LMFDB - Embedded newform 1344.4.b.h.895.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(895,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1344.895");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 2x^{10} - 6x^{9} + 56x^{7} - 448x^{6} + 448x^{5} - 3072x^{3} - 8192x^{2} - 32768x + 262144 $ x^12 - x^11 - 2x^10 - 6x^9 + 56x^7 - 448x^6 + 448x^5 - 3072x^3 - 8192x^2 - 32768x + 262144
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{30} $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			895.8
Root			$-2.78362 + 0.501431i$ of defining polynomial
Character	$\chi$	$=$	1344.895
Dual form			1344.4.b.h.895.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +4.57514i q^{5} +(2.93118 + 18.2868i) q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +4.57514i q^{5} +(2.93118 + 18.2868i) q^{7} +9.00000 q^{9} +26.0206i q^{11} +75.0905i q^{13} +13.7254i q^{15} -115.976i q^{17} +119.051 q^{19} +(8.79355 + 54.8605i) q^{21} -61.4339i q^{23} +104.068 q^{25} +27.0000 q^{27} +71.9146 q^{29} +231.587 q^{31} +78.0617i q^{33} +(-83.6647 + 13.4106i) q^{35} -13.3306 q^{37} +225.271i q^{39} +144.133i q^{41} +288.638i q^{43} +41.1762i q^{45} -343.549 q^{47} +(-325.816 + 107.204i) q^{49} -347.927i q^{51} -142.666 q^{53} -119.048 q^{55} +357.152 q^{57} -403.919 q^{59} +21.7765i q^{61} +(26.3807 + 164.581i) q^{63} -343.549 q^{65} +598.674i q^{67} -184.302i q^{69} +589.524i q^{71} +6.85314i q^{73} +312.204 q^{75} +(-475.834 + 76.2711i) q^{77} -972.231i q^{79} +81.0000 q^{81} +429.745 q^{83} +530.605 q^{85} +215.744 q^{87} +1083.91i q^{89} +(-1373.17 + 220.104i) q^{91} +694.761 q^{93} +544.673i q^{95} +1031.01i q^{97} +234.185i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 36 q^{3} + 10 q^{7} + 108 q^{9}+O(q^{10}) $ 12 * q + 36 * q^3 + 10 * q^7 + 108 * q^9	$ 12 q + 36 q^{3} + 10 q^{7} + 108 q^{9} + 84 q^{19} + 30 q^{21} - 216 q^{25} + 324 q^{27} - 200 q^{29} - 384 q^{31} - 84 q^{35} + 244 q^{37} + 280 q^{47} - 424 q^{49} + 16 q^{53} + 212 q^{55} + 252 q^{57} - 1168 q^{59} + 90 q^{63} + 280 q^{65} - 648 q^{75} - 968 q^{77} + 972 q^{81} + 968 q^{83} + 852 q^{85} - 600 q^{87} - 1648 q^{91} - 1152 q^{93}+O(q^{100}) $ 12 * q + 36 * q^3 + 10 * q^7 + 108 * q^9 + 84 * q^19 + 30 * q^21 - 216 * q^25 + 324 * q^27 - 200 * q^29 - 384 * q^31 - 84 * q^35 + 244 * q^37 + 280 * q^47 - 424 * q^49 + 16 * q^53 + 212 * q^55 + 252 * q^57 - 1168 * q^59 + 90 * q^63 + 280 * q^65 - 648 * q^75 - 968 * q^77 + 972 * q^81 + 968 * q^83 + 852 * q^85 - 600 * q^87 - 1648 * q^91 - 1152 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$-1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$		0			0
$2$		0			0
$3$	3.00000			0.577350
$3$	3.00000			0.577350
$4$		0			0
$5$			4.57514i			0.409213i	0.978844	+	0.204606i	$0.0655914\pi$
$5$			4.57514i			0.409213i	−0.978844	+	0.204606i	$0.934409\pi$
$6$		0			0
$7$	2.93118	+	18.2868i	0.158269	+	0.987396i
$7$	2.93118	+	18.2868i	0.158269	+	0.987396i
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$			26.0206i			0.713227i	0.934252	+	0.356613i	$0.116069\pi$
$11$			26.0206i			0.713227i	−0.934252	+	0.356613i	$0.883931\pi$
$12$		0			0
$13$			75.0905i			1.60203i	0.598647	+	0.801013i	$0.295705\pi$
$13$			75.0905i			1.60203i	−0.598647	+	0.801013i	$0.704295\pi$
$14$		0			0
$15$			13.7254i			0.236259i
$16$		0			0
$17$		−	115.976i		−	1.65460i	−0.561758	−	0.827302i	$-0.689875\pi$
$17$		−	115.976i		−	1.65460i	0.561758	−	0.827302i	$-0.310125\pi$
$18$		0			0
$19$	119.051			1.43748			0.718739	−	0.695280i	$-0.244720\pi$
$19$	119.051			1.43748			0.718739	+	0.695280i	$0.244720\pi$
$20$		0			0
$21$	8.79355	+	54.8605i	0.0913767	+	0.570073i
$22$		0			0
$23$		−	61.4339i		−	0.556950i	−0.960443	−	0.278475i	$-0.910171\pi$
$23$		−	61.4339i		−	0.556950i	0.960443	−	0.278475i	$-0.0898290\pi$
$24$		0			0
$25$	104.068			0.832545
$26$		0			0
$27$	27.0000			0.192450
$28$		0			0
$29$	71.9146			0.460490			0.230245	−	0.973133i	$-0.426047\pi$
$29$	71.9146			0.460490			0.230245	+	0.973133i	$0.426047\pi$
$30$		0			0
$31$	231.587			1.34175			0.670875	−	0.741571i	$-0.265919\pi$
$31$	231.587			1.34175			0.670875	+	0.741571i	$0.265919\pi$
$32$		0			0
$33$			78.0617i			0.411782i
$34$		0			0
$35$	−83.6647	+	13.4106i	−0.404055	+	0.0647657i
$36$		0			0
$37$	−13.3306			−0.0592309			−0.0296155	−	0.999561i	$-0.509428\pi$
$37$	−13.3306			−0.0592309			−0.0296155	+	0.999561i	$0.509428\pi$
$38$		0			0
$39$			225.271i			0.924931i
$40$		0			0
$41$			144.133i			0.549018i	0.961584	+	0.274509i	$0.0885154\pi$
$41$			144.133i			0.549018i	−0.961584	+	0.274509i	$0.911485\pi$
$42$		0			0
$43$			288.638i			1.02365i	0.859090	+	0.511825i	$0.171030\pi$
$43$			288.638i			1.02365i	−0.859090	+	0.511825i	$0.828970\pi$
$44$		0			0
$45$			41.1762i			0.136404i
$46$		0			0
$47$	−343.549			−1.06621			−0.533104	−	0.846050i	$-0.678975\pi$
$47$	−343.549			−1.06621			−0.533104	+	0.846050i	$0.678975\pi$
$48$		0			0
$49$	−325.816	+	107.204i	−0.949902	+	0.312548i
$50$		0			0
$51$		−	347.927i		−	0.955286i
$52$		0			0
$53$	−142.666			−0.369749			−0.184875	−	0.982762i	$-0.559188\pi$
$53$	−142.666			−0.369749			−0.184875	+	0.982762i	$0.559188\pi$
$54$		0			0
$55$	−119.048			−0.291861
$56$		0			0
$57$	357.152			0.829928
$58$		0			0
$59$	−403.919			−0.891284			−0.445642	−	0.895211i	$-0.647025\pi$
$59$	−403.919			−0.891284			−0.445642	+	0.895211i	$0.647025\pi$
$60$		0			0
$61$			21.7765i			0.0457082i	0.999739	+	0.0228541i	$0.00727532\pi$
$61$			21.7765i			0.0457082i	−0.999739	+	0.0228541i	$0.992725\pi$
$62$		0			0
$63$	26.3807	+	164.581i	0.0527564	+	0.329132i
$64$		0			0
$65$	−343.549			−0.655570
$66$		0			0
$67$			598.674i			1.09164i	0.837904	+	0.545818i	$0.183781\pi$
$67$			598.674i			1.09164i	−0.837904	+	0.545818i	$0.816219\pi$
$68$		0			0
$69$		−	184.302i		−	0.321555i
$70$		0			0
$71$			589.524i			0.985403i	0.870198	+	0.492701i	$0.163991\pi$
$71$			589.524i			0.985403i	−0.870198	+	0.492701i	$0.836009\pi$
$72$		0			0
$73$			6.85314i			0.0109877i	0.999985	+	0.00549383i	$0.00174875\pi$
$73$			6.85314i			0.0109877i	−0.999985	+	0.00549383i	$0.998251\pi$
$74$		0			0
$75$	312.204			0.480670
$76$		0			0
$77$	−475.834	+	76.2711i	−0.704237	+	0.112882i
$78$		0			0
$79$		−	972.231i		−	1.38461i	−0.721603	−	0.692307i	$-0.756594\pi$
$79$		−	972.231i		−	1.38461i	0.721603	−	0.692307i	$-0.243406\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	429.745			0.568321			0.284161	−	0.958777i	$-0.408285\pi$
$83$	429.745			0.568321			0.284161	+	0.958777i	$0.408285\pi$
$84$		0			0
$85$	530.605			0.677085
$86$		0			0
$87$	215.744			0.265864
$88$		0			0
$89$			1083.91i			1.29095i	0.763781	+	0.645476i	$0.223341\pi$
$89$			1083.91i			1.29095i	−0.763781	+	0.645476i	$0.776659\pi$
$90$		0			0
$91$	−1373.17	+	220.104i	−1.58183	+	0.253551i
$92$		0			0
$93$	694.761			0.774660
$94$		0			0
$95$			544.673i			0.588234i
$96$		0			0
$97$			1031.01i			1.07921i	0.841920	+	0.539603i	$0.181426\pi$
$97$			1031.01i			1.07921i	−0.841920	+	0.539603i	$0.818574\pi$
$98$		0			0
$99$			234.185i			0.237742i
$100$		0			0
$101$			207.988i			0.204907i	0.994738	+	0.102454i	$0.0326693\pi$
$101$			207.988i			0.204907i	−0.994738	+	0.102454i	$0.967331\pi$
$102$		0			0
$103$	−1229.81			−1.17647			−0.588237	−	0.808689i	$-0.700178\pi$
$103$	−1229.81			−1.17647			−0.588237	+	0.808689i	$0.700178\pi$
$104$		0			0
$105$	−250.994	+	40.2317i	−0.233281	+	0.0373925i
$106$		0			0
$107$		−	1502.68i		−	1.35766i	−0.734294	−	0.678832i	$-0.762487\pi$
$107$		−	1502.68i		−	1.35766i	0.734294	−	0.678832i	$-0.237513\pi$
$108$		0			0
$109$	−579.301			−0.509055			−0.254527	−	0.967066i	$-0.581920\pi$
$109$	−579.301			−0.509055			−0.254527	+	0.967066i	$0.581920\pi$
$110$		0			0
$111$	−39.9919			−0.0341970
$112$		0			0
$113$	614.637			0.511683			0.255842	−	0.966719i	$-0.417647\pi$
$113$	614.637			0.511683			0.255842	+	0.966719i	$0.417647\pi$
$114$		0			0
$115$	281.069			0.227911
$116$		0			0
$117$			675.814i			0.534009i
$118$		0			0
$119$	2120.83	−	339.946i	1.63375	−	0.261873i
$120$		0			0
$121$	653.930			0.491308
$122$		0			0
$123$			432.398i			0.316976i
$124$		0			0
$125$			1048.02i			0.749901i
$126$		0			0
$127$			273.874i			0.191358i	0.995412	+	0.0956788i	$0.0305022\pi$
$127$			273.874i			0.191358i	−0.995412	+	0.0956788i	$0.969498\pi$
$128$		0			0
$129$			865.915i			0.591004i
$130$		0			0
$131$	2187.84			1.45918			0.729591	−	0.683884i	$-0.239710\pi$
$131$	2187.84			1.45918			0.729591	+	0.683884i	$0.239710\pi$
$132$		0			0
$133$	348.959	+	2177.06i	0.227508	+	1.41936i
$134$		0			0
$135$			123.529i			0.0787530i
$136$		0			0
$137$	−2535.39			−1.58112			−0.790558	−	0.612388i	$-0.790209\pi$
$137$	−2535.39			−1.58112			−0.790558	+	0.612388i	$0.790209\pi$
$138$		0			0
$139$	−1007.32			−0.614675			−0.307338	−	0.951601i	$-0.599438\pi$
$139$	−1007.32			−0.614675			−0.307338	+	0.951601i	$0.599438\pi$
$140$		0			0
$141$	−1030.65			−0.615576
$142$		0			0
$143$	−1953.90			−1.14261
$144$		0			0
$145$			329.019i			0.188438i
$146$		0			0
$147$	−977.449	+	321.612i	−0.548426	+	0.180450i
$148$		0			0
$149$	−2342.59			−1.28800			−0.644002	−	0.765024i	$-0.722727\pi$
$149$	−2342.59			−1.28800			−0.644002	+	0.765024i	$0.722727\pi$
$150$		0			0
$151$		−	561.022i		−	0.302353i	−0.988507	−	0.151177i	$-0.951694\pi$
$151$		−	561.022i		−	0.302353i	0.988507	−	0.151177i	$-0.0483062\pi$
$152$		0			0
$153$		−	1043.78i		−	0.551534i
$154$		0			0
$155$			1059.54i			0.549061i
$156$		0			0
$157$			459.220i			0.233438i	0.993165	+	0.116719i	$0.0372377\pi$
$157$			459.220i			0.233438i	−0.993165	+	0.116719i	$0.962762\pi$
$158$		0			0
$159$	−427.998			−0.213475
$160$		0			0
$161$	1123.43	−	180.074i	0.549931	−	0.0881480i
$162$		0			0
$163$			558.186i			0.268224i	0.990966	+	0.134112i	$0.0428182\pi$
$163$			558.186i			0.268224i	−0.990966	+	0.134112i	$0.957182\pi$
$164$		0			0
$165$	−357.143			−0.168506
$166$		0			0
$167$	2494.32			1.15579			0.577893	−	0.816113i	$-0.303875\pi$
$167$	2494.32			1.15579			0.577893	+	0.816113i	$0.303875\pi$
$168$		0			0
$169$	−3441.58			−1.56649
$170$		0			0
$171$	1071.45			0.479159
$172$		0			0
$173$			2881.62i			1.26639i	0.773993	+	0.633195i	$0.218257\pi$
$173$			2881.62i			1.26639i	−0.773993	+	0.633195i	$0.781743\pi$
$174$		0			0
$175$	305.043	+	1903.08i	0.131766	+	0.822052i
$176$		0			0
$177$	−1211.76			−0.514583
$178$		0			0
$179$		−	2072.63i		−	0.865452i	−0.901525	−	0.432726i	$-0.857552\pi$
$179$		−	2072.63i		−	0.865452i	0.901525	−	0.432726i	$-0.142448\pi$
$180$		0			0
$181$			2764.12i			1.13511i	0.823335	+	0.567556i	$0.192111\pi$
$181$			2764.12i			1.13511i	−0.823335	+	0.567556i	$0.807889\pi$
$182$		0			0
$183$			65.3296i			0.0263896i
$184$		0			0
$185$		−	60.9895i		−	0.0242380i
$186$		0			0
$187$	3017.76			1.18011
$188$		0			0
$189$	79.1420	+	493.744i	0.0304589	+	0.190024i
$190$		0			0
$191$		−	4118.94i		−	1.56040i	−0.625530	−	0.780200i	$-0.715117\pi$
$191$		−	4118.94i		−	1.56040i	0.625530	−	0.780200i	$-0.284883\pi$
$192$		0			0
$193$	4537.82			1.69243			0.846216	−	0.532840i	$-0.178875\pi$
$193$	4537.82			1.69243			0.846216	+	0.532840i	$0.178875\pi$
$194$		0			0
$195$	−1030.65			−0.378493
$196$		0			0
$197$	−846.178			−0.306029			−0.153014	−	0.988224i	$-0.548898\pi$
$197$	−846.178			−0.306029			−0.153014	+	0.988224i	$0.548898\pi$
$198$		0			0
$199$	−1218.87			−0.434188			−0.217094	−	0.976151i	$-0.569658\pi$
$199$	−1218.87			−0.434188			−0.217094	+	0.976151i	$0.569658\pi$
$200$		0			0
$201$			1796.02i			0.630257i
$202$		0			0
$203$	210.795	+	1315.09i	0.0728813	+	0.454686i
$204$		0			0
$205$	−659.427			−0.224665
$206$		0			0
$207$		−	552.905i		−	0.185650i
$208$		0			0
$209$			3097.76i			1.02525i
$210$		0			0
$211$		−	5064.81i		−	1.65249i	−0.563308	−	0.826247i	$-0.690472\pi$
$211$		−	5064.81i		−	1.65249i	0.563308	−	0.826247i	$-0.309528\pi$
$212$		0			0
$213$			1768.57i			0.568923i
$214$		0			0
$215$	−1320.56			−0.418890
$216$		0			0
$217$	678.824	+	4234.99i	0.212357	+	1.32484i
$218$		0			0
$219$			20.5594i			0.00634373i
$220$		0			0
$221$	8708.67			2.65072
$222$		0			0
$223$	3894.06			1.16935			0.584676	−	0.811267i	$-0.301222\pi$
$223$	3894.06			1.16935			0.584676	+	0.811267i	$0.301222\pi$
$224$		0			0
$225$	936.613			0.277515
$226$		0			0
$227$	−1165.95			−0.340910			−0.170455	−	0.985365i	$-0.554524\pi$
$227$	−1165.95			−0.340910			−0.170455	+	0.985365i	$0.554524\pi$
$228$		0			0
$229$		−	2537.92i		−	0.732361i	−0.930544	−	0.366181i	$-0.880665\pi$
$229$		−	2537.92i		−	0.732361i	0.930544	−	0.366181i	$-0.119335\pi$
$230$		0			0
$231$	−1427.50	+	228.813i	−0.406592	+	0.0651723i
$232$		0			0
$233$	1345.53			0.378321			0.189160	−	0.981946i	$-0.439423\pi$
$233$	1345.53			0.378321			0.189160	+	0.981946i	$0.439423\pi$
$234$		0			0
$235$		−	1571.78i		−	0.436306i
$236$		0			0
$237$		−	2916.69i		−	0.799408i
$238$		0			0
$239$		−	6239.81i		−	1.68878i	−0.535725	−	0.844392i	$-0.679962\pi$
$239$		−	6239.81i		−	1.68878i	0.535725	−	0.844392i	$-0.320038\pi$
$240$		0			0
$241$			4546.01i			1.21508i	0.794289	+	0.607540i	$0.207844\pi$
$241$			4546.01i			1.21508i	−0.794289	+	0.607540i	$0.792156\pi$
$242$		0			0
$243$	243.000			0.0641500
$244$		0			0
$245$	−490.474	−	1490.65i	−0.127899	−	0.388712i
$246$		0			0
$247$			8939.56i			2.30288i
$248$		0			0
$249$	1289.24			0.328121
$250$		0			0
$251$	−6775.91			−1.70395			−0.851976	−	0.523582i	$-0.824596\pi$
$251$	−6775.91			−1.70395			−0.851976	+	0.523582i	$0.824596\pi$
$252$		0			0
$253$	1598.55			0.397232
$254$		0			0
$255$	1591.82			0.390915
$256$		0			0
$257$			7170.16i			1.74032i	0.492770	+	0.870160i	$0.335984\pi$
$257$			7170.16i			1.74032i	−0.492770	+	0.870160i	$0.664016\pi$
$258$		0			0
$259$	−39.0746	−	243.775i	−0.00937442	−	0.0584844i
$260$		0			0
$261$	647.231			0.153497
$262$		0			0
$263$		−	2093.31i		−	0.490795i	−0.969422	−	0.245398i	$-0.921081\pi$
$263$		−	2093.31i		−	0.490795i	0.969422	−	0.245398i	$-0.0789185\pi$
$264$		0			0
$265$		−	652.717i		−	0.151306i
$266$		0			0
$267$			3251.74i			0.745331i
$268$		0			0
$269$		−	1573.92i		−	0.356742i	−0.983963	−	0.178371i	$-0.942917\pi$
$269$		−	1573.92i		−	0.356742i	0.983963	−	0.178371i	$-0.0570828\pi$
$270$		0			0
$271$	−1974.18			−0.442519			−0.221260	−	0.975215i	$-0.571017\pi$
$271$	−1974.18			−0.442519			−0.221260	+	0.975215i	$0.571017\pi$
$272$		0			0
$273$	−4119.50	+	660.312i	−0.913273	+	0.146388i
$274$		0			0
$275$			2707.91i			0.593793i
$276$		0			0
$277$	−4758.46			−1.03216			−0.516079	−	0.856541i	$-0.672609\pi$
$277$	−4758.46			−1.03216			−0.516079	+	0.856541i	$0.672609\pi$
$278$		0			0
$279$	2084.28			0.447250
$280$		0			0
$281$	1753.46			0.372252			0.186126	−	0.982526i	$-0.440407\pi$
$281$	1753.46			0.372252			0.186126	+	0.982526i	$0.440407\pi$
$282$		0			0
$283$	3603.19			0.756846			0.378423	−	0.925633i	$-0.376466\pi$
$283$	3603.19			0.756846			0.378423	+	0.925633i	$0.376466\pi$
$284$		0			0
$285$			1634.02i			0.339617i
$286$		0			0
$287$	−2635.73	+	422.480i	−0.542099	+	0.0868926i
$288$		0			0
$289$	−8537.38			−1.73771
$290$		0			0
$291$			3093.02i			0.623080i
$292$		0			0
$293$			8854.04i			1.76539i	0.469949	+	0.882693i	$0.344272\pi$
$293$			8854.04i			1.76539i	−0.469949	+	0.882693i	$0.655728\pi$
$294$		0			0
$295$		−	1847.98i		−	0.364725i
$296$		0			0
$297$			702.555i			0.137261i
$298$		0			0
$299$	4613.10			0.892249
$300$		0			0
$301$	−5278.28	+	846.052i	−1.01075	+	0.162012i
$302$		0			0
$303$			623.965i			0.118303i
$304$		0			0
$305$	−99.6307			−0.0187044
$306$		0			0
$307$	−4372.30			−0.812836			−0.406418	−	0.913687i	$-0.633222\pi$
$307$	−4372.30			−0.812836			−0.406418	+	0.913687i	$0.633222\pi$
$308$		0			0
$309$	−3689.43			−0.679237
$310$		0			0
$311$	−1956.21			−0.356676			−0.178338	−	0.983969i	$-0.557072\pi$
$311$	−1956.21			−0.356676			−0.178338	+	0.983969i	$0.557072\pi$
$312$		0			0
$313$		−	6314.07i		−	1.14023i	−0.821565	−	0.570115i	$-0.806899\pi$
$313$		−	6314.07i		−	1.14023i	0.821565	−	0.570115i	$-0.193101\pi$
$314$		0			0
$315$	−752.983	+	120.695i	−0.134685	+	0.0215886i
$316$		0			0
$317$	−4962.78			−0.879298			−0.439649	−	0.898170i	$-0.644897\pi$
$317$	−4962.78			−0.879298			−0.439649	+	0.898170i	$0.644897\pi$
$318$		0			0
$319$			1871.26i			0.328434i
$320$		0			0
$321$		−	4508.05i		−	0.783848i
$322$		0			0
$323$		−	13807.0i		−	2.37846i
$324$		0			0
$325$			7814.52i			1.33376i
$326$		0			0
$327$	−1737.90			−0.293903
$328$		0			0
$329$	−1007.01	−	6282.42i	−0.168748	−	1.05277i
$330$		0			0
$331$			6941.96i			1.15276i	0.817181	+	0.576381i	$0.195536\pi$
$331$			6941.96i			1.15276i	−0.817181	+	0.576381i	$0.804464\pi$
$332$		0			0
$333$	−119.976			−0.0197436
$334$		0			0
$335$	−2739.01			−0.446711
$336$		0			0
$337$	−1809.91			−0.292558			−0.146279	−	0.989243i	$-0.546730\pi$
$337$	−1809.91			−0.292558			−0.146279	+	0.989243i	$0.546730\pi$
$338$		0			0
$339$	1843.91			0.295420
$340$		0			0
$341$			6026.02i			0.956972i
$342$		0			0
$343$	−2915.45	−	5643.91i	−0.458949	−	0.888463i
$344$		0			0
$345$	843.206			0.131585
$346$		0			0
$347$		−	3542.86i		−	0.548100i	−0.961716	−	0.274050i	$-0.911637\pi$
$347$		−	3542.86i		−	0.548100i	0.961716	−	0.274050i	$-0.0883633\pi$
$348$		0			0
$349$		−	1318.04i		−	0.202158i	−0.994878	−	0.101079i	$-0.967771\pi$
$349$		−	1318.04i		−	0.202158i	0.994878	−	0.101079i	$-0.0322295\pi$
$350$		0			0
$351$			2027.44i			0.308310i
$352$		0			0
$353$			1254.21i			0.189107i	0.995520	+	0.0945534i	$0.0301423\pi$
$353$			1254.21i			0.189107i	−0.995520	+	0.0945534i	$0.969858\pi$
$354$		0			0
$355$	−2697.15			−0.403239
$356$		0			0
$357$	6362.49	−	1019.84i	0.943245	−	0.151192i
$358$		0			0
$359$			2350.70i			0.345585i	0.984958	+	0.172793i	$0.0552791\pi$
$359$			2350.70i			0.345585i	−0.984958	+	0.172793i	$0.944721\pi$
$360$		0			0
$361$	7314.03			1.06634
$362$		0			0
$363$	1961.79			0.283657
$364$		0			0
$365$	−31.3540			−0.00449629
$366$		0			0
$367$	2941.18			0.418333			0.209167	−	0.977880i	$-0.432925\pi$
$367$	2941.18			0.418333			0.209167	+	0.977880i	$0.432925\pi$
$368$		0			0
$369$			1297.19i			0.183006i
$370$		0			0
$371$	−418.181	−	2608.91i	−0.0585198	−	0.365089i
$372$		0			0
$373$	7367.26			1.02269			0.511343	−	0.859377i	$-0.329148\pi$
$373$	7367.26			1.02269			0.511343	+	0.859377i	$0.329148\pi$
$374$		0			0
$375$			3144.05i			0.432955i
$376$		0			0
$377$			5400.10i			0.737717i
$378$		0			0
$379$			8022.57i			1.08731i	0.839308	+	0.543657i	$0.182961\pi$
$379$			8022.57i			1.08731i	−0.839308	+	0.543657i	$0.817039\pi$
$380$		0			0
$381$			821.623i			0.110480i
$382$		0			0
$383$	13469.5			1.79702			0.898511	−	0.438950i	$-0.144650\pi$
$383$	13469.5			1.79702			0.898511	+	0.438950i	$0.144650\pi$
$384$		0			0
$385$	−348.950	−	2177.00i	−0.0461926	−	0.288183i
$386$		0			0
$387$			2597.74i			0.341216i
$388$		0			0
$389$	642.221			0.0837067			0.0418533	−	0.999124i	$-0.486674\pi$
$389$	642.221			0.0837067			0.0418533	+	0.999124i	$0.486674\pi$
$390$		0			0
$391$	−7124.85			−0.921532
$392$		0			0
$393$	6563.53			0.842459
$394$		0			0
$395$	4448.09			0.566602
$396$		0			0
$397$		−	6101.12i		−	0.771301i	−0.922645	−	0.385650i	$-0.873977\pi$
$397$		−	6101.12i		−	0.771301i	0.922645	−	0.385650i	$-0.126023\pi$
$398$		0			0
$399$	1046.88	+	6531.17i	0.131352	+	0.819468i
$400$		0			0
$401$	6300.98			0.784679			0.392339	−	0.919821i	$-0.371666\pi$
$401$	6300.98			0.784679			0.392339	+	0.919821i	$0.371666\pi$
$402$		0			0
$403$			17390.0i			2.14952i
$404$		0			0
$405$			370.586i			0.0454681i
$406$		0			0
$407$		−	346.871i		−	0.0422451i
$408$		0			0
$409$		−	897.858i		−	0.108548i	−0.998526	−	0.0542741i	$-0.982716\pi$
$409$		−	897.858i		−	0.108548i	0.998526	−	0.0542741i	$-0.0172845\pi$
$410$		0			0
$411$	−7606.16			−0.912857
$412$		0			0
$413$	−1183.96	−	7386.40i	−0.141063	−	0.880051i
$414$		0			0
$415$			1966.14i			0.232564i
$416$		0			0
$417$	−3021.96			−0.354883
$418$		0			0
$419$	8654.42			1.00906			0.504530	−	0.863394i	$-0.331666\pi$
$419$	8654.42			1.00906			0.504530	+	0.863394i	$0.331666\pi$
$420$		0			0
$421$	13325.5			1.54263			0.771313	−	0.636456i	$-0.219600\pi$
$421$	13325.5			1.54263			0.771313	+	0.636456i	$0.219600\pi$
$422$		0			0
$423$	−3091.94			−0.355403
$424$		0			0
$425$		−	12069.4i		−	1.37753i
$426$		0			0
$427$	−398.224	+	63.8311i	−0.0451321	+	0.00723420i
$428$		0			0
$429$	−5861.69			−0.659685
$430$		0			0
$431$			4512.65i			0.504332i	0.967684	+	0.252166i	$0.0811428\pi$
$431$			4512.65i			0.504332i	−0.967684	+	0.252166i	$0.918857\pi$
$432$		0			0
$433$			4597.71i			0.510282i	0.966904	+	0.255141i	$0.0821219\pi$
$433$			4597.71i			0.510282i	−0.966904	+	0.255141i	$0.917878\pi$
$434$		0			0
$435$			987.057i			0.108795i
$436$		0			0
$437$		−	7313.74i		−	0.800604i
$438$		0			0
$439$	14584.7			1.58562			0.792811	−	0.609467i	$-0.208617\pi$
$439$	14584.7			1.58562			0.792811	+	0.609467i	$0.208617\pi$
$440$		0			0
$441$	−2932.35	+	964.837i	−0.316634	+	0.104183i
$442$		0			0
$443$		−	10578.3i		−	1.13452i	−0.823539	−	0.567259i	$-0.808004\pi$
$443$		−	10578.3i		−	1.13452i	0.823539	−	0.567259i	$-0.191996\pi$
$444$		0			0
$445$	−4959.06			−0.528274
$446$		0			0
$447$	−7027.78			−0.743630
$448$		0			0
$449$	−833.584			−0.0876152			−0.0438076	−	0.999040i	$-0.513949\pi$
$449$	−833.584			−0.0876152			−0.0438076	+	0.999040i	$0.513949\pi$
$450$		0			0
$451$	−3750.42			−0.391575
$452$		0			0
$453$		−	1683.07i		−	0.174564i
$454$		0			0
$455$	−1007.01	−	6282.42i	−0.103756	−	0.647307i
$456$		0			0
$457$	−6462.47			−0.661492			−0.330746	−	0.943720i	$-0.607300\pi$
$457$	−6462.47			−0.661492			−0.330746	+	0.943720i	$0.607300\pi$
$458$		0			0
$459$		−	3131.35i		−	0.318429i
$460$		0			0
$461$		−	10142.3i		−	1.02467i	−0.858786	−	0.512334i	$-0.828781\pi$
$461$		−	10142.3i		−	1.02467i	0.858786	−	0.512334i	$-0.171219\pi$
$462$		0			0
$463$		−	1428.43i		−	0.143380i	−0.997427	−	0.0716898i	$-0.977161\pi$
$463$		−	1428.43i		−	0.143380i	0.997427	−	0.0716898i	$-0.0228391\pi$
$464$		0			0
$465$			3178.63i			0.317001i
$466$		0			0
$467$	−2039.86			−0.202128			−0.101064	−	0.994880i	$-0.532225\pi$
$467$	−2039.86			−0.202128			−0.101064	+	0.994880i	$0.532225\pi$
$468$		0			0
$469$	−10947.8	+	1754.82i	−1.07788	+	0.172772i
$470$		0			0
$471$			1377.66i			0.134775i
$472$		0			0
$473$	−7510.53			−0.730094
$474$		0			0
$475$	12389.4			1.19676
$476$		0			0
$477$	−1284.00			−0.123250
$478$		0			0
$479$	13044.0			1.24425			0.622126	−	0.782917i	$-0.286269\pi$
$479$	13044.0			1.24425			0.622126	+	0.782917i	$0.286269\pi$
$480$		0			0
$481$		−	1001.00i		−	0.0948895i
$482$		0			0
$483$	3370.30	−	540.222i	0.317503	−	0.0508923i
$484$		0			0
$485$	−4717.00			−0.441625
$486$		0			0
$487$			6444.14i			0.599614i	0.954000	+	0.299807i	$0.0969223\pi$
$487$			6444.14i			0.599614i	−0.954000	+	0.299807i	$0.903078\pi$
$488$		0			0
$489$			1674.56i			0.154859i
$490$		0			0
$491$			8032.35i			0.738279i	0.929374	+	0.369140i	$0.120348\pi$
$491$			8032.35i			0.738279i	−0.929374	+	0.369140i	$0.879652\pi$
$492$		0			0
$493$		−	8340.35i		−	0.761928i
$494$		0			0
$495$	−1071.43			−0.0972871
$496$		0			0
$497$	−10780.5	+	1728.00i	−0.972983	+	0.155959i
$498$		0			0
$499$		−	1184.12i		−	0.106230i	−0.998588	−	0.0531148i	$-0.983085\pi$
$499$		−	1184.12i		−	0.106230i	0.998588	−	0.0531148i	$-0.0169149\pi$
$500$		0			0
$501$	7482.95			0.667293
$502$		0			0
$503$	−19759.9			−1.75159			−0.875794	−	0.482684i	$-0.839662\pi$
$503$	−19759.9			−1.75159			−0.875794	+	0.482684i	$0.839662\pi$
$504$		0			0
$505$	−951.576			−0.0838506
$506$		0			0
$507$	−10324.7			−0.904413
$508$		0			0
$509$		−	14150.4i		−	1.23223i	−0.787654	−	0.616117i	$-0.788705\pi$
$509$		−	14150.4i		−	1.23223i	0.787654	−	0.616117i	$-0.211295\pi$
$510$		0			0
$511$	−125.322	+	20.0878i	−0.0108492	+	0.00173901i
$512$		0			0
$513$	3214.36			0.276643
$514$		0			0
$515$		−	5626.55i		−	0.481428i
$516$		0			0
$517$		−	8939.34i		−	0.760448i
$518$		0			0
$519$			8644.85i			0.731150i
$520$		0			0
$521$		−	18031.5i		−	1.51626i	−0.652101	−	0.758132i	$-0.726112\pi$
$521$		−	18031.5i		−	1.51626i	0.652101	−	0.758132i	$-0.273888\pi$
$522$		0			0
$523$	−5733.64			−0.479378			−0.239689	−	0.970850i	$-0.577045\pi$
$523$	−5733.64			−0.479378			−0.239689	+	0.970850i	$0.577045\pi$
$524$		0			0
$525$	915.128	+	5709.23i	0.0760752	+	0.474612i
$526$		0			0
$527$		−	26858.5i		−	2.22006i
$528$		0			0
$529$	8392.87			0.689806
$530$		0			0
$531$	−3635.27			−0.297095
$532$		0			0
$533$	−10823.0			−0.879542
$534$		0			0
$535$	6874.99			0.555573
$536$		0			0
$537$		−	6217.90i		−	0.499669i
$538$		0			0
$539$	−2789.51	−	8477.92i	−0.222918	−	0.677495i
$540$		0			0
$541$	3887.89			0.308971			0.154486	−	0.987995i	$-0.450628\pi$
$541$	3887.89			0.308971			0.154486	+	0.987995i	$0.450628\pi$
$542$		0			0
$543$			8292.35i			0.655357i
$544$		0			0
$545$		−	2650.38i		−	0.208312i
$546$		0			0
$547$		−	4722.04i		−	0.369104i	−0.982823	−	0.184552i	$-0.940917\pi$
$547$		−	4722.04i		−	0.369104i	0.982823	−	0.184552i	$-0.0590834\pi$
$548$		0			0
$549$			195.989i			0.0152361i
$550$		0			0
$551$	8561.47			0.661943
$552$		0			0
$553$	17779.0	−	2849.79i	1.36716	−	0.219142i
$554$		0			0
$555$		−	182.969i		−	0.0139938i
$556$		0			0
$557$	13987.0			1.06400			0.532000	−	0.846745i	$-0.321441\pi$
$557$	13987.0			1.06400			0.532000	+	0.846745i	$0.321441\pi$
$558$		0			0
$559$	−21674.0			−1.63991
$560$		0			0
$561$	9053.27			0.681335
$562$		0			0
$563$	−6757.50			−0.505852			−0.252926	−	0.967486i	$-0.581393\pi$
$563$	−6757.50			−0.505852			−0.252926	+	0.967486i	$0.581393\pi$
$564$		0			0
$565$			2812.05i			0.209387i
$566$		0			0
$567$	237.426	+	1481.23i	0.0175855	+	0.109711i
$568$		0			0
$569$	−7220.86			−0.532011			−0.266005	−	0.963972i	$-0.585704\pi$
$569$	−7220.86			−0.532011			−0.266005	+	0.963972i	$0.585704\pi$
$570$		0			0
$571$			365.261i			0.0267700i	0.999910	+	0.0133850i	$0.00426071\pi$
$571$			365.261i			0.0267700i	−0.999910	+	0.0133850i	$0.995739\pi$
$572$		0			0
$573$		−	12356.8i		−	0.900897i
$574$		0			0
$575$		−	6393.31i		−	0.463686i
$576$		0			0
$577$			2258.22i			0.162931i	0.996676	+	0.0814653i	$0.0259600\pi$
$577$			2258.22i			0.162931i	−0.996676	+	0.0814653i	$0.974040\pi$
$578$		0			0
$579$	13613.5			0.977126
$580$		0			0
$581$	1259.66	+	7858.68i	0.0899477	+	0.561158i
$582$		0			0
$583$		−	3712.25i		−	0.263715i
$584$		0			0
$585$	−3091.94			−0.218523
$586$		0			0
$587$	−1086.09			−0.0763677			−0.0381839	−	0.999271i	$-0.512157\pi$
$587$	−1086.09			−0.0763677			−0.0381839	+	0.999271i	$0.512157\pi$
$588$		0			0
$589$	27570.6			1.92873
$590$		0			0
$591$	−2538.53			−0.176686
$592$		0			0
$593$			7013.68i			0.485695i	0.970064	+	0.242848i	$0.0780815\pi$
$593$			7013.68i			0.485695i	−0.970064	+	0.242848i	$0.921919\pi$
$594$		0			0
$595$	1555.30	+	9703.08i	0.107162	+	0.668551i
$596$		0			0
$597$	−3656.61			−0.250679
$598$		0			0
$599$		−	25452.7i		−	1.73618i	−0.496409	−	0.868089i	$-0.665348\pi$
$599$		−	25452.7i		−	1.73618i	0.496409	−	0.868089i	$-0.334652\pi$
$600$		0			0
$601$		−	26422.8i		−	1.79336i	−0.442682	−	0.896679i	$-0.645973\pi$
$601$		−	26422.8i		−	1.79336i	0.442682	−	0.896679i	$-0.354027\pi$
$602$		0			0
$603$			5388.06i			0.363879i
$604$		0			0
$605$			2991.82i			0.201049i
$606$		0			0
$607$	6179.28			0.413195			0.206597	−	0.978426i	$-0.433761\pi$
$607$	6179.28			0.413195			0.206597	+	0.978426i	$0.433761\pi$
$608$		0			0
$609$	632.384	+	3945.27i	0.0420780	+	0.262513i
$610$		0			0
$611$		−	25797.3i		−	1.70809i
$612$		0			0
$613$	6512.10			0.429072			0.214536	−	0.976716i	$-0.431176\pi$
$613$	6512.10			0.429072			0.214536	+	0.976716i	$0.431176\pi$
$614$		0			0
$615$	−1978.28			−0.129711
$616$		0			0
$617$	15211.8			0.992551			0.496275	−	0.868165i	$-0.334701\pi$
$617$	15211.8			0.992551			0.496275	+	0.868165i	$0.334701\pi$
$618$		0			0
$619$	−11712.0			−0.760496			−0.380248	−	0.924885i	$-0.624161\pi$
$619$	−11712.0			−0.760496			−0.380248	+	0.924885i	$0.624161\pi$
$620$		0			0
$621$		−	1658.72i		−	0.107185i
$622$		0			0
$623$	−19821.4	+	3177.15i	−1.27468	+	0.204318i
$624$		0			0
$625$	8213.69			0.525676
$626$		0			0
$627$			9293.29i			0.591927i
$628$		0			0
$629$			1546.03i			0.0980037i
$630$		0			0
$631$			16130.7i			1.01768i	0.860862	+	0.508839i	$0.169925\pi$
$631$			16130.7i			1.01768i	−0.860862	+	0.508839i	$0.830075\pi$
$632$		0			0
$633$		−	15194.4i		−	0.954068i
$634$		0			0
$635$	−1253.01			−0.0783059
$636$		0			0
$637$	−8050.01	−	24465.7i	−0.500711	−	1.52177i
$638$		0			0
$639$			5305.71i			0.328468i
$640$		0			0
$641$	20295.1			1.25056			0.625280	−	0.780401i	$-0.284985\pi$
$641$	20295.1			1.25056			0.625280	+	0.780401i	$0.284985\pi$
$642$		0			0
$643$	−10255.3			−0.628976			−0.314488	−	0.949262i	$-0.601833\pi$
$643$	−10255.3			−0.628976			−0.314488	+	0.949262i	$0.601833\pi$
$644$		0			0
$645$	−3961.68			−0.241846
$646$		0			0
$647$	15162.9			0.921352			0.460676	−	0.887568i	$-0.347607\pi$
$647$	15162.9			0.921352			0.460676	+	0.887568i	$0.347607\pi$
$648$		0			0
$649$		−	10510.2i		−	0.635688i
$650$		0			0
$651$	2036.47	+	12705.0i	0.122605	+	0.764896i
$652$		0			0
$653$	7126.05			0.427050			0.213525	−	0.976938i	$-0.431505\pi$
$653$	7126.05			0.427050			0.213525	+	0.976938i	$0.431505\pi$
$654$		0			0
$655$			10009.7i			0.597116i
$656$		0			0
$657$			61.6783i			0.00366255i
$658$		0			0
$659$			2646.23i			0.156422i	0.996937	+	0.0782112i	$0.0249208\pi$
$659$			2646.23i			0.156422i	−0.996937	+	0.0782112i	$0.975079\pi$
$660$		0			0
$661$			13839.8i			0.814381i	0.913343	+	0.407191i	$0.133492\pi$
$661$			13839.8i			0.814381i	−0.913343	+	0.407191i	$0.866508\pi$
$662$		0			0
$663$	26126.0			1.53039
$664$		0			0
$665$	−9960.33	+	1596.54i	−0.580820	+	0.0930992i
$666$		0			0
$667$		−	4417.99i		−	0.256470i
$668$		0			0
$669$	11682.2			0.675126
$670$		0			0
$671$	−566.638			−0.0326003
$672$		0			0
$673$	−32235.3			−1.84633			−0.923165	−	0.384404i	$-0.874407\pi$
$673$	−32235.3			−1.84633			−0.923165	+	0.384404i	$0.874407\pi$
$674$		0			0
$675$	2809.84			0.160223
$676$		0			0
$677$			9.30221i			0.000528084i	1.00000		0.000264042i	$8.40472e-5\pi$
$677$			9.30221i			0.000528084i	−1.00000		0.000264042i	$0.999916\pi$
$678$		0			0
$679$	−18853.9	+	3022.07i	−1.06560	+	0.170805i
$680$		0			0
$681$	−3497.84			−0.196825
$682$		0			0
$683$		−	26622.3i		−	1.49147i	−0.666244	−	0.745734i	$-0.732099\pi$
$683$		−	26622.3i		−	1.49147i	0.666244	−	0.745734i	$-0.267901\pi$
$684$		0			0
$685$		−	11599.7i		−	0.647012i
$686$		0			0
$687$		−	7613.77i		−	0.422829i
$688$		0			0
$689$		−	10712.9i		−	0.592348i
$690$		0			0
$691$	20064.5			1.10462			0.552309	−	0.833639i	$-0.313747\pi$
$691$	20064.5			1.10462			0.552309	+	0.833639i	$0.313747\pi$
$692$		0			0
$693$	−4282.50	+	686.439i	−0.234746	+	0.0376272i
$694$		0			0
$695$		−	4608.63i		−	0.251533i
$696$		0			0
$697$	16715.9			0.908408
$698$		0			0
$699$	4036.60			0.218424
$700$		0			0
$701$	−32384.8			−1.74487			−0.872436	−	0.488728i	$-0.837461\pi$
$701$	−32384.8			−1.74487			−0.872436	+	0.488728i	$0.837461\pi$
$702$		0			0
$703$	−1587.02			−0.0851431
$704$		0			0
$705$		−	4715.35i		−	0.251901i
$706$		0			0
$707$	−3803.45	+	609.652i	−0.202325	+	0.0324305i
$708$		0			0
$709$	−34258.5			−1.81468			−0.907338	−	0.420403i	$-0.861889\pi$
$709$	−34258.5			−1.81468			−0.907338	+	0.420403i	$0.861889\pi$
$710$		0			0
$711$		−	8750.08i		−	0.461538i
$712$		0			0
$713$		−	14227.3i		−	0.747288i
$714$		0			0
$715$		−	8939.34i		−	0.467570i
$716$		0			0
$717$		−	18719.4i		−	0.975020i
$718$		0			0
$719$	10303.7			0.534443			0.267221	−	0.963635i	$-0.413895\pi$
$719$	10303.7			0.534443			0.267221	+	0.963635i	$0.413895\pi$
$720$		0			0
$721$	−3604.80	−	22489.3i	−0.186199	−	1.16165i
$722$		0			0
$723$			13638.0i			0.701527i
$724$		0			0
$725$	7484.01			0.383378
$726$		0			0
$727$	−1468.60			−0.0749209			−0.0374605	−	0.999298i	$-0.511927\pi$
$727$	−1468.60			−0.0749209			−0.0374605	+	0.999298i	$0.511927\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	33475.1			1.69373
$732$		0			0
$733$			5917.44i			0.298179i	0.988824	+	0.149090i	$0.0476343\pi$
$733$			5917.44i			0.298179i	−0.988824	+	0.149090i	$0.952366\pi$
$734$		0			0
$735$	−1471.42	−	4471.96i	−0.0738424	−	0.224423i
$736$		0			0
$737$	−15577.8			−0.778584
$738$		0			0
$739$		−	8120.94i		−	0.404240i	−0.979361	−	0.202120i	$-0.935217\pi$
$739$		−	8120.94i		−	0.404240i	0.979361	−	0.202120i	$-0.0647831\pi$
$740$		0			0
$741$			26818.7i			1.32957i
$742$		0			0
$743$		−	7975.20i		−	0.393784i	−0.980425	−	0.196892i	$-0.936915\pi$
$743$		−	7975.20i		−	0.393784i	0.980425	−	0.196892i	$-0.0630849\pi$
$744$		0			0
$745$		−	10717.7i		−	0.527068i
$746$		0			0
$747$	3867.71			0.189440
$748$		0			0
$749$	27479.3	−	4404.65i	1.34055	−	0.214876i
$750$		0			0
$751$			7417.00i			0.360387i	0.983631	+	0.180193i	$0.0576723\pi$
$751$			7417.00i			0.360387i	−0.983631	+	0.180193i	$0.942328\pi$
$752$		0			0
$753$	−20327.7			−0.983777
$754$		0			0
$755$	2566.75			0.123727
$756$		0			0
$757$	−2056.45			−0.0987358			−0.0493679	−	0.998781i	$-0.515721\pi$
$757$	−2056.45			−0.0987358			−0.0493679	+	0.998781i	$0.515721\pi$
$758$		0			0
$759$	4795.64			0.229342
$760$		0			0
$761$		−	28244.9i		−	1.34544i	−0.739898	−	0.672719i	$-0.765126\pi$
$761$		−	28244.9i		−	1.34544i	0.739898	−	0.672719i	$-0.234874\pi$
$762$		0			0
$763$	−1698.04	−	10593.6i	−0.0805676	−	0.502639i
$764$		0			0
$765$	4775.45			0.225695
$766$		0			0
$767$		−	30330.5i		−	1.42786i
$768$		0			0
$769$			27026.1i			1.26734i	0.773603	+	0.633671i	$0.218453\pi$
$769$			27026.1i			1.26734i	−0.773603	+	0.633671i	$0.781547\pi$
$770$		0			0
$771$			21510.5i			1.00477i
$772$		0			0
$773$		−	21487.5i		−	0.999810i	−0.866080	−	0.499905i	$-0.833368\pi$
$773$		−	21487.5i		−	0.999810i	0.866080	−	0.499905i	$-0.166632\pi$
$774$		0			0
$775$	24100.8			1.11707
$776$		0			0
$777$	−117.224	−	731.326i	−0.00541233	−	0.0337660i
$778$		0			0
$779$			17159.1i			0.789202i
$780$		0			0
$781$	−15339.7			−0.702816
$782$		0			0
$783$	1941.69			0.0886213
$784$		0			0
$785$	−2100.99			−0.0955257
$786$		0			0
$787$	13905.7			0.629843			0.314921	−	0.949118i	$-0.398022\pi$
$787$	13905.7			0.629843			0.314921	+	0.949118i	$0.398022\pi$
$788$		0			0
$789$		−	6279.94i		−	0.283361i
$790$		0			0
$791$	1801.61	+	11239.8i	0.0809836	+	0.505234i
$792$		0			0
$793$	−1635.21			−0.0732258
$794$		0			0
$795$		−	1958.15i		−	0.0873566i
$796$		0			0
$797$		−	21571.7i		−	0.958730i	−0.877616	−	0.479365i	$-0.840867\pi$
$797$		−	21571.7i		−	0.958730i	0.877616	−	0.479365i	$-0.159133\pi$
$798$		0			0
$799$			39843.4i			1.76415i
$800$		0			0
$801$			9755.23i			0.430317i
$802$		0			0
$803$	−178.323			−0.00783669
$804$		0			0
$805$	823.864	+	5139.85i	0.0360713	+	0.225039i
$806$		0			0
$807$		−	4721.77i		−	0.205965i
$808$		0			0
$809$	33518.7			1.45668			0.728339	−	0.685217i	$-0.240292\pi$
$809$	33518.7			1.45668			0.728339	+	0.685217i	$0.240292\pi$
$810$		0			0
$811$	−35133.3			−1.52120			−0.760602	−	0.649218i	$-0.775096\pi$
$811$	−35133.3			−1.52120			−0.760602	+	0.649218i	$0.775096\pi$
$812$		0			0
$813$	−5922.53			−0.255489
$814$		0			0
$815$	−2553.78			−0.109761
$816$		0			0
$817$			34362.5i			1.47147i
$818$		0			0
$819$	−12358.5	+	1980.94i	−0.527278	+	0.0845171i
$820$		0			0
$821$	21025.2			0.893768			0.446884	−	0.894592i	$-0.352534\pi$
$821$	21025.2			0.893768			0.446884	+	0.894592i	$0.352534\pi$
$822$		0			0
$823$		−	20491.2i		−	0.867894i	−0.900938	−	0.433947i	$-0.857120\pi$
$823$		−	20491.2i		−	0.867894i	0.900938	−	0.433947i	$-0.142880\pi$
$824$		0			0
$825$			8123.73i			0.342827i
$826$		0			0
$827$		−	18080.6i		−	0.760245i	−0.924936	−	0.380122i	$-0.875882\pi$
$827$		−	18080.6i		−	0.760245i	0.924936	−	0.380122i	$-0.124118\pi$
$828$		0			0
$829$		−	38666.1i		−	1.61994i	−0.586471	−	0.809970i	$-0.699483\pi$
$829$		−	38666.1i		−	1.61994i	0.586471	−	0.809970i	$-0.300517\pi$
$830$		0			0
$831$	−14275.4			−0.595917
$832$		0			0
$833$	12433.1	+	37786.8i	0.517144	+	1.57171i
$834$		0			0
$835$			11411.8i			0.472962i
$836$		0			0
$837$	6252.85			0.258220
$838$		0			0
$839$	36214.5			1.49018			0.745091	−	0.666962i	$-0.232406\pi$
$839$	36214.5			1.49018			0.745091	+	0.666962i	$0.232406\pi$
$840$		0			0
$841$	−19217.3			−0.787949
$842$		0			0
$843$	5260.39			0.214920
$844$		0			0
$845$		−	15745.7i		−	0.641027i
$846$		0			0
$847$	1916.79	+	11958.3i	0.0777588	+	0.485115i
$848$		0			0
$849$	10809.6			0.436965
$850$		0			0
$851$			818.954i			0.0329887i
$852$		0			0
$853$			1180.57i			0.0473881i	0.999719	+	0.0236941i	$0.00754276\pi$
$853$			1180.57i			0.0473881i	−0.999719	+	0.0236941i	$0.992457\pi$
$854$		0			0
$855$			4902.05i			0.196078i
$856$		0			0
$857$			29570.1i			1.17864i	0.807900	+	0.589320i	$0.200604\pi$
$857$			29570.1i			1.17864i	−0.807900	+	0.589320i	$0.799396\pi$
$858$		0			0
$859$	18516.7			0.735483			0.367742	−	0.929928i	$-0.380131\pi$
$859$	18516.7			0.735483			0.367742	+	0.929928i	$0.380131\pi$
$860$		0			0
$861$	−7907.19	+	1267.44i	−0.312981	+	0.0501675i
$862$		0			0
$863$			35215.9i			1.38907i	0.719461	+	0.694533i	$0.244389\pi$
$863$			35215.9i			1.38907i	−0.719461	+	0.694533i	$0.755611\pi$
$864$		0			0
$865$	−13183.8			−0.518223
$866$		0			0
$867$	−25612.1			−1.00327
$868$		0			0
$869$	25298.0			0.987544
$870$		0			0
$871$	−44954.7			−1.74883
$872$		0			0
$873$			9279.07i			0.359735i
$874$		0			0
$875$	−19164.9	+	3071.93i	−0.740449	+	0.118686i
$876$		0			0
$877$	20910.2			0.805115			0.402558	−	0.915395i	$-0.368121\pi$
$877$	20910.2			0.805115			0.402558	+	0.915395i	$0.368121\pi$
$878$		0			0
$879$			26562.1i			1.01925i
$880$		0			0
$881$		−	19841.3i		−	0.758762i	−0.925241	−	0.379381i	$-0.876137\pi$
$881$		−	19841.3i		−	0.758762i	0.925241	−	0.379381i	$-0.123863\pi$
$882$		0			0
$883$			37682.4i			1.43614i	0.695971	+	0.718070i	$0.254974\pi$
$883$			37682.4i			1.43614i	−0.695971	+	0.718070i	$0.745026\pi$
$884$		0			0
$885$		−	5543.95i		−	0.210574i
$886$		0			0
$887$	−16665.2			−0.630850			−0.315425	−	0.948950i	$-0.602147\pi$
$887$	−16665.2			−0.630850			−0.315425	+	0.948950i	$0.602147\pi$
$888$		0			0
$889$	−5008.29	+	802.776i	−0.188946	+	0.0302860i
$890$		0			0
$891$			2107.67i			0.0792474i
$892$		0			0
$893$	−40899.7			−1.53265
$894$		0			0
$895$	9482.58			0.354154
$896$		0			0
$897$	13839.3			0.515140
$898$		0			0
$899$	16654.5			0.617862
$900$		0			0
$901$			16545.8i			0.611788i
$902$		0			0
$903$	−15834.8	+	2538.16i	−0.583555	+	0.0935377i
$904$		0			0
$905$	−12646.2			−0.464502
$906$		0			0
$907$		−	20577.7i		−	0.753331i	−0.926349	−	0.376665i	$-0.877071\pi$
$907$		−	20577.7i		−	0.753331i	0.926349	−	0.376665i	$-0.122929\pi$
$908$		0			0
$909$			1871.90i			0.0683024i
$910$		0			0
$911$		−	35684.4i		−	1.29778i	−0.760883	−	0.648889i	$-0.775234\pi$
$911$		−	35684.4i		−	1.29778i	0.760883	−	0.648889i	$-0.224766\pi$
$912$		0			0
$913$			11182.2i			0.405342i
$914$		0			0
$915$	−298.892			−0.0107990
$916$		0			0
$917$	6412.97	+	40008.7i	0.230943	+	1.44079i
$918$		0			0
$919$		−	2549.71i		−	0.0915205i	−0.998952	−	0.0457602i	$-0.985429\pi$
$919$		−	2549.71i		−	0.0915205i	0.998952	−	0.0457602i	$-0.0145710\pi$
$920$		0			0
$921$	−13116.9			−0.469291
$922$		0			0
$923$	−44267.6			−1.57864
$924$		0			0
$925$	−1387.30			−0.0493124
$926$		0			0
$927$	−11068.3			−0.392158
$928$		0			0
$929$			24991.8i			0.882620i	0.897355	+	0.441310i	$0.145486\pi$
$929$			24991.8i			0.882620i	−0.897355	+	0.441310i	$0.854514\pi$
$930$		0			0
$931$	−38788.6	+	12762.7i	−1.36546	+	0.449281i
$932$		0			0
$933$	−5868.62			−0.205927
$934$		0			0
$935$			13806.6i			0.482915i
$936$		0			0
$937$		−	36329.3i		−	1.26662i	−0.773896	−	0.633312i	$-0.781695\pi$
$937$		−	36329.3i		−	1.26662i	0.773896	−	0.633312i	$-0.218305\pi$
$938$		0			0
$939$		−	18942.2i		−	0.658312i
$940$		0			0
$941$			21505.5i			0.745015i	0.928029	+	0.372507i	$0.121502\pi$
$941$			21505.5i			0.745015i	−0.928029	+	0.372507i	$0.878498\pi$
$942$		0			0
$943$	8854.64			0.305776
$944$		0			0
$945$	−2258.95	+	362.085i	−0.0777604	+	0.0124642i
$946$		0			0
$947$		−	17365.5i		−	0.595886i	−0.954584	−	0.297943i	$-0.903699\pi$
$947$		−	17365.5i		−	0.595886i	0.954584	−	0.297943i	$-0.0963005\pi$
$948$		0			0
$949$	−514.605			−0.0176025
$950$		0			0
$951$	−14888.3			−0.507663
$952$		0			0
$953$	22686.6			0.771133			0.385566	−	0.922680i	$-0.374006\pi$
$953$	22686.6			0.771133			0.385566	+	0.922680i	$0.374006\pi$
$954$		0			0
$955$	18844.7			0.638535
$956$		0			0
$957$			5613.77i			0.189621i
$958$		0			0
$959$	−7431.69	−	46364.2i	−0.250242	−	1.56119i
$960$		0			0
$961$	23841.5			0.800293
$962$		0			0
$963$		−	13524.2i		−	0.452555i
$964$		0			0
$965$			20761.1i			0.692565i
$966$		0			0
$967$		−	30178.5i		−	1.00359i	−0.864985	−	0.501797i	$-0.832672\pi$
$967$		−	30178.5i		−	1.00359i	0.864985	−	0.501797i	$-0.167328\pi$
$968$		0			0
$969$		−	41420.9i		−	1.37320i
$970$		0			0
$971$	54731.9			1.80889			0.904445	−	0.426591i	$-0.140286\pi$
$971$	54731.9			1.80889			0.904445	+	0.426591i	$0.140286\pi$
$972$		0			0
$973$	−2952.64	−	18420.7i	−0.0972841	−	0.606928i
$974$		0			0
$975$			23443.6i			0.770046i
$976$		0			0
$977$	−29860.7			−0.977817			−0.488908	−	0.872335i	$-0.662605\pi$
$977$	−29860.7			−0.977817			−0.488908	+	0.872335i	$0.662605\pi$
$978$		0			0
$979$	−28204.1			−0.920741
$980$		0			0
$981$	−5213.71			−0.169685
$982$		0			0
$983$	−10417.7			−0.338018			−0.169009	−	0.985615i	$-0.554057\pi$
$983$	−10417.7			−0.338018			−0.169009	+	0.985615i	$0.554057\pi$
$984$		0			0
$985$		−	3871.38i		−	0.125231i
$986$		0			0
$987$	−3021.02	−	18847.3i	−0.0974266	−	0.607817i
$988$		0			0
$989$	17732.2			0.570122
$990$		0			0
$991$			3803.09i			0.121906i	0.998141	+	0.0609531i	$0.0194140\pi$
$991$			3803.09i			0.121906i	−0.998141	+	0.0609531i	$0.980586\pi$
$992$		0			0
$993$			20825.9i			0.665548i
$994$		0			0
$995$		−	5576.50i		−	0.177675i
$996$		0			0
$997$			53386.9i			1.69587i	0.530103	+	0.847933i	$0.322153\pi$
$997$			53386.9i			1.69587i	−0.530103	+	0.847933i	$0.677847\pi$
$998$		0			0
$999$	−359.927			−0.0113990

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.b.h.895.8		12
4.3	odd	2		1344.4.b.g.895.8		12
7.6	odd	2		1344.4.b.g.895.5		12
8.3	odd	2		84.4.b.b.55.2	yes	12
8.5	even	2		84.4.b.a.55.1	✓	12
24.5	odd	2		252.4.b.f.55.12		12
24.11	even	2		252.4.b.e.55.11		12
28.27	even	2	inner	1344.4.b.h.895.5		12
56.13	odd	2		84.4.b.b.55.1	yes	12
56.27	even	2		84.4.b.a.55.2	yes	12
168.83	odd	2		252.4.b.f.55.11		12
168.125	even	2		252.4.b.e.55.12		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.4.b.a.55.1	✓	12	8.5	even	2
84.4.b.a.55.2	yes	12	56.27	even	2
84.4.b.b.55.1	yes	12	56.13	odd	2
84.4.b.b.55.2	yes	12	8.3	odd	2
252.4.b.e.55.11		12	24.11	even	2
252.4.b.e.55.12		12	168.125	even	2
252.4.b.f.55.11		12	168.83	odd	2
252.4.b.f.55.12		12	24.5	odd	2
1344.4.b.g.895.5		12	7.6	odd	2
1344.4.b.g.895.8		12	4.3	odd	2
1344.4.b.h.895.5		12	28.27	even	2	inner
1344.4.b.h.895.8		12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +4.57514i q^{5} +(2.93118 + 18.2868i) q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +4.57514i q^{5} +(2.93118 + 18.2868i) q^{7} +9.00000 q^{9} +26.0206i q^{11} +75.0905i q^{13} +13.7254i q^{15} -115.976i q^{17} +119.051 q^{19} +(8.79355 + 54.8605i) q^{21} -61.4339i q^{23} +104.068 q^{25} +27.0000 q^{27} +71.9146 q^{29} +231.587 q^{31} +78.0617i q^{33} +(-83.6647 + 13.4106i) q^{35} -13.3306 q^{37} +225.271i q^{39} +144.133i q^{41} +288.638i q^{43} +41.1762i q^{45} -343.549 q^{47} +(-325.816 + 107.204i) q^{49} -347.927i q^{51} -142.666 q^{53} -119.048 q^{55} +357.152 q^{57} -403.919 q^{59} +21.7765i q^{61} +(26.3807 + 164.581i) q^{63} -343.549 q^{65} +598.674i q^{67} -184.302i q^{69} +589.524i q^{71} +6.85314i q^{73} +312.204 q^{75} +(-475.834 + 76.2711i) q^{77} -972.231i q^{79} +81.0000 q^{81} +429.745 q^{83} +530.605 q^{85} +215.744 q^{87} +1083.91i q^{89} +(-1373.17 + 220.104i) q^{91} +694.761 q^{93} +544.673i q^{95} +1031.01i q^{97} +234.185i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 36 q^{3} + 10 q^{7} + 108 q^{9}+O(q^{10}) \) 12 * q + 36 * q^3 + 10 * q^7 + 108 * q^9	\( 12 q + 36 q^{3} + 10 q^{7} + 108 q^{9} + 84 q^{19} + 30 q^{21} - 216 q^{25} + 324 q^{27} - 200 q^{29} - 384 q^{31} - 84 q^{35} + 244 q^{37} + 280 q^{47} - 424 q^{49} + 16 q^{53} + 212 q^{55} + 252 q^{57} - 1168 q^{59} + 90 q^{63} + 280 q^{65} - 648 q^{75} - 968 q^{77} + 972 q^{81} + 968 q^{83} + 852 q^{85} - 600 q^{87} - 1648 q^{91} - 1152 q^{93}+O(q^{100}) \) 12 * q + 36 * q^3 + 10 * q^7 + 108 * q^9 + 84 * q^19 + 30 * q^21 - 216 * q^25 + 324 * q^27 - 200 * q^29 - 384 * q^31 - 84 * q^35 + 244 * q^37 + 280 * q^47 - 424 * q^49 + 16 * q^53 + 212 * q^55 + 252 * q^57 - 1168 * q^59 + 90 * q^63 + 280 * q^65 - 648 * q^75 - 968 * q^77 + 972 * q^81 + 968 * q^83 + 852 * q^85 - 600 * q^87 - 1648 * q^91 - 1152 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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