LMFDB - Embedded newform 784.6.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,6,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, 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("784.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	784.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:	$125.740914733$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{109}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 27 $ x^2 - x - 27
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 28)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.72015$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-24.4403 q^{3} +83.6418 q^{5} +354.329 q^{9} +O(q^{10})$	$q-24.4403 q^{3} +83.6418 q^{5} +354.329 q^{9} -549.246 q^{11} -823.135 q^{13} -2044.23 q^{15} +145.567 q^{17} -1768.04 q^{19} +1522.73 q^{23} +3870.96 q^{25} -2720.90 q^{27} -741.943 q^{29} +3204.18 q^{31} +13423.8 q^{33} -3549.55 q^{37} +20117.7 q^{39} -6461.29 q^{41} -6716.00 q^{43} +29636.7 q^{45} +19176.8 q^{47} -3557.71 q^{51} -21278.6 q^{53} -45940.0 q^{55} +43211.4 q^{57} -36528.7 q^{59} +43433.8 q^{61} -68848.5 q^{65} -5028.15 q^{67} -37215.9 q^{69} +6311.32 q^{71} -42597.9 q^{73} -94607.4 q^{75} -95069.1 q^{79} -19602.1 q^{81} +34978.2 q^{83} +12175.5 q^{85} +18133.3 q^{87} +100553. q^{89} -78311.0 q^{93} -147882. q^{95} +76794.5 q^{97} -194614. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 28 q^{3} + 42 q^{5} + 124 q^{9}+O(q^{10}) $ 2 * q - 28 * q^3 + 42 * q^5 + 124 * q^9	$ 2 q - 28 q^{3} + 42 q^{5} + 124 q^{9} - 660 q^{11} - 644 q^{13} - 1896 q^{15} - 210 q^{17} - 3724 q^{19} - 24 q^{23} + 2480 q^{25} - 1036 q^{27} + 5532 q^{29} - 2800 q^{31} + 13818 q^{33} - 13238 q^{37} + 19480 q^{39} + 4116 q^{41} - 13432 q^{43} + 39228 q^{45} - 8064 q^{47} - 2292 q^{51} - 53958 q^{53} - 41328 q^{55} + 50174 q^{57} - 36036 q^{59} + 83986 q^{61} - 76308 q^{65} + 2660 q^{67} - 31710 q^{69} - 71568 q^{71} + 31318 q^{73} - 89656 q^{75} - 51136 q^{79} + 30370 q^{81} - 6216 q^{83} + 26982 q^{85} - 4200 q^{87} + 166278 q^{89} - 56938 q^{93} - 66432 q^{95} + 8260 q^{97} - 169104 q^{99}+O(q^{100}) $ 2 * q - 28 * q^3 + 42 * q^5 + 124 * q^9 - 660 * q^11 - 644 * q^13 - 1896 * q^15 - 210 * q^17 - 3724 * q^19 - 24 * q^23 + 2480 * q^25 - 1036 * q^27 + 5532 * q^29 - 2800 * q^31 + 13818 * q^33 - 13238 * q^37 + 19480 * q^39 + 4116 * q^41 - 13432 * q^43 + 39228 * q^45 - 8064 * q^47 - 2292 * q^51 - 53958 * q^53 - 41328 * q^55 + 50174 * q^57 - 36036 * q^59 + 83986 * q^61 - 76308 * q^65 + 2660 * q^67 - 31710 * q^69 - 71568 * q^71 + 31318 * q^73 - 89656 * q^75 - 51136 * q^79 + 30370 * q^81 - 6216 * q^83 + 26982 * q^85 - 4200 * q^87 + 166278 * q^89 - 56938 * q^93 - 66432 * q^95 + 8260 * q^97 - 169104 * 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
$3$	−24.4403	−1.56785	−0.783923	−	0.620858i	$-0.786784\pi$
$3$	−24.4403	−1.56785	−0.783923	+	0.620858i	$0.786784\pi$
$4$	0	0
$5$	83.6418	1.49623	0.748115	−	0.663569i	$-0.230959\pi$
$5$	83.6418	1.49623	0.748115	+	0.663569i	$0.230959\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	354.329	1.45814
$10$	0	0
$11$	−549.246	−1.36863	−0.684314	−	0.729187i	$-0.739898\pi$
$11$	−549.246	−1.36863	−0.684314	+	0.729187i	$0.739898\pi$
$12$	0	0
$13$	−823.135	−1.35087	−0.675433	−	0.737421i	$-0.736043\pi$
$13$	−823.135	−1.35087	−0.675433	+	0.737421i	$0.736043\pi$
$14$	0	0
$15$	−2044.23	−2.34586
$16$	0	0
$17$	145.567	0.122164	0.0610818	−	0.998133i	$-0.480545\pi$
$17$	145.567	0.122164	0.0610818	+	0.998133i	$0.480545\pi$
$18$	0	0
$19$	−1768.04	−1.12359	−0.561794	−	0.827277i	$-0.689889\pi$
$19$	−1768.04	−1.12359	−0.561794	+	0.827277i	$0.689889\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1522.73	0.600208	0.300104	−	0.953906i	$-0.402979\pi$
$23$	1522.73	0.600208	0.300104	+	0.953906i	$0.402979\pi$
$24$	0	0
$25$	3870.96	1.23871
$26$	0	0
$27$	−2720.90	−0.718297
$28$	0	0
$29$	−741.943	−0.163823	−0.0819116	−	0.996640i	$-0.526103\pi$
$29$	−741.943	−0.163823	−0.0819116	+	0.996640i	$0.526103\pi$
$30$	0	0
$31$	3204.18	0.598842	0.299421	−	0.954121i	$-0.403207\pi$
$31$	3204.18	0.598842	0.299421	+	0.954121i	$0.403207\pi$
$32$	0	0
$33$	13423.8	2.14580
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3549.55	−0.426254	−0.213127	−	0.977024i	$-0.568365\pi$
$37$	−3549.55	−0.426254	−0.213127	+	0.977024i	$0.568365\pi$
$38$	0	0
$39$	20117.7	2.11795
$40$	0	0
$41$	−6461.29	−0.600288	−0.300144	−	0.953894i	$-0.597035\pi$
$41$	−6461.29	−0.600288	−0.300144	+	0.953894i	$0.597035\pi$
$42$	0	0
$43$	−6716.00	−0.553910	−0.276955	−	0.960883i	$-0.589325\pi$
$43$	−6716.00	−0.553910	−0.276955	+	0.960883i	$0.589325\pi$
$44$	0	0
$45$	29636.7	2.18172
$46$	0	0
$47$	19176.8	1.26629	0.633143	−	0.774035i	$-0.281765\pi$
$47$	19176.8	1.26629	0.633143	+	0.774035i	$0.281765\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−3557.71	−0.191534
$52$	0	0
$53$	−21278.6	−1.04053	−0.520263	−	0.854006i	$-0.674166\pi$
$53$	−21278.6	−1.04053	−0.520263	+	0.854006i	$0.674166\pi$
$54$	0	0
$55$	−45940.0	−2.04778
$56$	0	0
$57$	43211.4	1.76161
$58$	0	0
$59$	−36528.7	−1.36617	−0.683083	−	0.730340i	$-0.739361\pi$
$59$	−36528.7	−1.36617	−0.683083	+	0.730340i	$0.739361\pi$
$60$	0	0
$61$	43433.8	1.49452	0.747262	−	0.664530i	$-0.231368\pi$
$61$	43433.8	1.49452	0.747262	+	0.664530i	$0.231368\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−68848.5	−2.02121
$66$	0	0
$67$	−5028.15	−0.136842	−0.0684212	−	0.997657i	$-0.521796\pi$
$67$	−5028.15	−0.136842	−0.0684212	+	0.997657i	$0.521796\pi$
$68$	0	0
$69$	−37215.9	−0.941034
$70$	0	0
$71$	6311.32	0.148585	0.0742923	−	0.997237i	$-0.476330\pi$
$71$	6311.32	0.148585	0.0742923	+	0.997237i	$0.476330\pi$
$72$	0	0
$73$	−42597.9	−0.935580	−0.467790	−	0.883840i	$-0.654950\pi$
$73$	−42597.9	−0.935580	−0.467790	+	0.883840i	$0.654950\pi$
$74$	0	0
$75$	−94607.4	−1.94210
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−95069.1	−1.71385	−0.856923	−	0.515445i	$-0.827627\pi$
$79$	−95069.1	−1.71385	−0.856923	+	0.515445i	$0.827627\pi$
$80$	0	0
$81$	−19602.1	−0.331963
$82$	0	0
$83$	34978.2	0.557318	0.278659	−	0.960390i	$-0.410110\pi$
$83$	34978.2	0.557318	0.278659	+	0.960390i	$0.410110\pi$
$84$	0	0
$85$	12175.5	0.182785
$86$	0	0
$87$	18133.3	0.256850
$88$	0	0
$89$	100553.	1.34562	0.672809	−	0.739816i	$-0.265088\pi$
$89$	100553.	1.34562	0.672809	+	0.739816i	$0.265088\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−78311.0	−0.938892
$94$	0	0
$95$	−147882.	−1.68115
$96$	0	0
$97$	76794.5	0.828707	0.414353	−	0.910116i	$-0.364008\pi$
$97$	76794.5	0.828707	0.414353	+	0.910116i	$0.364008\pi$
$98$	0	0
$99$	−194614.	−1.99565
$100$	0	0
$101$	114040.	1.11239	0.556193	−	0.831053i	$-0.312261\pi$
$101$	114040.	1.11239	0.556193	+	0.831053i	$0.312261\pi$
$102$	0	0
$103$	123637.	1.14830	0.574150	−	0.818750i	$-0.305333\pi$
$103$	123637.	1.14830	0.574150	+	0.818750i	$0.305333\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	23340.2	0.197081	0.0985404	−	0.995133i	$-0.468583\pi$
$107$	23340.2	0.197081	0.0985404	+	0.995133i	$0.468583\pi$
$108$	0	0
$109$	−27203.2	−0.219307	−0.109654	−	0.993970i	$-0.534974\pi$
$109$	−27203.2	−0.219307	−0.109654	+	0.993970i	$0.534974\pi$
$110$	0	0
$111$	86752.1	0.668302
$112$	0	0
$113$	−157394.	−1.15955	−0.579777	−	0.814775i	$-0.696860\pi$
$113$	−157394.	−1.15955	−0.579777	+	0.814775i	$0.696860\pi$
$114$	0	0
$115$	127364.	0.898050
$116$	0	0
$117$	−291660.	−1.96976
$118$	0	0
$119$	0	0
$120$	0	0
$121$	140621.	0.873144
$122$	0	0
$123$	157916.	0.941159
$124$	0	0
$125$	62393.2	0.357160
$126$	0	0
$127$	253263.	1.39336	0.696679	−	0.717383i	$-0.254660\pi$
$127$	253263.	1.39336	0.696679	+	0.717383i	$0.254660\pi$
$128$	0	0
$129$	164141.	0.868446
$130$	0	0
$131$	115545.	0.588266	0.294133	−	0.955764i	$-0.404969\pi$
$131$	115545.	0.588266	0.294133	+	0.955764i	$0.404969\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−227581.	−1.07474
$136$	0	0
$137$	−207808.	−0.945933	−0.472966	−	0.881080i	$-0.656817\pi$
$137$	−207808.	−0.945933	−0.472966	+	0.881080i	$0.656817\pi$
$138$	0	0
$139$	28993.1	0.127279	0.0636395	−	0.997973i	$-0.479729\pi$
$139$	28993.1	0.127279	0.0636395	+	0.997973i	$0.479729\pi$
$140$	0	0
$141$	−468687.	−1.98534
$142$	0	0
$143$	452104.	1.84883
$144$	0	0
$145$	−62057.5	−0.245117
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−330980.	−1.22134	−0.610668	−	0.791886i	$-0.709099\pi$
$149$	−330980.	−1.22134	−0.610668	+	0.791886i	$0.709099\pi$
$150$	0	0
$151$	−108857.	−0.388519	−0.194260	−	0.980950i	$-0.562230\pi$
$151$	−108857.	−0.388519	−0.194260	+	0.980950i	$0.562230\pi$
$152$	0	0
$153$	51578.7	0.178132
$154$	0	0
$155$	268003.	0.896005
$156$	0	0
$157$	−497745.	−1.61160	−0.805801	−	0.592186i	$-0.798265\pi$
$157$	−497745.	−1.61160	−0.805801	+	0.592186i	$0.798265\pi$
$158$	0	0
$159$	520055.	1.63139
$160$	0	0
$161$	0	0
$162$	0	0
$163$	359103.	1.05864	0.529321	−	0.848421i	$-0.322447\pi$
$163$	359103.	1.05864	0.529321	+	0.848421i	$0.322447\pi$
$164$	0	0
$165$	1.12279e6	3.21061
$166$	0	0
$167$	50645.2	0.140523	0.0702614	−	0.997529i	$-0.477617\pi$
$167$	50645.2	0.140523	0.0702614	+	0.997529i	$0.477617\pi$
$168$	0	0
$169$	306258.	0.824841
$170$	0	0
$171$	−626466.	−1.63835
$172$	0	0
$173$	303189.	0.770191	0.385095	−	0.922877i	$-0.374169\pi$
$173$	303189.	0.770191	0.385095	+	0.922877i	$0.374169\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	892772.	2.14194
$178$	0	0
$179$	682543.	1.59220	0.796100	−	0.605165i	$-0.206893\pi$
$179$	682543.	1.59220	0.796100	+	0.605165i	$0.206893\pi$
$180$	0	0
$181$	182455.	0.413961	0.206980	−	0.978345i	$-0.433636\pi$
$181$	182455.	0.413961	0.206980	+	0.978345i	$0.433636\pi$
$182$	0	0
$183$	−1.06153e6	−2.34318
$184$	0	0
$185$	−296891.	−0.637775
$186$	0	0
$187$	−79952.4	−0.167197
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−683543.	−1.35576	−0.677879	−	0.735173i	$-0.737101\pi$
$191$	−683543.	−1.35576	−0.677879	+	0.735173i	$0.737101\pi$
$192$	0	0
$193$	946503.	1.82906	0.914532	−	0.404515i	$-0.132559\pi$
$193$	946503.	1.82906	0.914532	+	0.404515i	$0.132559\pi$
$194$	0	0
$195$	1.68268e6	3.16894
$196$	0	0
$197$	−184209.	−0.338178	−0.169089	−	0.985601i	$-0.554082\pi$
$197$	−184209.	−0.338178	−0.169089	+	0.985601i	$0.554082\pi$
$198$	0	0
$199$	−68206.6	−0.122094	−0.0610469	−	0.998135i	$-0.519444\pi$
$199$	−68206.6	−0.122094	−0.0610469	+	0.998135i	$0.519444\pi$
$200$	0	0
$201$	122889.	0.214548
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−540434.	−0.898169
$206$	0	0
$207$	539545.	0.875189
$208$	0	0
$209$	971088.	1.53778
$210$	0	0
$211$	1.19270e6	1.84427	0.922134	−	0.386871i	$-0.126444\pi$
$211$	1.19270e6	1.84427	0.922134	+	0.386871i	$0.126444\pi$
$212$	0	0
$213$	−154250.	−0.232958
$214$	0	0
$215$	−561739.	−0.828778
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.04111e6	1.46685
$220$	0	0
$221$	−119822.	−0.165027
$222$	0	0
$223$	−200502.	−0.269995	−0.134998	−	0.990846i	$-0.543103\pi$
$223$	−200502.	−0.269995	−0.134998	+	0.990846i	$0.543103\pi$
$224$	0	0
$225$	1.37159e6	1.80621
$226$	0	0
$227$	−290769.	−0.374527	−0.187264	−	0.982310i	$-0.559962\pi$
$227$	−290769.	−0.374527	−0.187264	+	0.982310i	$0.559962\pi$
$228$	0	0
$229$	944942.	1.19074	0.595369	−	0.803452i	$-0.297006\pi$
$229$	944942.	1.19074	0.595369	+	0.803452i	$0.297006\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	216384.	0.261117	0.130558	−	0.991441i	$-0.458323\pi$
$233$	216384.	0.261117	0.130558	+	0.991441i	$0.458323\pi$
$234$	0	0
$235$	1.60398e6	1.89465
$236$	0	0
$237$	2.32352e6	2.68705
$238$	0	0
$239$	668807.	0.757366	0.378683	−	0.925526i	$-0.376377\pi$
$239$	668807.	0.757366	0.378683	+	0.925526i	$0.376377\pi$
$240$	0	0
$241$	1.15601e6	1.28209	0.641047	−	0.767501i	$-0.278500\pi$
$241$	1.15601e6	1.28209	0.641047	+	0.767501i	$0.278500\pi$
$242$	0	0
$243$	1.14026e6	1.23876
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.45533e6	1.51782
$248$	0	0
$249$	−854879.	−0.873788
$250$	0	0
$251$	77574.9	0.0777207	0.0388604	−	0.999245i	$-0.487627\pi$
$251$	77574.9	0.0777207	0.0388604	+	0.999245i	$0.487627\pi$
$252$	0	0
$253$	−836351.	−0.821462
$254$	0	0
$255$	−297573.	−0.286579
$256$	0	0
$257$	−280202.	−0.264630	−0.132315	−	0.991208i	$-0.542241\pi$
$257$	−280202.	−0.264630	−0.132315	+	0.991208i	$0.542241\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−262892.	−0.238878
$262$	0	0
$263$	269966.	0.240669	0.120334	−	0.992733i	$-0.461603\pi$
$263$	269966.	0.240669	0.120334	+	0.992733i	$0.461603\pi$
$264$	0	0
$265$	−1.77978e6	−1.55687
$266$	0	0
$267$	−2.45756e6	−2.10972
$268$	0	0
$269$	890267.	0.750135	0.375068	−	0.926997i	$-0.377620\pi$
$269$	890267.	0.750135	0.375068	+	0.926997i	$0.377620\pi$
$270$	0	0
$271$	−384279.	−0.317851	−0.158926	−	0.987291i	$-0.550803\pi$
$271$	−384279.	−0.317851	−0.158926	+	0.987291i	$0.550803\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.12611e6	−1.69533
$276$	0	0
$277$	−1.09193e6	−0.855061	−0.427530	−	0.904001i	$-0.640616\pi$
$277$	−1.09193e6	−0.855061	−0.427530	+	0.904001i	$0.640616\pi$
$278$	0	0
$279$	1.13533e6	0.873196
$280$	0	0
$281$	771640.	0.582974	0.291487	−	0.956575i	$-0.405850\pi$
$281$	771640.	0.582974	0.291487	+	0.956575i	$0.405850\pi$
$282$	0	0
$283$	541805.	0.402140	0.201070	−	0.979577i	$-0.435558\pi$
$283$	541805.	0.402140	0.201070	+	0.979577i	$0.435558\pi$
$284$	0	0
$285$	3.61428e6	2.63578
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.39867e6	−0.985076
$290$	0	0
$291$	−1.87688e6	−1.29929
$292$	0	0
$293$	1.69804e6	1.15552	0.577762	−	0.816205i	$-0.303926\pi$
$293$	1.69804e6	1.15552	0.577762	+	0.816205i	$0.303926\pi$
$294$	0	0
$295$	−3.05532e6	−2.04410
$296$	0	0
$297$	1.49445e6	0.983081
$298$	0	0
$299$	−1.25341e6	−0.810801
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−2.78718e6	−1.74405
$304$	0	0
$305$	3.63288e6	2.23615
$306$	0	0
$307$	288376.	0.174627	0.0873137	−	0.996181i	$-0.472172\pi$
$307$	288376.	0.174627	0.0873137	+	0.996181i	$0.472172\pi$
$308$	0	0
$309$	−3.02173e6	−1.80036
$310$	0	0
$311$	−347601.	−0.203789	−0.101894	−	0.994795i	$-0.532490\pi$
$311$	−347601.	−0.203789	−0.101894	+	0.994795i	$0.532490\pi$
$312$	0	0
$313$	−770403.	−0.444485	−0.222242	−	0.974991i	$-0.571338\pi$
$313$	−770403.	−0.444485	−0.222242	+	0.974991i	$0.571338\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.01752e6	−1.12764	−0.563819	−	0.825898i	$-0.690669\pi$
$317$	−2.01752e6	−1.12764	−0.563819	+	0.825898i	$0.690669\pi$
$318$	0	0
$319$	407510.	0.224213
$320$	0	0
$321$	−570441.	−0.308993
$322$	0	0
$323$	−257369.	−0.137262
$324$	0	0
$325$	−3.18632e6	−1.67333
$326$	0	0
$327$	664854.	0.343840
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−425932.	−0.213683	−0.106842	−	0.994276i	$-0.534074\pi$
$331$	−425932.	−0.213683	−0.106842	+	0.994276i	$0.534074\pi$
$332$	0	0
$333$	−1.25771e6	−0.621540
$334$	0	0
$335$	−420563.	−0.204748
$336$	0	0
$337$	1.15035e6	0.551767	0.275884	−	0.961191i	$-0.411030\pi$
$337$	1.15035e6	0.551767	0.275884	+	0.961191i	$0.411030\pi$
$338$	0	0
$339$	3.84675e6	1.81800
$340$	0	0
$341$	−1.75988e6	−0.819592
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−3.11280e6	−1.40800
$346$	0	0
$347$	313136.	0.139608	0.0698039	−	0.997561i	$-0.477763\pi$
$347$	313136.	0.139608	0.0698039	+	0.997561i	$0.477763\pi$
$348$	0	0
$349$	−3.01459e6	−1.32484	−0.662422	−	0.749131i	$-0.730471\pi$
$349$	−3.01459e6	−1.32484	−0.662422	+	0.749131i	$0.730471\pi$
$350$	0	0
$351$	2.23967e6	0.970323
$352$	0	0
$353$	4.38460e6	1.87281	0.936404	−	0.350925i	$-0.114133\pi$
$353$	4.38460e6	1.87281	0.936404	+	0.350925i	$0.114133\pi$
$354$	0	0
$355$	527890.	0.222317
$356$	0	0
$357$	0	0
$358$	0	0
$359$	192656.	0.0788943	0.0394471	−	0.999222i	$-0.487440\pi$
$359$	192656.	0.0788943	0.0394471	+	0.999222i	$0.487440\pi$
$360$	0	0
$361$	649857.	0.262452
$362$	0	0
$363$	−3.43681e6	−1.36896
$364$	0	0
$365$	−3.56297e6	−1.39984
$366$	0	0
$367$	2.33981e6	0.906809	0.453405	−	0.891305i	$-0.350209\pi$
$367$	2.33981e6	0.906809	0.453405	+	0.891305i	$0.350209\pi$
$368$	0	0
$369$	−2.28942e6	−0.875305
$370$	0	0
$371$	0	0
$372$	0	0
$373$	3.92433e6	1.46047	0.730237	−	0.683194i	$-0.239410\pi$
$373$	3.92433e6	1.46047	0.730237	+	0.683194i	$0.239410\pi$
$374$	0	0
$375$	−1.52491e6	−0.559972
$376$	0	0
$377$	610719.	0.221303
$378$	0	0
$379$	3.26556e6	1.16778	0.583889	−	0.811834i	$-0.301530\pi$
$379$	3.26556e6	1.16778	0.583889	+	0.811834i	$0.301530\pi$
$380$	0	0
$381$	−6.18983e6	−2.18457
$382$	0	0
$383$	4.42975e6	1.54306	0.771529	−	0.636194i	$-0.219492\pi$
$383$	4.42975e6	1.54306	0.771529	+	0.636194i	$0.219492\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.37967e6	−0.807680
$388$	0	0
$389$	−5.11467e6	−1.71373	−0.856867	−	0.515537i	$-0.827592\pi$
$389$	−5.11467e6	−1.71373	−0.856867	+	0.515537i	$0.827592\pi$
$390$	0	0
$391$	221659.	0.0733236
$392$	0	0
$393$	−2.82396e6	−0.922311
$394$	0	0
$395$	−7.95176e6	−2.56431
$396$	0	0
$397$	−1.54189e6	−0.490995	−0.245497	−	0.969397i	$-0.578951\pi$
$397$	−1.54189e6	−0.490995	−0.245497	+	0.969397i	$0.578951\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.44227e6	1.69013	0.845064	−	0.534666i	$-0.179563\pi$
$401$	5.44227e6	1.69013	0.845064	+	0.534666i	$0.179563\pi$
$402$	0	0
$403$	−2.63747e6	−0.808955
$404$	0	0
$405$	−1.63956e6	−0.496694
$406$	0	0
$407$	1.94958e6	0.583384
$408$	0	0
$409$	5.18482e6	1.53259	0.766294	−	0.642490i	$-0.222099\pi$
$409$	5.18482e6	1.53259	0.766294	+	0.642490i	$0.222099\pi$
$410$	0	0
$411$	5.07888e6	1.48308
$412$	0	0
$413$	0	0
$414$	0	0
$415$	2.92564e6	0.833876
$416$	0	0
$417$	−708599.	−0.199554
$418$	0	0
$419$	2.17093e6	0.604102	0.302051	−	0.953292i	$-0.402329\pi$
$419$	2.17093e6	0.604102	0.302051	+	0.953292i	$0.402329\pi$
$420$	0	0
$421$	6.61967e6	1.82025	0.910125	−	0.414334i	$-0.135986\pi$
$421$	6.61967e6	1.82025	0.910125	+	0.414334i	$0.135986\pi$
$422$	0	0
$423$	6.79489e6	1.84642
$424$	0	0
$425$	563485.	0.151325
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.10496e7	−2.89869
$430$	0	0
$431$	2.89984e6	0.751937	0.375968	−	0.926632i	$-0.377310\pi$
$431$	2.89984e6	0.751937	0.375968	+	0.926632i	$0.377310\pi$
$432$	0	0
$433$	4.71803e6	1.20932	0.604659	−	0.796484i	$-0.293309\pi$
$433$	4.71803e6	1.20932	0.604659	+	0.796484i	$0.293309\pi$
$434$	0	0
$435$	1.51670e6	0.384306
$436$	0	0
$437$	−2.69223e6	−0.674387
$438$	0	0
$439$	3.47700e6	0.861081	0.430540	−	0.902571i	$-0.358323\pi$
$439$	3.47700e6	0.861081	0.430540	+	0.902571i	$0.358323\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7.29658e6	−1.76649	−0.883243	−	0.468916i	$-0.844645\pi$
$443$	−7.29658e6	−1.76649	−0.883243	+	0.468916i	$0.844645\pi$
$444$	0	0
$445$	8.41047e6	2.01336
$446$	0	0
$447$	8.08924e6	1.91487
$448$	0	0
$449$	3.65867e6	0.856462	0.428231	−	0.903669i	$-0.359137\pi$
$449$	3.65867e6	0.856462	0.428231	+	0.903669i	$0.359137\pi$
$450$	0	0
$451$	3.54884e6	0.821571
$452$	0	0
$453$	2.66049e6	0.609139
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.44062e6	1.44257	0.721286	−	0.692638i	$-0.243552\pi$
$457$	6.44062e6	1.44257	0.721286	+	0.692638i	$0.243552\pi$
$458$	0	0
$459$	−396075.	−0.0877497
$460$	0	0
$461$	4.47898e6	0.981583	0.490791	−	0.871277i	$-0.336708\pi$
$461$	4.47898e6	0.981583	0.490791	+	0.871277i	$0.336708\pi$
$462$	0	0
$463$	2.11214e6	0.457899	0.228949	−	0.973438i	$-0.426471\pi$
$463$	2.11214e6	0.457899	0.228949	+	0.973438i	$0.426471\pi$
$464$	0	0
$465$	−6.55008e6	−1.40480
$466$	0	0
$467$	−7.85838e6	−1.66740	−0.833702	−	0.552214i	$-0.813783\pi$
$467$	−7.85838e6	−1.66740	−0.833702	+	0.552214i	$0.813783\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	1.21650e7	2.52675
$472$	0	0
$473$	3.68874e6	0.758098
$474$	0	0
$475$	−6.84400e6	−1.39180
$476$	0	0
$477$	−7.53961e6	−1.51724
$478$	0	0
$479$	−8.48745e6	−1.69020	−0.845101	−	0.534607i	$-0.820460\pi$
$479$	−8.48745e6	−1.69020	−0.845101	+	0.534607i	$0.820460\pi$
$480$	0	0
$481$	2.92176e6	0.575813
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.42324e6	1.23994
$486$	0	0
$487$	−2.75616e6	−0.526602	−0.263301	−	0.964714i	$-0.584811\pi$
$487$	−2.75616e6	−0.526602	−0.263301	+	0.964714i	$0.584811\pi$
$488$	0	0
$489$	−8.77658e6	−1.65979
$490$	0	0
$491$	−2.45533e6	−0.459628	−0.229814	−	0.973235i	$-0.573812\pi$
$491$	−2.45533e6	−0.459628	−0.229814	+	0.973235i	$0.573812\pi$
$492$	0	0
$493$	−108003.	−0.0200132
$494$	0	0
$495$	−1.62778e7	−2.98596
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4.45849e6	0.801561	0.400781	−	0.916174i	$-0.368739\pi$
$499$	4.45849e6	0.801561	0.400781	+	0.916174i	$0.368739\pi$
$500$	0	0
$501$	−1.23778e6	−0.220318
$502$	0	0
$503$	−2.64080e6	−0.465389	−0.232694	−	0.972550i	$-0.574754\pi$
$503$	−2.64080e6	−0.465389	−0.232694	+	0.972550i	$0.574754\pi$
$504$	0	0
$505$	9.53856e6	1.66439
$506$	0	0
$507$	−7.48503e6	−1.29322
$508$	0	0
$509$	4.71041e6	0.805868	0.402934	−	0.915229i	$-0.367990\pi$
$509$	4.71041e6	0.805868	0.402934	+	0.915229i	$0.367990\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.81066e6	0.807070
$514$	0	0
$515$	1.03412e7	1.71812
$516$	0	0
$517$	−1.05328e7	−1.73307
$518$	0	0
$519$	−7.41003e6	−1.20754
$520$	0	0
$521$	−2.80269e6	−0.452356	−0.226178	−	0.974086i	$-0.572623\pi$
$521$	−2.80269e6	−0.452356	−0.226178	+	0.974086i	$0.572623\pi$
$522$	0	0
$523$	2.99755e6	0.479195	0.239598	−	0.970872i	$-0.422985\pi$
$523$	2.99755e6	0.479195	0.239598	+	0.970872i	$0.422985\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	466423.	0.0731566
$528$	0	0
$529$	−4.11765e6	−0.639750
$530$	0	0
$531$	−1.29431e7	−1.99207
$532$	0	0
$533$	5.31851e6	0.810909
$534$	0	0
$535$	1.95221e6	0.294878
$536$	0	0
$537$	−1.66816e7	−2.49632
$538$	0	0
$539$	0	0
$540$	0	0
$541$	3.30504e6	0.485493	0.242747	−	0.970090i	$-0.421952\pi$
$541$	3.30504e6	0.485493	0.242747	+	0.970090i	$0.421952\pi$
$542$	0	0
$543$	−4.45926e6	−0.649027
$544$	0	0
$545$	−2.27532e6	−0.328135
$546$	0	0
$547$	−3.51435e6	−0.502201	−0.251100	−	0.967961i	$-0.580792\pi$
$547$	−3.51435e6	−0.502201	−0.251100	+	0.967961i	$0.580792\pi$
$548$	0	0
$549$	1.53898e7	2.17923
$550$	0	0
$551$	1.31178e6	0.184070
$552$	0	0
$553$	0	0
$554$	0	0
$555$	7.25610e6	0.999933
$556$	0	0
$557$	−5.98187e6	−0.816956	−0.408478	−	0.912768i	$-0.633940\pi$
$557$	−5.98187e6	−0.816956	−0.408478	+	0.912768i	$0.633940\pi$
$558$	0	0
$559$	5.52817e6	0.748259
$560$	0	0
$561$	1.95406e6	0.262138
$562$	0	0
$563$	−1.07384e7	−1.42781	−0.713903	−	0.700245i	$-0.753074\pi$
$563$	−1.07384e7	−1.42781	−0.713903	+	0.700245i	$0.753074\pi$
$564$	0	0
$565$	−1.31647e7	−1.73496
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.59548e6	−0.854016	−0.427008	−	0.904248i	$-0.640432\pi$
$569$	−6.59548e6	−0.854016	−0.427008	+	0.904248i	$0.640432\pi$
$570$	0	0
$571$	2.61577e6	0.335745	0.167873	−	0.985809i	$-0.446310\pi$
$571$	2.61577e6	0.335745	0.167873	+	0.985809i	$0.446310\pi$
$572$	0	0
$573$	1.67060e7	2.12562
$574$	0	0
$575$	5.89440e6	0.743482
$576$	0	0
$577$	−1.23898e6	−0.154926	−0.0774630	−	0.996995i	$-0.524682\pi$
$577$	−1.23898e6	−0.154926	−0.0774630	+	0.996995i	$0.524682\pi$
$578$	0	0
$579$	−2.31328e7	−2.86769
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.16872e7	1.42409
$584$	0	0
$585$	−2.43950e7	−2.94721
$586$	0	0
$587$	−1.02896e7	−1.23254	−0.616270	−	0.787535i	$-0.711357\pi$
$587$	−1.02896e7	−1.23254	−0.616270	+	0.787535i	$0.711357\pi$
$588$	0	0
$589$	−5.66510e6	−0.672852
$590$	0	0
$591$	4.50212e6	0.530211
$592$	0	0
$593$	1.53178e7	1.78879	0.894394	−	0.447280i	$-0.147607\pi$
$593$	1.53178e7	1.78879	0.894394	+	0.447280i	$0.147607\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.66699e6	0.191424
$598$	0	0
$599$	−2.69496e6	−0.306892	−0.153446	−	0.988157i	$-0.549037\pi$
$599$	−2.69496e6	−0.306892	−0.153446	+	0.988157i	$0.549037\pi$
$600$	0	0
$601$	1.54704e6	0.174709	0.0873547	−	0.996177i	$-0.472159\pi$
$601$	1.54704e6	0.174709	0.0873547	+	0.996177i	$0.472159\pi$
$602$	0	0
$603$	−1.78162e6	−0.199536
$604$	0	0
$605$	1.17618e7	1.30642
$606$	0	0
$607$	−3.79138e6	−0.417662	−0.208831	−	0.977952i	$-0.566966\pi$
$607$	−3.79138e6	−0.417662	−0.208831	+	0.977952i	$0.566966\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.57851e7	−1.71058
$612$	0	0
$613$	6.19207e6	0.665556	0.332778	−	0.943005i	$-0.392014\pi$
$613$	6.19207e6	0.665556	0.332778	+	0.943005i	$0.392014\pi$
$614$	0	0
$615$	1.32084e7	1.40819
$616$	0	0
$617$	−1.59349e7	−1.68514	−0.842571	−	0.538585i	$-0.818959\pi$
$617$	−1.59349e7	−1.68514	−0.842571	+	0.538585i	$0.818959\pi$
$618$	0	0
$619$	1.15732e7	1.21402	0.607011	−	0.794693i	$-0.292368\pi$
$619$	1.15732e7	1.21402	0.607011	+	0.794693i	$0.292368\pi$
$620$	0	0
$621$	−4.14319e6	−0.431128
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−6.87806e6	−0.704313
$626$	0	0
$627$	−2.37337e7	−2.41100
$628$	0	0
$629$	−516699.	−0.0520728
$630$	0	0
$631$	−900933.	−0.0900781	−0.0450390	−	0.998985i	$-0.514341\pi$
$631$	−900933.	−0.0900781	−0.0450390	+	0.998985i	$0.514341\pi$
$632$	0	0
$633$	−2.91499e7	−2.89153
$634$	0	0
$635$	2.11834e7	2.08479
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2.23628e6	0.216658
$640$	0	0
$641$	−848770.	−0.0815915	−0.0407958	−	0.999168i	$-0.512989\pi$
$641$	−848770.	−0.0815915	−0.0407958	+	0.999168i	$0.512989\pi$
$642$	0	0
$643$	1.37977e7	1.31607	0.658036	−	0.752986i	$-0.271387\pi$
$643$	1.37977e7	1.31607	0.658036	+	0.752986i	$0.271387\pi$
$644$	0	0
$645$	1.37291e7	1.29940
$646$	0	0
$647$	3.65350e6	0.343122	0.171561	−	0.985174i	$-0.445119\pi$
$647$	3.65350e6	0.343122	0.171561	+	0.985174i	$0.445119\pi$
$648$	0	0
$649$	2.00632e7	1.86977
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.17912e7	1.08212	0.541058	−	0.840985i	$-0.318024\pi$
$653$	1.17912e7	1.08212	0.541058	+	0.840985i	$0.318024\pi$
$654$	0	0
$655$	9.66442e6	0.880182
$656$	0	0
$657$	−1.50937e7	−1.36421
$658$	0	0
$659$	1.15208e7	1.03340	0.516700	−	0.856166i	$-0.327160\pi$
$659$	1.15208e7	1.03340	0.516700	+	0.856166i	$0.327160\pi$
$660$	0	0
$661$	5.37215e6	0.478238	0.239119	−	0.970990i	$-0.423141\pi$
$661$	5.37215e6	0.478238	0.239119	+	0.970990i	$0.423141\pi$
$662$	0	0
$663$	2.92848e6	0.258737
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.12978e6	−0.0983281
$668$	0	0
$669$	4.90032e6	0.423311
$670$	0	0
$671$	−2.38558e7	−2.04545
$672$	0	0
$673$	1.48042e7	1.25993	0.629966	−	0.776623i	$-0.283069\pi$
$673$	1.48042e7	1.25993	0.629966	+	0.776623i	$0.283069\pi$
$674$	0	0
$675$	−1.05325e7	−0.889759
$676$	0	0
$677$	−5.90707e6	−0.495336	−0.247668	−	0.968845i	$-0.579664\pi$
$677$	−5.90707e6	−0.495336	−0.247668	+	0.968845i	$0.579664\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	7.10648e6	0.587201
$682$	0	0
$683$	−2.20159e7	−1.80586	−0.902932	−	0.429784i	$-0.858590\pi$
$683$	−2.20159e7	−1.80586	−0.902932	+	0.429784i	$0.858590\pi$
$684$	0	0
$685$	−1.73814e7	−1.41533
$686$	0	0
$687$	−2.30947e7	−1.86690
$688$	0	0
$689$	1.75151e7	1.40561
$690$	0	0
$691$	−1.30651e7	−1.04092	−0.520460	−	0.853886i	$-0.674239\pi$
$691$	−1.30651e7	−1.04092	−0.520460	+	0.853886i	$0.674239\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.42503e6	0.190439
$696$	0	0
$697$	−940553.	−0.0733333
$698$	0	0
$699$	−5.28848e6	−0.409391
$700$	0	0
$701$	1.20043e7	0.922658	0.461329	−	0.887229i	$-0.347373\pi$
$701$	1.20043e7	0.922658	0.461329	+	0.887229i	$0.347373\pi$
$702$	0	0
$703$	6.27574e6	0.478935
$704$	0	0
$705$	−3.92018e7	−2.97053
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1.64794e7	−1.23119	−0.615597	−	0.788061i	$-0.711085\pi$
$709$	−1.64794e7	−1.23119	−0.615597	+	0.788061i	$0.711085\pi$
$710$	0	0
$711$	−3.36857e7	−2.49903
$712$	0	0
$713$	4.87908e6	0.359430
$714$	0	0
$715$	3.78148e7	2.76628
$716$	0	0
$717$	−1.63458e7	−1.18743
$718$	0	0
$719$	7.32353e6	0.528322	0.264161	−	0.964479i	$-0.414905\pi$
$719$	7.32353e6	0.528322	0.264161	+	0.964479i	$0.414905\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−2.82533e7	−2.01013
$724$	0	0
$725$	−2.87203e6	−0.202929
$726$	0	0
$727$	−1.83777e7	−1.28960	−0.644798	−	0.764353i	$-0.723059\pi$
$727$	−1.83777e7	−1.28960	−0.644798	+	0.764353i	$0.723059\pi$
$728$	0	0
$729$	−2.31050e7	−1.61023
$730$	0	0
$731$	−977630.	−0.0676677
$732$	0	0
$733$	−1.18498e7	−0.814609	−0.407305	−	0.913292i	$-0.633531\pi$
$733$	−1.18498e7	−0.814609	−0.407305	+	0.913292i	$0.633531\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.76169e6	0.187286
$738$	0	0
$739$	2.09511e7	1.41123	0.705613	−	0.708598i	$-0.250672\pi$
$739$	2.09511e7	1.41123	0.705613	+	0.708598i	$0.250672\pi$
$740$	0	0
$741$	−3.55688e7	−2.37971
$742$	0	0
$743$	1.42524e7	0.947141	0.473570	−	0.880756i	$-0.342965\pi$
$743$	1.42524e7	0.947141	0.473570	+	0.880756i	$0.342965\pi$
$744$	0	0
$745$	−2.76837e7	−1.82740
$746$	0	0
$747$	1.23938e7	0.812648
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1.31969e6	−0.0853832	−0.0426916	−	0.999088i	$-0.513593\pi$
$751$	−1.31969e6	−0.0853832	−0.0426916	+	0.999088i	$0.513593\pi$
$752$	0	0
$753$	−1.89595e6	−0.121854
$754$	0	0
$755$	−9.10497e6	−0.581315
$756$	0	0
$757$	102138.	0.00647812	0.00323906	−	0.999995i	$-0.498969\pi$
$757$	102138.	0.00647812	0.00323906	+	0.999995i	$0.498969\pi$
$758$	0	0
$759$	2.04407e7	1.28793
$760$	0	0
$761$	2.19193e7	1.37203	0.686016	−	0.727586i	$-0.259358\pi$
$761$	2.19193e7	1.37203	0.686016	+	0.727586i	$0.259358\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	4.31414e6	0.266526
$766$	0	0
$767$	3.00680e7	1.84551
$768$	0	0
$769$	6.44098e6	0.392768	0.196384	−	0.980527i	$-0.437080\pi$
$769$	6.44098e6	0.392768	0.196384	+	0.980527i	$0.437080\pi$
$770$	0	0
$771$	6.84823e6	0.414899
$772$	0	0
$773$	2.91160e6	0.175260	0.0876302	−	0.996153i	$-0.472071\pi$
$773$	2.91160e6	0.175260	0.0876302	+	0.996153i	$0.472071\pi$
$774$	0	0
$775$	1.24032e7	0.741789
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.14238e7	0.674477
$780$	0	0
$781$	−3.46647e6	−0.203357
$782$	0	0
$783$	2.01876e6	0.117674
$784$	0	0
$785$	−4.16323e7	−2.41133
$786$	0	0
$787$	−2.54579e7	−1.46516	−0.732582	−	0.680679i	$-0.761685\pi$
$787$	−2.54579e7	−1.46516	−0.732582	+	0.680679i	$0.761685\pi$
$788$	0	0
$789$	−6.59805e6	−0.377331
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−3.57518e7	−2.01890
$794$	0	0
$795$	4.34984e7	2.44093
$796$	0	0
$797$	−2.04344e7	−1.13950	−0.569752	−	0.821817i	$-0.692961\pi$
$797$	−2.04344e7	−1.13950	−0.569752	+	0.821817i	$0.692961\pi$
$798$	0	0
$799$	2.79152e6	0.154694
$800$	0	0
$801$	3.56290e7	1.96210
$802$	0	0
$803$	2.33968e7	1.28046
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.17584e7	−1.17610
$808$	0	0
$809$	2.82831e7	1.51934	0.759672	−	0.650306i	$-0.225359\pi$
$809$	2.82831e7	1.51934	0.759672	+	0.650306i	$0.225359\pi$
$810$	0	0
$811$	−1.58852e7	−0.848087	−0.424043	−	0.905642i	$-0.639390\pi$
$811$	−1.58852e7	−0.848087	−0.424043	+	0.905642i	$0.639390\pi$
$812$	0	0
$813$	9.39191e6	0.498342
$814$	0	0
$815$	3.00360e7	1.58397
$816$	0	0
$817$	1.18741e7	0.622368
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.54440e7	0.799652	0.399826	−	0.916591i	$-0.369071\pi$
$821$	1.54440e7	0.799652	0.399826	+	0.916591i	$0.369071\pi$
$822$	0	0
$823$	−2.97886e7	−1.53303	−0.766516	−	0.642225i	$-0.778011\pi$
$823$	−2.97886e7	−1.53303	−0.766516	+	0.642225i	$0.778011\pi$
$824$	0	0
$825$	5.19628e7	2.65801
$826$	0	0
$827$	−1.66831e7	−0.848230	−0.424115	−	0.905608i	$-0.639415\pi$
$827$	−1.66831e7	−0.848230	−0.424115	+	0.905608i	$0.639415\pi$
$828$	0	0
$829$	−2.21137e7	−1.11757	−0.558785	−	0.829313i	$-0.688732\pi$
$829$	−2.21137e7	−1.11757	−0.558785	+	0.829313i	$0.688732\pi$
$830$	0	0
$831$	2.66872e7	1.34060
$832$	0	0
$833$	0	0
$834$	0	0
$835$	4.23606e6	0.210255
$836$	0	0
$837$	−8.71826e6	−0.430146
$838$	0	0
$839$	−539166.	−0.0264434	−0.0132217	−	0.999913i	$-0.504209\pi$
$839$	−539166.	−0.0264434	−0.0132217	+	0.999913i	$0.504209\pi$
$840$	0	0
$841$	−1.99607e7	−0.973162
$842$	0	0
$843$	−1.88591e7	−0.914013
$844$	0	0
$845$	2.56160e7	1.23415
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.32419e7	−0.630493
$850$	0	0
$851$	−5.40499e6	−0.255841
$852$	0	0
$853$	2.55452e6	0.120209	0.0601044	−	0.998192i	$-0.480857\pi$
$853$	2.55452e6	0.120209	0.0601044	+	0.998192i	$0.480857\pi$
$854$	0	0
$855$	−5.23988e7	−2.45135
$856$	0	0
$857$	240078.	0.0111661	0.00558303	−	0.999984i	$-0.498223\pi$
$857$	240078.	0.0111661	0.00558303	+	0.999984i	$0.498223\pi$
$858$	0	0
$859$	−6.67911e6	−0.308841	−0.154421	−	0.988005i	$-0.549351\pi$
$859$	−6.67911e6	−0.308841	−0.154421	+	0.988005i	$0.549351\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.05060e7	0.937245	0.468622	−	0.883399i	$-0.344751\pi$
$863$	2.05060e7	0.937245	0.468622	+	0.883399i	$0.344751\pi$
$864$	0	0
$865$	2.53593e7	1.15238
$866$	0	0
$867$	3.41839e7	1.54445
$868$	0	0
$869$	5.22164e7	2.34562
$870$	0	0
$871$	4.13884e6	0.184856
$872$	0	0
$873$	2.72105e7	1.20837
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.52775e7	1.10978	0.554888	−	0.831925i	$-0.312761\pi$
$877$	2.52775e7	1.10978	0.554888	+	0.831925i	$0.312761\pi$
$878$	0	0
$879$	−4.15006e7	−1.81168
$880$	0	0
$881$	−2.45932e7	−1.06752	−0.533759	−	0.845637i	$-0.679221\pi$
$881$	−2.45932e7	−1.06752	−0.533759	+	0.845637i	$0.679221\pi$
$882$	0	0
$883$	−2.78301e7	−1.20120	−0.600598	−	0.799551i	$-0.705071\pi$
$883$	−2.78301e7	−1.20120	−0.600598	+	0.799551i	$0.705071\pi$
$884$	0	0
$885$	7.46731e7	3.20484
$886$	0	0
$887$	−3.49856e7	−1.49307	−0.746534	−	0.665347i	$-0.768284\pi$
$887$	−3.49856e7	−1.49307	−0.746534	+	0.665347i	$0.768284\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.07664e7	0.454334
$892$	0	0
$893$	−3.39053e7	−1.42278
$894$	0	0
$895$	5.70892e7	2.38230
$896$	0	0
$897$	3.06337e7	1.27121
$898$	0	0
$899$	−2.37732e6	−0.0981042
$900$	0	0
$901$	−3.09747e6	−0.127114
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.52609e7	0.619381
$906$	0	0
$907$	4.06601e7	1.64116	0.820578	−	0.571534i	$-0.193652\pi$
$907$	4.06601e7	1.64116	0.820578	+	0.571534i	$0.193652\pi$
$908$	0	0
$909$	4.04078e7	1.62202
$910$	0	0
$911$	2.87014e7	1.14579	0.572897	−	0.819627i	$-0.305820\pi$
$911$	2.87014e7	1.14579	0.572897	+	0.819627i	$0.305820\pi$
$912$	0	0
$913$	−1.92117e7	−0.762761
$914$	0	0
$915$	−8.87887e7	−3.50594
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−2.94829e7	−1.15155	−0.575773	−	0.817609i	$-0.695299\pi$
$919$	−2.94829e7	−1.15155	−0.575773	+	0.817609i	$0.695299\pi$
$920$	0	0
$921$	−7.04799e6	−0.273789
$922$	0	0
$923$	−5.19506e6	−0.200718
$924$	0	0
$925$	−1.37402e7	−0.528004
$926$	0	0
$927$	4.38081e7	1.67438
$928$	0	0
$929$	−1.58822e7	−0.603772	−0.301886	−	0.953344i	$-0.597616\pi$
$929$	−1.58822e7	−0.603772	−0.301886	+	0.953344i	$0.597616\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	8.49549e6	0.319510
$934$	0	0
$935$	−6.68736e6	−0.250165
$936$	0	0
$937$	1.04452e7	0.388656	0.194328	−	0.980937i	$-0.437747\pi$
$937$	1.04452e7	0.388656	0.194328	+	0.980937i	$0.437747\pi$
$938$	0	0
$939$	1.88289e7	0.696884
$940$	0	0
$941$	−2.68483e7	−0.988421	−0.494211	−	0.869342i	$-0.664543\pi$
$941$	−2.68483e7	−0.988421	−0.494211	+	0.869342i	$0.664543\pi$
$942$	0	0
$943$	−9.83877e6	−0.360298
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.52409e7	0.552251	0.276126	−	0.961122i	$-0.410949\pi$
$947$	1.52409e7	0.552251	0.276126	+	0.961122i	$0.410949\pi$
$948$	0	0
$949$	3.50638e7	1.26384
$950$	0	0
$951$	4.93088e7	1.76796
$952$	0	0
$953$	−1.29112e7	−0.460504	−0.230252	−	0.973131i	$-0.573955\pi$
$953$	−1.29112e7	−0.460504	−0.230252	+	0.973131i	$0.573955\pi$
$954$	0	0
$955$	−5.71728e7	−2.02853
$956$	0	0
$957$	−9.95966e6	−0.351532
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.83624e7	−0.641389
$962$	0	0
$963$	8.27009e6	0.287372
$964$	0	0
$965$	7.91672e7	2.73670
$966$	0	0
$967$	−4.60406e7	−1.58334	−0.791671	−	0.610948i	$-0.790789\pi$
$967$	−4.60406e7	−1.58334	−0.791671	+	0.610948i	$0.790789\pi$
$968$	0	0
$969$	6.29017e6	0.215205
$970$	0	0
$971$	−1.87316e7	−0.637568	−0.318784	−	0.947827i	$-0.603274\pi$
$971$	−1.87316e7	−0.637568	−0.318784	+	0.947827i	$0.603274\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	7.78746e7	2.62352
$976$	0	0
$977$	−2.89204e7	−0.969321	−0.484660	−	0.874702i	$-0.661057\pi$
$977$	−2.89204e7	−0.969321	−0.484660	+	0.874702i	$0.661057\pi$
$978$	0	0
$979$	−5.52286e7	−1.84165
$980$	0	0
$981$	−9.63886e6	−0.319781
$982$	0	0
$983$	−2.17667e7	−0.718470	−0.359235	−	0.933247i	$-0.616962\pi$
$983$	−2.17667e7	−0.718470	−0.359235	+	0.933247i	$0.616962\pi$
$984$	0	0
$985$	−1.54076e7	−0.505992
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.02266e7	−0.332462
$990$	0	0
$991$	1.87867e7	0.607667	0.303833	−	0.952725i	$-0.401733\pi$
$991$	1.87867e7	0.607667	0.303833	+	0.952725i	$0.401733\pi$
$992$	0	0
$993$	1.04099e7	0.335023
$994$	0	0
$995$	−5.70492e6	−0.182680
$996$	0	0
$997$	−1.47676e7	−0.470514	−0.235257	−	0.971933i	$-0.575593\pi$
$997$	−1.47676e7	−0.470514	−0.235257	+	0.971933i	$0.575593\pi$
$998$	0	0
$999$	9.65799e6	0.306177

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.o.1.1		2
4.3	odd	2		196.6.a.j.1.2		2
7.2	even	3		112.6.i.e.81.2		4
7.4	even	3		112.6.i.e.65.2		4
7.6	odd	2		784.6.a.bd.1.2		2
28.3	even	6		196.6.e.k.177.2		4
28.11	odd	6		28.6.e.b.9.1	✓	4
28.19	even	6		196.6.e.k.165.2		4
28.23	odd	6		28.6.e.b.25.1	yes	4
28.27	even	2		196.6.a.h.1.1		2
84.11	even	6		252.6.k.d.37.2		4
84.23	even	6		252.6.k.d.109.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.6.e.b.9.1	✓	4	28.11	odd	6
28.6.e.b.25.1	yes	4	28.23	odd	6
112.6.i.e.65.2		4	7.4	even	3
112.6.i.e.81.2		4	7.2	even	3
196.6.a.h.1.1		2	28.27	even	2
196.6.a.j.1.2		2	4.3	odd	2
196.6.e.k.165.2		4	28.19	even	6
196.6.e.k.177.2		4	28.3	even	6
252.6.k.d.37.2		4	84.11	even	6
252.6.k.d.109.2		4	84.23	even	6
784.6.a.o.1.1		2	1.1	even	1	trivial
784.6.a.bd.1.2		2	7.6	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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q-24.4403 q^{3} +83.6418 q^{5} +354.329 q^{9} +O(q^{10})\)	\(q-24.4403 q^{3} +83.6418 q^{5} +354.329 q^{9} -549.246 q^{11} -823.135 q^{13} -2044.23 q^{15} +145.567 q^{17} -1768.04 q^{19} +1522.73 q^{23} +3870.96 q^{25} -2720.90 q^{27} -741.943 q^{29} +3204.18 q^{31} +13423.8 q^{33} -3549.55 q^{37} +20117.7 q^{39} -6461.29 q^{41} -6716.00 q^{43} +29636.7 q^{45} +19176.8 q^{47} -3557.71 q^{51} -21278.6 q^{53} -45940.0 q^{55} +43211.4 q^{57} -36528.7 q^{59} +43433.8 q^{61} -68848.5 q^{65} -5028.15 q^{67} -37215.9 q^{69} +6311.32 q^{71} -42597.9 q^{73} -94607.4 q^{75} -95069.1 q^{79} -19602.1 q^{81} +34978.2 q^{83} +12175.5 q^{85} +18133.3 q^{87} +100553. q^{89} -78311.0 q^{93} -147882. q^{95} +76794.5 q^{97} -194614. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 28 q^{3} + 42 q^{5} + 124 q^{9}+O(q^{10}) \) 2 * q - 28 * q^3 + 42 * q^5 + 124 * q^9	\( 2 q - 28 q^{3} + 42 q^{5} + 124 q^{9} - 660 q^{11} - 644 q^{13} - 1896 q^{15} - 210 q^{17} - 3724 q^{19} - 24 q^{23} + 2480 q^{25} - 1036 q^{27} + 5532 q^{29} - 2800 q^{31} + 13818 q^{33} - 13238 q^{37} + 19480 q^{39} + 4116 q^{41} - 13432 q^{43} + 39228 q^{45} - 8064 q^{47} - 2292 q^{51} - 53958 q^{53} - 41328 q^{55} + 50174 q^{57} - 36036 q^{59} + 83986 q^{61} - 76308 q^{65} + 2660 q^{67} - 31710 q^{69} - 71568 q^{71} + 31318 q^{73} - 89656 q^{75} - 51136 q^{79} + 30370 q^{81} - 6216 q^{83} + 26982 q^{85} - 4200 q^{87} + 166278 q^{89} - 56938 q^{93} - 66432 q^{95} + 8260 q^{97} - 169104 q^{99}+O(q^{100}) \) 2 * q - 28 * q^3 + 42 * q^5 + 124 * q^9 - 660 * q^11 - 644 * q^13 - 1896 * q^15 - 210 * q^17 - 3724 * q^19 - 24 * q^23 + 2480 * q^25 - 1036 * q^27 + 5532 * q^29 - 2800 * q^31 + 13818 * q^33 - 13238 * q^37 + 19480 * q^39 + 4116 * q^41 - 13432 * q^43 + 39228 * q^45 - 8064 * q^47 - 2292 * q^51 - 53958 * q^53 - 41328 * q^55 + 50174 * q^57 - 36036 * q^59 + 83986 * q^61 - 76308 * q^65 + 2660 * q^67 - 31710 * q^69 - 71568 * q^71 + 31318 * q^73 - 89656 * q^75 - 51136 * q^79 + 30370 * q^81 - 6216 * q^83 + 26982 * q^85 - 4200 * q^87 + 166278 * q^89 - 56938 * q^93 - 66432 * q^95 + 8260 * q^97 - 169104 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more