^{37} +(-2.32653 - 4.04257i) q^{39} +0.990338i q^{41} -0.560979 q^{43} +3.71224i q^{45} +7.19080i q^{47} -1.00000 q^{49} -8.99892 q^{51} +1.17830 q^{53} -2.89054 q^{55} +3.28288i q^{57} -11.9234i q^{59} +13.8469 q^{61} +1.32653i q^{63} +(-8.74515 + 5.03291i) q^{65} -13.7811i q^{67} +4.95635 q^{69} -3.80432i q^{71} +3.19080i q^{73} -3.66272 q^{75} -1.03291 q^{77} -7.33240 q^{79} -3.26072 q^{81} +16.6260i q^{83} +19.4670i q^{85} -6.59975 q^{87} -11.8964i q^{89} +(-3.12499 + 1.79846i) q^{91} +4.90128i q^{93} +7.10175 q^{95} -8.60355i q^{97} +1.37018i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^9	\( 8 q + 8 q^{9} + 2 q^{13} + 4 q^{17} + 6 q^{23} + 2 q^{25} - 12 q^{27} - 2 q^{29} - 6 q^{35} + 6 q^{43} - 8 q^{49} + 8 q^{51} + 22 q^{53} + 20 q^{55} + 8 q^{61} - 6 q^{65} - 20 q^{69} + 20 q^{75} + 26 q^{79} - 24 q^{81} + 32 q^{87} + 10 q^{91} + 18 q^{95}+O(q^{100}) \) 8 * q + 8 * q^9 + 2 * q^13 + 4 * q^17 + 6 * q^23 + 2 * q^25 - 12 * q^27 - 2 * q^29 - 6 * q^35 + 6 * q^43 - 8 * q^49 + 8 * q^51 + 22 * q^53 + 20 * q^55 + 8 * q^61 - 6 * q^65 - 20 * q^69 + 20 * q^75 + 26 * q^79 - 24 * q^81 + 32 * q^87 + 10 * q^91 + 18 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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