LMFDB - Embedded newform 3380.2.c.a.2029.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(2029,3380)]

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, 0]))

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

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

chi := DirichletCharacter("3380.2029");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.839056.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 6x^{4} + 8x^{2} + 1 $ x^6 + 6x^4 + 8x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2029.4
Root			$-2.02852i$ of defining polynomial
Character	$\chi$	$=$	3380.2029
Dual form			3380.2.c.a.2029.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.233093i q^{3} +(-2.11491 + 0.726062i) q^{5} -4.29014i q^{7} +2.94567 q^{9} +O(q^{10})$	$q+0.233093i q^{3} +(-2.11491 + 0.726062i) q^{5} -4.29014i q^{7} +2.94567 q^{9} -1.00000 q^{11} +(-0.169240 - 0.492969i) q^{15} -3.82395i q^{17} -2.28415 q^{19} +1.00000 q^{21} -5.74226i q^{23} +(3.94567 - 3.07111i) q^{25} +1.38589i q^{27} +2.28415 q^{29} -6.22982 q^{31} -0.233093i q^{33} +(3.11491 + 9.07325i) q^{35} +6.72820i q^{37} +2.66152 q^{41} +6.89506i q^{43} +(-6.22982 + 2.13874i) q^{45} +6.60840i q^{47} -11.4053 q^{49} +0.891336 q^{51} -5.04299i q^{53} +(2.11491 - 0.726062i) q^{55} -0.532418i q^{57} -11.1212 q^{59} +4.17548 q^{61} -12.6373i q^{63} +4.34371i q^{67} +1.33848 q^{69} -6.74378 q^{71} -13.5697i q^{73} +(0.715853 + 0.919706i) q^{75} +4.29014i q^{77} -11.4053 q^{79} +8.51396 q^{81} -10.0324i q^{83} +(2.77643 + 8.08731i) q^{85} +0.532418i q^{87} +11.1212 q^{89} -1.45212i q^{93} +(4.83076 - 1.65843i) q^{95} -14.8423i q^{97} -2.94567 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{9}+O(q^{10}) $ 6 * q - 4 * q^9	$ 6 q - 4 q^{9} - 6 q^{11} - 10 q^{15} - 10 q^{19} + 6 q^{21} + 2 q^{25} + 10 q^{29} - 12 q^{31} + 6 q^{35} - 2 q^{41} - 12 q^{45} + 4 q^{49} - 38 q^{51} + 2 q^{59} - 22 q^{61} + 26 q^{69} + 14 q^{71} + 8 q^{75} + 4 q^{79} + 22 q^{81} - 14 q^{85} - 2 q^{89} + 20 q^{95} + 4 q^{99}+O(q^{100}) $ 6 * q - 4 * q^9 - 6 * q^11 - 10 * q^15 - 10 * q^19 + 6 * q^21 + 2 * q^25 + 10 * q^29 - 12 * q^31 + 6 * q^35 - 2 * q^41 - 12 * q^45 + 4 * q^49 - 38 * q^51 + 2 * q^59 - 22 * q^61 + 26 * q^69 + 14 * q^71 + 8 * q^75 + 4 * q^79 + 22 * q^81 - 14 * q^85 - 2 * q^89 + 20 * q^95 + 4 * q^99

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
$5$	−2.11491	+	0.726062i	−0.945815	+	0.324705i
$5$	−2.11491	+	0.726062i	−0.945815	+	0.324705i
$6$		0			0
$7$		−	4.29014i		−	1.62152i	−0.585378	−	0.810760i	$-0.699054\pi$
$7$		−	4.29014i		−	1.62152i	0.585378	−	0.810760i	$-0.300946\pi$
$8$		0			0
$9$	2.94567			0.981889
$10$		0			0
$11$	−1.00000			−0.301511			−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000			−0.301511			−0.150756	+	0.988571i	$0.548171\pi$
$12$		0			0
$13$		0			0
$13$		0			0
$14$		0			0
$15$	−0.169240	−	0.492969i	−0.0436975	−	0.127284i
$16$		0			0
$17$		−	3.82395i		−	0.927445i	−0.885980	−	0.463723i	$-0.846513\pi$
$17$		−	3.82395i		−	0.927445i	0.885980	−	0.463723i	$-0.153487\pi$
$18$		0			0
$19$	−2.28415			−0.524019			−0.262010	−	0.965065i	$-0.584385\pi$
$19$	−2.28415			−0.524019			−0.262010	+	0.965065i	$0.584385\pi$
$20$		0			0
$21$	1.00000			0.218218
$22$		0			0
$23$		−	5.74226i		−	1.19734i	−0.800994	−	0.598672i	$-0.795695\pi$
$23$		−	5.74226i		−	1.19734i	0.800994	−	0.598672i	$-0.204305\pi$
$24$		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
$31$	−6.22982			−1.11891			−0.559454	−	0.828861i	$-0.688989\pi$
$31$	−6.22982			−1.11891			−0.559454	+	0.828861i	$0.688989\pi$
$32$		0			0
$33$		−	0.233093i		−	0.0405762i
$34$		0			0
$35$	3.11491	+	9.07325i	0.526515	+	1.53366i
$36$		0			0
$37$			6.72820i			1.10611i	0.833145	+	0.553055i	$0.186538\pi$
$37$			6.72820i			1.10611i	−0.833145	+	0.553055i	$0.813462\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.66152			0.415660			0.207830	−	0.978165i	$-0.433360\pi$
$41$	2.66152			0.415660			0.207830	+	0.978165i	$0.433360\pi$
$42$		0			0
$43$			6.89506i			1.05149i	0.850643	+	0.525744i	$0.176213\pi$
$43$			6.89506i			1.05149i	−0.850643	+	0.525744i	$0.823787\pi$
$44$		0			0
$45$	−6.22982	+	2.13874i	−0.928686	+	0.318824i
$46$		0			0
$47$			6.60840i			0.963934i	0.876189	+	0.481967i	$0.160078\pi$
$47$			6.60840i			0.963934i	−0.876189	+	0.481967i	$0.839922\pi$
$48$		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
$57$		−	0.532418i		−	0.0705205i
$58$		0			0
$59$	−11.1212			−1.44785			−0.723925	−	0.689878i	$-0.757664\pi$
$59$	−11.1212			−1.44785			−0.723925	+	0.689878i	$0.757664\pi$
$60$		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
$63$		−	12.6373i		−	1.59215i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			4.34371i			0.530668i	0.964157	+	0.265334i	$0.0854823\pi$
$67$			4.34371i			0.530668i	−0.964157	+	0.265334i	$0.914518\pi$
$68$		0			0
$69$	1.33848			0.161134
$70$		0			0
$71$	−6.74378			−0.800339			−0.400170	−	0.916441i	$-0.631049\pi$
$71$	−6.74378			−0.800339			−0.400170	+	0.916441i	$0.631049\pi$
$72$		0			0
$73$		−	13.5697i		−	1.58821i	−0.607779	−	0.794106i	$-0.707939\pi$
$73$		−	13.5697i		−	1.58821i	0.607779	−	0.794106i	$-0.292061\pi$
$74$		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
$83$		−	10.0324i		−	1.10120i	−0.834769	−	0.550600i	$-0.814399\pi$
$83$		−	10.0324i		−	1.10120i	0.834769	−	0.550600i	$-0.185601\pi$
$84$		0			0
$85$	2.77643	+	8.08731i	0.301146	+	0.877192i
$86$		0			0
$87$			0.532418i			0.0570812i
$88$		0			0
$89$	11.1212			1.17884			0.589420	−	0.807827i	$-0.299356\pi$
$89$	11.1212			1.17884			0.589420	+	0.807827i	$0.299356\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		−	1.45212i		−	0.150578i
$94$		0			0
$95$	4.83076	−	1.65843i	0.495626	−	0.170152i
$96$		0			0
$97$		−	14.8423i		−	1.50701i	−0.657444	−	0.753503i	$-0.728362\pi$
$97$		−	14.8423i		−	1.50701i	0.657444	−	0.753503i	$-0.271638\pi$
$98$		0			0
$99$	−2.94567			−0.296051
$100$		0			0
$101$	−18.9736			−1.88794			−0.943972	−	0.330027i	$-0.892942\pi$
$101$	−18.9736			−1.88794			−0.943972	+	0.330027i	$0.892942\pi$
$102$		0			0
$103$			17.9796i			1.77159i	0.464080	+	0.885793i	$0.346385\pi$
$103$			17.9796i			1.77159i	−0.464080	+	0.885793i	$0.653615\pi$
$104$		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
$109$	0.147558			0.0141335			0.00706676	−	0.999975i	$-0.497751\pi$
$109$	0.147558			0.0141335			0.00706676	+	0.999975i	$0.497751\pi$
$110$		0			0
$111$	−1.56829			−0.148856
$112$		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
$115$	4.16924	+	12.1444i	0.388784	+	1.13247i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−16.4053			−1.50387
$120$		0			0
$121$	−10.0000			−0.909091
$122$		0			0
$123$			0.620381i			0.0559379i
$124$		0			0
$125$	−6.11491	+	9.35991i	−0.546934	+	0.837176i
$126$		0			0
$127$			10.3788i			0.920968i	0.887668	+	0.460484i	$0.152324\pi$
$127$			10.3788i			0.920968i	−0.887668	+	0.460484i	$0.847676\pi$
$128$		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
$135$	−1.00624	−	2.93103i	−0.0866036	−	0.252263i
$136$		0			0
$137$			6.42888i			0.549256i	0.961551	+	0.274628i	$0.0885547\pi$
$137$			6.42888i			0.549256i	−0.961551	+	0.274628i	$0.911445\pi$
$138$		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
$141$	−1.54037			−0.129723
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−4.83076	+	1.65843i	−0.401173	+	0.137725i
$146$		0			0
$147$		−	2.65849i		−	0.219269i
$148$		0			0
$149$	−15.6894			−1.28533			−0.642665	−	0.766148i	$-0.722171\pi$
$149$	−15.6894			−1.28533			−0.642665	+	0.766148i	$0.722171\pi$
$150$		0			0
$151$	−1.17548			−0.0956594			−0.0478297	−	0.998856i	$-0.515230\pi$
$151$	−1.17548			−0.0956594			−0.0478297	+	0.998856i	$0.515230\pi$
$152$		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
$161$	−24.6351			−1.94152
$162$		0			0
$163$		−	4.64303i		−	0.363670i	−0.983329	−	0.181835i	$-0.941796\pi$
$163$		−	4.64303i		−	0.363670i	0.983329	−	0.181835i	$-0.0582037\pi$
$164$		0			0
$165$	0.169240	+	0.492969i	0.0131753	+	0.0383776i
$166$		0			0
$167$		−	3.19091i		−	0.246920i	−0.992350	−	0.123460i	$-0.960601\pi$
$167$		−	3.19091i		−	0.246920i	0.992350	−	0.123460i	$-0.0393991\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$	−6.72834			−0.514529
$172$		0			0
$173$		−	3.47106i		−	0.263900i	−0.991256	−	0.131950i	$-0.957876\pi$
$173$		−	3.47106i		−	0.263900i	0.991256	−	0.131950i	$-0.0421238\pi$
$174$		0			0
$175$	−13.1755	−	16.9275i	−0.995973	−	1.27960i
$176$		0			0
$177$		−	2.59226i		−	0.194846i
$178$		0			0
$179$	−5.12115			−0.382773			−0.191386	−	0.981515i	$-0.561298\pi$
$179$	−5.12115			−0.382773			−0.191386	+	0.981515i	$0.561298\pi$
$180$		0			0
$181$	−4.14756			−0.308286			−0.154143	−	0.988049i	$-0.549262\pi$
$181$	−4.14756			−0.308286			−0.154143	+	0.988049i	$0.549262\pi$
$182$		0			0
$183$			0.973274i			0.0719465i
$184$		0			0
$185$	−4.88509	−	14.2295i	−0.359159	−	1.04618i
$186$		0			0
$187$			3.82395i			0.279635i
$188$		0			0
$189$	5.94567			0.432484
$190$		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
$193$			9.63245i			0.693359i	0.937984	+	0.346679i	$0.112691\pi$
$193$			9.63245i			0.693359i	−0.937984	+	0.346679i	$0.887309\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	16.2409i		−	1.15711i	−0.815642	−	0.578556i	$-0.803616\pi$
$197$		−	16.2409i		−	1.15711i	0.815642	−	0.578556i	$-0.196384\pi$
$198$		0			0
$199$	−4.13659			−0.293235			−0.146618	−	0.989193i	$-0.546839\pi$
$199$	−4.13659			−0.293235			−0.146618	+	0.989193i	$0.546839\pi$
$200$		0			0
$201$	−1.01249			−0.0714153
$202$		0			0
$203$		−	9.79931i		−	0.687777i
$204$		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
$213$		−	1.57192i		−	0.107707i
$214$		0			0
$215$	−5.00624	−	14.5824i	−0.341423	−	0.994513i
$216$		0			0
$217$			26.7268i			1.81433i
$218$		0			0
$219$	3.16300			0.213735
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	15.0627i		−	1.00867i	−0.863507	−	0.504337i	$-0.831737\pi$
$223$		−	15.0627i		−	1.00867i	0.863507	−	0.504337i	$-0.168263\pi$
$224$		0			0
$225$	11.6226	−	9.04646i	0.774842	−	0.603098i
$226$		0			0
$227$			20.1119i			1.33487i	0.744667	+	0.667436i	$0.232608\pi$
$227$			20.1119i			1.33487i	−0.744667	+	0.667436i	$0.767392\pi$
$228$		0			0
$229$	−12.5683			−0.830536			−0.415268	−	0.909699i	$-0.636312\pi$
$229$	−12.5683			−0.830536			−0.415268	+	0.909699i	$0.636312\pi$
$230$		0			0
$231$	−1.00000			−0.0657952
$232$		0			0
$233$		−	11.3712i		−	0.744954i	−0.928041	−	0.372477i	$-0.878508\pi$
$233$		−	11.3712i		−	0.744954i	0.928041	−	0.372477i	$-0.121492\pi$
$234$		0			0
$235$	−4.79811	−	13.9762i	−0.312994	−	0.911704i
$236$		0			0
$237$		−	2.65849i		−	0.172688i
$238$		0			0
$239$	−18.3510			−1.18703			−0.593513	−	0.804825i	$-0.702259\pi$
$239$	−18.3510			−1.18703			−0.593513	+	0.804825i	$0.702259\pi$
$240$		0			0
$241$	−1.37737			−0.0887244			−0.0443622	−	0.999016i	$-0.514126\pi$
$241$	−1.37737			−0.0887244			−0.0443622	+	0.999016i	$0.514126\pi$
$242$		0			0
$243$			6.14222i			0.394023i
$244$		0			0
$245$	24.1212	−	8.28095i	1.54104	−	0.529051i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	2.33848			0.148195
$250$		0			0
$251$	−25.2298			−1.59249			−0.796246	−	0.604973i	$-0.793184\pi$
$251$	−25.2298			−1.59249			−0.796246	+	0.604973i	$0.793184\pi$
$252$		0			0
$253$			5.74226i			0.361013i
$254$		0			0
$255$	−1.88509	+	0.647165i	−0.118049	+	0.0405270i
$256$		0			0
$257$		−	30.4374i		−	1.89864i	−0.314318	−	0.949318i	$-0.601776\pi$
$257$		−	30.4374i		−	1.89864i	0.314318	−	0.949318i	$-0.398224\pi$
$258$		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
$265$	3.66152	+	10.6654i	0.224925	+	0.655173i
$266$		0			0
$267$			2.59226i			0.158644i
$268$		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
$271$	−30.1755			−1.83303			−0.916515	−	0.400000i	$-0.869010\pi$
$271$	−30.1755			−1.83303			−0.916515	+	0.400000i	$0.869010\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.94567	+	3.07111i	−0.237933	+	0.185195i
$276$		0			0
$277$			25.9739i			1.56062i	0.625392	+	0.780311i	$0.284939\pi$
$277$			25.9739i			1.56062i	−0.625392	+	0.780311i	$0.715061\pi$
$278$		0			0
$279$	−18.3510			−1.09864
$280$		0			0
$281$	17.7827			1.06083			0.530413	−	0.847740i	$-0.322037\pi$
$281$	17.7827			1.06083			0.530413	+	0.847740i	$0.322037\pi$
$282$		0			0
$283$		−	7.36125i		−	0.437581i	−0.975772	−	0.218790i	$-0.929789\pi$
$283$		−	7.36125i		−	0.437581i	0.975772	−	0.218790i	$-0.0702111\pi$
$284$		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
$291$	3.45963			0.202807
$292$		0			0
$293$		−	16.7950i		−	0.981174i	−0.871392	−	0.490587i	$-0.836782\pi$
$293$		−	16.7950i		−	0.981174i	0.871392	−	0.490587i	$-0.163218\pi$
$294$		0			0
$295$	23.5202	−	8.07465i	1.36940	−	0.470124i
$296$		0			0
$297$		−	1.38589i		−	0.0804176i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	29.5808			1.70501
$302$		0			0
$303$		−	4.42260i		−	0.254072i
$304$		0			0
$305$	−8.83076	+	3.03166i	−0.505648	+	0.173592i
$306$		0			0
$307$		−	3.12468i		−	0.178335i	−0.996017	−	0.0891674i	$-0.971579\pi$
$307$		−	3.12468i		−	0.178335i	0.996017	−	0.0891674i	$-0.0284206\pi$
$308$		0			0
$309$	−4.19092			−0.238413
$310$		0			0
$311$	−18.6072			−1.05512			−0.527558	−	0.849519i	$-0.676892\pi$
$311$	−18.6072			−1.05512			−0.527558	+	0.849519i	$0.676892\pi$
$312$		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
$317$		−	22.2633i		−	1.25043i	−0.780453	−	0.625215i	$-0.785011\pi$
$317$		−	22.2633i		−	1.25043i	0.780453	−	0.625215i	$-0.214989\pi$
$318$		0			0
$319$	−2.28415			−0.127888
$320$		0			0
$321$	−1.06977			−0.0597088
$322$		0			0
$323$			8.73447i			0.485999i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			0.0343947i			0.00190203i
$328$		0			0
$329$	28.3510			1.56304
$330$		0			0
$331$	28.4053			1.56130			0.780648	−	0.624971i	$-0.214889\pi$
$331$	28.4053			1.56130			0.780648	+	0.624971i	$0.214889\pi$
$332$		0			0
$333$			19.8190i			1.08608i
$334$		0			0
$335$	−3.15380	−	9.18654i	−0.172311	−	0.501914i
$336$		0			0
$337$			8.87960i			0.483703i	0.970313	+	0.241851i	$0.0777547\pi$
$337$			8.87960i			0.483703i	−0.970313	+	0.241851i	$0.922245\pi$
$338$		0			0
$339$	3.75475			0.203930
$340$		0			0
$341$	6.22982			0.337363
$342$		0			0
$343$			18.8993i			1.02047i
$344$		0			0
$345$	−2.83076	+	0.971819i	−0.152403	+	0.0523210i
$346$		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
$349$	10.7438			0.575101			0.287551	−	0.957765i	$-0.407159\pi$
$349$	10.7438			0.575101			0.287551	+	0.957765i	$0.407159\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			32.1697i			1.71222i	0.516793	+	0.856111i	$0.327126\pi$
$353$			32.1697i			1.71222i	−0.516793	+	0.856111i	$0.672874\pi$
$354$		0			0
$355$	14.2625	−	4.89640i	0.756973	−	0.259874i
$356$		0			0
$357$		−	3.82395i		−	0.202385i
$358$		0			0
$359$	13.8913			0.733157			0.366578	−	0.930387i	$-0.380529\pi$
$359$	13.8913			0.733157			0.366578	+	0.930387i	$0.380529\pi$
$360$		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
$369$	7.83996			0.408132
$370$		0			0
$371$	−21.6351			−1.12324
$372$		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
$375$	−2.18173	−	1.42534i	−0.112664	−	0.0736042i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	24.4178			1.25426			0.627129	−	0.778916i	$-0.284230\pi$
$379$	24.4178			1.25426			0.627129	+	0.778916i	$0.284230\pi$
$380$		0			0
$381$	−2.41922			−0.123940
$382$		0			0
$383$		−	29.9774i		−	1.53177i	−0.642975	−	0.765887i	$-0.722300\pi$
$383$		−	29.9774i		−	1.53177i	0.642975	−	0.765887i	$-0.277700\pi$
$384$		0			0
$385$	−3.11491	−	9.07325i	−0.158750	−	0.462416i
$386$		0			0
$387$			20.3106i			1.03244i
$388$		0			0
$389$	3.80908			0.193128			0.0965640	−	0.995327i	$-0.469215\pi$
$389$	3.80908			0.193128			0.0965640	+	0.995327i	$0.469215\pi$
$390$		0			0
$391$	−21.9582			−1.11047
$392$		0			0
$393$		−	4.11062i		−	0.207353i
$394$		0			0
$395$	24.1212	−	8.28095i	1.21367	−	0.416660i
$396$		0			0
$397$		−	11.5852i		−	0.581442i	−0.956808	−	0.290721i	$-0.906105\pi$
$397$		−	11.5852i		−	0.581442i	0.956808	−	0.290721i	$-0.0938952\pi$
$398$		0			0
$399$	−2.28415			−0.114350
$400$		0			0
$401$	−19.4860			−0.973086			−0.486543	−	0.873657i	$-0.661742\pi$
$401$	−19.4860			−0.973086			−0.486543	+	0.873657i	$0.661742\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−18.0062	+	6.18166i	−0.894737	+	0.307169i
$406$		0			0
$407$		−	6.72820i		−	0.333505i
$408$		0			0
$409$	4.59622			0.227268			0.113634	−	0.993523i	$-0.463751\pi$
$409$	4.59622			0.227268			0.113634	+	0.993523i	$0.463751\pi$
$410$		0			0
$411$	−1.49852			−0.0739167
$412$		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
$421$	−16.1087			−0.785088			−0.392544	−	0.919733i	$-0.628405\pi$
$421$	−16.1087			−0.785088			−0.392544	+	0.919733i	$0.628405\pi$
$422$		0			0
$423$			19.4662i			0.946477i
$424$		0			0
$425$	−11.7438	−	15.0881i	−0.569657	−	0.731878i
$426$		0			0
$427$		−	17.9134i		−	0.866890i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−2.25774			−0.108751			−0.0543757	−	0.998521i	$-0.517317\pi$
$431$	−2.25774			−0.108751			−0.0543757	+	0.998521i	$0.517317\pi$
$432$		0			0
$433$		−	9.27956i		−	0.445947i	−0.974824	−	0.222974i	$-0.928424\pi$
$433$		−	9.27956i		−	0.445947i	0.974824	−	0.222974i	$-0.0715763\pi$
$434$		0			0
$435$	−0.386568	−	1.12601i	−0.0185345	−	0.0539883i
$436$		0			0
$437$			13.1162i			0.627432i
$438$		0			0
$439$	25.9317			1.23765			0.618827	−	0.785527i	$-0.287608\pi$
$439$	25.9317			1.23765			0.618827	+	0.785527i	$0.287608\pi$
$440$		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
$447$		−	3.65709i		−	0.172975i
$448$		0			0
$449$	25.4721			1.20210			0.601052	−	0.799210i	$-0.294749\pi$
$449$	25.4721			1.20210			0.601052	+	0.799210i	$0.294749\pi$
$450$		0			0
$451$	−2.66152			−0.125326
$452$		0			0
$453$		−	0.273996i		−	0.0128735i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			19.5516i			0.914584i	0.889317	+	0.457292i	$0.151180\pi$
$457$			19.5516i			0.914584i	−0.889317	+	0.457292i	$0.848820\pi$
$458$		0			0
$459$	5.29959			0.247363
$460$		0			0
$461$	32.3355			1.50602			0.753008	−	0.658012i	$-0.228602\pi$
$461$	32.3355			1.50602			0.753008	+	0.658012i	$0.228602\pi$
$462$		0			0
$463$			28.8119i			1.33900i	0.742810	+	0.669502i	$0.233493\pi$
$463$			28.8119i			1.33900i	−0.742810	+	0.669502i	$0.766507\pi$
$464$		0			0
$465$	1.05433	+	3.07111i	0.0488935	+	0.142419i
$466$		0			0
$467$			24.8085i			1.14800i	0.818856	+	0.573999i	$0.194609\pi$
$467$			24.8085i			1.14800i	−0.818856	+	0.573999i	$0.805391\pi$
$468$		0			0
$469$	18.6351			0.860490
$470$		0			0
$471$	1.27166			0.0585950
$472$		0			0
$473$		−	6.89506i		−	0.317035i
$474$		0			0
$475$	−9.01249	+	7.01486i	−0.413521	+	0.321864i
$476$		0			0
$477$		−	14.8550i		−	0.680162i
$478$		0			0
$479$	−6.37889			−0.291459			−0.145729	−	0.989324i	$-0.546553\pi$
$479$	−6.37889			−0.291459			−0.145729	+	0.989324i	$0.546553\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		−	5.74226i		−	0.261282i
$484$		0			0
$485$	10.7764	+	31.3901i	0.489332	+	1.42535i
$486$		0			0
$487$			18.3452i			0.831300i	0.909525	+	0.415650i	$0.136446\pi$
$487$			18.3452i			0.831300i	−0.909525	+	0.415650i	$0.863554\pi$
$488$		0			0
$489$	1.08226			0.0489413
$490$		0			0
$491$	3.91926			0.176874			0.0884369	−	0.996082i	$-0.471813\pi$
$491$	3.91926			0.176874			0.0884369	+	0.996082i	$0.471813\pi$
$492$		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
$499$	23.6087			1.05687			0.528435	−	0.848974i	$-0.322779\pi$
$499$	23.6087			1.05687			0.528435	+	0.848974i	$0.322779\pi$
$500$		0			0
$501$	0.743777			0.0332295
$502$		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
$505$	40.1274	−	13.7760i	1.78565	−	0.613024i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−13.6770			−0.606221			−0.303110	−	0.952955i	$-0.598025\pi$
$509$	−13.6770			−0.606221			−0.303110	+	0.952955i	$0.598025\pi$
$510$		0			0
$511$	−58.2159			−2.57532
$512$		0			0
$513$		−	3.16558i		−	0.139764i
$514$		0			0
$515$	−13.0543	−	38.0253i	−0.575243	−	1.67559i
$516$		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
$525$	3.94567	−	3.07111i	0.172203	−	0.134034i
$526$		0			0
$527$			23.8225i			1.03773i
$528$		0			0
$529$	−9.97359			−0.433634
$530$		0			0
$531$	−32.7592			−1.42163
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−3.33224	−	9.70629i	−0.144065	−	0.419640i
$536$		0			0
$537$		−	1.19370i		−	0.0515121i
$538$		0			0
$539$	11.4053			0.491261
$540$		0			0
$541$	22.8370			0.981839			0.490920	−	0.871205i	$-0.336661\pi$
$541$	22.8370			0.981839			0.490920	+	0.871205i	$0.336661\pi$
$542$		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
$551$	−5.21733			−0.222266
$552$		0			0
$553$			48.9303i			2.08073i
$554$		0			0
$555$	3.31680	−	1.13868i	0.140790	−	0.0483342i
$556$		0			0
$557$			18.4332i			0.781038i	0.920595	+	0.390519i	$0.127704\pi$
$557$			18.4332i			0.781038i	−0.920595	+	0.390519i	$0.872296\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−0.891336			−0.0376322
$562$		0			0
$563$		−	9.83371i		−	0.414441i	−0.978294	−	0.207221i	$-0.933558\pi$
$563$		−	9.83371i		−	0.414441i	0.978294	−	0.207221i	$-0.0664418\pi$
$564$		0			0
$565$	11.6957	+	34.0677i	0.492041	+	1.43324i
$566$		0			0
$567$		−	36.5261i		−	1.53395i
$568$		0			0
$569$	43.2034			1.81118			0.905591	−	0.424153i	$-0.139428\pi$
$569$	43.2034			1.81118			0.905591	+	0.424153i	$0.139428\pi$
$570$		0			0
$571$	−17.3106			−0.724424			−0.362212	−	0.932096i	$-0.617978\pi$
$571$	−17.3106			−0.724424			−0.362212	+	0.932096i	$0.617978\pi$
$572$		0			0
$573$		−	0.188590i		−	0.00787847i
$574$		0			0
$575$	−17.6351	−	22.6571i	−0.735435	−	0.944865i
$576$		0			0
$577$			0.245757i			0.0102310i	0.999987	+	0.00511550i	$0.00162832\pi$
$577$			0.245757i			0.0102310i	−0.999987	+	0.00511550i	$0.998372\pi$
$578$		0			0
$579$	−2.24525			−0.0933095
$580$		0			0
$581$	−43.0404			−1.78562
$582$		0			0
$583$			5.04299i			0.208859i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	18.3604i		−	0.757815i	−0.925434	−	0.378908i	$-0.876300\pi$
$587$		−	18.3604i		−	0.757815i	0.925434	−	0.378908i	$-0.123700\pi$
$588$		0			0
$589$	14.2298			0.586329
$590$		0			0
$591$	3.78562			0.155720
$592$		0			0
$593$			20.7514i			0.852159i	0.904686	+	0.426079i	$0.140106\pi$
$593$			20.7514i			0.852159i	−0.904686	+	0.426079i	$0.859894\pi$
$594$		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
$603$			12.7951i			0.521058i
$604$		0			0
$605$	21.1491	−	7.26062i	0.859832	−	0.295186i
$606$		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
$609$	2.28415			0.0925583
$610$		0			0
$611$		0			0
$612$		0			0
$613$		−	5.88995i		−	0.237893i	−0.992901	−	0.118946i	$-0.962048\pi$
$613$		−	5.88995i		−	0.237893i	0.992901	−	0.118946i	$-0.0379517\pi$
$614$		0			0
$615$	−0.450435	−	1.31205i	−0.0181633	−	0.0529069i
$616$		0			0
$617$		−	5.85556i		−	0.235736i	−0.993029	−	0.117868i	$-0.962394\pi$
$617$		−	5.85556i		−	0.235736i	0.993029	−	0.117868i	$-0.0376059\pi$
$618$		0			0
$619$	−29.6740			−1.19270			−0.596350	−	0.802725i	$-0.703383\pi$
$619$	−29.6740			−1.19270			−0.596350	+	0.802725i	$0.703383\pi$
$620$		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
$629$	25.7283			1.02586
$630$		0			0
$631$	30.2966			1.20609			0.603045	−	0.797707i	$-0.293954\pi$
$631$	30.2966			1.20609			0.603045	+	0.797707i	$0.293954\pi$
$632$		0			0
$633$			2.23936i			0.0890068i
$634$		0			0
$635$	−7.53564	−	21.9502i	−0.299043	−	0.871066i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−19.8649			−0.785844
$640$		0			0
$641$	−17.9721			−0.709854			−0.354927	−	0.934894i	$-0.615494\pi$
$641$	−17.9721			−0.709854			−0.354927	+	0.934894i	$0.615494\pi$
$642$		0			0
$643$			26.7333i			1.05426i	0.849785	+	0.527129i	$0.176732\pi$
$643$			26.7333i			1.05426i	−0.849785	+	0.527129i	$0.823268\pi$
$644$		0			0
$645$	3.39905	−	1.16692i	0.133838	−	0.0459474i
$646$		0			0
$647$			22.9756i			0.903263i	0.892205	+	0.451631i	$0.149158\pi$
$647$			22.9756i			0.903263i	−0.892205	+	0.451631i	$0.850842\pi$
$648$		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
$655$	37.2966	−	12.8042i	1.45730	−	0.500301i
$656$		0			0
$657$		−	39.9718i		−	1.55945i
$658$		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
$661$	−8.20189			−0.319017			−0.159508	−	0.987197i	$-0.550991\pi$
$661$	−8.20189			−0.319017			−0.159508	+	0.987197i	$0.550991\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−7.11491	−	20.7246i	−0.275904	−	0.803667i
$666$		0			0
$667$		−	13.1162i		−	0.507860i
$668$		0			0
$669$	3.51101			0.135744
$670$		0			0
$671$	−4.17548			−0.161193
$672$		0			0
$673$		−	8.50789i		−	0.327955i	−0.986464	−	0.163978i	$-0.947568\pi$
$673$		−	8.50789i		−	0.327955i	0.986464	−	0.163978i	$-0.0524325\pi$
$674$		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
$681$	−4.68793			−0.179642
$682$		0			0
$683$		−	29.3190i		−	1.12186i	−0.827863	−	0.560931i	$-0.810443\pi$
$683$		−	29.3190i		−	1.12186i	0.827863	−	0.560931i	$-0.189557\pi$
$684$		0			0
$685$	−4.66776	−	13.5965i	−0.178346	−	0.519495i
$686$		0			0
$687$		−	2.92958i		−	0.111770i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	37.2159			1.41576			0.707880	−	0.706333i	$-0.249652\pi$
$691$	37.2159			1.41576			0.707880	+	0.706333i	$0.249652\pi$
$692$		0			0
$693$			12.6373i			0.480052i
$694$		0			0
$695$	−31.6678	+	10.8718i	−1.20123	+	0.412389i
$696$		0			0
$697$		−	10.1775i		−	0.385502i
$698$		0			0
$699$	2.65055			0.100253
$700$		0			0
$701$	−49.9472			−1.88648			−0.943240	−	0.332113i	$-0.892238\pi$
$701$	−49.9472			−1.88648			−0.943240	+	0.332113i	$0.892238\pi$
$702$		0			0
$703$		−	15.3682i		−	0.579623i
$704$		0			0
$705$	3.25774	−	1.11840i	0.122694	−	0.0421215i
$706$		0			0
$707$			81.3994i			3.06134i
$708$		0			0
$709$	30.4876			1.14498			0.572492	−	0.819910i	$-0.305977\pi$
$709$	30.4876			1.14498			0.572492	+	0.819910i	$0.305977\pi$
$710$		0			0
$711$	−33.5962			−1.25996
$712$		0			0
$713$			35.7732i			1.33972i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	4.27748i		−	0.159745i
$718$		0			0
$719$	1.63360			0.0609228			0.0304614	−	0.999536i	$-0.490302\pi$
$719$	1.63360			0.0609228			0.0304614	+	0.999536i	$0.490302\pi$
$720$		0			0
$721$	77.1352			2.87266
$722$		0			0
$723$		−	0.321056i		−	0.0119402i
$724$		0			0
$725$	9.01249	−	7.01486i	0.334715	−	0.260525i
$726$		0			0
$727$		−	17.7936i		−	0.659928i	−0.943993	−	0.329964i	$-0.892963\pi$
$727$		−	17.7936i		−	0.659928i	0.943993	−	0.329964i	$-0.107037\pi$
$728$		0			0
$729$	24.1102			0.892970
$730$		0			0
$731$	26.3664			0.975197
$732$		0			0
$733$			22.1438i			0.817901i	0.912557	+	0.408950i	$0.134105\pi$
$733$			22.1438i			0.817901i	−0.912557	+	0.408950i	$0.865895\pi$
$734$		0			0
$735$	1.93023	+	5.62246i	0.0711976	+	0.207388i
$736$		0			0
$737$		−	4.34371i		−	0.160003i
$738$		0			0
$739$	17.9846			0.661573			0.330786	−	0.943706i	$-0.392686\pi$
$739$	17.9846			0.661573			0.330786	+	0.943706i	$0.392686\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			15.3429i			0.562876i	0.959579	+	0.281438i	$0.0908113\pi$
$743$			15.3429i			0.562876i	−0.959579	+	0.281438i	$0.909189\pi$
$744$		0			0
$745$	33.1817	−	11.3915i	1.21568	−	0.417353i
$746$		0			0
$747$		−	29.5521i		−	1.08126i
$748$		0			0
$749$	19.6894			0.719437
$750$		0			0
$751$	38.3525			1.39950			0.699751	−	0.714387i	$-0.253294\pi$
$751$	38.3525			1.39950			0.699751	+	0.714387i	$0.253294\pi$
$752$		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
$759$	−1.33848			−0.0485837
$760$		0			0
$761$	24.6351			0.893022			0.446511	−	0.894778i	$-0.352666\pi$
$761$	24.6351			0.893022			0.446511	+	0.894778i	$0.352666\pi$
$762$		0			0
$763$		−	0.633045i		−	0.0229178i
$764$		0			0
$765$	8.17843	+	23.8225i	0.295692	+	0.861305i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	14.1102			0.508826			0.254413	−	0.967096i	$-0.418118\pi$
$769$	14.1102			0.508826			0.254413	+	0.967096i	$0.418118\pi$
$770$		0			0
$771$	7.09474			0.255511
$772$		0			0
$773$			15.1416i			0.544606i	0.962212	+	0.272303i	$0.0877854\pi$
$773$			15.1416i			0.544606i	−0.962212	+	0.272303i	$0.912215\pi$
$774$		0			0
$775$	−24.5808	+	19.1324i	−0.882968	+	0.687257i
$776$		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
$785$	3.96111	+	11.5381i	0.141378	+	0.411812i
$786$		0			0
$787$		−	32.0564i		−	1.14269i	−0.820711	−	0.571344i	$-0.806422\pi$
$787$		−	32.0564i		−	1.14269i	0.820711	−	0.571344i	$-0.193578\pi$
$788$		0			0
$789$	−0.756264			−0.0269237
$790$		0			0
$791$	−69.1072			−2.45717
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−2.48604	+	0.853474i	−0.0881707	+	0.0302696i
$796$		0			0
$797$		−	43.2478i		−	1.53191i	−0.642891	−	0.765957i	$-0.722265\pi$
$797$		−	43.2478i		−	1.53191i	0.642891	−	0.765957i	$-0.277735\pi$
$798$		0			0
$799$	25.2702			0.893996
$800$		0			0
$801$	32.7592			1.15749
$802$		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
$811$	32.4068			1.13796			0.568979	−	0.822352i	$-0.307339\pi$
$811$	32.4068			1.13796			0.568979	+	0.822352i	$0.307339\pi$
$812$		0			0
$813$		−	7.03368i		−	0.246682i
$814$		0			0
$815$	3.37113	+	9.81959i	0.118086	+	0.343965i
$816$		0			0
$817$		−	15.7493i		−	0.551000i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	32.1755			1.12293			0.561466	−	0.827500i	$-0.310238\pi$
$821$	32.1755			1.12293			0.561466	+	0.827500i	$0.310238\pi$
$822$		0			0
$823$		−	26.5129i		−	0.924180i	−0.886833	−	0.462090i	$-0.847100\pi$
$823$		−	26.5129i		−	0.924180i	0.886833	−	0.462090i	$-0.152900\pi$
$824$		0			0
$825$	−0.715853	−	0.919706i	−0.0249228	−	0.0320201i
$826$		0			0
$827$			50.5431i			1.75756i	0.477229	+	0.878779i	$0.341641\pi$
$827$			50.5431i			1.75756i	−0.477229	+	0.878779i	$0.658359\pi$
$828$		0			0
$829$	−6.89134			−0.239346			−0.119673	−	0.992813i	$-0.538185\pi$
$829$	−6.89134			−0.239346			−0.119673	+	0.992813i	$0.538185\pi$
$830$		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
$837$		−	8.63385i		−	0.298429i
$838$		0			0
$839$	−46.2269			−1.59593			−0.797964	−	0.602705i	$-0.794090\pi$
$839$	−46.2269			−1.59593			−0.797964	+	0.602705i	$0.794090\pi$
$840$		0			0
$841$	−23.7827			−0.820092
$842$		0			0
$843$			4.14501i			0.142762i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			42.9014i			1.47411i
$848$		0			0
$849$	1.71585			0.0588879
$850$		0			0
$851$	38.6351			1.32439
$852$		0			0
$853$		−	17.3274i		−	0.593280i	−0.954989	−	0.296640i	$-0.904134\pi$
$853$		−	17.3274i		−	0.593280i	0.954989	−	0.296640i	$-0.0958661\pi$
$854$		0			0
$855$	14.2298	−	4.88519i	0.486649	−	0.167070i
$856$		0			0
$857$		−	0.186033i		−	0.00635478i	−0.999995	−	0.00317739i	$-0.998989\pi$
$857$		−	0.186033i		−	0.00635478i	0.999995	−	0.00317739i	$-0.00101140\pi$
$858$		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
$863$		−	17.5388i		−	0.597027i	−0.954405	−	0.298514i	$-0.903509\pi$
$863$		−	17.5388i		−	0.597027i	0.954405	−	0.298514i	$-0.0964908\pi$
$864$		0			0
$865$	2.52021	+	7.34097i	0.0856896	+	0.249601i
$866$		0			0
$867$			0.554148i			0.0188199i
$868$		0			0
$869$	11.4053			0.386898
$870$		0			0
$871$		0			0
$872$		0			0
$873$		−	43.7205i		−	1.47971i
$874$		0			0
$875$	40.1553	+	26.2338i	1.35750	+	0.886865i
$876$		0			0
$877$			12.4234i			0.419509i	0.977754	+	0.209754i	$0.0672664\pi$
$877$			12.4234i			0.419509i	−0.977754	+	0.209754i	$0.932734\pi$
$878$		0			0
$879$	3.91479			0.132043
$880$		0			0
$881$	−7.86341			−0.264925			−0.132463	−	0.991188i	$-0.542288\pi$
$881$	−7.86341			−0.264925			−0.132463	+	0.991188i	$0.542288\pi$
$882$		0			0
$883$			18.9717i			0.638450i	0.947679	+	0.319225i	$0.103423\pi$
$883$			18.9717i			0.638450i	−0.947679	+	0.319225i	$0.896577\pi$
$884$		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
$889$	44.5264			1.49337
$890$		0			0
$891$	−8.51396			−0.285228
$892$		0			0
$893$		−	15.0946i		−	0.505120i
$894$		0			0
$895$	10.8308	−	3.71827i	0.362032	−	0.124288i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−14.2298			−0.474591
$900$		0			0
$901$	−19.2841			−0.642448
$902$		0			0
$903$			6.89506i			0.229453i
$904$		0			0
$905$	8.77170	−	3.01138i	0.291581	−	0.100102i
$906$		0			0
$907$		−	5.96885i		−	0.198192i	−0.995078	−	0.0990962i	$-0.968405\pi$
$907$		−	5.96885i		−	0.198192i	0.995078	−	0.0990962i	$-0.0315951\pi$
$908$		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
$915$	−0.706658	−	2.05839i	−0.0233614	−	0.0680481i
$916$		0			0
$917$			75.6571i			2.49842i
$918$		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
$921$	0.728339			0.0239996
$922$		0			0
$923$		0			0
$924$		0			0
$925$	20.6630	+	26.5473i	0.679396	+	0.872868i
$926$		0			0
$927$			52.9620i			1.73950i
$928$		0			0
$929$	−0.700414			−0.0229799			−0.0114899	−	0.999934i	$-0.503657\pi$
$929$	−0.700414			−0.0229799			−0.0114899	+	0.999934i	$0.503657\pi$
$930$		0			0
$931$	26.0514			0.853800
$932$		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
$941$	−26.8370			−0.874861			−0.437431	−	0.899252i	$-0.644111\pi$
$941$	−26.8370			−0.874861			−0.437431	+	0.899252i	$0.644111\pi$
$942$		0			0
$943$		−	15.2832i		−	0.497688i
$944$		0			0
$945$	−12.5745	+	4.31692i	−0.409050	+	0.140430i
$946$		0			0
$947$		−	0.100627i		−	0.00326995i	−0.999999	−	0.00163498i	$-0.999480\pi$
$947$		−	0.100627i		−	0.00326995i	0.999999	−	0.00163498i	$-0.000520429\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	5.18940			0.168278
$952$		0			0
$953$			54.8394i			1.77642i	0.459434	+	0.888212i	$0.348052\pi$
$953$			54.8394i			1.77642i	−0.459434	+	0.888212i	$0.651948\pi$
$954$		0			0
$955$	1.71113	−	0.587441i	0.0553708	−	0.0190092i
$956$		0			0
$957$		−	0.532418i		−	0.0172106i
$958$		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
$967$		−	57.0788i		−	1.83553i	−0.397123	−	0.917765i	$-0.629991\pi$
$967$		−	57.0788i		−	1.83553i	0.397123	−	0.917765i	$-0.370009\pi$
$968$		0			0
$969$	−2.03594			−0.0654039
$970$		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
$973$		−	64.2388i		−	2.05940i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	10.3918i		−	0.332463i	−0.986087	−	0.166232i	$-0.946840\pi$
$977$		−	10.3918i		−	0.332463i	0.986087	−	0.166232i	$-0.0531600\pi$
$978$		0			0
$979$	−11.1212			−0.355434
$980$		0			0
$981$	0.434657			0.0138775
$982$		0			0
$983$		−	45.8531i		−	1.46249i	−0.682117	−	0.731243i	$-0.738941\pi$
$983$		−	45.8531i		−	1.46249i	0.682117	−	0.731243i	$-0.261059\pi$
$984$		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
$993$			6.62107i			0.210113i
$994$		0			0
$995$	8.74850	−	3.00342i	0.277346	−	0.0952148i
$996$		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
$999$	−9.32456			−0.295016

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.c.a.2029.4		6
5.4	even	2	inner	3380.2.c.a.2029.3		6
13.3	even	3		260.2.ba.a.9.3	✓	12
13.5	odd	4		3380.2.d.c.1689.8		12
13.8	odd	4		3380.2.d.c.1689.7		12
13.9	even	3		260.2.ba.a.29.4	yes	12
13.12	even	2		3380.2.c.b.2029.4		6
39.29	odd	6		2340.2.de.a.2089.5		12
39.35	odd	6		2340.2.de.a.289.5		12
52.3	odd	6		1040.2.dh.c.529.4		12
52.35	odd	6		1040.2.dh.c.289.3		12
65.3	odd	12		1300.2.i.i.1101.4		12
65.9	even	6		260.2.ba.a.29.3	yes	12
65.22	odd	12		1300.2.i.i.601.3		12
65.29	even	6		260.2.ba.a.9.4	yes	12
65.34	odd	4		3380.2.d.c.1689.6		12
65.42	odd	12		1300.2.i.i.1101.3		12
65.44	odd	4		3380.2.d.c.1689.5		12
65.48	odd	12		1300.2.i.i.601.4		12
65.64	even	2		3380.2.c.b.2029.3		6
195.29	odd	6		2340.2.de.a.2089.6		12
195.74	odd	6		2340.2.de.a.289.6		12
260.139	odd	6		1040.2.dh.c.289.4		12
260.159	odd	6		1040.2.dh.c.529.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.ba.a.9.3	✓	12	13.3	even	3
260.2.ba.a.9.4	yes	12	65.29	even	6
260.2.ba.a.29.3	yes	12	65.9	even	6
260.2.ba.a.29.4	yes	12	13.9	even	3
1040.2.dh.c.289.3		12	52.35	odd	6
1040.2.dh.c.289.4		12	260.139	odd	6
1040.2.dh.c.529.3		12	260.159	odd	6
1040.2.dh.c.529.4		12	52.3	odd	6
1300.2.i.i.601.3		12	65.22	odd	12
1300.2.i.i.601.4		12	65.48	odd	12
1300.2.i.i.1101.3		12	65.42	odd	12
1300.2.i.i.1101.4		12	65.3	odd	12
2340.2.de.a.289.5		12	39.35	odd	6
2340.2.de.a.289.6		12	195.74	odd	6
2340.2.de.a.2089.5		12	39.29	odd	6
2340.2.de.a.2089.6		12	195.29	odd	6
3380.2.c.a.2029.3		6	5.4	even	2	inner
3380.2.c.a.2029.4		6	1.1	even	1	trivial
3380.2.c.b.2029.3		6	65.64	even	2
3380.2.c.b.2029.4		6	13.12	even	2
3380.2.d.c.1689.5		12	65.44	odd	4
3380.2.d.c.1689.6		12	65.34	odd	4
3380.2.d.c.1689.7		12	13.8	odd	4
3380.2.d.c.1689.8		12	13.5	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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.233093i q^{3} +(-2.11491 + 0.726062i) q^{5} -4.29014i q^{7} +2.94567 q^{9} +O(q^{10})\)	\(q+0.233093i q^{3} +(-2.11491 + 0.726062i) q^{5} -4.29014i q^{7} +2.94567 q^{9} -1.00000 q^{11} +(-0.169240 - 0.492969i) q^{15} -3.82395i q^{17} -2.28415 q^{19} +1.00000 q^{21} -5.74226i q^{23} +(3.94567 - 3.07111i) q^{25} +1.38589i q^{27} +2.28415 q^{29} -6.22982 q^{31} -0.233093i q^{33} +(3.11491 + 9.07325i) q^{35} +6.72820i q^{37} +2.66152 q^{41} +6.89506i q^{43} +(-6.22982 + 2.13874i) q^{45} +6.60840i q^{47} -11.4053 q^{49} +0.891336 q^{51} -5.04299i q^{53} +(2.11491 - 0.726062i) q^{55} -0.532418i q^{57} -11.1212 q^{59} +4.17548 q^{61} -12.6373i q^{63} +4.34371i q^{67} +1.33848 q^{69} -6.74378 q^{71} -13.5697i q^{73} +(0.715853 + 0.919706i) q^{75} +4.29014i q^{77} -11.4053 q^{79} +8.51396 q^{81} -10.0324i q^{83} +(2.77643 + 8.08731i) q^{85} +0.532418i q^{87} +11.1212 q^{89} -1.45212i q^{93} +(4.83076 - 1.65843i) q^{95} -14.8423i q^{97} -2.94567 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{9}+O(q^{10}) \) 6 * q - 4 * q^9	\( 6 q - 4 q^{9} - 6 q^{11} - 10 q^{15} - 10 q^{19} + 6 q^{21} + 2 q^{25} + 10 q^{29} - 12 q^{31} + 6 q^{35} - 2 q^{41} - 12 q^{45} + 4 q^{49} - 38 q^{51} + 2 q^{59} - 22 q^{61} + 26 q^{69} + 14 q^{71} + 8 q^{75} + 4 q^{79} + 22 q^{81} - 14 q^{85} - 2 q^{89} + 20 q^{95} + 4 q^{99}+O(q^{100}) \) 6 * q - 4 * q^9 - 6 * q^11 - 10 * q^15 - 10 * q^19 + 6 * q^21 + 2 * q^25 + 10 * q^29 - 12 * q^31 + 6 * q^35 - 2 * q^41 - 12 * q^45 + 4 * q^49 - 38 * q^51 + 2 * q^59 - 22 * q^61 + 26 * q^69 + 14 * q^71 + 8 * q^75 + 4 * q^79 + 22 * q^81 - 14 * q^85 - 2 * q^89 + 20 * q^95 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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