LMFDB - Embedded newform 2912.2.h.b.2575.41

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2912,2,Mod(2575,2912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2912, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 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("2912.2575");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2912 = 2^{5} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2912.h (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:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 728)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2575.41
Character	$\chi$	$=$	2912.2575
Dual form			2912.2.h.b.2575.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.38356i q^{3} +1.68057 q^{5} +(1.10198 - 2.40533i) q^{7} -2.68138 q^{9} -6.45771 q^{11} +1.00000 q^{13} +4.00574i q^{15} -0.816850i q^{17} +5.75603i q^{19} +(5.73327 + 2.62665i) q^{21} +0.878221i q^{23} -2.17570 q^{25} +0.759458i q^{27} -0.973834i q^{29} -3.89496 q^{31} -15.3924i q^{33} +(1.85195 - 4.04232i) q^{35} +2.19900i q^{37} +2.38356i q^{39} +4.21071i q^{41} -3.91987 q^{43} -4.50623 q^{45} -9.01192 q^{47} +(-4.57127 - 5.30128i) q^{49} +1.94702 q^{51} +9.57202i q^{53} -10.8526 q^{55} -13.7199 q^{57} +0.722834i q^{59} -1.45484 q^{61} +(-2.95483 + 6.44961i) q^{63} +1.68057 q^{65} +5.83289 q^{67} -2.09330 q^{69} +5.00020i q^{71} +8.53527i q^{73} -5.18592i q^{75} +(-7.11628 + 15.5329i) q^{77} +14.3010i q^{79} -9.85435 q^{81} +10.7679i q^{83} -1.37277i q^{85} +2.32119 q^{87} -17.9213i q^{89} +(1.10198 - 2.40533i) q^{91} -9.28389i q^{93} +9.67338i q^{95} +13.1424i q^{97} +17.3155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{9} + 4 q^{11} + 48 q^{13} + 48 q^{25} - 12 q^{35} + 4 q^{43} - 24 q^{45} + 40 q^{51} + 20 q^{63} + 4 q^{67} + 20 q^{77} + 64 q^{81} - 40 q^{87} - 20 q^{99}+O(q^{100}) $ 48 * q - 48 * q^9 + 4 * q^11 + 48 * q^13 + 48 * q^25 - 12 * q^35 + 4 * q^43 - 24 * q^45 + 40 * q^51 + 20 * q^63 + 4 * q^67 + 20 * q^77 + 64 * q^81 - 40 * q^87 - 20 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2912\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$2017$	$2367$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$			2.38356i			1.37615i	0.725639	+	0.688076i	$0.241544\pi$
$3$			2.38356i			1.37615i	−0.725639	+	0.688076i	$0.758456\pi$
$4$		0			0
$5$	1.68057			0.751572			0.375786	−	0.926707i	$-0.377373\pi$
$5$	1.68057			0.751572			0.375786	+	0.926707i	$0.377373\pi$
$6$		0			0
$7$	1.10198	−	2.40533i	0.416510	−	0.909131i
$7$	1.10198	−	2.40533i	0.416510	−	0.909131i
$8$		0			0
$9$	−2.68138			−0.893792
$10$		0			0
$11$	−6.45771			−1.94707			−0.973536	−	0.228535i	$-0.926606\pi$
$11$	−6.45771			−1.94707			−0.973536	+	0.228535i	$0.926606\pi$
$12$		0			0
$13$	1.00000			0.277350
$13$	1.00000			0.277350
$14$		0			0
$15$			4.00574i			1.03428i
$16$		0			0
$17$		−	0.816850i		−	0.198115i	−0.995082	−	0.0990577i	$-0.968417\pi$
$17$		−	0.816850i		−	0.198115i	0.995082	−	0.0990577i	$-0.0315828\pi$
$18$		0			0
$19$			5.75603i			1.32052i	0.751036	+	0.660262i	$0.229555\pi$
$19$			5.75603i			1.32052i	−0.751036	+	0.660262i	$0.770445\pi$
$20$		0			0
$21$	5.73327	+	2.62665i	1.25110	+	0.573181i
$22$		0			0
$23$			0.878221i			0.183122i	0.995799	+	0.0915609i	$0.0291856\pi$
$23$			0.878221i			0.183122i	−0.995799	+	0.0915609i	$0.970814\pi$
$24$		0			0
$25$	−2.17570			−0.435140
$26$		0			0
$27$			0.759458i			0.146158i
$28$		0			0
$29$		−	0.973834i		−	0.180836i	−0.995904	−	0.0904182i	$-0.971180\pi$
$29$		−	0.973834i		−	0.180836i	0.995904	−	0.0904182i	$-0.0288204\pi$
$30$		0			0
$31$	−3.89496			−0.699555			−0.349778	−	0.936833i	$-0.613743\pi$
$31$	−3.89496			−0.699555			−0.349778	+	0.936833i	$0.613743\pi$
$32$		0			0
$33$		−	15.3924i		−	2.67946i
$34$		0			0
$35$	1.85195	−	4.04232i	0.313037	−	0.683277i
$36$		0			0
$37$			2.19900i			0.361514i	0.983528	+	0.180757i	$0.0578547\pi$
$37$			2.19900i			0.361514i	−0.983528	+	0.180757i	$0.942145\pi$
$38$		0			0
$39$			2.38356i			0.381676i
$40$		0			0
$41$			4.21071i			0.657602i	0.944399	+	0.328801i	$0.106645\pi$
$41$			4.21071i			0.657602i	−0.944399	+	0.328801i	$0.893355\pi$
$42$		0			0
$43$	−3.91987			−0.597775			−0.298887	−	0.954288i	$-0.596616\pi$
$43$	−3.91987			−0.597775			−0.298887	+	0.954288i	$0.596616\pi$
$44$		0			0
$45$	−4.50623			−0.671749
$46$		0			0
$47$	−9.01192			−1.31452			−0.657262	−	0.753662i	$-0.728285\pi$
$47$	−9.01192			−1.31452			−0.657262	+	0.753662i	$0.728285\pi$
$48$		0			0
$49$	−4.57127	−	5.30128i	−0.653038	−	0.757325i
$50$		0			0
$51$	1.94702			0.272637
$52$		0			0
$53$			9.57202i			1.31482i	0.753534	+	0.657409i	$0.228348\pi$
$53$			9.57202i			1.31482i	−0.753534	+	0.657409i	$0.771652\pi$
$54$		0			0
$55$	−10.8526			−1.46336
$56$		0			0
$57$	−13.7199			−1.81724
$58$		0			0
$59$			0.722834i			0.0941050i	0.998892	+	0.0470525i	$0.0149828\pi$
$59$			0.722834i			0.0941050i	−0.998892	+	0.0470525i	$0.985017\pi$
$60$		0			0
$61$	−1.45484			−0.186273			−0.0931365	−	0.995653i	$-0.529689\pi$
$61$	−1.45484			−0.186273			−0.0931365	+	0.995653i	$0.529689\pi$
$62$		0			0
$63$	−2.95483	+	6.44961i	−0.372274	+	0.812574i
$64$		0			0
$65$	1.68057			0.208449
$66$		0			0
$67$	5.83289			0.712600			0.356300	−	0.934372i	$-0.384038\pi$
$67$	5.83289			0.712600			0.356300	+	0.934372i	$0.384038\pi$
$68$		0			0
$69$	−2.09330			−0.252003
$70$		0			0
$71$			5.00020i			0.593414i	0.954969	+	0.296707i	$0.0958885\pi$
$71$			5.00020i			0.593414i	−0.954969	+	0.296707i	$0.904111\pi$
$72$		0			0
$73$			8.53527i			0.998978i	0.866320	+	0.499489i	$0.166479\pi$
$73$			8.53527i			0.998978i	−0.866320	+	0.499489i	$0.833521\pi$
$74$		0			0
$75$		−	5.18592i		−	0.598818i
$76$		0			0
$77$	−7.11628	+	15.5329i	−0.810976	+	1.77014i
$78$		0			0
$79$			14.3010i			1.60899i	0.593962	+	0.804493i	$0.297563\pi$
$79$			14.3010i			1.60899i	−0.593962	+	0.804493i	$0.702437\pi$
$80$		0			0
$81$	−9.85435			−1.09493
$82$		0			0
$83$			10.7679i			1.18193i	0.806696	+	0.590967i	$0.201254\pi$
$83$			10.7679i			1.18193i	−0.806696	+	0.590967i	$0.798746\pi$
$84$		0			0
$85$		−	1.37277i		−	0.148898i
$86$		0			0
$87$	2.32119			0.248858
$88$		0			0
$89$		−	17.9213i		−	1.89965i	−0.312782	−	0.949825i	$-0.601261\pi$
$89$		−	17.9213i		−	1.89965i	0.312782	−	0.949825i	$-0.398739\pi$
$90$		0			0
$91$	1.10198	−	2.40533i	0.115519	−	0.252148i
$92$		0			0
$93$		−	9.28389i		−	0.962694i
$94$		0			0
$95$			9.67338i			0.992468i
$96$		0			0
$97$			13.1424i			1.33441i	0.744874	+	0.667206i	$0.232510\pi$
$97$			13.1424i			1.33441i	−0.744874	+	0.667206i	$0.767490\pi$
$98$		0			0
$99$	17.3155			1.74028
$100$		0			0
$101$	−16.8120			−1.67286			−0.836429	−	0.548075i	$-0.815361\pi$
$101$	−16.8120			−1.67286			−0.836429	+	0.548075i	$0.815361\pi$
$102$		0			0
$103$	7.83407			0.771914			0.385957	−	0.922517i	$-0.373871\pi$
$103$	7.83407			0.771914			0.385957	+	0.922517i	$0.373871\pi$
$104$		0			0
$105$	9.63513	+	4.41425i	0.940293	+	0.430787i
$106$		0			0
$107$	−9.20742			−0.890115			−0.445057	−	0.895502i	$-0.646817\pi$
$107$	−9.20742			−0.890115			−0.445057	+	0.895502i	$0.646817\pi$
$108$		0			0
$109$		−	2.51531i		−	0.240923i	−0.992718	−	0.120461i	$-0.961563\pi$
$109$		−	2.51531i		−	0.240923i	0.992718	−	0.120461i	$-0.0384374\pi$
$110$		0			0
$111$	−5.24146			−0.497498
$112$		0			0
$113$	−4.54621			−0.427671			−0.213836	−	0.976870i	$-0.568596\pi$
$113$	−4.54621			−0.427671			−0.213836	+	0.976870i	$0.568596\pi$
$114$		0			0
$115$			1.47591i			0.137629i
$116$		0			0
$117$	−2.68138			−0.247893
$118$		0			0
$119$	−1.96480	−	0.900155i	−0.180113	−	0.0825171i
$120$		0			0
$121$	30.7020			2.79109
$122$		0			0
$123$	−10.0365			−0.904960
$124$		0			0
$125$	−12.0592			−1.07861
$126$		0			0
$127$		−	15.5388i		−	1.37884i	−0.724360	−	0.689422i	$-0.757865\pi$
$127$		−	15.5388i		−	1.37884i	0.724360	−	0.689422i	$-0.242135\pi$
$128$		0			0
$129$		−	9.34327i		−	0.822629i
$130$		0			0
$131$		−	13.0448i		−	1.13973i	−0.821738	−	0.569865i	$-0.806995\pi$
$131$		−	13.0448i		−	1.13973i	0.821738	−	0.569865i	$-0.193005\pi$
$132$		0			0
$133$	13.8452	+	6.34304i	1.20053	+	0.550012i
$134$		0			0
$135$			1.27632i			0.109848i
$136$		0			0
$137$	5.11894			0.437341			0.218670	−	0.975799i	$-0.429828\pi$
$137$	5.11894			0.437341			0.218670	+	0.975799i	$0.429828\pi$
$138$		0			0
$139$			0.246340i			0.0208943i	0.999945	+	0.0104471i	$0.00332549\pi$
$139$			0.246340i			0.0208943i	−0.999945	+	0.0104471i	$0.996675\pi$
$140$		0			0
$141$		−	21.4805i		−	1.80898i
$142$		0			0
$143$	−6.45771			−0.540020
$144$		0			0
$145$		−	1.63659i		−	0.135912i
$146$		0			0
$147$	12.6359	−	10.8959i	1.04219	−	0.898679i
$148$		0			0
$149$			0.552841i			0.0452905i	0.999744	+	0.0226452i	$0.00720882\pi$
$149$			0.552841i			0.0452905i	−0.999744	+	0.0226452i	$0.992791\pi$
$150$		0			0
$151$		−	5.10249i		−	0.415235i	−0.978210	−	0.207617i	$-0.933429\pi$
$151$		−	5.10249i		−	0.415235i	0.978210	−	0.207617i	$-0.0665709\pi$
$152$		0			0
$153$			2.19028i			0.177074i
$154$		0			0
$155$	−6.54574			−0.525766
$156$		0			0
$157$	19.7641			1.57735			0.788675	−	0.614810i	$-0.210767\pi$
$157$	19.7641			1.57735			0.788675	+	0.614810i	$0.210767\pi$
$158$		0			0
$159$	−22.8155			−1.80939
$160$		0			0
$161$	2.11242	+	0.967785i	0.166482	+	0.0762721i
$162$		0			0
$163$	9.64780			0.755674			0.377837	−	0.925872i	$-0.376668\pi$
$163$	9.64780			0.755674			0.377837	+	0.925872i	$0.376668\pi$
$164$		0			0
$165$		−	25.8679i		−	2.01381i
$166$		0			0
$167$	20.9545			1.62150			0.810752	−	0.585390i	$-0.199058\pi$
$167$	20.9545			1.62150			0.810752	+	0.585390i	$0.199058\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$		−	15.4341i		−	1.18027i
$172$		0			0
$173$	12.0936			0.919456			0.459728	−	0.888060i	$-0.347947\pi$
$173$	12.0936			0.919456			0.459728	+	0.888060i	$0.347947\pi$
$174$		0			0
$175$	−2.39758	+	5.23328i	−0.181240	+	0.395599i
$176$		0			0
$177$	−1.72292			−0.129503
$178$		0			0
$179$	−2.49636			−0.186587			−0.0932934	−	0.995639i	$-0.529739\pi$
$179$	−2.49636			−0.186587			−0.0932934	+	0.995639i	$0.529739\pi$
$180$		0			0
$181$	−13.0487			−0.969903			−0.484951	−	0.874541i	$-0.661163\pi$
$181$	−13.0487			−0.969903			−0.484951	+	0.874541i	$0.661163\pi$
$182$		0			0
$183$		−	3.46770i		−	0.256340i
$184$		0			0
$185$			3.69557i			0.271704i
$186$		0			0
$187$			5.27498i			0.385745i
$188$		0			0
$189$	1.82675	+	0.836909i	0.132876	+	0.0608762i
$190$		0			0
$191$			5.23967i			0.379129i	0.981868	+	0.189565i	$0.0607076\pi$
$191$			5.23967i			0.379129i	−0.981868	+	0.189565i	$0.939292\pi$
$192$		0			0
$193$	−16.4515			−1.18420			−0.592102	−	0.805863i	$-0.701702\pi$
$193$	−16.4515			−1.18420			−0.592102	+	0.805863i	$0.701702\pi$
$194$		0			0
$195$			4.00574i			0.286857i
$196$		0			0
$197$			18.0122i			1.28331i	0.766991	+	0.641657i	$0.221753\pi$
$197$			18.0122i			1.28331i	−0.766991	+	0.641657i	$0.778247\pi$
$198$		0			0
$199$	−9.12392			−0.646778			−0.323389	−	0.946266i	$-0.604822\pi$
$199$	−9.12392			−0.646778			−0.323389	+	0.946266i	$0.604822\pi$
$200$		0			0
$201$			13.9031i			0.980646i
$202$		0			0
$203$	−2.34240	−	1.07315i	−0.164404	−	0.0753202i
$204$		0			0
$205$			7.07637i			0.494235i
$206$		0			0
$207$		−	2.35484i		−	0.163673i
$208$		0			0
$209$		−	37.1707i		−	2.57115i
$210$		0			0
$211$	−17.3894			−1.19713			−0.598567	−	0.801073i	$-0.704263\pi$
$211$	−17.3894			−1.19713			−0.598567	+	0.801073i	$0.704263\pi$
$212$		0			0
$213$	−11.9183			−0.816628
$214$		0			0
$215$	−6.58760			−0.449271
$216$		0			0
$217$	−4.29218	+	9.36868i	−0.291372	+	0.635987i
$218$		0			0
$219$	−20.3444			−1.37474
$220$		0			0
$221$		−	0.816850i		−	0.0549473i
$222$		0			0
$223$	−2.74027			−0.183502			−0.0917510	−	0.995782i	$-0.529246\pi$
$223$	−2.74027			−0.183502			−0.0917510	+	0.995782i	$0.529246\pi$
$224$		0			0
$225$	5.83387			0.388925
$226$		0			0
$227$		−	16.9588i		−	1.12559i	−0.826595	−	0.562797i	$-0.809725\pi$
$227$		−	16.9588i		−	1.12559i	0.826595	−	0.562797i	$-0.190275\pi$
$228$		0			0
$229$	20.6306			1.36331			0.681655	−	0.731674i	$-0.261261\pi$
$229$	20.6306			1.36331			0.681655	+	0.731674i	$0.261261\pi$
$230$		0			0
$231$	−37.0238	−	16.9621i	−2.43598	−	1.11603i
$232$		0			0
$233$	20.2350			1.32564			0.662821	−	0.748778i	$-0.269359\pi$
$233$	20.2350			1.32564			0.662821	+	0.748778i	$0.269359\pi$
$234$		0			0
$235$	−15.1451			−0.987959
$236$		0			0
$237$	−34.0873			−2.21421
$238$		0			0
$239$		−	18.8318i		−	1.21813i	−0.793122	−	0.609063i	$-0.791546\pi$
$239$		−	18.8318i		−	1.21813i	0.793122	−	0.609063i	$-0.208454\pi$
$240$		0			0
$241$			4.30150i			0.277084i	0.990357	+	0.138542i	$0.0442415\pi$
$241$			4.30150i			0.277084i	−0.990357	+	0.138542i	$0.955758\pi$
$242$		0			0
$243$		−	21.2101i		−	1.36063i
$244$		0			0
$245$	−7.68231	−	8.90914i	−0.490805	−	0.569184i
$246$		0			0
$247$			5.75603i			0.366247i
$248$		0			0
$249$	−25.6661			−1.62652
$250$		0			0
$251$		−	7.40762i		−	0.467565i	−0.972289	−	0.233782i	$-0.924890\pi$
$251$		−	7.40762i		−	0.467565i	0.972289	−	0.233782i	$-0.0751103\pi$
$252$		0			0
$253$		−	5.67129i		−	0.356551i
$254$		0			0
$255$	3.27209			0.204906
$256$		0			0
$257$			8.82159i			0.550276i	0.961405	+	0.275138i	$0.0887235\pi$
$257$			8.82159i			0.550276i	−0.961405	+	0.275138i	$0.911277\pi$
$258$		0			0
$259$	5.28933	+	2.42326i	0.328663	+	0.150574i
$260$		0			0
$261$			2.61122i			0.161630i
$262$		0			0
$263$		−	17.8426i		−	1.10022i	−0.835092	−	0.550110i	$-0.814586\pi$
$263$		−	17.8426i		−	1.10022i	0.835092	−	0.550110i	$-0.185414\pi$
$264$		0			0
$265$			16.0864i			0.988181i
$266$		0			0
$267$	42.7165			2.61421
$268$		0			0
$269$	−22.2065			−1.35395			−0.676976	−	0.736005i	$-0.736710\pi$
$269$	−22.2065			−1.35395			−0.676976	+	0.736005i	$0.736710\pi$
$270$		0			0
$271$	10.1511			0.616635			0.308318	−	0.951283i	$-0.400234\pi$
$271$	10.1511			0.616635			0.308318	+	0.951283i	$0.400234\pi$
$272$		0			0
$273$	5.73327	+	2.62665i	0.346993	+	0.158972i
$274$		0			0
$275$	14.0500			0.847248
$276$		0			0
$277$			18.4917i			1.11106i	0.831497	+	0.555530i	$0.187484\pi$
$277$			18.4917i			1.11106i	−0.831497	+	0.555530i	$0.812516\pi$
$278$		0			0
$279$	10.4439			0.625257
$280$		0			0
$281$	−2.67691			−0.159691			−0.0798456	−	0.996807i	$-0.525443\pi$
$281$	−2.67691			−0.159691			−0.0798456	+	0.996807i	$0.525443\pi$
$282$		0			0
$283$		−	22.6633i		−	1.34719i	−0.739100	−	0.673596i	$-0.764749\pi$
$283$		−	22.6633i		−	1.34719i	0.739100	−	0.673596i	$-0.235251\pi$
$284$		0			0
$285$	−23.0571			−1.36579
$286$		0			0
$287$	10.1282	+	4.64013i	0.597846	+	0.273898i
$288$		0			0
$289$	16.3328			0.960750
$290$		0			0
$291$	−31.3258			−1.83635
$292$		0			0
$293$	−33.5920			−1.96247			−0.981234	−	0.192821i	$-0.938236\pi$
$293$	−33.5920			−1.96247			−0.981234	+	0.192821i	$0.938236\pi$
$294$		0			0
$295$			1.21477i			0.0707267i
$296$		0			0
$297$		−	4.90435i		−	0.284579i
$298$		0			0
$299$			0.878221i			0.0507888i
$300$		0			0
$301$	−4.31963	+	9.42860i	−0.248980	+	0.543456i
$302$		0			0
$303$		−	40.0725i		−	2.30211i
$304$		0			0
$305$	−2.44495			−0.139997
$306$		0			0
$307$			22.5430i			1.28660i	0.765616	+	0.643298i	$0.222434\pi$
$307$			22.5430i			1.28660i	−0.765616	+	0.643298i	$0.777566\pi$
$308$		0			0
$309$			18.6730i			1.06227i
$310$		0			0
$311$	14.5078			0.822663			0.411332	−	0.911486i	$-0.365064\pi$
$311$	14.5078			0.822663			0.411332	+	0.911486i	$0.365064\pi$
$312$		0			0
$313$		−	28.4797i		−	1.60977i	−0.593433	−	0.804883i	$-0.702228\pi$
$313$		−	28.4797i		−	1.60977i	0.593433	−	0.804883i	$-0.297772\pi$
$314$		0			0
$315$	−4.96579	+	10.8390i	−0.279791	+	0.610708i
$316$		0			0
$317$			9.99512i			0.561382i	0.959798	+	0.280691i	$0.0905636\pi$
$317$			9.99512i			0.561382i	−0.959798	+	0.280691i	$0.909436\pi$
$318$		0			0
$319$			6.28873i			0.352101i
$320$		0			0
$321$		−	21.9465i		−	1.22493i
$322$		0			0
$323$	4.70181			0.261616
$324$		0			0
$325$	−2.17570			−0.120686
$326$		0			0
$327$	5.99540			0.331546
$328$		0			0
$329$	−9.93098	+	21.6767i	−0.547513	+	1.19507i
$330$		0			0
$331$	5.10543			0.280620			0.140310	−	0.990108i	$-0.455190\pi$
$331$	5.10543			0.280620			0.140310	+	0.990108i	$0.455190\pi$
$332$		0			0
$333$		−	5.89635i		−	0.323118i
$334$		0			0
$335$	9.80255			0.535570
$336$		0			0
$337$	12.5235			0.682200			0.341100	−	0.940027i	$-0.389200\pi$
$337$	12.5235			0.682200			0.341100	+	0.940027i	$0.389200\pi$
$338$		0			0
$339$		−	10.8362i		−	0.588540i
$340$		0			0
$341$	25.1525			1.36208
$342$		0			0
$343$	−17.7888	+	5.15351i	−0.960505	+	0.278263i
$344$		0			0
$345$	−3.51792			−0.189399
$346$		0			0
$347$	−27.2021			−1.46029			−0.730143	−	0.683295i	$-0.760546\pi$
$347$	−27.2021			−1.46029			−0.730143	+	0.683295i	$0.760546\pi$
$348$		0			0
$349$	16.8525			0.902096			0.451048	−	0.892500i	$-0.351050\pi$
$349$	16.8525			0.902096			0.451048	+	0.892500i	$0.351050\pi$
$350$		0			0
$351$			0.759458i			0.0405369i
$352$		0			0
$353$			5.53986i			0.294857i	0.989073	+	0.147429i	$0.0470996\pi$
$353$			5.53986i			0.294857i	−0.989073	+	0.147429i	$0.952900\pi$
$354$		0			0
$355$			8.40316i			0.445993i
$356$		0			0
$357$	2.14558	−	4.68322i	0.113556	−	0.247862i
$358$		0			0
$359$			18.2949i			0.965567i	0.875740	+	0.482784i	$0.160374\pi$
$359$			18.2949i			0.965567i	−0.875740	+	0.482784i	$0.839626\pi$
$360$		0			0
$361$	−14.1319			−0.743782
$362$		0			0
$363$			73.1801i			3.84096i
$364$		0			0
$365$			14.3441i			0.750803i
$366$		0			0
$367$	32.4218			1.69240			0.846202	−	0.532862i	$-0.178884\pi$
$367$	32.4218			1.69240			0.846202	+	0.532862i	$0.178884\pi$
$368$		0			0
$369$		−	11.2905i		−	0.587760i
$370$		0			0
$371$	23.0239	+	10.5482i	1.19534	+	0.547636i
$372$		0			0
$373$		−	12.6232i		−	0.653604i	−0.945093	−	0.326802i	$-0.894029\pi$
$373$		−	12.6232i		−	0.653604i	0.945093	−	0.326802i	$-0.105971\pi$
$374$		0			0
$375$		−	28.7440i		−	1.48433i
$376$		0			0
$377$		−	0.973834i		−	0.0501550i
$378$		0			0
$379$	−18.1976			−0.934746			−0.467373	−	0.884060i	$-0.654800\pi$
$379$	−18.1976			−0.934746			−0.467373	+	0.884060i	$0.654800\pi$
$380$		0			0
$381$	37.0377			1.89750
$382$		0			0
$383$	18.4321			0.941835			0.470917	−	0.882177i	$-0.343923\pi$
$383$	18.4321			0.941835			0.470917	+	0.882177i	$0.343923\pi$
$384$		0			0
$385$	−11.9594	+	26.1041i	−0.609506	+	1.33039i
$386$		0			0
$387$	10.5107			0.534287
$388$		0			0
$389$		−	19.8567i		−	1.00678i	−0.864060	−	0.503388i	$-0.832086\pi$
$389$		−	19.8567i		−	1.00678i	0.864060	−	0.503388i	$-0.167914\pi$
$390$		0			0
$391$	0.717375			0.0362792
$392$		0			0
$393$	31.0932			1.56844
$394$		0			0
$395$			24.0337i			1.20927i
$396$		0			0
$397$	−7.46775			−0.374796			−0.187398	−	0.982284i	$-0.560005\pi$
$397$	−7.46775			−0.374796			−0.187398	+	0.982284i	$0.560005\pi$
$398$		0			0
$399$	−15.1191	+	33.0008i	−0.756899	+	1.65211i
$400$		0			0
$401$	−3.95952			−0.197729			−0.0988644	−	0.995101i	$-0.531521\pi$
$401$	−3.95952			−0.197729			−0.0988644	+	0.995101i	$0.531521\pi$
$402$		0			0
$403$	−3.89496			−0.194022
$404$		0			0
$405$	−16.5609			−0.822917
$406$		0			0
$407$		−	14.2005i		−	0.703893i
$408$		0			0
$409$			18.5513i			0.917303i	0.888616	+	0.458652i	$0.151667\pi$
$409$			18.5513i			0.917303i	−0.888616	+	0.458652i	$0.848333\pi$
$410$		0			0
$411$			12.2013i			0.601847i
$412$		0			0
$413$	1.73866	+	0.796551i	0.0855538	+	0.0391957i
$414$		0			0
$415$			18.0962i			0.888309i
$416$		0			0
$417$	−0.587167			−0.0287537
$418$		0			0
$419$		−	40.1301i		−	1.96048i	−0.197800	−	0.980242i	$-0.563380\pi$
$419$		−	40.1301i		−	1.96048i	0.197800	−	0.980242i	$-0.436620\pi$
$420$		0			0
$421$			7.51607i			0.366311i	0.983084	+	0.183155i	$0.0586311\pi$
$421$			7.51607i			0.366311i	−0.983084	+	0.183155i	$0.941369\pi$
$422$		0			0
$423$	24.1644			1.17491
$424$		0			0
$425$			1.77722i			0.0862079i
$426$		0			0
$427$	−1.60321	+	3.49937i	−0.0775846	+	0.169346i
$428$		0			0
$429$		−	15.3924i		−	0.743150i
$430$		0			0
$431$			23.2557i			1.12019i	0.828429	+	0.560095i	$0.189235\pi$
$431$			23.2557i			1.12019i	−0.828429	+	0.560095i	$0.810765\pi$
$432$		0			0
$433$		−	12.0026i		−	0.576809i	−0.957509	−	0.288404i	$-0.906875\pi$
$433$		−	12.0026i		−	0.576809i	0.957509	−	0.288404i	$-0.0931247\pi$
$434$		0			0
$435$	3.90092			0.187035
$436$		0			0
$437$	−5.05506			−0.241817
$438$		0			0
$439$	19.8600			0.947867			0.473933	−	0.880561i	$-0.342834\pi$
$439$	19.8600			0.947867			0.473933	+	0.880561i	$0.342834\pi$
$440$		0			0
$441$	12.2573	+	14.2147i	0.583681	+	0.676891i
$442$		0			0
$443$	36.9563			1.75585			0.877924	−	0.478800i	$-0.158928\pi$
$443$	36.9563			1.75585			0.877924	+	0.478800i	$0.158928\pi$
$444$		0			0
$445$		−	30.1179i		−	1.42772i
$446$		0			0
$447$	−1.31773			−0.0623266
$448$		0			0
$449$	−27.0181			−1.27506			−0.637531	−	0.770425i	$-0.720044\pi$
$449$	−27.0181			−1.27506			−0.637531	+	0.770425i	$0.720044\pi$
$450$		0			0
$451$		−	27.1915i		−	1.28040i
$452$		0			0
$453$	12.1621			0.571426
$454$		0			0
$455$	1.85195	−	4.04232i	0.0868210	−	0.189507i
$456$		0			0
$457$	−18.2396			−0.853214			−0.426607	−	0.904437i	$-0.640291\pi$
$457$	−18.2396			−0.853214			−0.426607	+	0.904437i	$0.640291\pi$
$458$		0			0
$459$	0.620363			0.0289561
$460$		0			0
$461$	29.4658			1.37236			0.686179	−	0.727432i	$-0.259287\pi$
$461$	29.4658			1.37236			0.686179	+	0.727432i	$0.259287\pi$
$462$		0			0
$463$		−	34.1247i		−	1.58591i	−0.609280	−	0.792955i	$-0.708541\pi$
$463$		−	34.1247i		−	1.58591i	0.609280	−	0.792955i	$-0.291459\pi$
$464$		0			0
$465$		−	15.6022i		−	0.723534i
$466$		0			0
$467$			10.3935i			0.480952i	0.970655	+	0.240476i	$0.0773036\pi$
$467$			10.3935i			0.480952i	−0.970655	+	0.240476i	$0.922696\pi$
$468$		0			0
$469$	6.42774	−	14.0300i	0.296806	−	0.647847i
$470$		0			0
$471$			47.1091i			2.17067i
$472$		0			0
$473$	25.3134			1.16391
$474$		0			0
$475$		−	12.5234i		−	0.574612i
$476$		0			0
$477$		−	25.6662i		−	1.17517i
$478$		0			0
$479$	25.5721			1.16842			0.584210	−	0.811602i	$-0.301404\pi$
$479$	25.5721			1.16842			0.584210	+	0.811602i	$0.301404\pi$
$480$		0			0
$481$			2.19900i			0.100266i
$482$		0			0
$483$	−2.30678	+	5.03508i	−0.104962	+	0.229104i
$484$		0			0
$485$			22.0867i			1.00291i
$486$		0			0
$487$			37.3038i			1.69040i	0.534453	+	0.845198i	$0.320518\pi$
$487$			37.3038i			1.69040i	−0.534453	+	0.845198i	$0.679482\pi$
$488$		0			0
$489$			22.9961i			1.03992i
$490$		0			0
$491$	−36.8751			−1.66415			−0.832075	−	0.554663i	$-0.812847\pi$
$491$	−36.8751			−1.66415			−0.832075	+	0.554663i	$0.812847\pi$
$492$		0			0
$493$	−0.795476			−0.0358265
$494$		0			0
$495$	29.0999			1.30794
$496$		0			0
$497$	12.0271	+	5.51013i	0.539491	+	0.247163i
$498$		0			0
$499$	26.1293			1.16971			0.584854	−	0.811139i	$-0.301152\pi$
$499$	26.1293			1.16971			0.584854	+	0.811139i	$0.301152\pi$
$500$		0			0
$501$			49.9463i			2.23144i
$502$		0			0
$503$	24.5290			1.09370			0.546848	−	0.837232i	$-0.315828\pi$
$503$	24.5290			1.09370			0.546848	+	0.837232i	$0.315828\pi$
$504$		0			0
$505$	−28.2537			−1.25727
$506$		0			0
$507$			2.38356i			0.105858i
$508$		0			0
$509$	−9.69607			−0.429771			−0.214885	−	0.976639i	$-0.568938\pi$
$509$	−9.69607			−0.429771			−0.214885	+	0.976639i	$0.568938\pi$
$510$		0			0
$511$	20.5302	+	9.40572i	0.908201	+	0.416085i
$512$		0			0
$513$	−4.37146			−0.193005
$514$		0			0
$515$	13.1657			0.580149
$516$		0			0
$517$	58.1963			2.55947
$518$		0			0
$519$			28.8258i			1.26531i
$520$		0			0
$521$			23.8145i			1.04333i	0.853149	+	0.521667i	$0.174690\pi$
$521$			23.8145i			1.04333i	−0.853149	+	0.521667i	$0.825310\pi$
$522$		0			0
$523$		−	12.8009i		−	0.559746i	−0.960037	−	0.279873i	$-0.909708\pi$
$523$		−	12.8009i		−	0.559746i	0.960037	−	0.279873i	$-0.0902923\pi$
$524$		0			0
$525$	−12.4739	−	5.71479i	−0.544404	−	0.249414i
$526$		0			0
$527$			3.18160i			0.138593i
$528$		0			0
$529$	22.2287			0.966466
$530$		0			0
$531$		−	1.93819i		−	0.0841103i
$532$		0			0
$533$			4.21071i			0.182386i
$534$		0			0
$535$	−15.4737			−0.668985
$536$		0			0
$537$		−	5.95023i		−	0.256772i
$538$		0			0
$539$	29.5199	+	34.2341i	1.27151	+	1.47457i
$540$		0			0
$541$			42.8431i			1.84197i	0.389597	+	0.920985i	$0.372614\pi$
$541$			42.8431i			1.84197i	−0.389597	+	0.920985i	$0.627386\pi$
$542$		0			0
$543$		−	31.1024i		−	1.33473i
$544$		0			0
$545$		−	4.22714i		−	0.181071i
$546$		0			0
$547$	−34.8819			−1.49144			−0.745721	−	0.666258i	$-0.767895\pi$
$547$	−34.8819			−1.49144			−0.745721	+	0.666258i	$0.767895\pi$
$548$		0			0
$549$	3.90097			0.166489
$550$		0			0
$551$	5.60541			0.238799
$552$		0			0
$553$	34.3986	+	15.7594i	1.46278	+	0.670159i
$554$		0			0
$555$	−8.80862			−0.373905
$556$		0			0
$557$			21.8138i			0.924281i	0.886807	+	0.462141i	$0.152918\pi$
$557$			21.8138i			0.924281i	−0.886807	+	0.462141i	$0.847082\pi$
$558$		0			0
$559$	−3.91987			−0.165793
$560$		0			0
$561$	−12.5733			−0.530843
$562$		0			0
$563$		−	40.3058i		−	1.69869i	−0.527842	−	0.849343i	$-0.676999\pi$
$563$		−	40.3058i		−	1.69869i	0.527842	−	0.849343i	$-0.323001\pi$
$564$		0			0
$565$	−7.64020			−0.321426
$566$		0			0
$567$	−10.8593	+	23.7030i	−0.456049	+	0.995432i
$568$		0			0
$569$	−43.5399			−1.82529			−0.912644	−	0.408756i	$-0.865963\pi$
$569$	−43.5399			−1.82529			−0.912644	+	0.408756i	$0.865963\pi$
$570$		0			0
$571$	3.15835			0.132173			0.0660865	−	0.997814i	$-0.478949\pi$
$571$	3.15835			0.132173			0.0660865	+	0.997814i	$0.478949\pi$
$572$		0			0
$573$	−12.4891			−0.521739
$574$		0			0
$575$		−	1.91074i		−	0.0796836i
$576$		0			0
$577$			8.84506i			0.368225i	0.982905	+	0.184112i	$0.0589410\pi$
$577$			8.84506i			0.368225i	−0.982905	+	0.184112i	$0.941059\pi$
$578$		0			0
$579$		−	39.2132i		−	1.62964i
$580$		0			0
$581$	25.9005	+	11.8661i	1.07453	+	0.492288i
$582$		0			0
$583$		−	61.8133i		−	2.56005i
$584$		0			0
$585$	−4.50623			−0.186310
$586$		0			0
$587$			25.8252i			1.06592i	0.846140	+	0.532960i	$0.178920\pi$
$587$			25.8252i			1.06592i	−0.846140	+	0.532960i	$0.821080\pi$
$588$		0			0
$589$		−	22.4195i		−	0.923779i
$590$		0			0
$591$	−42.9332			−1.76604
$592$		0			0
$593$			2.47895i			0.101798i	0.998704	+	0.0508992i	$0.0162087\pi$
$593$			2.47895i			0.101798i	−0.998704	+	0.0508992i	$0.983791\pi$
$594$		0			0
$595$	−3.30197	−	1.51277i	−0.135368	−	0.0620175i
$596$		0			0
$597$		−	21.7474i		−	0.890064i
$598$		0			0
$599$		−	4.73676i		−	0.193539i	−0.995307	−	0.0967695i	$-0.969149\pi$
$599$		−	4.73676i		−	0.193539i	0.995307	−	0.0967695i	$-0.0308510\pi$
$600$		0			0
$601$			13.4825i			0.549963i	0.961450	+	0.274981i	$0.0886717\pi$
$601$			13.4825i			0.549963i	−0.961450	+	0.274981i	$0.911328\pi$
$602$		0			0
$603$	−15.6402			−0.636917
$604$		0			0
$605$	51.5967			2.09770
$606$		0			0
$607$	12.6426			0.513148			0.256574	−	0.966525i	$-0.417406\pi$
$607$	12.6426			0.513148			0.256574	+	0.966525i	$0.417406\pi$
$608$		0			0
$609$	2.55792	−	5.58325i	0.103652	−	0.226245i
$610$		0			0
$611$	−9.01192			−0.364583
$612$		0			0
$613$		−	11.8220i		−	0.477487i	−0.971083	−	0.238743i	$-0.923265\pi$
$613$		−	11.8220i		−	0.477487i	0.971083	−	0.238743i	$-0.0767355\pi$
$614$		0			0
$615$	−16.8670			−0.680142
$616$		0			0
$617$	26.8587			1.08129			0.540645	−	0.841251i	$-0.318180\pi$
$617$	26.8587			1.08129			0.540645	+	0.841251i	$0.318180\pi$
$618$		0			0
$619$			22.1068i			0.888546i	0.895891	+	0.444273i	$0.146538\pi$
$619$			22.1068i			0.888546i	−0.895891	+	0.444273i	$0.853462\pi$
$620$		0			0
$621$	−0.666972			−0.0267647
$622$		0			0
$623$	−43.1066	−	19.7489i	−1.72703	−	0.791224i
$624$		0			0
$625$	−9.38784			−0.375514
$626$		0			0
$627$	88.5988			3.53830
$628$		0			0
$629$	1.79626			0.0716214
$630$		0			0
$631$			24.0898i			0.958999i	0.877542	+	0.479500i	$0.159182\pi$
$631$			24.0898i			0.958999i	−0.877542	+	0.479500i	$0.840818\pi$
$632$		0			0
$633$		−	41.4487i		−	1.64744i
$634$		0			0
$635$		−	26.1139i		−	1.03630i
$636$		0			0
$637$	−4.57127	−	5.30128i	−0.181120	−	0.210044i
$638$		0			0
$639$		−	13.4074i		−	0.530389i
$640$		0			0
$641$	−41.5721			−1.64200			−0.821000	−	0.570929i	$-0.806583\pi$
$641$	−41.5721			−1.64200			−0.821000	+	0.570929i	$0.806583\pi$
$642$		0			0
$643$			32.7452i			1.29134i	0.763615	+	0.645672i	$0.223423\pi$
$643$			32.7452i			1.29134i	−0.763615	+	0.645672i	$0.776577\pi$
$644$		0			0
$645$		−	15.7020i		−	0.618265i
$646$		0			0
$647$	−48.6201			−1.91145			−0.955727	−	0.294254i	$-0.904929\pi$
$647$	−48.6201			−1.91145			−0.955727	+	0.294254i	$0.904929\pi$
$648$		0			0
$649$		−	4.66785i		−	0.183229i
$650$		0			0
$651$	−22.3309	−	10.2307i	−0.875215	−	0.400972i
$652$		0			0
$653$			29.9679i			1.17273i	0.810045	+	0.586367i	$0.199442\pi$
$653$			29.9679i			1.17273i	−0.810045	+	0.586367i	$0.800558\pi$
$654$		0			0
$655$		−	21.9227i		−	0.856590i
$656$		0			0
$657$		−	22.8863i		−	0.892879i
$658$		0			0
$659$	2.82799			0.110163			0.0550814	−	0.998482i	$-0.482458\pi$
$659$	2.82799			0.110163			0.0550814	+	0.998482i	$0.482458\pi$
$660$		0			0
$661$	15.6250			0.607742			0.303871	−	0.952713i	$-0.401721\pi$
$661$	15.6250			0.607742			0.303871	+	0.952713i	$0.401721\pi$
$662$		0			0
$663$	1.94702			0.0756158
$664$		0			0
$665$	23.2677	+	10.6599i	0.902283	+	0.413373i
$666$		0			0
$667$	0.855241			0.0331151
$668$		0			0
$669$		−	6.53161i		−	0.252527i
$670$		0			0
$671$	9.39491			0.362687
$672$		0			0
$673$	−33.9631			−1.30918			−0.654590	−	0.755984i	$-0.727159\pi$
$673$	−33.9631			−1.30918			−0.654590	+	0.755984i	$0.727159\pi$
$674$		0			0
$675$		−	1.65235i		−	0.0635990i
$676$		0			0
$677$	11.9305			0.458525			0.229262	−	0.973365i	$-0.426369\pi$
$677$	11.9305			0.458525			0.229262	+	0.973365i	$0.426369\pi$
$678$		0			0
$679$	31.6119	+	14.4827i	1.21315	+	0.555796i
$680$		0			0
$681$	40.4224			1.54899
$682$		0			0
$683$	−16.8447			−0.644545			−0.322273	−	0.946647i	$-0.604447\pi$
$683$	−16.8447			−0.644545			−0.322273	+	0.946647i	$0.604447\pi$
$684$		0			0
$685$	8.60272			0.328693
$686$		0			0
$687$			49.1744i			1.87612i
$688$		0			0
$689$			9.57202i			0.364665i
$690$		0			0
$691$			14.3162i			0.544616i	0.962210	+	0.272308i	$0.0877869\pi$
$691$			14.3162i			0.544616i	−0.962210	+	0.272308i	$0.912213\pi$
$692$		0			0
$693$	19.0814	−	41.6497i	0.724844	−	1.58214i
$694$		0			0
$695$			0.413991i			0.0157036i
$696$		0			0
$697$	3.43952			0.130281
$698$		0			0
$699$			48.2315i			1.82428i
$700$		0			0
$701$		−	14.5603i		−	0.549935i	−0.961454	−	0.274967i	$-0.911333\pi$
$701$		−	14.5603i		−	0.549935i	0.961454	−	0.274967i	$-0.0886670\pi$
$702$		0			0
$703$	−12.6575			−0.477387
$704$		0			0
$705$		−	36.0994i		−	1.35958i
$706$		0			0
$707$	−18.5266	+	40.4385i	−0.696763	+	1.52085i
$708$		0			0
$709$		−	35.4973i		−	1.33313i	−0.745448	−	0.666564i	$-0.767764\pi$
$709$		−	35.4973i		−	1.33313i	0.745448	−	0.666564i	$-0.232236\pi$
$710$		0			0
$711$		−	38.3463i		−	1.43810i
$712$		0			0
$713$		−	3.42064i		−	0.128104i
$714$		0			0
$715$	−10.8526			−0.405864
$716$		0			0
$717$	44.8867			1.67633
$718$		0			0
$719$	23.4197			0.873406			0.436703	−	0.899606i	$-0.356146\pi$
$719$	23.4197			0.873406			0.436703	+	0.899606i	$0.356146\pi$
$720$		0			0
$721$	8.63301	−	18.8436i	0.321510	−	0.701771i
$722$		0			0
$723$	−10.2529			−0.381309
$724$		0			0
$725$			2.11877i			0.0786891i
$726$		0			0
$727$	19.4068			0.719757			0.359879	−	0.932999i	$-0.382818\pi$
$727$	19.4068			0.719757			0.359879	+	0.932999i	$0.382818\pi$
$728$		0			0
$729$	20.9926			0.777503
$730$		0			0
$731$			3.20195i			0.118428i
$732$		0			0
$733$	13.5804			0.501603			0.250802	−	0.968038i	$-0.419306\pi$
$733$	13.5804			0.501603			0.250802	+	0.968038i	$0.419306\pi$
$734$		0			0
$735$	21.2355	−	18.3113i	0.783284	−	0.675422i
$736$		0			0
$737$	−37.6671			−1.38748
$738$		0			0
$739$	−17.3365			−0.637734			−0.318867	−	0.947800i	$-0.603302\pi$
$739$	−17.3365			−0.637734			−0.318867	+	0.947800i	$0.603302\pi$
$740$		0			0
$741$	−13.7199			−0.504012
$742$		0			0
$743$		−	9.07974i		−	0.333104i	−0.986033	−	0.166552i	$-0.946737\pi$
$743$		−	9.07974i		−	0.333104i	0.986033	−	0.166552i	$-0.0532633\pi$
$744$		0			0
$745$			0.929085i			0.0340390i
$746$		0			0
$747$		−	28.8729i		−	1.05640i
$748$		0			0
$749$	−10.1464	+	22.1469i	−0.370742	+	0.809231i
$750$		0			0
$751$			7.78993i			0.284259i	0.989848	+	0.142129i	$0.0453949\pi$
$751$			7.78993i			0.284259i	−0.989848	+	0.142129i	$0.954605\pi$
$752$		0			0
$753$	17.6565			0.643440
$754$		0			0
$755$		−	8.57507i		−	0.312079i
$756$		0			0
$757$			44.2936i			1.60988i	0.593357	+	0.804939i	$0.297802\pi$
$757$			44.2936i			1.60988i	−0.593357	+	0.804939i	$0.702198\pi$
$758$		0			0
$759$	13.5179			0.490668
$760$		0			0
$761$			8.33939i			0.302303i	0.988511	+	0.151151i	$0.0482981\pi$
$761$			8.33939i			0.302303i	−0.988511	+	0.151151i	$0.951702\pi$
$762$		0			0
$763$	−6.05016	−	2.77183i	−0.219030	−	0.100347i
$764$		0			0
$765$			3.68092i			0.133084i
$766$		0			0
$767$			0.722834i			0.0261000i
$768$		0			0
$769$			25.5540i			0.921500i	0.887530	+	0.460750i	$0.152419\pi$
$769$			25.5540i			0.921500i	−0.887530	+	0.460750i	$0.847581\pi$
$770$		0			0
$771$	−21.0268			−0.757262
$772$		0			0
$773$	−15.7192			−0.565381			−0.282690	−	0.959211i	$-0.591227\pi$
$773$	−15.7192			−0.565381			−0.282690	+	0.959211i	$0.591227\pi$
$774$		0			0
$775$	8.47426			0.304404
$776$		0			0
$777$	−5.77600	+	12.6075i	−0.207213	+	0.452290i
$778$		0			0
$779$	−24.2369			−0.868379
$780$		0			0
$781$		−	32.2898i		−	1.15542i
$782$		0			0
$783$	0.739585			0.0264306
$784$		0			0
$785$	33.2149			1.18549
$786$		0			0
$787$			40.7884i			1.45395i	0.686665	+	0.726974i	$0.259074\pi$
$787$			40.7884i			1.45395i	−0.686665	+	0.726974i	$0.740926\pi$
$788$		0			0
$789$	42.5289			1.51407
$790$		0			0
$791$	−5.00984	+	10.9351i	−0.178129	+	0.388809i
$792$		0			0
$793$	−1.45484			−0.0516628
$794$		0			0
$795$	−38.3430			−1.35989
$796$		0			0
$797$	−29.9750			−1.06177			−0.530884	−	0.847444i	$-0.678140\pi$
$797$	−29.9750			−1.06177			−0.530884	+	0.847444i	$0.678140\pi$
$798$		0			0
$799$			7.36139i			0.260427i
$800$		0			0
$801$			48.0537i			1.69789i
$802$		0			0
$803$		−	55.1182i		−	1.94508i
$804$		0			0
$805$	3.55005	+	1.62643i	0.125123	+	0.0573240i
$806$		0			0
$807$		−	52.9306i		−	1.86324i
$808$		0			0
$809$	28.3634			0.997205			0.498603	−	0.866831i	$-0.333847\pi$
$809$	28.3634			0.997205			0.498603	+	0.866831i	$0.333847\pi$
$810$		0			0
$811$		−	17.8449i		−	0.626619i	−0.949651	−	0.313309i	$-0.898562\pi$
$811$		−	17.8449i		−	0.626619i	0.949651	−	0.313309i	$-0.101438\pi$
$812$		0			0
$813$			24.1958i			0.848584i
$814$		0			0
$815$	16.2138			0.567943
$816$		0			0
$817$		−	22.5629i		−	0.789376i
$818$		0			0
$819$	−2.95483	+	6.44961i	−0.103250	+	0.225368i
$820$		0			0
$821$			4.62357i			0.161364i	0.996740	+	0.0806818i	$0.0257098\pi$
$821$			4.62357i			0.161364i	−0.996740	+	0.0806818i	$0.974290\pi$
$822$		0			0
$823$			47.4508i			1.65403i	0.562178	+	0.827016i	$0.309964\pi$
$823$			47.4508i			1.65403i	−0.562178	+	0.827016i	$0.690036\pi$
$824$		0			0
$825$			33.4891i			1.16594i
$826$		0			0
$827$	21.4189			0.744807			0.372404	−	0.928071i	$-0.378534\pi$
$827$	21.4189			0.744807			0.372404	+	0.928071i	$0.378534\pi$
$828$		0			0
$829$	15.5176			0.538947			0.269474	−	0.963008i	$-0.413150\pi$
$829$	15.5176			0.538947			0.269474	+	0.963008i	$0.413150\pi$
$830$		0			0
$831$	−44.0762			−1.52899
$832$		0			0
$833$	−4.33035	+	3.73404i	−0.150038	+	0.129377i
$834$		0			0
$835$	35.2153			1.21868
$836$		0			0
$837$		−	2.95806i		−	0.102245i
$838$		0			0
$839$	8.76582			0.302630			0.151315	−	0.988486i	$-0.451649\pi$
$839$	8.76582			0.302630			0.151315	+	0.988486i	$0.451649\pi$
$840$		0			0
$841$	28.0516			0.967298
$842$		0			0
$843$		−	6.38059i		−	0.219759i
$844$		0			0
$845$	1.68057			0.0578132
$846$		0			0
$847$	33.8330	−	73.8485i	1.16252	−	2.53746i
$848$		0			0
$849$	54.0193			1.85394
$850$		0			0
$851$	−1.93121			−0.0662010
$852$		0			0
$853$	−21.4369			−0.733985			−0.366992	−	0.930224i	$-0.619613\pi$
$853$	−21.4369			−0.733985			−0.366992	+	0.930224i	$0.619613\pi$
$854$		0			0
$855$		−	25.9380i		−	0.887060i
$856$		0			0
$857$			48.0166i			1.64021i	0.572210	+	0.820107i	$0.306086\pi$
$857$			48.0166i			1.64021i	−0.572210	+	0.820107i	$0.693914\pi$
$858$		0			0
$859$		−	11.4813i		−	0.391738i	−0.980630	−	0.195869i	$-0.937247\pi$
$859$		−	11.4813i		−	0.391738i	0.980630	−	0.195869i	$-0.0627527\pi$
$860$		0			0
$861$	−11.0600	+	24.1411i	−0.376925	+	0.822727i
$862$		0			0
$863$			14.5463i			0.495163i	0.968867	+	0.247581i	$0.0796358\pi$
$863$			14.5463i			0.495163i	−0.968867	+	0.247581i	$0.920364\pi$
$864$		0			0
$865$	20.3240			0.691037
$866$		0			0
$867$			38.9302i			1.32214i
$868$		0			0
$869$		−	92.3515i		−	3.13281i
$870$		0			0
$871$	5.83289			0.197640
$872$		0			0
$873$		−	35.2398i		−	1.19269i
$874$		0			0
$875$	−13.2891	+	29.0065i	−0.449253	+	0.980598i
$876$		0			0
$877$		−	48.2621i		−	1.62969i	−0.579675	−	0.814847i	$-0.696821\pi$
$877$		−	48.2621i		−	1.62969i	0.579675	−	0.814847i	$-0.303179\pi$
$878$		0			0
$879$		−	80.0688i		−	2.70065i
$880$		0			0
$881$			3.34804i			0.112798i	0.998408	+	0.0563991i	$0.0179619\pi$
$881$			3.34804i			0.112798i	−0.998408	+	0.0563991i	$0.982038\pi$
$882$		0			0
$883$	39.7732			1.33847			0.669237	−	0.743049i	$-0.266621\pi$
$883$	39.7732			1.33847			0.669237	+	0.743049i	$0.266621\pi$
$884$		0			0
$885$	−2.89548			−0.0973306
$886$		0			0
$887$	1.92704			0.0647036			0.0323518	−	0.999477i	$-0.489700\pi$
$887$	1.92704			0.0647036			0.0323518	+	0.999477i	$0.489700\pi$
$888$		0			0
$889$	−37.3760	−	17.1235i	−1.25355	−	0.574303i
$890$		0			0
$891$	63.6365			2.13190
$892$		0			0
$893$		−	51.8729i		−	1.73586i
$894$		0			0
$895$	−4.19530			−0.140233
$896$		0			0
$897$	−2.09330			−0.0698931
$898$		0			0
$899$			3.79304i			0.126505i
$900$		0			0
$901$	7.81891			0.260486
$902$		0			0
$903$	−22.4737	−	10.2961i	−0.747877	−	0.342633i
$904$		0			0
$905$	−21.9292			−0.728952
$906$		0			0
$907$	47.2726			1.56966			0.784831	−	0.619710i	$-0.212750\pi$
$907$	47.2726			1.56966			0.784831	+	0.619710i	$0.212750\pi$
$908$		0			0
$909$	45.0794			1.49519
$910$		0			0
$911$			57.2072i			1.89536i	0.319222	+	0.947680i	$0.396578\pi$
$911$			57.2072i			1.89536i	−0.319222	+	0.947680i	$0.603422\pi$
$912$		0			0
$913$		−	69.5362i		−	2.30131i
$914$		0			0
$915$		−	5.82770i		−	0.192658i
$916$		0			0
$917$	−31.3771	−	14.3752i	−1.03616	−	0.474710i
$918$		0			0
$919$		−	11.7893i		−	0.388894i	−0.980913	−	0.194447i	$-0.937709\pi$
$919$		−	11.7893i		−	0.388894i	0.980913	−	0.194447i	$-0.0622912\pi$
$920$		0			0
$921$	−53.7326			−1.77055
$922$		0			0
$923$			5.00020i			0.164584i
$924$		0			0
$925$		−	4.78437i		−	0.157309i
$926$		0			0
$927$	−21.0061			−0.689931
$928$		0			0
$929$		−	6.11545i		−	0.200641i	−0.994955	−	0.100321i	$-0.968013\pi$
$929$		−	6.11545i		−	0.200641i	0.994955	−	0.100321i	$-0.0319869\pi$
$930$		0			0
$931$	30.5143	−	26.3123i	1.00007	−	0.862352i
$932$		0			0
$933$			34.5803i			1.13211i
$934$		0			0
$935$			8.86495i			0.289915i
$936$		0			0
$937$		−	13.1924i		−	0.430978i	−0.976506	−	0.215489i	$-0.930866\pi$
$937$		−	13.1924i		−	0.430978i	0.976506	−	0.215489i	$-0.0691345\pi$
$938$		0			0
$939$	67.8831			2.21528
$940$		0			0
$941$	−1.98071			−0.0645694			−0.0322847	−	0.999479i	$-0.510278\pi$
$941$	−1.98071			−0.0645694			−0.0322847	+	0.999479i	$0.510278\pi$
$942$		0			0
$943$	−3.69793			−0.120421
$944$		0			0
$945$	3.06997	+	1.40648i	0.0998662	+	0.0457528i
$946$		0			0
$947$	−37.0962			−1.20547			−0.602733	−	0.797943i	$-0.705921\pi$
$947$	−37.0962			−1.20547			−0.602733	+	0.797943i	$0.705921\pi$
$948$		0			0
$949$			8.53527i			0.277067i
$950$		0			0
$951$	−23.8240			−0.772547
$952$		0			0
$953$	1.46318			0.0473971			0.0236985	−	0.999719i	$-0.492456\pi$
$953$	1.46318			0.0473971			0.0236985	+	0.999719i	$0.492456\pi$
$954$		0			0
$955$			8.80561i			0.284943i
$956$		0			0
$957$	−14.9896			−0.484545
$958$		0			0
$959$	5.64098	−	12.3128i	0.182157	−	0.397600i
$960$		0			0
$961$	−15.8293			−0.510622
$962$		0			0
$963$	24.6886			0.795578
$964$		0			0
$965$	−27.6478			−0.890014
$966$		0			0
$967$			0.345097i			0.0110976i	0.999985	+	0.00554878i	$0.00176624\pi$
$967$			0.345097i			0.0110976i	−0.999985	+	0.00554878i	$0.998234\pi$
$968$		0			0
$969$			11.2071i			0.360023i
$970$		0			0
$971$			6.24436i			0.200391i	0.994968	+	0.100195i	$0.0319468\pi$
$971$			6.24436i			0.200391i	−0.994968	+	0.100195i	$0.968053\pi$
$972$		0			0
$973$	0.592530	+	0.271462i	0.0189956	+	0.00870269i
$974$		0			0
$975$		−	5.18592i		−	0.166082i
$976$		0			0
$977$	21.5410			0.689157			0.344578	−	0.938758i	$-0.388022\pi$
$977$	21.5410			0.689157			0.344578	+	0.938758i	$0.388022\pi$
$978$		0			0
$979$			115.730i			3.69875i
$980$		0			0
$981$			6.74449i			0.215335i
$982$		0			0
$983$	−31.1862			−0.994684			−0.497342	−	0.867555i	$-0.665691\pi$
$983$	−31.1862			−0.994684			−0.497342	+	0.867555i	$0.665691\pi$
$984$		0			0
$985$			30.2707i			0.964503i
$986$		0			0
$987$	−51.6678	−	23.6711i	−1.64460	−	0.753461i
$988$		0			0
$989$		−	3.44251i		−	0.109466i
$990$		0			0
$991$			7.02945i			0.223298i	0.993748	+	0.111649i	$0.0356132\pi$
$991$			7.02945i			0.223298i	−0.993748	+	0.111649i	$0.964387\pi$
$992$		0			0
$993$			12.1691i			0.386175i
$994$		0			0
$995$	−15.3333			−0.486100
$996$		0			0
$997$	−30.9954			−0.981636			−0.490818	−	0.871262i	$-0.663302\pi$
$997$	−30.9954			−0.981636			−0.490818	+	0.871262i	$0.663302\pi$
$998$		0			0
$999$	−1.67005			−0.0528380

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.h.b.2575.41		48
4.3	odd	2		728.2.h.b.27.17	yes	48
7.6	odd	2		2912.2.h.a.2575.8		48
8.3	odd	2		2912.2.h.a.2575.41		48
8.5	even	2		728.2.h.a.27.18	yes	48
28.27	even	2		728.2.h.a.27.17	✓	48
56.13	odd	2		728.2.h.b.27.18	yes	48
56.27	even	2	inner	2912.2.h.b.2575.8		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.h.a.27.17	✓	48	28.27	even	2
728.2.h.a.27.18	yes	48	8.5	even	2
728.2.h.b.27.17	yes	48	4.3	odd	2
728.2.h.b.27.18	yes	48	56.13	odd	2
2912.2.h.a.2575.8		48	7.6	odd	2
2912.2.h.a.2575.41		48	8.3	odd	2
2912.2.h.b.2575.8		48	56.27	even	2	inner
2912.2.h.b.2575.41		48	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2912 = 2^{5} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2912.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.38356i q^{3} +1.68057 q^{5} +(1.10198 - 2.40533i) q^{7} -2.68138 q^{9} -6.45771 q^{11} +1.00000 q^{13} +4.00574i q^{15} -0.816850i q^{17} +5.75603i q^{19} +(5.73327 + 2.62665i) q^{21} +0.878221i q^{23} -2.17570 q^{25} +0.759458i q^{27} -0.973834i q^{29} -3.89496 q^{31} -15.3924i q^{33} +(1.85195 - 4.04232i) q^{35} +2.19900i q^{37} +2.38356i q^{39} +4.21071i q^{41} -3.91987 q^{43} -4.50623 q^{45} -9.01192 q^{47} +(-4.57127 - 5.30128i) q^{49} +1.94702 q^{51} +9.57202i q^{53} -10.8526 q^{55} -13.7199 q^{57} +0.722834i q^{59} -1.45484 q^{61} +(-2.95483 + 6.44961i) q^{63} +1.68057 q^{65} +5.83289 q^{67} -2.09330 q^{69} +5.00020i q^{71} +8.53527i q^{73} -5.18592i q^{75} +(-7.11628 + 15.5329i) q^{77} +14.3010i q^{79} -9.85435 q^{81} +10.7679i q^{83} -1.37277i q^{85} +2.32119 q^{87} -17.9213i q^{89} +(1.10198 - 2.40533i) q^{91} -9.28389i q^{93} +9.67338i q^{95} +13.1424i q^{97} +17.3155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{9} + 4 q^{11} + 48 q^{13} + 48 q^{25} - 12 q^{35} + 4 q^{43} - 24 q^{45} + 40 q^{51} + 20 q^{63} + 4 q^{67} + 20 q^{77} + 64 q^{81} - 40 q^{87} - 20 q^{99}+O(q^{100}) \) 48 * q - 48 * q^9 + 4 * q^11 + 48 * q^13 + 48 * q^25 - 12 * q^35 + 4 * q^43 - 24 * q^45 + 40 * q^51 + 20 * q^63 + 4 * q^67 + 20 * q^77 + 64 * q^81 - 40 * q^87 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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