LMFDB - Embedded newform 3100.3.f.b.1549.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3100,3,Mod(1549,3100)]

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, 1, 1]))

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

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

chi := DirichletCharacter("3100.1549");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3100 = 2^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3100.f (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:	$84.4688819517$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.256992219136.8
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 105x^{4} + 64 $ x^8 + 105*x^4 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 124)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1549.7
Root			$2.26020 - 2.26020i$ of defining polynomial
Character	$\chi$	$=$	3100.1549
Dual form			3100.3.f.b.1549.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.52040 q^{3} -3.21699i q^{7} +11.4340 q^{9} +O(q^{10})$	$q+4.52040 q^{3} -3.21699i q^{7} +11.4340 q^{9} +8.05991i q^{11} -15.5230 q^{13} -25.5446 q^{17} +15.6510 q^{19} -14.5421i q^{21} -23.5829 q^{23} +11.0026 q^{27} +38.7218i q^{29} +(4.56602 - 30.6619i) q^{31} +36.4340i q^{33} -2.94265 q^{37} -70.1699 q^{39} +6.51893 q^{41} +4.52040 q^{43} +72.1699i q^{47} +38.6510 q^{49} -115.472 q^{51} -71.3455 q^{53} +70.7486 q^{57} -42.9529 q^{59} +83.9258i q^{61} -36.7830i q^{63} +82.8680i q^{67} -106.604 q^{69} -74.2549 q^{71} -5.11726 q^{73} +25.9287 q^{77} +8.23109i q^{79} -53.1699 q^{81} -103.585 q^{83} +175.038i q^{87} -114.759i q^{89} +49.9372i q^{91} +(20.6402 - 138.604i) q^{93} -21.8209i q^{97} +92.1568i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{9}+O(q^{10}) $ 8 * q + 16 * q^9	$ 8 q + 16 q^{9} + 12 q^{19} + 112 q^{31} - 184 q^{39} - 212 q^{41} + 196 q^{49} - 320 q^{51} - 4 q^{59} - 400 q^{69} - 28 q^{71} - 48 q^{81}+O(q^{100}) $ 8 * q + 16 * q^9 + 12 * q^19 + 112 * q^31 - 184 * q^39 - 212 * q^41 + 196 * q^49 - 320 * q^51 - 4 * q^59 - 400 * q^69 - 28 * q^71 - 48 * q^81

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
$3$	4.52040			1.50680			0.753399	−	0.657563i	$-0.228413\pi$
$3$	4.52040			1.50680			0.753399	+	0.657563i	$0.228413\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	3.21699i		−	0.459570i	−0.973241	−	0.229785i	$-0.926198\pi$
$7$		−	3.21699i		−	0.459570i	0.973241	−	0.229785i	$-0.0738023\pi$
$8$		0			0
$9$	11.4340			1.27044
$10$		0			0
$11$			8.05991i			0.732719i	0.930473	+	0.366359i	$0.119396\pi$
$11$			8.05991i			0.732719i	−0.930473	+	0.366359i	$0.880604\pi$
$12$		0			0
$13$	−15.5230			−1.19407			−0.597037	−	0.802214i	$-0.703655\pi$
$13$	−15.5230			−1.19407			−0.597037	+	0.802214i	$0.703655\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−25.5446			−1.50263			−0.751313	−	0.659946i	$-0.770579\pi$
$17$	−25.5446			−1.50263			−0.751313	+	0.659946i	$0.770579\pi$
$18$		0			0
$19$	15.6510			0.823735			0.411868	−	0.911244i	$-0.364877\pi$
$19$	15.6510			0.823735			0.411868	+	0.911244i	$0.364877\pi$
$20$		0			0
$21$		−	14.5421i		−	0.692480i
$22$		0			0
$23$	−23.5829			−1.02534			−0.512671	−	0.858585i	$-0.671344\pi$
$23$	−23.5829			−1.02534			−0.512671	+	0.858585i	$0.671344\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	11.0026			0.407502
$28$		0			0
$29$			38.7218i			1.33523i	0.744505	+	0.667617i	$0.232686\pi$
$29$			38.7218i			1.33523i	−0.744505	+	0.667617i	$0.767314\pi$
$30$		0			0
$31$	4.56602	−	30.6619i	0.147291	−	0.989093i
$31$	4.56602	−	30.6619i	0.147291	−	0.989093i
$32$		0			0
$33$			36.4340i			1.10406i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2.94265			−0.0795311			−0.0397655	−	0.999209i	$-0.512661\pi$
$37$	−2.94265			−0.0795311			−0.0397655	+	0.999209i	$0.512661\pi$
$38$		0			0
$39$	−70.1699			−1.79923
$40$		0			0
$41$	6.51893			0.158998			0.0794992	−	0.996835i	$-0.474668\pi$
$41$	6.51893			0.158998			0.0794992	+	0.996835i	$0.474668\pi$
$42$		0			0
$43$	4.52040			0.105125			0.0525627	−	0.998618i	$-0.483261\pi$
$43$	4.52040			0.105125			0.0525627	+	0.998618i	$0.483261\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			72.1699i			1.53553i	0.640732	+	0.767765i	$0.278631\pi$
$47$			72.1699i			1.53553i	−0.640732	+	0.767765i	$0.721369\pi$
$48$		0			0
$49$	38.6510			0.788795
$50$		0			0
$51$	−115.472			−2.26415
$52$		0			0
$53$	−71.3455			−1.34614			−0.673070	−	0.739578i	$-0.735025\pi$
$53$	−71.3455			−1.34614			−0.673070	+	0.739578i	$0.735025\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	70.7486			1.24120
$58$		0			0
$59$	−42.9529			−0.728016			−0.364008	−	0.931396i	$-0.618592\pi$
$59$	−42.9529			−0.728016			−0.364008	+	0.931396i	$0.618592\pi$
$60$		0			0
$61$			83.9258i			1.37583i	0.725790	+	0.687916i	$0.241474\pi$
$61$			83.9258i			1.37583i	−0.725790	+	0.687916i	$0.758526\pi$
$62$		0			0
$63$		−	36.7830i		−	0.583857i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			82.8680i			1.23684i	0.785850	+	0.618418i	$0.212226\pi$
$67$			82.8680i			1.23684i	−0.785850	+	0.618418i	$0.787774\pi$
$68$		0			0
$69$	−106.604			−1.54498
$70$		0			0
$71$	−74.2549			−1.04584			−0.522922	−	0.852381i	$-0.675158\pi$
$71$	−74.2549			−1.04584			−0.522922	+	0.852381i	$0.675158\pi$
$72$		0			0
$73$	−5.11726			−0.0700994			−0.0350497	−	0.999386i	$-0.511159\pi$
$73$	−5.11726			−0.0700994			−0.0350497	+	0.999386i	$0.511159\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	25.9287			0.336736
$78$		0			0
$79$			8.23109i			0.104191i	0.998642	+	0.0520955i	$0.0165900\pi$
$79$			8.23109i			0.104191i	−0.998642	+	0.0520955i	$0.983410\pi$
$80$		0			0
$81$	−53.1699			−0.656419
$82$		0			0
$83$	−103.585			−1.24801			−0.624007	−	0.781419i	$-0.714496\pi$
$83$	−103.585			−1.24801			−0.624007	+	0.781419i	$0.714496\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			175.038i			2.01193i
$88$		0			0
$89$		−	114.759i		−	1.28943i	−0.764425	−	0.644713i	$-0.776977\pi$
$89$		−	114.759i		−	1.28943i	0.764425	−	0.644713i	$-0.223023\pi$
$90$		0			0
$91$			49.9372i			0.548760i
$92$		0			0
$93$	20.6402	−	138.604i	0.221938	−	1.49036i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	21.8209i		−	0.224957i	−0.993654	−	0.112479i	$-0.964121\pi$
$97$		−	21.8209i		−	0.224957i	0.993654	−	0.112479i	$-0.0358790\pi$
$98$		0			0
$99$			92.1568i			0.930877i
$100$		0			0
$101$	−121.387			−1.20185			−0.600925	−	0.799305i	$-0.705201\pi$
$101$	−121.387			−1.20185			−0.600925	+	0.799305i	$0.705201\pi$
$102$		0			0
$103$			145.387i			1.41152i	0.708449	+	0.705762i	$0.249395\pi$
$103$			145.387i			1.41152i	−0.708449	+	0.705762i	$0.750605\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	119.651i		−	1.11823i	−0.829089	−	0.559117i	$-0.811140\pi$
$107$		−	119.651i		−	1.11823i	0.829089	−	0.559117i	$-0.188860\pi$
$108$		0			0
$109$	59.9908			0.550374			0.275187	−	0.961391i	$-0.411260\pi$
$109$	59.9908			0.550374			0.275187	+	0.961391i	$0.411260\pi$
$110$		0			0
$111$	−13.3019			−0.119837
$112$		0			0
$113$		−	66.8587i		−	0.591670i	−0.955239	−	0.295835i	$-0.904402\pi$
$113$		−	66.8587i		−	0.591670i	0.955239	−	0.295835i	$-0.0955979\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−177.489			−1.51700
$118$		0			0
$119$			82.1768i			0.690562i
$120$		0			0
$121$	56.0379			0.463123
$122$		0			0
$123$	29.4682			0.239579
$124$		0			0
$125$		0			0
$126$		0			0
$127$	31.4716			0.247808			0.123904	−	0.992294i	$-0.460459\pi$
$127$	31.4716			0.247808			0.123904	+	0.992294i	$0.460459\pi$
$128$		0			0
$129$	20.4340			0.158403
$130$		0			0
$131$	−93.1320			−0.710932			−0.355466	−	0.934689i	$-0.615678\pi$
$131$	−93.1320			−0.710932			−0.355466	+	0.934689i	$0.615678\pi$
$132$		0			0
$133$		−	50.3490i		−	0.378564i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	45.5880			0.332759			0.166379	−	0.986062i	$-0.446792\pi$
$137$	45.5880			0.332759			0.166379	+	0.986062i	$0.446792\pi$
$138$		0			0
$139$		−	224.613i		−	1.61592i	−0.589236	−	0.807961i	$-0.700571\pi$
$139$		−	224.613i		−	1.61592i	0.589236	−	0.807961i	$-0.299429\pi$
$140$		0			0
$141$			326.237i			2.31373i
$142$		0			0
$143$		−	125.114i		−	0.874920i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	174.718			1.18856
$148$		0			0
$149$	247.548			1.66139			0.830697	−	0.556725i	$-0.187942\pi$
$149$	247.548			1.66139			0.830697	+	0.556725i	$0.187942\pi$
$150$		0			0
$151$			7.88873i			0.0522433i	0.999659	+	0.0261216i	$0.00831572\pi$
$151$			7.88873i			0.0522433i	−0.999659	+	0.0261216i	$0.991684\pi$
$152$		0			0
$153$	−292.077			−1.90900
$154$		0			0
$155$		0			0
$156$		0			0
$157$			204.689i			1.30375i	0.758326	+	0.651875i	$0.226017\pi$
$157$			204.689i			1.30375i	−0.758326	+	0.651875i	$0.773983\pi$
$158$		0			0
$159$	−322.510			−2.02836
$160$		0			0
$161$			75.8659i			0.471216i
$162$		0			0
$163$			148.179i			0.909074i	0.890728	+	0.454537i	$0.150195\pi$
$163$			148.179i			0.909074i	−0.890728	+	0.454537i	$0.849805\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	69.5965			0.416746			0.208373	−	0.978049i	$-0.433183\pi$
$167$	69.5965			0.416746			0.208373	+	0.978049i	$0.433183\pi$
$168$		0			0
$169$	71.9621			0.425811
$170$		0			0
$171$	178.953			1.04651
$172$		0			0
$173$			114.340i			0.660924i	0.943819	+	0.330462i	$0.107205\pi$
$173$			114.340i			0.660924i	−0.943819	+	0.330462i	$0.892795\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−194.164			−1.09697
$178$		0			0
$179$		−	80.7703i		−	0.451231i	−0.974216	−	0.225615i	$-0.927561\pi$
$179$		−	80.7703i		−	0.451231i	0.974216	−	0.225615i	$-0.0724392\pi$
$180$		0			0
$181$		−	3.32667i		−	0.0183794i	−0.999958	−	0.00918970i	$-0.997075\pi$
$181$		−	3.32667i		−	0.0183794i	0.999958	−	0.00918970i	$-0.00292521\pi$
$182$		0			0
$183$			379.378i			2.07310i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	205.887i		−	1.10100i
$188$		0			0
$189$		−	35.3951i		−	0.187276i
$190$		0			0
$191$	−95.5568			−0.500297			−0.250149	−	0.968207i	$-0.580480\pi$
$191$	−95.5568			−0.500297			−0.250149	+	0.968207i	$0.580480\pi$
$192$		0			0
$193$		−	77.5568i		−	0.401849i	−0.979607	−	0.200924i	$-0.935605\pi$
$193$		−	77.5568i		−	0.401849i	0.979607	−	0.200924i	$-0.0643945\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−305.638			−1.55146			−0.775731	−	0.631064i	$-0.782619\pi$
$197$	−305.638			−1.55146			−0.775731	+	0.631064i	$0.782619\pi$
$198$		0			0
$199$			181.158i			0.910343i	0.890404	+	0.455171i	$0.150422\pi$
$199$			181.158i			0.910343i	−0.890404	+	0.455171i	$0.849578\pi$
$200$		0			0
$201$			374.596i			1.86366i
$202$		0			0
$203$	124.568			0.613634
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−269.646			−1.30264
$208$		0			0
$209$			126.145i			0.603567i
$210$		0			0
$211$	68.0092			0.322319			0.161159	−	0.986928i	$-0.448477\pi$
$211$	68.0092			0.322319			0.161159	+	0.986928i	$0.448477\pi$
$212$		0			0
$213$	−335.661			−1.57587
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−98.6390	−	14.6888i	−0.454558	−	0.0676905i
$218$		0			0
$219$	−23.1320			−0.105626
$220$		0			0
$221$	396.528			1.79425
$222$		0			0
$223$	263.293			1.18069			0.590344	−	0.807152i	$-0.298992\pi$
$223$	263.293			1.18069			0.590344	+	0.807152i	$0.298992\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			164.416i			0.724298i	0.932120	+	0.362149i	$0.117957\pi$
$227$			164.416i			0.724298i	−0.932120	+	0.362149i	$0.882043\pi$
$228$		0			0
$229$		−	298.559i		−	1.30375i	−0.758326	−	0.651876i	$-0.773982\pi$
$229$		−	298.559i		−	1.30375i	0.758326	−	0.651876i	$-0.226018\pi$
$230$		0			0
$231$	117.208			0.507393
$232$		0			0
$233$		−	65.3112i		−	0.280305i	−0.990130	−	0.140153i	$-0.955241\pi$
$233$		−	65.3112i		−	0.280305i	0.990130	−	0.140153i	$-0.0447593\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			37.2078i			0.156995i
$238$		0			0
$239$			192.202i			0.804194i	0.915597	+	0.402097i	$0.131719\pi$
$239$			192.202i			0.804194i	−0.915597	+	0.402097i	$0.868281\pi$
$240$		0			0
$241$		−	38.5506i		−	0.159961i	−0.996796	−	0.0799805i	$-0.974514\pi$
$241$		−	38.5506i		−	0.159961i	0.996796	−	0.0799805i	$-0.0254858\pi$
$242$		0			0
$243$	−339.372			−1.39659
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−242.949			−0.983601
$248$		0			0
$249$	−468.246			−1.88050
$250$		0			0
$251$		−	139.281i		−	0.554904i	−0.960739	−	0.277452i	$-0.910510\pi$
$251$		−	139.281i		−	0.554904i	0.960739	−	0.277452i	$-0.0894900\pi$
$252$		0			0
$253$		−	190.076i		−	0.751287i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			273.217i			1.06310i	0.847027	+	0.531551i	$0.178390\pi$
$257$			273.217i			1.06310i	−0.847027	+	0.531551i	$0.821610\pi$
$258$		0			0
$259$			9.46648i			0.0365501i
$260$		0			0
$261$			442.744i			1.69634i
$262$		0			0
$263$	−388.287			−1.47638			−0.738188	−	0.674595i	$-0.764318\pi$
$263$	−388.287			−1.47638			−0.738188	+	0.674595i	$0.764318\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	518.755i		−	1.94290i
$268$		0			0
$269$		−	192.374i		−	0.715144i	−0.933886	−	0.357572i	$-0.883605\pi$
$269$		−	192.374i		−	0.715144i	0.933886	−	0.357572i	$-0.116395\pi$
$270$		0			0
$271$			276.299i			1.01955i	0.860306	+	0.509777i	$0.170272\pi$
$271$			276.299i			1.01955i	−0.860306	+	0.509777i	$0.829728\pi$
$272$		0			0
$273$			225.736i			0.826872i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−220.731			−0.796864			−0.398432	−	0.917198i	$-0.630446\pi$
$277$	−220.731			−0.796864			−0.398432	+	0.917198i	$0.630446\pi$
$278$		0			0
$279$	52.2078	−	350.587i	0.187125	−	1.25659i
$280$		0			0
$281$	287.293			1.02239			0.511197	−	0.859464i	$-0.329202\pi$
$281$	287.293			1.02239			0.511197	+	0.859464i	$0.329202\pi$
$282$		0			0
$283$		−	24.2641i		−	0.0857388i	−0.999081	−	0.0428694i	$-0.986350\pi$
$283$		−	24.2641i		−	0.0857388i	0.999081	−	0.0428694i	$-0.0136499\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	20.9713i		−	0.0730709i
$288$		0			0
$289$	363.528			1.25788
$290$		0			0
$291$		−	98.6390i		−	0.338966i
$292$		0			0
$293$		−	187.359i		−	0.639451i	−0.947510	−	0.319726i	$-0.896409\pi$
$293$		−	187.359i		−	0.639451i	0.947510	−	0.319726i	$-0.103591\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			88.6796i			0.298585i
$298$		0			0
$299$	366.076			1.22433
$300$		0			0
$301$		−	14.5421i		−	0.0483125i
$302$		0			0
$303$	−548.717			−1.81095
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	491.123i		−	1.59975i	−0.600168	−	0.799874i	$-0.704899\pi$
$307$		−	491.123i		−	1.59975i	0.600168	−	0.799874i	$-0.295101\pi$
$308$		0			0
$309$			657.206i			2.12688i
$310$		0			0
$311$	4.19756			0.0134970			0.00674848	−	0.999977i	$-0.497852\pi$
$311$	4.19756			0.0134970			0.00674848	+	0.999977i	$0.497852\pi$
$312$		0			0
$313$	−309.816			−0.989828			−0.494914	−	0.868942i	$-0.664800\pi$
$313$	−309.816			−0.989828			−0.494914	+	0.868942i	$0.664800\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	196.859i		−	0.621006i	−0.950572	−	0.310503i	$-0.899503\pi$
$317$		−	196.859i		−	0.621006i	0.950572	−	0.310503i	$-0.100497\pi$
$318$		0			0
$319$	−312.094			−0.978352
$320$		0			0
$321$		−	540.870i		−	1.68495i
$322$		0			0
$323$	−399.798			−1.23777
$324$		0			0
$325$		0			0
$326$		0			0
$327$	271.182			0.829303
$328$		0			0
$329$	232.170			0.705684
$330$		0			0
$331$			495.152i			1.49593i	0.663739	+	0.747964i	$0.268969\pi$
$331$			495.152i			1.49593i	−0.663739	+	0.747964i	$0.731031\pi$
$332$		0			0
$333$	−33.6462			−0.101040
$334$		0			0
$335$		0			0
$336$		0			0
$337$	271.608			0.805958			0.402979	−	0.915209i	$-0.367975\pi$
$337$	271.608			0.805958			0.402979	+	0.915209i	$0.367975\pi$
$338$		0			0
$339$		−	302.228i		−	0.891528i
$340$		0			0
$341$	247.132	+	36.8017i	0.724727	+	0.107923i
$342$		0			0
$343$		−	281.972i		−	0.822077i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−300.562			−0.866174			−0.433087	−	0.901352i	$-0.642576\pi$
$347$	−300.562			−0.866174			−0.433087	+	0.901352i	$0.642576\pi$
$348$		0			0
$349$	617.472			1.76926			0.884630	−	0.466293i	$-0.154411\pi$
$349$	617.472			1.76926			0.884630	+	0.466293i	$0.154411\pi$
$350$		0			0
$351$	−170.792			−0.486588
$352$		0			0
$353$	449.652			1.27380			0.636901	−	0.770946i	$-0.280216\pi$
$353$	449.652			1.27380			0.636901	+	0.770946i	$0.280216\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			371.472i			1.04054i
$358$		0			0
$359$	294.312			0.819811			0.409906	−	0.912128i	$-0.365562\pi$
$359$	294.312			0.819811			0.409906	+	0.912128i	$0.365562\pi$
$360$		0			0
$361$	−116.047			−0.321460
$362$		0			0
$363$	253.313			0.697833
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−59.7044			−0.162682			−0.0813411	−	0.996686i	$-0.525920\pi$
$367$	−59.7044			−0.162682			−0.0813411	+	0.996686i	$0.525920\pi$
$368$		0			0
$369$	74.5374			0.201998
$370$		0			0
$371$			229.518i			0.618646i
$372$		0			0
$373$			149.727i			0.401412i	0.979652	+	0.200706i	$0.0643236\pi$
$373$			149.727i			0.401412i	−0.979652	+	0.200706i	$0.935676\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	601.077i		−	1.59437i
$378$		0			0
$379$	−221.284			−0.583862			−0.291931	−	0.956439i	$-0.594298\pi$
$379$	−221.284			−0.583862			−0.291931	+	0.956439i	$0.594298\pi$
$380$		0			0
$381$	142.264			0.373397
$382$		0			0
$383$	−186.275			−0.486359			−0.243179	−	0.969981i	$-0.578190\pi$
$383$	−186.275			−0.486359			−0.243179	+	0.969981i	$0.578190\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	51.6861			0.133556
$388$		0			0
$389$		−	140.174i		−	0.360344i	−0.983635	−	0.180172i	$-0.942335\pi$
$389$		−	140.174i		−	0.360344i	0.983635	−	0.180172i	$-0.0576655\pi$
$390$		0			0
$391$	602.416			1.54070
$392$		0			0
$393$	−420.994			−1.07123
$394$		0			0
$395$		0			0
$396$		0			0
$397$			106.085i			0.267217i	0.991034	+	0.133608i	$0.0426564\pi$
$397$			106.085i			0.267217i	−0.991034	+	0.133608i	$0.957344\pi$
$398$		0			0
$399$		−	227.598i		−	0.570420i
$400$		0			0
$401$			97.9543i			0.244275i	0.992513	+	0.122138i	$0.0389749\pi$
$401$			97.9543i			0.244275i	−0.992513	+	0.122138i	$0.961025\pi$
$402$		0			0
$403$	−70.8781	+	475.963i	−0.175876	+	1.18105i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	23.7175i		−	0.0582739i
$408$		0			0
$409$			11.7289i			0.0286771i	0.999897	+	0.0143386i	$0.00456426\pi$
$409$			11.7289i			0.0286771i	−0.999897	+	0.0143386i	$0.995436\pi$
$410$		0			0
$411$	206.076			0.501401
$412$		0			0
$413$			138.179i			0.334574i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	1015.34i		−	2.43487i
$418$		0			0
$419$	−5.74514			−0.0137116			−0.00685578	−	0.999976i	$-0.502182\pi$
$419$	−5.74514			−0.0137116			−0.00685578	+	0.999976i	$0.502182\pi$
$420$		0			0
$421$	35.8209			0.0850852			0.0425426	−	0.999095i	$-0.486454\pi$
$421$	35.8209			0.0850852			0.0425426	+	0.999095i	$0.486454\pi$
$422$		0			0
$423$			825.189i			1.95080i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	269.988			0.632291
$428$		0			0
$429$		−	565.563i		−	1.31833i
$430$		0			0
$431$	216.170			0.501554			0.250777	−	0.968045i	$-0.419314\pi$
$431$	216.170			0.501554			0.250777	+	0.968045i	$0.419314\pi$
$432$		0			0
$433$	−274.847			−0.634750			−0.317375	−	0.948300i	$-0.602801\pi$
$433$	−274.847			−0.634750			−0.317375	+	0.948300i	$0.602801\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−369.095			−0.844610
$438$		0			0
$439$	−549.463			−1.25162			−0.625812	−	0.779974i	$-0.715232\pi$
$439$	−549.463			−1.25162			−0.625812	+	0.779974i	$0.715232\pi$
$440$		0			0
$441$	441.934			1.00212
$442$		0			0
$443$		−	151.669i		−	0.342369i	−0.985239	−	0.171184i	$-0.945241\pi$
$443$		−	151.669i		−	0.342369i	0.985239	−	0.171184i	$-0.0547594\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	1119.01			2.50339
$448$		0			0
$449$		−	345.854i		−	0.770277i	−0.922859	−	0.385138i	$-0.874154\pi$
$449$		−	345.854i		−	0.770277i	0.922859	−	0.385138i	$-0.125846\pi$
$450$		0			0
$451$			52.5420i			0.116501i
$452$		0			0
$453$			35.6602i			0.0787201i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	707.060			1.54718			0.773589	−	0.633688i	$-0.218460\pi$
$457$	707.060			1.54718			0.773589	+	0.633688i	$0.218460\pi$
$458$		0			0
$459$	−281.056			−0.612323
$460$		0			0
$461$		−	860.967i		−	1.86761i	−0.357787	−	0.933803i	$-0.616469\pi$
$461$		−	860.967i		−	1.86761i	0.357787	−	0.933803i	$-0.383531\pi$
$462$		0			0
$463$	336.471			0.726719			0.363360	−	0.931649i	$-0.381630\pi$
$463$	336.471			0.726719			0.363360	+	0.931649i	$0.381630\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	429.727i		−	0.920186i	−0.887871	−	0.460093i	$-0.847816\pi$
$467$		−	429.727i		−	0.920186i	0.887871	−	0.460093i	$-0.152184\pi$
$468$		0			0
$469$	266.585			0.568412
$470$		0			0
$471$			925.275i			1.96449i
$472$		0			0
$473$			36.4340i			0.0770274i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−815.763			−1.71019
$478$		0			0
$479$	−232.652			−0.485703			−0.242852	−	0.970063i	$-0.578083\pi$
$479$	−232.652			−0.485703			−0.242852	+	0.970063i	$0.578083\pi$
$480$		0			0
$481$	45.6786			0.0949660
$482$		0			0
$483$			342.944i			0.710028i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−293.955			−0.603604			−0.301802	−	0.953371i	$-0.597588\pi$
$487$	−293.955			−0.603604			−0.301802	+	0.953371i	$0.597588\pi$
$488$		0			0
$489$			669.828i			1.36979i
$490$		0			0
$491$		−	554.348i		−	1.12902i	−0.825427	−	0.564509i	$-0.809066\pi$
$491$		−	554.348i		−	1.12902i	0.825427	−	0.564509i	$-0.190934\pi$
$492$		0			0
$493$		−	989.134i		−	2.00636i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			238.877i			0.480638i
$498$		0			0
$499$		−	49.4237i		−	0.0990454i	−0.998773	−	0.0495227i	$-0.984230\pi$
$499$		−	49.4237i		−	0.0990454i	0.998773	−	0.0495227i	$-0.0157700\pi$
$500$		0			0
$501$	314.604			0.627952
$502$		0			0
$503$		−	302.048i		−	0.600493i	−0.953862	−	0.300247i	$-0.902931\pi$
$503$		−	302.048i		−	0.600493i	0.953862	−	0.300247i	$-0.0970690\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	325.297			0.641612
$508$		0			0
$509$			904.593i			1.77720i	0.458687	+	0.888598i	$0.348320\pi$
$509$			904.593i			1.77720i	−0.458687	+	0.888598i	$0.651680\pi$
$510$		0			0
$511$			16.4622i			0.0322156i
$512$		0			0
$513$	172.201			0.335674
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−581.683			−1.12511
$518$		0			0
$519$			516.861i			0.995879i
$520$		0			0
$521$	238.340			0.457466			0.228733	−	0.973489i	$-0.426542\pi$
$521$	238.340			0.457466			0.228733	+	0.973489i	$0.426542\pi$
$522$		0			0
$523$	−328.027			−0.627203			−0.313601	−	0.949555i	$-0.601536\pi$
$523$	−328.027			−0.627203			−0.313601	+	0.949555i	$0.601536\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−116.637	+	783.247i	−0.221323	+	1.48624i
$528$		0			0
$529$	27.1515			0.0513260
$530$		0			0
$531$	−491.123			−0.924902
$532$		0			0
$533$	−101.193			−0.189856
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	365.114i		−	0.679914i
$538$		0			0
$539$			311.523i			0.577965i
$540$		0			0
$541$	−842.067			−1.55650			−0.778250	−	0.627955i	$-0.783892\pi$
$541$	−842.067			−1.55650			−0.778250	+	0.627955i	$0.783892\pi$
$542$		0			0
$543$		−	15.0379i		−	0.0276940i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			700.803i			1.28118i	0.767885	+	0.640588i	$0.221309\pi$
$547$			700.803i			1.28118i	−0.767885	+	0.640588i	$0.778691\pi$
$548$		0			0
$549$			959.606i			1.74792i
$550$		0			0
$551$			606.034i			1.09988i
$552$		0			0
$553$	26.4793			0.0478831
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−37.2274			−0.0668355			−0.0334178	−	0.999441i	$-0.510639\pi$
$557$	−37.2274			−0.0668355			−0.0334178	+	0.999441i	$0.510639\pi$
$558$		0			0
$559$	−70.1699			−0.125528
$560$		0			0
$561$		−	930.693i		−	1.65899i
$562$		0			0
$563$			240.896i			0.427879i	0.976847	+	0.213939i	$0.0686295\pi$
$563$			240.896i			0.427879i	−0.976847	+	0.213939i	$0.931371\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$			171.047i			0.301670i
$568$		0			0
$569$		−	129.301i		−	0.227242i	−0.993524	−	0.113621i	$-0.963755\pi$
$569$		−	129.301i		−	0.227242i	0.993524	−	0.113621i	$-0.0362450\pi$
$570$		0			0
$571$			1092.40i			1.91314i	0.291499	+	0.956571i	$0.405846\pi$
$571$			1092.40i			1.91314i	−0.291499	+	0.956571i	$0.594154\pi$
$572$		0			0
$573$	−431.955			−0.753847
$574$		0			0
$575$		0			0
$576$		0			0
$577$			1052.62i			1.82431i	0.409850	+	0.912153i	$0.365581\pi$
$577$			1052.62i			1.82431i	−0.409850	+	0.912153i	$0.634419\pi$
$578$		0			0
$579$		−	350.587i		−	0.605505i
$580$		0			0
$581$			333.232i			0.573549i
$582$		0			0
$583$		−	575.038i		−	0.986343i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−625.906			−1.06628			−0.533140	−	0.846027i	$-0.678988\pi$
$587$	−625.906			−1.06628			−0.533140	+	0.846027i	$0.678988\pi$
$588$		0			0
$589$	71.4626	−	479.888i	0.121329	−	0.814751i
$590$		0			0
$591$	−1381.60			−2.33774
$592$		0			0
$593$		−	500.971i		−	0.844808i	−0.906408	−	0.422404i	$-0.861186\pi$
$593$		−	500.971i		−	0.844808i	0.906408	−	0.422404i	$-0.138814\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			818.907i			1.37170i
$598$		0			0
$599$	−305.991			−0.510836			−0.255418	−	0.966831i	$-0.582213\pi$
$599$	−305.991			−0.510836			−0.255418	+	0.966831i	$0.582213\pi$
$600$		0			0
$601$			592.727i			0.986235i	0.869963	+	0.493117i	$0.164143\pi$
$601$			592.727i			0.986235i	−0.869963	+	0.493117i	$0.835857\pi$
$602$		0			0
$603$			947.511i			1.57133i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			853.095i			1.40543i	0.711472	+	0.702714i	$0.248029\pi$
$607$			853.095i			1.40543i	−0.711472	+	0.702714i	$0.751971\pi$
$608$		0			0
$609$	563.095			0.924623
$610$		0			0
$611$		−	1120.29i		−	1.83354i
$612$		0			0
$613$	−129.856			−0.211837			−0.105919	−	0.994375i	$-0.533778\pi$
$613$	−129.856			−0.211837			−0.105919	+	0.994375i	$0.533778\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	368.907i		−	0.597904i	−0.954268	−	0.298952i	$-0.903363\pi$
$617$		−	368.907i		−	0.597904i	0.954268	−	0.298952i	$-0.0966371\pi$
$618$		0			0
$619$		−	454.339i		−	0.733989i	−0.930223	−	0.366995i	$-0.880387\pi$
$619$		−	454.339i		−	0.733989i	0.930223	−	0.366995i	$-0.119613\pi$
$620$		0			0
$621$	−259.472			−0.417829
$622$		0			0
$623$	−369.178			−0.592581
$624$		0			0
$625$		0			0
$626$		0			0
$627$			570.227i			0.909453i
$628$		0			0
$629$	75.1689			0.119505
$630$		0			0
$631$		−	962.590i		−	1.52550i	−0.646694	−	0.762749i	$-0.723849\pi$
$631$		−	962.590i		−	1.52550i	0.646694	−	0.762749i	$-0.276151\pi$
$632$		0			0
$633$	307.429			0.485669
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−599.977			−0.941880
$638$		0			0
$639$	−849.029			−1.32868
$640$		0			0
$641$			53.6434i			0.0836870i	0.999124	+	0.0418435i	$0.0133231\pi$
$641$			53.6434i			0.0836870i	−0.999124	+	0.0418435i	$0.986677\pi$
$642$		0			0
$643$	−285.086			−0.443368			−0.221684	−	0.975119i	$-0.571155\pi$
$643$	−285.086			−0.443368			−0.221684	+	0.975119i	$0.571155\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	997.134			1.54116			0.770582	−	0.637340i	$-0.219965\pi$
$647$	997.134			1.54116			0.770582	+	0.637340i	$0.219965\pi$
$648$		0			0
$649$		−	346.197i		−	0.533431i
$650$		0			0
$651$	−445.887	−	66.3994i	−0.684927	−	0.101996i
$652$		0			0
$653$		−	205.660i		−	0.314947i	−0.987523	−	0.157473i	$-0.949665\pi$
$653$		−	205.660i		−	0.314947i	0.987523	−	0.157473i	$-0.0503348\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−58.5106			−0.0890573
$658$		0			0
$659$	−1070.54			−1.62449			−0.812245	−	0.583316i	$-0.801755\pi$
$659$	−1070.54			−1.62449			−0.812245	+	0.583316i	$0.801755\pi$
$660$		0			0
$661$	−773.162			−1.16968			−0.584842	−	0.811147i	$-0.698844\pi$
$661$	−773.162			−1.16968			−0.584842	+	0.811147i	$0.698844\pi$
$662$		0			0
$663$	1792.46			2.70357
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	913.171i		−	1.36907i
$668$		0			0
$669$	1190.19			1.77906
$670$		0			0
$671$	−676.434			−1.00810
$672$		0			0
$673$	1126.90			1.67445			0.837223	−	0.546862i	$-0.184178\pi$
$673$	1126.90			1.67445			0.837223	+	0.546862i	$0.184178\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	116.207			0.171650			0.0858250	−	0.996310i	$-0.472647\pi$
$677$	116.207			0.171650			0.0858250	+	0.996310i	$0.472647\pi$
$678$		0			0
$679$	−70.1976			−0.103384
$680$		0			0
$681$			743.223i			1.09137i
$682$		0			0
$683$		−	1016.16i		−	1.48779i	−0.668296	−	0.743895i	$-0.732976\pi$
$683$		−	1016.16i		−	1.48779i	0.668296	−	0.743895i	$-0.267024\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	1349.60i		−	1.96449i
$688$		0			0
$689$	1107.49			1.60739
$690$		0			0
$691$	1008.84			1.45997			0.729986	−	0.683462i	$-0.239527\pi$
$691$	1008.84			1.45997			0.729986	+	0.683462i	$0.239527\pi$
$692$		0			0
$693$	296.468			0.427803
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−166.524			−0.238915
$698$		0			0
$699$		−	295.232i		−	0.422364i
$700$		0			0
$701$	−594.632			−0.848262			−0.424131	−	0.905601i	$-0.639420\pi$
$701$	−594.632			−0.848262			−0.424131	+	0.905601i	$0.639420\pi$
$702$		0			0
$703$	−46.0553			−0.0655126
$704$		0			0
$705$		0			0
$706$		0			0
$707$			390.500i			0.552335i
$708$		0			0
$709$			249.172i			0.351442i	0.984440	+	0.175721i	$0.0562257\pi$
$709$			249.172i			0.351442i	−0.984440	+	0.175721i	$0.943774\pi$
$710$		0			0
$711$			94.1141i			0.132369i
$712$		0			0
$713$	−107.680	+	723.095i	−0.151024	+	1.01416i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			868.831i			1.21176i
$718$		0			0
$719$			667.700i			0.928651i	0.885665	+	0.464325i	$0.153703\pi$
$719$			667.700i			0.928651i	−0.885665	+	0.464325i	$0.846297\pi$
$720$		0			0
$721$	467.708			0.648694
$722$		0			0
$723$		−	174.264i		−	0.241029i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			393.217i			0.540876i	0.962737	+	0.270438i	$0.0871685\pi$
$727$			393.217i			0.540876i	−0.962737	+	0.270438i	$0.912831\pi$
$728$		0			0
$729$	−1055.57			−1.44797
$730$		0			0
$731$	−115.472			−0.157964
$732$		0			0
$733$			584.445i			0.797333i	0.917096	+	0.398667i	$0.130527\pi$
$733$			584.445i			0.797333i	−0.917096	+	0.398667i	$0.869473\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−667.908			−0.906253
$738$		0			0
$739$		−	301.506i		−	0.407992i	−0.978972	−	0.203996i	$-0.934607\pi$
$739$		−	301.506i		−	0.407992i	0.978972	−	0.203996i	$-0.0653930\pi$
$740$		0			0
$741$	−1098.23			−1.48209
$742$		0			0
$743$	988.819			1.33085			0.665423	−	0.746466i	$-0.268251\pi$
$743$	988.819			1.33085			0.665423	+	0.746466i	$0.268251\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1184.39			−1.58553
$748$		0			0
$749$	−384.916			−0.513907
$750$		0			0
$751$	−315.708			−0.420384			−0.210192	−	0.977660i	$-0.567409\pi$
$751$	−315.708			−0.420384			−0.210192	+	0.977660i	$0.567409\pi$
$752$		0			0
$753$		−	629.605i		−	0.836129i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	425.088			0.561543			0.280772	−	0.959775i	$-0.409410\pi$
$757$	425.088			0.561543			0.280772	+	0.959775i	$0.409410\pi$
$758$		0			0
$759$		−	859.218i		−	1.13204i
$760$		0			0
$761$		−	148.234i		−	0.194788i	−0.995246	−	0.0973941i	$-0.968949\pi$
$761$		−	148.234i		−	0.194788i	0.995246	−	0.0973941i	$-0.0310507\pi$
$762$		0			0
$763$		−	192.990i		−	0.252935i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	666.756			0.869304
$768$		0			0
$769$	311.933			0.405635			0.202818	−	0.979217i	$-0.434990\pi$
$769$	311.933			0.405635			0.202818	+	0.979217i	$0.434990\pi$
$770$		0			0
$771$			1235.05i			1.60188i
$772$		0			0
$773$	−864.247			−1.11804			−0.559021	−	0.829153i	$-0.688823\pi$
$773$	−864.247			−1.11804			−0.559021	+	0.829153i	$0.688823\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$			42.7922i			0.0550737i
$778$		0			0
$779$	102.028			0.130973
$780$		0			0
$781$		−	598.487i		−	0.766309i
$782$		0			0
$783$			426.039i			0.544111i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	1348.53			1.71350			0.856751	−	0.515730i	$-0.172479\pi$
$787$	1348.53			1.71350			0.856751	+	0.515730i	$0.172479\pi$
$788$		0			0
$789$	−1755.21			−2.22460
$790$		0			0
$791$	−215.084			−0.271914
$792$		0			0
$793$		−	1302.78i		−	1.64284i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−349.056			−0.437962			−0.218981	−	0.975729i	$-0.570273\pi$
$797$	−349.056			−0.437962			−0.218981	+	0.975729i	$0.570273\pi$
$798$		0			0
$799$		−	1843.55i		−	2.30733i
$800$		0			0
$801$		−	1312.15i		−	1.63814i
$802$		0			0
$803$		−	41.2446i		−	0.0513632i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	869.605i		−	1.07758i
$808$		0			0
$809$			285.974i			0.353491i	0.984257	+	0.176745i	$0.0565569\pi$
$809$			285.974i			0.353491i	−0.984257	+	0.176745i	$0.943443\pi$
$810$		0			0
$811$	165.775			0.204408			0.102204	−	0.994763i	$-0.467411\pi$
$811$	165.775			0.204408			0.102204	+	0.994763i	$0.467411\pi$
$812$		0			0
$813$			1248.98i			1.53626i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	70.7486			0.0865956
$818$		0			0
$819$			570.981i			0.697168i
$820$		0			0
$821$			424.496i			0.517048i	0.966005	+	0.258524i	$0.0832360\pi$
$821$			424.496i			0.517048i	−0.966005	+	0.258524i	$0.916764\pi$
$822$		0			0
$823$	1338.93			1.62689			0.813445	−	0.581642i	$-0.197590\pi$
$823$	1338.93			1.62689			0.813445	+	0.581642i	$0.197590\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	699.125			0.845375			0.422688	−	0.906275i	$-0.361087\pi$
$827$	699.125			0.845375			0.422688	+	0.906275i	$0.361087\pi$
$828$		0			0
$829$			1419.19i			1.71193i	0.517032	+	0.855966i	$0.327037\pi$
$829$			1419.19i			1.71193i	−0.517032	+	0.855966i	$0.672963\pi$
$830$		0			0
$831$	−997.793			−1.20071
$832$		0			0
$833$	−987.325			−1.18526
$834$		0			0
$835$		0			0
$836$		0			0
$837$	50.2379	−	337.359i	0.0600214	−	0.403058i
$838$		0			0
$839$	411.114			0.490004			0.245002	−	0.969523i	$-0.421211\pi$
$839$	411.114			0.490004			0.245002	+	0.969523i	$0.421211\pi$
$840$		0			0
$841$	−658.378			−0.782851
$842$		0			0
$843$	1298.68			1.54054
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	180.273i		−	0.212837i
$848$		0			0
$849$		−	109.683i		−	0.129191i
$850$		0			0
$851$	69.3961			0.0815465
$852$		0			0
$853$		−	683.171i		−	0.800904i	−0.916318	−	0.400452i	$-0.868853\pi$
$853$		−	683.171i		−	0.800904i	0.916318	−	0.400452i	$-0.131147\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			540.227i			0.630370i	0.949030	+	0.315185i	$0.102067\pi$
$857$			540.227i			0.630370i	−0.949030	+	0.315185i	$0.897933\pi$
$858$		0			0
$859$		−	1137.82i		−	1.32458i	−0.749246	−	0.662291i	$-0.769584\pi$
$859$		−	1137.82i		−	1.32458i	0.749246	−	0.662291i	$-0.230416\pi$
$860$		0			0
$861$		−	94.7988i		−	0.110103i
$862$		0			0
$863$	−436.646			−0.505963			−0.252982	−	0.967471i	$-0.581411\pi$
$863$	−436.646			−0.505963			−0.252982	+	0.967471i	$0.581411\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	1643.29			1.89538
$868$		0			0
$869$	−66.3418			−0.0763427
$870$		0			0
$871$		−	1286.36i		−	1.47687i
$872$		0			0
$873$		−	249.500i		−	0.285796i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	73.7636i		−	0.0841090i	−0.999115	−	0.0420545i	$-0.986610\pi$
$877$		−	73.7636i		−	0.0841090i	0.999115	−	0.0420545i	$-0.0133903\pi$
$878$		0			0
$879$		−	846.938i		−	0.963524i
$880$		0			0
$881$		−	109.341i		−	0.124110i	−0.998073	−	0.0620550i	$-0.980235\pi$
$881$		−	109.341i		−	0.124110i	0.998073	−	0.0620550i	$-0.0197654\pi$
$882$		0			0
$883$	1598.39			1.81018			0.905089	−	0.425221i	$-0.139804\pi$
$883$	1598.39			1.81018			0.905089	+	0.425221i	$0.139804\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			1514.29i			1.70721i	0.520922	+	0.853604i	$0.325588\pi$
$887$			1514.29i			1.70721i	−0.520922	+	0.853604i	$0.674412\pi$
$888$		0			0
$889$		−	101.244i		−	0.113885i
$890$		0			0
$891$		−	428.545i		−	0.480970i
$892$		0			0
$893$			1129.53i			1.26487i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	1654.81			1.84482
$898$		0			0
$899$	1187.28	+	176.804i	1.32067	+	0.196668i
$900$		0			0
$901$	1822.49			2.02275
$902$		0			0
$903$		−	65.7359i		−	0.0727973i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	1501.96i		−	1.65596i	−0.560757	−	0.827980i	$-0.689490\pi$
$907$		−	1501.96i		−	1.65596i	0.560757	−	0.827980i	$-0.310510\pi$
$908$		0			0
$909$	−1387.94			−1.52688
$910$		0			0
$911$		−	426.662i		−	0.468344i	−0.972195	−	0.234172i	$-0.924762\pi$
$911$		−	426.662i		−	0.468344i	0.972195	−	0.234172i	$-0.0752379\pi$
$912$		0			0
$913$		−	834.886i		−	0.914443i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			299.605i			0.326723i
$918$		0			0
$919$	1692.49			1.84167			0.920834	−	0.389954i	$-0.127509\pi$
$919$	1692.49			1.84167			0.920834	+	0.389954i	$0.127509\pi$
$920$		0			0
$921$		−	2220.07i		−	2.41050i
$922$		0			0
$923$	1152.65			1.24881
$924$		0			0
$925$		0			0
$926$		0			0
$927$			1662.35i			1.79326i
$928$		0			0
$929$			1196.29i			1.28772i	0.765144	+	0.643859i	$0.222668\pi$
$929$			1196.29i			1.28772i	−0.765144	+	0.643859i	$0.777332\pi$
$930$		0			0
$931$	604.925			0.649759
$932$		0			0
$933$	18.9746			0.0203372
$934$		0			0
$935$		0			0
$936$		0			0
$937$			1226.11i			1.30855i	0.756255	+	0.654277i	$0.227027\pi$
$937$			1226.11i			1.30855i	−0.756255	+	0.654277i	$0.772973\pi$
$938$		0			0
$939$	−1400.49			−1.49147
$940$		0			0
$941$			387.181i			0.411457i	0.978609	+	0.205728i	$0.0659563\pi$
$941$			387.181i			0.411457i	−0.978609	+	0.205728i	$0.934044\pi$
$942$		0			0
$943$	−153.735			−0.163028
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−385.636			−0.407218			−0.203609	−	0.979052i	$-0.565267\pi$
$947$	−385.636			−0.407218			−0.203609	+	0.979052i	$0.565267\pi$
$948$		0			0
$949$	79.4350			0.0837039
$950$		0			0
$951$		−	889.880i		−	0.935730i
$952$		0			0
$953$	708.546			0.743490			0.371745	−	0.928335i	$-0.378760\pi$
$953$	708.546			0.743490			0.371745	+	0.928335i	$0.378760\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−1410.79			−1.47418
$958$		0			0
$959$		−	146.656i		−	0.152926i
$960$		0			0
$961$	−919.303	−	280.006i	−0.956611	−	0.291369i
$962$		0			0
$963$		−	1368.09i		−	1.42065i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−699.130			−0.722988			−0.361494	−	0.932374i	$-0.617733\pi$
$967$	−699.130			−0.722988			−0.361494	+	0.932374i	$0.617733\pi$
$968$		0			0
$969$	−1807.25			−1.86506
$970$		0			0
$971$	872.452			0.898509			0.449255	−	0.893404i	$-0.351690\pi$
$971$	872.452			0.898509			0.449255	+	0.893404i	$0.351690\pi$
$972$		0			0
$973$	−722.579			−0.742630
$974$		0			0
$975$		0			0
$976$		0			0
$977$			827.633i			0.847116i	0.905869	+	0.423558i	$0.139219\pi$
$977$			827.633i			0.847116i	−0.905869	+	0.423558i	$0.860781\pi$
$978$		0			0
$979$	924.946			0.944786
$980$		0			0
$981$	685.933			0.699219
$982$		0			0
$983$	−265.857			−0.270454			−0.135227	−	0.990815i	$-0.543176\pi$
$983$	−265.857			−0.270454			−0.135227	+	0.990815i	$0.543176\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	1049.50			1.06332
$988$		0			0
$989$	−106.604			−0.107790
$990$		0			0
$991$			1209.46i			1.22045i	0.792229	+	0.610223i	$0.208920\pi$
$991$			1209.46i			1.22045i	−0.792229	+	0.610223i	$0.791080\pi$
$992$		0			0
$993$			2238.28i			2.25406i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	870.333i		−	0.872951i	−0.899716	−	0.436476i	$-0.856226\pi$
$997$		−	870.333i		−	0.872951i	0.899716	−	0.436476i	$-0.143774\pi$
$998$		0			0
$999$	−32.3767			−0.0324091

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3100.3.f.b.1549.7		8
5.2	odd	4		3100.3.d.b.1301.1		4
5.3	odd	4		124.3.c.b.61.4	yes	4
5.4	even	2	inner	3100.3.f.b.1549.2		8
15.8	even	4		1116.3.h.d.433.1		4
20.3	even	4		496.3.e.e.433.1		4
31.30	odd	2	inner	3100.3.f.b.1549.1		8
155.92	even	4		3100.3.d.b.1301.4		4
155.123	even	4		124.3.c.b.61.1	✓	4
155.154	odd	2	inner	3100.3.f.b.1549.8		8
465.278	odd	4		1116.3.h.d.433.2		4
620.123	odd	4		496.3.e.e.433.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.3.c.b.61.1	✓	4	155.123	even	4
124.3.c.b.61.4	yes	4	5.3	odd	4
496.3.e.e.433.1		4	20.3	even	4
496.3.e.e.433.4		4	620.123	odd	4
1116.3.h.d.433.1		4	15.8	even	4
1116.3.h.d.433.2		4	465.278	odd	4
3100.3.d.b.1301.1		4	5.2	odd	4
3100.3.d.b.1301.4		4	155.92	even	4
3100.3.f.b.1549.1		8	31.30	odd	2	inner
3100.3.f.b.1549.2		8	5.4	even	2	inner
3100.3.f.b.1549.7		8	1.1	even	1	trivial
3100.3.f.b.1549.8		8	155.154	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3100 = 2^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3100.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.52040 q^{3} -3.21699i q^{7} +11.4340 q^{9} +O(q^{10})\)	\(q+4.52040 q^{3} -3.21699i q^{7} +11.4340 q^{9} +8.05991i q^{11} -15.5230 q^{13} -25.5446 q^{17} +15.6510 q^{19} -14.5421i q^{21} -23.5829 q^{23} +11.0026 q^{27} +38.7218i q^{29} +(4.56602 - 30.6619i) q^{31} +36.4340i q^{33} -2.94265 q^{37} -70.1699 q^{39} +6.51893 q^{41} +4.52040 q^{43} +72.1699i q^{47} +38.6510 q^{49} -115.472 q^{51} -71.3455 q^{53} +70.7486 q^{57} -42.9529 q^{59} +83.9258i q^{61} -36.7830i q^{63} +82.8680i q^{67} -106.604 q^{69} -74.2549 q^{71} -5.11726 q^{73} +25.9287 q^{77} +8.23109i q^{79} -53.1699 q^{81} -103.585 q^{83} +175.038i q^{87} -114.759i q^{89} +49.9372i q^{91} +(20.6402 - 138.604i) q^{93} -21.8209i q^{97} +92.1568i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{9}+O(q^{10}) \) 8 * q + 16 * q^9	\( 8 q + 16 q^{9} + 12 q^{19} + 112 q^{31} - 184 q^{39} - 212 q^{41} + 196 q^{49} - 320 q^{51} - 4 q^{59} - 400 q^{69} - 28 q^{71} - 48 q^{81}+O(q^{100}) \) 8 * q + 16 * q^9 + 12 * q^19 + 112 * q^31 - 184 * q^39 - 212 * q^41 + 196 * q^49 - 320 * q^51 - 4 * q^59 - 400 * q^69 - 28 * q^71 - 48 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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