LMFDB - Embedded newform 252.6.k.f.37.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,6,Mod(37,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("252.37");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	252.k (of order $3$, degree $2$, 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:	$40.4167225929$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 703x^{6} + 2770x^{5} + 427565x^{4} + 718170x^{3} + 42175732x^{2} - 40929504x + 3559792896 $ x^8 - 2x^7 + 703x^6 + 2770x^5 + 427565x^4 + 718170x^3 + 42175732x^2 - 40929504*x + 3559792896
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{3}\cdot 7 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.4
Root			$4.59067 + 7.95128i$ of defining polynomial
Character	$\chi$	$=$	252.37
Dual form			252.6.k.f.109.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(39.3359 - 68.1317i) q^{5} +(-100.606 - 81.7641i) q^{7} +(345.759 + 598.873i) q^{11} +818.732 q^{13} +(554.638 + 960.661i) q^{17} +(286.523 - 496.273i) q^{19} +(1258.80 - 2180.30i) q^{23} +(-1532.12 - 2653.71i) q^{25} +3258.19 q^{29} +(-5059.55 - 8763.40i) q^{31} +(-9528.17 + 3638.23i) q^{35} +(-2434.81 + 4217.21i) q^{37} +13094.3 q^{41} -9303.64 q^{43} +(6452.90 - 11176.8i) q^{47} +(3436.28 + 16452.0i) q^{49} +(-9770.12 - 16922.3i) q^{53} +54403.0 q^{55} +(-12560.2 - 21755.0i) q^{59} +(15681.1 - 27160.5i) q^{61} +(32205.6 - 55781.7i) q^{65} +(-27971.9 - 48448.8i) q^{67} -20501.0 q^{71} +(33825.0 + 58586.6i) q^{73} +(14180.7 - 88521.1i) q^{77} +(-7039.95 + 12193.5i) q^{79} +77129.1 q^{83} +87268.6 q^{85} +(160.396 - 277.815i) q^{89} +(-82369.7 - 66942.9i) q^{91} +(-22541.3 - 39042.6i) q^{95} -112009. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 42 q^{7} + 462 q^{11} - 1204 q^{13} - 228 q^{17} + 358 q^{19} + 2148 q^{23} - 5454 q^{25} + 11064 q^{29} + 830 q^{31} - 7692 q^{35} - 3914 q^{37} + 16632 q^{41} - 29036 q^{43} - 41700 q^{47} + 41876 q^{49}+ \cdots - 433356 q^{97}+O(q^{100}) $ 8 * q - 42 * q^7 + 462 * q^11 - 1204 * q^13 - 228 * q^17 + 358 * q^19 + 2148 * q^23 - 5454 * q^25 + 11064 * q^29 + 830 * q^31 - 7692 * q^35 - 3914 * q^37 + 16632 * q^41 - 29036 * q^43 - 41700 * q^47 + 41876 * q^49 - 22164 * q^53 + 7784 * q^55 - 32886 * q^59 + 83732 * q^61 + 93192 * q^65 - 80034 * q^67 - 89544 * q^71 - 22470 * q^73 - 138732 * q^77 - 75286 * q^79 + 34836 * q^83 + 278504 * q^85 - 28944 * q^89 - 12058 * q^91 + 144120 * q^95 - 433356 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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			0
$3$		0			0
$4$		0			0
$5$	39.3359	−	68.1317i	0.703662	−	1.21878i	−0.263511	−	0.964656i	$-0.584880\pi$
$5$	39.3359	−	68.1317i	0.703662	−	1.21878i	0.967172	−	0.254121i	$-0.0817862\pi$
$6$		0			0
$7$	−100.606	−	81.7641i	−0.776033	−	0.630692i
$7$	−100.606	−	81.7641i	−0.776033	−	0.630692i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	345.759	+	598.873i	0.861574	+	1.49229i	0.870410	+	0.492328i	$0.163854\pi$
$11$	345.759	+	598.873i	0.861574	+	1.49229i	−0.00883597	+	0.999961i	$0.502813\pi$
$12$		0			0
$13$	818.732			1.34364			0.671821	−	0.740714i	$-0.265512\pi$
$13$	818.732			1.34364			0.671821	+	0.740714i	$0.265512\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	554.638	+	960.661i	0.465465	+	0.806209i	0.999222	−	0.0394286i	$-0.0125538\pi$
$17$	554.638	+	960.661i	0.465465	+	0.806209i	−0.533757	+	0.845638i	$0.679220\pi$
$18$		0			0
$19$	286.523	−	496.273i	0.182086	−	0.315382i	−0.760505	−	0.649332i	$-0.775049\pi$
$19$	286.523	−	496.273i	0.182086	−	0.315382i	0.942591	+	0.333951i	$0.108382\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1258.80	−	2180.30i	0.496177	−	0.859403i	−0.503814	−	0.863812i	$-0.668070\pi$
$23$	1258.80	−	2180.30i	0.496177	−	0.859403i	0.999990	+	0.00440926i	$0.00140352\pi$
$24$		0			0
$25$	−1532.12	−	2653.71i	−0.490279	−	0.849189i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3258.19			0.719419			0.359710	−	0.933064i	$-0.382876\pi$
$29$	3258.19			0.719419			0.359710	+	0.933064i	$0.382876\pi$
$30$		0			0
$31$	−5059.55	−	8763.40i	−0.945601	−	1.63783i	−0.754544	−	0.656249i	$-0.772142\pi$
$31$	−5059.55	−	8763.40i	−0.945601	−	1.63783i	−0.191056	−	0.981579i	$-0.561191\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−9528.17	+	3638.23i	−1.31474	+	0.502018i
$36$		0			0
$37$	−2434.81	+	4217.21i	−0.292389	+	0.506432i	−0.974374	−	0.224934i	$-0.927783\pi$
$37$	−2434.81	+	4217.21i	−0.292389	+	0.506432i	0.681985	+	0.731366i	$0.261117\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	13094.3			1.21653			0.608265	−	0.793734i	$-0.291866\pi$
$41$	13094.3			1.21653			0.608265	+	0.793734i	$0.291866\pi$
$42$		0			0
$43$	−9303.64			−0.767329			−0.383664	−	0.923473i	$-0.625338\pi$
$43$	−9303.64			−0.767329			−0.383664	+	0.923473i	$0.625338\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6452.90	−	11176.8i	0.426099	−	0.738025i	−0.570423	−	0.821351i	$-0.693221\pi$
$47$	6452.90	−	11176.8i	0.426099	−	0.738025i	0.996522	+	0.0833256i	$0.0265542\pi$
$48$		0			0
$49$	3436.28	+	16452.0i	0.204455	+	0.978876i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9770.12	−	16922.3i	−0.477760	−	0.827505i	0.521915	−	0.852998i	$-0.325218\pi$
$53$	−9770.12	−	16922.3i	−0.477760	−	0.827505i	−0.999675	+	0.0254926i	$0.991885\pi$
$54$		0			0
$55$	54403.0			2.42503
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−12560.2	−	21755.0i	−0.469751	−	0.813632i	0.529651	−	0.848216i	$-0.322323\pi$
$59$	−12560.2	−	21755.0i	−0.469751	−	0.813632i	−0.999402	+	0.0345835i	$0.988990\pi$
$60$		0			0
$61$	15681.1	−	27160.5i	0.539575	−	0.934571i	−0.459352	−	0.888254i	$-0.651918\pi$
$61$	15681.1	−	27160.5i	0.539575	−	0.934571i	0.998927	−	0.0463168i	$-0.0147484\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	32205.6	−	55781.7i	0.945469	−	1.63760i
$66$		0			0
$67$	−27971.9	−	48448.8i	−0.761264	−	1.31855i	−0.942200	−	0.335052i	$-0.891246\pi$
$67$	−27971.9	−	48448.8i	−0.761264	−	1.31855i	0.180936	−	0.983495i	$-0.442087\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−20501.0			−0.482647			−0.241323	−	0.970445i	$-0.577581\pi$
$71$	−20501.0			−0.482647			−0.241323	+	0.970445i	$0.577581\pi$
$72$		0			0
$73$	33825.0	+	58586.6i	0.742900	+	1.28674i	0.951170	+	0.308669i	$0.0998835\pi$
$73$	33825.0	+	58586.6i	0.742900	+	1.28674i	−0.208270	+	0.978071i	$0.566783\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	14180.7	−	88521.1i	0.272565	−	1.70145i
$78$		0			0
$79$	−7039.95	+	12193.5i	−0.126912	+	0.219818i	−0.922479	−	0.386048i	$-0.873840\pi$
$79$	−7039.95	+	12193.5i	−0.126912	+	0.219818i	0.795567	+	0.605866i	$0.207173\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	77129.1			1.22892			0.614459	−	0.788949i	$-0.289374\pi$
$83$	77129.1			1.22892			0.614459	+	0.788949i	$0.289374\pi$
$84$		0			0
$85$	87268.6			1.31012
$86$		0			0
$87$		0			0
$88$		0			0
$89$	160.396	−	277.815i	0.00214644	−	0.00371775i	−0.864950	−	0.501858i	$-0.832650\pi$
$89$	160.396	−	277.815i	0.00214644	−	0.00371775i	0.867097	+	0.498140i	$0.165983\pi$
$90$		0			0
$91$	−82369.7	−	66942.9i	−1.04271	−	0.847424i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−22541.3	−	39042.6i	−0.256253	−	0.443844i
$96$		0			0
$97$	−112009.			−1.20871			−0.604355	−	0.796715i	$-0.706569\pi$
$97$	−112009.			−1.20871			−0.604355	+	0.796715i	$0.706569\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	33627.0	+	58243.7i	0.328008	+	0.568127i	0.982117	−	0.188274i	$-0.0602893\pi$
$101$	33627.0	+	58243.7i	0.328008	+	0.568127i	−0.654108	+	0.756401i	$0.726956\pi$
$102$		0			0
$103$	58317.6	−	101009.i	0.541635	−	0.938140i	−0.457175	−	0.889377i	$-0.651139\pi$
$103$	58317.6	−	101009.i	0.541635	−	0.938140i	0.998810	−	0.0487630i	$-0.0155279\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−80322.6	+	139123.i	−0.678232	+	1.17473i	0.297281	+	0.954790i	$0.403920\pi$
$107$	−80322.6	+	139123.i	−0.678232	+	1.17473i	−0.975513	+	0.219942i	$0.929413\pi$
$108$		0			0
$109$	−59668.3	−	103349.i	−0.481036	−	0.833179i	0.518727	−	0.854940i	$-0.326406\pi$
$109$	−59668.3	−	103349.i	−0.481036	−	0.833179i	−0.999763	+	0.0217610i	$0.993073\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	110893.			0.816972			0.408486	−	0.912765i	$-0.366057\pi$
$113$	110893.			0.816972			0.408486	+	0.912765i	$0.366057\pi$
$114$		0			0
$115$	−99031.8	−	171528.i	−0.698281	−	1.20946i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	22747.4	−	141998.i	0.147253	−	0.919210i
$120$		0			0
$121$	−158574.	+	274658.i	−0.984618	+	1.70541i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4779.64			0.0273603
$126$		0			0
$127$	315184.			1.73402			0.867012	−	0.498287i	$-0.166038\pi$
$127$	315184.			1.73402			0.867012	+	0.498287i	$0.166038\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	42832.3	−	74187.6i	0.218068	−	0.377705i	−0.736149	−	0.676819i	$-0.763358\pi$
$131$	42832.3	−	74187.6i	0.218068	−	0.377705i	0.954217	+	0.299114i	$0.0966911\pi$
$132$		0			0
$133$	−69403.3	+	26500.9i	−0.340213	+	0.129907i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−16316.1	−	28260.3i	−0.0742703	−	0.128640i	0.826498	−	0.562939i	$-0.190329\pi$
$137$	−16316.1	−	28260.3i	−0.0742703	−	0.128640i	−0.900769	+	0.434299i	$0.856996\pi$
$138$		0			0
$139$	206590.			0.906925			0.453462	−	0.891275i	$-0.350189\pi$
$139$	206590.			0.906925			0.453462	+	0.891275i	$0.350189\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	283084.	+	490317.i	1.15765	+	2.00510i
$144$		0			0
$145$	128164.	−	221986.i	0.506228	−	0.876812i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	241855.	−	418906.i	0.892463	−	1.54579i	0.0555495	−	0.998456i	$-0.482309\pi$
$149$	241855.	−	418906.i	0.892463	−	1.54579i	0.836913	−	0.547335i	$-0.184358\pi$
$150$		0			0
$151$	83610.4	+	144818.i	0.298413	+	0.516867i	0.975773	−	0.218785i	$-0.0702093\pi$
$151$	83610.4	+	144818.i	0.298413	+	0.516867i	−0.677360	+	0.735652i	$0.736876\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−796087.			−2.66153
$156$		0			0
$157$	7054.26	+	12218.3i	0.0228403	+	0.0395606i	0.877220	−	0.480089i	$-0.159396\pi$
$157$	7054.26	+	12218.3i	0.0228403	+	0.0395606i	−0.854379	+	0.519650i	$0.826062\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−304913.	+	116428.i	−0.927068	+	0.353991i
$162$		0			0
$163$	−327188.	+	566706.i	−0.964558	+	1.67066i	−0.253762	+	0.967267i	$0.581668\pi$
$163$	−327188.	+	566706.i	−0.964558	+	1.67066i	−0.710797	+	0.703397i	$0.751665\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−182945.			−0.507610			−0.253805	−	0.967255i	$-0.581682\pi$
$167$	−182945.			−0.507610			−0.253805	+	0.967255i	$0.581682\pi$
$168$		0			0
$169$	299030.			0.805374
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−146004.	+	252887.i	−0.370894	+	0.642407i	−0.989703	−	0.143134i	$-0.954282\pi$
$173$	−146004.	+	252887.i	−0.370894	+	0.642407i	0.618809	+	0.785541i	$0.287615\pi$
$174$		0			0
$175$	−62837.2	+	392253.i	−0.155103	+	0.968214i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	97383.8	+	168674.i	0.227172	+	0.393473i	0.956969	−	0.290191i	$-0.0937188\pi$
$179$	97383.8	+	168674.i	0.227172	+	0.393473i	−0.729797	+	0.683664i	$0.760385\pi$
$180$		0			0
$181$	−256634.			−0.582261			−0.291130	−	0.956683i	$-0.594031\pi$
$181$	−256634.			−0.582261			−0.291130	+	0.956683i	$0.594031\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	191551.	+	331776.i	0.411486	+	0.712714i
$186$		0			0
$187$	−383542.	+	664315.i	−0.802065	+	1.38922i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−146975.	+	254567.i	−0.291513	+	0.504916i	−0.974168	−	0.225825i	$-0.927492\pi$
$191$	−146975.	+	254567.i	−0.291513	+	0.504916i	0.682654	+	0.730741i	$0.260825\pi$
$192$		0			0
$193$	34565.0	+	59868.4i	0.0667950	+	0.115692i	0.897489	−	0.441037i	$-0.145389\pi$
$193$	34565.0	+	59868.4i	0.0667950	+	0.115692i	−0.830694	+	0.556730i	$0.812056\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−331748.			−0.609037			−0.304518	−	0.952506i	$-0.598495\pi$
$197$	−331748.			−0.609037			−0.304518	+	0.952506i	$0.598495\pi$
$198$		0			0
$199$	−287214.	−	497470.i	−0.514130	−	0.890500i	−0.999866	−	0.0163939i	$-0.994781\pi$
$199$	−287214.	−	497470.i	−0.514130	−	0.890500i	0.485735	−	0.874106i	$-0.338552\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−327795.	−	266403.i	−0.558293	−	0.453732i
$204$		0			0
$205$	515076.	−	892138.i	0.856025	−	1.48268i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	396272.			0.627521
$210$		0			0
$211$	−383999.			−0.593777			−0.296889	−	0.954912i	$-0.595949\pi$
$211$	−383999.			−0.593777			−0.296889	+	0.954912i	$0.595949\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−365967.	+	633873.i	−0.539940	+	0.935203i
$216$		0			0
$217$	−207508.	+	1.29534e6i	−0.299148	+	1.86739i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	454100.	+	786524.i	0.625418	+	1.08326i
$222$		0			0
$223$	−347165.			−0.467492			−0.233746	−	0.972298i	$-0.575098\pi$
$223$	−347165.			−0.467492			−0.233746	+	0.972298i	$0.575098\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	180641.	+	312879.i	0.232676	+	0.403006i	0.958595	−	0.284774i	$-0.0919186\pi$
$227$	180641.	+	312879.i	0.232676	+	0.403006i	−0.725919	+	0.687780i	$0.758585\pi$
$228$		0			0
$229$	376492.	−	652102.i	0.474424	−	0.821726i	−0.525147	−	0.851011i	$-0.675990\pi$
$229$	376492.	−	652102.i	0.474424	−	0.821726i	0.999571	+	0.0292852i	$0.00932309\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	357199.	−	618686.i	0.431042	−	0.746587i	−0.565921	−	0.824460i	$-0.691479\pi$
$233$	357199.	−	618686.i	0.431042	−	0.746587i	0.996963	+	0.0778721i	$0.0248126\pi$
$234$		0			0
$235$	−507661.	−	879295.i	−0.599659	−	1.03864i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−343587.			−0.389083			−0.194541	−	0.980894i	$-0.562322\pi$
$239$	−343587.			−0.389083			−0.194541	+	0.980894i	$0.562322\pi$
$240$		0			0
$241$	38152.5	+	66082.0i	0.0423136	+	0.0732893i	0.886407	−	0.462908i	$-0.153194\pi$
$241$	38152.5	+	66082.0i	0.0423136	+	0.0732893i	−0.844093	+	0.536197i	$0.819860\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.25607e6	+	413033.i	1.33690	+	0.439612i
$246$		0			0
$247$	234586.	−	406314.i	0.244658	−	0.423760i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−196249.			−0.196618			−0.0983090	−	0.995156i	$-0.531343\pi$
$251$	−196249.			−0.196618			−0.0983090	+	0.995156i	$0.531343\pi$
$252$		0			0
$253$	1.74096e6			1.70997
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−854554.	+	1.48013e6i	−0.807062	+	1.39787i	0.107828	+	0.994170i	$0.465610\pi$
$257$	−854554.	+	1.48013e6i	−0.807062	+	1.39787i	−0.914890	+	0.403703i	$0.867723\pi$
$258$		0			0
$259$	589774.	−	225199.i	0.546306	−	0.208601i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−395762.	−	685481.i	−0.352814	−	0.611091i	0.633928	−	0.773392i	$-0.281442\pi$
$263$	−395762.	−	685481.i	−0.352814	−	0.611091i	−0.986741	+	0.162301i	$0.948108\pi$
$264$		0			0
$265$	−1.53726e6			−1.34473
$266$		0			0
$267$		0			0
$268$		0			0
$269$	579502.	+	1.00373e6i	0.488286	+	0.845736i	0.999909	−	0.0134740i	$-0.00428905\pi$
$269$	579502.	+	1.00373e6i	0.488286	+	0.845736i	−0.511623	+	0.859210i	$0.670956\pi$
$270$		0			0
$271$	−111567.	+	193240.i	−0.0922811	+	0.159836i	−0.908471	−	0.417949i	$-0.862749\pi$
$271$	−111567.	+	193240.i	−0.0922811	+	0.159836i	0.816190	+	0.577784i	$0.196083\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.05949e6	−	1.83509e6i	0.844823	−	1.46328i
$276$		0			0
$277$	234523.	+	406205.i	0.183648	+	0.318087i	0.943120	−	0.332452i	$-0.107876\pi$
$277$	234523.	+	406205.i	0.183648	+	0.318087i	−0.759472	+	0.650540i	$0.774543\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.51447e6			1.14418			0.572090	−	0.820191i	$-0.306133\pi$
$281$	1.51447e6			1.14418			0.572090	+	0.820191i	$0.306133\pi$
$282$		0			0
$283$	535303.	+	927171.i	0.397314	+	0.688167i	0.993393	−	0.114758i	$-0.0366092\pi$
$283$	535303.	+	927171.i	0.397314	+	0.688167i	−0.596080	+	0.802925i	$0.703276\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.31737e6	−	1.07064e6i	−0.944068	−	0.767256i
$288$		0			0
$289$	94682.6	−	163995.i	0.0666846	−	0.115501i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.45450e6			−0.989794			−0.494897	−	0.868952i	$-0.664794\pi$
$293$	−1.45450e6			−0.989794			−0.494897	+	0.868952i	$0.664794\pi$
$294$		0			0
$295$	−1.97627e6			−1.32218
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.03062e6	−	1.78508e6i	0.666684	−	1.15473i
$300$		0			0
$301$	936005.	+	760703.i	0.595473	+	0.483948i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.23366e6	−	2.13676e6i	−0.759356	−	1.31524i
$306$		0			0
$307$	−2.23869e6			−1.35565			−0.677824	−	0.735224i	$-0.737077\pi$
$307$	−2.23869e6			−1.35565			−0.677824	+	0.735224i	$0.737077\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−750695.	−	1.30024e6i	−0.440111	−	0.762295i	0.557586	−	0.830119i	$-0.311728\pi$
$311$	−750695.	−	1.30024e6i	−0.440111	−	0.762295i	−0.997697	+	0.0678240i	$0.978394\pi$
$312$		0			0
$313$	890067.	−	1.54164e6i	0.513525	−	0.889452i	−0.486352	−	0.873763i	$-0.661673\pi$
$313$	890067.	−	1.54164e6i	0.513525	−	0.889452i	0.999877	−	0.0156888i	$-0.00499410\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	735500.	−	1.27392e6i	0.411088	−	0.712025i	−0.583921	−	0.811810i	$-0.698482\pi$
$317$	735500.	−	1.27392e6i	0.411088	−	0.712025i	0.995009	+	0.0997854i	$0.0318156\pi$
$318$		0			0
$319$	1.12655e6	+	1.95124e6i	0.619832	+	1.07358i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	635666.			0.339018
$324$		0			0
$325$	−1.25440e6	−	2.17268e6i	−0.658760	−	1.14101i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.56306e6	+	596837.i	−0.796134	+	0.303995i
$330$		0			0
$331$	914873.	−	1.58461e6i	0.458977	−	0.794971i	−0.539930	−	0.841710i	$-0.681549\pi$
$331$	914873.	−	1.58461e6i	0.458977	−	0.794971i	0.998907	+	0.0467386i	$0.0148828\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4.40120e6			−2.14269
$336$		0			0
$337$	3.18627e6			1.52830			0.764148	−	0.645041i	$-0.223160\pi$
$337$	3.18627e6			1.52830			0.764148	+	0.645041i	$0.223160\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.49878e6	−	6.06006e6i	1.62941	−	2.82222i
$342$		0			0
$343$	999468.	−	1.93614e6i	0.458705	−	0.888589i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.23895e6	+	2.14592e6i	0.552370	+	0.956732i	0.998103	+	0.0615664i	$0.0196096\pi$
$347$	1.23895e6	+	2.14592e6i	0.552370	+	0.956732i	−0.445733	+	0.895166i	$0.647057\pi$
$348$		0			0
$349$	1.81720e6			0.798618			0.399309	−	0.916816i	$-0.369250\pi$
$349$	1.81720e6			0.798618			0.399309	+	0.916816i	$0.369250\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.57678e6	+	2.73106e6i	0.673493	+	1.16652i	0.976907	+	0.213666i	$0.0685403\pi$
$353$	1.57678e6	+	2.73106e6i	0.673493	+	1.16652i	−0.303413	+	0.952859i	$0.598126\pi$
$354$		0			0
$355$	−806425.	+	1.39677e6i	−0.339620	+	0.588239i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.35105e6	−	2.34008e6i	0.553266	−	0.958285i	−0.444770	−	0.895645i	$-0.646715\pi$
$359$	1.35105e6	−	2.34008e6i	0.553266	−	0.958285i	0.998036	−	0.0626404i	$-0.0199521\pi$
$360$		0			0
$361$	1.07386e6	+	1.85998e6i	0.433690	+	0.751173i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	5.32214e6			2.09100
$366$		0			0
$367$	1.80573e6	+	3.12762e6i	0.699824	+	1.21213i	0.968527	+	0.248908i	$0.0800716\pi$
$367$	1.80573e6	+	3.12762e6i	0.699824	+	1.21213i	−0.268703	+	0.963223i	$0.586595\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−400703.	+	2.50134e6i	−0.151143	+	0.943491i
$372$		0			0
$373$	216759.	−	375438.i	0.0806688	−	0.139722i	−0.822869	−	0.568232i	$-0.807628\pi$
$373$	216759.	−	375438.i	0.0806688	−	0.139722i	0.903537	+	0.428509i	$0.140961\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.66759e6			0.966642
$378$		0			0
$379$	−2.12163e6			−0.758704			−0.379352	−	0.925252i	$-0.623853\pi$
$379$	−2.12163e6			−0.758704			−0.379352	+	0.925252i	$0.623853\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2.12339e6	+	3.67782e6i	−0.739662	+	1.28113i	0.212985	+	0.977055i	$0.431681\pi$
$383$	−2.12339e6	+	3.67782e6i	−0.739662	+	1.28113i	−0.952647	+	0.304077i	$0.901652\pi$
$384$		0			0
$385$	−5.47329e6	−	4.44821e6i	−1.88190	−	1.52944i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.76088e6	−	3.04994e6i	−0.590006	−	1.02192i	−0.994231	−	0.107261i	$-0.965792\pi$
$389$	−1.76088e6	−	3.04994e6i	−0.590006	−	1.02192i	0.404224	−	0.914660i	$-0.367541\pi$
$390$		0			0
$391$	2.79271e6			0.923811
$392$		0			0
$393$		0			0
$394$		0			0
$395$	553845.	+	959288.i	0.178606	+	0.309354i
$396$		0			0
$397$	367025.	−	635706.i	0.116874	−	0.202432i	−0.801653	−	0.597790i	$-0.796046\pi$
$397$	367025.	−	635706.i	0.116874	−	0.202432i	0.918527	+	0.395357i	$0.129379\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−111135.	+	192492.i	−0.0345136	+	0.0597793i	−0.882766	−	0.469812i	$-0.844322\pi$
$401$	−111135.	+	192492.i	−0.0345136	+	0.0597793i	0.848253	+	0.529592i	$0.177655\pi$
$402$		0			0
$403$	−4.14242e6	−	7.17488e6i	−1.27055	−	2.20065i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.36743e6			−1.00766
$408$		0			0
$409$	1.75379e6	+	3.03765e6i	0.518405	+	0.897904i	0.999771	+	0.0213846i	$0.00680745\pi$
$409$	1.75379e6	+	3.03765e6i	0.518405	+	0.897904i	−0.481366	+	0.876520i	$0.659859\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−515134.	+	3.21566e6i	−0.148609	+	0.927674i
$414$		0			0
$415$	3.03394e6	−	5.25494e6i	0.864742	−	1.49778i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.97040e6			0.548302			0.274151	−	0.961687i	$-0.411603\pi$
$419$	1.97040e6			0.548302			0.274151	+	0.961687i	$0.411603\pi$
$420$		0			0
$421$	−7.19318e6			−1.97795			−0.988976	−	0.148079i	$-0.952691\pi$
$421$	−7.19318e6			−1.97795			−0.988976	+	0.148079i	$0.952691\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.69955e6	−	2.94370e6i	0.456416	−	0.790535i
$426$		0			0
$427$	−3.79837e6	+	1.45036e6i	−1.00815	+	0.384953i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	262091.	+	453955.i	0.0679608	+	0.117712i	0.898004	−	0.439988i	$-0.145017\pi$
$431$	262091.	+	453955.i	0.0679608	+	0.117712i	−0.830043	+	0.557700i	$0.811684\pi$
$432$		0			0
$433$	1.49326e6			0.382750			0.191375	−	0.981517i	$-0.438705\pi$
$433$	1.49326e6			0.382750			0.191375	+	0.981517i	$0.438705\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−721349.	−	1.24941e6i	−0.180693	−	0.312970i
$438$		0			0
$439$	−1.32835e6	+	2.30077e6i	−0.328966	+	0.569786i	−0.982307	−	0.187278i	$-0.940034\pi$
$439$	−1.32835e6	+	2.30077e6i	−0.328966	+	0.569786i	0.653341	+	0.757064i	$0.273367\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.34399e6	+	4.05991e6i	−0.567474	+	0.982895i	0.429340	+	0.903143i	$0.358746\pi$
$443$	−2.34399e6	+	4.05991e6i	−0.567474	+	0.982895i	−0.996815	+	0.0797518i	$0.974587\pi$
$444$		0			0
$445$	−12618.7	−	21856.2i	−0.00302074	−	0.00523207i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	3.45899e6			0.809716			0.404858	−	0.914379i	$-0.367321\pi$
$449$	3.45899e6			0.809716			0.404858	+	0.914379i	$0.367321\pi$
$450$		0			0
$451$	4.52748e6	+	7.84182e6i	1.04813	+	1.81541i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−7.80102e6	+	2.97873e6i	−1.76654	+	0.674533i
$456$		0			0
$457$	−359140.	+	622048.i	−0.0804401	+	0.139326i	−0.903439	−	0.428717i	$-0.858966\pi$
$457$	−359140.	+	622048.i	−0.0804401	+	0.139326i	0.822999	+	0.568043i	$0.192299\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−5.07998e6			−1.11329			−0.556647	−	0.830749i	$-0.687912\pi$
$461$	−5.07998e6			−1.11329			−0.556647	+	0.830749i	$0.687912\pi$
$462$		0			0
$463$	3.40607e6			0.738416			0.369208	−	0.929347i	$-0.379629\pi$
$463$	3.40607e6			0.738416			0.369208	+	0.929347i	$0.379629\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	921342.	−	1.59581e6i	0.195492	−	0.338602i	−0.751570	−	0.659654i	$-0.770703\pi$
$467$	921342.	−	1.59581e6i	0.195492	−	0.338602i	0.947062	+	0.321052i	$0.104036\pi$
$468$		0			0
$469$	−1.14722e6	+	7.16135e6i	−0.240831	+	1.50336i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.21682e6	−	5.57170e6i	−0.661110	−	1.14508i
$474$		0			0
$475$	−1.75595e6			−0.357091
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.55714e6	+	7.89319e6i	0.907514	+	1.57186i	0.817507	+	0.575919i	$0.195356\pi$
$479$	4.55714e6	+	7.89319e6i	0.907514	+	1.57186i	0.0900071	+	0.995941i	$0.471311\pi$
$480$		0			0
$481$	−1.99346e6	+	3.45277e6i	−0.392866	+	0.680464i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.40596e6	+	7.63134e6i	−0.850523	+	1.47315i
$486$		0			0
$487$	−2.19220e6	−	3.79701e6i	−0.418850	−	0.725469i	0.576974	−	0.816762i	$-0.304233\pi$
$487$	−2.19220e6	−	3.79701e6i	−0.418850	−	0.725469i	−0.995824	+	0.0912933i	$0.970900\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9.50592e6			1.77947			0.889735	−	0.456478i	$-0.150889\pi$
$491$	9.50592e6			1.77947			0.889735	+	0.456478i	$0.150889\pi$
$492$		0			0
$493$	1.80712e6	+	3.13002e6i	0.334864	+	0.580002i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.06253e6	+	1.67625e6i	0.374550	+	0.304401i
$498$		0			0
$499$	−1.68783e6	+	2.92341e6i	−0.303444	+	0.525580i	−0.976914	−	0.213635i	$-0.931470\pi$
$499$	−1.68783e6	+	2.92341e6i	−0.303444	+	0.525580i	0.673470	+	0.739215i	$0.264803\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−7.82645e6			−1.37926			−0.689628	−	0.724164i	$-0.742226\pi$
$503$	−7.82645e6			−1.37926			−0.689628	+	0.724164i	$0.742226\pi$
$504$		0			0
$505$	5.29099e6			0.923227
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3.71219e6	+	6.42969e6i	−0.635090	+	1.10001i	0.351406	+	0.936223i	$0.385704\pi$
$509$	−3.71219e6	+	6.42969e6i	−0.635090	+	1.10001i	−0.986496	+	0.163785i	$0.947630\pi$
$510$		0			0
$511$	1.38727e6	−	8.65985e6i	0.235022	−	1.46709i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.58795e6	−	7.94656e6i	−0.762256	−	1.32027i
$516$		0			0
$517$	8.92461e6			1.46846
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−530710.	−	919217.i	−0.0856570	−	0.148362i	0.820014	−	0.572344i	$-0.193966\pi$
$521$	−530710.	−	919217.i	−0.0856570	−	0.148362i	−0.905671	+	0.423981i	$0.860632\pi$
$522$		0			0
$523$	956760.	−	1.65716e6i	0.152950	−	0.264917i	−0.779361	−	0.626575i	$-0.784456\pi$
$523$	956760.	−	1.65716e6i	0.152950	−	0.264917i	0.932311	+	0.361659i	$0.117789\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	5.61243e6	−	9.72102e6i	0.880288	−	1.52470i
$528$		0			0
$529$	49029.1	+	84921.0i	0.00761755	+	0.0131940i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.07207e7			1.63458
$534$		0			0
$535$	6.31912e6	+	1.09450e7i	0.954492	+	1.65323i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−8.66451e6	+	7.74632e6i	−1.28461	+	1.14848i
$540$		0			0
$541$	−5.83140e6	+	1.01003e7i	−0.856603	+	1.48368i	0.0185472	+	0.999828i	$0.494096\pi$
$541$	−5.83140e6	+	1.01003e7i	−0.856603	+	1.48368i	−0.875150	+	0.483852i	$0.839237\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−9.38842e6			−1.35395
$546$		0			0
$547$	−6.36659e6			−0.909785			−0.454893	−	0.890546i	$-0.650322\pi$
$547$	−6.36659e6			−0.909785			−0.454893	+	0.890546i	$0.650322\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	933548.	−	1.61695e6i	0.130996	−	0.226891i
$552$		0			0
$553$	1.70526e6	−	651134.i	0.237125	−	0.0905435i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.86221e6	+	8.42159e6i	0.664042	+	1.15015i	0.979544	+	0.201230i	$0.0644937\pi$
$557$	4.86221e6	+	8.42159e6i	0.664042	+	1.15015i	−0.315502	+	0.948925i	$0.602173\pi$
$558$		0			0
$559$	−7.61719e6			−1.03102
$560$		0			0
$561$		0			0
$562$		0			0
$563$	3.15133e6	+	5.45826e6i	0.419008	+	0.725744i	0.995840	−	0.0911193i	$-0.0290445\pi$
$563$	3.15133e6	+	5.45826e6i	0.419008	+	0.725744i	−0.576832	+	0.816863i	$0.695711\pi$
$564$		0			0
$565$	4.36207e6	−	7.55532e6i	0.574872	−	0.995708i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	4.08244e6	−	7.07099e6i	0.528614	−	0.915586i	−0.470829	−	0.882224i	$-0.656045\pi$
$569$	4.08244e6	−	7.07099e6i	0.528614	−	0.915586i	0.999443	−	0.0333620i	$-0.0106214\pi$
$570$		0			0
$571$	6.43857e6	+	1.11519e7i	0.826416	+	1.43139i	0.900832	+	0.434168i	$0.142957\pi$
$571$	6.43857e6	+	1.11519e7i	0.826416	+	1.43139i	−0.0744158	+	0.997227i	$0.523709\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−7.71453e6			−0.973061
$576$		0			0
$577$	−5.70887e6	−	9.88805e6i	−0.713856	−	1.23643i	−0.963399	−	0.268071i	$-0.913614\pi$
$577$	−5.70887e6	−	9.88805e6i	−0.713856	−	1.23643i	0.249543	−	0.968364i	$-0.419720\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−7.75968e6	−	6.30639e6i	−0.953681	−	0.775069i
$582$		0			0
$583$	6.75622e6	−	1.17021e7i	0.823251	−	1.42591i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−2.12781e6			−0.254882			−0.127441	−	0.991846i	$-0.540676\pi$
$587$	−2.12781e6			−0.254882			−0.127441	+	0.991846i	$0.540676\pi$
$588$		0			0
$589$	−5.79871e6			−0.688721
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−4.28689e6	+	7.42511e6i	−0.500617	+	0.867094i	0.499383	+	0.866382i	$0.333560\pi$
$593$	−4.28689e6	+	7.42511e6i	−0.500617	+	0.867094i	−1.00000		0.000712836i	$0.999773\pi$
$594$		0			0
$595$	−8.77978e6	−	7.13544e6i	−1.01670	−	0.826282i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	5.64372e6	+	9.77521e6i	0.642686	+	1.11316i	0.984831	+	0.173517i	$0.0555131\pi$
$599$	5.64372e6	+	9.77521e6i	0.642686	+	1.11316i	−0.342145	+	0.939647i	$0.611154\pi$
$600$		0			0
$601$	−1.58260e7			−1.78725			−0.893625	−	0.448814i	$-0.851847\pi$
$601$	−1.58260e7			−1.78725			−0.893625	+	0.448814i	$0.851847\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.24753e7	+	2.16078e7i	1.38568	+	2.40006i
$606$		0			0
$607$	−4.87183e6	+	8.43826e6i	−0.536686	+	0.929568i	0.462393	+	0.886675i	$0.346991\pi$
$607$	−4.87183e6	+	8.43826e6i	−0.536686	+	0.929568i	−0.999080	+	0.0428932i	$0.986342\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5.28320e6	−	9.15077e6i	0.572525	−	0.991642i
$612$		0			0
$613$	−3.07524e6	−	5.32647e6i	−0.330543	−	0.572517i	0.652075	−	0.758154i	$-0.273899\pi$
$613$	−3.07524e6	−	5.32647e6i	−0.330543	−	0.572517i	−0.982618	+	0.185637i	$0.940565\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.36084e6			0.355414			0.177707	−	0.984083i	$-0.443132\pi$
$617$	3.36084e6			0.355414			0.177707	+	0.984083i	$0.443132\pi$
$618$		0			0
$619$	−3.44329e6	−	5.96396e6i	−0.361200	−	0.625616i	0.626959	−	0.779052i	$-0.284299\pi$
$619$	−3.44329e6	−	5.96396e6i	−0.361200	−	0.625616i	−0.988159	+	0.153436i	$0.950966\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−38852.1	+	14835.3i	−0.00401046	+	0.00153135i
$624$		0			0
$625$	4.97590e6	−	8.61850e6i	0.509532	−	0.882535i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−5.40175e6			−0.544387
$630$		0			0
$631$	3.59545e6			0.359484			0.179742	−	0.983714i	$-0.442474\pi$
$631$	3.59545e6			0.359484			0.179742	+	0.983714i	$0.442474\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.23980e7	−	2.14740e7i	1.22017	−	2.11339i
$636$		0			0
$637$	2.81339e6	+	1.34698e7i	0.274715	+	1.31526i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9.91433e6	−	1.71721e7i	−0.953056	−	1.65074i	−0.738756	−	0.673973i	$-0.764587\pi$
$641$	−9.91433e6	−	1.71721e7i	−0.953056	−	1.65074i	−0.214299	−	0.976768i	$-0.568747\pi$
$642$		0			0
$643$	−8.10851e6			−0.773417			−0.386708	−	0.922202i	$-0.626388\pi$
$643$	−8.10851e6			−0.773417			−0.386708	+	0.922202i	$0.626388\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.01373e6	+	1.75583e6i	0.0952053	+	0.164900i	0.909694	−	0.415279i	$-0.136316\pi$
$647$	1.01373e6	+	1.75583e6i	0.0952053	+	0.164900i	−0.814489	+	0.580179i	$0.802983\pi$
$648$		0			0
$649$	8.68564e6	−	1.50440e7i	0.809450	−	1.40201i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	757133.	−	1.31139e6i	0.0694847	−	0.120351i	−0.829190	−	0.558967i	$-0.811198\pi$
$653$	757133.	−	1.31139e6i	0.0694847	−	0.120351i	0.898675	+	0.438616i	$0.144531\pi$
$654$		0			0
$655$	−3.36969e6	−	5.83647e6i	−0.306893	−	0.531554i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	3.22358e6			0.289151			0.144576	−	0.989494i	$-0.453818\pi$
$659$	3.22358e6			0.289151			0.144576	+	0.989494i	$0.453818\pi$
$660$		0			0
$661$	7.19538e6	+	1.24628e7i	0.640546	+	1.10946i	0.985311	+	0.170769i	$0.0546253\pi$
$661$	7.19538e6	+	1.24628e7i	0.640546	+	1.10946i	−0.344765	+	0.938689i	$0.612041\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−924488.	+	5.77100e6i	−0.0810676	+	0.506054i
$666$		0			0
$667$	4.10141e6	−	7.10384e6i	0.356959	−	0.618271i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	2.16875e7			1.85953
$672$		0			0
$673$	7.33478e6			0.624237			0.312118	−	0.950043i	$-0.398961\pi$
$673$	7.33478e6			0.624237			0.312118	+	0.950043i	$0.398961\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.52642e6	+	2.64384e6i	−0.127998	+	0.221699i	−0.922901	−	0.385038i	$-0.874188\pi$
$677$	−1.52642e6	+	2.64384e6i	−0.127998	+	0.221699i	0.794903	+	0.606737i	$0.207522\pi$
$678$		0			0
$679$	1.12688e7	+	9.15828e6i	0.937999	+	0.762324i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	7.98793e6	+	1.38355e7i	0.655213	+	1.13486i	0.981840	+	0.189709i	$0.0607546\pi$
$683$	7.98793e6	+	1.38355e7i	0.655213	+	1.13486i	−0.326627	+	0.945153i	$0.605912\pi$
$684$		0			0
$685$	−2.56724e6			−0.209045
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−7.99911e6	−	1.38549e7i	−0.641939	−	1.11187i
$690$		0			0
$691$	1.23267e6	−	2.13504e6i	0.0982087	−	0.170103i	−0.812735	−	0.582634i	$-0.802022\pi$
$691$	1.23267e6	−	2.13504e6i	0.0982087	−	0.170103i	0.910943	+	0.412532i	$0.135355\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.12638e6	−	1.40753e7i	0.638168	−	1.10534i
$696$		0			0
$697$	7.26259e6	+	1.25792e7i	0.566252	+	0.980777i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.74893e7			−1.34424			−0.672122	−	0.740440i	$-0.734617\pi$
$701$	−1.74893e7			−1.34424			−0.672122	+	0.740440i	$0.734617\pi$
$702$		0			0
$703$	1.39526e6	+	2.41666e6i	0.106480	+	0.184428i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.37915e6	−	8.60916e6i	0.103768	−	0.647757i
$708$		0			0
$709$	1.09216e7	−	1.89167e7i	0.815961	−	1.41329i	−0.0926752	−	0.995696i	$-0.529542\pi$
$709$	1.09216e7	−	1.89167e7i	0.815961	−	1.41329i	0.908636	−	0.417589i	$-0.137125\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.54758e7			−1.87674
$714$		0			0
$715$	4.45415e7			3.25837
$716$		0			0
$717$		0			0
$718$		0			0
$719$	9.55868e6	−	1.65561e7i	0.689566	−	1.19436i	−0.282413	−	0.959293i	$-0.591135\pi$
$719$	9.55868e6	−	1.65561e7i	0.689566	−	1.19436i	0.971978	−	0.235070i	$-0.0755319\pi$
$720$		0			0
$721$	−1.41260e7	+	5.39387e6i	−1.01200	+	0.386423i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.99195e6	−	8.64632e6i	−0.352716	−	0.610923i
$726$		0			0
$727$	−4.08900e6			−0.286933			−0.143467	−	0.989655i	$-0.545825\pi$
$727$	−4.08900e6			−0.286933			−0.143467	+	0.989655i	$0.545825\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−5.16015e6	−	8.93764e6i	−0.357165	−	0.618628i
$732$		0			0
$733$	1.81634e6	−	3.14599e6i	0.124864	−	0.216271i	−0.796816	−	0.604222i	$-0.793484\pi$
$733$	1.81634e6	−	3.14599e6i	0.124864	−	0.216271i	0.921680	+	0.387952i	$0.126817\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.93431e7	−	3.35032e7i	1.31177	−	2.27205i
$738$		0			0
$739$	−1.08978e6	−	1.88756e6i	−0.0734056	−	0.127142i	0.826986	−	0.562222i	$-0.190053\pi$
$739$	−1.08978e6	−	1.88756e6i	−0.0734056	−	0.127142i	−0.900392	+	0.435080i	$0.856720\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.21595e7			−0.808060			−0.404030	−	0.914746i	$-0.632391\pi$
$743$	−1.21595e7			−0.808060			−0.404030	+	0.914746i	$0.632391\pi$
$744$		0			0
$745$	−1.90272e7	−	3.29561e7i	−1.25598	−	2.17543i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.94562e7	−	7.42914e6i	1.26722	−	0.483876i
$750$		0			0
$751$	−1.16248e7	+	2.01347e7i	−0.752115	+	1.30270i	0.194681	+	0.980867i	$0.437633\pi$
$751$	−1.16248e7	+	2.01347e7i	−0.752115	+	1.30270i	−0.946796	+	0.321835i	$0.895701\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.31556e7			0.839928
$756$		0			0
$757$	−1.49939e7			−0.950985			−0.475492	−	0.879720i	$-0.657730\pi$
$757$	−1.49939e7			−0.950985			−0.475492	+	0.879720i	$0.657730\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6.36596e6	+	1.10262e7i	−0.398476	+	0.690181i	−0.993538	−	0.113499i	$-0.963794\pi$
$761$	−6.36596e6	+	1.10262e7i	−0.398476	+	0.690181i	0.595062	+	0.803680i	$0.297127\pi$
$762$		0			0
$763$	−2.44719e6	+	1.52762e7i	−0.152179	+	0.949960i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.02835e7	−	1.78115e7i	−0.631177	−	1.09323i
$768$		0			0
$769$	2.34653e6			0.143090			0.0715451	−	0.997437i	$-0.477207\pi$
$769$	2.34653e6			0.143090			0.0715451	+	0.997437i	$0.477207\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.35931e7	+	2.35440e7i	0.818222	+	1.41720i	0.906991	+	0.421150i	$0.138373\pi$
$773$	1.35931e7	+	2.35440e7i	0.818222	+	1.41720i	−0.0887694	+	0.996052i	$0.528293\pi$
$774$		0			0
$775$	−1.55037e7	+	2.68532e7i	−0.927217	+	1.60599i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.75182e6	−	6.49834e6i	0.221513	−	0.383671i
$780$		0			0
$781$	−7.08842e6	−	1.22775e7i	−0.415836	−	0.720249i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.10994e6			0.0642874
$786$		0			0
$787$	−2.15137e6	−	3.72628e6i	−0.123816	−	0.214456i	0.797453	−	0.603381i	$-0.206180\pi$
$787$	−2.15137e6	−	3.72628e6i	−0.123816	−	0.214456i	−0.921270	+	0.388925i	$0.872847\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1.11565e7	−	9.06705e6i	−0.633998	−	0.515258

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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	252.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(39.3359 - 68.1317i) q^{5} +(-100.606 - 81.7641i) q^{7} +(345.759 + 598.873i) q^{11} +818.732 q^{13} +(554.638 + 960.661i) q^{17} +(286.523 - 496.273i) q^{19} +(1258.80 - 2180.30i) q^{23} +(-1532.12 - 2653.71i) q^{25} +3258.19 q^{29} +(-5059.55 - 8763.40i) q^{31} +(-9528.17 + 3638.23i) q^{35} +(-2434.81 + 4217.21i) q^{37} +13094.3 q^{41} -9303.64 q^{43} +(6452.90 - 11176.8i) q^{47} +(3436.28 + 16452.0i) q^{49} +(-9770.12 - 16922.3i) q^{53} +54403.0 q^{55} +(-12560.2 - 21755.0i) q^{59} +(15681.1 - 27160.5i) q^{61} +(32205.6 - 55781.7i) q^{65} +(-27971.9 - 48448.8i) q^{67} -20501.0 q^{71} +(33825.0 + 58586.6i) q^{73} +(14180.7 - 88521.1i) q^{77} +(-7039.95 + 12193.5i) q^{79} +77129.1 q^{83} +87268.6 q^{85} +(160.396 - 277.815i) q^{89} +(-82369.7 - 66942.9i) q^{91} +(-22541.3 - 39042.6i) q^{95} -112009. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 42 q^{7} + 462 q^{11} - 1204 q^{13} - 228 q^{17} + 358 q^{19} + 2148 q^{23} - 5454 q^{25} + 11064 q^{29} + 830 q^{31} - 7692 q^{35} - 3914 q^{37} + 16632 q^{41} - 29036 q^{43} - 41700 q^{47} + 41876 q^{49}+ \cdots - 433356 q^{97}+O(q^{100}) \) 8 * q - 42 * q^7 + 462 * q^11 - 1204 * q^13 - 228 * q^17 + 358 * q^19 + 2148 * q^23 - 5454 * q^25 + 11064 * q^29 + 830 * q^31 - 7692 * q^35 - 3914 * q^37 + 16632 * q^41 - 29036 * q^43 - 41700 * q^47 + 41876 * q^49 - 22164 * q^53 + 7784 * q^55 - 32886 * q^59 + 83732 * q^61 + 93192 * q^65 - 80034 * q^67 - 89544 * q^71 - 22470 * q^73 - 138732 * q^77 - 75286 * q^79 + 34836 * q^83 + 278504 * q^85 - 28944 * q^89 - 12058 * q^91 + 144120 * q^95 - 433356 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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