LMFDB - Embedded newform 528.6.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,6,Mod(1,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("528.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 528 = 2^{4} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	528.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:	$84.6826568613$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{313}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 78 $ x^2 - x - 78
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$9.34590$ of defining polynomial
Character	$\chi$	$=$	528.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} -107.459 q^{5} +26.6918 q^{7} +81.0000 q^{9} +O(q^{10})$	$q+9.00000 q^{3} -107.459 q^{5} +26.6918 q^{7} +81.0000 q^{9} -121.000 q^{11} +904.666 q^{13} -967.131 q^{15} -495.384 q^{17} +1501.38 q^{19} +240.226 q^{21} -2393.01 q^{23} +8422.44 q^{25} +729.000 q^{27} -5132.34 q^{29} +410.309 q^{31} -1089.00 q^{33} -2868.28 q^{35} +5828.69 q^{37} +8141.99 q^{39} +18561.8 q^{41} +788.920 q^{43} -8704.18 q^{45} -3979.11 q^{47} -16094.5 q^{49} -4458.45 q^{51} +15130.5 q^{53} +13002.5 q^{55} +13512.5 q^{57} -41623.5 q^{59} -29989.8 q^{61} +2162.04 q^{63} -97214.5 q^{65} -25890.3 q^{67} -21537.1 q^{69} -58163.0 q^{71} +23875.3 q^{73} +75802.0 q^{75} -3229.71 q^{77} -27402.8 q^{79} +6561.00 q^{81} -24056.3 q^{83} +53233.4 q^{85} -46191.1 q^{87} -55820.0 q^{89} +24147.2 q^{91} +3692.78 q^{93} -161337. q^{95} -101188. q^{97} -9801.00 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 18 q^{3} - 38 q^{5} + 18 q^{7} + 162 q^{9}+O(q^{10}) $ 2 * q + 18 * q^3 - 38 * q^5 + 18 * q^7 + 162 * q^9	$ 2 q + 18 q^{3} - 38 q^{5} + 18 q^{7} + 162 q^{9} - 242 q^{11} - 66 q^{13} - 342 q^{15} - 920 q^{17} + 2932 q^{19} + 162 q^{21} - 5246 q^{23} + 10122 q^{25} + 1458 q^{27} - 12600 q^{29} - 9936 q^{31} - 2178 q^{33} - 3472 q^{35} + 5996 q^{37} - 594 q^{39} + 24244 q^{41} - 20360 q^{43} - 3078 q^{45} + 5806 q^{47} - 32826 q^{49} - 8280 q^{51} + 40770 q^{53} + 4598 q^{55} + 26388 q^{57} - 18212 q^{59} - 11398 q^{61} + 1458 q^{63} - 164636 q^{65} - 65368 q^{67} - 47214 q^{69} - 61446 q^{71} + 53412 q^{73} + 91098 q^{75} - 2178 q^{77} - 17122 q^{79} + 13122 q^{81} + 14304 q^{83} + 23740 q^{85} - 113400 q^{87} - 58140 q^{89} + 32584 q^{91} - 89424 q^{93} - 61968 q^{95} - 183056 q^{97} - 19602 q^{99}+O(q^{100}) $ 2 * q + 18 * q^3 - 38 * q^5 + 18 * q^7 + 162 * q^9 - 242 * q^11 - 66 * q^13 - 342 * q^15 - 920 * q^17 + 2932 * q^19 + 162 * q^21 - 5246 * q^23 + 10122 * q^25 + 1458 * q^27 - 12600 * q^29 - 9936 * q^31 - 2178 * q^33 - 3472 * q^35 + 5996 * q^37 - 594 * q^39 + 24244 * q^41 - 20360 * q^43 - 3078 * q^45 + 5806 * q^47 - 32826 * q^49 - 8280 * q^51 + 40770 * q^53 + 4598 * q^55 + 26388 * q^57 - 18212 * q^59 - 11398 * q^61 + 1458 * q^63 - 164636 * q^65 - 65368 * q^67 - 47214 * q^69 - 61446 * q^71 + 53412 * q^73 + 91098 * q^75 - 2178 * q^77 - 17122 * q^79 + 13122 * q^81 + 14304 * q^83 + 23740 * q^85 - 113400 * q^87 - 58140 * q^89 + 32584 * q^91 - 89424 * q^93 - 61968 * q^95 - 183056 * q^97 - 19602 * 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
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	−107.459	−1.92229	−0.961143	−	0.276052i	$-0.910974\pi$
$5$	−107.459	−1.92229	−0.961143	+	0.276052i	$0.910974\pi$
$6$	0	0
$7$	26.6918	0.205889	0.102944	−	0.994687i	$-0.467174\pi$
$7$	26.6918	0.205889	0.102944	+	0.994687i	$0.467174\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	904.666	1.48467	0.742335	−	0.670029i	$-0.233718\pi$
$13$	904.666	1.48467	0.742335	+	0.670029i	$0.233718\pi$
$14$	0	0
$15$	−967.131	−1.10983
$16$	0	0
$17$	−495.384	−0.415738	−0.207869	−	0.978157i	$-0.566653\pi$
$17$	−495.384	−0.415738	−0.207869	+	0.978157i	$0.566653\pi$
$18$	0	0
$19$	1501.38	0.954130	0.477065	−	0.878868i	$-0.341701\pi$
$19$	1501.38	0.954130	0.477065	+	0.878868i	$0.341701\pi$
$20$	0	0
$21$	240.226	0.118870
$22$	0	0
$23$	−2393.01	−0.943245	−0.471622	−	0.881801i	$-0.656331\pi$
$23$	−2393.01	−0.943245	−0.471622	+	0.881801i	$0.656331\pi$
$24$	0	0
$25$	8422.44	2.69518
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	−5132.34	−1.13324	−0.566618	−	0.823980i	$-0.691749\pi$
$29$	−5132.34	−1.13324	−0.566618	+	0.823980i	$0.691749\pi$
$30$	0	0
$31$	410.309	0.0766844	0.0383422	−	0.999265i	$-0.487792\pi$
$31$	410.309	0.0766844	0.0383422	+	0.999265i	$0.487792\pi$
$32$	0	0
$33$	−1089.00	−0.174078
$34$	0	0
$35$	−2868.28	−0.395777
$36$	0	0
$37$	5828.69	0.699949	0.349975	−	0.936759i	$-0.386190\pi$
$37$	5828.69	0.699949	0.349975	+	0.936759i	$0.386190\pi$
$38$	0	0
$39$	8141.99	0.857174
$40$	0	0
$41$	18561.8	1.72449	0.862245	−	0.506491i	$-0.169058\pi$
$41$	18561.8	1.72449	0.862245	+	0.506491i	$0.169058\pi$
$42$	0	0
$43$	788.920	0.0650671	0.0325336	−	0.999471i	$-0.489642\pi$
$43$	788.920	0.0650671	0.0325336	+	0.999471i	$0.489642\pi$
$44$	0	0
$45$	−8704.18	−0.640762
$46$	0	0
$47$	−3979.11	−0.262749	−0.131375	−	0.991333i	$-0.541939\pi$
$47$	−3979.11	−0.262749	−0.131375	+	0.991333i	$0.541939\pi$
$48$	0	0
$49$	−16094.5	−0.957610
$50$	0	0
$51$	−4458.45	−0.240026
$52$	0	0
$53$	15130.5	0.739886	0.369943	−	0.929055i	$-0.379377\pi$
$53$	15130.5	0.739886	0.369943	+	0.929055i	$0.379377\pi$
$54$	0	0
$55$	13002.5	0.579591
$56$	0	0
$57$	13512.5	0.550867
$58$	0	0
$59$	−41623.5	−1.55671	−0.778357	−	0.627822i	$-0.783947\pi$
$59$	−41623.5	−1.55671	−0.778357	+	0.627822i	$0.783947\pi$
$60$	0	0
$61$	−29989.8	−1.03193	−0.515964	−	0.856610i	$-0.672566\pi$
$61$	−29989.8	−1.03193	−0.515964	+	0.856610i	$0.672566\pi$
$62$	0	0
$63$	2162.04	0.0686296
$64$	0	0
$65$	−97214.5	−2.85396
$66$	0	0
$67$	−25890.3	−0.704613	−0.352307	−	0.935885i	$-0.614603\pi$
$67$	−25890.3	−0.704613	−0.352307	+	0.935885i	$0.614603\pi$
$68$	0	0
$69$	−21537.1	−0.544582
$70$	0	0
$71$	−58163.0	−1.36931	−0.684654	−	0.728869i	$-0.740046\pi$
$71$	−58163.0	−1.36931	−0.684654	+	0.728869i	$0.740046\pi$
$72$	0	0
$73$	23875.3	0.524375	0.262187	−	0.965017i	$-0.415556\pi$
$73$	23875.3	0.524375	0.262187	+	0.965017i	$0.415556\pi$
$74$	0	0
$75$	75802.0	1.55606
$76$	0	0
$77$	−3229.71	−0.0620778
$78$	0	0
$79$	−27402.8	−0.494000	−0.247000	−	0.969016i	$-0.579445\pi$
$79$	−27402.8	−0.494000	−0.247000	+	0.969016i	$0.579445\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−24056.3	−0.383296	−0.191648	−	0.981464i	$-0.561383\pi$
$83$	−24056.3	−0.383296	−0.191648	+	0.981464i	$0.561383\pi$
$84$	0	0
$85$	53233.4	0.799166
$86$	0	0
$87$	−46191.1	−0.654274
$88$	0	0
$89$	−55820.0	−0.746990	−0.373495	−	0.927632i	$-0.621841\pi$
$89$	−55820.0	−0.746990	−0.373495	+	0.927632i	$0.621841\pi$
$90$	0	0
$91$	24147.2	0.305677
$92$	0	0
$93$	3692.78	0.0442737
$94$	0	0
$95$	−161337.	−1.83411
$96$	0	0
$97$	−101188.	−1.09194	−0.545970	−	0.837805i	$-0.683839\pi$
$97$	−101188.	−1.09194	−0.545970	+	0.837805i	$0.683839\pi$
$98$	0	0
$99$	−9801.00	−0.100504
$100$	0	0
$101$	−75190.1	−0.733428	−0.366714	−	0.930334i	$-0.619517\pi$
$101$	−75190.1	−0.733428	−0.366714	+	0.930334i	$0.619517\pi$
$102$	0	0
$103$	−176059.	−1.63518	−0.817590	−	0.575801i	$-0.804690\pi$
$103$	−176059.	−1.63518	−0.817590	+	0.575801i	$0.804690\pi$
$104$	0	0
$105$	−25814.5	−0.228502
$106$	0	0
$107$	124963.	1.05517	0.527586	−	0.849501i	$-0.323097\pi$
$107$	124963.	1.05517	0.527586	+	0.849501i	$0.323097\pi$
$108$	0	0
$109$	48001.5	0.386980	0.193490	−	0.981102i	$-0.438019\pi$
$109$	48001.5	0.386980	0.193490	+	0.981102i	$0.438019\pi$
$110$	0	0
$111$	52458.2	0.404116
$112$	0	0
$113$	38955.6	0.286995	0.143497	−	0.989651i	$-0.454165\pi$
$113$	38955.6	0.286995	0.143497	+	0.989651i	$0.454165\pi$
$114$	0	0
$115$	257150.	1.81319
$116$	0	0
$117$	73277.9	0.494890
$118$	0	0
$119$	−13222.7	−0.0855957
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	167056.	0.995635
$124$	0	0
$125$	−569258.	−3.25862
$126$	0	0
$127$	−186757.	−1.02746	−0.513732	−	0.857951i	$-0.671738\pi$
$127$	−186757.	−1.02746	−0.513732	+	0.857951i	$0.671738\pi$
$128$	0	0
$129$	7100.28	0.0375665
$130$	0	0
$131$	363603.	1.85118	0.925592	−	0.378523i	$-0.123568\pi$
$131$	363603.	1.85118	0.925592	+	0.378523i	$0.123568\pi$
$132$	0	0
$133$	40074.6	0.196445
$134$	0	0
$135$	−78337.6	−0.369944
$136$	0	0
$137$	−74002.6	−0.336857	−0.168429	−	0.985714i	$-0.553869\pi$
$137$	−74002.6	−0.336857	−0.168429	+	0.985714i	$0.553869\pi$
$138$	0	0
$139$	−70042.3	−0.307485	−0.153742	−	0.988111i	$-0.549133\pi$
$139$	−70042.3	−0.307485	−0.153742	+	0.988111i	$0.549133\pi$
$140$	0	0
$141$	−35812.0	−0.151698
$142$	0	0
$143$	−109465.	−0.447645
$144$	0	0
$145$	551516.	2.17840
$146$	0	0
$147$	−144851.	−0.552876
$148$	0	0
$149$	327767.	1.20948	0.604741	−	0.796422i	$-0.293276\pi$
$149$	327767.	1.20948	0.604741	+	0.796422i	$0.293276\pi$
$150$	0	0
$151$	−519.828	−0.00185531	−0.000927656	−	1.00000i	$-0.500295\pi$
$151$	−519.828	−0.00185531	−0.000927656		1.00000i	$0.500295\pi$
$152$	0	0
$153$	−40126.1	−0.138579
$154$	0	0
$155$	−44091.4	−0.147409
$156$	0	0
$157$	−119470.	−0.386821	−0.193411	−	0.981118i	$-0.561955\pi$
$157$	−119470.	−0.386821	−0.193411	+	0.981118i	$0.561955\pi$
$158$	0	0
$159$	136175.	0.427173
$160$	0	0
$161$	−63873.7	−0.194204
$162$	0	0
$163$	−314012.	−0.925716	−0.462858	−	0.886432i	$-0.653176\pi$
$163$	−314012.	−0.925716	−0.462858	+	0.886432i	$0.653176\pi$
$164$	0	0
$165$	117023.	0.334627
$166$	0	0
$167$	−281877.	−0.782111	−0.391056	−	0.920367i	$-0.627890\pi$
$167$	−281877.	−0.782111	−0.391056	+	0.920367i	$0.627890\pi$
$168$	0	0
$169$	447127.	1.20424
$170$	0	0
$171$	121612.	0.318043
$172$	0	0
$173$	154131.	0.391538	0.195769	−	0.980650i	$-0.437280\pi$
$173$	154131.	0.391538	0.195769	+	0.980650i	$0.437280\pi$
$174$	0	0
$175$	224810.	0.554908
$176$	0	0
$177$	−374612.	−0.898770
$178$	0	0
$179$	−603292.	−1.40733	−0.703664	−	0.710533i	$-0.748454\pi$
$179$	−603292.	−1.40733	−0.703664	+	0.710533i	$0.748454\pi$
$180$	0	0
$181$	−560418.	−1.27150	−0.635749	−	0.771896i	$-0.719309\pi$
$181$	−560418.	−1.27150	−0.635749	+	0.771896i	$0.719309\pi$
$182$	0	0
$183$	−269909.	−0.595784
$184$	0	0
$185$	−626345.	−1.34550
$186$	0	0
$187$	59941.4	0.125350
$188$	0	0
$189$	19458.3	0.0396233
$190$	0	0
$191$	−337255.	−0.668921	−0.334461	−	0.942410i	$-0.608554\pi$
$191$	−337255.	−0.668921	−0.334461	+	0.942410i	$0.608554\pi$
$192$	0	0
$193$	427002.	0.825158	0.412579	−	0.910922i	$-0.364628\pi$
$193$	427002.	0.825158	0.412579	+	0.910922i	$0.364628\pi$
$194$	0	0
$195$	−874931.	−1.64773
$196$	0	0
$197$	387552.	0.711483	0.355741	−	0.934584i	$-0.384228\pi$
$197$	387552.	0.711483	0.355741	+	0.934584i	$0.384228\pi$
$198$	0	0
$199$	−564409.	−1.01033	−0.505163	−	0.863024i	$-0.668568\pi$
$199$	−564409.	−1.01033	−0.505163	+	0.863024i	$0.668568\pi$
$200$	0	0
$201$	−233013.	−0.406809
$202$	0	0
$203$	−136991.	−0.233321
$204$	0	0
$205$	−1.99463e6	−3.31496
$206$	0	0
$207$	−193834.	−0.314415
$208$	0	0
$209$	−181667.	−0.287681
$210$	0	0
$211$	−495448.	−0.766112	−0.383056	−	0.923725i	$-0.625128\pi$
$211$	−495448.	−0.766112	−0.383056	+	0.923725i	$0.625128\pi$
$212$	0	0
$213$	−523467.	−0.790570
$214$	0	0
$215$	−84776.5	−0.125078
$216$	0	0
$217$	10951.9	0.0157885
$218$	0	0
$219$	214878.	0.302748
$220$	0	0
$221$	−448157.	−0.617233
$222$	0	0
$223$	−1.23517e6	−1.66327	−0.831637	−	0.555320i	$-0.812596\pi$
$223$	−1.23517e6	−1.66327	−0.831637	+	0.555320i	$0.812596\pi$
$224$	0	0
$225$	682218.	0.898394
$226$	0	0
$227$	802179.	1.03325	0.516627	−	0.856211i	$-0.327188\pi$
$227$	802179.	1.03325	0.516627	+	0.856211i	$0.327188\pi$
$228$	0	0
$229$	−162526.	−0.204802	−0.102401	−	0.994743i	$-0.532652\pi$
$229$	−162526.	−0.204802	−0.102401	+	0.994743i	$0.532652\pi$
$230$	0	0
$231$	−29067.4	−0.0358407
$232$	0	0
$233$	−1.45802e6	−1.75944	−0.879718	−	0.475496i	$-0.842269\pi$
$233$	−1.45802e6	−1.75944	−0.879718	+	0.475496i	$0.842269\pi$
$234$	0	0
$235$	427592.	0.505079
$236$	0	0
$237$	−246625.	−0.285211
$238$	0	0
$239$	582475.	0.659603	0.329802	−	0.944050i	$-0.393018\pi$
$239$	582475.	0.659603	0.329802	+	0.944050i	$0.393018\pi$
$240$	0	0
$241$	434564.	0.481961	0.240980	−	0.970530i	$-0.422531\pi$
$241$	434564.	0.481961	0.240980	+	0.970530i	$0.422531\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	1.72950e6	1.84080
$246$	0	0
$247$	1.35825e6	1.41657
$248$	0	0
$249$	−216507.	−0.221296
$250$	0	0
$251$	948184.	0.949967	0.474983	−	0.879995i	$-0.342454\pi$
$251$	948184.	0.949967	0.474983	+	0.879995i	$0.342454\pi$
$252$	0	0
$253$	289554.	0.284399
$254$	0	0
$255$	479101.	0.461399
$256$	0	0
$257$	1.14089e6	1.07749	0.538744	−	0.842470i	$-0.318899\pi$
$257$	1.14089e6	1.07749	0.538744	+	0.842470i	$0.318899\pi$
$258$	0	0
$259$	155578.	0.144112
$260$	0	0
$261$	−415720.	−0.377745
$262$	0	0
$263$	−280915.	−0.250430	−0.125215	−	0.992130i	$-0.539962\pi$
$263$	−280915.	−0.250430	−0.125215	+	0.992130i	$0.539962\pi$
$264$	0	0
$265$	−1.62591e6	−1.42227
$266$	0	0
$267$	−502380.	−0.431275
$268$	0	0
$269$	1.08759e6	0.916397	0.458198	−	0.888850i	$-0.348495\pi$
$269$	1.08759e6	0.916397	0.458198	+	0.888850i	$0.348495\pi$
$270$	0	0
$271$	−1.17929e6	−0.975431	−0.487716	−	0.873003i	$-0.662170\pi$
$271$	−1.17929e6	−0.975431	−0.487716	+	0.873003i	$0.662170\pi$
$272$	0	0
$273$	217324.	0.176483
$274$	0	0
$275$	−1.01912e6	−0.812628
$276$	0	0
$277$	−621357.	−0.486566	−0.243283	−	0.969955i	$-0.578224\pi$
$277$	−621357.	−0.486566	−0.243283	+	0.969955i	$0.578224\pi$
$278$	0	0
$279$	33235.0	0.0255615
$280$	0	0
$281$	−243988.	−0.184333	−0.0921666	−	0.995744i	$-0.529379\pi$
$281$	−243988.	−0.184333	−0.0921666	+	0.995744i	$0.529379\pi$
$282$	0	0
$283$	1.84579e6	1.36999	0.684993	−	0.728549i	$-0.259805\pi$
$283$	1.84579e6	1.36999	0.684993	+	0.728549i	$0.259805\pi$
$284$	0	0
$285$	−1.45204e6	−1.05892
$286$	0	0
$287$	495448.	0.355053
$288$	0	0
$289$	−1.17445e6	−0.827162
$290$	0	0
$291$	−910690.	−0.630431
$292$	0	0
$293$	−1.34899e6	−0.917995	−0.458998	−	0.888437i	$-0.651791\pi$
$293$	−1.34899e6	−0.917995	−0.458998	+	0.888437i	$0.651791\pi$
$294$	0	0
$295$	4.47283e6	2.99245
$296$	0	0
$297$	−88209.0	−0.0580259
$298$	0	0
$299$	−2.16487e6	−1.40041
$300$	0	0
$301$	21057.7	0.0133966
$302$	0	0
$303$	−676711.	−0.423445
$304$	0	0
$305$	3.22268e6	1.98366
$306$	0	0
$307$	1.20545e6	0.729970	0.364985	−	0.931013i	$-0.381074\pi$
$307$	1.20545e6	0.729970	0.364985	+	0.931013i	$0.381074\pi$
$308$	0	0
$309$	−1.58453e6	−0.944072
$310$	0	0
$311$	−636875.	−0.373382	−0.186691	−	0.982419i	$-0.559776\pi$
$311$	−636875.	−0.373382	−0.186691	+	0.982419i	$0.559776\pi$
$312$	0	0
$313$	1.86526e6	1.07616	0.538081	−	0.842893i	$-0.319150\pi$
$313$	1.86526e6	1.07616	0.538081	+	0.842893i	$0.319150\pi$
$314$	0	0
$315$	−232330.	−0.131926
$316$	0	0
$317$	2.35417e6	1.31580	0.657900	−	0.753105i	$-0.271445\pi$
$317$	2.35417e6	1.31580	0.657900	+	0.753105i	$0.271445\pi$
$318$	0	0
$319$	621013.	0.341684
$320$	0	0
$321$	1.12467e6	0.609204
$322$	0	0
$323$	−743761.	−0.396668
$324$	0	0
$325$	7.61950e6	4.00145
$326$	0	0
$327$	432013.	0.223423
$328$	0	0
$329$	−106210.	−0.0540972
$330$	0	0
$331$	−1.97887e6	−0.992768	−0.496384	−	0.868103i	$-0.665339\pi$
$331$	−1.97887e6	−0.992768	−0.496384	+	0.868103i	$0.665339\pi$
$332$	0	0
$333$	472124.	0.233316
$334$	0	0
$335$	2.78215e6	1.35447
$336$	0	0
$337$	2.86225e6	1.37288	0.686439	−	0.727187i	$-0.259173\pi$
$337$	2.86225e6	1.37288	0.686439	+	0.727187i	$0.259173\pi$
$338$	0	0
$339$	350601.	0.165697
$340$	0	0
$341$	−49647.4	−0.0231212
$342$	0	0
$343$	−878202.	−0.403050
$344$	0	0
$345$	2.31435e6	1.04684
$346$	0	0
$347$	1.81514e6	0.809257	0.404629	−	0.914481i	$-0.367401\pi$
$347$	1.81514e6	0.809257	0.404629	+	0.914481i	$0.367401\pi$
$348$	0	0
$349$	−1.77907e6	−0.781861	−0.390930	−	0.920420i	$-0.627847\pi$
$349$	−1.77907e6	−0.781861	−0.390930	+	0.920420i	$0.627847\pi$
$350$	0	0
$351$	659501.	0.285725
$352$	0	0
$353$	−4.18817e6	−1.78890	−0.894452	−	0.447164i	$-0.852434\pi$
$353$	−4.18817e6	−1.78890	−0.894452	+	0.447164i	$0.852434\pi$
$354$	0	0
$355$	6.25014e6	2.63220
$356$	0	0
$357$	−119004.	−0.0494187
$358$	0	0
$359$	−1.13143e6	−0.463329	−0.231665	−	0.972796i	$-0.574417\pi$
$359$	−1.13143e6	−0.463329	−0.231665	+	0.972796i	$0.574417\pi$
$360$	0	0
$361$	−221946.	−0.0896355
$362$	0	0
$363$	131769.	0.0524864
$364$	0	0
$365$	−2.56562e6	−1.00800
$366$	0	0
$367$	−1.31924e6	−0.511279	−0.255639	−	0.966772i	$-0.582286\pi$
$367$	−1.31924e6	−0.511279	−0.255639	+	0.966772i	$0.582286\pi$
$368$	0	0
$369$	1.50351e6	0.574830
$370$	0	0
$371$	403861.	0.152334
$372$	0	0
$373$	741.945	0.000276121 0	0.000138061	−	1.00000i	$-0.499956\pi$
$373$	741.945	0.000276121 0	0.000138061		1.00000i	$0.499956\pi$
$374$	0	0
$375$	−5.12332e6	−1.88137
$376$	0	0
$377$	−4.64305e6	−1.68248
$378$	0	0
$379$	567437.	0.202917	0.101459	−	0.994840i	$-0.467649\pi$
$379$	567437.	0.202917	0.101459	+	0.994840i	$0.467649\pi$
$380$	0	0
$381$	−1.68081e6	−0.593207
$382$	0	0
$383$	−3.54829e6	−1.23601	−0.618006	−	0.786174i	$-0.712059\pi$
$383$	−3.54829e6	−1.23601	−0.618006	+	0.786174i	$0.712059\pi$
$384$	0	0
$385$	347061.	0.119331
$386$	0	0
$387$	63902.5	0.0216890
$388$	0	0
$389$	4.09335e6	1.37153	0.685764	−	0.727824i	$-0.259468\pi$
$389$	4.09335e6	1.37153	0.685764	+	0.727824i	$0.259468\pi$
$390$	0	0
$391$	1.18546e6	0.392142
$392$	0	0
$393$	3.27243e6	1.06878
$394$	0	0
$395$	2.94468e6	0.949609
$396$	0	0
$397$	153511.	0.0488835	0.0244417	−	0.999701i	$-0.492219\pi$
$397$	153511.	0.0488835	0.0244417	+	0.999701i	$0.492219\pi$
$398$	0	0
$399$	360672.	0.113417
$400$	0	0
$401$	474424.	0.147335	0.0736674	−	0.997283i	$-0.476530\pi$
$401$	474424.	0.147335	0.0736674	+	0.997283i	$0.476530\pi$
$402$	0	0
$403$	371193.	0.113851
$404$	0	0
$405$	−705039.	−0.213587
$406$	0	0
$407$	−705271.	−0.211043
$408$	0	0
$409$	2.85246e6	0.843164	0.421582	−	0.906790i	$-0.361475\pi$
$409$	2.85246e6	0.843164	0.421582	+	0.906790i	$0.361475\pi$
$410$	0	0
$411$	−666023.	−0.194484
$412$	0	0
$413$	−1.11101e6	−0.320510
$414$	0	0
$415$	2.58507e6	0.736805
$416$	0	0
$417$	−630381.	−0.177526
$418$	0	0
$419$	−6.61913e6	−1.84190	−0.920950	−	0.389681i	$-0.872585\pi$
$419$	−6.61913e6	−1.84190	−0.920950	+	0.389681i	$0.872585\pi$
$420$	0	0
$421$	1.67579e6	0.460802	0.230401	−	0.973096i	$-0.425996\pi$
$421$	1.67579e6	0.460802	0.230401	+	0.973096i	$0.425996\pi$
$422$	0	0
$423$	−322308.	−0.0875831
$424$	0	0
$425$	−4.17234e6	−1.12049
$426$	0	0
$427$	−800483.	−0.212463
$428$	0	0
$429$	−985181.	−0.258448
$430$	0	0
$431$	4.53856e6	1.17686	0.588430	−	0.808548i	$-0.299746\pi$
$431$	4.53856e6	1.17686	0.588430	+	0.808548i	$0.299746\pi$
$432$	0	0
$433$	379605.	0.0972999	0.0486500	−	0.998816i	$-0.484508\pi$
$433$	379605.	0.0972999	0.0486500	+	0.998816i	$0.484508\pi$
$434$	0	0
$435$	4.96365e6	1.25770
$436$	0	0
$437$	−3.59282e6	−0.899978
$438$	0	0
$439$	1.97778e6	0.489797	0.244899	−	0.969549i	$-0.421245\pi$
$439$	1.97778e6	0.489797	0.244899	+	0.969549i	$0.421245\pi$
$440$	0	0
$441$	−1.30366e6	−0.319203
$442$	0	0
$443$	−3.58373e6	−0.867613	−0.433807	−	0.901006i	$-0.642830\pi$
$443$	−3.58373e6	−0.867613	−0.433807	+	0.901006i	$0.642830\pi$
$444$	0	0
$445$	5.99836e6	1.43593
$446$	0	0
$447$	2.94990e6	0.698295
$448$	0	0
$449$	−1.05735e6	−0.247515	−0.123758	−	0.992312i	$-0.539495\pi$
$449$	−1.05735e6	−0.247515	−0.123758	+	0.992312i	$0.539495\pi$
$450$	0	0
$451$	−2.24598e6	−0.519954
$452$	0	0
$453$	−4678.45	−0.00107117
$454$	0	0
$455$	−2.59483e6	−0.587598
$456$	0	0
$457$	1.39016e6	0.311368	0.155684	−	0.987807i	$-0.450242\pi$
$457$	1.39016e6	0.311368	0.155684	+	0.987807i	$0.450242\pi$
$458$	0	0
$459$	−361135.	−0.0800087
$460$	0	0
$461$	−1.09360e6	−0.239666	−0.119833	−	0.992794i	$-0.538236\pi$
$461$	−1.09360e6	−0.239666	−0.119833	+	0.992794i	$0.538236\pi$
$462$	0	0
$463$	−1.94891e6	−0.422513	−0.211256	−	0.977431i	$-0.567756\pi$
$463$	−1.94891e6	−0.422513	−0.211256	+	0.977431i	$0.567756\pi$
$464$	0	0
$465$	−396823.	−0.0851068
$466$	0	0
$467$	4.69017e6	0.995169	0.497584	−	0.867416i	$-0.334220\pi$
$467$	4.69017e6	0.995169	0.497584	+	0.867416i	$0.334220\pi$
$468$	0	0
$469$	−691060.	−0.145072
$470$	0	0
$471$	−1.07523e6	−0.223331
$472$	0	0
$473$	−95459.3	−0.0196185
$474$	0	0
$475$	1.26453e7	2.57155
$476$	0	0
$477$	1.22557e6	0.246629
$478$	0	0
$479$	−4.15576e6	−0.827584	−0.413792	−	0.910372i	$-0.635796\pi$
$479$	−4.15576e6	−0.827584	−0.413792	+	0.910372i	$0.635796\pi$
$480$	0	0
$481$	5.27302e6	1.03919
$482$	0	0
$483$	−574863.	−0.112123
$484$	0	0
$485$	1.08735e7	2.09902
$486$	0	0
$487$	2.37242e6	0.453282	0.226641	−	0.973978i	$-0.427226\pi$
$487$	2.37242e6	0.453282	0.226641	+	0.973978i	$0.427226\pi$
$488$	0	0
$489$	−2.82611e6	−0.534463
$490$	0	0
$491$	1.18797e6	0.222384	0.111192	−	0.993799i	$-0.464533\pi$
$491$	1.18797e6	0.222384	0.111192	+	0.993799i	$0.464533\pi$
$492$	0	0
$493$	2.54248e6	0.471129
$494$	0	0
$495$	1.05321e6	0.193197
$496$	0	0
$497$	−1.55248e6	−0.281925
$498$	0	0
$499$	−710493.	−0.127735	−0.0638673	−	0.997958i	$-0.520343\pi$
$499$	−710493.	−0.127735	−0.0638673	+	0.997958i	$0.520343\pi$
$500$	0	0
$501$	−2.53689e6	−0.451552
$502$	0	0
$503$	−4.75381e6	−0.837765	−0.418882	−	0.908041i	$-0.637578\pi$
$503$	−4.75381e6	−0.837765	−0.418882	+	0.908041i	$0.637578\pi$
$504$	0	0
$505$	8.07986e6	1.40986
$506$	0	0
$507$	4.02414e6	0.695270
$508$	0	0
$509$	−5.05479e6	−0.864787	−0.432393	−	0.901685i	$-0.642331\pi$
$509$	−5.05479e6	−0.864787	−0.432393	+	0.901685i	$0.642331\pi$
$510$	0	0
$511$	637275.	0.107963
$512$	0	0
$513$	1.09451e6	0.183622
$514$	0	0
$515$	1.89191e7	3.14328
$516$	0	0
$517$	481473.	0.0792219
$518$	0	0
$519$	1.38718e6	0.226055
$520$	0	0
$521$	−4.69775e6	−0.758220	−0.379110	−	0.925352i	$-0.623770\pi$
$521$	−4.69775e6	−0.758220	−0.379110	+	0.925352i	$0.623770\pi$
$522$	0	0
$523$	1.97083e6	0.315062	0.157531	−	0.987514i	$-0.449647\pi$
$523$	1.97083e6	0.315062	0.157531	+	0.987514i	$0.449647\pi$
$524$	0	0
$525$	2.02329e6	0.320376
$526$	0	0
$527$	−203260.	−0.0318806
$528$	0	0
$529$	−709863.	−0.110290
$530$	0	0
$531$	−3.37151e6	−0.518905
$532$	0	0
$533$	1.67922e7	2.56030
$534$	0	0
$535$	−1.34285e7	−2.02834
$536$	0	0
$537$	−5.42963e6	−0.812521
$538$	0	0
$539$	1.94744e6	0.288730
$540$	0	0
$541$	−5.76437e6	−0.846757	−0.423378	−	0.905953i	$-0.639156\pi$
$541$	−5.76437e6	−0.846757	−0.423378	+	0.905953i	$0.639156\pi$
$542$	0	0
$543$	−5.04377e6	−0.734100
$544$	0	0
$545$	−5.15819e6	−0.743886
$546$	0	0
$547$	4.65202e6	0.664772	0.332386	−	0.943143i	$-0.392146\pi$
$547$	4.65202e6	0.664772	0.332386	+	0.943143i	$0.392146\pi$
$548$	0	0
$549$	−2.42918e6	−0.343976
$550$	0	0
$551$	−7.70561e6	−1.08126
$552$	0	0
$553$	−731430.	−0.101709
$554$	0	0
$555$	−5.63711e6	−0.776826
$556$	0	0
$557$	−4.24834e6	−0.580204	−0.290102	−	0.956996i	$-0.593689\pi$
$557$	−4.24834e6	−0.580204	−0.290102	+	0.956996i	$0.593689\pi$
$558$	0	0
$559$	713709.	0.0966032
$560$	0	0
$561$	539473.	0.0723706
$562$	0	0
$563$	3.09512e6	0.411534	0.205767	−	0.978601i	$-0.434031\pi$
$563$	3.09512e6	0.411534	0.205767	+	0.978601i	$0.434031\pi$
$564$	0	0
$565$	−4.18613e6	−0.551686
$566$	0	0
$567$	175125.	0.0228765
$568$	0	0
$569$	−7.60568e6	−0.984821	−0.492411	−	0.870363i	$-0.663884\pi$
$569$	−7.60568e6	−0.984821	−0.492411	+	0.870363i	$0.663884\pi$
$570$	0	0
$571$	2.28582e6	0.293395	0.146697	−	0.989181i	$-0.453136\pi$
$571$	2.28582e6	0.293395	0.146697	+	0.989181i	$0.453136\pi$
$572$	0	0
$573$	−3.03529e6	−0.386202
$574$	0	0
$575$	−2.01550e7	−2.54222
$576$	0	0
$577$	−6.27770e6	−0.784984	−0.392492	−	0.919755i	$-0.628387\pi$
$577$	−6.27770e6	−0.784984	−0.392492	+	0.919755i	$0.628387\pi$
$578$	0	0
$579$	3.84302e6	0.476405
$580$	0	0
$581$	−642107.	−0.0789164
$582$	0	0
$583$	−1.83079e6	−0.223084
$584$	0	0
$585$	−7.87437e6	−0.951320
$586$	0	0
$587$	1.48651e6	0.178063	0.0890314	−	0.996029i	$-0.471623\pi$
$587$	1.48651e6	0.178063	0.0890314	+	0.996029i	$0.471623\pi$
$588$	0	0
$589$	616031.	0.0731669
$590$	0	0
$591$	3.48797e6	0.410775
$592$	0	0
$593$	4.72874e6	0.552216	0.276108	−	0.961127i	$-0.410955\pi$
$593$	4.72874e6	0.552216	0.276108	+	0.961127i	$0.410955\pi$
$594$	0	0
$595$	1.42090e6	0.164539
$596$	0	0
$597$	−5.07968e6	−0.583312
$598$	0	0
$599$	325927.	0.0371153	0.0185577	−	0.999828i	$-0.494093\pi$
$599$	325927.	0.0371153	0.0185577	+	0.999828i	$0.494093\pi$
$600$	0	0
$601$	−1.30291e7	−1.47140	−0.735698	−	0.677310i	$-0.763146\pi$
$601$	−1.30291e7	−1.47140	−0.735698	+	0.677310i	$0.763146\pi$
$602$	0	0
$603$	−2.09712e6	−0.234871
$604$	0	0
$605$	−1.57331e6	−0.174753
$606$	0	0
$607$	1.45956e6	0.160787	0.0803933	−	0.996763i	$-0.474382\pi$
$607$	1.45956e6	0.160787	0.0803933	+	0.996763i	$0.474382\pi$
$608$	0	0
$609$	−1.23292e6	−0.134708
$610$	0	0
$611$	−3.59977e6	−0.390096
$612$	0	0
$613$	1.26574e7	1.36048	0.680241	−	0.732989i	$-0.261875\pi$
$613$	1.26574e7	1.36048	0.680241	+	0.732989i	$0.261875\pi$
$614$	0	0
$615$	−1.79517e7	−1.91390
$616$	0	0
$617$	2.57472e6	0.272281	0.136140	−	0.990690i	$-0.456530\pi$
$617$	2.57472e6	0.272281	0.136140	+	0.990690i	$0.456530\pi$
$618$	0	0
$619$	1.82556e7	1.91500	0.957500	−	0.288434i	$-0.0931345\pi$
$619$	1.82556e7	1.91500	0.957500	+	0.288434i	$0.0931345\pi$
$620$	0	0
$621$	−1.74450e6	−0.181527
$622$	0	0
$623$	−1.48994e6	−0.153797
$624$	0	0
$625$	3.48518e7	3.56882
$626$	0	0
$627$	−1.63501e6	−0.166093
$628$	0	0
$629$	−2.88744e6	−0.290995
$630$	0	0
$631$	−7.91613e6	−0.791480	−0.395740	−	0.918363i	$-0.629512\pi$
$631$	−7.91613e6	−0.791480	−0.395740	+	0.918363i	$0.629512\pi$
$632$	0	0
$633$	−4.45903e6	−0.442315
$634$	0	0
$635$	2.00687e7	1.97508
$636$	0	0
$637$	−1.45602e7	−1.42173
$638$	0	0
$639$	−4.71120e6	−0.456436
$640$	0	0
$641$	−3.40205e6	−0.327036	−0.163518	−	0.986540i	$-0.552284\pi$
$641$	−3.40205e6	−0.327036	−0.163518	+	0.986540i	$0.552284\pi$
$642$	0	0
$643$	6.92901e6	0.660912	0.330456	−	0.943821i	$-0.392797\pi$
$643$	6.92901e6	0.660912	0.330456	+	0.943821i	$0.392797\pi$
$644$	0	0
$645$	−762989.	−0.0722136
$646$	0	0
$647$	−1.11125e7	−1.04364	−0.521819	−	0.853056i	$-0.674746\pi$
$647$	−1.11125e7	−1.04364	−0.521819	+	0.853056i	$0.674746\pi$
$648$	0	0
$649$	5.03645e6	0.469367
$650$	0	0
$651$	98567.0	0.00911547
$652$	0	0
$653$	6.61748e6	0.607309	0.303654	−	0.952782i	$-0.401793\pi$
$653$	6.61748e6	0.607309	0.303654	+	0.952782i	$0.401793\pi$
$654$	0	0
$655$	−3.90725e7	−3.55850
$656$	0	0
$657$	1.93390e6	0.174792
$658$	0	0
$659$	−1.88838e7	−1.69385	−0.846926	−	0.531711i	$-0.821549\pi$
$659$	−1.88838e7	−1.69385	−0.846926	+	0.531711i	$0.821549\pi$
$660$	0	0
$661$	7.10304e6	0.632326	0.316163	−	0.948705i	$-0.397605\pi$
$661$	7.10304e6	0.632326	0.316163	+	0.948705i	$0.397605\pi$
$662$	0	0
$663$	−4.03341e6	−0.356360
$664$	0	0
$665$	−4.30638e6	−0.377623
$666$	0	0
$667$	1.22817e7	1.06892
$668$	0	0
$669$	−1.11165e7	−0.960291
$670$	0	0
$671$	3.62877e6	0.311138
$672$	0	0
$673$	−1.79739e7	−1.52970	−0.764849	−	0.644209i	$-0.777187\pi$
$673$	−1.79739e7	−1.52970	−0.764849	+	0.644209i	$0.777187\pi$
$674$	0	0
$675$	6.13996e6	0.518688
$676$	0	0
$677$	−617034.	−0.0517413	−0.0258706	−	0.999665i	$-0.508236\pi$
$677$	−617034.	−0.0517413	−0.0258706	+	0.999665i	$0.508236\pi$
$678$	0	0
$679$	−2.70088e6	−0.224818
$680$	0	0
$681$	7.21961e6	0.596549
$682$	0	0
$683$	−924287.	−0.0758150	−0.0379075	−	0.999281i	$-0.512069\pi$
$683$	−924287.	−0.0758150	−0.0379075	+	0.999281i	$0.512069\pi$
$684$	0	0
$685$	7.95225e6	0.647535
$686$	0	0
$687$	−1.46273e6	−0.118242
$688$	0	0
$689$	1.36881e7	1.09849
$690$	0	0
$691$	−2.33550e7	−1.86074	−0.930368	−	0.366627i	$-0.880513\pi$
$691$	−2.33550e7	−1.86074	−0.930368	+	0.366627i	$0.880513\pi$
$692$	0	0
$693$	−261606.	−0.0206926
$694$	0	0
$695$	7.52668e6	0.591073
$696$	0	0
$697$	−9.19522e6	−0.716936
$698$	0	0
$699$	−1.31222e7	−1.01581
$700$	0	0
$701$	1.15300e7	0.886206	0.443103	−	0.896471i	$-0.353878\pi$
$701$	1.15300e7	0.886206	0.443103	+	0.896471i	$0.353878\pi$
$702$	0	0
$703$	8.75110e6	0.667843
$704$	0	0
$705$	3.84832e6	0.291608
$706$	0	0
$707$	−2.00696e6	−0.151005
$708$	0	0
$709$	−1.13035e7	−0.844497	−0.422248	−	0.906480i	$-0.638759\pi$
$709$	−1.13035e7	−0.844497	−0.422248	+	0.906480i	$0.638759\pi$
$710$	0	0
$711$	−2.21962e6	−0.164667
$712$	0	0
$713$	−981872.	−0.0723321
$714$	0	0
$715$	1.17630e7	0.860501
$716$	0	0
$717$	5.24228e6	0.380822
$718$	0	0
$719$	−406909.	−0.0293545	−0.0146773	−	0.999892i	$-0.504672\pi$
$719$	−406909.	−0.0293545	−0.0146773	+	0.999892i	$0.504672\pi$
$720$	0	0
$721$	−4.69934e6	−0.336665
$722$	0	0
$723$	3.91108e6	0.278260
$724$	0	0
$725$	−4.32268e7	−3.05428
$726$	0	0
$727$	−1.93239e6	−0.135600	−0.0678000	−	0.997699i	$-0.521598\pi$
$727$	−1.93239e6	−0.135600	−0.0678000	+	0.997699i	$0.521598\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−390818.	−0.0270509
$732$	0	0
$733$	357394.	0.0245690	0.0122845	−	0.999925i	$-0.496090\pi$
$733$	357394.	0.0245690	0.0122845	+	0.999925i	$0.496090\pi$
$734$	0	0
$735$	1.55655e7	1.06279
$736$	0	0
$737$	3.13273e6	0.212449
$738$	0	0
$739$	−2.22117e7	−1.49613	−0.748066	−	0.663624i	$-0.769017\pi$
$739$	−2.22117e7	−1.49613	−0.748066	+	0.663624i	$0.769017\pi$
$740$	0	0
$741$	1.22243e7	0.817856
$742$	0	0
$743$	2.10894e7	1.40149	0.700747	−	0.713410i	$-0.252850\pi$
$743$	2.10894e7	1.40149	0.700747	+	0.713410i	$0.252850\pi$
$744$	0	0
$745$	−3.52215e7	−2.32497
$746$	0	0
$747$	−1.94856e6	−0.127765
$748$	0	0
$749$	3.33550e6	0.217248
$750$	0	0
$751$	3.08277e7	1.99453	0.997267	−	0.0738774i	$-0.0235374\pi$
$751$	3.08277e7	1.99453	0.997267	+	0.0738774i	$0.0235374\pi$
$752$	0	0
$753$	8.53366e6	0.548464
$754$	0	0
$755$	55860.2	0.00356644
$756$	0	0
$757$	3.26795e6	0.207270	0.103635	−	0.994615i	$-0.466953\pi$
$757$	3.26795e6	0.207270	0.103635	+	0.994615i	$0.466953\pi$
$758$	0	0
$759$	2.60598e6	0.164198
$760$	0	0
$761$	1.01120e7	0.632961	0.316480	−	0.948599i	$-0.397499\pi$
$761$	1.01120e7	0.632961	0.316480	+	0.948599i	$0.397499\pi$
$762$	0	0
$763$	1.28125e6	0.0796748
$764$	0	0
$765$	4.31191e6	0.266389
$766$	0	0
$767$	−3.76554e7	−2.31121
$768$	0	0
$769$	2.14392e7	1.30735	0.653676	−	0.756775i	$-0.273226\pi$
$769$	2.14392e7	1.30735	0.653676	+	0.756775i	$0.273226\pi$
$770$	0	0
$771$	1.02680e7	0.622088
$772$	0	0
$773$	−1.89394e7	−1.14003	−0.570017	−	0.821633i	$-0.693063\pi$
$773$	−1.89394e7	−1.14003	−0.570017	+	0.821633i	$0.693063\pi$
$774$	0	0
$775$	3.45580e6	0.206678
$776$	0	0
$777$	1.40020e6	0.0832030
$778$	0	0
$779$	2.78684e7	1.64539
$780$	0	0
$781$	7.03772e6	0.412862
$782$	0	0
$783$	−3.74148e6	−0.218091
$784$	0	0
$785$	1.28381e7	0.743580
$786$	0	0
$787$	1.10351e7	0.635096	0.317548	−	0.948242i	$-0.397141\pi$
$787$	1.10351e7	0.635096	0.317548	+	0.948242i	$0.397141\pi$
$788$	0	0
$789$	−2.52824e6	−0.144586
$790$	0	0
$791$	1.03980e6	0.0590890
$792$	0	0
$793$	−2.71308e7	−1.53207
$794$	0	0
$795$	−1.46332e7	−0.821149
$796$	0	0
$797$	−1.22483e7	−0.683015	−0.341507	−	0.939879i	$-0.610937\pi$
$797$	−1.22483e7	−0.683015	−0.341507	+	0.939879i	$0.610937\pi$
$798$	0	0
$799$	1.97119e6	0.109235
$800$	0	0
$801$	−4.52142e6	−0.248997
$802$	0	0
$803$	−2.88891e6	−0.158105
$804$	0	0
$805$	6.86380e6	0.373315
$806$	0	0
$807$	9.78829e6	0.529082
$808$	0	0
$809$	−2.42528e7	−1.30284	−0.651421	−	0.758717i	$-0.725827\pi$
$809$	−2.42528e7	−1.30284	−0.651421	+	0.758717i	$0.725827\pi$
$810$	0	0
$811$	−3.64893e7	−1.94811	−0.974055	−	0.226311i	$-0.927334\pi$
$811$	−3.64893e7	−1.94811	−0.974055	+	0.226311i	$0.927334\pi$
$812$	0	0
$813$	−1.06136e7	−0.563166
$814$	0	0
$815$	3.37435e7	1.77949
$816$	0	0
$817$	1.18447e6	0.0620825
$818$	0	0
$819$	1.95592e6	0.101892
$820$	0	0
$821$	2.40297e7	1.24420	0.622100	−	0.782938i	$-0.286280\pi$
$821$	2.40297e7	1.24420	0.622100	+	0.782938i	$0.286280\pi$
$822$	0	0
$823$	1.16283e7	0.598432	0.299216	−	0.954185i	$-0.403275\pi$
$823$	1.16283e7	0.598432	0.299216	+	0.954185i	$0.403275\pi$
$824$	0	0
$825$	−9.17204e6	−0.469171
$826$	0	0
$827$	−1.01425e7	−0.515680	−0.257840	−	0.966188i	$-0.583011\pi$
$827$	−1.01425e7	−0.515680	−0.257840	+	0.966188i	$0.583011\pi$
$828$	0	0
$829$	1.51928e7	0.767804	0.383902	−	0.923374i	$-0.374580\pi$
$829$	1.51928e7	0.767804	0.383902	+	0.923374i	$0.374580\pi$
$830$	0	0
$831$	−5.59221e6	−0.280919
$832$	0	0
$833$	7.97298e6	0.398114
$834$	0	0
$835$	3.02902e7	1.50344
$836$	0	0
$837$	299115.	0.0147579
$838$	0	0
$839$	9.62729e6	0.472171	0.236085	−	0.971732i	$-0.424136\pi$
$839$	9.62729e6	0.472171	0.236085	+	0.971732i	$0.424136\pi$
$840$	0	0
$841$	5.82977e6	0.284225
$842$	0	0
$843$	−2.19590e6	−0.106425
$844$	0	0
$845$	−4.80478e7	−2.31490
$846$	0	0
$847$	390795.	0.0187172
$848$	0	0
$849$	1.66121e7	0.790962
$850$	0	0
$851$	−1.39481e7	−0.660223
$852$	0	0
$853$	7.49543e6	0.352715	0.176358	−	0.984326i	$-0.443568\pi$
$853$	7.49543e6	0.352715	0.176358	+	0.984326i	$0.443568\pi$
$854$	0	0
$855$	−1.30683e7	−0.611370
$856$	0	0
$857$	−2.69493e6	−0.125342	−0.0626708	−	0.998034i	$-0.519962\pi$
$857$	−2.69493e6	−0.125342	−0.0626708	+	0.998034i	$0.519962\pi$
$858$	0	0
$859$	3.28791e7	1.52033	0.760164	−	0.649731i	$-0.225118\pi$
$859$	3.28791e7	1.52033	0.760164	+	0.649731i	$0.225118\pi$
$860$	0	0
$861$	4.45904e6	0.204990
$862$	0	0
$863$	−3.03331e7	−1.38640	−0.693202	−	0.720744i	$-0.743801\pi$
$863$	−3.03331e7	−1.38640	−0.693202	+	0.720744i	$0.743801\pi$
$864$	0	0
$865$	−1.65627e7	−0.752648
$866$	0	0
$867$	−1.05701e7	−0.477562
$868$	0	0
$869$	3.31574e6	0.148947
$870$	0	0
$871$	−2.34221e7	−1.04612
$872$	0	0
$873$	−8.19621e6	−0.363980
$874$	0	0
$875$	−1.51945e7	−0.670914
$876$	0	0
$877$	−3.81826e7	−1.67636	−0.838178	−	0.545396i	$-0.816379\pi$
$877$	−3.81826e7	−1.67636	−0.838178	+	0.545396i	$0.816379\pi$
$878$	0	0
$879$	−1.21409e7	−0.530005
$880$	0	0
$881$	−8.09402e6	−0.351338	−0.175669	−	0.984449i	$-0.556209\pi$
$881$	−8.09402e6	−0.351338	−0.175669	+	0.984449i	$0.556209\pi$
$882$	0	0
$883$	−1.90914e7	−0.824018	−0.412009	−	0.911180i	$-0.635173\pi$
$883$	−1.90914e7	−0.824018	−0.412009	+	0.911180i	$0.635173\pi$
$884$	0	0
$885$	4.02554e7	1.72769
$886$	0	0
$887$	−1.40640e7	−0.600205	−0.300102	−	0.953907i	$-0.597021\pi$
$887$	−1.40640e7	−0.600205	−0.300102	+	0.953907i	$0.597021\pi$
$888$	0	0
$889$	−4.98487e6	−0.211543
$890$	0	0
$891$	−793881.	−0.0335013
$892$	0	0
$893$	−5.97417e6	−0.250697
$894$	0	0
$895$	6.48292e7	2.70529
$896$	0	0
$897$	−1.94838e7	−0.808525
$898$	0	0
$899$	−2.10585e6	−0.0869015
$900$	0	0
$901$	−7.49542e6	−0.307598
$902$	0	0
$903$	189519.	0.00773453
$904$	0	0
$905$	6.02220e7	2.44418
$906$	0	0
$907$	−1.37464e7	−0.554845	−0.277423	−	0.960748i	$-0.589480\pi$
$907$	−1.37464e7	−0.554845	−0.277423	+	0.960748i	$0.589480\pi$
$908$	0	0
$909$	−6.09040e6	−0.244476
$910$	0	0
$911$	−3.83931e7	−1.53270	−0.766350	−	0.642423i	$-0.777929\pi$
$911$	−3.83931e7	−1.53270	−0.766350	+	0.642423i	$0.777929\pi$
$912$	0	0
$913$	2.91082e6	0.115568
$914$	0	0
$915$	2.90041e7	1.14527
$916$	0	0
$917$	9.70523e6	0.381138
$918$	0	0
$919$	−1.85935e7	−0.726227	−0.363114	−	0.931745i	$-0.618286\pi$
$919$	−1.85935e7	−0.726227	−0.363114	+	0.931745i	$0.618286\pi$
$920$	0	0
$921$	1.08491e7	0.421448
$922$	0	0
$923$	−5.26181e7	−2.03297
$924$	0	0
$925$	4.90918e7	1.88649
$926$	0	0
$927$	−1.42608e7	−0.545060
$928$	0	0
$929$	−2.77987e7	−1.05678	−0.528391	−	0.849001i	$-0.677204\pi$
$929$	−2.77987e7	−1.05678	−0.528391	+	0.849001i	$0.677204\pi$
$930$	0	0
$931$	−2.41641e7	−0.913684
$932$	0	0
$933$	−5.73187e6	−0.215572
$934$	0	0
$935$	−6.44125e6	−0.240958
$936$	0	0
$937$	4.55794e7	1.69598	0.847988	−	0.530015i	$-0.177814\pi$
$937$	4.55794e7	1.69598	0.847988	+	0.530015i	$0.177814\pi$
$938$	0	0
$939$	1.67873e7	0.621322
$940$	0	0
$941$	−2.38944e7	−0.879676	−0.439838	−	0.898077i	$-0.644964\pi$
$941$	−2.38944e7	−0.879676	−0.439838	+	0.898077i	$0.644964\pi$
$942$	0	0
$943$	−4.44186e7	−1.62662
$944$	0	0
$945$	−2.09097e6	−0.0761674
$946$	0	0
$947$	−3.52637e7	−1.27777	−0.638885	−	0.769302i	$-0.720604\pi$
$947$	−3.52637e7	−1.27777	−0.638885	+	0.769302i	$0.720604\pi$
$948$	0	0
$949$	2.15992e7	0.778523
$950$	0	0
$951$	2.11875e7	0.759678
$952$	0	0
$953$	5.08550e7	1.81385	0.906924	−	0.421293i	$-0.138424\pi$
$953$	5.08550e7	1.81385	0.906924	+	0.421293i	$0.138424\pi$
$954$	0	0
$955$	3.62411e7	1.28586
$956$	0	0
$957$	5.58912e6	0.197271
$958$	0	0
$959$	−1.97526e6	−0.0693551
$960$	0	0
$961$	−2.84608e7	−0.994120
$962$	0	0
$963$	1.01220e7	0.351724
$964$	0	0
$965$	−4.58853e7	−1.58619
$966$	0	0
$967$	−2.33318e7	−0.802384	−0.401192	−	0.915994i	$-0.631404\pi$
$967$	−2.33318e7	−0.802384	−0.401192	+	0.915994i	$0.631404\pi$
$968$	0	0
$969$	−6.69385e6	−0.229016
$970$	0	0
$971$	2.99725e7	1.02017	0.510087	−	0.860123i	$-0.329613\pi$
$971$	2.99725e7	1.02017	0.510087	+	0.860123i	$0.329613\pi$
$972$	0	0
$973$	−1.86956e6	−0.0633077
$974$	0	0
$975$	6.85755e7	2.31024
$976$	0	0
$977$	3.06238e7	1.02642	0.513208	−	0.858264i	$-0.328457\pi$
$977$	3.06238e7	1.02642	0.513208	+	0.858264i	$0.328457\pi$
$978$	0	0
$979$	6.75422e6	0.225226
$980$	0	0
$981$	3.88812e6	0.128993
$982$	0	0
$983$	−2.10228e7	−0.693915	−0.346958	−	0.937881i	$-0.612785\pi$
$983$	−2.10228e7	−0.693915	−0.346958	+	0.937881i	$0.612785\pi$
$984$	0	0
$985$	−4.16460e7	−1.36767
$986$	0	0
$987$	−955887.	−0.0312330
$988$	0	0
$989$	−1.88789e6	−0.0613742
$990$	0	0
$991$	2.43651e7	0.788103	0.394052	−	0.919088i	$-0.371073\pi$
$991$	2.43651e7	0.788103	0.394052	+	0.919088i	$0.371073\pi$
$992$	0	0
$993$	−1.78098e7	−0.573175
$994$	0	0
$995$	6.06508e7	1.94213
$996$	0	0
$997$	−1.04094e7	−0.331655	−0.165827	−	0.986155i	$-0.553029\pi$
$997$	−1.04094e7	−0.331655	−0.165827	+	0.986155i	$0.553029\pi$
$998$	0	0
$999$	4.24911e6	0.134705

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.6.a.q.1.1		2
4.3	odd	2		33.6.a.d.1.1	✓	2
12.11	even	2		99.6.a.e.1.2		2
20.19	odd	2		825.6.a.d.1.2		2
44.43	even	2		363.6.a.g.1.2		2
132.131	odd	2		1089.6.a.o.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.a.d.1.1	✓	2	4.3	odd	2
99.6.a.e.1.2		2	12.11	even	2
363.6.a.g.1.2		2	44.43	even	2
528.6.a.q.1.1		2	1.1	even	1	trivial
825.6.a.d.1.2		2	20.19	odd	2
1089.6.a.o.1.1		2	132.131	odd	2

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:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	528.a (trivial)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} -107.459 q^{5} +26.6918 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q+9.00000 q^{3} -107.459 q^{5} +26.6918 q^{7} +81.0000 q^{9} -121.000 q^{11} +904.666 q^{13} -967.131 q^{15} -495.384 q^{17} +1501.38 q^{19} +240.226 q^{21} -2393.01 q^{23} +8422.44 q^{25} +729.000 q^{27} -5132.34 q^{29} +410.309 q^{31} -1089.00 q^{33} -2868.28 q^{35} +5828.69 q^{37} +8141.99 q^{39} +18561.8 q^{41} +788.920 q^{43} -8704.18 q^{45} -3979.11 q^{47} -16094.5 q^{49} -4458.45 q^{51} +15130.5 q^{53} +13002.5 q^{55} +13512.5 q^{57} -41623.5 q^{59} -29989.8 q^{61} +2162.04 q^{63} -97214.5 q^{65} -25890.3 q^{67} -21537.1 q^{69} -58163.0 q^{71} +23875.3 q^{73} +75802.0 q^{75} -3229.71 q^{77} -27402.8 q^{79} +6561.00 q^{81} -24056.3 q^{83} +53233.4 q^{85} -46191.1 q^{87} -55820.0 q^{89} +24147.2 q^{91} +3692.78 q^{93} -161337. q^{95} -101188. q^{97} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 18 q^{3} - 38 q^{5} + 18 q^{7} + 162 q^{9}+O(q^{10}) \) 2 * q + 18 * q^3 - 38 * q^5 + 18 * q^7 + 162 * q^9	\( 2 q + 18 q^{3} - 38 q^{5} + 18 q^{7} + 162 q^{9} - 242 q^{11} - 66 q^{13} - 342 q^{15} - 920 q^{17} + 2932 q^{19} + 162 q^{21} - 5246 q^{23} + 10122 q^{25} + 1458 q^{27} - 12600 q^{29} - 9936 q^{31} - 2178 q^{33} - 3472 q^{35} + 5996 q^{37} - 594 q^{39} + 24244 q^{41} - 20360 q^{43} - 3078 q^{45} + 5806 q^{47} - 32826 q^{49} - 8280 q^{51} + 40770 q^{53} + 4598 q^{55} + 26388 q^{57} - 18212 q^{59} - 11398 q^{61} + 1458 q^{63} - 164636 q^{65} - 65368 q^{67} - 47214 q^{69} - 61446 q^{71} + 53412 q^{73} + 91098 q^{75} - 2178 q^{77} - 17122 q^{79} + 13122 q^{81} + 14304 q^{83} + 23740 q^{85} - 113400 q^{87} - 58140 q^{89} + 32584 q^{91} - 89424 q^{93} - 61968 q^{95} - 183056 q^{97} - 19602 q^{99}+O(q^{100}) \) 2 * q + 18 * q^3 - 38 * q^5 + 18 * q^7 + 162 * q^9 - 242 * q^11 - 66 * q^13 - 342 * q^15 - 920 * q^17 + 2932 * q^19 + 162 * q^21 - 5246 * q^23 + 10122 * q^25 + 1458 * q^27 - 12600 * q^29 - 9936 * q^31 - 2178 * q^33 - 3472 * q^35 + 5996 * q^37 - 594 * q^39 + 24244 * q^41 - 20360 * q^43 - 3078 * q^45 + 5806 * q^47 - 32826 * q^49 - 8280 * q^51 + 40770 * q^53 + 4598 * q^55 + 26388 * q^57 - 18212 * q^59 - 11398 * q^61 + 1458 * q^63 - 164636 * q^65 - 65368 * q^67 - 47214 * q^69 - 61446 * q^71 + 53412 * q^73 + 91098 * q^75 - 2178 * q^77 - 17122 * q^79 + 13122 * q^81 + 14304 * q^83 + 23740 * q^85 - 113400 * q^87 - 58140 * q^89 + 32584 * q^91 - 89424 * q^93 - 61968 * q^95 - 183056 * q^97 - 19602 * q^99

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