LMFDB - Embedded newform 304.5.e.f.113.16

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			113.16
Root			$9.35107i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.5.e.f.113.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.35107i q^{3} +21.0112 q^{5} +9.13929 q^{7} -6.44248 q^{9} +O(q^{10})$	$q+9.35107i q^{3} +21.0112 q^{5} +9.13929 q^{7} -6.44248 q^{9} -229.573 q^{11} -31.9450i q^{13} +196.477i q^{15} -200.710 q^{17} +(-351.060 + 84.1286i) q^{19} +85.4622i q^{21} +317.677 q^{23} -183.531 q^{25} +697.192i q^{27} +222.798i q^{29} +1614.51i q^{31} -2146.75i q^{33} +192.027 q^{35} -1600.94i q^{37} +298.720 q^{39} +1955.66i q^{41} -1409.89 q^{43} -135.364 q^{45} -2214.61 q^{47} -2317.47 q^{49} -1876.86i q^{51} -1737.29i q^{53} -4823.60 q^{55} +(-786.693 - 3282.79i) q^{57} -3707.10i q^{59} -44.0139 q^{61} -58.8797 q^{63} -671.201i q^{65} -3062.04i q^{67} +2970.62i q^{69} +5451.07i q^{71} -729.395 q^{73} -1716.21i q^{75} -2098.14 q^{77} -9503.97i q^{79} -7041.34 q^{81} +1488.80 q^{83} -4217.16 q^{85} -2083.40 q^{87} +3854.22i q^{89} -291.955i q^{91} -15097.4 q^{93} +(-7376.19 + 1767.64i) q^{95} +11385.4i q^{97} +1479.02 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			9.35107i			1.03901i	0.854468	+	0.519504i	$0.173883\pi$
$3$			9.35107i			1.03901i	−0.854468	+	0.519504i	$0.826117\pi$
$4$		0			0
$5$	21.0112			0.840447			0.420223	−	0.907421i	$-0.361952\pi$
$5$	21.0112			0.840447			0.420223	+	0.907421i	$0.361952\pi$
$6$		0			0
$7$	9.13929			0.186516			0.0932581	−	0.995642i	$-0.470272\pi$
$7$	9.13929			0.186516			0.0932581	+	0.995642i	$0.470272\pi$
$8$		0			0
$9$	−6.44248			−0.0795368
$10$		0			0
$11$	−229.573			−1.89730			−0.948649	−	0.316330i	$-0.897549\pi$
$11$	−229.573			−1.89730			−0.948649	+	0.316330i	$0.897549\pi$
$12$		0			0
$13$		−	31.9450i		−	0.189024i	−0.995524	−	0.0945118i	$-0.969871\pi$
$13$		−	31.9450i		−	0.189024i	0.995524	−	0.0945118i	$-0.0301290\pi$
$14$		0			0
$15$			196.477i			0.873231i
$16$		0			0
$17$	−200.710			−0.694499			−0.347250	−	0.937773i	$-0.612884\pi$
$17$	−200.710			−0.694499			−0.347250	+	0.937773i	$0.612884\pi$
$18$		0			0
$19$	−351.060	+	84.1286i	−0.972466	+	0.233043i
$19$	−351.060	+	84.1286i	−0.972466	+	0.233043i
$20$		0			0
$21$			85.4622i			0.193792i
$22$		0			0
$23$	317.677			0.600524			0.300262	−	0.953857i	$-0.402926\pi$
$23$	317.677			0.600524			0.300262	+	0.953857i	$0.402926\pi$
$24$		0			0
$25$	−183.531			−0.293649
$26$		0			0
$27$			697.192i			0.956368i
$28$		0			0
$29$			222.798i			0.264920i	0.991188	+	0.132460i	$0.0422876\pi$
$29$			222.798i			0.264920i	−0.991188	+	0.132460i	$0.957712\pi$
$30$		0			0
$31$			1614.51i			1.68003i	0.542562	+	0.840016i	$0.317454\pi$
$31$			1614.51i			1.68003i	−0.542562	+	0.840016i	$0.682546\pi$
$32$		0			0
$33$		−	2146.75i		−	1.97131i
$34$		0			0
$35$	192.027			0.156757
$36$		0			0
$37$		−	1600.94i		−	1.16942i	−0.811241	−	0.584712i	$-0.801208\pi$
$37$		−	1600.94i		−	1.16942i	0.811241	−	0.584712i	$-0.198792\pi$
$38$		0			0
$39$	298.720			0.196397
$40$		0			0
$41$			1955.66i			1.16339i	0.813407	+	0.581695i	$0.197610\pi$
$41$			1955.66i			1.16339i	−0.813407	+	0.581695i	$0.802390\pi$
$42$		0			0
$43$	−1409.89			−0.762516			−0.381258	−	0.924469i	$-0.624509\pi$
$43$	−1409.89			−0.762516			−0.381258	+	0.924469i	$0.624509\pi$
$44$		0			0
$45$	−135.364			−0.0668464
$46$		0			0
$47$	−2214.61			−1.00254			−0.501270	−	0.865291i	$-0.667134\pi$
$47$	−2214.61			−1.00254			−0.501270	+	0.865291i	$0.667134\pi$
$48$		0			0
$49$	−2317.47			−0.965212
$50$		0			0
$51$		−	1876.86i		−	0.721590i
$52$		0			0
$53$		−	1737.29i		−	0.618472i	−0.950985	−	0.309236i	$-0.899927\pi$
$53$		−	1737.29i		−	0.618472i	0.950985	−	0.309236i	$-0.100073\pi$
$54$		0			0
$55$	−4823.60			−1.59458
$56$		0			0
$57$	−786.693	−	3282.79i	−0.242134	−	1.01040i
$58$		0			0
$59$		−	3707.10i		−	1.06495i	−0.846445	−	0.532477i	$-0.821261\pi$
$59$		−	3707.10i		−	1.06495i	0.846445	−	0.532477i	$-0.178739\pi$
$60$		0			0
$61$	−44.0139			−0.0118285			−0.00591425	−	0.999983i	$-0.501883\pi$
$61$	−44.0139			−0.0118285			−0.00591425	+	0.999983i	$0.501883\pi$
$62$		0			0
$63$	−58.8797			−0.0148349
$64$		0			0
$65$		−	671.201i		−	0.158864i
$66$		0			0
$67$		−	3062.04i		−	0.682121i	−0.940041	−	0.341061i	$-0.889214\pi$
$67$		−	3062.04i		−	0.682121i	0.940041	−	0.341061i	$-0.110786\pi$
$68$		0			0
$69$			2970.62i			0.623949i
$70$		0			0
$71$			5451.07i			1.08135i	0.841233	+	0.540673i	$0.181830\pi$
$71$			5451.07i			1.08135i	−0.841233	+	0.540673i	$0.818170\pi$
$72$		0			0
$73$	−729.395			−0.136873			−0.0684364	−	0.997655i	$-0.521801\pi$
$73$	−729.395			−0.136873			−0.0684364	+	0.997655i	$0.521801\pi$
$74$		0			0
$75$		−	1716.21i		−	0.305104i
$76$		0			0
$77$	−2098.14			−0.353877
$78$		0			0
$79$		−	9503.97i		−	1.52283i	−0.648266	−	0.761414i	$-0.724505\pi$
$79$		−	9503.97i		−	1.52283i	0.648266	−	0.761414i	$-0.275495\pi$
$80$		0			0
$81$	−7041.34			−1.07321
$82$		0			0
$83$	1488.80			0.216113			0.108056	−	0.994145i	$-0.465537\pi$
$83$	1488.80			0.216113			0.108056	+	0.994145i	$0.465537\pi$
$84$		0			0
$85$	−4217.16			−0.583690
$86$		0			0
$87$	−2083.40			−0.275254
$88$		0			0
$89$			3854.22i			0.486582i	0.969953	+	0.243291i	$0.0782271\pi$
$89$			3854.22i			0.486582i	−0.969953	+	0.243291i	$0.921773\pi$
$90$		0			0
$91$		−	291.955i		−	0.0352560i
$92$		0			0
$93$	−15097.4			−1.74557
$94$		0			0
$95$	−7376.19	+	1767.64i	−0.817306	+	0.195861i
$96$		0			0
$97$			11385.4i			1.21006i	0.796203	+	0.605029i	$0.206839\pi$
$97$			11385.4i			1.21006i	−0.796203	+	0.605029i	$0.793161\pi$
$98$		0			0
$99$	1479.02			0.150905
$100$		0			0
$101$	10346.6			1.01428			0.507139	−	0.861864i	$-0.330703\pi$
$101$	10346.6			1.01428			0.507139	+	0.861864i	$0.330703\pi$
$102$		0			0
$103$			1907.40i			0.179790i	0.995951	+	0.0898952i	$0.0286532\pi$
$103$			1907.40i			0.179790i	−0.995951	+	0.0898952i	$0.971347\pi$
$104$		0			0
$105$			1795.66i			0.162872i
$106$		0			0
$107$		−	13780.8i		−	1.20367i	−0.798621	−	0.601834i	$-0.794437\pi$
$107$		−	13780.8i		−	1.20367i	0.798621	−	0.601834i	$-0.205563\pi$
$108$		0			0
$109$			12384.6i			1.04239i	0.853438	+	0.521194i	$0.174513\pi$
$109$			12384.6i			1.04239i	−0.853438	+	0.521194i	$0.825487\pi$
$110$		0			0
$111$	14970.5			1.21504
$112$		0			0
$113$			3626.51i			0.284009i	0.989866	+	0.142005i	$0.0453548\pi$
$113$			3626.51i			0.284009i	−0.989866	+	0.142005i	$0.954645\pi$
$114$		0			0
$115$	6674.77			0.504709
$116$		0			0
$117$			205.805i			0.0150343i
$118$		0			0
$119$	−1834.35			−0.129535
$120$		0			0
$121$	38062.8			2.59974
$122$		0			0
$123$	−18287.5			−1.20877
$124$		0			0
$125$	−16988.2			−1.08724
$126$		0			0
$127$			30783.8i			1.90860i	0.298851	+	0.954300i	$0.403397\pi$
$127$			30783.8i			1.90860i	−0.298851	+	0.954300i	$0.596603\pi$
$128$		0			0
$129$		−	13184.0i		−	0.792260i
$130$		0			0
$131$	−14089.5			−0.821018			−0.410509	−	0.911856i	$-0.634649\pi$
$131$	−14089.5			−0.821018			−0.410509	+	0.911856i	$0.634649\pi$
$132$		0			0
$133$	−3208.44	+	768.876i	−0.181381	+	0.0434664i
$134$		0			0
$135$			14648.8i			0.803777i
$136$		0			0
$137$	8305.06			0.442488			0.221244	−	0.975218i	$-0.428988\pi$
$137$	8305.06			0.442488			0.221244	+	0.975218i	$0.428988\pi$
$138$		0			0
$139$	7891.88			0.408461			0.204231	−	0.978923i	$-0.434531\pi$
$139$	7891.88			0.408461			0.204231	+	0.978923i	$0.434531\pi$
$140$		0			0
$141$		−	20709.0i		−	1.04165i
$142$		0			0
$143$			7333.71i			0.358634i
$144$		0			0
$145$			4681.24i			0.222651i
$146$		0			0
$147$		−	21670.9i		−	1.00286i
$148$		0			0
$149$	−5757.60			−0.259340			−0.129670	−	0.991557i	$-0.541392\pi$
$149$	−5757.60			−0.259340			−0.129670	+	0.991557i	$0.541392\pi$
$150$		0			0
$151$		−	35174.7i		−	1.54268i	−0.636423	−	0.771340i	$-0.719587\pi$
$151$		−	35174.7i		−	1.54268i	0.636423	−	0.771340i	$-0.280413\pi$
$152$		0			0
$153$	1293.07			0.0552382
$154$		0			0
$155$			33922.8i			1.41198i
$156$		0			0
$157$	−12092.6			−0.490591			−0.245296	−	0.969448i	$-0.578885\pi$
$157$	−12092.6			−0.490591			−0.245296	+	0.969448i	$0.578885\pi$
$158$		0			0
$159$	16245.5			0.642597
$160$		0			0
$161$	2903.35			0.112007
$162$		0			0
$163$	33645.2			1.26633			0.633166	−	0.774016i	$-0.281755\pi$
$163$	33645.2			1.26633			0.633166	+	0.774016i	$0.281755\pi$
$164$		0			0
$165$		−	45105.8i		−	1.65678i
$166$		0			0
$167$			1536.23i			0.0550836i	0.999621	+	0.0275418i	$0.00876794\pi$
$167$			1536.23i			0.0550836i	−0.999621	+	0.0275418i	$0.991232\pi$
$168$		0			0
$169$	27540.5			0.964270
$170$		0			0
$171$	2261.70	−	541.997i	0.0773469	−	0.0185355i
$172$		0			0
$173$		−	20084.8i		−	0.671080i	−0.942026	−	0.335540i	$-0.891081\pi$
$173$		−	20084.8i		−	0.671080i	0.942026	−	0.335540i	$-0.108919\pi$
$174$		0			0
$175$	−1677.34			−0.0547703
$176$		0			0
$177$	34665.4			1.10649
$178$		0			0
$179$			46069.6i			1.43783i	0.695097	+	0.718916i	$0.255362\pi$
$179$			46069.6i			1.43783i	−0.695097	+	0.718916i	$0.744638\pi$
$180$		0			0
$181$			8071.82i			0.246385i	0.992383	+	0.123193i	$0.0393133\pi$
$181$			8071.82i			0.246385i	−0.992383	+	0.123193i	$0.960687\pi$
$182$		0			0
$183$		−	411.577i		−	0.0122899i
$184$		0			0
$185$		−	33637.6i		−	0.982838i
$186$		0			0
$187$	46077.7			1.31767
$188$		0			0
$189$			6371.85i			0.178378i
$190$		0			0
$191$	44830.8			1.22888			0.614440	−	0.788964i	$-0.289382\pi$
$191$	44830.8			1.22888			0.614440	+	0.788964i	$0.289382\pi$
$192$		0			0
$193$			20671.5i			0.554954i	0.960732	+	0.277477i	$0.0894982\pi$
$193$			20671.5i			0.554954i	−0.960732	+	0.277477i	$0.910502\pi$
$194$		0			0
$195$	6276.45			0.165061
$196$		0			0
$197$	30908.5			0.796427			0.398214	−	0.917293i	$-0.369630\pi$
$197$	30908.5			0.796427			0.398214	+	0.917293i	$0.369630\pi$
$198$		0			0
$199$	−36927.4			−0.932486			−0.466243	−	0.884657i	$-0.654393\pi$
$199$	−36927.4			−0.932486			−0.466243	+	0.884657i	$0.654393\pi$
$200$		0			0
$201$	28633.4			0.708729
$202$		0			0
$203$			2036.21i			0.0494118i
$204$		0			0
$205$			41090.7i			0.977768i
$206$		0			0
$207$	−2046.63			−0.0477638
$208$		0			0
$209$	80594.0	−	19313.7i	1.84506	−	0.442153i
$210$		0			0
$211$			11150.4i			0.250453i	0.992128	+	0.125226i	$0.0399657\pi$
$211$			11150.4i			0.250453i	−0.992128	+	0.125226i	$0.960034\pi$
$212$		0			0
$213$	−50973.3			−1.12353
$214$		0			0
$215$	−29623.5			−0.640854
$216$		0			0
$217$			14755.5i			0.313353i
$218$		0			0
$219$		−	6820.62i		−	0.142212i
$220$		0			0
$221$			6411.69i			0.131277i
$222$		0			0
$223$		−	69281.6i		−	1.39318i	−0.717468	−	0.696592i	$-0.754699\pi$
$223$		−	69281.6i		−	1.39318i	0.717468	−	0.696592i	$-0.245301\pi$
$224$		0			0
$225$	1182.39			0.0233559
$226$		0			0
$227$			45527.5i			0.883532i	0.897130	+	0.441766i	$0.145648\pi$
$227$			45527.5i			0.883532i	−0.897130	+	0.441766i	$0.854352\pi$
$228$		0			0
$229$	−52567.8			−1.00242			−0.501209	−	0.865326i	$-0.667111\pi$
$229$	−52567.8			−1.00242			−0.501209	+	0.865326i	$0.667111\pi$
$230$		0			0
$231$		−	19619.8i		−	0.367681i
$232$		0			0
$233$	−23490.8			−0.432699			−0.216349	−	0.976316i	$-0.569415\pi$
$233$	−23490.8			−0.432699			−0.216349	+	0.976316i	$0.569415\pi$
$234$		0			0
$235$	−46531.6			−0.842582
$236$		0			0
$237$	88872.3			1.58223
$238$		0			0
$239$	58344.0			1.02141			0.510706	−	0.859756i	$-0.329384\pi$
$239$	58344.0			1.02141			0.510706	+	0.859756i	$0.329384\pi$
$240$		0			0
$241$			110136.i			1.89626i	0.317889	+	0.948128i	$0.397026\pi$
$241$			110136.i			1.89626i	−0.317889	+	0.948128i	$0.602974\pi$
$242$		0			0
$243$		−	9371.42i		−	0.158706i
$244$		0			0
$245$	−48692.8			−0.811209
$246$		0			0
$247$	2687.49	+	11214.6i	0.0440507	+	0.183819i
$248$		0			0
$249$			13921.9i			0.224543i
$250$		0			0
$251$	−31833.5			−0.505285			−0.252642	−	0.967560i	$-0.581300\pi$
$251$	−31833.5			−0.505285			−0.252642	+	0.967560i	$0.581300\pi$
$252$		0			0
$253$	−72930.1			−1.13937
$254$		0			0
$255$		−	39434.9i		−	0.606458i
$256$		0			0
$257$			96608.9i			1.46268i	0.682011	+	0.731342i	$0.261106\pi$
$257$			96608.9i			1.46268i	−0.682011	+	0.731342i	$0.738894\pi$
$258$		0			0
$259$		−	14631.5i		−	0.218116i
$260$		0			0
$261$		−	1435.37i		−	0.0210709i
$262$		0			0
$263$	94714.1			1.36931			0.684657	−	0.728865i	$-0.259952\pi$
$263$	94714.1			1.36931			0.684657	+	0.728865i	$0.259952\pi$
$264$		0			0
$265$		−	36502.5i		−	0.519793i
$266$		0			0
$267$	−36041.1			−0.505563
$268$		0			0
$269$			100277.i			1.38578i	0.721042	+	0.692891i	$0.243663\pi$
$269$			100277.i			1.38578i	−0.721042	+	0.692891i	$0.756337\pi$
$270$		0			0
$271$	−12553.9			−0.170938			−0.0854692	−	0.996341i	$-0.527239\pi$
$271$	−12553.9			−0.170938			−0.0854692	+	0.996341i	$0.527239\pi$
$272$		0			0
$273$	2730.09			0.0366312
$274$		0			0
$275$	42133.7			0.557140
$276$		0			0
$277$	−124443.			−1.62185			−0.810926	−	0.585149i	$-0.801036\pi$
$277$	−124443.			−1.62185			−0.810926	+	0.585149i	$0.801036\pi$
$278$		0			0
$279$		−	10401.5i		−	0.133624i
$280$		0			0
$281$		−	1365.30i		−	0.0172908i	−0.999963	−	0.00864539i	$-0.997248\pi$
$281$		−	1365.30i		−	0.0172908i	0.999963	−	0.00864539i	$-0.00275195\pi$
$282$		0			0
$283$	−27823.0			−0.347401			−0.173700	−	0.984799i	$-0.555572\pi$
$283$	−27823.0			−0.347401			−0.173700	+	0.984799i	$0.555572\pi$
$284$		0			0
$285$	−16529.3	−	68975.3i	−0.203501	−	0.849187i
$286$		0			0
$287$			17873.3i			0.216991i
$288$		0			0
$289$	−43236.4			−0.517671
$290$		0			0
$291$	−106466.			−1.25726
$292$		0			0
$293$		−	61993.6i		−	0.722124i	−0.932542	−	0.361062i	$-0.882414\pi$
$293$		−	61993.6i		−	0.722124i	0.932542	−	0.361062i	$-0.117586\pi$
$294$		0			0
$295$		−	77890.6i		−	0.895037i
$296$		0			0
$297$		−	160057.i		−	1.81452i
$298$		0			0
$299$		−	10148.2i		−	0.113513i
$300$		0			0
$301$	−12885.4			−0.142222
$302$		0			0
$303$			96752.2i			1.05384i
$304$		0			0
$305$	−924.783			−0.00994123
$306$		0			0
$307$		−	63488.0i		−	0.673619i	−0.941573	−	0.336810i	$-0.890652\pi$
$307$		−	63488.0i		−	0.673619i	0.941573	−	0.336810i	$-0.109348\pi$
$308$		0			0
$309$	−17836.2			−0.186804
$310$		0			0
$311$	−114012.			−1.17877			−0.589385	−	0.807852i	$-0.700630\pi$
$311$	−114012.			−1.17877			−0.589385	+	0.807852i	$0.700630\pi$
$312$		0			0
$313$	−46918.7			−0.478914			−0.239457	−	0.970907i	$-0.576969\pi$
$313$	−46918.7			−0.478914			−0.239457	+	0.970907i	$0.576969\pi$
$314$		0			0
$315$	−1237.13			−0.0124679
$316$		0			0
$317$			136358.i			1.35695i	0.734625	+	0.678474i	$0.237358\pi$
$317$			136358.i			1.35695i	−0.734625	+	0.678474i	$0.762642\pi$
$318$		0			0
$319$		−	51148.3i		−	0.502632i
$320$		0			0
$321$	128865.			1.25062
$322$		0			0
$323$	70461.4	−	16885.5i	0.675377	−	0.161848i
$324$		0			0
$325$			5862.88i			0.0555066i
$326$		0			0
$327$	−115809.			−1.08305
$328$		0			0
$329$	−20240.0			−0.186990
$330$		0			0
$331$		−	51167.2i		−	0.467020i	−0.972354	−	0.233510i	$-0.924979\pi$
$331$		−	51167.2i		−	0.467020i	0.972354	−	0.233510i	$-0.0750211\pi$
$332$		0			0
$333$			10314.0i			0.0930122i
$334$		0			0
$335$		−	64337.1i		−	0.573287i
$336$		0			0
$337$			44236.7i			0.389514i	0.980852	+	0.194757i	$0.0623918\pi$
$337$			44236.7i			0.389514i	−0.980852	+	0.194757i	$0.937608\pi$
$338$		0			0
$339$	−33911.8			−0.295088
$340$		0			0
$341$		−	370648.i		−	3.18752i
$342$		0			0
$343$	−43123.5			−0.366544
$344$		0			0
$345$			62416.2i			0.524396i
$346$		0			0
$347$	41775.7			0.346948			0.173474	−	0.984838i	$-0.444501\pi$
$347$	41775.7			0.346948			0.173474	+	0.984838i	$0.444501\pi$
$348$		0			0
$349$	−51237.4			−0.420665			−0.210332	−	0.977630i	$-0.567455\pi$
$349$	−51237.4			−0.420665			−0.210332	+	0.977630i	$0.567455\pi$
$350$		0			0
$351$	22271.8			0.180776
$352$		0			0
$353$	135060.			1.08387			0.541935	−	0.840420i	$-0.317692\pi$
$353$	135060.			1.08387			0.541935	+	0.840420i	$0.317692\pi$
$354$		0			0
$355$			114533.i			0.908814i
$356$		0			0
$357$		−	17153.1i		−	0.134588i
$358$		0			0
$359$	60848.3			0.472128			0.236064	−	0.971738i	$-0.424143\pi$
$359$	60848.3			0.472128			0.236064	+	0.971738i	$0.424143\pi$
$360$		0			0
$361$	116166.	−	59068.5i	0.891382	−	0.453254i
$362$		0			0
$363$			355928.i			2.70115i
$364$		0			0
$365$	−15325.4			−0.115034
$366$		0			0
$367$	−196722.			−1.46056			−0.730282	−	0.683145i	$-0.760612\pi$
$367$	−196722.			−1.46056			−0.730282	+	0.683145i	$0.760612\pi$
$368$		0			0
$369$		−	12599.3i		−	0.0925323i
$370$		0			0
$371$		−	15877.6i		−	0.115355i
$372$		0			0
$373$		−	78353.6i		−	0.563172i	−0.959536	−	0.281586i	$-0.909139\pi$
$373$		−	78353.6i		−	0.563172i	0.959536	−	0.281586i	$-0.0908605\pi$
$374$		0			0
$375$		−	158858.i		−	1.12965i
$376$		0			0
$377$	7117.26			0.0500761
$378$		0			0
$379$		−	186817.i		−	1.30058i	−0.759686	−	0.650290i	$-0.774647\pi$
$379$		−	186817.i		−	1.30058i	0.759686	−	0.650290i	$-0.225353\pi$
$380$		0			0
$381$	−287861.			−1.98305
$382$		0			0
$383$			251623.i			1.71535i	0.514190	+	0.857676i	$0.328092\pi$
$383$			251623.i			1.71535i	−0.514190	+	0.857676i	$0.671908\pi$
$384$		0			0
$385$	−44084.3			−0.297415
$386$		0			0
$387$	9083.20			0.0606481
$388$		0			0
$389$	160443.			1.06028			0.530142	−	0.847909i	$-0.322139\pi$
$389$	160443.			1.06028			0.530142	+	0.847909i	$0.322139\pi$
$390$		0			0
$391$	−63761.1			−0.417064
$392$		0			0
$393$		−	131752.i		−	0.853044i
$394$		0			0
$395$		−	199690.i		−	1.27986i
$396$		0			0
$397$	−258787.			−1.64196			−0.820978	−	0.570960i	$-0.806571\pi$
$397$	−258787.			−1.64196			−0.820978	+	0.570960i	$0.806571\pi$
$398$		0			0
$399$	−7189.82	−	30002.4i	−0.0451619	−	0.188456i
$400$		0			0
$401$		−	294964.i		−	1.83434i	−0.398497	−	0.917170i	$-0.630468\pi$
$401$		−	294964.i		−	1.83434i	0.398497	−	0.917170i	$-0.369532\pi$
$402$		0			0
$403$	51575.5			0.317566
$404$		0			0
$405$	−147947.			−0.901977
$406$		0			0
$407$			367533.i			2.21874i
$408$		0			0
$409$			175061.i			1.04651i	0.852177	+	0.523254i	$0.175282\pi$
$409$			175061.i			1.04651i	−0.852177	+	0.523254i	$0.824718\pi$
$410$		0			0
$411$			77661.2i			0.459749i
$412$		0			0
$413$		−	33880.3i		−	0.198631i
$414$		0			0
$415$	31281.4			0.181631
$416$		0			0
$417$			73797.5i			0.424394i
$418$		0			0
$419$	−218082.			−1.24220			−0.621100	−	0.783731i	$-0.713314\pi$
$419$	−218082.			−1.24220			−0.621100	+	0.783731i	$0.713314\pi$
$420$		0			0
$421$			168830.i			0.952544i	0.879298	+	0.476272i	$0.158012\pi$
$421$			168830.i			0.952544i	−0.879298	+	0.476272i	$0.841988\pi$
$422$		0			0
$423$	14267.6			0.0797389
$424$		0			0
$425$	36836.5			0.203939
$426$		0			0
$427$	−402.256			−0.00220621
$428$		0			0
$429$	−68578.0			−0.372623
$430$		0			0
$431$			126599.i			0.681517i	0.940151	+	0.340758i	$0.110684\pi$
$431$			126599.i			0.681517i	−0.940151	+	0.340758i	$0.889316\pi$
$432$		0			0
$433$			75284.7i			0.401542i	0.979638	+	0.200771i	$0.0643447\pi$
$433$			75284.7i			0.401542i	−0.979638	+	0.200771i	$0.935655\pi$
$434$		0			0
$435$	−43774.6			−0.231336
$436$		0			0
$437$	−111524.	+	26725.7i	−0.583989	+	0.139948i
$438$		0			0
$439$			271248.i			1.40747i	0.710465	+	0.703733i	$0.248485\pi$
$439$			271248.i			1.40747i	−0.710465	+	0.703733i	$0.751515\pi$
$440$		0			0
$441$	14930.3			0.0767698
$442$		0			0
$443$	299553.			1.52639			0.763196	−	0.646167i	$-0.223629\pi$
$443$	299553.			1.52639			0.763196	+	0.646167i	$0.223629\pi$
$444$		0			0
$445$			80981.7i			0.408947i
$446$		0			0
$447$		−	53839.7i		−	0.269456i
$448$		0			0
$449$			362585.i			1.79853i	0.437408	+	0.899263i	$0.355897\pi$
$449$			362585.i			1.79853i	−0.437408	+	0.899263i	$0.644103\pi$
$450$		0			0
$451$		−	448967.i		−	2.20730i
$452$		0			0
$453$	328921.			1.60286
$454$		0			0
$455$		−	6134.31i		−	0.0296308i
$456$		0			0
$457$	319617.			1.53037			0.765187	−	0.643808i	$-0.222646\pi$
$457$	319617.			1.53037			0.765187	+	0.643808i	$0.222646\pi$
$458$		0			0
$459$		−	139934.i		−	0.664197i
$460$		0			0
$461$	328733.			1.54683			0.773414	−	0.633901i	$-0.218547\pi$
$461$	328733.			1.54683			0.773414	+	0.633901i	$0.218547\pi$
$462$		0			0
$463$	333761.			1.55694			0.778472	−	0.627679i	$-0.215995\pi$
$463$	333761.			1.55694			0.778472	+	0.627679i	$0.215995\pi$
$464$		0			0
$465$	−317214.			−1.46706
$466$		0			0
$467$	−41272.8			−0.189248			−0.0946238	−	0.995513i	$-0.530165\pi$
$467$	−41272.8			−0.189248			−0.0946238	+	0.995513i	$0.530165\pi$
$468$		0			0
$469$		−	27984.9i		−	0.127227i
$470$		0			0
$471$		−	113079.i		−	0.509728i
$472$		0			0
$473$	323673.			1.44672
$474$		0			0
$475$	64430.3	−	15440.2i	0.285564	−	0.0684329i
$476$		0			0
$477$			11192.4i			0.0491913i
$478$		0			0
$479$	276599.			1.20553			0.602767	−	0.797917i	$-0.294065\pi$
$479$	276599.			1.20553			0.602767	+	0.797917i	$0.294065\pi$
$480$		0			0
$481$	−51142.0			−0.221048
$482$		0			0
$483$			27149.4i			0.116377i
$484$		0			0
$485$			239222.i			1.01699i
$486$		0			0
$487$		−	62193.7i		−	0.262233i	−0.991367	−	0.131117i	$-0.958144\pi$
$487$		−	62193.7i		−	0.262233i	0.991367	−	0.131117i	$-0.0418563\pi$
$488$		0			0
$489$			314619.i			1.31573i
$490$		0			0
$491$	−37569.0			−0.155836			−0.0779178	−	0.996960i	$-0.524827\pi$
$491$	−37569.0			−0.155836			−0.0779178	+	0.996960i	$0.524827\pi$
$492$		0			0
$493$		−	44717.8i		−	0.183987i
$494$		0			0
$495$	31075.9			0.126828
$496$		0			0
$497$			49818.9i			0.201689i
$498$		0			0
$499$	−211710.			−0.850238			−0.425119	−	0.905137i	$-0.639768\pi$
$499$	−211710.			−0.850238			−0.425119	+	0.905137i	$0.639768\pi$
$500$		0			0
$501$	−14365.4			−0.0572323
$502$		0			0
$503$	−66632.2			−0.263359			−0.131680	−	0.991292i	$-0.542037\pi$
$503$	−66632.2			−0.263359			−0.131680	+	0.991292i	$0.542037\pi$
$504$		0			0
$505$	217395.			0.852447
$506$		0			0
$507$			257533.i			1.00188i
$508$		0			0
$509$			89169.0i			0.344174i	0.985082	+	0.172087i	$0.0550511\pi$
$509$			89169.0i			0.344174i	−0.985082	+	0.172087i	$0.944949\pi$
$510$		0			0
$511$	−6666.16			−0.0255290
$512$		0			0
$513$	−58653.8	−	244757.i	−0.222875	−	0.930036i
$514$		0			0
$515$			40076.6i			0.151104i
$516$		0			0
$517$	508415.			1.90212
$518$		0			0
$519$	187814.			0.697257
$520$		0			0
$521$		−	393870.i		−	1.45103i	−0.688204	−	0.725517i	$-0.741601\pi$
$521$		−	393870.i		−	1.45103i	0.688204	−	0.725517i	$-0.258399\pi$
$522$		0			0
$523$		−	218962.i		−	0.800508i	−0.916404	−	0.400254i	$-0.868922\pi$
$523$		−	218962.i		−	0.800508i	0.916404	−	0.400254i	$-0.131078\pi$
$524$		0			0
$525$		−	15684.9i		−	0.0569068i
$526$		0			0
$527$		−	324049.i		−	1.16678i
$528$		0			0
$529$	−178922.			−0.639371
$530$		0			0
$531$			23882.9i			0.0847030i
$532$		0			0
$533$	62473.5			0.219908
$534$		0			0
$535$		−	289551.i		−	1.01162i
$536$		0			0
$537$	−430800.			−1.49392
$538$		0			0
$539$	532030.			1.83129
$540$		0			0
$541$	−321582.			−1.09874			−0.549372	−	0.835578i	$-0.685133\pi$
$541$	−321582.			−1.09874			−0.549372	+	0.835578i	$0.685133\pi$
$542$		0			0
$543$	−75480.2			−0.255996
$544$		0			0
$545$			260215.i			0.876072i
$546$		0			0
$547$			150696.i			0.503648i	0.967773	+	0.251824i	$0.0810304\pi$
$547$			150696.i			0.503648i	−0.967773	+	0.251824i	$0.918970\pi$
$548$		0			0
$549$	283.558			0.000940801
$550$		0			0
$551$	−18743.7	−	78215.4i	−0.0617378	−	0.257626i
$552$		0			0
$553$		−	86859.6i		−	0.284032i
$554$		0			0
$555$	314548.			1.02118
$556$		0			0
$557$	314856.			1.01485			0.507425	−	0.861696i	$-0.330598\pi$
$557$	314856.			1.01485			0.507425	+	0.861696i	$0.330598\pi$
$558$		0			0
$559$			45039.0i			0.144133i
$560$		0			0
$561$			430876.i			1.36907i
$562$		0			0
$563$		−	424297.i		−	1.33861i	−0.742988	−	0.669304i	$-0.766592\pi$
$563$		−	424297.i		−	1.33861i	0.742988	−	0.669304i	$-0.233408\pi$
$564$		0			0
$565$			76197.3i			0.238695i
$566$		0			0
$567$	−64352.8			−0.200171
$568$		0			0
$569$		−	334927.i		−	1.03449i	−0.855838	−	0.517243i	$-0.826958\pi$
$569$		−	334927.i		−	1.03449i	0.855838	−	0.517243i	$-0.173042\pi$
$570$		0			0
$571$	166355.			0.510229			0.255114	−	0.966911i	$-0.417887\pi$
$571$	166355.			0.510229			0.255114	+	0.966911i	$0.417887\pi$
$572$		0			0
$573$			419216.i			1.27682i
$574$		0			0
$575$	−58303.5			−0.176343
$576$		0			0
$577$	−74405.0			−0.223486			−0.111743	−	0.993737i	$-0.535643\pi$
$577$	−74405.0			−0.223486			−0.111743	+	0.993737i	$0.535643\pi$
$578$		0			0
$579$	−193300.			−0.576601
$580$		0			0
$581$	13606.6			0.0403085
$582$		0			0
$583$			398835.i			1.17343i
$584$		0			0
$585$			4324.20i			0.0126356i
$586$		0			0
$587$	581715.			1.68824			0.844120	−	0.536155i	$-0.180124\pi$
$587$	581715.			1.68824			0.844120	+	0.536155i	$0.180124\pi$
$588$		0			0
$589$	−135827.	−	566791.i	−0.391520	−	1.63377i
$590$		0			0
$591$			289028.i			0.827494i
$592$		0			0
$593$	197285.			0.561027			0.280513	−	0.959850i	$-0.409495\pi$
$593$	197285.			0.561027			0.280513	+	0.959850i	$0.409495\pi$
$594$		0			0
$595$	−38541.9			−0.108868
$596$		0			0
$597$		−	345310.i		−	0.968860i
$598$		0			0
$599$		−	468344.i		−	1.30530i	−0.757659	−	0.652651i	$-0.773657\pi$
$599$		−	468344.i		−	1.30530i	0.757659	−	0.652651i	$-0.226343\pi$
$600$		0			0
$601$		−	618764.i		−	1.71307i	−0.516086	−	0.856537i	$-0.672612\pi$
$601$		−	618764.i		−	1.71307i	0.516086	−	0.856537i	$-0.327388\pi$
$602$		0			0
$603$			19727.1i			0.0542537i
$604$		0			0
$605$	799744.			2.18494
$606$		0			0
$607$			115181.i			0.312611i	0.987709	+	0.156305i	$0.0499584\pi$
$607$			115181.i			0.312611i	−0.987709	+	0.156305i	$0.950042\pi$
$608$		0			0
$609$	−19040.8			−0.0513393
$610$		0			0
$611$			70745.7i			0.189504i
$612$		0			0
$613$	282047.			0.750586			0.375293	−	0.926906i	$-0.377542\pi$
$613$	282047.			0.750586			0.375293	+	0.926906i	$0.377542\pi$
$614$		0			0
$615$	−384242.			−1.01591
$616$		0			0
$617$	−746763.			−1.96161			−0.980805	−	0.194992i	$-0.937532\pi$
$617$	−746763.			−1.96161			−0.980805	+	0.194992i	$0.937532\pi$
$618$		0			0
$619$	−688558.			−1.79704			−0.898522	−	0.438928i	$-0.855358\pi$
$619$	−688558.			−1.79704			−0.898522	+	0.438928i	$0.855358\pi$
$620$		0			0
$621$			221482.i			0.574322i
$622$		0			0
$623$			35224.8i			0.0907555i
$624$		0			0
$625$	−242235.			−0.620121
$626$		0			0
$627$	180603.	+	753640.i	0.459400	+	1.91703i
$628$		0			0
$629$			321325.i			0.812164i
$630$		0			0
$631$	416684.			1.04652			0.523260	−	0.852173i	$-0.324716\pi$
$631$	416684.			1.04652			0.523260	+	0.852173i	$0.324716\pi$
$632$		0			0
$633$	−104268.			−0.260222
$634$		0			0
$635$			646804.i			1.60408i
$636$		0			0
$637$			74031.6i			0.182448i
$638$		0			0
$639$		−	35118.4i		−	0.0860068i
$640$		0			0
$641$		−	454238.i		−	1.10552i	−0.833340	−	0.552761i	$-0.813574\pi$
$641$		−	454238.i		−	1.10552i	0.833340	−	0.552761i	$-0.186426\pi$
$642$		0			0
$643$	−660566.			−1.59770			−0.798848	−	0.601533i	$-0.794557\pi$
$643$	−660566.			−1.59770			−0.798848	+	0.601533i	$0.794557\pi$
$644$		0			0
$645$		−	277011.i		−	0.665852i
$646$		0			0
$647$	−498199.			−1.19013			−0.595065	−	0.803678i	$-0.702874\pi$
$647$	−498199.			−1.19013			−0.595065	+	0.803678i	$0.702874\pi$
$648$		0			0
$649$			851051.i			2.02053i
$650$		0			0
$651$	−137980.			−0.325576
$652$		0			0
$653$	23925.7			0.0561099			0.0280549	−	0.999606i	$-0.491069\pi$
$653$	23925.7			0.0561099			0.0280549	+	0.999606i	$0.491069\pi$
$654$		0			0
$655$	−296037.			−0.690022
$656$		0			0
$657$	4699.11			0.0108864
$658$		0			0
$659$		−	51119.1i		−	0.117710i	−0.998267	−	0.0588549i	$-0.981255\pi$
$659$		−	51119.1i		−	0.117710i	0.998267	−	0.0588549i	$-0.0187449\pi$
$660$		0			0
$661$		−	212473.i		−	0.486295i	−0.969989	−	0.243148i	$-0.921820\pi$
$661$		−	212473.i		−	0.486295i	0.969989	−	0.243148i	$-0.0781800\pi$
$662$		0			0
$663$	−59956.1			−0.136397
$664$		0			0
$665$	−67413.2	+	16155.0i	−0.152441	+	0.0365312i
$666$		0			0
$667$			70777.7i			0.159091i
$668$		0			0
$669$	647857.			1.44753
$670$		0			0
$671$	10104.4			0.0224422
$672$		0			0
$673$		−	715200.i		−	1.57906i	−0.613714	−	0.789528i	$-0.710325\pi$
$673$		−	715200.i		−	1.57906i	0.613714	−	0.789528i	$-0.289675\pi$
$674$		0			0
$675$		−	127956.i		−	0.280837i
$676$		0			0
$677$			294047.i			0.641564i	0.947153	+	0.320782i	$0.103946\pi$
$677$			294047.i			0.641564i	−0.947153	+	0.320782i	$0.896054\pi$
$678$		0			0
$679$			104055.i			0.225696i
$680$		0			0
$681$	−425731.			−0.917996
$682$		0			0
$683$			696159.i			1.49234i	0.665757	+	0.746168i	$0.268109\pi$
$683$			696159.i			1.49234i	−0.665757	+	0.746168i	$0.731891\pi$
$684$		0			0
$685$	174499.			0.371888
$686$		0			0
$687$		−	491565.i		−	1.04152i
$688$		0			0
$689$	−55497.6			−0.116906
$690$		0			0
$691$	150153.			0.314469			0.157234	−	0.987561i	$-0.449742\pi$
$691$	150153.			0.314469			0.157234	+	0.987561i	$0.449742\pi$
$692$		0			0
$693$	13517.2			0.0281462
$694$		0			0
$695$	165818.			0.343290
$696$		0			0
$697$		−	392521.i		−	0.807974i
$698$		0			0
$699$		−	219664.i		−	0.449577i
$700$		0			0
$701$	405313.			0.824811			0.412405	−	0.911000i	$-0.364689\pi$
$701$	405313.			0.824811			0.412405	+	0.911000i	$0.364689\pi$
$702$		0			0
$703$	134685.	+	562027.i	0.272526	+	1.13722i
$704$		0			0
$705$		−	435120.i		−	0.875449i
$706$		0			0
$707$	94561.1			0.189179
$708$		0			0
$709$	−633939.			−1.26112			−0.630558	−	0.776142i	$-0.717174\pi$
$709$	−633939.			−1.26112			−0.630558	+	0.776142i	$0.717174\pi$
$710$		0			0
$711$			61229.1i			0.121121i
$712$		0			0
$713$			512893.i			1.00890i
$714$		0			0
$715$			154090.i			0.301413i
$716$		0			0
$717$			545579.i			1.06125i
$718$		0			0
$719$	526722.			1.01888			0.509441	−	0.860506i	$-0.329852\pi$
$719$	526722.			1.01888			0.509441	+	0.860506i	$0.329852\pi$
$720$		0			0
$721$			17432.3i			0.0335338i
$722$		0			0
$723$	−1.02989e6			−1.97022
$724$		0			0
$725$		−	40890.2i		−	0.0777935i
$726$		0			0
$727$	125183.			0.236852			0.118426	−	0.992963i	$-0.462215\pi$
$727$	125183.			0.236852			0.118426	+	0.992963i	$0.462215\pi$
$728$		0			0
$729$	−482715.			−0.908314
$730$		0			0
$731$	282980.			0.529567
$732$		0			0
$733$	−636516.			−1.18468			−0.592340	−	0.805688i	$-0.701796\pi$
$733$	−636516.			−1.18468			−0.592340	+	0.805688i	$0.701796\pi$
$734$		0			0
$735$		−	455330.i		−	0.842852i
$736$		0			0
$737$			702962.i			1.29419i
$738$		0			0
$739$	−288112.			−0.527561			−0.263781	−	0.964583i	$-0.584969\pi$
$739$	−288112.			−0.527561			−0.263781	+	0.964583i	$0.584969\pi$
$740$		0			0
$741$	−104869.	+	25130.9i	−0.190989	+	0.0457690i
$742$		0			0
$743$			114608.i			0.207605i	0.994598	+	0.103803i	$0.0331010\pi$
$743$			114608.i			0.207605i	−0.994598	+	0.103803i	$0.966899\pi$
$744$		0			0
$745$	−120974.			−0.217961
$746$		0			0
$747$	−9591.56			−0.0171889
$748$		0			0
$749$		−	125947.i		−	0.224504i
$750$		0			0
$751$		−	75490.9i		−	0.133849i	−0.997758	−	0.0669245i	$-0.978681\pi$
$751$		−	75490.9i		−	0.133849i	0.997758	−	0.0669245i	$-0.0213187\pi$
$752$		0			0
$753$		−	297677.i		−	0.524995i
$754$		0			0
$755$		−	739061.i		−	1.29654i
$756$		0			0
$757$	−97240.8			−0.169690			−0.0848451	−	0.996394i	$-0.527040\pi$
$757$	−97240.8			−0.169690			−0.0848451	+	0.996394i	$0.527040\pi$
$758$		0			0
$759$		−	681975.i		−	1.18382i
$760$		0			0
$761$	−877960.			−1.51602			−0.758011	−	0.652242i	$-0.773829\pi$
$761$	−877960.			−1.51602			−0.758011	+	0.652242i	$0.773829\pi$
$762$		0			0
$763$			113187.i			0.194422i
$764$		0			0
$765$	27169.0			0.0464248
$766$		0			0
$767$	−118423.			−0.201301
$768$		0			0
$769$	−475997.			−0.804917			−0.402459	−	0.915438i	$-0.631844\pi$
$769$	−475997.			−0.804917			−0.402459	+	0.915438i	$0.631844\pi$
$770$		0			0
$771$	−903396.			−1.51974
$772$		0			0
$773$		−	755574.i		−	1.26450i	−0.774765	−	0.632249i	$-0.782132\pi$
$773$		−	755574.i		−	1.26450i	0.774765	−	0.632249i	$-0.217868\pi$
$774$		0			0
$775$		−	296312.i		−	0.493340i
$776$		0			0
$777$	136820.			0.226625
$778$		0			0
$779$	−164527.	−	686554.i	−0.271120	−	1.13136i
$780$		0			0
$781$		−	1.25142e6i		−	2.05164i
$782$		0			0
$783$	−155333.			−0.253361
$784$		0			0
$785$	−254079.			−0.412316
$786$		0			0
$787$			787554.i			1.27154i	0.771878	+	0.635771i	$0.219318\pi$
$787$			787554.i			1.27154i	−0.771878	+	0.635771i	$0.780682\pi$
$788$		0			0
$789$			885678.i			1.42273i
$790$		0			0
$791$			33143.8i			0.0529723i
$792$		0			0
$793$			1406.02i			0.00223587i
$794$		0			0
$795$	341337.			0.540069
$796$		0			0
$797$			177253.i			0.279047i	0.990219	+	0.139523i	$0.0445571\pi$
$797$			177253.i			0.279047i	−0.990219	+	0.139523i	$0.955443\pi$
$798$		0			0
$799$	444495.			0.696264
$800$		0			0
$801$		−	24830.7i		−	0.0387012i
$802$		0			0
$803$	167449.			0.259688
$804$		0			0
$805$	61002.7			0.0941363
$806$		0			0
$807$	−937693.			−1.43984
$808$		0			0
$809$	−444982.			−0.679901			−0.339951	−	0.940443i	$-0.610410\pi$
$809$	−444982.			−0.679901			−0.339951	+	0.940443i	$0.610410\pi$
$810$		0			0
$811$		−	160700.i		−	0.244329i	−0.992510	−	0.122164i	$-0.961017\pi$
$811$		−	160700.i		−	0.244329i	0.992510	−	0.122164i	$-0.0389835\pi$
$812$		0			0
$813$		−	117392.i		−	0.177606i
$814$		0			0
$815$	706925.			1.06429
$816$		0			0
$817$	494957.	−	118612.i	0.741521	−	0.177699i
$818$		0			0
$819$			1880.91i			0.00280415i
$820$		0			0
$821$	72962.3			0.108246			0.0541230	−	0.998534i	$-0.482764\pi$
$821$	72962.3			0.108246			0.0541230	+	0.998534i	$0.482764\pi$
$822$		0			0
$823$	−1.16476e6			−1.71964			−0.859821	−	0.510595i	$-0.829425\pi$
$823$	−1.16476e6			−1.71964			−0.859821	+	0.510595i	$0.829425\pi$
$824$		0			0
$825$			393995.i			0.578873i
$826$		0			0
$827$		−	689314.i		−	1.00787i	−0.863740	−	0.503937i	$-0.831884\pi$
$827$		−	689314.i		−	1.00787i	0.863740	−	0.503937i	$-0.168116\pi$
$828$		0			0
$829$			996944.i			1.45065i	0.688408	+	0.725323i	$0.258310\pi$
$829$			996944.i			1.45065i	−0.688408	+	0.725323i	$0.741690\pi$
$830$		0			0
$831$		−	1.16368e6i		−	1.68512i
$832$		0			0
$833$	465141.			0.670339
$834$		0			0
$835$			32277.9i			0.0462949i
$836$		0			0
$837$	−1.12562e6			−1.60673
$838$		0			0
$839$		−	801785.i		−	1.13903i	−0.821982	−	0.569514i	$-0.807132\pi$
$839$		−	801785.i		−	1.13903i	0.821982	−	0.569514i	$-0.192868\pi$
$840$		0			0
$841$	657642.			0.929817
$842$		0			0
$843$	12767.0			0.0179653
$844$		0			0
$845$	578659.			0.810418
$846$		0			0
$847$	347867.			0.484894
$848$		0			0
$849$		−	260175.i		−	0.360952i
$850$		0			0
$851$		−	508582.i		−	0.702267i
$852$		0			0
$853$	955672.			1.31344			0.656721	−	0.754134i	$-0.271943\pi$
$853$	955672.			1.31344			0.656721	+	0.754134i	$0.271943\pi$
$854$		0			0
$855$	47521.0	−	11388.0i	0.0650059	−	0.0155781i
$856$		0			0
$857$			327073.i			0.445331i	0.974895	+	0.222666i	$0.0714758\pi$
$857$			327073.i			0.445331i	−0.974895	+	0.222666i	$0.928524\pi$
$858$		0			0
$859$	580229.			0.786344			0.393172	−	0.919465i	$-0.371378\pi$
$859$	580229.			0.786344			0.393172	+	0.919465i	$0.371378\pi$
$860$		0			0
$861$	−167135.			−0.225455
$862$		0			0
$863$		−	163125.i		−	0.219028i	−0.993985	−	0.109514i	$-0.965071\pi$
$863$		−	163125.i		−	0.219028i	0.993985	−	0.109514i	$-0.0349294\pi$
$864$		0			0
$865$		−	422004.i		−	0.564007i
$866$		0			0
$867$		−	404306.i		−	0.537864i
$868$		0			0
$869$			2.18186e6i			2.88926i
$870$		0			0
$871$	−97816.9			−0.128937
$872$		0			0
$873$		−	73350.5i		−	0.0962442i
$874$		0			0
$875$	−155260.			−0.202789
$876$		0			0
$877$		−	934932.i		−	1.21557i	−0.794101	−	0.607786i	$-0.792058\pi$
$877$		−	934932.i		−	1.21557i	0.794101	−	0.607786i	$-0.207942\pi$
$878$		0			0
$879$	579707.			0.750293
$880$		0			0
$881$	−512798.			−0.660685			−0.330343	−	0.943861i	$-0.607164\pi$
$881$	−512798.			−0.660685			−0.330343	+	0.943861i	$0.607164\pi$
$882$		0			0
$883$	−94763.1			−0.121540			−0.0607698	−	0.998152i	$-0.519356\pi$
$883$	−94763.1			−0.121540			−0.0607698	+	0.998152i	$0.519356\pi$
$884$		0			0
$885$	728360.			0.929950
$886$		0			0
$887$			972322.i			1.23584i	0.786241	+	0.617921i	$0.212025\pi$
$887$			972322.i			1.23584i	−0.786241	+	0.617921i	$0.787975\pi$
$888$		0			0
$889$			281342.i			0.355985i
$890$		0			0
$891$	1.61650e6			2.03620
$892$		0			0
$893$	777463.	−	186312.i	0.974937	−	0.233635i
$894$		0			0
$895$			967976.i			1.20842i
$896$		0			0
$897$	94896.4			0.117941
$898$		0			0
$899$	−359709.			−0.445074
$900$		0			0
$901$			348692.i			0.429529i
$902$		0			0
$903$		−	120492.i		−	0.147769i
$904$		0			0
$905$			169598.i			0.207074i
$906$		0			0
$907$			356734.i			0.433640i	0.976212	+	0.216820i	$0.0695685\pi$
$907$			356734.i			0.433640i	−0.976212	+	0.216820i	$0.930431\pi$
$908$		0			0
$909$	−66658.1			−0.0806724
$910$		0			0
$911$			1.54115e6i			1.85698i	0.371360	+	0.928489i	$0.378892\pi$
$911$			1.54115e6i			1.85698i	−0.371360	+	0.928489i	$0.621108\pi$
$912$		0			0
$913$	−341788.			−0.410030
$914$		0			0
$915$		−	8647.71i		−	0.0103290i
$916$		0			0
$917$	−128768.			−0.153133
$918$		0			0
$919$	−1.17655e6			−1.39309			−0.696544	−	0.717514i	$-0.745280\pi$
$919$	−1.17655e6			−1.39309			−0.696544	+	0.717514i	$0.745280\pi$
$920$		0			0
$921$	593680.			0.699896
$922$		0			0
$923$	174134.			0.204400
$924$		0			0
$925$			293822.i			0.343400i
$926$		0			0
$927$		−	12288.4i		−	0.0142999i
$928$		0			0
$929$	726950.			0.842312			0.421156	−	0.906988i	$-0.361625\pi$
$929$	726950.			0.842312			0.421156	+	0.906988i	$0.361625\pi$
$930$		0			0
$931$	813573.	−	194966.i	0.938636	−	0.224936i
$932$		0			0
$933$		−	1.06613e6i		−	1.22475i
$934$		0			0
$935$	968146.			1.10743
$936$		0			0
$937$	−1.15653e6			−1.31727			−0.658637	−	0.752461i	$-0.728867\pi$
$937$	−1.15653e6			−1.31727			−0.658637	+	0.752461i	$0.728867\pi$
$938$		0			0
$939$		−	438740.i		−	0.497596i
$940$		0			0
$941$		−	1.04979e6i		−	1.18556i	−0.805365	−	0.592779i	$-0.798031\pi$
$941$		−	1.04979e6i		−	1.18556i	0.805365	−	0.592779i	$-0.201969\pi$
$942$		0			0
$943$			621268.i			0.698644i
$944$		0			0
$945$			133880.i			0.149917i
$946$		0			0
$947$	740785.			0.826023			0.413011	−	0.910726i	$-0.364477\pi$
$947$	740785.			0.826023			0.413011	+	0.910726i	$0.364477\pi$
$948$		0			0
$949$			23300.5i			0.0258722i
$950$		0			0
$951$	−1.27510e6			−1.40988
$952$		0			0
$953$			51255.2i			0.0564354i	0.999602	+	0.0282177i	$0.00898317\pi$
$953$			51255.2i			0.0564354i	−0.999602	+	0.0282177i	$0.991017\pi$
$954$		0			0
$955$	941947.			1.03281
$956$		0			0
$957$	478292.			0.522238
$958$		0			0
$959$	75902.4			0.0825313
$960$		0			0
$961$	−1.68312e6			−1.82251
$962$		0			0
$963$			88782.5i			0.0957359i
$964$		0			0
$965$			434332.i			0.466409i
$966$		0			0
$967$	−961151.			−1.02787			−0.513936	−	0.857829i	$-0.671813\pi$
$967$	−961151.			−1.02787			−0.513936	+	0.857829i	$0.671813\pi$
$968$		0			0
$969$	157897.	+	658890.i	0.168162	+	0.701722i
$970$		0			0
$971$			1.49433e6i			1.58493i	0.609921	+	0.792463i	$0.291201\pi$
$971$			1.49433e6i			1.58493i	−0.609921	+	0.792463i	$0.708799\pi$
$972$		0			0
$973$	72126.2			0.0761846
$974$		0			0
$975$	−54824.2			−0.0576718
$976$		0			0
$977$		−	833901.i		−	0.873625i	−0.899553	−	0.436813i	$-0.856107\pi$
$977$		−	833901.i		−	0.873625i	0.899553	−	0.436813i	$-0.143893\pi$
$978$		0			0
$979$		−	884825.i		−	0.923192i
$980$		0			0
$981$		−	79787.6i		−	0.0829082i
$982$		0			0
$983$			1.06793e6i			1.10518i	0.833453	+	0.552591i	$0.186361\pi$
$983$			1.06793e6i			1.10518i	−0.833453	+	0.552591i	$0.813639\pi$
$984$		0			0
$985$	649425.			0.669355
$986$		0			0
$987$		−	189266.i		−	0.194284i
$988$		0			0
$989$	−447890.			−0.457909
$990$		0			0
$991$			1.11485e6i			1.13519i	0.823309	+	0.567594i	$0.192126\pi$
$991$			1.11485e6i			1.13519i	−0.823309	+	0.567594i	$0.807874\pi$
$992$		0			0
$993$	478468.			0.485237
$994$		0			0
$995$	−775887.			−0.783705
$996$		0			0
$997$	−609672.			−0.613347			−0.306673	−	0.951815i	$-0.599216\pi$
$997$	−609672.			−0.613347			−0.306673	+	0.951815i	$0.599216\pi$
$998$		0			0
$999$	1.11616e6			1.11840

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.5.e.f.113.16		20
4.3	odd	2		152.5.e.a.113.5	✓	20
12.11	even	2		1368.5.o.a.721.3		20
19.18	odd	2	inner	304.5.e.f.113.5		20
76.75	even	2		152.5.e.a.113.16	yes	20
228.227	odd	2		1368.5.o.a.721.4		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.5.e.a.113.5	✓	20	4.3	odd	2
152.5.e.a.113.16	yes	20	76.75	even	2
304.5.e.f.113.5		20	19.18	odd	2	inner
304.5.e.f.113.16		20	1.1	even	1	trivial
1368.5.o.a.721.3		20	12.11	even	2
1368.5.o.a.721.4		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)

\(f(q)\)	\(=\)	\(q+9.35107i q^{3} +21.0112 q^{5} +9.13929 q^{7} -6.44248 q^{9} +O(q^{10})\)	\(q+9.35107i q^{3} +21.0112 q^{5} +9.13929 q^{7} -6.44248 q^{9} -229.573 q^{11} -31.9450i q^{13} +196.477i q^{15} -200.710 q^{17} +(-351.060 + 84.1286i) q^{19} +85.4622i q^{21} +317.677 q^{23} -183.531 q^{25} +697.192i q^{27} +222.798i q^{29} +1614.51i q^{31} -2146.75i q^{33} +192.027 q^{35} -1600.94i q^{37} +298.720 q^{39} +1955.66i q^{41} -1409.89 q^{43} -135.364 q^{45} -2214.61 q^{47} -2317.47 q^{49} -1876.86i q^{51} -1737.29i q^{53} -4823.60 q^{55} +(-786.693 - 3282.79i) q^{57} -3707.10i q^{59} -44.0139 q^{61} -58.8797 q^{63} -671.201i q^{65} -3062.04i q^{67} +2970.62i q^{69} +5451.07i q^{71} -729.395 q^{73} -1716.21i q^{75} -2098.14 q^{77} -9503.97i q^{79} -7041.34 q^{81} +1488.80 q^{83} -4217.16 q^{85} -2083.40 q^{87} +3854.22i q^{89} -291.955i q^{91} -15097.4 q^{93} +(-7376.19 + 1767.64i) q^{95} +11385.4i q^{97} +1479.02 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 32 q^{7} - 372 q^{9}+O(q^{10}) \) 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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