LMFDB - Embedded newform 1360.2.c.f.1121.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1360,2,Mod(1121,1360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1360, 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("1360.1121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1360 = 2^{4} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1360.c (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:	$10.8596546749$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 85)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.4
Root			$-0.854638 - 0.854638i$ of defining polynomial
Character	$\chi$	$=$	1360.1121
Dual form			1360.2.c.f.1121.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.539189i q^{3} +1.00000i q^{5} +4.87936i q^{7} +2.70928 q^{9} +3.17009i q^{11} +2.63090 q^{13} -0.539189 q^{15} +(-3.24846 + 2.53919i) q^{17} -1.07838 q^{19} -2.63090 q^{21} -5.21953i q^{23} -1.00000 q^{25} +3.07838i q^{27} -2.92162i q^{29} -4.09171i q^{31} -1.70928 q^{33} -4.87936 q^{35} +5.26180i q^{37} +1.41855i q^{39} -5.60197i q^{41} -3.36910 q^{43} +2.70928i q^{45} +6.78765 q^{47} -16.8082 q^{49} +(-1.36910 - 1.75154i) q^{51} +3.75872 q^{53} -3.17009 q^{55} -0.581449i q^{57} +2.34017 q^{59} +12.2557i q^{61} +13.2195i q^{63} +2.63090i q^{65} -10.2062 q^{67} +2.81432 q^{69} -4.06505i q^{71} +11.0784i q^{73} -0.539189i q^{75} -15.4680 q^{77} -6.92881i q^{79} +6.46800 q^{81} -8.23287 q^{83} +(-2.53919 - 3.24846i) q^{85} +1.57531 q^{87} +7.15449 q^{89} +12.8371i q^{91} +2.20620 q^{93} -1.07838i q^{95} -8.18342i q^{97} +8.58864i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{9} + 8 q^{13} - 2 q^{17} - 8 q^{21} - 6 q^{25} + 4 q^{33} - 4 q^{35} - 28 q^{43} + 20 q^{47} - 14 q^{49} - 16 q^{51} - 28 q^{53} - 8 q^{55} - 8 q^{59} - 12 q^{67} - 28 q^{77} - 26 q^{81} - 4 q^{83}+ \cdots - 36 q^{93}+O(q^{100}) $ 6 * q + 2 * q^9 + 8 * q^13 - 2 * q^17 - 8 * q^21 - 6 * q^25 + 4 * q^33 - 4 * q^35 - 28 * q^43 + 20 * q^47 - 14 * q^49 - 16 * q^51 - 28 * q^53 - 8 * q^55 - 8 * q^59 - 12 * q^67 - 28 * q^77 - 26 * q^81 - 4 * q^83 - 12 * q^85 - 32 * q^87 + 4 * q^89 - 36 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1360\mathbb{Z}\right)^\times$.

$n$	$241$	$341$	$511$	$817$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$			0.539189i			0.311301i	0.987812	+	0.155650i	$0.0497473\pi$
$3$			0.539189i			0.311301i	−0.987812	+	0.155650i	$0.950253\pi$
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$			4.87936i			1.84423i	0.386921	+	0.922113i	$0.373538\pi$
$7$			4.87936i			1.84423i	−0.386921	+	0.922113i	$0.626462\pi$
$8$		0			0
$9$	2.70928			0.903092
$10$		0			0
$11$			3.17009i			0.955817i	0.878410	+	0.477909i	$0.158605\pi$
$11$			3.17009i			0.955817i	−0.878410	+	0.477909i	$0.841395\pi$
$12$		0			0
$13$	2.63090			0.729680			0.364840	−	0.931070i	$-0.381124\pi$
$13$	2.63090			0.729680			0.364840	+	0.931070i	$0.381124\pi$
$14$		0			0
$15$	−0.539189			−0.139218
$16$		0			0
$17$	−3.24846	+	2.53919i	−0.787868	+	0.615844i
$17$	−3.24846	+	2.53919i	−0.787868	+	0.615844i
$18$		0			0
$19$	−1.07838			−0.247397			−0.123698	−	0.992320i	$-0.539476\pi$
$19$	−1.07838			−0.247397			−0.123698	+	0.992320i	$0.539476\pi$
$20$		0			0
$21$	−2.63090			−0.574109
$22$		0			0
$23$		−	5.21953i		−	1.08835i	−0.838972	−	0.544174i	$-0.816843\pi$
$23$		−	5.21953i		−	1.08835i	0.838972	−	0.544174i	$-0.183157\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$			3.07838i			0.592434i
$28$		0			0
$29$		−	2.92162i		−	0.542532i	−0.962504	−	0.271266i	$-0.912558\pi$
$29$		−	2.92162i		−	0.542532i	0.962504	−	0.271266i	$-0.0874422\pi$
$30$		0			0
$31$		−	4.09171i		−	0.734893i	−0.930045	−	0.367446i	$-0.880232\pi$
$31$		−	4.09171i		−	0.734893i	0.930045	−	0.367446i	$-0.119768\pi$
$32$		0			0
$33$	−1.70928			−0.297547
$34$		0			0
$35$	−4.87936			−0.824763
$36$		0			0
$37$			5.26180i			0.865034i	0.901626	+	0.432517i	$0.142374\pi$
$37$			5.26180i			0.865034i	−0.901626	+	0.432517i	$0.857626\pi$
$38$		0			0
$39$			1.41855i			0.227150i
$40$		0			0
$41$		−	5.60197i		−	0.874880i	−0.899247	−	0.437440i	$-0.855885\pi$
$41$		−	5.60197i		−	0.874880i	0.899247	−	0.437440i	$-0.144115\pi$
$42$		0			0
$43$	−3.36910			−0.513783			−0.256892	−	0.966440i	$-0.582698\pi$
$43$	−3.36910			−0.513783			−0.256892	+	0.966440i	$0.582698\pi$
$44$		0			0
$45$			2.70928i			0.403875i
$46$		0			0
$47$	6.78765			0.990081			0.495040	−	0.868870i	$-0.335153\pi$
$47$	6.78765			0.990081			0.495040	+	0.868870i	$0.335153\pi$
$48$		0			0
$49$	−16.8082			−2.40117
$50$		0			0
$51$	−1.36910	−	1.75154i	−0.191713	−	0.245264i
$52$		0			0
$53$	3.75872			0.516300			0.258150	−	0.966105i	$-0.416887\pi$
$53$	3.75872			0.516300			0.258150	+	0.966105i	$0.416887\pi$
$54$		0			0
$55$	−3.17009			−0.427454
$56$		0			0
$57$		−	0.581449i		−	0.0770148i
$58$		0			0
$59$	2.34017			0.304665			0.152332	−	0.988329i	$-0.451322\pi$
$59$	2.34017			0.304665			0.152332	+	0.988329i	$0.451322\pi$
$60$		0			0
$61$			12.2557i			1.56918i	0.620018	+	0.784588i	$0.287125\pi$
$61$			12.2557i			1.56918i	−0.620018	+	0.784588i	$0.712875\pi$
$62$		0			0
$63$			13.2195i			1.66550i
$64$		0			0
$65$			2.63090i			0.326323i
$66$		0			0
$67$	−10.2062			−1.24689			−0.623443	−	0.781869i	$-0.714267\pi$
$67$	−10.2062			−1.24689			−0.623443	+	0.781869i	$0.714267\pi$
$68$		0			0
$69$	2.81432			0.338804
$70$		0			0
$71$		−	4.06505i		−	0.482432i	−0.970471	−	0.241216i	$-0.922454\pi$
$71$		−	4.06505i		−	0.482432i	0.970471	−	0.241216i	$-0.0775463\pi$
$72$		0			0
$73$			11.0784i			1.29663i	0.761374	+	0.648313i	$0.224525\pi$
$73$			11.0784i			1.29663i	−0.761374	+	0.648313i	$0.775475\pi$
$74$		0			0
$75$		−	0.539189i		−	0.0622602i
$76$		0			0
$77$	−15.4680			−1.76274
$78$		0			0
$79$		−	6.92881i		−	0.779552i	−0.920910	−	0.389776i	$-0.872552\pi$
$79$		−	6.92881i		−	0.779552i	0.920910	−	0.389776i	$-0.127448\pi$
$80$		0			0
$81$	6.46800			0.718667
$82$		0			0
$83$	−8.23287			−0.903674			−0.451837	−	0.892100i	$-0.649231\pi$
$83$	−8.23287			−0.903674			−0.451837	+	0.892100i	$0.649231\pi$
$84$		0			0
$85$	−2.53919	−	3.24846i	−0.275414	−	0.352345i
$86$		0			0
$87$	1.57531			0.168891
$88$		0			0
$89$	7.15449			0.758374			0.379187	−	0.925320i	$-0.376204\pi$
$89$	7.15449			0.758374			0.379187	+	0.925320i	$0.376204\pi$
$90$		0			0
$91$			12.8371i			1.34569i
$92$		0			0
$93$	2.20620			0.228773
$94$		0			0
$95$		−	1.07838i		−	0.110639i
$96$		0			0
$97$		−	8.18342i		−	0.830900i	−0.909616	−	0.415450i	$-0.863624\pi$
$97$		−	8.18342i		−	0.830900i	0.909616	−	0.415450i	$-0.136376\pi$
$98$		0			0
$99$			8.58864i			0.863191i
$100$		0			0
$101$	−2.47414			−0.246186			−0.123093	−	0.992395i	$-0.539281\pi$
$101$	−2.47414			−0.246186			−0.123093	+	0.992395i	$0.539281\pi$
$102$		0			0
$103$	19.6514			1.93631			0.968156	−	0.250348i	$-0.0805452\pi$
$103$	19.6514			1.93631			0.968156	+	0.250348i	$0.0805452\pi$
$104$		0			0
$105$		−	2.63090i		−	0.256749i
$106$		0			0
$107$			12.6381i			1.22177i	0.791719	+	0.610885i	$0.209186\pi$
$107$			12.6381i			1.22177i	−0.791719	+	0.610885i	$0.790814\pi$
$108$		0			0
$109$		−	8.15676i		−	0.781275i	−0.920544	−	0.390638i	$-0.872255\pi$
$109$		−	8.15676i		−	0.781275i	0.920544	−	0.390638i	$-0.127745\pi$
$110$		0			0
$111$	−2.83710			−0.269286
$112$		0			0
$113$			17.0205i			1.60116i	0.599229	+	0.800578i	$0.295474\pi$
$113$			17.0205i			1.60116i	−0.599229	+	0.800578i	$0.704526\pi$
$114$		0			0
$115$	5.21953			0.486724
$116$		0			0
$117$	7.12783			0.658968
$118$		0			0
$119$	−12.3896	−	15.8504i	−1.13575	−	1.45301i
$120$		0			0
$121$	0.950552			0.0864138
$122$		0			0
$123$	3.02052			0.272351
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	8.04945			0.714273			0.357137	−	0.934052i	$-0.383753\pi$
$127$	8.04945			0.714273			0.357137	+	0.934052i	$0.383753\pi$
$128$		0			0
$129$		−	1.81658i		−	0.159941i
$130$		0			0
$131$		−	11.6937i		−	1.02168i	−0.859675	−	0.510841i	$-0.829334\pi$
$131$		−	11.6937i		−	1.02168i	0.859675	−	0.510841i	$-0.170666\pi$
$132$		0			0
$133$		−	5.26180i		−	0.456256i
$134$		0			0
$135$	−3.07838			−0.264945
$136$		0			0
$137$	1.95055			0.166647			0.0833234	−	0.996523i	$-0.473447\pi$
$137$	1.95055			0.166647			0.0833234	+	0.996523i	$0.473447\pi$
$138$		0			0
$139$			2.00719i			0.170247i	0.996370	+	0.0851237i	$0.0271286\pi$
$139$			2.00719i			0.170247i	−0.996370	+	0.0851237i	$0.972871\pi$
$140$		0			0
$141$			3.65983i			0.308213i
$142$		0			0
$143$			8.34017i			0.697440i
$144$		0			0
$145$	2.92162			0.242628
$146$		0			0
$147$		−	9.06278i		−	0.747485i
$148$		0			0
$149$	−14.2823			−1.17005			−0.585026	−	0.811014i	$-0.698916\pi$
$149$	−14.2823			−1.17005			−0.585026	+	0.811014i	$0.698916\pi$
$150$		0			0
$151$	−12.8638			−1.04684			−0.523419	−	0.852075i	$-0.675344\pi$
$151$	−12.8638			−1.04684			−0.523419	+	0.852075i	$0.675344\pi$
$152$		0			0
$153$	−8.80098	+	6.87936i	−0.711517	+	0.556163i
$154$		0			0
$155$	4.09171			0.328654
$156$		0			0
$157$	3.75872			0.299979			0.149989	−	0.988688i	$-0.452076\pi$
$157$	3.75872			0.299979			0.149989	+	0.988688i	$0.452076\pi$
$158$		0			0
$159$			2.02666i			0.160725i
$160$		0			0
$161$	25.4680			2.00716
$162$		0			0
$163$			8.69594i			0.681119i	0.940223	+	0.340559i	$0.110616\pi$
$163$			8.69594i			0.681119i	−0.940223	+	0.340559i	$0.889384\pi$
$164$		0			0
$165$		−	1.70928i		−	0.133067i
$166$		0			0
$167$		−	1.37629i		−	0.106501i	−0.998581	−	0.0532503i	$-0.983042\pi$
$167$		−	1.37629i		−	0.106501i	0.998581	−	0.0532503i	$-0.0169581\pi$
$168$		0			0
$169$	−6.07838			−0.467568
$170$		0			0
$171$	−2.92162			−0.223422
$172$		0			0
$173$		−	17.3607i		−	1.31991i	−0.751306	−	0.659954i	$-0.770576\pi$
$173$		−	17.3607i		−	1.31991i	0.751306	−	0.659954i	$-0.229424\pi$
$174$		0			0
$175$		−	4.87936i		−	0.368845i
$176$		0			0
$177$			1.26180i			0.0948423i
$178$		0			0
$179$	−6.83710			−0.511029			−0.255514	−	0.966805i	$-0.582245\pi$
$179$	−6.83710			−0.511029			−0.255514	+	0.966805i	$0.582245\pi$
$180$		0			0
$181$			15.0205i			1.11647i	0.829684	+	0.558233i	$0.188521\pi$
$181$			15.0205i			1.11647i	−0.829684	+	0.558233i	$0.811479\pi$
$182$		0			0
$183$	−6.60811			−0.488486
$184$		0			0
$185$	−5.26180			−0.386855
$186$		0			0
$187$	−8.04945	−	10.2979i	−0.588634	−	0.753058i
$188$		0			0
$189$	−15.0205			−1.09258
$190$		0			0
$191$	24.2823			1.75701			0.878503	−	0.477736i	$-0.158543\pi$
$191$	24.2823			1.75701			0.878503	+	0.477736i	$0.158543\pi$
$192$		0			0
$193$		−	11.8576i		−	0.853530i	−0.904363	−	0.426765i	$-0.859653\pi$
$193$		−	11.8576i		−	0.853530i	0.904363	−	0.426765i	$-0.140347\pi$
$194$		0			0
$195$	−1.41855			−0.101585
$196$		0			0
$197$		−	18.2557i		−	1.30066i	−0.759651	−	0.650331i	$-0.774630\pi$
$197$		−	18.2557i		−	1.30066i	0.759651	−	0.650331i	$-0.225370\pi$
$198$		0			0
$199$		−	3.72487i		−	0.264049i	−0.991246	−	0.132025i	$-0.957852\pi$
$199$		−	3.72487i		−	0.264049i	0.991246	−	0.132025i	$-0.0421478\pi$
$200$		0			0
$201$		−	5.50307i		−	0.388157i
$202$		0			0
$203$	14.2557			1.00055
$204$		0			0
$205$	5.60197			0.391258
$206$		0			0
$207$		−	14.1412i		−	0.982878i
$208$		0			0
$209$		−	3.41855i		−	0.236466i
$210$		0			0
$211$			22.2485i			1.53165i	0.643051	+	0.765824i	$0.277668\pi$
$211$			22.2485i			1.53165i	−0.643051	+	0.765824i	$0.722332\pi$
$212$		0			0
$213$	2.19183			0.150182
$214$		0			0
$215$		−	3.36910i		−	0.229771i
$216$		0			0
$217$	19.9649			1.35531
$218$		0			0
$219$	−5.97334			−0.403641
$220$		0			0
$221$	−8.54638	+	6.68035i	−0.574892	+	0.449369i
$222$		0			0
$223$	2.19183			0.146776			0.0733878	−	0.997303i	$-0.476619\pi$
$223$	2.19183			0.146776			0.0733878	+	0.997303i	$0.476619\pi$
$224$		0			0
$225$	−2.70928			−0.180618
$226$		0			0
$227$		−	9.55971i		−	0.634500i	−0.948342	−	0.317250i	$-0.897241\pi$
$227$		−	9.55971i		−	0.634500i	0.948342	−	0.317250i	$-0.102759\pi$
$228$		0			0
$229$	7.36910			0.486964			0.243482	−	0.969905i	$-0.421710\pi$
$229$	7.36910			0.486964			0.243482	+	0.969905i	$0.421710\pi$
$230$		0			0
$231$		−	8.34017i		−	0.548743i
$232$		0			0
$233$			9.44521i			0.618776i	0.950936	+	0.309388i	$0.100124\pi$
$233$			9.44521i			0.618776i	−0.950936	+	0.309388i	$0.899876\pi$
$234$		0			0
$235$			6.78765i			0.442778i
$236$		0			0
$237$	3.73594			0.242675
$238$		0			0
$239$	−6.25565			−0.404644			−0.202322	−	0.979319i	$-0.564849\pi$
$239$	−6.25565			−0.404644			−0.202322	+	0.979319i	$0.564849\pi$
$240$		0			0
$241$		−	2.49693i		−	0.160841i	−0.996761	−	0.0804207i	$-0.974374\pi$
$241$		−	2.49693i		−	0.160841i	0.996761	−	0.0804207i	$-0.0256264\pi$
$242$		0			0
$243$			12.7226i			0.816156i
$244$		0			0
$245$		−	16.8082i		−	1.07383i
$246$		0			0
$247$	−2.83710			−0.180520
$248$		0			0
$249$		−	4.43907i		−	0.281315i
$250$		0			0
$251$	11.8166			0.745856			0.372928	−	0.927860i	$-0.378354\pi$
$251$	11.8166			0.745856			0.372928	+	0.927860i	$0.378354\pi$
$252$		0			0
$253$	16.5464			1.04026
$254$		0			0
$255$	1.75154	−	1.36910i	0.109685	−	0.0857365i
$256$		0			0
$257$	14.9444			0.932207			0.466103	−	0.884730i	$-0.345658\pi$
$257$	14.9444			0.932207			0.466103	+	0.884730i	$0.345658\pi$
$258$		0			0
$259$	−25.6742			−1.59532
$260$		0			0
$261$		−	7.91548i		−	0.489956i
$262$		0			0
$263$	12.9444			0.798186			0.399093	−	0.916910i	$-0.369325\pi$
$263$	12.9444			0.798186			0.399093	+	0.916910i	$0.369325\pi$
$264$		0			0
$265$			3.75872i			0.230897i
$266$		0			0
$267$			3.85762i			0.236083i
$268$		0			0
$269$		−	7.47641i		−	0.455845i	−0.973679	−	0.227922i	$-0.926807\pi$
$269$		−	7.47641i		−	0.455845i	0.973679	−	0.227922i	$-0.0731932\pi$
$270$		0			0
$271$	2.15676			0.131014			0.0655068	−	0.997852i	$-0.479134\pi$
$271$	2.15676			0.131014			0.0655068	+	0.997852i	$0.479134\pi$
$272$		0			0
$273$	−6.92162			−0.418916
$274$		0			0
$275$		−	3.17009i		−	0.191163i
$276$		0			0
$277$			12.1568i			0.730429i	0.930923	+	0.365214i	$0.119004\pi$
$277$			12.1568i			0.730429i	−0.930923	+	0.365214i	$0.880996\pi$
$278$		0			0
$279$		−	11.0856i		−	0.663675i
$280$		0			0
$281$	13.1194			0.782639			0.391319	−	0.920255i	$-0.372019\pi$
$281$	13.1194			0.782639			0.391319	+	0.920255i	$0.372019\pi$
$282$		0			0
$283$			13.9577i			0.829701i	0.909889	+	0.414851i	$0.136166\pi$
$283$			13.9577i			0.829701i	−0.909889	+	0.414851i	$0.863834\pi$
$284$		0			0
$285$	0.581449			0.0344421
$286$		0			0
$287$	27.3340			1.61348
$288$		0			0
$289$	4.10504	−	16.4969i	0.241473	−	0.970408i
$290$		0			0
$291$	4.41241			0.258660
$292$		0			0
$293$	4.73820			0.276809			0.138404	−	0.990376i	$-0.455803\pi$
$293$	4.73820			0.276809			0.138404	+	0.990376i	$0.455803\pi$
$294$		0			0
$295$			2.34017i			0.136250i
$296$		0			0
$297$	−9.75872			−0.566259
$298$		0			0
$299$		−	13.7321i		−	0.794146i
$300$		0			0
$301$		−	16.4391i		−	0.947532i
$302$		0			0
$303$		−	1.33403i		−	0.0766380i
$304$		0			0
$305$	−12.2557			−0.701757
$306$		0			0
$307$	21.5936			1.23241			0.616205	−	0.787586i	$-0.288669\pi$
$307$	21.5936			1.23241			0.616205	+	0.787586i	$0.288669\pi$
$308$		0			0
$309$			10.5958i			0.602775i
$310$		0			0
$311$		−	24.2628i		−	1.37582i	−0.725796	−	0.687910i	$-0.758528\pi$
$311$		−	24.2628i		−	1.37582i	0.725796	−	0.687910i	$-0.241472\pi$
$312$		0			0
$313$			4.07223i			0.230176i	0.993355	+	0.115088i	$0.0367151\pi$
$313$			4.07223i			0.230176i	−0.993355	+	0.115088i	$0.963285\pi$
$314$		0			0
$315$	−13.2195			−0.744836
$316$		0			0
$317$			8.05786i			0.452574i	0.974061	+	0.226287i	$0.0726588\pi$
$317$			8.05786i			0.452574i	−0.974061	+	0.226287i	$0.927341\pi$
$318$		0			0
$319$	9.26180			0.518561
$320$		0			0
$321$	−6.81432			−0.380338
$322$		0			0
$323$	3.50307	−	2.73820i	0.194916	−	0.152358i
$324$		0			0
$325$	−2.63090			−0.145936
$326$		0			0
$327$	4.39803			0.243212
$328$		0			0
$329$			33.1194i			1.82593i
$330$		0			0
$331$	10.0722			0.553620			0.276810	−	0.960925i	$-0.410723\pi$
$331$	10.0722			0.553620			0.276810	+	0.960925i	$0.410723\pi$
$332$		0			0
$333$			14.2557i			0.781205i
$334$		0			0
$335$		−	10.2062i		−	0.557624i
$336$		0			0
$337$		−	11.2351i		−	0.612017i	−0.952029	−	0.306008i	$-0.901006\pi$
$337$		−	11.2351i		−	0.612017i	0.952029	−	0.306008i	$-0.0989936\pi$
$338$		0			0
$339$	−9.17727			−0.498441
$340$		0			0
$341$	12.9711			0.702423
$342$		0			0
$343$		−	47.8576i		−	2.58407i
$344$		0			0
$345$			2.81432i			0.151518i
$346$		0			0
$347$			8.74927i			0.469685i	0.972033	+	0.234843i	$0.0754575\pi$
$347$			8.74927i			0.469685i	−0.972033	+	0.234843i	$0.924543\pi$
$348$		0			0
$349$	−26.9093			−1.44042			−0.720212	−	0.693754i	$-0.755955\pi$
$349$	−26.9093			−1.44042			−0.720212	+	0.693754i	$0.755955\pi$
$350$		0			0
$351$			8.09890i			0.432287i
$352$		0			0
$353$	18.3135			0.974730			0.487365	−	0.873198i	$-0.337958\pi$
$353$	18.3135			0.974730			0.487365	+	0.873198i	$0.337958\pi$
$354$		0			0
$355$	4.06505			0.215750
$356$		0			0
$357$	8.54638	−	6.68035i	0.452322	−	0.353561i
$358$		0			0
$359$	9.57531			0.505365			0.252683	−	0.967549i	$-0.418687\pi$
$359$	9.57531			0.505365			0.252683	+	0.967549i	$0.418687\pi$
$360$		0			0
$361$	−17.8371			−0.938795
$362$		0			0
$363$			0.512527i			0.0269007i
$364$		0			0
$365$	−11.0784			−0.579869
$366$		0			0
$367$		−	12.1145i		−	0.632371i	−0.948697	−	0.316186i	$-0.897598\pi$
$367$		−	12.1145i		−	0.632371i	0.948697	−	0.316186i	$-0.102402\pi$
$368$		0			0
$369$		−	15.1773i		−	0.790097i
$370$		0			0
$371$			18.3402i			0.952174i
$372$		0			0
$373$	30.4619			1.57726			0.788628	−	0.614871i	$-0.210792\pi$
$373$	30.4619			1.57726			0.788628	+	0.614871i	$0.210792\pi$
$374$		0			0
$375$	0.539189			0.0278436
$376$		0			0
$377$		−	7.68649i		−	0.395874i
$378$		0			0
$379$			0.986669i			0.0506818i	0.999679	+	0.0253409i	$0.00806712\pi$
$379$			0.986669i			0.0506818i	−0.999679	+	0.0253409i	$0.991933\pi$
$380$		0			0
$381$			4.34017i			0.222354i
$382$		0			0
$383$	24.9588			1.27533			0.637667	−	0.770312i	$-0.279900\pi$
$383$	24.9588			1.27533			0.637667	+	0.770312i	$0.279900\pi$
$384$		0			0
$385$		−	15.4680i		−	0.788322i
$386$		0			0
$387$	−9.12783			−0.463993
$388$		0			0
$389$	33.8082			1.71414			0.857071	−	0.515198i	$-0.172282\pi$
$389$	33.8082			1.71414			0.857071	+	0.515198i	$0.172282\pi$
$390$		0			0
$391$	13.2534	+	16.9555i	0.670252	+	0.857475i
$392$		0			0
$393$	6.30510			0.318050
$394$		0			0
$395$	6.92881			0.348626
$396$		0			0
$397$			28.5236i			1.43156i	0.698327	+	0.715779i	$0.253928\pi$
$397$			28.5236i			1.43156i	−0.698327	+	0.715779i	$0.746072\pi$
$398$		0			0
$399$	2.83710			0.142033
$400$		0			0
$401$		−	33.0928i		−	1.65257i	−0.563250	−	0.826287i	$-0.690449\pi$
$401$		−	33.0928i		−	1.65257i	0.563250	−	0.826287i	$-0.309551\pi$
$402$		0			0
$403$		−	10.7649i		−	0.536236i
$404$		0			0
$405$			6.46800i			0.321397i
$406$		0			0
$407$	−16.6803			−0.826814
$408$		0			0
$409$	−30.1978			−1.49318			−0.746592	−	0.665282i	$-0.768311\pi$
$409$	−30.1978			−1.49318			−0.746592	+	0.665282i	$0.768311\pi$
$410$		0			0
$411$			1.05172i			0.0518773i
$412$		0			0
$413$			11.4186i			0.561870i
$414$		0			0
$415$		−	8.23287i		−	0.404135i
$416$		0			0
$417$	−1.08225			−0.0529982
$418$		0			0
$419$		−	18.1639i		−	0.887367i	−0.896184	−	0.443683i	$-0.853671\pi$
$419$		−	18.1639i		−	0.887367i	0.896184	−	0.443683i	$-0.146329\pi$
$420$		0			0
$421$	0.760991			0.0370884			0.0185442	−	0.999828i	$-0.494097\pi$
$421$	0.760991			0.0370884			0.0185442	+	0.999828i	$0.494097\pi$
$422$		0			0
$423$	18.3896			0.894134
$424$		0			0
$425$	3.24846	−	2.53919i	0.157574	−	0.123169i
$426$		0			0
$427$	−59.7998			−2.89391
$428$		0			0
$429$	−4.49693			−0.217114
$430$		0			0
$431$		−	6.34736i		−	0.305742i	−0.988246	−	0.152871i	$-0.951148\pi$
$431$		−	6.34736i		−	0.305742i	0.988246	−	0.152871i	$-0.0488518\pi$
$432$		0			0
$433$	3.62475			0.174195			0.0870973	−	0.996200i	$-0.472241\pi$
$433$	3.62475			0.174195			0.0870973	+	0.996200i	$0.472241\pi$
$434$		0			0
$435$			1.57531i			0.0755302i
$436$		0			0
$437$			5.62863i			0.269254i
$438$		0			0
$439$			6.40522i			0.305704i	0.988249	+	0.152852i	$0.0488459\pi$
$439$			6.40522i			0.305704i	−0.988249	+	0.152852i	$0.951154\pi$
$440$		0			0
$441$	−45.5380			−2.16847
$442$		0			0
$443$	−27.4824			−1.30573			−0.652864	−	0.757476i	$-0.726432\pi$
$443$	−27.4824			−1.30573			−0.652864	+	0.757476i	$0.726432\pi$
$444$		0			0
$445$			7.15449i			0.339155i
$446$		0			0
$447$		−	7.70086i		−	0.364238i
$448$		0			0
$449$		−	28.7526i		−	1.35692i	−0.734638	−	0.678459i	$-0.762648\pi$
$449$		−	28.7526i		−	1.35692i	0.734638	−	0.678459i	$-0.237352\pi$
$450$		0			0
$451$	17.7587			0.836226
$452$		0			0
$453$		−	6.93600i		−	0.325882i
$454$		0			0
$455$	−12.8371			−0.601813
$456$		0			0
$457$	−31.4101			−1.46930			−0.734652	−	0.678444i	$-0.762655\pi$
$457$	−31.4101			−1.46930			−0.734652	+	0.678444i	$0.762655\pi$
$458$		0			0
$459$	−7.81658	−	10.0000i	−0.364847	−	0.466760i
$460$		0			0
$461$	−7.75872			−0.361360			−0.180680	−	0.983542i	$-0.557830\pi$
$461$	−7.75872			−0.361360			−0.180680	+	0.983542i	$0.557830\pi$
$462$		0			0
$463$	−24.2329			−1.12620			−0.563098	−	0.826390i	$-0.690391\pi$
$463$	−24.2329			−1.12620			−0.563098	+	0.826390i	$0.690391\pi$
$464$		0			0
$465$			2.20620i			0.102310i
$466$		0			0
$467$	12.3174			0.569981			0.284990	−	0.958530i	$-0.408010\pi$
$467$	12.3174			0.569981			0.284990	+	0.958530i	$0.408010\pi$
$468$		0			0
$469$		−	49.7998i		−	2.29954i
$470$		0			0
$471$			2.02666i			0.0933837i
$472$		0			0
$473$		−	10.6803i		−	0.491083i
$474$		0			0
$475$	1.07838			0.0494794
$476$		0			0
$477$	10.1834			0.466267
$478$		0			0
$479$		−	2.14957i		−	0.0982162i	−0.998793	−	0.0491081i	$-0.984362\pi$
$479$		−	2.14957i		−	0.0982162i	0.998793	−	0.0491081i	$-0.0156379\pi$
$480$		0			0
$481$			13.8432i			0.631198i
$482$		0			0
$483$			13.7321i			0.624830i
$484$		0			0
$485$	8.18342			0.371590
$486$		0			0
$487$			40.0833i			1.81635i	0.418593	+	0.908174i	$0.362523\pi$
$487$			40.0833i			1.81635i	−0.418593	+	0.908174i	$0.637477\pi$
$488$		0			0
$489$	−4.68876			−0.212033
$490$		0			0
$491$	2.25565			0.101796			0.0508981	−	0.998704i	$-0.483792\pi$
$491$	2.25565			0.101796			0.0508981	+	0.998704i	$0.483792\pi$
$492$		0			0
$493$	7.41855	+	9.49079i	0.334115	+	0.427443i
$494$		0			0
$495$	−8.58864			−0.386031
$496$		0			0
$497$	19.8348			0.889714
$498$		0			0
$499$			42.5452i			1.90458i	0.305190	+	0.952291i	$0.401280\pi$
$499$			42.5452i			1.90458i	−0.305190	+	0.952291i	$0.598720\pi$
$500$		0			0
$501$	0.742080			0.0331537
$502$		0			0
$503$			9.55971i			0.426246i	0.977025	+	0.213123i	$0.0683636\pi$
$503$			9.55971i			0.426246i	−0.977025	+	0.213123i	$0.931636\pi$
$504$		0			0
$505$		−	2.47414i		−	0.110098i
$506$		0			0
$507$		−	3.27739i		−	0.145554i
$508$		0			0
$509$	28.3545			1.25679			0.628397	−	0.777893i	$-0.283712\pi$
$509$	28.3545			1.25679			0.628397	+	0.777893i	$0.283712\pi$
$510$		0			0
$511$	−54.0554			−2.39127
$512$		0			0
$513$		−	3.31965i		−	0.146566i
$514$		0			0
$515$			19.6514i			0.865945i
$516$		0			0
$517$			21.5174i			0.946336i
$518$		0			0
$519$	9.36069			0.410889
$520$		0			0
$521$			15.1050i			0.661764i	0.943672	+	0.330882i	$0.107346\pi$
$521$			15.1050i			0.661764i	−0.943672	+	0.330882i	$0.892654\pi$
$522$		0			0
$523$	10.8865			0.476036			0.238018	−	0.971261i	$-0.423502\pi$
$523$	10.8865			0.476036			0.238018	+	0.971261i	$0.423502\pi$
$524$		0			0
$525$	2.63090			0.114822
$526$		0			0
$527$	10.3896	+	13.2918i	0.452579	+	0.578999i
$528$		0			0
$529$	−4.24354			−0.184502
$530$		0			0
$531$	6.34017			0.275140
$532$		0			0
$533$		−	14.7382i		−	0.638383i
$534$		0			0
$535$	−12.6381			−0.546392
$536$		0			0
$537$		−	3.68649i		−	0.159084i
$538$		0			0
$539$		−	53.2834i		−	2.29508i
$540$		0			0
$541$			6.86830i			0.295291i	0.989040	+	0.147646i	$0.0471695\pi$
$541$			6.86830i			0.295291i	−0.989040	+	0.147646i	$0.952830\pi$
$542$		0			0
$543$	−8.09890			−0.347557
$544$		0			0
$545$	8.15676			0.349397
$546$		0			0
$547$			5.89988i			0.252261i	0.992014	+	0.126130i	$0.0402558\pi$
$547$			5.89988i			0.252261i	−0.992014	+	0.126130i	$0.959744\pi$
$548$		0			0
$549$			33.2039i			1.41711i
$550$		0			0
$551$			3.15061i			0.134221i
$552$		0			0
$553$	33.8082			1.43767
$554$		0			0
$555$		−	2.83710i		−	0.120428i
$556$		0			0
$557$	−17.7359			−0.751496			−0.375748	−	0.926722i	$-0.622614\pi$
$557$	−17.7359			−0.751496			−0.375748	+	0.926722i	$0.622614\pi$
$558$		0			0
$559$	−8.86376			−0.374897
$560$		0			0
$561$	5.55252	−	4.34017i	0.234428	−	0.183242i
$562$		0			0
$563$	38.8020			1.63531			0.817655	−	0.575708i	$-0.195274\pi$
$563$	38.8020			1.63531			0.817655	+	0.575708i	$0.195274\pi$
$564$		0			0
$565$	−17.0205			−0.716059
$566$		0			0
$567$			31.5597i			1.32538i
$568$		0			0
$569$	12.1568			0.509638			0.254819	−	0.966989i	$-0.417984\pi$
$569$	12.1568			0.509638			0.254819	+	0.966989i	$0.417984\pi$
$570$		0			0
$571$			15.4569i			0.646853i	0.946253	+	0.323426i	$0.104835\pi$
$571$			15.4569i			0.646853i	−0.946253	+	0.323426i	$0.895165\pi$
$572$		0			0
$573$			13.0928i			0.546958i
$574$		0			0
$575$			5.21953i			0.217670i
$576$		0			0
$577$	−2.36296			−0.0983713			−0.0491856	−	0.998790i	$-0.515663\pi$
$577$	−2.36296			−0.0983713			−0.0491856	+	0.998790i	$0.515663\pi$
$578$		0			0
$579$	6.39350			0.265705
$580$		0			0
$581$		−	40.1711i		−	1.66658i
$582$		0			0
$583$			11.9155i			0.493489i
$584$		0			0
$585$			7.12783i			0.294699i
$586$		0			0
$587$	3.65142			0.150710			0.0753550	−	0.997157i	$-0.475991\pi$
$587$	3.65142			0.150710			0.0753550	+	0.997157i	$0.475991\pi$
$588$		0			0
$589$			4.41241i			0.181810i
$590$		0			0
$591$	9.84324			0.404897
$592$		0			0
$593$	−1.38735			−0.0569718			−0.0284859	−	0.999594i	$-0.509069\pi$
$593$	−1.38735			−0.0569718			−0.0284859	+	0.999594i	$0.509069\pi$
$594$		0			0
$595$	15.8504	−	12.3896i	0.649804	−	0.507925i
$596$		0			0
$597$	2.00841			0.0821988
$598$		0			0
$599$	0.451356			0.0184419			0.00922095	−	0.999957i	$-0.497065\pi$
$599$	0.451356			0.0184419			0.00922095	+	0.999957i	$0.497065\pi$
$600$		0			0
$601$			22.1301i			0.902705i	0.892346	+	0.451353i	$0.149058\pi$
$601$			22.1301i			0.902705i	−0.892346	+	0.451353i	$0.850942\pi$
$602$		0			0
$603$	−27.6514			−1.12605
$604$		0			0
$605$			0.950552i			0.0386454i
$606$		0			0
$607$		−	10.2667i		−	0.416713i	−0.978053	−	0.208357i	$-0.933189\pi$
$607$		−	10.2667i		−	0.416713i	0.978053	−	0.208357i	$-0.0668114\pi$
$608$		0			0
$609$			7.68649i			0.311472i
$610$		0			0
$611$	17.8576			0.722442
$612$		0			0
$613$	−9.05172			−0.365595			−0.182798	−	0.983151i	$-0.558515\pi$
$613$	−9.05172			−0.365595			−0.182798	+	0.983151i	$0.558515\pi$
$614$		0			0
$615$			3.02052i			0.121799i
$616$		0			0
$617$			17.8166i			0.717269i	0.933478	+	0.358634i	$0.116757\pi$
$617$			17.8166i			0.717269i	−0.933478	+	0.358634i	$0.883243\pi$
$618$		0			0
$619$		−	38.3884i		−	1.54296i	−0.636254	−	0.771480i	$-0.719517\pi$
$619$		−	38.3884i		−	1.54296i	0.636254	−	0.771480i	$-0.280483\pi$
$620$		0			0
$621$	16.0677			0.644775
$622$		0			0
$623$			34.9093i			1.39861i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	1.84324			0.0736121
$628$		0			0
$629$	−13.3607	−	17.0928i	−0.532726	−	0.681533i
$630$		0			0
$631$	27.4863			1.09421			0.547105	−	0.837064i	$-0.315730\pi$
$631$	27.4863			1.09421			0.547105	+	0.837064i	$0.315730\pi$
$632$		0			0
$633$	−11.9961			−0.476803
$634$		0			0
$635$			8.04945i			0.319433i
$636$		0			0
$637$	−44.2206			−1.75208
$638$		0			0
$639$		−	11.0133i		−	0.435681i
$640$		0			0
$641$			9.79976i			0.387067i	0.981094	+	0.193534i	$0.0619949\pi$
$641$			9.79976i			0.387067i	−0.981094	+	0.193534i	$0.938005\pi$
$642$		0			0
$643$		−	16.9372i		−	0.667939i	−0.942584	−	0.333969i	$-0.891612\pi$
$643$		−	16.9372i		−	0.667939i	0.942584	−	0.333969i	$-0.108388\pi$
$644$		0			0
$645$	1.81658			0.0715279
$646$		0			0
$647$	2.98545			0.117370			0.0586850	−	0.998277i	$-0.481309\pi$
$647$	2.98545			0.117370			0.0586850	+	0.998277i	$0.481309\pi$
$648$		0			0
$649$			7.41855i			0.291204i
$650$		0			0
$651$			10.7649i			0.421908i
$652$		0			0
$653$			40.1978i			1.57306i	0.617551	+	0.786531i	$0.288125\pi$
$653$			40.1978i			1.57306i	−0.617551	+	0.786531i	$0.711875\pi$
$654$		0			0
$655$	11.6937			0.456910
$656$		0			0
$657$			30.0144i			1.17097i
$658$		0			0
$659$	−43.9832			−1.71334			−0.856671	−	0.515864i	$-0.827471\pi$
$659$	−43.9832			−1.71334			−0.856671	+	0.515864i	$0.827471\pi$
$660$		0			0
$661$	26.1133			1.01569			0.507844	−	0.861449i	$-0.330443\pi$
$661$	26.1133			1.01569			0.507844	+	0.861449i	$0.330443\pi$
$662$		0			0
$663$	−3.60197	−	4.60811i	−0.139889	−	0.178964i
$664$		0			0
$665$	5.26180			0.204044
$666$		0			0
$667$	−15.2495			−0.590463
$668$		0			0
$669$			1.18181i			0.0456914i
$670$		0			0
$671$	−38.8515			−1.49984
$672$		0			0
$673$		−	13.3340i		−	0.513989i	−0.966413	−	0.256995i	$-0.917268\pi$
$673$		−	13.3340i		−	0.513989i	0.966413	−	0.256995i	$-0.0827322\pi$
$674$		0			0
$675$		−	3.07838i		−	0.118487i
$676$		0			0
$677$		−	26.5113i		−	1.01891i	−0.860497	−	0.509456i	$-0.829847\pi$
$677$		−	26.5113i		−	1.01891i	0.860497	−	0.509456i	$-0.170153\pi$
$678$		0			0
$679$	39.9299			1.53237
$680$		0			0
$681$	5.15449			0.197520
$682$		0			0
$683$		−	5.71646i		−	0.218734i	−0.994001	−	0.109367i	$-0.965118\pi$
$683$		−	5.71646i		−	0.218734i	0.994001	−	0.109367i	$-0.0348824\pi$
$684$		0			0
$685$			1.95055i			0.0745267i
$686$		0			0
$687$			3.97334i			0.151592i
$688$		0			0
$689$	9.88882			0.376734
$690$		0			0
$691$			43.8504i			1.66815i	0.551652	+	0.834075i	$0.313998\pi$
$691$			43.8504i			1.66815i	−0.551652	+	0.834075i	$0.686002\pi$
$692$		0			0
$693$	−41.9071			−1.59192
$694$		0			0
$695$	−2.00719			−0.0761370
$696$		0			0
$697$	14.2245	+	18.1978i	0.538790	+	0.689291i
$698$		0			0
$699$	−5.09275			−0.192626
$700$		0			0
$701$	0.0806452			0.00304593			0.00152296	−	0.999999i	$-0.499515\pi$
$701$	0.0806452			0.00304593			0.00152296	+	0.999999i	$0.499515\pi$
$702$		0			0
$703$		−	5.67420i		−	0.214007i
$704$		0			0
$705$	−3.65983			−0.137837
$706$		0			0
$707$		−	12.0722i		−	0.454023i
$708$		0			0
$709$		−	10.8227i		−	0.406456i	−0.979131	−	0.203228i	$-0.934857\pi$
$709$		−	10.8227i		−	0.406456i	0.979131	−	0.203228i	$-0.0651433\pi$
$710$		0			0
$711$		−	18.7721i		−	0.704007i
$712$		0			0
$713$	−21.3568			−0.799819
$714$		0			0
$715$	−8.34017			−0.311905
$716$		0			0
$717$		−	3.37298i		−	0.125966i
$718$		0			0
$719$			43.7659i			1.63219i	0.577916	+	0.816097i	$0.303866\pi$
$719$			43.7659i			1.63219i	−0.577916	+	0.816097i	$0.696134\pi$
$720$		0			0
$721$			95.8864i			3.57100i
$722$		0			0
$723$	1.34632			0.0500700
$724$		0			0
$725$			2.92162i			0.108506i
$726$		0			0
$727$	3.59809			0.133446			0.0667229	−	0.997772i	$-0.478746\pi$
$727$	3.59809			0.133446			0.0667229	+	0.997772i	$0.478746\pi$
$728$		0			0
$729$	12.5441			0.464597
$730$		0			0
$731$	10.9444	−	8.55479i	0.404794	−	0.316410i
$732$		0			0
$733$	−39.8264			−1.47102			−0.735511	−	0.677513i	$-0.763058\pi$
$733$	−39.8264			−1.47102			−0.735511	+	0.677513i	$0.763058\pi$
$734$		0			0
$735$	9.06278			0.334286
$736$		0			0
$737$		−	32.3545i		−	1.19180i
$738$		0			0
$739$	−13.7587			−0.506123			−0.253061	−	0.967450i	$-0.581437\pi$
$739$	−13.7587			−0.506123			−0.253061	+	0.967450i	$0.581437\pi$
$740$		0			0
$741$		−	1.52973i		−	0.0561962i
$742$		0			0
$743$			9.34963i			0.343005i	0.985184	+	0.171502i	$0.0548621\pi$
$743$			9.34963i			0.343005i	−0.985184	+	0.171502i	$0.945138\pi$
$744$		0			0
$745$		−	14.2823i		−	0.523264i
$746$		0			0
$747$	−22.3051			−0.816101
$748$		0			0
$749$	−61.6658			−2.25322
$750$		0			0
$751$		−	53.7392i		−	1.96097i	−0.196586	−	0.980487i	$-0.562986\pi$
$751$		−	53.7392i		−	1.96097i	0.196586	−	0.980487i	$-0.437014\pi$
$752$		0			0
$753$			6.37137i			0.232186i
$754$		0			0
$755$		−	12.8638i		−	0.468160i
$756$		0			0
$757$	−41.5136			−1.50884			−0.754418	−	0.656394i	$-0.772081\pi$
$757$	−41.5136			−1.50884			−0.754418	+	0.656394i	$0.772081\pi$
$758$		0			0
$759$			8.92162i			0.323834i
$760$		0			0
$761$	−18.0372			−0.653847			−0.326923	−	0.945051i	$-0.606012\pi$
$761$	−18.0372			−0.653847			−0.326923	+	0.945051i	$0.606012\pi$
$762$		0			0
$763$	39.7998			1.44085
$764$		0			0
$765$	−6.87936	−	8.80098i	−0.248724	−	0.318200i
$766$		0			0
$767$	6.15676			0.222308
$768$		0			0
$769$	−23.0843			−0.832443			−0.416221	−	0.909263i	$-0.636646\pi$
$769$	−23.0843			−0.832443			−0.416221	+	0.909263i	$0.636646\pi$
$770$		0			0
$771$			8.05786i			0.290197i
$772$		0			0
$773$	−27.4101			−0.985874			−0.492937	−	0.870065i	$-0.664077\pi$
$773$	−27.4101			−0.985874			−0.492937	+	0.870065i	$0.664077\pi$
$774$		0			0
$775$			4.09171i			0.146979i
$776$		0			0
$777$		−	13.8432i		−	0.496624i
$778$		0			0
$779$			6.04104i			0.216443i
$780$		0			0
$781$	12.8865			0.461117
$782$		0			0
$783$	8.99386			0.321414
$784$		0			0
$785$			3.75872i			0.134155i
$786$		0			0
$787$		−	30.6069i		−	1.09102i	−0.838105	−	0.545509i	$-0.816336\pi$
$787$		−	30.6069i		−	1.09102i	0.838105	−	0.545509i	$-0.183664\pi$
$788$		0			0
$789$			6.97948i			0.248476i
$790$		0			0
$791$	−83.0493			−2.95289
$792$		0			0
$793$			32.2434i			1.14500i
$794$		0			0
$795$	−2.02666			−0.0718783
$796$		0			0
$797$	22.9770			0.813888			0.406944	−	0.913453i	$-0.366594\pi$
$797$	22.9770			0.813888			0.406944	+	0.913453i	$0.366594\pi$
$798$		0			0
$799$	−22.0494	+	17.2351i	−0.780053	+	0.609735i
$800$		0			0
$801$	19.3835			0.684882
$802$		0			0
$803$	−35.1194			−1.23934
$804$		0			0
$805$			25.4680i			0.897629i
$806$		0			0
$807$	4.03120			0.141905
$808$		0			0
$809$			42.7670i			1.50361i	0.659388	+	0.751803i	$0.270816\pi$
$809$			42.7670i			1.50361i	−0.659388	+	0.751803i	$0.729184\pi$
$810$		0			0
$811$		−	9.41136i		−	0.330478i	−0.986254	−	0.165239i	$-0.947161\pi$
$811$		−	9.41136i		−	0.330478i	0.986254	−	0.165239i	$-0.0528395\pi$
$812$		0			0
$813$			1.16290i			0.0407846i
$814$		0			0
$815$	−8.69594			−0.304606
$816$		0			0
$817$	3.63317			0.127108
$818$		0			0
$819$			34.7792i			1.21529i
$820$		0			0
$821$		−	32.6681i		−	1.14012i	−0.821602	−	0.570062i	$-0.806919\pi$
$821$		−	32.6681i		−	1.14012i	0.821602	−	0.570062i	$-0.193081\pi$
$822$		0			0
$823$			15.9265i			0.555164i	0.960702	+	0.277582i	$0.0895331\pi$
$823$			15.9265i			0.555164i	−0.960702	+	0.277582i	$0.910467\pi$
$824$		0			0
$825$	1.70928			0.0595093
$826$		0			0
$827$			6.01560i			0.209183i	0.994515	+	0.104591i	$0.0333535\pi$
$827$			6.01560i			0.209183i	−0.994515	+	0.104591i	$0.966647\pi$
$828$		0			0
$829$	11.0472			0.383684			0.191842	−	0.981426i	$-0.438554\pi$
$829$	11.0472			0.383684			0.191842	+	0.981426i	$0.438554\pi$
$830$		0			0
$831$	−6.55479			−0.227383
$832$		0			0
$833$	54.6007	−	42.6791i	1.89180	−	1.47874i
$834$		0			0
$835$	1.37629			0.0476285
$836$		0			0
$837$	12.5958			0.435375
$838$		0			0
$839$			35.9805i			1.24219i	0.783737	+	0.621093i	$0.213311\pi$
$839$			35.9805i			1.24219i	−0.783737	+	0.621093i	$0.786689\pi$
$840$		0			0
$841$	20.4641			0.705659
$842$		0			0
$843$			7.07384i			0.243636i
$844$		0			0
$845$		−	6.07838i		−	0.209103i
$846$		0			0
$847$			4.63809i			0.159367i
$848$		0			0
$849$	−7.52586			−0.258287
$850$		0			0
$851$	27.4641			0.941458
$852$		0			0
$853$		−	28.7792i		−	0.985382i	−0.870204	−	0.492691i	$-0.836013\pi$
$853$		−	28.7792i		−	0.985382i	0.870204	−	0.492691i	$-0.163987\pi$
$854$		0			0
$855$		−	2.92162i		−	0.0999174i
$856$		0			0
$857$		−	17.0661i		−	0.582967i	−0.956576	−	0.291483i	$-0.905851\pi$
$857$		−	17.0661i		−	0.582967i	0.956576	−	0.291483i	$-0.0941488\pi$
$858$		0			0
$859$	18.9360			0.646088			0.323044	−	0.946384i	$-0.395294\pi$
$859$	18.9360			0.646088			0.323044	+	0.946384i	$0.395294\pi$
$860$		0			0
$861$			14.7382i			0.502277i
$862$		0			0
$863$	−30.8332			−1.04958			−0.524788	−	0.851233i	$-0.675855\pi$
$863$	−30.8332			−1.04958			−0.524788	+	0.851233i	$0.675855\pi$
$864$		0			0
$865$	17.3607			0.590281
$866$		0			0
$867$	8.89496	+	2.21339i	0.302089	+	0.0751707i
$868$		0			0
$869$	21.9649			0.745109
$870$		0			0
$871$	−26.8515			−0.909828
$872$		0			0
$873$		−	22.1711i		−	0.750379i
$874$		0			0
$875$	4.87936			0.164953
$876$		0			0
$877$			4.30898i			0.145504i	0.997350	+	0.0727519i	$0.0231781\pi$
$877$			4.30898i			0.145504i	−0.997350	+	0.0727519i	$0.976822\pi$
$878$		0			0
$879$			2.55479i			0.0861708i
$880$		0			0
$881$		−	12.2245i		−	0.411852i	−0.978568	−	0.205926i	$-0.933979\pi$
$881$		−	12.2245i		−	0.411852i	0.978568	−	0.205926i	$-0.0660207\pi$
$882$		0			0
$883$	28.2329			0.950112			0.475056	−	0.879956i	$-0.342428\pi$
$883$	28.2329			0.950112			0.475056	+	0.879956i	$0.342428\pi$
$884$		0			0
$885$	−1.26180			−0.0424148
$886$		0			0
$887$		−	5.17396i		−	0.173725i	−0.996220	−	0.0868623i	$-0.972316\pi$
$887$		−	5.17396i		−	0.173725i	0.996220	−	0.0868623i	$-0.0276840\pi$
$888$		0			0
$889$			39.2762i			1.31728i
$890$		0			0
$891$			20.5041i			0.686914i
$892$		0			0
$893$	−7.31965			−0.244943
$894$		0			0
$895$		−	6.83710i		−	0.228539i
$896$		0			0
$897$	7.40417			0.247218
$898$		0			0
$899$	−11.9544			−0.398702
$900$		0			0
$901$	−12.2101	+	9.54411i	−0.406777	+	0.317960i
$902$		0			0
$903$	8.86376			0.294968
$904$		0			0
$905$	−15.0205			−0.499299
$906$		0			0
$907$			6.32457i			0.210004i	0.994472	+	0.105002i	$0.0334849\pi$
$907$			6.32457i			0.210004i	−0.994472	+	0.105002i	$0.966515\pi$
$908$		0			0
$909$	−6.70313			−0.222329
$910$		0			0
$911$		−	27.6526i		−	0.916173i	−0.888908	−	0.458086i	$-0.848535\pi$
$911$		−	27.6526i		−	0.916173i	0.888908	−	0.458086i	$-0.151465\pi$
$912$		0			0
$913$		−	26.0989i		−	0.863747i
$914$		0			0
$915$		−	6.60811i		−	0.218457i
$916$		0			0
$917$	57.0577			1.88421
$918$		0			0
$919$	47.5318			1.56793			0.783965	−	0.620805i	$-0.213194\pi$
$919$	47.5318			1.56793			0.783965	+	0.620805i	$0.213194\pi$
$920$		0			0
$921$			11.6430i			0.383650i
$922$		0			0
$923$		−	10.6947i		−	0.352021i
$924$		0			0
$925$		−	5.26180i		−	0.173007i
$926$		0			0
$927$	53.2411			1.74867
$928$		0			0
$929$		−	43.7875i		−	1.43662i	−0.695723	−	0.718310i	$-0.744916\pi$
$929$		−	43.7875i		−	1.43662i	0.695723	−	0.718310i	$-0.255084\pi$
$930$		0			0
$931$	18.1256			0.594041
$932$		0			0
$933$	13.0823			0.428294
$934$		0			0
$935$	10.2979	−	8.04945i	0.336778	−	0.263245i
$936$		0			0
$937$	14.2367			0.465094			0.232547	−	0.972585i	$-0.425294\pi$
$937$	14.2367			0.465094			0.232547	+	0.972585i	$0.425294\pi$
$938$		0			0
$939$	−2.19570			−0.0716541
$940$		0			0
$941$			35.2183i			1.14808i	0.818826	+	0.574042i	$0.194625\pi$
$941$			35.2183i			1.14808i	−0.818826	+	0.574042i	$0.805375\pi$
$942$		0			0
$943$	−29.2397			−0.952175
$944$		0			0
$945$		−	15.0205i		−	0.488618i
$946$		0			0
$947$		−	49.1227i		−	1.59627i	−0.602476	−	0.798137i	$-0.705819\pi$
$947$		−	49.1227i		−	1.59627i	0.602476	−	0.798137i	$-0.294181\pi$
$948$		0			0
$949$			29.1461i			0.946122i
$950$		0			0
$951$	−4.34471			−0.140887
$952$		0			0
$953$	−17.0556			−0.552485			−0.276242	−	0.961088i	$-0.589089\pi$
$953$	−17.0556			−0.552485			−0.276242	+	0.961088i	$0.589089\pi$
$954$		0			0
$955$			24.2823i			0.785757i
$956$		0			0
$957$			4.99386i			0.161428i
$958$		0			0
$959$			9.51745i			0.307334i
$960$		0			0
$961$	14.2579			0.459933
$962$		0			0
$963$			34.2401i			1.10337i
$964$		0			0
$965$	11.8576			0.381710
$966$		0			0
$967$	25.1955			0.810233			0.405117	−	0.914265i	$-0.367231\pi$
$967$	25.1955			0.810233			0.405117	+	0.914265i	$0.367231\pi$
$968$		0			0
$969$	1.47641	+	1.88882i	0.0474291	+	0.0606776i
$970$		0			0
$971$	−1.67420			−0.0537277			−0.0268639	−	0.999639i	$-0.508552\pi$
$971$	−1.67420			−0.0537277			−0.0268639	+	0.999639i	$0.508552\pi$
$972$		0			0
$973$	−9.79380			−0.313975
$974$		0			0
$975$		−	1.41855i		−	0.0454300i
$976$		0			0
$977$	−39.9109			−1.27686			−0.638432	−	0.769678i	$-0.720417\pi$
$977$	−39.9109			−1.27686			−0.638432	+	0.769678i	$0.720417\pi$
$978$		0			0
$979$			22.6803i			0.724867i
$980$		0			0
$981$		−	22.0989i		−	0.705563i
$982$		0			0
$983$			5.46081i			0.174173i	0.996201	+	0.0870864i	$0.0277556\pi$
$983$			5.46081i			0.174173i	−0.996201	+	0.0870864i	$0.972244\pi$
$984$		0			0
$985$	18.2557			0.581673
$986$		0			0
$987$	−17.8576			−0.568414
$988$		0			0
$989$			17.5851i			0.559175i
$990$		0			0
$991$			42.0749i			1.33655i	0.743913	+	0.668276i	$0.232968\pi$
$991$			42.0749i			1.33655i	−0.743913	+	0.668276i	$0.767032\pi$
$992$		0			0
$993$			5.43084i			0.172342i
$994$		0			0
$995$	3.72487			0.118086
$996$		0			0
$997$		−	54.6681i		−	1.73135i	−0.500602	−	0.865677i	$-0.666888\pi$
$997$		−	54.6681i		−	1.73135i	0.500602	−	0.865677i	$-0.333112\pi$
$998$		0			0
$999$	−16.1978			−0.512476

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1360.2.c.f.1121.4		6
4.3	odd	2		85.2.d.a.16.1	✓	6
12.11	even	2		765.2.g.b.271.5		6
17.16	even	2	inner	1360.2.c.f.1121.3		6
20.3	even	4		425.2.c.a.424.6		6
20.7	even	4		425.2.c.b.424.1		6
20.19	odd	2		425.2.d.c.101.6		6
68.47	odd	4		1445.2.a.j.1.3		3
68.55	odd	4		1445.2.a.k.1.3		3
68.67	odd	2		85.2.d.a.16.2	yes	6
204.203	even	2		765.2.g.b.271.6		6
340.67	even	4		425.2.c.a.424.1		6
340.203	even	4		425.2.c.b.424.6		6
340.259	odd	4		7225.2.a.r.1.1		3
340.319	odd	4		7225.2.a.q.1.1		3
340.339	odd	2		425.2.d.c.101.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.d.a.16.1	✓	6	4.3	odd	2
85.2.d.a.16.2	yes	6	68.67	odd	2
425.2.c.a.424.1		6	340.67	even	4
425.2.c.a.424.6		6	20.3	even	4
425.2.c.b.424.1		6	20.7	even	4
425.2.c.b.424.6		6	340.203	even	4
425.2.d.c.101.5		6	340.339	odd	2
425.2.d.c.101.6		6	20.19	odd	2
765.2.g.b.271.5		6	12.11	even	2
765.2.g.b.271.6		6	204.203	even	2
1360.2.c.f.1121.3		6	17.16	even	2	inner
1360.2.c.f.1121.4		6	1.1	even	1	trivial
1445.2.a.j.1.3		3	68.47	odd	4
1445.2.a.k.1.3		3	68.55	odd	4
7225.2.a.q.1.1		3	340.319	odd	4
7225.2.a.r.1.1		3	340.259	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:	\( N \)	\(=\)	\( 1360 = 2^{4} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1360.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.539189i q^{3} +1.00000i q^{5} +4.87936i q^{7} +2.70928 q^{9} +3.17009i q^{11} +2.63090 q^{13} -0.539189 q^{15} +(-3.24846 + 2.53919i) q^{17} -1.07838 q^{19} -2.63090 q^{21} -5.21953i q^{23} -1.00000 q^{25} +3.07838i q^{27} -2.92162i q^{29} -4.09171i q^{31} -1.70928 q^{33} -4.87936 q^{35} +5.26180i q^{37} +1.41855i q^{39} -5.60197i q^{41} -3.36910 q^{43} +2.70928i q^{45} +6.78765 q^{47} -16.8082 q^{49} +(-1.36910 - 1.75154i) q^{51} +3.75872 q^{53} -3.17009 q^{55} -0.581449i q^{57} +2.34017 q^{59} +12.2557i q^{61} +13.2195i q^{63} +2.63090i q^{65} -10.2062 q^{67} +2.81432 q^{69} -4.06505i q^{71} +11.0784i q^{73} -0.539189i q^{75} -15.4680 q^{77} -6.92881i q^{79} +6.46800 q^{81} -8.23287 q^{83} +(-2.53919 - 3.24846i) q^{85} +1.57531 q^{87} +7.15449 q^{89} +12.8371i q^{91} +2.20620 q^{93} -1.07838i q^{95} -8.18342i q^{97} +8.58864i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{9} + 8 q^{13} - 2 q^{17} - 8 q^{21} - 6 q^{25} + 4 q^{33} - 4 q^{35} - 28 q^{43} + 20 q^{47} - 14 q^{49} - 16 q^{51} - 28 q^{53} - 8 q^{55} - 8 q^{59} - 12 q^{67} - 28 q^{77} - 26 q^{81} - 4 q^{83}+ \cdots - 36 q^{93}+O(q^{100}) \) 6 * q + 2 * q^9 + 8 * q^13 - 2 * q^17 - 8 * q^21 - 6 * q^25 + 4 * q^33 - 4 * q^35 - 28 * q^43 + 20 * q^47 - 14 * q^49 - 16 * q^51 - 28 * q^53 - 8 * q^55 - 8 * q^59 - 12 * q^67 - 28 * q^77 - 26 * q^81 - 4 * q^83 - 12 * q^85 - 32 * q^87 + 4 * q^89 - 36 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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