LMFDB - Embedded newform 225.12.b.d.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,12,Mod(199,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("225.199");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	225.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:	$172.877215626$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.12.b.d.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+24.0000i q^{2} +1472.00 q^{4} -16744.0i q^{7} +84480.0i q^{8} +O(q^{10})$	$q+24.0000i q^{2} +1472.00 q^{4} -16744.0i q^{7} +84480.0i q^{8} -534612. q^{11} +577738. i q^{13} +401856. q^{14} +987136. q^{16} +6.90593e6i q^{17} -1.06614e7 q^{19} -1.28307e7i q^{22} +1.86433e7i q^{23} -1.38657e7 q^{26} -2.46472e7i q^{28} +1.28407e8 q^{29} -5.28432e7 q^{31} +1.96706e8i q^{32} -1.65742e8 q^{34} -1.82213e8i q^{37} -2.55874e8i q^{38} -3.08120e8 q^{41} +1.71257e7i q^{43} -7.86949e8 q^{44} -4.47439e8 q^{46} -2.68735e9i q^{47} +1.69697e9 q^{49} +8.50430e8i q^{52} -1.59606e9i q^{53} +1.41453e9 q^{56} +3.08176e9i q^{58} -5.18920e9 q^{59} +6.95648e9 q^{61} -1.26824e9i q^{62} -2.69930e9 q^{64} -1.54818e10i q^{67} +1.01655e10i q^{68} -9.79149e9 q^{71} -1.46379e9i q^{73} +4.37312e9 q^{74} -1.56936e10 q^{76} +8.95154e9i q^{77} -3.81168e10 q^{79} -7.39489e9i q^{82} -2.93351e10i q^{83} -4.11017e8 q^{86} -4.51640e10i q^{88} -2.49929e10 q^{89} +9.67365e9 q^{91} +2.74429e10i q^{92} +6.44964e10 q^{94} +7.50136e10i q^{97} +4.07272e10i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2944 q^{4}+O(q^{10}) $ 2 * q + 2944 * q^4	$ 2 q + 2944 q^{4} - 1069224 q^{11} + 803712 q^{14} + 1974272 q^{16} - 21322840 q^{19} - 27731424 q^{26} + 256813260 q^{29} - 105686336 q^{31} - 331484832 q^{34} - 616240884 q^{41} - 1573897728 q^{44} - 894877056 q^{46} + 3393930414 q^{49} + 2829066240 q^{56} - 10378407480 q^{59} + 13912957324 q^{61} - 5398593536 q^{64} - 19582970544 q^{71} + 8746239072 q^{74} - 31387220480 q^{76} - 76233691360 q^{79} - 822033984 q^{86} - 49985834220 q^{89} + 19347290144 q^{91} + 128992727808 q^{94}+O(q^{100}) $ 2 * q + 2944 * q^4 - 1069224 * q^11 + 803712 * q^14 + 1974272 * q^16 - 21322840 * q^19 - 27731424 * q^26 + 256813260 * q^29 - 105686336 * q^31 - 331484832 * q^34 - 616240884 * q^41 - 1573897728 * q^44 - 894877056 * q^46 + 3393930414 * q^49 + 2829066240 * q^56 - 10378407480 * q^59 + 13912957324 * q^61 - 5398593536 * q^64 - 19582970544 * q^71 + 8746239072 * q^74 - 31387220480 * q^76 - 76233691360 * q^79 - 822033984 * q^86 - 49985834220 * q^89 + 19347290144 * q^91 + 128992727808 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	24.0000i	0.530330i	0.964203	+	0.265165i	$0.0854264\pi$
$2$	24.0000i	0.530330i	−0.964203	+	0.265165i	$0.914574\pi$
$3$	0	0
$3$	0	0
$4$	1472.00	0.718750
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 16744.0i	− 0.376548i	−0.982117	−	0.188274i	$-0.939711\pi$
$7$	− 16744.0i	− 0.376548i	0.982117	−	0.188274i	$-0.0602893\pi$
$8$	84480.0i	0.911505i
$9$	0	0
$10$	0	0
$11$	−534612.	−1.00087	−0.500436	−	0.865773i	$-0.666827\pi$
$11$	−534612.	−1.00087	−0.500436	+	0.865773i	$0.666827\pi$
$12$	0	0
$13$	577738.i	0.431561i	0.976442	+	0.215781i	$0.0692296\pi$
$13$	577738.i	0.431561i	−0.976442	+	0.215781i	$0.930770\pi$
$14$	401856.	0.199695
$15$	0	0
$16$	987136.	0.235352
$17$	6.90593e6i	1.17965i	0.807531	+	0.589825i	$0.200803\pi$
$17$	6.90593e6i	1.17965i	−0.807531	+	0.589825i	$0.799197\pi$
$18$	0	0
$19$	−1.06614e7	−0.987803	−0.493901	−	0.869518i	$-0.664430\pi$
$19$	−1.06614e7	−0.987803	−0.493901	+	0.869518i	$0.664430\pi$
$20$	0	0
$21$	0	0
$22$	− 1.28307e7i	− 0.530793i
$23$	1.86433e7i	0.603975i	0.953312	+	0.301988i	$0.0976501\pi$
$23$	1.86433e7i	0.603975i	−0.953312	+	0.301988i	$0.902350\pi$
$24$	0	0
$25$	0	0
$26$	−1.38657e7	−0.228870
$27$	0	0
$28$	− 2.46472e7i	− 0.270644i
$29$	1.28407e8	1.16251	0.581257	−	0.813720i	$-0.302561\pi$
$29$	1.28407e8	1.16251	0.581257	+	0.813720i	$0.302561\pi$
$30$	0	0
$31$	−5.28432e7	−0.331512	−0.165756	−	0.986167i	$-0.553006\pi$
$31$	−5.28432e7	−0.331512	−0.165756	+	0.986167i	$0.553006\pi$
$32$	1.96706e8i	1.03632i
$33$	0	0
$34$	−1.65742e8	−0.625604
$35$	0	0
$36$	0	0
$37$	− 1.82213e8i	− 0.431987i	−0.976395	−	0.215993i	$-0.930701\pi$
$37$	− 1.82213e8i	− 0.431987i	0.976395	−	0.215993i	$-0.0692990\pi$
$38$	− 2.55874e8i	− 0.523862i
$39$	0	0
$40$	0	0
$41$	−3.08120e8	−0.415345	−0.207673	−	0.978198i	$-0.566589\pi$
$41$	−3.08120e8	−0.415345	−0.207673	+	0.978198i	$0.566589\pi$
$42$	0	0
$43$	1.71257e7i	0.0177653i	0.999961	+	0.00888264i	$0.00282747\pi$
$43$	1.71257e7i	0.0177653i	−0.999961	+	0.00888264i	$0.997173\pi$
$44$	−7.86949e8	−0.719377
$45$	0	0
$46$	−4.47439e8	−0.320306
$47$	− 2.68735e9i	− 1.70917i	−0.519310	−	0.854586i	$-0.673811\pi$
$47$	− 2.68735e9i	− 1.70917i	0.519310	−	0.854586i	$-0.326189\pi$
$48$	0	0
$49$	1.69697e9	0.858212
$50$	0	0
$51$	0	0
$52$	8.50430e8i	0.310185i
$53$	− 1.59606e9i	− 0.524241i	−0.965035	−	0.262120i	$-0.915578\pi$
$53$	− 1.59606e9i	− 0.524241i	0.965035	−	0.262120i	$-0.0844217\pi$
$54$	0	0
$55$	0	0
$56$	1.41453e9	0.343225
$57$	0	0
$58$	3.08176e9i	0.616517i
$59$	−5.18920e9	−0.944963	−0.472481	−	0.881341i	$-0.656642\pi$
$59$	−5.18920e9	−0.944963	−0.472481	+	0.881341i	$0.656642\pi$
$60$	0	0
$61$	6.95648e9	1.05457	0.527285	−	0.849689i	$-0.323210\pi$
$61$	6.95648e9	1.05457	0.527285	+	0.849689i	$0.323210\pi$
$62$	− 1.26824e9i	− 0.175811i
$63$	0	0
$64$	−2.69930e9	−0.314240
$65$	0	0
$66$	0	0
$67$	− 1.54818e10i	− 1.40091i	−0.713696	−	0.700456i	$-0.752980\pi$
$67$	− 1.54818e10i	− 1.40091i	0.713696	−	0.700456i	$-0.247020\pi$
$68$	1.01655e10i	0.847874i
$69$	0	0
$70$	0	0
$71$	−9.79149e9	−0.644062	−0.322031	−	0.946729i	$-0.604366\pi$
$71$	−9.79149e9	−0.644062	−0.322031	+	0.946729i	$0.604366\pi$
$72$	0	0
$73$	− 1.46379e9i	− 0.0826425i	−0.999146	−	0.0413212i	$-0.986843\pi$
$73$	− 1.46379e9i	− 0.0826425i	0.999146	−	0.0413212i	$-0.0131567\pi$
$74$	4.37312e9	0.229096
$75$	0	0
$76$	−1.56936e10	−0.709983
$77$	8.95154e9i	0.376876i
$78$	0	0
$79$	−3.81168e10	−1.39370	−0.696848	−	0.717219i	$-0.745415\pi$
$79$	−3.81168e10	−1.39370	−0.696848	+	0.717219i	$0.745415\pi$
$80$	0	0
$81$	0	0
$82$	− 7.39489e9i	− 0.220270i
$83$	− 2.93351e10i	− 0.817444i	−0.912659	−	0.408722i	$-0.865975\pi$
$83$	− 2.93351e10i	− 0.817444i	0.912659	−	0.408722i	$-0.134025\pi$
$84$	0	0
$85$	0	0
$86$	−4.11017e8	−0.00942146
$87$	0	0
$88$	− 4.51640e10i	− 0.912300i
$89$	−2.49929e10	−0.474430	−0.237215	−	0.971457i	$-0.576235\pi$
$89$	−2.49929e10	−0.474430	−0.237215	+	0.971457i	$0.576235\pi$
$90$	0	0
$91$	9.67365e9	0.162503
$92$	2.74429e10i	0.434107i
$93$	0	0
$94$	6.44964e10	0.906425
$95$	0	0
$96$	0	0
$97$	7.50136e10i	0.886942i	0.896289	+	0.443471i	$0.146253\pi$
$97$	7.50136e10i	0.886942i	−0.896289	+	0.443471i	$0.853747\pi$
$98$	4.07272e10i	0.455136i
$99$	0	0
$100$	0	0
$101$	−8.17430e10	−0.773896	−0.386948	−	0.922101i	$-0.626471\pi$
$101$	−8.17430e10	−0.773896	−0.386948	+	0.922101i	$0.626471\pi$
$102$	0	0
$103$	2.25755e11i	1.91881i	0.282025	+	0.959407i	$0.408994\pi$
$103$	2.25755e11i	1.91881i	−0.282025	+	0.959407i	$0.591006\pi$
$104$	−4.88073e10	−0.393370
$105$	0	0
$106$	3.83053e10	0.278021
$107$	− 9.02413e10i	− 0.622006i	−0.950409	−	0.311003i	$-0.899335\pi$
$107$	− 9.02413e10i	− 0.622006i	0.950409	−	0.311003i	$-0.100665\pi$
$108$	0	0
$109$	−7.34827e10	−0.457445	−0.228723	−	0.973492i	$-0.573455\pi$
$109$	−7.34827e10	−0.457445	−0.228723	+	0.973492i	$0.573455\pi$
$110$	0	0
$111$	0	0
$112$	− 1.65286e10i	− 0.0886211i
$113$	− 8.51469e10i	− 0.434748i	−0.976088	−	0.217374i	$-0.930251\pi$
$113$	− 8.51469e10i	− 0.434748i	0.976088	−	0.217374i	$-0.0697491\pi$
$114$	0	0
$115$	0	0
$116$	1.89015e11	0.835557
$117$	0	0
$118$	− 1.24541e11i	− 0.501142i
$119$	1.15633e11	0.444195
$120$	0	0
$121$	4.98320e8	0.00174658
$122$	1.66955e11i	0.559270i
$123$	0	0
$124$	−7.77851e10	−0.238274
$125$	0	0
$126$	0	0
$127$	− 2.62717e11i	− 0.705615i	−0.935696	−	0.352808i	$-0.885227\pi$
$127$	− 2.62717e11i	− 0.705615i	0.935696	−	0.352808i	$-0.114773\pi$
$128$	3.38071e11i	0.869668i
$129$	0	0
$130$	0	0
$131$	−6.31529e11	−1.43021	−0.715107	−	0.699015i	$-0.753622\pi$
$131$	−6.31529e11	−1.43021	−0.715107	+	0.699015i	$0.753622\pi$
$132$	0	0
$133$	1.78515e11i	0.371955i
$134$	3.71564e11	0.742946
$135$	0	0
$136$	−5.83413e11	−1.07526
$137$	2.97199e11i	0.526119i	0.964780	+	0.263059i	$0.0847315\pi$
$137$	2.97199e11i	0.526119i	−0.964780	+	0.263059i	$0.915268\pi$
$138$	0	0
$139$	−5.96794e11	−0.975535	−0.487767	−	0.872974i	$-0.662189\pi$
$139$	−5.96794e11	−0.975535	−0.487767	+	0.872974i	$0.662189\pi$
$140$	0	0
$141$	0	0
$142$	− 2.34996e11i	− 0.341565i
$143$	− 3.08866e11i	− 0.431938i
$144$	0	0
$145$	0	0
$146$	3.51310e10	0.0438278
$147$	0	0
$148$	− 2.68218e11i	− 0.310491i
$149$	−1.11543e12	−1.24428	−0.622142	−	0.782905i	$-0.713737\pi$
$149$	−1.11543e12	−1.24428	−0.622142	+	0.782905i	$0.713737\pi$
$150$	0	0
$151$	−8.24447e11	−0.854653	−0.427326	−	0.904097i	$-0.640544\pi$
$151$	−8.24447e11	−0.854653	−0.427326	+	0.904097i	$0.640544\pi$
$152$	− 9.00677e11i	− 0.900387i
$153$	0	0
$154$	−2.14837e11	−0.199869
$155$	0	0
$156$	0	0
$157$	1.31512e12i	1.10031i	0.835062	+	0.550156i	$0.185432\pi$
$157$	1.31512e12i	1.10031i	−0.835062	+	0.550156i	$0.814568\pi$
$158$	− 9.14804e11i	− 0.739119i
$159$	0	0
$160$	0	0
$161$	3.12163e11	0.227425
$162$	0	0
$163$	3.57833e11i	0.243584i	0.992556	+	0.121792i	$0.0388640\pi$
$163$	3.57833e11i	0.243584i	−0.992556	+	0.121792i	$0.961136\pi$
$164$	−4.53553e11	−0.298529
$165$	0	0
$166$	7.04042e11	0.433515
$167$	− 2.75483e12i	− 1.64117i	−0.571521	−	0.820587i	$-0.693646\pi$
$167$	− 2.75483e12i	− 1.64117i	0.571521	−	0.820587i	$-0.306354\pi$
$168$	0	0
$169$	1.45838e12	0.813755
$170$	0	0
$171$	0	0
$172$	2.52090e10i	0.0127688i
$173$	− 9.50387e11i	− 0.466280i	−0.972443	−	0.233140i	$-0.925100\pi$
$173$	− 9.50387e11i	− 0.466280i	0.972443	−	0.233140i	$-0.0749001\pi$
$174$	0	0
$175$	0	0
$176$	−5.27735e11	−0.235557
$177$	0	0
$178$	− 5.99830e11i	− 0.251604i
$179$	1.68138e12	0.683873	0.341936	−	0.939723i	$-0.388917\pi$
$179$	1.68138e12	0.683873	0.341936	+	0.939723i	$0.388917\pi$
$180$	0	0
$181$	−9.96774e11	−0.381386	−0.190693	−	0.981650i	$-0.561073\pi$
$181$	−9.96774e11	−0.381386	−0.190693	+	0.981650i	$0.561073\pi$
$182$	2.32167e11i	0.0861804i
$183$	0	0
$184$	−1.57498e12	−0.550526
$185$	0	0
$186$	0	0
$187$	− 3.69200e12i	− 1.18068i
$188$	− 3.95578e12i	− 1.22847i
$189$	0	0
$190$	0	0
$191$	−2.76240e12	−0.786328	−0.393164	−	0.919468i	$-0.628619\pi$
$191$	−2.76240e12	−0.786328	−0.393164	+	0.919468i	$0.628619\pi$
$192$	0	0
$193$	− 5.44239e12i	− 1.46293i	−0.681878	−	0.731466i	$-0.738836\pi$
$193$	− 5.44239e12i	− 1.46293i	0.681878	−	0.731466i	$-0.261164\pi$
$194$	−1.80033e12	−0.470372
$195$	0	0
$196$	2.49793e12	0.616840
$197$	2.87609e12i	0.690619i	0.938489	+	0.345309i	$0.112226\pi$
$197$	2.87609e12i	0.690619i	−0.938489	+	0.345309i	$0.887774\pi$
$198$	0	0
$199$	−7.28391e11	−0.165452	−0.0827262	−	0.996572i	$-0.526363\pi$
$199$	−7.28391e11	−0.165452	−0.0827262	+	0.996572i	$0.526363\pi$
$200$	0	0
$201$	0	0
$202$	− 1.96183e12i	− 0.410421i
$203$	− 2.15004e12i	− 0.437742i
$204$	0	0
$205$	0	0
$206$	−5.41812e12	−1.01760
$207$	0	0
$208$	5.70306e11i	0.101569i
$209$	5.69972e12	0.988665
$210$	0	0
$211$	−6.79317e12	−1.11820	−0.559099	−	0.829101i	$-0.688853\pi$
$211$	−6.79317e12	−1.11820	−0.559099	+	0.829101i	$0.688853\pi$
$212$	− 2.34939e12i	− 0.376798i
$213$	0	0
$214$	2.16579e12	0.329868
$215$	0	0
$216$	0	0
$217$	8.84806e11i	0.124830i
$218$	− 1.76358e12i	− 0.242597i
$219$	0	0
$220$	0	0
$221$	−3.98982e12	−0.509092
$222$	0	0
$223$	− 7.33486e12i	− 0.890667i	−0.895365	−	0.445333i	$-0.853085\pi$
$223$	− 7.33486e12i	− 0.890667i	0.895365	−	0.445333i	$-0.146915\pi$
$224$	3.29365e12	0.390223
$225$	0	0
$226$	2.04352e12	0.230560
$227$	1.35984e12i	0.149743i	0.997193	+	0.0748713i	$0.0238546\pi$
$227$	1.35984e12i	0.149743i	−0.997193	+	0.0748713i	$0.976145\pi$
$228$	0	0
$229$	1.18244e13	1.24075	0.620375	−	0.784305i	$-0.286980\pi$
$229$	1.18244e13	1.24075	0.620375	+	0.784305i	$0.286980\pi$
$230$	0	0
$231$	0	0
$232$	1.08478e13i	1.05964i
$233$	− 1.75634e13i	− 1.67552i	−0.546038	−	0.837761i	$-0.683865\pi$
$233$	− 1.75634e13i	− 1.67552i	0.546038	−	0.837761i	$-0.316135\pi$
$234$	0	0
$235$	0	0
$236$	−7.63851e12	−0.679192
$237$	0	0
$238$	2.77519e12i	0.235570i
$239$	−7.13958e12	−0.592221	−0.296111	−	0.955154i	$-0.595690\pi$
$239$	−7.13958e12	−0.592221	−0.296111	+	0.955154i	$0.595690\pi$
$240$	0	0
$241$	−2.31307e11	−0.0183271	−0.00916357	−	0.999958i	$-0.502917\pi$
$241$	−2.31307e11	−0.0183271	−0.00916357	+	0.999958i	$0.502917\pi$
$242$	1.19597e10i	0 0.000926264i
$243$	0	0
$244$	1.02399e13	0.757972
$245$	0	0
$246$	0	0
$247$	− 6.15951e12i	− 0.426297i
$248$	− 4.46419e12i	− 0.302175i
$249$	0	0
$250$	0	0
$251$	−1.29831e13	−0.822567	−0.411284	−	0.911507i	$-0.634919\pi$
$251$	−1.29831e13	−0.822567	−0.411284	+	0.911507i	$0.634919\pi$
$252$	0	0
$253$	− 9.96692e12i	− 0.604502i
$254$	6.30521e12	0.374209
$255$	0	0
$256$	−1.36419e13	−0.775451
$257$	− 2.39612e13i	− 1.33314i	−0.745442	−	0.666571i	$-0.767761\pi$
$257$	− 2.39612e13i	− 1.33314i	0.745442	−	0.666571i	$-0.232239\pi$
$258$	0	0
$259$	−3.05098e12	−0.162664
$260$	0	0
$261$	0	0
$262$	− 1.51567e13i	− 0.758485i
$263$	− 2.42737e13i	− 1.18954i	−0.803895	−	0.594771i	$-0.797243\pi$
$263$	− 2.42737e13i	− 1.18954i	0.803895	−	0.594771i	$-0.202757\pi$
$264$	0	0
$265$	0	0
$266$	−4.28436e12	−0.197259
$267$	0	0
$268$	− 2.27892e13i	− 1.00691i
$269$	2.58377e13	1.11845	0.559225	−	0.829016i	$-0.311099\pi$
$269$	2.58377e13	1.11845	0.559225	+	0.829016i	$0.311099\pi$
$270$	0	0
$271$	−3.76793e12	−0.156593	−0.0782964	−	0.996930i	$-0.524948\pi$
$271$	−3.76793e12	−0.156593	−0.0782964	+	0.996930i	$0.524948\pi$
$272$	6.81710e12i	0.277633i
$273$	0	0
$274$	−7.13277e12	−0.279017
$275$	0	0
$276$	0	0
$277$	− 1.64189e13i	− 0.604931i	−0.953160	−	0.302466i	$-0.902190\pi$
$277$	− 1.64189e13i	− 0.604931i	0.953160	−	0.302466i	$-0.0978098\pi$
$278$	− 1.43230e13i	− 0.517355i
$279$	0	0
$280$	0	0
$281$	−2.10357e13	−0.716263	−0.358132	−	0.933671i	$-0.616586\pi$
$281$	−2.10357e13	−0.716263	−0.358132	+	0.933671i	$0.616586\pi$
$282$	0	0
$283$	− 1.67132e13i	− 0.547310i	−0.961828	−	0.273655i	$-0.911767\pi$
$283$	− 1.67132e13i	− 0.547310i	0.961828	−	0.273655i	$-0.0882327\pi$
$284$	−1.44131e13	−0.462920
$285$	0	0
$286$	7.41278e12	0.229070
$287$	5.15917e12i	0.156397i
$288$	0	0
$289$	−1.34200e13	−0.391575
$290$	0	0
$291$	0	0
$292$	− 2.15470e12i	− 0.0593993i
$293$	− 2.39269e13i	− 0.647312i	−0.946175	−	0.323656i	$-0.895088\pi$
$293$	− 2.39269e13i	− 0.647312i	0.946175	−	0.323656i	$-0.104912\pi$
$294$	0	0
$295$	0	0
$296$	1.53934e13	0.393758
$297$	0	0
$298$	− 2.67704e13i	− 0.659881i
$299$	−1.07709e13	−0.260652
$300$	0	0
$301$	2.86753e11	0.00668947
$302$	− 1.97867e13i	− 0.453248i
$303$	0	0
$304$	−1.05243e13	−0.232481
$305$	0	0
$306$	0	0
$307$	1.53111e13i	0.320439i	0.987081	+	0.160219i	$0.0512202\pi$
$307$	1.53111e13i	0.320439i	−0.987081	+	0.160219i	$0.948780\pi$
$308$	1.31767e13i	0.270880i
$309$	0	0
$310$	0	0
$311$	−4.98752e13	−0.972080	−0.486040	−	0.873936i	$-0.661559\pi$
$311$	−4.98752e13	−0.972080	−0.486040	+	0.873936i	$0.661559\pi$
$312$	0	0
$313$	9.94808e13i	1.87174i	0.352345	+	0.935870i	$0.385384\pi$
$313$	9.94808e13i	1.87174i	−0.352345	+	0.935870i	$0.614616\pi$
$314$	−3.15628e13	−0.583529
$315$	0	0
$316$	−5.61080e13	−1.00172
$317$	− 8.33692e13i	− 1.46278i	−0.681958	−	0.731392i	$-0.738871\pi$
$317$	− 8.33692e13i	− 1.46278i	0.681958	−	0.731392i	$-0.261129\pi$
$318$	0	0
$319$	−6.86477e13	−1.16353
$320$	0	0
$321$	0	0
$322$	7.49191e12i	0.120611i
$323$	− 7.36271e13i	− 1.16526i
$324$	0	0
$325$	0	0
$326$	−8.58799e12	−0.129180
$327$	0	0
$328$	− 2.60300e13i	− 0.378589i
$329$	−4.49970e13	−0.643585
$330$	0	0
$331$	−6.35840e13	−0.879618	−0.439809	−	0.898091i	$-0.644954\pi$
$331$	−6.35840e13	−0.879618	−0.439809	+	0.898091i	$0.644954\pi$
$332$	− 4.31813e13i	− 0.587538i
$333$	0	0
$334$	6.61160e13	0.870364
$335$	0	0
$336$	0	0
$337$	1.21001e14i	1.51644i	0.651997	+	0.758221i	$0.273931\pi$
$337$	1.21001e14i	1.51644i	−0.651997	+	0.758221i	$0.726069\pi$
$338$	3.50011e13i	0.431559i
$339$	0	0
$340$	0	0
$341$	2.82506e13	0.331802
$342$	0	0
$343$	− 6.15223e13i	− 0.699705i
$344$	−1.44678e12	−0.0161931
$345$	0	0
$346$	2.28093e13	0.247283
$347$	1.55662e14i	1.66100i	0.557020	+	0.830499i	$0.311945\pi$
$347$	1.55662e14i	1.66100i	−0.557020	+	0.830499i	$0.688055\pi$
$348$	0	0
$349$	2.56430e13	0.265112	0.132556	−	0.991176i	$-0.457682\pi$
$349$	2.56430e13	0.265112	0.132556	+	0.991176i	$0.457682\pi$
$350$	0	0
$351$	0	0
$352$	− 1.05162e14i	− 1.03722i
$353$	2.49098e13i	0.241885i	0.992659	+	0.120943i	$0.0385917\pi$
$353$	2.49098e13i	0.241885i	−0.992659	+	0.120943i	$0.961408\pi$
$354$	0	0
$355$	0	0
$356$	−3.67896e13	−0.340996
$357$	0	0
$358$	4.03532e13i	0.362678i
$359$	1.57584e14	1.39474	0.697370	−	0.716712i	$-0.254354\pi$
$359$	1.57584e14	1.39474	0.697370	+	0.716712i	$0.254354\pi$
$360$	0	0
$361$	−2.82438e12	−0.0242457
$362$	− 2.39226e13i	− 0.202260i
$363$	0	0
$364$	1.42396e13	0.116799
$365$	0	0
$366$	0	0
$367$	− 1.77901e14i	− 1.39481i	−0.716676	−	0.697406i	$-0.754338\pi$
$367$	− 1.77901e14i	− 1.39481i	0.716676	−	0.697406i	$-0.245662\pi$
$368$	1.84034e13i	0.142146i
$369$	0	0
$370$	0	0
$371$	−2.67244e13	−0.197402
$372$	0	0
$373$	5.51617e13i	0.395585i	0.980244	+	0.197792i	$0.0633772\pi$
$373$	5.51617e13i	0.395585i	−0.980244	+	0.197792i	$0.936623\pi$
$374$	8.86079e13	0.626150
$375$	0	0
$376$	2.27027e14	1.55792
$377$	7.41854e13i	0.501696i
$378$	0	0
$379$	−1.46463e14	−0.962083	−0.481042	−	0.876698i	$-0.659741\pi$
$379$	−1.46463e14	−0.962083	−0.481042	+	0.876698i	$0.659741\pi$
$380$	0	0
$381$	0	0
$382$	− 6.62977e13i	− 0.417013i
$383$	2.31450e14i	1.43504i	0.696539	+	0.717519i	$0.254722\pi$
$383$	2.31450e14i	1.43504i	−0.696539	+	0.717519i	$0.745278\pi$
$384$	0	0
$385$	0	0
$386$	1.30617e14	0.775837
$387$	0	0
$388$	1.10420e14i	0.637490i
$389$	−1.49872e14	−0.853093	−0.426547	−	0.904466i	$-0.640270\pi$
$389$	−1.49872e14	−0.853093	−0.426547	+	0.904466i	$0.640270\pi$
$390$	0	0
$391$	−1.28749e14	−0.712480
$392$	1.43360e14i	0.782264i
$393$	0	0
$394$	−6.90262e13	−0.366256
$395$	0	0
$396$	0	0
$397$	2.08111e14i	1.05912i	0.848271	+	0.529562i	$0.177644\pi$
$397$	2.08111e14i	1.05912i	−0.848271	+	0.529562i	$0.822356\pi$
$398$	− 1.74814e13i	− 0.0877443i
$399$	0	0
$400$	0	0
$401$	1.33408e14	0.642521	0.321261	−	0.946991i	$-0.395893\pi$
$401$	1.33408e14	0.642521	0.321261	+	0.946991i	$0.395893\pi$
$402$	0	0
$403$	− 3.05295e13i	− 0.143068i
$404$	−1.20326e14	−0.556238
$405$	0	0
$406$	5.16010e13	0.232148
$407$	9.74134e13i	0.432364i
$408$	0	0
$409$	2.06168e14	0.890722	0.445361	−	0.895351i	$-0.353075\pi$
$409$	2.06168e14	0.890722	0.445361	+	0.895351i	$0.353075\pi$
$410$	0	0
$411$	0	0
$412$	3.32312e14i	1.37915i
$413$	8.68880e13i	0.355824i
$414$	0	0
$415$	0	0
$416$	−1.13645e14	−0.447235
$417$	0	0
$418$	1.36793e14i	0.524319i
$419$	7.34035e13	0.277677	0.138838	−	0.990315i	$-0.455663\pi$
$419$	7.34035e13	0.277677	0.138838	+	0.990315i	$0.455663\pi$
$420$	0	0
$421$	1.71112e14	0.630563	0.315282	−	0.948998i	$-0.397901\pi$
$421$	1.71112e14	0.630563	0.315282	+	0.948998i	$0.397901\pi$
$422$	− 1.63036e14i	− 0.593014i
$423$	0	0
$424$	1.34835e14	0.477848
$425$	0	0
$426$	0	0
$427$	− 1.16479e14i	− 0.397096i
$428$	− 1.32835e14i	− 0.447067i
$429$	0	0
$430$	0	0
$431$	7.17758e13	0.232463	0.116231	−	0.993222i	$-0.462919\pi$
$431$	7.17758e13	0.232463	0.116231	+	0.993222i	$0.462919\pi$
$432$	0	0
$433$	− 9.98812e13i	− 0.315356i	−0.987491	−	0.157678i	$-0.949599\pi$
$433$	− 9.98812e13i	− 0.315356i	0.987491	−	0.157678i	$-0.0504007\pi$
$434$	−2.12353e13	−0.0662012
$435$	0	0
$436$	−1.08166e14	−0.328789
$437$	− 1.98764e14i	− 0.596608i
$438$	0	0
$439$	2.90312e13	0.0849788	0.0424894	−	0.999097i	$-0.486471\pi$
$439$	2.90312e13	0.0849788	0.0424894	+	0.999097i	$0.486471\pi$
$440$	0	0
$441$	0	0
$442$	− 9.57557e13i	− 0.269987i
$443$	3.28370e14i	0.914414i	0.889360	+	0.457207i	$0.151150\pi$
$443$	3.28370e14i	0.914414i	−0.889360	+	0.457207i	$0.848850\pi$
$444$	0	0
$445$	0	0
$446$	1.76037e14	0.472347
$447$	0	0
$448$	4.51970e13i	0.118326i
$449$	−6.12368e14	−1.58364	−0.791822	−	0.610752i	$-0.790867\pi$
$449$	−6.12368e14	−1.58364	−0.791822	+	0.610752i	$0.790867\pi$
$450$	0	0
$451$	1.64725e14	0.415708
$452$	− 1.25336e14i	− 0.312475i
$453$	0	0
$454$	−3.26361e13	−0.0794130
$455$	0	0
$456$	0	0
$457$	3.03483e14i	0.712189i	0.934450	+	0.356095i	$0.115892\pi$
$457$	3.03483e14i	0.712189i	−0.934450	+	0.356095i	$0.884108\pi$
$458$	2.83786e14i	0.658007i
$459$	0	0
$460$	0	0
$461$	7.29308e14	1.63138	0.815691	−	0.578487i	$-0.196357\pi$
$461$	7.29308e14	1.63138	0.815691	+	0.578487i	$0.196357\pi$
$462$	0	0
$463$	− 1.22188e14i	− 0.266891i	−0.991056	−	0.133445i	$-0.957396\pi$
$463$	− 1.22188e14i	− 0.266891i	0.991056	−	0.133445i	$-0.0426041\pi$
$464$	1.26755e14	0.273600
$465$	0	0
$466$	4.21520e14	0.888579
$467$	6.17381e14i	1.28621i	0.765780	+	0.643103i	$0.222353\pi$
$467$	6.17381e14i	1.28621i	−0.765780	+	0.643103i	$0.777647\pi$
$468$	0	0
$469$	−2.59228e14	−0.527510
$470$	0	0
$471$	0	0
$472$	− 4.38384e14i	− 0.861338i
$473$	− 9.15561e12i	− 0.0177808i
$474$	0	0
$475$	0	0
$476$	1.70212e14	0.319265
$477$	0	0
$478$	− 1.71350e14i	− 0.314073i
$479$	1.05084e15	1.90410	0.952052	−	0.305938i	$-0.0989700\pi$
$479$	1.05084e15	1.90410	0.952052	+	0.305938i	$0.0989700\pi$
$480$	0	0
$481$	1.05272e14	0.186429
$482$	− 5.55137e12i	− 0.00971944i
$483$	0	0
$484$	7.33527e11	0.00125536
$485$	0	0
$486$	0	0
$487$	− 2.19910e14i	− 0.363777i	−0.983319	−	0.181889i	$-0.941779\pi$
$487$	− 2.19910e14i	− 0.363777i	0.983319	−	0.181889i	$-0.0582210\pi$
$488$	5.87683e14i	0.961246i
$489$	0	0
$490$	0	0
$491$	4.83863e14	0.765199	0.382599	−	0.923914i	$-0.375029\pi$
$491$	4.83863e14	0.765199	0.382599	+	0.923914i	$0.375029\pi$
$492$	0	0
$493$	8.86768e14i	1.37136i
$494$	1.47828e14	0.226078
$495$	0	0
$496$	−5.21634e13	−0.0780219
$497$	1.63949e14i	0.242520i
$498$	0	0
$499$	1.08878e14	0.157538	0.0787691	−	0.996893i	$-0.474901\pi$
$499$	1.08878e14	0.157538	0.0787691	+	0.996893i	$0.474901\pi$
$500$	0	0
$501$	0	0
$502$	− 3.11593e14i	− 0.436232i
$503$	5.06588e14i	0.701506i	0.936468	+	0.350753i	$0.114074\pi$
$503$	5.06588e14i	0.701506i	−0.936468	+	0.350753i	$0.885926\pi$
$504$	0	0
$505$	0	0
$506$	2.39206e14	0.320586
$507$	0	0
$508$	− 3.86720e14i	− 0.507161i
$509$	8.57534e13	0.111251	0.0556254	−	0.998452i	$-0.482285\pi$
$509$	8.57534e13	0.111251	0.0556254	+	0.998452i	$0.482285\pi$
$510$	0	0
$511$	−2.45097e13	−0.0311188
$512$	3.64965e14i	0.458423i
$513$	0	0
$514$	5.75069e14	0.707005
$515$	0	0
$516$	0	0
$517$	1.43669e15i	1.71066i
$518$	− 7.32235e13i	− 0.0862654i
$519$	0	0
$520$	0	0
$521$	−9.27575e14	−1.05862	−0.529312	−	0.848428i	$-0.677550\pi$
$521$	−9.27575e14	−1.05862	−0.529312	+	0.848428i	$0.677550\pi$
$522$	0	0
$523$	2.18187e13i	0.0243820i	0.999926	+	0.0121910i	$0.00388061\pi$
$523$	2.18187e13i	0.0243820i	−0.999926	+	0.0121910i	$0.996119\pi$
$524$	−9.29610e14	−1.02797
$525$	0	0
$526$	5.82569e14	0.630850
$527$	− 3.64931e14i	− 0.391069i
$528$	0	0
$529$	6.05238e14	0.635214
$530$	0	0
$531$	0	0
$532$	2.62774e14i	0.267343i
$533$	− 1.78013e14i	− 0.179247i
$534$	0	0
$535$	0	0
$536$	1.30790e15	1.27694
$537$	0	0
$538$	6.20105e14i	0.593147i
$539$	−9.07218e14	−0.858961
$540$	0	0
$541$	−1.69527e15	−1.57273	−0.786363	−	0.617765i	$-0.788038\pi$
$541$	−1.69527e15	−1.57273	−0.786363	+	0.617765i	$0.788038\pi$
$542$	− 9.04304e13i	− 0.0830459i
$543$	0	0
$544$	−1.35844e15	−1.22249
$545$	0	0
$546$	0	0
$547$	7.52145e14i	0.656706i	0.944555	+	0.328353i	$0.106494\pi$
$547$	7.52145e14i	0.656706i	−0.944555	+	0.328353i	$0.893506\pi$
$548$	4.37477e14i	0.378148i
$549$	0	0
$550$	0	0
$551$	−1.36900e15	−1.14834
$552$	0	0
$553$	6.38228e14i	0.524793i
$554$	3.94054e14	0.320813
$555$	0	0
$556$	−8.78480e14	−0.701166
$557$	− 1.87489e14i	− 0.148174i	−0.997252	−	0.0740870i	$-0.976396\pi$
$557$	− 1.87489e14i	− 0.148174i	0.997252	−	0.0740870i	$-0.0236043\pi$
$558$	0	0
$559$	−9.89417e12	−0.00766681
$560$	0	0
$561$	0	0
$562$	− 5.04857e14i	− 0.379856i
$563$	2.44971e14i	0.182524i	0.995827	+	0.0912618i	$0.0290900\pi$
$563$	2.44971e14i	0.182524i	−0.995827	+	0.0912618i	$0.970910\pi$
$564$	0	0
$565$	0	0
$566$	4.01116e14	0.290255
$567$	0	0
$568$	− 8.27185e14i	− 0.587066i
$569$	1.35243e15	0.950596	0.475298	−	0.879825i	$-0.342340\pi$
$569$	1.35243e15	0.950596	0.475298	+	0.879825i	$0.342340\pi$
$570$	0	0
$571$	1.43223e15	0.987447	0.493723	−	0.869619i	$-0.335636\pi$
$571$	1.43223e15	0.987447	0.493723	+	0.869619i	$0.335636\pi$
$572$	− 4.54650e14i	− 0.310455i
$573$	0	0
$574$	−1.23820e14	−0.0829422
$575$	0	0
$576$	0	0
$577$	− 8.77659e14i	− 0.571293i	−0.958335	−	0.285647i	$-0.907792\pi$
$577$	− 8.77659e14i	− 0.571293i	0.958335	−	0.285647i	$-0.0922083\pi$
$578$	− 3.22081e14i	− 0.207664i
$579$	0	0
$580$	0	0
$581$	−4.91187e14	−0.307807
$582$	0	0
$583$	8.53271e14i	0.524698i
$584$	1.23661e14	0.0753290
$585$	0	0
$586$	5.74245e14	0.343289
$587$	2.43425e15i	1.44164i	0.693124	+	0.720818i	$0.256234\pi$
$587$	2.43425e15i	1.44164i	−0.693124	+	0.720818i	$0.743766\pi$
$588$	0	0
$589$	5.63383e14	0.327469
$590$	0	0
$591$	0	0
$592$	− 1.79869e14i	− 0.101669i
$593$	− 3.03318e14i	− 0.169863i	−0.996387	−	0.0849313i	$-0.972933\pi$
$593$	− 3.03318e14i	− 0.169863i	0.996387	−	0.0849313i	$-0.0270671\pi$
$594$	0	0
$595$	0	0
$596$	−1.64192e15	−0.894329
$597$	0	0
$598$	− 2.58502e14i	− 0.138232i
$599$	−1.70198e15	−0.901795	−0.450898	−	0.892576i	$-0.648896\pi$
$599$	−1.70198e15	−0.901795	−0.450898	+	0.892576i	$0.648896\pi$
$600$	0	0
$601$	2.33922e15	1.21692	0.608458	−	0.793586i	$-0.291788\pi$
$601$	2.33922e15	1.21692	0.608458	+	0.793586i	$0.291788\pi$
$602$	6.88207e12i	0.00354763i
$603$	0	0
$604$	−1.21359e15	−0.614282
$605$	0	0
$606$	0	0
$607$	− 2.49607e15i	− 1.22947i	−0.788732	−	0.614737i	$-0.789262\pi$
$607$	− 2.49607e15i	− 1.22947i	0.788732	−	0.614737i	$-0.210738\pi$
$608$	− 2.09717e15i	− 1.02368i
$609$	0	0
$610$	0	0
$611$	1.55258e15	0.737612
$612$	0	0
$613$	− 2.47301e15i	− 1.15397i	−0.816756	−	0.576983i	$-0.804230\pi$
$613$	− 2.47301e15i	− 1.15397i	0.816756	−	0.576983i	$-0.195770\pi$
$614$	−3.67466e14	−0.169938
$615$	0	0
$616$	−7.56226e14	−0.343525
$617$	− 2.43368e13i	− 0.0109571i	−0.999985	−	0.00547854i	$-0.998256\pi$
$617$	− 2.43368e13i	− 0.0109571i	0.999985	−	0.00547854i	$-0.00174388\pi$
$618$	0	0
$619$	−4.22545e15	−1.86885	−0.934425	−	0.356160i	$-0.884086\pi$
$619$	−4.22545e15	−1.86885	−0.934425	+	0.356160i	$0.884086\pi$
$620$	0	0
$621$	0	0
$622$	− 1.19700e15i	− 0.515523i
$623$	4.18481e14i	0.178645i
$624$	0	0
$625$	0	0
$626$	−2.38754e15	−0.992640
$627$	0	0
$628$	1.93585e15i	0.790850i
$629$	1.25835e15	0.509594
$630$	0	0
$631$	−4.26326e15	−1.69660	−0.848302	−	0.529513i	$-0.822375\pi$
$631$	−4.26326e15	−1.69660	−0.848302	+	0.529513i	$0.822375\pi$
$632$	− 3.22011e15i	− 1.27036i
$633$	0	0
$634$	2.00086e15	0.775758
$635$	0	0
$636$	0	0
$637$	9.80401e14i	0.370371i
$638$	− 1.64755e15i	− 0.617055i
$639$	0	0
$640$	0	0
$641$	−1.00830e15	−0.368018	−0.184009	−	0.982925i	$-0.558908\pi$
$641$	−1.00830e15	−0.368018	−0.184009	+	0.982925i	$0.558908\pi$
$642$	0	0
$643$	− 3.03982e14i	− 0.109066i	−0.998512	−	0.0545328i	$-0.982633\pi$
$643$	− 3.03982e14i	− 0.109066i	0.998512	−	0.0545328i	$-0.0173670\pi$
$644$	4.59504e14	0.163462
$645$	0	0
$646$	1.76705e15	0.617974
$647$	− 3.43583e15i	− 1.19140i	−0.803207	−	0.595700i	$-0.796875\pi$
$647$	− 3.43583e15i	− 1.19140i	0.803207	−	0.595700i	$-0.203125\pi$
$648$	0	0
$649$	2.77421e15	0.945788
$650$	0	0
$651$	0	0
$652$	5.26730e14i	0.175076i
$653$	− 1.18539e15i	− 0.390695i	−0.980734	−	0.195347i	$-0.937417\pi$
$653$	− 1.18539e15i	− 0.390695i	0.980734	−	0.195347i	$-0.0625834\pi$
$654$	0	0
$655$	0	0
$656$	−3.04157e14	−0.0977522
$657$	0	0
$658$	− 1.07993e15i	− 0.341312i
$659$	−2.26510e15	−0.709934	−0.354967	−	0.934879i	$-0.615508\pi$
$659$	−2.26510e15	−0.709934	−0.354967	+	0.934879i	$0.615508\pi$
$660$	0	0
$661$	−5.33012e15	−1.64297	−0.821484	−	0.570232i	$-0.806853\pi$
$661$	−5.33012e15	−1.64297	−0.821484	+	0.570232i	$0.806853\pi$
$662$	− 1.52602e15i	− 0.466488i
$663$	0	0
$664$	2.47823e15	0.745104
$665$	0	0
$666$	0	0
$667$	2.39392e15i	0.702130i
$668$	− 4.05512e15i	− 1.17959i
$669$	0	0
$670$	0	0
$671$	−3.71902e15	−1.05549
$672$	0	0
$673$	− 4.74120e15i	− 1.32375i	−0.749615	−	0.661874i	$-0.769761\pi$
$673$	− 4.74120e15i	− 1.32375i	0.749615	−	0.661874i	$-0.230239\pi$
$674$	−2.90403e15	−0.804215
$675$	0	0
$676$	2.14673e15	0.584886
$677$	1.41307e15i	0.381880i	0.981602	+	0.190940i	$0.0611535\pi$
$677$	1.41307e15i	0.381880i	−0.981602	+	0.190940i	$0.938846\pi$
$678$	0	0
$679$	1.25603e15	0.333976
$680$	0	0
$681$	0	0
$682$	6.78014e14i	0.175964i
$683$	− 3.03116e15i	− 0.780359i	−0.920739	−	0.390180i	$-0.872413\pi$
$683$	− 3.03116e15i	− 0.780359i	0.920739	−	0.390180i	$-0.127587\pi$
$684$	0	0
$685$	0	0
$686$	1.47654e15	0.371075
$687$	0	0
$688$	1.69054e13i	0.00418109i
$689$	9.22102e14	0.226242
$690$	0	0
$691$	−2.74731e15	−0.663405	−0.331703	−	0.943384i	$-0.607623\pi$
$691$	−2.74731e15	−0.663405	−0.331703	+	0.943384i	$0.607623\pi$
$692$	− 1.39897e15i	− 0.335139i
$693$	0	0
$694$	−3.73588e15	−0.880878
$695$	0	0
$696$	0	0
$697$	− 2.12786e15i	− 0.489962i
$698$	6.15433e14i	0.140597i
$699$	0	0
$700$	0	0
$701$	−5.72747e15	−1.27795	−0.638974	−	0.769228i	$-0.720641\pi$
$701$	−5.72747e15	−1.27795	−0.638974	+	0.769228i	$0.720641\pi$
$702$	0	0
$703$	1.94265e15i	0.426718i
$704$	1.44308e15	0.314514
$705$	0	0
$706$	−5.97836e14	−0.128279
$707$	1.36870e15i	0.291409i
$708$	0	0
$709$	−6.98326e14	−0.146388	−0.0731938	−	0.997318i	$-0.523319\pi$
$709$	−6.98326e14	−0.146388	−0.0731938	+	0.997318i	$0.523319\pi$
$710$	0	0
$711$	0	0
$712$	− 2.11140e15i	− 0.432445i
$713$	− 9.85170e14i	− 0.200225i
$714$	0	0
$715$	0	0
$716$	2.47500e15	0.491534
$717$	0	0
$718$	3.78202e15i	0.739672i
$719$	9.70979e15	1.88452	0.942260	−	0.334882i	$-0.108696\pi$
$719$	9.70979e15	1.88452	0.942260	+	0.334882i	$0.108696\pi$
$720$	0	0
$721$	3.78004e15	0.722525
$722$	− 6.77852e13i	− 0.0128582i
$723$	0	0
$724$	−1.46725e15	−0.274121
$725$	0	0
$726$	0	0
$727$	2.46469e15i	0.450114i	0.974346	+	0.225057i	$0.0722568\pi$
$727$	2.46469e15i	0.450114i	−0.974346	+	0.225057i	$0.927743\pi$
$728$	8.17230e14i	0.148123i
$729$	0	0
$730$	0	0
$731$	−1.18269e14	−0.0209568
$732$	0	0
$733$	− 7.91285e15i	− 1.38121i	−0.723230	−	0.690607i	$-0.757343\pi$
$733$	− 7.91285e15i	− 1.38121i	0.723230	−	0.690607i	$-0.242657\pi$
$734$	4.26963e15	0.739711
$735$	0	0
$736$	−3.66725e15	−0.625911
$737$	8.27677e15i	1.40213i
$738$	0	0
$739$	8.40694e15	1.40312	0.701558	−	0.712613i	$-0.252488\pi$
$739$	8.40694e15	1.40312	0.701558	+	0.712613i	$0.252488\pi$
$740$	0	0
$741$	0	0
$742$	− 6.41385e14i	− 0.104688i
$743$	1.36287e15i	0.220809i	0.993887	+	0.110404i	$0.0352146\pi$
$743$	1.36287e15i	0.220809i	−0.993887	+	0.110404i	$0.964785\pi$
$744$	0	0
$745$	0	0
$746$	−1.32388e15	−0.209790
$747$	0	0
$748$	− 5.43462e15i	− 0.848614i
$749$	−1.51100e15	−0.234215
$750$	0	0
$751$	6.81722e15	1.04133	0.520664	−	0.853762i	$-0.325684\pi$
$751$	6.81722e15	1.04133	0.520664	+	0.853762i	$0.325684\pi$
$752$	− 2.65278e15i	− 0.402256i
$753$	0	0
$754$	−1.78045e15	−0.266065
$755$	0	0
$756$	0	0
$757$	− 6.67049e14i	− 0.0975282i	−0.998810	−	0.0487641i	$-0.984472\pi$
$757$	− 6.67049e14i	− 0.0975282i	0.998810	−	0.0487641i	$-0.0155283\pi$
$758$	− 3.51511e15i	− 0.510222i
$759$	0	0
$760$	0	0
$761$	7.74408e15	1.09990	0.549951	−	0.835197i	$-0.314646\pi$
$761$	7.74408e15	1.09990	0.549951	+	0.835197i	$0.314646\pi$
$762$	0	0
$763$	1.23039e15i	0.172250i
$764$	−4.06626e15	−0.565173
$765$	0	0
$766$	−5.55479e15	−0.761043
$767$	− 2.99800e15i	− 0.407809i
$768$	0	0
$769$	−2.52411e15	−0.338465	−0.169232	−	0.985576i	$-0.554129\pi$
$769$	−2.52411e15	−0.338465	−0.169232	+	0.985576i	$0.554129\pi$
$770$	0	0
$771$	0	0
$772$	− 8.01119e15i	− 1.05148i
$773$	− 1.11453e16i	− 1.45246i	−0.687453	−	0.726229i	$-0.741271\pi$
$773$	− 1.11453e16i	− 1.45246i	0.687453	−	0.726229i	$-0.258729\pi$
$774$	0	0
$775$	0	0
$776$	−6.33715e15	−0.808452
$777$	0	0
$778$	− 3.59692e15i	− 0.452421i
$779$	3.28500e15	0.410279
$780$	0	0
$781$	5.23465e15	0.644624
$782$	− 3.08998e15i	− 0.377849i
$783$	0	0
$784$	1.67514e15	0.201981
$785$	0	0
$786$	0	0
$787$	1.32271e16i	1.56172i	0.624705	+	0.780861i	$0.285219\pi$
$787$	1.32271e16i	1.56172i	−0.624705	+	0.780861i	$0.714781\pi$
$788$	4.23361e15i	0.496382i
$789$	0	0
$790$	0	0
$791$	−1.42570e15	−0.163703
$792$	0	0
$793$	4.01902e15i	0.455112i
$794$	−4.99466e15	−0.561685
$795$	0	0
$796$	−1.07219e15	−0.118919
$797$	− 2.30248e15i	− 0.253615i	−0.991927	−	0.126807i	$-0.959527\pi$
$797$	− 2.30248e15i	− 0.253615i	0.991927	−	0.126807i	$-0.0404730\pi$
$798$	0	0
$799$	1.85587e16	2.01623
$800$	0	0
$801$	0	0
$802$	3.20179e15i	0.340748i
$803$	7.82560e14i	0.0827146i
$804$	0	0
$805$	0	0
$806$	7.32708e14	0.0758732
$807$	0	0
$808$	− 6.90565e15i	− 0.705410i
$809$	5.60472e15	0.568639	0.284320	−	0.958730i	$-0.408232\pi$
$809$	5.60472e15	0.568639	0.284320	+	0.958730i	$0.408232\pi$
$810$	0	0
$811$	−5.08516e15	−0.508968	−0.254484	−	0.967077i	$-0.581906\pi$
$811$	−5.08516e15	−0.508968	−0.254484	+	0.967077i	$0.581906\pi$
$812$	− 3.16486e15i	− 0.314627i
$813$	0	0
$814$	−2.33792e15	−0.229296
$815$	0	0
$816$	0	0
$817$	− 1.82584e14i	− 0.0175486i
$818$	4.94802e15i	0.472377i
$819$	0	0
$820$	0	0
$821$	−2.79111e14	−0.0261150	−0.0130575	−	0.999915i	$-0.504156\pi$
$821$	−2.79111e14	−0.0261150	−0.0130575	+	0.999915i	$0.504156\pi$
$822$	0	0
$823$	1.35265e16i	1.24878i	0.781112	+	0.624391i	$0.214653\pi$
$823$	1.35265e16i	1.24878i	−0.781112	+	0.624391i	$0.785347\pi$
$824$	−1.90718e16	−1.74901
$825$	0	0
$826$	−2.08531e15	−0.188704
$827$	− 2.72544e14i	− 0.0244994i	−0.999925	−	0.0122497i	$-0.996101\pi$
$827$	− 2.72544e14i	− 0.0244994i	0.999925	−	0.0122497i	$-0.00389930\pi$
$828$	0	0
$829$	−1.80459e16	−1.60077	−0.800385	−	0.599486i	$-0.795372\pi$
$829$	−1.80459e16	−1.60077	−0.800385	+	0.599486i	$0.795372\pi$
$830$	0	0
$831$	0	0
$832$	− 1.55949e15i	− 0.135614i
$833$	1.17191e16i	1.01239i
$834$	0	0
$835$	0	0
$836$	8.38999e15	0.710603
$837$	0	0
$838$	1.76168e15i	0.147260i
$839$	−7.96183e15	−0.661184	−0.330592	−	0.943774i	$-0.607248\pi$
$839$	−7.96183e15	−0.661184	−0.330592	+	0.943774i	$0.607248\pi$
$840$	0	0
$841$	4.28775e15	0.351440
$842$	4.10669e15i	0.334407i
$843$	0	0
$844$	−9.99954e15	−0.803705
$845$	0	0
$846$	0	0
$847$	− 8.34387e12i	0 0.000657671i
$848$	− 1.57552e15i	− 0.123381i
$849$	0	0
$850$	0	0
$851$	3.39705e15	0.260909
$852$	0	0
$853$	1.49826e16i	1.13598i	0.823037	+	0.567988i	$0.192278\pi$
$853$	1.49826e16i	1.13598i	−0.823037	+	0.567988i	$0.807722\pi$
$854$	2.79550e15	0.210592
$855$	0	0
$856$	7.62358e15	0.566961
$857$	2.22561e16i	1.64458i	0.569068	+	0.822290i	$0.307304\pi$
$857$	2.22561e16i	1.64458i	−0.569068	+	0.822290i	$0.692696\pi$
$858$	0	0
$859$	−5.44237e15	−0.397032	−0.198516	−	0.980098i	$-0.563612\pi$
$859$	−5.44237e15	−0.397032	−0.198516	+	0.980098i	$0.563612\pi$
$860$	0	0
$861$	0	0
$862$	1.72262e15i	0.123282i
$863$	1.08110e16i	0.768787i	0.923169	+	0.384393i	$0.125589\pi$
$863$	1.08110e16i	0.768787i	−0.923169	+	0.384393i	$0.874411\pi$
$864$	0	0
$865$	0	0
$866$	2.39715e15	0.167243
$867$	0	0
$868$	1.30243e15i	0.0897217i
$869$	2.03777e16	1.39491
$870$	0	0
$871$	8.94444e15	0.604579
$872$	− 6.20782e15i	− 0.416964i
$873$	0	0
$874$	4.77033e15	0.316399
$875$	0	0
$876$	0	0
$877$	− 2.81024e16i	− 1.82914i	−0.404431	−	0.914568i	$-0.632530\pi$
$877$	− 2.81024e16i	− 1.82914i	0.404431	−	0.914568i	$-0.367470\pi$
$878$	6.96749e14i	0.0450668i
$879$	0	0
$880$	0	0
$881$	−4.22209e15	−0.268016	−0.134008	−	0.990980i	$-0.542785\pi$
$881$	−4.22209e15	−0.268016	−0.134008	+	0.990980i	$0.542785\pi$
$882$	0	0
$883$	− 5.16092e14i	− 0.0323551i	−0.999869	−	0.0161776i	$-0.994850\pi$
$883$	− 5.16092e14i	− 0.0323551i	0.999869	−	0.0161776i	$-0.00514970\pi$
$884$	−5.87302e15	−0.365910
$885$	0	0
$886$	−7.88088e15	−0.484941
$887$	− 5.71906e15i	− 0.349740i	−0.984592	−	0.174870i	$-0.944050\pi$
$887$	− 5.71906e15i	− 0.349740i	0.984592	−	0.174870i	$-0.0559504\pi$
$888$	0	0
$889$	−4.39894e15	−0.265698
$890$	0	0
$891$	0	0
$892$	− 1.07969e16i	− 0.640167i
$893$	2.86510e16i	1.68832i
$894$	0	0
$895$	0	0
$896$	5.66067e15	0.327472
$897$	0	0
$898$	− 1.46968e16i	− 0.839854i
$899$	−6.78541e15	−0.385388
$900$	0	0
$901$	1.10223e16	0.618421
$902$	3.95340e15i	0.220462i
$903$	0	0
$904$	7.19321e15	0.396275
$905$	0	0
$906$	0	0
$907$	− 8.43778e13i	− 0.00456445i	−0.999997	−	0.00228222i	$-0.999274\pi$
$907$	− 8.43778e13i	− 0.00456445i	0.999997	−	0.00228222i	$-0.000726455\pi$
$908$	2.00168e15i	0.107628i
$909$	0	0
$910$	0	0
$911$	1.10091e16	0.581298	0.290649	−	0.956830i	$-0.406129\pi$
$911$	1.10091e16	0.581298	0.290649	+	0.956830i	$0.406129\pi$
$912$	0	0
$913$	1.56829e16i	0.818158i
$914$	−7.28359e15	−0.377695
$915$	0	0
$916$	1.74055e16	0.891789
$917$	1.05743e16i	0.538544i
$918$	0	0
$919$	4.86351e15	0.244746	0.122373	−	0.992484i	$-0.460950\pi$
$919$	4.86351e15	0.244746	0.122373	+	0.992484i	$0.460950\pi$
$920$	0	0
$921$	0	0
$922$	1.75034e16i	0.865171i
$923$	− 5.65691e15i	− 0.277952i
$924$	0	0
$925$	0	0
$926$	2.93252e15	0.141540
$927$	0	0
$928$	2.52584e16i	1.20474i
$929$	3.57534e15	0.169524	0.0847620	−	0.996401i	$-0.472987\pi$
$929$	3.57534e15	0.169524	0.0847620	+	0.996401i	$0.472987\pi$
$930$	0	0
$931$	−1.80921e16	−0.847744
$932$	− 2.58533e16i	− 1.20428i
$933$	0	0
$934$	−1.48171e16	−0.682113
$935$	0	0
$936$	0	0
$937$	3.86373e16i	1.74759i	0.486295	+	0.873795i	$0.338348\pi$
$937$	3.86373e16i	1.74759i	−0.486295	+	0.873795i	$0.661652\pi$
$938$	− 6.22147e15i	− 0.279754i
$939$	0	0
$940$	0	0
$941$	3.48997e16	1.54198	0.770991	−	0.636846i	$-0.219761\pi$
$941$	3.48997e16	1.54198	0.770991	+	0.636846i	$0.219761\pi$
$942$	0	0
$943$	− 5.74437e15i	− 0.250858i
$944$	−5.12245e15	−0.222398
$945$	0	0
$946$	2.19735e14	0.00942969
$947$	2.85123e16i	1.21649i	0.793751	+	0.608243i	$0.208125\pi$
$947$	2.85123e16i	1.21649i	−0.793751	+	0.608243i	$0.791875\pi$
$948$	0	0
$949$	8.45688e14	0.0356653
$950$	0	0
$951$	0	0
$952$	9.76867e15i	0.404886i
$953$	4.00334e16i	1.64973i	0.565332	+	0.824863i	$0.308748\pi$
$953$	4.00334e16i	1.64973i	−0.565332	+	0.824863i	$0.691252\pi$
$954$	0	0
$955$	0	0
$956$	−1.05095e16	−0.425659
$957$	0	0
$958$	2.52201e16i	1.00980i
$959$	4.97630e15	0.198109
$960$	0	0
$961$	−2.26161e16	−0.890100
$962$	2.52652e15i	0.0988688i
$963$	0	0
$964$	−3.40484e14	−0.0131726
$965$	0	0
$966$	0	0
$967$	1.84953e16i	0.703422i	0.936109	+	0.351711i	$0.114400\pi$
$967$	1.84953e16i	0.703422i	−0.936109	+	0.351711i	$0.885600\pi$
$968$	4.20981e13i	0.00159202i
$969$	0	0
$970$	0	0
$971$	2.14877e16	0.798884	0.399442	−	0.916759i	$-0.369204\pi$
$971$	2.14877e16	0.798884	0.399442	+	0.916759i	$0.369204\pi$
$972$	0	0
$973$	9.99271e15i	0.367335i
$974$	5.27784e15	0.192922
$975$	0	0
$976$	6.86699e15	0.248195
$977$	8.73880e15i	0.314074i	0.987593	+	0.157037i	$0.0501942\pi$
$977$	8.73880e15i	0.314074i	−0.987593	+	0.157037i	$0.949806\pi$
$978$	0	0
$979$	1.33615e16	0.474844
$980$	0	0
$981$	0	0
$982$	1.16127e16i	0.405808i
$983$	1.18924e16i	0.413263i	0.978419	+	0.206631i	$0.0662501\pi$
$983$	1.18924e16i	0.413263i	−0.978419	+	0.206631i	$0.933750\pi$
$984$	0	0
$985$	0	0
$986$	−2.12824e16	−0.727274
$987$	0	0
$988$	− 9.06679e15i	− 0.306401i
$989$	−3.19279e14	−0.0107298
$990$	0	0
$991$	2.34409e16	0.779056	0.389528	−	0.921015i	$-0.372638\pi$
$991$	2.34409e16	0.779056	0.389528	+	0.921015i	$0.372638\pi$
$992$	− 1.03946e16i	− 0.343552i
$993$	0	0
$994$	−3.93477e15	−0.128616
$995$	0	0
$996$	0	0
$997$	− 2.14004e16i	− 0.688016i	−0.938967	−	0.344008i	$-0.888215\pi$
$997$	− 2.14004e16i	− 0.688016i	0.938967	−	0.344008i	$-0.111785\pi$
$998$	2.61307e15i	0.0835473i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.12.b.d.199.2		2
3.2	odd	2		25.12.b.b.24.1		2
5.2	odd	4		225.12.a.b.1.1		1
5.3	odd	4		9.12.a.b.1.1		1
5.4	even	2	inner	225.12.b.d.199.1		2
15.2	even	4		25.12.a.b.1.1		1
15.8	even	4		1.12.a.a.1.1	✓	1
15.14	odd	2		25.12.b.b.24.2		2
20.3	even	4		144.12.a.d.1.1		1
45.13	odd	12		81.12.c.b.55.1		2
45.23	even	12		81.12.c.d.55.1		2
45.38	even	12		81.12.c.d.28.1		2
45.43	odd	12		81.12.c.b.28.1		2
60.23	odd	4		16.12.a.a.1.1		1
105.23	even	12		49.12.c.b.18.1		2
105.38	odd	12		49.12.c.c.30.1		2
105.53	even	12		49.12.c.b.30.1		2
105.68	odd	12		49.12.c.c.18.1		2
105.83	odd	4		49.12.a.a.1.1		1
120.53	even	4		64.12.a.b.1.1		1
120.83	odd	4		64.12.a.f.1.1		1
165.98	odd	4		121.12.a.b.1.1		1
195.38	even	4		169.12.a.a.1.1		1
240.53	even	4		256.12.b.e.129.1		2
240.83	odd	4		256.12.b.c.129.1		2
240.173	even	4		256.12.b.e.129.2		2
240.203	odd	4		256.12.b.c.129.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.12.a.a.1.1	✓	1	15.8	even	4
9.12.a.b.1.1		1	5.3	odd	4
16.12.a.a.1.1		1	60.23	odd	4
25.12.a.b.1.1		1	15.2	even	4
25.12.b.b.24.1		2	3.2	odd	2
25.12.b.b.24.2		2	15.14	odd	2
49.12.a.a.1.1		1	105.83	odd	4
49.12.c.b.18.1		2	105.23	even	12
49.12.c.b.30.1		2	105.53	even	12
49.12.c.c.18.1		2	105.68	odd	12
49.12.c.c.30.1		2	105.38	odd	12
64.12.a.b.1.1		1	120.53	even	4
64.12.a.f.1.1		1	120.83	odd	4
81.12.c.b.28.1		2	45.43	odd	12
81.12.c.b.55.1		2	45.13	odd	12
81.12.c.d.28.1		2	45.38	even	12
81.12.c.d.55.1		2	45.23	even	12
121.12.a.b.1.1		1	165.98	odd	4
144.12.a.d.1.1		1	20.3	even	4
169.12.a.a.1.1		1	195.38	even	4
225.12.a.b.1.1		1	5.2	odd	4
225.12.b.d.199.1		2	5.4	even	2	inner
225.12.b.d.199.2		2	1.1	even	1	trivial
256.12.b.c.129.1		2	240.83	odd	4
256.12.b.c.129.2		2	240.203	odd	4
256.12.b.e.129.1		2	240.53	even	4
256.12.b.e.129.2		2	240.173	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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+24.0000i q^{2} +1472.00 q^{4} -16744.0i q^{7} +84480.0i q^{8} +O(q^{10})\)	\(q+24.0000i q^{2} +1472.00 q^{4} -16744.0i q^{7} +84480.0i q^{8} -534612. q^{11} +577738. i q^{13} +401856. q^{14} +987136. q^{16} +6.90593e6i q^{17} -1.06614e7 q^{19} -1.28307e7i q^{22} +1.86433e7i q^{23} -1.38657e7 q^{26} -2.46472e7i q^{28} +1.28407e8 q^{29} -5.28432e7 q^{31} +1.96706e8i q^{32} -1.65742e8 q^{34} -1.82213e8i q^{37} -2.55874e8i q^{38} -3.08120e8 q^{41} +1.71257e7i q^{43} -7.86949e8 q^{44} -4.47439e8 q^{46} -2.68735e9i q^{47} +1.69697e9 q^{49} +8.50430e8i q^{52} -1.59606e9i q^{53} +1.41453e9 q^{56} +3.08176e9i q^{58} -5.18920e9 q^{59} +6.95648e9 q^{61} -1.26824e9i q^{62} -2.69930e9 q^{64} -1.54818e10i q^{67} +1.01655e10i q^{68} -9.79149e9 q^{71} -1.46379e9i q^{73} +4.37312e9 q^{74} -1.56936e10 q^{76} +8.95154e9i q^{77} -3.81168e10 q^{79} -7.39489e9i q^{82} -2.93351e10i q^{83} -4.11017e8 q^{86} -4.51640e10i q^{88} -2.49929e10 q^{89} +9.67365e9 q^{91} +2.74429e10i q^{92} +6.44964e10 q^{94} +7.50136e10i q^{97} +4.07272e10i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2944 q^{4}+O(q^{10}) \) 2 * q + 2944 * q^4	\( 2 q + 2944 q^{4} - 1069224 q^{11} + 803712 q^{14} + 1974272 q^{16} - 21322840 q^{19} - 27731424 q^{26} + 256813260 q^{29} - 105686336 q^{31} - 331484832 q^{34} - 616240884 q^{41} - 1573897728 q^{44} - 894877056 q^{46} + 3393930414 q^{49} + 2829066240 q^{56} - 10378407480 q^{59} + 13912957324 q^{61} - 5398593536 q^{64} - 19582970544 q^{71} + 8746239072 q^{74} - 31387220480 q^{76} - 76233691360 q^{79} - 822033984 q^{86} - 49985834220 q^{89} + 19347290144 q^{91} + 128992727808 q^{94}+O(q^{100}) \) 2 * q + 2944 * q^4 - 1069224 * q^11 + 803712 * q^14 + 1974272 * q^16 - 21322840 * q^19 - 27731424 * q^26 + 256813260 * q^29 - 105686336 * q^31 - 331484832 * q^34 - 616240884 * q^41 - 1573897728 * q^44 - 894877056 * q^46 + 3393930414 * q^49 + 2829066240 * q^56 - 10378407480 * q^59 + 13912957324 * q^61 - 5398593536 * q^64 - 19582970544 * q^71 + 8746239072 * q^74 - 31387220480 * q^76 - 76233691360 * q^79 - 822033984 * q^86 - 49985834220 * q^89 + 19347290144 * q^91 + 128992727808 * q^94

Introduction

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