LMFDB - Embedded newform 1200.4.f.v.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(49,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1200.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1200.f (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:	$70.8022920069$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 25 $ x^4 - 9*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$-2.17945 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1200.49
Dual form			1200.4.f.v.49.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +30.4356i q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} +30.4356i q^{7} -9.00000 q^{9} -31.4356 q^{11} -60.7424i q^{13} +121.178i q^{17} -14.4356 q^{19} +91.3068 q^{21} -13.6932i q^{23} +27.0000i q^{27} +76.0492 q^{29} -183.049 q^{31} +94.3068i q^{33} +37.3864i q^{37} -182.227 q^{39} -30.6627 q^{41} -327.564i q^{43} -449.485i q^{47} -583.325 q^{49} +363.534 q^{51} -301.951i q^{53} +43.3068i q^{57} +340.970 q^{59} +619.098 q^{61} -273.920i q^{63} +256.890i q^{67} -41.0796 q^{69} -499.178 q^{71} -19.1288i q^{73} -956.761i q^{77} +257.424 q^{79} +81.0000 q^{81} -914.909i q^{83} -228.148i q^{87} +1059.68 q^{89} +1848.73 q^{91} +549.148i q^{93} -521.000i q^{97} +282.920 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{9}+O(q^{10}) $ 4 * q - 36 * q^9	$ 4 q - 36 q^{9} - 56 q^{11} + 12 q^{19} + 156 q^{21} - 184 q^{29} - 244 q^{31} + 108 q^{39} + 784 q^{41} - 520 q^{49} + 408 q^{51} + 248 q^{59} + 1500 q^{61} - 792 q^{69} - 1648 q^{71} - 1760 q^{79} + 324 q^{81} + 1728 q^{89} + 4396 q^{91} + 504 q^{99}+O(q^{100}) $ 4 * q - 36 * q^9 - 56 * q^11 + 12 * q^19 + 156 * q^21 - 184 * q^29 - 244 * q^31 + 108 * q^39 + 784 * q^41 - 520 * q^49 + 408 * q^51 + 248 * q^59 + 1500 * q^61 - 792 * q^69 - 1648 * q^71 - 1760 * q^79 + 324 * q^81 + 1728 * q^89 + 4396 * q^91 + 504 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	30.4356i	1.64337i	0.569944	+	0.821684i	$0.306965\pi$
$7$	30.4356i	1.64337i	−0.569944	+	0.821684i	$0.693035\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−31.4356	−0.861654	−0.430827	−	0.902435i	$-0.641778\pi$
$11$	−31.4356	−0.861654	−0.430827	+	0.902435i	$0.641778\pi$
$12$	0	0
$13$	− 60.7424i	− 1.29592i	−0.761676	−	0.647958i	$-0.775623\pi$
$13$	− 60.7424i	− 1.29592i	0.761676	−	0.647958i	$-0.224377\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	121.178i	1.72882i	0.502786	+	0.864411i	$0.332308\pi$
$17$	121.178i	1.72882i	−0.502786	+	0.864411i	$0.667692\pi$
$18$	0	0
$19$	−14.4356	−0.174303	−0.0871514	−	0.996195i	$-0.527776\pi$
$19$	−14.4356	−0.174303	−0.0871514	+	0.996195i	$0.527776\pi$
$20$	0	0
$21$	91.3068	0.948799
$22$	0	0
$23$	− 13.6932i	− 0.124141i	−0.998072	−	0.0620703i	$-0.980230\pi$
$23$	− 13.6932i	− 0.124141i	0.998072	−	0.0620703i	$-0.0197703\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	76.0492	0.486965	0.243482	−	0.969905i	$-0.421710\pi$
$29$	76.0492	0.486965	0.243482	+	0.969905i	$0.421710\pi$
$30$	0	0
$31$	−183.049	−1.06054	−0.530268	−	0.847830i	$-0.677909\pi$
$31$	−183.049	−1.06054	−0.530268	+	0.847830i	$0.677909\pi$
$32$	0	0
$33$	94.3068i	0.497476i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	37.3864i	0.166116i	0.996545	+	0.0830580i	$0.0264687\pi$
$37$	37.3864i	0.166116i	−0.996545	+	0.0830580i	$0.973531\pi$
$38$	0	0
$39$	−182.227	−0.748197
$40$	0	0
$41$	−30.6627	−0.116798	−0.0583990	−	0.998293i	$-0.518600\pi$
$41$	−30.6627	−0.116798	−0.0583990	+	0.998293i	$0.518600\pi$
$42$	0	0
$43$	− 327.564i	− 1.16170i	−0.814011	−	0.580850i	$-0.802720\pi$
$43$	− 327.564i	− 1.16170i	0.814011	−	0.580850i	$-0.197280\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 449.485i	− 1.39498i	−0.716594	−	0.697490i	$-0.754300\pi$
$47$	− 449.485i	− 1.39498i	0.716594	−	0.697490i	$-0.245700\pi$
$48$	0	0
$49$	−583.325	−1.70066
$50$	0	0
$51$	363.534	0.998136
$52$	0	0
$53$	− 301.951i	− 0.782569i	−0.920270	−	0.391284i	$-0.872031\pi$
$53$	− 301.951i	− 0.782569i	0.920270	−	0.391284i	$-0.127969\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	43.3068i	0.100634i
$58$	0	0
$59$	340.970	0.752381	0.376190	−	0.926542i	$-0.377234\pi$
$59$	340.970	0.752381	0.376190	+	0.926542i	$0.377234\pi$
$60$	0	0
$61$	619.098	1.29947	0.649733	−	0.760163i	$-0.274881\pi$
$61$	619.098	1.29947	0.649733	+	0.760163i	$0.274881\pi$
$62$	0	0
$63$	− 273.920i	− 0.547789i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	256.890i	0.468419i	0.972186	+	0.234210i	$0.0752502\pi$
$67$	256.890i	0.468419i	−0.972186	+	0.234210i	$0.924750\pi$
$68$	0	0
$69$	−41.0796	−0.0716726
$70$	0	0
$71$	−499.178	−0.834388	−0.417194	−	0.908818i	$-0.636986\pi$
$71$	−499.178	−0.834388	−0.417194	+	0.908818i	$0.636986\pi$
$72$	0	0
$73$	− 19.1288i	− 0.0306693i	−0.999882	−	0.0153346i	$-0.995119\pi$
$73$	− 19.1288i	− 0.0306693i	0.999882	−	0.0153346i	$-0.00488136\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 956.761i	− 1.41601i
$78$	0	0
$79$	257.424	0.366613	0.183307	−	0.983056i	$-0.441320\pi$
$79$	257.424	0.366613	0.183307	+	0.983056i	$0.441320\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 914.909i	− 1.20993i	−0.796252	−	0.604965i	$-0.793187\pi$
$83$	− 914.909i	− 1.20993i	0.796252	−	0.604965i	$-0.206813\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 228.148i	− 0.281149i
$88$	0	0
$89$	1059.68	1.26209	0.631045	−	0.775746i	$-0.282626\pi$
$89$	1059.68	1.26209	0.631045	+	0.775746i	$0.282626\pi$
$90$	0	0
$91$	1848.73	2.12967
$92$	0	0
$93$	549.148i	0.612300i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 521.000i	− 0.545356i	−0.962105	−	0.272678i	$-0.912091\pi$
$97$	− 521.000i	− 0.545356i	0.962105	−	0.272678i	$-0.0879094\pi$
$98$	0	0
$99$	282.920	0.287218
$100$	0	0
$101$	347.080	0.341938	0.170969	−	0.985276i	$-0.445310\pi$
$101$	347.080	0.341938	0.170969	+	0.985276i	$0.445310\pi$
$102$	0	0
$103$	770.749i	0.737322i	0.929564	+	0.368661i	$0.120184\pi$
$103$	770.749i	0.737322i	−0.929564	+	0.368661i	$0.879816\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1415.37i	− 1.27878i	−0.768883	−	0.639390i	$-0.779187\pi$
$107$	− 1415.37i	− 1.27878i	0.768883	−	0.639390i	$-0.220813\pi$
$108$	0	0
$109$	−908.386	−0.798235	−0.399118	−	0.916900i	$-0.630683\pi$
$109$	−908.386	−0.798235	−0.399118	+	0.916900i	$0.630683\pi$
$110$	0	0
$111$	112.159	0.0959071
$112$	0	0
$113$	− 2049.94i	− 1.70657i	−0.521447	−	0.853283i	$-0.674608\pi$
$113$	− 2049.94i	− 1.70657i	0.521447	−	0.853283i	$-0.325392\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	546.681i	0.431972i
$118$	0	0
$119$	−3688.12	−2.84109
$120$	0	0
$121$	−342.803	−0.257553
$122$	0	0
$123$	91.9882i	0.0674333i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	281.644i	0.196786i	0.995148	+	0.0983932i	$0.0313703\pi$
$127$	281.644i	0.196786i	−0.995148	+	0.0983932i	$0.968630\pi$
$128$	0	0
$129$	−982.693	−0.670708
$130$	0	0
$131$	−243.056	−0.162106	−0.0810531	−	0.996710i	$-0.525828\pi$
$131$	−243.056	−0.162106	−0.0810531	+	0.996710i	$0.525828\pi$
$132$	0	0
$133$	− 439.356i	− 0.286444i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 909.386i	− 0.567110i	−0.958956	−	0.283555i	$-0.908486\pi$
$137$	− 909.386i	− 0.567110i	0.958956	−	0.283555i	$-0.0915139\pi$
$138$	0	0
$139$	−2049.52	−1.25063	−0.625317	−	0.780371i	$-0.715030\pi$
$139$	−2049.52	−1.25063	−0.625317	+	0.780371i	$0.715030\pi$
$140$	0	0
$141$	−1348.45	−0.805392
$142$	0	0
$143$	1909.47i	1.11663i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1749.98i	0.981875i
$148$	0	0
$149$	−3601.14	−1.97998	−0.989990	−	0.141136i	$-0.954925\pi$
$149$	−3601.14	−1.97998	−0.989990	+	0.141136i	$0.954925\pi$
$150$	0	0
$151$	−1383.38	−0.745550	−0.372775	−	0.927922i	$-0.621594\pi$
$151$	−1383.38	−0.745550	−0.372775	+	0.927922i	$0.621594\pi$
$152$	0	0
$153$	− 1090.60i	− 0.576274i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	131.749i	0.0669729i	0.999439	+	0.0334864i	$0.0106611\pi$
$157$	131.749i	0.0669729i	−0.999439	+	0.0334864i	$0.989339\pi$
$158$	0	0
$159$	−905.852	−0.451816
$160$	0	0
$161$	416.761	0.204009
$162$	0	0
$163$	− 2897.74i	− 1.39244i	−0.717827	−	0.696222i	$-0.754863\pi$
$163$	− 2897.74i	− 1.39244i	0.717827	−	0.696222i	$-0.245137\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	260.283i	0.120607i	0.998180	+	0.0603034i	$0.0192068\pi$
$167$	260.283i	0.120607i	−0.998180	+	0.0603034i	$0.980793\pi$
$168$	0	0
$169$	−1492.64	−0.679398
$170$	0	0
$171$	129.920	0.0581009
$172$	0	0
$173$	− 1935.83i	− 0.850742i	−0.905019	−	0.425371i	$-0.860144\pi$
$173$	− 1935.83i	− 0.850742i	0.905019	−	0.425371i	$-0.139856\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1022.91i	− 0.434387i
$178$	0	0
$179$	576.627	0.240777	0.120389	−	0.992727i	$-0.461586\pi$
$179$	576.627	0.240777	0.120389	+	0.992727i	$0.461586\pi$
$180$	0	0
$181$	−1962.04	−0.805733	−0.402866	−	0.915259i	$-0.631986\pi$
$181$	−1962.04	−0.805733	−0.402866	+	0.915259i	$0.631986\pi$
$182$	0	0
$183$	− 1857.30i	− 0.750247i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 3809.30i	− 1.48965i
$188$	0	0
$189$	−821.761	−0.316266
$190$	0	0
$191$	4318.75	1.63609	0.818047	−	0.575152i	$-0.195057\pi$
$191$	4318.75	1.63609	0.818047	+	0.575152i	$0.195057\pi$
$192$	0	0
$193$	2.97647i	0.00111011i	1.00000		0.000555054i	$0.000176679\pi$
$193$	2.97647i	0.00111011i	−1.00000		0.000555054i	$0.999823\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 569.705i	− 0.206040i	−0.994679	−	0.103020i	$-0.967149\pi$
$197$	− 569.705i	− 0.206040i	0.994679	−	0.103020i	$-0.0328505\pi$
$198$	0	0
$199$	−3050.73	−1.08674	−0.543368	−	0.839494i	$-0.682851\pi$
$199$	−3050.73	−1.08674	−0.543368	+	0.839494i	$0.682851\pi$
$200$	0	0
$201$	770.670	0.270442
$202$	0	0
$203$	2314.60i	0.800262i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	123.239i	0.0413802i
$208$	0	0
$209$	453.792	0.150189
$210$	0	0
$211$	50.5104	0.0164800	0.00824000	−	0.999966i	$-0.497377\pi$
$211$	50.5104	0.0164800	0.00824000	+	0.999966i	$0.497377\pi$
$212$	0	0
$213$	1497.53i	0.481734i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 5571.21i	− 1.74285i
$218$	0	0
$219$	−57.3864	−0.0177069
$220$	0	0
$221$	7360.64	2.24041
$222$	0	0
$223$	− 5453.55i	− 1.63765i	−0.574040	−	0.818827i	$-0.694625\pi$
$223$	− 5453.55i	− 1.63765i	0.574040	−	0.818827i	$-0.305375\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 4777.14i	− 1.39678i	−0.715715	−	0.698392i	$-0.753899\pi$
$227$	− 4777.14i	− 1.39678i	0.715715	−	0.698392i	$-0.246101\pi$
$228$	0	0
$229$	2085.51	0.601808	0.300904	−	0.953654i	$-0.402712\pi$
$229$	2085.51	0.601808	0.300904	+	0.953654i	$0.402712\pi$
$230$	0	0
$231$	−2870.28	−0.817536
$232$	0	0
$233$	6484.53i	1.82324i	0.411030	+	0.911622i	$0.365169\pi$
$233$	6484.53i	1.82324i	−0.411030	+	0.911622i	$0.634831\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 772.271i	− 0.211664i
$238$	0	0
$239$	2234.62	0.604792	0.302396	−	0.953182i	$-0.402214\pi$
$239$	2234.62	0.604792	0.302396	+	0.953182i	$0.402214\pi$
$240$	0	0
$241$	−2393.01	−0.639616	−0.319808	−	0.947482i	$-0.603618\pi$
$241$	−2393.01	−0.639616	−0.319808	+	0.947482i	$0.603618\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	876.852i	0.225882i
$248$	0	0
$249$	−2744.73	−0.698554
$250$	0	0
$251$	612.661	0.154067	0.0770335	−	0.997029i	$-0.475455\pi$
$251$	612.661	0.154067	0.0770335	+	0.997029i	$0.475455\pi$
$252$	0	0
$253$	430.454i	0.106966i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	306.112i	0.0742987i	0.999310	+	0.0371493i	$0.0118277\pi$
$257$	306.112i	0.0742987i	−0.999310	+	0.0371493i	$0.988172\pi$
$258$	0	0
$259$	−1137.88	−0.272990
$260$	0	0
$261$	−684.443	−0.162322
$262$	0	0
$263$	− 283.839i	− 0.0665484i	−0.999446	−	0.0332742i	$-0.989407\pi$
$263$	− 283.839i	− 0.0665484i	0.999446	−	0.0332742i	$-0.0105935\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 3179.04i	− 0.728668i
$268$	0	0
$269$	−2426.21	−0.549920	−0.274960	−	0.961456i	$-0.588665\pi$
$269$	−2426.21	−0.549920	−0.274960	+	0.961456i	$0.588665\pi$
$270$	0	0
$271$	174.946	0.0392148	0.0196074	−	0.999808i	$-0.493758\pi$
$271$	174.946	0.0392148	0.0196074	+	0.999808i	$0.493758\pi$
$272$	0	0
$273$	− 5546.19i	− 1.22956i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7807.07i	1.69344i	0.532042	+	0.846718i	$0.321425\pi$
$277$	7807.07i	1.69344i	−0.532042	+	0.846718i	$0.678575\pi$
$278$	0	0
$279$	1647.44	0.353512
$280$	0	0
$281$	584.171	0.124017	0.0620084	−	0.998076i	$-0.480249\pi$
$281$	584.171	0.124017	0.0620084	+	0.998076i	$0.480249\pi$
$282$	0	0
$283$	5897.31i	1.23872i	0.785106	+	0.619362i	$0.212609\pi$
$283$	5897.31i	1.23872i	−0.785106	+	0.619362i	$0.787391\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 933.239i	− 0.191942i
$288$	0	0
$289$	−9771.10	−1.98883
$290$	0	0
$291$	−1563.00	−0.314861
$292$	0	0
$293$	1609.73i	0.320960i	0.987039	+	0.160480i	$0.0513042\pi$
$293$	1609.73i	0.320960i	−0.987039	+	0.160480i	$0.948696\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 848.761i	− 0.165825i
$298$	0	0
$299$	−831.758	−0.160876
$300$	0	0
$301$	9969.62	1.90910
$302$	0	0
$303$	− 1041.24i	− 0.197418i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	234.473i	0.0435898i	0.999762	+	0.0217949i	$0.00693809\pi$
$307$	234.473i	0.0435898i	−0.999762	+	0.0217949i	$0.993062\pi$
$308$	0	0
$309$	2312.25	0.425693
$310$	0	0
$311$	1795.25	0.327329	0.163665	−	0.986516i	$-0.447669\pi$
$311$	1795.25	0.327329	0.163665	+	0.986516i	$0.447669\pi$
$312$	0	0
$313$	− 8440.61i	− 1.52425i	−0.647427	−	0.762127i	$-0.724155\pi$
$313$	− 8440.61i	− 1.52425i	0.647427	−	0.762127i	$-0.275845\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 10551.7i	− 1.86953i	−0.355264	−	0.934766i	$-0.615609\pi$
$317$	− 10551.7i	− 1.86953i	0.355264	−	0.934766i	$-0.384391\pi$
$318$	0	0
$319$	−2390.65	−0.419595
$320$	0	0
$321$	−4246.12	−0.738304
$322$	0	0
$323$	− 1749.28i	− 0.301339i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2725.16i	0.460861i
$328$	0	0
$329$	13680.3	2.29247
$330$	0	0
$331$	−6743.17	−1.11975	−0.559876	−	0.828576i	$-0.689151\pi$
$331$	−6743.17	−1.11975	−0.559876	+	0.828576i	$0.689151\pi$
$332$	0	0
$333$	− 336.478i	− 0.0553720i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8437.26i	1.36382i	0.731437	+	0.681909i	$0.238850\pi$
$337$	8437.26i	1.36382i	−0.731437	+	0.681909i	$0.761150\pi$
$338$	0	0
$339$	−6149.82	−0.985287
$340$	0	0
$341$	5754.26	0.913814
$342$	0	0
$343$	− 7314.45i	− 1.15144i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1848.85i	− 0.286027i	−0.989721	−	0.143013i	$-0.954321\pi$
$347$	− 1848.85i	− 0.286027i	0.989721	−	0.143013i	$-0.0456792\pi$
$348$	0	0
$349$	1148.38	0.176136	0.0880678	−	0.996114i	$-0.471931\pi$
$349$	1148.38	0.176136	0.0880678	+	0.996114i	$0.471931\pi$
$350$	0	0
$351$	1640.04	0.249399
$352$	0	0
$353$	− 5753.60i	− 0.867516i	−0.901029	−	0.433758i	$-0.857187\pi$
$353$	− 5753.60i	− 0.867516i	0.901029	−	0.433758i	$-0.142813\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	11064.4i	1.64030i
$358$	0	0
$359$	5452.01	0.801521	0.400761	−	0.916183i	$-0.368746\pi$
$359$	5452.01	0.801521	0.400761	+	0.916183i	$0.368746\pi$
$360$	0	0
$361$	−6650.61	−0.969619
$362$	0	0
$363$	1028.41i	0.148698i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8385.93i	1.19276i	0.802703	+	0.596379i	$0.203394\pi$
$367$	8385.93i	1.19276i	−0.802703	+	0.596379i	$0.796606\pi$
$368$	0	0
$369$	275.965	0.0389327
$370$	0	0
$371$	9190.05	1.28605
$372$	0	0
$373$	2728.30i	0.378730i	0.981907	+	0.189365i	$0.0606429\pi$
$373$	2728.30i	0.378730i	−0.981907	+	0.189365i	$0.939357\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4619.41i	− 0.631065i
$378$	0	0
$379$	3348.99	0.453895	0.226947	−	0.973907i	$-0.427125\pi$
$379$	3348.99	0.453895	0.226947	+	0.973907i	$0.427125\pi$
$380$	0	0
$381$	844.932	0.113615
$382$	0	0
$383$	− 10430.3i	− 1.39155i	−0.718261	−	0.695774i	$-0.755062\pi$
$383$	− 10430.3i	− 1.39155i	0.718261	−	0.695774i	$-0.244938\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2948.08i	0.387233i
$388$	0	0
$389$	−9827.23	−1.28088	−0.640438	−	0.768010i	$-0.721247\pi$
$389$	−9827.23	−1.28088	−0.640438	+	0.768010i	$0.721247\pi$
$390$	0	0
$391$	1659.32	0.214617
$392$	0	0
$393$	729.168i	0.0935921i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	436.382i	0.0551672i	0.999619	+	0.0275836i	$0.00878125\pi$
$397$	436.382i	0.0551672i	−0.999619	+	0.0275836i	$0.991219\pi$
$398$	0	0
$399$	−1318.07	−0.165378
$400$	0	0
$401$	−14501.5	−1.80591	−0.902955	−	0.429736i	$-0.858607\pi$
$401$	−14501.5	−1.80591	−0.902955	+	0.429736i	$0.858607\pi$
$402$	0	0
$403$	11118.8i	1.37436i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1175.26i	− 0.143134i
$408$	0	0
$409$	12058.4	1.45782	0.728911	−	0.684609i	$-0.240027\pi$
$409$	12058.4	1.45782	0.728911	+	0.684609i	$0.240027\pi$
$410$	0	0
$411$	−2728.16	−0.327421
$412$	0	0
$413$	10377.6i	1.23644i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6148.57i	0.722054i
$418$	0	0
$419$	6042.95	0.704577	0.352288	−	0.935892i	$-0.385404\pi$
$419$	6042.95	0.704577	0.352288	+	0.935892i	$0.385404\pi$
$420$	0	0
$421$	−9994.67	−1.15703	−0.578516	−	0.815671i	$-0.696368\pi$
$421$	−9994.67	−1.15703	−0.578516	+	0.815671i	$0.696368\pi$
$422$	0	0
$423$	4045.36i	0.464994i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	18842.6i	2.13550i
$428$	0	0
$429$	5728.42	0.644687
$430$	0	0
$431$	−9327.32	−1.04242	−0.521208	−	0.853430i	$-0.674518\pi$
$431$	−9327.32	−1.04242	−0.521208	+	0.853430i	$0.674518\pi$
$432$	0	0
$433$	− 7861.22i	− 0.872485i	−0.899829	−	0.436243i	$-0.856309\pi$
$433$	− 7861.22i	− 0.872485i	0.899829	−	0.436243i	$-0.143691\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	197.670i	0.0216380i
$438$	0	0
$439$	7412.06	0.805828	0.402914	−	0.915238i	$-0.367997\pi$
$439$	7412.06	0.805828	0.402914	+	0.915238i	$0.367997\pi$
$440$	0	0
$441$	5249.93	0.566886
$442$	0	0
$443$	3043.66i	0.326430i	0.986591	+	0.163215i	$0.0521864\pi$
$443$	3043.66i	0.326430i	−0.986591	+	0.163215i	$0.947814\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	10803.4i	1.14314i
$448$	0	0
$449$	9547.87	1.00355	0.501773	−	0.864999i	$-0.332681\pi$
$449$	9547.87	1.00355	0.501773	+	0.864999i	$0.332681\pi$
$450$	0	0
$451$	963.902	0.100639
$452$	0	0
$453$	4150.14i	0.430443i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 13401.9i	− 1.37180i	−0.727695	−	0.685901i	$-0.759408\pi$
$457$	− 13401.9i	− 1.37180i	0.727695	−	0.685901i	$-0.240592\pi$
$458$	0	0
$459$	−3271.81	−0.332712
$460$	0	0
$461$	−4137.03	−0.417962	−0.208981	−	0.977920i	$-0.567015\pi$
$461$	−4137.03	−0.417962	−0.208981	+	0.977920i	$0.567015\pi$
$462$	0	0
$463$	13976.3i	1.40288i	0.712729	+	0.701439i	$0.247459\pi$
$463$	13976.3i	1.40288i	−0.712729	+	0.701439i	$0.752541\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 10796.5i	− 1.06982i	−0.844910	−	0.534908i	$-0.820346\pi$
$467$	− 10796.5i	− 1.06982i	0.844910	−	0.534908i	$-0.179654\pi$
$468$	0	0
$469$	−7818.60	−0.769785
$470$	0	0
$471$	395.248	0.0386668
$472$	0	0
$473$	10297.2i	1.00098i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2717.56i	0.260856i
$478$	0	0
$479$	14568.4	1.38966	0.694830	−	0.719174i	$-0.255479\pi$
$479$	14568.4	1.38966	0.694830	+	0.719174i	$0.255479\pi$
$480$	0	0
$481$	2270.94	0.215272
$482$	0	0
$483$	− 1250.28i	− 0.117784i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 11456.6i	− 1.06601i	−0.846113	−	0.533004i	$-0.821063\pi$
$487$	− 11456.6i	− 1.06601i	0.846113	−	0.533004i	$-0.178937\pi$
$488$	0	0
$489$	−8693.21	−0.803928
$490$	0	0
$491$	19666.5	1.80761	0.903804	−	0.427948i	$-0.140763\pi$
$491$	19666.5	1.80761	0.903804	+	0.427948i	$0.140763\pi$
$492$	0	0
$493$	9215.48i	0.841875i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 15192.8i	− 1.37121i
$498$	0	0
$499$	8379.31	0.751722	0.375861	−	0.926676i	$-0.377347\pi$
$499$	8379.31	0.751722	0.375861	+	0.926676i	$0.377347\pi$
$500$	0	0
$501$	780.850	0.0696323
$502$	0	0
$503$	15678.1i	1.38976i	0.719124	+	0.694881i	$0.244543\pi$
$503$	15678.1i	1.38976i	−0.719124	+	0.694881i	$0.755457\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4477.91i	0.392251i
$508$	0	0
$509$	17037.3	1.48363	0.741813	−	0.670606i	$-0.233966\pi$
$509$	17037.3	1.48363	0.741813	+	0.670606i	$0.233966\pi$
$510$	0	0
$511$	582.197	0.0504009
$512$	0	0
$513$	− 389.761i	− 0.0335446i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	14129.8i	1.20199i
$518$	0	0
$519$	−5807.49	−0.491176
$520$	0	0
$521$	−8776.12	−0.737982	−0.368991	−	0.929433i	$-0.620297\pi$
$521$	−8776.12	−0.737982	−0.368991	+	0.929433i	$0.620297\pi$
$522$	0	0
$523$	− 11120.4i	− 0.929753i	−0.885375	−	0.464877i	$-0.846099\pi$
$523$	− 11120.4i	− 0.929753i	0.885375	−	0.464877i	$-0.153901\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 22181.5i	− 1.83348i
$528$	0	0
$529$	11979.5	0.984589
$530$	0	0
$531$	−3068.73	−0.250794
$532$	0	0
$533$	1862.53i	0.151360i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 1729.88i	− 0.139013i
$538$	0	0
$539$	18337.2	1.46538
$540$	0	0
$541$	21730.6	1.72693	0.863467	−	0.504405i	$-0.168288\pi$
$541$	21730.6	1.72693	0.863467	+	0.504405i	$0.168288\pi$
$542$	0	0
$543$	5886.13i	0.465190i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 6926.17i	− 0.541392i	−0.962665	−	0.270696i	$-0.912746\pi$
$547$	− 6926.17i	− 0.541392i	0.962665	−	0.270696i	$-0.0872539\pi$
$548$	0	0
$549$	−5571.89	−0.433155
$550$	0	0
$551$	−1097.82	−0.0848793
$552$	0	0
$553$	7834.85i	0.602480i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6589.22i	0.501246i	0.968085	+	0.250623i	$0.0806355\pi$
$557$	6589.22i	0.501246i	−0.968085	+	0.250623i	$0.919364\pi$
$558$	0	0
$559$	−19897.0	−1.50547
$560$	0	0
$561$	−11427.9	−0.860047
$562$	0	0
$563$	− 3839.63i	− 0.287426i	−0.989619	−	0.143713i	$-0.954096\pi$
$563$	− 3839.63i	− 0.287426i	0.989619	−	0.143713i	$-0.0459043\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2465.28i	0.182596i
$568$	0	0
$569$	20874.0	1.53794	0.768968	−	0.639287i	$-0.220771\pi$
$569$	20874.0	1.53794	0.768968	+	0.639287i	$0.220771\pi$
$570$	0	0
$571$	−21175.9	−1.55199	−0.775994	−	0.630740i	$-0.782752\pi$
$571$	−21175.9	−1.55199	−0.775994	+	0.630740i	$0.782752\pi$
$572$	0	0
$573$	− 12956.3i	− 0.944599i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14924.2i	1.07678i	0.842695	+	0.538391i	$0.180968\pi$
$577$	14924.2i	1.07678i	−0.842695	+	0.538391i	$0.819032\pi$
$578$	0	0
$579$	8.92941	0.000640922 0
$580$	0	0
$581$	27845.8	1.98836
$582$	0	0
$583$	9492.00i	0.674303i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25218.0i	1.77318i	0.462552	+	0.886592i	$0.346934\pi$
$587$	25218.0i	1.77318i	−0.462552	+	0.886592i	$0.653066\pi$
$588$	0	0
$589$	2642.42	0.184854
$590$	0	0
$591$	−1709.11	−0.118957
$592$	0	0
$593$	5011.77i	0.347063i	0.984828	+	0.173532i	$0.0555179\pi$
$593$	5011.77i	0.347063i	−0.984828	+	0.173532i	$0.944482\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9152.19i	0.627428i
$598$	0	0
$599$	4943.41	0.337199	0.168600	−	0.985685i	$-0.446075\pi$
$599$	4943.41	0.337199	0.168600	+	0.985685i	$0.446075\pi$
$600$	0	0
$601$	−24334.8	−1.65164	−0.825821	−	0.563932i	$-0.809288\pi$
$601$	−24334.8	−1.65164	−0.825821	+	0.563932i	$0.809288\pi$
$602$	0	0
$603$	− 2312.01i	− 0.156140i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	28973.8i	1.93741i	0.248207	+	0.968707i	$0.420159\pi$
$607$	28973.8i	1.93741i	−0.248207	+	0.968707i	$0.579841\pi$
$608$	0	0
$609$	6943.81	0.462032
$610$	0	0
$611$	−27302.8	−1.80778
$612$	0	0
$613$	15139.1i	0.997490i	0.866749	+	0.498745i	$0.166206\pi$
$613$	15139.1i	0.997490i	−0.866749	+	0.498745i	$0.833794\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 13894.8i	− 0.906617i	−0.891354	−	0.453309i	$-0.850244\pi$
$617$	− 13894.8i	− 0.906617i	0.891354	−	0.453309i	$-0.149756\pi$
$618$	0	0
$619$	−4589.69	−0.298021	−0.149011	−	0.988836i	$-0.547609\pi$
$619$	−4589.69	−0.298021	−0.149011	+	0.988836i	$0.547609\pi$
$620$	0	0
$621$	369.717	0.0238909
$622$	0	0
$623$	32252.0i	2.07408i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1361.37i	− 0.0867114i
$628$	0	0
$629$	−4530.41	−0.287185
$630$	0	0
$631$	−3005.77	−0.189632	−0.0948160	−	0.995495i	$-0.530226\pi$
$631$	−3005.77	−0.189632	−0.0948160	+	0.995495i	$0.530226\pi$
$632$	0	0
$633$	− 151.531i	− 0.00951473i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	35432.6i	2.20391i
$638$	0	0
$639$	4492.60	0.278129
$640$	0	0
$641$	−5631.47	−0.347004	−0.173502	−	0.984834i	$-0.555508\pi$
$641$	−5631.47	−0.347004	−0.173502	+	0.984834i	$0.555508\pi$
$642$	0	0
$643$	− 11305.1i	− 0.693358i	−0.937984	−	0.346679i	$-0.887309\pi$
$643$	− 11305.1i	− 0.693358i	0.937984	−	0.346679i	$-0.112691\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 8614.30i	− 0.523436i	−0.965144	−	0.261718i	$-0.915711\pi$
$647$	− 8614.30i	− 0.523436i	0.965144	−	0.261718i	$-0.0842891\pi$
$648$	0	0
$649$	−10718.6	−0.648291
$650$	0	0
$651$	−16713.6	−1.00623
$652$	0	0
$653$	12639.8i	0.757479i	0.925503	+	0.378739i	$0.123642\pi$
$653$	12639.8i	0.757479i	−0.925503	+	0.378739i	$0.876358\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	172.159i	0.0102231i
$658$	0	0
$659$	13640.4	0.806306	0.403153	−	0.915133i	$-0.367914\pi$
$659$	13640.4	0.806306	0.403153	+	0.915133i	$0.367914\pi$
$660$	0	0
$661$	−17052.0	−1.00340	−0.501699	−	0.865042i	$-0.667291\pi$
$661$	−17052.0	−1.00340	−0.501699	+	0.865042i	$0.667291\pi$
$662$	0	0
$663$	− 22081.9i	− 1.29350i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1041.36i	− 0.0604521i
$668$	0	0
$669$	−16360.7	−0.945500
$670$	0	0
$671$	−19461.7	−1.11969
$672$	0	0
$673$	− 16419.7i	− 0.940467i	−0.882542	−	0.470234i	$-0.844170\pi$
$673$	− 16419.7i	− 0.940467i	0.882542	−	0.470234i	$-0.155830\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8670.47i	0.492221i	0.969242	+	0.246110i	$0.0791525\pi$
$677$	8670.47i	0.492221i	−0.969242	+	0.246110i	$0.920847\pi$
$678$	0	0
$679$	15856.9	0.896220
$680$	0	0
$681$	−14331.4	−0.806434
$682$	0	0
$683$	− 5973.36i	− 0.334647i	−0.985902	−	0.167324i	$-0.946488\pi$
$683$	− 5973.36i	− 0.334647i	0.985902	−	0.167324i	$-0.0535125\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 6256.52i	− 0.347454i
$688$	0	0
$689$	−18341.2	−1.01414
$690$	0	0
$691$	15316.3	0.843212	0.421606	−	0.906779i	$-0.361467\pi$
$691$	15316.3	0.843212	0.421606	+	0.906779i	$0.361467\pi$
$692$	0	0
$693$	8610.85i	0.472005i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 3715.65i	− 0.201923i
$698$	0	0
$699$	19453.6	1.05265
$700$	0	0
$701$	34583.1	1.86332	0.931660	−	0.363333i	$-0.118361\pi$
$701$	34583.1	1.86332	0.931660	+	0.363333i	$0.118361\pi$
$702$	0	0
$703$	− 539.695i	− 0.0289545i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10563.6i	0.561929i
$708$	0	0
$709$	−11194.1	−0.592955	−0.296477	−	0.955040i	$-0.595812\pi$
$709$	−11194.1	−0.592955	−0.296477	+	0.955040i	$0.595812\pi$
$710$	0	0
$711$	−2316.81	−0.122204
$712$	0	0
$713$	2506.53i	0.131655i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 6703.85i	− 0.349177i
$718$	0	0
$719$	−15491.9	−0.803549	−0.401774	−	0.915739i	$-0.631606\pi$
$719$	−15491.9	−0.803549	−0.401774	+	0.915739i	$0.631606\pi$
$720$	0	0
$721$	−23458.2	−1.21169
$722$	0	0
$723$	7179.04i	0.369283i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 6272.72i	− 0.320003i	−0.987117	−	0.160002i	$-0.948850\pi$
$727$	− 6272.72i	− 0.320003i	0.987117	−	0.160002i	$-0.0511500\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	39693.6	2.00837
$732$	0	0
$733$	24980.5i	1.25877i	0.777095	+	0.629383i	$0.216692\pi$
$733$	24980.5i	1.25877i	−0.777095	+	0.629383i	$0.783308\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8075.49i	− 0.403615i
$738$	0	0
$739$	−30660.7	−1.52621	−0.763107	−	0.646272i	$-0.776327\pi$
$739$	−30660.7	−1.52621	−0.763107	+	0.646272i	$0.776327\pi$
$740$	0	0
$741$	2630.56	0.130413
$742$	0	0
$743$	17205.7i	0.849551i	0.905299	+	0.424776i	$0.139647\pi$
$743$	17205.7i	0.849551i	−0.905299	+	0.424776i	$0.860353\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8234.18i	0.403310i
$748$	0	0
$749$	43077.8	2.10151
$750$	0	0
$751$	−18397.1	−0.893901	−0.446950	−	0.894559i	$-0.647490\pi$
$751$	−18397.1	−0.893901	−0.446950	+	0.894559i	$0.647490\pi$
$752$	0	0
$753$	− 1837.98i	− 0.0889506i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 22305.1i	− 1.07093i	−0.844557	−	0.535465i	$-0.820136\pi$
$757$	− 22305.1i	− 1.07093i	0.844557	−	0.535465i	$-0.179864\pi$
$758$	0	0
$759$	1291.36	0.0617569
$760$	0	0
$761$	14458.5	0.688727	0.344364	−	0.938836i	$-0.388095\pi$
$761$	14458.5	0.688727	0.344364	+	0.938836i	$0.388095\pi$
$762$	0	0
$763$	− 27647.3i	− 1.31179i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 20711.3i	− 0.975022i
$768$	0	0
$769$	39897.6	1.87093	0.935463	−	0.353424i	$-0.114983\pi$
$769$	39897.6	1.87093	0.935463	+	0.353424i	$0.114983\pi$
$770$	0	0
$771$	918.337	0.0428964
$772$	0	0
$773$	20070.2i	0.933863i	0.884294	+	0.466931i	$0.154641\pi$
$773$	20070.2i	0.933863i	−0.884294	+	0.466931i	$0.845359\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3413.63i	0.157611i
$778$	0	0
$779$	442.635	0.0203582
$780$	0	0
$781$	15692.0	0.718953
$782$	0	0
$783$	2053.33i	0.0937164i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	10733.0i	0.486137i	0.970009	+	0.243069i	$0.0781540\pi$
$787$	10733.0i	0.486137i	−0.970009	+	0.243069i	$0.921846\pi$
$788$	0	0
$789$	−851.516	−0.0384218
$790$	0	0
$791$	62391.1	2.80452
$792$	0	0
$793$	− 37605.5i	− 1.68400i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14335.8i	0.637140i	0.947899	+	0.318570i	$0.103203\pi$
$797$	14335.8i	0.637140i	−0.947899	+	0.318570i	$0.896797\pi$
$798$	0	0
$799$	54467.7	2.41167
$800$	0	0
$801$	−9537.13	−0.420697
$802$	0	0
$803$	601.325i	0.0264263i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7278.63i	0.317497i
$808$	0	0
$809$	−20920.7	−0.909187	−0.454593	−	0.890699i	$-0.650215\pi$
$809$	−20920.7	−0.909187	−0.454593	+	0.890699i	$0.650215\pi$
$810$	0	0
$811$	−12816.5	−0.554931	−0.277465	−	0.960736i	$-0.589494\pi$
$811$	−12816.5	−0.554931	−0.277465	+	0.960736i	$0.589494\pi$
$812$	0	0
$813$	− 524.838i	− 0.0226407i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4728.59i	0.202488i
$818$	0	0
$819$	−16638.6	−0.709889
$820$	0	0
$821$	7253.55	0.308344	0.154172	−	0.988044i	$-0.450729\pi$
$821$	7253.55	0.308344	0.154172	+	0.988044i	$0.450729\pi$
$822$	0	0
$823$	− 35288.1i	− 1.49461i	−0.664479	−	0.747307i	$-0.731346\pi$
$823$	− 35288.1i	− 1.49461i	0.664479	−	0.747307i	$-0.268654\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 32205.9i	− 1.35418i	−0.735899	−	0.677092i	$-0.763240\pi$
$827$	− 32205.9i	− 1.35418i	0.735899	−	0.677092i	$-0.236760\pi$
$828$	0	0
$829$	−29993.3	−1.25659	−0.628294	−	0.777976i	$-0.716246\pi$
$829$	−29993.3	−1.25659	−0.628294	+	0.777976i	$0.716246\pi$
$830$	0	0
$831$	23421.2	0.977706
$832$	0	0
$833$	− 70686.2i	− 2.94013i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 4942.33i	− 0.204100i
$838$	0	0
$839$	−35608.7	−1.46526	−0.732628	−	0.680629i	$-0.761706\pi$
$839$	−35608.7	−1.46526	−0.732628	+	0.680629i	$0.761706\pi$
$840$	0	0
$841$	−18605.5	−0.762865
$842$	0	0
$843$	− 1752.51i	− 0.0716011i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 10433.4i	− 0.423255i
$848$	0	0
$849$	17691.9	0.715177
$850$	0	0
$851$	511.940	0.0206217
$852$	0	0
$853$	− 11229.3i	− 0.450744i	−0.974273	−	0.225372i	$-0.927640\pi$
$853$	− 11229.3i	− 0.450744i	0.974273	−	0.225372i	$-0.0723598\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 22136.1i	− 0.882327i	−0.897427	−	0.441164i	$-0.854566\pi$
$857$	− 22136.1i	− 0.882327i	0.897427	−	0.441164i	$-0.145434\pi$
$858$	0	0
$859$	−820.727	−0.0325994	−0.0162997	−	0.999867i	$-0.505189\pi$
$859$	−820.727	−0.0325994	−0.0162997	+	0.999867i	$0.505189\pi$
$860$	0	0
$861$	−2799.72	−0.110818
$862$	0	0
$863$	245.223i	0.00967264i	0.999988	+	0.00483632i	$0.00153945\pi$
$863$	245.223i	0.00967264i	−0.999988	+	0.00483632i	$0.998461\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	29313.3i	1.14825i
$868$	0	0
$869$	−8092.27	−0.315894
$870$	0	0
$871$	15604.1	0.607032
$872$	0	0
$873$	4689.00i	0.181785i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 37727.8i	− 1.45265i	−0.687350	−	0.726326i	$-0.741226\pi$
$877$	− 37727.8i	− 1.45265i	0.687350	−	0.726326i	$-0.258774\pi$
$878$	0	0
$879$	4829.19	0.185306
$880$	0	0
$881$	21738.8	0.831326	0.415663	−	0.909519i	$-0.363550\pi$
$881$	21738.8	0.831326	0.415663	+	0.909519i	$0.363550\pi$
$882$	0	0
$883$	44340.4i	1.68989i	0.534852	+	0.844946i	$0.320368\pi$
$883$	44340.4i	1.68989i	−0.534852	+	0.844946i	$0.679632\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 681.008i	− 0.0257790i	−0.999917	−	0.0128895i	$-0.995897\pi$
$887$	− 681.008i	− 0.0257790i	0.999917	−	0.0128895i	$-0.00410298\pi$
$888$	0	0
$889$	−8572.00	−0.323392
$890$	0	0
$891$	−2546.28	−0.0957393
$892$	0	0
$893$	6488.58i	0.243149i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2495.28i	0.0928816i
$898$	0	0
$899$	−13920.7	−0.516443
$900$	0	0
$901$	36589.8	1.35292
$902$	0	0
$903$	− 29908.9i	− 1.10222i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	5348.01i	0.195786i	0.995197	+	0.0978929i	$0.0312103\pi$
$907$	5348.01i	0.195786i	−0.995197	+	0.0978929i	$0.968790\pi$
$908$	0	0
$909$	−3123.72	−0.113979
$910$	0	0
$911$	−14488.7	−0.526930	−0.263465	−	0.964669i	$-0.584865\pi$
$911$	−14488.7	−0.526930	−0.263465	+	0.964669i	$0.584865\pi$
$912$	0	0
$913$	28760.7i	1.04254i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 7397.56i	− 0.266400i
$918$	0	0
$919$	−22546.9	−0.809308	−0.404654	−	0.914470i	$-0.632608\pi$
$919$	−22546.9	−0.809308	−0.404654	+	0.914470i	$0.632608\pi$
$920$	0	0
$921$	703.419	0.0251666
$922$	0	0
$923$	30321.3i	1.08130i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 6936.74i	− 0.245774i
$928$	0	0
$929$	13980.7	0.493747	0.246874	−	0.969048i	$-0.420597\pi$
$929$	13980.7	0.493747	0.246874	+	0.969048i	$0.420597\pi$
$930$	0	0
$931$	8420.65	0.296429
$932$	0	0
$933$	− 5385.75i	− 0.188984i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 26362.6i	− 0.919133i	−0.888143	−	0.459567i	$-0.848005\pi$
$937$	− 26362.6i	− 0.919133i	0.888143	−	0.459567i	$-0.151995\pi$
$938$	0	0
$939$	−25321.8	−0.880029
$940$	0	0
$941$	−14715.6	−0.509792	−0.254896	−	0.966968i	$-0.582041\pi$
$941$	−14715.6	−0.509792	−0.254896	+	0.966968i	$0.582041\pi$
$942$	0	0
$943$	419.871i	0.0144994i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 11818.2i	− 0.405532i	−0.979227	−	0.202766i	$-0.935007\pi$
$947$	− 11818.2i	− 0.405532i	0.979227	−	0.202766i	$-0.0649931\pi$
$948$	0	0
$949$	−1161.93	−0.0397448
$950$	0	0
$951$	−31655.1	−1.07937
$952$	0	0
$953$	− 43832.3i	− 1.48989i	−0.667125	−	0.744945i	$-0.732476\pi$
$953$	− 43832.3i	− 1.48989i	0.667125	−	0.744945i	$-0.267524\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	7171.95i	0.242253i
$958$	0	0
$959$	27677.7	0.931971
$960$	0	0
$961$	3716.00	0.124736
$962$	0	0
$963$	12738.4i	0.426260i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	10696.2i	0.355706i	0.984057	+	0.177853i	$0.0569152\pi$
$967$	10696.2i	0.355706i	−0.984057	+	0.177853i	$0.943085\pi$
$968$	0	0
$969$	−5247.83	−0.173978
$970$	0	0
$971$	−27933.0	−0.923187	−0.461593	−	0.887092i	$-0.652722\pi$
$971$	−27933.0	−0.923187	−0.461593	+	0.887092i	$0.652722\pi$
$972$	0	0
$973$	− 62378.4i	− 2.05525i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 24341.7i	− 0.797094i	−0.917148	−	0.398547i	$-0.869515\pi$
$977$	− 24341.7i	− 0.797094i	0.917148	−	0.398547i	$-0.130485\pi$
$978$	0	0
$979$	−33311.7	−1.08748
$980$	0	0
$981$	8175.48	0.266078
$982$	0	0
$983$	− 12553.0i	− 0.407301i	−0.979044	−	0.203651i	$-0.934719\pi$
$983$	− 12553.0i	− 0.407301i	0.979044	−	0.203651i	$-0.0652807\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 41041.0i	− 1.32356i
$988$	0	0
$989$	−4485.41	−0.144214
$990$	0	0
$991$	45631.8	1.46271	0.731353	−	0.681999i	$-0.238889\pi$
$991$	45631.8	1.46271	0.731353	+	0.681999i	$0.238889\pi$
$992$	0	0
$993$	20229.5i	0.646490i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	34499.0i	1.09588i	0.836517	+	0.547941i	$0.184588\pi$
$997$	34499.0i	1.09588i	−0.836517	+	0.547941i	$0.815412\pi$
$998$	0	0
$999$	−1009.43	−0.0319690

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.f.v.49.2		4
4.3	odd	2		75.4.b.c.49.2		4
5.2	odd	4		1200.4.a.bl.1.1		2
5.3	odd	4		1200.4.a.bu.1.2		2
5.4	even	2	inner	1200.4.f.v.49.3		4
12.11	even	2		225.4.b.h.199.3		4
20.3	even	4		75.4.a.e.1.1	yes	2
20.7	even	4		75.4.a.d.1.2	✓	2
20.19	odd	2		75.4.b.c.49.3		4
60.23	odd	4		225.4.a.j.1.2		2
60.47	odd	4		225.4.a.n.1.1		2
60.59	even	2		225.4.b.h.199.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.a.d.1.2	✓	2	20.7	even	4
75.4.a.e.1.1	yes	2	20.3	even	4
75.4.b.c.49.2		4	4.3	odd	2
75.4.b.c.49.3		4	20.19	odd	2
225.4.a.j.1.2		2	60.23	odd	4
225.4.a.n.1.1		2	60.47	odd	4
225.4.b.h.199.2		4	60.59	even	2
225.4.b.h.199.3		4	12.11	even	2
1200.4.a.bl.1.1		2	5.2	odd	4
1200.4.a.bu.1.2		2	5.3	odd	4
1200.4.f.v.49.2		4	1.1	even	1	trivial
1200.4.f.v.49.3		4	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +30.4356i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} +30.4356i q^{7} -9.00000 q^{9} -31.4356 q^{11} -60.7424i q^{13} +121.178i q^{17} -14.4356 q^{19} +91.3068 q^{21} -13.6932i q^{23} +27.0000i q^{27} +76.0492 q^{29} -183.049 q^{31} +94.3068i q^{33} +37.3864i q^{37} -182.227 q^{39} -30.6627 q^{41} -327.564i q^{43} -449.485i q^{47} -583.325 q^{49} +363.534 q^{51} -301.951i q^{53} +43.3068i q^{57} +340.970 q^{59} +619.098 q^{61} -273.920i q^{63} +256.890i q^{67} -41.0796 q^{69} -499.178 q^{71} -19.1288i q^{73} -956.761i q^{77} +257.424 q^{79} +81.0000 q^{81} -914.909i q^{83} -228.148i q^{87} +1059.68 q^{89} +1848.73 q^{91} +549.148i q^{93} -521.000i q^{97} +282.920 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{9}+O(q^{10}) \) 4 * q - 36 * q^9	\( 4 q - 36 q^{9} - 56 q^{11} + 12 q^{19} + 156 q^{21} - 184 q^{29} - 244 q^{31} + 108 q^{39} + 784 q^{41} - 520 q^{49} + 408 q^{51} + 248 q^{59} + 1500 q^{61} - 792 q^{69} - 1648 q^{71} - 1760 q^{79} + 324 q^{81} + 1728 q^{89} + 4396 q^{91} + 504 q^{99}+O(q^{100}) \) 4 * q - 36 * q^9 - 56 * q^11 + 12 * q^19 + 156 * q^21 - 184 * q^29 - 244 * q^31 + 108 * q^39 + 784 * q^41 - 520 * q^49 + 408 * q^51 + 248 * q^59 + 1500 * q^61 - 792 * q^69 - 1648 * q^71 - 1760 * q^79 + 324 * q^81 + 1728 * q^89 + 4396 * q^91 + 504 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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