LMFDB - Embedded newform 304.7.e.f.113.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,7,Mod(113,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("304.113");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	304.e (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$69.9364414204$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.1
Character	$\chi$	$=$	304.113
Dual form			304.7.e.f.113.30

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-50.6470i q^{3} +207.318 q^{5} -216.532 q^{7} -1836.12 q^{9} +O(q^{10})$	$q-50.6470i q^{3} +207.318 q^{5} -216.532 q^{7} -1836.12 q^{9} -1269.69 q^{11} +2823.01i q^{13} -10500.0i q^{15} +3849.22 q^{17} +(-5534.11 + 4052.10i) q^{19} +10966.7i q^{21} -626.828 q^{23} +27355.6 q^{25} +56072.3i q^{27} +22150.7i q^{29} +5366.25i q^{31} +64306.2i q^{33} -44890.9 q^{35} +40725.2i q^{37} +142977. q^{39} +26044.7i q^{41} +134156. q^{43} -380660. q^{45} -156750. q^{47} -70762.8 q^{49} -194951. i q^{51} -243943. i q^{53} -263230. q^{55} +(205227. + 280286. i) q^{57} +361351. i q^{59} -434726. q^{61} +397579. q^{63} +585260. i q^{65} -217638. i q^{67} +31747.0i q^{69} +125018. i q^{71} +130570. q^{73} -1.38548e6i q^{75} +274930. q^{77} +431829. i q^{79} +1.50137e6 q^{81} +618585. q^{83} +798010. q^{85} +1.12186e6 q^{87} +189521. i q^{89} -611273. i q^{91} +271784. q^{93} +(-1.14732e6 + 840072. i) q^{95} -589806. i q^{97} +2.33131e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 720 q^{7} - 8670 q^{9}+O(q^{10}) $ 30 * q + 720 * q^7 - 8670 * q^9	$ 30 q + 720 q^{7} - 8670 q^{9} + 2524 q^{11} + 9700 q^{17} - 4014 q^{19} + 39376 q^{23} + 110742 q^{25} + 19976 q^{35} + 266500 q^{39} + 106788 q^{43} - 91360 q^{45} - 222756 q^{47} + 593586 q^{49} - 540936 q^{55} - 545972 q^{57} - 242640 q^{61} + 377716 q^{63} + 545964 q^{73} - 272356 q^{77} + 2189926 q^{81} - 1542652 q^{83} - 826908 q^{85} + 2729572 q^{87} - 2139912 q^{93} - 2142716 q^{95} + 293012 q^{99}+O(q^{100}) $ 30 * q + 720 * q^7 - 8670 * q^9 + 2524 * q^11 + 9700 * q^17 - 4014 * q^19 + 39376 * q^23 + 110742 * q^25 + 19976 * q^35 + 266500 * q^39 + 106788 * q^43 - 91360 * q^45 - 222756 * q^47 + 593586 * q^49 - 540936 * q^55 - 545972 * q^57 - 242640 * q^61 + 377716 * q^63 + 545964 * q^73 - 272356 * q^77 + 2189926 * q^81 - 1542652 * q^83 - 826908 * q^85 + 2729572 * q^87 - 2139912 * q^93 - 2142716 * q^95 + 293012 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$1$	$1$

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
$3$		−	50.6470i		−	1.87582i	−0.346885	−	0.937908i	$-0.612761\pi$
$3$		−	50.6470i		−	1.87582i	0.346885	−	0.937908i	$-0.387239\pi$
$4$		0			0
$5$	207.318			1.65854			0.829270	−	0.558848i	$-0.188756\pi$
$5$	207.318			1.65854			0.829270	+	0.558848i	$0.188756\pi$
$6$		0			0
$7$	−216.532			−0.631289			−0.315645	−	0.948878i	$-0.602221\pi$
$7$	−216.532			−0.631289			−0.315645	+	0.948878i	$0.602221\pi$
$8$		0			0
$9$	−1836.12			−2.51868
$10$		0			0
$11$	−1269.69			−0.953940			−0.476970	−	0.878920i	$-0.658265\pi$
$11$	−1269.69			−0.953940			−0.476970	+	0.878920i	$0.658265\pi$
$12$		0			0
$13$			2823.01i			1.28494i	0.766311	+	0.642470i	$0.222090\pi$
$13$			2823.01i			1.28494i	−0.766311	+	0.642470i	$0.777910\pi$
$14$		0			0
$15$		−	10500.0i		−	3.11112i
$16$		0			0
$17$	3849.22			0.783476			0.391738	−	0.920077i	$-0.371874\pi$
$17$	3849.22			0.783476			0.391738	+	0.920077i	$0.371874\pi$
$18$		0			0
$19$	−5534.11	+	4052.10i	−0.806839	+	0.590771i
$19$	−5534.11	+	4052.10i	−0.806839	+	0.590771i
$20$		0			0
$21$			10966.7i			1.18418i
$22$		0			0
$23$	−626.828			−0.0515187			−0.0257593	−	0.999668i	$-0.508200\pi$
$23$	−626.828			−0.0515187			−0.0257593	+	0.999668i	$0.508200\pi$
$24$		0			0
$25$	27355.6			1.75076
$26$		0			0
$27$			56072.3i			2.84877i
$28$		0			0
$29$			22150.7i			0.908223i	0.890945	+	0.454112i	$0.150043\pi$
$29$			22150.7i			0.908223i	−0.890945	+	0.454112i	$0.849957\pi$
$30$		0			0
$31$			5366.25i			0.180130i	0.995936	+	0.0900649i	$0.0287074\pi$
$31$			5366.25i			0.180130i	−0.995936	+	0.0900649i	$0.971293\pi$
$32$		0			0
$33$			64306.2i			1.78942i
$34$		0			0
$35$	−44890.9			−1.04702
$36$		0			0
$37$			40725.2i			0.804004i	0.915639	+	0.402002i	$0.131686\pi$
$37$			40725.2i			0.804004i	−0.915639	+	0.402002i	$0.868314\pi$
$38$		0			0
$39$	142977.			2.41031
$40$		0			0
$41$			26044.7i			0.377892i	0.981987	+	0.188946i	$0.0605071\pi$
$41$			26044.7i			0.377892i	−0.981987	+	0.188946i	$0.939493\pi$
$42$		0			0
$43$	134156.			1.68734			0.843672	−	0.536860i	$-0.180390\pi$
$43$	134156.			1.68734			0.843672	+	0.536860i	$0.180390\pi$
$44$		0			0
$45$	−380660.			−4.17734
$46$		0			0
$47$	−156750.			−1.50978			−0.754891	−	0.655850i	$-0.772310\pi$
$47$	−156750.			−1.50978			−0.754891	+	0.655850i	$0.772310\pi$
$48$		0			0
$49$	−70762.8			−0.601474
$50$		0			0
$51$		−	194951.i		−	1.46966i
$52$		0			0
$53$		−	243943.i		−	1.63855i	−0.573397	−	0.819277i	$-0.694375\pi$
$53$		−	243943.i		−	1.63855i	0.573397	−	0.819277i	$-0.305625\pi$
$54$		0			0
$55$	−263230.			−1.58215
$56$		0			0
$57$	205227.	+	280286.i	1.10818	+	1.51348i
$58$		0			0
$59$			361351.i			1.75944i	0.475495	+	0.879718i	$0.342269\pi$
$59$			361351.i			1.75944i	−0.475495	+	0.879718i	$0.657731\pi$
$60$		0			0
$61$	−434726.			−1.91525			−0.957627	−	0.288012i	$-0.907006\pi$
$61$	−434726.			−1.91525			−0.957627	+	0.288012i	$0.907006\pi$
$62$		0			0
$63$	397579.			1.59002
$64$		0			0
$65$			585260.i			2.13112i
$66$		0			0
$67$		−	217638.i		−	0.723621i	−0.932252	−	0.361810i	$-0.882159\pi$
$67$		−	217638.i		−	0.723621i	0.932252	−	0.361810i	$-0.117841\pi$
$68$		0			0
$69$			31747.0i			0.0966395i
$70$		0			0
$71$			125018.i			0.349299i	0.984631	+	0.174650i	$0.0558793\pi$
$71$			125018.i			0.349299i	−0.984631	+	0.174650i	$0.944121\pi$
$72$		0			0
$73$	130570.			0.335641			0.167821	−	0.985818i	$-0.446327\pi$
$73$	130570.			0.335641			0.167821	+	0.985818i	$0.446327\pi$
$74$		0			0
$75$		−	1.38548e6i		−	3.28410i
$76$		0			0
$77$	274930.			0.602212
$78$		0			0
$79$			431829.i			0.875851i	0.899011	+	0.437926i	$0.144287\pi$
$79$			431829.i			0.875851i	−0.899011	+	0.437926i	$0.855713\pi$
$80$		0			0
$81$	1.50137e6			2.82508
$82$		0			0
$83$	618585.			1.08184			0.540922	−	0.841073i	$-0.318075\pi$
$83$	618585.			1.08184			0.540922	+	0.841073i	$0.318075\pi$
$84$		0			0
$85$	798010.			1.29943
$86$		0			0
$87$	1.12186e6			1.70366
$88$		0			0
$89$			189521.i			0.268835i	0.990925	+	0.134418i	$0.0429164\pi$
$89$			189521.i			0.268835i	−0.990925	+	0.134418i	$0.957084\pi$
$90$		0			0
$91$		−	611273.i		−	0.811168i
$92$		0			0
$93$	271784.			0.337890
$94$		0			0
$95$	−1.14732e6	+	840072.i	−1.33818	+	0.979818i
$96$		0			0
$97$		−	589806.i		−	0.646240i	−0.946358	−	0.323120i	$-0.895268\pi$
$97$		−	589806.i		−	0.646240i	0.946358	−	0.323120i	$-0.104732\pi$
$98$		0			0
$99$	2.33131e6			2.40267
$100$		0			0
$101$	−929404.			−0.902070			−0.451035	−	0.892506i	$-0.648945\pi$
$101$	−929404.			−0.902070			−0.451035	+	0.892506i	$0.648945\pi$
$102$		0			0
$103$		−	111841.i		−	0.102350i	−0.998690	−	0.0511750i	$-0.983703\pi$
$103$		−	111841.i		−	0.102350i	0.998690	−	0.0511750i	$-0.0162966\pi$
$104$		0			0
$105$			2.27359e6i			1.96401i
$106$		0			0
$107$			1.19588e6i			0.976192i	0.872790	+	0.488096i	$0.162308\pi$
$107$			1.19588e6i			0.976192i	−0.872790	+	0.488096i	$0.837692\pi$
$108$		0			0
$109$			1.72605e6i			1.33282i	0.745584	+	0.666412i	$0.232171\pi$
$109$			1.72605e6i			1.33282i	−0.745584	+	0.666412i	$0.767829\pi$
$110$		0			0
$111$	2.06261e6			1.50816
$112$		0			0
$113$		−	245033.i		−	0.169820i	−0.996389	−	0.0849101i	$-0.972940\pi$
$113$		−	245033.i		−	0.169820i	0.996389	−	0.0849101i	$-0.0270603\pi$
$114$		0			0
$115$	−129952.			−0.0854458
$116$		0			0
$117$		−	5.18339e6i		−	3.23636i
$118$		0			0
$119$	−833479.			−0.494600
$120$		0			0
$121$	−159437.			−0.0899981
$122$		0			0
$123$	1.31909e6			0.708856
$124$		0			0
$125$	2.43196e6			1.24516
$126$		0			0
$127$			62044.4i			0.0302895i	0.999885	+	0.0151447i	$0.00482090\pi$
$127$			62044.4i			0.0302895i	−0.999885	+	0.0151447i	$0.995179\pi$
$128$		0			0
$129$		−	6.79458e6i		−	3.16514i
$130$		0			0
$131$	−784348.			−0.348895			−0.174448	−	0.984666i	$-0.555814\pi$
$131$	−784348.			−0.348895			−0.174448	+	0.984666i	$0.555814\pi$
$132$		0			0
$133$	1.19831e6	−	877410.i	0.509349	−	0.372948i
$134$		0			0
$135$			1.16248e7i			4.72480i
$136$		0			0
$137$	3.76986e6			1.46610			0.733049	−	0.680176i	$-0.238097\pi$
$137$	3.76986e6			1.46610			0.733049	+	0.680176i	$0.238097\pi$
$138$		0			0
$139$	−3.94187e6			−1.46777			−0.733886	−	0.679273i	$-0.762295\pi$
$139$	−3.94187e6			−1.46777			−0.733886	+	0.679273i	$0.762295\pi$
$140$		0			0
$141$			7.93892e6i			2.83207i
$142$		0			0
$143$		−	3.58436e6i		−	1.22576i
$144$		0			0
$145$			4.59222e6i			1.50633i
$146$		0			0
$147$			3.58393e6i			1.12825i
$148$		0			0
$149$	2.65430e6			0.802400			0.401200	−	0.915990i	$-0.368593\pi$
$149$	2.65430e6			0.802400			0.401200	+	0.915990i	$0.368593\pi$
$150$		0			0
$151$		−	4.60086e6i		−	1.33631i	−0.744021	−	0.668157i	$-0.767084\pi$
$151$		−	4.60086e6i		−	1.33631i	0.744021	−	0.668157i	$-0.232916\pi$
$152$		0			0
$153$	−7.06762e6			−1.97333
$154$		0			0
$155$			1.11252e6i			0.298753i
$156$		0			0
$157$	−1.82786e6			−0.472328			−0.236164	−	0.971713i	$-0.575890\pi$
$157$	−1.82786e6			−0.472328			−0.236164	+	0.971713i	$0.575890\pi$
$158$		0			0
$159$	−1.23550e7			−3.07363
$160$		0			0
$161$	135728.			0.0325232
$162$		0			0
$163$	3.74464e6			0.864664			0.432332	−	0.901715i	$-0.357691\pi$
$163$	3.74464e6			0.864664			0.432332	+	0.901715i	$0.357691\pi$
$164$		0			0
$165$			1.33318e7i			2.96782i
$166$		0			0
$167$			5.86421e6i			1.25910i	0.776960	+	0.629550i	$0.216761\pi$
$167$			5.86421e6i			1.25910i	−0.776960	+	0.629550i	$0.783239\pi$
$168$		0			0
$169$	−3.14259e6			−0.651069
$170$		0			0
$171$	1.01613e7	−	7.44015e6i	2.03217	−	1.48797i
$172$		0			0
$173$		−	2.69121e6i		−	0.519767i	−0.965640	−	0.259884i	$-0.916316\pi$
$173$		−	2.69121e6i		−	0.519767i	0.965640	−	0.259884i	$-0.0836842\pi$
$174$		0			0
$175$	−5.92336e6			−1.10523
$176$		0			0
$177$	1.83014e7			3.30038
$178$		0			0
$179$		−	7.16599e6i		−	1.24944i	−0.780847	−	0.624722i	$-0.785212\pi$
$179$		−	7.16599e6i		−	1.24944i	0.780847	−	0.624722i	$-0.214788\pi$
$180$		0			0
$181$		−	8.09581e6i		−	1.36529i	−0.730751	−	0.682645i	$-0.760830\pi$
$181$		−	8.09581e6i		−	1.36529i	0.730751	−	0.682645i	$-0.239170\pi$
$182$		0			0
$183$			2.20176e7i			3.59266i
$184$		0			0
$185$			8.44306e6i			1.33347i
$186$		0			0
$187$	−4.88733e6			−0.747389
$188$		0			0
$189$		−	1.21415e7i		−	1.79840i
$190$		0			0
$191$	−1.15110e7			−1.65202			−0.826009	−	0.563658i	$-0.809394\pi$
$191$	−1.15110e7			−1.65202			−0.826009	+	0.563658i	$0.809394\pi$
$192$		0			0
$193$		−	3.66284e6i		−	0.509502i	−0.967007	−	0.254751i	$-0.918006\pi$
$193$		−	3.66284e6i		−	0.509502i	0.967007	−	0.254751i	$-0.0819935\pi$
$194$		0			0
$195$	2.96417e7			3.99760
$196$		0			0
$197$	913305.			0.119459			0.0597293	−	0.998215i	$-0.480976\pi$
$197$	913305.			0.119459			0.0597293	+	0.998215i	$0.480976\pi$
$198$		0			0
$199$	502819.			0.0638046			0.0319023	−	0.999491i	$-0.489843\pi$
$199$	502819.			0.0638046			0.0319023	+	0.999491i	$0.489843\pi$
$200$		0			0
$201$	−1.10227e7			−1.35738
$202$		0			0
$203$		−	4.79633e6i		−	0.573351i
$204$		0			0
$205$			5.39953e6i			0.626750i
$206$		0			0
$207$	1.15093e6			0.129759
$208$		0			0
$209$	7.02663e6	−	5.14493e6i	0.769676	−	0.563561i
$210$		0			0
$211$		−	1.82973e6i		−	0.194778i	−0.995246	−	0.0973890i	$-0.968951\pi$
$211$		−	1.82973e6i		−	0.194778i	0.995246	−	0.0973890i	$-0.0310491\pi$
$212$		0			0
$213$	6.33179e6			0.655221
$214$		0			0
$215$	2.78128e7			2.79853
$216$		0			0
$217$		−	1.16196e6i		−	0.113714i
$218$		0			0
$219$		−	6.61299e6i		−	0.629601i
$220$		0			0
$221$			1.08664e7i			1.00672i
$222$		0			0
$223$		−	3.68141e6i		−	0.331970i	−0.986128	−	0.165985i	$-0.946920\pi$
$223$		−	3.68141e6i		−	0.331970i	0.986128	−	0.165985i	$-0.0530804\pi$
$224$		0			0
$225$	−5.02281e7			−4.40960
$226$		0			0
$227$			8.73952e6i			0.747154i	0.927599	+	0.373577i	$0.121869\pi$
$227$			8.73952e6i			0.747154i	−0.927599	+	0.373577i	$0.878131\pi$
$228$		0			0
$229$	−1.13097e7			−0.941767			−0.470884	−	0.882195i	$-0.656065\pi$
$229$	−1.13097e7			−0.941767			−0.470884	+	0.882195i	$0.656065\pi$
$230$		0			0
$231$		−	1.39244e7i		−	1.12964i
$232$		0			0
$233$	1.32060e7			1.04401			0.522004	−	0.852943i	$-0.325185\pi$
$233$	1.32060e7			1.04401			0.522004	+	0.852943i	$0.325185\pi$
$234$		0			0
$235$	−3.24970e7			−2.50403
$236$		0			0
$237$	2.18708e7			1.64293
$238$		0			0
$239$	9.46638e6			0.693410			0.346705	−	0.937974i	$-0.387301\pi$
$239$	9.46638e6			0.693410			0.346705	+	0.937974i	$0.387301\pi$
$240$		0			0
$241$			8.38759e6i			0.599220i	0.954062	+	0.299610i	$0.0968566\pi$
$241$			8.38759e6i			0.599220i	−0.954062	+	0.299610i	$0.903143\pi$
$242$		0			0
$243$		−	3.51629e7i		−	2.45056i
$244$		0			0
$245$	−1.46704e7			−0.997569
$246$		0			0
$247$	−1.14391e7	−	1.56229e7i	−0.759105	−	1.03674i
$248$		0			0
$249$		−	3.13295e7i		−	2.02934i
$250$		0			0
$251$	1.09671e7			0.693540			0.346770	−	0.937950i	$-0.387278\pi$
$251$	1.09671e7			0.693540			0.346770	+	0.937950i	$0.387278\pi$
$252$		0			0
$253$	795880.			0.0491457
$254$		0			0
$255$		−	4.04168e7i		−	2.43748i
$256$		0			0
$257$			2.47916e7i			1.46051i	0.683172	+	0.730257i	$0.260600\pi$
$257$			2.47916e7i			1.46051i	−0.683172	+	0.730257i	$0.739400\pi$
$258$		0			0
$259$		−	8.81832e6i		−	0.507559i
$260$		0			0
$261$		−	4.06713e7i		−	2.28753i
$262$		0			0
$263$	7.52662e6			0.413745			0.206873	−	0.978368i	$-0.433671\pi$
$263$	7.52662e6			0.413745			0.206873	+	0.978368i	$0.433671\pi$
$264$		0			0
$265$		−	5.05737e7i		−	2.71761i
$266$		0			0
$267$	9.59865e6			0.504285
$268$		0			0
$269$			2.94467e7i			1.51280i	0.654112	+	0.756398i	$0.273043\pi$
$269$			2.94467e7i			1.51280i	−0.654112	+	0.756398i	$0.726957\pi$
$270$		0			0
$271$	−1.81001e7			−0.909436			−0.454718	−	0.890635i	$-0.650260\pi$
$271$	−1.81001e7			−0.909436			−0.454718	+	0.890635i	$0.650260\pi$
$272$		0			0
$273$	−3.09591e7			−1.52160
$274$		0			0
$275$	−3.47332e7			−1.67012
$276$		0			0
$277$	1.52481e7			0.717425			0.358712	−	0.933448i	$-0.383216\pi$
$277$	1.52481e7			0.717425			0.358712	+	0.933448i	$0.383216\pi$
$278$		0			0
$279$		−	9.85307e6i		−	0.453690i
$280$		0			0
$281$			3.64567e7i			1.64308i	0.570153	+	0.821538i	$0.306884\pi$
$281$			3.64567e7i			1.64308i	−0.570153	+	0.821538i	$0.693116\pi$
$282$		0			0
$283$	−4.34796e7			−1.91834			−0.959171	−	0.282828i	$-0.908727\pi$
$283$	−4.34796e7			−1.91834			−0.959171	+	0.282828i	$0.908727\pi$
$284$		0			0
$285$	4.25471e7	+	5.81082e7i	1.83796	+	2.51017i
$286$		0			0
$287$		−	5.63952e6i		−	0.238559i
$288$		0			0
$289$	−9.32111e6			−0.386166
$290$		0			0
$291$	−2.98719e7			−1.21223
$292$		0			0
$293$			1.53729e7i			0.611159i	0.952167	+	0.305579i	$0.0988501\pi$
$293$			1.53729e7i			0.611159i	−0.952167	+	0.305579i	$0.901150\pi$
$294$		0			0
$295$			7.49145e7i			2.91810i
$296$		0			0
$297$		−	7.11947e7i		−	2.71756i
$298$		0			0
$299$		−	1.76954e6i		−	0.0661984i
$300$		0			0
$301$	−2.90490e7			−1.06520
$302$		0			0
$303$			4.70715e7i			1.69212i
$304$		0			0
$305$	−9.01264e7			−3.17653
$306$		0			0
$307$			3.49880e7i			1.20922i	0.796522	+	0.604609i	$0.206671\pi$
$307$			3.49880e7i			1.20922i	−0.796522	+	0.604609i	$0.793329\pi$
$308$		0			0
$309$	−5.66439e6			−0.191990
$310$		0			0
$311$	−3.67122e7			−1.22048			−0.610238	−	0.792218i	$-0.708926\pi$
$311$	−3.67122e7			−1.22048			−0.610238	+	0.792218i	$0.708926\pi$
$312$		0			0
$313$	6.60849e6			0.215511			0.107755	−	0.994177i	$-0.465634\pi$
$313$	6.60849e6			0.215511			0.107755	+	0.994177i	$0.465634\pi$
$314$		0			0
$315$	8.24251e7			2.63711
$316$		0			0
$317$		−	2.55203e7i		−	0.801140i	−0.916266	−	0.400570i	$-0.868812\pi$
$317$		−	2.55203e7i		−	0.801140i	0.916266	−	0.400570i	$-0.131188\pi$
$318$		0			0
$319$		−	2.81246e7i		−	0.866391i
$320$		0			0
$321$	6.05676e7			1.83116
$322$		0			0
$323$	−2.13020e7	+	1.55974e7i	−0.632139	+	0.462855i
$324$		0			0
$325$			7.72251e7i			2.24962i
$326$		0			0
$327$	8.74191e7			2.50013
$328$		0			0
$329$	3.39414e7			0.953109
$330$		0			0
$331$			6.09069e7i			1.67951i	0.542966	+	0.839755i	$0.317301\pi$
$331$			6.09069e7i			1.67951i	−0.542966	+	0.839755i	$0.682699\pi$
$332$		0			0
$333$		−	7.47764e7i		−	2.02503i
$334$		0			0
$335$		−	4.51203e7i		−	1.20015i
$336$		0			0
$337$			1.76978e7i			0.462414i	0.972905	+	0.231207i	$0.0742674\pi$
$337$			1.76978e7i			0.462414i	−0.972905	+	0.231207i	$0.925733\pi$
$338$		0			0
$339$	−1.24102e7			−0.318551
$340$		0			0
$341$		−	6.81349e6i		−	0.171833i
$342$		0			0
$343$	4.07972e7			1.01099
$344$		0			0
$345$			6.58170e6i			0.160281i
$346$		0			0
$347$	2.63162e7			0.629846			0.314923	−	0.949117i	$-0.398021\pi$
$347$	2.63162e7			0.629846			0.314923	+	0.949117i	$0.398021\pi$
$348$		0			0
$349$	−1.99981e7			−0.470450			−0.235225	−	0.971941i	$-0.575583\pi$
$349$	−1.99981e7			−0.470450			−0.235225	+	0.971941i	$0.575583\pi$
$350$		0			0
$351$	−1.58293e8			−3.66050
$352$		0			0
$353$	−8.34077e7			−1.89619			−0.948095	−	0.317987i	$-0.896993\pi$
$353$	−8.34077e7			−1.89619			−0.948095	+	0.317987i	$0.896993\pi$
$354$		0			0
$355$			2.59184e7i			0.579327i
$356$		0			0
$357$			4.22132e7i			0.927778i
$358$		0			0
$359$	−2.37139e7			−0.512531			−0.256266	−	0.966606i	$-0.582492\pi$
$359$	−2.37139e7			−0.512531			−0.256266	+	0.966606i	$0.582492\pi$
$360$		0			0
$361$	1.42068e7	−	4.48495e7i	0.301978	−	0.953315i
$362$		0			0
$363$			8.07502e6i			0.168820i
$364$		0			0
$365$	2.70695e7			0.556675
$366$		0			0
$367$	2.20658e7			0.446397			0.223198	−	0.974773i	$-0.428350\pi$
$367$	2.20658e7			0.446397			0.223198	+	0.974773i	$0.428350\pi$
$368$		0			0
$369$		−	4.78212e7i		−	0.951791i
$370$		0			0
$371$			5.28215e7i			1.03440i
$372$		0			0
$373$		−	1.03427e8i		−	1.99300i	−0.0836126	−	0.996498i	$-0.526646\pi$
$373$		−	1.03427e8i		−	1.99300i	0.0836126	−	0.996498i	$-0.473354\pi$
$374$		0			0
$375$		−	1.23171e8i		−	2.33569i
$376$		0			0
$377$	−6.25316e7			−1.16701
$378$		0			0
$379$			3.73678e7i			0.686403i	0.939262	+	0.343202i	$0.111511\pi$
$379$			3.73678e7i			0.686403i	−0.939262	+	0.343202i	$0.888489\pi$
$380$		0			0
$381$	3.14236e6			0.0568174
$382$		0			0
$383$			3.48185e7i			0.619746i	0.950778	+	0.309873i	$0.100287\pi$
$383$			3.48185e7i			0.619746i	−0.950778	+	0.309873i	$0.899713\pi$
$384$		0			0
$385$	5.69978e7			0.998793
$386$		0			0
$387$	−2.46326e8			−4.24988
$388$		0			0
$389$	−3.56140e7			−0.605023			−0.302511	−	0.953146i	$-0.597825\pi$
$389$	−3.56140e7			−0.605023			−0.302511	+	0.953146i	$0.597825\pi$
$390$		0			0
$391$	−2.41280e6			−0.0403636
$392$		0			0
$393$			3.97249e7i			0.654463i
$394$		0			0
$395$			8.95257e7i			1.45263i
$396$		0			0
$397$	3.55638e7			0.568377			0.284188	−	0.958768i	$-0.408276\pi$
$397$	3.55638e7			0.568377			0.284188	+	0.958768i	$0.408276\pi$
$398$		0			0
$399$	−4.44382e7	−	6.06910e7i	−0.699581	−	0.955444i
$400$		0			0
$401$		−	1.06933e8i		−	1.65836i	−0.558985	−	0.829178i	$-0.688809\pi$
$401$		−	1.06933e8i		−	1.65836i	0.558985	−	0.829178i	$-0.311191\pi$
$402$		0			0
$403$	−1.51490e7			−0.231456
$404$		0			0
$405$	3.11259e8			4.68552
$406$		0			0
$407$		−	5.17086e7i		−	0.766972i
$408$		0			0
$409$			1.09518e8i			1.60072i	0.599517	+	0.800362i	$0.295359\pi$
$409$			1.09518e8i			1.60072i	−0.599517	+	0.800362i	$0.704641\pi$
$410$		0			0
$411$		−	1.90932e8i		−	2.75013i
$412$		0			0
$413$		−	7.82442e7i		−	1.11071i
$414$		0			0
$415$	1.28243e8			1.79428
$416$		0			0
$417$			1.99644e8i			2.75327i
$418$		0			0
$419$	−4.88494e7			−0.664076			−0.332038	−	0.943266i	$-0.607736\pi$
$419$	−4.88494e7			−0.664076			−0.332038	+	0.943266i	$0.607736\pi$
$420$		0			0
$421$		−	1.20911e6i		−	0.0162039i	−0.999967	−	0.00810195i	$-0.997421\pi$
$421$		−	1.20911e6i		−	0.0162039i	0.999967	−	0.00810195i	$-0.00257896\pi$
$422$		0			0
$423$	2.87812e8			3.80266
$424$		0			0
$425$	1.05298e8			1.37168
$426$		0			0
$427$	9.41322e7			1.20908
$428$		0			0
$429$	−1.81537e8			−2.29929
$430$		0			0
$431$			4.01231e7i			0.501144i	0.968098	+	0.250572i	$0.0806187\pi$
$431$			4.01231e7i			0.501144i	−0.968098	+	0.250572i	$0.919381\pi$
$432$		0			0
$433$		−	9.93814e7i		−	1.22417i	−0.790792	−	0.612084i	$-0.790331\pi$
$433$		−	9.93814e7i		−	1.22417i	0.790792	−	0.612084i	$-0.209669\pi$
$434$		0			0
$435$	2.32582e8			2.82559
$436$		0			0
$437$	3.46893e6	−	2.53997e6i	0.0415673	−	0.0304358i
$438$		0			0
$439$			3.06555e6i			0.0362339i	0.999836	+	0.0181169i	$0.00576711\pi$
$439$			3.06555e6i			0.0362339i	−0.999836	+	0.0181169i	$0.994233\pi$
$440$		0			0
$441$	1.29929e8			1.51492
$442$		0			0
$443$	6.19279e7			0.712320			0.356160	−	0.934425i	$-0.384086\pi$
$443$	6.19279e7			0.712320			0.356160	+	0.934425i	$0.384086\pi$
$444$		0			0
$445$			3.92909e7i			0.445874i
$446$		0			0
$447$		−	1.34432e8i		−	1.50515i
$448$		0			0
$449$			3.61066e7i			0.398884i	0.979910	+	0.199442i	$0.0639130\pi$
$449$			3.61066e7i			0.398884i	−0.979910	+	0.199442i	$0.936087\pi$
$450$		0			0
$451$		−	3.30688e7i		−	0.360487i
$452$		0			0
$453$	−2.33020e8			−2.50668
$454$		0			0
$455$		−	1.26728e8i		−	1.34536i
$456$		0			0
$457$	1.29468e8			1.35648			0.678239	−	0.734841i	$-0.262743\pi$
$457$	1.29468e8			1.35648			0.678239	+	0.734841i	$0.262743\pi$
$458$		0			0
$459$			2.15835e8i			2.23194i
$460$		0			0
$461$	2.41857e7			0.246863			0.123431	−	0.992353i	$-0.460610\pi$
$461$	2.41857e7			0.246863			0.123431	+	0.992353i	$0.460610\pi$
$462$		0			0
$463$	−1.01576e7			−0.102340			−0.0511701	−	0.998690i	$-0.516295\pi$
$463$	−1.01576e7			−0.102340			−0.0511701	+	0.998690i	$0.516295\pi$
$464$		0			0
$465$	5.63457e7			0.560405
$466$		0			0
$467$	5.84261e7			0.573662			0.286831	−	0.957981i	$-0.407398\pi$
$467$	5.84261e7			0.573662			0.286831	+	0.957981i	$0.407398\pi$
$468$		0			0
$469$			4.71257e7i			0.456814i
$470$		0			0
$471$			9.25757e7i			0.886001i
$472$		0			0
$473$	−1.70337e8			−1.60962
$474$		0			0
$475$	−1.51389e8	+	1.10848e8i	−1.41258	+	1.03430i
$476$		0			0
$477$			4.47909e8i			4.12700i
$478$		0			0
$479$	−4.78622e7			−0.435498			−0.217749	−	0.976005i	$-0.569871\pi$
$479$	−4.78622e7			−0.435498			−0.217749	+	0.976005i	$0.569871\pi$
$480$		0			0
$481$	−1.14968e8			−1.03310
$482$		0			0
$483$		−	6.87424e6i		−	0.0610075i
$484$		0			0
$485$		−	1.22277e8i		−	1.07181i
$486$		0			0
$487$			2.04784e8i			1.77301i	0.462723	+	0.886503i	$0.346872\pi$
$487$			2.04784e8i			1.77301i	−0.462723	+	0.886503i	$0.653128\pi$
$488$		0			0
$489$		−	1.89655e8i		−	1.62195i
$490$		0			0
$491$	5.50285e7			0.464883			0.232441	−	0.972610i	$-0.425329\pi$
$491$	5.50285e7			0.464883			0.232441	+	0.972610i	$0.425329\pi$
$492$		0			0
$493$			8.52627e7i			0.711571i
$494$		0			0
$495$	4.83322e8			3.98493
$496$		0			0
$497$		−	2.70704e7i		−	0.220509i
$498$		0			0
$499$	−8.15743e7			−0.656526			−0.328263	−	0.944586i	$-0.606463\pi$
$499$	−8.15743e7			−0.656526			−0.328263	+	0.944586i	$0.606463\pi$
$500$		0			0
$501$	2.97005e8			2.36184
$502$		0			0
$503$	−9.81839e7			−0.771501			−0.385750	−	0.922603i	$-0.626057\pi$
$503$	−9.81839e7			−0.771501			−0.385750	+	0.922603i	$0.626057\pi$
$504$		0			0
$505$	−1.92682e8			−1.49612
$506$		0			0
$507$			1.59163e8i			1.22129i
$508$		0			0
$509$			8.31898e7i			0.630836i	0.948953	+	0.315418i	$0.102145\pi$
$509$			8.31898e7i			0.630836i	−0.948953	+	0.315418i	$0.897855\pi$
$510$		0			0
$511$	−2.82726e7			−0.211887
$512$		0			0
$513$	−2.27211e8	−	3.10310e8i	−1.68297	−	2.29850i
$514$		0			0
$515$		−	2.31865e7i		−	0.169751i
$516$		0			0
$517$	1.99025e8			1.44024
$518$		0			0
$519$	−1.36302e8			−0.974987
$520$		0			0
$521$			1.28939e8i			0.911739i	0.890047	+	0.455869i	$0.150672\pi$
$521$			1.28939e8i			0.911739i	−0.890047	+	0.455869i	$0.849328\pi$
$522$		0			0
$523$		−	1.11556e8i		−	0.779810i	−0.920855	−	0.389905i	$-0.872508\pi$
$523$		−	1.11556e8i		−	0.779810i	0.920855	−	0.389905i	$-0.127492\pi$
$524$		0			0
$525$			3.00001e8i			2.07321i
$526$		0			0
$527$			2.06558e7i			0.141127i
$528$		0			0
$529$	−1.47643e8			−0.997346
$530$		0			0
$531$		−	6.63484e8i		−	4.43146i
$532$		0			0
$533$	−7.35245e7			−0.485569
$534$		0			0
$535$			2.47926e8i			1.61905i
$536$		0			0
$537$	−3.62936e8			−2.34373
$538$		0			0
$539$	8.98472e7			0.573770
$540$		0			0
$541$	2.87933e8			1.81844			0.909221	−	0.416314i	$-0.136678\pi$
$541$	2.87933e8			1.81844			0.909221	+	0.416314i	$0.136678\pi$
$542$		0			0
$543$	−4.10029e8			−2.56103
$544$		0			0
$545$			3.57840e8i			2.21054i
$546$		0			0
$547$			6.05275e7i			0.369820i	0.982755	+	0.184910i	$0.0591994\pi$
$547$			6.05275e7i			0.369820i	−0.982755	+	0.184910i	$0.940801\pi$
$548$		0			0
$549$	7.98209e8			4.82392
$550$		0			0
$551$	−8.97567e7	−	1.22584e8i	−0.536552	−	0.732790i
$552$		0			0
$553$		−	9.35048e7i		−	0.552915i
$554$		0			0
$555$	4.27616e8			2.50135
$556$		0			0
$557$	−1.66203e8			−0.961775			−0.480888	−	0.876782i	$-0.659686\pi$
$557$	−1.66203e8			−0.961775			−0.480888	+	0.876782i	$0.659686\pi$
$558$		0			0
$559$			3.78723e8i			2.16813i
$560$		0			0
$561$			2.47529e8i			1.40196i
$562$		0			0
$563$			3.66687e7i			0.205480i	0.994708	+	0.102740i	$0.0327610\pi$
$563$			3.66687e7i			0.205480i	−0.994708	+	0.102740i	$0.967239\pi$
$564$		0			0
$565$		−	5.07997e7i		−	0.281654i
$566$		0			0
$567$	−3.25094e8			−1.78344
$568$		0			0
$569$			8.46961e7i			0.459755i	0.973220	+	0.229878i	$0.0738326\pi$
$569$			8.46961e7i			0.459755i	−0.973220	+	0.229878i	$0.926167\pi$
$570$		0			0
$571$	4.03766e6			0.0216881			0.0108441	−	0.999941i	$-0.496548\pi$
$571$	4.03766e6			0.0216881			0.0108441	+	0.999941i	$0.496548\pi$
$572$		0			0
$573$			5.83000e8i			3.09888i
$574$		0			0
$575$	−1.71472e7			−0.0901967
$576$		0			0
$577$	2.70898e7			0.141019			0.0705095	−	0.997511i	$-0.477537\pi$
$577$	2.70898e7			0.141019			0.0705095	+	0.997511i	$0.477537\pi$
$578$		0			0
$579$	−1.85512e8			−0.955732
$580$		0			0
$581$	−1.33943e8			−0.682957
$582$		0			0
$583$			3.09733e8i			1.56308i
$584$		0			0
$585$		−	1.07461e9i		−	5.36763i
$586$		0			0
$587$	−2.60715e7			−0.128900			−0.0644499	−	0.997921i	$-0.520529\pi$
$587$	−2.60715e7			−0.128900			−0.0644499	+	0.997921i	$0.520529\pi$
$588$		0			0
$589$	−2.17446e7	−	2.96974e7i	−0.106416	−	0.145336i
$590$		0			0
$591$		−	4.62562e7i		−	0.224082i
$592$		0			0
$593$	−3.35390e8			−1.60837			−0.804184	−	0.594380i	$-0.797398\pi$
$593$	−3.35390e8			−1.60837			−0.804184	+	0.594380i	$0.797398\pi$
$594$		0			0
$595$	−1.72795e8			−0.820314
$596$		0			0
$597$		−	2.54663e7i		−	0.119686i
$598$		0			0
$599$			5.93154e6i			0.0275986i	0.999905	+	0.0137993i	$0.00439259\pi$
$599$			5.93154e6i			0.0275986i	−0.999905	+	0.0137993i	$0.995607\pi$
$600$		0			0
$601$		−	1.64737e8i		−	0.758872i	−0.925218	−	0.379436i	$-0.876118\pi$
$601$		−	1.64737e8i		−	0.758872i	0.925218	−	0.379436i	$-0.123882\pi$
$602$		0			0
$603$			3.99610e8i			1.82257i
$604$		0			0
$605$	−3.30541e7			−0.149266
$606$		0			0
$607$		−	1.41818e8i		−	0.634110i	−0.948407	−	0.317055i	$-0.897306\pi$
$607$		−	1.41818e8i		−	0.634110i	0.948407	−	0.317055i	$-0.102694\pi$
$608$		0			0
$609$	−2.42920e8			−1.07550
$610$		0			0
$611$		−	4.42507e8i		−	1.93998i
$612$		0			0
$613$	4.56254e8			1.98073			0.990364	−	0.138487i	$-0.0442240\pi$
$613$	4.56254e8			1.98073			0.990364	+	0.138487i	$0.0442240\pi$
$614$		0			0
$615$	2.73470e8			1.17567
$616$		0			0
$617$	5.11590e7			0.217804			0.108902	−	0.994052i	$-0.465266\pi$
$617$	5.11590e7			0.217804			0.108902	+	0.994052i	$0.465266\pi$
$618$		0			0
$619$	6.62419e7			0.279294			0.139647	−	0.990201i	$-0.455403\pi$
$619$	6.62419e7			0.279294			0.139647	+	0.990201i	$0.455403\pi$
$620$		0			0
$621$		−	3.51477e7i		−	0.146765i
$622$		0			0
$623$		−	4.10373e7i		−	0.169713i
$624$		0			0
$625$	7.67563e7			0.314394
$626$		0			0
$627$	−2.60575e8	−	3.55878e8i	−1.05714	−	1.44377i
$628$		0			0
$629$			1.56760e8i			0.629918i
$630$		0			0
$631$	−2.56916e8			−1.02259			−0.511296	−	0.859404i	$-0.670835\pi$
$631$	−2.56916e8			−1.02259			−0.511296	+	0.859404i	$0.670835\pi$
$632$		0			0
$633$	−9.26704e7			−0.365368
$634$		0			0
$635$			1.28629e7i			0.0502363i
$636$		0			0
$637$		−	1.99764e8i		−	0.772858i
$638$		0			0
$639$		−	2.29548e8i		−	0.879775i
$640$		0			0
$641$		−	2.61269e8i		−	0.992004i	−0.868321	−	0.496002i	$-0.834801\pi$
$641$		−	2.61269e8i		−	0.992004i	0.868321	−	0.496002i	$-0.165199\pi$
$642$		0			0
$643$	3.05976e8			1.15094			0.575471	−	0.817822i	$-0.304819\pi$
$643$	3.05976e8			1.15094			0.575471	+	0.817822i	$0.304819\pi$
$644$		0			0
$645$		−	1.40864e9i		−	5.24952i
$646$		0			0
$647$	−2.62598e8			−0.969570			−0.484785	−	0.874633i	$-0.661102\pi$
$647$	−2.62598e8			−0.969570			−0.484785	+	0.874633i	$0.661102\pi$
$648$		0			0
$649$		−	4.58806e8i		−	1.67840i
$650$		0			0
$651$	−5.88500e7			−0.213306
$652$		0			0
$653$	−4.21431e8			−1.51351			−0.756757	−	0.653696i	$-0.773217\pi$
$653$	−4.21431e8			−1.51351			−0.756757	+	0.653696i	$0.773217\pi$
$654$		0			0
$655$	−1.62609e8			−0.578657
$656$		0			0
$657$	−2.39743e8			−0.845374
$658$		0			0
$659$		−	2.62225e6i		−	0.00916258i	−0.999990	−	0.00458129i	$-0.998542\pi$
$659$		−	2.62225e6i		−	0.00916258i	0.999990	−	0.00458129i	$-0.00145827\pi$
$660$		0			0
$661$			2.37860e8i			0.823601i	0.911274	+	0.411800i	$0.135100\pi$
$661$			2.37860e8i			0.823601i	−0.911274	+	0.411800i	$0.864900\pi$
$662$		0			0
$663$	5.50350e8			1.88842
$664$		0			0
$665$	2.48431e8	−	1.81903e8i	0.844775	−	0.618549i
$666$		0			0
$667$		−	1.38846e7i		−	0.0467905i
$668$		0			0
$669$	−1.86452e8			−0.622715
$670$		0			0
$671$	5.51969e8			1.82704
$672$		0			0
$673$		−	1.99653e7i		−	0.0654985i	−0.999464	−	0.0327493i	$-0.989574\pi$
$673$		−	1.99653e7i		−	0.0654985i	0.999464	−	0.0327493i	$-0.0104263\pi$
$674$		0			0
$675$			1.53389e9i			4.98750i
$676$		0			0
$677$		−	4.40423e7i		−	0.141940i	−0.997478	−	0.0709699i	$-0.977391\pi$
$677$		−	4.40423e7i		−	0.141940i	0.997478	−	0.0709699i	$-0.0226094\pi$
$678$		0			0
$679$			1.27712e8i			0.407964i
$680$		0			0
$681$	4.42631e8			1.40152
$682$		0			0
$683$		−	5.57046e8i		−	1.74835i	−0.485610	−	0.874176i	$-0.661402\pi$
$683$		−	5.57046e8i		−	1.74835i	0.485610	−	0.874176i	$-0.338598\pi$
$684$		0			0
$685$	7.81557e8			2.43158
$686$		0			0
$687$			5.72801e8i			1.76658i
$688$		0			0
$689$	6.88654e8			2.10544
$690$		0			0
$691$	−5.39941e6			−0.0163649			−0.00818243	−	0.999967i	$-0.502605\pi$
$691$	−5.39941e6			−0.0163649			−0.00818243	+	0.999967i	$0.502605\pi$
$692$		0			0
$693$	−5.04804e8			−1.51678
$694$		0			0
$695$	−8.17220e8			−2.43436
$696$		0			0
$697$			1.00252e8i			0.296069i
$698$		0			0
$699$		−	6.68845e8i		−	1.95837i
$700$		0			0
$701$	−1.53388e8			−0.445283			−0.222642	−	0.974900i	$-0.571468\pi$
$701$	−1.53388e8			−0.445283			−0.222642	+	0.974900i	$0.571468\pi$
$702$		0			0
$703$	−1.65023e8	−	2.25378e8i	−0.474983	−	0.648702i
$704$		0			0
$705$			1.64588e9i			4.69711i
$706$		0			0
$707$	2.01246e8			0.569467
$708$		0			0
$709$	−1.06669e8			−0.299296			−0.149648	−	0.988739i	$-0.547814\pi$
$709$	−1.06669e8			−0.299296			−0.149648	+	0.988739i	$0.547814\pi$
$710$		0			0
$711$		−	7.92890e8i		−	2.20599i
$712$		0			0
$713$		−	3.36371e6i		−	0.00928005i
$714$		0			0
$715$		−	7.43101e8i		−	2.03297i
$716$		0			0
$717$		−	4.79444e8i		−	1.30071i
$718$		0			0
$719$	1.76934e8			0.476020			0.238010	−	0.971263i	$-0.423505\pi$
$719$	1.76934e8			0.476020			0.238010	+	0.971263i	$0.423505\pi$
$720$		0			0
$721$			2.42171e7i			0.0646124i
$722$		0			0
$723$	4.24807e8			1.12403
$724$		0			0
$725$			6.05944e8i			1.59008i
$726$		0			0
$727$	−3.67103e8			−0.955400			−0.477700	−	0.878523i	$-0.658529\pi$
$727$	−3.67103e8			−0.955400			−0.477700	+	0.878523i	$0.658529\pi$
$728$		0			0
$729$	−6.86402e8			−1.77172
$730$		0			0
$731$	5.16394e8			1.32199
$732$		0			0
$733$	3.10562e8			0.788563			0.394282	−	0.918990i	$-0.370993\pi$
$733$	3.10562e8			0.788563			0.394282	+	0.918990i	$0.370993\pi$
$734$		0			0
$735$			7.43011e8i			1.87126i
$736$		0			0
$737$			2.76334e8i			0.690291i
$738$		0			0
$739$	4.60766e7			0.114169			0.0570843	−	0.998369i	$-0.481820\pi$
$739$	4.60766e7			0.114169			0.0570843	+	0.998369i	$0.481820\pi$
$740$		0			0
$741$	−7.91251e8	+	5.79358e8i	−1.94473	+	1.42394i
$742$		0			0
$743$			4.63207e7i			0.112930i	0.998405	+	0.0564649i	$0.0179829\pi$
$743$			4.63207e7i			0.112930i	−0.998405	+	0.0564649i	$0.982017\pi$
$744$		0			0
$745$	5.50283e8			1.33081
$746$		0			0
$747$	−1.13580e9			−2.72482
$748$		0			0
$749$		−	2.58946e8i		−	0.616259i
$750$		0			0
$751$		−	1.32812e7i		−	0.0313557i	−0.999877	−	0.0156778i	$-0.995009\pi$
$751$		−	1.32812e7i		−	0.0313557i	0.999877	−	0.0156778i	$-0.00499061\pi$
$752$		0			0
$753$		−	5.55452e8i		−	1.30095i
$754$		0			0
$755$		−	9.53840e8i		−	2.21633i
$756$		0			0
$757$	6.21173e7			0.143194			0.0715970	−	0.997434i	$-0.477190\pi$
$757$	6.21173e7			0.143194			0.0715970	+	0.997434i	$0.477190\pi$
$758$		0			0
$759$		−	4.03089e7i		−	0.0921883i
$760$		0			0
$761$	7.55965e8			1.71533			0.857665	−	0.514209i	$-0.171915\pi$
$761$	7.55965e8			1.71533			0.857665	+	0.514209i	$0.171915\pi$
$762$		0			0
$763$		−	3.73744e8i		−	0.841397i
$764$		0			0
$765$	−1.46524e9			−3.27284
$766$		0			0
$767$	−1.02010e9			−2.26077
$768$		0			0
$769$	−3.46140e8			−0.761154			−0.380577	−	0.924749i	$-0.624275\pi$
$769$	−3.46140e8			−0.761154			−0.380577	+	0.924749i	$0.624275\pi$
$770$		0			0
$771$	1.25562e9			2.73966
$772$		0			0
$773$		−	5.49258e7i		−	0.118915i	−0.998231	−	0.0594577i	$-0.981063\pi$
$773$		−	5.49258e7i		−	0.118915i	0.998231	−	0.0594577i	$-0.0189371\pi$
$774$		0			0
$775$			1.46797e8i			0.315363i
$776$		0			0
$777$	−4.46622e8			−0.952087
$778$		0			0
$779$	−1.05536e8	−	1.44134e8i	−0.223248	−	0.304898i
$780$		0			0
$781$		−	1.58735e8i		−	0.333211i
$782$		0			0
$783$	−1.24204e9			−2.58732
$784$		0			0
$785$	−3.78948e8			−0.783376
$786$		0			0
$787$		−	3.93944e8i		−	0.808184i	−0.914718	−	0.404092i	$-0.867587\pi$
$787$		−	3.93944e8i		−	0.808184i	0.914718	−	0.404092i	$-0.132413\pi$
$788$		0			0
$789$		−	3.81201e8i		−	0.776110i
$790$		0			0
$791$			5.30576e7i			0.107206i
$792$		0			0
$793$		−	1.22724e9i		−	2.46098i
$794$		0			0
$795$	−2.56141e9			−5.09773
$796$		0			0
$797$		−	3.48989e8i		−	0.689345i	−0.938723	−	0.344673i	$-0.887990\pi$
$797$		−	3.48989e8i		−	0.689345i	0.938723	−	0.344673i	$-0.112010\pi$
$798$		0			0
$799$	−6.03365e8			−1.18288
$800$		0			0
$801$		−	3.47983e8i		−	0.677111i
$802$		0			0
$803$	−1.65784e8			−0.320182
$804$		0			0
$805$	2.81389e7			0.0539410
$806$		0			0
$807$	1.49139e9			2.83773
$808$		0			0
$809$	2.19019e8			0.413654			0.206827	−	0.978378i	$-0.433686\pi$
$809$	2.19019e8			0.413654			0.206827	+	0.978378i	$0.433686\pi$
$810$		0			0
$811$			6.55534e8i			1.22895i	0.788938	+	0.614473i	$0.210631\pi$
$811$			6.55534e8i			1.22895i	−0.788938	+	0.614473i	$0.789369\pi$
$812$		0			0
$813$			9.16715e8i			1.70593i
$814$		0			0
$815$	7.76330e8			1.43408
$816$		0			0
$817$	−7.42432e8	+	5.43612e8i	−1.36141	+	0.996834i
$818$		0			0
$819$			1.12237e9i			2.04308i
$820$		0			0
$821$	−6.49817e8			−1.17425			−0.587126	−	0.809495i	$-0.699741\pi$
$821$	−6.49817e8			−1.17425			−0.587126	+	0.809495i	$0.699741\pi$
$822$		0			0
$823$	2.95114e8			0.529408			0.264704	−	0.964330i	$-0.414726\pi$
$823$	2.95114e8			0.529408			0.264704	+	0.964330i	$0.414726\pi$
$824$		0			0
$825$			1.75913e9i			3.13283i
$826$		0			0
$827$			2.18244e8i			0.385857i	0.981213	+	0.192929i	$0.0617985\pi$
$827$			2.18244e8i			0.385857i	−0.981213	+	0.192929i	$0.938201\pi$
$828$		0			0
$829$			4.05407e8i			0.711586i	0.934565	+	0.355793i	$0.115789\pi$
$829$			4.05407e8i			0.711586i	−0.934565	+	0.355793i	$0.884211\pi$
$830$		0			0
$831$		−	7.72271e8i		−	1.34576i
$832$		0			0
$833$	−2.72381e8			−0.471240
$834$		0			0
$835$			1.21575e9i			2.08827i
$836$		0			0
$837$	−3.00898e8			−0.513148
$838$		0			0
$839$			8.40387e8i			1.42296i	0.702705	+	0.711481i	$0.251975\pi$
$839$			8.40387e8i			1.42296i	−0.702705	+	0.711481i	$0.748025\pi$
$840$		0			0
$841$	1.04172e8			0.175131
$842$		0			0
$843$	1.84642e9			3.08211
$844$		0			0
$845$	−6.51514e8			−1.07982
$846$		0			0
$847$	3.45233e7			0.0568148
$848$		0			0
$849$			2.20211e9i			3.59845i
$850$		0			0
$851$		−	2.55277e7i		−	0.0414212i
$852$		0			0
$853$	−2.54245e8			−0.409642			−0.204821	−	0.978799i	$-0.565661\pi$
$853$	−2.54245e8			−0.409642			−0.204821	+	0.978799i	$0.565661\pi$
$854$		0			0
$855$	2.10661e9	−	1.54247e9i	3.37044	−	2.46785i
$856$		0			0
$857$		−	1.03869e9i		−	1.65022i	−0.564971	−	0.825111i	$-0.691113\pi$
$857$		−	1.03869e9i		−	1.65022i	0.564971	−	0.825111i	$-0.308887\pi$
$858$		0			0
$859$	8.27729e8			1.30590			0.652948	−	0.757403i	$-0.273532\pi$
$859$	8.27729e8			1.30590			0.652948	+	0.757403i	$0.273532\pi$
$860$		0			0
$861$	−2.85625e8			−0.447493
$862$		0			0
$863$		−	6.74705e8i		−	1.04974i	−0.851182	−	0.524870i	$-0.824114\pi$
$863$		−	6.74705e8i		−	1.04974i	0.851182	−	0.524870i	$-0.175886\pi$
$864$		0			0
$865$		−	5.57935e8i		−	0.862055i
$866$		0			0
$867$			4.72086e8i			0.724376i
$868$		0			0
$869$		−	5.48291e8i		−	0.835510i
$870$		0			0
$871$	6.14396e8			0.929809
$872$		0			0
$873$			1.08295e9i			1.62767i
$874$		0			0
$875$	−5.26597e8			−0.786057
$876$		0			0
$877$			9.31066e8i			1.38033i	0.723654	+	0.690163i	$0.242461\pi$
$877$			9.31066e8i			1.38033i	−0.723654	+	0.690163i	$0.757539\pi$
$878$		0			0
$879$	7.78593e8			1.14642
$880$		0			0
$881$	6.20596e8			0.907573			0.453786	−	0.891110i	$-0.350073\pi$
$881$	6.20596e8			0.907573			0.453786	+	0.891110i	$0.350073\pi$
$882$		0			0
$883$	1.01427e9			1.47323			0.736615	−	0.676312i	$-0.236423\pi$
$883$	1.01427e9			1.47323			0.736615	+	0.676312i	$0.236423\pi$
$884$		0			0
$885$	3.79419e9			5.47381
$886$		0			0
$887$		−	9.17328e7i		−	0.131448i	−0.997838	−	0.0657240i	$-0.979064\pi$
$887$		−	9.17328e7i		−	0.131448i	0.997838	−	0.0657240i	$-0.0209357\pi$
$888$		0			0
$889$		−	1.34346e7i		−	0.0191214i
$890$		0			0
$891$	−1.90627e9			−2.69496
$892$		0			0
$893$	8.67472e8	−	6.35167e8i	1.21815	−	0.891936i
$894$		0			0
$895$		−	1.48563e9i		−	2.07225i
$896$		0			0
$897$	−8.96220e7			−0.124176
$898$		0			0
$899$	−1.18866e8			−0.163598
$900$		0			0
$901$		−	9.38990e8i		−	1.28377i
$902$		0			0
$903$			1.47125e9i			1.99812i
$904$		0			0
$905$		−	1.67840e9i		−	2.26439i
$906$		0			0
$907$		−	1.04604e9i		−	1.40193i	−0.713198	−	0.700963i	$-0.752754\pi$
$907$		−	1.04604e9i		−	1.40193i	0.713198	−	0.700963i	$-0.247246\pi$
$908$		0			0
$909$	1.70650e9			2.27203
$910$		0			0
$911$		−	7.62867e8i		−	1.00901i	−0.863410	−	0.504503i	$-0.831676\pi$
$911$		−	7.62867e8i		−	1.00901i	0.863410	−	0.504503i	$-0.168324\pi$
$912$		0			0
$913$	−7.85413e8			−1.03201
$914$		0			0
$915$			4.56463e9i			5.95858i
$916$		0			0
$917$	1.69837e8			0.220254
$918$		0			0
$919$	7.32028e8			0.943151			0.471576	−	0.881826i	$-0.343685\pi$
$919$	7.32028e8			0.943151			0.471576	+	0.881826i	$0.343685\pi$
$920$		0			0
$921$	1.77204e9			2.26827
$922$		0			0
$923$	−3.52928e8			−0.448829
$924$		0			0
$925$			1.11406e9i			1.40762i
$926$		0			0
$927$			2.05353e8i			0.257787i
$928$		0			0
$929$	−7.20405e8			−0.898524			−0.449262	−	0.893400i	$-0.648313\pi$
$929$	−7.20405e8			−0.898524			−0.449262	+	0.893400i	$0.648313\pi$
$930$		0			0
$931$	3.91609e8	−	2.86738e8i	0.485293	−	0.355334i
$932$		0			0
$933$			1.85936e9i			2.28939i
$934$		0			0
$935$	−1.01323e9			−1.23957
$936$		0			0
$937$	−2.36992e8			−0.288081			−0.144040	−	0.989572i	$-0.546010\pi$
$937$	−2.36992e8			−0.288081			−0.144040	+	0.989572i	$0.546010\pi$
$938$		0			0
$939$		−	3.34700e8i		−	0.404259i
$940$		0			0
$941$		−	3.30813e8i		−	0.397021i	−0.980099	−	0.198511i	$-0.936390\pi$
$941$		−	3.30813e8i		−	0.397021i	0.980099	−	0.198511i	$-0.0636105\pi$
$942$		0			0
$943$		−	1.63255e7i		−	0.0194685i
$944$		0			0
$945$		−	2.51714e9i		−	2.98272i
$946$		0			0
$947$	3.47137e8			0.408743			0.204372	−	0.978893i	$-0.434485\pi$
$947$	3.47137e8			0.408743			0.204372	+	0.978893i	$0.434485\pi$
$948$		0			0
$949$			3.68601e8i			0.431279i
$950$		0			0
$951$	−1.29253e9			−1.50279
$952$		0			0
$953$		−	5.97394e8i		−	0.690211i	−0.938564	−	0.345106i	$-0.887843\pi$
$953$		−	5.97394e8i		−	0.690211i	0.938564	−	0.345106i	$-0.112157\pi$
$954$		0			0
$955$	−2.38644e9			−2.73994
$956$		0			0
$957$	−1.42443e9			−1.62519
$958$		0			0
$959$	−8.16295e8			−0.925532
$960$		0			0
$961$	8.58707e8			0.967553
$962$		0			0
$963$		−	2.19577e9i		−	2.45872i
$964$		0			0
$965$		−	7.59371e8i		−	0.845030i
$966$		0			0
$967$	−5.15343e8			−0.569924			−0.284962	−	0.958539i	$-0.591981\pi$
$967$	−5.15343e8			−0.569924			−0.284962	+	0.958539i	$0.591981\pi$
$968$		0			0
$969$	7.89962e8	+	1.07888e9i	0.868231	+	1.18578i
$970$		0			0
$971$			5.60668e8i			0.612418i	0.951964	+	0.306209i	$0.0990606\pi$
$971$			5.60668e8i			0.612418i	−0.951964	+	0.306209i	$0.900939\pi$
$972$		0			0
$973$	8.53543e8			0.926588
$974$		0			0
$975$	3.91122e9			4.21987
$976$		0			0
$977$			1.16776e9i			1.25219i	0.779747	+	0.626095i	$0.215348\pi$
$977$			1.16776e9i			1.25219i	−0.779747	+	0.626095i	$0.784652\pi$
$978$		0			0
$979$		−	2.40633e8i		−	0.256453i
$980$		0			0
$981$		−	3.16923e9i		−	3.35696i
$982$		0			0
$983$		−	1.18203e9i		−	1.24442i	−0.782850	−	0.622210i	$-0.786235\pi$
$983$		−	1.18203e9i		−	1.24442i	0.782850	−	0.622210i	$-0.213765\pi$
$984$		0			0
$985$	1.89344e8			0.198127
$986$		0			0
$987$		−	1.71903e9i		−	1.78786i
$988$		0			0
$989$	−8.40925e7			−0.0869297
$990$		0			0
$991$			3.82194e8i			0.392702i	0.980534	+	0.196351i	$0.0629092\pi$
$991$			3.82194e8i			0.392702i	−0.980534	+	0.196351i	$0.937091\pi$
$992$		0			0
$993$	3.08475e9			3.15045
$994$		0			0
$995$	1.04243e8			0.105823
$996$		0			0
$997$	−1.32684e8			−0.133885			−0.0669427	−	0.997757i	$-0.521324\pi$
$997$	−1.32684e8			−0.133885			−0.0669427	+	0.997757i	$0.521324\pi$
$998$		0			0
$999$	−2.28356e9			−2.29042

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.7.e.f.113.1		30
4.3	odd	2		152.7.e.a.113.30	yes	30
19.18	odd	2	inner	304.7.e.f.113.30		30
76.75	even	2		152.7.e.a.113.1	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.7.e.a.113.1	✓	30	76.75	even	2
152.7.e.a.113.30	yes	30	4.3	odd	2
304.7.e.f.113.1		30	1.1	even	1	trivial
304.7.e.f.113.30		30	19.18	odd	2	inner

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:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	304.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-50.6470i q^{3} +207.318 q^{5} -216.532 q^{7} -1836.12 q^{9} +O(q^{10})\)	\(q-50.6470i q^{3} +207.318 q^{5} -216.532 q^{7} -1836.12 q^{9} -1269.69 q^{11} +2823.01i q^{13} -10500.0i q^{15} +3849.22 q^{17} +(-5534.11 + 4052.10i) q^{19} +10966.7i q^{21} -626.828 q^{23} +27355.6 q^{25} +56072.3i q^{27} +22150.7i q^{29} +5366.25i q^{31} +64306.2i q^{33} -44890.9 q^{35} +40725.2i q^{37} +142977. q^{39} +26044.7i q^{41} +134156. q^{43} -380660. q^{45} -156750. q^{47} -70762.8 q^{49} -194951. i q^{51} -243943. i q^{53} -263230. q^{55} +(205227. + 280286. i) q^{57} +361351. i q^{59} -434726. q^{61} +397579. q^{63} +585260. i q^{65} -217638. i q^{67} +31747.0i q^{69} +125018. i q^{71} +130570. q^{73} -1.38548e6i q^{75} +274930. q^{77} +431829. i q^{79} +1.50137e6 q^{81} +618585. q^{83} +798010. q^{85} +1.12186e6 q^{87} +189521. i q^{89} -611273. i q^{91} +271784. q^{93} +(-1.14732e6 + 840072. i) q^{95} -589806. i q^{97} +2.33131e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 720 q^{7} - 8670 q^{9}+O(q^{10}) \) 30 * q + 720 * q^7 - 8670 * q^9	\( 30 q + 720 q^{7} - 8670 q^{9} + 2524 q^{11} + 9700 q^{17} - 4014 q^{19} + 39376 q^{23} + 110742 q^{25} + 19976 q^{35} + 266500 q^{39} + 106788 q^{43} - 91360 q^{45} - 222756 q^{47} + 593586 q^{49} - 540936 q^{55} - 545972 q^{57} - 242640 q^{61} + 377716 q^{63} + 545964 q^{73} - 272356 q^{77} + 2189926 q^{81} - 1542652 q^{83} - 826908 q^{85} + 2729572 q^{87} - 2139912 q^{93} - 2142716 q^{95} + 293012 q^{99}+O(q^{100}) \) 30 * q + 720 * q^7 - 8670 * q^9 + 2524 * q^11 + 9700 * q^17 - 4014 * q^19 + 39376 * q^23 + 110742 * q^25 + 19976 * q^35 + 266500 * q^39 + 106788 * q^43 - 91360 * q^45 - 222756 * q^47 + 593586 * q^49 - 540936 * q^55 - 545972 * q^57 - 242640 * q^61 + 377716 * q^63 + 545964 * q^73 - 272356 * q^77 + 2189926 * q^81 - 1542652 * q^83 - 826908 * q^85 + 2729572 * q^87 - 2139912 * q^93 - 2142716 * q^95 + 293012 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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