# Properties

 Label 784.4.a.k.1.1 Level $784$ Weight $4$ Character 784.1 Self dual yes Analytic conductor $46.257$ Analytic rank $0$ Dimension $1$ CM discriminant -7 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 784.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$46.2574974445$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 49) Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 784.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-27.0000 q^{9} +O(q^{10})$$ $$q-27.0000 q^{9} +68.0000 q^{11} +40.0000 q^{23} -125.000 q^{25} -166.000 q^{29} +450.000 q^{37} +180.000 q^{43} +590.000 q^{53} +740.000 q^{67} -688.000 q^{71} +1384.00 q^{79} +729.000 q^{81} -1836.00 q^{99} +O(q^{100})$$

## 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
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ 0 0
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −27.0000 −1.00000
$$10$$ 0 0
$$11$$ 68.0000 1.86389 0.931944 0.362602i $$-0.118111\pi$$
0.931944 + 0.362602i $$0.118111\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 40.0000 0.362634 0.181317 0.983425i $$-0.441964\pi$$
0.181317 + 0.983425i $$0.441964\pi$$
$$24$$ 0 0
$$25$$ −125.000 −1.00000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −166.000 −1.06295 −0.531473 0.847075i $$-0.678361\pi$$
−0.531473 + 0.847075i $$0.678361\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 450.000 1.99945 0.999724 0.0235113i $$-0.00748457\pi$$
0.999724 + 0.0235113i $$0.00748457\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 180.000 0.638366 0.319183 0.947693i $$-0.396592\pi$$
0.319183 + 0.947693i $$0.396592\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 590.000 1.52911 0.764554 0.644560i $$-0.222959\pi$$
0.764554 + 0.644560i $$0.222959\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 740.000 1.34933 0.674667 0.738122i $$-0.264287\pi$$
0.674667 + 0.738122i $$0.264287\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −688.000 −1.15001 −0.575004 0.818151i $$-0.695000\pi$$
−0.575004 + 0.818151i $$0.695000\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 1384.00 1.97104 0.985520 0.169559i $$-0.0542343\pi$$
0.985520 + 0.169559i $$0.0542343\pi$$
$$80$$ 0 0
$$81$$ 729.000 1.00000
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$98$$ 0 0
$$99$$ −1836.00 −1.86389
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1580.00 1.42752 0.713759 0.700392i $$-0.246991\pi$$
0.713759 + 0.700392i $$0.246991\pi$$
$$108$$ 0 0
$$109$$ −54.0000 −0.0474519 −0.0237260 0.999718i $$-0.507553\pi$$
−0.0237260 + 0.999718i $$0.507553\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −670.000 −0.557773 −0.278886 0.960324i $$-0.589965\pi$$
−0.278886 + 0.960324i $$0.589965\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 3293.00 2.47408
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 2000.00 1.39741 0.698706 0.715409i $$-0.253760\pi$$
0.698706 + 0.715409i $$0.253760\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 3110.00 1.93945 0.969727 0.244191i $$-0.0785224\pi$$
0.969727 + 0.244191i $$0.0785224\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 814.000 0.447554 0.223777 0.974640i $$-0.428161\pi$$
0.223777 + 0.974640i $$0.428161\pi$$
$$150$$ 0 0
$$151$$ 2952.00 1.59093 0.795465 0.606000i $$-0.207227\pi$$
0.795465 + 0.606000i $$0.207227\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −1780.00 −0.855340 −0.427670 0.903935i $$-0.640665\pi$$
−0.427670 + 0.903935i $$0.640665\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −2197.00 −1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 2084.00 0.870198 0.435099 0.900383i $$-0.356713\pi$$
0.435099 + 0.900383i $$0.356713\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 4072.00 1.54262 0.771308 0.636462i $$-0.219603\pi$$
0.771308 + 0.636462i $$0.219603\pi$$
$$192$$ 0 0
$$193$$ −4590.00 −1.71189 −0.855947 0.517064i $$-0.827025\pi$$
−0.855947 + 0.517064i $$0.827025\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −2210.00 −0.799269 −0.399634 0.916675i $$-0.630863\pi$$
−0.399634 + 0.916675i $$0.630863\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −1080.00 −0.362634
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −5868.00 −1.91455 −0.957274 0.289181i $$-0.906617\pi$$
−0.957274 + 0.289181i $$0.906617\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 3375.00 1.00000
$$226$$ 0 0
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −4730.00 −1.32993 −0.664963 0.746877i $$-0.731553\pi$$
−0.664963 + 0.746877i $$0.731553\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 7376.00 1.99629 0.998146 0.0608655i $$-0.0193861\pi$$
0.998146 + 0.0608655i $$0.0193861\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 2720.00 0.675909
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 4482.00 1.06295
$$262$$ 0 0
$$263$$ −7520.00 −1.76313 −0.881565 0.472063i $$-0.843509\pi$$
−0.881565 + 0.472063i $$0.843509\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −8500.00 −1.86389
$$276$$ 0 0
$$277$$ 7310.00 1.58561 0.792807 0.609472i $$-0.208619\pi$$
0.792807 + 0.609472i $$0.208619\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 4342.00 0.921786 0.460893 0.887456i $$-0.347529\pi$$
0.460893 + 0.887456i $$0.347529\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −4913.00 −1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −6970.00 −1.23493 −0.617467 0.786597i $$-0.711841\pi$$
−0.617467 + 0.786597i $$0.711841\pi$$
$$318$$ 0 0
$$319$$ −11288.0 −1.98121
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −10908.0 −1.81135 −0.905677 0.423969i $$-0.860636\pi$$
−0.905677 + 0.423969i $$0.860636\pi$$
$$332$$ 0 0
$$333$$ −12150.0 −1.99945
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −3330.00 −0.538269 −0.269135 0.963103i $$-0.586738\pi$$
−0.269135 + 0.963103i $$0.586738\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 4100.00 0.634293 0.317146 0.948377i $$-0.397275\pi$$
0.317146 + 0.948377i $$0.397275\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 8104.00 1.19140 0.595700 0.803207i $$-0.296875\pi$$
0.595700 + 0.803207i $$0.296875\pi$$
$$360$$ 0 0
$$361$$ −6859.00 −1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −13970.0 −1.93925 −0.969624 0.244602i $$-0.921343\pi$$
−0.969624 + 0.244602i $$0.921343\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −11916.0 −1.61500 −0.807498 0.589870i $$-0.799179\pi$$
−0.807498 + 0.589870i $$0.799179\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −4860.00 −0.638366
$$388$$ 0 0
$$389$$ −10526.0 −1.37195 −0.685976 0.727624i $$-0.740625\pi$$
−0.685976 + 0.727624i $$0.740625\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 1598.00 0.199003 0.0995016 0.995037i $$-0.468275\pi$$
0.0995016 + 0.995037i $$0.468275\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 30600.0 3.72675
$$408$$ 0 0
$$409$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 15262.0 1.76680 0.883402 0.468616i $$-0.155247\pi$$
0.883402 + 0.468616i $$0.155247\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 8608.00 0.962025 0.481012 0.876714i $$-0.340269\pi$$
0.481012 + 0.876714i $$0.340269\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −18580.0 −1.99269 −0.996346 0.0854102i $$-0.972780\pi$$
−0.996346 + 0.0854102i $$0.972780\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −2686.00 −0.282317 −0.141158 0.989987i $$-0.545083\pi$$
−0.141158 + 0.989987i $$0.545083\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 8010.00 0.819895 0.409947 0.912109i $$-0.365547\pi$$
0.409947 + 0.912109i $$0.365547\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ 8440.00 0.847171 0.423585 0.905856i $$-0.360771\pi$$
0.423585 + 0.905856i $$0.360771\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 12240.0 1.18984
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −15930.0 −1.52911
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −21240.0 −1.97634 −0.988169 0.153371i $$-0.950987\pi$$
−0.988169 + 0.153371i $$0.950987\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −20372.0 −1.87246 −0.936228 0.351394i $$-0.885708\pi$$
−0.936228 + 0.351394i $$0.885708\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 7236.00 0.649154 0.324577 0.945859i $$-0.394778\pi$$
0.324577 + 0.945859i $$0.394778\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −10567.0 −0.868497
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 15878.0 1.26183 0.630914 0.775853i $$-0.282680\pi$$
0.630914 + 0.775853i $$0.282680\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −12980.0 −1.01460 −0.507299 0.861770i $$-0.669356\pi$$
−0.507299 + 0.861770i $$0.669356\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 20470.0 1.55717 0.778583 0.627541i $$-0.215939\pi$$
0.778583 + 0.627541i $$0.215939\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −26906.0 −1.98235 −0.991176 0.132553i $$-0.957683\pi$$
−0.991176 + 0.132553i $$0.957683\pi$$
$$570$$ 0 0
$$571$$ 6788.00 0.497494 0.248747 0.968569i $$-0.419981\pi$$
0.248747 + 0.968569i $$0.419981\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −5000.00 −0.362634
$$576$$ 0 0
$$577$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 40120.0 2.85009
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 24736.0 1.68729 0.843644 0.536903i $$-0.180406\pi$$
0.843644 + 0.536903i $$0.180406\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$602$$ 0 0
$$603$$ −19980.0 −1.34933
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 15010.0 0.988986 0.494493 0.869182i $$-0.335354\pi$$
0.494493 + 0.869182i $$0.335354\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 30550.0 1.99335 0.996675 0.0814823i $$-0.0259654\pi$$
0.996675 + 0.0814823i $$0.0259654\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 15625.0 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 26192.0 1.65244 0.826218 0.563351i $$-0.190488\pi$$
0.826218 + 0.563351i $$0.190488\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 18576.0 1.15001
$$640$$ 0 0
$$641$$ 8878.00 0.547051 0.273526 0.961865i $$-0.411810\pi$$
0.273526 + 0.961865i $$0.411810\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 27050.0 1.62105 0.810527 0.585701i $$-0.199181\pi$$
0.810527 + 0.585701i $$0.199181\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 1804.00 0.106637 0.0533186 0.998578i $$-0.483020\pi$$
0.0533186 + 0.998578i $$0.483020\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −6640.00 −0.385460
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −33570.0 −1.92278 −0.961388 0.275196i $$-0.911257\pi$$
−0.961388 + 0.275196i $$0.911257\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 34060.0 1.90815 0.954077 0.299560i $$-0.0968400\pi$$
0.954077 + 0.299560i $$0.0968400\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −4198.00 −0.226186 −0.113093 0.993584i $$-0.536076\pi$$
−0.113093 + 0.993584i $$0.536076\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 12546.0 0.664563 0.332281 0.943180i $$-0.392182\pi$$
0.332281 + 0.943180i $$0.392182\pi$$
$$710$$ 0 0
$$711$$ −37368.0 −1.97104
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 20750.0 1.06295
$$726$$ 0 0
$$727$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$728$$ 0 0
$$729$$ −19683.0 −1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 50320.0 2.51501
$$738$$ 0 0
$$739$$ 25324.0 1.26057 0.630283 0.776365i $$-0.282939\pi$$
0.630283 + 0.776365i $$0.282939\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −25160.0 −1.24230 −0.621151 0.783691i $$-0.713335\pi$$
−0.621151 + 0.783691i $$0.713335\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 2448.00 0.118946 0.0594732 0.998230i $$-0.481058\pi$$
0.0594732 + 0.998230i $$0.481058\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −34830.0 −1.67228 −0.836141 0.548514i $$-0.815194\pi$$
−0.836141 + 0.548514i $$0.815194\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −46784.0 −2.14349
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 37354.0 1.62336 0.811679 0.584104i $$-0.198554\pi$$
0.811679 + 0.584104i $$0.198554\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −43538.0 −1.85078 −0.925388 0.379022i $$-0.876261\pi$$
−0.925388 + 0.379022i $$0.876261\pi$$
$$822$$ 0 0
$$823$$ 46240.0 1.95848 0.979238 0.202716i $$-0.0649768\pi$$
0.979238 + 0.202716i $$0.0649768\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 23980.0 1.00830 0.504151 0.863615i $$-0.331805\pi$$
0.504151 + 0.863615i $$0.331805\pi$$
$$828$$ 0 0
$$829$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 3167.00 0.129854
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 18000.0 0.725067
$$852$$ 0 0
$$853$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 0 0
$$859$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 20200.0 0.796774 0.398387 0.917217i $$-0.369570\pi$$
0.398387 + 0.917217i $$0.369570\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 94112.0 3.67380
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −6550.00 −0.252198 −0.126099 0.992018i $$-0.540246\pi$$
−0.126099 + 0.992018i $$0.540246\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ −30060.0 −1.14564 −0.572820 0.819681i $$-0.694150\pi$$
−0.572820 + 0.819681i $$0.694150\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 49572.0 1.86389
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −52740.0 −1.93076 −0.965382 0.260840i $$-0.916000\pi$$
−0.965382 + 0.260840i $$0.916000\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 39632.0 1.44135 0.720673 0.693275i $$-0.243833\pi$$
0.720673 + 0.693275i $$0.243833\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −21744.0 −0.780488 −0.390244 0.920711i $$-0.627609\pi$$
−0.390244 + 0.920711i $$0.627609\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −56250.0 −1.99945
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −48820.0 −1.67522 −0.837612 0.546266i $$-0.816049\pi$$
−0.837612 + 0.546266i $$0.816049\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 29290.0 0.995589 0.497794 0.867295i $$-0.334143\pi$$
0.497794 + 0.867295i $$0.334143\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −29791.0 −1.00000
$$962$$ 0 0
$$963$$ −42660.0 −1.42752
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −52040.0 −1.73060 −0.865302 0.501251i $$-0.832873\pi$$
−0.865302 + 0.501251i $$0.832873\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −37490.0 −1.22765 −0.613824 0.789443i $$-0.710369\pi$$
−0.613824 + 0.789443i $$0.710369\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 1458.00 0.0474519
$$982$$ 0 0
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 7200.00 0.231493
$$990$$ 0 0
$$991$$ −57528.0 −1.84403 −0.922017 0.387150i $$-0.873460\pi$$
−0.922017 + 0.387150i $$0.873460\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 784.4.a.k.1.1 1
4.3 odd 2 49.4.a.a.1.1 1
7.6 odd 2 CM 784.4.a.k.1.1 1
12.11 even 2 441.4.a.m.1.1 1
20.19 odd 2 1225.4.a.l.1.1 1
28.3 even 6 49.4.c.d.30.1 2
28.11 odd 6 49.4.c.d.30.1 2
28.19 even 6 49.4.c.d.18.1 2
28.23 odd 6 49.4.c.d.18.1 2
28.27 even 2 49.4.a.a.1.1 1
84.11 even 6 441.4.e.a.226.1 2
84.23 even 6 441.4.e.a.361.1 2
84.47 odd 6 441.4.e.a.361.1 2
84.59 odd 6 441.4.e.a.226.1 2
84.83 odd 2 441.4.a.m.1.1 1
140.139 even 2 1225.4.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
49.4.a.a.1.1 1 4.3 odd 2
49.4.a.a.1.1 1 28.27 even 2
49.4.c.d.18.1 2 28.19 even 6
49.4.c.d.18.1 2 28.23 odd 6
49.4.c.d.30.1 2 28.3 even 6
49.4.c.d.30.1 2 28.11 odd 6
441.4.a.m.1.1 1 12.11 even 2
441.4.a.m.1.1 1 84.83 odd 2
441.4.e.a.226.1 2 84.11 even 6
441.4.e.a.226.1 2 84.59 odd 6
441.4.e.a.361.1 2 84.23 even 6
441.4.e.a.361.1 2 84.47 odd 6
784.4.a.k.1.1 1 1.1 even 1 trivial
784.4.a.k.1.1 1 7.6 odd 2 CM
1225.4.a.l.1.1 1 20.19 odd 2
1225.4.a.l.1.1 1 140.139 even 2