LMFDB - Embedded newform 400.4.c.j.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(49,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	400.c (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:	$23.6007640023$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.4.c.j.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{3} +16.0000i q^{7} +11.0000 q^{9} +O(q^{10})$	$q+4.00000i q^{3} +16.0000i q^{7} +11.0000 q^{9} +60.0000 q^{11} -86.0000i q^{13} +18.0000i q^{17} +44.0000 q^{19} -64.0000 q^{21} +48.0000i q^{23} +152.000i q^{27} +186.000 q^{29} -176.000 q^{31} +240.000i q^{33} +254.000i q^{37} +344.000 q^{39} +186.000 q^{41} -100.000i q^{43} -168.000i q^{47} +87.0000 q^{49} -72.0000 q^{51} +498.000i q^{53} +176.000i q^{57} -252.000 q^{59} -58.0000 q^{61} +176.000i q^{63} +1036.00i q^{67} -192.000 q^{69} -168.000 q^{71} -506.000i q^{73} +960.000i q^{77} +272.000 q^{79} -311.000 q^{81} +948.000i q^{83} +744.000i q^{87} +1014.00 q^{89} +1376.00 q^{91} -704.000i q^{93} -766.000i q^{97} +660.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 22 q^{9}+O(q^{10}) $ 2 * q + 22 * q^9	$ 2 q + 22 q^{9} + 120 q^{11} + 88 q^{19} - 128 q^{21} + 372 q^{29} - 352 q^{31} + 688 q^{39} + 372 q^{41} + 174 q^{49} - 144 q^{51} - 504 q^{59} - 116 q^{61} - 384 q^{69} - 336 q^{71} + 544 q^{79} - 622 q^{81} + 2028 q^{89} + 2752 q^{91} + 1320 q^{99}+O(q^{100}) $ 2 * q + 22 * q^9 + 120 * q^11 + 88 * q^19 - 128 * q^21 + 372 * q^29 - 352 * q^31 + 688 * q^39 + 372 * q^41 + 174 * q^49 - 144 * q^51 - 504 * q^59 - 116 * q^61 - 384 * q^69 - 336 * q^71 + 544 * q^79 - 622 * q^81 + 2028 * q^89 + 2752 * q^91 + 1320 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	4.00000i	0.769800i	0.922958	+	0.384900i	$0.125764\pi$
$3$	4.00000i	0.769800i	−0.922958	+	0.384900i	$0.874236\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	16.0000i	0.863919i	0.901893	+	0.431959i	$0.142178\pi$
$7$	16.0000i	0.863919i	−0.901893	+	0.431959i	$0.857822\pi$
$8$	0	0
$9$	11.0000	0.407407
$10$	0	0
$11$	60.0000	1.64461	0.822304	−	0.569049i	$-0.192689\pi$
$11$	60.0000	1.64461	0.822304	+	0.569049i	$0.192689\pi$
$12$	0	0
$13$	− 86.0000i	− 1.83478i	−0.397992	−	0.917389i	$-0.630293\pi$
$13$	− 86.0000i	− 1.83478i	0.397992	−	0.917389i	$-0.369707\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	18.0000i	0.256802i	0.991722	+	0.128401i	$0.0409845\pi$
$17$	18.0000i	0.256802i	−0.991722	+	0.128401i	$0.959015\pi$
$18$	0	0
$19$	44.0000	0.531279	0.265639	−	0.964072i	$-0.414417\pi$
$19$	44.0000	0.531279	0.265639	+	0.964072i	$0.414417\pi$
$20$	0	0
$21$	−64.0000	−0.665045
$22$	0	0
$23$	48.0000i	0.435161i	0.976042	+	0.217580i	$0.0698164\pi$
$23$	48.0000i	0.435161i	−0.976042	+	0.217580i	$0.930184\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	152.000i	1.08342i
$28$	0	0
$29$	186.000	1.19101	0.595506	−	0.803351i	$-0.296952\pi$
$29$	186.000	1.19101	0.595506	+	0.803351i	$0.296952\pi$
$30$	0	0
$31$	−176.000	−1.01969	−0.509847	−	0.860265i	$-0.670298\pi$
$31$	−176.000	−1.01969	−0.509847	+	0.860265i	$0.670298\pi$
$32$	0	0
$33$	240.000i	1.26602i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	254.000i	1.12858i	0.825578	+	0.564288i	$0.190849\pi$
$37$	254.000i	1.12858i	−0.825578	+	0.564288i	$0.809151\pi$
$38$	0	0
$39$	344.000	1.41241
$40$	0	0
$41$	186.000	0.708496	0.354248	−	0.935152i	$-0.384737\pi$
$41$	186.000	0.708496	0.354248	+	0.935152i	$0.384737\pi$
$42$	0	0
$43$	− 100.000i	− 0.354648i	−0.984153	−	0.177324i	$-0.943256\pi$
$43$	− 100.000i	− 0.354648i	0.984153	−	0.177324i	$-0.0567440\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 168.000i	− 0.521390i	−0.965421	−	0.260695i	$-0.916048\pi$
$47$	− 168.000i	− 0.521390i	0.965421	−	0.260695i	$-0.0839517\pi$
$48$	0	0
$49$	87.0000	0.253644
$50$	0	0
$51$	−72.0000	−0.197687
$52$	0	0
$53$	498.000i	1.29067i	0.763899	+	0.645335i	$0.223282\pi$
$53$	498.000i	1.29067i	−0.763899	+	0.645335i	$0.776718\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	176.000i	0.408978i
$58$	0	0
$59$	−252.000	−0.556061	−0.278031	−	0.960572i	$-0.589682\pi$
$59$	−252.000	−0.556061	−0.278031	+	0.960572i	$0.589682\pi$
$60$	0	0
$61$	−58.0000	−0.121740	−0.0608700	−	0.998146i	$-0.519388\pi$
$61$	−58.0000	−0.121740	−0.0608700	+	0.998146i	$0.519388\pi$
$62$	0	0
$63$	176.000i	0.351967i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1036.00i	1.88907i	0.328414	+	0.944534i	$0.393486\pi$
$67$	1036.00i	1.88907i	−0.328414	+	0.944534i	$0.606514\pi$
$68$	0	0
$69$	−192.000	−0.334987
$70$	0	0
$71$	−168.000	−0.280816	−0.140408	−	0.990094i	$-0.544841\pi$
$71$	−168.000	−0.280816	−0.140408	+	0.990094i	$0.544841\pi$
$72$	0	0
$73$	− 506.000i	− 0.811272i	−0.914035	−	0.405636i	$-0.867050\pi$
$73$	− 506.000i	− 0.811272i	0.914035	−	0.405636i	$-0.132950\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	960.000i	1.42081i
$78$	0	0
$79$	272.000	0.387372	0.193686	−	0.981064i	$-0.437956\pi$
$79$	272.000	0.387372	0.193686	+	0.981064i	$0.437956\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	948.000i	1.25369i	0.779143	+	0.626846i	$0.215655\pi$
$83$	948.000i	1.25369i	−0.779143	+	0.626846i	$0.784345\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	744.000i	0.916841i
$88$	0	0
$89$	1014.00	1.20768	0.603841	−	0.797104i	$-0.293636\pi$
$89$	1014.00	1.20768	0.603841	+	0.797104i	$0.293636\pi$
$90$	0	0
$91$	1376.00	1.58510
$92$	0	0
$93$	− 704.000i	− 0.784961i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 766.000i	− 0.801809i	−0.916120	−	0.400905i	$-0.868696\pi$
$97$	− 766.000i	− 0.801809i	0.916120	−	0.400905i	$-0.131304\pi$
$98$	0	0
$99$	660.000	0.670025
$100$	0	0
$101$	−1314.00	−1.29453	−0.647267	−	0.762264i	$-0.724088\pi$
$101$	−1314.00	−1.29453	−0.647267	+	0.762264i	$0.724088\pi$
$102$	0	0
$103$	− 448.000i	− 0.428570i	−0.976771	−	0.214285i	$-0.931258\pi$
$103$	− 448.000i	− 0.428570i	0.976771	−	0.214285i	$-0.0687422\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1548.00i	− 1.39861i	−0.714826	−	0.699303i	$-0.753494\pi$
$107$	− 1548.00i	− 1.39861i	0.714826	−	0.699303i	$-0.246506\pi$
$108$	0	0
$109$	−278.000	−0.244290	−0.122145	−	0.992512i	$-0.538977\pi$
$109$	−278.000	−0.244290	−0.122145	+	0.992512i	$0.538977\pi$
$110$	0	0
$111$	−1016.00	−0.868779
$112$	0	0
$113$	558.000i	0.464533i	0.972652	+	0.232266i	$0.0746141\pi$
$113$	558.000i	0.464533i	−0.972652	+	0.232266i	$0.925386\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 946.000i	− 0.747502i
$118$	0	0
$119$	−288.000	−0.221856
$120$	0	0
$121$	2269.00	1.70473
$122$	0	0
$123$	744.000i	0.545400i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 344.000i	− 0.240355i	−0.992752	−	0.120177i	$-0.961654\pi$
$127$	− 344.000i	− 0.240355i	0.992752	−	0.120177i	$-0.0383463\pi$
$128$	0	0
$129$	400.000	0.273008
$130$	0	0
$131$	−780.000	−0.520221	−0.260110	−	0.965579i	$-0.583759\pi$
$131$	−780.000	−0.520221	−0.260110	+	0.965579i	$0.583759\pi$
$132$	0	0
$133$	704.000i	0.458982i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	666.000i	0.415330i	0.978200	+	0.207665i	$0.0665864\pi$
$137$	666.000i	0.415330i	−0.978200	+	0.207665i	$0.933414\pi$
$138$	0	0
$139$	884.000	0.539424	0.269712	−	0.962941i	$-0.413072\pi$
$139$	884.000	0.539424	0.269712	+	0.962941i	$0.413072\pi$
$140$	0	0
$141$	672.000	0.401366
$142$	0	0
$143$	− 5160.00i	− 3.01749i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	348.000i	0.195255i
$148$	0	0
$149$	114.000	0.0626795	0.0313397	−	0.999509i	$-0.490023\pi$
$149$	114.000	0.0626795	0.0313397	+	0.999509i	$0.490023\pi$
$150$	0	0
$151$	40.0000	0.0215573	0.0107787	−	0.999942i	$-0.496569\pi$
$151$	40.0000	0.0215573	0.0107787	+	0.999942i	$0.496569\pi$
$152$	0	0
$153$	198.000i	0.104623i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 154.000i	− 0.0782837i	−0.999234	−	0.0391418i	$-0.987538\pi$
$157$	− 154.000i	− 0.0782837i	0.999234	−	0.0391418i	$-0.0124624\pi$
$158$	0	0
$159$	−1992.00	−0.993559
$160$	0	0
$161$	−768.000	−0.375943
$162$	0	0
$163$	2180.00i	1.04755i	0.851856	+	0.523775i	$0.175477\pi$
$163$	2180.00i	1.04755i	−0.851856	+	0.523775i	$0.824523\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3696.00i	− 1.71261i	−0.516474	−	0.856303i	$-0.672756\pi$
$167$	− 3696.00i	− 1.71261i	0.516474	−	0.856303i	$-0.327244\pi$
$168$	0	0
$169$	−5199.00	−2.36641
$170$	0	0
$171$	484.000	0.216447
$172$	0	0
$173$	− 1302.00i	− 0.572192i	−0.958201	−	0.286096i	$-0.907642\pi$
$173$	− 1302.00i	− 0.572192i	0.958201	−	0.286096i	$-0.0923576\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1008.00i	− 0.428056i
$178$	0	0
$179$	−4308.00	−1.79885	−0.899427	−	0.437070i	$-0.856016\pi$
$179$	−4308.00	−1.79885	−0.899427	+	0.437070i	$0.856016\pi$
$180$	0	0
$181$	1550.00	0.636523	0.318261	−	0.948003i	$-0.396901\pi$
$181$	1550.00	0.636523	0.318261	+	0.948003i	$0.396901\pi$
$182$	0	0
$183$	− 232.000i	− 0.0937155i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1080.00i	0.422339i
$188$	0	0
$189$	−2432.00	−0.935989
$190$	0	0
$191$	−48.0000	−0.0181841	−0.00909204	−	0.999959i	$-0.502894\pi$
$191$	−48.0000	−0.0181841	−0.00909204	+	0.999959i	$0.502894\pi$
$192$	0	0
$193$	− 1058.00i	− 0.394593i	−0.980344	−	0.197297i	$-0.936784\pi$
$193$	− 1058.00i	− 0.394593i	0.980344	−	0.197297i	$-0.0632162\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3714.00i	− 1.34321i	−0.740911	−	0.671603i	$-0.765606\pi$
$197$	− 3714.00i	− 1.34321i	0.740911	−	0.671603i	$-0.234394\pi$
$198$	0	0
$199$	−1768.00	−0.629800	−0.314900	−	0.949125i	$-0.601971\pi$
$199$	−1768.00	−0.629800	−0.314900	+	0.949125i	$0.601971\pi$
$200$	0	0
$201$	−4144.00	−1.45421
$202$	0	0
$203$	2976.00i	1.02894i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	528.000i	0.177288i
$208$	0	0
$209$	2640.00	0.873745
$210$	0	0
$211$	4036.00	1.31682	0.658412	−	0.752658i	$-0.271229\pi$
$211$	4036.00	1.31682	0.658412	+	0.752658i	$0.271229\pi$
$212$	0	0
$213$	− 672.000i	− 0.216172i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2816.00i	− 0.880933i
$218$	0	0
$219$	2024.00	0.624517
$220$	0	0
$221$	1548.00	0.471175
$222$	0	0
$223$	680.000i	0.204198i	0.994774	+	0.102099i	$0.0325559\pi$
$223$	680.000i	0.204198i	−0.994774	+	0.102099i	$0.967444\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2388.00i	− 0.698225i	−0.937081	−	0.349113i	$-0.886483\pi$
$227$	− 2388.00i	− 0.698225i	0.937081	−	0.349113i	$-0.113517\pi$
$228$	0	0
$229$	3874.00	1.11791	0.558954	−	0.829198i	$-0.311203\pi$
$229$	3874.00	1.11791	0.558954	+	0.829198i	$0.311203\pi$
$230$	0	0
$231$	−3840.00	−1.09374
$232$	0	0
$233$	− 3162.00i	− 0.889054i	−0.895766	−	0.444527i	$-0.853372\pi$
$233$	− 3162.00i	− 0.889054i	0.895766	−	0.444527i	$-0.146628\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1088.00i	0.298199i
$238$	0	0
$239$	5424.00	1.46799	0.733995	−	0.679155i	$-0.237654\pi$
$239$	5424.00	1.46799	0.733995	+	0.679155i	$0.237654\pi$
$240$	0	0
$241$	−3886.00	−1.03867	−0.519335	−	0.854571i	$-0.673820\pi$
$241$	−3886.00	−1.03867	−0.519335	+	0.854571i	$0.673820\pi$
$242$	0	0
$243$	2860.00i	0.755017i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 3784.00i	− 0.974778i
$248$	0	0
$249$	−3792.00	−0.965093
$250$	0	0
$251$	5100.00	1.28251	0.641253	−	0.767329i	$-0.278415\pi$
$251$	5100.00	1.28251	0.641253	+	0.767329i	$0.278415\pi$
$252$	0	0
$253$	2880.00i	0.715668i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2178.00i	0.528638i	0.964435	+	0.264319i	$0.0851471\pi$
$257$	2178.00i	0.528638i	−0.964435	+	0.264319i	$0.914853\pi$
$258$	0	0
$259$	−4064.00	−0.974999
$260$	0	0
$261$	2046.00	0.485227
$262$	0	0
$263$	− 6144.00i	− 1.44051i	−0.693707	−	0.720257i	$-0.744024\pi$
$263$	− 6144.00i	− 1.44051i	0.693707	−	0.720257i	$-0.255976\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4056.00i	0.929675i
$268$	0	0
$269$	−822.000	−0.186313	−0.0931566	−	0.995651i	$-0.529696\pi$
$269$	−822.000	−0.186313	−0.0931566	+	0.995651i	$0.529696\pi$
$270$	0	0
$271$	−8480.00	−1.90082	−0.950412	−	0.310994i	$-0.899338\pi$
$271$	−8480.00	−1.90082	−0.950412	+	0.310994i	$0.899338\pi$
$272$	0	0
$273$	5504.00i	1.22021i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1138.00i	− 0.246844i	−0.992354	−	0.123422i	$-0.960613\pi$
$277$	− 1138.00i	− 0.246844i	0.992354	−	0.123422i	$-0.0393869\pi$
$278$	0	0
$279$	−1936.00	−0.415431
$280$	0	0
$281$	5706.00	1.21136	0.605679	−	0.795709i	$-0.292902\pi$
$281$	5706.00	1.21136	0.605679	+	0.795709i	$0.292902\pi$
$282$	0	0
$283$	− 3028.00i	− 0.636028i	−0.948086	−	0.318014i	$-0.896984\pi$
$283$	− 3028.00i	− 0.636028i	0.948086	−	0.318014i	$-0.103016\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2976.00i	0.612083i
$288$	0	0
$289$	4589.00	0.934053
$290$	0	0
$291$	3064.00	0.617233
$292$	0	0
$293$	− 3390.00i	− 0.675925i	−0.941160	−	0.337962i	$-0.890262\pi$
$293$	− 3390.00i	− 0.675925i	0.941160	−	0.337962i	$-0.109738\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	9120.00i	1.78180i
$298$	0	0
$299$	4128.00	0.798423
$300$	0	0
$301$	1600.00	0.306387
$302$	0	0
$303$	− 5256.00i	− 0.996532i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4156.00i	0.772624i	0.922368	+	0.386312i	$0.126251\pi$
$307$	4156.00i	0.772624i	−0.922368	+	0.386312i	$0.873749\pi$
$308$	0	0
$309$	1792.00	0.329914
$310$	0	0
$311$	−6552.00	−1.19463	−0.597315	−	0.802007i	$-0.703766\pi$
$311$	−6552.00	−1.19463	−0.597315	+	0.802007i	$0.703766\pi$
$312$	0	0
$313$	1366.00i	0.246680i	0.992364	+	0.123340i	$0.0393606\pi$
$313$	1366.00i	0.246680i	−0.992364	+	0.123340i	$0.960639\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2598.00i	0.460310i	0.973154	+	0.230155i	$0.0739233\pi$
$317$	2598.00i	0.460310i	−0.973154	+	0.230155i	$0.926077\pi$
$318$	0	0
$319$	11160.0	1.95875
$320$	0	0
$321$	6192.00	1.07665
$322$	0	0
$323$	792.000i	0.136434i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 1112.00i	− 0.188054i
$328$	0	0
$329$	2688.00	0.450438
$330$	0	0
$331$	3292.00	0.546661	0.273330	−	0.961920i	$-0.411875\pi$
$331$	3292.00	0.546661	0.273330	+	0.961920i	$0.411875\pi$
$332$	0	0
$333$	2794.00i	0.459791i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	6194.00i	1.00121i	0.865675	+	0.500606i	$0.166890\pi$
$337$	6194.00i	1.00121i	−0.865675	+	0.500606i	$0.833110\pi$
$338$	0	0
$339$	−2232.00	−0.357598
$340$	0	0
$341$	−10560.0	−1.67700
$342$	0	0
$343$	6880.00i	1.08305i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10020.0i	1.55015i	0.631870	+	0.775075i	$0.282288\pi$
$347$	10020.0i	1.55015i	−0.631870	+	0.775075i	$0.717712\pi$
$348$	0	0
$349$	3130.00	0.480072	0.240036	−	0.970764i	$-0.422841\pi$
$349$	3130.00	0.480072	0.240036	+	0.970764i	$0.422841\pi$
$350$	0	0
$351$	13072.0	1.98784
$352$	0	0
$353$	− 4194.00i	− 0.632363i	−0.948699	−	0.316181i	$-0.897599\pi$
$353$	− 4194.00i	− 0.632363i	0.948699	−	0.316181i	$-0.102401\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1152.00i	− 0.170785i
$358$	0	0
$359$	−4104.00	−0.603345	−0.301672	−	0.953412i	$-0.597545\pi$
$359$	−4104.00	−0.603345	−0.301672	+	0.953412i	$0.597545\pi$
$360$	0	0
$361$	−4923.00	−0.717743
$362$	0	0
$363$	9076.00i	1.31230i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 7496.00i	− 1.06618i	−0.846059	−	0.533090i	$-0.821031\pi$
$367$	− 7496.00i	− 1.06618i	0.846059	−	0.533090i	$-0.178969\pi$
$368$	0	0
$369$	2046.00	0.288646
$370$	0	0
$371$	−7968.00	−1.11503
$372$	0	0
$373$	5842.00i	0.810958i	0.914104	+	0.405479i	$0.132895\pi$
$373$	5842.00i	0.810958i	−0.914104	+	0.405479i	$0.867105\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 15996.0i	− 2.18524i
$378$	0	0
$379$	−412.000	−0.0558391	−0.0279195	−	0.999610i	$-0.508888\pi$
$379$	−412.000	−0.0558391	−0.0279195	+	0.999610i	$0.508888\pi$
$380$	0	0
$381$	1376.00	0.185025
$382$	0	0
$383$	2568.00i	0.342607i	0.985218	+	0.171304i	$0.0547979\pi$
$383$	2568.00i	0.342607i	−0.985218	+	0.171304i	$0.945202\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1100.00i	− 0.144486i
$388$	0	0
$389$	−13086.0	−1.70562	−0.852810	−	0.522221i	$-0.825104\pi$
$389$	−13086.0	−1.70562	−0.852810	+	0.522221i	$0.825104\pi$
$390$	0	0
$391$	−864.000	−0.111750
$392$	0	0
$393$	− 3120.00i	− 0.400466i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10454.0i	1.32159i	0.750566	+	0.660795i	$0.229781\pi$
$397$	10454.0i	1.32159i	−0.750566	+	0.660795i	$0.770219\pi$
$398$	0	0
$399$	−2816.00	−0.353324
$400$	0	0
$401$	−10830.0	−1.34869	−0.674345	−	0.738417i	$-0.735574\pi$
$401$	−10830.0	−1.34869	−0.674345	+	0.738417i	$0.735574\pi$
$402$	0	0
$403$	15136.0i	1.87091i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	15240.0i	1.85607i
$408$	0	0
$409$	8566.00	1.03560	0.517801	−	0.855501i	$-0.326751\pi$
$409$	8566.00	1.03560	0.517801	+	0.855501i	$0.326751\pi$
$410$	0	0
$411$	−2664.00	−0.319721
$412$	0	0
$413$	− 4032.00i	− 0.480392i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	3536.00i	0.415249i
$418$	0	0
$419$	13884.0	1.61880	0.809401	−	0.587257i	$-0.199792\pi$
$419$	13884.0	1.61880	0.809401	+	0.587257i	$0.199792\pi$
$420$	0	0
$421$	4286.00	0.496168	0.248084	−	0.968738i	$-0.420199\pi$
$421$	4286.00	0.496168	0.248084	+	0.968738i	$0.420199\pi$
$422$	0	0
$423$	− 1848.00i	− 0.212418i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 928.000i	− 0.105173i
$428$	0	0
$429$	20640.0	2.32286
$430$	0	0
$431$	−6336.00	−0.708108	−0.354054	−	0.935225i	$-0.615197\pi$
$431$	−6336.00	−0.708108	−0.354054	+	0.935225i	$0.615197\pi$
$432$	0	0
$433$	8974.00i	0.995988i	0.867180	+	0.497994i	$0.165930\pi$
$433$	8974.00i	0.995988i	−0.867180	+	0.497994i	$0.834070\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2112.00i	0.231191i
$438$	0	0
$439$	−2968.00	−0.322676	−0.161338	−	0.986899i	$-0.551581\pi$
$439$	−2968.00	−0.322676	−0.161338	+	0.986899i	$0.551581\pi$
$440$	0	0
$441$	957.000	0.103337
$442$	0	0
$443$	− 12372.0i	− 1.32689i	−0.748226	−	0.663444i	$-0.769094\pi$
$443$	− 12372.0i	− 1.32689i	0.748226	−	0.663444i	$-0.230906\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	456.000i	0.0482507i
$448$	0	0
$449$	−11394.0	−1.19759	−0.598793	−	0.800904i	$-0.704353\pi$
$449$	−11394.0	−1.19759	−0.598793	+	0.800904i	$0.704353\pi$
$450$	0	0
$451$	11160.0	1.16520
$452$	0	0
$453$	160.000i	0.0165948i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 358.000i	− 0.0366445i	−0.999832	−	0.0183222i	$-0.994168\pi$
$457$	− 358.000i	− 0.0366445i	0.999832	−	0.0183222i	$-0.00583248\pi$
$458$	0	0
$459$	−2736.00	−0.278226
$460$	0	0
$461$	−7530.00	−0.760753	−0.380376	−	0.924832i	$-0.624206\pi$
$461$	−7530.00	−0.760753	−0.380376	+	0.924832i	$0.624206\pi$
$462$	0	0
$463$	− 13768.0i	− 1.38197i	−0.722868	−	0.690986i	$-0.757177\pi$
$463$	− 13768.0i	− 1.38197i	0.722868	−	0.690986i	$-0.242823\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 13380.0i	− 1.32581i	−0.748704	−	0.662904i	$-0.769324\pi$
$467$	− 13380.0i	− 1.32581i	0.748704	−	0.662904i	$-0.230676\pi$
$468$	0	0
$469$	−16576.0	−1.63200
$470$	0	0
$471$	616.000	0.0602628
$472$	0	0
$473$	− 6000.00i	− 0.583256i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5478.00i	0.525829i
$478$	0	0
$479$	−6336.00	−0.604383	−0.302191	−	0.953247i	$-0.597718\pi$
$479$	−6336.00	−0.604383	−0.302191	+	0.953247i	$0.597718\pi$
$480$	0	0
$481$	21844.0	2.07069
$482$	0	0
$483$	− 3072.00i	− 0.289401i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5008.00i	0.465984i	0.972479	+	0.232992i	$0.0748516\pi$
$487$	5008.00i	0.465984i	−0.972479	+	0.232992i	$0.925148\pi$
$488$	0	0
$489$	−8720.00	−0.806405
$490$	0	0
$491$	−12900.0	−1.18568	−0.592840	−	0.805320i	$-0.701993\pi$
$491$	−12900.0	−1.18568	−0.592840	+	0.805320i	$0.701993\pi$
$492$	0	0
$493$	3348.00i	0.305855i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2688.00i	− 0.242602i
$498$	0	0
$499$	−8116.00	−0.728100	−0.364050	−	0.931379i	$-0.618606\pi$
$499$	−8116.00	−0.728100	−0.364050	+	0.931379i	$0.618606\pi$
$500$	0	0
$501$	14784.0	1.31836
$502$	0	0
$503$	− 4944.00i	− 0.438255i	−0.975696	−	0.219127i	$-0.929679\pi$
$503$	− 4944.00i	− 0.438255i	0.975696	−	0.219127i	$-0.0703210\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 20796.0i	− 1.82166i
$508$	0	0
$509$	5466.00	0.475985	0.237992	−	0.971267i	$-0.423511\pi$
$509$	5466.00	0.475985	0.237992	+	0.971267i	$0.423511\pi$
$510$	0	0
$511$	8096.00	0.700873
$512$	0	0
$513$	6688.00i	0.575599i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 10080.0i	− 0.857481i
$518$	0	0
$519$	5208.00	0.440474
$520$	0	0
$521$	10074.0	0.847121	0.423560	−	0.905868i	$-0.360780\pi$
$521$	10074.0	0.847121	0.423560	+	0.905868i	$0.360780\pi$
$522$	0	0
$523$	− 13828.0i	− 1.15613i	−0.815991	−	0.578065i	$-0.803808\pi$
$523$	− 13828.0i	− 1.15613i	0.815991	−	0.578065i	$-0.196192\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 3168.00i	− 0.261860i
$528$	0	0
$529$	9863.00	0.810635
$530$	0	0
$531$	−2772.00	−0.226543
$532$	0	0
$533$	− 15996.0i	− 1.29993i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 17232.0i	− 1.38476i
$538$	0	0
$539$	5220.00	0.417145
$540$	0	0
$541$	−15226.0	−1.21001	−0.605006	−	0.796221i	$-0.706829\pi$
$541$	−15226.0	−1.21001	−0.605006	+	0.796221i	$0.706829\pi$
$542$	0	0
$543$	6200.00i	0.489995i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	13228.0i	1.03398i	0.855991	+	0.516991i	$0.172948\pi$
$547$	13228.0i	1.03398i	−0.855991	+	0.516991i	$0.827052\pi$
$548$	0	0
$549$	−638.000	−0.0495978
$550$	0	0
$551$	8184.00	0.632759
$552$	0	0
$553$	4352.00i	0.334658i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 8490.00i	− 0.645840i	−0.946426	−	0.322920i	$-0.895336\pi$
$557$	− 8490.00i	− 0.645840i	0.946426	−	0.322920i	$-0.104664\pi$
$558$	0	0
$559$	−8600.00	−0.650700
$560$	0	0
$561$	−4320.00	−0.325117
$562$	0	0
$563$	− 10284.0i	− 0.769838i	−0.922950	−	0.384919i	$-0.874229\pi$
$563$	− 10284.0i	− 0.769838i	0.922950	−	0.384919i	$-0.125771\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 4976.00i	− 0.368558i
$568$	0	0
$569$	−1770.00	−0.130408	−0.0652041	−	0.997872i	$-0.520770\pi$
$569$	−1770.00	−0.130408	−0.0652041	+	0.997872i	$0.520770\pi$
$570$	0	0
$571$	−6068.00	−0.444725	−0.222362	−	0.974964i	$-0.571377\pi$
$571$	−6068.00	−0.444725	−0.222362	+	0.974964i	$0.571377\pi$
$572$	0	0
$573$	− 192.000i	− 0.0139981i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	21506.0i	1.55166i	0.630943	+	0.775829i	$0.282668\pi$
$577$	21506.0i	1.55166i	−0.630943	+	0.775829i	$0.717332\pi$
$578$	0	0
$579$	4232.00	0.303758
$580$	0	0
$581$	−15168.0	−1.08309
$582$	0	0
$583$	29880.0i	2.12265i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 12108.0i	− 0.851364i	−0.904873	−	0.425682i	$-0.860034\pi$
$587$	− 12108.0i	− 0.851364i	0.904873	−	0.425682i	$-0.139966\pi$
$588$	0	0
$589$	−7744.00	−0.541742
$590$	0	0
$591$	14856.0	1.03400
$592$	0	0
$593$	− 15474.0i	− 1.07157i	−0.844354	−	0.535785i	$-0.820016\pi$
$593$	− 15474.0i	− 1.07157i	0.844354	−	0.535785i	$-0.179984\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 7072.00i	− 0.484820i
$598$	0	0
$599$	−2520.00	−0.171894	−0.0859469	−	0.996300i	$-0.527392\pi$
$599$	−2520.00	−0.171894	−0.0859469	+	0.996300i	$0.527392\pi$
$600$	0	0
$601$	−12790.0	−0.868078	−0.434039	−	0.900894i	$-0.642912\pi$
$601$	−12790.0	−0.868078	−0.434039	+	0.900894i	$0.642912\pi$
$602$	0	0
$603$	11396.0i	0.769620i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 11576.0i	− 0.774062i	−0.922067	−	0.387031i	$-0.873501\pi$
$607$	− 11576.0i	− 0.774062i	0.922067	−	0.387031i	$-0.126499\pi$
$608$	0	0
$609$	−11904.0	−0.792076
$610$	0	0
$611$	−14448.0	−0.956634
$612$	0	0
$613$	− 20126.0i	− 1.32607i	−0.748588	−	0.663035i	$-0.769268\pi$
$613$	− 20126.0i	− 1.32607i	0.748588	−	0.663035i	$-0.230732\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 27942.0i	− 1.82318i	−0.411100	−	0.911590i	$-0.634855\pi$
$617$	− 27942.0i	− 1.82318i	0.411100	−	0.911590i	$-0.365145\pi$
$618$	0	0
$619$	−22540.0	−1.46358	−0.731792	−	0.681528i	$-0.761316\pi$
$619$	−22540.0	−1.46358	−0.731792	+	0.681528i	$0.761316\pi$
$620$	0	0
$621$	−7296.00	−0.471463
$622$	0	0
$623$	16224.0i	1.04334i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	10560.0i	0.672609i
$628$	0	0
$629$	−4572.00	−0.289821
$630$	0	0
$631$	5128.00	0.323522	0.161761	−	0.986830i	$-0.448283\pi$
$631$	5128.00	0.323522	0.161761	+	0.986830i	$0.448283\pi$
$632$	0	0
$633$	16144.0i	1.01369i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 7482.00i	− 0.465381i
$638$	0	0
$639$	−1848.00	−0.114406
$640$	0	0
$641$	−12798.0	−0.788597	−0.394298	−	0.918982i	$-0.629012\pi$
$641$	−12798.0	−0.788597	−0.394298	+	0.918982i	$0.629012\pi$
$642$	0	0
$643$	− 21148.0i	− 1.29704i	−0.761198	−	0.648519i	$-0.775389\pi$
$643$	− 21148.0i	− 1.29704i	0.761198	−	0.648519i	$-0.224611\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 16464.0i	− 1.00041i	−0.865906	−	0.500206i	$-0.833258\pi$
$647$	− 16464.0i	− 1.00041i	0.865906	−	0.500206i	$-0.166742\pi$
$648$	0	0
$649$	−15120.0	−0.914502
$650$	0	0
$651$	11264.0	0.678143
$652$	0	0
$653$	24234.0i	1.45230i	0.687538	+	0.726148i	$0.258691\pi$
$653$	24234.0i	1.45230i	−0.687538	+	0.726148i	$0.741309\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 5566.00i	− 0.330518i
$658$	0	0
$659$	−22836.0	−1.34987	−0.674935	−	0.737877i	$-0.735828\pi$
$659$	−22836.0	−1.34987	−0.674935	+	0.737877i	$0.735828\pi$
$660$	0	0
$661$	26318.0	1.54864	0.774320	−	0.632794i	$-0.218092\pi$
$661$	26318.0	1.54864	0.774320	+	0.632794i	$0.218092\pi$
$662$	0	0
$663$	6192.00i	0.362711i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8928.00i	0.518281i
$668$	0	0
$669$	−2720.00	−0.157192
$670$	0	0
$671$	−3480.00	−0.200214
$672$	0	0
$673$	− 28802.0i	− 1.64968i	−0.565365	−	0.824841i	$-0.691265\pi$
$673$	− 28802.0i	− 1.64968i	0.565365	−	0.824841i	$-0.308735\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2526.00i	0.143400i	0.997426	+	0.0717002i	$0.0228425\pi$
$677$	2526.00i	0.143400i	−0.997426	+	0.0717002i	$0.977158\pi$
$678$	0	0
$679$	12256.0	0.692698
$680$	0	0
$681$	9552.00	0.537494
$682$	0	0
$683$	− 23076.0i	− 1.29279i	−0.763001	−	0.646397i	$-0.776275\pi$
$683$	− 23076.0i	− 1.29279i	0.763001	−	0.646397i	$-0.223725\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	15496.0i	0.860567i
$688$	0	0
$689$	42828.0	2.36809
$690$	0	0
$691$	−7868.00	−0.433159	−0.216579	−	0.976265i	$-0.569490\pi$
$691$	−7868.00	−0.433159	−0.216579	+	0.976265i	$0.569490\pi$
$692$	0	0
$693$	10560.0i	0.578847i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3348.00i	0.181943i
$698$	0	0
$699$	12648.0	0.684394
$700$	0	0
$701$	21510.0	1.15895	0.579473	−	0.814991i	$-0.303258\pi$
$701$	21510.0	1.15895	0.579473	+	0.814991i	$0.303258\pi$
$702$	0	0
$703$	11176.0i	0.599589i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 21024.0i	− 1.11837i
$708$	0	0
$709$	−30014.0	−1.58984	−0.794922	−	0.606712i	$-0.792488\pi$
$709$	−30014.0	−1.58984	−0.794922	+	0.606712i	$0.792488\pi$
$710$	0	0
$711$	2992.00	0.157818
$712$	0	0
$713$	− 8448.00i	− 0.443731i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	21696.0i	1.13006i
$718$	0	0
$719$	−816.000	−0.0423250	−0.0211625	−	0.999776i	$-0.506737\pi$
$719$	−816.000	−0.0423250	−0.0211625	+	0.999776i	$0.506737\pi$
$720$	0	0
$721$	7168.00	0.370250
$722$	0	0
$723$	− 15544.0i	− 0.799568i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	9952.00i	0.507702i	0.967243	+	0.253851i	$0.0816973\pi$
$727$	9952.00i	0.507702i	−0.967243	+	0.253851i	$0.918303\pi$
$728$	0	0
$729$	−19837.0	−1.00782
$730$	0	0
$731$	1800.00	0.0910744
$732$	0	0
$733$	33946.0i	1.71054i	0.518185	+	0.855269i	$0.326608\pi$
$733$	33946.0i	1.71054i	−0.518185	+	0.855269i	$0.673392\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	62160.0i	3.10677i
$738$	0	0
$739$	23420.0	1.16579	0.582895	−	0.812548i	$-0.301920\pi$
$739$	23420.0	1.16579	0.582895	+	0.812548i	$0.301920\pi$
$740$	0	0
$741$	15136.0	0.750384
$742$	0	0
$743$	− 14592.0i	− 0.720496i	−0.932857	−	0.360248i	$-0.882692\pi$
$743$	− 14592.0i	− 0.720496i	0.932857	−	0.360248i	$-0.117308\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	10428.0i	0.510764i
$748$	0	0
$749$	24768.0	1.20828
$750$	0	0
$751$	−9056.00	−0.440024	−0.220012	−	0.975497i	$-0.570610\pi$
$751$	−9056.00	−0.440024	−0.220012	+	0.975497i	$0.570610\pi$
$752$	0	0
$753$	20400.0i	0.987274i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 17554.0i	− 0.842815i	−0.906871	−	0.421408i	$-0.861536\pi$
$757$	− 17554.0i	− 0.842815i	0.906871	−	0.421408i	$-0.138464\pi$
$758$	0	0
$759$	−11520.0	−0.550922
$760$	0	0
$761$	−36438.0	−1.73571	−0.867856	−	0.496816i	$-0.834502\pi$
$761$	−36438.0	−1.73571	−0.867856	+	0.496816i	$0.834502\pi$
$762$	0	0
$763$	− 4448.00i	− 0.211046i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	21672.0i	1.02025i
$768$	0	0
$769$	9022.00	0.423071	0.211536	−	0.977370i	$-0.432154\pi$
$769$	9022.00	0.423071	0.211536	+	0.977370i	$0.432154\pi$
$770$	0	0
$771$	−8712.00	−0.406946
$772$	0	0
$773$	− 1470.00i	− 0.0683987i	−0.999415	−	0.0341994i	$-0.989112\pi$
$773$	− 1470.00i	− 0.0683987i	0.999415	−	0.0341994i	$-0.0108881\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 16256.0i	− 0.750554i
$778$	0	0
$779$	8184.00	0.376409
$780$	0	0
$781$	−10080.0	−0.461832
$782$	0	0
$783$	28272.0i	1.29037i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 5252.00i	− 0.237883i	−0.992901	−	0.118941i	$-0.962050\pi$
$787$	− 5252.00i	− 0.237883i	0.992901	−	0.118941i	$-0.0379500\pi$
$788$	0	0
$789$	24576.0	1.10891
$790$	0	0
$791$	−8928.00	−0.401319
$792$	0	0
$793$	4988.00i	0.223366i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12294.0i	0.546394i	0.961958	+	0.273197i	$0.0880810\pi$
$797$	12294.0i	0.546394i	−0.961958	+	0.273197i	$0.911919\pi$
$798$	0	0
$799$	3024.00	0.133894
$800$	0	0
$801$	11154.0	0.492019
$802$	0	0
$803$	− 30360.0i	− 1.33422i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 3288.00i	− 0.143424i
$808$	0	0
$809$	−15546.0	−0.675610	−0.337805	−	0.941216i	$-0.609684\pi$
$809$	−15546.0	−0.675610	−0.337805	+	0.941216i	$0.609684\pi$
$810$	0	0
$811$	−19364.0	−0.838424	−0.419212	−	0.907888i	$-0.637694\pi$
$811$	−19364.0	−0.838424	−0.419212	+	0.907888i	$0.637694\pi$
$812$	0	0
$813$	− 33920.0i	− 1.46326i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 4400.00i	− 0.188417i
$818$	0	0
$819$	15136.0	0.645781
$820$	0	0
$821$	−7314.00	−0.310914	−0.155457	−	0.987843i	$-0.549685\pi$
$821$	−7314.00	−0.310914	−0.155457	+	0.987843i	$0.549685\pi$
$822$	0	0
$823$	11984.0i	0.507577i	0.967260	+	0.253789i	$0.0816767\pi$
$823$	11984.0i	0.507577i	−0.967260	+	0.253789i	$0.918323\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 13500.0i	− 0.567643i	−0.958877	−	0.283822i	$-0.908398\pi$
$827$	− 13500.0i	− 0.567643i	0.958877	−	0.283822i	$-0.0916024\pi$
$828$	0	0
$829$	44602.0	1.86863	0.934313	−	0.356453i	$-0.116014\pi$
$829$	44602.0	1.86863	0.934313	+	0.356453i	$0.116014\pi$
$830$	0	0
$831$	4552.00	0.190021
$832$	0	0
$833$	1566.00i	0.0651365i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 26752.0i	− 1.10476i
$838$	0	0
$839$	35448.0	1.45864	0.729321	−	0.684172i	$-0.239836\pi$
$839$	35448.0	1.45864	0.729321	+	0.684172i	$0.239836\pi$
$840$	0	0
$841$	10207.0	0.418508
$842$	0	0
$843$	22824.0i	0.932503i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	36304.0i	1.47275i
$848$	0	0
$849$	12112.0	0.489615
$850$	0	0
$851$	−12192.0	−0.491112
$852$	0	0
$853$	− 12590.0i	− 0.505362i	−0.967550	−	0.252681i	$-0.918688\pi$
$853$	− 12590.0i	− 0.505362i	0.967550	−	0.252681i	$-0.0813122\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	24906.0i	0.992734i	0.868113	+	0.496367i	$0.165333\pi$
$857$	24906.0i	0.992734i	−0.868113	+	0.496367i	$0.834667\pi$
$858$	0	0
$859$	23204.0	0.921665	0.460833	−	0.887487i	$-0.347551\pi$
$859$	23204.0	0.921665	0.460833	+	0.887487i	$0.347551\pi$
$860$	0	0
$861$	−11904.0	−0.471181
$862$	0	0
$863$	19848.0i	0.782890i	0.920202	+	0.391445i	$0.128025\pi$
$863$	19848.0i	0.782890i	−0.920202	+	0.391445i	$0.871975\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	18356.0i	0.719034i
$868$	0	0
$869$	16320.0	0.637075
$870$	0	0
$871$	89096.0	3.46602
$872$	0	0
$873$	− 8426.00i	− 0.326663i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	27542.0i	1.06046i	0.847852	+	0.530232i	$0.177895\pi$
$877$	27542.0i	1.06046i	−0.847852	+	0.530232i	$0.822105\pi$
$878$	0	0
$879$	13560.0	0.520327
$880$	0	0
$881$	−20718.0	−0.792290	−0.396145	−	0.918188i	$-0.629652\pi$
$881$	−20718.0	−0.792290	−0.396145	+	0.918188i	$0.629652\pi$
$882$	0	0
$883$	25172.0i	0.959349i	0.877446	+	0.479675i	$0.159245\pi$
$883$	25172.0i	0.959349i	−0.877446	+	0.479675i	$0.840755\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	12864.0i	0.486957i	0.969906	+	0.243478i	$0.0782885\pi$
$887$	12864.0i	0.486957i	−0.969906	+	0.243478i	$0.921711\pi$
$888$	0	0
$889$	5504.00	0.207647
$890$	0	0
$891$	−18660.0	−0.701609
$892$	0	0
$893$	− 7392.00i	− 0.277003i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	16512.0i	0.614626i
$898$	0	0
$899$	−32736.0	−1.21447
$900$	0	0
$901$	−8964.00	−0.331447
$902$	0	0
$903$	6400.00i	0.235857i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	23092.0i	0.845377i	0.906275	+	0.422689i	$0.138914\pi$
$907$	23092.0i	0.845377i	−0.906275	+	0.422689i	$0.861086\pi$
$908$	0	0
$909$	−14454.0	−0.527403
$910$	0	0
$911$	14208.0	0.516720	0.258360	−	0.966049i	$-0.416818\pi$
$911$	14208.0	0.516720	0.258360	+	0.966049i	$0.416818\pi$
$912$	0	0
$913$	56880.0i	2.06183i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 12480.0i	− 0.449428i
$918$	0	0
$919$	−26584.0	−0.954217	−0.477108	−	0.878844i	$-0.658315\pi$
$919$	−26584.0	−0.954217	−0.477108	+	0.878844i	$0.658315\pi$
$920$	0	0
$921$	−16624.0	−0.594766
$922$	0	0
$923$	14448.0i	0.515235i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 4928.00i	− 0.174603i
$928$	0	0
$929$	−162.000	−0.00572126	−0.00286063	−	0.999996i	$-0.500911\pi$
$929$	−162.000	−0.00572126	−0.00286063	+	0.999996i	$0.500911\pi$
$930$	0	0
$931$	3828.00	0.134756
$932$	0	0
$933$	− 26208.0i	− 0.919626i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 29734.0i	− 1.03668i	−0.855175	−	0.518339i	$-0.826551\pi$
$937$	− 29734.0i	− 1.03668i	0.855175	−	0.518339i	$-0.173449\pi$
$938$	0	0
$939$	−5464.00	−0.189894
$940$	0	0
$941$	17142.0	0.593850	0.296925	−	0.954901i	$-0.404039\pi$
$941$	17142.0	0.593850	0.296925	+	0.954901i	$0.404039\pi$
$942$	0	0
$943$	8928.00i	0.308309i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 26436.0i	− 0.907133i	−0.891223	−	0.453566i	$-0.850152\pi$
$947$	− 26436.0i	− 0.907133i	0.891223	−	0.453566i	$-0.149848\pi$
$948$	0	0
$949$	−43516.0	−1.48850
$950$	0	0
$951$	−10392.0	−0.354347
$952$	0	0
$953$	− 27882.0i	− 0.947730i	−0.880598	−	0.473865i	$-0.842858\pi$
$953$	− 27882.0i	− 0.947730i	0.880598	−	0.473865i	$-0.157142\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	44640.0i	1.50784i
$958$	0	0
$959$	−10656.0	−0.358811
$960$	0	0
$961$	1185.00	0.0397771
$962$	0	0
$963$	− 17028.0i	− 0.569802i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 12656.0i	− 0.420879i	−0.977607	−	0.210439i	$-0.932511\pi$
$967$	− 12656.0i	− 0.420879i	0.977607	−	0.210439i	$-0.0674894\pi$
$968$	0	0
$969$	−3168.00	−0.105027
$970$	0	0
$971$	−2916.00	−0.0963737	−0.0481869	−	0.998838i	$-0.515344\pi$
$971$	−2916.00	−0.0963737	−0.0481869	+	0.998838i	$0.515344\pi$
$972$	0	0
$973$	14144.0i	0.466018i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 6894.00i	− 0.225751i	−0.993609	−	0.112875i	$-0.963994\pi$
$977$	− 6894.00i	− 0.225751i	0.993609	−	0.112875i	$-0.0360061\pi$
$978$	0	0
$979$	60840.0	1.98616
$980$	0	0
$981$	−3058.00	−0.0995254
$982$	0	0
$983$	− 45264.0i	− 1.46866i	−0.678790	−	0.734332i	$-0.737495\pi$
$983$	− 45264.0i	− 1.46866i	0.678790	−	0.734332i	$-0.262505\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	10752.0i	0.346748i
$988$	0	0
$989$	4800.00	0.154329
$990$	0	0
$991$	−52016.0	−1.66735	−0.833674	−	0.552256i	$-0.813767\pi$
$991$	−52016.0	−1.66735	−0.833674	+	0.552256i	$0.813767\pi$
$992$	0	0
$993$	13168.0i	0.420820i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 13858.0i	− 0.440208i	−0.975476	−	0.220104i	$-0.929360\pi$
$997$	− 13858.0i	− 0.440208i	0.975476	−	0.220104i	$-0.0706397\pi$
$998$	0	0
$999$	−38608.0	−1.22273

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.4.c.j.49.2		2
4.3	odd	2		100.4.c.a.49.1		2
5.2	odd	4		400.4.a.o.1.1		1
5.3	odd	4		80.4.a.c.1.1		1
5.4	even	2	inner	400.4.c.j.49.1		2
12.11	even	2		900.4.d.k.649.1		2
15.8	even	4		720.4.a.k.1.1		1
20.3	even	4		20.4.a.a.1.1	✓	1
20.7	even	4		100.4.a.a.1.1		1
20.19	odd	2		100.4.c.a.49.2		2
40.3	even	4		320.4.a.d.1.1		1
40.13	odd	4		320.4.a.k.1.1		1
40.27	even	4		1600.4.a.bl.1.1		1
40.37	odd	4		1600.4.a.p.1.1		1
60.23	odd	4		180.4.a.a.1.1		1
60.47	odd	4		900.4.a.m.1.1		1
60.59	even	2		900.4.d.k.649.2		2
80.3	even	4		1280.4.d.n.641.2		2
80.13	odd	4		1280.4.d.c.641.1		2
80.43	even	4		1280.4.d.n.641.1		2
80.53	odd	4		1280.4.d.c.641.2		2
140.3	odd	12		980.4.i.n.961.1		2
140.23	even	12		980.4.i.e.361.1		2
140.83	odd	4		980.4.a.c.1.1		1
140.103	odd	12		980.4.i.n.361.1		2
140.123	even	12		980.4.i.e.961.1		2
180.23	odd	12		1620.4.i.j.541.1		2
180.43	even	12		1620.4.i.d.1081.1		2
180.83	odd	12		1620.4.i.j.1081.1		2
180.103	even	12		1620.4.i.d.541.1		2
220.43	odd	4		2420.4.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.4.a.a.1.1	✓	1	20.3	even	4
80.4.a.c.1.1		1	5.3	odd	4
100.4.a.a.1.1		1	20.7	even	4
100.4.c.a.49.1		2	4.3	odd	2
100.4.c.a.49.2		2	20.19	odd	2
180.4.a.a.1.1		1	60.23	odd	4
320.4.a.d.1.1		1	40.3	even	4
320.4.a.k.1.1		1	40.13	odd	4
400.4.a.o.1.1		1	5.2	odd	4
400.4.c.j.49.1		2	5.4	even	2	inner
400.4.c.j.49.2		2	1.1	even	1	trivial
720.4.a.k.1.1		1	15.8	even	4
900.4.a.m.1.1		1	60.47	odd	4
900.4.d.k.649.1		2	12.11	even	2
900.4.d.k.649.2		2	60.59	even	2
980.4.a.c.1.1		1	140.83	odd	4
980.4.i.e.361.1		2	140.23	even	12
980.4.i.e.961.1		2	140.123	even	12
980.4.i.n.361.1		2	140.103	odd	12
980.4.i.n.961.1		2	140.3	odd	12
1280.4.d.c.641.1		2	80.13	odd	4
1280.4.d.c.641.2		2	80.53	odd	4
1280.4.d.n.641.1		2	80.43	even	4
1280.4.d.n.641.2		2	80.3	even	4
1600.4.a.p.1.1		1	40.37	odd	4
1600.4.a.bl.1.1		1	40.27	even	4
1620.4.i.d.541.1		2	180.103	even	12
1620.4.i.d.1081.1		2	180.43	even	12
1620.4.i.j.541.1		2	180.23	odd	12
1620.4.i.j.1081.1		2	180.83	odd	12
2420.4.a.d.1.1		1	220.43	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{3} +16.0000i q^{7} +11.0000 q^{9} +O(q^{10})\)	\(q+4.00000i q^{3} +16.0000i q^{7} +11.0000 q^{9} +60.0000 q^{11} -86.0000i q^{13} +18.0000i q^{17} +44.0000 q^{19} -64.0000 q^{21} +48.0000i q^{23} +152.000i q^{27} +186.000 q^{29} -176.000 q^{31} +240.000i q^{33} +254.000i q^{37} +344.000 q^{39} +186.000 q^{41} -100.000i q^{43} -168.000i q^{47} +87.0000 q^{49} -72.0000 q^{51} +498.000i q^{53} +176.000i q^{57} -252.000 q^{59} -58.0000 q^{61} +176.000i q^{63} +1036.00i q^{67} -192.000 q^{69} -168.000 q^{71} -506.000i q^{73} +960.000i q^{77} +272.000 q^{79} -311.000 q^{81} +948.000i q^{83} +744.000i q^{87} +1014.00 q^{89} +1376.00 q^{91} -704.000i q^{93} -766.000i q^{97} +660.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 22 q^{9}+O(q^{10}) \) 2 * q + 22 * q^9	\( 2 q + 22 q^{9} + 120 q^{11} + 88 q^{19} - 128 q^{21} + 372 q^{29} - 352 q^{31} + 688 q^{39} + 372 q^{41} + 174 q^{49} - 144 q^{51} - 504 q^{59} - 116 q^{61} - 384 q^{69} - 336 q^{71} + 544 q^{79} - 622 q^{81} + 2028 q^{89} + 2752 q^{91} + 1320 q^{99}+O(q^{100}) \) 2 * q + 22 * q^9 + 120 * q^11 + 88 * q^19 - 128 * q^21 + 372 * q^29 - 352 * q^31 + 688 * q^39 + 372 * q^41 + 174 * q^49 - 144 * q^51 - 504 * q^59 - 116 * q^61 - 384 * q^69 - 336 * q^71 + 544 * q^79 - 622 * q^81 + 2028 * q^89 + 2752 * q^91 + 1320 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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