Properties

 Label 1200.4.f.b.49.2 Level $1200$ Weight $4$ Character 1200.49 Analytic conductor $70.802$ Analytic rank $0$ Dimension $2$ Inner twists $2$

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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: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 49.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1200.49 Dual form 1200.4.f.b.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000i q^{3} +24.0000i q^{7} -9.00000 q^{9} +O(q^{10})$$ $$q+3.00000i q^{3} +24.0000i q^{7} -9.00000 q^{9} -52.0000 q^{11} -22.0000i q^{13} -14.0000i q^{17} -20.0000 q^{19} -72.0000 q^{21} -168.000i q^{23} -27.0000i q^{27} -230.000 q^{29} +288.000 q^{31} -156.000i q^{33} -34.0000i q^{37} +66.0000 q^{39} +122.000 q^{41} -188.000i q^{43} -256.000i q^{47} -233.000 q^{49} +42.0000 q^{51} +338.000i q^{53} -60.0000i q^{57} +100.000 q^{59} +742.000 q^{61} -216.000i q^{63} +84.0000i q^{67} +504.000 q^{69} +328.000 q^{71} +38.0000i q^{73} -1248.00i q^{77} -240.000 q^{79} +81.0000 q^{81} +1212.00i q^{83} -690.000i q^{87} -330.000 q^{89} +528.000 q^{91} +864.000i q^{93} +866.000i q^{97} +468.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{9}+O(q^{10})$$ 2 * q - 18 * q^9 $$2 q - 18 q^{9} - 104 q^{11} - 40 q^{19} - 144 q^{21} - 460 q^{29} + 576 q^{31} + 132 q^{39} + 244 q^{41} - 466 q^{49} + 84 q^{51} + 200 q^{59} + 1484 q^{61} + 1008 q^{69} + 656 q^{71} - 480 q^{79} + 162 q^{81} - 660 q^{89} + 1056 q^{91} + 936 q^{99}+O(q^{100})$$ 2 * q - 18 * q^9 - 104 * q^11 - 40 * q^19 - 144 * q^21 - 460 * q^29 + 576 * q^31 + 132 * q^39 + 244 * q^41 - 466 * q^49 + 84 * q^51 + 200 * q^59 + 1484 * q^61 + 1008 * q^69 + 656 * q^71 - 480 * q^79 + 162 * q^81 - 660 * q^89 + 1056 * q^91 + 936 * q^99

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 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000i 0.577350i
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 24.0000i 1.29588i 0.761692 + 0.647939i $$0.224369\pi$$
−0.761692 + 0.647939i $$0.775631\pi$$
$$8$$ 0 0
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ −52.0000 −1.42533 −0.712663 0.701506i $$-0.752511\pi$$
−0.712663 + 0.701506i $$0.752511\pi$$
$$12$$ 0 0
$$13$$ − 22.0000i − 0.469362i −0.972072 0.234681i $$-0.924595\pi$$
0.972072 0.234681i $$-0.0754045\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 14.0000i − 0.199735i −0.995001 0.0998676i $$-0.968158\pi$$
0.995001 0.0998676i $$-0.0318419\pi$$
$$18$$ 0 0
$$19$$ −20.0000 −0.241490 −0.120745 0.992684i $$-0.538528\pi$$
−0.120745 + 0.992684i $$0.538528\pi$$
$$20$$ 0 0
$$21$$ −72.0000 −0.748176
$$22$$ 0 0
$$23$$ − 168.000i − 1.52306i −0.648129 0.761531i $$-0.724448\pi$$
0.648129 0.761531i $$-0.275552\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 27.0000i − 0.192450i
$$28$$ 0 0
$$29$$ −230.000 −1.47276 −0.736378 0.676570i $$-0.763465\pi$$
−0.736378 + 0.676570i $$0.763465\pi$$
$$30$$ 0 0
$$31$$ 288.000 1.66859 0.834296 0.551317i $$-0.185875\pi$$
0.834296 + 0.551317i $$0.185875\pi$$
$$32$$ 0 0
$$33$$ − 156.000i − 0.822913i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 34.0000i − 0.151069i −0.997143 0.0755347i $$-0.975934\pi$$
0.997143 0.0755347i $$-0.0240664\pi$$
$$38$$ 0 0
$$39$$ 66.0000 0.270986
$$40$$ 0 0
$$41$$ 122.000 0.464712 0.232356 0.972631i $$-0.425357\pi$$
0.232356 + 0.972631i $$0.425357\pi$$
$$42$$ 0 0
$$43$$ − 188.000i − 0.666738i −0.942796 0.333369i $$-0.891815\pi$$
0.942796 0.333369i $$-0.108185\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 256.000i − 0.794499i −0.917711 0.397249i $$-0.869965\pi$$
0.917711 0.397249i $$-0.130035\pi$$
$$48$$ 0 0
$$49$$ −233.000 −0.679300
$$50$$ 0 0
$$51$$ 42.0000 0.115317
$$52$$ 0 0
$$53$$ 338.000i 0.875998i 0.898976 + 0.437999i $$0.144313\pi$$
−0.898976 + 0.437999i $$0.855687\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 60.0000i − 0.139424i
$$58$$ 0 0
$$59$$ 100.000 0.220659 0.110330 0.993895i $$-0.464809\pi$$
0.110330 + 0.993895i $$0.464809\pi$$
$$60$$ 0 0
$$61$$ 742.000 1.55743 0.778716 0.627376i $$-0.215871\pi$$
0.778716 + 0.627376i $$0.215871\pi$$
$$62$$ 0 0
$$63$$ − 216.000i − 0.431959i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 84.0000i 0.153168i 0.997063 + 0.0765838i $$0.0244013\pi$$
−0.997063 + 0.0765838i $$0.975599\pi$$
$$68$$ 0 0
$$69$$ 504.000 0.879340
$$70$$ 0 0
$$71$$ 328.000 0.548260 0.274130 0.961693i $$-0.411610\pi$$
0.274130 + 0.961693i $$0.411610\pi$$
$$72$$ 0 0
$$73$$ 38.0000i 0.0609255i 0.999536 + 0.0304628i $$0.00969810\pi$$
−0.999536 + 0.0304628i $$0.990302\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 1248.00i − 1.84705i
$$78$$ 0 0
$$79$$ −240.000 −0.341799 −0.170899 0.985288i $$-0.554667\pi$$
−0.170899 + 0.985288i $$0.554667\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 1212.00i 1.60282i 0.598114 + 0.801411i $$0.295917\pi$$
−0.598114 + 0.801411i $$0.704083\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 690.000i − 0.850296i
$$88$$ 0 0
$$89$$ −330.000 −0.393033 −0.196516 0.980501i $$-0.562963\pi$$
−0.196516 + 0.980501i $$0.562963\pi$$
$$90$$ 0 0
$$91$$ 528.000 0.608236
$$92$$ 0 0
$$93$$ 864.000i 0.963362i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 866.000i 0.906484i 0.891387 + 0.453242i $$0.149733\pi$$
−0.891387 + 0.453242i $$0.850267\pi$$
$$98$$ 0 0
$$99$$ 468.000 0.475109
$$100$$ 0 0
$$101$$ −1218.00 −1.19996 −0.599978 0.800017i $$-0.704824\pi$$
−0.599978 + 0.800017i $$0.704824\pi$$
$$102$$ 0 0
$$103$$ − 88.0000i − 0.0841835i −0.999114 0.0420917i $$-0.986598\pi$$
0.999114 0.0420917i $$-0.0134022\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 36.0000i − 0.0325257i −0.999868 0.0162629i $$-0.994823\pi$$
0.999868 0.0162629i $$-0.00517686\pi$$
$$108$$ 0 0
$$109$$ 970.000 0.852378 0.426189 0.904634i $$-0.359856\pi$$
0.426189 + 0.904634i $$0.359856\pi$$
$$110$$ 0 0
$$111$$ 102.000 0.0872199
$$112$$ 0 0
$$113$$ − 1042.00i − 0.867461i −0.901043 0.433731i $$-0.857197\pi$$
0.901043 0.433731i $$-0.142803\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 198.000i 0.156454i
$$118$$ 0 0
$$119$$ 336.000 0.258833
$$120$$ 0 0
$$121$$ 1373.00 1.03156
$$122$$ 0 0
$$123$$ 366.000i 0.268302i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 1936.00i − 1.35269i −0.736583 0.676347i $$-0.763562\pi$$
0.736583 0.676347i $$-0.236438\pi$$
$$128$$ 0 0
$$129$$ 564.000 0.384941
$$130$$ 0 0
$$131$$ −732.000 −0.488207 −0.244104 0.969749i $$-0.578494\pi$$
−0.244104 + 0.969749i $$0.578494\pi$$
$$132$$ 0 0
$$133$$ − 480.000i − 0.312942i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 2214.00i − 1.38069i −0.723479 0.690346i $$-0.757458\pi$$
0.723479 0.690346i $$-0.242542\pi$$
$$138$$ 0 0
$$139$$ 20.0000 0.0122042 0.00610208 0.999981i $$-0.498058\pi$$
0.00610208 + 0.999981i $$0.498058\pi$$
$$140$$ 0 0
$$141$$ 768.000 0.458704
$$142$$ 0 0
$$143$$ 1144.00i 0.668994i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 699.000i − 0.392194i
$$148$$ 0 0
$$149$$ 1330.00 0.731261 0.365630 0.930760i $$-0.380853\pi$$
0.365630 + 0.930760i $$0.380853\pi$$
$$150$$ 0 0
$$151$$ 1208.00 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 0 0
$$153$$ 126.000i 0.0665784i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 3514.00i − 1.78629i −0.449768 0.893146i $$-0.648493\pi$$
0.449768 0.893146i $$-0.351507\pi$$
$$158$$ 0 0
$$159$$ −1014.00 −0.505757
$$160$$ 0 0
$$161$$ 4032.00 1.97370
$$162$$ 0 0
$$163$$ − 2068.00i − 0.993732i −0.867827 0.496866i $$-0.834484\pi$$
0.867827 0.496866i $$-0.165516\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 24.0000i 0.0111208i 0.999985 + 0.00556041i $$0.00176994\pi$$
−0.999985 + 0.00556041i $$0.998230\pi$$
$$168$$ 0 0
$$169$$ 1713.00 0.779700
$$170$$ 0 0
$$171$$ 180.000 0.0804967
$$172$$ 0 0
$$173$$ 618.000i 0.271593i 0.990737 + 0.135797i $$0.0433594\pi$$
−0.990737 + 0.135797i $$0.956641\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 300.000i 0.127398i
$$178$$ 0 0
$$179$$ 3340.00 1.39466 0.697328 0.716752i $$-0.254372\pi$$
0.697328 + 0.716752i $$0.254372\pi$$
$$180$$ 0 0
$$181$$ −178.000 −0.0730974 −0.0365487 0.999332i $$-0.511636\pi$$
−0.0365487 + 0.999332i $$0.511636\pi$$
$$182$$ 0 0
$$183$$ 2226.00i 0.899184i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 728.000i 0.284688i
$$188$$ 0 0
$$189$$ 648.000 0.249392
$$190$$ 0 0
$$191$$ 1888.00 0.715240 0.357620 0.933867i $$-0.383588\pi$$
0.357620 + 0.933867i $$0.383588\pi$$
$$192$$ 0 0
$$193$$ − 1922.00i − 0.716832i −0.933562 0.358416i $$-0.883317\pi$$
0.933562 0.358416i $$-0.116683\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2526.00i 0.913554i 0.889581 + 0.456777i $$0.150996\pi$$
−0.889581 + 0.456777i $$0.849004\pi$$
$$198$$ 0 0
$$199$$ −1160.00 −0.413217 −0.206609 0.978424i $$-0.566243\pi$$
−0.206609 + 0.978424i $$0.566243\pi$$
$$200$$ 0 0
$$201$$ −252.000 −0.0884314
$$202$$ 0 0
$$203$$ − 5520.00i − 1.90851i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 1512.00i 0.507687i
$$208$$ 0 0
$$209$$ 1040.00 0.344202
$$210$$ 0 0
$$211$$ 4468.00 1.45777 0.728886 0.684635i $$-0.240039\pi$$
0.728886 + 0.684635i $$0.240039\pi$$
$$212$$ 0 0
$$213$$ 984.000i 0.316538i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 6912.00i 2.16229i
$$218$$ 0 0
$$219$$ −114.000 −0.0351754
$$220$$ 0 0
$$221$$ −308.000 −0.0937481
$$222$$ 0 0
$$223$$ 6032.00i 1.81136i 0.423965 + 0.905678i $$0.360638\pi$$
−0.423965 + 0.905678i $$0.639362\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 2636.00i − 0.770738i −0.922763 0.385369i $$-0.874074\pi$$
0.922763 0.385369i $$-0.125926\pi$$
$$228$$ 0 0
$$229$$ −4830.00 −1.39378 −0.696889 0.717179i $$-0.745433\pi$$
−0.696889 + 0.717179i $$0.745433\pi$$
$$230$$ 0 0
$$231$$ 3744.00 1.06639
$$232$$ 0 0
$$233$$ − 2682.00i − 0.754093i −0.926194 0.377046i $$-0.876940\pi$$
0.926194 0.377046i $$-0.123060\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 720.000i − 0.197338i
$$238$$ 0 0
$$239$$ 2320.00 0.627901 0.313950 0.949439i $$-0.398347\pi$$
0.313950 + 0.949439i $$0.398347\pi$$
$$240$$ 0 0
$$241$$ 2002.00 0.535104 0.267552 0.963543i $$-0.413785\pi$$
0.267552 + 0.963543i $$0.413785\pi$$
$$242$$ 0 0
$$243$$ 243.000i 0.0641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 440.000i 0.113346i
$$248$$ 0 0
$$249$$ −3636.00 −0.925390
$$250$$ 0 0
$$251$$ −132.000 −0.0331943 −0.0165971 0.999862i $$-0.505283\pi$$
−0.0165971 + 0.999862i $$0.505283\pi$$
$$252$$ 0 0
$$253$$ 8736.00i 2.17086i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 7614.00i − 1.84805i −0.382335 0.924024i $$-0.624880\pi$$
0.382335 0.924024i $$-0.375120\pi$$
$$258$$ 0 0
$$259$$ 816.000 0.195767
$$260$$ 0 0
$$261$$ 2070.00 0.490919
$$262$$ 0 0
$$263$$ − 4888.00i − 1.14603i −0.819543 0.573017i $$-0.805773\pi$$
0.819543 0.573017i $$-0.194227\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 990.000i − 0.226918i
$$268$$ 0 0
$$269$$ −1270.00 −0.287856 −0.143928 0.989588i $$-0.545973\pi$$
−0.143928 + 0.989588i $$0.545973\pi$$
$$270$$ 0 0
$$271$$ −1072.00 −0.240293 −0.120146 0.992756i $$-0.538336\pi$$
−0.120146 + 0.992756i $$0.538336\pi$$
$$272$$ 0 0
$$273$$ 1584.00i 0.351165i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 5394.00i − 1.17001i −0.811028 0.585007i $$-0.801092\pi$$
0.811028 0.585007i $$-0.198908\pi$$
$$278$$ 0 0
$$279$$ −2592.00 −0.556197
$$280$$ 0 0
$$281$$ 2442.00 0.518425 0.259213 0.965820i $$-0.416537\pi$$
0.259213 + 0.965820i $$0.416537\pi$$
$$282$$ 0 0
$$283$$ 2772.00i 0.582255i 0.956684 + 0.291128i $$0.0940305\pi$$
−0.956684 + 0.291128i $$0.905970\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 2928.00i 0.602210i
$$288$$ 0 0
$$289$$ 4717.00 0.960106
$$290$$ 0 0
$$291$$ −2598.00 −0.523359
$$292$$ 0 0
$$293$$ − 4542.00i − 0.905619i −0.891607 0.452810i $$-0.850422\pi$$
0.891607 0.452810i $$-0.149578\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 1404.00i 0.274304i
$$298$$ 0 0
$$299$$ −3696.00 −0.714867
$$300$$ 0 0
$$301$$ 4512.00 0.864011
$$302$$ 0 0
$$303$$ − 3654.00i − 0.692795i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 5116.00i − 0.951093i −0.879691 0.475546i $$-0.842250\pi$$
0.879691 0.475546i $$-0.157750\pi$$
$$308$$ 0 0
$$309$$ 264.000 0.0486034
$$310$$ 0 0
$$311$$ 2808.00 0.511984 0.255992 0.966679i $$-0.417598\pi$$
0.255992 + 0.966679i $$0.417598\pi$$
$$312$$ 0 0
$$313$$ 7318.00i 1.32153i 0.750594 + 0.660763i $$0.229767\pi$$
−0.750594 + 0.660763i $$0.770233\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2246.00i 0.397943i 0.980005 + 0.198971i $$0.0637601\pi$$
−0.980005 + 0.198971i $$0.936240\pi$$
$$318$$ 0 0
$$319$$ 11960.0 2.09916
$$320$$ 0 0
$$321$$ 108.000 0.0187787
$$322$$ 0 0
$$323$$ 280.000i 0.0482341i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 2910.00i 0.492120i
$$328$$ 0 0
$$329$$ 6144.00 1.02957
$$330$$ 0 0
$$331$$ −1332.00 −0.221188 −0.110594 0.993866i $$-0.535275\pi$$
−0.110594 + 0.993866i $$0.535275\pi$$
$$332$$ 0 0
$$333$$ 306.000i 0.0503564i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 11534.0i − 1.86438i −0.361966 0.932191i $$-0.617894\pi$$
0.361966 0.932191i $$-0.382106\pi$$
$$338$$ 0 0
$$339$$ 3126.00 0.500829
$$340$$ 0 0
$$341$$ −14976.0 −2.37829
$$342$$ 0 0
$$343$$ 2640.00i 0.415588i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 11956.0i − 1.84966i −0.380382 0.924830i $$-0.624207\pi$$
0.380382 0.924830i $$-0.375793\pi$$
$$348$$ 0 0
$$349$$ −4870.00 −0.746949 −0.373474 0.927640i $$-0.621834\pi$$
−0.373474 + 0.927640i $$0.621834\pi$$
$$350$$ 0 0
$$351$$ −594.000 −0.0903287
$$352$$ 0 0
$$353$$ − 10722.0i − 1.61664i −0.588742 0.808321i $$-0.700377\pi$$
0.588742 0.808321i $$-0.299623\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 1008.00i 0.149437i
$$358$$ 0 0
$$359$$ 120.000 0.0176417 0.00882083 0.999961i $$-0.497192\pi$$
0.00882083 + 0.999961i $$0.497192\pi$$
$$360$$ 0 0
$$361$$ −6459.00 −0.941682
$$362$$ 0 0
$$363$$ 4119.00i 0.595569i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 3936.00i − 0.559830i −0.960025 0.279915i $$-0.909694\pi$$
0.960025 0.279915i $$-0.0903063\pi$$
$$368$$ 0 0
$$369$$ −1098.00 −0.154904
$$370$$ 0 0
$$371$$ −8112.00 −1.13519
$$372$$ 0 0
$$373$$ − 3022.00i − 0.419499i −0.977755 0.209750i $$-0.932735\pi$$
0.977755 0.209750i $$-0.0672649\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 5060.00i 0.691255i
$$378$$ 0 0
$$379$$ −13340.0 −1.80799 −0.903997 0.427539i $$-0.859381\pi$$
−0.903997 + 0.427539i $$0.859381\pi$$
$$380$$ 0 0
$$381$$ 5808.00 0.780979
$$382$$ 0 0
$$383$$ − 1008.00i − 0.134481i −0.997737 0.0672407i $$-0.978580\pi$$
0.997737 0.0672407i $$-0.0214195\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 1692.00i 0.222246i
$$388$$ 0 0
$$389$$ −9630.00 −1.25517 −0.627584 0.778549i $$-0.715956\pi$$
−0.627584 + 0.778549i $$0.715956\pi$$
$$390$$ 0 0
$$391$$ −2352.00 −0.304209
$$392$$ 0 0
$$393$$ − 2196.00i − 0.281867i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 7126.00i 0.900866i 0.892810 + 0.450433i $$0.148730\pi$$
−0.892810 + 0.450433i $$0.851270\pi$$
$$398$$ 0 0
$$399$$ 1440.00 0.180677
$$400$$ 0 0
$$401$$ −8718.00 −1.08568 −0.542838 0.839837i $$-0.682650\pi$$
−0.542838 + 0.839837i $$0.682650\pi$$
$$402$$ 0 0
$$403$$ − 6336.00i − 0.783173i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 1768.00i 0.215323i
$$408$$ 0 0
$$409$$ 10870.0 1.31415 0.657074 0.753826i $$-0.271794\pi$$
0.657074 + 0.753826i $$0.271794\pi$$
$$410$$ 0 0
$$411$$ 6642.00 0.797143
$$412$$ 0 0
$$413$$ 2400.00i 0.285947i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 60.0000i 0.00704607i
$$418$$ 0 0
$$419$$ −9700.00 −1.13097 −0.565484 0.824759i $$-0.691311\pi$$
−0.565484 + 0.824759i $$0.691311\pi$$
$$420$$ 0 0
$$421$$ 862.000 0.0997893 0.0498947 0.998754i $$-0.484111\pi$$
0.0498947 + 0.998754i $$0.484111\pi$$
$$422$$ 0 0
$$423$$ 2304.00i 0.264833i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 17808.0i 2.01824i
$$428$$ 0 0
$$429$$ −3432.00 −0.386244
$$430$$ 0 0
$$431$$ −15792.0 −1.76490 −0.882452 0.470402i $$-0.844109\pi$$
−0.882452 + 0.470402i $$0.844109\pi$$
$$432$$ 0 0
$$433$$ − 11602.0i − 1.28766i −0.765169 0.643830i $$-0.777345\pi$$
0.765169 0.643830i $$-0.222655\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 3360.00i 0.367805i
$$438$$ 0 0
$$439$$ −440.000 −0.0478361 −0.0239181 0.999714i $$-0.507614\pi$$
−0.0239181 + 0.999714i $$0.507614\pi$$
$$440$$ 0 0
$$441$$ 2097.00 0.226433
$$442$$ 0 0
$$443$$ − 10188.0i − 1.09266i −0.837571 0.546328i $$-0.816025\pi$$
0.837571 0.546328i $$-0.183975\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 3990.00i 0.422194i
$$448$$ 0 0
$$449$$ 13310.0 1.39897 0.699485 0.714647i $$-0.253413\pi$$
0.699485 + 0.714647i $$0.253413\pi$$
$$450$$ 0 0
$$451$$ −6344.00 −0.662367
$$452$$ 0 0
$$453$$ 3624.00i 0.375873i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 3226.00i 0.330210i 0.986276 + 0.165105i $$0.0527963\pi$$
−0.986276 + 0.165105i $$0.947204\pi$$
$$458$$ 0 0
$$459$$ −378.000 −0.0384391
$$460$$ 0 0
$$461$$ 6582.00 0.664977 0.332488 0.943107i $$-0.392112\pi$$
0.332488 + 0.943107i $$0.392112\pi$$
$$462$$ 0 0
$$463$$ 15072.0i 1.51286i 0.654073 + 0.756431i $$0.273059\pi$$
−0.654073 + 0.756431i $$0.726941\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 476.000i − 0.0471663i −0.999722 0.0235831i $$-0.992493\pi$$
0.999722 0.0235831i $$-0.00750744\pi$$
$$468$$ 0 0
$$469$$ −2016.00 −0.198487
$$470$$ 0 0
$$471$$ 10542.0 1.03132
$$472$$ 0 0
$$473$$ 9776.00i 0.950319i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 3042.00i − 0.291999i
$$478$$ 0 0
$$479$$ −19680.0 −1.87725 −0.938624 0.344941i $$-0.887899\pi$$
−0.938624 + 0.344941i $$0.887899\pi$$
$$480$$ 0 0
$$481$$ −748.000 −0.0709062
$$482$$ 0 0
$$483$$ 12096.0i 1.13952i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 5944.00i 0.553077i 0.961003 + 0.276538i $$0.0891873\pi$$
−0.961003 + 0.276538i $$0.910813\pi$$
$$488$$ 0 0
$$489$$ 6204.00 0.573731
$$490$$ 0 0
$$491$$ −10772.0 −0.990089 −0.495044 0.868868i $$-0.664848\pi$$
−0.495044 + 0.868868i $$0.664848\pi$$
$$492$$ 0 0
$$493$$ 3220.00i 0.294161i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 7872.00i 0.710478i
$$498$$ 0 0
$$499$$ 8140.00 0.730253 0.365127 0.930958i $$-0.381026\pi$$
0.365127 + 0.930958i $$0.381026\pi$$
$$500$$ 0 0
$$501$$ −72.0000 −0.00642060
$$502$$ 0 0
$$503$$ − 13768.0i − 1.22045i −0.792229 0.610223i $$-0.791080\pi$$
0.792229 0.610223i $$-0.208920\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 5139.00i 0.450160i
$$508$$ 0 0
$$509$$ −22150.0 −1.92884 −0.964422 0.264368i $$-0.914837\pi$$
−0.964422 + 0.264368i $$0.914837\pi$$
$$510$$ 0 0
$$511$$ −912.000 −0.0789521
$$512$$ 0 0
$$513$$ 540.000i 0.0464748i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 13312.0i 1.13242i
$$518$$ 0 0
$$519$$ −1854.00 −0.156805
$$520$$ 0 0
$$521$$ 1562.00 0.131348 0.0656741 0.997841i $$-0.479080\pi$$
0.0656741 + 0.997841i $$0.479080\pi$$
$$522$$ 0 0
$$523$$ − 668.000i − 0.0558501i −0.999610 0.0279250i $$-0.991110\pi$$
0.999610 0.0279250i $$-0.00888997\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 4032.00i − 0.333276i
$$528$$ 0 0
$$529$$ −16057.0 −1.31972
$$530$$ 0 0
$$531$$ −900.000 −0.0735531
$$532$$ 0 0
$$533$$ − 2684.00i − 0.218118i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 10020.0i 0.805205i
$$538$$ 0 0
$$539$$ 12116.0 0.968225
$$540$$ 0 0
$$541$$ −6138.00 −0.487788 −0.243894 0.969802i $$-0.578425\pi$$
−0.243894 + 0.969802i $$0.578425\pi$$
$$542$$ 0 0
$$543$$ − 534.000i − 0.0422028i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 10484.0i 0.819494i 0.912199 + 0.409747i $$0.134383\pi$$
−0.912199 + 0.409747i $$0.865617\pi$$
$$548$$ 0 0
$$549$$ −6678.00 −0.519144
$$550$$ 0 0
$$551$$ 4600.00 0.355656
$$552$$ 0 0
$$553$$ − 5760.00i − 0.442930i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 3606.00i 0.274311i 0.990550 + 0.137155i $$0.0437960\pi$$
−0.990550 + 0.137155i $$0.956204\pi$$
$$558$$ 0 0
$$559$$ −4136.00 −0.312941
$$560$$ 0 0
$$561$$ −2184.00 −0.164365
$$562$$ 0 0
$$563$$ 12252.0i 0.917159i 0.888654 + 0.458579i $$0.151641\pi$$
−0.888654 + 0.458579i $$0.848359\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 1944.00i 0.143986i
$$568$$ 0 0
$$569$$ 14550.0 1.07200 0.536000 0.844218i $$-0.319935\pi$$
0.536000 + 0.844218i $$0.319935\pi$$
$$570$$ 0 0
$$571$$ 25468.0 1.86655 0.933277 0.359157i $$-0.116936\pi$$
0.933277 + 0.359157i $$0.116936\pi$$
$$572$$ 0 0
$$573$$ 5664.00i 0.412944i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 12866.0i 0.928282i 0.885761 + 0.464141i $$0.153637\pi$$
−0.885761 + 0.464141i $$0.846363\pi$$
$$578$$ 0 0
$$579$$ 5766.00 0.413863
$$580$$ 0 0
$$581$$ −29088.0 −2.07706
$$582$$ 0 0
$$583$$ − 17576.0i − 1.24858i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 14844.0i 1.04374i 0.853024 + 0.521872i $$0.174766\pi$$
−0.853024 + 0.521872i $$0.825234\pi$$
$$588$$ 0 0
$$589$$ −5760.00 −0.402948
$$590$$ 0 0
$$591$$ −7578.00 −0.527440
$$592$$ 0 0
$$593$$ − 20402.0i − 1.41283i −0.707797 0.706416i $$-0.750311\pi$$
0.707797 0.706416i $$-0.249689\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 3480.00i − 0.238571i
$$598$$ 0 0
$$599$$ 10760.0 0.733959 0.366980 0.930229i $$-0.380392\pi$$
0.366980 + 0.930229i $$0.380392\pi$$
$$600$$ 0 0
$$601$$ 14282.0 0.969343 0.484671 0.874696i $$-0.338939\pi$$
0.484671 + 0.874696i $$0.338939\pi$$
$$602$$ 0 0
$$603$$ − 756.000i − 0.0510559i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 11056.0i − 0.739290i −0.929173 0.369645i $$-0.879479\pi$$
0.929173 0.369645i $$-0.120521\pi$$
$$608$$ 0 0
$$609$$ 16560.0 1.10188
$$610$$ 0 0
$$611$$ −5632.00 −0.372907
$$612$$ 0 0
$$613$$ 16418.0i 1.08176i 0.841101 + 0.540878i $$0.181908\pi$$
−0.841101 + 0.540878i $$0.818092\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 10374.0i − 0.676891i −0.940986 0.338445i $$-0.890099\pi$$
0.940986 0.338445i $$-0.109901\pi$$
$$618$$ 0 0
$$619$$ −5260.00 −0.341546 −0.170773 0.985310i $$-0.554627\pi$$
−0.170773 + 0.985310i $$0.554627\pi$$
$$620$$ 0 0
$$621$$ −4536.00 −0.293113
$$622$$ 0 0
$$623$$ − 7920.00i − 0.509323i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 3120.00i 0.198725i
$$628$$ 0 0
$$629$$ −476.000 −0.0301739
$$630$$ 0 0
$$631$$ −21352.0 −1.34708 −0.673542 0.739149i $$-0.735228\pi$$
−0.673542 + 0.739149i $$0.735228\pi$$
$$632$$ 0 0
$$633$$ 13404.0i 0.841645i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 5126.00i 0.318838i
$$638$$ 0 0
$$639$$ −2952.00 −0.182753
$$640$$ 0 0
$$641$$ −29118.0 −1.79422 −0.897108 0.441812i $$-0.854336\pi$$
−0.897108 + 0.441812i $$0.854336\pi$$
$$642$$ 0 0
$$643$$ 5772.00i 0.354005i 0.984210 + 0.177003i $$0.0566401\pi$$
−0.984210 + 0.177003i $$0.943360\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 14264.0i 0.866732i 0.901218 + 0.433366i $$0.142674\pi$$
−0.901218 + 0.433366i $$0.857326\pi$$
$$648$$ 0 0
$$649$$ −5200.00 −0.314511
$$650$$ 0 0
$$651$$ −20736.0 −1.24840
$$652$$ 0 0
$$653$$ − 6902.00i − 0.413623i −0.978381 0.206812i $$-0.933691\pi$$
0.978381 0.206812i $$-0.0663088\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 342.000i − 0.0203085i
$$658$$ 0 0
$$659$$ 20140.0 1.19051 0.595253 0.803539i $$-0.297052\pi$$
0.595253 + 0.803539i $$0.297052\pi$$
$$660$$ 0 0
$$661$$ −3218.00 −0.189358 −0.0946790 0.995508i $$-0.530182\pi$$
−0.0946790 + 0.995508i $$0.530182\pi$$
$$662$$ 0 0
$$663$$ − 924.000i − 0.0541255i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 38640.0i 2.24310i
$$668$$ 0 0
$$669$$ −18096.0 −1.04579
$$670$$ 0 0
$$671$$ −38584.0 −2.21985
$$672$$ 0 0
$$673$$ 7518.00i 0.430606i 0.976547 + 0.215303i $$0.0690739\pi$$
−0.976547 + 0.215303i $$0.930926\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 18114.0i − 1.02833i −0.857692 0.514164i $$-0.828102\pi$$
0.857692 0.514164i $$-0.171898\pi$$
$$678$$ 0 0
$$679$$ −20784.0 −1.17469
$$680$$ 0 0
$$681$$ 7908.00 0.444986
$$682$$ 0 0
$$683$$ − 23868.0i − 1.33716i −0.743638 0.668582i $$-0.766901\pi$$
0.743638 0.668582i $$-0.233099\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 14490.0i − 0.804699i
$$688$$ 0 0
$$689$$ 7436.00 0.411160
$$690$$ 0 0
$$691$$ −172.000 −0.00946916 −0.00473458 0.999989i $$-0.501507\pi$$
−0.00473458 + 0.999989i $$0.501507\pi$$
$$692$$ 0 0
$$693$$ 11232.0i 0.615683i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 1708.00i − 0.0928194i
$$698$$ 0 0
$$699$$ 8046.00 0.435376
$$700$$ 0 0
$$701$$ −22138.0 −1.19278 −0.596391 0.802694i $$-0.703399\pi$$
−0.596391 + 0.802694i $$0.703399\pi$$
$$702$$ 0 0
$$703$$ 680.000i 0.0364818i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 29232.0i − 1.55500i
$$708$$ 0 0
$$709$$ −3070.00 −0.162618 −0.0813091 0.996689i $$-0.525910\pi$$
−0.0813091 + 0.996689i $$0.525910\pi$$
$$710$$ 0 0
$$711$$ 2160.00 0.113933
$$712$$ 0 0
$$713$$ − 48384.0i − 2.54137i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 6960.00i 0.362519i
$$718$$ 0 0
$$719$$ 15600.0 0.809154 0.404577 0.914504i $$-0.367419\pi$$
0.404577 + 0.914504i $$0.367419\pi$$
$$720$$ 0 0
$$721$$ 2112.00 0.109092
$$722$$ 0 0
$$723$$ 6006.00i 0.308943i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 20696.0i − 1.05581i −0.849304 0.527904i $$-0.822978\pi$$
0.849304 0.527904i $$-0.177022\pi$$
$$728$$ 0 0
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ −2632.00 −0.133171
$$732$$ 0 0
$$733$$ 30778.0i 1.55090i 0.631408 + 0.775451i $$0.282478\pi$$
−0.631408 + 0.775451i $$0.717522\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 4368.00i − 0.218314i
$$738$$ 0 0
$$739$$ 11740.0 0.584388 0.292194 0.956359i $$-0.405615\pi$$
0.292194 + 0.956359i $$0.405615\pi$$
$$740$$ 0 0
$$741$$ −1320.00 −0.0654405
$$742$$ 0 0
$$743$$ 2632.00i 0.129958i 0.997887 + 0.0649789i $$0.0206980\pi$$
−0.997887 + 0.0649789i $$0.979302\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 10908.0i − 0.534274i
$$748$$ 0 0
$$749$$ 864.000 0.0421494
$$750$$ 0 0
$$751$$ 20528.0 0.997440 0.498720 0.866763i $$-0.333804\pi$$
0.498720 + 0.866763i $$0.333804\pi$$
$$752$$ 0 0
$$753$$ − 396.000i − 0.0191647i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 21646.0i 1.03928i 0.854384 + 0.519642i $$0.173934\pi$$
−0.854384 + 0.519642i $$0.826066\pi$$
$$758$$ 0 0
$$759$$ −26208.0 −1.25335
$$760$$ 0 0
$$761$$ 18282.0 0.870857 0.435428 0.900223i $$-0.356597\pi$$
0.435428 + 0.900223i $$0.356597\pi$$
$$762$$ 0 0
$$763$$ 23280.0i 1.10458i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 2200.00i − 0.103569i
$$768$$ 0 0
$$769$$ 24190.0 1.13435 0.567174 0.823598i $$-0.308037\pi$$
0.567174 + 0.823598i $$0.308037\pi$$
$$770$$ 0 0
$$771$$ 22842.0 1.06697
$$772$$ 0 0
$$773$$ 25698.0i 1.19572i 0.801600 + 0.597861i $$0.203982\pi$$
−0.801600 + 0.597861i $$0.796018\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 2448.00i 0.113026i
$$778$$ 0 0
$$779$$ −2440.00 −0.112223
$$780$$ 0 0
$$781$$ −17056.0 −0.781449
$$782$$ 0 0
$$783$$ 6210.00i 0.283432i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 33436.0i − 1.51444i −0.653160 0.757220i $$-0.726557\pi$$
0.653160 0.757220i $$-0.273443\pi$$
$$788$$ 0 0
$$789$$ 14664.0 0.661663
$$790$$ 0 0
$$791$$ 25008.0 1.12412
$$792$$ 0 0
$$793$$ − 16324.0i − 0.730999i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 37594.0i − 1.67083i −0.549623 0.835413i $$-0.685229\pi$$
0.549623 0.835413i $$-0.314771\pi$$
$$798$$ 0 0
$$799$$ −3584.00 −0.158689
$$800$$ 0 0
$$801$$ 2970.00 0.131011
$$802$$ 0 0
$$803$$ − 1976.00i − 0.0868388i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 3810.00i − 0.166194i
$$808$$ 0 0
$$809$$ −4730.00 −0.205560 −0.102780 0.994704i $$-0.532774\pi$$
−0.102780 + 0.994704i $$0.532774\pi$$
$$810$$ 0 0
$$811$$ 8748.00 0.378772 0.189386 0.981903i $$-0.439350\pi$$
0.189386 + 0.981903i $$0.439350\pi$$
$$812$$ 0 0
$$813$$ − 3216.00i − 0.138733i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 3760.00i 0.161011i
$$818$$ 0 0
$$819$$ −4752.00 −0.202745
$$820$$ 0 0
$$821$$ 44142.0 1.87645 0.938226 0.346024i $$-0.112468\pi$$
0.938226 + 0.346024i $$0.112468\pi$$
$$822$$ 0 0
$$823$$ 3992.00i 0.169079i 0.996420 + 0.0845397i $$0.0269420\pi$$
−0.996420 + 0.0845397i $$0.973058\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 14444.0i 0.607336i 0.952778 + 0.303668i $$0.0982114\pi$$
−0.952778 + 0.303668i $$0.901789\pi$$
$$828$$ 0 0
$$829$$ −42150.0 −1.76590 −0.882949 0.469468i $$-0.844446\pi$$
−0.882949 + 0.469468i $$0.844446\pi$$
$$830$$ 0 0
$$831$$ 16182.0 0.675508
$$832$$ 0 0
$$833$$ 3262.00i 0.135680i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 7776.00i − 0.321121i
$$838$$ 0 0
$$839$$ 13400.0 0.551394 0.275697 0.961245i $$-0.411091\pi$$
0.275697 + 0.961245i $$0.411091\pi$$
$$840$$ 0 0
$$841$$ 28511.0 1.16901
$$842$$ 0 0
$$843$$ 7326.00i 0.299313i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 32952.0i 1.33677i
$$848$$ 0 0
$$849$$ −8316.00 −0.336165
$$850$$ 0 0
$$851$$ −5712.00 −0.230088
$$852$$ 0 0
$$853$$ 8658.00i 0.347531i 0.984787 + 0.173766i $$0.0555935\pi$$
−0.984787 + 0.173766i $$0.944406\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 42826.0i 1.70701i 0.521084 + 0.853505i $$0.325528\pi$$
−0.521084 + 0.853505i $$0.674472\pi$$
$$858$$ 0 0
$$859$$ −35900.0 −1.42595 −0.712976 0.701189i $$-0.752653\pi$$
−0.712976 + 0.701189i $$0.752653\pi$$
$$860$$ 0 0
$$861$$ −8784.00 −0.347686
$$862$$ 0 0
$$863$$ − 3088.00i − 0.121804i −0.998144 0.0609019i $$-0.980602\pi$$
0.998144 0.0609019i $$-0.0193977\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 14151.0i 0.554317i
$$868$$ 0 0
$$869$$ 12480.0 0.487175
$$870$$ 0 0
$$871$$ 1848.00 0.0718910
$$872$$ 0 0
$$873$$ − 7794.00i − 0.302161i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 35274.0i − 1.35817i −0.734058 0.679087i $$-0.762376\pi$$
0.734058 0.679087i $$-0.237624\pi$$
$$878$$ 0 0
$$879$$ 13626.0 0.522860
$$880$$ 0 0
$$881$$ 25042.0 0.957646 0.478823 0.877911i $$-0.341064\pi$$
0.478823 + 0.877911i $$0.341064\pi$$
$$882$$ 0 0
$$883$$ 12572.0i 0.479141i 0.970879 + 0.239570i $$0.0770066\pi$$
−0.970879 + 0.239570i $$0.922993\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 21864.0i 0.827645i 0.910358 + 0.413823i $$0.135807\pi$$
−0.910358 + 0.413823i $$0.864193\pi$$
$$888$$ 0 0
$$889$$ 46464.0 1.75293
$$890$$ 0 0
$$891$$ −4212.00 −0.158370
$$892$$ 0 0
$$893$$ 5120.00i 0.191864i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 11088.0i − 0.412729i
$$898$$ 0 0
$$899$$ −66240.0 −2.45743
$$900$$ 0 0
$$901$$ 4732.00 0.174968
$$902$$ 0 0
$$903$$ 13536.0i 0.498837i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 31236.0i − 1.14352i −0.820420 0.571761i $$-0.806260\pi$$
0.820420 0.571761i $$-0.193740\pi$$
$$908$$ 0 0
$$909$$ 10962.0 0.399985
$$910$$ 0 0
$$911$$ −8272.00 −0.300838 −0.150419 0.988622i $$-0.548062\pi$$
−0.150419 + 0.988622i $$0.548062\pi$$
$$912$$ 0 0
$$913$$ − 63024.0i − 2.28455i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 17568.0i − 0.632657i
$$918$$ 0 0
$$919$$ 20200.0 0.725067 0.362533 0.931971i $$-0.381912\pi$$
0.362533 + 0.931971i $$0.381912\pi$$
$$920$$ 0 0
$$921$$ 15348.0 0.549114
$$922$$ 0 0
$$923$$ − 7216.00i − 0.257332i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 792.000i 0.0280612i
$$928$$ 0 0
$$929$$ −31010.0 −1.09516 −0.547581 0.836753i $$-0.684451\pi$$
−0.547581 + 0.836753i $$0.684451\pi$$
$$930$$ 0 0
$$931$$ 4660.00 0.164044
$$932$$ 0 0
$$933$$ 8424.00i 0.295594i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 39174.0i − 1.36580i −0.730510 0.682902i $$-0.760717\pi$$
0.730510 0.682902i $$-0.239283\pi$$
$$938$$ 0 0
$$939$$ −21954.0 −0.762984
$$940$$ 0 0
$$941$$ −4138.00 −0.143353 −0.0716764 0.997428i $$-0.522835\pi$$
−0.0716764 + 0.997428i $$0.522835\pi$$
$$942$$ 0 0
$$943$$ − 20496.0i − 0.707785i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 23676.0i − 0.812425i −0.913779 0.406213i $$-0.866849\pi$$
0.913779 0.406213i $$-0.133151\pi$$
$$948$$ 0 0
$$949$$ 836.000 0.0285961
$$950$$ 0 0
$$951$$ −6738.00 −0.229752
$$952$$ 0 0
$$953$$ − 18922.0i − 0.643173i −0.946880 0.321586i $$-0.895784\pi$$
0.946880 0.321586i $$-0.104216\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 35880.0i 1.21195i
$$958$$ 0 0
$$959$$ 53136.0 1.78921
$$960$$ 0 0
$$961$$ 53153.0 1.78420
$$962$$ 0 0
$$963$$ 324.000i 0.0108419i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 39656.0i − 1.31877i −0.751805 0.659385i $$-0.770817\pi$$
0.751805 0.659385i $$-0.229183\pi$$
$$968$$ 0 0
$$969$$ −840.000 −0.0278480
$$970$$ 0 0
$$971$$ 33228.0 1.09818 0.549092 0.835762i $$-0.314974\pi$$
0.549092 + 0.835762i $$0.314974\pi$$
$$972$$ 0 0
$$973$$ 480.000i 0.0158151i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 974.000i − 0.0318946i −0.999873 0.0159473i $$-0.994924\pi$$
0.999873 0.0159473i $$-0.00507640\pi$$
$$978$$ 0 0
$$979$$ 17160.0 0.560200
$$980$$ 0 0
$$981$$ −8730.00 −0.284126
$$982$$ 0 0
$$983$$ − 13608.0i − 0.441534i −0.975327 0.220767i $$-0.929144\pi$$
0.975327 0.220767i $$-0.0708560\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 18432.0i 0.594425i
$$988$$ 0 0
$$989$$ −31584.0 −1.01548
$$990$$ 0 0
$$991$$ −13472.0 −0.431839 −0.215919 0.976411i $$-0.569275\pi$$
−0.215919 + 0.976411i $$0.569275\pi$$
$$992$$ 0 0
$$993$$ − 3996.00i − 0.127703i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 3234.00i − 0.102730i −0.998680 0.0513650i $$-0.983643\pi$$
0.998680 0.0513650i $$-0.0163572\pi$$
$$998$$ 0 0
$$999$$ −918.000 −0.0290733
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1200.4.f.b.49.2 2
4.3 odd 2 75.4.b.b.49.2 2
5.2 odd 4 1200.4.a.t.1.1 1
5.3 odd 4 240.4.a.e.1.1 1
5.4 even 2 inner 1200.4.f.b.49.1 2
12.11 even 2 225.4.b.e.199.1 2
15.8 even 4 720.4.a.n.1.1 1
20.3 even 4 15.4.a.a.1.1 1
20.7 even 4 75.4.a.b.1.1 1
20.19 odd 2 75.4.b.b.49.1 2
40.3 even 4 960.4.a.b.1.1 1
40.13 odd 4 960.4.a.ba.1.1 1
60.23 odd 4 45.4.a.c.1.1 1
60.47 odd 4 225.4.a.f.1.1 1
60.59 even 2 225.4.b.e.199.2 2
140.83 odd 4 735.4.a.e.1.1 1
180.23 odd 12 405.4.e.i.136.1 2
180.43 even 12 405.4.e.g.271.1 2
180.83 odd 12 405.4.e.i.271.1 2
180.103 even 12 405.4.e.g.136.1 2
220.43 odd 4 1815.4.a.e.1.1 1
420.83 even 4 2205.4.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.a.1.1 1 20.3 even 4
45.4.a.c.1.1 1 60.23 odd 4
75.4.a.b.1.1 1 20.7 even 4
75.4.b.b.49.1 2 20.19 odd 2
75.4.b.b.49.2 2 4.3 odd 2
225.4.a.f.1.1 1 60.47 odd 4
225.4.b.e.199.1 2 12.11 even 2
225.4.b.e.199.2 2 60.59 even 2
240.4.a.e.1.1 1 5.3 odd 4
405.4.e.g.136.1 2 180.103 even 12
405.4.e.g.271.1 2 180.43 even 12
405.4.e.i.136.1 2 180.23 odd 12
405.4.e.i.271.1 2 180.83 odd 12
720.4.a.n.1.1 1 15.8 even 4
735.4.a.e.1.1 1 140.83 odd 4
960.4.a.b.1.1 1 40.3 even 4
960.4.a.ba.1.1 1 40.13 odd 4
1200.4.a.t.1.1 1 5.2 odd 4
1200.4.f.b.49.1 2 5.4 even 2 inner
1200.4.f.b.49.2 2 1.1 even 1 trivial
1815.4.a.e.1.1 1 220.43 odd 4
2205.4.a.l.1.1 1 420.83 even 4