LMFDB - Embedded newform 2312.2.b.g.577.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,2,Mod(577,2312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2312, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2312.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2312.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:	$18.4614129473$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 136)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	2312.577
Dual form			2312.2.b.g.577.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.23607i q^{3} +2.00000i q^{5} -3.23607i q^{7} -7.47214 q^{9} -3.23607i q^{11} -4.47214 q^{13} +6.47214 q^{15} -2.47214 q^{19} -10.4721 q^{21} -3.23607i q^{23} +1.00000 q^{25} +14.4721i q^{27} +2.00000i q^{29} -3.23607i q^{31} -10.4721 q^{33} +6.47214 q^{35} +6.94427i q^{37} +14.4721i q^{39} -2.00000i q^{41} +10.4721 q^{43} -14.9443i q^{45} -4.94427 q^{47} -3.47214 q^{49} +2.00000 q^{53} +6.47214 q^{55} +8.00000i q^{57} -5.52786 q^{59} +10.9443i q^{61} +24.1803i q^{63} -8.94427i q^{65} -12.0000 q^{67} -10.4721 q^{69} +4.76393i q^{71} -2.94427i q^{73} -3.23607i q^{75} -10.4721 q^{77} +1.70820i q^{79} +24.4164 q^{81} -10.4721 q^{83} +6.47214 q^{87} -16.4721 q^{89} +14.4721i q^{91} -10.4721 q^{93} -4.94427i q^{95} +2.00000i q^{97} +24.1803i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9} + 8 q^{15} + 8 q^{19} - 24 q^{21} + 4 q^{25} - 24 q^{33} + 8 q^{35} + 24 q^{43} + 16 q^{47} + 4 q^{49} + 8 q^{53} + 8 q^{55} - 40 q^{59} - 48 q^{67} - 24 q^{69} - 24 q^{77} + 44 q^{81} - 24 q^{83}+ \cdots - 24 q^{93}+O(q^{100}) $ 4 * q - 12 * q^9 + 8 * q^15 + 8 * q^19 - 24 * q^21 + 4 * q^25 - 24 * q^33 + 8 * q^35 + 24 * q^43 + 16 * q^47 + 4 * q^49 + 8 * q^53 + 8 * q^55 - 40 * q^59 - 48 * q^67 - 24 * q^69 - 24 * q^77 + 44 * q^81 - 24 * q^83 + 8 * q^87 - 48 * q^89 - 24 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times$.

$n$	$1157$	$1735$	$1737$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 3.23607i	− 1.86834i	−0.356822	−	0.934172i	$-0.616140\pi$
$3$	− 3.23607i	− 1.86834i	0.356822	−	0.934172i	$-0.383860\pi$
$4$	0	0
$5$	2.00000i	0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$	2.00000i	0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	− 3.23607i	− 1.22312i	−0.791199	−	0.611559i	$-0.790543\pi$
$7$	− 3.23607i	− 1.22312i	0.791199	−	0.611559i	$-0.209457\pi$
$8$	0	0
$9$	−7.47214	−2.49071
$10$	0	0
$11$	− 3.23607i	− 0.975711i	−0.872924	−	0.487856i	$-0.837779\pi$
$11$	− 3.23607i	− 0.975711i	0.872924	−	0.487856i	$-0.162221\pi$
$12$	0	0
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	6.47214	1.67110
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−2.47214	−0.567147	−0.283573	−	0.958951i	$-0.591520\pi$
$19$	−2.47214	−0.567147	−0.283573	+	0.958951i	$0.591520\pi$
$20$	0	0
$21$	−10.4721	−2.28521
$22$	0	0
$23$	− 3.23607i	− 0.674767i	−0.941367	−	0.337383i	$-0.890458\pi$
$23$	− 3.23607i	− 0.674767i	0.941367	−	0.337383i	$-0.109542\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	14.4721i	2.78516i
$28$	0	0
$29$	2.00000i	0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$	2.00000i	0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$	0	0
$31$	− 3.23607i	− 0.581215i	−0.956842	−	0.290607i	$-0.906143\pi$
$31$	− 3.23607i	− 0.581215i	0.956842	−	0.290607i	$-0.0938574\pi$
$32$	0	0
$33$	−10.4721	−1.82296
$34$	0	0
$35$	6.47214	1.09399
$36$	0	0
$37$	6.94427i	1.14163i	0.821078	+	0.570816i	$0.193373\pi$
$37$	6.94427i	1.14163i	−0.821078	+	0.570816i	$0.806627\pi$
$38$	0	0
$39$	14.4721i	2.31740i
$40$	0	0
$41$	− 2.00000i	− 0.312348i	−0.987730	−	0.156174i	$-0.950084\pi$
$41$	− 2.00000i	− 0.312348i	0.987730	−	0.156174i	$-0.0499160\pi$
$42$	0	0
$43$	10.4721	1.59699	0.798493	−	0.602004i	$-0.205631\pi$
$43$	10.4721	1.59699	0.798493	+	0.602004i	$0.205631\pi$
$44$	0	0
$45$	− 14.9443i	− 2.22776i
$46$	0	0
$47$	−4.94427	−0.721196	−0.360598	−	0.932721i	$-0.617427\pi$
$47$	−4.94427	−0.721196	−0.360598	+	0.932721i	$0.617427\pi$
$48$	0	0
$49$	−3.47214	−0.496019
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	6.47214	0.872703
$56$	0	0
$57$	8.00000i	1.05963i
$58$	0	0
$59$	−5.52786	−0.719667	−0.359833	−	0.933017i	$-0.617166\pi$
$59$	−5.52786	−0.719667	−0.359833	+	0.933017i	$0.617166\pi$
$60$	0	0
$61$	10.9443i	1.40127i	0.713520	+	0.700635i	$0.247100\pi$
$61$	10.9443i	1.40127i	−0.713520	+	0.700635i	$0.752900\pi$
$62$	0	0
$63$	24.1803i	3.04644i
$64$	0	0
$65$	− 8.94427i	− 1.10940i
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	−10.4721	−1.26070
$70$	0	0
$71$	4.76393i	0.565375i	0.959212	+	0.282687i	$0.0912259\pi$
$71$	4.76393i	0.565375i	−0.959212	+	0.282687i	$0.908774\pi$
$72$	0	0
$73$	− 2.94427i	− 0.344601i	−0.985044	−	0.172300i	$-0.944880\pi$
$73$	− 2.94427i	− 0.344601i	0.985044	−	0.172300i	$-0.0551200\pi$
$74$	0	0
$75$	− 3.23607i	− 0.373669i
$76$	0	0
$77$	−10.4721	−1.19341
$78$	0	0
$79$	1.70820i	0.192188i	0.995372	+	0.0960940i	$0.0306349\pi$
$79$	1.70820i	0.192188i	−0.995372	+	0.0960940i	$0.969365\pi$
$80$	0	0
$81$	24.4164	2.71293
$82$	0	0
$83$	−10.4721	−1.14947	−0.574733	−	0.818341i	$-0.694894\pi$
$83$	−10.4721	−1.14947	−0.574733	+	0.818341i	$0.694894\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.47214	0.693886
$88$	0	0
$89$	−16.4721	−1.74604	−0.873021	−	0.487682i	$-0.837843\pi$
$89$	−16.4721	−1.74604	−0.873021	+	0.487682i	$0.837843\pi$
$90$	0	0
$91$	14.4721i	1.51709i
$92$	0	0
$93$	−10.4721	−1.08591
$94$	0	0
$95$	− 4.94427i	− 0.507272i
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	0	0
$99$	24.1803i	2.43022i
$100$	0	0
$101$	13.4164	1.33498	0.667491	−	0.744618i	$-0.267368\pi$
$101$	13.4164	1.33498	0.667491	+	0.744618i	$0.267368\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	− 20.9443i	− 2.04395i
$106$	0	0
$107$	9.70820i	0.938527i	0.883058	+	0.469264i	$0.155481\pi$
$107$	9.70820i	0.938527i	−0.883058	+	0.469264i	$0.844519\pi$
$108$	0	0
$109$	− 10.0000i	− 0.957826i	−0.877862	−	0.478913i	$-0.841031\pi$
$109$	− 10.0000i	− 0.957826i	0.877862	−	0.478913i	$-0.158969\pi$
$110$	0	0
$111$	22.4721	2.13296
$112$	0	0
$113$	2.94427i	0.276974i	0.990364	+	0.138487i	$0.0442239\pi$
$113$	2.94427i	0.276974i	−0.990364	+	0.138487i	$0.955776\pi$
$114$	0	0
$115$	6.47214	0.603530
$116$	0	0
$117$	33.4164	3.08935
$118$	0	0
$119$	0	0
$120$	0	0
$121$	0.527864	0.0479876
$122$	0	0
$123$	−6.47214	−0.583573
$124$	0	0
$125$	12.0000i	1.07331i
$126$	0	0
$127$	6.47214	0.574309	0.287155	−	0.957884i	$-0.407291\pi$
$127$	6.47214	0.574309	0.287155	+	0.957884i	$0.407291\pi$
$128$	0	0
$129$	− 33.8885i	− 2.98372i
$130$	0	0
$131$	− 21.1246i	− 1.84567i	−0.385200	−	0.922833i	$-0.625868\pi$
$131$	− 21.1246i	− 1.84567i	0.385200	−	0.922833i	$-0.374132\pi$
$132$	0	0
$133$	8.00000i	0.693688i
$134$	0	0
$135$	−28.9443	−2.49113
$136$	0	0
$137$	−16.4721	−1.40731	−0.703655	−	0.710542i	$-0.748450\pi$
$137$	−16.4721	−1.40731	−0.703655	+	0.710542i	$0.748450\pi$
$138$	0	0
$139$	− 11.2361i	− 0.953031i	−0.879166	−	0.476515i	$-0.841900\pi$
$139$	− 11.2361i	− 0.953031i	0.879166	−	0.476515i	$-0.158100\pi$
$140$	0	0
$141$	16.0000i	1.34744i
$142$	0	0
$143$	14.4721i	1.21022i
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	11.2361i	0.926735i
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−14.4721	−1.17773	−0.588863	−	0.808233i	$-0.700424\pi$
$151$	−14.4721	−1.17773	−0.588863	+	0.808233i	$0.700424\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.47214	0.519854
$156$	0	0
$157$	23.8885	1.90651	0.953257	−	0.302162i	$-0.0977083\pi$
$157$	23.8885	1.90651	0.953257	+	0.302162i	$0.0977083\pi$
$158$	0	0
$159$	− 6.47214i	− 0.513274i
$160$	0	0
$161$	−10.4721	−0.825320
$162$	0	0
$163$	− 0.180340i	− 0.0141253i	−0.999975	−	0.00706266i	$-0.997752\pi$
$163$	− 0.180340i	− 0.0141253i	0.999975	−	0.00706266i	$-0.00224813\pi$
$164$	0	0
$165$	− 20.9443i	− 1.63051i
$166$	0	0
$167$	− 6.29180i	− 0.486874i	−0.969917	−	0.243437i	$-0.921725\pi$
$167$	− 6.29180i	− 0.486874i	0.969917	−	0.243437i	$-0.0782749\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	18.4721	1.41260
$172$	0	0
$173$	2.00000i	0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$	2.00000i	0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$	0	0
$175$	− 3.23607i	− 0.244624i
$176$	0	0
$177$	17.8885i	1.34459i
$178$	0	0
$179$	−5.52786	−0.413172	−0.206586	−	0.978428i	$-0.566235\pi$
$179$	−5.52786	−0.413172	−0.206586	+	0.978428i	$0.566235\pi$
$180$	0	0
$181$	− 10.0000i	− 0.743294i	−0.928374	−	0.371647i	$-0.878793\pi$
$181$	− 10.0000i	− 0.743294i	0.928374	−	0.371647i	$-0.121207\pi$
$182$	0	0
$183$	35.4164	2.61806
$184$	0	0
$185$	−13.8885	−1.02111
$186$	0	0
$187$	0	0
$188$	0	0
$189$	46.8328	3.40659
$190$	0	0
$191$	4.94427	0.357755	0.178877	−	0.983871i	$-0.442753\pi$
$191$	4.94427	0.357755	0.178877	+	0.983871i	$0.442753\pi$
$192$	0	0
$193$	2.94427i	0.211933i	0.994370	+	0.105967i	$0.0337937\pi$
$193$	2.94427i	0.211933i	−0.994370	+	0.105967i	$0.966206\pi$
$194$	0	0
$195$	−28.9443	−2.07274
$196$	0	0
$197$	10.9443i	0.779747i	0.920868	+	0.389874i	$0.127481\pi$
$197$	10.9443i	0.779747i	−0.920868	+	0.389874i	$0.872519\pi$
$198$	0	0
$199$	− 0.180340i	− 0.0127840i	−0.999980	−	0.00639198i	$-0.997965\pi$
$199$	− 0.180340i	− 0.0127840i	0.999980	−	0.00639198i	$-0.00203464\pi$
$200$	0	0
$201$	38.8328i	2.73906i
$202$	0	0
$203$	6.47214	0.454255
$204$	0	0
$205$	4.00000	0.279372
$206$	0	0
$207$	24.1803i	1.68065i
$208$	0	0
$209$	8.00000i	0.553372i
$210$	0	0
$211$	− 3.23607i	− 0.222780i	−0.993777	−	0.111390i	$-0.964470\pi$
$211$	− 3.23607i	− 0.222780i	0.993777	−	0.111390i	$-0.0355303\pi$
$212$	0	0
$213$	15.4164	1.05631
$214$	0	0
$215$	20.9443i	1.42839i
$216$	0	0
$217$	−10.4721	−0.710895
$218$	0	0
$219$	−9.52786	−0.643833
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−14.4721	−0.969126	−0.484563	−	0.874756i	$-0.661021\pi$
$223$	−14.4721	−0.969126	−0.484563	+	0.874756i	$0.661021\pi$
$224$	0	0
$225$	−7.47214	−0.498142
$226$	0	0
$227$	− 3.23607i	− 0.214785i	−0.994217	−	0.107393i	$-0.965750\pi$
$227$	− 3.23607i	− 0.214785i	0.994217	−	0.107393i	$-0.0342502\pi$
$228$	0	0
$229$	15.5279	1.02611	0.513055	−	0.858356i	$-0.328514\pi$
$229$	15.5279	1.02611	0.513055	+	0.858356i	$0.328514\pi$
$230$	0	0
$231$	33.8885i	2.22970i
$232$	0	0
$233$	11.8885i	0.778844i	0.921059	+	0.389422i	$0.127325\pi$
$233$	11.8885i	0.778844i	−0.921059	+	0.389422i	$0.872675\pi$
$234$	0	0
$235$	− 9.88854i	− 0.645057i
$236$	0	0
$237$	5.52786	0.359073
$238$	0	0
$239$	−17.8885	−1.15711	−0.578557	−	0.815642i	$-0.696384\pi$
$239$	−17.8885	−1.15711	−0.578557	+	0.815642i	$0.696384\pi$
$240$	0	0
$241$	− 18.9443i	− 1.22031i	−0.792283	−	0.610154i	$-0.791108\pi$
$241$	− 18.9443i	− 1.22031i	0.792283	−	0.610154i	$-0.208892\pi$
$242$	0	0
$243$	− 35.5967i	− 2.28353i
$244$	0	0
$245$	− 6.94427i	− 0.443653i
$246$	0	0
$247$	11.0557	0.703459
$248$	0	0
$249$	33.8885i	2.14760i
$250$	0	0
$251$	8.94427	0.564557	0.282279	−	0.959332i	$-0.408910\pi$
$251$	8.94427	0.564557	0.282279	+	0.959332i	$0.408910\pi$
$252$	0	0
$253$	−10.4721	−0.658378
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−22.3607	−1.39482	−0.697410	−	0.716672i	$-0.745665\pi$
$257$	−22.3607	−1.39482	−0.697410	+	0.716672i	$0.745665\pi$
$258$	0	0
$259$	22.4721	1.39635
$260$	0	0
$261$	− 14.9443i	− 0.925027i
$262$	0	0
$263$	−11.4164	−0.703966	−0.351983	−	0.936006i	$-0.614492\pi$
$263$	−11.4164	−0.703966	−0.351983	+	0.936006i	$0.614492\pi$
$264$	0	0
$265$	4.00000i	0.245718i
$266$	0	0
$267$	53.3050i	3.26221i
$268$	0	0
$269$	− 18.9443i	− 1.15505i	−0.816372	−	0.577526i	$-0.804018\pi$
$269$	− 18.9443i	− 1.15505i	0.816372	−	0.577526i	$-0.195982\pi$
$270$	0	0
$271$	17.8885	1.08665	0.543326	−	0.839522i	$-0.317165\pi$
$271$	17.8885	1.08665	0.543326	+	0.839522i	$0.317165\pi$
$272$	0	0
$273$	46.8328	2.83445
$274$	0	0
$275$	− 3.23607i	− 0.195142i
$276$	0	0
$277$	32.8328i	1.97273i	0.164564	+	0.986366i	$0.447378\pi$
$277$	32.8328i	1.97273i	−0.164564	+	0.986366i	$0.552622\pi$
$278$	0	0
$279$	24.1803i	1.44764i
$280$	0	0
$281$	15.8885	0.947831	0.473916	−	0.880570i	$-0.342840\pi$
$281$	15.8885	0.947831	0.473916	+	0.880570i	$0.342840\pi$
$282$	0	0
$283$	− 3.23607i	− 0.192364i	−0.995364	−	0.0961821i	$-0.969337\pi$
$283$	− 3.23607i	− 0.192364i	0.995364	−	0.0961821i	$-0.0306631\pi$
$284$	0	0
$285$	−16.0000	−0.947758
$286$	0	0
$287$	−6.47214	−0.382038
$288$	0	0
$289$	0	0
$290$	0	0
$291$	6.47214	0.379403
$292$	0	0
$293$	−10.0000	−0.584206	−0.292103	−	0.956387i	$-0.594355\pi$
$293$	−10.0000	−0.584206	−0.292103	+	0.956387i	$0.594355\pi$
$294$	0	0
$295$	− 11.0557i	− 0.643689i
$296$	0	0
$297$	46.8328	2.71752
$298$	0	0
$299$	14.4721i	0.836945i
$300$	0	0
$301$	− 33.8885i	− 1.95330i
$302$	0	0
$303$	− 43.4164i	− 2.49421i
$304$	0	0
$305$	−21.8885	−1.25333
$306$	0	0
$307$	−29.8885	−1.70583	−0.852915	−	0.522050i	$-0.825167\pi$
$307$	−29.8885	−1.70583	−0.852915	+	0.522050i	$0.825167\pi$
$308$	0	0
$309$	25.8885i	1.47275i
$310$	0	0
$311$	− 19.2361i	− 1.09078i	−0.838183	−	0.545389i	$-0.816382\pi$
$311$	− 19.2361i	− 1.09078i	0.838183	−	0.545389i	$-0.183618\pi$
$312$	0	0
$313$	− 27.8885i	− 1.57635i	−0.615449	−	0.788177i	$-0.711025\pi$
$313$	− 27.8885i	− 1.57635i	0.615449	−	0.788177i	$-0.288975\pi$
$314$	0	0
$315$	−48.3607	−2.72482
$316$	0	0
$317$	− 19.8885i	− 1.11705i	−0.829487	−	0.558526i	$-0.811367\pi$
$317$	− 19.8885i	− 1.11705i	0.829487	−	0.558526i	$-0.188633\pi$
$318$	0	0
$319$	6.47214	0.362370
$320$	0	0
$321$	31.4164	1.75349
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−4.47214	−0.248069
$326$	0	0
$327$	−32.3607	−1.78955
$328$	0	0
$329$	16.0000i	0.882109i
$330$	0	0
$331$	10.4721	0.575601	0.287800	−	0.957690i	$-0.407076\pi$
$331$	10.4721	0.575601	0.287800	+	0.957690i	$0.407076\pi$
$332$	0	0
$333$	− 51.8885i	− 2.84347i
$334$	0	0
$335$	− 24.0000i	− 1.31126i
$336$	0	0
$337$	− 23.8885i	− 1.30129i	−0.759381	−	0.650646i	$-0.774498\pi$
$337$	− 23.8885i	− 1.30129i	0.759381	−	0.650646i	$-0.225502\pi$
$338$	0	0
$339$	9.52786	0.517483
$340$	0	0
$341$	−10.4721	−0.567098
$342$	0	0
$343$	− 11.4164i	− 0.616428i
$344$	0	0
$345$	− 20.9443i	− 1.12760i
$346$	0	0
$347$	4.76393i	0.255741i	0.991791	+	0.127871i	$0.0408142\pi$
$347$	4.76393i	0.255741i	−0.991791	+	0.127871i	$0.959186\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	− 64.7214i	− 3.45457i
$352$	0	0
$353$	−23.8885	−1.27146	−0.635729	−	0.771912i	$-0.719301\pi$
$353$	−23.8885	−1.27146	−0.635729	+	0.771912i	$0.719301\pi$
$354$	0	0
$355$	−9.52786	−0.505687
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−22.4721	−1.18603	−0.593017	−	0.805190i	$-0.702063\pi$
$359$	−22.4721	−1.18603	−0.593017	+	0.805190i	$0.702063\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	0	0
$363$	− 1.70820i	− 0.0896575i
$364$	0	0
$365$	5.88854	0.308220
$366$	0	0
$367$	− 19.2361i	− 1.00411i	−0.864834	−	0.502057i	$-0.832577\pi$
$367$	− 19.2361i	− 1.00411i	0.864834	−	0.502057i	$-0.167423\pi$
$368$	0	0
$369$	14.9443i	0.777968i
$370$	0	0
$371$	− 6.47214i	− 0.336017i
$372$	0	0
$373$	3.52786	0.182666	0.0913329	−	0.995820i	$-0.470887\pi$
$373$	3.52786	0.182666	0.0913329	+	0.995820i	$0.470887\pi$
$374$	0	0
$375$	38.8328	2.00532
$376$	0	0
$377$	− 8.94427i	− 0.460653i
$378$	0	0
$379$	− 1.34752i	− 0.0692177i	−0.999401	−	0.0346088i	$-0.988981\pi$
$379$	− 1.34752i	− 0.0692177i	0.999401	−	0.0346088i	$-0.0110185\pi$
$380$	0	0
$381$	− 20.9443i	− 1.07301i
$382$	0	0
$383$	−3.41641	−0.174570	−0.0872851	−	0.996183i	$-0.527819\pi$
$383$	−3.41641	−0.174570	−0.0872851	+	0.996183i	$0.527819\pi$
$384$	0	0
$385$	− 20.9443i	− 1.06742i
$386$	0	0
$387$	−78.2492	−3.97763
$388$	0	0
$389$	25.4164	1.28866	0.644332	−	0.764746i	$-0.277136\pi$
$389$	25.4164	1.28866	0.644332	+	0.764746i	$0.277136\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−68.3607	−3.44834
$394$	0	0
$395$	−3.41641	−0.171898
$396$	0	0
$397$	− 30.9443i	− 1.55305i	−0.630087	−	0.776524i	$-0.716981\pi$
$397$	− 30.9443i	− 1.55305i	0.630087	−	0.776524i	$-0.283019\pi$
$398$	0	0
$399$	25.8885	1.29605
$400$	0	0
$401$	− 13.0557i	− 0.651972i	−0.945375	−	0.325986i	$-0.894304\pi$
$401$	− 13.0557i	− 0.651972i	0.945375	−	0.325986i	$-0.105696\pi$
$402$	0	0
$403$	14.4721i	0.720908i
$404$	0	0
$405$	48.8328i	2.42652i
$406$	0	0
$407$	22.4721	1.11390
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	53.3050i	2.62934i
$412$	0	0
$413$	17.8885i	0.880238i
$414$	0	0
$415$	− 20.9443i	− 1.02811i
$416$	0	0
$417$	−36.3607	−1.78059
$418$	0	0
$419$	20.7639i	1.01438i	0.861833	+	0.507192i	$0.169317\pi$
$419$	20.7639i	1.01438i	−0.861833	+	0.507192i	$0.830683\pi$
$420$	0	0
$421$	−38.3607	−1.86959	−0.934793	−	0.355194i	$-0.884415\pi$
$421$	−38.3607	−1.86959	−0.934793	+	0.355194i	$0.884415\pi$
$422$	0	0
$423$	36.9443	1.79629
$424$	0	0
$425$	0	0
$426$	0	0
$427$	35.4164	1.71392
$428$	0	0
$429$	46.8328	2.26111
$430$	0	0
$431$	11.5967i	0.558596i	0.960205	+	0.279298i	$0.0901017\pi$
$431$	11.5967i	0.558596i	−0.960205	+	0.279298i	$0.909898\pi$
$432$	0	0
$433$	19.5279	0.938449	0.469225	−	0.883079i	$-0.344533\pi$
$433$	19.5279	0.938449	0.469225	+	0.883079i	$0.344533\pi$
$434$	0	0
$435$	12.9443i	0.620630i
$436$	0	0
$437$	8.00000i	0.382692i
$438$	0	0
$439$	30.6525i	1.46296i	0.681861	+	0.731481i	$0.261171\pi$
$439$	30.6525i	1.46296i	−0.681861	+	0.731481i	$0.738829\pi$
$440$	0	0
$441$	25.9443	1.23544
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	− 32.9443i	− 1.56171i
$446$	0	0
$447$	− 19.4164i	− 0.918365i
$448$	0	0
$449$	− 27.8885i	− 1.31614i	−0.752956	−	0.658071i	$-0.771373\pi$
$449$	− 27.8885i	− 1.31614i	0.752956	−	0.658071i	$-0.228627\pi$
$450$	0	0
$451$	−6.47214	−0.304761
$452$	0	0
$453$	46.8328i	2.20040i
$454$	0	0
$455$	−28.9443	−1.35693
$456$	0	0
$457$	−10.5836	−0.495080	−0.247540	−	0.968878i	$-0.579622\pi$
$457$	−10.5836	−0.495080	−0.247540	+	0.968878i	$0.579622\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.88854	0.181108	0.0905538	−	0.995892i	$-0.471136\pi$
$461$	3.88854	0.181108	0.0905538	+	0.995892i	$0.471136\pi$
$462$	0	0
$463$	−22.8328	−1.06113	−0.530565	−	0.847644i	$-0.678020\pi$
$463$	−22.8328	−1.06113	−0.530565	+	0.847644i	$0.678020\pi$
$464$	0	0
$465$	− 20.9443i	− 0.971267i
$466$	0	0
$467$	−41.3050	−1.91137	−0.955683	−	0.294399i	$-0.904881\pi$
$467$	−41.3050	−1.91137	−0.955683	+	0.294399i	$0.904881\pi$
$468$	0	0
$469$	38.8328i	1.79313i
$470$	0	0
$471$	− 77.3050i	− 3.56202i
$472$	0	0
$473$	− 33.8885i	− 1.55820i
$474$	0	0
$475$	−2.47214	−0.113429
$476$	0	0
$477$	−14.9443	−0.684251
$478$	0	0
$479$	33.7082i	1.54017i	0.637943	+	0.770084i	$0.279786\pi$
$479$	33.7082i	1.54017i	−0.637943	+	0.770084i	$0.720214\pi$
$480$	0	0
$481$	− 31.0557i	− 1.41602i
$482$	0	0
$483$	33.8885i	1.54198i
$484$	0	0
$485$	−4.00000	−0.181631
$486$	0	0
$487$	− 43.2361i	− 1.95921i	−0.200925	−	0.979607i	$-0.564395\pi$
$487$	− 43.2361i	− 1.95921i	0.200925	−	0.979607i	$-0.435605\pi$
$488$	0	0
$489$	−0.583592	−0.0263909
$490$	0	0
$491$	−10.4721	−0.472601	−0.236300	−	0.971680i	$-0.575935\pi$
$491$	−10.4721	−0.472601	−0.236300	+	0.971680i	$0.575935\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−48.3607	−2.17365
$496$	0	0
$497$	15.4164	0.691520
$498$	0	0
$499$	15.8197i	0.708185i	0.935210	+	0.354093i	$0.115210\pi$
$499$	15.8197i	0.708185i	−0.935210	+	0.354093i	$0.884790\pi$
$500$	0	0
$501$	−20.3607	−0.909648
$502$	0	0
$503$	− 14.2918i	− 0.637240i	−0.947883	−	0.318620i	$-0.896781\pi$
$503$	− 14.2918i	− 0.637240i	0.947883	−	0.318620i	$-0.103219\pi$
$504$	0	0
$505$	26.8328i	1.19404i
$506$	0	0
$507$	− 22.6525i	− 1.00603i
$508$	0	0
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	0	0
$511$	−9.52786	−0.421488
$512$	0	0
$513$	− 35.7771i	− 1.57960i
$514$	0	0
$515$	− 16.0000i	− 0.705044i
$516$	0	0
$517$	16.0000i	0.703679i
$518$	0	0
$519$	6.47214	0.284095
$520$	0	0
$521$	− 11.8885i	− 0.520847i	−0.965495	−	0.260423i	$-0.916138\pi$
$521$	− 11.8885i	− 0.520847i	0.965495	−	0.260423i	$-0.0838621\pi$
$522$	0	0
$523$	−7.05573	−0.308525	−0.154263	−	0.988030i	$-0.549300\pi$
$523$	−7.05573	−0.308525	−0.154263	+	0.988030i	$0.549300\pi$
$524$	0	0
$525$	−10.4721	−0.457041
$526$	0	0
$527$	0	0
$528$	0	0
$529$	12.5279	0.544690
$530$	0	0
$531$	41.3050	1.79248
$532$	0	0
$533$	8.94427i	0.387419i
$534$	0	0
$535$	−19.4164	−0.839445
$536$	0	0
$537$	17.8885i	0.771948i
$538$	0	0
$539$	11.2361i	0.483972i
$540$	0	0
$541$	− 30.0000i	− 1.28980i	−0.764267	−	0.644900i	$-0.776899\pi$
$541$	− 30.0000i	− 1.28980i	0.764267	−	0.644900i	$-0.223101\pi$
$542$	0	0
$543$	−32.3607	−1.38873
$544$	0	0
$545$	20.0000	0.856706
$546$	0	0
$547$	1.70820i	0.0730375i	0.999333	+	0.0365188i	$0.0116269\pi$
$547$	1.70820i	0.0730375i	−0.999333	+	0.0365188i	$0.988373\pi$
$548$	0	0
$549$	− 81.7771i	− 3.49016i
$550$	0	0
$551$	− 4.94427i	− 0.210633i
$552$	0	0
$553$	5.52786	0.235069
$554$	0	0
$555$	44.9443i	1.90778i
$556$	0	0
$557$	21.4164	0.907442	0.453721	−	0.891144i	$-0.350096\pi$
$557$	21.4164	0.907442	0.453721	+	0.891144i	$0.350096\pi$
$558$	0	0
$559$	−46.8328	−1.98082
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−20.3607	−0.858100	−0.429050	−	0.903281i	$-0.641152\pi$
$563$	−20.3607	−0.858100	−0.429050	+	0.903281i	$0.641152\pi$
$564$	0	0
$565$	−5.88854	−0.247733
$566$	0	0
$567$	− 79.0132i	− 3.31824i
$568$	0	0
$569$	−35.8885	−1.50453	−0.752263	−	0.658863i	$-0.771038\pi$
$569$	−35.8885	−1.50453	−0.752263	+	0.658863i	$0.771038\pi$
$570$	0	0
$571$	− 3.23607i	− 0.135425i	−0.997705	−	0.0677126i	$-0.978430\pi$
$571$	− 3.23607i	− 0.135425i	0.997705	−	0.0677126i	$-0.0215701\pi$
$572$	0	0
$573$	− 16.0000i	− 0.668410i
$574$	0	0
$575$	− 3.23607i	− 0.134953i
$576$	0	0
$577$	−8.47214	−0.352700	−0.176350	−	0.984328i	$-0.556429\pi$
$577$	−8.47214	−0.352700	−0.176350	+	0.984328i	$0.556429\pi$
$578$	0	0
$579$	9.52786	0.395965
$580$	0	0
$581$	33.8885i	1.40593i
$582$	0	0
$583$	− 6.47214i	− 0.268048i
$584$	0	0
$585$	66.8328i	2.76320i
$586$	0	0
$587$	12.3607	0.510180	0.255090	−	0.966917i	$-0.417895\pi$
$587$	12.3607	0.510180	0.255090	+	0.966917i	$0.417895\pi$
$588$	0	0
$589$	8.00000i	0.329634i
$590$	0	0
$591$	35.4164	1.45684
$592$	0	0
$593$	−11.8885	−0.488204	−0.244102	−	0.969750i	$-0.578493\pi$
$593$	−11.8885	−0.488204	−0.244102	+	0.969750i	$0.578493\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−0.583592	−0.0238848
$598$	0	0
$599$	46.8328	1.91354	0.956768	−	0.290851i	$-0.0939383\pi$
$599$	46.8328	1.91354	0.956768	+	0.290851i	$0.0939383\pi$
$600$	0	0
$601$	− 38.9443i	− 1.58857i	−0.607545	−	0.794285i	$-0.707846\pi$
$601$	− 38.9443i	− 1.58857i	0.607545	−	0.794285i	$-0.292154\pi$
$602$	0	0
$603$	89.6656	3.65147
$604$	0	0
$605$	1.05573i	0.0429215i
$606$	0	0
$607$	− 6.29180i	− 0.255376i	−0.991814	−	0.127688i	$-0.959244\pi$
$607$	− 6.29180i	− 0.255376i	0.991814	−	0.127688i	$-0.0407556\pi$
$608$	0	0
$609$	− 20.9443i	− 0.848705i
$610$	0	0
$611$	22.1115	0.894534
$612$	0	0
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	0	0
$615$	− 12.9443i	− 0.521963i
$616$	0	0
$617$	34.0000i	1.36879i	0.729112	+	0.684394i	$0.239933\pi$
$617$	34.0000i	1.36879i	−0.729112	+	0.684394i	$0.760067\pi$
$618$	0	0
$619$	− 1.34752i	− 0.0541616i	−0.999633	−	0.0270808i	$-0.991379\pi$
$619$	− 1.34752i	− 0.0541616i	0.999633	−	0.0270808i	$-0.00862114\pi$
$620$	0	0
$621$	46.8328	1.87934
$622$	0	0
$623$	53.3050i	2.13562i
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	25.8885	1.03389
$628$	0	0
$629$	0	0
$630$	0	0
$631$	29.3050	1.16661	0.583306	−	0.812253i	$-0.301759\pi$
$631$	29.3050	1.16661	0.583306	+	0.812253i	$0.301759\pi$
$632$	0	0
$633$	−10.4721	−0.416230
$634$	0	0
$635$	12.9443i	0.513678i
$636$	0	0
$637$	15.5279	0.615236
$638$	0	0
$639$	− 35.5967i	− 1.40819i
$640$	0	0
$641$	− 7.88854i	− 0.311579i	−0.987790	−	0.155789i	$-0.950208\pi$
$641$	− 7.88854i	− 0.311579i	0.987790	−	0.155789i	$-0.0497921\pi$
$642$	0	0
$643$	48.5410i	1.91427i	0.289641	+	0.957135i	$0.406464\pi$
$643$	48.5410i	1.91427i	−0.289641	+	0.957135i	$0.593536\pi$
$644$	0	0
$645$	67.7771	2.66872
$646$	0	0
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	0	0
$649$	17.8885i	0.702187i
$650$	0	0
$651$	33.8885i	1.32820i
$652$	0	0
$653$	20.8328i	0.815251i	0.913149	+	0.407626i	$0.133643\pi$
$653$	20.8328i	0.815251i	−0.913149	+	0.407626i	$0.866357\pi$
$654$	0	0
$655$	42.2492	1.65081
$656$	0	0
$657$	22.0000i	0.858302i
$658$	0	0
$659$	18.8328	0.733622	0.366811	−	0.930295i	$-0.380450\pi$
$659$	18.8328	0.733622	0.366811	+	0.930295i	$0.380450\pi$
$660$	0	0
$661$	−39.8885	−1.55148	−0.775742	−	0.631050i	$-0.782624\pi$
$661$	−39.8885	−1.55148	−0.775742	+	0.631050i	$0.782624\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−16.0000	−0.620453
$666$	0	0
$667$	6.47214	0.250602
$668$	0	0
$669$	46.8328i	1.81066i
$670$	0	0
$671$	35.4164	1.36724
$672$	0	0
$673$	− 37.7771i	− 1.45620i	−0.685471	−	0.728100i	$-0.740404\pi$
$673$	− 37.7771i	− 1.45620i	0.685471	−	0.728100i	$-0.259596\pi$
$674$	0	0
$675$	14.4721i	0.557033i
$676$	0	0
$677$	43.8885i	1.68677i	0.537307	+	0.843387i	$0.319442\pi$
$677$	43.8885i	1.68677i	−0.537307	+	0.843387i	$0.680558\pi$
$678$	0	0
$679$	6.47214	0.248378
$680$	0	0
$681$	−10.4721	−0.401293
$682$	0	0
$683$	− 50.0689i	− 1.91583i	−0.287048	−	0.957916i	$-0.592674\pi$
$683$	− 50.0689i	− 1.91583i	0.287048	−	0.957916i	$-0.407326\pi$
$684$	0	0
$685$	− 32.9443i	− 1.25874i
$686$	0	0
$687$	− 50.2492i	− 1.91713i
$688$	0	0
$689$	−8.94427	−0.340750
$690$	0	0
$691$	11.5967i	0.441161i	0.975369	+	0.220581i	$0.0707952\pi$
$691$	11.5967i	0.441161i	−0.975369	+	0.220581i	$0.929205\pi$
$692$	0	0
$693$	78.2492	2.97244
$694$	0	0
$695$	22.4721	0.852417
$696$	0	0
$697$	0	0
$698$	0	0
$699$	38.4721	1.45515
$700$	0	0
$701$	−30.3607	−1.14671	−0.573354	−	0.819308i	$-0.694358\pi$
$701$	−30.3607	−1.14671	−0.573354	+	0.819308i	$0.694358\pi$
$702$	0	0
$703$	− 17.1672i	− 0.647473i
$704$	0	0
$705$	−32.0000	−1.20519
$706$	0	0
$707$	− 43.4164i	− 1.63284i
$708$	0	0
$709$	− 2.94427i	− 0.110574i	−0.998470	−	0.0552872i	$-0.982393\pi$
$709$	− 2.94427i	− 0.110574i	0.998470	−	0.0552872i	$-0.0176074\pi$
$710$	0	0
$711$	− 12.7639i	− 0.478685i
$712$	0	0
$713$	−10.4721	−0.392185
$714$	0	0
$715$	−28.9443	−1.08245
$716$	0	0
$717$	57.8885i	2.16189i
$718$	0	0
$719$	22.6525i	0.844795i	0.906411	+	0.422397i	$0.138811\pi$
$719$	22.6525i	0.844795i	−0.906411	+	0.422397i	$0.861189\pi$
$720$	0	0
$721$	25.8885i	0.964140i
$722$	0	0
$723$	−61.3050	−2.27996
$724$	0	0
$725$	2.00000i	0.0742781i
$726$	0	0
$727$	3.05573	0.113331	0.0566653	−	0.998393i	$-0.481953\pi$
$727$	3.05573	0.113331	0.0566653	+	0.998393i	$0.481953\pi$
$728$	0	0
$729$	−41.9443	−1.55349
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0.111456	0.00411673	0.00205836	−	0.999998i	$-0.499345\pi$
$733$	0.111456	0.00411673	0.00205836	+	0.999998i	$0.499345\pi$
$734$	0	0
$735$	−22.4721	−0.828897
$736$	0	0
$737$	38.8328i	1.43043i
$738$	0	0
$739$	16.5836	0.610037	0.305019	−	0.952346i	$-0.401337\pi$
$739$	16.5836	0.610037	0.305019	+	0.952346i	$0.401337\pi$
$740$	0	0
$741$	− 35.7771i	− 1.31430i
$742$	0	0
$743$	− 17.3475i	− 0.636419i	−0.948020	−	0.318209i	$-0.896918\pi$
$743$	− 17.3475i	− 0.636419i	0.948020	−	0.318209i	$-0.103082\pi$
$744$	0	0
$745$	12.0000i	0.439646i
$746$	0	0
$747$	78.2492	2.86299
$748$	0	0
$749$	31.4164	1.14793
$750$	0	0
$751$	− 22.2918i	− 0.813439i	−0.913553	−	0.406720i	$-0.866673\pi$
$751$	− 22.2918i	− 0.813439i	0.913553	−	0.406720i	$-0.133327\pi$
$752$	0	0
$753$	− 28.9443i	− 1.05479i
$754$	0	0
$755$	− 28.9443i	− 1.05339i
$756$	0	0
$757$	−16.4721	−0.598690	−0.299345	−	0.954145i	$-0.596768\pi$
$757$	−16.4721	−0.598690	−0.299345	+	0.954145i	$0.596768\pi$
$758$	0	0
$759$	33.8885i	1.23008i
$760$	0	0
$761$	−10.3607	−0.375574	−0.187787	−	0.982210i	$-0.560132\pi$
$761$	−10.3607	−0.375574	−0.187787	+	0.982210i	$0.560132\pi$
$762$	0	0
$763$	−32.3607	−1.17154
$764$	0	0
$765$	0	0
$766$	0	0
$767$	24.7214	0.892637
$768$	0	0
$769$	7.52786	0.271462	0.135731	−	0.990746i	$-0.456662\pi$
$769$	7.52786	0.271462	0.135731	+	0.990746i	$0.456662\pi$
$770$	0	0
$771$	72.3607i	2.60601i
$772$	0	0
$773$	35.3050	1.26983	0.634915	−	0.772582i	$-0.281035\pi$
$773$	35.3050	1.26983	0.634915	+	0.772582i	$0.281035\pi$
$774$	0	0
$775$	− 3.23607i	− 0.116243i
$776$	0	0
$777$	− 72.7214i	− 2.60886i
$778$	0	0
$779$	4.94427i	0.177147i
$780$	0	0
$781$	15.4164	0.551642
$782$	0	0
$783$	−28.9443	−1.03438
$784$	0	0
$785$	47.7771i	1.70524i
$786$	0	0
$787$	6.65248i	0.237135i	0.992946	+	0.118568i	$0.0378302\pi$
$787$	6.65248i	0.237135i	−0.992946	+	0.118568i	$0.962170\pi$
$788$	0	0
$789$	36.9443i	1.31525i
$790$	0	0
$791$	9.52786	0.338772
$792$	0	0
$793$	− 48.9443i	− 1.73806i
$794$	0	0
$795$	12.9443	0.459086
$796$	0	0
$797$	19.8885	0.704488	0.352244	−	0.935908i	$-0.385419\pi$
$797$	19.8885	0.704488	0.352244	+	0.935908i	$0.385419\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	123.082	4.34889
$802$	0	0
$803$	−9.52786	−0.336231
$804$	0	0
$805$	− 20.9443i	− 0.738189i
$806$	0	0
$807$	−61.3050	−2.15804
$808$	0	0
$809$	25.0557i	0.880912i	0.897774	+	0.440456i	$0.145183\pi$
$809$	25.0557i	0.880912i	−0.897774	+	0.440456i	$0.854817\pi$
$810$	0	0
$811$	− 13.1246i	− 0.460867i	−0.973088	−	0.230434i	$-0.925986\pi$
$811$	− 13.1246i	− 0.460867i	0.973088	−	0.230434i	$-0.0740145\pi$
$812$	0	0
$813$	− 57.8885i	− 2.03024i
$814$	0	0
$815$	0.360680	0.0126341
$816$	0	0
$817$	−25.8885	−0.905725
$818$	0	0
$819$	− 108.138i	− 3.77864i
$820$	0	0
$821$	50.0000i	1.74501i	0.488603	+	0.872506i	$0.337507\pi$
$821$	50.0000i	1.74501i	−0.488603	+	0.872506i	$0.662493\pi$
$822$	0	0
$823$	51.5967i	1.79855i	0.437384	+	0.899275i	$0.355905\pi$
$823$	51.5967i	1.79855i	−0.437384	+	0.899275i	$0.644095\pi$
$824$	0	0
$825$	−10.4721	−0.364593
$826$	0	0
$827$	20.7639i	0.722033i	0.932559	+	0.361016i	$0.117570\pi$
$827$	20.7639i	0.722033i	−0.932559	+	0.361016i	$0.882430\pi$
$828$	0	0
$829$	−8.11146	−0.281723	−0.140861	−	0.990029i	$-0.544987\pi$
$829$	−8.11146	−0.281723	−0.140861	+	0.990029i	$0.544987\pi$
$830$	0	0
$831$	106.249	3.68574
$832$	0	0
$833$	0	0
$834$	0	0
$835$	12.5836	0.435473
$836$	0	0
$837$	46.8328	1.61878
$838$	0	0
$839$	− 26.0689i	− 0.899998i	−0.893029	−	0.449999i	$-0.851424\pi$
$839$	− 26.0689i	− 0.899998i	0.893029	−	0.449999i	$-0.148576\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	− 51.4164i	− 1.77088i
$844$	0	0
$845$	14.0000i	0.481615i
$846$	0	0
$847$	− 1.70820i	− 0.0586946i
$848$	0	0
$849$	−10.4721	−0.359403
$850$	0	0
$851$	22.4721	0.770335
$852$	0	0
$853$	− 33.7771i	− 1.15651i	−0.815858	−	0.578253i	$-0.803735\pi$
$853$	− 33.7771i	− 1.15651i	0.815858	−	0.578253i	$-0.196265\pi$
$854$	0	0
$855$	36.9443i	1.26347i
$856$	0	0
$857$	− 27.8885i	− 0.952655i	−0.879268	−	0.476327i	$-0.841968\pi$
$857$	− 27.8885i	− 0.952655i	0.879268	−	0.476327i	$-0.158032\pi$
$858$	0	0
$859$	30.2492	1.03209	0.516045	−	0.856561i	$-0.327404\pi$
$859$	30.2492	1.03209	0.516045	+	0.856561i	$0.327404\pi$
$860$	0	0
$861$	20.9443i	0.713779i
$862$	0	0
$863$	22.8328	0.777238	0.388619	−	0.921399i	$-0.372952\pi$
$863$	22.8328	0.777238	0.388619	+	0.921399i	$0.372952\pi$
$864$	0	0
$865$	−4.00000	−0.136004
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.52786	0.187520
$870$	0	0
$871$	53.6656	1.81839
$872$	0	0
$873$	− 14.9443i	− 0.505787i
$874$	0	0
$875$	38.8328	1.31279
$876$	0	0
$877$	31.8885i	1.07680i	0.842690	+	0.538400i	$0.180971\pi$
$877$	31.8885i	1.07680i	−0.842690	+	0.538400i	$0.819029\pi$
$878$	0	0
$879$	32.3607i	1.09150i
$880$	0	0
$881$	27.8885i	0.939589i	0.882776	+	0.469794i	$0.155672\pi$
$881$	27.8885i	0.939589i	−0.882776	+	0.469794i	$0.844328\pi$
$882$	0	0
$883$	−37.8885	−1.27505	−0.637526	−	0.770429i	$-0.720042\pi$
$883$	−37.8885	−1.27505	−0.637526	+	0.770429i	$0.720042\pi$
$884$	0	0
$885$	−35.7771	−1.20263
$886$	0	0
$887$	− 29.1246i	− 0.977909i	−0.872309	−	0.488954i	$-0.837378\pi$
$887$	− 29.1246i	− 0.977909i	0.872309	−	0.488954i	$-0.162622\pi$
$888$	0	0
$889$	− 20.9443i	− 0.702448i
$890$	0	0
$891$	− 79.0132i	− 2.64704i
$892$	0	0
$893$	12.2229	0.409024
$894$	0	0
$895$	− 11.0557i	− 0.369552i
$896$	0	0
$897$	46.8328	1.56370
$898$	0	0
$899$	6.47214	0.215858
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−109.666	−3.64944
$904$	0	0
$905$	20.0000	0.664822
$906$	0	0
$907$	18.8754i	0.626747i	0.949630	+	0.313373i	$0.101459\pi$
$907$	18.8754i	0.626747i	−0.949630	+	0.313373i	$0.898541\pi$
$908$	0	0
$909$	−100.249	−3.32506
$910$	0	0
$911$	− 14.2918i	− 0.473508i	−0.971570	−	0.236754i	$-0.923916\pi$
$911$	− 14.2918i	− 0.473508i	0.971570	−	0.236754i	$-0.0760836\pi$
$912$	0	0
$913$	33.8885i	1.12155i
$914$	0	0
$915$	70.8328i	2.34166i
$916$	0	0
$917$	−68.3607	−2.25747
$918$	0	0
$919$	−6.11146	−0.201598	−0.100799	−	0.994907i	$-0.532140\pi$
$919$	−6.11146	−0.201598	−0.100799	+	0.994907i	$0.532140\pi$
$920$	0	0
$921$	96.7214i	3.18708i
$922$	0	0
$923$	− 21.3050i	− 0.701261i
$924$	0	0
$925$	6.94427i	0.228326i
$926$	0	0
$927$	59.7771	1.96334
$928$	0	0
$929$	9.05573i	0.297109i	0.988904	+	0.148554i	$0.0474620\pi$
$929$	9.05573i	0.297109i	−0.988904	+	0.148554i	$0.952538\pi$
$930$	0	0
$931$	8.58359	0.281316
$932$	0	0
$933$	−62.2492	−2.03795
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	0	0
$939$	−90.2492	−2.94517
$940$	0	0
$941$	− 45.7771i	− 1.49229i	−0.665783	−	0.746145i	$-0.731902\pi$
$941$	− 45.7771i	− 1.49229i	0.665783	−	0.746145i	$-0.268098\pi$
$942$	0	0
$943$	−6.47214	−0.210762
$944$	0	0
$945$	93.6656i	3.04694i
$946$	0	0
$947$	− 39.0132i	− 1.26776i	−0.773433	−	0.633879i	$-0.781462\pi$
$947$	− 39.0132i	− 1.26776i	0.773433	−	0.633879i	$-0.218538\pi$
$948$	0	0
$949$	13.1672i	0.427425i
$950$	0	0
$951$	−64.3607	−2.08704
$952$	0	0
$953$	−32.4721	−1.05188	−0.525938	−	0.850523i	$-0.676286\pi$
$953$	−32.4721	−1.05188	−0.525938	+	0.850523i	$0.676286\pi$
$954$	0	0
$955$	9.88854i	0.319986i
$956$	0	0
$957$	− 20.9443i	− 0.677032i
$958$	0	0
$959$	53.3050i	1.72131i
$960$	0	0
$961$	20.5279	0.662189
$962$	0	0
$963$	− 72.5410i	− 2.33760i
$964$	0	0
$965$	−5.88854	−0.189559
$966$	0	0
$967$	−6.47214	−0.208130	−0.104065	−	0.994571i	$-0.533185\pi$
$967$	−6.47214	−0.208130	−0.104065	+	0.994571i	$0.533185\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−18.4721	−0.592799	−0.296400	−	0.955064i	$-0.595786\pi$
$971$	−18.4721	−0.592799	−0.296400	+	0.955064i	$0.595786\pi$
$972$	0	0
$973$	−36.3607	−1.16567
$974$	0	0
$975$	14.4721i	0.463479i
$976$	0	0
$977$	−34.0000	−1.08776	−0.543878	−	0.839164i	$-0.683045\pi$
$977$	−34.0000	−1.08776	−0.543878	+	0.839164i	$0.683045\pi$
$978$	0	0
$979$	53.3050i	1.70363i
$980$	0	0
$981$	74.7214i	2.38567i
$982$	0	0
$983$	− 11.2361i	− 0.358375i	−0.983815	−	0.179187i	$-0.942653\pi$
$983$	− 11.2361i	− 0.358375i	0.983815	−	0.179187i	$-0.0573469\pi$
$984$	0	0
$985$	−21.8885	−0.697427
$986$	0	0
$987$	51.7771	1.64808
$988$	0	0
$989$	− 33.8885i	− 1.07759i
$990$	0	0
$991$	12.7639i	0.405460i	0.979235	+	0.202730i	$0.0649813\pi$
$991$	12.7639i	0.405460i	−0.979235	+	0.202730i	$0.935019\pi$
$992$	0	0
$993$	− 33.8885i	− 1.07542i
$994$	0	0
$995$	0.360680	0.0114343
$996$	0	0
$997$	− 46.9443i	− 1.48674i	−0.668880	−	0.743370i	$-0.733226\pi$
$997$	− 46.9443i	− 1.48674i	0.668880	−	0.743370i	$-0.266774\pi$
$998$	0	0
$999$	−100.498	−3.17963

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.2.b.g.577.1		4
17.4	even	4		136.2.a.c.1.1	✓	2
17.13	even	4		2312.2.a.m.1.2		2
17.16	even	2	inner	2312.2.b.g.577.4		4
51.38	odd	4		1224.2.a.i.1.2		2
68.47	odd	4		4624.2.a.h.1.1		2
68.55	odd	4		272.2.a.f.1.2		2
85.4	even	4		3400.2.a.i.1.2		2
85.38	odd	4		3400.2.e.f.2449.1		4
85.72	odd	4		3400.2.e.f.2449.4		4
119.55	odd	4		6664.2.a.i.1.2		2
136.21	even	4		1088.2.a.s.1.2		2
136.123	odd	4		1088.2.a.o.1.1		2
204.191	even	4		2448.2.a.u.1.1		2
340.259	odd	4		6800.2.a.bd.1.1		2
408.293	odd	4		9792.2.a.db.1.2		2
408.395	even	4		9792.2.a.da.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.a.c.1.1	✓	2	17.4	even	4
272.2.a.f.1.2		2	68.55	odd	4
1088.2.a.o.1.1		2	136.123	odd	4
1088.2.a.s.1.2		2	136.21	even	4
1224.2.a.i.1.2		2	51.38	odd	4
2312.2.a.m.1.2		2	17.13	even	4
2312.2.b.g.577.1		4	1.1	even	1	trivial
2312.2.b.g.577.4		4	17.16	even	2	inner
2448.2.a.u.1.1		2	204.191	even	4
3400.2.a.i.1.2		2	85.4	even	4
3400.2.e.f.2449.1		4	85.38	odd	4
3400.2.e.f.2449.4		4	85.72	odd	4
4624.2.a.h.1.1		2	68.47	odd	4
6664.2.a.i.1.2		2	119.55	odd	4
6800.2.a.bd.1.1		2	340.259	odd	4
9792.2.a.da.1.1		2	408.395	even	4
9792.2.a.db.1.2		2	408.293	odd	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:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2312.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.23607i q^{3} +2.00000i q^{5} -3.23607i q^{7} -7.47214 q^{9} -3.23607i q^{11} -4.47214 q^{13} +6.47214 q^{15} -2.47214 q^{19} -10.4721 q^{21} -3.23607i q^{23} +1.00000 q^{25} +14.4721i q^{27} +2.00000i q^{29} -3.23607i q^{31} -10.4721 q^{33} +6.47214 q^{35} +6.94427i q^{37} +14.4721i q^{39} -2.00000i q^{41} +10.4721 q^{43} -14.9443i q^{45} -4.94427 q^{47} -3.47214 q^{49} +2.00000 q^{53} +6.47214 q^{55} +8.00000i q^{57} -5.52786 q^{59} +10.9443i q^{61} +24.1803i q^{63} -8.94427i q^{65} -12.0000 q^{67} -10.4721 q^{69} +4.76393i q^{71} -2.94427i q^{73} -3.23607i q^{75} -10.4721 q^{77} +1.70820i q^{79} +24.4164 q^{81} -10.4721 q^{83} +6.47214 q^{87} -16.4721 q^{89} +14.4721i q^{91} -10.4721 q^{93} -4.94427i q^{95} +2.00000i q^{97} +24.1803i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9} + 8 q^{15} + 8 q^{19} - 24 q^{21} + 4 q^{25} - 24 q^{33} + 8 q^{35} + 24 q^{43} + 16 q^{47} + 4 q^{49} + 8 q^{53} + 8 q^{55} - 40 q^{59} - 48 q^{67} - 24 q^{69} - 24 q^{77} + 44 q^{81} - 24 q^{83}+ \cdots - 24 q^{93}+O(q^{100}) \) 4 * q - 12 * q^9 + 8 * q^15 + 8 * q^19 - 24 * q^21 + 4 * q^25 - 24 * q^33 + 8 * q^35 + 24 * q^43 + 16 * q^47 + 4 * q^49 + 8 * q^53 + 8 * q^55 - 40 * q^59 - 48 * q^67 - 24 * q^69 - 24 * q^77 + 44 * q^81 - 24 * q^83 + 8 * q^87 - 48 * q^89 - 24 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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