LMFDB - Embedded newform 300.5.g.f.101.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,5,Mod(101,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("300.101");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 300 = 2^{2} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	300.g (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:	$31.0109889252$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.9254440.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 67x^{2} + 40 $ x^4 + 67*x^2 + 40
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{2}\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			101.4
Root			$-8.14847i$ of defining polynomial
Character	$\chi$	$=$	300.101
Dual form			300.5.g.f.101.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(6.73293 + 5.97224i) q^{3} -15.3976 q^{7} +(9.66464 + 80.4214i) q^{9} +O(q^{10})$	$q+(6.73293 + 5.97224i) q^{3} -15.3976 q^{7} +(9.66464 + 80.4214i) q^{9} -50.5601i q^{11} +14.6024 q^{13} +467.801i q^{17} -194.976 q^{19} +(-103.671 - 91.9580i) q^{21} +145.300i q^{23} +(-415.225 + 599.191i) q^{27} +891.423i q^{29} -950.939 q^{31} +(301.957 - 340.418i) q^{33} -837.156 q^{37} +(98.3171 + 87.2093i) q^{39} +1998.33i q^{41} +2096.53 q^{43} +493.802i q^{47} -2163.91 q^{49} +(-2793.82 + 3149.67i) q^{51} -4887.12i q^{53} +(-1312.76 - 1164.44i) q^{57} -2224.65i q^{59} -5076.60 q^{61} +(-148.812 - 1238.29i) q^{63} -3403.17 q^{67} +(-867.769 + 978.297i) q^{69} +9566.09i q^{71} +7982.30 q^{73} +778.503i q^{77} -1522.22 q^{79} +(-6374.19 + 1554.49i) q^{81} +1142.30i q^{83} +(-5323.79 + 6001.89i) q^{87} +10931.2i q^{89} -224.842 q^{91} +(-6402.61 - 5679.24i) q^{93} +30.9915 q^{97} +(4066.12 - 488.646i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 5 q^{3} + 70 q^{7} - 71 q^{9}+O(q^{10}) $ 4 * q + 5 * q^3 + 70 * q^7 - 71 * q^9	$ 4 q + 5 q^{3} + 70 q^{7} - 71 q^{9} + 190 q^{13} + 536 q^{19} - 634 q^{21} + 620 q^{27} - 514 q^{31} - 1095 q^{33} - 980 q^{37} - 484 q^{39} + 4570 q^{43} - 4050 q^{49} - 1635 q^{51} - 6545 q^{57} + 1406 q^{61} - 4850 q^{63} - 980 q^{67} + 9030 q^{69} + 4690 q^{73} - 17932 q^{79} - 17711 q^{81} - 10110 q^{87} + 7654 q^{91} - 18680 q^{93} + 21310 q^{97} + 22515 q^{99}+O(q^{100}) $ 4 * q + 5 * q^3 + 70 * q^7 - 71 * q^9 + 190 * q^13 + 536 * q^19 - 634 * q^21 + 620 * q^27 - 514 * q^31 - 1095 * q^33 - 980 * q^37 - 484 * q^39 + 4570 * q^43 - 4050 * q^49 - 1635 * q^51 - 6545 * q^57 + 1406 * q^61 - 4850 * q^63 - 980 * q^67 + 9030 * q^69 + 4690 * q^73 - 17932 * q^79 - 17711 * q^81 - 10110 * q^87 + 7654 * q^91 - 18680 * q^93 + 21310 * q^97 + 22515 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$277$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	6.73293	+	5.97224i	0.748103	+	0.663582i
$3$	6.73293	+	5.97224i	0.748103	+	0.663582i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−15.3976			−0.314236			−0.157118	−	0.987580i	$-0.550220\pi$
$7$	−15.3976			−0.314236			−0.157118	+	0.987580i	$0.550220\pi$
$8$		0			0
$9$	9.66464	+	80.4214i	0.119317	+	0.992856i
$10$		0			0
$11$		−	50.5601i		−	0.417852i	−0.977931	−	0.208926i	$-0.933003\pi$
$11$		−	50.5601i		−	0.417852i	0.977931	−	0.208926i	$-0.0669969\pi$
$12$		0			0
$13$	14.6024			0.0864049			0.0432025	−	0.999066i	$-0.486244\pi$
$13$	14.6024			0.0864049			0.0432025	+	0.999066i	$0.486244\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			467.801i			1.61869i	0.587333	+	0.809345i	$0.300178\pi$
$17$			467.801i			1.61869i	−0.587333	+	0.809345i	$0.699822\pi$
$18$		0			0
$19$	−194.976			−0.540099			−0.270049	−	0.962846i	$-0.587040\pi$
$19$	−194.976			−0.540099			−0.270049	+	0.962846i	$0.587040\pi$
$20$		0			0
$21$	−103.671	−	91.9580i	−0.235081	−	0.208522i
$22$		0			0
$23$			145.300i			0.274670i	0.990525	+	0.137335i	$0.0438537\pi$
$23$			145.300i			0.274670i	−0.990525	+	0.137335i	$0.956146\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−415.225	+	599.191i	−0.569581	+	0.821935i
$28$		0			0
$29$			891.423i			1.05996i	0.848011	+	0.529978i	$0.177800\pi$
$29$			891.423i			1.05996i	−0.848011	+	0.529978i	$0.822200\pi$
$30$		0			0
$31$	−950.939			−0.989531			−0.494765	−	0.869027i	$-0.664746\pi$
$31$	−950.939			−0.989531			−0.494765	+	0.869027i	$0.664746\pi$
$32$		0			0
$33$	301.957	−	340.418i	0.277280	−	0.312597i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−837.156			−0.611509			−0.305755	−	0.952110i	$-0.598909\pi$
$37$	−837.156			−0.611509			−0.305755	+	0.952110i	$0.598909\pi$
$38$		0			0
$39$	98.3171	+	87.2093i	0.0646398	+	0.0573368i
$40$		0			0
$41$			1998.33i			1.18877i	0.804180	+	0.594386i	$0.202605\pi$
$41$			1998.33i			1.18877i	−0.804180	+	0.594386i	$0.797395\pi$
$42$		0			0
$43$	2096.53			1.13387			0.566936	−	0.823762i	$-0.308129\pi$
$43$	2096.53			1.13387			0.566936	+	0.823762i	$0.308129\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			493.802i			0.223541i	0.993734	+	0.111771i	$0.0356522\pi$
$47$			493.802i			0.223541i	−0.993734	+	0.111771i	$0.964348\pi$
$48$		0			0
$49$	−2163.91			−0.901256
$50$		0			0
$51$	−2793.82	+	3149.67i	−1.07413	+	1.21095i
$52$		0			0
$53$		−	4887.12i		−	1.73981i	−0.493223	−	0.869903i	$-0.664181\pi$
$53$		−	4887.12i		−	1.73981i	0.493223	−	0.869903i	$-0.335819\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1312.76	−	1164.44i	−0.404050	−	0.358400i
$58$		0			0
$59$		−	2224.65i		−	0.639083i	−0.947572	−	0.319541i	$-0.896471\pi$
$59$		−	2224.65i		−	0.639083i	0.947572	−	0.319541i	$-0.103529\pi$
$60$		0			0
$61$	−5076.60			−1.36431			−0.682155	−	0.731207i	$-0.738957\pi$
$61$	−5076.60			−1.36431			−0.682155	+	0.731207i	$0.738957\pi$
$62$		0			0
$63$	−148.812	−	1238.29i	−0.0374936	−	0.311991i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−3403.17			−0.758112			−0.379056	−	0.925374i	$-0.623751\pi$
$67$	−3403.17			−0.758112			−0.379056	+	0.925374i	$0.623751\pi$
$68$		0			0
$69$	−867.769	+	978.297i	−0.182266	+	0.205481i
$70$		0			0
$71$			9566.09i			1.89766i	0.315791	+	0.948829i	$0.397730\pi$
$71$			9566.09i			1.89766i	−0.315791	+	0.948829i	$0.602270\pi$
$72$		0			0
$73$	7982.30			1.49790			0.748949	−	0.662628i	$-0.230559\pi$
$73$	7982.30			1.49790			0.748949	+	0.662628i	$0.230559\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			778.503i			0.131304i
$78$		0			0
$79$	−1522.22			−0.243906			−0.121953	−	0.992536i	$-0.538916\pi$
$79$	−1522.22			−0.243906			−0.121953	+	0.992536i	$0.538916\pi$
$80$		0			0
$81$	−6374.19	+	1554.49i	−0.971527	+	0.236928i
$82$		0			0
$83$			1142.30i			0.165815i	0.996557	+	0.0829077i	$0.0264207\pi$
$83$			1142.30i			0.165815i	−0.996557	+	0.0829077i	$0.973579\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−5323.79	+	6001.89i	−0.703368	+	0.792956i
$88$		0			0
$89$			10931.2i			1.38003i	0.723795	+	0.690015i	$0.242396\pi$
$89$			10931.2i			1.38003i	−0.723795	+	0.690015i	$0.757604\pi$
$90$		0			0
$91$	−224.842			−0.0271515
$92$		0			0
$93$	−6402.61	−	5679.24i	−0.740271	−	0.656635i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	30.9915			0.00329381			0.00164691	−	0.999999i	$-0.499476\pi$
$97$	30.9915			0.00329381			0.00164691	+	0.999999i	$0.499476\pi$
$98$		0			0
$99$	4066.12	−	488.646i	0.414867	−	0.0498567i
$100$		0			0
$101$		−	5883.64i		−	0.576771i	−0.957514	−	0.288385i	$-0.906882\pi$
$101$		−	5883.64i		−	0.576771i	0.957514	−	0.288385i	$-0.0931184\pi$
$102$		0			0
$103$	10887.1			1.02621			0.513105	−	0.858326i	$-0.328495\pi$
$103$	10887.1			1.02621			0.513105	+	0.858326i	$0.328495\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			18168.7i			1.58692i	0.608622	+	0.793460i	$0.291723\pi$
$107$			18168.7i			1.58692i	−0.608622	+	0.793460i	$0.708277\pi$
$108$		0			0
$109$	−15676.1			−1.31943			−0.659714	−	0.751517i	$-0.729322\pi$
$109$	−15676.1			−1.31943			−0.659714	+	0.751517i	$0.729322\pi$
$110$		0			0
$111$	−5636.51	−	4999.70i	−0.457472	−	0.405787i
$112$		0			0
$113$			1310.11i			0.102600i	0.998683	+	0.0513002i	$0.0163365\pi$
$113$			1310.11i			0.102600i	−0.998683	+	0.0513002i	$0.983663\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	141.127	+	1174.35i	0.0103095	+	0.0857877i
$118$		0			0
$119$		−	7203.00i		−	0.508651i
$120$		0			0
$121$	12084.7			0.825399
$122$		0			0
$123$	−11934.5	+	13454.6i	−0.788849	+	0.889325i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	7838.62			0.485996			0.242998	−	0.970027i	$-0.421869\pi$
$127$	7838.62			0.485996			0.242998	+	0.970027i	$0.421869\pi$
$128$		0			0
$129$	14115.8	+	12521.0i	0.848253	+	0.752418i
$130$		0			0
$131$		−	12476.9i		−	0.727050i	−0.931584	−	0.363525i	$-0.881573\pi$
$131$		−	12476.9i		−	0.727050i	0.931584	−	0.363525i	$-0.118427\pi$
$132$		0			0
$133$	3002.15			0.169719
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	19266.2i		−	1.02649i	−0.858243	−	0.513244i	$-0.828443\pi$
$137$		−	19266.2i		−	1.02649i	0.858243	−	0.513244i	$-0.171557\pi$
$138$		0			0
$139$	6673.02			0.345376			0.172688	−	0.984977i	$-0.444755\pi$
$139$	6673.02			0.345376			0.172688	+	0.984977i	$0.444755\pi$
$140$		0			0
$141$	−2949.11	+	3324.73i	−0.148338	+	0.167232i
$142$		0			0
$143$		−	738.301i		−	0.0361045i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−14569.5	−	12923.4i	−0.674232	−	0.598057i
$148$		0			0
$149$		−	3613.85i		−	0.162779i	−0.996682	−	0.0813893i	$-0.974064\pi$
$149$		−	3613.85i		−	0.162779i	0.996682	−	0.0813893i	$-0.0259357\pi$
$150$		0			0
$151$	39087.7			1.71430			0.857149	−	0.515068i	$-0.172233\pi$
$151$	39087.7			1.71430			0.857149	+	0.515068i	$0.172233\pi$
$152$		0			0
$153$	−37621.2	+	4521.13i	−1.60713	+	0.193137i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	40667.8			1.64988			0.824939	−	0.565222i	$-0.191210\pi$
$157$	40667.8			1.64988			0.824939	+	0.565222i	$0.191210\pi$
$158$		0			0
$159$	29187.0	−	32904.6i	1.15450	−	1.30155i
$160$		0			0
$161$		−	2237.27i		−	0.0863112i
$162$		0			0
$163$	23364.7			0.879396			0.439698	−	0.898146i	$-0.355086\pi$
$163$	23364.7			0.879396			0.439698	+	0.898146i	$0.355086\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	41753.6i		−	1.49714i	−0.663058	−	0.748568i	$-0.730742\pi$
$167$		−	41753.6i		−	1.49714i	0.663058	−	0.748568i	$-0.269258\pi$
$168$		0			0
$169$	−28347.8			−0.992534
$170$		0			0
$171$	−1884.37	−	15680.2i	−0.0644427	−	0.536241i
$172$		0			0
$173$			30169.8i			1.00805i	0.863690	+	0.504023i	$0.168147\pi$
$173$			30169.8i			1.00805i	−0.863690	+	0.504023i	$0.831853\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	13286.1	−	14978.4i	0.424084	−	0.478100i
$178$		0			0
$179$		−	32585.4i		−	1.01699i	−0.861065	−	0.508495i	$-0.830202\pi$
$179$		−	32585.4i		−	1.01699i	0.861065	−	0.508495i	$-0.169798\pi$
$180$		0			0
$181$	43251.0			1.32020			0.660099	−	0.751179i	$-0.270514\pi$
$181$	43251.0			1.32020			0.660099	+	0.751179i	$0.270514\pi$
$182$		0			0
$183$	−34180.4	−	30318.7i	−1.02064	−	0.905332i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	23652.1			0.676374
$188$		0			0
$189$	6393.45	−	9226.08i	0.178983	−	0.258282i
$190$		0			0
$191$		−	34445.3i		−	0.944197i	−0.881546	−	0.472099i	$-0.843497\pi$
$191$		−	34445.3i		−	0.944197i	0.881546	−	0.472099i	$-0.156503\pi$
$192$		0			0
$193$	−3562.54			−0.0956412			−0.0478206	−	0.998856i	$-0.515228\pi$
$193$	−3562.54			−0.0956412			−0.0478206	+	0.998856i	$0.515228\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	27839.1i		−	0.717336i	−0.933465	−	0.358668i	$-0.883231\pi$
$197$		−	27839.1i		−	0.717336i	0.933465	−	0.358668i	$-0.116769\pi$
$198$		0			0
$199$	22102.2			0.558121			0.279061	−	0.960273i	$-0.409977\pi$
$199$	22102.2			0.558121			0.279061	+	0.960273i	$0.409977\pi$
$200$		0			0
$201$	−22913.3	−	20324.5i	−0.567146	−	0.503070i
$202$		0			0
$203$		−	13725.7i		−	0.333076i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−11685.3	+	1404.28i	−0.272708	+	0.0327727i
$208$		0			0
$209$			9858.00i			0.225682i
$210$		0			0
$211$	71143.7			1.59798			0.798990	−	0.601345i	$-0.205368\pi$
$211$	71143.7			1.59798			0.798990	+	0.601345i	$0.205368\pi$
$212$		0			0
$213$	−57131.0	+	64407.8i	−1.25925	+	1.41964i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	14642.2			0.310946
$218$		0			0
$219$	53744.2	+	47672.2i	1.12058	+	0.993979i
$220$		0			0
$221$			6831.04i			0.139863i
$222$		0			0
$223$	76917.3			1.54673			0.773365	−	0.633961i	$-0.218572\pi$
$223$	76917.3			1.54673			0.773365	+	0.633961i	$0.218572\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			75766.1i			1.47036i	0.677872	+	0.735180i	$0.262902\pi$
$227$			75766.1i			1.47036i	−0.677872	+	0.735180i	$0.737098\pi$
$228$		0			0
$229$	−57716.4			−1.10060			−0.550298	−	0.834968i	$-0.685486\pi$
$229$	−57716.4			−1.10060			−0.550298	+	0.834968i	$0.685486\pi$
$230$		0			0
$231$	−4649.41	+	5241.61i	−0.0871312	+	0.0982292i
$232$		0			0
$233$			65387.1i			1.20443i	0.798335	+	0.602213i	$0.205714\pi$
$233$			65387.1i			1.20443i	−0.798335	+	0.602213i	$0.794286\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−10249.0	−	9091.06i	−0.182467	−	0.161852i
$238$		0			0
$239$		−	2699.52i		−	0.0472597i	−0.999721	−	0.0236298i	$-0.992478\pi$
$239$		−	2699.52i		−	0.0472597i	0.999721	−	0.0236298i	$-0.00752231\pi$
$240$		0			0
$241$	−3271.59			−0.0563281			−0.0281640	−	0.999603i	$-0.508966\pi$
$241$	−3271.59			−0.0563281			−0.0281640	+	0.999603i	$0.508966\pi$
$242$		0			0
$243$	−52200.7	−	27602.0i	−0.884024	−	0.467442i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−2847.12			−0.0466672
$248$		0			0
$249$	−6822.10	+	7691.04i	−0.110032	+	0.124047i
$250$		0			0
$251$		−	91788.9i		−	1.45694i	−0.685076	−	0.728472i	$-0.740231\pi$
$251$		−	91788.9i		−	1.45694i	0.685076	−	0.728472i	$-0.259769\pi$
$252$		0			0
$253$	7346.41			0.114771
$254$		0			0
$255$		0			0
$256$		0			0
$257$			43758.3i			0.662513i	0.943541	+	0.331257i	$0.107473\pi$
$257$			43758.3i			0.662513i	−0.943541	+	0.331257i	$0.892527\pi$
$258$		0			0
$259$	12890.2			0.192158
$260$		0			0
$261$	−71689.4	+	8615.28i	−1.05238	+	0.126470i
$262$		0			0
$263$		−	85928.0i		−	1.24229i	−0.783696	−	0.621145i	$-0.786668\pi$
$263$		−	85928.0i		−	1.24229i	0.783696	−	0.621145i	$-0.213332\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−65283.9	+	73599.1i	−0.915764	+	1.03240i
$268$		0			0
$269$			78308.6i			1.08219i	0.840961	+	0.541096i	$0.181991\pi$
$269$			78308.6i			1.08219i	−0.840961	+	0.541096i	$0.818009\pi$
$270$		0			0
$271$	−36099.9			−0.491549			−0.245775	−	0.969327i	$-0.579042\pi$
$271$	−36099.9			−0.491549			−0.245775	+	0.969327i	$0.579042\pi$
$272$		0			0
$273$	−1513.84	−	1342.81i	−0.0203122	−	0.0180173i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	18413.4			0.239979			0.119990	−	0.992775i	$-0.461714\pi$
$277$	18413.4			0.239979			0.119990	+	0.992775i	$0.461714\pi$
$278$		0			0
$279$	−9190.49	−	76475.8i	−0.118067	−	0.982462i
$280$		0			0
$281$			131788.i			1.66903i	0.550989	+	0.834513i	$0.314251\pi$
$281$			131788.i			1.66903i	−0.550989	+	0.834513i	$0.685749\pi$
$282$		0			0
$283$	108394.			1.35342			0.676708	−	0.736252i	$-0.263406\pi$
$283$	108394.			1.35342			0.676708	+	0.736252i	$0.263406\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	30769.4i		−	0.373555i
$288$		0			0
$289$	−135317.			−1.62016
$290$		0			0
$291$	208.664	+	185.089i	0.00246411	+	0.00218572i
$292$		0			0
$293$			6856.42i			0.0798661i	0.999202	+	0.0399330i	$0.0127145\pi$
$293$			6856.42i			0.0798661i	−0.999202	+	0.0399330i	$0.987286\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	30295.2	+	20993.8i	0.343448	+	0.238001i
$298$		0			0
$299$			2121.74i			0.0237328i
$300$		0			0
$301$	−32281.5			−0.356304
$302$		0			0
$303$	35138.5	−	39614.1i	0.382735	−	0.431484i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−88860.5			−0.942827			−0.471413	−	0.881912i	$-0.656256\pi$
$307$	−88860.5			−0.942827			−0.471413	+	0.881912i	$0.656256\pi$
$308$		0			0
$309$	73301.8	+	65020.2i	0.767711	+	0.680975i
$310$		0			0
$311$			146598.i			1.51568i	0.652442	+	0.757839i	$0.273745\pi$
$311$			146598.i			1.51568i	−0.652442	+	0.757839i	$0.726255\pi$
$312$		0			0
$313$	31348.8			0.319987			0.159993	−	0.987118i	$-0.448853\pi$
$313$	31348.8			0.319987			0.159993	+	0.987118i	$0.448853\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	61530.0i		−	0.612306i	−0.951982	−	0.306153i	$-0.900958\pi$
$317$		−	61530.0i		−	0.612306i	0.951982	−	0.306153i	$-0.0990419\pi$
$318$		0			0
$319$	45070.5			0.442905
$320$		0			0
$321$	−108508.	+	122328.i	−1.05305	+	1.18718i
$322$		0			0
$323$		−	91209.9i		−	0.874253i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−105546.	−	93621.5i	−0.987067	−	0.875549i
$328$		0			0
$329$		−	7603.35i		−	0.0702447i
$330$		0			0
$331$	−127261.			−1.16156			−0.580779	−	0.814062i	$-0.697252\pi$
$331$	−127261.			−1.16156			−0.580779	+	0.814062i	$0.697252\pi$
$332$		0			0
$333$	−8090.81	−	67325.2i	−0.0729632	−	0.607141i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−23025.9			−0.202748			−0.101374	−	0.994848i	$-0.532324\pi$
$337$	−23025.9			−0.202748			−0.101374	+	0.994848i	$0.532324\pi$
$338$		0			0
$339$	−7824.27	+	8820.85i	−0.0680839	+	0.0767557i
$340$		0			0
$341$			48079.6i			0.413478i
$342$		0			0
$343$	70288.6			0.597443
$344$		0			0
$345$		0			0
$346$		0			0
$347$			118367.i			0.983042i	0.870866	+	0.491521i	$0.163559\pi$
$347$			118367.i			0.983042i	−0.870866	+	0.491521i	$0.836441\pi$
$348$		0			0
$349$	−80175.7			−0.658251			−0.329126	−	0.944286i	$-0.606754\pi$
$349$	−80175.7			−0.658251			−0.329126	+	0.944286i	$0.606754\pi$
$350$		0			0
$351$	−6063.29	+	8749.64i	−0.0492146	+	0.0710192i
$352$		0			0
$353$		−	73798.3i		−	0.592239i	−0.955151	−	0.296120i	$-0.904307\pi$
$353$		−	73798.3i		−	0.592239i	0.955151	−	0.296120i	$-0.0956927\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	43018.1	−	48497.3i	0.337532	−	0.380523i
$358$		0			0
$359$		−	184883.i		−	1.43453i	−0.696802	−	0.717264i	$-0.745394\pi$
$359$		−	184883.i		−	1.43453i	0.696802	−	0.717264i	$-0.254606\pi$
$360$		0			0
$361$	−92305.5			−0.708293
$362$		0			0
$363$	81365.2	+	72172.6i	0.617484	+	0.547721i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−7132.72			−0.0529570			−0.0264785	−	0.999649i	$-0.508429\pi$
$367$	−7132.72			−0.0529570			−0.0264785	+	0.999649i	$0.508429\pi$
$368$		0			0
$369$	−160708.	+	19313.1i	−1.18028	+	0.141840i
$370$		0			0
$371$			75249.7i			0.546710i
$372$		0			0
$373$	−90663.0			−0.651647			−0.325823	−	0.945431i	$-0.605641\pi$
$373$	−90663.0			−0.651647			−0.325823	+	0.945431i	$0.605641\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			13016.9i			0.0915854i
$378$		0			0
$379$	72641.4			0.505715			0.252857	−	0.967504i	$-0.418630\pi$
$379$	72641.4			0.505715			0.252857	+	0.967504i	$0.418630\pi$
$380$		0			0
$381$	52776.9	+	46814.2i	0.363575	+	0.322498i
$382$		0			0
$383$			18040.8i			0.122987i	0.998107	+	0.0614935i	$0.0195863\pi$
$383$			18040.8i			0.122987i	−0.998107	+	0.0614935i	$0.980414\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	20262.2	+	168606.i	0.135290	+	1.12577i
$388$		0			0
$389$			151605.i			1.00188i	0.865483	+	0.500939i	$0.167012\pi$
$389$			151605.i			1.00188i	−0.865483	+	0.500939i	$0.832988\pi$
$390$		0			0
$391$	−67971.7			−0.444605
$392$		0			0
$393$	74515.1	−	84006.1i	0.482458	−	0.543908i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−105906.			−0.671951			−0.335975	−	0.941871i	$-0.609066\pi$
$397$	−105906.			−0.671951			−0.335975	+	0.941871i	$0.609066\pi$
$398$		0			0
$399$	20213.3	+	17929.6i	0.126967	+	0.112622i
$400$		0			0
$401$		−	165722.i		−	1.03060i	−0.857010	−	0.515301i	$-0.827680\pi$
$401$		−	165722.i		−	1.03060i	0.857010	−	0.515301i	$-0.172320\pi$
$402$		0			0
$403$	−13886.0			−0.0855003
$404$		0			0
$405$		0			0
$406$		0			0
$407$			42326.7i			0.255521i
$408$		0			0
$409$	25834.5			0.154438			0.0772189	−	0.997014i	$-0.475396\pi$
$409$	25834.5			0.154438			0.0772189	+	0.997014i	$0.475396\pi$
$410$		0			0
$411$	115062.	−	129718.i	0.681160	−	0.767919i
$412$		0			0
$413$			34254.1i			0.200823i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	44928.9	+	39852.9i	0.258377	+	0.229186i
$418$		0			0
$419$			198893.i			1.13290i	0.824097	+	0.566449i	$0.191683\pi$
$419$			198893.i			1.13290i	−0.824097	+	0.566449i	$0.808317\pi$
$420$		0			0
$421$	106217.			0.599283			0.299641	−	0.954052i	$-0.403133\pi$
$421$	106217.			0.599283			0.299641	+	0.954052i	$0.403133\pi$
$422$		0			0
$423$	−39712.2	+	4772.42i	−0.221944	+	0.0266721i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	78167.3			0.428716
$428$		0			0
$429$	4409.31	−	4970.93i	0.0239583	−	0.0270099i
$430$		0			0
$431$			35487.9i			0.191040i	0.995427	+	0.0955202i	$0.0304515\pi$
$431$			35487.9i			0.191040i	−0.995427	+	0.0955202i	$0.969549\pi$
$432$		0			0
$433$	−138725.			−0.739911			−0.369955	−	0.929049i	$-0.620627\pi$
$433$	−138725.			−0.739911			−0.369955	+	0.929049i	$0.620627\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	28330.0i		−	0.148349i
$438$		0			0
$439$	235284.			1.22086			0.610428	−	0.792072i	$-0.290998\pi$
$439$	235284.			1.22086			0.610428	+	0.792072i	$0.290998\pi$
$440$		0			0
$441$	−20913.5	−	174025.i	−0.107535	−	0.894817i
$442$		0			0
$443$			293963.i			1.49791i	0.662622	+	0.748954i	$0.269443\pi$
$443$			293963.i			1.49791i	−0.662622	+	0.748954i	$0.730557\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	21582.8	−	24331.8i	0.108017	−	0.121775i
$448$		0			0
$449$			150257.i			0.745318i	0.927968	+	0.372659i	$0.121554\pi$
$449$			150257.i			0.745318i	−0.927968	+	0.372659i	$0.878446\pi$
$450$		0			0
$451$	101036.			0.496732
$452$		0			0
$453$	263175.	+	233441.i	1.28247	+	1.13758i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−161306.			−0.772356			−0.386178	−	0.922424i	$-0.626205\pi$
$457$	−161306.			−0.772356			−0.386178	+	0.922424i	$0.626205\pi$
$458$		0			0
$459$	−280302.	−	194243.i	−1.33046	−	0.921975i
$460$		0			0
$461$		−	339787.i		−	1.59884i	−0.600772	−	0.799421i	$-0.705140\pi$
$461$		−	339787.i		−	1.59884i	0.600772	−	0.799421i	$-0.294860\pi$
$462$		0			0
$463$	−298343.			−1.39173			−0.695863	−	0.718174i	$-0.744978\pi$
$463$	−298343.			−1.39173			−0.695863	+	0.718174i	$0.744978\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			70135.5i			0.321591i	0.986988	+	0.160796i	$0.0514060\pi$
$467$			70135.5i			0.321591i	−0.986988	+	0.160796i	$0.948594\pi$
$468$		0			0
$469$	52400.5			0.238226
$470$		0			0
$471$	273814.	+	242878.i	1.23428	+	1.09483i
$472$		0			0
$473$		−	106001.i		−	0.473791i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	393028.	−	47232.2i	1.72738	−	0.207588i
$478$		0			0
$479$		−	388819.i		−	1.69464i	−0.531085	−	0.847319i	$-0.678215\pi$
$479$		−	388819.i		−	1.69464i	0.531085	−	0.847319i	$-0.321785\pi$
$480$		0			0
$481$	−12224.5			−0.0528374
$482$		0			0
$483$	13361.5	−	15063.4i	0.0572746	−	0.0645697i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	206792.			0.871920			0.435960	−	0.899966i	$-0.356409\pi$
$487$	206792.			0.871920			0.435960	+	0.899966i	$0.356409\pi$
$488$		0			0
$489$	157313.	+	139539.i	0.657879	+	0.583552i
$490$		0			0
$491$		−	276164.i		−	1.14552i	−0.819722	−	0.572762i	$-0.805872\pi$
$491$		−	276164.i		−	1.14552i	0.819722	−	0.572762i	$-0.194128\pi$
$492$		0			0
$493$	−417009.			−1.71574
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	147295.i		−	0.596312i
$498$		0			0
$499$	187937.			0.754765			0.377383	−	0.926057i	$-0.376824\pi$
$499$	187937.			0.754765			0.377383	+	0.926057i	$0.376824\pi$
$500$		0			0
$501$	249363.	−	281124.i	0.993473	−	1.12001i
$502$		0			0
$503$		−	18554.5i		−	0.0733355i	−0.999328	−	0.0366677i	$-0.988326\pi$
$503$		−	18554.5i		−	0.0733355i	0.999328	−	0.0366677i	$-0.0116743\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−190863.	−	169300.i	−0.742518	−	0.658628i
$508$		0			0
$509$		−	412623.i		−	1.59264i	−0.604874	−	0.796321i	$-0.706777\pi$
$509$		−	412623.i		−	1.59264i	0.604874	−	0.796321i	$-0.293223\pi$
$510$		0			0
$511$	−122908.			−0.470694
$512$		0			0
$513$	80958.7	−	116828.i	0.307630	−	0.443926i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	24966.7			0.0934072
$518$		0			0
$519$	−180181.	+	203131.i	−0.668922	+	0.754122i
$520$		0			0
$521$			347649.i			1.28075i	0.768061	+	0.640377i	$0.221222\pi$
$521$			347649.i			1.28075i	−0.768061	+	0.640377i	$0.778778\pi$
$522$		0			0
$523$	−288154.			−1.05347			−0.526734	−	0.850030i	$-0.676584\pi$
$523$	−288154.			−1.05347			−0.526734	+	0.850030i	$0.676584\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	444851.i		−	1.60174i
$528$		0			0
$529$	258729.			0.924556
$530$		0			0
$531$	178909.	−	21500.4i	0.634517	−	0.0762531i
$532$		0			0
$533$			29180.4i			0.102716i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	194608.	−	219395.i	0.674857	−	0.760813i
$538$		0			0
$539$			109408.i			0.376592i
$540$		0			0
$541$	397537.			1.35826			0.679131	−	0.734018i	$-0.262357\pi$
$541$	397537.			1.35826			0.679131	+	0.734018i	$0.262357\pi$
$542$		0			0
$543$	291206.	+	258305.i	0.987644	+	0.876060i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	381872.			1.27627			0.638136	−	0.769924i	$-0.279706\pi$
$547$	381872.			1.27627			0.638136	+	0.769924i	$0.279706\pi$
$548$		0			0
$549$	−49063.5	−	408267.i	−0.162785	−	1.35456i
$550$		0			0
$551$		−	173806.i		−	0.572481i
$552$		0			0
$553$	23438.5			0.0766441
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	162015.i		−	0.522209i	−0.965311	−	0.261105i	$-0.915913\pi$
$557$		−	162015.i		−	0.522209i	0.965311	−	0.261105i	$-0.0840868\pi$
$558$		0			0
$559$	30614.4			0.0979721
$560$		0			0
$561$	159248.	+	141256.i	0.505997	+	0.448830i
$562$		0			0
$563$			306285.i			0.966292i	0.875540	+	0.483146i	$0.160506\pi$
$563$			306285.i			0.966292i	−0.875540	+	0.483146i	$0.839494\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	98147.0	−	23935.3i	0.305289	−	0.0744514i
$568$		0			0
$569$		−	2603.82i		−	0.00804241i	−0.999992	−	0.00402121i	$-0.998720\pi$
$569$		−	2603.82i		−	0.00804241i	0.999992	−	0.00402121i	$-0.00127999\pi$
$570$		0			0
$571$	−270421.			−0.829409			−0.414704	−	0.909956i	$-0.636115\pi$
$571$	−270421.			−0.829409			−0.414704	+	0.909956i	$0.636115\pi$
$572$		0			0
$573$	205715.	−	231917.i	0.626553	−	0.706357i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−569306.			−1.70999			−0.854996	−	0.518635i	$-0.826440\pi$
$577$	−569306.			−1.70999			−0.854996	+	0.518635i	$0.826440\pi$
$578$		0			0
$579$	−23986.3	−	21276.4i	−0.0715495	−	0.0634658i
$580$		0			0
$581$		−	17588.7i		−	0.0521052i
$582$		0			0
$583$	−247093.			−0.726982
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	232912.i		−	0.675952i	−0.941155	−	0.337976i	$-0.890258\pi$
$587$		−	232912.i		−	0.675952i	0.941155	−	0.337976i	$-0.109742\pi$
$588$		0			0
$589$	185410.			0.534445
$590$		0			0
$591$	166262.	−	187439.i	0.476012	−	0.536641i
$592$		0			0
$593$		−	56505.6i		−	0.160688i	−0.996767	−	0.0803438i	$-0.974398\pi$
$593$		−	56505.6i		−	0.160688i	0.996767	−	0.0803438i	$-0.0256018\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	148812.	+	132000.i	0.417532	+	0.370360i
$598$		0			0
$599$			408097.i			1.13739i	0.822548	+	0.568696i	$0.192552\pi$
$599$			408097.i			1.13739i	−0.822548	+	0.568696i	$0.807448\pi$
$600$		0			0
$601$	−227249.			−0.629147			−0.314574	−	0.949233i	$-0.601862\pi$
$601$	−227249.			−0.629147			−0.314574	+	0.949233i	$0.601862\pi$
$602$		0			0
$603$	−32890.4	−	273687.i	−0.0904554	−	0.752697i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	44956.0			0.122014			0.0610071	−	0.998137i	$-0.480569\pi$
$607$	44956.0			0.122014			0.0610071	+	0.998137i	$0.480569\pi$
$608$		0			0
$609$	81973.5	−	92414.4i	0.221024	−	0.249175i
$610$		0			0
$611$			7210.71i			0.0193150i
$612$		0			0
$613$	−512379.			−1.36355			−0.681773	−	0.731563i	$-0.738791\pi$
$613$	−512379.			−1.36355			−0.681773	+	0.731563i	$0.738791\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			554076.i			1.45546i	0.685866	+	0.727728i	$0.259424\pi$
$617$			554076.i			1.45546i	−0.685866	+	0.727728i	$0.740576\pi$
$618$		0			0
$619$	−19229.5			−0.0501865			−0.0250932	−	0.999685i	$-0.507988\pi$
$619$	−19229.5			−0.0501865			−0.0250932	+	0.999685i	$0.507988\pi$
$620$		0			0
$621$	−87062.6	−	60332.3i	−0.225761	−	0.156447i
$622$		0			0
$623$		−	168314.i		−	0.433655i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−58874.4	+	66373.2i	−0.149758	+	0.168833i
$628$		0			0
$629$		−	391623.i		−	0.989844i
$630$		0			0
$631$	−242825.			−0.609867			−0.304934	−	0.952374i	$-0.598634\pi$
$631$	−242825.			−0.609867			−0.304934	+	0.952374i	$0.598634\pi$
$632$		0			0
$633$	479005.	+	424887.i	1.19545	+	1.06039i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−31598.4			−0.0778729
$638$		0			0
$639$	−769318.	+	92452.8i	−1.88410	+	0.226422i
$640$		0			0
$641$			10523.4i			0.0256119i	0.999918	+	0.0128059i	$0.00407637\pi$
$641$			10523.4i			0.0256119i	−0.999918	+	0.0128059i	$0.995924\pi$
$642$		0			0
$643$	−16119.5			−0.0389879			−0.0194939	−	0.999810i	$-0.506206\pi$
$643$	−16119.5			−0.0389879			−0.0194939	+	0.999810i	$0.506206\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			388292.i			0.927576i	0.885946	+	0.463788i	$0.153510\pi$
$647$			388292.i			0.927576i	−0.885946	+	0.463788i	$0.846490\pi$
$648$		0			0
$649$	−112478.			−0.267042
$650$		0			0
$651$	98584.6	+	87446.5i	0.232620	+	0.206339i
$652$		0			0
$653$			93334.4i			0.218885i	0.993993	+	0.109442i	$0.0349065\pi$
$653$			93334.4i			0.218885i	−0.993993	+	0.109442i	$0.965093\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	77146.0	+	641947.i	0.178724	+	1.48720i
$658$		0			0
$659$			428636.i			0.987003i	0.869745	+	0.493501i	$0.164283\pi$
$659$			428636.i			0.987003i	−0.869745	+	0.493501i	$0.835717\pi$
$660$		0			0
$661$	40886.7			0.0935791			0.0467896	−	0.998905i	$-0.485101\pi$
$661$	40886.7			0.0935791			0.0467896	+	0.998905i	$0.485101\pi$
$662$		0			0
$663$	−40796.6	+	45992.9i	−0.0928105	+	0.104632i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−129524.			−0.291138
$668$		0			0
$669$	517879.	+	459369.i	1.15711	+	1.02638i
$670$		0			0
$671$			256674.i			0.570080i
$672$		0			0
$673$	670267.			1.47985			0.739925	−	0.672689i	$-0.234861\pi$
$673$	670267.			1.47985			0.739925	+	0.672689i	$0.234861\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			404058.i			0.881590i	0.897608	+	0.440795i	$0.145303\pi$
$677$			404058.i			0.881590i	−0.897608	+	0.440795i	$0.854697\pi$
$678$		0			0
$679$	−477.194			−0.00103504
$680$		0			0
$681$	−452494.	+	510128.i	−0.975705	+	1.09998i
$682$		0			0
$683$		−	416295.i		−	0.892400i	−0.894933	−	0.446200i	$-0.852777\pi$
$683$		−	416295.i		−	0.892400i	0.894933	−	0.446200i	$-0.147223\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−388600.	−	344696.i	−0.823359	−	0.730336i
$688$		0			0
$689$		−	71363.8i		−	0.150328i
$690$		0			0
$691$	−562500.			−1.17806			−0.589029	−	0.808112i	$-0.700490\pi$
$691$	−562500.			−1.17806			−0.589029	+	0.808112i	$0.700490\pi$
$692$		0			0
$693$	−62608.3	+	7523.95i	−0.130366	+	0.0156668i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−934820.			−1.92425
$698$		0			0
$699$	−390508.	+	440247.i	−0.799236	+	0.901035i
$700$		0			0
$701$		−	586463.i		−	1.19345i	−0.802445	−	0.596726i	$-0.796468\pi$
$701$		−	586463.i		−	1.19345i	0.802445	−	0.596726i	$-0.203532\pi$
$702$		0			0
$703$	163225.			0.330275
$704$		0			0
$705$		0			0
$706$		0			0
$707$			90593.7i			0.181242i
$708$		0			0
$709$	253496.			0.504288			0.252144	−	0.967690i	$-0.418864\pi$
$709$	253496.			0.504288			0.252144	+	0.967690i	$0.418864\pi$
$710$		0			0
$711$	−14711.7	−	122419.i	−0.0291021	−	0.242164i
$712$		0			0
$713$		−	138172.i		−	0.271794i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	16122.2	−	18175.7i	0.0313607	−	0.0353551i
$718$		0			0
$719$			769020.i			1.48758i	0.668414	+	0.743789i	$0.266973\pi$
$719$			769020.i			1.48758i	−0.668414	+	0.743789i	$0.733027\pi$
$720$		0			0
$721$	−167634.			−0.322472
$722$		0			0
$723$	−22027.4	−	19538.7i	−0.0421392	−	0.0373783i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	679811.			1.28623			0.643116	−	0.765769i	$-0.277641\pi$
$727$	679811.			1.28623			0.643116	+	0.765769i	$0.277641\pi$
$728$		0			0
$729$	−186618.	−	497597.i	−0.351155	−	0.936317i
$730$		0			0
$731$			980760.i			1.83539i
$732$		0			0
$733$	311430.			0.579633			0.289817	−	0.957082i	$-0.406406\pi$
$733$	311430.			0.579633			0.289817	+	0.957082i	$0.406406\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			172065.i			0.316779i
$738$		0			0
$739$	−425613.			−0.779339			−0.389669	−	0.920955i	$-0.627411\pi$
$739$	−425613.			−0.779339			−0.389669	+	0.920955i	$0.627411\pi$
$740$		0			0
$741$	−19169.4	−	17003.7i	−0.0349119	−	0.0309675i
$742$		0			0
$743$		−	240789.i		−	0.436173i	−0.975929	−	0.218087i	$-0.930018\pi$
$743$		−	240789.i		−	0.436173i	0.975929	−	0.218087i	$-0.0699816\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−91865.5	+	11039.9i	−0.164631	+	0.0197845i
$748$		0			0
$749$		−	279753.i		−	0.498668i
$750$		0			0
$751$	−767615.			−1.36102			−0.680508	−	0.732741i	$-0.738241\pi$
$751$	−767615.			−1.36102			−0.680508	+	0.732741i	$0.738241\pi$
$752$		0			0
$753$	548186.	−	618008.i	0.966802	−	1.08994i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−943107.			−1.64577			−0.822885	−	0.568208i	$-0.807637\pi$
$757$	−943107.			−1.64577			−0.822885	+	0.568208i	$0.807637\pi$
$758$		0			0
$759$	49462.8	+	43874.5i	0.0858609	+	0.0761603i
$760$		0			0
$761$			1.04405e6i			1.80281i	0.432977	+	0.901405i	$0.357463\pi$
$761$			1.04405e6i			1.80281i	−0.432977	+	0.901405i	$0.642537\pi$
$762$		0			0
$763$	241374.			0.414612
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	32485.2i		−	0.0552199i
$768$		0			0
$769$	−428633.			−0.724824			−0.362412	−	0.932018i	$-0.618047\pi$
$769$	−428633.			−0.724824			−0.362412	+	0.932018i	$0.618047\pi$
$770$		0			0
$771$	−261335.	+	294622.i	−0.439632	+	0.495628i
$772$		0			0
$773$			390545.i			0.653600i	0.945094	+	0.326800i	$0.105970\pi$
$773$			390545.i			0.653600i	−0.945094	+	0.326800i	$0.894030\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	86788.6	+	76983.2i	0.143754	+	0.127513i
$778$		0			0
$779$		−	389625.i		−	0.642055i
$780$		0			0
$781$	483663.			0.792941
$782$		0			0
$783$	−534132.	−	370141.i	−0.871215	−	0.603731i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	936753.			1.51243			0.756215	−	0.654323i	$-0.227046\pi$
$787$	936753.			1.51243			0.756215	+	0.654323i	$0.227046\pi$
$788$		0			0
$789$	513183.	−	578547.i	0.824362	−	0.929361i
$790$		0			0
$791$		−	20172.4i		−	0.0322408i
$792$		0			0
$793$	−74130.7			−0.117883
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	535580.i		−	0.843156i	−0.906792	−	0.421578i	$-0.861476\pi$
$797$		−	535580.i		−	0.843156i	0.906792	−	0.421578i	$-0.138524\pi$
$798$		0			0
$799$	−231001.			−0.361844
$800$		0			0
$801$	−879103.	+	105646.i	−1.37017	+	0.164660i
$802$		0			0
$803$		−	403586.i		−	0.625900i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−467678.	+	527246.i	−0.718124	+	0.809592i
$808$		0			0
$809$		−	357054.i		−	0.545552i	−0.962078	−	0.272776i	$-0.912058\pi$
$809$		−	357054.i		−	0.545552i	0.962078	−	0.272776i	$-0.0879419\pi$
$810$		0			0
$811$	96582.5			0.146844			0.0734221	−	0.997301i	$-0.476608\pi$
$811$	96582.5			0.146844			0.0734221	+	0.997301i	$0.476608\pi$
$812$		0			0
$813$	−243058.	−	215597.i	−0.367730	−	0.326184i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−408772.			−0.612403
$818$		0			0
$819$	−2173.02	−	18082.1i	−0.00323963	−	0.0269576i
$820$		0			0
$821$			183717.i			0.272560i	0.990670	+	0.136280i	$0.0435147\pi$
$821$			183717.i			0.272560i	−0.990670	+	0.136280i	$0.956485\pi$
$822$		0			0
$823$	715555.			1.05644			0.528218	−	0.849109i	$-0.322860\pi$
$823$	715555.			1.05644			0.528218	+	0.849109i	$0.322860\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1.19158e6i		−	1.74225i	−0.491059	−	0.871126i	$-0.663390\pi$
$827$		−	1.19158e6i		−	1.74225i	0.491059	−	0.871126i	$-0.336610\pi$
$828$		0			0
$829$	−146844.			−0.213671			−0.106836	−	0.994277i	$-0.534072\pi$
$829$	−146844.			−0.213671			−0.106836	+	0.994277i	$0.534072\pi$
$830$		0			0
$831$	123976.	+	109969.i	0.179529	+	0.159246i
$832$		0			0
$833$		−	1.01228e6i		−	1.45885i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	394853.	−	569794.i	0.563618	−	0.813330i
$838$		0			0
$839$			393256.i			0.558665i	0.960194	+	0.279332i	$0.0901131\pi$
$839$			393256.i			0.558665i	−0.960194	+	0.279332i	$0.909887\pi$
$840$		0			0
$841$	−87353.7			−0.123506
$842$		0			0
$843$	−787069.	+	887318.i	−1.10754	+	1.24860i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−186075.			−0.259370
$848$		0			0
$849$	729807.	+	647353.i	1.01249	+	0.898103i
$850$		0			0
$851$		−	121639.i		−	0.167963i
$852$		0			0
$853$	369154.			0.507352			0.253676	−	0.967289i	$-0.418360\pi$
$853$	369154.			0.507352			0.253676	+	0.967289i	$0.418360\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1.02980e6i			1.40214i	0.713095	+	0.701068i	$0.247293\pi$
$857$			1.02980e6i			1.40214i	−0.713095	+	0.701068i	$0.752707\pi$
$858$		0			0
$859$	−33080.0			−0.0448310			−0.0224155	−	0.999749i	$-0.507136\pi$
$859$	−33080.0			−0.0448310			−0.0224155	+	0.999749i	$0.507136\pi$
$860$		0			0
$861$	183762.	−	207168.i	0.247885	−	0.279458i
$862$		0			0
$863$			1.42924e6i			1.91903i	0.281653	+	0.959516i	$0.409117\pi$
$863$			1.42924e6i			1.91903i	−0.281653	+	0.959516i	$0.590883\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−911081.	−	808147.i	−1.21205	−	1.07511i
$868$		0			0
$869$			76963.6i			0.101917i
$870$		0			0
$871$	−49694.5			−0.0655046
$872$		0			0
$873$	299.522	+	2492.38i	0.000393007	+	0.00327028i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.09234e6			1.42023			0.710116	−	0.704085i	$-0.248642\pi$
$877$	1.09234e6			1.42023			0.710116	+	0.704085i	$0.248642\pi$
$878$		0			0
$879$	−40948.2	+	46163.8i	−0.0529977	+	0.0597481i
$880$		0			0
$881$		−	431951.i		−	0.556522i	−0.960505	−	0.278261i	$-0.910242\pi$
$881$		−	431951.i		−	0.556522i	0.960505	−	0.278261i	$-0.0897581\pi$
$882$		0			0
$883$	337546.			0.432924			0.216462	−	0.976291i	$-0.430548\pi$
$883$	337546.			0.432924			0.216462	+	0.976291i	$0.430548\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			297018.i			0.377516i	0.982024	+	0.188758i	$0.0604461\pi$
$887$			297018.i			0.377516i	−0.982024	+	0.188758i	$0.939554\pi$
$888$		0			0
$889$	−120696.			−0.152717
$890$		0			0
$891$	78595.1	+	322280.i	0.0990011	+	0.405955i
$892$		0			0
$893$		−	96279.4i		−	0.120734i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−12671.5	+	14285.5i	−0.0157487	+	0.0177546i
$898$		0			0
$899$		−	847689.i		−	1.04886i
$900$		0			0
$901$	2.28620e6			2.81621
$902$		0			0
$903$	−217349.	−	192793.i	−0.266552	−	0.236437i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	900767.			1.09496			0.547480	−	0.836819i	$-0.315587\pi$
$907$	900767.			1.09496			0.547480	+	0.836819i	$0.315587\pi$
$908$		0			0
$909$	473170.	−	56863.2i	0.572650	−	0.0688183i
$910$		0			0
$911$		−	1.04199e6i		−	1.25553i	−0.778405	−	0.627763i	$-0.783971\pi$
$911$		−	1.04199e6i		−	1.25553i	0.778405	−	0.627763i	$-0.216029\pi$
$912$		0			0
$913$	57754.9			0.0692863
$914$		0			0
$915$		0			0
$916$		0			0
$917$			192114.i			0.228465i
$918$		0			0
$919$	−644472.			−0.763085			−0.381542	−	0.924351i	$-0.624607\pi$
$919$	−644472.			−0.763085			−0.381542	+	0.924351i	$0.624607\pi$
$920$		0			0
$921$	−598291.	−	530696.i	−0.705331	−	0.625643i
$922$		0			0
$923$			139688.i			0.163967i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	105220.	+	875552.i	0.122444	+	1.01888i
$928$		0			0
$929$			650980.i			0.754287i	0.926155	+	0.377143i	$0.123094\pi$
$929$			650980.i			0.754287i	−0.926155	+	0.377143i	$0.876906\pi$
$930$		0			0
$931$	421911.			0.486767
$932$		0			0
$933$	−875518.	+	987033.i	−1.00578	+	1.13388i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−274195.			−0.312306			−0.156153	−	0.987733i	$-0.549909\pi$
$937$	−274195.			−0.312306			−0.156153	+	0.987733i	$0.549909\pi$
$938$		0			0
$939$	211069.	+	187223.i	0.239383	+	0.212338i
$940$		0			0
$941$		−	1.37521e6i		−	1.55307i	−0.630076	−	0.776534i	$-0.716976\pi$
$941$		−	1.37521e6i		−	1.55307i	0.630076	−	0.776534i	$-0.283024\pi$
$942$		0			0
$943$	−290358.			−0.326520
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	5205.62i		−	0.00580461i	−0.999996	−	0.00290230i	$-0.999076\pi$
$947$		−	5205.62i		−	0.00580461i	0.999996	−	0.00290230i	$-0.000923833\pi$
$948$		0			0
$949$	116561.			0.129426
$950$		0			0
$951$	367472.	−	414277.i	0.406315	−	0.458068i
$952$		0			0
$953$		−	914858.i		−	1.00732i	−0.863902	−	0.503660i	$-0.831986\pi$
$953$		−	914858.i		−	1.00732i	0.863902	−	0.503660i	$-0.168014\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	303456.	+	269172.i	0.331339	+	0.293904i
$958$		0			0
$959$			296652.i			0.322560i
$960$		0			0
$961$	−19235.6			−0.0208286
$962$		0			0
$963$	−1.46115e6	+	175594.i	−1.57558	+	0.189346i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	529106.			0.565835			0.282918	−	0.959144i	$-0.408698\pi$
$967$	529106.			0.565835			0.282918	+	0.959144i	$0.408698\pi$
$968$		0			0
$969$	544728.	−	614110.i	0.580139	−	0.654031i
$970$		0			0
$971$		−	130708.i		−	0.138632i	−0.997595	−	0.0693159i	$-0.977918\pi$
$971$		−	130708.i		−	0.138632i	0.997595	−	0.0693159i	$-0.0220816\pi$
$972$		0			0
$973$	−102748.			−0.108530
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	704930.i		−	0.738511i	−0.929328	−	0.369255i	$-0.879613\pi$
$977$		−	704930.i		−	0.738511i	0.929328	−	0.369255i	$-0.120387\pi$
$978$		0			0
$979$	552684.			0.576649
$980$		0			0
$981$	−151504.	−	1.26069e6i	−0.157429	−	1.31000i
$982$		0			0
$983$			1.79145e6i			1.85395i	0.375124	+	0.926975i	$0.377600\pi$
$983$			1.79145e6i			1.85395i	−0.375124	+	0.926975i	$0.622400\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	45409.1	−	51192.8i	0.0466131	−	0.0525503i
$988$		0			0
$989$			304626.i			0.311440i
$990$		0			0
$991$	1.41235e6			1.43811			0.719057	−	0.694951i	$-0.244574\pi$
$991$	1.41235e6			1.43811			0.719057	+	0.694951i	$0.244574\pi$
$992$		0			0
$993$	−856842.	−	760036.i	−0.868964	−	0.770789i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	859976.			0.865159			0.432579	−	0.901596i	$-0.357604\pi$
$997$	859976.			0.865159			0.432579	+	0.901596i	$0.357604\pi$
$998$		0			0
$999$	347608.	−	501616.i	0.348304	−	0.502621i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.5.g.f.101.4	yes	4
3.2	odd	2	inner	300.5.g.f.101.3	yes	4
5.2	odd	4		300.5.b.e.149.5		8
5.3	odd	4		300.5.b.e.149.4		8
5.4	even	2		300.5.g.e.101.1	✓	4
15.2	even	4		300.5.b.e.149.3		8
15.8	even	4		300.5.b.e.149.6		8
15.14	odd	2		300.5.g.e.101.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.5.b.e.149.3		8	15.2	even	4
300.5.b.e.149.4		8	5.3	odd	4
300.5.b.e.149.5		8	5.2	odd	4
300.5.b.e.149.6		8	15.8	even	4
300.5.g.e.101.1	✓	4	5.4	even	2
300.5.g.e.101.2	yes	4	15.14	odd	2
300.5.g.f.101.3	yes	4	3.2	odd	2	inner
300.5.g.f.101.4	yes	4	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:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	300.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(6.73293 + 5.97224i) q^{3} -15.3976 q^{7} +(9.66464 + 80.4214i) q^{9} +O(q^{10})\)	\(q+(6.73293 + 5.97224i) q^{3} -15.3976 q^{7} +(9.66464 + 80.4214i) q^{9} -50.5601i q^{11} +14.6024 q^{13} +467.801i q^{17} -194.976 q^{19} +(-103.671 - 91.9580i) q^{21} +145.300i q^{23} +(-415.225 + 599.191i) q^{27} +891.423i q^{29} -950.939 q^{31} +(301.957 - 340.418i) q^{33} -837.156 q^{37} +(98.3171 + 87.2093i) q^{39} +1998.33i q^{41} +2096.53 q^{43} +493.802i q^{47} -2163.91 q^{49} +(-2793.82 + 3149.67i) q^{51} -4887.12i q^{53} +(-1312.76 - 1164.44i) q^{57} -2224.65i q^{59} -5076.60 q^{61} +(-148.812 - 1238.29i) q^{63} -3403.17 q^{67} +(-867.769 + 978.297i) q^{69} +9566.09i q^{71} +7982.30 q^{73} +778.503i q^{77} -1522.22 q^{79} +(-6374.19 + 1554.49i) q^{81} +1142.30i q^{83} +(-5323.79 + 6001.89i) q^{87} +10931.2i q^{89} -224.842 q^{91} +(-6402.61 - 5679.24i) q^{93} +30.9915 q^{97} +(4066.12 - 488.646i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5 q^{3} + 70 q^{7} - 71 q^{9}+O(q^{10}) \) 4 * q + 5 * q^3 + 70 * q^7 - 71 * q^9	\( 4 q + 5 q^{3} + 70 q^{7} - 71 q^{9} + 190 q^{13} + 536 q^{19} - 634 q^{21} + 620 q^{27} - 514 q^{31} - 1095 q^{33} - 980 q^{37} - 484 q^{39} + 4570 q^{43} - 4050 q^{49} - 1635 q^{51} - 6545 q^{57} + 1406 q^{61} - 4850 q^{63} - 980 q^{67} + 9030 q^{69} + 4690 q^{73} - 17932 q^{79} - 17711 q^{81} - 10110 q^{87} + 7654 q^{91} - 18680 q^{93} + 21310 q^{97} + 22515 q^{99}+O(q^{100}) \) 4 * q + 5 * q^3 + 70 * q^7 - 71 * q^9 + 190 * q^13 + 536 * q^19 - 634 * q^21 + 620 * q^27 - 514 * q^31 - 1095 * q^33 - 980 * q^37 - 484 * q^39 + 4570 * q^43 - 4050 * q^49 - 1635 * q^51 - 6545 * q^57 + 1406 * q^61 - 4850 * q^63 - 980 * q^67 + 9030 * q^69 + 4690 * q^73 - 17932 * q^79 - 17711 * q^81 - 10110 * q^87 + 7654 * q^91 - 18680 * q^93 + 21310 * q^97 + 22515 * q^99

Introduction

L-functions

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