LMFDB - Embedded newform 784.6.a.bm.1.3

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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 167x^{3} - 387x^{2} + 1720x + 2340 $ x^5 - 167x^3 - 387x^2 + 1720*x + 2340
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{9}\cdot 7 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-10.9009$ 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)

$f(q)$	$=$	$q+6.22560 q^{3} -9.40551 q^{5} -204.242 q^{9} +O(q^{10})$	$q+6.22560 q^{3} -9.40551 q^{5} -204.242 q^{9} -765.115 q^{11} -732.172 q^{13} -58.5549 q^{15} -482.580 q^{17} -2617.07 q^{19} +535.506 q^{23} -3036.54 q^{25} -2784.35 q^{27} +3943.87 q^{29} +3663.45 q^{31} -4763.30 q^{33} +12639.0 q^{37} -4558.21 q^{39} -4814.47 q^{41} +4938.11 q^{43} +1921.00 q^{45} +17405.0 q^{47} -3004.35 q^{51} -4658.43 q^{53} +7196.30 q^{55} -16292.8 q^{57} +32500.8 q^{59} +43496.1 q^{61} +6886.45 q^{65} +32082.4 q^{67} +3333.85 q^{69} -15820.0 q^{71} -36515.4 q^{73} -18904.3 q^{75} -25472.1 q^{79} +32296.5 q^{81} +69443.5 q^{83} +4538.91 q^{85} +24553.0 q^{87} -108331. q^{89} +22807.1 q^{93} +24614.9 q^{95} -94569.1 q^{97} +156269. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 13 q^{3} - 31 q^{5} + 230 q^{9}+O(q^{10}) $ 5 * q + 13 * q^3 - 31 * q^5 + 230 * q^9	$ 5 q + 13 q^{3} - 31 q^{5} + 230 q^{9} + 351 q^{11} + 54 q^{13} - 607 q^{15} - 111 q^{17} + 1035 q^{19} - 3639 q^{23} + 1540 q^{25} + 3607 q^{27} - 734 q^{29} + 7677 q^{31} + 7439 q^{33} + 13595 q^{37} + 1406 q^{39} - 5310 q^{41} - 764 q^{43} - 38978 q^{45} + 6675 q^{47} - 20975 q^{51} - 30753 q^{53} + 28267 q^{55} - 14389 q^{57} + 87989 q^{59} - 19899 q^{61} + 119470 q^{65} + 33067 q^{67} - 100399 q^{69} + 108720 q^{71} - 141659 q^{73} + 108788 q^{75} - 118919 q^{79} - 143851 q^{81} + 211004 q^{83} - 143379 q^{85} + 302154 q^{87} + 55861 q^{89} + 410381 q^{93} + 26279 q^{95} - 135470 q^{97} + 300154 q^{99}+O(q^{100}) $ 5 * q + 13 * q^3 - 31 * q^5 + 230 * q^9 + 351 * q^11 + 54 * q^13 - 607 * q^15 - 111 * q^17 + 1035 * q^19 - 3639 * q^23 + 1540 * q^25 + 3607 * q^27 - 734 * q^29 + 7677 * q^31 + 7439 * q^33 + 13595 * q^37 + 1406 * q^39 - 5310 * q^41 - 764 * q^43 - 38978 * q^45 + 6675 * q^47 - 20975 * q^51 - 30753 * q^53 + 28267 * q^55 - 14389 * q^57 + 87989 * q^59 - 19899 * q^61 + 119470 * q^65 + 33067 * q^67 - 100399 * q^69 + 108720 * q^71 - 141659 * q^73 + 108788 * q^75 - 118919 * q^79 - 143851 * q^81 + 211004 * q^83 - 143379 * q^85 + 302154 * q^87 + 55861 * q^89 + 410381 * q^93 + 26279 * q^95 - 135470 * q^97 + 300154 * 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
$3$	6.22560	0.399372	0.199686	−	0.979860i	$-0.436008\pi$
$3$	6.22560	0.399372	0.199686	+	0.979860i	$0.436008\pi$
$4$	0	0
$5$	−9.40551	−0.168251	−0.0841254	−	0.996455i	$-0.526810\pi$
$5$	−9.40551	−0.168251	−0.0841254	+	0.996455i	$0.526810\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−204.242	−0.840502
$10$	0	0
$11$	−765.115	−1.90654	−0.953269	−	0.302124i	$-0.902304\pi$
$11$	−765.115	−1.90654	−0.953269	+	0.302124i	$0.902304\pi$
$12$	0	0
$13$	−732.172	−1.20159	−0.600793	−	0.799405i	$-0.705148\pi$
$13$	−732.172	−1.20159	−0.600793	+	0.799405i	$0.705148\pi$
$14$	0	0
$15$	−58.5549	−0.0671947
$16$	0	0
$17$	−482.580	−0.404993	−0.202496	−	0.979283i	$-0.564905\pi$
$17$	−482.580	−0.404993	−0.202496	+	0.979283i	$0.564905\pi$
$18$	0	0
$19$	−2617.07	−1.66315	−0.831576	−	0.555411i	$-0.812561\pi$
$19$	−2617.07	−1.66315	−0.831576	+	0.555411i	$0.812561\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	535.506	0.211079	0.105539	−	0.994415i	$-0.466343\pi$
$23$	535.506	0.211079	0.105539	+	0.994415i	$0.466343\pi$
$24$	0	0
$25$	−3036.54	−0.971692
$26$	0	0
$27$	−2784.35	−0.735046
$28$	0	0
$29$	3943.87	0.870819	0.435409	−	0.900233i	$-0.356604\pi$
$29$	3943.87	0.870819	0.435409	+	0.900233i	$0.356604\pi$
$30$	0	0
$31$	3663.45	0.684677	0.342338	−	0.939577i	$-0.388781\pi$
$31$	3663.45	0.684677	0.342338	+	0.939577i	$0.388781\pi$
$32$	0	0
$33$	−4763.30	−0.761418
$34$	0	0
$35$	0	0
$36$	0	0
$37$	12639.0	1.51778	0.758890	−	0.651219i	$-0.225742\pi$
$37$	12639.0	1.51778	0.758890	+	0.651219i	$0.225742\pi$
$38$	0	0
$39$	−4558.21	−0.479880
$40$	0	0
$41$	−4814.47	−0.447290	−0.223645	−	0.974671i	$-0.571796\pi$
$41$	−4814.47	−0.447290	−0.223645	+	0.974671i	$0.571796\pi$
$42$	0	0
$43$	4938.11	0.407277	0.203638	−	0.979046i	$-0.434723\pi$
$43$	4938.11	0.407277	0.203638	+	0.979046i	$0.434723\pi$
$44$	0	0
$45$	1921.00	0.141415
$46$	0	0
$47$	17405.0	1.14929	0.574644	−	0.818404i	$-0.305141\pi$
$47$	17405.0	1.14929	0.574644	+	0.818404i	$0.305141\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−3004.35	−0.161743
$52$	0	0
$53$	−4658.43	−0.227798	−0.113899	−	0.993492i	$-0.536334\pi$
$53$	−4658.43	−0.227798	−0.113899	+	0.993492i	$0.536334\pi$
$54$	0	0
$55$	7196.30	0.320776
$56$	0	0
$57$	−16292.8	−0.664217
$58$	0	0
$59$	32500.8	1.21553	0.607763	−	0.794118i	$-0.292067\pi$
$59$	32500.8	1.21553	0.607763	+	0.794118i	$0.292067\pi$
$60$	0	0
$61$	43496.1	1.49667	0.748335	−	0.663321i	$-0.230854\pi$
$61$	43496.1	1.49667	0.748335	+	0.663321i	$0.230854\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6886.45	0.202168
$66$	0	0
$67$	32082.4	0.873131	0.436565	−	0.899673i	$-0.356195\pi$
$67$	32082.4	0.873131	0.436565	+	0.899673i	$0.356195\pi$
$68$	0	0
$69$	3333.85	0.0842991
$70$	0	0
$71$	−15820.0	−0.372443	−0.186221	−	0.982508i	$-0.559624\pi$
$71$	−15820.0	−0.372443	−0.186221	+	0.982508i	$0.559624\pi$
$72$	0	0
$73$	−36515.4	−0.801991	−0.400996	−	0.916080i	$-0.631336\pi$
$73$	−36515.4	−0.801991	−0.400996	+	0.916080i	$0.631336\pi$
$74$	0	0
$75$	−18904.3	−0.388067
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−25472.1	−0.459196	−0.229598	−	0.973286i	$-0.573741\pi$
$79$	−25472.1	−0.459196	−0.229598	+	0.973286i	$0.573741\pi$
$80$	0	0
$81$	32296.5	0.546945
$82$	0	0
$83$	69443.5	1.10646	0.553231	−	0.833028i	$-0.313395\pi$
$83$	69443.5	1.10646	0.553231	+	0.833028i	$0.313395\pi$
$84$	0	0
$85$	4538.91	0.0681404
$86$	0	0
$87$	24553.0	0.347781
$88$	0	0
$89$	−108331.	−1.44970	−0.724848	−	0.688909i	$-0.758090\pi$
$89$	−108331.	−1.44970	−0.724848	+	0.688909i	$0.758090\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	22807.1	0.273441
$94$	0	0
$95$	24614.9	0.279827
$96$	0	0
$97$	−94569.1	−1.02052	−0.510258	−	0.860021i	$-0.670450\pi$
$97$	−94569.1	−1.02052	−0.510258	+	0.860021i	$0.670450\pi$
$98$	0	0
$99$	156269.	1.60245
$100$	0	0
$101$	−147529.	−1.43904	−0.719522	−	0.694470i	$-0.755639\pi$
$101$	−147529.	−1.43904	−0.719522	+	0.694470i	$0.755639\pi$
$102$	0	0
$103$	123876.	1.15052	0.575259	−	0.817971i	$-0.304901\pi$
$103$	123876.	1.15052	0.575259	+	0.817971i	$0.304901\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−70682.7	−0.596834	−0.298417	−	0.954436i	$-0.596459\pi$
$107$	−70682.7	−0.596834	−0.298417	+	0.954436i	$0.596459\pi$
$108$	0	0
$109$	−144132.	−1.16196	−0.580982	−	0.813916i	$-0.697331\pi$
$109$	−144132.	−1.16196	−0.580982	+	0.813916i	$0.697331\pi$
$110$	0	0
$111$	78685.4	0.606159
$112$	0	0
$113$	−129346.	−0.952919	−0.476459	−	0.879197i	$-0.658080\pi$
$113$	−129346.	−0.952919	−0.476459	+	0.879197i	$0.658080\pi$
$114$	0	0
$115$	−5036.71	−0.0355142
$116$	0	0
$117$	149540.	1.00994
$118$	0	0
$119$	0	0
$120$	0	0
$121$	424351.	2.63488
$122$	0	0
$123$	−29973.0	−0.178635
$124$	0	0
$125$	57952.4	0.331739
$126$	0	0
$127$	48800.6	0.268483	0.134241	−	0.990949i	$-0.457140\pi$
$127$	48800.6	0.268483	0.134241	+	0.990949i	$0.457140\pi$
$128$	0	0
$129$	30742.7	0.162655
$130$	0	0
$131$	46549.1	0.236991	0.118496	−	0.992955i	$-0.462193\pi$
$131$	46549.1	0.236991	0.118496	+	0.992955i	$0.462193\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	26188.2	0.123672
$136$	0	0
$137$	−202786.	−0.923075	−0.461538	−	0.887121i	$-0.652702\pi$
$137$	−202786.	−0.923075	−0.461538	+	0.887121i	$0.652702\pi$
$138$	0	0
$139$	146120.	0.641464	0.320732	−	0.947170i	$-0.396071\pi$
$139$	146120.	0.641464	0.320732	+	0.947170i	$0.396071\pi$
$140$	0	0
$141$	108356.	0.458994
$142$	0	0
$143$	560196.	2.29087
$144$	0	0
$145$	−37094.1	−0.146516
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−165201.	−0.609604	−0.304802	−	0.952416i	$-0.598590\pi$
$149$	−165201.	−0.609604	−0.304802	+	0.952416i	$0.598590\pi$
$150$	0	0
$151$	−117349.	−0.418830	−0.209415	−	0.977827i	$-0.567156\pi$
$151$	−117349.	−0.418830	−0.209415	+	0.977827i	$0.567156\pi$
$152$	0	0
$153$	98563.1	0.340397
$154$	0	0
$155$	−34456.6	−0.115197
$156$	0	0
$157$	−62907.4	−0.203682	−0.101841	−	0.994801i	$-0.532473\pi$
$157$	−62907.4	−0.203682	−0.101841	+	0.994801i	$0.532473\pi$
$158$	0	0
$159$	−29001.5	−0.0909763
$160$	0	0
$161$	0	0
$162$	0	0
$163$	52058.3	0.153469	0.0767346	−	0.997052i	$-0.475551\pi$
$163$	52058.3	0.153469	0.0767346	+	0.997052i	$0.475551\pi$
$164$	0	0
$165$	44801.3	0.128109
$166$	0	0
$167$	572591.	1.58874	0.794371	−	0.607433i	$-0.207801\pi$
$167$	572591.	1.58874	0.794371	+	0.607433i	$0.207801\pi$
$168$	0	0
$169$	164783.	0.443809
$170$	0	0
$171$	534516.	1.39788
$172$	0	0
$173$	169258.	0.429966	0.214983	−	0.976618i	$-0.431030\pi$
$173$	169258.	0.429966	0.214983	+	0.976618i	$0.431030\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	202337.	0.485448
$178$	0	0
$179$	−456336.	−1.06452	−0.532258	−	0.846582i	$-0.678656\pi$
$179$	−456336.	−1.06452	−0.532258	+	0.846582i	$0.678656\pi$
$180$	0	0
$181$	−243024.	−0.551383	−0.275692	−	0.961246i	$-0.588907\pi$
$181$	−243024.	−0.551383	−0.275692	+	0.961246i	$0.588907\pi$
$182$	0	0
$183$	270789.	0.597729
$184$	0	0
$185$	−118876.	−0.255368
$186$	0	0
$187$	369230.	0.772134
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−435574.	−0.863931	−0.431965	−	0.901890i	$-0.642180\pi$
$191$	−435574.	−0.863931	−0.431965	+	0.901890i	$0.642180\pi$
$192$	0	0
$193$	−1.02655e6	−1.98374	−0.991872	−	0.127237i	$-0.959389\pi$
$193$	−1.02655e6	−1.98374	−0.991872	+	0.127237i	$0.959389\pi$
$194$	0	0
$195$	42872.3	0.0807403
$196$	0	0
$197$	−147793.	−0.271325	−0.135662	−	0.990755i	$-0.543316\pi$
$197$	−147793.	−0.271325	−0.135662	+	0.990755i	$0.543316\pi$
$198$	0	0
$199$	−1.00414e6	−1.79748	−0.898739	−	0.438484i	$-0.855516\pi$
$199$	−1.00414e6	−1.79748	−0.898739	+	0.438484i	$0.855516\pi$
$200$	0	0
$201$	199732.	0.348704
$202$	0	0
$203$	0	0
$204$	0	0
$205$	45282.6	0.0752569
$206$	0	0
$207$	−109373.	−0.177412
$208$	0	0
$209$	2.00236e6	3.17086
$210$	0	0
$211$	95769.0	0.148088	0.0740438	−	0.997255i	$-0.476410\pi$
$211$	95769.0	0.148088	0.0740438	+	0.997255i	$0.476410\pi$
$212$	0	0
$213$	−98488.7	−0.148743
$214$	0	0
$215$	−46445.4	−0.0685247
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−227331.	−0.320293
$220$	0	0
$221$	353332.	0.486634
$222$	0	0
$223$	247314.	0.333033	0.166516	−	0.986039i	$-0.446748\pi$
$223$	247314.	0.333033	0.166516	+	0.986039i	$0.446748\pi$
$224$	0	0
$225$	620188.	0.816709
$226$	0	0
$227$	−935573.	−1.20507	−0.602536	−	0.798092i	$-0.705843\pi$
$227$	−935573.	−1.20507	−0.602536	+	0.798092i	$0.705843\pi$
$228$	0	0
$229$	−33820.2	−0.0426175	−0.0213087	−	0.999773i	$-0.506783\pi$
$229$	−33820.2	−0.0426175	−0.0213087	+	0.999773i	$0.506783\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	131568.	0.158767	0.0793835	−	0.996844i	$-0.474705\pi$
$233$	131568.	0.158767	0.0793835	+	0.996844i	$0.474705\pi$
$234$	0	0
$235$	−163703.	−0.193369
$236$	0	0
$237$	−158579.	−0.183390
$238$	0	0
$239$	−16742.1	−0.0189590	−0.00947948	−	0.999955i	$-0.503017\pi$
$239$	−16742.1	−0.0189590	−0.00947948	+	0.999955i	$0.503017\pi$
$240$	0	0
$241$	−73428.8	−0.0814374	−0.0407187	−	0.999171i	$-0.512965\pi$
$241$	−73428.8	−0.0814374	−0.0407187	+	0.999171i	$0.512965\pi$
$242$	0	0
$243$	877662.	0.953480
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.91615e6	1.99842
$248$	0	0
$249$	432327.	0.441890
$250$	0	0
$251$	1.53095e6	1.53383	0.766913	−	0.641751i	$-0.221792\pi$
$251$	1.53095e6	1.53383	0.766913	+	0.641751i	$0.221792\pi$
$252$	0	0
$253$	−409724.	−0.402430
$254$	0	0
$255$	28257.5	0.0272134
$256$	0	0
$257$	1.69062e6	1.59667	0.798333	−	0.602217i	$-0.205716\pi$
$257$	1.69062e6	1.59667	0.798333	+	0.602217i	$0.205716\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−805504.	−0.731925
$262$	0	0
$263$	−984916.	−0.878031	−0.439015	−	0.898480i	$-0.644673\pi$
$263$	−984916.	−0.878031	−0.439015	+	0.898480i	$0.644673\pi$
$264$	0	0
$265$	43814.9	0.0383272
$266$	0	0
$267$	−674424.	−0.578969
$268$	0	0
$269$	1.83701e6	1.54786	0.773929	−	0.633272i	$-0.218289\pi$
$269$	1.83701e6	1.54786	0.773929	+	0.633272i	$0.218289\pi$
$270$	0	0
$271$	489849.	0.405171	0.202586	−	0.979265i	$-0.435066\pi$
$271$	489849.	0.405171	0.202586	+	0.979265i	$0.435066\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.32330e6	1.85257
$276$	0	0
$277$	−933561.	−0.731044	−0.365522	−	0.930803i	$-0.619109\pi$
$277$	−933561.	−0.731044	−0.365522	+	0.930803i	$0.619109\pi$
$278$	0	0
$279$	−748229.	−0.575472
$280$	0	0
$281$	497412.	0.375794	0.187897	−	0.982189i	$-0.439833\pi$
$281$	497412.	0.375794	0.187897	+	0.982189i	$0.439833\pi$
$282$	0	0
$283$	1.41003e6	1.04656	0.523278	−	0.852162i	$-0.324709\pi$
$283$	1.41003e6	1.04656	0.523278	+	0.852162i	$0.324709\pi$
$284$	0	0
$285$	153242.	0.111755
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.18697e6	−0.835981
$290$	0	0
$291$	−588749.	−0.407566
$292$	0	0
$293$	2.71744e6	1.84923	0.924614	−	0.380905i	$-0.124387\pi$
$293$	2.71744e6	1.84923	0.924614	+	0.380905i	$0.124387\pi$
$294$	0	0
$295$	−305687.	−0.204513
$296$	0	0
$297$	2.13035e6	1.40139
$298$	0	0
$299$	−392083.	−0.253630
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−918456.	−0.574714
$304$	0	0
$305$	−409103.	−0.251816
$306$	0	0
$307$	−1.59907e6	−0.968324	−0.484162	−	0.874978i	$-0.660875\pi$
$307$	−1.59907e6	−0.968324	−0.484162	+	0.874978i	$0.660875\pi$
$308$	0	0
$309$	771202.	0.459485
$310$	0	0
$311$	−2.16973e6	−1.27205	−0.636027	−	0.771667i	$-0.719423\pi$
$311$	−2.16973e6	−1.27205	−0.636027	+	0.771667i	$0.719423\pi$
$312$	0	0
$313$	715556.	0.412841	0.206421	−	0.978463i	$-0.433818\pi$
$313$	715556.	0.412841	0.206421	+	0.978463i	$0.433818\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.04482e6	1.14290	0.571448	−	0.820638i	$-0.306382\pi$
$317$	2.04482e6	1.14290	0.571448	+	0.820638i	$0.306382\pi$
$318$	0	0
$319$	−3.01752e6	−1.66025
$320$	0	0
$321$	−440042.	−0.238359
$322$	0	0
$323$	1.26295e6	0.673565
$324$	0	0
$325$	2.22327e6	1.16757
$326$	0	0
$327$	−897305.	−0.464056
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−402485.	−0.201920	−0.100960	−	0.994890i	$-0.532191\pi$
$331$	−402485.	−0.201920	−0.100960	+	0.994890i	$0.532191\pi$
$332$	0	0
$333$	−2.58142e6	−1.27570
$334$	0	0
$335$	−301751.	−0.146905
$336$	0	0
$337$	3.01447e6	1.44589	0.722946	−	0.690905i	$-0.242788\pi$
$337$	3.01447e6	1.44589	0.722946	+	0.690905i	$0.242788\pi$
$338$	0	0
$339$	−805254.	−0.380569
$340$	0	0
$341$	−2.80296e6	−1.30536
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−31356.5	−0.0141834
$346$	0	0
$347$	2.32057e6	1.03460	0.517298	−	0.855806i	$-0.326938\pi$
$347$	2.32057e6	1.03460	0.517298	+	0.855806i	$0.326938\pi$
$348$	0	0
$349$	2.81397e6	1.23668	0.618338	−	0.785912i	$-0.287806\pi$
$349$	2.81397e6	1.23668	0.618338	+	0.785912i	$0.287806\pi$
$350$	0	0
$351$	2.03862e6	0.883221
$352$	0	0
$353$	−2.16741e6	−0.925772	−0.462886	−	0.886418i	$-0.653186\pi$
$353$	−2.16741e6	−0.925772	−0.462886	+	0.886418i	$0.653186\pi$
$354$	0	0
$355$	148795.	0.0626638
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.50959e6	1.43721	0.718605	−	0.695418i	$-0.244781\pi$
$359$	3.50959e6	1.43721	0.718605	+	0.695418i	$0.244781\pi$
$360$	0	0
$361$	4.37297e6	1.76607
$362$	0	0
$363$	2.64184e6	1.05230
$364$	0	0
$365$	343446.	0.134936
$366$	0	0
$367$	1.62655e6	0.630381	0.315190	−	0.949028i	$-0.397932\pi$
$367$	1.62655e6	0.630381	0.315190	+	0.949028i	$0.397932\pi$
$368$	0	0
$369$	983318.	0.375948
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−861811.	−0.320730	−0.160365	−	0.987058i	$-0.551267\pi$
$373$	−861811.	−0.320730	−0.160365	+	0.987058i	$0.551267\pi$
$374$	0	0
$375$	360788.	0.132487
$376$	0	0
$377$	−2.88759e6	−1.04636
$378$	0	0
$379$	652944.	0.233495	0.116748	−	0.993162i	$-0.462753\pi$
$379$	652944.	0.233495	0.116748	+	0.993162i	$0.462753\pi$
$380$	0	0
$381$	303813.	0.107225
$382$	0	0
$383$	2.34109e6	0.815496	0.407748	−	0.913094i	$-0.366314\pi$
$383$	2.34109e6	0.815496	0.407748	+	0.913094i	$0.366314\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.00857e6	−0.342317
$388$	0	0
$389$	−651295.	−0.218225	−0.109112	−	0.994029i	$-0.534801\pi$
$389$	−651295.	−0.218225	−0.109112	+	0.994029i	$0.534801\pi$
$390$	0	0
$391$	−258425.	−0.0854855
$392$	0	0
$393$	289796.	0.0946478
$394$	0	0
$395$	239578.	0.0772600
$396$	0	0
$397$	4.16665e6	1.32682	0.663408	−	0.748258i	$-0.269109\pi$
$397$	4.16665e6	1.32682	0.663408	+	0.748258i	$0.269109\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−482539.	−0.149855	−0.0749276	−	0.997189i	$-0.523873\pi$
$401$	−482539.	−0.149855	−0.0749276	+	0.997189i	$0.523873\pi$
$402$	0	0
$403$	−2.68227e6	−0.822698
$404$	0	0
$405$	−303765.	−0.0920239
$406$	0	0
$407$	−9.67030e6	−2.89370
$408$	0	0
$409$	−3.58360e6	−1.05928	−0.529640	−	0.848222i	$-0.677673\pi$
$409$	−3.58360e6	−1.05928	−0.529640	+	0.848222i	$0.677673\pi$
$410$	0	0
$411$	−1.26247e6	−0.368651
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−653151.	−0.186163
$416$	0	0
$417$	909684.	0.256183
$418$	0	0
$419$	−2.66040e6	−0.740307	−0.370154	−	0.928971i	$-0.620695\pi$
$419$	−2.66040e6	−0.740307	−0.370154	+	0.928971i	$0.620695\pi$
$420$	0	0
$421$	−4.78294e6	−1.31519	−0.657597	−	0.753370i	$-0.728427\pi$
$421$	−4.78294e6	−1.31519	−0.657597	+	0.753370i	$0.728427\pi$
$422$	0	0
$423$	−3.55483e6	−0.965978
$424$	0	0
$425$	1.46537e6	0.393528
$426$	0	0
$427$	0	0
$428$	0	0
$429$	3.48756e6	0.914910
$430$	0	0
$431$	5.70462e6	1.47922	0.739611	−	0.673034i	$-0.235009\pi$
$431$	5.70462e6	1.47922	0.739611	+	0.673034i	$0.235009\pi$
$432$	0	0
$433$	282700.	0.0724614	0.0362307	−	0.999343i	$-0.488465\pi$
$433$	282700.	0.0724614	0.0362307	+	0.999343i	$0.488465\pi$
$434$	0	0
$435$	−230933.	−0.0585144
$436$	0	0
$437$	−1.40146e6	−0.351056
$438$	0	0
$439$	−207276.	−0.0513319	−0.0256660	−	0.999671i	$-0.508171\pi$
$439$	−207276.	−0.0513319	−0.0256660	+	0.999671i	$0.508171\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.42321e6	−0.828751	−0.414376	−	0.910106i	$-0.636000\pi$
$443$	−3.42321e6	−0.828751	−0.414376	+	0.910106i	$0.636000\pi$
$444$	0	0
$445$	1.01891e6	0.243913
$446$	0	0
$447$	−1.02848e6	−0.243459
$448$	0	0
$449$	−6.36991e6	−1.49114	−0.745568	−	0.666430i	$-0.767822\pi$
$449$	−6.36991e6	−1.49114	−0.745568	+	0.666430i	$0.767822\pi$
$450$	0	0
$451$	3.68363e6	0.852775
$452$	0	0
$453$	−730569.	−0.167269
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.64168e6	−0.815665	−0.407832	−	0.913057i	$-0.633715\pi$
$457$	−3.64168e6	−0.815665	−0.407832	+	0.913057i	$0.633715\pi$
$458$	0	0
$459$	1.34367e6	0.297688
$460$	0	0
$461$	−3.65333e6	−0.800639	−0.400320	−	0.916376i	$-0.631101\pi$
$461$	−3.65333e6	−0.800639	−0.400320	+	0.916376i	$0.631101\pi$
$462$	0	0
$463$	8.28192e6	1.79547	0.897736	−	0.440533i	$-0.145211\pi$
$463$	8.28192e6	1.79547	0.897736	+	0.440533i	$0.145211\pi$
$464$	0	0
$465$	−214513.	−0.0460067
$466$	0	0
$467$	1.81432e6	0.384966	0.192483	−	0.981300i	$-0.438346\pi$
$467$	1.81432e6	0.384966	0.192483	+	0.981300i	$0.438346\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−391637.	−0.0813450
$472$	0	0
$473$	−3.77822e6	−0.776488
$474$	0	0
$475$	7.94684e6	1.61607
$476$	0	0
$477$	951447.	0.191465
$478$	0	0
$479$	3.24966e6	0.647142	0.323571	−	0.946204i	$-0.395116\pi$
$479$	3.24966e6	0.647142	0.323571	+	0.946204i	$0.395116\pi$
$480$	0	0
$481$	−9.25393e6	−1.82374
$482$	0	0
$483$	0	0
$484$	0	0
$485$	889470.	0.171703
$486$	0	0
$487$	2.70120e6	0.516101	0.258051	−	0.966131i	$-0.416920\pi$
$487$	2.70120e6	0.516101	0.258051	+	0.966131i	$0.416920\pi$
$488$	0	0
$489$	324094.	0.0612914
$490$	0	0
$491$	3.57623e6	0.669456	0.334728	−	0.942315i	$-0.391356\pi$
$491$	3.57623e6	0.669456	0.334728	+	0.942315i	$0.391356\pi$
$492$	0	0
$493$	−1.90324e6	−0.352675
$494$	0	0
$495$	−1.46979e6	−0.269613
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−9.88171e6	−1.77656	−0.888282	−	0.459299i	$-0.848101\pi$
$499$	−9.88171e6	−1.77656	−0.888282	+	0.459299i	$0.848101\pi$
$500$	0	0
$501$	3.56472e6	0.634500
$502$	0	0
$503$	−1.45199e6	−0.255884	−0.127942	−	0.991782i	$-0.540837\pi$
$503$	−1.45199e6	−0.255884	−0.127942	+	0.991782i	$0.540837\pi$
$504$	0	0
$505$	1.38759e6	0.242120
$506$	0	0
$507$	1.02588e6	0.177245
$508$	0	0
$509$	−7.30370e6	−1.24953	−0.624767	−	0.780811i	$-0.714806\pi$
$509$	−7.30370e6	−1.24953	−0.624767	+	0.780811i	$0.714806\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	7.28684e6	1.22249
$514$	0	0
$515$	−1.16512e6	−0.193576
$516$	0	0
$517$	−1.33168e7	−2.19116
$518$	0	0
$519$	1.05373e6	0.171716
$520$	0	0
$521$	−1.95720e6	−0.315894	−0.157947	−	0.987448i	$-0.550488\pi$
$521$	−1.95720e6	−0.315894	−0.157947	+	0.987448i	$0.550488\pi$
$522$	0	0
$523$	−356141.	−0.0569335	−0.0284667	−	0.999595i	$-0.509062\pi$
$523$	−356141.	−0.0569335	−0.0284667	+	0.999595i	$0.509062\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.76791e6	−0.277289
$528$	0	0
$529$	−6.14958e6	−0.955446
$530$	0	0
$531$	−6.63803e6	−1.02165
$532$	0	0
$533$	3.52503e6	0.537458
$534$	0	0
$535$	664807.	0.100418
$536$	0	0
$537$	−2.84097e6	−0.425138
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−9.79833e6	−1.43933	−0.719663	−	0.694324i	$-0.755704\pi$
$541$	−9.79833e6	−1.43933	−0.719663	+	0.694324i	$0.755704\pi$
$542$	0	0
$543$	−1.51297e6	−0.220207
$544$	0	0
$545$	1.35563e6	0.195501
$546$	0	0
$547$	1.13688e6	0.162460	0.0812301	−	0.996695i	$-0.474115\pi$
$547$	1.13688e6	0.162460	0.0812301	+	0.996695i	$0.474115\pi$
$548$	0	0
$549$	−8.88373e6	−1.25795
$550$	0	0
$551$	−1.03214e7	−1.44830
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−740076.	−0.101987
$556$	0	0
$557$	2.50975e6	0.342762	0.171381	−	0.985205i	$-0.445177\pi$
$557$	2.50975e6	0.342762	0.171381	+	0.985205i	$0.445177\pi$
$558$	0	0
$559$	−3.61555e6	−0.489378
$560$	0	0
$561$	2.29868e6	0.308369
$562$	0	0
$563$	8.65696e6	1.15105	0.575526	−	0.817784i	$-0.304798\pi$
$563$	8.65696e6	1.15105	0.575526	+	0.817784i	$0.304798\pi$
$564$	0	0
$565$	1.21656e6	0.160329
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.24657e6	0.938322	0.469161	−	0.883113i	$-0.344556\pi$
$569$	7.24657e6	0.938322	0.469161	+	0.883113i	$0.344556\pi$
$570$	0	0
$571$	8.01486e6	1.02874	0.514370	−	0.857568i	$-0.328026\pi$
$571$	8.01486e6	1.02874	0.514370	+	0.857568i	$0.328026\pi$
$572$	0	0
$573$	−2.71171e6	−0.345030
$574$	0	0
$575$	−1.62608e6	−0.205104
$576$	0	0
$577$	4.51956e6	0.565141	0.282571	−	0.959247i	$-0.408813\pi$
$577$	4.51956e6	0.565141	0.282571	+	0.959247i	$0.408813\pi$
$578$	0	0
$579$	−6.39087e6	−0.792253
$580$	0	0
$581$	0	0
$582$	0	0
$583$	3.56424e6	0.434305
$584$	0	0
$585$	−1.40650e6	−0.169922
$586$	0	0
$587$	2.07729e6	0.248830	0.124415	−	0.992230i	$-0.460295\pi$
$587$	2.07729e6	0.248830	0.124415	+	0.992230i	$0.460295\pi$
$588$	0	0
$589$	−9.58750e6	−1.13872
$590$	0	0
$591$	−920102.	−0.108360
$592$	0	0
$593$	1.26647e7	1.47897	0.739484	−	0.673174i	$-0.235070\pi$
$593$	1.26647e7	1.47897	0.739484	+	0.673174i	$0.235070\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.25140e6	−0.717863
$598$	0	0
$599$	−3.39852e6	−0.387011	−0.193505	−	0.981099i	$-0.561986\pi$
$599$	−3.39852e6	−0.387011	−0.193505	+	0.981099i	$0.561986\pi$
$600$	0	0
$601$	−8.79425e6	−0.993145	−0.496572	−	0.867995i	$-0.665408\pi$
$601$	−8.79425e6	−0.993145	−0.496572	+	0.867995i	$0.665408\pi$
$602$	0	0
$603$	−6.55256e6	−0.733868
$604$	0	0
$605$	−3.99123e6	−0.443321
$606$	0	0
$607$	6.14855e6	0.677331	0.338666	−	0.940907i	$-0.390024\pi$
$607$	6.14855e6	0.677331	0.338666	+	0.940907i	$0.390024\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.27434e7	−1.38097
$612$	0	0
$613$	1.42570e7	1.53242	0.766210	−	0.642590i	$-0.222140\pi$
$613$	1.42570e7	1.53242	0.766210	+	0.642590i	$0.222140\pi$
$614$	0	0
$615$	281911.	0.0300555
$616$	0	0
$617$	−1.26910e7	−1.34210	−0.671048	−	0.741414i	$-0.734156\pi$
$617$	−1.26910e7	−1.34210	−0.671048	+	0.741414i	$0.734156\pi$
$618$	0	0
$619$	4.80796e6	0.504352	0.252176	−	0.967681i	$-0.418854\pi$
$619$	4.80796e6	0.504352	0.252176	+	0.967681i	$0.418854\pi$
$620$	0	0
$621$	−1.49104e6	−0.155153
$622$	0	0
$623$	0	0
$624$	0	0
$625$	8.94410e6	0.915876
$626$	0	0
$627$	1.24659e7	1.26635
$628$	0	0
$629$	−6.09934e6	−0.614690
$630$	0	0
$631$	−1.89817e7	−1.89785	−0.948925	−	0.315501i	$-0.897827\pi$
$631$	−1.89817e7	−1.89785	−0.948925	+	0.315501i	$0.897827\pi$
$632$	0	0
$633$	596219.	0.0591421
$634$	0	0
$635$	−458995.	−0.0451724
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3.23110e6	0.313039
$640$	0	0
$641$	1.10255e7	1.05987	0.529935	−	0.848038i	$-0.322216\pi$
$641$	1.10255e7	1.05987	0.529935	+	0.848038i	$0.322216\pi$
$642$	0	0
$643$	4.19154e6	0.399803	0.199901	−	0.979816i	$-0.435938\pi$
$643$	4.19154e6	0.399803	0.199901	+	0.979816i	$0.435938\pi$
$644$	0	0
$645$	−289151.	−0.0273669
$646$	0	0
$647$	−1.59488e7	−1.49785	−0.748924	−	0.662655i	$-0.769429\pi$
$647$	−1.59488e7	−1.49785	−0.748924	+	0.662655i	$0.769429\pi$
$648$	0	0
$649$	−2.48669e7	−2.31745
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.80768e6	−0.900084	−0.450042	−	0.893007i	$-0.648591\pi$
$653$	−9.80768e6	−0.900084	−0.450042	+	0.893007i	$0.648591\pi$
$654$	0	0
$655$	−437817.	−0.0398740
$656$	0	0
$657$	7.45799e6	0.674075
$658$	0	0
$659$	1.12535e7	1.00943	0.504714	−	0.863287i	$-0.331598\pi$
$659$	1.12535e7	1.00943	0.504714	+	0.863287i	$0.331598\pi$
$660$	0	0
$661$	−3.76049e6	−0.334765	−0.167383	−	0.985892i	$-0.553532\pi$
$661$	−3.76049e6	−0.334765	−0.167383	+	0.985892i	$0.553532\pi$
$662$	0	0
$663$	2.19970e6	0.194348
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.11197e6	0.183811
$668$	0	0
$669$	1.53968e6	0.133004
$670$	0	0
$671$	−3.32796e7	−2.85346
$672$	0	0
$673$	4.12878e6	0.351386	0.175693	−	0.984445i	$-0.443783\pi$
$673$	4.12878e6	0.351386	0.175693	+	0.984445i	$0.443783\pi$
$674$	0	0
$675$	8.45478e6	0.714238
$676$	0	0
$677$	7.57640e6	0.635318	0.317659	−	0.948205i	$-0.397103\pi$
$677$	7.57640e6	0.635318	0.317659	+	0.948205i	$0.397103\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−5.82450e6	−0.481273
$682$	0	0
$683$	1.49945e7	1.22993	0.614964	−	0.788555i	$-0.289171\pi$
$683$	1.49945e7	1.22993	0.614964	+	0.788555i	$0.289171\pi$
$684$	0	0
$685$	1.90731e6	0.155308
$686$	0	0
$687$	−210551.	−0.0170202
$688$	0	0
$689$	3.41077e6	0.273719
$690$	0	0
$691$	1.10298e7	0.878764	0.439382	−	0.898300i	$-0.355197\pi$
$691$	1.10298e7	0.878764	0.439382	+	0.898300i	$0.355197\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.37433e6	−0.107927
$696$	0	0
$697$	2.32337e6	0.181149
$698$	0	0
$699$	819089.	0.0634072
$700$	0	0
$701$	1.77997e7	1.36810	0.684048	−	0.729437i	$-0.260218\pi$
$701$	1.77997e7	1.36810	0.684048	+	0.729437i	$0.260218\pi$
$702$	0	0
$703$	−3.30772e7	−2.52430
$704$	0	0
$705$	−1.01915e6	−0.0772261
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.22675e7	0.916518	0.458259	−	0.888819i	$-0.348473\pi$
$709$	1.22675e7	0.916518	0.458259	+	0.888819i	$0.348473\pi$
$710$	0	0
$711$	5.20248e6	0.385955
$712$	0	0
$713$	1.96180e6	0.144521
$714$	0	0
$715$	−5.26893e6	−0.385441
$716$	0	0
$717$	−104229.	−0.00757169
$718$	0	0
$719$	−1.66977e7	−1.20458	−0.602289	−	0.798278i	$-0.705744\pi$
$719$	−1.66977e7	−1.20458	−0.602289	+	0.798278i	$0.705744\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−457138.	−0.0325238
$724$	0	0
$725$	−1.19757e7	−0.846167
$726$	0	0
$727$	1.75983e6	0.123491	0.0617455	−	0.998092i	$-0.480333\pi$
$727$	1.75983e6	0.123491	0.0617455	+	0.998092i	$0.480333\pi$
$728$	0	0
$729$	−2.38409e6	−0.166151
$730$	0	0
$731$	−2.38304e6	−0.164944
$732$	0	0
$733$	9.70937e6	0.667469	0.333734	−	0.942667i	$-0.391691\pi$
$733$	9.70937e6	0.667469	0.333734	+	0.942667i	$0.391691\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.45467e7	−1.66466
$738$	0	0
$739$	−1.34819e7	−0.908117	−0.454058	−	0.890972i	$-0.650024\pi$
$739$	−1.34819e7	−0.908117	−0.454058	+	0.890972i	$0.650024\pi$
$740$	0	0
$741$	1.19292e7	0.798114
$742$	0	0
$743$	4.67927e6	0.310961	0.155481	−	0.987839i	$-0.450307\pi$
$743$	4.67927e6	0.310961	0.155481	+	0.987839i	$0.450307\pi$
$744$	0	0
$745$	1.55380e6	0.102566
$746$	0	0
$747$	−1.41833e7	−0.929982
$748$	0	0
$749$	0	0
$750$	0	0
$751$	2.26729e7	1.46692	0.733460	−	0.679732i	$-0.237904\pi$
$751$	2.26729e7	1.46692	0.733460	+	0.679732i	$0.237904\pi$
$752$	0	0
$753$	9.53107e6	0.612568
$754$	0	0
$755$	1.10373e6	0.0704684
$756$	0	0
$757$	1.81102e6	0.114864	0.0574320	−	0.998349i	$-0.481709\pi$
$757$	1.81102e6	0.114864	0.0574320	+	0.998349i	$0.481709\pi$
$758$	0	0
$759$	−2.55078e6	−0.160719
$760$	0	0
$761$	1.13955e7	0.713299	0.356650	−	0.934238i	$-0.383919\pi$
$761$	1.13955e7	0.713299	0.356650	+	0.934238i	$0.383919\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−927036.	−0.0572721
$766$	0	0
$767$	−2.37962e7	−1.46056
$768$	0	0
$769$	−3.12982e7	−1.90855	−0.954274	−	0.298933i	$-0.903369\pi$
$769$	−3.12982e7	−1.90855	−0.954274	+	0.298933i	$0.903369\pi$
$770$	0	0
$771$	1.05251e7	0.637664
$772$	0	0
$773$	2.55299e7	1.53674	0.768369	−	0.640008i	$-0.221069\pi$
$773$	2.55299e7	1.53674	0.768369	+	0.640008i	$0.221069\pi$
$774$	0	0
$775$	−1.11242e7	−0.665295
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.25998e7	0.743911
$780$	0	0
$781$	1.21041e7	0.710076
$782$	0	0
$783$	−1.09811e7	−0.640091
$784$	0	0
$785$	591676.	0.0342697
$786$	0	0
$787$	1.57573e7	0.906872	0.453436	−	0.891289i	$-0.350198\pi$
$787$	1.57573e7	0.906872	0.453436	+	0.891289i	$0.350198\pi$
$788$	0	0
$789$	−6.13169e6	−0.350661
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−3.18467e7	−1.79838
$794$	0	0
$795$	272774.	0.0153068
$796$	0	0
$797$	1.28373e7	0.715858	0.357929	−	0.933749i	$-0.383483\pi$
$797$	1.28373e7	0.715858	0.357929	+	0.933749i	$0.383483\pi$
$798$	0	0
$799$	−8.39930e6	−0.465453
$800$	0	0
$801$	2.21257e7	1.21847
$802$	0	0
$803$	2.79385e7	1.52903
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.14365e7	0.618172
$808$	0	0
$809$	1.16570e6	0.0626201	0.0313101	−	0.999510i	$-0.490032\pi$
$809$	1.16570e6	0.0626201	0.0313101	+	0.999510i	$0.490032\pi$
$810$	0	0
$811$	4.99526e6	0.266690	0.133345	−	0.991070i	$-0.457428\pi$
$811$	4.99526e6	0.266690	0.133345	+	0.991070i	$0.457428\pi$
$812$	0	0
$813$	3.04960e6	0.161814
$814$	0	0
$815$	−489635.	−0.0258213
$816$	0	0
$817$	−1.29234e7	−0.677363
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.80874e7	0.936523	0.468261	−	0.883590i	$-0.344881\pi$
$821$	1.80874e7	0.936523	0.468261	+	0.883590i	$0.344881\pi$
$822$	0	0
$823$	−1.20946e7	−0.622432	−0.311216	−	0.950339i	$-0.600736\pi$
$823$	−1.20946e7	−0.622432	−0.311216	+	0.950339i	$0.600736\pi$
$824$	0	0
$825$	1.44639e7	0.739864
$826$	0	0
$827$	−1.22010e7	−0.620345	−0.310172	−	0.950680i	$-0.600387\pi$
$827$	−1.22010e7	−0.620345	−0.310172	+	0.950680i	$0.600387\pi$
$828$	0	0
$829$	2.12155e7	1.07218	0.536088	−	0.844162i	$-0.319902\pi$
$829$	2.12155e7	1.07218	0.536088	+	0.844162i	$0.319902\pi$
$830$	0	0
$831$	−5.81198e6	−0.291959
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−5.38551e6	−0.267307
$836$	0	0
$837$	−1.02003e7	−0.503269
$838$	0	0
$839$	−4.68317e6	−0.229686	−0.114843	−	0.993384i	$-0.536637\pi$
$839$	−4.68317e6	−0.229686	−0.114843	+	0.993384i	$0.536637\pi$
$840$	0	0
$841$	−4.95703e6	−0.241675
$842$	0	0
$843$	3.09669e6	0.150082
$844$	0	0
$845$	−1.54987e6	−0.0746713
$846$	0	0
$847$	0	0
$848$	0	0
$849$	8.77829e6	0.417966
$850$	0	0
$851$	6.76827e6	0.320371
$852$	0	0
$853$	1.03858e7	0.488729	0.244364	−	0.969684i	$-0.421421\pi$
$853$	1.03858e7	0.488729	0.244364	+	0.969684i	$0.421421\pi$
$854$	0	0
$855$	−5.02739e6	−0.235195
$856$	0	0
$857$	9.43458e6	0.438804	0.219402	−	0.975635i	$-0.429589\pi$
$857$	9.43458e6	0.438804	0.219402	+	0.975635i	$0.429589\pi$
$858$	0	0
$859$	−7.08429e6	−0.327577	−0.163789	−	0.986495i	$-0.552371\pi$
$859$	−7.08429e6	−0.327577	−0.163789	+	0.986495i	$0.552371\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.10615e6	0.416205	0.208103	−	0.978107i	$-0.433271\pi$
$863$	9.10615e6	0.416205	0.208103	+	0.978107i	$0.433271\pi$
$864$	0	0
$865$	−1.59196e6	−0.0723421
$866$	0	0
$867$	−7.38962e6	−0.333868
$868$	0	0
$869$	1.94891e7	0.875474
$870$	0	0
$871$	−2.34898e7	−1.04914
$872$	0	0
$873$	1.93150e7	0.857746
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.84738e7	1.25010	0.625052	−	0.780583i	$-0.285078\pi$
$877$	2.84738e7	1.25010	0.625052	+	0.780583i	$0.285078\pi$
$878$	0	0
$879$	1.69177e7	0.738531
$880$	0	0
$881$	5.51673e6	0.239465	0.119732	−	0.992806i	$-0.461796\pi$
$881$	5.51673e6	0.239465	0.119732	+	0.992806i	$0.461796\pi$
$882$	0	0
$883$	1.82256e7	0.786647	0.393323	−	0.919400i	$-0.371325\pi$
$883$	1.82256e7	0.786647	0.393323	+	0.919400i	$0.371325\pi$
$884$	0	0
$885$	−1.90308e6	−0.0816770
$886$	0	0
$887$	3.74782e7	1.59944	0.799722	−	0.600370i	$-0.204980\pi$
$887$	3.74782e7	1.59944	0.799722	+	0.600370i	$0.204980\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−2.47106e7	−1.04277
$892$	0	0
$893$	−4.55501e7	−1.91144
$894$	0	0
$895$	4.29207e6	0.179106
$896$	0	0
$897$	−2.44095e6	−0.101293
$898$	0	0
$899$	1.44482e7	0.596229
$900$	0	0
$901$	2.24807e6	0.0922566
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.28577e6	0.0927707
$906$	0	0
$907$	−2.93095e7	−1.18302	−0.591508	−	0.806299i	$-0.701467\pi$
$907$	−2.93095e7	−1.18302	−0.591508	+	0.806299i	$0.701467\pi$
$908$	0	0
$909$	3.01316e7	1.20952
$910$	0	0
$911$	−2.86518e7	−1.14382	−0.571908	−	0.820318i	$-0.693797\pi$
$911$	−2.86518e7	−1.14382	−0.571908	+	0.820318i	$0.693797\pi$
$912$	0	0
$913$	−5.31323e7	−2.10951
$914$	0	0
$915$	−2.54691e6	−0.100568
$916$	0	0
$917$	0	0
$918$	0	0
$919$	4.42153e7	1.72697	0.863483	−	0.504378i	$-0.168278\pi$
$919$	4.42153e7	1.72697	0.863483	+	0.504378i	$0.168278\pi$
$920$	0	0
$921$	−9.95515e6	−0.386722
$922$	0	0
$923$	1.15829e7	0.447522
$924$	0	0
$925$	−3.83788e7	−1.47481
$926$	0	0
$927$	−2.53006e7	−0.967013
$928$	0	0
$929$	3.06558e7	1.16540	0.582698	−	0.812689i	$-0.301997\pi$
$929$	3.06558e7	1.16540	0.582698	+	0.812689i	$0.301997\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1.35079e7	−0.508023
$934$	0	0
$935$	−3.47279e6	−0.129912
$936$	0	0
$937$	1.26278e6	0.0469871	0.0234935	−	0.999724i	$-0.492521\pi$
$937$	1.26278e6	0.0469871	0.0234935	+	0.999724i	$0.492521\pi$
$938$	0	0
$939$	4.45477e6	0.164877
$940$	0	0
$941$	−7.51200e6	−0.276555	−0.138278	−	0.990394i	$-0.544157\pi$
$941$	−7.51200e6	−0.276555	−0.138278	+	0.990394i	$0.544157\pi$
$942$	0	0
$943$	−2.57818e6	−0.0944135
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.77330e7	−1.36724	−0.683622	−	0.729836i	$-0.739597\pi$
$947$	−3.77330e7	−1.36724	−0.683622	+	0.729836i	$0.739597\pi$
$948$	0	0
$949$	2.67356e7	0.963661
$950$	0	0
$951$	1.27302e7	0.456441
$952$	0	0
$953$	3.21132e7	1.14538	0.572692	−	0.819771i	$-0.305899\pi$
$953$	3.21132e7	1.14538	0.572692	+	0.819771i	$0.305899\pi$
$954$	0	0
$955$	4.09680e6	0.145357
$956$	0	0
$957$	−1.87859e7	−0.663057
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.52083e7	−0.531218
$962$	0	0
$963$	1.44364e7	0.501640
$964$	0	0
$965$	9.65520e6	0.333767
$966$	0	0
$967$	4.33133e7	1.48955	0.744775	−	0.667315i	$-0.232557\pi$
$967$	4.33133e7	1.48955	0.744775	+	0.667315i	$0.232557\pi$
$968$	0	0
$969$	7.86261e6	0.269003
$970$	0	0
$971$	7.34716e6	0.250076	0.125038	−	0.992152i	$-0.460095\pi$
$971$	7.34716e6	0.250076	0.125038	+	0.992152i	$0.460095\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.38412e7	0.466296
$976$	0	0
$977$	3.14544e7	1.05425	0.527126	−	0.849787i	$-0.323270\pi$
$977$	3.14544e7	1.05425	0.527126	+	0.849787i	$0.323270\pi$
$978$	0	0
$979$	8.28856e7	2.76390
$980$	0	0
$981$	2.94377e7	0.976633
$982$	0	0
$983$	1.09787e7	0.362383	0.181191	−	0.983448i	$-0.442005\pi$
$983$	1.09787e7	0.362383	0.181191	+	0.983448i	$0.442005\pi$
$984$	0	0
$985$	1.39007e6	0.0456506
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.64439e6	0.0859675
$990$	0	0
$991$	−5.46612e7	−1.76805	−0.884026	−	0.467438i	$-0.845177\pi$
$991$	−5.46612e7	−1.76805	−0.884026	+	0.467438i	$0.845177\pi$
$992$	0	0
$993$	−2.50571e6	−0.0806413
$994$	0	0
$995$	9.44449e6	0.302427
$996$	0	0
$997$	−5.57493e7	−1.77624	−0.888120	−	0.459612i	$-0.847988\pi$
$997$	−5.57493e7	−1.77624	−0.888120	+	0.459612i	$0.847988\pi$
$998$	0	0
$999$	−3.51914e7	−1.11564

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.bm.1.3		5
4.3	odd	2		392.6.a.i.1.3		5
7.3	odd	6		112.6.i.g.65.3		10
7.5	odd	6		112.6.i.g.81.3		10
7.6	odd	2		784.6.a.bj.1.3		5
28.3	even	6		56.6.i.a.9.3	✓	10
28.11	odd	6		392.6.i.p.177.3		10
28.19	even	6		56.6.i.a.25.3	yes	10
28.23	odd	6		392.6.i.p.361.3		10
28.27	even	2		392.6.a.l.1.3		5
84.47	odd	6		504.6.s.d.361.3		10
84.59	odd	6		504.6.s.d.289.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.6.i.a.9.3	✓	10	28.3	even	6
56.6.i.a.25.3	yes	10	28.19	even	6
112.6.i.g.65.3		10	7.3	odd	6
112.6.i.g.81.3		10	7.5	odd	6
392.6.a.i.1.3		5	4.3	odd	2
392.6.a.l.1.3		5	28.27	even	2
392.6.i.p.177.3		10	28.11	odd	6
392.6.i.p.361.3		10	28.23	odd	6
504.6.s.d.289.3		10	84.59	odd	6
504.6.s.d.361.3		10	84.47	odd	6
784.6.a.bj.1.3		5	7.6	odd	2
784.6.a.bm.1.3		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q+6.22560 q^{3} -9.40551 q^{5} -204.242 q^{9} +O(q^{10})\)	\(q+6.22560 q^{3} -9.40551 q^{5} -204.242 q^{9} -765.115 q^{11} -732.172 q^{13} -58.5549 q^{15} -482.580 q^{17} -2617.07 q^{19} +535.506 q^{23} -3036.54 q^{25} -2784.35 q^{27} +3943.87 q^{29} +3663.45 q^{31} -4763.30 q^{33} +12639.0 q^{37} -4558.21 q^{39} -4814.47 q^{41} +4938.11 q^{43} +1921.00 q^{45} +17405.0 q^{47} -3004.35 q^{51} -4658.43 q^{53} +7196.30 q^{55} -16292.8 q^{57} +32500.8 q^{59} +43496.1 q^{61} +6886.45 q^{65} +32082.4 q^{67} +3333.85 q^{69} -15820.0 q^{71} -36515.4 q^{73} -18904.3 q^{75} -25472.1 q^{79} +32296.5 q^{81} +69443.5 q^{83} +4538.91 q^{85} +24553.0 q^{87} -108331. q^{89} +22807.1 q^{93} +24614.9 q^{95} -94569.1 q^{97} +156269. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 13 q^{3} - 31 q^{5} + 230 q^{9}+O(q^{10}) \) 5 * q + 13 * q^3 - 31 * q^5 + 230 * q^9	\( 5 q + 13 q^{3} - 31 q^{5} + 230 q^{9} + 351 q^{11} + 54 q^{13} - 607 q^{15} - 111 q^{17} + 1035 q^{19} - 3639 q^{23} + 1540 q^{25} + 3607 q^{27} - 734 q^{29} + 7677 q^{31} + 7439 q^{33} + 13595 q^{37} + 1406 q^{39} - 5310 q^{41} - 764 q^{43} - 38978 q^{45} + 6675 q^{47} - 20975 q^{51} - 30753 q^{53} + 28267 q^{55} - 14389 q^{57} + 87989 q^{59} - 19899 q^{61} + 119470 q^{65} + 33067 q^{67} - 100399 q^{69} + 108720 q^{71} - 141659 q^{73} + 108788 q^{75} - 118919 q^{79} - 143851 q^{81} + 211004 q^{83} - 143379 q^{85} + 302154 q^{87} + 55861 q^{89} + 410381 q^{93} + 26279 q^{95} - 135470 q^{97} + 300154 q^{99}+O(q^{100}) \) 5 * q + 13 * q^3 - 31 * q^5 + 230 * q^9 + 351 * q^11 + 54 * q^13 - 607 * q^15 - 111 * q^17 + 1035 * q^19 - 3639 * q^23 + 1540 * q^25 + 3607 * q^27 - 734 * q^29 + 7677 * q^31 + 7439 * q^33 + 13595 * q^37 + 1406 * q^39 - 5310 * q^41 - 764 * q^43 - 38978 * q^45 + 6675 * q^47 - 20975 * q^51 - 30753 * q^53 + 28267 * q^55 - 14389 * q^57 + 87989 * q^59 - 19899 * q^61 + 119470 * q^65 + 33067 * q^67 - 100399 * q^69 + 108720 * q^71 - 141659 * q^73 + 108788 * q^75 - 118919 * q^79 - 143851 * q^81 + 211004 * q^83 - 143379 * q^85 + 302154 * q^87 + 55861 * q^89 + 410381 * q^93 + 26279 * q^95 - 135470 * q^97 + 300154 * q^99

Introduction

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