LMFDB - Embedded newform 6040.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.a (trivial)

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:	yes
Analytic conductor:	$48.2296428209$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 18 x^{10} + 54 x^{9} + 110 x^{8} - 335 x^{7} - 258 x^{6} + 825 x^{5} + 168 x^{4} + \cdots - 16 $ x^12 - 3x^11 - 18x^10 + 54x^9 + 110x^8 - 335x^7 - 258x^6 + 825x^5 + 168x^4 - 669x^3 + 39x^2 + 106*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.38247$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.38247 q^{3} -1.00000 q^{5} +3.11118 q^{7} +2.67614 q^{9} +O(q^{10})$	$q-2.38247 q^{3} -1.00000 q^{5} +3.11118 q^{7} +2.67614 q^{9} +0.421014 q^{11} +3.75654 q^{13} +2.38247 q^{15} -5.64771 q^{17} -1.71713 q^{19} -7.41228 q^{21} +0.413906 q^{23} +1.00000 q^{25} +0.771581 q^{27} +9.21271 q^{29} -3.25036 q^{31} -1.00305 q^{33} -3.11118 q^{35} -7.17074 q^{37} -8.94983 q^{39} +6.11100 q^{41} +6.53085 q^{43} -2.67614 q^{45} +7.84459 q^{47} +2.67944 q^{49} +13.4555 q^{51} +7.69162 q^{53} -0.421014 q^{55} +4.09100 q^{57} +1.58909 q^{59} -0.578276 q^{61} +8.32596 q^{63} -3.75654 q^{65} -2.21876 q^{67} -0.986117 q^{69} +7.37769 q^{71} +2.94319 q^{73} -2.38247 q^{75} +1.30985 q^{77} -3.73260 q^{79} -9.86669 q^{81} -7.52679 q^{83} +5.64771 q^{85} -21.9490 q^{87} -17.1318 q^{89} +11.6873 q^{91} +7.74387 q^{93} +1.71713 q^{95} -2.46964 q^{97} +1.12669 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{3} - 12 q^{5} + 5 q^{7} + 9 q^{9}+O(q^{10}) $ 12 * q + 3 * q^3 - 12 * q^5 + 5 * q^7 + 9 * q^9	$ 12 q + 3 q^{3} - 12 q^{5} + 5 q^{7} + 9 q^{9} + 10 q^{11} + 11 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} - q^{21} + 18 q^{23} + 12 q^{25} + 9 q^{27} + 16 q^{29} - q^{31} + 8 q^{33} - 5 q^{35} + 2 q^{37} + 6 q^{39} + 4 q^{41} + 7 q^{43} - 9 q^{45} + 3 q^{49} - 4 q^{51} + 39 q^{53} - 10 q^{55} - 15 q^{57} - 4 q^{59} - 32 q^{61} + 3 q^{63} - 11 q^{65} + 4 q^{67} + 12 q^{69} + 24 q^{71} - 10 q^{73} + 3 q^{75} + 38 q^{77} + 32 q^{79} - 8 q^{81} + 9 q^{83} + 4 q^{85} + 3 q^{87} + 15 q^{89} + 18 q^{91} + 36 q^{93} - 5 q^{95} + 15 q^{97} + 28 q^{99}+O(q^{100}) $ 12 * q + 3 * q^3 - 12 * q^5 + 5 * q^7 + 9 * q^9 + 10 * q^11 + 11 * q^13 - 3 * q^15 - 4 * q^17 + 5 * q^19 - q^21 + 18 * q^23 + 12 * q^25 + 9 * q^27 + 16 * q^29 - q^31 + 8 * q^33 - 5 * q^35 + 2 * q^37 + 6 * q^39 + 4 * q^41 + 7 * q^43 - 9 * q^45 + 3 * q^49 - 4 * q^51 + 39 * q^53 - 10 * q^55 - 15 * q^57 - 4 * q^59 - 32 * q^61 + 3 * q^63 - 11 * q^65 + 4 * q^67 + 12 * q^69 + 24 * q^71 - 10 * q^73 + 3 * q^75 + 38 * q^77 + 32 * q^79 - 8 * q^81 + 9 * q^83 + 4 * q^85 + 3 * q^87 + 15 * q^89 + 18 * q^91 + 36 * q^93 - 5 * q^95 + 15 * q^97 + 28 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−2.38247	−1.37552	−0.687759	−	0.725940i	$-0.741405\pi$
$3$	−2.38247	−1.37552	−0.687759	+	0.725940i	$0.741405\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.11118	1.17592	0.587958	−	0.808892i	$-0.299932\pi$
$7$	3.11118	1.17592	0.587958	+	0.808892i	$0.299932\pi$
$8$	0	0
$9$	2.67614	0.892047
$10$	0	0
$11$	0.421014	0.126941	0.0634703	−	0.997984i	$-0.479783\pi$
$11$	0.421014	0.126941	0.0634703	+	0.997984i	$0.479783\pi$
$12$	0	0
$13$	3.75654	1.04188	0.520939	−	0.853594i	$-0.325582\pi$
$13$	3.75654	1.04188	0.520939	+	0.853594i	$0.325582\pi$
$14$	0	0
$15$	2.38247	0.615150
$16$	0	0
$17$	−5.64771	−1.36977	−0.684886	−	0.728650i	$-0.740148\pi$
$17$	−5.64771	−1.36977	−0.684886	+	0.728650i	$0.740148\pi$
$18$	0	0
$19$	−1.71713	−0.393936	−0.196968	−	0.980410i	$-0.563110\pi$
$19$	−1.71713	−0.393936	−0.196968	+	0.980410i	$0.563110\pi$
$20$	0	0
$21$	−7.41228	−1.61749
$22$	0	0
$23$	0.413906	0.0863054	0.0431527	−	0.999068i	$-0.486260\pi$
$23$	0.413906	0.0863054	0.0431527	+	0.999068i	$0.486260\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0.771581	0.148491
$28$	0	0
$29$	9.21271	1.71076	0.855378	−	0.518004i	$-0.173325\pi$
$29$	9.21271	1.71076	0.855378	+	0.518004i	$0.173325\pi$
$30$	0	0
$31$	−3.25036	−0.583782	−0.291891	−	0.956452i	$-0.594284\pi$
$31$	−3.25036	−0.583782	−0.291891	+	0.956452i	$0.594284\pi$
$32$	0	0
$33$	−1.00305	−0.174609
$34$	0	0
$35$	−3.11118	−0.525885
$36$	0	0
$37$	−7.17074	−1.17886	−0.589431	−	0.807819i	$-0.700648\pi$
$37$	−7.17074	−1.17886	−0.589431	+	0.807819i	$0.700648\pi$
$38$	0	0
$39$	−8.94983	−1.43312
$40$	0	0
$41$	6.11100	0.954378	0.477189	−	0.878801i	$-0.341656\pi$
$41$	6.11100	0.954378	0.477189	+	0.878801i	$0.341656\pi$
$42$	0	0
$43$	6.53085	0.995946	0.497973	−	0.867193i	$-0.334078\pi$
$43$	6.53085	0.995946	0.497973	+	0.867193i	$0.334078\pi$
$44$	0	0
$45$	−2.67614	−0.398936
$46$	0	0
$47$	7.84459	1.14425	0.572126	−	0.820166i	$-0.306119\pi$
$47$	7.84459	1.14425	0.572126	+	0.820166i	$0.306119\pi$
$48$	0	0
$49$	2.67944	0.382777
$50$	0	0
$51$	13.4555	1.88414
$52$	0	0
$53$	7.69162	1.05652	0.528262	−	0.849081i	$-0.322844\pi$
$53$	7.69162	1.05652	0.528262	+	0.849081i	$0.322844\pi$
$54$	0	0
$55$	−0.421014	−0.0567696
$56$	0	0
$57$	4.09100	0.541866
$58$	0	0
$59$	1.58909	0.206882	0.103441	−	0.994636i	$-0.467015\pi$
$59$	1.58909	0.206882	0.103441	+	0.994636i	$0.467015\pi$
$60$	0	0
$61$	−0.578276	−0.0740407	−0.0370203	−	0.999315i	$-0.511787\pi$
$61$	−0.578276	−0.0740407	−0.0370203	+	0.999315i	$0.511787\pi$
$62$	0	0
$63$	8.32596	1.04897
$64$	0	0
$65$	−3.75654	−0.465942
$66$	0	0
$67$	−2.21876	−0.271065	−0.135533	−	0.990773i	$-0.543275\pi$
$67$	−2.21876	−0.271065	−0.135533	+	0.990773i	$0.543275\pi$
$68$	0	0
$69$	−0.986117	−0.118715
$70$	0	0
$71$	7.37769	0.875571	0.437785	−	0.899079i	$-0.355763\pi$
$71$	7.37769	0.875571	0.437785	+	0.899079i	$0.355763\pi$
$72$	0	0
$73$	2.94319	0.344475	0.172237	−	0.985055i	$-0.444900\pi$
$73$	2.94319	0.344475	0.172237	+	0.985055i	$0.444900\pi$
$74$	0	0
$75$	−2.38247	−0.275103
$76$	0	0
$77$	1.30985	0.149271
$78$	0	0
$79$	−3.73260	−0.419951	−0.209975	−	0.977707i	$-0.567338\pi$
$79$	−3.73260	−0.419951	−0.209975	+	0.977707i	$0.567338\pi$
$80$	0	0
$81$	−9.86669	−1.09630
$82$	0	0
$83$	−7.52679	−0.826172	−0.413086	−	0.910692i	$-0.635549\pi$
$83$	−7.52679	−0.826172	−0.413086	+	0.910692i	$0.635549\pi$
$84$	0	0
$85$	5.64771	0.612581
$86$	0	0
$87$	−21.9490	−2.35317
$88$	0	0
$89$	−17.1318	−1.81596	−0.907982	−	0.419008i	$-0.862378\pi$
$89$	−17.1318	−1.81596	−0.907982	+	0.419008i	$0.862378\pi$
$90$	0	0
$91$	11.6873	1.22516
$92$	0	0
$93$	7.74387	0.803002
$94$	0	0
$95$	1.71713	0.176174
$96$	0	0
$97$	−2.46964	−0.250754	−0.125377	−	0.992109i	$-0.540014\pi$
$97$	−2.46964	−0.250754	−0.125377	+	0.992109i	$0.540014\pi$
$98$	0	0
$99$	1.12669	0.113237
$100$	0	0
$101$	−15.4531	−1.53764	−0.768820	−	0.639466i	$-0.779156\pi$
$101$	−15.4531	−1.53764	−0.768820	+	0.639466i	$0.779156\pi$
$102$	0	0
$103$	−19.1961	−1.89145	−0.945725	−	0.324968i	$-0.894647\pi$
$103$	−19.1961	−1.89145	−0.945725	+	0.324968i	$0.894647\pi$
$104$	0	0
$105$	7.41228	0.723364
$106$	0	0
$107$	−2.16802	−0.209590	−0.104795	−	0.994494i	$-0.533419\pi$
$107$	−2.16802	−0.209590	−0.104795	+	0.994494i	$0.533419\pi$
$108$	0	0
$109$	−13.2648	−1.27054	−0.635268	−	0.772292i	$-0.719110\pi$
$109$	−13.2648	−1.27054	−0.635268	+	0.772292i	$0.719110\pi$
$110$	0	0
$111$	17.0840	1.62154
$112$	0	0
$113$	−7.58101	−0.713162	−0.356581	−	0.934264i	$-0.616058\pi$
$113$	−7.58101	−0.713162	−0.356581	+	0.934264i	$0.616058\pi$
$114$	0	0
$115$	−0.413906	−0.0385970
$116$	0	0
$117$	10.0530	0.929404
$118$	0	0
$119$	−17.5710	−1.61074
$120$	0	0
$121$	−10.8227	−0.983886
$122$	0	0
$123$	−14.5593	−1.31276
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	16.0922	1.42795	0.713977	−	0.700169i	$-0.246892\pi$
$127$	16.0922	1.42795	0.713977	+	0.700169i	$0.246892\pi$
$128$	0	0
$129$	−15.5595	−1.36994
$130$	0	0
$131$	18.5321	1.61916	0.809579	−	0.587012i	$-0.199696\pi$
$131$	18.5321	1.61916	0.809579	+	0.587012i	$0.199696\pi$
$132$	0	0
$133$	−5.34229	−0.463236
$134$	0	0
$135$	−0.771581	−0.0664071
$136$	0	0
$137$	19.3266	1.65118	0.825592	−	0.564267i	$-0.190841\pi$
$137$	19.3266	1.65118	0.825592	+	0.564267i	$0.190841\pi$
$138$	0	0
$139$	15.6656	1.32874	0.664370	−	0.747404i	$-0.268700\pi$
$139$	15.6656	1.32874	0.664370	+	0.747404i	$0.268700\pi$
$140$	0	0
$141$	−18.6895	−1.57394
$142$	0	0
$143$	1.58156	0.132257
$144$	0	0
$145$	−9.21271	−0.765074
$146$	0	0
$147$	−6.38367	−0.526516
$148$	0	0
$149$	13.6036	1.11445	0.557224	−	0.830362i	$-0.311866\pi$
$149$	13.6036	1.11445	0.557224	+	0.830362i	$0.311866\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	−15.1141	−1.22190
$154$	0	0
$155$	3.25036	0.261075
$156$	0	0
$157$	16.6353	1.32764	0.663822	−	0.747891i	$-0.268933\pi$
$157$	16.6353	1.32764	0.663822	+	0.747891i	$0.268933\pi$
$158$	0	0
$159$	−18.3250	−1.45327
$160$	0	0
$161$	1.28774	0.101488
$162$	0	0
$163$	−0.827251	−0.0647953	−0.0323976	−	0.999475i	$-0.510314\pi$
$163$	−0.827251	−0.0647953	−0.0323976	+	0.999475i	$0.510314\pi$
$164$	0	0
$165$	1.00305	0.0780875
$166$	0	0
$167$	10.9106	0.844291	0.422145	−	0.906528i	$-0.361277\pi$
$167$	10.9106	0.844291	0.422145	+	0.906528i	$0.361277\pi$
$168$	0	0
$169$	1.11162	0.0855090
$170$	0	0
$171$	−4.59528	−0.351410
$172$	0	0
$173$	9.69560	0.737143	0.368572	−	0.929599i	$-0.379847\pi$
$173$	9.69560	0.737143	0.368572	+	0.929599i	$0.379847\pi$
$174$	0	0
$175$	3.11118	0.235183
$176$	0	0
$177$	−3.78595	−0.284570
$178$	0	0
$179$	21.7348	1.62454	0.812268	−	0.583284i	$-0.198233\pi$
$179$	21.7348	1.62454	0.812268	+	0.583284i	$0.198233\pi$
$180$	0	0
$181$	−16.1369	−1.19945	−0.599725	−	0.800206i	$-0.704723\pi$
$181$	−16.1369	−1.19945	−0.599725	+	0.800206i	$0.704723\pi$
$182$	0	0
$183$	1.37772	0.101844
$184$	0	0
$185$	7.17074	0.527203
$186$	0	0
$187$	−2.37777	−0.173880
$188$	0	0
$189$	2.40053	0.174613
$190$	0	0
$191$	−7.55322	−0.546532	−0.273266	−	0.961939i	$-0.588104\pi$
$191$	−7.55322	−0.546532	−0.273266	+	0.961939i	$0.588104\pi$
$192$	0	0
$193$	−7.80287	−0.561663	−0.280831	−	0.959757i	$-0.590610\pi$
$193$	−7.80287	−0.561663	−0.280831	+	0.959757i	$0.590610\pi$
$194$	0	0
$195$	8.94983	0.640911
$196$	0	0
$197$	−24.5663	−1.75027	−0.875137	−	0.483875i	$-0.839229\pi$
$197$	−24.5663	−1.75027	−0.875137	+	0.483875i	$0.839229\pi$
$198$	0	0
$199$	6.57974	0.466425	0.233213	−	0.972426i	$-0.425076\pi$
$199$	6.57974	0.466425	0.233213	+	0.972426i	$0.425076\pi$
$200$	0	0
$201$	5.28613	0.372855
$202$	0	0
$203$	28.6624	2.01170
$204$	0	0
$205$	−6.11100	−0.426811
$206$	0	0
$207$	1.10767	0.0769885
$208$	0	0
$209$	−0.722935	−0.0500065
$210$	0	0
$211$	17.2300	1.18616	0.593082	−	0.805142i	$-0.297911\pi$
$211$	17.2300	1.18616	0.593082	+	0.805142i	$0.297911\pi$
$212$	0	0
$213$	−17.5771	−1.20436
$214$	0	0
$215$	−6.53085	−0.445401
$216$	0	0
$217$	−10.1125	−0.686478
$218$	0	0
$219$	−7.01205	−0.473831
$220$	0	0
$221$	−21.2159	−1.42713
$222$	0	0
$223$	3.42327	0.229239	0.114620	−	0.993409i	$-0.463435\pi$
$223$	3.42327	0.229239	0.114620	+	0.993409i	$0.463435\pi$
$224$	0	0
$225$	2.67614	0.178409
$226$	0	0
$227$	24.7541	1.64299	0.821494	−	0.570217i	$-0.193141\pi$
$227$	24.7541	1.64299	0.821494	+	0.570217i	$0.193141\pi$
$228$	0	0
$229$	−2.72251	−0.179908	−0.0899541	−	0.995946i	$-0.528672\pi$
$229$	−2.72251	−0.179908	−0.0899541	+	0.995946i	$0.528672\pi$
$230$	0	0
$231$	−3.12067	−0.205325
$232$	0	0
$233$	12.5400	0.821524	0.410762	−	0.911743i	$-0.365263\pi$
$233$	12.5400	0.821524	0.410762	+	0.911743i	$0.365263\pi$
$234$	0	0
$235$	−7.84459	−0.511725
$236$	0	0
$237$	8.89280	0.577650
$238$	0	0
$239$	6.25913	0.404869	0.202435	−	0.979296i	$-0.435115\pi$
$239$	6.25913	0.404869	0.202435	+	0.979296i	$0.435115\pi$
$240$	0	0
$241$	−19.8450	−1.27833	−0.639163	−	0.769071i	$-0.720719\pi$
$241$	−19.8450	−1.27833	−0.639163	+	0.769071i	$0.720719\pi$
$242$	0	0
$243$	21.1923	1.35949
$244$	0	0
$245$	−2.67944	−0.171183
$246$	0	0
$247$	−6.45047	−0.410433
$248$	0	0
$249$	17.9323	1.13641
$250$	0	0
$251$	14.5556	0.918742	0.459371	−	0.888244i	$-0.348075\pi$
$251$	14.5556	0.918742	0.459371	+	0.888244i	$0.348075\pi$
$252$	0	0
$253$	0.174260	0.0109557
$254$	0	0
$255$	−13.4555	−0.842615
$256$	0	0
$257$	30.1042	1.87785	0.938924	−	0.344124i	$-0.111824\pi$
$257$	30.1042	1.87785	0.938924	+	0.344124i	$0.111824\pi$
$258$	0	0
$259$	−22.3094	−1.38624
$260$	0	0
$261$	24.6545	1.52608
$262$	0	0
$263$	0.863700	0.0532580	0.0266290	−	0.999645i	$-0.491523\pi$
$263$	0.863700	0.0532580	0.0266290	+	0.999645i	$0.491523\pi$
$264$	0	0
$265$	−7.69162	−0.472492
$266$	0	0
$267$	40.8159	2.49789
$268$	0	0
$269$	−1.40819	−0.0858587	−0.0429293	−	0.999078i	$-0.513669\pi$
$269$	−1.40819	−0.0858587	−0.0429293	+	0.999078i	$0.513669\pi$
$270$	0	0
$271$	1.31229	0.0797161	0.0398580	−	0.999205i	$-0.487309\pi$
$271$	1.31229	0.0797161	0.0398580	+	0.999205i	$0.487309\pi$
$272$	0	0
$273$	−27.8445	−1.68523
$274$	0	0
$275$	0.421014	0.0253881
$276$	0	0
$277$	4.62566	0.277929	0.138964	−	0.990297i	$-0.455623\pi$
$277$	4.62566	0.277929	0.138964	+	0.990297i	$0.455623\pi$
$278$	0	0
$279$	−8.69842	−0.520761
$280$	0	0
$281$	21.5954	1.28827	0.644137	−	0.764910i	$-0.277217\pi$
$281$	21.5954	1.28827	0.644137	+	0.764910i	$0.277217\pi$
$282$	0	0
$283$	−21.3789	−1.27085	−0.635423	−	0.772164i	$-0.719174\pi$
$283$	−21.3789	−1.27085	−0.635423	+	0.772164i	$0.719174\pi$
$284$	0	0
$285$	−4.09100	−0.242330
$286$	0	0
$287$	19.0124	1.12227
$288$	0	0
$289$	14.8967	0.876275
$290$	0	0
$291$	5.88383	0.344916
$292$	0	0
$293$	24.0737	1.40640	0.703202	−	0.710990i	$-0.251753\pi$
$293$	24.0737	1.40640	0.703202	+	0.710990i	$0.251753\pi$
$294$	0	0
$295$	−1.58909	−0.0925204
$296$	0	0
$297$	0.324847	0.0188495
$298$	0	0
$299$	1.55486	0.0899197
$300$	0	0
$301$	20.3187	1.17115
$302$	0	0
$303$	36.8164	2.11505
$304$	0	0
$305$	0.578276	0.0331120
$306$	0	0
$307$	6.91259	0.394523	0.197261	−	0.980351i	$-0.436795\pi$
$307$	6.91259	0.394523	0.197261	+	0.980351i	$0.436795\pi$
$308$	0	0
$309$	45.7341	2.60172
$310$	0	0
$311$	−21.2053	−1.20244	−0.601220	−	0.799083i	$-0.705319\pi$
$311$	−21.2053	−1.20244	−0.601220	+	0.799083i	$0.705319\pi$
$312$	0	0
$313$	−24.9416	−1.40978	−0.704891	−	0.709316i	$-0.749004\pi$
$313$	−24.9416	−1.40978	−0.704891	+	0.709316i	$0.749004\pi$
$314$	0	0
$315$	−8.32596	−0.469115
$316$	0	0
$317$	16.1034	0.904456	0.452228	−	0.891902i	$-0.350629\pi$
$317$	16.1034	0.904456	0.452228	+	0.891902i	$0.350629\pi$
$318$	0	0
$319$	3.87868	0.217164
$320$	0	0
$321$	5.16523	0.288295
$322$	0	0
$323$	9.69785	0.539603
$324$	0	0
$325$	3.75654	0.208376
$326$	0	0
$327$	31.6029	1.74764
$328$	0	0
$329$	24.4059	1.34554
$330$	0	0
$331$	24.4269	1.34263	0.671313	−	0.741174i	$-0.265731\pi$
$331$	24.4269	1.34263	0.671313	+	0.741174i	$0.265731\pi$
$332$	0	0
$333$	−19.1899	−1.05160
$334$	0	0
$335$	2.21876	0.121224
$336$	0	0
$337$	17.8520	0.972461	0.486231	−	0.873831i	$-0.338372\pi$
$337$	17.8520	0.972461	0.486231	+	0.873831i	$0.338372\pi$
$338$	0	0
$339$	18.0615	0.980966
$340$	0	0
$341$	−1.36845	−0.0741056
$342$	0	0
$343$	−13.4420	−0.725802
$344$	0	0
$345$	0.986117	0.0530908
$346$	0	0
$347$	28.8851	1.55063	0.775317	−	0.631573i	$-0.217590\pi$
$347$	28.8851	1.55063	0.775317	+	0.631573i	$0.217590\pi$
$348$	0	0
$349$	−7.47754	−0.400263	−0.200132	−	0.979769i	$-0.564137\pi$
$349$	−7.47754	−0.400263	−0.200132	+	0.979769i	$0.564137\pi$
$350$	0	0
$351$	2.89848	0.154709
$352$	0	0
$353$	3.27705	0.174420	0.0872100	−	0.996190i	$-0.472205\pi$
$353$	3.27705	0.174420	0.0872100	+	0.996190i	$0.472205\pi$
$354$	0	0
$355$	−7.37769	−0.391567
$356$	0	0
$357$	41.8624	2.21559
$358$	0	0
$359$	−10.3992	−0.548846	−0.274423	−	0.961609i	$-0.588487\pi$
$359$	−10.3992	−0.548846	−0.274423	+	0.961609i	$0.588487\pi$
$360$	0	0
$361$	−16.0515	−0.844814
$362$	0	0
$363$	25.7848	1.35335
$364$	0	0
$365$	−2.94319	−0.154054
$366$	0	0
$367$	19.3724	1.01123	0.505615	−	0.862759i	$-0.331266\pi$
$367$	19.3724	1.01123	0.505615	+	0.862759i	$0.331266\pi$
$368$	0	0
$369$	16.3539	0.851351
$370$	0	0
$371$	23.9300	1.24238
$372$	0	0
$373$	−5.54279	−0.286995	−0.143497	−	0.989651i	$-0.545835\pi$
$373$	−5.54279	−0.286995	−0.143497	+	0.989651i	$0.545835\pi$
$374$	0	0
$375$	2.38247	0.123030
$376$	0	0
$377$	34.6079	1.78240
$378$	0	0
$379$	−1.05906	−0.0544002	−0.0272001	−	0.999630i	$-0.508659\pi$
$379$	−1.05906	−0.0544002	−0.0272001	+	0.999630i	$0.508659\pi$
$380$	0	0
$381$	−38.3392	−1.96418
$382$	0	0
$383$	−22.2473	−1.13678	−0.568392	−	0.822758i	$-0.692434\pi$
$383$	−22.2473	−1.13678	−0.568392	+	0.822758i	$0.692434\pi$
$384$	0	0
$385$	−1.30985	−0.0667562
$386$	0	0
$387$	17.4775	0.888431
$388$	0	0
$389$	3.47709	0.176295	0.0881477	−	0.996107i	$-0.471905\pi$
$389$	3.47709	0.176295	0.0881477	+	0.996107i	$0.471905\pi$
$390$	0	0
$391$	−2.33762	−0.118219
$392$	0	0
$393$	−44.1521	−2.22718
$394$	0	0
$395$	3.73260	0.187808
$396$	0	0
$397$	26.5616	1.33309	0.666545	−	0.745465i	$-0.267772\pi$
$397$	26.5616	1.33309	0.666545	+	0.745465i	$0.267772\pi$
$398$	0	0
$399$	12.7278	0.637188
$400$	0	0
$401$	20.5837	1.02790	0.513951	−	0.857820i	$-0.328181\pi$
$401$	20.5837	1.02790	0.513951	+	0.857820i	$0.328181\pi$
$402$	0	0
$403$	−12.2101	−0.608229
$404$	0	0
$405$	9.86669	0.490280
$406$	0	0
$407$	−3.01898	−0.149645
$408$	0	0
$409$	−24.1709	−1.19517	−0.597587	−	0.801804i	$-0.703874\pi$
$409$	−24.1709	−1.19517	−0.597587	+	0.801804i	$0.703874\pi$
$410$	0	0
$411$	−46.0450	−2.27123
$412$	0	0
$413$	4.94394	0.243276
$414$	0	0
$415$	7.52679	0.369476
$416$	0	0
$417$	−37.3228	−1.82771
$418$	0	0
$419$	12.6955	0.620216	0.310108	−	0.950701i	$-0.399635\pi$
$419$	12.6955	0.620216	0.310108	+	0.950701i	$0.399635\pi$
$420$	0	0
$421$	29.5056	1.43802	0.719008	−	0.695002i	$-0.244596\pi$
$421$	29.5056	1.43802	0.719008	+	0.695002i	$0.244596\pi$
$422$	0	0
$423$	20.9932	1.02073
$424$	0	0
$425$	−5.64771	−0.273954
$426$	0	0
$427$	−1.79912	−0.0870656
$428$	0	0
$429$	−3.76801	−0.181921
$430$	0	0
$431$	17.5800	0.846799	0.423400	−	0.905943i	$-0.360837\pi$
$431$	17.5800	0.846799	0.423400	+	0.905943i	$0.360837\pi$
$432$	0	0
$433$	10.0340	0.482201	0.241101	−	0.970500i	$-0.422492\pi$
$433$	10.0340	0.482201	0.241101	+	0.970500i	$0.422492\pi$
$434$	0	0
$435$	21.9490	1.05237
$436$	0	0
$437$	−0.710730	−0.0339988
$438$	0	0
$439$	−23.5618	−1.12454	−0.562272	−	0.826952i	$-0.690073\pi$
$439$	−23.5618	−1.12454	−0.562272	+	0.826952i	$0.690073\pi$
$440$	0	0
$441$	7.17055	0.341455
$442$	0	0
$443$	6.92709	0.329116	0.164558	−	0.986367i	$-0.447380\pi$
$443$	6.92709	0.329116	0.164558	+	0.986367i	$0.447380\pi$
$444$	0	0
$445$	17.1318	0.812124
$446$	0	0
$447$	−32.4101	−1.53294
$448$	0	0
$449$	−10.3423	−0.488084	−0.244042	−	0.969765i	$-0.578473\pi$
$449$	−10.3423	−0.488084	−0.244042	+	0.969765i	$0.578473\pi$
$450$	0	0
$451$	2.57282	0.121149
$452$	0	0
$453$	−2.38247	−0.111938
$454$	0	0
$455$	−11.6873	−0.547908
$456$	0	0
$457$	−16.7202	−0.782139	−0.391070	−	0.920361i	$-0.627895\pi$
$457$	−16.7202	−0.782139	−0.391070	+	0.920361i	$0.627895\pi$
$458$	0	0
$459$	−4.35767	−0.203399
$460$	0	0
$461$	−22.3421	−1.04058	−0.520288	−	0.853991i	$-0.674175\pi$
$461$	−22.3421	−1.04058	−0.520288	+	0.853991i	$0.674175\pi$
$462$	0	0
$463$	39.8434	1.85168	0.925840	−	0.377916i	$-0.123359\pi$
$463$	39.8434	1.85168	0.925840	+	0.377916i	$0.123359\pi$
$464$	0	0
$465$	−7.74387	−0.359113
$466$	0	0
$467$	37.4424	1.73263	0.866315	−	0.499498i	$-0.166482\pi$
$467$	37.4424	1.73263	0.866315	+	0.499498i	$0.166482\pi$
$468$	0	0
$469$	−6.90297	−0.318750
$470$	0	0
$471$	−39.6331	−1.82620
$472$	0	0
$473$	2.74958	0.126426
$474$	0	0
$475$	−1.71713	−0.0787872
$476$	0	0
$477$	20.5839	0.942470
$478$	0	0
$479$	−4.10695	−0.187651	−0.0938257	−	0.995589i	$-0.529910\pi$
$479$	−4.10695	−0.187651	−0.0938257	+	0.995589i	$0.529910\pi$
$480$	0	0
$481$	−26.9372	−1.22823
$482$	0	0
$483$	−3.06799	−0.139598
$484$	0	0
$485$	2.46964	0.112140
$486$	0	0
$487$	−20.6149	−0.934151	−0.467075	−	0.884217i	$-0.654692\pi$
$487$	−20.6149	−0.934151	−0.467075	+	0.884217i	$0.654692\pi$
$488$	0	0
$489$	1.97090	0.0891270
$490$	0	0
$491$	4.72366	0.213176	0.106588	−	0.994303i	$-0.466007\pi$
$491$	4.72366	0.213176	0.106588	+	0.994303i	$0.466007\pi$
$492$	0	0
$493$	−52.0307	−2.34335
$494$	0	0
$495$	−1.12669	−0.0506411
$496$	0	0
$497$	22.9533	1.02960
$498$	0	0
$499$	−8.26100	−0.369813	−0.184907	−	0.982756i	$-0.559198\pi$
$499$	−8.26100	−0.369813	−0.184907	+	0.982756i	$0.559198\pi$
$500$	0	0
$501$	−25.9942	−1.16134
$502$	0	0
$503$	8.57610	0.382390	0.191195	−	0.981552i	$-0.438764\pi$
$503$	8.57610	0.382390	0.191195	+	0.981552i	$0.438764\pi$
$504$	0	0
$505$	15.4531	0.687653
$506$	0	0
$507$	−2.64839	−0.117619
$508$	0	0
$509$	−35.9190	−1.59208	−0.796041	−	0.605243i	$-0.793076\pi$
$509$	−35.9190	−1.59208	−0.796041	+	0.605243i	$0.793076\pi$
$510$	0	0
$511$	9.15680	0.405073
$512$	0	0
$513$	−1.32490	−0.0584959
$514$	0	0
$515$	19.1961	0.845882
$516$	0	0
$517$	3.30268	0.145252
$518$	0	0
$519$	−23.0994	−1.01395
$520$	0	0
$521$	−42.1997	−1.84880	−0.924402	−	0.381420i	$-0.875435\pi$
$521$	−42.1997	−1.84880	−0.924402	+	0.381420i	$0.875435\pi$
$522$	0	0
$523$	24.8646	1.08725	0.543626	−	0.839327i	$-0.317051\pi$
$523$	24.8646	1.08725	0.543626	+	0.839327i	$0.317051\pi$
$524$	0	0
$525$	−7.41228	−0.323498
$526$	0	0
$527$	18.3571	0.799648
$528$	0	0
$529$	−22.8287	−0.992551
$530$	0	0
$531$	4.25263	0.184548
$532$	0	0
$533$	22.9562	0.994346
$534$	0	0
$535$	2.16802	0.0937316
$536$	0	0
$537$	−51.7825	−2.23458
$538$	0	0
$539$	1.12808	0.0485899
$540$	0	0
$541$	5.63079	0.242086	0.121043	−	0.992647i	$-0.461376\pi$
$541$	5.63079	0.242086	0.121043	+	0.992647i	$0.461376\pi$
$542$	0	0
$543$	38.4457	1.64986
$544$	0	0
$545$	13.2648	0.568201
$546$	0	0
$547$	42.1715	1.80312	0.901562	−	0.432650i	$-0.142421\pi$
$547$	42.1715	1.80312	0.901562	+	0.432650i	$0.142421\pi$
$548$	0	0
$549$	−1.54755	−0.0660478
$550$	0	0
$551$	−15.8194	−0.673929
$552$	0	0
$553$	−11.6128	−0.493827
$554$	0	0
$555$	−17.0840	−0.725177
$556$	0	0
$557$	26.4324	1.11998	0.559988	−	0.828501i	$-0.310806\pi$
$557$	26.4324	1.11998	0.559988	+	0.828501i	$0.310806\pi$
$558$	0	0
$559$	24.5334	1.03765
$560$	0	0
$561$	5.66495	0.239174
$562$	0	0
$563$	−27.3678	−1.15341	−0.576707	−	0.816951i	$-0.695663\pi$
$563$	−27.3678	−1.15341	−0.576707	+	0.816951i	$0.695663\pi$
$564$	0	0
$565$	7.58101	0.318936
$566$	0	0
$567$	−30.6970	−1.28915
$568$	0	0
$569$	−30.6911	−1.28664	−0.643319	−	0.765598i	$-0.722443\pi$
$569$	−30.6911	−1.28664	−0.643319	+	0.765598i	$0.722443\pi$
$570$	0	0
$571$	12.2960	0.514570	0.257285	−	0.966336i	$-0.417172\pi$
$571$	12.2960	0.514570	0.257285	+	0.966336i	$0.417172\pi$
$572$	0	0
$573$	17.9953	0.751764
$574$	0	0
$575$	0.413906	0.0172611
$576$	0	0
$577$	30.1197	1.25390	0.626950	−	0.779060i	$-0.284303\pi$
$577$	30.1197	1.25390	0.626950	+	0.779060i	$0.284303\pi$
$578$	0	0
$579$	18.5901	0.772577
$580$	0	0
$581$	−23.4172	−0.971509
$582$	0	0
$583$	3.23828	0.134116
$584$	0	0
$585$	−10.0530	−0.415642
$586$	0	0
$587$	−33.6624	−1.38939	−0.694697	−	0.719303i	$-0.744462\pi$
$587$	−33.6624	−1.38939	−0.694697	+	0.719303i	$0.744462\pi$
$588$	0	0
$589$	5.58128	0.229973
$590$	0	0
$591$	58.5283	2.40753
$592$	0	0
$593$	−20.1703	−0.828295	−0.414148	−	0.910210i	$-0.635920\pi$
$593$	−20.1703	−0.828295	−0.414148	+	0.910210i	$0.635920\pi$
$594$	0	0
$595$	17.5710	0.720343
$596$	0	0
$597$	−15.6760	−0.641576
$598$	0	0
$599$	18.6452	0.761821	0.380910	−	0.924612i	$-0.375611\pi$
$599$	18.6452	0.761821	0.380910	+	0.924612i	$0.375611\pi$
$600$	0	0
$601$	37.7059	1.53805	0.769027	−	0.639216i	$-0.220741\pi$
$601$	37.7059	1.53805	0.769027	+	0.639216i	$0.220741\pi$
$602$	0	0
$603$	−5.93773	−0.241803
$604$	0	0
$605$	10.8227	0.440007
$606$	0	0
$607$	19.5722	0.794410	0.397205	−	0.917730i	$-0.369980\pi$
$607$	19.5722	0.794410	0.397205	+	0.917730i	$0.369980\pi$
$608$	0	0
$609$	−68.2871	−2.76713
$610$	0	0
$611$	29.4685	1.19217
$612$	0	0
$613$	39.8575	1.60983	0.804915	−	0.593390i	$-0.202211\pi$
$613$	39.8575	1.60983	0.804915	+	0.593390i	$0.202211\pi$
$614$	0	0
$615$	14.5593	0.587086
$616$	0	0
$617$	38.1904	1.53749	0.768743	−	0.639558i	$-0.220883\pi$
$617$	38.1904	1.53749	0.768743	+	0.639558i	$0.220883\pi$
$618$	0	0
$619$	44.9131	1.80521	0.902605	−	0.430469i	$-0.141652\pi$
$619$	44.9131	1.80521	0.902605	+	0.430469i	$0.141652\pi$
$620$	0	0
$621$	0.319362	0.0128156
$622$	0	0
$623$	−53.3000	−2.13542
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.72237	0.0687848
$628$	0	0
$629$	40.4983	1.61477
$630$	0	0
$631$	0.374663	0.0149151	0.00745755	−	0.999972i	$-0.497626\pi$
$631$	0.374663	0.0149151	0.00745755	+	0.999972i	$0.497626\pi$
$632$	0	0
$633$	−41.0499	−1.63159
$634$	0	0
$635$	−16.0922	−0.638601
$636$	0	0
$637$	10.0654	0.398806
$638$	0	0
$639$	19.7437	0.781051
$640$	0	0
$641$	31.0776	1.22749	0.613746	−	0.789504i	$-0.289662\pi$
$641$	31.0776	1.22749	0.613746	+	0.789504i	$0.289662\pi$
$642$	0	0
$643$	−24.6865	−0.973541	−0.486771	−	0.873530i	$-0.661825\pi$
$643$	−24.6865	−0.973541	−0.486771	+	0.873530i	$0.661825\pi$
$644$	0	0
$645$	15.5595	0.612656
$646$	0	0
$647$	32.3401	1.27142	0.635710	−	0.771928i	$-0.280707\pi$
$647$	32.3401	1.27142	0.635710	+	0.771928i	$0.280707\pi$
$648$	0	0
$649$	0.669029	0.0262617
$650$	0	0
$651$	24.0926	0.944262
$652$	0	0
$653$	34.7854	1.36126	0.680629	−	0.732628i	$-0.261707\pi$
$653$	34.7854	1.36126	0.680629	+	0.732628i	$0.261707\pi$
$654$	0	0
$655$	−18.5321	−0.724109
$656$	0	0
$657$	7.87640	0.307288
$658$	0	0
$659$	−45.6238	−1.77725	−0.888625	−	0.458634i	$-0.848339\pi$
$659$	−45.6238	−1.77725	−0.888625	+	0.458634i	$0.848339\pi$
$660$	0	0
$661$	14.5480	0.565853	0.282926	−	0.959142i	$-0.408695\pi$
$661$	14.5480	0.565853	0.282926	+	0.959142i	$0.408695\pi$
$662$	0	0
$663$	50.5461	1.96305
$664$	0	0
$665$	5.34229	0.207165
$666$	0	0
$667$	3.81320	0.147648
$668$	0	0
$669$	−8.15583	−0.315322
$670$	0	0
$671$	−0.243463	−0.00939877
$672$	0	0
$673$	−23.3332	−0.899429	−0.449715	−	0.893172i	$-0.648474\pi$
$673$	−23.3332	−0.899429	−0.449715	+	0.893172i	$0.648474\pi$
$674$	0	0
$675$	0.771581	0.0296982
$676$	0	0
$677$	42.0721	1.61696	0.808480	−	0.588523i	$-0.200290\pi$
$677$	42.0721	1.61696	0.808480	+	0.588523i	$0.200290\pi$
$678$	0	0
$679$	−7.68348	−0.294865
$680$	0	0
$681$	−58.9758	−2.25996
$682$	0	0
$683$	26.5346	1.01532	0.507660	−	0.861558i	$-0.330511\pi$
$683$	26.5346	1.01532	0.507660	+	0.861558i	$0.330511\pi$
$684$	0	0
$685$	−19.3266	−0.738432
$686$	0	0
$687$	6.48628	0.247467
$688$	0	0
$689$	28.8939	1.10077
$690$	0	0
$691$	11.4360	0.435045	0.217523	−	0.976055i	$-0.430202\pi$
$691$	11.4360	0.435045	0.217523	+	0.976055i	$0.430202\pi$
$692$	0	0
$693$	3.50535	0.133157
$694$	0	0
$695$	−15.6656	−0.594231
$696$	0	0
$697$	−34.5132	−1.30728
$698$	0	0
$699$	−29.8762	−1.13002
$700$	0	0
$701$	−17.1678	−0.648419	−0.324210	−	0.945985i	$-0.605098\pi$
$701$	−17.1678	−0.648419	−0.324210	+	0.945985i	$0.605098\pi$
$702$	0	0
$703$	12.3131	0.464396
$704$	0	0
$705$	18.6895	0.703886
$706$	0	0
$707$	−48.0773	−1.80813
$708$	0	0
$709$	−25.3114	−0.950589	−0.475295	−	0.879827i	$-0.657659\pi$
$709$	−25.3114	−0.950589	−0.475295	+	0.879827i	$0.657659\pi$
$710$	0	0
$711$	−9.98898	−0.374616
$712$	0	0
$713$	−1.34534	−0.0503835
$714$	0	0
$715$	−1.58156	−0.0591469
$716$	0	0
$717$	−14.9122	−0.556905
$718$	0	0
$719$	−6.03671	−0.225131	−0.112566	−	0.993644i	$-0.535907\pi$
$719$	−6.03671	−0.225131	−0.112566	+	0.993644i	$0.535907\pi$
$720$	0	0
$721$	−59.7226	−2.22419
$722$	0	0
$723$	47.2799	1.75836
$724$	0	0
$725$	9.21271	0.342151
$726$	0	0
$727$	−4.82779	−0.179053	−0.0895265	−	0.995984i	$-0.528535\pi$
$727$	−4.82779	−0.179053	−0.0895265	+	0.995984i	$0.528535\pi$
$728$	0	0
$729$	−20.8899	−0.773699
$730$	0	0
$731$	−36.8844	−1.36422
$732$	0	0
$733$	21.8413	0.806726	0.403363	−	0.915040i	$-0.367841\pi$
$733$	21.8413	0.806726	0.403363	+	0.915040i	$0.367841\pi$
$734$	0	0
$735$	6.38367	0.235465
$736$	0	0
$737$	−0.934131	−0.0344092
$738$	0	0
$739$	24.7214	0.909392	0.454696	−	0.890647i	$-0.349748\pi$
$739$	24.7214	0.909392	0.454696	+	0.890647i	$0.349748\pi$
$740$	0	0
$741$	15.3680	0.564558
$742$	0	0
$743$	−27.4856	−1.00835	−0.504175	−	0.863602i	$-0.668203\pi$
$743$	−27.4856	−1.00835	−0.504175	+	0.863602i	$0.668203\pi$
$744$	0	0
$745$	−13.6036	−0.498397
$746$	0	0
$747$	−20.1428	−0.736985
$748$	0	0
$749$	−6.74510	−0.246460
$750$	0	0
$751$	2.65209	0.0967760	0.0483880	−	0.998829i	$-0.484592\pi$
$751$	2.65209	0.0967760	0.0483880	+	0.998829i	$0.484592\pi$
$752$	0	0
$753$	−34.6782	−1.26375
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	−21.7389	−0.790114	−0.395057	−	0.918657i	$-0.629275\pi$
$757$	−21.7389	−0.790114	−0.395057	+	0.918657i	$0.629275\pi$
$758$	0	0
$759$	−0.415169	−0.0150697
$760$	0	0
$761$	13.4317	0.486899	0.243449	−	0.969914i	$-0.421721\pi$
$761$	13.4317	0.486899	0.243449	+	0.969914i	$0.421721\pi$
$762$	0	0
$763$	−41.2691	−1.49404
$764$	0	0
$765$	15.1141	0.546451
$766$	0	0
$767$	5.96948	0.215546
$768$	0	0
$769$	45.9391	1.65661	0.828304	−	0.560280i	$-0.189306\pi$
$769$	45.9391	1.65661	0.828304	+	0.560280i	$0.189306\pi$
$770$	0	0
$771$	−71.7222	−2.58301
$772$	0	0
$773$	−7.72268	−0.277765	−0.138883	−	0.990309i	$-0.544351\pi$
$773$	−7.72268	−0.277765	−0.138883	+	0.990309i	$0.544351\pi$
$774$	0	0
$775$	−3.25036	−0.116756
$776$	0	0
$777$	53.1515	1.90680
$778$	0	0
$779$	−10.4934	−0.375964
$780$	0	0
$781$	3.10611	0.111145
$782$	0	0
$783$	7.10835	0.254032
$784$	0	0
$785$	−16.6353	−0.593740
$786$	0	0
$787$	−15.1471	−0.539935	−0.269967	−	0.962869i	$-0.587013\pi$
$787$	−15.1471	−0.539935	−0.269967	+	0.962869i	$0.587013\pi$
$788$	0	0
$789$	−2.05773	−0.0732573
$790$	0	0
$791$	−23.5859	−0.838618
$792$	0	0
$793$	−2.17232	−0.0771413
$794$	0	0
$795$	18.3250	0.649921
$796$	0	0
$797$	−6.53174	−0.231366	−0.115683	−	0.993286i	$-0.536906\pi$
$797$	−6.53174	−0.231366	−0.115683	+	0.993286i	$0.536906\pi$
$798$	0	0
$799$	−44.3040	−1.56736
$800$	0	0
$801$	−45.8471	−1.61993
$802$	0	0
$803$	1.23913	0.0437278
$804$	0	0
$805$	−1.28774	−0.0453868
$806$	0	0
$807$	3.35496	0.118100
$808$	0	0
$809$	−1.11319	−0.0391377	−0.0195689	−	0.999809i	$-0.506229\pi$
$809$	−1.11319	−0.0391377	−0.0195689	+	0.999809i	$0.506229\pi$
$810$	0	0
$811$	−2.45630	−0.0862523	−0.0431262	−	0.999070i	$-0.513732\pi$
$811$	−2.45630	−0.0862523	−0.0431262	+	0.999070i	$0.513732\pi$
$812$	0	0
$813$	−3.12649	−0.109651
$814$	0	0
$815$	0.827251	0.0289773
$816$	0	0
$817$	−11.2143	−0.392339
$818$	0	0
$819$	31.2768	1.09290
$820$	0	0
$821$	−21.5978	−0.753768	−0.376884	−	0.926260i	$-0.623005\pi$
$821$	−21.5978	−0.753768	−0.376884	+	0.926260i	$0.623005\pi$
$822$	0	0
$823$	−5.06499	−0.176554	−0.0882772	−	0.996096i	$-0.528136\pi$
$823$	−5.06499	−0.176554	−0.0882772	+	0.996096i	$0.528136\pi$
$824$	0	0
$825$	−1.00305	−0.0349218
$826$	0	0
$827$	29.5325	1.02695	0.513474	−	0.858105i	$-0.328358\pi$
$827$	29.5325	1.02695	0.513474	+	0.858105i	$0.328358\pi$
$828$	0	0
$829$	−24.9346	−0.866016	−0.433008	−	0.901390i	$-0.642548\pi$
$829$	−24.9346	−0.866016	−0.433008	+	0.901390i	$0.642548\pi$
$830$	0	0
$831$	−11.0205	−0.382296
$832$	0	0
$833$	−15.1327	−0.524317
$834$	0	0
$835$	−10.9106	−0.377578
$836$	0	0
$837$	−2.50792	−0.0866862
$838$	0	0
$839$	−33.3693	−1.15203	−0.576017	−	0.817437i	$-0.695394\pi$
$839$	−33.3693	−1.15203	−0.576017	+	0.817437i	$0.695394\pi$
$840$	0	0
$841$	55.8739	1.92669
$842$	0	0
$843$	−51.4503	−1.77204
$844$	0	0
$845$	−1.11162	−0.0382408
$846$	0	0
$847$	−33.6715	−1.15697
$848$	0	0
$849$	50.9346	1.74807
$850$	0	0
$851$	−2.96801	−0.101742
$852$	0	0
$853$	−13.6433	−0.467139	−0.233569	−	0.972340i	$-0.575041\pi$
$853$	−13.6433	−0.467139	−0.233569	+	0.972340i	$0.575041\pi$
$854$	0	0
$855$	4.59528	0.157155
$856$	0	0
$857$	4.75600	0.162462	0.0812309	−	0.996695i	$-0.474115\pi$
$857$	4.75600	0.162462	0.0812309	+	0.996695i	$0.474115\pi$
$858$	0	0
$859$	22.2598	0.759495	0.379747	−	0.925090i	$-0.376011\pi$
$859$	22.2598	0.759495	0.379747	+	0.925090i	$0.376011\pi$
$860$	0	0
$861$	−45.2965	−1.54370
$862$	0	0
$863$	−32.9006	−1.11995	−0.559974	−	0.828510i	$-0.689189\pi$
$863$	−32.9006	−1.11995	−0.559974	+	0.828510i	$0.689189\pi$
$864$	0	0
$865$	−9.69560	−0.329660
$866$	0	0
$867$	−35.4908	−1.20533
$868$	0	0
$869$	−1.57148	−0.0533088
$870$	0	0
$871$	−8.33488	−0.282417
$872$	0	0
$873$	−6.60910	−0.223684
$874$	0	0
$875$	−3.11118	−0.105177
$876$	0	0
$877$	19.8614	0.670672	0.335336	−	0.942099i	$-0.391150\pi$
$877$	19.8614	0.670672	0.335336	+	0.942099i	$0.391150\pi$
$878$	0	0
$879$	−57.3549	−1.93453
$880$	0	0
$881$	14.0105	0.472025	0.236013	−	0.971750i	$-0.424159\pi$
$881$	14.0105	0.472025	0.236013	+	0.971750i	$0.424159\pi$
$882$	0	0
$883$	12.2733	0.413030	0.206515	−	0.978443i	$-0.433788\pi$
$883$	12.2733	0.413030	0.206515	+	0.978443i	$0.433788\pi$
$884$	0	0
$885$	3.78595	0.127263
$886$	0	0
$887$	25.2724	0.848563	0.424282	−	0.905530i	$-0.360527\pi$
$887$	25.2724	0.848563	0.424282	+	0.905530i	$0.360527\pi$
$888$	0	0
$889$	50.0658	1.67915
$890$	0	0
$891$	−4.15402	−0.139165
$892$	0	0
$893$	−13.4702	−0.450762
$894$	0	0
$895$	−21.7348	−0.726515
$896$	0	0
$897$	−3.70439	−0.123686
$898$	0	0
$899$	−29.9446	−0.998709
$900$	0	0
$901$	−43.4400	−1.44720
$902$	0	0
$903$	−48.4085	−1.61093
$904$	0	0
$905$	16.1369	0.536410
$906$	0	0
$907$	−33.1964	−1.10227	−0.551135	−	0.834416i	$-0.685805\pi$
$907$	−33.1964	−1.10227	−0.551135	+	0.834416i	$0.685805\pi$
$908$	0	0
$909$	−41.3546	−1.37165
$910$	0	0
$911$	−28.2269	−0.935200	−0.467600	−	0.883940i	$-0.654881\pi$
$911$	−28.2269	−0.935200	−0.467600	+	0.883940i	$0.654881\pi$
$912$	0	0
$913$	−3.16889	−0.104875
$914$	0	0
$915$	−1.37772	−0.0455461
$916$	0	0
$917$	57.6567	1.90399
$918$	0	0
$919$	0.233541	0.00770380	0.00385190	−	0.999993i	$-0.498774\pi$
$919$	0.233541	0.00770380	0.00385190	+	0.999993i	$0.498774\pi$
$920$	0	0
$921$	−16.4690	−0.542673
$922$	0	0
$923$	27.7146	0.912238
$924$	0	0
$925$	−7.17074	−0.235772
$926$	0	0
$927$	−51.3716	−1.68726
$928$	0	0
$929$	16.3408	0.536125	0.268063	−	0.963402i	$-0.413617\pi$
$929$	16.3408	0.536125	0.268063	+	0.963402i	$0.413617\pi$
$930$	0	0
$931$	−4.60094	−0.150790
$932$	0	0
$933$	50.5208	1.65398
$934$	0	0
$935$	2.37777	0.0777613
$936$	0	0
$937$	−21.1920	−0.692312	−0.346156	−	0.938177i	$-0.612513\pi$
$937$	−21.1920	−0.692312	−0.346156	+	0.938177i	$0.612513\pi$
$938$	0	0
$939$	59.4224	1.93918
$940$	0	0
$941$	2.79390	0.0910785	0.0455392	−	0.998963i	$-0.485499\pi$
$941$	2.79390	0.0910785	0.0455392	+	0.998963i	$0.485499\pi$
$942$	0	0
$943$	2.52938	0.0823680
$944$	0	0
$945$	−2.40053	−0.0780891
$946$	0	0
$947$	−33.8253	−1.09918	−0.549588	−	0.835436i	$-0.685215\pi$
$947$	−33.8253	−1.09918	−0.549588	+	0.835436i	$0.685215\pi$
$948$	0	0
$949$	11.0562	0.358900
$950$	0	0
$951$	−38.3657	−1.24409
$952$	0	0
$953$	38.8498	1.25847	0.629234	−	0.777216i	$-0.283369\pi$
$953$	38.8498	1.25847	0.629234	+	0.777216i	$0.283369\pi$
$954$	0	0
$955$	7.55322	0.244417
$956$	0	0
$957$	−9.24082	−0.298713
$958$	0	0
$959$	60.1286	1.94165
$960$	0	0
$961$	−20.4352	−0.659199
$962$	0	0
$963$	−5.80193	−0.186964
$964$	0	0
$965$	7.80287	0.251183
$966$	0	0
$967$	15.1116	0.485957	0.242979	−	0.970032i	$-0.421875\pi$
$967$	15.1116	0.485957	0.242979	+	0.970032i	$0.421875\pi$
$968$	0	0
$969$	−23.1048	−0.742233
$970$	0	0
$971$	−43.3006	−1.38958	−0.694792	−	0.719211i	$-0.744504\pi$
$971$	−43.3006	−1.38958	−0.694792	+	0.719211i	$0.744504\pi$
$972$	0	0
$973$	48.7385	1.56249
$974$	0	0
$975$	−8.94983	−0.286624
$976$	0	0
$977$	−1.79619	−0.0574652	−0.0287326	−	0.999587i	$-0.509147\pi$
$977$	−1.79619	−0.0574652	−0.0287326	+	0.999587i	$0.509147\pi$
$978$	0	0
$979$	−7.21272	−0.230520
$980$	0	0
$981$	−35.4984	−1.13338
$982$	0	0
$983$	46.5646	1.48518	0.742590	−	0.669747i	$-0.233597\pi$
$983$	46.5646	1.48518	0.742590	+	0.669747i	$0.233597\pi$
$984$	0	0
$985$	24.5663	0.782746
$986$	0	0
$987$	−58.1463	−1.85082
$988$	0	0
$989$	2.70316	0.0859555
$990$	0	0
$991$	−8.70134	−0.276407	−0.138204	−	0.990404i	$-0.544133\pi$
$991$	−8.70134	−0.276407	−0.138204	+	0.990404i	$0.544133\pi$
$992$	0	0
$993$	−58.1963	−1.84681
$994$	0	0
$995$	−6.57974	−0.208592
$996$	0	0
$997$	−0.656913	−0.0208046	−0.0104023	−	0.999946i	$-0.503311\pi$
$997$	−0.656913	−0.0208046	−0.0104023	+	0.999946i	$0.503311\pi$
$998$	0	0
$999$	−5.53280	−0.175050

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.m.1.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.m.1.2	✓	12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q-2.38247 q^{3} -1.00000 q^{5} +3.11118 q^{7} +2.67614 q^{9} +O(q^{10})\)	\(q-2.38247 q^{3} -1.00000 q^{5} +3.11118 q^{7} +2.67614 q^{9} +0.421014 q^{11} +3.75654 q^{13} +2.38247 q^{15} -5.64771 q^{17} -1.71713 q^{19} -7.41228 q^{21} +0.413906 q^{23} +1.00000 q^{25} +0.771581 q^{27} +9.21271 q^{29} -3.25036 q^{31} -1.00305 q^{33} -3.11118 q^{35} -7.17074 q^{37} -8.94983 q^{39} +6.11100 q^{41} +6.53085 q^{43} -2.67614 q^{45} +7.84459 q^{47} +2.67944 q^{49} +13.4555 q^{51} +7.69162 q^{53} -0.421014 q^{55} +4.09100 q^{57} +1.58909 q^{59} -0.578276 q^{61} +8.32596 q^{63} -3.75654 q^{65} -2.21876 q^{67} -0.986117 q^{69} +7.37769 q^{71} +2.94319 q^{73} -2.38247 q^{75} +1.30985 q^{77} -3.73260 q^{79} -9.86669 q^{81} -7.52679 q^{83} +5.64771 q^{85} -21.9490 q^{87} -17.1318 q^{89} +11.6873 q^{91} +7.74387 q^{93} +1.71713 q^{95} -2.46964 q^{97} +1.12669 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{3} - 12 q^{5} + 5 q^{7} + 9 q^{9}+O(q^{10}) \) 12 * q + 3 * q^3 - 12 * q^5 + 5 * q^7 + 9 * q^9	\( 12 q + 3 q^{3} - 12 q^{5} + 5 q^{7} + 9 q^{9} + 10 q^{11} + 11 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} - q^{21} + 18 q^{23} + 12 q^{25} + 9 q^{27} + 16 q^{29} - q^{31} + 8 q^{33} - 5 q^{35} + 2 q^{37} + 6 q^{39} + 4 q^{41} + 7 q^{43} - 9 q^{45} + 3 q^{49} - 4 q^{51} + 39 q^{53} - 10 q^{55} - 15 q^{57} - 4 q^{59} - 32 q^{61} + 3 q^{63} - 11 q^{65} + 4 q^{67} + 12 q^{69} + 24 q^{71} - 10 q^{73} + 3 q^{75} + 38 q^{77} + 32 q^{79} - 8 q^{81} + 9 q^{83} + 4 q^{85} + 3 q^{87} + 15 q^{89} + 18 q^{91} + 36 q^{93} - 5 q^{95} + 15 q^{97} + 28 q^{99}+O(q^{100}) \) 12 * q + 3 * q^3 - 12 * q^5 + 5 * q^7 + 9 * q^9 + 10 * q^11 + 11 * q^13 - 3 * q^15 - 4 * q^17 + 5 * q^19 - q^21 + 18 * q^23 + 12 * q^25 + 9 * q^27 + 16 * q^29 - q^31 + 8 * q^33 - 5 * q^35 + 2 * q^37 + 6 * q^39 + 4 * q^41 + 7 * q^43 - 9 * q^45 + 3 * q^49 - 4 * q^51 + 39 * q^53 - 10 * q^55 - 15 * q^57 - 4 * q^59 - 32 * q^61 + 3 * q^63 - 11 * q^65 + 4 * q^67 + 12 * q^69 + 24 * q^71 - 10 * q^73 + 3 * q^75 + 38 * q^77 + 32 * q^79 - 8 * q^81 + 9 * q^83 + 4 * q^85 + 3 * q^87 + 15 * q^89 + 18 * q^91 + 36 * q^93 - 5 * q^95 + 15 * q^97 + 28 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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