LMFDB - Embedded newform 300.7.b.e.149.14

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			149.14
Root			$-3.23723i$ of defining polynomial
Character	$\chi$	$=$	300.149
Dual form			300.7.b.e.149.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(15.3993 + 22.1779i) q^{3} +437.053i q^{7} +(-254.720 + 683.051i) q^{9} +O(q^{10})$	$q+(15.3993 + 22.1779i) q^{3} +437.053i q^{7} +(-254.720 + 683.051i) q^{9} +1898.54i q^{11} +2670.19i q^{13} -4521.29 q^{17} +9502.28 q^{19} +(-9692.93 + 6730.33i) q^{21} +549.560 q^{23} +(-19071.2 + 4869.36i) q^{27} -43308.7i q^{29} +29588.2 q^{31} +(-42105.6 + 29236.2i) q^{33} -45794.0i q^{37} +(-59219.3 + 41119.2i) q^{39} +114611. i q^{41} +24917.0i q^{43} +18258.6 q^{47} -73366.3 q^{49} +(-69624.9 - 100273. i) q^{51} -174021. q^{53} +(146329. + 210741. i) q^{57} -67872.4i q^{59} +175028. q^{61} +(-298529. - 111326. i) q^{63} +16056.7i q^{67} +(8462.86 + 12188.1i) q^{69} -222821. i q^{71} +620790. i q^{73} -829761. q^{77} -525417. q^{79} +(-401676. - 347974. i) q^{81} +790761. q^{83} +(960496. - 666925. i) q^{87} -1.12565e6i q^{89} -1.16702e6 q^{91} +(455638. + 656204. i) q^{93} -624200. i q^{97} +(-1.29680e6 - 483596. i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	15.3993	+	22.1779i	0.570346	+	0.821405i
$3$	15.3993	+	22.1779i	0.570346	+	0.821405i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$			437.053i			1.27421i	0.770778	+	0.637103i	$0.219868\pi$
$7$			437.053i			1.27421i	−0.770778	+	0.637103i	$0.780132\pi$
$8$		0			0
$9$	−254.720	+	683.051i	−0.349411	+	0.936970i
$10$		0			0
$11$			1898.54i			1.42640i	0.700962	+	0.713199i	$0.252754\pi$
$11$			1898.54i			1.42640i	−0.700962	+	0.713199i	$0.747246\pi$
$12$		0			0
$13$			2670.19i			1.21538i	0.794174	+	0.607691i	$0.207904\pi$
$13$			2670.19i			1.21538i	−0.794174	+	0.607691i	$0.792096\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4521.29			−0.920270			−0.460135	−	0.887849i	$-0.652199\pi$
$17$	−4521.29			−0.920270			−0.460135	+	0.887849i	$0.652199\pi$
$18$		0			0
$19$	9502.28			1.38537			0.692687	−	0.721238i	$-0.256427\pi$
$19$	9502.28			1.38537			0.692687	+	0.721238i	$0.256427\pi$
$20$		0			0
$21$	−9692.93	+	6730.33i	−1.04664	+	0.726739i
$22$		0			0
$23$	549.560			0.0451680			0.0225840	−	0.999745i	$-0.492811\pi$
$23$	549.560			0.0451680			0.0225840	+	0.999745i	$0.492811\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−19071.2	+	4869.36i	−0.968916	+	0.247389i
$28$		0			0
$29$		−	43308.7i		−	1.77575i	−0.460089	−	0.887873i	$-0.652182\pi$
$29$		−	43308.7i		−	1.77575i	0.460089	−	0.887873i	$-0.347818\pi$
$30$		0			0
$31$	29588.2			0.993191			0.496596	−	0.867982i	$-0.334583\pi$
$31$	29588.2			0.993191			0.496596	+	0.867982i	$0.334583\pi$
$32$		0			0
$33$	−42105.6	+	29236.2i	−1.17165	+	0.813540i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	45794.0i		−	0.904074i	−0.891999	−	0.452037i	$-0.850698\pi$
$37$		−	45794.0i		−	0.904074i	0.891999	−	0.452037i	$-0.149302\pi$
$38$		0			0
$39$	−59219.3	+	41119.2i	−0.998320	+	0.693188i
$40$		0			0
$41$			114611.i			1.66294i	0.555570	+	0.831470i	$0.312500\pi$
$41$			114611.i			1.66294i	−0.555570	+	0.831470i	$0.687500\pi$
$42$		0			0
$43$			24917.0i			0.313394i	0.987647	+	0.156697i	$0.0500847\pi$
$43$			24917.0i			0.313394i	−0.987647	+	0.156697i	$0.949915\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	18258.6			0.175863			0.0879313	−	0.996127i	$-0.471974\pi$
$47$	18258.6			0.175863			0.0879313	+	0.996127i	$0.471974\pi$
$48$		0			0
$49$	−73366.3			−0.623603
$50$		0			0
$51$	−69624.9	−	100273.i	−0.524873	−	0.755914i
$52$		0			0
$53$	−174021.			−1.16889			−0.584444	−	0.811434i	$-0.698687\pi$
$53$	−174021.			−1.16889			−0.584444	+	0.811434i	$0.698687\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	146329.	+	210741.i	0.790143	+	1.13795i
$58$		0			0
$59$		−	67872.4i		−	0.330474i	−0.986254	−	0.165237i	$-0.947161\pi$
$59$		−	67872.4i		−	0.330474i	0.986254	−	0.165237i	$-0.0528389\pi$
$60$		0			0
$61$	175028.			0.771112			0.385556	−	0.922685i	$-0.374010\pi$
$61$	175028.			0.771112			0.385556	+	0.922685i	$0.374010\pi$
$62$		0			0
$63$	−298529.	−	111326.i	−1.19389	−	0.445222i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			16056.7i			0.0533865i	0.999644	+	0.0266932i	$0.00849773\pi$
$67$			16056.7i			0.0533865i	−0.999644	+	0.0266932i	$0.991502\pi$
$68$		0			0
$69$	8462.86	+	12188.1i	0.0257614	+	0.0371012i
$70$		0			0
$71$		−	222821.i		−	0.622561i	−0.950318	−	0.311281i	$-0.899242\pi$
$71$		−	222821.i		−	0.622561i	0.950318	−	0.311281i	$-0.100758\pi$
$72$		0			0
$73$			620790.i			1.59579i	0.602796	+	0.797896i	$0.294053\pi$
$73$			620790.i			1.59579i	−0.602796	+	0.797896i	$0.705947\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−829761.			−1.81753
$78$		0			0
$79$	−525417.			−1.06567			−0.532835	−	0.846219i	$-0.678873\pi$
$79$	−525417.			−1.06567			−0.532835	+	0.846219i	$0.678873\pi$
$80$		0			0
$81$	−401676.	−	347974.i	−0.755824	−	0.654775i
$82$		0			0
$83$	790761.			1.38296			0.691482	−	0.722394i	$-0.256958\pi$
$83$	790761.			1.38296			0.691482	+	0.722394i	$0.256958\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	960496.	−	666925.i	1.45861	−	1.01279i
$88$		0			0
$89$		−	1.12565e6i		−	1.59673i	−0.602172	−	0.798366i	$-0.705698\pi$
$89$		−	1.12565e6i		−	1.59673i	0.602172	−	0.798366i	$-0.294302\pi$
$90$		0			0
$91$	−1.16702e6			−1.54865
$92$		0			0
$93$	455638.	+	656204.i	0.566463	+	0.815812i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	624200.i		−	0.683925i	−0.939714	−	0.341963i	$-0.888908\pi$
$97$		−	624200.i		−	0.683925i	0.939714	−	0.341963i	$-0.111092\pi$
$98$		0			0
$99$	−1.29680e6	−	483596.i	−1.33649	−	0.498399i
$100$		0			0
$101$			136558.i			0.132542i	0.997802	+	0.0662709i	$0.0211101\pi$
$101$			136558.i			0.132542i	−0.997802	+	0.0662709i	$0.978890\pi$
$102$		0			0
$103$		−	244469.i		−	0.223724i	−0.993724	−	0.111862i	$-0.964319\pi$
$103$		−	244469.i		−	0.223724i	0.993724	−	0.111862i	$-0.0356814\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.66094e6			1.35582			0.677912	−	0.735143i	$-0.262885\pi$
$107$	1.66094e6			1.35582			0.677912	+	0.735143i	$0.262885\pi$
$108$		0			0
$109$	−1.14498e6			−0.884138			−0.442069	−	0.896981i	$-0.645755\pi$
$109$	−1.14498e6			−0.884138			−0.442069	+	0.896981i	$0.645755\pi$
$110$		0			0
$111$	1.01562e6	−	705198.i	0.742610	−	0.515635i
$112$		0			0
$113$	−1.03300e6			−0.715924			−0.357962	−	0.933736i	$-0.616528\pi$
$113$	−1.03300e6			−0.715924			−0.357962	+	0.933736i	$0.616528\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.82388e6	−	680153.i	−1.13878	−	0.424667i
$118$		0			0
$119$		−	1.97604e6i		−	1.17261i
$120$		0			0
$121$	−1.83288e6			−1.03461
$122$		0			0
$123$	−2.54184e6	+	1.76494e6i	−1.36595	+	0.948451i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			3.16679e6i			1.54600i	0.634408	+	0.772998i	$0.281244\pi$
$127$			3.16679e6i			1.54600i	−0.634408	+	0.772998i	$0.718756\pi$
$128$		0			0
$129$	−552608.	+	383706.i	−0.257423	+	0.178743i
$130$		0			0
$131$		−	2.06944e6i		−	0.920533i	−0.887781	−	0.460266i	$-0.847754\pi$
$131$		−	2.06944e6i		−	0.920533i	0.887781	−	0.460266i	$-0.152246\pi$
$132$		0			0
$133$			4.15300e6i			1.76525i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	520796.			0.202538			0.101269	−	0.994859i	$-0.467710\pi$
$137$	520796.			0.202538			0.101269	+	0.994859i	$0.467710\pi$
$138$		0			0
$139$	−3.89209e6			−1.44923			−0.724616	−	0.689153i	$-0.757983\pi$
$139$	−3.89209e6			−1.44923			−0.724616	+	0.689153i	$0.757983\pi$
$140$		0			0
$141$	281170.	+	404937.i	0.100303	+	0.144454i
$142$		0			0
$143$	−5.06946e6			−1.73362
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−1.12979e6	−	1.62711e6i	−0.355670	−	0.512230i
$148$		0			0
$149$			2.27030e6i			0.686318i	0.939277	+	0.343159i	$0.111497\pi$
$149$			2.27030e6i			0.686318i	−0.939277	+	0.343159i	$0.888503\pi$
$150$		0			0
$151$	1.27718e6			0.370954			0.185477	−	0.982649i	$-0.440617\pi$
$151$	1.27718e6			0.370954			0.185477	+	0.982649i	$0.440617\pi$
$152$		0			0
$153$	1.15166e6	−	3.08827e6i	0.321552	−	0.862265i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			1.40437e6i			0.362897i	0.983400	+	0.181448i	$0.0580786\pi$
$157$			1.40437e6i			0.362897i	−0.983400	+	0.181448i	$0.941921\pi$
$158$		0			0
$159$	−2.67980e6	−	3.85942e6i	−0.666671	−	0.960130i
$160$		0			0
$161$			240187.i			0.0575534i
$162$		0			0
$163$			4.84510e6i			1.11877i	0.828909	+	0.559384i	$0.188962\pi$
$163$			4.84510e6i			1.11877i	−0.828909	+	0.559384i	$0.811038\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	823298.			0.176770			0.0883848	−	0.996086i	$-0.471829\pi$
$167$	823298.			0.176770			0.0883848	+	0.996086i	$0.471829\pi$
$168$		0			0
$169$	−2.30312e6			−0.477152
$170$		0			0
$171$	−2.42043e6	+	6.49054e6i	−0.484065	+	1.29805i
$172$		0			0
$173$	−9.06349e6			−1.75048			−0.875240	−	0.483689i	$-0.839297\pi$
$173$	−9.06349e6			−1.75048			−0.875240	+	0.483689i	$0.839297\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	1.50527e6	−	1.04519e6i	0.271453	−	0.188485i
$178$		0			0
$179$		−	6.68808e6i		−	1.16612i	−0.812430	−	0.583059i	$-0.801856\pi$
$179$		−	6.68808e6i		−	1.16612i	0.812430	−	0.583059i	$-0.198144\pi$
$180$		0			0
$181$	−963367.			−0.162464			−0.0812318	−	0.996695i	$-0.525885\pi$
$181$	−963367.			−0.162464			−0.0812318	+	0.996695i	$0.525885\pi$
$182$		0			0
$183$	2.69531e6	+	3.88175e6i	0.439800	+	0.633395i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	8.58383e6i		−	1.31267i
$188$		0			0
$189$	−2.12817e6	−	8.33511e6i	−0.315225	−	1.23460i
$190$		0			0
$191$			1.37892e6i			0.197896i	0.995093	+	0.0989482i	$0.0315478\pi$
$191$			1.37892e6i			0.197896i	−0.995093	+	0.0989482i	$0.968452\pi$
$192$		0			0
$193$		−	549967.i		−	0.0765006i	−0.999268	−	0.0382503i	$-0.987822\pi$
$193$		−	549967.i		−	0.0765006i	0.999268	−	0.0382503i	$-0.0121784\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4.23680e6			−0.554166			−0.277083	−	0.960846i	$-0.589368\pi$
$197$	−4.23680e6			−0.554166			−0.277083	+	0.960846i	$0.589368\pi$
$198$		0			0
$199$	1.42388e7			1.80682			0.903411	−	0.428775i	$-0.141055\pi$
$199$	1.42388e7			1.80682			0.903411	+	0.428775i	$0.141055\pi$
$200$		0			0
$201$	−356104.	+	247262.i	−0.0438519	+	0.0304488i
$202$		0			0
$203$	1.89282e7			2.26267
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−139984.	+	375377.i	−0.0157822	+	0.0423211i
$208$		0			0
$209$			1.80404e7i			1.97610i
$210$		0			0
$211$	920077.			0.0979437			0.0489719	−	0.998800i	$-0.484406\pi$
$211$	920077.			0.0979437			0.0489719	+	0.998800i	$0.484406\pi$
$212$		0			0
$213$	4.94172e6	−	3.43130e6i	0.511374	−	0.355075i
$214$		0			0
$215$		0			0
$216$		0			0
$217$			1.29316e7i			1.26553i
$218$		0			0
$219$	−1.37678e7	+	9.55976e6i	−1.31079	+	0.910153i
$220$		0			0
$221$		−	1.20727e7i		−	1.11848i
$222$		0			0
$223$		−	1.23626e7i		−	1.11479i	−0.830246	−	0.557397i	$-0.811800\pi$
$223$		−	1.23626e7i		−	1.11479i	0.830246	−	0.557397i	$-0.188200\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	447256.			0.0382365			0.0191183	−	0.999817i	$-0.493914\pi$
$227$	447256.			0.0382365			0.0191183	+	0.999817i	$0.493914\pi$
$228$		0			0
$229$	1.85551e7			1.54510			0.772550	−	0.634954i	$-0.218981\pi$
$229$	1.85551e7			1.54510			0.772550	+	0.634954i	$0.218981\pi$
$230$		0			0
$231$	−1.27778e7	−	1.84024e7i	−1.03662	−	1.49292i
$232$		0			0
$233$	−1.33318e6			−0.105395			−0.0526976	−	0.998611i	$-0.516782\pi$
$233$	−1.33318e6			−0.105395			−0.0526976	+	0.998611i	$0.516782\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−8.09107e6	−	1.16526e7i	−0.607800	−	0.875346i
$238$		0			0
$239$		−	4.84514e6i		−	0.354905i	−0.984129	−	0.177453i	$-0.943214\pi$
$239$		−	4.84514e6i		−	0.354905i	0.984129	−	0.177453i	$-0.0567857\pi$
$240$		0			0
$241$	1.84505e7			1.31813			0.659063	−	0.752088i	$-0.270953\pi$
$241$	1.84505e7			1.31813			0.659063	+	0.752088i	$0.270953\pi$
$242$		0			0
$243$	1.53180e6	−	1.42669e7i	0.106754	−	0.994286i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			2.53729e7i			1.68376i
$248$		0			0
$249$	1.21772e7	+	1.75374e7i	0.788768	+	1.13597i
$250$		0			0
$251$			1.38133e7i			0.873526i	0.899577	+	0.436763i	$0.143875\pi$
$251$			1.38133e7i			0.873526i	−0.899577	+	0.436763i	$0.856125\pi$
$252$		0			0
$253$			1.04336e6i			0.0644276i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.89697e7			1.70665			0.853324	−	0.521381i	$-0.174583\pi$
$257$	2.89697e7			1.70665			0.853324	+	0.521381i	$0.174583\pi$
$258$		0			0
$259$	2.00144e7			1.15198
$260$		0			0
$261$	2.95820e7	+	1.10316e7i	1.66382	+	0.620465i
$262$		0			0
$263$	1.65865e7			0.911776			0.455888	−	0.890037i	$-0.349322\pi$
$263$	1.65865e7			0.911776			0.455888	+	0.890037i	$0.349322\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	2.49645e7	−	1.73342e7i	1.31156	−	0.910690i
$268$		0			0
$269$		−	230441.i		−	0.0118387i	−0.999982	−	0.00591934i	$-0.998116\pi$
$269$		−	230441.i		−	0.0118387i	0.999982	−	0.00591934i	$-0.00188420\pi$
$270$		0			0
$271$	−1.94774e7			−0.978638			−0.489319	−	0.872105i	$-0.662755\pi$
$271$	−1.94774e7			−0.978638			−0.489319	+	0.872105i	$0.662755\pi$
$272$		0			0
$273$	−1.79713e7	−	2.58820e7i	−0.883265	−	1.27207i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			140418.i			0.00660667i	0.999995	+	0.00330334i	$0.00105149\pi$
$277$			140418.i			0.00660667i	−0.999995	+	0.00330334i	$0.998949\pi$
$278$		0			0
$279$	−7.53671e6	+	2.02102e7i	−0.347032	+	0.930590i
$280$		0			0
$281$			1.34994e7i			0.608407i	0.952607	+	0.304203i	$0.0983902\pi$
$281$			1.34994e7i			0.608407i	−0.952607	+	0.304203i	$0.901610\pi$
$282$		0			0
$283$		−	2.01080e7i		−	0.887174i	−0.896231	−	0.443587i	$-0.853706\pi$
$283$		−	2.01080e7i		−	0.887174i	0.896231	−	0.443587i	$-0.146294\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5.00913e7			−2.11893
$288$		0			0
$289$	−3.69552e6			−0.153102
$290$		0			0
$291$	1.38435e7	−	9.61227e6i	0.561779	−	0.390074i
$292$		0			0
$293$	4.37099e7			1.73771			0.868854	−	0.495068i	$-0.164857\pi$
$293$	4.37099e7			1.73771			0.868854	+	0.495068i	$0.164857\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−9.24466e6	−	3.62073e7i	−0.352876	−	1.38206i
$298$		0			0
$299$			1.46743e6i			0.0548964i
$300$		0			0
$301$	−1.08901e7			−0.399329
$302$		0			0
$303$	−3.02857e6	+	2.10290e6i	−0.108870	+	0.0755947i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			4.79809e7i			1.65826i	0.559055	+	0.829131i	$0.311164\pi$
$307$			4.79809e7i			1.65826i	−0.559055	+	0.829131i	$0.688836\pi$
$308$		0			0
$309$	5.42181e6	−	3.76466e6i	0.183768	−	0.127600i
$310$		0			0
$311$			2.28475e7i			0.759553i	0.925078	+	0.379777i	$0.123999\pi$
$311$			2.28475e7i			0.759553i	−0.925078	+	0.379777i	$0.876001\pi$
$312$		0			0
$313$			2.38707e7i			0.778451i	0.921142	+	0.389226i	$0.127257\pi$
$313$			2.38707e7i			0.778451i	−0.921142	+	0.389226i	$0.872743\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.21703e7			−0.382054			−0.191027	−	0.981585i	$-0.561182\pi$
$317$	−1.21703e7			−0.382054			−0.191027	+	0.981585i	$0.561182\pi$
$318$		0			0
$319$	8.22230e7			2.53292
$320$		0			0
$321$	2.55774e7	+	3.68363e7i	0.773289	+	1.11368i
$322$		0			0
$323$	−4.29626e7			−1.27492
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−1.76320e7	−	2.53934e7i	−0.504265	−	0.726235i
$328$		0			0
$329$			7.97997e6i			0.224085i
$330$		0			0
$331$	−3.94247e6			−0.108714			−0.0543569	−	0.998522i	$-0.517311\pi$
$331$	−3.94247e6			−0.108714			−0.0543569	+	0.998522i	$0.517311\pi$
$332$		0			0
$333$	3.12797e7	+	1.16647e7i	0.847090	+	0.315893i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			3.45006e7i			0.901439i	0.892666	+	0.450720i	$0.148833\pi$
$337$			3.45006e7i			0.901439i	−0.892666	+	0.450720i	$0.851167\pi$
$338$		0			0
$339$	−1.59076e7	−	2.29099e7i	−0.408324	−	0.588063i
$340$		0			0
$341$			5.61742e7i			1.41669i
$342$		0			0
$343$			1.93539e7i			0.479607i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5.55712e7			−1.33003			−0.665015	−	0.746830i	$-0.731575\pi$
$347$	−5.55712e7			−1.33003			−0.665015	+	0.746830i	$0.731575\pi$
$348$		0			0
$349$	7.36309e7			1.73214			0.866071	−	0.499921i	$-0.166638\pi$
$349$	7.36309e7			1.73214			0.866071	+	0.499921i	$0.166638\pi$
$350$		0			0
$351$	−1.30021e7	−	5.09237e7i	−0.300672	−	1.17760i
$352$		0			0
$353$	−5.13378e7			−1.16711			−0.583557	−	0.812072i	$-0.698340\pi$
$353$	−5.13378e7			−1.16711			−0.583557	+	0.812072i	$0.698340\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	4.38245e7	−	3.04298e7i	0.963191	−	0.668796i
$358$		0			0
$359$			4.77258e7i			1.03150i	0.856739	+	0.515751i	$0.172487\pi$
$359$			4.77258e7i			1.03150i	−0.856739	+	0.515751i	$0.827513\pi$
$360$		0			0
$361$	4.32475e7			0.919263
$362$		0			0
$363$	−2.82251e7	−	4.06494e7i	−0.590086	−	0.849834i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			8.29285e7i			1.67767i	0.544388	+	0.838834i	$0.316762\pi$
$367$			8.29285e7i			1.67767i	−0.544388	+	0.838834i	$0.683238\pi$
$368$		0			0
$369$	−7.82854e7	−	2.91939e7i	−1.55812	−	0.581049i
$370$		0			0
$371$		−	7.60562e7i		−	1.48941i
$372$		0			0
$373$		−	4.09826e7i		−	0.789719i	−0.918741	−	0.394860i	$-0.870793\pi$
$373$		−	4.09826e7i		−	0.789719i	0.918741	−	0.394860i	$-0.129207\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.15642e8			2.15821
$378$		0			0
$379$	−9.54504e7			−1.75332			−0.876658	−	0.481114i	$-0.840232\pi$
$379$	−9.54504e7			−1.75332			−0.876658	+	0.481114i	$0.840232\pi$
$380$		0			0
$381$	−7.02329e7	+	4.87665e7i	−1.26989	+	0.881753i
$382$		0			0
$383$	4.90198e7			0.872520			0.436260	−	0.899821i	$-0.356303\pi$
$383$	4.90198e7			0.872520			0.436260	+	0.899821i	$0.356303\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.70196e7	−	6.34688e6i	−0.293641	−	0.109503i
$388$		0			0
$389$			6.92192e6i			0.117592i	0.998270	+	0.0587960i	$0.0187262\pi$
$389$			6.92192e6i			0.117592i	−0.998270	+	0.0587960i	$0.981274\pi$
$390$		0			0
$391$	−2.48472e6			−0.0415668
$392$		0			0
$393$	4.58959e7	−	3.18680e7i	0.756130	−	0.525022i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	1.04166e8i		−	1.66478i	−0.554191	−	0.832389i	$-0.686972\pi$
$397$		−	1.04166e8i		−	1.66478i	0.554191	−	0.832389i	$-0.313028\pi$
$398$		0			0
$399$	−9.21049e7	+	6.39535e7i	−1.44999	+	1.00681i
$400$		0			0
$401$		−	1.09926e8i		−	1.70477i	−0.522911	−	0.852387i	$-0.675154\pi$
$401$		−	1.09926e8i		−	1.70477i	0.522911	−	0.852387i	$-0.324846\pi$
$402$		0			0
$403$			7.90061e7i			1.20711i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	8.69416e7			1.28957
$408$		0			0
$409$	9.73956e7			1.42354			0.711770	−	0.702413i	$-0.247894\pi$
$409$	9.73956e7			1.42354			0.711770	+	0.702413i	$0.247894\pi$
$410$		0			0
$411$	8.01992e6	+	1.15502e7i	0.115517	+	0.166365i
$412$		0			0
$413$	2.96639e7			0.421092
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−5.99356e7	−	8.63184e7i	−0.826564	−	1.19041i
$418$		0			0
$419$		−	6.80064e7i		−	0.924501i	−0.886749	−	0.462251i	$-0.847042\pi$
$419$		−	6.80064e7i		−	0.924501i	0.886749	−	0.462251i	$-0.152958\pi$
$420$		0			0
$421$	3.17666e7			0.425720			0.212860	−	0.977083i	$-0.431722\pi$
$421$	3.17666e7			0.425720			0.212860	+	0.977083i	$0.431722\pi$
$422$		0			0
$423$	−4.65083e6	+	1.24715e7i	−0.0614483	+	0.164778i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			7.64964e7i			0.982556i
$428$		0			0
$429$	−7.80663e7	−	1.12430e8i	−0.988762	−	1.42400i
$430$		0			0
$431$		−	8.40240e7i		−	1.04947i	−0.851264	−	0.524737i	$-0.824164\pi$
$431$		−	8.40240e7i		−	1.04947i	0.851264	−	0.524737i	$-0.175836\pi$
$432$		0			0
$433$			8.82768e7i			1.08738i	0.839285	+	0.543692i	$0.182974\pi$
$433$			8.82768e7i			1.08738i	−0.839285	+	0.543692i	$0.817026\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5.22207e6			0.0625747
$438$		0			0
$439$	−1.03991e8			−1.22914			−0.614569	−	0.788863i	$-0.710670\pi$
$439$	−1.03991e8			−1.22914			−0.614569	+	0.788863i	$0.710670\pi$
$440$		0			0
$441$	1.86879e7	−	5.01129e7i	0.217894	−	0.584297i
$442$		0			0
$443$	−1.29638e8			−1.49115			−0.745575	−	0.666422i	$-0.767825\pi$
$443$	−1.29638e8			−1.49115			−0.745575	+	0.666422i	$0.767825\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−5.03506e7	+	3.49612e7i	−0.563745	+	0.391439i
$448$		0			0
$449$		−	9.46554e7i		−	1.04570i	−0.852425	−	0.522849i	$-0.824869\pi$
$449$		−	9.46554e7i		−	1.04570i	0.852425	−	0.522849i	$-0.175131\pi$
$450$		0			0
$451$	−2.17594e8			−2.37201
$452$		0			0
$453$	1.96677e7	+	2.83251e7i	0.211572	+	0.304703i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			5.45336e7i			0.571367i	0.958324	+	0.285684i	$0.0922206\pi$
$457$			5.45336e7i			0.571367i	−0.958324	+	0.285684i	$0.907779\pi$
$458$		0			0
$459$	8.62263e7	−	2.20158e7i	0.891665	−	0.227665i
$460$		0			0
$461$			7.37094e7i			0.752350i	0.926549	+	0.376175i	$0.122761\pi$
$461$			7.37094e7i			0.752350i	−0.926549	+	0.376175i	$0.877239\pi$
$462$		0			0
$463$			1.87082e7i			0.188490i	0.995549	+	0.0942452i	$0.0300438\pi$
$463$			1.87082e7i			0.188490i	−0.995549	+	0.0942452i	$0.969956\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.19481e8			−1.17313			−0.586567	−	0.809901i	$-0.699521\pi$
$467$	−1.19481e8			−1.17313			−0.586567	+	0.809901i	$0.699521\pi$
$468$		0			0
$469$	−7.01762e6			−0.0680254
$470$		0			0
$471$	−3.11461e7	+	2.16264e7i	−0.298085	+	0.206977i
$472$		0			0
$473$	−4.73059e7			−0.447025
$474$		0			0
$475$		0			0
$476$		0			0
$477$	4.43266e7	−	1.18865e8i	0.408422	−	1.09521i
$478$		0			0
$479$		−	1.84244e7i		−	0.167644i	−0.996481	−	0.0838220i	$-0.973287\pi$
$479$		−	1.84244e7i		−	0.167644i	0.996481	−	0.0838220i	$-0.0267127\pi$
$480$		0			0
$481$	1.22279e8			1.09879
$482$		0			0
$483$	−5.32684e6	+	3.69872e6i	−0.0472747	+	0.0328254i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			6.48625e7i			0.561574i	0.959770	+	0.280787i	$0.0905954\pi$
$487$			6.48625e7i			0.561574i	−0.959770	+	0.280787i	$0.909405\pi$
$488$		0			0
$489$	−1.07454e8	+	7.46113e7i	−0.918961	+	0.638085i
$490$		0			0
$491$			1.01202e8i			0.854961i	0.904025	+	0.427480i	$0.140599\pi$
$491$			1.01202e8i			0.854961i	−0.904025	+	0.427480i	$0.859401\pi$
$492$		0			0
$493$			1.95811e8i			1.63417i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	9.73848e7			0.793272
$498$		0			0
$499$	3.08259e7			0.248092			0.124046	−	0.992276i	$-0.460413\pi$
$499$	3.08259e7			0.248092			0.124046	+	0.992276i	$0.460413\pi$
$500$		0			0
$501$	1.26782e7	+	1.82590e7i	0.100820	+	0.145199i
$502$		0			0
$503$	1.31110e8			1.03022			0.515111	−	0.857123i	$-0.327751\pi$
$503$	1.31110e8			1.03022			0.515111	+	0.857123i	$0.327751\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−3.54665e7	−	5.10784e7i	−0.272142	−	0.391935i
$508$		0			0
$509$			1.77883e7i			0.134890i	0.997723	+	0.0674452i	$0.0214848\pi$
$509$			1.77883e7i			0.134890i	−0.997723	+	0.0674452i	$0.978515\pi$
$510$		0			0
$511$	−2.71318e8			−2.03337
$512$		0			0
$513$	−1.81220e8	+	4.62701e7i	−1.34231	+	0.342727i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			3.46646e7i			0.250850i
$518$		0			0
$519$	−1.39572e8	−	2.01009e8i	−0.998379	−	1.43785i
$520$		0			0
$521$			1.90916e8i			1.34999i	0.737823	+	0.674994i	$0.235854\pi$
$521$			1.90916e8i			1.34999i	−0.737823	+	0.674994i	$0.764146\pi$
$522$		0			0
$523$			1.68384e8i			1.17705i	0.808478	+	0.588527i	$0.200292\pi$
$523$			1.68384e8i			1.17705i	−0.808478	+	0.588527i	$0.799708\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.33777e8			−0.914005
$528$		0			0
$529$	−1.47734e8			−0.997960
$530$		0			0
$531$	4.63603e7	+	1.72885e7i	0.309644	+	0.115471i
$532$		0			0
$533$	−3.06035e8			−2.02111
$534$		0			0
$535$		0			0
$536$		0			0
$537$	1.48328e8	−	1.02992e8i	0.957855	−	0.665091i
$538$		0			0
$539$		−	1.39289e8i		−	0.889506i
$540$		0			0
$541$	2.49423e8			1.57523			0.787616	−	0.616167i	$-0.211315\pi$
$541$	2.49423e8			1.57523			0.787616	+	0.616167i	$0.211315\pi$
$542$		0			0
$543$	−1.48352e7	−	2.13655e7i	−0.0926605	−	0.133448i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			1.16411e8i			0.711268i	0.934625	+	0.355634i	$0.115735\pi$
$547$			1.16411e8i			0.711268i	−0.934625	+	0.355634i	$0.884265\pi$
$548$		0			0
$549$	−4.45831e7	+	1.19553e8i	−0.269435	+	0.722508i
$550$		0			0
$551$		−	4.11531e8i		−	2.46007i
$552$		0			0
$553$		−	2.29635e8i		−	1.35788i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.07511e8			−1.20082			−0.600408	−	0.799694i	$-0.704995\pi$
$557$	−2.07511e8			−1.20082			−0.600408	+	0.799694i	$0.704995\pi$
$558$		0			0
$559$	−6.65333e7			−0.380894
$560$		0			0
$561$	1.90371e8	−	1.32185e8i	1.07823	−	0.748677i
$562$		0			0
$563$	−8.47557e7			−0.474945			−0.237473	−	0.971394i	$-0.576319\pi$
$563$	−8.47557e7			−0.474945			−0.237473	+	0.971394i	$0.576319\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	1.52083e8	−	1.75554e8i	0.834318	−	0.963076i
$568$		0			0
$569$			1.84731e8i			1.00278i	0.865223	+	0.501388i	$0.167177\pi$
$569$			1.84731e8i			1.00278i	−0.865223	+	0.501388i	$0.832823\pi$
$570$		0			0
$571$	6.60597e7			0.354837			0.177418	−	0.984136i	$-0.443225\pi$
$571$	6.60597e7			0.354837			0.177418	+	0.984136i	$0.443225\pi$
$572$		0			0
$573$	−3.05815e7	+	2.12344e7i	−0.162553	+	0.112869i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			9.03538e7i			0.470348i	0.971953	+	0.235174i	$0.0755659\pi$
$577$			9.03538e7i			0.470348i	−0.971953	+	0.235174i	$0.924434\pi$
$578$		0			0
$579$	1.21971e7	−	8.46913e6i	0.0628379	−	0.0436318i
$580$		0			0
$581$			3.45604e8i			1.76218i
$582$		0			0
$583$		−	3.30384e8i		−	1.66730i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.74492e8			0.862704			0.431352	−	0.902184i	$-0.358037\pi$
$587$	1.74492e8			0.862704			0.431352	+	0.902184i	$0.358037\pi$
$588$		0			0
$589$	2.81155e8			1.37594
$590$		0			0
$591$	−6.52440e7	−	9.39635e7i	−0.316066	−	0.455194i
$592$		0			0
$593$	2.18826e8			1.04938			0.524692	−	0.851292i	$-0.324180\pi$
$593$	2.18826e8			1.04938			0.524692	+	0.851292i	$0.324180\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	2.19269e8	+	3.15788e8i	1.03051	+	1.48413i
$598$		0			0
$599$		−	3.01081e8i		−	1.40088i	−0.713709	−	0.700442i	$-0.752986\pi$
$599$		−	3.01081e8i		−	1.40088i	0.713709	−	0.700442i	$-0.247014\pi$
$600$		0			0
$601$	−2.54238e8			−1.17116			−0.585582	−	0.810614i	$-0.699134\pi$
$601$	−2.54238e8			−1.17116			−0.585582	+	0.810614i	$0.699134\pi$
$602$		0			0
$603$	−1.09675e7	−	4.08996e6i	−0.0500215	−	0.0186538i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	1.76940e8i		−	0.791153i	−0.918433	−	0.395576i	$-0.870545\pi$
$607$		−	1.76940e8i		−	0.791153i	0.918433	−	0.395576i	$-0.129455\pi$
$608$		0			0
$609$	2.91482e8	+	4.19788e8i	1.29050	+	1.85857i
$610$		0			0
$611$			4.87539e7i			0.213740i
$612$		0			0
$613$			1.16816e8i			0.507132i	0.967318	+	0.253566i	$0.0816034\pi$
$613$			1.16816e8i			0.507132i	−0.967318	+	0.253566i	$0.918397\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.32683e8			−0.564886			−0.282443	−	0.959284i	$-0.591145\pi$
$617$	−1.32683e8			−0.564886			−0.282443	+	0.959284i	$0.591145\pi$
$618$		0			0
$619$	−1.63767e7			−0.0690484			−0.0345242	−	0.999404i	$-0.510992\pi$
$619$	−1.63767e7			−0.0690484			−0.0345242	+	0.999404i	$0.510992\pi$
$620$		0			0
$621$	−1.04807e7	+	2.67601e6i	−0.0437641	+	0.0111741i
$622$		0			0
$623$	4.91967e8			2.03457
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−4.00099e8	+	2.77811e8i	−1.62317	+	1.12706i
$628$		0			0
$629$			2.07048e8i			0.831992i
$630$		0			0
$631$	−3.22391e7			−0.128320			−0.0641600	−	0.997940i	$-0.520437\pi$
$631$	−3.22391e7			−0.128320			−0.0641600	+	0.997940i	$0.520437\pi$
$632$		0			0
$633$	1.41686e7	+	2.04054e7i	0.0558618	+	0.0804514i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	1.95902e8i		−	0.757916i
$638$		0			0
$639$	1.52198e8	+	5.67572e7i	0.583321	+	0.217530i
$640$		0			0
$641$		−	1.81122e8i		−	0.687698i	−0.939025	−	0.343849i	$-0.888269\pi$
$641$		−	1.81122e8i		−	0.687698i	0.939025	−	0.343849i	$-0.111731\pi$
$642$		0			0
$643$			7.65331e6i			0.0287883i	0.999896	+	0.0143942i	$0.00458196\pi$
$643$			7.65331e6i			0.0287883i	−0.999896	+	0.0143942i	$0.995418\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	2.31805e8			0.855875			0.427937	−	0.903808i	$-0.359240\pi$
$647$	2.31805e8			0.855875			0.427937	+	0.903808i	$0.359240\pi$
$648$		0			0
$649$	1.28858e8			0.471388
$650$		0			0
$651$	−2.86796e8	+	1.99138e8i	−1.03951	+	0.721791i
$652$		0			0
$653$	3.95202e8			1.41932			0.709660	−	0.704545i	$-0.248849\pi$
$653$	3.95202e8			1.41932			0.709660	+	0.704545i	$0.248849\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−4.24031e8	−	1.58128e8i	−1.49521	−	0.557587i
$658$		0			0
$659$			5.10657e8i			1.78432i	0.451719	+	0.892160i	$0.350811\pi$
$659$			5.10657e8i			1.78432i	−0.451719	+	0.892160i	$0.649189\pi$
$660$		0			0
$661$	4.10564e8			1.42160			0.710799	−	0.703395i	$-0.248333\pi$
$661$	4.10564e8			1.42160			0.710799	+	0.703395i	$0.248333\pi$
$662$		0			0
$663$	2.67748e8	−	1.85912e8i	0.918724	−	0.637920i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	2.38007e7i		−	0.0802070i
$668$		0			0
$669$	2.74176e8	−	1.90376e8i	0.915696	−	0.635818i
$670$		0			0
$671$			3.32296e8i			1.09991i
$672$		0			0
$673$		−	9.19215e7i		−	0.301559i	−0.988567	−	0.150779i	$-0.951822\pi$
$673$		−	9.19215e7i		−	0.301559i	0.988567	−	0.150779i	$-0.0481783\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.47302e8			0.474726			0.237363	−	0.971421i	$-0.423717\pi$
$677$	1.47302e8			0.474726			0.237363	+	0.971421i	$0.423717\pi$
$678$		0			0
$679$	2.72809e8			0.871462
$680$		0			0
$681$	6.88744e6	+	9.91920e6i	0.0218080	+	0.0314076i
$682$		0			0
$683$	−1.01111e8			−0.317347			−0.158673	−	0.987331i	$-0.550722\pi$
$683$	−1.01111e8			−0.317347			−0.158673	+	0.987331i	$0.550722\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	2.85736e8	+	4.11513e8i	0.881241	+	1.26915i
$688$		0			0
$689$		−	4.64669e8i		−	1.42065i
$690$		0			0
$691$	2.65597e8			0.804987			0.402493	−	0.915423i	$-0.368144\pi$
$691$	2.65597e8			0.804987			0.402493	+	0.915423i	$0.368144\pi$
$692$		0			0
$693$	2.11357e8	−	5.66769e8i	0.635063	−	1.70297i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	5.18191e8i		−	1.53035i
$698$		0			0
$699$	−2.05301e7	−	2.95671e7i	−0.0601117	−	0.0865721i
$700$		0			0
$701$		−	4.56699e8i		−	1.32579i	−0.748711	−	0.662897i	$-0.769327\pi$
$701$		−	4.56699e8i		−	1.32579i	0.748711	−	0.662897i	$-0.230673\pi$
$702$		0			0
$703$		−	4.35148e8i		−	1.25248i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5.96830e7			−0.168886
$708$		0			0
$709$	2.21150e7			0.0620509			0.0310255	−	0.999519i	$-0.490123\pi$
$709$	2.21150e7			0.0620509			0.0310255	+	0.999519i	$0.490123\pi$
$710$		0			0
$711$	1.33834e8	−	3.58886e8i	0.372356	−	0.998500i
$712$		0			0
$713$	1.62605e7			0.0448605
$714$		0			0
$715$		0			0
$716$		0			0
$717$	1.07455e8	−	7.46119e7i	0.291521	−	0.202419i
$718$		0			0
$719$			1.60221e8i			0.431054i	0.976498	+	0.215527i	$0.0691469\pi$
$719$			1.60221e8i			0.431054i	−0.976498	+	0.215527i	$0.930853\pi$
$720$		0			0
$721$	1.06846e8			0.285070
$722$		0			0
$723$	2.84126e8	+	4.09194e8i	0.751788	+	1.08271i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	3.41495e8i		−	0.888753i	−0.895840	−	0.444376i	$-0.853425\pi$
$727$		−	3.41495e8i		−	0.888753i	0.895840	−	0.444376i	$-0.146575\pi$
$728$		0			0
$729$	3.39999e8	−	1.85729e8i	0.877597	−	0.479399i
$730$		0			0
$731$		−	1.12657e8i		−	0.288407i
$732$		0			0
$733$		−	4.06271e8i		−	1.03158i	−0.856714	−	0.515791i	$-0.827498\pi$
$733$		−	4.06271e8i		−	1.03158i	0.856714	−	0.515791i	$-0.172502\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.04842e7			−0.0761503
$738$		0			0
$739$	−1.06192e7			−0.0263123			−0.0131562	−	0.999913i	$-0.504188\pi$
$739$	−1.06192e7			−0.0263123			−0.0131562	+	0.999913i	$0.504188\pi$
$740$		0			0
$741$	−5.62719e8	+	3.90726e8i	−1.38305	+	0.960325i
$742$		0			0
$743$	−5.86011e8			−1.42869			−0.714347	−	0.699791i	$-0.753276\pi$
$743$	−5.86011e8			−1.42869			−0.714347	+	0.699791i	$0.753276\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−2.01423e8	+	5.40130e8i	−0.483222	+	1.29579i
$748$		0			0
$749$			7.25920e8i			1.72760i
$750$		0			0
$751$	4.96057e7			0.117115			0.0585574	−	0.998284i	$-0.481350\pi$
$751$	4.96057e7			0.117115			0.0585574	+	0.998284i	$0.481350\pi$
$752$		0			0
$753$	−3.06350e8	+	2.12715e8i	−0.717518	+	0.498212i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			4.58919e7i			0.105791i	0.998600	+	0.0528955i	$0.0168450\pi$
$757$			4.58919e7i			0.105791i	−0.998600	+	0.0528955i	$0.983155\pi$
$758$		0			0
$759$	−2.31395e7	+	1.60670e7i	−0.0529211	+	0.0367460i
$760$		0			0
$761$			6.06778e8i			1.37682i	0.725323	+	0.688408i	$0.241690\pi$
$761$			6.06778e8i			1.37682i	−0.725323	+	0.688408i	$0.758310\pi$
$762$		0			0
$763$		−	5.00419e8i		−	1.12658i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.81233e8			0.401652
$768$		0			0
$769$	−4.78906e8			−1.05310			−0.526552	−	0.850143i	$-0.676516\pi$
$769$	−4.78906e8			−1.05310			−0.526552	+	0.850143i	$0.676516\pi$
$770$		0			0
$771$	4.46114e8	+	6.42487e8i	0.973380	+	1.40185i
$772$		0			0
$773$	3.49686e7			0.0757076			0.0378538	−	0.999283i	$-0.487948\pi$
$773$	3.49686e7			0.0757076			0.0378538	+	0.999283i	$0.487948\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	3.08209e8	+	4.43878e8i	0.657025	+	0.946239i
$778$		0			0
$779$			1.08907e9i			2.30379i
$780$		0			0
$781$	4.23034e8			0.888020
$782$		0			0
$783$	2.10886e8	+	8.25947e8i	0.439301	+	1.72055i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			3.11411e8i			0.638866i	0.947609	+	0.319433i	$0.103492\pi$
$787$			3.11411e8i			0.638866i	−0.947609	+	0.319433i	$0.896508\pi$
$788$		0			0
$789$	2.55422e8	+	3.67855e8i	0.520028	+	0.748937i
$790$		0			0
$791$		−	4.51478e8i		−	0.912235i
$792$		0			0
$793$			4.67358e8i			0.937195i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−4.48813e8			−0.886525			−0.443262	−	0.896392i	$-0.646179\pi$
$797$	−4.48813e8			−0.886525			−0.443262	+	0.896392i	$0.646179\pi$
$798$		0			0
$799$	−8.25523e7			−0.161841
$800$		0			0
$801$	7.68874e8	+	2.86725e8i	1.49609	+	0.557915i
$802$		0			0
$803$	−1.17859e9			−2.27623
$804$		0			0
$805$		0			0
$806$		0			0
$807$	5.11071e6	−	3.54864e6i	0.00972435	−	0.00675215i
$808$		0			0
$809$		−	4.94261e8i		−	0.933493i	−0.884391	−	0.466746i	$-0.845426\pi$
$809$		−	4.94261e8i		−	0.933493i	0.884391	−	0.466746i	$-0.154574\pi$
$810$		0			0
$811$	−2.89935e8			−0.543549			−0.271774	−	0.962361i	$-0.587610\pi$
$811$	−2.89935e8			−0.543549			−0.271774	+	0.962361i	$0.587610\pi$
$812$		0			0
$813$	−2.99939e8	−	4.31967e8i	−0.558162	−	0.803858i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			2.36769e8i			0.434168i
$818$		0			0
$819$	2.97263e8	−	7.97131e8i	0.541114	−	1.45104i
$820$		0			0
$821$		−	1.20406e8i		−	0.217579i	−0.994065	−	0.108789i	$-0.965303\pi$
$821$		−	1.20406e8i		−	0.217579i	0.994065	−	0.108789i	$-0.0346974\pi$
$822$		0			0
$823$		−	2.61929e8i		−	0.469878i	−0.972010	−	0.234939i	$-0.924511\pi$
$823$		−	2.61929e8i		−	0.469878i	0.972010	−	0.234939i	$-0.0754890\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.59768e8			−0.282471			−0.141236	−	0.989976i	$-0.545107\pi$
$827$	−1.59768e8			−0.282471			−0.141236	+	0.989976i	$0.545107\pi$
$828$		0			0
$829$	−7.51308e7			−0.131873			−0.0659363	−	0.997824i	$-0.521003\pi$
$829$	−7.51308e7			−0.131873			−0.0659363	+	0.997824i	$0.521003\pi$
$830$		0			0
$831$	−3.11417e6	+	2.16234e6i	−0.00542675	+	0.00376809i
$832$		0			0
$833$	3.31710e8			0.573884
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−5.64281e8	+	1.44076e8i	−0.962319	+	0.245705i
$838$		0			0
$839$		−	7.46323e8i		−	1.26369i	−0.775094	−	0.631846i	$-0.782298\pi$
$839$		−	7.46323e8i		−	1.26369i	0.775094	−	0.631846i	$-0.217702\pi$
$840$		0			0
$841$	−1.28082e9			−2.15327
$842$		0			0
$843$	−2.99388e8	+	2.07881e8i	−0.499748	+	0.347002i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	8.01064e8i		−	1.31831i
$848$		0			0
$849$	4.45953e8	−	3.09649e8i	0.728729	−	0.505996i
$850$		0			0
$851$		−	2.51666e7i		−	0.0408352i
$852$		0			0
$853$			7.32618e8i			1.18040i	0.807256	+	0.590201i	$0.200952\pi$
$853$			7.32618e8i			1.18040i	−0.807256	+	0.590201i	$0.799048\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	4.41490e8			0.701420			0.350710	−	0.936484i	$-0.385940\pi$
$857$	4.41490e8			0.701420			0.350710	+	0.936484i	$0.385940\pi$
$858$		0			0
$859$	−8.02777e8			−1.26653			−0.633265	−	0.773935i	$-0.718286\pi$
$859$	−8.02777e8			−1.26653			−0.633265	+	0.773935i	$0.718286\pi$
$860$		0			0
$861$	−7.71373e8	−	1.11092e9i	−1.20852	−	1.74050i
$862$		0			0
$863$	8.01287e8			1.24668			0.623341	−	0.781950i	$-0.285775\pi$
$863$	8.01287e8			1.24668			0.623341	+	0.781950i	$0.285775\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−5.69085e7	−	8.19589e7i	−0.0873213	−	0.125759i
$868$		0			0
$869$		−	9.97522e8i		−	1.52007i
$870$		0			0
$871$	−4.28744e7			−0.0648849
$872$		0			0
$873$	4.26360e8	+	1.58997e8i	0.640817	+	0.238971i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	7.69001e8i		−	1.14006i	−0.821623	−	0.570031i	$-0.806931\pi$
$877$		−	7.69001e8i		−	1.14006i	0.821623	−	0.570031i	$-0.193069\pi$
$878$		0			0
$879$	6.73104e8	+	9.69394e8i	0.991095	+	1.42736i
$880$		0			0
$881$			9.64066e8i			1.40987i	0.709272	+	0.704935i	$0.249024\pi$
$881$			9.64066e8i			1.40987i	−0.709272	+	0.704935i	$0.750976\pi$
$882$		0			0
$883$		−	4.12143e8i		−	0.598639i	−0.954153	−	0.299320i	$-0.903240\pi$
$883$		−	4.12143e8i		−	0.598639i	0.954153	−	0.299320i	$-0.0967597\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.16317e8			−0.739854			−0.369927	−	0.929061i	$-0.620617\pi$
$887$	−5.16317e8			−0.739854			−0.369927	+	0.929061i	$0.620617\pi$
$888$		0			0
$889$	−1.38406e9			−1.96992
$890$		0			0
$891$	6.60641e8	−	7.62596e8i	0.933969	−	1.07811i
$892$		0			0
$893$	1.73498e8			0.243636
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−3.25446e7	+	2.25975e7i	−0.0450922	+	0.0313099i
$898$		0			0
$899$		−	1.28142e9i		−	1.76366i
$900$		0			0
$901$	7.86798e8			1.07569
$902$		0			0
$903$	−1.67700e8	−	2.41519e8i	−0.227756	−	0.328011i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	1.20289e9i		−	1.61214i	−0.591817	−	0.806072i	$-0.701589\pi$
$907$		−	1.20289e9i		−	1.61214i	0.591817	−	0.806072i	$-0.298411\pi$
$908$		0			0
$909$	−9.32760e7	−	3.47841e7i	−0.124188	−	0.0463115i
$910$		0			0
$911$			2.04819e8i			0.270903i	0.990784	+	0.135452i	$0.0432485\pi$
$911$			2.04819e8i			0.270903i	−0.990784	+	0.135452i	$0.956751\pi$
$912$		0			0
$913$			1.50129e9i			1.97266i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	9.04456e8			1.17295
$918$		0			0
$919$	−1.07525e9			−1.38536			−0.692682	−	0.721243i	$-0.743571\pi$
$919$	−1.07525e9			−1.38536			−0.692682	+	0.721243i	$0.743571\pi$
$920$		0			0
$921$	−1.06412e9	+	7.38874e8i	−1.36210	+	0.945783i
$922$		0			0
$923$	5.94976e8			0.756649
$924$		0			0
$925$		0			0
$926$		0			0
$927$	1.66985e8	+	6.22712e7i	0.209622	+	0.0781714i
$928$		0			0
$929$			3.27972e8i			0.409063i	0.978860	+	0.204532i	$0.0655671\pi$
$929$			3.27972e8i			0.409063i	−0.978860	+	0.204532i	$0.934433\pi$
$930$		0			0
$931$	−6.97147e8			−0.863924
$932$		0			0
$933$	−5.06711e8	+	3.51837e8i	−0.623900	+	0.433208i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			1.00558e8i			0.122235i	0.998131	+	0.0611177i	$0.0194665\pi$
$937$			1.00558e8i			0.122235i	−0.998131	+	0.0611177i	$0.980533\pi$
$938$		0			0
$939$	−5.29402e8	+	3.67592e8i	−0.639423	+	0.443987i
$940$		0			0
$941$		−	1.31841e9i		−	1.58227i	−0.611642	−	0.791134i	$-0.709491\pi$
$941$		−	1.31841e9i		−	1.58227i	0.611642	−	0.791134i	$-0.290509\pi$
$942$		0			0
$943$			6.29858e7i			0.0751117i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	7.08841e8			0.834640			0.417320	−	0.908760i	$-0.362969\pi$
$947$	7.08841e8			0.834640			0.417320	+	0.908760i	$0.362969\pi$
$948$		0			0
$949$	−1.65763e9			−1.93949
$950$		0			0
$951$	−1.87415e8	−	2.69913e8i	−0.217903	−	0.313821i
$952$		0			0
$953$	−4.08239e8			−0.471668			−0.235834	−	0.971793i	$-0.575782\pi$
$953$	−4.08239e8			−0.471668			−0.235834	+	0.971793i	$0.575782\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	1.26618e9	+	1.82354e9i	1.44464	+	2.08055i
$958$		0			0
$959$			2.27615e8i			0.258075i
$960$		0			0
$961$	−1.20442e7			−0.0135709
$962$		0			0
$963$	−4.23076e8	+	1.13451e9i	−0.473740	+	1.27037i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			6.53392e8i			0.722594i	0.932451	+	0.361297i	$0.117666\pi$
$967$			6.53392e8i			0.722594i	−0.932451	+	0.361297i	$0.882334\pi$
$968$		0			0
$969$	−6.61595e8	−	9.52821e8i	−0.727145	−	1.04722i
$970$		0			0
$971$			2.53885e8i			0.277319i	0.990340	+	0.138659i	$0.0442793\pi$
$971$			2.53885e8i			0.277319i	−0.990340	+	0.138659i	$0.955721\pi$
$972$		0			0
$973$		−	1.70105e9i		−	1.84662i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−4.70410e8			−0.504420			−0.252210	−	0.967672i	$-0.581157\pi$
$977$	−4.70410e8			−0.504420			−0.252210	+	0.967672i	$0.581157\pi$
$978$		0			0
$979$	2.13708e9			2.27758
$980$		0			0
$981$	2.91651e8	−	7.82083e8i	0.308928	−	0.828411i
$982$		0			0
$983$	5.68355e8			0.598355			0.299178	−	0.954197i	$-0.403288\pi$
$983$	5.68355e8			0.598355			0.299178	+	0.954197i	$0.403288\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−1.76979e8	+	1.22886e8i	−0.184065	+	0.127806i
$988$		0			0
$989$			1.36934e7i			0.0141554i
$990$		0			0
$991$	−3.44684e8			−0.354161			−0.177080	−	0.984196i	$-0.556665\pi$
$991$	−3.44684e8			−0.354161			−0.177080	+	0.984196i	$0.556665\pi$
$992$		0			0
$993$	−6.07115e7	−	8.74358e7i	−0.0620045	−	0.0892980i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	4.23493e8i		−	0.427327i	−0.976907	−	0.213664i	$-0.931460\pi$
$997$		−	4.23493e8i		−	0.427327i	0.976907	−	0.213664i	$-0.0685397\pi$
$998$		0			0
$999$	2.22988e8	+	8.73346e8i	0.223658	+	0.875972i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.7.b.e.149.14		16
3.2	odd	2	inner	300.7.b.e.149.4		16
5.2	odd	4		300.7.g.h.101.5		8
5.3	odd	4		60.7.g.a.41.4	yes	8
5.4	even	2	inner	300.7.b.e.149.3		16
15.2	even	4		300.7.g.h.101.6		8
15.8	even	4		60.7.g.a.41.3	✓	8
15.14	odd	2	inner	300.7.b.e.149.13		16
20.3	even	4		240.7.l.c.161.5		8
60.23	odd	4		240.7.l.c.161.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.7.g.a.41.3	✓	8	15.8	even	4
60.7.g.a.41.4	yes	8	5.3	odd	4
240.7.l.c.161.5		8	20.3	even	4
240.7.l.c.161.6		8	60.23	odd	4
300.7.b.e.149.3		16	5.4	even	2	inner
300.7.b.e.149.4		16	3.2	odd	2	inner
300.7.b.e.149.13		16	15.14	odd	2	inner
300.7.b.e.149.14		16	1.1	even	1	trivial
300.7.g.h.101.5		8	5.2	odd	4
300.7.g.h.101.6		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)

\(f(q)\)	\(=\)	\(q+(15.3993 + 22.1779i) q^{3} +437.053i q^{7} +(-254.720 + 683.051i) q^{9} +O(q^{10})\)	\(q+(15.3993 + 22.1779i) q^{3} +437.053i q^{7} +(-254.720 + 683.051i) q^{9} +1898.54i q^{11} +2670.19i q^{13} -4521.29 q^{17} +9502.28 q^{19} +(-9692.93 + 6730.33i) q^{21} +549.560 q^{23} +(-19071.2 + 4869.36i) q^{27} -43308.7i q^{29} +29588.2 q^{31} +(-42105.6 + 29236.2i) q^{33} -45794.0i q^{37} +(-59219.3 + 41119.2i) q^{39} +114611. i q^{41} +24917.0i q^{43} +18258.6 q^{47} -73366.3 q^{49} +(-69624.9 - 100273. i) q^{51} -174021. q^{53} +(146329. + 210741. i) q^{57} -67872.4i q^{59} +175028. q^{61} +(-298529. - 111326. i) q^{63} +16056.7i q^{67} +(8462.86 + 12188.1i) q^{69} -222821. i q^{71} +620790. i q^{73} -829761. q^{77} -525417. q^{79} +(-401676. - 347974. i) q^{81} +790761. q^{83} +(960496. - 666925. i) q^{87} -1.12565e6i q^{89} -1.16702e6 q^{91} +(455638. + 656204. i) q^{93} -624200. i q^{97} +(-1.29680e6 - 483596. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2984 q^{9}+O(q^{10}) \) 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

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