LMFDB - Embedded newform 300.7.b.e.149.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,7,Mod(149,300)]

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, 1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("300.149");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 300 = 2^{2} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	300.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$69.0162250860$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 406 x^{14} + 67561 x^{12} + 5921226 x^{10} + 291565644 x^{8} + 7924637994 x^{6} + \cdots + 276002078881 $ x^16 + 406x^14 + 67561x^12 + 5921226x^10 + 291565644x^8 + 7924637994x^6 + 107459649193x^4 + 554332708870*x^2 + 276002078881
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{30}\cdot 3^{18}\cdot 5^{26} $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.12
Root			$7.89868i$ of defining polynomial
Character	$\chi$	$=$	300.149
Dual form			300.7.b.e.149.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(12.0655 + 24.1542i) q^{3} -436.815i q^{7} +(-437.848 + 582.864i) q^{9} +O(q^{10})$	$q+(12.0655 + 24.1542i) q^{3} -436.815i q^{7} +(-437.848 + 582.864i) q^{9} +1100.06i q^{11} -908.083i q^{13} +9747.84 q^{17} -7704.75 q^{19} +(10550.9 - 5270.39i) q^{21} -19320.0 q^{23} +(-19361.4 - 3543.30i) q^{27} +8106.42i q^{29} -768.733 q^{31} +(-26571.1 + 13272.8i) q^{33} +72219.1i q^{37} +(21934.0 - 10956.5i) q^{39} +14041.3i q^{41} +121342. i q^{43} -37066.0 q^{47} -73158.5 q^{49} +(117612. + 235451. i) q^{51} -173405. q^{53} +(-92961.6 - 186102. i) q^{57} -138336. i q^{59} -278669. q^{61} +(254604. + 191258. i) q^{63} -96214.3i q^{67} +(-233105. - 466658. i) q^{69} -338334. i q^{71} -37504.8i q^{73} +480524. q^{77} -721768. q^{79} +(-148020. - 510411. i) q^{81} -310971. q^{83} +(-195804. + 97808.0i) q^{87} +1.11509e6i q^{89} -396665. q^{91} +(-9275.14 - 18568.1i) q^{93} +306615. i q^{97} +(-641187. - 481660. i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2984 q^{9}+O(q^{10}) $ 16 * q - 2984 * q^9	$ 16 q - 2984 q^{9} + 30544 q^{19} - 1736 q^{21} + 70064 q^{31} - 79216 q^{39} - 405120 q^{49} + 858240 q^{51} - 271216 q^{61} + 509880 q^{69} - 1415408 q^{79} - 2396224 q^{81} - 4008224 q^{91} - 5300160 q^{99}+O(q^{100}) $ 16 * q - 2984 * q^9 + 30544 * q^19 - 1736 * q^21 + 70064 * q^31 - 79216 * q^39 - 405120 * q^49 + 858240 * q^51 - 271216 * q^61 + 509880 * q^69 - 1415408 * q^79 - 2396224 * q^81 - 4008224 * q^91 - 5300160 * 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
$3$	12.0655	+	24.1542i	0.446870	+	0.894599i
$3$	12.0655	+	24.1542i	0.446870	+	0.894599i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	436.815i		−	1.27351i	−0.771065	−	0.636757i	$-0.780276\pi$
$7$		−	436.815i		−	1.27351i	0.771065	−	0.636757i	$-0.219724\pi$
$8$		0			0
$9$	−437.848	+	582.864i	−0.600614	+	0.799539i
$10$		0			0
$11$			1100.06i			0.826494i	0.910619	+	0.413247i	$0.135605\pi$
$11$			1100.06i			0.826494i	−0.910619	+	0.413247i	$0.864395\pi$
$12$		0			0
$13$		−	908.083i		−	0.413329i	−0.978412	−	0.206664i	$-0.933739\pi$
$13$		−	908.083i		−	0.413329i	0.978412	−	0.206664i	$-0.0662608\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	9747.84			1.98409			0.992045	−	0.125882i	$-0.0401761\pi$
$17$	9747.84			1.98409			0.992045	+	0.125882i	$0.0401761\pi$
$18$		0			0
$19$	−7704.75			−1.12331			−0.561653	−	0.827373i	$-0.689834\pi$
$19$	−7704.75			−1.12331			−0.561653	+	0.827373i	$0.689834\pi$
$20$		0			0
$21$	10550.9	−	5270.39i	1.13928	−	0.569095i
$22$		0			0
$23$	−19320.0			−1.58790			−0.793951	−	0.607982i	$-0.791979\pi$
$23$	−19320.0			−1.58790			−0.793951	+	0.607982i	$0.791979\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−19361.4	−	3543.30i	−0.983663	−	0.180018i
$28$		0			0
$29$			8106.42i			0.332380i	0.986094	+	0.166190i	$0.0531465\pi$
$29$			8106.42i			0.332380i	−0.986094	+	0.166190i	$0.946853\pi$
$30$		0			0
$31$	−768.733			−0.0258042			−0.0129021	−	0.999917i	$-0.504107\pi$
$31$	−768.733			−0.0258042			−0.0129021	+	0.999917i	$0.504107\pi$
$32$		0			0
$33$	−26571.1	+	13272.8i	−0.739380	+	0.369335i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			72219.1i			1.42576i	0.701285	+	0.712881i	$0.252610\pi$
$37$			72219.1i			1.42576i	−0.701285	+	0.712881i	$0.747390\pi$
$38$		0			0
$39$	21934.0	−	10956.5i	0.369763	−	0.184704i
$40$		0			0
$41$			14041.3i			0.203731i	0.994798	+	0.101865i	$0.0324811\pi$
$41$			14041.3i			0.203731i	−0.994798	+	0.101865i	$0.967519\pi$
$42$		0			0
$43$			121342.i			1.52618i	0.646289	+	0.763092i	$0.276320\pi$
$43$			121342.i			1.52618i	−0.646289	+	0.763092i	$0.723680\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−37066.0			−0.357011			−0.178506	−	0.983939i	$-0.557126\pi$
$47$	−37066.0			−0.357011			−0.178506	+	0.983939i	$0.557126\pi$
$48$		0			0
$49$	−73158.5			−0.621837
$50$		0			0
$51$	117612.	+	235451.i	0.886631	+	1.77496i
$52$		0			0
$53$	−173405.			−1.16476			−0.582378	−	0.812918i	$-0.697878\pi$
$53$	−173405.			−1.16476			−0.582378	+	0.812918i	$0.697878\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−92961.6	−	186102.i	−0.501972	−	1.00491i
$58$		0			0
$59$		−	138336.i		−	0.673563i	−0.941583	−	0.336782i	$-0.890662\pi$
$59$		−	138336.i		−	0.673563i	0.941583	−	0.336782i	$-0.109338\pi$
$60$		0			0
$61$	−278669.			−1.22772			−0.613861	−	0.789414i	$-0.710384\pi$
$61$	−278669.			−1.22772			−0.613861	+	0.789414i	$0.710384\pi$
$62$		0			0
$63$	254604.	+	191258.i	1.01822	+	0.764890i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	96214.3i		−	0.319901i	−0.987125	−	0.159950i	$-0.948867\pi$
$67$		−	96214.3i		−	0.319901i	0.987125	−	0.159950i	$-0.0511334\pi$
$68$		0			0
$69$	−233105.	−	466658.i	−0.709586	−	1.42053i
$70$		0			0
$71$		−	338334.i		−	0.945302i	−0.881250	−	0.472651i	$-0.843297\pi$
$71$		−	338334.i		−	0.945302i	0.881250	−	0.472651i	$-0.156703\pi$
$72$		0			0
$73$		−	37504.8i		−	0.0964093i	−0.998837	−	0.0482046i	$-0.984650\pi$
$73$		−	37504.8i		−	0.0964093i	0.998837	−	0.0482046i	$-0.0153500\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	480524.			1.05255
$78$		0			0
$79$	−721768.			−1.46392			−0.731958	−	0.681350i	$-0.761393\pi$
$79$	−721768.			−1.46392			−0.731958	+	0.681350i	$0.761393\pi$
$80$		0			0
$81$	−148020.	−	510411.i	−0.278526	−	0.960429i
$82$		0			0
$83$	−310971.			−0.543858			−0.271929	−	0.962317i	$-0.587662\pi$
$83$	−310971.			−0.543858			−0.271929	+	0.962317i	$0.587662\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−195804.	+	97808.0i	−0.297347	+	0.148531i
$88$		0			0
$89$			1.11509e6i			1.58176i	0.611969	+	0.790882i	$0.290378\pi$
$89$			1.11509e6i			1.58176i	−0.611969	+	0.790882i	$0.709622\pi$
$90$		0			0
$91$	−396665.			−0.526380
$92$		0			0
$93$	−9275.14	−	18568.1i	−0.0115311	−	0.0230844i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			306615.i			0.335952i	0.985791	+	0.167976i	$0.0537232\pi$
$97$			306615.i			0.335952i	−0.985791	+	0.167976i	$0.946277\pi$
$98$		0			0
$99$	−641187.	−	481660.i	−0.660814	−	0.496404i
$100$		0			0
$101$		−	1.43599e6i		−	1.39376i	−0.717189	−	0.696879i	$-0.754571\pi$
$101$		−	1.43599e6i		−	1.39376i	0.717189	−	0.696879i	$-0.245429\pi$
$102$		0			0
$103$			1.55067e6i			1.41908i	0.704665	+	0.709540i	$0.251097\pi$
$103$			1.55067e6i			1.41908i	−0.704665	+	0.709540i	$0.748903\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.02996e6			0.840756			0.420378	−	0.907349i	$-0.361897\pi$
$107$	1.02996e6			0.840756			0.420378	+	0.907349i	$0.361897\pi$
$108$		0			0
$109$	−906603.			−0.700064			−0.350032	−	0.936738i	$-0.613829\pi$
$109$	−906603.			−0.700064			−0.350032	+	0.936738i	$0.613829\pi$
$110$		0			0
$111$	−1.74439e6	+	871359.i	−1.27548	+	0.637131i
$112$		0			0
$113$	−889378.			−0.616383			−0.308192	−	0.951324i	$-0.599724\pi$
$113$	−889378.			−0.616383			−0.308192	+	0.951324i	$0.599724\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	529289.	+	397602.i	0.330473	+	0.248251i
$118$		0			0
$119$		−	4.25800e6i		−	2.52677i
$120$		0			0
$121$	561422.			0.316908
$122$		0			0
$123$	−339156.	+	169415.i	−0.182257	+	0.0910411i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	2.88160e6i		−	1.40677i	−0.710811	−	0.703383i	$-0.751672\pi$
$127$		−	2.88160e6i		−	1.40677i	0.710811	−	0.703383i	$-0.248328\pi$
$128$		0			0
$129$	−2.93092e6	+	1.46406e6i	−1.36532	+	0.682007i
$130$		0			0
$131$			627919.i			0.279312i	0.990200	+	0.139656i	$0.0445997\pi$
$131$			627919.i			0.279312i	−0.990200	+	0.139656i	$0.955400\pi$
$132$		0			0
$133$			3.36555e6i			1.43054i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.00040e6			−1.55576			−0.777878	−	0.628415i	$-0.783704\pi$
$137$	−4.00040e6			−1.55576			−0.777878	+	0.628415i	$0.783704\pi$
$138$		0			0
$139$	−4.05700e6			−1.51064			−0.755319	−	0.655357i	$-0.772518\pi$
$139$	−4.05700e6			−1.51064			−0.755319	+	0.655357i	$0.772518\pi$
$140$		0			0
$141$	−447219.	−	895298.i	−0.159538	−	0.319382i
$142$		0			0
$143$	998949.			0.341614
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−882694.	−	1.76708e6i	−0.277880	−	0.556295i
$148$		0			0
$149$		−	3.19253e6i		−	0.965110i	−0.875866	−	0.482555i	$-0.839709\pi$
$149$		−	3.19253e6i		−	0.965110i	0.875866	−	0.482555i	$-0.160291\pi$
$150$		0			0
$151$	−1.95081e6			−0.566610			−0.283305	−	0.959030i	$-0.591431\pi$
$151$	−1.95081e6			−0.566610			−0.283305	+	0.959030i	$0.591431\pi$
$152$		0			0
$153$	−4.26807e6	+	5.68166e6i	−1.19167	+	1.58636i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			3.91636e6i			1.01201i	0.862532	+	0.506003i	$0.168878\pi$
$157$			3.91636e6i			1.01201i	−0.862532	+	0.506003i	$0.831122\pi$
$158$		0			0
$159$	−2.09222e6	−	4.18847e6i	−0.520495	−	1.04199i
$160$		0			0
$161$			8.43927e6i			2.02221i
$162$		0			0
$163$			3.66984e6i			0.847392i	0.905804	+	0.423696i	$0.139268\pi$
$163$			3.66984e6i			0.847392i	−0.905804	+	0.423696i	$0.860732\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.16638e6			0.679851			0.339925	−	0.940452i	$-0.389598\pi$
$167$	3.16638e6			0.679851			0.339925	+	0.940452i	$0.389598\pi$
$168$		0			0
$169$	4.00219e6			0.829159
$170$		0			0
$171$	3.37351e6	−	4.49082e6i	0.674673	−	0.898126i
$172$		0			0
$173$	−5.80835e6			−1.12180			−0.560899	−	0.827884i	$-0.689544\pi$
$173$	−5.80835e6			−1.12180			−0.560899	+	0.827884i	$0.689544\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	3.34139e6	−	1.66909e6i	0.602569	−	0.300995i
$178$		0			0
$179$			6.27743e6i			1.09452i	0.836963	+	0.547259i	$0.184329\pi$
$179$			6.27743e6i			1.09452i	−0.836963	+	0.547259i	$0.815671\pi$
$180$		0			0
$181$	1.24869e6			0.210581			0.105291	−	0.994442i	$-0.466423\pi$
$181$	1.24869e6			0.210581			0.105291	+	0.994442i	$0.466423\pi$
$182$		0			0
$183$	−3.36228e6	−	6.73103e6i	−0.548632	−	1.09832i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			1.07232e7i			1.63984i
$188$		0			0
$189$	−1.54777e6	+	8.45737e6i	−0.229256	+	1.25271i
$190$		0			0
$191$		−	7.38849e6i		−	1.06037i	−0.847883	−	0.530183i	$-0.822123\pi$
$191$		−	7.38849e6i		−	1.06037i	0.847883	−	0.530183i	$-0.177877\pi$
$192$		0			0
$193$			901327.i			0.125375i	0.998033	+	0.0626874i	$0.0199671\pi$
$193$			901327.i			0.125375i	−0.998033	+	0.0626874i	$0.980033\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−7.82439e6			−1.02341			−0.511707	−	0.859160i	$-0.670987\pi$
$197$	−7.82439e6			−1.02341			−0.511707	+	0.859160i	$0.670987\pi$
$198$		0			0
$199$	−4.89870e6			−0.621615			−0.310808	−	0.950473i	$-0.600599\pi$
$199$	−4.89870e6			−0.621615			−0.310808	+	0.950473i	$0.600599\pi$
$200$		0			0
$201$	2.32398e6	−	1.16087e6i	0.286183	−	0.142954i
$202$		0			0
$203$	3.54101e6			0.423291
$204$		0			0
$205$		0			0
$206$		0			0
$207$	8.45921e6	−	1.12609e7i	0.953716	−	1.26959i
$208$		0			0
$209$		−	8.47571e6i		−	0.928405i
$210$		0			0
$211$	−1.05764e7			−1.12587			−0.562935	−	0.826501i	$-0.690328\pi$
$211$	−1.05764e7			−1.12587			−0.562935	+	0.826501i	$0.690328\pi$
$212$		0			0
$213$	8.17218e6	−	4.08217e6i	0.845666	−	0.422427i
$214$		0			0
$215$		0			0
$216$		0			0
$217$			335794.i			0.0328620i
$218$		0			0
$219$	905898.	−	452515.i	0.0862476	−	0.0430824i
$220$		0			0
$221$		−	8.85185e6i		−	0.820082i
$222$		0			0
$223$		−	9.73263e6i		−	0.877639i	−0.898575	−	0.438819i	$-0.855397\pi$
$223$		−	9.73263e6i		−	0.877639i	0.898575	−	0.438819i	$-0.144603\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	5.26593e6			0.450192			0.225096	−	0.974337i	$-0.427730\pi$
$227$	5.26593e6			0.450192			0.225096	+	0.974337i	$0.427730\pi$
$228$		0			0
$229$	1.98690e7			1.65451			0.827255	−	0.561827i	$-0.189901\pi$
$229$	1.98690e7			1.65451			0.827255	+	0.561827i	$0.189901\pi$
$230$		0			0
$231$	5.79776e6	+	1.16067e7i	0.470354	+	0.941611i
$232$		0			0
$233$	2.42512e7			1.91719			0.958595	−	0.284774i	$-0.0919186\pi$
$233$	2.42512e7			1.91719			0.958595	+	0.284774i	$0.0919186\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−8.70849e6	−	1.74337e7i	−0.654181	−	1.30962i
$238$		0			0
$239$			9.38393e6i			0.687371i	0.939085	+	0.343685i	$0.111675\pi$
$239$			9.38393e6i			0.687371i	−0.939085	+	0.343685i	$0.888325\pi$
$240$		0			0
$241$	239568.			0.0171151			0.00855753	−	0.999963i	$-0.497276\pi$
$241$	239568.			0.0171151			0.00855753	+	0.999963i	$0.497276\pi$
$242$		0			0
$243$	1.05426e7	−	9.73366e6i	0.734734	−	0.678356i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			6.99656e6i			0.464294i
$248$		0			0
$249$	−3.75202e6	−	7.51124e6i	−0.243034	−	0.486534i
$250$		0			0
$251$			2.51087e6i			0.158783i	0.996844	+	0.0793913i	$0.0252977\pi$
$251$			2.51087e6i			0.158783i	−0.996844	+	0.0793913i	$0.974702\pi$
$252$		0			0
$253$		−	2.12532e7i		−	1.31239i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.15026e7			0.677634			0.338817	−	0.940852i	$-0.389973\pi$
$257$	1.15026e7			0.677634			0.338817	+	0.940852i	$0.389973\pi$
$258$		0			0
$259$	3.15464e7			1.81573
$260$		0			0
$261$	−4.72494e6	−	3.54938e6i	−0.265751	−	0.199632i
$262$		0			0
$263$	−8.74829e6			−0.480901			−0.240451	−	0.970661i	$-0.577295\pi$
$263$	−8.74829e6			−0.480901			−0.240451	+	0.970661i	$0.577295\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−2.69342e7	+	1.34542e7i	−1.41504	+	0.706843i
$268$		0			0
$269$			2.28060e7i			1.17163i	0.810444	+	0.585816i	$0.199226\pi$
$269$			2.28060e7i			1.17163i	−0.810444	+	0.585816i	$0.800774\pi$
$270$		0			0
$271$	1.15949e7			0.582584			0.291292	−	0.956634i	$-0.405915\pi$
$271$	1.15949e7			0.582584			0.291292	+	0.956634i	$0.405915\pi$
$272$		0			0
$273$	−4.78596e6	−	9.58110e6i	−0.235224	−	0.470899i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			1.44035e7i			0.677688i	0.940843	+	0.338844i	$0.110036\pi$
$277$			1.44035e7i			0.677688i	−0.940843	+	0.338844i	$0.889964\pi$
$278$		0			0
$279$	336588.	−	448067.i	0.0154984	−	0.0206315i
$280$		0			0
$281$			3.74318e7i			1.68703i	0.537108	+	0.843514i	$0.319517\pi$
$281$			3.74318e7i			1.68703i	−0.537108	+	0.843514i	$0.680483\pi$
$282$		0			0
$283$			2.16540e7i			0.955384i	0.878527	+	0.477692i	$0.158527\pi$
$283$			2.16540e7i			0.955384i	−0.878527	+	0.477692i	$0.841473\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6.13346e6			0.259454
$288$		0			0
$289$	7.08827e7			2.93661
$290$		0			0
$291$	−7.40602e6	+	3.69946e6i	−0.300543	+	0.150127i
$292$		0			0
$293$	−1.13642e6			−0.0451790			−0.0225895	−	0.999745i	$-0.507191\pi$
$293$	−1.13642e6			−0.0451790			−0.0225895	+	0.999745i	$0.507191\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	3.89785e6	−	2.12988e7i	0.148784	−	0.812992i
$298$		0			0
$299$			1.75442e7i			0.656325i
$300$		0			0
$301$	5.30042e7			1.94362
$302$		0			0
$303$	3.46852e7	−	1.73259e7i	1.24685	−	0.622829i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			5.32583e7i			1.84065i	0.391151	+	0.920326i	$0.372077\pi$
$307$			5.32583e7i			1.84065i	−0.391151	+	0.920326i	$0.627923\pi$
$308$		0			0
$309$	−3.74551e7	+	1.87096e7i	−1.26951	+	0.634145i
$310$		0			0
$311$		−	5.42770e7i		−	1.80441i	−0.431309	−	0.902204i	$-0.641948\pi$
$311$		−	5.42770e7i		−	1.80441i	0.431309	−	0.902204i	$-0.358052\pi$
$312$		0			0
$313$		−	5.24188e6i		−	0.170944i	−0.996341	−	0.0854721i	$-0.972760\pi$
$313$		−	5.24188e6i		−	0.170944i	0.996341	−	0.0854721i	$-0.0272398\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.90245e6			0.0911143			0.0455572	−	0.998962i	$-0.485494\pi$
$317$	2.90245e6			0.0911143			0.0455572	+	0.998962i	$0.485494\pi$
$318$		0			0
$319$	−8.91758e6			−0.274710
$320$		0			0
$321$	1.24270e7	+	2.48779e7i	0.375709	+	0.752140i
$322$		0			0
$323$	−7.51046e7			−2.22874
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−1.09386e7	−	2.18983e7i	−0.312838	−	0.626277i
$328$		0			0
$329$			1.61910e7i			0.454659i
$330$		0			0
$331$	−5.14619e6			−0.141906			−0.0709532	−	0.997480i	$-0.522604\pi$
$331$	−5.14619e6			−0.141906			−0.0709532	+	0.997480i	$0.522604\pi$
$332$		0			0
$333$	−4.20939e7	−	3.16210e7i	−1.13995	−	0.856333i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	1.69798e7i		−	0.443652i	−0.975086	−	0.221826i	$-0.928798\pi$
$337$		−	1.69798e7i		−	0.443652i	0.975086	−	0.221826i	$-0.0712016\pi$
$338$		0			0
$339$	−1.07308e7	−	2.14822e7i	−0.275443	−	0.551416i
$340$		0			0
$341$		−	845655.i		−	0.0213270i
$342$		0			0
$343$		−	1.94341e7i		−	0.481596i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.99529e7			−0.956225			−0.478112	−	0.878299i	$-0.658679\pi$
$347$	−3.99529e7			−0.956225			−0.478112	+	0.878299i	$0.658679\pi$
$348$		0			0
$349$	5.61186e6			0.132017			0.0660086	−	0.997819i	$-0.478974\pi$
$349$	5.61186e6			0.132017			0.0660086	+	0.997819i	$0.478974\pi$
$350$		0			0
$351$	−3.21761e6	+	1.75818e7i	−0.0744068	+	0.406576i
$352$		0			0
$353$	6.35649e7			1.44508			0.722542	−	0.691327i	$-0.242973\pi$
$353$	6.35649e7			1.44508			0.722542	+	0.691327i	$0.242973\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	1.02849e8	−	5.13749e7i	2.26044	−	1.12914i
$358$		0			0
$359$		−	2.08068e7i		−	0.449699i	−0.974393	−	0.224850i	$-0.927811\pi$
$359$		−	2.08068e7i		−	0.449699i	0.974393	−	0.224850i	$-0.0721891\pi$
$360$		0			0
$361$	1.23173e7			0.261814
$362$		0			0
$363$	6.77383e6	+	1.35607e7i	0.141617	+	0.283505i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			6.33583e7i			1.28176i	0.767642	+	0.640878i	$0.221430\pi$
$367$			6.33583e7i			1.28176i	−0.767642	+	0.640878i	$0.778570\pi$
$368$		0			0
$369$	−8.18418e6	−	6.14796e6i	−0.162891	−	0.122363i
$370$		0			0
$371$			7.57462e7i			1.48333i
$372$		0			0
$373$			7.53615e7i			1.45219i	0.687595	+	0.726094i	$0.258666\pi$
$373$			7.53615e7i			1.45219i	−0.687595	+	0.726094i	$0.741334\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	7.36131e6			0.137382
$378$		0			0
$379$	1.07995e8			1.98375			0.991877	−	0.127204i	$-0.0406004\pi$
$379$	1.07995e8			1.98375			0.991877	+	0.127204i	$0.0406004\pi$
$380$		0			0
$381$	6.96026e7	−	3.47679e7i	1.25849	−	0.628642i
$382$		0			0
$383$	8.99874e7			1.60171			0.800857	−	0.598855i	$-0.204378\pi$
$383$	8.99874e7			1.60171			0.800857	+	0.598855i	$0.204378\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−7.07261e7	−	5.31295e7i	−1.22024	−	0.916648i
$388$		0			0
$389$		−	5.17288e7i		−	0.878788i	−0.898295	−	0.439394i	$-0.855193\pi$
$389$		−	5.17288e7i		−	0.878788i	0.898295	−	0.439394i	$-0.144807\pi$
$390$		0			0
$391$	−1.88328e8			−3.15054
$392$		0			0
$393$	−1.51669e7	+	7.57615e6i	−0.249872	+	0.124816i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			5.90854e7i			0.944297i	0.881519	+	0.472149i	$0.156521\pi$
$397$			5.90854e7i			0.944297i	−0.881519	+	0.472149i	$0.843479\pi$
$398$		0			0
$399$	−8.12921e7	+	4.06071e7i	−1.27976	+	0.639268i
$400$		0			0
$401$			1.40805e7i			0.218366i	0.994022	+	0.109183i	$0.0348235\pi$
$401$			1.40805e7i			0.218366i	−0.994022	+	0.109183i	$0.965177\pi$
$402$		0			0
$403$			698073.i			0.0106656i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−7.94456e7			−1.17838
$408$		0			0
$409$	−1.05471e8			−1.54157			−0.770783	−	0.637097i	$-0.780135\pi$
$409$	−1.05471e8			−1.54157			−0.770783	+	0.637097i	$0.780135\pi$
$410$		0			0
$411$	−4.82668e7	−	9.66263e7i	−0.695221	−	1.39178i
$412$		0			0
$413$	−6.04272e7			−0.857792
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−4.89497e7	−	9.79934e7i	−0.675059	−	1.35141i
$418$		0			0
$419$		−	1.26383e7i		−	0.171809i	−0.996303	−	0.0859046i	$-0.972622\pi$
$419$		−	1.26383e7i		−	0.171809i	0.996303	−	0.0859046i	$-0.0273780\pi$
$420$		0			0
$421$	7.64436e6			0.102446			0.0512230	−	0.998687i	$-0.483688\pi$
$421$	7.64436e6			0.102446			0.0512230	+	0.998687i	$0.483688\pi$
$422$		0			0
$423$	1.62293e7	−	2.16044e7i	0.214426	−	0.285444i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			1.21727e8i			1.56352i
$428$		0			0
$429$	1.20528e7	+	2.41288e7i	0.152657	+	0.305607i
$430$		0			0
$431$		−	1.11755e8i		−	1.39583i	−0.716179	−	0.697917i	$-0.754111\pi$
$431$		−	1.11755e8i		−	1.39583i	0.716179	−	0.697917i	$-0.245889\pi$
$432$		0			0
$433$			5.93268e7i			0.730781i	0.930854	+	0.365390i	$0.119065\pi$
$433$			5.93268e7i			0.730781i	−0.930854	+	0.365390i	$0.880935\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.48856e8			1.78370
$438$		0			0
$439$	3.07400e6			0.0363338			0.0181669	−	0.999835i	$-0.494217\pi$
$439$	3.07400e6			0.0363338			0.0181669	+	0.999835i	$0.494217\pi$
$440$		0			0
$441$	3.20323e7	−	4.26415e7i	0.373484	−	0.497183i
$442$		0			0
$443$	7.68560e7			0.884029			0.442014	−	0.897008i	$-0.354264\pi$
$443$	7.68560e7			0.884029			0.442014	+	0.897008i	$0.354264\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	7.71130e7	−	3.85195e7i	0.863386	−	0.431279i
$448$		0			0
$449$		−	1.60168e8i		−	1.76945i	−0.466117	−	0.884723i	$-0.654347\pi$
$449$		−	1.60168e8i		−	1.76945i	0.466117	−	0.884723i	$-0.345653\pi$
$450$		0			0
$451$	−1.54463e7			−0.168382
$452$		0			0
$453$	−2.35375e7	−	4.71202e7i	−0.253201	−	0.506889i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	9.10136e7i		−	0.953581i	−0.879017	−	0.476790i	$-0.841800\pi$
$457$		−	9.10136e7i		−	0.953581i	0.879017	−	0.476790i	$-0.158200\pi$
$458$		0			0
$459$	−1.88732e8	−	3.45395e7i	−1.95168	−	0.357173i
$460$		0			0
$461$			5.62811e7i			0.574460i	0.957862	+	0.287230i	$0.0927344\pi$
$461$			5.62811e7i			0.574460i	−0.957862	+	0.287230i	$0.907266\pi$
$462$		0			0
$463$			7.87451e7i			0.793378i	0.917953	+	0.396689i	$0.129841\pi$
$463$			7.87451e7i			0.793378i	−0.917953	+	0.396689i	$0.870159\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.51383e7			0.345009			0.172504	−	0.985009i	$-0.444814\pi$
$467$	3.51383e7			0.345009			0.172504	+	0.985009i	$0.444814\pi$
$468$		0			0
$469$	−4.20279e7			−0.407398
$470$		0			0
$471$	−9.45963e7	+	4.72528e7i	−0.905340	+	0.452236i
$472$		0			0
$473$	−1.33484e8			−1.26138
$474$		0			0
$475$		0			0
$476$		0			0
$477$	7.59252e7	−	1.01072e8i	0.699569	−	0.931269i
$478$		0			0
$479$			1.04950e7i			0.0954942i	0.998859	+	0.0477471i	$0.0152042\pi$
$479$			1.04950e7i			0.0954942i	−0.998859	+	0.0477471i	$0.984796\pi$
$480$		0			0
$481$	6.55810e7			0.589309
$482$		0			0
$483$	−2.03843e8	+	1.01824e8i	−1.80907	+	0.903667i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			8.61647e7i			0.746007i	0.927830	+	0.373003i	$0.121672\pi$
$487$			8.61647e7i			0.746007i	−0.927830	+	0.373003i	$0.878328\pi$
$488$		0			0
$489$	−8.86420e7	+	4.42785e7i	−0.758076	+	0.378674i
$490$		0			0
$491$		−	8.04932e7i		−	0.680009i	−0.940424	−	0.340005i	$-0.889571\pi$
$491$		−	8.04932e7i		−	0.680009i	0.940424	−	0.340005i	$-0.110429\pi$
$492$		0			0
$493$			7.90201e7i			0.659472i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.47789e8			−1.20386
$498$		0			0
$499$	−2.12437e8			−1.70974			−0.854869	−	0.518845i	$-0.826362\pi$
$499$	−2.12437e8			−1.70974			−0.854869	+	0.518845i	$0.826362\pi$
$500$		0			0
$501$	3.82039e7	+	7.64813e7i	0.303805	+	0.608194i
$502$		0			0
$503$	−1.26961e8			−0.997625			−0.498812	−	0.866710i	$-0.666230\pi$
$503$	−1.26961e8			−0.997625			−0.498812	+	0.866710i	$0.666230\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	4.82884e7	+	9.66696e7i	0.370527	+	0.741765i
$508$		0			0
$509$			1.99418e8i			1.51221i	0.654452	+	0.756104i	$0.272899\pi$
$509$			1.99418e8i			1.51221i	−0.654452	+	0.756104i	$0.727101\pi$
$510$		0			0
$511$	−1.63827e7			−0.122779
$512$		0			0
$513$	1.49175e8	+	2.73002e7i	1.10495	+	0.202216i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	4.07749e7i		−	0.295068i
$518$		0			0
$519$	−7.00807e7	−	1.40296e8i	−0.501298	−	1.00356i
$520$		0			0
$521$			4.85935e7i			0.343609i	0.985131	+	0.171805i	$0.0549598\pi$
$521$			4.85935e7i			0.343609i	−0.985131	+	0.171805i	$0.945040\pi$
$522$		0			0
$523$			5.79820e7i			0.405311i	0.979250	+	0.202655i	$0.0649572\pi$
$523$			5.79820e7i			0.405311i	−0.979250	+	0.202655i	$0.935043\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7.49348e6			−0.0511979
$528$		0			0
$529$	2.25226e8			1.52143
$530$		0			0
$531$	8.06309e7	+	6.05700e7i	0.538540	+	0.404552i
$532$		0			0
$533$	1.27507e7			0.0842077
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−1.51626e8	+	7.57404e7i	−0.979155	+	0.489108i
$538$		0			0
$539$		−	8.04790e7i		−	0.513944i
$540$		0			0
$541$	4.20855e7			0.265791			0.132896	−	0.991130i	$-0.457572\pi$
$541$	4.20855e7			0.265791			0.132896	+	0.991130i	$0.457572\pi$
$542$		0			0
$543$	1.50661e7	+	3.01611e7i	0.0941024	+	0.188386i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			7.51378e7i			0.459088i	0.973298	+	0.229544i	$0.0737236\pi$
$547$			7.51378e7i			0.459088i	−0.973298	+	0.229544i	$0.926276\pi$
$548$		0			0
$549$	1.22015e8	−	1.62426e8i	0.737387	−	0.981611i
$550$		0			0
$551$		−	6.24580e7i		−	0.373364i
$552$		0			0
$553$			3.15279e8i			1.86432i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.38895e7			−0.0803750			−0.0401875	−	0.999192i	$-0.512796\pi$
$557$	−1.38895e7			−0.0803750			−0.0401875	+	0.999192i	$0.512796\pi$
$558$		0			0
$559$	1.10189e8			0.630816
$560$		0			0
$561$	−2.59011e8	+	1.29381e8i	−1.46700	+	0.732795i
$562$		0			0
$563$	−2.69248e8			−1.50878			−0.754392	−	0.656424i	$-0.772068\pi$
$563$	−2.69248e8			−1.50878			−0.754392	+	0.656424i	$0.772068\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−2.22955e8	+	6.46574e7i	−1.22312	+	0.354706i
$568$		0			0
$569$		−	1.49380e8i		−	0.810878i	−0.914122	−	0.405439i	$-0.867119\pi$
$569$		−	1.49380e8i		−	0.810878i	0.914122	−	0.405439i	$-0.132881\pi$
$570$		0			0
$571$	−1.01093e7			−0.0543015			−0.0271507	−	0.999631i	$-0.508643\pi$
$571$	−1.01093e7			−0.0543015			−0.0271507	+	0.999631i	$0.508643\pi$
$572$		0			0
$573$	1.78463e8	−	8.91458e7i	0.948602	−	0.473846i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	1.19578e8i		−	0.622477i	−0.950332	−	0.311238i	$-0.899256\pi$
$577$		−	1.19578e8i		−	0.622477i	0.950332	−	0.311238i	$-0.100744\pi$
$578$		0			0
$579$	−2.17708e7	+	1.08750e7i	−0.112160	+	0.0560263i
$580$		0			0
$581$			1.35837e8i			0.692610i
$582$		0			0
$583$		−	1.90757e8i		−	0.962664i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−2.77627e7			−0.137261			−0.0686305	−	0.997642i	$-0.521863\pi$
$587$	−2.77627e7			−0.137261			−0.0686305	+	0.997642i	$0.521863\pi$
$588$		0			0
$589$	5.92289e6			0.0289860
$590$		0			0
$591$	−9.44051e7	−	1.88992e8i	−0.457334	−	0.915546i
$592$		0			0
$593$	9.85398e7			0.472550			0.236275	−	0.971686i	$-0.424073\pi$
$593$	9.85398e7			0.472550			0.236275	+	0.971686i	$0.424073\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−5.91052e7	−	1.18324e8i	−0.277781	−	0.556096i
$598$		0			0
$599$			6.51155e7i			0.302973i	0.988459	+	0.151486i	$0.0484060\pi$
$599$			6.51155e7i			0.302973i	−0.988459	+	0.151486i	$0.951594\pi$
$600$		0			0
$601$	2.53264e8			1.16668			0.583339	−	0.812229i	$-0.301746\pi$
$601$	2.53264e8			1.16668			0.583339	+	0.812229i	$0.301746\pi$
$602$		0			0
$603$	5.60799e7	+	4.21272e7i	0.255773	+	0.192137i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	5.70547e7i		−	0.255109i	−0.991832	−	0.127554i	$-0.959287\pi$
$607$		−	5.70547e7i		−	0.255109i	0.991832	−	0.127554i	$-0.0407127\pi$
$608$		0			0
$609$	4.27240e7	+	8.55301e7i	0.189156	+	0.378675i
$610$		0			0
$611$			3.36590e7i			0.147563i
$612$		0			0
$613$		−	3.26889e8i		−	1.41912i	−0.704645	−	0.709560i	$-0.748894\pi$
$613$		−	3.26889e8i		−	1.41912i	0.704645	−	0.709560i	$-0.251106\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.01353e8			1.70872			0.854361	−	0.519680i	$-0.173949\pi$
$617$	4.01353e8			1.70872			0.854361	+	0.519680i	$0.173949\pi$
$618$		0			0
$619$	−1.37298e8			−0.578884			−0.289442	−	0.957195i	$-0.593470\pi$
$619$	−1.37298e8			−0.578884			−0.289442	+	0.957195i	$0.593470\pi$
$620$		0			0
$621$	3.74063e8	+	6.84566e7i	1.56196	+	0.285851i
$622$		0			0
$623$	4.87090e8			2.01440
$624$		0			0
$625$		0			0
$626$		0			0
$627$	2.04724e8	−	1.02264e8i	0.830550	−	0.414876i
$628$		0			0
$629$			7.03980e8i			2.82884i
$630$		0			0
$631$	−1.95608e8			−0.778572			−0.389286	−	0.921117i	$-0.627278\pi$
$631$	−1.95608e8			−0.778572			−0.389286	+	0.921117i	$0.627278\pi$
$632$		0			0
$633$	−1.27609e8	−	2.55463e8i	−0.503118	−	1.00720i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			6.64340e7i			0.257023i
$638$		0			0
$639$	1.97203e8	+	1.48139e8i	0.755806	+	0.567762i
$640$		0			0
$641$		−	1.86584e8i		−	0.708437i	−0.935163	−	0.354219i	$-0.884747\pi$
$641$		−	1.86584e8i		−	0.708437i	0.935163	−	0.354219i	$-0.115253\pi$
$642$		0			0
$643$		−	3.74191e8i		−	1.40754i	−0.710427	−	0.703770i	$-0.751498\pi$
$643$		−	3.74191e8i		−	1.40754i	0.710427	−	0.703770i	$-0.248502\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3.66258e8			1.35230			0.676152	−	0.736762i	$-0.263646\pi$
$647$	3.66258e8			1.35230			0.676152	+	0.736762i	$0.263646\pi$
$648$		0			0
$649$	1.52178e8			0.556696
$650$		0			0
$651$	−8.11083e6	+	4.05152e6i	−0.0293983	+	0.0146850i
$652$		0			0
$653$	−1.50492e8			−0.540472			−0.270236	−	0.962794i	$-0.587102\pi$
$653$	−1.50492e8			−0.540472			−0.270236	+	0.962794i	$0.587102\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	2.18602e7	+	1.64214e7i	0.0770830	+	0.0579048i
$658$		0			0
$659$		−	4.31848e8i		−	1.50895i	−0.656329	−	0.754475i	$-0.727891\pi$
$659$		−	4.31848e8i		−	1.50895i	0.656329	−	0.754475i	$-0.272109\pi$
$660$		0			0
$661$	−1.04838e8			−0.363007			−0.181504	−	0.983390i	$-0.558096\pi$
$661$	−1.04838e8			−0.363007			−0.181504	+	0.983390i	$0.558096\pi$
$662$		0			0
$663$	2.13809e8	−	1.06802e8i	0.733644	−	0.366470i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	1.56616e8i		−	0.527787i
$668$		0			0
$669$	2.35084e8	−	1.17429e8i	0.785135	−	0.392191i
$670$		0			0
$671$		−	3.06554e8i		−	1.01470i
$672$		0			0
$673$		−	2.07160e8i		−	0.679611i	−0.940496	−	0.339805i	$-0.889639\pi$
$673$		−	2.07160e8i		−	0.679611i	0.940496	−	0.339805i	$-0.110361\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−4.59856e8			−1.48203			−0.741013	−	0.671491i	$-0.765654\pi$
$677$	−4.59856e8			−1.48203			−0.741013	+	0.671491i	$0.765654\pi$
$678$		0			0
$679$	1.33934e8			0.427840
$680$		0			0
$681$	6.35361e7	+	1.27194e8i	0.201177	+	0.402741i
$682$		0			0
$683$	4.04044e8			1.26814			0.634069	−	0.773277i	$-0.281384\pi$
$683$	4.04044e8			1.26814			0.634069	+	0.773277i	$0.281384\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	2.39729e8	+	4.79919e8i	0.739351	+	1.48012i
$688$		0			0
$689$			1.57467e8i			0.481428i
$690$		0			0
$691$	−2.98324e8			−0.904179			−0.452090	−	0.891973i	$-0.649321\pi$
$691$	−2.98324e8			−0.904179			−0.452090	+	0.891973i	$0.649321\pi$
$692$		0			0
$693$	−2.10396e8	+	2.80080e8i	−0.632177	+	0.841556i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			1.36872e8i			0.404220i
$698$		0			0
$699$	2.92602e8	+	5.85767e8i	0.856735	+	1.71512i
$700$		0			0
$701$		−	4.10378e8i		−	1.19132i	−0.803235	−	0.595662i	$-0.796890\pi$
$701$		−	4.10378e8i		−	1.19132i	0.803235	−	0.595662i	$-0.203110\pi$
$702$		0			0
$703$		−	5.56430e8i		−	1.60157i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−6.27262e8			−1.77497
$708$		0			0
$709$	4.38412e8			1.23011			0.615054	−	0.788485i	$-0.289134\pi$
$709$	4.38412e8			1.23011			0.615054	+	0.788485i	$0.289134\pi$
$710$		0			0
$711$	3.16024e8	−	4.20693e8i	0.879249	−	1.17046i
$712$		0			0
$713$	1.48519e7			0.0409745
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−2.26661e8	+	1.13222e8i	−0.614921	+	0.307166i
$718$		0			0
$719$			1.35325e8i			0.364076i	0.983291	+	0.182038i	$0.0582694\pi$
$719$			1.35325e8i			0.364076i	−0.983291	+	0.182038i	$0.941731\pi$
$720$		0			0
$721$	6.77355e8			1.80722
$722$		0			0
$723$	2.89051e6	+	5.78657e6i	0.00764821	+	0.0153111i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			1.04885e7i			0.0272966i	0.999907	+	0.0136483i	$0.00434452\pi$
$727$			1.04885e7i			0.0272966i	−0.999907	+	0.0136483i	$0.995655\pi$
$728$		0			0
$729$	3.62311e8	+	1.37207e8i	0.935187	+	0.354155i
$730$		0			0
$731$			1.18283e9i			3.02809i
$732$		0			0
$733$			2.59655e7i			0.0659303i	0.999457	+	0.0329652i	$0.0104950\pi$
$733$			2.59655e7i			0.0659303i	−0.999457	+	0.0329652i	$0.989505\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.05842e8			0.264396
$738$		0			0
$739$	−1.95717e8			−0.484948			−0.242474	−	0.970158i	$-0.577959\pi$
$739$	−1.95717e8			−0.484948			−0.242474	+	0.970158i	$0.577959\pi$
$740$		0			0
$741$	−1.68996e8	+	8.44169e7i	−0.415357	+	0.207479i
$742$		0			0
$743$	−1.21923e8			−0.297247			−0.148624	−	0.988894i	$-0.547484\pi$
$743$	−1.21923e8			−0.297247			−0.148624	+	0.988894i	$0.547484\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1.36158e8	−	1.81254e8i	0.326648	−	0.434835i
$748$		0			0
$749$		−	4.49903e8i		−	1.07071i
$750$		0			0
$751$	−3.28855e8			−0.776399			−0.388199	−	0.921575i	$-0.626903\pi$
$751$	−3.28855e8			−0.776399			−0.388199	+	0.921575i	$0.626903\pi$
$752$		0			0
$753$	−6.06480e7	+	3.02949e7i	−0.142047	+	0.0709553i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	2.45441e8i		−	0.565797i	−0.959150	−	0.282898i	$-0.908704\pi$
$757$		−	2.45441e8i		−	0.565797i	0.959150	−	0.282898i	$-0.0912959\pi$
$758$		0			0
$759$	5.13354e8	−	2.56431e8i	1.17406	−	0.586468i
$760$		0			0
$761$			5.19490e8i			1.17875i	0.807858	+	0.589377i	$0.200627\pi$
$761$			5.19490e8i			1.17875i	−0.807858	+	0.589377i	$0.799373\pi$
$762$		0			0
$763$			3.96018e8i			0.891541i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.25620e8			−0.278403
$768$		0			0
$769$	−4.01913e8			−0.883797			−0.441899	−	0.897065i	$-0.645695\pi$
$769$	−4.01913e8			−0.883797			−0.441899	+	0.897065i	$0.645695\pi$
$770$		0			0
$771$	1.38784e8	+	2.77835e8i	0.302814	+	0.606211i
$772$		0			0
$773$	3.45715e8			0.748478			0.374239	−	0.927332i	$-0.377904\pi$
$773$	3.45715e8			0.748478			0.374239	+	0.927332i	$0.377904\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	3.80623e8	+	7.61977e8i	0.811394	+	1.62435i
$778$		0			0
$779$		−	1.08185e8i		−	0.228852i
$780$		0			0
$781$	3.72189e8			0.781286
$782$		0			0
$783$	2.87235e7	−	1.56952e8i	0.0598345	−	0.326950i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	7.70491e8i		−	1.58068i	−0.612670	−	0.790339i	$-0.709905\pi$
$787$		−	7.70491e8i		−	1.58068i	0.612670	−	0.790339i	$-0.290095\pi$
$788$		0			0
$789$	−1.05553e8	−	2.11308e8i	−0.214901	−	0.430214i
$790$		0			0
$791$			3.88494e8i			0.784973i
$792$		0			0
$793$			2.53055e8i			0.507453i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	2.84076e8			0.561125			0.280563	−	0.959836i	$-0.409479\pi$
$797$	2.84076e8			0.561125			0.280563	+	0.959836i	$0.409479\pi$
$798$		0			0
$799$	−3.61313e8			−0.708343
$800$		0			0
$801$	−6.49948e8	−	4.88241e8i	−1.26468	−	0.950029i
$802$		0			0
$803$	4.12577e7			0.0796817
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−5.50859e8	+	2.75165e8i	−1.04814	+	0.523568i
$808$		0			0
$809$			6.17256e8i			1.16579i	0.812548	+	0.582894i	$0.198080\pi$
$809$			6.17256e8i			1.16579i	−0.812548	+	0.582894i	$0.801920\pi$
$810$		0			0
$811$	5.32210e8			0.997748			0.498874	−	0.866675i	$-0.333747\pi$
$811$	5.32210e8			0.997748			0.498874	+	0.866675i	$0.333747\pi$
$812$		0			0
$813$	1.39898e8	+	2.80065e8i	0.260339	+	0.521179i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	9.34913e8i		−	1.71437i
$818$		0			0
$819$	1.73679e8	−	2.31202e8i	0.316151	−	0.420861i
$820$		0			0
$821$			3.60333e8i			0.651140i	0.945518	+	0.325570i	$0.105556\pi$
$821$			3.60333e8i			0.651140i	−0.945518	+	0.325570i	$0.894444\pi$
$822$		0			0
$823$		−	9.46142e7i		−	0.169729i	−0.996392	−	0.0848646i	$-0.972954\pi$
$823$		−	9.46142e7i		−	0.169729i	0.996392	−	0.0848646i	$-0.0270458\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−7.55768e8			−1.33620			−0.668101	−	0.744070i	$-0.732893\pi$
$827$	−7.55768e8			−1.33620			−0.668101	+	0.744070i	$0.732893\pi$
$828$		0			0
$829$	−4.78503e8			−0.839887			−0.419944	−	0.907550i	$-0.637950\pi$
$829$	−4.78503e8			−0.839887			−0.419944	+	0.907550i	$0.637950\pi$
$830$		0			0
$831$	−3.47906e8	+	1.73786e8i	−0.606259	+	0.302839i
$832$		0			0
$833$	−7.13137e8			−1.23378
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.48838e7	+	2.72385e6i	0.0253826	+	0.00464523i
$838$		0			0
$839$		−	1.45796e8i		−	0.246866i	−0.992353	−	0.123433i	$-0.960610\pi$
$839$		−	1.45796e8i		−	0.246866i	0.992353	−	0.123433i	$-0.0393903\pi$
$840$		0			0
$841$	5.29109e8			0.889523
$842$		0			0
$843$	−9.04135e8	+	4.51634e8i	−1.50921	+	0.753882i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	2.45238e8i		−	0.403587i
$848$		0			0
$849$	−5.23033e8	+	2.61266e8i	−0.854685	+	0.426933i
$850$		0			0
$851$		−	1.39527e9i		−	2.26397i
$852$		0			0
$853$		−	7.64922e8i		−	1.23245i	−0.787569	−	0.616226i	$-0.788661\pi$
$853$		−	7.64922e8i		−	1.23245i	0.787569	−	0.616226i	$-0.211339\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	4.79513e8			0.761830			0.380915	−	0.924610i	$-0.375609\pi$
$857$	4.79513e8			0.761830			0.380915	+	0.924610i	$0.375609\pi$
$858$		0			0
$859$	2.90859e8			0.458884			0.229442	−	0.973322i	$-0.426310\pi$
$859$	2.90859e8			0.458884			0.229442	+	0.973322i	$0.426310\pi$
$860$		0			0
$861$	7.40032e7	+	1.48149e8i	0.115942	+	0.232107i
$862$		0			0
$863$	−3.78195e8			−0.588415			−0.294207	−	0.955742i	$-0.595056\pi$
$863$	−3.78195e8			−0.588415			−0.294207	+	0.955742i	$0.595056\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	8.55235e8	+	1.71211e9i	1.31229	+	2.62709i
$868$		0			0
$869$		−	7.93990e8i		−	1.20992i
$870$		0			0
$871$	−8.73706e7			−0.132224
$872$		0			0
$873$	−1.78715e8	−	1.34250e8i	−0.268607	−	0.201778i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			3.32209e8i			0.492508i	0.969205	+	0.246254i	$0.0791997\pi$
$877$			3.32209e8i			0.492508i	−0.969205	+	0.246254i	$0.920800\pi$
$878$		0			0
$879$	−1.37115e7	−	2.74493e7i	−0.0201891	−	0.0404171i
$880$		0			0
$881$		−	1.22274e9i		−	1.78815i	−0.447914	−	0.894077i	$-0.647833\pi$
$881$		−	1.22274e9i		−	1.78815i	0.447914	−	0.894077i	$-0.352167\pi$
$882$		0			0
$883$			8.56383e8i			1.24390i	0.783056	+	0.621951i	$0.213660\pi$
$883$			8.56383e8i			1.24390i	−0.783056	+	0.621951i	$0.786340\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	5.70473e8			0.817456			0.408728	−	0.912656i	$-0.365973\pi$
$887$	5.70473e8			0.817456			0.408728	+	0.912656i	$0.365973\pi$
$888$		0			0
$889$	−1.25873e9			−1.79154
$890$		0			0
$891$	5.61485e8	−	1.62831e8i	0.793788	−	0.230200i
$892$		0			0
$893$	2.85584e8			0.401033
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−4.23765e8	+	2.11679e8i	−0.587148	+	0.293292i
$898$		0			0
$899$		−	6.23167e6i		−	0.00857680i
$900$		0			0
$901$	−1.69033e9			−2.31098
$902$		0			0
$903$	6.39522e8	+	1.28027e9i	0.868545	+	1.73876i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	2.31544e8i		−	0.310321i	−0.987889	−	0.155160i	$-0.950411\pi$
$907$		−	2.31544e8i		−	0.310321i	0.987889	−	0.155160i	$-0.0495895\pi$
$908$		0			0
$909$	8.36987e8	+	6.28745e8i	1.11436	+	0.837111i
$910$		0			0
$911$		−	4.23991e8i		−	0.560792i	−0.959884	−	0.280396i	$-0.909534\pi$
$911$		−	4.23991e8i		−	0.560792i	0.959884	−	0.280396i	$-0.0904658\pi$
$912$		0			0
$913$		−	3.42087e8i		−	0.449495i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	2.74285e8			0.355708
$918$		0			0
$919$	−1.08546e9			−1.39852			−0.699259	−	0.714869i	$-0.746486\pi$
$919$	−1.08546e9			−1.39852			−0.699259	+	0.714869i	$0.746486\pi$
$920$		0			0
$921$	−1.28641e9	+	6.42587e8i	−1.64665	+	0.822533i
$922$		0			0
$923$	−3.07236e8			−0.390721
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−9.03828e8	−	6.78956e8i	−1.13461	−	0.852319i
$928$		0			0
$929$		−	8.68457e7i		−	0.108318i	−0.998532	−	0.0541591i	$-0.982752\pi$
$929$		−	8.68457e7i		−	0.108318i	0.998532	−	0.0541591i	$-0.0172478\pi$
$930$		0			0
$931$	5.63668e8			0.698513
$932$		0			0
$933$	1.31102e9	−	6.54879e8i	1.61422	−	0.806336i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	3.79010e8i		−	0.460715i	−0.973106	−	0.230358i	$-0.926010\pi$
$937$		−	3.79010e8i		−	0.460715i	0.973106	−	0.230358i	$-0.0739896\pi$
$938$		0			0
$939$	1.26613e8	−	6.32459e7i	0.152926	−	0.0763898i
$940$		0			0
$941$			5.55151e6i			0.00666258i	0.999994	+	0.00333129i	$0.00106038\pi$
$941$			5.55151e6i			0.00666258i	−0.999994	+	0.00333129i	$0.998940\pi$
$942$		0			0
$943$		−	2.71278e8i		−	0.323504i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5.18303e8			0.610286			0.305143	−	0.952306i	$-0.401296\pi$
$947$	5.18303e8			0.610286			0.305143	+	0.952306i	$0.401296\pi$
$948$		0			0
$949$	−3.40575e7			−0.0398487
$950$		0			0
$951$	3.50195e7	+	7.01062e7i	0.0407163	+	0.0815108i
$952$		0			0
$953$	−3.23349e8			−0.373588			−0.186794	−	0.982399i	$-0.559810\pi$
$953$	−3.23349e8			−0.373588			−0.186794	+	0.982399i	$0.559810\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−1.07595e8	−	2.15397e8i	−0.122760	−	0.245755i
$958$		0			0
$959$			1.74743e9i			1.98128i
$960$		0			0
$961$	−8.86913e8			−0.999334
$962$		0			0
$963$	−4.50967e8	+	6.00328e8i	−0.504970	+	0.672218i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	1.56777e9i		−	1.73381i	−0.498470	−	0.866907i	$-0.666105\pi$
$967$		−	1.56777e9i		−	1.73381i	0.498470	−	0.866907i	$-0.333895\pi$
$968$		0			0
$969$	−9.06175e8	−	1.81409e9i	−0.995957	−	1.99383i
$970$		0			0
$971$			1.19138e9i			1.30135i	0.759358	+	0.650673i	$0.225513\pi$
$971$			1.19138e9i			1.30135i	−0.759358	+	0.650673i	$0.774487\pi$
$972$		0			0
$973$			1.77216e9i			1.92382i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	2.83913e8			0.304440			0.152220	−	0.988347i	$-0.451358\pi$
$977$	2.83913e8			0.304440			0.152220	+	0.988347i	$0.451358\pi$
$978$		0			0
$979$	−1.22667e9			−1.30732
$980$		0			0
$981$	3.96954e8	−	5.28427e8i	0.420468	−	0.559729i
$982$		0			0
$983$	−2.30425e8			−0.242587			−0.121294	−	0.992617i	$-0.538704\pi$
$983$	−2.30425e8			−0.242587			−0.121294	+	0.992617i	$0.538704\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−3.91080e8	+	1.95352e8i	−0.406737	+	0.203173i
$988$		0			0
$989$		−	2.34433e9i		−	2.42343i
$990$		0			0
$991$	−1.46001e8			−0.150015			−0.0750076	−	0.997183i	$-0.523898\pi$
$991$	−1.46001e8			−0.150015			−0.0750076	+	0.997183i	$0.523898\pi$
$992$		0			0
$993$	−6.20914e7	−	1.24302e8i	−0.0634138	−	0.126949i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			1.78931e8i			0.180551i	0.995917	+	0.0902756i	$0.0287748\pi$
$997$			1.78931e8i			0.180551i	−0.995917	+	0.0902756i	$0.971225\pi$
$998$		0			0
$999$	2.55894e8	−	1.39827e9i	0.256663	−	1.40247i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.7.b.e.149.12		16
3.2	odd	2	inner	300.7.b.e.149.6		16
5.2	odd	4		300.7.g.h.101.7		8
5.3	odd	4		60.7.g.a.41.2	yes	8
5.4	even	2	inner	300.7.b.e.149.5		16
15.2	even	4		300.7.g.h.101.8		8
15.8	even	4		60.7.g.a.41.1	✓	8
15.14	odd	2	inner	300.7.b.e.149.11		16
20.3	even	4		240.7.l.c.161.7		8
60.23	odd	4		240.7.l.c.161.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.7.g.a.41.1	✓	8	15.8	even	4
60.7.g.a.41.2	yes	8	5.3	odd	4
240.7.l.c.161.7		8	20.3	even	4
240.7.l.c.161.8		8	60.23	odd	4
300.7.b.e.149.5		16	5.4	even	2	inner
300.7.b.e.149.6		16	3.2	odd	2	inner
300.7.b.e.149.11		16	15.14	odd	2	inner
300.7.b.e.149.12		16	1.1	even	1	trivial
300.7.g.h.101.7		8	5.2	odd	4
300.7.g.h.101.8		8	15.2	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:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	300.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(12.0655 + 24.1542i) q^{3} -436.815i q^{7} +(-437.848 + 582.864i) q^{9} +O(q^{10})\)	\(q+(12.0655 + 24.1542i) q^{3} -436.815i q^{7} +(-437.848 + 582.864i) q^{9} +1100.06i q^{11} -908.083i q^{13} +9747.84 q^{17} -7704.75 q^{19} +(10550.9 - 5270.39i) q^{21} -19320.0 q^{23} +(-19361.4 - 3543.30i) q^{27} +8106.42i q^{29} -768.733 q^{31} +(-26571.1 + 13272.8i) q^{33} +72219.1i q^{37} +(21934.0 - 10956.5i) q^{39} +14041.3i q^{41} +121342. i q^{43} -37066.0 q^{47} -73158.5 q^{49} +(117612. + 235451. i) q^{51} -173405. q^{53} +(-92961.6 - 186102. i) q^{57} -138336. i q^{59} -278669. q^{61} +(254604. + 191258. i) q^{63} -96214.3i q^{67} +(-233105. - 466658. i) q^{69} -338334. i q^{71} -37504.8i q^{73} +480524. q^{77} -721768. q^{79} +(-148020. - 510411. i) q^{81} -310971. q^{83} +(-195804. + 97808.0i) q^{87} +1.11509e6i q^{89} -396665. q^{91} +(-9275.14 - 18568.1i) q^{93} +306615. i q^{97} +(-641187. - 481660. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2984 q^{9}+O(q^{10}) \) 16 * q - 2984 * q^9	\( 16 q - 2984 q^{9} + 30544 q^{19} - 1736 q^{21} + 70064 q^{31} - 79216 q^{39} - 405120 q^{49} + 858240 q^{51} - 271216 q^{61} + 509880 q^{69} - 1415408 q^{79} - 2396224 q^{81} - 4008224 q^{91} - 5300160 q^{99}+O(q^{100}) \) 16 * q - 2984 * q^9 + 30544 * q^19 - 1736 * q^21 + 70064 * q^31 - 79216 * q^39 - 405120 * q^49 + 858240 * q^51 - 271216 * q^61 + 509880 * q^69 - 1415408 * q^79 - 2396224 * q^81 - 4008224 * q^91 - 5300160 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more