LMFDB - Embedded newform 76.5.c.b.37.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,5,Mod(37,76)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("76.37");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	76.c (of order $2$, degree $1$, 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:	$7.85611719437$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 269x^{2} + 17592 $ x^4 + 269*x^2 + 17592
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.3
Root			$10.5914i$ of defining polynomial
Character	$\chi$	$=$	76.37
Dual form			76.5.c.b.37.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.5914i q^{3} -16.8215 q^{5} -38.6430 q^{7} -31.1785 q^{9} +O(q^{10})$	$q+10.5914i q^{3} -16.8215 q^{5} -38.6430 q^{7} -31.1785 q^{9} -70.8215 q^{11} -231.121i q^{13} -178.164i q^{15} -90.4279 q^{17} +(-319.751 + 167.573i) q^{19} -409.285i q^{21} -285.538 q^{23} -342.037 q^{25} +527.681i q^{27} +285.969i q^{29} +852.235i q^{31} -750.101i q^{33} +650.034 q^{35} -54.8476i q^{37} +2447.90 q^{39} -37.4456i q^{41} +691.677 q^{43} +524.469 q^{45} +2630.47 q^{47} -907.716 q^{49} -957.761i q^{51} +3833.35i q^{53} +1191.33 q^{55} +(-1774.83 - 3386.62i) q^{57} +3339.72i q^{59} -2877.84 q^{61} +1204.83 q^{63} +3887.81i q^{65} +4052.74i q^{67} -3024.26i q^{69} -3315.87i q^{71} -5544.37 q^{73} -3622.66i q^{75} +2736.76 q^{77} -10304.4i q^{79} -8114.36 q^{81} -3541.95 q^{83} +1521.13 q^{85} -3028.82 q^{87} -13273.7i q^{89} +8931.22i q^{91} -9026.39 q^{93} +(5378.69 - 2818.82i) q^{95} -4281.99i q^{97} +2208.11 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 22 q^{5} + 24 q^{7} - 214 q^{9}+O(q^{10}) $ 4 * q + 22 * q^5 + 24 * q^7 - 214 * q^9	$ 4 q + 22 q^{5} + 24 q^{7} - 214 q^{9} - 194 q^{11} - 1076 q^{17} - 654 q^{19} + 1090 q^{23} - 386 q^{25} + 4118 q^{35} - 2262 q^{39} + 4106 q^{43} - 3170 q^{45} + 5254 q^{47} - 1488 q^{49} + 926 q^{55} + 5490 q^{57} + 6078 q^{61} - 5270 q^{63} - 9856 q^{73} + 2822 q^{77} - 30136 q^{81} + 3868 q^{83} - 21862 q^{85} - 14526 q^{87} - 1284 q^{93} + 10354 q^{95} + 8386 q^{99}+O(q^{100}) $ 4 * q + 22 * q^5 + 24 * q^7 - 214 * q^9 - 194 * q^11 - 1076 * q^17 - 654 * q^19 + 1090 * q^23 - 386 * q^25 + 4118 * q^35 - 2262 * q^39 + 4106 * q^43 - 3170 * q^45 + 5254 * q^47 - 1488 * q^49 + 926 * q^55 + 5490 * q^57 + 6078 * q^61 - 5270 * q^63 - 9856 * q^73 + 2822 * q^77 - 30136 * q^81 + 3868 * q^83 - 21862 * q^85 - 14526 * q^87 - 1284 * q^93 + 10354 * q^95 + 8386 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$-1$	$1$

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
$3$			10.5914i			1.17683i	0.808561	+	0.588413i	$0.200247\pi$
$3$			10.5914i			1.17683i	−0.808561	+	0.588413i	$0.799753\pi$
$4$		0			0
$5$	−16.8215			−0.672861			−0.336430	−	0.941708i	$-0.609220\pi$
$5$	−16.8215			−0.672861			−0.336430	+	0.941708i	$0.609220\pi$
$6$		0			0
$7$	−38.6430			−0.788633			−0.394317	−	0.918975i	$-0.629019\pi$
$7$	−38.6430			−0.788633			−0.394317	+	0.918975i	$0.629019\pi$
$8$		0			0
$9$	−31.1785			−0.384920
$10$		0			0
$11$	−70.8215			−0.585302			−0.292651	−	0.956219i	$-0.594537\pi$
$11$	−70.8215			−0.585302			−0.292651	+	0.956219i	$0.594537\pi$
$12$		0			0
$13$		−	231.121i		−	1.36758i	−0.729679	−	0.683790i	$-0.760330\pi$
$13$		−	231.121i		−	1.36758i	0.729679	−	0.683790i	$-0.239670\pi$
$14$		0			0
$15$		−	178.164i		−	0.791840i
$16$		0			0
$17$	−90.4279			−0.312899			−0.156450	−	0.987686i	$-0.550005\pi$
$17$	−90.4279			−0.312899			−0.156450	+	0.987686i	$0.550005\pi$
$18$		0			0
$19$	−319.751	+	167.573i	−0.885736	+	0.464190i
$19$	−319.751	+	167.573i	−0.885736	+	0.464190i
$20$		0			0
$21$		−	409.285i		−	0.928084i
$22$		0			0
$23$	−285.538			−0.539769			−0.269885	−	0.962893i	$-0.586986\pi$
$23$	−285.538			−0.539769			−0.269885	+	0.962893i	$0.586986\pi$
$24$		0			0
$25$	−342.037			−0.547259
$26$		0			0
$27$			527.681i			0.723843i
$28$		0			0
$29$			285.969i			0.340034i	0.985441	+	0.170017i	$0.0543823\pi$
$29$			285.969i			0.340034i	−0.985441	+	0.170017i	$0.945618\pi$
$30$		0			0
$31$			852.235i			0.886821i	0.896319	+	0.443410i	$0.146232\pi$
$31$			852.235i			0.886821i	−0.896319	+	0.443410i	$0.853768\pi$
$32$		0			0
$33$		−	750.101i		−	0.688798i
$34$		0			0
$35$	650.034			0.530640
$36$		0			0
$37$		−	54.8476i		−	0.0400640i	−0.999799	−	0.0200320i	$-0.993623\pi$
$37$		−	54.8476i		−	0.0400640i	0.999799	−	0.0200320i	$-0.00637681\pi$
$38$		0			0
$39$	2447.90			1.60940
$40$		0			0
$41$		−	37.4456i		−	0.0222758i	−0.999938	−	0.0111379i	$-0.996455\pi$
$41$		−	37.4456i		−	0.0222758i	0.999938	−	0.0111379i	$-0.00354537\pi$
$42$		0			0
$43$	691.677			0.374082			0.187041	−	0.982352i	$-0.440110\pi$
$43$	691.677			0.374082			0.187041	+	0.982352i	$0.440110\pi$
$44$		0			0
$45$	524.469			0.258997
$46$		0			0
$47$	2630.47			1.19080			0.595398	−	0.803431i	$-0.296994\pi$
$47$	2630.47			1.19080			0.595398	+	0.803431i	$0.296994\pi$
$48$		0			0
$49$	−907.716			−0.378058
$50$		0			0
$51$		−	957.761i		−	0.368228i
$52$		0			0
$53$			3833.35i			1.36467i	0.731041	+	0.682333i	$0.239035\pi$
$53$			3833.35i			1.36467i	−0.731041	+	0.682333i	$0.760965\pi$
$54$		0			0
$55$	1191.33			0.393826
$56$		0			0
$57$	−1774.83	−	3386.62i	−0.546271	−	1.04236i
$58$		0			0
$59$			3339.72i			0.959414i	0.877429	+	0.479707i	$0.159257\pi$
$59$			3339.72i			0.959414i	−0.877429	+	0.479707i	$0.840743\pi$
$60$		0			0
$61$	−2877.84			−0.773405			−0.386702	−	0.922205i	$-0.626386\pi$
$61$	−2877.84			−0.773405			−0.386702	+	0.922205i	$0.626386\pi$
$62$		0			0
$63$	1204.83			0.303560
$64$		0			0
$65$			3887.81i			0.920191i
$66$		0			0
$67$			4052.74i			0.902815i	0.892318	+	0.451408i	$0.149078\pi$
$67$			4052.74i			0.902815i	−0.892318	+	0.451408i	$0.850922\pi$
$68$		0			0
$69$		−	3024.26i		−	0.635214i
$70$		0			0
$71$		−	3315.87i		−	0.657780i	−0.944368	−	0.328890i	$-0.893325\pi$
$71$		−	3315.87i		−	0.657780i	0.944368	−	0.328890i	$-0.106675\pi$
$72$		0			0
$73$	−5544.37			−1.04041			−0.520207	−	0.854040i	$-0.674145\pi$
$73$	−5544.37			−1.04041			−0.520207	+	0.854040i	$0.674145\pi$
$74$		0			0
$75$		−	3622.66i		−	0.644028i
$76$		0			0
$77$	2736.76			0.461588
$78$		0			0
$79$		−	10304.4i		−	1.65107i	−0.564348	−	0.825537i	$-0.690872\pi$
$79$		−	10304.4i		−	1.65107i	0.564348	−	0.825537i	$-0.309128\pi$
$80$		0			0
$81$	−8114.36			−1.23676
$82$		0			0
$83$	−3541.95			−0.514145			−0.257073	−	0.966392i	$-0.582758\pi$
$83$	−3541.95			−0.514145			−0.257073	+	0.966392i	$0.582758\pi$
$84$		0			0
$85$	1521.13			0.210538
$86$		0			0
$87$	−3028.82			−0.400161
$88$		0			0
$89$		−	13273.7i		−	1.67576i	−0.545851	−	0.837882i	$-0.683793\pi$
$89$		−	13273.7i		−	1.67576i	0.545851	−	0.837882i	$-0.316207\pi$
$90$		0			0
$91$			8931.22i			1.07852i
$92$		0			0
$93$	−9026.39			−1.04363
$94$		0			0
$95$	5378.69	−	2818.82i	0.595977	−	0.312335i
$96$		0			0
$97$		−	4281.99i		−	0.455096i	−0.973767	−	0.227548i	$-0.926929\pi$
$97$		−	4281.99i		−	0.455096i	0.973767	−	0.227548i	$-0.0730708\pi$
$98$		0			0
$99$	2208.11			0.225294
$100$		0			0
$101$	4391.56			0.430503			0.215252	−	0.976559i	$-0.430943\pi$
$101$	4391.56			0.430503			0.215252	+	0.976559i	$0.430943\pi$
$102$		0			0
$103$			13489.7i			1.27153i	0.771881	+	0.635767i	$0.219316\pi$
$103$			13489.7i			1.27153i	−0.771881	+	0.635767i	$0.780684\pi$
$104$		0			0
$105$			6884.80i			0.624471i
$106$		0			0
$107$			21349.0i			1.86470i	0.361553	+	0.932352i	$0.382247\pi$
$107$			21349.0i			1.86470i	−0.361553	+	0.932352i	$0.617753\pi$
$108$		0			0
$109$		−	15145.0i		−	1.27472i	−0.770564	−	0.637362i	$-0.780026\pi$
$109$		−	15145.0i		−	1.27472i	0.770564	−	0.637362i	$-0.219974\pi$
$110$		0			0
$111$	580.915			0.0471483
$112$		0			0
$113$			3251.23i			0.254619i	0.991863	+	0.127310i	$0.0406342\pi$
$113$			3251.23i			0.254619i	−0.991863	+	0.127310i	$0.959366\pi$
$114$		0			0
$115$	4803.18			0.363189
$116$		0			0
$117$			7206.01i			0.526409i
$118$		0			0
$119$	3494.41			0.246763
$120$		0			0
$121$	−9625.31			−0.657422
$122$		0			0
$123$	396.602			0.0262147
$124$		0			0
$125$	16267.0			1.04109
$126$		0			0
$127$		−	16952.7i		−	1.05107i	−0.850772	−	0.525535i	$-0.823865\pi$
$127$		−	16952.7i		−	1.05107i	0.850772	−	0.525535i	$-0.176135\pi$
$128$		0			0
$129$			7325.85i			0.440229i
$130$		0			0
$131$	2912.67			0.169726			0.0848632	−	0.996393i	$-0.472955\pi$
$131$	2912.67			0.169726			0.0848632	+	0.996393i	$0.472955\pi$
$132$		0			0
$133$	12356.1	−	6475.51i	0.698521	−	0.366076i
$134$		0			0
$135$		−	8876.40i		−	0.487045i
$136$		0			0
$137$	35986.2			1.91732			0.958660	−	0.284556i	$-0.0918460\pi$
$137$	35986.2			1.91732			0.958660	+	0.284556i	$0.0918460\pi$
$138$		0			0
$139$	1155.96			0.0598294			0.0299147	−	0.999552i	$-0.490476\pi$
$139$	1155.96			0.0598294			0.0299147	+	0.999552i	$0.490476\pi$
$140$		0			0
$141$			27860.4i			1.40136i
$142$		0			0
$143$			16368.3i			0.800447i
$144$		0			0
$145$		−	4810.43i		−	0.228796i
$146$		0			0
$147$		−	9614.02i		−	0.444908i
$148$		0			0
$149$	27655.0			1.24567			0.622833	−	0.782355i	$-0.285981\pi$
$149$	27655.0			1.24567			0.622833	+	0.782355i	$0.285981\pi$
$150$		0			0
$151$		−	10441.7i		−	0.457949i	−0.973432	−	0.228975i	$-0.926463\pi$
$151$		−	10441.7i		−	0.457949i	0.973432	−	0.228975i	$-0.0735373\pi$
$152$		0			0
$153$	2819.40			0.120441
$154$		0			0
$155$		−	14335.9i		−	0.596707i
$156$		0			0
$157$	−27020.1			−1.09620			−0.548098	−	0.836414i	$-0.684648\pi$
$157$	−27020.1			−1.09620			−0.548098	+	0.836414i	$0.684648\pi$
$158$		0			0
$159$	−40600.7			−1.60598
$160$		0			0
$161$	11034.0			0.425680
$162$		0			0
$163$	−47347.7			−1.78206			−0.891032	−	0.453941i	$-0.850018\pi$
$163$	−47347.7			−1.78206			−0.891032	+	0.453941i	$0.850018\pi$
$164$		0			0
$165$			12617.8i			0.463465i
$166$		0			0
$167$			9876.14i			0.354123i	0.984200	+	0.177062i	$0.0566592\pi$
$167$			9876.14i			0.354123i	−0.984200	+	0.177062i	$0.943341\pi$
$168$		0			0
$169$	−24856.0			−0.870277
$170$		0			0
$171$	9969.34	−	5224.66i	0.340937	−	0.178676i
$172$		0			0
$173$			43235.9i			1.44462i	0.691572	+	0.722308i	$0.256919\pi$
$173$			43235.9i			1.44462i	−0.691572	+	0.722308i	$0.743081\pi$
$174$		0			0
$175$	13217.3			0.431586
$176$		0			0
$177$	−35372.4			−1.12906
$178$		0			0
$179$		−	33775.1i		−	1.05412i	−0.849828	−	0.527060i	$-0.823294\pi$
$179$		−	33775.1i		−	1.05412i	0.849828	−	0.527060i	$-0.176706\pi$
$180$		0			0
$181$		−	1508.54i		−	0.0460469i	−0.999735	−	0.0230234i	$-0.992671\pi$
$181$		−	1508.54i		−	0.0460469i	0.999735	−	0.0230234i	$-0.00732923\pi$
$182$		0			0
$183$		−	30480.4i		−	0.910163i
$184$		0			0
$185$			922.620i			0.0269575i
$186$		0			0
$187$	6404.24			0.183140
$188$		0			0
$189$		−	20391.2i		−	0.570846i
$190$		0			0
$191$	16937.5			0.464284			0.232142	−	0.972682i	$-0.425427\pi$
$191$	16937.5			0.464284			0.232142	+	0.972682i	$0.425427\pi$
$192$		0			0
$193$		−	15699.3i		−	0.421469i	−0.977543	−	0.210735i	$-0.932414\pi$
$193$		−	15699.3i		−	0.421469i	0.977543	−	0.210735i	$-0.0675856\pi$
$194$		0			0
$195$	−41177.5			−1.08290
$196$		0			0
$197$	16437.6			0.423551			0.211776	−	0.977318i	$-0.432075\pi$
$197$	16437.6			0.423551			0.211776	+	0.977318i	$0.432075\pi$
$198$		0			0
$199$	−13901.6			−0.351042			−0.175521	−	0.984476i	$-0.556161\pi$
$199$	−13901.6			−0.351042			−0.175521	+	0.984476i	$0.556161\pi$
$200$		0			0
$201$	−42924.3			−1.06246
$202$		0			0
$203$		−	11050.7i		−	0.268162i
$204$		0			0
$205$			629.891i			0.0149885i
$206$		0			0
$207$	8902.64			0.207768
$208$		0			0
$209$	22645.2	−	11867.7i	0.518423	−	0.271691i
$210$		0			0
$211$		−	32124.3i		−	0.721554i	−0.932652	−	0.360777i	$-0.882512\pi$
$211$		−	32124.3i		−	0.721554i	0.932652	−	0.360777i	$-0.117488\pi$
$212$		0			0
$213$	35119.8			0.774093
$214$		0			0
$215$	−11635.1			−0.251705
$216$		0			0
$217$		−	32932.9i		−	0.699376i
$218$		0			0
$219$		−	58722.8i		−	1.22439i
$220$		0			0
$221$			20899.8i			0.427915i
$222$		0			0
$223$			69965.9i			1.40694i	0.710723	+	0.703472i	$0.248368\pi$
$223$			69965.9i			1.40694i	−0.710723	+	0.703472i	$0.751632\pi$
$224$		0			0
$225$	10664.2			0.210651
$226$		0			0
$227$			88463.3i			1.71677i	0.513008	+	0.858384i	$0.328531\pi$
$227$			88463.3i			1.71677i	−0.513008	+	0.858384i	$0.671469\pi$
$228$		0			0
$229$	92563.5			1.76510			0.882549	−	0.470221i	$-0.155826\pi$
$229$	92563.5			1.76510			0.882549	+	0.470221i	$0.155826\pi$
$230$		0			0
$231$			28986.2i			0.543209i
$232$		0			0
$233$	−80439.0			−1.48168			−0.740841	−	0.671681i	$-0.765573\pi$
$233$	−80439.0			−1.48168			−0.740841	+	0.671681i	$0.765573\pi$
$234$		0			0
$235$	−44248.5			−0.801240
$236$		0			0
$237$	109138.			1.94303
$238$		0			0
$239$	−87546.3			−1.53265			−0.766323	−	0.642455i	$-0.777916\pi$
$239$	−87546.3			−1.53265			−0.766323	+	0.642455i	$0.777916\pi$
$240$		0			0
$241$			96049.6i			1.65372i	0.562409	+	0.826859i	$0.309875\pi$
$241$			96049.6i			1.65372i	−0.562409	+	0.826859i	$0.690125\pi$
$242$		0			0
$243$		−	43200.5i		−	0.731605i
$244$		0			0
$245$	15269.2			0.254380
$246$		0			0
$247$	38729.6	+	73901.1i	0.634817	+	1.21132i
$248$		0			0
$249$		−	37514.3i		−	0.605059i
$250$		0			0
$251$	92400.5			1.46665			0.733325	−	0.679878i	$-0.237967\pi$
$251$	92400.5			1.46665			0.733325	+	0.679878i	$0.237967\pi$
$252$		0			0
$253$	20222.2			0.315928
$254$		0			0
$255$			16111.0i			0.247766i
$256$		0			0
$257$			9573.31i			0.144942i	0.997371	+	0.0724712i	$0.0230886\pi$
$257$			9573.31i			0.144942i	−0.997371	+	0.0724712i	$0.976911\pi$
$258$		0			0
$259$			2119.48i			0.0315958i
$260$		0			0
$261$		−	8916.07i		−	0.130886i
$262$		0			0
$263$	27549.0			0.398285			0.199142	−	0.979971i	$-0.436184\pi$
$263$	27549.0			0.398285			0.199142	+	0.979971i	$0.436184\pi$
$264$		0			0
$265$		−	64482.7i		−	0.918230i
$266$		0			0
$267$	140588.			1.97208
$268$		0			0
$269$			97069.5i			1.34146i	0.741701	+	0.670731i	$0.234019\pi$
$269$			97069.5i			1.34146i	−0.741701	+	0.670731i	$0.765981\pi$
$270$		0			0
$271$	−52509.9			−0.714994			−0.357497	−	0.933914i	$-0.616370\pi$
$271$	−52509.9			−0.714994			−0.357497	+	0.933914i	$0.616370\pi$
$272$		0			0
$273$	−94594.4			−1.26923
$274$		0			0
$275$	24223.6			0.320311
$276$		0			0
$277$	31441.0			0.409767			0.204883	−	0.978786i	$-0.434318\pi$
$277$	31441.0			0.409767			0.204883	+	0.978786i	$0.434318\pi$
$278$		0			0
$279$		−	26571.4i		−	0.341355i
$280$		0			0
$281$		−	153218.i		−	1.94043i	−0.242245	−	0.970215i	$-0.577884\pi$
$281$		−	153218.i		−	1.94043i	0.242245	−	0.970215i	$-0.422116\pi$
$282$		0			0
$283$	−18715.4			−0.233682			−0.116841	−	0.993151i	$-0.537277\pi$
$283$	−18715.4			−0.233682			−0.116841	+	0.993151i	$0.537277\pi$
$284$		0			0
$285$	29855.4	+	56968.0i	0.367564	+	0.701361i
$286$		0			0
$287$			1447.01i			0.0175674i
$288$		0			0
$289$	−75343.8			−0.902094
$290$		0			0
$291$	45352.5			0.535568
$292$		0			0
$293$		−	135321.i		−	1.57627i	−0.615502	−	0.788136i	$-0.711047\pi$
$293$		−	135321.i		−	1.57627i	0.615502	−	0.788136i	$-0.288953\pi$
$294$		0			0
$295$		−	56179.1i		−	0.645552i
$296$		0			0
$297$		−	37371.2i		−	0.423666i
$298$		0			0
$299$			65993.8i			0.738178i
$300$		0			0
$301$	−26728.5			−0.295013
$302$		0			0
$303$			46513.0i			0.506628i
$304$		0			0
$305$	48409.6			0.520393
$306$		0			0
$307$			8548.60i			0.0907023i	0.998971	+	0.0453512i	$0.0144407\pi$
$307$			8548.60i			0.0907023i	−0.998971	+	0.0453512i	$0.985559\pi$
$308$		0			0
$309$	−142875.			−1.49637
$310$		0			0
$311$	−149498.			−1.54566			−0.772830	−	0.634613i	$-0.781159\pi$
$311$	−149498.			−1.54566			−0.772830	+	0.634613i	$0.781159\pi$
$312$		0			0
$313$	32077.5			0.327425			0.163713	−	0.986508i	$-0.447653\pi$
$313$	32077.5			0.327425			0.163713	+	0.986508i	$0.447653\pi$
$314$		0			0
$315$	−20267.1			−0.204254
$316$		0			0
$317$			127709.i			1.27087i	0.772153	+	0.635437i	$0.219180\pi$
$317$			127709.i			1.27087i	−0.772153	+	0.635437i	$0.780820\pi$
$318$		0			0
$319$		−	20252.7i		−	0.199023i
$320$		0			0
$321$	−226116.			−2.19443
$322$		0			0
$323$	28914.4	−	15153.2i	0.277146	−	0.145245i
$324$		0			0
$325$			79051.9i			0.748420i
$326$		0			0
$327$	160407.			1.50013
$328$		0			0
$329$	−101649.			−0.939102
$330$		0			0
$331$		−	56091.4i		−	0.511965i	−0.966681	−	0.255983i	$-0.917601\pi$
$331$		−	56091.4i		−	0.511965i	0.966681	−	0.255983i	$-0.0823990\pi$
$332$		0			0
$333$			1710.06i			0.0154214i
$334$		0			0
$335$		−	68173.2i		−	0.607469i
$336$		0			0
$337$		−	21720.9i		−	0.191257i	−0.995417	−	0.0956286i	$-0.969514\pi$
$337$		−	21720.9i		−	0.191257i	0.995417	−	0.0956286i	$-0.0304861\pi$
$338$		0			0
$339$	−34435.2			−0.299643
$340$		0			0
$341$		−	60356.6i		−	0.519058i
$342$		0			0
$343$	127859.			1.08678
$344$		0			0
$345$			50872.6i			0.427411i
$346$		0			0
$347$	−83039.8			−0.689648			−0.344824	−	0.938667i	$-0.612061\pi$
$347$	−83039.8			−0.689648			−0.344824	+	0.938667i	$0.612061\pi$
$348$		0			0
$349$	−205331.			−1.68579			−0.842895	−	0.538079i	$-0.819150\pi$
$349$	−205331.			−1.68579			−0.842895	+	0.538079i	$0.819150\pi$
$350$		0			0
$351$	121958.			0.989913
$352$		0			0
$353$	−119932.			−0.962463			−0.481232	−	0.876594i	$-0.659810\pi$
$353$	−119932.			−0.962463			−0.481232	+	0.876594i	$0.659810\pi$
$354$		0			0
$355$			55778.0i			0.442594i
$356$		0			0
$357$			37010.8i			0.290397i
$358$		0			0
$359$	−138143.			−1.07187			−0.535933	−	0.844260i	$-0.680040\pi$
$359$	−138143.			−1.07187			−0.535933	+	0.844260i	$0.680040\pi$
$360$		0			0
$361$	74159.9	−	107163.i	0.569056	−	0.822299i
$362$		0			0
$363$		−	101946.i		−	0.773671i
$364$		0			0
$365$	93264.7			0.700054
$366$		0			0
$367$	138217.			1.02619			0.513096	−	0.858331i	$-0.328498\pi$
$367$	138217.			1.02619			0.513096	+	0.858331i	$0.328498\pi$
$368$		0			0
$369$			1167.50i			0.00857438i
$370$		0			0
$371$		−	148132.i		−	1.07622i
$372$		0			0
$373$		−	151421.i		−	1.08835i	−0.838972	−	0.544175i	$-0.816843\pi$
$373$		−	151421.i		−	1.08835i	0.838972	−	0.544175i	$-0.183157\pi$
$374$		0			0
$375$			172291.i			1.22518i
$376$		0			0
$377$	66093.4			0.465024
$378$		0			0
$379$			117526.i			0.818194i	0.912491	+	0.409097i	$0.134156\pi$
$379$			117526.i			0.818194i	−0.912491	+	0.409097i	$0.865844\pi$
$380$		0			0
$381$	179554.			1.23693
$382$		0			0
$383$		−	217885.i		−	1.48535i	−0.669651	−	0.742676i	$-0.733556\pi$
$383$		−	217885.i		−	1.48535i	0.669651	−	0.742676i	$-0.266444\pi$
$384$		0			0
$385$	−46036.4			−0.310585
$386$		0			0
$387$	−21565.5			−0.143991
$388$		0			0
$389$	64818.5			0.428351			0.214175	−	0.976795i	$-0.431294\pi$
$389$	64818.5			0.428351			0.214175	+	0.976795i	$0.431294\pi$
$390$		0			0
$391$	25820.6			0.168893
$392$		0			0
$393$			30849.4i			0.199738i
$394$		0			0
$395$			173335.i			1.11094i
$396$		0			0
$397$	89625.3			0.568656			0.284328	−	0.958727i	$-0.408230\pi$
$397$	89625.3			0.568656			0.284328	+	0.958727i	$0.408230\pi$
$398$		0			0
$399$	68584.9	+	130869.i	0.430807	+	0.822037i
$400$		0			0
$401$			123664.i			0.769052i	0.923114	+	0.384526i	$0.125635\pi$
$401$			123664.i			0.769052i	−0.923114	+	0.384526i	$0.874365\pi$
$402$		0			0
$403$	196970.			1.21280
$404$		0			0
$405$	136496.			0.832165
$406$		0			0
$407$			3884.39i			0.0234495i
$408$		0			0
$409$			67492.8i			0.403470i	0.979440	+	0.201735i	$0.0646579\pi$
$409$			67492.8i			0.403470i	−0.979440	+	0.201735i	$0.935342\pi$
$410$		0			0
$411$			381145.i			2.25635i
$412$		0			0
$413$		−	129057.i		−	0.756626i
$414$		0			0
$415$	59580.9			0.345948
$416$		0			0
$417$			12243.3i			0.0704088i
$418$		0			0
$419$	−210278.			−1.19775			−0.598874	−	0.800843i	$-0.704385\pi$
$419$	−210278.			−1.19775			−0.598874	+	0.800843i	$0.704385\pi$
$420$		0			0
$421$			87195.1i			0.491958i	0.969275	+	0.245979i	$0.0791094\pi$
$421$			87195.1i			0.491958i	−0.969275	+	0.245979i	$0.920891\pi$
$422$		0			0
$423$	−82014.1			−0.458361
$424$		0			0
$425$	30929.7			0.171237
$426$		0			0
$427$	111208.			0.609933
$428$		0			0
$429$	−173364.			−0.941987
$430$		0			0
$431$		−	285166.i		−	1.53512i	−0.640976	−	0.767561i	$-0.721470\pi$
$431$		−	285166.i		−	1.53512i	0.640976	−	0.767561i	$-0.278530\pi$
$432$		0			0
$433$			47714.6i			0.254493i	0.991871	+	0.127246i	$0.0406139\pi$
$433$			47714.6i			0.254493i	−0.991871	+	0.127246i	$0.959386\pi$
$434$		0			0
$435$	50949.3			0.269253
$436$		0			0
$437$	91300.9	−	47848.3i	0.478093	−	0.250555i
$438$		0			0
$439$		−	6771.27i		−	0.0351351i	−0.999846	−	0.0175675i	$-0.994408\pi$
$439$		−	6771.27i		−	0.0351351i	0.999846	−	0.0175675i	$-0.00559221\pi$
$440$		0			0
$441$	28301.2			0.145522
$442$		0			0
$443$	−305876.			−1.55861			−0.779306	−	0.626643i	$-0.784428\pi$
$443$	−305876.			−1.55861			−0.779306	+	0.626643i	$0.784428\pi$
$444$		0			0
$445$			223284.i			1.12756i
$446$		0			0
$447$			292907.i			1.46593i
$448$		0			0
$449$		−	133235.i		−	0.660883i	−0.943826	−	0.330442i	$-0.892802\pi$
$449$		−	133235.i		−	0.660883i	0.943826	−	0.330442i	$-0.107198\pi$
$450$		0			0
$451$			2651.95i			0.0130380i
$452$		0			0
$453$	110593.			0.538927
$454$		0			0
$455$		−	150237.i		−	0.725693i
$456$		0			0
$457$	295991.			1.41725			0.708625	−	0.705585i	$-0.249316\pi$
$457$	295991.			1.41725			0.708625	+	0.705585i	$0.249316\pi$
$458$		0			0
$459$		−	47717.1i		−	0.226490i
$460$		0			0
$461$	−142958.			−0.672675			−0.336337	−	0.941742i	$-0.609188\pi$
$461$	−142958.			−0.672675			−0.336337	+	0.941742i	$0.609188\pi$
$462$		0			0
$463$	−29639.1			−0.138262			−0.0691310	−	0.997608i	$-0.522023\pi$
$463$	−29639.1			−0.138262			−0.0691310	+	0.997608i	$0.522023\pi$
$464$		0			0
$465$	151838.			0.702220
$466$		0			0
$467$	−256137.			−1.17446			−0.587231	−	0.809419i	$-0.699782\pi$
$467$	−256137.			−1.17446			−0.587231	+	0.809419i	$0.699782\pi$
$468$		0			0
$469$		−	156610.i		−	0.711990i
$470$		0			0
$471$		−	286182.i		−	1.29003i
$472$		0			0
$473$	−48985.6			−0.218951
$474$		0			0
$475$	109366.	−	57315.9i	0.484727	−	0.254032i
$476$		0			0
$477$		−	119518.i		−	0.525287i
$478$		0			0
$479$	142256.			0.620010			0.310005	−	0.950735i	$-0.399669\pi$
$479$	142256.			0.620010			0.310005	+	0.950735i	$0.399669\pi$
$480$		0			0
$481$	−12676.4			−0.0547907
$482$		0			0
$483$			116866.i			0.500951i
$484$		0			0
$485$			72029.6i			0.306216i
$486$		0			0
$487$			83814.6i			0.353396i	0.984265	+	0.176698i	$0.0565416\pi$
$487$			83814.6i			0.353396i	−0.984265	+	0.176698i	$0.943458\pi$
$488$		0			0
$489$		−	501480.i		−	2.09718i
$490$		0			0
$491$	70233.1			0.291326			0.145663	−	0.989334i	$-0.453469\pi$
$491$	70233.1			0.291326			0.145663	+	0.989334i	$0.453469\pi$
$492$		0			0
$493$		−	25859.5i		−	0.106396i
$494$		0			0
$495$	−37143.7			−0.151592
$496$		0			0
$497$			128135.i			0.518747i
$498$		0			0
$499$	−382202.			−1.53494			−0.767471	−	0.641083i	$-0.778485\pi$
$499$	−382202.			−1.53494			−0.767471	+	0.641083i	$0.778485\pi$
$500$		0			0
$501$	−104602.			−0.416741
$502$		0			0
$503$	−454164.			−1.79505			−0.897525	−	0.440964i	$-0.854637\pi$
$503$	−454164.			−1.79505			−0.897525	+	0.440964i	$0.854637\pi$
$504$		0			0
$505$	−73872.8			−0.289669
$506$		0			0
$507$		−	263261.i		−	1.02416i
$508$		0			0
$509$			300805.i			1.16105i	0.814244	+	0.580523i	$0.197152\pi$
$509$			300805.i			1.16105i	−0.814244	+	0.580523i	$0.802848\pi$
$510$		0			0
$511$	214251.			0.820505
$512$		0			0
$513$	−88424.9	−	168726.i	−0.336000	−	0.641133i
$514$		0			0
$515$		−	226917.i		−	0.855565i
$516$		0			0
$517$	−186294.			−0.696975
$518$		0			0
$519$	−457930.			−1.70006
$520$		0			0
$521$			488582.i			1.79996i	0.435935	+	0.899978i	$0.356418\pi$
$521$			488582.i			1.79996i	−0.435935	+	0.899978i	$0.643582\pi$
$522$		0			0
$523$			4829.78i			0.0176573i	0.999961	+	0.00882864i	$0.00281028\pi$
$523$			4829.78i			0.0176573i	−0.999961	+	0.00882864i	$0.997190\pi$
$524$		0			0
$525$			139991.i			0.507902i
$526$		0			0
$527$		−	77065.8i		−	0.277486i
$528$		0			0
$529$	−198309.			−0.708649
$530$		0			0
$531$		−	104127.i		−	0.369297i
$532$		0			0
$533$	−8654.46			−0.0304639
$534$		0			0
$535$		−	359122.i		−	1.25469i
$536$		0			0
$537$	357726.			1.24052
$538$		0			0
$539$	64285.8			0.221278
$540$		0			0
$541$	−221949.			−0.758332			−0.379166	−	0.925329i	$-0.623789\pi$
$541$	−221949.			−0.758332			−0.379166	+	0.925329i	$0.623789\pi$
$542$		0			0
$543$	15977.6			0.0541891
$544$		0			0
$545$			254762.i			0.857712i
$546$		0			0
$547$			505454.i			1.68930i	0.535318	+	0.844651i	$0.320192\pi$
$547$			505454.i			1.68930i	−0.535318	+	0.844651i	$0.679808\pi$
$548$		0			0
$549$	89726.6			0.297699
$550$		0			0
$551$	−47920.5	−	91438.7i	−0.157840	−	0.301180i
$552$		0			0
$553$			398191.i			1.30209i
$554$		0			0
$555$	−9771.86			−0.0317243
$556$		0			0
$557$	389257.			1.25466			0.627329	−	0.778754i	$-0.284148\pi$
$557$	389257.			1.25466			0.627329	+	0.778754i	$0.284148\pi$
$558$		0			0
$559$		−	159861.i		−	0.511587i
$560$		0			0
$561$			67830.1i			0.215525i
$562$		0			0
$563$		−	75812.9i		−	0.239181i	−0.992823	−	0.119590i	$-0.961842\pi$
$563$		−	75812.9i		−	0.239181i	0.992823	−	0.119590i	$-0.0381581\pi$
$564$		0			0
$565$		−	54690.7i		−	0.171323i
$566$		0			0
$567$	313563.			0.975347
$568$		0			0
$569$		−	181331.i		−	0.560077i	−0.959989	−	0.280039i	$-0.909653\pi$
$569$		−	181331.i		−	0.560077i	0.959989	−	0.280039i	$-0.0903473\pi$
$570$		0			0
$571$	−37223.1			−0.114167			−0.0570835	−	0.998369i	$-0.518180\pi$
$571$	−37223.1			−0.114167			−0.0570835	+	0.998369i	$0.518180\pi$
$572$		0			0
$573$			179393.i			0.546381i
$574$		0			0
$575$	97664.4			0.295393
$576$		0			0
$577$	148028.			0.444624			0.222312	−	0.974976i	$-0.428640\pi$
$577$	148028.			0.444624			0.222312	+	0.974976i	$0.428640\pi$
$578$		0			0
$579$	166278.			0.495996
$580$		0			0
$581$	136872.			0.405472
$582$		0			0
$583$		−	271484.i		−	0.798742i
$584$		0			0
$585$		−	121216.i		−	0.354200i
$586$		0			0
$587$	34520.0			0.100183			0.0500915	−	0.998745i	$-0.484049\pi$
$587$	34520.0			0.100183			0.0500915	+	0.998745i	$0.484049\pi$
$588$		0			0
$589$	−142811.	−	272503.i	−0.411653	−	0.785489i
$590$		0			0
$591$			174098.i			0.498446i
$592$		0			0
$593$	297841.			0.846985			0.423492	−	0.905900i	$-0.360804\pi$
$593$	297841.			0.846985			0.423492	+	0.905900i	$0.360804\pi$
$594$		0			0
$595$	−58781.2			−0.166037
$596$		0			0
$597$		−	147238.i		−	0.413115i
$598$		0			0
$599$			76234.7i			0.212471i	0.994341	+	0.106235i	$0.0338797\pi$
$599$			76234.7i			0.212471i	−0.994341	+	0.106235i	$0.966120\pi$
$600$		0			0
$601$			103183.i			0.285667i	0.989747	+	0.142833i	$0.0456213\pi$
$601$			103183.i			0.285667i	−0.989747	+	0.142833i	$0.954379\pi$
$602$		0			0
$603$		−	126358.i		−	0.347511i
$604$		0			0
$605$	161912.			0.442353
$606$		0			0
$607$			445896.i			1.21020i	0.796150	+	0.605099i	$0.206866\pi$
$607$			445896.i			1.21020i	−0.796150	+	0.605099i	$0.793134\pi$
$608$		0			0
$609$	117043.			0.315580
$610$		0			0
$611$		−	607957.i		−	1.62851i
$612$		0			0
$613$	−74724.0			−0.198856			−0.0994281	−	0.995045i	$-0.531701\pi$
$613$	−74724.0			−0.198856			−0.0994281	+	0.995045i	$0.531701\pi$
$614$		0			0
$615$	−6671.45			−0.0176388
$616$		0			0
$617$	218592.			0.574202			0.287101	−	0.957900i	$-0.407308\pi$
$617$	218592.			0.574202			0.287101	+	0.957900i	$0.407308\pi$
$618$		0			0
$619$	556223.			1.45167			0.725835	−	0.687869i	$-0.241454\pi$
$619$	556223.			1.45167			0.725835	+	0.687869i	$0.241454\pi$
$620$		0			0
$621$		−	150673.i		−	0.390708i
$622$		0			0
$623$			512937.i			1.32156i
$624$		0			0
$625$	−59863.0			−0.153249
$626$		0			0
$627$	125696.	+	239845.i	0.319733	+	0.610093i
$628$		0			0
$629$			4959.75i			0.0125360i
$630$		0			0
$631$	−237884.			−0.597458			−0.298729	−	0.954338i	$-0.596563\pi$
$631$	−237884.			−0.597458			−0.298729	+	0.954338i	$0.596563\pi$
$632$		0			0
$633$	340242.			0.849143
$634$		0			0
$635$			285170.i			0.707224i
$636$		0			0
$637$			209792.i			0.517024i
$638$		0			0
$639$			103384.i			0.253192i
$640$		0			0
$641$			759808.i			1.84922i	0.380919	+	0.924608i	$0.375608\pi$
$641$			759808.i			1.84922i	−0.380919	+	0.924608i	$0.624392\pi$
$642$		0			0
$643$	−239060.			−0.578208			−0.289104	−	0.957298i	$-0.593357\pi$
$643$	−239060.			−0.578208			−0.289104	+	0.957298i	$0.593357\pi$
$644$		0			0
$645$		−	123232.i		−	0.296213i
$646$		0			0
$647$	−252454.			−0.603079			−0.301540	−	0.953454i	$-0.597501\pi$
$647$	−252454.			−0.603079			−0.301540	+	0.953454i	$0.597501\pi$
$648$		0			0
$649$		−	236524.i		−	0.561547i
$650$		0			0
$651$	348807.			0.823044
$652$		0			0
$653$	61290.4			0.143736			0.0718681	−	0.997414i	$-0.477104\pi$
$653$	61290.4			0.143736			0.0718681	+	0.997414i	$0.477104\pi$
$654$		0			0
$655$	−48995.6			−0.114202
$656$		0			0
$657$	172865.			0.400476
$658$		0			0
$659$			115501.i			0.265958i	0.991119	+	0.132979i	$0.0424543\pi$
$659$			115501.i			0.265958i	−0.991119	+	0.132979i	$0.957546\pi$
$660$		0			0
$661$		−	185699.i		−	0.425018i	−0.977159	−	0.212509i	$-0.931837\pi$
$661$		−	185699.i		−	0.425018i	0.977159	−	0.212509i	$-0.0681635\pi$
$662$		0			0
$663$	−221359.			−0.503582
$664$		0			0
$665$	−207849.	+	108928.i	−0.470007	+	0.246318i
$666$		0			0
$667$		−	81654.9i		−	0.183540i
$668$		0			0
$669$	−741040.			−1.65573
$670$		0			0
$671$	203813.			0.452675
$672$		0			0
$673$		−	288265.i		−	0.636446i	−0.948016	−	0.318223i	$-0.896914\pi$
$673$		−	288265.i		−	0.636446i	0.948016	−	0.318223i	$-0.103086\pi$
$674$		0			0
$675$		−	180486.i		−	0.396129i
$676$		0			0
$677$		−	474041.i		−	1.03428i	−0.855901	−	0.517141i	$-0.826996\pi$
$677$		−	474041.i		−	1.03428i	0.855901	−	0.517141i	$-0.173004\pi$
$678$		0			0
$679$			165469.i			0.358904i
$680$		0			0
$681$	−936953.			−2.02034
$682$		0			0
$683$			167789.i			0.359685i	0.983695	+	0.179843i	$0.0575589\pi$
$683$			167789.i			0.359685i	−0.983695	+	0.179843i	$0.942441\pi$
$684$		0			0
$685$	−605342.			−1.29009
$686$		0			0
$687$			980380.i			2.07721i
$688$		0			0
$689$	885968.			1.86629
$690$		0			0
$691$	519893.			1.08882			0.544412	−	0.838818i	$-0.316753\pi$
$691$	519893.			1.08882			0.544412	+	0.838818i	$0.316753\pi$
$692$		0			0
$693$	−85328.0			−0.177674
$694$		0			0
$695$	−19445.1			−0.0402569
$696$		0			0
$697$			3386.12i			0.00697007i
$698$		0			0
$699$		−	851964.i		−	1.74368i
$700$		0			0
$701$	−440143.			−0.895691			−0.447845	−	0.894111i	$-0.647808\pi$
$701$	−440143.			−0.895691			−0.447845	+	0.894111i	$0.647808\pi$
$702$		0			0
$703$	9190.95	+	17537.6i	0.0185973	+	0.0354861i
$704$		0			0
$705$		−	468655.i		−	0.942920i
$706$		0			0
$707$	−169703.			−0.339509
$708$		0			0
$709$	−61143.5			−0.121635			−0.0608174	−	0.998149i	$-0.519371\pi$
$709$	−61143.5			−0.121635			−0.0608174	+	0.998149i	$0.519371\pi$
$710$		0			0
$711$			321274.i			0.635531i
$712$		0			0
$713$		−	243345.i		−	0.478679i
$714$		0			0
$715$		−	275340.i		−	0.538590i
$716$		0			0
$717$		−	927241.i		−	1.80366i
$718$		0			0
$719$	313516.			0.606459			0.303229	−	0.952918i	$-0.401935\pi$
$719$	313516.			0.606459			0.303229	+	0.952918i	$0.401935\pi$
$720$		0			0
$721$		−	521283.i		−	1.00277i
$722$		0			0
$723$	−1.01730e6			−1.94614
$724$		0			0
$725$		−	97811.8i		−	0.186087i
$726$		0			0
$727$	213177.			0.403340			0.201670	−	0.979454i	$-0.435363\pi$
$727$	213177.			0.403340			0.201670	+	0.979454i	$0.435363\pi$
$728$		0			0
$729$	−199708.			−0.375785
$730$		0			0
$731$	−62546.9			−0.117050
$732$		0			0
$733$	412494.			0.767732			0.383866	−	0.923389i	$-0.374592\pi$
$733$	412494.			0.767732			0.383866	+	0.923389i	$0.374592\pi$
$734$		0			0
$735$			161722.i			0.299361i
$736$		0			0
$737$		−	287021.i		−	0.528419i
$738$		0			0
$739$	331782.			0.607524			0.303762	−	0.952748i	$-0.401757\pi$
$739$	331782.			0.607524			0.303762	+	0.952748i	$0.401757\pi$
$740$		0			0
$741$	−782719.	+	410202.i	−1.42551	+	0.747069i
$742$		0			0
$743$		−	127921.i		−	0.231721i	−0.993266	−	0.115860i	$-0.963038\pi$
$743$		−	127921.i		−	0.231721i	0.993266	−	0.115860i	$-0.0369625\pi$
$744$		0			0
$745$	−465200.			−0.838160
$746$		0			0
$747$	110433.			0.197905
$748$		0			0
$749$		−	824990.i		−	1.47057i
$750$		0			0
$751$		−	757273.i		−	1.34268i	−0.741149	−	0.671340i	$-0.765719\pi$
$751$		−	757273.i		−	1.34268i	0.741149	−	0.671340i	$-0.234281\pi$
$752$		0			0
$753$			978654.i			1.72599i
$754$		0			0
$755$			175645.i			0.308136i
$756$		0			0
$757$	−1.11167e6			−1.93991			−0.969957	−	0.243275i	$-0.921778\pi$
$757$	−1.11167e6			−1.93991			−0.969957	+	0.243275i	$0.921778\pi$
$758$		0			0
$759$			214182.i			0.371792i
$760$		0			0
$761$	485750.			0.838771			0.419386	−	0.907808i	$-0.362246\pi$
$761$	485750.			0.838771			0.419386	+	0.907808i	$0.362246\pi$
$762$		0			0
$763$			585249.i			1.00529i
$764$		0			0
$765$	−47426.7			−0.0810400
$766$		0			0
$767$	771880.			1.31208
$768$		0			0
$769$	73393.4			0.124109			0.0620546	−	0.998073i	$-0.480235\pi$
$769$	73393.4			0.124109			0.0620546	+	0.998073i	$0.480235\pi$
$770$		0			0
$771$	−101395.			−0.170572
$772$		0			0
$773$			836995.i			1.40076i	0.713770	+	0.700380i	$0.246986\pi$
$773$			836995.i			1.40076i	−0.713770	+	0.700380i	$0.753014\pi$
$774$		0			0
$775$		−	291496.i		−	0.485320i
$776$		0			0
$777$	−22448.3			−0.0371827
$778$		0			0
$779$	6274.85	+	11973.2i	0.0103402	+	0.0197304i
$780$		0			0
$781$			234835.i			0.385000i
$782$		0			0
$783$	−150900.			−0.246131
$784$		0			0
$785$	454519.			0.737587
$786$		0			0
$787$		−	722223.i		−	1.16606i	−0.812450	−	0.583031i	$-0.801867\pi$
$787$		−	722223.i		−	1.16606i	0.812450	−	0.583031i	$-0.198133\pi$
$788$		0			0
$789$			291783.i			0.468712i
$790$		0			0
$791$		−	125637.i		−	0.200801i
$792$		0			0
$793$			665129.i			1.05769i
$794$		0			0
$795$	682965.			1.08060
$796$		0			0
$797$			533741.i			0.840260i	0.907464	+	0.420130i	$0.138016\pi$
$797$			533741.i			0.840260i	−0.907464	+	0.420130i	$0.861984\pi$
$798$		0			0
$799$	−237868.			−0.372599
$800$		0			0
$801$			413855.i			0.645035i
$802$		0			0
$803$	392661.			0.608956
$804$		0			0
$805$	−185609.			−0.286423
$806$		0			0
$807$	−1.02811e6			−1.57867
$808$		0			0
$809$	932045.			1.42410			0.712049	−	0.702130i	$-0.247767\pi$
$809$	932045.			1.42410			0.712049	+	0.702130i	$0.247767\pi$
$810$		0			0
$811$			878103.i			1.33507i	0.744579	+	0.667534i	$0.232650\pi$
$811$			878103.i			1.33507i	−0.744579	+	0.667534i	$0.767350\pi$
$812$		0			0
$813$		−	556155.i		−	0.841424i
$814$		0			0
$815$	796459.			1.19908
$816$		0			0
$817$	−221164.	+	115906.i	−0.331338	+	0.173645i
$818$		0			0
$819$		−	278462.i		−	0.415143i
$820$		0			0
$821$	−698577.			−1.03640			−0.518201	−	0.855259i	$-0.673398\pi$
$821$	−698577.			−1.03640			−0.518201	+	0.855259i	$0.673398\pi$
$822$		0			0
$823$	−351984.			−0.519665			−0.259833	−	0.965654i	$-0.583667\pi$
$823$	−351984.			−0.519665			−0.259833	+	0.965654i	$0.583667\pi$
$824$		0			0
$825$			256562.i			0.376951i
$826$		0			0
$827$			237628.i			0.347445i	0.984795	+	0.173723i	$0.0555797\pi$
$827$			237628.i			0.347445i	−0.984795	+	0.173723i	$0.944420\pi$
$828$		0			0
$829$		−	818714.i		−	1.19131i	−0.803242	−	0.595653i	$-0.796893\pi$
$829$		−	818714.i		−	1.19131i	0.803242	−	0.595653i	$-0.203107\pi$
$830$		0			0
$831$			333005.i			0.482224i
$832$		0			0
$833$	82082.9			0.118294
$834$		0			0
$835$		−	166132.i		−	0.238275i
$836$		0			0
$837$	−449708.			−0.641919
$838$		0			0
$839$			78055.7i			0.110887i	0.998462	+	0.0554435i	$0.0176573\pi$
$839$			78055.7i			0.110887i	−0.998462	+	0.0554435i	$0.982343\pi$
$840$		0			0
$841$	625503.			0.884377
$842$		0			0
$843$	1.62280e6			2.28355
$844$		0			0
$845$	418115.			0.585575
$846$		0			0
$847$	371951.			0.518465
$848$		0			0
$849$		−	198223.i		−	0.275003i
$850$		0			0
$851$			15661.1i			0.0216253i
$852$		0			0
$853$	841991.			1.15720			0.578602	−	0.815610i	$-0.303599\pi$
$853$	841991.			1.15720			0.578602	+	0.815610i	$0.303599\pi$
$854$		0			0
$855$	−167699.	+	87886.7i	−0.229403	+	0.120224i
$856$		0			0
$857$			699074.i			0.951835i	0.879490	+	0.475917i	$0.157884\pi$
$857$			699074.i			0.951835i	−0.879490	+	0.475917i	$0.842116\pi$
$858$		0			0
$859$	−171528.			−0.232461			−0.116230	−	0.993222i	$-0.537081\pi$
$859$	−171528.			−0.232461			−0.116230	+	0.993222i	$0.537081\pi$
$860$		0			0
$861$	−15325.9			−0.0206738
$862$		0			0
$863$		−	674012.i		−	0.904994i	−0.891766	−	0.452497i	$-0.850533\pi$
$863$		−	674012.i		−	0.904994i	0.891766	−	0.452497i	$-0.149467\pi$
$864$		0			0
$865$		−	727293.i		−	0.972025i
$866$		0			0
$867$		−	797999.i		−	1.06161i
$868$		0			0
$869$			729770.i			0.966376i
$870$		0			0
$871$	936674.			1.23467
$872$		0			0
$873$			133506.i			0.175175i
$874$		0			0
$875$	−628607.			−0.821038
$876$		0			0
$877$		−	969510.i		−	1.26053i	−0.776380	−	0.630265i	$-0.782946\pi$
$877$		−	969510.i		−	1.26053i	0.776380	−	0.630265i	$-0.217054\pi$
$878$		0			0
$879$	1.43325e6			1.85500
$880$		0			0
$881$	1.14587e6			1.47633			0.738164	−	0.674622i	$-0.235693\pi$
$881$	1.14587e6			1.47633			0.738164	+	0.674622i	$0.235693\pi$
$882$		0			0
$883$	−1.46931e6			−1.88449			−0.942243	−	0.334930i	$-0.891287\pi$
$883$	−1.46931e6			−1.88449			−0.942243	+	0.334930i	$0.891287\pi$
$884$		0			0
$885$	595018.			0.759702
$886$		0			0
$887$			336843.i			0.428134i	0.976819	+	0.214067i	$0.0686711\pi$
$887$			336843.i			0.428134i	−0.976819	+	0.214067i	$0.931329\pi$
$888$		0			0
$889$			655104.i			0.828909i
$890$		0			0
$891$	574671.			0.723876
$892$		0			0
$893$	−841094.	+	440794.i	−1.05473	+	0.552756i
$894$		0			0
$895$			568148.i			0.709276i
$896$		0			0
$897$	−698969.			−0.868707
$898$		0			0
$899$	−243713.			−0.301549
$900$		0			0
$901$		−	346642.i		−	0.427003i
$902$		0			0
$903$		−	283093.i		−	0.347179i
$904$		0			0
$905$			25375.9i			0.0309831i
$906$		0			0
$907$			977089.i			1.18774i	0.804563	+	0.593868i	$0.202400\pi$
$907$			977089.i			1.18774i	−0.804563	+	0.593868i	$0.797600\pi$
$908$		0			0
$909$	−136922.			−0.165709
$910$		0			0
$911$		−	233208.i		−	0.281001i	−0.990081	−	0.140500i	$-0.955129\pi$
$911$		−	233208.i		−	0.281001i	0.990081	−	0.140500i	$-0.0448711\pi$
$912$		0			0
$913$	250846.			0.300930
$914$		0			0
$915$			512727.i			0.612413i
$916$		0			0
$917$	−112555.			−0.133852
$918$		0			0
$919$	−672066.			−0.795757			−0.397879	−	0.917438i	$-0.630253\pi$
$919$	−672066.			−0.795757			−0.397879	+	0.917438i	$0.630253\pi$
$920$		0			0
$921$	−90542.0			−0.106741
$922$		0			0
$923$	−766368.			−0.899568
$924$		0			0
$925$			18759.9i			0.0219254i
$926$		0			0
$927$		−	420589.i		−	0.489438i
$928$		0			0
$929$	199293.			0.230919			0.115460	−	0.993312i	$-0.463166\pi$
$929$	199293.			0.230919			0.115460	+	0.993312i	$0.463166\pi$
$930$		0			0
$931$	290243.	−	152108.i	0.334859	−	0.175491i
$932$		0			0
$933$		−	1.58340e6i		−	1.81897i
$934$		0			0
$935$	−107729.			−0.123228
$936$		0			0
$937$	−1.10180e6			−1.25494			−0.627471	−	0.778640i	$-0.715910\pi$
$937$	−1.10180e6			−1.25494			−0.627471	+	0.778640i	$0.715910\pi$
$938$		0			0
$939$			339747.i			0.385323i
$940$		0			0
$941$		−	743721.i		−	0.839906i	−0.907546	−	0.419953i	$-0.862047\pi$
$941$		−	743721.i		−	0.839906i	0.907546	−	0.419953i	$-0.137953\pi$
$942$		0			0
$943$			10692.1i			0.0120238i
$944$		0			0
$945$			343011.i			0.384100i
$946$		0			0
$947$	−957119.			−1.06725			−0.533625	−	0.845721i	$-0.679170\pi$
$947$	−957119.			−1.06725			−0.533625	+	0.845721i	$0.679170\pi$
$948$		0			0
$949$			1.28142e6i			1.42285i
$950$		0			0
$951$	−1.35262e6			−1.49560
$952$		0			0
$953$		−	253186.i		−	0.278775i	−0.990238	−	0.139387i	$-0.955487\pi$
$953$		−	253186.i		−	0.278775i	0.990238	−	0.139387i	$-0.0445133\pi$
$954$		0			0
$955$	−284915.			−0.312398
$956$		0			0
$957$	214506.			0.234215
$958$		0			0
$959$	−1.39061e6			−1.51206
$960$		0			0
$961$	197217.			0.213549
$962$		0			0
$963$		−	665629.i		−	0.717761i
$964$		0			0
$965$			264086.i			0.283590i
$966$		0			0
$967$	98571.4			0.105414			0.0527070	−	0.998610i	$-0.483215\pi$
$967$	98571.4			0.105414			0.0527070	+	0.998610i	$0.483215\pi$
$968$		0			0
$969$	160494.	+	306245.i	0.170928	+	0.326153i
$970$		0			0
$971$			1.58528e6i			1.68138i	0.541515	+	0.840691i	$0.317851\pi$
$971$			1.58528e6i			1.68138i	−0.541515	+	0.840691i	$0.682149\pi$
$972$		0			0
$973$	−44670.0			−0.0471835
$974$		0			0
$975$	−837273.			−0.880761
$976$		0			0
$977$		−	789090.i		−	0.826680i	−0.910577	−	0.413340i	$-0.864362\pi$
$977$		−	789090.i		−	0.826680i	0.910577	−	0.413340i	$-0.135638\pi$
$978$		0			0
$979$			940066.i			0.980828i
$980$		0			0
$981$			472198.i			0.490666i
$982$		0			0
$983$		−	1.54784e6i		−	1.60184i	−0.598773	−	0.800919i	$-0.704345\pi$
$983$		−	1.54784e6i		−	1.60184i	0.598773	−	0.800919i	$-0.295655\pi$
$984$		0			0
$985$	−276505.			−0.284991
$986$		0			0
$987$		−	1.07661e6i		−	1.10516i
$988$		0			0
$989$	−197500.			−0.201918
$990$		0			0
$991$		−	860729.i		−	0.876434i	−0.898869	−	0.438217i	$-0.855610\pi$
$991$		−	860729.i		−	0.876434i	0.898869	−	0.438217i	$-0.144390\pi$
$992$		0			0
$993$	594089.			0.602494
$994$		0			0
$995$	233846.			0.236202
$996$		0			0
$997$	770151.			0.774793			0.387396	−	0.921913i	$-0.373375\pi$
$997$	770151.			0.774793			0.387396	+	0.921913i	$0.373375\pi$
$998$		0			0
$999$	28942.0			0.0290000

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.5.c.b.37.3	yes	4
3.2	odd	2		684.5.h.c.37.3		4
4.3	odd	2		304.5.e.d.113.2		4
19.18	odd	2	inner	76.5.c.b.37.2	✓	4
57.56	even	2		684.5.h.c.37.4		4
76.75	even	2		304.5.e.d.113.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.5.c.b.37.2	✓	4	19.18	odd	2	inner
76.5.c.b.37.3	yes	4	1.1	even	1	trivial
304.5.e.d.113.2		4	4.3	odd	2
304.5.e.d.113.3		4	76.75	even	2
684.5.h.c.37.3		4	3.2	odd	2
684.5.h.c.37.4		4	57.56	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	76.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+10.5914i q^{3} -16.8215 q^{5} -38.6430 q^{7} -31.1785 q^{9} +O(q^{10})\)	\(q+10.5914i q^{3} -16.8215 q^{5} -38.6430 q^{7} -31.1785 q^{9} -70.8215 q^{11} -231.121i q^{13} -178.164i q^{15} -90.4279 q^{17} +(-319.751 + 167.573i) q^{19} -409.285i q^{21} -285.538 q^{23} -342.037 q^{25} +527.681i q^{27} +285.969i q^{29} +852.235i q^{31} -750.101i q^{33} +650.034 q^{35} -54.8476i q^{37} +2447.90 q^{39} -37.4456i q^{41} +691.677 q^{43} +524.469 q^{45} +2630.47 q^{47} -907.716 q^{49} -957.761i q^{51} +3833.35i q^{53} +1191.33 q^{55} +(-1774.83 - 3386.62i) q^{57} +3339.72i q^{59} -2877.84 q^{61} +1204.83 q^{63} +3887.81i q^{65} +4052.74i q^{67} -3024.26i q^{69} -3315.87i q^{71} -5544.37 q^{73} -3622.66i q^{75} +2736.76 q^{77} -10304.4i q^{79} -8114.36 q^{81} -3541.95 q^{83} +1521.13 q^{85} -3028.82 q^{87} -13273.7i q^{89} +8931.22i q^{91} -9026.39 q^{93} +(5378.69 - 2818.82i) q^{95} -4281.99i q^{97} +2208.11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 22 q^{5} + 24 q^{7} - 214 q^{9}+O(q^{10}) \) 4 * q + 22 * q^5 + 24 * q^7 - 214 * q^9	\( 4 q + 22 q^{5} + 24 q^{7} - 214 q^{9} - 194 q^{11} - 1076 q^{17} - 654 q^{19} + 1090 q^{23} - 386 q^{25} + 4118 q^{35} - 2262 q^{39} + 4106 q^{43} - 3170 q^{45} + 5254 q^{47} - 1488 q^{49} + 926 q^{55} + 5490 q^{57} + 6078 q^{61} - 5270 q^{63} - 9856 q^{73} + 2822 q^{77} - 30136 q^{81} + 3868 q^{83} - 21862 q^{85} - 14526 q^{87} - 1284 q^{93} + 10354 q^{95} + 8386 q^{99}+O(q^{100}) \) 4 * q + 22 * q^5 + 24 * q^7 - 214 * q^9 - 194 * q^11 - 1076 * q^17 - 654 * q^19 + 1090 * q^23 - 386 * q^25 + 4118 * q^35 - 2262 * q^39 + 4106 * q^43 - 3170 * q^45 + 5254 * q^47 - 1488 * q^49 + 926 * q^55 + 5490 * q^57 + 6078 * q^61 - 5270 * q^63 - 9856 * q^73 + 2822 * q^77 - 30136 * q^81 + 3868 * q^83 - 21862 * q^85 - 14526 * q^87 - 1284 * q^93 + 10354 * q^95 + 8386 * q^99

Introduction

L-functions

Modular forms

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