LMFDB - Embedded newform 675.5.d.d.674.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,5,Mod(674,675)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("675.674"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	675.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,6,0,-14,0,0,0,-138,0,0,0,0,0,0,0,-190,-828,0,608] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$69.7747250816$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			674.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	675.674
Dual form			675.5.d.d.674.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{2} -7.00000 q^{4} +19.0000i q^{7} -69.0000 q^{8} -123.000i q^{11} +302.000i q^{13} +57.0000i q^{14} -95.0000 q^{16} -414.000 q^{17} +304.000 q^{19} -369.000i q^{22} +300.000 q^{23} +906.000i q^{26} -133.000i q^{28} -678.000i q^{29} +239.000 q^{31} +819.000 q^{32} -1242.00 q^{34} -740.000i q^{37} +912.000 q^{38} -228.000i q^{41} -982.000i q^{43} +861.000i q^{44} +900.000 q^{46} -2166.00 q^{47} +2040.00 q^{49} -2114.00i q^{52} -1593.00 q^{53} -1311.00i q^{56} -2034.00i q^{58} -2922.00i q^{59} -316.000 q^{61} +717.000 q^{62} +3977.00 q^{64} -4622.00i q^{67} +2898.00 q^{68} -1818.00i q^{71} -3031.00i q^{73} -2220.00i q^{74} -2128.00 q^{76} +2337.00 q^{77} +10450.0 q^{79} -684.000i q^{82} +12633.0 q^{83} -2946.00i q^{86} +8487.00i q^{88} -7002.00i q^{89} -5738.00 q^{91} -2100.00 q^{92} -6498.00 q^{94} +6517.00i q^{97} +6120.00 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{2} - 14 q^{4} - 138 q^{8} - 190 q^{16} - 828 q^{17} + 608 q^{19} + 600 q^{23} + 478 q^{31} + 1638 q^{32} - 2484 q^{34} + 1824 q^{38} + 1800 q^{46} - 4332 q^{47} + 4080 q^{49} - 3186 q^{53} - 632 q^{61}+ \cdots + 12240 q^{98}+O(q^{100}) $ 2 * q + 6 * q^2 - 14 * q^4 - 138 * q^8 - 190 * q^16 - 828 * q^17 + 608 * q^19 + 600 * q^23 + 478 * q^31 + 1638 * q^32 - 2484 * q^34 + 1824 * q^38 + 1800 * q^46 - 4332 * q^47 + 4080 * q^49 - 3186 * q^53 - 632 * q^61 + 1434 * q^62 + 7954 * q^64 + 5796 * q^68 - 4256 * q^76 + 4674 * q^77 + 20900 * q^79 + 25266 * q^83 - 11476 * q^91 - 4200 * q^92 - 12996 * q^94 + 12240 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$-1$	$-1$

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 $

$2$	3.00000	0.750000	0.375000	−	0.927025i	$-0.377643\pi$
$2$	3.00000	0.750000	0.375000	+	0.927025i	$0.377643\pi$
$3$	0	0
$3$	0	0
$4$	−7.00000	−0.437500
$5$	0	0
$5$	0	0
$6$	0	0
$7$	19.0000i	0.387755i	0.981026	+	0.193878i	$0.0621064\pi$
$7$	19.0000i	0.387755i	−0.981026	+	0.193878i	$0.937894\pi$
$8$	−69.0000	−1.07812
$9$	0	0
$10$	0	0
$11$	− 123.000i	− 1.01653i	−0.861201	−	0.508264i	$-0.830287\pi$
$11$	− 123.000i	− 1.01653i	0.861201	−	0.508264i	$-0.169713\pi$
$12$	0	0
$13$	302.000i	1.78698i	0.449081	+	0.893491i	$0.351752\pi$
$13$	302.000i	1.78698i	−0.449081	+	0.893491i	$0.648248\pi$
$14$	57.0000i	0.290816i
$15$	0	0
$16$	−95.0000	−0.371094
$17$	−414.000	−1.43253	−0.716263	−	0.697830i	$-0.754149\pi$
$17$	−414.000	−1.43253	−0.716263	+	0.697830i	$0.754149\pi$
$18$	0	0
$19$	304.000	0.842105	0.421053	−	0.907036i	$-0.361661\pi$
$19$	304.000	0.842105	0.421053	+	0.907036i	$0.361661\pi$
$20$	0	0
$21$	0	0
$22$	− 369.000i	− 0.762397i
$23$	300.000	0.567108	0.283554	−	0.958956i	$-0.408487\pi$
$23$	300.000	0.567108	0.283554	+	0.958956i	$0.408487\pi$
$24$	0	0
$25$	0	0
$26$	906.000i	1.34024i
$27$	0	0
$28$	− 133.000i	− 0.169643i
$29$	− 678.000i	− 0.806183i	−0.915160	−	0.403092i	$-0.867936\pi$
$29$	− 678.000i	− 0.806183i	0.915160	−	0.403092i	$-0.132064\pi$
$30$	0	0
$31$	239.000	0.248699	0.124350	−	0.992238i	$-0.460316\pi$
$31$	239.000	0.248699	0.124350	+	0.992238i	$0.460316\pi$
$32$	819.000	0.799805
$33$	0	0
$34$	−1242.00	−1.07439
$35$	0	0
$36$	0	0
$37$	− 740.000i	− 0.540541i	−0.962784	−	0.270270i	$-0.912887\pi$
$37$	− 740.000i	− 0.540541i	0.962784	−	0.270270i	$-0.0871131\pi$
$38$	912.000	0.631579
$39$	0	0
$40$	0	0
$41$	− 228.000i	− 0.135634i	−0.997698	−	0.0678168i	$-0.978397\pi$
$41$	− 228.000i	− 0.135634i	0.997698	−	0.0678168i	$-0.0216033\pi$
$42$	0	0
$43$	− 982.000i	− 0.531098i	−0.964097	−	0.265549i	$-0.914447\pi$
$43$	− 982.000i	− 0.531098i	0.964097	−	0.265549i	$-0.0855532\pi$
$44$	861.000i	0.444731i
$45$	0	0
$46$	900.000	0.425331
$47$	−2166.00	−0.980534	−0.490267	−	0.871572i	$-0.663101\pi$
$47$	−2166.00	−0.980534	−0.490267	+	0.871572i	$0.663101\pi$
$48$	0	0
$49$	2040.00	0.849646
$50$	0	0
$51$	0	0
$52$	− 2114.00i	− 0.781805i
$53$	−1593.00	−0.567106	−0.283553	−	0.958957i	$-0.591513\pi$
$53$	−1593.00	−0.567106	−0.283553	+	0.958957i	$0.591513\pi$
$54$	0	0
$55$	0	0
$56$	− 1311.00i	− 0.418048i
$57$	0	0
$58$	− 2034.00i	− 0.604637i
$59$	− 2922.00i	− 0.839414i	−0.907660	−	0.419707i	$-0.862133\pi$
$59$	− 2922.00i	− 0.839414i	0.907660	−	0.419707i	$-0.137867\pi$
$60$	0	0
$61$	−316.000	−0.0849234	−0.0424617	−	0.999098i	$-0.513520\pi$
$61$	−316.000	−0.0849234	−0.0424617	+	0.999098i	$0.513520\pi$
$62$	717.000	0.186524
$63$	0	0
$64$	3977.00	0.970947
$65$	0	0
$66$	0	0
$67$	− 4622.00i	− 1.02963i	−0.857302	−	0.514814i	$-0.827861\pi$
$67$	− 4622.00i	− 1.02963i	0.857302	−	0.514814i	$-0.172139\pi$
$68$	2898.00	0.626730
$69$	0	0
$70$	0	0
$71$	− 1818.00i	− 0.360643i	−0.983608	−	0.180321i	$-0.942286\pi$
$71$	− 1818.00i	− 0.360643i	0.983608	−	0.180321i	$-0.0577138\pi$
$72$	0	0
$73$	− 3031.00i	− 0.568775i	−0.958709	−	0.284387i	$-0.908210\pi$
$73$	− 3031.00i	− 0.568775i	0.958709	−	0.284387i	$-0.0917902\pi$
$74$	− 2220.00i	− 0.405405i
$75$	0	0
$76$	−2128.00	−0.368421
$77$	2337.00	0.394164
$78$	0	0
$79$	10450.0	1.67441	0.837206	−	0.546888i	$-0.184188\pi$
$79$	10450.0	1.67441	0.837206	+	0.546888i	$0.184188\pi$
$80$	0	0
$81$	0	0
$82$	− 684.000i	− 0.101725i
$83$	12633.0	1.83379	0.916897	−	0.399125i	$-0.130686\pi$
$83$	12633.0	1.83379	0.916897	+	0.399125i	$0.130686\pi$
$84$	0	0
$85$	0	0
$86$	− 2946.00i	− 0.398323i
$87$	0	0
$88$	8487.00i	1.09595i
$89$	− 7002.00i	− 0.883979i	−0.897020	−	0.441990i	$-0.854273\pi$
$89$	− 7002.00i	− 0.883979i	0.897020	−	0.441990i	$-0.145727\pi$
$90$	0	0
$91$	−5738.00	−0.692911
$92$	−2100.00	−0.248110
$93$	0	0
$94$	−6498.00	−0.735401
$95$	0	0
$96$	0	0
$97$	6517.00i	0.692635i	0.938117	+	0.346317i	$0.112568\pi$
$97$	6517.00i	0.692635i	−0.938117	+	0.346317i	$0.887432\pi$
$98$	6120.00	0.637234
$99$	0	0
$100$	0	0
$101$	− 5919.00i	− 0.580237i	−0.956991	−	0.290119i	$-0.906305\pi$
$101$	− 5919.00i	− 0.580237i	0.956991	−	0.290119i	$-0.0936948\pi$
$102$	0	0
$103$	− 7654.00i	− 0.721463i	−0.932670	−	0.360731i	$-0.882527\pi$
$103$	− 7654.00i	− 0.721463i	0.932670	−	0.360731i	$-0.117473\pi$
$104$	− 20838.0i	− 1.92659i
$105$	0	0
$106$	−4779.00	−0.425329
$107$	513.000	0.0448074	0.0224037	−	0.999749i	$-0.492868\pi$
$107$	513.000	0.0448074	0.0224037	+	0.999749i	$0.492868\pi$
$108$	0	0
$109$	−2324.00	−0.195606	−0.0978032	−	0.995206i	$-0.531182\pi$
$109$	−2324.00	−0.195606	−0.0978032	+	0.995206i	$0.531182\pi$
$110$	0	0
$111$	0	0
$112$	− 1805.00i	− 0.143893i
$113$	−4920.00	−0.385308	−0.192654	−	0.981267i	$-0.561710\pi$
$113$	−4920.00	−0.385308	−0.192654	+	0.981267i	$0.561710\pi$
$114$	0	0
$115$	0	0
$116$	4746.00i	0.352705i
$117$	0	0
$118$	− 8766.00i	− 0.629560i
$119$	− 7866.00i	− 0.555469i
$120$	0	0
$121$	−488.000	−0.0333311
$122$	−948.000	−0.0636926
$123$	0	0
$124$	−1673.00	−0.108806
$125$	0	0
$126$	0	0
$127$	− 24995.0i	− 1.54969i	−0.632149	−	0.774847i	$-0.717827\pi$
$127$	− 24995.0i	− 1.54969i	0.632149	−	0.774847i	$-0.282173\pi$
$128$	−1173.00	−0.0715942
$129$	0	0
$130$	0	0
$131$	− 28461.0i	− 1.65847i	−0.558900	−	0.829235i	$-0.688777\pi$
$131$	− 28461.0i	− 1.65847i	0.558900	−	0.829235i	$-0.311223\pi$
$132$	0	0
$133$	5776.00i	0.326531i
$134$	− 13866.0i	− 0.772221i
$135$	0	0
$136$	28566.0	1.54444
$137$	−2454.00	−0.130748	−0.0653738	−	0.997861i	$-0.520824\pi$
$137$	−2454.00	−0.130748	−0.0653738	+	0.997861i	$0.520824\pi$
$138$	0	0
$139$	11884.0	0.615082	0.307541	−	0.951535i	$-0.400494\pi$
$139$	11884.0	0.615082	0.307541	+	0.951535i	$0.400494\pi$
$140$	0	0
$141$	0	0
$142$	− 5454.00i	− 0.270482i
$143$	37146.0	1.81652
$144$	0	0
$145$	0	0
$146$	− 9093.00i	− 0.426581i
$147$	0	0
$148$	5180.00i	0.236486i
$149$	− 21993.0i	− 0.990631i	−0.868713	−	0.495316i	$-0.835052\pi$
$149$	− 21993.0i	− 0.990631i	0.868713	−	0.495316i	$-0.164948\pi$
$150$	0	0
$151$	−2683.00	−0.117670	−0.0588351	−	0.998268i	$-0.518739\pi$
$151$	−2683.00	−0.117670	−0.0588351	+	0.998268i	$0.518739\pi$
$152$	−20976.0	−0.907895
$153$	0	0
$154$	7011.00	0.295623
$155$	0	0
$156$	0	0
$157$	32116.0i	1.30293i	0.758677	+	0.651467i	$0.225846\pi$
$157$	32116.0i	1.30293i	−0.758677	+	0.651467i	$0.774154\pi$
$158$	31350.0	1.25581
$159$	0	0
$160$	0	0
$161$	5700.00i	0.219899i
$162$	0	0
$163$	22790.0i	0.857767i	0.903360	+	0.428883i	$0.141093\pi$
$163$	22790.0i	0.857767i	−0.903360	+	0.428883i	$0.858907\pi$
$164$	1596.00i	0.0593397i
$165$	0	0
$166$	37899.0	1.37534
$167$	36078.0	1.29363	0.646814	−	0.762648i	$-0.276101\pi$
$167$	36078.0	1.29363	0.646814	+	0.762648i	$0.276101\pi$
$168$	0	0
$169$	−62643.0	−2.19331
$170$	0	0
$171$	0	0
$172$	6874.00i	0.232355i
$173$	19725.0	0.659060	0.329530	−	0.944145i	$-0.393110\pi$
$173$	19725.0	0.659060	0.329530	+	0.944145i	$0.393110\pi$
$174$	0	0
$175$	0	0
$176$	11685.0i	0.377228i
$177$	0	0
$178$	− 21006.0i	− 0.662984i
$179$	− 48915.0i	− 1.52664i	−0.646022	−	0.763319i	$-0.723569\pi$
$179$	− 48915.0i	− 1.52664i	0.646022	−	0.763319i	$-0.276431\pi$
$180$	0	0
$181$	−49552.0	−1.51253	−0.756265	−	0.654265i	$-0.772978\pi$
$181$	−49552.0	−1.51253	−0.756265	+	0.654265i	$0.772978\pi$
$182$	−17214.0	−0.519684
$183$	0	0
$184$	−20700.0	−0.611413
$185$	0	0
$186$	0	0
$187$	50922.0i	1.45620i
$188$	15162.0	0.428984
$189$	0	0
$190$	0	0
$191$	45390.0i	1.24421i	0.782934	+	0.622105i	$0.213722\pi$
$191$	45390.0i	1.24421i	−0.782934	+	0.622105i	$0.786278\pi$
$192$	0	0
$193$	35447.0i	0.951623i	0.879547	+	0.475811i	$0.157846\pi$
$193$	35447.0i	0.951623i	−0.879547	+	0.475811i	$0.842154\pi$
$194$	19551.0i	0.519476i
$195$	0	0
$196$	−14280.0	−0.371720
$197$	−35739.0	−0.920895	−0.460447	−	0.887687i	$-0.652311\pi$
$197$	−35739.0	−0.920895	−0.460447	+	0.887687i	$0.652311\pi$
$198$	0	0
$199$	31255.0	0.789248	0.394624	−	0.918843i	$-0.370875\pi$
$199$	31255.0	0.789248	0.394624	+	0.918843i	$0.370875\pi$
$200$	0	0
$201$	0	0
$202$	− 17757.0i	− 0.435178i
$203$	12882.0	0.312602
$204$	0	0
$205$	0	0
$206$	− 22962.0i	− 0.541097i
$207$	0	0
$208$	− 28690.0i	− 0.663138i
$209$	− 37392.0i	− 0.856024i
$210$	0	0
$211$	−15052.0	−0.338088	−0.169044	−	0.985609i	$-0.554068\pi$
$211$	−15052.0	−0.338088	−0.169044	+	0.985609i	$0.554068\pi$
$212$	11151.0	0.248109
$213$	0	0
$214$	1539.00	0.0336056
$215$	0	0
$216$	0	0
$217$	4541.00i	0.0964344i
$218$	−6972.00	−0.146705
$219$	0	0
$220$	0	0
$221$	− 125028.i	− 2.55990i
$222$	0	0
$223$	50174.0i	1.00895i	0.863427	+	0.504474i	$0.168314\pi$
$223$	50174.0i	1.00895i	−0.863427	+	0.504474i	$0.831686\pi$
$224$	15561.0i	0.310128i
$225$	0	0
$226$	−14760.0	−0.288981
$227$	−19266.0	−0.373887	−0.186943	−	0.982371i	$-0.559858\pi$
$227$	−19266.0	−0.373887	−0.186943	+	0.982371i	$0.559858\pi$
$228$	0	0
$229$	−34214.0	−0.652428	−0.326214	−	0.945296i	$-0.605773\pi$
$229$	−34214.0	−0.652428	−0.326214	+	0.945296i	$0.605773\pi$
$230$	0	0
$231$	0	0
$232$	46782.0i	0.869166i
$233$	−37386.0	−0.688648	−0.344324	−	0.938851i	$-0.611892\pi$
$233$	−37386.0	−0.688648	−0.344324	+	0.938851i	$0.611892\pi$
$234$	0	0
$235$	0	0
$236$	20454.0i	0.367244i
$237$	0	0
$238$	− 23598.0i	− 0.416602i
$239$	− 61800.0i	− 1.08191i	−0.841050	−	0.540957i	$-0.818062\pi$
$239$	− 61800.0i	− 1.08191i	0.841050	−	0.540957i	$-0.181938\pi$
$240$	0	0
$241$	41390.0	0.712625	0.356313	−	0.934367i	$-0.384034\pi$
$241$	41390.0	0.712625	0.356313	+	0.934367i	$0.384034\pi$
$242$	−1464.00	−0.0249983
$243$	0	0
$244$	2212.00	0.0371540
$245$	0	0
$246$	0	0
$247$	91808.0i	1.50483i
$248$	−16491.0	−0.268129
$249$	0	0
$250$	0	0
$251$	82818.0i	1.31455i	0.753651	+	0.657275i	$0.228291\pi$
$251$	82818.0i	1.31455i	−0.753651	+	0.657275i	$0.771709\pi$
$252$	0	0
$253$	− 36900.0i	− 0.576481i
$254$	− 74985.0i	− 1.16227i
$255$	0	0
$256$	−67151.0	−1.02464
$257$	19590.0	0.296598	0.148299	−	0.988943i	$-0.452620\pi$
$257$	19590.0	0.296598	0.148299	+	0.988943i	$0.452620\pi$
$258$	0	0
$259$	14060.0	0.209597
$260$	0	0
$261$	0	0
$262$	− 85383.0i	− 1.24385i
$263$	16692.0	0.241322	0.120661	−	0.992694i	$-0.461499\pi$
$263$	16692.0	0.241322	0.120661	+	0.992694i	$0.461499\pi$
$264$	0	0
$265$	0	0
$266$	17328.0i	0.244898i
$267$	0	0
$268$	32354.0i	0.450462i
$269$	120906.i	1.67087i	0.549587	+	0.835436i	$0.314785\pi$
$269$	120906.i	1.67087i	−0.549587	+	0.835436i	$0.685215\pi$
$270$	0	0
$271$	73739.0	1.00406	0.502029	−	0.864851i	$-0.332587\pi$
$271$	73739.0	1.00406	0.502029	+	0.864851i	$0.332587\pi$
$272$	39330.0	0.531601
$273$	0	0
$274$	−7362.00	−0.0980606
$275$	0	0
$276$	0	0
$277$	− 11996.0i	− 0.156342i	−0.996940	−	0.0781712i	$-0.975092\pi$
$277$	− 11996.0i	− 0.156342i	0.996940	−	0.0781712i	$-0.0249081\pi$
$278$	35652.0	0.461312
$279$	0	0
$280$	0	0
$281$	− 51126.0i	− 0.647484i	−0.946145	−	0.323742i	$-0.895059\pi$
$281$	− 51126.0i	− 0.647484i	0.946145	−	0.323742i	$-0.104941\pi$
$282$	0	0
$283$	− 1048.00i	− 0.0130854i	−0.999979	−	0.00654272i	$-0.997917\pi$
$283$	− 1048.00i	− 0.0130854i	0.999979	−	0.00654272i	$-0.00208263\pi$
$284$	12726.0i	0.157781i
$285$	0	0
$286$	111438.	1.36239
$287$	4332.00	0.0525926
$288$	0	0
$289$	87875.0	1.05213
$290$	0	0
$291$	0	0
$292$	21217.0i	0.248839i
$293$	64182.0	0.747615	0.373807	−	0.927506i	$-0.378052\pi$
$293$	64182.0	0.747615	0.373807	+	0.927506i	$0.378052\pi$
$294$	0	0
$295$	0	0
$296$	51060.0i	0.582770i
$297$	0	0
$298$	− 65979.0i	− 0.742973i
$299$	90600.0i	1.01341i
$300$	0	0
$301$	18658.0	0.205936
$302$	−8049.00	−0.0882527
$303$	0	0
$304$	−28880.0	−0.312500
$305$	0	0
$306$	0	0
$307$	− 154154.i	− 1.63560i	−0.575500	−	0.817802i	$-0.695193\pi$
$307$	− 154154.i	− 1.63560i	0.575500	−	0.817802i	$-0.304807\pi$
$308$	−16359.0	−0.172447
$309$	0	0
$310$	0	0
$311$	− 94080.0i	− 0.972695i	−0.873766	−	0.486347i	$-0.838329\pi$
$311$	− 94080.0i	− 0.972695i	0.873766	−	0.486347i	$-0.161671\pi$
$312$	0	0
$313$	− 25903.0i	− 0.264400i	−0.991223	−	0.132200i	$-0.957796\pi$
$313$	− 25903.0i	− 0.264400i	0.991223	−	0.132200i	$-0.0422041\pi$
$314$	96348.0i	0.977200i
$315$	0	0
$316$	−73150.0	−0.732555
$317$	96843.0	0.963717	0.481859	−	0.876249i	$-0.339962\pi$
$317$	96843.0	0.963717	0.481859	+	0.876249i	$0.339962\pi$
$318$	0	0
$319$	−83394.0	−0.819508
$320$	0	0
$321$	0	0
$322$	17100.0i	0.164924i
$323$	−125856.	−1.20634
$324$	0	0
$325$	0	0
$326$	68370.0i	0.643325i
$327$	0	0
$328$	15732.0i	0.146230i
$329$	− 41154.0i	− 0.380207i
$330$	0	0
$331$	−164854.	−1.50468	−0.752339	−	0.658776i	$-0.771074\pi$
$331$	−164854.	−1.50468	−0.752339	+	0.658776i	$0.771074\pi$
$332$	−88431.0	−0.802284
$333$	0	0
$334$	108234.	0.970221
$335$	0	0
$336$	0	0
$337$	− 148694.i	− 1.30928i	−0.755939	−	0.654642i	$-0.772820\pi$
$337$	− 148694.i	− 1.30928i	0.755939	−	0.654642i	$-0.227180\pi$
$338$	−187929.	−1.64498
$339$	0	0
$340$	0	0
$341$	− 29397.0i	− 0.252810i
$342$	0	0
$343$	84379.0i	0.717210i
$344$	67758.0i	0.572590i
$345$	0	0
$346$	59175.0	0.494295
$347$	107673.	0.894227	0.447114	−	0.894477i	$-0.352452\pi$
$347$	107673.	0.894227	0.447114	+	0.894477i	$0.352452\pi$
$348$	0	0
$349$	−127520.	−1.04695	−0.523477	−	0.852040i	$-0.675365\pi$
$349$	−127520.	−1.04695	−0.523477	+	0.852040i	$0.675365\pi$
$350$	0	0
$351$	0	0
$352$	− 100737.i	− 0.813025i
$353$	−142104.	−1.14040	−0.570200	−	0.821506i	$-0.693134\pi$
$353$	−142104.	−1.14040	−0.570200	+	0.821506i	$0.693134\pi$
$354$	0	0
$355$	0	0
$356$	49014.0i	0.386741i
$357$	0	0
$358$	− 146745.i	− 1.14498i
$359$	− 19422.0i	− 0.150697i	−0.997157	−	0.0753486i	$-0.975993\pi$
$359$	− 19422.0i	− 0.150697i	0.997157	−	0.0753486i	$-0.0240069\pi$
$360$	0	0
$361$	−37905.0	−0.290859
$362$	−148656.	−1.13440
$363$	0	0
$364$	40166.0	0.303149
$365$	0	0
$366$	0	0
$367$	151345.i	1.12366i	0.827252	+	0.561831i	$0.189903\pi$
$367$	151345.i	1.12366i	−0.827252	+	0.561831i	$0.810097\pi$
$368$	−28500.0	−0.210450
$369$	0	0
$370$	0	0
$371$	− 30267.0i	− 0.219898i
$372$	0	0
$373$	237506.i	1.70709i	0.521018	+	0.853546i	$0.325553\pi$
$373$	237506.i	1.70709i	−0.521018	+	0.853546i	$0.674447\pi$
$374$	152766.i	1.09215i
$375$	0	0
$376$	149454.	1.05714
$377$	204756.	1.44063
$378$	0	0
$379$	261952.	1.82366	0.911829	−	0.410571i	$-0.134670\pi$
$379$	261952.	1.82366	0.911829	+	0.410571i	$0.134670\pi$
$380$	0	0
$381$	0	0
$382$	136170.i	0.933157i
$383$	−87162.0	−0.594196	−0.297098	−	0.954847i	$-0.596019\pi$
$383$	−87162.0	−0.594196	−0.297098	+	0.954847i	$0.596019\pi$
$384$	0	0
$385$	0	0
$386$	106341.i	0.713717i
$387$	0	0
$388$	− 45619.0i	− 0.303028i
$389$	239343.i	1.58169i	0.612016	+	0.790845i	$0.290359\pi$
$389$	239343.i	1.58169i	−0.612016	+	0.790845i	$0.709641\pi$
$390$	0	0
$391$	−124200.	−0.812397
$392$	−140760.	−0.916025
$393$	0	0
$394$	−107217.	−0.690671
$395$	0	0
$396$	0	0
$397$	− 217154.i	− 1.37780i	−0.724855	−	0.688901i	$-0.758093\pi$
$397$	− 217154.i	− 1.37780i	0.724855	−	0.688901i	$-0.241907\pi$
$398$	93765.0	0.591936
$399$	0	0
$400$	0	0
$401$	256200.i	1.59327i	0.604458	+	0.796637i	$0.293390\pi$
$401$	256200.i	1.59327i	−0.604458	+	0.796637i	$0.706610\pi$
$402$	0	0
$403$	72178.0i	0.444421i
$404$	41433.0i	0.253854i
$405$	0	0
$406$	38646.0	0.234451
$407$	−91020.0	−0.549475
$408$	0	0
$409$	199291.	1.19135	0.595677	−	0.803224i	$-0.296884\pi$
$409$	199291.	1.19135	0.595677	+	0.803224i	$0.296884\pi$
$410$	0	0
$411$	0	0
$412$	53578.0i	0.315640i
$413$	55518.0	0.325487
$414$	0	0
$415$	0	0
$416$	247338.i	1.42924i
$417$	0	0
$418$	− 112176.i	− 0.642018i
$419$	251274.i	1.43126i	0.698478	+	0.715632i	$0.253861\pi$
$419$	251274.i	1.43126i	−0.698478	+	0.715632i	$0.746139\pi$
$420$	0	0
$421$	−30412.0	−0.171586	−0.0857928	−	0.996313i	$-0.527342\pi$
$421$	−30412.0	−0.171586	−0.0857928	+	0.996313i	$0.527342\pi$
$422$	−45156.0	−0.253566
$423$	0	0
$424$	109917.	0.611411
$425$	0	0
$426$	0	0
$427$	− 6004.00i	− 0.0329295i
$428$	−3591.00	−0.0196032
$429$	0	0
$430$	0	0
$431$	− 161730.i	− 0.870635i	−0.900277	−	0.435317i	$-0.856636\pi$
$431$	− 161730.i	− 0.870635i	0.900277	−	0.435317i	$-0.143364\pi$
$432$	0	0
$433$	− 213541.i	− 1.13895i	−0.822008	−	0.569476i	$-0.807146\pi$
$433$	− 213541.i	− 1.13895i	0.822008	−	0.569476i	$-0.192854\pi$
$434$	13623.0i	0.0723258i
$435$	0	0
$436$	16268.0	0.0855778
$437$	91200.0	0.477564
$438$	0	0
$439$	−66725.0	−0.346226	−0.173113	−	0.984902i	$-0.555383\pi$
$439$	−66725.0	−0.346226	−0.173113	+	0.984902i	$0.555383\pi$
$440$	0	0
$441$	0	0
$442$	− 375084.i	− 1.91992i
$443$	−274170.	−1.39705	−0.698526	−	0.715585i	$-0.746160\pi$
$443$	−274170.	−1.39705	−0.698526	+	0.715585i	$0.746160\pi$
$444$	0	0
$445$	0	0
$446$	150522.i	0.756711i
$447$	0	0
$448$	75563.0i	0.376490i
$449$	− 233784.i	− 1.15964i	−0.814746	−	0.579819i	$-0.803123\pi$
$449$	− 233784.i	− 1.15964i	0.814746	−	0.579819i	$-0.196877\pi$
$450$	0	0
$451$	−28044.0	−0.137875
$452$	34440.0	0.168572
$453$	0	0
$454$	−57798.0	−0.280415
$455$	0	0
$456$	0	0
$457$	90667.0i	0.434127i	0.976157	+	0.217064i	$0.0696479\pi$
$457$	90667.0i	0.434127i	−0.976157	+	0.217064i	$0.930352\pi$
$458$	−102642.	−0.489321
$459$	0	0
$460$	0	0
$461$	− 201957.i	− 0.950292i	−0.879907	−	0.475146i	$-0.842395\pi$
$461$	− 201957.i	− 0.950292i	0.879907	−	0.475146i	$-0.157605\pi$
$462$	0	0
$463$	− 323977.i	− 1.51131i	−0.654973	−	0.755653i	$-0.727320\pi$
$463$	− 323977.i	− 1.51131i	0.654973	−	0.755653i	$-0.272680\pi$
$464$	64410.0i	0.299170i
$465$	0	0
$466$	−112158.	−0.516486
$467$	−76941.0	−0.352796	−0.176398	−	0.984319i	$-0.556445\pi$
$467$	−76941.0	−0.352796	−0.176398	+	0.984319i	$0.556445\pi$
$468$	0	0
$469$	87818.0	0.399244
$470$	0	0
$471$	0	0
$472$	201618.i	0.904993i
$473$	−120786.	−0.539876
$474$	0	0
$475$	0	0
$476$	55062.0i	0.243018i
$477$	0	0
$478$	− 185400.i	− 0.811435i
$479$	193218.i	0.842125i	0.907032	+	0.421062i	$0.138343\pi$
$479$	193218.i	0.842125i	−0.907032	+	0.421062i	$0.861657\pi$
$480$	0	0
$481$	223480.	0.965936
$482$	124170.	0.534469
$483$	0	0
$484$	3416.00	0.0145823
$485$	0	0
$486$	0	0
$487$	34882.0i	0.147077i	0.997292	+	0.0735383i	$0.0234291\pi$
$487$	34882.0i	0.147077i	−0.997292	+	0.0735383i	$0.976571\pi$
$488$	21804.0	0.0915580
$489$	0	0
$490$	0	0
$491$	− 217047.i	− 0.900307i	−0.892951	−	0.450154i	$-0.851369\pi$
$491$	− 217047.i	− 0.900307i	0.892951	−	0.450154i	$-0.148631\pi$
$492$	0	0
$493$	280692.i	1.15488i
$494$	275424.i	1.12862i
$495$	0	0
$496$	−22705.0	−0.0922907
$497$	34542.0	0.139841
$498$	0	0
$499$	−464810.	−1.86670	−0.933350	−	0.358969i	$-0.883129\pi$
$499$	−464810.	−1.86670	−0.933350	+	0.358969i	$0.883129\pi$
$500$	0	0
$501$	0	0
$502$	248454.i	0.985913i
$503$	−167580.	−0.662348	−0.331174	−	0.943570i	$-0.607445\pi$
$503$	−167580.	−0.662348	−0.331174	+	0.943570i	$0.607445\pi$
$504$	0	0
$505$	0	0
$506$	− 110700.i	− 0.432361i
$507$	0	0
$508$	174965.i	0.677991i
$509$	35697.0i	0.137783i	0.997624	+	0.0688916i	$0.0219463\pi$
$509$	35697.0i	0.137783i	−0.997624	+	0.0688916i	$0.978054\pi$
$510$	0	0
$511$	57589.0	0.220545
$512$	−182685.	−0.696888
$513$	0	0
$514$	58770.0	0.222448
$515$	0	0
$516$	0	0
$517$	266418.i	0.996741i
$518$	42180.0	0.157198
$519$	0	0
$520$	0	0
$521$	42750.0i	0.157493i	0.996895	+	0.0787464i	$0.0250917\pi$
$521$	42750.0i	0.157493i	−0.996895	+	0.0787464i	$0.974908\pi$
$522$	0	0
$523$	− 176434.i	− 0.645028i	−0.946565	−	0.322514i	$-0.895472\pi$
$523$	− 176434.i	− 0.645028i	0.946565	−	0.322514i	$-0.104528\pi$
$524$	199227.i	0.725581i
$525$	0	0
$526$	50076.0	0.180991
$527$	−98946.0	−0.356268
$528$	0	0
$529$	−189841.	−0.678389
$530$	0	0
$531$	0	0
$532$	− 40432.0i	− 0.142857i
$533$	68856.0	0.242375
$534$	0	0
$535$	0	0
$536$	318918.i	1.11007i
$537$	0	0
$538$	362718.i	1.25315i
$539$	− 250920.i	− 0.863690i
$540$	0	0
$541$	−323836.	−1.10645	−0.553223	−	0.833033i	$-0.686602\pi$
$541$	−323836.	−1.10645	−0.553223	+	0.833033i	$0.686602\pi$
$542$	221217.	0.753043
$543$	0	0
$544$	−339066.	−1.14574
$545$	0	0
$546$	0	0
$547$	223390.i	0.746602i	0.927710	+	0.373301i	$0.121774\pi$
$547$	223390.i	0.746602i	−0.927710	+	0.373301i	$0.878226\pi$
$548$	17178.0	0.0572020
$549$	0	0
$550$	0	0
$551$	− 206112.i	− 0.678891i
$552$	0	0
$553$	198550.i	0.649261i
$554$	− 35988.0i	− 0.117257i
$555$	0	0
$556$	−83188.0	−0.269098
$557$	585027.	1.88567	0.942835	−	0.333261i	$-0.108149\pi$
$557$	585027.	1.88567	0.942835	+	0.333261i	$0.108149\pi$
$558$	0	0
$559$	296564.	0.949063
$560$	0	0
$561$	0	0
$562$	− 153378.i	− 0.485613i
$563$	−84075.0	−0.265247	−0.132623	−	0.991167i	$-0.542340\pi$
$563$	−84075.0	−0.265247	−0.132623	+	0.991167i	$0.542340\pi$
$564$	0	0
$565$	0	0
$566$	− 3144.00i	− 0.00981408i
$567$	0	0
$568$	125442.i	0.388818i
$569$	− 637392.i	− 1.96871i	−0.176192	−	0.984356i	$-0.556378\pi$
$569$	− 637392.i	− 1.96871i	0.176192	−	0.984356i	$-0.443622\pi$
$570$	0	0
$571$	80726.0	0.247595	0.123797	−	0.992308i	$-0.460493\pi$
$571$	80726.0	0.247595	0.123797	+	0.992308i	$0.460493\pi$
$572$	−260022.	−0.794727
$573$	0	0
$574$	12996.0	0.0394445
$575$	0	0
$576$	0	0
$577$	− 261182.i	− 0.784498i	−0.919859	−	0.392249i	$-0.871697\pi$
$577$	− 261182.i	− 0.784498i	0.919859	−	0.392249i	$-0.128303\pi$
$578$	263625.	0.789098
$579$	0	0
$580$	0	0
$581$	240027.i	0.711063i
$582$	0	0
$583$	195939.i	0.576479i
$584$	209139.i	0.613210i
$585$	0	0
$586$	192546.	0.560711
$587$	391305.	1.13564	0.567818	−	0.823154i	$-0.307788\pi$
$587$	391305.	1.13564	0.567818	+	0.823154i	$0.307788\pi$
$588$	0	0
$589$	72656.0	0.209431
$590$	0	0
$591$	0	0
$592$	70300.0i	0.200591i
$593$	302670.	0.860716	0.430358	−	0.902658i	$-0.358387\pi$
$593$	302670.	0.860716	0.430358	+	0.902658i	$0.358387\pi$
$594$	0	0
$595$	0	0
$596$	153951.i	0.433401i
$597$	0	0
$598$	271800.i	0.760059i
$599$	− 291498.i	− 0.812422i	−0.913779	−	0.406211i	$-0.866850\pi$
$599$	− 291498.i	− 0.812422i	0.913779	−	0.406211i	$-0.133150\pi$
$600$	0	0
$601$	402173.	1.11343	0.556716	−	0.830703i	$-0.312061\pi$
$601$	402173.	1.11343	0.556716	+	0.830703i	$0.312061\pi$
$602$	55974.0	0.154452
$603$	0	0
$604$	18781.0	0.0514807
$605$	0	0
$606$	0	0
$607$	378670.i	1.02774i	0.857868	+	0.513870i	$0.171789\pi$
$607$	378670.i	1.02774i	−0.857868	+	0.513870i	$0.828211\pi$
$608$	248976.	0.673520
$609$	0	0
$610$	0	0
$611$	− 654132.i	− 1.75220i
$612$	0	0
$613$	287570.i	0.765284i	0.923897	+	0.382642i	$0.124986\pi$
$613$	287570.i	0.765284i	−0.923897	+	0.382642i	$0.875014\pi$
$614$	− 462462.i	− 1.22670i
$615$	0	0
$616$	−161253.	−0.424958
$617$	−576264.	−1.51374	−0.756870	−	0.653566i	$-0.773272\pi$
$617$	−576264.	−1.51374	−0.756870	+	0.653566i	$0.773272\pi$
$618$	0	0
$619$	−223262.	−0.582685	−0.291342	−	0.956619i	$-0.594102\pi$
$619$	−223262.	−0.582685	−0.291342	+	0.956619i	$0.594102\pi$
$620$	0	0
$621$	0	0
$622$	− 282240.i	− 0.729521i
$623$	133038.	0.342767
$624$	0	0
$625$	0	0
$626$	− 77709.0i	− 0.198300i
$627$	0	0
$628$	− 224812.i	− 0.570033i
$629$	306360.i	0.774338i
$630$	0	0
$631$	43373.0	0.108933	0.0544667	−	0.998516i	$-0.482654\pi$
$631$	43373.0	0.108933	0.0544667	+	0.998516i	$0.482654\pi$
$632$	−721050.	−1.80522
$633$	0	0
$634$	290529.	0.722788
$635$	0	0
$636$	0	0
$637$	616080.i	1.51830i
$638$	−250182.	−0.614631
$639$	0	0
$640$	0	0
$641$	− 423420.i	− 1.03052i	−0.857035	−	0.515259i	$-0.827696\pi$
$641$	− 423420.i	− 1.03052i	0.857035	−	0.515259i	$-0.172304\pi$
$642$	0	0
$643$	− 546088.i	− 1.32081i	−0.750909	−	0.660406i	$-0.770384\pi$
$643$	− 546088.i	− 1.32081i	0.750909	−	0.660406i	$-0.229616\pi$
$644$	− 39900.0i	− 0.0962058i
$645$	0	0
$646$	−377568.	−0.904753
$647$	418932.	1.00077	0.500386	−	0.865803i	$-0.333192\pi$
$647$	418932.	1.00077	0.500386	+	0.865803i	$0.333192\pi$
$648$	0	0
$649$	−359406.	−0.853289
$650$	0	0
$651$	0	0
$652$	− 159530.i	− 0.375273i
$653$	703209.	1.64914	0.824571	−	0.565758i	$-0.191417\pi$
$653$	703209.	1.64914	0.824571	+	0.565758i	$0.191417\pi$
$654$	0	0
$655$	0	0
$656$	21660.0i	0.0503328i
$657$	0	0
$658$	− 123462.i	− 0.285155i
$659$	102021.i	0.234919i	0.993078	+	0.117460i	$0.0374751\pi$
$659$	102021.i	0.234919i	−0.993078	+	0.117460i	$0.962525\pi$
$660$	0	0
$661$	230720.	0.528059	0.264029	−	0.964515i	$-0.414948\pi$
$661$	230720.	0.528059	0.264029	+	0.964515i	$0.414948\pi$
$662$	−494562.	−1.12851
$663$	0	0
$664$	−871677.	−1.97706
$665$	0	0
$666$	0	0
$667$	− 203400.i	− 0.457193i
$668$	−252546.	−0.565962
$669$	0	0
$670$	0	0
$671$	38868.0i	0.0863271i
$672$	0	0
$673$	− 469369.i	− 1.03630i	−0.855291	−	0.518149i	$-0.826621\pi$
$673$	− 469369.i	− 1.03630i	0.855291	−	0.518149i	$-0.173379\pi$
$674$	− 446082.i	− 0.981963i
$675$	0	0
$676$	438501.	0.959571
$677$	343146.	0.748689	0.374345	−	0.927290i	$-0.377868\pi$
$677$	343146.	0.748689	0.374345	+	0.927290i	$0.377868\pi$
$678$	0	0
$679$	−123823.	−0.268573
$680$	0	0
$681$	0	0
$682$	− 88191.0i	− 0.189608i
$683$	24642.0	0.0528244	0.0264122	−	0.999651i	$-0.491592\pi$
$683$	24642.0	0.0528244	0.0264122	+	0.999651i	$0.491592\pi$
$684$	0	0
$685$	0	0
$686$	253137.i	0.537907i
$687$	0	0
$688$	93290.0i	0.197087i
$689$	− 481086.i	− 1.01341i
$690$	0	0
$691$	−266500.	−0.558137	−0.279069	−	0.960271i	$-0.590026\pi$
$691$	−266500.	−0.558137	−0.279069	+	0.960271i	$0.590026\pi$
$692$	−138075.	−0.288339
$693$	0	0
$694$	323019.	0.670670
$695$	0	0
$696$	0	0
$697$	94392.0i	0.194299i
$698$	−382560.	−0.785215
$699$	0	0
$700$	0	0
$701$	− 690309.i	− 1.40478i	−0.711794	−	0.702389i	$-0.752117\pi$
$701$	− 690309.i	− 1.40478i	0.711794	−	0.702389i	$-0.247883\pi$
$702$	0	0
$703$	− 224960.i	− 0.455192i
$704$	− 489171.i	− 0.986996i
$705$	0	0
$706$	−426312.	−0.855299
$707$	112461.	0.224990
$708$	0	0
$709$	105184.	0.209246	0.104623	−	0.994512i	$-0.466636\pi$
$709$	105184.	0.209246	0.104623	+	0.994512i	$0.466636\pi$
$710$	0	0
$711$	0	0
$712$	483138.i	0.953040i
$713$	71700.0	0.141039
$714$	0	0
$715$	0	0
$716$	342405.i	0.667904i
$717$	0	0
$718$	− 58266.0i	− 0.113023i
$719$	− 704988.i	− 1.36372i	−0.731485	−	0.681858i	$-0.761172\pi$
$719$	− 704988.i	− 1.36372i	0.731485	−	0.681858i	$-0.238828\pi$
$720$	0	0
$721$	145426.	0.279751
$722$	−113715.	−0.218144
$723$	0	0
$724$	346864.	0.661732
$725$	0	0
$726$	0	0
$727$	− 126089.i	− 0.238566i	−0.992860	−	0.119283i	$-0.961940\pi$
$727$	− 126089.i	− 0.238566i	0.992860	−	0.119283i	$-0.0380596\pi$
$728$	395922.	0.747045
$729$	0	0
$730$	0	0
$731$	406548.i	0.760812i
$732$	0	0
$733$	97736.0i	0.181906i	0.995855	+	0.0909529i	$0.0289913\pi$
$733$	97736.0i	0.181906i	−0.995855	+	0.0909529i	$0.971009\pi$
$734$	454035.i	0.842747i
$735$	0	0
$736$	245700.	0.453575
$737$	−568506.	−1.04665
$738$	0	0
$739$	857158.	1.56954	0.784769	−	0.619788i	$-0.212781\pi$
$739$	857158.	1.56954	0.784769	+	0.619788i	$0.212781\pi$
$740$	0	0
$741$	0	0
$742$	− 90801.0i	− 0.164924i
$743$	909966.	1.64834	0.824171	−	0.566340i	$-0.191641\pi$
$743$	909966.	1.64834	0.824171	+	0.566340i	$0.191641\pi$
$744$	0	0
$745$	0	0
$746$	712518.i	1.28032i
$747$	0	0
$748$	− 356454.i	− 0.637089i
$749$	9747.00i	0.0173743i
$750$	0	0
$751$	61223.0	0.108551	0.0542756	−	0.998526i	$-0.482715\pi$
$751$	61223.0	0.108551	0.0542756	+	0.998526i	$0.482715\pi$
$752$	205770.	0.363870
$753$	0	0
$754$	614268.	1.08048
$755$	0	0
$756$	0	0
$757$	− 782570.i	− 1.36562i	−0.730594	−	0.682812i	$-0.760757\pi$
$757$	− 782570.i	− 1.36562i	0.730594	−	0.682812i	$-0.239243\pi$
$758$	785856.	1.36774
$759$	0	0
$760$	0	0
$761$	− 701400.i	− 1.21115i	−0.795790	−	0.605573i	$-0.792944\pi$
$761$	− 701400.i	− 1.21115i	0.795790	−	0.605573i	$-0.207056\pi$
$762$	0	0
$763$	− 44156.0i	− 0.0758474i
$764$	− 317730.i	− 0.544342i
$765$	0	0
$766$	−261486.	−0.445647
$767$	882444.	1.50002
$768$	0	0
$769$	85045.0	0.143812	0.0719062	−	0.997411i	$-0.477092\pi$
$769$	85045.0	0.143812	0.0719062	+	0.997411i	$0.477092\pi$
$770$	0	0
$771$	0	0
$772$	− 248129.i	− 0.416335i
$773$	643122.	1.07630	0.538151	−	0.842848i	$-0.319123\pi$
$773$	643122.	1.07630	0.538151	+	0.842848i	$0.319123\pi$
$774$	0	0
$775$	0	0
$776$	− 449673.i	− 0.746747i
$777$	0	0
$778$	718029.i	1.18627i
$779$	− 69312.0i	− 0.114218i
$780$	0	0
$781$	−223614.	−0.366604
$782$	−372600.	−0.609297
$783$	0	0
$784$	−193800.	−0.315298
$785$	0	0
$786$	0	0
$787$	− 1.06855e6i	− 1.72522i	−0.505869	−	0.862610i	$-0.668828\pi$
$787$	− 1.06855e6i	− 1.72522i	0.505869	−	0.862610i	$-0.331172\pi$
$788$	250173.	0.402891
$789$	0	0
$790$	0	0
$791$	− 93480.0i	− 0.149405i
$792$	0	0
$793$	− 95432.0i	− 0.151757i
$794$	− 651462.i	− 1.03335i
$795$	0	0
$796$	−218785.	−0.345296
$797$	538935.	0.848437	0.424219	−	0.905560i	$-0.360549\pi$
$797$	538935.	0.848437	0.424219	+	0.905560i	$0.360549\pi$
$798$	0	0
$799$	896724.	1.40464
$800$	0	0
$801$	0	0
$802$	768600.i	1.19496i
$803$	−372813.	−0.578176
$804$	0	0
$805$	0	0
$806$	216534.i	0.333316i
$807$	0	0
$808$	408411.i	0.625568i
$809$	− 459594.i	− 0.702227i	−0.936333	−	0.351113i	$-0.885803\pi$
$809$	− 459594.i	− 0.702227i	0.936333	−	0.351113i	$-0.114197\pi$
$810$	0	0
$811$	−961360.	−1.46165	−0.730827	−	0.682563i	$-0.760865\pi$
$811$	−961360.	−1.46165	−0.730827	+	0.682563i	$0.760865\pi$
$812$	−90174.0	−0.136763
$813$	0	0
$814$	−273060.	−0.412106
$815$	0	0
$816$	0	0
$817$	− 298528.i	− 0.447240i
$818$	597873.	0.893516
$819$	0	0
$820$	0	0
$821$	− 105666.i	− 0.156765i	−0.996923	−	0.0783825i	$-0.975024\pi$
$821$	− 105666.i	− 0.156765i	0.996923	−	0.0783825i	$-0.0249755\pi$
$822$	0	0
$823$	− 493555.i	− 0.728678i	−0.931266	−	0.364339i	$-0.881295\pi$
$823$	− 493555.i	− 0.728678i	0.931266	−	0.364339i	$-0.118705\pi$
$824$	528126.i	0.777827i
$825$	0	0
$826$	166554.	0.244115
$827$	−192870.	−0.282003	−0.141001	−	0.990009i	$-0.545032\pi$
$827$	−192870.	−0.282003	−0.141001	+	0.990009i	$0.545032\pi$
$828$	0	0
$829$	−577226.	−0.839918	−0.419959	−	0.907543i	$-0.637956\pi$
$829$	−577226.	−0.839918	−0.419959	+	0.907543i	$0.637956\pi$
$830$	0	0
$831$	0	0
$832$	1.20105e6i	1.73507i
$833$	−844560.	−1.21714
$834$	0	0
$835$	0	0
$836$	261744.i	0.374511i
$837$	0	0
$838$	753822.i	1.07345i
$839$	− 70986.0i	− 0.100844i	−0.998728	−	0.0504219i	$-0.983943\pi$
$839$	− 70986.0i	− 0.100844i	0.998728	−	0.0504219i	$-0.0160566\pi$
$840$	0	0
$841$	247597.	0.350069
$842$	−91236.0	−0.128689
$843$	0	0
$844$	105364.	0.147913
$845$	0	0
$846$	0	0
$847$	− 9272.00i	− 0.0129243i
$848$	151335.	0.210449
$849$	0	0
$850$	0	0
$851$	− 222000.i	− 0.306545i
$852$	0	0
$853$	81206.0i	0.111607i	0.998442	+	0.0558033i	$0.0177720\pi$
$853$	81206.0i	0.111607i	−0.998442	+	0.0558033i	$0.982228\pi$
$854$	− 18012.0i	− 0.0246971i
$855$	0	0
$856$	−35397.0	−0.0483080
$857$	−1.08044e6	−1.47109	−0.735543	−	0.677478i	$-0.763073\pi$
$857$	−1.08044e6	−1.47109	−0.735543	+	0.677478i	$0.763073\pi$
$858$	0	0
$859$	−503348.	−0.682153	−0.341077	−	0.940035i	$-0.610792\pi$
$859$	−503348.	−0.682153	−0.341077	+	0.940035i	$0.610792\pi$
$860$	0	0
$861$	0	0
$862$	− 485190.i	− 0.652976i
$863$	−548100.	−0.735933	−0.367966	−	0.929839i	$-0.619946\pi$
$863$	−548100.	−0.735933	−0.367966	+	0.929839i	$0.619946\pi$
$864$	0	0
$865$	0	0
$866$	− 640623.i	− 0.854214i
$867$	0	0
$868$	− 31787.0i	− 0.0421901i
$869$	− 1.28535e6i	− 1.70209i
$870$	0	0
$871$	1.39584e6	1.83993
$872$	160356.	0.210888
$873$	0	0
$874$	273600.	0.358173
$875$	0	0
$876$	0	0
$877$	− 700034.i	− 0.910165i	−0.890449	−	0.455082i	$-0.849610\pi$
$877$	− 700034.i	− 0.910165i	0.890449	−	0.455082i	$-0.150390\pi$
$878$	−200175.	−0.259669
$879$	0	0
$880$	0	0
$881$	− 806634.i	− 1.03926i	−0.854391	−	0.519631i	$-0.826070\pi$
$881$	− 806634.i	− 1.03926i	0.854391	−	0.519631i	$-0.173930\pi$
$882$	0	0
$883$	342704.i	0.439539i	0.975552	+	0.219770i	$0.0705306\pi$
$883$	342704.i	0.439539i	−0.975552	+	0.219770i	$0.929469\pi$
$884$	875196.i	1.11996i
$885$	0	0
$886$	−822510.	−1.04779
$887$	−16122.0	−0.0204914	−0.0102457	−	0.999948i	$-0.503261\pi$
$887$	−16122.0	−0.0204914	−0.0102457	+	0.999948i	$0.503261\pi$
$888$	0	0
$889$	474905.	0.600901
$890$	0	0
$891$	0	0
$892$	− 351218.i	− 0.441415i
$893$	−658464.	−0.825713
$894$	0	0
$895$	0	0
$896$	− 22287.0i	− 0.0277610i
$897$	0	0
$898$	− 701352.i	− 0.869728i
$899$	− 162042.i	− 0.200497i
$900$	0	0
$901$	659502.	0.812394
$902$	−84132.0	−0.103407
$903$	0	0
$904$	339480.	0.415410
$905$	0	0
$906$	0	0
$907$	− 1.56950e6i	− 1.90786i	−0.300028	−	0.953931i	$-0.596996\pi$
$907$	− 1.56950e6i	− 1.90786i	0.300028	−	0.953931i	$-0.403004\pi$
$908$	134862.	0.163575
$909$	0	0
$910$	0	0
$911$	500898.i	0.603549i	0.953379	+	0.301775i	$0.0975790\pi$
$911$	500898.i	0.603549i	−0.953379	+	0.301775i	$0.902421\pi$
$912$	0	0
$913$	− 1.55386e6i	− 1.86410i
$914$	272001.i	0.325595i
$915$	0	0
$916$	239498.	0.285437
$917$	540759.	0.643080
$918$	0	0
$919$	−1.03715e6	−1.22804	−0.614019	−	0.789291i	$-0.710448\pi$
$919$	−1.03715e6	−1.22804	−0.614019	+	0.789291i	$0.710448\pi$
$920$	0	0
$921$	0	0
$922$	− 605871.i	− 0.712719i
$923$	549036.	0.644462
$924$	0	0
$925$	0	0
$926$	− 971931.i	− 1.13348i
$927$	0	0
$928$	− 555282.i	− 0.644789i
$929$	128076.i	0.148401i	0.997243	+	0.0742004i	$0.0236405\pi$
$929$	128076.i	0.148401i	−0.997243	+	0.0742004i	$0.976360\pi$
$930$	0	0
$931$	620160.	0.715491
$932$	261702.	0.301283
$933$	0	0
$934$	−230823.	−0.264597
$935$	0	0
$936$	0	0
$937$	879451.i	1.00169i	0.865538	+	0.500844i	$0.166977\pi$
$937$	879451.i	1.00169i	−0.865538	+	0.500844i	$0.833023\pi$
$938$	263454.	0.299433
$939$	0	0
$940$	0	0
$941$	− 718257.i	− 0.811149i	−0.914062	−	0.405574i	$-0.867071\pi$
$941$	− 718257.i	− 0.811149i	0.914062	−	0.405574i	$-0.132929\pi$
$942$	0	0
$943$	− 68400.0i	− 0.0769188i
$944$	277590.i	0.311501i
$945$	0	0
$946$	−362358.	−0.404907
$947$	−73005.0	−0.0814053	−0.0407026	−	0.999171i	$-0.512960\pi$
$947$	−73005.0	−0.0814053	−0.0407026	+	0.999171i	$0.512960\pi$
$948$	0	0
$949$	915362.	1.01639
$950$	0	0
$951$	0	0
$952$	542754.i	0.598865i
$953$	309168.	0.340415	0.170208	−	0.985408i	$-0.445556\pi$
$953$	309168.	0.340415	0.170208	+	0.985408i	$0.445556\pi$
$954$	0	0
$955$	0	0
$956$	432600.i	0.473337i
$957$	0	0
$958$	579654.i	0.631594i
$959$	− 46626.0i	− 0.0506980i
$960$	0	0
$961$	−866400.	−0.938149
$962$	670440.	0.724452
$963$	0	0
$964$	−289730.	−0.311774
$965$	0	0
$966$	0	0
$967$	366187.i	0.391607i	0.980643	+	0.195803i	$0.0627314\pi$
$967$	366187.i	0.391607i	−0.980643	+	0.195803i	$0.937269\pi$
$968$	33672.0	0.0359350
$969$	0	0
$970$	0	0
$971$	1.43410e6i	1.52105i	0.649311	+	0.760523i	$0.275057\pi$
$971$	1.43410e6i	1.52105i	−0.649311	+	0.760523i	$0.724943\pi$
$972$	0	0
$973$	225796.i	0.238501i
$974$	104646.i	0.110307i
$975$	0	0
$976$	30020.0	0.0315145
$977$	311802.	0.326655	0.163328	−	0.986572i	$-0.447777\pi$
$977$	311802.	0.326655	0.163328	+	0.986572i	$0.447777\pi$
$978$	0	0
$979$	−861246.	−0.898591
$980$	0	0
$981$	0	0
$982$	− 651141.i	− 0.675231i
$983$	690162.	0.714240	0.357120	−	0.934059i	$-0.383759\pi$
$983$	690162.	0.714240	0.357120	+	0.934059i	$0.383759\pi$
$984$	0	0
$985$	0	0
$986$	842076.i	0.866159i
$987$	0	0
$988$	− 642656.i	− 0.658362i
$989$	− 294600.i	− 0.301190i
$990$	0	0
$991$	981875.	0.999790	0.499895	−	0.866086i	$-0.333372\pi$
$991$	981875.	0.999790	0.499895	+	0.866086i	$0.333372\pi$
$992$	195741.	0.198911
$993$	0	0
$994$	103626.	0.104881
$995$	0	0
$996$	0	0
$997$	− 241946.i	− 0.243404i	−0.992567	−	0.121702i	$-0.961165\pi$
$997$	− 241946.i	− 0.243404i	0.992567	−	0.121702i	$-0.0388353\pi$
$998$	−1.39443e6	−1.40002
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.5.d.d.674.2		2
3.2	odd	2		675.5.d.a.674.2		2
5.2	odd	4		27.5.b.c.26.2	yes	2
5.3	odd	4		675.5.c.h.26.1		2
5.4	even	2		675.5.d.a.674.1		2
15.2	even	4		27.5.b.c.26.1	✓	2
15.8	even	4		675.5.c.h.26.2		2
15.14	odd	2	inner	675.5.d.d.674.1		2
20.7	even	4		432.5.e.e.161.2		2
45.2	even	12		81.5.d.b.53.1		4
45.7	odd	12		81.5.d.b.53.2		4
45.22	odd	12		81.5.d.b.26.1		4
45.32	even	12		81.5.d.b.26.2		4
60.47	odd	4		432.5.e.e.161.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.5.b.c.26.1	✓	2	15.2	even	4
27.5.b.c.26.2	yes	2	5.2	odd	4
81.5.d.b.26.1		4	45.22	odd	12
81.5.d.b.26.2		4	45.32	even	12
81.5.d.b.53.1		4	45.2	even	12
81.5.d.b.53.2		4	45.7	odd	12
432.5.e.e.161.1		2	60.47	odd	4
432.5.e.e.161.2		2	20.7	even	4
675.5.c.h.26.1		2	5.3	odd	4
675.5.c.h.26.2		2	15.8	even	4
675.5.d.a.674.1		2	5.4	even	2
675.5.d.a.674.2		2	3.2	odd	2
675.5.d.d.674.1		2	15.14	odd	2	inner
675.5.d.d.674.2		2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	675.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000 q^{2} -7.00000 q^{4} +19.0000i q^{7} -69.0000 q^{8} -123.000i q^{11} +302.000i q^{13} +57.0000i q^{14} -95.0000 q^{16} -414.000 q^{17} +304.000 q^{19} -369.000i q^{22} +300.000 q^{23} +906.000i q^{26} -133.000i q^{28} -678.000i q^{29} +239.000 q^{31} +819.000 q^{32} -1242.00 q^{34} -740.000i q^{37} +912.000 q^{38} -228.000i q^{41} -982.000i q^{43} +861.000i q^{44} +900.000 q^{46} -2166.00 q^{47} +2040.00 q^{49} -2114.00i q^{52} -1593.00 q^{53} -1311.00i q^{56} -2034.00i q^{58} -2922.00i q^{59} -316.000 q^{61} +717.000 q^{62} +3977.00 q^{64} -4622.00i q^{67} +2898.00 q^{68} -1818.00i q^{71} -3031.00i q^{73} -2220.00i q^{74} -2128.00 q^{76} +2337.00 q^{77} +10450.0 q^{79} -684.000i q^{82} +12633.0 q^{83} -2946.00i q^{86} +8487.00i q^{88} -7002.00i q^{89} -5738.00 q^{91} -2100.00 q^{92} -6498.00 q^{94} +6517.00i q^{97} +6120.00 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{2} - 14 q^{4} - 138 q^{8} - 190 q^{16} - 828 q^{17} + 608 q^{19} + 600 q^{23} + 478 q^{31} + 1638 q^{32} - 2484 q^{34} + 1824 q^{38} + 1800 q^{46} - 4332 q^{47} + 4080 q^{49} - 3186 q^{53} - 632 q^{61}+ \cdots + 12240 q^{98}+O(q^{100}) \) 2 * q + 6 * q^2 - 14 * q^4 - 138 * q^8 - 190 * q^16 - 828 * q^17 + 608 * q^19 + 600 * q^23 + 478 * q^31 + 1638 * q^32 - 2484 * q^34 + 1824 * q^38 + 1800 * q^46 - 4332 * q^47 + 4080 * q^49 - 3186 * q^53 - 632 * q^61 + 1434 * q^62 + 7954 * q^64 + 5796 * q^68 - 4256 * q^76 + 4674 * q^77 + 20900 * q^79 + 25266 * q^83 - 11476 * q^91 - 4200 * q^92 - 12996 * q^94 + 12240 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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