LMFDB - Embedded newform 304.7.e.d.113.2

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 483x^{6} + 75582x^{4} + 4242376x^{2} + 71047680 $ x^8 + 483x^6 + 75582x^4 + 4242376*x^2 + 71047680
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{11}\cdot 29 $
Twist minimal:	no (minimal twist has level 19)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.2
Root			$14.9269i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.7.e.d.113.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-33.0827i q^{3} +159.622 q^{5} -445.794 q^{7} -365.466 q^{9} +O(q^{10})$	$q-33.0827i q^{3} +159.622 q^{5} -445.794 q^{7} -365.466 q^{9} +1700.61 q^{11} -439.288i q^{13} -5280.74i q^{15} +569.044 q^{17} +(-6482.51 + 2241.20i) q^{19} +14748.1i q^{21} +2746.30 q^{23} +9854.28 q^{25} -12026.7i q^{27} -26932.8i q^{29} -29892.3i q^{31} -56260.9i q^{33} -71158.6 q^{35} -52516.0i q^{37} -14532.9 q^{39} +27399.8i q^{41} -67448.4 q^{43} -58336.5 q^{45} +66319.6 q^{47} +81083.1 q^{49} -18825.5i q^{51} -59959.3i q^{53} +271456. q^{55} +(74145.1 + 214459. i) q^{57} -396359. i q^{59} -73557.7 q^{61} +162922. q^{63} -70120.2i q^{65} +160211. i q^{67} -90855.2i q^{69} +259957. i q^{71} +291024. q^{73} -326006. i q^{75} -758123. q^{77} +917363. i q^{79} -664300. q^{81} -681097. q^{83} +90832.1 q^{85} -891010. q^{87} -242644. i q^{89} +195832. i q^{91} -988920. q^{93} +(-1.03475e6 + 357746. i) q^{95} -1.24569e6i q^{97} -621516. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 108 q^{5} + 140 q^{7} - 1052 q^{9}+O(q^{10}) $ 8 * q + 108 * q^5 + 140 * q^7 - 1052 * q^9	$ 8 q + 108 q^{5} + 140 q^{7} - 1052 q^{9} + 2024 q^{11} + 6008 q^{17} - 20552 q^{19} + 50252 q^{23} + 78492 q^{25} - 210800 q^{35} - 43724 q^{39} - 260800 q^{43} - 191012 q^{45} + 100248 q^{47} - 301872 q^{49} + 52480 q^{55} - 186860 q^{57} - 54548 q^{61} + 137408 q^{63} + 479968 q^{73} - 1755300 q^{77} - 4279648 q^{81} - 483040 q^{83} + 2111780 q^{85} - 2802652 q^{87} + 1507528 q^{93} + 2383888 q^{95} - 528224 q^{99}+O(q^{100}) $ 8 * q + 108 * q^5 + 140 * q^7 - 1052 * q^9 + 2024 * q^11 + 6008 * q^17 - 20552 * q^19 + 50252 * q^23 + 78492 * q^25 - 210800 * q^35 - 43724 * q^39 - 260800 * q^43 - 191012 * q^45 + 100248 * q^47 - 301872 * q^49 + 52480 * q^55 - 186860 * q^57 - 54548 * q^61 + 137408 * q^63 + 479968 * q^73 - 1755300 * q^77 - 4279648 * q^81 - 483040 * q^83 + 2111780 * q^85 - 2802652 * q^87 + 1507528 * q^93 + 2383888 * q^95 - 528224 * 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
$3$		−	33.0827i		−	1.22529i	−0.790360	−	0.612643i	$-0.790106\pi$
$3$		−	33.0827i		−	1.22529i	0.790360	−	0.612643i	$-0.209894\pi$
$4$		0			0
$5$	159.622			1.27698			0.638489	−	0.769631i	$-0.279560\pi$
$5$	159.622			1.27698			0.638489	+	0.769631i	$0.279560\pi$
$6$		0			0
$7$	−445.794			−1.29969			−0.649845	−	0.760067i	$-0.725166\pi$
$7$	−445.794			−1.29969			−0.649845	+	0.760067i	$0.725166\pi$
$8$		0			0
$9$	−365.466			−0.501325
$10$		0			0
$11$	1700.61			1.27770			0.638848	−	0.769333i	$-0.279411\pi$
$11$	1700.61			1.27770			0.638848	+	0.769333i	$0.279411\pi$
$12$		0			0
$13$		−	439.288i		−	0.199949i	−0.994990	−	0.0999746i	$-0.968124\pi$
$13$		−	439.288i		−	0.199949i	0.994990	−	0.0999746i	$-0.0318762\pi$
$14$		0			0
$15$		−	5280.74i		−	1.56466i
$16$		0			0
$17$	569.044			0.115824			0.0579121	−	0.998322i	$-0.481556\pi$
$17$	569.044			0.115824			0.0579121	+	0.998322i	$0.481556\pi$
$18$		0			0
$19$	−6482.51	+	2241.20i	−0.945110	+	0.326754i
$19$	−6482.51	+	2241.20i	−0.945110	+	0.326754i
$20$		0			0
$21$			14748.1i			1.59249i
$22$		0			0
$23$	2746.30			0.225717			0.112859	−	0.993611i	$-0.463999\pi$
$23$	2746.30			0.225717			0.112859	+	0.993611i	$0.463999\pi$
$24$		0			0
$25$	9854.28			0.630674
$26$		0			0
$27$		−	12026.7i		−	0.611019i
$28$		0			0
$29$		−	26932.8i		−	1.10430i	−0.833745	−	0.552150i	$-0.813807\pi$
$29$		−	26932.8i		−	1.10430i	0.833745	−	0.552150i	$-0.186193\pi$
$30$		0			0
$31$		−	29892.3i		−	1.00340i	−0.865041	−	0.501701i	$-0.832708\pi$
$31$		−	29892.3i		−	1.00340i	0.865041	−	0.501701i	$-0.167292\pi$
$32$		0			0
$33$		−	56260.9i		−	1.56554i
$34$		0			0
$35$	−71158.6			−1.65968
$36$		0			0
$37$		−	52516.0i		−	1.03678i	−0.855144	−	0.518390i	$-0.826532\pi$
$37$		−	52516.0i		−	1.03678i	0.855144	−	0.518390i	$-0.173468\pi$
$38$		0			0
$39$	−14532.9			−0.244995
$40$		0			0
$41$			27399.8i			0.397554i	0.980045	+	0.198777i	$0.0636970\pi$
$41$			27399.8i			0.397554i	−0.980045	+	0.198777i	$0.936303\pi$
$42$		0			0
$43$	−67448.4			−0.848332			−0.424166	−	0.905584i	$-0.639433\pi$
$43$	−67448.4			−0.848332			−0.424166	+	0.905584i	$0.639433\pi$
$44$		0			0
$45$	−58336.5			−0.640181
$46$		0			0
$47$	66319.6			0.638776			0.319388	−	0.947624i	$-0.396523\pi$
$47$	66319.6			0.638776			0.319388	+	0.947624i	$0.396523\pi$
$48$		0			0
$49$	81083.1			0.689195
$50$		0			0
$51$		−	18825.5i		−	0.141918i
$52$		0			0
$53$		−	59959.3i		−	0.402744i	−0.979515	−	0.201372i	$-0.935460\pi$
$53$		−	59959.3i		−	0.402744i	0.979515	−	0.201372i	$-0.0645399\pi$
$54$		0			0
$55$	271456.			1.63159
$56$		0			0
$57$	74145.1	+	214459.i	0.400367	+	1.15803i
$58$		0			0
$59$		−	396359.i		−	1.92989i	−0.262452	−	0.964945i	$-0.584531\pi$
$59$		−	396359.i		−	1.92989i	0.262452	−	0.964945i	$-0.415469\pi$
$60$		0			0
$61$	−73557.7			−0.324070			−0.162035	−	0.986785i	$-0.551806\pi$
$61$	−73557.7			−0.324070			−0.162035	+	0.986785i	$0.551806\pi$
$62$		0			0
$63$	162922.			0.651567
$64$		0			0
$65$		−	70120.2i		−	0.255331i
$66$		0			0
$67$			160211.i			0.532681i	0.963879	+	0.266341i	$0.0858146\pi$
$67$			160211.i			0.532681i	−0.963879	+	0.266341i	$0.914185\pi$
$68$		0			0
$69$		−	90855.2i		−	0.276568i
$70$		0			0
$71$			259957.i			0.726317i	0.931727	+	0.363158i	$0.118302\pi$
$71$			259957.i			0.726317i	−0.931727	+	0.363158i	$0.881698\pi$
$72$		0			0
$73$	291024.			0.748101			0.374051	−	0.927408i	$-0.377969\pi$
$73$	291024.			0.748101			0.374051	+	0.927408i	$0.377969\pi$
$74$		0			0
$75$		−	326006.i		−	0.772755i
$76$		0			0
$77$	−758123.			−1.66061
$78$		0			0
$79$			917363.i			1.86063i	0.366761	+	0.930315i	$0.380467\pi$
$79$			917363.i			1.86063i	−0.366761	+	0.930315i	$0.619533\pi$
$80$		0			0
$81$	−664300.			−1.25000
$82$		0			0
$83$	−681097.			−1.19117			−0.595586	−	0.803291i	$-0.703080\pi$
$83$	−681097.			−1.19117			−0.595586	+	0.803291i	$0.703080\pi$
$84$		0			0
$85$	90832.1			0.147905
$86$		0			0
$87$	−891010.			−1.35308
$88$		0			0
$89$		−	242644.i		−	0.344191i	−0.985080	−	0.172095i	$-0.944946\pi$
$89$		−	242644.i		−	0.344191i	0.985080	−	0.172095i	$-0.0550537\pi$
$90$		0			0
$91$			195832.i			0.259872i
$92$		0			0
$93$	−988920.			−1.22945
$94$		0			0
$95$	−1.03475e6	+	357746.i	−1.20688	+	0.417257i
$96$		0			0
$97$		−	1.24569e6i		−	1.36488i	−0.730941	−	0.682440i	$-0.760919\pi$
$97$		−	1.24569e6i		−	1.36488i	0.730941	−	0.682440i	$-0.239081\pi$
$98$		0			0
$99$	−621516.			−0.640541
$100$		0			0
$101$	−1.34311e6			−1.30361			−0.651807	−	0.758385i	$-0.725989\pi$
$101$	−1.34311e6			−1.30361			−0.651807	+	0.758385i	$0.725989\pi$
$102$		0			0
$103$		−	1.37650e6i		−	1.25969i	−0.776719	−	0.629847i	$-0.783117\pi$
$103$		−	1.37650e6i		−	1.25969i	0.776719	−	0.629847i	$-0.216883\pi$
$104$		0			0
$105$			2.35412e6i			2.03358i
$106$		0			0
$107$		−	920616.i		−	0.751497i	−0.926722	−	0.375748i	$-0.877386\pi$
$107$		−	920616.i		−	0.751497i	0.926722	−	0.375748i	$-0.122614\pi$
$108$		0			0
$109$			385747.i			0.297868i	0.988847	+	0.148934i	$0.0475841\pi$
$109$			385747.i			0.297868i	−0.988847	+	0.148934i	$0.952416\pi$
$110$		0			0
$111$	−1.73737e6			−1.27035
$112$		0			0
$113$		−	620853.i		−	0.430282i	−0.976583	−	0.215141i	$-0.930979\pi$
$113$		−	620853.i		−	0.430282i	0.976583	−	0.215141i	$-0.0690212\pi$
$114$		0			0
$115$	438371.			0.288236
$116$		0			0
$117$			160545.i			0.100240i
$118$		0			0
$119$	−253676.			−0.150536
$120$		0			0
$121$	1.12052e6			0.632506
$122$		0			0
$123$	906461.			0.487118
$124$		0			0
$125$	−921136.			−0.471622
$126$		0			0
$127$			1.13559e6i			0.554385i	0.960814	+	0.277193i	$0.0894041\pi$
$127$			1.13559e6i			0.554385i	−0.960814	+	0.277193i	$0.910596\pi$
$128$		0			0
$129$			2.23138e6i			1.03945i
$130$		0			0
$131$	370425.			0.164773			0.0823866	−	0.996600i	$-0.473746\pi$
$131$	370425.			0.164773			0.0823866	+	0.996600i	$0.473746\pi$
$132$		0			0
$133$	2.88986e6	−	999114.i	1.22835	−	0.424679i
$134$		0			0
$135$		−	1.91973e6i		−	0.780258i
$136$		0			0
$137$	−2.35314e6			−0.915136			−0.457568	−	0.889175i	$-0.651279\pi$
$137$	−2.35314e6			−0.915136			−0.457568	+	0.889175i	$0.651279\pi$
$138$		0			0
$139$	−2.91982e6			−1.08720			−0.543602	−	0.839343i	$-0.682940\pi$
$139$	−2.91982e6			−1.08720			−0.543602	+	0.839343i	$0.682940\pi$
$140$		0			0
$141$		−	2.19403e6i		−	0.782683i
$142$		0			0
$143$		−	747059.i		−	0.255474i
$144$		0			0
$145$		−	4.29907e6i		−	1.41017i
$146$		0			0
$147$		−	2.68245e6i		−	0.844461i
$148$		0			0
$149$	−1.34951e6			−0.407961			−0.203980	−	0.978975i	$-0.565388\pi$
$149$	−1.34951e6			−0.407961			−0.203980	+	0.978975i	$0.565388\pi$
$150$		0			0
$151$			5.87074e6i			1.70515i	0.522607	+	0.852574i	$0.324960\pi$
$151$			5.87074e6i			1.70515i	−0.522607	+	0.852574i	$0.675040\pi$
$152$		0			0
$153$	−207966.			−0.0580656
$154$		0			0
$155$		−	4.77148e6i		−	1.28132i
$156$		0			0
$157$	−673724.			−0.174094			−0.0870469	−	0.996204i	$-0.527743\pi$
$157$	−673724.			−0.174094			−0.0870469	+	0.996204i	$0.527743\pi$
$158$		0			0
$159$	−1.98362e6			−0.493476
$160$		0			0
$161$	−1.22429e6			−0.293363
$162$		0			0
$163$	4.36326e6			1.00751			0.503754	−	0.863847i	$-0.331952\pi$
$163$	4.36326e6			1.00751			0.503754	+	0.863847i	$0.331952\pi$
$164$		0			0
$165$		−	8.98049e6i		−	1.99916i
$166$		0			0
$167$			2.14529e6i			0.460613i	0.973118	+	0.230306i	$0.0739728\pi$
$167$			2.14529e6i			0.460613i	−0.973118	+	0.230306i	$0.926027\pi$
$168$		0			0
$169$	4.63383e6			0.960020
$170$		0			0
$171$	2.36914e6	−	819084.i	0.473807	−	0.163810i
$172$		0			0
$173$			3.44402e6i			0.665161i	0.943075	+	0.332581i	$0.107919\pi$
$173$			3.44402e6i			0.665161i	−0.943075	+	0.332581i	$0.892081\pi$
$174$		0			0
$175$	−4.39297e6			−0.819680
$176$		0			0
$177$	−1.31126e7			−2.36467
$178$		0			0
$179$		−	9.69997e6i		−	1.69126i	−0.533767	−	0.845632i	$-0.679224\pi$
$179$		−	9.69997e6i		−	1.69126i	0.533767	−	0.845632i	$-0.320776\pi$
$180$		0			0
$181$			6.17944e6i			1.04211i	0.853523	+	0.521055i	$0.174461\pi$
$181$			6.17944e6i			1.04211i	−0.853523	+	0.521055i	$0.825539\pi$
$182$		0			0
$183$			2.43349e6i			0.397078i
$184$		0			0
$185$		−	8.38272e6i		−	1.32395i
$186$		0			0
$187$	967724.			0.147988
$188$		0			0
$189$			5.36142e6i			0.794136i
$190$		0			0
$191$	−6.71701e6			−0.963998			−0.481999	−	0.876172i	$-0.660089\pi$
$191$	−6.71701e6			−0.963998			−0.481999	+	0.876172i	$0.660089\pi$
$192$		0			0
$193$		−	5.48205e6i		−	0.762555i	−0.924461	−	0.381278i	$-0.875484\pi$
$193$		−	5.48205e6i		−	0.762555i	0.924461	−	0.381278i	$-0.124516\pi$
$194$		0			0
$195$	−2.31977e6			−0.312853
$196$		0			0
$197$	−8.07614e6			−1.05634			−0.528172	−	0.849137i	$-0.677122\pi$
$197$	−8.07614e6			−1.05634			−0.528172	+	0.849137i	$0.677122\pi$
$198$		0			0
$199$	7.76482e6			0.985309			0.492654	−	0.870225i	$-0.336027\pi$
$199$	7.76482e6			0.985309			0.492654	+	0.870225i	$0.336027\pi$
$200$		0			0
$201$	5.30021e6			0.652687
$202$		0			0
$203$			1.20065e7i			1.43525i
$204$		0			0
$205$			4.37363e6i			0.507668i
$206$		0			0
$207$	−1.00368e6			−0.113158
$208$		0			0
$209$	−1.10242e7	+	3.81142e6i	−1.20756	+	0.417492i
$210$		0			0
$211$		−	6.10034e6i		−	0.649392i	−0.945818	−	0.324696i	$-0.894738\pi$
$211$		−	6.10034e6i		−	0.649392i	0.945818	−	0.324696i	$-0.105262\pi$
$212$		0			0
$213$	8.60008e6			0.889946
$214$		0			0
$215$	−1.07663e7			−1.08330
$216$		0			0
$217$			1.33258e7i			1.30411i
$218$		0			0
$219$		−	9.62787e6i		−	0.916638i
$220$		0			0
$221$		−	249974.i		−	0.0231589i
$222$		0			0
$223$		−	3.09709e6i		−	0.279279i	−0.990202	−	0.139640i	$-0.955406\pi$
$223$		−	3.09709e6i		−	0.279279i	0.990202	−	0.139640i	$-0.0445944\pi$
$224$		0			0
$225$	−3.60140e6			−0.316173
$226$		0			0
$227$		−	1.26462e6i		−	0.108114i	−0.998538	−	0.0540569i	$-0.982785\pi$
$227$		−	1.26462e6i		−	0.108114i	0.998538	−	0.0540569i	$-0.0172152\pi$
$228$		0			0
$229$	1.06635e7			0.887958			0.443979	−	0.896037i	$-0.353566\pi$
$229$	1.06635e7			0.887958			0.443979	+	0.896037i	$0.353566\pi$
$230$		0			0
$231$			2.50808e7i			2.03472i
$232$		0			0
$233$	2.85242e6			0.225500			0.112750	−	0.993623i	$-0.464034\pi$
$233$	2.85242e6			0.225500			0.112750	+	0.993623i	$0.464034\pi$
$234$		0			0
$235$	1.05861e7			0.815703
$236$		0			0
$237$	3.03489e7			2.27980
$238$		0			0
$239$	1.78416e7			1.30689			0.653445	−	0.756974i	$-0.273323\pi$
$239$	1.78416e7			1.30689			0.653445	+	0.756974i	$0.273323\pi$
$240$		0			0
$241$		−	5.72052e6i		−	0.408681i	−0.978900	−	0.204340i	$-0.934495\pi$
$241$		−	5.72052e6i		−	0.408681i	0.978900	−	0.204340i	$-0.0655049\pi$
$242$		0			0
$243$			1.32094e7i			0.920586i
$244$		0			0
$245$	1.29427e7			0.880087
$246$		0			0
$247$	984535.	+	2.84769e6i	0.0653341	+	0.188974i
$248$		0			0
$249$			2.25325e7i			1.45953i
$250$		0			0
$251$	−7.46125e6			−0.471835			−0.235918	−	0.971773i	$-0.575810\pi$
$251$	−7.46125e6			−0.471835			−0.235918	+	0.971773i	$0.575810\pi$
$252$		0			0
$253$	4.67040e6			0.288398
$254$		0			0
$255$		−	3.00497e6i		−	0.181226i
$256$		0			0
$257$			8.46675e6i			0.498790i	0.968402	+	0.249395i	$0.0802317\pi$
$257$			8.46675e6i			0.498790i	−0.968402	+	0.249395i	$0.919768\pi$
$258$		0			0
$259$			2.34113e7i			1.34749i
$260$		0			0
$261$			9.84302e6i			0.553614i
$262$		0			0
$263$	3.14002e7			1.72610			0.863049	−	0.505121i	$-0.168552\pi$
$263$	3.14002e7			1.72610			0.863049	+	0.505121i	$0.168552\pi$
$264$		0			0
$265$		−	9.57083e6i		−	0.514295i
$266$		0			0
$267$	−8.02731e6			−0.421732
$268$		0			0
$269$		−	6.30864e6i		−	0.324100i	−0.986783	−	0.162050i	$-0.948189\pi$
$269$		−	6.30864e6i		−	0.324100i	0.986783	−	0.162050i	$-0.0518105\pi$
$270$		0			0
$271$	−1.43606e7			−0.721545			−0.360772	−	0.932654i	$-0.617487\pi$
$271$	−1.43606e7			−0.721545			−0.360772	+	0.932654i	$0.617487\pi$
$272$		0			0
$273$	6.47865e6			0.318417
$274$		0			0
$275$	1.67583e7			0.805809
$276$		0			0
$277$	9.38558e6			0.441593			0.220796	−	0.975320i	$-0.429134\pi$
$277$	9.38558e6			0.441593			0.220796	+	0.975320i	$0.429134\pi$
$278$		0			0
$279$			1.09246e7i			0.503030i
$280$		0			0
$281$		−	3.93600e7i		−	1.77393i	−0.461837	−	0.886965i	$-0.652810\pi$
$281$		−	3.93600e7i		−	1.77393i	0.461837	−	0.886965i	$-0.347190\pi$
$282$		0			0
$283$	2.61094e7			1.15196			0.575980	−	0.817464i	$-0.304621\pi$
$283$	2.61094e7			1.15196			0.575980	+	0.817464i	$0.304621\pi$
$284$		0			0
$285$	1.18352e7	+	3.42324e7i	0.511259	+	1.47878i
$286$		0			0
$287$		−	1.22147e7i		−	0.516698i
$288$		0			0
$289$	−2.38138e7			−0.986585
$290$		0			0
$291$	−4.12108e7			−1.67237
$292$		0			0
$293$			1.58306e7i			0.629354i	0.949199	+	0.314677i	$0.101896\pi$
$293$			1.58306e7i			0.629354i	−0.949199	+	0.314677i	$0.898104\pi$
$294$		0			0
$295$		−	6.32677e7i		−	2.46443i
$296$		0			0
$297$		−	2.04527e7i		−	0.780697i
$298$		0			0
$299$		−	1.20642e6i		−	0.0451320i
$300$		0			0
$301$	3.00681e7			1.10257
$302$		0			0
$303$			4.44339e7i			1.59730i
$304$		0			0
$305$	−1.17414e7			−0.413830
$306$		0			0
$307$		−	5.21895e7i		−	1.80371i	−0.432034	−	0.901857i	$-0.642204\pi$
$307$		−	5.21895e7i		−	1.80371i	0.432034	−	0.901857i	$-0.357796\pi$
$308$		0			0
$309$	−4.55384e7			−1.54349
$310$		0			0
$311$	5.40593e7			1.79717			0.898585	−	0.438800i	$-0.144596\pi$
$311$	5.40593e7			1.79717			0.898585	+	0.438800i	$0.144596\pi$
$312$		0			0
$313$	2.48919e7			0.811754			0.405877	−	0.913928i	$-0.366966\pi$
$313$	2.48919e7			0.811754			0.405877	+	0.913928i	$0.366966\pi$
$314$		0			0
$315$	2.60061e7			0.832038
$316$		0			0
$317$		−	3.78290e7i		−	1.18754i	−0.804636	−	0.593769i	$-0.797639\pi$
$317$		−	3.78290e7i		−	1.18754i	0.804636	−	0.593769i	$-0.202361\pi$
$318$		0			0
$319$		−	4.58022e7i		−	1.41096i
$320$		0			0
$321$	−3.04565e7			−0.920798
$322$		0			0
$323$	−3.68883e6	+	1.27534e6i	−0.109466	+	0.0378460i
$324$		0			0
$325$		−	4.32887e6i		−	0.126103i
$326$		0			0
$327$	1.27616e7			0.364973
$328$		0			0
$329$	−2.95649e7			−0.830211
$330$		0			0
$331$			6.04318e7i			1.66641i	0.552964	+	0.833205i	$0.313497\pi$
$331$			6.04318e7i			1.66641i	−0.552964	+	0.833205i	$0.686503\pi$
$332$		0			0
$333$			1.91928e7i			0.519764i
$334$		0			0
$335$			2.55732e7i			0.680222i
$336$		0			0
$337$			5.31644e7i			1.38909i	0.719449	+	0.694546i	$0.244395\pi$
$337$			5.31644e7i			1.38909i	−0.719449	+	0.694546i	$0.755605\pi$
$338$		0			0
$339$	−2.05395e7			−0.527219
$340$		0			0
$341$		−	5.08353e7i		−	1.28204i
$342$		0			0
$343$	1.63009e7			0.403950
$344$		0			0
$345$		−	1.45025e7i		−	0.353172i
$346$		0			0
$347$	1.73896e7			0.416200			0.208100	−	0.978108i	$-0.433272\pi$
$347$	1.73896e7			0.416200			0.208100	+	0.978108i	$0.433272\pi$
$348$		0			0
$349$	5.87325e7			1.38166			0.690832	−	0.723015i	$-0.257244\pi$
$349$	5.87325e7			1.38166			0.690832	+	0.723015i	$0.257244\pi$
$350$		0			0
$351$	−5.28319e6			−0.122173
$352$		0			0
$353$	7.21844e7			1.64104			0.820520	−	0.571618i	$-0.193684\pi$
$353$	7.21844e7			1.64104			0.820520	+	0.571618i	$0.193684\pi$
$354$		0			0
$355$			4.14949e7i			0.927491i
$356$		0			0
$357$			8.39230e6i			0.184449i
$358$		0			0
$359$	−2.58410e6			−0.0558504			−0.0279252	−	0.999610i	$-0.508890\pi$
$359$	−2.58410e6			−0.0558504			−0.0279252	+	0.999610i	$0.508890\pi$
$360$		0			0
$361$	3.69999e7	−	2.90572e7i	0.786464	−	0.617636i
$362$		0			0
$363$		−	3.70700e7i		−	0.775001i
$364$		0			0
$365$	4.64539e7			0.955309
$366$		0			0
$367$	−2.11836e7			−0.428551			−0.214275	−	0.976773i	$-0.568739\pi$
$367$	−2.11836e7			−0.428551			−0.214275	+	0.976773i	$0.568739\pi$
$368$		0			0
$369$		−	1.00137e7i		−	0.199304i
$370$		0			0
$371$			2.67295e7i			0.523442i
$372$		0			0
$373$		−	5.14538e6i		−	0.0991496i	−0.998770	−	0.0495748i	$-0.984213\pi$
$373$		−	5.14538e6i		−	0.0991496i	0.998770	−	0.0495748i	$-0.0157866\pi$
$374$		0			0
$375$			3.04737e7i			0.577871i
$376$		0			0
$377$	−1.18313e7			−0.220804
$378$		0			0
$379$		−	7.33294e7i		−	1.34698i	−0.739197	−	0.673489i	$-0.764795\pi$
$379$		−	7.33294e7i		−	1.34698i	0.739197	−	0.673489i	$-0.235205\pi$
$380$		0			0
$381$	3.75685e7			0.679281
$382$		0			0
$383$		−	110690.i		−	0.00197020i	−1.00000		0.000985101i	$-0.999686\pi$
$383$		−	110690.i		−	0.00197020i	1.00000		0.000985101i	$-0.000313567\pi$
$384$		0			0
$385$	−1.21013e8			−2.12056
$386$		0			0
$387$	2.46501e7			0.425290
$388$		0			0
$389$	5.72855e7			0.973186			0.486593	−	0.873629i	$-0.338239\pi$
$389$	5.72855e7			0.973186			0.486593	+	0.873629i	$0.338239\pi$
$390$		0			0
$391$	1.56277e6			0.0261435
$392$		0			0
$393$		−	1.22547e7i		−	0.201894i
$394$		0			0
$395$			1.46432e8i			2.37598i
$396$		0			0
$397$	−8.20397e7			−1.31115			−0.655575	−	0.755130i	$-0.727574\pi$
$397$	−8.20397e7			−1.31115			−0.655575	+	0.755130i	$0.727574\pi$
$398$		0			0
$399$	−3.30534e7	−	9.56044e7i	−0.520353	−	1.50508i
$400$		0			0
$401$			6.27155e6i			0.0972617i	0.998817	+	0.0486309i	$0.0154858\pi$
$401$			6.27155e6i			0.0972617i	−0.998817	+	0.0486309i	$0.984514\pi$
$402$		0			0
$403$	−1.31314e7			−0.200629
$404$		0			0
$405$	−1.06037e8			−1.59622
$406$		0			0
$407$		−	8.93094e7i		−	1.32469i
$408$		0			0
$409$			2.96714e7i			0.433678i	0.976207	+	0.216839i	$0.0695747\pi$
$409$			2.96714e7i			0.433678i	−0.976207	+	0.216839i	$0.930425\pi$
$410$		0			0
$411$			7.78482e7i			1.12130i
$412$		0			0
$413$			1.76694e8i			2.50826i
$414$		0			0
$415$	−1.08718e8			−1.52110
$416$		0			0
$417$			9.65955e7i			1.33214i
$418$		0			0
$419$	−6.59835e7			−0.897002			−0.448501	−	0.893782i	$-0.648042\pi$
$419$	−6.59835e7			−0.897002			−0.448501	+	0.893782i	$0.648042\pi$
$420$		0			0
$421$			1.93760e7i			0.259668i	0.991536	+	0.129834i	$0.0414444\pi$
$421$			1.93760e7i			0.259668i	−0.991536	+	0.129834i	$0.958556\pi$
$422$		0			0
$423$	−2.42376e7			−0.320234
$424$		0			0
$425$	5.60752e6			0.0730472
$426$		0			0
$427$	3.27916e7			0.421190
$428$		0			0
$429$	−2.47148e7			−0.313029
$430$		0			0
$431$		−	1.45675e8i		−	1.81951i	−0.415147	−	0.909755i	$-0.636270\pi$
$431$		−	1.45675e8i		−	1.81951i	0.415147	−	0.909755i	$-0.363730\pi$
$432$		0			0
$433$		−	9.54378e7i		−	1.17559i	−0.809009	−	0.587796i	$-0.799996\pi$
$433$		−	9.54378e7i		−	1.17559i	0.809009	−	0.587796i	$-0.200004\pi$
$434$		0			0
$435$	−1.42225e8			−1.72786
$436$		0			0
$437$	−1.78029e7	+	6.15503e6i	−0.213328	+	0.0737540i
$438$		0			0
$439$			1.14060e7i			0.134816i	0.997726	+	0.0674078i	$0.0214729\pi$
$439$			1.14060e7i			0.134816i	−0.997726	+	0.0674078i	$0.978527\pi$
$440$		0			0
$441$	−2.96331e7			−0.345511
$442$		0			0
$443$	−1.16201e7			−0.133659			−0.0668294	−	0.997764i	$-0.521288\pi$
$443$	−1.16201e7			−0.133659			−0.0668294	+	0.997764i	$0.521288\pi$
$444$		0			0
$445$		−	3.87313e7i		−	0.439524i
$446$		0			0
$447$			4.46456e7i			0.499869i
$448$		0			0
$449$		−	2.46346e7i		−	0.272148i	−0.990699	−	0.136074i	$-0.956551\pi$
$449$		−	2.46346e7i		−	0.272148i	0.990699	−	0.136074i	$-0.0434486\pi$
$450$		0			0
$451$			4.65965e7i			0.507954i
$452$		0			0
$453$	1.94220e8			2.08929
$454$		0			0
$455$			3.12592e7i			0.331851i
$456$		0			0
$457$	−2.23961e7			−0.234652			−0.117326	−	0.993093i	$-0.537432\pi$
$457$	−2.23961e7			−0.234652			−0.117326	+	0.993093i	$0.537432\pi$
$458$		0			0
$459$		−	6.84372e6i		−	0.0707708i
$460$		0			0
$461$	−6.33431e7			−0.646541			−0.323271	−	0.946307i	$-0.604782\pi$
$461$	−6.33431e7			−0.646541			−0.323271	+	0.946307i	$0.604782\pi$
$462$		0			0
$463$	1.35759e8			1.36781			0.683903	−	0.729573i	$-0.260281\pi$
$463$	1.35759e8			1.36781			0.683903	+	0.729573i	$0.260281\pi$
$464$		0			0
$465$	−1.57854e8			−1.56999
$466$		0			0
$467$	1.60120e8			1.57215			0.786075	−	0.618131i	$-0.212110\pi$
$467$	1.60120e8			1.57215			0.786075	+	0.618131i	$0.212110\pi$
$468$		0			0
$469$		−	7.14210e7i		−	0.692321i
$470$		0			0
$471$			2.22886e7i			0.213315i
$472$		0			0
$473$	−1.14704e8			−1.08391
$474$		0			0
$475$	−6.38804e7	+	2.20854e7i	−0.596056	+	0.206075i
$476$		0			0
$477$			2.19131e7i			0.201906i
$478$		0			0
$479$	−1.15981e8			−1.05531			−0.527654	−	0.849460i	$-0.676928\pi$
$479$	−1.15981e8			−1.05531			−0.527654	+	0.849460i	$0.676928\pi$
$480$		0			0
$481$	−2.30697e7			−0.207303
$482$		0			0
$483$			4.05027e7i			0.359453i
$484$		0			0
$485$		−	1.98840e8i		−	1.74292i
$486$		0			0
$487$			9.87991e7i			0.855394i	0.903922	+	0.427697i	$0.140675\pi$
$487$			9.87991e7i			0.855394i	−0.903922	+	0.427697i	$0.859325\pi$
$488$		0			0
$489$		−	1.44349e8i		−	1.23449i
$490$		0			0
$491$	1.66715e8			1.40841			0.704206	−	0.709996i	$-0.251303\pi$
$491$	1.66715e8			1.40841			0.704206	+	0.709996i	$0.251303\pi$
$492$		0			0
$493$		−	1.53259e7i		−	0.127905i
$494$		0			0
$495$	−9.92078e7			−0.817957
$496$		0			0
$497$		−	1.15887e8i		−	0.943987i
$498$		0			0
$499$	−1.89230e8			−1.52296			−0.761479	−	0.648189i	$-0.775527\pi$
$499$	−1.89230e8			−1.52296			−0.761479	+	0.648189i	$0.775527\pi$
$500$		0			0
$501$	7.09719e7			0.564382
$502$		0			0
$503$	−1.32915e8			−1.04441			−0.522203	−	0.852821i	$-0.674890\pi$
$503$	−1.32915e8			−1.04441			−0.522203	+	0.852821i	$0.674890\pi$
$504$		0			0
$505$	−2.14391e8			−1.66469
$506$		0			0
$507$		−	1.53300e8i		−	1.17630i
$508$		0			0
$509$			1.41670e8i			1.07430i	0.843488	+	0.537148i	$0.180499\pi$
$509$			1.41670e8i			1.07430i	−0.843488	+	0.537148i	$0.819501\pi$
$510$		0			0
$511$	−1.29737e8			−0.972300
$512$		0			0
$513$	2.69543e7	+	7.79631e7i	0.199653	+	0.577480i
$514$		0			0
$515$		−	2.19720e8i		−	1.60860i
$516$		0			0
$517$	1.12784e8			0.816161
$518$		0			0
$519$	1.13937e8			0.815013
$520$		0			0
$521$		−	6.06347e7i		−	0.428754i	−0.976751	−	0.214377i	$-0.931228\pi$
$521$		−	6.06347e7i		−	0.428754i	0.976751	−	0.214377i	$-0.0687721\pi$
$522$		0			0
$523$		−	7.90109e7i		−	0.552309i	−0.961113	−	0.276154i	$-0.910940\pi$
$523$		−	7.90109e7i		−	0.552309i	0.961113	−	0.276154i	$-0.0890601\pi$
$524$		0			0
$525$			1.45332e8i			1.00434i
$526$		0			0
$527$		−	1.70101e7i		−	0.116218i
$528$		0			0
$529$	−1.40494e8			−0.949052
$530$		0			0
$531$			1.44856e8i			0.967502i
$532$		0			0
$533$	1.20364e7			0.0794907
$534$		0			0
$535$		−	1.46951e8i		−	0.959645i
$536$		0			0
$537$	−3.20901e8			−2.07228
$538$		0			0
$539$	1.37891e8			0.880581
$540$		0			0
$541$	1.11613e8			0.704895			0.352448	−	0.935832i	$-0.385349\pi$
$541$	1.11613e8			0.704895			0.352448	+	0.935832i	$0.385349\pi$
$542$		0			0
$543$	2.04433e8			1.27688
$544$		0			0
$545$			6.15738e7i			0.380370i
$546$		0			0
$547$		−	1.81019e8i		−	1.10602i	−0.833176	−	0.553008i	$-0.813480\pi$
$547$		−	1.81019e8i		−	1.10602i	0.833176	−	0.553008i	$-0.186520\pi$
$548$		0			0
$549$	2.68828e7			0.162464
$550$		0			0
$551$	6.03618e7	+	1.74592e8i	0.360834	+	1.04368i
$552$		0			0
$553$		−	4.08955e8i		−	2.41824i
$554$		0			0
$555$	−2.77323e8			−1.62221
$556$		0			0
$557$	2.25990e8			1.30775			0.653873	−	0.756604i	$-0.273143\pi$
$557$	2.25990e8			1.30775			0.653873	+	0.756604i	$0.273143\pi$
$558$		0			0
$559$			2.96293e7i			0.169623i
$560$		0			0
$561$		−	3.20149e7i		−	0.181328i
$562$		0			0
$563$		−	3.26109e8i		−	1.82742i	−0.406370	−	0.913709i	$-0.633206\pi$
$563$		−	3.26109e8i		−	1.82742i	0.406370	−	0.913709i	$-0.366794\pi$
$564$		0			0
$565$		−	9.91020e7i		−	0.549461i
$566$		0			0
$567$	2.96141e8			1.62461
$568$		0			0
$569$			1.86517e8i			1.01247i	0.862396	+	0.506234i	$0.168963\pi$
$569$			1.86517e8i			1.01247i	−0.862396	+	0.506234i	$0.831037\pi$
$570$		0			0
$571$	−8.66919e7			−0.465662			−0.232831	−	0.972517i	$-0.574799\pi$
$571$	−8.66919e7			−0.465662			−0.232831	+	0.972517i	$0.574799\pi$
$572$		0			0
$573$			2.22217e8i			1.18117i
$574$		0			0
$575$	2.70628e7			0.142354
$576$		0			0
$577$	−1.74442e8			−0.908079			−0.454039	−	0.890982i	$-0.650017\pi$
$577$	−1.74442e8			−0.908079			−0.454039	+	0.890982i	$0.650017\pi$
$578$		0			0
$579$	−1.81361e8			−0.934348
$580$		0			0
$581$	3.03629e8			1.54816
$582$		0			0
$583$		−	1.01967e8i		−	0.514584i
$584$		0			0
$585$			2.56266e7i			0.128004i
$586$		0			0
$587$	6.77908e6			0.0335164			0.0167582	−	0.999860i	$-0.494665\pi$
$587$	6.77908e6			0.0335164			0.0167582	+	0.999860i	$0.494665\pi$
$588$		0			0
$589$	6.69948e7	+	1.93777e8i	0.327865	+	0.948324i
$590$		0			0
$591$			2.67181e8i			1.29432i
$592$		0			0
$593$	4.99086e7			0.239338			0.119669	−	0.992814i	$-0.461817\pi$
$593$	4.99086e7			0.239338			0.119669	+	0.992814i	$0.461817\pi$
$594$		0			0
$595$	−4.04924e7			−0.192231
$596$		0			0
$597$		−	2.56881e8i		−	1.20728i
$598$		0			0
$599$			1.34986e8i			0.628072i	0.949411	+	0.314036i	$0.101681\pi$
$599$			1.34986e8i			0.628072i	−0.949411	+	0.314036i	$0.898319\pi$
$600$		0			0
$601$			9.31195e7i			0.428960i	0.976728	+	0.214480i	$0.0688057\pi$
$601$			9.31195e7i			0.428960i	−0.976728	+	0.214480i	$0.931194\pi$
$602$		0			0
$603$		−	5.85516e7i		−	0.267047i
$604$		0			0
$605$	1.78860e8			0.807697
$606$		0			0
$607$			2.08533e8i			0.932415i	0.884675	+	0.466207i	$0.154380\pi$
$607$			2.08533e8i			0.932415i	−0.884675	+	0.466207i	$0.845620\pi$
$608$		0			0
$609$	3.97207e8			1.75859
$610$		0			0
$611$		−	2.91334e7i		−	0.127723i
$612$		0			0
$613$	3.14675e8			1.36609			0.683047	−	0.730374i	$-0.260654\pi$
$613$	3.14675e8			1.36609			0.683047	+	0.730374i	$0.260654\pi$
$614$		0			0
$615$	1.44691e8			0.622039
$616$		0			0
$617$	−4.21738e7			−0.179551			−0.0897753	−	0.995962i	$-0.528615\pi$
$617$	−4.21738e7			−0.179551			−0.0897753	+	0.995962i	$0.528615\pi$
$618$		0			0
$619$	2.07102e7			0.0873198			0.0436599	−	0.999046i	$-0.486098\pi$
$619$	2.07102e7			0.0873198			0.0436599	+	0.999046i	$0.486098\pi$
$620$		0			0
$621$		−	3.30290e7i		−	0.137918i
$622$		0			0
$623$			1.08169e8i			0.447341i
$624$		0			0
$625$	−3.01007e8			−1.23292
$626$		0			0
$627$	1.26092e8	+	3.64712e8i	0.511547	+	1.47961i
$628$		0			0
$629$		−	2.98839e7i		−	0.120084i
$630$		0			0
$631$	−2.34658e8			−0.934000			−0.467000	−	0.884257i	$-0.654665\pi$
$631$	−2.34658e8			−0.934000			−0.467000	+	0.884257i	$0.654665\pi$
$632$		0			0
$633$	−2.01816e8			−0.795691
$634$		0			0
$635$			1.81266e8i			0.707938i
$636$		0			0
$637$		−	3.56189e7i		−	0.137804i
$638$		0			0
$639$		−	9.50054e7i		−	0.364121i
$640$		0			0
$641$			4.48685e8i			1.70360i	0.523868	+	0.851799i	$0.324488\pi$
$641$			4.48685e8i			1.70360i	−0.523868	+	0.851799i	$0.675512\pi$
$642$		0			0
$643$	2.63984e7			0.0992991			0.0496495	−	0.998767i	$-0.484190\pi$
$643$	2.63984e7			0.0992991			0.0496495	+	0.998767i	$0.484190\pi$
$644$		0			0
$645$			3.56177e8i			1.32735i
$646$		0			0
$647$	−2.06171e8			−0.761229			−0.380614	−	0.924734i	$-0.624287\pi$
$647$	−2.06171e8			−0.761229			−0.380614	+	0.924734i	$0.624287\pi$
$648$		0			0
$649$		−	6.74053e8i		−	2.46581i
$650$		0			0
$651$	4.40854e8			1.59791
$652$		0			0
$653$	−1.04864e8			−0.376606			−0.188303	−	0.982111i	$-0.560299\pi$
$653$	−1.04864e8			−0.376606			−0.188303	+	0.982111i	$0.560299\pi$
$654$		0			0
$655$	5.91281e7			0.210412
$656$		0			0
$657$	−1.06359e8			−0.375042
$658$		0			0
$659$		−	1.00063e8i		−	0.349638i	−0.984601	−	0.174819i	$-0.944066\pi$
$659$		−	1.00063e8i		−	0.349638i	0.984601	−	0.174819i	$-0.0559340\pi$
$660$		0			0
$661$			3.79284e8i			1.31329i	0.754201	+	0.656644i	$0.228024\pi$
$661$			3.79284e8i			1.31329i	−0.754201	+	0.656644i	$0.771976\pi$
$662$		0			0
$663$	−8.26983e6			−0.0283763
$664$		0			0
$665$	4.61286e8	−	1.59481e8i	1.56858	−	0.542305i
$666$		0			0
$667$		−	7.39656e7i		−	0.249260i
$668$		0			0
$669$	−1.02460e8			−0.342197
$670$		0			0
$671$	−1.25093e8			−0.414063
$672$		0			0
$673$			3.66445e8i			1.20216i	0.799187	+	0.601082i	$0.205264\pi$
$673$			3.66445e8i			1.20216i	−0.799187	+	0.601082i	$0.794736\pi$
$674$		0			0
$675$		−	1.18514e8i		−	0.385354i
$676$		0			0
$677$			2.76020e8i			0.889560i	0.895640	+	0.444780i	$0.146718\pi$
$677$			2.76020e8i			0.889560i	−0.895640	+	0.444780i	$0.853282\pi$
$678$		0			0
$679$			5.55321e8i			1.77392i
$680$		0			0
$681$	−4.18370e7			−0.132470
$682$		0			0
$683$			1.05100e8i			0.329868i	0.986305	+	0.164934i	$0.0527412\pi$
$683$			1.05100e8i			0.329868i	−0.986305	+	0.164934i	$0.947259\pi$
$684$		0			0
$685$	−3.75613e8			−1.16861
$686$		0			0
$687$		−	3.52777e8i		−	1.08800i
$688$		0			0
$689$	−2.63394e7			−0.0805283
$690$		0			0
$691$	3.03677e8			0.920402			0.460201	−	0.887815i	$-0.347777\pi$
$691$	3.03677e8			0.920402			0.460201	+	0.887815i	$0.347777\pi$
$692$		0			0
$693$	2.77068e8			0.832505
$694$		0			0
$695$	−4.66068e8			−1.38834
$696$		0			0
$697$			1.55917e7i			0.0460464i
$698$		0			0
$699$		−	9.43659e7i		−	0.276302i
$700$		0			0
$701$	−4.26003e8			−1.23668			−0.618342	−	0.785909i	$-0.712195\pi$
$701$	−4.26003e8			−1.23668			−0.618342	+	0.785909i	$0.712195\pi$
$702$		0			0
$703$	1.17699e8	+	3.40435e8i	0.338772	+	0.979870i
$704$		0			0
$705$		−	3.50217e8i		−	0.999469i
$706$		0			0
$707$	5.98752e8			1.69429
$708$		0			0
$709$	4.72028e8			1.32443			0.662216	−	0.749313i	$-0.269616\pi$
$709$	4.72028e8			1.32443			0.662216	+	0.749313i	$0.269616\pi$
$710$		0			0
$711$		−	3.35265e8i		−	0.932781i
$712$		0			0
$713$		−	8.20935e7i		−	0.226485i
$714$		0			0
$715$		−	1.19247e8i		−	0.326235i
$716$		0			0
$717$		−	5.90247e8i		−	1.60131i
$718$		0			0
$719$	−7.72222e7			−0.207757			−0.103879	−	0.994590i	$-0.533125\pi$
$719$	−7.72222e7			−0.207757			−0.103879	+	0.994590i	$0.533125\pi$
$720$		0			0
$721$			6.13636e8i			1.63721i
$722$		0			0
$723$	−1.89250e8			−0.500751
$724$		0			0
$725$		−	2.65403e8i		−	0.696453i
$726$		0			0
$727$	4.32964e8			1.12680			0.563402	−	0.826183i	$-0.309492\pi$
$727$	4.32964e8			1.12680			0.563402	+	0.826183i	$0.309492\pi$
$728$		0			0
$729$	−4.72721e7			−0.122018
$730$		0			0
$731$	−3.83811e7			−0.0982574
$732$		0			0
$733$	−4.94099e8			−1.25459			−0.627296	−	0.778781i	$-0.715838\pi$
$733$	−4.94099e8			−1.25459			−0.627296	+	0.778781i	$0.715838\pi$
$734$		0			0
$735$		−	4.28179e8i		−	1.07836i
$736$		0			0
$737$			2.72457e8i			0.680605i
$738$		0			0
$739$	6.00444e8			1.48778			0.743890	−	0.668302i	$-0.232978\pi$
$739$	6.00444e8			1.48778			0.743890	+	0.668302i	$0.232978\pi$
$740$		0			0
$741$	9.42093e7	−	3.25711e7i	0.231547	−	0.0800530i
$742$		0			0
$743$			1.76594e8i			0.430536i	0.976555	+	0.215268i	$0.0690625\pi$
$743$			1.76594e8i			0.430536i	−0.976555	+	0.215268i	$0.930937\pi$
$744$		0			0
$745$	−2.15412e8			−0.520957
$746$		0			0
$747$	2.48918e8			0.597165
$748$		0			0
$749$			4.10405e8i			0.976713i
$750$		0			0
$751$			4.85737e8i			1.14678i	0.819281	+	0.573392i	$0.194373\pi$
$751$			4.85737e8i			1.14678i	−0.819281	+	0.573392i	$0.805627\pi$
$752$		0			0
$753$			2.46838e8i			0.578133i
$754$		0			0
$755$			9.37101e8i			2.17744i
$756$		0			0
$757$	6.87374e8			1.58455			0.792274	−	0.610166i	$-0.208897\pi$
$757$	6.87374e8			1.58455			0.792274	+	0.610166i	$0.208897\pi$
$758$		0			0
$759$		−	1.54510e8i		−	0.353370i
$760$		0			0
$761$	7.71555e8			1.75070			0.875352	−	0.483485i	$-0.160629\pi$
$761$	7.71555e8			1.75070			0.875352	+	0.483485i	$0.160629\pi$
$762$		0			0
$763$		−	1.71964e8i		−	0.387136i
$764$		0			0
$765$	−3.31960e7			−0.0741485
$766$		0			0
$767$	−1.74116e8			−0.385880
$768$		0			0
$769$	−1.01084e8			−0.222281			−0.111141	−	0.993805i	$-0.535450\pi$
$769$	−1.01084e8			−0.222281			−0.111141	+	0.993805i	$0.535450\pi$
$770$		0			0
$771$	2.80103e8			0.611160
$772$		0			0
$773$		−	4.04573e8i		−	0.875909i	−0.898997	−	0.437954i	$-0.855703\pi$
$773$		−	4.04573e8i		−	0.875909i	0.898997	−	0.437954i	$-0.144297\pi$
$774$		0			0
$775$		−	2.94567e8i		−	0.632819i
$776$		0			0
$777$	7.74510e8			1.65106
$778$		0			0
$779$	−6.14086e7	−	1.77620e8i	−0.129902	−	0.375732i
$780$		0			0
$781$			4.42086e8i			0.928012i
$782$		0			0
$783$	−3.23912e8			−0.674749
$784$		0			0
$785$	−1.07541e8			−0.222314
$786$		0			0
$787$		−	4.48190e8i		−	0.919470i	−0.888056	−	0.459735i	$-0.847944\pi$
$787$		−	4.48190e8i		−	0.919470i	0.888056	−	0.459735i	$-0.152056\pi$
$788$		0			0
$789$		−	1.03880e9i		−	2.11496i
$790$		0			0
$791$			2.76772e8i			0.559234i
$792$		0			0
$793$			3.23130e7i			0.0647975i
$794$		0			0
$795$	−3.16629e8			−0.630158
$796$		0			0
$797$		−	8.28284e8i		−	1.63608i	−0.575162	−	0.818040i	$-0.695061\pi$
$797$		−	8.28284e8i		−	1.63608i	0.575162	−	0.818040i	$-0.304939\pi$
$798$		0			0
$799$	3.77388e7			0.0739856
$800$		0			0
$801$			8.86780e7i			0.172551i
$802$		0			0
$803$	4.94919e8			0.955846
$804$		0			0
$805$	−1.95423e8			−0.374618
$806$		0			0
$807$	−2.08707e8			−0.397115
$808$		0			0
$809$	5.49140e8			1.03714			0.518570	−	0.855035i	$-0.326464\pi$
$809$	5.49140e8			1.03714			0.518570	+	0.855035i	$0.326464\pi$
$810$		0			0
$811$		−	4.97409e8i		−	0.932505i	−0.884652	−	0.466252i	$-0.845604\pi$
$811$		−	4.97409e8i		−	0.932505i	0.884652	−	0.466252i	$-0.154396\pi$
$812$		0			0
$813$			4.75086e8i			0.884099i
$814$		0			0
$815$	6.96474e8			1.28657
$816$		0			0
$817$	4.37234e8	−	1.51166e8i	0.801767	−	0.277196i
$818$		0			0
$819$		−	7.15700e7i		−	0.130280i
$820$		0			0
$821$	−2.38989e8			−0.431865			−0.215932	−	0.976408i	$-0.569279\pi$
$821$	−2.38989e8			−0.431865			−0.215932	+	0.976408i	$0.569279\pi$
$822$		0			0
$823$	6.68644e8			1.19949			0.599743	−	0.800193i	$-0.295269\pi$
$823$	6.68644e8			1.19949			0.599743	+	0.800193i	$0.295269\pi$
$824$		0			0
$825$		−	5.54410e8i		−	0.987346i
$826$		0			0
$827$		−	2.00405e8i		−	0.354317i	−0.984182	−	0.177158i	$-0.943310\pi$
$827$		−	2.00405e8i		−	0.354317i	0.984182	−	0.177158i	$-0.0566905\pi$
$828$		0			0
$829$			5.83575e8i			1.02431i	0.858892	+	0.512157i	$0.171154\pi$
$829$			5.83575e8i			1.02431i	−0.858892	+	0.512157i	$0.828846\pi$
$830$		0			0
$831$		−	3.10501e8i		−	0.541077i
$832$		0			0
$833$	4.61398e7			0.0798254
$834$		0			0
$835$			3.42435e8i			0.588192i
$836$		0			0
$837$	−3.59506e8			−0.613098
$838$		0			0
$839$		−	2.41664e8i		−	0.409190i	−0.978847	−	0.204595i	$-0.934412\pi$
$839$		−	2.41664e8i		−	0.409190i	0.978847	−	0.204595i	$-0.0655878\pi$
$840$		0			0
$841$	−1.30551e8			−0.219479
$842$		0			0
$843$	−1.30214e9			−2.17357
$844$		0			0
$845$	7.39663e8			1.22593
$846$		0			0
$847$	−4.99522e8			−0.822062
$848$		0			0
$849$		−	8.63770e8i		−	1.41148i
$850$		0			0
$851$		−	1.44225e8i		−	0.234019i
$852$		0			0
$853$	3.27850e8			0.528236			0.264118	−	0.964490i	$-0.414919\pi$
$853$	3.27850e8			0.528236			0.264118	+	0.964490i	$0.414919\pi$
$854$		0			0
$855$	3.78167e8	−	1.30744e8i	0.605042	−	0.209182i
$856$		0			0
$857$		−	5.60607e8i		−	0.890668i	−0.895364	−	0.445334i	$-0.853085\pi$
$857$		−	5.60607e8i		−	0.890668i	0.895364	−	0.445334i	$-0.146915\pi$
$858$		0			0
$859$	−1.07374e9			−1.69402			−0.847010	−	0.531576i	$-0.821600\pi$
$859$	−1.07374e9			−1.69402			−0.847010	+	0.531576i	$0.821600\pi$
$860$		0			0
$861$	−4.04095e8			−0.633102
$862$		0			0
$863$			5.66605e8i			0.881552i	0.897617	+	0.440776i	$0.145297\pi$
$863$			5.66605e8i			0.881552i	−0.897617	+	0.440776i	$0.854703\pi$
$864$		0			0
$865$			5.49742e8i			0.849397i
$866$		0			0
$867$			7.87824e8i			1.20885i
$868$		0			0
$869$			1.56008e9i			2.37732i
$870$		0			0
$871$	7.03788e7			0.106509
$872$		0			0
$873$			4.55257e8i			0.684249i
$874$		0			0
$875$	4.10637e8			0.612962
$876$		0			0
$877$		−	2.64799e8i		−	0.392571i	−0.980547	−	0.196285i	$-0.937112\pi$
$877$		−	2.64799e8i		−	0.392571i	0.980547	−	0.196285i	$-0.0628879\pi$
$878$		0			0
$879$	5.23720e8			0.771139
$880$		0			0
$881$	−3.11993e8			−0.456265			−0.228133	−	0.973630i	$-0.573262\pi$
$881$	−3.11993e8			−0.456265			−0.228133	+	0.973630i	$0.573262\pi$
$882$		0			0
$883$	−7.80065e8			−1.13305			−0.566525	−	0.824045i	$-0.691712\pi$
$883$	−7.80065e8			−1.13305			−0.566525	+	0.824045i	$0.691712\pi$
$884$		0			0
$885$	−2.09307e9			−3.01963
$886$		0			0
$887$		−	2.78539e8i		−	0.399131i	−0.979885	−	0.199565i	$-0.936047\pi$
$887$		−	2.78539e8i		−	0.399131i	0.979885	−	0.199565i	$-0.0639530\pi$
$888$		0			0
$889$		−	5.06241e8i		−	0.720529i
$890$		0			0
$891$	−1.12972e9			−1.59712
$892$		0			0
$893$	−4.29917e8	+	1.48636e8i	−0.603713	+	0.208722i
$894$		0			0
$895$		−	1.54833e9i		−	2.15971i
$896$		0			0
$897$	−3.99116e7			−0.0552996
$898$		0			0
$899$	−8.05084e8			−1.10806
$900$		0			0
$901$		−	3.41195e7i		−	0.0466474i
$902$		0			0
$903$		−	9.94733e8i		−	1.35096i
$904$		0			0
$905$			9.86376e8i			1.33075i
$906$		0			0
$907$			1.08363e9i			1.45231i	0.687533	+	0.726153i	$0.258693\pi$
$907$			1.08363e9i			1.45231i	−0.687533	+	0.726153i	$0.741307\pi$
$908$		0			0
$909$	4.90863e8			0.653534
$910$		0			0
$911$		−	3.58900e8i		−	0.474699i	−0.971424	−	0.237350i	$-0.923721\pi$
$911$		−	3.58900e8i		−	0.474699i	0.971424	−	0.237350i	$-0.0762787\pi$
$912$		0			0
$913$	−1.15828e9			−1.52196
$914$		0			0
$915$			3.88439e8i			0.507060i
$916$		0			0
$917$	−1.65133e8			−0.214154
$918$		0			0
$919$	−3.96276e8			−0.510565			−0.255282	−	0.966867i	$-0.582168\pi$
$919$	−3.96276e8			−0.510565			−0.255282	+	0.966867i	$0.582168\pi$
$920$		0			0
$921$	−1.72657e9			−2.21007
$922$		0			0
$923$	1.14196e8			0.145226
$924$		0			0
$925$		−	5.17507e8i		−	0.653870i
$926$		0			0
$927$			5.03065e8i			0.631517i
$928$		0			0
$929$	7.38996e8			0.921711			0.460855	−	0.887475i	$-0.347543\pi$
$929$	7.38996e8			0.921711			0.460855	+	0.887475i	$0.347543\pi$
$930$		0			0
$931$	−5.25622e8	+	1.81724e8i	−0.651365	+	0.225197i
$932$		0			0
$933$		−	1.78843e9i		−	2.20205i
$934$		0			0
$935$	1.54470e8			0.188977
$936$		0			0
$937$	−2.42474e8			−0.294745			−0.147372	−	0.989081i	$-0.547082\pi$
$937$	−2.42474e8			−0.294745			−0.147372	+	0.989081i	$0.547082\pi$
$938$		0			0
$939$		−	8.23490e8i		−	0.994630i
$940$		0			0
$941$			7.76759e8i			0.932218i	0.884727	+	0.466109i	$0.154345\pi$
$941$			7.76759e8i			0.932218i	−0.884727	+	0.466109i	$0.845655\pi$
$942$		0			0
$943$			7.52483e7i			0.0897350i
$944$		0			0
$945$			8.55803e8i			1.01409i
$946$		0			0
$947$	7.91274e8			0.931702			0.465851	−	0.884863i	$-0.345748\pi$
$947$	7.91274e8			0.931702			0.465851	+	0.884863i	$0.345748\pi$
$948$		0			0
$949$		−	1.27844e8i		−	0.149582i
$950$		0			0
$951$	−1.25149e9			−1.45507
$952$		0			0
$953$		−	1.79542e8i		−	0.207437i	−0.994607	−	0.103718i	$-0.966926\pi$
$953$		−	1.79542e8i		−	0.207437i	0.994607	−	0.103718i	$-0.0330741\pi$
$954$		0			0
$955$	−1.07219e9			−1.23100
$956$		0			0
$957$	−1.51526e9			−1.72883
$958$		0			0
$959$	1.04901e9			1.18939
$960$		0			0
$961$	−6.04825e6			−0.00681490
$962$		0			0
$963$			3.36454e8i			0.376744i
$964$		0			0
$965$		−	8.75058e8i		−	0.973767i
$966$		0			0
$967$	7.59012e8			0.839400			0.419700	−	0.907663i	$-0.362135\pi$
$967$	7.59012e8			0.839400			0.419700	+	0.907663i	$0.362135\pi$
$968$		0			0
$969$	4.21918e7	+	1.22037e8i	0.0463721	+	0.134128i
$970$		0			0
$971$			9.72178e8i			1.06191i	0.847400	+	0.530956i	$0.178167\pi$
$971$			9.72178e8i			1.06191i	−0.847400	+	0.530956i	$0.821833\pi$
$972$		0			0
$973$	1.30164e9			1.41303
$974$		0			0
$975$	−1.43211e8			−0.154512
$976$		0			0
$977$		−	3.54587e8i		−	0.380224i	−0.981762	−	0.190112i	$-0.939115\pi$
$977$		−	3.54587e8i		−	0.380224i	0.981762	−	0.190112i	$-0.0608851\pi$
$978$		0			0
$979$		−	4.12643e8i		−	0.439771i
$980$		0			0
$981$		−	1.40977e8i		−	0.149329i
$982$		0			0
$983$			7.87276e8i			0.828832i	0.910088	+	0.414416i	$0.136014\pi$
$983$			7.87276e8i			0.828832i	−0.910088	+	0.414416i	$0.863986\pi$
$984$		0			0
$985$	−1.28913e9			−1.34893
$986$		0			0
$987$			9.78086e8i			1.01725i
$988$		0			0
$989$	−1.85234e8			−0.191483
$990$		0			0
$991$		−	1.87947e8i		−	0.193114i	−0.995327	−	0.0965571i	$-0.969217\pi$
$991$		−	1.87947e8i		−	0.193114i	0.995327	−	0.0965571i	$-0.0307830\pi$
$992$		0			0
$993$	1.99925e9			2.04183
$994$		0			0
$995$	1.23944e9			1.25822
$996$		0			0
$997$	−3.26970e8			−0.329930			−0.164965	−	0.986299i	$-0.552751\pi$
$997$	−3.26970e8			−0.329930			−0.164965	+	0.986299i	$0.552751\pi$
$998$		0			0
$999$	−6.31594e8			−0.633492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.7.e.d.113.2		8
4.3	odd	2		19.7.b.b.18.8	yes	8
12.11	even	2		171.7.c.d.37.1		8
19.18	odd	2	inner	304.7.e.d.113.7		8
76.75	even	2		19.7.b.b.18.1	✓	8
228.227	odd	2		171.7.c.d.37.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.7.b.b.18.1	✓	8	76.75	even	2
19.7.b.b.18.8	yes	8	4.3	odd	2
171.7.c.d.37.1		8	12.11	even	2
171.7.c.d.37.8		8	228.227	odd	2
304.7.e.d.113.2		8	1.1	even	1	trivial
304.7.e.d.113.7		8	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)

\(f(q)\)	\(=\)	\(q-33.0827i q^{3} +159.622 q^{5} -445.794 q^{7} -365.466 q^{9} +O(q^{10})\)	\(q-33.0827i q^{3} +159.622 q^{5} -445.794 q^{7} -365.466 q^{9} +1700.61 q^{11} -439.288i q^{13} -5280.74i q^{15} +569.044 q^{17} +(-6482.51 + 2241.20i) q^{19} +14748.1i q^{21} +2746.30 q^{23} +9854.28 q^{25} -12026.7i q^{27} -26932.8i q^{29} -29892.3i q^{31} -56260.9i q^{33} -71158.6 q^{35} -52516.0i q^{37} -14532.9 q^{39} +27399.8i q^{41} -67448.4 q^{43} -58336.5 q^{45} +66319.6 q^{47} +81083.1 q^{49} -18825.5i q^{51} -59959.3i q^{53} +271456. q^{55} +(74145.1 + 214459. i) q^{57} -396359. i q^{59} -73557.7 q^{61} +162922. q^{63} -70120.2i q^{65} +160211. i q^{67} -90855.2i q^{69} +259957. i q^{71} +291024. q^{73} -326006. i q^{75} -758123. q^{77} +917363. i q^{79} -664300. q^{81} -681097. q^{83} +90832.1 q^{85} -891010. q^{87} -242644. i q^{89} +195832. i q^{91} -988920. q^{93} +(-1.03475e6 + 357746. i) q^{95} -1.24569e6i q^{97} -621516. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 108 q^{5} + 140 q^{7} - 1052 q^{9}+O(q^{10}) \) 8 * q + 108 * q^5 + 140 * q^7 - 1052 * q^9	\( 8 q + 108 q^{5} + 140 q^{7} - 1052 q^{9} + 2024 q^{11} + 6008 q^{17} - 20552 q^{19} + 50252 q^{23} + 78492 q^{25} - 210800 q^{35} - 43724 q^{39} - 260800 q^{43} - 191012 q^{45} + 100248 q^{47} - 301872 q^{49} + 52480 q^{55} - 186860 q^{57} - 54548 q^{61} + 137408 q^{63} + 479968 q^{73} - 1755300 q^{77} - 4279648 q^{81} - 483040 q^{83} + 2111780 q^{85} - 2802652 q^{87} + 1507528 q^{93} + 2383888 q^{95} - 528224 q^{99}+O(q^{100}) \) 8 * q + 108 * q^5 + 140 * q^7 - 1052 * q^9 + 2024 * q^11 + 6008 * q^17 - 20552 * q^19 + 50252 * q^23 + 78492 * q^25 - 210800 * q^35 - 43724 * q^39 - 260800 * q^43 - 191012 * q^45 + 100248 * q^47 - 301872 * q^49 + 52480 * q^55 - 186860 * q^57 - 54548 * q^61 + 137408 * q^63 + 479968 * q^73 - 1755300 * q^77 - 4279648 * q^81 - 483040 * q^83 + 2111780 * q^85 - 2802652 * q^87 + 1507528 * q^93 + 2383888 * q^95 - 528224 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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