LMFDB - Embedded newform 304.5.e.c.113.2

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:	$4$
Coefficient field:	4.0.12107488.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 35x^{2} + 142 $ x^4 + 35*x^2 + 142
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 19)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.2
Root			$-5.50600i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.5.e.c.113.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.25064i q^{3} -6.22800 q^{5} -8.36799 q^{7} +62.9321 q^{9} +O(q^{10})$	$q-4.25064i q^{3} -6.22800 q^{5} -8.36799 q^{7} +62.9321 q^{9} -128.124 q^{11} -101.817i q^{13} +26.4730i q^{15} -27.5280 q^{17} +(215.004 + 289.990i) q^{19} +35.5693i q^{21} -463.236 q^{23} -586.212 q^{25} -611.803i q^{27} -1132.50i q^{29} +1094.42i q^{31} +544.609i q^{33} +52.1158 q^{35} +1858.74i q^{37} -432.787 q^{39} +2520.60i q^{41} -2245.97 q^{43} -391.941 q^{45} +1531.38 q^{47} -2330.98 q^{49} +117.012i q^{51} +855.433i q^{53} +797.956 q^{55} +(1232.64 - 913.905i) q^{57} +2664.44i q^{59} -691.012 q^{61} -526.615 q^{63} +634.116i q^{65} +2998.41i q^{67} +1969.05i q^{69} +1183.34i q^{71} -5253.02 q^{73} +2491.78i q^{75} +1072.14 q^{77} +3439.44i q^{79} +2496.94 q^{81} -7199.74 q^{83} +171.445 q^{85} -4813.83 q^{87} +10651.2i q^{89} +852.004i q^{91} +4651.97 q^{93} +(-1339.05 - 1806.06i) q^{95} -3226.29i q^{97} -8063.11 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 42 q^{5} - 136 q^{7} - 278 q^{9}+O(q^{10}) $ 4 * q - 42 * q^5 - 136 * q^7 - 278 * q^9	$ 4 q - 42 q^{5} - 136 q^{7} - 278 q^{9} - 222 q^{11} + 300 q^{17} - 114 q^{19} + 78 q^{23} - 1986 q^{25} + 1866 q^{35} - 6362 q^{39} - 2986 q^{43} + 5182 q^{45} + 7578 q^{47} - 2352 q^{49} + 1090 q^{55} + 10450 q^{57} + 158 q^{61} + 23030 q^{63} - 15168 q^{73} + 102 q^{77} + 40712 q^{81} - 33276 q^{83} - 4902 q^{85} + 7214 q^{87} - 40004 q^{93} + 5358 q^{95} - 23042 q^{99}+O(q^{100}) $ 4 * q - 42 * q^5 - 136 * q^7 - 278 * q^9 - 222 * q^11 + 300 * q^17 - 114 * q^19 + 78 * q^23 - 1986 * q^25 + 1866 * q^35 - 6362 * q^39 - 2986 * q^43 + 5182 * q^45 + 7578 * q^47 - 2352 * q^49 + 1090 * q^55 + 10450 * q^57 + 158 * q^61 + 23030 * q^63 - 15168 * q^73 + 102 * q^77 + 40712 * q^81 - 33276 * q^83 - 4902 * q^85 + 7214 * q^87 - 40004 * q^93 + 5358 * q^95 - 23042 * 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
$3$		−	4.25064i		−	0.472293i	−0.971717	−	0.236147i	$-0.924115\pi$
$3$		−	4.25064i		−	0.472293i	0.971717	−	0.236147i	$-0.0758846\pi$
$4$		0			0
$5$	−6.22800			−0.249120			−0.124560	−	0.992212i	$-0.539752\pi$
$5$	−6.22800			−0.249120			−0.124560	+	0.992212i	$0.539752\pi$
$6$		0			0
$7$	−8.36799			−0.170775			−0.0853876	−	0.996348i	$-0.527213\pi$
$7$	−8.36799			−0.170775			−0.0853876	+	0.996348i	$0.527213\pi$
$8$		0			0
$9$	62.9321			0.776939
$10$		0			0
$11$	−128.124			−1.05888			−0.529438	−	0.848349i	$-0.677597\pi$
$11$	−128.124			−1.05888			−0.529438	+	0.848349i	$0.677597\pi$
$12$		0			0
$13$		−	101.817i		−	0.602468i	−0.953550	−	0.301234i	$-0.902602\pi$
$13$		−	101.817i		−	0.602468i	0.953550	−	0.301234i	$-0.0973985\pi$
$14$		0			0
$15$			26.4730i			0.117658i
$16$		0			0
$17$	−27.5280			−0.0952528			−0.0476264	−	0.998865i	$-0.515166\pi$
$17$	−27.5280			−0.0952528			−0.0476264	+	0.998865i	$0.515166\pi$
$18$		0			0
$19$	215.004	+	289.990i	0.595579	+	0.803297i
$19$	215.004	+	289.990i	0.595579	+	0.803297i
$20$		0			0
$21$			35.5693i			0.0806560i
$22$		0			0
$23$	−463.236			−0.875683			−0.437841	−	0.899052i	$-0.644257\pi$
$23$	−463.236			−0.875683			−0.437841	+	0.899052i	$0.644257\pi$
$24$		0			0
$25$	−586.212			−0.937939
$26$		0			0
$27$		−	611.803i		−	0.839236i
$28$		0			0
$29$		−	1132.50i		−	1.34661i	−0.739367	−	0.673303i	$-0.764875\pi$
$29$		−	1132.50i		−	1.34661i	0.739367	−	0.673303i	$-0.235125\pi$
$30$		0			0
$31$			1094.42i			1.13883i	0.822050	+	0.569415i	$0.192830\pi$
$31$			1094.42i			1.13883i	−0.822050	+	0.569415i	$0.807170\pi$
$32$		0			0
$33$			544.609i			0.500100i
$34$		0			0
$35$	52.1158			0.0425435
$36$		0			0
$37$			1858.74i			1.35773i	0.734261	+	0.678867i	$0.237529\pi$
$37$			1858.74i			1.35773i	−0.734261	+	0.678867i	$0.762471\pi$
$38$		0			0
$39$	−432.787			−0.284541
$40$		0			0
$41$			2520.60i			1.49947i	0.661741	+	0.749733i	$0.269818\pi$
$41$			2520.60i			1.49947i	−0.661741	+	0.749733i	$0.730182\pi$
$42$		0			0
$43$	−2245.97			−1.21470			−0.607348	−	0.794436i	$-0.707767\pi$
$43$	−2245.97			−1.21470			−0.607348	+	0.794436i	$0.707767\pi$
$44$		0			0
$45$	−391.941			−0.193551
$46$		0			0
$47$	1531.38			0.693246			0.346623	−	0.938005i	$-0.387328\pi$
$47$	1531.38			0.693246			0.346623	+	0.938005i	$0.387328\pi$
$48$		0			0
$49$	−2330.98			−0.970836
$50$		0			0
$51$			117.012i			0.0449872i
$52$		0			0
$53$			855.433i			0.304533i	0.988339	+	0.152266i	$0.0486572\pi$
$53$			855.433i			0.304533i	−0.988339	+	0.152266i	$0.951343\pi$
$54$		0			0
$55$	797.956			0.263787
$56$		0			0
$57$	1232.64	−	913.905i	0.379392	−	0.281288i
$58$		0			0
$59$			2664.44i			0.765425i	0.923868	+	0.382712i	$0.125010\pi$
$59$			2664.44i			0.765425i	−0.923868	+	0.382712i	$0.874990\pi$
$60$		0			0
$61$	−691.012			−0.185706			−0.0928530	−	0.995680i	$-0.529599\pi$
$61$	−691.012			−0.185706			−0.0928530	+	0.995680i	$0.529599\pi$
$62$		0			0
$63$	−526.615			−0.132682
$64$		0			0
$65$			634.116i			0.150087i
$66$		0			0
$67$			2998.41i			0.667947i	0.942582	+	0.333973i	$0.108390\pi$
$67$			2998.41i			0.667947i	−0.942582	+	0.333973i	$0.891610\pi$
$68$		0			0
$69$			1969.05i			0.413579i
$70$		0			0
$71$			1183.34i			0.234744i	0.993088	+	0.117372i	$0.0374470\pi$
$71$			1183.34i			0.234744i	−0.993088	+	0.117372i	$0.962553\pi$
$72$		0			0
$73$	−5253.02			−0.985743			−0.492872	−	0.870102i	$-0.664053\pi$
$73$	−5253.02			−0.985743			−0.492872	+	0.870102i	$0.664053\pi$
$74$		0			0
$75$			2491.78i			0.442982i
$76$		0			0
$77$	1072.14			0.180830
$78$		0			0
$79$			3439.44i			0.551104i	0.961286	+	0.275552i	$0.0888607\pi$
$79$			3439.44i			0.551104i	−0.961286	+	0.275552i	$0.911139\pi$
$80$		0			0
$81$	2496.94			0.380573
$82$		0			0
$83$	−7199.74			−1.04511			−0.522553	−	0.852607i	$-0.675020\pi$
$83$	−7199.74			−1.04511			−0.522553	+	0.852607i	$0.675020\pi$
$84$		0			0
$85$	171.445			0.0237294
$86$		0			0
$87$	−4813.83			−0.635993
$88$		0			0
$89$			10651.2i			1.34468i	0.740243	+	0.672340i	$0.234711\pi$
$89$			10651.2i			1.34468i	−0.740243	+	0.672340i	$0.765289\pi$
$90$		0			0
$91$			852.004i			0.102887i
$92$		0			0
$93$	4651.97			0.537862
$94$		0			0
$95$	−1339.05	−	1806.06i	−0.148371	−	0.200117i
$96$		0			0
$97$		−	3226.29i		−	0.342894i	−0.985193	−	0.171447i	$-0.945156\pi$
$97$		−	3226.29i		−	0.342894i	0.985193	−	0.171447i	$-0.0548443\pi$
$98$		0			0
$99$	−8063.11			−0.822682
$100$		0			0
$101$	−13626.2			−1.33578			−0.667888	−	0.744262i	$-0.732801\pi$
$101$	−13626.2			−1.33578			−0.667888	+	0.744262i	$0.732801\pi$
$102$		0			0
$103$			19402.7i			1.82889i	0.404709	+	0.914445i	$0.367373\pi$
$103$			19402.7i			1.82889i	−0.404709	+	0.914445i	$0.632627\pi$
$104$		0			0
$105$		−	221.526i		−	0.0200930i
$106$		0			0
$107$		−	20844.6i		−	1.82065i	−0.413898	−	0.910323i	$-0.635833\pi$
$107$		−	20844.6i		−	1.82065i	0.413898	−	0.910323i	$-0.364167\pi$
$108$		0			0
$109$		−	4563.13i		−	0.384069i	−0.981388	−	0.192035i	$-0.938491\pi$
$109$		−	4563.13i		−	0.384069i	0.981388	−	0.192035i	$-0.0615086\pi$
$110$		0			0
$111$	7900.82			0.641249
$112$		0			0
$113$		−	368.758i		−	0.0288792i	−0.999896	−	0.0144396i	$-0.995404\pi$
$113$		−	368.758i		−	0.0288792i	0.999896	−	0.0144396i	$-0.00459643\pi$
$114$		0			0
$115$	2885.03			0.218150
$116$		0			0
$117$		−	6407.55i		−	0.468081i
$118$		0			0
$119$	230.354			0.0162668
$120$		0			0
$121$	1774.77			0.121219
$122$		0			0
$123$	10714.2			0.708188
$124$		0			0
$125$	7543.43			0.482779
$126$		0			0
$127$		−	23143.0i		−	1.43487i	−0.696625	−	0.717435i	$-0.745316\pi$
$127$		−	23143.0i		−	1.43487i	0.696625	−	0.717435i	$-0.254684\pi$
$128$		0			0
$129$			9546.82i			0.573693i
$130$		0			0
$131$	17911.4			1.04373			0.521863	−	0.853030i	$-0.325237\pi$
$131$	17911.4			1.04373			0.521863	+	0.853030i	$0.325237\pi$
$132$		0			0
$133$	−1799.15	−	2426.63i	−0.101710	−	0.137183i
$134$		0			0
$135$			3810.31i			0.209071i
$136$		0			0
$137$	−32580.6			−1.73587			−0.867937	−	0.496674i	$-0.834555\pi$
$137$	−32580.6			−1.73587			−0.867937	+	0.496674i	$0.834555\pi$
$138$		0			0
$139$	−17676.3			−0.914877			−0.457439	−	0.889241i	$-0.651233\pi$
$139$	−17676.3			−0.914877			−0.457439	+	0.889241i	$0.651233\pi$
$140$		0			0
$141$		−	6509.34i		−	0.327415i
$142$		0			0
$143$			13045.2i			0.637939i
$144$		0			0
$145$			7053.18i			0.335466i
$146$		0			0
$147$			9908.14i			0.458519i
$148$		0			0
$149$	29045.3			1.30829			0.654143	−	0.756371i	$-0.273029\pi$
$149$	29045.3			1.30829			0.654143	+	0.756371i	$0.273029\pi$
$150$		0			0
$151$		−	21573.4i		−	0.946162i	−0.881019	−	0.473081i	$-0.843142\pi$
$151$		−	21573.4i		−	0.946162i	0.881019	−	0.473081i	$-0.156858\pi$
$152$		0			0
$153$	−1732.40			−0.0740056
$154$		0			0
$155$		−	6816.02i		−	0.283705i
$156$		0			0
$157$	25952.2			1.05287			0.526435	−	0.850215i	$-0.323528\pi$
$157$	25952.2			1.05287			0.526435	+	0.850215i	$0.323528\pi$
$158$		0			0
$159$	3636.14			0.143829
$160$		0			0
$161$	3876.36			0.149545
$162$		0			0
$163$	623.620			0.0234717			0.0117359	−	0.999931i	$-0.496264\pi$
$163$	623.620			0.0234717			0.0117359	+	0.999931i	$0.496264\pi$
$164$		0			0
$165$		−	3391.82i		−	0.124585i
$166$		0			0
$167$		−	18386.9i		−	0.659287i	−0.944106	−	0.329643i	$-0.893071\pi$
$167$		−	18386.9i		−	0.659287i	0.944106	−	0.329643i	$-0.106929\pi$
$168$		0			0
$169$	18194.3			0.637033
$170$		0			0
$171$	13530.7	+	18249.7i	0.462729	+	0.624112i
$172$		0			0
$173$		−	27268.7i		−	0.911112i	−0.890207	−	0.455556i	$-0.849441\pi$
$173$		−	27268.7i		−	0.911112i	0.890207	−	0.455556i	$-0.150559\pi$
$174$		0			0
$175$	4905.42			0.160177
$176$		0			0
$177$	11325.6			0.361505
$178$		0			0
$179$		−	41377.5i		−	1.29139i	−0.763594	−	0.645697i	$-0.776567\pi$
$179$		−	41377.5i		−	1.29139i	0.763594	−	0.645697i	$-0.223433\pi$
$180$		0			0
$181$		−	61346.6i		−	1.87255i	−0.351268	−	0.936275i	$-0.614249\pi$
$181$		−	61346.6i		−	1.87255i	0.351268	−	0.936275i	$-0.385751\pi$
$182$		0			0
$183$			2937.24i			0.0877077i
$184$		0			0
$185$		−	11576.2i		−	0.338238i
$186$		0			0
$187$	3527.00			0.100861
$188$		0			0
$189$			5119.56i			0.143321i
$190$		0			0
$191$	46322.2			1.26976			0.634882	−	0.772609i	$-0.281049\pi$
$191$	46322.2			1.26976			0.634882	+	0.772609i	$0.281049\pi$
$192$		0			0
$193$			44629.2i			1.19813i	0.800700	+	0.599066i	$0.204461\pi$
$193$			44629.2i			1.19813i	−0.800700	+	0.599066i	$0.795539\pi$
$194$		0			0
$195$	2695.40			0.0708849
$196$		0			0
$197$	39670.0			1.02219			0.511093	−	0.859525i	$-0.329241\pi$
$197$	39670.0			1.02219			0.511093	+	0.859525i	$0.329241\pi$
$198$		0			0
$199$	−5913.07			−0.149316			−0.0746581	−	0.997209i	$-0.523787\pi$
$199$	−5913.07			−0.149316			−0.0746581	+	0.997209i	$0.523787\pi$
$200$		0			0
$201$	12745.2			0.315467
$202$		0			0
$203$			9476.71i			0.229967i
$204$		0			0
$205$		−	15698.3i		−	0.373547i
$206$		0			0
$207$	−29152.4			−0.680352
$208$		0			0
$209$	−27547.2	−	37154.7i	−0.630645	−	0.850592i
$210$		0			0
$211$			40515.6i			0.910033i	0.890483	+	0.455017i	$0.150367\pi$
$211$			40515.6i			0.910033i	−0.890483	+	0.455017i	$0.849633\pi$
$212$		0			0
$213$	5029.97			0.110868
$214$		0			0
$215$	13987.9			0.302605
$216$		0			0
$217$		−	9158.06i		−	0.194484i
$218$		0			0
$219$			22328.7i			0.465560i
$220$		0			0
$221$			2802.82i			0.0573867i
$222$		0			0
$223$		−	39758.3i		−	0.799499i	−0.916624	−	0.399749i	$-0.869097\pi$
$223$		−	39758.3i		−	0.799499i	0.916624	−	0.399749i	$-0.130903\pi$
$224$		0			0
$225$	−36891.5			−0.728722
$226$		0			0
$227$		−	22114.8i		−	0.429172i	−0.976705	−	0.214586i	$-0.931160\pi$
$227$		−	22114.8i		−	0.429172i	0.976705	−	0.214586i	$-0.0688403\pi$
$228$		0			0
$229$	−39748.2			−0.757961			−0.378980	−	0.925405i	$-0.623725\pi$
$229$	−39748.2			−0.757961			−0.378980	+	0.925405i	$0.623725\pi$
$230$		0			0
$231$		−	4557.28i		−	0.0854048i
$232$		0			0
$233$	−32762.2			−0.603477			−0.301739	−	0.953391i	$-0.597567\pi$
$233$	−32762.2			−0.603477			−0.301739	+	0.953391i	$0.597567\pi$
$234$		0			0
$235$	−9537.43			−0.172701
$236$		0			0
$237$	14619.8			0.260283
$238$		0			0
$239$	−54843.2			−0.960123			−0.480062	−	0.877235i	$-0.659386\pi$
$239$	−54843.2			−0.960123			−0.480062	+	0.877235i	$0.659386\pi$
$240$		0			0
$241$			61855.8i			1.06499i	0.846432	+	0.532496i	$0.178746\pi$
$241$			61855.8i			1.06499i	−0.846432	+	0.532496i	$0.821254\pi$
$242$		0			0
$243$		−	60169.7i		−	1.01898i
$244$		0			0
$245$	14517.3			0.241855
$246$		0			0
$247$	29525.9	−	21891.1i	0.483960	−	0.358817i
$248$		0			0
$249$			30603.5i			0.493597i
$250$		0			0
$251$	40940.2			0.649835			0.324917	−	0.945742i	$-0.394664\pi$
$251$	40940.2			0.649835			0.324917	+	0.945742i	$0.394664\pi$
$252$		0			0
$253$	59351.7			0.927240
$254$		0			0
$255$		−	728.749i		−	0.0112072i
$256$		0			0
$257$			80245.9i			1.21494i	0.794341	+	0.607472i	$0.207816\pi$
$257$			80245.9i			1.21494i	−0.794341	+	0.607472i	$0.792184\pi$
$258$		0			0
$259$		−	15553.9i		−	0.231867i
$260$		0			0
$261$		−	71270.3i		−	1.04623i
$262$		0			0
$263$	−1735.04			−0.0250841			−0.0125421	−	0.999921i	$-0.503992\pi$
$263$	−1735.04			−0.0250841			−0.0125421	+	0.999921i	$0.503992\pi$
$264$		0			0
$265$		−	5327.64i		−	0.0758652i
$266$		0			0
$267$	45274.4			0.635083
$268$		0			0
$269$		−	71811.0i		−	0.992399i	−0.868209	−	0.496199i	$-0.834729\pi$
$269$		−	71811.0i		−	0.992399i	0.868209	−	0.496199i	$-0.165271\pi$
$270$		0			0
$271$	−87030.1			−1.18503			−0.592517	−	0.805558i	$-0.701866\pi$
$271$	−87030.1			−1.18503			−0.592517	+	0.805558i	$0.701866\pi$
$272$		0			0
$273$	3621.56			0.0485926
$274$		0			0
$275$	75107.8			0.993162
$276$		0			0
$277$	−24386.7			−0.317829			−0.158915	−	0.987292i	$-0.550799\pi$
$277$	−24386.7			−0.317829			−0.158915	+	0.987292i	$0.550799\pi$
$278$		0			0
$279$			68873.8i			0.884801i
$280$		0			0
$281$			41279.7i			0.522786i	0.965232	+	0.261393i	$0.0841819\pi$
$281$			41279.7i			0.522786i	−0.965232	+	0.261393i	$0.915818\pi$
$282$		0			0
$283$	−104967.			−1.31063			−0.655314	−	0.755357i	$-0.727464\pi$
$283$	−104967.			−1.31063			−0.655314	+	0.755357i	$0.727464\pi$
$284$		0			0
$285$	−7676.90	+	5691.80i	−0.0945140	+	0.0700745i
$286$		0			0
$287$		−	21092.4i		−	0.256072i
$288$		0			0
$289$	−82763.2			−0.990927
$290$		0			0
$291$	−13713.8			−0.161947
$292$		0			0
$293$		−	12502.2i		−	0.145630i	−0.997345	−	0.0728148i	$-0.976802\pi$
$293$		−	12502.2i		−	0.145630i	0.997345	−	0.0728148i	$-0.0231982\pi$
$294$		0			0
$295$		−	16594.1i		−	0.190683i
$296$		0			0
$297$			78386.7i			0.888648i
$298$		0			0
$299$			47165.3i			0.527570i
$300$		0			0
$301$	18794.3			0.207440
$302$		0			0
$303$			57920.3i			0.630878i
$304$		0			0
$305$	4303.62			0.0462631
$306$		0			0
$307$			73242.9i			0.777122i	0.921423	+	0.388561i	$0.127028\pi$
$307$			73242.9i			0.777122i	−0.921423	+	0.388561i	$0.872972\pi$
$308$		0			0
$309$	82473.9			0.863773
$310$		0			0
$311$	−18640.6			−0.192725			−0.0963627	−	0.995346i	$-0.530721\pi$
$311$	−18640.6			−0.192725			−0.0963627	+	0.995346i	$0.530721\pi$
$312$		0			0
$313$	−148964.			−1.52052			−0.760262	−	0.649616i	$-0.774930\pi$
$313$	−148964.			−1.52052			−0.760262	+	0.649616i	$0.774930\pi$
$314$		0			0
$315$	3279.76			0.0330537
$316$		0			0
$317$		−	10951.5i		−	0.108982i	−0.998514	−	0.0544911i	$-0.982646\pi$
$317$		−	10951.5i		−	0.108982i	0.998514	−	0.0544911i	$-0.0173537\pi$
$318$		0			0
$319$			145100.i			1.42589i
$320$		0			0
$321$	−88602.8			−0.859879
$322$		0			0
$323$	−5918.64	−	7982.86i	−0.0567306	−	0.0765162i
$324$		0			0
$325$			59686.4i			0.565078i
$326$		0			0
$327$	−19396.2			−0.181393
$328$		0			0
$329$	−12814.6			−0.118389
$330$		0			0
$331$			142816.i			1.30353i	0.758420	+	0.651766i	$0.225972\pi$
$331$			142816.i			1.30353i	−0.758420	+	0.651766i	$0.774028\pi$
$332$		0			0
$333$			116974.i			1.05488i
$334$		0			0
$335$		−	18674.1i		−	0.166399i
$336$		0			0
$337$			80979.6i			0.713043i	0.934287	+	0.356521i	$0.116037\pi$
$337$			80979.6i			0.713043i	−0.934287	+	0.356521i	$0.883963\pi$
$338$		0			0
$339$	−1567.46			−0.0136395
$340$		0			0
$341$		−	140221.i		−	1.20588i
$342$		0			0
$343$	39597.1			0.336570
$344$		0			0
$345$		−	12263.2i		−	0.103031i
$346$		0			0
$347$	−155079.			−1.28794			−0.643969	−	0.765052i	$-0.722713\pi$
$347$	−155079.			−1.28794			−0.643969	+	0.765052i	$0.722713\pi$
$348$		0			0
$349$	122697.			1.00736			0.503680	−	0.863890i	$-0.331979\pi$
$349$	122697.			1.00736			0.503680	+	0.863890i	$0.331979\pi$
$350$		0			0
$351$	−62292.0			−0.505613
$352$		0			0
$353$	192560.			1.54532			0.772658	−	0.634822i	$-0.218927\pi$
$353$	192560.			1.54532			0.772658	+	0.634822i	$0.218927\pi$
$354$		0			0
$355$		−	7369.87i		−	0.0584794i
$356$		0			0
$357$		−	979.154i		−	0.00768271i
$358$		0			0
$359$	−69072.7			−0.535942			−0.267971	−	0.963427i	$-0.586353\pi$
$359$	−69072.7			−0.535942			−0.267971	+	0.963427i	$0.586353\pi$
$360$		0			0
$361$	−37867.5	+	124698.i	−0.290571	+	0.956854i
$362$		0			0
$363$		−	7543.90i		−	0.0572509i
$364$		0			0
$365$	32715.8			0.245568
$366$		0			0
$367$	180912.			1.34319			0.671593	−	0.740920i	$-0.265610\pi$
$367$	180912.			1.34319			0.671593	+	0.740920i	$0.265610\pi$
$368$		0			0
$369$			158627.i			1.16499i
$370$		0			0
$371$		−	7158.25i		−	0.0520067i
$372$		0			0
$373$		−	171731.i		−	1.23433i	−0.786835	−	0.617163i	$-0.788282\pi$
$373$		−	171731.i		−	1.23433i	0.786835	−	0.617163i	$-0.211718\pi$
$374$		0			0
$375$		−	32064.4i		−	0.228013i
$376$		0			0
$377$	−115307.			−0.811286
$378$		0			0
$379$		−	95123.2i		−	0.662229i	−0.943591	−	0.331114i	$-0.892575\pi$
$379$		−	95123.2i		−	0.662229i	0.943591	−	0.331114i	$-0.107425\pi$
$380$		0			0
$381$	−98372.7			−0.677680
$382$		0			0
$383$			20992.7i			0.143110i	0.997437	+	0.0715552i	$0.0227962\pi$
$383$			20992.7i			0.143110i	−0.997437	+	0.0715552i	$0.977204\pi$
$384$		0			0
$385$	−6677.29			−0.0450483
$386$		0			0
$387$	−141344.			−0.943745
$388$		0			0
$389$	51020.3			0.337166			0.168583	−	0.985687i	$-0.446081\pi$
$389$	51020.3			0.337166			0.168583	+	0.985687i	$0.446081\pi$
$390$		0			0
$391$	12752.0			0.0834112
$392$		0			0
$393$		−	76134.8i		−	0.492944i
$394$		0			0
$395$		−	21420.8i		−	0.137291i
$396$		0			0
$397$	−204149.			−1.29529			−0.647645	−	0.761943i	$-0.724246\pi$
$397$	−204149.			−1.29529			−0.647645	+	0.761943i	$0.724246\pi$
$398$		0			0
$399$	−10314.7	+	7647.55i	−0.0647907	+	0.0480371i
$400$		0			0
$401$			92725.5i			0.576648i	0.957533	+	0.288324i	$0.0930980\pi$
$401$			92725.5i			0.576648i	−0.957533	+	0.288324i	$0.906902\pi$
$402$		0			0
$403$	111430.			0.686108
$404$		0			0
$405$	−15550.9			−0.0948084
$406$		0			0
$407$		−	238149.i		−	1.43767i
$408$		0			0
$409$		−	49018.9i		−	0.293033i	−0.989208	−	0.146517i	$-0.953194\pi$
$409$		−	49018.9i		−	0.293033i	0.989208	−	0.146517i	$-0.0468062\pi$
$410$		0			0
$411$			138489.i			0.819842i
$412$		0			0
$413$		−	22296.0i		−	0.130716i
$414$		0			0
$415$	44839.9			0.260357
$416$		0			0
$417$			75135.8i			0.432090i
$418$		0			0
$419$	19020.8			0.108343			0.0541716	−	0.998532i	$-0.482748\pi$
$419$	19020.8			0.108343			0.0541716	+	0.998532i	$0.482748\pi$
$420$		0			0
$421$			311710.i			1.75868i	0.476196	+	0.879339i	$0.342015\pi$
$421$			311710.i			1.75868i	−0.476196	+	0.879339i	$0.657985\pi$
$422$		0			0
$423$	96372.9			0.538610
$424$		0			0
$425$	16137.3			0.0893413
$426$		0			0
$427$	5782.38			0.0317140
$428$		0			0
$429$	55450.5			0.301294
$430$		0			0
$431$			285443.i			1.53661i	0.640083	+	0.768306i	$0.278900\pi$
$431$			285443.i			1.53661i	−0.640083	+	0.768306i	$0.721100\pi$
$432$		0			0
$433$		−	168848.i		−	0.900573i	−0.892884	−	0.450287i	$-0.851322\pi$
$433$		−	168848.i		−	0.900573i	0.892884	−	0.450287i	$-0.148678\pi$
$434$		0			0
$435$	29980.5			0.158439
$436$		0			0
$437$	−99597.7	−	134334.i	−0.521539	−	0.703433i
$438$		0			0
$439$			18598.0i			0.0965023i	0.998835	+	0.0482512i	$0.0153648\pi$
$439$			18598.0i			0.0965023i	−0.998835	+	0.0482512i	$0.984635\pi$
$440$		0			0
$441$	−146693.			−0.754280
$442$		0			0
$443$	13329.7			0.0679223			0.0339612	−	0.999423i	$-0.489188\pi$
$443$	13329.7			0.0679223			0.0339612	+	0.999423i	$0.489188\pi$
$444$		0			0
$445$		−	66335.7i		−	0.334986i
$446$		0			0
$447$		−	123461.i		−	0.617895i
$448$		0			0
$449$		−	303597.i		−	1.50593i	−0.658060	−	0.752965i	$-0.728623\pi$
$449$		−	303597.i		−	1.50593i	0.658060	−	0.752965i	$-0.271377\pi$
$450$		0			0
$451$		−	322950.i		−	1.58775i
$452$		0			0
$453$	−91700.9			−0.446866
$454$		0			0
$455$		−	5306.28i		−	0.0256311i
$456$		0			0
$457$	5953.52			0.0285064			0.0142532	−	0.999898i	$-0.495463\pi$
$457$	5953.52			0.0285064			0.0142532	+	0.999898i	$0.495463\pi$
$458$		0			0
$459$			16841.8i			0.0799396i
$460$		0			0
$461$	347201.			1.63373			0.816864	−	0.576830i	$-0.195711\pi$
$461$	347201.			1.63373			0.816864	+	0.576830i	$0.195711\pi$
$462$		0			0
$463$	−198439.			−0.925690			−0.462845	−	0.886439i	$-0.653171\pi$
$463$	−198439.			−0.925690			−0.462845	+	0.886439i	$0.653171\pi$
$464$		0			0
$465$	−28972.4			−0.133992
$466$		0			0
$467$	234121.			1.07351			0.536755	−	0.843738i	$-0.319650\pi$
$467$	234121.			1.07351			0.536755	+	0.843738i	$0.319650\pi$
$468$		0			0
$469$		−	25090.7i		−	0.114069i
$470$		0			0
$471$		−	110313.i		−	0.497263i
$472$		0			0
$473$	287763.			1.28621
$474$		0			0
$475$	−126038.	−	169996.i	−0.558617	−	0.753443i
$476$		0			0
$477$			53834.2i			0.236604i
$478$		0			0
$479$	−81975.6			−0.357284			−0.178642	−	0.983914i	$-0.557170\pi$
$479$	−81975.6			−0.357284			−0.178642	+	0.983914i	$0.557170\pi$
$480$		0			0
$481$	189251.			0.817990
$482$		0			0
$483$		−	16477.0i		−	0.0706291i
$484$		0			0
$485$			20093.3i			0.0854218i
$486$		0			0
$487$			115801.i			0.488265i	0.969742	+	0.244132i	$0.0785031\pi$
$487$			115801.i			0.488265i	−0.969742	+	0.244132i	$0.921497\pi$
$488$		0			0
$489$		−	2650.78i		−	0.0110855i
$490$		0			0
$491$	−200845.			−0.833102			−0.416551	−	0.909112i	$-0.636761\pi$
$491$	−200845.			−0.833102			−0.416551	+	0.909112i	$0.636761\pi$
$492$		0			0
$493$			31175.4i			0.128268i
$494$		0			0
$495$	50217.0			0.204947
$496$		0			0
$497$		−	9902.22i		−	0.0400885i
$498$		0			0
$499$	−51620.4			−0.207310			−0.103655	−	0.994613i	$-0.533054\pi$
$499$	−51620.4			−0.207310			−0.103655	+	0.994613i	$0.533054\pi$
$500$		0			0
$501$	−78155.9			−0.311377
$502$		0			0
$503$	442820.			1.75022			0.875108	−	0.483928i	$-0.160790\pi$
$503$	442820.			1.75022			0.875108	+	0.483928i	$0.160790\pi$
$504$		0			0
$505$	84864.2			0.332768
$506$		0			0
$507$		−	77337.4i		−	0.300866i
$508$		0			0
$509$		−	207679.i		−	0.801599i	−0.916166	−	0.400799i	$-0.868732\pi$
$509$		−	207679.i		−	0.801599i	0.916166	−	0.400799i	$-0.131268\pi$
$510$		0			0
$511$	43957.3			0.168341
$512$		0			0
$513$	177417.	−	131540.i	0.674156	−	0.499832i
$514$		0			0
$515$		−	120840.i		−	0.455613i
$516$		0			0
$517$	−196207.			−0.734061
$518$		0			0
$519$	−115909.			−0.430312
$520$		0			0
$521$		−	169750.i		−	0.625367i	−0.949857	−	0.312684i	$-0.898772\pi$
$521$		−	169750.i		−	0.625367i	0.949857	−	0.312684i	$-0.101228\pi$
$522$		0			0
$523$			67720.2i			0.247579i	0.992308	+	0.123790i	$0.0395048\pi$
$523$			67720.2i			0.247579i	−0.992308	+	0.123790i	$0.960495\pi$
$524$		0			0
$525$		−	20851.2i		−	0.0756505i
$526$		0			0
$527$		−	30127.1i		−	0.108477i
$528$		0			0
$529$	−65253.2			−0.233180
$530$		0			0
$531$			167679.i			0.594688i
$532$		0			0
$533$	256640.			0.903379
$534$		0			0
$535$			129820.i			0.453559i
$536$		0			0
$537$	−175881.			−0.609917
$538$		0			0
$539$	298654.			1.02800
$540$		0			0
$541$	498611.			1.70360			0.851800	−	0.523867i	$-0.175511\pi$
$541$	498611.			1.70360			0.851800	+	0.523867i	$0.175511\pi$
$542$		0			0
$543$	−260762.			−0.884393
$544$		0			0
$545$			28419.1i			0.0956793i
$546$		0			0
$547$			223692.i			0.747612i	0.927507	+	0.373806i	$0.121947\pi$
$547$			223692.i			0.747612i	−0.927507	+	0.373806i	$0.878053\pi$
$548$		0			0
$549$	−43486.8			−0.144282
$550$		0			0
$551$	328412.	−	243491.i	1.08172	−	0.802010i
$552$		0			0
$553$		−	28781.2i		−	0.0941150i
$554$		0			0
$555$	−49206.3			−0.159748
$556$		0			0
$557$	−127998.			−0.412565			−0.206282	−	0.978492i	$-0.566137\pi$
$557$	−127998.			−0.412565			−0.206282	+	0.978492i	$0.566137\pi$
$558$		0			0
$559$			228678.i			0.731815i
$560$		0			0
$561$		−	14992.0i		−	0.0476359i
$562$		0			0
$563$		−	411175.i		−	1.29721i	−0.761126	−	0.648604i	$-0.775353\pi$
$563$		−	411175.i		−	1.29721i	0.761126	−	0.648604i	$-0.224647\pi$
$564$		0			0
$565$			2296.63i			0.00719438i
$566$		0			0
$567$	−20894.4			−0.0649925
$568$		0			0
$569$		−	233842.i		−	0.722268i	−0.932514	−	0.361134i	$-0.882390\pi$
$569$		−	233842.i		−	0.722268i	0.932514	−	0.361134i	$-0.117610\pi$
$570$		0			0
$571$	227761.			0.698567			0.349283	−	0.937017i	$-0.386425\pi$
$571$	227761.			0.698567			0.349283	+	0.937017i	$0.386425\pi$
$572$		0			0
$573$		−	196899.i		−	0.599701i
$574$		0			0
$575$	271555.			0.821337
$576$		0			0
$577$	−79801.9			−0.239696			−0.119848	−	0.992792i	$-0.538241\pi$
$577$	−79801.9			−0.239696			−0.119848	+	0.992792i	$0.538241\pi$
$578$		0			0
$579$	189703.			0.565870
$580$		0			0
$581$	60247.3			0.178478
$582$		0			0
$583$		−	109602.i		−	0.322463i
$584$		0			0
$585$			39906.2i			0.116608i
$586$		0			0
$587$	−104327.			−0.302775			−0.151388	−	0.988474i	$-0.548374\pi$
$587$	−104327.			−0.302775			−0.151388	+	0.988474i	$0.548374\pi$
$588$		0			0
$589$	−317370.	+	235304.i	−0.914818	+	0.678263i
$590$		0			0
$591$		−	168623.i		−	0.482772i
$592$		0			0
$593$	98337.5			0.279647			0.139823	−	0.990176i	$-0.455347\pi$
$593$	98337.5			0.279647			0.139823	+	0.990176i	$0.455347\pi$
$594$		0			0
$595$	−1434.65			−0.00405239
$596$		0			0
$597$			25134.3i			0.0705210i
$598$		0			0
$599$		−	336387.i		−	0.937531i	−0.883323	−	0.468766i	$-0.844699\pi$
$599$		−	336387.i		−	0.937531i	0.883323	−	0.468766i	$-0.155301\pi$
$600$		0			0
$601$			352338.i			0.975464i	0.872994	+	0.487732i	$0.162176\pi$
$601$			352338.i			0.975464i	−0.872994	+	0.487732i	$0.837824\pi$
$602$		0			0
$603$			188696.i			0.518954i
$604$		0			0
$605$	−11053.2			−0.0301981
$606$		0			0
$607$		−	154178.i		−	0.418450i	−0.977868	−	0.209225i	$-0.932906\pi$
$607$		−	154178.i		−	0.418450i	0.977868	−	0.209225i	$-0.0670941\pi$
$608$		0			0
$609$	40282.1			0.108612
$610$		0			0
$611$		−	155921.i		−	0.417658i
$612$		0			0
$613$	−337444.			−0.898010			−0.449005	−	0.893529i	$-0.648221\pi$
$613$	−337444.			−0.898010			−0.449005	+	0.893529i	$0.648221\pi$
$614$		0			0
$615$	−66727.8			−0.176424
$616$		0			0
$617$	250707.			0.658561			0.329281	−	0.944232i	$-0.393194\pi$
$617$	250707.			0.658561			0.329281	+	0.944232i	$0.393194\pi$
$618$		0			0
$619$	103827.			0.270974			0.135487	−	0.990779i	$-0.456740\pi$
$619$	103827.			0.270974			0.135487	+	0.990779i	$0.456740\pi$
$620$		0			0
$621$			283409.i			0.734905i
$622$		0			0
$623$		−	89129.2i		−	0.229638i
$624$		0			0
$625$	319402.			0.817669
$626$		0			0
$627$	−157931.	+	117093.i	−0.401729	+	0.297849i
$628$		0			0
$629$		−	51167.4i		−	0.129328i
$630$		0			0
$631$	−627818.			−1.57680			−0.788398	−	0.615166i	$-0.789089\pi$
$631$	−627818.			−1.57680			−0.788398	+	0.615166i	$0.789089\pi$
$632$		0			0
$633$	172217.			0.429803
$634$		0			0
$635$			144135.i			0.357455i
$636$		0			0
$637$			237333.i			0.584897i
$638$		0			0
$639$			74470.3i			0.182382i
$640$		0			0
$641$		−	346575.i		−	0.843492i	−0.906714	−	0.421746i	$-0.861417\pi$
$641$		−	346575.i		−	0.843492i	0.906714	−	0.421746i	$-0.138583\pi$
$642$		0			0
$643$	−198701.			−0.480594			−0.240297	−	0.970699i	$-0.577245\pi$
$643$	−198701.			−0.480594			−0.240297	+	0.970699i	$0.577245\pi$
$644$		0			0
$645$		−	59457.6i		−	0.142918i
$646$		0			0
$647$	243382.			0.581407			0.290704	−	0.956813i	$-0.406111\pi$
$647$	243382.			0.581407			0.290704	+	0.956813i	$0.406111\pi$
$648$		0			0
$649$		−	341379.i		−	0.810490i
$650$		0			0
$651$	−38927.6			−0.0918535
$652$		0			0
$653$	−535192.			−1.25511			−0.627557	−	0.778570i	$-0.715945\pi$
$653$	−535192.			−1.25511			−0.627557	+	0.778570i	$0.715945\pi$
$654$		0			0
$655$	−111552.			−0.260013
$656$		0			0
$657$	−330584.			−0.765862
$658$		0			0
$659$		−	194311.i		−	0.447430i	−0.974655	−	0.223715i	$-0.928181\pi$
$659$		−	194311.i		−	0.447430i	0.974655	−	0.223715i	$-0.0718186\pi$
$660$		0			0
$661$			637741.i			1.45963i	0.683647	+	0.729813i	$0.260393\pi$
$661$			637741.i			1.45963i	−0.683647	+	0.729813i	$0.739607\pi$
$662$		0			0
$663$	11913.8			0.0271034
$664$		0			0
$665$	11205.1	+	15113.1i	0.0253380	+	0.0341751i
$666$		0			0
$667$			524613.i			1.17920i
$668$		0			0
$669$	−168998.			−0.377598
$670$		0			0
$671$	88535.3			0.196640
$672$		0			0
$673$			162571.i			0.358933i	0.983764	+	0.179466i	$0.0574371\pi$
$673$			162571.i			0.358933i	−0.983764	+	0.179466i	$0.942563\pi$
$674$		0			0
$675$			358647.i			0.787153i
$676$		0			0
$677$			65775.3i			0.143511i	0.997422	+	0.0717555i	$0.0228601\pi$
$677$			65775.3i			0.143511i	−0.997422	+	0.0717555i	$0.977140\pi$
$678$		0			0
$679$			26997.6i			0.0585579i
$680$		0			0
$681$	−94002.1			−0.202695
$682$		0			0
$683$		−	254941.i		−	0.546510i	−0.961942	−	0.273255i	$-0.911900\pi$
$683$		−	254941.i		−	0.546510i	0.961942	−	0.273255i	$-0.0881003\pi$
$684$		0			0
$685$	202912.			0.432441
$686$		0			0
$687$			168955.i			0.357980i
$688$		0			0
$689$	87097.6			0.183471
$690$		0			0
$691$	−265734.			−0.556534			−0.278267	−	0.960504i	$-0.589760\pi$
$691$	−265734.			−0.556534			−0.278267	+	0.960504i	$0.589760\pi$
$692$		0			0
$693$	67472.0			0.140494
$694$		0			0
$695$	110088.			0.227914
$696$		0			0
$697$		−	69387.2i		−	0.142828i
$698$		0			0
$699$			139260.i			0.285018i
$700$		0			0
$701$	−538778.			−1.09641			−0.548206	−	0.836343i	$-0.684689\pi$
$701$	−538778.			−1.09641			−0.548206	+	0.836343i	$0.684689\pi$
$702$		0			0
$703$	−539015.	+	399636.i	−1.09066	+	0.808638i
$704$		0			0
$705$			40540.2i			0.0815657i
$706$		0			0
$707$	114024.			0.228117
$708$		0			0
$709$	212545.			0.422824			0.211412	−	0.977397i	$-0.432194\pi$
$709$	212545.			0.422824			0.211412	+	0.977397i	$0.432194\pi$
$710$		0			0
$711$			216451.i			0.428175i
$712$		0			0
$713$		−	506973.i		−	0.997254i
$714$		0			0
$715$		−	81245.5i		−	0.158923i
$716$		0			0
$717$			233119.i			0.453460i
$718$		0			0
$719$	−708265.			−1.37006			−0.685028	−	0.728517i	$-0.740210\pi$
$719$	−708265.			−1.37006			−0.685028	+	0.728517i	$0.740210\pi$
$720$		0			0
$721$		−	162362.i		−	0.312329i
$722$		0			0
$723$	262927.			0.502989
$724$		0			0
$725$			663882.i			1.26303i
$726$		0			0
$727$	148970.			0.281857			0.140929	−	0.990020i	$-0.454991\pi$
$727$	148970.			0.281857			0.140929	+	0.990020i	$0.454991\pi$
$728$		0			0
$729$	−53507.4			−0.100684
$730$		0			0
$731$	61827.2			0.115703
$732$		0			0
$733$	−558167.			−1.03886			−0.519429	−	0.854514i	$-0.673855\pi$
$733$	−558167.			−1.03886			−0.519429	+	0.854514i	$0.673855\pi$
$734$		0			0
$735$		−	61707.9i		−	0.114226i
$736$		0			0
$737$		−	384169.i		−	0.707273i
$738$		0			0
$739$	204028.			0.373594			0.186797	−	0.982399i	$-0.440189\pi$
$739$	204028.			0.373594			0.186797	+	0.982399i	$0.440189\pi$
$740$		0			0
$741$	−93051.1	−	125504.i	−0.169467	−	0.228571i
$742$		0			0
$743$			306386.i			0.554997i	0.960726	+	0.277499i	$0.0895054\pi$
$743$			306386.i			0.554997i	−0.960726	+	0.277499i	$0.910495\pi$
$744$		0			0
$745$	−180894.			−0.325920
$746$		0			0
$747$	−453094.			−0.811984
$748$		0			0
$749$			174427.i			0.310921i
$750$		0			0
$751$			1.08343e6i			1.92098i	0.278319	+	0.960489i	$0.410223\pi$
$751$			1.08343e6i			1.92098i	−0.278319	+	0.960489i	$0.589777\pi$
$752$		0			0
$753$		−	174022.i		−	0.306913i
$754$		0			0
$755$			134359.i			0.235708i
$756$		0			0
$757$	−360494.			−0.629081			−0.314541	−	0.949244i	$-0.601851\pi$
$757$	−360494.			−0.629081			−0.314541	+	0.949244i	$0.601851\pi$
$758$		0			0
$759$		−	252283.i		−	0.437929i
$760$		0			0
$761$	985986.			1.70256			0.851278	−	0.524715i	$-0.175828\pi$
$761$	985986.			1.70256			0.851278	+	0.524715i	$0.175828\pi$
$762$		0			0
$763$			38184.2i			0.0655895i
$764$		0			0
$765$	10789.4			0.0184363
$766$		0			0
$767$	271286.			0.461144
$768$		0			0
$769$	300892.			0.508813			0.254406	−	0.967097i	$-0.418120\pi$
$769$	300892.			0.508813			0.254406	+	0.967097i	$0.418120\pi$
$770$		0			0
$771$	341096.			0.573810
$772$		0			0
$773$			391472.i			0.655152i	0.944825	+	0.327576i	$0.106232\pi$
$773$			391472.i			0.655152i	−0.944825	+	0.327576i	$0.893768\pi$
$774$		0			0
$775$		−	641560.i		−	1.06815i
$776$		0			0
$777$	−66114.0			−0.109509
$778$		0			0
$779$	−730949.	+	541940.i	−1.20452	+	0.893051i
$780$		0			0
$781$		−	151615.i		−	0.248565i
$782$		0			0
$783$	−692865.			−1.13012
$784$		0			0
$785$	−161630.			−0.262291
$786$		0			0
$787$		−	977441.i		−	1.57812i	−0.614314	−	0.789062i	$-0.710567\pi$
$787$		−	977441.i		−	1.57812i	0.614314	−	0.789062i	$-0.289433\pi$
$788$		0			0
$789$			7375.05i			0.0118471i
$790$		0			0
$791$			3085.77i			0.00493185i
$792$		0			0
$793$			70356.8i			0.111882i
$794$		0			0
$795$	−22645.9			−0.0358306
$796$		0			0
$797$			1.05182e6i			1.65586i	0.560828	+	0.827932i	$0.310483\pi$
$797$			1.05182e6i			1.65586i	−0.560828	+	0.827932i	$0.689517\pi$
$798$		0			0
$799$	−42155.9			−0.0660336
$800$		0			0
$801$			670302.i			1.04473i
$802$		0			0
$803$	673039.			1.04378
$804$		0			0
$805$	−24141.9			−0.0372546
$806$		0			0
$807$	−305243.			−0.468703
$808$		0			0
$809$	748664.			1.14391			0.571953	−	0.820287i	$-0.306186\pi$
$809$	748664.			1.14391			0.571953	+	0.820287i	$0.306186\pi$
$810$		0			0
$811$			1.22865e6i			1.86804i	0.357221	+	0.934020i	$0.383724\pi$
$811$			1.22865e6i			1.86804i	−0.357221	+	0.934020i	$0.616276\pi$
$812$		0			0
$813$			369934.i			0.559684i
$814$		0			0
$815$	−3883.90			−0.00584727
$816$		0			0
$817$	−482893.	−	651310.i	−0.723448	−	0.975761i
$818$		0			0
$819$			53618.3i			0.0799366i
$820$		0			0
$821$	−481515.			−0.714370			−0.357185	−	0.934034i	$-0.616263\pi$
$821$	−481515.			−0.714370			−0.357185	+	0.934034i	$0.616263\pi$
$822$		0			0
$823$	1.29558e6			1.91278			0.956389	−	0.292097i	$-0.0943531\pi$
$823$	1.29558e6			1.91278			0.956389	+	0.292097i	$0.0943531\pi$
$824$		0			0
$825$		−	319256.i		−	0.469064i
$826$		0			0
$827$		−	223200.i		−	0.326350i	−0.986597	−	0.163175i	$-0.947827\pi$
$827$		−	223200.i		−	0.326350i	0.986597	−	0.163175i	$-0.0521734\pi$
$828$		0			0
$829$			902881.i			1.31378i	0.753988	+	0.656888i	$0.228128\pi$
$829$			902881.i			1.31378i	−0.753988	+	0.656888i	$0.771872\pi$
$830$		0			0
$831$			103659.i			0.150109i
$832$		0			0
$833$	64167.2			0.0924748
$834$		0			0
$835$			114513.i			0.164242i
$836$		0			0
$837$	669567.			0.955748
$838$		0			0
$839$		−	565904.i		−	0.803932i	−0.915655	−	0.401966i	$-0.868327\pi$
$839$		−	565904.i		−	0.803932i	0.915655	−	0.401966i	$-0.131673\pi$
$840$		0			0
$841$	−575265.			−0.813347
$842$		0			0
$843$	175465.			0.246908
$844$		0			0
$845$	−113314.			−0.158698
$846$		0			0
$847$	−14851.2			−0.0207012
$848$		0			0
$849$			446176.i			0.619001i
$850$		0			0
$851$		−	861034.i		−	1.18894i
$852$		0			0
$853$	−377911.			−0.519388			−0.259694	−	0.965691i	$-0.583622\pi$
$853$	−377911.			−0.519388			−0.259694	+	0.965691i	$0.583622\pi$
$854$		0			0
$855$	−84268.9	−	113659.i	−0.115275	−	0.155479i
$856$		0			0
$857$		−	397067.i		−	0.540632i	−0.962772	−	0.270316i	$-0.912872\pi$
$857$		−	397067.i		−	0.540632i	0.962772	−	0.270316i	$-0.0871282\pi$
$858$		0			0
$859$	−1.16166e6			−1.57432			−0.787159	−	0.616750i	$-0.788449\pi$
$859$	−1.16166e6			−1.57432			−0.787159	+	0.616750i	$0.788449\pi$
$860$		0			0
$861$	−89656.1			−0.120941
$862$		0			0
$863$			35924.8i			0.0482362i	0.999709	+	0.0241181i	$0.00767777\pi$
$863$			35924.8i			0.0482362i	−0.999709	+	0.0241181i	$0.992322\pi$
$864$		0			0
$865$			169829.i			0.226976i
$866$		0			0
$867$			351797.i			0.468008i
$868$		0			0
$869$		−	440675.i		−	0.583551i
$870$		0			0
$871$	305290.			0.402416
$872$		0			0
$873$		−	203037.i		−	0.266408i
$874$		0			0
$875$	−63123.3			−0.0824468
$876$		0			0
$877$		−	1.27993e6i		−	1.66413i	−0.554678	−	0.832065i	$-0.687158\pi$
$877$		−	1.27993e6i		−	1.66413i	0.554678	−	0.832065i	$-0.312842\pi$
$878$		0			0
$879$	−53142.2			−0.0687799
$880$		0			0
$881$	−99651.0			−0.128390			−0.0641948	−	0.997937i	$-0.520448\pi$
$881$	−99651.0			−0.128390			−0.0641948	+	0.997937i	$0.520448\pi$
$882$		0			0
$883$	472047.			0.605430			0.302715	−	0.953081i	$-0.402107\pi$
$883$	472047.			0.605430			0.302715	+	0.953081i	$0.402107\pi$
$884$		0			0
$885$	−70535.8			−0.0900581
$886$		0			0
$887$		−	1.43385e6i		−	1.82245i	−0.411910	−	0.911224i	$-0.635138\pi$
$887$		−	1.43385e6i		−	1.82245i	0.411910	−	0.911224i	$-0.364862\pi$
$888$		0			0
$889$			193661.i			0.245040i
$890$		0			0
$891$	−319918.			−0.402980
$892$		0			0
$893$	329253.	+	444085.i	0.412883	+	0.556882i
$894$		0			0
$895$			257699.i			0.321712i
$896$		0			0
$897$	200483.			0.249168
$898$		0			0
$899$	1.23942e6			1.53355
$900$		0			0
$901$		−	23548.4i		−	0.0290076i
$902$		0			0
$903$		−	79887.7i		−	0.0979725i
$904$		0			0
$905$			382067.i			0.466490i
$906$		0			0
$907$			725228.i			0.881577i	0.897611	+	0.440788i	$0.145301\pi$
$907$			725228.i			0.881577i	−0.897611	+	0.440788i	$0.854699\pi$
$908$		0			0
$909$	−857528.			−1.03782
$910$		0			0
$911$			938716.i			1.13109i	0.824717	+	0.565545i	$0.191334\pi$
$911$			938716.i			1.13109i	−0.824717	+	0.565545i	$0.808666\pi$
$912$		0			0
$913$	922459.			1.10664
$914$		0			0
$915$		−	18293.2i		−	0.0218497i
$916$		0			0
$917$	−149882.			−0.178242
$918$		0			0
$919$	781727.			0.925602			0.462801	−	0.886462i	$-0.346844\pi$
$919$	781727.			0.925602			0.462801	+	0.886462i	$0.346844\pi$
$920$		0			0
$921$	311329.			0.367029
$922$		0			0
$923$	120485.			0.141426
$924$		0			0
$925$		−	1.08961e6i		−	1.27347i
$926$		0			0
$927$			1.22105e6i			1.42094i
$928$		0			0
$929$	−1.39816e6			−1.62003			−0.810017	−	0.586406i	$-0.800542\pi$
$929$	−1.39816e6			−1.62003			−0.810017	+	0.586406i	$0.800542\pi$
$930$		0			0
$931$	−501170.	−	675960.i	−0.578210	−	0.779869i
$932$		0			0
$933$			79234.4i			0.0910229i
$934$		0			0
$935$	−21966.2			−0.0251265
$936$		0			0
$937$	1.37375e6			1.56470			0.782348	−	0.622842i	$-0.214022\pi$
$937$	1.37375e6			1.56470			0.782348	+	0.622842i	$0.214022\pi$
$938$		0			0
$939$			633194.i			0.718134i
$940$		0			0
$941$		−	732949.i		−	0.827741i	−0.910336	−	0.413870i	$-0.864177\pi$
$941$		−	732949.i		−	0.827741i	0.910336	−	0.413870i	$-0.135823\pi$
$942$		0			0
$943$		−	1.16763e6i		−	1.31306i
$944$		0			0
$945$		−	31884.6i		−	0.0357041i
$946$		0			0
$947$	−896416.			−0.999562			−0.499781	−	0.866152i	$-0.666586\pi$
$947$	−896416.			−0.999562			−0.499781	+	0.866152i	$0.666586\pi$
$948$		0			0
$949$			534847.i			0.593878i
$950$		0			0
$951$	−46551.0			−0.0514716
$952$		0			0
$953$		−	625093.i		−	0.688270i	−0.938920	−	0.344135i	$-0.888172\pi$
$953$		−	625093.i		−	0.688270i	0.938920	−	0.344135i	$-0.111828\pi$
$954$		0			0
$955$	−288495.			−0.316323
$956$		0			0
$957$	616767.			0.673438
$958$		0			0
$959$	272634.			0.296444
$960$		0			0
$961$	−274224.			−0.296933
$962$		0			0
$963$		−	1.31179e6i		−	1.41453i
$964$		0			0
$965$		−	277951.i		−	0.298479i
$966$		0			0
$967$	−778159.			−0.832177			−0.416088	−	0.909324i	$-0.636599\pi$
$967$	−778159.			−0.832177			−0.416088	+	0.909324i	$0.636599\pi$
$968$		0			0
$969$	−33932.3	+	25158.0i	−0.0361381	+	0.0267935i
$970$		0			0
$971$		−	981293.i		−	1.04078i	−0.853928	−	0.520392i	$-0.825786\pi$
$971$		−	981293.i		−	1.04078i	0.853928	−	0.520392i	$-0.174214\pi$
$972$		0			0
$973$	147915.			0.156238
$974$		0			0
$975$	253705.			0.266883
$976$		0			0
$977$		−	603058.i		−	0.631786i	−0.948795	−	0.315893i	$-0.897696\pi$
$977$		−	603058.i		−	0.631786i	0.948795	−	0.315893i	$-0.102304\pi$
$978$		0			0
$979$		−	1.36468e6i		−	1.42385i
$980$		0			0
$981$		−	287167.i		−	0.298398i
$982$		0			0
$983$		−	142961.i		−	0.147948i	−0.997260	−	0.0739740i	$-0.976432\pi$
$983$		−	142961.i		−	0.147948i	0.997260	−	0.0739740i	$-0.0235682\pi$
$984$		0			0
$985$	−247065.			−0.254647
$986$		0			0
$987$			54470.1i			0.0559144i
$988$		0			0
$989$	1.04042e6			1.06369
$990$		0			0
$991$			396267.i			0.403498i	0.979437	+	0.201749i	$0.0646624\pi$
$991$			396267.i			0.403498i	−0.979437	+	0.201749i	$0.935338\pi$
$992$		0			0
$993$	607061.			0.615650
$994$		0			0
$995$	36826.6			0.0371976
$996$		0			0
$997$	731044.			0.735451			0.367725	−	0.929934i	$-0.380137\pi$
$997$	731044.			0.735451			0.367725	+	0.929934i	$0.380137\pi$
$998$		0			0
$999$	1.13718e6			1.13946

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.5.e.c.113.2		4
4.3	odd	2		19.5.b.b.18.4	yes	4
12.11	even	2		171.5.c.c.37.1		4
19.18	odd	2	inner	304.5.e.c.113.3		4
76.75	even	2		19.5.b.b.18.1	✓	4
228.227	odd	2		171.5.c.c.37.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.5.b.b.18.1	✓	4	76.75	even	2
19.5.b.b.18.4	yes	4	4.3	odd	2
171.5.c.c.37.1		4	12.11	even	2
171.5.c.c.37.4		4	228.227	odd	2
304.5.e.c.113.2		4	1.1	even	1	trivial
304.5.e.c.113.3		4	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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	304.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.25064i q^{3} -6.22800 q^{5} -8.36799 q^{7} +62.9321 q^{9} +O(q^{10})\)	\(q-4.25064i q^{3} -6.22800 q^{5} -8.36799 q^{7} +62.9321 q^{9} -128.124 q^{11} -101.817i q^{13} +26.4730i q^{15} -27.5280 q^{17} +(215.004 + 289.990i) q^{19} +35.5693i q^{21} -463.236 q^{23} -586.212 q^{25} -611.803i q^{27} -1132.50i q^{29} +1094.42i q^{31} +544.609i q^{33} +52.1158 q^{35} +1858.74i q^{37} -432.787 q^{39} +2520.60i q^{41} -2245.97 q^{43} -391.941 q^{45} +1531.38 q^{47} -2330.98 q^{49} +117.012i q^{51} +855.433i q^{53} +797.956 q^{55} +(1232.64 - 913.905i) q^{57} +2664.44i q^{59} -691.012 q^{61} -526.615 q^{63} +634.116i q^{65} +2998.41i q^{67} +1969.05i q^{69} +1183.34i q^{71} -5253.02 q^{73} +2491.78i q^{75} +1072.14 q^{77} +3439.44i q^{79} +2496.94 q^{81} -7199.74 q^{83} +171.445 q^{85} -4813.83 q^{87} +10651.2i q^{89} +852.004i q^{91} +4651.97 q^{93} +(-1339.05 - 1806.06i) q^{95} -3226.29i q^{97} -8063.11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 42 q^{5} - 136 q^{7} - 278 q^{9}+O(q^{10}) \) 4 * q - 42 * q^5 - 136 * q^7 - 278 * q^9	\( 4 q - 42 q^{5} - 136 q^{7} - 278 q^{9} - 222 q^{11} + 300 q^{17} - 114 q^{19} + 78 q^{23} - 1986 q^{25} + 1866 q^{35} - 6362 q^{39} - 2986 q^{43} + 5182 q^{45} + 7578 q^{47} - 2352 q^{49} + 1090 q^{55} + 10450 q^{57} + 158 q^{61} + 23030 q^{63} - 15168 q^{73} + 102 q^{77} + 40712 q^{81} - 33276 q^{83} - 4902 q^{85} + 7214 q^{87} - 40004 q^{93} + 5358 q^{95} - 23042 q^{99}+O(q^{100}) \) 4 * q - 42 * q^5 - 136 * q^7 - 278 * q^9 - 222 * q^11 + 300 * q^17 - 114 * q^19 + 78 * q^23 - 1986 * q^25 + 1866 * q^35 - 6362 * q^39 - 2986 * q^43 + 5182 * q^45 + 7578 * q^47 - 2352 * q^49 + 1090 * q^55 + 10450 * q^57 + 158 * q^61 + 23030 * q^63 - 15168 * q^73 + 102 * q^77 + 40712 * q^81 - 33276 * q^83 - 4902 * q^85 + 7214 * q^87 - 40004 * q^93 + 5358 * q^95 - 23042 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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