LMFDB - Embedded newform 3380.2.d.c.1689.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(1689,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3380.1689");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.d (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:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.180227832610816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 $ x^12 + x^10 - 8x^6 + 16x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1689.7
Root			$-1.37729 - 0.321037i$ of defining polynomial
Character	$\chi$	$=$	3380.1689
Dual form			3380.2.d.c.1689.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.233093i q^{3} +(-0.726062 - 2.11491i) q^{5} -4.29014 q^{7} +2.94567 q^{9} +O(q^{10})$	$q+0.233093i q^{3} +(-0.726062 - 2.11491i) q^{5} -4.29014 q^{7} +2.94567 q^{9} +1.00000i q^{11} +(0.492969 - 0.169240i) q^{15} +3.82395i q^{17} -2.28415i q^{19} -1.00000i q^{21} +5.74226i q^{23} +(-3.94567 + 3.07111i) q^{25} +1.38589i q^{27} +2.28415 q^{29} -6.22982i q^{31} -0.233093 q^{33} +(3.11491 + 9.07325i) q^{35} +6.72820 q^{37} +2.66152i q^{41} -6.89506i q^{43} +(-2.13874 - 6.22982i) q^{45} +6.60840 q^{47} +11.4053 q^{49} -0.891336 q^{51} -5.04299i q^{53} +(2.11491 - 0.726062i) q^{55} +0.532418 q^{57} +11.1212i q^{59} +4.17548 q^{61} -12.6373 q^{63} -4.34371 q^{67} -1.33848 q^{69} -6.74378i q^{71} -13.5697 q^{73} +(-0.715853 - 0.919706i) q^{75} -4.29014i q^{77} -11.4053 q^{79} +8.51396 q^{81} +10.0324 q^{83} +(8.08731 - 2.77643i) q^{85} +0.532418i q^{87} -11.1212i q^{89} +1.45212 q^{93} +(-4.83076 + 1.65843i) q^{95} +14.8423 q^{97} +2.94567i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{9}+O(q^{10}) $ 12 * q - 8 * q^9	$ 12 q - 8 q^{9} - 4 q^{25} + 20 q^{29} + 12 q^{35} - 8 q^{49} + 76 q^{51} - 44 q^{61} - 52 q^{69} - 16 q^{75} + 8 q^{79} + 44 q^{81} - 40 q^{95}+O(q^{100}) $ 12 * q - 8 * q^9 - 4 * q^25 + 20 * q^29 + 12 * q^35 - 8 * q^49 + 76 * q^51 - 44 * q^61 - 52 * q^69 - 16 * q^75 + 8 * q^79 + 44 * q^81 - 40 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$.

$n$	$677$	$1691$	$1861$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			0.233093i			0.134576i	0.997734	+	0.0672881i	$0.0214346\pi$
$3$			0.233093i			0.134576i	−0.997734	+	0.0672881i	$0.978565\pi$
$4$		0			0
$5$	−0.726062	−	2.11491i	−0.324705	−	0.945815i
$5$	−0.726062	−	2.11491i	−0.324705	−	0.945815i
$6$		0			0
$7$	−4.29014			−1.62152			−0.810760	−	0.585378i	$-0.800946\pi$
$7$	−4.29014			−1.62152			−0.810760	+	0.585378i	$0.800946\pi$
$8$		0			0
$9$	2.94567			0.981889
$10$		0			0
$11$			1.00000i			0.301511i	0.988571	+	0.150756i	$0.0481707\pi$
$11$			1.00000i			0.301511i	−0.988571	+	0.150756i	$0.951829\pi$
$12$		0			0
$13$		0			0
$13$		0			0
$14$		0			0
$15$	0.492969	−	0.169240i	0.127284	−	0.0436975i
$16$		0			0
$17$			3.82395i			0.927445i	0.885980	+	0.463723i	$0.153487\pi$
$17$			3.82395i			0.927445i	−0.885980	+	0.463723i	$0.846513\pi$
$18$		0			0
$19$		−	2.28415i		−	0.524019i	−0.965065	−	0.262010i	$-0.915615\pi$
$19$		−	2.28415i		−	0.524019i	0.965065	−	0.262010i	$-0.0843852\pi$
$20$		0			0
$21$		−	1.00000i		−	0.218218i
$22$		0			0
$23$			5.74226i			1.19734i	0.800994	+	0.598672i	$0.204305\pi$
$23$			5.74226i			1.19734i	−0.800994	+	0.598672i	$0.795695\pi$
$24$		0			0
$25$	−3.94567	+	3.07111i	−0.789134	+	0.614222i
$26$		0			0
$27$			1.38589i			0.266715i
$28$		0			0
$29$	2.28415			0.424155			0.212078	−	0.977253i	$-0.431977\pi$
$29$	2.28415			0.424155			0.212078	+	0.977253i	$0.431977\pi$
$30$		0			0
$31$		−	6.22982i		−	1.11891i	−0.828861	−	0.559454i	$-0.811011\pi$
$31$		−	6.22982i		−	1.11891i	0.828861	−	0.559454i	$-0.188989\pi$
$32$		0			0
$33$	−0.233093			−0.0405762
$34$		0			0
$35$	3.11491	+	9.07325i	0.526515	+	1.53366i
$36$		0			0
$37$	6.72820			1.10611			0.553055	−	0.833145i	$-0.313462\pi$
$37$	6.72820			1.10611			0.553055	+	0.833145i	$0.313462\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			2.66152i			0.415660i	0.978165	+	0.207830i	$0.0666400\pi$
$41$			2.66152i			0.415660i	−0.978165	+	0.207830i	$0.933360\pi$
$42$		0			0
$43$		−	6.89506i		−	1.05149i	−0.850643	−	0.525744i	$-0.823787\pi$
$43$		−	6.89506i		−	1.05149i	0.850643	−	0.525744i	$-0.176213\pi$
$44$		0			0
$45$	−2.13874	−	6.22982i	−0.318824	−	0.928686i
$46$		0			0
$47$	6.60840			0.963934			0.481967	−	0.876189i	$-0.339922\pi$
$47$	6.60840			0.963934			0.481967	+	0.876189i	$0.339922\pi$
$48$		0			0
$49$	11.4053			1.62933
$50$		0			0
$51$	−0.891336			−0.124812
$52$		0			0
$53$		−	5.04299i		−	0.692707i	−0.938104	−	0.346354i	$-0.887420\pi$
$53$		−	5.04299i		−	0.692707i	0.938104	−	0.346354i	$-0.112580\pi$
$54$		0			0
$55$	2.11491	−	0.726062i	0.285174	−	0.0979022i
$56$		0			0
$57$	0.532418			0.0705205
$58$		0			0
$59$			11.1212i			1.44785i	0.689878	+	0.723925i	$0.257664\pi$
$59$			11.1212i			1.44785i	−0.689878	+	0.723925i	$0.742336\pi$
$60$		0			0
$61$	4.17548			0.534616			0.267308	−	0.963611i	$-0.413866\pi$
$61$	4.17548			0.534616			0.267308	+	0.963611i	$0.413866\pi$
$62$		0			0
$63$	−12.6373			−1.59215
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−4.34371			−0.530668			−0.265334	−	0.964157i	$-0.585482\pi$
$67$	−4.34371			−0.530668			−0.265334	+	0.964157i	$0.585482\pi$
$68$		0			0
$69$	−1.33848			−0.161134
$70$		0			0
$71$		−	6.74378i		−	0.800339i	−0.916441	−	0.400170i	$-0.868951\pi$
$71$		−	6.74378i		−	0.800339i	0.916441	−	0.400170i	$-0.131049\pi$
$72$		0			0
$73$	−13.5697			−1.58821			−0.794106	−	0.607779i	$-0.792061\pi$
$73$	−13.5697			−1.58821			−0.794106	+	0.607779i	$0.792061\pi$
$74$		0			0
$75$	−0.715853	−	0.919706i	−0.0826596	−	0.106199i
$76$		0			0
$77$		−	4.29014i		−	0.488907i
$78$		0			0
$79$	−11.4053			−1.28320			−0.641598	−	0.767041i	$-0.721728\pi$
$79$	−11.4053			−1.28320			−0.641598	+	0.767041i	$0.721728\pi$
$80$		0			0
$81$	8.51396			0.945996
$82$		0			0
$83$	10.0324			1.10120			0.550600	−	0.834769i	$-0.314399\pi$
$83$	10.0324			1.10120			0.550600	+	0.834769i	$0.314399\pi$
$84$		0			0
$85$	8.08731	−	2.77643i	0.877192	−	0.301146i
$86$		0			0
$87$			0.532418i			0.0570812i
$88$		0			0
$89$		−	11.1212i		−	1.17884i	−0.807827	−	0.589420i	$-0.799356\pi$
$89$		−	11.1212i		−	1.17884i	0.807827	−	0.589420i	$-0.200644\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	1.45212			0.150578
$94$		0			0
$95$	−4.83076	+	1.65843i	−0.495626	+	0.170152i
$96$		0			0
$97$	14.8423			1.50701			0.753503	−	0.657444i	$-0.228362\pi$
$97$	14.8423			1.50701			0.753503	+	0.657444i	$0.228362\pi$
$98$		0			0
$99$			2.94567i			0.296051i
$100$		0			0
$101$	18.9736			1.88794			0.943972	−	0.330027i	$-0.107058\pi$
$101$	18.9736			1.88794			0.943972	+	0.330027i	$0.107058\pi$
$102$		0			0
$103$		−	17.9796i		−	1.77159i	−0.464080	−	0.885793i	$-0.653615\pi$
$103$		−	17.9796i		−	1.77159i	0.464080	−	0.885793i	$-0.346385\pi$
$104$		0			0
$105$	−2.11491	+	0.726062i	−0.206394	+	0.0708564i
$106$		0			0
$107$			4.58946i			0.443680i	0.975083	+	0.221840i	$0.0712063\pi$
$107$			4.58946i			0.443680i	−0.975083	+	0.221840i	$0.928794\pi$
$108$		0			0
$109$			0.147558i			0.0141335i	0.999975	+	0.00706676i	$0.00224944\pi$
$109$			0.147558i			0.0141335i	−0.999975	+	0.00706676i	$0.997751\pi$
$110$		0			0
$111$			1.56829i			0.148856i
$112$		0			0
$113$		−	16.1084i		−	1.51535i	−0.652632	−	0.757675i	$-0.726335\pi$
$113$		−	16.1084i		−	1.51535i	0.652632	−	0.757675i	$-0.273665\pi$
$114$		0			0
$115$	12.1444	−	4.16924i	1.13247	−	0.388784i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	16.4053i		−	1.50387i
$120$		0			0
$121$	10.0000			0.909091
$122$		0			0
$123$	−0.620381			−0.0559379
$124$		0			0
$125$	9.35991	+	6.11491i	0.837176	+	0.546934i
$126$		0			0
$127$		−	10.3788i		−	0.920968i	−0.887668	−	0.460484i	$-0.847676\pi$
$127$		−	10.3788i		−	0.920968i	0.887668	−	0.460484i	$-0.152324\pi$
$128$		0			0
$129$	1.60719			0.141505
$130$		0			0
$131$	−17.6351			−1.54079			−0.770393	−	0.637569i	$-0.779940\pi$
$131$	−17.6351			−1.54079			−0.770393	+	0.637569i	$0.779940\pi$
$132$		0			0
$133$			9.79931i			0.849708i
$134$		0			0
$135$	2.93103	−	1.00624i	0.252263	−	0.0866036i
$136$		0			0
$137$	6.42888			0.549256			0.274628	−	0.961551i	$-0.411445\pi$
$137$	6.42888			0.549256			0.274628	+	0.961551i	$0.411445\pi$
$138$		0			0
$139$	14.9736			1.27004			0.635022	−	0.772494i	$-0.280991\pi$
$139$	14.9736			1.27004			0.635022	+	0.772494i	$0.280991\pi$
$140$		0			0
$141$			1.54037i			0.129723i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−1.65843	−	4.83076i	−0.137725	−	0.401173i
$146$		0			0
$147$			2.65849i			0.219269i
$148$		0			0
$149$		−	15.6894i		−	1.28533i	−0.766148	−	0.642665i	$-0.777829\pi$
$149$		−	15.6894i		−	1.28533i	0.766148	−	0.642665i	$-0.222171\pi$
$150$		0			0
$151$			1.17548i			0.0956594i	0.998856	+	0.0478297i	$0.0152305\pi$
$151$			1.17548i			0.0956594i	−0.998856	+	0.0478297i	$0.984770\pi$
$152$		0			0
$153$			11.2641i			0.910648i
$154$		0			0
$155$	−13.1755	+	4.52323i	−1.05828	+	0.363315i
$156$		0			0
$157$		−	5.45560i		−	0.435405i	−0.976015	−	0.217702i	$-0.930144\pi$
$157$		−	5.45560i		−	0.435405i	0.976015	−	0.217702i	$-0.0698562\pi$
$158$		0			0
$159$	1.17548			0.0932219
$160$		0			0
$161$		−	24.6351i		−	1.94152i
$162$		0			0
$163$	−4.64303			−0.363670			−0.181835	−	0.983329i	$-0.558204\pi$
$163$	−4.64303			−0.363670			−0.181835	+	0.983329i	$0.558204\pi$
$164$		0			0
$165$	0.169240	+	0.492969i	0.0131753	+	0.0383776i
$166$		0			0
$167$	−3.19091			−0.246920			−0.123460	−	0.992350i	$-0.539399\pi$
$167$	−3.19091			−0.246920			−0.123460	+	0.992350i	$0.539399\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$		−	6.72834i		−	0.514529i
$172$		0			0
$173$			3.47106i			0.263900i	0.991256	+	0.131950i	$0.0421238\pi$
$173$			3.47106i			0.263900i	−0.991256	+	0.131950i	$0.957876\pi$
$174$		0			0
$175$	16.9275	−	13.1755i	1.27960	−	0.995973i
$176$		0			0
$177$	−2.59226			−0.194846
$178$		0			0
$179$	5.12115			0.382773			0.191386	−	0.981515i	$-0.438702\pi$
$179$	5.12115			0.382773			0.191386	+	0.981515i	$0.438702\pi$
$180$		0			0
$181$	4.14756			0.308286			0.154143	−	0.988049i	$-0.450738\pi$
$181$	4.14756			0.308286			0.154143	+	0.988049i	$0.450738\pi$
$182$		0			0
$183$			0.973274i			0.0719465i
$184$		0			0
$185$	−4.88509	−	14.2295i	−0.359159	−	1.04618i
$186$		0			0
$187$	−3.82395			−0.279635
$188$		0			0
$189$		−	5.94567i		−	0.432484i
$190$		0			0
$191$	−0.809079			−0.0585429			−0.0292714	−	0.999571i	$-0.509319\pi$
$191$	−0.809079			−0.0585429			−0.0292714	+	0.999571i	$0.509319\pi$
$192$		0			0
$193$	9.63245			0.693359			0.346679	−	0.937984i	$-0.387309\pi$
$193$	9.63245			0.693359			0.346679	+	0.937984i	$0.387309\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	16.2409			1.15711			0.578556	−	0.815642i	$-0.303616\pi$
$197$	16.2409			1.15711			0.578556	+	0.815642i	$0.303616\pi$
$198$		0			0
$199$	4.13659			0.293235			0.146618	−	0.989193i	$-0.453161\pi$
$199$	4.13659			0.293235			0.146618	+	0.989193i	$0.453161\pi$
$200$		0			0
$201$		−	1.01249i		−	0.0714153i
$202$		0			0
$203$	−9.79931			−0.687777
$204$		0			0
$205$	5.62887	−	1.93243i	0.393137	−	0.134967i
$206$		0			0
$207$			16.9148i			1.17566i
$208$		0			0
$209$	2.28415			0.157998
$210$		0			0
$211$	9.60719			0.661386			0.330693	−	0.943738i	$-0.392718\pi$
$211$	9.60719			0.661386			0.330693	+	0.943738i	$0.392718\pi$
$212$		0			0
$213$	1.57192			0.107707
$214$		0			0
$215$	−14.5824	+	5.00624i	−0.994513	+	0.341423i
$216$		0			0
$217$			26.7268i			1.81433i
$218$		0			0
$219$		−	3.16300i		−	0.213735i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	15.0627			1.00867			0.504337	−	0.863507i	$-0.331737\pi$
$223$	15.0627			1.00867			0.504337	+	0.863507i	$0.331737\pi$
$224$		0			0
$225$	−11.6226	+	9.04646i	−0.774842	+	0.603098i
$226$		0			0
$227$	−20.1119			−1.33487			−0.667436	−	0.744667i	$-0.732608\pi$
$227$	−20.1119			−1.33487			−0.667436	+	0.744667i	$0.732608\pi$
$228$		0			0
$229$			12.5683i			0.830536i	0.909699	+	0.415268i	$0.136312\pi$
$229$			12.5683i			0.830536i	−0.909699	+	0.415268i	$0.863688\pi$
$230$		0			0
$231$	1.00000			0.0657952
$232$		0			0
$233$			11.3712i			0.744954i	0.928041	+	0.372477i	$0.121492\pi$
$233$			11.3712i			0.744954i	−0.928041	+	0.372477i	$0.878508\pi$
$234$		0			0
$235$	−4.79811	−	13.9762i	−0.312994	−	0.911704i
$236$		0			0
$237$		−	2.65849i		−	0.172688i
$238$		0			0
$239$		−	18.3510i		−	1.18703i	−0.804825	−	0.593513i	$-0.797741\pi$
$239$		−	18.3510i		−	1.18703i	0.804825	−	0.593513i	$-0.202259\pi$
$240$		0			0
$241$			1.37737i			0.0887244i	0.999016	+	0.0443622i	$0.0141256\pi$
$241$			1.37737i			0.0887244i	−0.999016	+	0.0443622i	$0.985874\pi$
$242$		0			0
$243$			6.14222i			0.394023i
$244$		0			0
$245$	−8.28095	−	24.1212i	−0.529051	−	1.54104i
$246$		0			0
$247$		0			0
$248$		0			0
$249$			2.33848i			0.148195i
$250$		0			0
$251$	25.2298			1.59249			0.796246	−	0.604973i	$-0.206816\pi$
$251$	25.2298			1.59249			0.796246	+	0.604973i	$0.206816\pi$
$252$		0			0
$253$	−5.74226			−0.361013
$254$		0			0
$255$	0.647165	+	1.88509i	0.0405270	+	0.118049i
$256$		0			0
$257$			30.4374i			1.89864i	0.314318	+	0.949318i	$0.398224\pi$
$257$			30.4374i			1.89864i	−0.314318	+	0.949318i	$0.601776\pi$
$258$		0			0
$259$	−28.8649			−1.79358
$260$		0			0
$261$	6.72834			0.416474
$262$		0			0
$263$			3.24448i			0.200063i	0.994984	+	0.100032i	$0.0318944\pi$
$263$			3.24448i			0.200063i	−0.994984	+	0.100032i	$0.968106\pi$
$264$		0			0
$265$	−10.6654	+	3.66152i	−0.655173	+	0.224925i
$266$		0			0
$267$	2.59226			0.158644
$268$		0			0
$269$	1.10866			0.0675965			0.0337982	−	0.999429i	$-0.489240\pi$
$269$	1.10866			0.0675965			0.0337982	+	0.999429i	$0.489240\pi$
$270$		0			0
$271$			30.1755i			1.83303i	0.400000	+	0.916515i	$0.369010\pi$
$271$			30.1755i			1.83303i	−0.400000	+	0.916515i	$0.630990\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.07111	−	3.94567i	−0.185195	−	0.237933i
$276$		0			0
$277$		−	25.9739i		−	1.56062i	−0.625392	−	0.780311i	$-0.715061\pi$
$277$		−	25.9739i		−	1.56062i	0.625392	−	0.780311i	$-0.284939\pi$
$278$		0			0
$279$		−	18.3510i		−	1.09864i
$280$		0			0
$281$		−	17.7827i		−	1.06083i	−0.847740	−	0.530413i	$-0.822037\pi$
$281$		−	17.7827i		−	1.06083i	0.847740	−	0.530413i	$-0.177963\pi$
$282$		0			0
$283$			7.36125i			0.437581i	0.975772	+	0.218790i	$0.0702111\pi$
$283$			7.36125i			0.437581i	−0.975772	+	0.218790i	$0.929789\pi$
$284$		0			0
$285$	−0.386568	−	1.12601i	−0.0228983	−	0.0666994i
$286$		0			0
$287$		−	11.4183i		−	0.674001i
$288$		0			0
$289$	2.37737			0.139845
$290$		0			0
$291$			3.45963i			0.202807i
$292$		0			0
$293$	−16.7950			−0.981174			−0.490587	−	0.871392i	$-0.663218\pi$
$293$	−16.7950			−0.981174			−0.490587	+	0.871392i	$0.663218\pi$
$294$		0			0
$295$	23.5202	−	8.07465i	1.36940	−	0.470124i
$296$		0			0
$297$	−1.38589			−0.0804176
$298$		0			0
$299$		0			0
$300$		0			0
$301$			29.5808i			1.70501i
$302$		0			0
$303$			4.42260i			0.254072i
$304$		0			0
$305$	−3.03166	−	8.83076i	−0.173592	−	0.505648i
$306$		0			0
$307$	−3.12468			−0.178335			−0.0891674	−	0.996017i	$-0.528421\pi$
$307$	−3.12468			−0.178335			−0.0891674	+	0.996017i	$0.528421\pi$
$308$		0			0
$309$	4.19092			0.238413
$310$		0			0
$311$	18.6072			1.05512			0.527558	−	0.849519i	$-0.323108\pi$
$311$	18.6072			1.05512			0.527558	+	0.849519i	$0.323108\pi$
$312$		0			0
$313$			17.5732i			0.993295i	0.867952	+	0.496647i	$0.165436\pi$
$313$			17.5732i			0.993295i	−0.867952	+	0.496647i	$0.834564\pi$
$314$		0			0
$315$	9.17548	+	26.7268i	0.516980	+	1.50588i
$316$		0			0
$317$	22.2633			1.25043			0.625215	−	0.780453i	$-0.285011\pi$
$317$	22.2633			1.25043			0.625215	+	0.780453i	$0.285011\pi$
$318$		0			0
$319$			2.28415i			0.127888i
$320$		0			0
$321$	−1.06977			−0.0597088
$322$		0			0
$323$	8.73447			0.485999
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−0.0343947			−0.00190203
$328$		0			0
$329$	−28.3510			−1.56304
$330$		0			0
$331$			28.4053i			1.56130i	0.624971	+	0.780648i	$0.285111\pi$
$331$			28.4053i			1.56130i	−0.624971	+	0.780648i	$0.714889\pi$
$332$		0			0
$333$	19.8190			1.08608
$334$		0			0
$335$	3.15380	+	9.18654i	0.172311	+	0.501914i
$336$		0			0
$337$		−	8.87960i		−	0.483703i	−0.970313	−	0.241851i	$-0.922245\pi$
$337$		−	8.87960i		−	0.483703i	0.970313	−	0.241851i	$-0.0777547\pi$
$338$		0			0
$339$	3.75475			0.203930
$340$		0			0
$341$	6.22982			0.337363
$342$		0			0
$343$	−18.8993			−1.02047
$344$		0			0
$345$	0.971819	+	2.83076i	0.0523210	+	0.152403i
$346$		0			0
$347$		−	17.2804i		−	0.927658i	−0.885925	−	0.463829i	$-0.846475\pi$
$347$		−	17.2804i		−	0.927658i	0.885925	−	0.463829i	$-0.153525\pi$
$348$		0			0
$349$		−	10.7438i		−	0.575101i	−0.957765	−	0.287551i	$-0.907159\pi$
$349$		−	10.7438i		−	0.575101i	0.957765	−	0.287551i	$-0.0928410\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−32.1697			−1.71222			−0.856111	−	0.516793i	$-0.827126\pi$
$353$	−32.1697			−1.71222			−0.856111	+	0.516793i	$0.827126\pi$
$354$		0			0
$355$	−14.2625	+	4.89640i	−0.756973	+	0.259874i
$356$		0			0
$357$	3.82395			0.202385
$358$		0			0
$359$		−	13.8913i		−	0.733157i	−0.930387	−	0.366578i	$-0.880529\pi$
$359$		−	13.8913i		−	0.733157i	0.930387	−	0.366578i	$-0.119471\pi$
$360$		0			0
$361$	13.7827			0.725404
$362$		0			0
$363$			2.33093i			0.122342i
$364$		0			0
$365$	9.85244	+	28.6987i	0.515700	+	1.50216i
$366$		0			0
$367$		−	18.0459i		−	0.941987i	−0.882137	−	0.470993i	$-0.843896\pi$
$367$		−	18.0459i		−	0.941987i	0.882137	−	0.470993i	$-0.156104\pi$
$368$		0			0
$369$			7.83996i			0.408132i
$370$		0			0
$371$			21.6351i			1.12324i
$372$		0			0
$373$			20.9309i			1.08376i	0.840455	+	0.541882i	$0.182288\pi$
$373$			20.9309i			1.08376i	−0.840455	+	0.541882i	$0.817712\pi$
$374$		0			0
$375$	−1.42534	+	2.18173i	−0.0736042	+	0.112664i
$376$		0			0
$377$		0			0
$378$		0			0
$379$			24.4178i			1.25426i	0.778916	+	0.627129i	$0.215770\pi$
$379$			24.4178i			1.25426i	−0.778916	+	0.627129i	$0.784230\pi$
$380$		0			0
$381$	2.41922			0.123940
$382$		0			0
$383$	29.9774			1.53177			0.765887	−	0.642975i	$-0.222300\pi$
$383$	29.9774			1.53177			0.765887	+	0.642975i	$0.222300\pi$
$384$		0			0
$385$	−9.07325	+	3.11491i	−0.462416	+	0.158750i
$386$		0			0
$387$		−	20.3106i		−	1.03244i
$388$		0			0
$389$	−3.80908			−0.193128			−0.0965640	−	0.995327i	$-0.530785\pi$
$389$	−3.80908			−0.193128			−0.0965640	+	0.995327i	$0.530785\pi$
$390$		0			0
$391$	−21.9582			−1.11047
$392$		0			0
$393$		−	4.11062i		−	0.207353i
$394$		0			0
$395$	8.28095	+	24.1212i	0.416660	+	1.21367i
$396$		0			0
$397$	−11.5852			−0.581442			−0.290721	−	0.956808i	$-0.593895\pi$
$397$	−11.5852			−0.581442			−0.290721	+	0.956808i	$0.593895\pi$
$398$		0			0
$399$	−2.28415			−0.114350
$400$		0			0
$401$			19.4860i			0.973086i	0.873657	+	0.486543i	$0.161742\pi$
$401$			19.4860i			0.973086i	−0.873657	+	0.486543i	$0.838258\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−6.18166	−	18.0062i	−0.307169	−	0.894737i
$406$		0			0
$407$			6.72820i			0.333505i
$408$		0			0
$409$			4.59622i			0.227268i	0.993523	+	0.113634i	$0.0362492\pi$
$409$			4.59622i			0.227268i	−0.993523	+	0.113634i	$0.963751\pi$
$410$		0			0
$411$			1.49852i			0.0739167i
$412$		0			0
$413$		−	47.7113i		−	2.34772i
$414$		0			0
$415$	−7.28415	−	21.2176i	−0.357565	−	1.04153i
$416$		0			0
$417$			3.49023i			0.170918i
$418$		0			0
$419$	−6.77018			−0.330745			−0.165373	−	0.986231i	$-0.552883\pi$
$419$	−6.77018			−0.330745			−0.165373	+	0.986231i	$0.552883\pi$
$420$		0			0
$421$		−	16.1087i		−	0.785088i	−0.919733	−	0.392544i	$-0.871595\pi$
$421$		−	16.1087i		−	0.785088i	0.919733	−	0.392544i	$-0.128405\pi$
$422$		0			0
$423$	19.4662			0.946477
$424$		0			0
$425$	−11.7438	−	15.0881i	−0.569657	−	0.731878i
$426$		0			0
$427$	−17.9134			−0.866890
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	2.25774i		−	0.108751i	−0.998521	−	0.0543757i	$-0.982683\pi$
$431$		−	2.25774i		−	0.108751i	0.998521	−	0.0543757i	$-0.0173169\pi$
$432$		0			0
$433$			9.27956i			0.445947i	0.974824	+	0.222974i	$0.0715763\pi$
$433$			9.27956i			0.445947i	−0.974824	+	0.222974i	$0.928424\pi$
$434$		0			0
$435$	1.12601	−	0.386568i	0.0539883	−	0.0185345i
$436$		0			0
$437$	13.1162			0.627432
$438$		0			0
$439$	−25.9317			−1.23765			−0.618827	−	0.785527i	$-0.712392\pi$
$439$	−25.9317			−1.23765			−0.618827	+	0.785527i	$0.712392\pi$
$440$		0			0
$441$	33.5962			1.59982
$442$		0			0
$443$		−	19.4318i		−	0.923231i	−0.887080	−	0.461615i	$-0.847270\pi$
$443$		−	19.4318i		−	0.923231i	0.887080	−	0.461615i	$-0.152730\pi$
$444$		0			0
$445$	−23.5202	+	8.07465i	−1.11496	+	0.382775i
$446$		0			0
$447$	3.65709			0.172975
$448$		0			0
$449$		−	25.4721i		−	1.20210i	−0.799210	−	0.601052i	$-0.794749\pi$
$449$		−	25.4721i		−	1.20210i	0.799210	−	0.601052i	$-0.205251\pi$
$450$		0			0
$451$	−2.66152			−0.125326
$452$		0			0
$453$	−0.273996			−0.0128735
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−19.5516			−0.914584			−0.457292	−	0.889317i	$-0.651180\pi$
$457$	−19.5516			−0.914584			−0.457292	+	0.889317i	$0.651180\pi$
$458$		0			0
$459$	−5.29959			−0.247363
$460$		0			0
$461$			32.3355i			1.50602i	0.658012	+	0.753008i	$0.271398\pi$
$461$			32.3355i			1.50602i	−0.658012	+	0.753008i	$0.728602\pi$
$462$		0			0
$463$	28.8119			1.33900			0.669502	−	0.742810i	$-0.266507\pi$
$463$	28.8119			1.33900			0.669502	+	0.742810i	$0.266507\pi$
$464$		0			0
$465$	−1.05433	−	3.07111i	−0.0488935	−	0.142419i
$466$		0			0
$467$		−	24.8085i		−	1.14800i	−0.818856	−	0.573999i	$-0.805391\pi$
$467$		−	24.8085i		−	1.14800i	0.818856	−	0.573999i	$-0.194609\pi$
$468$		0			0
$469$	18.6351			0.860490
$470$		0			0
$471$	1.27166			0.0585950
$472$		0			0
$473$	6.89506			0.317035
$474$		0			0
$475$	7.01486	+	9.01249i	0.321864	+	0.413521i
$476$		0			0
$477$		−	14.8550i		−	0.680162i
$478$		0			0
$479$			6.37889i			0.291459i	0.989324	+	0.145729i	$0.0465529\pi$
$479$			6.37889i			0.291459i	−0.989324	+	0.145729i	$0.953447\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	5.74226			0.261282
$484$		0			0
$485$	−10.7764	−	31.3901i	−0.489332	−	1.42535i
$486$		0			0
$487$	−18.3452			−0.831300			−0.415650	−	0.909525i	$-0.636446\pi$
$487$	−18.3452			−0.831300			−0.415650	+	0.909525i	$0.636446\pi$
$488$		0			0
$489$		−	1.08226i		−	0.0489413i
$490$		0			0
$491$	−3.91926			−0.176874			−0.0884369	−	0.996082i	$-0.528187\pi$
$491$	−3.91926			−0.176874			−0.0884369	+	0.996082i	$0.528187\pi$
$492$		0			0
$493$			8.73447i			0.393381i
$494$		0			0
$495$	6.22982	−	2.13874i	0.280009	−	0.0961291i
$496$		0			0
$497$			28.9317i			1.29777i
$498$		0			0
$499$			23.6087i			1.05687i	0.848974	+	0.528435i	$0.177221\pi$
$499$			23.6087i			1.05687i	−0.848974	+	0.528435i	$0.822779\pi$
$500$		0			0
$501$		−	0.743777i		−	0.0332295i
$502$		0			0
$503$			18.4332i			0.821894i	0.911659	+	0.410947i	$0.134802\pi$
$503$			18.4332i			0.821894i	−0.911659	+	0.410947i	$0.865198\pi$
$504$		0			0
$505$	−13.7760	−	40.1274i	−0.613024	−	1.78565i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	13.6770i		−	0.606221i	−0.952955	−	0.303110i	$-0.901975\pi$
$509$		−	13.6770i		−	0.606221i	0.952955	−	0.303110i	$-0.0980251\pi$
$510$		0			0
$511$	58.2159			2.57532
$512$		0			0
$513$	3.16558			0.139764
$514$		0			0
$515$	−38.0253	+	13.0543i	−1.67559	+	0.575243i
$516$		0			0
$517$			6.60840i			0.290637i
$518$		0			0
$519$	−0.809079			−0.0355146
$520$		0			0
$521$	25.9472			1.13677			0.568383	−	0.822764i	$-0.307569\pi$
$521$	25.9472			1.13677			0.568383	+	0.822764i	$0.307569\pi$
$522$		0			0
$523$		−	32.8219i		−	1.43520i	−0.696454	−	0.717601i	$-0.745240\pi$
$523$		−	32.8219i		−	1.43520i	0.696454	−	0.717601i	$-0.254760\pi$
$524$		0			0
$525$	3.07111	+	3.94567i	0.134034	+	0.172203i
$526$		0			0
$527$	23.8225			1.03773
$528$		0			0
$529$	−9.97359			−0.433634
$530$		0			0
$531$			32.7592i			1.42163i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	9.70629	−	3.33224i	0.419640	−	0.144065i
$536$		0			0
$537$			1.19370i			0.0515121i
$538$		0			0
$539$			11.4053i			0.491261i
$540$		0			0
$541$		−	22.8370i		−	0.981839i	−0.871205	−	0.490920i	$-0.836661\pi$
$541$		−	22.8370i		−	0.981839i	0.871205	−	0.490920i	$-0.163339\pi$
$542$		0			0
$543$			0.966765i			0.0414879i
$544$		0			0
$545$	0.312072	−	0.107136i	0.0133677	−	0.00458922i
$546$		0			0
$547$			31.4513i			1.34476i	0.740207	+	0.672379i	$0.234728\pi$
$547$			31.4513i			1.34476i	−0.740207	+	0.672379i	$0.765272\pi$
$548$		0			0
$549$	12.2996			0.524934
$550$		0			0
$551$		−	5.21733i		−	0.222266i
$552$		0			0
$553$	48.9303			2.08073
$554$		0			0
$555$	3.31680	−	1.13868i	0.140790	−	0.0483342i
$556$		0			0
$557$	18.4332			0.781038			0.390519	−	0.920595i	$-0.372296\pi$
$557$	18.4332			0.781038			0.390519	+	0.920595i	$0.372296\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		−	0.891336i		−	0.0376322i
$562$		0			0
$563$			9.83371i			0.414441i	0.978294	+	0.207221i	$0.0664418\pi$
$563$			9.83371i			0.414441i	−0.978294	+	0.207221i	$0.933558\pi$
$564$		0			0
$565$	−34.0677	+	11.6957i	−1.43324	+	0.492041i
$566$		0			0
$567$	−36.5261			−1.53395
$568$		0			0
$569$	−43.2034			−1.81118			−0.905591	−	0.424153i	$-0.860572\pi$
$569$	−43.2034			−1.81118			−0.905591	+	0.424153i	$0.860572\pi$
$570$		0			0
$571$	17.3106			0.724424			0.362212	−	0.932096i	$-0.382022\pi$
$571$	17.3106			0.724424			0.362212	+	0.932096i	$0.382022\pi$
$572$		0			0
$573$		−	0.188590i		−	0.00787847i
$574$		0			0
$575$	−17.6351	−	22.6571i	−0.735435	−	0.944865i
$576$		0			0
$577$	−0.245757			−0.0102310			−0.00511550	−	0.999987i	$-0.501628\pi$
$577$	−0.245757			−0.0102310			−0.00511550	+	0.999987i	$0.501628\pi$
$578$		0			0
$579$			2.24525i			0.0933095i
$580$		0			0
$581$	−43.0404			−1.78562
$582$		0			0
$583$	5.04299			0.208859
$584$		0			0
$585$		0			0
$586$		0			0
$587$	18.3604			0.757815			0.378908	−	0.925434i	$-0.376300\pi$
$587$	18.3604			0.757815			0.378908	+	0.925434i	$0.376300\pi$
$588$		0			0
$589$	−14.2298			−0.586329
$590$		0			0
$591$			3.78562i			0.155720i
$592$		0			0
$593$	20.7514			0.852159			0.426079	−	0.904686i	$-0.359894\pi$
$593$	20.7514			0.852159			0.426079	+	0.904686i	$0.359894\pi$
$594$		0			0
$595$	−34.6957	+	11.9113i	−1.42238	+	0.488314i
$596$		0			0
$597$			0.964208i			0.0394624i
$598$		0			0
$599$	−1.80908			−0.0739170			−0.0369585	−	0.999317i	$-0.511767\pi$
$599$	−1.80908			−0.0739170			−0.0369585	+	0.999317i	$0.511767\pi$
$600$		0			0
$601$	−15.7717			−0.643341			−0.321671	−	0.946852i	$-0.604244\pi$
$601$	−15.7717			−0.643341			−0.321671	+	0.946852i	$0.604244\pi$
$602$		0			0
$603$	−12.7951			−0.521058
$604$		0			0
$605$	−7.26062	−	21.1491i	−0.295186	−	0.859832i
$606$		0			0
$607$		−	25.7879i		−	1.04670i	−0.852118	−	0.523349i	$-0.824682\pi$
$607$		−	25.7879i		−	1.04670i	0.852118	−	0.523349i	$-0.175318\pi$
$608$		0			0
$609$		−	2.28415i		−	0.0925583i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	5.88995			0.237893			0.118946	−	0.992901i	$-0.462048\pi$
$613$	5.88995			0.237893			0.118946	+	0.992901i	$0.462048\pi$
$614$		0			0
$615$	0.450435	+	1.31205i	0.0181633	+	0.0529069i
$616$		0			0
$617$	5.85556			0.235736			0.117868	−	0.993029i	$-0.462394\pi$
$617$	5.85556			0.235736			0.117868	+	0.993029i	$0.462394\pi$
$618$		0			0
$619$			29.6740i			1.19270i	0.802725	+	0.596350i	$0.203383\pi$
$619$			29.6740i			1.19270i	−0.802725	+	0.596350i	$0.796617\pi$
$620$		0			0
$621$	−7.95815			−0.319350
$622$		0			0
$623$			47.7113i			1.91151i
$624$		0			0
$625$	6.13659	−	24.2351i	0.245464	−	0.969406i
$626$		0			0
$627$			0.532418i			0.0212627i
$628$		0			0
$629$			25.7283i			1.02586i
$630$		0			0
$631$		−	30.2966i		−	1.20609i	−0.797707	−	0.603045i	$-0.793954\pi$
$631$		−	30.2966i		−	1.20609i	0.797707	−	0.603045i	$-0.206046\pi$
$632$		0			0
$633$			2.23936i			0.0890068i
$634$		0			0
$635$	−21.9502	+	7.53564i	−0.871066	+	0.299043i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		−	19.8649i		−	0.785844i
$640$		0			0
$641$	17.9721			0.709854			0.354927	−	0.934894i	$-0.384506\pi$
$641$	17.9721			0.709854			0.354927	+	0.934894i	$0.384506\pi$
$642$		0			0
$643$	−26.7333			−1.05426			−0.527129	−	0.849785i	$-0.676732\pi$
$643$	−26.7333			−1.05426			−0.527129	+	0.849785i	$0.676732\pi$
$644$		0			0
$645$	−1.16692	−	3.39905i	−0.0459474	−	0.133838i
$646$		0			0
$647$		−	22.9756i		−	0.903263i	−0.892205	−	0.451631i	$-0.850842\pi$
$647$		−	22.9756i		−	0.903263i	0.892205	−	0.451631i	$-0.149158\pi$
$648$		0			0
$649$	−11.1212			−0.436543
$650$		0			0
$651$	−6.22982			−0.244166
$652$		0			0
$653$		−	17.8001i		−	0.696572i	−0.937388	−	0.348286i	$-0.886764\pi$
$653$		−	17.8001i		−	0.696572i	0.937388	−	0.348286i	$-0.113236\pi$
$654$		0			0
$655$	12.8042	+	37.2966i	0.500301	+	1.45730i
$656$		0			0
$657$	−39.9718			−1.55945
$658$		0			0
$659$	−32.7174			−1.27449			−0.637244	−	0.770662i	$-0.719926\pi$
$659$	−32.7174			−1.27449			−0.637244	+	0.770662i	$0.719926\pi$
$660$		0			0
$661$			8.20189i			0.319017i	0.987197	+	0.159508i	$0.0509909\pi$
$661$			8.20189i			0.319017i	−0.987197	+	0.159508i	$0.949009\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	20.7246	−	7.11491i	0.803667	−	0.275904i
$666$		0			0
$667$			13.1162i			0.507860i
$668$		0			0
$669$			3.51101i			0.135744i
$670$		0			0
$671$			4.17548i			0.161193i
$672$		0			0
$673$			8.50789i			0.327955i	0.986464	+	0.163978i	$0.0524325\pi$
$673$			8.50789i			0.327955i	−0.986464	+	0.163978i	$0.947568\pi$
$674$		0			0
$675$	−4.25622	−	5.46827i	−0.163822	−	0.210474i
$676$		0			0
$677$		−	12.9710i		−	0.498518i	−0.968437	−	0.249259i	$-0.919813\pi$
$677$		−	12.9710i		−	0.498518i	0.968437	−	0.249259i	$-0.0801870\pi$
$678$		0			0
$679$	−63.6755			−2.44364
$680$		0			0
$681$		−	4.68793i		−	0.179642i
$682$		0			0
$683$	−29.3190			−1.12186			−0.560931	−	0.827863i	$-0.689557\pi$
$683$	−29.3190			−1.12186			−0.560931	+	0.827863i	$0.689557\pi$
$684$		0			0
$685$	−4.66776	−	13.5965i	−0.178346	−	0.519495i
$686$		0			0
$687$	−2.92958			−0.111770
$688$		0			0
$689$		0			0
$690$		0			0
$691$			37.2159i			1.41576i	0.706333	+	0.707880i	$0.250348\pi$
$691$			37.2159i			1.41576i	−0.706333	+	0.707880i	$0.749652\pi$
$692$		0			0
$693$		−	12.6373i		−	0.480052i
$694$		0			0
$695$	−10.8718	−	31.6678i	−0.412389	−	1.20123i
$696$		0			0
$697$	−10.1775			−0.385502
$698$		0			0
$699$	−2.65055			−0.100253
$700$		0			0
$701$	49.9472			1.88648			0.943240	−	0.332113i	$-0.107762\pi$
$701$	49.9472			1.88648			0.943240	+	0.332113i	$0.107762\pi$
$702$		0			0
$703$		−	15.3682i		−	0.579623i
$704$		0			0
$705$	3.25774	−	1.11840i	0.122694	−	0.0421215i
$706$		0			0
$707$	−81.3994			−3.06134
$708$		0			0
$709$		−	30.4876i		−	1.14498i	−0.819910	−	0.572492i	$-0.805977\pi$
$709$		−	30.4876i		−	1.14498i	0.819910	−	0.572492i	$-0.194023\pi$
$710$		0			0
$711$	−33.5962			−1.25996
$712$		0			0
$713$	35.7732			1.33972
$714$		0			0
$715$		0			0
$716$		0			0
$717$	4.27748			0.159745
$718$		0			0
$719$	−1.63360			−0.0609228			−0.0304614	−	0.999536i	$-0.509698\pi$
$719$	−1.63360			−0.0609228			−0.0304614	+	0.999536i	$0.509698\pi$
$720$		0			0
$721$			77.1352i			2.87266i
$722$		0			0
$723$	−0.321056			−0.0119402
$724$		0			0
$725$	−9.01249	+	7.01486i	−0.334715	+	0.260525i
$726$		0			0
$727$			17.7936i			0.659928i	0.943993	+	0.329964i	$0.107037\pi$
$727$			17.7936i			0.659928i	−0.943993	+	0.329964i	$0.892963\pi$
$728$		0			0
$729$	24.1102			0.892970
$730$		0			0
$731$	26.3664			0.975197
$732$		0			0
$733$	−22.1438			−0.817901			−0.408950	−	0.912557i	$-0.634105\pi$
$733$	−22.1438			−0.817901			−0.408950	+	0.912557i	$0.634105\pi$
$734$		0			0
$735$	5.62246	−	1.93023i	0.207388	−	0.0711976i
$736$		0			0
$737$		−	4.34371i		−	0.160003i
$738$		0			0
$739$		−	17.9846i		−	0.661573i	−0.943706	−	0.330786i	$-0.892686\pi$
$739$		−	17.9846i		−	0.661573i	0.943706	−	0.330786i	$-0.107314\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−15.3429			−0.562876			−0.281438	−	0.959579i	$-0.590811\pi$
$743$	−15.3429			−0.562876			−0.281438	+	0.959579i	$0.590811\pi$
$744$		0			0
$745$	−33.1817	+	11.3915i	−1.21568	+	0.417353i
$746$		0			0
$747$	29.5521			1.08126
$748$		0			0
$749$		−	19.6894i		−	0.719437i
$750$		0			0
$751$	−38.3525			−1.39950			−0.699751	−	0.714387i	$-0.746706\pi$
$751$	−38.3525			−1.39950			−0.699751	+	0.714387i	$0.746706\pi$
$752$		0			0
$753$			5.88088i			0.214311i
$754$		0			0
$755$	2.48604	−	0.853474i	0.0904762	−	0.0310611i
$756$		0			0
$757$			2.64583i			0.0961642i	0.998843	+	0.0480821i	$0.0153109\pi$
$757$			2.64583i			0.0961642i	−0.998843	+	0.0480821i	$0.984689\pi$
$758$		0			0
$759$		−	1.33848i		−	0.0485837i
$760$		0			0
$761$		−	24.6351i		−	0.893022i	−0.894778	−	0.446511i	$-0.852666\pi$
$761$		−	24.6351i		−	0.893022i	0.894778	−	0.446511i	$-0.147334\pi$
$762$		0			0
$763$		−	0.633045i		−	0.0229178i
$764$		0			0
$765$	23.8225	−	8.17843i	0.861305	−	0.295692i
$766$		0			0
$767$		0			0
$768$		0			0
$769$			14.1102i			0.508826i	0.967096	+	0.254413i	$0.0818823\pi$
$769$			14.1102i			0.508826i	−0.967096	+	0.254413i	$0.918118\pi$
$770$		0			0
$771$	−7.09474			−0.255511
$772$		0			0
$773$	−15.1416			−0.544606			−0.272303	−	0.962212i	$-0.587785\pi$
$773$	−15.1416			−0.544606			−0.272303	+	0.962212i	$0.587785\pi$
$774$		0			0
$775$	19.1324	+	24.5808i	0.687257	+	0.882968i
$776$		0			0
$777$		−	6.72820i		−	0.241373i
$778$		0			0
$779$	6.07930			0.217814
$780$		0			0
$781$	6.74378			0.241311
$782$		0			0
$783$			3.16558i			0.113129i
$784$		0			0
$785$	−11.5381	+	3.96111i	−0.411812	+	0.141378i
$786$		0			0
$787$	−32.0564			−1.14269			−0.571344	−	0.820711i	$-0.693578\pi$
$787$	−32.0564			−1.14269			−0.571344	+	0.820711i	$0.693578\pi$
$788$		0			0
$789$	−0.756264			−0.0269237
$790$		0			0
$791$			69.1072i			2.45717i
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−0.853474	−	2.48604i	−0.0302696	−	0.0881707i
$796$		0			0
$797$			43.2478i			1.53191i	0.642891	+	0.765957i	$0.277735\pi$
$797$			43.2478i			1.53191i	−0.642891	+	0.765957i	$0.722265\pi$
$798$		0			0
$799$			25.2702i			0.893996i
$800$		0			0
$801$		−	32.7592i		−	1.15749i
$802$		0			0
$803$		−	13.5697i		−	0.478864i
$804$		0			0
$805$	−52.1010	+	17.8866i	−1.83632	+	0.630421i
$806$		0			0
$807$			0.258422i			0.00909687i
$808$		0			0
$809$	11.7547			0.413275			0.206637	−	0.978418i	$-0.433748\pi$
$809$	11.7547			0.413275			0.206637	+	0.978418i	$0.433748\pi$
$810$		0			0
$811$			32.4068i			1.13796i	0.822352	+	0.568979i	$0.192661\pi$
$811$			32.4068i			1.13796i	−0.822352	+	0.568979i	$0.807339\pi$
$812$		0			0
$813$	−7.03368			−0.246682
$814$		0			0
$815$	3.37113	+	9.81959i	0.118086	+	0.343965i
$816$		0			0
$817$	−15.7493			−0.551000
$818$		0			0
$819$		0			0
$820$		0			0
$821$			32.1755i			1.12293i	0.827500	+	0.561466i	$0.189762\pi$
$821$			32.1755i			1.12293i	−0.827500	+	0.561466i	$0.810238\pi$
$822$		0			0
$823$			26.5129i			0.924180i	0.886833	+	0.462090i	$0.152900\pi$
$823$			26.5129i			0.924180i	−0.886833	+	0.462090i	$0.847100\pi$
$824$		0			0
$825$	0.919706	−	0.715853i	0.0320201	−	0.0249228i
$826$		0			0
$827$	50.5431			1.75756			0.878779	−	0.477229i	$-0.158359\pi$
$827$	50.5431			1.75756			0.878779	+	0.477229i	$0.158359\pi$
$828$		0			0
$829$	6.89134			0.239346			0.119673	−	0.992813i	$-0.461815\pi$
$829$	6.89134			0.239346			0.119673	+	0.992813i	$0.461815\pi$
$830$		0			0
$831$	6.05433			0.210022
$832$		0			0
$833$			43.6133i			1.51111i
$834$		0			0
$835$	2.31680	+	6.74848i	0.0801761	+	0.233541i
$836$		0			0
$837$	8.63385			0.298429
$838$		0			0
$839$			46.2269i			1.59593i	0.602705	+	0.797964i	$0.294090\pi$
$839$			46.2269i			1.59593i	−0.602705	+	0.797964i	$0.705910\pi$
$840$		0			0
$841$	−23.7827			−0.820092
$842$		0			0
$843$	4.14501			0.142762
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−42.9014			−1.47411
$848$		0			0
$849$	−1.71585			−0.0588879
$850$		0			0
$851$			38.6351i			1.32439i
$852$		0			0
$853$	−17.3274			−0.593280			−0.296640	−	0.954989i	$-0.595866\pi$
$853$	−17.3274			−0.593280			−0.296640	+	0.954989i	$0.595866\pi$
$854$		0			0
$855$	−14.2298	+	4.88519i	−0.486649	+	0.167070i
$856$		0			0
$857$			0.186033i			0.00635478i	0.999995	+	0.00317739i	$0.00101140\pi$
$857$			0.186033i			0.00635478i	−0.999995	+	0.00317739i	$0.998989\pi$
$858$		0			0
$859$	−22.0947			−0.753863			−0.376931	−	0.926241i	$-0.623021\pi$
$859$	−22.0947			−0.753863			−0.376931	+	0.926241i	$0.623021\pi$
$860$		0			0
$861$	2.66152			0.0907044
$862$		0			0
$863$	17.5388			0.597027			0.298514	−	0.954405i	$-0.403509\pi$
$863$	17.5388			0.597027			0.298514	+	0.954405i	$0.403509\pi$
$864$		0			0
$865$	7.34097	−	2.52021i	0.249601	−	0.0856896i
$866$		0			0
$867$			0.554148i			0.0188199i
$868$		0			0
$869$		−	11.4053i		−	0.386898i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	43.7205			1.47971
$874$		0			0
$875$	−40.1553	−	26.2338i	−1.35750	−	0.886865i
$876$		0			0
$877$	−12.4234			−0.419509			−0.209754	−	0.977754i	$-0.567266\pi$
$877$	−12.4234			−0.419509			−0.209754	+	0.977754i	$0.567266\pi$
$878$		0			0
$879$		−	3.91479i		−	0.132043i
$880$		0			0
$881$	7.86341			0.264925			0.132463	−	0.991188i	$-0.457712\pi$
$881$	7.86341			0.264925			0.132463	+	0.991188i	$0.457712\pi$
$882$		0			0
$883$		−	18.9717i		−	0.638450i	−0.947679	−	0.319225i	$-0.896577\pi$
$883$		−	18.9717i		−	0.638450i	0.947679	−	0.319225i	$-0.103423\pi$
$884$		0			0
$885$	1.88214	+	5.48239i	0.0632675	+	0.184288i
$886$		0			0
$887$			11.8309i			0.397243i	0.980076	+	0.198622i	$0.0636465\pi$
$887$			11.8309i			0.397243i	−0.980076	+	0.198622i	$0.936354\pi$
$888$		0			0
$889$			44.5264i			1.49337i
$890$		0			0
$891$			8.51396i			0.285228i
$892$		0			0
$893$		−	15.0946i		−	0.505120i
$894$		0			0
$895$	−3.71827	−	10.8308i	−0.124288	−	0.362032i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	14.2298i		−	0.474591i
$900$		0			0
$901$	19.2841			0.642448
$902$		0			0
$903$	−6.89506			−0.229453
$904$		0			0
$905$	−3.01138	−	8.77170i	−0.100102	−	0.291581i
$906$		0			0
$907$			5.96885i			0.198192i	0.995078	+	0.0990962i	$0.0315951\pi$
$907$			5.96885i			0.198192i	−0.995078	+	0.0990962i	$0.968405\pi$
$908$		0			0
$909$	55.8899			1.85375
$910$		0			0
$911$	29.7952			0.987158			0.493579	−	0.869701i	$-0.335688\pi$
$911$	29.7952			0.987158			0.493579	+	0.869701i	$0.335688\pi$
$912$		0			0
$913$			10.0324i			0.332024i
$914$		0			0
$915$	2.05839	−	0.706658i	0.0680481	−	0.0233614i
$916$		0			0
$917$	75.6571			2.49842
$918$		0			0
$919$	24.0140			0.792149			0.396074	−	0.918218i	$-0.370372\pi$
$919$	24.0140			0.792149			0.396074	+	0.918218i	$0.370372\pi$
$920$		0			0
$921$		−	0.728339i		−	0.0239996i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−26.5473	+	20.6630i	−0.872868	+	0.679396i
$926$		0			0
$927$		−	52.9620i		−	1.73950i
$928$		0			0
$929$		−	0.700414i		−	0.0229799i	−0.999934	−	0.0114899i	$-0.996343\pi$
$929$		−	0.700414i		−	0.0229799i	0.999934	−	0.0114899i	$-0.00365744\pi$
$930$		0			0
$931$		−	26.0514i		−	0.853800i
$932$		0			0
$933$			4.33720i			0.141994i
$934$		0			0
$935$	2.77643	+	8.08731i	0.0907989	+	0.264483i
$936$		0			0
$937$		−	2.65849i		−	0.0868491i	−0.999057	−	0.0434246i	$-0.986173\pi$
$937$		−	2.65849i		−	0.0868491i	0.999057	−	0.0434246i	$-0.0138268\pi$
$938$		0			0
$939$	−4.09618			−0.133674
$940$		0			0
$941$		−	26.8370i		−	0.874861i	−0.899252	−	0.437431i	$-0.855889\pi$
$941$		−	26.8370i		−	0.874861i	0.899252	−	0.437431i	$-0.144111\pi$
$942$		0			0
$943$	−15.2832			−0.497688
$944$		0			0
$945$	−12.5745	+	4.31692i	−0.409050	+	0.140430i
$946$		0			0
$947$	−0.100627			−0.00326995			−0.00163498	−	0.999999i	$-0.500520\pi$
$947$	−0.100627			−0.00326995			−0.00163498	+	0.999999i	$0.500520\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$			5.18940i			0.168278i
$952$		0			0
$953$		−	54.8394i		−	1.77642i	−0.459434	−	0.888212i	$-0.651948\pi$
$953$		−	54.8394i		−	1.77642i	0.459434	−	0.888212i	$-0.348052\pi$
$954$		0			0
$955$	0.587441	+	1.71113i	0.0190092	+	0.0553708i
$956$		0			0
$957$	−0.532418			−0.0172106
$958$		0			0
$959$	−27.5808			−0.890630
$960$		0			0
$961$	−7.81060			−0.251955
$962$		0			0
$963$			13.5190i			0.435645i
$964$		0			0
$965$	−6.99376	−	20.3717i	−0.225137	−	0.655790i
$966$		0			0
$967$	57.0788			1.83553			0.917765	−	0.397123i	$-0.129991\pi$
$967$	57.0788			1.83553			0.917765	+	0.397123i	$0.129991\pi$
$968$		0			0
$969$			2.03594i			0.0654039i
$970$		0			0
$971$	−4.93470			−0.158362			−0.0791810	−	0.996860i	$-0.525231\pi$
$971$	−4.93470			−0.158362			−0.0791810	+	0.996860i	$0.525231\pi$
$972$		0			0
$973$	−64.2388			−2.05940
$974$		0			0
$975$		0			0
$976$		0			0
$977$	10.3918			0.332463			0.166232	−	0.986087i	$-0.446840\pi$
$977$	10.3918			0.332463			0.166232	+	0.986087i	$0.446840\pi$
$978$		0			0
$979$	11.1212			0.355434
$980$		0			0
$981$			0.434657i			0.0138775i
$982$		0			0
$983$	−45.8531			−1.46249			−0.731243	−	0.682117i	$-0.761059\pi$
$983$	−45.8531			−1.46249			−0.731243	+	0.682117i	$0.761059\pi$
$984$		0			0
$985$	−11.7919	−	34.3479i	−0.375720	−	1.09442i
$986$		0			0
$987$		−	6.60840i		−	0.210348i
$988$		0			0
$989$	39.5933			1.25899
$990$		0			0
$991$	−28.0140			−0.889894			−0.444947	−	0.895557i	$-0.646778\pi$
$991$	−28.0140			−0.889894			−0.444947	+	0.895557i	$0.646778\pi$
$992$		0			0
$993$	−6.62107			−0.210113
$994$		0			0
$995$	−3.00342	−	8.74850i	−0.0952148	−	0.277346i
$996$		0			0
$997$		−	12.1241i		−	0.383974i	−0.981398	−	0.191987i	$-0.938507\pi$
$997$		−	12.1241i		−	0.383974i	0.981398	−	0.191987i	$-0.0614931\pi$
$998$		0			0
$999$			9.32456i			0.295016i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.d.c.1689.7		12
5.4	even	2	inner	3380.2.d.c.1689.6		12
13.2	odd	12		260.2.ba.a.9.3	✓	12
13.5	odd	4		3380.2.c.a.2029.4		6
13.6	odd	12		260.2.ba.a.29.4	yes	12
13.8	odd	4		3380.2.c.b.2029.4		6
13.12	even	2	inner	3380.2.d.c.1689.8		12
39.2	even	12		2340.2.de.a.2089.5		12
39.32	even	12		2340.2.de.a.289.5		12
52.15	even	12		1040.2.dh.c.529.4		12
52.19	even	12		1040.2.dh.c.289.3		12
65.2	even	12		1300.2.i.i.1101.3		12
65.19	odd	12		260.2.ba.a.29.3	yes	12
65.28	even	12		1300.2.i.i.1101.4		12
65.32	even	12		1300.2.i.i.601.3		12
65.34	odd	4		3380.2.c.b.2029.3		6
65.44	odd	4		3380.2.c.a.2029.3		6
65.54	odd	12		260.2.ba.a.9.4	yes	12
65.58	even	12		1300.2.i.i.601.4		12
65.64	even	2	inner	3380.2.d.c.1689.5		12
195.119	even	12		2340.2.de.a.2089.6		12
195.149	even	12		2340.2.de.a.289.6		12
260.19	even	12		1040.2.dh.c.289.4		12
260.119	even	12		1040.2.dh.c.529.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.ba.a.9.3	✓	12	13.2	odd	12
260.2.ba.a.9.4	yes	12	65.54	odd	12
260.2.ba.a.29.3	yes	12	65.19	odd	12
260.2.ba.a.29.4	yes	12	13.6	odd	12
1040.2.dh.c.289.3		12	52.19	even	12
1040.2.dh.c.289.4		12	260.19	even	12
1040.2.dh.c.529.3		12	260.119	even	12
1040.2.dh.c.529.4		12	52.15	even	12
1300.2.i.i.601.3		12	65.32	even	12
1300.2.i.i.601.4		12	65.58	even	12
1300.2.i.i.1101.3		12	65.2	even	12
1300.2.i.i.1101.4		12	65.28	even	12
2340.2.de.a.289.5		12	39.32	even	12
2340.2.de.a.289.6		12	195.149	even	12
2340.2.de.a.2089.5		12	39.2	even	12
2340.2.de.a.2089.6		12	195.119	even	12
3380.2.c.a.2029.3		6	65.44	odd	4
3380.2.c.a.2029.4		6	13.5	odd	4
3380.2.c.b.2029.3		6	65.34	odd	4
3380.2.c.b.2029.4		6	13.8	odd	4
3380.2.d.c.1689.5		12	65.64	even	2	inner
3380.2.d.c.1689.6		12	5.4	even	2	inner
3380.2.d.c.1689.7		12	1.1	even	1	trivial
3380.2.d.c.1689.8		12	13.12	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.233093i q^{3} +(-0.726062 - 2.11491i) q^{5} -4.29014 q^{7} +2.94567 q^{9} +O(q^{10})\)	\(q+0.233093i q^{3} +(-0.726062 - 2.11491i) q^{5} -4.29014 q^{7} +2.94567 q^{9} +1.00000i q^{11} +(0.492969 - 0.169240i) q^{15} +3.82395i q^{17} -2.28415i q^{19} -1.00000i q^{21} +5.74226i q^{23} +(-3.94567 + 3.07111i) q^{25} +1.38589i q^{27} +2.28415 q^{29} -6.22982i q^{31} -0.233093 q^{33} +(3.11491 + 9.07325i) q^{35} +6.72820 q^{37} +2.66152i q^{41} -6.89506i q^{43} +(-2.13874 - 6.22982i) q^{45} +6.60840 q^{47} +11.4053 q^{49} -0.891336 q^{51} -5.04299i q^{53} +(2.11491 - 0.726062i) q^{55} +0.532418 q^{57} +11.1212i q^{59} +4.17548 q^{61} -12.6373 q^{63} -4.34371 q^{67} -1.33848 q^{69} -6.74378i q^{71} -13.5697 q^{73} +(-0.715853 - 0.919706i) q^{75} -4.29014i q^{77} -11.4053 q^{79} +8.51396 q^{81} +10.0324 q^{83} +(8.08731 - 2.77643i) q^{85} +0.532418i q^{87} -11.1212i q^{89} +1.45212 q^{93} +(-4.83076 + 1.65843i) q^{95} +14.8423 q^{97} +2.94567i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{9}+O(q^{10}) \) 12 * q - 8 * q^9	\( 12 q - 8 q^{9} - 4 q^{25} + 20 q^{29} + 12 q^{35} - 8 q^{49} + 76 q^{51} - 44 q^{61} - 52 q^{69} - 16 q^{75} + 8 q^{79} + 44 q^{81} - 40 q^{95}+O(q^{100}) \) 12 * q - 8 * q^9 - 4 * q^25 + 20 * q^29 + 12 * q^35 - 8 * q^49 + 76 * q^51 - 44 * q^61 - 52 * q^69 - 16 * q^75 + 8 * q^79 + 44 * q^81 - 40 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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