LMFDB - Embedded newform 252.9.c.a.197.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,9,Mod(197,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("252.197");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	252.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$102.659409735$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 4002260 x^{14} + 6534459751956 x^{12} + \cdots + 19\!\cdots\!00 $ x^16 + 4002260x^14 + 6534459751956x^12 + 5613923146579405376x^10 + 2733728904154246859079616x^8 + 757873148017661341349205888000x^6 + 113644318422397913452531577312640000x^4 + 8098650340007618970326973663348480000000*x^2 + 199066230990417435753898292645889849600000000
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{37}\cdot 7^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.4
Root			$-675.810i$ of defining polynomial
Character	$\chi$	$=$	252.197
Dual form			252.9.c.a.197.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-675.810i q^{5} +907.493 q^{7} +O(q^{10})$	$q-675.810i q^{5} +907.493 q^{7} -18942.7i q^{11} +6233.80 q^{13} +15144.2i q^{17} +181701. q^{19} -421094. i q^{23} -66094.2 q^{25} +190089. i q^{29} -198070. q^{31} -613293. i q^{35} +700460. q^{37} +628097. i q^{41} +5.91651e6 q^{43} -6.30001e6i q^{47} +823543. q^{49} +2.32657e6i q^{53} -1.28016e7 q^{55} +1.34161e7i q^{59} -9.18165e6 q^{61} -4.21286e6i q^{65} +6.29819e6 q^{67} +1.43966e7i q^{71} -4.04114e7 q^{73} -1.71903e7i q^{77} -3.00463e7 q^{79} -5.01957e7i q^{83} +1.02346e7 q^{85} -4.17353e6i q^{89} +5.65713e6 q^{91} -1.22795e8i q^{95} +1.41211e8 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 95480 q^{13} - 287560 q^{19} - 1754520 q^{25} - 3554264 q^{31} - 182920 q^{37} + 8472416 q^{43} + 13176688 q^{49} - 18692072 q^{55} + 34224568 q^{61} + 22683096 q^{67} + 2137296 q^{73} - 90245624 q^{79} - 56204456 q^{85} - 25661888 q^{91} + 134041152 q^{97}+O(q^{100}) $ 16 * q - 95480 * q^13 - 287560 * q^19 - 1754520 * q^25 - 3554264 * q^31 - 182920 * q^37 + 8472416 * q^43 + 13176688 * q^49 - 18692072 * q^55 + 34224568 * q^61 + 22683096 * q^67 + 2137296 * q^73 - 90245624 * q^79 - 56204456 * q^85 - 25661888 * q^91 + 134041152 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 675.810i	− 1.08130i	−0.841249	−	0.540648i	$-0.818179\pi$
$5$	− 675.810i	− 1.08130i	0.841249	−	0.540648i	$-0.181821\pi$
$6$	0	0
$7$	907.493	0.377964
$7$	907.493	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 18942.7i	− 1.29381i	−0.762571	−	0.646905i	$-0.776063\pi$
$11$	− 18942.7i	− 1.29381i	0.762571	−	0.646905i	$-0.223937\pi$
$12$	0	0
$13$	6233.80	0.218263	0.109131	−	0.994027i	$-0.465193\pi$
$13$	6233.80	0.218263	0.109131	+	0.994027i	$0.465193\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	15144.2i	0.181322i	0.995882	+	0.0906611i	$0.0288980\pi$
$17$	15144.2i	0.181322i	−0.995882	+	0.0906611i	$0.971102\pi$
$18$	0	0
$19$	181701.	1.39425	0.697127	−	0.716947i	$-0.254461\pi$
$19$	181701.	1.39425	0.697127	+	0.716947i	$0.254461\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 421094.i	− 1.50476i	−0.658728	−	0.752381i	$-0.728905\pi$
$23$	− 421094.i	− 1.50476i	0.658728	−	0.752381i	$-0.271095\pi$
$24$	0	0
$25$	−66094.2	−0.169201
$26$	0	0
$27$	0	0
$28$	0	0
$29$	190089.i	0.268760i	0.990930	+	0.134380i	$0.0429043\pi$
$29$	190089.i	0.268760i	−0.990930	+	0.134380i	$0.957096\pi$
$30$	0	0
$31$	−198070.	−0.214473	−0.107236	−	0.994234i	$-0.534200\pi$
$31$	−198070.	−0.214473	−0.107236	+	0.994234i	$0.534200\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 613293.i	− 0.408692i
$36$	0	0
$37$	700460.	0.373746	0.186873	−	0.982384i	$-0.440165\pi$
$37$	700460.	0.373746	0.186873	+	0.982384i	$0.440165\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	628097.i	0.222275i	0.993805	+	0.111138i	$0.0354495\pi$
$41$	628097.i	0.222275i	−0.993805	+	0.111138i	$0.964551\pi$
$42$	0	0
$43$	5.91651e6	1.73058	0.865290	−	0.501271i	$-0.167134\pi$
$43$	5.91651e6	1.73058	0.865290	+	0.501271i	$0.167134\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 6.30001e6i	− 1.29107i	−0.763730	−	0.645535i	$-0.776634\pi$
$47$	− 6.30001e6i	− 1.29107i	0.763730	−	0.645535i	$-0.223366\pi$
$48$	0	0
$49$	823543.	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.32657e6i	0.294858i	0.989073	+	0.147429i	$0.0470998\pi$
$53$	2.32657e6i	0.294858i	−0.989073	+	0.147429i	$0.952900\pi$
$54$	0	0
$55$	−1.28016e7	−1.39899
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.34161e7i	1.10718i	0.832788	+	0.553592i	$0.186743\pi$
$59$	1.34161e7i	1.10718i	−0.832788	+	0.553592i	$0.813257\pi$
$60$	0	0
$61$	−9.18165e6	−0.663134	−0.331567	−	0.943432i	$-0.607577\pi$
$61$	−9.18165e6	−0.663134	−0.331567	+	0.943432i	$0.607577\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 4.21286e6i	− 0.236007i
$66$	0	0
$67$	6.29819e6	0.312548	0.156274	−	0.987714i	$-0.450052\pi$
$67$	6.29819e6	0.312548	0.156274	+	0.987714i	$0.450052\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.43966e7i	0.566536i	0.959041	+	0.283268i	$0.0914186\pi$
$71$	1.43966e7i	0.566536i	−0.959041	+	0.283268i	$0.908581\pi$
$72$	0	0
$73$	−4.04114e7	−1.42302	−0.711512	−	0.702674i	$-0.751989\pi$
$73$	−4.04114e7	−1.42302	−0.711512	+	0.702674i	$0.751989\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.71903e7i	− 0.489014i
$78$	0	0
$79$	−3.00463e7	−0.771406	−0.385703	−	0.922623i	$-0.626041\pi$
$79$	−3.00463e7	−0.771406	−0.385703	+	0.922623i	$0.626041\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 5.01957e7i	− 1.05768i	−0.848722	−	0.528840i	$-0.822627\pi$
$83$	− 5.01957e7i	− 1.05768i	0.848722	−	0.528840i	$-0.177373\pi$
$84$	0	0
$85$	1.02346e7	0.196063
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 4.17353e6i	− 0.0665186i	−0.999447	−	0.0332593i	$-0.989411\pi$
$89$	− 4.17353e6i	− 0.0665186i	0.999447	−	0.0332593i	$-0.0105887\pi$
$90$	0	0
$91$	5.65713e6	0.0824955
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 1.22795e8i	− 1.50760i
$96$	0	0
$97$	1.41211e8	1.59507	0.797536	−	0.603272i	$-0.206136\pi$
$97$	1.41211e8	1.59507	0.797536	+	0.603272i	$0.206136\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.22032e6i	0.0597761i	0.999553	+	0.0298881i	$0.00951508\pi$
$101$	6.22032e6i	0.0597761i	−0.999553	+	0.0298881i	$0.990485\pi$
$102$	0	0
$103$	−5.46907e7	−0.485920	−0.242960	−	0.970036i	$-0.578118\pi$
$103$	−5.46907e7	−0.485920	−0.242960	+	0.970036i	$0.578118\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.90127e7i	0.526494i	0.964728	+	0.263247i	$0.0847935\pi$
$107$	6.90127e7i	0.526494i	−0.964728	+	0.263247i	$0.915207\pi$
$108$	0	0
$109$	8.93853e7	0.633228	0.316614	−	0.948554i	$-0.397454\pi$
$109$	8.93853e7	0.633228	0.316614	+	0.948554i	$0.397454\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 2.90457e8i	− 1.78143i	−0.454565	−	0.890714i	$-0.650205\pi$
$113$	− 2.90457e8i	− 1.78143i	0.454565	−	0.890714i	$-0.349795\pi$
$114$	0	0
$115$	−2.84580e8	−1.62709
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.37433e7i	0.0685334i
$120$	0	0
$121$	−1.44466e8	−0.673943
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 2.19321e8i	− 0.898339i
$126$	0	0
$127$	−3.37474e8	−1.29725	−0.648627	−	0.761107i	$-0.724656\pi$
$127$	−3.37474e8	−1.29725	−0.648627	+	0.761107i	$0.724656\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 1.80523e8i	− 0.612982i	−0.951874	−	0.306491i	$-0.900845\pi$
$131$	− 1.80523e8i	− 0.612982i	0.951874	−	0.306491i	$-0.0991549\pi$
$132$	0	0
$133$	1.64892e8	0.526979
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1.75533e8i	− 0.498283i	−0.968467	−	0.249141i	$-0.919852\pi$
$137$	− 1.75533e8i	− 0.498283i	0.968467	−	0.249141i	$-0.0801484\pi$
$138$	0	0
$139$	−1.14176e8	−0.305856	−0.152928	−	0.988237i	$-0.548870\pi$
$139$	−1.14176e8	−0.305856	−0.152928	+	0.988237i	$0.548870\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 1.18085e8i	− 0.282390i
$144$	0	0
$145$	1.28464e8	0.290609
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 5.90588e8i	− 1.19823i	−0.800663	−	0.599115i	$-0.795519\pi$
$149$	− 5.90588e8i	− 1.19823i	0.800663	−	0.599115i	$-0.204481\pi$
$150$	0	0
$151$	−9.52935e8	−1.83297	−0.916485	−	0.400069i	$-0.868986\pi$
$151$	−9.52935e8	−1.83297	−0.916485	+	0.400069i	$0.868986\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.33858e8i	0.231908i
$156$	0	0
$157$	−9.02887e8	−1.48605	−0.743027	−	0.669261i	$-0.766611\pi$
$157$	−9.02887e8	−1.48605	−0.743027	+	0.669261i	$0.766611\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 3.82140e8i	− 0.568747i
$162$	0	0
$163$	1.63169e8	0.231146	0.115573	−	0.993299i	$-0.463130\pi$
$163$	1.63169e8	0.231146	0.115573	+	0.993299i	$0.463130\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3.25869e8i	− 0.418964i	−0.977813	−	0.209482i	$-0.932822\pi$
$167$	− 3.25869e8i	− 0.418964i	0.977813	−	0.209482i	$-0.0671777\pi$
$168$	0	0
$169$	−7.76870e8	−0.952361
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1.01042e9i	− 1.12802i	−0.825768	−	0.564009i	$-0.809258\pi$
$173$	− 1.01042e9i	− 1.12802i	0.825768	−	0.564009i	$-0.190742\pi$
$174$	0	0
$175$	−5.99800e7	−0.0639520
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.92198e7i	0.0869059i	0.999055	+	0.0434530i	$0.0138359\pi$
$179$	8.92198e7i	0.0869059i	−0.999055	+	0.0434530i	$0.986164\pi$
$180$	0	0
$181$	−9.63231e8	−0.897463	−0.448731	−	0.893667i	$-0.648124\pi$
$181$	−9.63231e8	−0.897463	−0.448731	+	0.893667i	$0.648124\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 4.73378e8i	− 0.404130i
$186$	0	0
$187$	2.86872e8	0.234597
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 2.36076e9i	− 1.77386i	−0.461907	−	0.886928i	$-0.652835\pi$
$191$	− 2.36076e9i	− 1.77386i	0.461907	−	0.886928i	$-0.347165\pi$
$192$	0	0
$193$	2.96485e8	0.213684	0.106842	−	0.994276i	$-0.465926\pi$
$193$	2.96485e8	0.213684	0.106842	+	0.994276i	$0.465926\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.55058e9i	1.02951i	0.857338	+	0.514754i	$0.172117\pi$
$197$	1.55058e9i	1.02951i	−0.857338	+	0.514754i	$0.827883\pi$
$198$	0	0
$199$	2.31261e9	1.47465	0.737326	−	0.675537i	$-0.236088\pi$
$199$	2.31261e9	1.47465	0.737326	+	0.675537i	$0.236088\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.72504e8i	0.101582i
$204$	0	0
$205$	4.24474e8	0.240345
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 3.44190e9i	− 1.80390i
$210$	0	0
$211$	−3.86336e9	−1.94910	−0.974552	−	0.224163i	$-0.928035\pi$
$211$	−3.86336e9	−1.94910	−0.974552	+	0.224163i	$0.928035\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 3.99844e9i	− 1.87127i
$216$	0	0
$217$	−1.79747e8	−0.0810630
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9.44060e7i	0.0395759i
$222$	0	0
$223$	3.06779e9	1.24053	0.620264	−	0.784393i	$-0.287025\pi$
$223$	3.06779e9	1.24053	0.620264	+	0.784393i	$0.287025\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 5.63319e8i	− 0.212154i	−0.994358	−	0.106077i	$-0.966171\pi$
$227$	− 5.63319e8i	− 0.212154i	0.994358	−	0.106077i	$-0.0338290\pi$
$228$	0	0
$229$	1.34829e9	0.490278	0.245139	−	0.969488i	$-0.421166\pi$
$229$	1.34829e9	0.490278	0.245139	+	0.969488i	$0.421166\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.61906e8i	0.224581i	0.993675	+	0.112290i	$0.0358187\pi$
$233$	6.61906e8i	0.224581i	−0.993675	+	0.112290i	$0.964181\pi$
$234$	0	0
$235$	−4.25761e9	−1.39603
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4.96165e9i	1.52067i	0.649531	+	0.760335i	$0.274965\pi$
$239$	4.96165e9i	1.52067i	−0.649531	+	0.760335i	$0.725035\pi$
$240$	0	0
$241$	−3.38994e9	−1.00490	−0.502452	−	0.864605i	$-0.667569\pi$
$241$	−3.38994e9	−1.00490	−0.502452	+	0.864605i	$0.667569\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 5.56559e8i	− 0.154471i
$246$	0	0
$247$	1.13269e9	0.304314
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.26441e9i	0.570507i	0.958452	+	0.285253i	$0.0920777\pi$
$251$	2.26441e9i	0.570507i	−0.958452	+	0.285253i	$0.907922\pi$
$252$	0	0
$253$	−7.97665e9	−1.94688
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.00346e9i	1.37616i	0.725635	+	0.688080i	$0.241546\pi$
$257$	6.00346e9i	1.37616i	−0.725635	+	0.688080i	$0.758454\pi$
$258$	0	0
$259$	6.35662e8	0.141263
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5.31075e8i	0.111003i	0.998459	+	0.0555013i	$0.0176757\pi$
$263$	5.31075e8i	0.111003i	−0.998459	+	0.0555013i	$0.982324\pi$
$264$	0	0
$265$	1.57232e9	0.318829
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 3.46616e8i	− 0.0661972i	−0.999452	−	0.0330986i	$-0.989462\pi$
$269$	− 3.46616e8i	− 0.0661972i	0.999452	−	0.0330986i	$-0.0105375\pi$
$270$	0	0
$271$	7.88625e9	1.46216	0.731078	−	0.682294i	$-0.239018\pi$
$271$	7.88625e9	1.46216	0.731078	+	0.682294i	$0.239018\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.25200e9i	0.218914i
$276$	0	0
$277$	2.59726e9	0.441161	0.220580	−	0.975369i	$-0.429205\pi$
$277$	2.59726e9	0.441161	0.220580	+	0.975369i	$0.429205\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 3.57542e9i	− 0.573459i	−0.958012	−	0.286729i	$-0.907432\pi$
$281$	− 3.57542e9i	− 0.573459i	0.958012	−	0.286729i	$-0.0925681\pi$
$282$	0	0
$283$	−3.40830e9	−0.531365	−0.265682	−	0.964061i	$-0.585597\pi$
$283$	−3.40830e9	−0.531365	−0.265682	+	0.964061i	$0.585597\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.69994e8i	0.0840122i
$288$	0	0
$289$	6.74641e9	0.967122
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.14339e9i	0.833562i	0.909007	+	0.416781i	$0.136842\pi$
$293$	6.14339e9i	0.833562i	−0.909007	+	0.416781i	$0.863158\pi$
$294$	0	0
$295$	9.06676e9	1.19719
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 2.62502e9i	− 0.328433i
$300$	0	0
$301$	5.36919e9	0.654098
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.20505e9i	0.717045i
$306$	0	0
$307$	5.89041e9	0.663120	0.331560	−	0.943434i	$-0.392425\pi$
$307$	5.89041e9	0.663120	0.331560	+	0.943434i	$0.392425\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.47415e10i	1.57580i	0.615803	+	0.787900i	$0.288832\pi$
$311$	1.47415e10i	1.57580i	−0.615803	+	0.787900i	$0.711168\pi$
$312$	0	0
$313$	−2.41159e9	−0.251262	−0.125631	−	0.992077i	$-0.540096\pi$
$313$	−2.41159e9	−0.251262	−0.125631	+	0.992077i	$0.540096\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.73533e9i	0.270877i	0.990786	+	0.135438i	$0.0432443\pi$
$317$	2.73533e9i	0.270877i	−0.990786	+	0.135438i	$0.956756\pi$
$318$	0	0
$319$	3.60079e9	0.347725
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.75171e9i	0.252809i
$324$	0	0
$325$	−4.12018e8	−0.0369303
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 5.71722e9i	− 0.487979i
$330$	0	0
$331$	1.97672e9	0.164677	0.0823387	−	0.996604i	$-0.473761\pi$
$331$	1.97672e9	0.164677	0.0823387	+	0.996604i	$0.473761\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 4.25638e9i	− 0.337957i
$336$	0	0
$337$	5.00212e9	0.387824	0.193912	−	0.981019i	$-0.437882\pi$
$337$	5.00212e9	0.387824	0.193912	+	0.981019i	$0.437882\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.75197e9i	0.277487i
$342$	0	0
$343$	7.47359e8	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.03079e9i	0.0710971i	0.999368	+	0.0355485i	$0.0113178\pi$
$347$	1.03079e9i	0.0710971i	−0.999368	+	0.0355485i	$0.988682\pi$
$348$	0	0
$349$	5.68272e9	0.383049	0.191525	−	0.981488i	$-0.438657\pi$
$349$	5.68272e9	0.383049	0.191525	+	0.981488i	$0.438657\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7.38729e9i	0.475758i	0.971295	+	0.237879i	$0.0764522\pi$
$353$	7.38729e9i	0.475758i	−0.971295	+	0.237879i	$0.923548\pi$
$354$	0	0
$355$	9.72939e9	0.612593
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.75128e10i	1.05434i	0.849761	+	0.527168i	$0.176746\pi$
$359$	1.75128e10i	1.05434i	−0.849761	+	0.527168i	$0.823254\pi$
$360$	0	0
$361$	1.60316e10	0.943946
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.73104e10i	1.53871i
$366$	0	0
$367$	−9.61052e9	−0.529764	−0.264882	−	0.964281i	$-0.585333\pi$
$367$	−9.61052e9	−0.529764	−0.264882	+	0.964281i	$0.585333\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.11135e9i	0.111446i
$372$	0	0
$373$	−1.74568e10	−0.901837	−0.450919	−	0.892565i	$-0.648904\pi$
$373$	−1.74568e10	−0.901837	−0.450919	+	0.892565i	$0.648904\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.18498e9i	0.0586603i
$378$	0	0
$379$	3.39385e10	1.64489	0.822444	−	0.568846i	$-0.192610\pi$
$379$	3.39385e10	1.64489	0.822444	+	0.568846i	$0.192610\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.21374e10i	1.49354i	0.665084	+	0.746769i	$0.268396\pi$
$383$	3.21374e10i	1.49354i	−0.665084	+	0.746769i	$0.731604\pi$
$384$	0	0
$385$	−1.16174e10	−0.528769
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 1.89590e10i	− 0.827977i	−0.910282	−	0.413988i	$-0.864135\pi$
$389$	− 1.89590e10i	− 0.827977i	0.910282	−	0.413988i	$-0.135865\pi$
$390$	0	0
$391$	6.37714e9	0.272847
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.03056e10i	0.834119i
$396$	0	0
$397$	−2.73123e10	−1.09950	−0.549751	−	0.835328i	$-0.685278\pi$
$397$	−2.73123e10	−1.09950	−0.549751	+	0.835328i	$0.685278\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 2.05653e10i	− 0.795349i	−0.917527	−	0.397675i	$-0.869817\pi$
$401$	− 2.05653e10i	− 0.795349i	0.917527	−	0.397675i	$-0.130183\pi$
$402$	0	0
$403$	−1.23473e9	−0.0468113
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.32686e10i	− 0.483556i
$408$	0	0
$409$	4.15906e10	1.48629	0.743143	−	0.669133i	$-0.233334\pi$
$409$	4.15906e10	1.48629	0.743143	+	0.669133i	$0.233334\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.21751e10i	0.418476i
$414$	0	0
$415$	−3.39228e10	−1.14366
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8.09475e9i	0.262632i	0.991341	+	0.131316i	$0.0419202\pi$
$419$	8.09475e9i	0.262632i	−0.991341	+	0.131316i	$0.958080\pi$
$420$	0	0
$421$	−1.32776e10	−0.422660	−0.211330	−	0.977415i	$-0.567780\pi$
$421$	−1.32776e10	−0.422660	−0.211330	+	0.977415i	$0.567780\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 1.00095e9i	− 0.0306800i
$426$	0	0
$427$	−8.33228e9	−0.250641
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.74178e10i	1.37415i	0.726589	+	0.687073i	$0.241105\pi$
$431$	4.74178e10i	1.37415i	−0.726589	+	0.687073i	$0.758895\pi$
$432$	0	0
$433$	−4.31302e10	−1.22696	−0.613479	−	0.789711i	$-0.710230\pi$
$433$	−4.31302e10	−1.22696	−0.613479	+	0.789711i	$0.710230\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 7.65131e10i	− 2.09802i
$438$	0	0
$439$	9.81569e9	0.264279	0.132140	−	0.991231i	$-0.457815\pi$
$439$	9.81569e9	0.264279	0.132140	+	0.991231i	$0.457815\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.98569e10i	0.775229i	0.921822	+	0.387614i	$0.126701\pi$
$443$	2.98569e10i	0.775229i	−0.921822	+	0.387614i	$0.873299\pi$
$444$	0	0
$445$	−2.82051e9	−0.0719263
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 6.45404e10i	− 1.58798i	−0.607928	−	0.793992i	$-0.707999\pi$
$449$	− 6.45404e10i	− 1.58798i	0.607928	−	0.793992i	$-0.292001\pi$
$450$	0	0
$451$	1.18978e10	0.287582
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 3.82314e9i	− 0.0892021i
$456$	0	0
$457$	7.79905e10	1.78804	0.894019	−	0.448029i	$-0.147874\pi$
$457$	7.79905e10	1.78804	0.894019	+	0.448029i	$0.147874\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.74346e10i	0.828838i	0.910086	+	0.414419i	$0.136015\pi$
$461$	3.74346e10i	0.828838i	−0.910086	+	0.414419i	$0.863985\pi$
$462$	0	0
$463$	1.63866e10	0.356586	0.178293	−	0.983977i	$-0.442943\pi$
$463$	1.63866e10	0.356586	0.178293	+	0.983977i	$0.442943\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.06367e10i	− 0.223634i	−0.993729	−	0.111817i	$-0.964333\pi$
$467$	− 1.06367e10i	− 0.223634i	0.993729	−	0.111817i	$-0.0356671\pi$
$468$	0	0
$469$	5.71556e9	0.118132
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 1.12074e11i	− 2.23904i
$474$	0	0
$475$	−1.20094e10	−0.235910
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 6.94703e10i	− 1.31964i	−0.751422	−	0.659822i	$-0.770632\pi$
$479$	− 6.94703e10i	− 1.31964i	0.751422	−	0.659822i	$-0.229368\pi$
$480$	0	0
$481$	4.36653e9	0.0815747
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 9.54315e10i	− 1.72474i
$486$	0	0
$487$	−5.49782e10	−0.977405	−0.488703	−	0.872450i	$-0.662530\pi$
$487$	−5.49782e10	−0.977405	−0.488703	+	0.872450i	$0.662530\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 4.04762e10i	− 0.696424i	−0.937416	−	0.348212i	$-0.886789\pi$
$491$	− 4.04762e10i	− 0.696424i	0.937416	−	0.348212i	$-0.113211\pi$
$492$	0	0
$493$	−2.87875e9	−0.0487322
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.30648e10i	0.214130i
$498$	0	0
$499$	−4.67410e10	−0.753869	−0.376935	−	0.926240i	$-0.623022\pi$
$499$	−4.67410e10	−0.753869	−0.376935	+	0.926240i	$0.623022\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	4.28912e10i	0.670033i	0.942212	+	0.335017i	$0.108742\pi$
$503$	4.28912e10i	0.670033i	−0.942212	+	0.335017i	$0.891258\pi$
$504$	0	0
$505$	4.20376e9	0.0646357
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6.68892e10i	0.996517i	0.867028	+	0.498259i	$0.166027\pi$
$509$	6.68892e10i	0.996517i	−0.867028	+	0.498259i	$0.833973\pi$
$510$	0	0
$511$	−3.66730e10	−0.537853
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.69606e10i	0.525424i
$516$	0	0
$517$	−1.19339e11	−1.67040
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 3.09751e10i	− 0.420399i	−0.977659	−	0.210199i	$-0.932589\pi$
$521$	− 3.09751e10i	− 0.420399i	0.977659	−	0.210199i	$-0.0674113\pi$
$522$	0	0
$523$	1.58369e10	0.211672	0.105836	−	0.994384i	$-0.466248\pi$
$523$	1.58369e10	0.211672	0.105836	+	0.994384i	$0.466248\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 2.99961e9i	− 0.0388886i
$528$	0	0
$529$	−9.90093e10	−1.26431
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.91543e9i	0.0485144i
$534$	0	0
$535$	4.66395e10	0.569296
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1.56001e10i	− 0.184830i
$540$	0	0
$541$	−1.46603e11	−1.71141	−0.855705	−	0.517464i	$-0.826876\pi$
$541$	−1.46603e11	−1.71141	−0.855705	+	0.517464i	$0.826876\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 6.04075e10i	− 0.684707i
$546$	0	0
$547$	8.52569e10	0.952314	0.476157	−	0.879360i	$-0.342029\pi$
$547$	8.52569e10	0.952314	0.476157	+	0.879360i	$0.342029\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.45393e10i	0.374720i
$552$	0	0
$553$	−2.72668e10	−0.291564
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1.05196e11i	− 1.09289i	−0.837495	−	0.546445i	$-0.815981\pi$
$557$	− 1.05196e11i	− 1.09289i	0.837495	−	0.546445i	$-0.184019\pi$
$558$	0	0
$559$	3.68823e10	0.377721
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 4.36005e10i	− 0.433969i	−0.976175	−	0.216984i	$-0.930378\pi$
$563$	− 4.36005e10i	− 0.433969i	0.976175	−	0.216984i	$-0.0696220\pi$
$564$	0	0
$565$	−1.96294e11	−1.92625
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.63036e10i	0.155538i	0.996971	+	0.0777688i	$0.0247796\pi$
$569$	1.63036e10i	0.155538i	−0.996971	+	0.0777688i	$0.975220\pi$
$570$	0	0
$571$	6.58110e10	0.619090	0.309545	−	0.950885i	$-0.399823\pi$
$571$	6.58110e10	0.619090	0.309545	+	0.950885i	$0.399823\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.78319e10i	0.254608i
$576$	0	0
$577$	−1.39333e11	−1.25705	−0.628523	−	0.777791i	$-0.716340\pi$
$577$	−1.39333e11	−1.25705	−0.628523	+	0.777791i	$0.716340\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 4.55522e10i	− 0.399765i
$582$	0	0
$583$	4.40715e10	0.381490
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.50365e11i	− 1.26647i	−0.773961	−	0.633233i	$-0.781728\pi$
$587$	− 1.50365e11i	− 1.26647i	0.773961	−	0.633233i	$-0.218272\pi$
$588$	0	0
$589$	−3.59894e10	−0.299029
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 1.73734e11i	− 1.40497i	−0.711700	−	0.702484i	$-0.752074\pi$
$593$	− 1.73734e11i	− 1.40497i	0.711700	−	0.702484i	$-0.247926\pi$
$594$	0	0
$595$	9.28784e9	0.0741049
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 2.75985e10i	− 0.214377i	−0.994239	−	0.107189i	$-0.965815\pi$
$599$	− 2.75985e10i	− 0.214377i	0.994239	−	0.107189i	$-0.0341849\pi$
$600$	0	0
$601$	1.52880e11	1.17180	0.585898	−	0.810385i	$-0.300742\pi$
$601$	1.52880e11	1.17180	0.585898	+	0.810385i	$0.300742\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	9.76314e10i	0.728732i
$606$	0	0
$607$	1.35603e11	0.998884	0.499442	−	0.866347i	$-0.333538\pi$
$607$	1.35603e11	0.998884	0.499442	+	0.866347i	$0.333538\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 3.92730e10i	− 0.281793i
$612$	0	0
$613$	1.37425e11	0.973249	0.486625	−	0.873611i	$-0.338228\pi$
$613$	1.37425e11	0.973249	0.486625	+	0.873611i	$0.338228\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.23941e10i	0.499531i	0.968306	+	0.249765i	$0.0803535\pi$
$617$	7.23941e10i	0.499531i	−0.968306	+	0.249765i	$0.919647\pi$
$618$	0	0
$619$	−1.64909e10	−0.112327	−0.0561633	−	0.998422i	$-0.517887\pi$
$619$	−1.64909e10	−0.112327	−0.0561633	+	0.998422i	$0.517887\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 3.78744e9i	− 0.0251417i
$624$	0	0
$625$	−1.74038e11	−1.14057
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.06079e10i	0.0677684i
$630$	0	0
$631$	3.52602e10	0.222417	0.111208	−	0.993797i	$-0.464528\pi$
$631$	3.52602e10	0.222417	0.111208	+	0.993797i	$0.464528\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.28068e11i	1.40272i
$636$	0	0
$637$	5.13380e9	0.0311804
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1.76394e11i	− 1.04484i	−0.852687	−	0.522422i	$-0.825029\pi$
$641$	− 1.76394e11i	− 1.04484i	0.852687	−	0.522422i	$-0.174971\pi$
$642$	0	0
$643$	1.16256e11	0.680095	0.340048	−	0.940408i	$-0.389557\pi$
$643$	1.16256e11	0.680095	0.340048	+	0.940408i	$0.389557\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.89390e10i	0.450479i	0.974303	+	0.225239i	$0.0723164\pi$
$647$	7.89390e10i	0.450479i	−0.974303	+	0.225239i	$0.927684\pi$
$648$	0	0
$649$	2.54138e11	1.43248
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 1.76500e11i	− 0.970713i	−0.874316	−	0.485357i	$-0.838690\pi$
$653$	− 1.76500e11i	− 0.970713i	0.874316	−	0.485357i	$-0.161310\pi$
$654$	0	0
$655$	−1.21999e11	−0.662815
$656$	0	0
$657$	0	0
$658$	0	0
$659$	2.00334e11i	1.06222i	0.847304	+	0.531108i	$0.178224\pi$
$659$	2.00334e11i	1.06222i	−0.847304	+	0.531108i	$0.821776\pi$
$660$	0	0
$661$	−3.06069e11	−1.60330	−0.801648	−	0.597796i	$-0.796043\pi$
$661$	−3.06069e11	−1.60330	−0.801648	+	0.597796i	$0.796043\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1.11436e11i	− 0.569820i
$666$	0	0
$667$	8.00454e10	0.404420
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.73925e11i	0.857970i
$672$	0	0
$673$	1.06648e11	0.519867	0.259934	−	0.965626i	$-0.416299\pi$
$673$	1.06648e11	0.519867	0.259934	+	0.965626i	$0.416299\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.42785e11i	1.63180i	0.578192	+	0.815901i	$0.303759\pi$
$677$	3.42785e11i	1.63180i	−0.578192	+	0.815901i	$0.696241\pi$
$678$	0	0
$679$	1.28148e11	0.602880
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.20025e11i	0.551555i	0.961222	+	0.275777i	$0.0889352\pi$
$683$	1.20025e11i	0.551555i	−0.961222	+	0.275777i	$0.911065\pi$
$684$	0	0
$685$	−1.18627e11	−0.538791
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.45034e10i	0.0643565i
$690$	0	0
$691$	2.21637e11	0.972144	0.486072	−	0.873919i	$-0.338429\pi$
$691$	2.21637e11	0.972144	0.486072	+	0.873919i	$0.338429\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.71615e10i	0.330721i
$696$	0	0
$697$	−9.51204e9	−0.0403035
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 1.28038e11i	− 0.530233i	−0.964216	−	0.265116i	$-0.914590\pi$
$701$	− 1.28038e11i	− 0.530233i	0.964216	−	0.265116i	$-0.0854104\pi$
$702$	0	0
$703$	1.27274e11	0.521097
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.64490e9i	0.0225932i
$708$	0	0
$709$	1.99426e11	0.789217	0.394608	−	0.918849i	$-0.370880\pi$
$709$	1.99426e11	0.789217	0.394608	+	0.918849i	$0.370880\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.34061e10i	0.322730i
$714$	0	0
$715$	−7.98029e10	−0.305348
$716$	0	0
$717$	0	0
$718$	0	0
$719$	3.72421e11i	1.39354i	0.717296	+	0.696769i	$0.245380\pi$
$719$	3.72421e11i	1.39354i	−0.717296	+	0.696769i	$0.754620\pi$
$720$	0	0
$721$	−4.96314e10	−0.183661
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1.25638e10i	− 0.0454746i
$726$	0	0
$727$	3.45884e11	1.23821	0.619103	−	0.785310i	$-0.287496\pi$
$727$	3.45884e11	1.23821	0.619103	+	0.785310i	$0.287496\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.96009e10i	0.313793i
$732$	0	0
$733$	5.42390e11	1.87887	0.939433	−	0.342733i	$-0.111353\pi$
$733$	5.42390e11	1.87887	0.939433	+	0.342733i	$0.111353\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.19305e11i	− 0.404378i
$738$	0	0
$739$	−5.62333e11	−1.88545	−0.942727	−	0.333565i	$-0.891748\pi$
$739$	−5.62333e11	−1.88545	−0.942727	+	0.333565i	$0.891748\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	4.74150e11i	1.55582i	0.628374	+	0.777911i	$0.283721\pi$
$743$	4.74150e11i	1.55582i	−0.628374	+	0.777911i	$0.716279\pi$
$744$	0	0
$745$	−3.99126e11	−1.29564
$746$	0	0
$747$	0	0
$748$	0	0
$749$	6.26285e10i	0.198996i
$750$	0	0
$751$	1.37525e11	0.432336	0.216168	−	0.976356i	$-0.430644\pi$
$751$	1.37525e11	0.432336	0.216168	+	0.976356i	$0.430644\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6.44003e11i	1.98198i
$756$	0	0
$757$	−6.09476e11	−1.85598	−0.927990	−	0.372606i	$-0.878464\pi$
$757$	−6.09476e11	−1.85598	−0.927990	+	0.372606i	$0.878464\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 3.88452e11i	− 1.15824i	−0.815242	−	0.579121i	$-0.803396\pi$
$761$	− 3.88452e11i	− 1.15824i	0.815242	−	0.579121i	$-0.196604\pi$
$762$	0	0
$763$	8.11166e10	0.239338
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.36335e10i	0.241657i
$768$	0	0
$769$	6.73576e10	0.192611	0.0963056	−	0.995352i	$-0.469297\pi$
$769$	6.73576e10	0.192611	0.0963056	+	0.995352i	$0.469297\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 3.57274e10i	− 0.100065i	−0.998748	−	0.0500327i	$-0.984067\pi$
$773$	− 3.57274e10i	− 0.100065i	0.998748	−	0.0500327i	$-0.0159325\pi$
$774$	0	0
$775$	1.30913e10	0.0362890
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.14126e11i	0.309908i
$780$	0	0
$781$	2.72711e11	0.732990
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6.10180e11i	1.60686i
$786$	0	0
$787$	5.51060e11	1.43648	0.718240	−	0.695795i	$-0.244948\pi$
$787$	5.51060e11	1.43648	0.718240	+	0.695795i	$0.244948\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 2.63588e11i	− 0.673316i
$792$	0	0
$793$	−5.72366e10	−0.144737
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 6.25775e11i	− 1.55090i	−0.631406	−	0.775452i	$-0.717522\pi$
$797$	− 6.25775e11i	− 1.55090i	0.631406	−	0.775452i	$-0.282478\pi$
$798$	0	0
$799$	9.54088e10	0.234100
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.65499e11i	1.84112i
$804$	0	0
$805$	−2.58254e11	−0.614984
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3.34395e10i	0.0780667i	0.999238	+	0.0390334i	$0.0124279\pi$
$809$	3.34395e10i	0.0780667i	−0.999238	+	0.0390334i	$0.987572\pi$
$810$	0	0
$811$	5.98789e11	1.38417	0.692086	−	0.721815i	$-0.256692\pi$
$811$	5.98789e11	1.38417	0.692086	+	0.721815i	$0.256692\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1.10271e11i	− 0.249938i
$816$	0	0
$817$	1.07503e12	2.41287
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 4.98864e11i	− 1.09802i	−0.835816	−	0.549009i	$-0.815005\pi$
$821$	− 4.98864e11i	− 1.09802i	0.835816	−	0.549009i	$-0.184995\pi$
$822$	0	0
$823$	−1.03490e11	−0.225580	−0.112790	−	0.993619i	$-0.535979\pi$
$823$	−1.03490e11	−0.225580	−0.112790	+	0.993619i	$0.535979\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.44947e11i	1.37880i	0.724380	+	0.689401i	$0.242126\pi$
$827$	6.44947e11i	1.37880i	−0.724380	+	0.689401i	$0.757874\pi$
$828$	0	0
$829$	3.99994e11	0.846906	0.423453	−	0.905918i	$-0.360818\pi$
$829$	3.99994e11	0.846906	0.423453	+	0.905918i	$0.360818\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.24719e10i	0.0259032i
$834$	0	0
$835$	−2.20225e11	−0.453024
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 3.99830e11i	− 0.806914i	−0.914999	−	0.403457i	$-0.867809\pi$
$839$	− 3.99830e11i	− 0.806914i	0.914999	−	0.403457i	$-0.132191\pi$
$840$	0	0
$841$	4.64113e11	0.927768
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5.25017e11i	1.02978i
$846$	0	0
$847$	−1.31102e11	−0.254727
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 2.94959e11i	− 0.562398i
$852$	0	0
$853$	−1.79501e11	−0.339055	−0.169527	−	0.985525i	$-0.554224\pi$
$853$	−1.79501e11	−0.339055	−0.169527	+	0.985525i	$0.554224\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 9.54874e11i	− 1.77020i	−0.465398	−	0.885101i	$-0.654089\pi$
$857$	− 9.54874e11i	− 1.77020i	0.465398	−	0.885101i	$-0.345911\pi$
$858$	0	0
$859$	−3.02178e11	−0.554997	−0.277499	−	0.960726i	$-0.589505\pi$
$859$	−3.02178e11	−0.554997	−0.277499	+	0.960726i	$0.589505\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 9.97426e11i	− 1.79820i	−0.437745	−	0.899099i	$-0.644223\pi$
$863$	− 9.97426e11i	− 1.79820i	0.437745	−	0.899099i	$-0.355777\pi$
$864$	0	0
$865$	−6.82850e11	−1.21972
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.69158e11i	0.998053i
$870$	0	0
$871$	3.92617e10	0.0682176
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 1.99032e11i	− 0.339540i
$876$	0	0
$877$	−1.40216e11	−0.237028	−0.118514	−	0.992952i	$-0.537813\pi$
$877$	−1.40216e11	−0.237028	−0.118514	+	0.992952i	$0.537813\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.02525e12i	1.70187i	0.525268	+	0.850937i	$0.323965\pi$
$881$	1.02525e12i	1.70187i	−0.525268	+	0.850937i	$0.676035\pi$
$882$	0	0
$883$	3.77728e11	0.621350	0.310675	−	0.950516i	$-0.399445\pi$
$883$	3.77728e11	0.621350	0.310675	+	0.950516i	$0.399445\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.25328e11i	0.687115i	0.939132	+	0.343557i	$0.111632\pi$
$887$	4.25328e11i	0.687115i	−0.939132	+	0.343557i	$0.888368\pi$
$888$	0	0
$889$	−3.06255e11	−0.490316
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1.14472e12i	− 1.80008i
$894$	0	0
$895$	6.02957e10	0.0939710
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 3.76509e10i	− 0.0576417i
$900$	0	0
$901$	−3.52341e10	−0.0534644
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.50961e11i	0.970423i
$906$	0	0
$907$	−2.30935e11	−0.341241	−0.170621	−	0.985337i	$-0.554577\pi$
$907$	−2.30935e11	−0.341241	−0.170621	+	0.985337i	$0.554577\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1.01079e11i	− 0.146753i	−0.997304	−	0.0733764i	$-0.976623\pi$
$911$	− 1.01079e11i	− 0.146753i	0.997304	−	0.0733764i	$-0.0233774\pi$
$912$	0	0
$913$	−9.50841e11	−1.36844
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.63823e11i	− 0.231685i
$918$	0	0
$919$	−5.67287e11	−0.795317	−0.397659	−	0.917533i	$-0.630177\pi$
$919$	−5.67287e11	−0.795317	−0.397659	+	0.917533i	$0.630177\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	8.97457e10i	0.123654i
$924$	0	0
$925$	−4.62963e10	−0.0632382
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 8.04274e10i	− 0.107979i	−0.998541	−	0.0539897i	$-0.982806\pi$
$929$	− 8.04274e10i	− 0.107979i	0.998541	−	0.0539897i	$-0.0171938\pi$
$930$	0	0
$931$	1.49638e11	0.199179
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1.93871e11i	− 0.253668i
$936$	0	0
$937$	−1.25692e12	−1.63061	−0.815306	−	0.579030i	$-0.803431\pi$
$937$	−1.25692e12	−1.63061	−0.815306	+	0.579030i	$0.803431\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.50994e11i	0.320115i	0.987108	+	0.160057i	$0.0511679\pi$
$941$	2.50994e11i	0.320115i	−0.987108	+	0.160057i	$0.948832\pi$
$942$	0	0
$943$	2.64488e11	0.334472
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 3.27838e11i	− 0.407623i	−0.979010	−	0.203812i	$-0.934667\pi$
$947$	− 3.27838e11i	− 0.407623i	0.979010	−	0.203812i	$-0.0653330\pi$
$948$	0	0
$949$	−2.51916e11	−0.310593
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1.04603e12i	− 1.26816i	−0.773268	−	0.634080i	$-0.781379\pi$
$953$	− 1.04603e12i	− 1.26816i	0.773268	−	0.634080i	$-0.218621\pi$
$954$	0	0
$955$	−1.59543e12	−1.91806
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1.59295e11i	− 0.188333i
$960$	0	0
$961$	−8.13659e11	−0.954002
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 2.00367e11i	− 0.231056i
$966$	0	0
$967$	6.58893e11	0.753545	0.376773	−	0.926306i	$-0.377034\pi$
$967$	6.58893e11	0.753545	0.376773	+	0.926306i	$0.377034\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.55166e12i	1.74550i	0.488170	+	0.872749i	$0.337665\pi$
$971$	1.55166e12i	1.74550i	−0.488170	+	0.872749i	$0.662335\pi$
$972$	0	0
$973$	−1.03614e11	−0.115603
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 6.77778e11i	− 0.743891i	−0.928255	−	0.371945i	$-0.878691\pi$
$977$	− 6.77778e11i	− 0.743891i	0.928255	−	0.371945i	$-0.121309\pi$
$978$	0	0
$979$	−7.90577e10	−0.0860624
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1.15349e12i	− 1.23538i	−0.786423	−	0.617688i	$-0.788070\pi$
$983$	− 1.15349e12i	− 1.23538i	0.786423	−	0.617688i	$-0.211930\pi$
$984$	0	0
$985$	1.04790e12	1.11320
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 2.49141e12i	− 2.60411i
$990$	0	0
$991$	−6.26653e11	−0.649730	−0.324865	−	0.945760i	$-0.605319\pi$
$991$	−6.26653e11	−0.649730	−0.324865	+	0.945760i	$0.605319\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1.56288e12i	− 1.59454i
$996$	0	0
$997$	5.43417e11	0.549987	0.274994	−	0.961446i	$-0.411324\pi$
$997$	5.43417e11	0.549987	0.274994	+	0.961446i	$0.411324\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.9.c.a.197.4	✓	16
3.2	odd	2	inner	252.9.c.a.197.13	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.c.a.197.4	✓	16	1.1	even	1	trivial
252.9.c.a.197.13	yes	16	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	252.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-675.810i q^{5} +907.493 q^{7} +O(q^{10})\)	\(q-675.810i q^{5} +907.493 q^{7} -18942.7i q^{11} +6233.80 q^{13} +15144.2i q^{17} +181701. q^{19} -421094. i q^{23} -66094.2 q^{25} +190089. i q^{29} -198070. q^{31} -613293. i q^{35} +700460. q^{37} +628097. i q^{41} +5.91651e6 q^{43} -6.30001e6i q^{47} +823543. q^{49} +2.32657e6i q^{53} -1.28016e7 q^{55} +1.34161e7i q^{59} -9.18165e6 q^{61} -4.21286e6i q^{65} +6.29819e6 q^{67} +1.43966e7i q^{71} -4.04114e7 q^{73} -1.71903e7i q^{77} -3.00463e7 q^{79} -5.01957e7i q^{83} +1.02346e7 q^{85} -4.17353e6i q^{89} +5.65713e6 q^{91} -1.22795e8i q^{95} +1.41211e8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 95480 q^{13} - 287560 q^{19} - 1754520 q^{25} - 3554264 q^{31} - 182920 q^{37} + 8472416 q^{43} + 13176688 q^{49} - 18692072 q^{55} + 34224568 q^{61} + 22683096 q^{67} + 2137296 q^{73} - 90245624 q^{79} - 56204456 q^{85} - 25661888 q^{91} + 134041152 q^{97}+O(q^{100}) \) 16 * q - 95480 * q^13 - 287560 * q^19 - 1754520 * q^25 - 3554264 * q^31 - 182920 * q^37 + 8472416 * q^43 + 13176688 * q^49 - 18692072 * q^55 + 34224568 * q^61 + 22683096 * q^67 + 2137296 * q^73 - 90245624 * q^79 - 56204456 * q^85 - 25661888 * q^91 + 134041152 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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