LMFDB - Embedded newform 192.6.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,6,Mod(1,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("192.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	192.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:	$30.7936934041$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{31}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 31 $ x^2 - 31
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 96)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.56776$ of defining polynomial
Character	$\chi$	$=$	192.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} +71.0842 q^{5} +29.0842 q^{7} +81.0000 q^{9} +O(q^{10})$	$q+9.00000 q^{3} +71.0842 q^{5} +29.0842 q^{7} +81.0000 q^{9} +634.505 q^{11} -676.505 q^{13} +639.758 q^{15} +1872.51 q^{17} -926.842 q^{19} +261.758 q^{21} -752.527 q^{23} +1927.97 q^{25} +729.000 q^{27} -3413.40 q^{29} +5689.42 q^{31} +5710.55 q^{33} +2067.43 q^{35} -14057.1 q^{37} -6088.55 q^{39} +6536.21 q^{41} +14496.9 q^{43} +5757.82 q^{45} +18398.7 q^{47} -15961.1 q^{49} +16852.5 q^{51} +1818.98 q^{53} +45103.3 q^{55} -8341.58 q^{57} +46790.2 q^{59} +27446.0 q^{61} +2355.82 q^{63} -48088.9 q^{65} -61869.1 q^{67} -6772.74 q^{69} -20186.9 q^{71} +30256.5 q^{73} +17351.7 q^{75} +18454.1 q^{77} +46164.2 q^{79} +6561.00 q^{81} +74412.7 q^{83} +133106. q^{85} -30720.6 q^{87} -67234.2 q^{89} -19675.6 q^{91} +51204.8 q^{93} -65883.9 q^{95} +29200.9 q^{97} +51394.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 18 q^{3} - 36 q^{5} - 120 q^{7} + 162 q^{9}+O(q^{10}) $ 2 * q + 18 * q^3 - 36 * q^5 - 120 * q^7 + 162 * q^9	$ 2 q + 18 q^{3} - 36 q^{5} - 120 q^{7} + 162 q^{9} + 200 q^{11} - 284 q^{13} - 324 q^{15} + 2676 q^{17} - 72 q^{19} - 1080 q^{21} + 3840 q^{23} + 10270 q^{25} + 1458 q^{27} - 10212 q^{29} + 10488 q^{31} + 1800 q^{33} + 18032 q^{35} - 13148 q^{37} - 2556 q^{39} + 4164 q^{41} + 5832 q^{43} - 2916 q^{45} + 1520 q^{47} - 10542 q^{49} + 24084 q^{51} - 9012 q^{53} + 91632 q^{55} - 648 q^{57} + 55096 q^{59} + 63444 q^{61} - 9720 q^{63} - 90120 q^{65} - 36792 q^{67} + 34560 q^{69} + 37664 q^{71} - 37836 q^{73} + 92430 q^{75} + 83232 q^{77} + 144888 q^{79} + 13122 q^{81} + 109272 q^{83} + 47064 q^{85} - 91908 q^{87} - 32556 q^{89} - 78192 q^{91} + 94392 q^{93} - 157424 q^{95} + 69092 q^{97} + 16200 q^{99}+O(q^{100}) $ 2 * q + 18 * q^3 - 36 * q^5 - 120 * q^7 + 162 * q^9 + 200 * q^11 - 284 * q^13 - 324 * q^15 + 2676 * q^17 - 72 * q^19 - 1080 * q^21 + 3840 * q^23 + 10270 * q^25 + 1458 * q^27 - 10212 * q^29 + 10488 * q^31 + 1800 * q^33 + 18032 * q^35 - 13148 * q^37 - 2556 * q^39 + 4164 * q^41 + 5832 * q^43 - 2916 * q^45 + 1520 * q^47 - 10542 * q^49 + 24084 * q^51 - 9012 * q^53 + 91632 * q^55 - 648 * q^57 + 55096 * q^59 + 63444 * q^61 - 9720 * q^63 - 90120 * q^65 - 36792 * q^67 + 34560 * q^69 + 37664 * q^71 - 37836 * q^73 + 92430 * q^75 + 83232 * q^77 + 144888 * q^79 + 13122 * q^81 + 109272 * q^83 + 47064 * q^85 - 91908 * q^87 - 32556 * q^89 - 78192 * q^91 + 94392 * q^93 - 157424 * q^95 + 69092 * q^97 + 16200 * 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
$5$	71.0842	1.27159	0.635797	−	0.771857i	$-0.280672\pi$
$5$	71.0842	1.27159	0.635797	+	0.771857i	$0.280672\pi$
$6$	0	0
$7$	29.0842	0.224343	0.112171	−	0.993689i	$-0.464219\pi$
$7$	29.0842	0.224343	0.112171	+	0.993689i	$0.464219\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	634.505	1.58108	0.790540	−	0.612411i	$-0.209800\pi$
$11$	634.505	1.58108	0.790540	+	0.612411i	$0.209800\pi$
$12$	0	0
$13$	−676.505	−1.11023	−0.555115	−	0.831774i	$-0.687326\pi$
$13$	−676.505	−1.11023	−0.555115	+	0.831774i	$0.687326\pi$
$14$	0	0
$15$	639.758	0.734155
$16$	0	0
$17$	1872.51	1.57145	0.785725	−	0.618575i	$-0.212290\pi$
$17$	1872.51	1.57145	0.785725	+	0.618575i	$0.212290\pi$
$18$	0	0
$19$	−926.842	−0.589009	−0.294504	−	0.955650i	$-0.595155\pi$
$19$	−926.842	−0.589009	−0.294504	+	0.955650i	$0.595155\pi$
$20$	0	0
$21$	261.758	0.129524
$22$	0	0
$23$	−752.527	−0.296621	−0.148311	−	0.988941i	$-0.547384\pi$
$23$	−752.527	−0.296621	−0.148311	+	0.988941i	$0.547384\pi$
$24$	0	0
$25$	1927.97	0.616950
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	−3413.40	−0.753689	−0.376844	−	0.926277i	$-0.622991\pi$
$29$	−3413.40	−0.753689	−0.376844	+	0.926277i	$0.622991\pi$
$30$	0	0
$31$	5689.42	1.06332	0.531660	−	0.846958i	$-0.321568\pi$
$31$	5689.42	1.06332	0.531660	+	0.846958i	$0.321568\pi$
$32$	0	0
$33$	5710.55	0.912836
$34$	0	0
$35$	2067.43	0.285273
$36$	0	0
$37$	−14057.1	−1.68807	−0.844035	−	0.536287i	$-0.819826\pi$
$37$	−14057.1	−1.68807	−0.844035	+	0.536287i	$0.819826\pi$
$38$	0	0
$39$	−6088.55	−0.640991
$40$	0	0
$41$	6536.21	0.607249	0.303624	−	0.952792i	$-0.401803\pi$
$41$	6536.21	0.607249	0.303624	+	0.952792i	$0.401803\pi$
$42$	0	0
$43$	14496.9	1.19565	0.597827	−	0.801625i	$-0.296031\pi$
$43$	14496.9	1.19565	0.597827	+	0.801625i	$0.296031\pi$
$44$	0	0
$45$	5757.82	0.423864
$46$	0	0
$47$	18398.7	1.21490	0.607452	−	0.794356i	$-0.292192\pi$
$47$	18398.7	1.21490	0.607452	+	0.794356i	$0.292192\pi$
$48$	0	0
$49$	−15961.1	−0.949670
$50$	0	0
$51$	16852.5	0.907278
$52$	0	0
$53$	1818.98	0.0889484	0.0444742	−	0.999011i	$-0.485839\pi$
$53$	1818.98	0.0889484	0.0444742	+	0.999011i	$0.485839\pi$
$54$	0	0
$55$	45103.3	2.01049
$56$	0	0
$57$	−8341.58	−0.340064
$58$	0	0
$59$	46790.2	1.74995	0.874973	−	0.484171i	$-0.160879\pi$
$59$	46790.2	1.74995	0.874973	+	0.484171i	$0.160879\pi$
$60$	0	0
$61$	27446.0	0.944395	0.472198	−	0.881493i	$-0.343461\pi$
$61$	27446.0	0.944395	0.472198	+	0.881493i	$0.343461\pi$
$62$	0	0
$63$	2355.82	0.0747810
$64$	0	0
$65$	−48088.9	−1.41176
$66$	0	0
$67$	−61869.1	−1.68379	−0.841893	−	0.539645i	$-0.818559\pi$
$67$	−61869.1	−1.68379	−0.841893	+	0.539645i	$0.818559\pi$
$68$	0	0
$69$	−6772.74	−0.171254
$70$	0	0
$71$	−20186.9	−0.475252	−0.237626	−	0.971357i	$-0.576369\pi$
$71$	−20186.9	−0.475252	−0.237626	+	0.971357i	$0.576369\pi$
$72$	0	0
$73$	30256.5	0.664525	0.332263	−	0.943187i	$-0.392188\pi$
$73$	30256.5	0.664525	0.332263	+	0.943187i	$0.392188\pi$
$74$	0	0
$75$	17351.7	0.356196
$76$	0	0
$77$	18454.1	0.354704
$78$	0	0
$79$	46164.2	0.832218	0.416109	−	0.909315i	$-0.363393\pi$
$79$	46164.2	0.832218	0.416109	+	0.909315i	$0.363393\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	74412.7	1.18564	0.592819	−	0.805336i	$-0.298015\pi$
$83$	74412.7	1.18564	0.592819	+	0.805336i	$0.298015\pi$
$84$	0	0
$85$	133106.	1.99825
$86$	0	0
$87$	−30720.6	−0.435143
$88$	0	0
$89$	−67234.2	−0.899736	−0.449868	−	0.893095i	$-0.648529\pi$
$89$	−67234.2	−0.899736	−0.449868	+	0.893095i	$0.648529\pi$
$90$	0	0
$91$	−19675.6	−0.249072
$92$	0	0
$93$	51204.8	0.613908
$94$	0	0
$95$	−65883.9	−0.748980
$96$	0	0
$97$	29200.9	0.315114	0.157557	−	0.987510i	$-0.449638\pi$
$97$	29200.9	0.315114	0.157557	+	0.987510i	$0.449638\pi$
$98$	0	0
$99$	51394.9	0.527026
$100$	0	0
$101$	−178054.	−1.73680	−0.868399	−	0.495866i	$-0.834851\pi$
$101$	−178054.	−1.73680	−0.868399	+	0.495866i	$0.834851\pi$
$102$	0	0
$103$	196481.	1.82486	0.912428	−	0.409238i	$-0.134205\pi$
$103$	196481.	1.82486	0.912428	+	0.409238i	$0.134205\pi$
$104$	0	0
$105$	18606.9	0.164702
$106$	0	0
$107$	−62374.1	−0.526677	−0.263339	−	0.964703i	$-0.584824\pi$
$107$	−62374.1	−0.526677	−0.263339	+	0.964703i	$0.584824\pi$
$108$	0	0
$109$	−131641.	−1.06127	−0.530633	−	0.847602i	$-0.678046\pi$
$109$	−131641.	−1.06127	−0.530633	+	0.847602i	$0.678046\pi$
$110$	0	0
$111$	−126514.	−0.974608
$112$	0	0
$113$	12508.2	0.0921510	0.0460755	−	0.998938i	$-0.485329\pi$
$113$	12508.2	0.0921510	0.0460755	+	0.998938i	$0.485329\pi$
$114$	0	0
$115$	−53492.8	−0.377182
$116$	0	0
$117$	−54796.9	−0.370077
$118$	0	0
$119$	54460.4	0.352544
$120$	0	0
$121$	241546.	1.49981
$122$	0	0
$123$	58825.9	0.350595
$124$	0	0
$125$	−85090.1	−0.487084
$126$	0	0
$127$	−218002.	−1.19937	−0.599683	−	0.800237i	$-0.704707\pi$
$127$	−218002.	−1.19937	−0.599683	+	0.800237i	$0.704707\pi$
$128$	0	0
$129$	130473.	0.690311
$130$	0	0
$131$	−244537.	−1.24499	−0.622495	−	0.782624i	$-0.713881\pi$
$131$	−244537.	−1.24499	−0.622495	+	0.782624i	$0.713881\pi$
$132$	0	0
$133$	−26956.5	−0.132140
$134$	0	0
$135$	51820.4	0.244718
$136$	0	0
$137$	−127780.	−0.581649	−0.290824	−	0.956776i	$-0.593930\pi$
$137$	−127780.	−0.581649	−0.290824	+	0.956776i	$0.593930\pi$
$138$	0	0
$139$	−380263.	−1.66935	−0.834674	−	0.550744i	$-0.814344\pi$
$139$	−380263.	−1.66935	−0.834674	+	0.550744i	$0.814344\pi$
$140$	0	0
$141$	165588.	0.701425
$142$	0	0
$143$	−429246.	−1.75536
$144$	0	0
$145$	−242639.	−0.958386
$146$	0	0
$147$	−143650.	−0.548292
$148$	0	0
$149$	−385500.	−1.42252	−0.711261	−	0.702928i	$-0.751876\pi$
$149$	−385500.	−1.42252	−0.711261	+	0.702928i	$0.751876\pi$
$150$	0	0
$151$	−337018.	−1.20285	−0.601425	−	0.798930i	$-0.705400\pi$
$151$	−337018.	−1.20285	−0.601425	+	0.798930i	$0.705400\pi$
$152$	0	0
$153$	151673.	0.523817
$154$	0	0
$155$	404428.	1.35211
$156$	0	0
$157$	−124624.	−0.403510	−0.201755	−	0.979436i	$-0.564664\pi$
$157$	−124624.	−0.403510	−0.201755	+	0.979436i	$0.564664\pi$
$158$	0	0
$159$	16370.8	0.0513544
$160$	0	0
$161$	−21886.7	−0.0665449
$162$	0	0
$163$	−170015.	−0.501208	−0.250604	−	0.968090i	$-0.580629\pi$
$163$	−170015.	−0.501208	−0.250604	+	0.968090i	$0.580629\pi$
$164$	0	0
$165$	405930.	1.16076
$166$	0	0
$167$	118697.	0.329344	0.164672	−	0.986348i	$-0.447343\pi$
$167$	118697.	0.329344	0.164672	+	0.986348i	$0.447343\pi$
$168$	0	0
$169$	86366.5	0.232610
$170$	0	0
$171$	−75074.2	−0.196336
$172$	0	0
$173$	88797.7	0.225573	0.112786	−	0.993619i	$-0.464022\pi$
$173$	88797.7	0.225573	0.112786	+	0.993619i	$0.464022\pi$
$174$	0	0
$175$	56073.5	0.138408
$176$	0	0
$177$	421112.	1.01033
$178$	0	0
$179$	−659166.	−1.53767	−0.768833	−	0.639449i	$-0.779162\pi$
$179$	−659166.	−1.53767	−0.768833	+	0.639449i	$0.779162\pi$
$180$	0	0
$181$	801952.	1.81950	0.909750	−	0.415156i	$-0.136273\pi$
$181$	801952.	1.81950	0.909750	+	0.415156i	$0.136273\pi$
$182$	0	0
$183$	247014.	0.545247
$184$	0	0
$185$	−999236.	−2.14654
$186$	0	0
$187$	1.18811e6	2.48459
$188$	0	0
$189$	21202.4	0.0431748
$190$	0	0
$191$	165421.	0.328101	0.164050	−	0.986452i	$-0.447544\pi$
$191$	165421.	0.328101	0.164050	+	0.986452i	$0.447544\pi$
$192$	0	0
$193$	181158.	0.350077	0.175038	−	0.984562i	$-0.443995\pi$
$193$	181158.	0.350077	0.175038	+	0.984562i	$0.443995\pi$
$194$	0	0
$195$	−432800.	−0.815080
$196$	0	0
$197$	317852.	0.583525	0.291763	−	0.956491i	$-0.405758\pi$
$197$	317852.	0.583525	0.291763	+	0.956491i	$0.405758\pi$
$198$	0	0
$199$	−26059.4	−0.0466479	−0.0233240	−	0.999728i	$-0.507425\pi$
$199$	−26059.4	−0.0466479	−0.0233240	+	0.999728i	$0.507425\pi$
$200$	0	0
$201$	−556822.	−0.972134
$202$	0	0
$203$	−99276.1	−0.169085
$204$	0	0
$205$	464622.	0.772173
$206$	0	0
$207$	−60954.7	−0.0988738
$208$	0	0
$209$	−588086.	−0.931270
$210$	0	0
$211$	140625.	0.217449	0.108724	−	0.994072i	$-0.465323\pi$
$211$	140625.	0.217449	0.108724	+	0.994072i	$0.465323\pi$
$212$	0	0
$213$	−181682.	−0.274387
$214$	0	0
$215$	1.03050e6	1.52039
$216$	0	0
$217$	165472.	0.238548
$218$	0	0
$219$	272308.	0.383664
$220$	0	0
$221$	−1.26676e6	−1.74467
$222$	0	0
$223$	−1.30080e6	−1.75166	−0.875828	−	0.482623i	$-0.839684\pi$
$223$	−1.30080e6	−1.75166	−0.875828	+	0.482623i	$0.839684\pi$
$224$	0	0
$225$	156165.	0.205650
$226$	0	0
$227$	−353320.	−0.455096	−0.227548	−	0.973767i	$-0.573071\pi$
$227$	−353320.	−0.455096	−0.227548	+	0.973767i	$0.573071\pi$
$228$	0	0
$229$	208522.	0.262763	0.131381	−	0.991332i	$-0.458059\pi$
$229$	208522.	0.262763	0.131381	+	0.991332i	$0.458059\pi$
$230$	0	0
$231$	166087.	0.204788
$232$	0	0
$233$	645454.	0.778889	0.389444	−	0.921050i	$-0.372667\pi$
$233$	645454.	0.778889	0.389444	+	0.921050i	$0.372667\pi$
$234$	0	0
$235$	1.30786e6	1.54486
$236$	0	0
$237$	415477.	0.480481
$238$	0	0
$239$	103534.	0.117243	0.0586216	−	0.998280i	$-0.481329\pi$
$239$	103534.	0.117243	0.0586216	+	0.998280i	$0.481329\pi$
$240$	0	0
$241$	741387.	0.822247	0.411123	−	0.911580i	$-0.365137\pi$
$241$	741387.	0.822247	0.411123	+	0.911580i	$0.365137\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	−1.13458e6	−1.20759
$246$	0	0
$247$	627014.	0.653935
$248$	0	0
$249$	669714.	0.684528
$250$	0	0
$251$	869395.	0.871030	0.435515	−	0.900182i	$-0.356566\pi$
$251$	869395.	0.871030	0.435515	+	0.900182i	$0.356566\pi$
$252$	0	0
$253$	−477482.	−0.468982
$254$	0	0
$255$	1.19795e6	1.15369
$256$	0	0
$257$	−1.55618e6	−1.46970	−0.734848	−	0.678232i	$-0.762746\pi$
$257$	−1.55618e6	−1.46970	−0.734848	+	0.678232i	$0.762746\pi$
$258$	0	0
$259$	−408839.	−0.378707
$260$	0	0
$261$	−276485.	−0.251230
$262$	0	0
$263$	607357.	0.541446	0.270723	−	0.962657i	$-0.412737\pi$
$263$	607357.	0.541446	0.270723	+	0.962657i	$0.412737\pi$
$264$	0	0
$265$	129301.	0.113106
$266$	0	0
$267$	−605108.	−0.519463
$268$	0	0
$269$	389729.	0.328384	0.164192	−	0.986428i	$-0.447498\pi$
$269$	389729.	0.328384	0.164192	+	0.986428i	$0.447498\pi$
$270$	0	0
$271$	−159964.	−0.132312	−0.0661560	−	0.997809i	$-0.521074\pi$
$271$	−159964.	−0.132312	−0.0661560	+	0.997809i	$0.521074\pi$
$272$	0	0
$273$	−177081.	−0.143802
$274$	0	0
$275$	1.22331e6	0.975446
$276$	0	0
$277$	−742703.	−0.581589	−0.290794	−	0.956786i	$-0.593920\pi$
$277$	−742703.	−0.581589	−0.290794	+	0.956786i	$0.593920\pi$
$278$	0	0
$279$	460843.	0.354440
$280$	0	0
$281$	1.36848e6	1.03389	0.516944	−	0.856019i	$-0.327069\pi$
$281$	1.36848e6	1.03389	0.516944	+	0.856019i	$0.327069\pi$
$282$	0	0
$283$	1.28233e6	0.951770	0.475885	−	0.879508i	$-0.342128\pi$
$283$	1.28233e6	0.951770	0.475885	+	0.879508i	$0.342128\pi$
$284$	0	0
$285$	−592955.	−0.432424
$286$	0	0
$287$	190101.	0.136232
$288$	0	0
$289$	2.08642e6	1.46946
$290$	0	0
$291$	262809.	0.181931
$292$	0	0
$293$	−1.14182e6	−0.777013	−0.388506	−	0.921446i	$-0.627009\pi$
$293$	−1.14182e6	−0.777013	−0.388506	+	0.921446i	$0.627009\pi$
$294$	0	0
$295$	3.32604e6	2.22522
$296$	0	0
$297$	462554.	0.304279
$298$	0	0
$299$	509088.	0.329318
$300$	0	0
$301$	421633.	0.268237
$302$	0	0
$303$	−1.60249e6	−1.00274
$304$	0	0
$305$	1.95097e6	1.20089
$306$	0	0
$307$	−950764.	−0.575740	−0.287870	−	0.957669i	$-0.592947\pi$
$307$	−950764.	−0.575740	−0.287870	+	0.957669i	$0.592947\pi$
$308$	0	0
$309$	1.76833e6	1.05358
$310$	0	0
$311$	158153.	0.0927207	0.0463603	−	0.998925i	$-0.485238\pi$
$311$	158153.	0.0927207	0.0463603	+	0.998925i	$0.485238\pi$
$312$	0	0
$313$	−2.63454e6	−1.52000	−0.760000	−	0.649923i	$-0.774801\pi$
$313$	−2.63454e6	−1.52000	−0.760000	+	0.649923i	$0.774801\pi$
$314$	0	0
$315$	167462.	0.0950910
$316$	0	0
$317$	2.30913e6	1.29063	0.645313	−	0.763919i	$-0.276727\pi$
$317$	2.30913e6	1.29063	0.645313	+	0.763919i	$0.276727\pi$
$318$	0	0
$319$	−2.16582e6	−1.19164
$320$	0	0
$321$	−561367.	−0.304077
$322$	0	0
$323$	−1.73552e6	−0.925598
$324$	0	0
$325$	−1.30428e6	−0.684956
$326$	0	0
$327$	−1.18477e6	−0.612722
$328$	0	0
$329$	535111.	0.272555
$330$	0	0
$331$	−1.52689e6	−0.766013	−0.383007	−	0.923746i	$-0.625111\pi$
$331$	−1.52689e6	−0.766013	−0.383007	+	0.923746i	$0.625111\pi$
$332$	0	0
$333$	−1.13862e6	−0.562690
$334$	0	0
$335$	−4.39792e6	−2.14109
$336$	0	0
$337$	−3.49378e6	−1.67579	−0.837897	−	0.545829i	$-0.816215\pi$
$337$	−3.49378e6	−1.67579	−0.837897	+	0.545829i	$0.816215\pi$
$338$	0	0
$339$	112574.	0.0532034
$340$	0	0
$341$	3.60997e6	1.68119
$342$	0	0
$343$	−953035.	−0.437395
$344$	0	0
$345$	−481435.	−0.217766
$346$	0	0
$347$	−1.62207e6	−0.723177	−0.361589	−	0.932338i	$-0.617765\pi$
$347$	−1.62207e6	−0.723177	−0.361589	+	0.932338i	$0.617765\pi$
$348$	0	0
$349$	54162.2	0.0238030	0.0119015	−	0.999929i	$-0.496212\pi$
$349$	54162.2	0.0238030	0.0119015	+	0.999929i	$0.496212\pi$
$350$	0	0
$351$	−493172.	−0.213664
$352$	0	0
$353$	−2.94002e6	−1.25578	−0.627891	−	0.778301i	$-0.716082\pi$
$353$	−2.94002e6	−1.25578	−0.627891	+	0.778301i	$0.716082\pi$
$354$	0	0
$355$	−1.43497e6	−0.604327
$356$	0	0
$357$	490143.	0.203541
$358$	0	0
$359$	2.25727e6	0.924373	0.462186	−	0.886783i	$-0.347065\pi$
$359$	2.25727e6	0.924373	0.462186	+	0.886783i	$0.347065\pi$
$360$	0	0
$361$	−1.61706e6	−0.653069
$362$	0	0
$363$	2.17391e6	0.865916
$364$	0	0
$365$	2.15076e6	0.845006
$366$	0	0
$367$	2.45465e6	0.951317	0.475659	−	0.879630i	$-0.342210\pi$
$367$	2.45465e6	0.951317	0.475659	+	0.879630i	$0.342210\pi$
$368$	0	0
$369$	529433.	0.202416
$370$	0	0
$371$	52903.6	0.0199550
$372$	0	0
$373$	−534814.	−0.199036	−0.0995178	−	0.995036i	$-0.531730\pi$
$373$	−534814.	−0.199036	−0.0995178	+	0.995036i	$0.531730\pi$
$374$	0	0
$375$	−765811.	−0.281218
$376$	0	0
$377$	2.30918e6	0.836768
$378$	0	0
$379$	−5.43593e6	−1.94391	−0.971954	−	0.235171i	$-0.924435\pi$
$379$	−5.43593e6	−1.94391	−0.971954	+	0.235171i	$0.924435\pi$
$380$	0	0
$381$	−1.96202e6	−0.692455
$382$	0	0
$383$	322151.	0.112218	0.0561090	−	0.998425i	$-0.482131\pi$
$383$	322151.	0.112218	0.0561090	+	0.998425i	$0.482131\pi$
$384$	0	0
$385$	1.31180e6	0.451039
$386$	0	0
$387$	1.17425e6	0.398551
$388$	0	0
$389$	−3.08165e6	−1.03255	−0.516274	−	0.856424i	$-0.672681\pi$
$389$	−3.08165e6	−1.03255	−0.516274	+	0.856424i	$0.672681\pi$
$390$	0	0
$391$	−1.40911e6	−0.466126
$392$	0	0
$393$	−2.20083e6	−0.718796
$394$	0	0
$395$	3.28154e6	1.05824
$396$	0	0
$397$	3.01159e6	0.959003	0.479502	−	0.877541i	$-0.340817\pi$
$397$	3.01159e6	0.959003	0.479502	+	0.877541i	$0.340817\pi$
$398$	0	0
$399$	−242608.	−0.0762911
$400$	0	0
$401$	−621064.	−0.192875	−0.0964374	−	0.995339i	$-0.530745\pi$
$401$	−621064.	−0.192875	−0.0964374	+	0.995339i	$0.530745\pi$
$402$	0	0
$403$	−3.84892e6	−1.18053
$404$	0	0
$405$	466384.	0.141288
$406$	0	0
$407$	−8.91929e6	−2.66897
$408$	0	0
$409$	1.50114e6	0.443724	0.221862	−	0.975078i	$-0.428787\pi$
$409$	1.50114e6	0.443724	0.221862	+	0.975078i	$0.428787\pi$
$410$	0	0
$411$	−1.15002e6	−0.335815
$412$	0	0
$413$	1.36086e6	0.392588
$414$	0	0
$415$	5.28957e6	1.50765
$416$	0	0
$417$	−3.42237e6	−0.963798
$418$	0	0
$419$	−47684.5	−0.0132691	−0.00663457	−	0.999978i	$-0.502112\pi$
$419$	−47684.5	−0.0132691	−0.00663457	+	0.999978i	$0.502112\pi$
$420$	0	0
$421$	−770010.	−0.211734	−0.105867	−	0.994380i	$-0.533762\pi$
$421$	−770010.	−0.211734	−0.105867	+	0.994380i	$0.533762\pi$
$422$	0	0
$423$	1.49029e6	0.404968
$424$	0	0
$425$	3.61013e6	0.969506
$426$	0	0
$427$	798245.	0.211868
$428$	0	0
$429$	−3.86322e6	−1.01346
$430$	0	0
$431$	1.28838e6	0.334080	0.167040	−	0.985950i	$-0.446579\pi$
$431$	1.28838e6	0.334080	0.167040	+	0.985950i	$0.446579\pi$
$432$	0	0
$433$	776431.	0.199014	0.0995069	−	0.995037i	$-0.468273\pi$
$433$	776431.	0.199014	0.0995069	+	0.995037i	$0.468273\pi$
$434$	0	0
$435$	−2.18375e6	−0.553324
$436$	0	0
$437$	697474.	0.174713
$438$	0	0
$439$	−3.06381e6	−0.758753	−0.379377	−	0.925242i	$-0.623862\pi$
$439$	−3.06381e6	−0.758753	−0.379377	+	0.925242i	$0.623862\pi$
$440$	0	0
$441$	−1.29285e6	−0.316557
$442$	0	0
$443$	1.25137e6	0.302954	0.151477	−	0.988461i	$-0.451597\pi$
$443$	1.25137e6	0.302954	0.151477	+	0.988461i	$0.451597\pi$
$444$	0	0
$445$	−4.77929e6	−1.14410
$446$	0	0
$447$	−3.46950e6	−0.821294
$448$	0	0
$449$	4.03712e6	0.945053	0.472527	−	0.881316i	$-0.343342\pi$
$449$	4.03712e6	0.945053	0.472527	+	0.881316i	$0.343342\pi$
$450$	0	0
$451$	4.14726e6	0.960108
$452$	0	0
$453$	−3.03317e6	−0.694465
$454$	0	0
$455$	−1.39863e6	−0.316719
$456$	0	0
$457$	−421134.	−0.0943257	−0.0471629	−	0.998887i	$-0.515018\pi$
$457$	−421134.	−0.0943257	−0.0471629	+	0.998887i	$0.515018\pi$
$458$	0	0
$459$	1.36506e6	0.302426
$460$	0	0
$461$	3.91515e6	0.858017	0.429008	−	0.903301i	$-0.358863\pi$
$461$	3.91515e6	0.858017	0.429008	+	0.903301i	$0.358863\pi$
$462$	0	0
$463$	148684.	0.0322339	0.0161170	−	0.999870i	$-0.494870\pi$
$463$	148684.	0.0322339	0.0161170	+	0.999870i	$0.494870\pi$
$464$	0	0
$465$	3.63985e6	0.780641
$466$	0	0
$467$	−5.31611e6	−1.12798	−0.563991	−	0.825781i	$-0.690735\pi$
$467$	−5.31611e6	−1.12798	−0.563991	+	0.825781i	$0.690735\pi$
$468$	0	0
$469$	−1.79942e6	−0.377745
$470$	0	0
$471$	−1.12162e6	−0.232967
$472$	0	0
$473$	9.19839e6	1.89042
$474$	0	0
$475$	−1.78692e6	−0.363389
$476$	0	0
$477$	147337.	0.0296495
$478$	0	0
$479$	3.31269e6	0.659694	0.329847	−	0.944034i	$-0.393003\pi$
$479$	3.31269e6	0.659694	0.329847	+	0.944034i	$0.393003\pi$
$480$	0	0
$481$	9.50969e6	1.87415
$482$	0	0
$483$	−196980.	−0.0384197
$484$	0	0
$485$	2.07573e6	0.400697
$486$	0	0
$487$	1.51955e6	0.290331	0.145165	−	0.989407i	$-0.453629\pi$
$487$	1.51955e6	0.290331	0.145165	+	0.989407i	$0.453629\pi$
$488$	0	0
$489$	−1.53013e6	−0.289372
$490$	0	0
$491$	−3.26842e6	−0.611834	−0.305917	−	0.952058i	$-0.598963\pi$
$491$	−3.26842e6	−0.611834	−0.305917	+	0.952058i	$0.598963\pi$
$492$	0	0
$493$	−6.39161e6	−1.18439
$494$	0	0
$495$	3.65337e6	0.670163
$496$	0	0
$497$	−587120.	−0.106619
$498$	0	0
$499$	6.50109e6	1.16878	0.584392	−	0.811471i	$-0.301333\pi$
$499$	6.50109e6	1.16878	0.584392	+	0.811471i	$0.301333\pi$
$500$	0	0
$501$	1.06828e6	0.190147
$502$	0	0
$503$	8.31454e6	1.46527	0.732636	−	0.680620i	$-0.238290\pi$
$503$	8.31454e6	1.46527	0.732636	+	0.680620i	$0.238290\pi$
$504$	0	0
$505$	−1.26569e7	−2.20850
$506$	0	0
$507$	777299.	0.134298
$508$	0	0
$509$	−1.21402e6	−0.207697	−0.103848	−	0.994593i	$-0.533116\pi$
$509$	−1.21402e6	−0.207697	−0.103848	+	0.994593i	$0.533116\pi$
$510$	0	0
$511$	879987.	0.149082
$512$	0	0
$513$	−675668.	−0.113355
$514$	0	0
$515$	1.39667e7	2.32047
$516$	0	0
$517$	1.16741e7	1.92086
$518$	0	0
$519$	799179.	0.130234
$520$	0	0
$521$	−1.69552e6	−0.273658	−0.136829	−	0.990595i	$-0.543691\pi$
$521$	−1.69552e6	−0.273658	−0.136829	+	0.990595i	$0.543691\pi$
$522$	0	0
$523$	5.24134e6	0.837892	0.418946	−	0.908011i	$-0.362400\pi$
$523$	5.24134e6	0.837892	0.418946	+	0.908011i	$0.362400\pi$
$524$	0	0
$525$	504661.	0.0799101
$526$	0	0
$527$	1.06535e7	1.67095
$528$	0	0
$529$	−5.87005e6	−0.912016
$530$	0	0
$531$	3.79001e6	0.583316
$532$	0	0
$533$	−4.42178e6	−0.674185
$534$	0	0
$535$	−4.43381e6	−0.669720
$536$	0	0
$537$	−5.93249e6	−0.887772
$538$	0	0
$539$	−1.01274e7	−1.50150
$540$	0	0
$541$	−7.35480e6	−1.08038	−0.540191	−	0.841542i	$-0.681648\pi$
$541$	−7.35480e6	−1.08038	−0.540191	+	0.841542i	$0.681648\pi$
$542$	0	0
$543$	7.21757e6	1.05049
$544$	0	0
$545$	−9.35758e6	−1.34950
$546$	0	0
$547$	9.28167e6	1.32635	0.663174	−	0.748465i	$-0.269209\pi$
$547$	9.28167e6	1.32635	0.663174	+	0.748465i	$0.269209\pi$
$548$	0	0
$549$	2.22312e6	0.314798
$550$	0	0
$551$	3.16368e6	0.443929
$552$	0	0
$553$	1.34265e6	0.186702
$554$	0	0
$555$	−8.99313e6	−1.23931
$556$	0	0
$557$	1.14537e7	1.56426	0.782128	−	0.623117i	$-0.214134\pi$
$557$	1.14537e7	1.56426	0.782128	+	0.623117i	$0.214134\pi$
$558$	0	0
$559$	−9.80726e6	−1.32745
$560$	0	0
$561$	1.06930e7	1.43448
$562$	0	0
$563$	9.26801e6	1.23230	0.616149	−	0.787630i	$-0.288692\pi$
$563$	9.26801e6	1.23230	0.616149	+	0.787630i	$0.288692\pi$
$564$	0	0
$565$	889139.	0.117179
$566$	0	0
$567$	190822.	0.0249270
$568$	0	0
$569$	4.62229e6	0.598516	0.299258	−	0.954172i	$-0.403261\pi$
$569$	4.62229e6	0.598516	0.299258	+	0.954172i	$0.403261\pi$
$570$	0	0
$571$	435588.	0.0559095	0.0279547	−	0.999609i	$-0.491101\pi$
$571$	435588.	0.0559095	0.0279547	+	0.999609i	$0.491101\pi$
$572$	0	0
$573$	1.48879e6	0.189429
$574$	0	0
$575$	−1.45085e6	−0.183000
$576$	0	0
$577$	−5.38078e6	−0.672831	−0.336415	−	0.941714i	$-0.609215\pi$
$577$	−5.38078e6	−0.672831	−0.336415	+	0.941714i	$0.609215\pi$
$578$	0	0
$579$	1.63042e6	0.202117
$580$	0	0
$581$	2.16424e6	0.265989
$582$	0	0
$583$	1.15415e6	0.140635
$584$	0	0
$585$	−3.89520e6	−0.470587
$586$	0	0
$587$	4.39482e6	0.526436	0.263218	−	0.964736i	$-0.415216\pi$
$587$	4.39482e6	0.526436	0.263218	+	0.964736i	$0.415216\pi$
$588$	0	0
$589$	−5.27320e6	−0.626305
$590$	0	0
$591$	2.86067e6	0.336898
$592$	0	0
$593$	−2.98721e6	−0.348842	−0.174421	−	0.984671i	$-0.555805\pi$
$593$	−2.98721e6	−0.348842	−0.174421	+	0.984671i	$0.555805\pi$
$594$	0	0
$595$	3.87127e6	0.448293
$596$	0	0
$597$	−234535.	−0.0269322
$598$	0	0
$599$	−1.64115e7	−1.86888	−0.934438	−	0.356126i	$-0.884097\pi$
$599$	−1.64115e7	−1.86888	−0.934438	+	0.356126i	$0.884097\pi$
$600$	0	0
$601$	−4.12475e6	−0.465812	−0.232906	−	0.972499i	$-0.574824\pi$
$601$	−4.12475e6	−0.465812	−0.232906	+	0.972499i	$0.574824\pi$
$602$	0	0
$603$	−5.01140e6	−0.561262
$604$	0	0
$605$	1.71701e7	1.90715
$606$	0	0
$607$	1.41090e7	1.55427	0.777133	−	0.629336i	$-0.216673\pi$
$607$	1.41090e7	1.55427	0.777133	+	0.629336i	$0.216673\pi$
$608$	0	0
$609$	−893485.	−0.0976212
$610$	0	0
$611$	−1.24468e7	−1.34882
$612$	0	0
$613$	1.49191e6	0.160359	0.0801794	−	0.996780i	$-0.474451\pi$
$613$	1.49191e6	0.160359	0.0801794	+	0.996780i	$0.474451\pi$
$614$	0	0
$615$	4.18159e6	0.445814
$616$	0	0
$617$	−3.18931e6	−0.337275	−0.168637	−	0.985678i	$-0.553937\pi$
$617$	−3.18931e6	−0.337275	−0.168637	+	0.985678i	$0.553937\pi$
$618$	0	0
$619$	1.92333e6	0.201757	0.100878	−	0.994899i	$-0.467835\pi$
$619$	1.92333e6	0.201757	0.100878	+	0.994899i	$0.467835\pi$
$620$	0	0
$621$	−548592.	−0.0570848
$622$	0	0
$623$	−1.95545e6	−0.201849
$624$	0	0
$625$	−1.20735e7	−1.23632
$626$	0	0
$627$	−5.29278e6	−0.537669
$628$	0	0
$629$	−2.63219e7	−2.65272
$630$	0	0
$631$	−8.85496e6	−0.885346	−0.442673	−	0.896683i	$-0.645970\pi$
$631$	−8.85496e6	−0.885346	−0.442673	+	0.896683i	$0.645970\pi$
$632$	0	0
$633$	1.26563e6	0.125544
$634$	0	0
$635$	−1.54965e7	−1.52511
$636$	0	0
$637$	1.07978e7	1.05435
$638$	0	0
$639$	−1.63514e6	−0.158417
$640$	0	0
$641$	910224.	0.0874990	0.0437495	−	0.999043i	$-0.486070\pi$
$641$	910224.	0.0874990	0.0437495	+	0.999043i	$0.486070\pi$
$642$	0	0
$643$	−1.38109e6	−0.131733	−0.0658666	−	0.997828i	$-0.520981\pi$
$643$	−1.38109e6	−0.131733	−0.0658666	+	0.997828i	$0.520981\pi$
$644$	0	0
$645$	9.27454e6	0.877795
$646$	0	0
$647$	1.42411e7	1.33747	0.668733	−	0.743502i	$-0.266837\pi$
$647$	1.42411e7	1.33747	0.668733	+	0.743502i	$0.266837\pi$
$648$	0	0
$649$	2.96886e7	2.76680
$650$	0	0
$651$	1.48925e6	0.137726
$652$	0	0
$653$	−7.01719e6	−0.643991	−0.321996	−	0.946741i	$-0.604354\pi$
$653$	−7.01719e6	−0.643991	−0.321996	+	0.946741i	$0.604354\pi$
$654$	0	0
$655$	−1.73827e7	−1.58312
$656$	0	0
$657$	2.45078e6	0.221508
$658$	0	0
$659$	1.37665e7	1.23484	0.617421	−	0.786633i	$-0.288177\pi$
$659$	1.37665e7	1.23484	0.617421	+	0.786633i	$0.288177\pi$
$660$	0	0
$661$	1.96282e7	1.74734	0.873671	−	0.486517i	$-0.161733\pi$
$661$	1.96282e7	1.74734	0.873671	+	0.486517i	$0.161733\pi$
$662$	0	0
$663$	−1.14008e7	−1.00729
$664$	0	0
$665$	−1.91618e6	−0.168028
$666$	0	0
$667$	2.56868e6	0.223560
$668$	0	0
$669$	−1.17072e7	−1.01132
$670$	0	0
$671$	1.74146e7	1.49316
$672$	0	0
$673$	2.11019e7	1.79590	0.897952	−	0.440092i	$-0.145054\pi$
$673$	2.11019e7	1.79590	0.897952	+	0.440092i	$0.145054\pi$
$674$	0	0
$675$	1.40549e6	0.118732
$676$	0	0
$677$	−1.45083e7	−1.21659	−0.608295	−	0.793711i	$-0.708146\pi$
$677$	−1.45083e7	−1.21659	−0.608295	+	0.793711i	$0.708146\pi$
$678$	0	0
$679$	849287.	0.0706936
$680$	0	0
$681$	−3.17988e6	−0.262750
$682$	0	0
$683$	−1.37196e7	−1.12536	−0.562680	−	0.826675i	$-0.690230\pi$
$683$	−1.37196e7	−1.12536	−0.562680	+	0.826675i	$0.690230\pi$
$684$	0	0
$685$	−9.08313e6	−0.739621
$686$	0	0
$687$	1.87670e6	0.151706
$688$	0	0
$689$	−1.23055e6	−0.0987532
$690$	0	0
$691$	5.47338e6	0.436074	0.218037	−	0.975941i	$-0.430035\pi$
$691$	5.47338e6	0.436074	0.218037	+	0.975941i	$0.430035\pi$
$692$	0	0
$693$	1.49478e6	0.118235
$694$	0	0
$695$	−2.70307e7	−2.12273
$696$	0	0
$697$	1.22391e7	0.954261
$698$	0	0
$699$	5.80908e6	0.449692
$700$	0	0
$701$	−1.05258e7	−0.809018	−0.404509	−	0.914534i	$-0.632558\pi$
$701$	−1.05258e7	−0.809018	−0.404509	+	0.914534i	$0.632558\pi$
$702$	0	0
$703$	1.30287e7	0.994289
$704$	0	0
$705$	1.17707e7	0.891928
$706$	0	0
$707$	−5.17858e6	−0.389639
$708$	0	0
$709$	−1.26221e7	−0.943009	−0.471504	−	0.881864i	$-0.656289\pi$
$709$	−1.26221e7	−0.943009	−0.471504	+	0.881864i	$0.656289\pi$
$710$	0	0
$711$	3.73930e6	0.277406
$712$	0	0
$713$	−4.28144e6	−0.315403
$714$	0	0
$715$	−3.05126e7	−2.23211
$716$	0	0
$717$	931804.	0.0676903
$718$	0	0
$719$	−3.93982e6	−0.284220	−0.142110	−	0.989851i	$-0.545389\pi$
$719$	−3.93982e6	−0.284220	−0.142110	+	0.989851i	$0.545389\pi$
$720$	0	0
$721$	5.71451e6	0.409393
$722$	0	0
$723$	6.67248e6	0.474724
$724$	0	0
$725$	−6.58092e6	−0.464988
$726$	0	0
$727$	−2.62609e7	−1.84278	−0.921389	−	0.388642i	$-0.872944\pi$
$727$	−2.62609e7	−1.84278	−0.921389	+	0.388642i	$0.872944\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	2.71456e7	1.87891
$732$	0	0
$733$	1.85231e6	0.127337	0.0636683	−	0.997971i	$-0.479720\pi$
$733$	1.85231e6	0.127337	0.0636683	+	0.997971i	$0.479720\pi$
$734$	0	0
$735$	−1.02112e7	−0.697205
$736$	0	0
$737$	−3.92563e7	−2.66220
$738$	0	0
$739$	−2.34461e7	−1.57928	−0.789642	−	0.613568i	$-0.789734\pi$
$739$	−2.34461e7	−1.57928	−0.789642	+	0.613568i	$0.789734\pi$
$740$	0	0
$741$	5.64312e6	0.377550
$742$	0	0
$743$	2.98738e6	0.198527	0.0992634	−	0.995061i	$-0.468351\pi$
$743$	2.98738e6	0.198527	0.0992634	+	0.995061i	$0.468351\pi$
$744$	0	0
$745$	−2.74030e7	−1.80887
$746$	0	0
$747$	6.02743e6	0.395212
$748$	0	0
$749$	−1.81410e6	−0.118156
$750$	0	0
$751$	4.00447e6	0.259087	0.129543	−	0.991574i	$-0.458649\pi$
$751$	4.00447e6	0.259087	0.129543	+	0.991574i	$0.458649\pi$
$752$	0	0
$753$	7.82456e6	0.502889
$754$	0	0
$755$	−2.39567e7	−1.52954
$756$	0	0
$757$	1.91096e7	1.21202	0.606011	−	0.795456i	$-0.292769\pi$
$757$	1.91096e7	1.21202	0.606011	+	0.795456i	$0.292769\pi$
$758$	0	0
$759$	−4.29734e6	−0.270767
$760$	0	0
$761$	2.63961e7	1.65226	0.826131	−	0.563478i	$-0.190537\pi$
$761$	2.63961e7	1.65226	0.826131	+	0.563478i	$0.190537\pi$
$762$	0	0
$763$	−3.82867e6	−0.238087
$764$	0	0
$765$	1.07816e7	0.666082
$766$	0	0
$767$	−3.16538e7	−1.94284
$768$	0	0
$769$	1.94131e7	1.18380	0.591899	−	0.806012i	$-0.298378\pi$
$769$	1.94131e7	1.18380	0.591899	+	0.806012i	$0.298378\pi$
$770$	0	0
$771$	−1.40056e7	−0.848529
$772$	0	0
$773$	−2.36424e7	−1.42313	−0.711563	−	0.702622i	$-0.752012\pi$
$773$	−2.36424e7	−1.42313	−0.711563	+	0.702622i	$0.752012\pi$
$774$	0	0
$775$	1.09690e7	0.656015
$776$	0	0
$777$	−3.67955e6	−0.218646
$778$	0	0
$779$	−6.05804e6	−0.357675
$780$	0	0
$781$	−1.28087e7	−0.751410
$782$	0	0
$783$	−2.48837e6	−0.145048
$784$	0	0
$785$	−8.85883e6	−0.513101
$786$	0	0
$787$	−1.45794e7	−0.839078	−0.419539	−	0.907737i	$-0.637808\pi$
$787$	−1.45794e7	−0.839078	−0.419539	+	0.907737i	$0.637808\pi$
$788$	0	0
$789$	5.46622e6	0.312604
$790$	0	0
$791$	363793.	0.0206734
$792$	0	0
$793$	−1.85673e7	−1.04850
$794$	0	0
$795$	1.16371e6	0.0653019
$796$	0	0
$797$	3.35874e7	1.87297	0.936485	−	0.350709i	$-0.114059\pi$
$797$	3.35874e7	1.87297	0.936485	+	0.350709i	$0.114059\pi$
$798$	0	0
$799$	3.44516e7	1.90916
$800$	0	0
$801$	−5.44597e6	−0.299912
$802$	0	0
$803$	1.91979e7	1.05067
$804$	0	0
$805$	−1.55580e6	−0.0846181
$806$	0	0
$807$	3.50756e6	0.189593
$808$	0	0
$809$	8.87523e6	0.476769	0.238385	−	0.971171i	$-0.423382\pi$
$809$	8.87523e6	0.476769	0.238385	+	0.971171i	$0.423382\pi$
$810$	0	0
$811$	2.12583e6	0.113495	0.0567476	−	0.998389i	$-0.481927\pi$
$811$	2.12583e6	0.113495	0.0567476	+	0.998389i	$0.481927\pi$
$812$	0	0
$813$	−1.43968e6	−0.0763904
$814$	0	0
$815$	−1.20854e7	−0.637332
$816$	0	0
$817$	−1.34364e7	−0.704251
$818$	0	0
$819$	−1.59373e6	−0.0830241
$820$	0	0
$821$	−1.40986e7	−0.729990	−0.364995	−	0.931010i	$-0.618929\pi$
$821$	−1.40986e7	−0.729990	−0.364995	+	0.931010i	$0.618929\pi$
$822$	0	0
$823$	1.89611e7	0.975808	0.487904	−	0.872897i	$-0.337762\pi$
$823$	1.89611e7	0.975808	0.487904	+	0.872897i	$0.337762\pi$
$824$	0	0
$825$	1.10098e7	0.563174
$826$	0	0
$827$	2.59667e7	1.32024	0.660121	−	0.751159i	$-0.270505\pi$
$827$	2.59667e7	1.32024	0.660121	+	0.751159i	$0.270505\pi$
$828$	0	0
$829$	2.00387e7	1.01271	0.506354	−	0.862326i	$-0.330993\pi$
$829$	2.00387e7	1.01271	0.506354	+	0.862326i	$0.330993\pi$
$830$	0	0
$831$	−6.68433e6	−0.335780
$832$	0	0
$833$	−2.98873e7	−1.49236
$834$	0	0
$835$	8.43750e6	0.418791
$836$	0	0
$837$	4.14759e6	0.204636
$838$	0	0
$839$	−4.34988e6	−0.213340	−0.106670	−	0.994294i	$-0.534019\pi$
$839$	−4.34988e6	−0.213340	−0.106670	+	0.994294i	$0.534019\pi$
$840$	0	0
$841$	−8.85985e6	−0.431953
$842$	0	0
$843$	1.23163e7	0.596916
$844$	0	0
$845$	6.13930e6	0.295786
$846$	0	0
$847$	7.02518e6	0.336472
$848$	0	0
$849$	1.15409e7	0.549505
$850$	0	0
$851$	1.05783e7	0.500718
$852$	0	0
$853$	−868195.	−0.0408550	−0.0204275	−	0.999791i	$-0.506503\pi$
$853$	−868195.	−0.0408550	−0.0204275	+	0.999791i	$0.506503\pi$
$854$	0	0
$855$	−5.33659e6	−0.249660
$856$	0	0
$857$	−3.01726e7	−1.40333	−0.701667	−	0.712505i	$-0.747561\pi$
$857$	−3.01726e7	−1.40333	−0.701667	+	0.712505i	$0.747561\pi$
$858$	0	0
$859$	1.17823e7	0.544812	0.272406	−	0.962182i	$-0.412181\pi$
$859$	1.17823e7	0.544812	0.272406	+	0.962182i	$0.412181\pi$
$860$	0	0
$861$	1.71091e6	0.0786535
$862$	0	0
$863$	3.84607e7	1.75789	0.878943	−	0.476927i	$-0.158249\pi$
$863$	3.84607e7	1.75789	0.878943	+	0.476927i	$0.158249\pi$
$864$	0	0
$865$	6.31212e6	0.286837
$866$	0	0
$867$	1.87778e7	0.848392
$868$	0	0
$869$	2.92914e7	1.31580
$870$	0	0
$871$	4.18548e7	1.86939
$872$	0	0
$873$	2.36528e6	0.105038
$874$	0	0
$875$	−2.47478e6	−0.109274
$876$	0	0
$877$	1.83936e7	0.807547	0.403773	−	0.914859i	$-0.367698\pi$
$877$	1.83936e7	0.807547	0.403773	+	0.914859i	$0.367698\pi$
$878$	0	0
$879$	−1.02764e7	−0.448608
$880$	0	0
$881$	−2.35976e7	−1.02430	−0.512152	−	0.858895i	$-0.671151\pi$
$881$	−2.35976e7	−1.02430	−0.512152	+	0.858895i	$0.671151\pi$
$882$	0	0
$883$	1.51716e7	0.654831	0.327416	−	0.944880i	$-0.393822\pi$
$883$	1.51716e7	0.654831	0.327416	+	0.944880i	$0.393822\pi$
$884$	0	0
$885$	2.99344e7	1.28473
$886$	0	0
$887$	1.26690e7	0.540669	0.270335	−	0.962766i	$-0.412866\pi$
$887$	1.26690e7	0.540669	0.270335	+	0.962766i	$0.412866\pi$
$888$	0	0
$889$	−6.34043e6	−0.269070
$890$	0	0
$891$	4.16299e6	0.175675
$892$	0	0
$893$	−1.70527e7	−0.715589
$894$	0	0
$895$	−4.68563e7	−1.95529
$896$	0	0
$897$	4.58180e6	0.190132
$898$	0	0
$899$	−1.94203e7	−0.801412
$900$	0	0
$901$	3.40605e6	0.139778
$902$	0	0
$903$	3.79469e6	0.154866
$904$	0	0
$905$	5.70062e7	2.31366
$906$	0	0
$907$	−1.76981e7	−0.714345	−0.357172	−	0.934039i	$-0.616259\pi$
$907$	−1.76981e7	−0.714345	−0.357172	+	0.934039i	$0.616259\pi$
$908$	0	0
$909$	−1.44224e7	−0.578933
$910$	0	0
$911$	−3.14279e7	−1.25464	−0.627321	−	0.778761i	$-0.715848\pi$
$911$	−3.14279e7	−1.25464	−0.627321	+	0.778761i	$0.715848\pi$
$912$	0	0
$913$	4.72153e7	1.87459
$914$	0	0
$915$	1.75588e7	0.693332
$916$	0	0
$917$	−7.11216e6	−0.279305
$918$	0	0
$919$	−1.62509e7	−0.634729	−0.317364	−	0.948304i	$-0.602798\pi$
$919$	−1.62509e7	−0.634729	−0.317364	+	0.948304i	$0.602798\pi$
$920$	0	0
$921$	−8.55688e6	−0.332404
$922$	0	0
$923$	1.36565e7	0.527638
$924$	0	0
$925$	−2.71016e7	−1.04145
$926$	0	0
$927$	1.59150e7	0.608285
$928$	0	0
$929$	3.21230e7	1.22117	0.610587	−	0.791949i	$-0.290934\pi$
$929$	3.21230e7	1.22117	0.610587	+	0.791949i	$0.290934\pi$
$930$	0	0
$931$	1.47934e7	0.559364
$932$	0	0
$933$	1.42338e6	0.0535323
$934$	0	0
$935$	8.44562e7	3.15939
$936$	0	0
$937$	−3.35215e7	−1.24731	−0.623654	−	0.781700i	$-0.714353\pi$
$937$	−3.35215e7	−1.24731	−0.623654	+	0.781700i	$0.714353\pi$
$938$	0	0
$939$	−2.37108e7	−0.877573
$940$	0	0
$941$	−3.01791e7	−1.11105	−0.555524	−	0.831501i	$-0.687482\pi$
$941$	−3.01791e7	−1.11105	−0.555524	+	0.831501i	$0.687482\pi$
$942$	0	0
$943$	−4.91867e6	−0.180123
$944$	0	0
$945$	1.50716e6	0.0549008
$946$	0	0
$947$	7.61259e6	0.275840	0.137920	−	0.990443i	$-0.455958\pi$
$947$	7.61259e6	0.275840	0.137920	+	0.990443i	$0.455958\pi$
$948$	0	0
$949$	−2.04687e7	−0.737776
$950$	0	0
$951$	2.07822e7	0.745143
$952$	0	0
$953$	1.74093e6	0.0620940	0.0310470	−	0.999518i	$-0.490116\pi$
$953$	1.74093e6	0.0620940	0.0310470	+	0.999518i	$0.490116\pi$
$954$	0	0
$955$	1.17588e7	0.417211
$956$	0	0
$957$	−1.94924e7	−0.687995
$958$	0	0
$959$	−3.71638e6	−0.130489
$960$	0	0
$961$	3.74036e6	0.130649
$962$	0	0
$963$	−5.05230e6	−0.175559
$964$	0	0
$965$	1.28775e7	0.445155
$966$	0	0
$967$	4.54917e7	1.56447	0.782233	−	0.622986i	$-0.214081\pi$
$967$	4.54917e7	1.56447	0.782233	+	0.622986i	$0.214081\pi$
$968$	0	0
$969$	−1.56197e7	−0.534395
$970$	0	0
$971$	2.65561e6	0.0903893	0.0451946	−	0.998978i	$-0.485609\pi$
$971$	2.65561e6	0.0903893	0.0451946	+	0.998978i	$0.485609\pi$
$972$	0	0
$973$	−1.10597e7	−0.374506
$974$	0	0
$975$	−1.17385e7	−0.395459
$976$	0	0
$977$	−7.00179e6	−0.234678	−0.117339	−	0.993092i	$-0.537436\pi$
$977$	−7.00179e6	−0.234678	−0.117339	+	0.993092i	$0.537436\pi$
$978$	0	0
$979$	−4.26604e7	−1.42255
$980$	0	0
$981$	−1.06629e7	−0.353755
$982$	0	0
$983$	−4.74138e7	−1.56502	−0.782511	−	0.622636i	$-0.786062\pi$
$983$	−4.74138e7	−1.56502	−0.782511	+	0.622636i	$0.786062\pi$
$984$	0	0
$985$	2.25943e7	0.742007
$986$	0	0
$987$	4.81600e6	0.157360
$988$	0	0
$989$	−1.09093e7	−0.354657
$990$	0	0
$991$	−2.92129e7	−0.944911	−0.472456	−	0.881355i	$-0.656632\pi$
$991$	−2.92129e7	−0.944911	−0.472456	+	0.881355i	$0.656632\pi$
$992$	0	0
$993$	−1.37420e7	−0.442258
$994$	0	0
$995$	−1.85241e6	−0.0593172
$996$	0	0
$997$	3.96525e6	0.126338	0.0631688	−	0.998003i	$-0.479879\pi$
$997$	3.96525e6	0.126338	0.0631688	+	0.998003i	$0.479879\pi$
$998$	0	0
$999$	−1.02476e7	−0.324869

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.6.a.r.1.2		2
3.2	odd	2		576.6.a.bm.1.1		2
4.3	odd	2		192.6.a.q.1.2		2
8.3	odd	2		96.6.a.h.1.1	yes	2
8.5	even	2		96.6.a.g.1.1	✓	2
12.11	even	2		576.6.a.bn.1.1		2
16.3	odd	4		768.6.d.s.385.1		4
16.5	even	4		768.6.d.z.385.2		4
16.11	odd	4		768.6.d.s.385.4		4
16.13	even	4		768.6.d.z.385.3		4
24.5	odd	2		288.6.a.n.1.2		2
24.11	even	2		288.6.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.6.a.g.1.1	✓	2	8.5	even	2
96.6.a.h.1.1	yes	2	8.3	odd	2
192.6.a.q.1.2		2	4.3	odd	2
192.6.a.r.1.2		2	1.1	even	1	trivial
288.6.a.n.1.2		2	24.5	odd	2
288.6.a.o.1.2		2	24.11	even	2
576.6.a.bm.1.1		2	3.2	odd	2
576.6.a.bn.1.1		2	12.11	even	2
768.6.d.s.385.1		4	16.3	odd	4
768.6.d.s.385.4		4	16.11	odd	4
768.6.d.z.385.2		4	16.5	even	4
768.6.d.z.385.3		4	16.13	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	192.a (trivial)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} +71.0842 q^{5} +29.0842 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q+9.00000 q^{3} +71.0842 q^{5} +29.0842 q^{7} +81.0000 q^{9} +634.505 q^{11} -676.505 q^{13} +639.758 q^{15} +1872.51 q^{17} -926.842 q^{19} +261.758 q^{21} -752.527 q^{23} +1927.97 q^{25} +729.000 q^{27} -3413.40 q^{29} +5689.42 q^{31} +5710.55 q^{33} +2067.43 q^{35} -14057.1 q^{37} -6088.55 q^{39} +6536.21 q^{41} +14496.9 q^{43} +5757.82 q^{45} +18398.7 q^{47} -15961.1 q^{49} +16852.5 q^{51} +1818.98 q^{53} +45103.3 q^{55} -8341.58 q^{57} +46790.2 q^{59} +27446.0 q^{61} +2355.82 q^{63} -48088.9 q^{65} -61869.1 q^{67} -6772.74 q^{69} -20186.9 q^{71} +30256.5 q^{73} +17351.7 q^{75} +18454.1 q^{77} +46164.2 q^{79} +6561.00 q^{81} +74412.7 q^{83} +133106. q^{85} -30720.6 q^{87} -67234.2 q^{89} -19675.6 q^{91} +51204.8 q^{93} -65883.9 q^{95} +29200.9 q^{97} +51394.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 18 q^{3} - 36 q^{5} - 120 q^{7} + 162 q^{9}+O(q^{10}) \) 2 * q + 18 * q^3 - 36 * q^5 - 120 * q^7 + 162 * q^9	\( 2 q + 18 q^{3} - 36 q^{5} - 120 q^{7} + 162 q^{9} + 200 q^{11} - 284 q^{13} - 324 q^{15} + 2676 q^{17} - 72 q^{19} - 1080 q^{21} + 3840 q^{23} + 10270 q^{25} + 1458 q^{27} - 10212 q^{29} + 10488 q^{31} + 1800 q^{33} + 18032 q^{35} - 13148 q^{37} - 2556 q^{39} + 4164 q^{41} + 5832 q^{43} - 2916 q^{45} + 1520 q^{47} - 10542 q^{49} + 24084 q^{51} - 9012 q^{53} + 91632 q^{55} - 648 q^{57} + 55096 q^{59} + 63444 q^{61} - 9720 q^{63} - 90120 q^{65} - 36792 q^{67} + 34560 q^{69} + 37664 q^{71} - 37836 q^{73} + 92430 q^{75} + 83232 q^{77} + 144888 q^{79} + 13122 q^{81} + 109272 q^{83} + 47064 q^{85} - 91908 q^{87} - 32556 q^{89} - 78192 q^{91} + 94392 q^{93} - 157424 q^{95} + 69092 q^{97} + 16200 q^{99}+O(q^{100}) \) 2 * q + 18 * q^3 - 36 * q^5 - 120 * q^7 + 162 * q^9 + 200 * q^11 - 284 * q^13 - 324 * q^15 + 2676 * q^17 - 72 * q^19 - 1080 * q^21 + 3840 * q^23 + 10270 * q^25 + 1458 * q^27 - 10212 * q^29 + 10488 * q^31 + 1800 * q^33 + 18032 * q^35 - 13148 * q^37 - 2556 * q^39 + 4164 * q^41 + 5832 * q^43 - 2916 * q^45 + 1520 * q^47 - 10542 * q^49 + 24084 * q^51 - 9012 * q^53 + 91632 * q^55 - 648 * q^57 + 55096 * q^59 + 63444 * q^61 - 9720 * q^63 - 90120 * q^65 - 36792 * q^67 + 34560 * q^69 + 37664 * q^71 - 37836 * q^73 + 92430 * q^75 + 83232 * q^77 + 144888 * q^79 + 13122 * q^81 + 109272 * q^83 + 47064 * q^85 - 91908 * q^87 - 32556 * q^89 - 78192 * q^91 + 94392 * q^93 - 157424 * q^95 + 69092 * q^97 + 16200 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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