LMFDB - Embedded newform 3100.3.d.f.1301.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3100,3,Mod(1301,3100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3100, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("3100.1301");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3100 = 2^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3100.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$84.4688819517$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1301.11
Character	$\chi$	$=$	3100.1301
Dual form			3100.3.d.f.1301.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.340074i q^{3} -1.52585 q^{7} +8.88435 q^{9} +O(q^{10})$	$q-0.340074i q^{3} -1.52585 q^{7} +8.88435 q^{9} -13.2007i q^{11} -13.5738i q^{13} -8.93888i q^{17} +30.6245 q^{19} +0.518903i q^{21} -35.6970i q^{23} -6.08201i q^{27} -9.29891i q^{29} +(-15.5581 + 26.8131i) q^{31} -4.48924 q^{33} -18.7806i q^{37} -4.61611 q^{39} -2.34276 q^{41} +36.4447i q^{43} +43.2467 q^{47} -46.6718 q^{49} -3.03988 q^{51} +10.7344i q^{53} -10.4146i q^{57} -29.7744 q^{59} +40.1994i q^{61} -13.5562 q^{63} -102.334 q^{67} -12.1396 q^{69} +31.7735 q^{71} -16.8117i q^{73} +20.1424i q^{77} +1.45363i q^{79} +77.8908 q^{81} -15.8696i q^{83} -3.16232 q^{87} +13.9223i q^{89} +20.7117i q^{91} +(9.11846 + 5.29092i) q^{93} -65.0536 q^{97} -117.280i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 4 q^{7} - 72 q^{9}+O(q^{10}) $ 22 * q - 4 * q^7 - 72 * q^9	$ 22 q - 4 q^{7} - 72 q^{9} - 28 q^{19} - 18 q^{31} + 34 q^{33} - 62 q^{39} + 64 q^{41} + 96 q^{47} + 150 q^{49} - 130 q^{51} + 40 q^{59} + 4 q^{63} - 110 q^{67} + 100 q^{69} + 132 q^{71} + 234 q^{81} + 62 q^{87} - 16 q^{93} - 186 q^{97}+O(q^{100}) $ 22 * q - 4 * q^7 - 72 * q^9 - 28 * q^19 - 18 * q^31 + 34 * q^33 - 62 * q^39 + 64 * q^41 + 96 * q^47 + 150 * q^49 - 130 * q^51 + 40 * q^59 + 4 * q^63 - 110 * q^67 + 100 * q^69 + 132 * q^71 + 234 * q^81 + 62 * q^87 - 16 * q^93 - 186 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3100\mathbb{Z}\right)^\times$.

$n$	$1551$	$1801$	$2977$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	0.340074i		−	0.113358i	−0.998392	−	0.0566791i	$-0.981949\pi$
$3$		−	0.340074i		−	0.113358i	0.998392	−	0.0566791i	$-0.0180512\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.52585			−0.217979			−0.108989	−	0.994043i	$-0.534761\pi$
$7$	−1.52585			−0.217979			−0.108989	+	0.994043i	$0.534761\pi$
$8$		0			0
$9$	8.88435			0.987150
$10$		0			0
$11$		−	13.2007i		−	1.20007i	−0.799975	−	0.600034i	$-0.795154\pi$
$11$		−	13.2007i		−	1.20007i	0.799975	−	0.600034i	$-0.204846\pi$
$12$		0			0
$13$		−	13.5738i		−	1.04414i	−0.852902	−	0.522071i	$-0.825160\pi$
$13$		−	13.5738i		−	1.04414i	0.852902	−	0.522071i	$-0.174840\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	8.93888i		−	0.525816i	−0.964821	−	0.262908i	$-0.915318\pi$
$17$		−	8.93888i		−	0.525816i	0.964821	−	0.262908i	$-0.0846816\pi$
$18$		0			0
$19$	30.6245			1.61182			0.805908	−	0.592041i	$-0.201678\pi$
$19$	30.6245			1.61182			0.805908	+	0.592041i	$0.201678\pi$
$20$		0			0
$21$			0.518903i			0.0247097i
$22$		0			0
$23$		−	35.6970i		−	1.55204i	−0.630706	−	0.776022i	$-0.717235\pi$
$23$		−	35.6970i		−	1.55204i	0.630706	−	0.776022i	$-0.282765\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	6.08201i		−	0.225260i
$28$		0			0
$29$		−	9.29891i		−	0.320652i	−0.987064	−	0.160326i	$-0.948745\pi$
$29$		−	9.29891i		−	0.320652i	0.987064	−	0.160326i	$-0.0512546\pi$
$30$		0			0
$31$	−15.5581	+	26.8131i	−0.501875	+	0.864940i
$31$	−15.5581	+	26.8131i	−0.501875	+	0.864940i
$32$		0			0
$33$	−4.48924			−0.136037
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	18.7806i		−	0.507585i	−0.967259	−	0.253793i	$-0.918322\pi$
$37$		−	18.7806i		−	0.507585i	0.967259	−	0.253793i	$-0.0816781\pi$
$38$		0			0
$39$	−4.61611			−0.118362
$40$		0			0
$41$	−2.34276			−0.0571404			−0.0285702	−	0.999592i	$-0.509095\pi$
$41$	−2.34276			−0.0571404			−0.0285702	+	0.999592i	$0.509095\pi$
$42$		0			0
$43$			36.4447i			0.847552i	0.905767	+	0.423776i	$0.139296\pi$
$43$			36.4447i			0.847552i	−0.905767	+	0.423776i	$0.860704\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	43.2467			0.920143			0.460071	−	0.887882i	$-0.347824\pi$
$47$	43.2467			0.920143			0.460071	+	0.887882i	$0.347824\pi$
$48$		0			0
$49$	−46.6718			−0.952485
$50$		0			0
$51$	−3.03988			−0.0596055
$52$		0			0
$53$			10.7344i			0.202536i	0.994859	+	0.101268i	$0.0322900\pi$
$53$			10.7344i			0.202536i	−0.994859	+	0.101268i	$0.967710\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	10.4146i		−	0.182712i
$58$		0			0
$59$	−29.7744			−0.504650			−0.252325	−	0.967643i	$-0.581195\pi$
$59$	−29.7744			−0.504650			−0.252325	+	0.967643i	$0.581195\pi$
$60$		0			0
$61$			40.1994i			0.659006i	0.944155	+	0.329503i	$0.106881\pi$
$61$			40.1994i			0.659006i	−0.944155	+	0.329503i	$0.893119\pi$
$62$		0			0
$63$	−13.5562			−0.215178
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−102.334			−1.52737			−0.763686	−	0.645588i	$-0.776612\pi$
$67$	−102.334			−1.52737			−0.763686	+	0.645588i	$0.776612\pi$
$68$		0			0
$69$	−12.1396			−0.175937
$70$		0			0
$71$	31.7735			0.447514			0.223757	−	0.974645i	$-0.428168\pi$
$71$	31.7735			0.447514			0.223757	+	0.974645i	$0.428168\pi$
$72$		0			0
$73$		−	16.8117i		−	0.230297i	−0.993348	−	0.115148i	$-0.963266\pi$
$73$		−	16.8117i		−	0.230297i	0.993348	−	0.115148i	$-0.0367343\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			20.1424i			0.261590i
$78$		0			0
$79$			1.45363i			0.0184004i	0.999958	+	0.00920020i	$0.00292856\pi$
$79$			1.45363i			0.0184004i	−0.999958	+	0.00920020i	$0.997071\pi$
$80$		0			0
$81$	77.8908			0.961615
$82$		0			0
$83$		−	15.8696i		−	0.191200i	−0.995420	−	0.0956002i	$-0.969523\pi$
$83$		−	15.8696i		−	0.191200i	0.995420	−	0.0956002i	$-0.0304770\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−3.16232			−0.0363485
$88$		0			0
$89$			13.9223i			0.156431i	0.996936	+	0.0782154i	$0.0249222\pi$
$89$			13.9223i			0.156431i	−0.996936	+	0.0782154i	$0.975078\pi$
$90$		0			0
$91$			20.7117i			0.227601i
$92$		0			0
$93$	9.11846	+	5.29092i	0.0980480	+	0.0568916i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−65.0536			−0.670655			−0.335328	−	0.942102i	$-0.608847\pi$
$97$	−65.0536			−0.670655			−0.335328	+	0.942102i	$0.608847\pi$
$98$		0			0
$99$		−	117.280i		−	1.18465i
$100$		0			0
$101$	41.6565			0.412441			0.206220	−	0.978506i	$-0.433884\pi$
$101$	41.6565			0.412441			0.206220	+	0.978506i	$0.433884\pi$
$102$		0			0
$103$	−120.036			−1.16540			−0.582701	−	0.812687i	$-0.698004\pi$
$103$	−120.036			−1.16540			−0.582701	+	0.812687i	$0.698004\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−77.7317			−0.726465			−0.363232	−	0.931699i	$-0.618327\pi$
$107$	−77.7317			−0.726465			−0.363232	+	0.931699i	$0.618327\pi$
$108$		0			0
$109$	−59.5557			−0.546382			−0.273191	−	0.961960i	$-0.588079\pi$
$109$	−59.5557			−0.546382			−0.273191	+	0.961960i	$0.588079\pi$
$110$		0			0
$111$	−6.38682			−0.0575389
$112$		0			0
$113$	160.098			1.41680			0.708399	−	0.705812i	$-0.249418\pi$
$113$	160.098			1.41680			0.708399	+	0.705812i	$0.249418\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	120.595i		−	1.03072i
$118$		0			0
$119$			13.6394i			0.114617i
$120$		0			0
$121$	−53.2597			−0.440163
$122$		0			0
$123$			0.796712i			0.00647733i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	187.188i		−	1.47392i	−0.675935	−	0.736961i	$-0.736260\pi$
$127$		−	187.188i		−	1.47392i	0.675935	−	0.736961i	$-0.263740\pi$
$128$		0			0
$129$	12.3939			0.0960769
$130$		0			0
$131$	54.9223			0.419254			0.209627	−	0.977781i	$-0.432775\pi$
$131$	54.9223			0.419254			0.209627	+	0.977781i	$0.432775\pi$
$132$		0			0
$133$	−46.7285			−0.351342
$134$		0			0
$135$		0			0
$136$		0			0
$137$			26.8420i			0.195927i	0.995190	+	0.0979634i	$0.0312328\pi$
$137$			26.8420i			0.195927i	−0.995190	+	0.0979634i	$0.968767\pi$
$138$		0			0
$139$		−	228.652i		−	1.64498i	−0.568779	−	0.822490i	$-0.692584\pi$
$139$		−	228.652i		−	1.64498i	0.568779	−	0.822490i	$-0.307416\pi$
$140$		0			0
$141$		−	14.7071i		−	0.104306i
$142$		0			0
$143$	−179.185			−1.25304
$144$		0			0
$145$		0			0
$146$		0			0
$147$			15.8719i			0.107972i
$148$		0			0
$149$	−24.5243			−0.164593			−0.0822963	−	0.996608i	$-0.526225\pi$
$149$	−24.5243			−0.164593			−0.0822963	+	0.996608i	$0.526225\pi$
$150$		0			0
$151$			180.115i			1.19281i	0.802683	+	0.596406i	$0.203405\pi$
$151$			180.115i			1.19281i	−0.802683	+	0.596406i	$0.796595\pi$
$152$		0			0
$153$		−	79.4161i		−	0.519059i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−10.4074			−0.0662892			−0.0331446	−	0.999451i	$-0.510552\pi$
$157$	−10.4074			−0.0662892			−0.0331446	+	0.999451i	$0.510552\pi$
$158$		0			0
$159$	3.65050			0.0229591
$160$		0			0
$161$			54.4684i			0.338313i
$162$		0			0
$163$	75.0437			0.460391			0.230195	−	0.973144i	$-0.426063\pi$
$163$	75.0437			0.460391			0.230195	+	0.973144i	$0.426063\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	160.226i		−	0.959436i	−0.877423	−	0.479718i	$-0.840739\pi$
$167$		−	160.226i		−	0.959436i	0.877423	−	0.479718i	$-0.159261\pi$
$168$		0			0
$169$	−15.2491			−0.0902314
$170$		0			0
$171$	272.079			1.59110
$172$		0			0
$173$	222.032			1.28342			0.641710	−	0.766947i	$-0.278225\pi$
$173$	222.032			1.28342			0.641710	+	0.766947i	$0.278225\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			10.1255i			0.0572062i
$178$		0			0
$179$			107.633i			0.601304i	0.953734	+	0.300652i	$0.0972043\pi$
$179$			107.633i			0.601304i	−0.953734	+	0.300652i	$0.902796\pi$
$180$		0			0
$181$			98.5000i			0.544199i	0.962269	+	0.272099i	$0.0877179\pi$
$181$			98.5000i			0.544199i	−0.962269	+	0.272099i	$0.912282\pi$
$182$		0			0
$183$	13.6708			0.0747037
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−118.000			−0.631015
$188$		0			0
$189$			9.28025i			0.0491019i
$190$		0			0
$191$	−240.580			−1.25958			−0.629790	−	0.776765i	$-0.716859\pi$
$191$	−240.580			−1.25958			−0.629790	+	0.776765i	$0.716859\pi$
$192$		0			0
$193$	−112.238			−0.581546			−0.290773	−	0.956792i	$-0.593912\pi$
$193$	−112.238			−0.581546			−0.290773	+	0.956792i	$0.593912\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	41.6011i		−	0.211173i	−0.994410	−	0.105586i	$-0.966328\pi$
$197$		−	41.6011i		−	0.211173i	0.994410	−	0.105586i	$-0.0336720\pi$
$198$		0			0
$199$		−	315.904i		−	1.58746i	−0.608272	−	0.793728i	$-0.708137\pi$
$199$		−	315.904i		−	1.58746i	0.608272	−	0.793728i	$-0.291863\pi$
$200$		0			0
$201$			34.8011i			0.173140i
$202$		0			0
$203$			14.1888i			0.0698954i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	317.145i		−	1.53210i
$208$		0			0
$209$		−	404.266i		−	1.93429i
$210$		0			0
$211$	92.3371			0.437616			0.218808	−	0.975768i	$-0.429783\pi$
$211$	92.3371			0.437616			0.218808	+	0.975768i	$0.429783\pi$
$212$		0			0
$213$		−	10.8054i		−	0.0507294i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	23.7394	−	40.9129i	0.109398	−	0.188539i
$218$		0			0
$219$	−5.71721			−0.0261060
$220$		0			0
$221$	−121.335			−0.549027
$222$		0			0
$223$		−	264.258i		−	1.18502i	−0.805565	−	0.592508i	$-0.798138\pi$
$223$		−	264.258i		−	1.18502i	0.805565	−	0.592508i	$-0.201862\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−162.193			−0.714505			−0.357252	−	0.934008i	$-0.616286\pi$
$227$	−162.193			−0.714505			−0.357252	+	0.934008i	$0.616286\pi$
$228$		0			0
$229$			184.473i			0.805559i	0.915297	+	0.402779i	$0.131956\pi$
$229$			184.473i			0.805559i	−0.915297	+	0.402779i	$0.868044\pi$
$230$		0			0
$231$	6.84991			0.0296533
$232$		0			0
$233$	246.764			1.05907			0.529537	−	0.848287i	$-0.322366\pi$
$233$	246.764			1.05907			0.529537	+	0.848287i	$0.322366\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	0.494343			0.00208584
$238$		0			0
$239$			118.390i			0.495357i	0.968842	+	0.247679i	$0.0796677\pi$
$239$			118.390i			0.495357i	−0.968842	+	0.247679i	$0.920332\pi$
$240$		0			0
$241$			201.362i			0.835529i	0.908555	+	0.417764i	$0.137186\pi$
$241$			201.362i			0.835529i	−0.908555	+	0.417764i	$0.862814\pi$
$242$		0			0
$243$		−	81.2267i		−	0.334266i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	415.692i		−	1.68296i
$248$		0			0
$249$	−5.39686			−0.0216741
$250$		0			0
$251$			350.294i			1.39559i	0.716296	+	0.697796i	$0.245836\pi$
$251$			350.294i			1.39559i	−0.716296	+	0.697796i	$0.754164\pi$
$252$		0			0
$253$	−471.227			−1.86256
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−142.409			−0.554119			−0.277059	−	0.960853i	$-0.589360\pi$
$257$	−142.409			−0.554119			−0.277059	+	0.960853i	$0.589360\pi$
$258$		0			0
$259$			28.6565i			0.110643i
$260$		0			0
$261$		−	82.6148i		−	0.316532i
$262$		0			0
$263$		−	508.385i		−	1.93302i	−0.256623	−	0.966511i	$-0.582610\pi$
$263$		−	508.385i		−	1.93302i	0.256623	−	0.966511i	$-0.417390\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	4.73463			0.0177327
$268$		0			0
$269$		−	209.924i		−	0.780385i	−0.920733	−	0.390192i	$-0.872408\pi$
$269$		−	209.924i		−	0.780385i	0.920733	−	0.390192i	$-0.127592\pi$
$270$		0			0
$271$		−	243.635i		−	0.899022i	−0.893275	−	0.449511i	$-0.851598\pi$
$271$		−	243.635i		−	0.899022i	0.893275	−	0.449511i	$-0.148402\pi$
$272$		0			0
$273$	7.04351			0.0258004
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	310.453i		−	1.12077i	−0.828233	−	0.560384i	$-0.810654\pi$
$277$		−	310.453i		−	1.12077i	0.828233	−	0.560384i	$-0.189346\pi$
$278$		0			0
$279$	−138.224	+	238.217i	−0.495426	+	0.853826i
$280$		0			0
$281$	301.008			1.07120			0.535601	−	0.844471i	$-0.320085\pi$
$281$	301.008			1.07120			0.535601	+	0.844471i	$0.320085\pi$
$282$		0			0
$283$	−284.321			−1.00467			−0.502334	−	0.864673i	$-0.667525\pi$
$283$	−284.321			−1.00467			−0.502334	+	0.864673i	$0.667525\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.57470			0.0124554
$288$		0			0
$289$	209.096			0.723517
$290$		0			0
$291$			22.1231i			0.0760242i
$292$		0			0
$293$	−418.987			−1.42999			−0.714995	−	0.699130i	$-0.753571\pi$
$293$	−418.987			−1.42999			−0.714995	+	0.699130i	$0.753571\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−80.2870			−0.270327
$298$		0			0
$299$	−484.545			−1.62055
$300$		0			0
$301$		−	55.6093i		−	0.184749i
$302$		0			0
$303$		−	14.1663i		−	0.0467535i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−255.279			−0.831528			−0.415764	−	0.909473i	$-0.636486\pi$
$307$	−255.279			−0.831528			−0.415764	+	0.909473i	$0.636486\pi$
$308$		0			0
$309$			40.8213i			0.132108i
$310$		0			0
$311$	−103.066			−0.331403			−0.165702	−	0.986176i	$-0.552989\pi$
$311$	−103.066			−0.331403			−0.165702	+	0.986176i	$0.552989\pi$
$312$		0			0
$313$			345.362i			1.10339i	0.834045	+	0.551696i	$0.186019\pi$
$313$			345.362i			1.10339i	−0.834045	+	0.551696i	$0.813981\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−159.664			−0.503672			−0.251836	−	0.967770i	$-0.581034\pi$
$317$	−159.664			−0.503672			−0.251836	+	0.967770i	$0.581034\pi$
$318$		0			0
$319$	−122.753			−0.384804
$320$		0			0
$321$			26.4346i			0.0823507i
$322$		0			0
$323$		−	273.749i		−	0.847519i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			20.2534i			0.0619369i
$328$		0			0
$329$	−65.9881			−0.200572
$330$		0			0
$331$			197.921i			0.597948i	0.954261	+	0.298974i	$0.0966444\pi$
$331$			197.921i			0.597948i	−0.954261	+	0.298974i	$0.903356\pi$
$332$		0			0
$333$		−	166.854i		−	0.501063i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	379.929i		−	1.12739i	−0.825984	−	0.563693i	$-0.809380\pi$
$337$		−	379.929i		−	1.12739i	0.825984	−	0.563693i	$-0.190620\pi$
$338$		0			0
$339$		−	54.4453i		−	0.160606i
$340$		0			0
$341$	353.953	+	205.379i	1.03799	+	0.602284i
$342$		0			0
$343$	145.981			0.425601
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	222.832i		−	0.642167i	−0.947051	−	0.321083i	$-0.895953\pi$
$347$		−	222.832i		−	0.642167i	0.947051	−	0.321083i	$-0.104047\pi$
$348$		0			0
$349$	−424.518			−1.21638			−0.608192	−	0.793790i	$-0.708105\pi$
$349$	−424.518			−1.21638			−0.608192	+	0.793790i	$0.708105\pi$
$350$		0			0
$351$	−82.5562			−0.235203
$352$		0			0
$353$			502.515i			1.42355i	0.702405	+	0.711777i	$0.252110\pi$
$353$			502.515i			1.42355i	−0.702405	+	0.711777i	$0.747890\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	4.63841			0.0129928
$358$		0			0
$359$	−337.582			−0.940340			−0.470170	−	0.882576i	$-0.655807\pi$
$359$	−337.582			−0.940340			−0.470170	+	0.882576i	$0.655807\pi$
$360$		0			0
$361$	576.860			1.59795
$362$		0			0
$363$			18.1122i			0.0498960i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			263.801i			0.718803i	0.933183	+	0.359401i	$0.117019\pi$
$367$			263.801i			0.718803i	−0.933183	+	0.359401i	$0.882981\pi$
$368$		0			0
$369$	−20.8139			−0.0564062
$370$		0			0
$371$		−	16.3792i		−	0.0441487i
$372$		0			0
$373$	−197.433			−0.529312			−0.264656	−	0.964343i	$-0.585258\pi$
$373$	−197.433			−0.529312			−0.264656	+	0.964343i	$0.585258\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−126.222			−0.334806
$378$		0			0
$379$	302.263			0.797528			0.398764	−	0.917054i	$-0.369439\pi$
$379$	302.263			0.797528			0.398764	+	0.917054i	$0.369439\pi$
$380$		0			0
$381$	−63.6579			−0.167081
$382$		0			0
$383$			140.637i			0.367200i	0.983001	+	0.183600i	$0.0587750\pi$
$383$			140.637i			0.367200i	−0.983001	+	0.183600i	$0.941225\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			323.788i			0.836661i
$388$		0			0
$389$		−	458.622i		−	1.17898i	−0.807776	−	0.589489i	$-0.799329\pi$
$389$		−	458.622i		−	1.17898i	0.807776	−	0.589489i	$-0.200671\pi$
$390$		0			0
$391$	−319.091			−0.816090
$392$		0			0
$393$		−	18.6777i		−	0.0475259i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	101.401			0.255418			0.127709	−	0.991812i	$-0.459238\pi$
$397$	101.401			0.255418			0.127709	+	0.991812i	$0.459238\pi$
$398$		0			0
$399$			15.8912i			0.0398275i
$400$		0			0
$401$			725.566i			1.80939i	0.426058	+	0.904696i	$0.359902\pi$
$401$			725.566i			1.80939i	−0.426058	+	0.904696i	$0.640098\pi$
$402$		0			0
$403$	363.957	+	211.184i	0.903120	+	0.524029i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−247.919			−0.609137
$408$		0			0
$409$		−	668.691i		−	1.63494i	−0.575970	−	0.817471i	$-0.695376\pi$
$409$		−	668.691i		−	1.63494i	0.575970	−	0.817471i	$-0.304624\pi$
$410$		0			0
$411$	9.12827			0.0222099
$412$		0			0
$413$	45.4313			0.110003
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−77.7588			−0.186472
$418$		0			0
$419$	−409.743			−0.977906			−0.488953	−	0.872310i	$-0.662621\pi$
$419$	−409.743			−0.977906			−0.488953	+	0.872310i	$0.662621\pi$
$420$		0			0
$421$	674.520			1.60218			0.801092	−	0.598541i	$-0.204253\pi$
$421$	674.520			1.60218			0.801092	+	0.598541i	$0.204253\pi$
$422$		0			0
$423$	384.219			0.908319
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	61.3383i		−	0.143649i
$428$		0			0
$429$			60.9362i			0.142042i
$430$		0			0
$431$	202.735			0.470383			0.235192	−	0.971949i	$-0.424428\pi$
$431$	202.735			0.470383			0.235192	+	0.971949i	$0.424428\pi$
$432$		0			0
$433$		−	99.1343i		−	0.228947i	−0.993426	−	0.114474i	$-0.963482\pi$
$433$		−	99.1343i		−	0.228947i	0.993426	−	0.114474i	$-0.0365182\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	1093.20i		−	2.50161i
$438$		0			0
$439$	831.880			1.89494			0.947472	−	0.319839i	$-0.103629\pi$
$439$	831.880			1.89494			0.947472	+	0.319839i	$0.103629\pi$
$440$		0			0
$441$	−414.648			−0.940246
$442$		0			0
$443$	188.284			0.425021			0.212510	−	0.977159i	$-0.431836\pi$
$443$	188.284			0.425021			0.212510	+	0.977159i	$0.431836\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			8.34008i			0.0186579i
$448$		0			0
$449$		−	790.951i		−	1.76158i	−0.473503	−	0.880792i	$-0.657011\pi$
$449$		−	790.951i		−	1.76158i	0.473503	−	0.880792i	$-0.342989\pi$
$450$		0			0
$451$			30.9262i			0.0685724i
$452$		0			0
$453$	61.2523			0.135215
$454$		0			0
$455$		0			0
$456$		0			0
$457$			661.286i			1.44702i	0.690316	+	0.723508i	$0.257471\pi$
$457$			661.286i			1.44702i	−0.690316	+	0.723508i	$0.742529\pi$
$458$		0			0
$459$	−54.3663			−0.118445
$460$		0			0
$461$			469.975i			1.01947i	0.860332	+	0.509734i	$0.170256\pi$
$461$			469.975i			1.01947i	−0.860332	+	0.509734i	$0.829744\pi$
$462$		0			0
$463$		−	209.165i		−	0.451760i	−0.974155	−	0.225880i	$-0.927474\pi$
$463$		−	209.165i		−	0.451760i	0.974155	−	0.225880i	$-0.0725257\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8.87504			−0.0190044			−0.00950219	−	0.999955i	$-0.503025\pi$
$467$	−8.87504			−0.0190044			−0.00950219	+	0.999955i	$0.503025\pi$
$468$		0			0
$469$	156.146			0.332935
$470$		0			0
$471$			3.53929i			0.00751442i
$472$		0			0
$473$	481.098			1.01712
$474$		0			0
$475$		0			0
$476$		0			0
$477$			95.3684i			0.199934i
$478$		0			0
$479$	667.940			1.39445			0.697223	−	0.716854i	$-0.254419\pi$
$479$	667.940			1.39445			0.697223	+	0.716854i	$0.254419\pi$
$480$		0			0
$481$	−254.926			−0.529991
$482$		0			0
$483$	18.5233			0.0383505
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	126.384i		−	0.259515i	−0.991546	−	0.129757i	$-0.958580\pi$
$487$		−	126.384i		−	0.259515i	0.991546	−	0.129757i	$-0.0414198\pi$
$488$		0			0
$489$		−	25.5204i		−	0.0521890i
$490$		0			0
$491$		−	458.056i		−	0.932905i	−0.884546	−	0.466453i	$-0.845532\pi$
$491$		−	458.056i		−	0.932905i	0.884546	−	0.466453i	$-0.154468\pi$
$492$		0			0
$493$	−83.1218			−0.168604
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−48.4817			−0.0975487
$498$		0			0
$499$			554.614i			1.11145i	0.831366	+	0.555726i	$0.187559\pi$
$499$			554.614i			1.11145i	−0.831366	+	0.555726i	$0.812441\pi$
$500$		0			0
$501$	−54.4887			−0.108760
$502$		0			0
$503$	−378.314			−0.752115			−0.376057	−	0.926596i	$-0.622720\pi$
$503$	−378.314			−0.752115			−0.376057	+	0.926596i	$0.622720\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			5.18583i			0.0102285i
$508$		0			0
$509$		−	701.672i		−	1.37853i	−0.724509	−	0.689265i	$-0.757934\pi$
$509$		−	701.672i		−	1.37853i	0.724509	−	0.689265i	$-0.242066\pi$
$510$		0			0
$511$			25.6521i			0.0501998i
$512$		0			0
$513$		−	186.258i		−	0.363077i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	570.889i		−	1.10423i
$518$		0			0
$519$		−	75.5073i		−	0.145486i
$520$		0			0
$521$	99.8036			0.191562			0.0957809	−	0.995402i	$-0.469465\pi$
$521$	99.8036			0.191562			0.0957809	+	0.995402i	$0.469465\pi$
$522$		0			0
$523$		−	481.944i		−	0.921499i	−0.887530	−	0.460749i	$-0.847581\pi$
$523$		−	481.944i		−	0.921499i	0.887530	−	0.460749i	$-0.152419\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	239.679	+	139.072i	0.454800	+	0.263894i
$528$		0			0
$529$	−745.276			−1.40884
$530$		0			0
$531$	−264.526			−0.498166
$532$		0			0
$533$			31.8002i			0.0596627i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	36.6034			0.0681627
$538$		0			0
$539$			616.102i			1.14305i
$540$		0			0
$541$	−718.944			−1.32892			−0.664458	−	0.747325i	$-0.731338\pi$
$541$	−718.944			−1.32892			−0.664458	+	0.747325i	$0.731338\pi$
$542$		0			0
$543$	33.4973			0.0616893
$544$		0			0
$545$		0			0
$546$		0			0
$547$	366.679			0.670345			0.335173	−	0.942157i	$-0.391205\pi$
$547$	366.679			0.670345			0.335173	+	0.942157i	$0.391205\pi$
$548$		0			0
$549$			357.145i			0.650538i
$550$		0			0
$551$		−	284.775i		−	0.516832i
$552$		0			0
$553$		−	2.21803i		−	0.00401090i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	61.2414i		−	0.109949i	−0.998488	−	0.0549743i	$-0.982492\pi$
$557$		−	61.2414i		−	0.109949i	0.998488	−	0.0549743i	$-0.0175077\pi$
$558$		0			0
$559$	494.695			0.884964
$560$		0			0
$561$			40.1287i			0.0715307i
$562$		0			0
$563$	−825.335			−1.46596			−0.732980	−	0.680251i	$-0.761871\pi$
$563$	−825.335			−1.46596			−0.732980	+	0.680251i	$0.761871\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−118.850			−0.209612
$568$		0			0
$569$			713.272i			1.25355i	0.779199	+	0.626776i	$0.215626\pi$
$569$			713.272i			1.25355i	−0.779199	+	0.626776i	$0.784374\pi$
$570$		0			0
$571$		−	385.649i		−	0.675393i	−0.941255	−	0.337696i	$-0.890352\pi$
$571$		−	385.649i		−	0.675393i	0.941255	−	0.337696i	$-0.109648\pi$
$572$		0			0
$573$			81.8150i			0.142784i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−583.345			−1.01100			−0.505498	−	0.862828i	$-0.668691\pi$
$577$	−583.345			−1.01100			−0.505498	+	0.862828i	$0.668691\pi$
$578$		0			0
$579$			38.1694i			0.0659229i
$580$		0			0
$581$			24.2147i			0.0416777i
$582$		0			0
$583$	141.702			0.243057
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	875.985i		−	1.49231i	−0.665773	−	0.746154i	$-0.731898\pi$
$587$		−	875.985i		−	1.49231i	0.665773	−	0.746154i	$-0.268102\pi$
$588$		0			0
$589$	−476.460	+	821.139i	−0.808930	+	1.39412i
$590$		0			0
$591$	−14.1475			−0.0239382
$592$		0			0
$593$	−111.819			−0.188564			−0.0942821	−	0.995546i	$-0.530056\pi$
$593$	−111.819			−0.188564			−0.0942821	+	0.995546i	$0.530056\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−107.431			−0.179951
$598$		0			0
$599$	−761.791			−1.27177			−0.635886	−	0.771783i	$-0.719365\pi$
$599$	−761.791			−1.27177			−0.635886	+	0.771783i	$0.719365\pi$
$600$		0			0
$601$		−	293.840i		−	0.488919i	−0.969660	−	0.244459i	$-0.921390\pi$
$601$		−	293.840i		−	0.488919i	0.969660	−	0.244459i	$-0.0786105\pi$
$602$		0			0
$603$	−909.170			−1.50774
$604$		0			0
$605$		0			0
$606$		0			0
$607$	702.596			1.15749			0.578745	−	0.815509i	$-0.303543\pi$
$607$	702.596			1.15749			0.578745	+	0.815509i	$0.303543\pi$
$608$		0			0
$609$	4.82524			0.00792322
$610$		0			0
$611$		−	587.024i		−	0.960759i
$612$		0			0
$613$		−	286.276i		−	0.467008i	−0.972356	−	0.233504i	$-0.924981\pi$
$613$		−	286.276i		−	0.467008i	0.972356	−	0.233504i	$-0.0750192\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1030.59			1.67032			0.835158	−	0.550010i	$-0.185376\pi$
$617$	1030.59			1.67032			0.835158	+	0.550010i	$0.185376\pi$
$618$		0			0
$619$		−	419.567i		−	0.677815i	−0.940820	−	0.338907i	$-0.889943\pi$
$619$		−	419.567i		−	0.677815i	0.940820	−	0.338907i	$-0.110057\pi$
$620$		0			0
$621$	−217.109			−0.349613
$622$		0			0
$623$		−	21.2434i		−	0.0340986i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−137.481			−0.219267
$628$		0			0
$629$	−167.878			−0.266896
$630$		0			0
$631$		−	399.185i		−	0.632624i	−0.948655	−	0.316312i	$-0.897555\pi$
$631$		−	399.185i		−	0.632624i	0.948655	−	0.316312i	$-0.102445\pi$
$632$		0			0
$633$		−	31.4015i		−	0.0496074i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			633.515i			0.994529i
$638$		0			0
$639$	282.287			0.441764
$640$		0			0
$641$			714.201i			1.11420i	0.830446	+	0.557099i	$0.188086\pi$
$641$			714.201i			1.11420i	−0.830446	+	0.557099i	$0.811914\pi$
$642$		0			0
$643$		−	404.800i		−	0.629550i	−0.949166	−	0.314775i	$-0.898071\pi$
$643$		−	404.800i		−	0.629550i	0.949166	−	0.314775i	$-0.101929\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			203.623i			0.314719i	0.987541	+	0.157359i	$0.0502981\pi$
$647$			203.623i			0.314719i	−0.987541	+	0.157359i	$0.949702\pi$
$648$		0			0
$649$			393.044i			0.605615i
$650$		0			0
$651$	−13.9134	−	8.07317i	−0.0213724	−	0.0124012i
$652$		0			0
$653$	299.917			0.459291			0.229646	−	0.973274i	$-0.426243\pi$
$653$	299.917			0.459291			0.229646	+	0.973274i	$0.426243\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	149.361i		−	0.227337i
$658$		0			0
$659$	657.739			0.998087			0.499043	−	0.866577i	$-0.333685\pi$
$659$	657.739			0.998087			0.499043	+	0.866577i	$0.333685\pi$
$660$		0			0
$661$	−79.9227			−0.120912			−0.0604559	−	0.998171i	$-0.519255\pi$
$661$	−79.9227			−0.120912			−0.0604559	+	0.998171i	$0.519255\pi$
$662$		0			0
$663$			41.2629i			0.0622366i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−331.943			−0.497666
$668$		0			0
$669$	−89.8675			−0.134331
$670$		0			0
$671$	530.662			0.790852
$672$		0			0
$673$			1117.96i			1.66117i	0.556895	+	0.830583i	$0.311992\pi$
$673$			1117.96i			1.66117i	−0.556895	+	0.830583i	$0.688008\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			1035.61i			1.52970i	0.644206	+	0.764852i	$0.277188\pi$
$677$			1035.61i			1.52970i	−0.644206	+	0.764852i	$0.722812\pi$
$678$		0			0
$679$	99.2622			0.146189
$680$		0			0
$681$			55.1575i			0.0809949i
$682$		0			0
$683$	−640.892			−0.938348			−0.469174	−	0.883106i	$-0.655448\pi$
$683$	−640.892			−0.938348			−0.469174	+	0.883106i	$0.655448\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	62.7345			0.0913166
$688$		0			0
$689$	145.707			0.211477
$690$		0			0
$691$	242.241			0.350565			0.175283	−	0.984518i	$-0.443916\pi$
$691$	242.241			0.350565			0.175283	+	0.984518i	$0.443916\pi$
$692$		0			0
$693$			178.952i			0.258228i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			20.9416i			0.0300454i
$698$		0			0
$699$		−	83.9181i		−	0.120055i
$700$		0			0
$701$	369.274			0.526781			0.263391	−	0.964689i	$-0.415159\pi$
$701$	369.274			0.526781			0.263391	+	0.964689i	$0.415159\pi$
$702$		0			0
$703$		−	575.148i		−	0.818134i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−63.5617			−0.0899034
$708$		0			0
$709$		−	933.472i		−	1.31660i	−0.752754	−	0.658302i	$-0.771275\pi$
$709$		−	933.472i		−	1.31660i	0.752754	−	0.658302i	$-0.228725\pi$
$710$		0			0
$711$			12.9146i			0.0181640i
$712$		0			0
$713$	957.149	+	555.379i	1.34242	+	0.778932i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	40.2615			0.0561528
$718$		0			0
$719$			50.7169i			0.0705381i	0.999378	+	0.0352691i	$0.0112288\pi$
$719$			50.7169i			0.0705381i	−0.999378	+	0.0352691i	$0.988771\pi$
$720$		0			0
$721$	183.158			0.254033
$722$		0			0
$723$	68.4782			0.0947140
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−1248.02			−1.71667			−0.858337	−	0.513086i	$-0.828502\pi$
$727$	−1248.02			−1.71667			−0.858337	+	0.513086i	$0.828502\pi$
$728$		0			0
$729$	673.394			0.923723
$730$		0			0
$731$	325.775			0.445657
$732$		0			0
$733$	1216.57			1.65972			0.829859	−	0.557974i	$-0.188421\pi$
$733$	1216.57			1.65972			0.829859	+	0.557974i	$0.188421\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1350.88i			1.83295i
$738$		0			0
$739$			595.062i			0.805226i	0.915370	+	0.402613i	$0.131898\pi$
$739$			595.062i			0.805226i	−0.915370	+	0.402613i	$0.868102\pi$
$740$		0			0
$741$	−141.366			−0.190778
$742$		0			0
$743$			528.164i			0.710854i	0.934704	+	0.355427i	$0.115664\pi$
$743$			528.164i			0.710854i	−0.934704	+	0.355427i	$0.884336\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	140.991i		−	0.188744i
$748$		0			0
$749$	118.607			0.158354
$750$		0			0
$751$	−279.577			−0.372273			−0.186136	−	0.982524i	$-0.559597\pi$
$751$	−279.577			−0.372273			−0.186136	+	0.982524i	$0.559597\pi$
$752$		0			0
$753$	119.126			0.158202
$754$		0			0
$755$		0			0
$756$		0			0
$757$			271.281i			0.358363i	0.983816	+	0.179181i	$0.0573449\pi$
$757$			271.281i			0.358363i	−0.983816	+	0.179181i	$0.942655\pi$
$758$		0			0
$759$			160.252i			0.211136i
$760$		0			0
$761$		−	1009.53i		−	1.32658i	−0.748361	−	0.663292i	$-0.769159\pi$
$761$		−	1009.53i		−	1.32658i	0.748361	−	0.663292i	$-0.230841\pi$
$762$		0			0
$763$	90.8732			0.119100
$764$		0			0
$765$		0			0
$766$		0			0
$767$			404.152i			0.526926i
$768$		0			0
$769$	−58.0853			−0.0755336			−0.0377668	−	0.999287i	$-0.512024\pi$
$769$	−58.0853			−0.0755336			−0.0377668	+	0.999287i	$0.512024\pi$
$770$		0			0
$771$			48.4295i			0.0628139i
$772$		0			0
$773$			506.233i			0.654894i	0.944870	+	0.327447i	$0.106188\pi$
$773$			506.233i			0.654894i	−0.944870	+	0.327447i	$0.893812\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	9.74534			0.0125423
$778$		0			0
$779$	−71.7458			−0.0920999
$780$		0			0
$781$		−	419.434i		−	0.537047i
$782$		0			0
$783$	−56.5561			−0.0722300
$784$		0			0
$785$		0			0
$786$		0			0
$787$			570.563i			0.724984i	0.931987	+	0.362492i	$0.118074\pi$
$787$			570.563i			0.724984i	−0.931987	+	0.362492i	$0.881926\pi$
$788$		0			0
$789$	−172.889			−0.219124
$790$		0			0
$791$	−244.286			−0.308832
$792$		0			0
$793$	545.660			0.688095
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1091.70i			1.36976i	0.728657	+	0.684879i	$0.240145\pi$
$797$			1091.70i			1.36976i	−0.728657	+	0.684879i	$0.759855\pi$
$798$		0			0
$799$		−	386.577i		−	0.483826i
$800$		0			0
$801$			123.691i			0.154421i
$802$		0			0
$803$	−221.926			−0.276372
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−71.3896			−0.0884630
$808$		0			0
$809$		−	356.586i		−	0.440774i	−0.975413	−	0.220387i	$-0.929268\pi$
$809$		−	356.586i		−	0.440774i	0.975413	−	0.220387i	$-0.0707320\pi$
$810$		0			0
$811$	−572.137			−0.705471			−0.352735	−	0.935723i	$-0.614748\pi$
$811$	−572.137			−0.705471			−0.352735	+	0.935723i	$0.614748\pi$
$812$		0			0
$813$	−82.8540			−0.101911
$814$		0			0
$815$		0			0
$816$		0			0
$817$			1116.10i			1.36610i
$818$		0			0
$819$			184.010i			0.224676i
$820$		0			0
$821$		−	131.113i		−	0.159699i	−0.996807	−	0.0798493i	$-0.974556\pi$
$821$		−	131.113i		−	0.159699i	0.996807	−	0.0798493i	$-0.0254439\pi$
$822$		0			0
$823$		−	800.405i		−	0.972545i	−0.873807	−	0.486273i	$-0.838356\pi$
$823$		−	800.405i		−	0.972545i	0.873807	−	0.486273i	$-0.161644\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	534.697i		−	0.646550i	−0.946305	−	0.323275i	$-0.895216\pi$
$827$		−	534.697i		−	0.646550i	0.946305	−	0.323275i	$-0.104784\pi$
$828$		0			0
$829$			724.322i			0.873729i	0.899527	+	0.436865i	$0.143911\pi$
$829$			724.322i			0.873729i	−0.899527	+	0.436865i	$0.856089\pi$
$830$		0			0
$831$	−105.577			−0.127048
$832$		0			0
$833$			417.193i			0.500832i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	163.078	+	94.6247i	0.194836	+	0.113052i
$838$		0			0
$839$	1204.86			1.43606			0.718031	−	0.696011i	$-0.245044\pi$
$839$	1204.86			1.43606			0.718031	+	0.696011i	$0.245044\pi$
$840$		0			0
$841$	754.530			0.897182
$842$		0			0
$843$		−	102.365i		−	0.121430i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	81.2664			0.0959462
$848$		0			0
$849$			96.6904i			0.113887i
$850$		0			0
$851$	−670.413			−0.787794
$852$		0			0
$853$	56.6817			0.0664498			0.0332249	−	0.999448i	$-0.489422\pi$
$853$	56.6817			0.0664498			0.0332249	+	0.999448i	$0.489422\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	358.028			0.417769			0.208885	−	0.977940i	$-0.433017\pi$
$857$	358.028			0.417769			0.208885	+	0.977940i	$0.433017\pi$
$858$		0			0
$859$			1453.53i			1.69211i	0.533093	+	0.846056i	$0.321029\pi$
$859$			1453.53i			1.69211i	−0.533093	+	0.846056i	$0.678971\pi$
$860$		0			0
$861$		−	1.21567i		−	0.00141192i
$862$		0			0
$863$			1006.72i			1.16654i	0.812278	+	0.583270i	$0.198227\pi$
$863$			1006.72i			1.16654i	−0.812278	+	0.583270i	$0.801773\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	71.1084i		−	0.0820166i
$868$		0			0
$869$	19.1890			0.0220817
$870$		0			0
$871$			1389.06i			1.59479i
$872$		0			0
$873$	−577.959			−0.662037
$874$		0			0
$875$		0			0
$876$		0			0
$877$	768.169			0.875906			0.437953	−	0.898998i	$-0.355704\pi$
$877$	768.169			0.875906			0.437953	+	0.898998i	$0.355704\pi$
$878$		0			0
$879$			142.487i			0.162101i
$880$		0			0
$881$		−	176.601i		−	0.200455i	−0.994965	−	0.100228i	$-0.968043\pi$
$881$		−	176.601i		−	0.200455i	0.994965	−	0.100228i	$-0.0319571\pi$
$882$		0			0
$883$		−	965.805i		−	1.09378i	−0.837205	−	0.546888i	$-0.815812\pi$
$883$		−	965.805i		−	1.09378i	0.837205	−	0.546888i	$-0.184188\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1437.02			1.62009			0.810044	−	0.586369i	$-0.199443\pi$
$887$	1437.02			1.62009			0.810044	+	0.586369i	$0.199443\pi$
$888$		0			0
$889$			285.621i			0.321284i
$890$		0			0
$891$		−	1028.22i		−	1.15400i
$892$		0			0
$893$	1324.41			1.48310
$894$		0			0
$895$		0			0
$896$		0			0
$897$			164.781i			0.183703i
$898$		0			0
$899$	249.333	+	144.674i	0.277345	+	0.160927i
$900$		0			0
$901$	95.9537			0.106497
$902$		0			0
$903$	−18.9113			−0.0209428
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1075.56			1.18584			0.592922	−	0.805260i	$-0.297974\pi$
$907$	1075.56			1.18584			0.592922	+	0.805260i	$0.297974\pi$
$908$		0			0
$909$	370.091			0.407141
$910$		0			0
$911$		−	891.650i		−	0.978760i	−0.872071	−	0.489380i	$-0.837223\pi$
$911$		−	891.650i		−	0.978760i	0.872071	−	0.489380i	$-0.162777\pi$
$912$		0			0
$913$	−209.491			−0.229454
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−83.8034			−0.0913886
$918$		0			0
$919$	−641.956			−0.698538			−0.349269	−	0.937022i	$-0.613570\pi$
$919$	−641.956			−0.698538			−0.349269	+	0.937022i	$0.613570\pi$
$920$		0			0
$921$			86.8139i			0.0942605i
$922$		0			0
$923$		−	431.289i		−	0.467268i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−1066.45			−1.15043
$928$		0			0
$929$		−	871.804i		−	0.938432i	−0.883083	−	0.469216i	$-0.844537\pi$
$929$		−	871.804i		−	0.938432i	0.883083	−	0.469216i	$-0.155463\pi$
$930$		0			0
$931$	−1429.30			−1.53523
$932$		0			0
$933$			35.0503i			0.0375673i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1455.27			1.55312			0.776559	−	0.630045i	$-0.216963\pi$
$937$	1455.27			1.55312			0.776559	+	0.630045i	$0.216963\pi$
$938$		0			0
$939$	117.449			0.125078
$940$		0			0
$941$			601.162i			0.638855i	0.947611	+	0.319427i	$0.103491\pi$
$941$			601.162i			0.638855i	−0.947611	+	0.319427i	$0.896509\pi$
$942$		0			0
$943$			83.6294i			0.0886844i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			964.435i			1.01841i	0.860645	+	0.509205i	$0.170061\pi$
$947$			964.435i			1.01841i	−0.860645	+	0.509205i	$0.829939\pi$
$948$		0			0
$949$	−228.199			−0.240462
$950$		0			0
$951$			54.2976i			0.0570953i
$952$		0			0
$953$			1408.90i			1.47839i	0.673494	+	0.739193i	$0.264793\pi$
$953$			1408.90i			1.47839i	−0.673494	+	0.739193i	$0.735207\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			41.7450i			0.0436207i
$958$		0			0
$959$		−	40.9569i		−	0.0427079i
$960$		0			0
$961$	−476.889	−	834.325i	−0.496243	−	0.868184i
$962$		0			0
$963$	−690.596			−0.717129
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	863.679i		−	0.893153i	−0.894745	−	0.446577i	$-0.852643\pi$
$967$		−	863.679i		−	0.893153i	0.894745	−	0.446577i	$-0.147357\pi$
$968$		0			0
$969$	−93.0949			−0.0960732
$970$		0			0
$971$	345.448			0.355766			0.177883	−	0.984052i	$-0.443075\pi$
$971$	345.448			0.355766			0.177883	+	0.984052i	$0.443075\pi$
$972$		0			0
$973$			348.890i			0.358571i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1355.89			1.38781			0.693906	−	0.720066i	$-0.255888\pi$
$977$	1355.89			1.38781			0.693906	+	0.720066i	$0.255888\pi$
$978$		0			0
$979$	183.785			0.187727
$980$		0			0
$981$	−529.114			−0.539361
$982$		0			0
$983$			515.321i			0.524233i	0.965036	+	0.262116i	$0.0844205\pi$
$983$			515.321i			0.524233i	−0.965036	+	0.262116i	$0.915580\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$			22.4409i			0.0227364i
$988$		0			0
$989$	1300.97			1.31544
$990$		0			0
$991$		−	1390.90i		−	1.40353i	−0.712406	−	0.701767i	$-0.752395\pi$
$991$		−	1390.90i		−	1.40353i	0.712406	−	0.701767i	$-0.247605\pi$
$992$		0			0
$993$	67.3078			0.0677823
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−601.753			−0.603564			−0.301782	−	0.953377i	$-0.597581\pi$
$997$	−601.753			−0.603564			−0.301782	+	0.953377i	$0.597581\pi$
$998$		0			0
$999$	−114.224			−0.114338

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3100.3.d.f.1301.11	✓	22
5.2	odd	4		3100.3.f.d.1549.22		44
5.3	odd	4		3100.3.f.d.1549.23		44
5.4	even	2		3100.3.d.g.1301.12	yes	22
31.30	odd	2	inner	3100.3.d.f.1301.12	yes	22
155.92	even	4		3100.3.f.d.1549.24		44
155.123	even	4		3100.3.f.d.1549.21		44
155.154	odd	2		3100.3.d.g.1301.11	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3100.3.d.f.1301.11	✓	22	1.1	even	1	trivial
3100.3.d.f.1301.12	yes	22	31.30	odd	2	inner
3100.3.d.g.1301.11	yes	22	155.154	odd	2
3100.3.d.g.1301.12	yes	22	5.4	even	2
3100.3.f.d.1549.21		44	155.123	even	4
3100.3.f.d.1549.22		44	5.2	odd	4
3100.3.f.d.1549.23		44	5.3	odd	4
3100.3.f.d.1549.24		44	155.92	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3100 = 2^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3100.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.340074i q^{3} -1.52585 q^{7} +8.88435 q^{9} +O(q^{10})\)	\(q-0.340074i q^{3} -1.52585 q^{7} +8.88435 q^{9} -13.2007i q^{11} -13.5738i q^{13} -8.93888i q^{17} +30.6245 q^{19} +0.518903i q^{21} -35.6970i q^{23} -6.08201i q^{27} -9.29891i q^{29} +(-15.5581 + 26.8131i) q^{31} -4.48924 q^{33} -18.7806i q^{37} -4.61611 q^{39} -2.34276 q^{41} +36.4447i q^{43} +43.2467 q^{47} -46.6718 q^{49} -3.03988 q^{51} +10.7344i q^{53} -10.4146i q^{57} -29.7744 q^{59} +40.1994i q^{61} -13.5562 q^{63} -102.334 q^{67} -12.1396 q^{69} +31.7735 q^{71} -16.8117i q^{73} +20.1424i q^{77} +1.45363i q^{79} +77.8908 q^{81} -15.8696i q^{83} -3.16232 q^{87} +13.9223i q^{89} +20.7117i q^{91} +(9.11846 + 5.29092i) q^{93} -65.0536 q^{97} -117.280i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 4 q^{7} - 72 q^{9}+O(q^{10}) \) 22 * q - 4 * q^7 - 72 * q^9	\( 22 q - 4 q^{7} - 72 q^{9} - 28 q^{19} - 18 q^{31} + 34 q^{33} - 62 q^{39} + 64 q^{41} + 96 q^{47} + 150 q^{49} - 130 q^{51} + 40 q^{59} + 4 q^{63} - 110 q^{67} + 100 q^{69} + 132 q^{71} + 234 q^{81} + 62 q^{87} - 16 q^{93} - 186 q^{97}+O(q^{100}) \) 22 * q - 4 * q^7 - 72 * q^9 - 28 * q^19 - 18 * q^31 + 34 * q^33 - 62 * q^39 + 64 * q^41 + 96 * q^47 + 150 * q^49 - 130 * q^51 + 40 * q^59 + 4 * q^63 - 110 * q^67 + 100 * q^69 + 132 * q^71 + 234 * q^81 + 62 * q^87 - 16 * q^93 - 186 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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