LMFDB - Embedded newform 300.11.b.c.149.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,11,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, 11, names="a")

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

chi := DirichletCharacter("300.149");

S:= CuspForms(chi, 11);

N := Newforms(S);

Level:	$ N $	$=$	$ 300 = 2^{2} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 11 $
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:	$190.607175802$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{35})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 17x^{2} + 81 $ x^4 - 17*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.1
Root			$-2.95804 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	300.149
Dual form			300.11.b.c.149.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-212.979 - 117.000i) q^{3} -10318.0i q^{7} +(31671.0 + 49837.1i) q^{9} +O(q^{10})$	$q+(-212.979 - 117.000i) q^{3} -10318.0i q^{7} +(31671.0 + 49837.1i) q^{9} -292633. i q^{11} +256822. i q^{13} -557153. q^{17} -3.19611e6 q^{19} +(-1.20721e6 + 2.19752e6i) q^{21} +8.35474e6 q^{23} +(-914318. - 1.43197e7i) q^{27} +3.14906e7i q^{29} +2.31410e7 q^{31} +(-3.42381e7 + 6.23246e7i) q^{33} +2.97979e7i q^{37} +(3.00482e7 - 5.46977e7i) q^{39} -948182. i q^{41} -2.47523e8i q^{43} -3.29553e8 q^{47} +1.76014e8 q^{49} +(1.18662e8 + 6.51869e7i) q^{51} -5.51291e8 q^{53} +(6.80703e8 + 3.73944e8i) q^{57} +3.59094e8i q^{59} -1.05484e9 q^{61} +(5.14219e8 - 3.26781e8i) q^{63} -3.61186e8i q^{67} +(-1.77938e9 - 9.77504e8i) q^{69} +9.31536e8i q^{71} -3.74437e8i q^{73} -3.01939e9 q^{77} +1.13914e8 q^{79} +(-1.48068e9 + 3.15678e9i) q^{81} -4.91092e9 q^{83} +(3.68440e9 - 6.70684e9i) q^{87} +3.28204e9i q^{89} +2.64989e9 q^{91} +(-4.92854e9 - 2.70750e9i) q^{93} +2.80992e9i q^{97} +(1.45840e10 - 9.26798e9i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 126684 q^{9}+O(q^{10}) $ 4 * q + 126684 * q^9	$ 4 q + 126684 q^{9} - 12784424 q^{19} - 4828824 q^{21} + 92563976 q^{31} + 120192696 q^{39} + 704056500 q^{49} + 474647040 q^{51} - 4219359064 q^{61} - 7117528320 q^{69} + 455657848 q^{79} - 5922719676 q^{81} + 10599557584 q^{91} + 58335863040 q^{99}+O(q^{100}) $ 4 * q + 126684 * q^9 - 12784424 * q^19 - 4828824 * q^21 + 92563976 * q^31 + 120192696 * q^39 + 704056500 * q^49 + 474647040 * q^51 - 4219359064 * q^61 - 7117528320 * q^69 + 455657848 * q^79 - 5922719676 * q^81 + 10599557584 * q^91 + 58335863040 * 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
$3$	−212.979	−	117.000i	−0.876456	−	0.481481i
$3$	−212.979	−	117.000i	−0.876456	−	0.481481i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	10318.0i		−	0.613911i	−0.951724	−	0.306955i	$-0.900690\pi$
$7$		−	10318.0i		−	0.613911i	0.951724	−	0.306955i	$-0.0993103\pi$
$8$		0			0
$9$	31671.0	+	49837.1i	0.536351	+	0.843995i
$10$		0			0
$11$		−	292633.i		−	1.81702i	−0.417863	−	0.908510i	$-0.637221\pi$
$11$		−	292633.i		−	1.81702i	0.417863	−	0.908510i	$-0.362779\pi$
$12$		0			0
$13$			256822.i			0.691696i	0.938290	+	0.345848i	$0.112409\pi$
$13$			256822.i			0.691696i	−0.938290	+	0.345848i	$0.887591\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−557153.			−0.392401			−0.196200	−	0.980564i	$-0.562860\pi$
$17$	−557153.			−0.392401			−0.196200	+	0.980564i	$0.562860\pi$
$18$		0			0
$19$	−3.19611e6			−1.29078			−0.645391	−	0.763852i	$-0.723306\pi$
$19$	−3.19611e6			−1.29078			−0.645391	+	0.763852i	$0.723306\pi$
$20$		0			0
$21$	−1.20721e6	+	2.19752e6i	−0.295587	+	0.538066i
$22$		0			0
$23$	8.35474e6			1.29806			0.649028	−	0.760764i	$-0.275176\pi$
$23$	8.35474e6			1.29806			0.649028	+	0.760764i	$0.275176\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−914318.	−	1.43197e7i	−0.0637204	−	0.997968i
$28$		0			0
$29$			3.14906e7i			1.53529i	0.640873	+	0.767647i	$0.278572\pi$
$29$			3.14906e7i			1.53529i	−0.640873	+	0.767647i	$0.721428\pi$
$30$		0			0
$31$	2.31410e7			0.808302			0.404151	−	0.914692i	$-0.367567\pi$
$31$	2.31410e7			0.808302			0.404151	+	0.914692i	$0.367567\pi$
$32$		0			0
$33$	−3.42381e7	+	6.23246e7i	−0.874862	+	1.59254i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			2.97979e7i			0.429712i	0.976646	+	0.214856i	$0.0689283\pi$
$37$			2.97979e7i			0.429712i	−0.976646	+	0.214856i	$0.931072\pi$
$38$		0			0
$39$	3.00482e7	−	5.46977e7i	0.333039	−	0.606242i
$40$		0			0
$41$		−	948182.i		−	0.00818413i	−0.999992	−	0.00409206i	$-0.998697\pi$
$41$		−	948182.i		−	0.00818413i	0.999992	−	0.00409206i	$-0.00130255\pi$
$42$		0			0
$43$		−	2.47523e8i		−	1.68373i	−0.539687	−	0.841866i	$-0.681457\pi$
$43$		−	2.47523e8i		−	1.68373i	0.539687	−	0.841866i	$-0.318543\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.29553e8			−1.43693			−0.718466	−	0.695562i	$-0.755156\pi$
$47$	−3.29553e8			−1.43693			−0.718466	+	0.695562i	$0.755156\pi$
$48$		0			0
$49$	1.76014e8			0.623113
$50$		0			0
$51$	1.18662e8	+	6.51869e7i	0.343922	+	0.188934i
$52$		0			0
$53$	−5.51291e8			−1.31826			−0.659131	−	0.752028i	$-0.729076\pi$
$53$	−5.51291e8			−1.31826			−0.659131	+	0.752028i	$0.729076\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	6.80703e8	+	3.73944e8i	1.13131	+	0.621488i
$58$		0			0
$59$			3.59094e8i			0.502282i	0.967950	+	0.251141i	$0.0808058\pi$
$59$			3.59094e8i			0.502282i	−0.967950	+	0.251141i	$0.919194\pi$
$60$		0			0
$61$	−1.05484e9			−1.24893			−0.624464	−	0.781054i	$-0.714682\pi$
$61$	−1.05484e9			−1.24893			−0.624464	+	0.781054i	$0.714682\pi$
$62$		0			0
$63$	5.14219e8	−	3.26781e8i	0.518138	−	0.329272i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	3.61186e8i		−	0.267521i	−0.991014	−	0.133760i	$-0.957295\pi$
$67$		−	3.61186e8i		−	0.267521i	0.991014	−	0.133760i	$-0.0427052\pi$
$68$		0			0
$69$	−1.77938e9	−	9.77504e8i	−1.13769	−	0.624990i
$70$		0			0
$71$			9.31536e8i			0.516307i	0.966104	+	0.258154i	$0.0831140\pi$
$71$			9.31536e8i			0.516307i	−0.966104	+	0.258154i	$0.916886\pi$
$72$		0			0
$73$		−	3.74437e8i		−	0.180620i	−0.995914	−	0.0903098i	$-0.971214\pi$
$73$		−	3.74437e8i		−	0.180620i	0.995914	−	0.0903098i	$-0.0287857\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.01939e9			−1.11549
$78$		0			0
$79$	1.13914e8			0.0370206			0.0185103	−	0.999829i	$-0.494108\pi$
$79$	1.13914e8			0.0370206			0.0185103	+	0.999829i	$0.494108\pi$
$80$		0			0
$81$	−1.48068e9	+	3.15678e9i	−0.424655	+	0.905355i
$82$		0			0
$83$	−4.91092e9			−1.24673			−0.623365	−	0.781931i	$-0.714235\pi$
$83$	−4.91092e9			−1.24673			−0.623365	+	0.781931i	$0.714235\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	3.68440e9	−	6.70684e9i	0.739215	−	1.34562i
$88$		0			0
$89$			3.28204e9i			0.587752i	0.955844	+	0.293876i	$0.0949452\pi$
$89$			3.28204e9i			0.587752i	−0.955844	+	0.293876i	$0.905055\pi$
$90$		0			0
$91$	2.64989e9			0.424640
$92$		0			0
$93$	−4.92854e9	−	2.70750e9i	−0.708441	−	0.389182i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			2.80992e9i			0.327216i	0.986525	+	0.163608i	$0.0523133\pi$
$97$			2.80992e9i			0.327216i	−0.986525	+	0.163608i	$0.947687\pi$
$98$		0			0
$99$	1.45840e10	−	9.26798e9i	1.53356	−	0.974561i
$100$		0			0
$101$			1.28917e10i			1.22660i	0.789849	+	0.613301i	$0.210159\pi$
$101$			1.28917e10i			1.22660i	−0.789849	+	0.613301i	$0.789841\pi$
$102$		0			0
$103$		−	1.51220e10i		−	1.30443i	−0.758032	−	0.652217i	$-0.773839\pi$
$103$		−	1.51220e10i		−	1.30443i	0.758032	−	0.652217i	$-0.226161\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.32160e10			−0.942285			−0.471143	−	0.882057i	$-0.656158\pi$
$107$	−1.32160e10			−0.942285			−0.471143	+	0.882057i	$0.656158\pi$
$108$		0			0
$109$	−1.39846e9			−0.0908901			−0.0454451	−	0.998967i	$-0.514471\pi$
$109$	−1.39846e9			−0.0908901			−0.0454451	+	0.998967i	$0.514471\pi$
$110$		0			0
$111$	3.48636e9	−	6.34633e9i	0.206898	−	0.376624i
$112$		0			0
$113$	−1.15787e10			−0.628448			−0.314224	−	0.949349i	$-0.601744\pi$
$113$	−1.15787e10			−0.628448			−0.314224	+	0.949349i	$0.601744\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.27993e10	+	8.13381e9i	−0.583788	+	0.370992i
$118$		0			0
$119$			5.74870e9i			0.240899i
$120$		0			0
$121$	−5.96966e10			−2.30156
$122$		0			0
$123$	−1.10937e8	+	2.01943e8i	−0.00394051	+	0.00717303i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			8.89747e8i			0.0269307i	0.999909	+	0.0134654i	$0.00428628\pi$
$127$			8.89747e8i			0.0269307i	−0.999909	+	0.0134654i	$0.995714\pi$
$128$		0			0
$129$	−2.89602e10	+	5.27171e10i	−0.810686	+	1.47572i
$130$		0			0
$131$			1.05561e10i			0.273619i	0.990597	+	0.136809i	$0.0436848\pi$
$131$			1.05561e10i			0.273619i	−0.990597	+	0.136809i	$0.956315\pi$
$132$		0			0
$133$			3.29774e10i			0.792426i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	3.96577e10			0.821722			0.410861	−	0.911698i	$-0.365228\pi$
$137$	3.96577e10			0.821722			0.410861	+	0.911698i	$0.365228\pi$
$138$		0			0
$139$	−5.42044e9			−0.104463			−0.0522313	−	0.998635i	$-0.516633\pi$
$139$	−5.42044e9			−0.104463			−0.0522313	+	0.998635i	$0.516633\pi$
$140$		0			0
$141$	7.01879e10	+	3.85577e10i	1.25941	+	0.691856i
$142$		0			0
$143$	7.51546e10			1.25683
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−3.74873e10	−	2.05937e10i	−0.546132	−	0.300018i
$148$		0			0
$149$		−	7.71177e10i		−	1.05008i	−0.851077	−	0.525040i	$-0.824050\pi$
$149$		−	7.71177e10i		−	1.05008i	0.851077	−	0.525040i	$-0.175950\pi$
$150$		0			0
$151$	7.48173e10			0.953053			0.476527	−	0.879160i	$-0.341896\pi$
$151$	7.48173e10			0.953053			0.476527	+	0.879160i	$0.341896\pi$
$152$		0			0
$153$	−1.76456e10	−	2.77669e10i	−0.210465	−	0.331184i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			2.06739e10i			0.216732i	0.994111	+	0.108366i	$0.0345619\pi$
$157$			2.06739e10i			0.216732i	−0.994111	+	0.108366i	$0.965438\pi$
$158$		0			0
$159$	1.17413e11	+	6.45011e10i	1.15540	+	0.634719i
$160$		0			0
$161$		−	8.62042e10i		−	0.796891i
$162$		0			0
$163$		−	9.57605e10i		−	0.832239i	−0.909310	−	0.416120i	$-0.863390\pi$
$163$		−	9.57605e10i		−	0.832239i	0.909310	−	0.416120i	$-0.136610\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.46483e11			1.12773			0.563866	−	0.825866i	$-0.309313\pi$
$167$	1.46483e11			1.12773			0.563866	+	0.825866i	$0.309313\pi$
$168$		0			0
$169$	7.19010e10			0.521556
$170$		0			0
$171$	−1.01224e11	−	1.59285e11i	−0.692313	−	1.08941i
$172$		0			0
$173$	2.30251e11			1.48583			0.742917	−	0.669383i	$-0.233442\pi$
$173$	2.30251e11			1.48583			0.742917	+	0.669383i	$0.233442\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	4.20140e10	−	7.64794e10i	0.241840	−	0.440229i
$178$		0			0
$179$			2.27901e11i			1.24017i	0.784536	+	0.620084i	$0.212901\pi$
$179$			2.27901e11i			1.24017i	−0.784536	+	0.620084i	$0.787099\pi$
$180$		0			0
$181$	3.45325e11			1.77761			0.888803	−	0.458290i	$-0.151538\pi$
$181$	3.45325e11			1.77761			0.888803	+	0.458290i	$0.151538\pi$
$182$		0			0
$183$	2.24659e11	+	1.23416e11i	1.09463	+	0.601336i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			1.63041e11i			0.713000i
$188$		0			0
$189$	−1.47751e11	+	9.43394e9i	−0.612663	+	0.0391187i
$190$		0			0
$191$			2.05287e11i			0.807596i	0.914848	+	0.403798i	$0.132310\pi$
$191$			2.05287e11i			0.807596i	−0.914848	+	0.403798i	$0.867690\pi$
$192$		0			0
$193$			1.39964e11i			0.522674i	0.965248	+	0.261337i	$0.0841634\pi$
$193$			1.39964e11i			0.522674i	−0.965248	+	0.261337i	$0.915837\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4.25733e11			−1.43485			−0.717425	−	0.696636i	$-0.754679\pi$
$197$	−4.25733e11			−1.43485			−0.717425	+	0.696636i	$0.754679\pi$
$198$		0			0
$199$	8.32985e10			0.266914			0.133457	−	0.991055i	$-0.457392\pi$
$199$	8.32985e10			0.266914			0.133457	+	0.991055i	$0.457392\pi$
$200$		0			0
$201$	−4.22588e10	+	7.69250e10i	−0.128806	+	0.234470i
$202$		0			0
$203$	3.24920e11			0.942533
$204$		0			0
$205$		0			0
$206$		0			0
$207$	2.64603e11	+	4.16375e11i	0.696214	+	1.09555i
$208$		0			0
$209$			9.35286e11i			2.34538i
$210$		0			0
$211$	−6.18669e11			−1.47927			−0.739633	−	0.673011i	$-0.765001\pi$
$211$	−6.18669e11			−1.47927			−0.739633	+	0.673011i	$0.765001\pi$
$212$		0			0
$213$	1.08990e11	−	1.98398e11i	0.248592	−	0.452521i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	2.38769e11i		−	0.496225i
$218$		0			0
$219$	−4.38092e10	+	7.97473e10i	−0.0869650	+	0.158305i
$220$		0			0
$221$		−	1.43089e11i		−	0.271422i
$222$		0			0
$223$		−	8.45523e10i		−	0.153321i	−0.997057	−	0.0766604i	$-0.975574\pi$
$223$		−	8.45523e10i		−	0.153321i	0.997057	−	0.0766604i	$-0.0244257\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3.28465e11			−0.544954			−0.272477	−	0.962162i	$-0.587843\pi$
$227$	−3.28465e11			−0.544954			−0.272477	+	0.962162i	$0.587843\pi$
$228$		0			0
$229$	1.06468e12			1.69060			0.845302	−	0.534289i	$-0.179421\pi$
$229$	1.06468e12			1.69060			0.845302	+	0.534289i	$0.179421\pi$
$230$		0			0
$231$	6.43066e11	+	3.53268e11i	0.977677	+	0.537087i
$232$		0			0
$233$	2.81192e11			0.409471			0.204736	−	0.978817i	$-0.434367\pi$
$233$	2.81192e11			0.409471			0.204736	+	0.978817i	$0.434367\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−2.42614e10	−	1.33280e10i	−0.0324469	−	0.0178247i
$238$		0			0
$239$			4.81414e10i			0.0617347i	0.999523	+	0.0308674i	$0.00982695\pi$
$239$			4.81414e10i			0.0617347i	−0.999523	+	0.0308674i	$0.990173\pi$
$240$		0			0
$241$	1.09510e12			1.34700			0.673502	−	0.739186i	$-0.264789\pi$
$241$	1.09510e12			1.34700			0.673502	+	0.739186i	$0.264789\pi$
$242$		0			0
$243$	6.84697e11	−	4.99088e11i	0.808103	−	0.589041i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	8.20830e11i		−	0.892830i
$248$		0			0
$249$	1.04592e12	+	5.74577e11i	1.09270	+	0.600277i
$250$		0			0
$251$			1.10846e12i			1.11264i	0.830969	+	0.556318i	$0.187786\pi$
$251$			1.10846e12i			1.11264i	−0.830969	+	0.556318i	$0.812214\pi$
$252$		0			0
$253$		−	2.44487e12i		−	2.35859i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.66907e10			−0.0505646			−0.0252823	−	0.999680i	$-0.508048\pi$
$257$	−5.66907e10			−0.0505646			−0.0252823	+	0.999680i	$0.508048\pi$
$258$		0			0
$259$	3.07455e11			0.263805
$260$		0			0
$261$	−1.56940e12	+	9.97340e11i	−1.29578	+	0.823456i
$262$		0			0
$263$	−1.24412e11			−0.0988740			−0.0494370	−	0.998777i	$-0.515743\pi$
$263$	−1.24412e11			−0.0988740			−0.0494370	+	0.998777i	$0.515743\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	3.83999e11	−	6.99005e11i	0.282992	−	0.515139i
$268$		0			0
$269$			2.14333e12i			1.52170i	0.648930	+	0.760848i	$0.275217\pi$
$269$			2.14333e12i			1.52170i	−0.648930	+	0.760848i	$0.724783\pi$
$270$		0			0
$271$	5.35797e11			0.366568			0.183284	−	0.983060i	$-0.441327\pi$
$271$	5.35797e11			0.366568			0.183284	+	0.983060i	$0.441327\pi$
$272$		0			0
$273$	−5.64370e11	−	3.10037e11i	−0.372178	−	0.204456i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			1.17533e12i			0.720710i	0.932815	+	0.360355i	$0.117344\pi$
$277$			1.17533e12i			0.720710i	−0.932815	+	0.360355i	$0.882656\pi$
$278$		0			0
$279$	7.32898e11	+	1.15328e12i	0.433534	+	0.682203i
$280$		0			0
$281$			1.99322e12i			1.13769i	0.822445	+	0.568845i	$0.192610\pi$
$281$			1.99322e12i			1.13769i	−0.822445	+	0.568845i	$0.807390\pi$
$282$		0			0
$283$			1.41547e12i			0.779773i	0.920863	+	0.389886i	$0.127486\pi$
$283$			1.41547e12i			0.779773i	−0.920863	+	0.389886i	$0.872514\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−9.78334e9			−0.00502432
$288$		0			0
$289$	−1.70557e12			−0.846022
$290$		0			0
$291$	3.28760e11	−	5.98453e11i	0.157549	−	0.286791i
$292$		0			0
$293$	3.38242e12			1.56635			0.783176	−	0.621800i	$-0.213598\pi$
$293$	3.38242e12			1.56635			0.783176	+	0.621800i	$0.213598\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−4.19043e12	+	2.67560e11i	−1.81333	+	0.115781i
$298$		0			0
$299$			2.14568e12i			0.897861i
$300$		0			0
$301$	−2.55394e12			−1.03366
$302$		0			0
$303$	1.50833e12	−	2.74566e12i	0.590586	−	1.07506i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			2.72469e12i			0.999138i	0.866274	+	0.499569i	$0.166508\pi$
$307$			2.72469e12i			0.999138i	−0.866274	+	0.499569i	$0.833492\pi$
$308$		0			0
$309$	−1.76927e12	+	3.22066e12i	−0.628061	+	1.14328i
$310$		0			0
$311$			2.09948e12i			0.721622i	0.932639	+	0.360811i	$0.117500\pi$
$311$			2.09948e12i			0.721622i	−0.932639	+	0.360811i	$0.882500\pi$
$312$		0			0
$313$		−	2.15746e12i		−	0.718160i	−0.933307	−	0.359080i	$-0.883090\pi$
$313$		−	2.15746e12i		−	0.718160i	0.933307	−	0.359080i	$-0.116910\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.52312e12			0.788210			0.394105	−	0.919065i	$-0.371055\pi$
$317$	2.52312e12			0.788210			0.394105	+	0.919065i	$0.371055\pi$
$318$		0			0
$319$	9.21520e12			2.78966
$320$		0			0
$321$	2.81474e12	+	1.54628e12i	0.825872	+	0.453693i
$322$		0			0
$323$	1.78072e12			0.506504
$324$		0			0
$325$		0			0
$326$		0			0
$327$	2.97842e11	+	1.63619e11i	0.0796612	+	0.0437619i
$328$		0			0
$329$			3.40033e12i			0.882148i
$330$		0			0
$331$	2.23406e12			0.562283			0.281142	−	0.959666i	$-0.409287\pi$
$331$	2.23406e12			0.562283			0.281142	+	0.959666i	$0.409287\pi$
$332$		0			0
$333$	−1.48504e12	+	9.43731e11i	−0.362675	+	0.230477i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	5.32046e12i		−	1.22405i	−0.790838	−	0.612026i	$-0.790355\pi$
$337$		−	5.32046e12i		−	1.22405i	0.790838	−	0.612026i	$-0.209645\pi$
$338$		0			0
$339$	2.46603e12	+	1.35471e12i	0.550807	+	0.302586i
$340$		0			0
$341$		−	6.77182e12i		−	1.46870i
$342$		0			0
$343$		−	4.73069e12i		−	0.996447i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3.39317e11			0.0674463			0.0337232	−	0.999431i	$-0.489264\pi$
$347$	3.39317e11			0.0674463			0.0337232	+	0.999431i	$0.489264\pi$
$348$		0			0
$349$	2.82236e12			0.545111			0.272555	−	0.962140i	$-0.412131\pi$
$349$	2.82236e12			0.545111			0.272555	+	0.962140i	$0.412131\pi$
$350$		0			0
$351$	3.67763e12	−	2.34817e11i	0.690291	−	0.0440752i
$352$		0			0
$353$	7.34646e12			1.34031			0.670154	−	0.742222i	$-0.266228\pi$
$353$	7.34646e12			1.34031			0.670154	+	0.742222i	$0.266228\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	6.72598e11	−	1.22435e12i	0.115988	−	0.211137i
$358$		0			0
$359$		−	1.03382e11i		−	0.0173369i	−0.999962	−	0.00866847i	$-0.997241\pi$
$359$		−	1.03382e11i		−	0.0173369i	0.999962	−	0.00866847i	$-0.00275930\pi$
$360$		0			0
$361$	4.08403e12			0.666120
$362$		0			0
$363$	1.27141e13	+	6.98451e12i	2.01722	+	1.10816i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			1.09343e13i			1.64233i	0.570687	+	0.821167i	$0.306677\pi$
$367$			1.09343e13i			1.64233i	−0.570687	+	0.821167i	$0.693323\pi$
$368$		0			0
$369$	4.72546e10	−	3.00299e10i	0.00690736	−	0.00438957i
$370$		0			0
$371$			5.68822e12i			0.809295i
$372$		0			0
$373$			5.57238e12i			0.771785i	0.922544	+	0.385893i	$0.126107\pi$
$373$			5.57238e12i			0.771785i	−0.922544	+	0.385893i	$0.873893\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−8.08749e12			−1.06196
$378$		0			0
$379$	−1.34795e13			−1.72376			−0.861881	−	0.507110i	$-0.830714\pi$
$379$	−1.34795e13			−1.72376			−0.861881	+	0.507110i	$0.830714\pi$
$380$		0			0
$381$	1.04100e11	−	1.89497e11i	0.0129666	−	0.0236036i
$382$		0			0
$383$	−8.09167e11			−0.0981848			−0.0490924	−	0.998794i	$-0.515633\pi$
$383$	−8.09167e11			−0.0981848			−0.0490924	+	0.998794i	$0.515633\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1.23358e13	−	7.83929e12i	1.42106	−	0.903071i
$388$		0			0
$389$			1.43807e13i			1.61448i	0.590225	+	0.807239i	$0.299039\pi$
$389$			1.43807e13i			1.61448i	−0.590225	+	0.807239i	$0.700961\pi$
$390$		0			0
$391$	−4.65486e12			−0.509358
$392$		0			0
$393$	1.23506e12	−	2.24822e12i	0.131742	−	0.239815i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	2.50553e12i		−	0.254066i	−0.991898	−	0.127033i	$-0.959455\pi$
$397$		−	2.50553e12i		−	0.254066i	0.991898	−	0.127033i	$-0.0405454\pi$
$398$		0			0
$399$	3.85836e12	−	7.02349e12i	0.381538	−	0.694526i
$400$		0			0
$401$		−	4.50020e12i		−	0.434021i	−0.976169	−	0.217010i	$-0.930369\pi$
$401$		−	4.50020e12i		−	0.434021i	0.976169	−	0.217010i	$-0.0696305\pi$
$402$		0			0
$403$			5.94312e12i			0.559099i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	8.71986e12			0.780796
$408$		0			0
$409$	4.13779e12			0.361537			0.180768	−	0.983526i	$-0.442142\pi$
$409$	4.13779e12			0.361537			0.180768	+	0.983526i	$0.442142\pi$
$410$		0			0
$411$	−8.44626e12	−	4.63995e12i	−0.720203	−	0.395644i
$412$		0			0
$413$	3.70513e12			0.308357
$414$		0			0
$415$		0			0
$416$		0			0
$417$	1.15444e12	+	6.34191e11i	0.0915568	+	0.0502968i
$418$		0			0
$419$		−	3.33113e12i		−	0.257941i	−0.991648	−	0.128971i	$-0.958833\pi$
$419$		−	3.33113e12i		−	0.257941i	0.991648	−	0.128971i	$-0.0411673\pi$
$420$		0			0
$421$	1.73199e12			0.130959			0.0654795	−	0.997854i	$-0.479142\pi$
$421$	1.73199e12			0.130959			0.0654795	+	0.997854i	$0.479142\pi$
$422$		0			0
$423$	−1.04373e13	−	1.64240e13i	−0.770700	−	1.21276i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			1.08838e13i			0.766730i
$428$		0			0
$429$	−1.60063e13	−	8.79309e12i	−1.10155	−	0.605139i
$430$		0			0
$431$		−	8.04274e12i		−	0.540776i	−0.962751	−	0.270388i	$-0.912848\pi$
$431$		−	8.04274e12i		−	0.540776i	0.962751	−	0.270388i	$-0.0871520\pi$
$432$		0			0
$433$			1.88870e12i			0.124086i	0.998073	+	0.0620430i	$0.0197616\pi$
$433$			1.88870e12i			0.124086i	−0.998073	+	0.0620430i	$0.980238\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.67026e13			−1.67551
$438$		0			0
$439$	2.06976e13			1.26940			0.634698	−	0.772760i	$-0.281124\pi$
$439$	2.06976e13			1.26940			0.634698	+	0.772760i	$0.281124\pi$
$440$		0			0
$441$	5.57454e12	+	8.77203e12i	0.334208	+	0.525905i
$442$		0			0
$443$	−2.36413e12			−0.138565			−0.0692823	−	0.997597i	$-0.522071\pi$
$443$	−2.36413e12			−0.138565			−0.0692823	+	0.997597i	$0.522071\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−9.02277e12	+	1.64244e13i	−0.505594	+	0.920350i
$448$		0			0
$449$		−	1.70932e13i		−	0.936683i	−0.883547	−	0.468342i	$-0.844852\pi$
$449$		−	1.70932e13i		−	0.936683i	0.883547	−	0.468342i	$-0.155148\pi$
$450$		0			0
$451$	−2.77469e11			−0.0148707
$452$		0			0
$453$	−1.59345e13	−	8.75362e12i	−0.835309	−	0.458877i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	9.29963e12i		−	0.466536i	−0.972413	−	0.233268i	$-0.925058\pi$
$457$		−	9.29963e12i		−	0.466536i	0.972413	−	0.233268i	$-0.0749419\pi$
$458$		0			0
$459$	5.09415e11	+	7.97829e12i	0.0250039	+	0.391603i
$460$		0			0
$461$			7.49894e12i			0.360160i	0.983652	+	0.180080i	$0.0576356\pi$
$461$			7.49894e12i			0.360160i	−0.983652	+	0.180080i	$0.942364\pi$
$462$		0			0
$463$			1.01227e13i			0.475762i	0.971294	+	0.237881i	$0.0764528\pi$
$463$			1.01227e13i			0.475762i	−0.971294	+	0.237881i	$0.923547\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.34480e13			1.50587			0.752933	−	0.658097i	$-0.228638\pi$
$467$	3.34480e13			1.50587			0.752933	+	0.658097i	$0.228638\pi$
$468$		0			0
$469$	−3.72672e12			−0.164234
$470$		0			0
$471$	2.41884e12	−	4.40310e12i	0.104353	−	0.189956i
$472$		0			0
$473$	−7.24333e13			−3.05938
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.74599e13	−	2.74747e13i	−0.707051	−	1.11261i
$478$		0			0
$479$		−	4.02680e13i		−	1.59692i	−0.602050	−	0.798458i	$-0.705649\pi$
$479$		−	4.02680e13i		−	1.59692i	0.602050	−	0.798458i	$-0.294351\pi$
$480$		0			0
$481$	−7.65277e12			−0.297230
$482$		0			0
$483$	−1.00859e13	+	1.83597e13i	−0.383688	+	0.698440i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	3.91707e13i		−	1.42993i	−0.699158	−	0.714967i	$-0.746442\pi$
$487$		−	3.91707e13i		−	1.42993i	0.699158	−	0.714967i	$-0.253558\pi$
$488$		0			0
$489$	−1.12040e13	+	2.03950e13i	−0.400708	+	0.729421i
$490$		0			0
$491$			3.39572e13i			1.18994i	0.803749	+	0.594968i	$0.202835\pi$
$491$			3.39572e13i			1.18994i	−0.803749	+	0.594968i	$0.797165\pi$
$492$		0			0
$493$		−	1.75451e13i		−	0.602450i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	9.61159e12			0.316967
$498$		0			0
$499$	−4.89256e13			−1.58137			−0.790685	−	0.612224i	$-0.790275\pi$
$499$	−4.89256e13			−1.58137			−0.790685	+	0.612224i	$0.790275\pi$
$500$		0			0
$501$	−3.11979e13	−	1.71386e13i	−0.988408	−	0.542982i
$502$		0			0
$503$	1.19856e13			0.372237			0.186119	−	0.982527i	$-0.440409\pi$
$503$	1.19856e13			0.372237			0.186119	+	0.982527i	$0.440409\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.53134e13	−	8.41241e12i	−0.457121	−	0.251120i
$508$		0			0
$509$		−	3.91544e13i		−	1.14602i	−0.819549	−	0.573009i	$-0.805776\pi$
$509$		−	3.91544e13i		−	1.14602i	0.819549	−	0.573009i	$-0.194224\pi$
$510$		0			0
$511$	−3.86345e12			−0.110884
$512$		0			0
$513$	2.92226e12	+	4.57674e13i	0.0822492	+	1.28816i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			9.64382e13i			2.61094i
$518$		0			0
$519$	−4.90385e13	−	2.69393e13i	−1.30227	−	0.715402i
$520$		0			0
$521$		−	1.47353e13i		−	0.383857i	−0.981409	−	0.191928i	$-0.938526\pi$
$521$		−	1.47353e13i		−	0.383857i	0.981409	−	0.191928i	$-0.0614741\pi$
$522$		0			0
$523$			1.61321e13i			0.412271i	0.978523	+	0.206136i	$0.0660888\pi$
$523$			1.61321e13i			0.412271i	−0.978523	+	0.206136i	$0.933911\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.28931e13			−0.317178
$528$		0			0
$529$	2.83751e13			0.684950
$530$		0			0
$531$	−1.78962e13	+	1.13729e13i	−0.423924	+	0.269400i
$532$		0			0
$533$	2.43514e11			0.00566093
$534$		0			0
$535$		0			0
$536$		0			0
$537$	2.66644e13	−	4.85380e13i	0.597118	−	1.08695i
$538$		0			0
$539$		−	5.15075e13i		−	1.13221i
$540$		0			0
$541$	−4.23894e13			−0.914684			−0.457342	−	0.889291i	$-0.651198\pi$
$541$	−4.23894e13			−0.914684			−0.457342	+	0.889291i	$0.651198\pi$
$542$		0			0
$543$	−7.35469e13	−	4.04030e13i	−1.55799	−	0.855884i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	2.46963e13i		−	0.504308i	−0.967687	−	0.252154i	$-0.918861\pi$
$547$		−	2.46963e13i		−	0.504308i	0.967687	−	0.252154i	$-0.0811389\pi$
$548$		0			0
$549$	−3.34078e13	−	5.25701e13i	−0.669864	−	1.05409i
$550$		0			0
$551$		−	1.00647e14i		−	1.98173i
$552$		0			0
$553$		−	1.17537e12i		−	0.0227273i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.65465e13			0.868183			0.434091	−	0.900869i	$-0.357069\pi$
$557$	4.65465e13			0.868183			0.434091	+	0.900869i	$0.357069\pi$
$558$		0			0
$559$	6.35693e13			1.16463
$560$		0			0
$561$	1.90758e13	−	3.47243e13i	0.343296	−	0.624913i
$562$		0			0
$563$	−6.01479e13			−1.06335			−0.531677	−	0.846947i	$-0.678438\pi$
$563$	−6.01479e13			−1.06335			−0.531677	+	0.846947i	$0.678438\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	3.25716e13	+	1.52777e13i	0.555807	+	0.260700i
$568$		0			0
$569$		−	3.39778e13i		−	0.569684i	−0.958574	−	0.284842i	$-0.908059\pi$
$569$		−	3.39778e13i		−	0.569684i	0.958574	−	0.284842i	$-0.0919412\pi$
$570$		0			0
$571$	2.04262e13			0.336517			0.168258	−	0.985743i	$-0.446186\pi$
$571$	2.04262e13			0.336517			0.168258	+	0.985743i	$0.446186\pi$
$572$		0			0
$573$	2.40185e13	−	4.37217e13i	0.388842	−	0.707822i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			1.57333e13i			0.246003i	0.992407	+	0.123001i	$0.0392519\pi$
$577$			1.57333e13i			0.246003i	−0.992407	+	0.123001i	$0.960748\pi$
$578$		0			0
$579$	1.63758e13	−	2.98094e13i	0.251658	−	0.458101i
$580$		0			0
$581$			5.06708e13i			0.765381i
$582$		0			0
$583$			1.61326e14i			2.39531i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.45758e13			−0.926572			−0.463286	−	0.886209i	$-0.653330\pi$
$587$	−6.45758e13			−0.926572			−0.463286	+	0.886209i	$0.653330\pi$
$588$		0			0
$589$	−7.39611e13			−1.04334
$590$		0			0
$591$	9.06722e13	+	4.98108e13i	1.25758	+	0.690854i
$592$		0			0
$593$	9.71953e12			0.132548			0.0662738	−	0.997801i	$-0.478889\pi$
$593$	9.71953e12			0.132548			0.0662738	+	0.997801i	$0.478889\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1.77408e13	−	9.74593e12i	−0.233939	−	0.128514i
$598$		0			0
$599$			8.73750e13i			1.13306i	0.824041	+	0.566530i	$0.191715\pi$
$599$			8.73750e13i			1.13306i	−0.824041	+	0.566530i	$0.808285\pi$
$600$		0			0
$601$	2.77331e13			0.353693			0.176846	−	0.984238i	$-0.443410\pi$
$601$	2.77331e13			0.353693			0.176846	+	0.984238i	$0.443410\pi$
$602$		0			0
$603$	1.80005e13	−	1.14391e13i	0.225786	−	0.143485i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			9.77365e13i			1.18608i	0.805174	+	0.593039i	$0.202072\pi$
$607$			9.77365e13i			1.18608i	−0.805174	+	0.593039i	$0.797928\pi$
$608$		0			0
$609$	−6.92012e13	−	3.80157e13i	−0.826089	−	0.453812i
$610$		0			0
$611$		−	8.46365e13i		−	0.993921i
$612$		0			0
$613$			1.41138e14i			1.63058i	0.579056	+	0.815288i	$0.303421\pi$
$613$			1.41138e14i			1.63058i	−0.579056	+	0.815288i	$0.696579\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.85738e13			−0.543220			−0.271610	−	0.962407i	$-0.587556\pi$
$617$	−4.85738e13			−0.543220			−0.271610	+	0.962407i	$0.587556\pi$
$618$		0			0
$619$	−3.95021e13			−0.434677			−0.217339	−	0.976096i	$-0.569738\pi$
$619$	−3.95021e13			−0.434677			−0.217339	+	0.976096i	$0.569738\pi$
$620$		0			0
$621$	−7.63889e12	−	1.19638e14i	−0.0827127	−	1.29542i
$622$		0			0
$623$	3.38641e13			0.360827
$624$		0			0
$625$		0			0
$626$		0			0
$627$	1.09428e14	−	1.99196e14i	1.12926	−	2.05562i
$628$		0			0
$629$		−	1.66020e13i		−	0.168619i
$630$		0			0
$631$	−6.04370e13			−0.604166			−0.302083	−	0.953282i	$-0.597682\pi$
$631$	−6.04370e13			−0.604166			−0.302083	+	0.953282i	$0.597682\pi$
$632$		0			0
$633$	1.31763e14	+	7.23843e13i	1.29651	+	0.712239i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			4.52043e13i			0.431005i
$638$		0			0
$639$	−4.64250e13	+	2.95027e13i	−0.435761	+	0.276922i
$640$		0			0
$641$		−	1.57323e14i		−	1.45379i	−0.686748	−	0.726896i	$-0.740962\pi$
$641$		−	1.57323e14i		−	1.45379i	0.686748	−	0.726896i	$-0.259038\pi$
$642$		0			0
$643$		−	1.34823e14i		−	1.22662i	−0.789843	−	0.613310i	$-0.789838\pi$
$643$		−	1.34823e14i		−	1.22662i	0.789843	−	0.613310i	$-0.210162\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	4.02182e13			0.354733			0.177366	−	0.984145i	$-0.443242\pi$
$647$	4.02182e13			0.354733			0.177366	+	0.984145i	$0.443242\pi$
$648$		0			0
$649$	1.05083e14			0.912657
$650$		0			0
$651$	−2.79359e13	+	5.08527e13i	−0.238923	+	0.434920i
$652$		0			0
$653$	−1.56515e14			−1.31823			−0.659113	−	0.752044i	$-0.729068\pi$
$653$	−1.56515e14			−1.31823			−0.659113	+	0.752044i	$0.729068\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.86609e13	−	1.18588e13i	0.152442	−	0.0968755i
$658$		0			0
$659$			4.35191e12i			0.0350149i	0.999847	+	0.0175074i	$0.00557308\pi$
$659$			4.35191e12i			0.0350149i	−0.999847	+	0.0175074i	$0.994427\pi$
$660$		0			0
$661$	−2.58433e13			−0.204805			−0.102403	−	0.994743i	$-0.532653\pi$
$661$	−2.58433e13			−0.204805			−0.102403	+	0.994743i	$0.532653\pi$
$662$		0			0
$663$	−1.67414e13	+	3.04750e13i	−0.130685	+	0.237890i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			2.63096e14i			1.99290i
$668$		0			0
$669$	−9.89261e12	+	1.80078e13i	−0.0738211	+	0.134379i
$670$		0			0
$671$			3.08681e14i			2.26933i
$672$		0			0
$673$		−	9.33181e13i		−	0.675912i	−0.941162	−	0.337956i	$-0.890264\pi$
$673$		−	9.33181e13i		−	0.675912i	0.941162	−	0.337956i	$-0.109736\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7.39555e13			0.520029			0.260014	−	0.965605i	$-0.416273\pi$
$677$	7.39555e13			0.520029			0.260014	+	0.965605i	$0.416273\pi$
$678$		0			0
$679$	2.89927e13			0.200882
$680$		0			0
$681$	6.99561e13	+	3.84304e13i	0.477628	+	0.262385i
$682$		0			0
$683$	1.43414e14			0.964914			0.482457	−	0.875920i	$-0.339744\pi$
$683$	1.43414e14			0.964914			0.482457	+	0.875920i	$0.339744\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−2.26754e14	−	1.24568e14i	−1.48174	−	0.813995i
$688$		0			0
$689$		−	1.41584e14i		−	0.911837i
$690$		0			0
$691$	4.15781e13			0.263921			0.131961	−	0.991255i	$-0.457873\pi$
$691$	4.15781e13			0.263921			0.131961	+	0.991255i	$0.457873\pi$
$692$		0			0
$693$	−9.56270e13	−	1.50477e14i	−0.598294	−	0.941467i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			5.28282e11i			0.00321146i
$698$		0			0
$699$	−5.98880e13	−	3.28995e13i	−0.358884	−	0.197153i
$700$		0			0
$701$		−	1.23285e14i		−	0.728317i	−0.931337	−	0.364159i	$-0.881357\pi$
$701$		−	1.23285e14i		−	0.728317i	0.931337	−	0.364159i	$-0.118643\pi$
$702$		0			0
$703$		−	9.52374e13i		−	0.554665i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.33017e14			0.753024
$708$		0			0
$709$	2.77906e14			1.55120			0.775599	−	0.631225i	$-0.217448\pi$
$709$	2.77906e14			1.55120			0.775599	+	0.631225i	$0.217448\pi$
$710$		0			0
$711$	3.60778e12	+	5.67716e12i	0.0198560	+	0.0312452i
$712$		0			0
$713$	1.93337e14			1.04922
$714$		0			0
$715$		0			0
$716$		0			0
$717$	5.63255e12	−	1.02531e13i	0.0297241	−	0.0541078i
$718$		0			0
$719$			2.39434e14i			1.24607i	0.782195	+	0.623034i	$0.214100\pi$
$719$			2.39434e14i			1.24607i	−0.782195	+	0.623034i	$0.785900\pi$
$720$		0			0
$721$	−1.56029e14			−0.800807
$722$		0			0
$723$	−2.33233e14	−	1.28127e14i	−1.18059	−	0.648557i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	1.51463e14i		−	0.745823i	−0.927867	−	0.372911i	$-0.878360\pi$
$727$		−	1.51463e14i		−	0.745823i	0.927867	−	0.372911i	$-0.121640\pi$
$728$		0			0
$729$	−2.04219e14	+	2.61856e13i	−0.991879	+	0.127182i
$730$		0			0
$731$			1.37908e14i			0.660697i
$732$		0			0
$733$			2.37806e14i			1.12384i	0.827193	+	0.561918i	$0.189936\pi$
$733$			2.37806e14i			1.12384i	−0.827193	+	0.561918i	$0.810064\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.05695e14			−0.486090
$738$		0			0
$739$	3.58451e14			1.62633			0.813164	−	0.582035i	$-0.197743\pi$
$739$	3.58451e14			1.62633			0.813164	+	0.582035i	$0.197743\pi$
$740$		0			0
$741$	−9.60371e13	+	1.74820e14i	−0.429881	+	0.782526i
$742$		0			0
$743$	−2.11655e14			−0.934726			−0.467363	−	0.884065i	$-0.654796\pi$
$743$	−2.11655e14			−0.934726			−0.467363	+	0.884065i	$0.654796\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1.55534e14	−	2.44746e14i	−0.668685	−	1.05223i
$748$		0			0
$749$			1.36363e14i			0.578479i
$750$		0			0
$751$	1.08945e13			0.0456046			0.0228023	−	0.999740i	$-0.492741\pi$
$751$	1.08945e13			0.0456046			0.0228023	+	0.999740i	$0.492741\pi$
$752$		0			0
$753$	1.29690e14	−	2.36080e14i	0.535714	−	0.975177i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			3.51497e14i			1.41398i	0.707224	+	0.706989i	$0.249947\pi$
$757$			3.51497e14i			1.41398i	−0.707224	+	0.706989i	$0.750053\pi$
$758$		0			0
$759$	−2.86050e14	+	5.20706e14i	−1.13562	+	2.06721i
$760$		0			0
$761$			3.08205e14i			1.20758i	0.797144	+	0.603790i	$0.206343\pi$
$761$			3.08205e14i			1.20758i	−0.797144	+	0.603790i	$0.793657\pi$
$762$		0			0
$763$			1.44293e13i			0.0557984i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9.22232e13			−0.347427
$768$		0			0
$769$	2.78711e14			1.03639			0.518194	−	0.855263i	$-0.326605\pi$
$769$	2.78711e14			1.03639			0.518194	+	0.855263i	$0.326605\pi$
$770$		0			0
$771$	1.20739e13	+	6.63281e12i	0.0443176	+	0.0243459i
$772$		0			0
$773$	4.04348e14			1.46507			0.732534	−	0.680730i	$-0.238337\pi$
$773$	4.04348e14			1.46507			0.732534	+	0.680730i	$0.238337\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−6.54815e13	−	3.59723e13i	−0.231214	−	0.127017i
$778$		0			0
$779$			3.03049e12i			0.0105639i
$780$		0			0
$781$	2.72598e14			0.938141
$782$		0			0
$783$	4.50938e14	−	2.87925e13i	1.53217	−	0.0978295i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	1.25181e14i		−	0.414634i	−0.978274	−	0.207317i	$-0.933527\pi$
$787$		−	1.25181e14i		−	0.414634i	0.978274	−	0.207317i	$-0.0664731\pi$
$788$		0			0
$789$	2.64970e13	+	1.45562e13i	0.0866587	+	0.0476060i
$790$		0			0
$791$			1.19469e14i			0.385811i
$792$		0			0
$793$		−	2.70906e14i		−	0.863879i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.24667e14			−0.387668			−0.193834	−	0.981034i	$-0.562092\pi$
$797$	−1.24667e14			−0.387668			−0.193834	+	0.981034i	$0.562092\pi$
$798$		0			0
$799$	1.83612e14			0.563853
$800$		0			0
$801$	−1.63567e14	+	1.03946e14i	−0.496060	+	0.315241i
$802$		0			0
$803$	−1.09573e14			−0.328190
$804$		0			0
$805$		0			0
$806$		0			0
$807$	2.50770e14	−	4.56484e14i	0.732668	−	1.33370i
$808$		0			0
$809$			4.18260e14i			1.20699i	0.797366	+	0.603496i	$0.206226\pi$
$809$			4.18260e14i			1.20699i	−0.797366	+	0.603496i	$0.793774\pi$
$810$		0			0
$811$	3.92624e14			1.11911			0.559555	−	0.828793i	$-0.310972\pi$
$811$	3.92624e14			1.11911			0.559555	+	0.828793i	$0.310972\pi$
$812$		0			0
$813$	−1.14113e14	−	6.26883e13i	−0.321280	−	0.176495i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			7.91109e14i			2.17333i
$818$		0			0
$819$	8.39246e13	+	1.32063e14i	0.227756	+	0.358394i
$820$		0			0
$821$		−	1.90066e14i		−	0.509553i	−0.967000	−	0.254777i	$-0.917998\pi$
$821$		−	1.90066e14i		−	0.509553i	0.967000	−	0.254777i	$-0.0820020\pi$
$822$		0			0
$823$			2.02565e14i			0.536495i	0.963350	+	0.268248i	$0.0864445\pi$
$823$			2.02565e14i			0.536495i	−0.963350	+	0.268248i	$0.913555\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.20987e14			−0.312759			−0.156380	−	0.987697i	$-0.549982\pi$
$827$	−1.20987e14			−0.312759			−0.156380	+	0.987697i	$0.549982\pi$
$828$		0			0
$829$	2.34411e14			0.598694			0.299347	−	0.954144i	$-0.403231\pi$
$829$	2.34411e14			0.598694			0.299347	+	0.954144i	$0.403231\pi$
$830$		0			0
$831$	1.37513e14	−	2.50320e14i	0.347008	−	0.631671i
$832$		0			0
$833$	−9.80668e13			−0.244510
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−2.11582e13	−	3.31373e14i	−0.0515053	−	0.806659i
$838$		0			0
$839$			5.13382e14i			1.23490i	0.786611	+	0.617449i	$0.211834\pi$
$839$			5.13382e14i			1.23490i	−0.786611	+	0.617449i	$0.788166\pi$
$840$		0			0
$841$	−5.70953e14			−1.35713
$842$		0			0
$843$	2.33207e14	−	4.24514e14i	0.547777	−	0.997135i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			6.15950e14i			1.41295i
$848$		0			0
$849$	1.65610e14	−	3.01465e14i	0.375446	−	0.683437i
$850$		0			0
$851$			2.48954e14i			0.557791i
$852$		0			0
$853$		−	3.95975e14i		−	0.876844i	−0.898769	−	0.438422i	$-0.855538\pi$
$853$		−	3.95975e14i		−	0.876844i	0.898769	−	0.438422i	$-0.144462\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	2.33139e13			0.0504326			0.0252163	−	0.999682i	$-0.491973\pi$
$857$	2.33139e13			0.0504326			0.0252163	+	0.999682i	$0.491973\pi$
$858$		0			0
$859$	4.78130e14			1.02230			0.511152	−	0.859490i	$-0.329219\pi$
$859$	4.78130e14			1.02230			0.511152	+	0.859490i	$0.329219\pi$
$860$		0			0
$861$	2.08364e12	+	1.14465e12i	0.00440360	+	0.00241912i
$862$		0			0
$863$	8.30319e14			1.73457			0.867283	−	0.497815i	$-0.165864\pi$
$863$	8.30319e14			1.73457			0.867283	+	0.497815i	$0.165864\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	3.63251e14	+	1.99552e14i	0.741501	+	0.407344i
$868$		0			0
$869$		−	3.33351e13i		−	0.0672672i
$870$		0			0
$871$	9.27606e13			0.185043
$872$		0			0
$873$	−1.40038e14	+	8.89929e13i	−0.276169	+	0.175503i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			7.06641e13i			0.136207i	0.997678	+	0.0681037i	$0.0216949\pi$
$877$			7.06641e13i			0.136207i	−0.997678	+	0.0681037i	$0.978305\pi$
$878$		0			0
$879$	−7.20384e14	−	3.95743e14i	−1.37284	−	0.754169i
$880$		0			0
$881$		−	7.25515e14i		−	1.36699i	−0.729953	−	0.683497i	$-0.760458\pi$
$881$		−	7.25515e14i		−	1.36699i	0.729953	−	0.683497i	$-0.239542\pi$
$882$		0			0
$883$			1.22030e14i			0.227334i	0.993519	+	0.113667i	$0.0362597\pi$
$883$			1.22030e14i			0.227334i	−0.993519	+	0.113667i	$0.963740\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.77377e14			−0.323056			−0.161528	−	0.986868i	$-0.551642\pi$
$887$	−1.77377e14			−0.323056			−0.161528	+	0.986868i	$0.551642\pi$
$888$		0			0
$889$	9.18041e12			0.0165331
$890$		0			0
$891$	9.23778e14	+	4.33296e14i	1.64505	+	0.771607i
$892$		0			0
$893$	1.05329e15			1.85477
$894$		0			0
$895$		0			0
$896$		0			0
$897$	2.51045e14	−	4.56984e14i	0.432303	−	0.786936i
$898$		0			0
$899$			7.28724e14i			1.24098i
$900$		0			0
$901$	3.07153e14			0.517287
$902$		0			0
$903$	5.43935e14	+	2.98811e14i	0.905959	+	0.497689i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	3.42190e14i		−	0.557482i	−0.960366	−	0.278741i	$-0.910083\pi$
$907$		−	3.42190e14i		−	0.557482i	0.960366	−	0.278741i	$-0.0899171\pi$
$908$		0			0
$909$	−6.42485e14	+	4.08293e14i	−1.03525	+	0.657889i
$910$		0			0
$911$			5.94239e14i			0.947043i	0.880782	+	0.473521i	$0.157017\pi$
$911$			5.94239e14i			0.947043i	−0.880782	+	0.473521i	$0.842983\pi$
$912$		0			0
$913$			1.43710e15i			2.26533i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.08918e14			0.167978
$918$		0			0
$919$	6.28087e14			0.958170			0.479085	−	0.877769i	$-0.340969\pi$
$919$	6.28087e14			0.958170			0.479085	+	0.877769i	$0.340969\pi$
$920$		0			0
$921$	3.18789e14	−	5.80302e14i	0.481067	−	0.875701i
$922$		0			0
$923$	−2.39239e14			−0.357128
$924$		0			0
$925$		0			0
$926$		0			0
$927$	7.53635e14	−	4.78928e14i	1.10094	−	0.699635i
$928$		0			0
$929$			1.36253e14i			0.196910i	0.995141	+	0.0984551i	$0.0313901\pi$
$929$			1.36253e14i			0.196910i	−0.995141	+	0.0984551i	$0.968610\pi$
$930$		0			0
$931$	−5.62560e14			−0.804304
$932$		0			0
$933$	2.45639e14	−	4.47145e14i	0.347448	−	0.632471i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			6.59993e14i			0.913780i	0.889523	+	0.456890i	$0.151037\pi$
$937$			6.59993e14i			0.913780i	−0.889523	+	0.456890i	$0.848963\pi$
$938$		0			0
$939$	−2.52423e14	+	4.59494e14i	−0.345781	+	0.629436i
$940$		0			0
$941$		−	6.64694e14i		−	0.900894i	−0.892803	−	0.450447i	$-0.851265\pi$
$941$		−	6.64694e14i		−	0.900894i	0.892803	−	0.450447i	$-0.148735\pi$
$942$		0			0
$943$		−	7.92181e12i		−	0.0106235i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	7.98687e14			1.04864			0.524320	−	0.851521i	$-0.324319\pi$
$947$	7.98687e14			1.04864			0.524320	+	0.851521i	$0.324319\pi$
$948$		0			0
$949$	9.61638e13			0.124934
$950$		0			0
$951$	−5.37372e14	−	2.95205e14i	−0.690832	−	0.379509i
$952$		0			0
$953$	−1.82292e14			−0.231901			−0.115950	−	0.993255i	$-0.536991\pi$
$953$	−1.82292e14			−0.231901			−0.115950	+	0.993255i	$0.536991\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−1.96264e15	−	1.07818e15i	−2.44501	−	1.34317i
$958$		0			0
$959$		−	4.09188e14i		−	0.504464i
$960$		0			0
$961$	−2.84123e14			−0.346648
$962$		0			0
$963$	−4.18565e14	−	6.58648e14i	−0.505396	−	0.795284i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			1.13273e15i			1.33965i	0.742518	+	0.669826i	$0.233631\pi$
$967$			1.13273e15i			1.33965i	−0.742518	+	0.669826i	$0.766369\pi$
$968$		0			0
$969$	−3.79256e14	−	2.08344e14i	−0.443929	−	0.243872i
$970$		0			0
$971$			9.18843e14i			1.06450i	0.846588	+	0.532249i	$0.178653\pi$
$971$			9.18843e14i			1.06450i	−0.846588	+	0.532249i	$0.821347\pi$
$972$		0			0
$973$			5.59281e13i			0.0641307i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.73004e15			−1.94350			−0.971749	−	0.236017i	$-0.924158\pi$
$977$	−1.73004e15			−1.94350			−0.971749	+	0.236017i	$0.924158\pi$
$978$		0			0
$979$	9.60433e14			1.06796
$980$		0			0
$981$	−4.42905e13	−	6.96950e13i	−0.0487490	−	0.0767108i
$982$		0			0
$983$	−1.29362e15			−1.40941			−0.704707	−	0.709498i	$-0.748922\pi$
$983$	−1.29362e15			−1.40941			−0.704707	+	0.709498i	$0.748922\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	3.97839e14	−	7.24199e14i	0.424738	−	0.773165i
$988$		0			0
$989$		−	2.06799e15i		−	2.18558i
$990$		0			0
$991$	1.43410e15			1.50041			0.750206	−	0.661205i	$-0.229954\pi$
$991$	1.43410e15			1.50041			0.750206	+	0.661205i	$0.229954\pi$
$992$		0			0
$993$	−4.75808e14	−	2.61385e14i	−0.492817	−	0.270729i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			1.70539e15i			1.73120i	0.500736	+	0.865600i	$0.333063\pi$
$997$			1.70539e15i			1.73120i	−0.500736	+	0.865600i	$0.666937\pi$
$998$		0			0
$999$	4.26699e14	−	2.72448e13i	0.428839	−	0.0273814i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.11.b.c.149.1		4
3.2	odd	2	inner	300.11.b.c.149.3		4
5.2	odd	4		300.11.g.d.101.2		2
5.3	odd	4		12.11.c.b.5.1	✓	2
5.4	even	2	inner	300.11.b.c.149.4		4
15.2	even	4		300.11.g.d.101.1		2
15.8	even	4		12.11.c.b.5.2	yes	2
15.14	odd	2	inner	300.11.b.c.149.2		4
20.3	even	4		48.11.e.b.17.2		2
40.3	even	4		192.11.e.f.65.1		2
40.13	odd	4		192.11.e.c.65.2		2
45.13	odd	12		324.11.g.d.269.1		4
45.23	even	12		324.11.g.d.269.2		4
45.38	even	12		324.11.g.d.53.1		4
45.43	odd	12		324.11.g.d.53.2		4
60.23	odd	4		48.11.e.b.17.1		2
120.53	even	4		192.11.e.c.65.1		2
120.83	odd	4		192.11.e.f.65.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.11.c.b.5.1	✓	2	5.3	odd	4
12.11.c.b.5.2	yes	2	15.8	even	4
48.11.e.b.17.1		2	60.23	odd	4
48.11.e.b.17.2		2	20.3	even	4
192.11.e.c.65.1		2	120.53	even	4
192.11.e.c.65.2		2	40.13	odd	4
192.11.e.f.65.1		2	40.3	even	4
192.11.e.f.65.2		2	120.83	odd	4
300.11.b.c.149.1		4	1.1	even	1	trivial
300.11.b.c.149.2		4	15.14	odd	2	inner
300.11.b.c.149.3		4	3.2	odd	2	inner
300.11.b.c.149.4		4	5.4	even	2	inner
300.11.g.d.101.1		2	15.2	even	4
300.11.g.d.101.2		2	5.2	odd	4
324.11.g.d.53.1		4	45.38	even	12
324.11.g.d.53.2		4	45.43	odd	12
324.11.g.d.269.1		4	45.13	odd	12
324.11.g.d.269.2		4	45.23	even	12

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:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	300.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-212.979 - 117.000i) q^{3} -10318.0i q^{7} +(31671.0 + 49837.1i) q^{9} +O(q^{10})\)	\(q+(-212.979 - 117.000i) q^{3} -10318.0i q^{7} +(31671.0 + 49837.1i) q^{9} -292633. i q^{11} +256822. i q^{13} -557153. q^{17} -3.19611e6 q^{19} +(-1.20721e6 + 2.19752e6i) q^{21} +8.35474e6 q^{23} +(-914318. - 1.43197e7i) q^{27} +3.14906e7i q^{29} +2.31410e7 q^{31} +(-3.42381e7 + 6.23246e7i) q^{33} +2.97979e7i q^{37} +(3.00482e7 - 5.46977e7i) q^{39} -948182. i q^{41} -2.47523e8i q^{43} -3.29553e8 q^{47} +1.76014e8 q^{49} +(1.18662e8 + 6.51869e7i) q^{51} -5.51291e8 q^{53} +(6.80703e8 + 3.73944e8i) q^{57} +3.59094e8i q^{59} -1.05484e9 q^{61} +(5.14219e8 - 3.26781e8i) q^{63} -3.61186e8i q^{67} +(-1.77938e9 - 9.77504e8i) q^{69} +9.31536e8i q^{71} -3.74437e8i q^{73} -3.01939e9 q^{77} +1.13914e8 q^{79} +(-1.48068e9 + 3.15678e9i) q^{81} -4.91092e9 q^{83} +(3.68440e9 - 6.70684e9i) q^{87} +3.28204e9i q^{89} +2.64989e9 q^{91} +(-4.92854e9 - 2.70750e9i) q^{93} +2.80992e9i q^{97} +(1.45840e10 - 9.26798e9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 126684 q^{9}+O(q^{10}) \) 4 * q + 126684 * q^9	\( 4 q + 126684 q^{9} - 12784424 q^{19} - 4828824 q^{21} + 92563976 q^{31} + 120192696 q^{39} + 704056500 q^{49} + 474647040 q^{51} - 4219359064 q^{61} - 7117528320 q^{69} + 455657848 q^{79} - 5922719676 q^{81} + 10599557584 q^{91} + 58335863040 q^{99}+O(q^{100}) \) 4 * q + 126684 * q^9 - 12784424 * q^19 - 4828824 * q^21 + 92563976 * q^31 + 120192696 * q^39 + 704056500 * q^49 + 474647040 * q^51 - 4219359064 * q^61 - 7117528320 * q^69 + 455657848 * q^79 - 5922719676 * q^81 + 10599557584 * q^91 + 58335863040 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more