LMFDB - Embedded newform 315.4.d.c.64.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [315,4,Mod(64,315)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("315.64"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$315 = 3^{2} \cdot 5 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	315.d (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,-36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$18.5856016518$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{10} + 55x^{8} + 983x^{6} + 6409x^{4} + 13560x^{2} + 3600$ x^10 + 55x^8 + 983x^6 + 6409x^4 + 13560x^2 + 3600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{2}\cdot 7^{4}$
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.9
Root			$5.04851i$ of defining polynomial
Character	$\chi$	$=$	315.64
Dual form			315.4.d.c.64.2

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.04851i q^{2} -8.39045 q^{4} +(-6.15172 - 9.33576i) q^{5} +7.00000i q^{7} -1.58074i q^{8} +(37.7959 - 24.9053i) q^{10} -6.78210 q^{11} -48.9221i q^{13} -28.3396 q^{14} -60.7239 q^{16} -92.4381i q^{17} +125.574 q^{19} +(51.6157 + 78.3312i) q^{20} -27.4574i q^{22} -32.2681i q^{23} +(-49.3128 + 114.862i) q^{25} +198.062 q^{26} -58.7332i q^{28} +282.778 q^{29} +205.434 q^{31} -258.488i q^{32} +374.237 q^{34} +(65.3503 - 43.0620i) q^{35} +190.627i q^{37} +508.388i q^{38} +(-14.7574 + 9.72429i) q^{40} -123.269 q^{41} -35.0202i q^{43} +56.9049 q^{44} +130.638 q^{46} -419.030i q^{47} -49.0000 q^{49} +(-465.020 - 199.643i) q^{50} +410.478i q^{52} -0.365379i q^{53} +(41.7215 + 63.3160i) q^{55} +11.0652 q^{56} +1144.83i q^{58} +328.317 q^{59} -515.707 q^{61} +831.704i q^{62} +560.699 q^{64} +(-456.725 + 300.955i) q^{65} -828.957i q^{67} +775.597i q^{68} +(174.337 + 264.572i) q^{70} +496.231 q^{71} -701.132i q^{73} -771.757 q^{74} -1053.62 q^{76} -47.4747i q^{77} -199.388 q^{79} +(373.556 + 566.904i) q^{80} -499.056i q^{82} -194.923i q^{83} +(-862.980 + 568.653i) q^{85} +141.780 q^{86} +10.7208i q^{88} +137.406 q^{89} +342.454 q^{91} +270.744i q^{92} +1696.45 q^{94} +(-772.496 - 1172.33i) q^{95} +220.440i q^{97} -198.377i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$10 q - 36 q^{4} - 6 q^{5} - 16 q^{10} - 84 q^{11} + 56 q^{14} + 148 q^{16} + 72 q^{19} + 68 q^{20} - 362 q^{25} + 620 q^{26} - 88 q^{29} + 120 q^{31} + 964 q^{34} + 28 q^{35} + 1396 q^{40} + 852 q^{41}+ \cdots + 1628 q^{95}+O(q^{100})$ 10 * q - 36 * q^4 - 6 * q^5 - 16 * q^10 - 84 * q^11 + 56 * q^14 + 148 * q^16 + 72 * q^19 + 68 * q^20 - 362 * q^25 + 620 * q^26 - 88 * q^29 + 120 * q^31 + 964 * q^34 + 28 * q^35 + 1396 * q^40 + 852 * q^41 + 1424 * q^44 + 176 * q^46 - 490 * q^49 - 1644 * q^50 - 1136 * q^55 - 588 * q^56 - 864 * q^59 - 884 * q^61 - 2724 * q^64 - 520 * q^65 + 756 * q^70 - 880 * q^71 - 3024 * q^74 - 2672 * q^76 - 3580 * q^79 + 2316 * q^80 + 1572 * q^85 - 2432 * q^86 + 1492 * q^89 + 476 * q^91 + 1652 * q^94 + 1628 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$ .

$n$	$127$	$136$	$281$
$\chi(n)$	$-1$	$1$	$1$

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

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$			4.04851i			1.43137i	0.698426	+	0.715683i	$0.253884\pi$
$2$			4.04851i			1.43137i	−0.698426	+	0.715683i	$0.746116\pi$
$3$		0			0
$3$		0			0
$4$	−8.39045			−1.04881
$5$	−6.15172	−	9.33576i	−0.550226	−	0.835016i
$5$	−6.15172	−	9.33576i	−0.550226	−	0.835016i
$6$		0			0
$7$			7.00000i			0.377964i
$7$			7.00000i			0.377964i
$8$		−	1.58074i		−	0.0698597i
$9$		0			0
$10$	37.7959	−	24.9053i	1.19521	−	0.787575i
$11$	−6.78210			−0.185898			−0.0929491	−	0.995671i	$-0.529629\pi$
$11$	−6.78210			−0.185898			−0.0929491	+	0.995671i	$0.529629\pi$
$12$		0			0
$13$		−	48.9221i		−	1.04373i	−0.853027	−	0.521867i	$-0.825236\pi$
$13$		−	48.9221i		−	1.04373i	0.853027	−	0.521867i	$-0.174764\pi$
$14$	−28.3396			−0.541005
$15$		0			0
$16$	−60.7239			−0.948812
$17$		−	92.4381i		−	1.31880i	−0.751794	−	0.659398i	$-0.770811\pi$
$17$		−	92.4381i		−	1.31880i	0.751794	−	0.659398i	$-0.229189\pi$
$18$		0			0
$19$	125.574			1.51625			0.758123	−	0.652112i	$-0.226117\pi$
$19$	125.574			1.51625			0.758123	+	0.652112i	$0.226117\pi$
$20$	51.6157	+	78.3312i	0.577081	+	0.875770i
$21$		0			0
$22$		−	27.4574i		−	0.266088i
$23$		−	32.2681i		−	0.292538i	−0.989245	−	0.146269i	$-0.953274\pi$
$23$		−	32.2681i		−	0.292538i	0.989245	−	0.146269i	$-0.0467265\pi$
$24$		0			0
$25$	−49.3128	+	114.862i	−0.394502	+	0.918895i
$26$	198.062			1.49396
$27$		0			0
$28$		−	58.7332i		−	0.396412i
$29$	282.778			1.81071			0.905354	−	0.424659i	$-0.139606\pi$
$29$	282.778			1.81071			0.905354	+	0.424659i	$0.139606\pi$
$30$		0			0
$31$	205.434			1.19023			0.595115	−	0.803641i	$-0.297107\pi$
$31$	205.434			1.19023			0.595115	+	0.803641i	$0.297107\pi$
$32$		−	258.488i		−	1.42796i
$33$		0			0
$34$	374.237			1.88768
$35$	65.3503	−	43.0620i	0.315606	−	0.207966i
$36$		0			0
$37$			190.627i			0.846998i	0.905897	+	0.423499i	$0.139198\pi$
$37$			190.627i			0.846998i	−0.905897	+	0.423499i	$0.860802\pi$
$38$			508.388i			2.17030i
$39$		0			0
$40$	−14.7574	+	9.72429i	−0.0583339	+	0.0384386i
$41$	−123.269			−0.469546			−0.234773	−	0.972050i	$-0.575435\pi$
$41$	−123.269			−0.469546			−0.234773	+	0.972050i	$0.575435\pi$
$42$		0			0
$43$		−	35.0202i		−	0.124198i	−0.998070	−	0.0620991i	$-0.980221\pi$
$43$		−	35.0202i		−	0.124198i	0.998070	−	0.0620991i	$-0.0197795\pi$
$44$	56.9049			0.194971
$45$		0			0
$46$	130.638			0.418728
$47$		−	419.030i		−	1.30046i	−0.759736	−	0.650231i	$-0.774672\pi$
$47$		−	419.030i		−	1.30046i	0.759736	−	0.650231i	$-0.225328\pi$
$48$		0			0
$49$	−49.0000			−0.142857
$50$	−465.020	−	199.643i	−1.31527	−	0.564677i
$51$		0			0
$52$			410.478i			1.09467i
$53$		−	0.365379i		−	0.000946956i	−1.00000		0.000473478i	$-0.999849\pi$
$53$		−	0.365379i		−	0.000946956i	1.00000		0.000473478i	$-0.000150713\pi$
$54$		0			0
$55$	41.7215	+	63.3160i	0.102286	+	0.155228i
$56$	11.0652			0.0264045
$57$		0			0
$58$			1144.83i			2.59178i
$59$	328.317			0.724461			0.362231	−	0.932088i	$-0.382015\pi$
$59$	328.317			0.724461			0.362231	+	0.932088i	$0.382015\pi$
$60$		0			0
$61$	−515.707			−1.08245			−0.541226	−	0.840877i	$-0.682040\pi$
$61$	−515.707			−1.08245			−0.541226	+	0.840877i	$0.682040\pi$
$62$			831.704i			1.70365i
$63$		0			0
$64$	560.699			1.09511
$65$	−456.725	+	300.955i	−0.871534	+	0.574290i
$66$		0			0
$67$		−	828.957i		−	1.51154i	−0.654837	−	0.755770i	$-0.727263\pi$
$67$		−	828.957i		−	1.51154i	0.654837	−	0.755770i	$-0.272737\pi$
$68$			775.597i			1.38316i
$69$		0			0
$70$	174.337	+	264.572i	0.297675	+	0.451748i
$71$	496.231			0.829462			0.414731	−	0.909944i	$-0.363876\pi$
$71$	496.231			0.829462			0.414731	+	0.909944i	$0.363876\pi$
$72$		0			0
$73$		−	701.132i		−	1.12413i	−0.827094	−	0.562064i	$-0.810008\pi$
$73$		−	701.132i		−	1.12413i	0.827094	−	0.562064i	$-0.189992\pi$
$74$	−771.757			−1.21236
$75$		0			0
$76$	−1053.62			−1.59025
$77$		−	47.4747i		−	0.0702629i
$78$		0			0
$79$	−199.388			−0.283961			−0.141981	−	0.989869i	$-0.545347\pi$
$79$	−199.388			−0.283961			−0.141981	+	0.989869i	$0.545347\pi$
$80$	373.556	+	566.904i	0.522061	+	0.792273i
$81$		0			0
$82$		−	499.056i		−	0.672092i
$83$		−	194.923i		−	0.257778i	−0.991659	−	0.128889i	$-0.958859\pi$
$83$		−	194.923i		−	0.257778i	0.991659	−	0.128889i	$-0.0411411\pi$
$84$		0			0
$85$	−862.980	+	568.653i	−1.10122	+	0.725636i
$86$	141.780			0.177773
$87$		0			0
$88$			10.7208i			0.0129868i
$89$	137.406			0.163651			0.0818257	−	0.996647i	$-0.473925\pi$
$89$	137.406			0.163651			0.0818257	+	0.996647i	$0.473925\pi$
$90$		0			0
$91$	342.454			0.394494
$92$			270.744i			0.306815i
$93$		0			0
$94$	1696.45			1.86144
$95$	−772.496	−	1172.33i	−0.834278	−	1.26609i
$96$		0			0
$97$			220.440i			0.230745i	0.993322	+	0.115372i	$0.0368062\pi$
$97$			220.440i			0.230745i	−0.993322	+	0.115372i	$0.963194\pi$
$98$		−	198.377i		−	0.204481i
$99$		0			0
$100$	413.756	−	963.743i	0.413756	−	0.963743i
$101$	591.358			0.582597			0.291298	−	0.956632i	$-0.405913\pi$
$101$	591.358			0.582597			0.291298	+	0.956632i	$0.405913\pi$
$102$		0			0
$103$			476.494i			0.455829i	0.973681	+	0.227914i	$0.0731906\pi$
$103$			476.494i			0.455829i	−0.973681	+	0.227914i	$0.926809\pi$
$104$	−77.3332			−0.0729149
$105$		0			0
$106$	1.47924			0.00135544
$107$			225.584i			0.203813i	0.994794	+	0.101907i	$0.0324943\pi$
$107$			225.584i			0.203813i	−0.994794	+	0.101907i	$0.967506\pi$
$108$		0			0
$109$	1627.65			1.43028			0.715142	−	0.698979i	$-0.246362\pi$
$109$	1627.65			1.43028			0.715142	+	0.698979i	$0.246362\pi$
$110$	−256.336	+	168.910i	−0.222188	+	0.146409i
$111$		0			0
$112$		−	425.068i		−	0.358617i
$113$			357.040i			0.297235i	0.988895	+	0.148617i	$0.0474823\pi$
$113$			357.040i			0.297235i	−0.988895	+	0.148617i	$0.952518\pi$
$114$		0			0
$115$	−301.247	+	198.504i	−0.244274	+	0.160962i
$116$	−2372.63			−1.89908
$117$		0			0
$118$			1329.19i			1.03697i
$119$	647.067			0.498458
$120$		0			0
$121$	−1285.00			−0.965442
$122$		−	2087.85i		−	1.54938i
$123$		0			0
$124$	−1723.69			−1.24832
$125$	1375.68	−	246.225i	0.984357	−	0.176185i
$126$		0			0
$127$		−	1728.25i		−	1.20754i	−0.797158	−	0.603771i	$-0.793664\pi$
$127$		−	1728.25i		−	1.20754i	0.797158	−	0.603771i	$-0.206336\pi$
$128$			202.094i			0.139553i
$129$		0			0
$130$	−1218.42	−	1849.06i	−0.822018	−	1.24748i
$131$	−1461.19			−0.974543			−0.487272	−	0.873250i	$-0.662008\pi$
$131$	−1461.19			−0.974543			−0.487272	+	0.873250i	$0.662008\pi$
$132$		0			0
$133$			879.018i			0.573087i
$134$	3356.04			2.16357
$135$		0			0
$136$	−146.121			−0.0921306
$137$		−	1892.96i		−	1.18049i	−0.807226	−	0.590243i	$-0.799032\pi$
$137$		−	1892.96i		−	1.18049i	0.807226	−	0.590243i	$-0.200968\pi$
$138$		0			0
$139$	1715.03			1.04652			0.523262	−	0.852172i	$-0.324715\pi$
$139$	1715.03			1.04652			0.523262	+	0.852172i	$0.324715\pi$
$140$	−548.319	+	361.310i	−0.331010	+	0.218116i
$141$		0			0
$142$			2009.00i			1.18726i
$143$			331.794i			0.194028i
$144$		0			0
$145$	−1739.57	−	2639.94i	−0.996299	−	1.51197i
$146$	2838.54			1.60904
$147$		0			0
$148$		−	1599.45i		−	0.888337i
$149$	−2798.32			−1.53857			−0.769286	−	0.638904i	$-0.779388\pi$
$149$	−2798.32			−1.53857			−0.769286	+	0.638904i	$0.779388\pi$
$150$		0			0
$151$	−2867.49			−1.54538			−0.772692	−	0.634781i	$-0.781090\pi$
$151$	−2867.49			−1.54538			−0.772692	+	0.634781i	$0.781090\pi$
$152$		−	198.500i		−	0.105924i
$153$		0			0
$154$	192.202			0.100572
$155$	−1263.77	−	1917.89i	−0.654896	−	0.993860i
$156$		0			0
$157$		−	783.696i		−	0.398381i	−0.979961	−	0.199190i	$-0.936169\pi$
$157$		−	783.696i		−	0.398381i	0.979961	−	0.199190i	$-0.0638312\pi$
$158$		−	807.226i		−	0.406452i
$159$		0			0
$160$	−2413.18	+	1590.14i	−1.19237	+	0.785699i
$161$	225.877			0.110569
$162$		0			0
$163$		−	2416.93i		−	1.16140i	−0.814116	−	0.580702i	$-0.802778\pi$
$163$		−	2416.93i		−	1.16140i	0.814116	−	0.580702i	$-0.197222\pi$
$164$	1034.28			0.492463
$165$		0			0
$166$	789.149			0.368975
$167$			704.424i			0.326407i	0.986592	+	0.163204i	$0.0521827\pi$
$167$			704.424i			0.326407i	−0.986592	+	0.163204i	$0.947817\pi$
$168$		0			0
$169$	−196.369			−0.0893803
$170$	−2302.20	−	3493.78i	−1.03865	−	1.57624i
$171$		0			0
$172$			293.835i			0.130260i
$173$		−	1398.49i		−	0.614598i	−0.951613	−	0.307299i	$-0.900575\pi$
$173$		−	1398.49i		−	0.614598i	0.951613	−	0.307299i	$-0.0994252\pi$
$174$		0			0
$175$	−804.033	−	345.189i	−0.347310	−	0.149108i
$176$	411.836			0.176382
$177$		0			0
$178$			556.289i			0.234245i
$179$	368.688			0.153950			0.0769749	−	0.997033i	$-0.475474\pi$
$179$	368.688			0.153950			0.0769749	+	0.997033i	$0.475474\pi$
$180$		0			0
$181$	−315.621			−0.129613			−0.0648064	−	0.997898i	$-0.520643\pi$
$181$	−315.621			−0.129613			−0.0648064	+	0.997898i	$0.520643\pi$
$182$			1386.43i			0.564665i
$183$		0			0
$184$	−51.0076			−0.0204366
$185$	1779.65	−	1172.69i	0.707257	−	0.466041i
$186$		0			0
$187$			626.924i			0.245162i
$188$			3515.85i			1.36393i
$189$		0			0
$190$	4746.19	−	3127.46i	1.81224	−	1.19416i
$191$	151.629			0.0574424			0.0287212	−	0.999587i	$-0.490857\pi$
$191$	151.629			0.0574424			0.0287212	+	0.999587i	$0.490857\pi$
$192$		0			0
$193$			690.689i			0.257600i	0.991671	+	0.128800i	$0.0411126\pi$
$193$			690.689i			0.257600i	−0.991671	+	0.128800i	$0.958887\pi$
$194$	−892.453			−0.330280
$195$		0			0
$196$	411.132			0.149829
$197$			834.136i			0.301674i	0.988559	+	0.150837i	$0.0481968\pi$
$197$			834.136i			0.301674i	−0.988559	+	0.150837i	$0.951803\pi$
$198$		0			0
$199$	−387.269			−0.137954			−0.0689769	−	0.997618i	$-0.521973\pi$
$199$	−387.269			−0.137954			−0.0689769	+	0.997618i	$0.521973\pi$
$200$	181.567	+	77.9509i	0.0641937	+	0.0275598i
$201$		0			0
$202$			2394.12i			0.833909i
$203$			1979.44i			0.684383i
$204$		0			0
$205$	758.316	+	1150.81i	0.258357	+	0.392078i
$206$	−1929.09			−0.652457
$207$		0			0
$208$			2970.74i			0.990307i
$209$	−851.656			−0.281867
$210$		0			0
$211$	3070.54			1.00182			0.500912	−	0.865498i	$-0.332998\pi$
$211$	3070.54			1.00182			0.500912	+	0.865498i	$0.332998\pi$
$212$			3.06570i			0.000993174i
$213$		0			0
$214$	−913.278			−0.291731
$215$	−326.940	+	215.434i	−0.103707	+	0.0683371i
$216$		0			0
$217$			1438.04i			0.449865i
$218$			6589.58i			2.04726i
$219$		0			0
$220$	−350.063	−	531.250i	−0.107278	−	0.162804i
$221$	−4522.26			−1.37647
$222$		0			0
$223$		−	1004.46i		−	0.301631i	−0.988562	−	0.150816i	$-0.951810\pi$
$223$		−	1004.46i		−	0.301631i	0.988562	−	0.150816i	$-0.0481900\pi$
$224$	1809.41			0.539716
$225$		0			0
$226$	−1445.48			−0.425451
$227$			5374.22i			1.57136i	0.618631	+	0.785681i	$0.287687\pi$
$227$			5374.22i			1.57136i	−0.618631	+	0.785681i	$0.712313\pi$
$228$		0			0
$229$	−3650.97			−1.05355			−0.526775	−	0.850005i	$-0.676599\pi$
$229$	−3650.97			−1.05355			−0.526775	+	0.850005i	$0.676599\pi$
$230$	−803.647	−	1219.60i	−0.230395	−	0.349645i
$231$		0			0
$232$		−	446.999i		−	0.126495i
$233$			4582.88i			1.28856i	0.764789	+	0.644280i	$0.222843\pi$
$233$			4582.88i			1.28856i	−0.764789	+	0.644280i	$0.777157\pi$
$234$		0			0
$235$	−3911.96	+	2577.75i	−1.08591	+	0.715549i
$236$	−2754.73			−0.759820
$237$		0			0
$238$			2619.66i			0.713476i
$239$	696.769			0.188578			0.0942892	−	0.995545i	$-0.469942\pi$
$239$	696.769			0.188578			0.0942892	+	0.995545i	$0.469942\pi$
$240$		0			0
$241$	5082.82			1.35856			0.679281	−	0.733878i	$-0.262292\pi$
$241$	5082.82			1.35856			0.679281	+	0.733878i	$0.262292\pi$
$242$		−	5202.35i		−	1.38190i
$243$		0			0
$244$	4327.02			1.13528
$245$	301.434	+	457.452i	0.0786037	+	0.119288i
$246$		0			0
$247$		−	6143.34i		−	1.58256i
$248$		−	324.739i		−	0.0831490i
$249$		0			0
$250$	996.847	+	5569.46i	0.252185	+	1.40897i
$251$	1207.32			0.303606			0.151803	−	0.988411i	$-0.451492\pi$
$251$	1207.32			0.303606			0.151803	+	0.988411i	$0.451492\pi$
$252$		0			0
$253$			218.846i			0.0543822i
$254$	6996.86			1.72843
$255$		0			0
$256$	3667.41			0.895363
$257$		−	510.936i		−	0.124013i	−0.998076	−	0.0620064i	$-0.980250\pi$
$257$		−	510.936i		−	0.124013i	0.998076	−	0.0620064i	$-0.0197499\pi$
$258$		0			0
$259$	−1334.39			−0.320135
$260$	3832.13	−	2525.15i	0.914070	−	0.602319i
$261$		0			0
$262$		−	5915.67i		−	1.39493i
$263$			2269.04i			0.531997i	0.963974	+	0.265998i	$0.0857015\pi$
$263$			2269.04i			0.531997i	−0.963974	+	0.265998i	$0.914298\pi$
$264$		0			0
$265$	−3.41109	+	2.24771i	−0.000790723	+	0.000521040i
$266$	−3558.72			−0.820297
$267$		0			0
$268$			6955.32i			1.58531i
$269$	−7836.10			−1.77612			−0.888058	−	0.459731i	$-0.847946\pi$
$269$	−7836.10			−1.77612			−0.888058	+	0.459731i	$0.847946\pi$
$270$		0			0
$271$	1466.28			0.328673			0.164336	−	0.986404i	$-0.447452\pi$
$271$	1466.28			0.328673			0.164336	+	0.986404i	$0.447452\pi$
$272$			5613.21i			1.25129i
$273$		0			0
$274$	7663.67			1.68971
$275$	334.444	−	779.005i	0.0733372	−	0.170821i
$276$		0			0
$277$			4154.17i			0.901083i	0.892755	+	0.450542i	$0.148769\pi$
$277$			4154.17i			0.901083i	−0.892755	+	0.450542i	$0.851231\pi$
$278$			6943.32i			1.49796i
$279$		0			0
$280$	−68.0700	−	103.302i	−0.0145284	−	0.0220481i
$281$	3490.40			0.740996			0.370498	−	0.928833i	$-0.379187\pi$
$281$	3490.40			0.740996			0.370498	+	0.928833i	$0.379187\pi$
$282$		0			0
$283$			3125.29i			0.656464i	0.944597	+	0.328232i	$0.106453\pi$
$283$			3125.29i			0.656464i	−0.944597	+	0.328232i	$0.893547\pi$
$284$	−4163.60			−0.869945
$285$		0			0
$286$	−1343.27			−0.277725
$287$		−	862.883i		−	0.177472i
$288$		0			0
$289$	−3631.80			−0.739223
$290$	10687.8	−	7042.66i	2.16418	−	1.42607i
$291$		0			0
$292$			5882.81i			1.17899i
$293$		−	1447.69i		−	0.288652i	−0.989530	−	0.144326i	$-0.953899\pi$
$293$		−	1447.69i		−	0.288652i	0.989530	−	0.144326i	$-0.0461014\pi$
$294$		0			0
$295$	−2019.71	−	3065.09i	−0.398618	−	0.604936i
$296$	301.333			0.0591710
$297$		0			0
$298$		−	11329.0i		−	2.20226i
$299$	−1578.62			−0.305332
$300$		0			0
$301$	245.141			0.0469425
$302$		−	11609.1i		−	2.21201i
$303$		0			0
$304$	−7625.35			−1.43863
$305$	3172.48	+	4814.52i	0.595593	+	0.903864i
$306$		0			0
$307$			1591.43i			0.295856i	0.988998	+	0.147928i	$0.0472603\pi$
$307$			1591.43i			0.295856i	−0.988998	+	0.147928i	$0.952740\pi$
$308$			398.334i			0.0736922i
$309$		0			0
$310$	7764.59	−	5116.41i	1.42258	−	0.937395i
$311$	8584.92			1.56529			0.782647	−	0.622466i	$-0.213869\pi$
$311$	8584.92			1.56529			0.782647	+	0.622466i	$0.213869\pi$
$312$		0			0
$313$			7210.05i			1.30203i	0.759064	+	0.651016i	$0.225657\pi$
$313$			7210.05i			1.30203i	−0.759064	+	0.651016i	$0.774343\pi$
$314$	3172.80			0.570228
$315$		0			0
$316$	1672.96			0.297820
$317$		−	7787.24i		−	1.37973i	−0.723938	−	0.689865i	$-0.757670\pi$
$317$		−	7787.24i		−	1.37973i	0.723938	−	0.689865i	$-0.242330\pi$
$318$		0			0
$319$	−1917.83			−0.336607
$320$	−3449.26	−	5234.55i	−0.602561	−	0.914438i
$321$		0			0
$322$			914.465i			0.158264i
$323$		−	11607.8i		−	1.99962i
$324$		0			0
$325$	5619.28	+	2412.48i	0.959082	+	0.411755i
$326$	9784.98			1.66239
$327$		0			0
$328$			194.857i			0.0328023i
$329$	2933.21			0.491529
$330$		0			0
$331$	−1729.78			−0.287243			−0.143621	−	0.989633i	$-0.545875\pi$
$331$	−1729.78			−0.287243			−0.143621	+	0.989633i	$0.545875\pi$
$332$			1635.49i			0.270359i
$333$		0			0
$334$	−2851.87			−0.467208
$335$	−7738.94	+	5099.51i	−1.26216	+	0.831689i
$336$		0			0
$337$		−	7815.06i		−	1.26325i	−0.775276	−	0.631623i	$-0.782389\pi$
$337$		−	7815.06i		−	1.26325i	0.775276	−	0.631623i	$-0.217611\pi$
$338$		−	795.000i		−	0.127936i
$339$		0			0
$340$	7240.79	−	4771.26i	1.15496	−	0.761052i
$341$	−1393.28			−0.221262
$342$		0			0
$343$		−	343.000i		−	0.0539949i
$344$	−55.3579			−0.00867645
$345$		0			0
$346$	5661.82			0.879714
$347$			2359.77i			0.365070i	0.983199	+	0.182535i	$0.0584303\pi$
$347$			2359.77i			0.365070i	−0.983199	+	0.182535i	$0.941570\pi$
$348$		0			0
$349$	−3212.73			−0.492761			−0.246380	−	0.969173i	$-0.579241\pi$
$349$	−3212.73			−0.492761			−0.246380	+	0.969173i	$0.579241\pi$
$350$	1397.50	−	3255.14i	0.213428	−	0.497127i
$351$		0			0
$352$			1753.09i			0.265454i
$353$			5582.04i			0.841649i	0.907142	+	0.420824i	$0.138259\pi$
$353$			5582.04i			0.841649i	−0.907142	+	0.420824i	$0.861741\pi$
$354$		0			0
$355$	−3052.67	−	4632.69i	−0.456392	−	0.692614i
$356$	−1152.90			−0.171639
$357$		0			0
$358$			1492.64i			0.220358i
$359$	−10630.6			−1.56285			−0.781425	−	0.623999i	$-0.785507\pi$
$359$	−10630.6			−1.56285			−0.781425	+	0.623999i	$0.785507\pi$
$360$		0			0
$361$	8909.84			1.29900
$362$		−	1277.80i		−	0.185523i
$363$		0			0
$364$	−2873.35			−0.413748
$365$	−6545.60	+	4313.17i	−0.938664	+	0.618524i
$366$		0			0
$367$		−	4514.69i		−	0.642139i	−0.947056	−	0.321070i	$-0.895958\pi$
$367$		−	4514.69i		−	0.642139i	0.947056	−	0.321070i	$-0.104042\pi$
$368$			1959.45i			0.277563i
$369$		0			0
$370$	4747.63	+	7204.94i	0.667074	+	1.01234i
$371$	2.55765			0.000357916
$372$		0			0
$373$			11445.8i			1.58885i	0.607361	+	0.794426i	$0.292228\pi$
$373$			11445.8i			1.58885i	−0.607361	+	0.794426i	$0.707772\pi$
$374$	−2538.11			−0.350916
$375$		0			0
$376$	−662.378			−0.0908499
$377$		−	13834.1i		−	1.88990i
$378$		0			0
$379$	−8146.48			−1.10411			−0.552054	−	0.833809i	$-0.686156\pi$
$379$	−8146.48			−1.10411			−0.552054	+	0.833809i	$0.686156\pi$
$380$	6481.59	+	9836.37i	0.874996	+	1.32788i
$381$		0			0
$382$			613.872i			0.0822210i
$383$			5261.56i			0.701967i	0.936382	+	0.350983i	$0.114153\pi$
$383$			5261.56i			0.701967i	−0.936382	+	0.350983i	$0.885847\pi$
$384$		0			0
$385$	−443.212	+	292.051i	−0.0586706	+	0.0386605i
$386$	−2796.26			−0.368720
$387$		0			0
$388$		−	1849.59i		−	0.242007i
$389$	13207.1			1.72141			0.860706	−	0.509103i	$-0.170023\pi$
$389$	13207.1			1.72141			0.860706	+	0.509103i	$0.170023\pi$
$390$		0			0
$391$	−2982.80			−0.385798
$392$			77.4564i			0.00997995i
$393$		0			0
$394$	−3377.01			−0.431805
$395$	1226.58	+	1861.44i	0.156243	+	0.237112i
$396$		0			0
$397$			5663.15i			0.715933i	0.933734	+	0.357967i	$0.116530\pi$
$397$			5663.15i			0.715933i	−0.933734	+	0.357967i	$0.883470\pi$
$398$		−	1567.86i		−	0.197462i
$399$		0			0
$400$	2994.47	−	6974.87i	0.374308	−	0.871858i
$401$	11989.0			1.49302			0.746512	−	0.665372i	$-0.231727\pi$
$401$	11989.0			1.49302			0.746512	+	0.665372i	$0.231727\pi$
$402$		0			0
$403$		−	10050.3i		−	1.24228i
$404$	−4961.76			−0.611031
$405$		0			0
$406$	−8013.80			−0.979602
$407$		−	1292.85i		−	0.157455i
$408$		0			0
$409$	−5249.67			−0.634669			−0.317334	−	0.948314i	$-0.602788\pi$
$409$	−5249.67			−0.634669			−0.317334	+	0.948314i	$0.602788\pi$
$410$	−4659.07	+	3070.05i	−0.561207	+	0.369803i
$411$		0			0
$412$		−	3998.00i		−	0.478076i
$413$			2298.22i			0.273821i
$414$		0			0
$415$	−1819.76	+	1199.11i	−0.215249	+	0.141836i
$416$	−12645.7			−1.49041
$417$		0			0
$418$		−	3447.94i		−	0.403455i
$419$	−14948.9			−1.74297			−0.871484	−	0.490424i	$-0.836842\pi$
$419$	−14948.9			−1.74297			−0.871484	+	0.490424i	$0.836842\pi$
$420$		0			0
$421$	5840.31			0.676103			0.338051	−	0.941128i	$-0.390232\pi$
$421$	5840.31			0.676103			0.338051	+	0.941128i	$0.390232\pi$
$422$			12431.1i			1.43398i
$423$		0			0
$424$	−0.577571			−6.61540e−5
$425$	10617.6	+	4558.38i	1.21184	+	0.520268i
$426$		0			0
$427$		−	3609.95i		−	0.409128i
$428$		−	1892.75i		−	0.213760i
$429$		0			0
$430$	−872.188	−	1323.62i	−0.0978154	−	0.148443i
$431$	−7439.21			−0.831402			−0.415701	−	0.909501i	$-0.636464\pi$
$431$	−7439.21			−0.831402			−0.415701	+	0.909501i	$0.636464\pi$
$432$		0			0
$433$		−	877.657i		−	0.0974077i	−0.998813	−	0.0487038i	$-0.984491\pi$
$433$		−	877.657i		−	0.0974077i	0.998813	−	0.0487038i	$-0.0155090\pi$
$434$	−5821.93			−0.643920
$435$		0			0
$436$	−13656.8			−1.50009
$437$		−	4052.04i		−	0.443559i
$438$		0			0
$439$	10855.8			1.18023			0.590114	−	0.807320i	$-0.299083\pi$
$439$	10855.8			1.18023			0.590114	+	0.807320i	$0.299083\pi$
$440$	100.086	−	65.9511i	0.0108442	−	0.00714567i
$441$		0			0
$442$		−	18308.4i		−	1.97023i
$443$			10797.0i			1.15798i	0.815336	+	0.578988i	$0.196552\pi$
$443$			10797.0i			1.15798i	−0.815336	+	0.578988i	$0.803448\pi$
$444$		0			0
$445$	−845.281	−	1282.79i	−0.0900453	−	0.136651i
$446$	4066.58			0.431745
$447$		0			0
$448$			3924.89i			0.413914i
$449$	8621.70			0.906198			0.453099	−	0.891460i	$-0.350318\pi$
$449$	8621.70			0.906198			0.453099	+	0.891460i	$0.350318\pi$
$450$		0			0
$451$	836.023			0.0872878
$452$		−	2995.73i		−	0.311742i
$453$		0			0
$454$	−21757.6			−2.24919
$455$	−2106.68	−	3197.07i	−0.217061	−	0.329409i
$456$		0			0
$457$			3785.90i			0.387520i	0.981049	+	0.193760i	$0.0620684\pi$
$457$			3785.90i			0.387520i	−0.981049	+	0.193760i	$0.937932\pi$
$458$		−	14781.0i		−	1.50801i
$459$		0			0
$460$	2527.60	−	1665.54i	0.256196	−	0.168818i
$461$	−5760.18			−0.581948			−0.290974	−	0.956731i	$-0.593979\pi$
$461$	−5760.18			−0.581948			−0.290974	+	0.956731i	$0.593979\pi$
$462$		0			0
$463$			10760.9i			1.08014i	0.841621	+	0.540068i	$0.181602\pi$
$463$			10760.9i			1.08014i	−0.841621	+	0.540068i	$0.818398\pi$
$464$	−17171.4			−1.71802
$465$		0			0
$466$	−18553.9			−1.84440
$467$			2153.58i			0.213395i	0.994292	+	0.106698i	$0.0340277\pi$
$467$			2153.58i			0.213395i	−0.994292	+	0.106698i	$0.965972\pi$
$468$		0			0
$469$	5802.70			0.571309
$470$	−10436.1	−	15837.6i	−1.02421	−	1.55433i
$471$		0			0
$472$		−	518.985i		−	0.0506106i
$473$			237.510i			0.0230882i
$474$		0			0
$475$	−6192.40	+	14423.7i	−0.598162	+	1.39327i
$476$	−5429.18			−0.522786
$477$		0			0
$478$			2820.88i			0.269924i
$479$	−6890.26			−0.657253			−0.328626	−	0.944460i	$-0.606586\pi$
$479$	−6890.26			−0.657253			−0.328626	+	0.944460i	$0.606586\pi$
$480$		0			0
$481$	9325.88			0.884041
$482$			20577.9i			1.94460i
$483$		0			0
$484$	10781.8			1.01256
$485$	2057.97	−	1356.08i	0.192676	−	0.126962i
$486$		0			0
$487$			19006.4i			1.76850i	0.467011	+	0.884251i	$0.345331\pi$
$487$			19006.4i			1.76850i	−0.467011	+	0.884251i	$0.654669\pi$
$488$			815.201i			0.0756197i
$489$		0			0
$490$	−1852.00	+	1220.36i	−0.170745	+	0.112511i
$491$	−4530.05			−0.416371			−0.208186	−	0.978089i	$-0.566756\pi$
$491$	−4530.05			−0.416371			−0.208186	+	0.978089i	$0.566756\pi$
$492$		0			0
$493$		−	26139.4i		−	2.38795i
$494$	24871.4			2.26522
$495$		0			0
$496$	−12474.8			−1.12930
$497$			3473.62i			0.313507i
$498$		0			0
$499$	−3620.18			−0.324773			−0.162386	−	0.986727i	$-0.551919\pi$
$499$	−3620.18			−0.324773			−0.162386	+	0.986727i	$0.551919\pi$
$500$	−11542.6	+	2065.94i	−1.03240	+	0.184784i
$501$		0			0
$502$			4887.84i			0.434571i
$503$		−	11761.8i		−	1.04261i	−0.853372	−	0.521303i	$-0.825446\pi$
$503$		−	11761.8i		−	1.04261i	0.853372	−	0.521303i	$-0.174554\pi$
$504$		0			0
$505$	−3637.86	−	5520.77i	−0.320560	−	0.486478i
$506$	−885.999			−0.0778408
$507$		0			0
$508$			14500.8i			1.26648i
$509$	−5254.47			−0.457564			−0.228782	−	0.973478i	$-0.573474\pi$
$509$	−5254.47			−0.457564			−0.228782	+	0.973478i	$0.573474\pi$
$510$		0			0
$511$	4907.92			0.424880
$512$			16464.3i			1.42114i
$513$		0			0
$514$	2068.53			0.177508
$515$	4448.43	−	2931.26i	0.380624	−	0.250809i
$516$		0			0
$517$			2841.90i			0.241754i
$518$		−	5402.30i		−	0.458230i
$519$		0			0
$520$	475.732	+	721.964i	0.0401197	+	0.0608851i
$521$	15511.8			1.30439			0.652193	−	0.758053i	$-0.273849\pi$
$521$	15511.8			1.30439			0.652193	+	0.758053i	$0.273849\pi$
$522$		0			0
$523$			3814.73i			0.318942i	0.987203	+	0.159471i	$0.0509788\pi$
$523$			3814.73i			0.318942i	−0.987203	+	0.159471i	$0.949021\pi$
$524$	12260.1			1.02211
$525$		0			0
$526$	−9186.24			−0.761481
$527$		−	18990.0i		−	1.56967i
$528$		0			0
$529$	11125.8			0.914422
$530$	−9.09988	−	13.8098i	−0.000745799	−	0.00113181i
$531$		0			0
$532$		−	7375.36i		−	0.601057i
$533$			6030.58i			0.490081i
$534$		0			0
$535$	2106.00	−	1387.73i	0.170187	−	0.112143i
$536$	−1310.37			−0.105596
$537$		0			0
$538$		−	31724.5i		−	2.54227i
$539$	332.323			0.0265569
$540$		0			0
$541$	−4573.77			−0.363478			−0.181739	−	0.983347i	$-0.558173\pi$
$541$	−4573.77			−0.363478			−0.181739	+	0.983347i	$0.558173\pi$
$542$			5936.26i			0.470451i
$543$		0			0
$544$	−23894.1			−1.88318
$545$	−10012.9	−	15195.4i	−0.786980	−	1.19431i
$546$		0			0
$547$		−	13327.6i		−	1.04177i	−0.853627	−	0.520885i	$-0.825602\pi$
$547$		−	13327.6i		−	1.04177i	0.853627	−	0.520885i	$-0.174398\pi$
$548$			15882.8i			1.23810i
$549$		0			0
$550$	3153.81	+	1354.00i	0.244507	+	0.104972i
$551$	35509.5			2.74548
$552$		0			0
$553$		−	1395.72i		−	0.107327i
$554$	−16818.2			−1.28978
$555$		0			0
$556$	−14389.9			−1.09760
$557$			12096.1i			0.920155i	0.887879	+	0.460077i	$0.152178\pi$
$557$			12096.1i			0.920155i	−0.887879	+	0.460077i	$0.847822\pi$
$558$		0			0
$559$	−1713.26			−0.129630
$560$	−3968.33	+	2614.90i	−0.299451	+	0.197321i
$561$		0			0
$562$			14130.9i			1.06064i
$563$		−	22943.6i		−	1.71751i	−0.512387	−	0.858755i	$-0.671239\pi$
$563$		−	22943.6i		−	1.71751i	0.512387	−	0.858755i	$-0.328761\pi$
$564$		0			0
$565$	3333.24	−	2196.41i	0.248196	−	0.163546i
$566$	−12652.8			−0.939639
$567$		0			0
$568$		−	784.414i		−	0.0579459i
$569$	485.307			0.0357560			0.0178780	−	0.999840i	$-0.494309\pi$
$569$	485.307			0.0357560			0.0178780	+	0.999840i	$0.494309\pi$
$570$		0			0
$571$	12271.8			0.899401			0.449701	−	0.893179i	$-0.351531\pi$
$571$	12271.8			0.899401			0.449701	+	0.893179i	$0.351531\pi$
$572$		−	2783.90i		−	0.203498i
$573$		0			0
$574$	3493.39			0.254027
$575$	3706.38	+	1591.23i	0.268811	+	0.115407i
$576$		0			0
$577$			14122.8i			1.01896i	0.860484	+	0.509478i	$0.170162\pi$
$577$			14122.8i			1.01896i	−0.860484	+	0.509478i	$0.829838\pi$
$578$		−	14703.4i		−	1.05810i
$579$		0			0
$580$	14595.8	+	22150.3i	1.04492	+	1.58576i
$581$	1364.46			0.0974310
$582$		0			0
$583$			2.47804i			0.000176037i
$584$	−1108.31			−0.0785312
$585$		0			0
$586$	5861.00			0.413167
$587$			11005.3i			0.773826i	0.922116	+	0.386913i	$0.126459\pi$
$587$			11005.3i			0.773826i	−0.922116	+	0.386913i	$0.873541\pi$
$588$		0			0
$589$	25797.2			1.80468
$590$	12409.0	−	8176.83i	0.865885	−	0.570567i
$591$		0			0
$592$		−	11575.6i		−	0.803642i
$593$			11233.1i			0.777889i	0.921261	+	0.388944i	$0.127160\pi$
$593$			11233.1i			0.777889i	−0.921261	+	0.388944i	$0.872840\pi$
$594$		0			0
$595$	−3980.57	−	6040.86i	−0.274265	−	0.416220i
$596$	23479.2			1.61366
$597$		0			0
$598$		−	6391.08i		−	0.437041i
$599$	14855.4			1.01331			0.506656	−	0.862149i	$-0.330882\pi$
$599$	14855.4			1.01331			0.506656	+	0.862149i	$0.330882\pi$
$600$		0			0
$601$	25358.8			1.72115			0.860573	−	0.509327i	$-0.170106\pi$
$601$	25358.8			1.72115			0.860573	+	0.509327i	$0.170106\pi$
$602$			992.457i			0.0671919i
$603$		0			0
$604$	24059.5			1.62081
$605$	7904.97	+	11996.5i	0.531211	+	0.806159i
$606$		0			0
$607$		−	14393.9i		−	0.962487i	−0.876587	−	0.481243i	$-0.840185\pi$
$607$		−	14393.9i		−	0.962487i	0.876587	−	0.481243i	$-0.159815\pi$
$608$		−	32459.3i		−	2.16513i
$609$		0			0
$610$	−19491.6	+	12843.8i	−1.29376	+	0.852511i
$611$	−20499.8			−1.35734
$612$		0			0
$613$			4769.94i			0.314284i	0.987576	+	0.157142i	$0.0502280\pi$
$613$			4769.94i			0.314284i	−0.987576	+	0.157142i	$0.949772\pi$
$614$	−6442.92			−0.423477
$615$		0			0
$616$	−75.0453			−0.00490854
$617$		−	7769.86i		−	0.506974i	−0.967339	−	0.253487i	$-0.918423\pi$
$617$		−	7769.86i		−	0.506974i	0.967339	−	0.253487i	$-0.0815775\pi$
$618$		0			0
$619$	5680.75			0.368867			0.184433	−	0.982845i	$-0.440955\pi$
$619$	5680.75			0.368867			0.184433	+	0.982845i	$0.440955\pi$
$620$	10603.6	+	16091.9i	0.686859	+	1.04237i
$621$		0			0
$622$			34756.2i			2.24051i
$623$			961.840i			0.0618544i
$624$		0			0
$625$	−10761.5	−	11328.3i	−0.688736	−	0.725012i
$626$	−29190.0			−1.86368
$627$		0			0
$628$			6575.56i			0.417824i
$629$	17621.2			1.11702
$630$		0			0
$631$	−10395.8			−0.655864			−0.327932	−	0.944701i	$-0.606352\pi$
$631$	−10395.8			−0.655864			−0.327932	+	0.944701i	$0.606352\pi$
$632$			315.182i			0.0198374i
$633$		0			0
$634$	31526.7			1.97490
$635$	−16134.6	+	10631.7i	−1.00832	+	0.664421i
$636$		0			0
$637$			2397.18i			0.149105i
$638$		−	7764.34i		−	0.481808i
$639$		0			0
$640$	1886.70	−	1243.23i	0.116529	−	0.0767857i
$641$	8194.28			0.504921			0.252461	−	0.967607i	$-0.418760\pi$
$641$	8194.28			0.504921			0.252461	+	0.967607i	$0.418760\pi$
$642$		0			0
$643$		−	32118.2i		−	1.96985i	−0.172970	−	0.984927i	$-0.555336\pi$
$643$		−	32118.2i		−	1.96985i	0.172970	−	0.984927i	$-0.444664\pi$
$644$	−1895.21			−0.115965
$645$		0			0
$646$	46994.4			2.86218
$647$		−	22299.0i		−	1.35496i	−0.735539	−	0.677482i	$-0.763071\pi$
$647$		−	22299.0i		−	1.35496i	0.735539	−	0.677482i	$-0.236929\pi$
$648$		0			0
$649$	−2226.68			−0.134676
$650$	−9766.97	+	22749.7i	−0.589372	+	1.37280i
$651$		0			0
$652$			20279.1i			1.21809i
$653$		−	920.410i		−	0.0551584i	−0.999620	−	0.0275792i	$-0.991220\pi$
$653$		−	920.410i		−	0.0551584i	0.999620	−	0.0275792i	$-0.00877984\pi$
$654$		0			0
$655$	8988.86	+	13641.4i	0.536219	+	0.813759i
$656$	7485.38			0.445511
$657$		0			0
$658$			11875.1i			0.703557i
$659$	17824.3			1.05362			0.526812	−	0.849982i	$-0.323387\pi$
$659$	17824.3			1.05362			0.526812	+	0.849982i	$0.323387\pi$
$660$		0			0
$661$	11343.8			0.667510			0.333755	−	0.942660i	$-0.391684\pi$
$661$	11343.8			0.667510			0.333755	+	0.942660i	$0.391684\pi$
$662$		−	7003.03i		−	0.411149i
$663$		0			0
$664$	−308.123			−0.0180083
$665$	8206.30	−	5407.47i	0.478536	−	0.315327i
$666$		0			0
$667$		−	9124.71i		−	0.529700i
$668$		−	5910.44i		−	0.342338i
$669$		0			0
$670$	−20645.4	−	31331.2i	−1.19045	−	1.80661i
$671$	3497.58			0.201226
$672$		0			0
$673$		−	13422.9i		−	0.768816i	−0.923163	−	0.384408i	$-0.874406\pi$
$673$		−	13422.9i		−	0.768816i	0.923163	−	0.384408i	$-0.125594\pi$
$674$	31639.4			1.80817
$675$		0			0
$676$	1647.62			0.0937426
$677$		−	1066.77i		−	0.0605602i	−0.999541	−	0.0302801i	$-0.990360\pi$
$677$		−	1066.77i		−	0.0605602i	0.999541	−	0.0302801i	$-0.00963993\pi$
$678$		0			0
$679$	−1543.08			−0.0872134
$680$	898.895	+	1364.15i	0.0506927	+	0.0769305i
$681$		0			0
$682$		−	5640.70i		−	0.316706i
$683$			19090.6i			1.06952i	0.845005	+	0.534759i	$0.179598\pi$
$683$			19090.6i			1.06952i	−0.845005	+	0.534759i	$0.820402\pi$
$684$		0			0
$685$	−17672.2	+	11645.0i	−0.985724	+	0.649534i
$686$	1388.64			0.0772865
$687$		0			0
$688$			2126.56i			0.117841i
$689$	−17.8751			−0.000988370
$690$		0			0
$691$	−16878.4			−0.929208			−0.464604	−	0.885518i	$-0.653803\pi$
$691$	−16878.4			−0.929208			−0.464604	+	0.885518i	$0.653803\pi$
$692$			11734.0i			0.644594i
$693$		0			0
$694$	−9553.57			−0.522549
$695$	−10550.4	−	16011.1i	−0.575825	−	0.873864i
$696$		0			0
$697$			11394.8i			0.619236i
$698$		−	13006.8i		−	0.705321i
$699$		0			0
$700$	6746.20	+	2896.29i	0.364261	+	0.156385i
$701$	−30272.6			−1.63107			−0.815535	−	0.578707i	$-0.803557\pi$
$701$	−30272.6			−1.63107			−0.815535	+	0.578707i	$0.803557\pi$
$702$		0			0
$703$			23937.8i			1.28426i
$704$	−3802.71			−0.203580
$705$		0			0
$706$	−22599.0			−1.20471
$707$			4139.50i			0.220201i
$708$		0			0
$709$	6593.32			0.349248			0.174624	−	0.984635i	$-0.444129\pi$
$709$	6593.32			0.349248			0.174624	+	0.984635i	$0.444129\pi$
$710$	18755.5	−	12358.8i	0.991383	−	0.653263i
$711$		0			0
$712$		−	217.203i		−	0.0114326i
$713$		−	6628.99i		−	0.348187i
$714$		0			0
$715$	3097.55	−	2041.10i	0.162017	−	0.106759i
$716$	−3093.46			−0.161464
$717$		0			0
$718$		−	43038.2i		−	2.23701i
$719$	3293.72			0.170842			0.0854208	−	0.996345i	$-0.472777\pi$
$719$	3293.72			0.170842			0.0854208	+	0.996345i	$0.472777\pi$
$720$		0			0
$721$	−3335.46			−0.172287
$722$			36071.6i			1.85934i
$723$		0			0
$724$	2648.20			0.135939
$725$	−13944.6	+	32480.4i	−0.714328	+	1.66385i
$726$		0			0
$727$		−	27757.8i		−	1.41606i	−0.706181	−	0.708032i	$-0.749583\pi$
$727$		−	27757.8i		−	1.41606i	0.706181	−	0.708032i	$-0.250417\pi$
$728$		−	541.333i		−	0.0275592i
$729$		0			0
$730$	−17461.9	−	26499.9i	−0.885334	−	1.34357i
$731$	−3237.20			−0.163792
$732$		0			0
$733$			38324.6i			1.93117i	0.260080	+	0.965587i	$0.416251\pi$
$733$			38324.6i			1.93117i	−0.260080	+	0.965587i	$0.583749\pi$
$734$	18277.8			0.919136
$735$		0			0
$736$	−8340.91			−0.417731
$737$			5622.07i			0.280993i
$738$		0			0
$739$	−21957.3			−1.09298			−0.546490	−	0.837466i	$-0.684036\pi$
$739$	−21957.3			−1.09298			−0.546490	+	0.837466i	$0.684036\pi$
$740$	−14932.1	+	9839.36i	−0.741775	+	0.488786i
$741$		0			0
$742$			10.3547i			0.000512308i
$743$			14695.0i			0.725580i	0.931871	+	0.362790i	$0.118176\pi$
$743$			14695.0i			0.725580i	−0.931871	+	0.362790i	$0.881824\pi$
$744$		0			0
$745$	17214.5	+	26124.4i	0.846563	+	1.28473i
$746$	−46338.5			−2.27423
$747$		0			0
$748$		−	5260.18i		−	0.257127i
$749$	−1579.09			−0.0770341
$750$		0			0
$751$	−21439.1			−1.04171			−0.520855	−	0.853645i	$-0.674387\pi$
$751$	−21439.1			−1.04171			−0.520855	+	0.853645i	$0.674387\pi$
$752$			25445.1i			1.23389i
$753$		0			0
$754$	56007.4			2.70513
$755$	17640.0	+	26770.2i	0.850311	+	1.29042i
$756$		0			0
$757$			23896.8i			1.14735i	0.819083	+	0.573675i	$0.194483\pi$
$757$			23896.8i			1.14735i	−0.819083	+	0.573675i	$0.805517\pi$
$758$		−	32981.1i		−	1.58038i
$759$		0			0
$760$	−1853.15	+	1221.12i	−0.0884485	+	0.0582824i
$761$	24436.9			1.16404			0.582022	−	0.813173i	$-0.302262\pi$
$761$	24436.9			1.16404			0.582022	+	0.813173i	$0.302262\pi$
$762$		0			0
$763$			11393.6i			0.540597i
$764$	−1272.24			−0.0602459
$765$		0			0
$766$	−21301.5			−1.00477
$767$		−	16061.9i		−	0.756145i
$768$		0			0
$769$	30689.3			1.43912			0.719560	−	0.694430i	$-0.244343\pi$
$769$	30689.3			1.43912			0.719560	+	0.694430i	$0.244343\pi$
$770$	−1182.37	−	1794.35i	−0.0553373	−	0.0839791i
$771$		0			0
$772$		−	5795.19i		−	0.270173i
$773$			5110.74i			0.237801i	0.992906	+	0.118901i	$0.0379370\pi$
$773$			5110.74i			0.237801i	−0.992906	+	0.118901i	$0.962063\pi$
$774$		0			0
$775$	−10130.5	+	23596.6i	−0.469548	+	1.09370i
$776$	348.459			0.0161198
$777$		0			0
$778$			53469.3i			2.46397i
$779$	−15479.4			−0.711947
$780$		0			0
$781$	−3365.49			−0.154195
$782$		−	12075.9i		−	0.552217i
$783$		0			0
$784$	2975.47			0.135545
$785$	−7316.40	+	4821.08i	−0.332654	+	0.219199i
$786$		0			0
$787$		−	6991.30i		−	0.316662i	−0.987386	−	0.158331i	$-0.949389\pi$
$787$		−	6991.30i		−	0.316662i	0.987386	−	0.158331i	$-0.0506113\pi$
$788$		−	6998.78i		−	0.316397i
$789$		0			0
$790$	−7536.06	+	4965.82i	−0.339394	+	0.223641i
$791$	−2499.28			−0.112344
$792$		0			0
$793$			25229.5i			1.12979i
$794$	−22927.3			−1.02476
$795$		0			0
$796$	3249.36			0.144687
$797$			4020.31i			0.178678i	0.996001	+	0.0893392i	$0.0284755\pi$
$797$			4020.31i			0.178678i	−0.996001	+	0.0893392i	$0.971524\pi$
$798$		0			0
$799$	−38734.3			−1.71505
$800$	29690.4	+	12746.7i	1.31214	+	0.563332i
$801$		0			0
$802$			48537.7i			2.13706i
$803$			4755.15i			0.208973i
$804$		0			0
$805$	−1389.53	−	2108.73i	−0.0608379	−	0.0923267i
$806$	40688.7			1.77816
$807$		0			0
$808$		−	934.785i		−	0.0407000i
$809$	−41608.1			−1.80824			−0.904119	−	0.427281i	$-0.859472\pi$
$809$	−41608.1			−1.80824			−0.904119	+	0.427281i	$0.859472\pi$
$810$		0			0
$811$	42271.3			1.83027			0.915133	−	0.403152i	$-0.132086\pi$
$811$	42271.3			1.83027			0.915133	+	0.403152i	$0.132086\pi$
$812$		−	16608.4i		−	0.717785i
$813$		0			0
$814$	5234.13			0.225376
$815$	−22563.9	+	14868.3i	−0.969790	+	0.639035i
$816$		0			0
$817$		−	4397.62i		−	0.188315i
$818$		−	21253.4i		−	0.908443i
$819$		0			0
$820$	−6362.62	−	9655.82i	−0.270966	−	0.411214i
$821$	−27184.7			−1.15561			−0.577804	−	0.816176i	$-0.696090\pi$
$821$	−27184.7			−1.15561			−0.577804	+	0.816176i	$0.696090\pi$
$822$		0			0
$823$		−	12967.9i		−	0.549250i	−0.961551	−	0.274625i	$-0.911446\pi$
$823$		−	12967.9i		−	0.549250i	0.961551	−	0.274625i	$-0.0885538\pi$
$824$	753.215			0.0318440
$825$		0			0
$826$	−9304.36			−0.391937
$827$		−	33111.2i		−	1.39225i	−0.717921	−	0.696124i	$-0.754906\pi$
$827$		−	33111.2i		−	1.39225i	0.717921	−	0.696124i	$-0.245094\pi$
$828$		0			0
$829$	−3715.75			−0.155673			−0.0778367	−	0.996966i	$-0.524801\pi$
$829$	−3715.75			−0.155673			−0.0778367	+	0.996966i	$0.524801\pi$
$830$	−4854.62	−	7367.30i	−0.203020	−	0.308100i
$831$		0			0
$832$		−	27430.5i		−	1.14301i
$833$			4529.47i			0.188399i
$834$		0			0
$835$	6576.33	−	4333.42i	0.272555	−	0.179598i
$836$	7145.77			0.295624
$837$		0			0
$838$		−	60521.0i		−	2.49482i
$839$	−1460.56			−0.0601005			−0.0300502	−	0.999548i	$-0.509567\pi$
$839$	−1460.56			−0.0601005			−0.0300502	+	0.999548i	$0.509567\pi$
$840$		0			0
$841$	55574.2			2.27866
$842$			23644.6i			0.967750i
$843$		0			0
$844$	−25763.2			−1.05072
$845$	1208.00	+	1833.25i	0.0491794	+	0.0746339i
$846$		0			0
$847$		−	8995.02i		−	0.364903i
$848$			22.1873i			0.000898483i
$849$		0			0
$850$	−18454.7	+	42985.5i	−0.744694	+	1.73458i
$851$	6151.19			0.247779
$852$		0			0
$853$		−	36027.4i		−	1.44613i	−0.690777	−	0.723067i	$-0.742732\pi$
$853$		−	36027.4i		−	1.44613i	0.690777	−	0.723067i	$-0.257268\pi$
$854$	14614.9			0.585612
$855$		0			0
$856$	356.590			0.0142383
$857$			23500.7i			0.936719i	0.883538	+	0.468359i	$0.155155\pi$
$857$			23500.7i			0.936719i	−0.883538	+	0.468359i	$0.844845\pi$
$858$		0			0
$859$	−5551.11			−0.220491			−0.110245	−	0.993904i	$-0.535164\pi$
$859$	−5551.11			−0.220491			−0.110245	+	0.993904i	$0.535164\pi$
$860$	2743.17	−	1807.59i	0.108769	−	0.0716724i
$861$		0			0
$862$		−	30117.7i		−	1.19004i
$863$		−	27721.4i		−	1.09345i	−0.837312	−	0.546725i	$-0.815874\pi$
$863$		−	27721.4i		−	1.09345i	0.837312	−	0.546725i	$-0.184126\pi$
$864$		0			0
$865$	−13056.0	+	8603.13i	−0.513199	+	0.338168i
$866$	3553.21			0.139426
$867$		0			0
$868$		−	12065.8i		−	0.471821i
$869$	1352.27			0.0527878
$870$		0			0
$871$	−40554.3			−1.57765
$872$		−	2572.90i		−	0.0999192i
$873$		0			0
$874$	16404.7			0.634895
$875$	1723.58	+	9629.77i	0.0665915	+	0.372052i
$876$		0			0
$877$		−	47255.6i		−	1.81951i	−0.415149	−	0.909754i	$-0.636270\pi$
$877$		−	47255.6i		−	1.81951i	0.415149	−	0.909754i	$-0.363730\pi$
$878$			43950.0i			1.68934i
$879$		0			0
$880$	−2533.50	−	3844.80i	−0.0970502	−	0.147282i
$881$	−32267.0			−1.23394			−0.616972	−	0.786985i	$-0.711641\pi$
$881$	−32267.0			−1.23394			−0.616972	+	0.786985i	$0.711641\pi$
$882$		0			0
$883$			5062.08i			0.192925i	0.995337	+	0.0964623i	$0.0307527\pi$
$883$			5062.08i			0.192925i	−0.995337	+	0.0964623i	$0.969247\pi$
$884$	37943.8			1.44365
$885$		0			0
$886$	−43712.0			−1.65749
$887$			7626.08i			0.288679i	0.989528	+	0.144340i	$0.0461058\pi$
$887$			7626.08i			0.288679i	−0.989528	+	0.144340i	$0.953894\pi$
$888$		0			0
$889$	12097.8			0.456408
$890$	5193.38	−	3422.13i	0.195598	−	0.128888i
$891$		0			0
$892$			8427.89i			0.316353i
$893$		−	52619.2i		−	1.97182i
$894$		0			0
$895$	−2268.06	−	3441.98i	−0.0847072	−	0.128550i
$896$	−1414.66			−0.0527460
$897$		0			0
$898$			34905.0i			1.29710i
$899$	58092.3			2.15516
$900$		0			0
$901$	−33.7750			−0.00124884
$902$			3384.65i			0.124941i
$903$		0			0
$904$	564.389			0.0207647
$905$	1941.61	+	2946.56i	0.0713164	+	0.108229i
$906$		0			0
$907$			8546.31i			0.312873i	0.987688	+	0.156436i	$0.0500006\pi$
$907$			8546.31i			0.312873i	−0.987688	+	0.156436i	$0.949999\pi$
$908$		−	45092.1i		−	1.64806i
$909$		0			0
$910$	12943.4	−	8528.93i	0.471504	−	0.310694i
$911$	−8778.92			−0.319274			−0.159637	−	0.987176i	$-0.551032\pi$
$911$	−8778.92			−0.319274			−0.159637	+	0.987176i	$0.551032\pi$
$912$		0			0
$913$			1321.99i			0.0479205i
$914$	−15327.3			−0.554683
$915$		0			0
$916$	30633.3			1.10497
$917$		−	10228.4i		−	0.368343i
$918$		0			0
$919$	−21626.0			−0.776253			−0.388126	−	0.921606i	$-0.626878\pi$
$919$	−21626.0			−0.776253			−0.388126	+	0.921606i	$0.626878\pi$
$920$	313.784	+	476.195i	0.0112447	+	0.0170649i
$921$		0			0
$922$		−	23320.1i		−	0.832981i
$923$		−	24276.7i		−	0.865738i
$924$		0			0
$925$	−21895.8	−	9400.36i	−0.778302	−	0.334143i
$926$	−43565.8			−1.54607
$927$		0			0
$928$		−	73094.5i		−	2.58561i
$929$	12262.6			0.433073			0.216536	−	0.976275i	$-0.430524\pi$
$929$	12262.6			0.433073			0.216536	+	0.976275i	$0.430524\pi$
$930$		0			0
$931$	−6153.13			−0.216606
$932$		−	38452.5i		−	1.35145i
$933$		0			0
$934$	−8718.78			−0.305447
$935$	5852.82	−	3856.66i	0.204714	−	0.134894i
$936$		0			0
$937$			13362.6i			0.465889i	0.972490	+	0.232945i	$0.0748361\pi$
$937$			13362.6i			0.465889i	−0.972490	+	0.232945i	$0.925164\pi$
$938$			23492.3i			0.817751i
$939$		0			0
$940$	32823.1	−	21628.5i	1.13891	−	0.750472i
$941$	46330.1			1.60501			0.802506	−	0.596643i	$-0.203499\pi$
$941$	46330.1			1.60501			0.802506	+	0.596643i	$0.203499\pi$
$942$		0			0
$943$			3977.66i			0.137360i
$944$	−19936.7			−0.687377
$945$		0			0
$946$	−961.563			−0.0330477
$947$		−	17387.6i		−	0.596643i	−0.954465	−	0.298321i	$-0.903573\pi$
$947$		−	17387.6i		−	0.596643i	0.954465	−	0.298321i	$-0.0964267\pi$
$948$		0			0
$949$	−34300.8			−1.17329
$950$	−58394.4	−	25070.0i	−1.99428	−	0.856188i
$951$		0			0
$952$		−	1022.85i		−	0.0348221i
$953$			42384.6i			1.44068i	0.693619	+	0.720342i	$0.256015\pi$
$953$			42384.6i			1.44068i	−0.693619	+	0.720342i	$0.743985\pi$
$954$		0			0
$955$	−932.779	−	1415.57i	−0.0316063	−	0.0479653i
$956$	−5846.20			−0.197782
$957$		0			0
$958$		−	27895.3i		−	0.940769i
$959$	13250.7			0.446182
$960$		0			0
$961$	12412.3			0.416647
$962$			37755.9i			1.26539i
$963$		0			0
$964$	−42647.2			−1.42487
$965$	6448.10	−	4248.92i	0.215100	−	0.141738i
$966$		0			0
$967$			15771.6i			0.524489i	0.965001	+	0.262245i	$0.0844627\pi$
$967$			15771.6i			0.524489i	−0.965001	+	0.262245i	$0.915537\pi$
$968$			2031.26i			0.0674454i
$969$		0			0
$970$	5490.12	+	8331.72i	0.181729	+	0.275789i
$971$	−36370.5			−1.20204			−0.601022	−	0.799232i	$-0.705240\pi$
$971$	−36370.5			−1.20204			−0.601022	+	0.799232i	$0.705240\pi$
$972$		0			0
$973$			12005.2i			0.395549i
$974$	−76947.5			−2.53137
$975$		0			0
$976$	31315.8			1.02704
$977$		−	35122.8i		−	1.15013i	−0.818108	−	0.575065i	$-0.804977\pi$
$977$		−	35122.8i		−	1.15013i	0.818108	−	0.575065i	$-0.195023\pi$
$978$		0			0
$979$	−931.899			−0.0304225
$980$	−2529.17	−	3838.23i	−0.0824401	−	0.125110i
$981$		0			0
$982$		−	18340.0i		−	0.595979i
$983$			12798.9i			0.415282i	0.978205	+	0.207641i	$0.0665785\pi$
$983$			12798.9i			0.415282i	−0.978205	+	0.207641i	$0.933421\pi$
$984$		0			0
$985$	7787.30	−	5131.37i	0.251902	−	0.165989i
$986$	105826.			3.41803
$987$		0			0
$988$			51545.4i			1.65980i
$989$	−1130.03			−0.0363327
$990$		0			0
$991$	−46594.9			−1.49358			−0.746789	−	0.665061i	$-0.768406\pi$
$991$	−46594.9			−1.49358			−0.746789	+	0.665061i	$0.768406\pi$
$992$		−	53102.3i		−	1.69960i
$993$		0			0
$994$	−14063.0			−0.448743
$995$	2382.37	+	3615.45i	0.0759058	+	0.115194i
$996$		0			0
$997$		−	47534.1i		−	1.50995i	−0.655753	−	0.754975i	$-0.727649\pi$
$997$		−	47534.1i		−	1.50995i	0.655753	−	0.754975i	$-0.272351\pi$
$998$		−	14656.4i		−	0.464869i
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.4.d.c.64.9		10
3.2	odd	2		35.4.b.a.29.2	✓	10
5.2	odd	4		1575.4.a.bq.1.1		5
5.3	odd	4		1575.4.a.bn.1.5		5
5.4	even	2	inner	315.4.d.c.64.2		10
12.11	even	2		560.4.g.f.449.9		10
15.2	even	4		175.4.a.i.1.5		5
15.8	even	4		175.4.a.j.1.1		5
15.14	odd	2		35.4.b.a.29.9	yes	10
21.2	odd	6		245.4.j.e.214.2		20
21.5	even	6		245.4.j.f.214.2		20
21.11	odd	6		245.4.j.e.79.9		20
21.17	even	6		245.4.j.f.79.9		20
21.20	even	2		245.4.b.d.99.2		10
60.59	even	2		560.4.g.f.449.2		10
105.44	odd	6		245.4.j.e.214.9		20
105.59	even	6		245.4.j.f.79.2		20
105.62	odd	4		1225.4.a.be.1.5		5
105.74	odd	6		245.4.j.e.79.2		20
105.83	odd	4		1225.4.a.bh.1.1		5
105.89	even	6		245.4.j.f.214.9		20
105.104	even	2		245.4.b.d.99.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.b.a.29.2	✓	10	3.2	odd	2
35.4.b.a.29.9	yes	10	15.14	odd	2
175.4.a.i.1.5		5	15.2	even	4
175.4.a.j.1.1		5	15.8	even	4
245.4.b.d.99.2		10	21.20	even	2
245.4.b.d.99.9		10	105.104	even	2
245.4.j.e.79.2		20	105.74	odd	6
245.4.j.e.79.9		20	21.11	odd	6
245.4.j.e.214.2		20	21.2	odd	6
245.4.j.e.214.9		20	105.44	odd	6
245.4.j.f.79.2		20	105.59	even	6
245.4.j.f.79.9		20	21.17	even	6
245.4.j.f.214.2		20	21.5	even	6
245.4.j.f.214.9		20	105.89	even	6
315.4.d.c.64.2		10	5.4	even	2	inner
315.4.d.c.64.9		10	1.1	even	1	trivial
560.4.g.f.449.2		10	60.59	even	2
560.4.g.f.449.9		10	12.11	even	2
1225.4.a.be.1.5		5	105.62	odd	4
1225.4.a.bh.1.1		5	105.83	odd	4
1575.4.a.bn.1.5		5	5.3	odd	4
1575.4.a.bq.1.1		5	5.2	odd	4

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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	315.4.d.c.64.9

Level	$315$
Weight	$4$
Character	315.64
Analytic conductor	$18.586$
Analytic rank	$0$
Dimension	$10$
Inner twists	$2$