LMFDB - Embedded newform 450.6.c.q.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,6,Mod(199,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("450.199");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	450.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:	$72.1727189158$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{4081})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2041x^{2} + 1040400 $ x^4 + 2041*x^2 + 1040400
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$32.4414i$ of defining polynomial
Character	$\chi$	$=$	450.199
Dual form			450.6.c.q.199.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{2} -16.0000 q^{4} -141.648i q^{7} +64.0000i q^{8} +O(q^{10})$	$q-4.00000i q^{2} -16.0000 q^{4} -141.648i q^{7} +64.0000i q^{8} -113.296 q^{11} +61.7038i q^{13} -566.592 q^{14} +256.000 q^{16} -1670.48i q^{17} +662.241 q^{19} +453.185i q^{22} +86.4812i q^{23} +246.815 q^{26} +2266.37i q^{28} -3233.30 q^{29} -3812.24 q^{31} -1024.00i q^{32} -6681.92 q^{34} -10249.6i q^{37} -2648.96i q^{38} +13459.4 q^{41} -4697.57i q^{43} +1812.74 q^{44} +345.925 q^{46} +15264.8i q^{47} -3257.19 q^{49} -987.260i q^{52} +498.632i q^{53} +9065.48 q^{56} +12933.2i q^{58} -15294.3 q^{59} -31811.3 q^{61} +15249.0i q^{62} -4096.00 q^{64} -49274.9i q^{67} +26727.7i q^{68} -21057.7 q^{71} +39447.8i q^{73} -40998.2 q^{74} -10595.8 q^{76} +16048.2i q^{77} -72949.3 q^{79} -53837.8i q^{82} +100897. i q^{83} -18790.3 q^{86} -7250.96i q^{88} -146714. q^{89} +8740.22 q^{91} -1383.70i q^{92} +61059.2 q^{94} +43544.9i q^{97} +13028.8i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 64 q^{4}+O(q^{10}) $ 4 * q - 64 * q^4	$ 4 q - 64 q^{4} + 1080 q^{11} + 800 q^{14} + 1024 q^{16} - 1184 q^{19} + 7120 q^{26} - 11400 q^{29} - 11416 q^{31} + 3936 q^{34} + 30840 q^{41} - 17280 q^{44} - 29280 q^{46} - 89688 q^{49} - 12800 q^{56} - 101040 q^{59} - 58252 q^{61} - 16384 q^{64} + 12360 q^{71} - 90400 q^{74} + 18944 q^{76} - 15824 q^{79} + 50560 q^{86} - 329280 q^{89} - 382832 q^{91} - 62400 q^{94}+O(q^{100}) $ 4 * q - 64 * q^4 + 1080 * q^11 + 800 * q^14 + 1024 * q^16 - 1184 * q^19 + 7120 * q^26 - 11400 * q^29 - 11416 * q^31 + 3936 * q^34 + 30840 * q^41 - 17280 * q^44 - 29280 * q^46 - 89688 * q^49 - 12800 * q^56 - 101040 * q^59 - 58252 * q^61 - 16384 * q^64 + 12360 * q^71 - 90400 * q^74 + 18944 * q^76 - 15824 * q^79 + 50560 * q^86 - 329280 * q^89 - 382832 * q^91 - 62400 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 4.00000i	− 0.707107i
$2$	− 4.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 141.648i	− 1.09261i	−0.837586	−	0.546306i	$-0.816034\pi$
$7$	− 141.648i	− 1.09261i	0.837586	−	0.546306i	$-0.183966\pi$
$8$	64.0000i	0.353553i
$9$	0	0
$10$	0	0
$11$	−113.296	−0.282315	−0.141157	−	0.989987i	$-0.545082\pi$
$11$	−113.296	−0.282315	−0.141157	+	0.989987i	$0.545082\pi$
$12$	0	0
$13$	61.7038i	0.101264i	0.998717	+	0.0506318i	$0.0161235\pi$
$13$	61.7038i	0.101264i	−0.998717	+	0.0506318i	$0.983877\pi$
$14$	−566.592	−0.772593
$15$	0	0
$16$	256.000	0.250000
$17$	− 1670.48i	− 1.40191i	−0.713207	−	0.700954i	$-0.752758\pi$
$17$	− 1670.48i	− 1.40191i	0.713207	−	0.700954i	$-0.247242\pi$
$18$	0	0
$19$	662.241	0.420854	0.210427	−	0.977610i	$-0.432515\pi$
$19$	662.241	0.420854	0.210427	+	0.977610i	$0.432515\pi$
$20$	0	0
$21$	0	0
$22$	453.185i	0.199627i
$23$	86.4812i	0.0340880i	0.999855	+	0.0170440i	$0.00542554\pi$
$23$	86.4812i	0.0340880i	−0.999855	+	0.0170440i	$0.994574\pi$
$24$	0	0
$25$	0	0
$26$	246.815	0.0716042
$27$	0	0
$28$	2266.37i	0.546306i
$29$	−3233.30	−0.713922	−0.356961	−	0.934119i	$-0.616187\pi$
$29$	−3233.30	−0.713922	−0.356961	+	0.934119i	$0.616187\pi$
$30$	0	0
$31$	−3812.24	−0.712486	−0.356243	−	0.934393i	$-0.615942\pi$
$31$	−3812.24	−0.712486	−0.356243	+	0.934393i	$0.615942\pi$
$32$	− 1024.00i	− 0.176777i
$33$	0	0
$34$	−6681.92	−0.991298
$35$	0	0
$36$	0	0
$37$	− 10249.6i	− 1.23084i	−0.788200	−	0.615419i	$-0.788987\pi$
$37$	− 10249.6i	− 1.23084i	0.788200	−	0.615419i	$-0.211013\pi$
$38$	− 2648.96i	− 0.297589i
$39$	0	0
$40$	0	0
$41$	13459.4	1.25045	0.625227	−	0.780443i	$-0.285007\pi$
$41$	13459.4	1.25045	0.625227	+	0.780443i	$0.285007\pi$
$42$	0	0
$43$	− 4697.57i	− 0.387438i	−0.981057	−	0.193719i	$-0.937945\pi$
$43$	− 4697.57i	− 0.387438i	0.981057	−	0.193719i	$-0.0620550\pi$
$44$	1812.74	0.141157
$45$	0	0
$46$	345.925	0.0241039
$47$	15264.8i	1.00797i	0.863713	+	0.503984i	$0.168133\pi$
$47$	15264.8i	1.00797i	−0.863713	+	0.503984i	$0.831867\pi$
$48$	0	0
$49$	−3257.19	−0.193800
$50$	0	0
$51$	0	0
$52$	− 987.260i	− 0.0506318i
$53$	498.632i	0.0243832i	0.999926	+	0.0121916i	$0.00388080\pi$
$53$	498.632i	0.0243832i	−0.999926	+	0.0121916i	$0.996119\pi$
$54$	0	0
$55$	0	0
$56$	9065.48	0.386296
$57$	0	0
$58$	12933.2i	0.504819i
$59$	−15294.3	−0.572005	−0.286002	−	0.958229i	$-0.592327\pi$
$59$	−15294.3	−0.572005	−0.286002	+	0.958229i	$0.592327\pi$
$60$	0	0
$61$	−31811.3	−1.09460	−0.547302	−	0.836935i	$-0.684345\pi$
$61$	−31811.3	−1.09460	−0.547302	+	0.836935i	$0.684345\pi$
$62$	15249.0i	0.503803i
$63$	0	0
$64$	−4096.00	−0.125000
$65$	0	0
$66$	0	0
$67$	− 49274.9i	− 1.34103i	−0.741895	−	0.670516i	$-0.766073\pi$
$67$	− 49274.9i	− 1.34103i	0.741895	−	0.670516i	$-0.233927\pi$
$68$	26727.7i	0.700954i
$69$	0	0
$70$	0	0
$71$	−21057.7	−0.495752	−0.247876	−	0.968792i	$-0.579733\pi$
$71$	−21057.7	−0.495752	−0.247876	+	0.968792i	$0.579733\pi$
$72$	0	0
$73$	39447.8i	0.866394i	0.901299	+	0.433197i	$0.142615\pi$
$73$	39447.8i	0.866394i	−0.901299	+	0.433197i	$0.857385\pi$
$74$	−40998.2	−0.870333
$75$	0	0
$76$	−10595.8	−0.210427
$77$	16048.2i	0.308460i
$78$	0	0
$79$	−72949.3	−1.31508	−0.657542	−	0.753418i	$-0.728404\pi$
$79$	−72949.3	−1.31508	−0.657542	+	0.753418i	$0.728404\pi$
$80$	0	0
$81$	0	0
$82$	− 53837.8i	− 0.884204i
$83$	100897.i	1.60762i	0.594889	+	0.803808i	$0.297196\pi$
$83$	100897.i	1.60762i	−0.594889	+	0.803808i	$0.702804\pi$
$84$	0	0
$85$	0	0
$86$	−18790.3	−0.273960
$87$	0	0
$88$	− 7250.96i	− 0.0998134i
$89$	−146714.	−1.96334	−0.981671	−	0.190585i	$-0.938962\pi$
$89$	−146714.	−1.96334	−0.981671	+	0.190585i	$0.938962\pi$
$90$	0	0
$91$	8740.22	0.110642
$92$	− 1383.70i	− 0.0170440i
$93$	0	0
$94$	61059.2	0.712741
$95$	0	0
$96$	0	0
$97$	43544.9i	0.469903i	0.972007	+	0.234951i	$0.0754931\pi$
$97$	43544.9i	0.469903i	−0.972007	+	0.234951i	$0.924507\pi$
$98$	13028.8i	0.137037i
$99$	0	0
$100$	0	0
$101$	49966.3	0.487386	0.243693	−	0.969852i	$-0.421641\pi$
$101$	49966.3	0.487386	0.243693	+	0.969852i	$0.421641\pi$
$102$	0	0
$103$	− 50579.6i	− 0.469766i	−0.972024	−	0.234883i	$-0.924529\pi$
$103$	− 50579.6i	− 0.469766i	0.972024	−	0.234883i	$-0.0754708\pi$
$104$	−3949.04	−0.0358021
$105$	0	0
$106$	1994.53	0.0172415
$107$	191553.i	1.61744i	0.588193	+	0.808721i	$0.299840\pi$
$107$	191553.i	1.61744i	−0.588193	+	0.808721i	$0.700160\pi$
$108$	0	0
$109$	−71531.6	−0.576676	−0.288338	−	0.957529i	$-0.593103\pi$
$109$	−71531.6	−0.576676	−0.288338	+	0.957529i	$0.593103\pi$
$110$	0	0
$111$	0	0
$112$	− 36261.9i	− 0.273153i
$113$	163395.i	1.20377i	0.798583	+	0.601885i	$0.205583\pi$
$113$	163395.i	1.20377i	−0.798583	+	0.601885i	$0.794417\pi$
$114$	0	0
$115$	0	0
$116$	51732.7	0.356961
$117$	0	0
$118$	61177.2i	0.404468i
$119$	−236621.	−1.53174
$120$	0	0
$121$	−148215.	−0.920298
$122$	127245.i	0.774002i
$123$	0	0
$124$	60995.8	0.356243
$125$	0	0
$126$	0	0
$127$	34350.7i	0.188984i	0.995526	+	0.0944922i	$0.0301228\pi$
$127$	34350.7i	0.188984i	−0.995526	+	0.0944922i	$0.969877\pi$
$128$	16384.0i	0.0883883i
$129$	0	0
$130$	0	0
$131$	349293.	1.77833	0.889164	−	0.457588i	$-0.151287\pi$
$131$	349293.	1.77833	0.889164	+	0.457588i	$0.151287\pi$
$132$	0	0
$133$	− 93805.1i	− 0.459830i
$134$	−197100.	−0.948253
$135$	0	0
$136$	106911.	0.495649
$137$	274334.i	1.24876i	0.781122	+	0.624379i	$0.214648\pi$
$137$	274334.i	1.24876i	−0.781122	+	0.624379i	$0.785352\pi$
$138$	0	0
$139$	47842.2	0.210027	0.105013	−	0.994471i	$-0.466511\pi$
$139$	47842.2	0.210027	0.105013	+	0.994471i	$0.466511\pi$
$140$	0	0
$141$	0	0
$142$	84230.6i	0.350549i
$143$	− 6990.80i	− 0.0285882i
$144$	0	0
$145$	0	0
$146$	157791.	0.612633
$147$	0	0
$148$	163993.i	0.615419i
$149$	−335645.	−1.23855	−0.619276	−	0.785174i	$-0.712574\pi$
$149$	−335645.	−1.23855	−0.619276	+	0.785174i	$0.712574\pi$
$150$	0	0
$151$	336370.	1.20054	0.600268	−	0.799799i	$-0.295061\pi$
$151$	336370.	1.20054	0.600268	+	0.799799i	$0.295061\pi$
$152$	42383.4i	0.148794i
$153$	0	0
$154$	64192.8	0.218114
$155$	0	0
$156$	0	0
$157$	381573.i	1.23546i	0.786391	+	0.617729i	$0.211947\pi$
$157$	381573.i	1.23546i	−0.786391	+	0.617729i	$0.788053\pi$
$158$	291797.i	0.929905i
$159$	0	0
$160$	0	0
$161$	12249.9	0.0372450
$162$	0	0
$163$	− 579006.i	− 1.70692i	−0.521155	−	0.853462i	$-0.674499\pi$
$163$	− 579006.i	− 1.70692i	0.521155	−	0.853462i	$-0.325501\pi$
$164$	−215351.	−0.625227
$165$	0	0
$166$	403587.	1.13676
$167$	− 128108.i	− 0.355455i	−0.984080	−	0.177728i	$-0.943125\pi$
$167$	− 128108.i	− 0.355455i	0.984080	−	0.177728i	$-0.0568746\pi$
$168$	0	0
$169$	367486.	0.989746
$170$	0	0
$171$	0	0
$172$	75161.2i	0.193719i
$173$	487187.i	1.23760i	0.785548	+	0.618800i	$0.212381\pi$
$173$	487187.i	1.23760i	−0.785548	+	0.618800i	$0.787619\pi$
$174$	0	0
$175$	0	0
$176$	−29003.8	−0.0705787
$177$	0	0
$178$	586855.i	1.38829i
$179$	−848235.	−1.97872	−0.989359	−	0.145497i	$-0.953522\pi$
$179$	−848235.	−1.97872	−0.989359	+	0.145497i	$0.953522\pi$
$180$	0	0
$181$	−575423.	−1.30554	−0.652771	−	0.757555i	$-0.726394\pi$
$181$	−575423.	−1.30554	−0.652771	+	0.757555i	$0.726394\pi$
$182$	− 34960.9i	− 0.0782355i
$183$	0	0
$184$	−5534.79	−0.0120519
$185$	0	0
$186$	0	0
$187$	189259.i	0.395779i
$188$	− 244237.i	− 0.503984i
$189$	0	0
$190$	0	0
$191$	−18362.7	−0.0364211	−0.0182105	−	0.999834i	$-0.505797\pi$
$191$	−18362.7	−0.0364211	−0.0182105	+	0.999834i	$0.505797\pi$
$192$	0	0
$193$	− 349513.i	− 0.675414i	−0.941251	−	0.337707i	$-0.890349\pi$
$193$	− 349513.i	− 0.675414i	0.941251	−	0.337707i	$-0.109651\pi$
$194$	174180.	0.332272
$195$	0	0
$196$	52115.0	0.0968998
$197$	371791.i	0.682547i	0.939964	+	0.341274i	$0.110858\pi$
$197$	371791.i	0.682547i	−0.939964	+	0.341274i	$0.889142\pi$
$198$	0	0
$199$	−300569.	−0.538036	−0.269018	−	0.963135i	$-0.586699\pi$
$199$	−300569.	−0.538036	−0.269018	+	0.963135i	$0.586699\pi$
$200$	0	0
$201$	0	0
$202$	− 199865.i	− 0.344634i
$203$	457990.i	0.780039i
$204$	0	0
$205$	0	0
$206$	−202318.	−0.332175
$207$	0	0
$208$	15796.2i	0.0253159i
$209$	−75029.4	−0.118813
$210$	0	0
$211$	59499.2	0.0920036	0.0460018	−	0.998941i	$-0.485352\pi$
$211$	59499.2	0.0920036	0.0460018	+	0.998941i	$0.485352\pi$
$212$	− 7978.11i	− 0.0121916i
$213$	0	0
$214$	766210.	1.14370
$215$	0	0
$216$	0	0
$217$	539997.i	0.778470i
$218$	286127.i	0.407772i
$219$	0	0
$220$	0	0
$221$	103075.	0.141962
$222$	0	0
$223$	− 866124.i	− 1.16632i	−0.812357	−	0.583160i	$-0.801816\pi$
$223$	− 866124.i	− 1.16632i	0.812357	−	0.583160i	$-0.198184\pi$
$224$	−145048.	−0.193148
$225$	0	0
$226$	653581.	0.851194
$227$	737058.i	0.949374i	0.880155	+	0.474687i	$0.157439\pi$
$227$	737058.i	0.949374i	−0.880155	+	0.474687i	$0.842561\pi$
$228$	0	0
$229$	−846501.	−1.06669	−0.533346	−	0.845897i	$-0.679065\pi$
$229$	−846501.	−1.06669	−0.533346	+	0.845897i	$0.679065\pi$
$230$	0	0
$231$	0	0
$232$	− 206931.i	− 0.252409i
$233$	− 590760.i	− 0.712888i	−0.934317	−	0.356444i	$-0.883989\pi$
$233$	− 590760.i	− 0.712888i	0.934317	−	0.356444i	$-0.116011\pi$
$234$	0	0
$235$	0	0
$236$	244709.	0.286002
$237$	0	0
$238$	946482.i	1.08310i
$239$	1.60375e6	1.81611	0.908054	−	0.418853i	$-0.137568\pi$
$239$	1.60375e6	1.81611	0.908054	+	0.418853i	$0.137568\pi$
$240$	0	0
$241$	−347071.	−0.384925	−0.192463	−	0.981304i	$-0.561647\pi$
$241$	−347071.	−0.384925	−0.192463	+	0.981304i	$0.561647\pi$
$242$	592860.i	0.650749i
$243$	0	0
$244$	508981.	0.547302
$245$	0	0
$246$	0	0
$247$	40862.7i	0.0426172i
$248$	− 243983.i	− 0.251902i
$249$	0	0
$250$	0	0
$251$	−536438.	−0.537446	−0.268723	−	0.963217i	$-0.586602\pi$
$251$	−536438.	−0.537446	−0.268723	+	0.963217i	$0.586602\pi$
$252$	0	0
$253$	− 9797.99i	− 0.00962356i
$254$	137403.	0.133632
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 763565.i	− 0.721130i	−0.932734	−	0.360565i	$-0.882584\pi$
$257$	− 763565.i	− 0.721130i	0.932734	−	0.360565i	$-0.117416\pi$
$258$	0	0
$259$	−1.45183e6	−1.34483
$260$	0	0
$261$	0	0
$262$	− 1.39717e6i	− 1.25747i
$263$	− 341099.i	− 0.304082i	−0.988374	−	0.152041i	$-0.951415\pi$
$263$	− 341099.i	− 0.304082i	0.988374	−	0.152041i	$-0.0485846\pi$
$264$	0	0
$265$	0	0
$266$	−375221.	−0.325149
$267$	0	0
$268$	788399.i	0.670516i
$269$	750163.	0.632084	0.316042	−	0.948745i	$-0.397646\pi$
$269$	750163.	0.632084	0.316042	+	0.948745i	$0.397646\pi$
$270$	0	0
$271$	−1.64462e6	−1.36033	−0.680163	−	0.733061i	$-0.738091\pi$
$271$	−1.64462e6	−1.36033	−0.680163	+	0.733061i	$0.738091\pi$
$272$	− 427643.i	− 0.350477i
$273$	0	0
$274$	1.09734e6	0.883005
$275$	0	0
$276$	0	0
$277$	− 1.04631e6i	− 0.819337i	−0.912235	−	0.409668i	$-0.865644\pi$
$277$	− 1.04631e6i	− 0.819337i	0.912235	−	0.409668i	$-0.134356\pi$
$278$	− 191369.i	− 0.148511i
$279$	0	0
$280$	0	0
$281$	54503.6	0.0411774	0.0205887	−	0.999788i	$-0.493446\pi$
$281$	54503.6	0.0411774	0.0205887	+	0.999788i	$0.493446\pi$
$282$	0	0
$283$	150130.i	0.111429i	0.998447	+	0.0557147i	$0.0177437\pi$
$283$	150130.i	0.111429i	−0.998447	+	0.0557147i	$0.982256\pi$
$284$	336923.	0.247876
$285$	0	0
$286$	−27963.2	−0.0202149
$287$	− 1.90650e6i	− 1.36626i
$288$	0	0
$289$	−1.37065e6	−0.965344
$290$	0	0
$291$	0	0
$292$	− 631164.i	− 0.433197i
$293$	− 1.54001e6i	− 1.04799i	−0.851723	−	0.523993i	$-0.824442\pi$
$293$	− 1.54001e6i	− 1.04799i	0.851723	−	0.523993i	$-0.175558\pi$
$294$	0	0
$295$	0	0
$296$	655972.	0.435167
$297$	0	0
$298$	1.34258e6i	0.875788i
$299$	−5336.21	−0.00345188
$300$	0	0
$301$	−665402.	−0.423319
$302$	− 1.34548e6i	− 0.848907i
$303$	0	0
$304$	169534.	0.105214
$305$	0	0
$306$	0	0
$307$	914773.i	0.553946i	0.960878	+	0.276973i	$0.0893312\pi$
$307$	914773.i	0.553946i	−0.960878	+	0.276973i	$0.910669\pi$
$308$	− 256771.i	− 0.154230i
$309$	0	0
$310$	0	0
$311$	−1.59808e6	−0.936910	−0.468455	−	0.883487i	$-0.655189\pi$
$311$	−1.59808e6	−0.936910	−0.468455	+	0.883487i	$0.655189\pi$
$312$	0	0
$313$	− 2.38412e6i	− 1.37552i	−0.725938	−	0.687760i	$-0.758594\pi$
$313$	− 2.38412e6i	− 1.37552i	0.725938	−	0.687760i	$-0.241406\pi$
$314$	1.52629e6	0.873601
$315$	0	0
$316$	1.16719e6	0.657542
$317$	− 2.00013e6i	− 1.11792i	−0.829194	−	0.558960i	$-0.811200\pi$
$317$	− 2.00013e6i	− 1.11792i	0.829194	−	0.558960i	$-0.188800\pi$
$318$	0	0
$319$	366320.	0.201551
$320$	0	0
$321$	0	0
$322$	− 48999.6i	− 0.0263362i
$323$	− 1.10626e6i	− 0.589999i
$324$	0	0
$325$	0	0
$326$	−2.31602e6	−1.20698
$327$	0	0
$328$	861404.i	0.442102i
$329$	2.16223e6	1.10132
$330$	0	0
$331$	−1.48362e6	−0.744306	−0.372153	−	0.928171i	$-0.621380\pi$
$331$	−1.48362e6	−0.744306	−0.372153	+	0.928171i	$0.621380\pi$
$332$	− 1.61435e6i	− 0.803808i
$333$	0	0
$334$	−512432.	−0.251345
$335$	0	0
$336$	0	0
$337$	− 2.42490e6i	− 1.16311i	−0.813508	−	0.581553i	$-0.802445\pi$
$337$	− 2.42490e6i	− 1.16311i	0.813508	−	0.581553i	$-0.197555\pi$
$338$	− 1.46994e6i	− 0.699856i
$339$	0	0
$340$	0	0
$341$	431912.	0.201145
$342$	0	0
$343$	− 1.91931e6i	− 0.880864i
$344$	300645.	0.136980
$345$	0	0
$346$	1.94875e6	0.875116
$347$	− 1.26681e6i	− 0.564793i	−0.959298	−	0.282397i	$-0.908871\pi$
$347$	− 1.26681e6i	− 0.564793i	0.959298	−	0.282397i	$-0.0911294\pi$
$348$	0	0
$349$	−2.54257e6	−1.11740	−0.558701	−	0.829369i	$-0.688700\pi$
$349$	−2.54257e6	−1.11740	−0.558701	+	0.829369i	$0.688700\pi$
$350$	0	0
$351$	0	0
$352$	116015.i	0.0499067i
$353$	− 1.28509e6i	− 0.548906i	−0.961601	−	0.274453i	$-0.911503\pi$
$353$	− 1.28509e6i	− 0.548906i	0.961601	−	0.274453i	$-0.0884967\pi$
$354$	0	0
$355$	0	0
$356$	2.34742e6	0.981671
$357$	0	0
$358$	3.39294e6i	1.39916i
$359$	4.16251e6	1.70459	0.852294	−	0.523063i	$-0.175211\pi$
$359$	4.16251e6	1.70459	0.852294	+	0.523063i	$0.175211\pi$
$360$	0	0
$361$	−2.03754e6	−0.822882
$362$	2.30169e6i	0.923158i
$363$	0	0
$364$	−139844.	−0.0553209
$365$	0	0
$366$	0	0
$367$	1.13066e6i	0.438195i	0.975703	+	0.219098i	$0.0703113\pi$
$367$	1.13066e6i	0.438195i	−0.975703	+	0.219098i	$0.929689\pi$
$368$	22139.2i	0.00852201i
$369$	0	0
$370$	0	0
$371$	70630.3	0.0266413
$372$	0	0
$373$	− 312371.i	− 0.116251i	−0.998309	−	0.0581257i	$-0.981488\pi$
$373$	− 312371.i	− 0.116251i	0.998309	−	0.0581257i	$-0.0185124\pi$
$374$	757037.	0.279858
$375$	0	0
$376$	−976948.	−0.356371
$377$	− 199507.i	− 0.0722943i
$378$	0	0
$379$	2.19990e6	0.786692	0.393346	−	0.919390i	$-0.371317\pi$
$379$	2.19990e6	0.786692	0.393346	+	0.919390i	$0.371317\pi$
$380$	0	0
$381$	0	0
$382$	73450.8i	0.0257536i
$383$	− 3.93331e6i	− 1.37013i	−0.728482	−	0.685065i	$-0.759774\pi$
$383$	− 3.93331e6i	− 1.37013i	0.728482	−	0.685065i	$-0.240226\pi$
$384$	0	0
$385$	0	0
$386$	−1.39805e6	−0.477590
$387$	0	0
$388$	− 696719.i	− 0.234951i
$389$	4.20593e6	1.40925	0.704625	−	0.709580i	$-0.251115\pi$
$389$	4.20593e6	1.40925	0.704625	+	0.709580i	$0.251115\pi$
$390$	0	0
$391$	144465.	0.0477883
$392$	− 208460.i	− 0.0685185i
$393$	0	0
$394$	1.48716e6	0.482634
$395$	0	0
$396$	0	0
$397$	2.39484e6i	0.762606i	0.924450	+	0.381303i	$0.124525\pi$
$397$	2.39484e6i	0.762606i	−0.924450	+	0.381303i	$0.875475\pi$
$398$	1.20228e6i	0.380449i
$399$	0	0
$400$	0	0
$401$	−4.42161e6	−1.37315	−0.686577	−	0.727057i	$-0.740888\pi$
$401$	−4.42161e6	−1.37315	−0.686577	+	0.727057i	$0.740888\pi$
$402$	0	0
$403$	− 235230.i	− 0.0721488i
$404$	−799460.	−0.243693
$405$	0	0
$406$	1.83196e6	0.551571
$407$	1.16124e6i	0.347484i
$408$	0	0
$409$	2.61075e6	0.771717	0.385858	−	0.922558i	$-0.373905\pi$
$409$	2.61075e6	0.771717	0.385858	+	0.922558i	$0.373905\pi$
$410$	0	0
$411$	0	0
$412$	809273.i	0.234883i
$413$	2.16641e6i	0.624979i
$414$	0	0
$415$	0	0
$416$	63184.7	0.0179010
$417$	0	0
$418$	300117.i	0.0840138i
$419$	1.01624e6	0.282789	0.141395	−	0.989953i	$-0.454841\pi$
$419$	1.01624e6	0.282789	0.141395	+	0.989953i	$0.454841\pi$
$420$	0	0
$421$	5.96915e6	1.64137	0.820687	−	0.571379i	$-0.193591\pi$
$421$	5.96915e6	1.64137	0.820687	+	0.571379i	$0.193591\pi$
$422$	− 237997.i	− 0.0650564i
$423$	0	0
$424$	−31912.4	−0.00862076
$425$	0	0
$426$	0	0
$427$	4.50601e6i	1.19598i
$428$	− 3.06484e6i	− 0.808721i
$429$	0	0
$430$	0	0
$431$	2.97145e6	0.770504	0.385252	−	0.922811i	$-0.374115\pi$
$431$	2.97145e6	0.770504	0.385252	+	0.922811i	$0.374115\pi$
$432$	0	0
$433$	− 4.95237e6i	− 1.26938i	−0.772765	−	0.634692i	$-0.781127\pi$
$433$	− 4.95237e6i	− 1.26938i	0.772765	−	0.634692i	$-0.218873\pi$
$434$	2.15999e6	0.550461
$435$	0	0
$436$	1.14451e6	0.288338
$437$	57271.3i	0.0143461i
$438$	0	0
$439$	1.12115e6	0.277652	0.138826	−	0.990317i	$-0.455667\pi$
$439$	1.12115e6	0.277652	0.138826	+	0.990317i	$0.455667\pi$
$440$	0	0
$441$	0	0
$442$	− 412300.i	− 0.100382i
$443$	2.66233e6i	0.644544i	0.946647	+	0.322272i	$0.104447\pi$
$443$	2.66233e6i	0.644544i	−0.946647	+	0.322272i	$0.895553\pi$
$444$	0	0
$445$	0	0
$446$	−3.46449e6	−0.824713
$447$	0	0
$448$	580191.i	0.136576i
$449$	−3.88781e6	−0.910101	−0.455050	−	0.890466i	$-0.650379\pi$
$449$	−3.88781e6	−0.910101	−0.455050	+	0.890466i	$0.650379\pi$
$450$	0	0
$451$	−1.52490e6	−0.353022
$452$	− 2.61432e6i	− 0.601885i
$453$	0	0
$454$	2.94823e6	0.671309
$455$	0	0
$456$	0	0
$457$	− 2.52630e6i	− 0.565841i	−0.959143	−	0.282920i	$-0.908697\pi$
$457$	− 2.52630e6i	− 0.565841i	0.959143	−	0.282920i	$-0.0913032\pi$
$458$	3.38600e6i	0.754265i
$459$	0	0
$460$	0	0
$461$	−4.63589e6	−1.01597	−0.507985	−	0.861366i	$-0.669610\pi$
$461$	−4.63589e6	−1.01597	−0.507985	+	0.861366i	$0.669610\pi$
$462$	0	0
$463$	3.62380e6i	0.785619i	0.919620	+	0.392810i	$0.128497\pi$
$463$	3.62380e6i	0.785619i	−0.919620	+	0.392810i	$0.871503\pi$
$464$	−827724.	−0.178480
$465$	0	0
$466$	−2.36304e6	−0.504088
$467$	− 3.72757e6i	− 0.790922i	−0.918483	−	0.395461i	$-0.870585\pi$
$467$	− 3.72757e6i	− 0.790922i	0.918483	−	0.395461i	$-0.129415\pi$
$468$	0	0
$469$	−6.97970e6	−1.46523
$470$	0	0
$471$	0	0
$472$	− 978835.i	− 0.202234i
$473$	532217.i	0.109380i
$474$	0	0
$475$	0	0
$476$	3.78593e6	0.765870
$477$	0	0
$478$	− 6.41500e6i	− 1.28418i
$479$	−6.88374e6	−1.37084	−0.685418	−	0.728150i	$-0.740381\pi$
$479$	−6.88374e6	−1.37084	−0.685418	+	0.728150i	$0.740381\pi$
$480$	0	0
$481$	632436.	0.124639
$482$	1.38829e6i	0.272183i
$483$	0	0
$484$	2.37144e6	0.460149
$485$	0	0
$486$	0	0
$487$	8.58990e6i	1.64122i	0.571492	+	0.820608i	$0.306365\pi$
$487$	8.58990e6i	1.64122i	−0.571492	+	0.820608i	$0.693635\pi$
$488$	− 2.03593e6i	− 0.387001i
$489$	0	0
$490$	0	0
$491$	1.46999e6	0.275176	0.137588	−	0.990490i	$-0.456065\pi$
$491$	1.46999e6	0.275176	0.137588	+	0.990490i	$0.456065\pi$
$492$	0	0
$493$	5.40116e6i	1.00085i
$494$	163451.	0.0301349
$495$	0	0
$496$	−975934.	−0.178121
$497$	2.98278e6i	0.541664i
$498$	0	0
$499$	3.13043e6	0.562798	0.281399	−	0.959591i	$-0.409202\pi$
$499$	3.13043e6	0.562798	0.281399	+	0.959591i	$0.409202\pi$
$500$	0	0
$501$	0	0
$502$	2.14575e6i	0.380032i
$503$	2.29993e6i	0.405317i	0.979249	+	0.202658i	$0.0649581\pi$
$503$	2.29993e6i	0.405317i	−0.979249	+	0.202658i	$0.935042\pi$
$504$	0	0
$505$	0	0
$506$	−39192.0	−0.00680488
$507$	0	0
$508$	− 549611.i	− 0.0944922i
$509$	−5.16574e6	−0.883767	−0.441884	−	0.897072i	$-0.645690\pi$
$509$	−5.16574e6	−0.883767	−0.441884	+	0.897072i	$0.645690\pi$
$510$	0	0
$511$	5.58770e6	0.946632
$512$	− 262144.i	− 0.0441942i
$513$	0	0
$514$	−3.05426e6	−0.509916
$515$	0	0
$516$	0	0
$517$	− 1.72945e6i	− 0.284564i
$518$	5.80732e6i	0.950936i
$519$	0	0
$520$	0	0
$521$	1.12853e7	1.82146	0.910729	−	0.413004i	$-0.135520\pi$
$521$	1.12853e7	1.82146	0.910729	+	0.413004i	$0.135520\pi$
$522$	0	0
$523$	− 4.86022e6i	− 0.776966i	−0.921456	−	0.388483i	$-0.872999\pi$
$523$	− 4.86022e6i	− 0.776966i	0.921456	−	0.388483i	$-0.127001\pi$
$524$	−5.58869e6	−0.889164
$525$	0	0
$526$	−1.36440e6	−0.215019
$527$	6.36828e6i	0.998839i
$528$	0	0
$529$	6.42886e6	0.998838
$530$	0	0
$531$	0	0
$532$	1.50088e6i	0.229915i
$533$	830498.i	0.126625i
$534$	0	0
$535$	0	0
$536$	3.15360e6	0.474126
$537$	0	0
$538$	− 3.00065e6i	− 0.446951i
$539$	369027.	0.0547125
$540$	0	0
$541$	−5.23864e6	−0.769530	−0.384765	−	0.923015i	$-0.625717\pi$
$541$	−5.23864e6	−0.769530	−0.384765	+	0.923015i	$0.625717\pi$
$542$	6.57849e6i	0.961896i
$543$	0	0
$544$	−1.71057e6	−0.247825
$545$	0	0
$546$	0	0
$547$	− 2.88586e6i	− 0.412389i	−0.978511	−	0.206194i	$-0.933892\pi$
$547$	− 2.88586e6i	− 0.412389i	0.978511	−	0.206194i	$-0.0661079\pi$
$548$	− 4.38934e6i	− 0.624379i
$549$	0	0
$550$	0	0
$551$	−2.14122e6	−0.300457
$552$	0	0
$553$	1.03331e7i	1.43688i
$554$	−4.18525e6	−0.579359
$555$	0	0
$556$	−765476.	−0.105013
$557$	3.27936e6i	0.447869i	0.974604	+	0.223935i	$0.0718902\pi$
$557$	3.27936e6i	0.447869i	−0.974604	+	0.223935i	$0.928110\pi$
$558$	0	0
$559$	289858.	0.0392334
$560$	0	0
$561$	0	0
$562$	− 218014.i	− 0.0291168i
$563$	664023.i	0.0882901i	0.999025	+	0.0441451i	$0.0140564\pi$
$563$	664023.i	0.0882901i	−0.999025	+	0.0441451i	$0.985944\pi$
$564$	0	0
$565$	0	0
$566$	600518.	0.0787925
$567$	0	0
$568$	− 1.34769e6i	− 0.175275i
$569$	−3.00851e6	−0.389557	−0.194778	−	0.980847i	$-0.562399\pi$
$569$	−3.00851e6	−0.389557	−0.194778	+	0.980847i	$0.562399\pi$
$570$	0	0
$571$	1.07571e7	1.38072	0.690359	−	0.723467i	$-0.257453\pi$
$571$	1.07571e7	1.38072	0.690359	+	0.723467i	$0.257453\pi$
$572$	111853.i	0.0142941i
$573$	0	0
$574$	−7.62602e6	−0.966091
$575$	0	0
$576$	0	0
$577$	− 1.53859e7i	− 1.92390i	−0.273220	−	0.961952i	$-0.588089\pi$
$577$	− 1.53859e7i	− 1.92390i	0.273220	−	0.961952i	$-0.411911\pi$
$578$	5.48260e6i	0.682601i
$579$	0	0
$580$	0	0
$581$	1.42918e7	1.75650
$582$	0	0
$583$	− 56493.1i	− 0.00688374i
$584$	−2.52466e6	−0.306316
$585$	0	0
$586$	−6.16005e6	−0.741038
$587$	1.16896e7i	1.40025i	0.714020	+	0.700125i	$0.246872\pi$
$587$	1.16896e7i	1.40025i	−0.714020	+	0.700125i	$0.753128\pi$
$588$	0	0
$589$	−2.52462e6	−0.299853
$590$	0	0
$591$	0	0
$592$	− 2.62389e6i	− 0.307709i
$593$	2.29301e6i	0.267774i	0.990997	+	0.133887i	$0.0427459\pi$
$593$	2.29301e6i	0.267774i	−0.990997	+	0.133887i	$0.957254\pi$
$594$	0	0
$595$	0	0
$596$	5.37031e6	0.619276
$597$	0	0
$598$	21344.9i	0.00244084i
$599$	−1.58740e7	−1.80767	−0.903834	−	0.427884i	$-0.859259\pi$
$599$	−1.58740e7	−1.80767	−0.903834	+	0.427884i	$0.859259\pi$
$600$	0	0
$601$	−1.16506e7	−1.31571	−0.657855	−	0.753144i	$-0.728536\pi$
$601$	−1.16506e7	−1.31571	−0.657855	+	0.753144i	$0.728536\pi$
$602$	2.66161e6i	0.299332i
$603$	0	0
$604$	−5.38193e6	−0.600268
$605$	0	0
$606$	0	0
$607$	− 9.40797e6i	− 1.03639i	−0.855262	−	0.518196i	$-0.826604\pi$
$607$	− 9.40797e6i	− 1.03639i	0.855262	−	0.518196i	$-0.173396\pi$
$608$	− 678134.i	− 0.0743972i
$609$	0	0
$610$	0	0
$611$	−941896.	−0.102070
$612$	0	0
$613$	9.30509e6i	1.00016i	0.865979	+	0.500080i	$0.166696\pi$
$613$	9.30509e6i	1.00016i	−0.865979	+	0.500080i	$0.833304\pi$
$614$	3.65909e6	0.391699
$615$	0	0
$616$	−1.02708e6	−0.109057
$617$	− 268346.i	− 0.0283780i	−0.999899	−	0.0141890i	$-0.995483\pi$
$617$	− 268346.i	− 0.0283780i	0.999899	−	0.0141890i	$-0.00451665\pi$
$618$	0	0
$619$	7.97678e6	0.836760	0.418380	−	0.908272i	$-0.362598\pi$
$619$	7.97678e6	0.836760	0.418380	+	0.908272i	$0.362598\pi$
$620$	0	0
$621$	0	0
$622$	6.39233e6i	0.662496i
$623$	2.07817e7i	2.14517i
$624$	0	0
$625$	0	0
$626$	−9.53647e6	−0.972640
$627$	0	0
$628$	− 6.10516e6i	− 0.617729i
$629$	−1.71217e7	−1.72552
$630$	0	0
$631$	2.94679e6	0.294629	0.147314	−	0.989090i	$-0.452937\pi$
$631$	2.94679e6	0.294629	0.147314	+	0.989090i	$0.452937\pi$
$632$	− 4.66876e6i	− 0.464952i
$633$	0	0
$634$	−8.00053e6	−0.790489
$635$	0	0
$636$	0	0
$637$	− 200981.i	− 0.0196248i
$638$	− 1.46528e6i	− 0.142518i
$639$	0	0
$640$	0	0
$641$	−1.28989e7	−1.23996	−0.619981	−	0.784617i	$-0.712860\pi$
$641$	−1.28989e7	−1.23996	−0.619981	+	0.784617i	$0.712860\pi$
$642$	0	0
$643$	9.20262e6i	0.877777i	0.898542	+	0.438889i	$0.144628\pi$
$643$	9.20262e6i	0.877777i	−0.898542	+	0.438889i	$0.855372\pi$
$644$	−195998.	−0.0186225
$645$	0	0
$646$	−4.42504e6	−0.417192
$647$	177907.i	0.0167083i	0.999965	+	0.00835417i	$0.00265925\pi$
$647$	177907.i	0.0167083i	−0.999965	+	0.00835417i	$0.997341\pi$
$648$	0	0
$649$	1.73279e6	0.161485
$650$	0	0
$651$	0	0
$652$	9.26410e6i	0.853462i
$653$	1.11867e7i	1.02664i	0.858197	+	0.513321i	$0.171585\pi$
$653$	1.11867e7i	1.02664i	−0.858197	+	0.513321i	$0.828415\pi$
$654$	0	0
$655$	0	0
$656$	3.44562e6	0.312613
$657$	0	0
$658$	− 8.64893e6i	− 0.778749i
$659$	2.34591e6	0.210425	0.105213	−	0.994450i	$-0.466448\pi$
$659$	2.34591e6	0.210425	0.105213	+	0.994450i	$0.466448\pi$
$660$	0	0
$661$	−3.73844e6	−0.332803	−0.166401	−	0.986058i	$-0.553215\pi$
$661$	−3.73844e6	−0.332803	−0.166401	+	0.986058i	$0.553215\pi$
$662$	5.93446e6i	0.526304i
$663$	0	0
$664$	−6.45740e6	−0.568378
$665$	0	0
$666$	0	0
$667$	− 279619.i	− 0.0243362i
$668$	2.04973e6i	0.177728i
$669$	0	0
$670$	0	0
$671$	3.60410e6	0.309023
$672$	0	0
$673$	6.11773e6i	0.520658i	0.965520	+	0.260329i	$0.0838310\pi$
$673$	6.11773e6i	0.520658i	−0.965520	+	0.260329i	$0.916169\pi$
$674$	−9.69961e6	−0.822440
$675$	0	0
$676$	−5.87977e6	−0.494873
$677$	6.50221e6i	0.545242i	0.962121	+	0.272621i	$0.0878905\pi$
$677$	6.50221e6i	0.545242i	−0.962121	+	0.272621i	$0.912109\pi$
$678$	0	0
$679$	6.16805e6	0.513421
$680$	0	0
$681$	0	0
$682$	− 1.72765e6i	− 0.142231i
$683$	− 1.37285e7i	− 1.12609i	−0.826427	−	0.563044i	$-0.809630\pi$
$683$	− 1.37285e7i	− 1.12609i	0.826427	−	0.563044i	$-0.190370\pi$
$684$	0	0
$685$	0	0
$686$	−7.67722e6	−0.622865
$687$	0	0
$688$	− 1.20258e6i	− 0.0968595i
$689$	−30767.5	−0.00246913
$690$	0	0
$691$	1.36316e7	1.08606	0.543028	−	0.839715i	$-0.317278\pi$
$691$	1.36316e7	1.08606	0.543028	+	0.839715i	$0.317278\pi$
$692$	− 7.79499e6i	− 0.618800i
$693$	0	0
$694$	−5.06726e6	−0.399369
$695$	0	0
$696$	0	0
$697$	− 2.24837e7i	− 1.75302i
$698$	1.01703e7i	0.790123i
$699$	0	0
$700$	0	0
$701$	−2.08567e6	−0.160306	−0.0801531	−	0.996783i	$-0.525541\pi$
$701$	−2.08567e6	−0.160306	−0.0801531	+	0.996783i	$0.525541\pi$
$702$	0	0
$703$	− 6.78767e6i	− 0.518003i
$704$	464061.	0.0352894
$705$	0	0
$706$	−5.14037e6	−0.388135
$707$	− 7.07763e6i	− 0.532524i
$708$	0	0
$709$	−1.23349e7	−0.921554	−0.460777	−	0.887516i	$-0.652429\pi$
$709$	−1.23349e7	−0.921554	−0.460777	+	0.887516i	$0.652429\pi$
$710$	0	0
$711$	0	0
$712$	− 9.38968e6i	− 0.694146i
$713$	− 329687.i	− 0.0242872i
$714$	0	0
$715$	0	0
$716$	1.35718e7	0.989359
$717$	0	0
$718$	− 1.66500e7i	− 1.20533i
$719$	−2.83517e6	−0.204530	−0.102265	−	0.994757i	$-0.532609\pi$
$719$	−2.83517e6	−0.204530	−0.102265	+	0.994757i	$0.532609\pi$
$720$	0	0
$721$	−7.16450e6	−0.513272
$722$	8.15015e6i	0.581865i
$723$	0	0
$724$	9.20677e6	0.652771
$725$	0	0
$726$	0	0
$727$	1.19423e7i	0.838017i	0.907982	+	0.419009i	$0.137622\pi$
$727$	1.19423e7i	0.838017i	−0.907982	+	0.419009i	$0.862378\pi$
$728$	559374.i	0.0391178i
$729$	0	0
$730$	0	0
$731$	−7.84721e6	−0.543152
$732$	0	0
$733$	2.50999e7i	1.72549i	0.505642	+	0.862744i	$0.331256\pi$
$733$	2.50999e7i	1.72549i	−0.505642	+	0.862744i	$0.668744\pi$
$734$	4.52265e6	0.309851
$735$	0	0
$736$	88556.7	0.00602597
$737$	5.58267e6i	0.378593i
$738$	0	0
$739$	1.66189e6	0.111942	0.0559708	−	0.998432i	$-0.482175\pi$
$739$	1.66189e6	0.111942	0.0559708	+	0.998432i	$0.482175\pi$
$740$	0	0
$741$	0	0
$742$	− 282521.i	− 0.0188383i
$743$	988158.i	0.0656681i	0.999461	+	0.0328340i	$0.0104533\pi$
$743$	988158.i	0.0656681i	−0.999461	+	0.0328340i	$0.989547\pi$
$744$	0	0
$745$	0	0
$746$	−1.24948e6	−0.0822022
$747$	0	0
$748$	− 3.02815e6i	− 0.197890i
$749$	2.71331e7	1.76724
$750$	0	0
$751$	2.42260e7	1.56741	0.783704	−	0.621135i	$-0.213328\pi$
$751$	2.42260e7	1.56741	0.783704	+	0.621135i	$0.213328\pi$
$752$	3.90779e6i	0.251992i
$753$	0	0
$754$	−798026.	−0.0511198
$755$	0	0
$756$	0	0
$757$	8.03239e6i	0.509454i	0.967013	+	0.254727i	$0.0819857\pi$
$757$	8.03239e6i	0.509454i	−0.967013	+	0.254727i	$0.918014\pi$
$758$	− 8.79960e6i	− 0.556275i
$759$	0	0
$760$	0	0
$761$	8.44956e6	0.528899	0.264449	−	0.964400i	$-0.414810\pi$
$761$	8.44956e6	0.528899	0.264449	+	0.964400i	$0.414810\pi$
$762$	0	0
$763$	1.01323e7i	0.630083i
$764$	293803.	0.0182105
$765$	0	0
$766$	−1.57333e7	−0.968828
$767$	− 943716.i	− 0.0579232i
$768$	0	0
$769$	1.68674e7	1.02857	0.514285	−	0.857620i	$-0.328057\pi$
$769$	1.68674e7	1.02857	0.514285	+	0.857620i	$0.328057\pi$
$770$	0	0
$771$	0	0
$772$	5.59220e6i	0.337707i
$773$	− 2.37205e7i	− 1.42782i	−0.700236	−	0.713912i	$-0.746922\pi$
$773$	− 2.37205e7i	− 1.42782i	0.700236	−	0.713912i	$-0.253078\pi$
$774$	0	0
$775$	0	0
$776$	−2.78687e6	−0.166136
$777$	0	0
$778$	− 1.68237e7i	− 0.996490i
$779$	8.91339e6	0.526259
$780$	0	0
$781$	2.38575e6	0.139958
$782$	− 577861.i	− 0.0337914i
$783$	0	0
$784$	−833840.	−0.0484499
$785$	0	0
$786$	0	0
$787$	2.21145e7i	1.27274i	0.771384	+	0.636370i	$0.219565\pi$
$787$	2.21145e7i	1.27274i	−0.771384	+	0.636370i	$0.780435\pi$
$788$	− 5.94865e6i	− 0.341274i
$789$	0	0
$790$	0	0
$791$	2.31446e7	1.31525
$792$	0	0
$793$	− 1.96288e6i	− 0.110844i
$794$	9.57936e6	0.539244
$795$	0	0
$796$	4.80910e6	0.269018
$797$	− 1.86780e6i	− 0.104156i	−0.998643	−	0.0520779i	$-0.983416\pi$
$797$	− 1.86780e6i	− 0.104156i	0.998643	−	0.0520779i	$-0.0165844\pi$
$798$	0	0
$799$	2.54996e7	1.41308
$800$	0	0
$801$	0	0
$802$	1.76864e7i	0.970967i
$803$	− 4.46928e6i	− 0.244596i
$804$	0	0
$805$	0	0
$806$	−940918.	−0.0510169
$807$	0	0
$808$	3.19784e6i	0.172317i
$809$	−203197.	−0.0109156	−0.00545778	−	0.999985i	$-0.501737\pi$
$809$	−203197.	−0.0109156	−0.00545778	+	0.999985i	$0.501737\pi$
$810$	0	0
$811$	2.27341e7	1.21374	0.606870	−	0.794801i	$-0.292425\pi$
$811$	2.27341e7	1.21374	0.606870	+	0.794801i	$0.292425\pi$
$812$	− 7.32785e6i	− 0.390019i
$813$	0	0
$814$	4.64494e6	0.245708
$815$	0	0
$816$	0	0
$817$	− 3.11092e6i	− 0.163055i
$818$	− 1.04430e7i	− 0.545686i
$819$	0	0
$820$	0	0
$821$	−2.56056e7	−1.32579	−0.662897	−	0.748710i	$-0.730673\pi$
$821$	−2.56056e7	−1.32579	−0.662897	+	0.748710i	$0.730673\pi$
$822$	0	0
$823$	− 7.10878e6i	− 0.365844i	−0.983127	−	0.182922i	$-0.941444\pi$
$823$	− 7.10878e6i	− 0.365844i	0.983127	−	0.182922i	$-0.0585555\pi$
$824$	3.23709e6	0.166087
$825$	0	0
$826$	8.66563e6	0.441927
$827$	− 2.69334e7i	− 1.36939i	−0.728830	−	0.684695i	$-0.759935\pi$
$827$	− 2.69334e7i	− 1.36939i	0.728830	−	0.684695i	$-0.240065\pi$
$828$	0	0
$829$	−2.89570e7	−1.46341	−0.731707	−	0.681619i	$-0.761276\pi$
$829$	−2.89570e7	−1.46341	−0.731707	+	0.681619i	$0.761276\pi$
$830$	0	0
$831$	0	0
$832$	− 252739.i	− 0.0126579i
$833$	5.44107e6i	0.271689i
$834$	0	0
$835$	0	0
$836$	1.20047e6	0.0594067
$837$	0	0
$838$	− 4.06498e6i	− 0.199962i
$839$	1.95591e7	0.959279	0.479639	−	0.877466i	$-0.340767\pi$
$839$	1.95591e7	0.959279	0.479639	+	0.877466i	$0.340767\pi$
$840$	0	0
$841$	−1.00569e7	−0.490316
$842$	− 2.38766e7i	− 1.16063i
$843$	0	0
$844$	−951987.	−0.0460018
$845$	0	0
$846$	0	0
$847$	2.09944e7i	1.00553i
$848$	127650.i	0.00609580i
$849$	0	0
$850$	0	0
$851$	886393.	0.0419568
$852$	0	0
$853$	1.74251e7i	0.819981i	0.912090	+	0.409990i	$0.134468\pi$
$853$	1.74251e7i	0.819981i	−0.912090	+	0.409990i	$0.865532\pi$
$854$	1.80241e7	0.845684
$855$	0	0
$856$	−1.22594e7	−0.571852
$857$	− 2.03324e7i	− 0.945663i	−0.881153	−	0.472831i	$-0.843232\pi$
$857$	− 2.03324e7i	− 0.945663i	0.881153	−	0.472831i	$-0.156768\pi$
$858$	0	0
$859$	1.19607e7	0.553061	0.276530	−	0.961005i	$-0.410815\pi$
$859$	1.19607e7	0.553061	0.276530	+	0.961005i	$0.410815\pi$
$860$	0	0
$861$	0	0
$862$	− 1.18858e7i	− 0.544829i
$863$	− 3.87101e7i	− 1.76928i	−0.466274	−	0.884640i	$-0.654404\pi$
$863$	− 3.87101e7i	− 1.76928i	0.466274	−	0.884640i	$-0.345596\pi$
$864$	0	0
$865$	0	0
$866$	−1.98095e7	−0.897590
$867$	0	0
$868$	− 8.63995e6i	− 0.389235i
$869$	8.26488e6	0.371268
$870$	0	0
$871$	3.04045e6	0.135798
$872$	− 4.57802e6i	− 0.203886i
$873$	0	0
$874$	229085.	0.0101442
$875$	0	0
$876$	0	0
$877$	− 3.76410e7i	− 1.65258i	−0.563247	−	0.826289i	$-0.690448\pi$
$877$	− 3.76410e7i	− 1.65258i	0.563247	−	0.826289i	$-0.309552\pi$
$878$	− 4.48459e6i	− 0.196330i
$879$	0	0
$880$	0	0
$881$	−2.92452e6	−0.126945	−0.0634723	−	0.997984i	$-0.520217\pi$
$881$	−2.92452e6	−0.126945	−0.0634723	+	0.997984i	$0.520217\pi$
$882$	0	0
$883$	− 3.90249e7i	− 1.68438i	−0.539181	−	0.842190i	$-0.681266\pi$
$883$	− 3.90249e7i	− 1.68438i	0.539181	−	0.842190i	$-0.318734\pi$
$884$	−1.64920e6	−0.0709811
$885$	0	0
$886$	1.06493e7	0.455761
$887$	− 1.16063e7i	− 0.495318i	−0.968847	−	0.247659i	$-0.920339\pi$
$887$	− 1.16063e7i	− 0.495318i	0.968847	−	0.247659i	$-0.0796613\pi$
$888$	0	0
$889$	4.86571e6	0.206487
$890$	0	0
$891$	0	0
$892$	1.38580e7i	0.583160i
$893$	1.01090e7i	0.424208i
$894$	0	0
$895$	0	0
$896$	2.32076e6	0.0965741
$897$	0	0
$898$	1.55513e7i	0.643538i
$899$	1.23261e7	0.508659
$900$	0	0
$901$	832955.	0.0341830
$902$	6.09962e6i	0.249624i
$903$	0	0
$904$	−1.04573e7	−0.425597
$905$	0	0
$906$	0	0
$907$	3.80327e7i	1.53511i	0.640984	+	0.767554i	$0.278527\pi$
$907$	3.80327e7i	1.53511i	−0.640984	+	0.767554i	$0.721473\pi$
$908$	− 1.17929e7i	− 0.474687i
$909$	0	0
$910$	0	0
$911$	−4.76995e7	−1.90422	−0.952112	−	0.305748i	$-0.901093\pi$
$911$	−4.76995e7	−1.90422	−0.952112	+	0.305748i	$0.901093\pi$
$912$	0	0
$913$	− 1.14312e7i	− 0.453854i
$914$	−1.01052e7	−0.400110
$915$	0	0
$916$	1.35440e7	0.533346
$917$	− 4.94767e7i	− 1.94302i
$918$	0	0
$919$	3.06445e7	1.19692	0.598459	−	0.801154i	$-0.295780\pi$
$919$	3.06445e7	1.19692	0.598459	+	0.801154i	$0.295780\pi$
$920$	0	0
$921$	0	0
$922$	1.85436e7i	0.718400i
$923$	− 1.29934e6i	− 0.0502016i
$924$	0	0
$925$	0	0
$926$	1.44952e7	0.555517
$927$	0	0
$928$	3.31090e6i	0.126205i
$929$	4.20442e7	1.59833	0.799165	−	0.601112i	$-0.205275\pi$
$929$	4.20442e7	1.59833	0.799165	+	0.601112i	$0.205275\pi$
$930$	0	0
$931$	−2.15704e6	−0.0815614
$932$	9.45216e6i	0.356444i
$933$	0	0
$934$	−1.49103e7	−0.559267
$935$	0	0
$936$	0	0
$937$	− 2.09921e7i	− 0.781101i	−0.920582	−	0.390551i	$-0.872285\pi$
$937$	− 2.09921e7i	− 0.781101i	0.920582	−	0.390551i	$-0.127715\pi$
$938$	2.79188e7i	1.03607i
$939$	0	0
$940$	0	0
$941$	−1.06080e7	−0.390536	−0.195268	−	0.980750i	$-0.562558\pi$
$941$	−1.06080e7	−0.390536	−0.195268	+	0.980750i	$0.562558\pi$
$942$	0	0
$943$	1.16399e6i	0.0426255i
$944$	−3.91534e6	−0.143001
$945$	0	0
$946$	2.12887e6	0.0773430
$947$	2.66948e7i	0.967280i	0.875267	+	0.483640i	$0.160686\pi$
$947$	2.66948e7i	0.967280i	−0.875267	+	0.483640i	$0.839314\pi$
$948$	0	0
$949$	−2.43408e6	−0.0877341
$950$	0	0
$951$	0	0
$952$	− 1.51437e7i	− 0.541552i
$953$	− 2.68015e7i	− 0.955932i	−0.878378	−	0.477966i	$-0.841374\pi$
$953$	− 2.68015e7i	− 0.955932i	0.878378	−	0.477966i	$-0.158626\pi$
$954$	0	0
$955$	0	0
$956$	−2.56600e7	−0.908054
$957$	0	0
$958$	2.75350e7i	0.969328i
$959$	3.88589e7	1.36441
$960$	0	0
$961$	−1.40960e7	−0.492364
$962$	− 2.52974e6i	− 0.0881331i
$963$	0	0
$964$	5.55314e6	0.192463
$965$	0	0
$966$	0	0
$967$	3.43270e7i	1.18051i	0.807217	+	0.590255i	$0.200973\pi$
$967$	3.43270e7i	1.18051i	−0.807217	+	0.590255i	$0.799027\pi$
$968$	− 9.48576e6i	− 0.325375i
$969$	0	0
$970$	0	0
$971$	1.43606e7	0.488791	0.244396	−	0.969676i	$-0.421410\pi$
$971$	1.43606e7	0.488791	0.244396	+	0.969676i	$0.421410\pi$
$972$	0	0
$973$	− 6.77676e6i	− 0.229477i
$974$	3.43596e7	1.16051
$975$	0	0
$976$	−8.14370e6	−0.273651
$977$	1.16248e7i	0.389627i	0.980840	+	0.194813i	$0.0624102\pi$
$977$	1.16248e7i	0.389627i	−0.980840	+	0.194813i	$0.937590\pi$
$978$	0	0
$979$	1.66221e7	0.554280
$980$	0	0
$981$	0	0
$982$	− 5.87996e6i	− 0.194579i
$983$	− 3.11250e6i	− 0.102737i	−0.998680	−	0.0513683i	$-0.983642\pi$
$983$	− 3.11250e6i	− 0.102737i	0.998680	−	0.0513683i	$-0.0163582\pi$
$984$	0	0
$985$	0	0
$986$	2.16046e7	0.707709
$987$	0	0
$988$	− 653804.i	− 0.0213086i
$989$	406251.	0.0132070
$990$	0	0
$991$	−2.33325e7	−0.754704	−0.377352	−	0.926070i	$-0.623165\pi$
$991$	−2.33325e7	−0.754704	−0.377352	+	0.926070i	$0.623165\pi$
$992$	3.90373e6i	0.125951i
$993$	0	0
$994$	1.19311e7	0.383014
$995$	0	0
$996$	0	0
$997$	− 5.35497e7i	− 1.70616i	−0.521782	−	0.853079i	$-0.674733\pi$
$997$	− 5.35497e7i	− 1.70616i	0.521782	−	0.853079i	$-0.325267\pi$
$998$	− 1.25217e7i	− 0.397958i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.c.q.199.1		4
3.2	odd	2		450.6.c.p.199.3		4
5.2	odd	4		450.6.a.bd.1.2	yes	2
5.3	odd	4		450.6.a.ba.1.1	yes	2
5.4	even	2	inner	450.6.c.q.199.4		4
15.2	even	4		450.6.a.y.1.2	✓	2
15.8	even	4		450.6.a.bf.1.1	yes	2
15.14	odd	2		450.6.c.p.199.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.6.a.y.1.2	✓	2	15.2	even	4
450.6.a.ba.1.1	yes	2	5.3	odd	4
450.6.a.bd.1.2	yes	2	5.2	odd	4
450.6.a.bf.1.1	yes	2	15.8	even	4
450.6.c.p.199.2		4	15.14	odd	2
450.6.c.p.199.3		4	3.2	odd	2
450.6.c.q.199.1		4	1.1	even	1	trivial
450.6.c.q.199.4		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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} -16.0000 q^{4} -141.648i q^{7} +64.0000i q^{8} +O(q^{10})\)	\(q-4.00000i q^{2} -16.0000 q^{4} -141.648i q^{7} +64.0000i q^{8} -113.296 q^{11} +61.7038i q^{13} -566.592 q^{14} +256.000 q^{16} -1670.48i q^{17} +662.241 q^{19} +453.185i q^{22} +86.4812i q^{23} +246.815 q^{26} +2266.37i q^{28} -3233.30 q^{29} -3812.24 q^{31} -1024.00i q^{32} -6681.92 q^{34} -10249.6i q^{37} -2648.96i q^{38} +13459.4 q^{41} -4697.57i q^{43} +1812.74 q^{44} +345.925 q^{46} +15264.8i q^{47} -3257.19 q^{49} -987.260i q^{52} +498.632i q^{53} +9065.48 q^{56} +12933.2i q^{58} -15294.3 q^{59} -31811.3 q^{61} +15249.0i q^{62} -4096.00 q^{64} -49274.9i q^{67} +26727.7i q^{68} -21057.7 q^{71} +39447.8i q^{73} -40998.2 q^{74} -10595.8 q^{76} +16048.2i q^{77} -72949.3 q^{79} -53837.8i q^{82} +100897. i q^{83} -18790.3 q^{86} -7250.96i q^{88} -146714. q^{89} +8740.22 q^{91} -1383.70i q^{92} +61059.2 q^{94} +43544.9i q^{97} +13028.8i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 64 q^{4}+O(q^{10}) \) 4 * q - 64 * q^4	\( 4 q - 64 q^{4} + 1080 q^{11} + 800 q^{14} + 1024 q^{16} - 1184 q^{19} + 7120 q^{26} - 11400 q^{29} - 11416 q^{31} + 3936 q^{34} + 30840 q^{41} - 17280 q^{44} - 29280 q^{46} - 89688 q^{49} - 12800 q^{56} - 101040 q^{59} - 58252 q^{61} - 16384 q^{64} + 12360 q^{71} - 90400 q^{74} + 18944 q^{76} - 15824 q^{79} + 50560 q^{86} - 329280 q^{89} - 382832 q^{91} - 62400 q^{94}+O(q^{100}) \) 4 * q - 64 * q^4 + 1080 * q^11 + 800 * q^14 + 1024 * q^16 - 1184 * q^19 + 7120 * q^26 - 11400 * q^29 - 11416 * q^31 + 3936 * q^34 + 30840 * q^41 - 17280 * q^44 - 29280 * q^46 - 89688 * q^49 - 12800 * q^56 - 101040 * q^59 - 58252 * q^61 - 16384 * q^64 + 12360 * q^71 - 90400 * q^74 + 18944 * q^76 - 15824 * q^79 + 50560 * q^86 - 329280 * q^89 - 382832 * q^91 - 62400 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more