LMFDB - Embedded newform 304.7.e.e.113.5

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 5050x^{8} + 7354489x^{6} + 2475755792x^{4} + 232626987584x^{2} + 2900002611200 $ x^10 + 5050x^8 + 7354489x^6 + 2475755792x^4 + 232626987584x^2 + 2900002611200
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.5
Root			$-3.82791i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.7.e.e.113.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.65634i q^{3} +146.942 q^{5} +183.624 q^{7} +684.693 q^{9} +O(q^{10})$	$q-6.65634i q^{3} +146.942 q^{5} +183.624 q^{7} +684.693 q^{9} -1718.09 q^{11} -4162.60i q^{13} -978.092i q^{15} -5344.58 q^{17} +(-5783.77 + 3686.99i) q^{19} -1222.27i q^{21} -17446.9 q^{23} +5966.81 q^{25} -9410.02i q^{27} +21286.1i q^{29} -51794.2i q^{31} +11436.2i q^{33} +26982.0 q^{35} -81389.5i q^{37} -27707.7 q^{39} +98459.1i q^{41} +65509.9 q^{43} +100610. q^{45} +17280.8 q^{47} -83931.1 q^{49} +35575.3i q^{51} -39630.4i q^{53} -252459. q^{55} +(24541.9 + 38498.7i) q^{57} +293899. i q^{59} -80661.6 q^{61} +125726. q^{63} -611659. i q^{65} -351602. i q^{67} +116132. i q^{69} -98918.1i q^{71} -325874. q^{73} -39717.1i q^{75} -315484. q^{77} +137910. i q^{79} +436505. q^{81} +142009. q^{83} -785340. q^{85} +141687. q^{87} +379704. i q^{89} -764355. i q^{91} -344760. q^{93} +(-849875. + 541773. i) q^{95} +329181. i q^{97} -1.17637e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 112 q^{5} + 224 q^{7} - 2890 q^{9}+O(q^{10}) $ 10 * q - 112 * q^5 + 224 * q^7 - 2890 * q^9	$ 10 q - 112 q^{5} + 224 q^{7} - 2890 q^{9} - 3644 q^{11} - 10420 q^{17} + 17230 q^{19} - 37712 q^{23} - 52078 q^{25} + 161720 q^{35} + 78876 q^{39} - 6308 q^{43} + 309808 q^{45} - 322220 q^{47} - 235770 q^{49} + 377880 q^{55} + 24228 q^{57} + 426304 q^{61} + 517916 q^{63} - 786076 q^{73} + 2303716 q^{77} + 5261090 q^{81} + 101500 q^{83} - 1261380 q^{85} + 2460732 q^{87} - 2827032 q^{93} - 3106292 q^{95} - 1061428 q^{99}+O(q^{100}) $ 10 * q - 112 * q^5 + 224 * q^7 - 2890 * q^9 - 3644 * q^11 - 10420 * q^17 + 17230 * q^19 - 37712 * q^23 - 52078 * q^25 + 161720 * q^35 + 78876 * q^39 - 6308 * q^43 + 309808 * q^45 - 322220 * q^47 - 235770 * q^49 + 377880 * q^55 + 24228 * q^57 + 426304 * q^61 + 517916 * q^63 - 786076 * q^73 + 2303716 * q^77 + 5261090 * q^81 + 101500 * q^83 - 1261380 * q^85 + 2460732 * q^87 - 2827032 * q^93 - 3106292 * q^95 - 1061428 * 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
$3$		−	6.65634i		−	0.246531i	−0.992374	−	0.123266i	$-0.960663\pi$
$3$		−	6.65634i		−	0.246531i	0.992374	−	0.123266i	$-0.0393367\pi$
$4$		0			0
$5$	146.942			1.17553			0.587766	−	0.809031i	$-0.300008\pi$
$5$	146.942			1.17553			0.587766	+	0.809031i	$0.300008\pi$
$6$		0			0
$7$	183.624			0.535348			0.267674	−	0.963510i	$-0.413745\pi$
$7$	183.624			0.535348			0.267674	+	0.963510i	$0.413745\pi$
$8$		0			0
$9$	684.693			0.939222
$10$		0			0
$11$	−1718.09			−1.29083			−0.645415	−	0.763832i	$-0.723316\pi$
$11$	−1718.09			−1.29083			−0.645415	+	0.763832i	$0.723316\pi$
$12$		0			0
$13$		−	4162.60i		−	1.89468i	−0.320235	−	0.947338i	$-0.603762\pi$
$13$		−	4162.60i		−	1.89468i	0.320235	−	0.947338i	$-0.396238\pi$
$14$		0			0
$15$		−	978.092i		−	0.289805i
$16$		0			0
$17$	−5344.58			−1.08784			−0.543922	−	0.839136i	$-0.683061\pi$
$17$	−5344.58			−1.08784			−0.543922	+	0.839136i	$0.683061\pi$
$18$		0			0
$19$	−5783.77	+	3686.99i	−0.843238	+	0.537541i
$19$	−5783.77	+	3686.99i	−0.843238	+	0.537541i
$20$		0			0
$21$		−	1222.27i		−	0.131980i
$22$		0			0
$23$	−17446.9			−1.43395			−0.716974	−	0.697100i	$-0.754473\pi$
$23$	−17446.9			−1.43395			−0.716974	+	0.697100i	$0.754473\pi$
$24$		0			0
$25$	5966.81			0.381876
$26$		0			0
$27$		−	9410.02i		−	0.478079i
$28$		0			0
$29$			21286.1i			0.872775i	0.899759	+	0.436387i	$0.143742\pi$
$29$			21286.1i			0.872775i	−0.899759	+	0.436387i	$0.856258\pi$
$30$		0			0
$31$		−	51794.2i		−	1.73859i	−0.494298	−	0.869293i	$-0.664575\pi$
$31$		−	51794.2i		−	1.73859i	0.494298	−	0.869293i	$-0.335425\pi$
$32$		0			0
$33$			11436.2i			0.318230i
$34$		0			0
$35$	26982.0			0.629318
$36$		0			0
$37$		−	81389.5i		−	1.60681i	−0.595436	−	0.803403i	$-0.703021\pi$
$37$		−	81389.5i		−	1.60681i	0.595436	−	0.803403i	$-0.296979\pi$
$38$		0			0
$39$	−27707.7			−0.467097
$40$		0			0
$41$			98459.1i			1.42858i	0.699850	+	0.714289i	$0.253250\pi$
$41$			98459.1i			1.42858i	−0.699850	+	0.714289i	$0.746750\pi$
$42$		0			0
$43$	65509.9			0.823951			0.411975	−	0.911195i	$-0.364839\pi$
$43$	65509.9			0.823951			0.411975	+	0.911195i	$0.364839\pi$
$44$		0			0
$45$	100610.			1.10409
$46$		0			0
$47$	17280.8			0.166445			0.0832226	−	0.996531i	$-0.473479\pi$
$47$	17280.8			0.166445			0.0832226	+	0.996531i	$0.473479\pi$
$48$		0			0
$49$	−83931.1			−0.713403
$50$		0			0
$51$			35575.3i			0.268187i
$52$		0			0
$53$		−	39630.4i		−	0.266195i	−0.991103	−	0.133098i	$-0.957508\pi$
$53$		−	39630.4i		−	0.266195i	0.991103	−	0.133098i	$-0.0424924\pi$
$54$		0			0
$55$	−252459.			−1.51741
$56$		0			0
$57$	24541.9	+	38498.7i	0.132521	+	0.207884i
$58$		0			0
$59$			293899.i			1.43101i	0.698609	+	0.715503i	$0.253803\pi$
$59$			293899.i			1.43101i	−0.698609	+	0.715503i	$0.746197\pi$
$60$		0			0
$61$	−80661.6			−0.355367			−0.177683	−	0.984088i	$-0.556860\pi$
$61$	−80661.6			−0.355367			−0.177683	+	0.984088i	$0.556860\pi$
$62$		0			0
$63$	125726.			0.502811
$64$		0			0
$65$		−	611659.i		−	2.22725i
$66$		0			0
$67$		−	351602.i		−	1.16903i	−0.811381	−	0.584517i	$-0.801284\pi$
$67$		−	351602.i		−	1.16903i	0.811381	−	0.584517i	$-0.198716\pi$
$68$		0			0
$69$			116132.i			0.353513i
$70$		0			0
$71$		−	98918.1i		−	0.276376i	−0.990406	−	0.138188i	$-0.955872\pi$
$71$		−	98918.1i		−	0.276376i	0.990406	−	0.138188i	$-0.0441278\pi$
$72$		0			0
$73$	−325874.			−0.837685			−0.418843	−	0.908059i	$-0.637564\pi$
$73$	−325874.			−0.837685			−0.418843	+	0.908059i	$0.637564\pi$
$74$		0			0
$75$		−	39717.1i		−	0.0941443i
$76$		0			0
$77$	−315484.			−0.691043
$78$		0			0
$79$			137910.i			0.279714i	0.990172	+	0.139857i	$0.0446643\pi$
$79$			137910.i			0.279714i	−0.990172	+	0.139857i	$0.955336\pi$
$80$		0			0
$81$	436505.			0.821361
$82$		0			0
$83$	142009.			0.248360			0.124180	−	0.992260i	$-0.460370\pi$
$83$	142009.			0.248360			0.124180	+	0.992260i	$0.460370\pi$
$84$		0			0
$85$	−785340.			−1.27880
$86$		0			0
$87$	141687.			0.215166
$88$		0			0
$89$			379704.i			0.538610i	0.963055	+	0.269305i	$0.0867940\pi$
$89$			379704.i			0.538610i	−0.963055	+	0.269305i	$0.913206\pi$
$90$		0			0
$91$		−	764355.i		−	1.01431i
$92$		0			0
$93$	−344760.			−0.428615
$94$		0			0
$95$	−849875.	+	541773.i	−0.991253	+	0.631897i
$96$		0			0
$97$			329181.i			0.360678i	0.983604	+	0.180339i	$0.0577195\pi$
$97$			329181.i			0.360678i	−0.983604	+	0.180339i	$0.942280\pi$
$98$		0			0
$99$	−1.17637e6			−1.21238
$100$		0			0
$101$	−1.17948e6			−1.14480			−0.572398	−	0.819976i	$-0.693987\pi$
$101$	−1.17948e6			−1.14480			−0.572398	+	0.819976i	$0.693987\pi$
$102$		0			0
$103$		−	1.38891e6i		−	1.27105i	−0.772081	−	0.635524i	$-0.780784\pi$
$103$		−	1.38891e6i		−	1.27105i	0.772081	−	0.635524i	$-0.219216\pi$
$104$		0			0
$105$		−	179602.i		−	0.155147i
$106$		0			0
$107$		−	65991.3i		−	0.0538685i	−0.999637	−	0.0269343i	$-0.991426\pi$
$107$		−	65991.3i		−	0.0538685i	0.999637	−	0.0269343i	$-0.00857448\pi$
$108$		0			0
$109$		−	836576.i		−	0.645990i	−0.946401	−	0.322995i	$-0.895310\pi$
$109$		−	836576.i		−	0.645990i	0.946401	−	0.322995i	$-0.104690\pi$
$110$		0			0
$111$	−541756.			−0.396128
$112$		0			0
$113$			7470.62i			0.00517752i	0.999997	+	0.00258876i	$0.000824028\pi$
$113$			7470.62i			0.00517752i	−0.999997	+	0.00258876i	$0.999176\pi$
$114$		0			0
$115$	−2.56367e6			−1.68565
$116$		0			0
$117$		−	2.85011e6i		−	1.77952i
$118$		0			0
$119$	−981394.			−0.582375
$120$		0			0
$121$	1.18029e6			0.666242
$122$		0			0
$123$	655377.			0.352189
$124$		0			0
$125$	−1.41919e6			−0.726625
$126$		0			0
$127$		−	1.21723e6i		−	0.594239i	−0.954840	−	0.297119i	$-0.903974\pi$
$127$		−	1.21723e6i		−	0.594239i	0.954840	−	0.297119i	$-0.0960260\pi$
$128$		0			0
$129$		−	436056.i		−	0.203129i
$130$		0			0
$131$	−1.72205e6			−0.766007			−0.383003	−	0.923747i	$-0.625110\pi$
$131$	−1.72205e6			−0.766007			−0.383003	+	0.923747i	$0.625110\pi$
$132$		0			0
$133$	−1.06204e6	+	677022.i	−0.451425	+	0.287771i
$134$		0			0
$135$		−	1.38272e6i		−	0.561997i
$136$		0			0
$137$	2.87455e6			1.11791			0.558957	−	0.829197i	$-0.311202\pi$
$137$	2.87455e6			1.11791			0.558957	+	0.829197i	$0.311202\pi$
$138$		0			0
$139$	3.94916e6			1.47048			0.735241	−	0.677805i	$-0.237069\pi$
$139$	3.94916e6			1.47048			0.735241	+	0.677805i	$0.237069\pi$
$140$		0			0
$141$		−	115027.i		−	0.0410339i
$142$		0			0
$143$			7.15175e6i			2.44571i
$144$		0			0
$145$			3.12781e6i			1.02597i
$146$		0			0
$147$			558674.i			0.175876i
$148$		0			0
$149$	183584.			0.0554978			0.0277489	−	0.999615i	$-0.491166\pi$
$149$	183584.			0.0554978			0.0277489	+	0.999615i	$0.491166\pi$
$150$		0			0
$151$		−	3.51046e6i		−	1.01961i	−0.860291	−	0.509804i	$-0.829718\pi$
$151$		−	3.51046e6i		−	1.01961i	0.860291	−	0.509804i	$-0.170282\pi$
$152$		0			0
$153$	−3.65940e6			−1.02173
$154$		0			0
$155$		−	7.61072e6i		−	2.04376i
$156$		0			0
$157$	3.21864e6			0.831714			0.415857	−	0.909430i	$-0.363482\pi$
$157$	3.21864e6			0.831714			0.415857	+	0.909430i	$0.363482\pi$
$158$		0			0
$159$	−263793.			−0.0656254
$160$		0			0
$161$	−3.20366e6			−0.767661
$162$		0			0
$163$	−1.57947e6			−0.364711			−0.182356	−	0.983233i	$-0.558372\pi$
$163$	−1.57947e6			−0.364711			−0.182356	+	0.983233i	$0.558372\pi$
$164$		0			0
$165$			1.68046e6i			0.374089i
$166$		0			0
$167$		−	961133.i		−	0.206364i	−0.994662	−	0.103182i	$-0.967098\pi$
$167$		−	961133.i		−	0.206364i	0.994662	−	0.103182i	$-0.0329024\pi$
$168$		0			0
$169$	−1.25005e7			−2.58980
$170$		0			0
$171$	−3.96011e6	+	2.52446e6i	−0.791988	+	0.504871i
$172$		0			0
$173$		−	7.05698e6i		−	1.36295i	−0.731840	−	0.681476i	$-0.761338\pi$
$173$		−	7.05698e6i		−	1.36295i	0.731840	−	0.681476i	$-0.238662\pi$
$174$		0			0
$175$	1.09565e6			0.204436
$176$		0			0
$177$	1.95629e6			0.352788
$178$		0			0
$179$		−	2.56000e6i		−	0.446356i	−0.974778	−	0.223178i	$-0.928357\pi$
$179$		−	2.56000e6i		−	0.446356i	0.974778	−	0.223178i	$-0.0716431\pi$
$180$		0			0
$181$			1.09680e6i			0.184966i	0.995714	+	0.0924828i	$0.0294803\pi$
$181$			1.09680e6i			0.184966i	−0.995714	+	0.0924828i	$0.970520\pi$
$182$		0			0
$183$			536911.i			0.0876090i
$184$		0			0
$185$		−	1.19595e7i		−	1.88885i
$186$		0			0
$187$	9.18249e6			1.40422
$188$		0			0
$189$		−	1.72791e6i		−	0.255938i
$190$		0			0
$191$	526703.			0.0755903			0.0377951	−	0.999286i	$-0.487967\pi$
$191$	526703.			0.0755903			0.0377951	+	0.999286i	$0.487967\pi$
$192$		0			0
$193$		−	1.69440e6i		−	0.235691i	−0.993032	−	0.117846i	$-0.962401\pi$
$193$		−	1.69440e6i		−	0.235691i	0.993032	−	0.117846i	$-0.0375988\pi$
$194$		0			0
$195$	−4.07141e6			−0.549087
$196$		0			0
$197$	2.22350e6			0.290830			0.145415	−	0.989371i	$-0.453548\pi$
$197$	2.22350e6			0.290830			0.145415	+	0.989371i	$0.453548\pi$
$198$		0			0
$199$	−7.32774e6			−0.929846			−0.464923	−	0.885351i	$-0.653918\pi$
$199$	−7.32774e6			−0.929846			−0.464923	+	0.885351i	$0.653918\pi$
$200$		0			0
$201$	−2.34038e6			−0.288203
$202$		0			0
$203$			3.90864e6i			0.467238i
$204$		0			0
$205$			1.44677e7i			1.67934i
$206$		0			0
$207$	−1.19457e7			−1.34680
$208$		0			0
$209$	9.93706e6	−	6.33461e6i	1.08848	−	0.693874i
$210$		0			0
$211$			1.58228e7i			1.68437i	0.539192	+	0.842183i	$0.318730\pi$
$211$			1.58228e7i			1.68437i	−0.539192	+	0.842183i	$0.681270\pi$
$212$		0			0
$213$	−658432.			−0.0681353
$214$		0			0
$215$	9.62612e6			0.968581
$216$		0			0
$217$		−	9.51067e6i		−	0.930748i
$218$		0			0
$219$			2.16913e6i			0.206515i
$220$		0			0
$221$			2.22474e7i			2.06111i
$222$		0			0
$223$		−	9.02405e6i		−	0.813743i	−0.913485	−	0.406871i	$-0.866620\pi$
$223$		−	9.02405e6i		−	0.813743i	0.913485	−	0.406871i	$-0.133380\pi$
$224$		0			0
$225$	4.08544e6			0.358667
$226$		0			0
$227$		−	1.04551e7i		−	0.893819i	−0.894579	−	0.446909i	$-0.852525\pi$
$227$		−	1.04551e7i		−	0.893819i	0.894579	−	0.446909i	$-0.147475\pi$
$228$		0			0
$229$	2.28936e7			1.90637			0.953185	−	0.302387i	$-0.0977835\pi$
$229$	2.28936e7			1.90637			0.953185	+	0.302387i	$0.0977835\pi$
$230$		0			0
$231$			2.09997e6i			0.170364i
$232$		0			0
$233$	−1.82578e7			−1.44338			−0.721689	−	0.692218i	$-0.756634\pi$
$233$	−1.82578e7			−1.44338			−0.721689	+	0.692218i	$0.756634\pi$
$234$		0			0
$235$	2.53927e6			0.195662
$236$		0			0
$237$	917975.			0.0689582
$238$		0			0
$239$	7.13532e6			0.522660			0.261330	−	0.965249i	$-0.415839\pi$
$239$	7.13532e6			0.522660			0.261330	+	0.965249i	$0.415839\pi$
$240$		0			0
$241$			2.74824e7i			1.96338i	0.190495	+	0.981688i	$0.438991\pi$
$241$			2.74824e7i			1.96338i	−0.190495	+	0.981688i	$0.561009\pi$
$242$		0			0
$243$		−	9.76543e6i		−	0.680570i
$244$		0			0
$245$	−1.23330e7			−0.838628
$246$		0			0
$247$	1.53475e7	+	2.40755e7i	1.01847	+	1.59766i
$248$		0			0
$249$		−	945258.i		−	0.0612283i
$250$		0			0
$251$	9.86492e6			0.623839			0.311919	−	0.950109i	$-0.399028\pi$
$251$	9.86492e6			0.623839			0.311919	+	0.950109i	$0.399028\pi$
$252$		0			0
$253$	2.99753e7			1.85098
$254$		0			0
$255$			5.22749e6i			0.315263i
$256$		0			0
$257$		−	1.06898e7i		−	0.629751i	−0.949133	−	0.314876i	$-0.898037\pi$
$257$		−	1.06898e7i		−	0.629751i	0.949133	−	0.314876i	$-0.101963\pi$
$258$		0			0
$259$		−	1.49451e7i		−	0.860200i
$260$		0			0
$261$			1.45744e7i			0.819730i
$262$		0			0
$263$	−1.67268e6			−0.0919485			−0.0459742	−	0.998943i	$-0.514639\pi$
$263$	−1.67268e6			−0.0919485			−0.0459742	+	0.998943i	$0.514639\pi$
$264$		0			0
$265$		−	5.82335e6i		−	0.312921i
$266$		0			0
$267$	2.52744e6			0.132784
$268$		0			0
$269$		−	1.91376e6i		−	0.0983177i	−0.998791	−	0.0491588i	$-0.984346\pi$
$269$		−	1.91376e6i		−	0.0983177i	0.998791	−	0.0491588i	$-0.0156541\pi$
$270$		0			0
$271$	8.66253e6			0.435248			0.217624	−	0.976033i	$-0.430169\pi$
$271$	8.66253e6			0.435248			0.217624	+	0.976033i	$0.430169\pi$
$272$		0			0
$273$	−5.08781e6			−0.250059
$274$		0			0
$275$	−1.02516e7			−0.492937
$276$		0			0
$277$	3.50325e7			1.64828			0.824142	−	0.566384i	$-0.191658\pi$
$277$	3.50325e7			1.64828			0.824142	+	0.566384i	$0.191658\pi$
$278$		0			0
$279$		−	3.54631e7i		−	1.63292i
$280$		0			0
$281$			5.99966e6i			0.270401i	0.990818	+	0.135200i	$0.0431678\pi$
$281$			5.99966e6i			0.270401i	−0.990818	+	0.135200i	$0.956832\pi$
$282$		0			0
$283$	−2.33721e6			−0.103119			−0.0515594	−	0.998670i	$-0.516419\pi$
$283$	−2.33721e6			−0.103119			−0.0515594	+	0.998670i	$0.516419\pi$
$284$		0			0
$285$	3.60622e6	+	5.65706e6i	0.155782	+	0.244375i
$286$		0			0
$287$			1.80795e7i			0.764786i
$288$		0			0
$289$	4.42695e6			0.183405
$290$		0			0
$291$	2.19114e6			0.0889184
$292$		0			0
$293$		−	1.96602e7i		−	0.781603i	−0.920475	−	0.390801i	$-0.872198\pi$
$293$		−	1.96602e7i		−	0.781603i	0.920475	−	0.390801i	$-0.127802\pi$
$294$		0			0
$295$			4.31859e7i			1.68219i
$296$		0			0
$297$			1.61673e7i			0.617118i
$298$		0			0
$299$			7.26243e7i			2.71687i
$300$		0			0
$301$	1.20292e7			0.441100
$302$		0			0
$303$			7.85104e6i			0.282228i
$304$		0			0
$305$	−1.18525e7			−0.417745
$306$		0			0
$307$			1.81733e7i			0.628087i	0.949409	+	0.314043i	$0.101684\pi$
$307$			1.81733e7i			0.628087i	−0.949409	+	0.314043i	$0.898316\pi$
$308$		0			0
$309$	−9.24504e6			−0.313353
$310$		0			0
$311$	4.39266e7			1.46031			0.730157	−	0.683280i	$-0.239447\pi$
$311$	4.39266e7			1.46031			0.730157	+	0.683280i	$0.239447\pi$
$312$		0			0
$313$	−1.83265e6			−0.0597650			−0.0298825	−	0.999553i	$-0.509513\pi$
$313$	−1.83265e6			−0.0597650			−0.0298825	+	0.999553i	$0.509513\pi$
$314$		0			0
$315$	1.84744e7			0.591070
$316$		0			0
$317$			4.42788e6i			0.139001i	0.997582	+	0.0695005i	$0.0221406\pi$
$317$			4.42788e6i			0.139001i	−0.997582	+	0.0695005i	$0.977859\pi$
$318$		0			0
$319$		−	3.65715e7i		−	1.12660i
$320$		0			0
$321$	−439260.			−0.0132803
$322$		0			0
$323$	3.09118e7	−	1.97054e7i	0.917311	−	0.584761i
$324$		0			0
$325$		−	2.48375e7i		−	0.723532i
$326$		0			0
$327$	−5.56853e6			−0.159257
$328$		0			0
$329$	3.17318e6			0.0891061
$330$		0			0
$331$		−	1.52513e7i		−	0.420556i	−0.977642	−	0.210278i	$-0.932563\pi$
$331$		−	1.52513e7i		−	0.420556i	0.977642	−	0.210278i	$-0.0674369\pi$
$332$		0			0
$333$		−	5.57269e7i		−	1.50915i
$334$		0			0
$335$		−	5.16650e7i		−	1.37424i
$336$		0			0
$337$		−	4.46345e7i		−	1.16622i	−0.812393	−	0.583111i	$-0.801835\pi$
$337$		−	4.46345e7i		−	1.16622i	0.812393	−	0.583111i	$-0.198165\pi$
$338$		0			0
$339$	49727.0			0.00127642
$340$		0			0
$341$			8.89873e7i			2.24422i
$342$		0			0
$343$	−3.70150e7			−0.917266
$344$		0			0
$345$			1.70646e7i			0.415566i
$346$		0			0
$347$	2.82394e7			0.675876			0.337938	−	0.941168i	$-0.390271\pi$
$347$	2.82394e7			0.675876			0.337938	+	0.941168i	$0.390271\pi$
$348$		0			0
$349$	−1.29030e6			−0.0303539			−0.0151769	−	0.999885i	$-0.504831\pi$
$349$	−1.29030e6			−0.0303539			−0.0151769	+	0.999885i	$0.504831\pi$
$350$		0			0
$351$	−3.91702e7			−0.905804
$352$		0			0
$353$	−4.60022e7			−1.04581			−0.522907	−	0.852390i	$-0.675152\pi$
$353$	−4.60022e7			−1.04581			−0.522907	+	0.852390i	$0.675152\pi$
$354$		0			0
$355$		−	1.45352e7i		−	0.324889i
$356$		0			0
$357$			6.53249e6i			0.143573i
$358$		0			0
$359$	−4.28733e7			−0.926623			−0.463312	−	0.886195i	$-0.653339\pi$
$359$	−4.28733e7			−0.926623			−0.463312	+	0.886195i	$0.653339\pi$
$360$		0			0
$361$	1.98580e7	−	4.26494e7i	0.422099	−	0.906550i
$362$		0			0
$363$		−	7.85640e6i		−	0.164249i
$364$		0			0
$365$	−4.78844e7			−0.984726
$366$		0			0
$367$	1.41368e7			0.285991			0.142995	−	0.989723i	$-0.454327\pi$
$367$	1.41368e7			0.285991			0.142995	+	0.989723i	$0.454327\pi$
$368$		0			0
$369$			6.74143e7i			1.34175i
$370$		0			0
$371$		−	7.27710e6i		−	0.142507i
$372$		0			0
$373$			3.96596e7i			0.764226i	0.924116	+	0.382113i	$0.124803\pi$
$373$			3.96596e7i			0.764226i	−0.924116	+	0.382113i	$0.875197\pi$
$374$		0			0
$375$			9.44660e6i			0.179136i
$376$		0			0
$377$	8.86056e7			1.65363
$378$		0			0
$379$			2.72051e7i			0.499727i	0.968281	+	0.249863i	$0.0803857\pi$
$379$			2.72051e7i			0.499727i	−0.968281	+	0.249863i	$0.919614\pi$
$380$		0			0
$381$	−8.10229e6			−0.146498
$382$		0			0
$383$		−	2.60602e7i		−	0.463854i	−0.972733	−	0.231927i	$-0.925497\pi$
$383$		−	2.60602e7i		−	0.463854i	0.972733	−	0.231927i	$-0.0745031\pi$
$384$		0			0
$385$	−4.63577e7			−0.812343
$386$		0			0
$387$	4.48542e7			0.773873
$388$		0			0
$389$	3.80804e7			0.646923			0.323461	−	0.946241i	$-0.395153\pi$
$389$	3.80804e7			0.646923			0.323461	+	0.946241i	$0.395153\pi$
$390$		0			0
$391$	9.32461e7			1.55991
$392$		0			0
$393$			1.14626e7i			0.188844i
$394$		0			0
$395$			2.02647e7i			0.328813i
$396$		0			0
$397$	−3.09038e6			−0.0493901			−0.0246951	−	0.999695i	$-0.507861\pi$
$397$	−3.09038e6			−0.0493901			−0.0246951	+	0.999695i	$0.507861\pi$
$398$		0			0
$399$	4.50648e6	+	7.06930e6i	0.0709446	+	0.111290i
$400$		0			0
$401$		−	4.93134e7i		−	0.764772i	−0.924003	−	0.382386i	$-0.875102\pi$
$401$		−	4.93134e7i		−	0.764772i	0.924003	−	0.382386i	$-0.124898\pi$
$402$		0			0
$403$	−2.15599e8			−3.29406
$404$		0			0
$405$	6.41407e7			0.965537
$406$		0			0
$407$			1.39835e8i			2.07411i
$408$		0			0
$409$		−	6.10251e7i		−	0.891947i	−0.895046	−	0.445973i	$-0.852858\pi$
$409$		−	6.10251e7i		−	0.891947i	0.895046	−	0.445973i	$-0.147142\pi$
$410$		0			0
$411$		−	1.91340e7i		−	0.275600i
$412$		0			0
$413$			5.39669e7i			0.766086i
$414$		0			0
$415$	2.08670e7			0.291955
$416$		0			0
$417$		−	2.62869e7i		−	0.362520i
$418$		0			0
$419$	6.74617e6			0.0917097			0.0458549	−	0.998948i	$-0.485399\pi$
$419$	6.74617e6			0.0917097			0.0458549	+	0.998948i	$0.485399\pi$
$420$		0			0
$421$			5.49198e7i			0.736009i	0.929824	+	0.368004i	$0.119959\pi$
$421$			5.49198e7i			0.736009i	−0.929824	+	0.368004i	$0.880041\pi$
$422$		0			0
$423$	1.18321e7			0.156329
$424$		0			0
$425$	−3.18901e7			−0.415422
$426$		0			0
$427$	−1.48114e7			−0.190245
$428$		0			0
$429$	4.76044e7			0.602942
$430$		0			0
$431$		−	1.14851e8i		−	1.43451i	−0.696812	−	0.717254i	$-0.745399\pi$
$431$		−	1.14851e8i		−	1.43451i	0.696812	−	0.717254i	$-0.254601\pi$
$432$		0			0
$433$			9.64970e6i			0.118864i	0.998232	+	0.0594320i	$0.0189289\pi$
$433$			9.64970e6i			0.118864i	−0.998232	+	0.0594320i	$0.981071\pi$
$434$		0			0
$435$	2.08198e7			0.252935
$436$		0			0
$437$	1.00909e8	−	6.43264e7i	1.20916	−	0.770806i
$438$		0			0
$439$			7.17354e7i			0.847891i	0.905688	+	0.423945i	$0.139355\pi$
$439$			7.17354e7i			0.847891i	−0.905688	+	0.423945i	$0.860645\pi$
$440$		0			0
$441$	−5.74671e7			−0.670044
$442$		0			0
$443$	−9.91448e7			−1.14040			−0.570202	−	0.821504i	$-0.693135\pi$
$443$	−9.91448e7			−1.14040			−0.570202	+	0.821504i	$0.693135\pi$
$444$		0			0
$445$			5.57942e7i			0.633154i
$446$		0			0
$447$		−	1.22200e6i		−	0.0136819i
$448$		0			0
$449$		−	1.47824e8i		−	1.63307i	−0.577296	−	0.816535i	$-0.695892\pi$
$449$		−	1.47824e8i		−	1.63307i	0.577296	−	0.816535i	$-0.304108\pi$
$450$		0			0
$451$		−	1.69162e8i		−	1.84405i
$452$		0			0
$453$	−2.33668e7			−0.251365
$454$		0			0
$455$		−	1.12316e8i		−	1.19235i
$456$		0			0
$457$	−1.25418e8			−1.31405			−0.657023	−	0.753870i	$-0.728185\pi$
$457$	−1.25418e8			−1.31405			−0.657023	+	0.753870i	$0.728185\pi$
$458$		0			0
$459$			5.02926e7i			0.520075i
$460$		0			0
$461$	−5.25946e7			−0.536831			−0.268416	−	0.963303i	$-0.586500\pi$
$461$	−5.25946e7			−0.536831			−0.268416	+	0.963303i	$0.586500\pi$
$462$		0			0
$463$	−4.14145e7			−0.417262			−0.208631	−	0.977994i	$-0.566901\pi$
$463$	−4.14145e7			−0.417262			−0.208631	+	0.977994i	$0.566901\pi$
$464$		0			0
$465$	−5.06595e7			−0.503851
$466$		0			0
$467$	−6.02070e7			−0.591148			−0.295574	−	0.955320i	$-0.595511\pi$
$467$	−6.02070e7			−0.591148			−0.295574	+	0.955320i	$0.595511\pi$
$468$		0			0
$469$		−	6.45627e7i		−	0.625840i
$470$		0			0
$471$		−	2.14244e7i		−	0.205043i
$472$		0			0
$473$	−1.12552e8			−1.06358
$474$		0			0
$475$	−3.45107e7	+	2.19996e7i	−0.322012	+	0.205274i
$476$		0			0
$477$		−	2.71346e7i		−	0.250017i
$478$		0			0
$479$	−1.18008e8			−1.07375			−0.536876	−	0.843661i	$-0.680396\pi$
$479$	−1.18008e8			−1.07375			−0.536876	+	0.843661i	$0.680396\pi$
$480$		0			0
$481$	−3.38792e8			−3.04438
$482$		0			0
$483$			2.13247e7i			0.189252i
$484$		0			0
$485$			4.83704e7i			0.423989i
$486$		0			0
$487$		−	8.37555e7i		−	0.725148i	−0.931955	−	0.362574i	$-0.881898\pi$
$487$		−	8.37555e7i		−	0.725148i	0.931955	−	0.362574i	$-0.118102\pi$
$488$		0			0
$489$			1.05135e7i			0.0899126i
$490$		0			0
$491$	9.11236e7			0.769815			0.384908	−	0.922955i	$-0.374233\pi$
$491$	9.11236e7			0.769815			0.384908	+	0.922955i	$0.374233\pi$
$492$		0			0
$493$		−	1.13765e8i		−	0.949443i
$494$		0			0
$495$	−1.72857e8			−1.42519
$496$		0			0
$497$		−	1.81638e7i		−	0.147957i
$498$		0			0
$499$	7.45201e7			0.599752			0.299876	−	0.953978i	$-0.403055\pi$
$499$	7.45201e7			0.599752			0.299876	+	0.953978i	$0.403055\pi$
$500$		0			0
$501$	−6.39763e6			−0.0508752
$502$		0			0
$503$	2.21380e8			1.73954			0.869769	−	0.493460i	$-0.164268\pi$
$503$	2.21380e8			1.73954			0.869769	+	0.493460i	$0.164268\pi$
$504$		0			0
$505$	−1.73315e8			−1.34574
$506$		0			0
$507$			8.32073e7i			0.638466i
$508$		0			0
$509$			1.44062e8i			1.09244i	0.837642	+	0.546220i	$0.183934\pi$
$509$			1.44062e8i			1.09244i	−0.837642	+	0.546220i	$0.816066\pi$
$510$		0			0
$511$	−5.98383e7			−0.448453
$512$		0			0
$513$	3.46947e7	+	5.44254e7i	0.256987	+	0.403134i
$514$		0			0
$515$		−	2.04088e8i		−	1.49416i
$516$		0			0
$517$	−2.96901e7			−0.214852
$518$		0			0
$519$	−4.69736e7			−0.336010
$520$		0			0
$521$		−	1.83396e8i		−	1.29681i	−0.761296	−	0.648405i	$-0.775436\pi$
$521$		−	1.83396e8i		−	1.29681i	0.761296	−	0.648405i	$-0.224564\pi$
$522$		0			0
$523$		−	2.60056e8i		−	1.81786i	−0.416946	−	0.908931i	$-0.636900\pi$
$523$		−	2.60056e8i		−	1.81786i	0.416946	−	0.908931i	$-0.363100\pi$
$524$		0			0
$525$		−	7.29303e6i		−	0.0503999i
$526$		0			0
$527$			2.76818e8i			1.89131i
$528$		0			0
$529$	1.56357e8			1.05621
$530$		0			0
$531$			2.01230e8i			1.34403i
$532$		0			0
$533$	4.09846e8			2.70669
$534$		0			0
$535$		−	9.69686e6i		−	0.0633242i
$536$		0			0
$537$	−1.70402e7			−0.110041
$538$		0			0
$539$	1.44202e8			0.920882
$540$		0			0
$541$	2.68741e8			1.69723			0.848617	−	0.529008i	$-0.177436\pi$
$541$	2.68741e8			1.69723			0.848617	+	0.529008i	$0.177436\pi$
$542$		0			0
$543$	7.30066e6			0.0455998
$544$		0			0
$545$		−	1.22928e8i		−	0.759382i
$546$		0			0
$547$			1.38572e8i			0.846671i	0.905973	+	0.423335i	$0.139141\pi$
$547$			1.38572e8i			0.846671i	−0.905973	+	0.423335i	$0.860859\pi$
$548$		0			0
$549$	−5.52284e7			−0.333769
$550$		0			0
$551$	−7.84817e7	−	1.23114e8i	−0.469152	−	0.735956i
$552$		0			0
$553$			2.53236e7i			0.149744i
$554$		0			0
$555$	−7.96065e7			−0.465661
$556$		0			0
$557$	−1.94423e8			−1.12508			−0.562538	−	0.826771i	$-0.690175\pi$
$557$	−1.94423e8			−1.12508			−0.562538	+	0.826771i	$0.690175\pi$
$558$		0			0
$559$		−	2.72692e8i		−	1.56112i
$560$		0			0
$561$		−	6.11218e7i		−	0.346184i
$562$		0			0
$563$			3.21069e8i			1.79917i	0.436744	+	0.899586i	$0.356132\pi$
$563$			3.21069e8i			1.79917i	−0.436744	+	0.899586i	$0.643868\pi$
$564$		0			0
$565$			1.09774e6i			0.00608634i
$566$		0			0
$567$	8.01529e7			0.439714
$568$		0			0
$569$			1.83897e8i			0.998248i	0.866531	+	0.499124i	$0.166345\pi$
$569$			1.83897e8i			0.998248i	−0.866531	+	0.499124i	$0.833655\pi$
$570$		0			0
$571$	−9.74111e6			−0.0523239			−0.0261619	−	0.999658i	$-0.508329\pi$
$571$	−9.74111e6			−0.0523239			−0.0261619	+	0.999658i	$0.508329\pi$
$572$		0			0
$573$		−	3.50591e6i		−	0.0186353i
$574$		0			0
$575$	−1.04102e8			−0.547591
$576$		0			0
$577$	−1.67489e8			−0.871886			−0.435943	−	0.899974i	$-0.643585\pi$
$577$	−1.67489e8			−0.871886			−0.435943	+	0.899974i	$0.643585\pi$
$578$		0			0
$579$	−1.12785e7			−0.0581053
$580$		0			0
$581$	2.60763e7			0.132959
$582$		0			0
$583$			6.80887e7i			0.343613i
$584$		0			0
$585$		−	4.18799e8i		−	2.09189i
$586$		0			0
$587$	1.87247e8			0.925763			0.462882	−	0.886420i	$-0.346815\pi$
$587$	1.87247e8			0.925763			0.462882	+	0.886420i	$0.346815\pi$
$588$		0			0
$589$	1.90965e8	+	2.99566e8i	0.934561	+	1.46604i
$590$		0			0
$591$		−	1.48004e7i		−	0.0716986i
$592$		0			0
$593$	1.56747e8			0.751683			0.375842	−	0.926684i	$-0.377354\pi$
$593$	1.56747e8			0.751683			0.375842	+	0.926684i	$0.377354\pi$
$594$		0			0
$595$	−1.44208e8			−0.684600
$596$		0			0
$597$			4.87759e7i			0.229236i
$598$		0			0
$599$			1.42517e8i			0.663112i	0.943435	+	0.331556i	$0.107574\pi$
$599$			1.42517e8i			0.663112i	−0.943435	+	0.331556i	$0.892426\pi$
$600$		0			0
$601$			1.48027e8i			0.681894i	0.940083	+	0.340947i	$0.110748\pi$
$601$			1.48027e8i			0.681894i	−0.940083	+	0.340947i	$0.889252\pi$
$602$		0			0
$603$		−	2.40740e8i		−	1.09798i
$604$		0			0
$605$	1.73433e8			0.783189
$606$		0			0
$607$			7.58253e7i			0.339038i	0.985527	+	0.169519i	$0.0542214\pi$
$607$			7.58253e7i			0.339038i	−0.985527	+	0.169519i	$0.945779\pi$
$608$		0			0
$609$	2.60173e7			0.115189
$610$		0			0
$611$		−	7.19333e7i		−	0.315360i
$612$		0			0
$613$	−2.23802e8			−0.971589			−0.485794	−	0.874073i	$-0.661470\pi$
$613$	−2.23802e8			−0.971589			−0.485794	+	0.874073i	$0.661470\pi$
$614$		0			0
$615$	9.63021e7			0.414010
$616$		0			0
$617$	1.50027e8			0.638726			0.319363	−	0.947633i	$-0.396531\pi$
$617$	1.50027e8			0.638726			0.319363	+	0.947633i	$0.396531\pi$
$618$		0			0
$619$	−3.37666e7			−0.142369			−0.0711844	−	0.997463i	$-0.522678\pi$
$619$	−3.37666e7			−0.142369			−0.0711844	+	0.997463i	$0.522678\pi$
$620$		0			0
$621$			1.64175e8i			0.685540i
$622$		0			0
$623$			6.97228e7i			0.288344i
$624$		0			0
$625$	−3.01769e8			−1.23605
$626$		0			0
$627$	−4.21653e7	−	6.61444e7i	−0.171062	−	0.268343i
$628$		0			0
$629$			4.34993e8i			1.74795i
$630$		0			0
$631$	8.02120e7			0.319265			0.159633	−	0.987177i	$-0.448969\pi$
$631$	8.02120e7			0.319265			0.159633	+	0.987177i	$0.448969\pi$
$632$		0			0
$633$	1.05322e8			0.415248
$634$		0			0
$635$		−	1.78861e8i		−	0.698547i
$636$		0			0
$637$			3.49372e8i			1.35167i
$638$		0			0
$639$		−	6.77285e7i		−	0.259579i
$640$		0			0
$641$			2.30796e8i			0.876303i	0.898901	+	0.438151i	$0.144367\pi$
$641$			2.30796e8i			0.876303i	−0.898901	+	0.438151i	$0.855633\pi$
$642$		0			0
$643$	4.17210e8			1.56936			0.784678	−	0.619903i	$-0.212828\pi$
$643$	4.17210e8			1.56936			0.784678	+	0.619903i	$0.212828\pi$
$644$		0			0
$645$		−	6.40747e7i		−	0.238785i
$646$		0			0
$647$	−2.74624e8			−1.01397			−0.506986	−	0.861954i	$-0.669240\pi$
$647$	−2.74624e8			−1.01397			−0.506986	+	0.861954i	$0.669240\pi$
$648$		0			0
$649$		−	5.04946e8i		−	1.84719i
$650$		0			0
$651$	−6.33062e7			−0.229458
$652$		0			0
$653$	3.30922e8			1.18847			0.594233	−	0.804293i	$-0.297456\pi$
$653$	3.30922e8			1.18847			0.594233	+	0.804293i	$0.297456\pi$
$654$		0			0
$655$	−2.53041e8			−0.900466
$656$		0			0
$657$	−2.23124e8			−0.786773
$658$		0			0
$659$			4.03864e8i			1.41117i	0.708625	+	0.705585i	$0.249316\pi$
$659$			4.03864e8i			1.41117i	−0.708625	+	0.705585i	$0.750684\pi$
$660$		0			0
$661$		−	2.70497e8i		−	0.936608i	−0.883568	−	0.468304i	$-0.844865\pi$
$661$		−	2.70497e8i		−	0.936608i	0.883568	−	0.468304i	$-0.155135\pi$
$662$		0			0
$663$	1.48086e8			0.508128
$664$		0			0
$665$	−1.56058e8	+	9.94826e7i	−0.530665	+	0.338285i
$666$		0			0
$667$		−	3.71375e8i		−	1.25151i
$668$		0			0
$669$	−6.00671e7			−0.200613
$670$		0			0
$671$	1.38584e8			0.458718
$672$		0			0
$673$			7.07317e6i			0.0232043i	0.999933	+	0.0116022i	$0.00369317\pi$
$673$			7.07317e6i			0.0232043i	−0.999933	+	0.0116022i	$0.996307\pi$
$674$		0			0
$675$		−	5.61478e7i		−	0.182567i
$676$		0			0
$677$		−	3.47099e8i		−	1.11863i	−0.828955	−	0.559316i	$-0.811064\pi$
$677$		−	3.47099e8i		−	1.11863i	0.828955	−	0.559316i	$-0.188936\pi$
$678$		0			0
$679$			6.04457e7i			0.193088i
$680$		0			0
$681$	−6.95925e7			−0.220354
$682$		0			0
$683$			4.98305e7i			0.156399i	0.996938	+	0.0781994i	$0.0249171\pi$
$683$			4.98305e7i			0.156399i	−0.996938	+	0.0781994i	$0.975083\pi$
$684$		0			0
$685$	4.22391e8			1.31414
$686$		0			0
$687$		−	1.52387e8i		−	0.469979i
$688$		0			0
$689$	−1.64966e8			−0.504354
$690$		0			0
$691$	1.58794e8			0.481282			0.240641	−	0.970614i	$-0.422642\pi$
$691$	1.58794e8			0.481282			0.240641	+	0.970614i	$0.422642\pi$
$692$		0			0
$693$	−2.16010e8			−0.649043
$694$		0			0
$695$	5.80295e8			1.72860
$696$		0			0
$697$		−	5.26222e8i		−	1.55407i
$698$		0			0
$699$			1.21530e8i			0.355837i
$700$		0			0
$701$	5.55496e8			1.61260			0.806301	−	0.591506i	$-0.201466\pi$
$701$	5.55496e8			1.61260			0.806301	+	0.591506i	$0.201466\pi$
$702$		0			0
$703$	3.00083e8	+	4.70738e8i	0.863724	+	1.35492i
$704$		0			0
$705$		−	1.69023e7i		−	0.0482367i
$706$		0			0
$707$	−2.16582e8			−0.612864
$708$		0			0
$709$	1.67180e7			0.0469080			0.0234540	−	0.999725i	$-0.492534\pi$
$709$	1.67180e7			0.0469080			0.0234540	+	0.999725i	$0.492534\pi$
$710$		0			0
$711$			9.44260e7i			0.262714i
$712$		0			0
$713$			9.03646e8i			2.49304i
$714$		0			0
$715$			1.05089e9i			2.87501i
$716$		0			0
$717$		−	4.74951e7i		−	0.128852i
$718$		0			0
$719$	−3.04546e8			−0.819344			−0.409672	−	0.912233i	$-0.634357\pi$
$719$	−3.04546e8			−0.819344			−0.409672	+	0.912233i	$0.634357\pi$
$720$		0			0
$721$		−	2.55037e8i		−	0.680452i
$722$		0			0
$723$	1.82932e8			0.484033
$724$		0			0
$725$			1.27010e8i			0.333292i
$726$		0			0
$727$	6.18054e7			0.160851			0.0804254	−	0.996761i	$-0.474372\pi$
$727$	6.18054e7			0.160851			0.0804254	+	0.996761i	$0.474372\pi$
$728$		0			0
$729$	2.53210e8			0.653580
$730$		0			0
$731$	−3.50123e8			−0.896330
$732$		0			0
$733$	8.30071e7			0.210767			0.105384	−	0.994432i	$-0.466393\pi$
$733$	8.30071e7			0.210767			0.105384	+	0.994432i	$0.466393\pi$
$734$		0			0
$735$			8.20924e7i			0.206748i
$736$		0			0
$737$			6.04086e8i			1.50902i
$738$		0			0
$739$	2.09997e8			0.520332			0.260166	−	0.965564i	$-0.416223\pi$
$739$	2.09997e8			0.520332			0.260166	+	0.965564i	$0.416223\pi$
$740$		0			0
$741$	1.60255e8	−	1.02158e8i	0.393873	−	0.251084i
$742$		0			0
$743$			5.36457e8i			1.30788i	0.756545	+	0.653941i	$0.226886\pi$
$743$			5.36457e8i			1.30788i	−0.756545	+	0.653941i	$0.773114\pi$
$744$		0			0
$745$	2.69761e7			0.0652395
$746$		0			0
$747$	9.72324e7			0.233265
$748$		0			0
$749$		−	1.21176e7i		−	0.0288384i
$750$		0			0
$751$			6.93030e7i			0.163618i	0.996648	+	0.0818092i	$0.0260698\pi$
$751$			6.93030e7i			0.163618i	−0.996648	+	0.0818092i	$0.973930\pi$
$752$		0			0
$753$		−	6.56642e7i		−	0.153796i
$754$		0			0
$755$		−	5.15832e8i		−	1.19858i
$756$		0			0
$757$	−1.18269e8			−0.272636			−0.136318	−	0.990665i	$-0.543527\pi$
$757$	−1.18269e8			−0.272636			−0.136318	+	0.990665i	$0.543527\pi$
$758$		0			0
$759$		−	1.99526e8i		−	0.456325i
$760$		0			0
$761$	−1.69867e8			−0.385438			−0.192719	−	0.981254i	$-0.561731\pi$
$761$	−1.69867e8			−0.385438			−0.192719	+	0.981254i	$0.561731\pi$
$762$		0			0
$763$		−	1.53616e8i		−	0.345829i
$764$		0			0
$765$	−5.37717e8			−1.20107
$766$		0			0
$767$	1.22338e9			2.71129
$768$		0			0
$769$	7.59248e8			1.66957			0.834785	−	0.550577i	$-0.185592\pi$
$769$	7.59248e8			1.66957			0.834785	+	0.550577i	$0.185592\pi$
$770$		0			0
$771$	−7.11547e7			−0.155253
$772$		0			0
$773$		−	1.05343e8i		−	0.228070i	−0.993477	−	0.114035i	$-0.963622\pi$
$773$		−	1.05343e8i		−	0.228070i	0.993477	−	0.114035i	$-0.0363776\pi$
$774$		0			0
$775$		−	3.09046e8i		−	0.663924i
$776$		0			0
$777$	−9.94796e7			−0.212066
$778$		0			0
$779$	−3.63018e8	−	5.69464e8i	−0.767920	−	1.20463i
$780$		0			0
$781$			1.69951e8i			0.356755i
$782$		0			0
$783$	2.00303e8			0.417255
$784$		0			0
$785$	4.72952e8			0.977706
$786$		0			0
$787$		−	7.32535e7i		−	0.150281i	−0.997173	−	0.0751405i	$-0.976059\pi$
$787$		−	7.32535e7i		−	0.150281i	0.997173	−	0.0751405i	$-0.0239405\pi$
$788$		0			0
$789$			1.11339e7i			0.0226682i
$790$		0			0
$791$			1.37179e6i			0.00277177i
$792$		0			0
$793$			3.35762e8i			0.673305i
$794$		0			0
$795$	−3.87622e7			−0.0771448
$796$		0			0
$797$		−	5.46491e8i		−	1.07946i	−0.841837	−	0.539732i	$-0.818526\pi$
$797$		−	5.46491e8i		−	1.07946i	0.841837	−	0.539732i	$-0.181474\pi$
$798$		0			0
$799$	−9.23588e7			−0.181066
$800$		0			0
$801$			2.59981e8i			0.505875i
$802$		0			0
$803$	5.59882e8			1.08131
$804$		0			0
$805$	−4.70751e8			−0.902410
$806$		0			0
$807$	−1.27387e7			−0.0242384
$808$		0			0
$809$	4.63575e8			0.875536			0.437768	−	0.899088i	$-0.355769\pi$
$809$	4.63575e8			0.875536			0.437768	+	0.899088i	$0.355769\pi$
$810$		0			0
$811$			3.91098e8i			0.733201i	0.930378	+	0.366601i	$0.119478\pi$
$811$			3.91098e8i			0.733201i	−0.930378	+	0.366601i	$0.880522\pi$
$812$		0			0
$813$		−	5.76607e7i		−	0.107302i
$814$		0			0
$815$	−2.32090e8			−0.428730
$816$		0			0
$817$	−3.78894e8	+	2.41535e8i	−0.694786	+	0.442908i
$818$		0			0
$819$		−	5.23349e8i		−	0.952663i
$820$		0			0
$821$	9.37749e8			1.69456			0.847281	−	0.531145i	$-0.178238\pi$
$821$	9.37749e8			1.69456			0.847281	+	0.531145i	$0.178238\pi$
$822$		0			0
$823$	−8.49232e8			−1.52345			−0.761723	−	0.647903i	$-0.775646\pi$
$823$	−8.49232e8			−1.52345			−0.761723	+	0.647903i	$0.775646\pi$
$824$		0			0
$825$			6.82378e7i			0.121524i
$826$		0			0
$827$		−	5.72787e8i		−	1.01269i	−0.862331	−	0.506345i	$-0.830996\pi$
$827$		−	5.72787e8i		−	1.01269i	0.862331	−	0.506345i	$-0.169004\pi$
$828$		0			0
$829$			4.15850e8i			0.729916i	0.931024	+	0.364958i	$0.118917\pi$
$829$			4.15850e8i			0.729916i	−0.931024	+	0.364958i	$0.881083\pi$
$830$		0			0
$831$		−	2.33188e8i		−	0.406353i
$832$		0			0
$833$	4.48576e8			0.776071
$834$		0			0
$835$		−	1.41230e8i		−	0.242588i
$836$		0			0
$837$	−4.87384e8			−0.831180
$838$		0			0
$839$		−	2.72831e8i		−	0.461963i	−0.972958	−	0.230982i	$-0.925806\pi$
$839$		−	2.72831e8i		−	0.461963i	0.972958	−	0.230982i	$-0.0741937\pi$
$840$		0			0
$841$	1.41725e8			0.238264
$842$		0			0
$843$	3.99358e7			0.0666622
$844$		0			0
$845$	−1.83684e9			−3.04439
$846$		0			0
$847$	2.16730e8			0.356671
$848$		0			0
$849$			1.55572e7i			0.0254220i
$850$		0			0
$851$			1.41999e9i			2.30408i
$852$		0			0
$853$	2.48260e8			0.400000			0.200000	−	0.979796i	$-0.435906\pi$
$853$	2.48260e8			0.400000			0.200000	+	0.979796i	$0.435906\pi$
$854$		0			0
$855$	−5.81904e8	+	3.70948e8i	−0.931007	+	0.593492i
$856$		0			0
$857$		−	3.79494e8i		−	0.602923i	−0.953478	−	0.301462i	$-0.902525\pi$
$857$		−	3.79494e8i		−	0.602923i	0.953478	−	0.301462i	$-0.0974745\pi$
$858$		0			0
$859$	2.67792e8			0.422491			0.211245	−	0.977433i	$-0.432248\pi$
$859$	2.67792e8			0.422491			0.211245	+	0.977433i	$0.432248\pi$
$860$		0			0
$861$	1.20343e8			0.188544
$862$		0			0
$863$			3.57938e7i			0.0556898i	0.999612	+	0.0278449i	$0.00886446\pi$
$863$			3.57938e7i			0.0556898i	−0.999612	+	0.0278449i	$0.991136\pi$
$864$		0			0
$865$		−	1.03696e9i		−	1.60219i
$866$		0			0
$867$		−	2.94672e7i		−	0.0452150i
$868$		0			0
$869$		−	2.36942e8i		−	0.361063i
$870$		0			0
$871$	−1.46358e9			−2.21494
$872$		0			0
$873$			2.25388e8i			0.338757i
$874$		0			0
$875$	−2.60597e8			−0.388997
$876$		0			0
$877$		−	2.97741e8i		−	0.441407i	−0.975341	−	0.220704i	$-0.929165\pi$
$877$		−	2.97741e8i		−	0.441407i	0.975341	−	0.220704i	$-0.0708353\pi$
$878$		0			0
$879$	−1.30865e8			−0.192689
$880$		0			0
$881$	−1.09276e8			−0.159807			−0.0799036	−	0.996803i	$-0.525461\pi$
$881$	−1.09276e8			−0.159807			−0.0799036	+	0.996803i	$0.525461\pi$
$882$		0			0
$883$	−7.36749e8			−1.07013			−0.535066	−	0.844810i	$-0.679713\pi$
$883$	−7.36749e8			−1.07013			−0.535066	+	0.844810i	$0.679713\pi$
$884$		0			0
$885$	2.87460e8			0.414713
$886$		0			0
$887$		−	1.02599e9i		−	1.47019i	−0.677966	−	0.735094i	$-0.737138\pi$
$887$		−	1.02599e9i		−	1.47019i	0.677966	−	0.735094i	$-0.262862\pi$
$888$		0			0
$889$		−	2.23513e8i		−	0.318124i
$890$		0			0
$891$	−7.49957e8			−1.06024
$892$		0			0
$893$	−9.99484e7	+	6.37144e7i	−0.140353	+	0.0894711i
$894$		0			0
$895$		−	3.76171e8i		−	0.524706i
$896$		0			0
$897$	4.83412e8			0.669792
$898$		0			0
$899$	1.10250e9			1.51739
$900$		0			0
$901$			2.11808e8i			0.289579i
$902$		0			0
$903$		−	8.00704e7i		−	0.108745i
$904$		0			0
$905$			1.61165e8i			0.217433i
$906$		0			0
$907$			1.29958e9i			1.74174i	0.491516	+	0.870869i	$0.336443\pi$
$907$			1.29958e9i			1.74174i	−0.491516	+	0.870869i	$0.663557\pi$
$908$		0			0
$909$	−8.07584e8			−1.07522
$910$		0			0
$911$			7.29839e8i			0.965322i	0.875807	+	0.482661i	$0.160330\pi$
$911$			7.29839e8i			0.965322i	−0.875807	+	0.482661i	$0.839670\pi$
$912$		0			0
$913$	−2.43984e8			−0.320590
$914$		0			0
$915$			7.88945e7i			0.102987i
$916$		0			0
$917$	−3.16211e8			−0.410080
$918$		0			0
$919$	−1.07725e9			−1.38794			−0.693968	−	0.720006i	$-0.744139\pi$
$919$	−1.07725e9			−1.38794			−0.693968	+	0.720006i	$0.744139\pi$
$920$		0			0
$921$	1.20968e8			0.154843
$922$		0			0
$923$	−4.11757e8			−0.523643
$924$		0			0
$925$		−	4.85636e8i		−	0.613601i
$926$		0			0
$927$		−	9.50976e8i		−	1.19380i
$928$		0			0
$929$	−8.99955e7			−0.112247			−0.0561234	−	0.998424i	$-0.517874\pi$
$929$	−8.99955e7			−0.112247			−0.0561234	+	0.998424i	$0.517874\pi$
$930$		0			0
$931$	4.85438e8	−	3.09454e8i	0.601568	−	0.383483i
$932$		0			0
$933$		−	2.92390e8i		−	0.360013i
$934$		0			0
$935$	1.34929e9			1.65071
$936$		0			0
$937$	−1.92583e8			−0.234099			−0.117050	−	0.993126i	$-0.537344\pi$
$937$	−1.92583e8			−0.234099			−0.117050	+	0.993126i	$0.537344\pi$
$938$		0			0
$939$			1.21987e7i			0.0147339i
$940$		0			0
$941$		−	9.44507e8i		−	1.13354i	−0.823877	−	0.566769i	$-0.808193\pi$
$941$		−	9.44507e8i		−	1.13354i	0.823877	−	0.566769i	$-0.191807\pi$
$942$		0			0
$943$		−	1.71780e9i		−	2.04851i
$944$		0			0
$945$		−	2.53901e8i		−	0.300864i
$946$		0			0
$947$	5.88012e7			0.0692367			0.0346184	−	0.999401i	$-0.488978\pi$
$947$	5.88012e7			0.0692367			0.0346184	+	0.999401i	$0.488978\pi$
$948$		0			0
$949$			1.35648e9i			1.58714i
$950$		0			0
$951$	2.94735e7			0.0342681
$952$		0			0
$953$		−	6.20225e8i		−	0.716589i	−0.933609	−	0.358295i	$-0.883358\pi$
$953$		−	6.20225e8i		−	0.716589i	0.933609	−	0.358295i	$-0.116642\pi$
$954$		0			0
$955$	7.73946e7			0.0888588
$956$		0			0
$957$	−2.43433e8			−0.277743
$958$		0			0
$959$	5.27837e8			0.598473
$960$		0			0
$961$	−1.79513e9			−2.02268
$962$		0			0
$963$		−	4.51838e7i		−	0.0505945i
$964$		0			0
$965$		−	2.48978e8i		−	0.277063i
$966$		0			0
$967$	7.24092e8			0.800782			0.400391	−	0.916344i	$-0.368874\pi$
$967$	7.24092e8			0.800782			0.400391	+	0.916344i	$0.368874\pi$
$968$		0			0
$969$	−1.31166e8	−	2.05759e8i	−0.144162	−	0.226146i
$970$		0			0
$971$			3.05228e8i			0.333401i	0.986008	+	0.166701i	$0.0533114\pi$
$971$			3.05228e8i			0.333401i	−0.986008	+	0.166701i	$0.946689\pi$
$972$		0			0
$973$	7.25161e8			0.787220
$974$		0			0
$975$	−1.65327e8			−0.178373
$976$		0			0
$977$			7.55141e8i			0.809738i	0.914375	+	0.404869i	$0.132683\pi$
$977$			7.55141e8i			0.809738i	−0.914375	+	0.404869i	$0.867317\pi$
$978$		0			0
$979$		−	6.52367e8i		−	0.695254i
$980$		0			0
$981$		−	5.72798e8i		−	0.606728i
$982$		0			0
$983$			9.20545e8i			0.969135i	0.874754	+	0.484568i	$0.161023\pi$
$983$			9.20545e8i			0.969135i	−0.874754	+	0.484568i	$0.838977\pi$
$984$		0			0
$985$	3.26725e8			0.341880
$986$		0			0
$987$		−	2.11218e7i		−	0.0219674i
$988$		0			0
$989$	−1.14294e9			−1.18150
$990$		0			0
$991$			1.60808e9i			1.65229i	0.563457	+	0.826146i	$0.309471\pi$
$991$			1.60808e9i			1.65229i	−0.563457	+	0.826146i	$0.690529\pi$
$992$		0			0
$993$	−1.01518e8			−0.103680
$994$		0			0
$995$	−1.07675e9			−1.09306
$996$		0			0
$997$	−8.94189e8			−0.902286			−0.451143	−	0.892452i	$-0.648984\pi$
$997$	−8.94189e8			−0.902286			−0.451143	+	0.892452i	$0.648984\pi$
$998$		0			0
$999$	−7.65877e8			−0.768179

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.7.e.e.113.5		10
4.3	odd	2		38.7.b.a.37.3	✓	10
12.11	even	2		342.7.d.a.37.6		10
19.18	odd	2	inner	304.7.e.e.113.6		10
76.75	even	2		38.7.b.a.37.8	yes	10
228.227	odd	2		342.7.d.a.37.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.7.b.a.37.3	✓	10	4.3	odd	2
38.7.b.a.37.8	yes	10	76.75	even	2
304.7.e.e.113.5		10	1.1	even	1	trivial
304.7.e.e.113.6		10	19.18	odd	2	inner
342.7.d.a.37.1		10	228.227	odd	2
342.7.d.a.37.6		10	12.11	even	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:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	304.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-6.65634i q^{3} +146.942 q^{5} +183.624 q^{7} +684.693 q^{9} +O(q^{10})\)	\(q-6.65634i q^{3} +146.942 q^{5} +183.624 q^{7} +684.693 q^{9} -1718.09 q^{11} -4162.60i q^{13} -978.092i q^{15} -5344.58 q^{17} +(-5783.77 + 3686.99i) q^{19} -1222.27i q^{21} -17446.9 q^{23} +5966.81 q^{25} -9410.02i q^{27} +21286.1i q^{29} -51794.2i q^{31} +11436.2i q^{33} +26982.0 q^{35} -81389.5i q^{37} -27707.7 q^{39} +98459.1i q^{41} +65509.9 q^{43} +100610. q^{45} +17280.8 q^{47} -83931.1 q^{49} +35575.3i q^{51} -39630.4i q^{53} -252459. q^{55} +(24541.9 + 38498.7i) q^{57} +293899. i q^{59} -80661.6 q^{61} +125726. q^{63} -611659. i q^{65} -351602. i q^{67} +116132. i q^{69} -98918.1i q^{71} -325874. q^{73} -39717.1i q^{75} -315484. q^{77} +137910. i q^{79} +436505. q^{81} +142009. q^{83} -785340. q^{85} +141687. q^{87} +379704. i q^{89} -764355. i q^{91} -344760. q^{93} +(-849875. + 541773. i) q^{95} +329181. i q^{97} -1.17637e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 112 q^{5} + 224 q^{7} - 2890 q^{9}+O(q^{10}) \) 10 * q - 112 * q^5 + 224 * q^7 - 2890 * q^9	\( 10 q - 112 q^{5} + 224 q^{7} - 2890 q^{9} - 3644 q^{11} - 10420 q^{17} + 17230 q^{19} - 37712 q^{23} - 52078 q^{25} + 161720 q^{35} + 78876 q^{39} - 6308 q^{43} + 309808 q^{45} - 322220 q^{47} - 235770 q^{49} + 377880 q^{55} + 24228 q^{57} + 426304 q^{61} + 517916 q^{63} - 786076 q^{73} + 2303716 q^{77} + 5261090 q^{81} + 101500 q^{83} - 1261380 q^{85} + 2460732 q^{87} - 2827032 q^{93} - 3106292 q^{95} - 1061428 q^{99}+O(q^{100}) \) 10 * q - 112 * q^5 + 224 * q^7 - 2890 * q^9 - 3644 * q^11 - 10420 * q^17 + 17230 * q^19 - 37712 * q^23 - 52078 * q^25 + 161720 * q^35 + 78876 * q^39 - 6308 * q^43 + 309808 * q^45 - 322220 * q^47 - 235770 * q^49 + 377880 * q^55 + 24228 * q^57 + 426304 * q^61 + 517916 * q^63 - 786076 * q^73 + 2303716 * q^77 + 5261090 * q^81 + 101500 * q^83 - 1261380 * q^85 + 2460732 * q^87 - 2827032 * q^93 - 3106292 * q^95 - 1061428 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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