LMFDB - Embedded newform 7488.2.d.m.4031.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7488,2,Mod(4031,7488)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7488, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7488.4031");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7488 = 2^{6} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7488.d (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:	$59.7919810335$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.1279179096064000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 5x^{8} - 4x^{6} + 20x^{4} + 64 $ x^12 + 5x^8 - 4x^6 + 20*x^4 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 468)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4031.3
Root			$0.653376 + 1.25423i$ of defining polynomial
Character	$\chi$	$=$	7488.4031
Dual form			7488.2.d.m.4031.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.24195i q^{5} -1.84803i q^{7} +O(q^{10})$	$q-3.24195i q^{5} -1.84803i q^{7} +5.77578 q^{11} +1.00000 q^{13} +3.60272i q^{17} -0.235720i q^{19} -8.93806 q^{23} -5.51021 q^{25} -4.24264i q^{29} +2.62411i q^{31} -5.99120 q^{35} +9.67982 q^{37} -6.07037i q^{41} -6.08444i q^{43} +12.1003 q^{47} +3.58480 q^{49} -3.60272i q^{53} -18.7248i q^{55} -3.16228 q^{59} -3.41520 q^{61} -3.24195i q^{65} -4.23642i q^{67} -6.10914 q^{71} -4.51021 q^{73} -10.6738i q^{77} -0.471440i q^{79} +14.0471 q^{83} +11.6798 q^{85} -8.89880i q^{89} -1.84803i q^{91} -0.764192 q^{95} -3.48979 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q + 12 q^{13} - 52 q^{25} - 8 q^{37} - 12 q^{49} - 96 q^{61} - 40 q^{73} + 16 q^{85} - 56 q^{97}+O(q^{100}) $ 12 * q + 12 * q^13 - 52 * q^25 - 8 * q^37 - 12 * q^49 - 96 * q^61 - 40 * q^73 + 16 * q^85 - 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7488\mathbb{Z}\right)^\times$.

$n$	$703$	$5761$	$5825$	$6085$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$5$	− 3.24195i	− 1.44984i	−0.688832	−	0.724921i	$-0.741876\pi$
$5$	− 3.24195i	− 1.44984i	0.688832	−	0.724921i	$-0.258124\pi$
$6$	0	0
$7$	− 1.84803i	− 0.698488i	−0.937032	−	0.349244i	$-0.886438\pi$
$7$	− 1.84803i	− 0.698488i	0.937032	−	0.349244i	$-0.113562\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.77578	1.74146	0.870732	−	0.491759i	$-0.163646\pi$
$11$	5.77578	1.74146	0.870732	+	0.491759i	$0.163646\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.60272i	0.873788i	0.899513	+	0.436894i	$0.143921\pi$
$17$	3.60272i	0.873788i	−0.899513	+	0.436894i	$0.856079\pi$
$18$	0	0
$19$	− 0.235720i	− 0.0540779i	−0.999634	−	0.0270390i	$-0.991392\pi$
$19$	− 0.235720i	− 0.0540779i	0.999634	−	0.0270390i	$-0.00860782\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.93806	−1.86371	−0.931857	−	0.362826i	$-0.881812\pi$
$23$	−8.93806	−1.86371	−0.931857	+	0.362826i	$0.881812\pi$
$24$	0	0
$25$	−5.51021	−1.10204
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 4.24264i	− 0.787839i	−0.919145	−	0.393919i	$-0.871119\pi$
$29$	− 4.24264i	− 0.787839i	0.919145	−	0.393919i	$-0.128881\pi$
$30$	0	0
$31$	2.62411i	0.471304i	0.971837	+	0.235652i	$0.0757226\pi$
$31$	2.62411i	0.471304i	−0.971837	+	0.235652i	$0.924277\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−5.99120	−1.01270
$36$	0	0
$37$	9.67982	1.59135	0.795676	−	0.605722i	$-0.207116\pi$
$37$	9.67982	1.59135	0.795676	+	0.605722i	$0.207116\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 6.07037i	− 0.948033i	−0.880516	−	0.474016i	$-0.842804\pi$
$41$	− 6.07037i	− 0.948033i	0.880516	−	0.474016i	$-0.157196\pi$
$42$	0	0
$43$	− 6.08444i	− 0.927869i	−0.885870	−	0.463934i	$-0.846437\pi$
$43$	− 6.08444i	− 0.927869i	0.885870	−	0.463934i	$-0.153563\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.1003	1.76502	0.882508	−	0.470298i	$-0.155854\pi$
$47$	12.1003	1.76502	0.882508	+	0.470298i	$0.155854\pi$
$48$	0	0
$49$	3.58480	0.512115
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 3.60272i	− 0.494871i	−0.968904	−	0.247436i	$-0.920412\pi$
$53$	− 3.60272i	− 0.494871i	0.968904	−	0.247436i	$-0.0795879\pi$
$54$	0	0
$55$	− 18.7248i	− 2.52485i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.16228	−0.411693	−0.205847	−	0.978584i	$-0.565995\pi$
$59$	−3.16228	−0.411693	−0.205847	+	0.978584i	$0.565995\pi$
$60$	0	0
$61$	−3.41520	−0.437271	−0.218636	−	0.975807i	$-0.570161\pi$
$61$	−3.41520	−0.437271	−0.218636	+	0.975807i	$0.570161\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 3.24195i	− 0.402114i
$66$	0	0
$67$	− 4.23642i	− 0.517561i	−0.965936	−	0.258780i	$-0.916679\pi$
$67$	− 4.23642i	− 0.517561i	0.965936	−	0.258780i	$-0.0833206\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.10914	−0.725021	−0.362511	−	0.931980i	$-0.618080\pi$
$71$	−6.10914	−0.725021	−0.362511	+	0.931980i	$0.618080\pi$
$72$	0	0
$73$	−4.51021	−0.527880	−0.263940	−	0.964539i	$-0.585022\pi$
$73$	−4.51021	−0.527880	−0.263940	+	0.964539i	$0.585022\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 10.6738i	− 1.21639i
$78$	0	0
$79$	− 0.471440i	− 0.0530412i	−0.999648	−	0.0265206i	$-0.991557\pi$
$79$	− 0.471440i	− 0.0530412i	0.999648	−	0.0265206i	$-0.00844276\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	14.0471	1.54187	0.770936	−	0.636913i	$-0.219789\pi$
$83$	14.0471	1.54187	0.770936	+	0.636913i	$0.219789\pi$
$84$	0	0
$85$	11.6798	1.26685
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 8.89880i	− 0.943271i	−0.881794	−	0.471635i	$-0.843664\pi$
$89$	− 8.89880i	− 0.943271i	0.881794	−	0.471635i	$-0.156336\pi$
$90$	0	0
$91$	− 1.84803i	− 0.193726i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.764192	−0.0784045
$96$	0	0
$97$	−3.48979	−0.354334	−0.177167	−	0.984181i	$-0.556693\pi$
$97$	−3.48979	−0.354334	−0.177167	+	0.984181i	$0.556693\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 16.5705i	− 1.64883i	−0.565988	−	0.824413i	$-0.691505\pi$
$101$	− 16.5705i	− 1.64883i	0.565988	−	0.824413i	$-0.308495\pi$
$102$	0	0
$103$	2.85983i	0.281788i	0.990025	+	0.140894i	$0.0449976\pi$
$103$	2.85983i	0.281788i	−0.990025	+	0.140894i	$0.955002\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.60478	−0.928529	−0.464264	−	0.885697i	$-0.653681\pi$
$107$	−9.60478	−0.928529	−0.464264	+	0.885697i	$0.653681\pi$
$108$	0	0
$109$	−5.09501	−0.488014	−0.244007	−	0.969774i	$-0.578462\pi$
$109$	−5.09501	−0.488014	−0.244007	+	0.969774i	$0.578462\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 19.2934i	− 1.81497i	−0.420080	−	0.907487i	$-0.637998\pi$
$113$	− 19.2934i	− 1.81497i	0.420080	−	0.907487i	$-0.362002\pi$
$114$	0	0
$115$	28.9767i	2.70209i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.65791	0.610330
$120$	0	0
$121$	22.3596	2.03269
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.65407i	0.147945i
$126$	0	0
$127$	12.6403i	1.12165i	0.827935	+	0.560824i	$0.189515\pi$
$127$	12.6403i	1.12165i	−0.827935	+	0.560824i	$0.810485\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.89365	−0.427560	−0.213780	−	0.976882i	$-0.568578\pi$
$131$	−4.89365	−0.427560	−0.213780	+	0.976882i	$0.568578\pi$
$132$	0	0
$133$	−0.435617	−0.0377728
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.52040i	− 0.215332i	−0.994187	−	0.107666i	$-0.965662\pi$
$137$	− 2.52040i	− 0.215332i	0.994187	−	0.107666i	$-0.0343378\pi$
$138$	0	0
$139$	− 1.30766i	− 0.110914i	−0.998461	−	0.0554571i	$-0.982338\pi$
$139$	− 1.30766i	− 0.110914i	0.998461	−	0.0554571i	$-0.0176616\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.77578	0.482995
$144$	0	0
$145$	−13.7544	−1.14224
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.9974i	1.14671i	0.819308	+	0.573354i	$0.194358\pi$
$149$	13.9974i	1.14671i	−0.819308	+	0.573354i	$0.805642\pi$
$150$	0	0
$151$	13.2408i	1.07752i	0.842458	+	0.538761i	$0.181108\pi$
$151$	13.2408i	1.07752i	−0.842458	+	0.538761i	$0.818892\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.50722	0.683317
$156$	0	0
$157$	−12.6594	−1.01033	−0.505165	−	0.863023i	$-0.668568\pi$
$157$	−12.6594	−1.01033	−0.505165	+	0.863023i	$0.668568\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	16.5178i	1.30178i
$162$	0	0
$163$	3.46033i	0.271034i	0.990775	+	0.135517i	$0.0432695\pi$
$163$	3.46033i	0.271034i	−0.990775	+	0.135517i	$0.956730\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.4249	1.42576	0.712880	−	0.701286i	$-0.247390\pi$
$167$	18.4249	1.42576	0.712880	+	0.701286i	$0.247390\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.15269i	0.543809i	0.962324	+	0.271905i	$0.0876535\pi$
$173$	7.15269i	0.543809i	−0.962324	+	0.271905i	$0.912347\pi$
$174$	0	0
$175$	10.1830i	0.769763i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.43091	0.106951	0.0534756	−	0.998569i	$-0.482970\pi$
$179$	1.43091	0.106951	0.0534756	+	0.998569i	$0.482970\pi$
$180$	0	0
$181$	−1.48979	−0.110735	−0.0553676	−	0.998466i	$-0.517633\pi$
$181$	−1.48979	−0.110735	−0.0553676	+	0.998466i	$0.517633\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 31.3814i	− 2.30721i
$186$	0	0
$187$	20.8085i	1.52167i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.43091	−0.103537	−0.0517685	−	0.998659i	$-0.516486\pi$
$191$	−1.43091	−0.103537	−0.0517685	+	0.998659i	$0.516486\pi$
$192$	0	0
$193$	−4.07459	−0.293296	−0.146648	−	0.989189i	$-0.546848\pi$
$193$	−4.07459	−0.293296	−0.146648	+	0.989189i	$0.546848\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.06898i	0.289903i	0.989439	+	0.144951i	$0.0463026\pi$
$197$	4.06898i	0.289903i	−0.989439	+	0.144951i	$0.953697\pi$
$198$	0	0
$199$	23.2570i	1.64865i	0.566119	+	0.824324i	$0.308444\pi$
$199$	23.2570i	1.64865i	−0.566119	+	0.824324i	$0.691556\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.84051	−0.550296
$204$	0	0
$205$	−19.6798	−1.37450
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 1.36147i	− 0.0941747i
$210$	0	0
$211$	− 15.8649i	− 1.09219i	−0.837724	−	0.546093i	$-0.816114\pi$
$211$	− 15.8649i	− 1.09219i	0.837724	−	0.546093i	$-0.183886\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−19.7254	−1.34526
$216$	0	0
$217$	4.84942	0.329200
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.60272i	0.242345i
$222$	0	0
$223$	− 27.9048i	− 1.86864i	−0.356435	−	0.934320i	$-0.616008\pi$
$223$	− 27.9048i	− 1.86864i	0.356435	−	0.934320i	$-0.383992\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	16.1447	1.07156	0.535782	−	0.844356i	$-0.320017\pi$
$227$	16.1447	1.07156	0.535782	+	0.844356i	$0.320017\pi$
$228$	0	0
$229$	−16.1900	−1.06987	−0.534934	−	0.844894i	$-0.679663\pi$
$229$	−16.1900	−1.06987	−0.534934	+	0.844894i	$0.679663\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.6521i	1.09092i	0.838138	+	0.545458i	$0.183644\pi$
$233$	16.6521i	1.09092i	−0.838138	+	0.545458i	$0.816356\pi$
$234$	0	0
$235$	− 39.2286i	− 2.55899i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.38928	0.542658	0.271329	−	0.962487i	$-0.412537\pi$
$239$	8.38928	0.542658	0.271329	+	0.962487i	$0.412537\pi$
$240$	0	0
$241$	−15.6052	−1.00522	−0.502610	−	0.864513i	$-0.667627\pi$
$241$	−15.6052	−1.00522	−0.502610	+	0.864513i	$0.667627\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 11.6217i	− 0.742485i
$246$	0	0
$247$	− 0.235720i	− 0.0149985i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.28014	−0.143921	−0.0719607	−	0.997407i	$-0.522926\pi$
$251$	−2.28014	−0.143921	−0.0719607	+	0.997407i	$0.522926\pi$
$252$	0	0
$253$	−51.6243	−3.24559
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2.13576i	− 0.133225i	−0.997779	−	0.0666125i	$-0.978781\pi$
$257$	− 2.13576i	− 0.133225i	0.997779	−	0.0666125i	$-0.0212191\pi$
$258$	0	0
$259$	− 17.8885i	− 1.11154i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−14.9293	−0.920577	−0.460289	−	0.887769i	$-0.652254\pi$
$263$	−14.9293	−0.920577	−0.460289	+	0.887769i	$0.652254\pi$
$264$	0	0
$265$	−11.6798	−0.717485
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 25.5902i	− 1.56026i	−0.625616	−	0.780131i	$-0.715152\pi$
$269$	− 25.5902i	− 1.56026i	0.625616	−	0.780131i	$-0.284848\pi$
$270$	0	0
$271$	19.7366i	1.19891i	0.800408	+	0.599456i	$0.204616\pi$
$271$	19.7366i	1.19891i	−0.800408	+	0.599456i	$0.795384\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−31.8258	−1.91917
$276$	0	0
$277$	−7.09501	−0.426298	−0.213149	−	0.977020i	$-0.568372\pi$
$277$	−7.09501	−0.426298	−0.213149	+	0.977020i	$0.568372\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 22.2139i	− 1.32517i	−0.748987	−	0.662585i	$-0.769459\pi$
$281$	− 22.2139i	− 1.32517i	0.748987	−	0.662585i	$-0.230541\pi$
$282$	0	0
$283$	− 13.1118i	− 0.779413i	−0.920939	−	0.389707i	$-0.872576\pi$
$283$	− 13.1118i	− 0.779413i	0.920939	−	0.389707i	$-0.127424\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−11.2182	−0.662189
$288$	0	0
$289$	4.02042	0.236495
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 24.6950i	− 1.44270i	−0.692572	−	0.721349i	$-0.743522\pi$
$293$	− 24.6950i	− 1.44270i	0.692572	−	0.721349i	$-0.256478\pi$
$294$	0	0
$295$	10.2519i	0.596891i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8.93806	−0.516901
$300$	0	0
$301$	−11.2442	−0.648105
$302$	0	0
$303$	0	0
$304$	0	0
$305$	11.0719i	0.633974i
$306$	0	0
$307$	28.4363i	1.62295i	0.584389	+	0.811474i	$0.301334\pi$
$307$	28.4363i	1.62295i	−0.584389	+	0.811474i	$0.698666\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−18.2095	−1.03256	−0.516282	−	0.856418i	$-0.672684\pi$
$311$	−18.2095	−1.03256	−0.516282	+	0.856418i	$0.672684\pi$
$312$	0	0
$313$	−20.1492	−1.13890	−0.569450	−	0.822026i	$-0.692844\pi$
$313$	−20.1492	−1.13890	−0.569450	+	0.822026i	$0.692844\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.41491i	− 0.135635i	−0.997698	−	0.0678174i	$-0.978396\pi$
$317$	− 2.41491i	− 0.135635i	0.997698	−	0.0678174i	$-0.0216035\pi$
$318$	0	0
$319$	− 24.5046i	− 1.37199i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.849234	0.0472526
$324$	0	0
$325$	−5.51021	−0.305651
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 22.3617i	− 1.23284i
$330$	0	0
$331$	− 11.6285i	− 0.639161i	−0.947559	−	0.319581i	$-0.896458\pi$
$331$	− 11.6285i	− 0.639161i	0.947559	−	0.319581i	$-0.103542\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−13.7342	−0.750381
$336$	0	0
$337$	−33.4751	−1.82350	−0.911752	−	0.410742i	$-0.865270\pi$
$337$	−33.4751	−1.82350	−0.911752	+	0.410742i	$0.865270\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	15.1563i	0.820759i
$342$	0	0
$343$	− 19.5610i	− 1.05619i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	29.8585	1.60289	0.801444	−	0.598069i	$-0.204065\pi$
$347$	29.8585	1.60289	0.801444	+	0.598069i	$0.204065\pi$
$348$	0	0
$349$	13.7952	0.738443	0.369221	−	0.929341i	$-0.379624\pi$
$349$	13.7952	0.738443	0.369221	+	0.929341i	$0.379624\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9.62035i	0.512039i	0.966672	+	0.256020i	$0.0824112\pi$
$353$	9.62035i	0.512039i	−0.966672	+	0.256020i	$0.917589\pi$
$354$	0	0
$355$	19.8055i	1.05117i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.38921	−0.389987	−0.194994	−	0.980804i	$-0.562469\pi$
$359$	−7.38921	−0.389987	−0.194994	+	0.980804i	$0.562469\pi$
$360$	0	0
$361$	18.9444	0.997076
$362$	0	0
$363$	0	0
$364$	0	0
$365$	14.6219i	0.765343i
$366$	0	0
$367$	9.78049i	0.510537i	0.966870	+	0.255269i	$0.0821639\pi$
$367$	9.78049i	0.510537i	−0.966870	+	0.255269i	$0.917836\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.65791	−0.345662
$372$	0	0
$373$	−1.02042	−0.0528354	−0.0264177	−	0.999651i	$-0.508410\pi$
$373$	−1.02042	−0.0528354	−0.0264177	+	0.999651i	$0.508410\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4.24264i	− 0.218507i
$378$	0	0
$379$	21.8203i	1.12083i	0.828211	+	0.560417i	$0.189359\pi$
$379$	21.8203i	1.12083i	−0.828211	+	0.560417i	$0.810641\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12.6162	0.644658	0.322329	−	0.946628i	$-0.395534\pi$
$383$	12.6162	0.644658	0.322329	+	0.946628i	$0.395534\pi$
$384$	0	0
$385$	−34.6038	−1.76357
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 15.9306i	− 0.807712i	−0.914822	−	0.403856i	$-0.867670\pi$
$389$	− 15.9306i	− 0.807712i	0.914822	−	0.403856i	$-0.132330\pi$
$390$	0	0
$391$	− 32.2013i	− 1.62849i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.52838	−0.0769014
$396$	0	0
$397$	18.9649	0.951819	0.475909	−	0.879494i	$-0.342119\pi$
$397$	18.9649	0.951819	0.475909	+	0.879494i	$0.342119\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.2758i	0.662962i	0.943462	+	0.331481i	$0.107548\pi$
$401$	13.2758i	0.662962i	−0.943462	+	0.331481i	$0.892452\pi$
$402$	0	0
$403$	2.62411i	0.130716i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	55.9085	2.77128
$408$	0	0
$409$	−4.65940	−0.230392	−0.115196	−	0.993343i	$-0.536750\pi$
$409$	−4.65940	−0.230392	−0.115196	+	0.993343i	$0.536750\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.84397i	0.287563i
$414$	0	0
$415$	− 45.5400i	− 2.23547i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−37.2682	−1.82067	−0.910335	−	0.413872i	$-0.864176\pi$
$419$	−37.2682	−1.82067	−0.910335	+	0.413872i	$0.864176\pi$
$420$	0	0
$421$	31.5497	1.53764	0.768818	−	0.639467i	$-0.220845\pi$
$421$	31.5497	1.53764	0.768818	+	0.639467i	$0.220845\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 19.8517i	− 0.962951i
$426$	0	0
$427$	6.31137i	0.305429i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.8345	−1.19624	−0.598118	−	0.801408i	$-0.704084\pi$
$431$	−24.8345	−1.19624	−0.598118	+	0.801408i	$0.704084\pi$
$432$	0	0
$433$	1.34060	0.0644253	0.0322127	−	0.999481i	$-0.489745\pi$
$433$	1.34060	0.0644253	0.0322127	+	0.999481i	$0.489745\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.10688i	0.100786i
$438$	0	0
$439$	− 10.8612i	− 0.518378i	−0.965827	−	0.259189i	$-0.916545\pi$
$439$	− 10.8612i	− 0.518378i	0.965827	−	0.259189i	$-0.0834552\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−11.1332	−0.528952	−0.264476	−	0.964392i	$-0.585199\pi$
$443$	−11.1332	−0.528952	−0.264476	+	0.964392i	$0.585199\pi$
$444$	0	0
$445$	−28.8494	−1.36759
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5.51207i	0.260131i	0.991505	+	0.130065i	$0.0415187\pi$
$449$	5.51207i	0.260131i	−0.991505	+	0.130065i	$0.958481\pi$
$450$	0	0
$451$	− 35.0611i	− 1.65096i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.99120	−0.280872
$456$	0	0
$457$	−26.8494	−1.25596	−0.627982	−	0.778228i	$-0.716119\pi$
$457$	−26.8494	−1.25596	−0.627982	+	0.778228i	$0.716119\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 29.4193i	− 1.37020i	−0.728451	−	0.685098i	$-0.759760\pi$
$461$	− 29.4193i	− 1.37020i	0.728451	−	0.685098i	$-0.240240\pi$
$462$	0	0
$463$	14.0169i	0.651421i	0.945470	+	0.325710i	$0.105603\pi$
$463$	14.0169i	0.651421i	−0.945470	+	0.325710i	$0.894397\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	35.8373	1.65835	0.829176	−	0.558988i	$-0.188810\pi$
$467$	35.8373	1.65835	0.829176	+	0.558988i	$0.188810\pi$
$468$	0	0
$469$	−7.82900	−0.361510
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 35.1424i	− 1.61585i
$474$	0	0
$475$	1.29887i	0.0595962i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.73144	0.124803	0.0624014	−	0.998051i	$-0.480124\pi$
$479$	2.73144	0.124803	0.0624014	+	0.998051i	$0.480124\pi$
$480$	0	0
$481$	9.67982	0.441362
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11.3137i	0.513729i
$486$	0	0
$487$	34.3540i	1.55673i	0.627814	+	0.778364i	$0.283950\pi$
$487$	34.3540i	1.55673i	−0.627814	+	0.778364i	$0.716050\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−12.7342	−0.574684	−0.287342	−	0.957828i	$-0.592772\pi$
$491$	−12.7342	−0.574684	−0.287342	+	0.957828i	$0.592772\pi$
$492$	0	0
$493$	15.2850	0.688404
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11.2898i	0.506419i
$498$	0	0
$499$	− 23.7974i	− 1.06532i	−0.846330	−	0.532659i	$-0.821193\pi$
$499$	− 23.7974i	− 1.06532i	0.846330	−	0.532659i	$-0.178807\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	17.4453	0.777847	0.388923	−	0.921270i	$-0.372847\pi$
$503$	17.4453	0.777847	0.388923	+	0.921270i	$0.372847\pi$
$504$	0	0
$505$	−53.7207	−2.39054
$506$	0	0
$507$	0	0
$508$	0	0
$509$	20.5867i	0.912491i	0.889854	+	0.456246i	$0.150806\pi$
$509$	20.5867i	0.912491i	−0.889854	+	0.456246i	$0.849194\pi$
$510$	0	0
$511$	8.33498i	0.368718i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	9.27142	0.408547
$516$	0	0
$517$	69.8889	3.07371
$518$	0	0
$519$	0	0
$520$	0	0
$521$	25.1613i	1.10234i	0.834395	+	0.551168i	$0.185818\pi$
$521$	25.1613i	1.10234i	−0.834395	+	0.551168i	$0.814182\pi$
$522$	0	0
$523$	17.0523i	0.745646i	0.927902	+	0.372823i	$0.121610\pi$
$523$	17.0523i	0.745646i	−0.927902	+	0.372823i	$0.878390\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.45393	−0.411820
$528$	0	0
$529$	56.8889	2.47343
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 6.07037i	− 0.262937i
$534$	0	0
$535$	31.1382i	1.34622i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	20.7050	0.891829
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	16.5178i	0.707543i
$546$	0	0
$547$	35.0611i	1.49911i	0.661944	+	0.749553i	$0.269731\pi$
$547$	35.0611i	1.49911i	−0.661944	+	0.749553i	$0.730269\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.00008	−0.0426047
$552$	0	0
$553$	−0.871234	−0.0370486
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 42.6559i	− 1.80739i	−0.428180	−	0.903693i	$-0.640845\pi$
$557$	− 42.6559i	− 1.80739i	0.428180	−	0.903693i	$-0.359155\pi$
$558$	0	0
$559$	− 6.08444i	− 0.257344i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	25.7166	1.08383	0.541913	−	0.840434i	$-0.317700\pi$
$563$	25.7166	1.08383	0.541913	+	0.840434i	$0.317700\pi$
$564$	0	0
$565$	−62.5483	−2.63143
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 7.17656i	− 0.300857i	−0.988621	−	0.150428i	$-0.951935\pi$
$569$	− 7.17656i	− 0.300857i	0.988621	−	0.150428i	$-0.0480653\pi$
$570$	0	0
$571$	27.5624i	1.15345i	0.816939	+	0.576725i	$0.195669\pi$
$571$	27.5624i	1.15345i	−0.816939	+	0.576725i	$0.804331\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	49.2506	2.05389
$576$	0	0
$577$	−5.21045	−0.216914	−0.108457	−	0.994101i	$-0.534591\pi$
$577$	−5.21045	−0.216914	−0.108457	+	0.994101i	$0.534591\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 25.9594i	− 1.07698i
$582$	0	0
$583$	− 20.8085i	− 0.861800i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.34487	0.179332	0.0896660	−	0.995972i	$-0.471420\pi$
$587$	4.34487	0.179332	0.0896660	+	0.995972i	$0.471420\pi$
$588$	0	0
$589$	0.618556	0.0254872
$590$	0	0
$591$	0	0
$592$	0	0
$593$	31.5262i	1.29463i	0.762224	+	0.647313i	$0.224107\pi$
$593$	31.5262i	1.29463i	−0.762224	+	0.647313i	$0.775893\pi$
$594$	0	0
$595$	− 21.5846i	− 0.884882i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	23.1882	0.947443	0.473721	−	0.880675i	$-0.342910\pi$
$599$	23.1882	0.947443	0.473721	+	0.880675i	$0.342910\pi$
$600$	0	0
$601$	−6.83039	−0.278618	−0.139309	−	0.990249i	$-0.544488\pi$
$601$	−6.83039	−0.278618	−0.139309	+	0.990249i	$0.544488\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 72.4887i	− 2.94709i
$606$	0	0
$607$	− 1.30766i	− 0.0530763i	−0.999648	−	0.0265381i	$-0.991552\pi$
$607$	− 1.30766i	− 0.0530763i	0.999648	−	0.0265381i	$-0.00844834\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.1003	0.489527
$612$	0	0
$613$	1.82900	0.0738727	0.0369364	−	0.999318i	$-0.488240\pi$
$613$	1.82900	0.0738727	0.0369364	+	0.999318i	$0.488240\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 18.6639i	− 0.751381i	−0.926745	−	0.375691i	$-0.877406\pi$
$617$	− 18.6639i	− 0.751381i	0.926745	−	0.375691i	$-0.122594\pi$
$618$	0	0
$619$	23.7974i	0.956498i	0.878224	+	0.478249i	$0.158728\pi$
$619$	23.7974i	0.956498i	−0.878224	+	0.478249i	$0.841272\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−16.4452	−0.658863
$624$	0	0
$625$	−22.1886	−0.887545
$626$	0	0
$627$	0	0
$628$	0	0
$629$	34.8737i	1.39050i
$630$	0	0
$631$	2.62411i	0.104464i	0.998635	+	0.0522321i	$0.0166336\pi$
$631$	2.62411i	0.104464i	−0.998635	+	0.0522321i	$0.983366\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	40.9792	1.62621
$636$	0	0
$637$	3.58480	0.142035
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 45.3922i	− 1.79289i	−0.443159	−	0.896443i	$-0.646142\pi$
$641$	− 45.3922i	− 1.79289i	0.443159	−	0.896443i	$-0.353858\pi$
$642$	0	0
$643$	− 10.5478i	− 0.415964i	−0.978133	−	0.207982i	$-0.933310\pi$
$643$	− 10.5478i	− 0.415964i	0.978133	−	0.207982i	$-0.0666896\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−24.9524	−0.980981	−0.490491	−	0.871446i	$-0.663182\pi$
$647$	−24.9524	−0.980981	−0.490491	+	0.871446i	$0.663182\pi$
$648$	0	0
$649$	−18.2646	−0.716949
$650$	0	0
$651$	0	0
$652$	0	0
$653$	12.1696i	0.476234i	0.971236	+	0.238117i	$0.0765302\pi$
$653$	12.1696i	0.476234i	−0.971236	+	0.238117i	$0.923470\pi$
$654$	0	0
$655$	15.8649i	0.619894i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−11.9824	−0.466768	−0.233384	−	0.972385i	$-0.574980\pi$
$659$	−11.9824	−0.466768	−0.233384	+	0.972385i	$0.574980\pi$
$660$	0	0
$661$	−28.0190	−1.08981	−0.544907	−	0.838497i	$-0.683435\pi$
$661$	−28.0190	−1.08981	−0.544907	+	0.838497i	$0.683435\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.41225i	0.0547646i
$666$	0	0
$667$	37.9210i	1.46831i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−19.7254	−0.761492
$672$	0	0
$673$	26.7340	1.03052	0.515260	−	0.857034i	$-0.327695\pi$
$673$	26.7340	1.03052	0.515260	+	0.857034i	$0.327695\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	17.7449i	0.681990i	0.940065	+	0.340995i	$0.110764\pi$
$677$	17.7449i	0.681990i	−0.940065	+	0.340995i	$0.889236\pi$
$678$	0	0
$679$	6.44922i	0.247498i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−6.29165	−0.240743	−0.120372	−	0.992729i	$-0.538409\pi$
$683$	−6.29165	−0.240743	−0.120372	+	0.992729i	$0.538409\pi$
$684$	0	0
$685$	−8.17100	−0.312198
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 3.60272i	− 0.137253i
$690$	0	0
$691$	− 43.5717i	− 1.65755i	−0.559585	−	0.828773i	$-0.689040\pi$
$691$	− 43.5717i	− 1.65755i	0.559585	−	0.828773i	$-0.310960\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.23936	−0.160808
$696$	0	0
$697$	21.8698	0.828379
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 46.1107i	− 1.74158i	−0.491656	−	0.870789i	$-0.663608\pi$
$701$	− 46.1107i	− 1.74158i	0.491656	−	0.870789i	$-0.336392\pi$
$702$	0	0
$703$	− 2.28173i	− 0.0860570i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−30.6227	−1.15169
$708$	0	0
$709$	28.4884	1.06990	0.534952	−	0.844882i	$-0.320330\pi$
$709$	28.4884	1.06990	0.534952	+	0.844882i	$0.320330\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 23.4545i	− 0.878376i
$714$	0	0
$715$	− 18.7248i	− 0.700266i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−17.0269	−0.634996	−0.317498	−	0.948259i	$-0.602843\pi$
$719$	−17.0269	−0.634996	−0.317498	+	0.948259i	$0.602843\pi$
$720$	0	0
$721$	5.28504	0.196825
$722$	0	0
$723$	0	0
$724$	0	0
$725$	23.3778i	0.868231i
$726$	0	0
$727$	11.4393i	0.424261i	0.977241	+	0.212131i	$0.0680402\pi$
$727$	11.4393i	0.424261i	−0.977241	+	0.212131i	$0.931960\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	21.9205	0.810760
$732$	0	0
$733$	18.9050	0.698272	0.349136	−	0.937072i	$-0.386475\pi$
$733$	18.9050	0.698272	0.349136	+	0.937072i	$0.386475\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 24.4686i	− 0.901313i
$738$	0	0
$739$	− 11.6285i	− 0.427762i	−0.976860	−	0.213881i	$-0.931390\pi$
$739$	− 11.6285i	− 0.427762i	0.976860	−	0.213881i	$-0.0686105\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.2297	0.558724	0.279362	−	0.960186i	$-0.409877\pi$
$743$	15.2297	0.558724	0.279362	+	0.960186i	$0.409877\pi$
$744$	0	0
$745$	45.3787	1.66255
$746$	0	0
$747$	0	0
$748$	0	0
$749$	17.7499i	0.648566i
$750$	0	0
$751$	− 29.2212i	− 1.06630i	−0.846022	−	0.533148i	$-0.821009\pi$
$751$	− 29.2212i	− 1.06630i	0.846022	−	0.533148i	$-0.178991\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	42.9260	1.56224
$756$	0	0
$757$	23.8698	0.867564	0.433782	−	0.901018i	$-0.357179\pi$
$757$	23.8698	0.867564	0.433782	+	0.901018i	$0.357179\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 29.5248i	− 1.07027i	−0.844765	−	0.535137i	$-0.820260\pi$
$761$	− 29.5248i	− 1.07027i	0.844765	−	0.535137i	$-0.179740\pi$
$762$	0	0
$763$	9.41571i	0.340872i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.16228	−0.114183
$768$	0	0
$769$	28.4546	1.02610	0.513050	−	0.858358i	$-0.328515\pi$
$769$	28.4546	1.02610	0.513050	+	0.858358i	$0.328515\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	39.9329i	1.43629i	0.695895	+	0.718144i	$0.255008\pi$
$773$	39.9329i	1.43629i	−0.695895	+	0.718144i	$0.744992\pi$
$774$	0	0
$775$	− 14.4594i	− 0.519397i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.43091	−0.0512677
$780$	0	0
$781$	−35.2850	−1.26260
$782$	0	0
$783$	0	0
$784$	0	0
$785$	41.0411i	1.46482i
$786$	0	0
$787$	28.4363i	1.01365i	0.862050	+	0.506823i	$0.169180\pi$
$787$	28.4363i	1.01365i	−0.862050	+	0.506823i	$0.830820\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−35.6548	−1.26774
$792$	0	0
$793$	−3.41520	−0.121277
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 10.8320i	− 0.383690i	−0.981425	−	0.191845i	$-0.938553\pi$
$797$	− 10.8320i	− 0.383690i	0.981425	−	0.191845i	$-0.0614471\pi$
$798$	0	0
$799$	43.5941i	1.54225i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−26.0500	−0.919284
$804$	0	0
$805$	53.5497	1.88738
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0.505553i	0.0177743i	0.999961	+	0.00888715i	$0.00282891\pi$
$809$	0.505553i	0.0177743i	−0.999961	+	0.00888715i	$0.997171\pi$
$810$	0	0
$811$	− 38.6281i	− 1.35642i	−0.734870	−	0.678209i	$-0.762757\pi$
$811$	− 38.6281i	− 1.35642i	0.734870	−	0.678209i	$-0.237243\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	11.2182	0.392956
$816$	0	0
$817$	−1.43423	−0.0501772
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 23.2042i	− 0.809832i	−0.914354	−	0.404916i	$-0.867301\pi$
$821$	− 23.2042i	− 0.809832i	0.914354	−	0.404916i	$-0.132699\pi$
$822$	0	0
$823$	42.4532i	1.47983i	0.672702	+	0.739913i	$0.265134\pi$
$823$	42.4532i	1.47983i	−0.672702	+	0.739913i	$0.734866\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−27.2655	−0.948113	−0.474057	−	0.880494i	$-0.657211\pi$
$827$	−27.2655	−0.948113	−0.474057	+	0.880494i	$0.657211\pi$
$828$	0	0
$829$	−15.7207	−0.546001	−0.273000	−	0.962014i	$-0.588016\pi$
$829$	−15.7207	−0.546001	−0.273000	+	0.962014i	$0.588016\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	12.9150i	0.447479i
$834$	0	0
$835$	− 59.7325i	− 2.06713i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3.91404	−0.135128	−0.0675638	−	0.997715i	$-0.521523\pi$
$839$	−3.91404	−0.135128	−0.0675638	+	0.997715i	$0.521523\pi$
$840$	0	0
$841$	11.0000	0.379310
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 3.24195i	− 0.111526i
$846$	0	0
$847$	− 41.3212i	− 1.41981i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−86.5188	−2.96582
$852$	0	0
$853$	−14.9986	−0.513543	−0.256771	−	0.966472i	$-0.582659\pi$
$853$	−14.9986	−0.513543	−0.256771	+	0.966472i	$0.582659\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.85731i	0.0976037i	0.998808	+	0.0488018i	$0.0155403\pi$
$857$	2.85731i	0.0976037i	−0.998808	+	0.0488018i	$0.984460\pi$
$858$	0	0
$859$	− 4.89705i	− 0.167085i	−0.996504	−	0.0835426i	$-0.973377\pi$
$859$	− 4.89705i	− 0.167085i	0.996504	−	0.0835426i	$-0.0266235\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	37.7195	1.28399	0.641993	−	0.766710i	$-0.278108\pi$
$863$	37.7195	1.28399	0.641993	+	0.766710i	$0.278108\pi$
$864$	0	0
$865$	23.1886	0.788437
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 2.72294i	− 0.0923693i
$870$	0	0
$871$	− 4.23642i	− 0.143545i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.05677	0.103338
$876$	0	0
$877$	22.8494	0.771570	0.385785	−	0.922589i	$-0.373931\pi$
$877$	22.8494	0.771570	0.385785	+	0.922589i	$0.373931\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 7.55079i	− 0.254392i	−0.991878	−	0.127196i	$-0.959402\pi$
$881$	− 7.55079i	− 0.254392i	0.991878	−	0.127196i	$-0.0405978\pi$
$882$	0	0
$883$	39.3353i	1.32374i	0.749620	+	0.661869i	$0.230236\pi$
$883$	39.3353i	1.32374i	−0.749620	+	0.661869i	$0.769764\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.61350	0.0877528	0.0438764	−	0.999037i	$-0.486029\pi$
$887$	2.61350	0.0877528	0.0438764	+	0.999037i	$0.486029\pi$
$888$	0	0
$889$	23.3596	0.783457
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 2.85229i	− 0.0954484i
$894$	0	0
$895$	− 4.63893i	− 0.155062i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	11.1332	0.371312
$900$	0	0
$901$	12.9796	0.432413
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.82982i	0.160549i
$906$	0	0
$907$	20.9753i	0.696474i	0.937407	+	0.348237i	$0.113219\pi$
$907$	20.9753i	0.696474i	−0.937407	+	0.348237i	$0.886781\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−36.1706	−1.19839	−0.599193	−	0.800604i	$-0.704512\pi$
$911$	−36.1706	−1.19839	−0.599193	+	0.800604i	$0.704512\pi$
$912$	0	0
$913$	81.1331	2.68511
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.04358i	0.298645i
$918$	0	0
$919$	− 7.27183i	− 0.239876i	−0.992781	−	0.119938i	$-0.961730\pi$
$919$	− 7.27183i	− 0.239876i	0.992781	−	0.119938i	$-0.0382695\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−6.10914	−0.201085
$924$	0	0
$925$	−53.3378	−1.75374
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 47.6489i	− 1.56331i	−0.623711	−	0.781655i	$-0.714376\pi$
$929$	− 47.6489i	− 1.56331i	0.623711	−	0.781655i	$-0.285624\pi$
$930$	0	0
$931$	− 0.845011i	− 0.0276941i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	67.4600	2.20618
$936$	0	0
$937$	−12.9986	−0.424646	−0.212323	−	0.977200i	$-0.568103\pi$
$937$	−12.9986	−0.424646	−0.212323	+	0.977200i	$0.568103\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 15.0354i	− 0.490139i	−0.969505	−	0.245070i	$-0.921189\pi$
$941$	− 15.0354i	− 0.490139i	0.969505	−	0.245070i	$-0.0788108\pi$
$942$	0	0
$943$	54.2573i	1.76686i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−27.8788	−0.905940	−0.452970	−	0.891526i	$-0.649636\pi$
$947$	−27.8788	−0.905940	−0.452970	+	0.891526i	$0.649636\pi$
$948$	0	0
$949$	−4.51021	−0.146408
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 12.7279i	− 0.412298i	−0.978521	−	0.206149i	$-0.933907\pi$
$953$	− 12.7279i	− 0.412298i	0.978521	−	0.206149i	$-0.0660931\pi$
$954$	0	0
$955$	4.63893i	0.150112i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4.65776	−0.150407
$960$	0	0
$961$	24.1140	0.777872
$962$	0	0
$963$	0	0
$964$	0	0
$965$	13.2096i	0.425232i
$966$	0	0
$967$	− 46.8564i	− 1.50680i	−0.657561	−	0.753401i	$-0.728412\pi$
$967$	− 46.8564i	− 1.50680i	0.657561	−	0.753401i	$-0.271588\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−37.6015	−1.20669	−0.603345	−	0.797480i	$-0.706166\pi$
$971$	−37.6015	−1.20669	−0.603345	+	0.797480i	$0.706166\pi$
$972$	0	0
$973$	−2.41659	−0.0774723
$974$	0	0
$975$	0	0
$976$	0	0
$977$	12.0745i	0.386299i	0.981169	+	0.193149i	$0.0618702\pi$
$977$	12.0745i	0.386299i	−0.981169	+	0.193149i	$0.938130\pi$
$978$	0	0
$979$	− 51.3975i	− 1.64267i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	27.8788	0.889196	0.444598	−	0.895730i	$-0.353346\pi$
$983$	27.8788	0.889196	0.444598	+	0.895730i	$0.353346\pi$
$984$	0	0
$985$	13.1914	0.420314
$986$	0	0
$987$	0	0
$988$	0	0
$989$	54.3831i	1.72928i
$990$	0	0
$991$	46.6207i	1.48096i	0.672080	+	0.740478i	$0.265401\pi$
$991$	46.6207i	1.48096i	−0.672080	+	0.740478i	$0.734599\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	75.3980	2.39028
$996$	0	0
$997$	24.3392	0.770831	0.385415	−	0.922743i	$-0.374058\pi$
$997$	24.3392	0.770831	0.385415	+	0.922743i	$0.374058\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7488.2.d.m.4031.3		12
3.2	odd	2	inner	7488.2.d.m.4031.9		12
4.3	odd	2	inner	7488.2.d.m.4031.4		12
8.3	odd	2		468.2.c.c.287.7	yes	12
8.5	even	2		468.2.c.c.287.5	✓	12
12.11	even	2	inner	7488.2.d.m.4031.10		12
24.5	odd	2		468.2.c.c.287.8	yes	12
24.11	even	2		468.2.c.c.287.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
468.2.c.c.287.5	✓	12	8.5	even	2
468.2.c.c.287.6	yes	12	24.11	even	2
468.2.c.c.287.7	yes	12	8.3	odd	2
468.2.c.c.287.8	yes	12	24.5	odd	2
7488.2.d.m.4031.3		12	1.1	even	1	trivial
7488.2.d.m.4031.4		12	4.3	odd	2	inner
7488.2.d.m.4031.9		12	3.2	odd	2	inner
7488.2.d.m.4031.10		12	12.11	even	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:	\( N \)	\(=\)	\( 7488 = 2^{6} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7488.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.24195i q^{5} -1.84803i q^{7} +O(q^{10})\)	\(q-3.24195i q^{5} -1.84803i q^{7} +5.77578 q^{11} +1.00000 q^{13} +3.60272i q^{17} -0.235720i q^{19} -8.93806 q^{23} -5.51021 q^{25} -4.24264i q^{29} +2.62411i q^{31} -5.99120 q^{35} +9.67982 q^{37} -6.07037i q^{41} -6.08444i q^{43} +12.1003 q^{47} +3.58480 q^{49} -3.60272i q^{53} -18.7248i q^{55} -3.16228 q^{59} -3.41520 q^{61} -3.24195i q^{65} -4.23642i q^{67} -6.10914 q^{71} -4.51021 q^{73} -10.6738i q^{77} -0.471440i q^{79} +14.0471 q^{83} +11.6798 q^{85} -8.89880i q^{89} -1.84803i q^{91} -0.764192 q^{95} -3.48979 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q + 12 q^{13} - 52 q^{25} - 8 q^{37} - 12 q^{49} - 96 q^{61} - 40 q^{73} + 16 q^{85} - 56 q^{97}+O(q^{100}) \) 12 * q + 12 * q^13 - 52 * q^25 - 8 * q^37 - 12 * q^49 - 96 * q^61 - 40 * q^73 + 16 * q^85 - 56 * q^97

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