LMFDB - Embedded newform 704.6.a.be.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [704,6,Mod(1,704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(704, 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("704.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 704 = 2^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	704.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:	$112.910209148$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 1343x^{5} + 3732x^{4} + 459776x^{3} - 2085920x^{2} - 16548352x + 80811520 $ x^7 - 2x^6 - 1343x^5 + 3732x^4 + 459776x^3 - 2085920x^2 - 16548352x + 80811520
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{15} $
Twist minimal:	no (minimal twist has level 352)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$5.85632$ of defining polynomial
Character	$\chi$	$=$	704.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.85632 q^{3} -88.9891 q^{5} +0.0104569 q^{7} -195.991 q^{9} +O(q^{10})$	$q-6.85632 q^{3} -88.9891 q^{5} +0.0104569 q^{7} -195.991 q^{9} -121.000 q^{11} -401.821 q^{13} +610.138 q^{15} +1024.26 q^{17} +1549.75 q^{19} -0.0716960 q^{21} +1888.01 q^{23} +4794.07 q^{25} +3009.86 q^{27} -4166.32 q^{29} +8188.45 q^{31} +829.615 q^{33} -0.930552 q^{35} -3447.51 q^{37} +2755.01 q^{39} +4388.74 q^{41} -899.922 q^{43} +17441.1 q^{45} -5031.99 q^{47} -16807.0 q^{49} -7022.65 q^{51} +6994.19 q^{53} +10767.7 q^{55} -10625.6 q^{57} +36929.5 q^{59} -20688.1 q^{61} -2.04946 q^{63} +35757.7 q^{65} -24257.0 q^{67} -12944.8 q^{69} +12264.1 q^{71} +64629.8 q^{73} -32869.7 q^{75} -1.26529 q^{77} -80768.6 q^{79} +26989.2 q^{81} +14464.4 q^{83} -91147.9 q^{85} +28565.6 q^{87} +118460. q^{89} -4.20181 q^{91} -56142.7 q^{93} -137911. q^{95} +125566. q^{97} +23714.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 9 q^{3} - 61 q^{5} + 1000 q^{9}+O(q^{10}) $ 7 * q - 9 * q^3 - 61 * q^5 + 1000 * q^9	$ 7 q - 9 q^{3} - 61 q^{5} + 1000 q^{9} - 847 q^{11} + 18 q^{13} - 1331 q^{15} + 2432 q^{17} - 2362 q^{19} - 1344 q^{21} - 1835 q^{23} + 770 q^{25} - 579 q^{27} + 1168 q^{29} - 3983 q^{31} + 1089 q^{33} - 1872 q^{35} - 9931 q^{37} + 8786 q^{39} + 3706 q^{41} - 13642 q^{43} - 38706 q^{45} + 29044 q^{47} + 64559 q^{49} + 27540 q^{51} - 56262 q^{53} + 7381 q^{55} + 93306 q^{57} - 13059 q^{59} - 50364 q^{61} - 51744 q^{63} + 48714 q^{65} + 35765 q^{67} - 141305 q^{69} - 88685 q^{71} + 133222 q^{73} - 1372 q^{75} + 95350 q^{79} + 207791 q^{81} - 110918 q^{83} - 211748 q^{85} - 170340 q^{87} + 305299 q^{89} - 341184 q^{91} - 422325 q^{93} - 195782 q^{95} + 343445 q^{97} - 121000 q^{99}+O(q^{100}) $ 7 * q - 9 * q^3 - 61 * q^5 + 1000 * q^9 - 847 * q^11 + 18 * q^13 - 1331 * q^15 + 2432 * q^17 - 2362 * q^19 - 1344 * q^21 - 1835 * q^23 + 770 * q^25 - 579 * q^27 + 1168 * q^29 - 3983 * q^31 + 1089 * q^33 - 1872 * q^35 - 9931 * q^37 + 8786 * q^39 + 3706 * q^41 - 13642 * q^43 - 38706 * q^45 + 29044 * q^47 + 64559 * q^49 + 27540 * q^51 - 56262 * q^53 + 7381 * q^55 + 93306 * q^57 - 13059 * q^59 - 50364 * q^61 - 51744 * q^63 + 48714 * q^65 + 35765 * q^67 - 141305 * q^69 - 88685 * q^71 + 133222 * q^73 - 1372 * q^75 + 95350 * q^79 + 207791 * q^81 - 110918 * q^83 - 211748 * q^85 - 170340 * q^87 + 305299 * q^89 - 341184 * q^91 - 422325 * q^93 - 195782 * q^95 + 343445 * q^97 - 121000 * 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
$3$	−6.85632	−0.439833	−0.219917	−	0.975519i	$-0.570579\pi$
$3$	−6.85632	−0.439833	−0.219917	+	0.975519i	$0.570579\pi$
$4$	0	0
$5$	−88.9891	−1.59189	−0.795943	−	0.605371i	$-0.793025\pi$
$5$	−88.9891	−1.59189	−0.795943	+	0.605371i	$0.793025\pi$
$6$	0	0
$7$	0.0104569	8.06600e−5 0	4.03300e−5	−	1.00000i	$-0.499987\pi$
$7$	0.0104569	8.06600e−5 0	4.03300e−5		1.00000i	$0.499987\pi$
$8$	0	0
$9$	−195.991	−0.806547
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	−401.821	−0.659439	−0.329719	−	0.944079i	$-0.606954\pi$
$13$	−401.821	−0.659439	−0.329719	+	0.944079i	$0.606954\pi$
$14$	0	0
$15$	610.138	0.700165
$16$	0	0
$17$	1024.26	0.859582	0.429791	−	0.902928i	$-0.358587\pi$
$17$	1024.26	0.859582	0.429791	+	0.902928i	$0.358587\pi$
$18$	0	0
$19$	1549.75	0.984864	0.492432	−	0.870351i	$-0.336108\pi$
$19$	1549.75	0.984864	0.492432	+	0.870351i	$0.336108\pi$
$20$	0	0
$21$	−0.0716960	−3.54770e−5 0
$22$	0	0
$23$	1888.01	0.744191	0.372095	−	0.928194i	$-0.378639\pi$
$23$	1888.01	0.744191	0.372095	+	0.928194i	$0.378639\pi$
$24$	0	0
$25$	4794.07	1.53410
$26$	0	0
$27$	3009.86	0.794579
$28$	0	0
$29$	−4166.32	−0.919936	−0.459968	−	0.887935i	$-0.652139\pi$
$29$	−4166.32	−0.919936	−0.459968	+	0.887935i	$0.652139\pi$
$30$	0	0
$31$	8188.45	1.53037	0.765187	−	0.643808i	$-0.222646\pi$
$31$	8188.45	1.53037	0.765187	+	0.643808i	$0.222646\pi$
$32$	0	0
$33$	829.615	0.132615
$34$	0	0
$35$	−0.930552	−0.000128402 0
$36$	0	0
$37$	−3447.51	−0.414000	−0.207000	−	0.978341i	$-0.566370\pi$
$37$	−3447.51	−0.414000	−0.207000	+	0.978341i	$0.566370\pi$
$38$	0	0
$39$	2755.01	0.290043
$40$	0	0
$41$	4388.74	0.407737	0.203868	−	0.978998i	$-0.434649\pi$
$41$	4388.74	0.407737	0.203868	+	0.978998i	$0.434649\pi$
$42$	0	0
$43$	−899.922	−0.0742221	−0.0371111	−	0.999311i	$-0.511816\pi$
$43$	−899.922	−0.0742221	−0.0371111	+	0.999311i	$0.511816\pi$
$44$	0	0
$45$	17441.1	1.28393
$46$	0	0
$47$	−5031.99	−0.332273	−0.166137	−	0.986103i	$-0.553129\pi$
$47$	−5031.99	−0.332273	−0.166137	+	0.986103i	$0.553129\pi$
$48$	0	0
$49$	−16807.0	−1.00000
$50$	0	0
$51$	−7022.65	−0.378073
$52$	0	0
$53$	6994.19	0.342017	0.171008	−	0.985270i	$-0.445297\pi$
$53$	6994.19	0.342017	0.171008	+	0.985270i	$0.445297\pi$
$54$	0	0
$55$	10767.7	0.479972
$56$	0	0
$57$	−10625.6	−0.433176
$58$	0	0
$59$	36929.5	1.38116	0.690579	−	0.723257i	$-0.257356\pi$
$59$	36929.5	1.38116	0.690579	+	0.723257i	$0.257356\pi$
$60$	0	0
$61$	−20688.1	−0.711863	−0.355931	−	0.934512i	$-0.615836\pi$
$61$	−20688.1	−0.711863	−0.355931	+	0.934512i	$0.615836\pi$
$62$	0	0
$63$	−2.04946	−6.50561e−5 0
$64$	0	0
$65$	35757.7	1.04975
$66$	0	0
$67$	−24257.0	−0.660160	−0.330080	−	0.943953i	$-0.607076\pi$
$67$	−24257.0	−0.660160	−0.330080	+	0.943953i	$0.607076\pi$
$68$	0	0
$69$	−12944.8	−0.327320
$70$	0	0
$71$	12264.1	0.288728	0.144364	−	0.989525i	$-0.453886\pi$
$71$	12264.1	0.288728	0.144364	+	0.989525i	$0.453886\pi$
$72$	0	0
$73$	64629.8	1.41947	0.709734	−	0.704470i	$-0.248815\pi$
$73$	64629.8	1.41947	0.709734	+	0.704470i	$0.248815\pi$
$74$	0	0
$75$	−32869.7	−0.674749
$76$	0	0
$77$	−1.26529	−2.43199e−5 0
$78$	0	0
$79$	−80768.6	−1.45605	−0.728023	−	0.685553i	$-0.759560\pi$
$79$	−80768.6	−1.45605	−0.728023	+	0.685553i	$0.759560\pi$
$80$	0	0
$81$	26989.2	0.457064
$82$	0	0
$83$	14464.4	0.230465	0.115232	−	0.993339i	$-0.463239\pi$
$83$	14464.4	0.230465	0.115232	+	0.993339i	$0.463239\pi$
$84$	0	0
$85$	−91147.9	−1.36836
$86$	0	0
$87$	28565.6	0.404619
$88$	0	0
$89$	118460.	1.58524	0.792622	−	0.609713i	$-0.208715\pi$
$89$	118460.	1.58524	0.792622	+	0.609713i	$0.208715\pi$
$90$	0	0
$91$	−4.20181	−5.31903e−5 0
$92$	0	0
$93$	−56142.7	−0.673110
$94$	0	0
$95$	−137911.	−1.56779
$96$	0	0
$97$	125566.	1.35501	0.677504	−	0.735519i	$-0.263062\pi$
$97$	125566.	1.35501	0.677504	+	0.735519i	$0.263062\pi$
$98$	0	0
$99$	23714.9	0.243183
$100$	0	0
$101$	−16450.1	−0.160459	−0.0802297	−	0.996776i	$-0.525565\pi$
$101$	−16450.1	−0.160459	−0.0802297	+	0.996776i	$0.525565\pi$
$102$	0	0
$103$	−32828.2	−0.304897	−0.152449	−	0.988311i	$-0.548716\pi$
$103$	−32828.2	−0.304897	−0.152449	+	0.988311i	$0.548716\pi$
$104$	0	0
$105$	6.38016	5.64753e−5 0
$106$	0	0
$107$	−165956.	−1.40131	−0.700656	−	0.713500i	$-0.747109\pi$
$107$	−165956.	−1.40131	−0.700656	+	0.713500i	$0.747109\pi$
$108$	0	0
$109$	−133827.	−1.07889	−0.539445	−	0.842021i	$-0.681366\pi$
$109$	−133827.	−1.07889	−0.539445	+	0.842021i	$0.681366\pi$
$110$	0	0
$111$	23637.2	0.182091
$112$	0	0
$113$	−38902.3	−0.286602	−0.143301	−	0.989679i	$-0.545772\pi$
$113$	−38902.3	−0.286602	−0.143301	+	0.989679i	$0.545772\pi$
$114$	0	0
$115$	−168012.	−1.18467
$116$	0	0
$117$	78753.2	0.531868
$118$	0	0
$119$	10.7106	6.93339e−5 0
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	−30090.6	−0.179336
$124$	0	0
$125$	−148529.	−0.850229
$126$	0	0
$127$	−314751.	−1.73164	−0.865821	−	0.500354i	$-0.833203\pi$
$127$	−314751.	−1.73164	−0.865821	+	0.500354i	$0.833203\pi$
$128$	0	0
$129$	6170.15	0.0326454
$130$	0	0
$131$	−62894.2	−0.320208	−0.160104	−	0.987100i	$-0.551183\pi$
$131$	−62894.2	−0.320208	−0.160104	+	0.987100i	$0.551183\pi$
$132$	0	0
$133$	16.2056	7.94392e−5 0
$134$	0	0
$135$	−267845.	−1.26488
$136$	0	0
$137$	−87570.6	−0.398618	−0.199309	−	0.979937i	$-0.563870\pi$
$137$	−87570.6	−0.398618	−0.199309	+	0.979937i	$0.563870\pi$
$138$	0	0
$139$	−27685.9	−0.121541	−0.0607703	−	0.998152i	$-0.519356\pi$
$139$	−27685.9	−0.121541	−0.0607703	+	0.998152i	$0.519356\pi$
$140$	0	0
$141$	34501.0	0.146145
$142$	0	0
$143$	48620.3	0.198828
$144$	0	0
$145$	370757.	1.46443
$146$	0	0
$147$	115234.	0.439833
$148$	0	0
$149$	−93713.5	−0.345809	−0.172905	−	0.984939i	$-0.555315\pi$
$149$	−93713.5	−0.345809	−0.172905	+	0.984939i	$0.555315\pi$
$150$	0	0
$151$	449253.	1.60343	0.801713	−	0.597709i	$-0.203922\pi$
$151$	449253.	1.60343	0.801713	+	0.597709i	$0.203922\pi$
$152$	0	0
$153$	−200745.	−0.693293
$154$	0	0
$155$	−728683.	−2.43618
$156$	0	0
$157$	−109665.	−0.355073	−0.177537	−	0.984114i	$-0.556813\pi$
$157$	−109665.	−0.355073	−0.177537	+	0.984114i	$0.556813\pi$
$158$	0	0
$159$	−47954.4	−0.150430
$160$	0	0
$161$	19.7427	6.00265e−5 0
$162$	0	0
$163$	353482.	1.04207	0.521036	−	0.853535i	$-0.325546\pi$
$163$	353482.	1.04207	0.521036	+	0.853535i	$0.325546\pi$
$164$	0	0
$165$	−73826.7	−0.211108
$166$	0	0
$167$	−25150.8	−0.0697847	−0.0348923	−	0.999391i	$-0.511109\pi$
$167$	−25150.8	−0.0697847	−0.0348923	+	0.999391i	$0.511109\pi$
$168$	0	0
$169$	−209833.	−0.565141
$170$	0	0
$171$	−303736.	−0.794339
$172$	0	0
$173$	422495.	1.07326	0.536632	−	0.843816i	$-0.319696\pi$
$173$	422495.	1.07326	0.536632	+	0.843816i	$0.319696\pi$
$174$	0	0
$175$	50.1312	0.000123741 0
$176$	0	0
$177$	−253201.	−0.607479
$178$	0	0
$179$	592592.	1.38237	0.691183	−	0.722680i	$-0.257090\pi$
$179$	592592.	1.38237	0.691183	+	0.722680i	$0.257090\pi$
$180$	0	0
$181$	−671765.	−1.52413	−0.762063	−	0.647503i	$-0.775813\pi$
$181$	−671765.	−1.52413	−0.762063	+	0.647503i	$0.775813\pi$
$182$	0	0
$183$	141844.	0.313101
$184$	0	0
$185$	306791.	0.659041
$186$	0	0
$187$	−123935.	−0.259174
$188$	0	0
$189$	31.4739	6.40908e−5 0
$190$	0	0
$191$	−1.00017e6	−1.98377	−0.991886	−	0.127129i	$-0.959424\pi$
$191$	−1.00017e6	−1.98377	−0.991886	+	0.127129i	$0.959424\pi$
$192$	0	0
$193$	451893.	0.873258	0.436629	−	0.899642i	$-0.356172\pi$
$193$	451893.	0.873258	0.436629	+	0.899642i	$0.356172\pi$
$194$	0	0
$195$	−245166.	−0.461715
$196$	0	0
$197$	404618.	0.742814	0.371407	−	0.928470i	$-0.378875\pi$
$197$	404618.	0.742814	0.371407	+	0.928470i	$0.378875\pi$
$198$	0	0
$199$	424308.	0.759536	0.379768	−	0.925082i	$-0.376004\pi$
$199$	424308.	0.759536	0.379768	+	0.925082i	$0.376004\pi$
$200$	0	0
$201$	166314.	0.290360
$202$	0	0
$203$	−43.5669	−7.42021e−5 0
$204$	0	0
$205$	−390550.	−0.649071
$206$	0	0
$207$	−370032.	−0.600225
$208$	0	0
$209$	−187519.	−0.296948
$210$	0	0
$211$	−782758.	−1.21038	−0.605190	−	0.796081i	$-0.706903\pi$
$211$	−782758.	−1.21038	−0.605190	+	0.796081i	$0.706903\pi$
$212$	0	0
$213$	−84086.4	−0.126992
$214$	0	0
$215$	80083.2	0.118153
$216$	0	0
$217$	85.6260	0.000123440 0
$218$	0	0
$219$	−443123.	−0.624329
$220$	0	0
$221$	−411569.	−0.566841
$222$	0	0
$223$	−1.16642e6	−1.57069	−0.785347	−	0.619056i	$-0.787515\pi$
$223$	−1.16642e6	−1.57069	−0.785347	+	0.619056i	$0.787515\pi$
$224$	0	0
$225$	−939593.	−1.23732
$226$	0	0
$227$	575923.	0.741822	0.370911	−	0.928668i	$-0.379045\pi$
$227$	575923.	0.741822	0.370911	+	0.928668i	$0.379045\pi$
$228$	0	0
$229$	631491.	0.795753	0.397877	−	0.917439i	$-0.369747\pi$
$229$	631491.	0.795753	0.397877	+	0.917439i	$0.369747\pi$
$230$	0	0
$231$	8.67521	1.06967e−5 0
$232$	0	0
$233$	−1.07545e6	−1.29778	−0.648889	−	0.760883i	$-0.724766\pi$
$233$	−1.07545e6	−1.29778	−0.648889	+	0.760883i	$0.724766\pi$
$234$	0	0
$235$	447793.	0.528941
$236$	0	0
$237$	553776.	0.640417
$238$	0	0
$239$	591325.	0.669624	0.334812	−	0.942285i	$-0.391327\pi$
$239$	591325.	0.669624	0.334812	+	0.942285i	$0.391327\pi$
$240$	0	0
$241$	464766.	0.515456	0.257728	−	0.966218i	$-0.417026\pi$
$241$	464766.	0.515456	0.257728	+	0.966218i	$0.417026\pi$
$242$	0	0
$243$	−916443.	−0.995611
$244$	0	0
$245$	1.49564e6	1.59189
$246$	0	0
$247$	−622721.	−0.649458
$248$	0	0
$249$	−99172.5	−0.101366
$250$	0	0
$251$	379607.	0.380321	0.190160	−	0.981753i	$-0.439099\pi$
$251$	379607.	0.380321	0.190160	+	0.981753i	$0.439099\pi$
$252$	0	0
$253$	−228449.	−0.224382
$254$	0	0
$255$	624939.	0.601849
$256$	0	0
$257$	−1.98149e6	−1.87136	−0.935682	−	0.352846i	$-0.885214\pi$
$257$	−1.98149e6	−1.87136	−0.935682	+	0.352846i	$0.885214\pi$
$258$	0	0
$259$	−36.0503	−3.33933e−5 0
$260$	0	0
$261$	816561.	0.741971
$262$	0	0
$263$	−342106.	−0.304980	−0.152490	−	0.988305i	$-0.548729\pi$
$263$	−342106.	−0.304980	−0.152490	+	0.988305i	$0.548729\pi$
$264$	0	0
$265$	−622407.	−0.544452
$266$	0	0
$267$	−812199.	−0.697243
$268$	0	0
$269$	555873.	0.468377	0.234188	−	0.972191i	$-0.424757\pi$
$269$	555873.	0.468377	0.234188	+	0.972191i	$0.424757\pi$
$270$	0	0
$271$	−2.33716e6	−1.93315	−0.966573	−	0.256392i	$-0.917466\pi$
$271$	−2.33716e6	−1.93315	−0.966573	+	0.256392i	$0.917466\pi$
$272$	0	0
$273$	28.8090	2.33949e−5 0
$274$	0	0
$275$	−580082.	−0.462549
$276$	0	0
$277$	1.27147e6	0.995647	0.497824	−	0.867278i	$-0.334133\pi$
$277$	1.27147e6	0.995647	0.497824	+	0.867278i	$0.334133\pi$
$278$	0	0
$279$	−1.60486e6	−1.23432
$280$	0	0
$281$	125820.	0.0950569	0.0475284	−	0.998870i	$-0.484866\pi$
$281$	125820.	0.0950569	0.0475284	+	0.998870i	$0.484866\pi$
$282$	0	0
$283$	−2.59512e6	−1.92616	−0.963079	−	0.269219i	$-0.913235\pi$
$283$	−2.59512e6	−1.92616	−0.963079	+	0.269219i	$0.913235\pi$
$284$	0	0
$285$	945559.	0.689567
$286$	0	0
$287$	45.8926	3.28881e−5 0
$288$	0	0
$289$	−370752.	−0.261119
$290$	0	0
$291$	−860919.	−0.595978
$292$	0	0
$293$	1.32770e6	0.903505	0.451753	−	0.892143i	$-0.350799\pi$
$293$	1.32770e6	0.903505	0.451753	+	0.892143i	$0.350799\pi$
$294$	0	0
$295$	−3.28632e6	−2.19865
$296$	0	0
$297$	−364193.	−0.239575
$298$	0	0
$299$	−758642.	−0.490748
$300$	0	0
$301$	−9.41040	−5.98676e−6 0
$302$	0	0
$303$	112787.	0.0705754
$304$	0	0
$305$	1.84102e6	1.13320
$306$	0	0
$307$	−2.66798e6	−1.61561	−0.807805	−	0.589450i	$-0.799345\pi$
$307$	−2.66798e6	−1.61561	−0.807805	+	0.589450i	$0.799345\pi$
$308$	0	0
$309$	225080.	0.134104
$310$	0	0
$311$	1.09601e6	0.642562	0.321281	−	0.946984i	$-0.395887\pi$
$311$	1.09601e6	0.642562	0.321281	+	0.946984i	$0.395887\pi$
$312$	0	0
$313$	−2.18409e6	−1.26011	−0.630057	−	0.776549i	$-0.716969\pi$
$313$	−2.18409e6	−1.26011	−0.630057	+	0.776549i	$0.716969\pi$
$314$	0	0
$315$	182.380	0.000103562 0
$316$	0	0
$317$	−1.27349e6	−0.711782	−0.355891	−	0.934528i	$-0.615823\pi$
$317$	−1.27349e6	−0.711782	−0.355891	+	0.934528i	$0.615823\pi$
$318$	0	0
$319$	504125.	0.277371
$320$	0	0
$321$	1.13785e6	0.616343
$322$	0	0
$323$	1.58734e6	0.846572
$324$	0	0
$325$	−1.92636e6	−1.01165
$326$	0	0
$327$	917560.	0.474532
$328$	0	0
$329$	−52.6191	−2.68012e−5 0
$330$	0	0
$331$	598901.	0.300459	0.150230	−	0.988651i	$-0.451999\pi$
$331$	598901.	0.300459	0.150230	+	0.988651i	$0.451999\pi$
$332$	0	0
$333$	675680.	0.333911
$334$	0	0
$335$	2.15861e6	1.05090
$336$	0	0
$337$	1.99004e6	0.954526	0.477263	−	0.878761i	$-0.341629\pi$
$337$	1.99004e6	0.954526	0.477263	+	0.878761i	$0.341629\pi$
$338$	0	0
$339$	266727.	0.126057
$340$	0	0
$341$	−990803.	−0.461425
$342$	0	0
$343$	−351.499	−0.000161320 0
$344$	0	0
$345$	1.15195e6	0.521056
$346$	0	0
$347$	2.28963e6	1.02080	0.510401	−	0.859936i	$-0.329497\pi$
$347$	2.28963e6	1.02080	0.510401	+	0.859936i	$0.329497\pi$
$348$	0	0
$349$	2.80188e6	1.23136	0.615680	−	0.787996i	$-0.288881\pi$
$349$	2.80188e6	1.23136	0.615680	+	0.787996i	$0.288881\pi$
$350$	0	0
$351$	−1.20943e6	−0.523976
$352$	0	0
$353$	929917.	0.397198	0.198599	−	0.980081i	$-0.436361\pi$
$353$	929917.	0.397198	0.198599	+	0.980081i	$0.436361\pi$
$354$	0	0
$355$	−1.09137e6	−0.459622
$356$	0	0
$357$	−73.4352	−3.04954e−5 0
$358$	0	0
$359$	1.53352e6	0.627990	0.313995	−	0.949425i	$-0.398332\pi$
$359$	1.53352e6	0.627990	0.313995	+	0.949425i	$0.398332\pi$
$360$	0	0
$361$	−74386.9	−0.0300420
$362$	0	0
$363$	−100383.	−0.0399848
$364$	0	0
$365$	−5.75135e6	−2.25963
$366$	0	0
$367$	815086.	0.315892	0.157946	−	0.987448i	$-0.449513\pi$
$367$	815086.	0.315892	0.157946	+	0.987448i	$0.449513\pi$
$368$	0	0
$369$	−860152.	−0.328859
$370$	0	0
$371$	73.1376	2.75871e−5 0
$372$	0	0
$373$	−951641.	−0.354161	−0.177081	−	0.984196i	$-0.556665\pi$
$373$	−951641.	−0.354161	−0.177081	+	0.984196i	$0.556665\pi$
$374$	0	0
$375$	1.01836e6	0.373959
$376$	0	0
$377$	1.67412e6	0.606641
$378$	0	0
$379$	3.97579e6	1.42176	0.710879	−	0.703315i	$-0.248298\pi$
$379$	3.97579e6	1.42176	0.710879	+	0.703315i	$0.248298\pi$
$380$	0	0
$381$	2.15804e6	0.761634
$382$	0	0
$383$	2.30638e6	0.803404	0.401702	−	0.915771i	$-0.368419\pi$
$383$	2.30638e6	0.803404	0.401702	+	0.915771i	$0.368419\pi$
$384$	0	0
$385$	112.597	3.87145e−5 0
$386$	0	0
$387$	176376.	0.0598636
$388$	0	0
$389$	−3.53163e6	−1.18332	−0.591658	−	0.806189i	$-0.701527\pi$
$389$	−3.53163e6	−1.18332	−0.591658	+	0.806189i	$0.701527\pi$
$390$	0	0
$391$	1.93381e6	0.639693
$392$	0	0
$393$	431223.	0.140838
$394$	0	0
$395$	7.18753e6	2.31786
$396$	0	0
$397$	4.65786e6	1.48324	0.741618	−	0.670822i	$-0.234059\pi$
$397$	4.65786e6	1.48324	0.741618	+	0.670822i	$0.234059\pi$
$398$	0	0
$399$	−111.111	−3.49400e−5 0
$400$	0	0
$401$	−1.84367e6	−0.572563	−0.286281	−	0.958146i	$-0.592419\pi$
$401$	−1.84367e6	−0.572563	−0.286281	+	0.958146i	$0.592419\pi$
$402$	0	0
$403$	−3.29029e6	−1.00919
$404$	0	0
$405$	−2.40174e6	−0.727594
$406$	0	0
$407$	417148.	0.124826
$408$	0	0
$409$	−6.37722e6	−1.88505	−0.942526	−	0.334134i	$-0.891556\pi$
$409$	−6.37722e6	−1.88505	−0.942526	+	0.334134i	$0.891556\pi$
$410$	0	0
$411$	600412.	0.175325
$412$	0	0
$413$	386.169	0.000111404 0
$414$	0	0
$415$	−1.28717e6	−0.366874
$416$	0	0
$417$	189823.	0.0534576
$418$	0	0
$419$	2.80523e6	0.780609	0.390305	−	0.920686i	$-0.372370\pi$
$419$	2.80523e6	0.780609	0.390305	+	0.920686i	$0.372370\pi$
$420$	0	0
$421$	6.65201e6	1.82914	0.914571	−	0.404424i	$-0.132528\pi$
$421$	6.65201e6	1.82914	0.914571	+	0.404424i	$0.132528\pi$
$422$	0	0
$423$	986225.	0.267994
$424$	0	0
$425$	4.91036e6	1.31869
$426$	0	0
$427$	−216.334	−5.74189e−5 0
$428$	0	0
$429$	−333357.	−0.0874513
$430$	0	0
$431$	930197.	0.241202	0.120601	−	0.992701i	$-0.461518\pi$
$431$	930197.	0.241202	0.120601	+	0.992701i	$0.461518\pi$
$432$	0	0
$433$	−4.40769e6	−1.12977	−0.564887	−	0.825168i	$-0.691080\pi$
$433$	−4.40769e6	−1.12977	−0.564887	+	0.825168i	$0.691080\pi$
$434$	0	0
$435$	−2.54203e6	−0.644107
$436$	0	0
$437$	2.92593e6	0.732927
$438$	0	0
$439$	4.59326e6	1.13752	0.568761	−	0.822503i	$-0.307423\pi$
$439$	4.59326e6	1.13752	0.568761	+	0.822503i	$0.307423\pi$
$440$	0	0
$441$	3.29402e6	0.806547
$442$	0	0
$443$	5.66510e6	1.37151	0.685754	−	0.727833i	$-0.259472\pi$
$443$	5.66510e6	1.37151	0.685754	+	0.727833i	$0.259472\pi$
$444$	0	0
$445$	−1.05416e7	−2.52353
$446$	0	0
$447$	642530.	0.152098
$448$	0	0
$449$	2.39840e6	0.561443	0.280721	−	0.959789i	$-0.409426\pi$
$449$	2.39840e6	0.561443	0.280721	+	0.959789i	$0.409426\pi$
$450$	0	0
$451$	−531037.	−0.122937
$452$	0	0
$453$	−3.08023e6	−0.705240
$454$	0	0
$455$	373.915	8.46730e−5 0
$456$	0	0
$457$	6.58804e6	1.47559	0.737795	−	0.675025i	$-0.235867\pi$
$457$	6.58804e6	1.47559	0.737795	+	0.675025i	$0.235867\pi$
$458$	0	0
$459$	3.08288e6	0.683006
$460$	0	0
$461$	−2.02859e6	−0.444573	−0.222286	−	0.974981i	$-0.571352\pi$
$461$	−2.02859e6	−0.444573	−0.222286	+	0.974981i	$0.571352\pi$
$462$	0	0
$463$	−6.09155e6	−1.32061	−0.660306	−	0.750996i	$-0.729574\pi$
$463$	−6.09155e6	−1.32061	−0.660306	+	0.750996i	$0.729574\pi$
$464$	0	0
$465$	4.99609e6	1.07151
$466$	0	0
$467$	−7.13147e6	−1.51317	−0.756583	−	0.653897i	$-0.773133\pi$
$467$	−7.13147e6	−1.51317	−0.756583	+	0.653897i	$0.773133\pi$
$468$	0	0
$469$	−253.653	−5.32486e−5 0
$470$	0	0
$471$	751897.	0.156173
$472$	0	0
$473$	108891.	0.0223788
$474$	0	0
$475$	7.42959e6	1.51088
$476$	0	0
$477$	−1.37080e6	−0.275853
$478$	0	0
$479$	1.49383e6	0.297482	0.148741	−	0.988876i	$-0.452478\pi$
$479$	1.49383e6	0.297482	0.148741	+	0.988876i	$0.452478\pi$
$480$	0	0
$481$	1.38528e6	0.273008
$482$	0	0
$483$	−135.363	−2.64016e−5 0
$484$	0	0
$485$	−1.11740e7	−2.15702
$486$	0	0
$487$	−1.29051e6	−0.246568	−0.123284	−	0.992371i	$-0.539343\pi$
$487$	−1.29051e6	−0.246568	−0.123284	+	0.992371i	$0.539343\pi$
$488$	0	0
$489$	−2.42358e6	−0.458338
$490$	0	0
$491$	−4.83433e6	−0.904967	−0.452484	−	0.891773i	$-0.649462\pi$
$491$	−4.83433e6	−0.904967	−0.452484	+	0.891773i	$0.649462\pi$
$492$	0	0
$493$	−4.26739e6	−0.790760
$494$	0	0
$495$	−2.11037e6	−0.387120
$496$	0	0
$497$	128.244	2.32888e−5 0
$498$	0	0
$499$	−4.59558e6	−0.826207	−0.413104	−	0.910684i	$-0.635555\pi$
$499$	−4.59558e6	−0.826207	−0.413104	+	0.910684i	$0.635555\pi$
$500$	0	0
$501$	172442.	0.0306936
$502$	0	0
$503$	−3.34644e6	−0.589743	−0.294871	−	0.955537i	$-0.595277\pi$
$503$	−3.34644e6	−0.589743	−0.294871	+	0.955537i	$0.595277\pi$
$504$	0	0
$505$	1.46388e6	0.255433
$506$	0	0
$507$	1.43868e6	0.248568
$508$	0	0
$509$	28293.3	0.00484049	0.00242024	−	0.999997i	$-0.499230\pi$
$509$	28293.3	0.00484049	0.00242024	+	0.999997i	$0.499230\pi$
$510$	0	0
$511$	675.828	0.000114494 0
$512$	0	0
$513$	4.66452e6	0.782553
$514$	0	0
$515$	2.92135e6	0.485362
$516$	0	0
$517$	608871.	0.100184
$518$	0	0
$519$	−2.89676e6	−0.472057
$520$	0	0
$521$	−3.57403e6	−0.576852	−0.288426	−	0.957502i	$-0.593132\pi$
$521$	−3.57403e6	−0.576852	−0.288426	+	0.957502i	$0.593132\pi$
$522$	0	0
$523$	3.11186e6	0.497469	0.248735	−	0.968572i	$-0.419985\pi$
$523$	3.11186e6	0.497469	0.248735	+	0.968572i	$0.419985\pi$
$524$	0	0
$525$	−343.715	−5.44253e−5 0
$526$	0	0
$527$	8.38709e6	1.31548
$528$	0	0
$529$	−2.87177e6	−0.446180
$530$	0	0
$531$	−7.23784e6	−1.11397
$532$	0	0
$533$	−1.76349e6	−0.268877
$534$	0	0
$535$	1.47683e7	2.23073
$536$	0	0
$537$	−4.06300e6	−0.608011
$538$	0	0
$539$	2.03365e6	0.301511
$540$	0	0
$541$	−5.27977e6	−0.775571	−0.387786	−	0.921750i	$-0.626760\pi$
$541$	−5.27977e6	−0.775571	−0.387786	+	0.921750i	$0.626760\pi$
$542$	0	0
$543$	4.60584e6	0.670361
$544$	0	0
$545$	1.19091e7	1.71747
$546$	0	0
$547$	1.01043e7	1.44390	0.721950	−	0.691945i	$-0.243246\pi$
$547$	1.01043e7	1.44390	0.721950	+	0.691945i	$0.243246\pi$
$548$	0	0
$549$	4.05468e6	0.574151
$550$	0	0
$551$	−6.45674e6	−0.906012
$552$	0	0
$553$	−844.590	−0.000117445 0
$554$	0	0
$555$	−2.10346e6	−0.289868
$556$	0	0
$557$	−962698.	−0.131478	−0.0657389	−	0.997837i	$-0.520940\pi$
$557$	−962698.	−0.131478	−0.0657389	+	0.997837i	$0.520940\pi$
$558$	0	0
$559$	361607.	0.0489449
$560$	0	0
$561$	849740.	0.113993
$562$	0	0
$563$	7.33941e6	0.975866	0.487933	−	0.872881i	$-0.337751\pi$
$563$	7.33941e6	0.975866	0.487933	+	0.872881i	$0.337751\pi$
$564$	0	0
$565$	3.46188e6	0.456238
$566$	0	0
$567$	282.224	3.68668e−5 0
$568$	0	0
$569$	1.33115e7	1.72364	0.861819	−	0.507216i	$-0.169325\pi$
$569$	1.33115e7	1.72364	0.861819	+	0.507216i	$0.169325\pi$
$570$	0	0
$571$	−366749.	−0.0470738	−0.0235369	−	0.999723i	$-0.507493\pi$
$571$	−366749.	−0.0470738	−0.0235369	+	0.999723i	$0.507493\pi$
$572$	0	0
$573$	6.85751e6	0.872529
$574$	0	0
$575$	9.05124e6	1.14166
$576$	0	0
$577$	−1.28992e7	−1.61296	−0.806478	−	0.591264i	$-0.798629\pi$
$577$	−1.28992e7	−1.61296	−0.806478	+	0.591264i	$0.798629\pi$
$578$	0	0
$579$	−3.09833e6	−0.384088
$580$	0	0
$581$	151.253	1.85893e−5 0
$582$	0	0
$583$	−846297.	−0.103122
$584$	0	0
$585$	−7.00818e6	−0.846673
$586$	0	0
$587$	−3.09826e6	−0.371127	−0.185563	−	0.982632i	$-0.559411\pi$
$587$	−3.09826e6	−0.371127	−0.185563	+	0.982632i	$0.559411\pi$
$588$	0	0
$589$	1.26900e7	1.50721
$590$	0	0
$591$	−2.77419e6	−0.326714
$592$	0	0
$593$	−3.00158e6	−0.350520	−0.175260	−	0.984522i	$-0.556077\pi$
$593$	−3.00158e6	−0.350520	−0.175260	+	0.984522i	$0.556077\pi$
$594$	0	0
$595$	−953.126	−0.000110372 0
$596$	0	0
$597$	−2.90919e6	−0.334069
$598$	0	0
$599$	1.38210e7	1.57389	0.786944	−	0.617024i	$-0.211662\pi$
$599$	1.38210e7	1.57389	0.786944	+	0.617024i	$0.211662\pi$
$600$	0	0
$601$	−9.44913e6	−1.06710	−0.533550	−	0.845768i	$-0.679143\pi$
$601$	−9.44913e6	−1.06710	−0.533550	+	0.845768i	$0.679143\pi$
$602$	0	0
$603$	4.75414e6	0.532450
$604$	0	0
$605$	−1.30289e6	−0.144717
$606$	0	0
$607$	−1.69053e7	−1.86230	−0.931151	−	0.364635i	$-0.881194\pi$
$607$	−1.69053e7	−1.86230	−0.931151	+	0.364635i	$0.881194\pi$
$608$	0	0
$609$	298.708	3.26365e−5 0
$610$	0	0
$611$	2.02196e6	0.219114
$612$	0	0
$613$	8.58788e6	0.923070	0.461535	−	0.887122i	$-0.347299\pi$
$613$	8.58788e6	0.923070	0.461535	+	0.887122i	$0.347299\pi$
$614$	0	0
$615$	2.67774e6	0.285483
$616$	0	0
$617$	903145.	0.0955091	0.0477545	−	0.998859i	$-0.484793\pi$
$617$	903145.	0.0955091	0.0477545	+	0.998859i	$0.484793\pi$
$618$	0	0
$619$	−6.38444e6	−0.669724	−0.334862	−	0.942267i	$-0.608690\pi$
$619$	−6.38444e6	−0.669724	−0.334862	+	0.942267i	$0.608690\pi$
$620$	0	0
$621$	5.68265e6	0.591319
$622$	0	0
$623$	1238.72	0.000127866 0
$624$	0	0
$625$	−1.76400e6	−0.180634
$626$	0	0
$627$	1.28569e6	0.130608
$628$	0	0
$629$	−3.53114e6	−0.355867
$630$	0	0
$631$	1.32430e7	1.32408	0.662039	−	0.749469i	$-0.269691\pi$
$631$	1.32430e7	1.32408	0.662039	+	0.749469i	$0.269691\pi$
$632$	0	0
$633$	5.36684e6	0.532365
$634$	0	0
$635$	2.80094e7	2.75658
$636$	0	0
$637$	6.75341e6	0.659439
$638$	0	0
$639$	−2.40365e6	−0.232873
$640$	0	0
$641$	−2.78618e6	−0.267833	−0.133916	−	0.990993i	$-0.542755\pi$
$641$	−2.78618e6	−0.267833	−0.133916	+	0.990993i	$0.542755\pi$
$642$	0	0
$643$	1.59102e6	0.151757	0.0758783	−	0.997117i	$-0.475824\pi$
$643$	1.59102e6	0.151757	0.0758783	+	0.997117i	$0.475824\pi$
$644$	0	0
$645$	−549077.	−0.0519677
$646$	0	0
$647$	4.40368e6	0.413575	0.206788	−	0.978386i	$-0.433699\pi$
$647$	4.40368e6	0.413575	0.206788	+	0.978386i	$0.433699\pi$
$648$	0	0
$649$	−4.46847e6	−0.416435
$650$	0	0
$651$	−587.079	−5.42930e−5 0
$652$	0	0
$653$	−3.13978e6	−0.288149	−0.144074	−	0.989567i	$-0.546020\pi$
$653$	−3.13978e6	−0.288149	−0.144074	+	0.989567i	$0.546020\pi$
$654$	0	0
$655$	5.59690e6	0.509735
$656$	0	0
$657$	−1.26668e7	−1.14487
$658$	0	0
$659$	−1.19749e7	−1.07413	−0.537067	−	0.843540i	$-0.680468\pi$
$659$	−1.19749e7	−1.07413	−0.537067	+	0.843540i	$0.680468\pi$
$660$	0	0
$661$	6.91351e6	0.615453	0.307727	−	0.951475i	$-0.400432\pi$
$661$	6.91351e6	0.615453	0.307727	+	0.951475i	$0.400432\pi$
$662$	0	0
$663$	2.82185e6	0.249316
$664$	0	0
$665$	−1442.12	−0.000126458 0
$666$	0	0
$667$	−7.86605e6	−0.684608
$668$	0	0
$669$	7.99733e6	0.690843
$670$	0	0
$671$	2.50326e6	0.214635
$672$	0	0
$673$	−5.93001e6	−0.504682	−0.252341	−	0.967638i	$-0.581200\pi$
$673$	−5.93001e6	−0.504682	−0.252341	+	0.967638i	$0.581200\pi$
$674$	0	0
$675$	1.44295e7	1.21897
$676$	0	0
$677$	8.65327e6	0.725619	0.362810	−	0.931863i	$-0.381818\pi$
$677$	8.65327e6	0.725619	0.362810	+	0.931863i	$0.381818\pi$
$678$	0	0
$679$	1313.03	0.000109295 0
$680$	0	0
$681$	−3.94871e6	−0.326278
$682$	0	0
$683$	7.45086e6	0.611159	0.305580	−	0.952167i	$-0.401150\pi$
$683$	7.45086e6	0.611159	0.305580	+	0.952167i	$0.401150\pi$
$684$	0	0
$685$	7.79283e6	0.634555
$686$	0	0
$687$	−4.32970e6	−0.349999
$688$	0	0
$689$	−2.81041e6	−0.225539
$690$	0	0
$691$	5.05191e6	0.402495	0.201247	−	0.979540i	$-0.435500\pi$
$691$	5.05191e6	0.402495	0.201247	+	0.979540i	$0.435500\pi$
$692$	0	0
$693$	247.985	1.96151e−5 0
$694$	0	0
$695$	2.46374e6	0.193479
$696$	0	0
$697$	4.49520e6	0.350483
$698$	0	0
$699$	7.37363e6	0.570806
$700$	0	0
$701$	4.21918e6	0.324290	0.162145	−	0.986767i	$-0.448159\pi$
$701$	4.21918e6	0.324290	0.162145	+	0.986767i	$0.448159\pi$
$702$	0	0
$703$	−5.34276e6	−0.407734
$704$	0	0
$705$	−3.07021e6	−0.232646
$706$	0	0
$707$	−172.017	−1.29427e−5 0
$708$	0	0
$709$	−3.99207e6	−0.298251	−0.149126	−	0.988818i	$-0.547646\pi$
$709$	−3.99207e6	−0.298251	−0.149126	+	0.988818i	$0.547646\pi$
$710$	0	0
$711$	1.58299e7	1.17437
$712$	0	0
$713$	1.54599e7	1.13889
$714$	0	0
$715$	−4.32668e6	−0.316512
$716$	0	0
$717$	−4.05431e6	−0.294523
$718$	0	0
$719$	−1.76545e7	−1.27360	−0.636801	−	0.771028i	$-0.719743\pi$
$719$	−1.76545e7	−1.27360	−0.636801	+	0.771028i	$0.719743\pi$
$720$	0	0
$721$	−343.281	−2.45930e−5 0
$722$	0	0
$723$	−3.18658e6	−0.226715
$724$	0	0
$725$	−1.99736e7	−1.41128
$726$	0	0
$727$	−1.39538e7	−0.979166	−0.489583	−	0.871957i	$-0.662851\pi$
$727$	−1.39538e7	−0.979166	−0.489583	+	0.871957i	$0.662851\pi$
$728$	0	0
$729$	−274941.	−0.0191611
$730$	0	0
$731$	−921752.	−0.0638000
$732$	0	0
$733$	6.86172e6	0.471708	0.235854	−	0.971789i	$-0.424211\pi$
$733$	6.86172e6	0.471708	0.235854	+	0.971789i	$0.424211\pi$
$734$	0	0
$735$	−1.02546e7	−0.700165
$736$	0	0
$737$	2.93509e6	0.199046
$738$	0	0
$739$	1.31502e7	0.885774	0.442887	−	0.896578i	$-0.353954\pi$
$739$	1.31502e7	0.885774	0.442887	+	0.896578i	$0.353954\pi$
$740$	0	0
$741$	4.26957e6	0.285653
$742$	0	0
$743$	1.94159e7	1.29028	0.645142	−	0.764063i	$-0.276798\pi$
$743$	1.94159e7	1.29028	0.645142	+	0.764063i	$0.276798\pi$
$744$	0	0
$745$	8.33948e6	0.550489
$746$	0	0
$747$	−2.83489e6	−0.185881
$748$	0	0
$749$	−1735.39	−0.000113030 0
$750$	0	0
$751$	−1.23128e7	−0.796631	−0.398315	−	0.917248i	$-0.630405\pi$
$751$	−1.23128e7	−0.796631	−0.398315	+	0.917248i	$0.630405\pi$
$752$	0	0
$753$	−2.60271e6	−0.167278
$754$	0	0
$755$	−3.99787e7	−2.55247
$756$	0	0
$757$	533087.	0.0338110	0.0169055	−	0.999857i	$-0.494619\pi$
$757$	533087.	0.0338110	0.0169055	+	0.999857i	$0.494619\pi$
$758$	0	0
$759$	1.56632e6	0.0986907
$760$	0	0
$761$	−2.19162e7	−1.37184	−0.685921	−	0.727676i	$-0.740600\pi$
$761$	−2.19162e7	−1.37184	−0.685921	+	0.727676i	$0.740600\pi$
$762$	0	0
$763$	−1399.42	−8.70233e−5 0
$764$	0	0
$765$	1.78642e7	1.10364
$766$	0	0
$767$	−1.48391e7	−0.910789
$768$	0	0
$769$	1.03975e7	0.634035	0.317018	−	0.948420i	$-0.397319\pi$
$769$	1.03975e7	0.634035	0.317018	+	0.948420i	$0.397319\pi$
$770$	0	0
$771$	1.35857e7	0.823088
$772$	0	0
$773$	1.35084e7	0.813122	0.406561	−	0.913624i	$-0.366728\pi$
$773$	1.35084e7	0.813122	0.406561	+	0.913624i	$0.366728\pi$
$774$	0	0
$775$	3.92560e7	2.34775
$776$	0	0
$777$	247.172	1.46875e−5 0
$778$	0	0
$779$	6.80143e6	0.401565
$780$	0	0
$781$	−1.48395e6	−0.0870547
$782$	0	0
$783$	−1.25401e7	−0.730962
$784$	0	0
$785$	9.75898e6	0.565237
$786$	0	0
$787$	−1.74398e7	−1.00370	−0.501852	−	0.864954i	$-0.667348\pi$
$787$	−1.74398e7	−1.00370	−0.501852	+	0.864954i	$0.667348\pi$
$788$	0	0
$789$	2.34559e6	0.134141
$790$	0	0
$791$	−406.798	−2.31173e−5 0
$792$	0	0
$793$	8.31292e6	0.469430
$794$	0	0
$795$	4.26742e6	0.239468
$796$	0	0
$797$	−2.43089e7	−1.35556	−0.677781	−	0.735264i	$-0.737058\pi$
$797$	−2.43089e7	−1.35556	−0.677781	+	0.735264i	$0.737058\pi$
$798$	0	0
$799$	−5.15406e6	−0.285616
$800$	0	0
$801$	−2.32170e7	−1.27857
$802$	0	0
$803$	−7.82020e6	−0.427986
$804$	0	0
$805$	−1756.89	−9.55553e−5 0
$806$	0	0
$807$	−3.81125e6	−0.206008
$808$	0	0
$809$	−3.53077e7	−1.89670	−0.948348	−	0.317232i	$-0.897247\pi$
$809$	−3.53077e7	−1.89670	−0.948348	+	0.317232i	$0.897247\pi$
$810$	0	0
$811$	−2.67923e7	−1.43040	−0.715200	−	0.698920i	$-0.753664\pi$
$811$	−2.67923e7	−1.43040	−0.715200	+	0.698920i	$0.753664\pi$
$812$	0	0
$813$	1.60243e7	0.850262
$814$	0	0
$815$	−3.14560e7	−1.65886
$816$	0	0
$817$	−1.39465e6	−0.0730988
$818$	0	0
$819$	823.516	4.29005e−5 0
$820$	0	0
$821$	−2.44086e7	−1.26382	−0.631910	−	0.775042i	$-0.717729\pi$
$821$	−2.44086e7	−1.26382	−0.631910	+	0.775042i	$0.717729\pi$
$822$	0	0
$823$	−2.11851e7	−1.09026	−0.545132	−	0.838350i	$-0.683520\pi$
$823$	−2.11851e7	−1.09026	−0.545132	+	0.838350i	$0.683520\pi$
$824$	0	0
$825$	3.97723e6	0.203444
$826$	0	0
$827$	−1.13240e7	−0.575755	−0.287877	−	0.957667i	$-0.592950\pi$
$827$	−1.13240e7	−0.575755	−0.287877	+	0.957667i	$0.592950\pi$
$828$	0	0
$829$	−1.42114e6	−0.0718211	−0.0359105	−	0.999355i	$-0.511433\pi$
$829$	−1.42114e6	−0.0718211	−0.0359105	+	0.999355i	$0.511433\pi$
$830$	0	0
$831$	−8.71758e6	−0.437919
$832$	0	0
$833$	−1.72147e7	−0.859582
$834$	0	0
$835$	2.23815e6	0.111089
$836$	0	0
$837$	2.46461e7	1.21600
$838$	0	0
$839$	−7.08557e6	−0.347512	−0.173756	−	0.984789i	$-0.555590\pi$
$839$	−7.08557e6	−0.347512	−0.173756	+	0.984789i	$0.555590\pi$
$840$	0	0
$841$	−3.15292e6	−0.153718
$842$	0	0
$843$	−862662.	−0.0418092
$844$	0	0
$845$	1.86728e7	0.899640
$846$	0	0
$847$	153.100	7.33273e−6 0
$848$	0	0
$849$	1.77930e7	0.847189
$850$	0	0
$851$	−6.50892e6	−0.308095
$852$	0	0
$853$	−3.19098e7	−1.50159	−0.750796	−	0.660534i	$-0.770330\pi$
$853$	−3.19098e7	−1.50159	−0.750796	+	0.660534i	$0.770330\pi$
$854$	0	0
$855$	2.70292e7	1.26450
$856$	0	0
$857$	−1.11472e6	−0.0518460	−0.0259230	−	0.999664i	$-0.508252\pi$
$857$	−1.11472e6	−0.0518460	−0.0259230	+	0.999664i	$0.508252\pi$
$858$	0	0
$859$	−7.39926e6	−0.342141	−0.171070	−	0.985259i	$-0.554723\pi$
$859$	−7.39926e6	−0.342141	−0.171070	+	0.985259i	$0.554723\pi$
$860$	0	0
$861$	−314.655	−1.44653e−5 0
$862$	0	0
$863$	−3.53363e6	−0.161508	−0.0807540	−	0.996734i	$-0.525733\pi$
$863$	−3.53363e6	−0.161508	−0.0807540	+	0.996734i	$0.525733\pi$
$864$	0	0
$865$	−3.75975e7	−1.70851
$866$	0	0
$867$	2.54199e6	0.114849
$868$	0	0
$869$	9.77300e6	0.439014
$870$	0	0
$871$	9.74696e6	0.435335
$872$	0	0
$873$	−2.46097e7	−1.09288
$874$	0	0
$875$	−1553.15	−6.85795e−5 0
$876$	0	0
$877$	−3.15792e7	−1.38644	−0.693222	−	0.720724i	$-0.743810\pi$
$877$	−3.15792e7	−1.38644	−0.693222	+	0.720724i	$0.743810\pi$
$878$	0	0
$879$	−9.10314e6	−0.397392
$880$	0	0
$881$	1.75463e7	0.761634	0.380817	−	0.924650i	$-0.375643\pi$
$881$	1.75463e7	0.761634	0.380817	+	0.924650i	$0.375643\pi$
$882$	0	0
$883$	−3.57681e7	−1.54381	−0.771905	−	0.635738i	$-0.780696\pi$
$883$	−3.57681e7	−1.54381	−0.771905	+	0.635738i	$0.780696\pi$
$884$	0	0
$885$	2.25321e7	0.967038
$886$	0	0
$887$	2.99661e6	0.127886	0.0639428	−	0.997954i	$-0.479632\pi$
$887$	2.99661e6	0.127886	0.0639428	+	0.997954i	$0.479632\pi$
$888$	0	0
$889$	−3291.33	−0.000139674 0
$890$	0	0
$891$	−3.26569e6	−0.137810
$892$	0	0
$893$	−7.79831e6	−0.327244
$894$	0	0
$895$	−5.27342e7	−2.20057
$896$	0	0
$897$	5.20149e6	0.215847
$898$	0	0
$899$	−3.41157e7	−1.40785
$900$	0	0
$901$	7.16385e6	0.293991
$902$	0	0
$903$	64.5208	2.63318e−6 0
$904$	0	0
$905$	5.97798e7	2.42623
$906$	0	0
$907$	−3.68471e7	−1.48725	−0.743626	−	0.668596i	$-0.766896\pi$
$907$	−3.68471e7	−1.48725	−0.743626	+	0.668596i	$0.766896\pi$
$908$	0	0
$909$	3.22407e6	0.129418
$910$	0	0
$911$	−4.79651e7	−1.91483	−0.957414	−	0.288720i	$-0.906770\pi$
$911$	−4.79651e7	−1.91483	−0.957414	+	0.288720i	$0.906770\pi$
$912$	0	0
$913$	−1.75019e6	−0.0694878
$914$	0	0
$915$	−1.26226e7	−0.498421
$916$	0	0
$917$	−657.679	−2.58280e−5 0
$918$	0	0
$919$	−1.94973e7	−0.761528	−0.380764	−	0.924672i	$-0.624339\pi$
$919$	−1.94973e7	−0.761528	−0.380764	+	0.924672i	$0.624339\pi$
$920$	0	0
$921$	1.82925e7	0.710599
$922$	0	0
$923$	−4.92796e6	−0.190398
$924$	0	0
$925$	−1.65276e7	−0.635119
$926$	0	0
$927$	6.43402e6	0.245914
$928$	0	0
$929$	4.03735e7	1.53482	0.767409	−	0.641158i	$-0.221546\pi$
$929$	4.03735e7	1.53482	0.767409	+	0.641158i	$0.221546\pi$
$930$	0	0
$931$	−2.60466e7	−0.984864
$932$	0	0
$933$	−7.51462e6	−0.282620
$934$	0	0
$935$	1.10289e7	0.412575
$936$	0	0
$937$	2.95553e7	1.09973	0.549866	−	0.835253i	$-0.314679\pi$
$937$	2.95553e7	1.09973	0.549866	+	0.835253i	$0.314679\pi$
$938$	0	0
$939$	1.49748e7	0.554240
$940$	0	0
$941$	−4.27570e7	−1.57410	−0.787051	−	0.616887i	$-0.788393\pi$
$941$	−4.27570e7	−1.57410	−0.787051	+	0.616887i	$0.788393\pi$
$942$	0	0
$943$	8.28597e6	0.303434
$944$	0	0
$945$	−2800.83	−0.000102025 0
$946$	0	0
$947$	−4.20841e7	−1.52491	−0.762453	−	0.647044i	$-0.776005\pi$
$947$	−4.20841e7	−1.52491	−0.762453	+	0.647044i	$0.776005\pi$
$948$	0	0
$949$	−2.59696e7	−0.936052
$950$	0	0
$951$	8.73145e6	0.313065
$952$	0	0
$953$	2.36894e7	0.844933	0.422467	−	0.906378i	$-0.361164\pi$
$953$	2.36894e7	0.844933	0.422467	+	0.906378i	$0.361164\pi$
$954$	0	0
$955$	8.90046e7	3.15794
$956$	0	0
$957$	−3.45644e6	−0.121997
$958$	0	0
$959$	−915.718	−3.21525e−5 0
$960$	0	0
$961$	3.84216e7	1.34205
$962$	0	0
$963$	3.25259e7	1.13022
$964$	0	0
$965$	−4.02136e7	−1.39013
$966$	0	0
$967$	2.21518e7	0.761802	0.380901	−	0.924616i	$-0.375614\pi$
$967$	2.21518e7	0.761802	0.380901	+	0.924616i	$0.375614\pi$
$968$	0	0
$969$	−1.08833e7	−0.372350
$970$	0	0
$971$	5.49276e7	1.86957	0.934787	−	0.355208i	$-0.115590\pi$
$971$	5.49276e7	1.86957	0.934787	+	0.355208i	$0.115590\pi$
$972$	0	0
$973$	−289.509	−9.80347e−6 0
$974$	0	0
$975$	1.32077e7	0.444955
$976$	0	0
$977$	1.79300e7	0.600959	0.300480	−	0.953788i	$-0.402853\pi$
$977$	1.79300e7	0.600959	0.300480	+	0.953788i	$0.402853\pi$
$978$	0	0
$979$	−1.43336e7	−0.477969
$980$	0	0
$981$	2.62288e7	0.870175
$982$	0	0
$983$	−5.46366e7	−1.80343	−0.901717	−	0.432328i	$-0.857692\pi$
$983$	−5.46366e7	−1.80343	−0.901717	+	0.432328i	$0.857692\pi$
$984$	0	0
$985$	−3.60066e7	−1.18248
$986$	0	0
$987$	360.774	1.17881e−5 0
$988$	0	0
$989$	−1.69906e6	−0.0552354
$990$	0	0
$991$	1.02992e7	0.333135	0.166568	−	0.986030i	$-0.446732\pi$
$991$	1.02992e7	0.333135	0.166568	+	0.986030i	$0.446732\pi$
$992$	0	0
$993$	−4.10626e6	−0.132152
$994$	0	0
$995$	−3.77588e7	−1.20909
$996$	0	0
$997$	−2.72521e7	−0.868284	−0.434142	−	0.900845i	$-0.642948\pi$
$997$	−2.72521e7	−0.868284	−0.434142	+	0.900845i	$0.642948\pi$
$998$	0	0
$999$	−1.03765e7	−0.328956

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	704.6.a.be.1.3		7
4.3	odd	2		704.6.a.bf.1.5		7
8.3	odd	2		352.6.a.g.1.3	✓	7
8.5	even	2		352.6.a.h.1.5	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.6.a.g.1.3	✓	7	8.3	odd	2
352.6.a.h.1.5	yes	7	8.5	even	2
704.6.a.be.1.3		7	1.1	even	1	trivial
704.6.a.bf.1.5		7	4.3	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:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	704.a (trivial)

\(f(q)\)	\(=\)	\(q-6.85632 q^{3} -88.9891 q^{5} +0.0104569 q^{7} -195.991 q^{9} +O(q^{10})\)	\(q-6.85632 q^{3} -88.9891 q^{5} +0.0104569 q^{7} -195.991 q^{9} -121.000 q^{11} -401.821 q^{13} +610.138 q^{15} +1024.26 q^{17} +1549.75 q^{19} -0.0716960 q^{21} +1888.01 q^{23} +4794.07 q^{25} +3009.86 q^{27} -4166.32 q^{29} +8188.45 q^{31} +829.615 q^{33} -0.930552 q^{35} -3447.51 q^{37} +2755.01 q^{39} +4388.74 q^{41} -899.922 q^{43} +17441.1 q^{45} -5031.99 q^{47} -16807.0 q^{49} -7022.65 q^{51} +6994.19 q^{53} +10767.7 q^{55} -10625.6 q^{57} +36929.5 q^{59} -20688.1 q^{61} -2.04946 q^{63} +35757.7 q^{65} -24257.0 q^{67} -12944.8 q^{69} +12264.1 q^{71} +64629.8 q^{73} -32869.7 q^{75} -1.26529 q^{77} -80768.6 q^{79} +26989.2 q^{81} +14464.4 q^{83} -91147.9 q^{85} +28565.6 q^{87} +118460. q^{89} -4.20181 q^{91} -56142.7 q^{93} -137911. q^{95} +125566. q^{97} +23714.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 9 q^{3} - 61 q^{5} + 1000 q^{9}+O(q^{10}) \) 7 * q - 9 * q^3 - 61 * q^5 + 1000 * q^9	\( 7 q - 9 q^{3} - 61 q^{5} + 1000 q^{9} - 847 q^{11} + 18 q^{13} - 1331 q^{15} + 2432 q^{17} - 2362 q^{19} - 1344 q^{21} - 1835 q^{23} + 770 q^{25} - 579 q^{27} + 1168 q^{29} - 3983 q^{31} + 1089 q^{33} - 1872 q^{35} - 9931 q^{37} + 8786 q^{39} + 3706 q^{41} - 13642 q^{43} - 38706 q^{45} + 29044 q^{47} + 64559 q^{49} + 27540 q^{51} - 56262 q^{53} + 7381 q^{55} + 93306 q^{57} - 13059 q^{59} - 50364 q^{61} - 51744 q^{63} + 48714 q^{65} + 35765 q^{67} - 141305 q^{69} - 88685 q^{71} + 133222 q^{73} - 1372 q^{75} + 95350 q^{79} + 207791 q^{81} - 110918 q^{83} - 211748 q^{85} - 170340 q^{87} + 305299 q^{89} - 341184 q^{91} - 422325 q^{93} - 195782 q^{95} + 343445 q^{97} - 121000 q^{99}+O(q^{100}) \) 7 * q - 9 * q^3 - 61 * q^5 + 1000 * q^9 - 847 * q^11 + 18 * q^13 - 1331 * q^15 + 2432 * q^17 - 2362 * q^19 - 1344 * q^21 - 1835 * q^23 + 770 * q^25 - 579 * q^27 + 1168 * q^29 - 3983 * q^31 + 1089 * q^33 - 1872 * q^35 - 9931 * q^37 + 8786 * q^39 + 3706 * q^41 - 13642 * q^43 - 38706 * q^45 + 29044 * q^47 + 64559 * q^49 + 27540 * q^51 - 56262 * q^53 + 7381 * q^55 + 93306 * q^57 - 13059 * q^59 - 50364 * q^61 - 51744 * q^63 + 48714 * q^65 + 35765 * q^67 - 141305 * q^69 - 88685 * q^71 + 133222 * q^73 - 1372 * q^75 + 95350 * q^79 + 207791 * q^81 - 110918 * q^83 - 211748 * q^85 - 170340 * q^87 + 305299 * q^89 - 341184 * q^91 - 422325 * q^93 - 195782 * q^95 + 343445 * q^97 - 121000 * q^99

Introduction

L-functions

Modular forms

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Database

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