LMFDB - Embedded newform 304.5.e.f.113.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,5,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, 5, names="a")

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

chi := DirichletCharacter("304.113");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
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:	$31.4244687775$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 996 x^{18} + 408854 x^{16} + 89661524 x^{14} + 11414409521 x^{12} + 861580608848 x^{10} + \cdots + 34\!\cdots\!64 $ x^20 + 996x^18 + 408854x^16 + 89661524x^14 + 11414409521x^12 + 861580608848x^10 + 37911701888064x^8 + 915101881952256x^6 + 10365057927217152x^4 + 34336742150504448*x^2 + 34273490123096064
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{50} $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.12
Root			$1.51452i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.5.e.f.113.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.51452i q^{3} +15.8064 q^{5} +74.7269 q^{7} +78.7062 q^{9} +O(q^{10})$	$q+1.51452i q^{3} +15.8064 q^{5} +74.7269 q^{7} +78.7062 q^{9} -8.49946 q^{11} -283.924i q^{13} +23.9391i q^{15} -36.0953 q^{17} +(345.662 + 104.111i) q^{19} +113.175i q^{21} -488.441 q^{23} -375.157 q^{25} +241.878i q^{27} -492.463i q^{29} +1281.73i q^{31} -12.8726i q^{33} +1181.17 q^{35} -2139.58i q^{37} +430.008 q^{39} -2076.32i q^{41} +3051.37 q^{43} +1244.06 q^{45} -1218.81 q^{47} +3183.11 q^{49} -54.6669i q^{51} +4680.27i q^{53} -134.346 q^{55} +(-157.678 + 523.510i) q^{57} -3207.19i q^{59} +2169.77 q^{61} +5881.48 q^{63} -4487.83i q^{65} +6862.25i q^{67} -739.752i q^{69} -6267.19i q^{71} +2394.57 q^{73} -568.181i q^{75} -635.138 q^{77} -1312.83i q^{79} +6008.88 q^{81} -672.952 q^{83} -570.537 q^{85} +745.843 q^{87} +11177.3i q^{89} -21216.8i q^{91} -1941.21 q^{93} +(5463.67 + 1645.62i) q^{95} +10698.0i q^{97} -668.960 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 32 q^{7} - 372 q^{9}+O(q^{10}) $ 20 * q - 32 * q^7 - 372 * q^9	$ 20 q - 32 q^{7} - 372 q^{9} + 24 q^{11} + 216 q^{17} + 596 q^{19} - 576 q^{23} + 1412 q^{25} + 144 q^{35} + 520 q^{39} + 1256 q^{43} + 7232 q^{45} + 3768 q^{47} - 2740 q^{49} + 10128 q^{55} - 728 q^{57} + 352 q^{61} - 6104 q^{63} + 1352 q^{73} + 9288 q^{77} - 4220 q^{81} + 16104 q^{83} + 10232 q^{85} - 2936 q^{87} + 36432 q^{93} - 14232 q^{95} - 760 q^{99}+O(q^{100}) $ 20 * q - 32 * q^7 - 372 * q^9 + 24 * q^11 + 216 * q^17 + 596 * q^19 - 576 * q^23 + 1412 * q^25 + 144 * q^35 + 520 * q^39 + 1256 * q^43 + 7232 * q^45 + 3768 * q^47 - 2740 * q^49 + 10128 * q^55 - 728 * q^57 + 352 * q^61 - 6104 * q^63 + 1352 * q^73 + 9288 * q^77 - 4220 * q^81 + 16104 * q^83 + 10232 * q^85 - 2936 * q^87 + 36432 * q^93 - 14232 * q^95 - 760 * 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
$3$			1.51452i			0.168279i	0.996454	+	0.0841397i	$0.0268142\pi$
$3$			1.51452i			0.168279i	−0.996454	+	0.0841397i	$0.973186\pi$
$4$		0			0
$5$	15.8064			0.632257			0.316129	−	0.948716i	$-0.397617\pi$
$5$	15.8064			0.632257			0.316129	+	0.948716i	$0.397617\pi$
$6$		0			0
$7$	74.7269			1.52504			0.762520	−	0.646965i	$-0.223962\pi$
$7$	74.7269			1.52504			0.762520	+	0.646965i	$0.223962\pi$
$8$		0			0
$9$	78.7062			0.971682
$10$		0			0
$11$	−8.49946			−0.0702434			−0.0351217	−	0.999383i	$-0.511182\pi$
$11$	−8.49946			−0.0702434			−0.0351217	+	0.999383i	$0.511182\pi$
$12$		0			0
$13$		−	283.924i		−	1.68003i	−0.542566	−	0.840013i	$-0.682547\pi$
$13$		−	283.924i		−	1.68003i	0.542566	−	0.840013i	$-0.317453\pi$
$14$		0			0
$15$			23.9391i			0.106396i
$16$		0			0
$17$	−36.0953			−0.124897			−0.0624486	−	0.998048i	$-0.519891\pi$
$17$	−36.0953			−0.124897			−0.0624486	+	0.998048i	$0.519891\pi$
$18$		0			0
$19$	345.662	+	104.111i	0.957511	+	0.288396i
$19$	345.662	+	104.111i	0.957511	+	0.288396i
$20$		0			0
$21$			113.175i			0.256633i
$22$		0			0
$23$	−488.441			−0.923330			−0.461665	−	0.887054i	$-0.652748\pi$
$23$	−488.441			−0.923330			−0.461665	+	0.887054i	$0.652748\pi$
$24$		0			0
$25$	−375.157			−0.600251
$26$		0			0
$27$			241.878i			0.331794i
$28$		0			0
$29$		−	492.463i		−	0.585568i	−0.956179	−	0.292784i	$-0.905418\pi$
$29$		−	492.463i		−	0.585568i	0.956179	−	0.292784i	$-0.0945817\pi$
$30$		0			0
$31$			1281.73i			1.33375i	0.745170	+	0.666875i	$0.232369\pi$
$31$			1281.73i			1.33375i	−0.745170	+	0.666875i	$0.767631\pi$
$32$		0			0
$33$		−	12.8726i		−	0.0118205i
$34$		0			0
$35$	1181.17			0.964217
$36$		0			0
$37$		−	2139.58i		−	1.56288i	−0.623983	−	0.781438i	$-0.714486\pi$
$37$		−	2139.58i		−	1.56288i	0.623983	−	0.781438i	$-0.285514\pi$
$38$		0			0
$39$	430.008			0.282714
$40$		0			0
$41$		−	2076.32i		−	1.23517i	−0.786505	−	0.617584i	$-0.788112\pi$
$41$		−	2076.32i		−	1.23517i	0.786505	−	0.617584i	$-0.211888\pi$
$42$		0			0
$43$	3051.37			1.65028			0.825139	−	0.564929i	$-0.191097\pi$
$43$	3051.37			1.65028			0.825139	+	0.564929i	$0.191097\pi$
$44$		0			0
$45$	1244.06			0.614353
$46$		0			0
$47$	−1218.81			−0.551747			−0.275873	−	0.961194i	$-0.588967\pi$
$47$	−1218.81			−0.551747			−0.275873	+	0.961194i	$0.588967\pi$
$48$		0			0
$49$	3183.11			1.32575
$50$		0			0
$51$		−	54.6669i		−	0.0210176i
$52$		0			0
$53$			4680.27i			1.66617i	0.553145	+	0.833085i	$0.313428\pi$
$53$			4680.27i			1.66617i	−0.553145	+	0.833085i	$0.686572\pi$
$54$		0			0
$55$	−134.346			−0.0444119
$56$		0			0
$57$	−157.678	+	523.510i	−0.0485312	+	0.161129i
$58$		0			0
$59$		−	3207.19i		−	0.921342i	−0.887571	−	0.460671i	$-0.847609\pi$
$59$		−	3207.19i		−	0.921342i	0.887571	−	0.460671i	$-0.152391\pi$
$60$		0			0
$61$	2169.77			0.583115			0.291558	−	0.956553i	$-0.405826\pi$
$61$	2169.77			0.583115			0.291558	+	0.956553i	$0.405826\pi$
$62$		0			0
$63$	5881.48			1.48185
$64$		0			0
$65$		−	4487.83i		−	1.06221i
$66$		0			0
$67$			6862.25i			1.52868i	0.644813	+	0.764340i	$0.276935\pi$
$67$			6862.25i			1.52868i	−0.644813	+	0.764340i	$0.723065\pi$
$68$		0			0
$69$		−	739.752i		−	0.155377i
$70$		0			0
$71$		−	6267.19i		−	1.24324i	−0.783317	−	0.621622i	$-0.786474\pi$
$71$		−	6267.19i		−	1.24324i	0.783317	−	0.621622i	$-0.213526\pi$
$72$		0			0
$73$	2394.57			0.449348			0.224674	−	0.974434i	$-0.427868\pi$
$73$	2394.57			0.449348			0.224674	+	0.974434i	$0.427868\pi$
$74$		0			0
$75$		−	568.181i		−	0.101010i
$76$		0			0
$77$	−635.138			−0.107124
$78$		0			0
$79$		−	1312.83i		−	0.210356i	−0.994453	−	0.105178i	$-0.966459\pi$
$79$		−	1312.83i		−	0.210356i	0.994453	−	0.105178i	$-0.0335413\pi$
$80$		0			0
$81$	6008.88			0.915848
$82$		0			0
$83$	−672.952			−0.0976850			−0.0488425	−	0.998806i	$-0.515553\pi$
$83$	−672.952			−0.0976850			−0.0488425	+	0.998806i	$0.515553\pi$
$84$		0			0
$85$	−570.537			−0.0789671
$86$		0			0
$87$	745.843			0.0985391
$88$		0			0
$89$			11177.3i			1.41110i	0.708661	+	0.705550i	$0.249300\pi$
$89$			11177.3i			1.41110i	−0.708661	+	0.705550i	$0.750700\pi$
$90$		0			0
$91$		−	21216.8i		−	2.56211i
$92$		0			0
$93$	−1941.21			−0.224443
$94$		0			0
$95$	5463.67	+	1645.62i	0.605393	+	0.182341i
$96$		0			0
$97$			10698.0i			1.13700i	0.822684	+	0.568499i	$0.192476\pi$
$97$			10698.0i			1.13700i	−0.822684	+	0.568499i	$0.807524\pi$
$98$		0			0
$99$	−668.960			−0.0682543
$100$		0			0
$101$	17196.2			1.68574			0.842869	−	0.538119i	$-0.180865\pi$
$101$	17196.2			1.68574			0.842869	+	0.538119i	$0.180865\pi$
$102$		0			0
$103$		−	10868.1i		−	1.02442i	−0.858860	−	0.512210i	$-0.828827\pi$
$103$		−	10868.1i		−	1.02442i	0.858860	−	0.512210i	$-0.171173\pi$
$104$		0			0
$105$			1788.89i			0.162258i
$106$		0			0
$107$			17778.1i			1.55281i	0.630235	+	0.776404i	$0.282958\pi$
$107$			17778.1i			1.55281i	−0.630235	+	0.776404i	$0.717042\pi$
$108$		0			0
$109$			7047.87i			0.593205i	0.955001	+	0.296603i	$0.0958537\pi$
$109$			7047.87i			0.593205i	−0.955001	+	0.296603i	$0.904146\pi$
$110$		0			0
$111$	3240.42			0.263000
$112$		0			0
$113$		−	10151.4i		−	0.795001i	−0.917602	−	0.397501i	$-0.869878\pi$
$113$		−	10151.4i		−	0.795001i	0.917602	−	0.397501i	$-0.130122\pi$
$114$		0			0
$115$	−7720.51			−0.583782
$116$		0			0
$117$		−	22346.6i		−	1.63245i
$118$		0			0
$119$	−2697.29			−0.190473
$120$		0			0
$121$	−14568.8			−0.995066
$122$		0			0
$123$	3144.61			0.207853
$124$		0			0
$125$	−15808.9			−1.01177
$126$		0			0
$127$		−	1075.85i		−	0.0667025i	−0.999444	−	0.0333513i	$-0.989382\pi$
$127$		−	1075.85i		−	0.0667025i	0.999444	−	0.0333513i	$-0.0106180\pi$
$128$		0			0
$129$			4621.34i			0.277708i
$130$		0			0
$131$	−16149.5			−0.941057			−0.470528	−	0.882385i	$-0.655937\pi$
$131$	−16149.5			−0.941057			−0.470528	+	0.882385i	$0.655937\pi$
$132$		0			0
$133$	25830.2	+	7779.90i	1.46024	+	0.439816i
$134$		0			0
$135$			3823.22i			0.209779i
$136$		0			0
$137$	−31949.1			−1.70223			−0.851113	−	0.524983i	$-0.824072\pi$
$137$	−31949.1			−1.70223			−0.851113	+	0.524983i	$0.824072\pi$
$138$		0			0
$139$	6268.09			0.324419			0.162209	−	0.986756i	$-0.448138\pi$
$139$	6268.09			0.324419			0.162209	+	0.986756i	$0.448138\pi$
$140$		0			0
$141$		−	1845.90i		−	0.0928476i
$142$		0			0
$143$			2413.20i			0.118011i
$144$		0			0
$145$		−	7784.08i		−	0.370230i
$146$		0			0
$147$			4820.87i			0.223096i
$148$		0			0
$149$	25057.7			1.12868			0.564338	−	0.825544i	$-0.309132\pi$
$149$	25057.7			1.12868			0.564338	+	0.825544i	$0.309132\pi$
$150$		0			0
$151$			1553.62i			0.0681383i	0.999419	+	0.0340691i	$0.0108466\pi$
$151$			1553.62i			0.0681383i	−0.999419	+	0.0340691i	$0.989153\pi$
$152$		0			0
$153$	−2840.92			−0.121360
$154$		0			0
$155$			20259.6i			0.843273i
$156$		0			0
$157$	6295.54			0.255408			0.127704	−	0.991812i	$-0.459239\pi$
$157$	6295.54			0.255408			0.127704	+	0.991812i	$0.459239\pi$
$158$		0			0
$159$	−7088.34			−0.280382
$160$		0			0
$161$	−36499.7			−1.40811
$162$		0			0
$163$	−11459.2			−0.431299			−0.215650	−	0.976471i	$-0.569187\pi$
$163$	−11459.2			−0.431299			−0.215650	+	0.976471i	$0.569187\pi$
$164$		0			0
$165$		−	203.469i		−	0.00747361i
$166$		0			0
$167$			47855.8i			1.71594i	0.513700	+	0.857970i	$0.328274\pi$
$167$			47855.8i			1.71594i	−0.513700	+	0.857970i	$0.671726\pi$
$168$		0			0
$169$	−52052.0			−1.82249
$170$		0			0
$171$	27205.7	+	8194.19i	0.930396	+	0.280229i
$172$		0			0
$173$		−	21330.4i		−	0.712701i	−0.934352	−	0.356351i	$-0.884021\pi$
$173$		−	21330.4i		−	0.712701i	0.934352	−	0.356351i	$-0.115979\pi$
$174$		0			0
$175$	−28034.3			−0.915407
$176$		0			0
$177$	4857.34			0.155043
$178$		0			0
$179$		−	8485.13i		−	0.264821i	−0.991195	−	0.132411i	$-0.957728\pi$
$179$		−	8485.13i		−	0.264821i	0.991195	−	0.132411i	$-0.0422717\pi$
$180$		0			0
$181$		−	6499.58i		−	0.198394i	−0.995068	−	0.0991969i	$-0.968373\pi$
$181$		−	6499.58i		−	0.198394i	0.995068	−	0.0991969i	$-0.0316274\pi$
$182$		0			0
$183$			3286.15i			0.0981263i
$184$		0			0
$185$		−	33819.1i		−	0.988140i
$186$		0			0
$187$	306.790			0.00877321
$188$		0			0
$189$			18074.8i			0.505998i
$190$		0			0
$191$	−6129.39			−0.168016			−0.0840079	−	0.996465i	$-0.526772\pi$
$191$	−6129.39			−0.168016			−0.0840079	+	0.996465i	$0.526772\pi$
$192$		0			0
$193$			18414.1i			0.494353i	0.968970	+	0.247176i	$0.0795027\pi$
$193$			18414.1i			0.494353i	−0.968970	+	0.247176i	$0.920497\pi$
$194$		0			0
$195$	6796.88			0.178748
$196$		0			0
$197$	−13128.9			−0.338294			−0.169147	−	0.985591i	$-0.554101\pi$
$197$	−13128.9			−0.338294			−0.169147	+	0.985591i	$0.554101\pi$
$198$		0			0
$199$	−50573.2			−1.27707			−0.638534	−	0.769593i	$-0.720459\pi$
$199$	−50573.2			−1.27707			−0.638534	+	0.769593i	$0.720459\pi$
$200$		0			0
$201$	−10393.0			−0.257246
$202$		0			0
$203$		−	36800.2i		−	0.893015i
$204$		0			0
$205$		−	32819.2i		−	0.780944i
$206$		0			0
$207$	−38443.4			−0.897183
$208$		0			0
$209$	−2937.94	−	884.887i	−0.0672589	−	0.0202579i
$210$		0			0
$211$			42125.2i			0.946186i	0.881012	+	0.473093i	$0.156863\pi$
$211$			42125.2i			0.946186i	−0.881012	+	0.473093i	$0.843137\pi$
$212$		0			0
$213$	9491.76			0.209212
$214$		0			0
$215$	48231.2			1.04340
$216$		0			0
$217$			95780.0i			2.03402i
$218$		0			0
$219$			3626.62i			0.0756160i
$220$		0			0
$221$			10248.3i			0.209830i
$222$		0			0
$223$			22513.9i			0.452731i	0.974042	+	0.226366i	$0.0726844\pi$
$223$			22513.9i			0.452731i	−0.974042	+	0.226366i	$0.927316\pi$
$224$		0			0
$225$	−29527.2			−0.583253
$226$		0			0
$227$		−	29584.8i		−	0.574138i	−0.957910	−	0.287069i	$-0.907319\pi$
$227$		−	29584.8i		−	0.574138i	0.957910	−	0.287069i	$-0.0926809\pi$
$228$		0			0
$229$	−16727.2			−0.318973			−0.159486	−	0.987200i	$-0.550984\pi$
$229$	−16727.2			−0.318973			−0.159486	+	0.987200i	$0.550984\pi$
$230$		0			0
$231$		−	961.927i		−	0.0180268i
$232$		0			0
$233$	−1427.44			−0.0262933			−0.0131467	−	0.999914i	$-0.504185\pi$
$233$	−1427.44			−0.0262933			−0.0131467	+	0.999914i	$0.504185\pi$
$234$		0			0
$235$	−19265.0			−0.348846
$236$		0			0
$237$	1988.31			0.0353987
$238$		0			0
$239$	−69656.3			−1.21945			−0.609726	−	0.792613i	$-0.708720\pi$
$239$	−69656.3			−1.21945			−0.609726	+	0.792613i	$0.708720\pi$
$240$		0			0
$241$			17559.7i			0.302331i	0.988508	+	0.151166i	$0.0483027\pi$
$241$			17559.7i			0.302331i	−0.988508	+	0.151166i	$0.951697\pi$
$242$		0			0
$243$			28692.6i			0.485912i
$244$		0			0
$245$	50313.7			0.838212
$246$		0			0
$247$	29559.7	−	98141.7i	0.484513	−	1.60864i
$248$		0			0
$249$		−	1019.20i		−	0.0164384i
$250$		0			0
$251$	−73019.0			−1.15901			−0.579507	−	0.814968i	$-0.696755\pi$
$251$	−73019.0			−1.15901			−0.579507	+	0.814968i	$0.696755\pi$
$252$		0			0
$253$	4151.49			0.0648579
$254$		0			0
$255$		−	864.088i		−	0.0132885i
$256$		0			0
$257$		−	14644.7i		−	0.221724i	−0.993836	−	0.110862i	$-0.964639\pi$
$257$		−	14644.7i		−	0.221724i	0.993836	−	0.110862i	$-0.0353612\pi$
$258$		0			0
$259$		−	159884.i		−	2.38345i
$260$		0			0
$261$		−	38759.9i		−	0.568986i
$262$		0			0
$263$	94182.6			1.36163			0.680815	−	0.732455i	$-0.261626\pi$
$263$	94182.6			1.36163			0.680815	+	0.732455i	$0.261626\pi$
$264$		0			0
$265$			73978.4i			1.05345i
$266$		0			0
$267$	−16928.2			−0.237459
$268$		0			0
$269$			79750.0i			1.10211i	0.834468	+	0.551057i	$0.185775\pi$
$269$			79750.0i			1.10211i	−0.834468	+	0.551057i	$0.814225\pi$
$270$		0			0
$271$	68908.9			0.938289			0.469145	−	0.883121i	$-0.344562\pi$
$271$	68908.9			0.938289			0.469145	+	0.883121i	$0.344562\pi$
$272$		0			0
$273$	32133.2			0.431150
$274$		0			0
$275$	3188.63			0.0421637
$276$		0			0
$277$	90990.1			1.18586			0.592931	−	0.805253i	$-0.297971\pi$
$277$	90990.1			1.18586			0.592931	+	0.805253i	$0.297971\pi$
$278$		0			0
$279$			100880.i			1.29598i
$280$		0			0
$281$		−	72273.8i		−	0.915311i	−0.889130	−	0.457655i	$-0.848689\pi$
$281$		−	72273.8i		−	0.915311i	0.889130	−	0.457655i	$-0.151311\pi$
$282$		0			0
$283$	63191.7			0.789019			0.394509	−	0.918892i	$-0.370915\pi$
$283$	63191.7			0.789019			0.394509	+	0.918892i	$0.370915\pi$
$284$		0			0
$285$	−2492.32	+	8274.82i	−0.0306842	+	0.101875i
$286$		0			0
$287$		−	155157.i		−	1.88368i
$288$		0			0
$289$	−82218.1			−0.984401
$290$		0			0
$291$	−16202.3			−0.191333
$292$		0			0
$293$			26746.7i			0.311556i	0.987792	+	0.155778i	$0.0497884\pi$
$293$			26746.7i			0.311556i	−0.987792	+	0.155778i	$0.950212\pi$
$294$		0			0
$295$		−	50694.2i		−	0.582525i
$296$		0			0
$297$		−	2055.83i		−	0.0233063i
$298$		0			0
$299$			138680.i			1.55122i
$300$		0			0
$301$	228019.			2.51674
$302$		0			0
$303$			26043.9i			0.283675i
$304$		0			0
$305$	34296.3			0.368679
$306$		0			0
$307$			20047.9i			0.212713i	0.994328	+	0.106356i	$0.0339184\pi$
$307$			20047.9i			0.212713i	−0.994328	+	0.106356i	$0.966082\pi$
$308$		0			0
$309$	16459.9			0.172389
$310$		0			0
$311$	91473.1			0.945742			0.472871	−	0.881132i	$-0.343218\pi$
$311$	91473.1			0.945742			0.472871	+	0.881132i	$0.343218\pi$
$312$		0			0
$313$	3767.82			0.0384593			0.0192297	−	0.999815i	$-0.493879\pi$
$313$	3767.82			0.0384593			0.0192297	+	0.999815i	$0.493879\pi$
$314$		0			0
$315$	92965.1			0.936912
$316$		0			0
$317$		−	104893.i		−	1.04382i	−0.853000	−	0.521911i	$-0.825219\pi$
$317$		−	104893.i		−	1.04382i	0.853000	−	0.521911i	$-0.174781\pi$
$318$		0			0
$319$			4185.67i			0.0411323i
$320$		0			0
$321$	−26925.2			−0.261306
$322$		0			0
$323$	−12476.8	−	3757.92i	−0.119590	−	0.0360199i
$324$		0			0
$325$			106516.i			1.00844i
$326$		0			0
$327$	−10674.1			−0.0998242
$328$		0			0
$329$	−91077.8			−0.841435
$330$		0			0
$331$		−	58070.6i		−	0.530030i	−0.964244	−	0.265015i	$-0.914623\pi$
$331$		−	58070.6i		−	0.530030i	0.964244	−	0.265015i	$-0.0853769\pi$
$332$		0			0
$333$		−	168398.i		−	1.51862i
$334$		0			0
$335$			108468.i			0.966519i
$336$		0			0
$337$		−	27715.4i		−	0.244040i	−0.992528	−	0.122020i	$-0.961063\pi$
$337$		−	27715.4i		−	0.244040i	0.992528	−	0.122020i	$-0.0389372\pi$
$338$		0			0
$339$	15374.4			0.133782
$340$		0			0
$341$		−	10894.0i		−	0.0936872i
$342$		0			0
$343$	58445.0			0.496774
$344$		0			0
$345$		−	11692.8i		−	0.0982385i
$346$		0			0
$347$	−148847.			−1.23618			−0.618089	−	0.786108i	$-0.712093\pi$
$347$	−148847.			−1.23618			−0.618089	+	0.786108i	$0.712093\pi$
$348$		0			0
$349$	113003.			0.927764			0.463882	−	0.885897i	$-0.346456\pi$
$349$	113003.			0.927764			0.463882	+	0.885897i	$0.346456\pi$
$350$		0			0
$351$	68674.9			0.557422
$352$		0			0
$353$	−173058.			−1.38881			−0.694404	−	0.719585i	$-0.744332\pi$
$353$	−173058.			−1.38881			−0.694404	+	0.719585i	$0.744332\pi$
$354$		0			0
$355$		−	99061.9i		−	0.786050i
$356$		0			0
$357$		−	4085.09i		−	0.0320527i
$358$		0			0
$359$	33967.3			0.263555			0.131778	−	0.991279i	$-0.457932\pi$
$359$	33967.3			0.263555			0.131778	+	0.991279i	$0.457932\pi$
$360$		0			0
$361$	108643.	+	71974.3i	0.833655	+	0.552285i
$362$		0			0
$363$		−	22064.6i		−	0.167449i
$364$		0			0
$365$	37849.7			0.284103
$366$		0			0
$367$	−132869.			−0.986490			−0.493245	−	0.869891i	$-0.664189\pi$
$367$	−132869.			−0.986490			−0.493245	+	0.869891i	$0.664189\pi$
$368$		0			0
$369$		−	163419.i		−	1.20019i
$370$		0			0
$371$			349742.i			2.54097i
$372$		0			0
$373$		−	76178.5i		−	0.547539i	−0.961795	−	0.273769i	$-0.911730\pi$
$373$		−	76178.5i		−	0.547539i	0.961795	−	0.273769i	$-0.0882705\pi$
$374$		0			0
$375$		−	23942.8i		−	0.170260i
$376$		0			0
$377$	−139822.			−0.983770
$378$		0			0
$379$		−	83156.8i		−	0.578921i	−0.957190	−	0.289461i	$-0.906524\pi$
$379$		−	83156.8i		−	0.578921i	0.957190	−	0.289461i	$-0.0934759\pi$
$380$		0			0
$381$	1629.38			0.0112247
$382$		0			0
$383$			169744.i			1.15717i	0.815622	+	0.578586i	$0.196395\pi$
$383$			169744.i			1.15717i	−0.815622	+	0.578586i	$0.803605\pi$
$384$		0			0
$385$	−10039.3			−0.0677299
$386$		0			0
$387$	240162.			1.60355
$388$		0			0
$389$	15312.4			0.101191			0.0505956	−	0.998719i	$-0.483888\pi$
$389$	15312.4			0.101191			0.0505956	+	0.998719i	$0.483888\pi$
$390$		0			0
$391$	17630.4			0.115321
$392$		0			0
$393$		−	24458.6i		−	0.158360i
$394$		0			0
$395$		−	20751.2i		−	0.132999i
$396$		0			0
$397$	131462.			0.834100			0.417050	−	0.908884i	$-0.363064\pi$
$397$	131462.			0.834100			0.417050	+	0.908884i	$0.363064\pi$
$398$		0			0
$399$	−11782.8	+	39120.3i	−0.0740119	+	0.245729i
$400$		0			0
$401$			284670.i			1.77032i	0.465284	+	0.885162i	$0.345952\pi$
$401$			284670.i			1.77032i	−0.465284	+	0.885162i	$0.654048\pi$
$402$		0			0
$403$	363915.			2.24073
$404$		0			0
$405$	94978.9			0.579051
$406$		0			0
$407$			18185.3i			0.109782i
$408$		0			0
$409$		−	183642.i		−	1.09781i	−0.835886	−	0.548903i	$-0.815046\pi$
$409$		−	183642.i		−	1.09781i	0.835886	−	0.548903i	$-0.184954\pi$
$410$		0			0
$411$		−	48387.4i		−	0.286450i
$412$		0			0
$413$		−	239663.i		−	1.40508i
$414$		0			0
$415$	−10637.0			−0.0617620
$416$		0			0
$417$			9493.12i			0.0545930i
$418$		0			0
$419$	−82365.8			−0.469157			−0.234579	−	0.972097i	$-0.575371\pi$
$419$	−82365.8			−0.469157			−0.234579	+	0.972097i	$0.575371\pi$
$420$		0			0
$421$		−	293141.i		−	1.65391i	−0.562266	−	0.826957i	$-0.690070\pi$
$421$		−	293141.i		−	1.65391i	0.562266	−	0.826957i	$-0.309930\pi$
$422$		0			0
$423$	−95927.8			−0.536122
$424$		0			0
$425$	13541.4			0.0749697
$426$		0			0
$427$	162140.			0.889274
$428$		0			0
$429$	−3654.83			−0.0198588
$430$		0			0
$431$			66323.1i			0.357035i	0.983937	+	0.178517i	$0.0571301\pi$
$431$			66323.1i			0.357035i	−0.983937	+	0.178517i	$0.942870\pi$
$432$		0			0
$433$			137702.i			0.734452i	0.930132	+	0.367226i	$0.119692\pi$
$433$			137702.i			0.734452i	−0.930132	+	0.367226i	$0.880308\pi$
$434$		0			0
$435$	11789.1			0.0623020
$436$		0			0
$437$	−168835.	−	50852.1i	−0.884099	−	0.266285i
$438$		0			0
$439$		−	311507.i		−	1.61636i	−0.588935	−	0.808180i	$-0.700453\pi$
$439$		−	311507.i		−	1.61636i	0.588935	−	0.808180i	$-0.299547\pi$
$440$		0			0
$441$	250531.			1.28820
$442$		0			0
$443$	−320788.			−1.63460			−0.817299	−	0.576214i	$-0.804530\pi$
$443$	−320788.			−1.63460			−0.817299	+	0.576214i	$0.804530\pi$
$444$		0			0
$445$			176673.i			0.892177i
$446$		0			0
$447$			37950.3i			0.189933i
$448$		0			0
$449$			119322.i			0.591873i	0.955208	+	0.295936i	$0.0956316\pi$
$449$			119322.i			0.591873i	−0.955208	+	0.295936i	$0.904368\pi$
$450$		0			0
$451$			17647.6i			0.0867625i
$452$		0			0
$453$	−2352.98			−0.0114663
$454$		0			0
$455$		−	335362.i		−	1.61991i
$456$		0			0
$457$	44310.8			0.212167			0.106083	−	0.994357i	$-0.466169\pi$
$457$	44310.8			0.212167			0.106083	+	0.994357i	$0.466169\pi$
$458$		0			0
$459$		−	8730.64i		−	0.0414401i
$460$		0			0
$461$	−292702.			−1.37729			−0.688643	−	0.725101i	$-0.741793\pi$
$461$	−292702.			−1.37729			−0.688643	+	0.725101i	$0.741793\pi$
$462$		0			0
$463$	−293714.			−1.37013			−0.685067	−	0.728480i	$-0.740227\pi$
$463$	−293714.			−1.37013			−0.685067	+	0.728480i	$0.740227\pi$
$464$		0			0
$465$	−30683.5			−0.141906
$466$		0			0
$467$	−67231.3			−0.308275			−0.154137	−	0.988049i	$-0.549260\pi$
$467$	−67231.3			−0.308275			−0.154137	+	0.988049i	$0.549260\pi$
$468$		0			0
$469$			512795.i			2.33130i
$470$		0			0
$471$			9534.70i			0.0429799i
$472$		0			0
$473$	−25934.9			−0.115921
$474$		0			0
$475$	−129677.	−	39058.0i	−0.574747	−	0.173110i
$476$		0			0
$477$			368367.i			1.61899i
$478$		0			0
$479$	−104619.			−0.455973			−0.227986	−	0.973664i	$-0.573214\pi$
$479$	−104619.			−0.455973			−0.227986	+	0.973664i	$0.573214\pi$
$480$		0			0
$481$	−607478.			−2.62567
$482$		0			0
$483$		−	55279.4i		−	0.236957i
$484$		0			0
$485$			169097.i			0.718875i
$486$		0			0
$487$			432894.i			1.82526i	0.408789	+	0.912629i	$0.365951\pi$
$487$			432894.i			1.82526i	−0.408789	+	0.912629i	$0.634049\pi$
$488$		0			0
$489$		−	17355.1i		−	0.0725788i
$490$		0			0
$491$	−198985.			−0.825387			−0.412694	−	0.910870i	$-0.635412\pi$
$491$	−198985.			−0.825387			−0.412694	+	0.910870i	$0.635412\pi$
$492$		0			0
$493$			17775.6i			0.0731358i
$494$		0			0
$495$	−10573.9			−0.0431543
$496$		0			0
$497$		−	468328.i		−	1.89600i
$498$		0			0
$499$	−111012.			−0.445828			−0.222914	−	0.974838i	$-0.571557\pi$
$499$	−111012.			−0.445828			−0.222914	+	0.974838i	$0.571557\pi$
$500$		0			0
$501$	−72478.4			−0.288757
$502$		0			0
$503$	−413050.			−1.63255			−0.816276	−	0.577662i	$-0.803965\pi$
$503$	−413050.			−1.63255			−0.816276	+	0.577662i	$0.803965\pi$
$504$		0			0
$505$	271811.			1.06582
$506$		0			0
$507$		−	78833.6i		−	0.306687i
$508$		0			0
$509$		−	93319.0i		−	0.360193i	−0.983649	−	0.180096i	$-0.942359\pi$
$509$		−	93319.0i		−	0.360193i	0.983649	−	0.180096i	$-0.0576409\pi$
$510$		0			0
$511$	178939.			0.685273
$512$		0			0
$513$	−25182.1	+	83607.8i	−0.0956880	+	0.317696i
$514$		0			0
$515$		−	171785.i		−	0.647697i
$516$		0			0
$517$	10359.2			0.0387566
$518$		0			0
$519$	32305.3			0.119933
$520$		0			0
$521$			381971.i			1.40720i	0.710598	+	0.703598i	$0.248424\pi$
$521$			381971.i			1.40720i	−0.710598	+	0.703598i	$0.751576\pi$
$522$		0			0
$523$			395430.i			1.44566i	0.691027	+	0.722829i	$0.257159\pi$
$523$			395430.i			1.44566i	−0.691027	+	0.722829i	$0.742841\pi$
$524$		0			0
$525$		−	42458.4i		−	0.154044i
$526$		0			0
$527$		−	46264.6i		−	0.166582i
$528$		0			0
$529$	−41265.9			−0.147462
$530$		0			0
$531$		−	252426.i		−	0.895251i
$532$		0			0
$533$	−589517.			−2.07511
$534$		0			0
$535$			281008.i			0.981774i
$536$		0			0
$537$	12850.9			0.0445639
$538$		0			0
$539$	−27054.7			−0.0931249
$540$		0			0
$541$	263394.			0.899934			0.449967	−	0.893045i	$-0.351436\pi$
$541$	263394.			0.899934			0.449967	+	0.893045i	$0.351436\pi$
$542$		0			0
$543$	9843.71			0.0333856
$544$		0			0
$545$			111402.i			0.375058i
$546$		0			0
$547$		−	122704.i		−	0.410093i	−0.978752	−	0.205047i	$-0.934265\pi$
$547$		−	122704.i		−	0.410093i	0.978752	−	0.205047i	$-0.0657346\pi$
$548$		0			0
$549$	170775.			0.566603
$550$		0			0
$551$	51270.8	−	170225.i	0.168876	−	0.560688i
$552$		0			0
$553$		−	98104.1i		−	0.320802i
$554$		0			0
$555$	51219.5			0.166284
$556$		0			0
$557$	29020.6			0.0935397			0.0467698	−	0.998906i	$-0.485107\pi$
$557$	29020.6			0.0935397			0.0467698	+	0.998906i	$0.485107\pi$
$558$		0			0
$559$		−	866357.i		−	2.77251i
$560$		0			0
$561$			464.639i			0.00147635i
$562$		0			0
$563$			79705.3i			0.251461i	0.992064	+	0.125731i	$0.0401275\pi$
$563$			79705.3i			0.251461i	−0.992064	+	0.125731i	$0.959873\pi$
$564$		0			0
$565$		−	160457.i		−	0.502645i
$566$		0			0
$567$	449025.			1.39670
$568$		0			0
$569$			449751.i			1.38914i	0.719423	+	0.694572i	$0.244406\pi$
$569$			449751.i			1.38914i	−0.719423	+	0.694572i	$0.755594\pi$
$570$		0			0
$571$	−346869.			−1.06388			−0.531941	−	0.846781i	$-0.678537\pi$
$571$	−346869.			−1.06388			−0.531941	+	0.846781i	$0.678537\pi$
$572$		0			0
$573$		−	9283.05i		−	0.0282736i
$574$		0			0
$575$	183242.			0.554230
$576$		0			0
$577$	407905.			1.22520			0.612600	−	0.790393i	$-0.290124\pi$
$577$	407905.			1.22520			0.612600	+	0.790393i	$0.290124\pi$
$578$		0			0
$579$	−27888.5			−0.0831894
$580$		0			0
$581$	−50287.6			−0.148973
$582$		0			0
$583$		−	39779.8i		−	0.117038i
$584$		0			0
$585$		−	353220.i		−	1.03213i
$586$		0			0
$587$	−120671.			−0.350209			−0.175104	−	0.984550i	$-0.556026\pi$
$587$	−120671.			−0.350209			−0.175104	+	0.984550i	$0.556026\pi$
$588$		0			0
$589$	−133443.	+	443046.i	−0.384648	+	1.27708i
$590$		0			0
$591$		−	19883.9i		−	0.0569280i
$592$		0			0
$593$	671576.			1.90979			0.954896	−	0.296942i	$-0.0959668\pi$
$593$	671576.			1.90979			0.954896	+	0.296942i	$0.0959668\pi$
$594$		0			0
$595$	−42634.5			−0.120428
$596$		0			0
$597$		−	76593.9i		−	0.214904i
$598$		0			0
$599$			159147.i			0.443554i	0.975097	+	0.221777i	$0.0711856\pi$
$599$			159147.i			0.443554i	−0.975097	+	0.221777i	$0.928814\pi$
$600$		0			0
$601$		−	321439.i		−	0.889918i	−0.895551	−	0.444959i	$-0.853218\pi$
$601$		−	321439.i		−	0.889918i	0.895551	−	0.444959i	$-0.146782\pi$
$602$		0			0
$603$			540102.i			1.48539i
$604$		0			0
$605$	−230280.			−0.629137
$606$		0			0
$607$			440276.i			1.19494i	0.801890	+	0.597472i	$0.203828\pi$
$607$			440276.i			1.19494i	−0.801890	+	0.597472i	$0.796172\pi$
$608$		0			0
$609$	55734.5			0.150276
$610$		0			0
$611$			346049.i			0.926949i
$612$		0			0
$613$	−516045.			−1.37330			−0.686652	−	0.726986i	$-0.740920\pi$
$613$	−516045.			−1.37330			−0.686652	+	0.726986i	$0.740920\pi$
$614$		0			0
$615$	49705.1			0.131417
$616$		0			0
$617$	−646632.			−1.69858			−0.849292	−	0.527924i	$-0.822971\pi$
$617$	−646632.			−1.69858			−0.849292	+	0.527924i	$0.822971\pi$
$618$		0			0
$619$	202214.			0.527751			0.263876	−	0.964557i	$-0.414999\pi$
$619$	202214.			0.527751			0.263876	+	0.964557i	$0.414999\pi$
$620$		0			0
$621$		−	118143.i		−	0.306355i
$622$		0			0
$623$			835247.i			2.15198i
$624$		0			0
$625$	−15409.2			−0.0394476
$626$		0			0
$627$	1340.17	−	4449.55i	0.00340900	−	0.0113183i
$628$		0			0
$629$			77228.7i			0.195199i
$630$		0			0
$631$	338994.			0.851399			0.425700	−	0.904864i	$-0.360028\pi$
$631$	338994.			0.851399			0.425700	+	0.904864i	$0.360028\pi$
$632$		0			0
$633$	−63799.2			−0.159224
$634$		0			0
$635$		−	17005.3i		−	0.0421731i
$636$		0			0
$637$		−	903764.i		−	2.22729i
$638$		0			0
$639$		−	493267.i		−	1.20804i
$640$		0			0
$641$		−	633633.i		−	1.54213i	−0.636754	−	0.771067i	$-0.719723\pi$
$641$		−	633633.i		−	1.54213i	0.636754	−	0.771067i	$-0.280277\pi$
$642$		0			0
$643$	−203816.			−0.492965			−0.246483	−	0.969147i	$-0.579275\pi$
$643$	−203816.			−0.492965			−0.246483	+	0.969147i	$0.579275\pi$
$644$		0			0
$645$			73046.8i			0.175583i
$646$		0			0
$647$	−347996.			−0.831314			−0.415657	−	0.909521i	$-0.636448\pi$
$647$	−347996.			−0.831314			−0.415657	+	0.909521i	$0.636448\pi$
$648$		0			0
$649$			27259.4i			0.0647182i
$650$		0			0
$651$	−145060.			−0.342284
$652$		0			0
$653$	−645403.			−1.51358			−0.756789	−	0.653659i	$-0.773233\pi$
$653$	−645403.			−1.51358			−0.756789	+	0.653659i	$0.773233\pi$
$654$		0			0
$655$	−255265.			−0.594990
$656$		0			0
$657$	188468.			0.436623
$658$		0			0
$659$		−	373142.i		−	0.859217i	−0.903015	−	0.429609i	$-0.858651\pi$
$659$		−	373142.i		−	0.859217i	0.903015	−	0.429609i	$-0.141349\pi$
$660$		0			0
$661$		−	297605.i		−	0.681141i	−0.940219	−	0.340571i	$-0.889380\pi$
$661$		−	297605.i		−	0.681141i	0.940219	−	0.340571i	$-0.110620\pi$
$662$		0			0
$663$	−15521.3			−0.0353102
$664$		0			0
$665$	408284.	+	122972.i	0.923248	+	0.278076i
$666$		0			0
$667$			240539.i			0.540673i
$668$		0			0
$669$	−34097.6			−0.0761853
$670$		0			0
$671$	−18441.9			−0.0409600
$672$		0			0
$673$			410131.i			0.905509i	0.891635	+	0.452755i	$0.149559\pi$
$673$			410131.i			0.905509i	−0.891635	+	0.452755i	$0.850441\pi$
$674$		0			0
$675$		−	90742.0i		−	0.199159i
$676$		0			0
$677$			308448.i			0.672984i	0.941686	+	0.336492i	$0.109241\pi$
$677$			308448.i			0.672984i	−0.941686	+	0.336492i	$0.890759\pi$
$678$		0			0
$679$			799429.i			1.73397i
$680$		0			0
$681$	44806.6			0.0966156
$682$		0			0
$683$		−	461976.i		−	0.990325i	−0.868800	−	0.495162i	$-0.835109\pi$
$683$		−	461976.i		−	0.990325i	0.868800	−	0.495162i	$-0.164891\pi$
$684$		0			0
$685$	−505001.			−1.07624
$686$		0			0
$687$		−	25333.7i		−	0.0536765i
$688$		0			0
$689$	1.32884e6			2.79921
$690$		0			0
$691$	365047.			0.764526			0.382263	−	0.924054i	$-0.375145\pi$
$691$	365047.			0.764526			0.382263	+	0.924054i	$0.375145\pi$
$692$		0			0
$693$	−49989.4			−0.104090
$694$		0			0
$695$	99076.1			0.205116
$696$		0			0
$697$			74945.3i			0.154269i
$698$		0			0
$699$		−	2161.88i		−	0.00442463i
$700$		0			0
$701$	174714.			0.355544			0.177772	−	0.984072i	$-0.443111\pi$
$701$	174714.			0.355544			0.177772	+	0.984072i	$0.443111\pi$
$702$		0			0
$703$	222754.	−	739570.i	0.450728	−	1.49647i
$704$		0			0
$705$		−	29177.1i		−	0.0587036i
$706$		0			0
$707$	1.28502e6			2.57082
$708$		0			0
$709$	624205.			1.24175			0.620876	−	0.783909i	$-0.286777\pi$
$709$	624205.			1.24175			0.620876	+	0.783909i	$0.286777\pi$
$710$		0			0
$711$		−	103328.i		−	0.204400i
$712$		0			0
$713$		−	626052.i		−	1.23149i
$714$		0			0
$715$			38144.1i			0.0746132i
$716$		0			0
$717$		−	105495.i		−	0.205209i
$718$		0			0
$719$	−116573.			−0.225496			−0.112748	−	0.993624i	$-0.535965\pi$
$719$	−116573.			−0.225496			−0.112748	+	0.993624i	$0.535965\pi$
$720$		0			0
$721$		−	812138.i		−	1.56228i
$722$		0			0
$723$	−26594.4			−0.0508762
$724$		0			0
$725$			184751.i			0.351488i
$726$		0			0
$727$	632724.			1.19714			0.598571	−	0.801070i	$-0.295735\pi$
$727$	632724.			1.19714			0.598571	+	0.801070i	$0.295735\pi$
$728$		0			0
$729$	443264.			0.834079
$730$		0			0
$731$	−110140.			−0.206115
$732$		0			0
$733$	−672228.			−1.25115			−0.625574	−	0.780165i	$-0.715135\pi$
$733$	−672228.			−1.25115			−0.625574	+	0.780165i	$0.715135\pi$
$734$		0			0
$735$			76200.8i			0.141054i
$736$		0			0
$737$		−	58325.4i		−	0.107380i
$738$		0			0
$739$	850121.			1.55665			0.778327	−	0.627860i	$-0.216069\pi$
$739$	850121.			1.55665			0.778327	+	0.627860i	$0.216069\pi$
$740$		0			0
$741$	148637.	+	44768.5i	0.270702	+	0.0815336i
$742$		0			0
$743$		−	105660.i		−	0.191396i	−0.995410	−	0.0956980i	$-0.969492\pi$
$743$		−	105660.i		−	0.191396i	0.995410	−	0.0956980i	$-0.0305083\pi$
$744$		0			0
$745$	396073.			0.713613
$746$		0			0
$747$	−52965.5			−0.0949188
$748$		0			0
$749$			1.32850e6i			2.36809i
$750$		0			0
$751$			783330.i			1.38888i	0.719550	+	0.694441i	$0.244348\pi$
$751$			783330.i			1.38888i	−0.719550	+	0.694441i	$0.755652\pi$
$752$		0			0
$753$		−	110588.i		−	0.195038i
$754$		0			0
$755$			24557.2i			0.0430809i
$756$		0			0
$757$	813155.			1.41900			0.709499	−	0.704706i	$-0.248921\pi$
$757$	813155.			1.41900			0.709499	+	0.704706i	$0.248921\pi$
$758$		0			0
$759$			6287.49i			0.0109142i
$760$		0			0
$761$	74387.2			0.128448			0.0642242	−	0.997935i	$-0.479543\pi$
$761$	74387.2			0.128448			0.0642242	+	0.997935i	$0.479543\pi$
$762$		0			0
$763$			526666.i			0.904661i
$764$		0			0
$765$	−44904.9			−0.0767309
$766$		0			0
$767$	−910599.			−1.54788
$768$		0			0
$769$	−228356.			−0.386153			−0.193076	−	0.981184i	$-0.561846\pi$
$769$	−228356.			−0.386153			−0.193076	+	0.981184i	$0.561846\pi$
$770$		0			0
$771$	22179.6			0.0373117
$772$		0			0
$773$		−	732635.i		−	1.22611i	−0.790041	−	0.613054i	$-0.789941\pi$
$773$		−	732635.i		−	1.22611i	0.790041	−	0.613054i	$-0.210059\pi$
$774$		0			0
$775$		−	480851.i		−	0.800585i
$776$		0			0
$777$	242147.			0.401085
$778$		0			0
$779$	216168.	−	717703.i	0.356218	−	1.18269i
$780$		0			0
$781$			53267.7i			0.0873297i
$782$		0			0
$783$	119116.			0.194288
$784$		0			0
$785$	99510.0			0.161483
$786$		0			0
$787$		−	759258.i		−	1.22586i	−0.790138	−	0.612929i	$-0.789991\pi$
$787$		−	759258.i		−	1.22586i	0.790138	−	0.612929i	$-0.210009\pi$
$788$		0			0
$789$			142641.i			0.229134i
$790$		0			0
$791$		−	758581.i		−	1.21241i
$792$		0			0
$793$		−	616051.i		−	0.979649i
$794$		0			0
$795$	−112041.			−0.177274
$796$		0			0
$797$			1.10047e6i			1.73245i	0.499650	+	0.866227i	$0.333462\pi$
$797$			1.10047e6i			1.73245i	−0.499650	+	0.866227i	$0.666538\pi$
$798$		0			0
$799$	43993.2			0.0689116
$800$		0			0
$801$			879725.i			1.37114i
$802$		0			0
$803$	−20352.6			−0.0315637
$804$		0			0
$805$	−576930.			−0.890290
$806$		0			0
$807$	−120783.			−0.185463
$808$		0			0
$809$	−687697.			−1.05075			−0.525375	−	0.850871i	$-0.676075\pi$
$809$	−687697.			−1.05075			−0.525375	+	0.850871i	$0.676075\pi$
$810$		0			0
$811$			194733.i			0.296073i	0.988982	+	0.148036i	$0.0472953\pi$
$811$			194733.i			0.296073i	−0.988982	+	0.148036i	$0.952705\pi$
$812$		0			0
$813$			104364.i			0.157895i
$814$		0			0
$815$	−181129.			−0.272692
$816$		0			0
$817$	1.05474e6	+	317681.i	1.58016	+	0.475934i
$818$		0			0
$819$		−	1.66989e6i		−	2.48955i
$820$		0			0
$821$	341765.			0.507039			0.253519	−	0.967330i	$-0.418412\pi$
$821$	341765.			0.507039			0.253519	+	0.967330i	$0.418412\pi$
$822$		0			0
$823$	76669.7			0.113194			0.0565971	−	0.998397i	$-0.481975\pi$
$823$	76669.7			0.113194			0.0565971	+	0.998397i	$0.481975\pi$
$824$		0			0
$825$			4829.23i			0.00709529i
$826$		0			0
$827$		−	25522.2i		−	0.0373171i	−0.999826	−	0.0186585i	$-0.994060\pi$
$827$		−	25522.2i		−	0.0373171i	0.999826	−	0.0186585i	$-0.00593954\pi$
$828$		0			0
$829$		−	341385.i		−	0.496747i	−0.968664	−	0.248374i	$-0.920104\pi$
$829$		−	341385.i		−	0.496747i	0.968664	−	0.248374i	$-0.0798961\pi$
$830$		0			0
$831$			137806.i			0.199556i
$832$		0			0
$833$	−114895.			−0.165582
$834$		0			0
$835$			756430.i			1.08492i
$836$		0			0
$837$	−310023.			−0.442530
$838$		0			0
$839$		−	617409.i		−	0.877099i	−0.898707	−	0.438550i	$-0.855492\pi$
$839$		−	617409.i		−	0.877099i	0.898707	−	0.438550i	$-0.144508\pi$
$840$		0			0
$841$	464761.			0.657110
$842$		0			0
$843$	109460.			0.154028
$844$		0			0
$845$	−822757.			−1.15228
$846$		0			0
$847$	−1.08868e6			−1.51751
$848$		0			0
$849$			95704.8i			0.132776i
$850$		0			0
$851$			1.04506e6i			1.44305i
$852$		0			0
$853$	517750.			0.711577			0.355789	−	0.934566i	$-0.384212\pi$
$853$	517750.			0.711577			0.355789	+	0.934566i	$0.384212\pi$
$854$		0			0
$855$	430025.	+	129521.i	0.588250	+	0.177177i
$856$		0			0
$857$			1.44446e6i			1.96673i	0.181647	+	0.983364i	$0.441857\pi$
$857$			1.44446e6i			1.96673i	−0.181647	+	0.983364i	$0.558143\pi$
$858$		0			0
$859$	−836422.			−1.13355			−0.566773	−	0.823874i	$-0.691808\pi$
$859$	−836422.			−1.13355			−0.566773	+	0.823874i	$0.691808\pi$
$860$		0			0
$861$	234987.			0.316985
$862$		0			0
$863$			1.13241e6i			1.52048i	0.649641	+	0.760241i	$0.274919\pi$
$863$			1.13241e6i			1.52048i	−0.649641	+	0.760241i	$0.725081\pi$
$864$		0			0
$865$		−	337158.i		−	0.450610i
$866$		0			0
$867$		−	124521.i		−	0.165654i
$868$		0			0
$869$			11158.4i			0.0147762i
$870$		0			0
$871$	1.94836e6			2.56822
$872$		0			0
$873$			842000.i			1.10480i
$874$		0			0
$875$	−1.18135e6			−1.54299
$876$		0			0
$877$		−	1.06124e6i		−	1.37980i	−0.723905	−	0.689899i	$-0.757655\pi$
$877$		−	1.06124e6i		−	1.37980i	0.723905	−	0.689899i	$-0.242345\pi$
$878$		0			0
$879$	−40508.3			−0.0524284
$880$		0			0
$881$	38098.7			0.0490861			0.0245430	−	0.999699i	$-0.492187\pi$
$881$	38098.7			0.0490861			0.0245430	+	0.999699i	$0.492187\pi$
$882$		0			0
$883$	−300580.			−0.385513			−0.192756	−	0.981247i	$-0.561743\pi$
$883$	−300580.			−0.385513			−0.192756	+	0.981247i	$0.561743\pi$
$884$		0			0
$885$	76777.1			0.0980269
$886$		0			0
$887$		−	928047.i		−	1.17957i	−0.807561	−	0.589784i	$-0.799213\pi$
$887$		−	928047.i		−	1.17957i	0.807561	−	0.589784i	$-0.200787\pi$
$888$		0			0
$889$		−	80394.6i		−	0.101724i
$890$		0			0
$891$	−51072.2			−0.0643323
$892$		0			0
$893$	−421295.	−	126891.i	−0.528304	−	0.159122i
$894$		0			0
$895$		−	134120.i		−	0.167435i
$896$		0			0
$897$	−210034.			−0.261038
$898$		0			0
$899$	631206.			0.781002
$900$		0			0
$901$		−	168936.i		−	0.208100i
$902$		0			0
$903$			345338.i			0.423516i
$904$		0			0
$905$		−	102735.i		−	0.125436i
$906$		0			0
$907$			129940.i			0.157953i	0.996876	+	0.0789764i	$0.0251652\pi$
$907$			129940.i			0.157953i	−0.996876	+	0.0789764i	$0.974835\pi$
$908$		0			0
$909$	1.35345e6			1.63800
$910$		0			0
$911$		−	1.14347e6i		−	1.37781i	−0.724853	−	0.688904i	$-0.758092\pi$
$911$		−	1.14347e6i		−	1.37781i	0.724853	−	0.688904i	$-0.241908\pi$
$912$		0			0
$913$	5719.73			0.00686173
$914$		0			0
$915$			51942.3i			0.0620411i
$916$		0			0
$917$	−1.20680e6			−1.43515
$918$		0			0
$919$	723426.			0.856571			0.428285	−	0.903643i	$-0.359118\pi$
$919$	723426.			0.856571			0.428285	+	0.903643i	$0.359118\pi$
$920$		0			0
$921$	−30362.9			−0.0357952
$922$		0			0
$923$	−1.77941e6			−2.08868
$924$		0			0
$925$			802677.i			0.938118i
$926$		0			0
$927$		−	855385.i		−	0.995411i
$928$		0			0
$929$	−223643.			−0.259133			−0.129567	−	0.991571i	$-0.541359\pi$
$929$	−223643.			−0.259133			−0.129567	+	0.991571i	$0.541359\pi$
$930$		0			0
$931$	1.10028e6	+	331397.i	1.26942	+	0.382340i
$932$		0			0
$933$			138537.i			0.159149i
$934$		0			0
$935$	4849.26			0.00554692
$936$		0			0
$937$	−980898.			−1.11723			−0.558617	−	0.829425i	$-0.688668\pi$
$937$	−980898.			−1.11723			−0.558617	+	0.829425i	$0.688668\pi$
$938$		0			0
$939$			5706.42i			0.00647192i
$940$		0			0
$941$		−	482742.i		−	0.545175i	−0.962131	−	0.272588i	$-0.912121\pi$
$941$		−	482742.i		−	0.545175i	0.962131	−	0.272588i	$-0.0878794\pi$
$942$		0			0
$943$			1.01416e6i			1.14047i
$944$		0			0
$945$			285697.i			0.319921i
$946$		0			0
$947$	325880.			0.363377			0.181688	−	0.983356i	$-0.441844\pi$
$947$	325880.			0.363377			0.181688	+	0.983356i	$0.441844\pi$
$948$		0			0
$949$		−	679878.i		−	0.754916i
$950$		0			0
$951$	158861.			0.175654
$952$		0			0
$953$		−	1.78316e6i		−	1.96338i	−0.190491	−	0.981689i	$-0.561008\pi$
$953$		−	1.78316e6i		−	1.96338i	0.190491	−	0.981689i	$-0.438992\pi$
$954$		0			0
$955$	−96883.7			−0.106229
$956$		0			0
$957$	−6339.26			−0.00692173
$958$		0			0
$959$	−2.38746e6			−2.59596
$960$		0			0
$961$	−719321.			−0.778890
$962$		0			0
$963$			1.39925e6i			1.50884i
$964$		0			0
$965$			291062.i			0.312558i
$966$		0			0
$967$	−138031.			−0.147613			−0.0738063	−	0.997273i	$-0.523515\pi$
$967$	−138031.			−0.147613			−0.0738063	+	0.997273i	$0.523515\pi$
$968$		0			0
$969$	5691.42	−	18896.2i	0.00606140	−	0.0201246i
$970$		0			0
$971$		−	497346.i		−	0.527498i	−0.964591	−	0.263749i	$-0.915041\pi$
$971$		−	497346.i		−	0.527498i	0.964591	−	0.263749i	$-0.0849590\pi$
$972$		0			0
$973$	468395.			0.494751
$974$		0			0
$975$	−161320.			−0.169699
$976$		0			0
$977$		−	795595.i		−	0.833495i	−0.909022	−	0.416747i	$-0.863170\pi$
$977$		−	795595.i		−	0.833495i	0.909022	−	0.416747i	$-0.136830\pi$
$978$		0			0
$979$		−	95001.1i		−	0.0991205i
$980$		0			0
$981$			554711.i			0.576407i
$982$		0			0
$983$			501529.i			0.519026i	0.965740	+	0.259513i	$0.0835621\pi$
$983$			501529.i			0.519026i	−0.965740	+	0.259513i	$0.916438\pi$
$984$		0			0
$985$	−207520.			−0.213889
$986$		0			0
$987$		−	137939.i		−	0.141596i
$988$		0			0
$989$	−1.49041e6			−1.52375
$990$		0			0
$991$			1.71173e6i			1.74297i	0.490426	+	0.871483i	$0.336841\pi$
$991$			1.71173e6i			1.74297i	−0.490426	+	0.871483i	$0.663159\pi$
$992$		0			0
$993$	87948.8			0.0891931
$994$		0			0
$995$	−799381.			−0.807436
$996$		0			0
$997$	−1.42725e6			−1.43586			−0.717928	−	0.696118i	$-0.754909\pi$
$997$	−1.42725e6			−1.43586			−0.717928	+	0.696118i	$0.754909\pi$
$998$		0			0
$999$	517516.			0.518552

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.5.e.f.113.12		20
4.3	odd	2		152.5.e.a.113.9	✓	20
12.11	even	2		1368.5.o.a.721.7		20
19.18	odd	2	inner	304.5.e.f.113.9		20
76.75	even	2		152.5.e.a.113.12	yes	20
228.227	odd	2		1368.5.o.a.721.8		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.5.e.a.113.9	✓	20	4.3	odd	2
152.5.e.a.113.12	yes	20	76.75	even	2
304.5.e.f.113.9		20	19.18	odd	2	inner
304.5.e.f.113.12		20	1.1	even	1	trivial
1368.5.o.a.721.7		20	12.11	even	2
1368.5.o.a.721.8		20	228.227	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	304.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.51452i q^{3} +15.8064 q^{5} +74.7269 q^{7} +78.7062 q^{9} +O(q^{10})\)	\(q+1.51452i q^{3} +15.8064 q^{5} +74.7269 q^{7} +78.7062 q^{9} -8.49946 q^{11} -283.924i q^{13} +23.9391i q^{15} -36.0953 q^{17} +(345.662 + 104.111i) q^{19} +113.175i q^{21} -488.441 q^{23} -375.157 q^{25} +241.878i q^{27} -492.463i q^{29} +1281.73i q^{31} -12.8726i q^{33} +1181.17 q^{35} -2139.58i q^{37} +430.008 q^{39} -2076.32i q^{41} +3051.37 q^{43} +1244.06 q^{45} -1218.81 q^{47} +3183.11 q^{49} -54.6669i q^{51} +4680.27i q^{53} -134.346 q^{55} +(-157.678 + 523.510i) q^{57} -3207.19i q^{59} +2169.77 q^{61} +5881.48 q^{63} -4487.83i q^{65} +6862.25i q^{67} -739.752i q^{69} -6267.19i q^{71} +2394.57 q^{73} -568.181i q^{75} -635.138 q^{77} -1312.83i q^{79} +6008.88 q^{81} -672.952 q^{83} -570.537 q^{85} +745.843 q^{87} +11177.3i q^{89} -21216.8i q^{91} -1941.21 q^{93} +(5463.67 + 1645.62i) q^{95} +10698.0i q^{97} -668.960 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 32 q^{7} - 372 q^{9}+O(q^{10}) \) 20 * q - 32 * q^7 - 372 * q^9	\( 20 q - 32 q^{7} - 372 q^{9} + 24 q^{11} + 216 q^{17} + 596 q^{19} - 576 q^{23} + 1412 q^{25} + 144 q^{35} + 520 q^{39} + 1256 q^{43} + 7232 q^{45} + 3768 q^{47} - 2740 q^{49} + 10128 q^{55} - 728 q^{57} + 352 q^{61} - 6104 q^{63} + 1352 q^{73} + 9288 q^{77} - 4220 q^{81} + 16104 q^{83} + 10232 q^{85} - 2936 q^{87} + 36432 q^{93} - 14232 q^{95} - 760 q^{99}+O(q^{100}) \) 20 * q - 32 * q^7 - 372 * q^9 + 24 * q^11 + 216 * q^17 + 596 * q^19 - 576 * q^23 + 1412 * q^25 + 144 * q^35 + 520 * q^39 + 1256 * q^43 + 7232 * q^45 + 3768 * q^47 - 2740 * q^49 + 10128 * q^55 - 728 * q^57 + 352 * q^61 - 6104 * q^63 + 1352 * q^73 + 9288 * q^77 - 4220 * q^81 + 16104 * q^83 + 10232 * q^85 - 2936 * q^87 + 36432 * q^93 - 14232 * q^95 - 760 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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